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2024RNAAS...8..201B

Stellar age estimates are often calculated by interpolating a stars properties in a grid of models. However different model grids will give different ages for the same star. We used the open cluster M67 to compare four different model grids DSEP GARSTEC MIST and YREC 10.5281zenodo.12775242. Across ltAQ1gtall model grids age estimates for main sequence stars were consistently higher than the accepted age of M67 while age estimates for red giant stars were lower. We compared modelgenerated age and mass values to external constraints as an additional test of the reliability of each model grid. For stars near solar age and metallicity we recommend using the DSEP model grid to estimate the ages of main sequence stars and the GARSTEC model grid for red giant stars.

2024-08-01T00:00:00Z

['10.3847/2515-5172/ad7093', '2024arXiv240904581B', '10.48550/arXiv.2409.04581', '2024RNAAS...8..201B', 'arXiv:2409.04581']

['Stellar evolutionary models', 'Open star clusters', '2046', '1160', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Identifying Uncertainties in Stellar Evolution Models Using the Open Cluster M67

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https://arxiv.org/pdf/2409.04581.pdf

{'Identifying Uncertainties in Stellar Evolution Models Using the Open Cluster M67': 'Susan Byrom 1 and Jamie Tayar 2 \n1 Department of Physics, University of Illinois Urbana-Champaign, USA 2 Department of Astronomy, University of Florida, USA', 'ABSTRACT': "Stellar age estimates are often calculated by interpolating a star's properties in a grid of models. However, different model grids will give different ages for the same star. We used the open cluster M67 to compare four different model grids: DSEP, GARSTEC, MIST, and YREC a) . Across all model grids, age estimates for main sequence stars were consistently higher than the accepted age of M67, while age estimates for red giant stars were lower. We compared model-generated age and mass values to external constraints as an additional test of the reliability of each model grid. For stars near solar age and metallicity, we recommend using the DSEP model grid to estimate the ages of main sequence stars and the GARSTEC model grid for red giant stars. \nKeywords: Stellar evolutionary models (2046) - Open star clusters (1160)", '1. INTRODUCTION': "Stellar ages used in fields such as galactic archaeology and exoplanet evolution are commonly determined by fitting a star to a model grid. A stellar model grid is a set of evolutionary tracks generated by a modeling code at a range of initial masses and metallicities. These tracks predict physical parameters (e.g. luminosity and temperature) of a star as a function of age, initial mass, and initial composition. Therefore, given a model grid and sufficient observational constraints, the age and mass of a star can inferred. However, the assumptions and calibrations that go into creating a stellar model grid can cause significant differences between different grids' age estimates of a star, sometimes more than 30% (Tayar et al. 2022). \nOpen star clusters are commonly used to check the accuracy of stellar model grids. In particular, the cluster M67 is often used to calibrate stellar models (Choi et al. 2016) because M67 is a well-studied, nearby old open cluster that is approximately 4.0 Gyr old and is near solar metallicity ([ Fe/H ] = 0 . 00 ± . 05) (Myers et al. 2022). \nThe age of M67 has been calculated many times. Most commonly, the age of a cluster can be determined by plotting the stars on a color-magnitude diagram and fitting an isochrone to the main sequence turnoff. For M67, VictoriaRegina isochrones give an age estimate of 3.6-4.6 Gyr, with an average of 4.0 Gyr (VandenBerg & Stetson 2004). Sandquist et al. (2021) used MIST, PADOVA, and BASTI isochrones constrained by the eclisping binary WOCS 11028 to get an age of 3.5-4.0 Gyr. Stello et al. (2016) used asteroseismology of red giants to obtain an age estimate of 3.46 ± 0.13 Gyr. \nModel grids can be used to estimate the ages of individual M67 stars, essentially treating them as field stars, which are stars that do not belong to a cluster. With perfect models and data, this would return the same age for each star in M67. Systematic discrepancies in the age estimates can reveal flaws in the model grid or input data. We did this for the model grids DSEP, GARSTEC, MIST, and YREC, with model parameters as presented in Tayar et al. (2022). \nThe age of M67 has been previously estimated with some of the model grids used in this work. \n- · MIST isochrones were calibrated using M67 at 4.0 Gyr and solar metallicity (Choi et al. 2016).\n- · The DSEP model grid isochrone age estimate for M67 is 4.0 Gyr (Dotter et al. 2008).\n- · In Magic et al. (2010), GARSTEC's age estimate for M67 was determined to be 4.2 Gyr assuming Z/X = 0.0165 and 4.5 Gyr assuming Z/X = 0.0230. \n- · YREC model grids from Viani & Basu (2017), which are slightly different from those used here, estimate the age of M67 at 3.6-4.8 Gyr (best fit at 4.4 Gyr).", '2. APOGEE DATA': 'Data from the Apache Point Observatory Galactive Evolution Experiment (APOGEE) Data Release 17 (DR17), as compiled by the Open Cluster Chemical Analysis and Mapping Survey (OCCAM) (Myers et al. 2022) lists 663 members of M67, 608 of which have recorded surface gravity (log( g )), effective temperature, and metallicity values. We discarded stars with a membership probability of 0 in any of the four categories from OCCAM: proper motion ( 200 stars ), [Fe/H] ( 166 stars ), radial velocity ( 188 stars ), and Cantat-Gaudin et al. (2018) ( 232 stars ). Values from Cantat-Gaudin et al. (2018) exist for all stars considered in this step. For the remaining 311 stars, we used Kiauhoku (Claytor et al. 2020) to fit each star to the model grids based on log( g ), [M/H], and T eff . We discarded stars that did not fall within the parameter space covered by the model grids. APOGEE DR17 log( g )values are less reliable for low-mass dwarfs, so we discarded stars with log( g ) > 4 . 5. This left 141, 140, 143, and 140 star age and mass estimates for DSEP, GARSTEC, MIST, and YREC respectively.', '3. RESULTS': "The average model-generated age of M67 is 4.72, 4.85, 4.84, and 5.74 Gyr for DSEP, GARSTEC, MIST, and YREC respectively. In Figure 1e, the x-axis is divided into main sequence (MS), subgiant, and red giant evolutionary phases. In Figure 1a-d, the isochrones are close together on the MS, so a small difference in log( g ) between two stars results in a large difference in their estimated ages. We compared ages inferred from the models as a function of log( g ) for the models (Figure 1e). There is a wide spread of MS age estimates, with a model-wide average of 5.03 Gyr. The model-wide average for red giant stars is 2.64 Gyr, and red giant ages from YREC, DSEP, and MIST are lower than the accepted age of M67, while GARSTEC age estimates are more accurate. \nIn Figure 1f, the mass estimates from the models are compared to asteroseismic mass estimates from Stello et al. (2016). While there were no shared stars between our data and Stello et al. (2016), we assume that all red giants in a cluster that have evolved as single stars will have similar masses. Of the four model grids, the GARSTEC mass estimates for red giants are closest to the Stello et al. (2016) values. Combined with GARSTEC's reasonable red giant age estimates, this indicates that the GARSTEC-generated model grid is a good choice for estimating parameters of red giant stars near solar age and metallicity.", '4. CONCLUSIONS': "When using a model grid to estimate the age of a star, we find that the accuracy of each model grid varies based on the star's metallicity and evolutionary phase. For stars near solar age and metallicity, the GARSTEC model grid is recommended for estimating the age and mass of red giant stars. For main sequence stars, the DSEP model grid was found to be most accurate. More generally, our findings highlight the need for careful verification and calibration of models in the regime in which they are going to be used. \nWe acknowledge support from the National Science Foundation under grant No. 2243878 through the University of Florida 2023 REU. We used data from SDSS (https://www.sdss.org/collaboration/citing-sdss/).", 'REFERENCES': "Cantat-Gaudin, T., Jordi, C., Vallenari, A., et al. 2018, A&A, 618, A93, doi: 10.1051/0004-6361/201833476 Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102 \nClaytor, Z. R., van Saders, J. L., Santos, ˆ A. R. G., et al. 2020, kiauhoku: Stellar model grid interpolation, Astrophysics Source Code Library, record ascl:2011.027 Dotter, A., Chaboyer, B., Jevremovi'c, D., et al. 2008, ApJS, 178, 89, doi: 10.1086/589654 \nMagic, Z., Serenelli, A., Weiss, A., & Chaboyer, B. 2010, ApJ, 718, 1378, doi: 10.1088/0004-637X/718/2/1378 \nMyers, N., Donor, J., Spoo, T., et al. 2022, AJ, 164, 85, doi: 10.3847/1538-3881/ac7ce5 \nSandquist, E. L., Latham, D. W., Mathieu, R. D., et al. 2021, AJ, 161, 59, doi: 10.3847/1538-3881/abca8d \nStello, D., Vanderburg, A., Casagrande, L., et al. 2016, ApJ, 832, 133, doi: 10.3847/0004-637X/832/2/133 \nFigure 1. a , b , c , and d are Kiel diagrams of the open cluster M67. Isochrones generated at [M/H] = 0.00 are shown for 4, 5, and 6 Gyr. Stars from M67 were treated as field stars to estimate their ages, which are shown by the colorbar. 1. e shows age as a function of surface gravity for all four models. Vertical dashed lines indicate the approximate boundary between main sequence, subgiant, and red giant stars. The age estimates for each model are fitted with LOWESS non-parametric smooth curves. Notably, the GARSTEC red giant age estimates are near the predicted age of M67, and all models have similar behavior on the subgiant branch. In subfigure f , model-generated mass estimates for M67 red giants are compared to Stello et al. (2016) asteroseismic mass estimates for red giants. \n<!-- image --> \nTayar, J., Claytor, Z. R., Huber, D., & van Saders, J. 2022, \nApJ, 927, 31, doi: 10.3847/1538-4357/ac4bbc \nVandenBerg, D. A., & Stetson, P. B. 2004, PASP, 116, 997, \ndoi: 10.1086/426340 \nViani, L., & Basu, S. 2017, in European Physical Journal \nWeb of Conferences, Vol. 160, European Physical Journal Web of Conferences, 05005, \ndoi: 10.1051/epjconf/201716005005"}

2024arXiv240904522Y

Due to highcadence automated surveys we can now detect and classify supernovae SNe within a few days after explosion if not earlier. Earlytime spectra of young SNe directly probe the outermost layers of the ejecta providing insights into the extent of stripping in the progenitor star and the explosion mechanism in the case of corecollapse supernovae. However many SNe show overlapping observational characteristics at early time complicating the earlytime classification. In this paper we focus on the study and classification of Type Ib supernovae SNe Ib which are a subclass of corecollapse supernovae that lack strong hydrogen lines but show helium lines in their spectra. Here we present a spectral dataset of 8 SNe Ib chosen to have at least 3 premaximum spectra which we call early spectra. Our dataset was obtained mainly by the the Las Cumbres Observatory LCO and consists of a total of 82 optical photospheric spectra including 38 early spectra. This data set increases the number of published SNe Ib with at least three early spectra by 60. For our classification efforts we use early spectra in addition to spectra taken around maximum light. We also convert our spectra into SN Identification SNID templates and make them available to the community for easier identification of young SNe Ib. Our data set increases the number of publicly available SNID templates of early spectra of SNe Ib by 43. Almost half of our sample has SN types that change over time or are different from what is listed on the Transient Name Server TNS. We discuss the implications of our dataset and our findings for current and upcoming SN surveys and their classification efforts.

2024-09-01T00:00:00Z

['2024arXiv240904522Y', 'arXiv:2409.04522', '10.48550/arXiv.2409.04522']

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

Spectral Dataset of Young Type Ib Supernovae and their Timeevolution

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https://arxiv.org/pdf/2409.04522.pdf

{'Spectral Dataset of Young Type Ib Supernovae and their Time-evolution': 'N. Yesmin 1 , C. Pellegrino 1 , M. Modjaz 1 , R. Baer-Way 1 , D. A. Howell 2 , 3 , I. Arcavi 4 , J. Farah 2 , 3 , D. Hiramatsu 5 , 6 Hosseinzadeh 7 , 8 , C. McCully 2 , 3 , M. Newsome 2 , 3 , E. Padilla Gonzalez 2 , 3 , G. Terreran 2 , 3 , and S. Jha 9 \n- 1 Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA, USA\n- 2 Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA\n- 3 Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA\n- 4 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel\n- 5 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA\n- 6 The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA\n- 7 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA\n- 8 Department of Astronomy & Astrophysics, University of California, San Diego, 9500 Gilman Drive, MC 0424, La Jolla, CA 92093-0424, USA\n- 9 Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA \nSeptember 10, 2024', 'ABSTRACT': 'Due to high-cadence automated surveys, we can now detect and classify supernovae (SNe) within a few days after explosion, if not earlier. Early-time spectra of young SNe directly probe the outermost layers of the ejecta, providing insights into the extent of stripping in the progenitor star and the explosion mechanism in the case of core-collapse supernovae. However, many SNe show overlapping observational characteristics at early time, complicating the early-time classification. In this paper, we focus on the study and classification of Type Ib supernovae (SNe Ib), which are a subclass of core-collapse supernovae that lack strong hydrogen lines but show helium lines in their spectra. Here we present a spectral dataset of 8 SNe Ib, chosen to have at least 3 pre-maximum spectra, which we call early spectra. Our dataset was obtained mainly by the the Las Cumbres Observatory (LCO) and consists of a total of 82 optical photospheric spectra, including 38 early spectra. This data set increases the number of published SNe Ib with at least three early spectra by ∼ 60%. For our classification e ff orts, we use early spectra in addition to spectra taken around maximum light. We also convert our spectra into SN Identification (SNID) templates and make them available to the community for easier identification of young SNe Ib. Our data set increases the number of publicly available SNID templates of early spectra of SNe Ib by ∼ 43%. Almost half of our sample has SN types that change over time or are di ff erent from what is listed on the Transient Name Server (TNS). We discuss the implications of our dataset and our findings for current and upcoming SN surveys and their classification e ff orts. \nKey words. supernovae - spectroscopic - SN 2023ljf - SN 2022nyo - SN 2021ukt - SN 2021njk - SN 2021hen - SN 2020hvp SN 2019odp - SN 2016bau', '1. Introduction': "Most massive stars end their life cycles as core-collapse supernovae (CCSNe), which are categorized by their spectral features: Type I supernovae (SNe) lack H lines, while Type II SNe exhibit them. Type I CCSNe are further divided into subclasses, which are collectively called Stripped Envelope Supernovae (SESNe, Clocchiatti et al. 1997; Filippenko 1997; Gal-Yam 2017; Modjaz et al. 2019), as they have been stripped of their outer layers: Type Ib SNe (SNe Ib) are characterized by the presence of strong He lines and the absence of strong H lines (though even SNe Ib may have a weak H α line (Liu et al. 2016a)), and Type Ic SNe (SNe Ic) are devoid of both H and He lines. Transitional Type IIb SNe (SNe IIb) show diminishing H lines over time, but with emerging He lines akin to SNe Ib. Additionally, a subset of high-velocity SNe Ic with broad lines (SNe Ic-bl, e.g., Modjaz et al. 2016) has been interesting because they are the only kind of CCSNe observed to be connected to long-duration gamma-ray bursts (for reviews see e,g., Woosley & Bloom 2006; Modjaz 2011; Cano et al. 2017). \nDespite the understanding that SESNe result from the core collapse of massive stars, the diverse range of observed explosion properties poses a challenge in classifying SESNe into distinct types. As a result, it remains unclear if the classification types are distinct or share overlapping characteristics, and what properties of the progenitors or of the explosion cause this diversity in the observed SNe properties. Addressing these unresolved questions involves two approaches: (i) analyzing data of large samples to identify trends in di ff erent supernova (SN) types and (ii) studying data of SNe well before maximum light to directly probe the outermost layers of the progenitor envelope, as the photosphere lies in the outermost layers of the ejecta during this early phase and recedes into deeper layers over time. \nTraditionally, individual SNe have been thoroughly studied, such as Type IIb SN 1993J (Filippenko et al. 1993; Matheson et al. 2000b; Matheson et al. 2000a) and Type Ic SN 1994I (Filippenko et al. 1995; Richmond et al. 1996). The prevalence of population studies has increased in recent years. Statistical comparisons of SESNe spectra by Matheson et al. (2000a) revealed \n, G. \nTable 1: Overview of SNe Ib included in our Sample. \n- b The total number of photospheric spectra for the corresponding SN included in this work. \nheterogeneous properties, while subsequent studies by Fremling et al. (2018), Prentice et al. (2019), Liu et al. (2016a), and Holmbo et al. (2023) explored the spectral properties of di ff erent types of SESNe. To quantify and visualize the continuum of SESNe classification, Williamson et al. (2019a) presented a new quantitative and data-driven classification technique utilizing the machine learning tool of Support Vector Machine (SVM) after applying Principal Component Analysis (PCA) to the photospheric spectra of SESNe. They found that the SESNe types, in particular SNe Ib, are most distinguishable in later phase ranges, particularly 10 ± 5 days and 15 ± 5 days relative to V-band maximum. \nHowever, these studies lacked early spectra in their sample because they included data from earlier surveys, which had low cadences. Advancements in high-cadence wide-field surveys now enable prompt detection and observation of SNe. Through these early time observations, though, we are discovering an increasing number of SNe with evolving properties, such as SNe that change spectroscopic type throughout their evolution. These SNe pose a challenge to our classification e ff orts. The transitional SN 2022crv (Dong et al. 2023a; Gangopadhyay et al. 2023) is an example of an evolving SESN. \nNow is the perfect time to start conducting comprehensive population studies of young SESNe, in particular SNe Ib. This work constitutes the first step in these population studies, as it publishes optical spectra and spectral templates of young SNe Ib only days after explosion, well before maximum light. Studying the spectra of young SNe Ib over time will provide valuable insights into when the tell-tale He lines will appear. Consequently, it will o ff er significant constraints on 56 Ni mixing, as the He lines result from non-thermal excitation induced by fast electrons that are accelerated by energetic MeV photons emitted during Ni decay (Lucy 1991). Tracking the appearance of the He lines helps with inferring the distribution of Ni-mixing further out in the \nejecta. In turn, the exact details of the Ni-mixing can constrain explosion models, which, for objects where the He lines are visible shortly after explosion, would require the release of Ni close to the outermost He layer, potentially indicating large scale turbulence (e.g., Hammer et al. 2010). Furthermore, analyzing early spectra will inform the most appropriate timing to schedule spectroscopy to classify an SN Ib, which is essential for the new era of big-data transient science, in which thousands of SNe will be photometrically discovered every night with the Vera C. Rubin Observatory as part of the Legacy Survey of Space and Time (LSST; Ivezi'c et al. 2019). \nIn Section 2, we present a dataset of 82 photospheric spectra of 8 di ff erent SNe Ib, including 38 spectra taken before maximum light. All spectra will be made public via the Weizmann Interactive Supernova Data REPository (WISeREP 1 ; Yaron & Gal-Yam 2012). We discuss our spectrum phase calculation and the method for calculating date of maximum light in Section 3. In Section 4, we discuss the specifics of the Supernova Identification (SNID) code (Blondin & Tonry 2007) classification of each SN in the sample. In Section 5, we present our procedure for creating spectral templates for SNID for the SNe in our sample using our data as well as prior published data. Our SNID templates can be downloaded via our METAL GitHub repository 2 , formerly known as the SNYU github page.", '2. Spectroscopic Observations and Data Reduction': "Our sample consists of 8 SNe Ib with data taken between 2016 and 2023 mainly by the Las Cumbres Observatory (LCO) (Brown et al. 2013) as part of the Global Supernova Project (GSP) (Howell 2024). These 8 SNe Ib were selected from the \nGSP sample because a) they had observed photometry that covers the lightcurve peak in at least one band, so spectral phases could be calculated relative to the date of maximum (see Section 3 for details), and b) they had unpublished spectra, with at least three spectra before the date of V-band maximum light (t Vmax ), such that we could study their early-time behavior. Additional data were taken as part of the LCO Supernova Key Project (Howell 2017) for SN 2016bau and by GSP members with other observatories. \nThus, we present a total of 82 photospheric (t Vmax < 60 days) optical spectra of 8 SNe Ib. We exclude nebular-phase spectra for this study since we focus on photospheric-phase classification. Our data set includes a total of 38 spectra taken before the Vband maximum light, which we refer to as early spectra, increasing the number of published SNe Ib with at least 3 early spectra by ∼ 60% and the number of publicly available early spectra of SNe Ib that satisfy the same criterion of at least three early spectra by ∼ 37%. Table 1 shows a summary of our data set. Figure 1 shows the spectral time series of the 8 SNe Ib in our sample. \nAlmost all (80 out of 82) optical spectra were taken using the FLOYDS spectrographs on the 2m telescopes at Siding Spring Observatory (COJ 2m), Australia, and Haleakala (OGG 2m), Hawaii. A 2 arcsec slit was placed on the SN along the parallactic angle (Filippenko 1982). The spectra are extracted from the raw images with wavelength scale and flux calibration applied by the FLOYDS pipeline 3 , following standard procedures (Valenti et al. 2014). One optical spectrum was taken with the Southern African Large Telescope (SALT) using the Robert Stobe Spectrograph (RSS; Smi 2006) through Rutgers University program 2022-1-MLT-004 (PI: Jha). The observations were taken with the PG0900 grating and 1.5 '' wide longslit, giving a spectral resolution R = λ/ ∆ λ ≃ 1000. The data were reduced using a custom pipeline based on standard Pyraf spectral reduction routines and the PySALT package (Crawford et al. 2010). Additionally, one spectrum was obtained with the Goodman High Throughput Spectrograph on the Southern Astrophysical Research (SOAR) 4.1m telescope. The spectrum was taken using the blue camera, covering the wavelength range 3500 - 7000 Å, with a 1 '' slit and 400 mm -1 line grating. We used the Goodman Spectroscopic Data Reduction Pipeline to perform bias and flat corrections, cosmic-ray removal, and wavelength calibration. Fluxes were calibrated to a standard star observed on the same night with the same instrumental setup. \nThe details of the observation log are presented in Appendix A Table A.1. All the spectra presented in this work will be made available through WISeREP. \nFor the identification of the supernova type, we initially relied on the classification available from the Transient Name Server (TNS 4 ; Gal-Yam 2021), the o ffi cial mechanism of the International Astronomical Union (IAU) for reporting new astronomical transients such as SN candidates. To further solidify the classification of the SNe, we ran the spectra taken closest to the date of the V-band maximum light and the early spectra through SNID. Three out of eight SNe have classifications that are di ff erent from what is listed on the TNS. Details on the classification methodology are discussed in Section 4.", '3. Spectrum Phases and V-band Maximum Date Calculation': 'Determining the phase of each spectrum, or the epoch relative to maximum light, is essential for conducting a systematic analysis of di ff erent SN properties. Our phase calculations are based on the V-band maximum date ( tV max ), ensuring consistent tracking of the appearance of He lines across our sample. This approach also facilitates consistent comparisons with other population studies, such as those conducted by Liu et al. (2016a) and Holmbo et al. (2023), which similarly adopt the time of the V-band maximum date as their reference point for phase calculation. \nWhile all SNe in our sample have V-band light curves taken by LCO, 2 SNe lack a clear peak in the V-band light curve. For these two cases, namely SNe 2021hen and 2021ukt, we utilized the LCO i-band light curve to determine the maximum date in the i-band, as redder bands exhibit maxima at later times. We then convert the determined maximum date in the i-band to the maximum date in the V-band using Table 10 from Bianco et al. (2014), assuming these SN lightcurves have the same behavior as the SESNe in Bianco et al. (2014), which for SN 2021ukt is a large caveat (see section 4.3). This step adds a systematic uncertainty of ∼ 1.5 days. To measure the date of maximum light for the SNe in our sample, we follow Bianco et al. (2014) and use a Monte Carlo (MC) simulation process to fit a quadratic around the peak of the light curve for a given filter. The adopted date of maximum light is the mean of the maximum epochs across the MC realizations, and the error associated with the date of maximum light is the corresponding standard deviation. Table 1 shows the dates of maximum in the V band and their associated uncertainties that we calculated for each of the SNe in our data set. \nWe compare our calculated maximum dates for two SNe in our sample with previously published results and find general agreement. For SN 2019odp, we find a V-band maximum date of Modified Julian Date (MJD) 58735.19 ± 0.16, which is consistent with the findings of Schweyer et al. (2023), who reported a maximum date of MJD 58734 ± 1 day in the g-band. For SN 2016bau, Aryan et al. (2021) found the maximum date in the B-band to be MJD 57477.37 ± 1.99 days. For comparison, we converted our calculated V-band maximum date to the B-band equivalent by using Table 10 from Bianco et al. (2014), resulting in MJD 57475.87 ± 0.69 days, with an additional systematic uncertainty of ∼ 1.3 days. Thus, our measured date of max for SN 2016bau is consistent within the uncertainties with the published determination by Aryan et al. (2021). \n<!-- image --> \n<!-- image --> \n2 \nArticle number, page 4 of 12 \n<!-- image --> \n2 \n8 \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 1: Spectral time series for the eight SNe Ib presented in this work. The smoothed spectra (produced following the procedure described in Appendix B of Liu et al. (2016a), shown as solid black lines, are highlighted in the foreground, while the reduced spectra before smoothing are in gray in the background. Dashed vertical lines mark the HeI 4471, HeI 5876, HeI 6678, HeI 7065, H α 6563, and OI 7774 lines, and the phases are relative to the date of the V-band maximum. Note that we manually removed galaxy emission lines from the spectra at tV max -13.8d and -4.1d of SN 2022nyo. All SNe were chosen to have at least three spectra before maximum light in V-band. Thus, the observed change in the SN types for almost half of our sample could be uncovered with our large number of pre-maximum and post-maximum spectra. \n<!-- image --> \n<!-- image -->', '4. Supernova Classification via SNID': 'Ensuring a robust classification of SNe in our sample as Type Ib is an essential criterion for our data set. As discussed below in detail for each SN in our sample, we classify each SN by running the spectrum closest to the V-band peak ( tV max = 0 days) through SNID (Blondin & Tonry 2007). We also run SNID on the early spectra (i.e., tV max < 0), including the earliest spectra taken after discovery, to verify consistency and identify SNe showing uncertain or evolving early types. SNID operates by removing the continuum from the newly obtained spectrum (in order to focus on the spectral lines that define the most common SN types and to take out the e ff ect of any reddening) and by cross-correlating it with a library of previously observed and classified SN spectra, called individual "SN templates". SNID provides a quality parameter called rlap to measure the correlation; generally, correlations with rlap > rlapmin = 5.0 are considered a good match (Blondin & Tonry 2007). Therefore, the SNID matches reflect spectral behavior in only the absorption lines, with stronger absorption lines having higher weights, and do not take into account the shape of the continuum. For the SNID SESN template libraries, we use the most up-to-date ones based on the Modjaz Group Sample (MGS; Modjaz et al. 2014; Liu et al. 2016a, 2017; Williamson et al. 2019a, Williamson et al. 2023) to ensure a robust classification of SNe in our sample. \nClassifying early spectra has been di ffi cult due to the limited availability of previously observed young SNe Ib templates in the SNID library, or in the libraries of any of the other SN identification codes, e.g., Superfit (Howell et al. 2005), GELATO (Harutyunyan et al. 2008), for that matter. For SNID, the current MGS so far includes 235 spectral templates of 22 SNe Ib with tV max < 60 days, but only 91 spectra have tV max < 0 days. Additionally, spectra may not develop strong identifiable features, which is a well-known case for the He lines (e.g. Liu et al. 2016a, Williamson et al. 2019a,Williamson et al. 2021). Therefore, while the spectra closest to the V-band maximum light of the SNe in our sample all match those of normal SNe Ib in the SNID library (since they were chosen as such to be included in this paper), some of them may have uncertain or evolving-types before maximum light, which we are uncovering in detail in this paper. For example, as discussed below for the individual SNe, the earliest spectra of SN 2023ljf, SN 2021ukt, and SN 2021njk do not strongly match any specific SN type. Additionally, SN 2022nyo, SN 2021ukt, and SN 2019odp show evolving types in their early spectra, as described in detail below. Furthermore, we compared our classification with the TNS classification attempts: Table 2 provides the TNS classification in detail for each SN in the sample. SN 2021ukt and SN 2021njk have di ff erent classifications than Type Ib on TNS. Below, we detail the discovery and classification information for each of the SNe in our sample. For our classification information, we report the SNID classification results for the earliest spectrum and the spectrum closest to the V-band maximum light, unless otherwise mentioned.', '4.1. SN 2023ljf': 'SN 2023ljf was discovered by the Asteroid Terrestrial Impact Last Alert System (ATLAS; Tonry et al. 2018) on UT 202306-22 07:40:30 (MJD 60117.32) with a non-detection two days prior (Tonry et al. 2023). Pellegrino et al. (2023), as part of the GSP, obtained a classification spectrum of the object on UT 2023-06-23 09:29:05 (t Vmax = -14.5d) and used it to classify SN 2023ljf as a young SN Ib on TNS. We ran our spectrum closest to the V-band maximum light taken on UT 2023-07-03 07:23:36 \n(t Vmax = -4.6d) through SNID. SNID found that the best type is a normal SN Ib, with 55.7% of the templates with rlap ≥ rlapmin corresponding to this type. We also ran our earliest spectrum taken on UT 2023-06-23 09:29:05 (t Vmax = -14.5d) through SNID. SNID did not find any favored type or subtype for this spectrum. Note that Pellegrino et al. (2023) used the same spectrum to classify SN 2023ljf as a normal SN Ib on TNS. However, they do not mention the use of SNID for classification. For the subsequent spectrum taken on UT 2023-06-25 08:16:46 (t Vmax = -12.5d), SNID found the best match to be SN Ib, with 74.6% of the templates with rlap ≥ rlapmin corresponding to normal SN Ib.', '4.2. SN 2022nyo: transitioning from SN IIb to SN Ib': 'SN 2022nyo was discovered by the Distance Less Than 40 Mpc (DLT40) survey (Tartaglia et al. 2018) on 2022-06-30 06:22:01 (MJD 59760.27) with a non-detection 5 days prior (Pearson et al. 2022b). On TNS, 2022nyo is a Type Ib due to the classification of Pearson et al. (2022a), who used a spectrum taken on UT 2022-07-01 00:39:45 (MJD 59761.03) taken by the Gemini South telescope. We ran our spectrum closest to the V-band maximum light, taken on UT 2022-07-15 10:06:09 (t Vmax = -1.5d) through SNID. SNID did not find a favored type match for this spectrum but found the best matches with normal SN Ib. However, the earliest spectrum of SN 2022nyo (t Vmax = -13.8d) shows a prominent H α absorption feature around 6200 Å (implying an absorption velocity of ∼ -20,000 km / s, which is similar to other SNe IIb and SNe Ib at similar phases, see Fig. 2 in Liu et al. 2016b), which diminished in strength in the subsequent spectra starting at tV max = + 4.1 days. SNID confirms the H-like feature in the earliest spectrum by finding its best match with the Type IIb SN 2008cw (Modjaz et al. 2014) and 87.5% of the templates with rlap ≥ rlapmin corresponding to Type IIb. Although the feature at 6200 Å could be due to other lines, such as C II 6580, Ne I 6402, or Si II 6355 (e.g., Branch et al. 2002; Tominaga et al. 2005; Elmhamdi et al. 2006; Parrent et al. 2007), several studies suggest that is most likely some H in SNe Ib (e.g. Liu et al. 2016a). A qualitatively similar early-time transition from Type IIb to Type Ib has been noted in SN 2022crv (Dong et al. 2023a; Gangopadhyay et al. 2023), though a similarly detailed study of SN 2022nyo is outside the scope of this paper.', '4.3. SN 2021ukt: Peculiar SN Ib with IIn-like features': 'SN 2021ukt was discovered via the Zwicky Transient Facility (ZTF; Graham et al. 2019) on 2021-07-31 10:34:47 (MJD 59426.44) with a non-detection 7 days prior (De 2021). SN 2021ukt is a unique case within our sample given its peculiar evolution. TNS lists SN 2021ukt as a Type IIn due to the classification report of Hinkle (2021b), which was based on a quickreduction spectrum at t Vmax = -14.1 days (1.9 days before our first spectrum) showing a strong blue continuum and narrow H α and H β emission lines. For our earliest spectrum taken on UT 202108-03 11:41:04 (t Vmax = -12.2d), SNID did not find any significant classification type for this spectrum - nevertheless, a narrow H α that can be well-fit by a Lorenzian shape with a FWHM 1640 + 140 -130 km / s (see below for the fit) is visible, along with its blue continuum, fulfilling the classification criterion of a SN IIn. For the spectrum on UT 2021-08-06 12:17:32 (t Vmax = -9.1d), SNID found matches to 19 SN templates with rlap ≥ rlapmin, without a strong match to any single type. Nevertheless, the beginning of the He absorption lines (at ca 15,000 km / s) are seen \nTable 2: Supernova Type at initial classification and / or on TNS. \nNotes. a Phases are calculated with reference to the V-band maximum light. \nin the spectra. For the following spectrum taken on UT 2021-0809 11:57:22 (t Vmax = -6.2d), SNID found the favored type to be normal SN Ib, with 60.0% of the templates with rlap ≥ rlapmin corresponding to this type. In addition, we ran our spectrum closest to the V-band maximum, taken on UT 2021-08-15 13:30:07 (t Vmax = -0.1d) through SNID. SNID found that the best-fitting type is a normal SN Ib, with 71.1% of the templates with rlap ≥ rlapmin corresponding to normal SN Ib. Thus, we classify SN 2021ukt as a Type Ib based on the classification closest to the V-band maximum and mentions its type change in Tables 1 and 2. \nSN 2021ukt displays a broad HeI 5876Å absorption line which decreases in absorption velocity to ∼ 8000 km / s at 45.6 days, while the narrow H emission line persist throughout our spectra at varying strengths. As shown in Figure 2, the shape of the H α emission line evolves from a Lorentzian profile in our earliest spectrum, at tVmax = -12.2 days with FWHM 1640 + 140 -130 km / s to a two-component Gaussian profile in our latest spectrum at tVmax = + 45.8 days with a similarly narrow component( FWHM 2090 + 47 -46 km / s). There is also an intriguing redshifted broad H α component in the spectrum at tVmax = + 45.8 days centered at ∼ 2680 km / s, but its detailed interpretation is beyond the scope of this work. We obtained fits to the H α emission line by performing MCMC fits (10000 steps with a 1000-step burnin run) with emcee (Foreman-Mackey et al. 2013) and adopted the 16th and 84th percentiles of the posterior distribution as the upper and lower error bars on the FWHM. \nThe combination of broad He absorption with a narrow H emission line at the same time is something that has never been seen up to this point for an SN of any type. While SN 2014C (Milisavljevic et al. 2015) has been known to evolve from a SN Ib to an SN IIn and develop strong narrow emission lines, there has never been a case when a SN so strongly displayed characteristics of both subtypes at the same time. A detailed paper on this SN, focusing on modeling its light curve to infer the properties of its progenitor, will be presented by Pichay et al. (in prep).', '4.4. SN 2021njk': 'SN 2021njk was discovered by ATLAS on UT 2021-05-25 06:48:57 (MJD 59354.38) with a non-detection two days prior (Tonry et al. 2021) and classified by the GSP Hiramatsu et al. (2021). Given that the TNS classification spectrum, taken on UT 2021-05-30 09:51:19 (t Vmax = -11.3d), was obtained by LCO, this spectrum is also present in our study. SNID indicates the best match to SN 2005bf for this spectrum, in line with the TNS classification report by Hiramatsu et al. (2021). SN 2005bf is regarded as a peculiar Type Ib because it shows a light curve with two peaks and He lines with increasing velocity over time (Tominaga et al. 2005). However, for the purpose of spectroscopic classification, we list SN 2005bf as normal SN Ib. Hiramatsu et al. (2021) also found a match with SN 1999ex using SNID and Superfit. Although Hiramatsu et al. (2021) claimed SN 1999ex as peculiar Type Ib, Modjaz et al. (2014) argued that SN 1999ex is a normal Type Ib and not an intermediate Ib / c. Additionally, we also found top matches with other normal SN Ib (iPTF13bvn, SN 2001ai, SN 2009jf) via SNID. We also ran the spectrum closest to the V-band maximum date, taken on UT 2021-06-13 10:05:47 (t Vmax = 2.7d) through SNID. Although SNID did not find a favored type, the top 6 template matches are normal Ib. Thus, we classify SN 2021njk as a normal SN Ib.', '4.5. SN 2021hen': 'SN 2021hen was discovered by the Automatic Learning for the Rapid Classification of Events (ALeRCE) broker team (Förster et al. 2021) on UT 2021-03-28 08:07:19.004 (MJD 59301.54) in ZTF data with a non-detection four days prior (Forster et al. 2021). SN 2021hen was initially classified as an SN I using a GSP-obtained spectrum by Pellegrino et al. (2021), who found SNID matches to both Type Ia and Type Ib SNe. This SN was reclassified as an SN Ib by Sollerman (2022) using a spectrum obtained 2.5 days after the initial classification spectrum. We ran the spectrum closest to the V-band maximum light, taken on UT 2021-04-06 11:54:36 (t Vmax = -1.2d) through SNID. SNID classified this spectrum as a normal SN Ib, with 37.2% of the templates with rlap ≥ rlapmin corresponding to this type. For our earliest spectrum taken on UT 2021-03-30 13:01:37 (t Vmax = - \nFig. 2: Azoom-in of the H α emission line of SN 2021ukt at two epochs: the earliest one for which we have a spectrum ( -12.2 days before V -band maximum) and one of the latest. The results of the MCMC fitting performed on continuum-subtracted spectra are shown, with vC referring to the central velocity of the broad Gaussian in the second spectrum. These careful fits to the H α emission lines are important for the classification of SN 2021ukt as a SN Ib with IIn features - and at the earliest time, as a bona fide SN IIn. \n<!-- image --> \n-25 \n-20 \n-15 \n-10 \n-5 \n0 \n5 \n10 \n15 \n20 \n25 \nVelocity(10 3 km/s) with respect to rest H \n<!-- image --> \n8.2d), SNID classified this spectrum as normal Ib, with 41.8% of the templates with rlap ≥ rlapmin corresponding to this type. Pellegrino et al. (2021) used the same spectrum for their initial classification on TNS, reporting SN 2021hen as Type Ia or Type Ib. But for our classification based on the same SN spectrum, SNID found the best match with SN Ib by using the updated MGS SNID template library with significantly more SESNe templates.', '4.6. SN 2020hvp': 'SN 2020hvp was discovered by ATLAS on UT 2020-04-21 12:24:28 (MJD 58960.50) with a non-detection two days prior (Tonry et al. 2020). It was classified as an SN Ib by both Burke et al. (2020) and Dahiwale & Fremling (2020) using spectra obtained 2 and 4 days after discovery, respectively. We ran our spectrum closest to the V-band maximum, taken on UT 2020-0506 14:08:49 (t Vmax = -0.4d) through SNID. SNID found that the best match is a normal SN Ib, with 29.3% of the templates with rlap ≥ rlapmin corresponding to this type. For the earliest spectrum taken on UT 2020-04-23 17:50:18 (t Vmax = -13.2d), SNID did not find any favored type or subtype, but the top 6 template matches are normal SNe Ib.', '4.7. SN 2019odp: transitioning from SN Ic-bl to SN Ib': 'SN 2019odp is an interesting transitional object, which shows features like SNe Ic-bl before the V-band maximum light (t Vmax < 0) but transitions to Type Ib afterwards (see also Schweyer et al. 2023 with their own set of spectra for this SN). SN 2019odp was discovered (Nordin et al. 2019) by ZTF on UT 2019-08-21 09:19:53 (MJD 58716.39) with a non-detection 7 days prior and was initially classified as SNe Ic-bl on 23 August 2019 (MJD = 58718.2) by Brennan et al. (2019) as part of the ePESSTO + survey (Smartt et al. 2015). Using presumably the same spectrum, \nSchweyer et al. (2023) found that the spectrum was observed 16 days before the g-band peak (t gmax = -16d) and that it matches SNe Ic-bl. However, they reclassified SN 2019odp as Type Ib based on the spectrum taken 26 days after the g-band peak (t gmax = 26d). Thus, Schweyer et al. (2024) updated the classification of SN 2019odp as Type Ib on TNS. We ran the early spectra and the spectrum closest to the V-band peak date through SNID. SNID found SN 2007bi, a hydrogen-poor super-luminous SN Ic (SLSNe Ic; Liu et al. 2017), as the best match for the earliest spectrum (t Vmax = -15.7d). For the subsequent three early spectra (t Vmax = -11.6d, -8.7d, -5.6d) , SNID found the best match with SN 1999bw (SNe Ic-bl; Patat et al. 2001). Liu et al. (2017) found that the average absorption velocities are similar for SLSNe Ic and SNe Ic-bl, suggesting high velocity features in the earliest spectrum of SN 2019odp. The best match for our spectrum at -2.7d is a normal SN Ic, SN 2013ge. For the spectrum closest to the V-band maximum light, taken on UT 2019-09-10 13:47:02 (t Vmax = 1.4d), SNID found that the best-fitting type is normal Ib, with 55.9% of the templates with rlap ≥ rlapmin corresponding to this type. Thus, we re-classify SN 2019odp as a normal SN Ib based on spectra very close to maximum light (at t Vmax = 1.4d) - 3 weeks earlier than Schweyer et al. (2023) did - and indicate its transitional nature in Tables 1 and 2 with "Ic-bl -> Ib". This transition from a SN Ic-bl to a SN Ib has been observed once before, in the famous SN 2008D, whose spectra resembled that of Type Ic-bl shortly after explosion, i.e., 15-10 days before the date of maximum light, but developed narrow absorption lines of helium by the date of maximum light (Soderberg et al. 2008; Mazzali et al. 2008, Modjaz et al. 2009), similar to SN 2019odp.', '4.8. SN 2016bau': 'SN 2016bau was discovered on UT 2016-03-13 23:22:33 (MJD 57460.97) with a non-detection 9 days prior (Arbour 2016). It \nwas classified as an SN Ib on TNS using a classification spectrum obtained one day after discovery (Benetti et al. 2016). We ran our spectrum closest to the V-band maximum, taken on UT 2016-03-29 10:25:44 (t Vmax = -1.7d) through SNID. SNID found that the best-fitting type is normal SN Ib, with 48.1% of the templates with rlap ≥ rlapmin corresponding to this type. For the earliest spectrum taken on UT 2016-03-15 05:24:59 (t Vmax = -15.9d), SNID found that the best-fitting type is normal SN Ib, with 62.5% of the templates with rlap ≥ rlapmin corresponding to this type. We note that Aryan et al. (2021) published their own extensive data set on this SN (including eight optical spectra, two of which were taken before maximum light), with a date of maximum calculation that is consistent with ours (see Section 3).', '5. Constructing SNID Templates from our Sample of Young SNe Ib, Including Published Spectra': 'In order to improve the classification by SNID, it is crucial to enrich the SNID library by adding more classified SN templates of early SNe Ib. We created SNID templates using the 82 photospheric spectra of the 8 SNe Ib published in this work. In addition, we included the 19 spectra of SN 2019odp published in Schweyer et al. (2023) and 6 spectra of SN 2016bau published in Aryan et al. (2021) when creating the SNID templates of those SNe. We used the logwave 5 function in SNID to create spectral templates that are compatible with SNID 5.0. Our work adds 38 new SNe Ib spectra with tV max < 0 to the SNID library, increasing the early SNe Ib spectra by ∼ 43%. Including previously published spectra for SN 2019odp and SN 2016bau as mentioned above, the total increase is ∼ 54%. SNID templates for the SNe Ib presented in this paper are publicly available via the METAL GitHub repository 6 , along with the previously published Modjaz Group sample (MGS) SNID templates, including those of Young SN Ic from Williamson et al. (2023).', '6. Conclusions': "In the era of transient surveys with extensive sky coverage and rapid cadences such as ATLAS and ZTF, and with LSST on the horizon, prompt and accurate classification of young SNe holds significant importance for subsequent follow-up observations. In this paper, we present a data set of 82 photospheric spectra of 8 SNe Ib, which includes a total of 38 spectra observed before the V-band maximum. This spectral data set is publicly available via WISeREP. This data set increases the number of published SNe Ib with at least three premaximum spectra by 60% and the number of available premaximum SNe Ib spectra in the SNID template library by ∼ 43%. \nOur findings have a number of implications for the identification e ff orts of the SN community: \n1) As mentioned above, almost half of our sample had a different type than listed in TNS or an evolving type over time, similar to the sample of Young SNe Ic we published in Williamson et al. (2023). Thus, we suggest that the community keep observing and classifying SNe Ib over time when they are discovered young - and in general young CCSNe, including Young SNe Ic - so that the community can be alerted about any SNe that change types or develop peculiarities. In addition, since one \nthree of our SN sample had time-evolving types (SN 2019odp, SN 2021ukt, and 2022nyo), with additional ones in the literature (e.g., SN 2008D, SN 2022crv) we suggest that alternative SN identification methods are used that can capture the SN's amount of transitioning between di ff erent SESN subclasses - one such method was suggested by us in Williamson et al. (2019b), where we developed a PCA method for the photospheric spectra of SESNe which can naturally capture transition SNe (as applied to SN 2022crv, see Figure 17 in Dong et al. 2023b). \n2) Incorporating early SN spectra into classification libraries as well as training sets for machine learning tools can improve the classification of young SNe. We suggest that the spectra of young SNe Ib, such as those presented here, are included in the spectral libraries of the various SN identification codes, e.g. SNID (Blondin et al. 2006), Superfit (Howell et al. 2005) and GELATO (Harutyunyan et al. 2008). For the benefit of the community, we produce and make public the SNID templates of our Young SNe Ib spectral dataset in this paper. \nOur release of these young spectra, which o ff er information about the outermost layers of SNe Ib ejecta, will hopefully facilitate work on constraining the characteristics and mass-loss mechanism of the SN progenitor and on investigating di ff erent spectral characteristics, such as measuring line velocities and strength, to quantify diversity in SNe Ib at early times. \nAcknowledgements. This work makes use of observations from the Las Cumbres Observatory global telescope network. M.M. and the METAL group at UVa acknowledge support in part from ADAP program grant No. 80NSSC22K0486, from the NSF grant AST-2206657 and from the HST GO program HST-GO16656. The Las Cumbres Observatory group is supported by NSF grants AST1911225 and AST-1911151. 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J., Valenti, S., et al. 2018, ApJ, 853, 62 Tominaga, N., Tanaka, M., Nomoto, K., et al. 2005, ApJ, 633, L97 \n- Tonry, J., Denneau, L., Heinze, A., et al. 2020, Transient Name Server Discovery Report, 2020-1094, 1\n- Tonry, J., Denneau, L., Heinze, A., et al. 2021, Transient Name Server Discovery Report, 2021-1799, 1\n- Tonry, J., Denneau, L., Weiland, H., et al. 2023, Transient Name Server Discovery Report, 2023-1450, 1 \nArticle number, page 10 of 12 \nTonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505 \nValenti, S., Sand, D., Pastorello, A., et al. 2014, MNRAS, 438, L101 \n- Williamson, M., Kerzendorf, W., & Modjaz, M. 2021, The Astrophysical Journal, 908, 150\n- Williamson, M., Modjaz, M., & Bianco, F. B. 2019a, ApJ, 880, L22\n- Williamson, M., Modjaz, M., & Bianco, F. B. 2019b, ApJ, 880, L22\n- Williamson, M., Vogl, C., Modjaz, M., et al. 2023, ApJ, 944, L49\n- Woosley, S. E. & Bloom, J. S. 2006, ARA&A, 44, 507\n- Yaron, O. & Gal-Yam, A. 2012, PASP, 124, 668', 'Appendix A: Observation Log': 'Table A.1: Spectra Observation Log of Young Type Ib. \nA & A proofs: manuscript no. sources \nTable A.1: continued. \nNote: Host galaxy HII region emission lines are removed from these spectra. Figure 1 shows the spectra without the galaxy emission lines.'}

2024arXiv240909499L

The ultrahighenergy UHE gammaray source 1LHAASO J00077303u is positionally associated with the composite SNR CTA1 that is located at high Galactic Latitude bapprox 10.5circ. This provides a rare opportunity to spatially resolve the component of the pulsar wind nebula PWN and supernova remnant SNR at UHE. This paper conducted a dedicated data analysis of 1LHAASO J00077303u using the data collected from December 2019 to July 2023. This source is well detected with significances of 21sigma and 17sigma at 8100 TeV and gt100 TeV respectively. The corresponding extensions are determined to be 0.23circpm0.03circ and 0.17circpm0.03circ. The emission is proposed to originate from the relativistic electrons and positrons accelerated within the PWN of PSR J00077303. The energy spectrum is well described by a powerlaw with an exponential cutoff function dNdE 42.4pm4.1fracE20rm TeV2.31pm0.11expfracE110pm25rm TeV rm TeV1 cm2 s1in the energy range from 8 TeV to 300 TeV implying a steadystate parent electron spectrum dNedEepropto fracEe100rm TeV3.13pm0.16expfracEe373pm70rm TeV2 at energies above approx 50 rm TeV. The cutoff energy of the electron spectrum is roughly equal to the expected current maximum energy of particles accelerated at the PWN terminal shock. Combining the Xray and gammaray emission the current spaceaveraged magnetic field can be limited to approx 4.5rm mu G. To satisfy the multiwavelength spectrum and the gammaray extensions the transport of relativistic particles within the PWN is likely dominated by the advection process under the freeexpansion phase assumption.

2024-09-01T00:00:00Z

['arXiv:2409.09499', '2024arXiv240909499L', '10.48550/arXiv.2409.09499']

['Astrophysics - High Energy Astrophysical Phenomena']

Deep view of Composite SNR CTA1 with LHAASO in gammarays up to 300 TeV

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09499.pdf

{'No Header': '. Article . \nSPECIAL TOPIC: \nJanuary 2023 Vol. 66 No. 1: 000000 \nhttps: // doi.org / ??', 'Deep view of Composite SNR CTA1 with LHAASO in γ -rays up to 300 TeV': 'LHAASO Collaboration * \n(The LHAASO Collaboration authors and a ffi liations are listed after the references.) \nReceived ; accepted \nThe ultra-high-energy (UHE) gamma-ray source 1LHAASO J0007 + 7303u is positionally associated with the composite SNR CTA1 that is located at high Galactic Latitude b ≈ 10 . 5 · . This provides a rare opportunity to spatially resolve the component of the pulsar wind nebula (PWN) and supernova remnant (SNR) at UHE. This paper conducted a dedicated data analysis of 1LHAASO J0007 + 7303u using the data collected from December 2019 to July 2023. This source is well detected with significances of 21 σ and 17 σ at 8 -100 TeV and > 100 TeV, respectively. The corresponding extensions are determined to be 0.23 · ± 0.03 · and 0.17 · ± 0.03 · . The emission is proposed to originate from the relativistic electrons accelerated within the PWN of PSR J0007 + 7303. The energy spectrum is well described by a power-law with an exponential cuto ff function dN / dE = (42 . 4 ± 4 . 1)( E 20 TeV ) -2 . 31 ± 0 . 11 exp( -E 110 ± 25 TeV ) TeV -1 cm -2 s -1 in the energy range from 8 TeV to 300 TeV, implying a steady-state parent electron spectrum dNe / dEe ∝ ( Ee 100 TeV ) -3 . 13 ± 0 . 16 exp[( -Ee 373 ± 70 TeV ) 2 ] at energies above ≈ 50 TeV. The cuto ff energy of the electron spectrum is roughly equal to the expected current maximum energy of particles accelerated at the PWN terminal shock. Combining the X-ray and gamma-ray emission, the current space-averaged magnetic field can be limited to ≈ 4 . 5 µ G. To satisfy the multi-wavelength spectrum and the γ -ray extensions, the transport of relativistic particles within the PWN is likely dominated by the advection process under the free-expansion phase assumption. \nPWN, γ -rays , UHE \nPACS number(s): 97.60.Bw;95.85.Pw;95.85.Ry; \nCitation: \nZhen Cao, et al, \n, Sci. China-Phys. Mech. Astron. 66 , 000000 (2023), https: // doi.org / ??', '1 Introduction': 'CTA1 is a famous composite supernova remnant (SNR), characterized by a shell SNR and a central pulsar wind nebula (PWN). It was discovered in the radio band and first proposed as a SNR by Harris & Roberts (1960). The detailed radio observation further revealed a northwest-incomplete shell with a diameter ≈ 1 . 8 · and a bridge-like structure towards the center, with a kinematic distance of ≈ 1 . 4 kpc according to the associated HI shell (Pineault et al., 1993, 1997; Sun et al., 2011). The first X-ray detection in the direction of CTA1 was made by ROSATSeward et al. (1995), which revealed both thermal emission from the outer shock-heated region and a non-thermal component from the central region. Subsequent studies of the central X-ray source (RX J0007.0 + 7302) with ASCA, XMM-Newton, Chandra and Suzaku observatory have resolved the X-ray emission', 'Physics, Mechanics & Astronomy': "into a point-like source likely corresponding to the central pulsar, and a di ff use nebula, implying a X-ray PWNe driven by the central active pulsar (Halpern et al., 2004; Lin et al., 2012; Slane et al., 1997, 2004). \nThe point-like source has been firmly identified as a pulsar (PSR J0007 + 7303), owing to the successful search for pulsation in GeV band (Abdo et al., 2008) and X-ray band (Caraveo et al., 2010; Lin et al., 2010). PSR J0007 + 7303, with a period of ≈ 316 ms, has a su ffi cient spin-down power ˙ E = 4 . 5 × 10 35 erg s -1 to drive the X-ray PWN. In addition, the corresponding characteristic age τ c is ≈ 13 . 9 kyr,consistent with the SNR age derived by Slane et al. (2004). In GeV regime, searches for the extended emission associated with radio shell and / or X-ray nebulae are ongoing (Abdo et al., 2012; Acero et al., 2013; Li et al., 2016). The latest results of the E > 1 GeV extended emission searching shows a ≈ 0 . 98 · disk-shaped source, potentially associated with the SNR due to its overlap with the radio emission contours (Ackermann et al., 2018). In the E > 50 GeV energy range, Zhou et al. (2024) report a possible ∼ 0 . 4 · extended γ -ray emission from the PWN. \nComposite SNR CTA1 is associated with an extended TeV γ -ray source, VER J0006 + 729, discovered by VERITAS experiment in the energy range 0 . 6 TeV -17 . 8 TeV (Aliu et al., 2013). The TeV source VER J0006 + 729 shows an elliptical Gaussian morphology with 1 σ angular extension of 0 . 30 · along the major axis and 0 . 24 · along the minor axis, with the orientation of the major axis of ≈ 17 . 8 · west of north, centered at the position of the pulsar PSR J0007 + 7303. Due to the positional and morphological coincidence of the TeV emission with the X-ray PWN, the TeV source VER J0006 + 729 is discussed as TeV PWN in studies such as Aliu et al. (2013) and Torres et al. (2014a). Generally, PWNe are clouds of magnetised plasma that are created inside SNRs by the highly relativistic outflow ('wind') of a pulsar, observed via synchrotron emission produced when energetic electrons interact with the magnetic field, or through inverse Compton (IC) radiation generated by the scattering of electrons o ff the background photon fields (see Gaensler & Slane (2006) and Mitchell & Gelfand (2022), for a comprehensive review). The latter mechanism is believed to be relevant to the TeV γ -ray emission of the PWN. Alternatively, hadronic mechanisms could also be the origin of TeV emission, although unidentified by observation, in which case relativistic hadrons collide with the ambient medium, producing TeV emission through π 0 decay. The study of UHE γ -ray emission is crucial in determining whether the γ -ray emission is predominantly from hadronic processes or the IC process, as IC emission above 100 TeV energies undergoes suppression due to the Klein-Nishina e ff ect. \nIndeed, the vast majority of Galactic TeV emittiers have been identified in the population of PWNe (H. E. S. S. Collaboration et al., 2018). Among them, CTA1 holds a special status as it is located at a relatively high Galactic Latitude b ≈ 10 . 5 · , experiencing minimal influence from Galactic di ff use emission and nearby TeV sources. This positioning makes it an excellent candidate for accurate measuring characteristics of TeV emission, serving as a valuable object for probing the physical model of TeV PWNe and distinguishing between leptonic and hadronic processes. Thanks to the wide field view, high sensitivity, and broad energy range of the Large High Altitude Air Shower Observatory (LHAASO) (Cao, 2010), our understanding of CTA1 has made impressive progress. Using approximately 2 years of data, the LHAASO collaboration has reported an extended TeV source, 1LHAASO J0007 + 7303u, which is tentatively shaped by a Gaussian with 1 σ angular extension of ∼ 0 . 2 · at energies E > 25 TeV and is positionally coincident with the X-ray / TeV PWN in Composite SNR CTA1. Particularly noteworthy is the significant detection of the ultra-high-energy (UHE, E > 100 TeV) γ -ray emission from the source 1LHAASO J0007 + 7303u, at a 13 σ confidence level (Cao et al., 2023). \nThis paper presents a deep observation of CTA1 with LHAASO using approximately 3 years of data, and proposes some possible mechanisms for the UHE emission through dedicated data analysis and discussion. Section 2 provides a brief introduction to the LHAASO detector array and the analysis method. In Section 3, we report the analysis results based on LHAASO data. A discussion of the multi-wavelength observation and a model for CTA1 is presented in Section 4. Finally, we summarize the conclusions in Section 5.", '2 LHAASO Data Analysis': 'LHAASO,constructed on Mountain Haizi, in Sichuan province, China, is a complex extensive air shower (EAS) array with a high sensitivity ( ∼ 1% CU 1 year) and a wide field of view (FOV, ∼ 2.24 Sr for the maximum zenith angle of 50 · ) for CRs and γ -rays. It consists of three subarrays, i.e., Water Cherenkov Detector Array (WCDA), Kilomiter Square Array (KM2A) and Wide Field-of-view Cherenkov Telescope Array (WFCTA). By Combining WCDA and KM2A, LHAASO can cover the energy range from ∼ 1 TeV to > 1 PeV for γ -ray observation, with the low-energy threshold depending on the zenith angle distribution to some extent. The performance of WCDA and KM2A has been studied in detail employing the Monte Carlo simulations (Cao et al., 2024a), and calibrated using the measurements of Crab Nebula as a standard candle (Aharonian et al., 2021a,b).', '2.1 Data selection and binning': "The KM2A events utilized in this study were taken during the period from 17th December 2019 to 31st July 2023. The data quality control system and the long-term performance monitoring of KM2A data can be found in (Cao et al., 2024b). After rigorous data selection, the total live time amounts to 1216 days. These events were further reduced and reconstructed, according to the selection criteria and reconstructing methods outlined in Aharonian et al. (2021a). We applied a γ -ray / background discrimination cut to select out all γ -like events, with the survival fraction of cosmic ray background events at approximately 4 × 10 -4 at 50 TeV energy compared to the CR observation. For the analysis presented in this paper, only KM2A γ -like events with zenith angles less than 50 degrees and reconstructed energies ( Erec ) above 40 TeV were included. The low-energy threshold is dictated by the higher threshold for the higher zenith angle of the source. At the LHAASO site, the CTA1 (Decl. = 73 · ) culminates at a zenith angle θ = 43.7 · and lies at zenith angles θ < 50 · for 6.2 hours per sidereal day. We binned our selected data into 5 logarithmically spaced Erec bins per decade. Within each Erec bin, we generated an 'on map' by filling events into a grid of the spatial pixels with dimensions of ∆ R . A . × ∆ Decl . = 0 . 1 · × 0 . 1 · based on the reconstructed direction. The number of isotropic background events, still predominantly cosmic-ray events, in each spatial pixel was estimated using the 'direct integral method' (Fleysher et al., 2004), from which we derived the 'background map'. \nThe WCDA data used in this study were collected from 5th March 2021 to 31st July 2023, covering a total of 796 days of live time. To obtain the available γ -like event set for source analysis, we implemented quality cuts, direction reconstruction, and γ -ray / background discrimination, as detailed in Aharonian et al. (2021b). The number of hit ( Nhit ) for each event was selected as a shower energy estimator. We divided our data into 5 intervals of Nhit , namely [100,200], [200,300], [300,500], [500,800], and [800, 2000]. Within each Nhit bin, we generated the 'on map' and 'backgound map' following the aforementioned procedure for KM2A data. \nA region of interest (ROI) specific to the CTA1 was defined as a 6 · × 6 · square region centered at the position of pulsar PSR J0007 + 7303 (R.A. = 1.757 · ; Decl. = 73.052 · ). We selected the 'on map' and 'background map' within ROI and for all energy bins to following analysis. In total, there are 14 energy bins represented by Nhit bins for WCDA data and Erec bins for KM2A data. The angular and energy resolution are dependent on the zenith angle and the energy. Considering the position of pulsar PSR J0007 + 7303, the distribution of zenith angle is constant, and thus the median energy in each energy bin slightly varies with the spectral shape of the source.", '2.2 Maximum Likelihood Analysis and Statistic Test': "We utilized a maximum likelihood fit using parametric spatial and spectral models to determine the statistical significance of γ -ray emissions detected by LHAASO. This process involved generating expected source counts map ('src map') of γ -ray signals in our ROI through forward-folded models with the detector response. We then compared the sum of the model counts listed in 'src map' and the background events listed in 'background map' with the actual observed counts ( Nobs ) listed in 'on map'. Given a model θ with spatial and spectral parameters, we maximized the likelihood of the model comparing with Nobs as follows: \nln L ( Nobs | θ ) = Nbins X j = 0 lnP ( N j obs | θ ) (1) \nwhere Nbins is the number of the data bins in the ROI and the analysis energy, P is the Poisson probability of detecting N j obs events in each bin given the model parameters θ . \nThe ratio of the maximum likelihood defines a test statistic (TS): \nTS = 2 ln( L ( θ 1) / L ( θ 0)) (2) \nwhere θ 0 and θ 1 represent the parameters of the model of the null and alternative hypotheses, respectively. Based on Wilks' Theorem, the TS follows a χ 2 n distribution, where n is the degrees of freedom (dof.) derived from the di ff erence in the number of free parameters between the models. Typically, detection significance is determined by comparing the likelihood of the background-only model (null hypothesis) with that of the signal plus background model (alternative hypothesis). Unless otherwise mentioned, the TS utilized in this study is for detection significance. We employed the di ff erence in detection significance \n( ∆ TS) to compare the spatial and / or spectral models of the source. It is important to note that the di ff erence in TS cannot be used to quantitatively determine the preferred model when the models are not nested.Alternatively, the Akaike information criterion test (AIC, Akaike, 1974) can be considered. The AIC is defined as AIC = 2 k -2 ln L , where k is the number of parameters in the model. In this context, the best hypothesis is considered to be the one that minimizes the AIC. A qualitative strength of evidence rules to assess the significance of a model is based on the di ff erence in AIC ( ∆ AIC) values between the two models. If ∆ AIC > 5, then it is considered strong evidence against the model with higher AIC and ∆ AIC > 10 is considered as decisive evidence against the model with higher AIC (Kulkarni & Desai, 2017; Liddle, 2007).", '3.1 Morphology': 'We evaluated the TS value for a point-like signal with a power-law spectral shape dN / dE ∼ E -2 . 7 at each pixel in the map. In this case, the significance is represented by √ TS . The √ TS map is displayed within a 2 . 5 · × 2 . 5 · square area centered at the position of pulsar PSR J0007 + 7303 in the energy range of 8 TeV < E < 100 TeV and E > 100 TeV, respectively, as illustrated in Figure 1. In both the 8 TeV < E < 100 TeV and E > 100 TeV energy ranges, an obvious γ -ray emission excess structure is observed in the inner region of the radio shell. \nFigure 1 The significance maps of CTA1 in the energy ranges of 8 TeV < E < 100 TeV (a) and E > 100 TeV (b). Overlaid are the GB6 image countour (4850MHz) in gray. Cyan stars represent the position of the pulsar PSR J0007 + 7303. \n<!-- image --> \nWe studied the morphology of the γ -ray emission in the energy range of 8 TeV < E < 100 TeV and E > 100 TeV, respectively. Due to the impact of angular resolution, the actual morphology of the γ -ray emission in CTA1 should be addressed by using a forward-folded approach. We first explored the routine and mathematical-empirical geometrical models, such as point, Gaussian and disk. Motivated by the observations of VERITAS in the energy range of 0 . 6 TeV < E < 17 . 7 TeV, we also examined the elliptical Gaussian model. However, due to the limited angular resolution of LHAASO, which is 3-10 times larger than that of Image Atmosphere Cherenkove Telescope (IACT), we encountered convergence issues in the case of all parameters being free. To address this, we maintained the rotation angle fixed. For each energy range, the spectrum of the γ -ray emission in CTA1 region was initially modeled by a simple power-law for the 3D maximum likelihood analysis. Fit results are detailed in Table 1. \nComparing the Gaussian model with the point model, we find that the Gaussian model improved the fit with a significance of 6.7 σ ( ∆ TS = 45 . 4) at energies 8-100 TeV and 3.5 σ ( ∆ TS = 13 . 2) at energies > 100 TeV, respectively. These results confirm that the TeV source detected by LHAASO is indeed extended. Using an elliptical gaussian intensity distribution instead of a Gaussian profile only marginally improves the fit likelihood by 1 . 6 σ ( ∆ TS = 2 . 8 ) at energies 8-100 TeV and by a negligible amount at energies > 100 TeV. This suggests that LHAASO cannot claim an obviously asymmetric structure as VERITAS has previously reported. To quantitatively compare the Gaussian and disk models, we evaluated their AIC values since these models \nTable 1 Morphological Fit Results for CTA1. \nP 95 is the statistical positional uncertainty at a 95% confidence level. The Major Axis is the radius of 100% flux region for Disk model and the radius of 39% flux region for Gaussian model. The AIC values are subtracted from that of the point model for a clear comparison. For the elliptical Gaussian model, we fixed the rotation angle (from west to north 17.8 · ) determined from VERITAS observations to ensure fitting convergence. \nTable 2 Spectral Fitting Results for CTA1 above 8 TeV \nNote: PL stands for the power-law model defined by dN / dE = N 0( E / E 0) -α , PLC represents the power-law with exponential cuto ff defined by dN / dE = N 0 × ( E / E 0) -α exp [ -( E / Ec )], and LP represents the log-parabola model defined by F ( E ) = N 0( E / E 0) -( α + β log 10( E / E 0)) . For the LP model, the latter value listed in α column is the β parameter. The AIC value is subtracted from that of the PL spectrum for a clear comparison. \nare not nested. Due to a small di ff erence in AIC ( ∆ AIC < 4), we are unable to clearly distinguish between these two extended models. For the subsequent analysis, we tentatively consider the Gaussian profile as our benchmark spatial model. With this choice, the significance is estimated to be 21 σ and 17 σ at 8 -100 TeV and > 100 TeV, respectively, according to the TS values shown in Table 1.', '3.2 Spectrum': 'For the spectral analysis of CTA1, we performed a maximum likelihood fitting in the energy range above 8 TeV, using a Gaussian model described above. We compared three di ff erent spectral shapes for CTA1: a simple power-law (PL), a powerlaw with exponential cuto ff (PLC), and a log-parabola (LP). As shown in Table 2, the addition of a curvature in the spectrum is significantly improve the fit ( > 6 σ ). Comparing the LP and PLC spectrum, the best spectral model is PLC with an improvement in modeling with ∆ AIC = 8.4. Therefore, we can conclude that the best-fit spectral model is PLC model with an index of 2.31 ± 0.13 and a cuto ff energy of 110 ± 25 TeV. The integral energy flux above 8 TeV is F > 8 TeV ≈ 4 . 9 × 10 -12 erg cm -2 s -1 . We determined the spectral points by performing a maximum likelihood analysis in each Nhit or Erec bin considering the fixed PLC spectral shape. Upper limits on flux at a 95% confidence level were derived when CTA1 had TS < 4 (2 σ ) in a given bin. The di ff erential photon spectrum is shown in Figure 2, with spectral data points listed in Table 3. \nUsing the PLC spectrum presented in Table 2, the energy of the gamma-ray events from the CTA1 region were re-estimated event by event. We detailed the events with reconstruction energy above 250 TeV considering more stringent exclusion of cosmic ray background. Four gamma-like events with reconstructed energy larger than 250 TeV are listed in Table 4. The \nFigure 2 LHAASOdi ff erential γ -ray spectrum of the PWN CTA1. The red line and the orange butterfly represent the best-fit PLC model and its uncertainties, respectively. \n<!-- image --> \nmaximum energy of the events is ≈ 300 TeV. \nTable 3 Di ff erential Flux Measurements of CTA1Note: Mid-Energy represents the median energy corresponding to di ff erent bins. The error is statistic. The systematic uncertainties of the flux are estimated to 8%.', '3.3 Systematic Uncertainties': 'To assess the robustness of our results we performed a number of systematic checks similar to those employed in Cao et al. (2024c). The pointing error were estimated to 0 . 04 · for WCDA and KM2A data. A conservative estimation of the systematic error in Gaussian extension measurement was on the order of 0 . 08 · . The systematic uncertainties in flux were estimated to be 8%, which arise from the atmospheric model used in the Monte Carlo simulations. Furthermore, we assessed the uncertainties arising from the GDE by incorporating a dust template into our fitting. The impact on the flux was found to be less than 2%. \nTable 4 Photons with Erec > 250 TeV for CTA1 \nNote: Erec represents reconstructed energy based the PLC spectral shape. Ne is the number of electromagnetic particles. N µ is the number of muons detected in the region with a distance farther than 15 m from the core of the shower.', '4.1 The Nature of the γ -ray emission detected by LHAASO': "To provide a clear identification of the γ -ray emission in the energy range from 8 TeV to > 100 TeV detected by LHAASO, we compare its position and extension to observations in other bands, as shown in Figure 3. The γ -ray emission detected by LHAASO can be firmly associated with the X-ray PWN powered by the young pulsar PSR J0007 + 7303, as the position of the γ -ray emission coincides with that of X-ray PWN, while being clearly distinguished from the SNR radio shell. The nonthermal X-ray emission has an extension of approximately 0 . 3 · in ASCA observation (Slane et al., 1997) and 0 . 16 · in Suzaku observation (Lin et al., 2012). Due to the constraints of the small field of view of the X-ray telescope and the influence of thermal X-rays, it is possible that the size of the X-ray PWN is more extensive. Conservatively, we consider the lower limit (LL) size of the X-ray PWN at 0 . 3 · , and thus the 39% flux radius of the X-ray PWN is larger than 0 . 15 · assuming a Gaussian X-ray flux distribution. There is no conflict in extension between the X-ray and γ -ray observations. We favor a one-zone leptonic model in which the same population of electrons contributes to both the X-ray and γ -ray emission. Additionally, evidence for leptonic process includes possible energy-dependent morphology detection. The cooling of energetic electrons as they are transported away from the pulsar results in the size of the emission shrinks and its center moves closer to the pulsar at higher energies. This behavior is supported by observations from Fermi-LAT, VERITAS, and LHAASO, with a confidence level of 2.5 σ . \nIt is widely believed that the γ -ray radiation in the PWNe is produced in the leptonic scenario. However, as the IC emission above 100 TeV energies undergoes suppression due to the Klein-Nishina e ff ect, the hadronic emission is possibly identified in UHE band. Indeed, Crab spectrum detected by LHAASO implies the possible contribution of the hadronic process due to the hardening at PeV energies(Liu & Wang, 2021). In the case of CTA1, we specifically investigate whether the UHE photons are dominated by the π 0 -decay gamma rays generated by the CR protons interacting with the interstellar medium (ISM). To investigate the environment around the pulsar, Mart'ın et al. (2016) conducted a dedicated study based on a survey data of the intensity of CO line provided in Dame et al. (2001). Most of the data in the velocity range -26 km s -1 < v < -6 km s -1 , which corresponds to the distance of CTA1, is tagged as noise. Due to the absence of a molecular cloud, we disfavor the UHE γ -ray emission originating from protons escaping a shock around CTA1 SNR and illuminating the molecular cloud. Additionally, specific regions were assigned a density of 0.017 and 0.037 cm -3 based on the X-ray absorption observations along the line of sight of CTA1, indicating a lower density in the vicinity of CTA1 (Slane et al., 1997). Assuming an average density of 0 . 1 cm -3 , which is the upper limit from the ISM density of the Galaxy in the direction of CTA1, we can conservatively estimate a total energy of proton with energies > 1 PeV to be Wp > 1 × 10 47 ergs based on the UHE γ -ray flux observed. The energy budget is larger than the total energies from the pulsar spin-down power, roughly estimated by ˙ E τ c ≈ 1 × 10 46 ergs. we can also exclude that the CTA1 PWN accelerates a su ffi cient number of PeV protons to dominate the UHE γ -ray emission observed.", '4.2 The broad-band spectrum: implications for the magnetic field and particle spectrum': "In the leptonic scenario, the broad-band nonthermal emission is dominated by two mechanisms: (1) synchrotron radiation for the emission from radio to X-ray, and (2) IC radiation for emission in the γ -ray band. In the one-zone model, the same population of relativistic electrons can produce the X-ray and γ -ray emission by interacting with the ambient magnetic field and the soft photon field, respectively. \nThe γ -ray emission is contributed by IC scattering of relativistic electrons interacting with several target photon fields, \nFigure 3 Multi-wavelength observations in CTA1 region. The circles represent 39% flux region. The dashed circle represents the lower limit of the extension. The size of the X-ray PWN has not been determined to date. The non-thermal X-ray emission is detected by ASCA in the 0 . 3 · region around the pulsar. Conservatively, the lower limit radius of the 39% X-ray flux region is estimated to 0 . 15 · , assuming a Gaussian flux distribution. \n<!-- image --> \nincluding the cosmic microwave background (CMB), infrared radiation field and star-light photon field. In the composite SNR CTA1 region, almost all of the VHE and UHE γ -ray fluxes come from the IC scattering with CMB, since the contributions from interactions with other photon fields are suppressed due to their low number densities and the Klein-Nishina e ff ect (Mart'ın et al., 2016; Torres et al., 2014a). Because the property of the CMB is well quantified, the γ -ray emission in LHAASO band could provide precise information about the spectrum of parent electrons. At energies above a few tens of TeV, the energy of the upscattered CMB photon E γ and that of the parent electron Ee could be linked through the following simple relation (LHAASO Collaboration et al., 2021), \nEe ≃ 0 . 85( E γ/ 300 TeV) 0 . 77 PeV . (3) \nThe energy range of γ -rays detected by LHAASO is from ≈ 8 TeV to ≈ 300 TeV, which corresponds to the parent electron energy ranging from ≈ 50 TeV to ≈ 850 TeV. We assume that the electron spectrum follows a steady-state power-law distribution terminated by a super-exponential cuto ff , i.e. E -α e exp[ -( Ee / Ee , c ) 2 ]. The spectral index and cuto ff energy of parent electron in the energy band 50 TeV ≲ Ee ≲ 880 TeV can be constrained to α = 3 . 13 ± 0 . 16 and Ee , c = 373 ± 70 TeV, respectively, by fitting to the LHAASO data as shown in Figure 4. \nThe mean energies of the synchrotron ( Esyn ) and IC photons produced by the same population of electrons in the ambient magnetic field and CMB are related by \nEsyn = 4 . 6( E γ/ 50 TeV) 1 . 5 ( B / 4 . 5 µ G ) keV . (4) \nThe energy range of the X-rays detected by ASCA is Esyn = 0 . 5 -10 KeV, which corresponds to the γ -ray energy range of E γ ≈ 11 -83 TeV considering a magnetic filed of 4.5 µ G. This implies that the observed X-ray and γ -ray emission from \nCTA1 are roughly produced by the population of electrons in the same energy range, in which the electron spectrum can be well constrained by LHAASO. Utilizing both the ASCA and LHAASO observations of the PWN, we can roughly constrain the current space-averaged magnetic field strength B to ∼ 4.5 µ G for the PWN CTA1, as shown in Figure 4. The deduced magnetic field is at the same level with the interstellar magnetic field. Such low magnetic field has also been reported in other PWNe by analysing their synchrotron spectra, like HESS J1825-137 and HESS J1809-193. The magnetic field in young PWNe decreases with time during the free-expansion phase, due to the expansion of the volume of PWNe. Therefore, the magnetic field inside PWNe is expected to reach its minimum at the end of the free-expansion phase. We also note that although the magnetic field in PWN CTA1 is as low as the interstellar magnetic field, this does not necessarily imply that the turbulence is also similar. The possible stronger turbulence in the PWN could prevent particles from escaping to the ISM. It's also worth noting that the non-thermal X-ray flux is not extracted from the entire TeV PWN region detected by LHAASO, but rather from a specific 0 . 3 · region within it. A more extended non-thermal X-ray survey around the entire TeV PWN is needed to provide a precise estimation of the current space-averaged magnetic field strength. \nIn the PWN CTA1, the energy loss of the electrons is primarily dominated by synchrotron radiation, with a cooling timescale given by tc , syn ≈ 13 . 0( Ee / 40 TeV) -1 ( B / 5 µ G ) -2 kyr. Considering a time-averaged magnetic field strength of B ∼ 5 µ G, the cooling time roughly equals the pulsar's characteristic age of 13.9 kyr. Electrons with an energy greater than ≈ 40 TeV are expected to be in the fast cooling regime. Assuming that particle escape and the time evolution of energy injection rate can be neglected, the expected spectral index of the injected electrons can be approximated as p ∼ α -1 = 2 . 13 ± 0 . 16 in the energy range from ≈ 40 TeV to ≈ 373 TeV, considering the spectral steeping as being due to only the synchrotron loss. Conversely, electrons with Ee ≲ 40 TeV are in the slow cooling regime, where the spectral index of the integrated electrons remains almost unchanged with respect to that of the injected spectrum. Therefore, a spectral break is expected in the steady-state electron spectrum at an energy around 40 TeV, which is indicated by the possible break observed in the γ -ray flux around the transition energy ( ∼ 8 TeV) between the VERITAS and LHAASO energy range (see Figure 4). In our time-dependent model, we take into account the decrease in energy injection rate and magnetic field strength over time, as well as the potential escape of particles. Through a detailed numerical simulation of the evolution of the particle distribution (see appendix), we find that an intrinsic injection spectrum with an index of p = 2 . 2 could indeed result in a present electron spectrum with index of α ≈ 3 . 11 in the energy range from ≈ 40 TeV to ≈ 350 TeV, and of α ≈ 2 . 18 below ≈ 40 TeV. This is consistent with our previous simple explanation of the parent electron spectral properties of the PWN CTA1. \nIn the case of standard shock acceleration in the Bohm di ff usion regime, a simple analytical presentation of the electron spectrum at shock over the entire energy range follows the form of a super-exponential cuto ff , i.e., Q ( Ee ) ∝ E -p e exp( -E 2 e / E 2 e , max ) (Zirakashvili & Aharonian, 2007). Considering a compression ratio of 3-4, the index p is estimated to be 2-2.25, which is consistent with the spectral index of the injected electrons inferred by the LHAASO observations. The maximum energy ( Ee , max ) of accelerated electrons is limited by the requirement of confinement of the particles inside the termination shock. This requires that the Larmor radius RL is smaller than the termination shock radius Rs , i.e., RL = ε Rs , where ε is the so-called containment factor with the value of 0 < ε < 1. The Larmor radius of the electron with energy Ee is \nRL = Ee / ( eBs ) , (5) \nwhere e is the electron charge, Bs is the post-shock magnetic field strength, defined as (Kennel & Coroniti, 1984) \nBs ∼ (3( η BL ( t ) / c ) 0 . 5 ) / Rs , (6) \nwhere L ( t ) is the spin-down luminosity of the associated pulsar at time t , η B is the magnetic energy fraction. Using Equation (5) and (6) and the condition RL = ε Rs , the maximum energy of the accelerated electrons is given by \nE e , max = 3 e ε p η B L ( t ) / c . (7) \nWe can see that the maximum energy of electrons evolves as the spin-down power with time. Taking into account the spindown power ˙ E = 4 . 5 × 10 35 erg / s at current time, the maximum energy can be estimated as Ee , max ≈ 430( ε 0 . 16 )( η B 0 . 6 ) 0 . 5 TeV. The cuto ff energy of the steady-state electron spectrum, approximately 373 TeV, should be the time-integrated value over the last few thousand years. However, the electrons injected in earlier stages experienced more energy losses, the time-integrated average maximum energy is expected to be close to the maximum energy of the recently injected electrons, which indicates that the maximum energy of electrons injected at current time could be estimated by the maximum energy inferred from our observation. Moreover, in our time-dependent modeling, the cuto ff energy of approximately 350 TeV in the accumulated electron spectrum \n(see Fig A1 in the Appendex) indeed roughly coincides with the maximum energy of 430 TeV in the currently injected electron spectrum. This implies that the cuto ff energy derived from the steady-state electron spectrum could be a good estimation of the current particle acceleration ability of the termination shock in PWNe similar to CTA1. \nFigure 4 The di ff erential energy spectrum of the PWN CTA1. The red line represents the expectation from a one-zone leptonic model, assuming a magnetic field strength of 4.5 µ G and considering only CMB target photons. The electron spectrum follows a broken power law distribution, where dNe / dEe ∝ E -3 . 13 e exp[ -( Ee / 373 TeV) 2 ] for energies E > 40 TeV and dNe / dEe ∝ E -2 . 13 e for energies E < 40 TeV. The time-dependent model (pure advection scenario) is referenced in the Appendix. \n<!-- image -->", '4.3 The Morphology: implications for propagation mechanism': "Relativistic particles accelerated at the termination shock propagate in PWN mainly by two transport mechanisms, i.e. advection and di ff usion. To investigate the e ff ects of particle advection and di ff usion in the PWN, we performed detailed simulations of the spectrum and morphology of CTA1 for each transport scenario. The description of our model can be referred to the Appendix. \nWe first tune the parameters for both pure advection and pure di ff usion scenarios to explain the multi-band non-thermal spectrum from the PWN CTA1. The parameters used to fit the flux data are almost similar in these two cases, except for those related to particle transport (see Table A1 in the Appendix). As a result, the calculated fluxes in the two cases are also similar, so we only plot the result in advection case for convenience (see Figure 4). \nThe morphology of the emission is an interplay between particle transport and energy losses in the PWN. Figure 5 shows the energy dependence of the γ -ray extension for di ff erent transport mechanisms. In our model, the γ -ray extension at a given energy refers to the 39% flux size, in which the γ -ray flux is 39% of the total flux in the PWN. It is clear that overall the extension of the γ -ray emission decreases with increasing photon energies in both pure advection and di ff usion scenarios (for Kolmogorov turbulence), as the electrons responsible for producing higher energy γ -ray photons have a shorter lifetime, hence a shorter travelling distance. We can also see that the energy dependence of the size is not strictly a power-law in the whole energy range, because the electrons responsible for the lower energy γ -rays (below few TeV) are not completely cooled at present age. The calculated size in di ff usion scenario is less dependent on energy than that in advection scenario (the curve is flatter), particularly in the higher energy range. This can be explained by the fact that higher energy electrons di ff use faster, which compensates the e ff ect of stronger energy losses on the propagation distance. \nFigure 5 shows that the size measured by Fermi-LAT, VERITAS and LHAASO could be well explained by the advection scenario, however, the extension calculated in the di ff usion scenario is much smaller. The reason for this di ff erence is that in the advection scenario, particles propagate faster in the inner nebula than in the outer region, causing them to accumulate at a larger radius. While in the case of di ff usion, the particle density decreases roughly exponentially with increasing distance from \nthe central source, resulting in a more compact distribution of particles. The di ff erent impacts of particle transport on particle distribution are illustrated in Figure A2 in the Appendix. For the di ff usion scenario, it is unlikely to fit the data by simply increasing the di ff usion coe ffi cient, since larger di ff usion coe ffi cient means greater escape losses, which will make it hard to explain the observed flux. Moreover, when particles start to escape the PWN, further increasing di ff usion coe ffi cient would have very limited impact on the extension, because escaped particles are assumed not to contribute any emission outside the PWN. In fact, recent works suggest that escaped particles also produce IC emissions by interacting with the CMB photons, an e ff ect which we did not consider here (Martin et al., 2024). However, the similar size of the X-ray and γ -ray emission region may indicate that particle escape is not important for PWN CTA1, if the magnetic field is confined in the PWN. \nIn our simulation, the radius of PWN is a crucial factor which can a ff ect the size of the emission. The PWN radius is mainly dependent on the mass of SN ejecta, the density of ISM, the initial spin-down luminosity and the age of the system. For simplicity, we assume that the evolution of PWN CTA1 is still in the free-expansion phase, and choose reasonable parameters to make the PWN radius as large as possible. For the chosen parameters (see Table A1 in the Appendix), the PWN radius at present time is about 11.3 pc (A larger radius requires extreme parameters). The largest 39% flux size of the γ -ray emission corresponding to this PWN radius in the pure di ff usion scenario is around 0.1 · , which is significantly smaller than the observed size (see Figure 5). On the other hand, we can estimate the upper limit of the PWN radius by the fact that it should be smaller than the corresponding SNR radius, which is ∼ 20.4 pc infered from the radio observation (Mart'ın et al., 2016). Even for such radius, the expected 39% flux size in the di ff usion scenario is ∼ 0.2 · , which still does not fit the data as well as the advection scenario. Therefore, the pure di ff usion scenario is unlikely to explain the observed PWN size, and we suggest that the particle transport is dominated by advection in the nebula of CTA1. \nThe radius of PWN can also be influenced by its evolutionary stage. While we have assumed free-expansion phase in our calculation, it remains unclear whether the PWN CTA1 is in the free-expansion phase or the compression phase. Its relatively large age ( ∼ 10 kyr) compared to the typical young PWNe indicates it may have already passed the free-expansion phase. Nonetheless, the large PWN radius ( ∼ 10 pc indicated by observations) seems not to favor compression by the reverse shock. The attempt to fit the CTA1 spectrum using the reverberation model by Mart'ın et al. (2016) also failed due to the incapability of fitting the VERITAS data. This may imply that the CTA1 is in an intermediate phase, where the PWN is currently under transition from the free-expansion phase to the compression phase. There is another possibility that some parts of the PWN are in free-expansion phase, while other parts in compression phase, due to di ff erent density of the ISM in the northwest and southeast direction to the PWN, as suggested by Mart'ın et al. (2016). If the PWN CTA1 is indeed in this intermediate phase, our modelling based on the assumption of free-expansion phase may still applies to some extent, since the compression e ff ect of the reverse shock may be relatively small at that time. Finally, a more sophisticated PWN model is needed to study particle transport during the reverberation phase. Additionally, future detailed observations in radio and X-rays may help distinguish between the di ff erent evolutionary stages of the PWN CTA1.", '5 Conclusions': 'Using about three years of KM2A data and two years of WCDA data, we have a deep view of γ -rays emission from composite SNR CTA1 in energy range from ≈ 8 TeV to ≈ 300 TeV. The LHAASO source is detected with a significance of 21 σ in the energy band 8-100 TeV and 17 σ at energies above 100 TeV. The source is a significant extended source with a 39% flux radius of ≈ 0 . 23 · at energy band 8-100 TeV and of ≈ 0 . 17 · at energies above 100 TeV, based on a Gaussian profile. The best spectral model is PLC with dN / dE = (42 . 4 ± 4 . 1)( E 20 TeV ) -2 . 31 ± 0 . 11 exp( -E 110 ± 25 TeV ) TeV -1 cm -2 s -1 in the energy range from ≈ 8 TeV to ≈ 300 TeV. The integral energy flux above 8 TeV, F > 8 TeV ≈ 4 . 9 × 10 -12 erg cm -2 s -1 . Given that the emission is extended, with a centroid near the pulsar PSR J0007 + 7303 and its X-ray PWN, we confirm that the γ -ray emission detected by LHAASO is physically associated with the PWN in the composite SNR CTA1. Additionally, the plausible morphological evolution in the γ -ray emission can be confirmed by combining Fermi-LAT, VERITAS and LHAASO observations, further implying the γ -ray leptonic origin. We also investigate whether or not the UHE γ -ray emission is from a hadronic process. Due to the absence of a molecular cloud and the lower density of the ISM, we can almost rule out the UHE γ -ray emission being dominated by a hadronic process. In the leptonic scenario, we can accurately determine the electron spectrum above 50 TeV energies. The steady-state electron spectrum index of approximately 3.13 implies that the intrinsic injected electrons index should be around 2.13, which is consistent with the expected acceleration mechanism at the terminal shock of PWNe. The cuto ff energy might be treated as a good indicator of the maximum energy of the current injected spectrum. Combined with X-ray observations, \nFigure 5 The γ -ray extensions of the PWN CTA1. The simulated extension corresponds to the 39% flux size, which means that the flux inside this region is 39% of the total flux in the PWN. The black and grey dashed lines represent the extension calculated by the pure advection and pure di ff usion scenarios. \n<!-- image --> \nthe current space-averaged magnetic field of the CTA1 PWN can be roughly limited to ∼ 4 . 5 µ G. Under the assumption of free-expansion phase, we favor advection dominating the particle transportation in the CTA1 PWN, in order to satisfy the multi-wavelength spectrum and the γ -ray extensions.', 'Acknowledgements': 'We would like to thank all sta ff members who work at the LHAASO site above 4400 meters above sea level year-round to maintain the detector and keep the water recycling system, electricity power supply and other components of the experiment operating smoothly. We are grateful to Chengdu Management Committee of Tianfu New Area for the constant financial support for research with LHAASO data. We deeply appreciate the computing and data service support provided by the National High Energy Physics Data Center for the data analysis in this paper. This research work is also supported by the following grants: in China by the National Natural Science Foundation of China NSFC No. 12393851, No.12393854, No.12393852, No.12393853, No.12022502, No.12205314, No.12105301, No.12261160362, No.12105294, No.U1931201, No.2024NSFJQ0060, and in Thailand by the National Science and Technology Development Agency (NSTDA) and the National Research Council of Thailand (NRCT) under the High-Potential Research Team Grant Program (N42A650868)..', 'AUTHOR CONTRIBUTIONS': 'S.Q. Xi, S.Z. Chen and B. Li led the drafting of the text. Y.Z. Li and S.Q. Xi conducted the data analysis, and Y. Huang performed the cross-check for the data analysis. B. Li and S.Q. Xi led the modeling e ff orts. S.Z. Chen and S.C. Hu provided the sky map and contributed to the background estimation. Zhen Cao, the spokesperson of the LHAASO Collaboration and the principal investigator of the LHAASO project. All other authors participated in data analysis, including detector calibration, data processing, event reconstruction, data quality checks, and various simulations, and provided valuable comments on the manuscript. \nτ \n0', 'Appendix': 'To obtain the particle spatial and energy distribution in PWN, we need to solve the advection-di ff usion equation (assuming spherical symmetry): \n∂ N e( r , E e , t ) ∂ t = Q e( E e , t ) + 1 r 2 ∂ ∂ r " r 2 D ( E e , t ) ∂ N e( r , E e , t ) ∂ r # -1 r 2 ∂ ∂ r h r 2 V ( r ) N e( r , E e , t ) i + ∂ ∂ E e h ˙ E e N e( r , E e , t ) i , (A1) \nwhere N e( r , E e , t ) is the di ff erential number density of electrons and positrons. The first term of the equation is the source term. The injection spectrum of electrons is assumed to follow a broken power-law with supper exponential cuto ff distribution. It\'s a good approximation to express the spectrum as (LHAASO Collaboration et al., 2021) \nQ e( E e , t ) = Q 0( t ) E -α 2 e h 1 + ( E e / E b) -( α 2 -α 1) i -1 exp h -GLYPH<0> E e / E e , max( t ) GLYPH<1> 2 i , (A2) \nwhere Q 0( t ) is the normalization factor, which can be determined by solving L ( t ) = R E e , max E e , min E e Q ( E e , t ) d E e. E e , min and E e , max are the minimum and maximum energy of injected electrons. The maximum energy of electrons accelerated at the termination shock could be estimated by Equation (7). E b is the break energy. α 1 and α 2 are the spectral indexes of the lower energy and higher energy part of the electron spectrum. \nAssuming a braking index n = 3, the spin-down luminosity L ( t ) of the pulsar evolves with time as (Gaensler & Slane, 2006) \nL ( t ) = ˙ E GLYPH<16> 1 + t age /τ 0 GLYPH<17> 2 (1 + t /τ 0) 2 , (A3) \nwhere ˙ E is the pulsar\'s spin-down power at its current age t age \nt age = τ c GLYPH<20> 1 -( P 0 P ) 2 GLYPH<21> , (A4) \nwith τ c = P / 2 ˙ P the characteristic age of pulsar. P 0 and P are the initial and current spin periods, respectively. The initial spin-down timescale of the pulsar is defined as \n= \nτ \nc( \nP \n0 \nP \n) \n. \n(A5) \nThe second term on the right hand side of Equation (A1) is the di ff usion term. We assume that the di ff usion coe ffi cient is homogeneous in the PWN, while it has time and energy dependence of D ( E e , t ) = D 0( t ) ( E e / 1 TeV) δ , with D 0( t ) being the normalization factor at 1 TeV and δ = 1 / 3 for the Kolmogorov turbulence. D 0( t ) is assumed to be inversely proportional to the magnetic field B ( t ). The evolution of magnetic field B ( t ) in PWN could be calculated by (Pacini & Salvati, 1973) \ndW B( t ) dt = η B L ( t ) -W B( t ) R pwn( t ) dR pwn( t ) dt , (A6) \nwhere W B = B 2 R 3 pwn / 6 is the total magnetic energy in PWN and η B the fraction of the spin-down luminosity converted into magnetic energy. \nThe third term describes particle advection by the flow downstream the termination shock. The bulk velocity of the downstream plasma flow is supposed to decreases with radius as V ( r ) = V 0( r / R ts) -β , where R ts( t ) is the radius of termination shock and β (0 ≤ β ≤ 2) describes the spatial dependence of advection velocity. \nThe last term represents the energy losses of relativistic electrons due to synchrotron radiation, IC scattering and adiabatic cooling, which are given by \n˙ E e = 4 3 σ T γ 2 U B + X Ui GLYPH<0> 1 + 4 γϵ 0 , i GLYPH<1> 3 / 2 + E e 3 1 r 2 ∂ ∂ r h r 2 V ( r ) i , (A7) \nwhere U B = B 2 / 8 π is the energy density of magnetic field and Ui is the energy density of the i -th component of the interstellar radiation field (ISRF). Following Torres et al. (2014b), we consider three components for the radiation field in the direction \n2 \nof CTA1: the cosmic microwave background (CMB, T = 2.73 K, U = 0.25 eV cm -3 ), far-infrared radiation field ( T = 70 K, U = 0.1 eV cm -3 ), and near-infrared radiation field ( T = 5000 K, U = 0.1 eV cm -3 ). ϵ 0 , i is the average photon energy of the radiation field, which equals to 2 . 8 k B Ti for blackbody / greybody radiation with Ti being the temperature. \nEquation (A1) is solved numerically in this work. The computational domain is the region between the termination shock and the outer boundary of the PWN. The evolution of the radii of termination shock and PWN ( R pwn( t ) and R ts( t )) could be obtained using the method given by Gelfand et al. (2009). At the termination shock, the inner boundary condition requires that the number of particles injected from the upper stream should equal to that transported downstream by advection and di ff usion (Peng et al., 2022). At the outer boundary, a free escape condition is imposed to simulate the particle escape from PWN. The model described above is then applied to the nebula of CTA1. The parameters of the model used to simulate the the particle distribution and emission from the nebula of CTA1 are shown in Table A1. \nTable A1 Parameters used to fit the SED and γ -ray extensions of the PWN CTA1 in the pure advection and pure di ff usion scenarios. \nThe time-integrated electron spectrum calculated by our model is shown in Figure A1. In order to compare the numerical result with our measurements, we use a broken power-law with an exponential cuto ff function to fit the spectrum from 0.5 TeV \nto 1 PeV, in which the break energy is fixed at 40 TeV. The spatial distribution of electrons in di ff erent transport scenarios are plotted in Figure A2. \nFigure A1 The spectral energy distribution of the electrons in PWN CTA1. \n<!-- image --> \n<!-- image --> \nFigure A2 Particle number densities as a function of the distance from the termination shock in di ff erent transport scenarios. The parameters used to plot are given in Table A1. The particle distribution for the advection case (a) is nearly constant for the flow velocity profile V ∝ r -1 . 9 . The bump near the PWN outer boundary is due to the higher pulsar spin-down power in the early times. In the di ff usion case (b), the particle distribution decreases nearly exponentially with the radius. \n<!-- image -->', 'References': "Abdo, A. A., Ackermann, M., Atwood, W. B., et al. 2008, Science, 322, 1218, doi: 10.1126 / science.1165572 Abdo, A. A., Wood, K. S., DeCesar, M. 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Zhao 13 , 12 , L.Z. Zhao 11 , S.P. Zhao 14 , X.H. Zhao 26 , F. Zheng 34 , W.J. Zhong 7 , B. Zhou 1 , 3 , H. Zhou 9 , J.N. Zhou 16 , M. Zhou 33 , P. Zhou 7 , R. Zhou 28 , X.X. Zhou 1 , 2 , 3 , X.X. Zhou 6 , B.Y. Zhu 12 , 14 , C.G. Zhu 22 , F.R. Zhu 6 , H. Zhu 20 , K.J. Zhu 1 , 2 , 3 , 13 , Y.C. Zou 35 , X. Zuo 1 , 3 , B. Li 7 (The LHAASO Collaboration) \n- 1 Key Laboratory of Particle Astrophysics & Experimental Physics Division & Computing Center, Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China\n- 2 University of Chinese Academy of Sciences, 100049 Beijing, China\n- 3 TIANFU Cosmic Ray Research Center, Chengdu, Sichuan, China\n- 4 Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, 2 Dublin, Ireland\n- 5 Max-Planck-Institut for Nuclear Physics, P.O. Box 103980, 69029 Heidelberg, Germany\n- 6 School of Physical Science and Technology & School of Information Science and Technology, Southwest Jiaotong University, 610031 Chengdu, Sichuan, China\n- 7 School of Astronomy and Space Science, Nanjing University, 210023 Nanjing, Jiangsu, China\n- 8 Center for Astrophysics, Guangzhou University, 510006 Guangzhou, Guangdong, China\n- 9 Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China\n- 10 Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russia\n- 11 Hebei Normal University, 050024 Shijiazhuang, Hebei, China\n- 12 University of Science and Technology of China, 230026 Hefei, Anhui, China\n- 13 State Key Laboratory of Particle Detection and Electronics, China\n- 14\n- Key Laboratory of Dark Matter and Space Astronomy & Key Laboratory of Radio Astronomy, Purple Mountain Observatory,\n- Chinese Academy of Sciences, 210023 Nanjing, Jiangsu, China\n- 15 Research Center for Astronomical Computing, Zhejiang Laboratory, 311121 Hangzhou, Zhejiang, China\n- 16 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 200030 Shanghai, China \n- 17 School of Physics and Astronomy, Yunnan University, 650091 Kunming, Yunnan, China\n- 18 Key Laboratory of Cosmic Rays (Tibet University), Ministry of Education, 850000 Lhasa, Tibet, China\n- 19 Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China\n- 20 Key Laboratory of Radio Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, 100101 Beijing, China \n21 \nSchool of Physics and Astronomy (Zhuhai) & School of Physics (Guangzhou) & Sino-French Institute of Nuclear Engineer- \n- ing and Technology (Zhuhai), Sun Yat-sen University, 519000 Zhuhai & 510275 Guangzhou, Guangdong, China\n- 22 Institute of Frontier and Interdisciplinary Science, Shandong University, 266237 Qingdao, Shandong, China\n- 23 APC, Universit'e Paris Cit'e, CNRS / IN2P3, CEA / IRFU, Observatoire de Paris, 119 75205 Paris, France\n- 24 Department of Engineering Physics & Department of Astronomy, Tsinghua University, 100084 Beijing, China\n- 25 School of Physics and Microelectronics, Zhengzhou University, 450001 Zhengzhou, Henan, China\n- 26 Yunnan Observatories, Chinese Academy of Sciences, 650216 Kunming, Yunnan, China\n- 27 China Center of Advanced Science and Technology, Beijing 100190, China\n- 28 College of Physics, Sichuan University, 610065 Chengdu, Sichuan, China\n- 29 School of Physics, Peking University, 100871 Beijing, China\n- 30 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University,\n- 530004 Nanning, Guangxi, China\n- 31 Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand\n- 32 Moscow Institute of Physics and Technology, 141700 Moscow, Russia\n- 33 Center for Relativistic Astrophysics and High Energy Physics, School of Physics and Materials Science & Institute of Space Science and Technology, Nanchang University, 330031 Nanchang, Jiangxi, China\n- 34 National Space Science Center, Chinese Academy of Sciences, 100190 Beijing, China\n- 35 School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China"}

2024arXiv240903371K

In this chapter we will cover how stars form from the stellar nurseries that are giant molecular clouds. We will first review the physical processes that compete to regulate star formation. We then review star formation in turbulent magnetized molecular clouds and the associated statistics giving rise to the star formation rate and the initial mass function of stars. We then present the protostellar stages in detail from an observational perspective. We will primarily discuss lowmass lt1.5msun stars. Finally we examine how multiplicity complicates the singlestar formation picture. This chapter will focus on star formation at redshift0

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.03371', '2024arXiv240903371K', 'arXiv:2409.03371']

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

Star Formation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.03371.pdf

{'Rajika Kuruwita ,a , Łukasz Tychoniec b and Christoph Federrath c': 'a Heidelberg Institute for Theoretical Studies, Stellar Evolution Theory, Schloß-Wolfsbrunnenweg 35, 69118 Heidelberg, \nGermany \nb Leiden University, Leiden Observatory, PO Box 9513, 2300RA, Leiden, The Netherlands \nc Australian National University, Research School of Astronomy and Astrophysics, ACT 2611, Australia \n© 20xx Elsevier Ltd. All rights reserved. \nChapter Article tagline: update of previous edition, reprint..', 'Glossary': 'Molecular cloud a region of space primarily composed of hydrogen in its molecular state (H2). Pre-stellar core an over-density of gas that does not have a star, but will collapse to form a star.', 'Nomenclature': 'MHD \nMagnetohydrodynamics \nISM \nInterstellar medium \nAU \nAstronomical units \nYSO \nYoung stellar object \nPDF Probability density function \nSED \nSpectral energy distribution', 'Abstract': 'In this chapter, we will cover how stars form from the stellar nurseries that are giant molecular clouds. We will first review the physical processes that compete to regulate star formation. We then review star formation in turbulent, magnetized molecular clouds and the associated statistics giving rise to the star formation rate and the initial mass function of stars. We then present the protostellar stages in detail from an observational perspective. We will primarily discuss low-mass ( < 1 . 5 M ⊙ ) stars. Finally, we examine how multiplicity complicates the single-star formation picture. This chapter will focus on star formation at redshift 0. \nKey words : Astrophysical processes: astrophysical magnetism - gravitation; Interstellar medium: interstellar dynamics, molecular clouds; Protostars: young stellar objects.', 'Learning Objectives': '- · Understand the role of gravity, hydrodynamics, turbulence, radiation, and magnetic fields in star formation.\n- · Look at how the star-forming environments set the initial conditions for star formation and the rate of star formation.\n- · Review what observations of young stellar objects (YSOs) tell us about the star formation process.\n- · Explore how the picture of isolated star formation is complicated by the fact that most stars are born with siblings.', '1 Introduction to the physics of star formation': "The most abundant element in the universe is hydrogen (see 'Big Bang Nucleosynthesis'), which typically exists in an ionized state as protons in low-density, hot regions (temperature T > 10 4 K), or as atomic hydrogen with a proton and electron in higher-density, cooler (100 K ≲ T < 10 4 K) regions (Ferri'ere, 2001; McClure-Gri ffi ths et al., 2023). However, when the density is high enough and the temperature is su ffi ciently low ( ∼ 10 -100 K; see e.g., Tacconi et al., 2020), hydrogen atoms can form the more stable H2 molecule. These regions of higher density, where molecular hydrogen exists, are called 'molecular clouds'. Stars form from over-densities of gas within these molecular clouds.", '1.1 Gravity, gas dynamics and the formation of the first hydrostatic core': "When a local over-density forms in a molecular cloud, it is typically called a 'pre-stellar core'. At this stage, a star has not formed, but the pre-stellar core will collapse under its own gravity to start the star-formation process. Assuming that the pre-stellar core is mostly spherical, with no rotation, gravity is the only force acting on the pre-stellar core. Also assuming that the gas is pressure-less, a star should form within a free-fall time t ff . The free-fall time for a spherical cloud of uniform density ρ is defined as \nt ff = s 3 π 32 G ρ , (1) \nwhere G is the gravitational constant. Observed pre-stellar cores have number densities of n H2 ≃ 10 5 cm -3 (Keto and Caselli, 2008), which translates to ρ ≃ 3 . 8 × 10 -19 g cm -3 . At this density, the free-fall time of a typical pre-stellar core is 0 . 1 Myr. \nHowever, gas is not a pressure-less fluid, and as the pre-stellar core collapses, the outward force of the pressure gradient will counteract the inward gravitational force. When the collapse progresses to gas densities of ≃ 10 -10 g cm -3 , the pressure force and the gravitational force are comparable and the fluid is approaching hydrostatic equilibrium. This is defined as \ndP ( r ) dr = GM ( r ) ρ ( r ) r 2 , (2) \nwhere P ( r ) , ρ ( r ) and M ( r ) are the pressure, density, and enclosed mass, at radius r . This hydrostatic object that forms from the pre-stellar collapse is called the 'first hydrostatic core' (Larson, 1969). This first hydrostatic core is not yet a star, because its radius is ∼ 4 AU. To describe the further collapse of the first hydrostatic core into a star, we must now understand how radiation plays a role.", '1.2 Radiation and the formation of the second hydrostatic core': "Electromagnetic radiation can be produced from two pathways: 1. Converting mass into radiative energy through stellar nucleosynthesis / nuclear fusion (using Einsteins E = mc 2 ; also see 'Evolution and final fates of low- and intermediate-mass stars', 'Evolution and final fates of massive stars'), or 2. Converting from a di ff erent type of energy into radiative energy. \nConversion between energy types is an essential physical process. During the pre-stellar core collapse described in Section 1.1, the gas has gravitational potential energy that is converted into thermal energy. This thermal energy is equivalent to the gas pressure, which eventually slows down the pre-stellar collapse, to form the first hydrostatic core. The question then is, how do we continue to collapse beyond the first hydrostatic core to form a star? \nThermal energy can be removed from the system via radiation. During the initial stages of pre-stellar collapse, the thermal energy from the gravitational collapse can be radiated away e ffi ciently because the gas has a low optical depth ( τ ). This e ffi cient radiative cooling means that the gas remains approximately isothermal until number densities of ∼ 10 10 cm -3 are reached. \nThe optical depth is a measure of how easily photons can pass through a medium, telling us if a medium is opaque (or 'optically thick'; τ ≫ 1) or transparent (or 'optically thin'; τ ≪ 1). The optical depth is defined as \nτλ = Z L 0 n σλ dl , (3) \nwhere σλ is the cross-section of interaction of the particles in the medium with photons of wavelength λ , and n is the number density of particles. The optical depth is calculated by integrating these values over some line of sight L . \nThe interactions that are quantified by σλ are absorption and scattering events. An atom or molecule is more likely to absorb a photon if its energy matches a di ff erence in electron shell energies because this can excite electrons into higher energy states. This process produces absorption-line spectra. Scattering events are dependent on the size of the particles ( rp ) and wavelength. In the Rayleigh scattering regime, longer wavelength photons can penetrate the medium further before an interaction than shorter wavelength photons. For a hydrogen molecule, which has a size of 1 . 2 × 10 -8 cm, any wavelength from the visible light ( ∼ 5 × 10 -5 cm) to radio waves ( > 10 2 cm) falls into the Rayleigh scattering regime. \nIn the optically thin regime, a photon of radiation can pass through gas unhindered allowing for e ffi cient radiative cooling. As stated previously, the temperature of molecular hydrogen in molecular clouds is quite cool, around 10 -40 K, which according to Wien's law would radiate a blackbody spectrum that peaks in the infrared ( ∼ 10 -2 cm). During the initial protostellar collapse, the number density of molecular hydrogen is low enough that the infrared radiation is optically thin, leading to e ffi cient cooling. \nIn the optically thick regime, a photon has many interactions with particles in the medium being absorbed, re-emitted, and scattered. The trapping of radiation makes cooling very ine ffi cient, and radiation is more likely to be converted to thermal energy increasing the temperature of the medium. This is what happens when the first hydrostatic core forms: the medium transitions from optically thin to optically thick. \nThe first hydrostatic core is still accreting mass from the surrounding pre-stellar core, which disturbs the hydrostatic equilibrium. With the increasing mass, the hydrostatic core will continue to contract under its own gravity. However, because the gas is optically thick, making radiative cooling ine ffi cient, the temperature of the core increases during contraction. \nOnce the temperature at the center of the first hydrostatic cores exceeds 2000 K, the H2 molecules separate into individual atoms. When this dissociation happens, thermal energy is used to break the chemical bonds in the H2 molecular to create atomic hydrogen. Because of this e ffi cient use of thermal energy for H2 dissociation, the system returns to a near isothermal state and a second isothermal collapse is triggered. As with the first isothermal collapse, the second isothermal collapse will halt when the internal pressure increases and a new hydrostatic equilibrium is established. \nThe second hydrostatic core is a few solar radii large and is embedded in the first hydrostatic core, which the second core accretes from. Further accretion makes the second hydrostatic core undergo another adiabatic contraction, which increases the central density and \ntemperature, ionizing the atomic hydrogen. When the central temperature exceeds 10 4 K the hydrogen is fully ionized. The higher density and temperature make the second core opaque to the radiation being produced in the center. Convection is triggered within the second hydrostatic core because of the high optical depth. \nWhen most of the mass from the pre-stellar core has been accreted by the second core, we essentially have a protostar. This protostar will continue to contract, following the Hayashi track, decreasing in luminosity, but maintaining the same surface temperature of ∼ 4000 K (Hayashi, 1966). For low-mass protostars ( M ⋆ < 0 . 6 M ⊙ ), they remain fully convective and will continue contracting along the Hayashi track until hydrogen fusion is triggered, and the star is on the main sequence. For more massive protostars, the central temperature of the protostar increases more during the Hayashi track contraction, and the core becomes radiative. These stars continue to contract slowly, but the surface temperature increases, e ff ectively making the protostar have a constant luminosity (Henyey et al., 1988), eventually also triggering hydrogen fusion and joining the main sequence. For a more detailed evolution of the later stages of star formation. For further details on the stellar evolution of stars, we refer the reader to 'Evolution and final fates of low- and intermediate-mass stars' and 'Evolution and final fates of massive stars'.", '1.3 Magnetic fields and angular momentum transport': "In the previous section, we focused on the collapse of a non-rotating cloud. However, the pre-stellar cores that form in giant molecular clouds inherit angular momentum from the turbulence in the parent cloud, such that the cores are always rotating. Even though these clouds initially have very low rotation, as they collapse, they rotationally flatten and spin up due to the conservation of angular momentum (Tscharnuter, 1987). The collapse halts when the radius of the cloud matches the centrifugal radius. This is defined as the radius where the rotation rate equals the Keplerian velocity and is given by \nrc = j 2 GM , (4) \nwhere M and j are the mass and mass-specific angular momentum of the pre-stellar core, respectively. However, based on observed rotation rates, for a 1 M ⊙ pre-stellar core, the centrifugal radius is 2 . 7 × 10 6 R ⊙ , which is significantly larger than a star, i.e. ∼ 1 R ⊙ (Spitzer, 1978; Tomisaka, 2000). Therefore, approximately 99.99% of the angular momentum must be removed, for the gas to collapse down to stellar radii. This angular is now understood to be removed via jets and outflows which are the result of magnetic fields. \nMagnetic fields permeate the universe and play an important role in the star formation process. Charged particles preferentially travel along magnetic fields. As stated earlier, most of the gas in the ISM exists in an ionized state, which is essentially full of positively charged protons, and negatively charged electrons. Due to the charged nature of the ISM, it is expected that there is a strong coupling between gas motions and magnetic fields. However, even in regions where the ionization fraction is very low, such as in the star-forming, molecular phase of the ISM, the coupling between ions and neutrals ensures that even the neutral gas is subject to the same Lorentz force as the ions. \nMagnetic field lines have tension, similar to a rubber band. If a magnetic field line is perturbed by charged gas moving perpendicular to the line, the magnetic tension makes the field line want to return to its unperturbed state. The magnetic field line, bouncing back after being perturbed, excites waves that move along the magnetic field. These waves are called Alfv'en waves, and they can remove angular momentum and energy from a collapsing pre-stellar core. As pre-stellar cores rotate and collapse, the gas drags the magnetic field along with the motion, building magnetic pressure. Magnetic pressure is defined as \nP B = B 2 2 µ 0 . (5) \nwhere B is the magnetic field strength, µ 0 is the vacuum permeability. The force from the magnetic pressure gradient is the actual restoring force for the magnetic field lines and acts as the 'tension' mentioned previously. This pressure gradient force F B opposes the gravitational force, and this pressure gradient causes the field lines to 'snap' back triggering Alfv'en waves to travel along magnetic field lines with velocity vA , and reduces the rotational speed of the pre-stellar core. The velocity of Alfv'en waves is given by \nvA = B √ µ 0 ρ , (6) \nρ is the mass density of the gas. \nThe energy for the Alfv'en waves is converted from the kinetic energy of the ionized material in the pre-stellar core, and this is how magnetism can remove energy and angular momentum during the initial stages of pre-stellar core collapse, however at later stages other magnetic mechanisms become more dominant. \nWhile magnetism can remove energy from moving ionizing gas in a collapsing pre-stellar core, strong magnetic fields can slow down, or halt collapse due to magnetic pressure. As described in Section 1.1, the collapse of the pre-stellar core due to gravity is counteracted by the gas pressure. Magnetic pressure also contributes to counteracting gravitational collapse. The mass-to-flux ratio ( M / Φ ) has historically been used to quantify whether a pre-stellar core will collapse, or if the magnetic pressure is too high that it prevents collapse, where Φ is the magnetic flux through the pre-stellar core. This ratio essentially quantifies the balance of the gravitational force against the force due to magnetic pressure. The critical mass-to-flux ratio (( M / Φ )crit) was calculated by Mouschovias and Spitzer (1976) to be 487 g cm -2 G -1 , where cores that are supercritical (i.e. ( M / Φ ) > ( M / Φ )crit) will collapse, while sub-critical cores (( M / Φ ) < ( M / Φ )crit) will not. \nAs supercritical pre-stellar cores collapse, the rotation leads to the formation of a disk due to the conservation of angular momentum, but the magnetic field is also dragged along with the gas and is wound up within the disk. Figure 1 shows this magnetic field morphology produced from simulations of the collapse of a rotating supercritical core. The left and middle panels show gas density projections of this core perpendicular and parallel to the rotation axis of the core, and the blue streamlines trace the magnetic field. The right panel shows a three-dimensional rendering of the magnetic field, with the color indicating magnetic field strength. We see that the magnetic field morphology is pinched inwards in the disk, and within the disk, the magnetic field is coiled up. A natural consequence of the coiling of magnetic fields is the production of protostellar jets and outflows. Multiple mechanisms have been proposed for launching outflows including the disk wind model (Blandford and Payne, 1982; Konigl and Pudritz, 2000), the magnetic tower (Lynden-Bell, 2003) and the 'X-wind' model (Shu et al., 1994). All of these models are likely to be present during the star formation process and act in di ff erent regimes. Figure 2 describes how the protostar and circumstellar disk are connected, and where di ff erent outflows are launched. \nThe disk wind model: Blandford and Payne (1982) calculated that centrifugally-driven outflows can be launched from a disk with a coiled-up magnetic field if the angle of the magnetic field to the disk mid-plane is less than 60 · . The velocity of the outflows reflects the rotation profile of the disk, with faster-velocity outflows being launched at smaller radii, and lower-velocity outflows being launched at larger radii. These outflows are often called 'winds' and are launched from the disk surface, as shown in Figure 2. \nThe magnetic tower describes the launching of outflows via a magnetic pressure gradient. Lynden-Bell (2003) describe these as highly coiled-up magnetic structures and the pinching of magnetic fields to produce strong pressure gradients away from the disk, producing a force that significantly overcomes the gravitational force. Many ideal MHD simulations of molecular core collapse and protostar formation find that this mechanism is what drives the initial jet launching (Banerjee and Pudritz, 2006; Machida and Matsumoto, 2012). The magnetic tower is also likely acting along with the magneto-centrifugally driven disk winds of Blandford and Payne (1982); Nolan et al. (2017). \nThe X-wind model mainly concerns regions in the inner disk where the magnetosphere of the star threads through the inner disk, as shown in Figure 2. Because of the heat from the protostar, the inner disk is fully ionized, and strong coupling between the gas and magnetic field. Due to the conservation of angular momentum, the protostar would typically rotate quickly, but because of this strong coupling between the protostar and inner disk which rotates at lower speeds, (magnetic) tension builds up between these two regions. This tension leads to the launching of jets by the X-wind mechanism. These jets have velocities of a few 100 kms, similar to velocities observed in protostellar jets. \nThe modern consensus is that it is a combination of the disk wind and X-wind model working together to produce the outflow features observed in protostars, with the X-wind producing the highly collimated jet and the disk-wind producing the lower-velocity outflow. Overall, magnetic fields play an important role in the removal of angular momentum, aiding pre-stellar collapse, and allowing protostars to accrete from their circumstellar disks.", '2 Stellar nurseries - molecular clouds': "Stars form in cold, turbulent, molecular clouds. We know this from molecular-line observations with radio and sub-mm telescopes (see also 'The interstellar medium'). These clouds consist mainly of molecular hydrogen, H2, with carbon monoxide, CO, being the second-most abundant molecule. CO is typically used to measure the turbulent velocities in molecular clouds, because at the low temperatures of about 10-50 K, H2 cannot emit photons due to its missing permanent dipole moment, while CO is easily excited, and the Doppler shift of its rotational lines can be used to measure the line-of-sight (LOS) velocity of the gas (Stahler and Palla, 2004). \nGiven the typical temperatures of molecular clouds, implying sound speeds of the order of c s ≈ 0 . 2-0 . 5 km s -1 , and measured velocity dispersion of σ v ≈ 0 . 5-10 kms -1 , the clouds are governed by supersonic turbulent motions with sonic Mach numbers of M = σ v / c s ≈ 1- \nFig. 1: Magnetic field morphology, side-on (left) and top-down (middle) in a simulation of protostar formation in an accretion disc. A three-dimensional rendering of this morphology is shown in the right panel. Adapted from figures 2, 5, and 11 of Kuruwita et al. (2017). \n<!-- image --> \nFig. 2: Schematic view of a young star accreting from a disk through the stellar magnetosphere. Jets are launched from the inner disk, while disk winds are launched at larger radii. Both mechanisms remove angular momentum, allowing the gas to move inwards through the disk. The protostellar magnetic field threads through the inner disk, allowing ionized material to be funneled along these lines onto the protostar. Figure 1 from Hartmann et al. (2016), used with permission. \n<!-- image --> \n50. When studied over di ff erent length scales, the velocity dispersion follows a power-law relation with scale, ℓ , \nσ v ( ℓ ) = σ v ( L ) ℓ L ! p ≈ 1 km s -1 ℓ pc ! p , (7) \nwith p ≈ 0 . 4-0 . 5 based on observations (e.g., Larson, 1981; Solomon et al., 1987; Ossenkopf and Mac Low, 2002; Heyer and Brunt, 2004; Rosolowsky and Blitz, 2005; Heyer et al., 2009; Roman-Duval et al., 2011). This power-law form is indeed similar to the power law obtained in the famous Kolmogorov model of turbulence ( p = 1 / 3) (Kolmogorov, 1941; Frisch, 1995), however, which strictly only applies to incompressible turbulence. Instead, the Burgers model of turbulence (Burgers, 1948) may be more applicable here, as it is based on an ensemble of discontinuities (shocks), which corresponds to p = 1 / 2. Reality likely sits in between those extremes, not to mention the added complication of intermittency and magnetic fields, with active research exploring turbulence models for this complex regime of compressible plasma turbulence (She and Leveque, 1994; Boldyrev et al., 2002; Brandenburg and Subramanian, 2005; Schekochihin et al., 2007; Schmidt et al., 2008; Konstandin et al., 2012; Federrath, 2016a; Seta and Federrath, 2021; Achikanath Chirakkara et al., 2021; Federrath et al., 2021; Beattie et al., 2023).", '2.1 Turbulence-regulated star formation': "Motivated by the fact that all star-forming clouds observed so far exhibit high levels of compressible turbulence, many authors have investigated star formation in turbulent media (Padoan, 1995; Klessen et al., 2000; Elmegreen et al., 2003; Krumholz and McKee, 2005; Padoan and Nordlund, 2011; Hennebelle and Chabrier, 2011; Federrath and Klessen, 2012; Federrath, 2018; Burkhart and Mocz, 2019). The compressible nature of turbulence gives rise to a characteristic gas density probability distribution function (PDF), enabling analytic estimates of the star formation rate (SFR) and the initial mass function (IMF; See also 'Stellar initial mass function') of stars.", '2.1.1 The gas density PDF': 'Turbulent isothermal gas can be well approximated by a log-normal density PDF (V\'azquez-Semadeni, 1994; Passot and V\'azquez-Semadeni, 1998; Padoan and Nordlund, 2002; Kritsuk et al., 2007), \np ( s ) = GLYPH<16> 2 πσ 2 s GLYPH<17> -1 / 2 exp " -( s - ⟨ s ⟩ ) 2 2 σ 2 s # , (8) \nwith the dimensionless logarithmic density contrast s = ln( ρ/ ⟨ ρ ⟩ ), mean density ⟨ ρ ⟩ , mean log-density ⟨ s ⟩ = -σ 2 s / 2 (Li et al., 2003; Federrath et al., 2008, 2010), and log-density variance (Padoan and Nordlund, 2011; Molina et al., 2012), \nσ 2 s = ln GLYPH<20> 1 + b 2 M 2 GLYPH<16> 1 + β -1 GLYPH<17> -1 GLYPH<21> , (9) \nwhere β is the plasma beta (ratio of thermal to magnetic pressure; note that β →∞ if the magnetic field is zero). A more recent modification of this relation accounts for strong magnetic guide fields (Beattie et al., 2021). The parameter b in Equation (9) is the turbulence driving parameter, which is controlled by the mixture of solenoidal vs. compressive modes in the driving mechanism of the turbulence (Federrath et al., 2008). Purely solenoidal (divergence-free) driving has b ∼ 1 / 3, while purely compressive (curl-free) driving has b ∼ 1 (Federrath et al., 2010; Dhawalikar et al., 2022; Gerrard et al., 2023). Modifications of Equation (9) can be made to account for non-isothermal gas conditions (Nolan et al., 2015; Federrath and Banerjee, 2015), and where intermittency plays a role (Hopkins, 2013b).', '2.1.2 The star formation rate': 'The modern theory of star formation is based on the turbulent density PDF. A key step in turbulence-regulated theories of the SFR and IMF is to estimate the fraction of dense gas that can form stars, and this is exactly what Equation (8) and (9) can provide. To derive a rate at which this dense gas turns into stars, we need to divide the dense gas fraction by the freefall time, which, using the definitions in Sec. 2.1.1, gives the basic expression for the SFR per average freefall time, ⟨ t ff ⟩ , for a cloud of mass M cloud (see Hennebelle and Chabrier, 2011; Federrath and Klessen, 2012), \nSFR ff = SFR ⟨ t ff ⟩ M cloud = ϵ ϕ t Z ∞ s crit ρ ⟨ ρ ⟩ ⟨ t ff ⟩ t ff ( ρ ) p ( s ) ds = ϵ ϕ t Z ∞ s crit exp 3 2 s ! p ( s ) ds = ϵ 2 ϕ t exp 3 8 σ 2 s ! " 1 + erf σ 2 s -s crit (2 σ 2 s ) 1 / 2 !# , (10) \nwhere t ff ( ρ ) = 3 π/ (32 G ρ ) defined in Equation (1), the star-to-core mass ratio ϵ ∼ 0 . 3-0 . 5 (Matzner and McKee, 2000; Federrath et al., 2014), and ϕ t ∼ 2 is a numerical correction factor, calibrated in simulations (see tab. 3 in Federrath and Klessen, 2012). Finally, the critical density for star formation, s crit, is given by \ns crit = ln GLYPH<20> ( π 2 / 5) ϕ 2 x α vir M 2 GLYPH<16> 1 + β -1 GLYPH<17> -1 GLYPH<21> , (11) \nwhich is obtained by comparing the Jeans length (Krumholz and McKee, 2005) with the turbulent sonic scale (Federrath et al., 2021), which marks the transition from supersonic turbulence on cloud scales, to subsonic turbulence inside the dense star-forming cores and accretion disks. Thus, the critical density is a result of the competition of gravity and turbulence, which gives rise to the virial parameter α vir = 2 E kin / E grav in Equation (11), the ratio of twice the kinetic to gravitational energy of the cloud. The numerical correction factor ϕ x ∼ 0 . 2 accounts for a slight mismatch between the sonic and Jeans scales when forming s crit, and can be determined by calibration with numerical simulations (see tab. 3 in Federrath and Klessen, 2012). \nA key prediction of this theory is that the SFR depends on 4 basic cloud parameters, namely the virial parameter ( α vir), the sonic Mach number ( M ), the turbulence driving mode ( b ), and the magnetic plasma beta ( β ). For instance, keeping all parameters fixed at typical cloud values ( α vir ∼ 1, M∼ 10, β ∼ 0 . 3), except for the driving mode, Equation (10) predicts an SFR that is a factor of ∼ 2 . 4 higher for compressive driving compared to solenoidal driving. With the associated reduction in α vir for compressive driving, due to the stronger local compressions leading to a higher overall binding energy of a cloud, compressive driving can yield an order of magnitude higher SFR than solenoidal driving (Federrath and Klessen, 2012).', '2.2 The initial mass function of stars': "The initial mass function (IMF) is the distribution of the birth mass of stars. 'Stellar initial mass function' in this series provides more information, so we focus here on the link between turbulence and the IMF and present a brief summary of the physics primarily involved in controlling the IMF.", '2.2.1 Basic characterization': 'The IMF is usually characterized by a power-law section for masses ≳ 1 M ⊙ (Salpeter, 1955; Hopkins, 2018), and a log-normal (or several power-law sections that may be approximated by a log-normal) turnover toward smaller scales, into the brown-dwarf regime (Kroupa, 2001; Chabrier, 2005), which is di ffi cult to constrain exactly, due to the uncertainties involved in observing low-mass (low-luminosity) stars. The peak (or characteristic mass) of the IMF is around 0 . 1-0 . 5 M ⊙ .', '2.2.2 Physical processes': 'The IMF is controlled by a combination of physical processes. Gravity and turbulence play a central role (Klessen et al., 2000; Padoan and Nordlund, 2002; Hennebelle and Chabrier, 2008; Hopkins, 2013a; Nam et al., 2021). However, magnetic fields and radiation (Price and Bate, 2009; Bate, 2009; Mathew and Federrath, 2020, 2021) are also key ingredients, as they tend to reduce fragmentation. Moreover, magnetic fields produce powerful jets and outflows from the accretion disk around a newborn star, removing mass from the disk and core, and thereby significantly contributing to setting the final mass of a young star (Federrath et al., 2014; Guszejnov et al., 2020). \nIn Section 2.1 we saw that the turbulent density distribution determines the amount of dense gas eligible for star formation, thereby determining the SFR. Similar holds for the IMF, with most modern theories of the IMF relying on the same underlying physics that gives rise to the turbulent gas density distribution described by Equation (8). Here we highlight one aspect of this distribution, namely, its width, which is crucially determined by the driving mode of the turbulence ( b parameter in Equation (9)). Using numerical simulations, Mathew et al. (2023) showed that the IMF depends on the driving mode (solenoidal vs. compressive) of the turbulence. This is shown in Figure 3. We see that compressive driving produces substantially stronger density fluctuations than solenoidal driving. The IMF resulting from several sets of simulations with di ff erent random seeds yields a total of 468 and 445 stars, with a median stellar mass of (0 . 4 ± 0 . 1) M ⊙ and (0 . 6 ± 0 . 2) M ⊙ for compressive and solenoidal driving, respectively. This shows that turbulence is a key ingredient for the IMF, and variations in the driving mode of the turbulence may produce significant variations in the IMF.', '2.3 Feedback processes': 'While the interplay of turbulence and gravity is a key controller of star formation, as discussed in the previous two subsections, stellar feedback processes also play a crucial role. We broadly distinguish mechanical and radiative forms of feedback. \nMechanical feedback is the redistribution of mass and momentum by jets and outflows from the accretion disk around protostars (see \nFig. 3: Top panels: gas column density in star-formation simulations with purely compressive (curl-free) turbulence driving (left) and purely solenoidal (divergence-free) turbulence driving (right), as defined in Sec. 2.1.1. Young stars are shown as circles, forming in dense gas, primarily at the intersection of filamentary structures (Schneider et al., 2012). Bottom panel: comparison of various observational IMFs together with the IMF obtained in several sets of simulations using the driving modes of turbulence shown in the top panels: compressive driving (blue histogram) and solenoidal driving (red histogram). The curves are the system IMF models based on observational surveys by Salpeter (1955) (dash-dotted), Chabrier (2005) (short-dotted), Parravano et al. (2011) (long-dotted), Da Rio et al. (2012) (solid), Kroupa et al. (2013) for brown dwarfs (long-dashed) and stars (short-dashed), and Damian et al. (2021) (dash-double-dotted). Adapted from figures 2 and 6 of Mathew et al. (2023). \n<!-- image --> \nSec. 1.3) or from supernova explosions. Jets and outflows are particularly relevant for the SFR and IMF, in that they limit the amount of material that can be accreted onto the protostar by about a factor of 2, therefore slowing down star formation (Padoan and Nordlund, 2011; Federrath et al., 2014; Federrath, 2015). Moreover, the mechanical nature of this type of feedback can cause coherent accretion streams to break, thereby inducing additional fragmentation, which, together with the direct limiting e ff ect on accretion, leads to an overall reduction of the average stellar mass by a factor of ∼ 3 (Federrath et al., 2014; Guszejnov et al., 2020; Mathew and Federrath, 2021). \nRadiative feedback describes the heating, ionization, and / or radiation pressure induced by stars. This form of feedback, in particular direct radiation pressure and reprocessed ionizing radiation from massive stars, also causes a mechanical e ff ect in that the radiation force can push on the dust (Menon et al., 2023), forming expanding shells around HII regions, sculpting dense structures such as the Pillars of Creation. Evolved stars drive winds throughout most of their lifetime, re-injecting material (in particular metals), momentum and energy into the ISM. While the aforementioned radiative feedback processes are primarily relevant for massive stars, heating feedback is crucial for all young stars, including low-mass stars. Accretion causes a local ( ≲ 0 . 1 pc) heating e ff ect around young stars, which limits fragmentation of the surrounding gas, thereby significantly controlling the low-mass end of IMF (O ff ner et al., 2009; Bate, 2009; Price and Bate, 2009; Federrath et al., 2017a; Guszejnov et al., 2018; Mathew and Federrath, 2020). \nFinally, all the mechanical feedback types, as well as the radiative ones that cause a mechanical e ff ect can drive turbulence (Elmegreen, 2009; Federrath et al., 2017b), thereby closing a feedback loop, in which turbulence is responsible for regulating star formation as described in Sec. 2.1 and 2.2 above. \nTable 1: Observational characteristics of protostellar classifications \nClass: protostellar class, α IR : the gradient of the spectral energy distribution in the infrared, L submm / L bol : the ratio of the luminosity in the sub-millimeter and the bolometric luminosity, T bol : bolometric temperature.', '3 Observational view of the star formation process': "In this section, we describe the observational constraints on the formation of a single Solar-like stellar system. Due to the embedded nature of protostellar sources, their studies are mostly conducted at infrared and longer wavelengths, as the young protostars are often in the densest part of the cores from which they form, where extinction inhibits observations at shorter wavelengths. The focus of this section is the protostellar stages (i.e., Class 0 and Class I objects). The evolution of later stages (pre-main sequence stars), protoplanetary disks, as well as planet formation, is covered in 'Protoplanetary disk origins and free-floating exoplanets', 'Protoplanetary disk chemistry and structure', and 'Planet formation mechanisms'.", '3.1 Protostellar evolutionary path and classifications': "The protostellar evolution is divided into classes, based on their observed properties. The di ff erent observational properties of protostellar evolutionary characterization are summarized in Table 1. These empirical classes of evolution were first introduced based on the observations of a near-infrared spectral index between 2 to 20 µ mdefined as \nα IR = d log( λ F λ ) d log( λ ) , (12) \nfor flux F λ at wavelength λ (Lada and Wilking, 1984). The spectral index changes as the protostar gains mass, disperses the envelope and forms a protostellar disk. The youngest protostars show a redder (positive) spectral index, with increasing brightness towards longer wavelengths, and more evolved sources have a negative spectral index as the stellar spectral energy distribution (SED) approaches that of a main-sequence star. \nHowever, this characteristic does not account for the existence of even younger objects, Class 0 protostars, which are often too cold and visually extinct to emit in the near-IR regime (Andr'e et al., 1993). Flat-spectrum sources were also later distinguished as a transition between Class I and Class II sources (Greene et al., 1994). \nProtostars have been also categorized by their bolometric temperature Tbol, which is established as the temperature of a blackbody with the same mean wavelength as the SED of a protostar (Myers and Ladd, 1993). Based on that classification, the Class 0 stage can be distinguished as having T bol ≤ 70 K. Both of the described methods, however, rely on observable properties of the system, where, for \nFig. 4: An illustrated overview of protostellar evolutionary classes. In the collapse stage, the infall motions create a dense central region in the prestellar core. Class 0 / I stage (protostellar stage) is associated with powerful outflows and jets accompanied by the most vigorous accretion; this is also the stage for protoplanetary disk forms. In the Class II stage, the disk is cold and quiescent, associated with disk winds, this is also where embedded planets are detected. In Class III, the disk disperses and a residual dusty debris disk is present. \n<!-- image --> \nFig. 5: Schematic of protostellar physical components with molecules and their associated emission / absorption lines that can be used to probe disk structure, chemistry, and dynamics through sub-millimeter spectroscopy. From Tychoniec et al. (2021), Reproduced with permission from Astronomy & Astrophysics, © ESO. \n<!-- image --> \nexample, the inclination of the protostellar disk can alter the measured infrared spectral index (Whitney et al., 2003). Another method involves comparing the contribution of sub-millimeter luminosity to the bolometric luminosity of the source, where most embedded sources have at least 0.5% of their luminosity above 350 µ mcontribution (Andr'e et al., 1993). \nAnother way to classify protostellar sources is by their physical parameters instead of observed properties, which provide more descriptive characteristics of the state of the system (Whitney et al., 2003; Robitaille et al., 2006). These physical classifications are illustrated in Figure 4. In Class 0, most of the system's mass is still in the envelope; Class I marks the transition where disk mass is comparable to or greater than the mass of the envelope, while most of the system's mass is already in the central star; Class II sources have a negligible envelope, with the gaseous disk still present, but its mass is much lower than the mass of the central star; by Class III the star is a pre-main sequence object and the disk is gas-less and of negligible mass. \nAprotostellar system comprises di ff erent physical components that can be observationally characterized using various molecular tracers. These tracers are summarised in Figure 5. In the following sections, we describe the key characteristics and evolutionary trends in each of those components.", '3.2 Protostellar envelope': 'Sub-millimeter single-dish and interferometric continuum observations, sensitive to cold grains, are widely used tools to recover the dust structure of the envelopes. Observations find that protostellar envelopes have radii of several 1000 AU. Inferred from dust emission, the density profiles of the protostellar envelopes are often found following a radial density profile close to ρ ∝ r -2 (Looney et al., 2003; Maury et al., 2019), which is consistent with theoretical predictions described in Larson (1969), but steeper profile closer to ρ ∝ r -3 / 2 of the insideout collapse (Shu, 1977) is also observed (Kristensen et al., 2012). Dust properties are typically similar to the interstellar medium; however, in the inner envelope, signatures of grain growth can be observed (Galametz et al., 2019). \nGas in the protostellar envelopes is traced almost exclusively at (sub-)millimeter wavelengths due to the very low temperatures of the order of 10 -20 K. Velocity-resolved observations of emission lines can trace the infall and rotation of the envelope and can be used to constrain the angular momentum (Gaudel et al., 2020). At densities of 10 4 -10 5 cm -3 , the freeze-out timescales of the gas become shorter than the envelope lifetime, and certain gas species sublimate onto the dust grains. For example, the freeze-out temperature of carbon monoxide (CO) is 20 -25 K, while water (H2O) freezes at temperatures below 100 K. The depletion of CO and H2O from the gas to the ice phase causes a rise of emission of molecules, which otherwise are e ffi ciently destroyed through reactions in the gas-phase with CO. Such tracers are DCO + and N2H + , which are tracers of CO freeze-out and H 13 CO + , which tracers H2O freeze-out (Hogerheijde et al., 1997; Tobin et al., 2011). The frozen molecules cannot be traced with emission spectroscopy, but they absorb the light, especially in the infrared regime. Combined with the laboratory characterization of ice mixtures, a detailed composition of the ice mantle in envelopes around protostars can be obtained (see Boogert et al., 2015, for review). \nThe envelope dissipates during protostellar evolution as the material is delivered to the disk and star. At the same time, protostellar outflows and jets open up a cavity wall and expel a large amount of material from the system. On the other hand, streamers of gas from the larger cloud scales can replenish the envelope with material at various stages of evolution (e.g., Pineda et al., 2020).', '3.3 Outflows and jets': 'Outflows are one of the first signs of the new star being born as they expel gas away from the deeply embedded protostar. As the outflow propagates at supersonic velocities, it creates shocks with the surrounding medium. Therefore, high-temperature traces such as H2 rotational \ntransitions are commonly used to study shocked gas. Shocks disrupt dust mantles and cores, releasing material that would, in quiescent ISM conditions, remain in the solid phase. Therefore, SiO molecular gas or atomic and ionized emission from Si, Fe, and Ni is observed in shocked gas. \nObservationally, outflows are typically divided into the high velocity ( > 30 km s -1 ), highly collimated component often called jets, and the low velocity ( < 30 km s -1 ) wind angle component sometimes called winds. The low-velocity component is expected to trace the envelope material entrained by the faster component, or the disk wind which is the gas directly released from the protoplanetary disk. The low-velocity outflow is traced by rotational transitions of CO. In some young outflows, the outflow can also be traced by more complex species such as CH3OH and H2CO, which trace the sputtering of grains in low-velocity shocks at the outflow cavity walls. \nDetailed studies of jet kinematics can inform about their precise physical origin and mechanism (see 1.3. The chemical content of the jets undergoes evolution. Molecules such as CO, SiO, and SO are mostly detected in very young Class 0 sources. This is likely because high number densities of the order of 10 6 cm -3 are required for e ffi cient gas-phase formation of molecules from initially atomic material. Further into the evolution, the neutral ([O I], [Ni I], [Cl I], [S I]) and ionized ([Fe II], [Ne II], [Ar II]) components of the jet become dominant (Nisini et al., 2015). Prominent refractory contents of the jet material suggest that jets either launch from the inner regions of the disk or that dust grains are launched and e ffi ciently destroyed in the jet. \nApart from chemical evolution, jets and outflows also significantly change their energetic and mass output during protostellar life. Young outflows are characterized by the most energetic outflows, and the total outflow force is found to be correlated with protostellar luminosity, indicating a strong relation between accretion and ejection activity of the protostar (Bontemps et al., 1996). This correlation between outflow and accretion rate is used to design simulation sub-resolution models of jets and outflows (Cunningham et al., 2011; Federrath et al., 2014; Guszejnov et al., 2020). \nSince the outflows are expected to remove angular momentum, observations of the rotational signature is one of the crucial observations. Rotation of the jet and wind has been observed (Bjerkeli et al., 2016; Lee et al., 2017), indicating that the angular momentum is indeed removed with the outflow. \nJets launched from the inner regions of the protoplanetary disks often form internal shocks, which are characterized by high densities, where molecules can e ffi ciently form. Those shocks are caused by internal variations of the jet velocity, which occur due to accretion variability. Because of that, jets are fossil records of the accretion process, revealing that the protostellar accretion process is highly variable in nature (Lee, 2020).', '3.4 Protostellar accretion': 'Most of the stellar mass is assembled during the early stages of evolution (Class 0 / I). Direct observations of the protostar remain a challenge for observations since the protostars are deeply embedded. Nevertheless, in recent years significant progress has been made to extract stellar properties (Fiorellino et al., 2023). Hydrogen recombination lines, which are tracers of high-density and high-temperature gas, are used to probe the accretion onto the protostar. With a combination of bolometric luminosity estimates and infrared photometry, stellar properties can be constrained. \nMeasured accretion rates are often lower than expected, considering the duration of accretion and the final masses of stars from the initial mass function (Kenyon et al., 1990). This discrepancy between the observed accretion rate of young stars being significantly lower than expected from models is called the Luminosity Problem. A solution to this problem is that protostars accrete a significant portion of their mass during periods of high accretion, such as outbursts or in the initial stages of protostar formation. Protostellar accretion is, therefore, a highly variable process that evolves dramatically during protostellar life (Fischer et al., 2023).', '3.5 Embedded disk': "In the inner regions of the envelope, the velocity profile changes as the forming circumstellar accretion disk follows Keplerian rotation. Several young disks have their Keplerian rotation characterized in observations. However, it remains a challenge as most of the dust disks are small, with radii ≤ 50 AU (Maury et al., 2019). With di ff erent tracers such as formaldehyde (H2CO) or optically-thin isotopologues of CO, it is possible to study the temperature of the disk (van 't Ho ff et al., 2020). \nDust masses of the young Class 0 and Class I disks are fundamental to estimating the total budget of building blocks of planets. However, they are di ffi cult to constrain as the young disks are optically thick and hard to discern from the surrounding envelope. Observations at longer wavelengths ∼ 1 cm can mitigate those issues and have been used to constrain masses of the order of 50 to 150 Earth masses (Tychoniec et al., 2020). This is a factor of 5 to 20 more than typical masses of Class II disks (Ansdell et al., 2017). \nThe available mass budget, grain growth observed in Class I systems, and substructures omnipresent in Class II disks suggest that planet formation should already begin early. These structures, such as gaps and spirals, are rarely observed in Class 0, while they appear to be more common in the Class I stage, suggesting an evolution of disks potentially shaped by planets (Ohashi et al., 2023). For further details on protoplanetary disks and planet formation, we refer to 'Protoplanetary disk chemistry and structure' and 'Planet formation mechanisms'.", '4 Multiplicity and the formation of binary/multiple star formation': "Section 1 and 3 have focused on the formation of a single star, however many stars exist in binary or multiple star systems (O ff ner et al. (2023); See also 'Observing binary stars'). Figure 6 compiles observational surveys of main sequence stars, with the left panel showing the fraction of stars of mass M that are in binary or higher (thick crosses), or triple or higher (thin crosses) star systems. The right panel shows \n<!-- image --> \nFig. 6: Left: observed multiplicity fraction as a function of star mass. The fraction of stars in binary or higher-order, or triple or higher-order systems is shown by the thick and thin crosses, respectively. Right: companion frequency, or the average number of stellar companions a star has as a function of star mass. Reproduced with permission from O ff ner et al. (2023). \n<!-- image --> \nFig. 7: Observed separation distribution of young multiple star systems, with at least one Class 0 object. This combines observations from the Perseus and Orion star-forming regions. The thin dotted line is the separation distribution of solar-type field stars (Raghavan et al., 2010). Reproduced with permission from Tobin et al. (2022). \n<!-- image --> \nthe companion frequency as a function of star mass. We see that many stars can exist in binary or multiple-star systems, with more massive stars being more likely to have companions. The actual fraction of all stars that are in multiple star systems is sensitive to the initial mass function (see Section 2.2), however, it is accepted that a significant number of stars are in multiple star systems, and their formation must not be ignored when understanding star and planet formation. \nWe also find that most stars are born with a companion, with multiplicity being highest in the protostellar Class 0 (see Table 1), decreasing as we look at more evolved protostars. This means that many of the stars in these Class 0 multiple-star systems will interact and get ejected as they evolve towards the main sequence, or maybe even merge to form more massive stars (Bally and Zinnecker, 2005). These interactions early on can a ff ect the disks around the protostars and a ff ect the sites of planet formation. Many stars that are single on the main sequence may have begun their life in a multiple-star system and were ejected through complex orbital dynamics. \nObservations of separations in young binary and multiple star systems in star-forming regions find a bimodal distribution with one peak at ∼ 100 AU and another at ∼ 3000 AU, as seen in Figure 7. When this bimodal distribution was first observed, the origin of the two peaks was attributed to two formation pathways for multiple star systems: 1. pre-stellar core fragmentation, and 2. circumstellar disk fragmentation.", '4.1 Core fragmentation': 'Core fragmentation was used to explain the separation peak at ∼ 3000 AU because this formation pathway acts on larger scales of 100s to 1000s of AU. As stated throughout this chapter, molecular clouds are turbulent, and this turbulence can create over-densities that make pre-stellar cores collapse to form a star. However, pre-stellar cores also have sub-sonic turbulence, which may seed further over-densities that can fragment to form stars. The description of pre-stellar collapse in Section 1 starts with a spherical cloud, however, turbulence in the ISM can seed the formation of filaments (Federrath, 2016b), from which most pre-stellar cores fragment. This is seen in Figure 8a, with an observed filament (leftmost panel) versus a modeled filament and cores (rightmost panel). This elongation adds asymmetry, which can seed \n(a) Filament fragmentation in the California molecular cloud. (a) the true observations, (b) the extracted filament, (c) the identified pre-stellar cores, and (d) the reconstructed observations with the modeled filament and cores. From (Zhang et al., 2020), reproduced with permission from Astronomy & Astrophysics, © ESO. \n<!-- image --> \n(b) Hub fragmentation in G333. Reproduced from (Li et al., 2024). (a) shows the low-resolution ATCA observations, (b) shows the low-resolution ALMA observations revealing further fragmentation, and the remaining panels show high-resolution ALMAobservations of parts of the hub, showing further hierarchical fragmentation. \n<!-- image --> \nFig. 8: Observations of fragmentation in di ff erent environments. \nfragmentation, along with the turbulent nature of the cores. \nFragmentation in hubs, where filaments intersect is also observed. This is seen in Figure 8b where at low resolution (center panel) fragmentation is observed, and at higher resolution, further hierarchical fragmentation is also observed. The filaments that feed these hubs inject turbulent energy, which can lead to fragmentation. \nThe fragments that form along a filament can dynamically fall towards each other because the relative velocity of the fragments to each other is low, and stars that form from core fragmentation in hubs will likely also experience complex dynamical interactions. While the larger peak in the separation distribution at ∼ 3000 AU is attributed to core fragmentation, many multiple star systems that form via this pathway often inspiral to smaller separations, even down to < 100 AU (Kuruwita and Haugbølle, 2023).', '4.2 Disk fragmentation': 'As described in Section 1.3, circumstellar disks are a natural consequence of pre-stellar core collapse, and under the right conditions, fragmentation can occur within these disks to form new stars. The separation peak at ∼ 100 AU has been attributed to disk fragmentation because this formation pathway acts on disk scales. Circumstellar disks can extend up to ∼ 600 AU, with mean disk sizes in the Class 0 / I stage being around ∼ 75 AU (Tsukamoto et al., 2022). \nThe Toomre Q (Toomre, 1964) is a quantity that is often used to measure the stability of a disk, defined as \nQ = c s Ω π G Σ , (13) \nwhere c s is the sound speed, Ω is the angular frequency, and Σ is the gas surface density of the disk. A parcel of gas is considered to be stable if Q ≫ 1, and unstable and prone to collapse if Q ≪ 1. The Toomre Q essentially measures a ratio of how pressure and rotationally supported the disk is against its own gravity. A disk that is Toomre unstable can become stable by either rotating faster (increase Ω ), reducing surface density, or being hotter; i.e., higher c s. This definition ignored other forms of support against fragmentation, specifically any type of non-thermal pressure. Sources of non-thermal pressure include magnetic pressure, turbulent pressure, and, radiation pressure. The latter two in particular, as well as magnetic tension, however, can have significant anisotropic e ff ects, so simply adding them as an isotropic pressure contribution to Q may be too simplistic. A Toomre Q that has magnetic pressure added has been derived from MHD simulations (Forgan et al., 2017), and is defined by multiplying the Toomre Q by a scaling factor that arises from adding the thermal and magnetic pressures (see eqs. 17 and 18 in Federrath and Klessen, 2012), \nQB = Q q 1 + β -1 , (14) \nwhere β is plasma beta, as defined in Section 2.1.1. \nThe ideal disks for fragmentation are massive, cold disks, which are relatively rare. Radiation feedback from the central star, is expected to provide thermal support against disk fragmentation (O ff ner, 2011). However, stars accrete episodically (see Section 3.4), therefore circumstellar disks can go through cycles of heating and cooling, and if the time between accretion events is su ffi ciently long, disks may cool temporarily to become unstable (Stamatellos et al., 2012).', '4.3 Evolution of young multiple star systems': "After fragmentation into binary or multiple-star systems, these stars also interact. Simulations of the formation of eccentric binaries find that accretion bursts can be triggered at periastron (the closest separation) because the companion star disrupts the circumstellar disk (Kuruwita et al., 2020). Accretion bursts can also be triggered by the flyby of unbound stars (Borchert et al., 2022), which is not uncommon in clustered star-forming environments. Most binaries have orbital periods that are significantly longer than a human lifetime, therefore, it has been di ffi cult to observe directly companion-triggered accretion, but there are a handful of short-period young binaries where companiontriggered accretion is observed over multiple orbits (Mathieu et al., 1997). \nInteractions between stars can also truncate circumstellar disks. Simulations find that the radius of circumstellar disks is truncated to approximately a third of the binary separation (Artymowicz and Lubow, 1994). Thus, truncation can shorten the lifetime of the disk and potentially hinder planet formation. \nWhile circumstellar disks (the disks around individual stars) can be truncated or destroyed by binary-star interactions, simulations find that the formation of circumbinary disks is ubiquitous. Circumbinary disks can form either via the inspiral of binaries formed through core-fragmentation (Kuruwita and Federrath, 2019), or through disk fragmentation (Tokovinin, 2021). Observations find that many of the largest protostellar disks are circumbinary disks (Harris et al., 2012), and some are unusually old ( > 10 Myr), for example, AK Sco (18 ± 1 Myr; Czekala et al. 2015), HD 98800 B (10 ± 5 Myr; Furlan et al. 2007) and V4046 Sgr (12-23 Myr; Rapson et al. 2015). The size and persistence of circumbinary disks may provide an ideal environment for planet formation. For details on accretion from circumbinary disks, we refer the reader to 'Circumbinary Disk Accretion'. \nYoung multiple-star systems can experience complex orbital dynamics such as higher-order systems ejecting companions, and new multiple-star systems forming through dynamical capture. Approximately one-third of binaries are estimated to not have been born together based on observations (Murillo et al., 2016) and simulations (Kuruwita and Haugbølle, 2023). Dynamical capture is likely to be easier in star-forming environments because these young stars are actively accreting from their gaseous environments, which produce dynamical drag. Simulations of binaries that formed via core fragmentation also find that in-spiraling halts when the binary is no longer embedded in a dense gaseous environment (Kuruwita and Haugbølle, 2023), highlighting that early stellar dynamics are strongly influenced by gas dynamics. Once these young multiple-star systems have accreted mass and are no longer embedded, they are not expected to evolve much dynamically. For details on the evolution of multiple are systems after their initial formation, we refer readers to 'Evolution of binary stars'.", '5 Conclusions': 'Stars form in turbulent environments with a complex interplay of di ff erent physics. At the beginning of this chapter, we reviewed the role of gravity, hydrodynamics, radiation, and magnetism in the collapse of a pre-stellar core into a star. We find that the collapse of a core by gravity is counteracted by gas pressure, radiation feedback, and magnetic pressure on di ff erent scales, but e ffi cient radiative cooling, and angular momentum removal by magnetic fields, jets, and outflows can also aid pre-stellar collapse. We provided a summary of the physics of molecular clouds in which stars form, and how the interplay of gravity, turbulence, magnetic fields, and stellar feedback in the form of jets / outflows and radiation controls the star formation rate (SFR) and the initial mass function (IMF) of stars. We then reviewed what observations can tell us about the star formation processes. Sub-millimeter and infrared studies reveal a complex interplay of di ff erent components of the protostellar systems. Observations of gas kinematics can constrain theoretical predictions on the origin of protostellar jets and outflows, while thermal dust continuum observations deliver constraints on the onset of planet formation. Finally, we highlighted how the formation of multiple star systems complicates our single-star picture of star formation. We emphasize that most stars are born with companions and why and how this may a ff ect planet formation.', 'Acknowledgments': 'R.L.K. acknowledges funding from the Klaus Tschira Foundation. C.F. acknowledges funding provided by the Australian Research Council (Discovery Project DP230102280), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). 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2024PhRvD.110k6001G

Open system dynamics of an electron is studied in the presence of radiation field confined between two parallel conducting pates. It has been suggested in previous works that the quantized zeropoint modes of this field lead to finite decoherence effects possibly due to the Casimir force. However in this work it is shown that the decoherence found in previous works is due to the sudden switching on of the systemenvironment interaction and due to the acceleration of the electron enforced by the background paths whose superposition was analyzed. The work discusses important theoretical aspects of the setup and shows that while coherence might be lost due to bremsstrahlung induced by an external or the image potential it cannot be lost due to the mere presence of the quantum vacuum fluctuations between the plates.

2024-12-01T00:00:00Z

['2024PhRvD.110k6001G', 'arXiv:2409.03866', '2024arXiv240903866G', '10.48550/arXiv.2409.03866', '10.1103/PhysRevD.110.116001']

['Phenomenological aspects of field theory', 'general methods', 'Quantum Physics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Decoherence due to the Casimir effect

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

https://arxiv.org/pdf/2409.03866.pdf

{'Decoherence due to the Casimir effect?': 'Anirudh Gundhi ∗ \nDepartment of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy \n(Dated: December 5, 2024) \nOpen system dynamics of an electron is studied in the presence of radiation field, confined between two parallel conducting pates. It has been suggested in previous works that the quantized zero-point modes of this field lead to finite decoherence effects, possibly due to the Casimir force. However, in this work it is shown that the decoherence found in previous works is due to the sudden switching on of the system-environment interaction and due to the acceleration of the electron enforced by the background paths whose superposition was analyzed. The work discusses important theoretical aspects of the setup and shows that while coherence might be lost due to bremsstrahlung induced by an external or the image potential, it cannot be lost due to the mere presence of the quantum vacuum fluctuations between the plates.', 'I. INTRODUCTION': "Previous works [1-7] have considered the intriguing possibility of observing decoherence effects solely due to the presence of vacuum fluctuations. 1 Any such loss of coherence would also compete with the models of wavefunction collapse [11-14] in predicting the quantum-toclassical transition of perfectly isolated systems. However, it was shown in [3, 6, 7] that there can be no observable decoherence due to vacuum fluctuations alone. The overall effect of the vacuum is merely to dress the bare electron with a cloud of virtual photons [3, 6]. The QED vacuum state depends upon the position of the charged particle but has no memory of its trajectory. It cannot acquire which-path information about the particle and results in no irreversible loss of coherence. Therefore, the presence of zero-point modes cannot be deduced by measuring the loss of coherence for the free electron. \nOne can infer the presence of vacuum fluctuations, however, by measuring the stress two parallel and perfectly conducting plates exert on each other [15-19]. The Simplest way to understand the Casimir effect is to notice that the boundary conditions imposed by the plates do not allow for all the zero-point modes to be present, since the allowed mode wavelengths are constrained by the separation between the plates. Thus, there is a favorable energy gradient in bringing the plates from infinity to a finite separation, resulting in the attractive Casimir force. A review of the subject can be found in [20, 21]. \nThe question then arises whether the Casimir force can also monitor the motion of a charged particle and thus lead to decoherence. In other words, it remains to be understood whether the role of the environment played by vacuum fluctuations changes from being trivial, in empty space, to something significant and observable, when the radiation field is confined between conducting \nplates. Such effects have been studied in [22-24] (single conducting plate) and in [25] (where the case of parallel conducting plates was also considered explicitly). It is concluded in [22, 24, 25] that the zero-point fluctuations of the radiation field result in some decoherence at the level of the electron, and that this effect might be related to the Casimir force. \nInstead, in this work it will be shown that decoherence found in previous works is due to the sudden switching on of the interaction between the electron and the radiation field [22], and due to the acceleration of the electron along the trajectories whose superposition is analyzed [24, 25]. This loss of coherence should not be interpreted due to the mere presence of the vacuum fluctuations. There is no physical mechanism through which the electron's whichpath information can be lost irreversibly to the environment, especially if the plates are treated as ideal conductors and effects such as ohmic resistance to the image currents, as considered in [26], are ignored. 2 Nevertheless, the initial jolt , resulting from the sudden switching on of the interaction, leads to peculiar features in the reduced density matrix which are typically not found in other open systems. These features are explained by realizing that part of the initial jolt in the radiation field would affect the dynamics even at later times, since it is reflected back and forth by the conducting plates on either side of the electron. Although it would have little observational relevance in typical scenarios, the dynamics might still serve as a special reference for this widely discussed technical aspect of open quantum systems, dealing with the transient effects due to the sudden switching on of the system-environment (S-E) interaction [31-38]. \nNevertheless, from a physical point of view in which such a jolt would be absent, decoherence can only result from bremsstrahlung induced by the Coulomb potential of the image charges or an external potential, but not from the quantum vacuum fluctuations alone. \nFIG. 1. Infinite number of positive (blue) and negative (red) image charges induced by the plates, located at x = -L/ 2 and x = L/ 2, to cancel the Coulomb field of the electron (center) along their surface. \n<!-- image --> \nThe electric field along the surface of an idealized perfect conductor is zero. It is well known from standard electrostatics that if an electron is placed between two parallel conducting plates, as shown in Fig. 1, this requirement can be met with the help of an infinite number of image charges, whose positions are determined by the position of the electron between the plates. If the electron is placed at a coordinate x , the position state of the image charges is given by \n| + ⟩ x = |· · · , -(2 n -1) L -x, · · · , (2 n -1) L -x, · · ·⟩ , |-⟩ x = |· · · , -2 nL + x, · · · , 2 nL + x, · · ·⟩ , (1) \nwhere n ≥ 1 is an integer. \nFrom an open quantum system point of view, these infinite image charges belong to the environment. One might therefore ask whether or not there would be any direct positional decoherence due to the image charges themselves, since the position state of the images is perfectly correlated to the position state of the electron. This is the subject of this section, while decoherence due to vacuum fluctuations is studied in the following ones. \nTo find an answer for the loss in coherence, one might look at the off-diagonal elements of the reduced density matrix ˆ ρ r . The standard expression for ˆ ρ r is given by \nˆ ρ r ( t ) = Σ E ⟨ E | Ψ t ⟩ ⟨ Ψ t | E ⟩ , (2) \nwhere | Ψ t ⟩ represents the full S-E state at time t , and | E ⟩ the environmental basis states. In general, due to the S-E interaction, the state of the environment |E⟩ becomes correlated to the position of the system such that | Ψ t ⟩ = ∫ dxψ t ( x ) |E x ⟩ | x ⟩ , where ψ t ( x ) is the probability amplitude for the system to be at a position x at time t . In such a scenario, the off-diagonal elements of ˆ ρ r are given by \n⟨ x | ˆ ρ r ( t ) | x ' ⟩ = ⟨E x ' |E x ⟩ ψ t ( x ) ψ ∗ t ( x ' ) . (3) \nIf the environmental states for two different states of the system are completely orthogonal, the system is generally understood to have decohered completely. If one follows this line of reasoning strictly, the decoherence kernel \nD ( x, x ' ) due to the image charges will be given by \nD ( x, x ' ) := ⟨E x ' |E x ⟩ = x ' ⟨ + , -| + , -⟩ x . (4) \n̸ \nIt is clear that D ( x, x ' ) = 0 whenever x ' = x and D ( x, x ' ) = 1 for x ' = x . Thus, one might conclude that the image charges measure the electron's position perfectly and instantaneously. \nHowever, this suppression of the off-diagonal elements represents false decoherence . In order to observe the loss of fringe contrast in a double-slit experiment, the environment must be able to acquire which-path information rather than which-position information about the system. The interference fringes appear because the electron reaches a given point on the detector screen while being in a superposition of different paths. As it approaches the screen and is detected at a given point, the state of the image charges would only be correlated to the position at which the electron is detected on the screen, and would not have any memory of which slit the electron passed through. 3 In other words, D → 1 for the two paths that pass through different slits but end at a given point on the screen where the electron is detected. Therefore, there would be no loss in fringe contrast due to the images. The situation here is analogous to empty space where it is understood why there are no decoherence effects due to the particle's Coulomb field itself [39, 40]. For a loss in fringe contrast, the environment should continuously monitor the system, like a thermal bath, where the photons scattering off the electron at different times can effectively measure its full trajectory. Instead, if D depends only on the latest position of the system, whichpath information cannot be acquired. \nWhile the image charges cannot directly decohere the electron, there is an indirect way by which they can. It is due to the acceleration caused by the image charges, resulting in bremsstrahlung, which carries away which-path information about the electron. The effective potential at the electron's location can be computed by summing over the Coulomb potential due to all the infinite images. In doing so, one might first notice that the distance between a given negative image charge and the electron is | ± 2 nL + x -x | = 2 nL . Since it is independent of the electron's position, the Coulomb potential due to the negative images cannot influence the electron's dynamics, as it simply adds a constant to the Hamiltonian. It is therefore sufficient to compute the Coulomb potential \ndue to the positive image charges only. It is given by \nV im = -α ℏ c ∞ ∑ n =1 [ 1 (2 n -1) L -2 x + 1 (2 n -1) L +2 x ] , (5) \nwhere α = e 2 / (4 πϵ 0 ℏ c ) is the fine structure constant. This sum is divergent, but not the dependence on x . To see this, the expression above is rewritten as \nV im = -α ℏ c ∞ ∑ n =0 2(2 n +1) L (2 n +1) 2 L 2 -4 x 2 = -α ℏ c ∞ ∑ n =0 [ 2 (2 n +1) L ( 1 -4 x 2 (2 n +1) 2 L 2 ) -1 ] . (6) \nFrom the setup shown in Fig. 1, it can be seen that -L/ 2 < x < L/ 2 and therefore the term ( 1 -4 x 2 / ((2 n +1) L ) 2 ) -1 can be written as a geometric series. The zero th order term in x of this expansion would again just give an irrelevant (divergent) constant contribution to the Hamiltonian and can therefore be ignored. The remaining x -dependent part of the potential is given by \nV im ( x ) = -2 α ℏ c L ∞ ∑ n =0 1 2 n +1 × 4 x 2 (2 n +1) 2 L 2 -4 x 2 = α ℏ c 2 L [ H ( -1 / 2 -x/L ) + H ( -1 / 2 + x/L ) + ln(16)] , (7) \nwhere, H ( x ) is the standard harmonic number function generalized to the domain of real numbers. Note that the expression for the effective potential V im has been previously derived in [41], using a more formal technique, but not by explicitly summing over the Coulomb potential due to all the image charges. \nFIG. 2. Exact Coulomb potential due to the infinite image charges, Eq. (7), at a location x between the plates. \n<!-- image --> \nIt can be seen from Fig. 2 that the electron would be in an unstable equilibrium at x = 0 and inevitably attracted to either one of the two plates. This is inevitable, because in quantum mechanics one cannot have both the \nposition and velocity to be zero simultaneously. 4 The effect of the images on the electron dynamics if far from being negligible. It is clear that the electron would experience strong acceleration, especially near x = ± L/ 2, and thus decohere due to which-path information being carried away by bremsstrahlung. To conclude this section, a simple scenario is analyzed in which the effects of the image charges can be computed easily, and contrasted with the predictions in empty space. \nTo study decoherence effects over large timescales, one might introduce an external harmonic potential potential, \nV ext ( x ) = 1 2 m Ω 2 x 2 , (8) \nwhich restricts the electron dynamics in the neighborhood of the origin. This is because the electron would otherwise run away towards the plates. In such a scenario, it is useful to expand V im ( x ) in powers of x/L and to keep the leading order term. The potential due to the images (7) can be written as \nV im ( x ) = α ℏ c L ∞ ∑ n =1 ψ (2 n, 1 / 2) (2 n )! ( x L ) 2 n , (9) \nwhere ψ (2 n, 1 / 2) := d 2 n ψ ( x ) /dx 2 n , evaluated at x = 1 / 2, ψ ( x ) being the digamma function. Therefore, for a large enough Ω, the overall effect of the image charges is to lower the frequency of the harmonic oscillator (since ψ (2 , 1 / 2) = -16 . 8288) \nV eff = V im + V ext = 1 2 m Ω 2 eff x 2 , Ω 2 eff = Ω 2 + α ℏ cψ (2 , 1 / 2) mL 3 , (10) \nand thus redshift the spectrum. To leading order in 1 /L , the emission of bremsstrahlung between the plates is the same as in empty space. This is simply because in the limit L →∞ we should recover the dynamics as in empty space. Therefore, to leading order, decoherence in the presence of parallel plates can be described by the master equation for bremsstrahlung [43] \n∂ t ˆ ρ r ( t ) = -i ℏ [ ˆ p 2 2 m + 1 2 m Ω 2 eff ˆ x 2 , ˆ ρ r ( t ) ] -iα Ω 2 eff 3 mc 2 [ˆ x, { ˆ p, ˆ ρ r ( t ) } ] -α Ω 3 eff 3 c 2 [ˆ x, [ˆ x, ˆ ρ r ( t )]] . (11) \nWe see that since Ω eff < Ω, the overall effect of the image charges is to increase the coherence length, simply because the plates effectively lower the frequency of the \nexternal harmonic potential V ext . Nevertheless, it is clear that the image potential exerts an additional force on the electron, which is absent in empty space, leading to decoherence rates which are different compared to those in empty space. The rate can be lower or higher, depending upon the setup. For other experimental scenarios, one might use the exact potential derived in Eq. (7) and compute the decoherence rate using the full noise kernel, presented in the next section. Further details concerning the effect of image charges on the electron dynamics will not be discussed. The main objective of this section is to highlight how the effective potential inevitably accelerates the electron, leading to irreversible decoherence.", 'III. ZERO-POINT MODES': 'The final aspect of the analysis is to understand whether vacuum fluctuations, confined between conducting plates, can lead to observable decoherence. To answer this question, one must study if the suppression of the offdiagonal elements of the reduced density matrix implies genuine decoherence, and therefore check if the environment remembers the state of the system, even when its interaction with the system is switched-off after some finite time. If it does not, the general implication would be that the environment is only correlated to the state of the system at a given time, and not to its history/trajectory. The situation would then be identical to that of Sec. II, where it was explained why environments of this type do not lead to a loss in the fringe contrast.', 'A. Master equation': "Up to second order in the charge e , using the influencefunctional formalism [44, 45], the time evolution of the reduced density matrix of the electron was recently derived in [7], starting from the nonrelativistic QED Lagrangian. It reads \n∂ t ˆ ρ r = -i ℏ [ ˆ H s , ˆ ρ r ( t ) ] -1 ℏ ∫ t -t i 0 dτ N ( t ; t -τ ) [ˆ x, [ˆ x H s ( -τ ) , ˆ ρ r ( t )]] + i 2 ℏ ∫ t -t i 0 dτD ( t ; t -τ ) [ˆ x, { ˆ x H s ( -τ ) , ˆ ρ r ( t ) } ] . (12) \nIn the equation above, H s is the system part of the full Hamiltonian H \nH := H s +H EM +H int , (13) \nwhere H EM governs the free evolution of the radiation field and H int governs the S-E interaction, \nH EM = ϵ 0 2 ∫ d 3 r ( Π 2 + c 2 B 2 ) , H int = e rΠ ( r , t ) , (14) \nwhere Π := -P /ϵ 0 , P being the conjugate momentum of the radiation field. The system Hamiltonian is given by \nˆ H s = ˆ p 2 2 m + V eff (ˆ x ) + V EM (ˆ x ) , (15) \nwhere V eff is the effective potential due to the images and some external potential that might be applied, while V EM = e 2 ω 3 max ˆ x 2 / (3 π 2 ϵ 0 c 3 ) is a divergent contribution, which scales with the UV cutoff ω max , originating from the Legendre transformation relating the QED Lagrangian and the Hamiltonian [1, 7]. The role played by V EM is to cancel the divergence coming from the dissipation kernel D in the master equation (12), as shown in [7]. \nFor the motion of the electron considered along the x axis, the noise kernel N and the dissipation kernel D in Eq. (12) are defined to be \nN ( t 1 ; t 2 ) := e 2 2 ℏ 〈 { ˆ Π x ( t 1 ) , ˆ Π x ( t 2 ) } 〉 0 , D ( t 1 ; t 2 ) := ie 2 ℏ 〈[ ˆ Π x ( t 1 ) , ˆ Π x ( t 2 ) ]〉 0 θ ( t 1 -t 2 ) . (16) \nIn Eq. (16), the operator ˆ Π x ( t ) is the freely evolved Heisenberg operator, the same as the transverse electric field operator of the free EM field, while the expectation value ⟨⟩ 0 is taken with respect to the initial state of the environment. To study decoherence due to vacuum fluctuations, the initial state of the environment is taken to be the ground state | 0 ⟩ of the free electromagnetic (EM) field, assuming that there are no photons at the initial time t i = 0 and that the initial S-E state is completely uncorrelated, i.e., ˆ ρ ( t i ) = ˆ ρ s ( t i ) ⊗ | 0 ⟩ ⟨ 0 | . The kernel N describes decoherence while D describes the energy lost by the system to the environment. Since it is important to distinguish between the decoherence effects due to bremsstrahlung and those due to zero-point modes of the EM field, the off-diagonal elements are computed with the electron kept steady in a superposition of x and x ' , so that decoherence due to bremsstrahlung does not show up. In such a scenario, ˆ x H s ( -τ ), which is the Heisenberg time-evolved position operator with respect to the system Hamiltonian ˆ H s , is given by ˆ x -ˆ pτ/m . Moreover, in the absence of acceleration, the dissipation kernel would not give any contribution to the master equation as the electron can only lose energy to the environment via radiation emission upon acceleration (to second order in e ). \nThus, the master equation describing decoherence reduces to \n∂ t ˆ ρ r | dec = -1 ℏ ∫ t -t i 0 dτ N ( t ; t -τ ) [ˆ x, [ˆ x, ˆ ρ r ( t )]] . (17) \nThe decoherence kernel in Eq. (4) for the problem at hand can also be computed more directly, without referring to the influence functional formalism, as it will \nbe shown later. However, computing decoherence effects via Eq. (17) has the advantage of discussing any possible connection between the Casimir force and decoherence in a straightforward way. Therefore, in what follows, the decoherence kernel D is derived using both approaches.", 'B. Decoherence': "In the presence of parallel conducting plates, the freely evolved operator ˆ Π , same as the transverse electric field operator of the free EM field, is given by [25, 46] \nˆ Π ( x ∥ , x, t ) = -i √ 2 ℏ ϵ 0 L ∞ ∑ n =0 f ( n ) ∫ d 2 k ∥ 2 π √ ω n 2 e i ( k ∥ · x ∥ -ω n t ) × [ ˆ a 1 ( k ∥ , n )( ˆ k ∥ × ˆ x ) sin ( nπ ( x + L/ 2) L ) + ˆ a 2 ( k ∥ , n ) { i ˆ k ∥ nπc ω n L sin ( nπ ( x + L/ 2) L ) -ˆ x k ∥ c ω n cos ( nπ ( x + L/ 2) L )}] +HC . (18) \nHere, f ( n ) is a weighing function such that f (0) = 1 / √ 2 and f ( n ) = 1 ∀ n > 0 [15], ˆ k ∥ is a unit vector along the plates, ω 2 n /c 2 := k 2 ∥ + n 2 π 2 /L 2 , and ˆ a 1 , ˆ a 2 are the annihilation operators corresponding to the two modes of the EMfield. It can also be seen that ˆ Π ∥ = 0 along the plates at x = ± L/ 2, thus satisfying the appropriate boundary conditions. \nWe are mainly interested in studying decoherence for the electron in a superposition along the ˆ x axis. Thus, only the ˆ x component of ˆ Π enters the noise kernel in Eq. (17). For electron superpositions prepared near the origin, on length scales | x -x ' | ≪ L (superpositions over a distance L will also be discussed later), the two-point correlation ⟨ 0 | ˆ Π x (1) ˆ Π x (2) | 0 ⟩ is given by \n⟨ 0 | ˆ Π x (1) ˆ Π x (2) | 0 ⟩ = ℏ c 2 ϵ 0 L ∞ ∑ n = -∞ ∫ d 2 k ∥ (2 π ) 2 k 2 y + k 2 z 2 ω n cos 2 ( nπ/ 2) e i ( k ∥ · ∆ x ∥ -ω n τ ) (19) \nwhere ∆ x ∥ := x (1) ∥ -x (2) ∥ and τ := t (1) -t (2) . The surface integral above can be turned into a volume integral by introducing δ n ( k x ) := δ ( k x -nπ/L ) such that \n⟨ 0 | ˆ Π x (1) ˆ Π x (2) | 0 ⟩ = ℏ c 2 ϵ 0 L ∑ n even ∫ d 3 k (2 π ) 2 k 2 -k 2 x 2 ω e i ( k ∥ · ∆ x ∥ -ωτ ) δ n ( k x ) . (20) \nUsing the properties of the Dirac comb, \n∑ n even δ n = ∞ ∑ n = -∞ δ 2 n = L 2 π ∞ ∑ m = -∞ e imk x L , (21) \nthe two-point correlation becomes \n⟨ 0 | ˆ Π x (1) ˆ Π x (2) | 0 ⟩ = ℏ c 2 2 πϵ 0 ∞ ∑ m = -∞ ˆ □ m ∫ d 3 k (2 π ) 2 2 ω e i ( k · ∆ x m -ωτ ) e -k/k max , (22) \nwhere ∆ x m := { x m , ∆ x ∥ } , x m := mL , ˆ □ m := -∂ 2 τ /c 2 + ∂ 2 x m , and the UV cutoff has been introduced in the calculations by inserting exp( -k/k max ) inside the integral. Further, the dependence on ∆ x ∥ can be ignored not only because we are mainly interested in the dynamics along the x axis, where ∆ x ∥ = 0, but also because ∆ x ∥ ≪ cτ for a nonrelativistic particle. The integral can now be easily evaluated and gives \n⟨ 0 | ˆ Π x (1) ˆ Π x (2) | 0 ⟩ = ℏ c π 2 ϵ 0 ∑ m 1 ( m 2 L 2 -c 2 ( τ -iϵ ) 2 ) 2 , (23) \nwhere ϵ := 1 / ( k max c ). The noise kernel is thus given by \nN ( τ ) = e 2 c 2 π 2 ϵ 0 ∞ ∑ m = -∞ 1 ( m 2 L 2 -c 2 ( τ -iϵ ) 2 ) 2 +HC . (24) \nSome important observations can already be made at this stage. The m = 0 term in the summation is independent of L and is the only piece that survives in the limit L →∞ . It must therefore correspond to the noise kernel N 0 of empty space without any conducting plates. Indeed, it can be seen that the m = 0 term matches the noise kernel derived in [7] (cf. Eqs. (47) - (49) therein). When the acceleration of the charged particle due to an external potential can be neglected, the suppression of the off-diagonal elements of the density matrix due to N 0 is false [3, 7]. It is thus not relevant to observable decoherence effects. Therefore, the observationally relevant part N ob ( τ ) is given by \nN ob ( τ ) = e 2 c ϵ 0 π 2 ∞ ∑ m =1 1 ( m 2 L 2 -c 2 ( τ -iϵ ) 2 ) 2 +HC . (25) \nAnother interesting aspect emerges if we consider the large L limit or, more correctly, the limit cτ ≪ L which is given by \nN Cas = 16 e 2 ℏ ϵ 0 × π 2 ℏ c 720 L 4 . (26) \nThe expression is written in a suggestive way since π 2 ℏ c/ (720 L 4 ) is nothing but the Casimir energy density (cf. pg. 13 in [21]). One might therefore attribute the leading order contribution to decoherence to the Casimir effect. This, however, should not be done. If expression (26) is used in Eq. (17), the off-diagonal elements of the density matrix would be suppressed as \n⟨ x ' | ˆ ρ r ( t ) | x ⟩ Cas. = ⟨ x ' | ˆ ρ r (0) | x ⟩ exp { -4 π 3 αc 2 t 2 ( x -x ' ) 2 90 L 4 } . (27) \nWe see that within the validity of the 1 /L expansion of N ob , i.e. ct ≪ L , there will be practically no loss of coherence over any length scale | x -x ' | ≤ L . One might, however, extrapolate the consequences to the time domain ct ≫ L and conclude that eventually coherence will be lost over time due to the plates. This should not be done, since the behavior of the full noise kernel is very different from the leading order term. \nTo see this, the full expression for N ob in Eq. (25) needs to be integrated. Doing that, the evolution of the off-diagonal elements, and thus the decoherence kernel (Eq. (4)), is obtained to be \nρ r ( x ' , x, t ) = exp ( -( x ' -x ) 2 ℏ N 2 ( t ) ) ρ r ( x ' , x, 0) , N 2 ( t ) := ∫ t 0 dt ' ∫ t ' 0 dτ N ob ( τ ) = ∞ ∑ m =1 e 2 2 π 2 ϵ 0 m 3 L 3 × t ln (∣ ∣ ∣ ∣ mL + ct mL -ct ∣ ∣ ∣ ∣ ) , (28) \nwhere the limit ϵ → 0 has been taken at the end. Before going into the properties of the decoherence kernel D , it is instructive to derive it using a more direct approach. As mentioned in Sec. II, for a particle held in a superposition of x and x ' , D ( x, x ' , t ) is given by the overlap of the environmental states corresponding to the two particle locations. Given the interaction Hamiltonian in Eq. (14), to leading order in e , the state of the environment (in the interaction picture) is given by \n|E ( x ) ⟩ t = exp ( -iex ℏ ∫ t 0 dt ' ˆ Π x ( x, t ' ) ) | 0 ⟩ . (29) \nSince the operator ˆ Π x given in Eq. (18) is simply a linear sum of creation and annihilation operators, |E ( x ) ⟩ t is a coherent state | α ( x, t ) ⟩ . To work out the overlap between the coherent states corresponding to different particle locations, the use of dipole approximation is well justified if the superpositions are prepared around x = 0. Therefore \n⟨ α ( x ' , t ) | α ( x, t ) ⟩ = ∏ k ∥ ∏ n 〈 α k ∥ n ( x ' , t ) ∣ ∣ α k ∥ n ( x, t ) 〉 = exp { -e 2 ( x ' -x ) 2 4 π 2 ℏ ϵ 0 L ∞ ∑ n = -∞ ∫ d k ∥ × × ( k ∥ c ) 2 cos 2 ( nπ/ 2) sin 2 ( ω n t/ 2) ω 3 n } , (30) \nwhere, the relation for the coherent states, \n⟨ β | α ⟩ = exp ( -( | β | 2 + | α | 2 -2 β ∗ α ) / 2 ) , (31) \nhas been used. Using the Dirac comb (21), the integral above can be evaluated exactly and gives \nD ( x, x ' , t ) = exp { -e 2 ( x ' -x ) 2 2 π 2 ℏ ϵ 0 L 3 ∞ ∑ m =1 t m 3 ln (∣ ∣ ∣ ∣ mL + ct mL -ct ∣ ∣ ∣ ∣ ) } , (32) \nwhere again, the contribution from the m = 0 term has been discarded for the reasons described before. We see that the expression for the decoherence kernel is indeed the same when derived using the influence functional formalism (Eq. (28)), or when it is obtained by computing the overlap between the environmental states correlated to different electron positions. \nFIG. 3. Time evolution of the decoherence kernel showing suppression of the off-diagonal elements at times t = mL/c , and an otherwise constant curve for ct ≫ L . \n<!-- image -->", 'IV. INITIAL JOLT': "In this section it will be argued that all the features in D , as depicted in Fig. 3, are due to the sudden switching on of the S-E interaction. These effects should not be considered physical, since in a realistic scenario there is no such sharp instant of time at which the electron starts interacting with the EM field vacuum. \nA sudden switching on of the S-E interaction results in a disturbance in the state of the environment, which is sometimes referred to as a jolt (cf. [35], the references therein, and [38]). In general, after the initial jolt has passed away, an equilibrium between the system and the environment is established. However, in the setup that is being considered in this work, part of the initial jolt would always influence the dynamics and the S-E equilibrium is never established. This can be seen from the periodic fall in D at times t m = mL/c . \nA possible explanation of these features is that the jolt in the EM field at time t = 0 propagates, gets reflected by the plates, and arrives at the location of the electron again. Since the superpositions are assumed to be near the center, this jolt would arrive at the electron location again at times t m , resulting in a temporary fall in coherence. 5 \nThe same features can also be equivalently described in terms of the image charges. It can be seen from Eq. (1) \nthat for an electron located at the center x = 0, the nearest images are at a distance L , the next nearest at 2 L , and so on. As the initial jolt is sourced by the electron at the center, the images must source a counterjolt in order to maintain the boundary conditions. That is, by the time the jolt from the electron arrives at one of the plates, a counterjolt must arrive at the same plate from the nearest image to nullify the electric field along the plate. Subsequently, the jolt from the nearest images reaches the electron at t = L/c , from the next nearest images at t = 2 L/c and so on, 6 explaining the fall in D at times t m . \nThe reasoning above indicates that the sharp falls in the off-diagonal elements would not be present in the absence of a jolt. Therefore, in such a more realistic scenario, the decoherence kernel D would then only have a steady value, given by the constant upper bound in Fig. 3. This asymptotic value can be calculated analytically. If ct ≫ L , the main contribution to D comes from the modes for which mL ≪ ct , since the contribution of the higher modes is greatly suppressed due to the 1 /mL dependence. In this limit ln { (1+ mL/ct ) / (1 -mL/ct ) } ≈ mL/ct and therefore \nD ( x, x ' , t ) ≈ exp { -2 α ( x ' -x ) 2 πL 2 ∞ ∑ m =1 1 m 2 } . (33) \nThe asymptotic expression for the coherence length is rather simple and involves both the fine structure constant α and the separation length L . Further, since ∑ m 1 /m 2 = π 2 / 6, the off-diagonal elements would be unaffected over length scales L/ √ α , 7 which basically means no decoherence at all. \nTherefore, it is important to realize that the environment under consideration does not have a typical behavior. For instance, in the environment of thermal photons, the off-diagonal elements decay exponentially in time (ignoring dissipation, cf. Fig. 3.8 in [8]). Thus, the extrapolation of Eq. (27) in thinking that the Casimir force leads to a continuous irreversible loss of coherence, like ordinary environments, does not hold. \nIt was shown in [7] that if one considers a time dependent coupling q ( t ) with the environment, i.e., q ( t ) = -ef ( t ) such that f ( t ) = 1 for most of the dynamics between the initial time t = 0 and the final time t = T , while f (0) = f ( T ) = 0, the noise kernel in Eq. (17), and defined in Eq. (16), transforms as N → ˜ N , with \n˜ N = f ( t 1 ) f ( t 2 ) N ( t 1 ; t 2 ) = f ( t 1 ) f ( t 2 ) N ( t 1 -t 2 ) . (34) \nIt was further shown at a very general level that if N 2 approaches a constant value on some timescale, which in this case is L/c , and if the S-E interaction is switched on adiabatically over this timescale, then N 2 transforms as N 2 → ˜ N 2 (cf. Eq. (75) in [7]) with \n˜ N 2 ( T ) = N 2 2 ( f 2 (0) + f 2 ( T ) ) = 0 , (35) \nimplying no irreversible decoherence. These considerations imply that in a typical scenario in which the S-E interaction is not switched on suddenly, there would be no irreversible loss of coherence at the level of the electron and thus no loss in the fringe contrast in a double slit type of experiment. The reasoning and the arguments presented in this section are confirmed in Sec. VI, where the decoherence kernel is computed after adiabatically switching on the S-E interaction.", 'V. SUPERPOSITIONS OVER LARGER LENGTH SCALES': "For particle superpositions near the center, the analysis shows that there is no loss in coherence over time and that the features in D ( x, x ' , t ) are solely due to the sudden switching on of the interactions. One may still question the validity of these conclusions for superpositions prepared over large length scales. It should first be clear that when considering superpositions along the x axis, the results will not be affected by the y and z dependence of the noise kernel. This is because the dependence on the difference between the y and z coordinates, of the nonrelativistic particle superpositions, can be ignored since ∆ x ∥ ≪ cτ . This can be seen easily from Eq. (22). 8 \nWith this in mind, the decoherence kernel for the electron in a superposition over large length scales, x = -L/ 2 and x ' = L/ 2, is now computed. In this case, the twopoint correlation is given by \n⟨ 0 | ˆ Π x ( -L/ 2 , t 1 ) ˆ Π x ( L/ 2 , t 2 ) | 0 ⟩ = ℏ c 2 ϵ 0 L ∞ ∑ n = -∞ ∫ d 2 k ∥ (2 π ) 2 k 2 y + k 2 z 2 ω n ( -1) n e i ( k ∥ · ∆ x ∥ -ω n τ ) . (36) \nAs before, the surface integral can be converted into a volume integral by inserting a Dirac comb, and then summing over ∑ n ( -1) n δ n . This sum can be written as \n∞ ∑ n = -∞ ( -1) n δ n = 2 ∞ ∑ n = -∞ δ 2 n -∞ ∑ n = -∞ δ n . (37) \nThe identity involving the Dirac comb can now be applied to both terms on the rhs of the equation above. With little algebra, it is easy to see that it gives \nN ( τ ) = e 2 c ϵ 0 π 2 ∞ ∑ m = -∞ 1 ( m 2 L 2 -c 2 ( τ -iϵ ) 2 ) 2 +HC -e 2 c ϵ 0 π 2 ∞ ∑ m = -∞ 1 ( m 2 (2 L ) 2 -c 2 ( τ -iϵ ) 2 ) 2 +HC . (38) \nThe decoherence kernel is then immediately obtained to be \nD ( -L/ 2 , L/ 2 , t ) = exp { -e 2 ( x ' -x ) 2 π 2 ℏ ϵ 0 ∞ ∑ m =1 t m 3 × [ 1 L 3 ln (∣ ∣ ∣ ∣ mL + ct mL -ct ∣ ∣ ∣ ∣ ) -1 (2 L ) 3 ln (∣ ∣ ∣ ∣ 2 mL + ct 2 mL -ct ∣ ∣ ∣ ∣ )]} . (39) \nConsistent with the description above, it can be seen that there will be falls in coherence not only at time intervals t = 2 mL/c , since it takes the time 2 L/c for the jolt sourced near one of the plates to return to the same plate, but also at time intervals t = mL/c , since the jolt sourced from x = -L/ 2 can temporarily become correlated to the electron at x ' = L/ 2. Again, even for the largest possible superposition, one sees that all the features in the off-diagonal elements can be ascribed to the sudden switching on of the interaction. The same and consistent conclusions are reached if D ( -L/ 2 , 0 , t ) or D (0 , L/ 2 , t ) is computed.", 'VI. ADIABATIC SWITCHING ON': "To confirm the physical interpretation, one can compute the off-diagonal elements of the density matrix, but this time after adiabatically switching on the S-E interaction. This can be modeled by evolving the state of the environment as \n|E ( x ) ⟩ ad t = exp { -iex ℏ ∫ t 0 dt ' ˆ Π x ( x, t ' ) ( 1 -e -t ' /T ) } | 0 ⟩ . (40) \nHere, T represents the timescale over which the S-E interaction is switched on and t is some late time at which the interaction is fully switched on. After computing the integral, one should first take the limit t ≫ T and then the limit T →∞ . If the limits are taken in the opposite order, the S-E interaction would never be switched on. After the interaction is fully switched on at some time t , \nthe decoherence kernel is given by \nD ad ( x, x ' , t ) = ⟨E ( x ' ) |E ( x ) ⟩ ad t = exp { -e 2 ( x ' -x ) 2 16 π 2 ℏ ϵ 0 L ∞ ∑ n = -∞ ∫ d k ∥ ( k ∥ c ) 2 cos 2 ( nπ/ 2) ω 3 n } . (41) \nHere, the calculations are performed for the case where the superpositions are prepared near the center. The main effect of a sudden jolt is not related to the length scale over which the superpositions are prepared and the conclusions of adiabatically switching on the interaction would also apply to D ( L/ 2 , -L/ 2 , t ). \nIt is already interesting to notice that D in Eq. (41) does not depend on time t , indicating that the oscillations in Fig. 3 do not appear in the absence of a sudden jolt. As before, the integral can be computed by inserting a Dirac comb and using the identity (21). It gives \nD ad ( x, x ' , t ) = exp { -e 2 ( x ' -x ) 2 8 π 2 ℏ cϵ 0 × × ∞ ∑ m =1 ∫ ∞ 0 dk k ∫ π 0 dθ sin θ (1 -cos 2 θ ) e imLk cos θ } = exp { -2 α ( x ' -x ) 2 πL 2 ∞ ∑ m =1 1 m 2 } . (42) \nIt can be clearly seen that the oscillations have disappeared from the decoherence kernel altogether (compared to Eq. (32)) and that its value is the same as that of the asymptotic curve in Fig. 3. The off-diagonal elements are not suppressed over any length scale | x -x ' | ≤ L . What is even more important to notice is that after the S-E interaction is fully switched on, the overlap between the environmental states is stationary in time. Thus, the vacuum of the interacting radiation field effectively acts like an environment which is only correlated to the position of the electron. The situation then becomes identical to the one described in Sec. II. This implies that there would be no loss in the fringe contrast due to vacuum fluctuations, as they are not able to resolve the two paths which end at the same point on the detector screen.", 'VII. COMPARISON WITH PREVIOUS WORKS': "In this section a comparison is made with previous works [22, 24, 25] where it was shown that vacuum fluctuations in the presence of a conducting plate can lead to finite decoherence, contrary to the conclusion reached in this work. \nRef. [22] studies decoherence due to vacuum fluctuations in the presence of a single conducting plate. However, since Eq. (57) in Ref. [22] resembles Eqs. (32) and (39) in the present work (though it is not the same), \nthe analysis in [22] is most likely performed without switching on the S-E interaction adiabatically, and therefore the results obtained therein represent decoherence due to the initial jolt described in Sec. IV. \nThe standard assumption of starting with ˆ ρ ( t i ) = ˆ ρ s ( t i ) ⊗ | 0 ⟩ ⟨ 0 | cannot be made when studying interaction with vacuum fluctuations, since there is no way to avoid the vacuum. One might work with this assumption, however, after switching off the S-E interaction by hand at the initial time and then adiabatically switching it on. As explained in the last section, when this is done, there would be no decoherence due to vacuum fluctuations in the presence of conducting plates. Having two parallel conducting plates introduces infinite number of image charges instead of one (as in the case of a single conducting plate). Thus, it is reasonable to assume that the conclusions reached in this article would also hold for the case in which only a single conducting plate is present [22]. \nRefs. [24, 25] also obtain a finite value of decoherence in the presence of a conducting plate, although their result is different to the one in [22]. However, the explanation for the nonzero value found in [24, 25] is different and a bit more subtle. \nIn order to avoid misinterpretations concerning the decay of the off-diagonal elements ρ r ( x ' , x, t ), such as the one described in Sec. II, one might analyze if the environment can decohere a superposition of different trajectories rather than different positions. However, even such an analysis, in general, is not free of misinterpretations. \nAccording to the axes-convention used in Fig. 1, [24, 25] study decoherence for a superposition of trajectories ( t, Re -t 2 /T 2 , v y t, 0) and ( t, -Re -t 2 /T 2 , v y t, 0). See, for example, the discussion below Eq. (11) in [24] and above Eq. (27) in [25]. Since the acceleration of the electron along the two paths is finite, of the order R/T 2 , in such an analysis, in addition to any possible decoherence due to vacuum fluctuations, one also gets decoherence due to photons emitted by the accelerating electron. In fact, the finite decoherence obtained in the two works is due to bremsstrahlung induced by the chosen background trajectories whose superposition is analyzed and not due to vacuum fluctuations. The reasoning is based on the following estimate: \nDecoherence due to bremsstrahlung, for an electron accelerated by an external harmonic potential, is given by \n˙ ˆ ρ r ( t ) | dec = -α Ω 3 3 c 2 [ˆ x, [ˆ x, ˆ ρ r ]] . (43) \nHere, only the double commutator term on the rhs of Eq. (11) is retained. This equation can be readily integrated to see that the off-diagonal elements decay as \nρ r ( x ' , x, t ) = exp { -α Ω 3 t ( x ' -x ) 2 3 c 2 } ρ r ( x ' , x, 0) . (44) \nNow, the main objective in [24, 25] is to study decoherence between two trajectories that diverge up to a \ndistance R , and then converge back again on a timescale T . This can be equivalently achieved with the trajectories ( t, ± R sin Ω t, v y t, 0) where t goes from 0 → π/ Ω. In this way, decoherence due to bremsstrahlung over trajectories considered in [24, 25] can be equivalently estimated using Eq. (44). Setting Ω = 1 /T and ( x ' -x ) 2 = R 2 , one obtains \nρ r ( x ' , x, T ) ∼ exp { -απR 2 3 c 2 T 2 } ρ r ( x ' , x, 0) . (45) \nThis estimate closely resembles the expression for decoherence exp { W 0 } , W 0 ∝ -e 2 R 2 / ( c 2 T 2 ), found in [24, 25] (cf. Eqs. (23, 31) in [25]). However, in [24, 25] this result is misinterpreted as decoherence due to vacuum fluctuations in empty space. To avoid decoherence due to photon emission, one could consider the adiabatic trajectories T →∞ , so that the acceleration along the paths can be ignored. As expected, there is no decoherence in this limit. \nAccording to [24, 25], there is decoherence due to vacuum fluctuations even in the absence of conducting plates, which contradicts previous works [3, 6, 7]. Refs. [24, 25] further show that the presence of plates may increase or reduce the magnitude of this decoherence. However, since [24, 25] use the same trajectories in their analyses of a conducting plate, what they are possibly computing are corrections to decoherence due to bremsstrahlung, induced by the acceleration of the electron that is enforced by the chosen background paths. The same issue prevails for the case of two parallel conducting plates examined in [25].", 'VIII. CONCLUSIONS': "In this work, decoherence due to vacuum fluctuations of the EM field confined between parallel conducting plates is analyzed. It is argued that the loss in coherence which has been found in previous works is either due to the sudden switching on of the S-E interaction [22], or due to bremsstrahlung induced by the chosen background paths for which decoherence is analyzed [24, 25]. It is further shown in this work that decoherence in the presence of conducting plates is not a consequence of the nature of the vacuum fluctuations themselves, and thus not related to the Casimir force. \nTo study the time evolution of the reduced density matrix, it is standard practice, for technical convenience, to start from an initially uncorrelated S-E state. However, this initial condition, in general, is only justified when the interaction between the system and the environment is also switched off initially, since S-E interaction leads to S-E correlations. \nTherefore, if one takes the vacuum state of the free EM field as the initial state of the environment, uncorrelated to the initial state of the electron, one should also switch off their interaction initially. It is well known from standard quantum mechanics that to go from the vacuum \nstate of the free EM field (i.e. the bare vacuum), to the vacuum state of the interacting radiation field (which is the main subject of the analysis), the interaction must be switched on adiabatically. In other words, the gap between the starting point (dictated by technical convenience), and the actual physical situation of interest, must be bridged by adiabatically switching on the interaction. When this is done, it is shown in this work that the zero-point modes of the vacuum do not lead to any decoherence effects at the level of the electron. Decoherence due to zero-point modes might still be relevant in a situation where the charged particle suddenly enters and leaves a region confined between conducting plates. \n- [1] P. M. V. B. Barone and A. O. Caldeira, Phys. Rev. A 43 , 57 (1991).\n- [2] E. Santos, Phys. Lett. A 188 , 198 (1994).\n- [3] L. Di'osi, Phys. Lett. A 197 , 183 (1995).\n- [4] L. H. Ford, Phys. Rev. A 56 , 1812 (1997).\n- [5] H.-P. Breuer and F. Petruccione, in Relativistic Quantum Measurement and Decoherence , edited by H.-P. Breuer and F. Petruccione (Springer, Berlin, 2000) pp. 31-65.\n- [6] W. G. Unruh, in Relativistic Quantum Measurement and Decoherence , edited by H.-P. Breuer and F. Petruccione (Springer, Berlin, 2000) pp. 125-140.\n- [7] A. Gundhi and A. Bassi, Phys. Rev. A 107 , 062801 (2023).\n- [8] E. Joos, H. Zeh, D. Giulini, C. Kiefer, J. Kupsch, and I. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory , Physics and Astronomy Online Library (Springer, New York, 2003).\n- [9] M. A. Schlosshauer, Decoherence and the Quantum-ToClassical Transition (Springer-Verlag, Berlin, 2007).\n- [10] C. Kiefer, Phys. Rev. D 46 , 1658 (1992).\n- [11] A. Bassi and G. Ghirardi, Phys. Rep. 379 , 257 (2003).\n- [12] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85 , 471 (2013).\n- [13] M. Arndt and K. Hornberger, Nat. Phys. 10 , 271 (2014).\n- [14] M. Carlesso, S. Donadi, L. Ferialdi, M. Paternostro, H. Ulbricht, and A. Bassi, Nat. Phys. 18 , 243 (2022).\n- [15] H. B. G. Casimir, Indag. Math. 10 , 261 (1948).\n- [16] E. S. Sabisky and C. H. Anderson, Phys. Rev. A 7 , 790 (1973).\n- [17] S. K. Lamoreaux, Phys. Rev. Lett. 78 , 5 (1997).\n- [18] S. K. Lamoreaux, Phys. Rev. Lett. 81 , 5475 (1998).\n- [19] S. K. Lamoreaux, Phys. Rev. A 59 , R3149 (1999).\n- [20] L. Spruch, An overview of long-range casimir interactions, in Long-Range Casimir Forces: Theory and Recent Experiments on Atomic Systems , edited by F. S. Levin and D. A. Micha (Springer US, Boston, MA, 1993) pp. 1-71.\n- [21] K. A. Milton, The Casimir Effect: Physical manifestations of zero-point energy (World Scientific, Singapore, 2001).\n- [22] L. H. Ford, Phys. Rev. D 47 , 5571 (1993).\n- [23] H. Yu and L. H. Ford, Phys. Rev. D 70 , 065009 (2004).\n- [24] F. D. Mazzitelli, J. P. Paz, and A. Villanueva, Phys. Rev. A 68 , 062106 (2003). \nHowever, in typical scenarios, the only irreversible loss of coherence in the presence of parallel plates would be due to the effective Coulomb potential of all the infinite image charges, which accelerates the electron resulting in bremsstrahlung.", 'IX. ACKNOWLEDGEMENTS': "I thank Angelo Bassi and fellow group members for several discussions. This work was financially supported by the University of Trieste, INFN and EIC Pathfinder project QuCoM (GA No. 101046973). \n- [25] J.-T. Hsiang and D.-S. Lee, Phys. Rev. D 73 , 065022 (2006).\n- [26] J. R. Anglin and W. H. Zurek, in 31st Rencontres de Moriond: Dark Matter and Cosmology, Quantum Measurements and Experimental Gravitation (1996) pp. 263270, arXiv:quant-ph/9611049.\n- [27] Y. Levinson, J. Phys. A 37 , 3003 (2004).\n- [28] D. A. R. Dalvit and P. A. Maia Neto, Phys. Rev. Lett. 84 , 798 (2000).\n- [29] C.-H. Wu and D.-S. Lee, Phys. Rev. D 71 , 125005 (2005).\n- [30] K. Sinha and Y. b. u. Suba¸s ı, Phys. Rev. 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2022ApJS..259...35A

This paper documents the seventeenth data release DR17 from the Sloan Digital Sky Surveys the fifth and final release from the fourth phase SDSSIV. DR17 contains the complete release of the Mapping Nearby Galaxies at Apache Point Observatory MaNGA survey which reached its goal of surveying over 10000 nearby galaxies. The complete release of the MaNGA Stellar Library accompanies this data providing observations of almost 30000 stars through the MaNGA instrument during bright time. DR17 also contains the complete release of the Apache Point Observatory Galactic Evolution Experiment 2 survey that publicly releases infrared spectra of over 650000 stars. The main sample from the Extended Baryon Oscillation Spectroscopic Survey eBOSS as well as the subsurvey Time Domain Spectroscopic Survey data were fully released in DR16. New singlefiber optical spectroscopy released in DR17 is from the SPectroscipic IDentification of ERosita Survey subsurvey and the eBOSSRM program. Along with the primary data sets DR17 includes 25 new or updated valueadded catalogs. This paper concludes the release of SDSSIV survey data. SDSS continues into its fifth phase with observations already underway for the Milky Way Mapper Local Volume Mapper and Black Hole Mapper surveys.

2022-04-01T00:00:00Z

['10.3847/1538-4365/ac4414', 'arXiv:2112.02026', '2021arXiv211202026A', '2022ApJS..259...35A', '10.48550/arXiv.2112.02026']

['Astronomy data acquisition', 'Astronomy databases', 'Surveys', '1860', '83', '1671', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Instrumentation and Methods for Astrophysics']

The Seventeenth Data Release of the Sloan Digital Sky Surveys Complete Release of MaNGA MaStar and APOGEE2 Data

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

https://arxiv.org/pdf/2112.02026.pdf

{'THE SEVENTEENTH DATA RELEASE OF THE SLOAN DIGITAL SKY SURVEYS: COMPLETE RELEASE OF MANGA, MASTAR AND APOGEE-2 DATA': "Abdurro'uf 1 , Katherine Accetta 2 , Conny Aerts 3 , V'ıctor Silva Aguirre 4 , Romina Ahumada 5 , Nikhil Ajgaonkar 6 , N. Filiz Ak 7 , Shadab Alam 8 , Carlos Allende Prieto 9,10 , Andr'es Almeida 11 , Friedrich Anders 12,13 , Scott F. Anderson 14 , Brett H. Andrews 15 , Borja Anguiano 11 , Erik Aquino-Ort'ız 16 , Alfonso Arag'on-Salamanca 17 , Maria Argudo-Fern'andez 18 , Metin Ata 19 , Marie Aubert 20 , Vladimir Avila-Reese 16 , Carles Badenes 15 , Rodolfo H. Barb'a 21 , Kat Barger 22 , Jorge K. Barrera-Ballesteros 16 , Rachael L. Beaton 23 , Timothy C. Beers 24 , Francesco Belfiore 25 , Chad F. Bender 26 , Mariangela Bernardi 27 , Matthew A. Bershady 28,29,30 , Florian Beutler 8 , Christian Moni Bidin 5 , Jonathan C. Bird 31 , Dmitry Bizyaev 32,33 , Guillermo A. Blanc 23 , Michael R. Blanton 34 , Nicholas Fraser Boardman 35,36 , Adam S. Bolton 37 , M'ed'eric Boquien 38 , Jura Borissova 39,40 , Jo Bovy 41,42 , W.N. Brandt 43,44,45 , Jordan Brown 46 , Joel R. Brownstein 35 , Marcella Brusa 47,48 , Johannes Buchner 49 , Kevin Bundy 50 , Joseph N. Burchett 51 , Martin Bureau 52 , Adam Burgasser 53 , Tuesday K. Cabang 46 , Stephanie Campbell 36 , Michele Cappellari 52 , Joleen K. Carlberg 54 , F'abio Carneiro Wanderley 55 , Ricardo Carrera 56 , Jennifer Cash 46 , Yan-Ping Chen 57 , Wei-Huai Chen 1,58 , Brian Cherinka 54 , Cristina Chiappini 12 , Peter Doohyun Choi 59 , S. Drew Chojnowski 51 , Haeun Chung 26 , Nicolas Clerc 60 , Roger E. Cohen 54 , Julia M. Comerford 61 , Johan Comparat 49 , Luiz da Costa 62 , Kevin Covey 63 , Jeffrey D. Crane 23 , Irene Cruz-Gonzalez 16 , Connor Culhane 63 , Katia Cunha 55,26 , Y. Sophia Dai ( 戴 昱 ) 64 , Guillermo Damke 65,66 , Jeremy Darling 61 , James W. Davidson Jr. 11 , Roger Davies 52 , Kyle Dawson 35 , Nathan De Lee 67 , Aleksandar M. Diamond-Stanic 68 , Mariana Cano-D'ıaz 16 , Helena Dom'ınguez S'anchez 69 , John Donor 22 , Chris Duckworth 36 , Tom Dwelly 49 , Daniel J. Eisenstein 70 , Yvonne P. Elsworth 71 , Eric Emsellem 72,73 , Mike Eracleous 43 , Stephanie Escoffier 20 , Xiaohui Fan 26 , Emily Farr 14 , Shuai Feng 74 , Jos'e G. Fern'andez-Trincado 75,5 , Diane Feuillet 76,77 , Andreas Filipp 78 , Sean P Fillingham 14 , Peter M. Frinchaboy 22 , Sebastien Fromenteau 79 , Llu'ıs Galbany 69 , Rafael A. Garc'ıa 80 , D. A. Garc'ıa-Hern'andez 9,10 , Junqiang Ge 64 , Doug Geisler 81,65,82 , Joseph Gelfand 34 , Tobias G'eron 52 , Benjamin J. Gibson 35 , Julian Goddy 83 , Diego Godoy-Rivera 84 , Kathleen Grabowski 32 , Paul J. Green 70 , Michael Greener 17 , Catherine J. Grier 26 , Emily Griffith 84 , Hong Guo 85 , Julien Guy 86 , Massinissa Hadjara 87,88 , Paul Harding 89 , Sten Hasselquist 35,90 , Christian R. Hayes 14 , Fred Hearty 43 , Jes'us Hern'andez 91 , Lewis Hill 92 , David W. Hogg 34 , Jon A. Holtzman 51 , Danny Horta 93 , Bau-Ching Hsieh 1 , Chin-Hao Hsu 1 , Yun-Hsin Hsu 1,94 , Daniel Huber 95 , Marc Huertas-Company 9,96 , Brian Hutchinson 97,98 , Ho Seong Hwang 99,100 , H'ector J. Ibarra-Medel 101 , Jacob Ider Chitham 49 , Gabriele S. Ilha 62,102 , Julie Imig 51 , Will Jaekle 68 , Tharindu Jayasinghe 84 , Xihan Ji 6 , Jennifer A. Johnson 84 , Amy Jones 54 , Henrik Jonsson 103 , Ivan Katkov 57,33 , Dr. Arman Khalatyan 12 , Karen Kinemuchi 32 , Shobhit Kisku 93 , Johan H. Knapen 9,10 , Jean-Paul Kneib 104 , Juna A. Kollmeier 23 , Miranda Kong 105 , Marina Kounkel 31,63 , Kathryn Kreckel 106 , Dhanesh Krishnarao 28 , Ivan Lacerna 75,40 , Richard R. Lane 107 , Rachel Langgin 105 , Ramon Lavender 46 , David R. Law 54 , Daniel Lazarz 6 , Henry W. Leung 41 , Ho-Hin Leung 36 , Hannah M. Lewis 11 , Cheng Li 108 , Ran Li 64 , Jianhui Lian 35 , Fu-Heng Liang 108,52 , Lihwai Lin ( 林 俐 暉 ) 1 , Yen-Ting Lin 1 , Sicheng Lin 34 , Chris Lintott 52 , Dan Long 32 , Pen'elope Longa-Pe˜na 38 , Carlos L'opez-Cob'a 1 , Shengdong Lu 108 , Britt F. Lundgren 109 , Yuanze Luo 110 , J. Ted Mackereth 111,42,41 , Axel de la Macorra 112 , Suvrath Mahadevan 43 , Steven R. Majewski 11 , Arturo Manchado 9,10,113 , Travis Mandeville 14 , Claudia Maraston 92 , Berta Margalef-Bentabol 27 , Thomas Masseron 9,10 , Karen L. Masters 83,114 , Savita Mathur 9,10 , Richard M. McDermid 115,116 , Myles Mckay 14 , Andrea Merloni 49 , Michael Merrifield 17 , Szabolcs Meszaros 117,118,119 , Andrea Miglio 47 , Francesco Di Mille 120 , Dante Minniti 121,153 , Rebecca Minsley 68 , Antonela Monachesi 65 , Jeongin Moon 59 , Benoit Mosser 122 , John Mulchaey 23 , Demitri Muna 84 , Ricardo R. Mu˜noz 87 , Adam D. Myers 123 , Natalie Myers 22 , Seshadri Nadathur 124 , Preethi Nair 125 , Kirpal Nandra 49 , Justus Neumann 92 , Jeffrey A. Newman 15 , David L. Nidever 126 , Farnik Nikakhtar 27 , Christian Nitschelm 38 , Julia E. O'Connell 22,81 , Luis Garma-Oehmichen 16 , Gabriel Luan Souza de Oliveira 102,62 , Richard Olney 63 , Daniel Oravetz 32 , Mario Ortigoza-Urdaneta 75 , Yeisson Osorio 9 , Justin Otter 110 , Zachary J. Pace 28 , Nelson Padilla 127 , Kaike Pan 32 , Hsi-An Pan 76 , Taniya Parikh 49 , James Parker 32 , Sebastien Peirani 128 , Karla Pe˜na Ram'ırez 38 , Samantha Penny 92 , Will J. Percival 129,130,131 , Ismael Perez-Fournon 9,10 , Marc Pinsonneault 84 , Fr'ed'erick Poidevin 9,10 , Vijith Jacob Poovelil 35 , Adrian M. Price-Whelan 132 , Anna B'arbara de Andrade Queiroz 12 , M. Jordan Raddick 110 , Amy Ray 22 , Sandro Barboza Rembold 102,62 , Nicole Riddle 22 , Rogemar A. Riffel 62,102 , Rog'erio Riffel 133,62 , Hans-Walter Rix 76 , Annie C. Robin 134 , Aldo Rodr'ıguez-Puebla 16 , Alexandre Roman-Lopes 21 , Carlos Rom'an-Z'u˜niga 91 , Benjamin Rose 24 , Ashley J. Ross 135 , Graziano Rossi 59 , Kate H. R. Rubin 136,53 , Mara Salvato 49 , Seb'astian F. S'anchez 16 , Jos'e R. S'anchez-Gallego 14 , Robyn Sanderson 27,132 , Felipe Antonio Santana Rojas 87 , Edgar Sarceno 68 , Regina Sarmiento 9,10 , Conor Sayres 14 , Elizaveta Sazonova 110 , Adam L. Schaefer 78 , David J Schlegel 86 , Donald P. Schneider 43,44 , Ricardo Schiavon 93 , Mathias Schultheis 137 , Axel Schwope 12 , Aldo Serenelli 69,138 , Javier Serna 16 , Zhengyi Shao 85 , Griffin Shapiro 139 , Anubhav Sharma 83 , Yue Shen 101 , Matthew Shetrone 50 , Yiping Shu 78 , Joshua D. Simon 23 , M. F. Skrutskie 11 , Rebecca Smethurst 52 , Verne Smith 37 , Jennifer Sobeck 14 , Taylor Spoo 22 , Dani Sprague 97 , David V. Stark 83 , Keivan G. Stassun 31 , Matthias Steinmetz 12 , Dennis Stello 140,141 , Alexander Stone-Martinez 51 , Thaisa Storchi-Bergmann 133,62 , Guy S. Stringfellow 61 , Amelia Stutz 81 , Yung-Chau Su 1,58 , Manuchehr Taghizadeh-Popp 110 , Michael S. Talbot 35 , Jamie Tayar 95,142 , Eduardo Telles 55 , Johanna Teske 143 , Ani Thakar 110 , Christopher Theissen 53 , Daniel Thomas 92 , Andrew Tkachenko 3 , Rita Tojeiro 36 , Hector Hernandez Toledo 16 , Nicholas W. Troup 144 , Jonathan R. Trump 145 , James Trussler 146,147 , Jacqueline Turner 83 , Sarah Tuttle 14 , Eduardo Unda-Sanzana 38 , Jos'e Antonio V'azquez-Mata 16,148 , Marica Valentini 12 , Octavio Valenzuela 16 , Jaime Vargas-Gonz'alez 149 , Mariana Vargas-Maga˜na 112 , Pablo Vera Alfaro 21 , Sandro Villanova 81 , Fiorenzo Vincenzo 84 , David Wake 109 , Jack T. \nWarfield 11 , Jessica Diane Washington 150 , Benjamin Alan Weaver 37 , Anne-Marie Weijmans 36 , David H. Weinberg 84 , Achim Weiss 78 , Kyle B. Westfall 50 , Vivienne Wild 36 , Matthew C. Wilde 14 , John C. Wilson 11 , Robert F. Wilson 11 , Mikayla Wilson 22 , Julien Wolf 49,151 , W. M. Wood-Vasey 15 , Renbin Yan ( 严人 斌 ) 152,6 , Olga Zamora 9 , Gail Zasowski 35 , Kai Zhang 86 , Cheng Zhao 104 , Zheng Zheng 35 , Zheng Zheng 64 , Kai Zhu 64 \nDraft version September 4, 2024", 'Abstract': "This paper documents the seventeenth data release (DR17) from the Sloan Digital Sky Surveys; the fifth and final release from the fourth phase (SDSS-IV). DR17 contains the complete release of the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey, which reached its goal of surveying over 10,000 nearby galaxies. The complete release of the MaNGA Stellar Library (MaStar) accompanies this data, providing observations of almost 30,000 stars through the MaNGA instrument during bright time. DR17 also contains the complete release of the Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2) survey which publicly releases infra-red spectra of over 650,000 stars. The main sample from the Extended Baryon Oscillation Spectroscopic Survey (eBOSS), as well as the sub-survey Time Domain Spectroscopic Survey (TDSS) data were fully released in DR16. New single-fiber optical spectroscopy released in DR17 is from the SPectroscipic IDentification of ERosita Survey (SPIDERS) sub-survey and the eBOSS-RM program. Along with the primary data sets, DR17 includes 25 new or updated Value Added Catalogs (VACs). This paper concludes the release of SDSSIV survey data. SDSS continues into its fifth phase with observations already underway for the Milky Way Mapper (MWM), Local Volume Mapper (LVM) and Black Hole Mapper (BHM) surveys. \nSubject headings: Atlases - Catalogs - Surveys \nspokesperson@sdss.org \n- 1 Academia Sinica Institute of Astronomy and Astrophysics, 11F of AS/NTU, Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei, 10617, Taiwan\n- 2 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA\n- 3 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium\n- 4 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark\n- 5 Instituto de Astronom'ıa, Universidad Cat'olica del Norte, Av. Angamos 0610, Antofagasta, Chile\n- 6 Department of Physics and Astronomy, University of Kentucky, 505 Rose St., Lexington, KY, 40506-0055, USA\n- 7 Department of Astronomy and Space Sciences, Erciyes University, 38039 Kayseri, Turkey\n- 8 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\n- 9 Instituto de Astrof'ısica de Canarias (IAC), C/ Via L'actea s/n, E-38205 La Laguna, Tenerife, Spain\n- 10 Universidad de La Laguna (ULL), Departamento de Astrof'ısica, E-38206 La Laguna, Tenerife Spain\n- 11 Department of Astronomy, University of Virginia, Charlottesville, VA 22904-4325, USA\n- 12 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany\n- 13 Institut de Ci'encies del Cosmos, Universitat de Barcelona (IEEC-UB), Carrer Mart'ı i Franqu'es 1, E-08028 Barcelona, Spain\n- 14 Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA\n- 15 PITT PACC, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA\n- 16 Instituto de Astronom'ıa, Universidad Nacional Aut'onoma de M'exico, A.P. 70-264, 04510, Mexico, D.F., M'exico\n- 17 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK\n- 18 Instituto de F'ısica, Pontificia Universidad Cat'olica de Valpara'ıso, Casilla 4059, Valpara'ıso, Chile\n- 19 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan 20 Aix Marseille Universit'e, CNRS/IN2P3, CPPM, Marseille, France\n- 21 Departamento de Astronom'ıa, Universidad de La Serena, Av. Juan Cisternas 1200 Norte, La Serena, Chile\n- 22 Department of Physics & Astronomy, Texas Christian University, Fort Worth, TX 76129, USA\n- 23 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA\n- 24 Department of Physics and JINA Center for the Evolution of the Elements, University of Notre Dame, Notre Dame, IN 46556, USA\n- 25 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy\n- 26 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA\n- 27 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA\n- 28 Department of Astronomy, University of WisconsinMadison, 475N. Charter St., Madison WI 53703, USA\n- 29 South African Astronomical Observatory, P.O. Box 9, Observatory 7935, Cape Town, South Africa\n- 30 Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa\n- 31 Department of Physics and Astronomy, Vanderbilt University, VU Station 1807, Nashville, TN 37235, USA\n- 32 Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349, USA\n- 33 Sternberg Astronomical Institute, Moscow State University, Moscow, 119992, Russia\n- 34 Center for Cosmology and Particle Physics, Department of Physics, 726 Broadway, Room 1005, New York University, New York, NY 10003, USA\n- 35 Department of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA\n- 36 School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK\n- 37 NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA 38 Centro de Astronom'ıa (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile\n- 39 Instituto de F'ısica y Astronom'ıa, Universidad de Valpara'ıso, Av. Gran Breta˜na 1111, Playa Ancha, Casilla 5030, Chile.\n- 40 Millennium Institute of Astrophysics, MAS, Nuncio Monsenor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile\n- 41 David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada\n- 42 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario M5S 3H4, Canada\n- 43 Department of Astronomy & Astrophysics, Eberly College of Science, The Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802, USA", '1. INTRODUCTION': "The Sloan Digital Sky Surveys (SDSS) have been almost continuously observing the skies for over 20 years, since the project began with a first phase in 1998 (SDSSI; York et al. 2000). SDSS has now completed four phases of operations (with a fifth ongoing; see § 8). Since 2017, SDSS has had a dual hemisphere view of the sky, observing from both Las Campanas Observatory (LCO), \n- 44 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA\n- 45 Department of Physics, Eberly College of Science, The Pennsylvania State University, 104 Davey Laboratory, University Park, PA 16802, USA\n- 46 Department of Biological and Physical Sciences, South Carolina State University, P.O. Box 7024, Orangeburg, SC 29117, USA\n- 47 Dipartimento di Fisica e Astronomia 'Augusto Righi', Universit'a di Bologna, via Gobetti 93/2, 40129 Bologna, Italy\n- 48 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, 40129 Bologna, Italy\n- 49 Max-Planck-Institut fur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany\n- 50 UCO/Lick Observatory, University of California, Santa Cruz, 1156 High St. Santa Cruz, CA 95064, USA\n- 51 Department of Astronomy, New Mexico State University, Las Cruces, NM 88003, USA\n- 52 Sub-department of Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK\n- 53 Center for Astrophysics and Space Science, University of California San Diego, La Jolla, CA 92093, USA\n- 54 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 55 Observat'orio Nacional, Rio de Janeiro, Brasil\n- 56 Astronomical Observatory of Padova, National Institute of Astrophysics, Vicolo Osservatorio 5 - 35122 - Padova, Italy\n- 57 NYU Abu Dhabi, PO Box 129188, Abu Dhabi, UAE\n- 58 Department of Physics, National Taiwan University, Taipei 10617, Taiwan\n- 59 Department of Astronomy and Space Science, Sejong University, 209, Neungdong-ro, Gwangjin-gu, Seoul, South Korea\n- 60 IRAP Institut de Recherche en Astrophysique et Plan'etologie, Universit'e de Toulouse, CNRS, UPS, CNES, Toulouse, France\n- 61 Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389, USA\n- 62 Laborat'orio Interinstitucional de e-Astronomia, 77 Rua General Jos'e Cristino, Rio de Janeiro, 20921-400, Brasil\n- 63 Department of Physics and Astronomy, Western Washington University, 516 High Street, Bellingham, WA 98225, USA\n- 64 National Astronomical Observatories of China, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China\n- 65 Instituto de Investigaci'on Multidisciplinario en Ciencia y Tecnolog'ıa, Universidad de La Serena. Avenida Ra'ul Bitr'an S/N, La Serena, Chile\n- 66 AURA Observatory in Chile, Avda. Juan Cisternas 1500, La Serena, Chile\n- 67 Department of Physics, Geology, and Engineering Tech, Northern Kentucky University, Highland Heights, KY 41099, USA\n- 68 Department of Physics and Astronomy, Bates College, 44 Campus Avenue, Lewiston ME 04240, USA\n- 69 Institute of Space Sciences (ICE, CSIC), Carrer de Can Magrans S/N, Campus UAB, Barcelona, E-08193, Spain\n- 70 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 20, Cambridge, MA 02138, USA\n- 71 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK\n- 72 European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany\n- 73 Univ Lyon, Univ Lyon1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-Genis-Laval France \nusing the du Pont Telescope and the Sloan Foundation 2.5-m Telescope, (Gunn et al. 2006) at Apache Point Observatory (APO). This paper describes data taken during the fourth phase of SDSS (SDSS-IV; Blanton et al. 2017), which consisted of three main surveys; the Extended Baryon Oscillation Spectroscopic Survey (eBOSS; Dawson et al. 2016), Mapping Nearby Galaxies at APO (MaNGA; Bundy et al. 2015), and the APO Galactic \n- 74 College of Physics, Hebei Normal University, Shijiazhuang 050024, China\n- 75 Instituto de Astronom'ıa y Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiap'o, Chile\n- 76 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D69117 Heidelberg, Germany\n- 77 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-22100 Lund, Sweden\n- 78 Max-Planck-Institut fur Astrophysik, Karl-SchwarzschildStr. 1, D-85748 Garching, Germany\n- 79 Instituto de Ciencias F'sicas (ICF), Universidad Nacional Aut'onoma de M'exico, Av. Universidad s/n, Col. Chamilpa, Cuernavaca, Morelos, 62210, M'exico\n- 80 AIM, CEA, CNRS, Universit'e Paris-Saclay, Universit'e Paris Diderot, Sorbonne Paris Cit'e, F-91191 Gif-sur-Yvette, France\n- 81 Departmento de Astronom'ıa, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile\n- 82 Departamento de F'ısica y Astronom'ıa, Facultad de Ciencias, Universidad de La Serena. Av. Juan Cisternas 1200, La Serena, Chile\n- 83 Departments of Physics and Astronomy, Haverford College, 370 Lancaster Ave, Haverford, PA 19041, USA\n- 84 Department of Astronomy and Center for Cosmology and AstroParticle Physics, The Ohio State University, 140 W. 18th Ave, Columbus, OH, 43210, USA\n- 85 Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China\n- 86 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 87 Departamento de Astronom'ıa, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Chile\n- 88 Chinese Academy of Sciences South America Center for Astronomy, National Astronomical Observatories, CAS, Beijing 100101, China\n- 89 Department of Astronomy, Case Western Reserve University, Cleveland, OH 44106, USA\n- 90 NSF Astronomy and Astrophysics Postdoctoral Fellow\n- 91 Universidad Nacional Aut'onoma de M'exico, Instituto de Astronom'ıa, AP 106, Ensenada 22800, BC, Mexico\n- 92 Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK\n- 93 Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK \n94 \nInstitute of Astronomy, National Tsing Hua University, \nNo. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan \n- 95 Institute for Astronomy, University of Hawai'i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\n- 96 LERMA, UMR 8112, PSL University, University of Paris, 75014, Paris, France\n- 97 Computer Science Department, Western Washington University, 516 High Street, Bellingham, WA 98225, USA\n- 98 Computing & Analytics Division, Pacific Northwest, Richland, WA USA\n- 99 Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 305-348, Republic of Korea\n- 100 Astronomy Program, Department of Physics and Astronomy, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea\n- 101 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA\n- 102 Departamento de F'ısica, Centro de Ciˆencias Naturais e Exatas, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil\n- 103 Materials Science and Applied Mathematics, Malmo Uni- \nEvolution Experiment 2 (APOGEE-2; Majewski et al. 2017). Within eBOSS, SDSS-IV also conducted two smaller programs: the SPectroscopic IDentification of ERosita Sources (SPIDERS; Clerc et al. 2016; Dwelly et al. 2017) and the Time Domain Spectroscopic Survey (TDSS; Morganson et al. 2015), and continued the SDSS Reverberation Mapping (SDSS-RM) program to measure black hole masses out to redshifts z ∼ 1-2 us- \nversity, SE-205 06 Malmo, Sweden \n- 104 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique F'ed'erale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 105 Bryn Mawr College, 101 North Merion Ave, Bryn Mawr, PA 19010, USA\n- 106 Astronomisches Rechen-Institut, Zentrum fur Astronomie der Universitat Heidelberg, Monchhofstraße 12-14, D-69120 Heidelberg, Germany\n- 107 Centro de Investigaci'on en Astronom'ıa, Universidad Bernardo O'Higgins, Avenida Viel 1497, Santiago, Chile.\n- 108 Department of Astronomy, Tsinghua University, Beijing 100084, China\n- 109 Department of Physics and Astronomy, University of North Carolina Asheville, One University Heights, Asheville, NC 28804, USA\n- 110 Center for Astrophysical Sciences, Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA\n- 111 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 3H8, Canada\n- 112 Instituto de F'ısica Universidad Nacional Aut'onoma de M'exico, Cd. de M'exico 04510, M'exico\n- 113 CSIC, Spain\n- 114 SDSS-IV Spokesperson\n- 115 Department of Physics and Astronomy, Macquarie University, Sydney NSW 2109, Australia\n- 116 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia\n- 117 ELTE Eotvos Lor'and University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre H. st. 112, Hungary 118 MTA-ELTE Lendulet Milky Way Research Group, Hungary\n- 119 MTA-ELTE Exoplanet Research Group, Hungary\n- 120 Las Campanas Observatory, Colina El Pino Casilla 601 La Serena, Chile\n- 121 Departamento de Ciencias Fısicas, Universidad Andres Bello, Av. Republica 220, Santiago, Chile\n- 122 LESIA, Observatoire de Paris, Universit'e PSL, CNRS, Sorbonne Universit'e, Universit'e de Paris, 5 place Jules Janssen, 92195 Meudon, France\n- 123 Department of Physics and Astronomy, University of Wyoming, Laramie, WY 82071, USA\n- 124 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK\n- 125 Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA\n- 126 Department of Physics, Montana State University, P.O. Box 173840, Bozeman, MT 59717-3840, USA\n- 127 Instituto de Astrof'ısica, Pontificia Universidad Cat'olica de Chile, Av. Vicuna Mackenna 4860, 782-0436 Macul, Santiago, Chile\n- 128 Institut d'Astrophysique de Paris, UMR 7095, SU-CNRS, 98bis bd Arago, 75014 Paris, France\n- 129 Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada\n- 130 Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada\n- 131 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada\n- 132 Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY, 10010\n- 133 Departamento de Astronomia, Instituto de F'ısica, Universidade Federal do Rio Grande do Sul. Av. Bento Goncalves 9500, 91501-970, Porto Alegre, RS, Brazil\n- 134 Institut UTINAM, CNRS, OSU THETA Franche-Comt'e Bourgogne, Univ. Bourgogne Franche-Comt'e, 25000 Besan¸con, \ning single fiber spectra. Finally, the use of dual observing modes with the MaNGA and APOGEE instruments (Drory et al. 2015; Wilson et al. 2019) facilitated the development of the MaNGA Stellar Library (MaStar; Yan et al. 2019), which observed stars using the MaNGA fiber bundles during APOGEE-led bright time observing. \nThis suite of SDSS-IV programs was developed to map the Universe on a range of scales, from stars in the Milky Way and nearby satellites in APOGEE-2, to nearby galaxies in MaNGA, and out to cosmological scales with eBOSS. SPIDERS provided follow-up observations of Xray emitting sources, especially from eROSITA (Merloni et al. 2012; Predehl et al. 2014), and TDSS and SDSSRM provided a spectroscopic view of the variable sky. \nThe final year's schedule for SDSS-IV was substantially altered due to the COVID-19 pandemic. Originally, the SDSS-IV observations were scheduled to end at APO on the night of June 30, 2020 and at LCO on the night of September 8, 2020. Closures in response to COVID-19 altered this plan. APO closed on the morning of March 24, 2020 and the 2.5-m Sloan Foundation Telescope reopened for science observations the night of June 2, 2020. The summer shutdown ordinarily scheduled in July and August was delayed and instead SDSS-IV observations continued through the morning of August 24, 2020. LCO closed on the morning of March 17, 2020 and the du Pont Telescope reopened for science observations the night of October 20, 2020. The du Pont Telescope was used almost continuously for SDSS-IV through the morning of January 21, 2021. These changes led to different sky coverages than were originally planned for SDSS-IV but still", 'France': "- 135 Department of Physics and Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA\n- 136 Department of Astronomy, San Diego State University, San Diego, CA 92182, USA\n- 137 Observatoire de la Cˆote d'Azur, Laboratoire Lagrange, 06304 Nice Cedex 4, France\n- 138 Institut d'Estudis Espacials de Catalunya, C. Gran Capita 2-4, Barcelona, Spain\n- 139 Middlebury College, Middlebury, Vermont 05753, USA\n- 140 Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia\n- 141 School of Physics, UNSW Sydney, NSW 2052, Australia 142 Hubble Fellow\n- 143 Carnegie Institution for Science, Earth and Planets Laboratory, 5241 Broad Branch Road NW, Washington, DC 20015, USA\n- 144 Department of Physics, Salisbury University, 1101 Camden Ave., Salisbury, MD 21804, USA\n- 145 Department of Physics, University of Connecticut, 2152 Hillside Road, Unit 3046, Storrs, CT 06269, USA\n- 146 Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, UK\n- 147 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom\n- 148 Departamento de F'ısica, Facultad de Ciencias, Universidad Nacional Aut'onoma de M'exico, Ciudad Universitaria, CDMX, 04510, M'exico\n- 149 Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK\n- 150 Wellesley College Address: 106 Central St, Wellesley, MA 02481, USA\n- 151 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany\n- 152 Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, China\n- 153 Vatican Observatory, V00120 Vatican City State, Italy \nFig. 1.The growth in data volume hosted by the SDSS Science Archive Server (SAS) since DR8. For a more detailed break down of data volume see https://sdss.org/dr17/data\\_access/volume \n<!-- image --> \nallowed it to achieve or exceed all of its original goals. \nThis paper documents the seventeenth data release (DR17) from SDSS overall, and is the fifth and final annual release from SDSS-IV (following DR13: Albareti et al. 2017; DR14: Abolfathi et al. 2018, DR15: Aguado et al. 2019 and DR16: Ahumada et al. 2020). With this release SDSS-IV has completed the goals set out in Blanton et al. (2017). \nA complete overview of the scope of DR17 is provided in § 2, and information on how to access the data can be found in § 3. We have separate sections on MaNGA ( § 5), MaStar ( § 6) and APOGEE-2 ( § 4), and while there is no new main eBOSS survey or TDSS data in this release, we document releases from SPIDERS and the eBOSS-RM program as well as eBOSS related value added cataloges (VACs) in § 7. We conclude with a summary of the current status of SDSS-V now in active operations along with describing plans for future data releases ( § 8).", '2. SCOPE OF DR17': 'SDSS data releases have always been cumulative, and DR17 follows that tradition, meaning that the most upto-date reduction of data in all previous data releases are included in DR17. The exact data products and catalogs of previous releases also remain accessible on our servers. However, we emphatically advise users to access any SDSS data from the most recent SDSS data release, because data may have been reprocessed using updated data reduction pipelines, and catalogs may have been updated with new entries and/or improved analysis methods. Changes between the processing methods used in DR17 compared to previous data releases are documented on the DR17 version of the SDSS website https://www.sdss.org/dr17 as well as in this article. \nThis data release itself includes over 46 million new files totalling over 222 TB. Although many of these files replace previous versions, the total volume of all SDSS files including all previous versions now exceeds 623 TB on the Science Archive Server (SAS). The growth of the volume of data on the SAS since DR8 (which was the first data release of SDSS-III) is shown in Figure 1. \nTable 1 shows the growth of SDSS-IV data separated by survey and target types across our five annual data re- \nleases. These numbers are a mixture of counts of unique spectra and unique objects, and while correct to the best of our ability, can be subject to change based on which quality control flags are implemented. We also summarize these information below: \n- 1. APOGEE-2 is including 879,437 new infrared spectra. 154 These data come from observations taken from MJD 58302 to MJD 59160 (i.e., from July 2, 2018 to November 07, 2020) for APOGEE2 North (APOGEE-2N) at APO and from MJD 58358 to MJD 59234 (August 29, 2018 to January 20, 2021) for APOGEE-2 South (APOGEE-2S) at LCO and the new spectra comprise both observations of 260,594 new targets and additional epochs on targets included in previous DRs. The majority of the targets are in the Milky Way galaxy, but DR17 also contains observations of stars in the Large and Small Magellanic Clouds and eight dwarf spheroidal satellites as well as integrated light observations of both M33 and M31. Notably, DR17 contains 408,118 new spectra taken with the APOGEE-S spectrograph at LCO; this brings the total APOGEE-2S observations to 671,379 spectra of 204,193 unique stars. DR17 also includes all previously released APOGEE and APOGEE-2 spectra for a cumulative total of 2,659,178 individual spectra, all of which have been re-reduced with the latest version of the APOGEE data reduction and analysis pipeline (J. Holtzman et al. in prep.). In addition to the reduced spectra, element abundances and stellar parameters are included in this data release. APOGEE-2 is also releasing a number of VACs, which are listed in Table 2.\n- 2. MaNGA and MaStar are releasing all scientific data products from the now-completed surveys. This contains a substantial number of new galaxy and star observations respectively, along with updated products for all observations previously released in DR15 and before. These updated data products include modifications to achieve sub-percent accuracy in the spectral line-spread function, revised flux calibration, and Data Analysis Pipeline (DAP) products that now use stellar templates constructed from the MaStar observations to model the MaNGA galaxy stellar continuum throughout the full optical and near-infrared (NIR) wavelength range. MaNGA reached its target goal of observing more than 10,000 nearby galaxies, as well as a small number of non-galaxy targets, while bright time observations enable MaStar to collect spectra for almost 30,000 stars through the MaNGA instrument. MaNGA is also releasing a number of VACs (Table 2).\n- 3. There is no change in the main survey eBOSS data released since DR16, when a total of 1.4 million eBOSS spectra were released, completing its main survey goals. However, a number of Value Added Catalogs (VACs) useful for cosmological and other\n- 154 The number of spectra are tallied as the number of new entries in the AllVisit file. Table 1 conveys the numbers of unique targets that come from the AllStar file. \nTABLE 1 SDSS-IV spectroscopic data in all releases (DR13-DR17) \napplications are released in DR17. The TDSS survey also released its complete dataset in DR16. However, on-going eBOSS-like observations of Xray sources under the SPIDERS program and continued monitoring of quasars under the reverberation mapping program (SDSS-RM) are released in DR17. \n- 4. DR17 also includes data from all previous SDSS data releases. All MaNGA, BOSS, eBOSS, APOGEE and APOGEE-2 spectra that were previously released have all been reprocessed with the latest reduction and analysis pipelines. eBOSS main survey data were last released in DR16 (Ahumada et al. 2020), SDSS-III MARVELS spectra were finalized in DR12 (Alam et al. 2015). SDSS Legacy Spectra were released in its final form in DR8 (Aihara et al. 2011), and the SEGUE-1 and SEGUE-2 surveys had their final reductions released with DR9 (Ahn et al. 2012). The SDSS imaging had its most recent release in DR13 (Albareti et al. 2017), when it was recalibrated for eBOSS imaging purposes. \nA numerical overview of the total content of DR17 is given in Table 1. An overview of the value-added catalogs that are new or updated in DR17 can be found in Table 2; adding these to the VACs previously released in SDSS, the total number of VACs in SDSS as of DR17 is now 63 (DR17 updates 14 existing VACs and introduces 11 new ones). DR17 also contains the VACs that were first published in the mini-data release DR16+ on 20 June 2020. DR16+ did not contain any new spectra, and consisted of VACs only. Most of the VACs in \nDR16+ were based on the final eBOSS DR16 spectra, and these include large scale structure and quasar catalogs. In addition, DR16+ contained three VACs based on DR15 MaNGA sample. The DR16+ VACs can be found in Table 2, and are described in more detail in the sections listed there.', '3. DATA ACCESS': "There are various ways to access the SDSS DR17 data products, and an overview of all these methods is available on the SDSS website https://www.sdss. org/dr17/data\\_access/ , and in Table 3. In general, the best way to access a data product will depend on the particular data product and what the data product will be used for. We give an overview of all different access methods below, and also refer to tutorials and examples on data access available on this website: https://www.sdss.org/dr17/tutorials/ . \nFor those users interested in the reduced images and spectra of the SDSS, we recommend that they access these data products through the SDSS Science Archive Server (SAS, https://data.sdss.org/sas/ ). These data products were all derived through the official SDSS data reduction pipelines, which are also publicly available through SVN or GitHub ( https://www.sdss.org/ dr17/software/ ). The SAS also contains the VACs that science team members have contributed to the data releases (see Table 2), as well as raw and intermediate data products. All files available through the SAS have a data model that explains their content ( https://data.sdss. org/datamodel/ ). Data products can be downloaded from the SAS either directly through browsing, or by using methods such as wget, rsync and Globus Online (see https://www.sdss.org/dr17/data\\_access/bulk , \nNew or Updated Value Added Catalogs (DR16+ where noted, otherwise new or updated for DR17) \nTABLE 3 Summary of Methods for Accessing SDSS Data \nTABLE 2 \nfor more details and examples). For large data downloads, we recommend the use of Globus Online. Since SDSS data releases are cumulative, in that data products released in earlier data releases are included in DR17, and will have been processed by the latest available pipelines, we reiterate that users should always use the latest data release, as pipelines have often been updated to improve their output and fix previously known bugs. \nThe Science Archive Webservers (SAW) provides visu- \nalisations of most of the reduced images and data products available on the SAS. The SAW offers the option to display spectra with their model fits, and to search spectra based on a variety of parameters (e.g. observing program, redshift, coordinates). These searches can be saved as permalinks, so that they can be consulted again in the future and be shared with collaborators. All SAW webapps are available from https://dr17.sdss.org/ , and allow for displaying and searching of images (SDSS- \nI/II), optical single-fiber spectra (SDSS-I/II, SEGUE, BOSS and eBOSS), infrared spectra (APOGEE-1 and APOGEE-2), and MaStar stellar library spectra. Images and spectra can be downloaded through the SAW, and previous data releases are available back to DR8. The SAW also offers direct links to SkyServer Explore pages (see below). \nThe MaNGA integral-field data is not incorporated in the SAW due to its more complex data structure, and can instead be accessed through Marvin ( https://dr17. sdss.org/marvin/ ; Cherinka et al. 2019). Marvin offers not only visualisation options through its web interface, but also allows the user to query the data and analyze data products remotely through a suite of Python tools. Marvin also offers access to various MaNGA value added catalogs, as described in § 5.5. Marvin 's Python tools are available through pip-install, and installation instructions as well as tutorials and examples are available here: https://sdss-marvin.readthedocs.io/en/ stable/ . No installation is required to use Marvin 's Python tools in SciServer, as described later in this section and in § 5.3. \nCatalogs of derived data products are available on the SAS, but can be accessed more directly through the Catalog Archive Server (CAS, Thakar et al. 2008). These include photometric and spectroscopic properties, as well as some value added catalogs. The SkyServer webapp ( https://skyserver.sdss.org ) allows for visual inspection of objects using e.g. the QuickLook and Explore tools, and is also suitable for synchronous SQL queries in the browser. Tutorials and examples explaining the SQL syntax and how to query in SkyServer are available at http://skyserver.sdss.org/en/help/ docs/docshome.aspx . For DR17, the SkyServer underwent a significant upgrade, which includes a completely redesigned user interface as well as migration of the back end to a platform independent, modular architecture. Although SkyServer is optimal for smaller queries that can run in the browser, for larger ones we recommend using CASJobs ( https://skyserver.sdss. org/casjobs ). CASJobs allows for asynchronous queries in batch mode, and offers the user free storage space for query results in a personal database (MyDB) for server-side analysis that minimizes data movement (Li & Thakar 2008). \nSkyServer and CASJobs are now part of the SciServer science platform (Taghizadeh-Popp et al. 2020, https:// www.sciserver.org ), which is accessible with free registration on a single-sign-on portal, and offers server-side analysis with Jupyter notebooks in both interactive and batch mode, via SciServer Compute. SciServer is fully integrated with the CAS, and users will be able to access the data and store their notebooks in their personal account (shared with CASJobs). SciServer offers data and resource sharing via its Groups functionality that greatly facilitates its use in the classroom, to organize classes with student, teacher and teaching assistant privileges. Several SciServer Jupyter notebooks with use cases of SDSS data are available through the SDSS education webpages ( https://www.sdss.org/education/ ), some of which have been used by SDSS members in collegelevel based courses as an introduction to working with astronomical data. SciServer has prominently featured in the 'SDSS in the Classroom' workshops at AAS meet- \nings. \nUsers can now analyze the MaNGA DR17 data in SciServer, using the Marvin suite of Python tools. SciServer integration enables users to use the access and analysis capabilities of Marvin without having a local installation. In the SciServer Compute system 155 , the MaNGA dataset is available as an attachable MaNGA Data Volume, with the Marvin toolkit available as a loadable Marvin Compute Image. Once loaded, the Marvin package along with a set of Marvin Jupyter example notebooks and tutorials are available on the compute platform. \nWith DR17, we are also releasing in SciServer a new feature called SpecDash (Taghizadeh-Popp 2021) to interactively analyze and visualize one-dimensional optical spectra from SDSS Legacy and eBOSS surveys, and soon from APOGEE as well. SpecDash is available both as stand-alone website 156 , and as a Jupyter notebook widget in SciServer. \nUsers can load and compare multiple spectra at the same time, smooth them with several kernels, overlay error bars, spectral masks and lines, and show individual exposure frames, sky background and model spectra. For analysis and modeling, spectral regions can be interactively selected for fitting the continuum or spectral lines with several predefined models. All spectra and models shown in SpecDash can be downloaded, shared, and then uploaded again for subsequent analysis and reproducibility. Although the web-based version shares the same functionality as the Jupyter widget version, the latter has the advantage that users can use the SpecDash python library to programmatically load any kind of 1D spectra, and analyze or model them using their own models and kernels. \nAll tools and data access points described above are designed to serve a wide range of users from undergraduate level to expert users with significant programming experience. In addition, Voyages ( https://voyages. sdss.org/ ) provides an introduction to astronomical concepts and the SDSS data for less experienced users, and can also be used by teachers in a classroom setting. The Voyages activities were specifically developed around pointers to K-12 US science standards, and a Spanish language version of the site is available at https://voyages.sdss.org/es/ .", '4. APOGEE-2 : FULL RELEASE': "The central goal of APOGEE is to map the chemodynamics of all structural components of the Milky Way Galaxy via near-twin, multiplexed NIR high-resolution spectrographs operating simultaneously in both hemispheres (APOGEE-N and APOGEE-S spectrographs respectively; both described in Wilson et al. 2019). DR17 constitutes the sixth release of data from APOGEE, which has run in two phases (APOGEE-1 and APOGEE2) spanning both SDSS-III and SDSS-IV. As part of SDSS-III, the APOGEE-1 survey operated for approximately 3 years from August 2011 to July 2014 using the 2.5-m Sloan Foundation Telescope at APO. At the start of of SDSS-IV, APOGEE-2 continued its operations in the Northern Hemisphere by initiating a ∼ 6-year sur- \ny (APOGEE-2N). Thanks to unanticipated on-sky efficiency, APOGEE-2N operations concluded in November 2020 with an effective ∼ 7.5 years of bright time observations, with many programs expanded from their original 6-year baseline. In April 2017, operations began with the newly built APOGEE-S spectrograph and associated fiber plugplate infrastructure on the 2.5-m Ir'en'ee du Pont Telescope at LCO; APOGEE-2S observations concluded in January 2021. A full overview of the APOGEE-1 scientific portfolio and operations was given in Majewski et al. (2017) and a parallel overview for APOGEE-2 is forthcoming (S. Majewski et al., in prep.). \nThe APOGEE data in DR17 encompass all SDSS-III APOGEE-1 and SDSS-IV APOGEE-2 observations acquired with both instruments from the start of operations at APO in SDSS-III (September 2011) through the conclusion of SDSS-IV operations at APO and LCO (in November 2020 and January 2021, respectively). Compared to the previous APOGEE data release (DR16), DR17 contains roughly two additional years of observations in both hemispheres; this doubles the number of targets observed from APOGEE-2S (see Table 1). \nDR17 contains APOGEE data and information for 657,135 unique targets, with 372,458 of these (57%) as part of the main red star sample that uses a simple selection function based on de-reddened colors and magnitudes (for more details see Zasowski et al. 2013, 2017). The primary data products are: (1) reduced visit and visit-combined spectra, (2) radial velocity measurements, (3) atmospheric parameters (eight in total), and (4) individual element abundances (up to 20 species). Approximately 2.6 million individual visit spectra are included in DR17; 399,505 sources have three or more visits (54%) and 35,009 sources (5%) have ten or more visits. \nThe final APOGEE survey map is shown in Figure 2, where each circle represents a single 'field' that is colorcoded by survey phase: APOGEE-1 (cyan), APOGEE2N (blue), or APOGEE-2S (red). The difference in fieldof-view between APOGEE-N and APOGEE-S is visible by the size of the symbol, with each APOGEE-S field spanning 2.8 deg 2 and APOGEE-N spanning 7 deg 2 (for the instrument descriptions, see Wilson et al. 2019). Those fields with any new data in DR17 are encircled in black; new data can either be fields observed for the first time or fields receiving additional epochs. The irregular high Galactic latitude coverage is largely due to piggyback 'co-observing' with MaNGA during dark time. Notably, these cooperative operations resulted in observations of an additional 162,817 targets, or 22% of the total DR17 targets ( ∼ 30% of targets in APOGEE-2), which is a comparable number of targets as were observed in all of APOGEE-1. \nA different visualization of the final field plan is given in Figure 3, where now each field is color-coded by the number of unique stars targeted in each field. APOGEE plates have 300 fibers, but APOGEE targeting uses a 'cohorting' strategy by which exposure is accumulated over many visits for the faintest targets in a field while brighter targets are swapped in and out over time (for a schematic see Zasowski et al. 2013, Figure 1 therein). Moreover, some fields were included in multiple programs, like those in the Kepler footprint, and as many as 1600 unique targets were accommodated in a single 7 deg 2 APOGEE-2N field over the full span of the \nAPOGEE-1 and APOGEE-2 observing programs. \nExtensive descriptions of the target selection and strategy are found in Zasowski et al. (2013) for APOGEE1 and in Zasowski et al. (2017) for APOGEE-2. Details about the final target selection schemes used for APOGEE-2N and APOGEE-2S, which evolved over time, are presented in Beaton et al. (2021) and Santana et al. (2021), respectively.", '4.1. DR17 Sample Highlights': 'DR17 represents the culmination of the APOGEE-2 program (and, indeed, all of APOGEE) and presents a number of large, focused subsamples that are worth noting briefly. DR17 contains over 18,000 targets in the TESS Northern Continuous Viewing Zone (CVZ) and over 35,000 targets in the TESS Southern CVZ (Ricker et al. 2016). In DR17, there are over 35,000 targets which are part of 13 of the Kepler K2 Campaigns and over 20,000 in the primary Kepler field. In total, over 100,000 targets are also found in high-cadence, spacebased photometry programs. Among all scientific targeting programs, there are more than 13,000 targets that have more than 18 individual epochs, spanning all parts of the Galaxy. \nDR17 includes extensive APOGEE coverage for numerous star clusters, including 29 open clusters, 35 globular clusters, and 18 young clusters. However, detailed membership characterization identifies at least one possible member in as many as 126 open clusters and 48 globular clusters, after accounting for targets in Contributed and Ancillary Science programs (N. Myers et al., in prep, R. Schiavon et al., in prep.). Thus, some observations exist in DR17 for approximately 200 star clusters spanning a range of ages and physical properties. \nIn addition, DR17 contains measurements of resolved stars from ten dwarf satellite galaxies of the Milky Way (including the dwarf spheroidal systems Bootes I, Sextans, Carina, Fornax, Sculptor, Sagittarius, Draco, and Ursa Minor, as well as the Large and Small Magellanic Clouds); 14,000 of the over 20,000 targets toward dwarf satellites are in the Magellanic System. In addition, DR17 contains integrated light observations of star clusters in Fornax, M31, and M33 and of the central regions of M31 and of its highest-surface brightness dwarf satellites.', '4.2. APOGEE DR17 Data Products': 'The basic procedure for processing and analysis of APOGEE data is similar to that from previous data releases (Abolfathi et al. 2018; Holtzman et al. 2018; Jonsson et al. 2020), but a few notable differences are highlighted here. More details are presented in J. Holtzman et al. (in prep.).', '4.2.1. Spectral Reduction and Radial Velocity Determinations': 'Nidever et al. (2015) describes the original reduction procedure for APOGEE data, and the various APOGEE Data Release papers present updates (Abolfathi et al. 2018; Holtzman et al. 2018; Jonsson et al. 2020, J. Holtzman et al. in prep.). For DR17, at the visit reduction level, a small change was made to the criteria by which pixels are flagged as being potentially affected by poor sky subtraction. \nFig. 2.The DR17 final APOGEE sky coverage shown in Galactic coordinates with fields color-coded by the survey phase in which the field was observed: APOGEE-1 (cyan), APOGEE-2N (blue), and APOGEE-2S (red). The fiber plugplates used with the APOGEE-N spectrograph have a 7 square degree field-of-view while those used with the APOGEE-S spectrograph have a 2.8 square degree field of view. Those fields with any new observations in DR17 are highlighted with a black outline. \n<!-- image --> \nFig. 3.A sky map in Galactic coordinates showing the number of stars per APOGEE field. The disk is targeted with a more or less systematic grid of pointings within | b | < 15 deg. For glyph[lscript] < 30 deg there is more dense coverage of the bulge and inner Galaxy. The circle sizes reflect the different field-of-view of APOGEE-N and APOGEE-S. The dense coverage at the North Galactic Cap is due to co-observing with the MaNGA survey, which contributed 22% of the targets in DR17. \n<!-- image --> \nThe routines for combination of the individual visit spectra were rewritten for DR17 to incorporate a new radial velocity analysis, called Doppler (Nidever et al. 2021). Doppler performs a least squares fit to a set of visit spectra, solving simultaneously for basic stellar parameters ( T eff , log g , and [M/H]) and the radial velocity for each visit. The fitting is accomplished by using a series of Cannon (Ness et al. 2015; Casey et al. 2016) models to generate spectra for arbitrary choices of stellar parameters across the Hertzsprung-Russell diagram (from 3500 K to 20,000 K in T eff ); the Cannon models were trained on a grid of spectra produced using Synspec (e.g., Hubeny & Lanz 2017; Hubeny et al. 2021) with Kurucz model atmospheres (Kurucz 1979; Castelli &Kurucz 2003; Munari et al. 2005). The primary output of Doppler are the radial velocities; while the stellar parameters from Doppler are stored, they are not adopted as the final values (see ASPCAP, § 4.2.2 below). The Doppler routine produces slightly better results for radial velocities in most cases, as judged by scatter across repeated visits of stars. Details will be given in J. Holtzman et al. (in prep), but, for example, for ∼ 85,000 stars that have more than 3 visits, VSCATTER < 1 km/s, TEFF < 6000 K, and no additional data since DR17, the median VSCATTER is reduced from 128 m/s to 96 m/s. \nIn addition to the new methodology, the radial velocities for faint stars were improved. This was accomplished by making an initial combination of the visit spectra using only the barycentric correction. This initial combination provided a combined spectrum from which a radial velocity was determined. The radial velocity for each individual visit was then determined separately, but was required to be within 50 km/s of the original estimate. This yielded a higher fraction of successful radial velocities for faint stars, as judged by looking at targets in nearby dwarf spheroidal galaxies.', '4.2.2. Atmospheric Parameter and Element Abundance Derivations': "Stellar parameters and abundances are determined using the APOGEE Stellar Parameters and Chemical Abundance Pipeline (ASPCAP, Garc'ıa P'erez et al. 2016) that relies on the FERRE optimization code (Allende Prieto et al. 2006). 157 \nThe basic methodology of ASPCAP remained the same for DR17 as in previous releases, but new synthetic spectral grids were created. These took advantage of new, non-local thermodynamic equilibrium (NLTE) population calculations by Osorio et al. (2020) for four elements: Na, Mg, K, and Ca; as discussed in Osorio et al. (2020) the H-band abundance differences between LTE and NLTE were always less than 0.1 dex. Adopting these calculations, however, required the adoption of a different spectral synthesis code from that used in the last several APOGEE data releases: for DR17, the Synspec code (e.g., Hubeny & Lanz 2017; Hubeny et al. 2021) was adopted for the primary analysis instead of the Turbospectrum code (Alvarez & Plez 1998; Plez 2012) used in previous releases. This was not a straightforward choice because, while Synspec allows the NLTE levels to be used, it calculates the synthetic spectra under the assumption of plane parallel geometry, which becomes \nless valid for the largest giant stars. On the other hand, Turbospectrum can use spherical geometry, but does not accommodate NLTE populations to be specified. \nDR17 uses multiple sub-grids to span from T eff =3000 K (M dwarf) to T eff =20,000 K (BA), with log g ranges from 0 to 5 (3 to 5 for the BA grid). The full details of these grids and the reliability of the parameters as a function of stellar type are provided in J. Holtzman et al. (in prep.). Modifications to the linelists used for the syntheses are described in Smith et al. (2021), which is an augmentation to prior linelist work for APOGEE (Shetrone et al. 2015; Hasselquist et al. 2016; Cunha et al. 2017). \nThe ASPCAP results from the new Synspec grid are the primary APOGEE DR17 results and the majority of users will likely be satisfied with the results in this catalog; only this primary catalog will be loaded into the CAS. However, unlike prior releases, DR17 also includes supplemental analyses constructed using alternate libraries that have different underlying physical assumptions. The different analyses in DR17 are provided in separate summary files and include: \n- 1. the primary library using Synspec including NLTE calculations for Na, Mg, K, and Ca (with files on the SAS under dr17/synspec rev1) 158 ;\n- 2. one created using Synspec, but assuming LTE for all elements (files under dr17/synspec lte);\n- 3. another created using Turbospectrum 20 (files under dr17/turbo20), using spherical geometry for log g< =3;\n- 4. one created with Turbospectrum, but with plane parallel geometry (files under dr17/turbo20 pp) for all stars. \nAll of the libraries use the same underlying MARCS stellar atmospheres for stars with T eff < 8000 K, computed with spherical geometry for log g< =3. A full description of these spectral grids will be presented in J. Holtzman et al. (in prep.) and a focused discussion on the differences between the libraries and the physical implications will be presented in Y. Osorio et al. (in prep.). In summary, however, the differences are subtle in most cases. We encourage those using the APOGEE DR17 results to clearly specify the catalog version that they are using in their analyses 159 . \nFor DR17, we present 20 elemental abundances: C, C I, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, Ti II, V, Cr, Mn, Fe, Co, Ni, and Ce. In DR16, we attempted to measure the abundances of Ge, Rb, and Yb, but given the poor results for extremely weak lines, we did not attempt these in DR17. While we attempted measurements of P, Cu, Nd, and 13 C in DR17, these were judged to be unsuccessful. Overall, the spectral windows used to measure \n158 This is a revised version of the dr17/synspec directories, correcting a minor problem with the LSF convolution for a subset of stars observed at LCO, however, since Value Added Catalogs were constructed with the original dr17/synspec we have retained it for completness. \n159 Users may find the library version in the name of the summary file, as well as in the ASPCAP ID tag provided for each source in these files. \nthe abundances were largely unchanged, but several additional windows were added for Cerium, such that the results for Ce appear to be significantly improved over those in DR16. \nAs in DR16, both the raw spectroscopic stellar parameters as well as calibrated parameters and abundances are provided. Calibrated effective temperatures are determined by a comparison to photometric effective temperatures, as determined from the relations of (Gonz'alez Hern'andez & Bonifacio 2009), using stars with low reddening. Calibrated surface gravities are provided by comparison to a set of surface gravities from asteroseismology (Serenelli et al. 2017, M. Pinsonneault et al. in prep.) and isochrones (Berger et al. 2020). For DR17, the surface gravity calibration was applied using a neural network, unlike previous data releases where separate calibrations were derived and applied for different groups (red giants, red clump, and main sequence) of stars. The new approach eliminates small discontinuities that were previously apparent, and is described in more detail in J. Holtzman et al. (in prep.). For the elemental abundances, calibration just consists of a zeropoint offset (separately for dwarfs and giants), determined by setting the median abundance [X/M] of solar metallicity stars in the solar neighborhood with thin disk kinematics such that [X/M]=0. \nAdditional details on the ASPCAP changes are described in J. Holtzman et al. (in prep.).", '4.2.3. Additional data': "Several other modifications were made for DR17. \n- 1. The summary data files for APOGEE that are available on the Science Archive Server now include data from the Gaia Early Data Release 3 (EDR3) for the APOGEE targets (Gaia Collaboration et al. 2021, 2016). Positional matches were performed by the APOGEE team. More specifically, the following data are included:\n- · Gaia EDR3 identifiers (Gaia Collaboration et al. 2021),\n- · Gaia EDR3 parallaxes and proper motions (Lindegren et al. 2021),\n- · Gaia EDR3 photometry (Riello et al. 2021),\n- · Gaia EDR3 RVs (Seabroke et al. 2021),\n- · Distances and uncertainties following BailerJones et al. (2021).\n- 2. Likely membership for a set of open clusters, globular clusters, and dwarf spheroidal galaxies, as determined from position, radial velocity, proper motion, and distance, is provided in a MEMBERS column. More specifically, initial memberships were computed based on position and literature RVs, and these are then used to determine proper motion and distance criteria. Literature RVs were taken from:\n- · APOGEE-based mean RVs for the wellsampled 'calibration clusters' in Holtzman et al. (2018),\n- · mean RVs for globular clusters from Harris (2010) 160 , and\n- · mean RVs for dwarf spheroidal galaxies from McConnachie (2012). \nUsers interested in the properties of the clusters or satellite galaxies are encouraged to do more detailed membership characterization and probabilities (e.g., Masseron et al. 2019; M'esz'aros et al. 2020; Hasselquist et al. 2021, Schiavon et al., in prep., Shetrone et al., in prep.) \n- 3. Some spectroscopic binary identification is provided through bits in the STARFLAG and ASPCAPFLAG bitmasks. A more comprehensive analysis of spectroscopic binaries is provided in a VAC (see § 4.4.1 below) . \nWe encourage those utilizing these data in our summary catalogs to cite the original references as given above.", '4.3. Data Quality': "The overall quality of the DR17 results for radial velocities, stellar parameters, and chemical abundances is similar to that of previous APOGEE data releases (full evaluation will be provided in Holtzman et al. in prep.). 161 As in DR16, uncertainties for stellar parameters and abundances are estimated by analyzing the scatter in repeat observations of a set of targets. \nUsers should be aware that deriving consistent abundances across a wide range of parameter space is challenging, so some systematic features and trends arise. Users should be careful when comparing abundances of stars with significantly different stellar parameters. Also, the quality of the abundance measurements varies between different elements, across parameter space, and with signal-to-noise. \nSome regions of parameter space present larger challenges than others. In particular, it is challenging to model the spectra of the coolest stars and, while abundances are derived for the coolest stars in DR17, there seem to be significant systematic issues for the dwarfs with T eff < 3500 K such that although we provide calibrated results in the PARAM array, we do not populate the 'named tags.' Separately, for warm/hot stars ( T eff > 7000), information on many abundances is lacking in the spectra, and uncertainties in the model grids at these temperatures may lead to systematic issues with the DR17 stellar parameters. \nAs a demonstration of the quality and scientific potential of the data, Figure 4 shows a set of [Mg/Fe] versus [Fe/H] diagrams for different three-dimensional spatial zones within the disk of the Milky Way, restricted to giant-stars with 1 < log g< 2.5 to minimize potential systematics or sampling bias. Spectrophotometric distances to individual stars are determined from Value Added Catalogs 162 and then are used with stellar positions to determine the Galactocentric radius ( R G ) and \n- 160 This is the 2010 update to the Harris (1996) catalog.\n- 161 The web documentation contains the details of the data model. Morevoer, the documentation communicates how data was flagged, including a brief list of changes relative to prior releases.\n- 162 In this visualization, from the DistMass VAC to be released in 2022 that uses a Neural Net at the parameter level to determine spectroscopic distances. \nheight above the plane ( z ) for each individual star; this highlights the scientific potential enabled via the analyses in the Value Added Catalogs. The color coding indicates the orbital eccentricity based on calculations from GalPy (Bovy 2015) using Gaia EDR3 proper motions (Gaia Collaboration et al. 2021) and APOGEE DR17 radial velocities. Figure 4 is a merging of similar visualizations previously presented in Hayden et al. (2015) and Mackereth et al. (2019b), such that the spatial zones of the former are merged with the dynamical inference of the latter. The stars of the solar neighborhood (middle panel, 7 < R G < 9) show two distinct chemical sequences, commonly referred to the the low- and high- [ α /Fe] sequences that are also somewhat dynamically distinct (apparent in the color-coding by orbital eccentricity). The inner Galaxy, however, is dominated both by high-eccentricity (bulge-like orbits) stars on the high-[ α /Fe] sequence just as the outer galaxy is dominated by low-eccentricity (near circular orbits) stars on the low-[ α /Fe] sequence, with some slight dependence on z . The relative contributions of low-eccentricity versus high-eccentricity and low-[ α /Fe] versus high-[ α /Fe] sequences shift throughout the Galaxy. These spatial, chemical, and dynamical groupings provide evidence for various disk-formation and disk-evolution scenarios (e.g., as discussed in Hayden et al. 2015; Mackereth et al. 2019b, among others) that add complexity and nuance to the canonical schemes. .", '4.4. APOGEE Value Added Catalogs': 'There are a large number of APOGEE-associated VACs in DR17. In what follows we provide brief descriptions of each VAC along with references where the reader can find more detail. Broadly speaking, APOGEE VACs can be split into characterising special subsamples, like binary stars, open clusters, and photometric variables, those which calculate stellar or orbital parameters for all (or most) APOGEE target stars (e.g. Starhorse, APOGEEnet and others). We also document the release of a mock catalog of APOGEE based on a hydrodynamical simulation.', '4.4.1. VACs Describing Categories of Objects in APOGEE': "The first set of APOGEE VACs describe special categories of objects in APOGEE data and in most cases provide additional information/characteristics for these objects. They are: \n- 1. Open Cluster Chemical Abundances and Mapping catalog (OCCAM) : The goal of OCCAM is to leverage the APOGEE survey to create a large, uniform catalog of open cluster chemical abundances and use these clusters to study Galactic chemical evolution. The catalog contains average chemical abundances for each cluster and membership probability estimates for APOGEE stars in the cluster area. We combine proper motion (PM) and radial velocity (RV) measurements from Gaia EDR3 (Gaia Collaboration et al. 2021) with RV and metallicity measurements from APOGEE to establish cluster membership probabilities for each star observed by APOGEE. The VAC includes 26,699 stars in the areas of 153 cataloged disk clusters. Detailed descriptions of the OCCAM survey, including tar- \ngeting and the methodology for membership determinations, are presented in Frinchaboy et al. (2013), Donor et al. (2018), and Donor et al. (2020). This third catalog from the OCCAM survey includes 44 new open clusters, including many in the Southern hemisphere and those targeted specifically in GC size ( R GC ) ranges with little coverage in the DR16 catalog (specific targeting described in Beaton et al. 2021; Santana et al. 2021). Average RV, PM, and abundances for reliable ASPCAP elements are provided for each cluster, along with the visual quality determination. Membership probabilities based individually upon PM, RV, and [Fe/H] are provided for each star, stars are considered 3 σ members if they have probability > 0 . 01 in all three membership dimensions 163 . The results and caveats from this VAC will be discussed thoroughly in N. Myers et al. (in prep.). \n- 2. APOGEE Red-Clump (RC) Catalog : DR17 contains an updated version of the APOGEE redclump (APOGEE-RC) catalog. This catalog is created in the same way as the previous DR14 and DR16 versions of the catalog, with a more stringent log g cut compared to the original version of the catalog (Bovy et al. 2014). The catalog contains 50,837 unique stars, about 30% more than in DR16. The catalog is created using a spectrophotometric technique first presented in Bovy et al. (2014) that results in a rather pure sample of red-clump stars (e.g., minimal contamination from red-giant-branch, secondary-red-clump, and asymptotic-giant-branch stars that have similar CMD and H-R positions). Bovy et al. estimated a purity of ∼ 95%. The narrowness of the RC locus in color-metallicity-luminosity space allows distances to the stars to be assigned with an accuracy of 5%-10%, which exceeds the precision of spectrophotometric distances in other parts of the H-R diagram. We recommend users adopt the most recent catalog (DR17) for their analyses; additional discussion on how to use the catalog is given in Bovy et al. (2014). While the overall datamodel is similar to previous versions of the catalog, the proper motions are from Gaia EDR3 (Gaia Collaboration et al. 2021; ? ).\n- 3. APOGEE-Joker : The APOGEE-Joker VAC contains posterior samples for binary-star orbital parameters (Keplerian orbital elements) for 358,350 sources with three or more APOGEE visit spectra that pass a set of quality cuts as described in A. Price-Whelan et al. (in prep.). The posterior samples are generated using The Joker , a custom Monte Carlo sampler designed to handle the multimodal likelihood functions that arise when inferring orbital parameters with sparsely-sampled or noisy radial velocity time data (Price-Whelan et al. 2017). This VAC deprecates the previous iterations of the catalog (Price-Whelan et al. 2018, 2020). \n163 However, some stars near the main sequence turn-off may 'fail' the [Fe/H] cut due to evolutionary diffusion effects (Souto et al. 2018, 2019) \nFig. 4.A series of [Mg/Fe] vs [Fe/H] plots from APOGEE DR17 for different zones in the Milky Way. Distances from the DistMass VAC are used to determine Galactocentric radius ( R G ) and height above the plane ( z ). Points are color-coded by orbital eccentricities as computed with GalPy (Bovy 2015) using Gaia EDR3 proper motions and APOGEE radial velocities. \n<!-- image --> \nFor 2,819 stars, the orbital parameters are well constrained, and the returned samples are effectively unimodal in period. For these cases, we use the sample(s) returned from The Joker to initialize standard MCMC sampling of the Keplerian parameters using the time-series optimized MCMC code known as exoplanet 164 (Foreman-Mackey et al. 2021) and provide these MCMC samples. For all stars, we provide a catalog containing metadata about the samplings, such as the maximum a posteriori (MAP) parameter values and sample statistics for the MAP sample. A. Price-Whelan et al. (in prep.) describes the data analysis procedure in more detail, and defines and analyzes a catalog of glyph[greaterorsimilar] 40,000 binary star systems selected using the raw orbital parameter samples released in this VAC. \n- 4. Double lined spectroscopic binaries in APOGEE spectra : Generally, APOGEE fibers capture a spectrum of single stars. Sometimes, however, there may be multiple stars of comparable brightness with the sky separations closer than the fiber radius whose individual spectra are captured by the same recorded spectrum. Most often, these stars are double-lined spectroscopic binaries or higher order multiples (SBs), but on an occasion they may also be chance line-of-sight alignments of random field stars (most often observed towards the Galactic center). Through analyzing the cross-correlation function (CCF) of the APOGEE spectra, Kounkel et al. (2021) have developed a routine to automatically identify these SBs using Gaussian deconvolution of the CCFs (Kounkel 2021) 165 , and to measure RVs of the individual stars. The catalog of these sources and the sub-component RVs are presented here as a VAC. For the subset of sources \nthat had a sufficient number of measurements to fully characterize the motion of both stars, the orbit is also constructed. \nThe data obtained though April/May 2020 were processed with the DR16 version of the APOGEE radial velocity pipeline and this processing was made available internally to the collaboration as an intermediate data release. All of the SBs identified in this internal data release have undergone rigorous visual vetting to ensure that every component that can be detected is included and that spurious detections have been removed. However, the final DR17 radial velocity pipeline is distinct from that used for DR16 (summarized above; J. Holtzman et al. in prep.) and the reductions are sufficiently different that they introduce minor discrepancies within the catalog. In comparison to DR16, the DR17 pipeline limits the span of the CCF for some stars to a velocity range around the mean radial velocity to ensure a more stable overall set of RV measurements; on the other hand the DR16 pipeline itself may fail on a larger number of individual visit spectra and thus not produce a full set of outputs. For the sources that have both good parameters and a complete CCF coverage for both DR16 and DR17, the widely resolved components of SBs are generally consistent with one another; close companions that have only small RV separations are not always identified in both datasets. For this reason, SBs that could be identified in both the DR16 and DR17 reductions are kept as separate entries in the catalog. Visual vetting was limited only to the data processed with the DR16 pipeline (e.g., data through April/May 2020); the full automatic deconvolutions of the DR17 CCFs are presented as-is. \nVACs providing distances and other properties (mostly related to orbital parameters) are released (or rereleased): \n- 1. StarHorse distances, extinctions, and stellar parameters for APOGEE DR17 + Gaia EDR3: We combine high-resolution spectroscopic data from APOGEE DR17 with broad-band photometric data from 2MASS, unWISE and PanSTARRS-1, as well as parallaxes from Gaia EDR3. Using the Bayesian isochrone-fitting code StarHorse (Santiago et al. 2016; Queiroz et al. 2018), we derive distances, extinctions, and astrophysical parameters. We achieve typical distance uncertainties of ∼ 5 % and extinction uncertainties in V-band amount to ∼ 0.05 mag for stars with available PanSTARRS-1 photometry, and ∼ 0.17 mag for stars with only infra-red photometry. The estimated StarHorse parameters are robust to changes in the Galactic priors assumed and corrections for Gaia parallax zero-point offset. This work represents an update of DR16-based results presented in Queiroz et al. (2020).\n- 2. APOGEEastroNN : The APOGEEastroNN valueadded catalog holds the results from applying the astroNN deep-learning code to APOGEE spectra to determine stellar parameters, individual stellar abundances (Leung & Bovy 2019a), distances (Leung & Bovy 2019b), and ages (Mackereth et al. 2019a). For DR17, we have retrained all neural networks using the latest data, i.e., APOGEE DR17 results for the abundances, Gaia EDR3 parallax measurements, and an intermediate APOKASC data set with stellar ages (v6.6.1, March 2020 using DR16 ASPCAP). Additionally, we augmented the APOKASC age data with low-metallicity asteroseismic ages from Montalb'an et al. (2021) to improve the accuracy of ages at low metallicities; the Montalb'an et al. (2021) analysis is similar to that of APOKASC, but performed by an independent team. As in DR16, we correct for systematic differences between spectra taken at LCO and APO by applying the median difference between stars observed at both observatories. In addition to abundances, distances, and ages, properties of the orbits in the Milky Way (and their uncertainties) for all stars are computed using the fast method of Mackereth & Bovy (2018) assuming the MWPotential2014 gravitational potential from Bovy (2015). Typical uncertainties in the parameters are 35 K in T eff , 0.1 dex in log g , 0.05 dex in elemental abundances, 5 % in distance, and 30 % in age. Orbital properties such as the eccentricity, maximum height above the mid-plane, radial, and vertical action are typically precise to 4 to 8 %.", '4.4.3. APOGEE Net: a unified spectral model': 'A number of different pipelines are available for extracting spectral parameters from the APOGEE spectra. These pipelines generally manage to achieve optimal performance for red giants and, increasingly, G & K dwarfs, which compose the bulk of the stars in the catalog. However, the APOGEE2 catalog contains a number \nof parameter spaces that are often not well characterized by the primary pipelines. Such parameter spaces include pre-main sequence stars and low mass stars, with their measured parameters showing systematic T eff & log g deviations making them inconsistent from the isochrones and the main sequence. OBA stars are also less well constrained and in prior data releases many were classified as F dwarfs (due to grid-edge effects) and have their T eff underestimated in the formal results. By using data-driven techniques, we attempt to fill in those gaps to construct a unified model of APOGEE spectra. In the past, we have developed a neural network, APOGEE Net (Olney et al. 2020), which was shown to perform well to extract T eff , log g , & [Fe/H] on all stars with T eff < 6,500 K, including pre-main sequence stars. We now expand these efforts to also characterize hotter stars with 6,500 < T eff < 50,000 K. APOGEE NET II is described in Sprague et al. (2022).', '4.4.4. APOGEE FIRE VAC': 'Mock catalogs made by making simulated observations of sophisticated galaxy simulations provide unique opportunities for observational projects, in particular, the ability to test for or constrain the impact of selection functions, field plans, and algorithms on scientific inferences. One of the most realistic galaxy simulations to date is the Latte simulation suite, which uses FIRE-2 (Hopkins et al. 2018) to produce galaxies in Milky Waymass halos in a cosmological framework (Wetzel et al. 2016). Sanderson et al. (2020) translated three of the simulations into realistic mock catalogs (using three solar locations, resulting in nine catalogs), known as the Ananke simulations 166 . Ananke contains key Gaia measureables for the star particles in the simulations and these include radial velocity, proper motion, parallax, and photometry in the Gaia bands as well as chemistry (10 chemical elements are tracked in the simulation), and other stellar properties. Because the input physics and the global structure of the model galaxy are known, these mock catalogs provide an experimental laboratory to make connections between the resolved stellar populations and global galaxy studies. \nIn this VAC, Ananke is expanded to permit APOGEEstyle sampling of the mock-catalogs. For all observed quantities both the intrinsic, e.g., error-free, and the observed values are reported; the observed values are the intrinsic values convolved with an error-model derived from observational data for similar object types. As described in Nikakhtar et al. (2021), Ananke mock-catalogs now contain: (i) 2MASS ( JHK s ) photometry and reddening, (ii) abundance uncertainties following APOGEE DR16 performance (following Poovelil et al. 2020; Jonsson et al. 2020), and (iii) a column that applies a basic survey map (Zasowski et al. 2013, 2017; Beaton et al. 2021; Santana et al. 2021). The full mock-catalogs are released such that users can impose their own selection function to constructs a mock APOGEE survey in the simulation. Mock-surveys can then be used to test the performance of methods and algorithms to recover the true underlying galactic physics as demonstrated in Nikakhtar et al. (2021). \n166 For data access see: https://fire.northwestern.edu/ ananke/#dm', '5. MANGA: FULL RELEASE OF FINAL SAMPLE': "The MaNGA survey (Bundy et al. 2015) uses a custombuilt set of hexagonal integral field unit (IFU) fiber bundles (Drory et al. 2015) to feed spectroscopic fibers into the BOSS spectrograph (Smee et al. 2013). Over its operational lifetime, MaNGA has successfully met its goal of obtaining integral field spectroscopy for ∼ 10,000 nearby galaxies (Law et al. 2015; Yan et al. 2016a) at redshift z ∼ 0 . 03 with a nearly flat distribution in stellar mass (Wake et al. 2017). \nDR17 contains all MaNGA observations taken throughout SDSS-IV, and more than doubles the sample size of fully reduced galaxy data products previously released in DR15 (Aguado et al. 2019). These data products include raw data, intermediate reductions such as flux-calibrated spectra from individual exposures, and final calibrated data cubes and row-stacked spectra (RSS) produced using the MaNGA Data Reduction Pipeline (DRP; Law et al. 2016, 2021a; Yan et al. 2016b). \nDR17 includes DRP data products (see § 5.1) for 11,273 MaNGA cubes distributed amongst 674 plates. 10,296 of these data cubes are for 'traditional' MaNGA type galaxies, and 977 represent data cubes associated with non-standard ancillary programs (targeting a variety of objects including globular clusters, faint galaxies and intracluster light in the Coma cluster, background reference sky, and also tiling of the large nearby galaxies M31 and IC342; see § 5.4 for more details). Of the 10,296 galaxy cubes, 10,145 have the highest data quality with no warning flags indicating significant issues with the data reduction process. These 10,145 data cubes correspond to 10,010 unique targets (as identified via their MANGAID ) with a small number of repeat observations taken for cross-calibration purposes (each has an individual plate-ifu code, MANGAID needs to be used to identify unique galaxies). As in previous releases, DR17 also includes the release of derived spectroscopic products (e.g., stellar kinematics, emission-line diagnostic maps, etc.) from the MaNGA Data Analysis Pipeline (DAP; Belfiore et al. 2019; Westfall et al. 2019); see § 5.2. Additionally, DR17 contains the final data release for the MaNGA Stellar Library (MaStar; Yan et al. 2019, and § 6), which includes calibrated 1D spectra for 28,124 unique stars spanning a wide range of stellar types. \nWe illustrate the sky footprint of MaNGA galaxies released in DR17 in Figure 5, along with colored boxes indicating the locations of a selection of other galaxy surveys, namely the HI surveys Apertif (K. Hess et al. in prep) and ALFALFA (or Arecibo Legacy Fast ALFA, Haynes et al. 2018; also see § 5.5.4 for more HI followup); IR surveys like Herschel-ATLAS, (H-ATLAS, Smith et al. 2017), the UKIRT Infrared Deep Sky Survey, (UKIDSS, Lawrence et al. 2007), and other optical surveys, like Galaxy and Mass Assembly Survey (GAMA, Liske et al. 2015), the footprint of which includes most of the SAMI IFU observations, (Croom et al. 2021, in total, 74 galaxies are observed by both MaNGA and SAMI) and Hyper Suprime-Cam (HSC, Aihara et al. 2019). In some cases the prioritization of which MaNGA plates to observe was driven by the availability of these ancillary data (e.g. note how observed plates fill in parts of the UKIDSS footprint). MaNGA plates in an earlier projected footprint of Apertif were also prioritized but changes in Apertif \nobservation plans has significantly reduced the final overlap.", '5.1. MaNGA Data Reduction Pipeline and Products': "The MaNGA DRP has evolved substantially throughout the survey across a variety of both public (DR) and internal ('MaNGA Product Launch', or MPL) data releases. A summary of these various DRP versions and the number of unique galaxies in each is given by Law et al. (2021a, see their Table 1). These authors also provide a detailed description of the differences in the DRP for DR17 compared to previous releases. 167 In brief, changes in the DR17 data products compared to DR15 include: \n- 1. Updated spectral line-spread function (LSF): Many stages of the pipeline have been rewritten to further improve the accuracy of the LSF estimate, which is now good to better than 1%. As demonstrated by Law et al. (2021a) by comparison against observations with higher-resolution spectrographs, this allows MaNGA emission-line velocity dispersions to be reliable down to 20 km s -1 at signal-to-noise ratio (SNR) above 50, which is well below the 70 km s -1 instrumental resolution.\n- 2. Multiple pipeline changes have affected the overall MaNGA survey flux calibration. The most significant changes included adoption of a different extinction model for the calibration standard stars and correction for a few-percent scale error in lab measurements of the MaNGA fiber bundle metrology using on-sky self calibrations (see Law et al. 2021a, their Appendix A).\n- 3. New data quality flags have been defined to better identify potential reduction problems. These include a new UNUSUAL data quality bit to identify cubes that are different from ordinary data quality but still useful for many analyzes (e.g., that may be missing a fraction of the field of view due to hardware problems). These are distinct from the previously-defined CRITICAL data quality bit that indicates data with significant problems that should preclude it from most scientific analyzes ( < 1% of the total sample).\n- 4. Introduction of a new processing step to detect and subtract bright electronic artifacts (dubbed the 'blowtorch') arising from a persistent electronic artifact within the Charge-coupled devices (CCDs) in one of the red cameras during the final year of survey operations (see Law et al. 2021a, their Appendix B).", '5.2. MaNGA Data Analysis Pipeline and Products': 'In this section we describe two specific changes to the DAP analysis between MaNGA data released in DR15 and DR17. The first is a change in the stellar continuum templates used for the emission line measurements; this change only affects emission line measurements and does \n- 167 Strictly Law et al. (2021a) describe the team-internal data release MPL-10, but these data are practically identical to the final public data release DR17 (which is the team internal release MPL11) in everything except the total number of galaxies. \nFig. 5.DR17 final MaNGA survey area; blue tiles indicate observed fields (plates), grey tiles indicate potential fields from which the MaNGA final sample was drawn. Colored boxes indicate the regions observed by a variety of other surveys as described in the text. \n<!-- image --> \nnot affect stellar kinematic measurements. The second is the addition of new spectral index measurements more appropriate for stacking analyzes and coaddition of spaxels; the previously existing spectral index measurements are not affected by this addition. \nThe MaNGA Data Analysis Pipeline (DAP) as a whole is discussed extensively in the DR15 paper (Aguado et al. 2019) and in Westfall et al. (2019), Belfiore et al. (2019), and Law et al. (2021a). The last provides a summary of other improvements made to the DAP since DR15. \nThe SDSS data release website ( https://www.sdss. org/ ) provides information on data access and changes to the DAP data models in DR17 for its major output products. Further information can be found in the documentation of the code base. 168', '5.2.1. Stellar Continuum Templates': "In DR17, we use different spectral templates to model the galaxy continuum for emission line measurements than we use for stellar kinematics measurements. In DR15, we used the same templates in both cases, but as discussed by Law et al. (2021a), these template sets diverged starting with our ninth internal data set (MPL-9; between DR15 and DR17). For the emission line measurements, the new templates are based on the MaStar survey, allowing us to take advantage of the full MaNGA spectral range (3600-10000 ˚ A) and, e.g., model the [S III ] λλ 9071,9533 ˚ A doublet and some of the blue Paschen lines. For the stellar kinematics measurements, we have continued to use the same templates used in DR15, the MILES-HC library, taking advantage of its modestly higher spectral resolution than MaStar. Since MILES only spans between 3575 to 7400 ˚ A, this means \nMaNGA stellar kinematics do not include, e.g., contributions from the calcium near-infrared triplet near 8600 ˚ A. \nIn DR17, we provide DAP emission line measurements based on two different continuum template sets, both based on the MaStar Survey (Yan et al. 2019, and § 6), and referred to as MASTARSSP and MASTARHC2 . There are four different analysis approaches, indicated by DAPTYPE . Three use MASTARSSP , with three different spatial binning approaches, and the fourth uses MASTARHC2 . \nThe template set referred to as the MASTARSSP library by the DAP are a subset of simple-stellar-population (SSP) models provided by Maraston et al. (2020). Largely to decrease execution time, we down-selected templates from the larger library provided by Maraston et al. (2020) to only those spectra with a Salpeter Initial Mass Function (IMF) and the following grid in SSP age and metallicity, for a total of 54 spectra: \n- 1. Age/[1 Gyr] = 0.003, 0.01, 0.03, 0.1, 0.3, 1, 3, 9, 14\n- 2. log( Z/Z glyph[circledot] ) = -1.35, -1., -0.7, -0.33, 0, 0.35. \nExtensive testing was done to check differences in stellarcontinuum fits based on this choice; small differences that were found are well within the limits described by Belfiore et al. (2019). Section 5.3 of Law et al. (2021b) show further analysis, including a direct comparison of results for the BPT emission-line diagnostics plots when using either the MASTARHC2 or MASTARSSP templates showing that the templates have a limited effect on their analysis. Importantly, note that the DAP places no constraints on how these templates can be combined (e.g., unlike methods which use the Penalized PiXelFitting, or pPXF; Cappellari & Emsellem 2004; Cappellari 2017, implementation of regularized weights), and \nthe weight applied to each template is not used to construct luminosity-weighted ages or metallicities for the fitted spectra. The use of the SSP models, as opposed to spectra of single stars, is meant only to impose a physically relevant prior on the best-fitting continua, even if minimally so compared to more sophisticated stellarpopulation modeling. \nThe template set referred to as the MASTARHC2 169 library by the DAP is a set of 65 hierarchically clustered templates based on ∼ 2800 MaStar spectra from MPL-10. Only one of the four DAPTYPE s provided in DR17 uses these templates; however, we note that the results based on these templates are the primary data sets used by Law et al. (2021b,a) to improve the DRP (see above). The approach used to build the MASTARHC2 library is inspired by, but different in many details, from the hierarchical clustering method used to build the MILESHC library (cf., Westfall et al. 2019, Section 5), as described below. \nThe principles of the hierarchical clustering approach used by Westfall et al. (2019) to construct the MILESHC library are maintained, except we perform the clustering for the MASTARHC2 library in two steps. The first step clusters spectra based on their low-order continuum differences, leading to a set of 'base clusters.' We use pPXF (Cappellari & Emsellem 2004; Cappellari 2017) to perform a least-squares fit of each spectrum using every other spectrum; however, we do not include Gaussian kernel terms or polynomial continuum optimization, meaning the least-squares fit simply optimizes the scaling between the two spectra. We use the rms difference between the best-fit spectra as the clustering 'distance,' and the distance matrix is used to construct eight base clusters. The choice of eight clusters was based on an a qualitative assessment of the appropriate number which separated MaStar spectra into distinct types. The second step uses pPXF to fit each spectrum using every other spectrum within its base cluster . In this step, we modestly degrade the resolution of the template being fit with σ = 1 pixel, and then our pPXF fit includes a freely fit Gaussian kernel with bounds of a ± 1 pixel shift and a 0 . 1 -2 pixel broadening. This was done in the same way across all parts of the spectra. We also include a multiplicative Legendre polynomial of order 100 to optimize the continuum match between the two templates. The very high-order fit (the choice of the exact number of 100 was arbitrary) acts like a high-pass filter on the differences between the two spectra, ensuring that the optimized rms difference between the two spectra is driven by the high-order (line) structure differences. The spectra within each base cluster are organized into 'template clusters' and visually inspected. The visual inspection leads to iterations on the number of template clusters in each base cluster, as well as removing some of the spectra from the analysis. The number of template clusters per base cluster ranged from 6 to 16, depending on a by-eye assessment of the spectra in each template cluster. The final assignment of each MaStar spectrum (identified by its MANGAID ) to a template cluster is pro- \n169 MASTARHC2 was the second of two library versions based on hierarchical clustering (HC) of MaStar spectra. MASTARHC1 is also available from the DAP code repository, but it was only used in the processing for MPL-9. \nvided in the DAP code repository. 170 Note that 34 of 99 clusters were not included in the MASTARHC2 library because they were either composed of single stars, resulted in noisy spectral stacks, contained isolated specific datareduction artifacts, or contained a set of spectra that were considered too disparate for a single cluster. For the vetted set of 65 template clusters, the median number of spectra per cluster is 14, but the range is from 2 to more than 300. \nWith the assignments in hand, we combine spectra in each template cluster as follows. We first scale each spectrum by their median flux and create an initial stack, weighting each spectrum by its median SNR. We then calculate the ratio of each spectrum to the stacked spectrum and fit this with an order-14 Legendre polynomial, which provides a low-order correction function to the continuum shape of each spectrum. The specific choice of order 14 was driven by a desire to match the choice made in the DAP fitting of galaxy spectra, which was justified in Westfall et al. (2019) We constrain the correction function to be no more than a factor of 2, which is particularly important to the stacks of late-type stars with very little flux toward the blue end of MaStar's spectral range. The low-order correction function is then applied to each spectrum in the template cluster before the final S/N weighted stack. The error vector for each stack is the quadrature sum of the propagated error from the stacking operations and the, typically much more significant, standard deviation measured for the spectra in the stack. The final spectra in the MASTARHC2 library are shown in Figure 6.", '5.2.2. Spectral Index Measurements': "In DR17, we have added spectral index measurements that are more suited to stacking analyzes and coaddition of spectra among spaxels, such as those based on definitions of Burstein et al. (1984) and Faber et al. (1985). These measurements are particularly useful for low SNR spaxels. \nThe motivation for this change emerges from the fact that in DAP's hybrid binning scheme, the spectral index measurements are performed on individual spaxels, which can have very low S/N (cf. Westfall et al. 2019, Section 9). Westfall et al. (2019) recommend improving the precision of the spectral index measurements using specific aggregation calculations that closely match the results obtained by performing the measurements on stacked spectra over the same spatial regions (specifically, see their Section 10.3.3). However, the comparison between an aggregated index and an index measured using a stacked spectrum is not mathematically identical for the index definitions used by Westfall et al. (2019). Motivated by the analysis of Molina et al. (2020), the DAP calculates the spectral indices (specifically the absorption line indices) using two definitions for DR17: (1) those definitions provided by Worthey (1994) and Trager et al. (1998) and (2) earlier definitions provided by Burstein et al. (1984) and Faber et al. (1985). The advantage of the definitions provided by Burstein et al. (1984) and Faber et al. (1985) is that they allow for a mathematically rigorous aggregation of spectral indices, \n∨∐√˜⌈˜{˜√[({GLYPH<197> } \nFig. 6.Spectra in the MASTARHC2 template library. Spectra are arranged and colored according to the membership in one of eight base clusters (the first clustering step used in the process to generate the template library from individual MaStar spectra). \n<!-- image --> \nas we derive below. \nFollowing the derivation by Westfall et al. (2019), we define a utility function, which is a sum of pixel values, multiplied by pixel width, \nS ( y ) ≡ ∫ λ 2 λ 1 y dλ ≈ ∑ i y i d p i d λ i , (1) \nwhere y is usually, but not always a function describing the flux in the spectrum, f ( λ ), d p i is the fraction of spectral pixel i (with width d λ i ) in the passband defined by λ 1 < λ < λ 2 . Note that masked pixels in the passband are excluded by setting d p i = 0, and S (1) = ∆ λ ≡ λ 2 -λ 1 if no pixels are masked. We can then define a linear continuum between two sidebands, referred to as the blue and red sidebands, as \nC ( λ ) = ( 〈 f 〉 red -〈 f 〉 blue ) λ -λ blue λ red -λ blue + 〈 f 〉 blue , (2) \nwhere f is the spectrum flux density, λ blue and λ red are the wavelengths at the center of the two sidebands, and 〈 f 〉 = S ( f ) /S (1). \nThe absorption-line index definitions used by Worthey (1994) and Trager et al. (1998) are: \nI WT = { S (1 -f/C ) , for ˚ A units -2 . 5 log [ 〈 f/C 〉 ] , for magnitude units , (3) \nwhere the measurements are made on a rest-wavelength spectrum. 171 Under this definition, the integration is performed over the ratio of the flux to a linear continuum, which means that the sum of, say, two index measurements is not identical to a single index measurement made using the sum of two spectra. In contrast, Burstein et al. (1984) and Faber et al. (1985) define: \nI BF = { S (1) -S ( f ) /C 0 , for ˚ A units -2 . 5 log [ 〈 f 〉 /C 0 ] , for magnitude units , (4) \nwhere C 0 is the value of the continuum, C ( λ ), at the center of the main passband. Note that, given that C ( λ ) is linear and assuming no pixels are masked, \nS ( C ) = C 0 ∆ λ . Using the definition in Equation 4, we can calculate a weighted sum of indices using the value of the continuum, C 0 , for each index as the weight to obtain \n∑ i C 0 ,i I BF ∑ i C 0 ,i = ∆ λ -∑ i S ( f ) i ∑ i C 0 ,i , (5) \nassuming no pixels are masked such that S (1) = ∆ λ . That is, the weighted sum of the individual indices is mathematically identical (to within the limits of how error affects the construction of the linear continuum) to the index measured for the sum (or mean) of the individual spectra. Similarly, for the indices in magnitude units, we find: \n-2 . 5 log [∑ i C 0 ,i 10 -0 . 4 I BF ∑ i C 0 ,i ] = -2 . 5 log [ ∑ i 〈 f 〉 i ∑ i C 0 ,i ] . (6) \n171 Note the subtle difference between Equation 3 and Equation 22 from Westfall et al. (2019); the latter has an error in the expression for the index in magnitudes units. \nGiven the ease with which one can combine indices in the latter definition, we provide both I BF (in the SPECINDEX BF extension of the DAP MAPS file) and C 0 (in SPECINDEX WGT ) for all absorption-line indices in DR17, along with the original definitions ( I WT ; SPECINDEX ) provided in DR15/DR16.", '5.3. Marvin Visualization and Analysis Tools': "Marvin (Cherinka et al. 2019) was developed as the tool for streamlined access to the MaNGA data, optimized for overcoming the challenges of searching, accessing, and visualizing the complexity of the MaNGA dataset. Besides patches and internal optimizations, the DR17 updates to Marvin include several enhancements such as querying targets by MaNGA quality and target bitmasks and values, full support for installation on Windows machines, as well as updates to the web interface. The Marvin Web Galaxy Page (Figure 7) now includes data quality indicators for the DAP Maps, as well as toggle-able features for the spectrum display. For how to use the web or Python tools, see the Marvin documentation 172 . See the Marvin Changelog for a complete list of what has changed since the last released version. Contributions to Marvin are welcome and encouraged. Please see the contribution guidelines 173 for more details. \nMarvin now includes access for many of the MaNGA VACs 174 , that have been integrated into the Marvin ecosystem. Each integrated VAC is accessible either as the full catalog through the new Marvin VACs Tool, or on a per-target basis through the existing Marvin Tools, e.g. Cube or Maps, via a new 'vacs' attribute attached to each Marvin Tool instance. Check the VACs section of the DR17 Datamodel in the Marvin documentation to see which VACs are available in this release.", '5.4. Ancillary Programs': "As described in detail in Wake et al. (2017), the MaNGA galaxy sample is comprised of a Primary sample covering galaxies to 1.5 r e (where r e is the effective radius; the radius containing 50% of the light), a Secondary sample covering galaxies to 2.5 r e , and a color-enhanced sample designed to fill in underrepresented locations in the galaxy color-magnitude plane. However the number density of these main sample targets was not uniform on the sky, and in regions of lower-than-average target density, not all of the IFUs on a plate could be assigned to core target categories. In order to fill the remaining ∼ 5% of MaNGA bundles, MaNGA held two competitive calls for ancillary targets (in July 2014 and January 2017), and a variety of ancillary programs targeting assorted kinds of galaxies or other targets were selected. \nWe document here the final set of ancillary targets, with detail on how to identify them for use, or to exclude them from studies of the primary, secondary and/or color-enhanced samples. We provide in Table 4 an updated list of the number of bundles available in each documented sample, along with the binary bit mask digit \n- 172 https://sdss-marvin.rtfd.io/en/latest/index.html \n173 \nhttps://sdss-marvin.rtfd.io/en/latest/contributing/ \ncontributing.html \n174 \nhttps://sdss-marvin.rtfd.io/en/latest/tools/vacs. \nstored in MANGA TARGET3 (sometimes MNGTARG3 ) which can be used to identify the sample. 175 \nAncillary programs in general were designed to increase the numbers of specific types of galaxies in the MaNGA sample. We provide a short summary of all programs here (also see http://www.sdss.org/dr17/manga/ manga-target-selection/ancillary-targets ): \n- 1. Luminous AGN: various luminous active galactic nuclear (AGN) samples were targeted, either selected from Swift BAT ( AGN BAT ), [OIII] emission selected (Mullaney et al. 2013, AGN OIII) , Widefield Infrared Survey Explorer (WISE; Wright et al. 2010) colors ( AGN WISE ), or other AGN from the Palomar survey ( AGN PALOMAR , Ho 1995). The goal of this program was to increase the range of luminosities of AGN observed by MaNGA.\n- 2. Void Galaxies: ( VOID ) this program targeted rare void galaxies located in low-density large scale environments. Targets were selected from the Void Galaxy Survey (VGS; Kreckel et al. 2011).\n- 3. Edge-On SF Galaxies: a set of edge-on star forming (SF) galaxies were selected, using WISE data to estimate star-formation rates (SFRs), and S'ersic axial ratios ( b/a ) from the NASA Sloan Atlas (NSA; Blanton et al. 2011) to estimate inclinations. The BITNAME is EDGE ON WINDS .\n- 4. Close Pairs and Mergers: a set of close pairs and/or mergers were observed. These were either selected to be in larger bundles than typical ( PAIR ENLARGE ), re-centered bundles (i.e. not centered on one of the pair, but somewhere in the middle, ( PAIR RECENTER )), or in overlapping tiles, sometimes two bundles are assigned - one to each of the pair ( PAIR 2IFU ). In addition one bundle was assigned to a merger simulated by the Galaxy Zoo: Mergers program ( PAIR SIM , Holincheck et al. 2016).\n- 5. Writing MaNGA: two bundles were assigned to an education/public outreach (EPO) program to obtain MaNGA data for galaxies in the shape of letters in the word MaNGA (selected from the Galaxy Zoo 'Alphabet' 176 ). These are an 'M' (plate-ifu = 8721-6102) and a 'g' (9499-9102). See LETTERS .\n- 6. Massive Nearby Galaxies: very massive nearby galaxies are underrepresented in MaNGA as they are too large in angular size to fit in the bundles. This program targeted bundles at the central regions of very massive nearby galaxies ( MASSIVE ).\n- 7. Milky Way Analogs: two different sets of Milky Way Analog galaxies are included. These galaxies are identified using the method described in Licquia & Newman (2015): one set matched on stellar mass and SFR ( MWA ), the other matched on stellar mass and bulge-to-total ratio ( MW ANALOG ). \n175 For advice on using bitmasks see https://www.sdss.org/ \ndr17/algorithms/bitmasks 176 \nhttps://writing.galaxyzoo.org/ \nFig. 7.A screenshot of the galaxy maps view of the Marvin Web for the MaNGA galaxy 12-193481 (Mrk 848). The SDSS three-color image of the galaxy is shown in the top left part of the figure. The upper right panel shows the spectrum of the spaxel at the position (37,37), which corresponds to the center of the bundle. The maps show: (lower left) stellar kinematics; (lower middle) H α emission line flux; and (lower right) D4000 spectral index for this galaxy based on its 'hybrid-binned' spectral data cube from the MaNGA DAP. \n<!-- image --> \n- 8. Dwarf Galaxies: a sample of dwarf galaxies selected from the Geha et al. (2012) catalog ( DWARF ).\n- 9. ETGs with Radio Jets: a sample of early-type galaxies (ETGs) with radio jets and evidence of suppressed SF ( RADIO JETS Lin et al. 2010) .\n- 10. DiskMass Sample: a sample of face-on disc galaxies which had previously been observed in the DiskMass survey (Bershady et al. 2011) with the goal of providing a cross-calibration set ( DISKMASS ).\n- 11. Brightest Cluster Galaxies: a sample of Brightest Cluster Galaxies (BCGs; BCG ) from the Yang et al. (2007) catalog. This type of galaxy is otherwise underrepresented in MaNGA.\n- 12. Resolved Stellar Populations: observations of very nearby galaxies in the ACS Nearby Galaxy Survey Treasury (ANGST, ANGST ; Dalcanton et al. 2009) survey, as well as in M31 ( M31 ), to facilitate detailed studies of the resolved stellar populations.\n- 13. Coma Plates: a set of very deep observations of the Coma cluster ( DEEP COMA ). Each dedicated plate used for this program observes the central massive cD galaxies (varying placement between the central regions and the galaxy outskirts), a selection of ordinary galaxies, 3 ultrafaint galaxies, and 3 regions\n- of intracluster light (ICL). The goal was to provide very high quality spectra to enable detailed stellar population analysis.\n- 14. IC 342: a mosaic of 49 MaNGA plates, covering 5 ' × 5 ' (5 kpc × 5 kpc) across the disk of the nearby galaxy IC342 with ∼ 30 pc spatial resolution ( IC342 ). This project provides test data for the Local Volume Mapper (LVM) in SDSS-V (see § 8).\n- 15. SN Hosts: observations of the host galaxies of known Supernova, both SN Type 1a under SN1A HOST and other types of SN under SN ENV .\n- 16. Giant LSB galaxies: a set of giant low surface brightness (GLSB, GLSB ) galaxies identified in the NSA.\n- 17. Globular clusters: a set of dithered observations of the cores of eight globular clusters (GCs) and 19 bulge/background fields around 3 GCs (NGC6316, NGC6522, and NGC6528) taken to help with MaStar ( GLOBULAR CLUSTER ).", '5.5. MaNGA Related VACs': "A large number of MaNGA related VACs are presented in DR17, and will be summarized in brief below. \nTABLE 4 Summary of MaNGA Ancillary Programs and Targeting Bits. See § 5.4 for an explanation of each program and the BITNAMES \n1 8 GC targets, plus 19 bulge background fields \n5.5.1. DR16+ VACs \nTwo MaNGA related VACs were released in DR16+ (a mini-data release which happened in July 2020). In addition a version of the 'Visual Morphology from DECaLS Images' VAC, which is updated for DR17, was also released in DR16+. We document those first. \n- 1. NASA Sloan Atlas Images and Image Analysis : This VAC contains the underlying image and image analysis for the NASA Sloan Atlas (NSA). The methods used are described in Blanton et al. (2011) and Wake et al. (2017). Briefly, for a set of nearby galaxies of known redshift ( z < 0 . 15) within the SDSS imaging area, we have created and analyzed GALEX (Morrissey et al. 2007) and SDSS images. This analysis forms the basis for the MaNGA targeting, and resulted in the v1 0 1 NSA catalog released originally with DR14. We are now releasing the images which were analyzed to create those parameters. The data set includes the original catalogs from which the NSA sample was drawn, the mosaic images and inverse variance images that were analyzed, the deblending results for each object, the curve-of-growth and aperture corrections for each object, and other intermediate outputs. We expect that this data set may be useful for reanalysis of the GALEX or SDSS imaging. The full data set is large (15 terabytes) and therefore any users interested in using a large fraction of it should transfer the data through Globus (see § 3 for details on how to use Globus 177 ). \nhttps://www.sdss.org/dr16/data\\_access/bulk/ \n- 2. MaNGA SWIFT VAC : The Swift +MaNGA (SwiM) value added catalog comprises 150 galaxies with both SDSS-IV/MaNGA IFU spectroscopy and archival Swift Ultraviolet Optical Telescope (UVOT) near-ultraviolet (NUV) images, and is presented in Molina et al. (2020). The similar angular resolution ( ∼ 3 '' ) between the Swift /UVOT three NUV filters and the MaNGA IFU maps allows for spatially-resolved comparisons of optical and NUV star formation indicators, which is crucial for constraining attenuation and star formation quenching in the local universe. The UVOT NUV images, SDSS optical images, and MaNGA emission line and spectral index maps have been spatially matched and re-projected so that all of the data match the pixel sampling, resolution and coordinate system of the UVOT uvw2 image for each galaxy. The spectral index maps utilize the definition given in Burstein et al. (1984), which allows users to more easily compute spectral indices when binning the maps. Spatial covariance is properly accounted for in the propagated uncertainties. In addition to the spatially-matched maps, Molina et al. (2020) also provides a catalog with PSF-matched aperture photometry for the SDSS optical and Swift /UVOT NUV bands.", '5.5.2. Galaxy Morphology VACs': "A variety of galaxy morphology catalogs are provided as VACs, with analysis done in a variety of ways, using a variety of images. We provide a short summary of each here - for more details please see the appropriate paper. \n- 1. Galaxy Zoo: 3D : (GZ:3D; Masters et al. 2021) provides crowdsourced spaxel masks locating galaxy centers, foreground stars, bars and spirals in the SDSS images of MaNGA target galaxies. Available for use within Marvin , these masks can be used to pick out spectra, or map quantities likely associated with the different structures (see Peterken et al. 2019b,a; Fraser-McKelvie et al. 2019, 2020; Greener et al. 2020; Krishnarao et al. 2020, for example use cases).\n- 2. Galaxy Zoo Morphologies from SDSS, DECaLS and UKIDSS : The Galaxy Zoo method, which involves combining classifications from a large number of classifiers collected via an online interface, has been applied to a variety of images, including the original SDSS images (Willett et al. 2013; Hart et al. 2016), the UK Infrared Telescope Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2007; Galloway 2018) and most recently the Dark Energy Camera Legacy Survey (DECaLS; Dey et al. 2019; Walmsley et al. 2022). This latter analysis combines Machine Learning (ML) methods with crowdsourcing in an active loop (for details see Walmsley et al. 2022). We collect together all these crowdsourced morphologies for as many MaNGA galaxies as possible in this VAC.\n- 3. Visual Morphology from DECaLS Images: This VAC contains a direct visual morphological classification, based on the inspection of image mosaics generated from a combination of SDSS and DECaLs (Dey et al. 2019) images, for the MaNGA galaxies. The DR16+ version contains the classification for the first half of MaNGA galaxies (4600, MaNGA DR15) while the DR17 version contains the classification for the full MaNGA DR17 with unique MaNGAID . Through a digital image postprocessing, we exploit the advantages of this combination of images to identify inner structures, as well as external low surface brightness features for an homogeneous classification, following an empirical implementation of the methods in Hern'andezToledo et al. (2010) and Cheng et al. (2011). The visual morphological classification is carried out by two classifiers inspecting three-panel image mosaics, containing a gray logarithmic-scaled r -band image, a filter-enhanced r -band image and the corresponding RGB color composite image from SDSS and a similar mosaic using DECaLS images incorporating the residual image after subtraction of a best surface brightness model from the DESI legacypipeline 178 . The catalog contains the TType morphology, a variety of visual morphological attributes (bars, bar families, tidal debris, etc.) and our estimate of the non-parametric structural, Concentration, Asymmetry and Clumpiness (CAS; Conselice 2003) parameters from the DECaLS images. For more detail in see V'azquez-Mata et al. (2021). An updated version including morphologies for all DR17 MaNGA galaxies in being prepared, see V'azquez-Mata et al. (in prep.). \n- 4. MaNGA PyMorph DR17 photometric catalog: (MPP-VAC, see Fischer et al. 2019; Dom'ınguez S'anchez et al. 2022 for details) provides photometric parameters obtained from S'ersic and S'ersic+Exponential fits to the 2D surface brightness profiles of the final MaNGA DR17 galaxy sample (e.g. total fluxes, half light radii, bulge-disk fractions, ellipticities, position angles, etc.). It extends the MaNGA PyMorph DR15 photometric VAC to now include all MaNGA galaxies in DR17.\n- 5. MaNGA Morphology Deep Learning DR17 catalog: (MDLM-VAC, see Dom'ınguez S'anchez et al. 2022 for details) provides morphological classifications for the final MaNGA DR17 galaxy sample using Convolutional Neural Networks (CNN). The catalog provides a T-Type value (trained in regression mode) plus four binary classifications: P LTG (separates early type galaxies, or ETGs, from late types, or LTGs), P S0 (separates ellipticals from S0) , P edge -on (identifies edge-on galaxies) , P bar (identifies barred galaxies). It extends the 'MaNGA Deep Learning Morphology DR15 VAC' (Fischer et al. 2019) to now include galaxies which were added to make the final DR17. There are some differences with respect to the previous version - namely, the low-end of the T-Types are better recovered in this new version. In addition, the P LTG classification separates ETGs from LTGs in a cleaner way, especially at the intermediate types ( -1 < T-Type < 2), where the T-Type values show a large scatter. Moreover, the value provided in the catalog is the average of 10 models trained with k-folding for each classification task (15 for the T-Type classification). The standard deviation, which can be used as a proxy for the uncertainty in the classification, is also reported.", '5.5.3. Stellar Population Modeling VACs': "There are a variety of stellar population modeling based VACs released. \n- 1. Principle Component Analysis (PCA) VAC (DR17): this VAC includes measurements of resolved and integrated galaxy stellar masses, obtained using a low-dimensional, PCA-derived fit to the stellar continuum and subsequent matches to simulated star-formation histories (SFHs). The general methodology for obtaining the principal component basis set, the stellar continuum fitting routine, and the process of inferring stellar population properties such as mass-to-light ratio are discussed in Pace et al. (2019a). The aggregation of pixel-based mass estimates and adopted aperture-correction procedure are described in Pace et al. (2019b). This procedure yields estimates of galaxy-wide, integrated stellar masses also provided as part of this VAC. Key VAC characteristics remain unchanged in comparison to DR16 (Ahumada et al. 2020), where a holistic description of the VAC can be found. The principal enhancement in this release is in the sample size: the number of galaxies has been expanded to include all MaNGA galaxies to which the analysis \ncould be readily applied, a total of 10223 unique plate-ifu designations (this number differs from unique galaxy counts, as some MaNGA galaxies were observed multiple times, so have multiple plate-ifus, each of which are analyzed separately in this VAC). \n- 2. Firefly Stellar Populations: This VAC provides measurements of spatially resolved stellar population properties of MaNGA galaxies employing the Firefly 179 (Wilkinson et al. 2017) full spectral fitting code. For DR17, Firefly v1.0.1 was run over all 10735 datacubes that had been processed by both the DRP and the DAP 180 . The major difference to the DR15 VAC is that we now provide the catalog in two versions. The first employs the stellar population models of Maraston & Stromback (2011) based on the MILES stellar library (S'anchez-Bl'azquez et al. 2006). The second version uses new MaStar models described in Maraston et al. (2020). Both model libraries assume a Kroupa (2001) IMF. Compared to the Firefly VAC in DR15, the radius (stored in HDU4 in the file) is now given in elliptical coordinates and the azimuth is added. Masses ( HDU11 and HDU12 ) are given per spaxel and per Voronoi cell. We do not provide absorption index measurements anymore. Each version of the VAC is offered as a single FITS file ( ∼ 6 GB) comprising the whole catalog of global and spatially resolved parameters and, additionally, as a small version ( ∼ 3 MB) that contains only global galaxy stellar population parameters. A detailed description can be found in J. Neumann et al. (in prep.) and Goddard et al. (2017).\n- 3. Pipe3D : This VAC containes the Pipe3D (S'anchez et al. 2016) analysis of the full MaNGA dataset comprises the main properties of the stellar populations and emission lines for more than 10,000 galaxies, both spatial resolved and integrated across the entire field-of-view (FoV) of the IFUs. The content of the released distribution was originally described in S'anchez et al. (2018), and updated in S. F. S'anchez et al in prep. The new releases include considerable modifications from the previous ones, the most important ones being (i) the use of an updated version of the code fully transcribed to python (E. Lacerda et al. in prep.), (ii) the use of a new stellar population library based on the MaStar stellar library (A. Mejia-Narvaez et al. in prep.), and (iii) an update on the list of analyzed emission lines.", '5.5.4. HI-MaNGA DR3': 'HI-MaNGA is a HI 21cm line followup program, to provide estimates of total atomic hydrogen content for galaxies in the MaNGA survey (Masters et al. 2019). It makes use of both previously published HI data (primarily from the ALFALFA survey; Haynes et al. 2018) and new observations using the Robert C. Bryd Green \n180 This number differs from the total number of cubes listed in Table 1, as 538 data cubes did not run through the DAP for various reasons \nBank Telescope (GBT; to date under observing codes GBT16A 095 , GBT17A 012 , GBT19A 127 , GBT20B 033 and GBT21B 130 ). This VAC comprises the third data release (DR3) of HI 21cm detections or upper limits for 6358 galaxies in the MaNGA sample. In some cases both GBT and ALFALFA data exist for a single galaxy, and we provide both observations separately, so the total number of rows is 6632, with 3358 coming from our GBT observations. The observation and reduction strategy are documented in Masters et al. (2019); Stark et al. (2021). As part of this program a 20% offset between actual, and estimated L-band calibration at GBT was noticed (see Goddy et al. 2020). Stark et al. (2021) provide guidance on dealing with confusion, and including upper limits into statistical analysis. Observations are ongoing (under proposal code GBT21B 130 ), with the program on track to observe, or homogenise HI data for all MaNGA galaxies at z < 0 . 05, with no pre-selection on color or morphology. This is expected to result in a final HI-MaNGA sample size of around 7000 MaNGA galaxies, with over 6800 already having at least some data in hand.', '5.5.5. The MaNGA AGN Catalog': 'The MaNGA AGN Catalog presents AGN in the DR15 sample of MaNGA that are identified via mid-infrared WISE colors, Swift/BAT ultrahard X-ray detections, NRAO VLA Sky Survey (NVSS) and Faint Images of the Radio Sky at Twenty-cm (FIRST) radio observations, and broad emission lines in SDSS spectra. The catalog further divides the radio AGN into quasar-mode and radio-mode subpopulations, and provides estimates of the AGN bolometric luminosities. Full details of the AGN selection and luminosity measurements are described in Comerford et al. (2020). It is intended that this will be updated to include all MaNGA galaxies in the future.', '5.5.6. GEMA-VAC: Galaxy Environment for MaNGA Value Added Catalog': "The Galaxy Environment for MaNGA (GEMA) VAC (M. Argudo-Fern'adez et al. in prep.) provides a variety of different measures of environment for galaxies in MaNGA. The combination of mass-dependent and mass-independent parameters provided in the catalog can be used to explore the effects of the local and largescale environments on the spatial distribution of star formation enhancement/quenching, in the interaction of AGN with galaxies, or in the connection with kinematics or galaxy morphology, for instance, in an homogenous way allowing comparisons between different studies. In DR17, we present the final and updated version of GEMA-VAC for the final MaNGA sample. The quantifications of the environments are based on the methods described in Argudo-Fern'andez et al. (2015) to estimate tidal strengths and projected number densities, as well as that in Etherington & Thomas (2015) to estimate overdensity-corrected local densities (MaNGA galaxies in the SDSS-DR15 only), and Wang et al. (2016) for an estimation of the cosmic web environment. To better explore the environment of galaxies located in dense local environments, for instance, galaxies in compact groups or with strong interactions (close paris/mergers), but not necessarily a high density environment at larger scale, \nwe also provide these same quantifications considering MaNGA galaxies in groups, according to an updated version of the catalog of groups compiled by Yang et al. (2007); and MaNGA galaxies in close pairs, according to the sample used in Pan et al. (2019).", '5.5.7. MaNGA Spectroscopic Redshifts for DR17': 'We present a catalog of precise spectroscopic redshifts M. Talbot et al. (in prep.) for the RSS and spaxels in MaNGA, updating the previous version of this catalog (Talbot et al. 2018) to include the completed sample of MaNGA galaxies. These spectroscopic redshifts are computed using the spec1d - zfind code from the publicly available BOSS pipeline (Bolton et al. 2012), in which the NSA catalog provides the initial redshift. Once spectroscopic redshifts were determined for the high signal-tonoise region within the galaxy half-light radius, a second pass attempted to determine the remaining redshifts in the low signal-to-noise spectra using the mean spectroscopic redshift as a prior. The spectroscopic redshifts and a foreground model are presented for each spectrum with sufficient SNR to model, in this VAC.', '5.5.8. MaNGA Strong Gravitational Lens Candidate catalog': 'We present six likely, 12 probable, and 74 possible candidate strong galaxy-galaxy scale gravitational lenses found within the completed MaNGA survey. The lens candidates are found by the Spectroscopic Identification of Lensing Object program (SILO; Talbot et al. 2018, M. Talbot et al. in prep) , which was adapted from the BELLS (Brownstein et al. 2012) spectroscopic detection method to find background emission-lines within coadded foreground-subtracted row-stacked-spectra of the MaNGAIFU, in which the co-added residuals are stacked across exposures from the same fiber at the same dither position. Visual inspection of any background emissionline detected was performed, including the position of detections in proximity to an estimated Einstein radius. Narrowband images were constructed from the co-added residuals to search for any lensing features.', '6. MASTAR: THE MANGA STELLAR LIBRARY': 'The MaNGA Stellar Library (MaStar) is a project in SDSS-IV to build a large library of well-calibrated empirical stellar spectra, covering a wide range in stellar parameter space, roughly from 2,500K to 35,000K in effective temperature ( T eff ), from -1 to 5.5 in surface gravity (log g ), and from -2.5 to 0.5 in metallicity ([Fe/H]). It is conducted using the same instrument as MaNGA but during bright time (Yan et al. 2019). Most of the observations were done by piggybacking on APOGEE2N, in the sense that the field centers of those plates, the time spent on the field, and the number of visits were determined according to the science need of APOGEE2N. Only in a small number of fields were observational parameters determined by the science needs of MaStar. DR17 presents data for all of the stars observed in the MaStar program, along with complementary analysis of all the standard stars targeted on MaNGA-led plates. \nThe MaStar targets were selected to cover a wide range in the 4-dimensional stellar parameter space ( T eff , log g , [Fe/H], [ α /Fe].) In the part of parameter space covered by APOGEE (Majewski et al. 2017), APOGEE-2 ( § 4), \nSEGUE (Yanny et al. 2009), and the Large-sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST Luo et al. 2015), we make use of the stellar parameters derived from these surveys to select targets, aiming to evenly sample the parameter space. However, due to availability of stars of certain parameters, the constraints of the fields selected by APOGEE-2N and the evolving field choices, the stellar parameter space coverage cannot be completely even. In addition to these selections, we further use photometry data from the Panoramic Survey Telescope and Rapid Response System (PanSTARRS1; Kaiser et al. 2010) and the American Association of Variable Star Observers Photometric All-Sky Survey (APASS; Munari et al. 2014) to select stars that are more likely to have extreme temperatures, either very hot or very cool. Further details of the MaStar target selection have been described in Yan et al. (2019). Once Gaia Data Release 2 (Gaia Collaboration et al. 2018, 2016) was available, we made use of Gaia color and absolute magnitudes (derived using distances from Bailer-Jones et al. 2018) to select stars to fill up the parts of the color-magnitude space that were not sufficiently sampled, including hot main sequence stars, blue supergiants, yellow supergiants, stars at the tip of red giant branch, and red supergiants, Carbon stars and other asymptotic giant branch (AGB) stars, white dwarfs (WD), extreme horizontal branch stars, metal poor dwarfs, and late Mdwarfs. These recent changes to the target selection involving Gaia photometry will be described by R. Yan et al. (in prep.). \nWithin the APOGEE-2 bright time extension ancillary call, the MaStar project was given a small number of hours to observe stars that could not be targeted by piggybacking on APOGEE-2N. During these times, we targeted a number of star-forming fields with a large number of hot main sequence stars, blue and red supergiants, a number of fields with known metal-poor late M dwarfs, and a number of open cluster and globular cluster fields. For the dedicated globular cluster fields, we conducted dithered observations to obtain integrated spectra for the core regions of the globular clusters with some fibers targeting relatively isolated stars in the outskirts of globular clusters.', '6.1. MaStar-specific Changes to the MaNGA DRP': "The MaStar data are obtained using the same MaNGA fiber feed system and the BOSS spectrographs as the main MaNGA survey. The data reductions for MaStar are done with the MaNGA DRP through its 2D phase. The details of this were described by Law et al. (2016) with DR17 updates described in § 5.1. In the 2D phase, the only difference is in how we correct for the extinction of the standard stars in the flux calibration module, which we describe below. The reduction for the 3D phase is done differently from MaNGA. The basics of the data reduction were described by Yan et al. (2019). We briefly describe the updates since DR15/16 below. More details of these will be presented in R. Yan et al. (in prep.). \n- · Flux calibration for both MaNGA and MaStar plates are done using a set of 12 standard stars observed simultaneously with the science targets. By comparing the spectra of these 12 stars with the theoretical spectra we determine the through- \n- t ratio between the observed spectra and the expected spectra above the atmosphere. This ratio is then applied to all the spectra from all the fibers to determine the per-fiber spectra. The theoretical spectra used in the comparison need to have galactic extinction applied. For MaNGA-led plates, we use the values from Schlegel et al. (1998) dust map as the standards in MaNGA-led fields are at high galactic latitude and at a far enough distance to be beyond most of the dust in the Milky Way in those line of sights. But this is not always the case for MaStar plates. On MaStar plates, we use the spectra themselves (relative to their respective models) to estimate the relative extinction difference between different standard stars. Then we use the broadband colors to estimate the absolute extinction level for all the stars. With that, we can then determine the combined throughput curve of the atmosphere, the telescope, and the instrument. This throughput curve is then applied to calibrate spectra from all science fibers.\n- · Standard stars targeted with mini-bundles on MaNGA-led plates are also treated like other MaStar targets. This adds a significant number of F stars to the library.\n- · We have updated the template set used in the determination of stellar radial velocity search, which is a selected subset from the BOSZ templates (Bohlin et al. 2017). We expanded the subset to include templates with temperatures between 3500K and 35000K, with surface gravity between 1 and 5 in log (g cm -1 s 2 ), and with two different [ α /Fe] settings (0 and 0.5). We also included the Koester white dwarf templates for DA-type white dwarfs (Koester 2010).\n- · We changed the method used to select the fiber on which the final spectrum is based, for stars that saturate the central fiber in a bundle. We also changed the criteria of determining whether to combine spectra of multiple fibers together in each exposure. When combining spectra, the risk of 'red upturn' is evaluated and taken into consideration. The 'red upturn' refers to the artificial extra flux at the extreme red wavelengths in some of the spectra, which is likely introduced by crosstalk between adjacent spectral traces and imperfect 1D extraction at the extreme wavelengths. This will be discussed in more detail in R. Yan et al. (in prep.).\n- · For some MaStar plates (usually those done in APOGEE-2 time and therefore led by APOGEE2), we adopted exposure times much shorter than 900s in order to target bright stars. Three exposures time settings were adopted: 28s, 83s, and 250s. For those exposures shorter than 180s, the flexure-compensation algorithm adopted by DRP for MaNGA-length exposures no longer works due to the faintness of sky emission lines. In these cases, we measure the radial velocities for the standard stars, separately for the blue and red cameras, to figure out the median relative shifts between the \ntwo cameras. This is then used to adjust the blue cameras' wavelength solutions to match those of the red cameras'. The flexure could also cause the wavelength solution to differ from exposure to exposure. In this case, we use the radial velocities derived for all stars (both science targets and standard stars) to shift the wavelength solutions of all exposures to be consistent with the first exposure on a given visit, which is closest in time to the arc calibration frame. \nWe also added many quality checks and quality flagging in this data release. \n- · We added checks in the DRP to indicate the risk of red upturns. A subset of per-exposure spectra with significant upturn or downturn risks were visually inspected. The results were stored in a metadata file and read in by the pipeline to flag those per-exposure spectra. We exclude those perexposure spectra with UPTURNRISK , REDUPTURN , or REDDOWNTURN set in their quality bitmask, if possible, when producing the per-visit spectra.\n- · We run all MaStar spectra through an emissionline measurement code. The results were used to select a subset for visual inspection. The visual inspection results were combined with the automated measurements to decide which spectra to be flagged as having emission lines. Spectra with H α equivalent widths (EW) greater than 0.6 Angstrom are flagged with the EMLINE bit in EXPQUAL and MJDQUAL . We note that usual convention has positive EW for absorption not emission, but for this context (MaStar emission-line identification), we define emission lines to be positive in EW.\n- · We evaluated the quality of flux calibration for both the spectra from individual exposures and the combined spectra per visit. For individual exposure spectra, we flag them according to the chisquare produced when fitting for the flux ratios between the central fiber and the surrounding fibers. When the chi-square is greater than 50, we flag the BADFLUX bit of the quality bitmask of this exposure ( EXPQUAL ). This threshold corresponds to an uncertainty larger than 0.15 mag ( ∼ 14%) uncertainty in the relative flux calibration between the BP and RP bands (evaluated by comparing the synthetic color with Gaia photometry). This only flags the 1% worst cases on a per-exposure per-star basis..\n- · When we combine spectra from multiple exposures together to construct the combined exposure per visit, the quality of flux calibration is going to significantly improve due to averaging and due to the dominance by spectra with higher signal-to-noise ratio which tend to have better calibration. The evaluation of flux calibration quality for the combined spectra per-visit is based on uncertainty determined through Jackknife resampling technique. If the uncertainty on the synthetic BP-RP color is more than 0.05 mag, which corresponds to 5% relative calibration error, we flag the BADFLUX bit of the quality bitmask for this visit ( MJDQUAL ). \n- · We exclude those per-exposure spectra with BADSKYSUB , POORCAL , or SEVERBT set, if possible, when producing the per-visit spectra. \nAll of these changes and updates will be discussed in more detail by R. Yan et al. (in prep.).", '6.2. Changes to the MaStar Post-processing Pipeline': 'The MaStar post-processing pipeline (mastarproc) is updated for several purposes. It processes the result from a preliminary DRP run to identify candidates with emission line risks and red upturn/downturn risks for visual inspection. It is also updated to give more information in the MaStar summary files. We evaluate the variations of heliocentric radial velocities among all visits of a star (same MaNGAID ) and provide both the median velocity per visit and that across all visits. The significance of the variation is provided and if it is more than 3 σ , we flag the VELVARFLAG column in the summary file. We also added several useful columns to the summary files to indicate the SNR, bad pixel fraction, etc. See more details in R. Yan et al. (in prep.).', '6.3. MaStar Summary files': 'In this data release, we provide several summary files. The mastarall file contains only metadata information about the stars and visits, but no spectra. It has four extensions containing four tables: GOODSTARS , GOODVISITS , ALLSTARS , and ALLVISITS . The GOODSTARS table contains the summary information for all stars with at least one good quality visit spectrum, which we define as good stars. It has one entry per unique MaNGAID . The GOODVISITS table lists out all of the good quality visits for the good stars. It has one entry per unique visit. The ALLSTARS table contains the summary information for all of the stars observed in MaStar, regardless of the quality of the visits, with one entry per unique MaNGAID . The ALLVISITS table contains the information for all of the visits of all of the stars. \nThe mastar-goodspec file contains all of the good quality visit-spectra. It matches row-to-row to the GOODVISITS table in the mastarall file. The file mastar-badspec-v3 1 1-v1 7 7.fits.gz contains all the other visit-spectra, that are excluded from the GOODVISITS table. \nIn addition, we also provide two sets of files that contain spectra with unified spectral resolution curves. Because each spectrum in mastar-goodspec files can have different spectral resolution curves or line spread function curves, it could be cumbersome for the users. We thus defined four resolution curves based on the distribution of the spectral resolution at each wavelength among all good visit-spectra. For each resolution curve, we select visit-spectra that have higher resolution at all wavelengths and broadened their line spread function by convolution to match the common resolution curve. We provide four files containing four subsets of visit-spectra convolved to these four uniform resolution curves, respectively. \nWith resolution curves unified, we could easily combine multiple visit-spectra for the same star to improve signal-to-noise. These combined spectra are also provided corresponding to the four subsets defined by the four resolution curves. \nThe detailed data models for these files can be found in R. Yan et al. (in prep.) or on the SDSS data release website (see § 3).', '6.4. Photometry crossmatch': "With this data release, we also provide a valueadded catalog giving crossmatch information between MaStar and a few other catalogs. We crossmatch it with Gaia DR2 (Gaia Collaboration et al. 2018, 2016), Gaia EDR3 (Gaia Collaboration et al. 2021), PanSTARRS-1 (Flewelling et al. 2020), 2MASS (Skrutskie et al. 2006), and Simbad (Wenger et al. 2000). \nThe crossmatches between MaStar and Gaia (DR2 or EDR3) are performed in a few steps. For each MaStar target, given its coordinates and epoch of the coordinates, we select all Gaia targets within 40 '' of the star, apply proper motion correction to shift them to the epoch for which the MaStar coordinates were given. Then we search for the corresponding match within 3 '' . If there is only one match, the match is considered to be the correct one. If there is more than one candidate within the search radius, we compute a positional matching probability and a photometry-matching probability for all candidates. The photometry-matching probability is the average among probabilities in multiple bands available for the MaStar targets, computed using empirical relationships we established between Gaia photometry bands and the photometry bands of the MaStar targets. Both probabilities take into account their respective uncertainties. The product of the two probabilities are used to determine the best match among the multiple matches within 3 '' . Nearly all MaStar good stars have a match with Gaia. Among 24,290 good stars with unique MaNGAID, all but one have a match in Gaia EDR3; all but 14 have a match in Gaia DR2. \nSometimes, a single MaStar target based on groundbased photometry is resolved by Gaia into multiple sources. Our algorithm tends to choose the brighter and more dominant source as the match. In such cases, the MaStar spectra could also be affected in two ways. First, the spectrum would contain light from both stars. Second, if the separation between the two stars is large enough to make the combined image non-circular, then the fiber-aperture correction could be significantly affected resulting in a poor flux calibration for the final spectrum. These can be identified or excluded by checking the gaia cleanmatch column in the two tables. Sources with gaia cleanmatch =1 181 are considered cleanly isolated, for which the flux correction should be sufficiently accurate, according to Gaia astrometry and photometry. \nWith most stars crossmatched with Gaia , the crossmatch with PanSTARRS-1 is done through Gaia astrometry since it has more accurate coordinates and epoch information. Applying Gaia proper motion, we shift the Gaia -provided coordinates for MaStar targets to Epoch 2012.3 which is the approximate average epoch of the PanSTARRS-1 photometry catalog we used. Then we \n181 This corresponds to a value lower than 0.0084 in the 'contamination' column, whose meaning is defined in Appendix D.8 of Yan et al. (2019). The threshold adopted here is 3 times larger than that in Yan et al. (2019). The threshold adopted by Yan et al. (2019) is more conservative for targeting purposes. \nsearch for crossmatches in the PanSTARRS-1 catalog using a search radius of 2 '' . All sources with a unique match within 1 '' are considered a secure match. For those with multiple candidates within 2 '' , we choose the candidate with the largest product of the positional-matching probability and the photometry-matching probability. \nThe Simbad catalog (Wenger et al. 2000) contains useful spectral type and object type information for a small fraction of our targets. For crossmatching to Simbad, we shift the coordinates of all our targets to epoch 2000.0, then use a search radius of 3 '' . Sometimes, the same source appears as multiple entries in Simbad with different object types. In these cases, we choose the one that is most relevant or more informative for the star. About 13.3% of all the good stars and 23.4% of the science targets have a Simbad crossmatch with object type information. About 6.1% of all good stars and 9.9% of the science targets have a Simbad match with spectral type information. \nFor 2MASS, the crossmatch is done through the crossmatch table provided by Gaia (Marrese et al. 2019), using the astrometry solution provided by Gaia Data Processing and Analysis Consortium (DPAC Lindegren et al. 2018, 2021). We also derive extinction-corrected absolute magnitude and colors based on the Gaia photometry using a 3D dust map (Green et al. 2019) and Bailer-Jones distances (Bailer-Jones et al. 2018 for DR2 and BailerJones et al. 2021 for EDR3). We also include spectral type and object type information available from Simbad. We provide two files. Both contain information from MaStar, PanSTARRS-1, 2MASS, and Simbad. The only difference is that one file is based on Gaia DR2, while the other file is based on Gaia EDR3.", '6.5. MaStar Stellar Parameters VAC': 'Accurate stellar parameter labeling of the stars is essential for a stellar library. Although a significant fraction of the stars targeted in our library have been observed by other surveys with parameters derived, a large fraction still lack such information. Furthermore, the parameter derivations from previous surveys were inhomogeneous and some were based on data with poorer quality than we have. Thus, we initiated multiple parallel efforts to determine the stellar parameters for MaStar based on MaStar spectra themselves. Within DR17, we include a VAC giving four sets of stellar parameter measurements based on different methods along with the median values of them when available and deemed robust. \nThe four sets of parameters are described below. More details can be found in the respective papers. A comparison between the parameters will be presented in R. Yan et al. (in prep.). \n- DL: This parameter set (D. Lazarz et al. in prep.) is derived using full-spectrum fitting with an Markov Chain Monte Carlo (MCMC) sampler using interpolated BOSZ model spectra with continuum shape information included in the chi-square calculation. Extinction is fitted as a by-product. No photometry prior is used. \nJI: This parameter set (Imig et al. 2021) is derived using a neural network which models flux as a function of labels and is trained on a combination of empirical MaStar spectra with parameters from the \nAPOGEE Stellar Parameters and Chemical Abundance Pipeline (ASPCAP, see § 4.2.2 below) and the model spectra produced by Allende Prieto et al. (2018). \n- LH: This parameter set (Hill et al. 2022) is derived using full-spectrum, single-template, pPXF fitting with an MCMC sampler, using interpolated BOSZ and MARCS model spectra, with a flat prior based on Gaia color-magnitude diagram. The continuum is modeled with a multiplicative polynomial.\n- YC: This parameter set (Y. Chen et al in prep.) is derived using full-spectrum fitting using both the BOSZ and MARCS model spectra without interpolation, with the result produced by Bayesian average and a flat prior based on Gaia color-magnitude diagram. The continuum is modeled with a multiplicative polynomial. \nAll four methods provide T eff , log g , and [Fe/H]. On top of that, the methods by DL, LH and JI also provide [ α /Fe], and the method by JI additionally provides micro turbulence velocity ( v micro ). All methods have been applied to all spectra with quality control applied differently for different methods. Each method flags the spectra for which the parameters are considered invalid due to poor fitting quality. When we take the median, we only take the median among those methods that provide a valid measurement for a given spectrum. The uncertainties of the median values are also computed accordingly. Which methods are used in the median calculation are indicated by the INPUT GROUPS and INPUT GROUPS NAME columns for T eff , log g , and [Fe/H]. The quality control is more strict for [ α /Fe], for which the contributing methods are indicated by the INPUT ALPHA GROUPS and INPUT ALPHA GROUPS NAME columns in the VAC. The details of these will be provided in R. Yan et al. in prep. \nMetallicity measurements are crucial for assigning the right library spectra to the right metallicity bin when building stellar population models. We found some of the parameter sets could have slight systematic bias in metallicity measurements. Therefore, we calibrated the metallicity measurements for three of the four sets against APOGEE ASPCAP [Fe/H] measurements. In the VAC, in addition to the straight median among the four sets, we also provide a calibrated metallicity for each set and the median calibrated one among the four. We consider this to be a more accurate representation of the true metallicities of the stars. The stellar parameters are more reliable when at least two of the four groups have valid measurements for the given star. The users could choose to apply similar cuts to select a set of stars with more reliable parameters. \nIn the right panel of Figure 8, we show the median effective temperature vs. median surface gravity for the subset of good science stars in MaStar with at least two of four groups providing valid measurements. This includes 91% of all good science stars.', '6.6. MaStar Sample Statistics': 'In total, the MaStar library includes 24,130 unique good quality stars with 59,266 good quality visits. Among these, 11,817 unique stars were targeted as science targets and 12,345 unique stars were targeted as \nflux standards, with some overlap between the two categories. The 24,130 unique stars correspond to 24,290 unique MaNGAID s as some stars correspond to more than one MaNGAID s when taken from different source catalogs. \nIn Figure 8, we show a Hertzsprung-Russell diagram for all good stars in MaStar, based on Gaia EDR3 photometry after correcting for dust using the 3D dust map. The points are color-coded according to the median calibrated metallicity. This illustrates the comprehensive stellar parameter coverage of our library.', '7. EBOSS LIKE DATA': 'While both the main eBOSS as well as the co-observed TDSS made their final, full catalog release in DR16, new eBOSS like data is released for both the SPIDERS sub-survey, and the eBOSS-RM program. A number of eBOSS related VACs are also released.', '7.1. eBOSS VACs': "eBOSS (Dawson et al. 2016) concluded its observations of galaxies and quasars as tracers of large-scale structure on March 1, 2019. The goal of these measurements was to measure the distance-redshift relation with the baryon acoustic oscillation (BAO) feature that appears at a scale of roughly 150 Mpc. These data were also used to measure the growth of structure through redshift space distortions (RSD; Kaiser 1987). \nThe final eBOSS cosmology measurements were presented in a series of papers submitted in July, 2020. These results included consensus measurements of BAO and RSD for luminous red galaxies (LRG; Bautista et al. 2021; Gil-Mar'ın et al. 2020) over 0 . 6 < z < 1 . 0, emission line galaxies (ELG; Tamone et al. 2020; de Mattia et al. 2021) over 0 . 6 < z < 1 . 1, and quasars (Hou et al. 2021; Neveux et al. 2020) over 0 . 8 < z < 2 . 2. In addition, measurements of BAO were performed at z > 2 . 1 using clustering in the Lymanα forest and cross-correlations between quasars and the forest (du Mas des Bourboux et al. 2020). These measurements were combined with the final SDSS and BOSS (Dawson et al. 2013) BAO and RSD measurements spanning redshifts 0 . 07 < z < 0 . 6 (Ross et al. 2015; Howlett et al. 2015; Alam et al. 2017) to form a final sample of distinct clustering measurements over roughly ten billion years. The aggregate precision of the expansion history measurements is 0.70% at redshifts z < 1 and 1.19% at redshifts z > 1, while the aggregate precision of the growth measurements is 4.78% over the redshift interval 0 < z < 1 . 5. Using the BAO technique by itself, with no other constraints, SDSS has built up a clear picture of the distance-redshift relationship revealing a clear need for dark energy with a detection significance of 8 σ (Alam et al. 2021). \nThe full cosmological interpretation of these data are described in Alam et al. (2021) and demonstrate the power of BAO for constraining curvature and providing robust estimates of H 0 The analysis also demonstrates the ability of RSD data to complement weak lensing and cosmic microwave background measurements in providing independent evidence for a flat cosmological model with dark energy described by a cosmological constant. The combined BAO and RSD measurements indicate σ 8 = 0 . 85 ± 0 . 03, implying a growth rate that is consistent with predictions from Planck temperature and polarization data (Planck Collaboration et al. 2020) and \nwith General Relativity. Combining these results with Planck , Pantheon Type Ia supernovae (SNe Ia; Scolnic et al. 2018), and weak lensing and clustering measurements (Troxel et al. 2018) from the Dark Energy Survey (DES) leads to significant advances in cosmological constraints relative to the prior generation of experiments. Each of the three parameters, Ω Λ , H 0 , and σ 8 is constrained at roughly 1% precision, even in a model that allows free curvature and a time-evolving equation of state for dark energy. In total, the data are best described by a flat Λ CDM model with H 0 = 68 . 18 ± 0 . 79 kms -1 Mpc -1 (for full details of this analysis see Alam et al. 2021). The Dark Energy Task Force Figure of Merit (Albrecht et al. 2006) of these data sets together is 94 182 . \nNew value-added catalogs derived from eBOSS data were released publicly at the same time as the cosmology results. These catalogs contain the redshifts and weights for each of the LRG, ELG, quasar, and Lyα forest samples, as well as the properties of all quasars observed during the four generations of SDSS. In addition, the mock catalogs used to characterize covariance in the clustering measurements are being released in coordination with DR17. A description of each of these cosmology value-added catalogs is as follows:", '7.1.1. eBOSS Large Scale Structure Catalogs': 'DR16 included full reductions of the completed set of observed eBOSS spectra. An additional series of redshift estimates for the eBOSS galaxy samples was produced by an algorithm known as redrock 183 . The galaxy redshifts derived from redrock are described in Section 4 of Ross et al. (2020). An additional series of classifications and redshift estimates was also performed for the BOSS and eBOSS quasar samples. The origin of quasar redshift estimates is described in Lyke et al. (2020), while a summary of how those redshifts were used in the clustering measurements is also found in Section 4 of Ross et al. (2020). From these updated redshift estimates largescale structure (LSS) VACs are created, which, together map the three-dimensional structure of the Universe using galaxies and quasars over redshifts 0 . 6 < z < 2 . 2. These maps are carefully constructed with corrections for observational systematic errors and random positions that sample the survey selection function to allow unbiased cosmological inference. Three distinct samples were observed by SDSS-IV and used to produce LSS catalogs: LRG (Prakash et al. 2016); ELG (Raichoor et al. 2017); and quasars (Myers et al. 2015). A value-added catalog for each of these samples was released in July 2020. The LSS catalogs for the LRG and quasar samples are described in Ross et al. (2020) while the LSS catalog for the ELG sample is described in Raichoor et al. (2021).', '7.1.2. eBOSS DR16 Large-scale structure multi-tracer EZmock catalogs': 'We present 1000 realizations of multi-tracer EZmock catalogs, with redshift evolution and observational systematics, for each sample of the DR16 LSS data. These \n182 The measurements and cosmological impact are summarised in two web pages https://www.sdss.org/science/ final-bao-and-rsd-measurements/ and https://www.sdss.org/ science/cosmology-results-from-eboss/ \nFig. 8.Left: Extinction-corrected G-band absolute magnitude vs. BP-RP color for all good science targets in the MaStar library. The color coding indicates the median calibrated metallicity. Stars without valid metallicity measurements are marked as red crosses. Right: Median effective temperature vs. median surface gravity for MaStar science targets with valid measurements from at least two of the four groups. Standard stars are also included in the library but are not shown in these plots. \n<!-- image --> \nmock catalogs are generated using the EZmock method (Chuang et al. 2015), and applied the survey footprints and redshift distributions extracted from the corresponding data. They accurately reproduce the two- and threepoint clustering statistics of the DR16 data, including cross correlations between different tracers, down to the scale of a few h -1 Mpc, and provide reliable estimates of covariance matrices and analyzes on the robustness of the cosmological results. Details on the construction and clustering properties of the EZmock catalogs are presented in Zhao et al. (2021).', '7.1.3. eBOSS Quasar Catalog': 'Beginning with SDSS-I, SDSS has maintained a tradition of releasing a visually-inspected quasar (or quasistellar object; QSO) catalog alongside major data releases. The new SDSS-DR16Q catalog (DR16Q; Lyke et al. 2020) represents the most recent, and largest, catalog of known unique quasars within SDSS. To ensure completeness, quasars from previous catalog releases (DR7Q; Schneider et al. 2010, DR12Q; Pˆaris et al. 2017) have been combined with observations from eBOSS in SDSS-IV. The catalog contains data for more than 750,000 unique quasars, including redshifts from visual inspections, principle component analysis (PCA), and the SDSS automated pipeline. Additionally, the catalog is the first from SDSS to contain both the Hewett & Wild (2010) DR6 redshift estimates and the Shen et al. (2011) DR7 redshift estimates that are based on the Hewett & Wild (2010) algorithm. Where applicable, the catalog also contains information about broad absorption line (BAL) troughs, damped Lymanα (DLA) absorbers, and emission line redshifts (via PCA). As in previous releases, DR16Q also contains properties for each quasar from GALEX (Martin et al. 2005), UKIDSS (Lawrence et al. 2007), WISE (Wright et al. 2010), FIRST (Becker et al. 1995), 2MASS (Skrutskie et al. 2006), ROSAT/2RXS (Boller et al. 2016), XMMNewton (Rosen et al. 2016), and Gaia (Gaia Collaboration et al. 2018), when available. To facilitate analyzes of pipeline accuracy and automated classification, a super- \nt was also released. This sample contained ∼ 1 . 4 million unique observations for objects targeted as quasars from SDSS-I/II/III/IV.', '7.1.4. Lymanα Forest Transmission VAC': 'This VAC contains the estimated fluctuations of transmitted flux fraction in the pixels across the Lymanα and Lymanβ spectra region of DR16Q quasars. In total, 211,375 line-of-sights contribute to the Lymanα spectral regions and 70,626 to the Lymanβ one. This VAC contains everything needed to compute the threedimensional auto-correlation of Lymanα absorption in two different spectral regions as in du Mas des Bourboux et al. (2020). When combined with the DR16Q quasar catalog, this VAC also provides the information to compute the three-dimensional quasar × Lymanα crosscorrelation. These two measurements are used to measure the location of the BAO as reported in du Mas des Bourboux et al. (2020).', '7.2. Other VACs based on Single-Fiber Optical Spectra': 'The final eBOSS data sample contains more than one million spectra of stars, galaxies, and quasars obtained during SDSS-IV. This catalog has been used for a range of studies of astrophysical processes beyond the BAO and RSD measurements described above. Demonstrating the impact of these data for additional studies, this release includes a value-added catalog of strong lensing systems (originally released in July 2020 in a mini data release), a new catalog of lensed Lymanα emitting (LAE) galaxies, a new catalog of the cosmic web, and a catalog of metal absorbers. A description of these value-added catalogs is as follows:', '7.2.1. eBOSS Strong Gravitational Lens Detection Catalog': "A value-added catalog of 838 likely, 448 probable, and 265 possible candidate strong galaxy gravitational lens systems was released along with the batch of cosmology results and value-added catalogs in July 2020 (aka DR16+). These systems were discovered by the presence of higher redshift background emission-lines in DR16 \neBOSS galaxy spectra. The methodology, including quantitative explanations of the 'likely', 'probably' and 'possible' categorisation is described in full in Talbot et al. (2021); also see Talbot et al. (2018). This Spectroscopic Identification of Lensing Objects (SILO) program extends the method of the BOSS Emission-Line Lens Survey (BELLS; Brownstein et al. 2012) and Sloan Lens ACS (SLACS; Bolton et al. 2006) survey to higher redshift, and has recently been applied to the spectroscopic discovery of strongly lensed galaxies in MaNGA (SILO; Talbot et al. 2018, also see § 5.5.8). Although these candidates have not been studied through a dedicated imaging program, an analysis of existing imaging from the SDSS Legacy Survey and the DESI Legacy Surveys (Dey et al. 2019) provides additional information on these systems. This catalog includes the results of a manual inspection process, including grades and comments for each candidate, consideration of sky contamination, low signalto-noise emission-lines, improper calibration, weak target emission-lines, systematic errors, Gaussian modeling, and potential lensing features visually identified within the available imaging.", '7.2.2. eBOSS ELG-LAE Strong Lens Catalog': 'The eBOSS ELG-LAE Strong Lens Catalog contains roughly 150 candidate strong lens systems selected from the ELG sample of the eBOSS survey released by DR17 using the method presented in Shu et al. (2016). By construction, the lensing galaxies in this catalog are ELGs up to z glyph[similarequal] 1 . 1, and the source galaxies are Ly α emitters at z > 2. A full description of the catalog will be presented in A. Filipp et al. (in prep.). This catalog is complementary to the eBOSS Strong Gravitational Lens Detection Catalog constructed by Talbot et al. (2021, § 7.2.1 above), in which the source galaxies are all [O ii ] emitters at z < 1 . 7.', '7.2.3. Cosmic Web Environmental Densities from Monte Carlo Physarum Machine': "The Monte Carlo Physarum Machine (MCPM) cosmic web reconstruction algorithm (Burchett et al. 2020; Elek et al. 2021) was employed to characterize the matter density field from galaxies spectroscopically observed in SDSS, including those previously included in the NSA (Blanton et al. 2011) and LOWZ and eBOSS LRG catalogs. The MCPM framework, which is particularly sensitive to the filamentary structure of the cosmic web, requires as inputs galaxy coordinates, redshifts, and halo masses, and we leverage stellar mass measurements included in VACs from previous Data Releases, the NASASloan Atlas itself and the eBOSS Firefly Value-Added Catalog (Comparat et al. 2017). Halo masses were then derived by the halo abundance matching relations from Moster et al. (2013). Using the Polyphorm software (Elek et al. 2020), in which MCPM is implemented, reconstructions were produced in various redshift ranges corresponding to the varying depth/completeness of the galaxy samples. MCPM then provides a proxy metric (known as the trace) for the environmental density at each point in the fitted volume. These arbitrary density values are then calibrated to the cosmological matter density relative to the mean matter density by performing MCPM fits to the Bolshoi-Planck (Klypin et al. 2016) dark matter-only cosmological simulation's halo catalog \n(Behroozi et al. 2013) and producing a mapping between the MCPM trace and simulation 3D matter density field. The data products are: (1) galaxy catalogs with local environmental density values evaluated at their locations from the full 3D density field and (2) the 3D density field itself, which users may in turn query for density values at locations away from known galaxy positions.", '7.3. SPIDERS': 'Within eBOSS, the SPectroscopic IDentification of ERosita Sources (SPIDERS) program, dedicated to the characterization of the X-ray sources, was the largest and most complete spectroscopic follow up of the best all-sky X-ray surveys available at the time (mainly the ROSAT all-sky survey; Clerc et al. 2016; Dwelly et al. 2017; Salvato et al. 2018). As the acronym suggests, the original motivation and plans for SPIDERS were driven by the capabilities of the eROSITA X-ray telescope (on board the Spectrum Roentgen Gamma, or SRG, satellite), which launched on July 13, 2019 (Predehl et al. 2021), significantly later than the planned date at the start of SDSS-IV. While waiting for eROSITA to collect data, the SPIDERS program targeted ROSAT/1RXS and XMMNewton Slew Survey sources within the eBOSS footprint. These targets are expected to make up the bright X-ray flux end of the eROSITA population. Much of the SDSS spectroscopy from the SPIDERS program was released as part of SDSS DR16 (Ahumada et al. 2020), including the largest spectroscopic redshift catalogs of X-ray selected AGN and clusters (Comparat et al. 2020b; Clerc et al. 2020; Kirkpatrick et al. 2021). \nBelow, we highlight some recent preeROSITA SPIDERS results that are based on the SDSS DR16 dataset. Then, in § 7.5 we present a new SPIDERS dataset (to be released as part of SDSS DR17 184 ), dedicated to the spectroscopic follow-up of X-ray sources discovered in early SRG/ eROSITA performance verification observations. In § 7.5.6 we discuss future plans for eROSITA followup in SDSS-V (also see § 8).', '7.4. SPIDERS Galaxy Cluster Highlights': 'In this section we describe the SPIDERS cluster sample, the largest catalog of spectroscopically confirmed Xray detected clusters to date. While no new data is released in DR17, several new results have come out since DR16. We refer the reader to the below papers for details of the ancillary data created and how to access it. \nThe bulk of the galaxy cluster population targeted by SPIDERS up to and including DR16 originated from the ROSAT-based CODEX survey (Finoguenov et al. 2020), with additions of low-mass systems from the XMMNewton-based X-CLASS catalog. The initial SPIDERS CODEX and X-CLASS targeting plans and survey strategy were described by Clerc et al. (2016). Clerc et al. (2020) provide complementary details of targeting updates that occurred during the course of the project. Clerc et al. (2020) also describe the full sample of acquired targets, the achieved spectral quality, as well as the redshift measurement success rate (nearly 98%) and precision (20 km s -1 at z = 0 . 2 after averaging all galax- \nies in a cluster). In particular, Clerc et al. (2020) highlight the very homogeneous nature of targets observed throughout the program; this specific feature supports a minimally biased spectroscopic follow-up of clusters. \nGalaxy clusters confirmed and validated with spectroscopy from SPIDERS are presented in Kirkpatrick et al. (2021, 2740 CODEX clusters) and Koulouridis et al. (2021, 124 X-CLASS clusters). These two catalog papers list confirmed clusters up to z glyph[lessorsimilar] 0 . 6, and present derived X-ray properties that exploit high precision cluster redshifts, obtained by averaging multiple SPIDERS member galaxy spectra per cluster. Kirkpatrick et al. (2021) also provide velocity dispersion estimates for those clusters with more than 15 members, confirming correlations with X-ray luminosity and optical richness found by other studies. A galaxy cluster number count analysis performed in the velocity dispersion-redshift parameter space yields constraints on cosmological parameters Ω m and σ 8 consistent with the main cosmological analysis of the sample. The main cosmological analysis of the SPIDERS clusters program is presented by Ider Chitham et al. (2020); it involves the modeling of the redshiftrichness relation of galaxy clusters spectroscopically confirmed by SPIDERS. The best-fit parameters are found to be Ω m = 0 . 34 +0 . 09 -0 . 05 and σ 8 = 0 . 73 ± 0 . 03. As a central ingredient in those cosmological studies, the calibration of the mass-observable relations is presented in two papers: Capasso et al. (2019) focus on the mass-richness relation and Capasso et al. (2020) on the mass-X-ray luminosity relation. The large scale clustering of clusters - highly biased tracers of the underlying dark matter density field - is presented and discussed by Lindholm et al. (2021). Taking advantage of the dense and precise spectroscopic sampling of cluster members, Mpetha et al. (2021) extract the gravitational redshift signal from stacks of SPIDERS confirmed galaxy clusters. Brightest SPIDERS cluster galaxies are studied by Furnell et al. (2018) and Erfanianfar et al. (2019), in particular, the scaling of their stellar mass with the host halo properties is shown to lack evolution since z ∼ 0 . 6.', '7.5. The eROSITA Final Equatorial-Depth Survey': "During the SRG performance verification phase (November 2019), eROSITA was used to survey a contiguous region of ∼ 140 deg 2 . This eROSITA Final Equatorial-Depth Survey (eFEDS; Brunner et al. 2021; Liu et al. 2021c) is intended as an early representative demonstration of the capabilities of the eROSITA allsky survey, which will not be completed until late 2023. In order to exploit the availability of these X-ray data, we allocated a dedicated set of 12 SDSS-IV/SPIDERS plates to follow up counterparts of eFEDS X-ray sources, to be observed at APO during the first quarter of 2020. These special eFEDS plates, documented here, are released as part of SDSS DR17. Unfortunately, because of the COVID19-related closure of APO (see § 1), only a small fraction of the originally designed plates were observed, most to shallower than expected depth. Nevertheless, this program did collect useful data, and the experience aqcuired has greatly helped the preparation of the eROSITA followup component of the SDSS-V Black Hole Mapper program (Kollmeier et al. 2017, also see § 8). \nThe target selection procedure for eFEDS broadly followed the steps carried out for follow up of ROSAT sources in SPIDERS (Clerc et al. 2016; Dwelly et al. 2017). First, the X-ray source detection process (described in detail by Brunner et al. 2021), provides a catalog, including parameters that describe the detection likelihood, extent likelihood (to distinguish point sources from extended ones), and astrometrically corrected Xray position and positional uncertainty. For the SDSS-IV targeting of eFEDS sources, we used the preliminary data processing software version available at that date (late 2019, 'c940'), and only considered eFEDS sources with detection likelihood > 8, so that the fraction of spurious detections is kept to a relatively low rate. Based on simulations (Comparat et al. 2019, 2020a; Liu et al. 2021b), we expect only around 2% of these X-ray detections to be spurious. Optical/IR counterparts were associated with the X-ray sources using the methods described below. Finally, each of these counterparts is identified as being in one or more target classes (recorded using flags in the targeting bitmask, Sec. 7.5.4), and then the final target list is created which includes only objects suited for spectroscopic observations with the BOSS spectrographs.", '7.5.1. eFEDS: Optical counterparts to extended sources': 'The identification of SPIDERS clusters and their membership, as well as the detailed target selection process are described fully by A. Merloni et al. (in prep.) and J. Ider Chitham et al. (in prep.). Here we document the resulting targeting bits that describe the contents of the SPIDERS cluster target catalog: \n- 1. EFEDS EROSITA CLUS : These are galaxies associated with extended X-ray emission. A combination of all member galaxies selected using redMaPPer in scanning-mode (Rykoff et al. 2014) as well as additional BCGs identified using the multicomponent matched filter cluster confirmation tool (Klein et al. 2019). BCGs are allocated the highest target priority (0), while all other members are given a lower priority (33-130). The selection method on eFEDS extended sources is described fully in Liu et al. (2021a); Klein et al. (2021). The redMaPPer based selection uses positional X-ray information to estimate optical properties such as photometric redshifts, centering information and member selection in a method analogous to that of as described by Clerc et al. (2016), as used in the SPIDERS DR16 data release. The method itself is described by Ider Chitham et al. (2020), J. Ider Chitham (in prep.). The photometric selections of BCGs and cluster members are based on the eighth data release of the DESI Legacy Imaging Surveys (using g -r and r -z colors) as well as HSC photometric ( grizy ) data (Dey et al. 2019; Aihara et al. 2019). Cluster BCG photometric redshifts are within 0 . 0 < z < 1 . 3, a total of 85 were observed.\n- 2. EFEDS SDSS REDMAPPER : Publicly available optically selected cluster member catalog based on SDSS DR8 and CODEX data (Rykoff et al. 2014; Finoguenov et al. 2020). Cluster photometric redshifts are within 0 . 0 < z < 0 . 6, a total of 1031 potential cluster galaxies were observed. \n- 3. EFEDS HSC REDMAPPER : Optically selected cluster member catalog (Rykoff et al. 2014) based on HSC DR2 data (Aihara et al. 2019). Cluster photometric redshifts are within 0 . 3 < z < 1 . 1, a total of 13 cluster galaxies were observed.', '7.5.2. eFEDS: Optical counterparts to point sources': 'The method used to associate optical/IR counterparts to the eFEDS X-ray point-sources is presented by Salvato et al. (2021). In short, the optical counterparts to the X-ray sources were identified using a Bayesian association algorithm NWAY (Salvato et al. 2018), which accounts for both position and photometric properties of the counterpart. The association algorithm was informed by a large training sample of XMM-Newton and Chandra X-ray serendipitous sources with secure optical identifications. Supplementary optical/IR counterparts were also provided via a Likelihood Ratio approach (Sutherland & Saunders 1992; Ruiz et al. 2018). Note that at the time of selection of targets for the SDSS-IV special plates, only very early versions of the X-ray catalog and counterpart selection algorithms were available - so the pool of targets is different from the counterparts presented by Salvato et al. (2021). The main eFEDS point-like target class is EFEDS EROSITA AGN with 2149 observed targets. Despite the name, we expect this target set to also include a number of stars and (inactive) galaxies.', '7.5.3. eFEDS: additional targets': 'To ensure that all fibers available in the designed plates could be allocated to interesting scientific targets, beyond the eROSITA -selected sources described above, several additional target classes were defined before the arrival of the eROSITA data. The largest of these were special classes of AGN drawn from optical targets selected by a combination of the HSC Subaru Strategic Program (SSP) imaging and other multi-wavelength catalogs. Note that due to the high sky density seen in the available eROSITA /eFEDS source catalog, very few fibers were left for filler targets, and so most of the defined target classes were allocated zero fibers (unless the same astrophysical object was also selected as an eROSITA target). However, since these target selection classes are defined in the DR17 data products, for completeness of the documentation, we give a brief description here of each target class. \n- 1. EFEDS HSC HIZQSO : Optical/NIR High-redshift QSO candidates selected from HSC photometric data witha drop-out technique targeting z > 4.\n- 2. EFEDS HSC WERGS : FIRST-detected bright radio galaxies selected for their extreme red Optical-NIR colors\n- 3. EFEDS HSC REDAGN : WISE-detected obscured AGN candidates selected on the basis of their extremely red Optical-NIR colors.\n- 4. EFEDS COSMOQSO : UV excess/variability selected QSO - following prototype selection criteria of 4MOST Cosmology Survey.\n- 5. EFEDS WISE AGN : AGN candidate selected via location in WISE color-magnitude space, following \ncriteria of Assef et al. (2018) - with optical counterparts selected from DESI Legacy Imaging Survey DR8. \n- 6. EFEDS WISE VARAGN : AGN candidate selected via variability signature in WISE data - with optical counterparts selected from DESI Legacy Imaging Survey DR8.\n- 7. EFEDS XMMATLAS : Optical counterparts to X-ray sources from the 3XMM/dr8 catalog (Rosen et al. 2016) and from the XMM-ATLAS survey field (Ranalli et al. 2015).\n- 8. EFEDS CSC2 : X-ray sources from the ChandraCSC2.0 catalog (Evans et al. 2010) - with optical counterparts selected from DESI Legacy Imaging Survey (DR8).\n- 9. EFEDS SDSSWISE QSO : QSO candidates from the Clarke et al. (2020) ML classification of SDSSWISE photometric sources\n- 10. EFEDS GAIAWISE QSO : QSO candidate from the Shu et al. (2019) Random Forest classification of Gaia -unWISE sources\n- 11. EFEDS KIDS QSO : QSO candidate from the Nakoneczny et al. (2019) classification of KiDS+VIKING DR3 sources\n- 12. EFEDS GAIA WD : Gaia -selected WD binary candidate in the region between the zero age main sequence and the WD sequence.\n- 13. EFEDS GAIAGALEX WD : Gaia -GALEX UV excess source - expected to be compact binary harbouring a WD\n- 14. EFEDS VAR WD : WD candidate selected on the basis of its optical variability \nFurther details of the eFEDS target selection process will be presented in A. Merloni et al. (in prep.).', '7.5.4. Target classes, target bits': 'The targets described above for the eFEDS field are assigned to 19 different target classes. Table 5 describes each of the target classes designed for the 12 plates covering eFEDS. The maskbits used range from 19 to 47 and can be found in the EBOSS TARGET1 target bitmask. For clarity, only the bits for target classes that resulted in observations are reported.', '7.5.5. eFEDS Observations': "Observations were carried out in March 2020, but due to a combination of poor weather and Covid-19 shut down (on MJD 58932), the program could unfortunately not be completed. The initial success criteria for a plate to be considered complete were SN2 G1,2 > 20, SN2 I1,2 > 40, where the SN2 are estimates of the squared SNR, at fiducial 2 '' fiber magnitudes of g = 21 . 2 and i = 20 . 2, averaged over targets in each of the four BOSS spectrograph arms (G1, I1, G2, I2). These SN2 thresholds are approximately double those adopted for regular eBOSS observations (Dawson et al. 2016). This \nTABLE 5 \nDifferent target classes created for the SDSS-IV/eFEDS plates. \nTable 6 details the plate numbers and the meta data related to their observations. Only seven out of 12 plates were actually observed in the period MJD 58928-58932; of these seven, only one plate reached nominal depth, three received significant exposure beyond the typical eBOSS limit, and three were only partially exposed. In Table 6 these are marked as good (G), fair (F) and bad (B), respectively. \nThese bits are located within the EBOSS TARGET1 targeting bitmask. In the column 'N', we report the number of observed targets falling in a category. Note that targets can belong to several categories. \nguarantees a highly complete redshift measurements for faint AGN, and allows the measurement of the properties of the AGN and of the host galaxy. \nThe redshifts from eFEDS special plates are part of the catalog that describes the multiwavelength properties of the eROSITA point sources (Salvato et al. 2021), currently available at https://erosita.mpe.mpg.de/edr/ eROSITAObservations/Catalogues/ and expected to be available in Vizier in the future.", '7.5.6. eFEDs in SDSS-V': "The spectroscopic observations of the eROSITA sources in the eFEDS PV field, reduced at the end of the SDSS-IV program because of the March 2020 closure at APO, has been resumed within the early (plate-mode operations) phase of SDSS-V (see § 8). A much more complete description of the eFEDS program (SDSS-IV and SDSS-V) will be provided in A. Merloni et al., (in prep.). Among others, the more homogeneous coverage of the combined program will enable X-ray AGN clustering studies and galaxy-galaxy lensing measurements (J. Comparat et al. in prep) on an unprecedentedly large sample. \nEventually, with the commissioning of the robotic fibre positioners on both northern and southern SDSSV sites, SPIDERS will deliver on its original promises of massive, systematic spectroscopic observations of the sources detected in the eROSITA all-sky survey as part of the SDSS-V 'Black Hole Mapper' (BHM) program (Kollmeier et al. 2017). The experience and the scientific results obtained by the SPIDERS team within SDSS-IV, that we have briefly described above, represent a major \nmilestone for the planning, execution and exploitation of the BHM survey program.", '7.6. eBOSS-RM': "The Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project is a dedicated multi-object optical reverberation mapping program (Shen et al. 2015b) that has monitored a single 7 deg 2 field (R.A. J2000=213.704, decl. J2000=+53.083) since 2014 during both SDSS-III and SDSS-IV, using the BOSS spectrographs (Smee et al. 2013) at APO. The first-season SDSS-RM spectroscopic data were taken during January-July 2014 in SDSS-III (Eisenstein et al. 2011) and consist of a total of 32 epochs with an average cadence of ∼ 4 days; each epoch had a typical exposure time of 2 hr. The SDSS-RM program continued in SDSS-IV (Blanton et al. 2017), with ∼ 12 epochs per year (2 per month) with a nominal exposure time of 1 hr each during 2015-2017, and ∼ 6 epochs per year (monthly cadence) during 2018-2020. As of July 2020, SDSS-RM has obtained a total of 90 spectroscopic epochs over a spectroscopic baseline of 7 years (20142020). All SDSS spectroscopic data (pipeline reduced and calibrated) from SDSS-RM are included in the data releases of SDSS-III and SDSS-IV. The full technical details of SDSS-RM are provided in Shen et al. (2015b). \nIn addition to optical BOSS spectroscopy, accompanying photometric data in the g and i bands are acquired with the 3.6-m Canada-France-Hawaii Telescope (CFHT) and the Steward Observatory 2.3-m Bok telescope (Kinemuchi et al. 2020). The final photometric baseline spans 11 years (2010-2020) when including the 2010-2013 photometric light curves from the PanSTARRS 1 (Kaiser et al. 2010) Medium Deep survey that covers the entire SDSS-RM field. The SDSS-RM sample includes 849 broad-line quasars with i PSF < 21 . 7 and 0 . 1 < z < 4 . 5 without any constraints on quasar properties. The detailed sample properties are described in Shen et al. (2019b). \nThe primary science goal of SDSS-RM is to measure reverberation mapping lags of different broad emission \nTABLE 6 Summary of eFEDS observations. \nPlate: plate number. MJD: Modified Julian Date of the observations. RA: Right Ascension of the center of the plate (degrees). Dec: Declination of the center of the plate (degrees). Quality: G: good; F: fair; B: bad. SN2: signal to noise reached in each spectrograph's arm. \nlines covered by optical spectroscopy across the full range of quasar luminosity and redshift probed by the sample. SDSS-RM has successfully measured short ( < 6 months) lags for the low-ionization broad lines (e.g., H α , H β , and Mg II ; Shen et al. 2016a; Grier et al. 2017) based on the 2014 data. Grier et al. (2019) reported results on C IV lags using the first four years (2014-2017) of imaging and spectroscopy from SDSS-RM (and for Mg II lags in Homayouni et al. 2020), where the lags are typically longer than one season in the observed frame. Lag measurements based on more extended light curves are reported in Shen et al. (2019a), and the analyzes of the final SDSS-RM dataset are underway. \nIn addition to the main science goal of RM measurements, SDSS-RM also enables a diverse range of quasar science. Some notable examples are: measurements of continuum lags to constrain quasar accretion disk sizes (Homayouni et al. 2019); constraints on the black hole mass - host galaxy correlations at z > 0 . 3 (Shen et al. 2015a; Matsuoka et al. 2015; Yue et al. 2018; Li et al. 2021); spectral and variability properties of quasar emission and absorption lines (Shen et al. 2016b; Sun et al. 2018; Hemler et al. 2019; Wang et al. 2019, 2020); and extreme variability in quasars (Dexter et al. 2019). \nThe SDSS-RM field continues to be monitored with optical imaging and spectroscopy as part of the BHM program in SDSS-V (Kollmeier et al. 2017, and § 8 below). With more extended light curves, this program will be able to measure broad-line RM lags for the most luminous quasars at high redshift.", '8. CONCLUSIONS AND FUTURE PLANS': 'This paper documents the final data release from the SDSS-IV collaboration, and the seventeenth data release from SDSS programs as a whole (DR17). With this paper we make all SDSS-IV data public, concluding the program described in Blanton et al. (2017). In the rest of this section, we give an update of SDSS-V which is now actively observing.', '8.1. SDSS-V': "As SDSS-IV was wrapping up observations (in late 2020 at APO and early 2021 at LCO), the next generation of SDSS began - SDSS-V 185 (Kollmeier et al. 2017), with its first data taken using the existing plug \nplate system at APO in late 2020. Building on the legacy of earlier generations, SDSS-V is a multi-epoch spectroscopic survey to observe nearly six million sources using the existing BOSS/APOGEE spectrographs and 2.5 m telescopes, as well as very large swathes of the interstellar medium (ISM) in the Milky Way and Local Group using new optical spectrographs and small telescopes. SDSS-V operates at both APO and LCO and will provide the first 'panoptic' spectroscopic view of the entire sky, spanning a wide variety of target types, observing cadences, and science goals. \nThe scientific program is overseen by three 'Mappers': \n- 1. The Milky Way Mapper (MWM) is targeting millions of stars and stellar remnants with both the APOGEE and BOSS spectrographs, probing stellar populations from the immediate solar neighborhood across the MW, to the far side of the Galactic disk and in the MW's satellite companions. The MWM's primary goals are to explore (1) the formation and evolution of the MW, (2) the interior physics and evolutionary pathways of stars across all T eff regimes, from combined asteroseismology and spectroscopy (e.g. Aerts 2021), and (3) the architecture of multi-star and planetary systems.\n- 2. The Black Hole Mapper (BHM) is targeting nearly half a million accreting SMBHs and other X-ray sources, including newly discovered systems from the SRG/eROSITA mission, with the optical BOSS spectrographs. The BHM seeks to characterize the X-ray sky, improve our understanding of accretion physics, and trace the evolution and impact of supermassive black holes across cosmic time.\n- 3. The Local Volume Mapper is using a wide-field optical IFU, with new optical spectrographs (R ∼ 4000, λ = 3600 -9800 ˚ A) fed by a 16cm telescope, to map ∼ 2800 deg 2 of sky. With their tens of millions of spectra sampling the ISM and embedded stellar populations in the MW and satellite galaxies, the LVM's maps will reveal the physics of star formation and the ISM, the complex interplay between stars and the ISM, and the connections between interstellar processes on parsec-sized to galaxy-wide scales. \nSDSS-V expands upon the operational infrastructure and data legacy of earlier SDSS iterations with sev- \neral key developments. Among these are the installation of robotic fiber positioners in the focal planes of both 2.5 m telescopes at APO and LCO, which replace the now-retired SDSS plug plate system. In comparison to earlier SDSS surveys, these focal plane systems (FPS) enable improved observing efficiency, larger target densities, and more complex observing cadences for the MWM and BHM programs. In addition, the LVM is constructing additional small telescopes at LCO (with longer-term plans to expand to APO) that are linked to new optical spectrographs based on the DESI design (Martini et al. 2018). SDSS-V is continuing the decadeslong SDSS legacy of open data policies and efficient, welldocumented public data access, boosted with the development of improved data distribution systems to serve its expansive time-domain, multi-object and integral-field data set to the world. The eighteenth data release of the SDSS (DR18), which will include the first SDSS-V data, mostly targeting and other information, is currently anticipated for 2022. \nSDSS-V will also build on the rich educational legacy established by previous SDSS generations. This will include continuing to distribute as many SDSS plug plates to schools and science centers as possible, and broadening the reach of the SDSS Voyages suite of educational activities. SDSS's large online datasets are an ideal resource for supporting hands-on activities using real archival data, at a wide range of ability levels, and have the potential to act as a gateway for numerous data science and analysis topics. \nAfter twenty-one years of Sloan Digital Sky Surveys, the data released by SDSS-IV in DR17 has made significant contributions to our understanding of the MW, galaxy evolution, and the Universe as a whole, and it will continue to enable new research for years to come. The SDSS-IV project is now complete, but the new technology and exciting new surveys coming as part of SDSSV mean that the future remains bright for the SDSS legacy.", '9. ACKNOWLEDGEMENTS': "Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. \nSDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Center for Astrophysics - Harvard & Smithsonian, Instituto de Astrof'ısica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut fur Astrophysik Potsdam (AIP), Max-Planck-Institut fur Astronomie (MPIA Heidelberg), Max-Planck-Institut fur Astrophysik (MPA Garching), Max-Planck-Institut fur Extraterrestrische Physik (MPE), National Astronomical Observatories of \nChina, New Mexico State University, New York University, University of Notre Dame, Observat'ario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut'onoma de M'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \nCo-authorship on SDSS-IV data papers is alphabetical by last name and offered to all collaboration members who have contributed at least 1 month FTE towards any of the surveys during the period up to the end of data collection; as well as to any formally approved external collaborator who has contributed at least 1 month FTE to work critical to the data release. \nThe documentation workshop for DR17 was held over Zoom ('DocuZoom') in May 2021. This event was the main venue for documentation updates for DR17 (including the start of this paper) and was attended by AnneMarie Weijmans, Joel Brownstein, Mike Blanton, Karen Masters, Renbin Yan, Sten Hasselquist, Michael Talbot, Hannah lewis, Christian Hayes, Ani Thaker, Gail Zasowski, David Law, Brian Cherinka, Kyle Westfall, Amy Jones, Lewis Hill, Jon Holtzman, Jos'e S'anchez-Gallego, Rachael Beaton, Scott Anderson, Jennifer Johnson, Caroline Swartz, Jordan Raddick, Julie Imig and a Llama. \nFigure 1 made by Joel Browstein; Figure 2 and 3 made by Christian Hayes; Figure 4 made by Jon Holtzman. Figure 5 by Jos'e S'anchez-Gallego; Figure 6 made by Kyle Westfall; Figure 7 made by Brian Cherinka; Figure 8 made by Renbin Yan; \nThis publication uses data generated via the Zooniverse.org platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. \nThis publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. \nThis work is based, in part, on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. \nThis publication makes use of data products from the Wide-field Infrared Survey Explorer (Wright et al. 2010), which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration \nThis work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/ dpac/consortium ). Funding for the DPAC has been \nprovided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \nThis research has made use of NASA's Astrophysics Data System. \nThis research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. The original description of the SIMBAD service was published in Wenger et al. (2000). \nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: \n10.26093/cds/vizier). The original description of the VizieR service was published Ochsenbein et al. (2000). \nThis research makes use of data from the Green Bank Observatory. 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2024arXiv240909181O

Starting from the WheelerDeWitt equation for the Schwarzschild black hole interior which is derived from a Hamiltonian formulated in terms of canonical phase space coordinates we show that by applying a simple reparametrization this equation can be expressed as the eigenvalue equation of a quantum linear harmonic oscillator. Within the standard quantization framework we find that the resulting wave function diverges in the region of the classical singularity and the expectation value of the Kretschmann scalar is undefined for all states within the black hole. However when we apply the minimal uncertainty approach to the quantization process we obtain a wave function that is both welldefined and squareintegrable. Additionally the expectation value of the Kretschmann scalar for these states remains finite throughout the black holes interior suggesting that the classical singularity is resolved in this approach replaced it by a minimum radius.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.09181', 'arXiv:2409.09181', '2024arXiv240909181O']

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

Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09181.pdf

{'Octavio Obregón, a Wilfredo Yupanqui. a': '- a Departamento de Física, División de Ciencias e Ingenierías, Universidad de Guanajuato Loma del Bosque 103, León 37150, Guanajuato, México. \nE-mail: octavio@fisica.ugto.mx , w.yupanquicarpio@ugto.mx', 'Abstract:': "Starting from the Wheeler-DeWitt equation for the Schwarzschild black hole interior, which is derived from a Hamiltonian formulated in terms of canonical phase space coordinates, we show that by applying a simple reparametrization, this equation can be expressed as the eigenvalue equation of a quantum linear harmonic oscillator. Within the standard quantization framework, we find that the resulting wave function diverges in the region of the classical singularity, and the expectation value of the Kretschmann scalar is undefined for all states within the black hole. However, when we apply the minimal uncertainty approach to the quantization process, we obtain a wave function that is both well-defined and square-integrable. Additionally, the expectation value of the Kretschmann scalar for these states remains finite throughout the black hole's interior, suggesting that the classical singularity is resolved in this approach, replaced it by a minimum radius.", '1 Introduction': "A quantum black hole is a theoretical concept that arises from the formulation of a quantized Hamiltonian, which is defined within the framework of general relativity. This concept is central to the ongoing efforts to develop a theory of quantum gravity, which would reconcile the two fundamental theories of modern physics: quantum mechanics and general relativity. \nSeveral approaches have been explored to investigate the interior of a black hole within the framework of quantum gravity. For instance, in Loop Quantum Gravity (LQG) [1], which is one of the leading non-perturbative approaches to the quantization of gravity, numerous studies have been conducted on both the interior and the entire spacetime of black holes. In these studies, the Hamiltonian describing the black hole's interior, expressed in terms of Ashtekar-Barbero variables, is quantized using the socalled polymer quantization [2-7]. This quantization method effectively introduces a parameter that establishes a minimal scale in the model, which allows for the avoidance of the singularity and also introduces a bounce from a black hole to a white hole in the vacuum case [8-14]. Polymer quantization also induces certain modifications in the algebra of the theory at the quantum level, which can be viewed as an effective modification of the classical algebra. Other methods for quantizing the interior of black holes can be found in the works [15-24]. \nWhen gravity is incorporated into quantum measurement processes, led to the generalization of Heisenberg's uncertainty relation (HUR). This modification to the HUR is known as the Generalized Uncertainty Principle (GUP), which implements a minimal uncertainty in the position by modifying the ordinary uncertainty relation of quantum mechanics to accommodate deformations at high energies, typically at the Planck scale [25]. On the other hand, GUP can be understood as an alternative quantization procedure that imposes a modified commutator between the position and a generalized momentum (or between generalized position and momentum [25], depending on the chosen representation) [26, 27], resulting in a minimum uncertainty in position or momentum [28]. In [29], it is demonstrated that the GUP arises from the consideration of non-extensive entropies that depend only on the probabilities. GUP is also derived from different proposals: in [30], the scattering of strings at ultra-high energies is considered to analyze the divergences of quantum gravity at the Planck scale; in [31], a gedanken experiment is proposed to measure the area of the apparent horizon of black holes in the context of quantum gravity; and [32] explores the idea that spacetime in the Planck region fluctuates, leading to the possibility of virtual micro-black holes affecting the measurement process. \nAs mentioned, the effects of GUP are significant in systems with energies close to the Planck scale. Particularly relevant examples of such systems include the early universe and the interiors of black holes, where quantum gravity effects are expected to dominate [33-36]. Therefore, quantum cosmology, the branch of physics that studies these systems, is the appropriate field where this modified quantization rule is expected to have a considerable impact. In this context, taking advantage of the fact that the interior of a Schwarzschild black hole is isometric to the Kantowski-Sachs cosmological model, for the first time, the quantization based on the minimal uncertainty approach has been applied to the minisuperspace variables that describe the dynamics inside the black hole [37]. This implies a modification of the Wheeler-DeWitt equation, which governs the quantum cosmological model, thereby characterizing a modified dynamic of the solution. \nFollowing this line of applying the minimal uncertainty approach to Quantum Cosmology, several works have been published. In [38], the classical Hamiltonian of the Schwarzschild black hole interior is considered within the Ashtekar-Barbero connection formalism. Inspired by models based on the Generalized Uncertainty Principle, the canonical algebra of the model is deformed, leading to the derivation of the effective dynamics. This deformation results in the resolution of the black hole singularity by introducing a minimum nonzero radius for the infalling two-spheres. Recently, in [39], starting from the proposal of a new reduced Hamiltonian, the classical black hole singularity is resolved by replacing it with an effective bounce that connects the interior \nof a black hole with the interior of a white hole. This bounce occurs in the region near the Planck scale, where a new event horizon emerges. Crossing this horizon changes the nature of the interval from spatial to temporal outside the white hole. Finally, in [40], the interior of a Schwarzschild black hole was quantized using the minimal uncertainty approach suitable for the Ashtekar-Barbero connection variables. As a result, it was found that all interior states remain well-defined and square-integrable. Moreover, the expectation value of the Kretschmann scalar remains finite throughout the entire interior region of the black hole, particularly in the area where the classical singularity used to reside, indicating the resolution of the black hole singularity. Additionally, a minimum value for the radius of the 2-spheres was also identified. \nIn this work, we focus on the Wheeler-DeWitt equation derived in [41] from a Hamiltonian that describes the spherically symmetric spacetime within the interior of the Reissner-Nordström black hole. The phase space characterizing this spacetime is parameterized by the charge Q , the mass M , and their respective conjugate momenta. Through a canonical transformation, configuration variables are obtained that naturally describe the dynamical properties of the black hole's interior. In [42], the simplest case of this model is considered, taking Q = 0 , and interestingly, by reparametrizing the black hole's radial coordinate, the eigenvalue equation of a linear harmonic oscillator is obtained. As a result, it is found that the area spectrum, and therefore the Schwarzschild radius, is discrete and proportional to the square of the Planck length [41, 43]. Upon solving the eigenvalue equation for the black hole, modeled as a linear harmonic oscillator, we find that the black hole singularity persists in the standard quantization model. To address this issue, we introduce quantization under the minimum uncertainty approach, which resolves the singularity by imposing a minimum radius on the 2-sphere. Furthermore, this approach modifies the area spectrum and the black hole's radius. \nThe paper is organized as follows: In section 2, we provide a brief overview of the interior of the Schwarzschild black hole from the Hamiltonian perspective derived in [41, 43]. In section 3, we apply an appropriate transformation to express the eigenvalue equation for the black hole mass as a quantum harmonic oscillator-like eigenvalue equation. Then, following the standard quantization procedure, i.e., using the usual commutation relation between canonical variables, we quantize the interior of the black hole. Section 4 is dedicated to the quantization of the black hole interior using the minimal uncertainty approach, in which the usual commutation relation is modified. We demonstrate that the resulting wave function is finite within the black hole, as is the expectation value of the Kretschmann scalar. Finally in section 5 we summarize our results and conclude.", '2 Hamiltonian description of the black hole interior': 'The Reissner-Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M . In spherical coordinates ( t, r, θ, φ ) , this metric is \nds 2 = -( 1 -R s r + R 2 Q r 2 ) c 2 dt 2 + ( 1 -R s r + R 2 Q r 2 ) -1 dr 2 + r 2 d Ω 2 , (2.1) \nwhere R s = 2 GM/c 2 is the Schwarzschild radius and R 2 Q = Q 2 G/ 4 πϵ 0 c 4 the characteristic length scale, in which Q is the electric charge. In the limit that the charge Q (or equivalently, the length scale R Q ) goes to zero, one recovers the Schwarzschild metric \nds 2 = -( 1 -R s r ) c 2 dt 2 + ( 1 -R s r ) -1 dr 2 + r 2 d Ω 2 , (2.2) \nwith r ∈ (0 , ∞ ) being the radial coordinate. It is well-known that upon crossing the event horizon of the Schwarzschild black hole which is located at R s , the timelike and spacelike curves switch their causal nature. This is because when an object crosses the event horizon of a Schwarzschild black hole, its timelike trajectory inside the event horizon becomes spacelike, and it can no longer escape the gravitational attraction of the black hole. So, one can obtain the interior metric by switching t ↔ r in (2.2) [44] \nds 2 = -( R s t -1 ) -1 dt 2 + ( R s t -1 ) dr 2 + t 2 d Ω 2 . (2.3) \nHere t is the Schwarzschild time coordinate which has a range t ∈ (0 , R s ) in the interior. Consider the interior solution as measured by an observer at rest relatively to the space coordinates, i.e., dr = d Ω = 0 . In this case, from the metric (2.3), we have [44] \n( dt dτ ) 2 = R s t -1 , (2.4) \nwhere τ is the proper time. \nIn [41, 45], a classical Hamiltonian corresponding to the metric in equation (2.1) is derived from the Einstein-Maxwell action. The constraint equations derived in this model allow us to define the Hamiltonian of the Reissner-Nordström black hole in terms of the variables m and q , which can be identified as the mass M and the charge Q of the hole. These variables, along with their conjugate canonical momenta p m and p q , form a finite phase space. If the charge Q is fixed as an external parameter (in our case, we set Q = 0 ), it is possible to perform a canonical transformation from the phase space \nvariables ( m,p m ) to the new phase space variables ( a, p a ) , which naturally describe the dynamic properties inside the Reissner-Nordström black hole. Consequently, the classical Hamiltonian, in terms of the variables a and p a , takes the form [42] \nH = p 2 a 2 a + 1 2 a, (2.5) \nwhere the new variable a and its conjugate momentum p a satisfy the classical algebra \n{ a, p a } = 1 . (2.6) \nAs noted in [41], the geometric interpretation of a is related to the radial coordinate of the black hole. In our case, where an uncharged hole ( R Q = 0 ) is considered, we conclude that 0 < a < R s . Therefore, we interpret the variable a as describing the dynamics exclusively within the interior of the Schwarzschild black hole. \nSince the Hamiltonian, (2.5), describing the dynamic properties of the interior of the Schwarzschild spacetime is known, we can proceed with the canonical quantization of the spacetime in question. For this process, we choose a Hilbert space of the type L 2 ( R n , d x ) , in which the variable a acts as a multiplicative operator and its conjugate momentum p a is represented as a differential operator, p a = -i∂/∂a . In [42], a simplified form of the Hamiltonian is considered by setting Q = 0 , and the full-dimensional form of its differential equation is derived \nℏ 2 G 2 c 6 a -s -1 d da ( a s d da ) Ψ( a ) = ( a -R s ) Ψ( a ) , (2.7) \nwhere s is an arbitrary factor ordering parameter. It is important to note that (2.7) is a Wheeler-DeWitt-type equation. Therefore, the wave function in this model represents a quantum black hole. Additionally, the phase space coordinates represented by the pair ( a, p a ) are not affected at the quantum level by their classical dependence on time t or radius r . Thus, they can be interpreted as the variables of a minisuperspace. \nThe equation of motion for a , in the case where Q = 0 , is given by [41] \n˙ a 2 = R s a -1 , (2.8) \nand by comparing this expression with (2.4), it becomes evident that the Schwarzschild time, t , and the dynamical variable, a , exhibit similar behavior. Therefore, we can express the interval (2.3) in terms of a \nds 2 = -( R s a -1 ) -1 da 2 + ( R s a -1 ) dr 2 + a 2 d Ω 2 . (2.9) \nThis justification is based on the fact that both variables t and a are confined to the interior of the black hole, where t ∈ (0 , R s ) and a ∈ (0 , R s ) . Furthermore, they describe the interior dynamics of the black hole in the same manner, as can be verified by comparing equations (2.4) and (2.8). \nA way of detecting singularities is to find where the energy density or the spacetime curvature become infinite and the usual description of the spacetime breaks down. However, to be sure that there is an essential singularity which cannot be transformed away by a coordinate transformation, invariants are constructed from the curvature tensor. These quantities, particularly the Kretschmann scalar K = R µναβ R µναβ , are coordinate invariant measures of whether a singularity exists in some region, given that the said region is not an infinite affine parameter away. For the black hole region, given by the metric (2.9), the scalar Kretschmann polynomial, K , is given by \nK = 48 G 2 M 2 a 6 . (2.10) \nOn the horizon a → R s = 2 GM and K = 3 4 G 4 M 4 , which indicates that in this region there is no real singularity. In a → 0 we have that K →∞ indicating the presence of a physical singularity there.', '3 Black hole as harmonic oscillator': "It is interesting and somewhat surprising to note that the Wheeler-DeWitt equation for a Schwarzschild black hole, as expressed in (2.7), can be reparametrized as follows [42] \nψ ( a ) = U ( a ) a , x = a -R s 2 , (3.1) \nwhich leads to an eigenvalue equation resembling that of a linear harmonic oscillator \n-ℏ 2 G 2 c 6 ( d 2 dx 2 + s -2 x -R s / 2 d dx -s -2 ( x -R s / 2) 2 ) U ( x ) + x 2 U ( x ) = R 2 s 4 U ( x ) , (3.2) \nin particular, by selecting the factor ordering parameter as s = 2 , the expression in (3.2) transforms into a quantum linear oscillator \n( -l 2 Pl d 2 dx 2 + x 2 l 2 Pl ) U ( x ) = R 2 s 4 l 2 Pl U ( x ) , (3.3) \nwhere l 2 Pl = ℏ G/c 3 denotes the Planck length. \nFor our purposes, we can define the momentum operator, conjugate to x , as a differential operator \nˆ p x = -il Pl d dx , (3.4) \nwhich clearly corresponds to the differential operator on the left-hand side of equation (3.3). Therefore, we can rewrite (3.3) as follows \n( ˆ p 2 x + ˆ x 2 l 2 Pl ) U ( x ) = R 2 s 4 l 2 Pl U ( x ) . (3.5) \nOur goal is to solve this eigenvalue equation in the momentum representation, where ˆ p x acts as a multiplicative operator and ˆ x as a differential operator, meaning that \nˆ p x = p x , ˆ x = il Pl d dp x . (3.6) \nOn the other hand, it can be verified that ˆ x and ˆ p x satisfy a Heisenberg-type algebra, taking the form \n[ˆ x, ˆ p x ] = il Pl . (3.7) \nWith this in mind, we express the differential equation (3.5) in momentum representation \nd 2 ¯ U ( p x ) dp 2 x +( κ 2 -p 2 x ) ¯ U ( p x ) = 0 , (3.8) \nwhere κ 2 = R 2 s / 4 l 2 Pl . Equation (3.8) is known as Weber's differential equation, and its solution is given by \n¯ U n ( p x ) = N n e -p 2 x / 2 H n ( p x ) , (3.9) \nwhere H n ( z ) is the Hermite polynomials and N n a normalization constant \nN n = √ 1 2 n √ πn ! . (3.10) \nAccording to this solution, the possible eigenvalues of the black hole horizon area are of the form \nA s ( n ) = 32 π ( n + 1 2 ) l 2 Pl . (3.11) \nThe equation (3.11) implies that the area of the quantum black hole is quantized and proportional to the Planck length, where n is a quantum number with values n = 0 , 1 , 2 , · · · . This result coincides with those obtained in [41, 42, 46]. On the other hand, given the quantization of the area (3.11), the Schwarzschild radius must also be \nquantized due to the relation A s ( n ) = 4 πR 2 s ( n ) , and consequently the mass M ( n ) of the black hole. Thus, from (3.11) we obtain \nR s ( n ) = 2 √ 2 ( n + 1 2 ) l Pl , (3.12) \nwhich implies that the radial coordinate inside the black hole will be constrained between 0 and R s ( n ) . \nAlthough we have found the wave function of the interior of a Schwarzschild black hole in standard quantum mechanics as a function of the momentum variable p x , more direct physical information about the black hole can be obtained from configuration variable x , as the variable x is related to the radius of the black hole by (3.1). It is therefore convenient to change the representation by performing a standard Fourier transform \nϕ ( x ) = 1 √ 2 πl Pl ∫ e ip x x/l Pl ¯ ϕ ( p x ) dp x . (3.13) \nThe Fourier transformation of (3.9) into position space can be achieved using the exponential generating function of the Hermite polynomials. Then, from (3.13) one have \nU n ( x ) = ( i ) n N n √ l Pl e -x 2 / 2 l Pl H n ( x/l Pl ) . (3.14) \nIt is our interest to derive the wave function in terms of the variable a , arising from the modification in (3.1), given that the metric components (2.9) are expressed in terms of this variable. So, we obtain \nΨ n ( x ) = ( i ) n N n √ l Pl e -x 2 / 2 l Pl x + R s ( n ) 2 H n ( x/l Pl ) , (3.15) \nThe domain of x , according to (3.1), is x ∈ ( -R s ( n ) / 2 , R s ( n ) / 2) . It is evident that the integral of the square of the wave function (3.15) within the integration domain of x does not converge, indicating it is not square-integrable in the interior of the black hole. This behavior can be attributed to the presence of the physical singularity at a = 0 (or in x = -R s ( n ) / 2 ); in this region, the wave function (3.15) diverges. This suggests that the classical singularity remains unresolved within the context of the standard quantization approach employed above. \nIn Figure 1, we depict the square of the wave function (3.15) plotted against the variable x , which is associated with the radius of the 2-sphere. As the quantum number n escalates, the number of probable states also rises. Notably, the probability density | Ψ n ( a ) | 2 is more pronounced in the vicinity of x = -R s ( n ) / 2 (or a = 0 ) and diminishes \nFigure 1 . Graph showing the square of the wave function (3.15) plotted against different values of the quantum number n . \n<!-- image --> \n(a) Square wave function for \nn \n= 0 \n<!-- image --> \n- (c) Square wave function for n = 5 \n<!-- image --> \n(b) Square wave function for \nn \n= 2 \n<!-- image --> \n(d) Square wave function for \nn \n= 10 \nas we approach the event horizon. In the region where a equals zero, the probability density becomes infinite due to the classical singularity present in that region. \nTo investigate the fate of the singularity in the quantum regime, we calculate the expectation value of K , as defined in (2.10), with respect to the quantum state given in (3.15). Upon quantization, K becomes an operator due to its dependence on the operator ˆ a . For simplicity, we evaluate its expectation value in the position representation, where ˆ a acts multiplicatively. Therefore, we utilize the wave function given in (3.15) to compute the expectation value of the Kretschmann scalar \n⟨ ˆ K ⟩ n = ∫ Ψ ∗ n ( x ) ˆ K Ψ n ( x ) dx ∫ | Ψ n ( x ) | 2 dx . (3.16) \nSince the wave function in (3.15) is not square-integrable, the expectation value of the Kretschmann scalar is indeterminate in the standard quantization process. This outcome was anticipated due to the unresolved physical singularity inside the black hole in the approach taken.", '4 Quantization employing the minimum uncertainty approach': "As observed in the previous section, standard quantization of the interior of the Schwarzschild black hole does not resolve the physical singularity present in the region a = 0 . In this section, we will implement quantization based on the minimum uncertainty approach, which introduces a minimal length description, typically at high energies. \nTo introduce the quantum effects of gravity in the measurement process, the uncertainty relation for position and momentum is extended to [25] \n∆ q ∆ p ≥ ℏ 2 ( 1 + β (∆ p ) 2 ) , (4.1) \nwhere β is know as the deformed parameter. In ordinary quantum mechanics, ∆ q can be made arbitrarily small by letting ∆ p grow correspondingly, this is no longer the case when (4.1) is considered. If for decreasing ∆ q , ∆ p increases, the new term β (∆ p ) 2 on the right-hand side of (4.1) will eventually grow faster than the left-hand side. Hence ∆ q can no longer be made arbitrarily small, searching a minimal uncertainty of the order ℏ √ β . It allows one to express the idea that a minimal length l min should quantum theoretically be described as a minimal uncertainty in position measurements. The modified commutation relation for the ˆ q and ˆ p operators associated with (4.1) is expressed as [25] \n[ˆ q, ˆ p ] = i ℏ ( 1 + β ˆ p 2 ) . (4.2) \nDue to the deformed commutator (4.2) the operators ˆ q and ˆ p are not conjugates anymore. Now, these fundamental variables are to be high energy operators valid, in particular, at or near the Planck scale. They have non-linear representations, ˆ q = q (ˆ q 0 ) , ˆ p = p (ˆ p 0 ) in terms of the variables ˆ q 0 , ˆ p 0 which are position and momentum operators at low energies, obeying the standard Heisenberg algebra [ˆ q 0 , ˆ p 0 ] = i ℏ . \nThis minimal uncertainty approach has yielded intriguing results when considering the quantum effects of gravity. For instance, we can cite some works [37, 38]. Motivated by these successes, we will apply this quantization approach in this section to resolve the singularity of the black hole under consideration. To implement the minimal uncertainty approach in the quantization procedure, the algebra (3.7) will be modified according to (4.2) in order to achieve \n[ˆ x, ˆ p x ] = il Pl ( 1 + β ˆ p 2 x ) , (4.3) \nfrom which one can find the generalized uncertainty relation \n∆ x ∆ p x ≥ l Pl 2 [ 1 + β (∆ p x ) 2 ] , (4.4) \nwhich correspond to minimal uncertainty in x of the order l Pl √ β . Therefore, β effectively defines the magnitude of the minimal uncertainty effects. \nTo simplify our calculations and proceed with a standard quantization procedure, it is convenient to introduce a new variable, p x 0 , conjugate to x , that satisfies the usual commutation relation [26, 27] \n[ˆ x, ˆ p x 0 ] = il Pl . (4.5) \nFor simplicity in the calculations, we will work in the representation where ˆ p x 0 acts as a multiplicative operator and the position operator is represented as a differential operator ˆ x = il Pl ∂/∂p x 0 . The reason for choosing this representation is that it is not possible to clearly define the momentum operator ˆ p x as a differential operator that satisfies the commutation relation (4.3) [25]. Therefore, from (4.3), we can find the relation between the physical variable p x and the auxiliary p x 0 by \np x = 1 √ β tan ( √ βp x 0 ) , (4.6) \nwhich satisfies the modified commutation relation (4.3). Also, the domain of p x 0 is restricted to -π/ 2 √ β < p x 0 < π/ 2 √ β [26]. \nIn this approach, the differential equation (3.8) is expressed in terms of the new canonically conjugate variables x and p x 0 . Thus, in the p x 0 space, it reads \nd 2 ¯ U ( p x 0 ) dp 2 x 0 + ( κ 2 -tan 2 ( √ βp x 0 ) β ) ¯ U ( p x 0 ) = 0 . (4.7) \nUsing the new variable ξ by the change ξ = √ βp x 0 , the above equation can be written as \nd 2 ¯ U ( ξ ) dξ 2 + ( ϵ -tan 2 ξ β 2 ) ¯ U ( ξ ) = 0 , (4.8) \nhere we denoted ϵ = κ 2 /β = R 2 s / 4 βl 2 Pl . This differential equation reduces to the hypergeometric one through the transformation z = sin 2 ξ and y = ¯ U cos m ξ [47] \nz ( z -1) y '' ( z ) + [ (1 -m ) z -1 2 ] y ' ( z ) -1 4 ( m + ϵ ) y ( z ) = 0 , (4.9) \nwhere m is a root of the quadratic equation m 2 + m -1 β 2 = 0 , from which one have for m \nm = -β ± √ 4 + β 2 2 β . (4.10) \nThen, the general solution of (4.9) is given by \ny ( z ) = C 1 2 F 1 ( β + √ β 2 +4+2 √ ϵβ 2 +1 4 β , β + √ β 2 +4 -2 √ ϵβ 2 +1 4 β ; 1 2 ; z ) + C 2 z 1 / 2 2 F 1 ( β + √ β 2 +4+2 √ ϵβ 2 +1 4 β + 1 2 , β + √ β 2 +4 -2 √ ϵβ 2 +1 4 β + 1 2 ; 3 2 ; z ) . (4.11) \nHere, 2 F 1 ( A, B ; C ; z ) represents the hypergeometric function, which converges if one of the first two arguments, A or B , is a non-positive integer. For example, if we consider the hypergeometric function appearing in the first line of equation (4.11), this condition for the first argument results in \nβ + √ β 2 +4+2 √ ϵβ 2 +1 4 β = -n, with n = 0 , 1 , 2 , · · · . (4.12) \nFrom this, one can derive the eigenvalues of the black hole's area, considering that ϵ = R 2 s / 4 βl 2 Pl and A s = 4 πR 2 s , within the framework of the minimum uncertainty approach. That is \nA GUP s (2 n ) = 32 π ( 2 n + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16(2 n ) 2 πβl 2 Pl . (4.13) \nIf, on the other hand, we apply the convergence condition to the second argument in the hypergeometric function 2 F 1 ( A, -n ; C ; z ) , we obtain exactly (4.13), thus ensuring the convergence of the first term in the wave function (4.11). \nWhen the same convergence criterion is applied to the hypergeometric function appearing in the second line of equation (4.11), we find that the spectrum of the area, in this case, is \nA GUP s (2 n +1) = 32 π ( (2 n +1) + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16(2 n +1) 2 πβl 2 Pl , (4.14) \nand combining both expressions, (4.13) and (4.14), we obtain the discrete spectrum of the black hole's area \nA GUP s ( n ) = 32 π ( n + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16 n 2 πβl 2 Pl . (4.15) \nIt can be readily observed from this expression that in the limit where β → 0 , the usual area spectrum in (3.11) is recovered. Conversely, if β remains finite, the area \nlevels depend on the square of the quantum number n , and for large n , they grow like n 2 . Just as the area is modified, the Schwarzschild radius will also be, by \nR GUP s ( n ) = l Pl √ √ √ √ 8 ( n + 1 2 ) ( √ β 2 4 +1+ β 2 ) +4 βn 2 . (4.16) \nAccording to these convergence conditions, (4.12), the wave function (4.11) is written, in terms of the original variable ξ , as \ny n ( ξ ) = C 1 2 F 1 ( -n, n + α ; 1 2 ; sin 2 ξ ) + C 2 sin ξ 2 F 1 ( -n, n + α +1; 3 2 ; sin 2 ξ ) , (4.17) \nwhere we have denoted \nα = β + √ 4 + β 2 2 β . (4.18) \nHere, if the negative sign is chosen instead of the positive in (4.10), we find that -m = α . Additionally, considering the special doublen formulas of the Gegenbauer polynomials [48] \nC ( α ) 2 n (sin ξ ) =( -1) n ( n + α -1 n ) 2 F 1 ( -n, n + α ; 1 2 ; sin 2 ξ ) , (4.19) \nC ( α ) 2 n +1 (sin ξ ) =2( -1) n α ( n + α n ) sin ξ 2 F 1 ( -n, n + α +1; 3 2 ; sin 2 ξ ) , (4.20) \nwe can write the solution (4.17) in a compact form \n¯ U n ( p 0 x ) = N n C ( α ) n (sin ( √ βp 0 x )) cos α ( √ βp 0 x ) , (4.21) \nwhere the substitution ξ = √ βp x 0 has been used, and N n is a constant that can be determined from the normalization condition of the wave function (4.21). That is \nN 2 n ∫ π 2 √ β -π 2 √ β ( 1 -sin 2 ( √ βp 0 x ) ) α [ C ( α ) n (sin ( √ βp 0 x )) ] 2 dp 0 x = 1 , (4.22) \nand considering that the Gegenbauer polynomials are normalized by [49] \n∫ 1 -1 (1 -h 2 ) α -1 / 2 [ C ( α ) n ( h )] 2 dh = 2 1 -2 α π Γ( n +2 α ) ( n + α )Γ 2 ( α )Γ( n +1) , (4.23) \nwe obtain \nN n = [ √ β ( n + α )Γ 2 ( α )Γ( n +1) 2 1 -2 α π Γ( n +2 α ) ] 1 / 2 . (4.24) \nUnlike the constant in (3.10), this integration constant is expressed not only in terms of the quantum number n but also in terms of the deformation parameter β , as shown in (4.18).", '4.1 Limit β → 0': 'In the standard limit, where β → 0 , the usual solution (3.9) should be recovered. To verify this, consider from (4.18) that for small values of the deformation parameter β , we have α = 1 /β , which implies that when β → 0 , α →∞ . On the other hand, from the relationship between the Gegenbauer polynomial C ( α ) n and the relativistic Hermite polynomial H ( α ) n [50], given by \nH ( α ) n ( √ αu ) = n ! α n/ 2 (1 + u 2 ) n/ 2 C ( α ) n ( u √ 1 + u 2 ) , (4.25) \nwe can express (4.21) as follows \n¯ U n ( p 0 x ) = [ √ β ( n + α )Γ 2 ( α ) α n 2 1 -2 α π Γ( n +2 α ) n ! ] 1 / 2 cos n + α ( √ βp 0 x ) H ( α ) n ( √ α tan ( √ βp 0 x )) , (4.26) \nwhere u = tan ( √ βp 0 x ) was consider. Using the Stirling and asymptotic formulas [49] \nΓ( α ) ∼ √ 2 πe -α α α -1 / 2 , (4.27) \nΓ( n +2 α ) ∼ √ 2 πe -2 α (2 α ) 2 α + n -1 / 2 , (4.28) \nis easy to verify that the term in brackets in (4.26) reduces to \nlim α →∞ [ √ β ( n + α )Γ 2 ( α ) α n 2 1 -2 α π Γ( n +2 α ) n ! ] 1 / 2 = 1 √ π 1 / 2 2 n n ! . (4.29) \nOn the other hand, using the expansion ln cos x = -x 2 / 2 -x 4 / 12 -··· , we can make the following approximation \nlim β → 0 cos n + α ( √ βp 0 x ) = lim β → 0 e n ln cos ( √ βp 0 x ) e α ln cos ( √ βp 0 x ) ∼ e -p 2 0 x / 2 . (4.30) \nIn the limit α → ∞ (non-relativistic limit) the relativistic Hermite polynomial H ( α ) n turns into the Hermite polynomial H n ( ξ ) [50], and equivalently, its argument reduces to lim β → 0 tan ( √ βp 0 x ) / √ β = p x 0 . So, we have that \nlim α → 0 H ( α ) n ( √ α tan ( √ βp 0 x )) = H n ( p 0 x ) . (4.31) \nCombining this limit with those found in (4.29) and (4.30), we find that the wave function (4.21), derived using the minimal uncertainty approach, reduces to, as expected, the wave function (3.9) in the limit β → 0 .', '4.2 Fourier transform': 'The wave function in (4.21) is defined in momentum space, in terms of the auxiliary variable p 0 x . To extract physical information from our results, we need to express the wave function inside the black hole in terms of the radial coordinate, as the components of the metric (2.9) depend on this variable. This can be accomplished using the Fourier transform (3.13), which in this case would be \nU n ( x ) = N n √ 2 πl Pl ∫ π 2 √ β -π 2 √ β e ip 0 x x/l Pl cos α ( √ βp 0 x ) C ( α ) n (sin ( √ βp 0 x )) dp 0 x . (4.32) \nTo perform this integration, we will use the explicit expression of the Gegenbauer polynomials [49] \nC ( α ) n (sin ( √ βp 0 x )) = [ n 2 ] ∑ k =0 ( -1) k Γ( n -k + α ) Γ( α ) k !( n -2 k )! (2 sin ( √ βp 0 x )) n -2 k , (4.33) \nso, (4.32) becomes in \nU n ( x ) = [ n 2 ] ∑ k =0 W n,k ∫ π 2 √ β -π 2 √ β e ip 0 x x/l Pl ( 1 -sin 2 ( √ βp 0 x ) ) α 2 sin n -2 k ( √ βp 0 x ) dp 0 x , (4.34) \nwhere W n,k is a constant denoted as \nW n,k = ( -1) k N n 2 n -2 k Γ( n -k + α ) √ 2 πl Pl Γ( α ) k !( n -2 k )! . (4.35) \nThe Fourier transform in (4.34) yields the wave function in x space, which can be expressed as \nΨ \nGUP n ( x ) = -i √ β ( x + R s 2 ) [ n 2 ] ∑ k =0 W n,k e -1 2 iπ (2 α +10 k -5 n ) × { Γ( α +1) [ e 1 2 iπ ( α +6 k -x √ βl Pl -3 n ) Γ ( -l Pl ( α -2 k + n ) + x √ β 2 l Pl ) × 2 ˜ F 1 ( α +1 , 1 2 ( α -2 k + n -x l Pl √ β +2 ) ; 1 2 ( α +2 k -n -x l Pl √ β +2 ) ; -1 ) + e 1 2 iπ ( α +10 k + x √ βl Pl -5 n ) Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) × 2 ˜ F 1 ( α +1 , 1 2 ( α -2 k + x √ βl Pl + n +2 ) ; 1 2 ( α +2 k + x √ βl Pl -n +2 ) ; -1 ) ] - \nΓ( -2 k + n +1) [ e iπ ( α +2 k -n ) Γ ( -l Pl ( α -2 k + n ) + x √ β 2 l Pl ) × 2 ˜ F 1 ( -2 k + n +1 , 1 2 ( α -2 k -x √ βl Pl + n +2 ) ; 1 2 ( -α -2 k -x √ βl Pl + n +2 ) ; -1 ) + e iπ ( α +4 k -2 n ) Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) × 2 ˜ F 1 ( -2 k + n +1 , 1 2 ( α -2 k + x √ βl Pl + n +2 ) ; 1 2 ( -α -2 k + x √ βl Pl + n +2 ) ; -1 ) ]} , (4.36) \n̸ \nwith the conditions β > 0 and α > -1 . Here, 2 ˜ F 1 ( A, B ; C ; z ) = 2 F 1 ( A,B ; C ; z ) Γ( C ) is the regularized hypergeometric function. To ensure that the wave function is convergent, both the gamma functions and the hypergeometric functions appearing in (4.36) must be finite. Based on the definition of Γ( z ) , where z cannot be a negative integer, we find from Γ ( -l Pl ( α -2 k + n )+ x √ β 2 l Pl ) that x = l Pl √ β (2 l -( α -2 k + n )) , where l = 0 , 1 , 2 , . . . . Similarly, from Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) , we find that x = -l Pl √ β (2 l -( α -2 k + n )) . These convergence conditions for x can be summarized as follows \n̸ \n̸ \nx = ± l Pl √ β (2 l -( α -2 k + n )) , (4.37) \nand, as shown in (3.1), the variable x is related to the radius of the 2-sphere, with the condition a > 0 . This leads us to conclude, from (4.37), that the physically acceptable condition for x is \nx > ∣ ∣ ∣ l Pl √ β (2 l -( α -2 k + n )) ∣ ∣ ∣ . (4.38) \nFor the ground state n = 0 , the floor function ⌊ n 2 ⌋ restricts the summation in (4.36) to k = 0 , which simplifies condition (4.38) to x > l Pl √ β α . Similarly, for n = 1 , the condition becomes x > l Pl √ β ( α +1) , with l = 0 in both cases. This shows that the minimal uncertainty approach introduces a minimum radius on the 2-sphere, which is related to the quantum parameter β . Notice that this is a universal bound, meaning that it is independent of the mass of the black hole and purely a quantum effect, which vanishes for β → 0 . This can also be seen as a weaker argument for the resolution of the black hole singularity in this approach. On the other hand, due to condition (4.38), the domain of x is restricted between \n∣ ∣ ∣ l Pl √ β (2 l -( α -2 k + n )) ∣ ∣ ∣ < x < R GUP s ( n ) , (4.39) \nwhich clearly differs from the domain in the standard case. \nMoreover, it is noteworthy that condition (4.38) ensures the convergence of the hypergeometric functions appearing in (4.36). This guarantees that the wave function, derived from the quantization based on the minimal uncertainty approach, is wellbehaved within the domain of interest (4.39). Figure 2 shows the plots of the wave function squared for various values of n . \nBy comparing the wave function obtained in the standard quantization scheme (3.15) with the one derived from the minimal uncertainty approach (4.36), we observe that they behave differently both at the horizon and in the region where the classical singularity of the black hole is located. For instance, in the standard case, the classical singularity is found at x = -R s ( n ) / 2 , where the wave function Ψ n ( x ) diverges. In contrast, due to condition (4.38), which imposes a minimum radius for the black hole, x cannot reach the region where the classical singularity resides, and therefore the wave function Ψ GUP n remains finite in that region and is square-integrable (see Table 1). This result suggests that the singularity is resolved in the quantization approach implemented in this section. A strong indication that the singularity has been resolved is demonstrating that the expectation value of the Kretschmann scalar is finite within the interior of the black hole.', '4.3 Expectation value of Kretschmann scalar': 'As in the standard case, to determine whether the singularity is truly resolved in the minimal uncertainty approach, we must calculate the expectation value of the Kretschmann scalar for the states characterizing the interior of the modified black hole. Given the complex form of the wave function (4.36) for each state characterized by the quantum number n , we perform the calculation of ⟨ K ⟩ n in (3.16) numerically. Table 1 shows the expected values for various n . \nTable 1 . Expected values of the Kretschmann scalar for different quantum numbers n . \nClearly, the expected value of the Kretschmann scalar is finite inside the black hole for all the states that characterize it, which is not the case in the standard approach, where the expected value of the Kretschmann scalar is undefined due to the presence of the singularity. This result allows us to argue that the classical singularity that once \nFigure 2 . Graphical representation of the wave function (4.36), obtained using the minimal uncertainty quantization scheme, for different values of n . \n<!-- image --> \n(a) Square wave function for \nn \n= 0 \n0.04 \n0.03 \n0.02 \n0.01 \n0.00 \n2 \nP \nU \nG \n2 \nΨ \n0 \n1 \n2 \n3 \n4 \nx \n- (c) Square wave function for n = 2 \nexisted inside the black hole has been resolved as a direct consequence of implementing quantization based on the minimal uncertainty approach. \nIn a previous work [40], the interior of a Schwarzschild black hole was quantized using the minimal uncertainty approach appropriate for the Ashtekar-Barbero connection variables. Some of these results align with those found in this work, derived from a different formulation, yet physically consistent with each other.', '5 Discussion and conclusion': "In this work, we explore the interior of a black hole through the perspective of two quantization approaches. We begin with the Wheeler-DeWitt equation (2.7), which describes the dynamics within the Schwarzschild black hole's interior. By rescaling the wave function and the radial coordinate, as shown in (3.1), we reformulate this equation as the eigenvalue equation of a quantum linear harmonic oscillator (3.3) [42]. \nn \n= \n2, \nl \nPl \n2 \nP \nU \nG \n1 \nΨ \n= \n1, \nR \ns \n0.05 \n0.04 \n0.03 \n0.02 \n0.01 \n0.00 \n0.0 \n0.5 \n1.0 \n1.5 \n2.0 \n2.5 \n3.0 \n3.5 \nx \n(b) Square wave function for \nn \n= 1 \nPl \n, G \n= \n1 \n= \n4.7 \nl \nn \n= \n1, \nl \nPl \n= \n1, \nR \ns \n= \n3.6 \nl \nPl \n, G \n= \n1 \nThe first quantization approach we apply is based on standard quantization, in which the usual commutation relation for a pair of canonical variables is satisfied. In this approach, we find that the wave function (3.15) is not square-integrable within the black hole, due to the fact that the wave function diverges in the region where the classical singularity is located. This result can be interpreted as a sign that the singularity still persists after the standard quantization process. Additionally, it is concluded that the expectation value of the Kretschmann scalar is undefined within the black hole's interior. \nThe second quantization method we implement is based on the minimal uncertainty approach (4.3). Using this method, we find that the wave function (4.36), and therefore its squared modulus, remain finite throughout the entire interior of the black hole, implying that all states within it are square-integrable. To ensure the convergence of the wave function (4.36), we discover that the black hole's radius acquires a minimum value (4.38) that depends only on the quantum number n , which determines the possible states within the black hole, and the deformation parameter β , which defines the magnitude of minimal uncertainty effects. This minimum radius is universal as it does not depend on the black hole's mass. \nIn order to investigate the fate of the singularity within the minimal uncertainty approach, we calculate the expectation value of the Kretschmann scalar (3.16). In this case, unlike in the standard quantization scheme, we find that the expectation value is well-defined and finite throughout the interior of the black hole (see Table 1), bounded by the domain (4.39). Ultimately, based on these results, we can conclude that both the finiteness of the wave function and the expectation value of the Kretschmann scalar indicate that the classical singularity is resolved and replaced by a minimum radius on the 2-sphere in the minimal uncertainty quantization scheme. \nWe want to emphasize that the methods applied in this work, as well as the results obtained by treating the black hole as a harmonic oscillator, both in standard quantization and in the minimal uncertainty approach, are novel in the literature. 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Platania, Towards conditions for black-hole singularity-resolution in asymptotically safe quantum gravity , Classical and Quantum Gravity 35 (2018) 225007. \n- [16] G. J. Olmo, D. Rubiera-Garcia and A. Sanchez-Puente, Classical resolution of black hole singularities via wormholes , The European Physical Journal C 76 (2016) 1.\n- [17] V. Husain and O. Winkler, Quantum resolution of black hole singularities , Classical and Quantum Gravity 22 (2005) L127.\n- [18] D. Cartin and G. Khanna, Wave functions for the schwarzschild black hole interior , Phys. Rev. D 73 (2006) 104009.\n- [19] S. Jalalzadeh and B. Vakili, Quantization of the interior schwarzschild black hole , International Journal of Theoretical Physics 51 (2012) 263.\n- [20] T. Brotz, Quantization of black holes in the wheeler-dewitt approach , Phys. Rev. D 57 (1998) 2349.\n- [21] C. R. Almeida and D. C. Rodrigues, Quantization of a black-hole gravity: geometrodynamics and the quantum , Classical and Quantum Gravity 40 (2023) 035004.\n- [22] C. Vaz and L. Witten, Mass quantization of the schwarzschild black hole , Phys. Rev. D 60 (1999) 024009.\n- [23] A. Corichi, J. Díaz-Polo and E. Fernández-Borja, Black hole entropy quantization , Phys. Rev. Lett. 98 (2007) 181301.\n- [24] A. Bina, S. Jalalzadeh and A. Moslehi, Quantum black hole in the generalized uncertainty principle framework , Phys. Rev. D 81 (2010) 023528.\n- [25] A. Kempf, G. Mangano and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation , Phys. Rev. D 52 (1995) 1108.\n- [26] P. Pedram, New approach to nonperturbative quantum mechanics with minimal length uncertainty , Phys. Rev. D 85 (2012) 024016.\n- [27] P. Bosso, L. Petruzziello and F. Wagner, Minimal length: A cut-off in disguise? , Phys. Rev. D 107 (2023) 126009.\n- [28] P. Bosso, S. Das and R. B. Mann, Potential tests of the generalized uncertainty principle in the advanced ligo experiment , Physics Letters B 785 (2018) 498.\n- [29] N. C. Bizet, O. Obregón and W. Yupanqui, Modified entropies as the origin of generalized uncertainty principles , Physics Letters B 836 (2023) 137636.\n- [30] G. Veneziano, A stringy nature needs just two constants , Europhysics Letters 2 (1986) 199.\n- [31] M. Maggiore, A generalized uncertainty principle in quantum gravity , Physics Letters B 304 (1993) 65.\n- [32] F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment , Physics Letters B 452 (1999) 39.\n- [33] M. P. Ryan, Hamiltonian Cosmology , Lecture Notes in Physics. Springer-Verlag, 1972. \n- [34] J. B. Hartle and S. W. Hawking, Wave function of the universe , Phys. Rev. D 28 (1983) 2960.\n- [35] M. P. Ryan and L. C. Shepley, Homogeneous relativistic cosmologies , vol. 59. Princeton University Press, 2015.\n- [36] B. S. DeWitt, Quantum theory of gravity. i. the canonical theory , Phys. Rev. 160 (1967) 1113.\n- [37] P. Bosso and O. Obregón, Minimal length effects on quantum cosmology and quantum black hole models , Classical and Quantum Gravity 37 (2020) 045003.\n- [38] P. Bosso, O. Obregón, S. Rastgoo and W. Yupanqui, Deformed algebra and the effective dynamics of the interior of black holes , Classical and Quantum Gravity 38 (2021) 145006.\n- [39] B. Melchor, R. Perca and W. Yupanqui, Semiclassical resolution of the black hole singularity inspired in the minimal uncertainty approach , Nuclear Physics B 1004 (2024) 116584.\n- [40] P. Bosso, O. Obregón, S. Rastgoo and W. Yupanqui, Black hole interior quantization: a minimal uncertainty approach , Classical and Quantum Gravity 41 (2024) 135011.\n- [41] J. Mäkelä and P. Repo, Quantum-mechanical model of the reissner-nordström black hole , Phys. Rev. D 57 (1998) 4899.\n- [42] O. Obregon, M. Sabido and V. Tkach, Entropy using path integrals for quantum black hole models , General Relativity and Gravitation 33 (2001) 913.\n- [43] J. D. Bekenstein, Generalized second law of thermodynamics in black-hole physics , Phys. Rev. D 9 (1974) 3292.\n- [44] R. Doran, F. S. Lobo and P. Crawford, Interior of a schwarzschild black hole revisited , Foundations of Physics 38 (2008) 160.\n- [45] J. Louko and S. N. Winters-Hilt, Hamiltonian thermodynamics of the reissner-nordström-anti-de sitter black hole , Phys. Rev. D 54 (1996) 2647.\n- [46] J. D. Bekenstein, Quantum black holes as atoms , 1998.\n- [47] V. F. Zaitsev and A. D. Polyanin, Handbook of exact solutions for ordinary differential equations . Chapman and Hall/CRC, 2002, 10.1201/9781420035339.\n- [48] W. O. Frank, NIST handbook of mathematical functions . Cambridge University Press, 2010.\n- [49] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables , vol. 55. 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2024RNAAS...8..220C

It was recently claimed httpsarxiv.orgpdf2408.10799 that solar system ephemeris data can exclude primordial black hole PBH dark matter in the mass range 10SUP18SUP10SUP22SUP g. I show that this conclusion is based on an implausible implicit assumption namely the uncertainty on the solar system mass within 50 au is as small as the uncertainty on the mass of the Sun. Correcting for this error I find that ephemeris data can only constrain PBHs with mass below 10SUP16SUP g which is already excluded by constraints on their evaporation via Hawking radiation. Correcting a further error concerning the timeaveraged rate of such fluctuations nullifies even this weaker constraint.

2024-09-01T00:00:00Z

['2024arXiv240901993C', 'arXiv:2409.01993', '10.48550/arXiv.2409.01993', '10.3847/2515-5172/ad7674', '2024RNAAS...8..220C']

['Primordial black holes', '1292', 'High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology']

Comment on Excluding Primordial Black Holes as Dark Matter Based on Solar System Ephemeris

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

https://arxiv.org/pdf/2409.01993.pdf

{"Comment on 'Excluding Primordial Black Holes as Dark Matter Based on Solar System Ephemeris'": "James M. Cline ∗ \nMcGill University Department of Physics & Trottier Space Institute, 3600 Rue University, Montr'eal, QC, H3A 2T8, Canada and CERN, Theoretical Physics Department, Geneva, Switzerland \nIt was recently claimed ( https://arxiv.org/pdf/2408.10799 ) that solar system ephemeris data can exclude primordial black hole (PBH) dark matter in the mass range 10 18 -10 22 g. We show that this conclusion is based on an implausible, implicit assumption; namely the uncertainty on the solar system mass within 50 au is as small as the uncertainty on the mass of the sun. Correcting for this error, we find that ephemeris data can only constrain PBH's with mass below 10 16 g, which is already excluded by constraints on their evaporation via Hawking radiation. Correcting a further error concerning the time-averaged rate of such fluctuations nullifies even this weaker constraint. \nRef. [1] recently argued that if primordial black holes (PBHs) in the mass range 10 18 -10 22 g constitute the dark matter of the Universe, they would give rise to Poisson fluctuations in the total mass of the solar system contained within some assumed fiducial volume, and that observational limits on such fluctuations can be used to exclude this PBH mass range. It was claimed that fluctuations due to crossing PBHs would show up as an effective time rate of change of the solar mass M /circledot , by their perturbation of planetary orbits. These orbits have been monitored over more than 50 years, and the search for deviations was used by Ref. [2] to set strong limits on the time variation of GM /circledot , at the level of a few parts in 10 14 per year. \nIn Ref. [1], Loeb assumed the relevant volume is the region within 50 au, the orbital radius of Pluto. However, Ref. [3] shows that the ephemeris data are vastly dominated (94 %) by measurements of the inner planets. Therefore a fiducial radius of 1.5 au (encompassing Mars) would be more appropriate for the data at hand. To have at least one PBH on average within this volume, in order to cause a fluctuation, it should have a mass no greater than 3 × 10 16 g, assuming a monochromatic spectrum of PBHs making up the total dark matter density. Such a light PBH as dark matter is already ruled out by constraints on its Hawking radiation evaporation products, namely photons and e + -e -pairs [4]. \nHowever, even this weakened limit on the PBH mass cannot be a valid inference from the ephemeris data. The limit on d/dt ( GM /circledot ) derived by Ref. [2] relies upon a long duration, ∼ 50y of data taking, in order to be so stringent. In contrast, the crossing time of the 1.5 au \nfiducial volume by a PBH is less than a few weeks, hence any perturbation produced on planetary orbits would average to zero during the observation time. Moreover, the direct limit on the dark matter density in the solar system from ephemeris data is 14,000 times weaker than the accepted value 0 . 4 GeV/cm 3 [3], underscoring the implausibility of these data being sensitive to such a weak perturbation as a PBH crossing. \nNot discussed in Ref. [1] is the possibility that a close encounter of a PBH could strongly perturb an inner planet. An order of magnitude estimate shows that this is highly unlikely. The ephemeris bound on the integrated perturbation to the gravitational force on a planet of mass m is of order \nδF ∼ G ˙ M /circledot Tm R 2 , (1) \nwhere ˙ M /circledot is the limit on the time variation of the solar mass, T is the observing time, and R ∼ 1 au is the orbital radius. The maximum force exerted by a transiting PBH is F = GM bh m/b 2 , where b is the distance of closest approach. Thus one needs \nb /lessorsimilar R ( M bh ˙ MT ) 1 / 2 ∼ 0 . 01 au (2) \nfor M bh = 10 18 g, to have an observable effect. On the other hand, Ref. [1] computes the rate of such crossings to be \nΓ ∼ 1000 ( b 50 au ) 2 yr -1 < 4 × 10 -5 yr -1 . (3) \nAcknowledgment. I thank Gabriele Franciolini for helpful discussions. \n- [1] A. Loeb, 'Excluding Primordial Black Holes as Dark Matter Based on Solar System Ephemeris,' arXiv:2408.10799 [hep-ph] .\n- [2] E. V. Pitjeva, N. P. Pitjev, D. A. Pavlov, and C. C. Turygin, 'Estimates of the change rate of solar mass and gravitational constant based on the dynamics of the Solar System,' Astron. Astrophys. 647 (2021) A141, arXiv:2201.09804 [astro-ph.EP] .\n- [3] N. P. Pitjev and E. V. Pitjeva, 'Constraints on\n- dark matter in the solar system,' Astron. Lett. 39 (2013) 141-149, arXiv:1306.5534 [astro-ph.EP] \n. \n- [4] B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, 'Constraints on primordial black holes,'\n- Rept. Prog. Phys. 84 no. 11, (2021) 116902, arXiv:2002.12778 [astro-ph.CO] ."}

2024arXiv240913468G

Nextgeneration gravitationalwave instruments are expected to constrain the equation of state of dense nuclear matter by observing binaries involving neutron stars. We highlight a problematic systematic error in finitetemperature merger simulations where shock heating associated with the neutronstar surface gives rise to elevated temperatures. We demonstrate the severe implications of this artificial heating by computing static and dynamical tidal parameters for neutron stars immersed in simulation temperature profiles. The induced systematic errors must be addressed if we want to build robust gravitationalwave signal models for neutronstar or indeed neutron starblack hole binaries.

2024-09-01T00:00:00Z

['2024arXiv240913468G', '10.48550/arXiv.2409.13468', 'arXiv:2409.13468']

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

Problematic systematics in neutronstar merger simulations

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13468.pdf

{'Problematic systematics in neutron-star merger simulations': 'F. Gittins, R. Matur, N. Andersson, and I. Hawke Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom (Dated: September 23, 2024) \n∗ \nNext-generation gravitational-wave instruments are expected to constrain the equation of state of dense nuclear matter by observing binaries involving neutron stars. We highlight a problematic systematic error in finite-temperature merger simulations, where shock heating associated with the neutron-star surface gives rise to elevated temperatures. We demonstrate the severe implications of this artificial heating by computing static and dynamical tidal parameters for neutron stars immersed in simulation temperature profiles. The induced systematic errors must be addressed if we want to build robust gravitational-wave signal models for neutron-star, or indeed neutron star-black hole, binaries.', 'I. INTRODUCTION': "Gravitational-wave astronomy has the potential to provide precise constraints on the extreme physics represented by neutron stars. The observation of the first binary neutron-star merger event, GW170817, in LIGOVirgo highlighted this promise through the absence of a distinguishable imprint of tidal interaction in the gravitational-wave signal [1-3]. This, in turn, placed constraints on the neutron-star tidal deformability and the permitted range of neutron-star radii. To date, GW170817 is the only observed event for which such constraints have been obtained. This is expected to change with the development of more sensitive gravitationalwave interferometers, like the Einstein Telescope [4] and Cosmic Explorer [5]. Neutron-star physics provides a key science driver for these advanced instruments. \nDesigned to be about a factor of 50 more sensitive than the interferometers that were in operation in 2017, the next-generation detectors should allow us to put much tighter bounds on the equation of state of matter at supranuclear densities. This will involve precision measurements of the response of a neutron star to the tidal interaction with a companion during the late stages of compact binary inspiral. The tide induced on a neutron star has two aspects: At lower frequencies, the dominant feature is the static tide , represented by the tidal deformability [6-8]. As the binary orbit shrinks and the system approaches merger, the dynamical tide starts to play a role [9-12]. The dynamical tide has various features, but it is dominated by the contribution from the fundamental mode of oscillation of the star (the f -mode) [13]. Moreover, the tidal deformability and the f -mode frequency are known to be linked by a quasi-universal relation [14], which means that the two features may not be truly 'independent' from the parameter-extraction point of view. There are convincing arguments that we need to include both static and dynamical aspects in our tidal models if we want to maximise the information extracted from observations [15-17]. \nAdditional contributions to the dynamical tidee.g. , from low-frequency gravity g -modes arising from composition and entropy gradients [10, 18] or interface i -modes associated with sharp phase transitions [19, 20]-are expected to leave a weak imprint on the gravitational-wave signal. Detections of such fine-print contributions would shed light on the composition and state of the highdensity matter. This is an important issue, but not the focus of the discussion here. \nGravitational-wave astronomy relies, in general, on a robust link between observations and the underlying theory. In particular, for the problem of neutron-star tides, waveform models need to be calibrated against two distinct bodies of work [21-27]. At low frequencies (early stages of inspiral), the problem is well described by perturbation theory coupled to a post-Newtonian model for orbital evolution. At higher frequencies-as the system approaches merger-the dynamics become nonlinear and require numerical simulations. Each approach to the problem comes with its own set of computational issues. The specific feature we highlight here relates to the late stages of evolution where numerical simulations are necessary. \nA variety of approaches are currently used to build waveform models for binaries involving neutron stars. For example, the time-domain effective-one-body framework [21-23] calibrates the model using numericalrelativity simulations of merging black-hole binaries, while the frequency-domain closed-form tidal approximant method [24-27] normalises with respect to neutronstar simulations. There are also hybrid models that combine the two techniques, see for instance Ref. [28]. Arguably, the current state of the art relies on data from neutron-star merger simulations which implement fits of one-parameter nuclear-matter models [29], implicitly assuming the stellar material to be cold and the nuclear reactions to maintain equilibrium. However, realistic neutron-star mergers are hot, out-of-equilibrium events, so we ultimately need to work towards calibrations based on realistic, finite-temperature simulations [30-32] to capture the anticipated physics. Such simulations are essential if we want to connect the pre-merger behaviour with the violent dynamics, including electro- \nmagnetic counterpart emission, of the post-merger remnant. \nThe issue we bring to the fore relates to a (known, but somewhat ignored) systematic error present in all current (grid-based) finite-temperature neutron-star merger simulations involving shock-capturing schemes: The temperature in the simulated stars is much too high before merger. The problem is due to the simple fact that the employed numerical schemes treat the sharp drop in density at the neutron-star surface as a 'shock' leading to generation of (numerical) entropy and artificial heating. The level of heating varies between simulation codes, but it is generally the case that the temperature is ramped up to 5-10 MeV throughout the star very soon after the beginning of the simulation (see Fig. 1 and Appendix A). As a result, the simulated neutron star is several orders of magnitude too hot. In fact, at this level the thermal pressure is significant, effectively making the star much less compressed than it ought to be. Based on a comparison to proto-neutron stars, which typically reach similar temperatures, one would estimate that the neutron-star radius can change by as much as a factor of 2. This obviously impacts on both the tidal deformability and the f -mode frequency and hence may render debatable any waveform calibration involving such numerical simulation data. This is clearly problematic. \nFIG. 1. The peak temperature reached in the neutron-star matter during a numerical simulation. Results correspond to the DD2 equation of state and are provided for both single isolated neutron stars (at three different resolutions; 369 m in black, 185 m in green and 92 m in purple) and one of the companions in a neutron-star binary simulation (with grid resolution 221 . 5 m in red). In the binary simulation, the initial separation is set to 45 km. The isolated neutron-star simulations suggest that the issue with the spurious heating remains for affordable levels of resolution. The comparison to the binary neutron-star case shows that the artificial heating is enhanced when the star is moving across the numerical grid. The temperatures indicated here are representative of those seen in all current finite-temperature (grid-based) neutron-star simulations. \n<!-- image --> \nAs a first step towards a better quantitative under- \nstanding of the issue, we calculate both static and dynamical tidal parameters for two specific realistic finitetemperature equations of state. Our results provide a more precise illustration of the impact of the artificial heating and a clear indication of the systematics associated with current finite-temperature neutron-star simulations.", 'II. SYSTEMATICS IN FINITE-TEMPERATURE SIMULATIONS': 'The fluid equations that determine the neutron-star structure are closed by a thermodynamical equation of state . For our calculations, we assume a three-parameter model for the nuclear matter-for simplicity, only comprising neutrons, protons and electrons-with the temperature T , baryon-number density n b and electron fraction Y e as the natural thermodynamical variables. Such models represent the current state of the art in nuclear astrophysics [33] and are regularly implemented in numerical simulations [30-32]. \nFor the assumed three-parameter model, the first law of thermodynamics may be expressed as \ndf = -n b s dT + f + p n b dn b + n b µ ∆ dY e , (1) \nwhere s is the entropy per baryon, p is the isotropic pressure, µ ∆ = µ p + µ e -µ n and { µ n , µ p , µ e } are the neutron, proton and electron chemical potentials, respectively. The various quantities are as measured in the local inertial reference frame of the fluid. The free-energy density f = f ( T, n b , Y e ) represents the fundamental equation of state from which all the thermodynamical information about the stellar material can be derived. \nIn order to illustrate the artificial surface heating in binary simulations, we use temperature results from two separate inspiral-merger simulations with different prescriptions for the finite-temperature matter: APR [34] and DD2 [35] (as implemented in the CompOSE library [33, 36, 37]). The first simulation is taken from Refs. [31, 38], which uses the Einstein Toolkit codebase to collide neutron stars described by the APR nuclear-matter equation of state. The simulation clearly exhibits the characteristic artificial surface heating, reaching temperatures of order 10 MeV, see Fig. 1 of Ref. [31]. In addition, mainly in order to demonstrate the behaviour with a different numerical set-up, we have performed simulations with the WhiskyTHC framework adopting the DD2 model for the microphysics. The details of the simulations are provided in Appendix A. We consider two physical situations: an isolated neutron star and a binary merger for which the two stars (obviously) move across the numerical grid. \nThe results show that, at typical simulation resolutions, the bulk of an isolated neutron star heats up to temperatures of ∼ 1 -4 MeV (with the peak temperature slightly higher, see Figure 1) due to fluid shocks on \nthe grid. The evolution of the peak temperature reached in each simulation is displayed in Fig. 1. For the DD2 merger simulation, the results show that the artificial heating is further enhanced when the stars move across the numerical grid.', 'III. IMPACT ON TIDES': "The main question we want to answer is: What is the impact of the observed artificial heating on the main tidal parameters that one would aim to extract from gravitational-wave observations? In order to explore this issue, we solve for the linear perturbations of a spherically symmetric, perfect-fluid relativistic star. Our perturbation calculation closely follows Ref. [39] and we provide further details in Appendix B. From the outset, we assume that the star is immersed in a temperature profile T = T ( n b ). (This will be either uniform or lifted from a numerical simulation.) To determine the corresponding matter composition, we assume that the background fluid is in chemical equilibrium such that \nµ ∆ ( T, n b , Y e ) = 0 . (2) \nThis fixes Y e = Y e ( n b ) and reduces the thermodynamical state to depend solely on n b . The structure of the equilibrium neutron star is then straightforwardly obtained by solving the standard equations supplemented with the functions ε = ε ( n b ) and p = p ( n b ) obtained from the equation of state. This provides the background on which the linear perturbations are computed. Although our setup is similar to Ref. [39], here we are concerned with different temperature profiles, in particular those extracted from merger simulations. \nAs already mentioned, there are two regimes in a binary inspiral: the early regime, which is characterised by the static tide, and the late regime, where the dynamics become important (and non-linear aspects come into play). The static tide is commonly represented by the neutron star's tidal deformability Λ. This quantity enters the inspiral waveform through its mass-weighted average for the binary at fifth order in the post-Newtonian approximation [1, 40]. The tidal deformability provides a dimensionless measure of the star's susceptibility to a companion's gravitational field. If the nuclear matter is particularly stiff, the neutron star can support large deformations (large values of Λ). Meanwhile, if the stellar material is comparatively soft, the neutron star will be more compact and have a small Λ. These qualities make Λ an attractive observable with which to constrain the properties of dense nuclear matter [40], as demonstrated in the case of GW170817 [1-3]. \nWe determine the static, quadrupolar deformations using the formalism detailed in Ref. [6] to obtain Λ. The required static perturbations depend on an additional aspect of the nuclear matter, the equilibrium adiabatic in- \nex \nΓ = ε + p p dp dε , (3) \nwhich is determined from partial derivatives of the thermodynamical functions (see Ref. [39]). \nDuring the later, dynamical regime, the tidal driving frequency increases as the binary inspirals, radiating gravitational waves. As the orbital frequency increases, it will eventually become resonant with the low-frequency g -modes of the neutron star, as well as implicate the other higher frequency oscillations, like the f -and p -modes [9, 10]. Indeed, the f -mode is expected to provide the dominant contribution to the tide [13]. At this point, the assumption of a static tide breaks down and the problem becomes dynamical. In the absence of dissipation, the Newtonian oscillation problem is self-adjoint, implying that the modes form a complete basis in terms of which the tide can be decomposed. This mode-sum representation elegantly addresses the mathematical challenges of solving the full time-dependent, tidal response problem. Although motion in general relativity is inherently dissipative due to gravitational radiation, the natural vibrational modes of neutron stars are still expected to dominate the dynamical tide. \nTo represent the dynamical tide, we calculate the quadrupolar, quasi-normal oscillation modes of a neutron star using the equations provided in Refs. [41, 42]. The equation of state now enters the fluid perturbations through the adiabatic index \nΓ 1 = ε + p p ( ∂p ∂ε ) s,Y e , (4) \nassuming frozen composition during an oscillation ( i.e. , that the mode dynamics are fast enough that we may ignore nuclear reactions). Similarly to Γ, the determination of Γ 1 involves thermodynamical derivatives (for more detail, see Ref. [39]). \nThe two adiabatic indices Γ and Γ 1 characterise stratification in the fluid. In general, stellar material can support both entropy and composition gradients. When Γ 1 ≥ Γ, the star is convectively stable and supports lowfrequency g -mode oscillations. Meanwhile, when Γ 1 = Γ, the g -modes vanish. For realistic nuclear-matter models, the neutron star is expected to be stably stratified throughout, except possibly at low densities, and will therefore support g -mode oscillations. Indeed, a recent study has shown that these low-frequency fluid oscillations, which likely become resonant with the tide as the binary inspirals, may be within reach of next-generation observatories [43]. \nWe present a sample of numerical results in Table I. For each of the two nuclear-matter models we consider, we show the tidal parameters (static and dynamical) when the star has cold, uniform temperature and when the star is placed in a temperature profile obtained from a numerical-relativity merger simulation (modelled on the \nresults in Fig. 4 in Appendix A for DD2), assuming the same matter model (and stars with the same baryon mass). This provides an immediate quantitative measure of the impact of the artificial simulation temperatures on the tidal parameters. \nTABLE I. The effects on the tidal parameters due to artificial temperatures in numerical-relativity merger simulations. The table lists the stellar radius R , tidal deformability Λ, f -mode frequency ω f and first three g -mode frequencies ω g 1 , ω g 2 and ω g 3 . For each nuclear-matter model, APR and DD2, the parameters are shown for two neutron stars with different temperatures; one neutron star has uniform temperature, whereas the other is immersed in a merger simulation profile. The stars described by APR have baryon mass M b = 1 . 40 M ⊙ and those for DD2 have M b = 1 . 55 M ⊙ . It is evident that the neutron stars with simulation-level temperatures have significantly altered tidal parameters compared to the cold stars. \nFor the APR matter model, the numerical results show that Λ is 16% larger due to the artificially high temperature in the numerical simulation than for the corresponding cold neutron star. The difference is even starker with DD2, for which the deformability increases by more than 25% with the simulation temperature profile. It should be noted that the two equation-of-state cases are not intended to be directly compared to each other. Indeed, the involved neutron stars have different masses, as well as different numerical treatments for the hydrodynamics and the spacetime evolution. Rather, the point of the analysis is to illustrate that the artificial temperatures in two separate merger simulations lead to the same qualitative behaviour: The tidal parameters are significantly distorted. \nThe results for the tidal deformability are quite intuitive. The artificially high simulation temperatures contribute to a strong thermal pressure, which leads to larger stellar radii. In fact, from Table I, we see that the neutron-star radii approximately double. This leads to an effective stiffening of the nuclear matter and means that Λ-which scales with the areal radius R as ∝ R 5 -increases. It is not particularly surprising that higher temperatures lead to larger tidal deformabilities (this feature is generic). What is notable is the extent to which the values change. If we were to calibrate gravitationalwaveform models with these kinds of simulations, these aspects would inevitably manifest as systematic errors that bias parameter inference. \nWe now turn our attention to the effect on the dynamics. Presented in Table I are the (real) oscillation \nfrequencies of the f -mode and first three g -modes. The results show that these are also drastically altered by the artificial temperature. We see that, when the neutron star is placed in the temperature profile of a simulation, the spectrum shifts considerably. The f -mode is the most weakly affected of the oscillations, decreasing by 16% for the APR model, but only 5% for DD2. In contrast, the low-frequency g -modes are substantially altered, increasing in frequency due to enhanced entropy gradients caused by the high temperatures. \nFinally, we calculate the tidal parameters' evolution with respect to uniform temperature. The results are presented in Fig. 2 for APR nuclear matter and are consistent with the behaviour we have just discussed: The tidal deformability and g -mode frequencies increase with temperature, while the f -mode gradually oscillates slower. We see that the f -mode is the least affected by temperature. In each panel of Fig. 2, we show the tidal values from Table I using the simulation profile for comparison. The tidal deformability and g -mode frequencies in the simulation are very roughly similar to that of a star with uniform T ∼ 10 MeV. However, even at T = 15MeV, the frequency of the f -mode remains higher than that of the simulated neutron star.", 'IV. WORDS OF CAUTION': "We have demonstrated that the key tidal parameters of neutron stars in finite-temperature, merger simulations are severely distorted due to artificial heating arising from numerical shocks close to the stellar surface. Using data from two numerical-relativity simulations, with different numerical methodologies and matter prescriptions, we showed how quantities associated with both the static and dynamical tide are substantially shifted by the artificially high temperatures. Specifically, this includes the neutron-star tidal deformability, already constrained by the GW170817 observation, as well as the oscillation modes that dominate the dynamical tide, which we hope to probe with upcoming, next-generation gravitationalwave interferometers, the Einstein Telescope and Cosmic Explorer. Although we considered only two simulations, we expect these features to be generic for all grid-based numerical-relativity codes, as they all reach unphysically high temperatures. \nThe main take-away message from this work is cautionary. If we were to use results from finite-temperature non-linear simulations to calibrate gravitational waveforms, we may incur considerable systematic error in the parameter inference. Current instruments are not sensitive enough to the tide for these issues to manifest, but the problem will need to be addressed if we want to make precision observations with the Einstein Telescope and Cosmic Explorer. \nFuture work will need to be dedicated to either reducing the temperature systematics or correcting for them in the gravitational-wave analyses. As a first step, we \n<!-- image --> \nFIG. 2. The tidal deformability (left panel) and mode eigenfrequencies (right panel) against uniform temperature of an M b = 1 . 40 M ⊙ neutron star described by the APR equation of state. The frequencies correspond to the f - and first three g -modes, as indicated by the legend. The horizontal, dashed lines correspond to the parameters listed in Table I calculated using the simulation temperature profile. As the neutron star is heated up, the f -mode frequency decreases with larger neutron-star radius. The other quantities, including the tidal deformability and g -mode frequencies, increase. \n<!-- image --> \nneed to better understand the origin of the problem. This includes reconciling the results we have presented here with previous work using simpler equation-of-state models (like polytropes) and a phenomenological thermal (Γ-law) representation. For example, the results in Ref. [44] demonstrate much better agreement between simulations and perturbation theory. To some extent, the difference in the results likely stems from the fact that the phenomenological equation-of-state models assume chemical equilibrium. To illustrate this, consider Eq. (1) as a relation for the numerical errors. By enforcing chemical equilibrium we have µ ∆ = 0 and the final term does not contribute. Away from chemical equilibrium we would need the error dY e to 'match' the other errors to give a sensible temperature. However, in current simulations the evolved quantity is (roughly) n b Y e , and hence the accuracy will be poor at low densities. This is likely related to the observation-see for example the results presented in Fig. 3 of [31]-that finite-temperature equation-of-state simulations are prone to large errors at low temperatures. Of course, these arguments only hint at the origin of the problem we are discussing. We do not yet have a solution. \nIn order to make progress, it would be helpful to understand to what extent the issue we have raised is relevant for particle-based simulations, like those in Refs. [45, 46]. In principle, such simulations provide a better representation of the neutron-star surface, but it remains to be established to what extent this alleviates the artificial heating issue. \nLet us make two final remarks. First, and this is important in order to keep the discussion in the proper context, the problem we have discussed does not influence current gravitational-waveform models for neutron-star \ntides. These models, like the work in Ref. [27], do not (yet) involve thermodynamically consistent matter models and hence do not suffer the artifical heating problem (at least not at the level we have indicated here). Having said that, if we want to do better in the future (and we do!), then we need to get a handle on the temperature problem. Second, the artificial heating of low-density matter may have considerable impact on the related problem of matter ejecta, for which finite-temperature simulations are being used [47]. In this problem, the additional thermal pressure may tend to unbind matter and an elevated temperature will affect the composition (the electron fraction) of the outflows, which may in turn alter the nuclear reaction rates and the associated kilonova signature. Similarly, an artifically high temperature will impact on neutrino opacities, and hence need to be carefully considered in efforts to implement realistic neutrino transport [48]. If this concern is justified remains to be explored.", 'ACKNOWLEDGMENTS': 'NA and IH acknowledge support from STFC via grant number ST/R00045X/1. The authors thank Peter Hammond for sharing temperature data from a numericalrelativity merger simulation and Tim Dietrich and Nick Stergioulas for useful discussions. We acknowledge the use of the IRIDIS High Performance Computing Facility and associated support services at the University of Southampton. The software developed to support this article is available in a GitHub repository [49] and is written in the Julia programming language [50-54]. The figures were generated using matplotlib [55, 56].', 'Appendix A: Numerical simulations': 'For the finite-temperature inspiral-merger simulations we considered two distinct models. The first simulation used the APR matter model [31, 38] and was performed using the Einstein Toolkit [57]. The initial data was created using Lorene [58], while the hydrodynamical and spacetime evolution was performed using GRHydro [59-62] and McLachlan [63, 64], respectively. McLachlan uses the BSSN formulation [65-67] of the Einstein equations. The simulation was performed by setting the atmosphere temperature and rest-mass density to 0 . 02 MeV and 6 . 2 × 10 6 g cm -3 , respectively. A fourth-order Runge-Kutta method was employed, with the Courant-Friedrichs-Lewy condition set to 0 . 25. Neutron stars were tracked using NSTracker . \nThe second set of simulations, for the DD2 matter model, involved initial data generated using the Fuka solver [68] and the evolution of the hydrodynamics and spacetime are performed with WhiskyTHC [69-72] and CTGamma [73], respectively. We used the z4c formulation [74] of the Einstein equations, the Carpet adaptive mesh refinement driver (as in the APR model) [75] of Cactus [76], and track the neutron stars with the BNSTrackerGen component of WhiskyTHC . The atmosphere temperature and rest-mass density are 0 . 02 MeV and 6 . 2 × 10 3 g cm -3 , respectively. The time evolution was carried out using a fourth-order Runge-Kutta method and the Courant-Friedrichs-Lewy condition is set to 0 . 15. We used the Sophie Kowalevski release of the Einstein Toolkit [57, 77]. \nFirst of all, the artificial heating for simulations of isolated (static) neutron stars and three different numerical resolutions, corresponding to scales of ∼ 369 m, ∼ 185 m and ∼ 92 m, respectively, is provided in Fig. 3. The \n- [1] B. P. Abbott et al. , GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 , 161101 (2017).\n- [2] B. P. Abbott et al. , GW170817: Measurements of Neutron Star Radii and Equation of State, Phys. Rev. Lett. 121 , 161101 (2018).\n- [3] B. P. Abbott et al. , Properties of the Binary Neutron Star Merger GW170817, Phys. Rev. X 9 , 011001 (2019).\n- [4] M. Punturo et al. , The Einstein Telescope: a thirdgeneration gravitational wave observatory, Classical Quant. Grav. 27 , 194002 (2010).\n- [5] D. Reitze, R. X. Adhikari, S. Ballmer, B. Barish, L. Barsotti, G. Billingsley, D. A. Brown, Y. Chen, D. Coyne, R. Eisenstein, M. Evans, P. Fritschel, E. D. Hall, A. Lazzarini, G. Lovelace, J. Read, B. S. Sathyaprakash, D. Shoemaker, J. Smith, C. Torrie, S. Vitale, R. Weiss, C. Wipf, and M. Zucker, Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, in Bull. Am. Astron. Soc. , Vol. 51 (2019) p. 35, \nresults show that the star heats up to temperatures of ∼ 1 -4 MeV due to fluid shocks on the grid. It is notable that, while the shock originate at the surface, the heat rapidly propagates to the stellar interior as the simulation progresses. \nThe inspiral-merger simulation shows that the artificial heating is enhanced when the stars move across the numerical grid. The typical temperature profile extracted from our DD2 simulation, demonstrating that the peak temperatures reaches well above 10 MeV, is shown in Fig. 4.', 'Appendix B: Perturbative mode calculations': "The mode calculations closely follow the analysis in Ref. [39]. Specifically, to address noise with lowfrequency oscillations, we use an augment developed in Ref. [78] and perturbations in the exterior are calculated using the method from Ref. [79]. A mode with discrete (complex) frequency ω corresponds to a solution such that the ingoing gravitational waves in the exterior, with amplitude ˜ A in , vanish. \nThe mode-calculations show that, at low temperatures in the APR neutron star, the first g -mode-the g 1 -mode-is an interfacial i -mode that arises due to the core-crust phase transition [39]. Such oscillations appear when there is a discontinuity in the energy density of the equation of state. This identification is made clear by examining the mode's eigenfunctions, which possess a sharp cusp at the point in the star where the discontinuity lies. It is notable that this mode oscillates at a relatively high frequency of Re( ω/ 2 π ) = 508 . 5 Hz. To illustrate the changes, we show in Fig. 5 the neutron-star oscillation spectra for the finite-temperature DD2 equation of state. The qualitative results for APR, summarised in Table I, are the same. \narXiv:1907.04833 [astro-ph.IM]. \n- [6] T. 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The results are obtained from the evolution of a single isolated neutron star and the three panels represent different simulation resolutions. The left, middle and right panels show the low-resolution (LR), medium-resolution (MR) and high-resolution (HR) simulation results, with a grid spacing of ∼ 369 m, ∼ 185 m and ∼ 92 m in the finest grid, respectively. The neutron stars are modelled using the finite-temperature, composition-dependent DD2 equation of state. While the temperature in the core of the NS reaches ∼ 4 MeV in the low-resolution simulation, it reduces to ∼ 1 -2 MeV for the high-resolution simulation within nearly ∼ 1 . 6 ms. \n<!-- image --> \nFIG. 4. The typical temperature profile (in the equatorial plane) extracted from the DD2 binary neutron-star simulation. The histogram illustrates the distribution of baryon mass in the density-temperature plane. The data is extracted after 3 . 9 ms of evolution. \n<!-- image --> \n- [13] N. Andersson and P. 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The oscillation spectra of an M b = 1 . 55 M ⊙ neutron star described by the finite-temperature DD2 equation of state for two different temperature profiles; uniform temperature T = 0 . 2 MeV (left panel) and the simulation profile (right panel). Here, ˜ A in represents the amplitude of ingoing gravitational radiation and ω is the (real) frequency of the perturbations. A mode of the neutron star corresponds to when ˜ A in = 0. The weakly damped oscillations can be identified by the singularities in the spectrum. (Rapidly damped, spacetime w -modes are not visible.) The vertical, dotted lines indicate the mode eigenfrequencies listed in Table I. The high temperatures in the merger simulation lead to a substantial distortion of the mode spectrum. The f -and p -mode frequencies decrease. We see in the right panel how the first p -mode becomes visible. 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2024A&A...691A..26W

Context. Magnetohydrodynamic MHD turbulence plays a critical role in many key astrophysical processes such as star formation acceleration of cosmic rays and heat conduction. However its properties are still poorly understood. Aims. In this work we explore how to extract the intermittency of compressible MHD turbulence from synthetic and real observations. Methods. We used three statistical methods namely the probability distribution function kurtosis and scaling exponent of the multiorder structure function to reveal the intermittency of MHD turbulence. Results. Our numerical results demonstrate that 1 the synchrotron polarization intensity statistics can be used to probe the intermittency of magnetic turbulence by which we can distinguish different turbulence regimes 2 the intermittency of MHD turbulence is dominated by the slow mode in the subAlfvnic turbulence regime and 3 the Galactic interstellar medium ISM in the low latitude region corresponds to the subAlfvnic and supersonic turbulence regime. Conclusions.We have successfully measured the intermittency of the Galactic ISM from synthetic and realistic observations.

2024-11-01T00:00:00Z

['10.1051/0004-6361/202450414', 'arXiv:2409.05739', '2024arXiv240905739W', '10.48550/arXiv.2409.05739', '2024A&A...691A..26W']

['magnetohydrodynamics (MHD)', 'polarization', 'radiation mechanisms: non-thermal', 'ISM: magnetic fields', 'ISM: structure', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Exploring the intermittency of magnetohydrodynamic turbulence by synchrotron polarization radiation

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

https://arxiv.org/pdf/2409.05739.pdf

{'Exploring the intermittency of magnetohydrodynamic turbulence by synchrotron polarization radiation': "Ru-Yue Wang 1 , Jian-Fu Zhang 1 , 2 , 3 , Fang Lu 1 , Fu-Yuan Xiang 1 , 2 \n- 1 Department of Physics, Xiangtan University, Xiangtan, Hunan 411105, People's Republic of China, e-mail: jfzhang@xtu.edu.cn; flu@xtu.edu.cn\n- 2 Key Laboratory of Stars and Interstellar Medium, Xiangtan University, Xiangtan 411105, People's Republic of China\n- 3 Department of Astronomy and Space Science, Chungnam National University, Daejeon, Republic of Korea \nSeptember 10, 2024", 'ABSTRACT': 'Context. Magnetohydrodynamic (MHD) turbulence plays a critical role in many key astrophysical processes such as star formation, acceleration of cosmic rays, and heat conduction. However, its properties are still poorly understood. \nAims. We explore how to extract the intermittency of compressible MHD turbulence from the synthetic and real observations. \nMethods. The three statistical methods, namely the probability distribution function, kurtosis, and scaling exponent of the multi-order structure function, are used to reveal the intermittency of MHD turbulence. \nResults. Our numerical results demonstrate that: (1) the synchrotron polarization intensity statistics can be used to probe the intermittency of magnetic turbulence, by which we can distinguish di ff erent turbulence regimes; (2) the intermittency of MHD turbulence is dominated by the slow mode in the sub-Alfvénic turbulence regime; (3) the Galactic interstellar medium (ISM) at the low latitude region corresponds to the sub-Alfvénic and supersonic turbulence regime. \nConclusions. We have successfully measured the intermittency of the Galactic ISM from the synthetic and realistic observations. \nKey words. Magnetohydrodynamics-ISM: magnetic field-ISM: numerical-methods: synchrotron emission-methods', '1. Introduction': 'Intermittency is one of the properties of MHD turbulence, which has been studied in the solar wind (Veltri 1999; Carbone et al. 2004; Chen et al. 2014; Wang et al. 2015), interplanetary medium (Bruno et al. 2007; Zelenyi et al. 2015), magnetosphere (Xu et al. 2023) and interstellar medium (McKee & Ostriker 1977; Rickett 2011; Falgarone et al. 2011; Fraternale et al. 2019). The intermittency, associated with a small-scale coherent structure, can influence interstellar gas heating (Osman et al. 2011, 2012a,b; Wu et al. 2013; Chen et al. 2020; Phillips et al. 2023), energy dissipation (Zhdankin et al. 2016; Wan et al. 2012; Huang et al. 2022), and increased temperature anisotropy (Servidio et al. 2012; Osman et al. 2012b) in plasma turbulence. Moreover, this coherent structure plays an important role in particle acceleration (Décamp & Malara 2006; Lemoine 2021; Vega et al. 2023) and scattering (Butsky et al. 2023). Therefore, studying intermittency is significant for understanding and interpreting several astrophysical processes. \nWhen neglecting the intermittency e ff ect, Kolmogorov (1941) predicted that the power-law index ζ p of structure function of velocity fluctuations is proportional to its order p in the inertial range, namely ζ p = p / 3. The intermittency is ubiquitous in turbulence environments such as the solar wind (e.g., Osman et al. 2014) and di ff usion ISM (e.g., Falgarone et al. 2011). With intermittency, the relation between the order and the powerlaw index is expected to be a non-linear behaviour. Indeed, the later studies found that the scaling exponent ζ p gradually deviates from linear relation of ζ p = p / 3 as the order p increases (Anselmet et al. 1984; Meneveau & Sreenivasan 1987; Białas & Seixas 1990; Vincent & Meneguzzi 1991). To understand this \nphenomenon, two modified models have been proposed to describe incompressible hydrodynamic (She & Leveque 1994) and MHD turbulence (Müller & Biskamp 2000), respectively (see Section 2.1, and also Biskamp 2003; Beresnyak & Lazarian 2019 for more details). \nThe intermittency of MHD turbulence has been extensively investigated in numerical simulations (Falgarone & Passot 2008; Biskamp 2003; Esquivel & Lazarian 2010). For the incompressible homogeneous MHD turbulence, Yoshimatsu et al. (2011) concluded that the magnetic field is more intermittent than the velocity, consistent with other works (Cho et al. 2003; Haugen et al. 2004; Mallet & Schekochihin 2017). For the compressible MHDturbulence, Kowal et al. (2007) claimed that the density intermittency strongly depends on Alfvénic and sonic Mach numbers, with the velocity intermittency being di ff erent from density one. \nThe later simulations confirmed that the intermittency of MHD turbulence is closely related to its compressibility, scaledependent anisotropy, and magnetization. For instance, it was found that the intermittency of three plasma modes (Alfvén, slow, and fast) increases with the sonic Mach number in the weak magnetic field (Kowal & Lazarian 2010). The viscosity-damped MHD turbulence shows the scale-dependent intermittency, i.e., the more intermittency at much smaller scales (Cho et al. 2003). For the latter, Davis et al. (2023) concluded that with increasing magnetization, the velocity fluctuations display an inverse trend of the co-dimension of structures compared to the magnetic field. Yang et al. (2018) also presented that the e ff ect of MHDturbulence amplitude on the distribution of magnetic field is preferred to that of the angle to the local magnetic field. Besides, it is claimed that the intermittency is associated with the \ndriving way of turbulence (Federrath et al. 2008, 2009; Beattie et al. 2022). Based on statistics of centroid velocity, Federrath et al. (2010) found that the intermittency for compressive forcing is stronger than that for solenoidal one. \nNote that there are studies on measuring MHD turbulence intermittency in solar physics and astrophysics. For the former, with the solar wind data from the STEREO spacecraft, Osman et al. (2014) measured the intermittency of magnetic and Elsa sser field fluctuations in the solar wind and found that the intermittency in the direction perpendicular to the local magnetic field is stronger than that in the parallel direction. Considering density fluctuations, Chen et al. (2014) also found strong intermittency ranging from ion to electron scales. For the latter, the turbulence in molecular clouds exhibits small-scale and inertial-range intermittency (Hily-Blant et al. 2008). In addition, Falgarone et al. (2011) found the non-Gaussian statistic results and the existence of coherent structures in the di ff use interstellar medium (ISM). \nAs mentioned above, the measurement of intermittency has been performed using spectroscopic data of ISM and in situ data of the solar wind. Can we use synchrotron polarization observations to extract the intermittency of MHD turbulence? One purpose of our numerical studies is to explore the intermittency of magnetic fields and densities via synthetic observations. Another purpose is to study the intermittency of the Galactic ISM using realistic observations. Specifically, we first synthesize synchrotron polarization observations using numerical simulation data to study the intermittency of the magnetic field and density, and then adopt realistic observations from the Canadian Galactic Plane Survey (CGPS) to explore the intermittency of the Galactic ISM. From an observational perspective, this work will be dedicated to advancing the understanding of the intermittency of compressible MHD turbulence. \nThe structure of this paper is organised as follows. In Section 2, we give descriptions of theoretical models of intermittency, synchrotron radiative processes, and methods to characterise intermittency. Section 3 introduces numerical setup and decomposed method of compressible MHD turbulence. Section 4 presents numerical results, followed by the studies of intermittency from the observational data in Section 5. The discussion and summary are provided in Sections 6 and 7, respectively.', '2.1. Theoretical models related to intermittency': 'In the framework of incompressible hydrodynamic turbulence, Kolmogorov (1941) assumed that the turbulence is self-similar within the inertial range, in which the relationship between the velocity fluctuations δ u l and the scale l exhibits a simple scaling behaviour as follows \n⟨ δ u p l ⟩ ∼ l ζ p , ζ p = p 3 . (1) \nWhen the scaling exponent ζ p deviates from this relation, it will imply the appearance of intermittency phenomenon, which reflects the inhomogeneous distribution of fluctuations (see Chap. 7 in Biskamp 2003). Later, taking the scaling of velocity u l ∼ l 1 /g and energy cascade rate t -1 ∼ l -x into account, She & Leveque (1994) analytically proposed a classical non-linear scaling and expressed by \nζ p = p g (1 -x ) + C (1 -(1 -x / C ) p /g ) , \nArticle number, page 2 of 13 \nwhere C denotes the co-dimension of the dissipative structures related to the dimension of a dissipative structure D via the relation of C = 3 -D . \nIn the case of hydrodynamic turbulence, one usually considers the parameters g = 3 and x = 2 / 3 (according to Kolmogorov scaling). For the 1D vortex filament ( C = 3 -D = 2), Equation (2) can be simplified as \nζ p = p 9 + 2[1 -(2 / 3) p / 3 ] , (3) \nwhich is called the She & Leveque (SL) model in this paper. For the 2D sheet-like structure, it can be rewritten as (Müller & Biskamp 2000) \nζ p = p 9 + 1 -(1 / 3) p / 3 , (4) \nwhich is called the Müller & Biskamp (MB) model.', '2.2. Synchrotron radiative processes': "The production of synchrotron radiation requires two key factors, namely the relativistic electrons and the magnetic field. In this paper, we assume that the relativistic electron population follows isotropic pitch-angle distribution and has the following power-law relationship \nN ( E ) dE = N 0 E 2 α -1 dE , (5) \nwhere N ( E ) dE is the number density of relativistic electrons in the energy interval E and E + dE , N 0 a normalisation constant, and α = (1 -p ) / 2 the photon spectral index related to the electron index p . In the simulation below, we set the photon spectral index of α = -1 . 0 for simplicity. \nThe synchrotron radiation intensity is expressed as (Ginzburg & Syrovatskii 1965) \nI ( X ) ∝ Z L 0 B 1 -α ⊥ ( X , z ) dz , (6) \nwhere B ⊥ is the component of the magnetic field perpendicular to the line of sight (LOS), X = ( x , y ) a 2D vector in the plane of the sky (POS), and L the spatial length of emitting region. \nConsidering the linearly polarized properties of synchrotron radiation, we have the intrinsic polarization intensity \nP 0( X ) = p 0 I ( X ) , (7) \nwhere p 0 = (3 -3 α ) / (5 -3 α ) is the fraction polarization degree. The observable Stokes parameters Q , U can be expressed as Q ( X ) = P 0( X ) cos 2 ϕ and U ( X ) = P 0( X ) sin 2 ϕ , respectively. Here, the angle ϕ = ϕ 0 = π/ 2 + arctan( B y / B x) represents the polarization angle. When involving Faraday rotation e ff ect, this angle can be expressed as ϕ = ϕ 0 + λ 2 φ , with Faraday rotation measure (RM) of φ ( X , z ) = 0 . 81 R z 0 n e( X , z ' ) B ∥ ( X , z ' ) dz ' rad m -2 , where n e represents the number density of thermal electrons and B ∥ the component of the magnetic field along the LOS. Defining the complex polarization vector of P = Q + iU , we have the synchrotron polarization intensity (SPI) of \n(2) P = p Q 2 + U 2 . (8)", '2.3. Methods to characterise intermittency': 'Firstly, the appearance of intermittency can be revealed by the probability distribution function (PDF). To characterise the statistical behaviour at a specific separation R , we can calculate the PDF of the dispersion δ F ( R ), which is defined as \nδ F ( R ) = F ( X + R ) -F ( X ) (9) \nfor any fluctuation quantity F , where X represents a 2D position vector in the POS. In general, the PDFs of fluctuations exhibit a non-Gaussian distribution with two extended tails when the intermittency occurs. \nSecondly, to further understand the intermittency over the whole spatial scale, we need to use another method such as kurtosis and scaling exponent. The multi-order structure function can be defined as \nSF p ( R ) = ⟨| F ( X + R ) -F ( X ) | p ⟩ , (10) \nwhere ⟨ ... ⟩ represents a spatial average of the system. Using the second- and fourth-order structure functions, we can define the kurtosis as (Bruno et al. 2003) \nK = SF4( R ) (SF2( R )) 2 , (11) \nthe value of which will reflect the distribution of fluctuations. If K , 3, the fluctuations have a non-Gaussian distribution. In addition, the K change with R can characterise the level of intermittency (Frisch 1995). When K grows faster, the fluctuations are more intermittent. When K remains a constant within a certain scale range, the fluctuations are self-similar and not intermittent. \nThirdly, intermittency can also be measured by the scaling exponent of the multi-order structure function. The multi-order structure function is related to the separation scale R within the inertial range, and described by a power-law relation of \nSF p ( R ) ∝ R ζ ( p ) , (12) \nwhere ζ ( p ) is the absolute scaling exponent related to the order of structure function. In this paper, we adopt the extended selfsimilarity hypothesis (Benzi et al. 1993), that is, the power-law scaling can be extended from the inertial range to the dissipation scale. Under this hypothesis, we explore the scaling exponent ξ ( p ) between the 3rd- and p th-order structure functions, by which we distinguish the intermittency level. When the relation between the scaling exponent ξ ( p ) and the order p is nonlinear, it represents the presence of intermittency with the multi-fractal feature.', '3. MHD Turbulence Simulation': 'The second-order-accurate hybrid essentially non-oscillatory code (see Cho & Lazarian 2003) is used to solve the ideal singlefluid MHD equations (i.e., only including the proton component ρ to simulate MHD turbulence) \n∂ρ ∂ t + ∇ · ( ρ u ) = 0 , (13) \nρ [ ∂ u ∂ t + ( u · ∇ ) u ] + ∇ p g -J × B 4 π = f , (14) \n∂ B - ∇ × ( u × B ) = 0 , \n∂ t (15) \n∇ · B = 0 , (16) \nwhere t is the evolution time of turbulence, p g = c 2 s ρ the thermal gas pressure, J = ∇ × B the current density, and f a random driving force. These physical quantities are dimensionless. The computation domain is a cube with a side length of 2 π . Periodic boundary conditions are applied at the computational boundaries. \nUsing a numerical resolution of 512 3 , we drive the turbulence by a solenoidal driving force acting on the wavenumber of k ≈ 2 . 5, with a continuous injection of energy. We use a threestage Runge-Kutta method for time integration, in units of the large eddy turnover time of ∼ L /δ V . Meanwhile, we also set the initial magnetic field ( B init) along the x axis and the gas pressure ( P init). To characterise di ff erent models, we define three parameters: Alfvénic Mach number M A = V L / V A, sonic Mach number M s = V L / c s and plasma parameter β = 2 M A 2 / M s 2 , where V L is the injection velocity, and V A = B init / p 4 πρ is the Alfvénic velocity. The first two parameters characterise the strength of magnetic field and compressibility, respectively. The latter indicates the ratio of thermal to magnetic pressure, in which the magnetic field is dynamically important ( β < 1) or unimportant ( β > 1). The related parameters are listed in Table 1. \nBased on data cubes, we decompose compressible MHD turbulence into three modes in Fourier space, the unit vectors of which are defined by (Cho & Lazarian 2002) \nˆ Ξ f ∝ (1 + β 2 + √ D )( k ⊥ ˆ k ⊥ ) + ( -1 + β 2 + √ D )( k ∥ ˆ k ∥ ) , (17) \nˆ Ξ s ∝ (1 + β 2 -√ D )( k ⊥ ˆ k ⊥ ) + ( -1 + β 2 -√ D )( k ∥ ˆ k ∥ ) , (18) \nˆ Ξ A ∝ -ˆ k ⊥ × ˆ k ∥ , (19) \nwith D = (1 + β 2 ) 2 -2 β cos 2 θ and cos θ = ˆ k ∥ · ˆ B . When projecting the magnetic field onto these unit vector directions, we obtain the magnetic field components of each mode in Fourier space. These projection quantities are then transformed into a real space to recover the corresponding magnetic field.', '4.1. The measurement of intermittency arising from different turbulence regimes': 'To generate synthetic observations, we calculate the SPI via Equation (8) using the above data cubes, with the assumption of the thermal electron density n e proportional to the plasma density ρ , i.e., n e = ρ when involving Faraday rotation measure. We use the typical values of the Galactic ISM to parameterise dimensionless physical quantities. Here, we just provide three key parameters such as the spatial length of L = 100 pc along the LOS, the thermal electron density n e = 0 . 1 cm -3 , and the magnetic field strength B = 1 . 23 µ G. With the numerical resolution of 512 pixels, we have a mesh grid of 100 pc / 512 ∼ 0 . 2 pc, corresponding to the smallest resolved spatial length. \nWe first analyse the PDFs of SPI arising from four turbulence regimes (see Table 1). The resulting finding is shown in Fig. 1, from which we see that PDFs at di ff erent scales R exhibit other characteristics that deviate from the Gaussian distribution. In general, this deviation mainly occurs in the two tail parts of the Gaussian distribution. With the decreasing scale R , we find that the level of deviation from both tails increases (except for panel (a)), indicating intermittent enhancement. Comparing all \nfour scenarios, the PDFs can qualitatively reveal the presence \nor disappearance of intermittency. Next, we will quantitatively \nevaluate the intermittency level using the kurtosis. \nTable 1. Di ff erent models of compressible MHD turbulence. \nNotes. B init- magnetic field strength; P init- gas pressure; M A - Alfvénic Mach number; M s-sonic Mach number; β -plasma parameter; δ B rms-root mean square of the random magnetic field. B init and P init are our initial parameter setting, respectively. The other resulting values are obtained from the final snapshot data. \n(a) Run1 \n<!-- image --> \nFig. 1. The PDFs of SPI normalised by its standard deviation σ at di ff erent scales R , arising from four di ff erent turbulence regimes. The black dashed lines represent the Gaussian distributions. \n<!-- image --> \nThe kurtosis distributions of SPI over the separation scale R are presented in Fig. 2, where the horizontal dashed line corresponds to K = 3 representing the kurtosis value of Gaussian distribution shown in Fig. 1. As shown in Fig. 2, the kurtosis values of Run2 and Run3 decrease faster than those of Run1 and Run4 as the separation scale increases. This reflects the more obvious intermittency of both Run2 and Run3 at small scales. In other words, the greater the deviation from K = 3, the more the intermittency. The kurtosis of SPI shows an irregular coupling between the Mach numbers M A and M s. Although there is intermittency at a small scale, we cannot find a significant correlation between the kurtosis distribution and Mach numbers. However, it is apparent that the most intermittency corresponds to the largest deviations of β from unity. \nIn addition to the methods discussed above, the scaling exponent of multi-order structure functions is the third method for measuring intermittency. Specifically, we use the extended selfsimilarity (Benzi et al. 1993) to obtain the scaling exponent between the 3rd- and p th-order structure function. Here, we first explore how di ff erent fitting ranges for R a ff ect the scaling exponent. Our results are shown in Fig. 3 (a) describing the relation between the scaling exponent and the order at di ff erent upper limits of the inertial range. In this figure, the dotted, dashed, \nFig. 2. The kurtosis of SPI as a function of the separation scale R in di ff erent turbulence regimes. The horizontal dashed line corresponds to the kurtosis values of the Gaussian distribution. \n<!-- image --> \nand dash-dotted lines indicate theoretical results provided by the Kolmogorov, SL, and MB models, respectively. The error bar represents the standard deviation. From panel (a), we know that di ff erent upper limits of the expected inertial range have little e ff ect on the measurement of the scaling index, thus we fix the \nseparation scale R = 15 . 6 pc as an upper limit in Fig. 3 (b). Similarly, Fig. 3 (b) explores the influence of di ff erent lower limits of the approximate dissipation scale on the scaling index. It is clear that the distribution of ξ ( p ) with p behaves similarly except for the results in the range of 0 . 2 -15 . 6 pc. This may be a ff ected by the numerical dissipation. \nBased on the above exploration, we fix the lower and upper limits of spatial scales as R = 0 . 6 and R = 15 . 6 pc, respectively. As is shown in Fig. 3 (c), the scaling exponents in di ff erent turbulence regimes have di ff erent deviations from the Kolmogorov model. For Run1, i.e., the sub-Alfvénic and subsonic turbulence, the scaling exponent is close to linear, revealing weak intermittency. For Run4, corresponding to the super-Alfvénic and supersonic case, we see that there is a significant deviation from the Kolmogorov model at large order p , reflecting the presence of intermittency. For the other turbulence regimes (see Run2 and Run3), the distributions of the scaling exponent are almost close to the SL model, characterising more intermittency.', '4.2. The measurement of intermittency at different frequencies': 'Based on Run1, we explore the influence of frequency on the kurtosis and scaling exponent of SPI, respectively. The numerical results are shown in Fig. 4, from panel (a) of which we see that the frequency has a significant e ff ect on the kurtosis profiles. At low frequencies, the kurtosis shows a dramatic rise toward the small scales, while at high frequencies, the kurtosis steadily increases over the small scales. \nThis reflects that the intermittency at low frequencies becomes more significant than that at high frequencies, which may be due to the Faraday depolarization e ff ect at low frequencies making more inhomogeneous structures. It can be seen that the most significant change for kurtosis occurs at the frequency of 0 . 4 GHz, indicating strong intermittency. Fig. 4 (b) shows that the scaling exponent of SPI displays di ff erent behaviours at different frequencies. At the frequency of 0 . 4 GHz, there is an evident nonlinear relation close to the SL model. At the frequency of 0 . 5 GHz, the scaling exponent is slight departure from the Kolmogorov model, indicating weak intermittency. Moreover, the scaling index is close to linear relation at high frequencies of 1.4 and 10 GHz, indicating a weaker intermittency. As a result, the two methods consistently demonstrate that in the lowfrequency range explored in this paper, the SPI statistics can probe the intermittency of magnetic turbulence.', '4.3. The measurement of intermittency using Faraday rotation measure': 'The kurtosis of RMs as a function of the separation scale R is presented in Fig. 5 (a), from which we see that there are large kurtosis values of RMs at small R , while small values at large R . In addition, we also find that both supersonic turbulence cases (see Run2 and Run4) show much larger kurtosis values than subsonic ones (Run1 and Run3). This reveals that the RMs in the supersonic turbulence regime are more intermittent than those in the subsonic one. The reason may be that the formation of shocks in the supersonic turbulence increases the intermittency of MHD turbulence. Amongst four cases, kurtosis values for Run2 are the largest, exhibiting the strongest intermittency. \nFig. 5 (b) shows the scaling exponent for RMs as a function of the order in four di ff erent turbulence regimes. As shown in this panel, although all the profiles show multifractal features, the \nTable 2. The CGPS archive data observed at 1 . 42 GHz. \nscaling exponent ξ ( p ) varying with the order p behaves di ff erently in di ff erent sonic turbulence regimes. In the subsonic turbulence regimes, the profile of ξ ( p ) almost follows the SL model, while it deviates far from the three theoretical models in the supersonic turbulence regimes. This reveals that the RMs in supersonic turbulence regimes are more intermittent than those in subsonic turbulence ones. Compared with three theoretical models, the scaling exponent in the sub-Alfvénic and supersonic turbulence regime displays the largest deviation, which reveals the largest intermittency in this turbulence regime. Consequently, we find that the statistics of the Faraday rotation measure can recover the intermittency of MHD turbulence.', '4.4. The measurement of intermittency of plasma modes': 'We first explore the kurtosis and scaling exponent for three plasma modes in the case of sub-Alfvénic and subsonic turbulence (i.e., Run1). The results are shown in the upper row of Fig. 6, from panel (a) of which we see that the kurtosis of Alfvén and slow modes decreases with the separation scale R , while that of fast mode remains almost unchanged. Note that the kurtosis variations of slow mode are the largest at small scales. This reflects the fact that the intermittency of SPI in the sub-Alfvénic and subsonic turbulence regime is dominated by slow mode. From Fig. 6 (b), we see that the scaling exponents for fast and slow modes follow the Kolmogorov and SL models, respectively, while for Alfvén mode the distribution of scaling exponents lies in these two models. This may be related to the anisotropic level of three modes in the sub-Alfvénic and subsonic turbulence regime. As demonstrated by Wang et al. (2020), slow mode results in inhomogeneous structures because of its strong anisotropy, while fast mode produces uniform fluctuations due to its isotropy. This suggests that slow mode has the strongest intermittency, while fast mode has no intermittency. \nMoreover, the lower row of Fig. 6 explores the case of subAlfvénic and supersonic turbulence (i.e., Run2). Fig. 6 (c) shows that the kurtosis of SPI for three modes exhibits di ff erent increasing levels as the separation scale decreases. It can be seen that the kurtosis of slow and fast modes shows a dramatic rise at small separation scales, while that of Alfvén mode rises slowly. This indicates that the former two have stronger intermittency, while the latter does not manifest significant intermittency. This should be caused by the compressible nature of slow and fast modes in this turbulence regime. From Fig. 6 (d), we see that the SPI for three modes displays nonlinear scaling exponents. The scaling exponents for Alfvén mode are consistent with the MB model, while those for the other two modes deviate from this model. As a consequence of these two methods, the SPI for slow mode dominates the intermittency of MHD turbulence. In addition, compared with the results of Fig. 3 (c), we find that the intermittency of SPI for postdecomposition MHD modes is stronger than that for predecomposition MHD modes in the sub-Alfvénic and supersonic turbulence regime, which may be weakened by the coupling of three modes. \nFig. 3. The scaling exponent as a function of the order for the SPI at three scenarios: the lower limits of the fixed R (panel (a)), the upper limits of the fixed R (panel (b)), and di ff erent turbulence models (panel (c)). The results of panels (a) and (b) are obtained by Run2. \n<!-- image --> \n<!-- image --> \nFig. 4. The kurtosis (panel (a)) and scaling exponent (panel (b)) of SPI at di ff erent frequencies for the simulation of Run1. The horizontal dashed line plotted in panel (a) corresponds to the kurtosis value of Gaussian distribution. \n<!-- image --> \n<!-- image --> \nFig. 5. The kurtosis (panel (a)) and scaling exponent (panel (b)) of Faraday rotation measure in di ff erent turbulence regimes. The horizontal dashed line plotted in panel (a) corresponds to the kurtosis value of Gaussian distribution. \n<!-- image --> \n(a) \n(b) \nFig. 6. The kurtosis (left column) and scaling exponent (right column) of SPI for three modes. The results obtained by Run1 and Run2 are shown in the upper and lower rows, respectively. The horizontal dashed lines plotted in the left column correspond to the kurtosis values of Gaussian distribution. \n<!-- image --> \nFig. 7. The PDFs of SPI are normalised by its standard deviation σ at di ff erent angles θ , arising from four sets of CGPS data. The black dashed lines represent the Gaussian distributions. \n<!-- image --> \n(a) \n<!-- image --> \nFig. 8. The kurtosis (panel (a)) and scaling exponent (panel (b)) of SPI for four sets of CGPS data at 1.42 GHz. The horizontal dashed line plotted in panel (a) corresponds to the kurtosis values of Gaussian distribution. \n<!-- image -->', '5. Application to observations': 'In this section, we explore the intermittency of the Galactic ISM using the archive data from the CGPS at 1 . 42 GHz. 1 The CGPS is a project involving radio, millimeter, and infrared surveys of the Galactic plane to provide arcminute-scale images of all major components of the ISM over a large part of the Galactic disk (Taylor et al. 2003). The synchrotron radio surveys are carried out at the Dominion Radio Astrophysical Observatory (DRAO). The DRAO Synthesis Telescope surveys have imaged a 73 · section of the Galactic plane between April 1995 and June 2000. The surveys cover the region with the longitude range of 74 · . 2 < l < 147 · . 3 and the latitude extent of -3 · . 6 < b < + 5 · . 6. The full area of the CGPS is covered by 36 mosaics, each mosaic of which has a resolution of 1024 × 1024 pixels corresponding to the 5 · . 12 × 5 · . 12 region on the POS. \nWe explore the ISM turbulence intermittency by extracting the resolution of 512 × 512 pixels from the 1024 × 1024 mosaic image to avoid the margin of images. As a representative example, we firstly provide PDFs from four mosaic images with the coordinate information listed in Table 2, as shown in Fig. 7. In practice, we filter noise-like structures of data by a Gaussian kernel of σ = 2 pixels. From this figure, we can see that PDFs at different angle scales present di ff erent deviation levels of two tails from the Gaussian distribution. As the angle scale decreases, the deviation from the normal distribution increases, revealing an increment of intermittency. Moreover, it is found that the PDFs in four scenarios exhibit the presence of intermittency; which scenario has more abundant intermittency still needs to further be explored. \nNext, we will quantitatively explore the degree of intermittency in four scenarios. Fig. 8 (a) depicts the kurtosis of SPI as a function of the separation angle for four sets of CGPS data. It can be seen that the kurtosis of SPI for MA2 and MC2 varies notably with the separation angle, while the kurtosis for MB2 and ME2changes slowly. In this regard, we conclude that the SPI for MA2 and MC2 displays more intermittency than that for MB2 and ME2. Fig. 8 (b) presents the relation between the scaling ex- \nponent and order for the same data sets. It clearly shows that all the curves of the scaling exponent are nonlinear, characterising the presence of strong intermittency. For MB2 and ME2, the profiles of the scaling exponent coincide with those of MB models, while for MA2 and MC2, the scaling exponent deviates greatly from this model. This proves that the latter is more intermittent than the former. Compared with the results of synthetic observations (see Section 4), we predict that the ISM around the Galactic plane corresponds to the sub-Alfvénic and supersonic regimes (Falceta-Gonçalves et al. 2008; Heyer et al. 2008; Burkhart et al. 2009).', '6. Discussion': 'In this paper, we mainly explore the intermittency of MHD turbulence by the SPI and RM statistics. At su ffi ciently high frequencies, SPI statistics can capture the intermittency of the projected magnetic fields B ⊥ on the plane of the sky, while at low frequencies, it can provide insights into the intermittency properties of more physical quantities such as B ∥ , B ⊥ , and n e. At lower frequencies, the intermittency of SPI can also be a ff ected by the noise-like structures. On the other hand, RM statistics can directly reflect the total intermittency for both n e and B ∥ , without involving the e ff ect of B ⊥ . We also tested the contribution of n e and B ∥ to the intermittency of RM separately, and found that the former contributes more than the latter. \nWe have utilized three common statistical methods - the PDFs, the kurtosis, and the scaling exponent of the multi-order structure function - to explore the intermittency of MHD turbulence. The PDFs act as an indicator for the presence of intermittency when they deviate from a Gaussian distribution. Note that this method can only qualitatively reflect the intermittency level at a certain separation scale. Di ff erently, the kurtosis and scaling exponent can provide a quantitative estimation of intermittency. The former can display the intermittency over all the separation scales. If the kurtosis varies faster with the separation scale, it implies that the fluid-structure is more intermittent. For the latter, when the relation between the scaling exponent and the order p becomes nonlinear, this means the multifractal feature of the fluctuations and the presence of intermittency. Our \nstudies demonstrated that the results from the three methods explored are self-consistent, the synergy of which can provide a more comprehensive understanding of MHD turbulence intermittency. \nThis work is carried out in the framework of the modern understanding of MHD turbulence theory (Goldreich & Sridhar 1995). Considering that the magnetic field and velocity retain the same cascade properties, we use three theoretical models related to velocity, i.e., the Kolmogorov, SL, and MB models, to characterise the intermittency levels of magnetic turbulence. We also tested the intermittency of the 3D magnetic field and velocity and found that they have slightly stronger intermittency than that revealed by the statistics of the SPI. Therefore, we speculate that the measured intermittency should be slightly weaker than the underlying MHD turbulence intermittency. We think that the projection e ff ect, i.e., integration along the LOS, attenuates the intrinsic intermittency amplitude of the Galactic ISM. Similarly, the direct numerical simulations also claimed that the 3D simulation shows more intermittency than the 2D one (e.g., Schmidt et al. 2008; Brunt et al. 2003; Brunt & Mac Low 2004); the latter is a projection from the 3D case. \nCompared to the results from sub-Alfvénic and subsonic turbulence (see upper panels of Fig. 6), our results demonstrated that in the case of sub-Alfvénic and supersonic (with the presence of shocks) turbulence (see lower panels of Fig. 6), the intermittency of solenoidal mode is intensified. At the same time, we also see the intensification from intermittency of compressive (slow and fast) modes. Note that these results we obtained are limited in the framework of solenoidal driving used in this paper, which would cause more kinetic energy to be in solenoidal motions (e.g., Federrath et al. 2011). In any case, this suggests that shocks may be a source of intermittency. \nEarlier studies claimed that the change of spectral index only a ff ects the amplitude of the structure function, which is only limited to the second-order structure function (Lazarian & Pogosyan 2016; Zhang et al. 2018). In this paper, we find that the change of spectral index ( α ) a ff ects not only the amplitude but also the scaling exponent for higher-order structures. As the spectral index increases, the intermittency of SPI becomes abundant. This is a new point. For the studies of other cases, we set the spectral index α = -1 for the calculation of SPI. Theoretically, one expects a bottleneck e ff ect in the power spectra of the velocity and magnetic field at large wavenumbers. However, the power spectra obtained by our data cubes do not significantly show this e ff ect (Wang et al. 2022; Kowal & Lazarian 2010). This e ff ect does not influence the measurement of intermittency. However, the numerical dissipation at the smallest resolved spatial length 0 . 2 pc a ff ects the results. \nFor our numerical studies, we provided the results up to the order p = 8. We found that with increasing the order, the distributions of the scaling exponent are self-similar extending, that is, the increase in the order does not change our numerical results. However, when we use the higher order ( p > 12), there are significant fluctuations due to the limitation of the numerical resolution. For the realistic observational data, the maximum order of the structure function is only taken to the 6th order. When the 8th order is reached, the results will show abnormal fluctuations due to the denoising of the real data. \nRecently, the properties of MHD turbulence have been studied using the synchrotron polarization statistics (see Zhang & Wang 2022 for a recent review), including the spatial and frequency analysis techniques (Lazarian & Pogosyan 2016; Zhang et al. 2016; Lee et al. 2016), gradient techniques (e.g., Lazarian &Yuen 2018; Zhang et al. 2019), and quadrupole ratio modulus \n(Lee et al. 2019; Wang et al. 2020). It is stressed that these works mainly focused on the inertial range of turbulence cascade. Differently, our current work covers a wide range of spatial scales, particularly involving the small-scale non-noise structure, to understand the properties of compressible MHD turbulence. \nCho & Lazarian (2010) proposed that analysing intermittent features can separate foreground signals from cosmic microwave background signals via the high-order structure function. The intermittency can also explain the observed strong and rapid variations from pulsar magnetosphere (Zelenyi et al. 2015). We expect that measuring intermittency may distinguish the di ff erence between Goldreich & Sridhar (1995) and Boldyrev (2006) theories, which will be discussed elsewhere.', '7. Summary': 'Using real observational data from the CGPS together with MHD turbulence simulation, we have investigated how to recover the intermittency of the magnetized ISM. The main results are briefly summarised as follows. \n- -The SPI statistics can be used to probe the intermittency of MHDturbulence. The most significant intermittency appears in the sub-Alfvénic and supersonic turbulence regime, while the least intermittency in the sub-Alfvénic and subsonic one.\n- -The intermittency measured by the SPI depends on the level of the Faraday depolarization. The intermittency measured by the RM shows a strong dependence on the sonic Mach number, with significant intermittency occurring in the supersonic turbulence regime. Therefore, RM statistics can recover the intermittency of thermal electron density and magnetic field component along the LOS.\n- -Slow mode dominates the intermittency of MHD turbulence in the sub-Alfvénic turbulence regime, where Alfvén (for supersonic) and fast (for subsonic) modes almost present a negligible intermittency.\n- -With realistic observations from the CGPS, we find that the Galactic ISM at the low latitude region corresponds to the sub-Alfvénic and supersonic turbulence regime. \nAcknowledgements. Wewould like to thank the anonymous referee for constructive comments that have significantly improved our manuscript. 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V. 2015, Journal of Plasma Physics, 81, 395810401\n- Zhang, J.-F., Lazarian, A., Lee, H., & Cho, J. 2016, ApJ, 825, 154\n- Zhang, J.-F., Lazarian, A., & Xiang, F.-Y. 2018, ApJ, 863, 197\n- Zhang, J.-F., Liu, Q., & Lazarian, A. 2019, ApJ, 886, 63 \nZhang, J.-F. & Wang, R.-Y. 2022, Frontiers in Astronomy and Space Sciences, \n9, 869370 \nZhdankin, V., Boldyrev, S., & Chen, C. H. K. 2016, MNRAS, 457, L69", 'Appendix A: Multi-order structure functions': 'In this appendix, we provide the multi-order structure functions of the SPI and RM in Figs. A.1 and A.2, respectively. These figures exhibit the structure functions with di ff erent orders as a function of SF3( R ). As is shown in these figures, we can see a good power-law relation. For quantitative characterization, we perform a linear fitting process in the interval between R = 0 . 6 and R = 15 . 6 pc. According to the fitting from these two figures, we can further obtain the results of Figs. 3 (c) and 5, respectively, where the uncertainties arising from the linear fitting are reflected by the error bars, i.e., the standard deviations. \n(a) Run1 \n(b) Run2 \nFig. A.1. Structure functions of the SPI with di ff erent orders (from p = 1 to 8) as a function of SF3( R ) under the extended self-similarity hypothesis. The vertical blue, green, and red dotted lines denote the values of third-order structure function in the smallest resolved spatial length, transition scale, and dissipation scale, respectively. The black dashed lines show a linear fit to the structure function on log-log scales. \n<!-- image --> \nWang et al.: Exploring the intermittency by synchrotron polarizationFig. A.2. Same as Fig. A.1, but for the multi-order structure functions of the RM statistics. \n<!-- image -->'}

2024arXiv240907248H

Extracting information about the gravitational background from black hole images is both important and challenging. In this study we use a physically motivated plasma model typically applied to stationary axisymmetric spacetimes to demonstrate that in a rotating black hole spacetime the polarizations of emitted light near the event horizon depend solely on the spacetime geometry independent of the plasma flow geometry. We confirm that the framedragging effect of a rotating black hole governs the observed polarization structure in the nearhorizon image. This finding indicates a unique imprint of the black hole spin on the polarization of the nearhorizon image. We anticipate that refined observations of nearhorizon emissions by the nextgeneration Event Horizon Telescope will enable us to determine the black hole spin in a straightforward manner.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.07248', 'arXiv:2409.07248', '2024arXiv240907248H']

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

Unique Imprint of Black Hole Spin on the Polarization of NearHorizon Images

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07248.pdf

{'Unique Imprint of Black Hole Spin on the Polarization of Near-Horizon Images': "Yehui Hou ¶ , 1 Jiewei Huang ¶ , 2 Yosuke Mizuno, 3, 4, ∗ Minyong Guo, 2, 5, † and Bin Chen 6, 7, 1, 8, ‡ \n1 School of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China \n2 School of physics and astronomy, Beijing Normal University, Beijing 100875, P. R. China \n3 \nTsung-Dao Lee Institute, Shanghai Jiao-Tong University, Shanghai, 201210, P. R. China \n4 School of Physics & Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240, P. R. China \n- 5 Key Laboratory of Multiscale Spin Physics, Ministry of Education, Beijing 100875, P. R. China 6 Institute of Fundamental Physics and Quantum Technology, \nNingbo University, Ningbo, Zhejiang 315211, China \n7 School of Physical Science and Technology, Ningbo University, Ningbo, Zhejiang 315211, China 8 Center for High Energy Physics, Peking University, \nNo.5 Yiheyuan Rd, Beijing 100871, P. R. China \nExtracting information about the gravitational background from black hole images is both important and challenging. In this study, we use a physically motivated plasma model, typically applied to stationary, axisymmetric spacetimes, to demonstrate that in a rotating black hole spacetime, the polarizations of emitted light near the event horizon depend solely on the spacetime geometry, independent of the plasma flow geometry. We confirm that the frame-dragging effect of a rotating black hole governs the observed polarization structure in the near-horizon image. This finding indicates a unique imprint of the black hole spin on the polarization of the near-horizon image. We anticipate that refined observations of near-horizon emissions by the next-generation Event Horizon Telescope will enable us to determine the black hole spin in a straightforward manner. \nIntroduction -The images of the supermassive black holes, recently released by the Event Horizon Telescope (EHT) Collaboration [1-9], provide a new window to study various strong-field problems in astrophysics. One of the most crucial tasks is to determine the parameters of the central black hole embedded in the surrounding spacetime. The mass and spin of the black hole play a central role in this determination. Regarding spin measurements, in addition to the well-known indicators such as the photon ring [2, 10] and sub-images that could be resolved by the next-generation EHT [11-13] and space very-long baseline interferometry (VLBI) projects [14, 15], there are other methods available. These include brightness asymmetry [16], photon ring auto-correlation [17], inner shadow measurement [18, 19], and the polarization structure [4, 5]. Among these methods, the timeaveraged polarization structure of the image is particularly regarded as one of the most reliable for estimating black hole spin [20-23]. \nA rotating black hole theoretically exhibits the framedragging effect [24], which causes the accretion flow, threaded with magnetic fields, to adopt a spiral shape. This spiral shape imparts a distinctive signature in linear polarizations [25]. Consequently, measuring the helical behavior of the polarization structure could offer valuable insights into the black hole spin [23]. The method for determining the spin of black holes using polarized \nimages typically involves comparing observational data with the polarized images generated from general relativistic magnetohydrodynamics (GRMHD) simulations and general relativistic radiative transfer (GRRT) calculations [26-29]. However, these simulations rely on several assumptions. In particular, the uncertainty of electron temperature significantly affects the resultant polarized images. Moreover, the GRMHD simulations are computationally intensive, making it challenging to explore larger parameter ranges. Additionally, concerns have been raised about the validity of fluid treatment at the horizon scale [30], underscoring the need for a more comprehensive approach to model the accretion flows. Alternative methods, such as Vlasov dynamics or particle-in-cell simulations, have been suggested [31, 32], although performing large-scale simulations remains difficult due to the spatial scale being determined by the ion Larmor radius. Despite these advancements, it remains a significant challenge to simulate the plasma environment around black holes accurately. \nIndeed, the primary challenge arises from the unknown properties of the accreting plasma [33, 34]. It is particularly difficult to distinguish whether the observed polarization pattern is due to gravitational effects or the intrinsic electromagnetic behavior of the accreting plasma. This ambiguity complicates the process of inferring the black hole's spin from the overall polarization image. To simplify the measurement of black hole spin and reduce the reliance on complex physical models, we must ask: Is it possible to identify a polarization quantity that depends solely on gravitational effects, independent \nof plasma flow geometry? Such a polarization quantity could provide more direct insights into black hole characteristics, thereby enhancing our understanding of their properties. \nIt is hypothesized that near the event horizon, the magnetic field becomes highly toroidal due to the corotating plasma having an angular velocity that matches the event horizon's rotation. Furthermore, the emission from this near-horizon corotating plasma is believed to be detectable. Current research supports the magnetically arrested disk (MAD) model for both M87* and SgrA* [2, 5, 9, 35-37]. The simulation results suggest an increasing emission as the event horizon is approached [2, 38, 39]. Consequently, the polarization of the near-horizon image (NHI), which results from the synchrotron radiation of the near-horizon plasma, can be detected. This polarization has the potential to encode precisely the information about the black hole spin, primarily due to the extreme frame-dragging effects that overwhelm other influences. This enables us to read black hole spin from the polarizations of the NHI of black holes. \nIn this letter, we confirm that there is a distinctive signature of spin on the polarization pattern of the NHI for a rotating black hole, independent of the plasma flow geometry. We begin by discussing the general properties of horizon-scale plasma within the context of a stationary, axisymmetric black hole spacetime. The plasma is highly conductive and exhibits a geometrically thin emission profile at the equatorial plane. The synchrotron radiations emitted by the electrons within the plasma are predominantly perpendicular to the magnetic field. For computational convenience, we then narrow our focus to Petrov Type D black holes. We reveal that the ratio of the real part to the imaginary part of the complex Penrose-Walker constant for the polarized light emitted from the equatorial near-horizon source depends solely on the black hole parameters and the conserved impact parameters. Most importantly, we show the unique imprint of Kerr spin on the polarization electric vector position angle (EVPA) of the NHI. Throughout the study, we use the units G = c = 1. \nSetup -Plasma, a state of matter composed of charged particles such as electrons and ions, is attracted to black holes. In this study, we examine a general stationary axisymmetric spacetime that contains a rotating black hole. This spacetime is characterized by a line element, which can be expressed as d s 2 = g tt dt 2 +2 g tϕ dtdϕ + g ϕϕ dϕ 2 + g rr dr 2 + g θθ dθ 2 in the Boyer-Lindquist coordinates. The magnetic field is coupled with the plasma flow, which is required to obey the high conductivity condition so that any possible electric field will be eliminated by the fast rearrangement of the electrons, E µ = F µν U ν = 0 where U µ is the bulk velocity of the plasma, F µν is the electromagnetic tensor. Using the Bianchi identity ∇ µ ∗ F µν = 0, the magnetic field B µ = ∗ F µν U ν has a \nsimple form[40, 41]: \nB µ = Ψ √ -g U r ( U t U µ + δ µ t ) , (1) \nwhere g is the determinant of the spacetime metric; Ψ denotes the overall strength, which is conserved along the bulk velocity, U µ ∂ µ Ψ ≈ U r ∂ r Ψ = 0. Note that we do not consider the region with U r = 0, where the accretion towards the black hole stops and Eq. (1) cannot be applied. This magnetic field matches the simulational results in [42] for the regions close to a Kerr black hole. The millimeter-wavelength emission is dominated by the collective synchrotron radiation of the thermal or nonthermal electrons within the plasma, whose polarization vector f µ is largely perpendicular to the global magnetic field [43]. Without imposing any specific distribution of electrons within the plasma, the covariant expression for f µ must take the following form \nf µ = ϵ µναβ U ν p α ( B ⊥ ) β ω √ B 2 ⊥ , (2) \nwhere ϵ µναβ is the Levi-Civita tensor; p µ denotes the four-momentum of photon; ω = -U µ p µ is the photon frequency in the plasma frame; B µ ⊥ = B µ -ω -2 ( B · p ) p µ ⊥ denotes the spatial magnetic field in the plasma frame with p µ ⊥ = p µ -ωU µ being the spatial four-momentum of photon. The expression Eq. (2) naturally satisfies the gauge condition f t = 0. \nThe photon emitted from the synchrotron radiation of the plasma, as observed on the screen, is described by the coordinates ( α, β ) ∝ ( -sin θ o p ϕ , p θ ) [44] in asymptotic flat spacetime, where θ o represents the observational angle. Given the position ( α, β ), the electric vector polarization angle (EVPA) is computed from the components of f µ projected along the α and β directions on the observer's screen \nχ = tan -1 f β f α , (3) \nwhich describes the angle between the polarization vector and the α -axis. \nPolarizations of NHI -In the following discussion, for computational simplicity (to make an analytical study), we restrict our attention to Petrov Type D spacetime. Neglecting in-medium effects, the polarization evolution can be determined analytically using the Penrose-Walker constant [45, 46], which is a complex number conserved along null geodesics. By choosing a null basis ( l, n, m, ¯ m ) with l and n as the principal null directions, the complex Penrose-Walker constant is given by: \nκ = κ 1 + iκ 2 = 2Ψ -1 3 2 p µ f ν ( l [ µ n ν ] -m [ µ ¯ m ν ] ) , (4) \nwhere p µ denotes the photon four-momentum, Ψ 2 is the only non-vanishing Weyl scalar in this framework [47], \nand κ 1 , κ 2 represent the real and imagine part of κ , respectively. \nThe Kerr spacetime, classified as a Petrov Type D spacetime, is of significant astrophysical relevance. In this context, we aim to use the Kerr geometry as an example to demonstrate the unique imprint of black hole rotation on the polarizations of photons emitted very near the event horizon. Specifically, Eq. (4) can be rewritten as \nκ ≡ κ 1 + iκ 2 = ( Ai B )( r -ia cos θ ) , A = 2 p [ t f r ] +2 a sin 2 θp [ r f ϕ ] , B = 2sin θ [ ( r 2 + a 2 ) p [ ϕ f θ ] -ap [ t f θ ] ] , (5) \nwhere a is the Kerr spin. For a Kerr black hole, the event horizon is determined by the larger root r + of the equation ∆ = ( r -r + )( r -r -) = 0. For simplicity, we set the ADM mass M = 1. To analyze the near-horizon behavior of the polarizations of emissions from the equatorial plasma, we need to take the near-horizon limit, r → r + of Eq. (2). Considering the fact that the accretion is concentrated on the equatorial plane and the near-horizon plasma flow is dominated by gravitational attraction, the bulk velocity satisfies | rU θ | ≪ | U r | and U t , U ϕ should be finite and vary slowly outside the horizon. This assumption is consistent with the simulated results of MADs presented in [39, 42]. Without loss of generality, we will set U θ = 0 and keep U t and U ϕ invariant when performing the near-horizon expansion. Consequently, for the bulk velocity of the equatorial plasma near r = r + , we obtain the following relations: \nU t = -4 Y ∆ -( 1 + 2 r + ) U t + aU ϕ r 2 + ( r + -1) + o (∆ 0 ) , U r = -2 | Y | r + + o (∆ 0 ) , U θ = 0 , U ϕ = -2 aY r + ∆ + aU t + U ϕ r 2 + ( r + -1) + o (∆ 0 ) , (6) \nwhere Y = U t +Ω h U ϕ , Ω h = a/ (2 r + ) is the angular velocity of the Kerr black hole. We have chosen U r < 0 for the accreting plasma. To the lowest order approximation, all components of U µ are proportional to Y . Specifically, we have: \nU r U t = Ω h ∆ a | Y | Y , U ϕ U t = Ω h , (7) \nwhich is a general relationship for the accretion velocity near the horizon. The near-horizon expansion of the magnetic field, described by Eq. (1), is given as follows: \nB t = 2Ψ U t r + ∆ Y | Y | + o (∆ -1 ) , B r = Ψ r 2 + U t , B θ = 0 , B ϕ = Ψ U t a r 2 ∆ Y | Y | + o (∆ -1 ) . (8) \n+ \nHere, B θ = 0 arises from U θ = 0; the plunging matter must have Y < 0, consistent with the second law of black hole thermodynamics [48]. Specifically, if a probe particle has Y > 0, it cannot classically reach the event horizon. Furthermore, the ratio B ϕ /B r ≈ -a ∆ -1 aligns with the results from the simulation presented in [42]. Expanding f µ near r = r + in the Kerr metric, we obtain: \nX E f t = -2 aσ θ √ η r + ∆ + aσ θ √ η r 2 + ( r + -1) + o (∆ 0 ) , X E f r = aσ θ √ η r 2 + + R 1 ∆+ o (∆) , X E f θ = a ( aλ -2 r + ) -σ r a | aλ -2 r + | r 2 + ∆ +Θ 1 + o (∆ 0 ) , X E f ϕ = -a 2 σ θ √ η r 2 + ∆ + σ θ √ η r 2 + ( r + -1) + o (∆ 0 ) , (9) \nwhere X = -Ψ -1 √ -g ( U µ p µ ) √ B 2 ⊥ , E = -p t , λ and η are conserved impact parameters that indicate the direction of photon emission [49]. The term σ θ denotes the sign of p θ . The terms R 1 and Θ 1 depend on the bulk velocity of the plasma, and their forms are rather complicated and thus are omitted here. \nNote that the overall factor X/ E involves the bulk velocity, meaning that κ in Eq. (5) depends on the plasma motion. However, by combining Eqs. (5) and (9) and expanding the photon four-momentum, we find that the ratio \nz ≡ κ 1 κ 2 = z 0 + z 1 ∆+ O (∆ 2 ) (10) \nis independent of the plasma motion near the horizon. Here, we have introduced the following expressions \nz 0 = σ θ √ η λ -a , z 1 = σ θ ( 1 + z 2 0 ) √ η Ω h 2 a 2 (Ω h λ -1) . (11) \nThe ratio z is directly related to the EVPA on the observer's screen, as described by the equation \nχ = tan -1 ( βz + µ β -µz ) = tan -1 ( µ β ) +tan -1 z , (12) \nbased on Eq. (3), where µ = -( α + a sin θ o ). From this, we can derive the following equation: \nχ = tan -1 ( µ β ) +tan -1 z 0 + z 1 ∆ 1 + z 2 0 + o (∆ 1 ) , (13) \nwhich represents a fundamental and universal relationship. This equation indicates that the EVPA of the NHI is independent of the plasma flow and is solely influenced by the spacetime geometry. Specifically, it highlights a unique imprint of black hole spin on the polarization of the NHI. \nVerification -In Fig. 1, we compare the results of the near-horizon expansion given by Eq. (10) with the exact value of z for the plasma with three different flow \ntypes: the plasma infalling from infinity (Type I), the plasma plunging from the prograde innermost stable circular orbit (ISCO) (Type II), and the fitting result from the simulation of MAD introduced in [39] (Type III). We have chosen a typical direction for the emitted photons with ( λ, η ) = (2 , 8) for a = 0 . 94 and a = 0 . 2. \nFIG. 1. Compare the universal relationship of z with the exact z for the plasma under three different flow conditions. \n<!-- image --> \nIt is evident that the expansion up to the next-toleading order of ∆ works quite well for 0 ≤ ∆ ≲ 10 -2 (corresponding to r = 1 . 01 r + for a = 0 . 94 and r = 1 . 003 r + for a = 0 . 2) for all three types of plasma flow. The inclusion of the next-next-to-leading term only improves the expansion in a very small interval around ∆ ∼ 5 × 10 -2 (corresponding to r = 1 . 05 r + for a = 0 . 94 and r = 1 . 01 r + for a = 0 . 2). Therefore, the validity of the near-horizon expansion is manifested. \nIn Fig. 2, we present the polarization vectors at the event horizon for a Kerr black hole with spin parameters a = 0 . 2 and a = 0 . 94, observed at inclination angles θ o = 17 · and θ o = 80 · . In the plots, the projection of the black hole's rotation direction onto the screen aligns with the β axis. The EVPAs in the NHIs exhibit significant variations at χ = π/ 4 and χ = -π/ 4 when either the spin parameter a or the observation angle θ o is altered, indicating a mutual dependence on both the black hole's spin and the observation angle. By integrating these observations, we can infer the values of the black hole's spin and the observation angle. \nConclusion and discussion -In this letter, we show that the EVPA in the NHI of a rotating black hole pro- \nFIG. 2. The EVPAs in the NHIs of Kerr black holes are represented by different symbols: · indicates χ = π/ 4, and △ indicates χ = -π/ 4. The dashed lines represent the NHIs. \n<!-- image --> \nvides a novel measure, which is independent of the photon ring and inner shadow and can serve as an alternative means of detecting supermassive black hole spins. Our result shows a unique imprint of black hole spin on the polarization of NHI. The discussions can be readily extended to any Petrov Type D spacetime. In fact, in a companion paper [50], we have demonstrated that Eq. (10) remains invariant and have derived a similar expression for Eq. (13) in a Kerr-Newman-Taub-NUT spacetime. The universal relationship illustrates that the EVPA in the NHI is governed exclusively by spacetime geometry, irrespective of the plasma flow geometry. This phenomenon results from the frame-dragging effect near the event horizon of rotating black holes. Therefore, this relationship is expected to be valid in all rotating spacetimes. \nOur study did not account for the Faraday rotation effects caused by plasma, whereas current polarization results from the EHT are depolarized by the Faraday rotation [22]. This raises the question whether our results can be useful when taking into account of the Faraday rotation. The influence of the Faraday rotation could be quantified through upcoming multi-frequency observations, which would allow it to be separated from the polarization signal. Specifically, the measurements at higher frequencies can distinguish the Faraday rotation from the underlying magnetic field structure [20, 30], thereby isolating the gravitational effects from the nearhorizon polarization. This issue could also be addressed by incorporating the frequency dependence of the emitted photons in our calculations, which was not considered in our current work. Additionally, it would be beneficial \nto use GRMHD simulations and GRRT calculations to test realistic scenarios, including the presence of a Faraday screen. \nThe work is partly supported by NSFC Grant Nos. 12205013 and 12275004. YM is supported by the National Key R&D Program of China (Grant No. 2023YFE0101200), the National Natural Science Foundation of China (Grant No. 12273022), and the Shanghai Municipality Orientation Program of Basic Research for International Scientists (Grant No. 22JC1410600). \n- ∗ Corresponding author: mizuno@sjtu.edu.cn\n- † Corresponding author: minyongguo@bnu.edu.cn\n- ‡ Corresponding author: bchen01@pku.edu.cn\n- [1] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,' Astrophys. J. Lett. 875 (2019) L1, arXiv:1906.11238 [astro-ph.GA] .\n- [2] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,' Astrophys. J. Lett. 875 no. 1, (2019) L5, arXiv:1906.11242 [astro-ph.GA] .\n- [3] Event Horizon Telescope Collaboration, K. Akiyama et al. , 'First M87 Event Horizon Telescope Results. VI. 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2024EPJC...84.1125E

Motivated by the impact of the phantom field or antiMaxwell field on the structure of threedimensional black holes in the presence of the cosmological constant we present the first extraction of solutions for the phantom BTZ AdS black hole. In this study we analyze the effect of the phantom field on the horizon structure. Furthermore we compare the BTZ black holes in the presence of both the phantom and Maxwell fields. Additionally we calculate the conserved and thermodynamic quantities of the phantom BTZ black holes demonstrating their compliance with the first law of thermodynamics. Subsequently we assess the effects of the electrical charge and the cosmological constant on the local stability in the canonical ensemble by considering these fields with respect to the heat capacity. We then investigate the global stability area of the BTZ black holes with phantom and Maxwell fields within the grand canonical ensemble using Gibbs free energy. In this analysis we evaluate the influence of the electrical charge and the cosmological constant on this area.

2024-10-01T00:00:00Z

['2024arXiv240912214E', 'arXiv:2409.12214', '10.48550/arXiv.2409.12214', '10.1140/epjc/s10052-024-13485-z', '2024EPJC...84.1125E']

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

Phantom BTZ black holes

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

https://arxiv.org/pdf/2409.12214.pdf

{'Phantom BTZ black holes': "B. Eslam Panah 1 ∗ , and M. E. Rodrigues 2 , 3 † \n- 1 Department of Theoretical Physics, Faculty of Science, \nUniversity of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran \n- 2 Faculdade de Ciˆencias Exatas e Tecnologia, Universidade Federal do Par'a \nCampus Universit'ario de Abaetetuba, 68440-000, Abaetetuba, Par'a, Brazil \n- 3 Faculdade de F'ısica, Programa de P'os-Gradua¸c˜ao em F'ısica, Brazil \nMotivated by the impact of the phantom field (or anti-Maxwell field) on the structure of threedimensional black holes in the presence of the cosmological constant, we present the first extraction of solutions for the phantom BTZ (A)dS black hole. In this study, we analyze the effect of the phantom field on the horizon structure. Furthermore, we compare the BTZ black holes in the presence of both the phantom and Maxwell fields. Additionally, we calculate the conserved and thermodynamic quantities of the phantom BTZ black holes, demonstrating their compliance with the first law of thermodynamics. Subsequently, we assess the effects of the electrical charge and the cosmological constant on the local stability in the canonical ensemble by considering these fields with respect to the heat capacity. We then investigate the global stability area of the BTZ black holes with phantom and Maxwell fields within the grand canonical ensemble using Gibbs free energy. In this analysis, we evaluate the influence of the electrical charge and the cosmological constant on this area.", 'I. INTRODUCTION': 'Black holes are celestial objects characterized by strong gravitational fields and intricate structures. They play an important role in the investigation of theories of gravity. According to the theory of general relativity (GR), a black hole is defined as a region in spacetime encompassed by an event horizon. Hawking further expanded on this concept by demonstrating that the area of an event horizon cannot decrease over time [1]. In this regard, Bekenstein also formulated the concept of black hole entropy [2-4]. It has also been established that there is a correlation between surface gravity and temperature, indicating that surface gravity can be utilized to determine the temperature of a black hole [5]. Hawking proposed a mechanism through which black holes can emit thermal radiation, which is known as Hawking temperature, proportional to their surface gravity. This radiation is caused by quantum fluctuations near the event horizon, which allow for creating virtual particle pairs and observing positive energy flux in distant regions [6]. On the other hand, recent observations have provided compelling evidence supporting the existence of black holes in our universe [7-10]. So, Black holes were regarded as thermodynamic systems, wherein the temperature and entropy are denoted by the surface gravity and area of the black hole horizon, respectively. The study focused on the phase transition of a black hole as a thermodynamic system. Specifically, it was demonstrated that a Hawking-Page phase transition exists between the pure radiation phase and the stable large Schwarzschild black hole phase in AdS space [11, 12]. Additionally, a first-order phase transition, resembling the liquid-gas system, was observed between the small and large black hole phases for charged AdS black holes [13]. Numerous studies have been conducted to explore this phenomenon, considering different types of black holes [14-27]. \nThe first three-dimensional black hole solutions were introduced by Banados, Teitelboim, and Zanelli which are known as BTZ black holes [28]. The significance of BTZ black holes lies in their ability to provide an elegant framework for understanding the interactions of lower-dimensional gravitational systems [29]. They have also been utilized to establish a link with string theory [12], study different thermal properties of black holes [30, 31], and see Refs. [28], for more details. Since then, different theories of gravity to investigate various three-dimensional black hole solutions have been explored in much literature [32-60]. The study of black holes in three dimensions can help us understand the phenomena that occur in black hole physics in four dimensions. We have three main motivations for doing a study in a lower dimension: 1- While gravity is not perturbatively renormalizable in four spacetime dimensions, it is in three dimensions. Notably, three-dimensional gravity possesses black hole solutions and is exactly solvable, paving the way for the exploration of quantum black holes. This makes three-dimensional gravity an intriguing system in its own right; 2- The AdS/CFT correspondence establishes a connection between conformal field theories (CFT) in two dimensions and gravity in three dimensions. CFTs play a significant role in condensed matter physics, and studying 3D gravity can provide valuable insights into these theories; 3- Gravity in three dimensions is simpler to handle \ncompared to gravity in four dimensions, making it an effective toy model for understanding four-dimensional gravity. In quantum gravity, models in 2 + 1 dimensions are of great importance for understanding new phenomena [61-67]. According to the importance of BTZ black holes, we study the phantom BTZ black holes. \nObservational evidence indicates that our Universe is accelerating, and the concept of dark energy was proposed to explain this accelerated expansion [68-71]. Considering that our Universe is filled with a perfect fluid of pressure p and density ρ , an important property of this fluid is that by the equation of state (EoS) p = ωρ , the fluid has ω ∼ -1 today. So the fluid has exotic characteristics with negative pressure and attractive gravity. By applying the SevenYear Wilkinson Microwave Anisotropy Probe (WMAP) observations data, the latest distance measurements from the BAO in the distribution of galaxies, and the Hubble constant measurement, for a flat universe, the current value of a constant EoS for dark energy has been estimated as ω = -1 . 10 ± 0 . 14 (68% confidence level) [72]. The model most consistent with the observations is ΛCDM [73], where we can have a phantom phase with ω < -1, asymptotically de Sitter phantom universe [74]. This phantom phase can be modeled using a scalar field with a kinetic term in the action, opposite to the conventional canonical one. In this case, the pressure of the field is negative [75, 76]. One specific type of dark energy, called the phantom field, exists in the Einstein-Maxwell-dilaton system [77] and is studied in relation to ghost branes in string theory [78]. It should be noted that the phantom field may give rise to quantum instabilities [68, 69], presenting a challenge to the theory. However, in Refs. [79, 80] argued that these instabilities can be avoided. Therefore, the phantom field has emerged as a potential candidate for the dark energy model from a theoretical perspective. Given the significance of the phantom field, and particularly its effects on the background of the black hole spacetime, this investigation is important because dark energy will inevitably influence black holes in the Universe. on the other hand, a new black hole solution known as the anti-Reissner-Nordstrom-(A)dS solution or phantom Reissner-Nordstrom-(A)dS black hole solution [81] was extracted in the Einstein-anti-Maxwell theory in the presence of a cosmological constant. These black holes possess a phantom charge, making them fundamentally distinct from the usual Reissner-Nordstrom-(A)dS black holes. Therefore, studying the characteristics of these black holes is of great interest. Historically, Einstein and Rosen proposed the concept of changing q → -q in 1935 [82]. They suggested adding a purely imaginary charge, to the Reissner-Nordstrom solution to describe what is known as the quasicharged bridge. The literature extensively discusses phantom black hole solutions in general relativity and other modified theories of gravity [83-87]. So, in this study, our objective is to investigate the phantom BTZ black hole. To achieve this, we will first extract the exact solution of the phantom BTZ black hole. Subsequently, we will analyze other properties, including conserved and thermodynamic quantities, heat capacity, and phase transition.', 'II. BLACK HOLE SOLUTION': "The action of this theory in three-dimensional spacetime is given by \nI = 1 2 κ 2 ∫ ∂ M d 3 x √ -g [ R -2Λ + η F ] , (1) \nwhere R and Λ are, respectively, the Ricci scalar curvature and the cosmological constant. The third term is the coupling with the Maxwell field, when η = 1, or a phantom field of spin 1, when η = -1. Also, F = F µν F µν is the Maxwell invariant. In addition F µν = ∂ µ A ν -∂ ν A µ is the electromagnetic tensor field, and A µ is the gauge potential. In the above action, κ 2 = 8 πG , and G is the Newtonian gravitational constant. Hereafter, we consider G = c = 1. Meanwhile, g = det ( g µν ) is the determinant of metric tensor g µν . \nMaking the functional variation of the above action (1), with respect to the gauge field A µ and the gravitational field g µν , we get the following field equations in the following forms \nG µν +Λ g µν = 2 η ( 1 4 g µν F F α µ F να ) , (2) \nwhere G µν is the Einstein tensor. \n∂ µ ( √ -gF µν ) = 0 , (3) \nWe consider a three-dimensional static spacetime in the following form \nds 2 = -g ( r ) dt 2 + dr 2 g ( r ) + r 2 dϕ 2 , (4) \nwhere g ( r ) is the metric function that we have to find it. In order to obtain phantom black hole solutions, we consider a radial electric field which its related gauge potential is given \nA µ = h ( r ) δ t µ = ( h ( r ) , 0 , 0) . (5) \nr \n-→ \n0 \nUsing the Maxwell equations (3) with the metric (4), and the mentioned gauge potential (5), we can find the following differential equation by \nrh '' ( r ) + h ' ( r ) = 0 , (6) \nwhere the prime and double prime are, respectively, the first and the second derivatives with respect to r . We extract the solution of the equation (6) in the following form \nh ( r ) = -q ln ( r l ) , (7) \nwhere q is a integration constant which is related to the electric charge. To have logarithmic arguments dimensionless, we introdcue l in which is an arbitrary constant with length dimension. Considering the equation (7), the electromagnetic field tensor is given \nF tr = ∂ t A r -∂ r A t = q r . (8) \nSubstituting (8) into the equation (2), we get \neq tt = eq rr = rg ' ( r ) + 2Λ r 2 -2 ηq 2 , (9) \neq ϕϕ = r 2 g '' ( r ) + 2Λ r 2 +2 ηq 2 , (10) \nwhere eq tt , eq rr and eq ϕϕ , are components of tt , rr and ϕϕ \nof field equations (2), respectively. \nUsing the equations (9) and (10), we obtain the metric function in the following form \ng ( r ) = -m 0 -Λ r 2 +2 ηq 2 ln ( r l ) , (11) \nwhere m 0 is integration constant related to the mass of the black hole. \nWe calculate the Ricci and Kretschmann scalars for the solution (11) and get them as \nR = 6Λ -2 ηq 2 r 2 , (12) \nR αβγδ R αβγδ = 12Λ 2 -8Λ ηq 2 r 2 + 12 η 2 q 4 r 4 , (13) \nwhich show that there is a curvature singularity located at r = 0, i.e., \nlim \nR \n, \n-→ ∞ \nlim r -→ 0 R αβγδ R αβγδ -→ ∞ , (14) \nand also it is finite for r = 0. The asymptotical behavior of them are given by \nr \nlim r -→∞ R -→ 6Λ , lim -→∞ R αβγδ R αβγδ -→ 12Λ 2 , (15) \nwhich indicates the spacetime is independent of η , and it will be asymptotically (A)dS. \nHorizon Structure: To evaluate the effects of different parameters of these black holes, we plot the metric function (11) versus r in Fig. 1-3. This figure gives us some information about the behavior of horizons. In addition, by studying Figs. 1-3, we can compare the BTZ black holes in the presence of Maxwell and phantom (anti-Maxwell) fields together. \nConsidering some values of parameters of this solution, we want to investigate the effects of mass, the electrical charge, and the cosmological constant on BTZ black holes in the presence of Maxwell and phantom fields. \n/negationslash \nFIG. 1: The metric function g ( r ) versus r for different values of the mass. Left panels for phantom ( η = 1), and right panels for Maxwell case ( η = -1). \n<!-- image --> \nFIG. 2: The metric function g ( r ) versus r for different values of the electrical charge. Left panels for phantom ( η = 1), and right panels for Maxwell case ( η = -1). \n<!-- image --> \nMass: Here, we investigate the impact of the mass parameter on the event horizon. The findings reveal that as the mass increases, both types of black holes become larger. Indeed, the radii of black holes increase as m 0 increases (see Fig. 1). However, there is a significant distinction between charged BTZ black holes in the phantom case in terms of the number of roots. In other words, the phantom BTZ black holes exhibit only one root (or event horizon), as shown in the left panel of Fig. 1. Conversely, BTZ black holes in the presence of the Maxwell field have two roots (inner and outer roots), one root (extremal black holes), and without root (naked singularities), as illustrated in the right panel of Fig. 1. This discrepancy arises from the presence of the phantom effect. \nElectrical Charge: Another interesting result is related to the effect of the electrical charge on the event horizon. Our findings show that the higher-charged phantom BTZ black holes have large radii. In other words, by increasing the electrical charge ( q ), the radius of the phantom black hole increases (see the left panel in Fig. 2). But in the presence of Maxwell field when the electrical charge increases, we first encounter with small BTZ black holes and then the number of roots decreases (see the right panel in Fig. 2). \nCosmological Constant: Phantom BTZ black holes exist for three cases of the cosmological constant (Λ > 0, Λ < 0, and Λ = 0). Indeed, there is an event horizon for BTZ black holes in the presence of a phantom field when the values of the cosmological constant are zero, negative and positive (see the left panel in Fig. 3). But, for Λ > 0, there are two roots which the smaller root is relsted to the event horizon and another one is the coslomogical horizon. \nand so the Hawking temperature leads to \nFIG. 3: The metric function g ( r ) versus r for different values of the cosmological constant. Left panels for phantom ( η = 1), and right panels for Maxwell case ( η = -1). \n<!-- image --> \nIn other words, the small black holes can be exist for Λ > 0. However, the charged BTZ black holes (Maxwell case) can exist when the value of the cosmological constant is only negative (see the right panel in Fig. 3). \nIn general, there are three essentially different behaviors of BTZ black holes between Maxwell and phantom fields. i) There is only one root for phantom BTZ black holes. However, the roots of BTZ black holes change in the presence of the Maxwell field. ii) The radius of the phantom BTZ black hole increases with the increase of electric charge, which is different from the Maxwell case. Moreover, the number of roots decreases as the electric charge increases. In fact, BTZ black holes can have two roots (inner and outer roots), one root (extremal black holes) and no root (naked singularities) as the electric charge increases. iii) Phantom BTZ black holes can exist for λ < 0, Λ > 0, and λ = 0. However, the BTZ black hole with Maxwell field only exists in the case Λ < 0.", 'III. THERMODYNAMICS': "Now, we are interested in to calculate the conserved and thermodynamic quantities of BTZ black holes in the presence of Maxwell and phantom fields to check the first law of thermodynamics. \nUsing the Hawking temperature, which is given by T = κ 2 π (where κ is the superficial gravity), we can obtain the Hawking temperature of phantom BTZ black holes. For this purpose, by considering g ( r ) = 0, we first express the mass ( m 0 ) in terms of the radius of the event horizon ( r + ), the cosmological constant and the charge q in the following form \nm 0 = 2 ηq 2 ln ( r + l ) -Λ r 2 + , (16) \nThen, we calculate the superficial gravity for the mentioned spacetime (4), which leads to \nκ = g ' ( r ) 2 ∣ ∣ ∣ ∣ r = r + , (17) \nby considering BTZ black holes (11), and by substituting the mass (16) within Eq. (17), one can calculate the superficial gravity as \nκ = ηq 2 r + -Λ r + , (18) \nT = ηq 2 2 πr + -Λ r + 2 π . (19) \nThe electric charge of a black hole can be obtained by using the Gauss law in the following form \nQ = 1 4 π ∫ 2 π 0 F tr ( r ) √ gdϕ ∣ ∣ ∣ ∣ r = r + = q 2 (20) \nwhere F tr ( r ) = q r , and g = det ( g µν ) = r 2 . \nUsing F µν = ∂ µ A ν -∂ ν A µ , one can find the nonzero component of the gauge potential in which is A t = -∫ F tr ( r ) dr . So, we can get the electric potential ( U ) at the event's horizon with respect to the reference ( r →∞ ) as \nU = -∫ + ∞ r + F tr ( r ) dr = q ln ( r + l ) . (21) \nTo obtain the entropy of BTZ black holes, one can use of the area law \nS = A 4 , (22) \nwhere A is the horizon area and is defined by \nA = ∫ 2 π 0 √ g ϕϕ dϕ ∣ ∣ ∣ ∣ r = r + = 2 πr | r = r + = 2 πr + , (23) \nwhere g ϕϕ = r 2 . So, the entropy of BTZ black holes in the presence Maxwell and phantom fields is given by \nS = πr + 2 . (24) \nAnother interesting quantity is related to the total mass of the black hole. To obtain the total mass we use of the Ashtekar-Magnon-Das (AMD) approach [88, 89], and we have \nM = m 0 8 , (25) \nsubstituting the mass (16) within the equation (25), yields \nM = ηq 2 4 ln ( r + l ) -Λ r 2 + 8 . (26) \nFIG. 4: M versus r + for different values of the cosmological constant. Left panels for phantom ( η = 1), and right panels for Maxwell case ( η = -1). \n<!-- image --> \nTo have a positive mass for the phantom case ( η > 0), the cosmological constant can be positive, negative, or zero, as long as Λ < 2 ηq 2 r 2 + ln ( r + l ) holds. This constraint indicates that the mass of phantom BTZ black holes is always positive when Λ < 0 and Λ = 0 (see dotted and dashed lines in the left panel of Fig. 4). However, for Λ > 0 to have a positive mass, we need to satisfy Λ < 2 ηq 2 r 2 + ln ( r + l ) . This reveals that the total mass of black holes cannot be positive for large phantom dS BTZ black holes (see the continuous line in the left panel of Fig. 4). On the other hand, the mass of the Maxwell AdS BTZ black hole with a large radius is positive only for a negative cosmological constant (see the dotted line in the right panel of Fig. 4). This result is consistent with the behavior of the mass ( m 0 ) for the metric function obtained (compare Fig. 3 with Fig. 4). \nIt is straightforward to show that the conserved and thermodynamics quantities satisfy the first law of thermodynamics in the following form \ndM = TdS + ηUdQ, (27) \nand one can define the intensive parameters conjugate to S and Q . These quantities are the electric potential ( U = ( ∂M ∂Q ) S ) , and the temperature ( T = ( ∂M ∂S ) Q ) which are the same as those calculated for the electric potential (21), and the temperature (19).", 'A. Local Stability': "Heat capacity can study a thermodynamic system's local stability in the canonical ensemble context. Therefore, we evaluate the heat capacity to find the local stability for phantom BTZ black holes. Indeed, we study the effect of phantom field on local stability of three-dimensional black holes and compare it with Maxwell field. The heat capacity is given by \nC Q = T ( ∂T ∂S ) Q . (28) \nBy replacing Eq. (24) in Eq. (19), we can find the Hawking temperature ( T ) versus S in the following form \nT = ηQ S -Λ S π 2 , (29) \nand after some calculation, we can extract the heat capacity of phantom BTZ black holes, which leads to \nC Q = ( Λ S 2 -ηπ 2 Q 2 ) S Λ S 2 + ηπ 2 Q 2 , (30) \nWe solve C Q to find the roots of the heat capacity, which are \nS root 1 = πQ √ η Λ Λ , & S root 2 = -πQ √ η Λ Λ . (31) \nTo have real root, we must respect to the condition η Λ > 0. This imposes the following conditions for phantom and Maxwell fields, which are: \nPhantom case ( η = 1 ): to have a positive value of η Λ (i.e., η Λ > 0), the cosmological constant must be positive. In this case, S root 1 > 0 and S root 2 < 0. Furthermore, there are two roots for the mass. The mass is positive between these roots. Our analysis reveals that the phantom dS BTZ black hole satisfies local stability when it is between S root 1 of the heat capacity and the second root of the mass. In other words, medium phantom black holes with a positive value of the cosmological constant (i.e. phantom dS black holes) justify the local stability condition because the mass and the heat capacity are positive (see the hatched area in the left panel of Fig. 5). \nMaxwell case ( η = -1 ): the cosmological constant is negative (Λ < 0) when we consider the condition η Λ > 0 for the Maxwell case. Therefore, S root 1 < 0, and S root 2 > 0. In other words, only S root 2 represents the real root of the heat capacity. Our findings indicate that charged AdS BTZ black holes with a large radius (entropy) can be stable. In other words, after the second root of the mass, both the heat capacity and mass are simultaneously positive. Therefore, large BTZ black holes with a negative value of the cosmological constant satisfy local stability (see the hatched area in the right panel of Fig. 5). \nFIG. 5: The heat capacity C Q and mass ( M ) versus S for different values of the mass. Left panel for phantom ( η = 1), and right panel for Maxwell case ( η = -1). \n<!-- image --> \nTo find the heat capacity divergence points, we solve the denominator of Eq. (30) in terms of entropy. We extract them in the following forms \nS div 1 = πQ √ -η Λ Λ , & S div 2 = -πQ √ -η Λ Λ . (32) \nTo have real root, we must respect to the condition -η Λ > 0. This imposes the following conditions for phantom and Maxwell fields, which are: \nPhantom case ( η = 1 ): by applying the condition -η Λ > 0, the cosmological constant must be negative (i.e., Λ < 0), and it implies that S div 1 < 0, and S div 2 > 0. Therefore, there is a divergence point ( S div 2 ) for the heat capacity. The behavior of the mass and the heat capacity indicate that phantom AdS black holes with a large radius (entropy) satisfy local stability. In other words, large phantom AdS black holes are stable (see the hatched area in the left panel of Fig. 6). \nMaxwell case ( η = -1 ): for this case, the cosmological constant must be positive (Λ > 0) when considering the condition -η Λ > 0. This condition leads to S div 1 > 0 and S div 2 < 0. The results in the right panel of Fig. 6 demonstrate that there is no local stability region for charged BTZ dS black holes, as the mass and heat capacity are not simultaneously positive. In other words, the BTZ black hole in the presence of the Maxwell field with a positive cosmological constant value cannot satisfy local stability. \nBriefly, our results reveal that: \ni) for Λ > 0 (dS case), medium phantom BTZ black holes are stable. However, charged BTZ black holes in the presence of the Maxwell field are not stable. \nii) for Λ < 0 (AdS case), large black holes in the presence of phantom and Maxwell fields exhibit local stability. However, there is a distinct stable region. In other words, phantom BTZ black holes have a larger stable area compared to the Maxwell case.", 'B. Global Stability': "Hawking and Page were the first to suggest studying the global stability of black holes [90]. They proposed that the global stability of a black hole can be assessed in the grand canonical ensemble by calculating the Gibbs free energy. A black hole is deemed globally stable if it possesses negative Gibbs free energy [91, 92]. Here, our objective is to analyze the global stability of phantom BTZ black holes using the Gibbs free energy approach. \nIt is notable that in the context of the black holes, the Gibbs free energy is defined in the following form \nG = M ( S, Q, Λ) -TS -QU. (33) \nBy replacing Eq. (20), and Eq. (24), in Eq. (26), we rewrite the mass of the phantom BTZ black holes as \nM ( S, Q, Λ) = ηQ 2 ln ( 2 S πl ) -Λ S 2 2 π 2 . (34) \nFIG. 6: The heat capacity C Q and mass ( M ) versus S for different values of the mass. Left panel for phantom ( η = 1), and right panel for Maxwell case ( η = -1). \n<!-- image --> \nUsing Eqs. (21), (20), (29), and (34) in Eq. (33), we can obtain the Gibbs free energy in the following form \nG = ( η -2) Q 2 ln ( 2 S πl ) + Λ S 2 2 π 2 -ηQ 2 . (35) \nThe global stability areas are deduced from the negative area of the Gibbs free energy (i.e., G < 0). We apply the constraint G < 0 to find the global stability areas. This constraint imposes the following two conditions: Condition I: by applying G < 0 for phantom BTZ black holes ( η = 1), we find that \nΛ < 2 π 2 Q 2 S 2 ( ln ( 2 S πl ) +1 ) , (36) \nwhere by adjusting thermodynamic quantities, we can satisfy the above condition. This means that global stability can be achieved for positive, negative, and zero values of the cosmological constant. \nCondition II: by considering G < 0 for Maxwell BTZ black holes ( η = -1), we can extract the following constraint \nΛ < 2 π 2 Q 2 S 2 ( 3 ln ( 2 S πl ) -1 ) , (37) \nso, we can find the global stability areas for Λ > 0, Λ < 0, and Λ = 0 by adjusting the suitable parameters of the above condition. \nIn order to gain a clear insight into the two conditions mentioned above, we plot the Gibbs free energy versus entropy in Fig. 7 to determine the areas of global stability. Our findings confirm that there is a global stability area for different values of the cosmological constant for BTZ black holes in the presence of both phantom and Maxwell fields. However, the global stability area is more extensive for phantom BTZ black holes compared to Maxwell's case. Furthermore, the global stability area changes with the variation of the electrical charge ( Q ). These changes are: \ni) In the case of Λ < 0 (AdS case); the global stability area decreases as the electrical charge increases (see the up panels in Fig. 7). \nii) In the case of Λ > 0 (dS case); there exists a critical value for the electrical charge ( Q critical ). For Q < Q critical , there is no global stability area, but for Q > Q critical , this area appears and increases as the charge increases (see the middle panels in Fig. 7). \niii) For Λ = 0; large BTZ black holes satisfy the global stability condition. Furthermore, the global stability area is independent of the electrical charge (see the down panels in Fig. 7).", 'V. CONCLUSIONS': "In this work, the phantom BTZ black hole solutions were extracted in three-dimensional spacetime for the first time. Then, the Ricci and Kretschmann scalars of the obtained solutions were studied to find the curvature singularity. \nFIG. 7: The Gibbs free energy G versus S for different values of the charge. Left panel for phantom ( η = 1), and right panel for Maxwell case ( η = -1). Up panels for Λ < 0, middle panels for Λ > 0, and down panels for Λ = 0 \n<!-- image --> \nAlso, the asymptotical behavior of phantom BTZ black holes was investigated. Our findings indicated that a curvature singularity exists at r = 0, and the asymptotical behavior could be (A)dS. To extract different behaviors between the BTZ black holes in the presence of Maxwell and phantom (anti-Maxwell) fields, the effects of different parameters (mass, charged, and the cosmological constant) on the horizon of phantom BTZ black holes were studied. Indeed, phantom BTZ black holes with charged BTZ black holes were compared. Our findings indicated that there were three substantially different behaviors of BTZ black holes between Maxwell and phantom fields, which were: i) There was \nonly one root for phantom (anti-Maxwell) BTZ black holes. However, the roots of BTZ black holes changed in the presence of the Maxwell field. ii) The radius of the phantom BTZ black hole increased with the increase of electric charge, which was different from the Maxwell case. Moreover, the number of roots decreased with increasing electric charge. Our analysis indicated that BTZ black holes can have two roots (inner and outer roots), one root (extremal black holes), and no root (naked singularities) as the electric charge increases. iii) We found that phantom BTZ black holes can exist for λ < 0, Λ > 0 and λ = 0. However, the BTZ black hole with Maxwell field only exists for the case Λ < 0. \nThe study focused on obtaining the thermodynamic quantities of BTZ black holes with Maxwell and phantom fields. It was found that these black holes satisfied the first law of thermodynamics. Interestingly, the results showed that the obtained phantom BTZ black holes had a positive mass for all values of the cosmological constant (Λ = 0, Λ > 0, and Λ < 0). Specifically, a constraint was identified in the form of Λ < 2 ηq 2 r 2 + ln ( r + l ) , to have a positive mass for the BTZ black holes. This constraint implies that the mass of the phantom BTZ black holes is always positive when Λ < 0 or Λ = 0. However, for Λ > 0, we must respect this constraint. On the other hand, the mass of the Maxwell BTZ black hole was found to be positive only for negative values of the cosmological constant. \nThe local stability of three-dimensional black holes has been studied in the context of the canonical ensemble using the heat capacity. Our analysis revealed two different behaviors for the heat capacity of the BTZ black hole in the presence of phantom and Maxwell fields. These are as follows: \n- i) For Λ > 0 (dS case), medium-sized phantom BTZ black holes were found to be stable. However, charged BTZ black holes in the presence of the Maxwell field were found to be unstable. \nii) For Λ < 0 (AdS case), large black holes in the presence of phantom and Maxwell fields exhibited local stability. However, there was a distinct stable region. In other words, phantom BTZ black holes had a larger stable area compared to the Maxwell case. \nThe global stability of BTZ black holes in the presence of phantom and Maxwell fields has been evaluated in the context of the grand canonical ensemble by calculating the Gibbs free energy. Our analysis indicated that the BTZ black holes can satisfy the global stability condition in the presence of phantom and Maxwell fields for all values of the cosmological constant (Λ = 0, Λ > 0, and Λ < 0). However, the global stability area was more extensive for phantom BTZ black holes compared to Maxwell's case. In addition, we have studied the effect of electrical charge on the global stability area. 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2024arXiv240911467C

The reduced speed of light approximation has been employed to speed up radiative transfer simulations of reionization by a factor of gtrsim 510. However it has been shown to cause significant errors in the HIionizing background near reionizations end in simulations of representative cosmological volumes. This can bias inferences on the galaxy ionizing emissivity required to match observables such as the Lyalpha forest. In this work we show that using a reduced speed of light is to a good approximation equivalent to rescaling the global ionizing emissivity in a redshiftdependent way. We derive this rescaling and show that it can be used to correct the emissivity in reduced speed of light simulations. This approach of rescaling the emissivity after the simulation has been run is useful in contexts where the emissivity is a free parameter. We test our method by running full speed of light simulations using these rescaled emissivities and comparing them with their reduced speed of light counterparts. We find that for reduced speeds of light tildec geq 0.2 the 21 cm power spectrum at 0.1 leq k hrm Mpc1 leq 0.2 and key Lyalpha forest observables agree to within 20 throughout reionization and often better than 10. Positiondependent timedelay effects cause inaccuracies in reionizations morphology on large scales that produce errors up to a factor of 2 for tildec leq 0.1. Our method enables a factor of 5 speedup of radiative transfer simulations of reionization in situations where the emissivity can be treated as a free parameter.

2024-09-01T00:00:00Z

['2024arXiv240911467C', '10.48550/arXiv.2409.11467', 'arXiv:2409.11467']

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

Accurate simulations of reionization using the reduced speed of light approximation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11467.pdf

{'Christopher Cain a': 'a School of Earth and Space exploration, Arizona State University, Tempe, AZ 85281, USA \nE-mail: clcain3@asu.edu', 'Abstract.': "The reduced speed of light approximation has been employed to speed up radiative transfer simulations of reionization by a factor of ≳ 5 -10 . However, it has been shown to cause significant errors in the HI-ionizing background near reionization's end in simulations of representative cosmological volumes. This can bias inferences on the galaxy ionizing emissivity required to match observables, such as the Ly α forest. In this work, we show that using a reduced speed of light is, to a good approximation, equivalent to re-scaling the global ionizing emissivity in a redshift-dependent way. We derive this re-scaling and show that it can be used to 'correct' the emissivity in reduced speed of light simulations. This approach of re-scaling the emissivity after the simulation has been run is useful in contexts where the emissivity is a free parameter. We test our method by running full speed of light simulations using these re-scaled emissivities and comparing them with their reduced speed of light counterparts. We find that for reduced speeds of light ˜ c ≥ 0 . 2 , the 21 cm power spectrum at 0 . 1 ≤ k/ [ h Mpc -1 ] ≤ 0 . 2 and key Ly α forest observables agree to within 20% throughout reionization, and often better than 10% . Position-dependent time-delay effects cause inaccuracies in reionization's morphology on large scales that produce errors up to a factor of 2 for ˜ c ≤ 0 . 1 . Our method enables a factor of 5 speedup of radiative transfer simulations of reionization in situations where the emissivity can be treated as a free parameter.", '1 Introduction': "Radiative transfer (RT) simulations are a crucial tool for studying the Epoch of Reionization (EoR). During this period, the first generation of hydrogen ionizing sources drove ionization fronts (I-fronts) into the neutral intergalactic medium (IGM), which later overlapped and produced a fully ionized IGM by z = 5 -6 [1-3]. RT simulations can self-consistently track the growth and overlap of ionized regions during reionization, and model in detail the physical conditions in the intergalactic medium (IGM) during the process [4-7]. They also allow the physics of galaxies to be directly linked to IGM conditions, enabling observables that probe reionization to constrain the physics of high-redshift galaxies. \nDespite being the most physically accurate method available to study reionization, RT simulations are also the most computationally expensive. Formally, the RT equation is a 7 -dimensional problem - three spatial, two angular, one frequency, and one time. Moreover, representative cosmological volumes ( ≳ [100Mpc] 3 ) typically contain millions of ionizing sources, further increasing the cost of RT. Several methods, including moment-based RT [8-10] and adaptive ray tracing [11, 12] have been developed to speed up the angular component of the problem and eliminate the direct scaling of computation time with the number of sources. For many applications, only one to a few frequency bins are required to accurately model IGM conditions [e.g. 6, 13, 14]. Sub-grid treatments of the clumping of the IGM on ∼ kpc scales [e.g. 15, 16] can allow for simulations to be run at relatively coarse spatial resolution, which helps to counter the computational scaling with number of spatial dimensions. \nAn common trick used to speed up the time-dependence of the RT equation is the reduced speed of light approximation (RSLA). The time step required to resolve the transfer \nof radiation on an Eulerian grid is \nt rad ≤ ∆ x cell c (1.1) \nwhere ∆ x cell is the cell width and c is the speed of light. Often, t rad is much smaller than any other timescale (such as hydro-dynamical or chemical timescales), meaning that it could be much larger without affecting the treatment of these other processes. Indeed, if the RT is done in post-processing (as in many applications), the only relevant timescales are those associated with the RT equation itself. In addition to t rad , these are (1) the cell-crossing time of ionization fronts (I-fronts) and (2) the timescale over which the source population and underlying density field evolve. The typical speed of I-fronts ( v IF ) is 1 -2 orders of magnitude lower than c [17], and the galaxy population evolves on timescales of 10 s -100 s of Myr, much longer than typical values of t rad ( ≲ 1 Myr). \nThe RSLA introduces a smaller speed of light, ˜ c < c , chosen as small as possible so that t rad remains the smallest timescale in the problem. Because the number of time-steps is inversely proportional to t rad , this gives a direct computational speedup by a factor of c/ ˜ c . The RSLA has been used in a number of contexts to dramatically increase computational efficiency of reionization RT simulations at various spatial scales [18, 19]. \nHowever, several recent works [20-23] have noted that the RSLA fails in certain important regimes of reionization modeling. Ref. [20] argued that the RSLA should be used only to model local fluctuations in the ionizing background during reionization, but never the mean ionizing background itself. Supporting this point, Ref. [22] found that using the RSLA leads to an under-estimate of the ionizing background close to and after the end of reionization by as much as a factor of ˜ c/c . Most recently, Ref. [23] found that reionization models calibrated to reproduce recent Ly α forest observations at 5 < z < 6 (from Ref. [24]) using the RSLA yielded biased results for the galaxy ionizing emissivity required to match these measurements. In this work, we seek to understand the reasons for these issues and find a way to account for them. \nThe findings of Ref. [23] hint at a possible pathway to a solution. They found that in simulations with different values of ˜ c , the global ionizing emissivity ˙ N em ( z ) could be calibrated to reproduce measurements of the Ly α forest transmission at z ≤ 6 (their Fig. 10). This suggests that perhaps the RSLA is, at least approximately, equivalent to a redshift-dependent re-scaling of ˙ N em . In this work, we will show that this is indeed the case, and we will provide a formula for this re-scaling that can be applied to ˙ N em for any simulation that uses the RSLA. In situations where ˙ N em is being set by hand (as in Ref. [23]), or is a function of some set of free parameters, this re-scaling can be applied to ˙ N em in 'post-processing'. Then, the simulation can be treated as if it had used the full speed of light and the re-scaled ˙ N em . \nThis work is organized as follows: in §2, we use a simple analytic model to understand why the RSLA fails in certain situations, and derive the aforementioned re-scaling of ˙ N em . In §3, we briefly describe our numerical methods for running RT simulations of reionization. In §4-5 we use simulations to test how well our approach works, and we conclude in §6. Throughout, we assume the following cosmological parameters: Ω m = 0 . 305 , Ω Λ = 1 -Ω m , Ω b = 0 . 048 , h = 0 . 68 , n s = 0 . 9667 and σ 8 = 0 . 82 , consistent with Ref. [25] results. All distances are quoted in co-moving units unless otherwise specified. Throughout, we will work in units in which the speed of light is unity ( c = 1 ).", '2.1 When and why does the RSLA fail?': "Wewill first build intuition for when and why the RSLA fails using a simple analytic argument. In the early stages of reionization, the universe contains a collection of ionized 'bubbles' surrounding isolated clusters of ionizing sources, and is otherwise fully neutral. The mean photo-ionization rate, Γ HI , inside one of these bubbles is \nΓ HI = N γ cσ HI (2.1) \nwhere N γ is the mean number density of ionizing photons inside the bubble, c is the speed of light, and σ HI is the HI-ionizing cross-section averaged over the ionizing spectrum. At some time t after reionization starts, N γ is given by \nN γ ( t ) = ∫ t 0 dt ' [ ˙ N em ( t ' ) -˙ N abs ( t ' )] (2.2) \nwhere ˙ N em is the emission rate of photons inside the bubble by sources and ˙ N abs is the absorption rate. The latter can be written as two terms, \n˙ N abs = ˙ N ion abs + ˙ N rec abs (2.3) \nwhere ˙ N ion abs is the absorption rate by neutral gas at the edges of the bubble, and ˙ N rec abs is due to recombinations within the bubble. \nWe will first consider the limit where the absorption rate is dominated by ionizations of neutral H atoms, such that ˙ N ion abs >> ˙ N rec abs . This limit is a good approximation early in reionization when the ionized bubbles are much smaller than their recombination-limited Stromgren volumes and most ionized regions have not yet overlapped. The absorption rate can then be written as \n˙ N abs ( t ) = ˙ N ion abs ( t ) ≈ ˙ N em ( t -R/c ) (2.4) \nwhere we have approximated the ionized region as a spherical bubble with radius R . The quantity t -R/c is the so-called 'retarded time' at the boundary of the ionized region (the ionization front or I-front) relative to the sources near the bubble's center. That is, photons emitted at time t -R/c will reach the bubble's edge and be absorbed at time t . \nWe next assume that R/c is sufficiently small that both ˙ N em and R can be approximated as constant across this length of time. In this limit, Eq. 2.2 evaluates to \nN γ = ˙ N em × ( t -[ t -R/c ]) = ˙ N em R c (2.5) \nPutting this into Eq. 2.1, we get \nΓ HI = ˙ N em R c cσ HI = ˙ N em Rσ HI (2.6) \nIn this limit, Γ HI is independent of the speed of light, and we can get the right Γ HI inside the bubble using the RSLA. Indeed, replacing R in Eq. 2.6 with the mean free path to ionizing photons, λ , reveals that we have reproduced the so-called 'local source approximation' with its well-known relationship between Γ HI , λ , and ˙ N em [e.g. Ref. 26]. \nNext, we consider the opposite limit, in which ˙ N ion abs = 0 . In this case, either the bubble has reached its limiting Stromgren volume (in which case ˙ N rec abs ≈ ˙ N em ), or the universe is entirely ionized. Either way, the right-hand side of Eq. 2.2 will be independent of the speed of light, since neither ˙ N em nor ˙ N rec abs depend on it 1 . Thus, N γ is independent of c , and we have \nΓ HI ∝ c (2.7) \nThus, using the RSLA in this limit would result in Γ HI being incorrect by the same factor by which c is reduced. \nThe first approximation - that ˙ N ion abs dominates the absorption rate - is valid near the beginning of reionization, as we mentioned earlier. The reason Γ HI is independent of c in this limit is that the time required for photons to reach neutral gas is proportional to 1 /c , such that N γ inside the bubble(s) is also proportional to this factor. This cancels the factor of c in Eq. 2.1, which gives the correct Γ HI if a reduced c is used, even though N γ is incorrect. In the other limit, the fact that N γ is independent of c means that it will still be correct when a reduced c is used, causing Γ HI to be incorrect. The first limit is expected to apply near the beginning of reionization, and the second near and after its end. The limiting behavior described here is exactly what was found in Ref. [22] (see also Ref. [21]).", '2.2 Fixing the problem': "In this section, we will show that, to a good approximation, the use of the RSLA equivalent to a redshift-dependent re-scaling of ˙ N em . Consider two simulations, one using the full speed of light ˜ c = c = 1 and the other a reduced speed of light ˜ c < 1 . Let these simulations have global ionizing emissivities ˙ N c em and ˙ N ˜ c em , respectively. Our goal is to find a relationship between these four quantities such that these two simulations have identical physical properties including reionization history, net absorption rate, Γ HI , etc. In particular, if they have the same absorption rate (that is, ˙ N c abs = ˙ N ˜ c abs ), then from Eq. 2.2, we have \nN c γ ( t ) -∫ t 0 dt ' ˙ N c em ( t ' ) = N ˜ c γ ( t ) -∫ t 0 dt ' ˙ N ˜ c em ( t ' ) (2.8) \nwhere we have used the same speed-of-light notation for N γ that we did for ˙ N em . If we also assume that both simulations have the same mean Γ HI , then from Eq. 2.1 we have \nN c γ cσ HI = N ˜ c γ ˜ cσ HI → N c γ = N ˜ c γ ˜ c c (2.9) \nProvided Eq. 2.9 holds at all times, we can differentiate both sides with respect to time to get \n˙ N c γ = ˙ N ˜ c γ ˜ c c (2.10) \nDifferentiating both sides of Eq. 2.8 with respect to time and substituting for N c γ using Eq. 2.10 yields the main result of this paper, \n˙ N c em ( t ) = ˙ N ˜ c em ( t ) -˙ N ˜ c γ ( t ) ( 1 -˜ c c ) (2.11) \nWe have explicitly written the time dependence to emphasize that Eq. 2.11 holds at all times. \nEq. 2.11 expresses the emissivity in the simulation with the full speed of light in terms of c , ˜ c , N ˜ c em , and ˙ N ˜ c γ . The first two of these are just numbers, and the other two are known after the simulation with the reduced speed of light has been run. This allows us to calculate ˙ N c em without running any simulations with the full speed of light . Again, ˙ N c em is the emissivity would produce (approximately) the same ionization history, Γ HI , etc. as the reduced speed of light run, had the full speed of light been used. Of course, since we had to assume that this relationship exists in order to derive Eq. 2.11, we have not yet proven that this statement is true. We will do so using RT simulations in §4-5. \nThe usefulness of Eq. 2.11 is found in situations where ˙ N em is being treated as a 'free parameter', which can be 'corrected' after the simulation has been run. Indeed, in works such as Refs. [3, 6, 16], ˙ N em is a 'free function' of redshift that is set by hand to match some set of observables - in those works, the mean transmission of Ly α forest at z ≤ 6 . In many semi-numerical frameworks [e.g. 21cmFAST, Refs. 29, 30], ˙ N em is uniquely determined by a set of free parameters that are being constrained using a set of observations. In this case, a re-scaling of ˙ N em corresponds to changing the mapping between these parameters and ˙ N em in a way that can be quantified. We emphasize the Eq. 2.11 does not constitute a correction to the RSLA simulation results themselves, but rather simply a re-scaling of the global emissivity after the simulation has been run. As such, it is unfortunately not directly applicable in situations where ˙ N em is a self-consistent prediction of e.g. an underlying galaxy model [5, 7, 19]. In this situation, one would require a method to correct the simulation quantities themselves, which is beyond the scope of this work.", '3 Numerical Methods': "Our simulations are run with the adaptive ray-tracing RT code FlexRT, first described in [16, 31] and tested in Ref. [32]. Here, we will briefly discuss the relevant features of the code, referring the reader to these works for details. FlexRT solves the RT equation in post-processing on a time-series of cosmological density fields. We model the ionizing sources in a simple manner by extracting halos from an N-body simulation with the same large-scale structure as the density field and assigning them ionizing emissivities (see below). The RT equation itself is solved using an adaptive ray tracing method similar to the one described in Ref. [11] and employed in the code of Ref. [12]. Sub-resolved ionization fronts are tracked using the 'moving-screen' approximation, and the temperature behind them ( T reion ) is estimated using the flux-based method described in Ref. [17] 2 . FlexRT solves for the opacity of the ionized IGM to ionizing photons using a novel sub-grid model based on high-resolution hydro/RT simulations of IGM gas dynamics similar to those described in Refs. [18, 27]. Recent improvements to this model, and details about forward-modeled observables are detailed in Ref. [23]. \nIn §4, we will use the 'Reference' setup in Ref. [23], which assumes a fully dynamically relaxed IGM and mono-chromatic RT. The physical properties of this model are summarized in their Fig. 2 and associated text. Throughout in §4, the only parameters we will change relative to their Reference model are ˙ N em and ˜ c . We will generalize and test Eq. 2.11 in the context of multi-frequency RT simulations in §5. Our simulations have a box size of L = 200 h -1 Mpc with N RT = 200 3 RT cells. Ly α forest properties are calculated (following §3.7 of \nTable 1 . Summary of different kinds of simulations referred to in this work, and the names used for each (left-most column). The second column from left gives the emissivity used in that simulation, and the third column the speed of light. The right-most column briefly describes each type of simulation. \nRef. [23]) on a high-resolution ( N = 2048 3 ) hydrodynamics simulation with the same largescale initial conditions used for the N-body simulation from which we get the halos. In this work, we will consider reduced speeds of light of ˜ c = 0 . 05 , 0 . 1 , 0 . 2 , 0 . 3 , and 0 . 5 , alongside simulations that use the full speed of light ( ˜ c = c = 1 ).", '4.1 Strategy & Terminology': 'The claim of Eq. 2.11 can be summarized as follows: Given an RT simulation run with a reduced speed of light ˜ c and emissivity ˙ N ˜ c em ( z ) that has a mean photon number density N ˜ c γ ( z ) , a simulation using the full speed of light and emissivity ˙ N c em ( z ) should have the same reionization history, ionization morphology, IGM transmission properties, etc. as the original simulation. We can verify whether this claim is true by comparing two types of simulations. We first run simulations using the RSLA, with ˜ c < 1 . We refer to these simply as reducedc runs, with emissivity ˙ N ˜ c em . Next, we apply the re-scaling in Eq. 2.11 to ˙ N ˜ c em to obtain ˙ N c em , and run another simulation with the full speed of light using ˙ N c em . We refer to these as fullc , re-scaled ˙ N em runs. Eq. 2.11 is accurate to the extent that the properties of these two types of simulations agree with each other. Lastly, for purposes of comparison, we will also include a third type of simulation, in which we use the full speed of light but do not apply any re-scaling to ˙ N ˜ c em . We refer to these as fullc , un-scaled ˙ N em runs. We summarize the properties of these three types of simulations in Table 1.', '4.2 Ionization history and morphology': "We begin by studying how well the reionization history and morphology (shapes and sizes) of ionized regions match in the reducedc and fullc , re-scaled ˙ N em runs. For this section, we scale up ˙ N em from the reference model of Ref. [23] until reionization ends at z ≈ 6 when c = 1 . This is our fullc , un-scaled ˙ N em run. Next, we use the same ˙ N em to run a series of reducedc runs with ˜ c = 0 . 05 , 0 . 1 , 0 . 2 , 0 . 3 , and 0 . 5 . Lastly, we evaluate Eq. 2.11 for \nFigure 1 . Summary of fullc , un-scaled ˙ N em and reducedc simulations used to study how well Eq. 2.11 works with respect to the reionization history and morphology. Left: the reionization history for each value of ˜ c , using a scaled-up version of ˙ N em from the reference model from Ref. [23], which ends reionization at z = 6 for ˜ c = c = 1 (black solid curve, the fullc , un-scaled ˙ N em run). Reducing ˜ c pushes the end of reionization later, such that it has not finished by z = 4 . 8 for ˜ c = 0 . 05 . Middle: number density of ionizing photons, N γ . The thick lines denote the actual values, while the faded lines show ˜ cN γ . Consistent with our analytic argument in §2.1, at the beginning of reionization ˙ N γ is a factor of 1 / ˜ c higher in the reducedc runs than in the fullc , un-scaled ˙ N em case (Eq. 2.5). Near reionization's end, they approach similar values. Right: Ionizing emissivity used in all these simulations ( ˙ N ˜ c em ). \n<!-- image --> \neach of these and then run a set of fullc , re-scaled ˙ N em simulations. Again, the degree to which the reducedc and corresponding fullc , re-scaled ˙ N em runs agree with each other determines how accurate is Eq. 2.11.", '4.2.1 Ionization history': "In Figure 1, we show the volume-averaged ionized fraction x V ion (left), the ionizing photon number density ˙ N γ (middle), and the ionizing emissivity ˙ N em (right) for our fullc , unscaled ˙ N em run (black solid curve) and our reducedc runs (colored curves, see legend). In the left panel, we see that the reionization histories are initially similar, but start to diverge as reionization enters its later stages. The end of reionization is delayed by ∆ z ≈ 1 . 5 in the ˜ c = 0 . 05 simulation relative to the ˜ c = 1 case. The bold curves in the middle panel show ˙ N γ for each simulation, while the faded curves show ˜ cN γ . Early in reionization, ˙ N γ scales linearly with 1 / ˜ c , consistent with the early reionization limit described by Eq. 2.5. However, as reionization ends, ˙ N γ for all simulations approaches the same value, reflecting the behavior expected near reionization's end in §2.1. The right-most panel shows ˙ N ˜ c em , which is the same for all the simulations shown here. \nIn the left panels Figure 2, we compare each of our reducedc runs to their fullc , re-scaled ˙ N em counterparts. We show the former as bold curves and the latter as faded curves, with the color and line style identifying the value of ˜ c . The upper left panel shows x V ion , zoomed in on the second half of the reionization history for clarity. We find excellent agreement between the two even for our lowest value of ˜ c = 0 . 05 , such that the different sets of curves are difficult to see on the plot. The bottom left panel shows the linear differences of x V ion in the reducedc and fullc , re-scaled ˙ N em runs over the entire reionization history. For all but the ˜ c = 0 . 05 case, the difference is always ≲ 0 . 01 , and for ˜ c ≥ 0 . 3 it is < 0 . 005 . \nIn the upper right panel, we show ˙ N c em calculated using Eq. 2.11, and the bottom right panel shows the ratio with ˙ N ˜ c em (the black solid curve). This ratio differs from unity the most at the end of reionization, dropping to ≈ 0 . 5 ( 0 . 8 ) by z = 5 for ˜ c = 0 . 05 ( 0 . 5 ). The fact \nFigure 2 . Comparison of the reionization history and emissivity in our reducedc and fullc , rescaled ˙ N em simulations. Upper Left: the same reionization histories (again), but zoomed in on the last half of reionization and including the fullc , re-scaled ˙ N em results for each value of ˜ c (faded curves). The bold and faded curves agree very well for all values of ˜ c , such that they are difficult to tell apart on the plot. Lower Left: the linear differences between x V ion for the reducedc and corresponding fullc , re-scaled ˙ N em runs. This is less than 0 . 01 for all except ˜ c = 0 . 05 . Upper Right: emissivities for each of the fullc , re-scaled ˙ N em runs ( ˙ N c em from Eq. 2.11, colored curves) compared to ˙ N ˜ c em (black solid curve). Bottom Right: ratio of ˙ N c em and ˙ N ˜ c em for each ˜ c . The ratio is smallest at the end of reionization, dropping to 0 . 5 ( 0 . 8 ) for ˜ c = 0 . 05 ( 0 . 5 ). \n<!-- image --> \nthat the reducedc and fullc , re-scaled ˙ N em runs have such different ˙ N em but similar reionization histories validates Eq. 2.11. As we will see, the small differences in reionization history arise because Eq. 2.11 applies to globally averaged quantities, and does not account for spatial fluctuations in the distribution of photons in the IGM. \nThese results show that, at least with respect to the global ionization history, the claim of Eq. 2.11 is reasonably accurate. Namely, that using the RSLA is equivalent to a redshiftdependent re-scaling of ˙ N em . We emphasize again that Eq. 2.11 does not prescribe a way to directly correct the reducedc simulation results for the effects of the RSLA. Put another way, we do not prescribe a way to convert the reducedc results into those of the fullc , un-scaled ˙ N em results. As such, our method is not directly applicable when ˙ N em is a prediction of e.g. some underlying galaxy model.", '4.2.2 Ionization morphology': 'Next, we apply the same analysis to the morphology of ionized and neutral regions, and statistics that depend on it. Figure 3 shows maps of the ionization field when the universe is 80% ionized for reducedc (top row) and fullc , un-scaled ˙ N em runs (middle row). Dark regions are neutral and white ones are ionized. The bottom row shows the difference maps for the top and middle rows. Red (blue) denotes regions that are more neutral (ionized) in the reducedc simulations. \nFrom right to left (decreasing ˜ c ) in the top row, the neutral regions (islands) grow slightly more extended and porous. Equivalently, the smallest ionized bubbles grow larger, and the largest ones smaller, when ˜ c decreases. We see no such trend, however, in the middle row. From this, we deduce that the morphological differences in the top row are not due to the reducedc runs having different reionization histories, since such differences would appear \nFigure 3 . Visualization of the ionization field at 80% ionized in reducedc (top row) and fullc , re-scaled ˙ N em runs (middle row) for ˜ c = 0 . 05 , 0 . 1 , 0 . 2 , and 0 . 3 (left to right). We show the difference maps for each value of ˜ c in the bottom row. We see in the top row that as ˜ c decreases (right to left), the neutral regions (islands) become more extended and porous at fixed ionized fraction. We do not see this in the middle row, indicating that it is not due to differences in the reionization history between reducedc runs. Instead, it is because the RSLA induces a longer delay between emission and absorption for photons emitted in large vs. small ionized bubbles. The re-scaling in Eq. 2.11 corrects for the delay only on average, and does not account for position dependence. The result is that the morphologies in the reducedc and fullc , re-scaled ˙ N em runs diverge as ˜ c decreases. \n<!-- image --> \nfor the fullc , re-scaled ˙ N em also. They are instead a position-dependent artifact of the RSLA. Photons emitted in large ionized regions experience a longer delay between emission and absorption than do those emitted in small bubbles. Eq. 2.11 accounts for this time-delay only on average, and thus misses the differences between photons emitted in different regions. As a result, the largest (smallest) bubbles are too small (large) in the reducedc runs. We can most clearly see this effect in the difference maps, where it is clearly visible even for ˜ c = 0 . 3 . In this work, we do not attempt to correct for this position-dependent effect, since doing so would be complicated by the non-locality of the ionizing radiation field. As we will see, this effect will set the minimum value of ˜ c for which Eq. 2.11 can be used. \nFigure 4 . Power spectrum of the ionization field and 21 cm signal for simulations with ˜ c = 0 . 05 , 0 . 1 , 0 . 2 , and 0 . 3 (columns from left to right). Top Row: power spectrum of the ionization field, ∆ ion , vs. wavenumber k at an ionized fraction of 80% . The bold colored curves show results for the reducedc simulations, and the faded colored curves the fullc re-scaled runs. The faded gray curves shows (at fixed redshift) the fullc , un-scaled ˙ N em runs. 2nd Row: the same, but for the 21 cm power spectrum, ∆ 21 . The fullc , un-scaled ˙ N em results differ dramatically from the other two, reflecting its lower neutral fraction at fixed redshift. However, the reducedc and fullc , re-scaled ˙ N em runs differ in their shape: the former has significantly less power on large scales (small k ) and slightly more on small-scales (large k ). 3rd Row: ∆ 21 vs redshift at k = 0 . 2 and 0 . 1 h Mpc -1 (top and bottom set of curves, respectively). The bold colored curves are noticeably below the faded ones, reflecting the deficit in large-scale power in the reducedc runs. 4th & 5th Rows: ratios of the reducedc and fullc re-scaled results in the 2nd and 3rd rows, respectively. The shaded regions denote ± 20% from unity. In the bottom row, we show the ratio for k = 0 . 1 h Mpc -1 . Decreasing ˜ c increases differences at low k , so that for ˜ c = 0 . 1 the difference is ≈ 20% at all redshifts at k = 0 . 1 h Mpc -1 . At ˜ c ≥ 0 . 2 , we find ≲ 15% differences at k = 0 . 1 h Mpc -1 , which get smaller at larger k . \n<!-- image --> \nFigure 4 quantitatively compares the large-scale fluctuations in the ionization field in all three types of simulations. In the top row, we show the dimensionless ionization power spectrum ∆ ion ( k ) at 80% ionized for ˜ c = 0 . 05 , 0 . 1 , 0 . 2 , and 0 . 3 (left to right). The bold colored lines, faded colored lines, and faded gray lines indicate results for reducedc , fullc , re-scaled ˙ N em , and fullc , un-scaled ˙ N em runs, respectively. The second row is the same thing, but instead showing the 21 cm power spectrum 3 , calculated using the tools21cm \npackage of Ref. [33]. \nThe power in the fullc , un-scaled ˙ N em run is much lower than the others - increasingly so at lower ˜ c . This reflects the fact that the reducedc runs are more delayed relative to the fullc , un-scaled ˙ N em case the smaller ˜ c is. So, at a fixed neutral fraction in the former, the neutral fraction in the latter decreases for smaller ˜ c . The fullc , re-scaled ˙ N em sims are much closer to the reducedc results, but differ noticeably in their shape. The latter have significantly less power at large scales (small k ), and slightly more power at small scales (large k ). This is a direct result of the morphological errors shown in Figure 3, with the reduced sizes of the largest bubbles driving the decrease in smallk power. In the third row, we show ∆ 21 vs. redshift at fixed k = 0 . 2 and 0 . 1 h Mpc -1 (top and bottom set of curves, respectively), key targets for experiments such as HERA [34-36]. The suppression in power is consistent across the entire reionization history, indicating that even early in reionization the RSLA can cause significant errors in ionization morphology. \nThe fourth row shows the ratio of the bold and faded colored curves in the second row, with the shaded region denoting ± 20% . For ˜ c = 0 . 3 , the differences between the reducedc and fullc re-scaled runs never exceeds 20% even at the largest scales at 80% ionized. For lower ˜ c , the drop in power at small k increases, reaching a factor of ≈ 2 at the box scale for ˜ c = 0 . 05 . For ˜ c = 0 . 2 the deviation is still only ∼ 10% at k = 0 . 1 h Mpc -1 , but at ˜ c = 0 . 1 this increases to 20% , and to 30% for ˜ c = 0 . 05 . The bottom row shows the ratio of the k = 0 . 1 h Mpc -1 curves in the third row. Here we see that for ˜ c = 0 . 2 , the differences are never larger than 15% at any redshift. For ˜ c = 0 . 1 , the difference exceeds 20% at z ∼ 9 , and for ˜ c = 0 . 05 , it reaches nearly a factor of 2 . \nWe see that the utility of Eq. 2.11 is limited by the fact that it does not account for the spatial fluctuations in the delay between emission and ionization of ionizing photons in the reducedc simulations. To summarize, ∆ 21 at k ≤ 0 . 1 h Mpc -1 can deviate by 20% between the reducedc and fullc , re-scaled ˙ N em runs for ˜ c = 0 . 1 , and by up to 40% for ˜ c = 0 . 05 . These differences drop to ≤ 15% for ˜ c = 0 . 2 and ≤ 10% for ˜ c = 0 . 3 . As such, we caution against applying our method for ˜ c < 0 . 2 in contexts where the large-scale reionization morphology is important.', '4.3 QSO observations at z ≤ 6': 'In this section, we apply the same kind of test to observables derived from high-redshift quasar absorption spectra at z ≤ 6 . These include the Ly α forest, which has recently yielded detailed insights into the tail end of reionization at 5 < z < 6 [3, 6, 24, 30, 37], and the ionizing photon mean free path [38-41]. These observables are sensitive not only to ionization morphology, but also to the large-scale spatial fluctuations in the ionizing background Γ HI and the IGM temperature, which are affected by the RSLA.', '4.3.1 Global Observables': "The evolution of the mean transmission of the Ly α forest is a tight boundary condition on the end of reionization. Several studies [3, 14, 23, 42] have 'calibrated' their reionization simulations to agree by construction with this observable, then asked whether other observables predicted by the simulation agree well with observations. Here, we will follow this procedure and assess how applicable Eq. 2.11 is in the context of forest-anchored studies. We first run a set of reducedc simulations with ˙ N ˜ c em ( z ) calibrated so that the mean transmission of the Ly α forest matches the recent observations of Ref. [24]. Note that unlike in the previous section, this procedure yields a different ˙ N ˜ c em for each value of ˜ c , since we are requiring all \nFigure 5 . Tests with simulations calibrated to match the Ly α forest at z = 6 . Upper Left: ⟨ F Ly α ⟩ vs. redshift for all three types of simulations described in Table 1 for ˜ c = 0 . 1 , 0 . 2 , and 0 . 3 . The formatting of the curves is the same used in Figure 4 (see text and legend). By construction, all the reducedc results agree with each other and the measurements from Ref. [24]. The fullc , re-scaled ˙ N em , ˜ c = 0 . 3 run is indistinguishable from these. The ˜ c = 0 . 2 run is slightly above the others, but still in reasonable agreement with the measurements. However, the ˜ c = 0 . 1 run diverges significantly from the measurements. All the fullc , un-scaled ˙ N em have too much transmission, since they end reionization earlier than the reducedc runs. Bottom Left: ratio of ⟨ F Ly α ⟩ in the fullc , re-scaled ˙ N em and reducedc runs, with the shaded region denoting ± 20% . We find ≲ 5% agreement for ˜ c = 0 . 3 and ≲ 20% agreement for ˜ c = 0 . 2 , but for ˜ c = 0 . 1 , the simulations disagree by a factor of 1 . 3 -2 . Upper Right: ˙ N em for the reducedc (bold curves) and fullc , re-scaled ˙ N em (faded curves) simulations. As previously mentioned, the reducedc runs have different ˙ N em since they are all separately calibrated to match the forest. The fullc , re-scaled ˙ N em runs have similar ˙ N em , which is expected since they use the full speed of light and are supposed to match the same observable, ⟨ F Ly α ⟩ . Bottom Right: ratio of ˙ N em in the fullc , re-scaled ˙ N em and reducedc runs. They are similar early in reionization but diverge by up to a factor of 3 by its end. \n<!-- image --> \nreducedc runs to match the same observable. For each ˜ c , we apply Eq. 2.11 to ˙ N ˜ c em and run fullc , re-scaled ˙ N em simulations using c = 1 . In some plots, we will also include results from fullc , un-scaled ˙ N em runs. We will judge the accuracy of Eq. 2.11 in the same way as in the previous section - by how well the reducedc and fullc , re-scaled ˙ N em runs agree with each other. \nWe first compare the mean transmission of the Ly α forest, ⟨ F Ly α ⟩ , with measurements from Ref. [24] in the top left panel of Figure 5. The bold colored lines show reducedc runs for ˜ c = 0 . 1 , 0 . 2 , and 0 . 3 . Faded colored lines (with matching line styles) show the fullc , re-scaled ˙ N em results, and faded gray lines (again, with matching line styles) show the fullc , un-scaled ˙ N em results. The bottom panel shows the ratios of the fullc , rescaled ˙ N em and reducedc results (with the shaded line again denoting ± 20% from unity). All the reducedc results agree well with the measurements and each other by construction. The fullc , re-scaled ˙ N em run with ˜ c = 0 . 3 is indistinguishable from its reducedc counterpart. For ˜ c = 0 . 2 , ⟨ F Ly α ⟩ for the fullc , re-scaled ˙ N em run slightly above the measurements, but is still within at most 25% of its reducedc counterpart. However, for \nFigure 6 . The same as Figure 5, but showing λ 912 mfp (left) and T 0 (right). To avoid cluttering the plot, we omit the fullc , un-scaled ˙ N em results. For both ˜ c = 0 . 3 and 0 . 2 , we see agreement to within a few percent in both quantities between the reducedc and fullc , re-scaled ˙ N em runs. However, for ˜ c = 0 . 1 , two diverge by up to ≈ 35% for λ 912 mfp and ≈ 6% for T 0 . These results further confirm our earlier observations that our method should only be applied for ˜ c ≥ 0 . 2 . \n<!-- image --> \n˜ c = 0 . 1 , the fullc , re-scaled ˙ N em run diverges by as much as a factor of 2 . All three of the fullc , un-scaled ˙ N em lie well above the measurements, since their reionization histories end too early. \nIn the upper right panel, we show the emissivities for the reducedc and fullc , rescaled ˙ N em runs ( ˙ N ˜ c em and ˙ N c em , respectively). The bottom panel shows their ratio. We see that ˙ N ˜ c em is quite different for different values of ˜ c , since all runs were calibrated to agree with the same observable. On the other hand, ˙ N c em is quite similar between the different values of ˜ c . This is indeed to be expected if Eq. 2.11 is accurate, since each ˙ N c em is supposed to match the same observable, ⟨ F Ly α ⟩ , with the full speed of light. In the lower right panel, we see that the ˙ N c em and ˙ N ˜ c em are similar early in reionization, but differ by a factor of 2 -3 by its end. \nIn Figure 6, we make the same comparison for the lyman limit mean free path 4 ( λ mfp 912 , MFP, left) and the IGM temperature at mean density T 0 (right). To avoid cluttering the plot, we omit results from the fullc , un-scaled ˙ N em runs. We also include several sets of observations for comparison (see references in caption). For ˜ c = 0 . 2 and 0 . 3 , the reducedc and fullc , re-scaled ˙ N em agree to within a few percent in both λ 912 mfp and T 0 , as the bottom panels show. For ˜ c = 0 . 1 , the disagreement is much larger - up to 35% for λ 912 mfp and ≈ 6% for T 0 . Thus, the trend here is qualitatively similar to that observed in Figures 4 and 5 namely, that Eq. 2.11 gives a reasonable match between reducedc and fullc , re-scaled ˙ N em simulations for ˜ c = 0 . 2 , but not for ˜ c = 0 . 1 . This further confirms that our approach can only be reliably used down to ˜ c = 0 . 2 .", '4.3.2 Large-scale fluctuations': 'We saw in §4.2.2 and §4.3.1 that the applicability of Eq. 2.11 is likely limited to ˜ c ≳ 0 . 2 by spatially-varying time-delay effects caused by the RSLA on large scales. In this section, \nFigure 7 . Distribution of Ly α forest effective optical depths over 50 h -1 Mpc segments of the forest, P ( < τ 50 eff ) , at z = 5 . 4 , 5 . 6 , 5 . 8 , and 6 . 0 . The line styles and colors have the same meanings as in Figure 6. At z = 5 . 8 and 6 , when the neutral fraction is > 10% , the reducedc results differ from each other at the few percent level. This is unsurprising since, as we showed earlier, they differ in large-scale ionization morphology. The ˜ c = 0 . 2 and 0 . 3 fullc , re-scaled ˙ N em agree with their reducedc counterparts to within a few percent, but this deviation grows noticeably larger for ˜ c = 0 . 1 , especially at z = 5 . 4 and 5 . 6 . This suggests that the effect of the RSLA on fluctuations in Γ HI and T may be affecting these differences when the neutral fraction is < 10% . \n<!-- image --> \nwe study this effect in more detail in the context of the Ly α forest. At the end of reionization, large-scale fluctuations in forest properties are set by three quantities that the RSLA affects: the ionization morphology [3], large-scale fluctuations in Γ HI [44], and fluctuations in temperature [45]. We will focus on the latter two in this section. \nIn Figure 7, we show the cumulative distribution function (CDF) of τ 50 eff , P ( < τ 50 eff ) , the effective optical depth over 50 h -1 Mpc segments of the forest, at z = 5 . 4 , 5 . 6 , and 5 . 8 , and 6 . 0 . This statistic has been used in a number of works to argue that reionization must have ended later than z = 6 [3, 6, 30, 37, 42, 46]. The format of the curves is the same as that in Figure 6. All the reducedc runs, and the fullc , re-scaled ˙ N em runs with ˜ c = 0 . 2 and 0 . 3 (faded blue-dotted and green dot-dashed curves) mutually agree to within a few percent at all redshifts. We do see few-percent level differences between different reducedc runs at z = 6 and 5 . 8 , which get smaller at z = 5 . 6 and 5 . 4 . This is unsurprising, since we have already shown that the reducedc runs differ from each other in large-scale ionization morphology (Figures 3-4), which also affects the τ 50 eff CDF when the IGM is still partially neutral. We \n<!-- image --> \n<!-- image --> \nFigure 8 . Visualization of the effect of the RSLA on large-scale fluctuations in Γ HI (in units of 10 -12 s -1 ). We show maps of the reducedc , fullc , re-scaled ˙ N em , and fullc , un-scaled ˙ N em runs in the upper left, upper right, and lower left panels, respectively, for ˜ c = 0 . 2 . The fullc , un-scaled ˙ N em run has a higher mean Γ HI , fewer neutral islands (white regions), and weaker spatial fluctuations in Γ HI , since reionization ends earlier in that run than in the others. The reducedc and fullc , re-scaled ˙ N em runs are in reasonably good visual agreement. However, the former has noticeably larger Γ HI fluctuations, with the bright (faint) regions being brighter (fainter) than in the fullc , re-scaled ˙ N em run. In the lower right, we plot log-space PDFs of Γ HI normalized by its mean. We show results for all three values of ˜ c , with the same line styles and colors used in Figure 5. We have offset the ˜ c = 0 . 1 ( 0 . 3 ) PDFs by 1 . 2 dex to the left (right) for clarity. In all three cases, the PDFs are much narrower in the fullc , un-scaled ˙ N em runs than in the others. As ˜ c decreases, the PDFs in the reducedc become grow wider than their fullc , re-scaled ˙ N em counterparts, owing to time-delay effects (see text). \n<!-- image --> \nalso see that the fullc , re-scaled ˙ N em ˜ c = 0 . 1 run differs significantly from the others, especially at z = 5 . 6 and 5 . 4 , when the neutral fraction is < 10% . This suggests that the effect of the RSLA on large-scale fluctuations in Γ HI and/or T may also play a significant role in driving these differences. \nWe visualize the effect of the RSLA on large-scale fluctuations in Γ HI in Figure 8. We show maps of Γ HI (in units of 10 -12 s -1 ) at z = 5 . 65 , when the universe is ≈ 10% neutral in the reducedc and fullc , re-scaled ˙ N em runs. The top left, top right, and lower left \n<!-- image --> \npanels show maps for the reducedc , fullc , re-scaled ˙ N em , and fullc , un-scaled ˙ N em runs, respectively, for ˜ c = 0 . 2 . We see that the fullc , un-scaled ˙ N em run has a much higher mean Γ HI and weaker Γ HI fluctuations than the other two, owing to its earlier end to reionization. Compared to this run, the reducedc and fullc , re-scaled ˙ N em runs are in relatively good agreement. However, the spatial fluctuations in Γ HI are noticeably stronger in the reducedc run. Once again, this is due to time-delay effects caused by the RSLA. Because photons are delayed longest in reaching the edges of the largest ionized regions, the differences in N γ between them and the smaller bubbles are amplified. In addition, photons far from sources were emitted at a retarded time t ret = R/ ˜ c , where R is the ionized bubble size. Again, these effects are accounted for only on average by the re-scaling of Eq. 2.11, which results in larger spatial Γ HI fluctuations in the reducedc runs than in their fullc , re-scaled ˙ N em counterparts. \nIn the lower right panel, we show the PDFs of Γ HI , normalized by its mean (in log space). Here, we show results for all three values of ˜ c , using the same color and line style convention adopted in Figure 5. To make the plot readable, we have offset the ˜ c = 0 . 1 ( 0 . 3 ) PDFs to the left (right) by 1 . 2 dex. In all cases, the gray curves are much narrower than the others, reflecting the reduced Γ HI fluctuations in the fullc , un-scaled ˙ N em runs. As ˜ c decreases, the PDFs of the reducedc runs grow noticeably wider than those of their fullc , re-scaled ˙ N em counterparts, with the difference becoming very significant for ˜ c = 0 . 1 . These differences contribute to those in P ( < τ 50 eff ) seen in Figure 7. \nFigure 9 shows the same thing as Figure 8, but for IGM temperature. In the lower right panel, we offset the ˜ c = 0 . 1 and 0 . 3 results by ± 30000 K for clarity. The voids, which have ionized most recently, are hottest in the fullc , un-scaled ˙ N em run, which ends reionization the earliest and most rapidly. The reducedc run displays the smallest such T enhancements. The fastest I-fronts around the largest ionized regions, which produce the largest T reion [17, 47], grow move more slowly than they should in the reducedc runs for aforementioned reasons. We see in the lower right panel that the difference between the reducedc and fullc , re-scaled ˙ N em in the highT tail of the PDF becomes significant for ˜ c = 0 . 1 , explaining the corresponding differences in T 0 in Figure 6. \nWe have repeated the tests in this section, in whole or in part, for several variations of the properties of the ionized sources and the IGM assumed in this work. We have first varied the clustering of ionizing sources assumed in our fiducial scenario, which assumes that the ionizing emissivity of halos scales with their UV luminosity. We have tested models wherein the bulk of the ionizing photons are emitted by the only the least massive (least clustered) and most massive (most clustered) halos. We have also applied our method to multi-frequency RT simulations, for which we show results in the next section. Lastly, we tested models that assume more sub-grid recombinations than the Reference model of Ref. [23], and we have varied the reionization history. We find that our results remain essentially the same for all of these variations of our fiducial scenario - namely, that Eq. 2.11 is accurate (to 20% or better) for ˜ c = 0 . 2 , but that using ˜ c = 0 . 1 leads to much larger errors. \nThese results further corroborate our conclusions from §4.2. We find that, for ˜ c ≥ 0 . 2 , reducedc simulations calibrated to match the Ly α forest match well the properties of their fullc , re-scaled ˙ N em counterparts. This confirms that Eq. 2.11 can be applied to simulations using the RSLA with ˜ c ≥ 0 . 2 . However, for ˜ c = 0 . 1 , we see significant differences between the reducedc and fullc , re-scaled ˙ N em runs. These can largely be attributed to position-dependent time-delay effects that arise from using a reduced speed of light, for which Eq. 2.11 does not account. As such, for applications in which the morphology of \nFigure 9 . The same as Figure 8, but for the IGM gas temperature. The main difference between the three maps is the strength of the temperature enhancements near neutral islands, where gas has been recently reionized and is hottest. The fullc , un-scaled ˙ N em run has the hottest temperatures in the voids, since reionization ends earliest and most rapidly. The reducedc run, by contrast, has the weakest T fluctuations. This is because the slowed growth of the largest ionized regions (see §4.2.2) reduces the speed of the fastest I-fronts, which lowers T reion around the largest bubbles. In the lower right panel, we see a significant difference between reducedc and fullc , re-scaled ˙ N em runs for ˜ c = 0 . 1 at the highT end of the PDF, which explains the differences in T 0 seen in Figure 6. \n<!-- image --> \nreionization and/or quasar-based observables at z ≤ 6 are important, we caution against applying our re-scaling method for ˜ c smaller than 0 . 2 .', '5 Generalization to multi-frequency simulations': "In the previous section, we tested Eq. 2.11 in mono-chromatic simulations. Here, we will generalize Eq. 2.11 to apply to multi-frequency simulations and demonstrate that the method works just as well in this case, but with one key additional caveat. For our purposes, the main effect of including multi-frequency RT is to change the temperature and ionizing background in the ionized IGM [23], but not the morphology of ionized regions. As such, we will only show our Ly α forest-focused tests (§4.3) in this section. \nFigure 10 . Same as Figure 5, but showing results for our tests with multi-frequency RT, using Eq. 5.1 to re-scale the emissivity. We find very similar results in all panels to those in Figure 5, showing that our method works nearly as well for multi-frequency RT simulations as for single-frequency ones. See text for details. \n<!-- image --> \nA straightforward multi-frequency generalization of Eq. 2.11 is to simply apply the arguments of §2.2 in each frequency bin separately. Then we have \n˙ N c em ( t, ν ) = ˙ N ˜ c em ( t, ν ) -˙ N ˜ c γ ( t, ν ) ( 1 -˜ c c ) (5.1) \nA subtlety of Eq. 5.1 is that in general, ˙ N ˜ c em ( t, ν ) and ˙ N ˜ c γ ( t, ν ) do not share the same shape in frequency space. Indeed, this is the case during reionization because the absorption crosssection of HI is frequency-dependent, which results in hardening of the radiation field by the IGM. As such ˙ N c em ( t, ν ) will have a softer ionizing spectrum than ˙ N ˜ c em ( t, ν ) , since ˙ N ˜ c γ ( t, ν ) has a harder spectrum. This complicates somewhat the physical interpretation of the re-scaling. \nIn this section, we use the same setup used for multi-frequency RT simulations described in Ref. [23], assuming that the intrinsic spectrum of ionizing sources is a power law of the form J ν ∝ ν -α with α = 1 . 5 . We use 5 frequency bins, with central frequencies chosen such that (initially) all bins contain an equal fraction of the emitted ionizing photons. The condition for choosing the frequency bins is that the average HI-ionizing cross-section, σ HI , should be the same as that of an α = 1 . 5 power law. As in the previous section, we ran reducedc simulations with ˜ c = 0 . 1 , 0 . 2 , and 0 . 3 , and we apply Eq. 5.1 to run their fullc , re-scaled ˙ N em counterparts. In Figure 10, we show ⟨ F Ly α ⟩ in the same format as Figure 5 for our multifrequency tests, except that we omit the fullc , un-scaled ˙ N em results. We find results very similar to those in Figure 5. The mean flux in the reducedc and fullc , re-scaled ˙ N em runs remains within 10% for ˜ c = 0 . 3 , and within 30% or better for ˜ c = 0 . 2 . For ˜ c = 0 . 1 , we again find differences as large as a factor of 2 in ⟨ F Ly α ⟩ . In the right column, we see that ˙ N ˜ c em and ˙ N c em , and the relationship between them, are very similar in the multi-frequency runs as in the mono-chromatic case. This shows that our method works just as well when applied to multi-frequency simulations with respect to the mean Ly α forest transmission. \nFigure 11 . Same as Figure 7, but showing P ( < τ 50 eff ) for our multi-frequency RT tests. We find results very similar to those of the mono-chromatic runs, showing that our re-scaling procedure works well in the multi-frequency case. See text for details. \n<!-- image --> \nIn Figure 11, we show P ( < τ 50 eff ) , in the same format as Figure 7, for our multi-frequency tests. Again, we see results very similar to those in Figure 7. The differences between the bold and faded red curves are slightly larger than they are in Figure 7, suggesting that spatial fluctuations in the spectrum of the ionizing background may be slightly worsening the timedelay effects discussed in §4.3.2. However, this effect is not significant enough to change the level of accuracy obtained with ˜ c = 0 . 2 and 0 . 3 , further validating that for these values of ˜ c , our method can safely be applied to simulations with multi-frequency RT. \nAs mentioned, applying Eq. 5.1 in each frequency bin results in ˙ N c em ( ν ) having a different spectral shape than ˙ N ˜ c em ( ν ) , in a way that depends on redshift. We illustrate this in Figure 12, where we quantify the spectral shape of ˙ N em before and after the re-scaling. The black solid curve in the left panel shows the fraction of the total emissivity in each bin in our multifrequency reducedc run with ˜ c = 0 . 2 (recall that all bins share the same fraction of the photons in this case). In the right panel, the black dotted curve shows 1 N freq ˙ N c em , the fraction that would be in each bin if ˙ N c em and ˙ N ˜ c em shared the same frequency dependence. The colored dot-dashed lines show ˙ N c em ( ν ) for each bin, with redder (bluer) colors denoting lower (higher) photon energies (see annotations). The re-scaling removes significantly more photons at higher energies, causing ˙ N c em ( ν ) to get softer with decreasing redshift. \nIn the right panel, we calculate the 'effective' power law index, α eff , for each case vs. \nFigure 12 . Illustration of how our re-scaling procedure changes the intrinsic spectrum of the sources when applied to multi-frequency RT simulations. Left : Ionizing emissivity in each bin in the reducedc run (black solid) compared to the spectrum in the fullc , re-scaled ˙ N em run. The black dotted line shows the average fraction of photons in each bin in the latter, while the colored dot-dashed lines show the actual fraction in each of the 5 frequency bins after re-scaling. Redder (bluer) lines indicate lower (higher) photon energies, as the annotations indicate. The re-scaling removes more high-energy photons from ˙ N c em due to IGM filtering. Right : effective spectral index α eff (see text) for the multi-frequency reducedc and fullc , re-scaled ˙ N em simulations. The latter has a softer spectrum (larger α eff ) which gets softer with time, reaching 2 . 5 by z = 5 . \n<!-- image --> \nredshift. We define α eff such that a power law spectrum with that index would have the same σ HI as the source spectrum in the simulation. For the reducedc run, this is trivially 1 . 5 , but for the fullc , re-scaled ˙ N em run, α eff increases with time, reaching ≈ 2 . 5 by z = 5 . Note that this effect arises from the fact that the IGM absorbs (filters) low-energy photons more readily than high-energy ones, such that ˙ N γ has a harder spectrum than ˙ N em . This effect will thus be less significant in simulations with less filtering. We showed in Appendix B of Ref. [23] that our simulations are likely predict levels of filtering on the high end of expectations, so the effect may be somewhat exaggerated here. Still, it represents a complication to applying Eq. 5.1 to multi-frequency simulations that should be carefully considered and quantified in any studies that use this approach.", '6 Conclusions': "In this work, we have studied the effect of the reduced speed of light approximation (RSLA) on radiative transfer simulations of reionization. We used a simple analytic model to show (1) when and why the RSLA produces inaccurate results, especially near reionization's end, and (2) that using the RSLA is, to a good approximation, equivalent to a redshift-dependent re-scaling of the global ionizing emissivity of reionization's sources (Eq. 2.11). We have run simulations with the reduced speed of light ( reducedc runs) and with the full speed of light after applying the re-scaling prescribed by Eq. 2.11 ( fullc , re-scaled ˙ N em runs). We have assessed how accurate Eq. 2.11 is by comparing these two sets of simulations in a number of different physical properties and observables. Our main findings are summarized below: \n- · For ˜ c as small as 0 . 05 , our reducedc and fullc , re-scaled ˙ N em simulations agree in the global reionization history within a linear difference of ≲ 0 . 01 at all redshifts. \nThis confirms that that the re-scaling of ˙ N em prescribed by Eq. 2.11 is equivalent to using the RSLA with respect to the global reionization history. \n- · We have compared the morphology of ionized regions and the 21 cm power spectrum in our reducedc and fullc , re-scaled ˙ N em runs. We find that the sizes of the largest ionized regions are suppressed in the reducedc runs relative to their fullc , re-scaled ˙ N em counterparts. This is caused by position-dependent time-delay effects caused by the RSLA for which Eq. 2.11 does not account. The net effect is to suppress ionization power in the reducedc simulations. We find that differences in the 21 cm power spectrum at k = 0 . 1 h Mpc -1 is at most 15% ( 10% ) for ˜ c = 0 . 2 ( 0 . 3 ), but can exceed 20% for ˜ c = 0 . 1 and 30% for ˜ c = 0 . 05 . As such, we recommend applying Eq. 2.11 only for ˜ c ≥ 0 . 2 when ionization morphology is important.\n- · We have made the same comparison with respect to observables derived from highredshift quasar spectra, with a focus on the Ly α forest at z ≤ 6 . We find that when ˙ N em is calibrated to reproduce the mean transmission of the Ly α forest at z ≤ 6 , the resulting ˙ N em histories are very different for different ˜ c . For reducedc runs calibrated in this way with ˜ c = 0 . 2 , and 0 . 3 , we find that their fullc , re-scaled ˙ N em agree in the mean transmission to 20% or better. However, for ˜ c = 0 . 1 the difference can be as large as a factor of 2 . We find qualitatively similar results for the ionizing photon mean free path and the IGM temperature at mean density. Both agree to within ≤ 5% for ˜ c = 0 . 2 and 0 . 3 , but differ by up to 35% and 6% , respectively, for ˜ c = 0 . 1 .\n- · We looked at the large-scale fluctuations in the Ly α forest opacity in our two sets of simulations. For ˜ c ≥ 0 . 2 , the distribution of effective optical depths over 50 h -1 Mpc segments of the forest agrees to within a few percent between reducedc and fullc , re-scaled ˙ N em simulations. Once again, we find that these differences are considerably larger for ˜ c = 0 . 1 . We explored this result in more depth by looking at the large-scale fluctuations in Γ HI and T during the end stages of reionization. reducedc simulations over-produce large-scale fluctuations in Γ HI relative to their fullc , re-scaled ˙ N em counterparts thanks to the same position-dependent time-delay effects responsible for affecting the ionization morphology. In addition, reducedc simulations slightly underestimate the temperatures of recently ionized, hot voids near reionization's end, owing to their lower mean I-front speeds.\n- · Lastly, we generalized our method to apply to multi-frequency RT simulations. We found that with respect to the Ly α forest, the method works just as well with multifrequency simulations as with single-frequency ones. However, because of IGM filtering effects, ˙ N c em ends up having a softer spectrum than ˙ N ˜ c em , complicating the physical interpretation of the re-scaling procedure. We caution that when applying our method to multi-frequency simulations, this effect should be quantified and its implications carefully considered. \nThe approach described in this work is potentially useful in the following situations: either (1) when ˙ N em is entirely a free function of redshift that is calibrated to match some observable (as in Refs. [14, 23]) or (2) when ˙ N em is uniquely determined by some set of free parameters that are being marginalized over, as in Ref. [30]. In the first case, ˙ N em can be calibrated using the RSLA at a reduced computational cost, and Eq. 2.11 can be applied to the end result to recover the 'true' ˙ N em . In the second case, the re-scaling prescribed by \nEq. 2.11 would be equivalent to changing the mapping between ˙ N em and the free parameters that determine it in a quantifiable way. Unfortunately, our method is not directly applicable for situations where ˙ N em cannot simply be re-scaled after the simulation has been run. This is the case if ˙ N em is a self-consistent prediction a model that has no or very few free parameters, such as the THESAN [7, 19], CoDa [5, 13], or SPHINX [48, 49] simulations. Our results further suggest that for the first two applications, the RSLA should only be used (in conjunction with Eq. 2.11/5.1) for ˜ c ≥ 0 . 2 , to ensure that the effects of the RSLA on large-scale fluctuations in the ionization and radiation fields are minimized. This corresponds to a speed-up factor of up to 5 , which will help enable considerably faster searches of the reionization parameter space using RT simulations.", 'Acknowledgments': "The author acknowledges support from the Beus Center for Cosmic Foundations while this work was ongoing. He also acknowledges helpful conversations with Anson D'Aloisio, Garett Lopez, and Shikhar Asthana.", 'References': "- [1] X. Fan, M.A. Strauss, R.H. Becker, R.L. White, J.E. Gunn, G.R. Knapp et al., Constraining the Evolution of the Ionizing Background and the Epoch of Reionization with z ~ 6 Quasars. II. A Sample of 19 Quasars , The Astronomical Journal 132 (2006) 117 [ astro-ph/0512082 ].\n- [2] B.E. Robertson, R.S. Ellis, S.R. Furlanetto and J.S. Dunlop, Cosmic Reionization and Early Star-forming Galaxies: A Joint Analysis of New Constraints from Planck and the Hubble Space Telescope , The Astrophysical Journal Letters 802 (2015) L19 [ 1502.02024 ].\n- [3] G. Kulkarni, L.C. Keating, M.G. Haehnelt, S.E.I. Bosman, E. Puchwein, J. Chardin et al., Large Ly α opacity fluctuations and low CMB τ in models of late reionization with large islands of neutral hydrogen extending to z < 5.5 , Monthly Notices of the Royal Astronomical Society 485 (2019) L24 [ 1809.06374 ].\n- [4] N.Y. Gnedin, Cosmic Reionization on Computers. I. Design and Calibration of Simulations , The Astrophysical Journal 793 (2014) 29 [ 1403.4245 ].\n- [5] P. Ocvirk, D. Aubert, J.G. Sorce, P.R. Shapiro, N. Deparis, T. Dawoodbhoy et al., Cosmic Dawn II (CoDa II): a new radiation-hydrodynamics simulation of the self-consistent coupling of galaxy formation and reionization , Monthly Notices of the Royal Astronomical Society 496 (2020) 4087 [ 1811.11192 ].\n- [6] L.C. Keating, L.H. Weinberger, G. Kulkarni, M.G. Haehnelt, J. Chardin and D. Aubert, Long troughs in the Lymanα forest below redshift 6 due to islands of neutral hydrogen , Monthly Notices of the Royal Astronomical Society 491 (2020) 1736 [ 1905.12640 ].\n- [7] E. Garaldi, R. Kannan, A. Smith, V. Springel, R. Pakmor, M. Vogelsberger et al., The THESAN project: properties of the intergalactic medium and its connection to reionization-era galaxies , Monthly Notices of the Royal Astronomical Society (2022) [ 2110.01628 ].\n- [8] N.Y. Gnedin and T. Abel, Multi-dimensional cosmological radiative transfer with a Variable Eddington Tensor formalism , New Astronomy 6 (2001) 437 [ astro-ph/0106278 ].\n- [9] D. Aubert and R. Teyssier, A radiative transfer scheme for cosmological reionization based on a local Eddington tensor , Monthly Notices of the Royal Astronomical Society 387 (2008) 295 [ 0709.1544 ]. \n- [42] L.C. Keating, G. Kulkarni, M.G. Haehnelt, J. Chardin and D. Aubert, Constraining the second half of reionization with the Ly β forest , Monthly Notices of the Royal Astronomical Society 497 (2020) 906 [ 1912.05582 ].\n- [43] J. Chardin, M.G. Haehnelt, D. Aubert and E. Puchwein, Calibrating cosmological radiative transfer simulations with Ly α forest data: evidence for large spatial UV background fluctuations at z ∼ 5.6-5.8 due to rare bright sources , Monthly Notices of the Royal Astronomical Society 453 (2015) 2943 [ 1505.01853 ].\n- [44] F.B. Davies and S.R. Furlanetto, Large fluctuations in the hydrogen-ionizing background and mean free path following the epoch of reionization , Monthly Notices of the Royal Astronomical Society 460 (2016) 1328 [ 1509.07131 ].\n- [45] A. D'Aloisio, M. McQuinn and H. Trac, LARGE OPACITY VARIATIONS IN THE HIGH-REDSHIFT LY α FOREST: THE SIGNATURE OF RELIC TEMPERATURE FLUCTUATIONS FROM PATCHY REIONIZATION , The Astrophysical Journal 813 (2015) L38.\n- [46] T.R. Choudhury, A. Paranjape and S.E.I. Bosman, Studying the Lyman α optical depth fluctuations at z ∼ 5.5 using fast semi-numerical methods , Monthly Notices of the Royal Astronomical Society 501 (2021) 5782 [ 2003.08958 ].\n- [47] C. Zeng and C.M. Hirata, Nonequilibrium Temperature Evolution of Ionization Fronts during the Epoch of Reionization , The Astrophysical Journal 906 (2021) 124 [ 2007.02940 ].\n- [48] J. Rosdahl, H. Katz, J. Blaizot, T. Kimm, L. Michel-Dansac, T. Garel et al., The SPHINX cosmological simulations of the first billion years: the impact of binary stars on reionization , Monthly Notices of the Royal Astronomical Society 479 (2018) 994 [ 1801.07259 ].\n- [49] H. Katz, S. Martin-Alvarez, J. Rosdahl, T. Kimm, J. Blaizot, M.G. Haehnelt et al., Introducing SPHINX-MHD: the impact of primordial magnetic fields on the first galaxies, reionization, and the global 21-cm signal , Monthly Notices of the Royal Astronomical Society 507 (2021) 1254 [ 2101.11624 ]."}

2024arXiv240913584M

We explore the cosmological and astrophysical implications of a realistic hybrid inflation model based on flipped SU5. The model contains superheavy metastable cosmic strings arising from a waterfall field that encounters a limited number of efoldings during the inflationary phase. In addition to the gravitational waves emitted by the metastable strings there also appear scalar induced gravitational waves linked to the waterfall phase transition. These two independent sources of gravitational waves can yield a combined spectrum that is compatible with the recent PTA measurements and with additional features that can be probed in future experiments. We also show the appearance of primordial black holes with mass on the order of 1026 g from the waterfall phase transition and with an abundance that can be tested in the gravitational lensing experiments.

2024-09-01T00:00:00Z

['arXiv:2409.13584', '2024arXiv240913584M', '10.48550/arXiv.2409.13584']

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

Induced gravitational waves metastable cosmic strings and primordial black holes in GUTs

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13584.pdf

{'Rinku Maji, a Ahmad Moursy, b and Qaisar Shafi c': '- a Cosmology, Gravity and Astroparticle Physics Group, Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon 34126, Republic of Korea\n- b Department of Basic Sciences, Faculty of Computers and Artificial Intelligence, Cairo University, Giza 12613, Egypt\n- c Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA \nE-mail: rinkumaji9792@gmail.com, a.moursy@fci-cu.edu.eg \nAbstract. We explore the cosmological and astrophysical implications of a realistic hybrid inflation model based on flipped SU (5) . The model contains superheavy metastable cosmic strings arising from a waterfall field that encounters a limited number of e -foldings during the inflationary phase. In addition to the gravitational waves emitted by the metastable strings, there also appear scalar induced gravitational waves linked to the waterfall phase transition. These two independent sources of gravitational waves can yield a combined spectrum that is compatible with the recent PTA measurements, and with additional features that can be probed in future experiments. We also show the appearance of primordial black holes with mass on the order of 10 26 g from the waterfall phase transition, and with an abundance that can be tested in the gravitational lensing experiments. \nDedicated to the memory of our dear friend and collaborator George Lazarides. George was a larger than life figure as well as an outstanding theoretical physicist, and he will be sorely missed.', '1 Introduction': 'Grand unified theories (GUTs) predict proton decay [1-3], and superheavy magnetic monopoles [4, 5] that may survive inflation and occur at an observable level [6, 7]. Other topological defects such as domain walls, cosmic strings, intermediate mass monopoles and various composite structures [8-13] can appear in GUTs, depending on the symmetry breaking patterns, with rank greater than four. Detecting such topological defects would be one of the most spectacular confirmation of the physics beyond the Standard Model (BSM). Topologically stable magnetic monopoles can be present at an observable level provided that their density is diluted during inflation by a suitable number of e -foldings in order to satisfy observational constraints from experiments such as MACRO [14], IceCube [15] and ANTARES [16]. For example, intermediate scale monopoles in GUTs such as SO (10) should experience N M ≳ 13 number of e -foldings during inflation [6, 7]. Superheavy metastable cosmic strings (MSS) [13, 17-24] with a dimensionless string tension parameter Gµ ∼ 10 -6 emit a gravitational wave (GW) spectrum that can explain the NANOGrav 15 year [25-27] and other pulsar timing array data [28-31]. However, to evade the LIGO-VIRGO constraint on the GW spectrum at the decaHertz frequencies [32], the strings should be partially inflated by about 20-45 e -foldings [18] or experience an early matter dominated era [19]. Other hybrid topological structures, such as quasistable strings (QSS) and walls bounded by strings (WBS), can emit also gravitational waves compatible with the NANOGrav 15 year results provided that the superheavy strings and monopoles experience an appropriate amount of inflation [33, 34]. \nA recent hybrid inflation model based on flipped SU (5) [18] provides a recipe to entirely inflate the SU (5) superheavy monopoles, but the associated cosmic strings, produced during the waterfall phase, only experience a limited number of e -foldings. This kind of intermediate waterfall phase also may produce enhanced primordial curvature perturbations [7, 35-37], which can result in the generation of an additional source of GWs, commonly referred to as \nthe scalar induced gravitational waves (SIGW). Finally, the enhanced scalar perturbations can give rise to primordial black holes (PBHs) which may be present at an observable level. \nIn this article, we investigate the possibility that the stochastic gravitational wave background accessible in the pulsar timing array experiments is a superposition of the emission from the two distinct sources above, namely the SIGW and the metastable cosmic string gravitational waves (MSSGW). We focus on the flipped SU (5) hybrid inflation model [18], where the necessity that the cosmic strings are partially inflated during an intermediate waterfall phase, motivates us to look for another region of the parameter space that allows for the appearance of both SIGWs and MSSGW in explaining the PTA results. The SU (5) symmetry breaking scale is ≈ 5 × 10 16 GeV, and the corresponding proton lifetime is estimated to be of order 10 37 yrs, which is beyond the reach of the Hyper-Kamiokande experiment [38]. We explore a scenario where the PBHs are produced with an abundance that makes them detectable in the gravitational lensing experiments. Although we focus here on a specific GUT gauge group, we should emphasize that our considerations based on hybrid inflation and waterfall phase transitions can be readily extended to other GUT models. The emission of gravitational waves from two distinct sources and the appearance of primordial black holes is a salient feature of our hybrid inflation model. \nThe paper is structured as follows. In section 2 we briefly review the flipped SU (5) hybrid inflation model in Ref. [18] and the production of metastable strings during the waterfall transition. We study the inflationary dynamics and observables in section 3, and present representative benchmark points. In section 4, we discuss the scenario with two distinct sources of GWs and compatibility with the PTA data as well as the LIGO-VIRGO data. In section 5, we study the production of PBHs during the waterfall phase transition, and our conclusions are summarized in section 6.', '2 Metastable cosmic strings from flipped SU (5) model and hybrid inflation': 'The flipped SU (5) ( SU (5) × U (1) X ) model in Ref. [18] is an interesting candidate for our study of multiple sources of stochastic gravitational waves background, with the following symmetry breaking chain: 1 \nSU (5) × U (1) X ⟨ Φ ⟩ --→ SU (3) c × SU (2) L × U (1) Z × U (1) X ⟨ Ψ ⟩ --→ SU (3) c × SU (2) L × U (1) Y , (2.1) \nwhere Φ ≡ 24 H (0) and Ψ ≡ 10 H ( -1) . The gauge invariant terms in the scalar potential, realizing the symmetry breaking chain (2.1) and relevant to inflation, are given as follows: \nV ⊃ V 0 -µ 2 Φ tr (Φ 2 ) -µ 1 3 tr (Φ 3 ) + λ 1 4 tr (Φ 4 ) + λ 2 [ tr (Φ 2 )] 2 -µ 2 Ψ 2 tr (Ψ † Ψ) + λ 3 4 [ tr (Ψ † Ψ)] 2 + λ 4 4 tr (Ψ † ΨΨ † Ψ) + λ 5 tr (Ψ † Φ 2 Ψ) + λ 6 tr (Ψ † Ψ) tr (Φ 2 ) + µ 2 tr (Ψ † ΦΨ) + m 2 2 S 2 + λ 7 S 2 tr (Ψ † Ψ) -λ 8 S 2 tr (Φ 2 ) , (2.2) \nwhere the adjoint representation Φ α β ≡ ϕ a ( T a ) α β , with T a being the SU (5) generators, and the 10-plet Ψ is a 5 × 5 complex antisymmetric matrix Ψ αβ . The sum over repeated indices is understood, with a, b, c, · · · = 1 , 2 , · · · 24 , and α, β, · · · = 1 , 2 , · · · 5 . The real singlet scalar S represents the inflaton, and we do not include linear, cubic and quartic terms in S assuming that they are adequately small, and hence do not affect the inflation dynamics. The constant vacuum energy V 0 is added in order to guarantee a zero cosmological constant in the desired potential minimum. \nWith the symmetry breaking chain (2.1), SU (5) is broken first via the 24-plet Φ yielding magnetic monopoles carrying SU (3) c , SU (2) L and U (1) Z magnetic fluxes, which are entirely inflated away. The subsequent symmetry breaking U (1) Z × U (1) X → U (1) Y yields cosmic strings which can decay via the quantum-tunneling of monopole-antimonopole pairs [18]. The decay width per unit length of the string is expressed as [39] \nΓ d = µ 2 π exp ( -πm 2 M /µ ) , (2.3) \nwith m M being the monopole mass and µ the string tension. The superheavy metastable strings can produce gravitational waves compatible with the NANOGrav 15 year data and the constraint from LIGO-VIRGO third run results if: i) the dimensionless string tension parameter Gµ is of order 10 -6 with a metastability factor √ κ ≡ m M / √ µ ∼ 8 , and ii) the strings experience a limited number of e -foldings before the end of inflation [18]. \nAs explained in Ref. [18], the scalar potential (2.2) provides a hybrid inflation model where the Higgs field Ψ plays the role of the waterfall field that is frozen at the origin during inflation until S reaches a critical value S c , at which point the waterfall phase transition is triggered. The Higgs field Φ is shifted from the origin, following a field dependent minimum during inflation until the time at which S = S c . The inflationary potential during this phase has the form \nV inf = V 0 -m 2 ϕ 2 ϕ 2 + β ϕ 4 ϕ 4 -m 2 ψ 2 ψ 2 + β ψ 4 ψ 4 + β ψϕ 2 ψ 2 ϕ 2 + m 2 2 S 2 + β Sψ 2 S 2 ψ 2 -β Sϕ 2 S 2 ϕ 2 , (2.4) \nwhere ϕ and ψ represent the real canonically normalized components of the scalar fields Φ and Ψ which acquire the VEVs. The superrenormalizable terms with coefficients µ 1 and µ 2 are suitably controlled such that their effects on the dynamics are negligible, and for simplicity we \nset µ 1 = µ 2 = 0 in Eq. (2.2). Also, we assume that all of the remaining coefficients are real. The components of Φ and Ψ are fixed at zero during and after inflation, and the parameters m ϕ , m ψ , β ϕ , β ψ , β ψϕ , β Sϕ , β Sψ can be expressed in terms of the original potential parameters in Eq. (2.2). The constant vacuum energy V 0 satisfies \nV 0 = 1 4 ( m 2 ψ v 2 ψ + m 2 ϕ v 2 ϕ ) , (2.5) \nwhere v ψ and v ϕ are the vacuum expectation values of ψ and ϕ respectively. The inflation trajectory in the ( ψ, ϕ ) plane is given by [7, 18, 40] \n( ψ, ϕ ) = 0 , √ m 2 ϕ + β Sϕ S 2 β ϕ . (2.6) \nTh standard hybrid inflation tree level potential [41] is then modified to a hill-top shaped one [7, 18, 40], with the tree level effective potential given by \nV inf ( ˜ S ) = ˜ V 0 ( 1 + ˜ S 2 -γ ˜ S 4 ) , (2.7) \nwhere \n˜ V 0 ≡ V 0 -m 4 ϕ 4 β ϕ , ˜ S ≡ √ η 0 2 S , η 0 ≡ m 2 β ϕ -m 2 ϕ β Sϕ ˜ V 0 β ϕ , γ ≡ β 2 Sϕ η 2 0 ˜ V 0 β ϕ , (2.8) \nand the critical value of the inflaton field S c , at which the waterfall is triggered is given by \nS c = √ β ϕ m 2 ψ -β ψϕ m 2 ϕ β ψϕ β Sϕ + β ϕ β Sψ . (2.9) \nThe 1-loop Coleman-Weinberg (CW) correction to the tree level inflation potential (2.7) may yield a significant change to the inflation observables, since the waterfall fields have inflaton dependent masses during inflation. However, it was shown in Ref.[18] that the CW radiative correction is under control and yields a tiny contribution to the total potential if we introduce an extra pair of fermionic 10 F 1 , 10 F 2 , with X = 4 , -4 respectively. In our numerical calculations we choose values of the Yukawa couplings Y S S 10 F 1 10 F 2 and Y Φ Φ10 F 1 10 F 2 , such that the CW correction is very small compared to the tree level inflation potential (2.7). \nAfter reaching the global minimum of the scalar potential (2.4), the inflaton field S and the waterfall fields ϕ and ψ start to oscillate around their respective minima and decay, such that the reheating phase starts. The inflaton field S and the adjoint scalar field ϕ decay to the SM Higgs doublets H via the couplings δ 1 S H † H and δ 2 ϕ H † H . On the other hand, the scalar field ψ decay produces right handed neutrinos via the non-renormalizable coupling f M Pl 10 F 10 F 10 † H 10 † H . The reheating temperature T r consistent with the observables is less than or of order 10 12 GeV [18]. \nOur aim is to shed light on representative points in an interesting region of the parameter space with intermediate waterfall phase [7, 18, 36, 42, 43] that continues for ∼ 20 -40 e -foldings, with a significant enhancement of the curvature power spectrum from the waterfall \nphase transition. In this case, SIGWs are generated by second order effects in perturbation theory. Moreover, primordial black holes may be produced with a significant abundance. Moreover, the metastable cosmic strings produced during the waterfall phase are partially inflated, and they constitute another source of gravitational waves (MSSGW) in addition to the SIGWs. We explore the interplay between the two sources of gravitational waves and the potential production of PBHs in the next sections.', '3 Inflationary dynamics and observables': "We define the Hubble slow-roll parameters as follows \nϵ H ≡ 1 2 H 2 3 ∑ n =1 ˙ φ 2 n = 1 2 3 ∑ n =1 φ ' n 2 , η H ≡ ϵ ' H ϵ H , (3.1) \nwith the Hubble parameter H given by \nH 2 = ( ˙ a a ) 2 = 1 3 M 2 Pl [ 1 2 3 ∑ n =1 ˙ φ 2 n + V ( φ n ) ] , (3.2) \nwhere a is the scale factor. Here, a dot denotes the derivative with respect to the cosmic time t , and a prime denotes the derivative with respect to the number of e -foldings variable N . The classical field dynamics is governed by the Klein-Gordon equations which can be recast in the form \nφ '' n +(3 -ϵ H ) φ ' n +(3 -ϵ H ) V n V = 0 , (3.3) \nwhere n stands for S, ψ, ϕ , and V n is the derivative of the potential with respect to φ n . Denoting the Bardeen potentials by Φ B and Ψ B , the perturbed metric takes the form \nds 2 = a ( τ ) 2 [ -(1 + 2Φ B ) dτ 2 +(1 -2Ψ B ) δ ij dx i dx j ] , (3.4) \nwhere τ is the conformal time defined by dt ≡ adτ . The evolution of the scalar field perturbations δφ n and Φ B is described by the following equations [7, 44, 45] \nδφ '' n +(3 -ϵ H ) δφ ' n + 1 H 2 3 ∑ m =1 V nm δφ m + k 2 a 2 H 2 δφ n = 4Φ ' B φ ' n -2 Φ B H 2 V n , (3.5) \nΦ '' B +(7 -ϵ H ) Φ ' B + ( 2 V H 2 + k 2 a 2 H 2 ) Φ B = -1 H 2 3 ∑ m =1 V m δφ m , (3.6) \nwhere k is the co-moving wave vector, and the initial conditions at N = N ic are given by [7, 44, 45]: \nδφ n ( k, N ic ) = 1 a √ 2 k , (3.7) \nδφ ' n ( k, N ic ) = -1 a √ 2 k ( 1 + i k aH ) , (3.8) \nΦ B ( k, N ic ) = 1 2 ( ϵ H -k 2 a 2 H 2 ) 3 ∑ m =1 ( φ ' m δφ ' m +3 φ ' m δφ m + 1 H 2 V m δφ m ) , (3.9) \nΦ ' B ( k, N ic ) = 3 ∑ m =1 1 2 φ ' m δφ m -Φ B . (3.10) \nThe scalar power spectrum P ζ ( k ) is then given by [7, 44, 45] \nP ζ ( k ) = k 3 2 π 2 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ Φ B + 3 ∑ m =1 φ ' m δφ m 3 ∑ m =1 φ ' 2 m ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 . (3.11) \nIn our numerical simulations, integrating the perturbation equations and calculation of the \nTable 1 : Values of the parameters in the inflation potential (2.7). \nTable 2 : Values of the parameters of the potential in Eq. (2.4), with dimensionful parameters given in GeV. \nprimordial power spectrum, we use the method described in [7, 36, 44, 45]. In addition, we assume an initial displacement ψ 0 ∼ H/ (2 π ) at the time when S = S c [7, 36, 43]. The power spectrum is normalized at the pivot scale k ∗ = 0 . 05 Mpc -1 to satisfy the Planck constraints P ζ ( k ∗ ) = (2 . 099 ± 0 . 101) × 10 -9 [46, 47]. The total number of e -foldings during inflation is calculated, taking into account the thermal history of the universe, from the relation [48-50]: \n∆ N ∗ ≃ 61 . 5 + 1 2 ln ρ ∗ M 4 Pl -1 3(1 + ω r ) ln ρ e M 4 Pl + ( 1 3(1 + ω r ) -1 4 ) ln ρ r M 4 Pl , (3.12) \nwhere M Pl ≈ 2 . 44 × 10 18 GeV is the reduced Planck mass, ρ ∗ = V ( ˜ S ∗ ) stands for the energy density of the universe when the pivot scale exits the horizon, ρ e = V ( ˜ S e ) denotes the energy density at the end of inflation. The energy density at the reheating time is given by ρ r = ( π 2 / 30) g ∗ T 4 r , with g ∗ = 106 . 75 being the effective number of massless degrees of freedom, corresponding to the SM spectrum. The quantity w r denotes the effective equationof-state parameter from the end of inflation until reheating and we set it equal to 1 / 3 [51]. The time at reheating t r is computed from the relation [52, 53] \nT 2 r ≈ √ 45 2 π 2 g ∗ M Pl t r . (3.13) \nTable 3 : Values of vevs and physical masses of the inflaton and waterfall fields in GeV. M ϕ ' and M ψ ' are the eigenvalues of the mass matrix. \nTable 4 : Predictions of CMB observables, and the total number of e -foldings during inflation ∆ N ∗ , and during waterfall ∆ N c . \nTable 5 : Time scales (in seconds) at different phases, and value of the equation of state parameter w r . t c , t e , t r denote the cosmic times when S = S c , at the end of inflation, and at reheating, respectively. \nThe benchmark points BP1 and BP2 correspond to metastable cosmic strings with dimensionless string tension Gµ = 8 . 44 × 10 -6 and Gµ = 1 . 44 × 10 -5 respectively. The Hubble parameter during the waterfall is H ≈ 2 . × 10 -6 M Pl . The curvature power spectrum is enhanced significantly at small scales due to the waterfall phase transition. The potential parameter values accounting for both SIGW and MSSGW are given in Tables (1) and (2), and the scalar fields vevs, physical masses and inflation observables are listed in Tables 3 and 4. The time scales at different phases are given in Table 5.", '4 Gravitational waves from strings and scalar perturbations': 'As advocated above, the waterfall phase transition plays a dual role in our scenario. First, it is important to partially dilute the cosmic strings density such that the GW spectrum emitted from the sting network is compatible with the LVK bound. Secondly, as stated earlier, it enhances the curvature perturbations at small scales, and hence sources second order tensor perturbations, leading to the scalar induced gravitational waves. Assuming that the scalar induced GWs are produced during the radiation-dominated epoch, the spectrum is calculated from the formula [54-58] \nΩ SI GW h 2 ≈ 4 . 6 × 10 -4 ( g 4 ∗ ,s g -3 ∗ 100 ) -1 3 ∫ 1 -1 d x ∫ ∞ 1 d y P ζ ( y -x 2 k ) P ζ ( x + y 2 k ) F ( x, y ) ∣ ∣ ∣ ∣ k =2 πf , (4.1) \nwith g ∗ ,s ≈ g ∗ , and the function F is defined as \nF ( x, y ) = ( x 2 + y 2 -6) 2 ( x 2 -1) 2 ( y 2 -1) 2 ( x -y ) 8 ( x + y ) 8 × { [ x 2 -y 2 + x 2 + y 2 -6 2 ln ∣ ∣ ∣ ∣ y 2 -3 x 2 -3 ∣ ∣ ∣ ∣ ] 2 + π 2 ( x 2 + y 2 -6) 2 4 θ ( y -√ 3) } . (4.2) \nThe gravitational waves from the string network have been extensively studied in the literature [59-74]. The metastable cosmic string network produces string loops with the number distribution given by [72] \nn ( l, t ) = 0 . 18 t 3 / 2 ( l +Γ Gµt ) 5 / 2 Θ(0 . 18 t -l ) for t < t s e -Γ d [ l ( t -t s )+ 1 2 Γ Gµ ( t -t s ) 2 ] Θ(0 . 18 t s -¯ l ) for t > t s , (4.3) \nwhere ¯ l = l +Γ Gµ ( t -t s ) , Γ ≃ 50 is a numerical factor, and t s = 1 / √ Γ d . The loops oscillate and radiate gravitational waves. The gravitational wave burst rate per unit spacetime is given by \nd 2 R dz dl = N B H -3 0 ϕ V ( z ) 2 n ( l, t ) l (1 + z ) ∆ B ( f, l, z ) , (4.4) \nwhere N B is the average number of burst events per oscillation, H 0 denotes the Hubble parameter today, ∆ B ( f, l, z ) is the observable fraction of the bursts [62, 66, 67], and \nϕ V ( z ) = 4 πH 3 0 r 2 (1 + z ) 3 H ( z ) , (4.5) \nwith r denoting the proper distance at redshift z . We assume that the cusp events provide the dominant contribution to the gravitational wave background. The wave form of the bursts from a cusp is expressed as [62], \nh c ( f, l, z ) = g c Gµl 2 / 3 (1 + z ) 1 / 3 r ( z ) f -4 / 3 , (4.6) \nFigure 1 : Gravitational waves from the metastable cosmic strings and the associated scalar induced gravitational waves. The gray region depicts the Big Bang Nucleosynthesis (BBN) bound [75] on the gravitational wave background for the BP1. The power-law integrated sensitivity curves [76, 77] for the ongoing and several proposed experiments, namely, HLVK [78], CE [79], ET [80], DECIGO [81], BBO [82, 83], LISA [84, 85] and SKA [86, 87] are shown. \n<!-- image --> \nwhere g c ≃ 0 . 85 [32]. The gravitational wave background can be expressed as [66, 67] \nΩ MSS GW ( f ) = 4 π 2 3 H 2 0 f 3 ∫ z F z ∗ dz ∫ d H 0 dl h 2 c ( f, l, z ) d 2 R dz dl , (4.7) \nwhere z F and z ∗ denote the redshifts at the time of string network formation and disappearance, and we have taken the particle horizon d H at redshift z as the upper limit of integration on l . \nFig. 1 depicts the stochastic gravitational wave backgrounds sourced by the scalar perturbations and metastable strings. In the case of BP1, the metastable strings provide the dominant contribution to the gravitational wave background and can explain the NANOGrav 15 year data. Moreover, there will be a significant PBH dark matter abundance which can be observed in future lensing data (see Fig. 2). On the other hand, if the strings experience about 40 e -foldings of inflation and the PTA data can be explained around the nHz frequencies from a combination of the two sources. Moreover, the gravitational wave background displays a UV tail varying as Ω GW ∝ f -1 / 3 starting from the µ Hz frequencies which can be probed in several planned experiments such as µ Ares [88], LISA [84, 85], DECIGO [81], BBO [82, 83], CE [79], and ET [80]. The f -1 / 3 UV tail starts around the mHz frequency in the former case, and the background can be detected in the proposed and ongoing experiments including HLVK [78].', '5 Primordial black holes': "Figure 2 : Primordial black holes fractional abundance for BP1. PBHs generated from BP2 have negligibly small abundance. \n<!-- image --> \nThe enhancement in the power spectrum may result in the formation of PBHs if the density contrast δ ≡ δρ/ρ exceeds a critical value δ c ( k ) [89]. In this case, the PBHs mass fraction β compared to the total mass of the universe can be evaluated using the PressSchechter formalism, namely, \nβ ( k ) = 1 √ 2 πσ 2 ( k ) ∫ ∞ δ c ( k ) dδ exp ( -δ 2 2 σ 2 ( k ) ) , (5.1) \nwhere σ is the variance of the curvature perturbations which can be computed using the power spectrum and a window function with Gaussian distribution function ˜ W ( x ) = e -x 2 / 2 [90, 91], \nσ 2 ( k ) = 16 81 ∫ dk ' k ' ( k ' k ) 4 P ζ ( k ' ) ˜ W ( k ' k ) , (5.2) \nwith 0 . 4 ≲ δ c ≲ 0 . 6 [90, 92-100]. In terms of the co-moving wave number k , the PBH mass (in grams) is given by [101] \nM PBH ( k ) = 10 18 ( γ 0 . 2 ) ( g ∗ ( T f ) 106 . 75 ) -1 / 6 ( k 7 × 10 13 Mpc -1 ) -2 , (5.3) \nwhere we assume that PBHs are created during the radiation epoch, and T f represents the temperature at the time of their formation. The fractional abundance of PBHs, f PBH ≡ Ω PBH / Ω DM has the form \nf PBH ( M PBH ) = β ( M PBH ) 8 × 10 -16 ( γ 0 . 2 ) 3 / 2 ( g ∗ ( T f ) 106 . 75 ) -1 / 4 ( M PBH 10 -18 grams ) -1 / 2 , (5.4) \nwhere the factor γ ∼ 0 . 2 represents the dependence on the gravitational collapse [102]. \nFigure 2 shows the predicted dark matter abundance of PBH with masses M = 2 . 3 × 10 26 g ( 1 . 15 × 10 -7 M ⊙ ) for BP1, that constitutes about 2% of the observed dark matter density which can be detected in the next generation of lensing experiments. For BP2, where the PTA results are explained in terms of SIGW and MSSGW, the corresponding PBH fractional abundance is extremely small. The shaded regions represent the various observational constraints on PBHs from black hole evaporation, accretion and GWs [103109], and microlensing, including HSC, EROS and OGLE experiments [110].", '6 Conclusions': 'We have explored the cosmological implications of a realistic flipped SU(5) model of hybrid inflation that yields superheavy metastable cosmic strings during the waterfall phase transition. We show how two sources of gravitational waves (GWs) appear, namely from the metastable cosmic strings (MSSGW), as well as from scalar perturbations associated with the waterfall field, often referred to as scalar induced gravitational waves (SIGW). We display representative benchmark points that allow for both types of GWs compatible with the recent PTA/NANOGrav results, and also with the LIGO-VIRGO third run. The spectrum features a f -1 / 3 UV tail that can be detected in the proposed and ongoing experiments including HLVK. We also show that PBHs can be produced with mass of around 10 26 g for the case where the MSSGW spectrum is the dominant one. Despite the relatively small dark matter fractional abundance of PBHs, their presence can be tested in the future lensing experiments.', 'Acknowledgments': 'R.M. is supported by Institute for Basic Science under the project code: IBS-R018-D3. R.M. and Q.S. would like to thank Professor Masahide Yamaguchi and his colleagues, students and staff for the hospitality provided at the IBS-CTPU-CGA, Tokyo Tech, USTC 2024 Summer Workshop and School on Cosmology, Gravity and Particle Physics, which gave us the opportunity to discuss this project in person.', 'References': "```\n[1] H. Georgi and S.L. Glashow, Unity of All Elementary Particle Forces , Phys. Rev. Lett. 32 (1974) 438. [2] H. Fritzsch and P. Minkowski, Unified Interactions of Leptons and Hadrons , Annals Phys. 93 (1975) 193. [3] Q. Shafi, E(6) as a Unifying Gauge Symmetry , Phys. Lett. B 79 (1978) 301. 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2024arXiv240906281L

We revisit the gravitational lensing of light or gravitational waves by Schwarzschild black hole in geometric optics. Instead of a single massless particle we investigate the collective behavior of a congruence of lightgravitational rays described by the geodesic deviation equation GDE. By projecting on the NewmanPenrose tetrad GDE is decoupled and we find an analytical Dysonlike series solution in the weak deflection and thin lens limits. Based on such a solution we study the evolution of crosssectional area and axis ratio. Finally we reproduce the magnification and axis ratio of the lensing images up to the second order of weak deflection approximation and improve some missing corrections in previous works.

2024-09-01T00:00:00Z

['arXiv:2409.06281', '2024arXiv240906281L', '10.48550/arXiv.2409.06281']

['General Relativity and Quantum Cosmology']

Schwarzschild Lensing From Geodesic Deviation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06281.pdf

{'No Header': '2', 'Schwarzschild Lensing From Geodesic Deviation': 'Zhao Li, 1, 2, 3, ∗ Xiao Guo, 4 Tan Liu, 4, 5 Tao Zhu, 6 and Wen Zhao 1, 2, † \n- 1 Department of Astronomy, University of Science and Technology of China, Hefei, Anhui 230026, China \nSchool of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China \n3 \nDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan \n4 School of Fundamental Physics and Mathematical Sciences, \nHangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, \nNo.1 Xiangshan Branch, Hangzhou 310024, China \n5 University of Chinese Academy of Sciences, 100049/100190 Beijing, China \n6 Institute for Theoretical Physics and Cosmology, \nZhejiang University of Technology, Hangzhou, 310032, China, \nUnited Center for Gravitational Wave Physics (UCGWP), \nZhejiang University of Technology, Hangzhou, 310032, China \nWe revisit the gravitational lensing of light or gravitational waves by Schwarzschild black hole in geometric optics. Instead of a single massless particle, we investigate the collective behavior of a congruence of light/gravitational rays, described by the geodesic deviation equation (GDE). By projecting on the Newman-Penrose tetrad, GDE is decoupled, and we find an analytical Dyson-like series solution in the weak deflection and thin lens limits. Based on such a solution, we study the evolution of cross-sectional area and axis ratio. Finally, we reproduce the magnification and axis ratio of the lensing images up to the second order of weak deflection approximation and improve some missing corrections in previous works.', 'I. INTRODUCTION': "Gravitational lensing is one of the most important phenomena predicted by general relativity (GR), which occurs when a massive object lies between a distant source and Earth [1-3]. The gravitational field bends the light rays from the source, creating multiple images, or producing arcs and rings [4]. A direct consequence is magnification, allowing us to observe objects that would otherwise be too far away and too faint to be seen at high redshift [5]. Gravitational lensing offers a powerful tool to probe the Universe and test gravity at the astrophysical scale [6-9]. Because the lensing effect directly depends on the lens object's mass distribution, including both visible and dark matter, it allows researchers to measure the dynamical mass of the galaxy and galaxy cluster, and study the distribution of dark matter [10]. \nLike light, gravitational waves (GWs) are lensed when passing through massive objects [7, 11-17]. Several studies have begun to search the lensed GW signal in the current catalog by LIGO/Virgo/KAGRA network, however, finding no strong evidence for lensing imprint [18-24]. As predicted, the future third-generation ground-based GW observatory is expected to detect hundreds of lensing GW events, providing rich information on the gravitational theory, cosmic structure, dark matter, and so on [25-27]. \nAn ideal situation is that the typical wavelength of the lensed signal is much shorter than the background curvature scale. Through eikonal expansion, one finds the behavior of light and GWs are the same as the massless particle in the lowest order. The photon and graviton paths are nothing but the null geodesics [28-30]. In this framework, the lensing process is described by the lens equation [31, 32], for a given surface mass density model of the lens object. The main observational quantities of the lensing events consist of magnification and shear, which can be calculated by the Jacobian matrix [31, 32]. \nIn this work, we revisit the lensing process by a Schwarzschild black hole through a different viewpoint. The lensed signal is not made up of a single photon/graviton, but a congruence of light/gravitational rays [33]. During the propagation, the congruence cross-section area and axis ratio are significantly affected by the tidal of the Schwarzschild lens [34, 35]. We study the evolution of geodesic congruence by solving the geodesic deviation equation (GDE) [2]. Under the weak deflection limit, where the impact distance is much larger than the gravitational radius of the lens [31, 32], we find the analytical solution through a Dyson-like series expansion [36]. Based on that, we analytically \n∗ \n† \nreproduce the magnification and axis ratio of the images. In addition, we improve the previous lens equation and the corresponding results, in which some higher-order corrections are missed. \nThis paper is organized as follows. In Sec II, we review the GDE and project it onto the Newman-Penrose (NP) tetrad. We present the Dyson-like series solution to the GDE and the physical meaning of the optical scalars in Sec III. In Sec IV, we investigate the Schwarzschild lensing, especially, the evolution of the deviation vector and the optical scalars, and most importantly, we reproduce the magnification and shear of the Schwarzschild lens. We take a brief conclusion and discussion in Sec V. We do not consider the cosmological background and redshift in this work. Throughout the paper, we work in geometric units in which c = G = 1, where c is the speed of light in the vacuum and G is the gravitational constant.", 'II. GEODESIC CONGRUENCE': 'In this section, we investigate the behavior of the geodesic congruence [33]. The arm length of GW detectors and the aperture of optical telescopes are much smaller than the scale of lens objects. Therefore, the detected light and gravitational rays are very close to each other and approximately located inside a common geodesic congruence. \nFor simplicity, we only study two neighboring points, whose trajectories are denoted by \nx α ( λ, µ a ) , and x α ( λ + δλ, µ a + δµ a ) , (1) \nwhere λ is the affine parameter and µ a is the a -th motion constant of massless particle in the given background. The difference δλ and δµ a are kept sufficiently small. The geodesic deviation vector is defined as the difference between such two trajectories, and can be expanded in terms of small parameters δµ a and δλ , \nξ α ≡ x α ( λ + δλ ; µ a + δµ a ) -x α ( λ ; µ a ) = ∂x α ∂λ δλ + ∑ a ∂x α ∂µ a δµ a + O (2) . (2) \nwhere O (2) means the second and higher order of δλ , ξ α , and δµ a . We denote the tangent vector of the geodesics as k µ ≡ ∂x µ /∂λ , which satisfies the geodesic equation, k α ( ∇ α k ν ) = 0. Then the difference in wavevectors associated with each trajectory is expanded by the deviation vector, \nδk ν = ( ∂ α k ν ) ξ α + O (2) . (3) \nMeanwhile, this difference can also be expanded in terms of small quantities δλ and δµ a , \nδk ν = ∑ a ∂k ν ∂µ a δµ a + ∂k ν ∂λ δλ + O (2) = ∑ a ∂ 2 x ν ∂µ a ∂λ δµ a + ∂ 2 x ν ∂λ 2 δλ + O (2) = ∂ ∂λ [ ∑ a ∂x ν ∂µ a δµ a + ∂x ν ∂λ δλ ] + O (2) = ∂ξ ν ∂λ + O (2) = k α ∂ α ξ ν + O (2) , (4) \nCombining Eqs. (3) and (4) up to the linear order of δλ and δµ α , we get [34, 35] \nξ α ∇ α k ν = k α ∇ α ξ ν , (5) \nwith ∇ α being the covariant derivative operator compatible with the background metric. An equivalent explanation to Eq. (5) is the Lie derivatives of deviation vector along geodesic vanish, i.e., L k ξ ν = k α ( ∇ α ξ ν ) -( ∇ α k ν ) ξ α = k α ( ∂ α ξ ν ) -( ∂ α k ν ) ξ α = 0. Multiplying k ν on both sides of Eq. (5), and combining the geodesic equation, one gets conservation law that \nk α ∇ α ( k ν ξ ν ) = 0 , (6) \nSetting the inner product k ν ξ ν as zero means that the points x α and x α + ξ α belong to the same wavefront, with equal phases, defined by k µ ≡ -∇ µ Φ [2]. This is proved by \nΦ( x α + ξ α ) ≈ Φ( x α ) + ( ∂ α Φ) δx α = Φ( x α ) -k α ξ α = Φ( x α ) . (7) \nThe GDE dominates the evolution of the congruence, that is \nD 2 ξ ν Dλ 2 = -R ν ραβ k ρ k β ξ α , (8) \nwith D/Dλ ≡ k α ∇ α and R ν ραβ being the background Riemann tensor. To solve Eq. (8) more conveniently, we construct the NP tetrad along the null geodesics, whose four legs are denoted as [3, 37] \ne ( a ) = { k , n , m , ¯ m } . (9) \nThe subscript ( a ) is the tetrad indices. The first leg is the wavevector of the lensed waves, defined as the gradient of a scalar field denoted by Φ. The second one is along the spatially opposite direction. The third and fourth legs are orthogonal to the propagation direction of the waves, they are complex conjugate to each other. The plane spanned by m and ¯ m is generally called the polarization plane, representing the oscillation direction of light or GWs. The orthogonality of the NP tetrad is presented as -k · n = m · ¯ m = 1, and other possible inner products vanish. Most importantly, all of the tetrad legs should be required to be parallel-transported along the geodesics, k α ∇ α e µ ( a ) = 0. We project the deviation vector onto the NP tetrad, \nξ α = ξ ( a ) e α ( a ) , with ξ ( a ) = ξ α e ( a ) α . (10) \nSetting ξ α to satisfy the equal-phase condition, we have ξ n = k α ξ α = 0 [53]. Substituting Eq. (10) into Eqs. (5, 8), we obtain [34, 35, 37-41] \nD ξ m = ρξ m + ¯ σξ ¯ m , (11) \nand \nD 2 ξ m = ¯ Ψ 0 ξ ¯ m , (12) \nfor the m -components. The derivative operator is D ≡ k µ ∂ µ = d/dλ . One obtains similar equations for ¯ m -component by taking complex conjugation of these two equations. In Eqs. (11, 12), the Greek letters ρ and σ represent the spin coefficients, defined as ρ ≡ m µ ( ∇ µ k ν ) ¯ m ν and σ ≡ m µ ( ∇ µ k ν ) m ν . These two scalars are also known as optical scalars, describing the geometry and evolution of the geodesic congruence. The Weyl scalar Ψ 0 is defined as Ψ 0 ≡ -R µναβ k µ m ν k α m β . We have assumed the spacetime to be a vacuum throughout this work, and then the Weyl tensor is identical to the Riemann tensor.', 'III. SOLUTION TO DEVIATION VECTOR': "In this section, we present the solution to the geodesic vector. The Weyl scalar Ψ 0 is a real number for Schwarzschild lensing, which simplifies the GDE (12). We consider Eq. (12) and its complex conjunction. By adding and subtracting this pair of equations, and defining ξ ± ≡ ξ m ± ξ ¯ m , we get \nD 2 ξ ± = ± Ψ 0 ξ ± . (13) \nThus, Eq. (13) for ξ ± are decoupled. Transforming these two equations into a set of first-order differential equations, we get the equivalent matrix form \nDY = A · Y , with Y ≡ ( ξ ± D ξ ± ) , and A ≡ ( 0 1 ± Ψ 0 0 ) . (14) \nIn a typical lensing system, the impact parameter of the photon/graviton paths is much larger than the gravitational radius of the lens object. Meanwhile, the source and observer are far from the lens, and the background curvature is sufficiently approaching zero at these two points. These two facts are usually called weak deflection and thin lens limits. We assume that the lensed wave is emitted from a point source, and propagates freely near the source. Therefore, the initial conditions of the deviation vector can be set as ξ ± = 0 and D ξ ± = k ± . The initial derivative is proportional to the open angle of the congruence near the wave source. In these two limits, the right-hand side of Eq. (14) is always perturbation, and its 0-th solution can be set as \nY 0 = ( k ± λ k ± ) . (15) \nThen the Dyson-like series solution is written as [36, 40, 42] \nY 1 = ∫ λ 0 dλ ' A ( λ ' ) · Y 0 ( λ ' ) , (16) \nY 2 = ∫ λ 0 dλ ' A ( λ ' ) · Y 1 ( λ ' ) = ∫ λ 0 dλ ' ∫ λ ' 0 dλ '' A ( λ ' ) · A ( λ '' ) · Y 0 ( λ '' ) , (17) \n· · · \n. \nY n +1 = ∫ λ 0 dλ ' A ( λ ' ) · Y n ( λ ' ) (18) \nThe solution to ξ ± is finally iteratively given by, \nξ ± = k ± · ˜ ξ ± , ˜ ξ ± ≡ ∞ ∑ n =0 ( ± 1) n · I n , (19) \nwhere the integrations I n are \nI n = ∫ λ 0 dλ ' ∫ λ ' 0 dλ '' Ψ 0 ( λ '' ) I n -1 ( λ '' ) = ∫ λ 0 ( λ -λ ' )Ψ 0 ( λ ' ) I n -1 ( λ ' ) dλ ' , with I 0 = λ. (20) \nand ˜ ξ ± are real. Because ξ + is real and ξ -is purely imaginary, such that the integration constants k + and k -are real and purely imaginary, respectively. \nThe GDE (12) is a second-order equation, there should be two independent solutions. Supposing ξ m 1 and ξ ¯ m 2 to be two independent solutions of deviation vector, with different k ± . Therefore, from Eq. (11), the optical scalars are determined by [34] \n( ρ σ ) = ( ξ ¯ m 1 ξ m 1 ξ ¯ m 2 ξ m 2 ) -1 ( D ( ξ ¯ m 1 ) D ( ξ ¯ m 2 ) ) , (21) \nand explicitly [43], \nρ = D ln √ ∣ ∣ ∣ ˜ ξ + ˜ ξ -∣ ∣ ∣ , and σ = D ln √ ∣ ∣ ∣ ˜ ξ + / ˜ ξ -∣ ∣ ∣ . (22) \nwhich is independent of the integration constant k ± . \nIt is noted that the null tetrad leg m relates a spacelike tetrad by m = ( e x + e y ) / √ 2. Therefore, ˜ ξ ± is identified to be the scale of geodesic congruence on the e x and e y directions. Such that, the product ˜ ξ + ˜ ξ -is proportional to the cross-sectional area, and ˜ ξ + / ˜ ξ -represents the axis ratio of the congruence. We can define such area and axis ratio as \nA ≡ ˜ ξ + ˜ ξ -, and R≡ ˜ ξ -/ ˜ ξ + , (23) \nThe corresponding evolution equations are derived as [34, 43, 44] \nD A = 2 ρ A , and D R = -2 σ R . (24) \nThe physical meanings of the optical scalars are the relative variation rate of the cross-sectional area and the axis ratio of the null congruences.", 'A. Null tetrad and Weyl scalars': 'In coordinate { t, r, θ, φ } , the line element of Schwarzschild black hole with mass M is [1-3] \nds 2 = -( 1 -2 M r ) dt 2 + ( 1 -2 M r ) -1 dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (25) \nand the tangent vector of the null geodesics equations are \nk µ = { ( 1 -2 M r ) -1 , U r , 0 , L r 2 } , (26) \nDue to the spherical symmetry of the Schwarzschild background, the orbits of the massless test particle are always on the equatorial plane of the black hole, with θ = π/ 2. L is the angular momentum of the photon or graviton. The radial potential U r is defined as \nU r ≡ ± √ 1 -L 2 r 2 ( 1 -2 M r ) , (27) \nwhere the overall sign is negative and positive when the particle moves toward or backward to the pericenter, respectively. Following the steps presented in Ref. [45] and Appendix A, the null tetrad is constructed as \nn µ = { r 2 2 L 2 [ -2 λ r U r + ( 1 + λ 2 r 2 )]( 1 -2 M r ) -1 , r 2 L 2 [ U r 2 ( 1 + λ 2 r 2 ) + λ r ] , 0 , -1 2 L ( 1 -λ 2 r 2 ) } , (28) \nand \nm µ = 1 √ 2 { r L ( λ r + U r )( 1 -2 M r ) -1 , -r L ( 1 + λ r U r ) , i r , -λ r 2 } . (29) \nThe Weyl scalar Ψ 0 can be computed as \nΨ 0 = 3 ML 2 r 5 , (30) \nwhich is a real number, and ensures that we can apply the framework in Sec III to the Schwarzschild lensing.', 'B. Analytical solution under the weak deflection limit': "We use the Dyson-like series to re-express the solution to the geodesic deviation vector and optical scalars. This work analytically integrates these integrations under weak deflection and thin-lens assumptions. In these limits, the angular momentum L (with mass dimension) is much larger than the Schwarzschild radius, therefore, the impact parameter b ≡ L/M ≫ 1. The distances of the emitter and observer are much larger than the impact distance, i.e. r s , r o ≫ L . Ref. [3] and Appendix B discuss the solution to the null geodesic equation. \nTo perform the integration (20), it is convenient to express the affine parameter in terms of a new parameter x , defined as x ≡ u/u m , with u ≡ M/r and u m ≡ M/r m , where r m is the pericenter radius of the photon/graviton path. The solution to λ = λ ( x ) is presented in Appendix C, with two different forms in intervals [ λ s , λ m ] and [ λ m , λ o ], denoted by λ I ( x ) and λ II ( x ). λ s , λ m , λ o , are the affine parameter λ evaluated at the source, pericenter, and observer positions. Such that the integration becomes different forms in these two intervals. When λ s < λ < λ m , \nI n ( x ) = ∫ x x s [ λ I ( x ) -λ I ( x ' )]Ψ 0 [ λ I ( x ' )] I n -1 [ λ I ( x ' )] dλ I ( x ) dx ' dx ' , ( n = 1 , 2 , 3 , · · · ) . (31) \nWhen λ m < λ < λ o , \nI n ( x ) = ∫ 1 x s [ λ II ( x ) -λ I ( x ' )]Ψ 0 [ λ I ( x ' )] I n -1 [ λ I ( x ' )] dλ I ( x ) dx ' dx ' + ∫ x 1 [ λ II ( x ) -λ II ( x ' )]Ψ 0 [ λ II ( x ' )] I n -1 [ λ II ( x ' )] dλ II ( x ) dx ' dx ' , ( n = 1 , 2 , 3 , · · · ) . (32) \nThe analytical expressions of integrations I n are shown in Appendix D. Due to the complex form of these expressions, we performed numerical calculations to more clearly demonstrate their behavior. \nThe typical configuration of gravitational lensing under the weak deflection and thin lens limits is depicted in Fig. 1. The two panels correspond to the two possible photon/graviton paths, forming two images in the geometric optics. \nFor convenience, we denote the left case as image 1 and the right one as image 2. In Fig. 1, O, L, and S represent the observer, lens object, and wave source. The light blue line approximately characterizes the photon/graviton path, and I is the lensing image on the source plane. Correspondingly, OS is the free-propagation path. And α , β , and θ are the deflection angle, the angular coordinate of the source and image, that is related by lens equation, discussed in Appendix B. D L , D S , and D LS are the distances between observer, lens plane, and source plane. In our numerical calculation, we set D L = 780 M and D LS = 750 M . The well-known Einstein angle is defined by [4] \nθ E ≡ √ 4 M D LS D S D L , (33) \nthe value in our setup is about 0 . 0501381 rad ≈ 2 . 8727 · , which is a small angle, consistent with the weak deflection assumption. After giving the distances D L and D LS (in the unit of lens mass), the unique free parameter is the angular coordinate of the wave source, β . For convenience, the normalized coordinate is β 0 ≡ β/θ E . We set β 0 = { 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 } and the corresponding values of the image position, deflection angle, and impact parameter are listed in Table I. In the weak deflection limit, such quantities satisfy β, θ, α, u m ≪ 1 and b ≫ 1. \n) \nFIG. 1: Geometrical configuration of gravitational lensing system (left: for image 1, right: for image 2). The points O, L, and S represent the observer, lens object, and wave source. The straight line ON is the optical axis. The planes passing through the lens and source, and being vertical to the optical axis are the lens plane and source plane. The distances from the observer to the lens and source plane are denoted by D L and D S , and then D LS = D S -D L . α is the deflection angle. β and θ are the angular coordinates of the source and image. \n<!-- image --> \nTABLE I: We list all of the example values of the lensing parameters involved in our discussion. We have fixed D L = 780 M and D LS = 750 M , and the system is completely determined by source position. The first two columns are the (normalized) source coordinates. The rest of the columns are the normalized image coordinates, deflection angle, impact parameter, magnification, and axis ratio, for the two images produced by the Schwarzschild lensing. \nIn Fig. 2, we display the evolution of the first four terms in the Dyson-like series in Eq. (20). The first two terms I 0 = λ , shown as a straight line with slope 1, and I 1 are the dominant terms of the geodesic deviation vector. There \nis an obvious turning point at λ/M ∼ 750, because the external gravitational field deflects the photon/graviton path by slightly different angles. The subsequent two terms, I 2 and I 3 , are almost always zero because they represent highorder weak-deflection correction. This also indicates that the Dyson-like series converges under the weak-deflection limit. \nFIG. 2: The evolution of each term in Dyson-like series (19) along the geodesic. These curves are plotted based on the analytical solution shown in Eqs. (C2, C4, D1, D2, D3). \n<!-- image --> \nFig. 3 shows the evolution of the transversal components of the geodesic deviation vector along the null geodesics. Before entering the external gravitational field, the evolutions of ˜ ξ + and ˜ ξ -are almost equal. However, the slopes of ˜ ξ + and ˜ ξ -are obviously different after leaving the gravitational field. From the expressions of tetrad leg m µ , we conclude that ˜ ξ + lies on the equatorial plane of the Schwarzschild black hole, and ˜ ξ -is vertical to that. Due to the tidal force, ˜ ξ + always increases, and the increasing rate gets larger after leaving the lens object than before entering. Distinguishingly, the growth of ˜ ξ -is always suppressed. For relatively large b , the tidal force is strong enough to make ˜ ξ -decrease rapidly enough, as shown in the second line of Fig. 3, ˜ ξ -arrives at zero at the moment, where the cross-sectional area and axis ratio tend to 0. This point is usually called a caustic point, denoted by λ c and determined by ˜ ξ -( λ c ) = 0. In this case, the detected lensing image is inverted (negative parity). Conversely, when the impact parameter is relatively small, there is no caustic point, and ˜ ξ -remains positive on the entire path, ultimately resulting in an upright (positive parity) image. Besides the approximate analytical solution, we implement the numerical solution to Eq. (13), with initial values being ξ ± ( λ = 0) = 0 and D ξ ± ( λ = 0) = 1. These results are also plotted in Fig. 3, verifying that the Dyson-like series solution is valid under the weak deflection and thin lens limits. \nAs shown in Eq. (24) and around it, we can define the cross-sectional area ( A ) and axis ratio ( R ) from the deviation vector. Fig. 4 depicts their evolution along the path. The area and axis ratio are always positive for image 1 (the left panels), but meet zero at the caustic point for image 2 (the right panels). The relative changing rates of A and R are described by optical scalars, ρ and σ , respectively, which are shown in Fig. 5. When closing to the wave source (point like as we have assumed), all of the geodesics in congruence focus on a point, and subsequently, ρ is infinity. At the starting point, ˜ ξ + and ˜ ξ -are set to the same initial conditions, such that σ is keeping to zero before approaching the lens object. More importantly, in image 1, these two optical scalars are continuous, but there is a discontinuity in image 2, caused by the caustic point. \nThe weak deflection and thin-lens approximation bring errors in the analytical solutions. For example, one can find slight deviations between the slopes of the numerical and analytical solutions to ˜ ξ ± (see Fig. 3). Such error is amplified when calculating σ , especially when approaching the observer (see the left-bottom panel in Fig. 5). It is noted that the example parameters in Table I significantly deviate from the realistic situation. A typical lens object has mass M ∼ 10 8 M ⊙ and redshift z ∼ 1. The angular separation of the image and lens is about θ ∼ 1 '' . We can estimate the distance and impact parameter for such a system as D L ≈ 4 . 3 × 10 3 Mpc ≈ 9 . 0 × 10 14 M and b ∼ 4 . 8 × 10 -6 D L . This \nestimation indicates that the impact distance is much greater than the Schwarzschild radius, and D L is much greater than the impact distance, ensuring the validity of the weak deflection and thin lens approximations. \nFIG. 3: The evolution of the transversal components of the geodesic deviation vector, ˜ ξ ± from different source positions and images. The red curves show the evaluated results based on the analytical expressions from the Dyson-like series (20). The blue curves are obtained by numerically solving the GDE (13), together with the radial geodesic equation in Eq. (26). \n<!-- image -->", 'C. The magnification and axis ratio of the lensing images': 'In the last subsection, we investigate the behaviors of the geodesic deviation vector, cross-sectional area/axis ratio, and optical scalars along the lensing path. In this subsection, we study the observational imprints by such evolution. The magnification of the lensing image is defined as the ratio between the lensed and unlensed energy flux at the observer. In geometric optics, the energy flux of light and GWs is covariantly conserved within the congruence, and then proportional to 1 / A . Therefore, the magnification is rewritten as \nµ = ̂ A ( λ o ) A ( λ o ) . (34) \n̂ A means the cross-sectional area of a congruence freely propagating along the unlensed path (OS, see Fig. 1). By setting the initial conditions as ̂ A ( λ = 0) = 0 and D ̂ A ( λ = 0) = 1, its solution is given by \n̂ A ( λ ) = 1 OS 2 , with OS = D S √ 1 + tan 2 β = r o √ 1 + tan 2 β [ 1 + √ 1 -(1 + tan 2 β ) ( 1 -r 2 s r 2 o ) ] . (35) \nWe recall that r s and r o are the radial Schwarzschild coordinates of the source and observer, relating D L and D S by \nr o = D L , and , r s = D 2 S (1 + tan 2 β ) -2 D L D S + D 2 L (36) \nThe area of the lensed congruence is calculated from the Dyson-like series (20). At the observer position, we get the expressions of the first four terms in this series, \nI 0 ( λ o ) = λ o ≈ M u m ( 1 x s + 1 x o )[ 1 -1 2 x s x o ( 1 -4 u m x s + x o )] , (37) \nand \nI 3 ( λ o ) ≈ M x s x o · 16 5 u 2 m . (40) \nUnder the weak deflection and thin-lens assumptions, u m ≪ 1, and x s,o ≡ r m /r s,o ≪ 1 with the same order of u m , we have expanded the expressions in terms of small parameters u m , x s , and x o , up to the zeroth order, in Eqs. (37 40). Inserting Eqs. (37 - 40) into Eq. (19), and then into the definition Eq. (23), the area A ( λ o ) is obtained. Then, combining this result with the unlensed area in Eq. (35), the magnification is given from its definition Eq. (34). \nHowever, the parameters used here are different with those in the standard textbook [4], and we transform them into the commonly used lensing parameter as follows. (1) Transform u m into b by Eq. (B3), and then, from the geometry shown in Fig. 1, b = ± D L sin θ/M for image 1 and 2, respectively. (2) Transform x s and x o into β , D L , and D LS by their definition and Eq. (36). (3) Substitute the relation between β and θ from lens equation (B12, B12). (4) In the weak deflection and thin lens limit, we expand the image position as \nθ = θ E [ θ 0 + θ 1 ϵ + θ 2 ϵ 2 + O ( ϵ 3 ) ] , (41) \nwhere the bookkeeper is of order, \nFIG. 4: The evolution of the area and axis ratio of the geodesic congruence. Same with Fig. 3, the blue and red curves are obtained from the analytical solutions and corresponding numerical solutions to Eq. (13). \n<!-- image --> \nI 1 ( λ o ) ≈ 2 M x s x o { 2 + ( 45 π 16 -4 ) u m + [( 192 5 -135 π 16 ) u 2 m +2 u m ( x s + x o ) -( x 2 s -x s x o -x 2 o ) ]} , (38) \nI 2 ( λ o ) ≈ M x s x o { 15 π 8 u m + u m [( 48 -45 π 8 ) u m -4( x s + x o ) ]} , (39) \nϵ ∼ θ E 4 D , with D ≡ D LS D S . (42) \nFIG. 5: The evolution of the spin coefficients ρ and σ . Same with Fig. 3, the blue and red curves are obtained from the analytical solutions and corresponding numerical solutions to Eq. (13). The vertical dotted lines in the right panels represent the caustic points. The corresponding values are listed in Table I. \n<!-- image --> \nAfter the above operations, we obtain the final result of the magnification, which is expanded in powers of ϵ as \nµ = θ 4 0 θ 4 0 -1 ∓ 15 πθ 3 0 16(1 + θ 2 0 ) 3 · ϵ + θ 2 0 384(1 -θ 2 0 ) 2 (1 + θ 2 0 ) 5 { -1024(3 -2 D 2 ) +1024(9 + 12 D14 D 2 ) θ 2 0 +[4096(3 -D 2 ) -2025 π 2 ] θ 4 0 -[ 2048(6 + 15 D17 D 2 ) -2025 π 2 ] θ 6 0 -1024(9 + 6 D8 D 2 ) θ 8 0 +1024 D (3 + 18 D20 D 2 ) θ 10 0 +6144 D (1 -D ) θ 12 0 } ϵ 2 + O ( ϵ 3 ) , (43) \nThe leading-order result can be found in several standard textbooks, e.g., [4]. For the case where θ = θ E , and θ 0 = 1, the magnification tends to infinity. The positions with infinite magnification is defined as the critical curve. When the image is located at the outside/inside of the critical curve, the magnification is positive/negative, as shown in Table I, representing whether the image is upright (positive parity) or inverted (negative parity). In the second-order terms, the ± corresponds to image 1 and image 2. The higher-order corrections are derived by Ref. [31], through the Jacobian matrix. In Appendix E, we compare our result (43) with the previous work [31]. \nAnother direct observational quantity is the axis ratio of the images, describing the shape distortion of the background wave source. Similar to the deduction of magnification, the axis ratio R ( λ o ) is given by, \nR = θ 2 0 -1 θ 2 0 +1 ∓ 15 π ( θ 2 0 -1)(1 + 3 θ 2 0 ) 16 θ 0 (1 + θ 2 0 ) 3 · ϵ + ( θ 2 0 -1) 768 θ 2 0 (1 + θ 2 0 ) 5 { -9(2048 -225 π 2 ) -[4096(18 + 3 D4 D 2 ) -8100 π 2 ] θ 2 0 -3[4096(6 + D-D 2 ) -3375 π 2 ] θ 4 0 -8193 D 2 θ 6 0 +6144(3 -2 D +2 D 2 ) θ 8 0 -4096 D (3 -4 D ) θ 10 0 } ϵ 2 + O ( ϵ 3 ) . (44) \nWhen the lensing image is located at the critical curve, with θ 0 = 1, the axis ratio is zero. This means the extended \nsource is distorted to be a line segment, with zero area and infinite amplitude. Similar to the magnification, the axis ratio is positive/negative for the images located outside/inside of the critical curve, representing whether the image is upright (positive parity) or inverted (negative parity). The numerical results of R are listed in Table I. \nThe magnification and axis ratio are related to the convergence ( κ ) and shear ( γ 1 and γ 2 ) by \nµ = 1 (1 -κ ) 2 -γ 2 , and R = 1 -κ -γ 1 -κ + γ , (45) \nwith γ = √ γ 2 1 + γ 2 2 . The convergence depends on the lens mass density within the geodesic congruence. The shear γ 1 characterizes the stretching of the lensing images and γ 2 characterizes its orientation. For the Schwarzschild lens, that is vacuum and axisymmetric, κ = 0, γ 2 = 0, and γ = γ 1 . Through the Jacobian matrix [4], the shear of the point-mass lens is calculated as γ = 1 /θ 2 0 at the leading order of weak deflection approximation, resulting in µ = θ 4 0 / ( θ 4 0 -1) and R = ( θ 2 0 -1) / ( θ 2 0 +1), as shown in the leading terms of Eqs. (43) and (44).', 'V. CONCLUSION': 'In this work, we revisit the Schwarzschild lensing of light or GWs from geodesic deviation. The detected lensed signal consists of a congruence of light/gravitational rays, and their collective behavior is described by the GDE. To solve it, we project the GDE onto the NP tetrad constructed along the null geodesics and get a pair of decoupled equations for ξ + and ξ -(13) for real Weyls scalar Ψ 0 . Similar to the Dyson series, we present the solution to the transversal components of the deviation vector in Eqs. (19) and (20). In addition, the relationship between the deviation vector and the optical scalars, ρ and σ , is presented in Eq. (11). The physical meaning of these two scalars is the relative changing rate of the congruence cross section and the axis ratio [see Eq. (22)]. \nSubsequently, we consider the lensing process in the weak deflection limit, where the impact parameter is sufficiently large. The general signature is shown in Fig. 1, two images are formed by Schwarzschild lensing. In this assumption, the Dyson-like series is calculated from Eq. (31, 32) analytically. We depict such results in Fig. 2, and compare them with the numerical solution for a group of parameters listed in Table I. And we find that I 0 and I 1 are dominant and I 2 , I 3 are close to zero, implying the series (19) is convergent in weak deflection limit. Fig. 3 shows the evolution of the deviation vector. ˜ ξ + increases more rapidly after lensing and keeps positive. And ˜ ξ -increases more slowly or even decreases after lensing, meeting the caustic point, and finally forms an inverted image at the observer. Correspondingly, Figs. 4 and 5 show the cross-sectional area, axis ratio, and optical scalars, defined in Eqs. (23, 24). \nIn subsection IV C, we reproduce the magnification from the geodesic deviation [see Eq. (43)], consistent with that from the Jacobian matrix in Ref. [31]. Meanwhile, we find some missing corrections in the previous calculation, e.g., the deflection point deviates from the lens plane and the lens equations are slightly different for the two images. Additionally, we present the result of the axis ratio, equivalently, the shear, of lensing images up to the second weak deflection approximation [see Eq. (44)]. \nThis work is potentially worthwhile for future studies. As discussed in Ref. [35, 44], the calculation of spin coefficients is indispensable when investigating the lensing effects beyond geometric optics, in which richer information on light and GW polarization is presented [46-51]. In this work, we calculate ρ and σ following the procedure in Ref. [34] in the simplest case, the Schwarzschild lensing. Furthermore, the astrophysical lens objects are more complex than a Schwarzschild black hole [52]. Extending the computation to a more general lens model is helpful to the lensing observation and gravitational testing in the future.', 'Acknowledgments': 'We would like to thank Takahiro Tanaka, Emanuel Gallo, Shaoqi Hou, Sam. R. Dolan, Haofu Zhu, Xinyue Jiang, and Donglin Gao for their helpful discussions and comments. This work is supported by Strategic Priority Research Program of the Chinese Academy of Science (Grant No. XDB0550300), the National Key R&D Program of China Grant No.2022YFC2200100 and 2021YFC2203102, NSFC No.12273035 and 12325301 the Fundamental Research Funds for the Central Universities under Grant No. WK3440000004, and the science research grants from the China Manned Space Project with No.CMS-CSST-2021-B01, and China Scholarship Council, No. 202306340128. X. G. acknowledges the fellowship of China National Postdoctoral Program for Innovative Talents (Grant No. BX20230104). T.L. is supported by NSFC No. 12003008. T.Z. is supported in part by the National Key Research and Development Program of China under Grant No. 2020YFC2201503, the Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LR21A050001 and LY20A050002, the National Natural Science Foundation of China under Grant No. 12275238.', 'Appendix A: Null tetrad in Kerr spacetime': 'In this section, we review the main procedure to construct the null tetrad in Kerr spacetime, shown in Ref. [45]. Let us denote the Boyer-Lindquist coordinate as { t, r, θ, φ } , the background metric as g µν , the black hole mass as M , and spin as a in this section. Before construction, it is necessary to summarize several important tensors. The first one is the principal tensor, \nγ µν = 0 r a 2 sin θ cos θ 0 -r 0 0 ar sin 2 θ -a 2 sin θ cos θ 0 0 a ( r 2 + a 2 ) sin θ cos θ 0 -ar sin 2 θ -a ( r 2 + a 2 ) sin θ cos θ 0 , (A1) \nsatisfying ∇ α γ µν = g αµ ξ ν -g αν ξ µ , where ξ µ = (1 , 0 , 0 , 0) is the Killing vector, indicating the energy conservation of the particle moving along the geodesics. Its Hoge dual defines the Killing-Yano tensor by \nf µν ≡ -ˆ γ µν = -1 2 √ -gϵ αβµν γ αβ = 0 -a cos θ ar sin θ 0 a cos θ 0 0 -a 2 sin 2 θ cos θ -ar sin θ 0 0 r ( r 2 + a 2 ) sin θ 0 a 2 sin 2 θ cos θ -r ( r 2 + a 2 ) sin θ 0 , (A2) \nsatisfying ∇ ( α f µν ) = 0. ϵ αβµν is Levi-Civita symbol. \nFirstly, we need to construct a set of basis vectors, denoted by { λ (1) , λ (2) , λ (3) , λ (4) } , as follows, \nλ α (1) = k α , (A3) \nλ α (3) = γ αβ λ (1) β -A ( r, θ ) λ α (1) , (A4) \nλ α (2) = γ αβ λ (3) β -B ( r, θ ) λ α (1) , (A5) \nλ α (4) = f α β λ β (1) , (A6) \nThe wavevector in Kerr spacetime is presented by \nλ α (1) = k α = { ( r 2 + a 2 ) 2 -a 2 ∆sin 2 θ -2 Mξar ∆Σ , R Σ , Θ Σ , 2 Mar +(Σ -2 Mr ) ξ csc 2 θ ∆Σ } , (A7) \nwith ∆ ≡ r 2 (1 -2 M/r + a 2 /r 2 ) and Σ ≡ r 2 + a 2 cos 2 θ . A ( r, θ ) and B ( r, θ ) are two underdetermined functions. ξ, η being the conserved angular momentum, and Carter constant. The first leg is parallel-transported. Letting the third leg be parallel-transported, we get the constraint on the unknown functions, \ndA dλ = ξ β k β , and dB dλ = h γβ ξ γ k β -1 2 dA 2 dλ . (A8) \nFrom the above equation, one can find that the function B is determined by directly integrating on the right-hand side once A is given. Based on the above discussion, we can list all of the expressions of the tetrad. The third one is \nλ α (3) = { a 2 sin 2 θ/ Σ -[( r 2 + a 2 ) 2 -2 Maξr ] / ∆Σ } A -r ( r 2 + a 2 ) R / ∆Σ -(Θ / Σ) a 2 sin θ cos θ -r ( r 2 + a 2 -aξ ) / Σ -( R / Σ) A a cot θ ( ξ -a sin 2 θ ) / Σ -(Θ / Σ) A -[2 Mar +(Σ -2 Mr ) ξ csc 2 θ ] A/ (∆Σ) + a [(Θ / Σ)cot θ +( r/ ∆)( R / Σ)] , (A9) \nUsing the known Killing vector, the constraint on A ( r, θ ) becomes dA/dλ = -1. We shall suppose that the function A is variable-separable, which gives the form A ( r, θ ) = F ( r ) + G ( θ ). An obvious set of solutions is \nF ( r ) = -∫ r 2 R dr, G ( θ ) = -∫ a 2 cos 2 θ Θ dθ. (A10) \nUsing the principal tensor (A1) and Killing vector, the constraint on B becomes \nB ( r, θ ) = 1 2 [ r 2 -a 2 cos 2 θ -A 2 ( r, θ )] . (A11) \nAnd then the second leg is \nλ t (2) = -(1 / 2)( r 2 -a 2 cos 2 θ -A 2 ) { ( r 2 + a 2 ) 2 -a 2 ∆sin 2 θ -2 aMξr } / (∆Σ) + r ( r 2 + a 2 )( R A + a 2 r -aξr + r 3 ) / (Σ∆) + a 2 cos θ [Θsin θA + a cos θ ( a sin 2 θ -ξ )] / Σ , λ r (2) = { 2 rA [( r 2 + a 2 ) -aξ ] + R (Σ + A 2 ) } / (2Σ) , λ θ (2) = [ -2 a cot θA ( ξ -a sin 2 θ ) + Θ A 2 -ΣΘ ] / (2Σ) , λ φ (2) = ( A 2 + a 2 cos 2 θ -r 2 )[ a 2 ξ cot 2 θ +2 aMr + ξr csc 2 θ ( r -2 M )] / (2∆Σ) +2 ar ( R A + a 2 r -aξr + r 3 ) / (∆Σ) -a cot θ ( a cot θ ( ξ -a sin 2 θ ) -A Θ) / Σ , (A12) \nand the fourth leg is \nλ α (4) = a [ R ( r 2 + a 2 ) cos θ -r Θsin θ ∆] / (Σ∆) a cos θ [( r 2 + a 2 ) -aξ ] / Σ r csc θ ( ξ -a sin 2 θ ) / Σ csc θ ( a 2 R sin θ cos θ -r ∆Θ) / (∆Σ) . (A13) \nAfter constructing { λ (1) , λ (2) , λ (3) , λ (4) } , the NP tetrad is defined by \ne α ( a ) = { λ α (1) , 1 C 2 λ α (2) , 1 C 1 √ 2 [ λ α (3) + iλ α (4) ] , 1 C 1 √ 2 [ λ α (3) -iλ α (4) ] } , (A14) \nwhich satisfies the orthogonality and parallel-transported condition. The normalization constant is \nC ≡ √ ( a -ξ ) 2 + η. (A15)', 'Appendix B: Schwarzschild lensing': 'One defines u ≡ M/r , and then the photon/graviton trajectory is dominated by [3] \n( du dφ ) 2 = 1 b 2 -u 2 (1 -2 u ) ≡ f ( u ) , (B1) \nThe impact parameter is b = | L | /M . When b > 3 √ 3, the function f ( u ) possesses two different zero points, denoted by 0 < u 2 < u 3 , the allowed orbits should be located at interval [0 , u 2 ] and [ u 3 , + ∞ ], where f ( u ) > 0. These first kinds of orbits are open, in which the massless particle moves between an infinite region and a pericenter, corresponding to the gravitational lensing situation. The radial range of allowed motion is r 0 < r < max { r o , r s } , or equivalently, min { u o , u s } < u < u m ≡ M/r m . r o and r s denote the radial coordinates of the wave source and observer. correspondingly, u o ≡ M/r o and u s ≡ M/r s . \nIn this work, we only focus on the weak deflection limit and thin lens approximation, in which the impact distance ( Mb ) is much larger than the gravitational radius of the lens black hole, and the distances from emitter/observer to lens object are very larger than the impact distance. Using a bookkeeper, ϵ , the order of magnitude impact parameter, source/observer distance are \nu m ∼ 1 b ∼ ϵ, r m r s , r m r o ∼ ϵ. (B2) \nThe following calculations are based on the Taylor expansion by the bookkeeper ϵ . Additionally, the explicit relationship between u m and b is given by f ( u m ) = 0, \nb = 1 u m 1 √ 1 -2 u m = 1 u m [ 1 + u m + 3 2 u 2 m + 5 2 u 3 m + O ( u 4 m ) ] . (B3) \nFollowing the standard steps, the trajectory equation u = u ( φ ) is \nu = u m cos φ + u 2 m ( 1 -cos φ +sin 2 φ ) + u 3 m [( 2 -3 4 sin 2 φ ) cos φ -2(1 + sin 2 φ ) + 15 4 φ sin φ ] + O ( u 3 m ) , (B4) \nUp to order O ( u 3 m ). The deflection angle, α , is obtained from the asymptotic behavior of solution (B4), given by \nα = 4 u m + ( -4 + 15 4 π ) u 2 m + ( 122 3 -15 π 2 ) u 3 m . (B5) \nAfter giving the deflection angle (B5), we re-drive the lens equation in this appendix. We first investigate image 1 shown in Fig. 1a, in which the source and image lie on the same side of the optical axis. In △ ASI, applying the sine theorem gives, \nSI sin α = AI sin ∠ ASI ⇒ NI -SN sin α = OI -OA sin[ π/ 2 -( α -θ )] = OI -OA cos( α -θ ) , (B6) \nwhere we use ∠ ASI = π -α -∠ AIS = π -α -( π/ 2 -θ ) = π/ 2 -( α -θ ) and then \n( NI OI -SN OI ) cos( α -θ ) = ( 1 -OA OI ) sin α, (B7) \nor, equivalently \n( D S tan θ -D S tan β D S sec θ ) cos( α -θ ) = ( 1 -OA D S sec θ ) sin α. (B8) \nThen the length of line OA can be expressed from the △ OAL by sine theorem, \nOL sin ∠ OAL = OA sin ∠ OLA . (B9) \nHere, the line AL divides ∠ OAS equally, such that the above equation becomes \nD L sin( π -α ) / 2 = OA sin[ π -θ -( π -α ) / 2] . (B10) \nUsing the expression of OA in terms of D L , θ , and α , \nOA = D L cos( α/ 2 -θ ) cos( α/ 2) . (B11) \nWe finally arrived at \n(tan θ -tan β ) cos θ cos( α -θ ) = sin α -2 D L D S cos ( α 2 -θ ) cos θ sin α 2 . (B12) \nBy replacing θ by -θ , and β by -β (equivalently, replace NI -SN by NI+SN) in Eq.(B12), we get the lens equation for image 2, \n-(tan θ -tan β ) cos θ cos( α + θ ) = sin α -2 D L D S cos ( α 2 + θ ) cos θ sin α 2 , (B13) \nin which the source and image lie on the different sides of the optical axis. \nTo solve the lens equation (B12, B13), we rescale the source position and image position as \nβ = β 0 θ E , θ = θ E [ θ 0 + θ 1 ϵ + θ 2 ϵ 2 + O ( ϵ 3 ) ] , (B14) \nwhere the bookkeeper ϵ is defined in Eq. (42). Then the lens equation at leading order is simplified as \nβ 0 = θ 0 -1 θ 0 , (B15) \nat the leading order, whose solution is \nθ 0 = 1 2 [ β 0 ± √ β 2 0 +4 ] . (B16) \nIn the second and third orders, the solutions to θ 1 and θ 2 are \nθ 1 = ± 15 π 16(1 + θ 2 0 ) , (B17) \nand \nθ 2 = 1 3 θ 0 (1 + θ 2 0 ) 3 { 48 -675 256 π 2 -16 D 2 + [ 8(9 -3 D +2 D 2 ) -675 128 π 2 ] θ 2 0 +40 D 2 θ 4 0 -8(3 -9 D +4 D 2 ) θ 6 0 +8 D (6 -5 D ) θ 8 0 } . (B18) \nThe plus-minus symbol corresponds to the image 1 and 2, respectively. The leading-order result (B16) is consistent with Ref. [31], but there is an extra plus-minus symbol in first-order correction (B17). And the second-order result (B18) is different, because the deflection point deviates from the lens plane.', 'Appendix C: Solution to the radial motion': 'In weak deflection limit, the radial potential reduces to \nU r = ± √ 1 -x 2 { 1 -u m ( x 2 1 + x ) -u 2 m [ x 2 (2 + x ) 2 2(1 + x ) 2 ]} + O ( u 3 m ) . (C1) \nwhere x ≡ u/u m , u = M/r ⩽ u m , u m = M/r m ≪ 1, with r m is the radius of pericenter. When the particle moves toward the pericenter, the above equation is integrated by \nM -1 λ I = M -1 ∫ r r s U -1 r dr ≈ -1 u m ( ν x -ν s x s ) -( ν 1 + x -ν s 1 + x s ) + u m 2 [ xν (1 + x ) 2 -x s ν s (1 + x s ) 2 +3arcsin x -3 arcsin x s ] , (C2) \nwith initial condition λ s = 0. For simplicity, we define ν = √ 1 -x 2 , and ν s = √ 1 -x 2 s . When arriving the pericenter, r = r m , where x = 1, the affine parameter is \nλ m = 1 u m M x s [ 1 + x s ( u m -x s 2 )] , (C3) \nAnd then, when moving backward, the integration should be \nM -1 λ II = M -1 { λ m + ∫ r r m U -1 r dr } ≈ 1 u m ( ν x + ν s x s ) + ( ν 1 + x + ν s 1 + x s ) -u m 2 [ xν (1 + x ) 2 + x s ν s (1 + x s ) 2 +3arcsin x +3arcsin x s -3 π ] . (C4) \nAnd the affine parameter at the observer position has been shown in Eq. (37).', 'Appendix D: The analytical expressions of Eqs. (31) and (32)': 'In this appendix, we present the analytical solution of Eqs. (31) and (32). The first three orders of the Dyson-like series are \nM -1 I 1 = ± ( 1 + x s x -2 xx s ) ν ± u m 8 xx s (1 + x ) {[ 16(2 + x ) -(61 + 61 x -15 x 2 -7 x 3 ) x s +8 xx 2 s +7(1 + x ) x 2 s ] ν +15(3 νν s ± xx s ) arccos x } ± u 2 m 40 xx s (1 + x ) 2 [ -8(192 + 364 x +167 x 2 ) +(835 + 1709 x +1143 x 2 +326 x 3 +119 x 4 +42 x 5 ) x s +(543 + 1086 x +523 x 2 ) x 2 s +35 x (1 + x ) x 3 s +42(1 + x ) 2 x 4 s ] ν ∓ 3 u 2 m 8 xx s (1 + x )(1 + x s ) { ± 15 [3 + 2 x +(2 + x ) x s ] νν s +(1 + x ) [ -8 x -(8 + 13 x -4 x 2 ) x s -(8 + x -4 x 2 ) x 2 s +4 xx 3 s ] } arccos x ∓ ( x ↔ x s ) , (D1) \nM -1 I 2 = ± u m 8 xx s { [ 16 x +(1 -3 x 2 ) x s -8 xx 2 s -3 x 2 s ] ν ± 3(5 νν s ∓ xx s ) arccos x } ± u 2 m 8 xx s (1 + x )(1 + x s ) [ -2(96 + 112 x + x 2 -7 x 3 ) -(179 + 151 x -60 x 2 -6 x 3 +5 x 4 ) x s +(139 + 199 x +47 x 2 -15 x 3 -5 x 4 ) x 2 s +(126 + 116 x -22 x 2 -7 x 3 ) x 3 s -(5 + 15 x +7 x 2 ) x 4 s -5(1 + x ) x 5 s ] ν + 3 u 2 m 8 xx s (1 + x )(1 + x s ) { -5 [ 3 + 5 x +3 x 2 +(5 + 7 x +3 x 2 ) x s +3(1 + x ) x 2 s ] νν s ± (1 + x ) [ 10 x +(10 + 11 x -5 x 2 ) x s +(10 -4 x -5 x 2 ) x 2 s -5 xx 3 s ] } arccos x ∓ ( x ↔ x s ) , (D2) \nand \nM -1 I 3 = ± u 2 m 40 xx s [ -2(32 + 15 x ) -x (19 -3 x ) x s +(107 + 15 x 2 ) x 2 s +15 xx 3 s +3 x 4 s ] ν + 3 u 2 m ( x + x s ) 8 xx s ( -5 νν s ∓ 2 ± xx s ) arccos x ∓ ( x ↔ x s ) . (D3) \nThe upper sign corresponds to the case where v s < v < v m , and the lower one to v m < v < v o . ( x ↔ x s ) represents exchanging all x and x s in the formula.', 'Appendix E: Comparison between Eq. (43) and previous result.': "In this appendix, we compare our result on magnification with that provided by Ref. [31] [Eq. (76-79), with A 1 = 4, A 2 = 15 π/ 4, and A 3 = 128 / 3] and discuss the difference. In [31], the magnification is defined by the Jacobian matrix, \nµ = ∣ ∣ ∣ ∣ sin βdβ sin θdθ ∣ ∣ ∣ ∣ -1 . (E1) \nIn the leading order, these two results are consistent. But there are some missing corrections when [31] driving the higher-order corrections. Firstly, the images formed by the lensing are at the same and opposite sides to the source, with positive and negative angular momentum. Therefore the lens equation for these two cases is slightly different. This is shown in Eqs. (B12, B13) and Fig. 1. This leads to an extra ± symbol in the first-order correction, which is missed in Ref. [31]. Secondly, the deflection point (denoted by A) deviates from the lens plane (see Fig. 1), resulting in inconsistency in the second order. Thirdly, the lensed and unlensed signals are emitted in slightly different directions. As the point-source assumption in this work, another correction factor cos β/ cos( α -θ ) is needed to consider. After taking these three extra corrections, one obtains the same magnification through geodesic deviation and Jacobian matrix. \n- [2] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1971).\n- [3] S. Chandrasekhar, The mathematical theory of black holes (1983).\n- [4] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (1992).\n- [5] K. C. Wong et al. , The Astrophysical Journal Letters 789 , L31 (2014).\n- [6] X.-H. Liu, Z.-H. Li, J.-Z. Qi, and X. Zhang, The Astrophysical Journal 927 , 28 (2022). \n[7] J. M. Ezquiaga \net al. \n, Phys. Rev. D \n103 \n, 064047 (2021). \n- [8] Q. Wang et al. , The Astrophysical Journal 969 , 119 (2024).\n- [9] A. Refregier, Annual Review of Astronomy and Astrophysics 41 , 645 (2003).\n- [10] L. V. E. Koopmans et al. , (2009), arXiv:0902.3186 [astro-ph.CO] .\n- [11] M. Grespan and M. Biesiada, Universe 9 , 10.3390/universe9050200 (2023).\n- [12] X. Guo and Y. Lu, Phys. Rev. D 102 , 124076 (2020).\n- [13] R. Takahashi and T. Nakamura, The Astrophysical Journal 595 , 1039 (2003).\n- [14] A. Mishra et al. , Monthly Notices of the Royal Astronomical Society 508 , 4869 (2021).\n- [15] A. K. Meena and J. S. Bagla, Monthly Notices of the Royal Astronomical Society 492 , 1127 (2019).\n- [16] L. Dai and T. Venumadhav, (2017), arXiv:1702.04724 [gr-qc] .\n- [17] Pagano, G., Hannuksela, O. A., and Li, T. G. F., Astronomy and Astrophysics 643 , A167 (2020).\n- [18] K. 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Keeton and A. O. Petters, Phys. Rev. D 72 , 104006 (2005).\n- [32] M. Sereno and F. De Luca, Phys. Rev. D\n- 74 , 123009 (2006).\n- [33] E. Poisson, A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, 2009).\n- [34] S. R. Dolan, International Journal of Modern Physics D 27 , 1843010 (2018).\n- [35] S. R. Dolan, (2018), arXiv:1801.02273 [gr-qc] .\n- [36] E. F. Boero and O. M. Moreschi, Monthly Notices of the Royal Astronomical Society 492 , 3763 (2019).\n- [37] E. Newman and R. Penrose, Journal of Mathematical Physics 3 , 566 (1962).\n- [38] S. Pineault and R. C. Roeder, Astrophys. J. 212 , 541 (1977).\n- [39] S. Seitz, P. Schneider, and J. Ehlers, Classical and Quantum Gravity 11 , 2345 (1994).\n- [40] E. Gallo and O. M. Moreschi, Phys. Rev. D 83 , 083007 (2011).\n- [41] C. C. Dyer, Monthly Notices of the Royal Astronomical Society 180 , 231 (1977).\n- [42] D. W. Berry and P. C. S. Costa, Quantum 8 , 1369 (2024).\n- [43] S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D 63 , 023007 (2000).\n- [44] J. O. Shipley, Strong-field gravitational lensing by black holes (2019), arXiv:1909.04691 [gr-qc] .\n- [45] P. Dahal, Eur. Phys. J. Plus 138 , 205 (2023).\n- [46] A. I. Harte, General Relativity and Gravitation 51 , 10.1007/s10714-018-2494-x (2019).\n- [47] A. I. Harte, General Relativity and Gravitation 51 , 10.1007/s10714-019-2646-7 (2019).\n- [48] G. Cusin and M. Lagos, Phys. Rev. D 101 , 044041 (2020).\n- [49] C. Dalang, G. Cusin, and M. Lagos, Phys. Rev. D 105 , 024005 (2022).\n- [50] Z. Li, J. Qiao, W. Zhao, and X. Er, Journal of Cosmology and Astroparticle Physics 2022 (10), 095.\n- [51] K.-i. Kubota, S. Arai, and S. Mukohyama, Phys. Rev. D 109 , 044027 (2024).\n- [52] C. R. Keeton, (2002), arXiv:astro-ph/0102341 [astro-ph] .\n- [53] The superscript n represents the components along n -axis. It is similar for the m and ¯ m superscript in Eq. (11) and (12)."}

2024arXiv240907876S

Intense bombardment of solar system planets in the immediate aftermath of protoplanetary disk dissipation has played a key role in their atmospheric evolution. During this epoch energetic collisions will have removed significant masses of gas from rocky planet atmospheres. Noble gases are powerful tracers of this early atmospheric history xenon in particular which on Mars and Earth shows significant depletions and isotopic fractionations relative to the lighter noble gasses. To evaluate the effect of impacts on the loss and fractionation of xenon we measure its ionization and recombination efficiency by laser shock and apply these constraints to model impactdriven atmospheric escape on Mars. We demonstrate that impact bombardment within the first 200 to 300textMyr of solar system history generates the observed Xe depletion and isotope fractionation of the modern martian atmosphere. This process may also explain the Xe depletion recorded in Earths deep mantle and provides a latest date for the timing of giant planet instability.

2024-09-01T00:00:00Z

['arXiv:2409.07876', '10.48550/arXiv.2409.07876', '2024arXiv240907876S']

['Astrophysics - Earth and Planetary Astrophysics']

Impact sculpting of the early martian atmosphere

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07876.pdf

{'Impact sculpting of the early martian atmosphere': "Oliver Shorttle 1,2,* , Homa Saeidfirozeh 3 , Paul Brandon Rimmer 2,4 , Vojt˘ech Laitl 3, 5 , Petr Kubel'ık 3 , Luk'a˘s Petera 3,6 , and Martin Ferus 3 \n- 1 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 2 Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, UK\n- 3 J. Heyrovsk'y Institute of Physical Chemistry, Czech Academy of Sciences, Dolejˇskova 3, CZ 18223 Prague\n- 8, Czech Republic\n- 4 Cavendish Laboratory, University of Cambridge, JJ Thompson Avenue, Cambridge CB3 0HE, UK\n- 5 University of Antwerp, Faculty of Science, Groenenborgerlaan 171, BE-2020 Antwerpen, Belgium\n- 6 Department of Inorganic Chemistry, Faculty of Science, Charles University, Hlavova 8, Prague, Czech Republic", 'ABSTRACT': "Intense bombardment of solar system planets in the immediate aftermath of protoplanetary disk dissipation has played a key role in their atmospheric evolution. During this epoch, energetic collisions will have removed significant masses of gas from rocky planet atmospheres. Noble gases are powerful tracers of this early atmospheric history, xenon in particular, which on Mars and Earth shows significant depletions and isotopic fractionations relative to the lighter noble gasses. To evaluate the effect of impacts on the loss and fractionation of xenon, we measure its ionization and recombination efficiency by laser shock and apply these constraints to model impact-driven atmospheric escape on Mars. We demonstrate that impact bombardment within the first 200 -300 Myr of solar system history generates the observed Xe depletion and isotope fractionation of the modern martian atmosphere. This process may also explain the Xe depletion recorded in Earth's deep mantle and provides a latest date for the timing of giant planet instability.", 'Introduction': "Noble gases are a key tracer of the early loss and gain of planetary atmospheres. Their inert nature makes their abundances and isotopic composition ideal archives of fractionation processes that occur during volatile loss and redistribution in planets. Xenon, in particular, has proven an enigmatic carrier of information on atmospheric evolution, showing anomalous depletions and mass fractionations in the atmospheres of both Earth and Mars ( 1 ). On Earth, \nthis has led to the identification of the 'missing Xe' problem, where Xe is noted to be unusually depleted in Earth's atmosphere given the element's high mass ( 2 ). Associated with this depletion is a large mass-dependent fractionation of Xe isotopes, greater than that seen among the isotopes of Kr despite their being lighter, and, remarkably, is a signal that has been constrained to have grown to its modern value over the first 2Gyr of Earth history ( 3, 4, 5 ). \nLike Earth, Mars's atmosphere evidences preferential Xe loss and isotopic fractionation compared with other noble gases (e.g., ref. 6, Fig. 1). The isotopic fractionation of Xe is of similar magnitude between the two atmospheres ( 7 ), however, the atmospheres began their evolution from different starting compositions, solar Xe in the case of Mars ( 8 ) compared with U-Xe for Earth (itself a mixture of cometary and chondritic Xe, ( 1, 9 )). Importantly, the isotopic fractionation of Xe in the martian atmosphere must have occurred early: data from two martian meteorites, ALH 84001 and NWA 11220, which sample Mars's atmosphere at ∼ 4 . 2Ga and ∼ 4 . 4Ga respectively, indicate that fractionation to the modern value occurred rapidly in the first few hundred million years of the planet's history ( 10,11 ). \nAn explanation for the progressive multi-billion year isotopic fractionation of Earth's atmospheric Xe has been provided by ref. 12, who suggest that Xe escaped Earth in a photo-ionized hydrogen wind. In this model, Xe loss and fractionation occur sporadically over the first several billion year's of Earth history and only terminate when atmospheric oxygenation sharply decreases the H2 mixing ratio of the atmosphere. Key to this model is the low ionization potential of Xe, the lowest among the noble gases, which means it is preferentially removed despite its high mass: ionized Xe couples to open magnetic field lines and is dragged to space by ionized hydrogen. A similar model of Xe escaping in an ionized wind has recently been advanced to explain Xe's loss from Mars's atmosphere ( 11 ). In the case of Mars, observations require that the fractionation terminates early (before 4 . 2Ga, ( 10 )), which is suggested to occur because of either waning extreme solar UV, diminished H2 sources on the planet, or cessation of the martian magnetic field ( 11 ). The latter scenario is disfavoured because of evidence of magnetism in rocks younger than 4.2 Ga ( 13 ). Diminishing H2 sources on the planet may be inconsistent with requirements for Mars's transiently wet early climate, which recent modelling has suggested needed episodic reducing conditions (i.e., high-H2) for over a billion years into the planet's life ( 14 ). This is long after fractionation of Xe in the atmosphere is observed to have ceased ( 10 ). \nIntriguingly, recent work has indicated that a separate epoch of preferential Xe loss must have also occurred much earlier in Earth's history, earlier than the slow loss recorded in its atmosphere over ∼ 2Gyr ( 15 ). Earth's deep mantle may, therefore, have captured evidence of a separate Xe (and wider volatile) loss episode within its first few hundred Myr ( 16 ); an episode contemporaneous with Mars's observed Xe loss ( 10 ). Together, these observations suggest that additional processes(s) may have operated across the solar system early in the life of rocky planets to drive the rapid loss of volatile elements. As Xe records preferential depletion by these processes, models other than the classic hydrodynamic escape scenarios are required (e.g., ( 17 )), as these would favour the preferential loss of light gases over \nFigure 1. Key characteristics of the martian noble gas reservoirs. (A) , Elemental abundances of noble gases in the martian interior (as represented by Chassigny ( 18 )) and atmosphere ( 1, 19 ), ratioed to the abundance of Xe, relative to the elemental ratio in CI chondrites ( 18 ). Xe is noticeably depleted compared to the other noble gases. The relative noble gas abundances that would be produced in an atmosphere formed from a two stage degassing process are shown as triangles. First, during the magma ocean stage, noble gases are distributed between the atmosphere and interior according to their relative solubilities ( 20 ), after which the initial atmosphere is lost. Second, subsequent volcanic degassing quantitatively removes noble gases from the melted sources region as magmas degas at low pressure. In this way, the most soluble gasses are dominant in the atmosphere that is restored. (B) , The isotopic fractionations of Xe (red circles) and Kr (blue squares) in Mars's atmosphere (data from ref. 6). Fractionations were calculated per atomic mass unit, considering only the heaviest four isotope ratios of Xe (131/130, 132/130, 134/130, and 136/130) and the 83/84 and 84/84 ratios of Kr. All reported errors are 1 sigma. \n<!-- image --> \nAlthough long considered in the context of atmospheric loss ( 21 ), impact bombardment is one mechanism for isotopic fractionation of atmospheres that has received little attention. The impact bombardment of planets is widely recognised as an important process shaping their formation and evolution ( 22 ). During planetary growth, accretionary impacts are the primary mechanism by which rocky planets gain mass ( 23,24,25 ), with the most dramatic and recent known example in the solar system being the collision of a Mars-sized impactor with the proto-Earth to form the Moon ( 26 ). One peak in impact rates onto planets likely occurs following the dissipation of their natal protoplanetary disk, which in the solar system occurred at ∼ 3Myr after its birth ( 27 ). Later peaks in impact fluxes would coincide with dynamical reorganisations of the solar system giant planets ( 28 ). \nThe abundance and rate of impacts in this early epoch have been constrained by cratering records on the Moon ( 29 ), Mars ( 30,29,31 ), and by the highly siderophile element (HSE) abundances of the terrestrial and martian mantles ( 32 ). However, the timing of any increased bombardment following giant planet instability remains uncertain: initially linked to clustered ages of lunar rocks between ∼ 3 . 5-4 . 2Ga ( 33,34 ), it now seems likely that the lunar observations at least can be explained with a gradual tailing-off of accretion ( 35 ) and that any instability occurred earlier ( 36,37,38 ). Impact \nbombardment could significantly affect atmospheric retention on rocky planets ( 39, 40, 41 ), being capable of both delivering and removing volatile elements depending on impactor size. In this case, the atmospheric evolution of the terrestrial planets may itself be a record of this early bombardment history. \nFigure 2. Model framework for describing Xe-loss and fractionation from the martian atmosphere. Impacts eject all material in a central cone, with an annulus of partially ionized material escaping the atmosphere according to whether ions survive to high altitude to connect with Mars's open magnetic field lines ( 42 ). \n<!-- image --> \nImpacts have been presumed to not affect the fractionation of Xe in planetary atmospheres because of their propensity to remove atmosphere from their target indiscriminately ( 43, 21, 40 ). However, whilst a hypervelocity impactor is moving through the atmosphere, it produces a high temperature ionized shock (a quasi-neutral plasma over 10 4 K) in a wide region around it ( 44 ). This plasma ends up thrown away from the planet's surface to high altitude (Fig. 2 and ref. 40). Whilst for a sufficiently large impactor some atmosphere in this cone-shaped region will be directly ejected, the planet will also retain a proportion of the ionized atmosphere. The transiently ionized atmosphere from hypervelocity impacts, therefore, represents a potential mechanism of creating ionized Xe at heights in the atmosphere where it may couple to magnetic field lines and escape the planet. This is a scenario similar to that envisaged by ref. 12, but with impacts rather than photons as both the source of the ionization and as the mechanism to move Xe up to high altitude (i.e., induce enhanced vertical mixing). Importantly, Xe loss by impacts would decouple the history of Xe from the H2 mixing ratio of a planetary atmosphere: whereas ref. 12,11 have emphasised the importance of hydrogen for ionising the Xe and transporting it vertically. Impacts achieve both of these processes, ionising Xe through high temperature plasma and ballistically lofting it in the atmosphere, without recourse to specific background atmospheric H2 mixing ratios. This is perhaps useful in the context of the requirements of Mars's early climate ( 14 ), and would present a new dependence for the timescale of Xe's evolution, occurring during epochs of intense impact bombardment. \nIndeed, such an association between impacts and Xe isotopic evolution has been noticed for Earth (Fig. 7c of ( 4 )), although our present work does not seek to explain Earth's more protracted Xe-loss history. \nFor such a scenario to explain the relative Xe abundance and isotope composition of Mars's atmosphere, it is key that Xe, compared with the other noble gases, be preferentially affected by impact-driven ionization and escape. Two conditions would contribute to preferential Xe loss and isotope fractionation: first, if the high temperatures of an impact plasma ionized Xe appreciably more than other noble gases; second, preferential Xe loss would be aided by the recombination rate of Xe being lower than that of the other noble gases, given that the ionized Xe will also need to be transported from the site of the shock to a height where it can couple to open magnetic field lines and be lost. In particular, for this second condition, we are interested in the rate of the reaction \nXe + + e -+ M → Xe + M , (1) \nwhere M is any neutral third body, versus the rate of equivalent reactions for other noble gases. If one or both of these conditions were met without the other actively disfavouring Xe ionization, then more Xe remains ionized once lofted up by impact than other noble gases; aiding Xe escape. \nExperiment and theory constrain the ionization energy for xenon to be lower than for any other noble gas ( 45 ) and that the rate at which ionized xenon recombines with its electron is slower than for the other noble gases ( 46 ). However, the experiments that give these rates either measure the thermal ionization rate and calculate the rate of the reverse reaction ( 47,48 ), or have only measured the forward electron-ion recombination rates ( 49,50 ). A final limitation of the current data is that no ion-electron recombination experiments have been performed for noble gases in the context of a planetary atmosphere; the role of geologically relevant background gasses in Xe recombination is therefore poorly constrained. \nTo provide new constraints on the key parameters of impact-driven ionization, we performed laser shock experiments to simulate impactor entry into planetary atmospheres. These give us direct experimental constraints on the thermal ionization of noble gases during an impact and the recombination rates for those ions during, and immediately after, the impact. We obtain new estimates of ionization fractions and recombination rates for Xe, using Ar as a reference noble gas with a higher ionization threshold. Combining the Xe ionization parameters with a simple model of impact driven atmospheric loss (based on ref. 40), we then explore the viability of impacts as the exclusive mechanism explaining Mars's history of Xe isotopic fractionation. We find that impacts can explain the early onset and magnitude of Xe isotopic fractionation in the martian atmosphere, subject to Mars's initial mass of the atmosphere and the specific bombardment history it experiences. Whilst other loss processes were likely simultaneously operating on planets \nto drive atmospheric loss and Xe fractionation ( 11 ), our results suggest that impact sculpting of the early martian atmosphere was likely significant. In this context, the recently discovered Xe-depletions in the deep Earth ( 15 ) indicate impact-driven loss processes may have operated early in its history as well.", 'Laser shock experiments': "Laser shock experiments were conducted as described in ref. 51 to estimate key parameters of atmospheric ionisation during hypervelocity impact. Mixtures of Xe-H2 and Ar-H2 were heated by short pulses of high energy laser to create a plasma. Ar, as a noble gas with higher ionization threshold than Xe ( 12 ), was chosen so that the relative ionisation efficiencies and recombination rates between the two gases could be compared. Spectra were recorded from ∼ 200ns after the initial laser-induced plasma was formed, and from these, the Xe and Ar ionisation fractions were estimated (See Materials and Methods for the complete description; Fig. 3). \nThe laser shock experiments show that Xe is more strongly ionised than Ar (Fig. 3a vs 3b): for a plasma temperature of 2 × 10 4 K, well within the range of temperatures atmosphere is expected to be heated to during impact ( 44 ), ∼ 40% of the Xe is present as either singly or doubly charged ions (Fig. 3a); this compares with less than 1% of Ar being present as charged ions at the same temperature (Fig. 3b). With increasing temperature, more Xe and Ar are ionised, but at all temperatures investigated, Xe remains over an order of magnitude more ionised than Ar. \nThe second important result from the experiments is an estimate of the relative recombination rates of Xe and Ar. Recombination rates were calculated by fitting the following equation to the data \nne ( t ) = ne ( 0 ) 1 + ne ( 0 ) krt , (2) \nwhere ne ( t ) (cm -3 ) is the electron density at time t (s), ne ( 0 ) is the electron density immediately following the laser pulse (at which time the gas is assumed to be fully ionised), and kr (cm 3 s -1 ) is the recombination rate. The fits of equation (2) to these time series are shown in Fig. 3c and 3d. A recombination rate of 6 ± 2 . 5 × 10 -11 cm 3 s -1 is found for Xe and a higher rate, 4 ± 1 × 10 -10 cm 3 s -1 , for Ar. Xe recombination is, therefore found to occur significantly more slowly than Ar recombination. \nThe results shown in Fig. 3a and 3b are for ionization of pure Xe or Ar gases. However, experiments were also conducted for various mixtures of Xe and Ar with H2 gas. Each of these species has different ionization energies, I (eV). Since H2 will have been much more abundant than Xe in Mars's early atmosphere ( 52,14 ), and that charge exchange between the ionization products of H2 ( I = 15 . 4 eV) and Xe ( I = 12 . 1 eV) has been proposed as a mechanism for \n<!-- image --> \n<!-- image --> \nFigure 3. Experimental results showing the relative ionisation efficiency and recombination rates of Xe and Ar during laser shock. (A) and (B) , represent the abundance of particular Xe/Ar states as a function of temperature for experiments run with pure Xe and Ar at a total pressure of ∼ 0 . 17bar at 1500ns following the laser pulse. Superscript star indicates neutral species, and superscript plus indicates charged or multiply charged species. The solid lines show the fit from simulations of the excited states, with shaded envelopes giving the 97.5% confidence interval from data uncertainty: simulation were based on modelling Saha ionisation equilibrium with varying temperature (see Materials and Methods for details). ( C ) and ( D ) Show the electron density variation with time for pure Xe/Ar mixtures following the initial laser pulse. Points were taken from experimental data, and solid curves show fits of equation 2 to the data to estimate kr , the recombination rate (full details in Materials and Methods). \n<!-- image --> \nionizing Xe ( 12 ), we look into the dependency of Xe and Ar ( I = 15 . 8 eV) thermal ionization yield and recombination rates on H2 concentrations. The results of these experiments (Fig. S2 and S3) show that increasing the H2 mixing ratio has only a small effect on the electron density and the recombination rate of Xe and Ar. This provides reassurance that in an early planetary atmosphere, whether it has a high mean molecular weight e.g., is CO2 ± H2O ± N2 dominated, or is H2-rich, Xe's ionization behaviour will be well described by the results for the pure gasses presented in Fig. 3. \nOverall, the laser shock experiments indicate that atmospheric plasma generated by hypervelocity impact will contain more ionized Xe than Ar and that the Xe will remain ionised for longer than the Ar. The ionization behaviour of Xe during impact compared to that of Ar therefore favour Xe's transport to high altitude whilst still ionized, and \n<!-- image --> \nhence its potential loss from the planet along magnetic field lines. We next take the parameters derived from these experiments and use them in a simple model of impact-driven Xe and Ar escape to investigate whether the observed isotopic fractionation of Xe can be generated and on what timescale (Fig. 1).", 'Modelling impact-driven Xe loss and fractionation': "A schematic of how we model impact-driven loss of Xe to occur is presented in Fig. 4. This treatment of impact-driven loss closely follows that developed by ref. 40, in which the atmosphere is ejected by planetesimal impact above a tangent plane with respect to the planet's surface. An atmospheric loss event involves an impactor of a given mass entering the atmosphere with some velocity. The impactor ejects all atmosphere, i.e., all gases equally, in a solid cone around its entry axis. A second wider cone extends around this central volume of the lost atmosphere; this outer cone is ionized and ejected upwards, but without sufficient velocity that it will be ballistically ejected from the planet's gravitational well. Instead, we model gas in this cone of material as being conditionally lost subject to the ionization state of the gases at the point they reach the homopause. \nBefore proceeding, we emphasise two important simplifications, or assumptions, of our model's approach. First, we assume the overall atmospheric pressure during impact bombardment is constant. In the context of explaining Xe loss and fractionation, this would correspond to a scenario in which impact processing of the martian surface releases major atmospheric constituents such as CO2 and H2O, by melting rock or destabilising ices, equivalent to the mass of atmosphere lost. Second, we assume that the impactors deliver no noble gases. This second assumption would correspond to either the impactors themselves containing negligible noble gases when compared to the amount removed (e.g., they may be differentiated and degassed), or, that the gases they do contain are volatilised and ejected during impact in the escaping cone (Fig. 4) along with the target atmosphere. We revisit both of these assumptions in the discussion. \nAs we saw in the previous section, Xe is ionized more completely and recombines more slowly than Ar. The key question for impact-driven loss and mass-dependent fractionation of Xe is then to quantify how much more efficiently Xe will be lost compared with the other noble gases (i.e., to explain the observations in Fig. 1A), and how much more efficiently will the lighter isotopes of Xe be lost than heavier isotopes (i.e., to explain the observations in Fig. 1B). In the framework of impact-driven escape, these efficiencies are set by the relative degrees of ionization of the noble gases at the point the impact plasma reaches the homopause. To calculate this our simplified model defines a single pressure in the atmosphere at which the plasma is made. From this point it calculates the mass of gas in the outer cone that will be ejected upwards but only conditionally lost, and how long it takes that gas to reach the homopause. Knowing the ionization yields (Fig. 3A & B) from impact and the recombination rates (Fig. 3C & D), equation (1) can be applied to convert ions back to neutral species during the time it takes to move the impact-induced plasma from its site of \nformation to the homopause. \nThe time to the homopause, therefore, emerges as key in setting the efficiency with which impacts can drive Xe (and Ar) loss. We assume that the ionized xenon and argon travel a distance h with a time of flight inversely proportional to the square root of their mean molecular weights. Ions, therefore, are mass segregated in the ionized gas during their time of flight and have more time to become neutralized and then settle back into the lower atmosphere, the heavier they are. We set the time of the flight to \nt fl = √ h 2 g √ µ ( X ) µ a , (3) \nwhere g is the gravitational acceleration, µ ( X ) is the mean molecular weight of species X, and µ a is the mean molecular weight of the atmosphere. As Xe is heavier than argon, equation (3) favours Xe retention and Ar loss. However, as we have already seen, the recombination rates and initial ionization fractions counteract this effect. The differential time of flight does, though, mean that the heavier isotopes of Xe may be separated from the lighter isotopes and preferentially retained (Fig. 1B). \nOnce the partially ionized gas has reached the homopause we assume that all species remaining ionized are lost. For the purposes of these calculations we are only tracking ionisation-dependent loss of Xe and Ar, but we also track the total atmospheric mass lost by direct ejection (central cone shown in Fig. 4). \nOur model considers the cone of atmosphere ejected past the homopause into the thermosphere, subtracting out the cone of atmosphere that is entirely ejected (Figure 4). The fraction of the non-ejected atmosphere that remains Xe + is removed from the atmosphere. This mass loss can be expressed as the depleted mass of Xenon ( M D ( Xe ) , kg): \nM D ( Xe ) = M 0 ( Xe ) f 0 ( Xe ) M T ( Xe ) M 0 ( Xe ) f 0 ( Xe ) + M T ( Xe ) , (4) \nwhere f 0 ( Xe ) is the initial mixing ratio of Xe below the homopause, M T ( Xe ) (kg) is the total mass of Xe below the homopause, and: \nM 0 ( Xe ) = 2 N 0 β -2 ( µ ( Xe ) µ a ) χ 0 ( Xe + ) e -Λ r ρ imp R 3 0 . (5) \nχ 0 ( Xe + ) is the initial fraction of Xe that is ionized, µ a is the mean molecular weight of the atmosphere, µ ( Xe ) is the mean molecular weight of Xenon, ρ imp (g cm -3 ) is the mass density of the impactor, and N 0 and R 0 (km) are the \nnumber and size of impactors found by integrating the impactor size distribution: \ndN = β N 0 ( R 0 r imp ) β dr imp r imp , (6) \nwhere r imp (km) is the size of a given impactor and β > 2 is an integer that sets the power-law of the size distribution. \nThe quantities R 0 and Λ r are parameterized in terms of the free and experimentally-constrained parameters of the height of the homopause ( h , km), the radius of the planet ( Rp , expressed in Earth radii, R ⊕ ), surface gravity ( g , m s -2 ), surface pressure ( p , bar), surface temperature ( T , K), ionization fraction ( fe ), and recombination rate ( kr , cm 3 s -1 ), as so: \nR 0 ( km ) = 28 . 45 ( µ a ) 1 / 3 ( 1 gcm -3 ρ imp ) 1 / 3 ( h 1km ) 1 / 6 ( R ⊕ Rp ) 1 / 6 ( 1 ms -2 g )( p 1bar ) 1 / 3 ( T 300K ) 2 / 3 (7) \nΛ r = ( 5 . 4 × 10 20 cm -3 s ) fekr ( p 1bar )( 300K T )( h 1km ) 1 / 2 ( 1 ms -2 g ) 1 / 2 ( µ ( Xe ) µ a ) 1 / 2 . (8) \nDetails of this model are given in the SI. \nThe forward model described above is combined with a Bayesian Monte Carlo inversion routine to estimate the values for the parameters of interest (see Table 1 for a complete list of parameters and the priors they were assigned in the modelling) by fitting the model to the Xe mass fractionation record (Fig. 5A and ref. 11). Bayesian inversion is performed using MultiNest, a Monte Carlo nested sampling algorithm ( 53,54 ) via pyMultiNest ( 55 ). The model includes eight parameters, four of which are included as nuisance parameters to be marginalised over for error propagation. Four parameters are of key geological interest as they speak to the solar system and martian history, and these are discussed in more detail below. The value these parameters take when matching the observations is, therefore, a key test of the model's validity; they offer potential for testing against independent geological records. \nThe first of these parameters is atmospheric surface pressure, p surf (bar). This defines the mass of the martian atmosphere, and therefore it's starting Xe inventory (via the prescribed Xe mixing ratio). In the context of impact-driven escape, more Xe in the atmosphere initially will require more impacts to remove. The surface pressure of the martian atmosphere is also of direct relevance to the planet's climate history. This leads to the second parameter, C peak , a factor modifying the exponent of the mass flux of impactors onto the planet. A highly simplified prescription for impactor flux is used, in which a constant background flux is perturbed by a single peak, of height 10 C peak above background (see Materials and Methods for full description). The timing of this peak in impactor flux is controlled by the third parameter, t peak , and its width by the fourth parameter ∆ t . As we will see, in this simple model the time of the peak impact flux is a key parameter in fitting the temporal evolution of martian Xe loss and fractionation ( 11 ) \nFigure 4. Diagram of xenon impact ejection and escape. Cross-sections are shown for the two cones of ejected (centre) and ionised and conditionally lost atmosphere (either side) for our model. Atmosphere in the solid grey cone subtended by θ esc is ejected from the atmosphere by the impact, is unbound by gravity and moved into the exosphere > r exo (km). Atmosphere in the gradient-grey cone subtended by θ th passes the homopause and reaches the thermosphere r th (km). Species that remain ionised at this point (e.g., Xe and Ar) will be confined by the stellar magnetic field and escape the atmosphere. \n<!-- image --> \nThe net effect of the range of t peak and ∆ t permitted in the inversion (Table 1) is for there to be a declining impactor flux over the first ∼ 1Gyr of solar system history. The integrated impactor flux gives the total mass gained by the planet during impact bombardment, and can in principle be tested against cratering records and highly siderophile element (HSE) derived estimates of post-core formation mass accretion. \nThe fit of the model to the time-resolved record of Xe isotopic fractionation data are shown in Fig. 5A. The red line records the solution calculated from the median of the parameters' posterior distributions and provides a close fit to the ∼ 4 Gyr of Mars's Xe mass fractionation. In particular, the model is able to reproduce the rapid in-growth of mass-fractionated Xe in Mars's atmosphere in the first few hundred million years of its history. \nThe single modern constraint on the factor by which the atmospheric Xe/Ar ratio has been decreased below its \nTable 1. Parameters estimated in the Bayesian inversion, including their prior values and distributions . Priors are prescribed as uniform ('U'), log uniform ('LU'), or normally ('N') distributed. \nFigure 5. The evolution of martian Xe during impact bombardment. (A) , The isotopic fractionation of Xe through time in the model, compared to modern and ancient estimates of the atmospheric Xe isotope fractionation ( 56,6,10,11 ). (B) , The evolution of the Xe/Ar ratio during impact-driven loss. Impact bombardment in the best fitting model leaves the Xe/Ar ratio below modern values, allowing for subsequent solar-wind driven loss and fractionation of Ar ( 57 ) to drive the atmosphere to modern values. ( C ), The cumulative mass delivered to Mars in impactors, compared to the estimate of accreted mass post-core closure from HSE's ( 58,59 ). ( D ) The impact rate histories of solutions consistent with the Xe mass fractionation data. In all cases, the best fitting model (maximum likelihood estimate) is shown in a dark red solid line. \n<!-- image --> \ninitial value, ( 130 Xe / 36 Ar ( t = 0 ) 130 Xe / 36 Ar ( t ) ) , was not included in the model fit, however solutions range from around the modern atmospheric value to significantly higher degrees of fractionation (Fig. 5B; median ( 130 Xe / 36 Ar ( t = 0 ) 130 Xe / 36 Ar ( t = 4 . 5Gyr ) ) of ∼ 100). \nThis wide distribution on the level of decrease of the Xe/Ar ratio, yet tightly constrained Xe mass fractionation histories, comes from the sensitivity of the loss of Xe and Ar to their relative recombination rates; a factor that does not affect consideration of Xe isotopes alone, where a given recombination rate can be compensated for through other parameters. Importantly for the model's consistency with the modern atmosphere, subsequent loss processes may remove atmospheric Ar over Mars's history and thus lower an initially high ( 130 Xe / 36 Ar ( t = 0 ) 130 Xe / 36 Ar ( t ) ) down to the observed value ( 60 ). Given the median ( 130 Xe / 36 Ar ( t = 0 ) 130 Xe / 36 Ar ( t = 4 . 5Gyr ) ) predicted in our model, a loss of ∼ 90% of the post-impact Ar inventory would evolve Mars's atmosphere to its modern Xe/Ar ratio. This is consistent with independent models explaining the modern 36 Ar/ 38 Ar ratio of the martian atmosphere (a ratio affected at less than the part per million level by impact bombardment), which suggest up to 95% of Mars's atmospheric argon inventory may have been removed gradually over its history ( 57 ). \nFigure 6. Posterior distributions of parameters consistent with highly siderophile element constraints on \n<!-- image --> \naccreted mass. (A) The posterior distribution of accreted mass to Mars ( M accreted ( M Mars %)), following impact bombardment sufficient to produce the observed Xe isotopic fractionation. Approximately 45% of solutions, given the choice of priors, fall at or below the ∼ 1% ( M Mars) limit on post-core closure mass addition inferred from \nHSE's ( 58,59 ), these are shown in light orange. In dark red are the solutions where less than 1% the mass of Mars was accreted, and the observed Xe isotopic fractionation was still produced. (B) The posterior distribution of the onset of impact bombardment-driven isotopic fractionation. The grey line shows the uniform prior, and the histograms show the resultant prior (light orange) and subset of the posterior that matches the accreted mass constraint (dark red). (C) The posterior distribution of surface pressure during the impact bombardment that produces the observed Xe isotopic mass fractionation (light orange). Dark red identifies the results consistent with constraints on accreted mass.", 'Discussion': "Our experimental and numerical results demonstrate that impact bombardment could have driven the preferential loss and isotopic fractionation of Xe in the atmosphere of Mars. Critically, this could only have occurred early in the lifetime of the solar system, whilst the mass flux of impactors onto planets was still significant: the ability of the models to match the observed rapid in-growth of Xe isotope fractionation followed by stasis (Fig. 5C), directly emerges from this history. Specifically, the models require an early peak in impactor flux followed by rapid decline in impact rate over the first few hundred million years of solar system history ( 61 ). \nTwo important assumptions the model makes are of a constant atmospheric pressure during bombardment and no Xe delivery. Atmospheric bombardment is expected to erode atmospheres ( 40 ), inferred from the same theoretical approach we take to predict Xenon escape. This erosion can be counter-balanced by delivery of volatiles ( 62 ), or by degassing from the magma generated at the impact site. For simplicity, we assume that these different sources (delivery, impact-generated magma degassing) and sinks (impact erosion), when summed over all species lead to a stable, albeit low pressure, atmosphere, and that the sources provide negligible Xenon (the convergent state of an atmosphere under bombarded ( 62 )). Below we consider the effect of relaxing each of these assumptions. \nAllowing changes in the atmospheric surface pressure can enhance impact-driven isotope fractionation. If the atmosphere is being eroded, then the Xenon depletion will be increased because, as we have shown (Fig. 6C), Xenon ionization and escape is more efficient when the surface atmospheric pressure is lower. If the atmosphere is growing, due to efficient delivery or efficient magma degassing, then our mechanism will be frustrated. Further work to constrain these sources and sinks, and explore potentially observable implications of these constraints, will be needed to inform whether and how this assumption should be relaxed. However, the evidence for a low pressure atmosphere on early Mars suggests significant atmospheric growth during this bombardment epoch was unlikely (see discussion below). \nWe consider the assumption of no Xe delivery during bombardment a reasonable starting point for modelling the following reasons. \n- 1. Impactor Xe will be lost along with the cone of atmosphere it is interacting with. The efficiency of impactor Xe partitioning between loss and delivery is difficult to predict with our present state of knowledge, because it will depend on the dynamics of the impact event. If the delivered Xenon is included within the cone of matter that is ejected from the atmosphere ( 40 ), then it will necessarily not contribute to the atmospheric budget. A complete physical model would be required to calculate the partitioning of impactor Xe between loss and delivery, which is beyond the scope of this work, but qualitatively has the effect of diminishing the potential of Xe delivery compared with the impactor's initial inventory; \n- 2. For a surface atmospheric pressure of 0 . 1bar, Xenon will only be added if the delivery brings more Xenon than 10 -11 g/g (i.e., more Xe contained in the meteorite than is lost above the homopause; ( 63 ). If we consider the higher surface pressures permitted by the model (Fig. 6C), then even higher impactor Xe concentrations are required to perturb the atmospheric Xe budget. By comparison to a conservative 10 -11 g/g value though, carbonaceous chondrites bracket this, having a range of Xenon concentrations between ( 0 . 01 -2 . 5 ) × 10 -11 g Xe/g total. If the more Xenon-rich Carbonaceous chondrites represent the average impactor composition during the tail end of accretion, then Xenon will be slowly increased by impact delivery. Ordinary chondrites, in contrast, have less than 10 -11 g/g Xenon on average ( 64 ), and so will not contribute enough Xenon to substantially change our results. E-Chondrites have even less Xenon, on average 5 × 10 -13 g/g ( 65 ). Evidence points to Mars's late accretion being dominated by ordinary- and enstatite chondrite-like material ( 66 ), suggesting Xe delivery would have been unimportant even before consideration of point (1) above; and,\n- 3. Mars's atmosphere is constrained to have started with a solar-like Xe isotopic composition ( 8 ), from which subsequent atmospheric evolution occurred. This observation rules out late delivery as a prominent contributor to Mars's atmospheric Xe, because this would have delivered non-solar chondritic Xe - impacts must have acted primarily to sculpt Mars's initial solar-Xe inventory, rather than add to it. \nIn addition to impactor delivery, impact-generated magma degassing is unlikely to provide significant Xenon to the atmosphere: Xenon's insolubility in magmas ( 20 ) makes it likely that the Xenon concentration in the Mars's crust/interior will be much lower than the Xenon abundance in the atmosphere. Overall then, we consider it most likely that Mars's Xe inventory during this bombardment epoch was dominated by loss and fractionation, not delivery or interior outgassing. \nAn important aspect of impact bombardment as a means of preferentially driving Xe loss and fractionation is that it does not depend on background H2 in a planet's atmosphere. This is in contrast to previous work for Earth and Mars ( 12,11 ), in which Xe ionisation by charge exchange and transport through the atmosphere is linked to atmospheric hydrogen content: such that in the case of the Earth the cessation of Xe fractionation is hypothesised to be linked to the oxidation of the atmosphere ( 12 ). Our experimental results showed limited sensitivity of Xe ionisation and recombination to the presence of background H2 and the impact mechanism directly provides ballistic transport of Xe through the atmosphere to the ionosphere. Hence, impact-driven fractionation and Xe loss does not make specific predictions, nor have specific requirements for, the evolution of the background atmospheric composition. A corollary of this is that end of Xe fractionation is also separated from atmospheric H2 evolution, instead presumably being linked to declining bombardment flux or a waning of the martian magnetic field. The latter scenario cannot explain when Xe fractionation ceases, because ALH84001, which records a Xe isotopic composition close to the modern martian \natmosphere also preserves a magnetic field ( 13, 11 ). However, this serves to emphasise that for both the model we propose, and those previously considered ( 11 ), the presence of a martian magnetic field is critical in guiding ionised Xe from the planet whilst leaving non-ionised gases behind. Future insights into the history of the martian dynamo will therefore provide tests of these models of Xe escape. \nA key geological prediction impact-driven Xe fractionation makes is of the mass of late delivery to Mars-. Estimates from HSE abundances in Mars's mantle, inferred from martian meteorites, place the mass of material delivered post-core closure to be ∼ 0 . 4 -1% of the mass of the planet ( 58,59 ). Fig. 5C calculates the integrated mass delivered over time for the median solution (red line) and a sample of the posteriors (light orange). Many of these solutions fall above the window of permissible mass delivery; however, a significant fraction ( ∼ 45%) have parameters governing the loss and fractionation of Xe that allow for smaller impactor mass fluxes. Impact-driven loss is, therefore, in principle, able to operate within the independent constraints on mass delivery to Mars to explain the history of its atmospheric Xe mass fractionation. \nWe can look more closely at the properties of those simulations that successfully match the mass-delivery constraint to understand the broader requirements for conditions in the early solar system and on Mars if impact bombardment is to explain the Xe fractionation. Fig. 6 highlights the posterior distributions for the mass accreted to Mars over the early period of intense bombardment ( M accreted ), the timing of the impact peak ( t peak ), and the atmospheric surface pressure during the bombardment interval ( p surf ). For Models accreting < 1% the mass of Mars ( ∼ 45% of simulations), in line with the HSE observations, the posteriors are shown in dark orange. The key insight from these posterior distributions is that a narrow timing of the onset of Xe fractionation and lower atmospheric pressure on Mars ( < 1bar) is favoured for successful model runs. The reasons for these parameter values being favoured are that (1) too early an onset of atmospheric escape leads to too much Xe isotopic fractionation occurring before the ∼ 4 . 4Ga NWA11220-derived constraint on martian atmospheric Xe ( 11 ), and (2) increased atmospheric pressure suppresses the degree of Xe fractionation observed, as a smaller fraction of the initial Xe pool ends up processed by impacts. Combined, these aspects of the impact-drive loss scenario favour Mars's atmosphere beginning to record fractionation from ∼ 200 Myr after the birth of the solar system, with a low pressure atmosphere at this time. \nEarly peaks in impactor rate have long been predicted by dynamical models seeking to match the architecture of the solar system ( 67,28 ). However, there has been much debate over the timing of such events. Measurements of Xe isotopes in martian meteorites NWA11220 and ALH84001, dated at ∼ 4 . 4 and 4 . 16Ga respectively, tightly constrain the timing of impact-driven fractionation. Between these two measurements, mass fractionation of Xe isotopes increases from ∼ 1 . 6 to ∼ 4 . 4% / amu, essentially the modern value of the Mars atmosphere ( 6, 10, 11 ). In our simplified model this epoch of Xe fractionation must be matched directly by a peak of impact bombardment. However, more complex \nmodels that couple a time evolving atmospheric mass with monotonically declining impactor flux (an 'accretion tail' scenario ( 29 )) could likely also be reconciled to the data: such models have been previously suggested for Mars's atmospheric evolution ( 39,68 ). In light of our impact-based description of the Xe isotope data, accretion-tail scenarios would need to be paired with an atmospheric mass stabilising at ∼ 4 . 35Ga to an initial solar composition from which isotopic fractionation could in-grow. \nEven if in principle the model cannot separate an accretion tail scenario from am impact peak, our results do place an upper limit on how long after solar system birth any giant planet instability and associated bombardment flux can occur. Xe fractionation, and therefore intense impact sculpting of the atmosphere, needs to end by 300-400Myr after CAI formation, else martian Xe would experience a more protracted period of isotopic evolution than it exhibits. This is an important bound on the timing of giant planet instability, given the difficulty of directly probing impact history this far back in time ( 69,70 ). \nAn important feature of impact-driven Xe loss and its relation to Mars's impact chronology is that it is most efficient for small impactors, those just above the threshold to eject atmosphere ( 40 ). Larger impactors, such as those responsible for large scale basin formation (e.g., Hellas basin) are both less efficient at removing atmosphere per unit mass, and eject all atmosphere above the tangent plane to the planet ( 40 ), thereby being less efficient at fractionating the atmosphere. Whereas it is the differential loss of atmosphere that is possible with smaller impactors that drives Xe fractionation. In this sense, Mars's history of Xe fractionation, if driven by impacts, is a complementary archive of impact bombardment to the surface geological record: whereas the latter best preserves and age dates ( 71,72 ) the large events, the former integrates the effects of the much more numerous small events. How these large basin forming impactors could have affected atmospheric Xe is through remelting of the crust and mantle, processes which may release trapped Xe. In fact, a late (i.e., young) age of these large basin forming events may help explain a peculiar feature of the martian Xe isotopic record: that in our quantification of the Xe fractionation (Fig. 5A), the martian meteorites record a peak fractionation at ALH84001 followed by a decline between then and the modern atmosphere (and, albeit with lower certainty, Nahkla). Large impacts liberating Xe from the martian mantle, or delivering some solar-like Xe, could explain this slight reversal in fractionation trend. \nAn impact solution to Mars's fractionated Xe also has implications for its climate. Our results favour a low pressure atmosphere, < 1bar, remaining after formation and impact bombardment (Fig. 6C). This result is consistent with independent constraints on Mars's early atmospheric pressure ( 73,68 ), and the observation of sulfur mass-independent isotopic fractionation in martian meteorite NWA11220 ( 74,75 ), which favours low atmospheric pressures. Importantly, a tenuous atmosphere is consistent with a climate history of punctuated warmth and episodes of surface liquid water, with the later tail of impacts contributing to this stochasticity ( 14 ). \nImpact-driven loss and fractionation of Xe may also be tested by NASA's planned DAVINCI mission to Venus ( 76 ), which has as a core science aim measurement of the heavy noble gases in the planet's atmosphere. If modern Venus is representative of the planet's past, then its Xe isotopes should not show preferential fractionation compared to Kr, and the Kr to Xe ratio should be sub-chondritic rather than super-chondritic as on Mars. This is because Venus's massive atmosphere would stifle the efficiency of impact-driven fractionation of Xe: First, we have shown that for a thick atmosphere too little Xe is removed to perturb the existing inventory; and second, both the model presented here and that of ref. 12 require a magnetic field to channel ions away from the planet, the thick atmosphere and resulting hot surface and slow mantle cooling may have suppressed Venus's dynamo for its entire history preventing this process. Conversely, in this paradigm, if DAVINCI's measurements do evidence preferential Xe loss and fractionation, then it may point to a more temperate early Venus (e.g., as propsed by ref. 77). \nThe history of Xe on Mars has importance for Earth's own Xe depletion. Our experiments and calculations show the potential of impactors to fractionate Xe early in a planet's history. This impact-driven fractionation occurs too early to explain the slow ingrowth of Xe mass fractionation seen in Earth's atmosphere ( 4 ). However, recent results have shown the presence of a more ancient history of Xe fractionation in Earth's deep mantle ( 15 ). This signal may point to Earth having once experienced similar impact loss processes to Mars, albeit resolvable only in mantle long hidden from subsequent volatile addition and recycling.", 'Apparatus': "The plasma UV-ViS measurements were performed inside a vacuum-sealed cylindrical glass cell. The sample was contained within the cell using three diagnostic quartz windows and an in-house SwagelokTM gas-vacuum handling system. The Nd:YAG 1064 nm laser, capable of delivering a maximum energy of 850 mJ, was employed to generate a plasma spark at the vessel's center. This laser radiation was focused using a coated plano-convex quartz lens (= 1.5 cm, f = 10.5 cm). The emission spectra of the laser-induced plasma were then captured by the ESA 4000 Echelle spectrograph (LLA Instruments GmbH, Germany) using a fiber optic cable. A photodiode was attached to detect the laser spark radiation. \nThe laser pulses were cumulated at a repetition frequency of 10 Hz to simulate an impact shock wave. The observation gate delay was adjusted to varying values to facilitate time-resolved screening while maintaining a constant gate width of 500 ns. The time gating was controlled using ESAWIN software (version 14.3.0). \nA vacuum pump and pressure gauge from Pfeiffer Vacuum Austria GmbH were employed to manage the gas flow and measure pressure within the cell. A schematic of the experimental setup can be found in the Supplementary Materials (Fig. S1).", 'Experimental Conditions': 'The vessel was filled with pure noble gases and their mixtures under a series of nominal pressures and concentrations investigated in this study: pressures of 40 - 700 Torr and hydrogen concentration ranges from 0 up to 90 %. For any measurements, certified gas samples, i.e., respectively 5.6 Linde Gas argon, 5.0 Linde Gas xenon, and 6.0 Linde Gas di-hydrogen were used. The spectra have been recorded by an Echelle spectrograph with delays of 10, 500, 1000, 1500, 2000, 2500, and 3000 ns with the gate width of 100 ns.', 'Experimental data acquisition and processing': 'Approximately 250 spectra were recorded in the wavelength range between 200 to 750 nm. A simple pre-processing procedure ( 51 ) was applied to all spectra to obtain basic plasma diagnostics. This analysis was performed by in-house programmed scripts in PYTHON-NUMPY, and PYTHON-SCIPY. Theoretical values for the calculations were extracted from the NIST database ( 78 ).', 'Acknowledgements': 'William Cassata is thanked for his guidance in interpreting the data reporting the noble gas isotope composition of the martian atmosphere. Guillaume Avice is thanked for the discussion of the terrestrial Xe record. This work is part of a research series funded by grant no. 21-11366S of the Czech Science Foundation', 'Funding': 'This work is part of a research series funded by grant no. 21-11366S of the Czech Science Foundation.', 'Author Contributions': 'All authors discussed the experimental plan. HS MF designed experiments and conducted them with VL PK and LP. All authors contributed to the drafting and editing of the manuscript.', 'Competing interests': 'All authors declare that they have no competing interests.', 'Data availability': 'All data needed to evaluate the conclusions in the paper are present in the paper and the Supplementary Materials.', 'References': "- 1. R. 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2024PhRvL.133g1501C

On 9 October 2022 the Large High Altitude Air Shower Observatory LHAASO reported the observation of the very early TeV afterglow of the brightestofalltime gammaray burst 221009A recording the highest photon statistics in the TeV band ever obtained from a gammaray burst. We use this unique observation to place stringent constraints on the energy dependence of the speed of light in vacuum a manifestation of Lorentz invariance violation LIV predicted by some quantum gravity QG theories. Our results show that the 95 confidence level lower limits on the QG energy scales are ESUBQG 1SUBgt10 times the Planck energy ESUBPlSUB for the linear LIV effect and ESUBQG 2SUBgt6 10SUP8SUPESUBPlSUB for the quadratic LIV effect. Our limits on the quadratic LIV case improve previous best bounds by factors of 57.

2024-08-01T00:00:00Z

['10.1103/PhysRevLett.133.071501', 'arXiv:2402.06009', '2024arXiv240206009T', '2024PhRvL.133g1501C', '10.48550/arXiv.2402.06009']

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

Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A

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

https://arxiv.org/pdf/2402.06009.pdf

{'Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A': "Zhen Cao, 1, 2, 3 F. Aharonian, 4, 5 Axikegu, 6 Y.X. Bai, 1, 3 Y.W. Bao, 7 D. Bastieri, 8 X.J. Bi, 1, 2, 3 Y.J. Bi, 1, 3 W. Bian, 9 A.V. Bukevich, 10 Q. Cao, 11 W.Y. Cao, 12 Zhe Cao, 13, 12 J. Chang, 14 J.F. Chang, 1, 3, 13 A.M. Chen, 9 E.S. Chen, 1, 2, 3 H.X. Chen, 15 Liang Chen, 16 Lin Chen, 6 Long Chen, 6 M.J. Chen, 1, 3 M.L. Chen, 1, 3, 13 Q.H. Chen, 6 S. Chen, 17 S.H. Chen, 1, 2, 3 S.Z. Chen, 1, 3 T.L. Chen, 18 Y. Chen, 7 N. Cheng, 1, 3 Y.D. Cheng, 1, 2, 3 M.Y. Cui, 14 S.W. Cui, 11 X.H. Cui, 19 Y.D. Cui, 20 B.Z. Dai, 17 H.L. Dai, 1, 3, 13 Z.G. Dai, 12 Danzengluobu, 18 X.Q. Dong, 1, 2, 3 K.K. Duan, 14 J.H. Fan, 8 Y.Z. Fan, 14 J. Fang, 17 J.H. Fang, 15 K. Fang, 1, 3 C.F. Feng, 21 H. Feng, 1 L. Feng, 14 S.H. Feng, 1, 3 X.T. Feng, 21 Y. Feng, 15 Y.L. Feng, 18 S. Gabici, 22 B. Gao, 1, 3 C.D. Gao, 21 Q. Gao, 18 W. Gao, 1, 3 W.K. Gao, 1, 2, 3 M.M. Ge, 17 L.S. Geng, 1, 3 G. Giacinti, 9 G.H. Gong, 23 Q.B. Gou, 1, 3 M.H. Gu, 1, 3, 13 F.L. Guo, 16 X.L. Guo, 6 Y.Q. Guo, 1, 3 Y.Y. Guo, 14 Y.A. Han, 24 M. Hasan, 1, 2, 3 H.H. He, 1, 2, 3 H.N. He, 14 J.Y. He, 14 Y. He, 6 Y.K. Hor, 20 B.W. Hou, 1, 2, 3 C. Hou, 1, 3 X. Hou, 25 H.B. Hu, 1, 2, 3 Q. Hu, 12, 14 S.C. Hu, 1, 3, 26 D.H. Huang, 6 T.Q. Huang, 1, 3 W.J. Huang, 20 X.T. Huang, 21 X.Y. Huang, 14 Y. Huang, 1, 2, 3 X.L. Ji, 1, 3, 13 H.Y. Jia, 6 K. Jia, 21 K. Jiang, 13, 12 X.W. Jiang, 1, 3 Z.J. Jiang, 17 M. Jin, 6 M.M. Kang, 27 I. Karpikov, 10 D. Kuleshov, 10 K. Kurinov, 10 B.B. Li, 11 C.M. Li, 7 Cheng Li, 13, 12 Cong Li, 1, 3 D. Li, 1, 2, 3 F. Li, 1, 3, 13 H.B. Li, 1, 3 H.C. Li, 1, 3 Jian Li, 12 Jie Li, 1, 3, 13 K. Li, 1, 3 S.D. Li, 16, 2 W.L. Li, 21 W.L. Li, 9 X.R. Li, 1, 3 Xin Li, 13, 12 Y.Z. Li, 1, 2, 3 Zhe Li, 1, 3 Zhuo Li, 28 E.W. Liang, 29 Y.F. Liang, 29 S.J. Lin, 20 B. Liu, 12 C. Liu, 1, 3 D. Liu, 21 D.B. Liu, 9 H. Liu, 6 H.D. Liu, 24 J. Liu, 1, 3 J.L. Liu, 1, 3 M.Y. Liu, 18 R.Y. Liu, 7 S.M. Liu, 6 W. Liu, 1, 3 Y. Liu, 8 Y.N. Liu, 23 Q. Luo, 20 Y. Luo, 9 H.K. Lv, 1, 3 B.Q. Ma, 28 L.L. Ma, 1, 3 X.H. Ma, 1, 3 J.R. Mao, 25 Z. Min, 1, 3 W. Mitthumsiri, 30 H.J. Mu, 24 Y.C. Nan, 1, 3 A. Neronov, 22 L.J. Ou, 8 P. Pattarakijwanich, 30 Z.Y. Pei, 8 J.C. Qi, 1, 2, 3 M.Y. Qi, 1, 3 B.Q. Qiao, 1, 3 J.J. Qin, 12 A. Raza, 1, 2, 3 D. Ru ff olo, 30 A. S'aiz, 30 M. Saeed, 1, 2, 3 D. Semikoz, 22 L. Shao, 11 O. Shchegolev, 10, 31 X.D. Sheng, 1, 3 F.W. Shu, 32 H.C. Song, 28 Yu.V. Stenkin, 10, 31 V. Stepanov, 10 Y. Su, 14 D.X. Sun, 12, 14 Q.N. Sun, 6 X.N. Sun, 29 Z.B. Sun, 33 J. Takata, 34 P.H.T. Tam, 20 Q.W. Tang, 32 R. Tang, 9 Z.B. Tang, 13, 12 W.W. Tian, 2, 19 C. Wang, 33 C.B. Wang, 6 G.W. Wang, 12 H.G. Wang, 8 H.H. Wang, 20 J.C. Wang, 25 Kai Wang, 7 Kai Wang, 34 L.P. Wang, 1, 2, 3 L.Y. Wang, 1, 3 P.H. Wang, 6 R. Wang, 21 W. Wang, 20 X.G. Wang, 29 X.Y. Wang, 7 Y. Wang, 6 Y.D. Wang, 1, 3 Y.J. Wang, 1, 3 Z.H. Wang, 27 Z.X. Wang, 17 Zhen Wang, 9 Zheng Wang, 1, 3, 13 D.M. Wei, 14 J.J. Wei, 14 Y.J. Wei, 1, 2, 3 T. Wen, 17 C.Y. Wu, 1, 3 H.R. Wu, 1, 3 Q.W. Wu, 34 S. Wu, 1, 3 X.F. Wu, 14 Y.S. Wu, 12 S.Q. Xi, 1, 3 J. Xia, 12, 14 G.M. Xiang, 16, 2 D.X. Xiao, 11 G. Xiao, 1, 3 Y.L. Xin, 6 Y. Xing, 16 D.R. Xiong, 25 Z. Xiong, 1, 2, 3 D.L. Xu, 9 R.F. Xu, 1, 2, 3 R.X. Xu, 28 W.L. Xu, 27 L. Xue, 21 D.H. Yan, 17 J.Z. Yan, 14 T. Yan, 1, 3 C.W. Yang, 27 C.Y. Yang, 25 F. Yang, 11 F.F. Yang, 1, 3, 13 L.L. Yang, 20 M.J. Yang, 1, 3 R.Z. Yang, 12 W.X. Yang, 8 Y.H. Yao, 1, 3 Z.G. Yao, 1, 3 L.Q. Yin, 1, 3 N. Yin, 21 X.H. You, 1, 3 Z.Y. You, 1, 3 Y.H. Yu, 12 Q. Yuan, 14 H. Yue, 1, 2, 3 H.D. Zeng, 14 T.X. Zeng, 1, 3, 13 W. Zeng, 17 M. Zha, 1, 3 B.B. Zhang, 7 F. Zhang, 6 H. Zhang, 9 H.M. Zhang, 7 H.Y. Zhang, 1, 3 J.L. Zhang, 19 Li Zhang, 17 P.F. Zhang, 17 P.P. Zhang, 12, 14 R. Zhang, 12, 14 S.B. Zhang, 2, 19 S.R. Zhang, 11 S.S. Zhang, 1, 3 X. Zhang, 7 X.P. Zhang, 1, 3 Y.F. Zhang, 6 Yi Zhang, 1, 14 Yong Zhang, 1, 3 B. Zhao, 6 J. Zhao, 1, 3 L. Zhao, 13, 12 L.Z. Zhao, 11 S.P. Zhao, 14 X.H. Zhao, 25 F. Zheng, 33 W.J. Zhong, 7 B. Zhou, 1, 3 H. Zhou, 9 J.N. Zhou, 16 M. Zhou, 32 P. Zhou, 7 R. Zhou, 27 X.X. Zhou, 1, 2, 3 X.X. Zhou, 6 B.Y. Zhu, 12, 14 C.G. Zhu, 21 F.R. Zhu, 6 H. Zhu, 19 K.J. Zhu, 1, 2, 3, 13 Y.C. Zou, 34 and X. Zuo 1, 3 \n(The LHAASO Collaboration) ∗ \n1 Key Laboratory of Particle Astrophysics & Experimental Physics Division & Computing Center, Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China 2 University of Chinese Academy of Sciences, 100049 Beijing, China 3 Tianfu Cosmic Ray Research Center, 610000 Chengdu, Sichuan, China 4 Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, 2 Dublin, Ireland 5 Max-Planck-Institut for Nuclear Physics, P.O. Box 103980, 69029 Heidelberg, Germany 6 School of Physical Science and Technology & School of Information Science and Technology, Southwest Jiaotong University, 610031 Chengdu, Sichuan, China 7 School of Astronomy and Space Science, Nanjing University, 210023 Nanjing, Jiangsu, China 8 Center for Astrophysics, Guangzhou University, 510006 Guangzhou, Guangdong, China 9 Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China 10 Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russia 11 Hebei Normal University, 050024 Shijiazhuang, Hebei, China 12 University of Science and Technology of China, 230026 Hefei, Anhui, China 13 State Key Laboratory of Particle Detection and Electronics, China 14 Key Laboratory of Dark Matter and Space Astronomy & Key Laboratory of Radio Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, 210023 Nanjing, Jiangsu, China 15 Research Center for Astronomical Computing, Zhejiang Laboratory, 311121 Hangzhou, Zhejiang, China 16 Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 200030 Shanghai, China 17 School of Physics and Astronomy, Yunnan University, 650091 Kunming, Yunnan, China 18 Key Laboratory of Cosmic Rays (Tibet University), Ministry of Education, 850000 Lhasa, Tibet, China 19 National Astronomical Observatories, Chinese Academy of Sciences, 100101 Beijing, China \n21 \n20 School of Physics and Astronomy (Zhuhai) & School of Physics (Guangzhou) \n& Sino-French Institute of Nuclear Engineering and Technology (Zhuhai), Sun Yat-sen University, 519000 Zhuhai & 510275 Guangzhou, Guangdong, China Institute of Frontier and Interdisciplinary Science, Shandong University, 266237 Qingdao, Shandong, China 22 APC, Universit'e Paris Cit'e, CNRS / IN2P3, CEA / IRFU, Observatoire de Paris, 119 75205 Paris, France 23 Department of Engineering Physics, Tsinghua University, 100084 Beijing, China 24 School of Physics and Microelectronics, Zhengzhou University, 450001 Zhengzhou, Henan, China 25 Yunnan Observatories, Chinese Academy of Sciences, 650216 Kunming, Yunnan, China 26 China Center of Advanced Science and Technology, Beijing 100190, China 27 College of Physics, Sichuan University, 610065 Chengdu, Sichuan, China 28 School of Physics, Peking University, 100871 Beijing, China 29 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, 530004 Nanning, Guangxi, China 30 Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand 31 Moscow Institute of Physics and Technology, 141700 Moscow, Russia 32 Center for Relativistic Astrophysics and High Energy Physics, School of Physics and Materials Science & Institute of Space Science and Technology, Nanchang University, 330031 Nanchang, Jiangxi, China 33 National Space Science Center, Chinese Academy of Sciences, 100190 Beijing, China 34 School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China \n(Dated: August 27, 2024) \nOn October 9, 2022, the Large High Altitude Air Shower Observatory (LHAASO) reported the observation of the very early TeV afterglow of the brightest-of-all-time GRB 221009A, recording the highest photon statistics in the TeV band ever obtained from a Gamma-ray burst. We use this unique observation to place stringent constraints on an energy dependence of the speed of light in vacuum, a manifestation of Lorentz invariance violation (LIV) predicted by some quantum gravity (QG) theories. Our results show that the 95% confidence level lower limits on the QG energy scales are E QG , 1 > 10 times of the Planck energy E Pl for the linear, and E QG , 2 > 6 × 10 -8 E Pl for the quadratic LIV e ff ects, respectively. Our limits on the quadratic LIV case improve previous best bounds by factors of 5-7.", 'I. INTRODUCTION': 'Lorentz invariance, the fundamental symmetry of Einstein\'s relativity, has withstood strict tests over the past century [1]. However, deviations from Lorentz invariance at energies approaching the Planck scale E Pl = p ℏ c 5 / G ≃ 1 . 22 × 10 19 GeV are predicted in many quantum gravity (QG) theories seeking to unify quantum theory and general relativity [2-15]. Although any signals of Lorentz invariance violation (LIV) are expected to be very tiny at attainable energies ≪ E Pl, they can increase with energy and accumulate to detectable levels over large propagation distances. Astrophysical observations involving high-energy radiation and long distances are therefore suitable for performing sensitive tests of Lorentz invariance. \nOne of the manifestations of LIV can be characterised as energy-dependent modifications to the photon dispersion relation in vacuum [16]: \nE 2 ≃ p 2 c 2 1 -∞ X n = 1 s E E QG , n ! n , (1) \nwhere E and p are the energy and momentum of a photon, s = ± 1 is the \'sign\' of the LIV e ff ect, corresponding to the \n\'subluminal\' or \'superluminal\' scenarios, and E QG , n denotes the hypothetical QG energy scale. Since the sum is dominated by the lowest-order term of the series at small energies E ≪ E QG , n , only the first two leading terms ( n = 1 or n = 2) are of interest for independently LIV tests. They are usually referred to as linear and quadratic LIV corrections, respectively. Taking into account only the leading LIV modification of order n , the photon group velocity is then given by \nυ ( E ) = ∂ E ∂ p ≈ c " 1 -s n + 1 2 E E QG , n ! n # . (2) \nBecause of the energy dependence of υ ( E ), two photons with di ff erent energies (denoted by Eh and El , where Eh > El ) emitted simultaneously from the same source at redshift z would reach us at di ff erent times. The energy-dependent time delay due to LIV e ff ects can be expressed as [17] \n∆ t LIV = s n + 1 2 E n h -E n l E n QG , n Z z 0 (1 + z \' ) n H ( z \' ) dz \' , (3) \nwhere H ( z ) = H 0 p Ω m(1 + z ) 3 + ΩΛ , assuming a flat Λ CDM cosmology with Hubble constant H 0 = 67 . 36 km s -1 Mpc -1 , matter density parameter Ω m = 0 . 315, and vacuum energy density ΩΛ = 1 -Ω m [18]. For convenience, in Eq. (3) we introduce the dimensionless LIV parameters \nη 1 = sE Pl / E QG , 1 (4) \nand \nη 2 = 10 -15 × sE 2 Pl / E 2 QG , 2 (5) \nfor linear ( n = 1) and quadratic ( n = 2) modifications, respectively, to replace E QG , 1 and E QG , 2. \nIt is obvious from Eq. (3) that the greatest sensitivities on η n (or E QG , n ) can be expected from those astrophysical sources with rapid signal variability, large distances, and high-energy emission. As the most violent explosions occurring at cosmological distances, Gamma-ray bursts (GRBs) have been deemed as excellent probes for searching for the LIV-induced vacuum dispersion [16, 19-22]. Indeed, the most stringent limits to date on η n (or E QG , n ), resulting from vacuum dispersion time-of-flight studies, have been obtained using the GeV emission by GRB 090510 observed by the Fermi-LAT. The limits set for the subluminal (superluminal) scenario are η 1 < 0 . 13, or equivalently E QG , 1 > 9 . 3 × 10 19 GeV ( η 1 > -0 . 09, or equivalently E QG , 1 > 1 . 3 × 10 20 GeV) and η 2 < 8 . 8, or equivalently E QG , 2 > 1 . 3 × 10 11 GeV ( η 2 > -16 . 8, or equivalently E QG , 2 > 9 . 4 × 10 10 GeV) for linear and quadratic LIV e ff ects, respectively [21]. Based on the detection of sub-TeV emission from GRB 190114C, MAGIC Collaboration (2020) [23] obtained competitive lower limits on the quadratic LIV energy scale, i.e., η 2 < 37 . 0, or equivalently E QG , 2 > 6 . 3 × 10 10 GeV ( η 2 > -48 . 0, or equivalently E QG , 2 > 5 . 6 × 10 10 GeV) for the subluminal (superluminal) case. \nThe Large High Altitude Air Shower Observatory (LHAASO) detected more than 64,000 photons in the energy range of 0.2-7 TeV from GRB 221009A within the first 4000 s after the MeV burst trigger [24]. This object is located at redshift z = 0 . 151 [25, 26]. In this Letter, we study Lorentzviolating e ff ects using the time-of-flight measurements of the unprecedentedly very-high-energy (VHE, > 100 GeV) photons from GRB 221009A.', 'II. LHAASO OBSERVATIONS OF GRB 221009A': 'At 13:16:59.99 UTC on 9 October 2022 (denoted as T 0), a long-duration GRB, numbered as GRB 221009A, triggered the Gamma-ray Burst Monitor (GBM) onboard the Fermi satellite [27, 28]. The subsequent detection with Fermi-LAT made clear that it is an extraordinarily bright burst [29, 30]. The Gamma-ray emission of GRB 221009A was also detected by several other space missions [31-45], and by the groundbased air shower detector LHAASO [46]. \nLHAASO [47] is a new generation Gamma-ray and cosmic-ray observatory situated in Daocheng, China, at an elevation of ∼ 4410 meters. Due to the large area, wide field of view, and broad energy coverage, the LHAASO detectors are meticulously designed to delve into new frontiers physics, including investigations into LIV, among other scientific objectives. \nAt T 0, GRB 221009A was observed by LHAASO at a zenith angle of 28 . 1 · , and remained within LHAASO\'s field of view for the next 6000 seconds. In the initial 4000 seconds, the Water Cherenkov Detector Array (WCDA) of LHAASO captured over 64,000 photons in the 0.2-7 TeV energy range, both the light curve and energy spectrum of VHE photons were measured [24]. The intrinsic light curve reveals a rise to a peak from 231 to 244 seconds after T 0, followed by a \ndecay lasting 650 seconds. \nThe light curve of energy flux in the specified time range, as detected by LHAASO-WCDA, can be well described by a smoothly broken power-law function [24], \nλ ( t ) ∝ " t t b ! -ωα 1 + t t b ! -ωα 2 # -1 /ω , (6) \nwhere all time-related variables are relative to a reference time T ∗ = T 0 + 226 s. Here, α 1 = 1 . 82 and α 2 = -1 . 115 denote the power-law indices before and after the break time t b = 15 . 37 s, and ω = 1 . 07 represents the sharpness of the break. The intrinsic time-resolved spectrum can be fitted with a power-law function, and the positive power-law spectral index varies with time following the expression [24] \nγ ( t ) = a log( t ) + b , (7) \nwhere the unit of t is seconds, and a = -0 . 135 and b = 2 . 579. When a time delay due to LIV is introduced, this λ ( t ) will be modified to λ GLYPH<2> t -∆ t LIV( E , η n ) GLYPH<3> . \nThe observed count rate light curve, characterizing the probability of observing a photon in a range of the number of hits ∆ N hit and the arrival time t from the GRB, can be converted from the energy flux light curve with \nf ( t , ∆ N hit | η n ) = Z + ∞ 0 λ GLYPH<2> t -∆ t LIV( E , η n ) GLYPH<3> × ζ ( t ) E -γ ( t ) P EBL( E ) S ( E , t , ∆ N hit) d E . (8) \nHere, ζ ( t ) = A / R E 2 E 1 E 1 -γ ( t ) d E serves as the conversion factor from energy flux (within the energy range from E 1 = 0 . 3 TeV to E 2 = 5 TeV) to flux coe ffi cient, where γ ( t ) represents the power-law index evolution as per Eq. (7), and A is a constant to be determined. The term P EBL( E ) denotes the survival probability of photons subject to extra-galactic background light (EBL) attenuation, adopting the model [48]. Lastly, S ( E , t , ∆ N hit) accounts for the e ff ective detection area of photons at energy E and time t , with the number of fired cells in a given segment ∆ N hit.', 'III. ANALYSIS METHODS AND RESULTS': 'We utilize two analysis methods to examine the LIV lags in the VHE Gamma-ray signals from GRB 221009A. The cross-correlation function (CCF) is employed to directly measure the time delays between di ff erent energy bands, while the maximum likelihood (ML) method is adopted to extract energy-dependent arrival time delays. These two methods are widely employed in similar LIV studies.', 'A. Cross-correlation Function Method': 'We segment the light curve of the count rate detected by LHAASO-WDA from GRB 221009A into ten intervals, covering the time span from 232 to 400 seconds after T 0. The segmentation is based on the number of fired cells ( N hit), with approximately the same number of events in each segment. The \nFIG. 1: Count rate light curves of GRB 221009A, as detected by LHAASO-WCDA, presented in ten N hit segments. The time binning of the light curves used for analysis is 0.1 s; however, they are depicted here with 2 s intervals for clarity. The blue-marked range 232-400 s after the GBM trigger T 0 is selected for calculating the time lags. \n<!-- image --> \nN hit segments roughly correspond to di ff erent energy ranges, and the median energies are 0.354, 0.375, 0.395, 0.419, 0.457, 0.486, 0.556, 0.658, 0.843, and 1.601 TeV, respectively, considering the spectral index evolution from [24] for the interested time span. The energy-dependent light curves of GRB 221009A for the ten N hit segments (denoted by Seg0-Seg9) are displayed in Fig. 1. \nIn our analysis, we use the the CCF method to calculate the time lags ∆ t between the lowest energy band (Seg0) and any of the other nine high energy bands (Seg1-Seg9) (see Section A of the Supplemental Material for details). Assuming that the observed time lags ∆ t are primarily caused by LIV e ff ects, we can establish a conservative limit on the LIV parameter η n ( n = 1 or 2). Utilizing the 9 pairs of CCF measurements, we conduct a global fit to constrain η n by minimizing \nχ 2 ( η n ) = X j h ∆ t j -∆ t LIV( η n ) i 2 σ 2 ( ∆ t j ) , (9) \nwhere σ ( ∆ t j ) is the uncertainty of ∆ t j , regarded as a statistical origin. This uncertainty is obtained by a bootstrapping method: mocking the data for this CCF pair 1000 times, and RMS of the obtained ∆ t j is set to σ ( ∆ t j ). \nIn Eq. (9), the median energy of photons is used to calculate the LIV time delay ∆ t LIV( η n ) for each N hit segment. However, \nFIG. 2: Relationship between the induced LIV value of η n (with η 1 in the left panel and η 2 in the right panel) and the input η n (denoted as η 1 and η 2 respectively), obtained through a simulation procedure. \n<!-- image --> \nFIG. 3: χ 2 as a function of the dimensionless LIV parameter η n for the linear case (on the left) and the quadratic case (on the right), utilizing the CCF method. The vertical dashed lines indicate the best fits, and the shaded areas represent the 95% confidence intervals. \n<!-- image --> \nthis method may introduce bias due to several factors: spectral index evolution, wide dispersion of photon energies within each N hit segment (mainly caused by air shower fluctuations), and significant overlap in energy ranges among adjacent segments (see Section B of the Supplemental Material). \nThe biases are estimated as follows. Assuming a LIV parameter η 1 or η 2, the intrinsic light curve of energy flux as Eq. (6) can be simulated, and then translated by Eq. (8) into ten count rate light curves. Employing simulated light curves, the same procedure as with the data is executed, leading to the determination of the measured LIV parameters, as shown in Fig. 2. This involves iterating over various η 1 and η 2 values as input, enabling a polynomial fitting. Using the fitted function, the bias η n -η n for the analysis results on experimental data can be evaluated and subtracted. \nThe χ 2 distribution as a function of unbiased η n is calculated and depicted in Fig. 3. The MINUIT package [49], along with its error estimation processor MINOS [50], is used for fitting and constructing the 95% confidence levels (by setting χ 2 value 3.84 up the minimum). MINOS accounts for parameter correlations and non-linearities, providing asymmetric er- \nintervals with correct probability coverage for both χ 2 and likelihood fits. The best-fit values of η n and their uncertainties are translated into limits on the QG energy scale E QG at the 95% confidence level. These limits are detailed in Table I.', 'B. Maximum Likelihood Method': "Another approach for inferring η n is the ML method, using Eq. (8) as the function to describe the count rate light curve. A binned Poissonian likelihood method is employed for the analysis, with a time binning set to 0.1 s and N hit split into 10 segments as illustrated in Fig. 1. The background distribution, as a function of time for each N hit segment, is determined using data from the same transit as the GRB, encompassing two sidereal days before and after the burst. The Poisson probability is given by \nP i , j = e -( µ b , i , j + µ s , i , j ) ( µ b , i , j + µ s , i , j ) N on , i , j N on , i , j ! , (10) \nwhere index i represents time bins, and j represents N hit segments. The number of signals from the GRB µ s , i , j is calculated using Eq. (8), the number of background events µ b , i , j is evaluated from the polynomial fitting of the background as a function of time (see Section C of the Supplemental Material), and N on , i , j is the number of observed events. The logarithmic likelihood ratio is defined as \nΛ = -2 ln L 0 L = -2 X i , j GLYPH<16> µ ' s , i , j -µ s , i , j GLYPH<17> + N on , i , j ln µ b , i , j + µ s , i , j µ b , i , j + µ ' s , i , j . (11) \nThis ratio is utilized for minimization to determine the LIV parameter η n and all other light curve parameters. The reference time ( T ∗ , see the context of Eq. (6)) and two parameters for spectral index evolution ( a and b , see Eq. (7)) are fixed to the values provided in [24]. In this equation, L = Q i , j P i , j is the likelihood function, and L 0 is the one for a null hypothesis that no LIV delay exists, where the number of signals corresponds to µ ' s i j through setting ∆ t LIV ( E , η n ) = 0 in Eq. (8). \n, , The MINUIT package and MINOS processor is utilized for the likelihood fitting and calculating errors. The best-fit values are η 1 = 0 . 066 and η 2 = 0 . 20. \nThere could be biases in this approach for η 1 and η 2. For instance, Eq. (8) does not fit the data well due to the stochastic nature of the afterglow process and various disruptive phenomena involved. To address this, we analyzed 1000 shu ffl ed data sets by randomly exchanging the time and N hit information of events to decouple the correlation between time and energy. To preserve the behavior of the spectral index evolution, we locally shu ffl ed the events within each time bin of 6 seconds and reshu ffl ed them after shifting half of the time bin. The chosen time binning for shu ffl ing is su ffi ciently large, as the maximum time delay is less than 6 seconds within the energy coverage of the data for | η 1 | < 0 . 15 or | η 2 | < 0 . 35. The means of the 1000 best-fit values of η n from the shu ffl ed \nFIG. 4: Distribution of best-fit values of η n from the shu ffl ed data, presented for both linear (on the left) and quadratic (on the right) cases. The vertical lines indicate the means. \n<!-- image --> \nFIG. 5: Profile likelihood distributions resulting from the analysis with the ML method, depicted for both linear (on the left) and quadratic (on the right) cases. The vertical dashed lines signify the best fits after bias subtraction, and the shaded areas correspond to the 95% confidence intervals. \n<!-- image --> \ndata are considered as the biases. As shown in Fig. 4, the mean values are η 1 , bias = 0 . 063 and η 2 , bias = 0 . 19. The profile likelihood curves after subtracting these biases are depicted in Fig. 5. \nAs no significant LIV time delay from GRB 221009A are detected in this approach, we set limits for η n by constructing 95% confidence intervals with the assistance of MINOS. The obtained intervals are subsequently used to compute the limits on the QG energy scale E QG. The corresponding results are listed in Table I. \nAn alternative approach via bootstrapping and shu ffl ing, as developed in [21, 23], is also employed to obtain the so-called 'calibrated' limits. As shown in the last rows of Table I, the results appear to be very similar. \nThe EBL model could introduce systematic uncertainties in our analysis. We conducted another two rounds of analyses for the same data, considering two extreme cases of the EBL models (corresponding to the upper and lower boundaries of the uncertainty of the EBL model in [48]). For the linear LIV e ff ect, we observe that the EBL models would enlarge the η 1 limits by 18% (12%) in subluminal (superluminal) scenario. In the quadratic case, the η 2 limits would be reduced by 6% (5%) in subluminal (superluminal) scenario. \nTABLE I: Values for the best fits (BF) and the 95% lower (LL) and upper (UL) limits, provided for the dimensionless LIV parameter η n using both the CCF and ML methods. Additionally, the 95% confidence level (CL) lower limits on the quantum gravity (QG) energy scale E QG for the linear ( n = 1) and quadratic ( n = 2) cases are listed. a", 'IV. SUMMARY': "LHAASO observed unprecedented large number of VHE photon events from the brightest GRB 221009A at the earliest epoch, marking the identification of the onset of a TeV GRB afterglow for the first time. These characteristics render this signal a unique opportunity to probe LIV in the photon sector. Utilizing both CCF and ML methods, we searched for LIVinduced lags in the arrival time of the energetic photons. 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D 109 , L081501", '(2024).': '- [52] T. N. Ukwatta, et al. , Astrophys. J. 711 , 1073 (2010), arXiv:0908.2370 [astro-ph.HE].\n- [53] R.-J. Lu, et al. , Astrophys. J. 865 , 153 (2018), arXiv:1808.00636 [astro-ph.HE].', 'Appendix A: The Cross-correlation Function and time Lag': 'For a given pair of light curves, we utilize the crosscorrelation function (CCF) to determine the time lag ∆ t . The CCF with a delay ∆ t is defined as \nFj ( ∆ t ) = P i R 0( ti ) Rj ( ti + ∆ t ) q P i R 2 0 ( ti ) P i R 2 j ( ti ) , (S1) \nwhere j = 1 , 2 , 3 , . . . , 9 represents the N hit segment number except the first one, i is the time bin number, and R 0( ti ) and Rj ( ti ) are the rates of segment 0 and j at time bin ti . A discrete analysis is performed with a time step 0.1 s. The corresponding CCF as a function of time delay for the two light curves between the lowest energy band (Seg0) and any of the other nine high energy bands (Seg1-Seg9) are shown in Figure S1. We define ∆ t as the time delay corresponding to the global maximum of the CCF. By fitting Fj ( ∆ t ) around the peak with a Gaussian function (see the orange dashed line), a more precise peak position ∆ t j can be obtained. From these plots we would say that the distribution around the peak behaves like a Gaussian, and the fitting is quite reasonable, where the obtained peak position reflects the mean behaviour. Some other similar analyses such as [52, 53] also took this kind of treatment. \nWe employ the bootstrap method to estimate the variance of the time lag ∆ t , whose square root is taken as the error. The details are as follows. Firstly, we randomly sample these 10 light curves of data using the Monte Carlo (MC) method to obtain 10 mocked distributions, where the size of each MC sample is same as the data. Secondly, using the same CCF analysis method, we calculate the ∆ t for the mocked distributions. Lastly, we repeat above two procedures for 1000 times, and then obtain 1000 ∆ t values, whose variance is finally calculated. This kind of treatment is also adopted by other similar analysis, such as [52]. We assume the systematic uncertainty of ∆ t plays negligible role in the fitting, and only statistical fluctuation takes part. This is the prerequisites of the bootstrap method that we applied. \nIn our CCF analysis method, we extract 10 energydependent light curves of GRB 221009A, and the time binning of the light curves used for analysis is 0.1 s. In order to understand the e ff ects of energy and time binning more fully, we have ever investigated the cases for three time-bin sizes of 0.1, 0.5, and 1.0 s, and two N hit-segment numbers of 5 and 10, \nand found that the di ff erences in the best-fitted η n (the dimensionless LIV parameters) values of the CCF are less than 2%, and the di ff erences in the uncertainty of the time lag are less than 5%. Tests show that the choice of energy and time bins for the light curves has negligible impact on the results.', 'Appendix B: Bias from the Energy Overlapping of Di ff erent N hit Segments': 'The energy ranges of ten N hit segments are listed in Table S1. Due to large fluctuation of the cascade processes of low energy particles / photons in the atmosphere, the energy TABLE S1: Energy ranges of 10 N hit segments. The lower and upper limits represent the 16% and 84% percentiles of the energy distribution, respectively. \nranges of neighbouring N hit segments are much overlapped. Evaluating the LIV time delay ∆ t LIV( η n ) based on the median energy may introduce bias, since photon energies exhibit wide dispersion within each N hit segment. The procedure of estimating this bias is presented in Section III of the main letter.', 'Appendix C: Polynomial Fitting of the Background': 'The polynomial fit to the background has also been adopted in the previous study that was published in [24]. Two examples for the background fitting of Segments 0 and 3 are shown in Figure S2. \n(a) Seg1 N Seg0 \n<!-- image --> \n(d) Seg4 N Seg0 \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure S1: CCF as a function of time delay for the two light curves between the lowest energy band (Seg0) and any of the other nine high energy bands (Seg1-Seg9). The orange line represents the best Gaussian fitting result. \n<!-- image --> \n(b) Seg2 N Seg0 \n<!-- image --> \n(e) Seg5 N Seg0 \n<!-- image --> \n(c) Seg3 N Seg0 \n<!-- image --> \nCCF \n(f) Seg6 N Seg0 \n<!-- image --> \n<!-- image --> \nFigure S2: Background count rate as a function of time and its polynomial fitting (red line). The upper and lower panel shows the fitting results for N hit segment 0 and 3, respectively. \n<!-- image -->'}

2024PhRvL.133f1603D

We demonstrate how tabletop settings combining hyperbolic lattices with nonlinear dynamics universally encode aspects of the bulkboundary correspondence between gravity in antideSitter AdS space and conformal field theory CFT. Our concrete and broadly applicable holographic toy model simulates gravitational selfinteractions in the bulk and features an emergent CFT with nontrivial correlations on the boundary. We measure the CFT data contained in the two and threepoint functions and clarify how a thermal CFT is simulated through an effective black hole geometry. As a concrete example we propose and simulate an experimentally feasible protocol to measure the holographic CFT using electrical circuits.

2024-08-01T00:00:00Z

['arXiv:2404.03062', '10.48550/arXiv.2404.03062', '10.1103/PhysRevLett.133.061603', '2024arXiv240403062D', '2024PhRvL.133f1603D']

['Condensed Matter - Mesoscale and Nanoscale Physics', 'Condensed Matter - Statistical Mechanics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematical Physics']

Simulating Holographic Conformal Field Theories on Hyperbolic Lattices

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

https://arxiv.org/pdf/2404.03062.pdf

{'Simulating Holographic Conformal Field Theories on Hyperbolic Lattices': "Santanu Dey , 1, 2, ∗ Anffany Chen , 1, 2, † Pablo Basteiro, 3, 4 Alexander Fritzsche, 3, 4, 5 Martin Greiter , 3, 4 Matthias Kaminski , 6 Patrick M. Lenggenhager , 4, 7, 8, 9, 10 Ren'e Meyer , 3, 4 Riccardo Sorbello, 3, 4 1, 2 \nAlexander Stegmaier , 3, 4 Ronny Thomale , 3, 4 Johanna Erdmenger , 3, 4 and Igor Boettcher \n1 Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada \n2 Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada \n3 Institute for Theoretical Physics and Astrophysics, \nJulius Maximilians University Wurzburg, Am Hubland, 97074 Wurzburg, Germany \n4 Wurzburg-Dresden Excellence Cluster ct.qmat, Julius Maximilians \nUniversity Wurzburg, Am Hubland, 97074 Wurzburg, Germany \n5 Institut fur Physik, Universitat Rostock, 18059 Rostock, Germany \n6 Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA \n7 Department of Physics, University of Zurich, 8057 Zurich, Switzerland \n8 Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen, Switzerland \n9 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland \n10 Max Planck Institute for the Physics of Complex Systems, Nothnitzer Str. 38, 01187 Dresden, Germany \nWe demonstrate how table-top settings combining hyperbolic lattices with nonlinear dynamics universally encode aspects of the bulk-boundary-correspondence between gravity in anti-de-Sitter (AdS) space and conformal field theory (CFT). Our concrete and broadly applicable holographic toy model simulates gravitational self-interactions in the bulk and features an emergent CFT with nontrivial correlations on the boundary. We measure the CFT data contained in the two- and three-point functions and clarify how a thermal CFT is simulated through an effective black hole geometry. As a concrete example, we propose and simulate an experimentally feasible protocol to measure the holographic CFT using electrical circuits. \nThe holographic principle as realized by the AdS/CFT correspondence postulates a deep connection between two of the most intriguing, yet unfathomable phenomena in modern physics [1-4]: scale-invariance close to second-order phase transitions, described by conformal field theories (CFTs) [5], and quantum gravity in curved spacetimes. The conjecture states that gravity in a negatively curved space is dual to a CFT on the boundary of that space. Furthermore, the presence of a black hole in the bulk results in a thermal CFT, with the temperature given by the Hawking temperature. While numerous calculations indicate the validity of the correspondence, an experimental verification is complicated in particular by the need to observe or simulate graviton-graviton interactions, since only these give rise to nonvanishing three- or four-point boundary correlation functions, and nontrivial CFT data. \nThe goal of this work is to construct a concrete holographic toy model for the AdS/CFT correspondence that can be realized in the laboratory. We show that, with the right four ingredients, a large class of realistic and experimentally feasible low-energy models features holography through conformal boundary correlations. As a concrete example, we demonstrate how electrical circuits with nonlinear circuit elements can achieve this milestone, but our universal model is applicable to other experimental platforms and constitutes a theoretically intriguing model in itself. With the setup, we are able to emulate three-point functions, characterize the CFT data, and clarify how thermal effects are incorporated. \nThe four ingredients of our holographic toy model are: \n(i) a lattice realization of AdS space through hyperbolic lattices [6-49], now routinely implemented in circuit quantum electrodynamics and coplanar waveguides [6, 50], topoelectrical circuits [20, 23, 29, 30], topological photonics [51], and mechanical elastic lattices [52]; (ii) nonlinear dynamical equations to emulate gravitational self-interactions, realized by local qubit-photon interactions [16, 53, 54], nonlinear or active circuit elements [5559], nonlinear optics [60, 61], or spring hardening; (iii) effective black hole geometries to emulate temperature by using type-II hyperbolic lattices [35, 62-64]; (iv) a theoretical framework and experimental protocol to compute and measure boundary correlation functions. We demonstrate that with (i)-(iv) even classical platforms are holographic, corresponding to the limit where weakly-coupled semiclassical bulk gravity is dual to a strongly coupled CFT, as reviewed in [4]. \nOur work considerably extends previous works on holographic aspects of hyperbolic lattices. Power-law scaling of two-point boundary correlators has been demonstrated at zero temperature in the important works [10, 13, 35], together with the massless contact-limit of the four-point function [13], the introduction of black holes through type-II lattices [35], and the emergence of the RyuTakayanagi formula [35]. However, our systematic study of higher-point correlation functions and their ensuing CFT data, precise characterization of thermal correlations, theoretical framework and concrete experimental protocol to measure boundary correlations, and simulation of an experimental setup satisfying (i)-(iv) is unprecedented. This is critical for a thorough investigation \nFIG. 1. Hyperbolic flakes and black hole geometry. (a) Starting from a Poincar'e disk (type-I geometry, coordinates z = re i θ ), we create a black hole by identifying points on two geodesics (shown in green). Equivalently, we squash the disk to form a strip of width w and then wrap it into a ring (type-II geometry, coordinates ˆ z = ˆ re i θ ). This identification creates two separate Universes, 1 and 2, each with a holographic boundary, connected through an Einstein-Rosen (ER) bridge (dashed line) [65]. This line corresponds to the bifurcation point of an eternal black hole horizon. (b) Applying the squash-and-wrap procedure to a hyperbolic { p, q } -flake creates a type-II flake, which emulates a black hole with Hawking temperature T = w/ 8 πℓ . Herein, w = kP , with k an integer representing the number of repeated cells in the strip ( k = 5 in the plot), and P = P ( p, q ) a constant. (c) Type-I and -II hyperbolic flakes of a { 3 , 7 } tessellation with some graph-shortest-distance paths connecting boundary sites. These paths are pivotal for the boundary-boundary correlations. (d) Schematic Penrose diagram with space and time as x- and y-axis, respectively. The hyperbolic lattice is located on a slice of constant time (gray line). \n<!-- image --> \nof both the duality, and consistency of the associated CFT. Here we achieve this milestone for the first time by studying interacting hyperbolic matter on the lattice. \nHolographic lattice model.-Classical or quantum theories of gravity are models where the metric tensor g and distance line element d s 2 = ∑ α,β g αβ d x α d x β fluctuate in time and space. In two imaginary-time spacetime dimensions, the metric can be parametrized by a single real field φ ( z ) as d s 2 = e φ ( z ) | d z | 2 (isothermal coordinates), where z = x + i y . Although the usual Einstein-Hilbert action is non-dynamical in two dimensions, a dynamical action principle for gravity results from minimizing the Liouville gravity action [66, 67] \nS LG [ φ ] = ∫ d 2 z [ 1 2 ( ∇ φ ) 2 + 2 ℓ 2 e φ ] . (1) \nThe term -1 /ℓ 2 plays the role of a negative cosmological constant. The stationary solution φ ⋆ ( z ) that minimizes the action is the hyperbolic Poincar'e disk metric d s 2 ⋆ = (2 ℓ ) 2 | d z | 2 / (1 -| z | 2 ) 2 . Fluctuations about this solution, φ = φ ⋆ + ϕ , follow the imaginary-time action [68] \nS grav [ ϕ ] = S LG [ φ ⋆ + ϕ ] -S LG [ φ ⋆ ] (2) \n= ∫ d 2 z (1 -| z | 2 ) 2 [ 1 2 ϕ ( -□ + m 2 ) ϕ + ϕ 3 3 ℓ 2 + ϕ 4 12 ℓ 2 + . . . ] . \nWe may interpret ϕ ( z ) as graviton-like mode, since it parametrizes small metric fluctuations. Importantly, dynamical gravity corresponds here to a real scalar field \nϕ propagating in its own hyperbolic background with □ = (2 ℓ ) -2 (1 - | z | 2 ) 2 ∇ 2 the Laplace-Beltrami operator [ingredient (i)], and nonlinear terms like ϕ 3 and ϕ 4 correspond to gravitational self-interactions [ingredient (ii)]. We choose m 2 ℓ 2 > -1 / 4 above the BreitenlohnerFreedman bound for a stable theory [33]. \nThe continuum action (2) can be simulated on discrete hyperbolic lattices using the dictionary of Ref. [11], which yields the universal holographic lattice action \nS ( { ϕ µ } ) = -1 2 ∑ µ,ν ϕ µ A µν ϕ ν + ∑ µ ( ˆ m 2 2 ϕ 2 µ + u 3! ϕ 3 µ ) . (3) \nHerein, ϕ µ = ϕ ( z µ ) is the field variable defined on the sites z µ of a graph or lattice G with adjacency matrix A . Equation (3) represents a generic tight-binding Hamiltonian realizable on the platforms discussed in the introduction, and is a universal low-energy limit of more complicated Hamiltonians. The parameters ˆ m 2 and u are tunable in experiment and can be matched to the Liouville action (1) [68]. We neglect higher-order interaction terms beyond ϕ 3 for simplicity, resulting in a model that deviates from Liouville gravity through this omission. \nThe choice of graph G determines the curved bulk space on which the action is simulated. Two-dimensional imaginary-time AdS space is emulated by hyperbolic { p, q } lattices with ( p -2)( q -2) > 4, labelled type-I \nhereafter. Black holes are realized by identifying points on two geodesics in a type-I geometry. This is equivalent to a type-II ring graph [35] obtained by a squashand-wrap procedure, see Fig. 1. The squash step is z µ ↦→ ζ µ = 2 π ln( 1+ z µ 1 -z µ ), which produces an infinite strip, from which a finite strip of width w is obtained with its truncated edges identified, followed by the wrap step ζ µ ↦→ ˆ z µ = e 2 π i( ζ µ +i) /w yielding a ring. When applied to the Poincar'e disk metric, this results in a timeslice of the three-dimensional Ba˜nados-TeitelboimZanelli (BTZ) black hole metric [35, 62-64, 68] given by \nd s 2 II = ( ℓw 4 ) 2 | dˆ z | 2 | ˆ z | 2 cos 2 [ π 2 (1 -ln | ˆ z | / ln ˆ r H )] , (4) \nwhich suggests the black hole interpretation on the lattice. The black hole temperature is T = w/ 8 πℓ with horizon radius ˆ r H = e -2 π/w . In our numerical construction [68], we have w = kP , where k is an integer that we can choose freely and P = P ( p, q ) is a lattice-dependent constant. By varying k, p, q , we can access a large, albeit discrete set of temperatures [ingredient (iii)]. \nBulk-boundary-correspondence.-To study holography, we divide the graph G , which may be either type-I or type-II, into its bulk and boundary components, ˚ G and ∂ G , such that G = ˚ G ∪ ∂ G . We label generic sites on G by Greek letters µ, ν , whereas bulk and boundary sites are labeled by Roman letters i, j and a, b , respectively. We then consider fixed boundary field values given by ϕ a = J a . The associated bulk partition function reads \nZ ( { J a } ) = ∫ ( ∏ i d ϕ i ) e -S ( { ϕ i ,ϕ a = J a } ) . (5) \nNote that we only integrate over the bulk field values. The bulk-boundary-correspondence asserts that Z ( { J a } ) is the generating function for a CFT on the boundary [2, 3]. The associated connected n -point correlation functions for some boundary field O a are given by \n⟨O a 1 . . . O a n ⟩ = ∂ n ln Z ∂J a 1 · · · ∂J a n ∣ ∣ ∣ J =0 (6) \n[ingredient (iv)]. Properties like spin or scaling behavior of O a are determined by the bulk theory, but no action underlying the CFT is specified by the duality per se . An important future research inquiry is to explore whether a representative CFT action can be constructed in the holographic toy model. Furthermore, adding more fields and symmetries to the bulk action, the consistency of richer dual CFTs can be probed. \nThe holographic properties of the lattice model (3) can be studied theoretically in the perturbative regime u ≪ 1, which mirrors the semiclassical limit of the AdS/CFT correspondence where only tree-level diagrams contribute [68]. The 2-point function for u = 0 reads \n⟨O a O b ⟩ = -M ab + ∑ i,j M ai G ij M jb . (7) \n| \n- \n| \n0 \nFIG. 2. CFT two-point function. Results are shown for { 3 , 7 } hyperbolic flakes. (a) Representative two-point function in type-I geometry with bulk mass m 2 ℓ 2 = 0 . 271. Data sets are computed from Eq. (7) (blue) and numerical simulation of the electrical circuit in Eqs. (12)-(14) using a realistic diode (pink). The data scatter results from binning the discrete lattice coordinates θ a . Fitted power-law formula (9) shown in orange. (b) Fitted conformal dimension ∆ versus m 2 ℓ 2 > -1 / 4 (blue squares with fit error). Due to lattice corrections captured by Eq. (11) (orange), ∆ deviates from the continuum formula m 2 ℓ 2 = ∆(∆ -1) (gray dashed). (c) Two-point function on type-II lattice for same mass as in a , showing thermal behavior according to Eq. (9). (d) The fitted type-II temperature T (circles) is approximately independent of m and described by Tℓ = kP/ 8 π (lines). Here, P ≈ 1 . 845 for { 3 , 7 } and k = 4 , 6 , 8 , 10 , 12 in the plot. \n<!-- image --> \nHere, M µν = ( G -1 ) µν = -A µν + ˆ m 2 δ µν is the bare inverse propagator, with ˆ m 2 = q + qh 2 m 2 ℓ 2 relating to the physical mass m through a dimensionless lattice constant h = (1 -sin 2 ( π/q ) cos 2 ( π/p ) ) 1 / 2 [11, 38]. Approximately, ⟨O a O b ⟩ ≈ G ab is the bulk propagator extrapolated to the boundary sites a, b . For the 3-point function, we have \n⟨O a O b O c ⟩ = u ∑ i B ai B bi B ci (8) \nwith B ai = ∑ j M aj G ji the boundary-to-bulk propagator. This is a Witten diagram, where the bulk site i connects the boundary sites a, b, c , see Fig. 3a, and the largest contribution comes from graph-geodesic paths. Remarkably, this perturbative formula captures the CFT correlations also on the lattice. All n -point functions with n ≥ 3 vanish for u = 0 in our model, emphasizing again the importance of interactions embodied by uϕ 3 in Eq. (3). A 4-point function appears at order u 2 [4]. \nFIG. 3. CFT three-point function. We visualize the three-point function on type-I (top row) and type-II geometries (bottom row). We place O a at angle θ a , O c at angle θ c , and vary the position of O b at angle θ b . (a) The 3-point function corresponds to the star-shaped Witten diagram of Eq. (8). The largest contribution to the sum comes from graph-geodesic paths connecting the boundary sites through an intermediate bulk site i . (b) Three-point functions on { 3 , 7 } type-I and type-II flakes with parameters as in Fig. 2. Data sets are computed from Eq. (8) (blue) and numerical simulation of the electrical circuit in Eqs. (12)-(14) using a realistic diode (pink). The solid orange lines constitute the theory prediction, where ∆ and T follow from the 2-point function, and only C 3 is a fitted parameter. Correlations diverge when any two of the angles coincide (arrows). (c) Scaling collapse according to the characteristic power-law behavior ⟨O a O b O c ⟩ ∝ ( d ab d ac d bc ) -∆ (orange), where d ab = | e i θ a -e i θ b | for type-I and d ab ( T ) = sinh( πTℓ | θ a -θ b | ) /πTℓ for type-II geometries, respectively. (d) Scaling dimensions ∆( m 2 ℓ 2 ) extracted from 2- and 3-point function on type-I graphs are consistent within the errors. (e) The normalized 3-point coefficient C 3 /u is determined by m 2 ℓ 2 and agrees for both type-I and type-II lattices, indicating their CFTs are equal. \n<!-- image --> \nCorrelation functions and CFT data.-The characteristic forms of the two- and higher-point correlation functions of a CFT distinguish it from a mere scale invariant theory. In particular, the 2- and 3-point functions are fully determined by two parameters as part of their CFT data, the scaling dimension ∆ and 3-point coefficient C 3 , \n⟨O a O b ⟩ ≃ 1 ( d ab ) 2∆ , ⟨O a O b O c ⟩ ≃ C 3 ( d ab d ac d bc ) ∆ . (9) \nHere we normalize the boundary fields O a such that the numerator of the 2-point function is unity. The function d ab determines the distance between the sites a, b in the CFT. A hallmark of the continuum AdS/CFT correspondence is that ∆ and C 3 are tunable by varying the bulk parameters m 2 ℓ 2 and u . We now show that the same is true for our holographic lattice model. \nThe boundary 2- and 3-point functions determined from Eqs. (7) and (8) are summarized in Figs. 2 and 3, respectively. They agree with the expected behavior through the identification [68] \nd ab = { | e i θ a -e i θ b | (type-I) sinh( πTℓ | θ a -θ b | ) πTℓ (type-II) . (10) \nHere θ a,b are the angle coordinates of the boundary \nsites. This emulates CFTs on a circle at zero and finite temperature. While Eqs. (9) are good approximations on type-II lattices, especially for the 2-point function, the proper quantitative formula requires to replace θ a -θ b → θ a -θ b + 2 πn in d ab with a subsequent sum over n ∈ Z to make the functions periodic [68]. This behavior, reminiscent of the method of images, also arises in the BTZ geometry and is thus expected here [64]. \nOur main novel findings from the analysis of correlations are that (a) the scaling dimension ∆( m 2 ℓ 2 ) extracted from the 2-point function also captures the scaling of the 3-point function, indicating a consistent CFT [Fig 3d], (b) the parameter k on type-II lattices determines the temperature consistent with the formula Tℓ = kP/ 8 π [Fig. 2d], but leaves the CFT data invariant, and (c) the CFT data ∆( m 2 ℓ 2 ) and C 3 ( m 2 ℓ 2 ) for the CFTs simulated on both type-I and type-II lattices is identical [Fig. 3d,e]. The latter finding indicates that they are the same CFTs, but at zero and finite temperature. These nontrivial results also solidify the interpretation of the type-II lattice as a geometry that emulates a black hole. \nThe holographic relation between ∆ and m 2 ℓ 2 is of the form m 2 ℓ 2 = f (∆(∆ -1)) with f cont ( X ) = X in the continuum, while a gradient expansion for the { 3 , 7 } - \nelds \nf ( X ) ≃ X + h 2 4 ( X 2 +2 X ) + h 4 36 ( X 3 +10 X 2 +12 X ) (11) \nwith h = 0 . 497 [19, 22, 68], see Fig. 2b. The value of h and form of f ( X ) depend on the { p, q } lattice, with the universal continuum limit recovered for h → 0. To extract the 3-point coefficient C 3 in a manner that reduces scatter due to the lattice discretization, we construct the function F abc = ⟨O a O b O c ⟩ / [ ⟨O a O b ⟩⟨O a O c ⟩⟨O b O c ⟩ ] 1 / 2 that we average over a, b, c to obtain F abc ≃ C 3 [68]. \nExperimental protocol for electrical circuits.-We propose to realize the holographic toy model in electrical circuits by implementing the equation of motion \n-∑ ν A µν ¯ V ν + ˆ m 2 ¯ V µ + u 2 ¯ V 2 µ = 0 , (12) \nwhere ϕ µ → ¯ V µ is the normalized local voltage at node z µ . By applying voltage sources on the boundary, fixed values ¯ V a = J a can be realized. The solution ¯ V µ parametrically depends on the J a chosen. By applying timedependent boundary conditions, J a ( t ), t -derivatives conveniently translate to J a -derivatives below. From the measured or simulated voltage signal ¯ V µ ( t ), we compute the circuit generating function \nW circ ( t ) = -S ( { ¯ V µ ( t ) } ) , (13) \nwith action S ( { ϕ µ } ) from Eq. (3). We have W circ ≃ ln Z in the saddle-point approximation, sufficient for the semiclassical holographic limit studied here. \nTo realize the terms in Eq. (12), we use resistors to generate the linear couplings ∑ ν ( -A µν + ˆ m 2 δ µν ) V ν , while diodes can be used for the nonlinear term uV 2 µ . Indeed, the current through a diode is approximated by the Shockley equation I ( V ) = I S (e V/V S -1) ≈ I S [ V/V S + ( V/V S ) 2 / 2], which yields the desired V 2 µ -term after absorbing the linear part into ˆ m 2 V µ . The concrete circuit parameters are listed in Suppl. Sec. S8 [68]. On short time scales, dissipative terms present in any realistic circuit lead to transient behavior. These effects are neglected here and the correspondence is realized in the steady state, assuming an instantaneous response to J a ( t ). \n̸ \nOur protocol to compute 2-point functions from W circ ( t ) is as follows. We apply a drive that linearly ramps ( J a ( t ) , J b ( t )) = ( K a , K b )( t -t 0 ), crossing zero at t = t 0 , while all other J µ ( t ) ≡ 0 for µ = a, b . Since all second time-derivatives vanish, we have \nd 2 W circ d t 2 ∣ ∣ ∣ t = t 0 = ∑ µ,ν ˙ J µ ˙ J ν ∂ 2 W circ ∂J µ ∂J ν ∣ ∣ ∣ J =0 (14) = K 2 a ⟨O 2 a ⟩ +2 K a K b ⟨O a O b ⟩ + K 2 b ⟨O 2 b ⟩ . \nBy choosing three linearly independent ramps in a series of measurements, e.g. ( K a , K b ) = (1 , 0) , (0 , 1) , (1 , 1), this set of three linear equations can be solved for ⟨O a O b ⟩ . Similarly, to measure n -point functions, we ramp n boundary sites J a 1 ( t ) , . . . , J a n ( t ) linearly. By using ten linearly independent ramps, we obtain ⟨O a O b O c ⟩ from d 3 W circ / d t 3 | t = t 0 . \nIn Fig. 2a we present results for the so-obtained 2point function using an LTspice simulation of an electrical circuit with a realistic Schottky diode (model RBE1VAM20A). In Fig. 3b,c we present the 3-point function for the same parameters from a Mathematica simulation of Eqs. (12)-(14). Since the signals in actual electrical circuits are known to be close to such numerical simulations [55-57], this serves as a proof of principle that our protocol is experimentally feasible. \nAcknowledgments.-We thank Joseph Maciejko for profound contributions especially in the early stages of the project and his insightful comments on the manuscript. We are grateful for fruitful discussions with Giuseppe Di Giulio, Dominik Neuenfeld, and Canon Sun. AC was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RGPIN-2020-06999, the Avadh Bhatia Fellowship, and the Faculty of Science at the University of Alberta. AC and IB acknowledge support through the University of Alberta startup fund UOFAB Startup Boettcher. SD was supported by the Faculty of Science at the University of Alberta. MK was supported, in part, by the U.S. Department of Energy grant DESC0012447. IB acknowledges funding from the NSERC Discovery Grants RGPIN-2021-02534 and DGECR202100043. PB, AF, MG, PML, RM, RS, AS, RT, and JE acknowledge support by Germany's Excellence Strategy through the Wurzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, project-id 390858490), and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center 'ToCoTronics', ProjectID 258499086-SFB 1170. PML was supported by the Ambizione grant No. 185806 by the Swiss National Science Foundation (SNSF) and the European Union (ERC, QuSimCtrl, 101113633). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. 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Seiberg, Notes on Quantum Liouville Theory and Quantum Gravity, Progress of Theoretical Physics Supplement 102 , 319 (1990).\n- [67] P. Ginsparg and G. Moore, Lectures on 2D gravity and 2D string theory (TASI 1992), hep-th/9304011 (1993).\n- [68] See Supplemental Material for construction of type-II lattices, relation to BTZ geometry, fit procedure, derivation of Eqs. (2), (7), (8), (10), and (11), and electrical circuit simulation.", 'Supplemental Material': 'This supplemental material details the algorithm for constructing hyperbolic lattice flakes, the reason for interpreting type-II as BTZ geometry, our fit procedure, the derivation of Eqs. (2), (7), (8), (10), and (11), and the electrical circuit simulation.', 'CONTENTS': 'References \n5 \n- S1. Derivation of Eq. (2) 8\n- S2. Construction of type-I and II flakes\n- 9\n- S3. Relation between type-II and BTZ geometry \n10 \n- S4. Derivation of Eqs. (7) and (8) 12 \nS6. Derivation of Eq. (10) 13 \nS5. Derivation of Eq. (11) 14 \nS7. Fit procedure \n15 \nType-I / Zero temperature \n- 15 \nType-II / Finite temperature \n17 \n3-pt coefficient \n17 \nS8. Simulation in electrical circuits 19', 'S1. DERIVATION OF EQ. (2)': "We write the metric in isothermal coordinates as \nd s 2 = g ij d x i d x j = e φ (d x 2 +d y 2 ) (S1) \nwith g ij = e φ δ ij and i, j = 1 , 2. We have g ij = e -φ δ ij and √ det( g ij ) = e φ . Define the Gaussian curvature \nK [ g ] = -1 2 e -φ ∇ 2 φ. (S2) \nThe Poincar'e disk metric \nd s 2 ⋆ = (2 ℓ ) 2 d x 2 +d y 2 (1 -| z | 2 ) 2 , (S3) \nφ ⋆ = ln ( (2 ℓ ) 2 (1 -x 2 -y 2 ) 2 ) (S4) \nsatisfies K [ g ⋆ ] = -1 ℓ 2 . \nThe equations of motion associated to the Liouville action in Eq. (1) are \n0 = δS LG δφ [ φ ⋆ ] = -∇ 2 φ ⋆ + 2 ℓ 2 e φ ⋆ , (S5) \nwhich is equivalent to K [ g ⋆ ] = -1 ℓ 2 , hence solved by the hyperbolic metric. We write S LG = S kin + S pot . Expanding φ = φ ⋆ + ϕ we obtain for the kinetic part \nS kin [ φ ⋆ + ϕ ] = ∫ d 2 z [ 1 2 [ ∇ ( φ ⋆ + ϕ )] 2 ] = ∫ d 2 z [ 1 2 ( ∇ φ ⋆ ) 2 + ∇ φ ⋆ · ∇ ϕ + 1 2 ( ∇ ϕ ) 2 ] = ∫ d 2 z [ -1 2 φ ⋆ ∇ 2 φ ⋆ -ϕ ∇ 2 φ ⋆ -1 2 ϕ ∇ 2 ϕ ] = ∫ d 2 z e φ ⋆ [ -1 2 φ ⋆ 2 ℓ 2 -ϕ 2 ℓ 2 -1 2 ϕ ( e -φ ⋆ ∇ 2 ) ϕ ] . \nFor the potential part we obtain \nS pot [ φ ⋆ + ϕ ] = ∫ d 2 z [ 2 ℓ 2 e φ ⋆ e ϕ ] = ∫ d 2 z e φ ⋆ [ 2 ℓ 2 e ϕ ] . \nAdding both expansions and neglecting terms that are independent of ϕ we obtain \nS grav [ ϕ ] = ∫ d 2 z e φ ⋆ [ -1 2 ϕ ( e -φ ⋆ ∇ 2 ) ϕ + 2 ℓ 2 ( e ϕ -ϕ ) ] . (S6) \nNote that this action is again of Liouville form [67], but for a hyperbolic fiducial metric, whereas the starting action S LG was based on a flat fiducial metric. This reflects the shift invariance of the Liouville action. However, due to the saddle-point expansion around φ ⋆ , fluctuations parametrized by ϕ are expected to be small, whereas this is not true for the field φ . Expanding the potential term in powers of ϕ we arrive at \nS grav [ ϕ ] = ∫ d 2 z e φ ⋆ [ -1 2 ϕ ( e -φ ⋆ ∇ 2 ) ϕ + 2 ℓ 2 ( 1 2 ϕ 2 + 1 3! ϕ 3 + 1 4! ϕ 4 + . . . )] (S7) = ∫ d 2 z (2 ℓ ) 2 (1 -| z | 2 ) 2 [ 1 2 ϕ ( -□ + 2 ℓ 2 ) ϕ + 1 3 ℓ 2 ϕ 3 + 1 12 ℓ 2 ϕ 4 + . . . ] . (S8) \nThis is of the form of Eq. (2) after appropriate choice of constants m,u,µ , and normalization of ϕ . Here we used \ne φ ⋆ = (2 ℓ ) 2 (1 -| z | 2 ) 2 , (S9) \n□ = e -φ ⋆ ∇ 2 = 1 (2 ℓ ) 2 (1 -| z | 2 ) 2 ∇ 2 . (S10) \nTo connect the Liouville action to the lattice action \nS ( { ϕ µ } ) = -1 2 ∑ µ,ν ϕ µ A µν ϕ ν + ∑ µ ( ˆ m 2 2 ϕ 2 µ + u 3! ϕ 3 µ ) , (S11) \nwe replace ∑ ν A µν ϕ ν → ( q + qh 2 ℓ 2 □ ) ϕ µ [11] and find \nS ( { ϕ µ } ) = ∑ µ ( 1 2 ϕ µ ( -qh 2 ℓ 2 □ -q + ˆ m 2 ) ϕ µ + u 3! ϕ 3 µ ) = ∑ µ ( qh 2 ℓ 2 1 2 ϕ µ ( -□ + m 2 ) ϕ µ + u 3! ϕ 3 µ ) . (S12) \nWe inserted ˆ m 2 = q + qh 2 ℓ 2 m 2 . Through this equation, the experimental parameters ˆ m 2 and u can be connected to the continuum Liouville action S grav [ ϕ ].", 'S2. CONSTRUCTION OF TYPE-I AND II FLAKES': "To construct hyperbolic flakes, we start from regular hyperbolic { p, q } tilings with ( p -2)( q -2) > 4, tessellated by p -sided polygons with q of them meeting at each site. We refer to type-I flakes as hyperbolic lattices with open boundary conditions, generated via vertex inflation [8, 12, 40] and embedded in the Poincar'e disk \nD = { z ∈ C , | z | < 1 } , (S13) \nd s 2 = (2 ℓ ) 2 | d z | 2 (1 -| z | 2 ) 2 . (S14) \nType-II flakes are constructed from the type-I flakes by identifying two geodesics related by a boost (as illustrated in Fig. 1), akin to the construction of BTZ black holes from the AdS space. While such identification can be done by modifying the adjacency matrix of the lattice without a change of site coordinates, as the adjacency matrix alone encodes the lattice connectivity and thereby dictates the subsequent computation of correlation functions, we perform two conformal transformations of the site coordinates to facilitate the construction of finitesized type-II flakes and clear visualization of their topologically nontrivial geometry. \nIn our analysis, we focus on the { 3 , 7 } tiling because using triangles as tiles allows every boundary site to be connected to a bulk site, leading to ample data for the correlation functions. We generate a finite-sized type-I { 3 , 7 } lattice, as depicted in Fig. S1a, via the vertex-inflation algorithm detailed in Ref. 40. Specifically, we implement six or seven iterations of vertex inflation (dubbed 'shells'), resulting in a total of N = 960 or 2523 sites, respectively. We denote the adjacency matrix of the type-I lattice as A and the Poincar'e disk coordinates as { z µ } N µ =1 . \nWe then proceed to construct type-II hyperbolic lattices from the 6-shell { 3 , 7 } type-I lattice. We first apply a conformal transformation C 1 from the Poincar'e disk to strip coordinates: \nC 1 : z ↦→ ζ = 2 π ln ( 1 + z 1 -z ) , (S15) \nwith domain and range \nC 1 : D → B = { ζ ∈ C , | Im( ζ ) | < 1 } . (S16) \nC 1 maps 0 to 0 and the upper and lower halves of the Poincar'e-disk boundary z = e i ϕ to the lines i + R and -i + R , respectively. The metric of the strip model is \nd s 2 = ℓ 2 ( π 2 ) 2 | d ζ | 2 cos 2 ( π 2 Im( ζ ) ) , (S17) \nwhich diverges at the asymptotic boundary at Im( ζ ) → ± 1. The strip model is periodic along the x -direction with the period dependent on p and q . Given that our type-I lattice is finite-sized, the resultant strip model with site coordinates { ζ µ } N µ =1 (Fig. S1b) is non-uniform with progressively less complete asymptotic boundary away from Re( ζ ) = 0. Upon closer inspection, we identify two sites (highlighted in red in Fig. S1b) which lie on the left and right vertical edges that define a periodic segment. The difference of their coordinates gives the period P ≈ 1 . 845 for the { 3 , 7 } tiling. \nOur strategy is to create a mostly homogeneous type-II flake is to take the central periodic segment between the x -range [ -P/ 2 , P/ 2), where the asymptotic boundaries at ± i + R are the most complete, and copy this small segment k times, joining them together to form a long strip of length kP . To achieve this, we single out those sites in Fig. S1b that lie outside the x -range [ -P/ 2 , P/ 2) and truncate them from the adjacency matrix A by removing the corresponding rows and columns. We also remove these sites from the list of all coordinates { ζ µ } N µ =1 . The truncated segment, shown in Fig. S1c, has N ' = 686 sites; its adjacency matrix is denoted by a . Next, we begin building the adjacency matrix of the long strip, denoted ˜ A , by computing the Kronecker product of the k × k identity matrix and a : ˜ A = 1 k × k ⊗ a . The coordinates of the long strip are copies of the small-segment coordinates, shifted horizontally by integer multiples of P : 0 , P, 2 P, . . . , ( k -1) P . \nAt this stage, our strip is formed from k disjoint segments. While the coordinates are correct, the adjacency matrix does not include edges connecting the neighboring segments to maintain a coordination number of 7 on every site. To connect the segments, we need to first determine the edges that were severed at the truncation step. The procedure is as follows. For each site µ in the truncated segment (Fig. S1c), we identify its nearest neighbors in the untruncated band (Fig. S1b) using the original adjacency matrix A . If a particular nearest neighbor ν is not included in the truncated segment, we numerically search for the site ν ' such that it belongs to the segment and equivalent to ν modulo horizontal translation defined by the periodicity. Note that during this identification process, we set the numerical tolerance in distance comparison to 10 -10 . We record all the severed edges on the right side of the truncated segment in a list \nFIG. S1. Construction of type-II hyperbolic lattices. (a) A type-I hyperbolic lattice with { 3 , 7 } tessellation generated via finite iterations of vertex inflation. (b) After the conformal transformation in Eq. (S15), the finite-sized lattice is 'squashed' into a finite strip with a discrete translation symmetry along the x -direction. The translation symmetry is incomplete due to the finite system size. The period P is given by the distance between the highlighted sites in red. (c) We isolate the central periodic segment in the range x ∈ [ -P/ 2 , P/ 2) (shown here), where the CFT boundaries are the most complete, and join together k copies of the segment to form a finite periodic strip of width w = kP . (d) After joining the ends of the strip and passing the coordinates through a second conformal map defined in Eq. (S18), we arrive at a type-II hyperbolic lattice that represents the discretized time-slice of a BTZ black hole with temperature T = kP/ 8 π . \n<!-- image --> \n( µ, ν ' ). We also check that the severed edges on the left side correspond to the same set of edges. Finally, we reintroduce these severed edges to the adjacency matrix ˜ A between every pair of neighboring segments centered at cP and ( c +1) P for c = 0 , ..., k -2, as well as the pair at the ends centered at ( k -1) P and 0 to respect the periodic boundary condition. At this stage, ˜ A correctly describes a type-II lattice with kN ' sites. \nWe apply another conformal transformation C 2 to wrap the strip model into an annulus geometry (Fig. S1d): \nC 2 : ζ ↦→ ˆ z = e 2 π i( ζ +i) / ( kP ) , (S18) \nwith domain and range \nC 2 : B → A = { ˆ z ∈ C , ˆ r 0 < | ˆ z | < 1 } . (S19) \nC 2 maps the upper boundary i + R in the strip model to the inner boundary of the annulus at radius ˆ r 0 = C 2 (i) = e -4 π/ ( kP ) and the lower boundary -i + R to the outer boundary at 1. The new metric is \nd s 2 = ( ℓkP 4 ) 2 | dˆ z | 2 | ˆ z | 2 cos 2 ( π 2 (1 + kP 2 π log | ˆ z | ) ) (S20) = ( ℓkP 4 ) 2 | dˆ z | 2 | ˆ z | 2 sin 2 ( kP 4 log | ˆ z | ) , \nwhich diverges at the boundaries | ˆ z | = ˆ r 0 and 1. The circle separating the inner and outer halves of the typeII lattice is at radius \nˆ r H = C 2 (0) = e -2 π/ ( kP ) = √ ˆ r 0 , (S21) \nwhich can be interpreted as the black hole horizon.", 'S3. RELATION BETWEEN TYPE-II AND BTZ GEOMETRY': "In this section, we show that timeslices of the threedimensional BTZ black hole correspond to the type-II geometry simulated on our hyperbolic lattices. For this purpose, we first recall some facts about the BTZ solution following Ref. [63], and then show explicitly how timeslices are embedded in the Poincar'e upper half-plane and disk models, H and D , respectively. \nWe consider the imaginary-time BTZ black hole, which is a solution to the imaginary-time Einstein equations in vacuum in 2 + 1 spacetime dimensions for a negative cosmological constant Λ = -1 /ℓ 2 , given by the metric \nd s 2 BTZ = ( ρ 2 ℓ 2 -M ) d τ 2 + ( ρ 2 ℓ 2 -M ) -1 d ρ 2 + ρ 2 d ϑ 2 . (S22) \nHere M is a dimensionless parameter and ρ ≥ ρ H is a non-compact radial coordinate that describes the region outside the event horizon, which is located at the singular radius \nρ H = √ Mℓ. (S23) \nThe imaginary time and radial coordinates, τ and ϑ , are compactified to the intervals τ ∈ [0 , 2 πℓ 2 /ρ H ) and ϑ ∈ [0 , 2 π ) through the identifications \n( ρ, τ, ϑ ) ∼ ( ρ, τ +2 π ℓ 2 ρ H , ϑ ) , (S24) \n( ρ, τ, ϑ ) ∼ ( ρ, τ, ϑ +2 π ) . (S25) \nThe periodicity in τ with period T -1 corresponds to a black hole temperature \nT = ρ H 2 πℓ 2 = √ M 2 πℓ . (S26) \nAs any solution to the vacuum Einstein field equations in 2+1 dimensions with negative cosmological constant, the BTZ black hole is equivalent to a quotient H 3 / ∼ , where H 3 is hyperbolic three-space with constant negative curvature, and ∼ is an equivalence relation. Here we denote H 3 in the upper half-space coordinates \nH 3 = { ( x, y, z ) ∈ R 3 , z > 0 } , (S27) \nd s 2 = ℓ 2 z 2 (d x 2 +d y 2 +d z 2 ) . (S28) \nThe equivalence is made explicit by the coordinate transformation \nx = ( 1 -ρ 2 H ρ 2 ) 1 / 2 cos ( ρ H τ ℓ 2 ) e √ Mϑ , y = ( 1 -ρ 2 H ρ 2 ) 1 / 2 sin ( ρ H τ ℓ 2 ) e √ Mϑ , (S29) z = ρ H ρ e √ Mϑ . \nThe identification in Eqs. (S24) and (S25) limits the image to a region S 3 / ∼ given by \nS 3 = { ( x, y, z ) ∈ H 3 , 1 ≤ √ x 2 + y 2 + z 2 < e 2 π √ M } (S30) \nwith the identification \n( x, y, z ) ∼ e 2 π √ M ( x, y, z ) . (S31) \nThis consists of the region between two hemispheres in the upper half-space, where the inner and outer hemisphere are identified such that identified points lie on rays in H 3 that go through the origin (0 , 0 , 0). Timeslices of the BTZ geometry with a fixed value of τ correspond to d τ = 0 in Eq. (S22), They are described by the static black hole metric \nd s 2 = ( ρ 2 ℓ 2 -M ) -1 d ρ 2 + ρ 2 d ϑ 2 . (S32) \nEmbedding timeslices into H . We define the twodimensional Poincar'e upper half-plane model through \nH = { x +i z ∈ C , z > 0 } , (S33) \nd s 2 = ℓ 2 z 2 (d x 2 +d z 2 ) . (S34) \nChoosing τ + = 0 or τ -= 1 2 T -1 = πℓ 2 ρ H in Eqs. (S29) yields \nx = ± ( 1 -ρ 2 H ρ 2 ) 1 / 2 e √ Mϑ , y = 0 , (S35) z = ρ H ρ e √ Mϑ , \nFIG. S2. Timeslice of the BTZ black hole with τ = 0 after mapping to the upper half-plane model H (left) and Poincar'e disk model D (right). Points along the geodesics indicated in green are identified; this reveals the quotient structure of the geometry. We show the points ± x H in D that are mapped to ± w/ 2 in the strip coordinates under C 2 , which gives the relation T = w/ (8 πℓ ). \n<!-- image --> \nwhere the plus (minus) sign in x corresponds to τ + ( τ -). Discarding the y -variable, this yields Universes I and II embedded into H given by S 2 , I / ∼ and S 2 , II / ∼ , respectively, with \nS 2 , I = { ( x, z ) ∈ H , 1 ≤ √ x 2 + z 2 < e 2 π √ M , x > 0 } , (S36) \nS 2 , II = { ( x, z ) ∈ H , 1 ≤ √ x 2 + z 2 < e 2 π √ M , x < 0 } (S37) \nand the identification \n( x, z ) ∼ e 2 π √ M ( x, z ) . (S38) \nThe event horizon ( ρ = ρ H ) corresponds to the topological circle S 2 , H / ∼ with \nS 2 , H = { (0 , z ) ∈ H , 1 ≤ z < e 2 π √ M } . (S39) \nThis corresponds to a region between two geodesics (circular arcs) that are identified. See Fig. S2a for a visualization. \nEmbedding timeslices into D . We achieved to show that timeslices τ ± corresponds to the region between two geodesics that are identified. This is the starting point of the squash-and-wrap procedure and thus proves the equivalence to the type-II geometry. To make the correspondence even clearer, we formulate the timeslices in the Poincar'e disk geometry. The mapping between H and D is facilitated by the Cayley transformation. \nH → D , (S40) \nw ↦→ z = w -i w +i . (S41) \nHere we work with the complex conjugate of the Cayley map for convenience and choose ϑ ∈ [ -π, π ) to obtain a more symmetric image. The mapping in Eqs. (S35) translate to Poincar'e disk coordinates z = x + i y ∈ D \nwith \nx = ρ ρ H sinh( √ Mϑ ) 1 + ρ ρ H cosh( √ Mϑ ) , (S42) \ny = ± 1 √ ρ 2 ρ 2 H -1 1 + ρ ρ H cosh( √ Mϑ ) . (S43) \nThe static BTZ metric in Eq. (S32) is mapped to the hyperbolic metric \nd s 2 = (2 ℓ ) 2 | d z | 2 (1 -| z | 2 ) 2 . (S44) \nThe image splits into Universes I and II in D with points along a geodesic identified, see Figs. 1b and S2b. Along the horizon with ρ = ρ H we have y = 0 and -x H ≤ x < x H with opposite ends identified, and \nx H = tanh ( ρ H ϑ max 2 ℓ ) = tanh ( ρ H π 2 ℓ ) . (S45) \nUnder the map C 2 from Eq. (S18), x H is mapped to the point w 2 in the strip geometry, and hence \nw 2 = C 2 ( x H ) = 2 π ln ( 1 + x H 1 -x H ) = 4 π ρ H π 2 ℓ = 2 ρ H ℓ . (S46) \nUsing Eq. (S26), we conclude that the relation between T and w in the type-II geometries is given by \nT = w 8 πℓ . (S47)", 'S4. DERIVATION OF EQS. (7) AND (8)': "We divide the graph G into bulk (interior) and boundary and use the following notation: \nThe quadratic part of the action is \nS 0 ( { ϕ µ } ) = 1 2 ∑ µ,ν M µν ϕ µ ϕ ν (S48) \nwith M = -A + ω 1 a real symmetric matrix. We set the values of ϕ a on ∂ G to J a and consider the generating function \nZ 0 ( { J a } ) = ∫ ( ∏ i d ϕ i ) e -S 0 ( { ϕ i ,ϕ a = J a } ) = e -S 0 , eff ( { J a } ) (S49) \nwith \nS 0 , eff = 1 2 ∑ a,b ( M ab -∑ i,j M ai G ij M jb ) J a J b (S50) \nand \nG \nij \n= ( \nM \n- \n1 \n) \nij \n. \nBoundary two-point function. We obtain the noninteracting two-point boundary correlator \n⟨O a O b ⟩ 0 = ∂ 2 ln Z 0 ∂J a ∂J b = -∂ 2 S eff ∂J a ∂J b = -M ab + ∑ i,j M ai ( M -1 ) ij M jb . (S51) \n̸ \nFor u = 0 this coincides with ⟨O a O b ⟩ and yields Eq. (7). We neglect here the subleading self-energy term that arises for u = 0, as is commonly done in holographic studies. \nBoundary three-point function. Note that the noninteracting three-point boundary correlator vanishes, \n⟨O a O b O c ⟩ 0 = ∂ 3 ln Z 0 ∂J a ∂J b ∂J c = 0 , (S52) \nas do higher n -point functions. Including the cubic interaction term we have \nS ( { ϕ µ } ) = S 0 ( { ϕ µ } ) + u 3! ∑ µ ϕ 3 µ . (S53) \nWe treat interactions perturbatively according to \nZ ( { J a } ) = e -u 3! ∑ a J 3 a ∫ ( ∏ i d ϕ i ) e -S 0 ( { ϕ i ,J a } ) -u 3! ∑ i ϕ 3 i ≃ e -u 3! ∑ a J 3 a ∫ ( ∏ i d ϕ i )( 1 -u 3! ∑ j ϕ 3 j ) e -S 0 ( { ϕ i ,J a } ) = e -u 3! ∑ a J 3 a Z 0 ( { J a } ) ( 1 -u 3! ∑ i ⟨ ϕ 3 i ⟩ 0 ) . (S54) \nWe have \n⟨ ϕ 3 i ⟩ 0 = 3 ⟨ ϕ 2 i ⟩ 0 , con ⟨ ϕ i ⟩ 0 + ⟨ ϕ i ⟩ 0 ⟨ ϕ i ⟩ 0 ⟨ ϕ i ⟩ 0 (S55) \n=: ∑ a J a w ( i ) a + ∑ a,b,c J a J b J c w ( i ) abc (S56) \nwith 'con' indicating the connected correlation function. We show below that \n⟨ ϕ i ⟩ 0 = -∑ a ∑ j J a M aj ( M -1 ) ji , (S57) \n⟨ ϕ i ϕ j ⟩ 0 = ( M -1 ) ij + ⟨ ϕ i ⟩ 0 ⟨ ϕ j ⟩ 0 , (S58) \n⟨ ϕ i ϕ j ⟩ 0 , con = ( M -1 ) ij , (S59) \nwhereas ⟨ ϕ 3 i ⟩ 0 , con = 0. We then obtain \nw ( i ) a = -3 M ii ∑ j M aj G ji , (S60) \nw ( i ) abc = -( ∑ j M aj G ji )( ∑ k M bk G ki )( ∑ l M cl G li ) , (S61) \nthe latter being symmetric in the indices a, b, c . The perturbative generating function becomes \nZ ≃ e -u 3! ∑ a J 3 a Z 0 ( 1 -u 3! ∑ i ∑ a J a w ( i ) a -u 3! ∑ i ∑ a,b,c J a J b J c w ( i ) abc ) . (S62) \nWe re-exponentiate this to obtain \nln Z ≃ -1 3! ∑ a uJ 3 a -S 0 , eff -u 3! ∑ i ∑ a J a w ( i ) a -u 3! ∑ i ∑ a,b,c J a J b J c w ( i ) abc . (S63) \nWe then find \n⟨O a O b O c ⟩ = ∂ 3 ln Z ∂J a ∂J b ∂J c = -uδ ab δ ac -u ∑ i w ( i ) abc . (S64) \nFor a = b we arrive at \n̸ \n⟨O a O b O c ⟩ = u ∑ i ( ∑ j M aj G ji )( ∑ k M bk G ki )( ∑ l M cl G li ) , (S65) \nwhich is Eq. (8). \nCorrelation functions of ϕ with sources. We now derive Eqs. (S57)-(S59). To compute correlation functions of bulk fields ϕ i in the presence of boundary sources J a , we introduce auxiliary bulk sources h i according to \nZ 0 ( { J a , h i } ) = ∫ ( ∏ i d ϕ i ) e -S 0 ( { ϕ i ,J a } )+ ∑ i h i ϕ i (S66) \n= e -S 0 , eff [ J ] -∑ i,j ∑ a M ai J a ( M -1 ) ij h j + 1 2 ∑ i,j ( M -1 ) ij h i h j \n(S67) \nand correlation functions are obtained from \n⟨ ϕ i 1 · · · ϕ i n ⟩ 0 = ∂ n Z 0 ∂h i 1 · · · ∂h i n ∣ ∣ ∣ h =0 , (S68) ⟨ ϕ i 1 · · · ϕ i n ⟩ 0 , con = ∂ n ln Z 0 ∂h i 1 · · · ∂h i n ∣ ∣ ∣ h =0 . (S69) \nThis yields Eqs. (S57)-(S58). All connected n -point functions of ϕ i vanish, because ln Z 0 is quadratic in h .", 'S6. DERIVATION OF EQ. (10)': "In this section we discuss the expected behavior of the 2-pt function ⟨O a O b ⟩ for both type-I and type-II geometries in the continuum. This enables the identification of the respective expressions for d ab in Eq. (10). Our \napproach is based on the observation that the two-point function asymptotically behaves like ⟨O a O b ⟩ ≃ e -∆ σ ab /ℓ , where σ ab is the large geodesic distance between two sites z a and z b . The latter, however, can be computed in the continuum for any two sites z and z ' . \nBoth type-I and type-II geometries are locally AdS, with a Riemannian (all-positive) signature. The equation of motion of real scalar fields in the bulk are expressed through the Laplace-Beltrami operator □ = (1 / √ det( g )) ∂ µ ( √ det( g ) g µν ∂ ν ). The bulk Green function satisfies \n( -□ + m 2 ) G ( Z ; Z ' ) = 1 √ det( g ) δ ( d +1) ( Z -Z ' ) . (S70) \nwhere Z denotes the generalized coordinate of the AdS 2 with d = 1. For general AdS d+1 , the solution is \nG ( Z ; Z ' ) = G ( ξ ( Z, Z ' )) , (S71) G ( ξ ) = ξ ∆ 2 F 1 ( ∆ , d 2 ; ∆ + 1 -d 2 ; ξ ) , \nwhere m 2 ℓ 2 = ∆(∆ -d ), and 0 ≤ ξ ( Z, Z ' ) -1 = cosh( σ ( Z, Z ' ) /ℓ ) ≤ 1 is a dimensionless parameter, determined by the geodesic distance σ ( Z, Z ' ) [4]. The hypergeometric function is denoted by 2 F 1 . For large σ , the term containing 2 F 1 can be neglected and we have G ( σ ) ≃ ξ ∆ ≃ e -∆ σ/ℓ . \nType-I geometry.-The geodesic distance in the Poincar'e disk with metric \nd s 2 I = (2 ℓ ) 2 (1 -| z | 2 ) 2 | d z | 2 (S72) \nis given by \nσ ( z, z ' ) = ℓ arcosh ( 1 + 2 | z -z ' | 2 (1 -r 2 )(1 -r ' 2 ) ) . (S73) \nFor sites close to the boundary, r, r ' → 1, we can use arcosh( x ) ≃ ln(2 x ) for x ≳ 4, to approximate this as \nσ ( z, z ' ) ≃ ℓ ln ( 4 | z -z ' | 2 (1 -r 2 )(1 -r ' 2 ) ) . (S74) \nThe corresponding correlation function reads \nG ( z ; z ' ) ≃ ( (1 -r 2 )(1 -r ' 2 ) 4 | z -z ' | 2 ) ∆ . (S75) \nFor two sites z = re i θ and z ' = r ' e i θ ' close to the boundary, we then obtain the boundary 2-pt function \nG ( θ, θ ' ) ≃ ˜ C 2 | e i θ -e i θ ' | 2∆ . (S76) \nComparing this with \nG ( θ a , θ b ) ≃ ¯ C 2 ( d ab ) 2∆ (S77) \nleads to the first line of Eq. (10). \nThe coefficient ˜ C 2 depends on r, r ' and, technically, vanishes as r, r ' → 1. In the continuum, choosing the boundary on a circle of radius R < 1, this can be absorbed by a redefinition of the field, while on the discrete lattice (due to the variation in the discrete r i ), there is a residual dependence of ⟨O a O b ⟩ on r a and r b . This effect, however, is small for large enough flakes. \nType-II geometry. The continuum geometry of the type-II lattice (Eq. 4 and Eq. S20) can be parametrized in angular coordinates ˆ z = ˆ re iθ through the metric \nd s 2 II = ( ℓw 4 ) 2 dˆ r 2 + ˆ r 2 d θ 2 ˆ r 2 sin 2 ( w 4 log ˆ r ) = ℓ 2 d[ w 4 log ˆ r ] 2 +d[ w 4 θ ] 2 sin 2 ( w 4 log ˆ r ) , (S78) \nwith w = kP = 8 πTℓ as defined in Eq. (S47). The locally-AdS geometry can also be described by the upper \nhalf-plane metric \nd s 2 II = ℓ 2 y 2 ( d x 2 +d y 2 ) , (S79) \nfrom which the angle-coordinate metric [Eq. (S78)] of the same geometry is obtained by the coordinate transformations, \nx = e wθ/ 4 cos ( w 4 log ˆ r ) , y = e wθ/ 4 sin ( w 4 log ˆ r ) . (S80) \nThe coordinate transformation also provides a map to the type-II Green function from the corresponding function expressed in terms of the upper half-plane geodesic distances. For two sites on the Poincar'e upper half-plane, ( x, y ) and ( x ' , y ' ), the Green function for long distances is given by [4] \nG ( x, y ; x ' , y ' ) ≃ ( 2 yy ' y 2 + y ' 2 +( x -x ' ) 2 ) ∆ . (S81) \nDefine χ = w 4 log ˆ r , hence ( x, y ) = e wθ/ 4 (cos χ, sin χ ), so that the Green function becomes \nG ( χ, θ ; χ ' , θ ' ) ≃ ( 2 sin χ sin χ ' e w 4 ( θ + θ ' ) e w 2 θ sin 2 χ + e w ' 2 θ ' sin 2 χ ' +( e w 4 θ cos χ -e w 4 θ ' cos χ ' ) 2 ) ∆ . (S82) \nTo approach the outer boundary of the type-II graph, ˆ r → 1, we introduce an infrared regulator ε according to ˆ r = e -4 πε/w or, equivalently, χ = -πε . The two-point function on the boundary becomes \nG ( θ, θ ' ) ≃ lim ε,ε ' → 0 ( 2 π 2 εε ' [sinh πTℓ ( θ -θ ' )] 2 ) ∆ . (S83) \nAs in the type-I case, the vanishing prefactor as ε, ε ' → 0 can be absorbed into a redefinition of the boundary field. By enforcing angular periodicity via the method of images, we obtain the 2-point function between between boundary sites θ a and θ b as \nG ( θ a , θ b ) ≃ ¯ C 2 ∑ n ∈ Z ( 1 sinh[ πTℓ ( θ a -θ b +2 πn )] ) 2∆ . (S84) \nThis form validates the choice of d ab for the type-II graphs in Eq. (10).", 'S5. DERIVATION OF EQ. (11)': "To derive Eq. (11), we extend the presentations in Refs. [11, 19]. We consider the non-interacting part of the lattice action S ( { ϕ µ } ) and consider the equation of motion for the field on a bulk site z i ∈ D with q neighbors given by \n-t ∑ µ A µi ϕ ( z i ) + ˆ m 2 ϕ ( z i ) = 0 . (S85) \nWe set t = 1. This can be written as \nq ∑ α =1 ϕ ( z i -w α 1 -w α ¯ z i ) = ˆ m 2 ϕ ( z i ) , (S86) \nwhere the sum extends over the neighbors of z i . Here w α = he 2 π ( α -1)i /q e i χ i and χ i a defect angle that depends on z i . The dimensionless parameter h is a lattice constant given by \nh = ( 1 -sin 2 ( π q ) cos 2 ( π p ) ) 1 / 2 . (S87) \nThe expression on the left-hand side of Eq. (S86) can be expanded in powers of h with the universal leading term given by [11] \n( q + qh 2 ℓ 2 □ + O ( h 3 ) ) ϕ ( z i ) = ˆ m 2 ϕ ( z i ) . (S88) \nThis is equivalent to the Klein-Gordon type equation \n( -□ + m 2 ) ϕ ( z ) = 0 (S89) \nthrough the identification \nˆ m 2 = q + qh 2 m 2 ℓ 2 . (S90) \nWe define the dimensionless Laplacian ˆ □ = ℓ 2 □ . In the Poincar'e disk with z = x + i y ∈ D we have ˆ □ = 1 4 (1 -| z | 2 ) 2 ( ∂ 2 x + ∂ 2 y ) = (1 -| z | 2 ) 2 ∂ z ¯ ∂ z . Eigenfunctions of ˆ □ are labelled by a real quantum number ∆ such that ˆ □ ϕ ∆ ( z ) = ∆(∆ -1) ϕ ∆ ( z ). (This is most easily seen in upper half-plane coordinates x + i y ∈ H with ˆ □ = y 2 ( ∂ 2 x + ∂ 2 y ) and ϕ ∆ ( x, y ) = y ∆ .) We have \n( q + qh 2 ∆(∆ -1) ℓ 2 + O ( h 3 ) ) ϕ ∆ ( z i ) = ˆ m 2 ϕ ∆ ( z i ) , (S91) \nor, \n∆(∆ -1) + O ( h ) = m 2 ℓ 2 , (S92) \nwhich recovers the universal continuum limit result. \nHigher orders in an expansion in the parameter h in Eq. (S88) can be derived. Importantly, terms that are not powers of ˆ □ appear at order h q , and so the power series needs to be stopped earlier. For the { 3 , 7 } tessellation with q = 7 we have \nq ( 1 + h 2 ˆ □ + h 4 4 ( ˆ □ 2 +2 ˆ □ ) (S93) \n+ h 6 36 ( ˆ □ 3 +10 ˆ □ 2 +12 ˆ □ ) + O ( h 7 ) ) ϕ ( z i ) = ˆ m 2 ϕ ( z i ) , (S94) \nwhich is of the form \nq + qh 2 f ( ˆ □ ) ϕ ( z i ) = ˆ m 2 ϕ ( z i ) (S95) \nwith f ( X ) given in Eq. (10). Acting on eigenfunctions ϕ ∆ we obtain \nf (∆(∆ -1)) ϕ ∆ = m 2 ℓ 2 ϕ ∆ , (S96) \nand hence the lattice relation between ∆ and m 2 ℓ 2 . For a general { p, q } lattice, the expansion reads \nf ( X ) = l max ∑ l =1 h 2( l -1) l ! 2 T l ( X ) + O ( h q -2 ) , (S97) \nFIG. S3. The power-law scaling of the 2-point boundary correlation functions obtained from Eq. (7), between two boundary points a and b . In (a) and (b) , the function on a type-I flake with 6 shells are shown with bulk scalar masses m 2 ℓ 2 = 0 . 01 and 0 . 74, respectively. The quoted values of ∆ are the fitted values. On the left panels, a fixed boundary site at θ a = 0 is chosen as a reference. On the right panels, three separate angular positions have been chosen for a , as the site b crawls along the boundary. The discreteness of the boundary introduces small amplitude variations but the scaling dimension remains largely unaffected, demonstrating the expected CFT behavior at zero temperature. The scatter (depicted as error bars) results from the binning of the boundary distances. Note that m = 0 is consistent with ∆ = 1. \n<!-- image --> \nwhich has to be stopped at order l max such that 2( l max -1) < q -1. For instance q = 3 means l max = 1 and q = 7 means l max = 3. The polynomials T l ( X ) are \nT 1 ( X ) = X, (S98) \nT 2 ( X ) = X 2 +2 X, (S99) \nT 3 ( X ) = X 3 +10 X 2 +12 X, (S100) \nT 4 ( X ) = X 4 +28 X 3 +156 X 2 +144 X. (S101) \nThey can be computed systematically to arbitrary order from a Taylor expansion of Eq. (S86).", 'Type-I / Zero temperature': "2-point functions.-We first check the scaling form of the 2-pt functions on the boundary by tuning the bulk \nscalar mass. In Fig. S3, we verify the scaling behavior \n⟨O a O b ⟩ ≃ ¯ C 2 | e i θ a -e i θ b | 2∆ (S102) \nfor ⟨O a O b ⟩ obtained from Eq. (7) in the main text, with the two boundary points θ a and θ b on the Poincar'e disk. Fig. S3 uses a binning of the correlation function dataset to reduce small-distance discretization effects. We consider different sample sizes for the binning and perform a least-square fitting of the data on a logarithmic grid. We pick the binning with the least net residual for each dataset depicted in Fig. S3 (a). While the binning reduces the scatter in the local magnitudes of the correlation function, the discrete lattice also introduces a ruggedness in the boundary geometry that requires further consideration. However, this effect is mitigated by sampling over several distinct pairs of boundary locations that share the same global distances. In effect, we fix the location of one site a , and let the other site b move all the way across the compact boundary. The right-hand panels in Fig. S3 depict the variation of the 2-point function for three such choices of the anchored site a . By averaging through many such boundary reference frames distinguished by the anchoring of their origin coordinates a , we obtain the mass parametrization of the scaling dimension ∆ that is depicted in the main text along with its related standard deviation. \n3-point functions.-For the 3-point function, we use Eq. (8) to obtain the boundary correlators and compare to the zero temperature scaling formula \n⟨O a O b O c ⟩ ≃ ¯ C 3 ( d ab d ac d bc ) ∆ (S103) \nwith d ab = | e i θ a -e i θ b | . Similar to the 2-point function, a linear least-square fit of the 3-point function over the product over the three relative distances d ab d ac d bc demonstrates the expected power-law behavior as seen in Fig. S4. In Fig. S4 (b) and (c), a robust evidence for the conformal symmetry emerges as the scaling dimension ∆ obtained from the scaling-collapse of the 3-point function for different masses shows good agreement with ∆obtained from the 2-point function. To average out the discreteness of the data, we consider independent sets of 3-point functions with one fixed angle θ a , and two varying angles θ b and θ c , the last of which parametrizes the data sets. A given realization with a fixed set of two angles θ a,c (Fig. S4 shows an example) contains 3-point function data that is binned and fitted, and the overall scaling dimension is obtained from averaging over the different data sets. This yields the comparison between the 2-point and 3-point fits of the scaling dimension shown in Fig 3d in the main text. \n<!-- image --> \nFIG. S4. The conformal scaling of the 3-point function computed from Eq. (8), applied to type-I graphs with 7 shells for masses m 2 ℓ 2 = 0 . 01, 0 . 48, and 0 . 90, plotted respectively in (a) , (b) , and (c) . The figures depict the correlation function for two fixed boundary sites a and c , with a third site b that moves along the boundary. In (a) , the function is plotted against the relative distance between the a and b coordinates, depicting the contact divergence when b coincides with the third coordinate c . In (b) and (c) , a scaling collapse over the product of the pairwise relative distances, fitted as the dashed grey lines, shows a remarkable agreement with the solid orange lines defined by the scaling dimension obtained from the 2-point function. \n<!-- image --> \nFIG. S5. Thermal boundary 2-point function obtained by applying Eq. (7) for a type-II graph with k = 12, obtained by squash-and-wrap from a flake with 6 shells, and bulk scalar mass m 2 ℓ 2 = 0 . 740. (a) The orange fit line is given by Eq. S104 with the sum extending over n ∈ [ -120 , 120]. In ( d ), the variation of the 2-point function over a set of varying anchoring coordinates θ a is shown to lead to only a mild variation of the fitted temperature. \n<!-- image --> \nFIG. S6. Type-II 2-point functions for m 2 ℓ 2 = 0 . 32 shown for ( a ) k = 6, ( b ) k = 8, ( c ) k = 10, and, ( d ) k = 12, plotted vs. θ a -θ b along with orange fit lines from Eq. (S104). \n<!-- image --> \nFIG. S7. Type-II 2-point functions for k = 6 , 8 , 10, and 12 ( (a)-(d) ) as in Fig. S6, plotted vs. d ab ( T ). \n<!-- image -->", 'Type-II / Finite temperature': "2-point functions.-We compute the 2-pt function through Eq. (7) on the type-II flakes for various values of k . To extract the temperature of the boundary CFT, we fit the boundary 2-pt correlator to the continuum limit form \n⟨O a O b ⟩ = ∑ n ∈ Z ¯ C ' 2 ( d ( n ) ab ) 2∆ (S104) \nwith \nd ( n ) ab = sinh( πTℓ | θ a -θ b +2 πn | ) πTℓ . (S105) \nThis continuum form is derived from the formula for a constant time-slice of the 3D BTZ black hole geometry. The sum over n is reminiscent of the method of images, and produces a periodic function. We find that the sum converges quickly with | n | ≲ 120 images and yields a very good agreement, see Fig. S5. While the scaling dimension ∆( m 2 ℓ 2 ) can be fitted for the type-II 2-pt function, we find that it is consistent with the type-I result ∆( m 2 ℓ 2 ). For this reason, to increase the accuracy of the temperature fit, we use the relation ∆( m 2 ℓ 2 ) from type-I for the fit of Tℓ in Eq. (S104) and confirm the relation Tℓ = kP/ 8 π . The rugged-boundary problem is \naddressed in the same way as for the type-I graphs discussed above. We average the temperature fits across various θ a -parametrized data sets, leading to the favorable comparison with the Hawking temperature associated with the BTZ geometry. \n3-point functions.-To produce Fig 3b in the main text, we fix θ a = 0 and θ c , and vary θ b . We compare the 3-point function obtained from applying Eq. (8) against the form anticipated for a timeslice of a 3D BTZ black hole given by \n⟨O a O b O c ⟩ = ∑ n ∈ Z ¯ C ' 3 ( d ab d ac d bc ) ∆ ∣ ∣ ∣ ∣ θ b → θ b +2 πn , (S106) \n= ∑ n ∈ Z ¯ C ' 3 ( d ( n ) ab d ac d ( -n ) bc ) ∆ . (S107) \nThe case of various masses is shown in Fig. S8, together with with their scaling collapse.", '3-pt coefficient': "The normalized 3-point coefficient C 3 is given by \nC 3 = ¯ C 3 ( ¯ C 2 ) 3 / 2 (S108) \nFIG. S8. The 3-point functions at the boundary of the type-II graph with k = 12, for m 2 ℓ 2 = 0 . 32 (a) and m 2 ℓ 2 = 0 . 90 (b) . The observed contact divergences (left panels) and asymptotic thermal scaling (right panels) are consistent with the expected conformal behavior. \n<!-- image --> \nFIG. S9. Normalized 3-point ratio (S109) plotted for fixed θ a = 0, and varying values of θ b and θ c on the boundary of the type-I graph obtained for the bulk scalar mass m 2 ℓ 2 = 0 . 11 (a) , 0 . 32 (b) , 0 . 63 (c) , and 0 . 95 (d) . The value of F abc approaches the saturation value C 3 away from the lines the contact divergences, given by θ c = θ a = 0, θ b = θ a = 0, and θ c = θ c . \n<!-- image --> \nin the type-I case, and similarly C 3 = ¯ C ' 3 / ( ¯ C ' 2 ) 3 / 2 in the type-II case. For a systematic way to fit C 3 , we construct \nFIG. S10. Asymptotic scaling form of the normalized ratio F abc for m 2 ℓ 2 = 0 . 06 plotted in (a) for the type-I boundary at zero-temperature, and for type-II boundary with k = 4, k = 8, and k = 10 (b)-(d) . The limit F abc ≃ C 3 (left region of the plots where the argument approaches zero) is approximately independent of T , see Fig. 3e in the maintext. \n<!-- image --> \nthe ratio \nF abc = ⟨O a O b O c ⟩ [ ⟨O a O b ⟩⟨O a O c ⟩⟨O b O c ⟩ ] 1 / 2 (S109) \nthat we sample over many values of a, b, c . The idea of using F abc is that, naively, one could simply sample \nlim d ab d bc d ca →∞ ⟨O a O b O c ⟩ ( d ab d ac d bc ) ∆ ≃ C 3 , (S110) \nand determine C 3 from this. However, for the type-II lattices we construct here as described in Sec. S2, the ring boundary is too rough and leads to a k -fold modulation of the 2-pt function already, which amplifies when considering the 3-pt function. (The type-I boundaries are also somewhat rough but less so than the type-II boundaries.) This could be incorporated by appropriate form-factors for both the 2-pt and 3-pt function. By considering F abc , instead, these effects are expected to cancel, and eventually yield a cleaner prediction of the actual physical type system, whose discrete boundary can be made as smooth as possible, in principle. As Fig. S9 shows, F abc only mildly depends on a, b, c and can be used to determine F abc ≃ C 3 as long as we stay away from the contact divergences that arise when any two of the boundary sites coincide. \nAs the comparison in Fig. S10 shows for a fixed m 2 ℓ 2 , this approach works satisfactory, with the long-distance \nasymptotic behavior of F abc yielding an estimate of C 3 . We find that C 3 remains consistently close across the type-I and type-II graphs. The coefficient depends on m 2 ℓ 2 and u as depicted in Fig. 3 (e) of the main text.", 'S8. SIMULATION IN ELECTRICAL CIRCUITS': "We first describe how to realize the equation of motion (EOM) in Eq. (12) in an electrical circuit network consisting of resistors and diodes. Resistors R 0 are connected between circuit nodes according to the adjacency graph of the respective lattice, resistors R g are connected between each node and ground, and a diode is placed between each node and ground. The resistors correspond to the linear terms of the EOM, whereas the diode introduces the on-site nonlinearity. \nThe current through a diode is approximately described by the Shockley equation I ( V ) = I S (e V V S -1), which, expanded to second order, reads I ( V ) = c 1 V + c 2 V 2 + O ( V 3 ), with c 1 , c 2 constants that depend on the specific diode. The values of ¯ V µ are represented by the voltages at the circuit nodes, with ¯ V i in the bulk and ¯ V a = J a is realized by applying a fixed voltage at the respective boundary sites. The EOM only needs to be implemented for the bulk sites ¯ V i . Starting from the general EOM in Eq. (12) for any site µ , \n∑ ν M µν ¯ V ν + u 2 ¯ V 2 µ = 0 (S111) \nwith M µν = -A µν + ˆ m 2 δ µν , we find \n∑ j M ij ¯ V j + u 2 ¯ V 2 i = I eff i (S112) \nfor the bulk sites, with an effective bulk source term \nI eff i = I eff i ( { J a } ) = -∑ a M ia J a . (S113) \nTo derive the EOM of the circuit, we add up all currents flowing out of one node, including up to second order in V i . According to Kirchhoff's current law, we then obtain \nI i = 1 R 0 ∑ j ( q δ ij -A ij ) V j + 1 R g V i + c 1 V i + c 2 V 2 i . (S114) \nHere, I i is any external current fed into the circuit nodes, for example from a connected voltage source. If no external supplies are added to a node then I i = 0, while a fixed voltage gives the corresponding current I i = I eff i ( { J a } ). Furthermore, in practice (and in numerical simulations), a small parasitic capacitance will be present at each node, contributing an additional term of C i ˙ V i ( t ). Much like the process of discharging a capacitor, the voltages at the circuit nodes will decay exponentially towards the equilibrium state where ˙ V i = 0, so that the solution for Eq. (12) is obtained as t →∞ . In the following, we assume that all parasitic capacitances are so small that the equilibrium (or a sufficient approximation thereof) is reached instantaneously. \nThe relationship between circuit parameters and those of Eq. (12) is found from matching \n∑ j ( -A ij + ˆ m 2 δ ij ) ¯ ϕ j + u 2 ¯ ϕ 2 i (S115) ! = c [ 1 R 0 ∑ j ( q δ ij -A ij ) V j + 1 R g V i + c 1 V i + c 2 V 2 i ] . (S116) \nHere, c = R 0 /V 0 is an arbitrary proportionality constant, and V 0 is a voltage scale that can be chosen freely (we set V 0 = 1 in the main text for simplicity). The matching yields the values shown this table: \n¯ ϕ µ ˆ m 2 u/ 2 V µ /V 0 q + R 0 ( R -1 g + c 1 ) V 0 R 0 c 2 \nWe chose a Schottky diode (model RBE1VAM20A) for its low threshold voltage and high switching speed. From a calibration fit of the diode we extract the parameters c 1 = 5 . 24 · 10 -4 and c 2 = 9 . 43 · 10 -3 . After arbitrarily fixing R 0 = 100Ω, we choose R g = 240 . 3Ω in order to obtain the target value of m 2 ℓ 2 = 1 qh 2 R 0 ( R -1 g + c 1 ) = 0 . 271, which is the mass used in Figs. 2 and 3 in the maintext. These parameters correspond to a non-linearity coefficient u/ 2 = 0 . 943 V 0 . \nThe linear-ramp protocol to compute W circ ( t ) is implemented in the analog electronic circuit simulator computer software LTspice for each set( a, b ) and ( a, b, c ) for the 2-pt and 3-pt correlation functions, respectively. The generating function of each run W circ ( t ) is exported to a Mathematica notebook where the corresponding linear system is solved giving the correlation functions. For the 3-pt function, while the LTspice simulation works in practice, we use Mathematica to simulate the circuit EOM directly to reduce noise of the data. A comparison for small flakes, where the deviations are smaller, is shown in Fig. S11. \nFIG. S11. Comparison of circuit simulation using Mathematica (green) and LTspice (pink) for a small flake. For larger flakes with more sites, as presented in Fig. 3b of the main text, the LTspice simulation becomes noisier or larger distances, while the Mathematica simulation is still reliable. Note that the circuit data is normalized differently (choice of C 3 ) compared to Fig. 3b to make both data sets agree better for small distances. \n<!-- image -->"}

2024arXiv240521054A

With stunning clarity JWST has revealed the Universes first billion years. The scientific community is analyzing a wealth of JWST imaging and spectroscopic data from that era and is in the process of rewriting the astronomy textbooks. Here 1.5 years into the JWST science mission we provide a snapshot of the great progress made towards understanding the initial chapters of our cosmic history. We highlight discoveries and breakthroughs topics and issues that are not yet understood and questions that will be addressed in the coming years as JWST continues its revolutionary observations of the Early Universe. While this compendium is written by a small number of authors invited to ISSI Bern in March 2024 as part of the 2024 ISSI Breakthrough Workshop we acknowledge the work of a large community that is advancing our collective understanding of the evolution of the Early Universe.

2024-05-01T00:00:00Z

['arXiv:2405.21054', '10.48550/arXiv.2405.21054', '2024arXiv240521054A']

['Astrophysics - Astrophysics of Galaxies']

The First Billion Years According to JWST

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2405.21054.pdf

{'No Header': 'Draft version June 3, 2024 \nTypeset using L A T E X twocolumn style in AASTeX631', 'The First Billion Years, According to JWST': "Participants of the ISSI Breakthrough Workshop 2024, 1 Angela Adamo, 2 Hakim Atek, 3 Micaela B. Bagley, 4 Eduardo Ba˜nados, 5 Kirk S. S. Barrow, 6 Danielle A. Berg, 4 Rachel Bezanson, 7 Maruˇsa Bradaˇc, 8, 9 Gabriel Brammer, 10, 11 Adam C. Carnall, 12 John Chisholm, 4 Dan Coe, 13, 14, 15 Pratika Dayal, 16 Daniel J. Eisenstein, 17 Jan J. Eldridge, 18 Andrea Ferrara, 19 Seiji Fujimoto, 20 Anna de Graaff, 21 Melanie Habouzit, 21 Taylor A. Hutchison, 22, ∗ Jeyhan S. Kartaltepe, 23 Susan A. Kassin, 24, 25 Mariska Kriek, 26 Ivo Labb'e, 27 Roberto Maiolino, 28, 29, 30 Rui Marques-Chaves, 31 Michael V. Maseda, 32 Charlotte Mason, 10, 11 Jorryt Matthee, 33 Kristen B. W. McQuinn, 34, 35 Georges Meynet, 31 Rohan P. Naidu, 36, † Pascal A. Oesch, 31, 10, 11 Laura Pentericci, 37 Pablo G. P'erez-Gonz'alez, 38 Jane R. Rigby, 22 Guido Roberts-Borsani, 31 Daniel Schaerer, 31 Alice E. Shapley, 39 Daniel P. Stark, 40 Massimo Stiavelli, 13 Allison L. Strom, 41, 42 Eros Vanzella, 43 Feige Wang, 44 Stephen M. Wilkins, 45, 46 Christina C. Williams, 47 Chris J. Willott, 48 Dominika Wylezalek, 49 and Antonella Nota 1, 13 \n1 \nInternational Space Science Institute, Hallerstrasse 6, 3012 Bern, Switzerland 2 Department of Astronomy, The Oskar Klein Centre, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden 3 Institut d'Astrophysique de Paris, CNRS, Sorbonne Universit'e, 98bis Boulevard Arago, 75014, Paris, France 4 Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA 5 Max Planck Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany 6 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W Green St, Urbana, IL 61801, USA 7 Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA 8 University of Ljubljana, Faculty of Mathematics and Physics, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia 9 Department of Physics and Astronomy, University of California Davis, 1 Shields Avenue, Davis, CA 95616, USA 10 Cosmic Dawn Center (DAWN) 11 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen N, Denmark 12 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK 13 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 14 Association of Universities for Research in Astronomy (AURA), Inc. for the European Space Agency (ESA) 15 Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD 21218, USA 16 Kapteyn Astronomical Institute, University of Groningen, 9700 AV Groningen, The Netherlands 17 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA 18 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 19 Scuola Normale Superiore, Piazza dei Cavalieri 7, 50126 Pisa, Italy 20 Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 21 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany 22 Astrophysics Science Division, Code 660, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771, USA 23 Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA 24 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21210, USA 25 Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA 26 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands 27 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122, Australia 28 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 OHA, UK 29 Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK 30 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK 31 Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland 32 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706, USA 33 Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria 34 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA \n35 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA \nCorresponding author: Antonella Nota \nantonella.nota@issibern.ch \n- 36 MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave., Cambridge, MA 02139, USA 37 INAF -OAR Osservatorio Astronomico di Roma, via di Frascati 33 00078 Monte Porzio Catone, Italy 38 Centro de Astrobiolog'ıa (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrej'on de Ardoz, E-28850, Madrid, Spain 39 Department of Physics and Astronomy, University of California, Los Angeles, 430 Portola Plaza, Los Angeles, CA 90095 40 Department of Astronomy, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721 41 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave., Evanston, IL, 60201, USA 42 Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA 43 INAF - OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy 44 Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85721, USA 45 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK 46 Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA\n- 47 48 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada\n- 49 Astronomisches Rechen-Institut, Zentrum fur Astronomie der Universitat Heidelberg, Monchhofstr. 12-14, D-69120 Heidelberg, Germany", 'ABSTRACT': "With stunning clarity, JWST has revealed the Universe's first billion years. The scientific community is analyzing a wealth of JWST imaging and spectroscopic data from that era, and is in the process of rewriting the astronomy textbooks. Here, 1.5 years into the JWST science mission, we provide a snapshot of the great progress made towards understanding the initial chapters of our cosmic history. We highlight discoveries and breakthroughs, topics and issues that are not yet understood, and questions that will be addressed in the coming years, as JWST continues its revolutionary observations of the Early Universe. While this compendium is written by a small number of authors, invited to ISSI Bern in March 2024 as part of the 2024 ISSI Breakthrough Workshop, we acknowledge the work of a large community that is advancing our collective understanding of the evolution of the Early Universe.", '1. INTRODUCTION': "The first billion years of cosmic history were most eventful, rich with firsts and transformations (see a graphic representation in Figure 1). After the Big Bang and initial accelerated expansion, the Universe's temperature and density constantly declined until photons and matter decoupled starting a period known as the cosmic 'Dark Ages'. It is during this period that the seeds leading to the formation of the first stars and primordial galaxies were growing. Slow perturbations of dark matter accreted baryons until the conditions for the ignition of the first stars were met; the very first sources of light in the Universe. The energy, momentum, and enriched material ejected by these stars initiated the sparks that would eventually lead to the last major phase-transition of the Universe, the epoch of Reionization (EoR), consisting in the ionization of the neutral hydrogen permeating the Universe. This time of infancy for galaxies, where star formation processes and black hole (BH) formation and growth take place, is fundamental for our understanding of galaxy growth and evolution. Unfortu- \nnately, we know the least about this early time - many questions remain about the basic facts and chronology. \nWhen did the first stars form? Were they very different from the stars today? Did any early galaxies have disks like our Milky Way, or were they disordered and bursting with star formation? When did the first supermassive black holes form and fuel active galactic nuclei (AGN) powerful enough to impact the star formation of entire galaxies? How quickly did supernovae enrich their surroundings with heavy elements and dust, the building blocks of planets and life? And what was the timeline and topology of the last major phase transition of the Universe, as light steadily reionized the hydrogen gas permeating the Universe? \nJWST is now enabling us to address these questions with direct observations of the light of these early galaxies emitted more than 13 billion years ago and redshifted into the infrared by the expansion of the Universe. This is the first and overarching success that JWST has unlocked: that it is possible to do these studies at all. With 100 × greater sensitivity than previous telescopes at similar wavelengths (Rigby et al. 2023), JWST 's four science instruments NIRCam, NIRSpec, NIRISS, and MIRI can complete in hours of observations what pre- \niously would have taken years, and therefore been unfeasible. \nWe will refer to lookback time and distance in units of redshift ( z ), starting at z = 5 . 6, one billion years after the Big Bang according to standard cosmological models (Planck Collaboration et al. 2020), and pushing back in time to the highest redshifts yet observed. \nWe dedicate this paper to the 20,000 people who spent decades to make JWST an incredible discovery machine.", '2.1. Extending the redshift frontier': 'JWST data reveal a robust population of high-redshift galaxies at z > 10, beyond the reach of the Hubble Space Telescope (HST, Figure 2). In most cases, the galaxies are selected through the signature, in multicolor imaging, of a sharp spectral cutoff blueward of the Lyman α line. This requires the galaxies to be reasonably luminous in the rest-frame UV. Many hundreds of such candidates have been identified at z > 8 (see next Section),and about twenty have been spectroscopically confirmed at z > 10, with the current high mark at z = 14 . 32 (Carniani et al. 2024). This record will surely not hold, as there is no technical limit on finding galaxies to z = 20.', '2.2. UV Luminosity Function': "From these samples, JWST has produced a precise determination of the galaxy UV luminosity function (LF) and luminosity density, ρ UV ( z ), at redshifts z ≳ 7 (see Fig. 2, e.g., Donnan et al. 2024; Bouwens et al. 2023; Harikane et al. 2023a; P'erez-Gonz'alez et al. 2023a; McLeod et al. 2024; Robertson et al. 2023a; Willott et al. 2023). The decrease of ρ UV ( z ) is slower than generally expected before JWST, implying a more sustained early production of UV photons. Moreover, we observe that the bright end of the LF does not significantly evolve between 8 < z < 12 (Naidu et al. 2022; Finkelstein et al. 2023; Leung et al. 2023; Chemerynska et al. 2023) due to an unexpectedly large number of luminous ( M UV ≃ -21) galaxies with blue spectral slopes β < -2 (Topping et al. 2023; Cullen et al. 2023a). The abundances are still limited in precision due to detected field-to-field variation attributed to large-scale structure (Willott et al. 2023), but these findings have important implications for cosmic reionization as discussed in Sec. 4. \nWhile it has been proposed that these results might not support the standard version of the ΛCDM cosmology, they can likely be explained with a better understanding of star formation and baryon physics. Vari- \nus physical scenarios, along with existing semi-analytic, semi-numerical models and simulations (e.g., Wilkins et al. 2023; Mauerhofer & Dayal 2023), have been proposed to explain the evolution of the LF. These include: (a) dust clearing by radiation-driven outflows (Ferrara et al. 2023; Ferrara 2023), (b) star formation variability resulting in a flattening of the LF (Mason et al. 2023; Mirocha & Furlanetto 2023; Pallottini & Ferrara 2023), (c) reduced feedback resulting in a higher star formation efficiency (Dekel et al. 2023; Li et al. 2023), and (d) a top-heavy IMF (Inayoshi et al. 2022a), although see (Rasmussen Cueto et al. 2023). \nThe faint end of the LF is also important, as it depends on the nature of feedback processes acting in lowmass galaxies. Pre-JWST data suggested a steepening slope at higher redshift, but early JWST results appear to indicate a convergence to a logarithmic slope of α = -2 in the LF, suggesting that extremely faint galaxies dominate the cosmic star-formation rate density. \nWhile the UV LF is based directly on observations, care must be taken in the interpretation. The UV LF contains several astrophysical degeneracies, as many processes can produce the same observed luminosity density. Viable models must simultaneously explain the tightly related stellar mass function evolution at lower redshifts (Harvey et al. 2024; Weibel et al. 2024).", '2.3. Stellar mass census': "In contrast to HST, JWST 's wavelength coverage and spectral sampling of the rest-frame optical allows tracking of the bulk of the stellar mass, rather than solely the young stellar populations probed by UV light. These constraints are resolving earlier degeneracies regarding the contribution of emission lines, versus stellar continuum, enabling more accurate characterization of galaxy physical properties even beyond z = 10 when the restframe optical emission of galaxies redshifts into the MIRI wavelength range. \nThe rest-frame optical observations enable improved constraints on stellar mass, placing the first robust constraints on the rate of mass buildup in early galaxies, back to the first 500 Myr (Harvey et al. 2024; Weibel et al. 2024). Pre-JWST observations with HST, Spitzer, and ALMA had hinted at the existence of rare, massive and red galaxies beyond z > 4 that were typically excluded from rest-frame UV selections. JWST data reveals that these previously missed galaxies also contribute significantly to the stellar census (Barrufet et al. 2023; Gottumukkala et al. 2023). An exciting frontier is access to the rest-frame infrared provided by MIRI, which is crucial to prevent overestimates of stellar mass \namong massive, red, dust obscured sources (Williams et al. 2023a; Wang et al. 2024). \nOverly massive galaxy candidates emerged from the first JWST data (e.g., Labb'e et al. 2023; Xiao et al. 2023). If real, these would require impossible or implausible star formation efficiencies (Boylan-Kolchin 2023). To date, most follow-up observations of the earliest ( z > 6) massive candidates allow for alternative interpretations (Desprez et al. 2023; Kocevski et al. 2023a). Nonetheless, early massive quiescent galaxies have been spectroscopically characterized, requiring rapid formation at z ∼ 10 (Carnall et al. 2023a; Glazebrook et al. 2023; Nanayakkara et al. 2024; Setton et al. 2024). The high number densities of these early quiescent galaxies, as implied by photometric samples, present severe challenges for theoretical galaxy formation models (e.g. Carnall et al. 2023b; P'erez-Gonz'alez et al. 2023b; Valentino et al. 2023). Larger spectroscopic samples, which enable more detailed stellar population modeling, will further constrain their formation histories and test the extent of this tension. \nAccurate characterisation of the optical rest-frame emission also paves the way to probe the growth of galaxies and the structural and kinematics of their stars and gas, and to map star formation sites and their feedback in terms of energy, momentum, metal enrichment injected in the galaxy medium, as discussed in detail below.", '2.4. Star-Formation Histories': "Modeling the unprecedented broadband spectral energy distributions from JWST reveals that z > 6 galaxies contain stars that are young, with typical inferred ages of ∼ 50 Myr, although younger ( < 10 Myr) (e.g., Whitler et al. 2023, 2024; Endsley et al. 2023b; Carnall et al. 2023c; Casey et al. 2023; P'erez-Gonz'alez et al. 2024) and older (Looser et al. 2023a; Strait et al. 2023) samples exist. Rest-frame optical ([OIII]+H β ) emissionline equivalent width distributions and UV slopes also indicate bursty star formation, with both on and off modes (Topping et al. 2022; Endsley et al. 2023a; Looser et al. 2023b), also referred in the literature as stochastic star formation episodes. Theoretical predictions suggest that stochasticity can be induced by various feedback processes, such as (i) photoevaporation of molecular hydrogen, (ii) supernova explosions, and (iii) cosmological accretion/merging which dominate low-, intermediate-, and high-mass systems, respectively, and act on different timescales (Pallottini & Ferrara 2023; Mirocha & Furlanetto 2023; Sun et al. 2023). These results have fueled a resurgence of attention to outshining by the youngest stellar populations, including examples from \nspatially resolved data that can isolate multiple stellar populations (Gim'enez-Arteaga et al. 2023; P'erezGonz'alez et al. 2023b; Fujimoto et al. 2024; Bradaˇc et al. 2024). Ultimately, as spectroscopic samples become more detailed and representative and analysis tools are optimized for the new datasets, we expect significant progress in this area.", '2.5. Structural and Kinematic Properties': "The dramatic improvement in the spatial resolution and wavelength coverage of JWST 's NIRCam instrument compared to its HST predecessors has enabled a sharper and more complete view of the distribution of stars in galaxies beyond z ∼ 7. The rest-frame optical size-mass relation has been measured into uncharted parameter space. At low masses, galaxies are more extended than conventional scaling relations would have predicted (e.g., Cutler et al. 2023). High-redshift galaxies at z ≳ 7 roughly extend earlier relations, more rapidly evolving for quiescent galaxies than their star forming counterparts (e.g., Yang et al. 2022; Morishita & Stiavelli 2023; Ito et al. 2023). \nThe same NIRCam imaging yields an unparalleled view of the stellar structures of those same galaxies. Although galaxies at high redshift ( z > 3) exhibit a broad diversity of morphologies, visual inspection has revealed that disk-like galaxies appear at early times ( ∼ 60% at z = 3, ∼ 30% at z ∼ 6 -9) (e.g., Kartaltepe et al. 2023; Robertson et al. 2023b; Treu et al. 2023; Pandya et al. 2023). These studies have included galaxies that were previously faint, or undetected in HST imaging (e.g., Nelson et al. 2023). Remarkably, the imaging also enables the identification of more detailed structures, including galactic bars at z ∼ 3 (Costantin et al. 2023), a disk plus proto-bulge at z ∼ 7 (Baker et al. 2023), and an enigmatic hint that some disk-like galaxies might actually be a population of elongated, or possibly prolate, systems (Vega-Ferrero et al. 2024; Pandya et al. 2023). Machine-learning methods, informed by comparisons with simulations, begin to address the scalability of morphological classification (e.g., Vega-Ferrero et al. 2024; Huertas-Company et al. 2023; Tohill et al. 2024). Although challenges remain in identifying ongoing mergers from imaging alone, the merger fraction of sub-populations have been quantified out to z ∼ 5 -7 (Asada et al. 2024). \nSimilarly, NIRSpec and the NIRCam grism have opened an unprecedented window into a range of kinematic structures, primarily from gas kinematics. In addition to a diversity of kinematic structures in smaller, Milky Way mass progenitors (e.g., de Graaff et al. 2023; Fujimoto et al. 2024), there are remarkable examples of \nmassive rotating disks in the early universe (Nelson et al. 2023; Arribas et al. 2023), and a host of non-virial structures in early galaxies driven by stellar and AGN-driven outflows observed in both emission and absorption (e.g., Carniani et al. 2023; D'Eugenio et al. 2023).", '2.6. Clustered star formation, proto-globular clusters, single stars': "JWST 's uniquely high spatial resolution has enabled us to map the physics of star formation within early galaxies (Figure 3). Early galaxies in deep cosmic fields show clumpy stellar structures and overall compact morphologies at scales of several hundred parsecs that dominate the galaxy's UV-blue optical light. Access to gravitational lensing enables us to resolve the galaxy light down to tens of parsecs or less. The clumpy structures observed in field galaxies break further down into smaller and denser stellar structures (e.g., Topping et al. 2024; Williams et al. 2023b; Hsiao et al. 2023; Bradaˇc et al. 2024; Claeyssens et al. 2023). In some cases, these data reach parsec scale resolutions, revealing young, massive star clusters in their interiors (Vanzella et al. 2023a; Adamo et al. 2024; Mowla et al. 2024). In general, more than 10% of the total stellar mass observed in these galaxies is in star clusters, which dominate the FUV light of the host galaxies. The recovered ages show a progression, with the youngest clusters coincident with extreme emission line equivalent widths and elevated ionizing photon production rate per UV photons ( ξ ion ). The latter is in agreement with predictions from stochastically populated evolutionary models of massive star clusters, which show that clustered star formation overall increases ξ ion (Stanway & Eldridge 2023). \nStar clusters detected in early galaxies are consistent with being gravitationally bound, thus, potential protoglobular clusters (GCs) (Vanzella et al. 2023a; Adamo et al. 2024; Mowla et al. 2024). Their stellar densities are comparable or higher than GCs, enabling runaway stellar and BH mergers in their cores (Antonini et al. 2023). Radiative and mechanical feedback from these proto-GCs is likely to be responsible for the extreme ionisation state of these early galaxies, as well as enhanced nitrogen abundances (Cameron et al. 2023; Topping et al. 2024), which are potentially linked to the chemical pattern of multiple stellar populations found in Milky Way GCs (Gratton et al. 2019). Star clusters within faint z > 6 galaxies might be important units for reionization. JWST observations of highly magnified galaxies will be fundamental in characterizing clustered star formation within these early galaxies, through cluster mass functions, cluster formation efficiency, and survival rates. \nRemarkably, gravitational lensing has also revealed individual stars as distant as Earendel at z = 6 (Welch et al. 2024). JWST observations have begun revealing their properties, showing most are hot and luminous O and B stars (Meena et al. 2023). Deeper JWST observations could deliver metallicities and outflow wind strengths of individual stars in the early universe (Lundqvist et al. 2024).", '2.7. Chemical Enrichment': "JWST 's spectroscopic capabilities have illuminated the buildup of metals within the first billion years, where direct oxygen abundances of roughly 2-30% solar have been measured for a dozen galaxies to date (Schaerer et al. 2022; Nakajima et al. 2023; Laseter et al. 2024; Sanders et al. 2024). These 'gold standard' direct oxygen abundances are being used to anchor methods based on much brighter emission lines (Sanders et al. 2023), which are detected in large samples of galaxies out to z ∼ 10 (see Figure 4). To date, the highest-redshift measurements of N, C, O, Ne, Ar, and S abundances reach as far as z = 12 . 5, only 350 Myr after the Big Bang (D'Eugenio et al. 2023). \nDozens of robust oxygen abundances based on electron temperatures have been measured for the first time between z ∼ 4 -10 (see, e.g., Morishita et al. 2024; Laseter et al. 2024, for compilations), spanning roughly 2-50% solar, with the promise of many more to come. Early JWST results demonstrate a continued gradual evolution towards lower total metallicity at fixed stellar mass beyond z = 4, and intriguing evidence for redshift evolution in the baryon cycle at z > 8 (e.g., Nakajima et al. 2023; Curti et al. 2023). Importantly, JWST has proven to be a powerful 'metal detector' for prominent highionization emission lines of C, N, and Ne (e.g., Bunker et al. 2023; Maiolino et al. 2023a; Chisholm et al. 2024; Topping et al. 2024); these observations strongly suggest non-solar abundance patterns (C/O and N/O; e.g., Cameron et al. 2023; Jones et al. 2023; D'Eugenio et al. 2023) that may be the unique nucleosynthetic legacy of early stars. \nTo date with JWST the lowest measured oxygen abundances reach a few percent of the solar value (e.g., Atek et al. 2024; Maseda et al. 2023), with evidence reported for one object of abundances below 1% (Vanzella et al. 2023b). It remains unclear if this observed metallicity floor is the result of very rapid chemical enrichment on galactic scales (with associated light-weighted galaxy ages of ≲ 5 Myr) or a selection effect. JWST has thus not conclusively identified any first-generation galaxies that are composed entirely of pristine gas and Population III stars. Nor have we yet seen other signatures of \nindividual first stars, such as those that have been gravitationally lensed, or caught when they explode as the hypothesized pair-instability supernovae.", '2.8. Interstellar Dust': "JWST has provided an unprecedented view of dust properties in early galaxies. NIRCam photometry and NIRSpec spectroscopy have shown that the rest-UV continuum slopes of star-forming galaxies in the early Universe are typically blue, as is characteristic of young, dust-poor systems (e.g., Topping et al. 2023; Cullen et al. 2023b,a; Roberts-Borsani et al. 2024). These results corroborate previous HST photometric studies of bright galaxies and, for the first time, extend our understanding to a more representative sample, even offering a tantalizing suggestion of dust-free stellar populations at z > 10 (Curtis-Lake et al. 2023). JWST has also delivered the first constraints on nebular attenuation using hydrogen recombination lines past z ∼ 2 . 5, revealing surprisingly little evolution at fixed stellar mass out to z ∼ 7 (Shapley et al. 2023; Sandles et al. 2023). Given the general gas and metal evolution of galaxies, the constancy of attenuation over such a large redshift range is puzzling. \nThe incredible sensitivity of JWST has enabled the identification of optically faint, dusty sources that had previously eluded detection (e.g., Kokorev et al. 2023a; Barrufet et al. 2023; P'erez-Gonz'alez et al. 2023b). Such sources are now included, approaching a more complete census of star formation within the first billion years (Xiao et al. 2023; Williams et al. 2023a). \nThe properties of dust grains reflect key processes of chemical enrichment, as well as heating and cooling in the interstellar medium. JWST has enabled characterization of the composition and size distribution of the dust grains, with the detection of the 2175 ˚ A feature at z = 6 . 7, suggesting a rapid formation timescale for carbonaceous grains (Witstok et al. 2023). The extinction curve seems to flatten at high redshifts, perhaps reflecting a change of the dust grain size distribution, and of the main dust production sources (Markov et al. 2024).", '2.9. Large-scale Environments': 'Early photometric and grism studies reveal substantially clustered megaparsec-scale distributions of galaxies and AGN beyond z ≳ 5 (Endsley et al. 2023b; Herard-Demanche et al. 2023; Kashino et al. 2023a; Wang et al. 2023). A typical JWST field contains multiple prominent peaks in the histogram of redshift, spanning over-densities δ = 1 -100 on arcmin scales. That high-redshift galaxies would be highly clustered was theorized, following mid-redshift measurements, but it indicates that care must be taken in interpreting the results \nof surveys in single sight-lines and narrow redshift intervals. Substantial field-to-field variations are already observed (e.g. Eilers et al. 2024; Helton et al. 2024; Willott et al. 2024). \nFurther, at low redshift, it is well known that galaxy properties differ between high and low density environments. It is not yet known what high-redshift properties will correlate substantially with environment.', '3. ACTIVE GALACTIC NUCLEI (AGN) IN EARLY GALAXIES': "Before the advent of JWST , observational studies were limited to detecting the most massive accreting BHs at high redshift. Such observations had confirmed the existence of extremely luminous and rare quasars ( L bol ⩾ 10 45 -46 erg / s, ∼ 1 Gpc -3 ) as early as z ∼ 7 . 6, powered by BHs of M BH ⩾ 10 8 M ⊙ . JWST has not only detected, for the first time, the stellar light of these quasars' host galaxies, but has also uncovered the presence of fainter AGN with L bol ∼ 10 43 -46 erg / s (Fig. 5). The demographics of BHs now encompass objects with masses of ∼ 10 6 -10 9 M ⊙ at z ∼ 3 -10.", '3.1. Demographics and Identification': "Many candidate AGN have been identified through JWST imaging (broadband, photometric SED, Onoue et al. 2023; Barro et al. 2023; Labbe et al. 2023; P'erezGonz'alez et al. 2024; Williams et al. 2023a; Kokorev et al. 2024, and references therein). Follow-up spectroscopy has confirmed the presence of AGN at high redshift, with many identifications occurring serendipitously during large spectroscopic surveys. AGN identifications primarily rely on the detection of a broad component of permitted lines, notably H α , without any counterpart in the forbidden lines (e.g., [OIII]5008), ruling out outflows (e.g. Kocevski et al. 2023b; Ubler et al. 2023a; Matthee et al. 2023; Maiolino et al. 2023b; Greene et al. 2023, and references therein). The SEDs of some broad-line or candidate AGN have raised additional puzzles: some are heavily obscured by dust ( A v ∼ 1 -4, Greene et al. 2023; Matthee et al. 2023), and some less extreme dust obscured AGN candidates have flatterthan-expected SEDs at rest-frame ⩾ 1 µ m. The latter might indicate that early obscured AGN could lack hot dust emission or that their rest-frame NIR is dominated by stars (Williams et al. 2023a; P'erez-Gonz'alez et al. 2024, with MIRI imaging). \nA large fraction of BH growth could occur in obscured phases; there are now efforts to find these. While the classical BPT diagram seems to break down at high redshift, several narrow-line AGN were recently successfully identified from high ionization UV emission lines (e.g. \nScholtz et al. 2023; Chisholm et al. 2024). Interestingly, most of these AGN could be X-ray weak (Yue et al. 2024; Ananna et al. 2024; Maiolino et al. 2024), and various reasons are being explored (e.g., intrinsic weakness, extreme absorption).", '3.1.1. A large population': "These newly discovered AGN with L bol ∼ 10 44 -10 46 erg / s appear more numerous than expected from the extrapolation of the luminous quasars' luminosity function, from X-ray selected AGN, or from some cosmological simulations (Habouzit 2024). Specifically, number densities range from 10 -5 to 10 -3 mag -1 cMpc -3 in the UV magnitude range -17 < M UV < -21 where M UV is the magnitude of the AGN and their hosts \nThe fraction of early galaxies hosting an AGN depends strongly on the sensitivity, AGN bolometric luminosity, and whether colour, size or spectroscopic criteria are used, such that the AGN fraction varies from ∼ 1% to ∼ 20%.", '3.1.2. BH masses': "JWST's wavelength range has made it possible to estimate the BH mass of type-1 AGN at high redshift with the same tracers as those used in the local Universe, namely Balmer broad lines such as H α and H β . This finally enables consistent comparisons across redshift. For the small number of high-redshift quasars measured to date, the Balmer-derived masses are consistent with previously-derived Mg II masses, within a scatter of 0.5 dex (e.g., Yang et al. 2023; Marshall et al. 2023; Bosman et al. 2023; Loiacono et al. 2024).", '3.2. Quasar and AGN hosts': 'JWST data have revealed, for the first time, the stellar light of a few quasar hosts at z ⩾ 6 (e.g., Ding et al. 2023; Stone et al. 2023; Yue et al. 2023), indicating that quasars reside in galaxies with stellar masses of 10 10 -11 . 5 M ⊙ . In addition, JWST allowed astronomers to delve into the host properties (stellar mass, velocity dispersion) of the recently discovered AGN at z ⩾ 4, and already raised intriguing questions. Such AGN seem to be embedded in galaxies with stellar masses of 10 8 -10 . 5 M ⊙ . Both these newly discovered AGN and high-redshift quasars have high BH-to-stellar mass ratios compared to those of the local Universe ( Ubler et al. 2023b; Harikane et al. 2023b; Maiolino et al. 2023a; Kokorev et al. 2023b; Juodˇzbalis et al. 2024). However, selection effects and uncertainties in both BH and stellar masses can play a role. \n3.3. AGN Large-scale environment \nOn larger scales, JWST data have not only spectroscopically confirmed that quasars have close companions (Wylezalek et al. 2022; Marshall et al. 2023), but also demonstrated that some quasars are embedded in overdense environments, with an enhanced number of [O iii ] emitters on Mpc scales (Wang et al. 2023). Ongoing JWST programs are now statistically evaluating whether quasars inhabit overdense, biased regions of the Universe. First results show that quasars live in diverse environments, including fields consistent with the average density of the Universe (Eilers et al. 2024). JWST will be key to disentangling whether this is due to a broad range of quasar dark matter haloes and/or the consequence of quasar feedback into their surroundings (e.g., suppression of nearby galaxy formation).', '3.4. Implications for Black Hole Seeding and Growth': "Prior to JWST , the existence of quasars at z ⩾ 6 suggested that their progenitors, referred to as BH seeds, must have formed with masses ranging from a few hundred solar masses ('light seeds' formed from the remnants of PopIII stars) to about a million solar masses ('heavy seeds') in the very early Universe. While JWST has not ruled out any of the existing theoretical models, the confirmation of overmassive BHs (with high BH to stellar mass ratios) could hint towards the existence of heavy seed channel(s). Those can form, e.g., from the collapse of high peak density fluctuations ('primordial BHs'), the collapse of super massive stars formed in atomic cooling haloes ('direct collapse' seeds with ∼ 10 3 -5 M ⊙ ) or through hierarchical mergers of massive stars or black holes in compact stellar clusters (seeds with ∼ 10 2 -3 M ⊙ ). \nAdditional solutions to the abundance of AGN and overmassive BHs include sustained super-Eddington accretion and BH mergers. Super-Eddington phases can account for a large fraction of BH growth under reduced BH feedback as shown in models (Inayoshi et al. 2022b; Schneider et al. 2023; Bennett et al. 2024) and supported by recent observational findings (Juodˇzbalis et al. 2024; Maiolino et al. 2023a). The contribution of BH mergers in their mass budget remains a longstanding question, tied to their mass, dynamics, and environment. Remarkably, JWST has already revealed the existence of dual AGN at high redshift with separations of a few/several kpc (Perna et al. 2023; Ishikawa et al. 2024); a population surprisingly larger (by about an order of magnitude) than what is predicted by cosmological simulations. Additionally, there are growing indications of dual AGN on even smaller scales of a few 100 pc, which are likely in the process of merging ( Ubler et al. 2023a; Maiolino et al. 2023b).", '4. REIONIZATION': "The collective energy of the first stars, galaxies and accreting black holes - including those too faint for even JWST to detect directly - transform the universe around them by heating and ionizing hydrogen (and later helium) gas in the intergalactic medium. This 'reionization' process sets the stage for all subsequent galaxy formation, as it impacts the ability of future dwarf galaxies to cool gas and form stars. We know that hydrogen reionization happened, but exactly when and how it happened has been a major missing piece in our understanding of the first billion years. \nAssuming reionization is driven by stars in early galaxies, the energy injection is commonly simply parameterised by the product of three terms. (1) The UV luminosity density of sources, ρ UV ( z ), which describes the total abundance of high-redshift ionizing sources, as discussed above. (2) The production rate of hydrogen ionizing photons per UV photons, ξ ion , determined by the stellar populations. (3) The fraction of ionizing photons which escape the dense ISM of galaxies into the IGM, f esc . The rate of ionization is determined by the evolution of these quantities with redshift and galaxy properties, and the rate of hydrogen recombination in the IGM. \nBefore JWST we knew reionization ended approximately one billion years after the Big Bang ( z ∼ 5 -6), but not when it started. Due to the limited observations of the UV luminosity function for galaxies and AGNs, and local Universe indications that the escape fraction of ionizing photons from galaxies was modest ( < 5%), the ionizing photon budget was tight. It was commonly assumed that reionization was accomplished by dwarf ( M UV ∼ -13) galaxies, by extrapolating a UV luminosity function steep at the faint end (slope ∼ -2), and assuming escape fractions 10 -20%. Due to the low observed number densities of UV-bright z > 6 AGN, they were commonly assumed to not play a major role in hydrogen reionization.", '4.1. Sources of reionization': "With JWST , we have made significant progress in understanding the reionization process. JWST has shown that the ionizing photon budget is not necessarily tight and that high average values ( > 20%) of the escape fraction are not necessary. As discussed above, we have now been able to measure the faint end of the UV LF at least up to z ∼ 7 -8, and obtained information on the UV luminosity density of M UV < -17 sources within the first billion years. We have also obtained hints on the variability of star formation of these sources. We have found spectroscopically confirmed galaxies up to z = 13 . 2, im- \nying reionization may have started just a few hundred million years after the Big Bang. \nJWST has for the first time spectroscopically measured the ionizing photon production of galaxies down to M UV = -15 . 5 (Atek et al. 2024). Direct measurements of ξ ion from photometry and spectra of Balmer lines up to z ∼ 6 -9 (Prieto-Lyon et al. 2023; Endsley et al. 2023b; Simmonds et al. 2023; Roberts-Borsani et al. 2024) find a range of values, log ξ ion ≈ 25 -26 erg -1 Hz, likely implying that galaxies have a range of star formation histories. Mean ξ ion values are ∼ 3 × higher than pre-JWST assumptions, reducing the need for high escape fractions. \nJWST 's discovery and characterisation of faint AGN at z > 5 (see above) has opened new discussions on the AGN contribution to reionization. However, as the majority of these high redshift AGN are likely UV-obscured or intrinsically faint, black hole accretion likely contributes no more than 25% to ionizing budget (Dayal et al. 2024). \nThere is still significant debate about the primary sources of reionization, in particular the contribution of faint galaxies ( M UV < -17). This is driven by uncertainties in the galaxy properties including the 'burstiness' of the star formation history, the IMF, and f esc . The foreground neutral IGM makes direct measurement of f esc impossible within the epoch of reionization. Therefore, JWST observations are being combined with those from ground-based telescopes and HST to identify galaxy properties that correlate with f esc in the z < 3 Universe, where direct observations of ionizing photons are possible. These correlations imply modest average f esc ( ≲ 10 %) values in samples of tens of z > 6 sources where direct comparisons with z < 3 galaxies are possible (e.g., Mascia et al. 2023; Lin et al. 2024). \nSignificant progress should be expected with future JWST observations by surveying the faint ( M UV > -17) galaxy population with deeper imaging and spectroscopy to understand their contribution to reionization.", '4.2. Morphology and growth of ionized regions': "JWST is also enabling us to observe the reionization process directly. Reionization is predicted to be 'patchy', with ionized bubbles growing around overdensities of galaxies (see Figure 6). The ionization state of the IGM can be traced by transmission of Lyman series photons - either through the Lymanα and β forest, probing the very end stages of reionization at z ∼ 6, or Lymanα emission from galaxies at all stages of reionization. JWST has shown that the z ∼ 6 Lymanα and β forest transmission measured in quasar spectra is \ncorrelated with distance from galaxies, suggesting that galaxies (and their overdensities) dominated the ionizing emissivity at z ∼ 6 (Kashino et al. 2023a). A downturn in the average Lymanα emission from galaxies at z ≳ 6 has been confirmed (Nakane et al. 2023; Chen et al. 2023), implying we are observing a predominantly neutral universe at higher redshifts. JWST spectroscopy enables us to infer the ionization state of the IGM at z > 10 (Umeda et al. 2023) hence providing an independent constraint on the total star formation rate density at early times, probing sources below the detection limits of even JWST. \nJWST also finds strong evidence for ionized regions, giving the first opportunity to understand how reionization is occurring on local scales. Early results have shown large field-to-field variations in Lymanα emission visibility, even in overdensities (Morishita et al. 2023; Chen et al. 2024; Napolitano et al. 2024), and also highlighted the importance of understanding Lymanα absorption by dense residual neutral gas within and around galaxies (Hsiao et al. 2023; Heintz et al. 2023; Chen et al. 2024). The sizes of ionized regions are predicted to be determined by the clustering strength, luminosity and star formation histories of the dominant reionizing sources (e.g., Lu et al. 2024) and can be probed directly via the spatial variations in Lymanα transmission. \nJWST has enabled the first measurements of Lymanα escape fractions (from Balmer lines), robust fluxes and velocity offsets from systemic in z > 7 galaxies. The discovery of sources with high Lymanα rest-frame equivalent width and Lymanα escape fraction, and low velocity offsets, in overdense regions implies these galaxies reside in ionized regions ≳ 1 pMpc (Saxena et al. 2023; Tang et al. 2024; Chen et al. 2024). By spectroscopically confirming galaxies in these regions, we are starting to build 3D maps (Figure 6, Chen et al. 2024; Witstok et al. 2024) which probe the contribution to reionization from faint sources below JWST 's detection limits (Whitler et al. 2024). \nFuture systematic spectroscopic studies of Lymanα visibility as a function of environment and galaxy properties will help us understand how ionized regions grew. This will enable us to constrain ionizing photon escape fractions and star formation in sources beyond JWST 's detection limits.", '5. CONCLUSIONS': "JWST 's sharp imaging, broad wavelength coverage, exquisite spectroscopy, and two-order-of-magnitude improvement in sensitivity have unlocked the study of how galaxies evolved in the first billion years. The highredshift Universe has emerged in stunning definition and \nthe initial discoveries of JWST are rewriting astronomy textbooks. \nJWST has revealed that galaxies and black holes assembled, formed significant stellar mass, dust, and synthesized the elements of the periodic table much earlier than was expected. The remarkable abundance of luminous galaxies at such early times hold the promise of upending theories of primordial galaxy formation. The discovery of very dense and massive star clusters urge us to decipher the role that these tiny stellar systems play in the evolution of early galaxies. The large numbers and apparent high masses of accreting supermassive black holes found by JWST have renewed interest in channels that generate intermediate-mass seeds. JWST has directly measured the abundance of chemical elements in these early galaxies, finding low ( ≳ 2%) but non-zero oxygen abundance, which indicates either that the Universe rapidly self-enriched, or that JWST has not yet reached the epoch of the very first galaxies. Whereas before JWST we knew that the Universe must have reionized, but did not know which sources were responsible, JWST has now found thousands of galaxies in the epoch of reionization, and associated individual ionized bubbles with star-forming galaxies, beginning to map out the topology of reionization. \nWe are gathering many pieces in the puzzle that describes the chronology of the first billion years of the universe, however, the emerging picture is still scattered. In the coming years, as many more discoveries are made, we will be able to connect these subfields, and comprehensively describe the first billion years of cosmic history. \nFigure 1. The cosmic timeline, from the origin of the Universe in the Big Bang, 13.8 billion years ago, till the present day. In the current standard picture, the Universe underwent a period of accelerated expansion called 'inflation' that expanded the Universe by about 60 orders of magnitude. The Universe then kept cooling and expanding until the next major epoch of 'recombination' about 4 × 10 5 yr later when the first hydrogen atoms formed. This was followed by the 'Dark ages' of the Universe that lasted for a few hundred million years. The emergence of the earliest galaxies, a few hundred million years after the Big Bang, marked the start of the era of 'cosmic dawn'. The first galaxies also produced the first photons capable of ionizing the neutral hydrogen atoms permeating space, starting the Epoch of Reionization (EoR), the last major phase transition in the Universe. In the initial stages of reionization, isolated galaxies (light yellow dots) produced ionized regions (gray patches) that grew and merged until the Universe was fully reionized. Image Credit: DELPHI project (ERC 717001). \n<!-- image --> \nFigure 2. a. The distribution of absolute magnitudes and redshifts of spectroscopically-confirmed galaxies from pre-JWST candidates (blue dots) and from public JWST data sets (orange squares), showing the power of JWST to detect galaxies beyond redshift 6. The latter include compilations (Roberts-Borsani et al. 2024) and single targets (Castellano et al. 2024; Carniani et al. 2024) observed with NIRSpec MSA observations, as well as NIRCam grism (FRESCO and EIGER; (Oesch et al. 2023) and (Kashino et al. 2023b), respectively). b. The cosmic SFR density over the first billion years (adapted from Figure 17 of (Harikane et al. 2024), as seen from HST/WFC3 samples (dark circles), compared to JWST/NIRCam estimates (light squares). A model of constant star formation efficiency is plotted in grey, for comparison. The model and all literature points are derived from (Harikane et al. 2024) (and references therein), where the latter are integrated down to M UV = -18 mag. \n<!-- image --> \nFigure 3. Resolved galaxy morphologies at redshift > 6 observed with JWST NIRCam (unless otherwise specified). Galaxies in the field (top row) show clumpy and dense structures (Kartaltepe et al. 2023). Thanks to gravitational lensing, the light from these compact galaxies is resolved into several stellar clumps down to scales of tens of parsecs ('The Cosmic Grapes'; Fujimoto et al. 2024). In some cases, these clumps show strong emission lines as showcased for M1149-JD1 observed with NIRISS and NIRCam (Bradaˇc et al. 2024), MIRI imaging and integral field spectroscopy ( ' Alvarez-M'arquez et al. 2023), and NIRSpec (GA-NIFS collab. in prep.) suggesting that intense episodes of star formation are concentrated within them. Near the critical lines, the galaxy light is stretched into long arcs revealing bright compact bound star clusters, with intrinsic sizes smaller than 10 parsecs ('Cosmic Gems arc', 'Firefly Sparkle', 'Sunrise arc' Adamo et al. 2024; Mowla et al. 2024; Vanzella et al. 2023a, respectively) and single stars ('Earendel' Welch et al. 2022). These stellar systems dominate the light of their galaxies, suggesting that star cluster might be a dominant star formation mode for young galaxies. \n<!-- image --> \nNIRSpec SPECTRUM \nFigure 4. A composite spectrum using all publicly available z ⩾ 5 low-resolution multi-object NIRSpec spectra. The unprecedented sensitivity and wavelength coverage of NIRSpec reveals a plethora of IGM, stellar, and ISM features, from Lyman and Balmer breaks to rest-frame UV and optical line emission. The plethora of features allow for characterizations of IGM opacity, stellar ages and masses, and gas-phase metallicities, to name a few. Plot adapted from Figure 3 of (Roberts-Borsani et al. 2024). \n<!-- image --> \nFigure 5. Distribution of bolometric luminosity of known AGN as a function of redshift, illustrating the discovery space opened by JWST at much lower luminosities and higher redshifts than probed by previous surveys (adapted from Scholtz et al. (2023)). The brown shaded region shows the range of L bol and redshift spanned by studies before JWST. The blue symbols show a compilation of AGN discovered by JWST, with stars showing type 1 (broad line) AGN and diamonds identifying type 2 (narrow line) AGN. \n<!-- image --> \nFigure 6. Spectroscopically-confirmed galaxies in the CEERS EGS field at z = 7 . 1-7 . 8, reproduced from (Chen et al. 2024). The presence of numerous Lyman-alpha emitting galaxies (red stars), including several with high equivalent widths ( > 200 ˚ A) and Lyman-alpha escape fractions ( ≳ 50%), in this field provides strong evidence for candidate ionized bubbles along the line of sight (shaded cyan regions - for illustration purposes only). These early observations, primarily of M UV < -19 HST-selected sources, highlight the potential of JWST to create tomographic maps of ionized regions to learn about the reionization process on local scales. \n<!-- image -->", '6. ACKNOWLEDGMENTS': "The authors wish to thank ISSI for sponsoring the 2024 Breakthrough Workshop, and the ISSI staff for their wonderful welcome and support. The authors are grateful to their collaborators, who made this paper possible. Collectively, we are grateful to the large community of scientists and engineers, worldwide, who designed, built and commissioned JWST and made a decades long astronomer dream a reality. \nAuthor Contributions: All of the authors were invited participants in a Breakthrough Workshop at the International Space Science Institute, titled 'The Chronology of the very early Universe according to JWST', which was held 11-15 March, 2024, in Bern, Switzerland. Most participants were successful JWST Cycle 1 PIs. All authors actively participated in workshop discussions and writing of the paper. The con- \nveners of the workshop were Antonella Nota, Angela Adamo, Gabriel Brammer, Dan Coe, Pascal Oesch, and Jane Rigby. The editors were the conveners and Daniel Eisenstein. The leads of the individual sections of the paper were Pascal Oesch, Dan Coe, Angela Adamo, Rachel Bezanson, Daniel Eisenstein, Andrea Ferrara, Melanie Habouzit, Roberto Maiolino, Charlotte Mason, Alice Shapley, and Massimo Stiavelli. The discussion moderators were Jane Rigby, Danielle Berg, John Chisholm, Pratika Dayal, Ivo Labbe Michael Maseda, Jorryt Matthee, Kristen McQuinn, Dan Stark, Allison Strom, Christina Williams, and Dominika Wylezalek. The discussion notetakers were Gabe Brammer, Anna de Graaff, Taylor Hutchison, Jeyhan Kartaltepe, Rui Marques and Rohan Naidu. The authors are grateful to Mark Dickinson for a careful read of the final manuscript, and to Fabio Crameri (ISSI) for his expert help designing the very best figures.", 'REFERENCES': "Bosman, S. E. I., ' Alvarez-M'arquez, J., Colina, L., et al. 2023, arXiv e-prints, arXiv:2307.14414, \ndoi: 10.48550/arXiv.2307.14414 \nBouwens, R. J., Stefanon, M., Brammer, G., et al. 2023, MNRAS, 523, 1036, doi: 10.1093/mnras/stad1145 \nBoylan-Kolchin, M. 2023, Nature Astronomy, 7, 731, doi: 10.1038/s41550-023-01937-7 \nBradaˇc, M., Strait, V., Mowla, L., et al. 2024, ApJL, 961, L21, doi: 10.3847/2041-8213/ad0e73 \nBunker, A. J., Cameron, A. J., Curtis-Lake, E., et al. 2023, arXiv e-prints, arXiv:2306.02467, \ndoi: 10.48550/arXiv.2306.02467 \nCameron, A. J., Katz, H., Rey, M. P., & Saxena, A. 2023, MNRAS, 523, 3516, doi: 10.1093/mnras/stad1579 \nCarnall, A. C., McLure, R. J., Dunlop, J. S., et al. 2023a, Nature, 619, 716, doi: 10.1038/s41586-023-06158-6 \nCarnall, A. C., McLeod, D. J., McLure, R. J., et al. 2023b, MNRAS, 520, 3974, doi: 10.1093/mnras/stad369 \nCarnall, A. C., Begley, R., McLeod, D. J., et al. 2023c, MNRAS, 518, L45, doi: 10.1093/mnrasl/slac136 \nCarniani, S., Venturi, G., Parlanti, E., et al. 2023, arXiv e-prints, arXiv:2306.11801, \ndoi: 10.48550/arXiv.2306.11801 \nCarniani, S., Hainline, K., D'Eugenio, F., et al. 2024, A shining cosmic dawn: spectroscopic confirmation of two luminous galaxies at z ∼ 14. \nhttps://arxiv.org/abs/2405.18485 \nCasey, C. M., Akins, H. B., Shuntov, M., et al. 2023, arXiv e-prints, arXiv:2308.10932, \ndoi: 10.48550/arXiv.2308.10932"}

2024arXiv240911994C

We review the advantages of fitting with a Two Component Advective Flow TCAF which uses only four physical parameters. We then present the results of hydrodynamic simulations to highlight the fact that the primary component of a black hole accretion remains the subKeplerian or the low angular momentum flow independent of whether we have a high intermediate or low mass Xray binary. Every aspect of spectral and timing properties including the diskjet connection could be understood well only if such a component is present along with a Keplerian component of variable size and accretion rate.

2024-09-01T00:00:00Z

['2024arXiv240911994C', '10.48550/arXiv.2409.11994', 'arXiv:2409.11994']

['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Black Hole Accretion is all about SubKeplerian Flows

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11994.pdf

{'Black Hole Accretion is all about Sub-Keplerian Flows': "Sandip K. Chakrabarti [ 0000 -0002 -0193 -1136 ] \nAbstract We review the advantages of fitting with a Two Component Advective Flow (TCAF) which uses only four physical parameters. We then present the results of hydrodynamic simulations to highlight the fact that the primary component of a black hole accretion remains the sub-Keplerian or the low angular momentum flow independent of whether we have a high, intermediate or low mass X-ray binary. Every aspect of spectral and timing properties, including the disk-jet connection could be understood well only if such a component is present along with a Keplerian component of variable size and accretion rate. \nTo be published in Astrophysics and Space Science Proceedings, 'The Relativistic Universe: From Classical to Quantum, Proceedings of the International Symposium on Recent Developments in Relativistic Astrophysics', Gangtok, December 11-13, 2023: to felicitate Prof. Banibrata Mukhopadhyay on his 50th Birth Anniversary' Editors S. Ghosh, A. R. Rao, Springer Nature", '1 Introduction': "Black holes are the simplest of all celestial bodies. It is no wonder that the behaviour of matter would also be very simple. The inner boundary condition of any flow onto black holes of any mass is universal, i.e., the flow enters through the horizon with the velocity of light. This supersonic flow at the inner boundary, forces it to pass through a sonic surface outside the horizon [1, 2, 3]. If the angular momentum is high enough so that the barrier ∼ 1 / /u1D45F 3 becomes comparable to the gravitational force ∝ 1 / /u1D45F 2 , the centrifugal barrier resists the nearly freely falling matter to slow it down, forcing it \nIndian Centre for Space Physics, Kolkata, West Bengal e-mail: sandip@csp.res.in \nto pass through a standing or oscillating shock wave, puffing it up and then entering through the horizon supersonically after passing through a second sonic surface. The sonic surface farther from the black hole is of Bondi type, while the one just outside of the horizon is the result of a strong gravity created by the general relativistic effects. This rapidly falling transonic, sub-Keplerian flow does not find time to cool down, carries all the specific energy and specific angular momentum (but not the total angular momentum) to its 'grave', i.e., the black hole, and is generally known as the advective accretion flow [2, 4]. A part of the incoming flow is pushed in the direction perpendicular to the disk plane to create the outflow, the core of which could become fast moving, collimated jets in presence of toroidal magnetic fields. The jet could be accelerated beyond hydrodynamic values in presence of radiative and magnetic fields. These are the exact solutions of radiatively inefficient advective flows, or advection dominated flows etc. Though originally these latter flows did not include any shocks and thus missed the most exciting component of the flow, slowly, they started appreciating the role of the shocks [2] in shaping the observed spectra as well [5]. \nA more complex situation occurs when the viscosity and radiative transfer are included. Depending on the viscosity, which transports angular momentum and weakens the centrifugal barrier, or the residual flow energy remained after radiating, the shock may or may not form, even though the flow remains transonic and enters the horizon supersonically. These transonic, advective, radiative, viscous disks are examplesof complete disk models and are capable to explaining observationalaspect of black hole accretion, except when strong magnetic fields are present which may produce an extra power-law component [2, 6, 7, 8]. \nIn contrast, the oft-used [9] viscous, black body radiating, standard Keplerian disk hardly moves towards the black hole. It is chopped off at the marginally stable radius (3 Schwarzschild radii in case of a non-rotating black hole). Being optically thick and geometrically thin, it radiates multi-colour black-body emission. It has no length scale of importance (like the shocks or sonic points in a transonic flow) apart from that due to optical depth depending on accretion rates. Strictly speaking, they are subsonic everywhere. The accretion rate /dotacc /u1D440 = 3 /u1D70B /uni03A3 /u1D708 [ 1 - ( /u1D45F /u1D45F /u1D456 ) -1 2 ] -1 depends on the kinematic viscosity coefficient /u1D708 and not directly on velocity. Any non-zero /u1D708 would make the flow Keplerian in this model. Thus a Keplerian disk formation is always guaranteed. \nIn our complete transonic/advective flow scenario, the flow above a critical viscosity parameter /u1D6FC , is shock-less, makes the disk Keplerian and forces the inner disk to become supersonic at a point between the marginally bound and marginally stable orbits. This is what a standard disk should have been. Its temperature distribution would always be modified black body type, independent of accretion rates at each disk annulus, as it could 'reflect' (absorb and re-emit) radiation of the Compton cloud (discussed below). \nSimilarly, a flow with high 'accretion rate' and strong radiative pressure OR a flow with low 'accretion rate' and strong ion pressure has angular momentum distribution which deviates from a Keplerian distribution [10, 11]. These doughnut shaped isolated rotating bodies produce the so-called 'Thick accretion disk' which \nare geometrically thick and optically 'thick' or 'thin' depending on the accretion rate but have no solution connecting to the black hole horizon, nor have a solution through which matter is supplied. In our complete transonic/advective flow scenario, their isolation is remedied by identifying them as the post-shock flow of a complete solution [12, 13].", '2 Two component Advective Flow or TCAF': "Fig. 1 Cartoon diagram (roughly in log scale) of a Two Component Advective Flow which seems to successfully fit X-ray data with only four physical flow parameters. From the normal companion both low and high angular momentum flow enters the Roche lobe of the compact object. In outbursting sources, high angular momentum matter halts at piling radius due to lack of viscosity. At outbursts, all or part of this stored matter is released due to enhanced viscosity, and a Keplerian disk moves in surrounded by the sub-Keplerian flow. The latter is slowed down at the shock front while the inner edge of the former is truncated there. The locations of successive slowing down, the strengths of the centrifugal barrier at these places, and the accretion rates in these components, decide all the spectral and timing properties. \n<!-- image --> \nIn our approach, a simple flow geometry is used for illustration (Fig. 1). The exact shape is unimportant as we use only the gross characteristics of the flow. A thick sub-Keplerian flow will initially have a height-dependent viscosity and will have higher viscosity on the equatorial plane due to higher temperature, turbulence etc. So, by definition, a disk is segregated into two components by an imaginary boundary away from the equatorial plane and it becomes a two component flow. It is irrelevant how the actual viscosity is sourced or is distributed or where the boundary lies [6, 7]. The boundary will not exist if the viscosity is very low or very high, as the viscosity parameter does not cross critical value which differentiates between shock and no-shock solutions. The region with high viscosity will behave like a Standard Keplerian disk, and the region surrounding it will behave as a subKeplerian transonic flow with a shock [14]. The Keplerian disk maintains the high viscosity, not because of high temperature, but because of high pressure (due to high density). The post-shock region will behave as a thick disk with proper advection as decided by the transonic flow solution. We need to concentrate only on four physical quantities to understand this flow: The accretion rates of the two components (i.e., exact viscosity or net accretion rate is not needed), the location of the shock (i.e., exact viscosity, energy/angular momentum etc. not needed) and most importantly, the strength of the shock. With these choice the parameters which 'embarrass' scientists, such as the exact viscosity, opacity of the torus not needed, net accretion rate, specific energy or angular momentum etc. are not needed. The compression ratio or the strength of the shock decides the post-shock density and thus the opacity is determined. Furthermore, the post-shock flow being hot, it is assumed to evaporate the inner region of the Keplerian disk inside the shock location and the truncated inner edge location becomes identical to the shock location. The post-shock hot torus is also identified as the Compton cloud. So, computing the degree of interception of soft photons, reprocessing of the soft photons from the Keplerian disk to re-emit as hard photons become simpler. Step-by-step fitting spectral generation procedure was presented in [7] and this is used widely while fitting the spectra of stellar mass and supermassive black holes. This torus is called the CENtrifugal barrier supported BOundary Layer or the CENBOL for a good reason. Like a star with a hard surface, CENBOLalso dissipates energy of the flow and is the primary source of high energy radiation of the accretion flow. Similarly, as a boundary layer, CENBOL is also the base of the outflow [15, 16, 17]. If the CENBOL is absent for excessive cooling, the hard radiation and the outflow are quenched [18, 19]. Most interestingly, the size of the CENBOL alone does not decide the outflow rate. For very large CENBOL, the outflow base area is high, but the temperature is low. For very small CENBOL, the base is smaller though the temperature could be higher. In both the cases, the outflow rate is low. Only when the CENBOL has an intermediate size, the outflow rate is high. \nCombination of these four parameters produces the complete spectrum including effects of reflection at a given time. Generally speaking, as the disk rate is raised the spectrum becomes softer. If the sub-Keplerian or halo rate is raised, the spectrum becomes harder. If the shock location is reduced, the spectrum becomes softer. If the shock strength is enhanced, the spectrum becomes harder. Fitting spectrum has \nbeen extensively done either directly running the code [8] several times [20, 21] or making a table model where results of running the code thousands of times are saved [22, 23] and interpolated parameters give the best fits. One can also use the code directly and iteratively change parameters to obtain the best fit [23, 24].", '3 Timing Properties, Outbursts etc.': "In the same spirit of simplicity, TCAF produces Quasi-Periodic Oscillations (QPOs) in radiation 'free of cost' without invoking any extra component [25, 26, 40]. The CENBOL itself oscillates when the cooling time scale roughly agrees with the infall time scale (Fig. 2). However, the CENBOL has a finite size and different part oscillates at a slightly different frequency. This produces 'quasi' periodicity and often higher harmonics. Depending on the mass of the black hole, radiation from CENBOL will be emitted at different wavelength. TCAF is the only solution which directly correlates the disk accretion rate with QPO frequency. Higher disk rate cools the CENBOL faster and increases the QPO frequency [26, 27, 40]. These are typically the type-C QPOs. This is observed in all the outbursts in X-ray binaries [28, 29, 30]. When the Rankine-Hugoniot relations at the shock is not satisfied, shock starts oscillating 'searching for' the solution. This type of shocks does not follow direct relation with accretion rate or viscosity and are responsible for Type A and Type B QPOs. \nOutbursts themselves are easy to understand in the TCAF scenario. In between two outbursts, much of the time, some matter with low-angular momentum will continuously reach the central black hole and the spectrum will show quiescence since the Keplerian disk is non-existent or exists only very far away and few soft photons are present for reprocessing by CENBOL. Assuming the net mass loss rate of the companion to be constant, clearly, most of the matter would be stuck at a 'pile-up' radius, accumulating steadily for months or years in this period (Fig. 1). The piled up matter, fully or partly, will eventually be driven inward triggered by enhanced viscosity on the hot equatorial plane. Both Keplerian and sub-Keplerian flow rates will suddenly go up but the sub-Keplerian component (halo) reaches the central black hole earlier than the Keplerian component (disk), causing sudden hardening of the spectrum before an outburst ensues [32, 36, 37]. This is typically the Hard state where type-C QPOs may form. The time delay of the soft peak with respect to hard peak is regularly observed and is proportional to the size of the Keplerian disk formed. The Keplerian disk will have its own outer boundary, since the supply of Keplerian matter is stopped with the drop of viscosity at the piling radius (after the convective viscosity is reduced, for example. As the Keplerian disk rate is increased, the spectrum softens from the hard state to the hard intermediate state with increasing QPO frequency (Fig. 2). In case the entire matter at piled up radius is evacuated in one shot, the outburst will last longer, and the next one will be expected with a large time gap. In case the piled up matter is partly evacuated, there could be secondary outbursts or several outbursts of smaller peaks.At some \nFig. 2 The 'optical depth' variation with shock location for a set of injected parameters when cooling is decreasing (from A to F) for the stellar mass black holes (top) and supermassive black holes (bottom) in two dimensional SPH simulations. The scatter in shock location and optical depth is lowest when there is very low cooling (F) or very high cooling (A) and highest when resonance happens. The non-monotonicity of the optical depth is important as it rightly explains the non-monotonicity of the time lag behaviour of QPOs [31]. \n<!-- image --> \ns \npoint of time, with a rise in Keplerian to sub-Keplerian flow rate ratio, the resonance condition is not fulfilled and only Type A/B QPOs are seen. This is the so-called Soft Intermediate State. Here the outflow rate is the highest. When the complete flow becomes Keplerian, the soft state is formed. Once the Keplerian supply from the pileup radius is stopped, the Keplerian disk can no longer sustain its identity or its angular momentum distribution. Its matter will slowly be entrained by the fast moving sub-Keplerian flow sandwiching it. In the rising stage of the outburst, the Keplerian disk is formed rapidly due to enhanced viscosity, but during the declining stage, the process of disappearance of the Keplerian disk is slower. This is the cause of hysteresis [38] effect. If one plots the dynamic hardness ratio or Comptonizing efficiency, the effect of hysteresis is seen clearly and is independent of the mass of the black hole [33, 32] unlike other diagrams (such as hardness ratio plots) which use rigid definition of hardness ratio. \nApart from the five spectral states (including the quiescent state with mainly a sub-Keplerian flow) mentioned above, there could be a super-soft state, when the Keplerian rate is very high and may produce radiation pressure supported thick disk at the inner edge from where radiation pressure driven outflow may form. In presence of magnetic field which could be brought by the flow, collimation and acceleration of the jet may occur when CENBOL is present through effects similar to Blandford-Payne (diverging fields) or Chakrabarti-Bhaskaran effects (divergingconverging fields) [34, 35]. In quiescent or soft states the kinematic luminosity of the jets would be negligible while it would be the highest in the soft-intermediate states. In the super-soft states, a part of the field lines would be blocked by puffed up thick disk, and thus the polarization is reduced and observed jets would emit at lower X-ray energies. Similar effects are expected in neutron stars and ULX sources with very high accretion rates. In Fig. 3, we show all the Spectral states of an outbursting X-ray binary. \nFig. 3 All possible configurations of TCAF (a) Quiescent State (QS) with only very weak outflow, no jets, and no Keplerian component and no QPOs; (b) Hard State (HS) with a low-disk rate Keplerian component and high halo-rate sub-Keplerian component with Type-C QPOS, moderate or intermediate strong jets and weak winds (ISJ/WW); (c) Hard Intermediate (HIM) state having Type-C QPO, stronger jets from the CENBOL, stronger magnetically driven outflows; Strongest jets at (c) to (d) transitions; (d) Soft Intermediate (SIM) states have sporadic Type A/B QPOs, jets and outflows; (e) Soft States (SS) have only magnetically driven weak outflows but no jets and no QPOs and finally (f) Super-soft states (SSS) with very high accretion rates where inner-jet is blocked by the puffed-up matter. Collimated radiation and magnetically driven strong jets at the core of a strong wind from the whole disk. No QPOs are expected. \n<!-- image --> \nOne added advantage of fitting a spectrum with TCAF, be it in the context of a stellar mass black hole or a supermassive black hole, is that the mass of the central black hole is derived from the fitting procedure 'free of cost' from each data across spectral states. This is because, all the physical quantities derived from the four physical fitted parameters are functions of the mass of the black hole in a nontrivial way. Similarly, the normalization constant, which is the ratio of the photon flux emitted by the theoretical TCAF solution at the object frame, and the photon flux detected by the instruments (after correction due to the absorption) is also obtained after fitting. This should give us certain combination of the distance and the inclination angle of the source which can be cross-checked with those obtained from other methods [28, 29, 30, 24]. \nSince TCAF is the quasi-exact solution (in the sense, it uses average flow properties while fitting) for the disk accretion, any X-ray generated at the outflows and jets other than its base, namely, the CENBOL (which is an integral part of TCAF), would be in excess of what TCAF predicts at the object frame. This excess should be correlated with the radio/IR data of the jets [39]. Since the jets necessarily originate from the accretion, disk and jet are connected across the spectral states. A transonic flow solution connects the inflow and outflow properties as a function of the physical flow parameters.", '4 Where would sub-Keplerian flows come from?': 'Since spectral fits by TCAF heavily depends on the existence of a low angular momentum flow (halo) component, it is pertinent to ask: where does the sub-Keplerian flow come from? In the case of an active galaxy with a super-massive black hole at the center, stellar winds from a large number of stars contribute. Ideally, if there are infinite number of stars, the tangential velocities will cancel totally and only the radial flow would dominate accretion. In reality, presence of a few stars would create a flow which will have very low net angular momentum as compared to the Keplerian value. In the worst case, the matter falling back from jets and outflows at larger radius, will have low angular momentum. In the case of a high mass X-ray binary, where the normal star is known to have profuse amount of winds, the net angular momentum would be lower than the Keplerian value at the Roche lobe. Matter may be accreted from both sides of the Roche-lobe also. But the compact object being lighter, high energy winds will not be gravitationally attracted from all directions. In both the cases, the flow will face the barrier due to centrifugal force close to the black hole and the size of this disk would be much smaller as compared to the accretion radius or Roche lobe distance. In the case of a low mass X-ray binary (LMXRB),it is believed without proof that disk is totally Keplerian (and the community is more worried about how to get rid of the excess angular momentum to create such a disk!). Surprisingly, the numerical simulations show that exactly the opposite situation prevails! We see that the average angular momentum still remains highly sub-Keplerian, though larger than that of a high mass X-ray binary system (Fig. 4 \nand Fig. 5). Because of this, the Keplerian disk is of larger size since the compact object is more massive and the /u1D43F 1 point is farther from the compact primary, and the average angular momentum of the injected matter is higher. The sub-Keplerian nature in LMXRB is because the low mass stars also have winds and high mass compact captures most of this wind from all directions! \nFigure 5 shows the average specific angular momentum distribution across the mass ratio of compact binaries. We observe that in all the cases, this is below the Keplerian value. In other words, a Keplerian disk could be formed only if sufficient viscosity transports angular momentum internally. This explains why low angular momentum flow is important in shaping the spectrum. Without this, fitting of a spectrum would require larger number of parameters.', '5 Caveats in fitting spectral data': 'In an evolving system, such as in an outburst, the matter approaches a black hole in a viscous time scale inside the Keplerian disk. It may take a few days to cross, say, 50 , 000 -100 , 000 Schwarzschild radii. What this implies is that in the rising phase, at a given instant when the observation is made, the entire disk does not have a constant accretion rate. A propagation front of gradually changing accretion rate in the disk component brings in the matter. The true range of variation of /dotacc /u1D440 ( /u1D45F ) is much higher than the average < /dotacc /u1D440 ( /u1D45F ) > . Hence fitted variation of < /dotacc /u1D440 ( /u1D45F ) > maybe much lower than the true variation (Fig. 6). In some occasions when violent changes in accretion rate takes place, such fitting with an average rate is not proper and would be misleading and one adds more components or parameters to get a good fit. This is true for any model which takes accretion rate or temperature as the parameter. In these cases spatially dependent mass flow has to be used to fit with TCAF and the code has to be re-written accordingly. Since the infall timescale of a sub-Keplerian flow is smaller, this problem is not so severe for the halo rate.', '6 Conclusions': 'TCAF provides a four parameter family of spectra which appear to fit most of the observed data in X-ray binaries or in AGNs. Its fundamental premise is that the black hole accretion always (except in soft or supersoft states) has the sub-Keplerian flow and Keplerian disks form only when the viscosity rises. The fact that a high viscosity is difficult to have, this suits the TCAF scenario. Since rising and falling of viscosity will affect the formation and disappearance of the Keplerian component differently, hysteresis effect is seen. TCAF also provides the QPO frequency from spectral fits itself, provide a resonance like condition [40] between the infall time from the shock and the cooling time scale inside the CENBOL prevails. We showed through numerical simulations that the TCAF can address the spectral and timing properties \n<!-- image --> \nFig. 4 Three dimensional hydrodynamic simulation of the accretion flow in a high mass X-ray binary (Top) where /u1D440 compact / /u1D440 companion = 0 . 1 and in a low mass X-ray binary (Bottom) where /u1D440 compact / /u1D440 companion = 10. In the former, wind matter is mostly passing through the Lagrange point as the light compact object is unable to attract winds from all directions. In the latter, even a little wind is attracted from all directions by the heavier compact resulting in a substantial amount of sub-Keplerian flow. \n<!-- image --> \nFig. 5 Average angular momentum of matter accreting from Roche lobe to the compact as obtained by three dimensional simulations of inviscid flow. For reference, Keplerian distribution is also plotted. Note that across the mass-ratio the average flow is always sub-Keplerian. Viscosity transports angular momentum of this matter and creates a standard Keplerian disk as a bi-product. This is the main reason why TCAF requires a sub-Keplerian component to fit the spectra. \n<!-- image --> \nFig. 6 A cartoon diagram showing propagation of matter in a Keplerian disk in viscous timescale towards the black hole in the rising phase of an outburst for a period of time (T) when the viscosity was high. As the viscosity is turned off, supply of Keplerian matter is also reduced, and this front continues to move in creating the declining state. /u1D447 0 < /u1D447 1 < /u1D447 2 < /u1D447 3. The true accretion rate ( /dotacc /u1D440 ( /u1D45F ) ) is shown in thick curve and the average ( < /dotacc /u1D440 ( /u1D45F ) > ) is shown in thin straight line for illustration. \n<!-- image --> \nDistance from black hole \nvery squarely. Large scale magnetic fields do not appear to be dynamically important in fitting the gross properties of the spectra except adding a power in certain states. However for the formation of collimated jets, it could be important. Power-law in soft states can be understood from the contribution by the bulk motion inside the inner sonic point of TCAF [7].', 'References': "- 1. S.K. Chakrabarti, ApJ 347 , 365 (1989).\n- 2. S.K. Chakrabarti, Theory of Transonic Astrophysical Flows (World Scientific: Singapore, 1990).\n- 3. S.K. Chakrabarti, ApJ 464 , 664 (1996).\n- 4. S.K. Chakrabarti, Phys. Rep. 266 , No. 5 & 6, 229-392 (1996).\n- 5. L. Truong, K.S. Wood, M.T. Wolff et al. ApJ 819 112 (2016).\n- 6. S.K. Chakrabarti, in Proceedings of 17th Texas Symposium (C94-12-12), p. 546, (New York Academy of Sciences: New York, 1995)\n- 7. S.K. Chakrabarti & L.G. Titarchuk, ApJ 455 , 623 (1995).\n- 8. S.K. Chakrabarti, ApJ 484 , 313 (1997).\n- 9. N.I. Shakura & R.A. Sunyaev, A&A 24 , 337 (1973).\n- 10. B. Paczy'nsky & P.J. Wiita, A&A 88 , 23 (1980).\n- 11. M.J. Rees, M.C. Begelman, R.D. Blandford & E.S. Phinney, Nature 295 , 17 (1982).\n- 12. D. Molteni, G. Lanzafame & S.K. Chakrabarti, ApJ 425 , 161 (1994).\n- 13. S.K. Chakrabarti, ApJ 471 , 237 (1996).\n- 14. K. Giri & S.K. Chakrabarti, MNRAS 430 2836 (2013).\n- 15. S.K. Chakrabarti, ApJ 288 , 1 (1985).\n- 16. S.K. Chakrabarti, ApJ 288 , 7 (1985).\n- 17. S.K. Chakrabarti, ApJ 303 , 582 (1986).\n- 18. S.K. Chakrabarti, A&A 351 , 185 (1999).\n- 19. S.K. Garain, H. Ghosh, S.K. Chakrabarti, ApJ 758 114 (2012).\n- 20. S.K. Chakrabarti, B. Dutta & P.S. Pal, MNRAS 394 , 1463 (2009).\n- 21. S. Mandal & S.K. Chakrabarti, ApJ 689 , 17.\n- 22. D. Debnath, S.K. Chakrabarti & S. Mondal, MNRAS 440 , 121 (2014).\n- 23. S. Mondal, S.K. Chakrabarti & D. Debnath, ApJ 798 , 57 (2015).\n- 24. P. Nandi, A. Chatterjee, S.K. Chakrabarti & B.G. Dutta, MNRAS, 506 , 3111 (2021).\n- 25. D. Molteni, H. Sponholz & S.K. Chakrabarti, ApJ 457 , 805.\n- 26. S.K. Chakrabarti & S.G. Manickam, ApJL 531 , L41 (2000).\n- 27. S.K. Garain, H. Ghosh & S.K. Chakrabarti, MNRAS 437 , 1329 (2014).\n- 28. D. Debnath, A.A. Molla, S.K. Chakrabarti & S. Mondal, ApJ 803 , 59 (2015).\n- 29. A.A. Molla, D. Debnath, S.K. Chakrabarti, S. Mondal & A. Jana, MNRAS 460 , 3163 (2016).\n- 30. D. Chatterjee, D. Debnath, S.K. Chakrabarti, S. Mondal & A. JanaApJ 827 , 88 (2016).\n- 31. B.G. Dutta & S.K. Chakrabarti, ApJ 828 , 101 (2016).\n- 32. A. Ghosh & S.K. Chakrabarti, MNRAS 479 , 1210 (2018).\n- 33. P.S. 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2024ApJ...975...73K

We present a multiwavelength spectral study of NGC 4151 based on five epochs of simultaneous AstroSat observations in the nearultraviolet NUV to hard Xray band 0.00580 keV during 20172018. We derived the intrinsic accretion disk continuum after correcting for internal and Galactic extinction contributions from broad and narrowline regions and emission from the host galaxy. We found a bluer continuum at brighter UV flux possibly due to variations in the accretion disk continuum or the UV reddening. We estimated the intrinsic reddening EB V 0.4 using highresolution Hubble Space Telescope HSTSTIS spectrum acquired in 2000 March. We used thermal Comptonization neutral and ionized absorption and Xray reflection to model the Xray spectra. We obtained the Xray absorbing neutral column varying between N SUB H SUB 1.2 and 3.4 10SUP23SUP cmSUP2SUP which are 100 times larger than that estimated from UV extinction assuming the Galactic dusttogas ratio. To reconcile this discrepancy we propose two plausible configurations of the obscurer a a twozone obscurer consisting of dustfree and dusty regions divided by the sublimation radius or b a twophase obscurer consisting of clumpy dense clouds embedded in a lowdensity medium resulting in a scenario where a few dense clouds obscure the compact Xray source substantially while the bulk of UV emission arising from the extended accretion disk passes through the lowdensity medium. Furthermore we find a positive correlation between the Xray absorption column and NUV farUV color and UV flux indicative of enhanced winds possibly driven by the bluerwhenbrighter UV continuum.

2024-11-01T00:00:00Z

['2024ApJ...975...73K', '10.48550/arXiv.2409.04762', '2024arXiv240904762K', 'arXiv:2409.04762', '10.3847/1538-4357/ad77ca']

['Seyfert galaxies', 'Ultraviolet spectroscopy', 'X-ray active galactic nuclei', '1447', '2284', '2035', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies']

Multiepoch UVXRay Spectral Study of NGC 4151 with AstroSat

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

https://arxiv.org/pdf/2409.04762.pdf

{'Multi-epoch UV - X-ray spectral study of NGC 4151 with AstroSat': 'Shrabani Kumar, 1 G. C. Dewangan , 1 P. Gandhi , 2 I. E. Papadakis, 3, 4 N. P. S. Mithun, 5 K. P. Singh, 6, 7 D. Bhattacharya, 8 A. A. Zdziarski , 9 G. C. Stewart , 10 S. Bhattacharyya, 7 and S. Chandra 11 \n- 1 Inter-University Centre for Astronomy and Astrophysics (IUCAA), PB No.4, Ganeshkhind, Pune-411007, India \n2 School of Physics and Astronomy, University of Southampton, Highfield, Southampton SO17 1BJ, UK \n- 3 Department of Physics and Institute of Theoretical and Computational Physics, University of Crete, 71003 Heraklion, Greece 4 Institute of Astrophysics, FORTH, GR-71110 Heraklion, Greece \n5 Physical Research Laboratory Thaltej, Ahmedabad, Gujarat 380009, India \n- 6 Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, Manauli P.O., SAS Nagar, 140306, Punjab, India\n- 7 Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India 8 Ashoka University, Dept. Of Physics, Sonipat, Haryana-131029, India \n9 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland 10 Department of Physics and Astronomy, The University of Leicester, University Road, Leicester LE1 7RH, UK 11 Center for Space Research, North-West University, Potchefstroom 2520, South Africa', 'ABSTRACT': "We present a multi-wavelength spectral study of NGC 4151 based on five epochs of simultaneous AstroSat observations in the near ultra-violet (NUV) to hard X-ray band ( ∼ 0 . 005 -80 keV) during 2017 -2018. We derived the intrinsic accretion disk continuum after correcting for internal and Galactic extinction, contributions from broad and narrow line regions, and emission from the host galaxy. We found a bluer continuum at brighter UV flux possibly due to variations in the accretion disk continuum or the UV reddening. We estimated the intrinsic reddening, E ( B -V ) ∼ 0 . 4, using high-resolution HST /STIS spectrum acquired in March 2000. We used thermal Comptonization, neutral and ionized absorption, and X-ray reflection to model the X-ray spectra. We obtained the X-ray absorbing neutral column varying between N H ∼ 1 . 2 -3 . 4 × 10 23 cm -2 , which are ∼ 100 times larger than that estimated from UV extinction, assuming the Galactic dust-to-gas ratio. To reconcile this discrepancy, we propose two plausible configurations of the obscurer: (a) a two-zone obscurer consisting of dust-free and dusty regions, divided by the sublimation radius, or (b) a two-phase obscurer consisting of clumpy, dense clouds embedded in a low-density medium, resulting in a scenario where a few dense clouds obscure the compact X-ray source substantially, while the bulk of UV emission arising from the extended accretion disk passes through the low-density medium. Furthermore, we find a positive correlation between X-ray absorption column and NUV -FUV color and UV flux, indicative of enhanced winds possibly driven by the 'bluer-when-brighter' UV continuum. \nKeywords: galaxies: Seyfert - ultraviolet: galaxies - X-rays: galaxies - techniques: spectroscopic - galaxies: individual: NGC 4151", '1. INTRODUCTION': "NGC 4151, a type 1.5 Seyfert galaxy, is one of the brightest and nearest (z ∼ 0 . 0033, distance ∼ 15 . 8 Mpc) active galactic nuclei (AGN). It has been studied quite extensively in different wavebands (Perola et al. 1986; Zdziarski & Magdziarz 1996; Ulrich 2000; Zoghbi et al. 2019; Mahmoud & Done 2020). The X-ray spectrum shows complex absorption due to multiple layers of ionized and neutral absorbing columns (Zdziarski & \nMagdziarz 1996; Puccetti et al. 2007; Zoghbi et al. 2019; Kraemer et al. 2020). The shape of the X-ray spectrum, particularly in the 1 -6 keV band (Beuchert et al. 2017), is observed to be significantly modified mostly due to the variation in the neutral absorption column density ( N H ∼ 10 22 -10 23 cm -2 ). Large changes in the column density (by a factor of ∼ 10) are observed to happen on timescales of days to months (Puccetti et al. 2007). The soft X-ray excess emission, observed below 2 keV, is nearly constant in flux (Edelson et al. 1996; Zoghbi et al. \n2019). This emission component, spatially resolved by Chandra, is observed to be arising from a few hundredparsec distance off an extended region (Yang et al. 2001). The X-ray photon index is observed to vary between 1 . 4 -1 . 8 (Zoghbi et al. 2019). The X-ray reflection spectrum shows the ubiquitous presence of a narrow Fe K α line at 6.4 keV, implying distant reflection (possibly due to the torus) of the primary X-ray continuum. Lubi'nski et al. (2010) observed the relative reflection strength increasing at lower flux. They attributed the variation in the reflection strength to the nearby reflection from the disk. In the bright flux state, weak relativistic reflection is observed, which could originate from the disk truncated around 10 r g (Szanecki et al. 2021). The distant reflection component is observed to be constant with a reflection fraction of ∼ 0 . 3 (Lubi'nski et al. 2010). \nThe UV/optical spectrum shows multiple broad and narrow emission and absorption lines. The broad emission lines (C IV , Mg II , H α , H β ) are associated with appearing and disappearing, blue and red shifted, wings, which also vary in strength over time (Ulrich & Horne 1996; Metzroth et al. 2006; Shapovalova et al. 2008). This indicates a change in the BLR kinematics in response to continuum strength (Ulrich et al. 1991; Ulrich 2000; Bon et al. 2012; Chen et al. 2023). The continuum flux shows an order of magnitude variability on a timescale of a few years. The optical continuum flux (at 5100 ˚ A) since 1993 has shown two minima, in 2000 -2001 and 2005 -2006 (Chen et al. 2023). The maximum UV/Optical flux was observed during 1995 -1996, and the other high flux states were observed during 2003 and 2010. From 1996 to 2006, the continuum and line flux varied by a factor of ∼ 6. The historical peak continuum flux (in UV) observed during 1995 is ∼ 5 × 10 -13 erg cm -2 s -1 ˚ A -1 , and the minimum flux observed in 2000 -2001 was ∼ 10 -14 erg cm -2 s -1 ˚ A -1 . During this flux change, the spectral state changed from Seyfert 1.5 (maximum flux) to 1.8 (minimum flux), the reason for which could be attributed to the change in BLR radiation pressure (Chen et al. 2023). \nFlux variation has similarly been observed in the Xray emission. Lubi'nski et al. (2010) categorized the source into three states: bright, medium, and dim, according to the level of X-ray flux. The X-ray flux was at its maximum (bright state) around 1993, 2003, and 2009 and at its minimum in 2000, 2006, and 2007 (dim state). The maximum X-ray flux observed during 1993 was ∼ 10 -10 erg cm -2 s -1 in the 2 -10 keV band. The electron temperature of the X-ray emitting hot corona is ∼ 50 -70 keV in the bright state and ∼ 180 -230 keV in the dim state (Lubi'nski et al. 2010). \nThe UV/optical and X-ray variations followed similar trends. The UV/optical and X-ray flux correlate in simultaneous UV/X-ray observations (Perola et al. 1986; Lubi'nski et al. 2010). Here, we investigate the connection between UV and X-ray spectral variability. We analyzed the five sets of UV/X-ray data simultaneously acquired by AstroSat during 2017 - 2018. We studied the effect of X-ray absorbing material on UV emission and the connection between the UV and X-ray emission. We describe the observation and data processing in Section 2, we present the spectral analysis in Section 3, 4 and 5, and describe our results and discuss them in Section 6.", '2. AstroSat OBSERVATIONS AND DATA REDUCTION': 'AstroSat (Singh et al. 2014) is a multi-wavelength space observatory launched in 2015 by the Indian Space Research Organization (ISRO). It carries four coaligned payloads, namely, Cadmium-Zinc-Telluride Imager (CZTI; Vadawale et al. 2016; Bhalerao et al. 2017), Large Area X-ray Proportional Counter (LAXPC; Yadav et al. 2016; Antia et al. 2017), Soft X-ray Telescope (SXT; Singh et al. 2016, 2017), and Ultra-violet Imaging Telescope (UVIT; Tandon et al. 2017, 2020). AstroSat observed NGC 4151 five times during February 2017 May 2018 with all four co-aligned payloads. We list the details of the five sets of simultaneous UV/Xray observations G06 117T01 9000001012 (obs1), G06 117T01 9000001046 (obs2), G06 117T01 9000001086 (obs3), G08 064T01 9000001814 (obs4) and G08 064T01 9000002070 (obs5) in Table 1.', '2.1. SXT': "The SXT is a focusing soft X-ray telescope with a CCD similar in design to the XRT onboard the Swift observatory. It observes mainly in photon counting (PC) mode in the 0 . 3 -7 keV band. The field of view is ∼ 40 ' , the half power diameter is ∼ 11 ' , and the energy resolution is 150 eV at 6 keV (Singh et al. 2017). We used level1 data and generated the level2 clean event files using SXT pipeline software AS1SXTLevel2-1.4b available at the SXT payload operation center (POC 1 ). For each observation, we merged the clean event files from every orbit using the SXT merger tool SXTMerger 2 , also available at the SXT POC website. The final cleaned image of the source corresponding to obs4 is shown in Figure 1. \nTable 1. Log of AstroSat observations of NGC 4151. The energy bands used to calculate the net count rates are 7 -9 . 5 eV (UVIT/Grating), 0 . 7 -7 keV (SXT), 4 -20 keV (LAXPC), and 22 -80 keV (CZTI). \nWe used the XSELECT tool within HEASoft (version 6.29) to extract the source spectrum from the circular region of 15 ' radius. We used the background spectrum (SkyBkg comb EL3p5 Cl Rd16p0 v01.pha), instrument response (RMF: sxt pc mat g0to12.rmf), and effective area (ARF: sxt pc excl00v04 20190608.arf) from the SXT POC website. We grouped each PHA spectral dataset with a minimum 25 counts bin -1 using the tool FTGROUPPHA available under the HEASoft package. The net source count rate varies from 0 . 18 -0 . 24 counts s -1 in the 0 . 7 -7 keV energy band (Table 1). \nThe LAXPC consists of three gas-proportional counters (LAXPC10, LAXPC20, and LAXPC30) and is sensitive to the 3 -80 keV energy band. We used the data acquired by LAXPC20 as the data from the other two instruments are not usable since LAXPC10 is unstable due to continuous variation of gain, and the LAXPC30 has significant gas leakage in the detector (Antia et al. 2017). We used the pipeline software LAXPCSOFT V3.4.4 3 to generate the spectrum and a suitable background. We used the response file lx20cshm08L1v1.0.rmf for our spectral analysis. We grouped the PHA spectral data to a minimum of 20 counts bin -1 using the tool FT- \nFigure 1. The SXT image of NGC 4151 (obs4), the yellow circle (radius = 15 ' ) shows the source extraction area. \n<!-- image --> \nGROUPPHA available in the HEASoft package. The net source count rate varied between 14 -22 counts s -1 in the 4 -20 keV band during our observations (see Table 1).", '2.3. CZTI': 'The CZTI is a hard X-ray telescope that uses coded mask imaging. The imager consists of four quadrants with 16 CZT detector modules in each of them. It operates in photon counting mode and observes in the 20 -100 keV energy band. The field of view is 4 . 6 · × 4 . 6 · and the energy resolution is 8% at 100 keV. We processed the CZTI data using the pipeline version 3.0 along with the associated CALDB. Following the standard pipeline procedure, we obtained the event files, which we used to generate background-subtracted source spectra for each quadrant (along with associated response matrices) by employing the mask-weighting technique.', '2.4. UVIT': "The UVIT comprises three channels: far ultraviolet (FUV: 1200 -1800 ˚ A), near ultraviolet (NUV: 2000 -3000 ˚ A), and visible (VIS: 3200 -5500 ˚ A). Both the FUV and NUV channels are equipped with several broadband filters that provide high-resolution (FWHM ∼ 1 -1 . 5 arcsec) images in 28 arcmin diameter field (Tandon et al. 2017, 2020). The FUV channel also includes two low spectral resolution slit-less gratings (hereafter FUV-G1 and FUV-G2) oriented orthogonal to each other; the NUV channel has one grating (NUV-G). The calibration of these gratings is described in Dewangan (2021), and the analysis tools are included in the UVIT- \nTools.jl package 4 with updated calibration. The spectral resolution of the FUV gratings in the -2 order is ∼ 14 . 3 ˚ A. We used the available broadband filter observation F154W (FUV-BaF 2 , λ m = 1541 ˚ A, ∆ λ = 380 ˚ A) in the FUV channel for all the five observations, and N219M (NUV-B15, λ m = 2197 ˚ A, ∆ λ = 270 ˚ A) in the NUV channel for four of our observations (excluding the obs5), since the NUV detector stopped functioning in 2018, March (Ghosh et al. 2021). Additionally, we used FUV grating observations, which were available for obs4 and obs5 only. \nFigure 2. NUV-B15 image of NGC 4151, the yellow circles correspond to a radius of 20 pixels (8 . 3 '' ; 1 pixel ∼ 0 . 416 '' ). The background is taken from within the galaxy with a relatively clean region. \n<!-- image --> \nWe processed the level1 UVIT data using the CCDLAB pipeline software (Postma & Leahy 2017). This pipeline generates a single image for each orbit, implements rotation and translation to co-align the orbit-wise images, considering one of the images as a reference, and then merges into one final image. We obtained the final images for both the broadband filters and gratings. We extracted the photometric flux for the broadband filter data using the UVITTools.jl package after correcting for the saturation effect on the source following Tandon et al. (2020) for all five observations. We calculated the flux within a circular region of radius 8 . 3 '' centered at the source centroid position. We obtained the background flux by using an identical circular area from within the host galaxy (see Figure 2). We purpose- \ny used the background region from within the galaxy as the diffuse emission from the galaxy may have contaminated the AGN emission as well. For obs1, CCDLAB could not generate the final image in the FUV filter. Therefore, we used one of the orbit images with the longest exposure to calculate the observed flux and generated the single-channel spectrum. In Figure 2, we show the NUV broadband filter image from one of the observations where yellow circles mark the source and background extraction regions used to obtain the photometric flux. \nWe extracted the count spectra from the FUV-G1 and FUV-G2 merged images for obs4 and obs5, for which grating data are available. We used a cross-dispersion width of 40 pixels in the -2 order in the FUV gratings to extract the source spectra. The background region for FUV-G1 images of both the observations (obs4 and obs5) is significantly contaminated with emissions from the spiral arms and the diffuse emission from the galaxy. In Fig. 3, we show the FUV-G1 image of obs4, where the emission in the background region is significantly contaminated. We investigated the image and the background spectra of the upper and lower adjacent regions to the source. The region above the source seemed more suitable for extracting the background spectrum (cyan rectangular box in Fig. 3). The adjacent regions close to the source in the FUV-G2 image are relatively contamination-free. Therefore, we extracted the background spectrum from a source-free region away from the diffuse galaxy emission. \nWeshow the count spectra from all five observations in Figure 4. The energy ranges covered by the data from different instruments are marked with shaded regions. We performed the spectral analysis and model fitting with XSPEC (version 12.12.0; Arnaud 1996) using χ 2 statistic for goodness of fit. We quote the errors at the 90% confidence level unless otherwise specified.", '3. UVIT SPECTRAL ANALYSIS': "We analyzed the UVIT spectral data (in the 5 . 5 -9 . 5 eV or 1305 -2254 ˚ A band), namely the FUV-G1/FUVG2 grating spectra, and FUV-BaF 2 , NUV-B15 photometric flux from obs4 (5 . 5 -9 . 5 eV or 1305 -2254 ˚ A) jointly and similarly for obs5 in the 7 -9 . 5 eV (1305 -1770 ˚ A) band (as NUV data were not available). We used a variable cross-normalization constant between the gratings to account for any difference in flux measurements. \nWe used the XSPEC model REDDEN (Cardelli et al. 1989) to account for the reddening due to our Galaxy. We calculated the color excess ( E ( B -V )) using the following linear relation provided by Guver & Ozel (2009): \nFigure 3. FUV-G1 image of NGC 4151. The background can be seen to be significantly contaminated by the emission from the spiral arms. The cyan rectangular box shows the background extraction region for FUV-G1 in obs4. \n<!-- image --> \nN H [ cm -2 ] = (2 . 21 ± 0 . 09) × 10 21 A V [mag] (1) \nwhere, A V = 3 . 1 × E ( B -V ). We used N H = 2 . 07 × 10 20 cm -2 obtained from the N H calculator available at the HEASARC website 5 , and derived E ( B -V ) = 0 . 03. We used this E ( B -V ) and kept fixed in the REDDEN model for the Galactic extinction in our spectral analysis. \nWe used a simple multi-temperature disk blackbody ( DISKBB ) to model the underlying continuum and Gaussian lines for the BLR/NLR emission lines (e.g., Si IV /O IV λ 1400 ˚ A, N IV ] λ 1486 ˚ A, C IV λ 1549 ˚ A, He II λ 1640 ˚ A and O III ] λ 1667 ˚ A). We observed some positive residuals near ∼ 1600 ˚ A and added a Gaussian line at the rest wavelength of 1599 ˚ A. This improved the statistic by ∆ χ 2 = 8 for two additional parameters: the line width and the normalization. This line could be the C IV satellite line observed during 1980 -1984 with the International Ultraviolet Explorer ( IUE ) and was denoted by L2 (Ulrich et al. 1985). The narrow satellite emission lines of C IV , L1 (1515 ˚ A), and L2 (1600 ˚ A) are usually observed in the low flux states when the narrow emission lines become more prominent (Crenshaw et al. 2000). We used a Gaussian profile ( GABS ) for the only absorption line near the rest wavelength of 1388 ˚ A with the best-fit values of width and strength being 1 . 4 × 10 -5 and 3 . 7 × 10 -5 keV, respectively. \nThe semi-forbidden emission line C III ] and C II lie around the edges of the NUV-B15 filter band. As the effective area around the edges decreases, we can treat the total NUV flux mostly due to the continuum. \n<!-- image --> \nFigure 4. The NUV photometric data and the FUV grating \n<!-- image --> \n(left), and SXT, LAXPC and CZTI (right) count spectra from all five observations. \n<!-- image --> \nFigure 5. Upper panels: For clarity, only the UVIT/FUV-G1 spectrum with the best-fit total model is shown (red). The model components are the intrinsic and Galactic absorbed power law (green solid line), Gaussian emission lines (cyan dashed line), and host galaxy template model (black dotted line). \n<!-- image --> \nLower panels: (data-model)/error. The green triangle and yellow star show the NUV-B15 and FUV-BaF 2 photometric flux points. \nTo account for the host galaxy contamination (spiral galaxy; Mahmoud & Done 2020), which is significant in our NUV band, we incorporated the Sb galaxy template (Kinney et al. 1996) in our fitting. Using optical observations with ground based telescopes, Shapovalova et al. (2008) calculated the host galaxy flux at 5100 ˚ A as ∼ 1 . 1 × 10 -14 erg cm -2 s -1 ˚ A -1 for an aperture of 4 . 2 × 19 . 8 arcsec 2 . We scaled the flux for our circular source extraction region of radius 8 . 3 '' , assuming the host galaxy emission to be nearly uniform. We used the scaled galaxy template as a table model (hereafter sb temp) and fixed the normalization (total \nmodel flux in 5 . 5 -10 eV band ( ∼ 1250 -2250 ˚ A) is 6 . 5 × 10 -12 erg cm -2 s -1 ). For obs4, the contribution of the host galaxy emission to the total continuum emission is 0 . 8% in the 5 . 5 -10 eV band. \nPrevious observations have found this AGN, a Seyfert 1.5, to be highly obscured (Zdziarski & Magdziarz 1996; Ricci & Trakhtenbrot 2023). Therefore, the UV emission may suffer high extinction. In the X-ray band, we observed a very high absorption column ( N H ∼ 10 23 cm -2 ) along the line of sight (discussed later); therefore, it is highly likely that the UV emission is also affected. To account for the intrinsic extinction in the UV band, \nwe created an XSPEC model (hereafter intr ext) using the empirical extinction curve provided by Czerny et al. (2004) for AGNs, \nA λ E ( B -V ) = -1 . 36 + 13 log 1 λ (2) \nwhere, 1 λ ranges from 1.5 to 8.5 µm -1 . We used the Balmer decrement (flux ratio of H α and H β ) as a free parameter to estimate the intrinsic reddening (see Eq. 3 in Dom'ınguez et al. 2013). As NGC 4151 went through episodes of transition between high and low flux states, the flux ratio of the Balmer lines was also observed to vary. It has been observed that for the highest flux state, based on the continuum flux measured at 5100 ˚ A, the Balmer decrement approaches ∼ 3 . 1, while for the low flux state, this ratio is as high as ∼ 6 (Shapovalova et al. 2010; Raki'c et al. 2017). We compared the observed UVIT/FUV flux at 1440 ˚ A with previous observations (Kraemer et al. 2006; Couto et al. 2016) and found that it is similar to the low flux state observations. To determine the intrinsic extinction, we used a high spectral resolution ( ∼ 45800) HST /STIS-E140M (Space Telescope Imaging Spectrograph) spectrum acquired during March 2000. The principal reason for choosing this data is that the observed flux at 1440 ˚ A is similar to that in UVIT/FUV grating spectrum from obs4 taken in January 2018 ( ∼ 3 × 10 -14 erg cm -2 s -1 ˚ A -1 ). We compare the STIS and FUV spectra in Figure A1 (left) of appendix A. \nWe used KERRD , a relativistic accretion disc model available in XSPEC, as the continuum component and Gaussian profiles to account for the emission/absorption lines in the STIS spectrum. We obtained the best-fit value of the Balmer decrement, 4 . 95 ± 0 . 09, for the intrinsic reddening. The details of the spectral analysis for STIS spectrum are provided in the appendix (A). We used this value of the Balmer decrement for the UVIT grating spectra and fixed it in our UV/X-ray joint fitting for each of the observations. \nThe final model expression for the UVIT grating spectral analysis is CONSTANT × REDDEN × [sb temp + intr ext × ( DISKBB + emission lines) × absorption line]. In obs4, we obtained the best fit χ 2 per degree of freedom (dof) = 308/274, and inner disk temperature, kT in = 9 . 4 +4 . 7 -2 . 9 eV. For obs5, we obtained the best fit χ 2 / dof = 309/273, and kT in = 5 . 9 +4 . 0 -1 . 7 eV. In Fig. 5, we show the data, best-fit model and model components, and the (data-model)/error for the UVIT grating spectra. We list the best-fit values of the emission line parameters in Table 2. The line centroids of all emission lines and the width of narrow or weak emission lines are fixed during error calculations.", '4. X-RAY SPECTRAL ANALYSIS': "We analyzed the SXT, LAXPC, and CZTI spectral data jointly for each observation. We used the 0 . 7 -7 keV SXT data, 3 -20 keV LAXPC data, and 22 -80 keV CZTI data. We began the analysis with only the SXT data with a simple power law model modified with the Galactic absorption. For the Galactic X-ray absorption, we used TBABS and fixed the equivalent hydrogen column density N H at 2 . 07 × 10 20 cm -2 obtained from the HEASoft N H calculator 6 . We also used a neutral partial covering absorption model available in XSPEC TBPCF to account for the absorbed primary continuum and unabsorbed scattered continuum. For all five observations, we detected the presence of narrow emission lines due to Fe K α at 6.4 keV, He-like Ne IX at 0.92 keV. We also observed positive residuals around 1.8 keV. We added a narrow ( σ = 1 eV) Gaussian emission line fixed at 1.84 keV. This improved the fit by ∆ χ 2 = 5 (obs1), 6 (obs2), 3 (obs3 and obs4), and 8 (obs5) for one additional free parameter, the normalization. This line is most likely the blend of Si K and triplet lines at ∼ 1 . 84 keV due to He-like Si i.e., Si XIII that have been detected in the Chandra /HETG spectrum by (Kraemer et al. 2020). Therefore, we kept the Gaussian line at 1 . 84 keV included in our spectral fitting. Removing the line does not change the derived continuum parameters in any significant way. \nIn obs3, we found excess emission at around six keV. Adding a narrow ( σ = 1 eV) Gaussian line at 6.01 keV improved the χ 2 by 28 for two additional parameters, namely, the line centroid and normalization. \nWe used ZBLACKBODY to account for the soft excess emission. For obs3, this improved the χ 2 by 15 for two additional parameters with the best-fit value of kT ∼ 0 . 09 keV. We fixed the kT at 0.09 keV, as the temperature is similar in all the observations, and let the normalization vary for the other observations. For these, including a soft excess component resulted in no significant improvement in the fit. We could constrain the normalization in obs3 and obtain an upper limit for the remaining four observations. The upper limits for the normalization are consistent with that of obs3. \nThe spectral model used for the SXT data can be expressed as TBABS × [ ZBLACKBODY + TBPCF × ( ZPOWERLAW + \nTable 2. Best-fit parameters of the emission lines in the UV grating spectra of obs4 and obs5. λ o is the observed wavelength in the unit of ˚ A. v fwhm is the full-width half maxima (FWHM) of the emission line in the unit of km s -1 . f line is the line flux in the unit of photon cm -2 s -1 . (f) - fixed during the error calculation.Table 3. Best-fit parameters of the model fitted only to the X-ray spectrum (0 . 7 -80 keV; SXT/LAXPC/CZTI data). N H and N WA H are in the unit of cm -2 , ξ is in erg cm s -1 . The DISKBB Norm is given by ( r in /D 10 ) 2 cosθ , where r in is the inner disk radius, D 10 is the distance in the unit of 10 kpc, and θ is the inclination angle. Emission lines are not shown here. \nFe K α + Si K α + Ne IX )]. We obtained the best fit χ 2 /dof of 108/72 (obs1), 85/72 (obs2), 68/68 (obs3), 79/74 (obs4), and 80/73 (obs5). We obtained a range for the unabsorbed 2 -10 keV primary X-ray continuum flux of 1 . 1 -2 . 1 × 10 -10 erg cm -2 s -1 . This is similar to that observed when the source is in a bright X-ray state (Lubi'nski et al. 2010). \nNext, we included the LAXPC and CZTI spectral data. We used a model systematic error of 3% to account for calibration issues. We replaced the simple powerlaw component with a thermal comptonization model THCOMP (Zdziarski et al. 2020), which we convolved with the DISKBB model. The parameters for the THCOMP are \nthe photon index (Γ), high energy electron temperature ( kT e ), the fraction of seed photons being Comptonized ( f THCOMP c ), and the redshift. This model can Comptonize the disk seed photons to generate the soft excess emission or the hard X-ray continuum depending upon the electron temperature of the Comptonizing medium. Here, we fixed the electron temperature at 100 keV, usually observed in the bright state (Lubi'nski et al. 2010). We fixed the f THCOMP c at 1. Additionally, we used a cross-normalization constant for the LAXPC and CZTI spectral data. For the CZTI spectra, we obtained similar values for this constant in all the observations. Therefore, we fixed this to the best-fit value of 0 . 77, as ob- \nined in obs2 and obs4, in all the observations. For the LAXPC, we found this constant to be 0 . 86 (obs1, obs2, and obs4) and 1 . 1 (obs3 and obs5), which we fixed during the subsequent analyses. We also fixed the inner disk temperature ( kT in ) in DISKBB at 5 eV and varied the normalization only since changing the kT in from 5 to 50 eV did not affect the X-ray continuum parameters. \nThe ubiquitous presence of a narrow Fe K α emission line in all the observations indicates the signature of distant reflection. We used a convolution model IREFLECT (Magdziarz & Zdziarski 1995) to model this reflection component. We extended the energy grid from 10 -4 to 1000 keV to use the convolution models. The parameters of this model are f refl (reflection fraction), the metal abundance, Fe abundance, inclination angle, disk temperature, and ξ (disk ionization parameter). Initially, we varied the reflection fraction for all the observations, but we could constrain this parameter for obs4 and obs5 only. For obs4, it lies between 0 . 03 -0 . 4, and for obs5, it lies between 0 . 3 -0 . 9. We fixed this parameter at 0.3 for all the observations, assuming the distant reflection to be nearly constant over time. The metal and iron abundances and inclination angle are fixed at solar and 45 · , respectively. We fixed the ionization parameter to 0, assuming the distant reflector to be neutral. Adding the reflection component improves the fit by ∆ χ 2 ∼ 5 -20 for no additional free parameter in four observations (except for obs1). We obtained the reduced χ 2 of 1.8 (obs1), 1.2 (obs2), 1.1 (obs3), 0.9 (obs4), and 0.9 (obs5). \nWe observed significant residuals in obs1 around 1 keV. We tested the presence of warm absorbers in this observation using the SPEX (Kaastra et al. 1996) model XABS (Steenbrugge et al. 2003) available as a table model for XSPEC (Parker et al. 2019). The parameters of this model are the logarithm of the ionization parameter (log ξ ), hydrogen column density ( N WA H ), root mean square (rms) velocity of the absorbing plasma (v), covering factor of the absorbing plasma ( f XABS c ) and the redshift. We used this table model for the X-ray spectra. This improved the χ 2 by 70 for four additional parameters. The data could not constrain the rms velocity; therefore, we fixed it to 200 km s -1 , which did not change the χ 2 . The best-fit values of XABS parameters are: log ξ < 1 . 02, N WA H ∼ 3 . 6 × 10 22 cm -2 , f XABS c ∼ 0 . 7. The fit residuals before and after adding the XABS component are shown in Figure 6. \nWe also tested the presence of warm absorbers in all other observations. The log ξ is fixed to the best-fit value obtained from obs1 ( -0 . 51). We fixed the turbulent velocity arbitrarily at 200 km s -1 . For obs2, adding this component improved the χ 2 by 33 for three additional \nFigure 6. Obs1: The (data-model)/error before (top) and after (bottom) using the XABS model component. \n<!-- image --> \nparameters. The best-fit parameters are listed in Table 3. Adding this model component did not improve the fit statistics significantly for obs3, obs4, and obs5. We tested the variability of the column density of the warm absorber by fixing the covering fraction to 0.7. We could only obtain the 90% upper limit of 5 × 10 21 cm -2 for obs3, 4 × 10 21 cm -2 for obs4, and 2 × 10 21 cm -2 for obs5. This could be due to the variation in the warm absorber along the line of sight, as observed earlier in this source (Schurch & Warwick 2002; Zoghbi et al. 2019). Therefore, we did not use this component for these observations. The final reduced χ 2 for obs1 and obs2 are 1.2 and 0.9, respectively. The final expression of the combination of model is, CONSTANT × TBABS × [ ZBLACKBODY + XABS × TBPCF × ( IREFLECT × THCOMP × DISKBB + Fe K α + Si K α + Ne IX )]. The N H in the TBPCF component varies from ∼ 1 . 2 -3 . 4 × 10 23 cm -2 and the Γ varies from ∼ 1 . 56 -1 . 87. Next, we proceeded to the UV/Xray broadband spectral modeling.", '5. UV/X-RAY JOINT SPECTRAL ANALYSIS': 'We started our UV/X-ray joint spectral analysis with obs4 and obs5, for which the FUV-G1 and FUV-G2 grating spectral data are available. We also used the FUV-BaF 2 and NUV-B15 single-channel spectral data wherever available. We put together the final models fitted separately to the UV and X-ray spectral datasets, our joint UV - X-ray final model expression reads as CONSTANT × TBABS × REDDEN × [ sb temp + intr ext × absorption line × { ZBLACKBODY + XABS × TBPCF × ( IREFLECT × THCOMP × DISKBB +Fe K α +Si K α +Ne IX \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. UV and X-ray joint final best-fit model for the five observations. For each figure, the upper panel shows the best-fit model and data. The dashed lines show the emission line components. The lower panel shows the (data-model)/error. The data are rebinned for plotting purposes. UVIT/FUV-G1 and FUV-G2 spectra are in orange and cyan, NUV-B15 and FUV-BaF 2 data points are represented by the triangle and star symbols, respectively, SXT spectrum in green, LAXPC spectrum in red, and CZTI spectrum in blue. \n<!-- image --> \nFigure 8. The black solid line shows the final best fit unabsorbed total model and the gray is the same for the absorbed model. The different unabsorbed components are shown by - green dash-dotted: accretion disk emission; red dashed: comptonized disk emission; blue dotted: soft excess and the Compton hump (reflection emission); fuchsia: host galaxy (sb temp) template. The dereddened and unabsorbed spectral data are shown by the orange star (NUV-B15), yellow squares (FUV-G1), green circles (SXT), blue diamonds (LAXPC), and red triangles (CZTI). \n<!-- image --> \n+ UV Emission lines) } ] . The XABS model is used only for the obs1 and obs2. \nAs before, we used relative normalizations for the spectral data from different instruments. We fixed the relative normalizations at 1 for the UVIT/FUV-G1, FUV-BaF 2 , NUV-B15, and SXT data. For the rest of the instruments, we fixed the relative normalizations at the values obtained earlier from the UVIT, and X-ray spectral analysis performed separately. We fixed the E(B-V) in REDDEN and Balmer decrement in intr red at 0 and 2.72, respectively, for the X-ray spectral data. We also fixed the partial covering fraction ( f TBPCF c ) and N H in TBPCF , N H in TBABS , N WA H and f XABS c in XABS at 0 for the UV spectral data. \nInitially, we varied the DISKBB temperature and normalizations. We found that the best-fit normalizations for obs4 and obs5 were similar within errors, while we could not constrain the normalizations in the case of obs1, obs2, and obs3 as only two UV data points from the broadband filters are available for these observations. Therefore, we fixed the DISKBB normalizations for obs1, obs2, obs3, and obs5 to the best-fit value derived for obs4. We varied the Γ and the f THCOMP c in THCOMP ; the N H and partial covering in TBPCF ; log ξ , N WA H and f XABS c in XABS ; and the normalization of the \nZBLACKBODY and emission lines in X-ray bands. For obs2, we fixed the log ξ at the best-fit value of -0 . 51 as obtained in obs1. We varied the normalizations of the UV emission lines and found them similar to those obtained during the separate UVIT spectral analysis. Therefore, we fixed all the UV emission or absorption line parameters. \nWe obtained a χ 2 /dof = 401 . 2 / 395 for obs4 and 400 . 4 / 395 for obs5. The total unabsorbed flux from the UV emission lines has increased by a factor of ∼ 1 . 3 in obs5 compared to those in obs4. \nFor obs1, obs2, and obs3, we have no information regarding the emission lines in FUV/NUV bands. We arbitrarily used the emission lines obtained from obs5 as a reference for the FUV band in obs1, obs2, and obs3. For obs1 and obs2, we used a constant multiplicative factor for the FUV emission lines to account for the increase in the line flux (if any) with the rise of the total UV flux. we obtained the best-fit value for this constant ∼ 1 . 23 in obs2 and ∼ 1 . 47 in obs1, assuming the same intrinsic reddening suffered during all the observations. The bestfit values are listed in Table 4. The data, the best-fit model, and the residuals in terms of (data-model)/error for all observations are shown in Fig. 7. For obs4, the final unabsorbed and de-reddened SED and its different model components are shown in Figure 8.', '6. RESULTS AND DISCUSSION': "We analyzed the AstroSat multi-wavelength data from the five observations performed during 2017 Feb/March and 2018 Jan/May. In the UV band, we derived the intrinsic continuum after removing the effect of intrinsic and Galactic extinction and correcting for emission from the BLR/NLR. We obtained the inner disk temperature varying between ∼ 8 . 7 -10 . 3 eV. The variation in the disk temperature may not be real but due to our assumption of constant intrinsic reddening (see below). The emission line widths are of the order of a few hundred to a few thousand km s -1 , typical of the NLR and BLR emission. During our observations (obs4), the observed UV flux is near the minimum level (at 1350 ˚ A, flux ∼ 4 × 10 -14 erg cm -2 s -1 ˚ A -1 ) observed in the past as reported in Kraemer et al. (2006). \nWe found the X-ray photon index varying from ∼ 1 . 56 -1 . 87, heavily absorbed by the neutral absorber of column 1 . 2 -3 . 4 × 10 23 cm -2 . This high absorption leads to a significant suppression of the soft excess. We found a weak presence of soft excess, which most likely originates in photo-ionization of regions far away from \nTable 4. Best-fit parameters of the final model fitted to the joint UV - X-ray spectra (UVIT/SXT/LAXPC/CZTI). The normalizations of the X-ray emission lines are in the unit of photons cm -2 s -1 . \nFigure 9. Total unabsorbed continuum model for all the observations. The shaded energy bands show the NUV-B15 (gray), FUV-G1/G2 (sky blue), SXT (green), LAXPC (yellow), and CZTI (pink). \n<!-- image --> \nthe central engine and is not directly correlated with the primary continuum flux. \nThe mass of the central black hole has some uncertainty. It could be between 2 . 5 × 10 6 -5 . 6 × 10 7 M ⊙ (Williams et al. 2023) from estimates made using different methods: circum-nuclear gas dynamics (Hicks \nFigure 10. Top: FUV-BaF 2 flux in the unit 10 -14 erg cm -2 s -1 ˚ A -1 . Middle: NUV-B15 filter flux in the unit 10 -14 erg cm -2 s -1 ˚ A -1 . Bottom: Unabsorbed X-ray Flux in the 2 -10 keV energy band in the unit of 10 -10 erg s -1 . The black vertical lines are the error on the FUV/NUV fluxes. \n<!-- image --> \n& Malkan 2008), stellar dynamics (Onken et al. 2014; Roberts et al. 2021), H β reverberation mapping (Bentz et al. 2006, 2022; De Rosa et al. 2018) and X-ray rever- \neration mapping (Zoghbi et al. 2019). The most recent measured mass of 1 . 7 ± 0 . 4 × 10 7 M ⊙ (Bentz et al. 2022), gives an Eddington ratio varying from 0 . 07 -0 . 33. For a mass of 2 . 5 × 10 6 M ⊙ , the Eddington ratio will be in the range of 0 . 4 -2 . 3 and for a mass of 5 . 6 × 10 7 M ⊙ , the Eddington ratio will be in the range of 0 . 02 -0 . 1. We find that the relative contribution of the intrinsic accretion disc emission to the bolometric emission ( L bol ) is higher than that of the X-ray emission (see Fig. 9). However, this bolometric luminosity is subject to some uncertainty because of the high intrinsic extinction. As mentioned earlier, we modeled the STIS spectrum, which has a continuum flux similar to the UVIT, for estimating intrinsic extinction. Although these non-simultaneous observed spectra are similar, the intrinsic shape of the spectra and the amount of obscuration could be different. At this point, we cannot rectify this systematic uncertainty. \nBased on our joint UV/X-ray spectroscopy, we derived the SED of NGC 4151 at five epochs. The absorptioncorrected SEDs are compared in Figure 9. We show the observed FUV and NUV flux and the unabsorbed 2 -10 keV X-ray flux in Figure 10. In the top and middle panels in Figure 10, we show the observed flux obtained from the circular area of radius 8 . 3 '' using the FUV and NUV filters at mean wavelengths of 1541 ˚ A and 2197 ˚ A, respectively. The variation in the flux in these two bands appears correlated. This is consistent with the previous observations where a strong correlation is observed within the UV/optical bands (Edelson et al. 2017). In the bottom panel of Fig. 10, we show the unabsorbed Xray flux in the 2 -10 keV energy band. The flux varies between 1 . 3 -3 . 2 × 10 -10 erg cm -2 s -1 . This is near the historical peak observed during the 1993 December ( ∼ 3 . 6 × 10 -10 erg cm -2 s -1 ; Edelson et al. 1996). Therefore, during our observations, the nuclear flux could be approaching the fourth maximum since 1993. The X-ray and UV flux variations are not correlated, and the flux in the X-ray band remains within ∼ 30% except for the obs3, where the flux drops by a factor of ∼ 2. Earlier works on short-term X-ray/UV/optical variability have shown a correlation in NGC 4151 where the X-rays are observed to lead UV emission by ∼ 3 days (Edelson et al. 2017). However, on timescales of months, we do not observe a correlation though the number of observations is only five in our case.", '6.1. Variations in the intrinsic reddening': "The variation in the intrinsic UV continuum ( ∼ 5 -10 × 10 -9 erg cm -2 s -1 ) could be caused by our assumption of constant intrinsic reddening. In Figure 11, we show the ratio of observed NUV and FUV flux, a \nmeasure of FUV -NUV color, as a function of observed FUV flux. The color decreases with increasing FUV flux, i.e., for higher FUV flux, the flux increase in the FUV band is more than that in the NUV band. Such a change can easily be caused by the variations in the intrinsic reddening as the FUV emission suffers more extinction than the NUV emission. Thus, the extinction appears to be anti-correlated with the continuum flux over a year during our AstroSat observations. Using optical spectroscopic monitoring of NGC 4151 during 1996-2001, Raki'c et al. (2017) found that the Balmer decrement approaches 3, as expected for a pure photoionization model with no reddening, and the Balmer decrement is also strongly anti-correlated with the continuum flux at 5100 ˚ A (see their Fig. 4). These findings clearly suggest that variations in the internal reddening play a significant role in flux variability. \nThe timescales of continuum flux variation over the years indicate that the variation in the UV flux we observed could be at least partly due to reddening. Since we lack information on the Balmer decrement during our observations and the quality of the UVIT data do not allow an exact estimate of the intrinsic reddening, we were unable to eliminate completely the effect of this extinction for all the observations. Therefore, we tested the possibility of varying intrinsic extinction with our UVIT data. As shown in Fig. A1 (left), our FUV-grating spectrum is nearly identical to that from HST /STIS, which we used to estimate the intrinsic extinction. Thus, the intrinsic continuum derived for obs4 is most likely a true representation of the disk emission. \nFigure 11. Variation in the UV color FUV -NUV with the observed FUV flux. \n<!-- image --> \nTo estimate the intrinsic emission for the other four observations, we assumed the disk spectrum to be con- \nstant, the same as that for obs4. We refitted the UVIT data from each observation, except obs4, by varying the intrinsic reddening only. We found that the Balmer decrement decreases with the increasing observed FUV or NUV flux, but the fit resulted in reduced χ 2 more than 6. Therefore, first, we varied the line fluxes in obs5 along with the Balmer decrement. We obtained the best fit Balmer decrement of ∼ 4 . 84 ( E ( B -V ) = 0 . 37). Next, we used these updated emission line fluxes for obs1, obs2, and obs3 with a variable relative normalization. This resulted in acceptable fits with the relative line normalizations ∼ 0 . 98 (obs3), 0.92 (obs2), and 0.86 (obs1). We obtained the Balmer decrement of 4.43 (obs1), 4.62 (obs2), 4.84 (obs3), and 4.83 (obs5). In Fig. 12, we show the variation in extinction with observed FUV flux, assuming a constant disk and variable line emission. The E ( B -V ) can be seen to anticorrelate with the observed FUV flux. \nThe total emission line fluxes increased by about 0 . 6% (obs1), 8% (obs2), 20% (obs3 and obs5) compared to that for obs4. On the other hand, from Fig. 10, it can be seen that the total FUV flux, from obs1 to obs4, decreased in a uniform manner, i.e., by ∼ 20%, between the consecutive observations. Therefore, the line fluxes we measured in obs1 and obs2 by fixing the disk continuum are increasing disproportionately with the increase in total observed flux. Usually, during the low to medium flux state, the emission line flux is observed to correlate with the continuum flux (Ulrich & Horne 1996; Shapovalova et al. 2008; Chen et al. 2023). The C IV emission line is observed to be delayed with respect to the continuum emission by ∼ 3 days (Ulrich & Horne 1996; Metzroth et al. 2006), which is less than the interval between our observations. This rules out the time delay between the continuum and the emission line as a possible cause of the non-uniform increase in emission line flux with the continuum. Therefore, it is likely that the apparent anti-correlation observed between line and continuum flux in obs1 and obs2 is an artifact due to the lack of emission line information in those data sets. The fall in the intrinsic extinction in obs1 and obs2 is not likely as steep as estimated. \nMany AGN, including the changing-look AGN, show the trend of 'bluer-when-brighter' (Green et al. 2022; Guo et al. 2024). We examined the difference spectrum using the obs4 and obs5 grating spectra. We modeled it with intr ext, DISKBB , and three emission lines (as these lines are variable between the observations). We fixed the Balmer decrement at 4.95. We found the disk temperature, kT in > 3 . 4, and normalization 7 . 3 +1800 -4 × 10 8 . As we could not constrain the temperature, it is inconclusive whether the difference spectrum is consistent \nwith the disk. Thus, we conclude that the observed variation in the UV flux could be a combined effect of the change in the intrinsic shape of the continuum ('bluerwhen-brighter') and internal reddening. \nFigure 12. Variation in the E ( B -V ) with the observed FUV flux when the intrinsic continuum flux is fixed as obtained in obs4. \n<!-- image -->", '6.2. UV reddening and X-ray absorption': 'We found a large neutral absorbing column ( ∼ 10 23 cm -2 ) existing along the line of sight to the Xray source. This would suggest much higher UV extinction (E(B -V) = 14 for N H = 10 23 cm -2 ) with the Galactic dust-to-gas ratio (Eq. 1). In our observations, the Balmer decrement of ∼ 4 . 95 (E(B -V) = 0 . 38; using Eq. 2) is equivalent to a column density of ∼ 2 . 6 × 10 21 cm -2 in the UV following Eq. 1. Apparently, the column estimated from the UV spectrum is much smaller than that obtained from the X-ray spectrum. In the case of the two intermediate Seyferts NGC 7582 and NGC 5506, Maccacaro et al. (1982) found that the X-ray absorbing column is ∼ 10 times larger than the N H measured using the reddening of optical continuum and Balmer lines. This discrepancy can be attributed to different possibilities. \nThe E ( B -V ) to N H conversion relation, Eq. 1, is obtained assuming a Galactic dust-to-gas ratio. The AGN environment could substantially differ from the Galactic dust compositions and grain sizes due to heating by the AGN (Maiolino et al. 2001; Czerny et al. 2004). Jaffarian & Gaskell (2020) studied the relation between the X-ray absorbing column and the N H estimated from Balmer decrement; they found that the obscuring column predicted from extinction assuming a Galactic dust-to-gas ratio is much lower than the column \nof the X-ray absorbing gas for most Seyfert 2. There is a large scatter between the E ( B -V ) calculated from the Balmer decrement and N H obtained from X-ray observations considering all the AGN in their sample. \nOne of the possibilities to reconcile the discrepancy between the obscuring columns N H we estimated from our UV and X-ray observations is to assume a substantial amount of dust-free gas within the dust sublimation radius either in the form of weakly ionized winds from the accretion disk or the inner regions of the obscuring torus. While the dust-free region within the dust sublimation radius will absorb X-rays, it will cause very little or no UV reddening. The dusty gas outside the dustsublimation radius will cause both X-ray absorption and UV reddening. This will result in effectively low column density for the UV reddening and large column density for the X-ray absorbing gas. This scenario requires a substantial amount of obscuring matter where the dust has been destroyed by the AGN heating. \nAnother possibility could be that the obscuration is caused by a two-phase medium where the dense dusty clumps are embedded in a low-density gas. This could be similar to an obscuring torus consisting of a number of compact dense clouds embedded in the low-density medium. A single or a small number of compact dense clouds along the line of sight can then cover the compact X-ray source, i.e., the hot corona, resulting in a high obscuring column. The UV emission arising from the accretion disk, which is extended, will largely be reddened by the low-density medium, thus resulting in a much lower obscuring column. Further, the dense clouds embedded in low-density medium responsible for the X-ray absorption are expected to be moving, which can then cause variations in the X-ray absorbing column, which has been observed. \nIn Figure 13, we show the variations of X-ray absorbing column N H with the NUV -FUV color. The apparent correlation between the X-ray N H and the UV color cannot be explained just by the change in the column obscuring both the X-ray source and the UV source as the increasing column will lead to redder colors contrary to the correlation. If the bluer when brighter trend seen in Fig. 11 is at least partly caused by intrinsic variability rather than a change in the UV reddening, then it is possible that increased UV flux may be driving stronger winds which would result in the increased X-ray absorption column. This would explain the trend seen in Fig. 13.', '6.3. X-ray spectral variability': 'We show the variation of the X-ray photon index with the intrinsic disk continuum emission in Figure 14. We \nFigure 13. Variation of intrinsic neutral absorbing column density with the observed NUV -FUV color. \n<!-- image --> \nFigure 14. Variation in the X-ray photon index with intrinsic disk flux ( DISKBB ) for all five observations. \n<!-- image --> \nobserve that the total disk continuum flux increases by a factor of ∼ 2, whereas the index remains fairly constant. The large time spans between the data points can wash out the effect of Compton cooling or X-ray reprocessing. Therefore, based on these observations, we could not establish any correlation on the variation in the UV/Xray flux.', '7. CONCLUSION': "We performed a broadband UV/X-ray spectral variability study of NGC 4151 based on five sets of AstroSat observations. The main results of our study are as follows. \n- 1. The broadband UV to X-ray spectrum of NGC 4151 primarily comprises of emission from the accretion disk ( kT in ∼ 8 . 7 -10 . 3 eV), emission lines from the broad and narrow-line regions in the UV band, the primary X-ray power-law emission Γ ∼ 1 . 56 -1 . 87 is produced by thermal Comptonization of disk photons and a distant X-ray reflection component.\n- 2. We obtained a high intrinsic reddening ( E ( B -V ) ∼ 0 . 4) in the UV band, generally observed in the low optical/UV flux state (Shapovalova et al. 2010; Raki'c et al. 2017).\n- 3. The FUV -NUV color becomes bluer with increasing FUV flux. This is similar to the 'bluerwhen-brighter' trend observed in the optical/UV spectra of many AGN and could be due to intrinsic variations in the accretion disk. Such a trend could also arise due to variations in the intrinsic reddening. We also observed increasing X-ray absorption column with UV flux or color, this could result due to stronger winds at intrinsically bluer and higher UV emission.\n- 4. X-ray emission from NGC 4151 undergoes strong absorption through a large column density ( N H ∼ 1 . 2 -3 . 4 × 10 23 cm -2 ). The UV emission is affected by internal reddening with E ( B -V ) ∼ 0 . 4, which is equivalent to an obscuring column of only ∼ 10 21 cm -2 assuming the Galactic dust-to-gas ratio. This ∼ 2 orders of magnitude difference in the obscuring columns may imply that the dustto-gas ratio of the obscurer could be very different than the Galactic value or the X-ray and the UV obscurers are not the same. We invoke two possible geometries to explain the observed discrepancy. Firstly, the obscuring medium may be divided in two zones by the dust sublimation radius. The gas within and outside the dust sublimation radius can cause X-ray absorption, while the dust outside the dust sublimation radius will be primarily responsible for the UV reddening. Secondly, the obscurer could be dense, clumpy clouds embedded within low-density gas and dust. The X-ray emission could be obscured substantially by a small number of dense clouds along the line of sight to the compact hot corona, while a significant amount of the UV emission arising from an\n- extended region may pass through the low-density regions.\n- 5. The X-ray fluxes during our observations are quite high ( ∼ 10 -10 erg cm -2 s -1 ), similar to that observed during previous periods of high flux (Lubi'nski et al. 2010). We also found the presence of variable neutral and warm absorbers, usually observed in this source (Puccetti et al. 2007; Zoghbi et al. 2019).\n- We sincerely thank an anonymous referee whose insight1\n- ful comments and suggestions significantly enhanced the 2\n- quality of this paper. This work uses data from Indian 3\n- Space Science Data Centre (ISSDC) of the AstroSat mis4\n- sion of the Indian Space Research Organisation (ISRO). 5\n- We acknowledge the SXT and LAXPC POCs at TIFR 6\n- (Mumbai), CZTI POC at IUCAA (Pune), and UVIT 7\n- POCat IIA (Bangalore) for providing the necessary soft8\n- ware tools for data processing. The UVIT data were re9\n- processed by CCDLAB pipeline (Postma & Leahy 2017). 10\n- This publication used archival STIS spectrum from HST 11\n- data archive (https://archive.stsci.edu/hst/search.php). 12\n- This research has used the Python and Julia packages. 13\n- This research has used the SIMBAD/NED database. 14\n- S.K. acknowledges the University Grant Commission 15 \n16 \n(UGC), Government of India, for financial support. \n- Kulinder Pal Singh thanks the Indian National Sci17 \n18 \n19 \n20 \n21 \nence Academy for support under the INSA Senior \nScientist \nProgramme. \nAAZ acknowledges support \nfrom the Polish National Science Center under the \ngrants 2019/35/B/ST9/03944, 2023/48/Q/ST9/00138, \n- and from the Copernicus Academy under the grant CBMK/01/24. 22 23 \n24 \n- Facility: AstroSat , HST . 25\n- Data: The HST /STIS spectrum presented in this arti26\n- cle was obtained from the Mikulski Archive for Space 27\n- Telescopes (MAST) at the Space Telescope Science 28\n- The specific observations analyzed can be\n- Institute. accessed via DOI. 29 30 \n31 \n- Software: CCDLAB (Postma & Leahy 2017), XSPEC 32\n- (Arnaud 1996), SAOImageDS9 (Joye & Mandel 2003), 33\n- Julia (Bezanson et al. 2017), Astropy (Collaboration et al. 2013). 34 35", 'REFERENCES': "Arnaud, K. 1996, in Astronomical Data Analysis Software and Systems V, ASP Conference Series, Vol. 101, 1996, George H. Jacoby and Jeannette Barnes, eds., p. 17., Vol. 101, 17 \nAntia, H., Yadav, J., Agrawal, P., et al. 2017, The Astrophysical Journal Supplement Series, 231, 10 \n- Singh, K., Stewart, G., Westergaard, N., et al. 2017, Journal of Astrophysics and Astronomy, 38, 1\n- Singh, K. P., Tandon, S., Agrawal, P., et al. 2014, in Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, Vol. 9144, SPIE, 517-531\n- Singh, K. P., Stewart, G. C., Chandra, S., et al. 2016, in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Vol. 9905, International Society for Optics and Photonics, 99051E\n- Steenbrugge, K., Kaastra, J., De Vries, C., & Edelson, R. 2003, Astronomy & Astrophysics, 402, 477\n- Szanecki, M., Nied'zwiecki, A., & Zdziarski, A. A. 2021, The Astrophysical Journal, 909, 205\n- Tandon, S., Subramaniam, A., Girish, V., et al. 2017, The Astronomical Journal, 154, 128\n- Tandon, S., Postma, J., Joseph, P., et al. 2020, The Astronomical Journal, 159, 158\n- Ulrich, M., Altamore, A., Boksenberg, A., et al. 1985, Nature, 313, 747\n- Ulrich, M.-H. 2000, The Astronomy and Astrophysics Review, 10, 135\n- Ulrich, M.-H., Boksenberg, A., Bromage, G., et al. 1991, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 382, Dec. 1, 1991, p. 483-500. Research supported by CNR and MURST., 382, 483\n- Ulrich, M.-H., & Horne, K. 1996, Monthly Notices of the Royal Astronomical Society, 283, 748\n- Vadawale, S. V., Rao, A. R., Bhattacharya, D., et al. 2016, SPIE Proceedings, 9905, 409, doi: 10.1117/12.2235373 Vestergaard, M., & Wilkes, B. J. 2001, The Astrophysical Journal Supplement Series, 134, 1, doi: 10.1086/320357\n- Williams, J. K., Gliozzi, M., Bockwoldt, K. A., & Shuvo, O. I. 2023, Monthly Notices of the Royal Astronomical Society, 521, 2897\n- Yadav, J., Agrawal, P., Antia, H., et al. 2016, in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Vol. 9905, SPIE, 374-388\n- Yang, Y., Wilson, A., & Ferruit, P. 2001, The Astrophysical Journal, 563, 124\n- Zdziarski, A. A., & Magdziarz, P. 1996, Monthly Notices of the Royal Astronomical Society, 279, L21\n- Zdziarski, A. A., Szanecki, M., Poutanen, J., Gierli'nski, M., & Biernacki, P. 2020, Monthly Notices of the Royal Astronomical Society, 492, 5234\n- Zoghbi, A., Miller, J., & Cackett, E. 2019, The Astrophysical Journal, 884, 26", 'A. STIS SPECTRAL ANALYSIS': "We used the STIS spectrum observed on March 3, 2000, when the source was in the low flux state. We averaged the six observations of 1.2 ks exposure time each, which are nearly continuous. We used the relativistic accretion disk model KERRD available in XSPEC for the underlying continuum. The model parameters are distance, color correction factor, black hole mass, inclination angle, inner and outer disk radius, and normalization. We fixed the mass, distance, color correction factor, inclination angle, and outer disk radius to 1 . 7 × 10 7 M ⊙ , 15 . 8 Mpc, 2 . 4, 45 · , and 10 5 r g respectively. We used multiple Gaussian lines to account for the emission/absorption lines from the BLR/NLR region (see Table A1). Also, there are thousands of Fe II emission lines that overlap with each other, making a pseudo continuum. We used Fe II template model from Vestergaard & Wilkes (2001), which improved the statistic by ∆ χ 2 of 59 for two additional free parameters, the amplitude and the Doppler broadening. To account for the Galactic reddening, we used the XSPEC model REDDEN where we fixed the E(B-V) to 0.03, as mentioned earlier. In addition to the Galactic extinction, we tested the presence of intrinsic extinction. We used the Czerny et al. (2004) extinction curve with Balmer decrement as the free parameter. We obtained the best-fit value of 4 . 95 ± 0 . 09 for the Balmer decrement, and the statistic improved by ∆ χ 2 of 28. As the contribution of host galaxy emission is negligible in the FUV band, we did not use this component. We obtained the best-fit inner disk radius < 4 r g and the mass accretion rate 2 . 2 +0 . 3 -0 . 2 × 10 25 g s -1 . The absorption lines used in the STIS spectrum are listed in Table A2. The line centroids of the absorption and emission lines are fixed during the error calculation. \n<!-- image --> \nFigure A1. Left: Comparison of the STIS and UVIT/FUV-G1 flux spectrum. Right: STIS spectrum: Best fit is shown in blue, reddened and absorbed continuum is shown in black dotted line, emission lines are shown in red, and Fe II emission is shown in green. The (data - model)/error is shown in the bottom panel. \n<!-- image --> \nTable A1. Best-fit parameters of the detected emission lines in STIS spectrum. λ o is the observed wavelength in the unit of ˚ A. v fwhm is the full-width half maxima (FWHM) of the emission line in the unit of km s -1 . f line is the line flux in the unit of photon cm -2 s -1 . '(f)' - the parameter is fixed during error calculation. \nTable A2. Best-fit parameters of the absorption lines detected in the STIS spectrum. '(f)' - the parameter is fixed during error calculation."}

2024arXiv240902989C

Recent JWST observations at z gt 6 may imply galactic ionizing photon production in excess of prior expectations. Under observationally motivated assumptions about escape fractions these suggest a z sim 89 end to reionization in strong tension with the z lt 6 end required by the Lyalpha forest. In this work we use radiative transfer simulations to understand what different observations tell us about when reionization ended and when it started. We consider a model that ends too early at z approx 8 alongside two more realistic scenarios that end late at z approx 5 one that starts late z sim 9 and another that starts early z sim 13. We find that the latter requires up to an orderofmagnitude evolution in galaxy ionizing properties at 6 lt z lt 12 perhaps in tension with recent measurements of xirm ion by JWST which indicate little evolution. We also study how these models compare to recent measurements of the Lyalpha forest opacity mean free path IGM thermal history visibility of z gt 8 Lyalpha emitters and the patchy kSZ signal from the CMB. We find that neither of the lateending scenarios is conclusively disfavored by any single data set. However a majority of these observables spanning several distinct types of observations prefer a late start. Not all probes agree with this conclusion hinting at a possible lack of concordance between observables. Observations by multiple experiments including JWST Roman and CMBS4 in the coming years will either establish a concordance picture of reionizations early stages or reveal systematics in data andor theoretical modeling.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.02989', '2024arXiv240902989C', 'arXiv:2409.02989']

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

Chasing the beginning of reionization in the JWST era

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.02989.pdf

{'Chasing the beginning of reionization in the JWST era': "Christopher Cain, 1 Garett Lopez, 2 Anson D'Aloisio, 2 Julian B. Mu˜noz, 3 Rolf A. Jansen, 1 Rogier A. Windhorst, 1 and Nakul Gangolli 2 \n1 \nSchool of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-6004, USA 2 Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA \n3", 'ABSTRACT': "Recent JWST observations at z > 6 may imply galactic ionizing photon production in excess of prior expectations. Under observationally motivated assumptions about escape fractions, these suggest a z ∼ 8 -9 end to reionization, in strong tension with the z < 6 end required by the Ly α forest. In this work, we use radiative transfer simulations to understand what different observations tell us about when reionization ended and when it started. We consider a model that ends too early (at z ≈ 8) alongside two more realistic scenarios that end late at z ≈ 5: one that starts late ( z ∼ 9) and another that starts early ( z ∼ 13). We find that the latter requires up to an order-of-magnitude evolution in galaxy ionizing properties at 6 < z < 12, perhaps in tension with recent measurements of ξ ion by JWST, which indicate little evolution. We also study how these models compare to recent measurements of the Ly α forest opacity, mean free path, IGM thermal history, visibility of z > 8 Ly α emitters, and the patchy kSZ signal from the CMB. We find that neither of the late-ending scenarios is conclusively disfavored by any single data set. However, a majority of these observables, spanning several distinct types of observations, prefer a late start. Not all probes agree with this conclusion, hinting at a possible lack of concordance between observables. Observations by multiple experiments (including JWST, Roman, and CMB-S4) in the coming years will either establish a concordance picture of reionization's early stages or reveal systematics in data and/or theoretical modeling.", '1. INTRODUCTION': "Despite an explosion of new data in the past decade probing cosmic reionization, little is known about how and when the process began. The ending of reionization, believed to occur at 5 < z < 6, has been constrained by observations of the Ly α forest of highredshift QSOs (Becker et al. 2015; Kulkarni et al. 2019; Keating et al. 2020; Nasir & D'Aloisio 2020; Qin et al. 2021; Bosman et al. 2022). Direct (and indirect) measurements of the mean free path from QSO spectra (Worseck et al. 2014; Becker et al. 2021; Zhu et al. 2023; Gaikwad et al. 2023; Davies et al. 2024b), and measurements of the IGM thermal history at z ≥ 5 . 5 (Gaikwad et al. 2020) have further corroborated this picture. The electron scattering optical depth to the CMB ( τ es , Planck Collaboration et al. 2020) constrains the midpoint of reionization to be z ∼ 7 . 5. Ev- \nCorresponding author: Christopher Cain \nclcain3@asu.edu \nidence of damping wings in high-redshift z ∼ 7 -8 QSOs (Davies et al. 2018; Wang et al. 2020; Yang et al. 2020) and measurements of the neutral fraction based on (non)detections of Ly α emitters (LAEs, e.g. Mason et al. 2018; Mason et al. 2019; Whitler et al. 2020; Jung et al. 2020) indicate that the IGM was partially neutral at these redshifts. \nUnderstanding the timeline of reionization is crucial for revealing the nature of the ionizing sources that drove it. Explaining how reionization could be driven by galaxies alone has been historically challenging thanks to the high early measurements of τ es (Dunkley et al. 2009; Komatsu et al. 2011) which required very early and/or extended reionization histories. This tension eased as the measured value of τ es steadily decreased. Several recent works (e.g. Bouwens et al. 2015; Robertson et al. 2015; Finkelstein et al. 2019; Matthee et al. 2022) have showed that galaxies can complete reionization by z ≈ 6 under physically reasonable assumptions about their ionizing properties. Recent evidence for an end later than z = 6 further relaxed demands on galaxy ionizing output (although see Davies et al. 2021, 2024a). \nConcurrent efforts demonstrated that AGN are unlikely to have contributed the majority of the ionizing budget responsible for reionization (e.g. Dayal et al. 2020; Trebitsch et al. 2023, although see Madau & Haardt 2015; Madau et al. 2024). \nThese findings paint a relatively simple, consistent picture of reionization: it ended at 5 < z < 6, was in progress at z ∼ 7 -8, and was likely driven by galaxies. Prior to JWST, the precise timing of reionization's midpoint and especially its early stages were not tightly constrained. Constraints on reionization's midpoint from τ es (Planck Collaboration et al. 2020) spanned a range of ± 0 . 75 in redshift (at 1 σ ), and few direct constraints on the first half of reionization existed. The space of models proposed by the aforementioned works span a wide range of possibilities 1 without contradicting observations. Indeed, one important goal for JWST is to probe the properties of galaxies at z > 7 -8 in hopes of learning more about reionization's early stages. \nHowever, the first JWST results may be complicating, as much as clarifying, our understanding of reionization. JWST has allowed for measurements of the UV luminosity function (UVLF) at much higher redshifts than HST, up to z ∼ 14 (Finkelstein et al. 2024; Adams et al. 2024; Donnan et al. 2024). It has also allowed us to measure the ionizing efficiency of galaxies, ξ ion , above z = 6 (Endsley et al. 2024; Simmonds et al. 2024; Pahl et al. 2024). Recently, Mu˜noz et al. 2024 pointed out that a face-value interpretation of recent UVLF and ξ ion measurements from JWST (Donnan et al. 2024; Simmonds et al. 2024) combined with observationally motivated assumptions about ionizing escape fractions ( f esc ) suggests reionization ended around z ∼ 8 -9, inconsistent at > 2 σ with the Planck τ es measurement 2 . This result represents a stark reversal from the historical problem of galaxies producing too few ionizing photons to complete reionization on time (see Robertson et al. 2015, and references therein). Several bright Ly α emitters (LAEs) at z > 8 (Zitrin et al. 2015; Larson et al. 2022; Bunker et al. 2023; Curti et al. 2024; Tang et al. 2024) have also been observed, with the highest-redshift detection to date at z = 10 . 6 (Bunker et al. 2023) by JWST. These observations may be surprising if the IGM is mostly neutral at these redshifts, since observing Ly α \nFigure 1. Summary of the reionization scenarios studied in this work. Broadly speaking, they represent the three types of possible reionization histories. Top: volume-averaged ionized fraction x V HII vs. redshift for the late start/late end , early start/late end , and early start/early end models. These models have z end ≈ 5, 5, 8 and z mid ≈ 6 . 5, 7 . 5, 8 . 5, respectively. Bottom: CMB electron scattering optical depth τ es for each model, compared to the results of Planck Collaboration et al. 2020; de Belsunce et al. 2021 (shaded regions denote ± 1 σ ). The late start/late end model is within 1 σ of Planck, while early start/late end is within 1 σ of the re-analysis by de Belsunce et al. 2021. \n<!-- image --> \nemission requires some level of ionization around galaxies (Mason & Gronke 2020). \nThe top panel of Figure 1 shows the volume-averaged ionized fraction ( x V HII ) for the three reionization models we will study in this work. The early start/early end model is motivated by the aforementioned findings of Mu˜noz et al. 2024, and has a midpoint (endpoint) of z mid = 8 . 5 ( z end = 8). The late start/late end model is motivated by Ly α forest observations at 5 < z < 6 and the Planck τ es measurement, has z mid = 6 . 5 and z end = 5. The early start/late end model also has z end = 5, but an earlier midpoint ( z mid = 7 . 5) and a start at z ∼ 13. These three models broadly represent the three types, or categories, of possible reionization histories. The bottom panel shows τ es for each, compared with the Planck Collaboration et al. 2020 measurement (black) and the recent re-analysis of Planck data from de Belsunce et al. 2021, with shaded regions denoting ± 1 σ uncertainties. The late start/late end model is consistent within 1 σ with Planck Collaboration et al. 2020, and the early start/late end case is similarly consistent with de Belsunce et al. 2021. \nIn this work, we study the observational properties of the three reionization models shown in Figure 1 us- \ning radiative transfer (RT) simulations. We will demonstrate, in accord with previous works, that observations from the z ≲ 6 Ly α forest strongly disfavor the early start/early end model, in agreement with τ es . We will also compare the two late-ending models to a wide range of observations with the goal of understanding whether existing data supports a late or an early start to reionization. This work is organized as follows. § 2 describes how we calibrate our three reionization models and discusses their implications for galaxy properties in light of recent JWST observations. In § 3, we describe our methods for running RT simulations of reionization and forward-modeling various observables. We compare our models to several sets of complementary observations in § 4, discuss the implications of our findings in § 5, and conclude in § 6. After discussing an observable, we will bold-face the name of the reionization model (if any) preferred by that observable. Throughout, we assume the following cosmological parameters: Ω m = 0 . 305, Ω Λ = 1 -Ω m , Ω b = 0 . 048, h = 0 . 68, n s = 0 . 9667 and σ 8 = 0 . 82, consistent with Planck Collaboration et al. 2020 results. All distances are in co-moving units unless otherwise specified.", '2.1. Reionization Model Calibration': "The main input to our RT simulations (described in § 3) is the globally averaged ionizing photon emissivity of sources verses redshift, ˙ N γ ( z ). In this section we describe how we use a combination of JWST observations and Ly α forest data to construct (or 'calibrate') ˙ N γ ( z ) for our three models. \nOur starting point is the amount of non-ionizing UV light produced by galaxies, which is quantified by the UV luminosity function (UVLF). This has been measured up to z ∼ 14 by JWST using both photometry and spectroscopy (e.g. Harikane et al. 2024; Finkelstein et al. 2024; Adams et al. 2024; Donnan et al. 2024). The top two panels of Figure 2 show two sets of UVLFs that we use in our analysis. In both panels, the dashed curves show the z < 8 UVLFs measured by Bouwens et al. 2021 with HST. In the left panel, the solid curves denote the double-power-law (DPL) fits from Adams et al. 2024 at 8 ≤ z ≤ 12, and the dotted curve is the measurement from Donnan et al. 2024 at z = 14 . 5. In the right panel, the dotted curves show results for the redshift-dependent UVLF parameters given in Eq. 3-6 of Donnan et al. 2024 (which is a best-fit to measurements from Bowler et al. 2016, 2020; Donnan et al. 2022; McLeod et al. 2023). At z > 8, the UVLFs in the left panel are a factor of ∼ 3 lower than those in the right \npanel. We will use these sets to roughly bracket observational uncertainties on the UVLF. \nThe lower left panel shows the integrated UV luminosity density, ρ UV , vs. redshift. The curves show the logarithmic average of ρ UV calculated using the two sets of UVLFs in the top panels, and the shaded regions show the spread between them. For illustration, we integrate the UVLF down to two limiting magnitudes - a bright cutoff of M cut UV = -17 (black) and a fainter M cut UV = -13 (magenta). The background shading denotes the redshift ranges covered by HST and JWST data. Note that at z > 8, the redshift evolution of ρ UV ( z ) is fairly insensitive to M cut UV , the main difference being normalization. This is because both sets of UVLFs have faint end slopes close to α = -2 . 1 at z ≥ 8, with little evolution across that redshift range (Kravtsov & Belokurov 2024). This is in contrast to some pre-JWST expectations, which predicted a steepening of the faint-end slope with redshift and a corresponding shallow evolution in ρ UV for faint M cut UV (e.g. Finkelstein et al. 2019). Instead, ρ UV evolves quickly with redshift, with a factor of ∼ 10 evolution between z = 8 and 13. \nThe lower right panel quantifies the redshift evolution of galaxy ionizing properties in our models using \n⟨ f esc ξ ion ⟩ L UV ≡ ˙ N γ ( z ) ρ UV ( z ) (1) \nThis is the UV-luminosity ( L UV )-weighted average of the product of f esc and ξ ion for the galaxy population. The black, red, and blue curves show this quantity for our three reionization models (see legend) assuming the average ρ UV curve for M cut UV = -13 (lower left panel). The shaded regions show the spread in this quantity (for fixed ˙ N γ ( z )) arising from observational uncertainty in ρ UV ( z ), as shown in the lower left. We calibrate ˙ N γ ( z ) for the early start/early end model such that its ⟨ f esc ξ ion ⟩ L UV agrees with the observationally motivated model from Mu˜noz et al. 2024 (which assumes M cut UV = -13), shown by the faded gray dashed curve. Consistent with the findings of Mu˜noz et al. 2024, this model ends reionization early at z ≈ 8 (see Figure 1). \nOur late start/late end scenario is motivated by the possibility that the tension with τ es and the Ly α forest in the early start/early end model can be solved with a simple redshift-independent re-scaling of ˙ N γ ( z ). This could be the case if the measurements of ξ ion assumed in Mu˜noz et al. 2024 (from Simmonds et al. 2024) are systematically biased high, and/or if the same is true of the f esc values they inferred from the results of Chisholm et al. 2022; Zhao & Furlanetto 2024. We find that scaling ˙ N γ ( z ) down by a factor of 5 from the early start/early end case brings the end of reion- \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 2. Evolution of galaxy properties and calibration of ˙ N γ ( z ) for our reionization scenarios. Top Left: measured UVLFs from Bouwens et al. 2021 at z < 8, Adams et al. 2024 at 8 < z < 12 . 5, and Donnan et al. 2024 at z = 14 . 5. Top Right: The same, but using UVLF parameters from Eq. 3-6 of Donnan et al. 2024 at z ≥ 8. These two sets are different by a factor of ∼ 3 at z > 8, roughly bracketing observational uncertainty in the UVLF. Lower left: integrated ρ UV ( z ) for cutoff magnitudes M cut UV = -17 and -13. The shaded regions denote the spread between the two sets of UVLFs in the top panels, and the curves denote the logarithmic averages. Lower Right: product of f esc and ξ ion , weighted by L UV and averaged over the galaxy population (Eq. 1). The colored lines show this quantity for our models, and the shaded regions show the spread resulting from observational uncertainty in the UVLF (at fixed ˙ N γ ( z )). The faded gray dashed curve shows one of the observationally motivated models assumed in Mu˜noz et al. 2024 (with M cut UV = -13) and the faded cyan dashed curve shows the model of Pahl et al. 2024 scaled up by a factor of 3 . 5. Our early start/early end scenario (by construction) approximately matches the Mu˜noz et al. 2024 model. The late start/late end case shares the same redshift evolution, but is scaled down by a factor of 5 to recover agreement with the z ≤ 6 Ly α forest. The early start/late end model is calibrated to agree with the Mu˜noz et al. 2024 result at z ≥ 10, but it also required to agree with the Ly α forest. This model demands a factor of ≈ 10 decline in ⟨ f esc ξ ion ⟩ L UV between z = 12 and 6 - much steeper evolution than suggested by the Mu˜noz et al. 2024 or Pahl et al. 2024 findings. \n<!-- image --> \nto z ∼ 5. We then further adjust ˙ N γ ( z ) at z < 7 at the few-percent level until we achieve good agreement with the Ly α forest at z ≤ 6 (as we will show in § 4.1.1). The cyan-dashed curve shows the observationally motivated model from Pahl et al. 2024, multiplied 3 . 5, which also has redshift evolution similar to this scenario. \nHowever, the late start/late end scenario is not the only possibility allowed by τ es and the Ly α forest. It could also be that ˙ N γ ( z ) is consistent with the Mu˜noz et al. 2024 model at the highest redshifts ( z ≳ 10), but \ndeclines at lower redshifts in such a way that reionization ends at z < 6, as required by the Ly α forest. This scenario is represented by our early start/late end model (red dashed curve). This model also ends reionization at z = 5, and by adjusting its ˙ N γ ( z ) at z < 10, we can also achieve agreement with the Ly α forest. This model requires a factor of ≈ 10 decline in ⟨ f esc ξ ion ⟩ L UV between z = 10 and 6, a decrease much steeper than suggested by the results of of Mu˜noz et al. 2024 and Pahl et al. 2024. This could be achieved if ξ ion , f esc , M cut UV , \nor some combination of these evolves significantly across this redshift range (see § 2.2).", '2.2. Is the early start/late end model plausible?': "Given the discrepancy between the evolution of ⟨ f esc ξ ion ⟩ L UV in the early start/late end model and in the Mu˜noz et al. 2024 and Pahl et al. 2024 models, it is natural to ask whether this scenario is plausible. The top panel of Figure 3 shows ⟨ f esc ξ ion ⟩ L UV for this model, alongside observationally and theoretically motivated scenarios that make various assumptions about the redshift evolution of ξ ion and/or f esc . The dotted curve is the Mu˜noz et al. 2024 model scaled down by 0 . 2, which agrees with the late start/late end case. The dashed curve is the same, except that we extrapolate ξ ion (Eq. 4 in Mu˜noz et al. 2024) outside the range of M UV and redshift within which it was fit to data. For the dot-dashed curve, we further replace the observationally motivated f esc prescription assumed in Mu˜noz et al. 2024 with the global f esc ( z ) from the flagship THESAN simulation (Yeh et al. 2023). We have re-scaled each of the gray curves by different constants to bring them as close as possible to the early start/late end model. We obtain the best agreement for the dot-dashed curve, which boosts the redshift evolution of both ξ ion and f esc relative to Mu˜noz et al. 2024. This shows that the early start/late end model is plausible given evolution in f esc and/or ξ ion , but only under 'favorable' assumptions about both. \nThe bottom panel of Figure 3 illustrates another mechanism that could work in the direction of making reionization start earlier: evolution in M cut UV . The red curve assumes that the M cut UV evolves from -10 at z = 12 to -19 at z = 6 ( M cut UV ( z ) = -19 + 3 2 ( z -6) ) , which causes ρ UV to decline much less rapidly with redshift than it does in the bottom left panel of Figure 2. This allows ⟨ f esc ξ ion ⟩ L UV in the early start/late end scenario to evolve less quickly, in better agreement with the Mu˜noz et al. 2024 model (dotted curve). This type of behavior in the galaxy population could arise from decreasing dust obscuration (Topping et al. 2024) and/or feedback from the IGM reducing or shutting off star formation in low-mass halos at lower redshifts (Wu et al. 2019; Ocvirk et al. 2021). The evolution in M cut UV assumed in this illustrative example is extreme, and likely ruled out by existing observations (Atek et al. 2018), but serves to show how an evolving M cut UV could support the early start/late end model. \nThese comparisons suggest that a factor of a few of evolution in each of ξ ion and f esc , and perhaps some in M cut UV , could explain the early start/late end scenario. Indeed, there is some observational and theoret- \nFigure 3. Summary of mechanisms that could allow for the early start/late end model. Top: ⟨ f esc ξ ion ⟩ L UV (as in Figure 2), alongside several observationally and theoretically motivated scenarios. The grey dotted curve shows the Mu˜noz et al. 2024 model multiplied by 0 . 2, which matches the late start/late end model . The gray dashed curve is the same model, but with caps on extrapolation of ξ ion with M UV and redshift imposed by Mu˜noz et al. 2024 removed (see text). The dot-dashed curve further replaces the model for f esc used in Mu˜noz et al. 2024 with the average f esc ( z ) measured in the flagship THESAN simulation (Yeh et al. 2023). This last curve reproduces evolution qualitatively consistent with the early start/late end model. Bottom: the same, but assuming M cut UV evolves linearly from -19 at z = 6 to -10 at z = 12 (see annotation). A trends towards fainter M cut UV at higher z causes ρ UV to decline less steeply, requiring shallower redshift evolution of ionizing properties. See text for further details. \n<!-- image --> \ncal support for the idea that ξ ion could increase with redshift and/or be larger in fainter galaxies (Atek et al. 2024a; Cameron et al. 2024; Simmonds et al. 2024), However, some works find that ξ ion is closer to constant with M UV , and perhaps redshift (Matthee et al. 2022; Pahl et al. 2024). Unfortunately, f esc at these redshifts cannot be directly measured, and there is no consensus in the literature from either indirect observational estimates (e.g. Naidu et al. 2022; Citro et al. 2024) or cosmological simulations (e.g. Trebitsch et al. 2018; Rosdahl et al. 2022; Kostyuk et al. 2023). Thus, galaxy observations alone cannot rule out the early \n<!-- image --> \nFigure 4. Top: ionizing photon emissivity for our models, alongside several others from the literature that are calibrated to match the Ly α forest observations at z ≤ 6. Our late start/late end model has the steepest evolution at z > 9, and our early start/late end model agrees well with the 'early' model from Asthana et al. 2024. Bottom: ionizing photon production per H atom by galaxies. The vertical dashed lines denote the end of reionization in each model, and the horizontal lines indicate the number of photons per H atom required to complete reionization. \n<!-- image --> \nstart/late end case, although such a model does require more significant evolution in galaxy ionizing properties than suggested by current observations. As such, we conclude that the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model seems more likely based on existing data.", '2.3. Ionizing photon budget': "We show the ˙ N γ ( z ) for our models in the top panel of Figure 4, alongside several others from the literature that also match the z < 6 Ly α forest. The cyansolid and magenta-dashed faded curves show the 'fiducial' and 'early' models from Asthana et al. 2024, and the dot-dashed gray curve shows the reference model 3 from Cain et al. 2024. Our late start/late end model drops off faster than the others at z > 9, and \nour early start/late end model closely matches the 'early' model from Asthana et al. 2024. The bottom panel of Figure 4 shows the integrated ionizing photon output of galaxies per H atom, with the vertical lines denoting the end of reionization and the horizontal lines the number of photons per H atom needed to complete reionization (the so-called 'ionizing photon budget'). We find a budget of ≈ 2 . 6, 2 . 8, and 2 . 7 photons per H atom for the late start/late end , early start/late end , and early start/early end models, respectively. \nAll our models are mildly 'absorption-dominated' meaning N γ/ H is (slightly) more than twice the number of baryons per H atom in the universe (1 . 082, counting helium). The photon budget depends mildly on the reionization history itself, but is determined mainly by the recombination rate predicted by our IGM opacity sub-grid model. Recently, Davies et al. 2024a made a first attempt to observationally measure the 'clumping factor', which quantifies the recombination rate in the in-homogeneous IGM (Gnedin & Ostriker 1997; Pawlik et al. 2010). They found C ∼ 12 at z > 5, higher than the commonly assumed C = 3 (e.g. Finlator et al. 2011; McQuinn et al. 2011). For a reionization history similar to our late start/late end model, they find this higher C increases the photon budget from ≈ 1 . 5 to ≈ 3. Assuming that the number of recombinations, N γ/ H -1 . 082, is ∝ C , the photon budget in the late start/late end model implies an effective C ∼ 9 . 5, slightly below but consistent at 1 σ with the Davies et al. 2024a measurements 4 . Using this crude approximation, a recombination rate high enough to delay the end of the early start/early end model to z ≈ 5 would require C ∼ 85, an unrealistically high number even compared to the most extreme simulations 5 .", '3. NUMERICAL METHODS': 'In this section, we discuss some of the relevant technical details of our RT simulations, and our methods for forward-modeling the observables discussed in the next section. The reader interested only in the results of this work may safely skip this section and pick up at § 4.', '3.1. Radiative Transfer Simulations': "We ran RT simulations of reionization using FlexRT (Cain & D'Aloisio in prep.). FlexRT is an adaptive ray \ntracing code that post-processes a time series of cosmological density fields to simulate the growth of ionized regions, the ionizing background, and the IGM thermal history. The code uses a sub-grid model for opacity to ionizing photons in ionized gas, which captures the effects of ∼ kpc-scale structure that cannot be directly resolved. Our sub-grid model is based on a suite of high-resolution, coupled hydro/RT simulations that can resolve the Jeans scale of cold, pre-ionized gas, run with the setup of (D'Aloisio et al. 2020; Nasir et al. 2021). We refer the reader to (Cain et al. 2021, 2023), and a forthcoming code paper (Cain & D'Aloisio in prep.) for further details about FlexRT. \nIonizing sources for the RT calculation are halos taken from a dark matter (DM)-only N-body simulation run with the particle-particle-particle-mesh (P 3 M) code of Trac et al. 2015. This run has a box size of L box = 200 h -1 Mpc and N = 3600 3 DM particles, and uses a spherical over-density halo finder to identify halos. We find that our halo mass function matches the halo mass function of Trac et al. 2015 to within ≈ 5 -10% (HMF) for M halo > 3 × 10 9 h -1 M ⊙ , and is ≈ 50% incomplete at M halo = 1 × 10 9 h -1 M ⊙ . To ensure we have a significant number of halos at the highest redshifts we simulate ( z > 15), we adopt a mass cutoff for our sources of M halo > 1 × 10 9 h -1 M ⊙ , even though our HMF is incomplete there. The density fields used in our RT calculation are taken from a high-resolution hydrodynamics simulation with the same large-scale initial conditions as the N-body run, which are re-binned to N RT = 200 3 for the RT. Our simulations start at z = 18 and end at z = 4 . 8. \nWe assign UV luminosities ( L UV ) to halos by abundance-matching to observed UV luminosity functions (UVLFs). We use the measurements of Bouwens et al. 2021 at z < 8, those of Adams et al. 2024 at 8 < z < 14, and that of Donnan et al. 2024 at z = 14 . 5 - those shown in the upper left panel of Figure 2. The ionizing emissivity is distributed between halos following ˙ n γ ∝ L UV 6 . The volume-averaged ionizing emissivity of sources, ˙ N γ = 1 L 3 box ∑ i ∈ halos ˙ n i γ , is set by hand at each redshift and used to normalize the ˙ n γ proportionality. As explained in § 2.1, we calibrated ˙ N γ ( z ) for each scenario using a combination of JWST data ( § 2.1) and \nmeasurements from the z < 6 Ly α forest ( § 3.2, § 4.1.1). We bin halos by the RT cell they occupy and treat cells containing one or more halos as sources. We use a single ionizing frequency 7 ( E γ = 19 eV), chosen to reproduce the same frequency-averaged HI cross-section, ⟨ σ HI ⟩ , as a power law spectrum of the form J ν ∝ ν -1 . 5 between 1 and 4 Rydbergs. Cells are assigned post ionization-front (I-front) temperatures using the flux-based method prescribed in D'Aloisio et al. 2019. The subsequent thermal history is calculated using their Eq. 6.", '3.2. Modeling the Ly α forest': "We model the Ly α forest in post-processing using the density fields from our aforementioned high-resolution hydrodynamics simulation, which has N = 2048 3 gas cells. Ionized fractions, photo-ionization rates (Γ HI ) and temperatures are mapped from the FlexRT simulations onto these density fields, and the residual neutral fraction in ionized regions is calculated assuming photo-ionization equilibrium and the case A recombination rate 8 . The native spatial resolution of our density field is ∆ x cell = 97 . 6 h -1 kpc, too coarse to fully resolve the low-density voids that set the transmission at 5 < z < 6 (Doughty et al. 2023). We apply the multiplicative correction prescribed in Appendix A of D'Aloisio et al. 2018 to our residual neutral fractions to get an effective resolution of ∆ x cell = 12 . 2 h -1 kpc. The Ly α voigt profile is approximated using the analytic fit from Tepper-Garc'ıa 2006. \nSince our gas temperatures are calculated on the coarse RT grid, we do not capture the temperaturedensity relation on scales smaller than the RT cell size. As temperature (usually) correlates positively with density in the IGM, low (high)-density hydro cells embedded in larger RT cells will be assigned temperatures that are too high (low). To correct for this, we assign a local temperature-density relation to each cell using the procedure described in Cain et al. 2024 (see also their Appendix E), which uses the IGM temperature model of McQuinn & Upton Sanderbeck 2016. We find this lowers the mean transmission by ≈ 10 -15%, since the correction cools the under-dense cells, which affect the mean transmission the most. \nWe compute Ly α forest statistics by casting 4000 sightlines from random locations and in random directions of length 50 h -1 Mpc, for a total path length of 200 h -1 Gpc. We calculate transmission statistics from \nz = 4 . 8 to z = 6 in ∆ z = 0 . 2 increments. Since our native resolution of 97 . 6 h -1 kpc is only 2 -3 × narrower than the typical width of Ly α line profile ( ≈ 15 km/s vs. 30 -50 km/s), we do the integration over the line at a velocity resolution 4 × higher than that of the hydro sim. We find this reduces the mean transmission by a few percent at most.", '3.3. Ly α transmission around galaxies': 'We have also modeled Ly α transmission on the red side of line center around halos that could host Ly α emitting galaxies (LAEs). This allows us to assess the statistics of LAE visibility in our models at z > 7. LAEs typically emit Ly α red-shifted to both the red and blue sides of line center (Verhamme, A. et al. 2006). Although any emission on the blue side would be absorbed by even the ionized part of the IGM at these redshifts, attenuating the red side requires damping wing absorption from the fully neutral IGM (see Mason & Gronke 2020, and references therein). This makes LAEs a potentially powerful probe of the IGM neutral fraction. However, interpreting observed red-side Ly α emission (or lack thereof) is complicated by uncertainties in modeling the intrinsic line profile of the LAEs themselves, and other factors such as surrounding inflows/outflows and proximate self-shielding systems (Park et al. 2021; Smith et al. 2022). \nIn this work, we will avoid modeling the intrinsic LAE line profile, instead focusing on the IGM transmission, T IGM . To calculate this, we trace 50 randomly oriented sightlines away from each halo and compute the Ly α transmission profiles along each sightline at ± 5 ˚ A from line center ( ≈ ± 800 km/s). We use the same N = 2048 3 high-resolution hydro simulation for this calculation as for the Ly α forest, and we calculate the ionization state of the ionized gas in the same way 9 . We begin integrating the Ly α opacity 500 h -1 kpc (5 hydro cells) away from the location of the halo to avoid gas within the halo itself contributing to T IGM . Gas velocities relative to the halo are computed by subtracting the halo velocity measured from the N-body simulation. \nThe gas around massive halos can have sharp line-ofsight velocity gradients owing to inflows near the halo. Sightlines pointing away from these halos see positive velocity gradients, which narrows the Ly α line in redshift space. We find that the resulting sharp jumps in velocity between adjacent cells can produce artifacts in our transmission spectra. To mitigate this, we linearly in- \nerpolate the gas velocities (and all other relevant quantities) onto a grid with 4 × higher resolution than that of the simulation when calculating T IGM (see Gangolli et al. 2024 for a description of a similar procedure). We find the transmission profiles to be well-converged at this resolution.', '3.4. Modeling the Patchy kSZ signal': "CMB photons can scatter off free electrons during reionization. If the electrons are moving relative the the CMB rest frame, this results in a Doppler shift of the photons. This can shift the blackbody spectrum and result in additional temperature anisotropies in the CMB (Sunyaev & Zeldovich 1980). This is known as the Sunyaev-Zel'dovich Effect, and its resultant temperature deviation along a line of sight is given by: \n∆ T T = -σ T n e, 0 ∫ e -τ es ˆ γ · ⃗q c ds a 2 (2) \nwhere ˆ γ is the line of sight direction, ⃗q = (1 + δ ) χ⃗v is the ionized momentum field, σ T is the Thompson scattering cross section and τ es is the CMB optical depth to the last scattering surface, and n e, 0 is the mean electron density at z =0. The integral can be broken up into a post-reionization, homogeneous kinetic SunyaevZel'dovich Effect (hkSZ), and a highz patchy kinetic Sunyaev-Zel'dovich Effect (pkSZ) while reionization is still occurring. Untangling these components is difficult, so observations use templates to subtract the hkSZ component from the measured total kSZ power. For the purposes of this work, we take any contributions from z ≳ 5 as part of the pkSZ, even in simulations where reionization ends at z > 5. This keeps the different scenarios directly comparable to each other. Note that actual measurements (such as the one from SPT by Reichardt et al. 2021, see § 4.3) must assume a fixed z end in their analysis. \nWe calculate the signal using a method similar to that first suggested by Park et al. 2013. The kSZ angular power spectrum can be calculated from the 3D power spectrum of ⃗q by: \nC ℓ = ( σ T n e, 0 c ) 2 ∫ ds s 2 a 4 e -2 τ P q ⊥ ( k = ℓ/s, s ) 2 (3) \nwhere (2 π ) 3 P q ⊥ ( k ) δ D ( ⃗ k -⃗ k ' ) = ⟨ ˜ q ⊥ ( ⃗ k ) · ˜ q ∗ ⊥ ( ⃗ k ' ) ⟩ is the power spectrum of the specific ionized momentum modes perpendicular to the Fourier wave vector, ˜ q ⊥ = ˜ q -ˆ k ( ˜ q · ˆ k ), δ D ( k -k ' ) is the Diracδ function and ˜ q = ∫ ⃗qe -i ⃗ k · ⃗ q d 3 ⃗x denotes the Fourier transform of ⃗q . Only the ˜ q ⊥ mode contributes significantly to the kSZ signal, due to ˜ q || contributions canceling out \nwhen integrating over the line of sight. We refer the reader to Ma & Fry 2002 and appendix A in Park et al. 2013 for details. In general, it is common to write the angular power spectrum in the dimensionless form: D ℓ = C ℓ ( ℓ +1) ℓ/ (2 π ). \nRunning RT simulations imposes practical limitations on the volume of our box. This limited box size means we miss large scale velocity modes that add to the kSZ power. To compensate for this, we use the same correction applied in Park et al. 2013 eq. (B1), and take advantage of linear theory to generate a correction term: \nP miss q ⊥ ( k, z ) = ∫ k<k box d 3 k ' (2 π ) 3 (1 -µ 2 ) P χ (1+ δ ) ( | ⃗ k -⃗ k ' | ) P lin vv ( k ' ) (4) \nwhere k box = 2 π/L box is the box-scale wavemode, µ = ˆ k · ˆ k ' , P χ (1+ δ ) ,χ (1+ δ ) ( k ) is the ionized matter power spectrum and P lin vv ( k ) = (˙ af/k ) P lin δδ ( k ) is the velocity power spectrum in the linear approximation taken from the public code CAMB (Lewis et al. 2000). \nIt was found by Alvarez (2016) that this kind of approach can still underestimate the D pkSZ 3000 by ∼ 10-20% due to the irreducible or connected component of the ⟨ δ χ v · δ χ v ⟩ term being non-negligible because of the nonGaussianity of reionization. So our calculations could be seen as conservative estimates of the power. As we will see in § 4.3, an increase in power could strengthen our conclusions. As such, we do not expect this missing term to affect our qualitative results.", '4. IMPLICATIONS OF OTHER REIONIZATION OBSERVABLES': 'In the rest of this work, we study three other observational windows into reionization: measurements from the spectra of high-redshift QSOs at z ≤ 6 . 5 ( § 4.1), observations of Ly α -emitting galaxies at z ≥ 8 ( § 4.2), and the patchy kSZ effect from reionization ( § 4.3). Our goal will be to see if these observations, together with aforementioned JWST data, reveal a consistent picture about when reionization started and when it ended.', '4.1. QSO Observations at 5 < z < 6 4.1.1. The Ly α Forest': "The 5 < z < 6 Ly α forest is perhaps the most compelling indicator that reionization ended at z < 6. This conclusion has emerged from studies of the mean transmission of the Ly α forest and the scatter in Ly α opacities (Kulkarni et al. 2019; Keating et al. 2020; Nasir & D'Aloisio 2020; Bosman et al. 2022). We will study both in this section. \nFigure 5 shows the mean transmission of the Ly α forest, ⟨ F Ly α ⟩ , at 5 ≤ z ≤ 6 in our models compared with \nFigure 5. Ly α forest mean transmission, compared with recent observations (see text). The early start/early end model is in severe disagreement with measurements, predicting transmission near unity at all redshifts. The other two models agree with the measurements of Bosman et al. 2022 by construction, since their ˙ N γ ( z ) histories were calibrated to match those measurements (see discussion in § 2.1). The fact that such a calibration is possible for both late-ending models indicates that ⟨ F Ly α ⟩ alone cannot distinguish between them. \n<!-- image --> \nrecent measurements from Becker & Bolton 2013; Eilers et al. 2018; Bosman et al. 2018; Yang et al. 2020; Bosman et al. 2022. The late start/late end and early start/late end models both agree well with the data this is by design, since both ˙ N γ histories were calibrated so that the simulations would match the Bosman et al. 2022 measurements of ⟨ F Ly α ⟩ . As such, these two models cannot be distinguished using the mean forest transmission alone. We see that the early start/early end model misses the measurements severely, predicting a mean transmission near unity at all redshifts. Indeed, the average HI photo-ionization rate, Γ HI , in ionized gas is ≈ 2 . 7 × 10 -11 s -1 at z = 6, ≈ 2 orders of magnitude higher than measurements at that redshift (D'Aloisio et al. 2018; Gaikwad et al. 2023). Thus, the forest transmission measurements strongly disfavor a z ∼ 8 end to reionization. Because of this, we will omit the early start/early end model from many of our subsequent comparisons, and focus instead on whether observations can distinguish the other two scenarios. \nFigure 6 shows the cumulative distribution function (CDF) of forest effective optical depths measured over intervals of 50 h -1 Mpc, P ( < τ 50 eff ). The τ 50 eff distribution contains more information than ⟨ F Ly α ⟩ , since it is sensitive to the spatial fluctuations in the IGM ionization state. This shape is sensitive to the IGM neutral fraction, since neutral islands produce highτ 50 eff sight- \nFigure 6. Distribution of effective Ly α optical depths over 50 h -1 Mpc segments of the Ly α forest, P ( < τ 50 eff ), compared to measurements from Bosman et al. 2022. We show results and measurements at z = 5, 5 . 2, 5 . 6, and 6 (see panel titles). For the early start/early end model, we have re-scaled the Ly α opacities along each all sightlines by a constant value such that the mean transmission matches the Bosman et al. 2022 measurements. This allows us to make a clean comparison of how the IGM neutral fraction affects the shape of P ( < τ 50 eff ) at fixed ⟨ F Ly α ⟩ . At z = 6, the re-scaled early start/early end model produces too little scatter in τ 50 eff , the late start/late end case produces too much, and the early start/late end model is a good match. At z = 5 . 6, none of the models match the observations particularly well. The observations lie between the blue dotted and red dashed curves, suggesting a non-zero neutral fraction smaller than that of the early start/late end model (7 . 5%). At z = 5 . 2, the re-scaled early start/early end matches the data best, suggesting that reionization should end slightly earlier than it does in both late-ending models. All models match the data at z = 5, when reionization is complete. \n<!-- image --> \nz < 6 in late reionization scenarios (Kulkarni et al. 2019; Nasir & D'Aloisio 2020; Qin et al. 2021). We show P ( < τ 50 eff ) at z = 5, 5 . 2, 5 . 6, and 6 for each of our models, alongside measurements from Bosman et al. 2022. To show what P ( < τ 50 eff ) would look like in a hypothetical early-ending model that is compatible with the forest, we re-scaled Γ HI in the early start/early end 10 model such that ⟨ F Lyα ⟩ agrees with the measurements shown in Figure 5. This allows us to cleanly compare how the shape of P ( < τ 50 eff ) is affected by the IGM neutral fraction at fixed ⟨ F Ly α ⟩ . \nAt z = 6 (lower right), the early start/late end model best matches the Bosman et al. 2022 data. The blue dotted curve shows that a z > 6 end to reionization results in a CDF that is too narrow, implying too little scatter in τ 50 eff , echoing the conclusions of Bosman et al. 2022. Conversely, the late start/late end model predicts too much scatter (the CDF is too wide). This is because it has a neutral fraction of ≈ 30% at z = 6, as compared to only 15% in the early start/late end case. At z = 5 . 6 (lower left), none of the models match the observations very well. Both of the lateending models produce too much scatter in τ 50 eff , and the early-ending one produces too little. This suggests that a model with a non-zero neutral fraction smaller than that in the early start/late end case (7 . 5%) would \nmatch the data best. At z = 5 . 2 (upper right), the early-ending scenario fits the data very well, and the other two models have too much scatter in P ( < τ 50 eff ). This indicates that both late-ending models end reionization slightly too late. At z = 5 (upper left), when the neutral fraction is < 1% in all three models, they agree well with the observations. \nAt face value, observations of P ( < τ 50 eff ) at 5 ≤ z ≤ 6 seem to prefer the early start/late end model. This scenario has a lower neutral fraction at z ≤ 6 than the late start/late end case, and as such better matches the observed scatter in P ( < τ 50 eff ). However, none of the scenarios in Figure 6 match the observed P ( < τ 50 eff ) at all redshifts. This is likely because our late-ending models finish reionization too late, as evidenced by the z = 5 . 2 comparison (upper right panel). The findings of Bosman et al. 2022, based on these same measurements, suggest that reionization should be complete by z = 5 . 3 (vs. z ≈ 5 in our models), a shift of ∆ z ≈ 0 . 3 from our models. In Appendix A, we estimate what P ( < τ 50 eff ) would look like if both late-ending models finished reionization at z = 5 . 3. We use the FlexRT outputs at z ' = z -0 . 3 to calculate P ( < τ 50 eff ), then re-scale Γ HI in ionized gas until ⟨ F Ly α ⟩ matches measurements. We show that this procedure brings our simulations into better agreement with the measurements, and that the ❡❛/a114❧② /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model remains preferred. \nIn Cain et al. 2024, we pointed out several factors that can affect the precise timing of reionization's end in models matched to measurements of ⟨ F Ly α ⟩ . First, lack of spatial resolution in the forest can lead to an underestimate of the mean transmission at fixed x HI (Doughty et al. 2023), resulting in a spuriously early end to reionization (by ∆ z ≈ 0 . 2, see Fig. 11 of Cain et al. 2024) when calibrating to measurements. Our forest calculations include the resolution correction prescribed Appendix A of D'Aloisio et al. 2018, so in principle they account for this effect. However, those corrections were derived in a 25 h -1 Mpc box, and it is unclear how they might change in a larger box. It is therefore possible that we have over-corrected for resolution. We also found in Cain et al. 2024 that harder ionizing spectra and less clustered ionizing sources result in an earlier end to reionization at fixed ⟨ F Ly α ⟩ . Moreover, the uncertain clustering of ionizing sources also affects large-scale fluctuations in the ionizing background, which could be particularly intense if quasars played a large role in the end-stages of reionization (Chardin et al. 2015; Madau et al. 2024). Any of these effects could affect when reionization needs to end to match ⟨ F Ly α ⟩ measurements, and the shape of P ( < τ 50 eff ) at fixed neutral fraction, poten- \ninterpretation of P ( < τ 50 eff ) measurements.", '4.1.2. The Mean Free Path': "Next, we will study the mean free path to ionizing photons (MFP, λ mfp ), the average distance an ionizing photon travels through the IGM before being absorbed. The MFP is sensitive to the distribution of neutral gas in the IGM and small-scale clumping in the ionized IGM (Emberson et al. 2013; Park et al. 2016; D'Aloisio et al. 2020; Chan et al. 2024). We calculate the Lyman limit MFP in our simulations using the definition in Appendix C of Chardin et al. 2015, \nλ 912 mfp = ⟨ ∫ xdf ⟩ ⟨ ∫ df ⟩ = -〈∫ 0 1 xdf 〉 (5) \nwhere x is the position along a randomly oriented sightline, f ( x ) is the transmission of 912 ˚ A photons, and the angle brackets denote an average over many sightlines. Roth et al. 2024 found that this definition matches well with forward-modeled direct MFP measurements from QSO spectra, even in a partially neutral IGM. We caution, however, that different ways of estimating the MFP from simulations can give modestly different results (Lewis et al. 2022). \nThe left panel of Figure 7 shows the MFP in our late-ending models, compared to measurements from Worseck et al. 2014; Becker et al. 2021; Zhu et al. 2023; Gaikwad et al. 2023. Both scenarios are in broad agreement with the measurements. However, the direct measurements using QSO Lyman Continuum (LyC) spectra (all but the Gaikwad et al. 2023 points) display a preference for the late start/late end model. This is largely due to the short z = 6 direct measurements, which prefer the rapid neutral fraction-driven decline in λ 912 mfp in the late start/late end case. By contrast, the early start/late end model is ≈ 2 σ away from the central values of the z = 6 measurements from Becker et al. 2021; Zhu et al. 2023. Both models are within 1 σ of the indirect, Ly α forest-based measurements from Gaikwad et al. 2023 (see also Davies et al. (2024b)). \nThis result is consistent with previous findings that ongoing reionization at z = 6 is needed to explain the direct QSO measurements (Cain et al. 2021; Lewis et al. 2022; Garaldi et al. 2022; Lewis et al. 2022; Satyavolu et al. 2024; Roth et al. 2024). Another effect at play is that the ionized IGM at z = 6 in the late start/late end was more recently ionized, and thus clumpier (Park et al. 2016; D'Aloisio et al. 2020). These factors result in direct MFP measurements preferring the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ scenario. Our earlier result that \nFigure 7. Other (non-Ly α forest) QSO-based measurements of global IGM properties at 5 < z < 6 in our late-ending models. Left: the mean free path to ionizing photons at 912 ˚ A compared to recent measurements. We show direct measurements from the Lyman Continuum spectra of QSOs (Worseck et al. 2014; Becker et al. 2021; Zhu et al. 2023), and the recent indirect measurements of Gaikwad et al. 2023 based on the Ly α forest. Both models are consistent with the forest-based measurements from Gaikwad et al. 2023, but the direct QSO-based measurements (particularly at z = 6) prefer the late start/late end scenario. The faster redshift evolution of λ mfp in that model, driven by the evolving neutral fraction, is in better agreement with the z = 6 direct measurements. Right: Average temperature at mean density ( T 0 ), compared to measurements from Becker et al. 2011; Boera et al. 2019; Walther et al. 2019; Gaikwad et al. 2020. Both models display reionization-driven temperature peaks at z ∼ 5 . 3, consistent with the Gaikwad et al. 2020 measurements. The late start/late end model peaks at a higher temperature, since a larger fraction of the IGM has been recently heated by fast-moving I-fronts (see annotation). The early start/late end model is more consistent with measurements at z ≥ 5, indicating a mild preference for that scenario (but with some caveats - see text). \n<!-- image --> \nP ( < τ 50 eff ) measurements prefer the early start/late end model hints at a possible tension between the Ly α forest and direct MFP measurements. This is consistent with the fact that the indirect MFP measurements from Gaikwad et al. 2023 (based on P ( < τ 50 eff ) itself) at z = 6 are a factor of ≈ 2 above the direct measurements. We emphasize that this tension is mild, since the 1 σ error bars of these measurements overlap.", '4.1.3. IGM Thermal History': "The IGM temperature at mean density, T 0 , is shown in the right panel of Figure 7, alongside measurements from Becker et al. 2011; Boera et al. 2019; Walther et al. 2019; Gaikwad et al. 2020. Both late-ending models display a 'bump' in temperature at z ≈ 5 . 3 due to the end of reionization. Heating by I-fronts (D'Aloisio et al. 2019; Zeng & Hirata 2021) increases T 0 until near reionization's end, after which cooling from the expansion of the universe and Compton scattering off the CMB set the evolution of T 0 (McQuinn & Upton Sanderbeck 2016). The peak in T 0 is higher in the late start/late end model because a larger fraction of the IGM is reionized at z < 6. The redshift of the bump suggested by the Gaikwad et al. 2020 measurements is closer to z ∼ 5 . 6 - this is consistent with our earlier finding (based \non P ( < τ 50 eff )) that reionization may end ∆ z ∼ 0 . 3 too late in our models. \nAt face value, the early start/late end model agrees best with T 0 measurements. Although both models are consistent with the Gaikwad et al. 2020 points, the late start/late end is too hot at z ≥ 5 for the measurements there. In the early start/late end case, a larger fraction of the IGM has re-ionized at higher redshift, giving it more time to cool by z = 5. That model is also agrees well with the reionization history in the best-fitting model of Villasenor et al. 2022, which fits a broad range of IGM temperature measurements down to z = 2. That model has ionized fractions of ≈ 35% ( ≈ 15%) at z = 8 (10), similar to our early start/late end scenario (which has 1 -x HI ≈ 40%, (20%)). An important caveat is that the thermal history is sensitive at the 20 -30% level to the spectrum of the ionizing radiation, through a combination of the post Ifront temperature ( T reion ) and photo-heating in ionized gas afterwards (D'Aloisio et al. 2019). For example, a much softer ionizing spectrum could shift the T 0 histories significantly lower at fixed reionization history (see e.g. the bottom middle panel of Fig. 3 in Asthana et al. 2024). This could bring the late start/late \nend model into agreement with z ≤ 5 T 0 measurements. However, this would also require a later reionization history at fixed ⟨ F Ly α ⟩ (Figure 5 of Cain et al. 2024), which would worsen the disagreement with the observed P ( < τ 50 eff ) in Figure 6. As such, we conclude that measurements of T 0 mildly prefer the ❡❛/a114❧② /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model.", '4.1.4. Neutral fraction constraints at z ≤ 6 . 5': 'Finally, we compare our reionization models to observational constraints on x HI at z ≤ 6 . 5 obtained using QSO spectra. Figure 8 compares our models with constraints from Ly α forest dark pixels (McGreer et al. 2015; Jin et al. 2023), dark gaps (Zhu et al. 2022), QSO damping wings (Greig et al. 2024), P ( < τ 50 eff ) (Choudhury et al. 2021; Gaikwad et al. 2023), and Ly α forest damping wings (Zhu et al. 2024; Spina et al. 2024). The bold lines show the reionization histories in our two lateending models, while the faded lines show these shifted to the right by ∆ z = 0 . 3, consistent with the discussion surrounding P ( < τ 50 eff ) in § 4.1.1. \nMost constraints are upper (lower) limits on the neutral (ionized) fraction. Several of these are in mild tension with the late start/late end model , and most are consistent with the early start/late end case. The z = 5 . 5 dark gap constraint from Zhu et al. 2022 and the recent QSO damping wing limits from Greig et al. 2024 disfavor the late start/late end model. The recent lower limit on the neutral fraction from Zhu et al. 2024, derived from Ly α forest damping wings at z = 5 . 8, disfavors an end to reionization early than this. The same can be said of the Spina et al. 2024 forest damping wing measurement at z = 5 . 6, although their measurement actually prefers the late start/late end case. An important caveat is that if reionization ends earlier by ∆ z = 0 . 3 (as hinted by P ( < τ 50 eff ) measurements), the tension with the late start/late end model disappears. In fact, the neutral fraction in the shifted early start/late end model cannot be much lower without being in tension with the Zhu et al. 2024 damping wing limit. We conclude that neutral fraction constraints at z < 6 . 5 mildly prefer the ❡❛/a114❧② /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model, but that relatively small, realistic shifts in the reionization history could change this conclusion.', '4.2. Ly α Emitters at z > 8': 'In this section, we study the Ly α transmission properties at z ≥ 8 around massive halos that could host bright LAEs, such as GN-z11 (Bunker et al. 2023). Our goal is to determine whether these observations prefer an early or late start to reionization. \nFigure 8. Constraints on the neutral fraction at z ≤ 6 . 5 from dark pixels, dark gaps, damping wings, P ( < τ 50 eff ), and forest damping wings (see text for references). The bold curves show our late-ending reionization histories, and the faded curves show these shifted to the right by ∆ z = 0 . 3 (see § 4.1.1). The upper limit at z = 5 . 8 from Zhu et al. 2024 from Ly α forest damping wings is inconsistent with an earlier ending to reionization, and the same can be said of the z = 5 . 6 measurement from Spina et al. 2024. Some of the lower limits on the ionized fraction are mildly inconsistent with the late start/late end model. However, shifting the histories earlier by ∆ z = 0 . 3 removes these tensions. \n<!-- image -->', '4.2.1. Examples of IGM transmission at z = 8': 'In Figure 9, we illustrate how Ly α transmission surrounding bright galaxies differs in the late start/late end and early start/late end models at z = 8. The solid curves show the IGM transmission ( T IGM ) vs. velocity offset ( v off ) on the red side of systemic averaged over halos with M UV < -17. The vertical magenta line denotes systemic Ly α , v off = 0. T IGM goes to 0 at systemic, and at v off > 0 displays a shape similar to the characteristic damping-wing profile. We see much higher Ly α transmission in the early start/late end case, owing to its much higher ionized fraction (indicated in the legend). At v off < 500 km/s, T IGM is a factor of 2 or more above the late start/late end model. The thin lines show individual transmission profiles for 20 sightlines surrounding the brightest galaxy in the box. These are much higher than the average at v off ≳ 200 -400 km/s, and drop below the mean at smaller v off due to fast in-flowing gas around this object (see discussion of inflows in § 3.3). The higher transmission at large v off owes to this object occupying a larger ionized region than the average galaxy. \nThe higher T IGM in the early start/late end model would seem to naturally explain the detection of bright galaxies hosting Ly α emission at z ≥ 8. However, \nFigure 9. Mean IGM Ly α transmission vs. velocity offset around galaxies with M UV < -17 at z = 8 in our lateending models. The thick curves show the mean transmission profiles, which go to 0 near systemic Ly α ( v off = 0, magenta line), and rise at v off > 0, displaying a shape similar to the damping wing absorption profile. The thin lines denote 20 profiles for individual sightlines surrounding the brightest galaxy (most massive halo) in our box. The early start/late end model has much more transmission on average than the late start/late end case, by a factor of 2 or more at v off < 500 km/s. This owes to its higher ionized fraction (see legend). The sightlines surrounding the brightest galaxy display more transmission at v off ≳ 200 -400 km/s than the average, and less at smaller v off owing to large-scale inflows surrounding that object. The higher transmission owes to the large, biased ionized bubble inhabited by the most massive halo in the box. This illustrates that even in models with low average Ly α transmission, individual sightlines can still be transmissive, especially towards the brightest galaxies. \n<!-- image --> \nit is important to note that even the late start/late end model displays some transmission, even with its 16% ionized fraction, and this can be fairly high around the most biased objects (as the thin black lines show). Indeed, even around this single halo there is significant sightline-to-sightline scatter. This suggests that a statistical sample of observations is required to judge conclusively which model is preferred (Smith et al. 2022; Perez et al. 2023). It also suggests that some LAE detections at z ≥ 8 could be explainable even if reionization starts relatively late.', '4.2.2. Visibility of LAEs': "For an LAE with an intrinsic equivalent width EW int , and an average IGM transmission over the emitted line, ⟨ T IGM ⟩ line , the observed equivalent width EW obs is \nEW obs = ⟨ T IGM ⟩ line EW int (6) \nAn object is detectable if EW obs is greater than some threshold EW min obs . This condition can be expressed as \nEW obs EW int = ⟨ T IGM ⟩ line > EW min obs EW int ≡ T thresh (7) \nwhere we have defined T thresh as the minimum IGM transmission that would make the LAE detectable. To avoid assumptions about the intrinsic properties of the LAE population, we will parameterize our visibility calculations in terms of T thresh . We will also adopt the common simplification that ⟨ T IGM ⟩ line can be approximated by T IGM at the v off of the line's emission peak 11 . This allows us to parameterize LAE visibility in the ( T thresh , v off ) parameter space. In this section, we calculate LAE visibility statistics at z = 8, 9, 10, and 11. \nRecently, Asthana et al. 2024 studied the distribution of ionized bubble sizes in reionization models similar to ours. Generally, it is expected that galaxies must inhabit an ionized bubble of radius ≳ 1 pMpc to guarantee a high level of Ly α transmission on the red side of systemic (Weinberger et al. 2018; Mason & Gronke 2020). They found that a model similar to our early start/late end scenario is required to produce a significant number of such ionized bubbles 8 ≤ z ≤ 10. In Figure 10, we perform a similar analysis using our visibility calculations. We show the fraction of LAEs with T IGM > T thresh vs. redshift for several choices of T thresh , v off , and UV magnitude range (faint vs. bright galaxies). The caption gives these parameter combinations for each curve as a brightness ( M UV ) and a combination of T thresh and v off . Visibility fractions increase with decreasing T thresh and increasing v off . The latter is true because T IGM increases with v off as the damping wing opacity decreases (Figure 9). Visibility is also higher for brighter galaxies, which inhabit the largest ionized bubbles. The left and right panels show results for the late start/late end and early start/late end models, respectively. \nThe solid curves show visibility for bright ( M UV < -21) LAEs with large velocity offsets (400 km/s) and low visibility thresholds ( T thresh = 0 . 2). Such objects are visible nearly 100% of the time in the early start/late end case, and 40% of the time in the late start/late end model even at z = 11. Reducing v off to 200 km/s (dashed curves) deceases visibility, especially in the late start/late end case, but even then 20 -40% of LAEs are visible at 8 < z < 11. In the early start/late end case, the visibility fraction counter-intuitively increases with redshift. This is be- \nFigure 10. Evolution of LAE visibility with redshift for several combinations of v off (in km/s), T thresh , and M UV range - denoted as ( v off , T thresh , M UV range) in the legend. The left and right panels show the late start/late end and early start/late end models, respectively. The solid, dot-dashed, and dashed curves show visibilities for bright galaxies with M UV < -21 - these three curves have ( T thresh , v off ) = (400 km/s , 0 . 2), (200 km/s , 0 . 2), and (200 km/s , 0 . 5), respectively. The dotted and double-dot dashed curves show faint galaxies ( -19 < M UV < -17), with ( T thresh , v off ) = (400 km/s , 0 . 2) and (200 km/s , 0 . 5), respectively. Most choices of these parameters yield ≈ 50 -100% visibility in the early start/late end model, the only exception being faint LAEs with low v off and high T thresh (double-dot dashed). By contrast, the late start/late end model displays a wide range of expected LAE visibility statistics for different types of LAEs. In the left panel, we show recent measurements of the LAE fraction from Tang et al. 2024. The magenta points show the fraction of galaxies observed to have Ly α EW > 25 ˚ A. Interpreting this as a fraction of LAEs that are visible means assuming X int Ly α = 1, so these points are shown as lower limits. The green points assume X int Ly α = 0 . 3, as measured at z = 5 by Tang et al. 2024. The low measured visibility fractions are consistent with range of visibility statistics found in the late start/late end case. However, in the early start/late end model, only the most 'restrictive' parameter combination that we show T thresh = 0 . 5, v off = 200 km/s, and -17 < M UV < -19 displays similar statistics. \n<!-- image --> \ncause the evolution in visibility is not being driven by the neutral fraction, but by inflows surrounding massive halos. At higher redshifts, brighter objects are found in less massive halos, which are surrounded by smaller inflows. This leads to increased transmission at v off = 200 km/s (Park et al. 2021). \nThe dotted curves show that increasing T thresh from 0 . 2 to 0 . 5 (for M UV < -21 and v off = 200 km/s) has a substantial effect on visibility. In the late start/late end model, < 20% of LAEs are visible at z = 8, and this drops to near-0 at z ≥ 9. However, in the early start/late end case, 40 -50% of such objects are visible across this redshift range. The dot dashed curve considers faint ( -19 < M UV < -17) galaxies with high v off = 400 km/s and low T thresh = 0 . 2. These objects are visible 50 -90% of the time in the early start/late end model, but < 20% of the time in the late start/late end case. Finally, the doubledot dashed curves also show faint galaxies, but with v off = 200 km/s and T thresh = 0 . 5, a parameter combination that minimizes LAE visibility. In the late start/late end case, fewer than 10% of such ob- \ncts are visible at any redshift, while in the early start/late end model, 17% (8%) of such objects are visible at z = 10 (11), and over half are visible at z = 8. \nIn the left panel, we show recent measurements of the fraction of galaxies hosting Ly α emitters, (the Ly α fraction, X obs Ly α ), at z = 8 . 7 and z = 10 from Tang et al. 2024. The purple points show X obs Ly α as measured from that work. Comparing these points directly with the fraction of LAEs that are visible assumes that the intrinsic fraction of galaxies hosting LAEs is unity - that is, X int Ly α = 1. For this reason, we display these points as lower limits on X obs Ly α /X int Ly α . The green points show X obs Ly α /X int Ly α assuming X int Ly α = 0 . 3, the Ly α fraction measured at z = 5 by Tang et al. 2024, which we do not show as limits. The curves in the left panel show wide spread that is broadly consistent with the measured visibilities. However, in the right panel, all the curves are on the high end of the measurements (except for the double-dot dashed curve). At face value, these findings indicate that the observed visibility of LAEs is too low for the early start/late end scenario, instead preferring the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model. \nFor the global LAE visibility fraction to evolve consistently with measurements from Tang et al. 2024 in the early start/late end model, the double-dot dashed curve in Figure 10 would have to characterize the bulk of the population. These are faint LAEs with emission at low v off that require high IGM transmission to observe. While it is true that faint LAEs tend to have low v off (Mason et al. 2018), they also tend to have fairly high EW intr (Dijkstra & Wyithe 2012; Tang et al. 2024). A majority of the LAEs observed at z = 5 by Tang et al. 2024 in that M UV range have EW intr > 50 ˚ A, and about half have EW intr > 100 ˚ A. With the visibility threshold of EW min obs = 25 ˚ A used in Tang et al. 2024, this would imply T thresh < 0 . 5 for the majority of faint objects, and T thresh < 0 . 25 for half of them. Moreover, a significant fraction of the faint objects observed at z = 5 -6 in Tang et al. 2024 have v off > 200 km/s. Brighter galaxies in their sample generally have smaller EW intr , but these also tend to have higher v off (see also Mason et al. 2018) and inhabit larger bubbles, such that they would remain visible in the early start/late end model even if they required higher T thresh to detect. By contrast, in the late start/late end model, LAEs with a wide range of properties have visibilities consistent with the Tang et al. 2024 measurements.", '4.2.3. Does GN-z11 require an early start?': 'GN-z11 is the highest-redshift LAE detected to date, at z = 10 . 6. It has a broad Ly α emission feature centered at v off ≈ 550 km/s, a full-width at half maximum (FWHM) of ∆ v ≈ 400 km/s, and an observed EW of 18 ˚ A. Using a Bayesian analysis based on reionization simulations and an empirically derived model for the intrinsic EW distribution of LAEs from Mason et al. 2018, Bruton et al. 2023 inferred that the IGM must be at least 12% ionized at 2 σ confidence (yellow point in Figure 12). This constraint, at face value, clearly favors the early start/late end model (see Figure 12 in the next section). Here, we consider whether the observed properties of GN-z11 require reionization to start early. \nWe can estimate the EW intr required to produce the observed GN-z11 emission line as follows. First, we model the intrinsic line as a Gaussian with some central velocity v intr off , FWHM ∆ v intr , and amplitude A . Then, the observed emission profile for a given sightline is given by the intrinsic profile multiplied by T IGM ( v off ). We also model the observed line as a Gaussian, with parameters given in the previous paragraph, and the continuum and normalization chosen to give the observed EW. For a sample of ∼ 2000 sightlines surrounding -22 < M UV < -21 galaxies, we fit for the parameters of the intrinsic line that, after attenuation by the IGM, \nFigure 11. Distribution of EW intr required to produce observed an LAE with the properties of GN-z11 at z = 10 . 6 (see text for details). The shaded region denotes EW intr < 60 ˚ A, the range observed for similarly bright galaxies at low redshift (see text for references). We see that nearly all sightlines to bright galaxies in the early start/late end model have EW intr within this shaded region, and with 20% (95%) of sightlines requiring EW intr < 30 ˚ A (60 ˚ A). By contrast, the late start/late end model has a wider distribution centered at higher EW intr . About 12% of sightlines in this model require EW intr < 60 ˚ A though, suggesting that a particularly bright LAE with relatively high EW viewed along a sightline with properties that occur ≈ 10% of the time could produce an observation like GN-z11 in the late start/late end scenario. \n<!-- image --> \ngives a best fit to the observed line. The distribution of EW intr recovered with this procedure, P (EW int | EW obs ), at z = 10 . 6 is shown in Figure 11 for our late-ending models. The shaded region denotes the range of EW observed in similarly bright galaxies at lower redshifts Endsley et al. 2022; Tang et al. 2023; Saxena et al. 2024; Tang et al. 2024, the highest of which is ≈ 60 ˚ A (Fig. 1 of Tang et al. 2024). \nWe see a stark contrast between P (EW int | EW obs ) in our models. Nearly all the sightlines in the early start/late end model require EW intr < 60 ˚ A, and about 20% require EW intr < 30 ˚ A. This suggests that bright LAEs with EWs on the high end of the observed distribution will produce a GN-z11-like observation most of the time in this scenario. In the late start/late end model, the distribution is much wider and shifted to much higher EW intr . Only 12% of sightlines allow for EW intr < 60 ˚ A, and none of them allow < 30 ˚ A. So, although objects such as GN-z11 are expected to be fairly rare in the late end/late start scenario, they would not be impossible to find. Note that the two M UV < -21 LAEs observed in Tang et al. 2024 with the highest EWs ( ≈ 30 and 60 ˚ A) were both detected in H α . Based off a \nFigure 12. Summary of neutral fraction measurements in the literature based on either Ly α emission or Ly α damping wings in galaxy spectra. Most of the measurements are at 7 < z < 8, and taken together they do not strongly prefer either late-ending model over the other. There is a dearth of constraints at 8 . 5 < z < 10 (the exception being the Tang et al. 2024 point at z ∼ 8 . 7), and those at z > 10 do not show a clear consensus either. The only measurement that shows a strong preference for the early start/late end model is the constraint from Bruton et al. 2023, based on GN-z11 (see § 4.2.3). Measurements are from Mason et al. 2018; Mason et al. 2019; Hoag et al. 2019; Whitler et al. 2020; Jung et al. 2020; Morales et al. 2021; Bolan et al. 2022; Wold et al. 2022; Morishita et al. 2023; Nakane et al. 2024; Bruton et al. 2023; Hsiao et al. 2023; Curtis-Lake et al. 2023; Tang et al. 2024. \n<!-- image --> \nclear detection of H γ emission in GN-z11, Bunker et al. 2023 estimated that it should have strong H α emission. As such, we can conclude that the detection of GN-z11 does not rule out the late start/late end scenario. However, if forthcoming observations reveal similar objects to be ubiquitous at z > 10, it would be strong evidence in favor of something similar to the early start/late end model.', '4.2.4. Constraints on x HI with galaxies at z ≥ 6 . 5': 'To conclude our discussion of LAEs, we look at measurements of x HI from observations of galaxies at z ≥ 6 . 5. These include constraints from the statistics of LAE detections, and those based on Ly α damping wing absorption in galaxy spectra. Figure 12 shows a collection of these measurements and limits compared to our lateending reionization models in the same format as Figure 8 (with references in the caption). These constraints are all model-dependent to some degree, so showing them on the same plot may not constitute a fair comparison. Our goal here is to illustrate the diversity of constraints obtained across multiple observations and inference techniques. \nUnlike in Figure 8, we see no clear preference for either scenario. Indeed, at z < 8, several constraints prefer each of the models, while some have error bars too large to distinguish them. The only consensus that these constraints give collectively is that reionization is in progress at 7 < z < 8. At z > 10, all the constraints are based on damping wings except that of Bruton et al. 2023, which is based on the detection of GN-z11. There is no clear consensus between these constraints either. There is a notable dearth of constraints at 8 . 5 < z < 10, the redshift range where the two models differ the most. The only exception is the Tang et al. 2024 point at z ∼ 8 . 7, which falls exactly between our models but has large error bars. It seems clear from this comparison that constraints on x HI from high-redshift galaxies do not, at present, display a clear preference for either a late or early start to reionization.', '4.3. Patchy kSZ from reionization': "In this section, we will turn again to the CMB to help distinguish our reionization models. We display the patchy kSZ power spectra for all three models (see § 3.4) in the left panel of Fig. 13, along with the 1 σ error bar and 95% confidence upper limits from Reichardt et al. 2021. We see that the late start/late end model alone lies within the 1 σ of the SPT measurement. The other two both fall outside this range but still within the 95% confidence upper limit, with the early start/late end case coming closest to the upper limit 12 . We also include the revised 2 σ upper limit from Gorce et al. 2022, which is somewhat lower than the SPT result and favors the late start/late end model even more. \nTo gain intuition for the origin of the differences in pkSZ power, we plot the differential contribution to D ℓ =3000 per z in the right panel. Both early-starting models begin contributing power as soon as reionization starts at z = 18, as ionized bubbles form and grow to sufficient scales. The two begin diverging at z ∼ 10, as the early start/early end case finishes reionization, causing the pkSZ power at ℓ = 3000 to drop abruptly at z = 8 (see annotation). This fall-off in power corresponds to the disappearance of large-scale ionization fluctuations, at which point the features in the kSZ signal on these scales are set by fluctuations in density and \nFigure 13. Patchy kSZ from reionization for all three of our model. Left: patchy kSZ power vs. ℓ , compared to the recent measurement and 2 σ upper limit at ℓ = 3000 from Reichardt et al. 2021. We also include the 2 σ upper limit from the re-analysis of the SPT data by Gorce et al. 2022 (offset for readability), which disfavors the early start/late end model. Only the late start/late end model falls within the 1 σ errors, and the early start/late end case has the most power and lies close to the 2 σ upper limit from Reichardt et al. 2021. Right: differential contribution to the total power at ℓ = 3000 as a function of redshift. The shaded red region shows the range where the universe is less than 20% ionized in the early start/late end model, from which nearly half the ℓ = 3000 power originates. The early start/early end model has slightly less power than the early start/late end case because it ends reionization earlier and thus has a shorter duration (see annotation). \n<!-- image --> \nvelocity only. In contrast, ionization fluctuations persist longer in the early start/late end model, and so continue to contribute power to the pkSZ signal at ℓ = 3000 at z < 8. \nIn the late start/late end case, reionization begins much later but still ends at z = 5, which makes the peak in dD pkSZ 3000 /dz narrower. The shaded red region shows that nearly half the power in the early start/late end case arises at z > 10 when the ionized fraction is < 20% in that model. In the late start/late end case, reionization is just starting around z = 10. Thus, we see that the pkSZ is highly sensitive to reionization's duration, and particularly its early stages (Battaglia et al. 2013; Chen et al. 2023). We see also that although the Reichardt et al. 2021 measurement does not rule out the early start/late end model at 2 σ , it clearly prefers the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ case. This finding is consistent with the recent limits on the duration of reionization from Raghunathan et al. 2024 using data from SPT and the Herschel -SPIRE experiment. They found that ∆ z 50 , the difference between redshifts at 25% and 75% ionized fractions, is \n< 4 . 5 at 95% confidence 13 - our early start/late end model has ∆ z 50 = 3 . 1.", "5.1. 'Face-value' interpretations of the data": "In the previous sections, we studied how the properties of our three models compare to a broad range of observables. These include measurements of the UVLF and ξ ion from JWST ( § 2), inferences from the spectra of high-redshift QSOs ( § 4.1), Ly α transmission from z > 8 galaxies ( § 4.2), and constraints from the CMB ( § 4.3). We concluded in § 4.1 that measurements of the Ly α forest at z ≤ 6 strongly disfavor the early start/early end scenario. However, the evolution of the ⟨ F Ly α ⟩ alone could not distinguish between an early and late start to reionization. Most of the other observables we studied individually displayed a preference for one or the other, but none could conclusively rule out either 14 . This motivates the key question in this work: \nTable 1. Summary of the 'face value' preferences of each observable we studied for a late or early start to reionization. The left-most column gives the category for each observable - whether it derives from the CMB (blue), high-redshift galaxies (red), or z < 6 . 5 QSOs (magenta). The second column lists the observables, grouped by category. The right two columns give the preference of each probe for a late or early start to reionization. Both columns list 'no preference' if an observable does not clearly lean towards either scenario. Importantly, neither scenario is 'confirmed' or 'ruled out' by any observables. We see that not all the probes prefer the same scenario. Three of the observables derived from QSO spectra ( P ( < τ 50 eff ), T 0 , and x HI ( z < 6 . 5), see § 4.1) prefer an early start, while the remaining probes all either prefer a late start or cannot distinguish the two scenarios. All three categories, which use largely independent data sets with vastly different analysis techniques, have at least one probe that prefers a late start. However, all the probes that prefer an early start derive from the same type of data (QSO spectra). These in particular are challenging to interpret due to uncertainties in modeling the z < 6 . 5 reionization history and connecting this to what happens at higher redshifts. As such, we conclude that observations display a mild preference for the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ scenario. \nwhen taken together, what story do these observables tell about reionization's early stages? Indeed, a major goal of the field is to synergize the constraining power of many observations to constrain reionization, and qualitative analyses like the one presented here can help chart the path for more detailed studies. \nWe summarize our findings in Table 1. The leftmost column lists the 'categories' of observables that we studied - the CMB (blue), high-redshift galaxies (red), and z < 6 . 5 QSOs (magenta). The second column from the left lists each of the observables, and the remaining two columns denote whether each observable prefers a late or early start to reionization. We find that τ es , ⟨ F Ly α ⟩ , and measurements of x HI at z > 6 . 5 display no preference for either case. JWST observations of the UVLF/ ξ ion , the MFP, LAE visibility at z > 8, and the SPT pkSZ measurement prefer the late start/late end case. By contrast, the Ly α forest P ( < τ 50 eff ), the IGM thermal history, and x HI measurements at z < 6 . 5 prefer the early start/late end model. \nA key finding is that not all these observables prefer the same scenario. This suggests a possible lack of concordance between different data sets with respect to reionization's early stages. We note, however, that all three observables that favor the early start/late end model are based QSO spectra at z ≤ 6 . 5. Indeed, nearly all the data associated with these three observables (with the exception of the QSO damping wings) arises, directly or indirectly, from the Ly α forest, and thus cannot be treated as fully independent 15 . As we explained in § 4.1, the conclusions we drew from these probes are sensitive to our modeling choices, which are necessary to link the late stages of reionization (which the forest probes directly) to its early stages. By contrast, the observables that support a late start are derived from different data sets using vastly different techniques, and at least one probe in every category prefers \na late start. As such, we judge that the consensus of these probes from a wide range of data sets indicates that the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ model is mildly preferred (overall) by observations. \nThe lack of consensus between different observables has several possible resolutions. Perhaps the most straightforward is that existing observations lack the accuracy and/or precision to achieve a unanimous consensus about the early stages of reionization. Indeed, nearly all the observables considered here still have large uncertainties. Theoretical modeling uncertainties, required to interpret the data, can similarly affect these conclusions. As mentioned earlier, the QSO-based observables that seem to support an early start probe only reionization's end stages, requiring a model to infer the early history. These issues will continue to improve with time as more (and better) observational data is acquired and reionization models become faster and more accurate. \nHowever, a more concerning possibility is that forthcoming observations and rigorous theoretical analysis will reveal a statistically significant tension between different observables with respect to reionization's early stages. In this case, the fault must lie with observational systematics and/or hidden deficiencies in theoretical modeling. Either possibility presents a potential pitfall for efforts to constrain reionization with multiple data sets. Such constraints may be artificially tight if 'tensions' between data sets exist. This could lead to the pre-mature conclusion that the reionization history is known to high precision. Forthcoming efforts should be aware of this potential pitfall, and take care to understand the effects of individual data sets on joint constraints.", '5.2. Forthcoming observational prospects': 'In this section, we will briefly discuss prospects for future observations that would help strengthen the constraining power of some of the probes discussed here. The first is to continue improving constraints on the UVLF and ξ ion , particularly for faint galaxies. Atek et al. 2024a demonstrated that ξ ion could be measured reliably for faint ( -17 < M UV < -15), lensed galaxies during reionization. Such studies, together with efforts to directly constrain the faint end of the UVLF, will be crucial for determining the redshift evolution of these quantities and whether there is a fall-off in ionizing output for the faintest galaxies (see bottom panel of Figure 3). Continued efforts to understand how f esc correlates with galaxy properties at low redshift, such as the Low-redshift Lyman Continuum Survey (LzLCS, Chisholm et al. 2022; Flury et al. 2022; Jaskot et al. 2024a,b) will also be crucial for placing rea- \nnable limits on the evolution of f esc (see also e.g. Smith et al. 2020; Pahl et al. 2021; Wang et al. 2023). \nThere is also further progress to be made with QSObased observations at z ∼ 6. Improved constraints on the mean free path and IGM thermal history may help distinguish an early vs. late start, as Figure 7 shows. Forthcoming observations with Euclid (Atek et al. 2024b) will dramatically increase the number of known quasars at these redshifts, allowing for spectroscopic follow-up that will improve statistical uncertainties on both sets of measurements. Efforts to measure the relationship between Ly α forest opacity and galaxy density (Christenson et al. 2021; Ishimoto et al. 2022; Christenson et al. 2023; Kashino et al. 2023) may also help tighten constraints on the reionization history at z < 6 . 5 (Garaldi et al. 2022; Gangolli et al. 2024). \nFurther observations with JWST will improve the statistics of z > 8 galaxies displaying significant Ly α emission. They will also yield constraints on x HI from Ly α damping wing absorption at redshifts where very few bright quasars are available. Forthcoming observations with the Nancy Grace Roman telescope (Wold et al. 2024) will also reveal bright LAEs over a much wider area than JWST, enabling improved constraints on the early reionization history (Perez et al. 2023). \nForthcoming improvements on CMB constraints from multiple experiments, including the Atacama Cosmology Telescope (ACT, Hlozek et al. 2012), SPT (Raghunathan et al. 2024), Simons Observatory (Bhimani et al. 2024), and CMB-S4 (Alvarez et al. 2021) will improve constraints on τ es and pkSZ. They will may also detect new signals that probe reionization, such as patchy τ (Coulton et al. 2024) and higher-order statistics and cross-correlations with other signals (e.g. La Plante et al. 2020). These will help constrain the early stages of reionization because of their sensitivity to its duration and morphology (Chen et al. 2023).', '6. CONCLUSIONS': "In this work, we have studied the observational properties of three representative reionization histories. In the first, reionization starts early and ends at z ∼ 8, earlier than suggested by the Ly α forest and τ es . This scenario is motivated by recent JWST observations of the UVLF and ξ ion at z > 6, which, when combined with observationally motivated assumptions about f esc , suggest copious ionizing photon output by high-redshift galaxies. We have investigated the observational properties of this model, alongside two others in which reionization ends much later at z ∼ 5, in agreement with the Ly α forest and τ es . One model starts reionization relatively late at z ∼ 9, and the other starts early at z ∼ 13. \n- · We find, consistent with previous work, that the early start/early end scenario severely violates high-redshift QSO observations, most notably the Ly α forest. These observations require reionization to end at 5 < z < 6, or at least not much sooner. This is consistent with recent measurements of τ es from Planck Collaboration et al. 2020; de Belsunce et al. 2021. Unfortunately, neither the mean transmission of the Ly α forest nor τ es display a clear preference for whether reionization started late or early. The former measures only the global average transmission of the IGM at reionization's end, and the latter is only an integrated constraint, and thus does not uniquely constrain reionization's early stages.\n- · Observations of the UVLF and ξ ion by JWST, direct measurements of the MFP from QSO spectra, the visibility of z ≥ 8 LAEs, and the recent SPT measurement of the patchy kSZ all prefer a late start to reionization. In light of the latest UVLF measurements at z ≥ 8 from JWST, our earlystarting model requires an order of magnitude of evolution in galaxy ionizing properties (quantified by ⟨ f esc ξ ion ⟩ L UV ) between z = 6 and 12. This is less compatible with observations than the flat evolution in our late-starting model. Direct measurements of the MFP from QSO spectra also prefer a late start, mainly because of the high neutral fraction needed to match direct measurements from QSO spectra at z = 6. The steep drop-off in LAE visibility at z > 8 observed by Tang et al. 2024 is more consistent with a late than an early start. Finally, the low central value of the SPT pkSZ measurement prefers a late start, and disfavors our early-starting model at almost 2 σ .\n- · By contrast, the distribution of Ly α forest opacities, the thermal history of the IGM, and measurements of x HI at z < 6 . 5 prefer an early start. The forest P ( < τ eff ) is too wide for the observations in our late start/late end model, preferring instead the lower neutral fraction in the early start/late end case. Constraints on x HI at z ≤ 6 . 5 from a variety of QSO-based inferences suggest a similar conclusion. The cooler IGM in the early-starting case at z ≤ 5 is also in better agreement with observations.\n- · Our findings suggest that no single probe can conclusively rule out either the late start/late end or early start/late end model in favor of the other. However, we do find that observations \nacross multiple independent datasets - JWST observations of galaxy properties, LAE detections, the CMB, and QSO absorption spectra - prefer a late start to reionization. The observables that prefer an early start are all derived from the same type of observations (QSO spectra) and only probe the tail end of reionization. As such, they are not fully independent probes, and they require a model to derive inferences about reionization's early stages. As such, we conclude that overall, existing observational data displays a mild preference for the ❧❛/a116❡ /a115/a116❛/a114/a116✴❧❛/a116❡ ❡♥❞ scenario. \n- · The face-value disagreement we find between different probes suggests that (1) present observations, and the models used to interpret them, are insufficiently precise/accurate to paint a consensus picture of reionization's early stages, and/or (2) there are systematic effects (in observations and/or theoretical modeling) leading to the appearance of tension. The second possibility motivates care in interpreting the results of analyses using multiple data sets. Joint analyses using many observables could lead to artificially tight constraints on the reionization history and other quantities if tensions arising from systematics are not carefully understood. \nForthcoming work, on the observational and theoretical side, should continue working to synergize the information available from many observables. Our work motivates further efforts targeting the early stages of reionization, which will yield key insights into the evolution of galaxy properties across the reionization epoch and into the cosmic dawn era. This work is also a cautionary tale that motivates careful understanding of potential systematics in both observations and theoretical modeling. Such systematics, if not studied carefully, could lead to premature conclusions about reionization. \nCC acknowledges helpful conversations with Seth Cohen, Timothy Carleton, Evan Scannapieco, Frederick Davies, and Kevin Croker, and support from the Beus Center for Cosmic Foundations. A.D.'s group was supported by grants NSF AST-2045600 and JWSTAR02608.001-A. RAW acknowledges support from NASA JWST Interdisciplinary Scientist grants NAG5-12460, NNX14AN10G and 80NSSC18K0200 from GSFC. JBM acknowledges support from NSF Grants AST-2307354 and AST-2408637, and thanks the Kavli Institute for Theoretical Physics for their hospitality. Computations were made possible by NSF ACCESS allocation TGPHY230158.", 'REFERENCES': "- Sunyaev, R. A., & Zeldovich, Y. B. 1980, Mon. Not. R. Astron. Soc., 190, 413\n- Tang, M., Stark, D. P., Topping, M. W., Mason, C., & Ellis, R. S. 2024, arXiv e-prints, arXiv:2408.01507, doi: 10.48550/arXiv.2408.01507\n- Tang, M., Stark, D. P., Chen, Z., et al. 2023, MNRAS, 526, 1657, doi: 10.1093/mnras/stad2763\n- Tang, M., Stark, D. P., Ellis, R. S., et al. 2024, Monthly Notices of the Royal Astronomical Society, 531, 2701, doi: 10.1093/mnras/stae1338\n- Tepper-Garc'ıa, T. 2006, MNRAS, 369, 2025, doi: 10.1111/j.1365-2966.2006.10450.x\n- Topping, M. W., Stark, D. 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R., & Puchwein, E. 2018, Monthly Notices of the Royal Astronomical Society, 479, 2564, doi: 10.1093/mnras/sty1563 \nWhitler, L. R., Mason, C. A., Ren, K., et al. 2020, \nMNRAS, 495, 3602, doi: 10.1093/mnras/staa1178 \n- Wold, I. G. B., Malhotra, S., Rhoads, J. E., Tilvi, V., & Gabrielpillai, A. 2024, AJ, 167, 157, \ndoi: 10.3847/1538-3881/ad2adf \n- Wold, I. G. B., Malhotra, S., Rhoads, J., et al. 2022, ApJ, 927, 36, doi: 10.3847/1538-4357/ac4997\n- Worseck, G., Prochaska, J. X., O'Meara, J. M., et al. 2014, MNRAS, 445, 1745, doi: 10.1093/mnras/stu1827 \nWu, X., Kannan, R., Marinacci, F., Vogelsberger, M., & \nHernquist, L. 2019, MNRAS, 488, 419, \ndoi: 10.1093/mnras/stz1726 \n- Yang, J., Wang, F., Fan, X., et al. 2020, ApJL, 897, L14, doi: 10.3847/2041-8213/ab9c26\n- Yeh, J. Y. C., Smith, A., Kannan, R., et al. 2023, MNRAS, 520, 2757, doi: 10.1093/mnras/stad210\n- Zeng, C., & Hirata, C. M. 2021, ApJ, 906, 124, doi: 10.3847/1538-4357/abca38\n- Zhao, J., & Furlanetto, S. R. 2024, arXiv e-prints, arXiv:2401.07893, doi: 10.48550/arXiv.2401.07893\n- Zhu, Y., Becker, G. D., Bosman, S. E. I., et al. 2022, ApJ, 932, 76, doi: 10.3847/1538-4357/ac6e60\n- Zhu, Y., Becker, G. D., Christenson, H. M., et al. 2023, ApJ, 955, 115, doi: 10.3847/1538-4357/aceef4\n- Zhu, Y., Becker, G. D., Bosman, S. E. I., et al. 2024, MNRAS, doi: 10.1093/mnrasl/slae061\n- Zitrin, A., Labb'e, I., Belli, S., et al. 2015, ApJL, 810, L12, doi: 10.1088/2041-8205/810/1/L12 \nFigure 14. Ly α forest P ( < τ 50 eff ) with the reionization histories in our late-ending models shifted earlier by ∆ z = 0 . 3, as described in § 4.1.1. This test estimates what these models would look like if reionization ended at z = 5 . 3 (as suggested by the analysis of Bosman et al. 2022) instead of at z = 5. At z = 5, 5 . 2, and 5 . 4, there is little difference between the two models, as reionization is complete (or nearly so) in both cases. At z = 5 . 6 and 5 . 8, P ( < τ 50 eff ) is narrower in the early start/late end model and agrees well with the observations, whereas P ( < τ 50 eff ) is still too wide in the late start/late end case. At z = 6, it is difficult to tell which model agrees best with the data. \n<!-- image -->", 'A. P ( < τ 50 EFF ) FOR SHIFTED REIONIZATION HISTORIES': 'In Figure 14, we show P ( < τ 50 eff ) for the late start/late end and early start/late end models with their reionization histories shifted earlier by ∆ z = 0 . 3, as described in § 4.1.1. We show 6 redshifts between z = 5 and 6 in intervals of 0 . 2. This exercise shows roughly what P ( < τ 50 eff ) would look like if these models ended reionization at z = 5 . 3 instead of 5. We see that at z = 5, 5 . 2, and 5 . 4, there is little difference between the models. At z = 5 . 6 and 5 . 8, P ( < τ 50 eff ) is slightly narrower in the early start/late end case, as it is in Figure 6. The difference is that the early start/late end model now seems to agree well with the observations, whereas in the late start/late end case, P ( < τ 50 eff ) is still too wide. At z = 6, it is difficult to tell which model is a better fit to the data due to the large number of non-detections in the data, which set the width of the green shaded region. This test demonstrates that even if reionization ends significantly earlier than it does in our models, the early start/late end scenario remains preferred by P ( < τ 50 eff ).', 'B. LY α VISIBILITY IN THE FULL ( T THRESH , V OFF ) PARAMETER SPACE': "In this appendix, we show complete results for our LAE visibility analysis in terms of v off and T thresh presented in § 4.2.2. Figure 15 shows the fraction of visible LAEs at T thresh < 0 . 5 and 0 < v off < 800 km/s, at the four redshifts (columns) and three magnitude ranges (rows) we considered in the late start/late end model. Red (blue) regions denote high (low) LAE visibility fractions. The white contour lines denote visibility fractions of 50%, 25%, and 10% \nFigure 15. Summary of LAE transmission statistics in the late start/late end model. The y axis is T thresh (Eq. 7) and the x axis is the v off at line center. The color plot shows the fraction of visible LAEs (with T IGM > T thresh ) across this parameter space. The panels denote different redshifts (columns) and ranges of M UV (rows). The white contour lines denote 'visibility fractions' of 10%, 25%, and 50%. The hatched region in the upper left of each panel denotes T thresh > 0 . 2 and v off < 500 km/s. \n<!-- image --> \n(see annotation in the left-most panel of the middle row). The hatched white box in the upper left corner of each panel denotes the region where T thresh > 0 . 2 and v off < 500 km/s. We expect a majority of z ≥ 8 LAEs to inhabit this region. Most LAEs in M UV < -17 galaxies observed at slightly lower redshifts ( z ∼ 5 -6, when T IGM is close to unity) have v off < 500 km/s and EW int < 500 ˚ A (Goovaerts et al. 2023; Tang et al. 2024). For T thresh = 0 . 2, the latter would correspond to EW min obs < 100 ˚ A. \nAt z = 8, a significant portion of the hatched region displays a high visibility fraction, especially for the brightest galaxies. For M UV < -21 galaxies, LAEs with v off > 200 km/s have a significant chance of being visible, provided they are bright enough to be observed with a factor of 2 IGM attenuation. Fainter galaxies require somewhat fainter detection thresholds, since on average they inhabit smaller ionized bubbles and are more sensitive to IGM attenuation. In the faintest M UV bin, v off > 400 kms/ and T thresh < 0 . 25 is required for half of LAEs to be visible. Still, we should expect to observe some LAEs at z = 8 in the late start/late end model, especially the brightest ones. \nAt z > 8, the transmission of Ly α declines rapidly, especially for fainter galaxies. At these redshifts, in all but the brightest M UV bin, the 50% contour line does not intersect the hatched region. As we showed in Figure 10, this drop-off in visibility is consistent with the observed decline in X Ly α observed by Tang et al. 2024 at z > 8. The brightest objects remain likely to be observed if v off > 400 km/s and T thresh < 0 . 25 all the way to z = 11. Notably, GN-z11 ( M UV = -21 . 5, z = 10 . 6, Bunker et al. 2023) has v off = 550 km/s, meeting this condition. The recently observed JADES-GS-z9-0 ( M UV = -20 . 43, z = 9 . 4, Curti et al. 2024) has v off = 450 km/s. These objects would be likely visible in the late start/late end model if their intrinsic EWs were in the neighborhood of 100 ˚ A, ≈ 4 × \nFigure 16. Same as Figure 15, but for the early start/late end model. A significant fraction of LAEs with typical properties (even faint ones) should remain visible in this scenario even at z = 11, in contrast with the observations Tang et al. 2024. \n<!-- image --> \nlarger than their observed EWs (see § 4.2.3). These results are consistent with a universe in which most LAEs at z > 8 are obscured by the IGM, but a small number of objects that are relatively bright, have high v off , and/or high intrinsic EWs remain visible. \nFigure 16 is the same as Figure 15, but for the early start/late end model. In contrast to the late start/late end case, a significant fraction of the parameter space displays high visibility, even at z = 11. Objects with M UV < -21 are likely to be visible up to z = 11 as long as T thresh ≤ 0 . 5 and v off > 200 km/s. Even the faintest galaxies have a significant chance of being observed at z = 11. It should then be expected that in such a scenario, a significant fraction of LAEs - even faint ones - with typical physical properties should remain visible up to z = 11. At face value, the observed sharp decline in LAE visibility across the population of LAEs up to this redshift does not prefer this scenario."}

2024arXiv240909023A

We study the contribution of large scalar perturbations sourced by a sharp feature during cosmic inflation to the stochastic gravitational wave background SGWB extending our previous work to include the SGWB sourced during the inflationary era. We focus in particular on threefield inflation since the third dynamical field is the first not privileged by the perturbations equations of motion and allows a more direct generalization to Nfield inflation. For the first time we study the threefield isocurvature perturbations sourced during the feature and include the effects of isocurvature masses. In addition to a twofield limit we find that the third fields dynamics during the feature can source large isocurvature transients which then later decay leaving an inflationaryerasourced SGWB as their only observable signature. We find that the inflationaryera signal shape near the peak is largely independent of the number of dynamical fields and has a greatly enhanced amplitude sourced by the large isocurvature transient suppressing the radiationera contribution and opening a new window of detectable parameter space with small adiabatic enhancement. The largest enhancements we study could easily violate backreaction constraints but much of parameter space remains under perturbative control. These SGWBs could be visible in LISA and other gravitational wave experiments leaving an almost universal signature of sharp features during multifield inflation even when the sourcing isocurvature decays to unobservability shortly afterwards.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.09023', '2024arXiv240909023A', 'arXiv:2409.09023']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Threefield Inflation II The Inflationary Era

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09023.pdf

{'Vikas Aragam a Sonia Paban b Robert Rosati c,d': "- a Medical Physics Division, Thomas Jefferson University, Philadelphia, PA 19107, USA b Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA c NASA Postdoctoral Program Fellow, NASA Marshall Space Flight Center, Huntsville, AL 35812, USA d Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA \nE-mail: aragam@utexas.edu, sppaban@fas.harvard.edu, robert.j.rosati@nasa.gov \nAbstract. We study the contribution of large scalar perturbations sourced by a sharp feature during cosmic inflation to the stochastic gravitational wave background (SGWB), extending our previous work to include the SGWB sourced during the inflationary era. We focus in particular on three-field inflation, since the third dynamical field is the first not privileged by the perturbations' equations of motion and allows a more direct generalization to N -field inflation. For the first time, we study the three-field isocurvature perturbations sourced during the feature and include the effects of isocurvature masses. In addition to a two-field limit, we find that the third field's dynamics during the feature can source large isocurvature transients which then later decay, leaving an inflationary-era-sourced SGWB as their only observable signature. We find that the inflationary-era signal shape near the peak is largely independent of the number of dynamical fields and has a greatly enhanced amplitude sourced by the large isocurvature transient, suppressing the radiation-era contribution and opening a new window of detectable parameter space with small adiabatic enhancement. The largest enhancements we study could easily violate backreaction constraints, but much of parameter space remains under perturbative control. These SGWBs could be visible in LISA and other gravitational wave experiments, leaving an almost universal signature of sharp features during multi-field inflation, even when the sourcing isocurvature decays to unobservability shortly afterwards.", '1 Introduction': 'Exploring the stochastic gravitational wave background (SGWB) could unlock invaluable information about astrophysical source populations and processes that remain out of reach through any other means. Gravitational waves can, for example, give us information all the way back to the onset of inflation [1-3]. But to learn from this complex data, we must have precise predictions for the observables commonly used to characterize it: fractional energy density spectrum (Ω GW ), characteristic strain, and distribution of gravitational wave power on the sky. \nInflation is among the cosmological sources of SGWB. Single-field slow-roll inflation, without features, predicts an Ω GW that is almost constant in frequency and whose magnitude is well below the sensitivity level of future detectors [4]. Single-field models with periods of ultra-slow rolling or rapid-turning multi-field models of inflation, however, can source detectable gravitational waves at second order in perturbation theory [5-13]. In [9], Fumagalli et al. derive the tensor power spectrum sourced at second order from excited states during inflation in a generic multi-field model. One concrete realization of this scenario, which can occur in SUGRA models [14, 15], is a brief rapid turn in field space. This turning is a departure from single-field behavior and sources both a feature in P ζ (at shorter scales than observed by CMB and LSS) and the Stochastic Gravitational-Wave Background (SGWB). The authors of [9] compute analytically and numerically the frequency profile of Ω GW for two fields in this scenario. Then, they hypothesize that if the contributions from all the fields are of the same order, the power spectrum is enhanced by a N 4 factor, where N is the number of fields. \nIn this work, we examine and extend the findings of Fumagalli et al. [9] to identify the conditions under which all fields contribute equally. Specifically, we calculate the contributions to Ω GW in three-field models by varying the relative values of the turn rate and torsion. This allows for a smooth transition from two to three active dynamic fields. In the evolution of the fluctuations, N = 3 is distinct because for N ≥ 3, only one linear combination of isocurvature fluctuations couples directly to the adiabatic fluctuations, enabling a more direct generalization to N fields. Enhanced power spectra and primordial black holes from broader features with three fields were also studied in [16]. \nSecondary gravitational waves are generated at two times: during inflation, Ω inf GW and during the radiation-dominated era as the features in the scalar spectrum reenter the horizon Ω rad GW . In [17], we examined the N dependence in the latter case, finding the frequency profile to depend on the number of active fields, and hinted that not all fields may contribute equally to Ω GW . \nWhen the mechanism that generates excited states is a brief turn in field space, the mode that crosses the horizon when the turn occurs becomes a reference scale k f . Fumagalli et al. [9] demonstrated that, for N = 2, the frequency profile of Ω inf GW has a principal peak at k max ≃ O (1) k f , followed by a series of order-one oscillations with frequency 2 /k f . Our findings indicate that the location of the maximum ( ∼ k f ) and the frequency of the oscillation (2 /k f ), as derived in [9], are the same in the two- and three-field limits. By contrast, the Ω rad GW envelope depends on N [17]. The Ω inf GW peak enhancement does not scale as N 4 , but it is notably larger than for N = 2, and is primarily caused by a large isocurvature transient. Our analysis reveals that isocurvature modes for low k grow rapidly when torsion is comparable to, or greater than, the turning. Therefore, in order to match the unobserved isocurvature in CMBdata, we assign them a mass of O ( H ). This has been an implicit assumption in previous \nliterature on sharp features. We have, for the first time, calculated all the massive Bogoliubov coefficients analytically and demonstrated that although the isocurvature perturbations decay quickly while the modes are beyond the horizon, they still leave an imprint on the SGWB that could be observable in future gravitational wave experiments. \nThe presentation of our work is organized as follows. In Section 2, we review the threefield inflation formalism, summarizing the concepts of torsion and turning rate of background motion and their effect on the time evolution of the linear quantum perturbations. In Subsection 2.4, we recap how to compute the contribution of the scalar perturbations to the tensor power spectrum. The results are presented in Section 3. The unexpected considerable growth in the perturbations demands a critical appraisal of the results we present in Section 4. In Section 5 we discuss the phenomenology of sharp features in CMB, LSS and SGWB.', '2 The Set-up': 'In this section, we give a self-contained summary of the set-up. We refer the reader to [9] for the general derivation of the contribution to Ω GW from excited states and to [16-18] for more details on the three field dynamics.', '2.1 Background Motion': "We consider scenarios where inflation is driven by three scalar fields minimally coupled to gravity. The action is: \nS = ∫ d 4 x √ -g [ M 2 Pl 2 R ( g ) -1 2 G ab g µν ∂ µ ϕ a ∂ ν ϕ b -V ( ϕ a ) ] (2.1) \nGreek letters label spacetime indices, and lower Latin indices label field-space indices, a = 1 , 2 , 3. g µν is the spacetime metric and G ab is the field space-metric. M Pl = √ 8 πG is the reduced Planck mass. We will work in units where M Pl = 1 . \nThis work uses the kinematic field basis, so called because it is defined from the fields' trajectory. The first unit vector in the basis is the velocity unit vector, ˆ σ a ≡ ˙ ¯ ϕ a / ˙ ¯ ϕ , and subsequent unit vectors are defined by additional covariant time derivatives as summarized in the Frenet-Serret system \nD N ˆ σ a ˆ s a ˆ b a = 0 Ω 0 -Ω 0 τ 0 -τ 0 ˆ σ a ˆ s a ˆ b a ≡ Ω a b ˆ σ b ˆ s b ˆ b b , (2.2) \nwhere D N A a ≡ ( dA a /dN ) + Γ a bc A b ( d ¯ ϕ c /dN ), and N is the number of e-folds. Ω and τ measure the turn rate and torsion of the trajectory respectively, in agreement with the literature [18, 19]. When Ω > 0, the trajectory undergoes turning, and when τ > 0 as well, that turning is non-planar. If both Ω and τ are constant, the trajectory follows a helix. In general, τ and the kinematic basis are not well-defined when Ω = 0. A schematic representation of the trajectory during such a feature is available in Figure 2.2. \nFigure 2.1 : The stochastic gravitational-wave backgrounds computed in this work. We show the range of possibilities for the spectrum in three fields, fixing the approximate effective two-field enhancement of the curvature perturbation (Ω 2 2f ≡ Ω 2 + τ 2 ) but varying the ratio of turning and torsion ρ ≡ Ω /τ . The inflationary-era background is shown in solid lines, while the corresponding radiation-era background is shown in the same color as dot-dashed, with the predicted observable signal being the sum of the two. As we discuss below, when ρ ≲ 1, the isocurvature power spectra are large at horizon-exit (and then rapidly decay) and this can enhance the inflationary-era SGWB without substantially enhancing the curvature perturbation P ζ or the radiation-era signal. We therefore expect highly dimensional field spaces to source these inflationary-era SGWBs much more easily than the radiation-era ones. \n<!-- image -->", '2.2 Perturbations': "The linear equations of motion for the Mukhanov-Sasaki variables in kinetic basis Q a ≡ { Q σ , Q s , Q b } can be written as [17, 19] \nD N ( Q ' ) a + F a ( Q ' ) b + C a Q b = 0 \nb b F a b ≡ (3 -ϵ ) δ a b -2Ω a b C a b = ( k aH ) 2 δ a b + 0 -2(3 -ϵ )Ω 0 0 M ss -Ω 2 -τ 2 M sb -τ (3 -ϵ ) 0 M sb + τ (3 -ϵ ) M bb -τ 2 + O ( ϵ 2 , η, ν, ν τ ) \n(2.3) \nThe prime denotes a derivative with respect to conformal time. ϵ ≡ -H ' / H , η ≡ ϵ ' /ϵ , ν ≡ Ω ' / Ω and ν τ ≡ τ ' /τ . {M ss , M sb , M bb } are defined as contractions with { ˆ s a , ˆ b a } of: \nM ab ≡ V ; ab H 2 -2 ϵR aσσb +2 ϵ (3 -ϵ )ˆ σ a ˆ σ b + √ 2 ϵ ˆ σ a V ,b + ˆ σ b V ,a H 2 M 2 Pl (2.4) \nTo simplify C a b , we have used that some mass matrix elements can be expressed purely in kinematic quantities using the background equations of motion. For three fields, these are \nM σσ = Ω 2 -1 4 η (6 -2 ϵ + η +2 ξ ) M σs = Ω( -3 + ϵ -η -ν ) M σb = -Ω τ. (2.5)", '2.3 Dynamics': "Instead of describing the brief rapid turn with a potential and field space metric, following [9, 17] we study a synthetic background evolution. We directly parameterize the time evolution of the turning rate Ω and torsion τ as \nT ( N e ) = [ θ ( N e -( N f -δ/ 2)) -θ ( N e -( N f + δ/ 2))] (2.6) \nwhere Ω( N e ) = Ω 0 T ( N e ) and τ ( N e ) = τ 0 T ( N e ), N e counts the e-folds after the beginning of inflation, and N f is the e-fold number at the center of the feature. For analytic convenience, we assume the turn is a top hat centered at time N f e-folds after the beginning of inflation, with width δ , and height either Ω 0 or τ 0 . Considering coincident profiles in Ω and τ is convenient, but it is not the most general case. Because torsion is not well defined if Ω equals zero, the most general situation would be a profile in τ inside the profile in Ω. \nWithout choosing a potential and a field metric, the remaining masses in (2.3) are unknown. We parametrize them in terms of the turn rates as \nM ss = ξ ss (Ω 2 + τ 2 ) + M ss, 0 M sb = ξ sb τ M bb = ξ bb τ 2 + M bb, 0 , (2.7) \nwhere ξ ss , ξ sb , ξ bb , M ss, 0 , M bb, 0 are assumed to be arbitrary real constants. We are unaware of any model where this parametrization of the masses is precisely valid, but take it as a natural generalization of the two-field scenario, where models with M ss an exact multiple of Ω 2 are known, e.g. [20]. It is also possible to build models with arbitrary mass parametrization through a superpotential method [16, 21]. In field spaces with sufficiently many isometries, [18] found that rapid-turn, slow-roll trajectories must have M sb ∼ (3 -ϵ ) τ . Because this is of order of the friction terms we have already dropped, we choose ξ sb = 0 in all of our analytic calculations. \nAlthough not an exact or approximate description of any concrete model's dynamics to our knowledge, the top-hat feature we study does qualitatively reflect many concrete models. For example, the two-field models in [8, 10, 15] are concrete microphysics realizations of sharp inflationary features and generate gravitational wave spectra similar to the ones studied here. We discuss several ways these features might be observable in Section 5. \nThe brief turn in the background field trajectory causes perturbations to change from a quasi-single field Bunch-Davies initial state before the turn (region I) into an excited quasisingle field state after the turn (region III). The simple form of (2.6) allows us to use the WKB approximation to describe analytically the behavior of the perturbations during the turn \nFigure 2.2 : The turn profile considered in this work. We show an exaggerated path in three-dimensional field space (left) and the turn rates as a function of time (right). On the left, the color changes from blue to red as a function of time. We label for future convenience three regions: I, II, and III, corresponding to before, during, and after the turn respectively. Note that the turn displayed on the left has a duration of δ = 1 . 1 efolds for illustrative purposes, while values considered below consider much briefer turns, δ ≤ 0 . 25. \n<!-- image --> \n(region II). The background metric and its first-time derivative are continuous at the junctions between regions I, II, and III. The corresponding matching conditions for the MukhanovSasaki variables were derived by Deruelle and Mukhanov [22] and are \n∆( Q i ) = 0 ∆( D N Q σ -2 Q s Ω) = 0 ∆( D N Q s -Q b τ ) = 0 ∆( D N Q b + Q s τ ) = 0 , (2.8) \nwhere the ∆ operator matches quantities from region A to those from region B at junction time t as ∆ x ≡ x A | t + -x B | t -. These matching conditions are valid for any set of turn profiles Ω( N ) and τ ( N ), even ones with a derivative discontinuity. \nThe perturbations in regions I and III are relatively simple to write down since they are quasi-single field. We describe region I in terms of a Bunch-Davies state: \nQ i, I = u ( ⃗ k, N ) ˆ a i +h . c . ( -⃗ k ) (2.9) u ( k, N ) ≡ ( iH √ 2 k 3 )( 1 -i k Ha ( N ) ) e ik/ ( Ha ( N )) \nu ( k, N ) ≡ ( iH √ 2 k 3 ) ( -ie i (2˜ ν +1) π/ 4 π √ 2 ) ( k Ha ( N ) ) 3 / 2 H (1) ˜ ν ( k Ha ( N ) ) for massive fields \nfor massless fields (2.10) √ . (2.11) \n˜ ν 2 ≡ 9 4 - M , and H (1) ˜ ν are Hankel functions of the first kind (˜ ν ≡ √ ˜ ν 2 if ˜ ν 2 > 0 and ˜ ν ≡ i √ -˜ ν 2 if ˜ ν 2 < 0.) Similarly, we describe region III in terms of an excited Bunch-Davies vacuum: \nQ i, III = ( α ij u ( k, N ) + β ij u ∗ ( k, N )) ˆ a j +h . c . ( -⃗ k ) (2.12) \nNote that our choice of including explicit constant contributions to the isocurvature masses is a first in the literature. As we later describe in Section 3, depending on the nature of the feature it can source large growths in the isocurvature power spectra. Without constant isocurvature masses, this isocurvature power would not decay, and any sharp feature would greatly enhance the adiabatic modes in its superhorizon. To maintain the known CMB amplitude, we would have to lower the scale of inflation very significantly, and would risk disturbing big bang nucleosynthesis and would remove any chance of sourcing a detectable SGWB. In fact the presence of these masses was already taken as a default assumption in the literature, including in [7, 21] and they were present during our numerical solutions of the perturbations' equations of motion in our previous work [17]. \nImposing the canonical commutation relations for the fields and momenta is equivalent to requiring [9]: \nα ik α ∗ jk -β ∗ ik β jk = δ ij α ik β ∗ jk -β ∗ ik α jk = 0 , (2.13) \nfor each pair of fields i, j . \nTo describe the α ij and β ij in terms of the initial state, we need to use the nontrivial matching procedure described above and a method to evaluate the perturbations in Region II. As noted above, we use the WKB approximation. Additionally, to simplify computations later on, we assume that the turn duration is so short that the effects of Hubble expansion on the perturbations can be ignored. In practice, we neglect all remaining terms proportional to (3 -ϵ ) in F a b and C a b , which is equivalent to deriving the equations of motion from the Lagrangian (2.4) and disregarding any time derivatives of a ( t ). In principle, this leaves C s b = M s b as the only remaining nonzero off-diagonal element of C a b . However, we also neglect this term since M sb ∼ (3 -ϵ ) τ in a wide range of models [18]. \nTo find the WKB approximation of the perturbations during the turn, we take the mode functions to have the form [7, 21, 23] \nQ i = Q i, 0 e iqN e (2.14) \nwhere the Q i, 0 are different for each field, and q labels all of the WKB frequencies. Plugging this into the frictionless equations of motion, we find that the Q i, 0 must be inter-related by \nQ s, 0 = i q 2 -k 2 k 2 f 2 q Ω 0 Q σ, 0 Q b, 0 = τ 0 Ω 0 q 2 -k 2 k 2 f k 2 k 2 f -q 2 +( ξ bb -1) τ 2 0 Q σ, 0 , (2.15) \nand that there are six possible solutions q = ± √ α i , where the α i are the three roots of the cubic polynomial. See [17] for details. In general, any solution in region II is a linear \nsuperposition of all six possible WKB exponents: \nQ i, II = ∑ j ∈ fields ˆ a j 3 ∑ k =1 ∑ ± Q ijk ± , 0 e ± i √ α k N e +h . c . ( -⃗ k ) (2.16) \nThe (2 N 3 = 54) Q ijk ± , 0 amplitudes are independent along the solution axis ( k, ± ), but they are inter-related via (2.15) along the field labels i , resulting in a total of 18 independent amplitudes. \nThe expressions for the nine α ij and the nine β ij in (2.12) derived from the matching are very cumbersome and not immediately enlightening, so we have chosen to omit them from the text. However we have released them as supplementary code available with this paper, in both Mathematica and Julia, available at https://github.com/rjrosati/ 3field-sharp-feature . In the uploaded notebooks, we include all α ij , β ij , the WKB roots α i , a verification of the identities (2.13), and plotting code to reproduce almost all figures in this paper. Our code for performing the integral (2.18) will be released at a later date.", '2.4 The Tensor Power Spectrum': 'The quantity of interest Ω GW is proportional to the tensor power spectrum [9], \nh 2 Ω GW = r i P t ( k, τ end ) ≡ r i P t ( k ) , r i ≡ h 2 0 . 0416 c g Ω r, 0 (2.17) \nwhere c g ≃ 0 . 4 and Ω r, 0 is the energy density fraction in radiation today. In the presence of excited states, and when non-Gaussian corrections can be neglected [24], \nP t ( k ) = H 4 8 π 4 M 4 Pl ∫ ∞ 0 dy ∫ 1+ y | 1 -y | dxµ ( x, y ) × ∑ i,j ∣ ∣ ∣ ∣ ∣ ∣ ∑ X,s 1 , 2 = ± α s 1 Xi ( xk ) α s 2 Xj ( yk ) G XX ( s 1 x, s 2 y, z out ) ∣ ∣ ∣ ∣ ∣ ∣ 2 (2.18) \nwhere, following the notation in [9], we defined α + ij ≡ α ij and α -ij ≡ β ij . The geometrical factor is \nµ ( x, y ) = ( (4 x 2 -(1 + x 2 -y 2 ) 2 ) 4 xy ) 2 (2.19) \nµ ( x, y ) varies between 0 and 1 and tends to zero as x approaches | 1 -y | and 1+ y . The equation (2.18) generalizes equation (4.31) in the reference [9] slightly to include non-vanishing zeromasses M ss, 0 and M bb, 0 . \nG XX ( x, y, z out ) ≡ ∫ 0 z out dz z 2 ζ X ( xz ) ζ X ( yz ) ζ σ ( z ) -ζ ∗ σ ( z ) 2 i (2.20) \nwhere \nζ σ ( z ) = (1 + iz ) e -iz ζ s ( z ) = -e i ( ν s +1 / 2)( π/ 2) i √ π 2 ( -z ) 3 / 2 H (1) ν s ( -z ) ζ b ( z ) = -e i ( ν b +1 / 2)( π/ 2) i √ π 2 ( -z ) 3 / 2 H (1) ν b ( -z ) (2.21)', '2.4.1 The Time Integral G XX': "The time integral G XX can be computed exactly for some masses. When both fields are massless, the complete expression was given in section 4.4 of [9]. \nG σσ ( x, y, z ) = K ( x, y ) -F ( x, y, z ) -F ∗ ( -x, -y, z ) (2.22) \nwhere \nK ( x, y ) = 1 -2 xy -( x + y ) 2 (1 -( x + y ) 2 ) 2 (2.23) \nF ( x, y, z ) = e -i (1+ x + y ) z 2(1 + x + y ) 2 × (2.24) ( i [ (1 + x + y ) 2 z -xy (1 + x + y ) z ] -[ x + y +( x + y ) 2 + xy (2 + x + y )] ) \nThe only other case we know of with an exact closed form is when the dimensionless constant mass M ss, 0 = 2 and M bb, 0 = 2 . In that case \nG ss ( x, y, z ) = ˜ K ( x, y ) -˜ F ( x, y, z ) -˜ F ∗ ( -x, -y, z ) (2.25) \nwhere \n˜ K ( x, y ) = -2 xy (1 -( x + y ) 2 ) 2 (2.26) \n˜ F ( x, y, z ) = -e -i (1+ x + y ) z 2(1 + x + y ) 2 ( i xy (1 + x + y ) z + xy (2 + x + y )) (2.27) \nAs expected, G σσ and G ss agree for large values of x and y where the mass becomes irrelevant, but they have different limits for small values of y ( x is constrained to be in the interval | 1 -y | ≤ x ≤ 1 + y ). In the y ≪ 1 limit \nG σσ = 3 z +2 i -2 iz 2 4 z + e -2 iz ( -2 i + z 4 z ) + O ( y, ( x -1)) (2.28) \nG ss = y ( -3 + 4 iz +2 z 2 8 + e -2 iz ( 3 + 2 iz 8 )) + O ( y 2 , y ( x -1)) (2.29) \n(2.30) \nRegardless of the mass, the time integral has the following properties: \nG XX ( x, y, z ) = G XX ( y, x, z ) (2.31) G XX ( -x, -y, z ) = G ∗ XX ( x, y, z ) G XX ( x, -y, z ) = G ∗ XX ( -x, y, z ) \n<!-- image --> \n<!-- image --> \nFigure 3.1 : A comparison of the Bogoliubov coefficients as a function of κ for three values of ρ ≡ Ω /τ and a fixed Ω 2 f = √ Ω 2 + τ 2 = 23 . 7 and δ = 0 . 225. The masses are parameterized by (2.7) with ξ ss = -3 , ξ bb = 2 , M ss, 0 = M bb, 0 = 2. Solid lines represent α ζζ and β ζζ . Dashed lines correspond to Bogoliubov coefficients for which at least one of the indices is an 's.' Dotdashed lines correspond to Bogoliubov coefficients with at least one index being a 'b.' The quasi-two-field, Ω-dominated case on the left has all coefficients contributing approximately equally and a sharp peak around k ∼ Ω 2 f k f , while a strong hierarchy develops as the torsion increases. The dominant coefficients rapidly become α ss , β ss and the peak vanishes, replaced by a steep powerlaw growth towards the superhorizon. \n<!-- image -->", '3 Results': "In this section we investigate the structure of the three-field excited state after the sharp feature, as well as the inflationary-era graviatational stochastic background it sources. \nIn Ref. [9], the scaling of the inflationary-era SGWB amplitude was predicted to go with the number of fields N as N 4 , under the assumption that the Bogoliubov coefficients' shape in k remains roughly constant and the additional fields enter at the same amplitude. We examine this conjecture in Fig. 3.1, which plots the values of the 2 N 2 = 18 Bogoliubov coefficients for three representative values of ρ = Ω /τ and a fixed √ Ω 2 + τ 2 = 23 . 7. \nAt superhorizon scales, the rapid turning causes both adiabatic and isocurvature modes to be activated, with the latter being more prominent, as seen in Figure 3.1. In order to match the Planck bounds on isocurvature power, the growth is restricted by giving the isocurvature modes a non-zero mass. For M ss, 0 = M bb, 0 = 2, the Bogoliubov coefficients growth is proportional to k -1 (for smaller masses, α ss ∼ k -g with g > 1), and the amplitude depends heavily on ρ , as shown in Figure 3.3. The k -1 momentum-dependent behavior leads to the isocurvature power spectrum being flat for low k because the massive wave functions (2.11) behave as 1 / √ k when k ≪ k f . As the modes move outside the horizon, the generated isocurvature power decreases rapidly. Figure 3.2 illustrates the behavior of the power spectra at three different time points, which is consistent with current constraints on isocurvature modes. \nFigure 3.2 : We show the approximate power spectra P i ( k, N ) = ζ i ( k, N ) 2 ∑ j | α ij ( k ) + β ij ( k ) | 2 at the time of the feature (left), 10 e-folds after (middle), and 23 e-folds after (right) for three values of the turn rate ratio ρ (rows). We plot the out-region estimates of the power spectrum shape, but scale them with the appropriate time dependence of the mode functions (2.21). These approximate power spectra are therefore a poor approximation for the modes that have not yet exited the horizon at the plotted time, regions we have shaded in red in these plots. The isocurvature power spectra decay rapidly outside the horizon. This plot corresponds to Ω 2 f = 50 . 0 , ξ ss = -3 , ξ bb = 2 , δ = 0 . 1. \n<!-- image --> \n<!-- image --> \nFigure 3.3 : This plot shows how the Bogoliubov coefficients vary as a function of ρ = Ω 0 /τ 0 in the interval of values used to plot Fig 3.1, fixing κ = Ω 2 f / 2 , ξ ss = -3 , Ω 2 f = 23 . 7 , δ = 0 . 225. We compare the variety of possible growths with two values of ξ bb . At high ρ (two-field limit), we see the ( ζ , s ) 2 × 2 block of Bogoliubov coefficients reaches approximately the same value. At lower ρ , the coefficients split into a strong hierarchy, with the relative amplitudes set by ξ bb and the ss -coefficients the highest amplitude. \n<!-- image --> \nThe left plot in Fig. 3.1 corresponds to Ω ≫ τ and matches the two-field results of Fig. 3 in [9] in the region the plots overlap. Initial states with momenta larger than 2 k ∗ = 2Ω 2 f k f are not excited and remain in the Bunch Davies vacuum. Below this momentum cutoff, there are always excited states, but how much each field is excited depends on the value of the momenta. In this regime, { α ζζ , β ζζ , α sζ , β sζ , α ζs , β ζs , α ss , β ss } are equally dominant around k ∗ . This gives the 2 4 enhancement with respect to the single field case found in [9] for the two-field case, but the factor of 2 4 does not translate into a N 4 behavior for N fields. Because torsion is subdominant to the turning rate, Bogoliubov coefficients with an index 'b' are subdominant, and hardly contribute to sourcing the SGWB (2.18). \nThe middle plot in Fig. 3.1 corresponds to Ω ∼ τ . As in the previous case, modes with k > 2 k ∗ remain in the initial Bunch Davies vacuum. While there is still a peak around k ∗ , the contributions from the different Bogoliubov coefficients develop a hierarchy. The largest contribution corresponds to { α ss , β ss } , followed by the contribution from { α ζs , β ζs , α sζ , β sζ } which is down by a factor of 2, followed by { α bs , β bs , α sb , β sb , α zz , β zz } down by approximately another factor of 2. Though more Bogoliubov coefficients are activated, there is a hierarchy and the power spectrum can't grow like N 4 . Overall, the maximum value of P t is larger for Ω ∼ τ than for Ω ≫ τ when keeping √ Ω 2 + τ 2 fixed. At superhorizon scales, the growth is dominated by { α ss , β ss } . These two coefficients have the same absolute value as Fig. 3.1 shows and is expected from (2.13). \nThe plot on the right in Fig. 3.1 corresponds to Ω ≪ τ . In this regime, the peak around k ∗ has nearly disappeared. The Bogoliubov coefficients { α ss , β ss } dominate over all others, but with the same small momenta dependence, k -1 . In this limit | α ζX | and | β ζX | are several orders of magnitude smaller than {| α ss | , | β ss |} , which is consistent with the observation in \n[18] that on superhorizon scales the evolution of the adiabatic modes and the isocurvature modes decouple. \nIn the previous paragraph, the results may not be consistent with first-order perturbation theory. In Figure 3.2, we have plotted the power spectrum for both adiabatic and isocurvature modes at three different times: when the feature is produced ( N extra = 0), ten efolds after the feature ( N extra = 10), and twenty-three e-folds after the feature ( N extra = 23), for three different values of the ratio ρ = Ω /τ . The power spectra at late times match experimental data in all cases, but for ρ = 0 . 1, the isocurvature power spectrum at ( N extra = 0) briefly reaches O (1), calling into question the validity of first-order perturbation theory. \nIn the second part of the results, we discuss the shape of the fractional energy density spectrum (Ω GW ). Scalar fluctuations source gravitational waves at two different times: during the radiation-dominated era as the perturbations re-enter the horizon and during inflation. In our previous work [17], we calculated the contribution from gravitational waves generated during the radiation-dominated era, which were caused by adiabatic scalar fluctuations reentering the horizon. The frequency profile of Ω rad GW changes substantially as a function of turning rate over torsion, ρ . When Ω ≫ τ , Ω rad GW peaks at k ∗ , while for Ω ≪ τ the peak has migrated below k f . Said differently, the profile depends on the number of active fields. \nOn the other hand, the profile of Ω inf GW is not greatly affected by the relative ratio of Ω to τ or the effective kinetic masses during the feature. In Figure 2.1, we compare the relative amplitudes and shapes of Ω inf GW as a function of ρ . The background peaks at approximately k f and, for k > k f , oscillates with a frequency 2 /k f . This result is independent of the effective number of acting fields. The amplitude depends on the number of fields, but not simply as a power of the number of active fields N . The actual dependence is more complicated. Figure 3.3 shows how the dominant α ss and β ss grow with the ratio of the turning rate over the torsion, ρ . This continuous variable interpolates between two active fields in the limit of vanishing torsion, and three active fields when the torsion dominates over the turning. The significant growth of the perturbations when ρ ≪ 1 raises questions about the reliability of perturbation theory to compute Ω GW as we discuss in the next section. \nIn Figure 3.4, we compare the variety of inflationary SGWB shapes that we are able to produce in this work. Despite the variations in ρ and the mass parameters leading to fairly variable structures of the excited state (cf. Figures 3.1, 3.3), the inflationary-era SGWBs remain approximately the same shape, at least around the main peak. As we discuss in Section 5, discerning these small differences would require resolving features of the signal at least 10 2 or 10 3 times lower amplitude than the peak, so except in the case of a very loud signal these shapes would be equivalent from a data analysis perspective.", '4 Backreaction': "To detect signals at LISA, the scalar perturbations must be enhanced by at least 10 4 , as illustrated in Fig 3.2. Such a significant enhancement raises doubts about the reliability of perturbation theory and has been a concern of previous work on this subject [7, 9, 2527]. There are two conditions for trusting perturbation theory [28-31]: (i) ensuring that the energy in the perturbations is much smaller than the energy in the background field not to disrupt the background equation of state and (ii) ensuring the reliability of the linear equations for the evolution of the perturbations. Both concerns were addressed in [7, 9] for the two-field equivalent of the model considered in this work and [17] for the three-field case. \nFigure 3.4 : We compare the possible shapes of Ω inf GW from our sharp feature, normalizing by the maximum amplitude to focus on the shapes of the signals. These signals all occur at the same feature scale and share the parameters ξ ss = -3 , δ = 0 . 225 , Ω 2 f = 23 . 7, while we vary the turn rate ratio and the kinetic coupling of the third mass ξ bb . The mass content and turn rate ratio ρ do not strongly affect the shape of the signal in frequency, especially near the peak. The ρ = 0 . 1 , ξ bb = 0 case does show some deviation in the envelope of the signal towards the subhorizon of the feature (high f ). We caution that the largest enhancements we present with ρ = 0 . 1 are likely subject to backreaction corrections. The relative similarity of these signals despite the large variation in the structure of the corresponding excited states leads us to call this SGWB profile 'universal'. \n<!-- image --> \nIt is easy to show that (i) is easily satisfied in the cases studied in this work. According to [28], the contribution of the perturbations to the energy density is: \nρ excited states = 1 a 4 ∑ i ∫ d 3 k (2 π ) 3 k ( | β ζi k | 2 + | β si k | 2 + | β bi k | 2 ) (4.1) \nWhen Ω ≲ τ , it is easy to compute this quantity. In the comparison shown in Fig 3.1, β ss dominates. For the mass choices made in this calculation, | β ss | ∝ 1 /k when k < Ω 2 f k f , and it vanishes when k > Ω 2 f k f . \nρ excited ∼ 3 a 4 k 4 f = 3 H 4 (4.2) \nThe constraint ρ excited ≪ ρ inf is always satisfied as \nρ excited ∼ 3 H 4 ≪ 3 M 2 Pl H 2 (4.3) \nIn the two-field case, [9] showed that the second constraint is always the most stringent. For two fields, the backreaction constraint (i) equals the linearity constraint (ii) times ϵ . \nThe ϵ difference between the two constraints may also explain the outcome of the lattice computations carried out in a different two-field model [32]. In this case, the energy density in the perturbations is approximately 10 -2 ρ inf . Still, the simulation shows that perturbation theory breaks down as the perturbations don't grow as much as expected in perturbation theory. Their effect is to shift the background to a new classical configuration. Because in [32] ϵ ≲ 10 -2 , the analysis presented in [7, 9] leads to the expectation that constraint (ii) is not satisfied even if (i) is. Non-perturbative effects have been also shown to be important in a single-field inflation model with a departure from slow-roll [33]. \nIn the three-field case, the constraints analysis performed in [17] only considered the enhancement in the adiabatic power spectrum as this was the only one to survive until radiation-dominated re-entry. But Fig. 3.2 shows that the isocurvature power spectrum is several orders of magnitude larger at the time of the feature, for example, when ρ = 0 . 1, P s ∼ O (1), and, hence, O ( Q s /M Pl ) ∼ 1. This behavior points to a brief breakdown of perturbation theory because the cubic Lagrangian [19] contains terms such as \nL 3 ⊃ a √ 2 ϵ Ω M Pl Q s ( ∂ζ ) 2 \nwhich is larger, for a brief time around the feature, than the quadratic term \nL 2 ⊃ aϵM Pl ( ∂ζ ) 2 \nWe therefore caution that the largest enhancements we present in this work with short periods of P S ∼ 1 would experience strong corrections from backreaction effects in full lattice simulations 1 . The brief high amplitude of the isocurvature power spectra around the time of the feature would likely be dampened, leading to less power in gravitational waves. Nonetheless, the results we present at lower enhancements remain valid and experience only small corrections from backreaction.", '5 Phenomenology': "In this section we discuss the observable consequences of sharp features during inflation, and how they impact the possible cosmological observables at CMB scales, LSS, and of course a possible SGWB. We also comment on the detectability of the observable probes, and how likely a possible detection is to well-constrain the sourcing feature. \nThe toy model of sharp feature we study in this work generates a linear in k oscillatory enhancement in the adiabatic power spectrum and a similar-magnitude temporary enhancement in the isocurvature power spectra, which decay to unobservability within a few e-folds when they have positive masses. When sufficiently strong, these features can generate radiation-era and inflationary-era SGWBs. But observing the effects of a sharp feature in any particular cosmological probe is very dependent on when exactly it occurs during inflation, or equivalently the value of the scale of the feature k f . Although not the focus of this work, we will briefly comment on non-GW observables from inflationary sharp features. Current constraints on the primordial adiabatic power spectrum rule out large enhancements at CMB scales, but perhaps sufficiently low-amplitude features could be visible at CMB scales as 'primordial clock' oscillations in the adiabatic power spectrum [34]. The current constraints from LSS place many limits on the primordial adiabatic power spectrum at scales \nsmaller than the CMB, including not disrupting baryogenesis P ζ ( k ) ≲ 10 -2 , and constraints from structure formation (see [35, 36] for a recent review). Otherwise, LSS allows for large scalar perturbations at small scales, which can source the radiation-era SGWB we studied in part I of this work [17]. We note that these cosmological probes also constrain and limit isocurvature power - the features we study do not predict any observable isocurvature, as the brief spike in isocurvature decays with a small positive mass long before undergoing reheating and contributing to either the CMB or LSS. Non-gaussianities will be another constraining observable. They can be significant in multifield models with sustained turning [37-40] and large-scale features [41]. A recent computation [42] found them to also be significant for small-scale large features. Although the models in this analysis differ from those presented in this work, we expect their result to hold. \nGravitational waves provide perhaps the most undistorted probe of the inflationary era, and the inflationary-era SGWB is uniquely sensitive to the entire evolution of the perturbations, even transients that do not leave a signature in the perturbations at the end of inflation. The signal we study in this work (cf. Figure 2.1) fits a roughly broken powerlaw shape, with numerically measured powerlaws of k 3 on the lowf tail and k -3 to k -4 fitting the first 3-4 peaks, and k 1 . 7 and k -8 fitting just the most prominent peak. The signal has O (10%) oscillations linear in k beginning at the peak and with frequency ω ∼ 2 /k f . The shape of the signal is surprisingly largely independent of the mass parameters (2.7) and even the torsion to turn rate ratio ρ for fixed Ω 2 f (cf. Figure 3.4). The amplitude, however, rises quite steeply as the torsion increases (see the growth of the Bogoliubov coefficients in Figure 3.3 as ρ → 0). We numerically measure Ω GW ∝ τ 24 (!) in the regime of steepest growth around ρ = 1 for a fixed Ω 2 f and ξ bb = 2, although we caution that the largest enhancements are almost certainly subject to backreaction corrections. The envelope of the inflationary-era SGWB at high k does begin to change at sufficiently high τ , but the peak remains unaffected. Because the shape of the inflationary-era SGWB near its peak does not depend on the details of the feature (and as argued below, this generalizes to more realistic models), we call it a universal signature of sharp features in multi-field inflation . Though a similar universal phenomenology was claimed in Ref. [9], they assumed the Bogoliubov coefficients were strongly peaked and approximately equal magnitude. We have found that the excited state takes on a significantly different form with a strong hierarchy in the coefficients (cf. Figure 3.1), and that the amplitude of the SGWB depends strongly on the entropic sector of the perturbations with growth that vastly outpaces the predicted N 4 . This enhancement is so strong that it opens a new window of observable parameter space: we may source a detectable inflationary-era SGWB without requiring a large enhancement in the adiabatic power spectrum. \nIf such a signal were detected, its shape would immediately reveal the feature scale from the oscillation frequency 2 /k f , and confirm that an excited state occurred during inflation in the early universe. Interpreting the signal in terms of multi-field dynamics would be difficult due to its somewhat universal nature, unless the radiation-era signal were also visible (cf. Figure 2.1). Because the radiation-era feature is only sourced by the adiabatic power spectrum, the ratio of the two amplitudes gives information about the effective amplitude of the turn Ω 2 f and the number of dynamically contributing fields. Parameter estimation studies on the recovery of templated SGWBs with similar features to the one presented here are available in [13, 43]. The equation of state of the universe could also modify the post-inflationary scalar-induced portion of the signal if different than radiation [44]. \nOf course physical inflationary backgrounds are quite unlikely to generate features ex- \ntremely similar to the top hat we study (2.6), though isolated single turns with a smoother profile can occur in some backgrounds [15]. One common source of sharp features in multifield inflation occurs when the trajectory becomes unstable (e.g., due to a sufficiently tachyonic entropic mass) 2 and the fields change direction abruptly. Several classes of models in the literature fit this description including waterfall mechanisms [45] and geometric destabilization [46]. Some of the present authors also saw similar features in a O (100)-field, random potential inflation model as the trajectory approached saddle points [47]. Multi-field models often give rise to attractor behavior in field space and sharp features can even lie on the attractor, as seen in [48, 49]. \nReferences [8, 10] studied several such two-field models that produce observable signals in LISA - these features typically are not one top hat as we've studied here, but a series of sharp turns with decaying amplitude, although the observed power spectra and radiation-era SGWBs (inflationary-era SGWBs are not studied in their work) are qualitatively very similar to the two-field limit of ours. Although we consider a full investigation outside the scope of this work, we have generalized some of the models in [8] to three fields and have found that the features' behavior persists, and we see decaying oscillations in both turn rate and torsion during the trajectory's realignment. Reference [16] has also studied three-field backgrounds with more broad features or constant turning and found large enhancements. \nThe features we study here, then, are by no means an exact quantitative prediction for a physical model's SGWB, but are qualitatively similar. It remains an open question how universal the shape we have found is to physical backgrounds with sharp features, but we suspect they will be qualitatively very similar. The Green's functions and kernel in (2.18) are fixed by the physics and set the behavior of the background at low- and highk . And despite a very different structure of excited state when τ ≫ Ω with a top hat, we find a similar shape of SGWB. \nSimilarly, we might speculate on how more dynamical fields might affect the SGWB. We do not have a strong expectation for the structure of the excited state, but the above arguments about the Green's function and kernel still apply so we expect the qualitative nature of the SGWB to remain the same. Because the entropic sector of the perturbations can experience a large enhancement during the feature and all fields contribute to the SGWB (2.18), all else equal, we expect more fields to increase the amplitude of the inflationary-era SGWB compared to the radiation-era one, and allow for a LISA-detectable SGWB without large scalar perturbations. The radiation SGWB can also be relatively suppressed depending on the duration of the feature [9], so its absence is not a smoking gun for more dynamical fields.", '6 Conclusions': "In this work we have completed the inflationary-era contribution of the stochastic gravitational wave background sourced by a sharp turn during three-field inflation, extending our previous work computing the radiation-era contribution in [17]. These two contributions together complete the calculation of the expected SGWB signal from a sharp feature up to tree-level in the perturbations. \nWe find that the calculation is not well-controlled at large scales without including a constant term in the entropic masses, so we recalculate the Bogoliubov coefficients of \nthe excited state including these masses. Due to the difficulty of this calculation and the unwieldiness of these coefficients, we do not present them in this text but have uploaded them as supplementary material in both Julia and Mathematica code so that further study is easily available to the community. They are available at https://github.com/rjrosati/ 3field-sharp-feature . \nCompared to the two-field case studied in [9], we find that the excited state has a very different structure for the modes at scales near the feature scale and in its superhorizon. As seen in Figure 3.1, as the torsion increases, the well-resolved peak in k vanishes, giving way to an approximately 1/ k powerlaw growth towards the superhorizon. Similarly, the approximately equal amplitudes of the Bogoliubov coefficients split into a strong hierarchy at high torsion, with the dominant coefficients chosen by the mass parameters. \nThis structure of the excited state results in a large enhancement in the inflationary-era contribution to the SGWB compared to the radiation-era one. Surprisingly, the frequency of oscillations and envelope approximately match the two-field signal unless ρ = Ω /τ ≪ 1 (several such spectra are visible in Figure 2.1). This state generates a large enhancement in the isocurvature perturbations in the feature's superhorizon, which then rapidly decays before the end of inflation. The inflationary-era SGWB, then, is a probe of the physical dynamics during inflation, unlike the radiation-era signal that only depends on the modes at its end. \nFrom a phenomenological perspective, the inflationary-era SGWB frequency profile, especially near its peak, is almost independent of the nature of the sharp feature, despite the very different structure of the excited state when torsion is present. The amplitude of the inflationary-era signal, however, is greatly enhanced compared to the radiation-era signal. The large transient growth and decay in the isocurvature modes would otherwise be unobservable after the end of inflation, but in our case is reflected in the enhancement of the inflationary-era signal. When the feature is not too strong, we open a new window of parameter space to detectability as only small adiabatic enhancements are necessary to source a detectable inflationary-era SGWB. \nHowever when the turn is strong and the maximum Q s /M Pl ∼ 1, this enhancement can be large enough to doubt perturbation theory. Lattice simulations have found backreaction plays an important role in geometrical destablilization and several other situations, although to our knowledge no such simulations have studied sharp features that correspond to brief violations of perturbation theory. We suspect that these corrections will further affect the shape of the inflationary-era SGWB and likely reduce the effective amplitude of the isocurvature perturbations when linear theory predicts them to be large. We stress, however, that due to the integrated nature of the signal (cf. (2.18)) even brief transient growths during inflation can be visible in the inflationary-era SGWB. \nThe background dynamics in (2.6) are likely unrealistic, and sharp features from concrete models (e.g. those in [8, 10]) often contain several subsequent peaks in the turn rate, affecting the structure of the feature and some of the details of its phenomenology. Note that the SGWBs from broad features and constant turns in three-field models have been studied in [16]. \nIt is our hope that future work will solve these issues, so that any future SGWB search can have accurate predictions for a wide array of inflationary features.", '7 Acknowledgments': "We would like to thank A. Ach'ucarro, P. Christodoulidis, J. Fumagalli, E. McDonough, G. Palma, L. Pinol, S. Renaux-Petel, K. Turzynsky, and I. Zavala for their valuable comments and discussions after presenting this work at the following workshops: Simons Center's 'Multifield Cosmology: Inflation, Dark Energy and More', MIAPbP's 'Quantum Aspects of Inflationary Cosmology', and ESI's 'The Landscape vs the Swampland.' RR is supported by an appointment to the NASA Postdoctoral Program at the NASA Marshall Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. The work of VA and SP was supported in part by the National Science Foundation under Grant No. PHY- 2210562.", 'References': "- [1] M. Maggiore, Gravitational wave experiments and early universe cosmology , Physics Reports 331 (2000) 283-367.\n- [2] J. D. Romano and N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: a unified treatment , Living Rev. Rel. 20 (2017) 2 [ 1608.06889 ].\n- [3] G. Dom'enech, Scalar Induced Gravitational Waves Review , Universe 7 (2021) 398 [ 2109.01398 ].\n- [4] M. S. 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2024ApJ...975...64G

The idea of composite dark energy DE is quite natural since on general grounds we expect that the vacuum energy density associated with the cosmological term may appear in combination with other effective forms of DE denoted X. Here we deal with model wXCDM a simplified version of the old XCDM model and exploit the possibility that X behaves as phantom matter PM which appears in stringy versions of the running vacuum model RVM. Unlike phantom DE the PM fluid satisfies the strong energy condition like usual matter hence bringing to bear positive pressure at the expense of negative energy. Bubbles of PM may appear in the manner of a transitory phantom vacuum tunneled into the late Universe before it heads toward a new de Sitter era thereby offering a crop field for the growing of structures earlier than expected. Using Type Ia supernovae cosmic chronometers transversal baryon acoustic oscillations BAO 2D largescale structure data and the full cosmic microwave background likelihood from Planck 2018 we find that the H SUB0SUB and growth tensions virtually disappear provided that BAO 2D are the only source of BAO data used in the fit. In contrast our preliminary analysis using exclusively anisotropic BAO BAO 3D indicates that the ability to ease the H SUB0SUB tension is significantly reduced as compared to the scenario with BAO 2D despite the fact that the overall fit to the cosmological data is still better than in the CDM. Finally our approach with BAO 2D favors quintessencelike behavior of the DE below z 1.5 at 3 confidence level which is compatible with the recent DESI measurements.

2024-11-01T00:00:00Z

['2024ApJ...975...64G', '10.3847/1538-4357/ad7a62', '10.48550/arXiv.2404.18845', 'arXiv:2404.18845', '2024arXiv240418845G']

['Cosmology', 'Cosmological models', 'Cosmological parameters', 'Cosmological evolution', 'Dark energy', 'Large-scale structure of the universe', '343', '337', '339', '336', '351', '902', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology', 'High Energy Physics - Theory']

Phantom Matter A Challenging Solution to the Cosmological Tensions

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

https://arxiv.org/pdf/2404.18845.pdf

{'Phantom matter: a challenging solution to the cosmological tensions': "Adri'a G'omez Valent 1 and Joan Sol'a Peracaula 1 \n1 \nDepartament de F'ısica Qu'antica i Astrof'ısica and Institute of Cosmos Sciences, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain", 'ABSTRACT': "The idea of composite dark energy (DE) is quite natural since on general grounds we expect that the vacuum energy density (associated with the cosmological term Λ) may appear in combination with other effective forms of DE, denoted X . Here we deal with model w XCDM, a simplified version of the old ΛXCDM model (Grande et al. (2006)), and exploit the possibility that X behaves as 'phantom matter' (PM), which appears in stringy versions of the running vacuum model (RVM). Unlike phantom DE, the PM fluid satisfies the strong energy condition like usual matter, hence bringing to bear positive pressure at the expense of negative energy. Bubbles of PM may appear in the manner of a transitory 'phantom vacuum' tunneled into the late universe before it heads towards a new de Sitter era, thereby offering a crop field for the growing of structures earlier than expected. Using SNIa, cosmic chronometers, transversal BAO (BAO 2D), LSS data and the full CMB likelihood from Planck 2018, we find that the H 0 and growth tensions virtually disappear, provided that BAO 2D are the only source of BAO data used in the fit. In contrast, our preliminary analysis using exclusively anisotropic BAO (BAO 3D) indicates that the ability to ease the H 0 tension is significantly reduced as compared to the scenario with BAO 2D, despite the fact that the overall fit to the cosmological data is still better than in the ΛCDM. Finally, our approach with BAO 2D favors quintessence-like behavior of the DE below z ≃ 1 . 5 at ≳ 3 σ CL, which is compatible with the recent DESI measurements. \nKeywords: stars: Cosmology - Cosmological models - Cosmological parameters - Cosmological evolution - Dark energy - Large-scale structure of the universe", '1. INTRODUCTION': "The standard (or concordance) cosmological model, aka ΛCDM, has been a rather successful paradigm for the description of the universe for more than three decades (Peebles 1993), especially since the late nineties (Turner 2022). Despite it constitutes the main theoretical pillar at our hands for the description of the universe's evolution within the General Relativity (GR) context, the ΛCDM is being pestered by a number of glitches and hitches which prove to be more and more difficult to iron out. While the model is presumably right in the main (at least at a pure phenomenological level), an increasing number of worrisome inconsistencies (or ' tensions') have been perturbing its reputation and its future prospects. For a long time the role played by a rigid cosmological constant (CC) Λ in the standard ΛCDM model has been rather successful since it has provided a fairly reasonable description of the overall cosmological data (Ade et al. 2016; Aghanim et al. 2020). But the use of the cosmological term has \nnever been fully clarified at the formal theoretical level since it is usually associated to the so-called 'cosmological constant problem', a serious conundrum which has been amply discussed in the literature, see e.g. (Weinberg 1989; Peebles & Ratra 2003; Padmanabhan 2003; Sol'a Peracaula 2022; Sol'a 2013) and references therein. Inasmuch as the vacuum energy density (VED) is related to Λ through ρ vac = Λ / (8 πG N ) ( G N being Newton's gravitational coupling), the conceptual fate of Λ is tied to our ultimate understanding of the VED on fundamental physical terms. Fortunately, recent theoretical developments on the VED in quantum field theory (QFT) are bringing new light for a potential alleviation of these theoretical difficulties. In fact, the possibility that the quantum vacuum (and hence that Λ itself) is actually dynamical (i.e. evolving with the cosmological expansion) rather than being stuck at a rigid value, has lately been substantiated in the context of QFT in curved spacetime (Sol'a Peracaula 2022; Sol'a 2013) as well as in the framework of low-energy effective string theory (Mavromatos & Sol'a Peracaula 2021a,b) \nand, very recently, in lattice quantum gravity (Dai et al. 2024). For a long time, the dynamics of the VED has been popularly addressed on phenomenological grounds using ad hoc scalar fields (quintessence and the like) which supplant the role of the Λ-term through the current value of some suitable effective potential (Peebles & Ratra 2003; Padmanabhan 2003), see e.g. (Avsajanishvili et al. 2024) for a recent review. Currently, the first data release of the Dark Energy Spectroscopic Instrument (DESI) suggests tantalizing evidence that the dark energy (DE) might be dynamical using some common parameterizations of the DE (Adame et al. 2024). More detailed analyses will be needed before getting a final confirmation, of course, but it is a fact that some anticipatory (and fairly robust) hints of dynamical DE in the literature already pointed out this possibility a few years ago from different perspectives and using a significant amount of cosmological data. The level of evidence put forward by these anticipatory studies was substantial and ranged 3 -4 σ . Some of these analysis are well-known (Sol'a et al. 2015, 2017; Sol'a Peracaula et al. 2017; Sol'a Peracaula et al. 2018a,b; G'omez-Valent & Sol'a Peracaula 2018) and involved the running vacuum model (RVM) 1 . Subsequent analysis around the same time supported also this possibility using different methods and parameterizations (Zhao et al. 2017; Sol'a Peracaula et al. 2019). \nDynamical DE could also impinge positively on the resolution of the cosmological tensions. Let us recall that these are mainly concerned with the measurement of the current Hubble parameter H 0 ≡ 100 h km/s/Mpc and the growth of large scale structures (LSS). The latter is usually monitored with S 8 or σ 8 , or even better using a parameter σ 12 defining the amplitude of the matter power spectrum at fixed spheres of radius 12 Mpc rather than 8 h -1 Mpc, thus avoiding artificial dependence on the value of h (S'anchez 2020; Semenaite et al. 2022, 2023) 2 . The first kind of tension involves a serious disagreement between the CMB observations, using fiducial ΛCDM cosmology, and the local direct (distance ladder) measurements of the Hubble parameter today. It is arguably the most puzzling open question and it leads to a severe inconsistency of ∼ 5 σ c.l. between the mentioned observables. The second kind of tension is related with the exceeding rate of large scale structure formation in the late universe predicted by the ΛCDM \nas compared to measurements, although the discordance here is moderate, at a confidence level of ∼ 2 -3 σ . More recently, data from the James Webb Space Telescope (JWST)(Gardner et al. 2006; Labb'e et al. 2023) have revealed the existence of an unexpectedly large population of extremely massive galaxies at large redshifts z ≳ 5 -10, a fact which is also strongly at odds with the prospects of the concordance model. \nThere is a wide panoply of strategies in the literature trying to mitigate some of the above tensions, although at present we may be still far away from a satisfactory resolution of the situation. We mention only a few. It has been argued that within the class of models where the DE is dealt with as a cosmic fluid with equation of state (EoS) w ( z ), solving the H 0 tension demands the phantom condition w ( z ) < -1 at some z , whereas solving both the H 0 and σ 8 tensions requires w ( z ) to cross the phantom divide and/or other sorts of exotic transitions, see e.g. (Heisenberg et al. 2022; Marra & Perivolaropoulos 2021; Alestas et al. 2021; Perivolaropoulos & Skara 2021; Alestas et al. 2022; Perivolaropoulos & Skara 2022b; G'omez-Valent et al. 2024). The possibility of a sign flip of the Λ term has been entertained in recent times, e.g. in (Calder'on et al. 2021). Let us also mention the model analyzed in (Akarsu et al. 2021, 2023) (based on the framework of Akarsu et al. (2020)), in which one considers a sudden transition from anti-de Sitter (AdS), hence Λ < 0, into de Sitter (dS) regime (Λ > 0) occurring near our time. Despite it being essentially ad hoc, the model yields a rather good fit to the data when transversal/angular BAO (BAO 2D for short) is employed in the fitting analysis, see also (G'omez-Valent et al. 2024). Actually, a more general framework already existed since long ago, namely the ΛXCDM model(Grande et al. 2006, 2007, 2009), where the running vacuum and an extra X component can exchange energy. In such a context, one can have Λ < 0 or Λ > 0. \nIn this work, we further exploit the virtues of the ΛXCDM, which is an enhanced family of RVM's(Sol'a Peracaula 2022). The linchpin of the RVM framework is the dynamical nature of the VED framed in the fundamental context of QFT in curved spacetime or in low-energy string theory. Appropriate renormalization in this context gets rid of the quartic mass terms ∼ m 4 responsible for the fine tuning troubles, and as a result the VED becomes a smooth function of even powers of the Hubble rate (Moreno-Pulido & Sol'a 2020; Moreno-Pulido & Sol'a Peracaula 2022; Moreno-Pulido &Sol'a Peracaula 2022; Moreno-Pulido et al. 2023). This leads to a fairly good description of the cosmological data and helps easing the tensions (Sol'a Peracaula et al. \n2021, 2023). Worth noticing is also the phenomenological performance of the stringy version of the RVM (G'omez-Valent et al. 2023). This said, we cannot exclude that the DE may be a composite fluid, a possibility which could further help in the task of alleviating these tensions. For this reason a good candidate is the aforementioned 'ΛXCDM model'(Grande et al. 2006, 2007, 2009), a RVM-born model which was initially motivated as a possible cure for the cosmic coincidence problem. In it, we have two DE components: one is the running VED, ρ vac = ρ vac ( H ), and the other is the mentioned X . As a matter of fact, the entity X need not be a fundamental field, as emphasized in (Grande et al. 2006): it may just be due to particular terms in the effective action which mimic a dynamical DE component. In this sense, X can have phantom-like behavior without causing uproar at the theoretical level. Not only so, X may even display 'phantom matter' (PM) behavior (Grande et al. 2006), namely an intriguing form of DE which, in stark contrast to the usual phantom DE, is characterized by a positive pressure ( p X > 0) at the expense of a negative energy density ( ρ X < 0). It is remarkable that there exists specific theoretical scenarios in the current literature which support the ΛXCDM structure with a PMlike component X . An example appears in the stringy RVM context (Mavromatos & Sol'a Peracaula 2021a,b). No less remarkable is the fact that the phenomenological performance attained by such a peculiar composite DE scenario may surpass by far that of the standard ΛCDM under appropriate conditions, as we shall shortly clarify. However, for the sake of simplicity, in this paper we shall address a reduced version of the ΛXCDM, which we call the w XCDMmodel (not to be confused with the conventional w CDMparameterization), in which the additional X component of the ΛXCDM is kept intact, but on the other hand we mimic the running Λ with another dynamical component which we call Y . While the analysis within the full ΛXCDM will be presented elsewhere, we shall nonetheless demonstrate here that even the simplified w XCDM model can be very efficient in dealing with the cosmological tensions. However, for this to be so the following two conditions must be met in our analysis: i) the X component must have the ability to behave as phantom matter; and ii) BAO 2D must be the only source of BAO data used in our global fit. The precise reasons for considering only this type of BAO data in the current work will be explained below, together with a preliminary discussion of the results obtained with the BAO 3D variant. The upshot is that by accepting these reasons for using BAO 2D only, and also taking advantage of the mentioned (stringy) theoretical framework to accommodate phantom matter, the possibility to quell \nthe cosmological tensions on fundamental grounds is viable and can be extremely effective.", '2. COMPOSITE DARK ENERGY': "We next mention three related types of composite models of the DE: i) ΛXCDM, ii) w XCDMand iii) Λ s CDM. As noted, model i) exists in different versions since long ago (Grande et al. 2006, 2007, 2009); model ii) will be analyzed here for the first time, it constitutes a simplified version of model i) and embodies the PM feature. Finally, model iii) was recently analyzed in (Akarsu et al. 2023). \n- · i) ΛXCDM model. Its definition and comprehensive discussion, including the background cosmological solution, are provided in utmost detail in (Grande et al. 2006). In addition, in (Grande et al. 2009) the corresponding cosmic perturbations equations are fully accounted for. Here we just present a very short qualitative description. Within the ΛXCDM, the cosmic fluid contains the usual matter energy density ρ m and a composite DE sector made out of two components, one is the running vacuum energy density ρ vac and the other is called X , with energy density ρ X . The VED here is treated within the QFT framework of the RVM(Moreno-Pulido & Sol'a 2020; Moreno-Pulido & Sol'a Peracaula 2022; Moreno-Pulido & Sol'a Peracaula 2022; Moreno-Pulido et al. 2023), in which ρ vac = ρ vac ( H ) evolves with the expansion rate. In the current universe, such an evolution reads (Sol'a Peracaula 2022): \nρ vac ( H ) = ρ 0 vac + 3 ν eff 8 π ( H 2 -H 2 0 ) m 2 Pl , (1) \nwith ρ 0 vac the current VED value, m Pl the Planck mass and | ν eff | ≪ 1 a small parameter which is formally computable in QFT, see the aforementioned papers. For ν eff > 0 the VED decreases with the expansion and hence the RVM mimics quintessence, whereas for ν eff < 0 the VED increases with the expansion and the RVM behaves effectively as phantom DE. The measured cosmological term is Λ = 8 πGρ vac , and hence Λ also runs with H since ρ vac = ρ vac ( H ). Such a running feature of the VED and Λ occurs in the ΛXCDM too, and in exactly the same form (1), but here we have also the dynamics of X and hence the cosmological solution of the model in terms of the redshift variable is considerably more complicated (Grande et al. 2006). Besides, as recently demonstrated in the QFT context, ρ vac has an EoS which departs from -1 (Moreno-Pulido & Sol'a Peracaula 2022), so the 'modern version of the ΛXCDM model' has actually two nontrivial EoS parameters, one for the running VED and the other for X , which we call ω X . These are basic ingredients in our analysis, and \nfor this reason the ΛXCDM fleshes out the theoretical basis inspiring the current work. However, the actual scenario which we will analyze here is model w XCDM (see below). It is simpler than the ΛXCDM but it has the same genetic ingredients. We will use it to emulate the basic properties of the latter. The analysis of the cosmic tensions within the full ΛXCDM framework is more demanding and will be presented in a separate work (G'omez-Valent & Sol'a Peracaula 2024). \n- · ii) w XCDMmodel. In order to illustrate the possibilities of our composite DE scenario in connection with the cosmological tensions, herein we will use the w XCDMmodel, a reduced (skeleton) version of the ΛXCDM. The two models share the X component, but w XCDMmimics the running vacuum feature of ΛXCDM through another dynamical component Y (replacing Λ) and whose EoS, w Y , can be different from -1. This means that Y does not act as a rigid CC, which is fair enough since, as mentioned before, in the RVM context the VED is dynamical and its EoS (hence that of Λ) actually departs from -1 owing to quantum effects (Moreno-Pulido & Sol'a Peracaula 2022). Thus, in the w XCDM, we have two DE components with potentially different EoS behaviors. Furthermore, as we will see in section 5, in the best-fit model the X component behaves phantom-like ( ω X ≲ -1) and the Y component behaves quintessence-like ( ω Y ≳ -1). An important point is that X and Y do not act simultaneously: X acts first in the cosmic expansion, and Y acts subsequently. To be precise, X enters only above a transition redshift z > z t (fitted from the data) whilst Y enters below that redshift until the current time. No less crucial is the fact that the phantom-like component X actually behaves as PM, therefore with negative energy density (Ω X = ρ X /ρ c < 0) and positive pressure ( p X > 0), whereas for the Y component we have Ω Y > 0, p Y < 0. Notice that the characteristic free parameters of model w XCDM are just ( z t , w X , w Y ). Indeed, the density parameters for the DE components are not free since e.g. the value of Ω 0 Y ≡ Ω Y ( z = 0) depends on the fitting values of H 0 , ω b , ω dm (cf. Table 1). In addition, the respective values of Ω X and of Ω Y immediately above and below z t are assumed to be equal in absolute value. This means that | Ω X ( z ) | = Ω Y ( z ) at z = z t . This kind of assumption aims at reducing the number of parameters and is entirely similar to the one made in the Λ s CDM model, see point iii) below. \nLast, but not least, we note that the existence of these two different phases of the DE separated by the redshift z t is not just an ad hoc assumption since it is found in theoretical contexts such as the stringy RVM approach (Mavromatos & Sol'a Peracaula 2021b), which \npoints to the existence of transitory domains or bubbles of PM whenever the universe is approaching a de Sitter epoch (see next sections for more elaboration). \n- · iii) Λ s CDM model. This is the model recently analyzed with considerable phenomenological success in (Akarsu et al. 2023). Strictly speaking, it is not a composite model since only Λ is involved, although it enters with two signs and in this sense it is a composition of two phases of Λ separated also by a transition redshift z t , which is the characteristic parameter of this model. It is assumed that at that point there is a sudden AdSdS transition from -Λ < 0 in the upper redshift range to +Λ > 0 in the lower range, i.e. an abrupt change of sign of Λ at z = z t , but keeping the same absolute value. We refer the reader to the quoted reference for more details. \nFor convenience, we shall simultaneously provide the fitting results for models ii) and iii) under the very same data sources. Together with the standard ΛCDM model, the simultaneous analysis of Λ s CDM will be useful as a benchmark for rating the performance of the phantom matter approach w XCDM to the cosmological tensions.", '3. PHANTOM MATTER AND THE ENERGY CONDITIONS OF THE COSMIC FLUIDS': "Even though phantom matter was first proposed phenomenologically in the old composite cosmological model ΛXCDM(Grande et al. 2006, 2007, 2009), more recently it has been put forward on more formal grounds in (Mavromatos & Sol'a Peracaula 2021b), specifically within the context of the stringy RVM approach, see (Mavromatos & Sol'a Peracaula 2021a) for a review. Obviously, we need not provide technical details here and we refer the reader to the aforesaid references. However, we can at least describe the conceptual design of this picture. In a nutshell, it is the following: in a framework of a string-inspired cosmology with primordial gravitational waves and gravitational anomalies, which lead to dynamical inflation of RVM type without the need for ad hoc inflaton fields (Sol'a & G'omezValent 2015; Mavromatos & Sol'a Peracaula 2021a), a crucial role is played by the fundamental axion field existing in the gravitational multiplet of string theory, viz. the Kalb-Ramond (KR) axion field b ( x ), which couples to the gravitational Chern-Simons (gChS) term: ∼ b ( x ) R µνρσ ˜ R µνρσ ≡ b ( x ) R ˜ R , where ˜ R denotes the dual of the Riemann tensor. The KR axion, when combined with the gChS contribution, obey together a peculiar equation of state of vacuum type p = -ρ but with negative energy density ρ < 0 (hence with positive pressure p > 0), which defines the 'phantom vacuum' (Mavromatos & Sol'a Peracaula 2021b). This form of vacuum, \nhowever, is merely transitory until the gChS condensates ⟨ R ˜ R ⟩ become activated through the condensation of primordial gravitational waves (GW), thereby making possible a positive-definite vacuum state energy with effective cosmological 'constant' Λ( H ) ∼ ⟨ bR ˜ R ⟩ . At this juncture, the total pressure and density already involve the combined effect from all the contributions, what makes possible a normal vacuum state with ρ total > 0 and hence with p total = -ρ total < 0, i.e. a standard de Sitter phase with positive energy density and negative pressure. The latter nevertheless may still be subdued to corrections owing to quantum effects. Interestingly enough, in such a framework the KR axion can be a candidate to Dark Matter, what provides an additional motivation for this cosmological picture since it could embrace the entire cosmic history. \nThus, a phantom-matter dominated era can be present in the early cosmic evolution near a de Sitter phase, but we should emphasize that it could also reflourish in the late universe. As shown in (Basilakos et al. 2019, 2020a,b), owing to the generation of chiral matter, which will dominate the universe at the exit from inflation, at large scales the universe can recover its FLRW background profile. This is because the chiral matter fields insure the cancellation of the gravitational anomalies during the FLRW regime, where they are indeed not observed as otherwise it would imply a glaring violation of general covariance. However, as soon as we are in the process of exiting the FLRW regime towards a new de Sitter era, we cannot exclude that in successive stages of the cosmic evolution the universe can be affected by the presence of lurking PM. The reason is that the chiral matter fields will get more and more diluted with the cosmic expansion and, then, owing to the incomplete cancellation of the gravitational anomalies, these may eventually re-surface near the current quasi-de Sitter era. It is therefore not inconceivable to think of the existence of phantom matter bubbles or domains tunneling now and then into our universe at relatively close redshift ranges before the ultimate de Sitter phase takes over. These bubbles of PM, being endowed with positive pressure p > 0, could obviously foster a larger rate of the structure formation, a fact which might explain the overproduction of large scale structures at unexpected places and times deep in our past. As we shall next show, this ideology provides an excellent framework for a fit to the overall cosmological data which proves to be (far) better than that of the standard ΛCDM model, provided that we exclusively focus on BAO 2D data in our fit. We refer once more the reader to (Mavromatos & Sol'a Peracaula 2021b) and the review (Mavromatos & Sol'a Peracaula 2021a) for the details underlying the \nFigure 1. EoS diagram for the energy conditions of the cosmic fluids. The quintessence (Q) region ( -1 < w < -1 / 3 with ρ > 0, p < 0) is marked cross-hatched, and the conventional phantom region ( w ≤ -1 with ρ > 0, p < 0) is the blue sector indicated as P. The gray-dotted area corresponds to the peculiar 'Phantom Matter' (PM) region: w < -1 with ρ < 0 and p > 0. Notice that PM satisfies the strong energy condition: ρ + p ≥ 0 , ρ +3 p ≥ 0 (all of the gray area) but not the weak energy condition: ρ ≥ 0 , ρ + p ≥ 0 (shaded area, except the P and PM sectors). The DE component X in our analysis behaves as PM, whereas the Y component behaves as quintessence. The EoS line w = -1 marks off the classical vacuum. The RVM fulfills this EoS only approximately near our time owing to quantum effects (MorenoPulido & Sol'a Peracaula 2022). See also (Grande et al. 2006), and (Mavromatos & Sol'a Peracaula 2021b) for the modern developments on PM. \n<!-- image --> \nPMscenario supporting our proposal for solving the tensions. See also (Mavromatos 2022) and the recent developments (Dorlis et al. 2024; Mavromatos et al. 2024). Here we have limited ourselves to a very succinct exposition of the theoretical background and in the rest of this work we focus on exploring the phenomenological implications. \nBefore closing this section and for the sake of a better contextualization of the PM option within the class of energy conditions, in Fig. 1 we show a few of the most common possibilities for the EoS of the cosmic fluids. In it, we particularly highlight the phantom DE (labelled P, in blue) and the phantom matter (PM) regions. The latter (marked gray-dotted) is far away from the usual phantom DE, it actually lies in its antipodes! These two phantom-like possibilities, both satisfying w < -1, are therefore dramatically different and must not be confused. It should also be emphasized that while phantom DE has been amply considered in the literature to describe different features of the DE, including a possible explanation for the H 0 and growth tensions in different frameworks (see the Introduction and references therein), in our approach phantom DE is not used at all. In point of fact, only PM is singled out in an optimal \nway within our specific proposal for a possible resolution of the cosmological tensions.", '4. DATA AND NUMERICAL ANALYSIS': "Let us now come back to the three composite DE models mentioned in Section 2. Leaving model i) for a separate study (G'omez-Valent & Sol'a Peracaula 2024), in this work we constrain the DE models ii) and iii) making use of the following cosmological data sets: \n- · The full Planck 2018 CMB temperature, polarization and lensing likelihoods (Aghanim et al. 2020).\n- · The SNIa contained in the Pantheon+ compilation (Brout et al. 2022), calibrated with the cosmic distance ladder measurements of the SH0ES Team (Riess et al. 2022). We will refer to this data set as SNIa+SH0ES for short.\n- · 33 data points on H ( z ) in the redshift range z ∈ [0 . 07 , 1 . 97] from cosmic chronometers (CCH) (Jim'enez et al. 2003; Simon et al. 2005; Stern et al. 2010; Zhang et al. 2014; Moresco et al. 2012; Moresco 2015; Moresco et al. 2016; Ratsimbazafy et al. 2017; Borghi et al. 2022; Tomasetti et al. 2023), see Appendix A of (Favale et al. 2024a). We employ the corresponding full covariance matrix, as described in (Moresco et al. 2020).\n- · Transverse (aka angular or 2D) BAO data from Refs. (Carvalho et al. 2016; Alcaniz et al. 2017; Carvalho et al. 2020; de Carvalho et al. 2018, 2021). This type of BAO data are claimed to be less subject to model-dependencies, since they are obtained without assuming any fiducial cosmology to convert angles and redshifts into distances to build up the tracer map. The BAO 2D data points are extracted from the two-point angular correlation function or its Fourier transform. In anisotropic (or 3D) BAO analyses, instead, a fiducial cosmology (the standard ΛCDM model) is employed to construct the 3D map in redshift space, \npotentially introducing some model-dependency 3 . It is also worth noticing that despite the fact of being constructed from the same parent catalogues of tracers, angular and anisotropic BAO data exhibit some degree of tension, see e.g. (Camarena & Marra 2020) and, particularly, the very recent analysis of Favale et al. (2024b). In the last study, it is shown that upon excluding the radial component of BAO 3D (which is of course not present in BAO 2D) the 'residual' tensions left between the two BAO types can still attain in between 2 σ to 4 . 6 σ , depending on the data set used. All in all, not surprisingly, these BAO data tensions may directly impinge in a significant way on the discussion of the cosmological tensions themselves, for the required solution to alleviate them may be conditioned to the specific BAO data type considered in the analysis (G'omez-Valent et al. 2024). We should also mention that, in contradistinction to the BAO 3D data, BAO 2D observations still offer room for low-redshift solutions to the Hubble tension while respecting the constancy of the absolute magnitude of SNIa (Akarsu et al. 2023; G'omez-Valent et al. 2024). Given, however, the persisting conflict between these two BAO types, our aim here is to remain as model-independent as possible in the light of the present knowledge, and for this reason we opt for using at this point the kind of BAO data which seem to be less subject to criticism at present (and in this sense that may be more trouble-free), to wit: the BAO 2D type. We leave the detailed comparison with the BAO 3D results for a more comprehensive exposition. A preview of such a comparative analysis will be \nnonetheless advanced in the conclusions (see Section 6) 4 . \n- · The data on large-scale structure (LSS) at z ≲ 1 . 5 from Refs. (Guzzo et al. 2008; Song & Percival 2009; Blake et al. 2011, 2013; Simpson et al. 2016; Gil-Mar'ın et al. 2017; Hou et al. 2020; Said et al. 2020; Avila et al. 2021; Mohammad et al. 2018; Okumura et al. 2016). These are data points on the observable f ( z ) σ 8 ( z ), with f ( z ) = -(1 + z ) d ln δ m /dz the growth rate, δ m = δρ m /ρ m the matter density contrast, and σ 8 ( z ) the rms mass fluctuations at a scale R 8 = 8 h -1 Mpc, see Table 3 of (Sol'a Peracaula et al. 2023). These data points, though, are taken by the observational groups using a fiducial cosmology with h ∼ 0 . 67, which translates into measurements at a characteristic scale of R ∼ 12 Mpc. Here, in contrast, we adhere to the reasoning and practice of Refs. (S'anchez 2020; Semenaite et al. 2022, 2023), and we treat these observations as data points on f ( z ) σ 12 ( z ), using a Fourier-transformed top-hat window function W ( kR 12 ) in the computation of σ 12 ( z ). The advantage, as emphasized by these authors, is that the scale R 12 = 12 Mpc is independent of the parameter h . \nWe compute all the cosmological observables using a modified version of the Einstein-Boltzmann code CLASS (Lesgourgues 2011; Blas et al. 2011), and explore and constrain the parameter space of the various models using the Metropolis-Hastings algorithm (Metropolis et al. 1953; Hastings 1970) implemented in MontePython (Audren et al. 2013; Brinckmann & Lesgourgues 2019). The resulting Monte Carlo Markov chains are analyzed with the Python code GetDist (Lewis 2019). Our main numerical results are presented in Table 1 and in the triangle plots of Figs. 2-3. Supplementary information on plots and tables is provided in the Appendix, which will be commented on in the main text. \nIn Table 1 we do not only list the mean of the various parameters, with their uncertainties and best-fit values, but also report the minimum values of χ 2 obtained for each model and the difference between the deviance (DIC) (Spiegelhalter et al. 2002) and Akaike (AIC) (Akaike 1974) information criteria found between the composite DE models and ΛCDM, which we treat \nas our fundamental benchmark model. These differences read ∆DIC ≡ DIC ΛCDM -DIC i and ∆AIC ≡ AIC ΛCDM -AIC i , respectively, with i referring to w XCDM or Λ s CDM. Both criteria penalize the use of additional parameters and can be regarded as a rigorous mathematical implementation of Occam's razor. The DIC is defined as \nDIC = χ 2 ( ¯ θ ) + 2 p D , (2) \nwith p D = χ 2 -χ 2 ( ¯ θ ) the effective number of parameters in the model, χ 2 the mean value of χ 2 , and ¯ θ the mean of the parameters that are left free in the Monte Carlo analysis. Similarly, AIC is defined as \nAIC = χ 2 min +2 n p , (3) \nwith n p the number of free parameters entering the fit. We note that the above formula is a good approximation when the number of data points entering the fit is much larger than n p , which is certainly the case here. With our definition, a positive difference of these information criteria implies that the composite DE models perform better than the ΛCDM, whereas negative differences imply just the opposite. According to the usual jargon of the information criteria, if 0 ≤ ∆DIC < 2 it is said that one finds weak evidence in favor of the new model under test (in this case, the composite DE models under scrutiny), as compared to the standard model. If 2 ≤ ∆DIC < 6 we speak, instead, of positive evidence . If 6 ≤ ∆DIC < 10, there is strong evidence in favor of the composite DE models, whilst for ∆DIC > 10 we may legitimately conclude that there is very strong statistical evidence supporting the new model or models against the standard ΛCDM. Analogous considerations can be made using AIC, of course. DIC is considered to be a more accurate information criterion, since it incorporates the information encapsulated in the full Markov chains. However, for the sake of generality and to explicitly show the consistency between these two criteria, we display the results obtained from both criteria at the same time in our Table 1.", '5. DISCUSSION': "As we have mentioned, bubbles of PM are connected with the process of attaining the de Sitter (dS) era both in the early and in the late universe. During the transit from the provisional phantom vacuum into the true and stable (dS) vacuum, the PM bubbles being tunneled in the late universe are characterized by positive pressure and hence offer a fertile field for the growing of unsuspected structures. This mechanism, therefore, provides a tantalizing theoretical framework capable of explaining physically the appearance of a late time AdS epoch \nTable 1. Mean values and uncertainties at 68% CL obtained with the full data set CMB+CCH+SNIa+SH0ES+BAO+ fσ 12 . We show the best-fit values in brackets. We use the standard notations for the ΛCDM parameters. In the last three lines, we display the values of the minimum χ 2 , ∆DIC and ∆AIC, as defined in Eqs. (2) and (3), respectively. Positive values of ∆DIC and ∆AIC denote preference of the new models over the ΛCDM. As can be seen, the preference is extraordinarily high. \n(or epochs) before the usual dS epoch is eventually attained 5 . In point of fact, we do not expect that these PM bubbles are perfect AdS phases (nor that the subsequent stage is necessarily characterized by a rigid Λ > 0) and for this reason we have left the EoS parameter w X as a free parameter, whose fitted value turns out to be w X ≃ -1 . 16 (cf. Table 1), although with ρ X < 0 (in \ncontrast to conventional phantom DE). Since the bubble is transitory, the ensuing DE phase below z t carries another EoS which we fit it to be quintessence-like: w Y ≃ -0 . 90 (see Table 1). We have performed a comparative fitting analysis of the w XCDM and Λ s CDM scenarios using the same data sets and have found a significantly better fit quality for the PM scenario, as shown in Table 1. We shall come back to these results in a moment. \nThe PM phase (or phases) with negative energy density and positive pressure are left behind during the cosmic evolution at relatively large redshifts of order 1 -10, \nand in these places they can leave a sort of oasis rich of large scale structures, whereas the consecutive evolution (closer to our time) flips into the quintessence regime. Although we have used an abrupt θ -function behavior to connect the two EoS regimes, in actual fact the process is continuous since the Chern-Simons condensates eventually dominate and restore the normal vacuum phase with positive energy. To better understand, in a quantitative way, how the PM bubbles with positive pressure may enhance the formation of (unsuspected) large scale structures in the relatively distant past, it is useful to analyze the differential equation for the matter density contrast in the presence of PM. This is a key aspect of the role played by PM for potentially helping to solve the cosmological tensions. Before the transition at z t (i.e. for z > z t ), the equation for δ m reads 6 \nδ '' m + 3 2 a (1 -Ω X ( a ) w X ) δ ' m -3 2 a 2 (1 -Ω X ( a )) δ m = 0 , (4) \nwith the primes denoting derivatives with respect to the scale factor. PM has negative energy density (Ω X < 0) and positive pressure (due to w X < -1 < 0) and, therefore, it induces a decrease of the friction term and an increase of the Poisson term (the last one) in Eq. (4). Both effects are in harmony and therefore bring about, given some fixed initial conditions, a net enhancement of the structure formation processes in the PM bubbles. Notice that this would not occur for ordinary phantom DE nor for quintessence (cf. Fig. 1), for which the friction term gets enhanced and the Poisson term suppressed, i.e. just the opposite situation to PM. This conspicuous impact on LSS is therefore unique to PM since there is no other cosmic fluid capable of matching such an achievement in the EoS diagram of Fig. 1. \nMoreover, and this is crucial to understanding how the H 0 tension can be potentially cured in the PM framework: because the energy of the X entity is negative, Ω X < 0 (and non-negligible at high redshift), it enforces a higher value of the expansion rate H in the quintessence stage in order to preserve the angular diameter distance to the last scattering surface, which is essentially fixed from the very precise measurement of θ ∗ (the angular size of the sound horizon) by Planck (Aghanim et al. 2020) and the standard physics before recombination (G'omez-Valent et al. 2024). This explains on plausible physical grounds why H 0 is found larger than in the ΛCDM in our PM scenario. Quantita- \nFigure 2. Contour plots at 68% and 95% CL and the corresponding one-dimensional posterior distributions for some of the parameters that are relevant in the discussion of the cosmological tensions, for all the models under study. H 0 is given in km/s/Mpc. The complete triangle plot is presented in Fig. 4 of the Appendix. \n<!-- image --> \nour value of H 0 (cf. Table 1) is in full agreement with that of SH0ES ( H 0 = 73 . 04 ± 1 . 04 km/s/Mpc) (Riess et al. 2022) to within ≲ 0 . 25 σ . The Hubble tension is therefore virtually washed out, also when it is formulated in terms of the absolute magnitude of SNIa: our fitted value of M and that of SH0ES ( M = 19 . 253 ± 0 . 027 mag) differ by only ∼ 0 . 6 σ . At the same time the rate of LSS formation becomes suppressed below z t ∼ 1 . 5 during the quintessence-like regime, as explained above, which is also in accordance with the observations. In Table 1 and, more graphically, in Fig. 2, we can see that the amplitude of the power spectrum at linear scales that is preferred by the data is pretty similar in all the models under study. Indeed, in all cases we find values of σ 12 ∼ 0 . 78. However, at a finer level of scrutiny, Table 2 in the Appendix tells us that the w XCDM is able to describe better the LSS data than the ΛCDM, what means that the composite DE model is able to produce lower values of f ( z ) σ 12 ( z ) and, hence, of f ( z ) since σ 12 remains stable in the various models. This can again be understood by looking at the equation for the density contrast, whose form at z < z t is identical to Eq. (4), but with the replacements Ω X → Ω Y and w X → w Y , \nδ '' m + 3 2 a (1 -Ω Y ( a ) w Y ) δ ' m -3 2 a 2 (1 -Ω Y ( a )) δ m = 0 . (5) \nIt is clear from Fig. 2 that the values of the matter density parameter Ω 0 m = Ω m ( a = 1) in the w XCDM are a way smaller than in ΛCDM. This obviously translates into larger values of Ω Y ( a ) > 0 (higher than Ω 0 Λ in the ΛCDM), which makes matter fluctuations to grow less efficiently in our model after the transition (at z < z t ), a welcome feature that is rightly aligned for potentially solving also the growth tension within the PM picture. As previously mentioned, despite the presence of PM in the past, the value of σ 12 in w XCDM is not significantly different ( ∼ 0 . 4 σ ) from that of ΛCDM. This is due to the slower increase of matter fluctuations in the latetime universe and the smaller value of the amplitude of primordial fluctuations ( A s ) found in w XCDM, which somehow compensate for the presence of PM at z > z t an keep the value of σ 12 stable (cf. again Table 1, and Fig. 4 in the Appendix). \nFrom our numerical analysis we can see that both models w XCDM and Λ s CDM offer a dramatic reduction of the cosmological tensions. This is apparent from Table 1. One could say that the Hubble and growth tensions disappear in these models. However, there are also important quantitative (and conceptual) differences between them. In terms of the information criteria, we find that the two models w XCDM and Λ s CDM are very strongly preferred over ΛCDM. Numerically, however, the w XCDM solution provides a significantly better global fit. For this model we find ∆DIC,∆AIC ≳ 55, whereas for Λ s CDM we obtain ∆DIC,∆AIC ≳ 40. The difference of 15 units is quite substantive and indicates (in the common parlance of the information criteria) that there is very strong evidence that model w XCDM is preferred over model Λ s CDM (for the same data). Worth noticing is also the fact that our own results for Λ s CDM agree with those originally reported in (Akarsu et al. 2023), despite the existing differences in the data sets employed in the two analyses. Apart from our using data on CCH, which were not considered in (Akarsu et al. 2023), we employ data on f ( z ) σ 12 ( z ) instead of the weak lensing (WL) data from KiDS. We do not use WL data in our analysis in order not to compromise the computation of the non-linear matter power spectrum, which might not be sufficiently well under control in the models under study. Our fitting results, though, are completely consistent with theirs, leading also to similar values of S 8 ∼ 0 . 78 (as we have checked), the latter being a more natural LSS parameter in the context of WL analyses, but not in our approach. \nPerhaps the most remarkable outcome of our analysis is that, in the light of the overall fit quality of our results, which lead to such an outstanding support to the composite DE models from the information criteria, \nit should be fair to conclude that the standard ΛCDM model appears to be comparatively ruled out at a very high confidence level. However, once more, we have to warn the reader that, in principle, such a conclusion ensues under the assumption that only BAO 2D have been employed in our analysis (see, however, the additional comments in the Conclusions, i.e. Section 6). In an attempt to further understand the great success of the numerical fits from the composite DE models in Table 1, let us analyze the degree of impact from each data source. The quantitative influence of each data source can be inferred directly from Table 2 in the Appendix, where a detailed breakdown of the different contributions to χ 2 is reckoned. The angular/transversal BAO data certainly contributes in a significant way, as could be expected from the analysis of (G'omez-Valent et al. 2024). However, it is by no means the leading contribution, as it is responsible for roughly ∼ 18% of the success. A close inspection of the mentioned table shows that the composite DE models beat actually the ΛCDM in every single observable, and not only in those driving the tensions. Particularly noticeable is the sizeable impact from such a solid asset of basic data as the CMB and SNIa+SH0ES observations, which are responsible for about 70% of our successful fitting result, whereas the influence from LSS data is less than 10%; and, finally, that of CCH is marginal (a few percent). The bulk of the successful fit, therefore, relies on the most fundamental cosmological data, which is remarkable. The ΛCDM is not only unable to explain the large value of H 0 measured by SH0ES or to further slow down the growth rate at low redshift, but it also introduces other tensions, e.g. with the value of the (reduced) cosmological matter parameter ω m = ω b + ω dm that is preferred by the Planck data in models which assume standard model physics before the decoupling of the CMB photons, ω m = 0 . 142 ± 0 . 001 (Aghanim et al. 2020). The latter is fully consistent with the w XCDM value ( ω m = 0 . 142 ± 0 . 001), but differs by 2 . 8 σ with the one inferred in the standard ΛCDM ( ω m = 0 . 138 ± 0 . 001). All that said, we remark that the BAO data play indeed a momentous role in the H 0 tension, for despite the fact that the overall fit with composite DE does substantially improve using any of the two variants of BAO, the specific relieve of the H 0 tension hinges dramatically upon the particular sort (2D or 3D) of BAO that is employed, see Section 6. \nWe should emphasize that the global outperformance of the composite DE models under study as compared to the ΛCDM is obtained after duly penalizing the use of extra parameters in them. This is of course implemented automatically by the information criteria, as \nFigure 3. Confidence regions in the w Y -w X plane of the w XCDM model, and the corresponding one-dimensional posterior distributions. The dotted lines are set at w Y = -1 and w X = -1. The intersection of the horizontal and vertical lines in the contour plot corresponds to the Λ s CDM model, which falls ≳ 3 σ away from the preferred region of the w XCDM. See the main text for further comments. \n<!-- image --> \nexplained above. Thus, for the Λ s CDM there is the transition redshift z t as a new parameter with respect to the ΛCDM, whereas for the w XCDM we have, in addition, the two EoS parameters w X and w Y for the two phases of the DE. Despite the presence of these extra parameters, Occam's razor (formalized through the verdict of the information criteria) still bestows exceptional preference for the composite models over the concordance model. This is the first important conclusion of our analysis, which suggests that the composite nature of the DE could be a fact. The second, is that the relative differences between the two composite models under scrutiny, namely DIC Λ s CDM -DIC w XCDM = 17 . 78 and AIC Λ s CDM -AIC w XCDM = 8 . 42, clearly point to a rather keen preference for w XCDM over Λ s CDM under the same data. \nFocusing now on the w XCDM parameters, in Fig. 3 we show the constraints obtained in the EoS plane w Y -w X , which involves two of the characteristic new parameters of the w XCDM model (the third one being z t , shared with Λ s CDM). The central value of w X = -1 . 16 falls in the phantom region (in fact, PM region since Ω X < 0) but is compatible with -1 at ∼ 1 σ c.l. There is, in contrast, a non-negligible ( ∼ 3 . 3 σ ) preference for a quintessence-like evolution of the DE for the low redshift range nearer to our time ( z < z t ): w Y = -0 . 90 ± 0 . 03 (see also Fig. 3). The preference that we find for quintessence-like behavior in this region can be contrasted with the situation in the Λ s CDM, \nwhere a rigid Λ-term is assumed below the transition redshift (also above that redshift, with Λ →-Λ). Now the fact that our fit with model w XCDM proves significantly better than with model Λ s CDM (as commented above) suggests that indeed the quintessence option in the low redshift range is strongly preferred. Our result aligns perfectly well with the one obtained from the analysis of the Pantheon+ data in the context of the flat w CDM parametrization (Brout et al. 2022), which showed that SNIa data alone lead to an EoS parameter w = -0 . 89 ± 0 . 13 after marginalizing over Ω 0 m . This marginalized result, though, is still compatible with a CC ( w = -1). To explain why in our case is more focused on the quintessence domain, let us note that for w XCDMwefind a tight constraint on the matter density parameter, strongly favoring the region of small values, Ω 0 m = 0 . 269 ± 0 . 005. This is induced by two facts: (i) CMB prefers values of ω m ∼ 0 . 142 in models with standard pre-recombination physics; and (ii) the large values of H 0 ∼ 73 km/s/Mpc measured by SH0ES. Combining the aforementioned tight constraint on Ω 0 m with the constraints in the w -Ω 0 m plane obtained for the w CDM from the analysis of the Pantheon+ data alone (cf. the contours in cyan of Fig. 9 of (Brout et al. 2022)) we can break the existing degeneracy in that plane and explain the ≳ 3 σ evidence for quintessence DE that we find within the w XCDM for the low redshift range. We point out that, for the conventional w CDM parameterization (i.e. without the X component), such low values of Ω 0 m are not favored by Planck, since they are in tension with the angular diameter distance to the last scattering surface, see again Fig. 9 of (Brout et al. 2022). Remarkably, this problem can be fully averted in the w XCDM model owing precisely to the concurrence of the X component, which can act as PM in the high redshift stretch z > z t and compensate for the decrease of the contribution to the angular diameter distance in the low redshift domain. On the other hand, in the Λ s CDM case we also find similar low values of Ω 0 m . However, to retrieve this model from the w XCDM we have to set w Y to -1 (rigid CC) 7 , so the Λ s CDM falls outside the 3 σ region obtained with w CDM using only SNIa data (Brout et al. 2022). This issue is also solved in the w XCDM and explains the improvement found with respect to the Λ s CDM model. The full 5-year data set of high-redshift SNIa from the Dark Energy Survey (Abbott et al. 2024) and the Union3 SNIa compilation (Rubin et al. 2023) will most likely increase the aforesaid \nsignal, since they also find constraints in the region of interest, which read (Ω 0 m , w ) = (0 . 264 +0 . 074 -0 . 096 , -0 . 80 +0 . 14 -0 . 16 ) and (Ω 0 m , w ) = (0 . 244 +0 . 092 -0 . 128 , -0 . 735 +0 . 169 -0 . 191 ), respectively. Hence, it will be of utmost importance to study the impact of these data sets in future analyses. In addition, our fitting results for w XCDM are compatible at roughly ∼ 1 σ with those recently reported by DESI using the w CDM parametrization(Adame et al. 2024). Although two of the twelve DESI BAO data points (specifically those obtained with Lyα quasars (QSO) at z eff = 2 . 33) lie outside the quintessence region identified in our w XCDM model, the bulk of the DESI points lie below z = 1 . 5. The exclusion of the Lyα QSO data points in their fitting analysis might even improve further the compatibility with our findings. Additionally, some recent analyses using DESI data to reconstruct the dark energy density have found hints of negative values at z > 1 . 5 (Calder'on et al. 2024; Orchard & C'ardenas 2024). Since the DESI data are derived from anisotropic (3D) BAO, exploring future angular BAO data from DESI (if available) and comparing them with the BAO 3D data would be an interesting avenue for further research.", '6. CONCLUSIONS': "In this paper, we have addressed a possible solution to the cosmological tensions within a simplified version of the old existing ΛXCDM framework (Grande et al. 2006, 2007, 2009), which was born in the theoretical arena of the RVM - see (Sol'a Peracaula 2022; Sol'a 2013) and references therein. Our modern approach benefits from the new theoretical developments in the context of the stringy version of the running vacuum model (StRVM) (Mavromatos & Sol'a Peracaula 2021b,a). The entity X that is involved in the ΛXCDM need not be a fundamental field but can display a phantom matter (PM) behaviour, which in the StRVM is predicted to appear in the cosmic transit to the de Sitter phase. The PM regime is, therefore, only transitory and acts as a temporary stage with negative energy and positive pressure. It can be a pure anti-de Sitter (AdS) phase but in general its EoS w X is not necessarily -1, although it must satisfy w X ≲ -1. In this work we have contented ourselves with a simplified version of the full ΛXCDM, which we have called the w XCDM. We have utilized a robust data set, which includes the Pantheon+ compilation of SNIa, cosmic chronometers, transverse BAO, large-scale structure data, and the full CMB likelihood from Planck 2018. As warned previously, our exclusive use of transverse (2D) BAO aims at maximally mitigating the model-dependent effects that may be introduced on using anisotropic (3D) BAO, according to some pub- \nd studies. No similar issues have been identified in BAO 2D and hence we have focused here on using only uncontested BAO data, with the hope that the situation with BAO 3D will be completely clarified in the near future (see, however, the additional comments below). \nUsing the w XCDM and the aforementioned data configuration, we find that the PM phase shows up above a transition redshift z t ≃ 1 . 5 (cf. Table 1). For z > z t the PM behavior is rapidly washed out since its energy density fraction becomes smaller and smaller in the past. And, on the other hand, its effect towards our time ( z < z t ) becomes erased by the progressive appearance of the Chern-Simons condensates, which are responsible for eventually establishing a steady Λ =const. > 0 regime (Mavromatos & Sol'a Peracaula 2021b). For this reason, in practice, the appearance of a PM phase is confined to a bubble of spacetime, which is left behind. However, in that space and time new structures can emerge, being completely unsuspected in the context of the standard model. The PM bubbles that are met within the stringy RVM formulation originate as a result of a 'phantom vacuum' phase, which emerges in the universe when the expansion heads towards the next de Sitter epoch. While the implications of this fact for the early universe were considered previously (Mavromatos & Sol'a Peracaula 2021b), here we have extended this phenomenon to the late universe. In both cases the PM bubbles are localized spacetime events. They can only occur as tunneling attempts to establish the phantom vacuum (Mavromatos & Sol'a Peracaula 2021b) with positive pressure and negative energy: p = -ρ > 0. Below the transition redshift, the PM phase ceases to occur and the field Y (viz. the would-be running VED in the full ΛXCDM(Grande et al. 2006)) takes its turn with a new effective EoS, which is fitted to be of quintessence type ( w Y ≳ -1) at ∼ 3 . 3 σ c.l. As a matter of fact, this is the effective form of DE that is expected in our most recent past ( z < 1 . 5) within the w XCDM, although from a more fundamental level it could just be running vacuum energy with ν eff > 0 in the full ΛXCDM model, see Eq. (1). The observed DE should thereupon be dynamical, and this important conclusion is in stark contrast with the alternative Λ s CDMscheme analyzed in(Akarsu et al. 2023), in which below the transition redshift there is a rigid and positive cosmological term Λ =const. Actually, no possible time dynamics for the DE is available for the Λ s CDM, neither above nor below z t . On this account, the picture emerging from the w XCDM looks more consistent with the first data release by DESI (Adame et al. 2024), which points to dynamical DE. As noted in the Introduction, the fashionable conclusion about potential evidence of time-evolving DE is actu- \nnsonance with previous phenomenological studies of the RVM from several years ago, which were based on a pretty large set of cosmological data of various sorts. These early detailed studies already anticipated significant signs of dynamical DE at 3 σ -4 σ c.l. (Sol'a et al. 2015, 2017; Sol'a Peracaula et al. 2017; Sol'a Peracaula et al. 2018a,b; G'omez-Valent & Sol'a Peracaula 2018). The current work aligns well with these studies, which were like a harbinger of the dynamical character of the DE emerging from the analysis of a large body of observational data. But at the same time it leads to further insight into possibly new qualitative properties of the DE, in particular its preference for undergoing a sign flip at a nearby transition redshift, thus adopting a sort of chameleonic nature around that transition point, for the DE appears phantom-like first (viz. 'phantom matter') and subsequently it shows up as effective quintessence near our time (as recently observed by DESI). This combined feature of the DE optimizes to a large extent the quality fit to the overall cosmological data as compared to the ΛCDM. Now while this positive feature holds true for the two sorts of BAO, it is not so for curing the H 0 tension (see below). \nAll in all, model w XCDM potentially offers a rich conceptual framework for our understanding of the physical phenomena that may be involved. In the w XCDM, the formation of PM bubbles is induced by quantum fluctuations associated to the nearing of the universe towards a de Sitter phase. The phenomenon need not be unique: the bubbles of PM could well be operating more than once at earlier times on the same physical grounds, what would trigger an anomalous outgrowth of structures at even higher redshifts, say in the range z ∼ 5 -10. This might explain the appearance of the large scale structures recently spotted at unusually high redshifts by the JWST mission(Labb'e et al. 2023; Adil et al. 2023; Menci et al. 2024). Such LSS 'anomalies', which find no explanation in the ΛCDM, might also be described within the current proposal since they can be conceived as being the earliest bubbles of PM popping up in the late universe, corresponding probably to the first tunneling attempts anticipating the eventual dS phase near our time. This is reasonable, given the fact that the tunneling process towards the final de Sitter era should be gradual, exactly as in the early universe (Mavromatos & Sol'a Peracaula 2021b). \nLast but not least, as promised, we would like to advance here, if only very briefly, the outcome of the preliminary analysis performed with BAO 3D data. Our extended study has revealed that the w XCDM model with BAO 3D (replacing BAO 2D) data improves once more (and quite significantly) the overall fit to the cosmologi- \ncal data as compared to the ΛCDM, see (G'omez-Valent & Sol'a Peracaula 2024). This should not come as a big surprise, if we take into account our discussion in the previous section, where we have shown by appealing to Table 2 in the Appendix that the composite DE models beat the ΛCDM in every single observable, not only in those driving the tensions. It means that the composite feature of the DE proves really fruitful to better accommodate the various sources of cosmological data. This is remarkable in the first place, for it means that the superiority of the composite DE models over the ΛCDM is not compromised at present by the particular type (2D or 3D) of BAO data used, despite them being currently under some tension (Camarena & Marra 2020; G'omezValent et al. 2024; Favale et al. 2024b). Notwithstanding this relevant fact, which holds good for the global fit involving all sorts of cosmological data, we should not underemphasize that, insofar as concerns the H 0 tension itself, the fit with BAO 3D data proves unable to match the efficiency of the BAO 2D fit. Specifically, with BAO 3D the H 0 tension persists at a level of ∼ 2 . 9 σ (no longer at the insignificant level of 0 . 25 σ like in the BAO 2D case). Thus, while it is true that with BAO 3D the H 0 tension gets somewhat reduced from the large initial value ( ∼ 5 σ c.l.), it remains still sizeable. In contrast, we find that the growth tension is less affected. All in all, a final solution to the cosmological tensions - above all to the H 0 tension - is still being compromised by a lack of consistency between the two sources of BAO, angular and anisotropic. The systematic exposition of the results with BAO 3D, along with the quantitative differences with respect to the analysis using BAO 2D, is beyond the scope of this work and its presentation is left for a separate publication (G'omez-Valent & Sol'a Peracaula 2024). \nAt the end of the day, it is encouraging to realize that the physics involved in the PM scenario presented here for a potential solution to the cosmological tensions, might ultimately stem from (dynamical) quantum vacuum phenomena, what gives hope for eventually achieving a deeper understanding of the cosmological evolution of the universe from fundamental physics. Whether this is actually the case or not will depend, of course, on the eventual resolution of the BAO tension itself, which appears to be the main stumbling block for the ultimate understanding of the problem.", '7. ACKNOWLEDGEMENTS': "This work is partially supported by grants PID2022136224NB-C21 and PID2019-105614GB-C21, from MCIN/AEI/10.13039/501100011033. AGV is funded by 'la Caixa' Foundation (ID 100010434) and the \nEuropean Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 847648, with fellowship code LCF/BQ/PI21/11830027. JSP is funded also by 2021SGR-249 (Generalitat de Catalunya) and CEX2019- \n000918-M (ICCUB, Barcelona). Both of us acknowledge networking support by the COST Association Action CA21136 ' Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) '.", 'FULL TRIANGLE PLOT AND BREAKDOWN OF χ 2 MIN CONTRIBUTIONS': 'Table 2. Individual χ 2 i contributing to χ 2 min , obtained in the fitting analyses for the various models with CMB+CCH+SNIa+SH0ES+BAO+ fσ 12 . χ 2 CMB contains the total CMB contribution, i.e. it is the sum of all the Planck χ 2 i , which we list in the upper half of the table. \nFigure 4. Full triangle plot for the various models studied in this paper. We show the constraints at 68% and 95% CL in all the relevant planes of the parameter spaces, together with the individual one-dimensional posterior distributions. H 0 is given in km/s/Mpc. \n<!-- image -->', 'REFERENCES': "Abbott, T. M. C., et al. 2024. https://arxiv.org/abs/2401.02929 Abdalla, E., et al. 2022, JHEAp, 34, 49, doi: 10.1016/j.jheap.2022.04.002 Adame, A. G., et al. 2024. \nhttps://arxiv.org/abs/2404.03002 \nAde, P. A. R., et al. 2016, Astron. Astrophys., 594, A13, \ndoi: 10.1051/0004-6361/201525830 \nAdil, S. A., Mukhopadhyay, U., Sen, A. 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Dark Univ., 25, 100311,\n- doi: 10.1016/j.dark.2019.100311\n- Sol'a Peracaula, J., G'omez-Valent, A., de Cruz P'erez, J., & Moreno-Pulido, C. 2021, EPL, 134, 19001, doi: 10.1209/0295-5075/134/19001 -. 2023, Universe, 9, 262, doi: 10.3390/universe9060262 Song, Y.-S., & Percival, W. J. 2009, JCAP, 0910, 004, doi: 10.1088/1475-7516/2009/10/004 Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & van der Linde, A. 2002, J. Roy. Stat. Soc., 64, 583, doi: 10.1111/1467-9868.00353 Stern, D., et al. 2010, JCAP, 1002, 008, doi: 10.1088/1475-7516/2010/02/008 Tomasetti, E., et al. 2023, Astron. Astrophys., 679, A96, doi: 10.1051/0004-6361/202346992 Turner, M. S. 2022, Annu. Rev. Nucl. Part. Sci. 2022, 72, 1, doi: 10.1146/annurev-nucl-111119-041046 Weinberg, S. 1989, Rev. Mod. Phys., 61, 1, doi: 10.1103/RevModPhys.61.1 Zhang, C., et al. 2014, Res. Astron. Astrophys., 14, 1221, doi: 10.1088/1674-4527/14/10/002 Zhao, G.-B., et al. 2017, Nat. Astron., 1, 627, doi: 10.1038/s41550-017-0216-z"}

2024arXiv240911470N

We introduce adaptive particle refinement for compressible smoothed particle hydrodynamics SPH. SPH calculations have the natural advantage that resolution follows mass but this is not always optimal. Our implementation allows the user to specify local regions of the simulation that can be more highly resolved. We test our implementation on practical applications including a circumbinary disc a planet embedded in a disc and a flyby. By comparing with equivalent globally high resolution calculations we show that our method is accurate and fast with errors in the mass accreted onto sinks of less than 9 percent and speed ups of 1.076.62 times for the examples shown. Our method is adaptable and easily extendable for example with multiple refinement regions or derefinement.

2024-09-01T00:00:00Z

['2024arXiv240911470N', 'arXiv:2409.11470', '10.48550/arXiv.2409.11470']

['Astrophysics - Instrumentation and Methods for Astrophysics']

Adaptive particle refinement for compressible smoothed particle hydrodynamics

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11470.pdf

{'No Header': '<!-- image -->', 'Adaptive particle refinement for compressible smoothed particle hydrodynamics': 'Rebecca Nealon 1,2 and Daniel J. Price 3 \n- 1 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, CV4 7AL Coventry, UK\n- 2 Department of Physics, University of Warwick, Gibbet Hill Road, CV4 7AL Coventry, UK\n- 3 School of Physics and Astronomy, Monash University, Clayton, Vic 3800, Australia\n- Author for correspondence: R. Nealon, Email: rebecca.nealon@warwick.ac.uk.', 'Abstract': "We introduce adaptive particle refinement for compressible smoothed particle hydrodynamics (SPH). SPH calculations have the natural advantage that resolution follows mass, but this is not always optimal. Our implementation allows the user to specify local regions of the simulation that can be more highly resolved. We test our implementation on practical applications including a circumbinary disc, a planet embedded in a disc and a flyby. By comparing with equivalent globally high resolution calculations we show that our method is accurate and fast, with errors in the mass accreted onto sinks of less than 9% and speed ups of 1.07 - 6.62 × for the examples shown. Our method is adaptable and easily extendable, for example with multiple refinement regions or derefinement. \nKeywords: hydrodynamics, methods: numerical \nThis paper considers the best way to locally adapt resolution in the simulation method known as smoothed particle hydrodynamics (SPH, Lucy 1977; Gingold and Monaghan 1977). Because resolution follows mass in SPH, the best resolved region often corresponds to the region of interest. However, this may be inefficient, as high densities correlate with short timesteps which are expensive. A potential solution is Adaptive Particle Refinement (APR): splitting and merging of particles. This method allows multiple resolutions to co-exist in the same simulation (Monaghan and Varnas 1988; Meglicki, Wickramasinghe, and Bicknell 1993; Kitsionas and Whitworth 2002, 2007; Vacondio et al. 2013; López, Roose, and Recarey Morfa 2013; Barcarolo et al. 2014; Chiron et al. 2018; Gao, Qiu, and Fu 2022). APR is conceptually similar to Adaptive Mesh Refinement (AMR, Berger and Colella 1989; Truelove et al. 1997) commonly used in simulations employing Eulerian meshes. \nAlthough less widely exploited than mesh refinement, the fundamentals of particle refinement and its applications are well developed. Monaghan and Varnas (1988) and Meglicki, Wickramasinghe, and Bicknell (1993) first applied particle splitting in SPH. They merged particles in their simulations to create fewer, more massive particles when the density in a cell exceeded a critical density threshold. Meglicki, Wickramasinghe, and Bicknell (1993) additionally included particle splitting in low density regions to increase the local resolution. In the context of cosmological simulations, based on the method outlined in Porter (1985), Katz and White (1993) (and later Navarro and White 1994) scaled the mass of the particles in spherical, nested layers throughout the computational domain such that the innermost layer had the smallest mass and thus highest resolution. Thacker and Couchman (2000) found that there was a small amount of noise generated at the boundary between the mass layers but that it did not affect the structures identified \nin their simulations. Bertschinger (2001) produced a method to accurately initialise simulations with multiple masses for a Gaussian random field. Klypin et al. (2001) demonstrated numerical convergence with this simulation style. This 'multiple mass' or 'zoom' method, the approach of nested regions is a core feature of cosmological simulations (e.g. Springel 2005). \nBørve, Omang, and Trulsen (2001) implemented an alternative multi-resolution scheme they called Regularised-SPH (RSPH). In this method, the smoothing length h was set to a piece-wise constant in steps of 2 with the contribution from neighbouring particles interpolated using a grid. In its most modern invocation (Børve, Omang, and Trulsen 2005), this method used auxiliary particles that exist at the boundary between different h regions and which are passively evolved. Børve, Omang, and Trulsen (2006) tested RSPH on a multidimensional MHD shock, showing excellent shock capturing properties. Børve, Speith, and Trulsen (2009) also successfully applied RSPH to planet-disc interactions in two dimensions. \nKitsionas and Whitworth (2002) considered the collapse of clumps in self gravitating filaments. They increased the resolution in high density regions to ensure that the Jean's criteria was met, finding their method was robust and comparable to results from AMR calculations. Importantly they used 13 children for each parent ( n child = 13 where we refer to more massive and less massive particles as 'parents' and 'children', respectively) to ensure spherical symmetry in 3D when splitting particles and established an empirical test to determine the radius of the sphere that the children are located on ( r sep). Kitsionas and Whitworth (2007) showed that using a mass weighted kernel (with 50 times the mass of the largest particle) was preferable to volume weighted. \nFeldman and Bonet (2007) introduced a general splitting procedure, placing child particles symmetrically in hexagons or \ntriangles around their parent to conserve angular momentum. Importantly, they quantified the error that is generated when a parent particle is split into children; when the children have equal mass this error is due to the new particle arrangement and is a function of the distance between the children and their parent. \nLastiwka, Quinlan, and Basa (2005) showed that this error could be reduced by slightly shifting the children. López, Roose, and Recarey Morfa (2013) extended this idea by measuring the error between the parent and child distributions as a function of the separation between the children when they are newly placed, r sep. Solutions to this error include shuffling the particles until the error is minimised (Vacondio et al. 2013), blending the region between the parent and children regions (Barcarolo et al. 2014), particle disordering (Chiron et al. 2018) and particle regularisation (Gao, Qiu, and Fu 2022). Vacondio et al. (2013) first introduced particle merging in incompressible applications but found it too expensive for practical use. Chiaki and Yoshida (2015) showed that the noise introduced by a split could be reduced by placing the children particles on the vertices of a Voronoi mesh. \nMost recently, Franchini, Lupi, and Sesana (2022) used particle splitting in a meshless-finite-volume method (Hopkins 2014) to better resolve circumstellar discs within the inner cavity of a circumbinary disc. Adopting the method from Anglés-Alcázar et al. (2021) they used n child = 2 and r sep = min(0.25 × h parent, 0.35 × d neighbour ) (where d neighbour is the distance to the nearest neighbour) to ensure that split particles were not inadvertently placed too close to existing particles (i.e. that the fluid elements did not overlap). The robustness of their method was demonstrated in Duffell et al. (2024) where their splitting method found agreement with comparable codes including P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc (Price, Wurster, et al. 2018). \nWhile the above implementations are promising, practical limitations remain. First, while most authors state that merging is possible, in practice it is not used - and never in simulations of compressible flow. On the rare occasion that merging is included it is found to be computationally prohibitive (Vacondio et al. 2013) which restricts what problems APR can be practically applied to. Second, typical implementations require hard-wiring of parameters like r sep in order to reduce noise when particles are split, but these values are derived from simple tests (e.g. Kitsionas and Whitworth 2002). Third, some of the above applications in astrophysics allow particles of different refinement levels to mix which may result in numerical instabilities (e.g. Chiron et al. 2018). \nIn this paper we introduce an APR implementation into the smoothed particle hydrodynamics code P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc that includes both splitting and merging, separation of refinement levels and which is accurate and fast. In Section 1 we outline the method. Section 2 applies our method to examples and practical calculations compared to equivalent global high resolution simulations. We measure speed, accuracy and disc storage. In Sections 3 and 4 we discuss and draw conclusions. Basic tests for the interested reader are summarised in Appendix 1.", '1. Method': 'Here we describe the core of our implementation including splitting, merging, relaxing and the order in which these are completed. The splitting and merging processes are summarised in Figure 1.', '1.1 Overall procedure': "We assume spherical refinement and de-refinement zones. Their size and location may be either fixed or dynamic and comoving with another particle. In some of our later applications the refinement zone is centred on a moving point mass particle but it may also be set by a particle property like density. Here we show examples with spherical zones but other volumes are possible. The main points of our implementation can be captured with five 'rules': \n- 1. We assign all particles a refinement level, ℓ , determined exclusively from the spatial position of the particle. The refinement level represents the number of refinements above the base resolution.\n- 2. When a particle enters a new refinement zone with a given refinement level\n- · it is split if the particle's current refinement level is less than that of the zone.\n- · it is merged if the particle's current refinement level is greater than that of the zone.\n- 3. In simulations where the goal is to locally increase the resolution the default refinement level is set to ℓ = 0. In simulations where we want to locally decrease the resolution the default refinement level is instead set to ℓ = ℓ max.\n- 4. We restrict the difference in mass between adjacent refinement levels to be strictly a factor of two.\n- 5. As P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc stores the mass by particle type, the refinement level is also used to relate the refined mass mp ( ℓ ) from the largest particle mass mp (0) with the following relation: \nmp ( ℓ ) = mp (0) 2 ℓ , (1) \nwhere ℓ >= 1. When we compare particles with different refinement levels we refer to the more highly resolved particles as 'children' and the lower resolution particles as 'parents'. In our scheme then, two children merge to form one parent and on a split one parent produces two children.", '1.2 Splitting': "A parent particle with refinement level ℓ is split into two children when it crosses the boundary into a refinement zone with a higher refinement level. Once a particle is confirmed to be split; \n- 1. A new child particle is made with identical properties to the parent.\n- 2. The parent itself is reassigned to be a second child particle. \nFigure 1. Schematic showing our refining and derefining process. The flow of the fluid is le/f\\_t to right and parent particles are larger and blue, children particles smaller and pink and the particle size is proportional to its mass. Particles that are split or merged in the time step shown are indicated with outlines and ℓ shows the refinement level. Le/f\\_t: As a parent particle crosses the boundary it is split into two children particles, their common centre of mass is at the parent's location and they are split tangentially to the boundary. Right: Children particles are paired according to our modified k -d tree grouping (indicated with the grey boxes). When the centre of mass of a cell crosses the boundary the particles are merged, with the parent adopting the average velocity and position of the children. Further details are in Section 1. \n<!-- image --> \n- 3. Both children have their refinement level increased and their smoothing lengths rescaled (assuming constant density) \nh ℓ +1 = ( 1 2 ) 1/3 h ℓ . (2) \n- 4. Children are placed on opposite sides of the parent at a distance r sep = 0.2 h ℓ .\n- 5. Children are additionally placed tangential to the boundary of the refinement region (as in Figure 1). \nThe scaling value in r sep has been determined empirically to reduce noise following the test from Kitsionas and Whitworth (2002), however our inclusion of relaxing (see Section 1.4) means that our implementation is not particularly sensitive to this choice (as demonstrated in Appendix 1.3). Additionally, as P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc calculates the density and smoothing lengths iteratively (2.1.4, Price, Wurster, et al. 2018), the smoothing lengths of surrounding particles are accurately adjusted to accommodate the new particles straight after the split (and later, merge) has occurred. \nTo ensure particles are split tangentially to the boundary in 3D we identify the vector ⃗ v between the original particle and the centre of the refinement zone. A vector ⃗ w represents the perpendicular bisector of ⃗ v and is always tangential to the boundary of the refinement zone. We rotate ⃗ w around ⃗ v through a random angle and place the children on opposite sides of the parent along ± ⃗ w . As in Franchini, Lupi, and Sesana (2022), if the distance shifted is more than 0.35 × the distance to the closest neighbour we adopt this value instead, however we have found that this only affects a handful of particles even in very well resolved simulations. \nFigure 2. Flowchart summarising the APR routine implemented in P/H.s/A.s/N.s/T.s/O.s/M.s. \n<!-- image -->", '1.3 Merging': "Wemake use of a modified k -d tree in P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc to make rapid on-the-fly merging possible. We have altered the existing k -d tree so that it always returns leaf nodes with exactly two particles (this modification is only used in the call from the APR routine and is not used elsewhere in the calculations). Every time the domain is split through the centre of mass we enforce an even number of particles on either side of the new 'subdomains'. When a split occurs that results in an odd number of particles in each new subdomain, we move the location of the split across one particle in the direction that makes the most balanced split by number of particles. The tree continues until finally all leaves have precisely two particles. \nAt each time-step where merging occurs; \n- 1. All particles at the same refinement level ℓ are isolated and then paired them using our modified k -d tree.\n- 2. For each leaf in this tree, we calculate what refinement level the leaf has based on it's centre of mass and compare it to the refinement level of the child particles, ℓ .\n- 3. The particles are merged when the refinement level based on the position of the leaf is less than the refinement level of the child particles in the leaf.\n- 4. One of these children particle is removed from the simulation and the remaining child particle is reassigned as a parent.\n- 5. The parent is given the position and velocity of the centre of mass of the leaf (i.e. the two children), the refinement level is decreased and the smoothing length is scaled by \nh ℓ -1 = 2 1/3 h ℓ . (3) \n- 6. If an odd number of particles are considered for merging, we discard the last particle that is listed at that timestep. In the next timestep when particles are again considered for merging this particle is then included. \nThe right panel of Figure 2 schematically shows the merging process. With our implementation it is important to note that although the boundary for where merging occurs is well defined, because we are merging based on the children particle distribution the practical boundary is fuzzy and a transition region typically about 10% of the larger smoothing length naturally forms. In well resolved simulations (e.g. Section 2) this region is negligible in width. A major benefit of our approach is we have at most a maximum of one spare particle per refinement level per timestep. As we do not consider ℓ = 0 particles for merging, the maximum error e in the total mass of these spare particles for ℓ max refinement levels is \ne = mp (0) ℓ max ∑ ℓ =1 1 2 ℓ , (4) \nwhich is conveniently always less than mp (0), the largest particle mass.", '1.4 Relaxing': "When splitting or merging a significant number of particles - for example the first time the splitting/merging routine is called - there is a discrepancy between the original and refined density distribution. This effect was identified by Feldman and Bonet (2007) and is due to how well the new particle arrangement can represent the original density distribution. \nTo combat this we have implemented a relaxing procedure which shuffles the introduced particles until they more accurately represent the original distribution. When splitting or merging an original set of particles into a new set, our algorithm \n- 1. Calculates the accelerations of the new set of particles at their current locations, a new.\n- 3. Shifts each particle by ∆ x calculated from\n- 2. Calculates the accelerations at the locations of the new set of particles, interpolated from the original reference particles, a ref . \n∆ x = 0.5 ∆ t 2 ( a new a ref ), (5) \nwhere ∆ t is calculated from the sound speed and smoothing length by ∆ t = 0.3 h / cs on the new particle. \nFor any given particle the magnitude of the shuffle is also capped to be less than the particles smoothing length. At each shuffle step, we estimate a 'kinetic energy' defined as the magnitude of the shift divided by the particle's timestep squared and summed across all the shuffled particles. For multiple shuffles we find that the kinetic energy decreases in an exponential fashion with each subsequent shuffle smaller than the previous. Our shuffling process is repeated until either the total kinetic energy for a shuffle has decreased to 0.5% of its original value or 50 shuffles have been completed. These limits are set to strike a balance between accuracy and computational expense, as we have found that more shuffles does not continue to dramatically improve the particle distribution. How quickly the kinetic energy limit is reached does depend on the resolution, with higher resolution simulations achieving it in fewer shuffles.", '1.5 With individual timesteps': "A key part of our implementation is compatibility with individual particle timesteps as these are used widely in P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc applications (see Price, Wurster, et al. 2018). Each particle is assigned to a timestep bin; these are arranged such that the zeroth bin has ∆ t = ∆ t max (which is limited to be the time between outputs) and particles are arranged in bins according to their local timestep constraint, where ∆ t decreases by factor of two in each bin. \nEach bin is then evolved separately, synchronising with the bin above when the appropriate timesteps are synchronised. Particles are then defined as 'active' (as in, they will be moved by the stepping routine) when their bin is being evolved. \nWhen implemented in conjunction with APR we restrict the splitting and merging procedure to occur over active particles only. This does mean that a merging particle's closest \nTable 1. Summary of the APR simulations shown in Section 2. Columns state the Name of the simulation, the radius of the central refinement region r ℓ , the steps into each refinement zone dr ℓ and the average number of particles used N . The error, speed up and storage are all compared to the high resolution reference cases. The method to measure the error is described in the text for each simulation and is measured according to Equation 6. The storage is calculated as a fraction compared to the high resolution reference calculation. All simulations have 3 levels of increased refinement except F6 which has ℓ = 6 . Here 'B' refers to binary, 'P' to planet-disc and 'F' to flyby simulations. \nneighbour could be on a different time-step, forcing the particle to merge with an active particle further away. In practice this is rare because the particles have been paired using our modified k -d tree. This pairing takes advantage of the fact that spatially associated particles tend to have the same acceleration and so are naturally on the same timestep bin such that the closest neighbour is also on the same timestep. Additionally, when a particle is either split, merged or a new particle introduced through either of those procedures it is conservatively assigned the shortest timestep (corresponding to the maximum timestep bin).", '1.6 Every timestep': 'Figure 2 summarises the process undertaken at each timestep on active particles. First the location, size and number of refinement regions are updated. If relaxing is required at this step the properties of all particles are also saved as a reference. \nWethen consider whether any particles should be split. Using the particles current position we calculate what refinement level it should have and compare this to its current refinement level. When the particle has a refinement level less than what it should have we split it according to Section 1.2. We repeat this process across for all active particles and for each refinement level, starting from the lowest refinement level. \nAfter all the particles have been split we then consider whether any should be merged. We isolate all particles that are at a particular refinement level and put these particles into our modified tree routine, merging as per Section 1.3. Mirroring the splitting procedure, this process is repeated from the highest to lowest refinement level. This order means that particles are only ever changed one refinement level each split/merge which we have found leads to a smoother distribution (e.g. \nAppendix 1.4). Only once all the particles have been split and merged do we consider the relaxing process, using the original particle distribution as the reference set. With the particles refinement levels updated the program then returns to the normal time stepping routine and continues.', '2. Example applications': "We demonstrate the accuracy, speed up and typical use cases of our APR implementation with three example applications: a circumbinary disc, planet disc interactions and a stellar flyby. In each case we include a low and high resolution comparison simulation, where the high resolution simulation has the same global resolution as the highest APR zone. Our examples also demonstrate the adaptability of the implementation and provide typical values for the size of the refinement region and the step width. Our examples include gravity from sink particles (stars and planets) as well as accretion onto those sinks. We also assume a vertically isothermal temperature profile for these applications, but we refer to Appendix 1 for tests with an adiabatic equation of state. Radiative cooling or source terms that are sensitive to temperature are left to future work. Our results are summarised in Table 1. In all cases we find that our implementation is accurate, fast and requires less storage space when compared to the high resolution reference cases. \nIn the examples shown here we choose to use n child = 2 and nested refinement levels and with 'relaxing' only implemented in the first step (see Appendix 1). We additionally ensure that the velocity of the fluid across the boundary is small (see Appendix 1.5) by choosing the location of our refinement regions carefully. These examples are presented with a Wendland C2 kernel with a kernel radius of 2 h and h fact = 1.3 (Wendland 1995; Price, Wurster, et al. 2018). As we are considering parti- \nFigure 3. Columndensity of the HD142527 simulations from Price, Cuello, et al. (2018) at t = 1137 years. The le/f\\_t and right panels have no APR and are separated by a factor of two in spatial resolution. The middle panel shows simulation B4 with three levels of refinement, locally matching the evolution of the high resolution reference case. The refinement zone is shown with the light green circles where the highest resolution region is indicated with a solid line and the nested regions with dotted lines. The similarities in the streamers and circumprimary disc confirm our APR implementation is capable of accurately locally increasing the resolution of a simulation. A movie of these simulations is available online. \n<!-- image --> \nFigure 4. Mass of each sink in the HD142527 simulations with di/fferent combinations of refinement region sizes. The mass accretion rate reflects the properties of the disc surrounding the sink, confirming the structure of circumprimary disc in the APR simulations is the same as the high resolution reference calculation (as seen in Figure 3). \n<!-- image --> \ncles of different masses co-existing in the same simulation it is prudent to consider kernels other than the typical M 4 cubic spline (e.g. Dehnen and Aly 2012). Our comparison of M 4 , Wendland C2, Wendland C4 and Wendland C6 is shown in Appendix 1. Here we find that the M 4 cubic spline is robust to pairing in these tests but that the higher order splines offer a slightly smoother transition between the refinement zones. We thus chose the Wendland C2 as a balance between computational cost and accuracy but note that the cubic spline is likely sufficient for most purposes.", '2.1 Circumbinary disc': 'We repeat the concluding simulation from Price, Cuello, et al. (2018) of a circumbinary disc nominally representing the protoplanetary disc HD 142527 (but see Nowak et al. 2024). The star and disc parameters are the same as used in these works and our low resolution reference simulation has N = 1 × 10 6 particles, as in their work. Our high resolution reference simulation is the same as the low except it has double the spatial resolution with N = 8 × 10 6 particles. In our APR versions (simulations B1-B4) the refinement region has three nested levels of refinement ( ℓ = 3) and is centred on the centre of mass of the two stars so that the accretion streams and circumstellar discs are more highly resolved. As the gas has a negative radial velocity, we find that this application mostly represents a test of splitting - little if any merging occurs in these simulations. This is similar to the published applications of APR in GIZMO (Franchini, Lupi, and Sesana 2022; Duffell et al. 2024). \nFigure 3 shows a representative APR simulation of HD142527 book-ended by reference calculations. At t = 1137 years in the low resolution reference calculation we find a poorly resolved circumprimary disc consistent with Price, Cuello, et al. (2018), but this disc is recovered in the high resolution case on the far right. Our APR simulation shows the same disc structure around the primary star as in the high resolution case. The spiral arms at the inner edge of the disc are well recovered when they are within the \nFigure 5. Column density of the region surrounding a planet embedded in a disc shown in the corotating frame. The le/f\\_t and right panels have no APR and are separated by a factor of two in spatial resolution. The middle panel shows simulation P6 with three levels of refinement, matching the circumprimary disc structure and spiral arms in the high resolution reference simulation. The refinement zone is shown as in Figure 3. This test demonstrates that structures like spiral arms are faithfully reproduced even when they cross the refinement boundaries. A movie of these simulations is available online. \n<!-- image --> \nFigure 6. Mass and radius of the planet in the planet-disc simulation with di/fferent combinations of refinement region sizes. The APR simulations and high resolution simulations are again distinct from the low resolution reference simulation. \n<!-- image --> \nhighest refinement region (e.g. simulations B3 and B4). The consistency between the APR and high resolution case is maintained through the end of the simulation at t = 3283 years. \nFigure 4 shows how the mass of the two sink particles changes throughout the reference simulations and simulations B1-B4. The evolution of the mass for the APR simulations \nclosely follows the high resolution reference cases where the refinement region covers the cavity, indicating that the disc structure and thus the accretion rate is the same across these simulations. The low resolution simulation with the poorly resolved circumprimary disc accretes faster, resulting in a larger final mass. Simulations B1 and B2 show distinct behaviour after t ∼ 1000 years with up to ∼ 3 times more mass accreted onto both sinks due to their small refinement regions; the spiral arms and even the secondary sink are not consistently resolved inside the highest refinement region. For both B1 and B2 the lower resolution of the inner edge of the cavity then leads to about 30% more mass falling onto the sinks and thus the higher accretion rate for both. \nWe calculate the error, E , using the largest percentage difference in mass between the APR versions and high resolution reference case on each sink as \nE = max [ m APR ( t ) -m ref ( t ) m ref ( t ) × 100 ] . (6) \nWerepeat this calculation across both sinks and take the largest of these as the error in our simulation when compared to the high resolution reference case, summarised in Table 1. We find the largest error of 9.0 × 10 -3 %lies with the secondary sink of the B1 simulation, possibly because it ventures closer to the edge of the refinement region each orbit. In contrast to our other example applications, our APR simulations here show a speed-up compared to the globally high resolution case of ∼ 1.1 - 2.3 × faster (Table 1). The speed up is inversely proportional to the size of the refinement zone because it is centred on the binary; the tightest and most computationally expensive orbits are located here and our APR simulations add more particles to this region.', '2.2 Planet-disc interaction': 'We simulate the interaction of a planet embedded in a disc where the region around the planet is refined using APR. The disc extends from R in = 1 au to R out = 10 au with a total \ndisc mass of 0.05M ⊙ . We adopt typical protoplanetary disc values with a surface density profile Σ ∝ (1 -√ R in / R ) R -p with p = 1, a sound speed c s ∝ R -q with q = 0.25, a disc aspect ratio of H / R = 0.05 at the inner edge R = R in where H ≡ c s( R )/ Ω ( R ) is the scale-height, Ω ( R ) = √ GM ∗ / R 3 is the Keplerian angular velocity, M ∗ = 1M ⊙ is the central mass and R is the cylindrical radius. We also assume a Shakura and Sunyaev (1973) viscosity of α = 0.005, modelled with the method outlined in Lodato and Price (2010). Into this disc we add a planet with 0.1M J initially located at R p = 5 au, which is large enough to generate spirals in the disc but not so massive that the wakes of the spirals interact strongly with the planet. We set the accretion radius of the sink to be 0.25 R H = 0.04 au where R H is the Hill radius (Nealon et al. 2018). Our low and high reference simulations use N = 1 × 10 6 and N = 8 × 10 6 respectively. We conduct six APR simulations (P1-P6, see Table 1) varying the size of the refinement region and the size of the step around it, all with a maximum refinement level of ℓ = 3 (corresponding to the high resolution reference). \nOur simulations with a planet show the formation of spiral arms, indications of a circumplanetary disc and a shallow gap being carved. Figure 5 compares the column density of our low and high resolution reference case to the representative example P6. For all of these simulations we find that the spiral arms generated by the planet are recovered faithfully even as they cross multiple refinement levels. The column density contrast of the circumplanetary disc and the spiral arms are visually almost identical between our APR simulations and the high resolution reference case. \nFigure 6 shows the mass of the planet and radius measured from the central star. As with the HD 142527 simulations, the high resolution reference and the APR versions are qualitatively distinct from the low resolution case. Using Equation 6 we calculate the error from the mass but also equivalently from the radius, taking the largest percentage difference as the error reported in Table 1. We find the largest difference of 9.1% in the mass and 3.06% in the radius, both for simulation P5. While this error is the largest we recover with our APR implementation in the examples shown here it is still about 1/3 of the largest difference between the low and high resolution reference cases. Additionally, Nealon et al. (2018) showed that the choice of accretion radius has a stronger impact on the planet location and mass (their Figure A1) than the difference we have found here. In general we find that the error increases with both the size of and the width of the refinement region.', '2.3 Flyby encounter': "Smallwood et al. (2024) showed that there is a direct and robust relationship between the orientation of the disc formed from captured material in a flyby encounter and the original disc. With a broad suite of simulations, their results showed that the disc that formed around the perturber was always tilted twice as much as the original inclination between the perturber and the disc around the primary star. However the perturber disc in their simulations was generally underesolved (see their Figure A2). Smallwood et al. (2024) conducted a resolution \nstudy with 5 × 10 5 and 4 × 10 6 particles and determined that in order to well resolve the disc around the perturber they would need to model the original disc with ∼ 144 × 10 6 particles. \nWe repeat the calculation from Smallwood et al. (2024) at 5 × 10 5 particles for our low resolution reference and 4 × 10 6 as our high resolution reference. We then conduct four APR variations (F1 - F4) with the refinement region centred on the perturber star. The upper panel of Figure 7 shows the column density of the two reference cases and simulations F1 with a refinement region radius of 20 au and F4 with 40 au, about 300 years after pericentre passage (as in Smallwood et al. 2024). The two APR cases and the high resolution share the same disc structure but the tidal stream is different. In F1 with the smallest refinement region the tidal stream more closely resembles that of the low resolution reference simulation, in line with our expectations. \nAs before, Figure 8 shows the mass of the perturber sink throughout the simulations. We find that the high resolution reference case and APR versions describe the same mass accretion pathway and that it is distinct from the low resolution reference case. The error measured from the mass of the perturber sink is ≲ 10 -3 %for all combinations of APR used here and with speed ups of between 5.28 - 6.62 × faster than the high resolution reference case. \nAs in Smallwood et al. (2024) we also measure the resolution of the perturber disc for the different simulations. Figure 9 shows the ⟨ h ⟩ / H resolution at t = 2500 years after the pericentre encounter where ⟨ h ⟩ / H is the shell averaged smoothing length. We find that the high resolution reference case and APR versions produce the same profile, confirming that they are the same local resolution. In particular, the profiles for F1 and F2 deviate around R = 25au and 30 au respectively, near to the edge of the refinement region for these simulations. The factor of 1/2 in ⟨ h ⟩ / H between the APR simulations and the low resolution case demonstrates that the resolution has doubled (as expected from three refinement levels). \nThese example applications so far have demonstrated the accuracy and speed of the APR implementation by comparing to a globally high resolution reference case, but this is not the most powerful or intended use of APR. Here we perform an additional two simulations using APR on the flyby problem: simulation F5 which initially has N = 4 × 10 6 particles and uses three levels of refinement ( ℓ = 3) and simulation F6 which initially has N = 5 × 10 5 particles and uses six levels of refinement ( ℓ = 6). Around the perturber these simulations achieve the same local resolution in terms of local particle number. \nThe lower row of Figure 7 shows the discs formed around the perturber star for F5 and F6, finding the same disc structure as in the previous cases. These two simulations are also included in Figures 8 and 9. The mass accreted onto the sink in simulations F5 and F6 is 9.2 × 10 -4 %lower than the high resolution reference case. This difference is about half that between the low and high resolution reference cases (1.89 × 10 -3 %) and consistent with our previous observation that increasing the resolution decreases the mass accretion rate. Figure 9 demonstrates that the linear resolution of these two simulations is double the previous F1 - F4 simulations. We have thus shown \nFigure 7. Comparison of discs formed from captured material around the perturber in the flyby simulation (see Smallwood et al. 2024). The 'Low resolution' and 'High resolution' panels do not have APR and are separated by a factor of two in resolution. Simulations F1 and F4 initially have N = 5 × 10 5 particles with 3 levels of APR, F5 has N = 4 × 10 6 particles with 3 levels of refinement and F6 has N = 5 × 10 5 particles with 6 levels of refinement. The refinement zone is shown as in Figure 3. The disc structure is similar irrespective of the base resolution of the simulation but the tidal stream onto the disc depends on the size of the refinement region and the number of levels. A movie of these simulations is available online. \n<!-- image --> \nFigure 8. Mass of the perturber star in the flyby simulation for di/fferent combinations of refinement regions and maximum refinement levels. As before, increasing the resolution using APR leads to a lower mass accretion rate and the similarities between the APR and high resolution reference evidence the similarity in the disc structure around the perturber star. \n<!-- image --> \nFigure 9. Quantifying the resolution of the perturber disc in the flyby calculations using ⟨ h ⟩ / H as a function of R . Three levels of refinement corresponds to a factor of two in linear resolution and this is recovered here by the ⟨ h ⟩ / H decreasing by about half for three refinement levels. Colour scheme is the same as in Figure 8. \n<!-- image --> \nthat with just 5.7 × 10 6 (F5) and 3.0 × 10 6 particles we can achieve an equivalent resolution around the perturber of simulation that uses 32 × 10 6 particles globally.", '3. Discussion': "Our results demonstrate that our method is accurate and fast (see Table 1) with the most rapid speed up seen in the flyby simulations and the slowest in the circumbinary disc simulations. In the flyby case the majority of the splitting/merging occurred during the actual flyby encounter and the time-step constraining orbits were around the primary star which we did not change in our APR simulations. Together, these led to a significant speed up in the simulation. By contrast, in the circumbinary disc we increased the resolution in the region of the simulation where particles had the largest acceleration and thus the shortest time-steps. We still found a speed up in that case because of our use of individual time-steps (Price, Wurster, et al. 2018). In the planet-disc interaction simulations we found constant splitting and merging occurred as the variation in the orbital speed meant particles were continually entering and exiting the refinement region. For those simulations the size of the refinement region generally dictated the speed up, with smaller regions corresponding to faster speed ups. \nThe circumbinary disc and flyby tests demonstrated the importance of where to set the refinement region. In the former, the largest error was measured when the inner edge of the circumbinary disc was not adequately resolved (simulations B1 and B2) such that material was entering the cavity with the same rate as the low resolution reference case and occasionally, the secondary star popped out of the refinement region. But when the inner edge of the circumbinary disc and the complete orbit of the secondary was captured in the refinement region (B3 and B4) the error decreased significantly as compared to the high resolution reference case. This effect was also noted by Franchini, Lupi, and Sesana (2022), who recommend setting the inner edge of the refinement region to 4 × the semi-major axis. \nOur implementation is also designed to be as adaptable as possible. Our use of the k -d tree in P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc for merging means that our APR method can naturally be used for derefinement, where the APR zone has a lower resolution than the global simulation. This is particularly useful when increases in density result in very small time-steps that effectively kill a simulation - commonly experienced in simulations including self-gravity (e.g. Longarini et al. 2023; Lau et al. 2022; Hall, Forgan, and Rice 2017). Our method is also easily extendable to multiple refinement regions (e.g. refining separately around two planets in a disc). \nAn important and desirable property of SPH is its conservation properties. Conservation of kinetic energy and both linear and angular momentum are respected with APR when the children particles inherit the velocity of the parent (Feldman and Bonet 2007; López, Roose, and Recarey Morfa 2013). While our inclusion of relaxing means that children may not necessarily be placed symmetrically around the parent particle this only affects the conservation of angular momentum. Our \nmethod perfectly conserves mass and in simulations with accretion (where we may get an odd number of particles to be merged) our implementation caps the number of unmerged particles to be ℓ max - 1 (Equation 4). In addition to this, APR in P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc is subjected to the regular conservation checks (see Section 2.2, Price, Wurster, et al. 2018). \nThe current limitation of our method is a ∼ 5% blip in density that occurs every time particles are either split or merged. In our 3D simulations this 'blip' in density is cosmetic (it can just be seen in the middle panel of Figure 5) but is more obvious in our tests on small amplitude linear sound waves outlined in Appendix 1 (although it does not seem to affect the propagation of the wave itself ). We tested different methods to mitigate this feature and found that reducing n child was ultimately the most effective measure, setting our limitation of n child = 2 and thus having nested refinement regions. Importantly, the 'blip' feature is also visibly present in the GIZMO implementation of APR (see density renderings in Duffell et al. 2024) which demonstrates that it is not a function of our method, but is inherent to the process of changing the particle mass. \nThe most promising method to remove this blip is the blending method outlined by Barcarolo et al. (2014) which allows particles to be introduced in a way that removes the noise before they contribute to the density summation. Although successful for incompressible flows, blending is practically very difficult to implement with compressible flows due to mass conservation in the blending zone. For example, we could successfully incorporate blending as in Barcarolo et al. (2014) in our initial conditions, but once particles started moving through the blending region we found that the solution deteriorated rapidly. We additionally implemented other forms of relaxing (e.g. Lind et al. 2012; Diehl et al. 2015; Sun et al. 2017) and other forms of blending (Chiron et al. 2018; Gao, Qiu, and Fu 2022) but did not find a satisfactory improvement on the blip found in the simulations.", '4. Conclusion': 'We introduce a live adaptive particle refinement implementation into the smoothed particle hydrodynamics code P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc. We have considered example applications of a circumbinary disc, planet-disc interaction and a flyby to demonstrate our method. With these examples we have shown that our implementation is \n- 1. Accurate: We measure the mass accreted onto sink particles as a proxy of the disc structure in our simulations. For the circumbinary and flyby examples, we find that the APR calculations are accurate to < 0.1%. For the planet-disc interaction example we measure both the mass and the radial location of the planet and find that it is accurate to at least 9% (but is generally more accurate than this).\n- 2. Fast: Every APR simulation showed a speed up, offering between 1.07-6.62 × than when compared to a simulation with the same resolution but globally. The speed up is application-dependent, with the flyby example being the most rapid and the circumbinary disc the least amenable to speedup in this manner. \n- 3. Uses less storage: Because our APR simulations use fewer particles in total, they require between 15 - 27% of the storage of an equivalent globally resolved calculation. \nOur example applications suggested optimal sizes of the refinement region as a guide for future calculations. We found accuracy of the implementation depends sensitively on whether or not key features were uniformly resolved. We also demonstrated that the location of the refinement region can be dynamic and note that derefinement is possible. Finally we showed that APR can increase the resolution of simulations at low cost; for the flyby example we achieved a local resolution of 32 million particles with totals of either 3.0 or 5.7 million particles. \nWe showed examples limited to hydrodynamical simulations, but P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc also includes dust (Guillaume Laibe and Daniel J. Price 2012; G. Laibe and D. J. Price 2014), magnetohydrodynamics (Tricco and Price 2012), self-gravity, general relativity (Liptai and Price 2019) and can be coupled with MCFOST (Pinte et al. 2006; Pinte et al. 2009; Nealon, Price, and Pinte 2020). For brevity we leave dedicated testing of our APR implementation with these features to future works.', 'Acknowledgement': 'The authors thank James Wurster for significant contributions in the early stages of this work and the referee for helpful suggestions. RN acknowledges Matthew Bate, Richard Booth, Terry Tricco and Alessia Franchini for discussions and Sahl Rowther for aesthetics assistance. \nFunding Statement R.N. acknowledges support from UKRI/EPSRC through a Stephen Hawking Fellowship (EP/T017287/1). This work was performed using Avon, the HPC clusters at the University of Warwick. DP is grateful for Australian Research Council funding via DP220103767 and DP240103290. 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AAp 459 (December): 797-804. https: //doi.org/10.1051/0004-6361:20053275. eprint: astro-ph/0606550. \nPorter, D. H. 1985. A Study of Hierarchical Clustering of Galaxies in an Expanding Universe. PhD diss., University of California, Berkeley, January. \nPrice, N. Cuello, C. Pinte, D. Mentiplay, S. Casassus, V. Christiaens, G. M. Kennedy, et al. 2018. Circumbinary, not transitional: on the spiral arms, cavity, shadows, fast radial flows, streamers, and horseshoe in the HD 142527 disc. MNRAS 477 (June): 1270-1284. https://doi.org/10.1093/ mnras/sty647. arXiv: 1803.02484 [astro-ph.SR] . \nPrice, Daniel J., David Liptai, Ilya Mandel, Joanna Shepherd, Giuseppe Lodato, and Yuri Levin. 2024. Eddington envelopes: The fate of stars on parabolic orbits tidally disrupted by supermassive black holes. arXiv e-prints (April): arXiv:2404.09381. https://doi.org/10.48550/arXiv.2404.09381. arXiv: 2404.09381 [astro-ph.HE] . \nPrice, James Wurster, Terrence S. Tricco, Chris Nixon, Stéven Toupin, Alex Pettitt, Conrad Chan, et al. 2018. Phantom: A Smoothed Particle Hydrodynamics and Magnetohydrodynamics Code for Astrophysics. PASA 35 (September): e031. https://doi.org/10.1017/pasa.2018.25. arXiv: 1702.03930 [astro-ph.IM] . \nShakura, N.I., and R.A. Sunyaev. 1973. Black Holes in Binary-Systems - Observational Appearance. AAp (17, AVE DU HOGGAR, PA COURTABOEUF, BP 112, F-91944 LES ULIS CEDEX A, FRANCE) 24 (3): 337-355. /I.sc/S.sc/S.sc/N.sc: 0004-6361. \nSmallwood, Jeremy L., Rebecca Nealon, Nicolás Cuello, Ruobing Dong, and Richard A. Booth. 2024. Formation of misaligned second-generation discs through fly-by encounters. MNRAS 527, no. 2 (January): 20942109. https://doi.org/10.1093/mnras/stad3057. arXiv: 2310.03327 [astro-ph.EP] . \nSpringel, Volker. 2005. The cosmological simulation code GADGET-2. MNRAS 364, no. 4 (December): 1105-1134. https://doi.org/10.1111/j.13652966.2005.09655.x. arXiv: astro-ph/0505010 [astro-ph] . \nSun, P.N., A. Colagrossi, S. Marrone, and A.M. Zhang. 2017. The δ plussph model: simple procedures for a further improvement of the sph scheme. Computer Methods in Applied Mechanics and Engineering 315:2549. /I.sc/S.sc/S.sc/N.sc: 0045-7825. https://doi.org/https://doi.org/10.1016/j.cma. 2016.10.028. https://www.sciencedirect.com/science/article/pii/ S0045782516309112. \nThacker, R. J., and H. M. P. Couchman. 2000. Implementing Feedback in Simulations of Galaxy Formation: A Survey of Methods. ApJ 545, no. 2 (December): 728-752. https : / / doi . org / 10 . 1086 / 317828. arXiv: astro-ph/0001276 [astro-ph] . \nTricco, Terrence S, and Daniel J Price. 2012. Constrained hyperbolic divergence cleaning for smoothed particle magnetohydrodynamics. Journal of Computational Physics 231 (21): 7214-7236. \nTruelove, J. Kelly, Richard I. Klein, Christopher F. McKee, II Holliman John H., Louis H. Howell, and Jeffrey A. Greenough. 1997. The Jeans Condition: A New Constraint on Spatial Resolution in Simulations of Isothermal Self-gravitational Hydrodynamics. ApJL 489, no. 2 (November): L179-L183. https://doi.org/10.1086/310975. \nVacondio, R., B. D. Rogers, P. K. Stansby, P. Mignosa, and J. Feldman. 2013. Variable resolution for SPH: A dynamic particle coalescing and splitting scheme. Computer Methods in Applied Mechanics and Engineering 256 (April): 132-148. https://doi.org/10.1016/j.cma.2012.12.014. \nWendland, H. 1995. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in computational Mathematics 4 (1): 389-396. \nYang, Xiufeng, and Song-Charng Kong. 2019. Adaptive resolution for multiphase smoothed particle hydrodynamics. Computer Physics Communications 239:112-125. /I.sc/S.sc/S.sc/N.sc: 0010-4655. https://doi.org/https://doi.org/10. 1016/j.cpc.2019.01.002. https://www.sciencedirect.com/science/article/ pii/S0010465519300037.", 'Appendix 1. Simple tests': 'To demonstrate some of the key choices of our implementation we use the straightforward example of a sound wave in a periodic box simulated with P/H.sc/A.sc/N.sc/T.sc/O.sc/M.sc. The density is set by \nρ ( x ) = ρ 0 + A sin ( 2 π x L ) , (7) \nwhere A = 0.02 is the amplitude of the perturbation and ρ 0 = 1. The thermal energy is perturbed in the same manner with the same amplitude A but scaled by P 0 / ρ 0 , the unperturbed pressure and density. The fluid has an unperturbed sound speed of c s,0 = 1.0 and we adopt an adiabatic equation of state. Unless otherwise stated the velocity is set to vx = 0.001, vy = vz = 0. The particles are initially set on a close-packed lattice with the above density distribution. We add noise to the lattice by randomly adjusting the x and y positions of the particles by 0.0001 × the length of the box. We use 64 particles across the box of width dx = 1.0 and dy = 1.0, corresponding to a total of N = 56832 particles initially. \nThe refinement region is a circle centred at x = 0.5, y = 0.5 with a radius of r = 0.10. In contrast to our example applications, these simulations only make use of one refinement level ( ℓ = 1) which corresponds to a doubling of the particle number inside this region. The first split occurs at t = 0.5 to allow the particles to settle a bit first and the simulation is then performed until t = 2.0, corresponding to five wave-crossings in total. After the split the simulation has around N = 63500 particles. \nOur results in this section are all shown with the cross section of the density, ρ ( x ), across the box divided by the initial average density, ρ IC , to avoid small variations in the background density caused by kernel bias. For clarity we also restrict the visualisation to only include particles between 0.4 < y < 0.6, essentially making it a cross section centred on y = 0.5. On each plot we also display the average density as a function of position across the box for both the APR simulation and a reference simulation without any refinement. This allows us to easily compare how much noise is introduced as a result of the splitting/merging and how accurate these simulations are compared to the simulations that do not include APR. Movies of some of the simulations in this section are available online.', "Appendix 1.1 Density discontinuity - the 'blip'": "A common feature in all of our simulations is a density discontinuity at the refinement boundary (i.e. 'the blip' between the different resolution regions). It appears as a slight increase in the density just outside the refinement region and a slight decrease immediately inside. This discontinuity does not introduce spurious waves, has a constant amplitude and does not appear to affect the density profile inside the refinement region. Its apparent prominence is because of our use of a comparatively small amplitude sound wave as our test case we chose it specifically so that any spurious features would be easily identifiable. This feature maintains a constant amplitude (here, of < 5% of the original density) and in our examples in Section 2 is barely visible in renderings (e.g. Figure 5), \nsimilar to the density profiles seen in examples with GIZMO (Anglés-Alcázar et al. 2021; Franchini, Lupi, and Sesana 2022; Duffell et al. 2024). Importantly, the blip does not seem to affect the propagation of the wave itself. \nThis density blip is a manifestation of a known issue in APR, where the splitting and merging procedure introduces small errors into the density distribution. Feldman and Bonet (2007) formally established that an error in the density distribution is introduced each time a parent particle is replaced by children particles when the children all have the same mass. When a split occurs and the children particles are introduced, they should be placed symmetrically around the parent to ensure conservation of angular momentum (e.g. López, Roose, and Recarey Morfa 2013). However, the introduction of these new particles also introduces this error, the magnitude of which is a function of the distance the children are placed at r sep. In turn, this means the error can be minimised by adjusting the locations of the children. \nLópez, Roose, and Recarey Morfa (2013) extended this idea to consider the error introduced in the derivative of the density, suggesting that this was the more relevant error to minimise for subsequent calculations. Further works looked at how best to minimise this error that is introduced every time particles are split or merged with various r sep (e.g. López, Roose, and Recarey Morfa 2013), particle shuffling/relaxing (e.g. Lind et al. 2012; Diehl et al. 2015; Sun et al. 2017), the use of ghost particles and blending techniques (e.g. Barcarolo et al. 2014; Chiron et al. 2018; Gao, Qiu, and Fu 2022) to mitigate this at the boundaries of refinement regions. Barcarolo et al. (2014) eliminated this density blip by employing a blending zone between each refinement level, where both children and parents co-exist but their contribution to the density summation is graduated by their distance across the blending zone. This approach was also adopted by Gao, Qiu, and Fu (2022) but only as new particles moved into the higher refinement level, allowing them to regularise first. While promising, blending is difficult to implement in compressible flows due to mass conservation in the blending zone. \nThe alternative is particle relaxation (shuffling): to minimise the error introduced when particles were split, Diehl et al. (2015) used a WVT shuffling method to rearrange the particles to a lower error state compared to the original parent distribution. Yang and Kong (2019) employed multiple, stepped refinement regions to control the ratio of the smoothing lengths across the boundaries and mitigate instabilities (similar to Børve, Omang, and Trulsen 2001). In our implementation we employ relaxation, but find that it is only important when a large number of particles are split or merged at once.", 'Appendix 1.2 Kernel choice': "Figure 10 shows the final snapshot of our test simulation comparing the use of the Cubic spline (Monaghan and Lattanzio 1985), Wendland C2, Wendland C4 and Wendland C6 kernels (Wendland 1995), summarised in Table 2. The density profile in the refinement region (blue line) is in excellent agreement \nFigure 10. Wave in a box test showing the e/ffect of di/fferent kernels with APR. The blue points show the particles within the y = 0.5 cross-section across the x dimension of the box at t = 2.0 . The dark blue line shows their average density, while the red line shows the average density of the same simulation without APR. The density discontinuity at the refinement boundary maintains a constant amplitude across all of the simulations. \n<!-- image --> \nwith the expected profile (red line) for all kernel choices. As might be expected, the kernels with more neighbours have less noise inside the refinement region and at the boundary. In particular, the cubic has the largest noise at the refinement region boundary, the largest spread in density inside the refinement region and has the least accurate density profile (although the difference here is marginal). \nFigure 11 shows our planet disc interaction test at t = 10 orbits but with different combinations of r sep and either a Cubic, Wendland C2 or Wendland C4 kernel. Here we have zoomed in to see the boundaries of the refinement zone to examine the effect of these choices. Moving between kernels, the WC4 in the lowest row is smoothest while the cubic has the largest noise at the boundaries. For any of the kernels, increasing r sep beyond 0.2 also corresponds to an increase in the noise at the boundary (seen most clearly in the WC4 case). Both of these tests confirm that a higher order kernel is more effective at mitigating the density 'blip' but that the difference is marginal. For our example applications we adopt the Wendland C2 kernel as a balance between accuracy and computational expense.", 'Appendix 1.3 Adjusting the split': 'Figure 12 demonstrates the effectiveness of both particle relaxing (as per Section 1.4) and directional splitting, where particles are split tangentially to the refinement boundary. Importantly, once the particles have settled after their initial split (Figure 12, top panel) there is no significant difference between the methods in the density profiles. However, their time to compute is quite different with the simulations that include relaxing \nFigure 11. Planet disc interaction test showing the e/ffect of di/fferent kernel choices and r sep on the noise introduced at the refinement boundary in a full simulation. Refinement boundaries are indicated in the same way as Figure 3. The boundary is smoothest for the Wendland C4 kernel when r sep = 0.2 but we note the di/fference is marginal. \n<!-- image --> \nTable 2. Summary of the h fact employed and the average number of neighbours N neigh for the kernels tested in Figure 10 (e.g. Price, Wurster, et al. 2018). \ntaking much longer. In practice, we thus adopt directional splitting as it is fast, gives an improved initial density estimate and prevents particles from being immediately placed across the boundary into a low resolution region when they are split. To improve computational time by default we only employ the relaxing routine when refinement regions are activated and there is a large number of particles that are split or merged at once.', 'Appendix 1.4 Number of children': 'In our tests so far we have used n child = 2, splitting parents into two children and merging two children to become a parent. In Figure 13 we show the impact of this choice by considering n child = 4 and n child = 8. We compare this to simulations with n child = 2 and ℓ = 2, 3 respectively which have equivalent local resolutions. \nTosplit one parent into four children we follow the method outlined in Section 1.2 and treat those two children as the diagonal corners of a square. We then add an additional two particles on the opposing diagonal which ensures that the face of the square of four children particles is parallel to the refinement boundary. For eight children we create two squares that are offset by 0.1 × the smoothing length of the original parent, with the centre of mass of the resultant cube centred on the original parent. For our merging routines we simply edit our modified k -d tree to return cells with n child = 4 and n child = 8. \nFigure 13 shows the planet disc interaction test at t = 4 orbits for these choices, where the centrally refined zone has the same width ( r ℓ = 0.35au and the steps into the refinement \nFigure 12. Wave in a box test showing the e/ffect of di/fferent particle placement options when a split or merge occurs shown at the initial split (upper, t = 0.5 ) and at the end of the simulation (lower, t = 2.0 ). The colours are the same as in Figure 10. Relaxing is most successful when the split first occurs but makes negligible di/fference during the course of the simulation. Directional splitting also makes little di/fference in the long term but does prevent particles from splitting across the boundary. \n<!-- image --> \nzone at increments of 0.10au). The noise introduced by the nested zones is visibly lower in both cases; in the n child = 8 case the noise is the largest and is also prominent across the whole refined region, even after it has had the opportunity to settle after the first split. We note that while this test may be improved by relaxing at each step, in practice this becomes computationally extremely expensive. This implies that using a smaller number of children - even if it means having nested refinement regions - is the preferred approach. In other words, even four children is too many. \nFigure 13. The planet disc interaction test, examining the e/ffect of having di/fferent n child . The le/f\\_t column has n child = 2 with nested refinement levels, the upper right has n child = 4 and one level of refinement, the lower right n child = 8 and one level. The refinement zones are shown as in Figure 3. Rows have the same local resolution around the planet. While the nested refinement zones do add noise at the boundaries, this is demonstrably less than bigger numbers of children. \n<!-- image -->', 'Appendix 1.5 Size of the APR zone': 'We test the effect of the size of the refinement region using the jet test presented in Price et al. (2024), Appendix E, Figure 14. In this test a jet of gas shoots in one direction from a fixed location in an empty domain, the gas flares out as it expands slightly. Based on the results of Price et al. (2024) we set the Mach number of the gas to 10 to ensure a fairly narrow jet. The particles are injected as a cylinder with radius R = 1 16 particles in each layer. The rest of the parameters in this simulation are scale free. \nWe apply four different APR zones to this jet test with ℓ = 1 for all but r ℓ = 3, 5, 10 and 15. The refinement boundary of each different zone has the same position of y = 5 so that particles are always split at the same location but they are merged at different distances from the launching point of the jet. Figure 14 shows this with the refinement zone indicated with a green dashed circle. We chose an injection velocity of vy = 37.5 and calculate how many sound crossing times the particles will experience between refinement and derefinement to be < 1, 1.3, 2.7 and 4.0. \nFigure 14 shows the column density of the jets at t = 50 with the refinement zones superimposed. For the tests where particles have fewer sound crossing times we find that there is a slight narrowing of the jet within the refinement zone but for the final test where particles have several sound crossing times this narrowing is not apparent. This test demonstrates that particles require several sound crossing times between the refinement boundaries to allow them to relax before they are derefined. \nFigure 14. Testing the size of the refinement zone, where the four simulations are characterised by the number of sound crossings that can occur between the refinement and derefinement boundaries. The refinement region is indicated with a green circle in the figure. \n<!-- image -->'}

2024ApJ...974L...8L

The largest geomagnetic storm in two decades occurred in 2024 May with a minimum D SUBstSUB of 412 nT. We examine its solar and interplanetary origins by combining multipoint imaging and in situ observations. The source active region NOAA AR 13664 exhibited extraordinary activity and produced successive halo eruptions which were responsible for two complex ejecta observed at the Earth. In situ measurements from STEREO A which was 12.6 apart allow us to compare the geoeffectiveness at the Earth and STEREO A. We obtain key findings concerning the formation of solar superstorms and how mesoscale variations of coronal mass ejections affect geoeffectiveness 1 the 2024 May storm supports the hypothesis that solar superstorms are perfect storms in nature i.e. a combination of circumstances resulting in an event of an unusual magnitude 2 the first complex ejecta which caused the geomagnetic superstorm shows considerable differences in the magnetic field and associated geoeffectiveness between the Earth and STEREO A despite a mesoscale separation and 3 two contrasting cases of complex ejecta are found in terms of the geoeffectiveness at the Earth which is largely due to different magnetic field configurations within the same active region.

2024-10-01T00:00:00Z

['arXiv:2409.11492', '2024arXiv240911492L', '2024ApJ...974L...8L', '10.48550/arXiv.2409.11492', '10.3847/2041-8213/ad7ba4']

['Shocks', 'Solar-terrestrial interactions', 'Solar wind', 'Solar coronal mass ejections', '2086', '1473', '1534', '310', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Physics - Space Physics']

A Pileup of Coronal Mass Ejections Produced the Largest Geomagnetic Storm in Two Decades

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

https://arxiv.org/pdf/2409.11492.pdf

{'A Pileup of Coronal Mass Ejections Produced the Largest Geomagnetic Storm in Two Decades': 'Ying D. Liu 1 , 2 , Huidong Hu 1 , Xiaowei Zhao 3 , 4 , Chong Chen 5 , and Rui Wang 1', 'ABSTRACT': "The largest geomagnetic storm in two decades occurred in 2024 May with a minimum D st of -412 nT. We examine its solar and interplanetary origins by combining multipoint imaging and in situ observations. The source active region, NOAA AR 13664, exhibited extraordinary activity and produced successive halo eruptions, which were responsible for two complex ejecta observed at the Earth. In situ measurements from STEREO A, which was 12 . 6 · apart, allow us to compare the 'geo-effectiveness' at the Earth and STEREO A. We obtain key findings concerning the formation of solar superstorms and how mesoscale variations of coronal mass ejections affect geo-effectiveness: (1) the 2024 May storm supports the hypothesis that solar superstorms are 'perfect storms' in nature, i.e., a combination of circumstances resulting in an event of an unusual magnitude; (2) the first complex ejecta, which caused the geomagnetic superstorm, shows considerable differences in the magnetic field and associated 'geo-effectiveness' between the Earth and STEREO A, despite a mesoscale separation; and (3) two contrasting cases of complex ejecta are found in terms of the geo-effectiveness at the Earth, which is largely due to different magnetic field configurations within the same active region. \nSubject headings: shock waves - solar-terrestrial relations - solar wind - Sun: coronal mass ejections (CMEs)", '1. Introduction': "In 2024 May the Sun exhibited substantial activity. A large complex active region resulted from the merging of NOAA AR 13664, which appeared on the east limb on April 30, and NOAA AR 13668, which emerged a few days later. Since the two active regions cannot be separated, we treat them as a single complex active region, referred to hereinafter as NOAA AR 13664. The complex active region disappeared on the west limb (i.e., rotated to the backside of the Sun) on May 13. As it rotated with the Sun from the east to west, it produced a series of M/X class solar flares and coronal mass ejections (CMEs). A prolonged geomagnetic storm began on May 10 and reached a minimum D st of -412 nT. The D st index measures the intensity of geomagnetic storms: the more negative value, the more intense the storm. The only storm in the 21st century stronger than the current one is the 2003 November storm, which had a minimum D st of -422 nT. \nThe 2024 May storm can be classified as an extreme space weather event, i.e., a lowprobability but high-consequence event otherwise called a solar superstorm (National Research Council 2008; Cannon et al. 2013; Riley et al. 2018). A well-known example of extreme space weather before the space era is the 1859 Carrington event (Carrington 1859), with a minimum D st estimated to be about -850 nT (Siscoe et al. 2006). The most severe event of the space age is the 1989 March storm with a minimum D st of -589 nT (Allen et al. 1989). However, if the term 'solar superstorms' includes extreme solar wind disturbances that do not necessarily hit the Earth, then we have witnessed more cases, and extreme space weather may be more frequent than we imagine. In the last solar cycle (solar cycle 24), two solar superstorms occurred despite it being a historically weak cycle. The first was the 2012 July event (e.g., Baker et al. 2013; Russell et al. 2013; Liu et al. 2014), which was Carrington-class, and the second was the 2017 July event (Liu et al. 2019), which could rival the 1989 March storm. Although they did not hit the Earth, we infer that these two events would have produced geomagnetic superstorms if they had been Earth directed. \nIn addition to creating a historic record, the 2024 May storm, with modern imaging and in situ observations from multiple vantage points, provides a significant opportunity to study the formation of solar superstorms. Important clues have already been learned from the 2012 July and 2017 July events. Based on the observations of the 2012 July event, Liu et al. (2014) suggest a 'perfect storm' scenario for the formation of an extreme event at 1 AU: in-transit interaction between successive eruptions to preserve the magnetic field, plus a preconditioning effect to prevent deceleration (i.e., one or several earlier eruptions clear the path). Liu et al. (2019) also performed a comparative study of the 2012 July and 2017 July events. They identify similar patterns in both cases: 'a long-lived eruptive nature of the source active region, a complex event composed of successive eruptions from \nthe same active region, and in-transit interaction between the successive eruptions resulting in exceptionally strong ejecta magnetic fields at 1 AU.' Liu et al. (2019) indicate that the concept of 'preconditioning' is a necessary condition for a Carrington-class storm to occur. They further propose a hypothesis that solar superstorms are essentially 'perfect storms,' i.e., a combination of circumstances that produces an event of an unusual magnitude. It is worth mentioning that historical records of some extreme events support this hypothesis. For instance, revisits of the 1859 Carrington and 1989 March events with modern perspectives suggest that both resulted from successive CMEs from the same active regions (Green & Boardsen 2006; Boteler 2019). The recent 2022 September 5 CME, which is also considered as an extreme event, occurred from a complex active region with high productivity of flares and CMEs (Paouris et al. 2023); the authors also highlight the importance of preconditioning as a key factor for a high speed at 1 AU. It is intriguing to see how the 2024 May case will fit or challenge this hypothesis. \nAnother merit of the 2024 May storm is that it allows us to investigate how the magnetic structure of CMEs varies across a mesoscale distance 1 , and in particular, how the mesoscale variations affect geo-effectiveness. It has in situ measurements at both the Earth and STEREO A (see below), which were separated in longitude by about 12 . 6 · at that time. The magnetic fields of the CMEs at the Earth and STEREO A could be quite different, even if their flux-rope orientations are the same at the two points. This may result in different D st values (i.e., different 'geo-effectiveness') at the two points. In the STEREO era, the first CME that was measured in situ by multiple spacecraft at a mesoscale separation was the 2007 May event (Liu et al. 2008; Kilpua et al. 2009; Mostl et al. 2009), during which the Earth and STEREO B were about 3 · apart. Liu et al. (2008) examined the magnetic field measurements of the CME at the Earth and STEREO B. They find that, while the measurements at the two points are consistent with a flux-rope geometry, they show a considerable difference in the magnetic field (both strength and components) even for a 3 · separation. Similar results have been obtained with multipoint measurements of a few degrees apart (e.g., Lugaz et al. 2018; Regnault et al. 2024; Palmerio et al. 2024). Regnault et al. (2024) warn that, if in situ measurements at a mesoscale separation are used to predict the southward \nfield component at the Earth, a large error may result. The current case is a superstorm, and we have the rare opportunity to compare the 'geo-effectiveness' of a superstorm at two points separated by a mesoscale distance. \nWhile this Letter may serve as a description of the timeline leading up to the geomagnetic superstorm, it also attempts to: (1) provide a timely analysis of the solar and interplanetary origins of the 2024 May geomagnetic superstorm; (2) examine if the 2024 May storm conforms to the hypothesis of Liu et al. (2019) that solar superstorms are 'perfect storms' in nature; and (3) investigate how the mesoscale variations of CME magnetic fields affect geo-effectiveness. We first identify full halo eruptions that are the most likely candidates impacting the Earth. We then perform reconstruction of the CMEs to determine their parameters near the Sun. Time-elongation maps from wide-angle imaging observations are constructed in order to see their propagation behaviors in the Sun-Earth space, which may help enhance geo-effectiveness. We also examine the solar wind signatures at 1 AU and their connections with the geomagnetic storm. Modeling of the D st index is carried out, which allows us to compare the 'geo-effectiveness' between two different points. We describe multipoint imaging and in situ observations in Section 2, and conclude in Section 3.", '2. Observations and Results': "Figure 1 shows the positions of the spacecraft in the ecliptic plane as well as the propagation directions of some full halo CMEs. STEREO A was about 0.96 AU from the Sun and about 12 . 6 · west of the Earth on May 10. The Parker Solar Probe (PSP) was near its aphelion, i.e., about 0.74 AU from the Sun and around 95 . 4 · west of the Earth. Solar Orbiter (SolO) was located at about 0.69 AU and about 167 . 3 · west of the Earth. At L1 upstream of the Earth we have imaging observations from SOHO and in situ measurements from Wind, while STEREO A provides both imaging and in situ observations. The longitudinal separation between the Earth and STEREO A at almost the same distance is appropriate for an investigation of mesoscale variations of CME magnetic structures and associated 'geo-effectiveness.'", '2.1. White-light Imaging Observations': 'We first identify full halo CMEs from LASCO imaging observations aboard SOHO. These are the most likely candidates that could have impacted the Earth. We start from May 1 when the active region appeared on the east limb, but all full halo CMEs occurred starting \nfrom May 8. Table 1 provides the information about the full halo eruptions, including the associated flares and source locations on the Sun. They are successive eruptions, all from NOAA AR 13664. The source location is determined from the brightest flare kernel on the Sun. The CME parameters are estimated with a graduated cylindrical shell (GCS) technique, which assumes a rope-like morphology for CMEs with two ends anchored at the Sun (Thernisien et al. 2006, 2009). Both views from SOHO and STEREO A are used in the modeling. Note that the CMEs also appeared as full halo events for STEREO A, since it was only 12 . 6 · apart from the Earth. For each event, the flux-rope tilt angle, aspect ratio and half angle are assumed to be fixed in the associated time series of GCS modeling; the propagation longitude and latitude are allowed to vary, but we observe little variations in these two parameters. The propagation longitude and latitude given in Table 1 are average values from the time series of the modeling. The CME propagation directions are generally consistent with the source locations on the Sun, although deviations are observed. For each CME, the velocity is derived from a linear fit of the leading-front distances in GCS reconstruction; projection effects are already removed. All the velocities in Table 1 are below 2000 km s -1 , so the CMEs are not extreme in terms of speeds. The flux-rope tilt angles listed in Table 1 may have large uncertainties, as the CMEs are full halos for both spacecraft. \nThe turbulent corona and inner heliosphere during the time period can be seen from the time-elongation maps in Figure 2, which are produced by stacking the running-difference images within a slit along the ecliptic plane (e.g., Sheeley et al. 2008; Davies et al. 2009; Liu et al. 2010). Also shown is the GOES X-ray flux (Figure 2 top), so we can associate the CMEs with the flares (EUV images are also examined for this association). Many CMEs occurred, but our focus is the full halo eruptions, which are revealed by the maps from both SOHO and STEREO A. Given the clustering of the CMEs from the same active region, some of them may interact during their propagation in the corona and inner heliosphere. We observe intersections of some tracks in HI1 and HI2 of STEREO A (Figure 2 bottom), which indicate CME-CME interactions. In particular, the CMEs from May 8 (at least) seem to merge into a large, bright front in HI2. According to our experience of connecting wide-angle imaging observations with in situ signatures, the merging of multiple tracks into a bright front in time-elongation maps implies in situ signatures of a forward shock followed by a complex ejecta (see Liu et al. 2012, 2020). This is also what we expect at the Earth (which was at an elongation of about 97 . 9 · ) in the current case. \nComparing the tracks to the observed shock arrival times at the Earth is helpful to establish the connections between the CMEs and the in situ signatures observed at the Earth. The CME parameters from GCS modeling also provide some clues. We observe an increasing trend in the CME velocities for CMEs 1 - 4 and another increasing trend for \nCMEs 5 - 7 (see Figure 1 and Table 1). The speed distributions, together with the timing, suggest that CMEs 1 - 4 would pile up and interact, and that CMEs 5 - 7 would do the same. If this is true, we may expect two complex ejecta at the Earth, one from the merging of CMEs 1 - 4 and the other from the merging of CMEs 5 - 7. CME 8 is not relevant, as its direction is far too westward and its launch time is far too late (compared to the occurrence of the geomagnetic storm). Note that the average direction of CMEs 1 - 4 is more towards STEREO A than towards the Earth (see Figure 1), so we may anticipate a more head-on collision with STEREO A.', '2.2. In Situ Measurements at Wind': "The in situ measurements at Wind are shown in Figure 3. We see complex magnetic field and plasma signatures. In general, two complex ejecta can be identified from the measurements. As defined by Burlaga et al. (2001, 2002), complex ejecta result from interactions between successive CMEs and usually do not have well ordered magnetic fields at 1 AU. Complex ejecta can be very geo-effective because of their prolonged durations and enhanced magnetic fields (e.g., Farrugia & Berdichevsky 2004; Zhang et al. 2007; Lugaz & Farrugia 2014; Liu et al. 2015, 2020; Mishra et al. 2015). Our identification of the complex ejecta is based on a combination of the depressed proton temperature, low proton β , and magnetic field signatures (such as enhancement, smoothness and indication of rotation), but this is a subjective undertaking. The second complex ejecta resembles a typical ICME (interplanetary counterpart of a CME), given the declining speed profile and the depressed temperature throughout its interval compared with the expected temperature. The magnetic field inside it, however, shows multiple polarities and the presence of a shock-like structure (around 23:21 UT on May 13), so we consider it as a complex ejecta. Each complex ejecta is associated with a forward shock, which passed Wind at 16:37 UT on May 10 and 09:09 UT on May 12, respectively. The second shock is propagating into the first complex ejecta, compressing the plasma and magnetic field. There is another shock passage at 19:02 UT on May 15 (not shown here), which was likely produced by CME 8 from May 13. The shock driven by the May 13 eruption must have a large longitudinal width (see Figure 1). \nThe observations of two complex ejecta at 1 AU are consistent with our expectation in Section 2.1. There are multiple dips in the temperature profile inside the first complex ejecta (Figure 3c), indicative of multiple CMEs. The speed profile is not declining monotonically across the interval but shows four major bumps (Figure 3b), so the merging of the CMEs is still in process at 1 AU. The velocity is not particularly high, with a peak value of about 1000 km s -1 within the complex ejecta. We suggest that the four major bumps correspond \nto CMEs 1 - 4 (see Figure 1 and Table 1), but other smaller events (not listed in Table 1) may also have contributed to the formation of the first complex ejecta. The second complex ejecta is presumably formed from the merging of CMEs 5 - 7, as mentioned earlier. Again, other smaller CMEs (not listed in Table 1) may also have contributed to its formation. \nOf particular interest is the extremely enhanced magnetic field inside the first complex ejecta. The magnetic field strength is as high as about 72 nT and the peak southward field component is about 59 nT, both of which are unusually large values at 1 AU. As discussed earlier, this large magnetic field must have been produced by the interactions between the successive CMEs during their transit to 1 AU, which amplify the magnetic field and help maintain a strong field. The enhanced southward field mainly takes place inside the first complex ejecta but not inside the second one, so we can contrast two cases of complex ejecta in terms of the magnetic field and geo-effectiveness. Wang et al. (2024) examine the active region magnetic field and suggest that CMEs 1 - 4 and CMEs 5 - 7 erupted from two different groups of polarity inversion lines (PILs): the first group of PILs, from which CMEs 1 - 4 occurred, is associated with a field distribution that implies strong southward fields; the opposite is true for the second group of PILs, from which CMEs 5 - 7 erupted. Readers are directed to Wang et al. (2024) for details. Also note that CMEs 5 - 7 are not as clustered as CMEs 1 - 4 (see Figure 2 bottom). Combined together, these may explain the much smaller magnetic field strength and much weaker southward field inside the second complex ejecta, although it is also a result of CME-CME interactions. \nWe notice a shift in the flux-rope tilt angles of CMEs 1 - 7 from about -60 · to about 60 · (see Table 1). This shift seems consistent with the alterations in the local B N component from being predominantly southward inside the first complex ejecta to being predominantly northward within the second complex ejecta. It also appears to agree with the two different configurations of the active region magnetic fields mentioned above. While this shift is intriguing, we warn that the flux-rope tilt angles may have large uncertainties. They are derived from white-light images, and the CMEs are full halos for both the Earth and STEREO A. \nThe geomagnetic superstorm with a minimum D st of -412 nT was caused by the first complex ejecta with its persistent, enhanced southward magnetic field (Figure 3i). It shows a prolonged recovery phase, during which the second complex ejecta only produced slight dips in the D st profile. We determine the D st index from the solar wind parameters using two empirical formulas (Burton et al. 1975; O'Brien & McPherron 2000). The two schemes assume different decays for the terrestrial ring current, which result in different D st profiles. According to our experience, the Burton et al. (1975) model tends to overestimate, while the O'Brien & McPherron (2000) formula tends to underestimate, the D st index. We therefore \naverage their results. The resulting D st profile is similar to the measured one. It has a minimum value of about -378 nT, only about 8% smaller than the actual minimum. The similarity between the estimated and measured D st index at the Earth can be considered as a 'calibration,' which enables the application of our approach to the solar wind measurements at STEREO A.", '2.3. In Situ Measurements at STEREO A': "The in situ measurements at STEREO A are displayed in Figure 4. The plasma parameters are from level 2 preliminary Maxwellian fits and have many data gaps. Note that we have shifted the data at Wind forward in time by about 2.6 hr in order to align the first shock at each spacecraft. Specifically, the shock associated with the first complex ejecta arrived at STEREO A at 14:03 UT on May 10, which is earlier by about 2.6 hr. The magnetic field profile is similar to that at Wind, but the field strength at STEREO A is generally stronger (Figure 4d). The solar wind velocity also seems higher at STEREO A (Figure 4b), although the data gaps bring some uncertainties. The stronger field strength and earlier arrival at STEREO A agree with a more head-on collision with STEREO A (see Figure 1), as we expected from coronagraph imaging observations. The data (particularly the magnetic field) indicate two other shock passages at STEREO A: 07:32 UT on May 12 and 00:05 UT on May 15 (with the latter corresponding to the May 13 CME). Both of them are also earlier than their counterparts at Wind. \nIn addition to the field strength, we also observe considerable differences in the magnetic field components between Wind and STEREO A (Figure 4f - h), despite a longitudinal separation of only 12 . 6 · . Note that radial variations are minimized since the two spacecraft are at similar distances from the Sun. These differences would affect geo-effectiveness associated with the magnetic structure. We evaluate the D st index using the same approach as we have done on the solar wind measurements at the Earth. We feed the models with the magnetic field measurements at STEREO A. Since the plasma data at STEREO A have a lot of data gaps, we take the shifted plasma data at Wind as input. This gives a lower limit for the 'geo-effectiveness,' as the solar wind velocity should be higher for a more head-on impact. The resulting D st profile shows multiple dips (Figure 4i), as we have seen at the Earth. It has a minimum value of about -494 nT, which is much larger than its counterpart at the Earth (i.e., -378 nT). Therefore, the geomagnetic storm would have been much stronger, if the portion of the solar wind disturbance measured at STEREO A had hit the Earth. Clearly, a mesoscale separation can result in significant differences in both the magnetic field and associated geo-effectiveness. It is worth mentioning that here we look at a complex ejecta. \nThe scenario for a single CME may (or may not) be different. \nNote that the result of a much stronger storm at STEREO A relies on the assumption about the solar wind velocity at STEREO A. The southward magnetic field component and the solar wind velocity, in order of importance, are two key elements controlling the intensity of geomagnetic storms. Measurements at STEREO A indicate a generally stronger southward field component (Figure 4h), but the velocity data are largely missing. Our approach using the shifted solar wind speed at Wind may carry a considerable uncertainty. However, the velocity measurements at STEREO A, although sparse, suggest that the speed at STEREO A is at least comparable to that at Wind (Figure 4b). Therefore, the conclusion of a stronger storm at STEREO A is valid, and this is consistent with a more head-on impact with STEREO A.", '3. Conclusions': "We have examined the solar and interplanetary sources of the 2024 May geomagnetic superstorm, the largest one in two decades. Key findings are revealed concerning the formation of solar superstorms and how CME mesoscale variations affect geo-effectiveness. We summarize the results as follows. \n- (1) The 2024 May storm supports the hypothesis of Liu et al. (2019) that solar superstorms are 'perfect storms' in nature. The active region NOAA AR 13664 kept erupting for a prolonged time period. Among the series of eruptions from the active region, four halo CMEs (CMEs 1 - 4) on May 8 - 9 interacted during their transit to 1 AU, which was responsible for the unusually large magnetic field inside the first complex ejecta observed at the Earth. This is essentially a 'perfect storm' scenario, specifically, a combination of circumstances resulting in an event of an unusual magnitude. It is also exactly what Liu et al. (2019) find in their comparative study of the 2012 July and 2017 July events: 'a long-lived eruptive nature of the source active region, a complex event composed of successive eruptions from the same active region, and in-transit interaction between the successive eruptions resulting in exceptionally strong ejecta magnetic fields at 1 AU.' Note that none of the present CMEs is extreme in terms of their velocity near the Sun. These results strengthen the idea that extreme events generally are not simple and can involve reinforcing factors. As in Liu et al. (2014, 2019), the results also point out that extreme events are not as rare as we imagine.\n- (2) The complex ejecta, which caused the geomagnetic storm, shows considerable differences in the magnetic field and associated 'geo-effectiveness' between the Earth and STEREO A, despite a longitudinal separation of only 12 . 6 · . An earlier arrival and stronger \nfield are observed at STEREO A, which is consistent with a more head-on collision with STEREO A as expected from coronagraph imaging observations. We evaluate the D st index using the same approach for the two points, which allows us to compare the 'geoeffectiveness' at the Earth and STEREO A. The simulated D st profile is similar to the measured one at the Earth, with a slightly underestimated minimum ( -378 nT from the model versus the actual -412 nT). At STEREO A we obtain a lower limit for the 'geoeffectiveness' with a minimum D st of -494 nT. The geomagnetic storm would be much stronger (compared with the counterpart -378 nT at the Earth), if the Earth were at the position of STEREO A. Therefore, a mesoscale separation can result in significant differences in CME magnetic field and associated geo-effectiveness. \n(3) We find two contrasting cases of complex ejecta in terms of their geo-effectiveness. The geomagnetic superstorm was caused by the first complex ejecta resulting from the merging of CMEs 1 - 4, due to its persistent, enhanced southward magnetic field. The second complex ejecta observed at the Earth, which was likely formed from the merging of CMEs 5 - 7, is not associated with a strong southward field. It only produced slight dips in the D st profile during the recovery phase of the geomagnetic storm. Wang et al. (2024) indicate two different distributions of the active region magnetic field around the erupting PILs: one implies strong southward fields, but the other does not. Also, CMEs 5 - 7 are not as clustered as CMEs 1 - 4. These may explain their distinct geo-effectiveness, although both were a result of CME-CME interactions. \nThe research was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB0560000), NSFC (under grants 42274201, 42204176, 12073032 and 42150105), and National Key R&D Program of China (No. 2021YFA0718600). We acknowledge the use of data from Wind, STEREO and SOHO and the D st index from WDC in Kyoto. Readers are directed to Kaiser et al. (2008) for the STEREO mission, Domingo et al. (1995) for the SOHO mission, Ogilvie et al. (1995) for Wind/SWE, and Lepping et al. (1995) for Wind/MFI.", 'REFERENCES': "Allen, J., Sauer, H., Frank, L., & Reiff, P. 1989, Eos, Transactions American Geophysical Union, 70, 1486 \nBaker, D. N., Li, X., Pulkkinen, A., et al. 2013, SpWea, 11, 585 \nBoteler, D. H. 2019, SpWea, 17, 1427 \nBurlaga, L. F., Skoug, R. M., Smith, C. W., Webb, D. F., Zurbuchen, T. H., & Reinard, A. 2001, JGRA, 106, 20957 \nBurlaga, L. F., Plunkett, S. P., & St. Cyr, O. C. 2002, JGRA, 107, 1266 \nBurton, R. K., McPherron, R. L., & Russell, C. T. 1975, JGRA, 80, 4204 \nCannon, P., Angling, M., Barclay, L., et al. 2013, Extreme Space Weather: Impacts on Engineered Systems and Infrastructure (Royal Academy of Engineering, London, U. K.) \nCarrington, R. C. 1859, MNRAS, 20, 13 \nDavies, J. A., Harrison, R. A., Rouillard, A. P., et al. 2009, GeoRL, 36, L02102 \nDomingo, V., Fleck, B., & Poland, A. I. 1995, SoPh, 162, 1 \n- Farrugia, C., & Berdichevsky, D. 2004, AnGeo, 22, 3679\n- Green, J. L., & Boardsen, S. 2006, AdSpR, 38, 130\n- Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, SSRv, 136, 5\n- Kilpua, E. K. J., Liewer, P. C., Farrugia, C., et al. 2009, SoPh, 254, 325\n- Lepping, R. P., Acuna, M. H., Burlaga, L. F., et al. 1995, SSRv, 71, 207\n- Liu, Y., Richardson, J. D., & Belcher, J. W. 2005, P&SS, 53, 3\n- Liu, Y., Luhmann, J. G., Huttunen, K. E. J., et al. 2008, ApJL, 677, L133\n- Liu, Y., Davies, J. A., Luhmann, J. G., et al. 2010, ApJL, 710, L82\n- Liu, Y. D., Luhmann, J. G., Mostl, C., et al. 2012, ApJL, 746, L15\n- Liu, Y. D., Luhmann, J. G., Kajdiˇc, P., et al. 2014, NatCo, 5, 3481\n- Liu, Y. D., Hu, H., Wang, R., et al. 2015, ApJL, 809, L34\n- Liu, Y. D., Zhao, X., Hu, H., et al. 2019, ApJS, 241, 15\n- Liu, Y. D., Chen, C., & Zhao, X. 2020, ApJL, 897, L11\n- Lopez, R. E. 1987, JGRA, 92, 11189 \nLugaz, N., & Farrugia, C. J. 2014, GeoRL, 41, 769 \nLugaz, N., Farrugia, C. J., Winslow, R. M., et al. 2018, ApJL, 864, L7 \nMishra, W., Srivastava, N., & Chakrabarty, D. 2015, SoPh, 290, 527 \nMostl, C., Farrugia, C. J., Miklenic, C., et al. 2009, JGR, 114, A04102 \nNational Research Council, 2008, Severe Space Weather Events - Understanding Societal and Economic Impacts: A Workshop Report (The National Academies Press, Washington, D. C.) \nO'Brien, T. P., & McPherron, R. L. 2000, JGRA, 105, 7707 \nOgilvie, K. W., Chornay, D. J., Fritzenreiter, R. J., et al. 1995, SSRv, 71, 55 \nPalmerio, E., Carcaboso, F., Khoo, L. Y., et al. 2024, ApJ, 963, 108 \nPaouris, E., Vourlidas, A., Kouloumvakos, A., et al. 2023, ApJ, 956, 58 \nRegnault, F., Al-Haddad, N., Lugaz, N., et al. 2024, ApJ, 962, 190 \nRichardson, I. G., & Cane, H. V. 1995, JGR, 100, 23397 \nRichardson, I. G., & Cane, H. V. 2010, SoPh, 264, 189 \nRiley, P., Baker, D., Liu, Y. D., et al. 2018, SSRv, 214, 21 \nRussell, C. T., Mewaldt, R. A., Luhmann, J. G., et al. 2013, ApJ, 770, 38 \nSheeley, N. R., Jr., Herbst, A. D., Palatchi, C. A., et al. 2008, ApJ, 675, 853 \nSiscoe, G., Crooker, N. U., & Clauer, C. R., 2006, AdSpR, 38, 173 \nThernisien, A. F. R., Howard, R. A., & Vourlidas, A. 2006, ApJ, 652, 763 \nThernisien, A., Vourlidas, A., & Howard, R. A. 2009, SoPh, 256, 111 \nWang, R., Liu, Y. D., Zhao, X., & Hu, H. 2024, ApJL, submitted \nZhang, J., Richardson, I. G., Webb, D. F., et al. 2007, JGR, 112, A10102 \nFig. 1.- Positions of the spacecraft in the ecliptic plane at 12:00 UT on 2024 May 10. The dashed lines indicate the longitudes of the Earth (upstream of which Wind and SOHO are located), STEREO A and SolO, respectively. The gray dashed curves are the orbits of the Earth and PSP. The dotted lines show Parker spiral magnetic fields created with a solar wind speed of 450 km s -1 . The arrows mark the propagation directions of the full halo CMEs from May 8 - 13 obtained through CME reconstruction, with the lengths of the arrows indicating CME velocities near the Sun. Red (blue) arrows represent CMEs that likely contributed to the first (second) complex ejecta at the Earth. The numbers label the CMEs as in Table 1. \n<!-- image --> \nTable 1. Information about the Full Halo CMEs Impacting the Earth \nFig. 2.- Full halo eruptions on May 8 - 13. Top: GOES X-ray flux at 1 - 8 ˚ A. Middle: Time-elongation maps from LASCO of SOHO (at a position angle of 90 · measured clockwise from the solar north). Bottom: Time-elongation maps from SECCHI of STEREO A (at a position angle of 270 · clockwise from the solar north). The vertical dashed lines indicate the peak times of the associated flares, with the numbers labelling the eruptions as in Table 1. The red vertical lines mark the observed arrival times of the shocks at the Earth. \n<!-- image --> \nFig. 3.- Solar wind parameters observed at Wind and measured D st index. From top to bottom, the panels show the proton density (a), bulk speed (b), proton temperature (c), magnetic field strength (d), proton β (e), magnetic field components (f-h), and D st index (i), respectively. The shaded regions indicate the intervals of two complex ejecta, and the vertical dashed lines mark the associated shocks. The red curve in panel (c) denotes the expected proton temperature calculated from the observed speed (Lopez 1987; Richardson & Cane 1995). The red curve in panel (i) represents the estimated D st index by combining the formulas of Burton et al. (1975) and O'Brien & McPherron (2000). \n<!-- image --> \nFig. 4.- Solar wind parameters observed at STEREO A and estimated D st index. Data at Wind, which are shifted forward in time by about 2.6 hr, are shown in gray for comparison. Similar to Figure 3. \n<!-- image -->"}

2020ARA&A..58..257G

We describe ongoing searches for intermediatemass black holes with MSUBBHSUB 1010SUP5SUP MSUBSUB. We review a range of search mechanisms both dynamical and those that rely on accretion signatures. We find the following conclusions Dynamical and accretion signatures alike point to a high fraction of 10SUP9SUP10SUP10SUP MSUBSUB galaxies hosting black holes with MSUBBHSUB 10SUP5SUP MSUBSUB. In contrast there are no solid detections of black holes in globular clusters. There are few observational constraints on black holes in any environment with MSUBBHSUB 10010SUP4SUP MSUBSUB. Considering lowmass galaxies with dynamical black hole masses and constraining limits we find that the MSUBBHSUBSUBSUB relation continues unbroken to MSUBBHSUB 10SUP5SUP MSUBSUB albeit with large scatter. We believe the scatter is at least partially driven by a broad range in black hole masses because the occupation fraction appears to be relatively high in these galaxies. We fold the observed scaling relations with our empirical limits on occupation fraction and the galaxy mass function to put observational bounds on the black hole mass function in galaxy nuclei. We are pessimistic that local demographic observations of galaxy nuclei alone could constrain seeding mechanisms although either highredshift luminosity functions or robust measurements of offnuclear black holes could begin to discriminate models.

2020-08-01T00:00:00Z

['10.48550/arXiv.1911.09678', '10.1146/annurev-astro-032620-021835', '2020ARA&A..58..257G', '2019arXiv191109678G', 'arXiv:1911.09678']

['Astrophysics - Astrophysics of Galaxies']

IntermediateMass Black Holes

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

https://arxiv.org/pdf/1911.09678.pdf

{'No Header': 'xxxxxx 0000. 00:1-69 Copyright c © 0000 by Annual Reviews. All rights reserved', 'Jenny E. Greene, 1 Jay Strader, 2 and Luis C. Ho 3': '1 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA; email: jgreene@astro.princeton.edu 2 Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA 3 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China; Department of Astronomy, School of Physics, Peking University, Beijing 100871, China', 'Keywords': 'black holes, active galactic nuclei, globular clusters, gravitational waves, tidal disruption, ultra-luminous X-ray sources', 'Abstract': 'We describe ongoing searches for intermediate-mass black holes with M BH ≈ 100 -10 5 M glyph[circledot] . We review a range of search mechanisms, both dynamical and those that rely on accretion signatures. We find: \n- · Dynamical and accretion signatures alike point to a high fraction of 10 9 -10 10 M glyph[circledot] galaxies hosting black holes with M BH ∼ 10 5 M glyph[circledot] . In contrast, there are no solid detections of black holes in globular clusters.\n- · There are few observational constraints on black holes in any environment with M BH ≈ 100 -10 4 M glyph[circledot] .\n- · Considering low-mass galaxies with dynamical black hole masses and constraining limits, we find that the M BH -σ ∗ relation continues unbroken to M BH ∼ 10 5 M glyph[circledot] , albeit with large scatter. We believe the scatter is at least partially driven by a broad range in black hole mass, since the occupation fraction appears to be relatively high in these galaxies.\n- · We fold the observed scaling relations with our empirical limits on occupation fraction and the galaxy mass function to put observational bounds on the black hole mass function in galaxy nuclei.\n- · We are pessimistic that local demographic observations of galaxy nuclei alone could constrain seeding mechanisms, although either highredshift luminosity functions or robust measurements of off-nuclear black holes could begin to discriminate models.', '1.1. Motivation': "'Intermediate-mass' black holes (IMBHs) are often introduced as what they are not. They are not stellar-mass black holes, which are formed in the deaths of massive stars and are historically thought to be ∼ 10 M glyph[circledot] (Remillard & McClintock 2006). They are not supermassive black holes, which are historically considered to have masses of 10 6 -10 10 M glyph[circledot] . The question is often framed: are there black holes with masses between these two classes? \nWe frame the central question differently. At some point in cosmic time, black holes between 10 and 10 6 M glyph[circledot] had to exist, in order to make the 10 9 M glyph[circledot] black holes that are seen only hundreds of millions of years after the Big Bang (e.g., Ba˜nados et al. 2018). The central question of this review is whether we can find evidence for these 'intermediate-mass' black holes. The corollary is whether, based on the mass distributions and environments of the black holes, we can determine how these IMBHs were formed. Currently we have no concrete evidence for black holes with masses ∼ > 100 -10 5 M glyph[circledot] , although there are some important candidates in this mass range that we will discuss. Finding objects, and characterizing the black hole mass function in this range, is of interest for many reasons. \nExtending scaling relations to this regime may provide unique insight into the evolution of black holes (e.g., Kormendy & Ho 2013, Shankar et al. 2016, Pacucci et al. 2018), along with the possible importance of feedback for dwarf galaxies (Silk 2017, Bradford et al. 2018, Penny et al. 2018, Dickey et al. 2019). Demographics of black holes at lower masses will help elucidate the dynamical evolution of dense stellar systems (e.g., Miller & Hamilton 2002, Portegies Zwart & McMillan 2002, Gurkan et al. 2004, Antonini & Rasio 2016). \nIntermediate-mass black holes will also be prime sources of gravitational radiation for upcoming gravitational wave detectors in space ( Laser Interferometer Space Antenna [LISA] , e.g., Amaro-Seoane et al. 2015). To determine the rates and interpret the gravitational wave signals, we need independent measurements of the black hole number densities (e.g., Stone & Metzger 2016, MacLeod et al. 2016a, Eracleous et al. 2019). Furthermore, ongoing and future time-domain surveys are detecting more and more tidal disruption events. In principle, lower-mass black holes should be significant contributors to the detected tidal disruption events, and they are starting to be detected (Maksym et al. 2013, Wevers et al. 2017, van Velzen 2018). \nFinally, we do not know exactly how emission from accretion flows onto black holes will evolve with black hole mass: at lower masses the accretion disk gets hotter and the bolometric luminosity drops, perhaps leading the phenomenology to look more like X-ray binaries and less like accreting supermassive black holes (e.g., Cann et al. 2019). The emergent spectrum of such sources is of interest not only to understand accretion, but also in thinking about the impact of 'mini-quasars' on the formation of the first galaxies and the reionization of the Universe (e.g., Madau & Haardt 2015). \nFor all of these reasons, the time is right to review the current state of our knowledge, and to prepare for the rich new upcoming data sets that will bear on these questions.", '1.2. Definition': "Observationally, we have compelling evidence for black holes with M BH ∼ < 100 M glyph[circledot] and with M BH ∼ > 10 5 M glyph[circledot] . Work over the past 25 years has established clearly that galaxy centers harbor 'normal' active galactic nuclei (AGN) with masses M BH ≈ 10 5 M glyph[circledot] and higher ( § 4). Existential discovery space lies in the mass range of M BH ≈ 100 -10 5 M glyph[circledot] . Nevertheless, \n0 \n. \n1 \n8 \n. \n0 \n6 \n. \n0 \nFigure 1 \n<!-- image --> \n0 \n. \n0 \n0 \n. \n0 \n0 \n. \n2 \n0 \n. \n4 \n0 \n. \n6 \n0 \n. \n8 \n1 \n. \n0 \nPossible observable differences between different seeding scenarios. Early formation through direct collapse (red) or Population III stars (blue) occur at z > 10, while gravitational runaway (green) can happen throughout cosmic time. As cosmic structures evolve, the seed black holes will suffer mergers (black ovals) leading to the emission of gravitational waves, as well as accretion episodes (blue disks) that could be observed as active galactic nuclei. At the present day, differences in black hole mass functions, occupation fractions, and black hole-galaxy scaling relations may ensue from different seeding channels, for simplicity here shown only for nuclear black holes. Grey bars in these relations show where we do not yet have observational constraints. \nfrom the perspective of understanding the growth and demographics of black holes, we will show that black holes in the range M BH ≈ 10 5 -10 6 M glyph[circledot] also encode important (and poorly quantified) information. Thus we cover them under the purview of this review as well.", '1.3. Historical Context': "The growth of supermassive black hole seeds, and the existence of IMBHs, have both long been discussed in the literature, even before we were completely sure of the existence of astrophysical black holes (e.g., Eardley & Press 1975). Quickly thereafter, the community developed theoretical ideas about how one might form non-stellar-mass black holes; the famous diagram from Rees (1978) includes all the formation mechanisms we consider today, including gravitational runaway (Bahcall & Ostriker 1975, Begelman & Rees 1978, Quinlan & Shapiro 1990, Lee 1993), the collapse of Population III stars (e.g., Bond et al. 1984, Madau & Rees 2001), and the idea of a 'direct collapse' into a black hole (e.g., Haehnelt & Rees 1993, Loeb & Rasio 1994, Koushiappas et al. 2004). There was also early interest in intermediate-mass black holes as a possible explanation for dark matter (e.g., Lacey & Ostriker 1985, Carr 1994), although there are now many limits on black holes as dark matter \n(e.g., Tisserand et al. 2007, Ali-Ha¨ımoud et al. 2017, Zumalac'arregui & Seljak 2018). \nAs observations of young quasars push to earlier and earlier times (e.g., Fan et al. 2006, Mortlock et al. 2011, Ba˜nados et al. 2018), the community has recognized the significant challenge of creating such massive black holes so quickly (e.g., Haiman 2013), leading to a desire to make relatively massive seed black holes, further motivating searches for IMBHs. Accordingly, theoretical work identified that local observations of BHs in dwarf galaxies, which suffer fewer mergers and less accretion, may provide an additional key to understanding seeding processes (e.g., Volonteri et al. 2008). \nIn parallel, the Einstein telescope enabled the discovery of the first ultra-luminous X-ray sources (ULXs; e.g., Long et al. 1981, Fabbiano 1989), whose luminosities are nominally above the Eddington limit for compact stellar objects. One natural explanation for the high X-ray luminosities of these sources is that they are powered by more massive compact objects, namely IMBHs (e.g., Kaaret et al. 2017, and references therein; § 4). The ultra-luminous X-ray sources show association with young star clusters and this spurred continuing interest in the formation of IMBHs in dense stellar clusters through gravitational runaway (e.g., Ebisuzaki et al. 2001, Portegies Zwart & McMillan 2002, Miller & Hamilton 2002). Puzzling suggestions of abnormal numbers of young stars in the Galactic Center also sparked interest in a possible IMBH to carry in the stars (e.g., Ghez et al. 2003, Hansen & Milosavljevi'c 2003). \nThanks to the successful detection of supermassive black holes in galaxy nuclei (e.g., Kormendy & Richstone 1995), dynamical searches for IMBHs at the centers of globular clusters like M15 were also underway, yielding conflicting results (e.g., Gebhardt et al. 2000, 2002, Gerssen et al. 2002, McNamara et al. 2003). Early dynamical searches for black holes in the nuclei of low-mass galaxies also reported non-detections (Gebhardt et al. 2001, Valluri et al. 2004). Comprehensive searches for AGN signatures in nearby galaxies likewise pointed to a low fraction of AGN in late-type and low-mass galaxies (e.g., Ho et al. 1997, Kauffmann et al. 2003). These facts together led the community to conclude that black holes are rarely found in the nuclei of < 10 10 M glyph[circledot] galaxies (i.e. galaxies less massive than the Milky Way). Even today, there is a severe dearth of information about black hole demographics in subL ∗ galaxies. \nReal progress in the last decade, based both on dynamical studies ( § 3) and accretionbased ones ( § 4), suggests instead that a high fraction ( > 50%) of 10 9 -10 10 M glyph[circledot] galaxies do harbor central black holes, likely with M BH > 10 5 M glyph[circledot] (Miller et al. 2015, Nguyen et al. 2019b, and § 9). We use this recent progress as a motivation to revisit both scaling relations ( § 8) and the allowed range of black hole mass functions for M BH < 10 6 M glyph[circledot] ( § 9). \nWe are not the first to attempt a review of the linked subjects of IMBH searches and seeding mechanisms (Volonteri 2010, Greene 2012, Reines & Comastri 2016, Mezcua 2017). There is a related review about seeding mechanisms in Inayoshi et al. this volume.", '2. Formation Paths: Where to Find IMBHs': 'We briefly review the primary theoretical channels for seeding supermassive black holes (Figure 1). We then describe the predictions for local observations in terms of number densities and environments of IMBHs.', '2.1. Seeding Models': "We know that black holes are made in the death of massive stars. This process happens at the present day, but at early times we expect theoretically that the first (so-called Population III) stars were quite massive, due to the inability of molecular hydrogen gas to cool (Bromm &Larson 2004, Karlsson et al. 2013, and references therein). Such massive stars would likely end their lives as black holes with M BH ∼ 100 M glyph[circledot] (e.g. Fryer et al. 2001), excepting first stars in the mass range of 140 -260 M glyph[circledot] that would explode as pair-instability supernovae and leave no remnant (Heger et al. 2003). \nAs shown by Madau & Rees (2001), if these first massive stars are rare, such that only ∼ one is made in each high density peak at very high redshift, then the number density of black holes would be well-matched to the number of supermassive black holes today. However, growing from ∼ 100 M glyph[circledot] to a billion suns in a fraction of a Gyr requires dramatic fueling (Haiman & Loeb 2001). Specifically, for a typical e -folding time of 45 million years, 14 such e -foldings are available at z = 7, but 17 are needed if the black hole grows at the Eddington limit (Madau et al. 2014, and references therein). Arranging this uninterrupted growth seems challenging even for these rare quasars (e.g., Johnson & Bromm 2007, Milosavljevi'c et al. 2009). On the other hand, a few episodes of superEddington accretion could largely alleviate the tension, and super-Eddington accretion now appears to arise naturally in numerical experiments (e.g., Jiang et al. 2019). \nA different channel, with no local analogs, is that of collapsing gas clouds forming a massive seed black hole without passing through all the phases of stellar evolution. These 'direct collapse' models form seed black holes with M BH ∼ 10 4 -10 6 M glyph[circledot] (e.g., Bromm & Loeb 2003, Loeb & Rasio 1994, Lodato & Natarajan 2006, Begelman et al. 2006), perhaps passing through a 'quasi-star' phase (e.g., Begelman 2010). This channel can only operate at very high redshift, since pristine gas is required to suppress cooling and fragmentation. Recent literature has discussed in detail how rare the conditions may be to ensure that the gas does not cool and fragment into smaller units (e.g., Visbal et al. 2014, Habouzit et al. 2016), typically assuming that an elevated Lyman-Werner background is needed (e.g., Omukai 2001). Some works instead argue that additional photons from star formation in nearby halos (e.g., Dijkstra et al. 2008, Visbal et al. 2014), additional gas heating through mergers (Yoshida et al. 2003, Wise et al. 2019), or even star formation within individual halos (Dunn et al. 2018), may provide sufficient heating or ionizing photons to make this channel viable. We emphasize that the number density of halos that suffer direct collapse is very uncertain, with predictions differing on the halo gas accretion rates as a function of mass, the required elevation of the ionizing background, and the possible sources of ionizing photons. See a comprehensive review in Inayoshi et al., this volume. \nA final class of models makes ∼ 10 3 -10 4 M glyph[circledot] black holes in a gravitational runaway event within a dense stellar cluster. Modern discussion divides globular cluster IMBH formation scenarios into 'slow' and 'fast' versions, depending on whether the formation occurs over ∼ 100 Myr to 1 Gyr or ∼ < a few Myr, respectively. Here we focus specifically on the formation of the seed at the centers of dense clusters. In any scenario where an IMBH does form, the present-day mass may well be higher due to accretion of gas and/or stars (e.g., Rosswog et al. 2009, MacLeod et al. 2016a, Sakurai et al. 2017; § 9). We also note that while normal single stars are not expected to produce black holes > 100 M glyph[circledot] , it is plausible that repeated mergers in star clusters could occasionally produce IMBHs larger than 100 M glyph[circledot] by this channel (Rodriguez et al. 2019). \nThe modern incarnation of the 'slow' scenario (Miller & Hamilton 2002) envisions a \nblack hole of mass ∼ > 50 M glyph[circledot] forming as the remnant of a massive star, then growing though mergers of mass-segregated black holes, leading to ∼ 10 3 M glyph[circledot] IMBHs in a few × 10% of massive globular clusters. In this scenario the occupation fraction is set by considering the central densities observed in Galactic globular clusters. Lower-mass ( ∼ 10-20 M glyph[circledot] ) black hole seeds will not lead to a runaway since early stellar interactions will tend to eject these lighter black holes from the cluster. A governing uncertainty in this scenario is whether even 50 M glyph[circledot] seeds can form, as such remnants might be rare or absent in single star evolution (Woosley 2017). \nThe 'fast' scenario was re-introduced in a contemporary form by Portegies Zwart & McMillan (2002), who used a combination of simulations and analytic calculations to argue that sufficiently dense star clusters will undergo a collisional runaway of massive stars, the product of which could eventually lead to the formation of an IMBH. Such supermassive stars could potentially also produce enough ejected material with unusual abundances to explain the multiple stellar populations seen in globular clusters (Gieles et al. 2018). Both this scenario and the direct collapse scenario may be sensitive to metallicity: at high metallicity, stellar winds may remove so much mass that collapse to a black hole is uncertain (e.g., Mapelli 2016). In the context of very low metallicity and very high redshift, some models predict that very dense and massive clusters may form black holes at their centers (e.g., Devecchi & Volonteri 2009, Katz et al. 2015, Sakurai et al. 2017). While clusters this dense are not found at lower redshift, hypothetical Pop III star clusters could optimistically be detectable with the James Webb Space Telescope ( JWST ).", '2.2. From Seeds to Local Predictions: Population III and Direct Collapse Channels': "Currently it is challenging to compare predictions of local demographics based on the seeding mechanism. To that end, we gather approximate predicted number densities and mass distributions of present day IMBH populations as a function of channel and modeling approach. Even with perfect knowledge of the number density and halo locations of black hole seeds at early times, there are many unknown factors at play as the seeds evolve towards the present day (e.g., Volonteri et al. 2008, Volonteri 2010, Mezcua 2017, Buchner et al. 2019, Mezcua et al. 2019). We outline some of these uncertainties and then compare the predictions from different types of models. \nWhen galaxies merge, the central black holes may also merge, and any differences in the mass and angular momentum of the two black holes can translate into anisotropic gravitational wave emission during the merger, which in turn can impart a linear momentum 'kick' to the merged remnant (e.g., Redmount & Rees 1989). A major uncertainty in going from initial seed models to local observations of black holes in galaxy nuclei comes from the unknown fraction of black holes that are ejected via gravitational recoil. These kicks may be more than sufficient to exceed the escape velocity of low-mass halos (e.g., Haiman 2004, Campanelli et al. 2007), but the most common mergers with much lower-mass black hole secondaries likely never lead to recoil (Volonteri & Rees 2006). In practice, the distribution of kick velocities will depend on the mass ratios and relative spins of the incoming black holes, for which we do not have complete models. There are other dynamical uncertainties to consider. Such low-mass black holes in such shallow potentials may never settle at the centers of their galaxies (Bellovary et al. 2019). Away from the center of the potential, the black holes may not grow efficiently, and would likely be missed with current searches \n(although see Reines et al. 2019). \nFinally, tracking accretion in a physically motivated way requires prohibitively high spatial resolution for cosmological simulations. In particular, if super-Eddington accretion is possible, then the differences between the Population III channel and the direct collapse channel may be washed out at early times (e.g., Volonteri & Gnedin 2009, Alexander & Natarajan 2014). Thus, it is possible that the high redshift luminosity function and number of gravitational wave detections from growing black holes may be more sensitive to seeding (e.g., Sesana et al. 2007, Tanaka & Haiman 2009). Furthermore, late-time growth contributes in an unconstrained way to the present-day black hole mass distribution. Nuclear star clusters, the hosts of at least some candidate IMBHs ( § 3), experience multiple bursts of star formation (e.g., Walcher et al. 2006, Carson et al. 2015, Kacharov et al. 2018). Such bursts may well feed the black hole, and additional accretion via stellar interactions may also occur (e.g., Stone et al. 2017, Alexander & Bar-Or 2017). \n2.2.1. Number Densities. The Population III and direct collapse channels are distinguished from the dynamical runaway channels in that they happen exclusively at very high redshift and rely on the presence of metal-free gas, so we will refer to them collectively as early seeding models. As described above, the direct collapse scenario requires finely tuned conditions. For those theoretical works that consider direct collapse viable, then both channels are generally thought to make at most one IMBH per halo (although as emphasized above, direct collapse black hole may be considerably rarer), and thus yield very similar number density predictions (Table 1). \nThe hope for observable differences between these two seeding mechanisms has hinged on the idea that 'heavy' seeding would leave lower numbers of more massive seeds while Population III remnants would be more common but of lower mass (e.g., van Wassenhove et al. 2010, Greene 2012). Two approaches have been taken to predict the observable signatures of seeding mechanisms on the black hole population today, semi-analytic modeling and hydrodynamic simulations. \nSemi-analytic modeling (Volonteri et al. 2008) can quickly track the evolution of black holes using merger trees, and explore the dependence on all the uncertain accretion and merging prescriptions (others include Volonteri & Natarajan 2009, van Wassenhove et al. 2010). Most of the semi-analytic models in the literature that compare Population III and heavy seeds adopt the heavy seeding model of Lodato & Natarajan (2006), which predicts a high number density of heavy seeds ∼ 0 . 1 Mpc -3 (Volonteri 2010), as compared to ∼ 10 -4 Mpc -3 quoted by Inayoshi et al. this volume. Thus, these are optimistic scenarios for heavy seeds, and likely depend in detail on the assumed ionizing background (e.g., Valiante et al. 2016). Recently, Bellovary et al. (2019) also used zoom-in hydrodynamical simulations to track seeds to the present day as formed in high-redshift via a direct collapse scenario (Dunn et al. 2018). We will take the recent work by Ricarte & Natarajan (2018) as representative of the semi-analytic approach, because they consider a range of fueling mechanisms. The two have comparable predicted occupation fractions in the context of direct collapse. However, Ricarte & Natarajan find a very wide range of acceptable occupation fractions for the Population III channel (see also § 9). We note that according to predictions from both models, the occupation fraction starts to change dramatically below galaxy host masses of 10 9 M glyph[circledot] , a regime that is not yet empirically constrained. \nWe translate these occupation fractions into number densities for the two channels (Table 1). To convert the occupation fractions into number-density predictions, we take a \ngalaxy stellar mass function (Wright et al. 2017) and integrate from 10 7 -10 10 M glyph[circledot] , incorporating the range of occupation fractions presented in Table 1. Note that we have observational constraints only down to 10 9 M glyph[circledot] , and so these predictions represent a significant extrapolation, but are consistent with existing observations in all cases. These numbers are also roughly consistent with the numbers quoted by Volonteri (2010), which again relies on the optimistic direct collapse channel from Lodato & Natarajan (2006). Although mass distributions may differ between the two, the full range of predicted occupation fractions for the direct collapse and Population III channels are quite similar. We emphasize that the accretion history will be tracked eventually with active galaxy luminosity functions (e.g., Civano et al. 2019), which will provide additional constraints on the fueling models and may allow us to distinguish between seeding models more effectively ( § 13).", '2.3. From Seeds to Local Predictions: Gravitational Runaway': "We now attempt to translate theory on the different channels of gravitational runaway into predictions for the number density and mass distribution of IMBHs at present (Table 1). Unique to this channel is the expectation of glyph[greatermuch] 1 massive black hole per galaxy, which will be quantifiable if we ever manage to detect or rule out non-nuclear black holes. \nRecent work has revisited the slow formation scenario described in § 2.1 in the context of dynamical formation of black hole-black hole binaries observable as gravitational wave mergers (e.g., Rodriguez et al. 2018). Indeed, there is some prospect for useful observational constraints for this channel using gravitational wave observations from the current generation of detectors (Kovetz et al. 2018). The predictions for the slow scenario depend in essential ways on whether the dynamics of close encounters are treated in a relativistic manner. While there is no simple expression for the gravitational recoil velocity as a function of the black hole masses, mergers with more extreme mass ratios tend to have lower velocities. The predictions also depend strongly on the unknown initial spins of black holes, which can lead to enormous velocities that will eject nearly all systems from clusters (Campanelli et al. 2007). Even if the initial spins are low, post-merger products will have higher spins, suggesting that successive mergers are more likely to eject black holes. HolleyBockelmann et al. (2008) found that in most cases the IMBH will be ejected over a wide range of seed masses, unless all of the encounters have mass ratios M low /M high ∼ < 0 . 05. To ensure that this condition is met would require that the initial seed be ∼ > 500 M glyph[circledot] , and that it is surrounded by a sub-cluster of ∼ < 25 M glyph[circledot] stellar-mass black holes. Seeds of lower mass would have a high likelihood of ejection if other black holes are present. \nIn the very densest clusters, the escape velocity might be high enough to allow merger products to remain bound (Antonini et al. 2019), which would limit IMBH retention to rare dense clusters ∼ > a few × 10 6 M glyph[circledot] . Some other studies (e.g., Giersz et al. 2015) that advocate for slow IMBH formation in star clusters sometimes miss the crucial contribution of recoil in the dynamics of IMBH formation through this channel. While the occupation fraction from the slow channel is uncertain, current simulations are consistent with a low or negligible value for typical globular clusters. \nFor the the fast channel, Portegies Zwart & McMillan (2002) argue that fast runaway only applies to clusters with short initial relaxation times. A short relaxation time excludes most typical globular clusters, unless perhaps they are born with primordial mass segregation. Gurkan et al. (2004) find that runaway collisions should be common in dense clusters, but the final product of the runaway is uncertain, since a 'supermassive' star formed by \ncollisions will evolve on a timescale comparable to the collision rate (Freitag et al. 2006). The metallicity of the stars and the possible presence of gas in the core add additional complications. Hence it is difficult to predict whether a supermassive star will form at all, its expected mass if it does form, and whether it would eventually collapse to an IMBH, or instead destroy itself entirely as a pair-instability supernova (Spera & Mapelli 2017). \nOther papers have discussed the possibility of fast seeds selectively forming only at high redshift in star clusters at the centers of dense and relatively massive halos (e.g., Sakurai et al. 2017). If so, then most of these clusters will preferentially migrate to galaxy centers. However, there is voluminous evidence that most globular clusters of all metallicities formed in the same way that massive clusters still form today, rather than in a unique channel only operating in the early universe (e.g., Brodie & Strader 2006, El-Badry et al. 2019). \nMiller & Davies (2012) argue that all clusters above a central velocity dispersion of ∼ 40 km s -1 will necessarily form an IMBH through some mechanism, since neither primordial nor dynamically formed binaries of stars or stellar remnants can prevent core collapse for these velocity dispersions. This analytic argument rests on efficient ejection of stellar-mass black holes. Breen & Heggie (2013) argue that black hole ejection is expected to be rather inefficient for typical globular clusters and hence core collapse will usually occur only after many relaxation times. This result is generally born out by numerical simulations (e.g., Kremer et al. 2019), though few simulations of ∼ > 10 6 M glyph[circledot] clusters have yet been published. \nOverall, there is little theoretical support for the idea that IMBH formation is favored in typical globular clusters. By contrast, many nuclear star clusters have high velocity dispersions and a range of relaxation times, and may be plausible sites for the formation of IMBHs. We return to this idea in § 5. \nWhile old globular clusters are currently too dynamically evolved to suffer recent gravitational runaway, young star clusters or growing nuclear star clusters should still be in a position to grow massive BHs if this channel is robust. To date, no such trustworthy sources have been found (although see § 5 for some candidates). One way out of this constraint is to note that few sensitive optical, radio, or X-ray surveys for massive BHs are currently possible outside of the Milky Way, which itself has scarcely any young massive clusters (see § 5). Another way out is that the channel may require low metallicities (Mapelli 2016), and then we would not expect it to operate in the local universe. However, metal-poor and metal-rich clusters have no observed substantial structural or dynamical differences, so if IMBHs form commonly but only in low-metallicity clusters, they have not yet been found to affect the clusters in any noticeable way. \n2.3.1. Number Densities. To translate these predictions into number densities, we first consider galaxy nuclei, then the cluster population. If gravitational runaway happens, one would think that nuclear star clusters, the most massive and densest clusters, may be the most likely to form and retain IMBHs. However, not all nuclear star clusters may successfully form massive black holes by gravitational collapse (Breen & Heggie 2013, § 5). We therefore assume that under this channel, there will be a wide range of occupation fractions in galaxy nuclei, between 10-100%. Adopting a measured range of nucleation fractions as a function of stellar mass from S'anchez-Janssen et al. (2019, see also Neumayer et al. in preparation), we find a final number density of 0.02-0.25 Mpc -3 (Table 1). This number could be higher than for early collapse models.", '2.4. From Seeds to Local Predictions: Wandering Black Holes': "In addition to black holes in galaxy centers, any seeding model will lead to a population of off-nuclear 'wandering' black holes. We divide wandering black hole channels into those where the black hole is made via gravitational runaway in star clusters and those where the black hole is formed in the nucleus of a satellite galaxy. \nIf gravitational runaway operates, Holley-Bockelmann et al. (2008) show that any IMBH formation scenario within globular clusters would lead to the majority of IMBHs with M BH < 3000 M glyph[circledot] being ejected. Fragione et al. (2018) use a semi-analytic model and explore a range of M BH / M cluster of 0.5-4% to find that only a small percentage (3%) of IMBHs would be retained in globular clusters today, while the majority of the ∼ 1000 IMBHs in the Milky Way would either be ejected during black hole-black hole mergers (70-90%) or have their cluster dissolve around them (4-30%), leaving a large population of wandering black holes with a low-mass dense stellar cluster remaining bound. \nTo find these wanderers, consider that at the present day ∼ 20-40 stars are expected to remain surrounding the M BH ∼ 10 4 M glyph[circledot] IMBHs and perhaps ∼ 10 3 stars surrounding rarer M BH ∼ 10 5 M glyph[circledot] (see also Komossa & Merritt 2008). The expected half-light radii of these remnant clusters are small, ∼ < 1 pc. Such clusters should be detectable as resolved stars in future wide-fields surveys such as the Large Synoptic Survey Telescope (LSST), since current observations are already discovering very low-mass globular clusters in the outer halo (e.g., HSC-1, with ∼ 250 stars; Homma et al. 2019). Follow-up spectroscopy of even a few stars in such objects could reveal their provenance via an unexpectedly large velocity dispersion. Closer to the Sun, remnant recoil clusters containing only a handful of stars could be identified using Gaia data as a population in a small patch of sky ( ∼ < a few × 10') with a net proper motion and a substantial dispersion around that value. These IMBHs could also pass through molecular clouds in the disk, sparking into detectability from the additional fuel as a faint radio source with high proper motion (Fender et al. 2013). \nEven if there are no black holes in globular clusters, disrupted infalling satellite galaxies that harbor IMBHs will deposit their black holes somewhere outside the nucleus of the larger halo. If some infalling satellites harbor a massive black hole (which is likely for at least some of the massive satellites; § 9) then we expect a population of wandering black holes (Governato et al. 1994, Schneider et al. 2002, Volonteri et al. 2003, Islam et al. 2004, Bellovary et al. 2010, Micic et al. 2011, Rashkov & Madau 2014). A related pathway comes from very early accretion of satellites. Mergers between satellites could have lead to ejections of black holes that might still be wandering around the Milky Way. The recoiling black hole will likely take with it a tightly bound cluster of stars (O'Leary & Loeb 2009, 2012). O'Leary and Loeb estimate ∼ > 100 black holes with M BH > 10 3 M glyph[circledot] (and ∼ > 10 black holes with M BH > 10 4 M glyph[circledot] ) in the Milky Way. That massive black holes have been detected in the centers of ultra-compact dwarf galaxies (Seth et al. 2014, Ahn et al. 2018), which are the dense remnants of tidally stripped galaxies, provides empirical evidence for this channel. \n2.4.1. Number Densities. We will make two different estimates for the number density of wanderers (non-nuclear IMBHs), first for those deposited by globular clusters, and then for those formed in satellites. We have decided to make our estimates as fully empirical as possible, by linking the number of wanderers from clusters directly to observed numbers of globular clusters per unit host stellar mass and number of wanderers from satellites to the number of ultra-compact dwarf galaxies per unit host stellar mass, assuming these are tidally stripped galaxies. \nWe start by calculating the number of globular clusters per galaxy using the stellar mass to globular cluster number relation from Harris (2016). We assume a V -band mass-to-light ratio of two for galaxies with M ∗ < 3 × 10 10 M glyph[circledot] , and three for more massive galaxies. Not all globular clusters host black holes. We assume that 10% of massive globular clusters with M ∗ > 10 6 M glyph[circledot] can form and retain massive black holes (Holley-Bockelmann et al. 2008, Fragione et al. 2018). To estimate the fraction of globular clusters with M ∗ > 10 6 M glyph[circledot] , we integrate the globular cluster luminosity functions from Jord'an et al. (2007) for different galaxy masses. Integrating these luminosity functions, we find that ∼ 5% of the globular clusters around a ∼ 2 × 10 10 M glyph[circledot] galaxy have M ∗ > 10 6 M glyph[circledot] , while ∼ 10% of the globular clusters around a 10 11 M glyph[circledot] galaxy are of this mass or higher. Folding that in with the number of massive clusters per galaxy, we find a number density of 0.31 IMBH per cubic Mpc (Table 1). Again, we emphasize that this is a lower limit, since it does not include black holes in dissolved clusters or those that may have been ejected from their cluster after a merger. \nWe further assume that massive clusters ( M ∗ > 10 6 M glyph[circledot] ) in a galaxy today comprise a mixture of a tail of truly massive objects that formed as clusters ( ∼ 80%) with the cores of stripped galaxies ( ∼ 20%) that we refer to as ultra-compact dwarfs (e.g., Pfeffer et al. 2016). With the number of such clusters calculated in the prior paragraph, we then expect somewhere between zero and two ultra-compact dwarfs for the Milky Way (e.g., M54 and perhaps ω Cen), and somewhere between 75-150 in a cluster like Virgo, as observed (Liu et al. 2015). Our assumption that ultra-compact dwarfs do not dominate the globular cluster luminosity function even at high mass is also supported by the Jordan et al. luminosity functions, which do not change if all clusters with sizes greater than 5 pc (a proxy for stripped dwarf galaxies) are excluded. \nTo calculate the number of wandering black holes due to disrupted satellites, we not only need the number of disrupted satellites, but also how often the satellite hosts a black hole. This is mass dependent. Based on work summarized in § 9, we take an occupation fraction between 10% and 50%, leading to at most one wandering black hole in a Milky Way-mass galaxy according to this scaling. Our numbers are somewhat lower than predictions from some theoretical works (Bellovary et al. 2010, Rashkov & Madau 2014) but slightly higher than what is presented by Volonteri & Perna (2005). Integrating over the galaxy luminosity function (Wright et al. 2017), we find a comparable number density of IMBHs in wanderers in L ∗ galaxies as there are in dwarf nuclei, ∼ 0 . 06 -0 . 3 per cubic Mpc (Table 1). \nAfar more conservative number density is given by Voggel et al. (2019). They investigate the number density of black holes associated with ultra-compact dwarf galaxies found in cluster environments and likely to host black holes detectable by current techniques. That number density is 2 × 10 -3 supermassive black holes per cubic Mpc. They then extend their methodology to include ultra-compact dwarf galaxies around all elliptical galaxies to find 7 -8 × 10 -3 supermassive black holes per cubic Mpc. Our much higher numbers, in contrast, are dominated by so-far unknown lower-mass black holes ( < 10 5 M glyph[circledot] ) that we expect to be found in the halos of ∼ L ∗ galaxies due to stripping of low-mass satellites like Sagittarius and the LMC (e.g., Rashkov & Madau 2014).", '2.5. Tests of Seeding Mechanisms': 'Clearly number densities alone are too blunt a tool to distinguish different early seeding mechanisms. Other means have been discussed in the literature. The black hole mass \nfunction could encode seeding history, particularly if there is a mass cut-off corresponding to a heavy seeding mechanism ( § 9). Occupation fractions and black hole scaling relations ( § 8) might reflect differences in seeding mechanisms, or depend more on fueling than seeding (e.g., Volonteri & Natarajan 2009, Ricarte & Natarajan 2018). Transients (tidal disruptions and gravitational waves; § 6) may provide our best hope to find wandering black holes, as well as to see direct evidence of early seeds from direct merger detections with LISA . \nIn summary, theoretical challenges continue to impede our ability to make definitive predictions for local IMBH populations based on the channel. It is still unclear from the theoretical literature whether enough gas can be in a prime condition to facilitate enough direct collapse black holes. If yes, then the differences in local number density and mass distribution between direct collapse and Population III channels will likely be insufficient to distinguish between them. Likewise it is unclear whether Population III seeds can grow efficiently in low-mass halos at early times (e.g., Pelupessy et al. 2007, Alvarez et al. 2009). Additional information on the early growth history of these black holes may come from active galaxy luminosity functions (e.g., Civano et al. 2019) and gravitational waves from black hole mergers at high redshift (Sesana et al. 2007).', '3.1. Dynamics From Integrated Light Measurements': "Dynamical modeling of supermassive black holes has a long history and has been reviewed elsewhere (e.g., Kormendy & Richstone 1995, Kormendy & Ho 2013). The kinematics can be measured from either stars or gas in the vicinity of the black hole. For stellar-dynamical modeling, state-of-the-art codes use Schwarzschild modeling (Schwarzschild 1979) to jointly model the mass density of the central black hole, stars, and dark matter by orbit superposition (e.g., Rix et al. 1997, Gebhardt et al. 2003). The stellar mass profile is modeled from the light, which is then converted to a mass profile by solving for the stellar mass-to-light ratio ( M/L ), and if needed a dark matter halo component is also independently fitted (e.g., Gebhardt & Thomas 2009). The orbit families are weighted to match the spatially resolved stellar kinematical measurements. A major issue for all dynamical models is to distinguish between orbital anisotropy and mass (e.g., a black hole). This mass-anisotropy degeneracy is well-known (e.g., Binney & Mamon 1982). Other (related) issues include: (1) degeneracy between the stellar M/L and the black hole mass (2) incomplete stellar orbit libraries and (3) the assumption of axisymmetry (van den Bosch & de Zeeuw 2010). The systematic uncertainties for black hole masses from gas dynamics are qualitatively different than for stellar dynamics, but given the large systematics for both kinds of analysis, we use them interchangeably in our discussion (although see Walsh et al. 2013, for a more complete discussion). \nLow-mass stellar systems (galaxies, nuclear star clusters and globular clusters) present additional challenges. One is spatial resolution. To detect a black hole, the region within which the gravity of the black hole dominates the stellar motions must be resolved. This gravitational sphere of influence, roughly where the mass enclosed in stars equals that of the black hole, is given by: SOI (pc) ≈ 0.0043 ( M BH /M glyph[circledot] ) ( σ/ kms -1 ) -2 . If we consider black holes with M BH ∼ 10 5 M glyph[circledot] , for typical σ ∗ ≈ 10 -30 km s -1 , then at the limiting distance of current samples (4 Mpc) the angular size is not more than ∼ 0 . 2' (e.g., Nguyen et al. 2018). Another challenge is M/L determinations, which are more complicated in low-mass galaxies, due to ongoing star formation and dust. A third is non-axisymmetric structures such as \nFigure 2 \n<!-- image --> \nSummary of existing dynamical measurements or limits on black hole mass in low-mass stellar systems. Black hole masses in galactic nuclei (green circles) and limits (grey) are from various sources (Neumayer & Walcher 2012, Nguyen et al. 2019b, and references therein). For the limits from Neumayer et al., we show their 'best' measurement (grey circles) and the 'maximum' allowed black hole mass (arrow top). In pink are Galactic globular clusters with controversial black hole mass measurements; the bottom of the bars represent upper limits from radio observations (Tremou et al. 2018) while the top are dynamical measurements (Lutzgendorf et al. 2013). The similarly contested measurement in G1 is shown in yellow (Gebhardt et al. 2005, van der Marel & Anderson 2010a). The maroon upper limit for NGC 6397 is representative of the best limits possible in the nearest systems (Kamann et al. 2016). Blue arrows are upper limits on three Milky Way satellite dwarf galaxies, the LMC (Boyce et al. 2017), Fornax (Jardel & Gebhardt 2012), and Ursa Minor (Lora et al. 2009). \ndisks and bars, and other irregular star-forming features (e.g., Kormendy & Kennicutt 2004). A fourth is that individual bright stars can dominate the total light output, biasing the measured stellar dispersion (e.g., Lutzgendorf et al. 2015). In galaxies with ongoing star formation, this issue is particularly important. In the Milky Way, Feldmeier et al. (2014) have shown that one blue supergiant could erase the signature of the BH. \nThe issue of identifying a robust galaxy center is a fifth significant concern. For instance, center determinations for the Large Magellanic Cloud (LMC) have an uncertainty that is larger than a square degree (van der Marel & Kallivayalil 2014). For more distant galaxies, the field has avoided issues of determining galactic centers by focusing on galaxies with known nuclear star clusters, which encompasses nearly all galaxies with M ∗ ∼ 10 9 -10 10 M glyph[circledot] (Georgiev & Boker 2014, S'anchez-Janssen et al. 2019). At lower galaxy masses, as the nucleation fraction drops and galaxies appear more irregular, we will be fundamentally \nlimited by centroid determinations. \nFinally, one of the most challenging limitations to dynamical masses for < 10 4 M glyph[circledot] black holes in dense stellar systems is confusion between a putative IMBH and a central cluster of stellar-mass remnants. As dense stellar clusters evolve, the compact remnants sink to the center of the cluster and segregate in mass. Thus, the detailed assumptions made about the presence of stellar remnants (especially stellar-mass black holes) have a crucial impact on the relative evidence for an IMBH (e.g., Mann et al. 2019, and references therein). A related issue is that because of mass segregation and the desire to constrain IMBHs of modest mass, there may be very few luminous stars within the sphere of influence to provide dynamical constraints (see next subsection). Studies that do not carefully consider a range of predictions for stellar remnants will not reach robust conclusions about IMBHs. Observations in galaxy nuclei are already approaching this limit for the nearest sources (Nguyen et al. 2019b).", '3.2. Proper Motions': "When possible, stellar dynamics can be more precisely measured through proper motions of individual stars, which can break the mass-anisotropy degeneracy (e.g., Zocchi et al. 2017). Nevertheless, there are still challenges with these measurements, which generally are looking for less massive black holes as well. \nAs a test case, consider the well-studied globular cluster 47 Tuc. The sphere of influence of a ∼ 1000 M glyph[circledot] black hole in 47 Tuc would only be ∼ 1', and few stars will be observable within this radius. For example, studies of the central 1' with the Hubble Space Telescope (HST) have only produced precise proper motions for 11-12 stars (McLaughlin et al. 2006, Mann et al. 2019) down to a main-sequence mass of ∼ 0 . 65 M glyph[circledot] . Since the central mass function is relatively depleted of low-mass stars due to mass segregation, the total number of stars observable in the sphere of influence of such an IMBH, even down to the hydrogen burning limit with next-generation extremely large telescopes, might still only be ∼ < 50 stars. With this modest sample, it is not clear that the advantages in breaking the massanisotropy degeneracy gained from proper motions will be definitive in proving the presence of ∼ 1000 M glyph[circledot] IMBHs. On the other hand, a ∼ 3000-4000 M glyph[circledot] IMBH would have a factor of ∼ 10-15 more stars within the sphere of influence, and hence be considerably easier to detect than a 1000 M glyph[circledot] IMBH. \nThe concerns about a misleading population of dense stellar remnants still apply to proper motion measurements. The current state-of-the-art is to use N-body codes to model the possible signal from such a cluster of stellar-mass compact remnants (e.g., Baumgardt 2017), which still leaves ambiguous cases. A possible way to alleviate this degeneracy is to determine indirect tests of the presence of stellar-mass compact objects such as expected numbers of pulsars or X-ray binaries. Perhaps more promising is the finding of MacLeod et al. (2016a) that IMBHs should typically acquire companions with orbital periods of years, corresponding to semi-major axes of ∼ 5-10 mas for typical globular cluster distances and ∼ 1000 M glyph[circledot] IMBHs. In the models, the companions undergo frequent exchanges, but about half of the time these companions would be main sequence stars or giants, most of which should be observable in data of sufficient depth with extremely large telescopes. The proper motion due to the orbit should be very high compared to other cluster members ( ∼ > a few mas per year), making such stars readily identifiable even in short-baseline observations, and distinguishable from foreground stars by their precise central location. The theoretical \npredictions in MacLeod et al. (2016a) cover a small range of parameter space and it would be worthwhile to see extensions to a wider set of IMBH masses and core densities. This method could not prove the absence of an IMBH in any particular cluster, since a visible companion cannot be guaranteed. However, if few hundred M glyph[circledot] black holes are common in clusters, companion studies could push down to lower masses than any other method. The timing of millisecond pulsars in even wider orbits could also reveal the presence of an IMBH, though the interpretation of these timing observations is not necessarily straightforward. We revisit some of these issues below in § 3.5.", '3.3. IMBH Demographics in Galaxy Nuclei From Dynamics': "The most definite existing dynamical measurements are for nucleated 10 9 -10 10 M glyph[circledot] galaxies within ∼ 4 Mpc of the Sun. Ten such galaxies have published black hole masses or limits from stellar and gas dynamics (see Figure 2). There are published detections in five of these galaxies: M32, NGC 5102, NGC 5206, NGC 205 (Nguyen et al. 2018, 2019b), and NGC4395 (den Brok et al. 2015). There are published upper limits for five additional galaxies: NGC 300 and NGC 7793 (Neumayer & Walcher 2012), NGC 404 (Nguyen et al. 2018), the LMC (Boyce et al. 2017) and M33 (Gebhardt et al. 2001). Taking these ten objects and five detections at face value, we have a lower limit on the occupation fraction of > 50%. This is a lower limit because the measurements are really only sensitive to black holes with M BH > 10 5 M glyph[circledot] for galaxies outside of the Local Group. \nIt is worth commenting specifically on the case of the LMC. Later in this section we discuss indirect evidence (from a hypervelocity star) for a massive black hole somewhere in the LMC. Because the LMC is so close to the Milky Way, we have high spatial resolution and thus sensitivity to even lower-mass black holes. However, the increased resolution is debilitating. No measurement of the center of the LMC is in good agreement with any other (van der Marel & Kallivayalil 2014), and because there is an offset bar it is very challenging to know where to search for a putative massive black hole. The best limit we have thus far was made with great effort by Boyce et al. (2017) using VLT/MUSE observations, but their limit of M BH < 10 7 M glyph[circledot] is not terribly constraining. \nFor galaxies with lower masses ( M ∗ < 10 9 M glyph[circledot] ) within the Local Group, there are three other interesting published upper limits to mention. In order of decreasing stellar mass, we have limits in the dwarf galaxies Sagittarius, Fornax, and Ursa Minor. In the case of Sagittarius, the nucleus is the globular cluster also known as M54. Since the galaxy is actively being disrupted, indirect means are needed to estimate the original mass of the galaxy. Summing the light in the tidal features (Niederste-Ostholt et al. 2010), modeling of the stream (Laporte et al. 2018), and stellar abundances and abundance ratios (de Boer et al. 2014) all suggest that the galaxy was one of the more massive satellites, comparable to the Small Magellanic Cloud (SMC), with a stellar mass of a few × 10 8 M glyph[circledot] . Dynamical measurements of the center of M54 are reviewed below, but there is no consensus on the presence of a black hole in this cluster (Ibata et al. 2009, Baumgardt 2017). \nMoving downward in mass to the Fornax dwarf galaxy, Jardel & Gebhardt (2012) present an orbit modeling limit of M BH < 10 5 M glyph[circledot] (3 σ limit) on any black hole. Finally, there is an interesting published limit on a massive black hole in the galaxy Ursa Minor. Lora et al. (2009) argue that any centrally located black hole with M BH > 3 × 10 4 M glyph[circledot] would dissolve observed stellar clumps. The major caveat here, also noted by the authors, is that the initial location of the black hole in the galaxy is unconstrained (Bellovary et al. 2019). To \nfind any 10 3 -10 4 M glyph[circledot] black holes that may be lurking in Local Group dwarfs will likely require proper motions with an extremely large telescope (e.g., Greene et al. 2019). \nFinally, at slightly larger distances out to 10 Mpc, Neumayer & Walcher (2012) perform Jeans modeling for a sample of nine nucleated spiral galaxies in the M ∗ = 10 9 -10 10 M glyph[circledot] range. There are published upper limits for NGC 3621 (Barth et al. 2009) and NGC 4474 (De Lorenzi et al. 2013) as well. Together, these limits will prove crucial in measuring scaling relations in this mass range ( § 8). \n3.3.1. Sample Completeness. In § 9 below, we will use the dynamical sample of ten galaxies with 10 9 < M ∗ /M glyph[circledot] < 10 10 and D < 4 Mpc as one constraint on the occupation fraction. Although there is not a truly volume-complete sample of galaxies with dynamical measurements, there is no obvious bias in the galaxies that have been targeted dynamically. The searches have focused on galaxies with known nuclear star clusters, but the nucleation fraction among galaxies in this stellar mass range is as high as ∼ 80 -90% (Georgiev & Boker 2014, S'anchez-Janssen et al. 2019). Neumayer et al. (in preparation) do not see any differences between the nucleation fractions of red and blue galaxies. On the other hand, the nucleation fractions measured for blue galaxies in the M ∗ = 10 9 -10 10 M glyph[circledot] range may be biased against galaxies with ongoing vigorous star formation. A case in point is the LMC, which has no readily-identified nuclear star cluster, although as discussed above its center is unknown. \nTo explore these issues a bit more quantitatively, we rely on the nearby galaxy catalog of Karachentsev et al. (2013). This catalog includes K -band magnitudes and we make the simplifying assumption that all galaxies have M K /M glyph[circledot] ≈ 1. There are 21 galaxies with 1 . 5 < D < 5 Mpc and 10 9 < M ∗ /M glyph[circledot] < 10 10 by this definition. Of these 21, 13 have known and well-studied nuclear star clusters, while one galaxy has none (NGC 55; Seth et al. 2006). Most of the remaining seven galaxies have imaging with HST ; it is a high priority to examine these for the presence of nuclei (Hoyer et al. in preparation). In terms of mass and size, we do not see obvious biases in the nuclear star cluster properties in this sample relative to Georgiev & Boker (2014), nor is there evidence for differences between red and blue galaxies (see also Foord et al. 2017). Finally it is worth noting again that dynamical measurements are particularly challenging in blue galaxies, due both to their complicated non axisymmetric kinematics and the shotnoise from individual young stars, which must be considered as more dynamical constraints become available.", '3.4. Dynamical Searches for IMBHs in the Milky Way': "Another special nucleus postulated to house an IMBH is the one within our own Galactic Center. The Milky Way may host a population of 'leftover' IMBHs from past accretion of dwarf galaxies (e.g., Rashkov & Madau 2014, and § 2.1). IMBHs have been invoked to explain several observational phenomena associated with the Galactic Center, although, to date, none of the evidence can be regarded as definitive. It is worth recalling that rather strict constraints exist from the small residual proper motion of Sgr A* perpendicular to the plane of the Galaxy: no dark object larger than 10 4 M glyph[circledot] is permitted within 10 3 -10 5 AU from Sgr A ∗ (Reid & Brunthaler 2004). \nThe 'paradox of youth' of the stars within the central parsec of the Galactic Center (Ghez et al. 2003) has inspired mechanisms to shepherd stars into the Galactic Center with the aid of an IMBH (Hansen & Milosavljevi'c 2003). The inward migration of an \nIMBH can also account for the origin of the kinematic distribution of these young stars (Yu et al. 2007). To this end, the bright infrared source IRS 13E, at a projected distance of only 0.13 pc from Sgr A ∗ , has been the subject of intense scrutiny. High-spatial resolution observations by Maillard et al. (2004) resolve the source into a compact group of several co-moving massive stars, prompting speculation that it constitutes the disrupted core of a young massive cluster in which an IMBH has coalesced by runaway growth (e.g., Portegies Zwart & McMillan 2002). The proper motion measurements of Schodel et al. (2005) would require a large black hole mass of > 10 4 M glyph[circledot] , which is difficult to reconcile with the absence of clear non-thermal radio and X-ray emission. A dark object of this mass scale, however, would satisfy the ionized gas kinematics recently reported by Tsuboi et al. (2017). \nGas kinematics can be notoriously tricky to interpret. This challenge is well exemplified by the compact cloud CO-0.40-0.22, whose large line-of-sight velocity and large internal velocity dispersion ( ∼ 100 km s -1 ) prompted Oka et al. (2016) to suggest that it experienced a gravitational kick from a dark 10 5 M glyph[circledot] object within 60 pc of the Galactic Center. The spectrum of its associated millimeter continuum and IR source, however, is more consistent with that of a protostellar disk instead of a scaled-down version of Sgr A ∗ (Ravi et al. 2018), and its detailed kinematics are more consistent with cloud-cloud collisions (Tanaka 2018) or supernova-driven interactions (Yalinewich & Beniamini 2018). Interest in this topic continues unabated (Takekawa et al. 2019, Tsuboi et al. 2019).", '3.5. IMBH Demographics in Globular Clusters From Stellar Dynamics': 'Many papers have presented dynamical evidence for IMBHs in globular clusters, but as of this writing there are no systems for which such evidence is unambiguous. \nPerhaps the best-studied system is ω Cen. There are claims of a massive IMBH ( ∼ 4 -5 × 10 4 M glyph[circledot] ) in this star cluster based on isotropic modeling of the radial velocity dispersion and surface brightness profiles (Noyola et al. 2010, Jalali et al. 2012, Baumgardt 2017). However, the velocity dispersion signature of an IMBH is not found in the proper motions of central stars (van der Marel & Anderson 2010b), and no resolution of this observational issue has been given in the literature. On the modeling side, anisotropy and the presence of dark remnants could account for most or all of the IMBH signature found in other studies (e.g., Zocchi et al. 2017, 2019, Baumgardt et al. 2019). Some papers have highlighted the need for more sophisticated modeling of the velocities and proper motions of individual stars (see § 3.2) rather than a binned dispersion profile in the context of ω Cen and other clusters (Lutzgendorf et al. 2012, Baumgardt et al. 2019, Mann et al. 2019). Additional observations and modeling are clearly needed for ω Cen, but in any case, the contrasting results highlight the challenges of this work. These challenges are only magnified as one considers IMBHs of lower mass. \nClaims of dynamical evidence for massive black holes have been made in multiple other clusters, including for NGC 6388 and M54. The case of NGC 6388 is similar to ω Cen, with contrasting results rooted partially in conflicting observational results (Lutzgendorf et al. 2011, Lanzoni et al. 2013, Lutzgendorf et al. 2015). Multiple studies working with similar data sets have found tentative evidence for a ∼ 10 4 M glyph[circledot] IMBH in M54, with the same interpretation caveats as for ω Cen, and the additional complication that the cluster is embedded in the remnants of the Sagittarius dwarf galaxy (Ibata et al. 2009, Baumgardt 2017). There is no accretion evidence for an IMBH in any of these clusters (Tremou et al. 2018, § 5). As a promising example of the dynamical limits possible using forefront instru-me \nfor nearby, dense globular clusters, Kamann et al. (2016) show that any IMBH in NGC 6397 must be ∼ < 600 M glyph[circledot] (Figure 2), consistent with the radio limit for this cluster (Tremou et al. 2018). \nIMBH searches have been extended outside the Milky Way to the nearest massive galaxy with a large globular cluster population, M31. The best-studied globular cluster in M31 is G1. At this distance ( D = 0 . 8 Mpc), the observational complication is that the putative sphere of influence of an ∼ 2 × 10 4 M glyph[circledot] IMBH is barely resolved with HST spectroscopy and imaging. G1 was notable not solely for the contested interpretation of the central kinematics (Gebhardt et al. 2002, Baumgardt et al. 2003, Gebhardt et al. 2005), which have the same issues already discussed, but also for the unique addition of X-ray and radio accretion evidence for a possible IMBH (Pooley & Rappaport 2006, Ulvestad et al. 2007, and § 4). Unfortunately, this radio emission was not confirmed in deeper observations, and the multi-wavelength data are consistent with a standard low-mass X-ray binary (MillerJones et al. 2012). Hence G1 belongs to a similar category as ω Cen and M54, with debated dynamical evidence for an IMBH. \nSeveral recent papers have taken a new dynamical tack: using the timing properties of millisecond pulsars in the cores of globular clusters to constrain the presence of an IMBH. Kızıltan et al. (2017) modeled the properties of a subset of pulsars in the central regions of 47 Tuc, finding evidence that pulsar accelerations were best-explained by the presence of a central IMBH of mass ∼ 2300 M glyph[circledot] . Subsequent analyses of similar pulsar data sets did not reproduce this result: they found no evidence for an IMBH, with a formal 99% upper limit of < 4000 M glyph[circledot] (Freire et al. 2017, Abbate et al. 2018). This non-detection is consistent with the results of several studies that modeled the detailed proper motions of individual stars in the core of 47 Tuc (McLaughlin et al. 2006, Mann et al. 2019), and found 1 σ upper limits in the range < 1000-1500 M glyph[circledot] , and also consistent with the 3 σ radio upper limit of < 1040 M glyph[circledot] (Tremou et al. 2018). We conclude that there is no compelling evidence for an IMBH in 47 Tuc based on available observations and modeling. \nPerera et al. (2017) argue that a single pulsar in NGC 6624 has timing properties consistent with being in a very long period, eccentric, loosely bound orbit around an IMBH of mass > 7500 M glyph[circledot] . Other interpretations of these data are possible, and recent dynamical modeling of stars constrains an IMBH to be ∼ < 1000 M glyph[circledot] (Baumgardt et al. 2019). Future observations and modeling will help distinguish among these possibilities.', '3.6. Hypervelocity stars': 'Hypervelocity stars are those with Galactocentric velocities in excess of the escape velocity at their present location. There is compelling evidence that some of the most extreme hypervelocity stars are due to interactions between stellar binaries and the supermassive black hole at the center of the Galaxy (Hills 1988, Brown et al. 2005, Zhang et al. 2013, Brown et al. 2018, Koposov et al. 2019). The remainder, including some of the stars with less extreme velocities, likely have a wide range of origins, including close binaries disrupted by the death of one of the stars, dynamical encounters in star clusters, and even tidal material from accreted satellites (e.g., Hirsch et al. 2005, Perets & ˇ Subr 2012, Abadi et al. 2009, Shen et al. 2018). \nIf star clusters or Galactic satellites host IMBHs, then extreme hypervelocity stars whose kinematics exclude a Galactic Center origin could provide evidence for IMBHs. Perhaps the best example of this is for the hypervelocity star HVS3, which may have been ejected from \nthe LMC (Edelmann et al. 2005) by an interaction with an IMBH (Gualandris & Portegies Zwart 2007). The star is at most ∼ 35 Myr old, so the ejection must have been relatively recent, and from a young stellar population (Edelmann et al. 2005). Erkal et al. (2019) add Gaia data to the analysis and show that an origin in the LMC is much more likely than from the Galactic Center (see also Lennon et al. 2017). They argue the relative velocity of HVS3 ( ∼ 870 km s -1 ) could only have originated in an interaction with a ∼ 4 × 10 3 -10 4 M glyph[circledot] IMBH. This should intensify efforts to search for other evidence of an IMBH near the center of the LMC or in its young star clusters. We note that the confirmation of an IMBH in one of these clusters (which typically have M glyph[star] ∼ < 10 5 M glyph[circledot] ) would be remarkable. \nHypervelocity stars may also prove to be the most robust probe of IMBHs in the Galactic Center (Yu & Tremaine 2003, Baumgardt et al. 2006). The observed spectrum of ejection velocities appears to be inconsistent with theoretical expectations for a supermassive black hole-intermediate-mass black hole binary, but the current data leave much room for improvement (Sesana et al. 2007). Even more powerful would be the eventual future detection of hypervelocity binary stars (Lu et al. 2007, Sesana et al. 2009). \nFuture missions such as LSST and WFIRST will improve the ability to search for hypervelocity stars as signposts to IMBHs in Galactic satellites or globular clusters, since they will be sensitive to more typical lower-mass stars. Because of the longer lifetimes of these stars, they may have more possible kinematic origins, and other information like chemical abundances and abundance ratios may be needed to interpret their origin.', '4. Searches for Accreting IMBHs in Galaxy Nuclei': 'Even with next-generation facilities, dynamical measurements will only reach ∼ 10 Mpc. Thus, to gain population statistics we must rely on accretion signatures to identify the presence of a black hole. We discuss the physics of accretion signatures at different wavelengths, and the samples that have resulted from searches so far.', 'IMBH Demographics in Globular Clusters From Microlensing': "There is also a promising alternative method for IMBH searches in globular clusters. Typically, the microlensing effects discussed are 'photometric', resulting in magnification of the background source by the lens. However, the optical depth for photometric microlensing by IMBHs is very low (Safonova & Stalin 2010), and detections are unlikely for reasonable monitoring campaigns of Galactic globular clusters. Kains et al. (2016) instead suggest the possibility of using astrometric microlensing, which is much less sensitive to the angular separation of the source and lens than standard microlensing. The detectability of this astrometric microlensing signal is maximized for clusters close to the Sun that also have high background densities. This methodology was used to search for evidence of a central IMBH in M22 with HST by Kains et al. (2018). Owing to gaps in the astrometric time series the resulting upper limit on an IMBH was not constraining compared to limits from other methods (Strader et al. 2012), but future observations with the JWST or the Wide Field Infrared Survey Telescope ( WFIRST ) should provide improved constraints to IMBH masses ∼ < 10 4 M glyph[circledot] in a subset of nearby clusters, including M22, M4, and 47 Tuc.", '4.1. Optical Spectroscopic Selection': "The two prototype low-mass active galactic nuclei (AGNs), NGC 4395 (Filippenko & Sargent 1989, Filippenko & Ho 2003) and POX 52 (Kunth et al. 1987, Barth et al. 2004), were both originally identified based on optical spectroscopic signatures. In light of the apparent rarity of AGNs in late-type hosts (Ho et al. 1997, Ho 2008), large spectroscopic surveys are needed to tease out any significant statistical sample. Greene & Ho (2004) performed the first systematic search for AGNs powered by low-mass black holes using SDSS DR1, producing 19 low-redshift ( z < 0 . 35) broad-line AGNs with estimated black hole masses ∼ < 10 6 M glyph[circledot] , subsequently boosted to a sample of ∼ 200 using SDSS DR4 (Greene & Ho 2007b,a). Although the AGN-based masses are uncertain (see later in this section), followup shows that the ensemble of these objects are powered by low-mass black holes. The host galaxies are sub -L ∗ , disky galaxies (Greene et al. 2008, Jiang et al. 2011), with the low gas-phase metallicities expected at these masses (Ludwig et al. 2012). Subsequent efforts have adopted variants of this strategy to enlarge and refine the broad-line sample, which to date stands at ∼ 500 sources (Dong et al. 2012b, Chilingarian et al. 2018, Liu et al. 2018). These objects can only be detected in SDSS when they radiate at close to their Eddington limits, and as such they are rare, comprising only f AGN ≈ 0 . 1% of the local galaxy population with M ∗ < 10 10 M glyph[circledot] . \nIn addition to selecting on broad-line properties, one can also pre-select low-mass galaxies and look for those whose emission lines classify them as AGN based on the 'BPT' diagram (Baldwin et al. 1981), which classifies objects based on line ratios between strong inter-stellar medium lines. AGNs have long been known to separate in these diagrams due to their hard ionizing spectra (e.g., Ho et al. 1997). A number of groups have started with a stellar-mass selected sample, M ∗ < 10 10 M glyph[circledot] or less, to search for AGN signatures in dwarf galaxies (Barth et al. 2008, Reines et al. 2013, Moran et al. 2014, Sartori et al. 2015). Moran et al. focus on 28 sources within 80 Mpc ( f AGN = 2 . 7%), while Reines et al. present 151 AGN candidates from a parent sample of ∼ 25 , 000 emission-line galaxies ( f AGN ≈ 0 . 5%). Most AGN uncovered in this manner are narrow-line objects, but a subset have broad H α emission indicating M BH ≈ 10 5 -10 6 M glyph[circledot] . The late-type spiral RGG118 (Baldassare et al. 2017) even has a black hole as small as 50,000 M glyph[circledot] (Baldassare et al. 2015). \nFor the local AGN selected from SDSS, there is substantial multi-wavelength follow-up that can inform further searches. Multi-wavelength follow-up of the Barth et al. sample confirm their obscured AGN nature (Thornton et al. 2009, Hood et al. 2017). The AGN selected based on broad emission lines behave like AGN powered by more massive black holes. In the radio, a small number of objects have deep follow-up observations but few are radio loud (Greene et al. 2006, Wrobel & Ho 2006, Wrobel et al. 2008). More attention has been devoted to the X-rays (Greene & Ho 2007c, Desroches et al. 2009, Dong et al. 2012a, Yuan et al. 2014, Plotkin et al. 2016), using Chandra and XMM-Newton observations of sufficient depth to perform detailed spectral and timing analysis (Moran et al. 2005, Dewangan et al. 2008, Thornton et al. 2008, Miniutti et al. 2009, Ai et al. 2011, Kamizasa et al. 2012, Jin et al. 2016). The results from X-ray timing are particularly critical, as they help to lend confidence to and independently confirm the black hole mass estimates from broad H α . As a class, these objects are among the most rapidly variable extragalactic X-ray sources (Dewangan et al. 2008, Miniutti et al. 2009, Ai et al. 2011, Kamizasa et al. 2012), pointing to low-mass black holes. \nIn general, optically selected AGNs appear to be rare in dwarf galaxies. The AGN detection rate for low-mass galaxies using traditional optical tracers seems to hover around \n- ∼ 1% (Supplemental Table 5). It so far has been prohibitive to correct for the major incompleteness in these samples stemming from star formation contamination, aperture dilution, and dust reddening (e.g., Greene & Ho 2007b, Trump et al. 2015). \n4.1.1. Reverberation Mapping. For objects with broad emission lines, 'reverberation mapping' yields information about the size scale of the broad-line region (BLR) by measuring the delay between the continuum and line light curve, emitted from the accretion disk and BLR respectively (Peterson 2014). Combining the BLR radius r with the line width ∆ V , yields a virial-like mass M BH = f vir r (∆ V ) 2 /G , with f vir the the virial constant. The low luminosities of low-mass AGNs suggests that their BLRs will be compact, and hence any attempt at reverberation mapping must have sufficiently high cadence to sample lags of less than a few days at most. We emphasize that the reverberation mapping-based masses are currently calibrated with the dynamical samples through f vir . Because the structure and kinematics of the broad-line region are unknown, reverberation mapping yields a 'virial product', which is currently scaled by f vir such that the AGN samples obey the same M BH -σ ∗ relations as inactive galaxies (although see Pancoast et al. 2014). This scaling has at least a factor of two ambiguity in it depending on what galaxy samples are used in the M BH -σ ∗ calibration sample (e.g., Ho & Kim 2014). \nThe initial effort to monitor the prototype low-mass AGN NGC 4395 gave only a marginally useful constraint on the lag for H β (Desroches et al. 2006), and preference has been given to the C IV λ 1549 measurement of Peterson et al. (2005), which led to a black hole mass estimate of M BH = (3 . 6 ± 1 . 1) × 10 5 M glyph[circledot] . This mass is consistent, within the considerable tolerance of the uncertainties, with the direct dynamical estimate of M BH = 4 +8 -3 × 10 5 M glyph[circledot] by den Brok et al. (2015). Woo et al. (2019) recently advocate a markedly lower value of M BH ≈ 9100 M glyph[circledot] , based on a short, 80-min lag detected from a narrow-band H α reverberation mapping campaign with rapid sampling. The published mass adopts a virial constant f vir = 4 . 5. Using a value calibrated to lower-mass spiral galaxies ( f vir = 3 . 2 Ho & Kim 2014), the mass formally drops to M BH ≈ 6500 M glyph[circledot] . The difference in black hole mass between Woo and Peterson is mainly due to differences between the linewidth of C IV and H β , while the two works recover consistent lags. Very few studies have done intercomparisons of multiple lines in the context of reverberation mapping, and certainly not in this mass and luminosity range (e.g., Park et al. 2017). \nSomewhat more massive but still in the neighborhood of 10 6 M glyph[circledot] or less is UGC 6728, whose H β lag of τ = 1 . 5 ± 0 . 8 days yields M BH = (5 . 2 ± 2 . 9) × 10 5 M glyph[circledot] (Bentz et al. 2016, scaled to our preferred f vir = 3 . 2). The published lag for SDSS J114008.71+030711.4 (GH08 from Greene & Ho 2004) of τ = 1 . 5 +4 -2 days is short and highly uncertain (Rafter et al. 2011), but it is likely less than 6 days, in which case M BH < 3 . 9 × 10 5 M glyph[circledot] . Lastly, we mention three Seyfert 1 nuclei with well-measured H β lags hosted in late-type spiral galaxies, as summarized in Ho & Kim (2014): NGC 4051 with τ = 2 . 5 ± 0 . 1 days and log ( M BH /M glyph[circledot] ) = 6 . 11 ± 0 . 04; NGC 4253 (Mrk 766) with τ = 5 . 4 0 . 2 -0 . 8 days and log ( M BH /M glyph[circledot] ) = 5 . 98 ± 0 . 29; and Mrk 202 with τ = 3 . 5 ± 0 . 1 days and log ( M BH /M glyph[circledot] ) = 5 . 98 ± 0 . 06. \nReverberation mapping also provides a relationship (the so-called 'radius-luminosity' relation) between the AGN luminosity and the typical size of the broad-line region (e.g., Bentz et al. 2013). Using the virial constant and the radius-luminosity relation, we can calculate 'single-epoch' virial black hole masses for broad-line AGN. It is hard to know what the systematic uncertainties are on these masses, particularly at low black hole mass where the radius-luminosity relation is not directly measured.", '4.2. Multi-wavelength Searches and Confusion With Star Formation': "Searching for concrete evidence of AGNs in low-mass galaxies poses a set of unique challenges. One of the major complications in applying the traditional optical BPT diagrams is that the AGNs become hopelessly intermingled with star-forming galaxies at low metallicity (Groves et al. 2006, Stasi'nska et al. 2006, Cann et al. 2019). Along with a BPT selection, Sartori et al. (2015) selected additional samples using both a He II λ 4686 diagnostic diagram and mid-IR color cuts that have proven effective in selecting luminous AGNs (Jarrett et al. 2011, Stern et al. 2012). Distressingly, almost none of the samples identified by the three methods overlap. Why? \nAs Hainline et al. (2016) emphasize, young starbursts in the low-metallicity environment of dwarf galaxies have red mid-IR colors that closely mimic those of AGNs. This, unfortunately, calls into question the usage of mid-IR color to select AGNs in late-type, low-mass galaxies (Satyapal et al. 2014, Marleau et al. 2017, Kaviraj et al. 2019). The lower metallicity environment of dwarf galaxies is characterized by higher electron temperatures and higher levels of excitation for the ISM. In theory, there may also be a more top-heavy stellar initial mass function (e.g., Bromm et al. 2002). The preponderance of massive stars profoundly affects the heating and ionization of the gas. High-mass X-ray binaries may be responsible for the ionizing photons for He II λ 4686 (Schaerer et al. 2019). If so, He II ceases to be a useful AGN indicator in dwarf galaxies (Sartori et al. 2015, Bar et al. 2017). Massive O stars and Wolf-Rayet stars can generate sufficient extreme UV radiation to excite high-ionization lines such as [O IV] 25.89 µ m (Lutz et al. 1998, Schaerer & Stasi'nska 1999), which renders moot any attempt to use this line to select low-mass AGNs (Georgakakis et al. 2011). [Ne V] 14.32 µ m, normally considered a robust AGN indicator because of its high ionization potential of 97.12 eV (Satyapal et al. 2007, 2008, 2009, Goulding et al. 2010) is not immune either, as it can be excited by fast-shocks produced by stellar winds from massive stars and supernovae (Contini 1997, Izotov et al. 2012). \nAs a case in point, we draw attention to the local spheroidal NGC 185, whose nuclear optical line emission, though feeble, technically qualifies it as a 'Seyfert 2' galaxy (Ho et al. 1997). However, the recent detailed spatially resolved optical and X-ray analysis of this object by Vuˇceti'c et al. (2019) clearly shows that the excitation of the central nebula is due to supernova remnants. NGC 185 is a fake AGN. This caveat may well impact larger samples of AGN selected based on narrow emission lines. \nEven broad emission lines are not sacrosanct. While the presence of broad H α is typically regarded as ironclad evidence of an AGN, high-velocity gas can also be of stellar origin. Wolf-Rayet galaxies, for example, often exhibit broad wings to the H α line (M'endez & Esteban 1997), and the optical spectra of some Type II supernovae bear an uncanny resemblance to Seyfert 1 nuclei (Filippenko 1989). Baldassare et al. (2016) obtained multiple-epoch observations of the type 1 sources from Reines et al. (2013) with evidence of star formation in their narrow-line spectra and discovered that in most of them the broad H α line is transient over a baseline of several years, suggesting a supernova origin. Even when broad H α is persistent and too strong to be explained easily by supernovae, as is the case in some blue compact dwarfs (Izotov & Thuan 2008, Izotov et al. 2010), no compelling, independent evidence for AGNs has yet surfaced. Follow-up Chandra observations of the metal-poor AGN candidates of Izotov & Thuan (2008) reveal that their X-ray emission is far weaker compared to their optical or mid-IR emission than expected for active galaxies (Simmonds et al. 2016). Even sensitive X-ray and radio observations do not find compelling evidence for AGN in blue compact dwarfs (Latimer et al. 2019). There is also the converse problem, \nthat broadened lines with ∼ 200 -400 km s -1 may arise from the narrow-line region of an obscured AGN, with the width reflecting non-virial motions associated with the AGN. In such cases the linewidth may not have any relation to the black hole mass. Studies that push to ambitiously low linewidth may suffer this contamination (Chilingarian et al. 2018). \nSeveral of the AGN candidates in late-type galaxies originally identified through detection of [O IV] 25.89 µ m or [Ne V] 14.32 µ m have since been followed up in X-rays (Gliozzi et al. 2009, McAlpine et al. 2011, Georgakakis et al. 2011, Secrest et al. 2012, Hebbar et al. 2019), and in general the X-rays are found to be weaker than expected. Diffuse, thermal emission is occasionally detected when the data are of sufficient quality. Certainly one can appeal to absorption to explain the deficit of hard X-rays, but we cannot rule out the possibility that these galaxies actually lack AGNs.", '4.3. Pushing Down the Luminosity Function with X-ray or Radio Observations': "Given the observation of central black holes in the Local Group with very low Edddington ratios (including Sgr A glyph[star] and M31 glyph[star] ), the minority population of highly accreting black holes identified by optical spectroscopy must be the tip of the iceberg. In addition to being faint, AGNs of low accretion rate have systematically lower ionization parameters (Ho 2009), and they may lack broad emission lines altogether (Elitzur & Ho 2009). X-ray observations provide a clean tool to overcome these problems in a wide range of circumstances. Black hole accretion invariably generates X-ray emission (Brandt & Alexander 2015), and low accretion rates have the virtue of producing proportionately even more hard X-rays (Ho 2008). The resolving power of Chandra and the low background of its ACIS instrument offer the ideal combination to detect faint compact sources in nearby galaxies, even with short exposures, and the excellent astrometric accuracy of the satellite can align the optical or near-IR nucleus of the galaxy to within 1' or better. The main source of confusion comes from X-ray binaries, but the degree of contamination can be estimated once the stellar mass and star formation rate are known (e.g., Miller et al. 2015). \nA number of studies have exploited this opportunity to evaluate the incidence of AGNs in nearby late-type galaxies, succeeding in identifying X-ray nuclei in star-forming (Ghosh et al. 2008, Zhang et al. 2009, Grier et al. 2011, She et al. 2017b), dwarf irregular (Lemons et al. 2015), and local Lyman-break analog (Jiang et al. 2011) galaxies. Desroches & Ho (2009) analyzed Chandra images of 64 Scd-Sm spirals within 30 Mpc and discovered an X-ray core in 17 of them ( f AGN = 27%). The Sc-Sm spirals in the sample of Zhang et al. (2009) yield a consistent result ( f AGN = 30%). She et al. (2017a) extended this effort to a more comprehensive census of more than 700 galaxies within 50 Mpc; among late-type, bulgeless spirals, the detection rate of X-ray cores is f AGN = 21%. Unlike the optically selected sources, these X-ray selected nuclei are all highly sub-Eddington, with median L/L bol ≈ 10 -4 . Archival work searching for X-rays in ultra-compact dwarfs has been far less conclusive (Pandya et al. 2016). \nThe above efforts, largely based on archival data, have been complemented by a series of experiments aimed at characterizing the incidence of X-ray nuclei using dedicated Chandra surveys of nearby, lower mass early-type galaxies, focused on the Virgo cluster (Ghosh et al. 2008, Gallo et al. 2010), the Fornax cluster (Lee et al. 2019), and in the field (Miller et al. 2012, Gallo & Sesana 2019). Among the ∼ 200 early-type galaxies within 30 Mpc uniformly observed to date, Miller et al. (2015) conclude that for galaxies with stellar mass M ∗ < 10 10 M glyph[circledot] , f AGN > 20%. Their observations in this stellar mass range are \nsensitive to L/L bol ≈ 10 -4 , but in more massive galaxies, the same depths probe much lower L/L bol . By assuming that the Eddington ratio distribution can be modeled as a smooth function of M ∗ , Miller et al. bracket the occupation fraction to fall between 30% and 100% at 1 σ (see also Gallo & Sesana 2019). Work by Aird et al. (2013) supports the idea that the Eddington ratio distribution varies only very mildly as a function of stellar mass. Finally, it is very encouraging that the active fractions uncovered from X-rays in both star-forming and quiescent low-mass galaxies are comparable. These X-ray results, combined with the dynamical ones above ( § 3), strongly suggest a high ( > 50%) occupation fraction in M ∗ = 10 9 -10 10 M glyph[circledot] galaxies. \nRecently Reines et al. (2019) used deep and high-resolution radio imaging with the VLA to follow up 111 dwarf galaxies (3 × 10 7 < M ∗ /M glyph[circledot] < 3 × 10 9 ) with prior radio detections in FIRST. Of these 13 are likely to be powered by AGN, due to their point-like nature and luminosity relative to their star formation rates. Only one of these 13 was also identified by optical spectroscopy. These data cannot be used to measure the AGN fraction in a straightforward way because of the sample selection, but they do highlight the promise of even existing radio telescopes to unveil lower luminosity AGN populations. Intriguingly, many of the sources are found offset from their galaxy nucleus, perhaps consistent with predictions (Bellovary et al. 2019, Pfister et al. 2019) that in low-mass galaxies the seed black holes may never settle at galaxy centers ( § 2.1). Another possible origin for some of these sources may be fueling of wandering black holes (see also § 5).", '4.4. Going Further with the Fundamental Plane of Radio Activity': "Combining radio emission with X-rays could be even more effective at probing AGNs with very low Eddington ratios. Merloni et al. (2003) and Falcke et al. (2004) found that both supermassive and stellar-mass black holes show fundamental similarities in their accretion flows in the radiatively inefficient low/hard state. Observationally, the X-ray luminosity ( L X ; a product of the accretion rate and radiative efficiency) and the radio continuum luminosity ( L R ; a measure of the jet power) scale with the mass of the black hole in a simple manner, such that a combination of these three quantities form a tolerably clean two-dimensional sequence (the fundamental plane) in three-dimensional space. \nThe physical cause of the fundamental plane is not necessarily straightforward, since the spectral energy distributions of accretion flows are expected to change systematically with black hole mass. There are no clear conclusions about whether most or all of the radio and X-ray emission is associated with a jet, or whether the X-ray emission might instead come from a corona that is separate from the jet, though still presumably linked to the accretion flow (see, e.g., the discussion in Plotkin et al. 2012). \nA major challenge in using the fundamental plane comes from the large scatter. This considerable scatter must be at least partially intrinsic, based on a number of arguments. One is that among X-ray binaries, L R /L X can vary by a factor of at least a few at fixed L X (e.g., Jonker et al. 2012). Another is that even in the newest modeling of the fundamental plane-with careful restriction to objects with high-quality dynamical masses and radio and X-ray data, and consideration of the measurement uncertainties-the scatter for determining black hole masses is still large at ∼ 1 dex. The best-fit plane from Gultekin et al. (2019) is: log ( M/ 10 8 M glyph[circledot] ) = (1 . 09 ± 0 . 10) log ( L R / 10 38 erg s -1 ) ( -0 . 59 ± 0 . 16) log ( L X / 10 40 erg s -1 ) + (0 . 55 ± 0 . 22). We note that the use of previous fits based on careful sample selection and fitting (e.g., Plotkin et al. 2012) would give similar results within the \nuncertainties in most cases. \nDespite these uncertainties, many works have moved forward in a observational spirit to use the fundamental plane to constrain black hole masses in circumstances where more direct measurements are prohibitively difficult or impossible. \nSoon after the discovery of the fundamental plane, Maccarone (2004) pointed out that this would be a powerful tool to search for IMBHs in globular clusters ( § 5). The first highprofile use of the fundamental plane to show evidence for a modest mass nuclear black hole was in the starbursting galaxy Henize 2-10 (Reines et al. 2011). They used the radio and X-ray data to argue for the presence of a log M/M glyph[circledot] = 6 . 3 ± 1 . 1 supermassive black hole. Follow-up very long baseline interferometry observations constrained the radio source size to be < 3 × 1 pc and confirmed a non-thermal origin (Reines & Deller 2012). Higher-resolution data revealed multiple components to the central X-ray source, moving the fundamental plane mass estimate up to log M/M glyph[circledot] ∼ 7 (Reines et al. 2016). Subsequent work showed that the X-ray spectrum of the nuclear source is more consistent with a supernova remnant than an AGN (Hebbar et al. 2019), although the possible presence of hour-scale variability would not favor this scenario (Reines et al. 2016, Hebbar et al. 2019). There is also no evidence for AGN ionization in the galaxy center (Cresci et al. 2017). In the end, given the stellar mass of 10 10 M glyph[circledot] (Nguyen et al. 2014), the detection of a black hole may be of limited relevance for IMBH studies, but it does show that use of the fundamental plane is challenging in star-forming galaxies. \nAnother use of the fundamental plane to find evidence for a low-mass central black hole is in the low-mass galaxy NGC 404, where the plane yielded a mass estimate of log M/M glyph[circledot] = 6 . 4 ± 1 . 1 (Nyland et al. 2012). Newer observations may support an AGN interpretation of this system, although the evidence is not yet conclusive (Nyland et al. 2017). On the other hand, dynamical studies of NGC 404 give a 3 σ upper limit of M BH < 1 . 5 × 10 5 M glyph[circledot] (Nguyen et al. 2017). These measurements are formally consistent given the large scatter in the fundamental plane, but cannot be taken as a vote of confidence for the success of the fundamental plane near the IMBH regime either. \nGoing forward, for most reasonable central black hole mass functions, volume-limited fundamental plane surveys are almost certainly needed to effectively address the question of whether IMBHs exist, due to the usual Malmquist bias that a flux-limited survey is much more likely to detect massive black holes at larger distances. If IMBHs are less common than central black holes of higher masses (see § 9), then this bias would be exacerbated by the scatter in the fundamental plane, producing an Eddington-like bias in the IMBH candidate sample. \nConsider the use of the fundamental plane to find candidate IMBHs well-suited for dynamical follow-up in a nearly fixed distance sample, e.g., the Chandra survey of Virgo galaxies by Gallo et al. (2008). This survey found a large number of central X-ray sources with luminosities near their detection limit of ∼ 4 × 10 38 erg s -1 . If a subset of these sources hosted IMBHs with masses in the range 10 4 -10 5 M glyph[circledot] and they obeyed the fundamental plane, the 5 GHz radio continuum flux densities would typically be only a few µ Jy. Such sensitivities are achievable with very long (10-20 hr) integrations on the Jansky Very Large Array, but the time investment for such a 'fishing expedition' would be challenging for more than a few galaxies. On the other hand, next generation facilities such as the next generation VLA (ngVLA), which has a factor of 10 higher sensitivity than the Jansky VLA, could carry out a survey of ∼ 100 galaxies with a reasonable time investment, and with a resolution of < 0 . 1', could pinpoint the location of any radio emission. Cross-matching \nwith existing high-resolution HST images could weed out possible non-nuclear contaminants such as star-forming regions, supernova remnants, or background galaxies.", '4.5. The Promise of Variability Selection': 'All AGN vary, and this has been used as a successful selection tool (e.g., Sarajedini et al. 2006, MacLeod et al. 2011). In the near future, a number of upcoming surveys will make variability selection very powerful for IMBH searches. \nOptical variability has been used to select low-mass black holes specifically (Morokuma et al. 2016), but the promise of this technique has not been fully explored yet. Heinis et al. (2016) find AGN in galaxies with stellar masses as low as M ∗ ≈ 10 9 . 5 M glyph[circledot] using a variability selection. Baldassare et al. (2018, 2019) use optical variability to find AGN candidates in galaxies with M ∗ spanning 10 7 -10 10 M glyph[circledot] , a large fraction of which are not uncovered through optical emission-line selection. With upcoming surveys like LSST, this discovery space should grow. \nAGN also vary in the radio, for diverse reasons both intrinsic (variations in the accretion rate; shocks in the relativistic jet) and extrinsic (scattering and/or magnification caused by interstellar plasma). Synoptic radio surveys have successfully used radio variability to select AGN (e.g., Mooley et al. 2016). Current and future radio continuum surveys such as the VLA Sky Survey and those with the Square Kilometer Array could identify candidate IMBHs as variable radio sources associated with low-mass galaxies. \nIMBHs should also leave distinct signatures in their X-ray variability signals. Specifically, because the X-ray emission region is very compact compared to more massive black holes with AGN, the variability timescales should be short. This is clearly seen with NGC 4395 (Moran et al. 2005). Kamizasa et al. (2012) leveraged this idea to search the XMMNewton archive for low-mass black hole candidates. Those they found have a median black hole mass of ∼ 10 6 M glyph[circledot] (Ho & Kim 2016), but in principle surveys like eROSITA may further such a search. The X-ray excess variance, while promising as a search tool, may not track black hole mass anymore below ∼ < 10 6 M glyph[circledot] , where the variance-mass relation seems to flatten (Ludlam et al. 2015, Pan et al. 2015).', '4.6. Moving to Higher Redshift': 'In addition to pushing to deeper limits for local samples, we could gain orthogonal demographic constraints by looking at high luminosity sources over much larger volumes; this work is in its infancy. So far, the effort to search for accreting black holes in dwarf galaxies beyond the local universe has focused on deep X-ray observations of well-studied extragalactic fields, finding candidates from z < 0 . 5 (Schramm et al. 2013, Pardo et al. 2016, Aird et al. 2018) out to z ≈ 2 . 4 (Mezcua et al. 2016, 2018a). We emphasize that these objects should be considered as candidates. In the case of the Mezcua objects in particular, many of the faint sources are proximate to more luminous objects, making the matching particularly challenging. The use of photometric redshifts with active galaxies also adds additional ambiguity. Finally, in these very high redshift sources, the implied Eddington ratios are substantially super-Eddington. Additional folow-up is needed. As high-redshift spectroscopic samples continue to grow (e.g., Takada et al. 2014) we will continue to build reliable luminosity functions for lower-mass black holes at intermediate and high redshifts. \nStudying the incidence of X-ray emission from low-mass host galaxies ( M ∗ = 5 × 10 9 -2 × 10 10 M glyph[circledot] ) out to z ≈ 1, Shi et al. (2008) placed a strong lower limit of 12% to the fraction \nof local low-mass galaxies harboring black holes. This statistical result agrees well with the surveys of nearby galaxies summarized above. In the future, along with next-generation X-ray missions, deep spectroscopic surveys with JWST and WFIRST will certainly provide complementary samples of optically selected AGN at moderate redshift.', '5. Searches for Accreting Black Holes Outside of Galaxy Nuclei': 'Perhaps the most promising tool to distinguish between seeding models comes from finding the black holes that are not in galaxy nuclei ( § 2). Thus far, dynamical searches in stellar clusters have been challenging to interpret ( § 3). However, some intriguing off-nuclear objects have surfaced due to their accretion signatures, which we review here.', '5.1. Ultra-Luminous X-ray Sources': "Ultra-luminous X-ray sources (ULXs) provided early impetus to think about black hole formation in stellar clusters (e.g., Ebisuzaki et al. 2001), but in recent years our understanding of these objects has evolved. There are many reviews of ultra-luminous X-ray sources (e.g., Kaaret et al. 2017), and we provide only a brief discussion here of the work most directly relevant to IMBHs. \nUsually ULX samples are constructed from objects for which the Eddington limit is exceeded for a typical stellar mass black hole of ∼ 10 M glyph[circledot] (although the mass of the compact object may vary). ULXs were initially interpreted as strong candidates for IMBHs, largely from the simple argument that inferred isotropic luminosities ∼ > 10 40 erg s -1 imply accretors above ∼ 100 M glyph[circledot] (e.g., Colbert & Mushotzky 1999). By definition, ULXs are non-nuclear, to rule out quiescent or low/hard emission from central supermassive black holes. \nEarly surveys for ULXs were primarily statistical, since contamination from background AGN can dominate candidate ULX samples (Zolotukhin et al. 2016). ULXs are overabundant in star-forming galaxies and indeed are primarily found near regions of recent star formation themselves (Swartz et al. 2009), consistent with ULXs being associated with young massive stars rather than old globular clusters or Population III remnants. ULXs are also much more common among metal-poor stellar populations than those of solar metallicity (per unit of star formation; Prestwich et al. 2013). \nX-ray spectra of ULXs show substantial variety, but overall are not congruent with the states observed for Galactic stellar-mass black holes scaled to the IMBH mass regime. Instead, some ULXs show distinct spectral states that can be physically interpreted as 'ultraluminous' states consistent with super-Eddington accretion onto stellar-mass black holes (see discussion in Kaaret et al. 2017). \nAnother breakthrough in the interpretation of ULXs was the observation of X-ray pulsations in some ULXs (Bachetti et al. 2014, Furst et al. 2016, Israel et al. 2017), proving that the accretors in these systems are neutron stars, likely with highly anisotropic accretion due to strong magnetic fields. Some ULXs mooted as IMBHs (e.g., M51 X-7; Earnshaw et al. 2016) have been proven to be neutron star ULXs (Rodr'ıguez Castillo et al. 2019). \nPutting this together, there is strong evidence that most ULXs are not IMBHs. There are a few important exceptions that we discuss below. \n5.1.1. HLX-1 and Friends. Even though most typical ULXs (with L X ∼ 10 39 -10 40 erg s -1 ) are unlikely to be IMBHs, some rare sources have been observed with L X ∼ > 10 41 erg s -1 \nthat cannot be easily explained even within a super-Eddington paradigm for neutron stars or stellar-mass black holes. The clearest example is HLX-1 (a so-called 'hyperluminous' X-ray source), located in the disk galaxy ESO 243-49 ( D ∼ 95 Mpc), at a projected distance ∼ 4 kpc from the galaxy's center (Farrell et al. 2009). \nThe isotropic-equivalent X-ray luminosity of HLX-1 ranges from L X ∼ 10 40 -10 42 erg s -1 . The association of HLX-1 with its host, and hence its extreme luminosity, have been confirmed via optical spectroscopy (Wiersema et al. 2010). Unlike typical ULXs, the source shows spectral behavior more similar to standard accretion disks than super-Eddington accretion, including typical high-thermal and low-hard states and state transitions (e.g., Servillat et al. 2011). \nModeling of the optical and X-ray emission through the state changes are consistent with a black hole mass of a few × 10 4 M glyph[circledot] , with a fair degree of uncertainty (Davis et al. 2011, Godet et al. 2012, Straub et al. 2014). Radio emission associated with the state changes gives similar mass estimates, in the range M BH ∼ 10 4 -10 5 M glyph[circledot] (Webb et al. 2012). The fundamental plane of X-ray/radio emission allows a mass as high as ∼ 3 × 10 6 M glyph[circledot] (Cseh et al. 2015) and hence is of limited utility. \nAn intriguing puzzle in HLX-1 is the nature of the X-ray luminosity variations and state changes. Since X-ray monitoring of the source began in 2008, the state changes appeared nearly periodic at intervals of ∼ 1 yr, leading to the idea that they were associated with the orbital period of a tidally captured companion star on an eccentric orbit (Lasota et al. 2011). Challenging this simple scenario, only a few years later the state change interval began elongating unpredictably, perhaps consistent with the unstable, tidally affected orbit of a compact donor such as a white dwarf (Godet et al. 2014). Other models, in which the luminosity variations do not directly reflect the orbital period of a donor star, have also been proposed (e.g., Soria et al. 2017). Future state changes, or a lack thereof, can provide additional constraints on these models. \nObservations of the surrounding stellar population give additional insight into HLX-1. While the modeling of the data is not conclusive due to optical emission associated with HLX-1 itself, the photometry is most consistent with a relatively massive (few ∼ × 10 6 M glyph[circledot] ) star cluster dominated by intermediate to old stellar population (Soria et al. 2017). No compelling evidence for a recent major merger has been observed around the galaxy (Webb et al. 2017), but nevertheless a self-consistent scenario is that HLX-1 represents a central massive black hole on the low-mass end of the mass distribution of nucleated ∼ 10 9 -10 10 M glyph[star] galaxies whose parent galaxy was accreted and tidally stripped, leaving only the bare nuclear star cluster (e.g., Mapelli et al. 2013). Star formation, perhaps due to the merger, could have increased the capture rate for a central black hole to acquire a companion star on which it is currently feeding. In this scenario HLX-1 does not represent a unique formation channel for IMBHs, but it is one of the best candidates we have for a (previously central) ∼ 10 4 M glyph[circledot] black hole. HLX-1 and similar systems could help inform our understanding of low-mass central black holes in galaxies. \nBesides HLX-1, no other very luminous ULXs are as convincing as IMBH candidates. Pasham et al. (2014) argue that M82 X-1 (which can reach L X ∼ 10 41 erg s -1 ) contains a ∼ 430 ± 100 M glyph[circledot] IMBH via an extrapolation of a stellar-mass black hole scaling relation for X-ray quasi-periodic oscillations. Brightman et al. (2016) model X-ray data of the source over a wide range of energies and prefer a model of super-Eddington accretion onto a stellarmass black hole, although depending on the spin, the black hole mass might be as high as ∼ 100 M glyph[circledot] . The interpretation of these data are not settled, and may require future X-ray \nmissions; confusion with the bright ULX pulsar M82 X-2 is a challenge for low-resolution observations. M82 is a starbursting dwarf only ∼ 3 . 5 Mpc distant, and a confirmation of a luminous IMBH at such a distance would suggest a high space density of IMBHs. \nAnother source of recent interest is an off-nuclear ULX in the Seyfert galaxy NGC 5252 at ∼ 100 Mpc (Kim et al. 2015). This ULX has an associated optical and radio source that has been studied at high resolution with the HST and very long baseline interferometry. Overall, these data are consistent with a massive black hole in the ∼ 10 5 -10 6 M glyph[circledot] regime whose stars have been stripped, though a somewhat lower mass in the IMBH range cannot be excluded (Mezcua et al. 2018b). \nAn even less settled case is that of NGC 2276-3c, an L X = 2 × 10 40 erg s -1 ULX in a face-on starbursting spiral at ∼ 33 Mpc. The source was associated with a large (100s of pc) radio nebula in Very Large Array (VLA) radio imaging (Mezcua et al. 2013). Follow-up quasi-simultaneous X-ray and very long baseline radio data were used to find unresolved hard X-ray emission and detect a compact pc-scale radio jet at modest significance (Mezcua et al. 2015). A compact, steady jet is consistent with an AGN in the low/hard state, and the fundamental plane was then used to estimate a mass of ∼ 5 × 10 4 M glyph[circledot] (all such mass estimates have uncertainties of at least 1 dex). However, Yang et al. (2017) independently reduced the radio data and did not confirm the high-resolution radio detection. Furthermore, there is no claimed optical counterpart to the ULX, which might be expected in a scenario in which the IMBH is associated with a star cluster or stripped nucleus. The large number of ULXs found in NGC 2276 (Wolter et al. 2015) suggests that NGC 2276-3c is truly associated with the galaxy. Nonetheless, given the galaxy's high star formation rate, the lack of an optical counterpart, and uncertain compact jet detection, it seems more likely that NGC 2276-3c is a super-Eddington stellar-mass black hole or a neutron star ULX than an IMBH. In this scenario the extended radio emission might well be associated with NGC 2276-3c, as radio nebulae due to ULXs are not uncommon (see the discussion in Urquhart et al. 2018).", '5.2. Fundamental Plane Searches in Globular Clusters': 'Accretion constraints on the presence of IMBHs in globular clusters rest on the assumption that, in a manner analogous to that of low-luminosity AGN, such IMBHs will accrete a fraction of the gas within their sphere of influence, resulting in detectable radio or X-ray emission from the accretion flow or jet (Maccarone 2004). To convert a measurement or upper limit in radio or X-ray luminosity to a corresponding IMBH mass or limit requires assumptions about the accretion rate and radiative efficiency of the accretion process. \nIn a globular cluster, the winds of evolved stars represent a source of low-velocity gas that is continually replenished: no long-term accumulation of gas is needed to produce an observable level of accretion. Assuming that the gas is ionized, the expected electron density is n e ∼ 0 . 05 -0 . 5 cm -3 (Pfahl & Rappaport 2001). The predicted level of ionized gas was first observed in the core of the globular cluster 47 Tuc by Freire et al. (2001), using the radial distribution of the dispersion measure of millisecond pulsars. This single observation has had little follow-up in subsequent years, since the method requires (a) a large population of millisecond pulsars, and (b) a low foreground of ionized material. This combination is, at present, still only satisfied for 47 Tuc. Nonetheless, Abbate et al. (2018) revisit the measurement for this well-studied cluster with updated pulsar parameters, finding an even larger gas density than in the original paper ( n e = 0 . 23 ± 0 . 05 cm -3 , compared to ∼ 0 . 07 cm -3 ). This measurement is consistent with theory and supports the basic underpinning \nof the accretion-based mass constraints: globular clusters should have some amount of gas that is available to be accreted by an IMBH. Other routes to fueling at higher rates are also available, such as winds or tidally stripped material from binary companion stars acquired dynamically (MacLeod et al. 2016b). \nSince the X-ray luminosity ( L X ) is observed to be a non-negligible fraction of the bolometric luminosity for known low-luminosity AGN, it is the most directly accessible tracer of the accretion rate. Given the expected gas density, IMBHs with 10 3 -10 4 M glyph[circledot] accreting at the Bondi rate with high ( glyph[epsilon1] ∼ 0 . 05-0.1) radiative efficiency would be X-ray sources with L X ∼ 10 34 -10 36 erg s -1 at the centers of globular clusters. X-ray sources are observed at this L X , but are identified as individual X-ray binaries rather than IMBHs. Hence it is clear that if IMBHs are present in Galactic globular clusters, they accrete at levels below that of the Bondi rate or with lower radiative efficiency. \nMaccarone (2004) first pointed out that, based on the fundamental plane, radio continuum observations put the most stringent accretion constraints on IMBHs accreting at low radiative efficiency, at least for typical observational sensitivities of radio and X-ray data. Assuming radiatively inefficient accretion and the sub-Bondi accretion rates observed for nearby low-luminosity AGN (e.g., Pellegrini 2005), many papers have used the fundamental plane and radio upper limits to set stringent constraints on the presence of accreting IMBHs in globular clusters (Maccarone et al. 2005, Maccarone & Servillat 2008, Cseh et al. 2010, Lu & Kong 2011, Strader et al. 2012). \nThese observational efforts culminated in Tremou et al. (2018), which uses a similar formalism but a much larger sample of deep radio continuum imaging of 50 Galactic globular clusters from the Jansky VLA and the Australia Telescope Compact Array. No emission consistent with an IMBH was observed for any cluster or for a cluster stack, including clusters with dynamical IMBH claims, such as ω Cen, M54, and NGC 6388 (see above). For the clusters with more sensitive VLA data, using the assumed formalism, the median 3 σ upper limit to an IMBH is ∼ < 1100 M glyph[circledot] , corresponding to very low accretion rate upper limits of ∼ < few × 10 -11 M glyph[circledot] yr -1 . \nThere are plausible mechanisms that could temporarily remove all gas from the immediate vicinity of the IMBH, rendering it invisible in radio continuum emission. Hence the lack of radio emission alone does not definitively prove the absence of an IMBH in a particular cluster. On the other hand, scatter in the accretion rate and efficiency is expected in both directions, so if IMBHs were common, at least some obvious sources would be observed. They are not. \nIt is reasonable to use the full sample of objects with radio constraints to set limits on the occupation fraction. The Tremou et al. (2018) work is sensitive to ∼ > 1000 M glyph[circledot] IMBHs and searches ∼ > 10 5 M glyph[circledot] globular clusters. Assuming the conservative accretion luminosities outlined by Tremou et al. (2018), their non-detections set a 3 σ upper limit of 10 -15% on the fraction of such massive globular clusters that could host ∼ 1000 M glyph[circledot] black holes just based on Poisson statistics. The modest size of the Galactic globular cluster system overall makes it challenging to substantially improve this limit on the occupation fraction with future data, except to push to fainter emission levels and hence lower (or more conservative) mass limits. Deep X-ray observations can also provide complementary constraints if the observing times are sufficiently long (Grindlay et al. 2001, Haggard et al. 2013). \nPerhaps the best way to improve the limit, especially if IMBHs closer to 10 4 M glyph[circledot] are being considered, is with observations of extragalactic globular clusters in nearby galaxies with next-generation radio continuum telescopes: the large number of clusters (hundreds to \nthousands per galaxy) should allow the detection of the high accretion tail of the distribution even if the occupation fraction is modest (Wrobel et al. 2016, 2019).', '6. IMBH Searches with Transient Phenomena': 'Thus far we have focused on search techniques specialized for either galaxy nuclei that can extend downward to the upper-end of the IMBH regime or stellar-cluster focused work aiming to identify sources in the 10 3 M glyph[circledot] realm. However, tidal disruptions and gravitational waves have the potential to work across these boundaries, and potentially find wandering black holes in lower-mass stellar systems, should they exist.', '6.1. Tidal Disruption Events': "Tidal disruption events (TDEs) are the electromagntic signature that may result if a star passes within its tidal radius of a black hole (e.g., Rees 1988). TDEs are powered by accretion onto massive black holes, but their rates and environments will provide an independent probe of the space density of IMBHs, since the rates depend on stellar (rather than gas) dynamics. In principle it is possible to derive M BH from modeling of the TDE light curve itself (Lodato et al. 2009, Guillochon & Ramirez-Ruiz 2013, Mockler et al. 2019) as the emission from these events depends on both the mass and radius of the star and the mass of the black hole (e.g., Law-Smith et al. 2017). The TDE literature merits its own review; we focus here only on aspects that may bear directly on IMBH demographics. \nWevers et al. (2017) present black hole masses for a sample of optically selected TDEs using stellar velocity dispersion measurements. We also note an X-ray-detected transient that is a likely TDE with an M BH -σ ∗ -based mass estimate of 1 . 3 -6 . 3 × 10 5 M glyph[circledot] (Maksym et al. 2013). If the M BH -σ ∗ relation can be extrapolated below σ ∗ ∼ 100 km s -1 (we provide direct support for this in § 8) then it is very likely some < 10 6 M glyph[circledot] black holes have already produced detectable TDEs. \nvan Velzen (2018) goes a step further and claims that a constant black hole occupation fraction from stellar masses M ∗ ≈ 5 × 10 9 M glyph[circledot] to 3 × 10 10 M glyph[circledot] is required to reproduce the observed TDE rate as a function of mass. This result highlights the promise of TDE observations to independently constrain the occupation fraction, and is consistent with our inferences from dynamics ( § 3) and X-ray observations of local galaxies ( § 4) above. That said, to fully utilize the promise of TDEs, some key assumptions need to be explicitly checked. We do not know how TDE rates in low-mass galaxies may depend on star formation rate or galaxy structure, although we know that TDE rates in more massive galaxies are a function of galaxy properties (e.g., Arcavi et al. 2014). Another open question is whether the TDE emission properties depend systematically on M BH in a way that biases the mass distributions of samples as a function of their selection (e.g., Strubbe & Quataert 2009). Finally, for white dwarf disruptions, there is an interesting literature comparing the emission signatures to those of Type Ia supernovae (Rosswog et al. 2009, MacLeod et al. 2016a, Anninos et al. 2018). These events are particularly exciting since they could be accompanied by the gravitational wave signature of an extreme mass-ratio inspiral ( § 6). \nTDE rates are so uncertain that in theory, all AGN activity in low-mass galaxies may \nbe powered by TDEs, and most black hole mass density at low-mass may be built up through tidal capture and TDEs (Milosavljevi'c et al. 2006, MacLeod et al. 2016a, Stone et al. 2017, Zubovas 2019). Zubovas in particular claims that the AGN fraction matches TDE expectations at low mass by assuming (a) rates from Stone & Metzger (2016) and (b) unity occupation fraction. It should be possible to use multi-epoch observations of active nuclei in low-mass galaxies to test whether they fade, as would be expected for TDEs, but at present our time baselines of ∼ 10 yr are too short to be discriminating (e.g., Baldassare et al. 2016, 2018). On the positive side, Jonker et al. (2019) present some hopeful evidence for persistent (5-10 year) X-ray emission from optically identified TDEs, suggesting that eROSITA may find as many as 1000s of TDEs powered by low-mass black holes. \n6.1.1. Peculiar explosions and possible off-nuclear TDEs. In theory there may be numerous off-nuclear IMBHs living in globular clusters, or wandering with only a small number of tightly bound stars (O'Leary & Loeb 2012, Fragione et al. 2018), which may give rise to off-nuclear TDEs observed as peculiar transients. For instance, there is a class of rapid blue transients (e.g., Drout et al. 2014, Vink'o et al. 2015, Tanaka et al. 2016, Pursiainen et al. 2018, Perley et al. 2019) that are not easily explained as standard classes of supernovae. These do not occur in galaxy centers, nor do they quite look like TDEs. Various groups have suggested that the object presented by Vink'o et al. (2015) may be powered by the tidal disruption of a white dwarf (e.g., Law-Smith et al. 2017). By contrast, Margutti et al. (2019) argue that AT2018cow (Perley et al. 2019) has too much circumstellar material to be explained by an IMBH TDE (although see also Kuin et al. 2019). \nAnother class of TDE candidate is associated with off-nuclear sources that show transient X-ray emission, often ascribed to a large temporary increase in the accretion rate of the putative IMBH associated with a TDE. Perhaps the most compelling of these is 3XMM J215022.4-055108, which is an X-ray source associated with a z ∼ 0 . 06 lenticular galaxy (Lin et al. 2018). This source shows a TDE-like X-ray light curve measured over 10 years, with a luminosity of at least 10 43 erg s -1 . The X-ray source matches the position of a compact star cluster (possibly a stripped nucleus) with an old stellar population and a mass of ∼ 10 7 M glyph[circledot] . Fitting the X-ray spectra suggests a black hole mass of 5 × 10 4 -10 5 M glyph[circledot] , depending on the unknown spin of the black hole. The position of the source on a diagram of luminosity and temperature for a standard thin accretion disk is very similar to that of HLX-1, suggesting a similar mass. \nOther similar sources have been discovered (e.g., Lin et al. 2016, 2017), although the evidence for a TDE-like decay (rather than typical AGN activity) is less compelling due to a smaller number of deep observations over time, and generally the masses of the associated black holes are more in the regime of 'normal' low-mass central black holes (masses 10 5 -10 6 M glyph[circledot] ) rather than in the IMBH regime. If TDEs associated with low-mass central black holes in stripped nuclei are as common as suggested by these recent works, then future sensitive X-ray missions such as Athena should confirm such sources in substantial quantities, with rates up to ∼ 100 yr -1 possible (Lin et al. 2018, Cassano et al. 2018). \nWhile these TDE-like sources decay over timescales of years, another type of source is worth a quick mention: Irwin et al. (2016) found extremely brief (100s of seconds), luminous ( L X ∼ 10 40 -10 41 erg s -1 ) X-ray flares from star clusters associated with two nearby earlytype galaxies. One of the clusters appears to be a typical massive globular cluster, while the other has properties consistent with being a stripped nucleus. These X-ray sources show properties most similar to flares from young pulsars with strong magnetic fields-sources not \nexpected to exist in old stellar populations. If the flares are Eddington-limited, then they could represent accretion onto IMBHs with masses of ∼ 100 -1000 M glyph[circledot] . So far, these results are only suggestive, and other evidence (such as radio emission from the sources in their lower state) would be necessary to provide stronger evidence for an IMBH classification. \nVery likely, these peculiar objects have diverse origins, and only with next-generation observations, along with detailed modeling, will we be in a position to find the most likely TDE candidates among all the rich stellar death phenomenology.", '6.2. Gravitational Waves': 'Given the range of mass and mass ratio that we can hope to detect in the upcoming decades, we summarize some of the expected discoveries related to IMBHs from LIGO and LISA . \n6.2.1. IMBH-IMBH Mergers at High Redshift. LISA will be sensitive to black holes mergers with mass ratios q ∼ 0 . 1 -1 for M BH ∼ 10 4 -10 5 M glyph[circledot] out to very high redshift ( z ≈ 20). The black hole masses and redshifts can be measured from high S/N gravitational waveforms (Amaro-Seoane et al. 2017). These measurements will provide insight into the seeding mechanisms and fueling rates (e.g., Sesana et al. 2011). However, in detail the rates will depend not only on the seeding mechanism(s) at play, but also on the accretion history and the dynamics driving the mergers of the black holes (e.g., Sesana et al. 2007, Klein et al. 2016). In practice it will not be trivial to disentangle these effects. Observations in the X-ray and optical/IR with missions like JWST and Lynx that should detect the actively accreting 10 5 -10 6 M glyph[circledot] black holes at similar epochs would be very complementary (e.g., Haiman et al. 2019, § 13).', '6.2.2. Low Redshift Constraints from Intermediate and Extreme Mass-ratio Inspirals. If': 'there are IMBHs floating around in more massive halos, then occasionally they should merge with the primary supermassive black hole in an intermediate-mass ratio inspiral event (e.g., Holley-Bockelmann et al. 2010). In addition, an IMBH embedded in a star cluster (nuclear or otherwise) could merge with a stellar-mass black hole in an extreme mass-ratio inspiral with even lower mass ratios (e.g., Gair et al. 2010). Again, the rates of both events are heavily dependent on a number of unknown factors, including the spin of the black hole, the dynamics of the surrounding stellar cluster, and the number density of IMBHs (e.g., Amaro-Seoane et al. 2015, Fragione et al. 2018, Berry et al. 2019), which hopefully can be partially determined through complementary electromagnetic observations. \n6.2.3. Gravitational Runaway Constraints at Low Redshift. As alluded to in § 2 above, if the gravitational runaway channel operates to make seeds, then it should be operating at present in young and forming star clusters. It should be possible to catch the early stages of this process with LIGO detections (Kovetz et al. 2018, Antonini et al. 2019). Current limits on the merger of ∼ 100 M glyph[circledot] black holes are not yet constraining (Abbate et al. 2018), but should become so as LIGO progresses. The one caveat relates to the possible subset of runaway processes involving stars (as opposed to black holes) that may require low metallicity. If so, this channel also may operate only at high redshift. We have argued, based on the similarity of observed properties of globular clusters as a function of metallicity, that metallicity dependence is in some tension with observations.', '7. IMBH Candidate Tables': "We summarize the prior three sections in three tables containing all of the credible IMBH candidates in the literature to date. For clarity, we separate measurements in nuclei (Table 2) from off-nuclear candidates (Table 3) and constraining limits (Table 4). To be included in the table, an object must have a published black hole mass or limit that falls below 10 6 M glyph[circledot] . Furthermore, the mass must be based on a primary or secondary black hole mass determination method. We include M BH estimates based on stellar and gas dynamics, reverberation mapping, and scaling from the M BH -σ ∗ relation in the specific case of TDE events that seem to have reliable determinations of the host galaxies. We also include a small number of measurements based on modeling of the X-ray emission or on the radio/X-ray fundamental plane. We do not include single-epoch virial black hole mass estimates for AGN (but see references in § 4). For the credible candidates, we also include other representative published measurements or limits, which in many cases are contradictory. However, we do not include off-nuclear sources that have only limits with no claimed detections. \nThe tables neatly summarize the state of the field. We have high confidence that nuclear black holes extend downward to ∼ 10 5 M glyph[circledot] . There is tantalizing evidence for objects below this limit in galaxy nuclei (NGC 205 and upper limits), but not definitive evidence to date. HLX-1 and objects like it provide additional circumstantial evidence for black holes with masses in the 10 4 -10 5 M glyph[circledot] range. There is no compelling evidence yet for any object with M BH ∼ 10 3 M glyph[circledot] , and it is likely that many of the candidates listed in the table will not be confirmed as IMBHs. \nWhile all the search techniques described above have their own challenges, there are a few themes worth drawing out that are special roadblocks when searching for IMBHs. Angular resolution is currently the limiting factor for dynamical methods, and enhances accretion searches by eliminating more contamination from stellar sources and enables one to better tease out the signal from low-level accretion (e.g., Dickey et al. 2019). \nHowever, even the order-of-magnitude improvements in angular resolution coming soon will not completely eliminate this confusion. Indeed, nearly all techniques suffer contamination from stars. Even at high Eddington ratio, IMBHs cannot be uniquely distinguished on the basis of their luminosity, and for a short period of time luminous supernovae can masquerade as accreting black holes (Filippenko 1989, Baldassare et al. 2015). We have already hit a confusion floor with X-ray observations since going deeper than ∼ 10 38 erg s -1 simply yields large samples of low-mass X-ray binaries. In principle incorporating radio observations could help. However, in nearly all accreting objects the X-ray and radio emission are linked in some manner, leading the fundamental plane to 'return' a seemingly reasonable mass even in cases where its use turns out later to have been not justified. For example, early discussions of ULXs argued that the fundamental plane might support their identification as IMBHs (e.g., Miller 2005, Kaaret et al. 2009), while at least some of these sources turned out to be pulsars. Even dynamical masses in nuclei for sources with M BH < 10 4 M glyph[circledot] will suffer from confusion with clusters of compact objects, and measurements in globular clusters are very challenging due to this confusion.", '8. Scaling Relations': 'Scaling relations between M BH and macroscopic galaxy properties are useful as a tool to estimate M BH for exciting objects (e.g., Wevers et al. 2017), or to calculate the black hole mass density in the universe (e.g., Marconi et al. 2004). These relations may also encode \n1 \nLeft : Relationship between M BH and M ∗ for dynamical early-type (red open circles), late-type (blue open squares), and dynamical upper limits (blue triangles). We show fits to the early and late-type galaxies (red and blue shaded regions) and the full sample (grey). It is clear that the late-type galaxies have a comparable slope, but lower normalization, compared to the early-type galaxies. We also see hints for increased scatter at low mass, in part perhaps due to non-unity occupation fraction at low mass. Right : Same as left , but for M BH versus σ ∗ . Here we also include the sample of active galactic nuclei from Xiao et al. (2011, grey dots). By construction through f vir , the AGN points are consistent with the late-type galaxies. \n<!-- image --> \nthe evolutionary history of black holes. In the case of IMBHs, scaling relations at low mass might differ based on seeding mechanisms (e.g., Volonteri et al. 2008, van Wassenhove et al. 2010), although accretion processes at early times may also wash out these signatures (e.g., Volonteri & Gnedin 2009).', '8.1. M BH -σ ∗': "We consider the low-mass end of the M BH -σ ∗ relation, based on dynamical masses from Kormendy & Ho (2013), supplemented with more recent work (Greene et al. 2016, Saglia et al. 2016, Krajnovi'c et al. 2018, Thater et al. 2019), particularly at low masses (den Brok et al. 2015, Nguyen et al. 2018, 2019b), although our results do not change if we focus exclusively on Kormendy & Ho augmented by low-mass objects. Crucially, we explore the importance of including upper limits (Boker et al. 1999, Barth et al. 2009, Neumayer & Walcher 2012, Nguyen et al. 2017). There has been considerable literature on the scatter in scaling relations as a function of galaxy properties (Hopkins et al. 2007, Kormendy & Ho 2013, Saglia et al. 2016). We do not attempt to address these issues, since the measurement uncertainties are still very large in the low-mass systems that concern us here. Finally, since we need it for our black hole mass function determinations below, we also split the sample into early (elliptical and S0) and late-type (spiral) galaxies, using the Hubble Types from Saglia et al. (2016) where available and the individual papers for all other galaxies (Supplemental Tables 6, 7, and 8). \nAs we will show, including the constraining upper limits on low-mass galaxies (Barth et al. 2009, Neumayer & Walcher 2012, De Lorenzi et al. 2013) will make a real difference in the fitted slopes of the relations. We note that Neumayer and Walcher provide both a \n'best' and 'maximum' allowed black hole mass, using the best and minimum M/L from population synthesis models. We use their maximum value as the upper limits in the fits. \nAssuming that log( M BH /M glyph[circledot] ) = α + β log( σ ∗ / 160 kms -1 ) + glyph[epsilon1] , where glyph[epsilon1] is the intrinsic scatter, we present our fits in Supplemental Table 9 and Figure 3. Our results are broadly consistent with prior work. However, we are able to explore some additional issues due to the inclusion of low-mass M BH measurements and limits. First of all, when only detections are included in the fitting, the slope we fit to the late-type galaxies alone is very shallow, likely due to the bias in M BH measurements towards the most massive black holes at a given galaxy property (see also Batcheldor 2010, Pacucci et al. 2018). Similar flattening or breaks have been reported based on AGN-based M BH values (e.g., Greene & Ho 2006, Mart'ın-Navarro & Mezcua 2018), perhaps suffering from a similar bias. When limits are included, the fit to late and early-type galaxies become much more similar. In either case, the limits mitigate the bias seen in the detections, and we see no evidence for a change in M BH -σ ∗ relations at low σ ∗ (as also concluded by Barth et al. 2005, Neumayer & Walcher 2012). \nEarly work raised the exciting prospect that the shape and scatter in the M BH -σ ∗ relation for low M BH objects might depend on the seeding mechanism (Volonteri et al. 2008, Volonteri & Natarajan 2009). Certainly, if all seeds were made at or above 10 5 M glyph[circledot] , then the scaling relations would flatten at low mass. As of now, we do not see evidence for flattening at 10 5 M glyph[circledot] , but rather NGC 205 and the published upper limits argue for a broad distribution of M BH at the low-mass end, particularly given the additional evidence for a high occupation fraction in this stellar mass range. Nominally, the observed M BH -σ ∗ relation disfavors heavy seed models that make exclusively M BH ∼ > 10 5 M glyph[circledot] .", '8.2. M BH -M ∗': "Unfortunately, there are not σ ∗ functions measured for low-mass galaxies, and bulge fractions are no longer meaningful in these galaxies either (e.g., MacArthur et al. 2003), although see Schutte et al. (2019). For these reasons, we revisit the correlation between M BH and M ∗ from dynamical studies including black holes with M BH < 10 6 M glyph[circledot] . We will then use this relation to estimate the black hole mass function in that same mass range. \nRecently, Reines & Volonteri (2015) presented a fit to the M BH -M ∗ relation using the dynamical sample of Kormendy & Ho (2013). We add additional galaxies that have been published since then as above (Greene et al. 2016, Saglia et al. 2016, Krajnovi'c et al. 2018, Thater et al. 2019), including low-mass black holes (den Brok et al. 2015, Nguyen et al. 2018, 2019b). \nAs for Kormendy & Ho (2013), we measure K -band magnitudes and B -V colors for the new galaxies. We use this single color and the fitting functions from Bell et al. (2003) to calculate M ∗ for all samples, so that all M ∗ estimates share a common IMF and stellar population assumptions. The one exception is the low-mass galaxies ( M ∗ < 10 10 M glyph[circledot] ), where we use the stellar masses from the dynamical papers. Nguyen et al. (2019b) gives a more detailed comparison between different colorM/L relations. All black hole mass measurements and stellar masses used in this fit are tabulated in Supplemental Tables (Tables 6, 7, & 8). \nIn Figure 3 (Supplemental Table 9), we present the updated M BH -M ∗ scaling relation. One thing that is immediately apparent is that, when plotted against M ∗ , the dearth of dynamical black hole mass measurements for M ∗ < 3 × 10 10 M glyph[circledot] is striking. A high priority \nfor understanding black hole demographics at low M ∗ is to study the supermassive black hole demographics in Milky-Way-mass galaxies (Krajnovi'c et al. 2018, Nguyen et al. 2019a). \nFocusing on the IMBH regime, again we find that without upper limits, the fit to latetype galaxies returns a very shallow relation, because the measured M BH values are biased high. When we include the limits, however, the slope of the relation becomes consistent between the red and blue populations. On the other hand, the overall normalization for the late-type galaxies is considerably lower than for the early-type galaxies (in agreement with Greene et al. 2010, Reines & Volonteri 2015, Greene et al. 2016, Lasker et al. 2016). This difference has been interpreted to mean that black hole mass does not correlate with disk mass (e.g., Kormendy & Richstone 1995), although the different trends have also been used to constrain the relationship between star formation and black hole growth (Caplar et al. 2015). We note that our fit to the full sample including limits has a lower normalization and slightly steeper slope than the Reines & Volonteri fit, while it is very similar to that presented recently by Gallo & Sesana (2019).", '8.3. Using AGN to Probe Scaling Relations': 'We overplot the sample of low-mass AGNs from Xiao et al. (2011) in the M BH -σ ∗ plane (Figure 3; excluding those with uncertain broad lines). We adjust the single-epoch virial masses to an f vir calibrated with late-type galaxies, as advocated by Ho & Kim (2014). By construction, the AGNs align with the late-type galaxy fit. We again see no evidence in the AGN sample for any flattening or offset at low mass, given the caveat that we do not know the absolute black hole masses of these objects. Given the large uncertainty in the single-epoch masses we cannot make any additional statements about scatter from this sample. \nMore generally, we would urge extreme caution in using AGN signatures to infer scaling relations, particularly in this low-mass regime. Even the nearby AGN NGC 4395 ( D ≈ 4 Mpc) has published reverberation mapping masses that differ by more than an order of magnitude (Peterson et al. 2005, Edri et al. 2012, Woo et al. 2019). There are legitimate reasons that we do not yet know to fully interpret the reverberation masses. One problem is the f vir uncertainty. Beyond that, in the case of NGC 4395, the disagreement in black hole mass is mostly attributable to different line widths between the UV resonance line CIV and the optical recombination line H β . Much more work is needed to effectively harness reverberation mapping for AGN with low-mass black holes.', '8.4. Scaling With Nuclear Star Clusters': "In addition to scaling relations between galaxy properties and black hole mass, we consider possible scaling relations between the central black hole and the surrounding nucear star cluster. We observe a high incidence of black holes in nuclear star clusters. As we will argue below ( § 9), a fraction > 50% of 10 9 -10 10 M glyph[circledot] galaxies harbor black holes, and nearly all such galaxies harbor nuclear star clusters. Nuclear star clusters do not seem to replace black holes as the 'central compact object' (Ferrarese et al. 2006). Rather the two appear to coexist often at low galaxy mass (Seth et al. 2008). In contrast, most highermass galaxies contain supermassive black holes (Gultekin et al. 2011) but show a very low incidence of nuclear star clusters (e.g., Graham & Spitler 2009). Likely the growing black hole contributes to the demise of the nuclear star cluster (Antonini et al. 2019). \nIt is not obvious what (if any) causal relationship exists between black holes and nuclear \nstar clusters. We have already discussed the possibility that black holes form via gravitational runaway processes in stellar clusters ( § 2), and the dearth of concrete observational evidence to date for black holes with M BH > 1000 M glyph[circledot] in globular clusters (Tremou et al. 2018, § 5). Nuclear star clusters are different from globular clusters in four key ways: they have higher stellar densities (and thus higher interaction rates), they have longer relaxation times, they have deeper potential wells (and hence higher escape velocities), and they have well-documented multi-age populations (e.g., Kacharov et al. 2018). While some theoretical work postulated a higher likelihood of gravitational runaway in clusters with the highest σ ∗ (Miller & Davies 2012), more recent work suggests that massive black holes likely can only form via rapid processes in nuclear star clusters (Breen & Heggie 2013, Stone et al. 2017). \nInstead, the high coincidence of black holes and nuclear star clusters may suggest that globular clusters with IMBHs are more successful at surviving and migrating to a galaxy center, where through continued gas accretion they can become a nuclear star cluster. Or, the black hole may form early via other means ( § 2) and then sit at the center of a cluster that grows through mergers and accretion to be a present-day nuclear star cluster. \nGoing beyond occupation fractions, the mass fraction of the cluster bound up in the black hole may constrain models. We extend the work of Seth et al. (2008), Georgiev et al. (2016), and Nguyen et al. (2018) based on the increased number of dynamical black hole masses available now in the literature (Figure 4 and Supplemental Table 11). Our full sample includes upper limits on M BH in globular clusters from Tremou et al. (2018), ultracompact dwarfs with dynamical black hole mass measurements or upper limits (Seth et al. 2014, Ahn et al. 2017, 2018, Afanasiev et al. 2018), and dynamical black holes in low-mass galaxies ( § 3), drawing nuclear star cluster masses from those papers. For the black hole masses and limits in Krajnovi'c et al. (2018) and Pechetti et al. (2017), we calculate the nuclear star cluster masses using color and luminosity information from Cˆot'e et al. (2006)", 'Feedback': "One plausible explanation of black hole-galaxy scaling relations more generally is that there is a feedback loop between black hole growth and star formation, such that black holes are able to remove or heat gas in galaxies when they reach a critical mass relative to the galaxy potential (e.g., Silk & Rees 1998). For low-mass galaxies in particular, AGN feedback has largely been ignored. However, recent theoretical interest and observational evidence have brought the possibility of AGN feedback in dwarf galaxies to the fore. \nSilk (2017) argues on theoretical grounds that AGN feedback in low-mass galaxies may be important to solve small-scale challenges for ΛCDM (see also Dashyan et al. 2018). Penny et al. (2018) present evidence for AGN activity in 10% of LMC-mass quenched galaxies. They interpret misaligned kinematics as signs of AGN-driven feedback. Dickey et al. (2019) examine 20 of the rare M ∗ ≈ 10 9 M glyph[circledot] galaxies that are both isolated and non-star-forming, and find evidence for AGN activity in 16 of them (see also Bradford et al. 2018). Dickey et al. conclude that AGN activity can 'self-quench' even relatively isolated low-mass galaxies that otherwise are blue (Geha et al. 2012). Nyland et al. (2017) also identify possible evidence for feedback in the jet in NGC 404. In short, there is now intriuging evidence that even dwarf galaxies with low-mass black holes may suffer the impacts of AGN feedback. We do not yet know whether these episodes are crucial in setting scaling relations in this regime. \nM NSC ( M /circledot ) \n<!-- image -->", 'Figure 4': "The ratio of black hole to cluster mass, including globular clusters from Tremou et al. (2018, red), dynamical black hole mass measurements with known nuclear star clusters from the literature (blue; building on Seth et al. 2008, Nguyen et al. 2018), limits. For the limits from Neumayer et al., we show their 'best' measurement as a symbol and the 'maximum' allowed black hole mass as the top of the arrow. We further include ultra-compact dwarfs with M BH masses or upper limits published (green). The median ratio of M BH to nuclear star cluster mass for objects with black holes detected is ∼ 25%, but with a factor of two scatter. The globular clusters are consistent with a 0 . 1% mass fraction for a small fraction of the Milky Way globular cluster system. \nand the relations of Bell et al. (2003). Finally, we include NGC 1023 and NGC 3384, two early-type galaxies that Lauer et al. (2005) identify as containing stellar nuclei, with the caveat that in some cases it can be challenging to determine whether these point sources are stellar or nonthermal in nature (e.g., Ravindranath et al. 2001). \nRoughly ∼ 0 . 1% mass fractions are predicted from gravitational runaway scenarios ( § 2). The observed mass fractions in nuclear star clusters far exceed these limits. We see from Figure 4 that in most cases M BH comprises a much higher fraction of the nuclear star cluster mass: the median value of M BH / M NSC for the objects with black hole detections is ∼ 0 . 25, with more than a factor of two scatter. There is even more scatter when the limits are considered. This scatter likely reflects both a lack of black holes in some clusters and intrinsic scatter in the growth histories of both constituents. Interestingly, the largest outliers with low ratios of black hole to nuclear star cluster mass are found in the low-mass \nand late-type spiral galaxies. \nGiven the wide range of M BH / M NSC that we observe, it seems possible that stochastic late-time fueling and/or merging plays a substantial role in the growth of each component (e.g., Naiman et al. 2015). Ultra-compact dwarfs provide an interesting testing ground for how much black hole growth occurs via later-time accretion versus initial formation. Since ultra-compact dwarfs were stripped, likely quite early in some cases (Pfeffer et al. 2014), we would expect less late-time growth, leaving their M BH / M NSC ratios closer to their value at formation, while there may be more scatter in the population due to different stripping times. So far, the ultra-compact dwarfs do not obviously segregate in this plane, but the constraint will be more interesting as their number with M BH constraints continues to grow.", '9.1. Existing limits on the occupation fraction': "After Gebhardt et al. (2001) published an upper limit of < 1500 M glyph[circledot] on any putative black hole in the nuclear star cluster in M33, the community assumed that massive black holes were rare in low-mass, late-type galaxies with little or no bulge component. In fact, black holes do not appear to be rare, at least in galaxies with M ∗ < 10 10 M glyph[circledot] . \nRecall that there are ten galaxies within 4 Mpc and 10 9 < M ∗ /M glyph[circledot] < 10 10 M glyph[circledot] with published dynamical masses or limits ( § 3). Of these, five are detections. From the dynamical measurements, we infer an occupation fraction > 50% (Figure 5, blue region). If the upper limit in M33 of M BH < 1500 M glyph[circledot] is taken at face value, it would suggest that the occupation fraction is not unity for M BH =10 3 -10 6 M glyph[circledot] black holes in galaxies with M ∗ ≈ 10 9 M glyph[circledot] . However, better statistics and more limits are needed. \nIn addition to dynamical constraints, X-ray surveys of local galaxies also allow us to make a concrete measurement of the occupation fraction. Just based on the number of detected X-ray sources in galaxies with M ∗ = 10 9 -10 10 M glyph[circledot] , Miller et al. (2015) and She et al. (2017a) both find a lower limit of 20% on the occupation fraction. Miller et al. argue for a likely occupation fraction of ∼ 70% based on jointly modeling the L X /M ∗ relation and the occupation fraction. Taking ∼ 20% as the lower limit, we find that sensitive Xray surveys of local galaxies already point towards relatively high occupation fractions, in agreement with the dynamics (Figure 5, yellow box). \nAs yet, we see no concrete evidence for differences in mass distribution between red and blue low-mass galaxies, but given the small numbers involved this is mostly just an assumption (which also appears to hold for the X-ray detection fractions described above § 4). Since blue galaxies are far more common than red at these masses, any difference could have significant ramifications for the black hole mass density. \nThe growing field of tidal disruptions will soon start to put competitive constraints on the occupation fraction if we can understand the rates ( § 6). van Velzen (2018) makes a first attempt. He finds that the observed luminosity function of tidal disruption events can only be explained if the occupation fraction is basically flat from LMC to Milky Way mass galaxies. While this result is hard to turn into an exact occupation fraction, it certainly argues that a high fraction of galaxies in this mass range host an IMBH. \nAs a point of comparison, we also plot the fraction of Virgo galaxies with M ∗ < 10 9 M glyph[circledot] containing nuclear star clusters (S'anchez-Janssen et al. 2019). It is interesting to see that if most nuclear star clusters host a central black hole, then the black hole occupation fraction would be within the limits derived by other methods (see also Seth et al. 2008, Foord \nM ∗ ( M /circledot ) \n<!-- image -->", 'Figure 5': "Multiple constraints are converging towards a relatively high value of occupation fraction > 50% of black holes with M BH ∼ > 10 5 M glyph[circledot] in galaxies with stellar masses 10 9 < M ∗ /M glyph[circledot] < 10 10 . Observational constraints from Miller et al. (2015, yellow box) are from X-ray observations of low-mass red galaxies. Dynamical constraints are summarized in Nguyen et al. (2019b, blue box) and suggest occupation fractions > 50%. Likewise, the TDE mass function suggests a high occupation fraction (van Velzen 2018) even below 10 10 M glyph[circledot] . Intriguingly, the nuclear star cluster occupation fraction is consistent with existing black hole limits as well, as exemplified by the green region derived from Virgo by S'anchez-Janssen et al. (2019). For reference, we include predictions from three models (red). We include Ricarte & Natarajan (2018, dotted) predictions for black holes more massive than 3 × 10 5 M glyph[circledot] . Their Population III models span the full range shown here, with the Light-Low case corresponding to their power-law accretion mode while the Light-High case is their main-sequence accretion mode. Their direct collapse seeds fall in between these two limiting cases. We also show the limits on occupation fraction from Bellovary et al. (2019, dashed), which is effectively a heavy seeding model. \net al. 2017). Below, when we calculate limits on the black hole mass function, we will use the measured nuclear star cluster fraction as one estimate of the black hole occupation fraction. We also show predictions from recent models for the occupation fraction (Figure 5). The Ricarte & Natarajan (2018) models with Population III seeds span nearly all possible occupation fractions, depending on their fueling model (red dotted lines in Figure 5), while the direct collapse models lie in between, consistent with Bellovary et al. (2019, \n1 \nFigure 6 \n<!-- image --> \n/circledot \n/circledot \n∗ \n/circledot \nLeft : Inferred black hole mass function based on the GAMA stellar mass function (Wright et al. 2017) and the M BH -M ∗ relation from § 8. We take two cartoon occupation fractions as illustrated on the right , with the optimistic 'nuclear cluster' model (blue) assuming that every nuclear star cluster houses an IMBH and the nucleation fraction coming from S'anchez-Janssen et al. (2019), while the more pessimistic line (red) is just meant to be consistent with existing observations. In the solid lines, we show the black hole mass density from nuclear sources alone, assuming a single power-law relation between M BH and M ∗ . We additionally add to the default power-law model a wandering black hole component with a number density tied to ultra compact dwarfs (dash-dot). \nred dashed line). \nIn summary, at least ∼ > 50% of galaxies with M ∗ ≈ 10 9 -10 10 M glyph[circledot] host a massive black hole with M BH ∼ 10 4 -10 6 M glyph[circledot] .", '9.2. Inferred Black Hole Mass Functions at Low Mass': "Predictions for the rates of events like tidal disruptions (e.g., Stone & Metzger 2016) or extreme mass-ratio inspirals (e.g., Gair et al. 2010) depend on the black hole mass function into the IMBH mass range. However, there are few observational constraints on the M BH mass function below ∼ 10 6 M glyph[circledot] (e.g., Marconi et al. 2004, Greene & Ho 2007a). We now have the ingredients needed to calculate a range of possible black hole mass functions down to ∼ 10 4 M glyph[circledot] , albeit with significant uncertainty. \nUsing the M BH -M ∗ relation derived in § 8 above, we convert the observed galaxy mass function into a black hole mass function (e.g., Marconi et al. 2004). The mass function of Wright et al. (2017) extends to M ∗ ≈ 10 6 M glyph[circledot] , and thus allows us to explore the ramifications of a non-zero occupation fraction to very low stellar mass. The galaxy stellar mass is converted into black hole mass density using the M BH -M ∗ relation that we fit in § 8, including intrinsic scatter. We consider the red and blue galaxies separately, since they have quite different scaling zeropoints (see also Shankar et al. 2016), and for this we take guidance from the red and blue galaxy luminosity functions presented by Blanton & Moustakas (2009). We thus input a galaxy mass function that is dominated by blue galaxies below, and red galaxies above, M ∗ = 10 10 M glyph[circledot] . \nBelow M ∗ ≈ 10 10 M glyph[circledot] , the occupation fraction may deviate from unity. We adopt two \ndifferent forms for the occupation fraction, both of which are consistent with the current data and models (Figure 6). As a pessimistic case, we assume that the occupation fraction drops linearly from unity at 10 10 to zero at 3 × 10 7 M glyph[circledot] . As an optimistic limit, we take the fraction of galaxies with nuclear star clusters and assume that every nuclear star cluster harbors a massive black hole (derived in Virgo by S'anchez-Janssen et al. 2019). In the default mass function, there are only nuclear black holes and a single power-law relation between stellar and black hole mass. The resulting range of possible mass functions are shown in Figure 6 (Supplemental Table 10). There is quite a range of possible M BH densities for M BH < 10 6 M glyph[circledot] . We note again that we have assumed comparable occupation fractions for red and blue galaxies, which is a systematic uncertainty that must be tested. \nWe also explore what a contribution from wandering black holes might look like ( § 2). These black holes may reveal themselves in our own galaxy through dynamical signatures, and they may be the sites of extreme mass-ratio inspirals or white dwarf TDEs as well. We tie the number of wandering black holes to the number of ultra-compact dwarfs, as described in § 2.1. In short, we take the number of ultra-compact dwarfs as a proxy for the number of disrupted satellites and then assume a 10-50% occupation fraction for them. We are not adopting a cosmologically motivated mass spectrum of satellites, nor do we have an empirical way to assign black hole masses to this population. The ultra-compact dwarfs with known black holes orbit more massive hosts and likely had more massive progenitors than those we consider here (Seth et al. 2014, Voggel et al. 2019). Thus, we simply take a typical black hole mass of 3 × 10 3 M glyph[circledot] , chosen to fall below current detection limits, and a log-normal width of 0.5 dex. These are arbitrary, but at least qualitatively demonstrate how much higher the black hole mass density might be for relatively conservative estimates on wandering populations. \nThere are a number of systematic uncertainties here, including the shape of the M BH -M ∗ relation and all the challenges inherent in calculating stellar masses (Conroy 2013). Gallo & Sesana (2019) nicely demonstrate the systematic effect of different scaling relation fits (see also Shankar et al. 2016). Furthermore, we do not know the functional form of the occupation fraction with stellar mass. Because the stellar mass function rises steeply, even a small occupation fraction in low-mass galaxies dramatically changes the black hole mass density at low mass. Nevertheless, the mass functions we present here may be used to bracket the expected event rates of TDEs and extreme mass-ratio inspirals.", '10. The Future': "In the coming two decades, we hope to break into the elusive ∼ 100 -10 4 M glyph[circledot] regime with robust dynamical constraints from stars, gas, and gravitational wave detections. We provide a short synopsis of the most promising ongoing and upcoming experiments, along with the distance distributions and available number of objects that they can each explore (Figure 7). We provide a full explanation for the numbers used in the figure in Supplementary Materials (I). If we are to take full advantage of these next-generation opportunities, we must begin to lay the groundwork now to be maximally ready to exploit upcoming surveys. We present here a list of our high priority items. \nNext-generation extremely large telescopes and ALMA at full capacity will have stunning spatial resolution, but we will only be able to find putatitive < 10 5 M glyph[circledot] black holes in < 10 9 M glyph[circledot] galaxies if we know where to look. We must use all the tools at our disposal, from astrometry with Gaia to gas kinematics to determine centers for Local Group dwarfs \n1 \n/circledot \nFigure 7 \n<!-- image --> \nLeft : Volume and number of targets to be reached by next-generation facilities. The color-bar reflects the black hole mass range (from 100-10 5 M glyph[circledot] , with darker color for more massive black holes, see scale bar). The red outline signals that we tabulate a rate per year rather than a number. Right : Same as left, but for more distant black holes. \n(van der Marel & Kallivayalil 2014). Moving out to larger radius, a complete sample of nuclear star clusters within 5 Mpc would be immensely useful to plan for next-generation dynamical searches. Likewise a survey of the molecular gas content of < 10 9 M glyph[circledot] galaxies in the South to identify good ALMA targets is needed. Finally, it is a high priority to start mining Gaia for potential wandering black hole candidates. \nWe have assumed throughout this article that red and blue low-mass galaxies have similar occupation fractions, which is consistent with the current X-ray and dynamical results. Even a larger sample of constraining limits for more nuclear star clusters (Neumayer & Walcher 2012), particularly including all the available red galaxies within 10 Mpc, could potentially shed light on this important issue before many more detections are in reach. Because red and blue galaxies have very different number density at low mass, it is important to know if there are differences in their occupation fractions. \nWe will not lose our reliance on accretion signatures, even as dynamical capabilities grow. We urgently need theoretical predictions for the spectral energy distributions and presence or absence of broad lines in black holes of 10 3 -10 5 M glyph[circledot] black holes (e.g., Cann et al. 2019) to search effectively for them with upcoming sensitive instruments like JWST and/or Lynx . 'Standard' AGN search techniques have identified targets with inferred masses pushing towards 10 4 M glyph[circledot] (e.g., Baldassare et al. 2015), but we may be unable to identify the majority of sources at this black hole mass because their accretion signatures do not match our expectations (Cann et al. 2019). This question is important not only for selection reasons, but also because the ionizing spectrum of low-mass black holes is needed to think about the impact of IMBHs on reionization (e.g., Madau & Haardt 2015). \nThere are some hints that the spectral energy distributions of accreting black holes may be changing in interesting ways as we push to low mass. Most concretely, AGN with inferred low black hole mass and high Eddington ratio show a very pronounced soft X-ray excess, that may even be the tail of emission from the accretion disk itself (e.g., Done et al. 2012, Jin et al. 2012, Yuan et al. 2014), as would be expected in the low-mass regime. Such \nobjects are also highly variable (Kamizasa et al. 2012). Along with this, the ratio of optical to X-ray emission also shows interesting behavior among the low-mass black holes (e.g., Dong et al. 2012a, Plotkin et al. 2016), but these changes may be tied more closely to the high Eddington ratios rather than low masses of this sample. On the flip side, Ludwig et al. (2012) did not find the ratio of He II/H β to depend on black hole mass, as one would expect if the big blue bump moves to higher temperature at lower masses. At the low accretion rate end, there is also need for theoretical guidance in terms of the expected radio and X-ray emission. We would like to see calculations like those of Ressler et al. (2019), built for the Galactic Center, applied to the globular cluster context to directly predict accretion rates based on the angular momentum of the stellar winds.", '11. Summary': 'By reviewing the observational literature, we draw the following conclusions about IMBH populations: \n- · There are clear and compelling cases for black holes in the mass range of 10 5 -10 6 M glyph[circledot] in the nuclei of low-mass galaxies. Moving downward into the true IMBH regime, there are tantalizing hints from NGC 205 and some constraining limits for lowermass black holes, while HLX-1 and a handful of similar objects provide additional circumstantial evidence that ∼ 10 4 M glyph[circledot] black holes exist. There is not even circumstantial evidence yet for 10 3 M glyph[circledot] black holes.\n- · Excepting HLX-1, most ultra-luminous X-ray sources are not powered by IMBHs.\n- · Among M ∗ > 10 5 M glyph[circledot] globular clusters in the Milky Way, observations limit the occupation fraction of ∼ 10 3 M glyph[circledot] black holes to 10-15%. In contrast, both dynamical measurements and X-ray observations in 10 9 -10 10 M glyph[circledot] galaxies argue for > 50% occupation fractions of black holes in these galaxies.\n- · The most promising avenues to find ∼ 100 -10 3 M glyph[circledot] black holes are either to identify stellar binaries, which are expected to be common or to detect them in gravitational radiation. Ground-based gravitational wave limits on the low end of this mass regime are continuing to improve.\n- · We do not see evidence for a change in the M BH -σ ∗ relation for σ ∗ < 100 km s -1 . However, the existing limits suggest there is a broad range of black hole masses in these galaxies just below our detection limit, particularly since observations suggest that a large fraction of these galaxies do host black holes with M BH ∼ > 10 4 M glyph[circledot] .\n- · Folding together the empirically allowed range of occupation fractions with the observed correlation between stellar mass and black hole mass, we are able to bracket the allowed range of black hole number density down to ∼ 10 4 M glyph[circledot] . More dynamical observations of black holes in < 10 10 M glyph[circledot] galaxies, which will be facilitated by ALMA at full capacity along with extremely large telescopes in the coming decade, are needed to determine the scaling relations and occupation fractions for both red and blue galaxies. \nBy putting together theoretical predictions of occupation fractions with galaxy luminosity functions, we evaluate the current constrains on seeding mechanisms: \n- · Early seeding mechanisms-direct collapse type models or Population III starswill be hard to distinguish using local observations alone, because varied accretion \nhistories can wash out the early differences in these models. However, early-time luminosity functions and gravitational wave observations at high redshift could help. \n- · Theoretical predictions are quite divided on whether direct collapse scenarios can produce sufficient black holes. If direct collapse models do operate, the typical formation mass must be < 10 5 M glyph[circledot] .\n- · From both a theoretical and observational perspective, gravitational runaway is unlikely to take off in typical globular clusters, but the most massive star clusters may still host such events. \nWe highlight a few concrete conclusions related to next-generation constraints here: \n- · Due to mass segregation, the total number of stars within the sphere of influence of a ∼ 1000 M glyph[circledot] black hole may be insufficient to place stringent dynamical constraints on objects in this mass range even with extremely large telescopes, while ∼ 3000 M glyph[circledot] black holes should be measurable.\n- · To unlock the potential of the fundamental plane for understanding black hole demographics, next generation radio telescopes will be crucial to detect low-mass, lowaccretion-rate black holes in dwarf galaxies.\n- · As the number of ultra-compact dwarfs with black holes grow, these may help illuminate the relative importance of birth and accretion for black holes in nuclear clusters, since their accretion histories were likely truncated by falling into a bigger halo.\n- · Nuclear tidal disruption events are on the cusp of providing important clues to black hole demographics, but we must understand the rates as a function of galactic environment.\n- · It should be possible to identify (or rule out) wandering black holes in our galaxy by searching for compact and low-mass star clusters with Gaia . Complementary offnuclear tidal disruption events offer a promising avenue to getting large-scale statistics of such objects, again provided we can understand the rates and emission mechanisms. \nWe are very excited about future prospects for finding true IMBHs in the coming decades, between improved sensitivity and frequency range for gravitational wave detection, the leap in angular resolution and sensitivity afforded by next-generation optical, radio, and X-ray telescopes, and the continuing reach of time-domain surveys.', '12. Acknowledgements': 'We thank Gillian Bellovary, Ruben Sanchez-Janssen, and Angelo Ricarte for sharing their data. We thank Ruancun Li for performing bespoke 2MASS fitting and providing luminosities and colors for more than 40 galaxies. We thank Joan Wrobel for catching an error in an earlier version of this manuscript. We are grateful to Suvi Gezari, Zoltan Haiman, Morgan MacLeod, Nadine Neumayer, Anil Seth, Sjoert van Velzen, Marta Volonteri for useful discussions. LH is supported by the National Science Foundation of China (11721303) and the National Key R&D Program of China (2016YFA0400702). JS acknowledges support from the Packard Foundation and from National Science Foundation grant AST-1514763. JEG acknowledges support from National Science Foundation grants AST-1713828 and AST-1815417.', ':arXiv:1902.03813': "Volonteri M, Rees MJ. 2006. Ap. J. 650:669-678 \nVuˇceti´c MM, Ili´c D, Egorov OV, Moiseev A, Oni´c D, et al. 2019. Astron. Astrophys. 628:A87 \nWalcher CJ, B¨oker T, Charlot S, Ho LC, Rix HW, et al. 2006. Ap. J. 649:692-708 \nWalsh JL, Barth AJ, Ho LC, Sarzi M. 2013. Ap. J. 770:86 \nWebb N, Cseh D, Lenc E, Godet O, Barret D, et al. 2012. Science 337:554 \nWebb NA, Gu´erou A, Ciambur B, Detoeuf A, Coriat M, et al. 2017. Astron. 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Lett. \n121:141101 \nTable 1 Seeds to Local Predictions \nWe show the range of occupation fractions and implied number densities for each seed formation channel. The direct collapse numbers are based on Bellovary et al. (2019) and the Population III numbers are based on Ricarte & Natarajan (2018). For the gravitational runaway channel, we have relied on a number of sources described in § 2.3. \nTable 2 Detections in Nuclei \nColumn (1): Object name. Column (2): Distance (Mpc). Column (3): Black hole mass determination method. Column (4): Black hole mass ( M glyph[circledot] ). Column (5): Star cluster mass, when relevant ( M glyph[circledot] ). Column (6): Stellar mass when galaxy host ( M glyph[circledot] ). Column (7): Stellar velocity dispersion, cluster or galaxy (km s -1 ). Column (8): (1) Nguyen et al. (2019b); (2) den Brok et al. (2015); (3) Peterson et al. (2005); (4) Woo et al. (2019); (5) Bentz et al. (2016); (6) Wevers et al. (2017); (7) Blagorodnova et al. (2017); (8) Holoien et al. (2014) ; (9) Arcavi et al. (2014); (10) Gezari et al. (2012); (11) Maksym et al. (2013, 2014). \nTable 3 IMBH Candidates Outside Nuclei \nColumn (1): Object name. Column (2): Distance (Mpc). Column (3): Black hole mass determination method. Column (4): Black hole mass ( M glyph[circledot] ). When limits are listed, an effort has been made to quote a 3 σ limit. Column (5): Star cluster mass, when relevant ( M glyph[circledot] ). Column (6): Stellar velocity dispersion, cluster or galaxy (km s -1 ). Column (7): (1) Kızıltan et al. (2017); (2) Mann et al. (2019); Tremou et al. (2018); (4) Noyola et al. (2010); (5) van der Marel & Anderson (2010a); (6) Lutzgendorf et al. (2013); (7) Perera et al. (2017); (8) Gieles et al. (2018); (9) Baumgardt et al. (2019); (10) Gerssen et al. (2002); (11) Baumgardt et al. (2003); (12) Lutzgendorf et al. (2015); (13) Lanzoni et al. (2013); (14) Ibata et al. (2009); (15) Baumgardt (2017); (16) Gebhardt et al. (2005); (17) Miller-Jones et al. (2012); (18) Straub et al. (2014); (19) Webb et al. (2012); (20) Cseh et al. (2015); (21) Pasham et al. (2014); (22) Brightman et al. (2016); (23) Mezcua et al. (2018b) \nTable 4 Constraining Upper Limits \nColumn (1): Object name. Column (2): Distance (Mpc). Column (3): Black hole mass determination method. Column (4): Black hole mass limit, 3 σ when possible ( M glyph[circledot] ). For the Neumayer & Walcher (2012) sample, we use their maximum black hole masses. Column (5): Star cluster mass, when relevant ( M glyph[circledot] ). Column (6): Stellar mass when galaxy host ( M glyph[circledot] ). Column (7): Stellar velocity dispersion, cluster or galaxy (km s -1 ). Column (8): (1) Jardel & Gebhardt (2012); (2) Lora et al. (2009); (3) Boker et al. (1999); (4) Neumayer & Walcher (2012); (5) Nguyen et al. (2017); (6) Barth et al. (2009); (7) De Lorenzi et al. (2013).", '13.1. Next-generation Opportunities': "We include here a full description of all the numbers displayed in Figure 7. Starting in the Local Volume with dynamical signatures, for the densest and most massive globular clusters, we expect to reach limits of a few × 10 3 M glyph[circledot] using the proper motions of stars with extremely large telescopes (e.g., Greene et al. 2019), although as mentioned in § 3 above, the sensitivity will be a function of the number of stars in the sphere of influence of the black hole. Moving out to the Local Volume, assuming stellar velocity dispersions of 10 -30 km s -1 for the stars, extremely large telescopes and ALMA at full capabilities should be able to detect or put robust limits on 10 4 M glyph[circledot] black holes out to 5-10 Mpc, although systematics in the modeling and in degeneracy with clusters of lower-mass compact objects will be limiting factors at this mass. A conservative number of targets would be the ∼ 20 -40 galaxies with masses less than the LMC in the Local Volume. \nBeyond the relatively modest number of galaxies and star clusters accessible for dynamical measurements, we will have to rely on accretion signatures to determine the presence of black holes. The proposed next-generation Very Large Array (ngVLA) instrument offers the largest discovery space (Murphy 2018). With the sensitivity of the ngVLA, it will become possible to detect or rule out IMBHs at 1000 M glyph[circledot] in all massive globular clusters in the Milky Way, as well as Local Volume galaxy nuclei and thousands of extragalactic globular clusters (Wrobel et al. 2019), even given the large scatter in the Fundamental Plane and the uncertain gas density and angular momentum content of that gas. That said, as emphasized in § 4, there are many sources of confusion from stellar processes when attempting to use radiation to find IMBHs. Multi-epoch and multi-wavelength follow-up will be needed. Complementary time-domain surveys like the Zwicky Transient Factory (ZTF; Bellm et al. 2019) and LSST (Ivezic et al. 2008), with next-generation X-ray missions like eROSITA and Athena , will begin to uncover yet more off-nuclear TDE candidates, optimistically at rates up to ∼ 100 yr -1 (Lin et al. 2018, Cassano et al. 2018). \nOur current best constraints on the occupation fraction from radiation come from X-ray observations of galaxies within 30 Mpc (Miller et al. 2015), and while the number of targets may increase by factors of a few (e.g., Gallo & Sesana 2019), the X-ray searches cannot go any deeper than current limits because they hit a confusion limit with X-ray binaries at luminosities below ∼ 10 38 erg s -1 (Sivakoff et al. 2007, Gallo et al. 2008, Ghosh et al. 2008). Going forward, radio studies with the ngVLA will be the most promising way to improve the existing constraints from Miller et al. As shown by Plotkin & Reines (2018), with relatively modest exposures times of an hour, one could systematically study 10 5 M glyph[circledot] black holes accreting at ∼ 10 -5 of their Eddington limits out to Gpc distances, and 10 4 M glyph[circledot] black holes out to the Virgo cluster. The black hole fundamental plane could be calibrated within 5 Mpc using dynamical measurements from extremely large telescopes, and then extended to map the occupation fraction of 10 4 M glyph[circledot] black holes out to 20 Mpc. Again, we note that deep and high-resolution radio imaging alone will be insufficient; we will need multi-wavelength information to determine the true nature of these radio sources ( § 4). \nAs advanced LIGO, VIRGO, and other gravitational wave experiments continue, the constraints on 100 M glyph[circledot] black hole mergers (which appear as bursts in the LIGO band) should get more stringent, putting yet more concrete constraints on that population. Unfortunately, those limits constrain a product of the number density and the merger rate, and we see no immediate prospect for independent number density measurements of ∼ 100 M glyph[circledot] \nblack holes. \nMoving beyond the Local Volume, we increasingly will rely on accretion at high Eddington ratio and gravitational wave signatures to find IMBHs. Ongoing and upcoming time-domain surveys have the prospect to uncover unprecedented numbers of tidal disruption events (e.g., van Velzen et al. 2011). Although at present only a very small fraction of the events have black hole mass primaries with M BH < 10 6 M glyph[circledot] (Maksym et al. 2014, Wevers et al. 2017, Mockler et al. 2019), the increased depth of LSST will improve the situation (although see Bricman & Gomboc 2019). \nA particularly exciting prospect is white dwarf TDEs, which happen uniquely around < 10 5 M glyph[circledot] black holes, and may be accompanied by a gravitational wave signal. As argued by Eracleous et al. (2019), based on depth, we can expect ∼ 10 events per year in the ZTF+LSST era (see also Hung et al. 2018), although we should note that the rates are highly uncertain, as is our ability to uniquely identify such events from electromagnetic signatures alone (e.g., MacLeod et al. 2016a). \nThe EMRI rates will include not only the white dwarf-black hole mergers, but also neutron star and stellar-mass black hole mergers with the central black hole. These rates are also uncertain, as they depend on both the unknown black hole mass function ( § 9) and the unknown stellar populations in the vicinity of low-mass black holes (see Berry et al. 2019, and references therein). However, the 'optimistic' number may be hundreds of events per year (Gair et al. 2010) with the pessimistic one being a couple of events per year. \nLISA will also be sensitive to black hole-black hole mergers, and here the event rate will depend on the seed formation mechanism, as well as on the early accretion history of the seeds. Ricarte & Natarajan (2018) present example rates for their different seeding and accretion prescriptions, used here. \nFinally, first JWST and then a Lynxlike mission in the X-ray, should each be sensitive to ∼ 10 5 M BH black holes even at z ≈ 10 (e.g., Barrow et al. 2018, Haiman et al. 2019). The challenge will be identifying such accreting low-mass black holes, which will require extensive multi-wavelength data. Nevertheless, we again use the predictions of Ricarte & Natarajan (2018) to estimate the numbers of accreting black holes that will be detectable by upcoming X-ray missions.", '14. Supplement II: Additional Tables': "Table 5 AGN Fractions \n(a) In the case of this sample, the number represents total number of galaxies searched, without any cut on morphology or stellar mass, thus these numbers cannot be compared with others in this table. (1) Sample (see § 4). (2) Type of selection. Broad-line region (BLR) means that a black hole mass cut was applied. BPT used the strong emission lines. X-ray was in all cases a search in Chandra data for X-ray point soures. (3) Total number of objects with M ∗ < 10 10 M glyph[circledot] searched. (4) Fraction of objects found to harbor black holes. (5) Approximate detection threshold, converted to a bolometric luminosity assuming a bolometric correction of 10 for both the X-ray and optical continuum. \nTable 6 Stellar Masses, σ ∗ and Black Hole Masses for Scaling Relations (New) \nFirst group of galaxies are added to Kormendy & Ho (2013), while the rest are taken from that paper. (1) Galaxy. (2) Distance (Mpc). (3) σ ∗ (km s -1 ). (4) K -band magnitude. (5) B -V color (mag). (6) Stellar mass derived using Bell et al. (2003). (7) Black hole mass ( M glyph[circledot] ). (8) Lower 1 σ limit on black hole mass ( M glyph[circledot] ). (9) Upper 1 σ limit on black hole mass ( M glyph[circledot] ). (10) Crude Hubble type. (11) Reference: (1) Greene et al. (2016); (2) Saglia et al. (2016); (3) Thater et al. (2019); (4) Thomas et al. (2016); (5) Krajnovi'c et al. (2018). \nTable 7 Stellar Masses, σ ∗ and Black Hole Masses for Scaling Relations (KHI) \nFirst group of galaxies are added to Kormendy & Ho (2013), while the rest are taken from that paper. (1) Galaxy. (2) Distance (Mpc). (3) σ ∗ (km s -1 ). (4) K -band magnitude. (5) B -V color (mag). (6) Stellar mass derived using Bell et al. (2003). (7) Black hole mass ( M glyph[circledot] ). (8) Lower 1 σ limit on black hole mass ( M glyph[circledot] ). (9) Upper 1 σ limit on black hole mass ( M glyph[circledot] ). (10) Crude Hubble type. \nTable 8 Stellar Masses, σ ∗ and Black Hole Masses for Scaling Relations (KHII) \nFirst group of galaxies are added to Kormendy & Ho (2013), while the rest are taken from that paper. (1) Galaxy. (2) Distance (Mpc). (3) σ ∗ (km s -1 ). (4) K -band magnitude. (5) B -V color (mag). (6) Stellar mass derived using Bell et al. (2003). (7) Black hole mass ( M glyph[circledot] ). (8) Lower 1 σ limit on black hole mass ( M glyph[circledot] ). (9) Upper 1 σ limit on black hole mass ( M glyph[circledot] ). (10) Crude Hubble type. \nTable 9 Scaling Relation Fits \n(1) Fit being presented. We fit log ( M BH ) = α + β log( σ ∗ /σ 0 ) + glyph[epsilon1] where σ 0 = 160 km s -1 ; log ( M BH )= α + β log( M ∗ /M 0 ) + glyph[epsilon1] for M 0 = 3 × 10 10 M glyph[circledot] . (2) Sample. (3) Intercept ( α ). (4) Slope ( β ). (5) Intrinsic scatter ( glyph[epsilon1] ). \nTable 10 Black Hole Mass Functions \n- (1) Black hole mass bin. (2) Number density of black holes, assuming the more pessimistic 'linear'\n- occupation fraction. (3) Lower 68% confidence limit on number density. (4) Upper 68% limit on number\n- density. (5) Number density of black holes, assuming that every nuclear star cluster hosts a black hole.\n- (6) Lower 68% confidence limit on number density. (7) Upper 68% limit on number density. \nTable 11 Other Nuclei \nColumn (1): Object name (galaxy or UCD). Column (2): Distance (Mpc). Column (3): Mass of the nuclear star cluster ( M glyph[circledot] ). Errors are dominated by systematics in M/L determinations and are likely 0.2-0.3 dex. Column (4): BH mass. Column (5): Reference for M BH and M NSC . (1) Saglia et al. (2016); (2) Lauer et al. (2005); (3) Pechetti et al. (2017); (4) Cˆot'e et al. (2006); (5) Krajnovi'c et al. (2018); (6) Schodel et al. (2007); (7) Boker et al. (1999); (8) Ahn et al. (2018); (9) Ahn et al. (2017); (10) Seth et al. (2014); (11) Afanasiev et al. (2018); (12) Norris & Kannappan (2011); (13) Voggel et al. (2019)."}

2023ApJ...956L..13M

The search for habitable environments and biomarkers in exoplanetary atmospheres is the holy grail of exoplanet science. The detection of atmospheric signatures of habitable Earthlike exoplanets is challenging owing to their small planetstar size contrast and thin atmospheres with high mean molecular weight. Recently a new class of habitable exoplanets called Hycean worlds has been proposed defined as temperate oceancovered worlds with HSUB2SUBrich atmospheres. Their large sizes and extended atmospheres compared to rocky planets of the same mass make Hycean worlds significantly more accessible to atmospheric spectroscopy with JWST. Here we report a transmission spectrum of the candidate Hycean world K218 b observed with the JWST NIRISS and NIRSpec instruments in the 0.95.2 m range. The spectrum reveals strong detections of methane CHSUB4SUB and carbon dioxide COSUB2SUB at 5 and 3 confidence respectively with high volume mixing ratios of 1 each in a HSUB2SUBrich atmosphere. The abundant CHSUB4SUB and COSUB2SUB along with the nondetection of ammonia NHSUB3SUB are consistent with chemical predictions for an ocean under a temperate HSUB2SUBrich atmosphere on K218 b. The spectrum also suggests potential signs of dimethyl sulfide DMS which has been predicted to be an observable biomarker in Hycean worlds motivating considerations of possible biological activity on the planet. The detection of CHSUB4SUB resolves the longstanding missing methane problem for temperate exoplanets and the degeneracy in the atmospheric composition of K218 b from previous observations. We discuss possible implications of the findings open questions and future observations to explore this new regime in the search for life elsewhere.

2023-10-01T00:00:00Z

['10.3847/2041-8213/acf577', '2023arXiv230905566M', 'arXiv:2309.05566', '2023ApJ...956L..13M', '10.48550/arXiv.2309.05566']

['Exoplanets', 'Habitable planets', 'Exoplanet atmospheres', 'Exoplanet atmospheric composition', 'James Webb Space Telescope', 'Infrared spectroscopy', 'Astrobiology', 'Biosignatures', '498', '695', '487', '2021', '2291', '2285', '74', '2018', 'Astrophysics - Earth and Planetary Astrophysics']

Carbonbearing Molecules in a Possible Hycean Atmosphere

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

https://arxiv.org/pdf/2309.05566.pdf

{'Carbon-bearing Molecules in a Possible Hycean Atmosphere': 'Nikku Madhusudhan, 1 Subhajit Sarkar, 2, ∗ Savvas Constantinou, 1, ∗ M˚ans Holmberg, 1, ∗ Anjali A. A. Piette, 3 and Julianne I. Moses 4 \n1 \nInstitute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 2 School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK 3 Earth & Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA 4 Space Science Institute, Boulder, CO 80301, USA', 'ABSTRACT': 'The search for habitable environments and biomarkers in exoplanetary atmospheres is the holy grail of exoplanet science. The detection of atmospheric signatures of habitable Earth-like exoplanets is challenging owing to their small planet-star size contrast and thin atmospheres with high mean molecular weight. Recently, a new class of habitable exoplanets, called Hycean worlds, has been proposed, defined as temperate ocean-covered worlds with H 2 -rich atmospheres. Their large sizes and extended atmospheres, compared to rocky planets of the same mass, make Hycean worlds significantly more accessible to atmospheric spectroscopy with JWST. Here we report a transmission spectrum of the candidate Hycean world K2-18 b, observed with the JWST NIRISS and NIRSpec instruments in the 0.9-5.2 µ m range. The spectrum reveals strong detections of methane (CH 4 ) and carbon dioxide (CO 2 ) at 5 σ and 3 σ confidence, respectively, with high volume mixing ratios of ∼ 1% each in a H 2 -rich atmosphere. The abundant CH 4 and CO 2 along with the nondetection of ammonia (NH 3 ) are consistent with chemical predictions for an ocean under a temperate H 2 -rich atmosphere on K2-18 b. The spectrum also suggests potential signs of dimethyl sulfide (DMS), which has been predicted to be an observable biomarker in Hycean worlds, motivating considerations of possible biological activity on the planet. The detection of CH 4 resolves the long-standing missing methane problem for temperate exoplanets and the degeneracy in the atmospheric composition of K2-18 b from previous observations. We discuss possible implications of the findings, open questions, and future observations to explore this new regime in the search for life elsewhere. \nKeywords: Exoplanets(498) - Habitable planets(695) - Exoplanet atmospheres(487) - Exoplanet atmospheric composition (2021) - JWST (2291) - Infrared spectroscopy(2285) - Astrobiology(74) - Biosignatures(2018)', '1. INTRODUCTION': "The detection and characterisation of habitable-zone exoplanets is a major frontier in modern astronomy. Until recently, the quest for exoplanetary habitability and biosignatures has been focused primarily on rocky exoplanets, naturally motivated by the terrestrial experience of life (Kasting et al. 1993; Meadows & Barnes 2018). The extreme diversity of exoplanetary systems witnessed over the past three decades motivates considerations of new avenues in the search for life elsewhere. \nCorrespondence: nmadhu@ast.cam.ac.uk \n- ∗ These authors contributed comparably to this work. \nSuch an endeavour may open the doors to a wider range of habitable environments that may be more numerous and more favourable to atmospheric characterisation. Hycean worlds, a recently proposed class of habitable exoplanets, represent one such avenue that is accessible to current observational facilities (Madhusudhan et al. 2021). \nHycean worlds are a class of water-rich sub-Neptunes with planet-wide oceans underlying H 2 -rich atmospheres. Such planets have a significantly wider habitable zone compared to terrestrial planets. With expected radii between 1-2.6 R ⊕ for masses between 110 M ⊕ , Hycean planets represent a habitable subset of temperate sub-Neptunes that allow for a vast diversity \nof atmospheric and internal structures (Madhusudhan et al. 2020, 2021; Piette & Madhusudhan 2020; Nixon & Madhusudhan 2021). Such planets are also potentially abundant in the exoplanet population given the predominance of exoplanets in the sub-Neptune regime (Fulton & Petigura 2018). The large volatile content in the interior of a Hycean world implies a lower density and, hence, larger radius and lower gravity, compared to a rocky planet of comparable mass. The low gravity and low atmospheric mean molecular weight (MMW) in turn result in a larger atmospheric scale height for a given temperature relative to terrestrial-like exoplanets with high-MMW atmospheres. These factors make Hycean worlds readily accessible for atmospheric characterisation, including potential biomarker detection, using modest observing time with JWST (Madhusudhan et al. 2021; Phillips et al. 2021, 2022; Leung et al. 2022). \nThe Hycean planet class was motivated by the demonstration that the bulk properties of the habitable-zone sub-Neptune K2-18 b (Montet et al. 2015; Cloutier et al. 2017, 2019; Benneke et al. 2019a) are consistent with the possibility of a water-rich interior and a liquid-water ocean at habitable temperatures and pressures underlying a H 2 -rich atmosphere (Madhusudhan et al. 2020). The planet has a mass of 8 . 63 ± 1 . 35 M ⊕ and radius of 2 . 61 ± 0 . 09 R ⊕ , with an equilibrium temperature of ∼ 250-300 K for an albedo between 0-0.3 (Cloutier et al. 2019; Benneke et al. 2019a). While a Hycean interpretation for K2-18 b is plausible and promising, a broad set of other internal structures and nonhabitable surface conditions are also compatible with its bulk properties (Madhusudhan et al. 2020; Piette & Madhusudhan 2020; Nixon & Madhusudhan 2021), especially when considering only cloud/haze-free atmospheres (e.g. Scheucher et al. 2020; Pierrehumbert 2023; Innes et al. 2023). Originally, the observed transmission spectrum of the planet in the near-infrared (1.1-1.7 µ m) with the Hubble Space Telescope (HST) WFC3 spectrograph suggested a H 2 -rich atmosphere with strong H 2 O absorption (Benneke et al. 2019a; Tsiaras et al. 2019; Madhusudhan et al. 2020). However, other studies highlighted the degeneracy between H 2 O and CH 4 in the observed HST spectrum (Blain et al. 2021; B'ezard et al. 2022), and potential contributions due to stellar heterogeneities (Barclay et al. 2021), rendering the previous H 2 O inference inconclusive. \nAtmospheric observations with JWST have the potential to provide important insights into the atmospheric, surface, and interior conditions of K2-18 b. The planet has been theoretically demonstrated to be accessible to detailed atmospheric characterisation with a modest amount of JWST time, including the possi- \nbility of detecting prominent CNO molecules, such as H 2 O, CH 4 , NH 3 , as well as several biomarkers, such as (CH 3 ) 2 S or dimethyl sulfide (DMS), methyl chloride (CH 3 Cl), carbonyl sulfide (OCS), and others (Madhusudhan et al. 2021). The major molecules are expected to be detectable even in the presence of highaltitude clouds (Constantinou & Madhusudhan 2022). Furthermore, several recent theoretical studies have demonstrated that atmospheric abundances of prominent CNO molecules can be used to infer the presence of surfaces beneath H 2 -rich atmospheres in temperate sub-Neptunes (Yu et al. 2021; Hu et al. 2021; Tsai et al. 2021). For example, the presence of an ocean underneath a shallow H 2 -rich atmosphere, as would be the case for a Hycean world, may be inferred by an enhanced abundance of CO 2 , H 2 O and/or CH 4 , but with a depletion of NH 3 (Hu et al. 2021; Tsai et al. 2021; Madhusudhan et al. 2023). \nIn this work, we report the first JWST transmission spectrum of K2-18 b. The spectrum was observed using NIRISS SOSS and NIRSpec G395H instruments in the 0.9-5.2 µ m wavelength range, which contains strong spectral features of multiple chemical species. The chemical constraints derived from the observed spectrum provide key insights into its atmospheric and surface conditions and pave the way for a new era of atmospheric characterisation of low-mass exoplanets with JWST. In what follows, we present our JWST observations and data reduction in section 2. We discuss our atmospheric retrievals of the transmission spectrum in section 3. We summarize our results and discuss the implications in section 4.", '2. OBSERVATIONS AND DATA REDUCTION': 'We report transmission spectroscopy of K2-18 b using the JWST NIRSpec (Ferruit et al. 2012; Birkmann et al. 2014) and NIRISS (Doyon et al. 2012; Doyon et al. 2023) instruments. We observed two primary transits of the planet in front of its host star, one with each instrument, as part of the JWST GO Program 2722 (PI: N. Madhusudhan). The first transit was observed using the NIRSpec G395H grating between Jan 20, 2023, 18:37:38 UTC and Jan 21, 2023, 01:11:32 UTC for a total exposure time of 5.3 hours, which is nearly twice the expected transit duration. The observation was made in the Bright Object Time Series (BOTS) mode with the F290LP filter, the SUB2048 subarray and the NRSRAPID readout pattern, with the spectra dispersed over two detectors (NRS1 and NRS2). The two detectors, NRS1 and NRS2, span wavelength ranges of 2.733.72 µ m and 3.82-5.17 µ m, respectively, with a gap in between at 3.72-3.82 µ m. The G395H grating offers \nthe highest-resolution mode of NIRSpec with R ∼ 2700. The spectroscopic time-series observation is composed of 1625 integrations, with 12 groups per integration. For NIRSpec, the host star K2-18 was too bright for target acquisition (TA). Therefore, another nearby target (2MASSJ11301306+0735116) within the splitting distance of the science target was used for TA. \nThe second observation was conducted using the NIRISS Single Object Slitless Spectroscopy (SOSS) instrument mode (Albert et al. 2023) between Jun 1, 2023, 13:49:20 UTC and Jun 1, 2023, 19:36:05 UTC, totalling an exposure time of 4.9 hours. The observation used the GR700XD grism (R ∼ 700), the CLEAR filter, the SUBSTRIP256 subarray and the NISRAPID readout pattern, giving a wavelength coverage of 0.85- 2.85 µ m for the first spectral order. The exposure consisted of 648 integrations, with 4 groups per integration. There were no tilt events or high-gain antenna movements during any of the observations.', '2.1. NIRSpec': 'The data reduction is conducted using a combination of the JWST Science Calibration Pipeline (Bushouse 2020) and our custom-built pipeline for the spectral extraction. We start with the raw 2D images (the .uncal files) of the spectroscopic time-series data which contain group-level counts for each integration. Stage 1 of the data reduction is performed mainly using the JWST Science Calibration Pipeline. This involves performing saturation flagging, superbias subtraction, reference pixel correction, linearity correction, dark current subtraction (where reference data is available), jump detection and linear fitting of the group-level ramps to obtain the count rate for each pixel per integration. This is repeated for each integration in the exposure. The jump detection threshold is set at 5 σ . Prior to ramp fitting, we also perform an additional step for background subtraction, at the group level, in order to mitigate 1/f noise, as common in previous works (JWST Transiting Exoplanet Community Early Release Science Team et al. 2023; Rustamkulov et al. 2023; Alderson et al. 2023). For this step we measure the background level for each detector column, as the mean of the pixels that are ± 10 pixels away from the midpoint of the curved trace, while masking bad pixels and cosmic-ray hits. The outputs of Stage 1 are 2D images of the count rate for each integration, saved as .rateints files. \nWe also use steps from the JWST Science Calibration Pipeline for Stage 2, which applies the wavelength calibration for the spectral trace. Following previous studies using NIRSpec (e.g. Alderson et al. 2023), we forgo the flat-field correction in Stage 2 for our differential tran- \nsit measurement. The resultant 2D images along with the wavelength calibration, which are saved as .calints files, are then used for the spectral extraction of the time series of 1D stellar spectra. \nWe conduct the spectral extraction, Stage 3, applying our custom-built pipeline to convert the 2D images into 1D spectra. This is conducted for each of the two detectors (NRS1 and NRS2) separately. We first create a bad pixel mask based on the data quality flags in the Stage 2 products. We then extract the 1D spectrum from the 2D image using an optimal extraction algorithm (Horne 1986). To perform the extraction, we obtain the pointspread functions (PSFs) as the sum of the first three principal components of the time series of the detector images 1 , inspired by the principal-component-analysisbased morphology analysis in Coulombe et al. (2023). This takes into account the wavelength and time dependence of the PSF. Outliers were iteratively rejected during the spectrum extraction, with the threshold set at 5 σ . Spectral channels with more than 20% of the flux masked were discarded from further analysis.', '2.2. NIRISS': 'To reduce the NIRISS data, we used the JWST Science Calibration Pipeline (Bushouse 2020) for Stages 12 and the JExoRES pipeline for the spectral extraction (Holmberg & Madhusudhan 2023). During Stage 1, we perform the standard saturation flagging, superbias subtraction, linearity correction, jump detection (threshold set at 5 σ ) and fitting of the group-level ramp to obtain the count rate. We perform a custom background subtraction step before the linearity correction and ramp fitting to reduce the effect of 1/f noise. In line with Radica et al. (2023) and Albert et al. (2023), this involves temporarily subtracting a model of the flux of the detector, containing both the background and stellar flux, to reveal the 1/f noise. Initially, we use the groupwise median stack of all integrations to model the flux. For each group and integration, the 1/f noise level is then estimated using the median of each column, while masking the traces of the spectral orders, as well as bad pixels. This 1/f noise estimate is subtracted from the data to reduce the level of correlated noise. We later repeat all stages of the NIRISS data reduction using updated models of the background level and stellar flux to refine the 1/f noise correction step, as described further below. Furthermore, we do not perform the dark current subtraction, given the quality of the dark reference file from the Calibration Reference Data System (CRDS) \nFigure 1. Left: white light curve for the transit of K2-18 b observed with NIRSpec G395H. The top panel shows the combined white light curve from NRS1 and NRS2, together with the 1 σ model interval in red. The bottom panel shows the residuals after subtracting the median model, together with line at zero. The standard deviation of the residuals is measured to be 123 ppm, which is 1.2 × the expected noise level. The red line is shown to indicate zero. Right: the top panel shows the normalised spectroscopic light curves (binned in wavelength for visual clarity). The detector gap is shown in white. The bottom panel depicts the transit depth precision at pixel resolution. \n<!-- image --> \nFigure 2. Left: white light curve for the transit of K2-18 b observed with NIRISS SOSS. The blue contour shows the 1 σ model interval. We find evidence of a spot occultation at the start of the transit. The bottom panel shows the residuals after subtracting the median model. The standard deviation of the residuals is measured to be 115 ppm, which is 2.2 × the expected noise level. The blue line is shown to indicate zero. Right: the top panel shows the normalised spectroscopic light curves (binned in wavelength for visual clarity). The bottom panel shows the transit depth precision obtained from the light-curve fitting, binned at two pixel columns per light curve. \n<!-- image --> \nis insufficient, so as not to contribute additional noise (Feinstein et al. 2023; Radica et al. 2023). \nFor Stage 2, we perform the flat-field correction before modelling the background flux in a two-step process. First, we use the background model available via JDOX, created from program 1541, scaled to the median of all integrations in a small rectangular region ( x ∈ [720 , 770], y ∈ [210 , 250]). By subtracting this we precisely remove the brighter of the two background components, caused by zodiacal light. However, this leaves residuals in the dimmer component of the background for columns up to the ∼ 700 pixel column. This is corrected by first generating a median image from all integrations, and then from this median image obtaining the median of each pixel column (while masking spectral traces, contami- \nand bad pixels). These column-wise medians are then subtracted from each integration image. This is only performed for columns up to the 700-pixel column. We use this additional background flux to update the (scaled) background model. Next, we perform the order tracing, PSF estimation, and spectral extraction (Stage 3) according to Holmberg & Madhusudhan (2023), with an extraction aperture of 35 pixels, leaving out the background refinement since we correct the 1/f noise at the group level. As in the case of NIRSpec, spectral channels with more than 20% of the flux masked were discarded from further analysis. \nFinally, we repeat the data reduction stages in order to improve the 1/f noise correction. This time, we model the flux at the group level using the updated background \nmodel from above and the groupwise out-of-transit median stellar flux. The background model is scaled using the integration time of each group while the outof-transit stellar flux (minus the background) is scaled with a model of the transit light curve, derived using the initial white light curve from the first spectral order. We note that in the remaining analysis we only consider the first spectral order given that the second order has a considerably lower flux level, meaning that it is more sensitive to systematics due to sources of contamination.', '2.3. Starspot Occultations': "Starspot and faculae crossings during exoplanet transits are known to affect the apparent transit depth (e.g. Pont et al. 2008; Czesla et al. 2009). If left uncorrected, starspot and faculae occultations effectively decrease and increase the observed transit depth, respectively. This effect is wavelength dependent and must therefore be accounted for so as not to impact the transmission spectrum. From the present observations we find evidence of starspot occultations, which is especially strong for the transit observed with NIRISS. Fortunately, correcting the transmission spectrum is possible by measuring the intensity ratio and size of the occulted feature from the transit light curves. \nWe perform a joint inference of the transit parameters and the properties of the active region using the semianalytical spot modelling code SPOTROD (B'eky et al. 2014). Hence, we also avoid having to discard the affected data. The code computes the transit light curve with arbitrary limb darkening (affecting the stellar photosphere and spots equally) and homogeneous circular star spots or faculae. The spot is represented by four parameters: the spot-to-star radius ratio R spot /R ∗ , the spot-to-unspotted stellar surface intensity ratio f , and the coordinates of the spot centre projected onto the stellar surface ( ϑ , r 2 ). Intensity ratios below and above 1 represent starspots and faculae, respectively. Similar to Espinoza et al. (2019), we use nested sampling (Skilling 2004), implemented via MultiNest (Feroz et al. 2009), to obtain the Bayesian model evidence and parameter estimation. This allows us to perform a Bayesian model comparison between transit models with or without stellar spots. For this, we use uniform priors between 0 and 0.3 for the spot-to-star radius ratio, 0 and 2 for the intensity ratio, 0 and 2 π for ϑ (with periodic boundary condition), and 0 and 1 for r 2 , ensuring uniform sampling of the disk. \nUsing the NIRISS white light curve from the first order, we find that a single starspot is strongly preferred over no spots with a Bayes factor of ln B = 21 . 6, corresponding to a significance of 6 . 9 σ . Fig. 2 shows the \nwhite light curve and the fitted model. We obtain an intensity ratio of f = 0 . 9329 +0 . 0082 -0 . 0091 and a spot-to-star radius ratio of R spot /R ∗ = 0 . 254 +0 . 033 -0 . 044 . For the NIRSpec white light curve (NRS1 and NRS2 combined), including a spot is also marginally preferred by the data compared to a model without spots, with a Bayes factor of ln B = 1 . 15, corresponding to a significance of 2 . 1 σ . The spot parameters in this case are f = 0 . 82 +0 . 60 -0 . 39 and R spot /R ∗ = 0 . 113 +0 . 094 -0 . 077 . For these reasons, we choose to use the spot modelling for both observations.", '2.4. Light-curve Analysis': "The 1D spectral time series from both observations are then used for light-curve fitting to derive the transit depths. This is done in three stages. The first stage uses the white light curves, from both observations, to derive the wavelength-independent system parameters at high precision. For NIRISS we use the white light curve from the first order and for NIRSpec we use the combined white light curve from both NRS1 and NRS2. In the next stage, we bin the light curves to R ∼ 20 and fit the wavelength-dependent limb-darkening coefficients (LDCs). Finally, we fix these LDCs in the respective R ∼ 20 bins and fit the transit depths at native resolution to obtain the final transmission spectrum of the planet. \nAs described above, we model the transit light curves using SPOTROD (B'eky et al. 2014) as we find evidence of starspot occultations. We assume a circular orbit with the orbital period from Benneke et al. (2019a). We adopt the two-parameter quadratic limb-darkening law, in line with previous JWST transmission spectroscopy studies of M dwarf systems (e.g. Moran et al. 2023; Lustig-Yaeger et al. 2023) and previous work on K218 b (Benneke et al. 2019a). For the LDCs, we use the parameterization and priors by Kipping (2013). To model the baseline flux we implement both a linear and a quadratic trend. We find that the NIRISS out-of-transit white light curve shows a preference for a linear trend, while the NIRSpec data show a weak preference for a quadratic trend in the case of NRS1 and a moderate preference in the case of NRS2. For the white light curves, we use a linear trend for both instruments, given that we find the derived system parameters to be insensitive to the choice of trend. However, for the spectrum, we consider both trends for NIRSpec. For NIRISS, we fix the trend to be linear. Moreover, we discard the first 5 minutes of both observations owing to a small settling ramp. Apart from starspot occultations, no other systematics can be identified, highlighting the excellent data quality provided by JWST. \nFigure 3. The transmission spectrum of K2-18 b. The observed JWST spectrum and retrieved model fits are shown for the one-offset retrieval case discussed in section 3. The data in orange show our NIRISS spectrum between 0.9 - 2.8 µ m and those in dark-red show our NIRSpec G395H spectrum between 2.8 - 5.2 µ m. The spectra are binned to R ≈ 25 and R ≈ 55 for NIRISS and NIRSpec, respectively, for visual clarity. The retrievals are conducted on the native resolution spectra, resulting in the best-fit reduced χ 2 ν = 1 . 080. The NIRSpec spectrum is vertically offset by -41 ppm, corresponding to the median retrieved offset in the one-offset retrieval case. The blue curve shows the median retrieved model spectrum, while the medium- and lighter-blue contours denote the 1 σ and 2 σ intervals, respectively. Yellow points correspond to the median spectrum binned to match the observations. The prominent molecules responsible for the features in different spectral regions are labelled. \n<!-- image --> \nTable 1. Parameter estimation from the white light curve analysis of our JWST NIRISS SOSS and NIRSpec G395H observations of K2-18 b. For the white light curve of NIRSpec, we combine NRS1 and NRS2. For both white light curves, we use a linear trend to model the baseline flux as well as the spot modelling described in section 2.3. The orbital period is held fixed at P = 32 . 940045 days, adopted from Benneke et al. (2019a). \nFor the white light curve of each observation (see left panels of Figs. 1 and 2), we fit for the midtransit time (T 0 ,), the normalised semi-major axis ( a/R ∗ ), the orbital inclination ( i ), the planet-to-star radius ratio ( R p /R ∗ ), the quadratic LDCs ( u 1 and u 2 ), two parameters representing the baseline flux, and four spot parameters (described in 2.3). In the likelihood, we also include a parameter to inflate the photometric uncertainties to match the residual scatter between the data and the transit light-curve model. We measure the precision to be 1.2 × and 2.2 × the expected noise level (photon and read noise, propagated using the Jacobian of the transit model) for the white light curves from NIRSpec (NRS1 and NRS2 combined) and NIRISS, respectively. To sample the posteriors and estimate the parameters we use MultiNest (Feroz et al. 2009), result- \nthe parameters given in Table 1. For all cases, we find that the derived system parameters are consistent within 1 σ , regardless of the trend and the choice to model the starspot or not. \nNext, we bin the spectroscopic light curves to R ∼ 20 for the purpose of fitting the wavelength-dependent LDCs. This resolution strikes a balance between precision and the expected wavelength variability of the limb darkening. We choose to fit the LDCs instead of using values from stellar atmospheric models in order to maximise accuracy (Csizmadia et al. 2013; Espinoza & Jord'an 2015). We fit these binned light curves using MultiNest and fix the system parameters (T 0 , a/R ∗ , i ) to the values obtained from the white light curve analysis (see Table 1), i.e. the weighted average a/R ∗ , i , and T 0 from each observation. We also fixed R spot /R ∗ , ϑ , \nand r 2 to the best-fit parameters for each white light curve. Equipped with empirical LDCs, we go on to fit the transit depths of the high-resolution light curves, while fixing the LDCs to the values within their respective R ∼ 20 bin. This leaves only R p /R ∗ , f , the 2-3 trend parameters, and the uncertainty scaling parameter as free parameters. We fit the light curves at the pixel level for NIRSpec and two pixels per bin for NIRISS, given the potential inaccuracy of the NIRISS SOSS wavelength calibration (Albert et al. 2023). To fit these high-resolution light curves we use the Levenberg-Marquardt algorithm, as applied in previous works (Alderson et al. 2023; Moran et al. 2023). The right panel of Figs. 1 and 2 show the precision of the resulting transmission spectrum at high resolution. In total, we have 1010 and 3401 spectral data points for NIRISS and NIRSpec, respectively, covering the 0.9 - 5.2 µ m wavelength range. \nAs mentioned above, we obtain two spectra in the case of NIRSpec, one with a linear trend and another with a quadratic trend. For the quadratic-trend case, we construct white light curves for NRS1 and NRS2, and fit these separately to obtain detector-averaged values for the quadratic trend component 2 . For each detector, we then fix the quadratic trend component to these values when fitting the spectroscopic light curves (Moran et al. 2023). Overall, we find that the NRS1 spectrum is almost agnostic to the choice of trend ( ∼ 10 ppm difference), whereas, for NRS2, there is a significant difference between the two spectra (approximately a 60 ppm offset). Since we only have one transit observation for NIRSpec, we use an offset as a free parameter in the atmospheric retrieval to account for potential baseline shifts.", '3. ATMOSPHERIC RETRIEVAL': "The observed transmission spectrum allows us to retrieve the atmospheric properties of K2-18 b at the daynight terminator region. We perform the retrieval using the AURA retrieval code (Pinhas et al. 2018) following a similar approach to previous retrieval studies of the planet considering HST and/or simulated JWST observations (e.g. Madhusudhan et al. 2020, 2021; Welbanks et al. 2019; Constantinou & Madhusudhan 2022). The planet's terminator is modelled as a plane-parallel atmosphere in hydrostatic equilibrium, with uniform chemical composition. The chemical abundances and \npressure-temperature ( P -T ) profile are free parameters in the model. The retrieval framework follows a free chemistry approach, whereby the individual mixing ratio of each chemical species is a free parameter. The atmospheric temperature structure is modelled with a parametric P -T profile (Madhusudhan & Seager 2009) with six free parameters. The model also considers inhomogeneous clouds/hazes at the day-night terminator region (MacDonald & Madhusudhan 2017; Pinhas et al. 2018). They are modelled as a combination of a grey cloud deck at a parametric cloud-top pressure P c , above which are hazes with Rayleigh-like spectral contributions, with an enhancement factor a and a scattering slope γ ; for H 2 Rayleigh scattering, a = 1 and γ = -4. The combined clouds and hazes cover a fraction ϕ of the atmosphere at the terminator region. \nThe model includes molecular opacity contributions from prominent CNO molecules expected in temperate H 2 -rich atmospheres as considered in previous works (Pinhas et al. 2018; Welbanks et al. 2019; Constantinou & Madhusudhan 2022). These include H 2 O, CH 4 , NH 3 , HCN, CO and CO 2 . The molecular absorption cross sections are obtained following recent works (Gandhi & Madhusudhan 2017; Gandhi et al. 2020) using line lists from the following sources: H 2 O(Polyansky et al. 2018), CH 4 (Hargreaves et al. 2020), NH 3 (Coles et al. 2019), HCN (Harris et al. 2006; Barber et al. 2014), CO (Rothman et al. 2010; Li et al. 2015) and CO 2 (Huang et al. 2013, 2017). Pressure broadening due to H 2 is considered for all these molecules as described in Gandhi et al. (2020). \nWe additionally consider five molecules that have been suggested to be promising biomarkers in habitable rocky exoplanets (Segura et al. 2005; Domagal-Goldman et al. 2011; Seager et al. 2013a,b; Catling et al. 2018; Schwieterman et al. 2018) as well as Hycean worlds (Madhusudhan et al. 2021): (CH 3 ) 2 S (or DMS), CS 2 , CH 3 Cl, OCS and N 2 O. The absorption cross sections of CH 3 Cl, OCS and N 2 O were computed from the corresponding line lists from the HITRAN database (Gordon et al. 2017): CH 3 Cl (Bray et al. 2011; Nikitin et al. 2016), OCS (Bouanich et al. 1986; Golebiowski et al. 2014; Muller et al. 2005; Auwera & Fayt 2006; Sung et al. 2009; Toth et al. 2010; R'egalia-Jarlot et al. 2002), and N 2 O(Daumont et al. 2001). As pressure broadening due to H 2 is not available for these molecules, we only use thermal broadening. For DMS and CS 2 , we use the absorption cross sections provided directly by HITRAN (Kochanov et al. 2019; Sharpe et al. 2004; Gordon et al. 2017) at 1 bar and 298 K, following Madhusudhan et al. (2021). In addition to molecular cross-sections, we also consider H 2 -H 2 and H 2 -He collision-induced absorption \n(Borysow et al. 1988; Orton et al. 2007; Abel et al. 2011; Richard et al. 2012).", '3.1. Retrieval Setup': 'Our canonical model comprises of 22 free parameters overall: 11 corresponding to the individual mixing ratios of the above chemical species, 6 for the P -T profile, 4 for the clouds/hazes and 1 for the reference pressure P ref , defined as the pressure at a fixed planetary radius of 2.61 R ⊕ . The Bayesian inference and parameter estimation is conducted using the MultiNest nested sampling algorithm (Feroz et al. 2009) implemented through PyMultiNest (Buchner et al. 2014). The retrieval setup and priors on the model parameters are similar to those in recent implementations of the AURA retrieval framework (Madhusudhan et al. 2020; Constantinou & Madhusudhan 2022) and are shown in Appendix C. We also consider variations to the canonical model, including a cloud/haze-free atmosphere, Mie scattering due to hazes, and the presence of stellar heterogeneities influencing the spectrum, as discussed below. We use the present JWST NIRISS and NIRSpec transmission spectra of K2-18 b in the 0.9 - 2.8 µ m and 2.8 - 5.2 µ m ranges, respectively, at their native resolution for the retrieval. As discussed in section 2.4, we derive two spectra for NIRSpec corresponding to the two trends (linear and quadratic) we use to model the baseline flux; the NIRISS spectrum has a robust preference for a linear trend. Furthermore, the NIRSpec G395H grating uses two detectors (NRS1 and NRS2) and recent studies have suggested the possibility of an offset in the spectrum derived from a given detector (Moran et al. 2023). Therefore, we conducted retrievals on different combinations of NIRSpec spectra obtained with different trends and offsets. \nWe consider two broad combinations of data: (1) NIRISS and linear-trend NIRSpec spectra, and (2) NIRISS and quadratic-trend NIRSpec spectra. For each combination, we consider a range of different spectral offsets as free parameters in the retrieval. We consider four cases: a baseline case with no offsets, one combined offset for the NIRSpec spectrum, one offset for the NIRISS spectrum, and two separate offsets for the NIRSpec NRS1 and NRS2 spectra. The last case is the most conservative and is motivated by the transit depth offset between NRS1 and NRS2 recently reported by Moran et al. (2023). The offset on either NIRISS or NIRSpec represents cases where we assume no offset between NIRSpec NRS1 and NRS2 (Alderson et al. 2023; Lustig-Yaeger et al. 2023; August et al. 2023) but instead allow for an offset between NIRISS and NIRSpec as a whole. In the no-offset case, we consider the data \nas is and do not perform any offsets. We conduct the four retrievals for each data combination and assess their relative Bayesian evidence. \nBy comparing the Bayesian evidence of the considered cases, we find that generally some offset is preferred over no offsets. For the linear-trend data, a single offset between the data sets (on either NIRISS or NIRSpec) is the most preferred. The two one-offset cases are comparable, with an offset on NIRSpec being marginally favoured over that on NIRISS. For the quadratic-trend data, the two-offset case is the most favoured. The fact that the configuration with separate offsets for NRS1 and NRS2 is strongly preferred for the quadratic trend while not being favoured for the linear trend suggests that, for the present observation, using a quadratic trend contributes to an offset between NRS1 and NRS2. Based on these findings, from across the different combinations we finally select three nominal cases for NIRSpec along with NIRISS: (a) NIRSpec with a linear trend and no offsets, representing the data without modification, (b) NIRSpec with a linear trend and one offset, and (c) NIRSpec with a quadratic trend and two separate offsets for NRS1 and NRS2, representing the most conservative case. These are the retrieval cases considered in the rest of this work. In what follows we report the atmospheric properties at the day-night terminator region of K2-18 b retrieved using the transmission spectrum in each of these cases, as shown in Tables 2 and 3.', '3.2. Prominent CNO Molecules': "Our atmospheric retrieval provides important constraints on the dominant CNO molecules expected in H 2 -rich atmospheres. The retrieved spectral fit is shown in Fig. 3, and the corresponding posterior distributions for several molecules are shown in Fig. 4. Amongst the prominent CNO molecules, we find strong spectral contributions from CH 4 and CO 2 in a H 2 -rich atmosphere. For our retrieval with no offset, we derive log volume mixing ratios of log(X CH 4 ) = -2 . 04 +0 . 61 -0 . 72 and log(X CO 2 ) = -1 . 75 +0 . 45 -1 . 03 . For the one-offset case, we obtain log(X CH 4 ) = -1 . 74 +0 . 59 -0 . 69 and log(X CO 2 ) = -2 . 09 +0 . 51 -0 . 94 . As shown in Table 2, these abundance estimates for both molecules are consistent across the three retrieval cases, with median abundances of ∼ 1% and average uncertainties below 1-dex, underscoring the robustness of the derived estimates. Both CH 4 and CO 2 are detected for the first time in a sub-Neptune exoplanet and the precision of their abundance estimates is the best measured for any molecule in a sub-Neptune atmosphere to date. \nWe do not find significant contributions due to H 2 O or NH 3 , but find 95% upper limits of -3.21 for log(X H 2 O ) \nFigure 4. Retrieved posterior probability distributions for the mixing ratios of important molecules for the three retrieval cases described in section 3. The horizontal error bar in the top row denotes each distribution's median and corresponding 1 σ interval for the three molecules with significant detections. The arrows in the bottom row indicate 95% upper limits. We find strong evidence for CH 4 and CO 2 , at a significance of 5 σ and 3 σ , respectively. We find marginal evidence for DMS and no significant evidence for the remaining molecules. The abundance estimates and detection significances are shown in Table 2. \n<!-- image --> \nTable 2. Retrieved molecular abundances and detection significances of prominent molecules in the atmosphere of K2-18 b. The three canonical atmospheric model cases are described in section 3, and pertain to different considerations for offsets between data from different instruments. The molecular abundances are shown as log 10 of volume mixing ratios. The retrieved median and 1 σ estimates are given for CH 4 , CO 2 and DMS which show strong to marginal detections, and 95% upper-limits are given for the remaining molecules. The quantities in brackets for CH 4 , CO 2 and DMS show the detection significances greater than 1 σ . The detection significances have a nominal statistical uncertainty of ∼ 0.1 σ due to the uncertainty on the Bayesian evidence estimated by the nested sampling algorithm. See section 3.4. \nand -4.46 for log(X NH 3 ) in the no-offset case. These upper limits are also consistent with those from the other retrieval cases, as shown in Table 2. The nondetections of both molecules are important considering their strong spectral features and detectability expected in the 0.95.2 µ m range (Madhusudhan et al. 2021; Constantinou & Madhusudhan 2022). The nondetection of H 2 O is at odds with its previous inference using the HST WFC3 spectrum in the 1.1-1.7 µ m range (Tsiaras et al. 2019; \nBenneke et al. 2019a; Madhusudhan et al. 2020). A strong degeneracy between H 2 O and CH 4 in the HST WFC3 band was noted previously (Blain et al. 2021; B'ezard et al. 2022). Our retrieved CH 4 abundance is consistent with previous predictions of stronger absorption due to CH 4 relative to H 2 O in the HST WFC3 band (Blain et al. 2021; B'ezard et al. 2022) and some upper bounds on the CH 4 abundance (Madhusudhan et al. 2021; Blain et al. 2021; B'ezard et al. 2022). Our \nTable 3. Retrieved temperature, cloud/haze properties and reference pressure for the atmosphere of K2-18 b, as well as retrieved dataset offsets where applicable. Similarly to Table 2, the three canonical atmospheric model cases are described in section 3, which pertain to different considerations for offsets (denoted OS above) between data from different instruments. The 1-offset retrieval considers a shift to the NIRSpec observations relative to NIRISS, while the 2-offset retrieval considers two shifts, applied to observations from the NIRSpec NRS1 and NRS2 detectors. The temperature constraints shown are at 10 mbar, which corresponds to the observed photosphere. In all cases, the best-fit reduced χ 2 ν is close to unity, with the degrees of freedom being 4389, 4388, and 4387 for no, 1, and 2 offsets, respectively. \nresults, therefore, resolve the degeneracy in the atmospheric composition of K2-18 b from previous observations. \nWhile our NIRISS spectrum is generally in good agreement with the previous HST WFC3 spectrum (Benneke et al. 2019a) in the 1.1-1.7 µ m range, as shown in Fig. 6, there is notable difference in two of the data points at the blue end of the WFC3 spectrum. A comparison between the new JWST spectrum and the HST spectrum is presented in Appendix A. Furthermore, the presence of multiple CH 4 features across our NIRISS and NIRSpec spectral range provides a very strong detection of CH 4 , as discussed in section 3.4. We note that our upper limit for H 2 O corresponds to the planet's stratosphere at pressures below ∼ 100 mbar. Water vapor may very well be abundant at deeper levels in the atmosphere, but condensation of H 2 O is expected in the upper troposphere of this temperate planet (Benneke et al. 2019a; Madhusudhan et al. 2023), resulting in a comparatively dry stratosphere, as on Earth. \nWe also do not detect CO or HCN despite their strong spectral features expected in the 0.9-5.2 µ mrange (Madhusudhan et al. 2023). The 95% upper limits for both molecules are shown in Table 2, with maximum values of -3.00 for log(X CO ) and -2.41 for log(X HCN ). Given the low-temperature H 2 -rich atmosphere, the nondetection of CO is not necessarily surprising, as CH 4 is expected to be the dominant equilibrium constituent in deep H 2 atmospheres on cooler planets (Moses et al. 2013). However, some CO is expected to be present from disequilibrium quenching in deep atmospheres or photochemistry at high altitudes, becoming especially important in thinner atmospheres (Yu et al. 2021; Tsai et al. 2021; Hu et al. 2021; Madhusudhan et al. 2023). The high abundances of CO 2 and CH 4 , along with the nondetection of NH 3 and CO, and a high CO 2 /CO ratio, are consistent with predictions for an ocean surface under a thin H 2 -rich atmosphere (Hu et al. 2021; Madhusudhan et al. 2023), as discussed further in section 4.", '3.3. Biosignature Molecules': 'The retrievals provide notable constraints on two methyl-group terrestrial biomarkers, DMS and CH 3 Cl, predicted to be detectable in Hycean atmospheres, especially for K2-18 b (Madhusudhan et al. 2021). We retrieve a DMS mixing ratio of log(X DMS ) = -4 . 46 +0 . 77 -0 . 88 in the no-offset case, -6 . 35 +1 . 59 -3 . 60 in the one-offset case, and -6 . 87 +1 . 87 -3 . 25 in the two-offset case. The weaker constraints on the DMS abundance with increasing number of offsets between the detectors are due to the DMS spectral feature being broad and the continuum level spanning multiple detectors, as discussed in section 3.4. The potential inference of DMS is of high importance as it is known to be a robust biomarker on Earth and has been extensively advocated to be a promising biomarker for exoplanets (Seager et al. 2013; Seager et al. 2016; Catling et al. 2018; Madhusudhan et al. 2021); this is discussed further in section 4. We also find a nominal peak in the posterior distribution of CH 3 Cl that is more significant than other nondetections such as those of H 2 O or NH 3 , as shown in Fig. 4. The retrieval in the no-offset case provides an abundance estimate of log(X CH 3 Cl ) = -6 . 62 +3 . 08 -3 . 40 and a 95% upper limit of -2.50, which are comparable within ∼ 1-dex to those from the other retrieval cases. \nWe note that CH 4 , DMS, and CH 3 Cl all have strong spectral features owing to the C-H bond in the 3-3.5 µ m range in the NIRSpec G395H band and are therefore degenerate to some extent, as shown in Fig. 5. However, thanks to the multiple strong CH 4 features in the NIRISS band, the degeneracies between the two stronger molecules, CH 4 and DMS, are somewhat mitigated, whereas CH 3 Cl is relatively unconstrained.', '3.4. Molecular Detection Significances': 'In addition to the abundance constraints discussed above, we determine the detection significances of the key molecules using Bayesian model comparisons (Trotta 2008; Benneke & Seager 2013; Pinhas et al. 2018). In the present context, we evaluate the detection \nsignificance of a molecule as a Bayesian preference for a model fit to the data while including that molecule, relative to the same model with the molecule absent (Benneke & Seager 2013; Pinhas et al. 2018). Naturally, such an evidence comparison depends to some extent on the specific model considerations and the combination of the data used, which are discussed in section 3.1. Therefore, we estimate detection significances for the prominent molecules for each of the three retrieval cases, as shown in Table 2. We note that the detection significances reported here have intrinsic statistical uncertainties of ∼ 0.1 σ owing to the uncertainty in Bayesian evidence obtained by the nested sampling algorithm for a given retrieval. \nAmong the prominent CNO molecules, we find the strongest detection for CH 4 at 4.7-5.0 σ across all three cases. The consistently high detection significance value independent of the offset(s) considered underscores the robustness of the detection, which is due to multiple features of CH 4 being present across the 1-5 µ m range of the observed spectrum as shown in Fig. 3. We also detect CO 2 robustly at ∼ 3 σ significance across all three cases. The strong detection of CO 2 is made possible by its prominent spectral feature around 4.3 µ mand the full feature along with the spectral baseline around it being on the same detector (NRS2) in the NIRSpec band. As discussed above, we do not find significant evidence for NH 3 , H 2 O, CO or HCN. \nAmong the biomarkers, we find some evidence for DMS depending on the retrieval case. The detection significance of DMS depends on the offsets considered. This is because, while DMS has a strong spectral feature around 3.3 µ m in the NIRSpec NRS1 detector, the feature is broad and the spectral baseline falls on neighbouring detectors - the NIRISS at shorter wavelengths and NIRSpec NRS2 at longer wavelengths. Therefore, the spectral amplitude and hence the detection significance and abundance estimate rely strongly on the relative offsets between the detectors; the detection significance lowers for each additional offset in the retrieval. We infer DMS at 2.4 σ confidence for the no-offset case but at only ∼ 1 σ for the one-offset case and no significant evidence for the two-offset case. Nevertheless, as shown in Fig. 4, in all three cases the retrieved posterior distributions for DMS show notable peaks within 1-dex of each other, except that for the cases with offsets the distributions contain long low-abundance tails owing to the degeneracy with the spectral baseline as discussed above. The posteriors are also notably different from the nondetections for other prominent molecules, such as H 2 O, NH 3 or HCN, and indicate a nonnegligible chance for DMS being present in the atmosphere. Upcoming \nobservations will be able to further constrain the presence of DMS, as discussed below and in section 4. \nThere could also be potential contributions from CH 3 Cl to the spectrum, albeit without any appreciable detection significance, as evident from Fig. 4. CH 3 Cl, being a methylated molecule like DMS, has some overlapping features with DMS as shown in Fig. 5. Therefore, its significance increases marginally in the absence of DMS. While we do not detect CH 3 Cl on its own in any of the retrieval cases, the combination of DMS and CH 3 Cl has a slightly higher detection significance of 2.7 σ than DMS alone (2.4 σ ) in the no-offset case. We find no significant evidence for any other biomarkers considered in the retrievals. \nOverall, we find CH 4 and CO 2 to be our most confident detections, followed by DMS, with the abundance estimates reported above. While our results provide important first insights into the chemical composition of K2-18 b, upcoming observations will be able to verify our present findings. These include observations of the transmission spectrum of K2-18 b with JWST MIRI between ∼ 5 and 10 µ m (JWST Program GO 2722; PI: N. Madhusudhan) and more observations with JWST NIRSpec G395H and G235H (JWST Program GO 2372, PI: R. Hu).', '3.5. Clouds/Hazes and Photospheric Temperature': 'The observed transmission spectrum provides nominal constraints on the presence of clouds/hazes in the atmosphere. The constraints on the cloud/haze parameters are shown in Table 3. The constraints on the cloud-top pressure for the gray clouds are relatively weak, mostly lying below the observable photosphere (e.g. cloud-top pressures ≳ 100 mbar). Even though the scattering slope ( γ ) is not well constrained, the enhancement factors ( a ) are generally higher than that expected for H 2 Rayleigh scattering ( a =1), albeit still consistent with the latter at the 3 σ uncertainties for the no-offset case. The haze coverage fraction at the day-night terminator region is constrained to ∼ 0.6, albeit with large uncertainties of ∼ 0.2. Based on Bayesian model comparisons as discussed above, we find that a model with clouds/hazes is preferred over a model without clouds/hazes by 2.8-3.2 σ across the three retrieval cases considered. However, more observations in the optical to near-infrared wavelengths would be needed to provide stronger constraints on the cloud/haze properties of K2-18 b, as discussed in Appendix B. We find that the abundance constraints on the molecules are consistent within the 1σ uncertainties between the retrievals with and without clouds/hazes. \nWe additionally consider retrievals in which the parametric clouds/hazes of the canonical model are replaced \nwith Mie scattering hazes, as described by Pinhas & Madhusudhan (2017) and Constantinou et al. (2023). We specifically include two forms of organic haze, using optical constants presented by Khare et al. (1984) and He et al. (2023). We find no evidence for this model, and a Bayesian model comparison shows a preference in favour of the parametric clouds/hazes considered in our canonical model. While the haze properties are unconstrained in this retrieval, the abundance constraints on the gaseous species remain consistent with those from our canonical retrieval cases. \nThe observations provide nominal constraints on the temperature in the planetary photosphere. We find the temperature at 10 mbar to range between 235 +78 -56 and 257 +127 -74 among the three cases, as shown in Table 3. We note that the retrieved temperature structure is typically less well constrained using transmission spectroscopy, compared to emission spectroscopy (Madhusudhan et al. 2016). Nevertheless, the retrieved temperature range and the nondetection of H 2 O allow for the possibility of H 2 O clouds in the deeper atmosphere. Considering the pressures probed by the spectral features across our observed range, we find that the photosphere, i.e. the τ =1 surface, lies between pressures of ∼ 0.1-100 mbar. The 10 mbar temperature estimates for the three retrieval cases are shown in Table 3. While the H 2 -rich atmosphere can result in a significant greenhouse effect and warm the ocean surface, clouds and/or hazes play a crucial role in cooling the atmosphere and decreasing the temperature gradient (Madhusudhan et al. 2020, 2021; Madhusudhan et al. 2023; Piette & Madhusudhan 2020). The possible presence of clouds can allow more temperate conditions at the ocean surface compared to those predicted by cloud-free models (see Innes et al. 2023).', '3.6. Stellar Heterogeneities': 'We also consider retrievals including the effects of unocculted stellar heterogeneities on the transmission spectrum, using our AURA retrieval framework (Pinhas et al. 2018). We do not find significant evidence for such effects in any of the three retrieval cases. The spot covering fraction in our retrieval is consistent with zero at the 2 σ uncertainties, and a model with stellar heterogeneity is not favored over a model without it across the three cases. We also find that the abundance constraints on the molecules are not significantly affected by the consideration of stellar heterogeneities in the retrieval. The nondetection of H 2 O is further evidence against the effects of unocculted stellar heterogeneities on the present transmission spectrum, given that H 2 O in cool unocculted starspots may contaminate the spec- \nrum, as recently reported (Barclay et al. 2021; Moran et al. 2023).', '4. SUMMARY AND DISCUSSION': 'We report a transmission spectrum of the candidate Hycean exoplanet K2-18 b observed with JWST. The spectrum observed with the JWST NIRISS and NIRSpec instruments spans the 0.9-5.2 µ m range containing strong absorption features of prominent CNO molecules and biomarkers predicted for Hycean worlds. We report strong detections of CH 4 and CO 2 in a H 2 -rich atmosphere at 5 σ and 3 σ confidence, respectively, with high volume mixing ratios ( ∼ 1%) for both. However, we do not detect H 2 O, NH 3 , CO or HCN, while obtaining upper limits on their abundances that are consistent with chemical expectations for an ocean under a cold and thin H 2 -rich atmosphere (Hu et al. 2021; Madhusudhan et al. 2023). We also find potential evidence for DMS, which has been predicted as a robust biomarker in both terrestrial and Hycean worlds. These findings support the Hycean nature of K2-18 b and the potential for biological activity on the planet. \nThe observed mass, radius, and equilibrium temperature of K2-18 b have been known to be consistent with a degenerate set of internal structures (Madhusudhan et al. 2020; Madhusudhan et al. 2023). These include (a) a Hycean world with a thin H 2 -rich atmosphere over a water-rich interior, (b) a mini-Neptune with a deep H 2 -rich atmosphere, or (c) a predominantly rocky superEarth interior with a deep H 2 -rich atmosphere. Our retrieved chemical composition of the atmosphere of K218 b helps distinguish between these scenarios. In what follows, we discuss the implications and possible explanations of our findings and future directions.', '4.1. A Potential Hycean World': "K2-18 b has been originally predicted to be the archetype of a Hycean world (Madhusudhan et al. 2021), one with habitable oceans underneath a H 2 -rich atmosphere. The currently derived chemical composition of the atmosphere is in agreement with previous theoretical predictions for the presence of an ocean under a shallow H 2 -rich atmosphere (Hu et al. 2021; Madhusudhan et al. 2023). In particular, Hu et al. (2021) predicted the abundance of CO 2 to range between 4 × 10 -4 and 10 -1 and that of CH 4 to range between 1.5-5.3 × 10 -2 , which are in agreement with our retrieved abundances. They also predicted relatively lower abundances of CO, NH 3 and stratospheric H 2 O, which are consistent with our nondetections. A low retrieved H 2 O gas-phase mixing ratio at pressures less than ∼ 100 mbar is consistent with condensation due to a tropospheric cold trap (Madhusudhan et al. 2023), as in Earth's stratosphere, and \nFigure 5. Spectral contributions of key molecular species in the 1-5 µ mrange. The different curves show individual contributions from different molecules to a nominal model transmission spectrum of K2-18 b shown in black and denoted as 'Combined'. The model assumes a mixing ratio of 10 -2 for CH 4 and CO 2 , 10 -4 for H 2 O, and 10 -5 for all the other species, consistent with our retrieval estimates discussed in section 3, and an isothermal temperature profile of 250 K. Each curve corresponds to a transmission spectrum with opacity contributions from a single molecule at a time, in addition to H 2 -H 2 and H 2 -He collisioninduced absorption. The spectral ranges of our JWST NIRISS and NIRSpec observations are also indicated; the NIRSpec range spans two detectors (NRS1 and NRS2), with a gap between them at 3.72-3.82 µ m. \n<!-- image --> \nwith the retrieved thermal structure in this work and previous studies (e.g. Benneke et al. 2019a; Piette & Madhusudhan 2020; Madhusudhan et al. 2023). That is, H 2 O could be abundant below its condensation region in the atmosphere, but the transit observations of the terminator region do not probe deep enough to detect it. \nThe main argument against K2-18 b being a potential Hycean world is based on climate considerations a greenhouse effect in a thick H 2 -rich atmosphere that is cloud/haze-free would result in temperatures being sufficiently elevated at pressures greater than ∼ 10 bar, such that a liquid ocean would instead be converted to a steam-dominated atmosphere that ultimately goes supercritical at depth for irradiation levels relevant to K2-18 b (Scheucher et al. 2020; Piette & Madhusudhan 2020; Innes et al. 2023; Pierrehumbert 2023). Any global ocean surface must then reside at pressures less than ∼ 10 bar. However, too shallow a H 2 atmosphere could be subject to escape over time (e.g., Kubyshkina et al. 2018a,b; Hu et al. 2023), so there is a limited parameter range over which a habitable ocean could exist on K2-18 b without significant clouds/hazes. As mentioned previously, high-albedo tropospheric water clouds or scattering hazes can help alleviate the steam and supercritical water problem by reducing the stellar energy absorbed by the planet (Madhusudhan et al. 2021; Piette & Madhusudhan 2020). \nOur retrieved atmospheric temperatures and upper limits for H 2 O are consistent with the possibility that H 2 O is condensing into clouds below the photosphere in K2-18 b, indicating a cold upper troposphere (see also Benneke et al. 2019a; Madhusudhan et al. 2023). Furthermore, as discussed in section 3.5, our retrievals show some evidence for scattering due to hazes at the day-night terminator region of the atmosphere; however, more optical observations are required to robustly confirm the same. If such clouds or hazes enshroud the planet or, in particular, are present on the dayside, they could provide the required albedo to sustain a habitable ocean in K2-18 b. Given the possible greenhouse effects of clouds themselves, a relatively shallow H 2 atmosphere might still be required to maintain low-enough temperatures for a liquid ocean.", '4.2. Is a Deep Atmosphere a Possibility?': "Our derived atmospheric composition in combination with K2-18 b's irradiation level is seemingly inconsistent with a deep-atmosphere, mini-Neptune scenario. In that scenario any photochemically produced carbon and nitrogen species would be recycled in the deep atmosphere back to their thermodynamically stable forms, CH 4 and NH 3 , with transport then returning these molecules back to the observable upper regions of the atmosphere (Yu et al. 2021; Tsai et al. 2021; Madhusudhan et al. 2023). Although CO 2 can have a substantial mixing ratio in a H 2 -rich atmosphere with a high metallicity and/or low C/O ratio (e.g., Moses et al. 2013), we find \nthat the simultaneous presence of ∼ 1% CO 2 and CH 4 in a thick H 2 -rich atmosphere requires a moderately high C/H metallicity, a very low C/O ratio, ( ∼ 0.02) and efficient vertical quenching. For example, C/H = 30 × solar, O/H = 690 × solar, and K zz ≳ 10 7 cm 2 s -1 combined with a temperature profile with T int = 60 K provides CH 4 and CO 2 mixing ratios of the right magnitude. Aside from the question of how such an oxygenenriched atmosphere would originate from a formation and evolution standpoint, this scenario would cause H 2 O to dominate over H 2 throughout much of the atmosphere below a few hundred mbar (increasing its mean molecular weight), and NH 3 and CO would also be present in nonnegligible quantities, none of which are consistent with the observations. \nNor can our observations be explained by a shallow, H 2 -rich atmosphere overlying a solid surface at the few-bar pressure level, despite the abundances of CH 4 and CO 2 being seemingly consistent with such models (Yu et al. 2021; Tsai et al. 2021; Madhusudhan et al. 2023). The planet's bulk density precludes a thin H 2 atmosphere overlying an extensive silicate mantle (Madhusudhan et al. 2020). Even a pure silicate interior would require a thick ( ≳ 10 3 bar) H 2 -rich envelope to explain the mass and radius. Although interaction of a thick H 2 atmosphere with a deep silicate mantle might explain the presence of some atmospheric CO 2 (e.g., Kite et al. 2020; Kite & Barnett 2020; Schlichting & Young 2022; Tian & Heng 2023), such models do not generally predict a resulting 1% CO 2 mixing ratio from high-pressure interaction with silicates at depth in a thick H 2 atmosphere, nor can that scenario explain the apparent lack of recycling of N 2 back to NH 3 . Overall, the bulk density of the planet, combined with our derived chemical composition for the atmosphere, presents strong evidence in support of K2-18 b as a Hycean world rather than a rocky or volatile-rich planet with a deep H 2 atmosphere, or a rocky planet with a thin H 2 atmosphere. \nOne caveat to the above discussion is that current photochemical models for mini-Neptune conditions assume ideal-gas behavior and do not consider how a primordial H 2 atmosphere with its expected reduced species might interact chemically with a supercritical water layer and/or silicate magma at depth in the atmosphere. Moreover, our understanding of the bulk composition and chemistry of super-Earth and mini-Neptune atmospheres is rudimentary at this stage, and the depleted NH 3 and CO could potentially have some chemical explanation that has not been considered up to this point. We also note that the retrieved abundances represent average chemical abundances in the observable photo- \nsphere ( ∼ 0.1-100 mbar) at the day-night terminator region. While the molecules with robust detections, CH 4 and CO 2 , have been predicted to be relatively uniform in this pressure range for K2-18 b in several cases (Hu et al. 2021; Yu et al. 2021; Madhusudhan et al. 2023) more precise observations in the future may be able to constrain nonuniform chemical abundances in the atmosphere.", '4.3. Possible Evidence of Life': "Our potential evidence for DMS in K2-18 b motivates consideration of possible biological activity on the planet. While the present evidence is not as strong as that for CH 4 or CO 2 , upcoming JWST observations of K2-18 b will be able to robustly constrain the presence and abundance of DMS, as discussed in section 4.5 and earlier work (Madhusudhan et al. 2021). Here we discuss the plausibility of our DMS abundance constraints from a potential biosphere on K2-18 b in order to inform future observations and retrieval studies. \nOn Earth, DMS is a by-product of living organisms, with the bulk of the nonanthropogenic DMS in the terrestrial atmosphere being emitted from phytoplankton in marine environments (Charlson et al. 1987; Barnes et al. 2006). Both DMS and CH 3 Cl are thought to be terrestrial biosignatures with no known false positives (Catling et al. 2018). On Earth these molecules are produced exclusively by life in relatively small quantities compared to more abundant by-products of life, such as O 2 , CH 4 and N 2 O; the latter are, therefore, more favoured as biosignatures for Earth-like planets. However, it has been suggested that in H 2 -rich environments with large biomass, molecules such as DMS and CH 3 Cl could be abundant and observable for habitable superEarths (Seager et al. 2013) and Hycean worlds (Madhusudhan et al. 2021). \nWhile we infer DMS with marginal confidence, our retrieved DMS abundance spans a relatively wide range across the cases considered, from log(X DMS ) = -4 . 46 +0 . 77 -0 . 88 at the higher end for the no-offset case to log(X DMS ) = -6 . 87 +1 . 87 -3 . 25 for the two-offset case. On Earth, the typical mixing ratios of DMS, found near the ocean surface, are a few hundred parts per trillion (Hopkins et al. 2023). DMS is rapidly depleted at higher altitudes, with a few-day lifetime, due to photochemical reactions with OH and other radicals, ultimately leading to the production of more oxidized sulfur molecules, such as SO 2 . The upper end of our retrieved abundance for DMS is significantly higher than that on Earth, and we do not detect other sulfur-bearing species; however, the lower end is more plausible (Seager et al. 2013). We note that the infrared absorption cross-sections of DMS \nare currently limited (Kochanov et al. 2019; Sharpe et al. 2004; Gordon et al. 2017). New cross-section data may revise our mixing ratio estimates, while future more intensive observations could be sensitive to other sulfur species. \nThe DMS abundance is also strongly dependent on the chromospheric activity of the host star. A quiescent Mdwarf host star with a lower ultraviolet flux compared to a sun-like star of the same bolometric flux could enable DMS to survive longer and be more abundant in the planetary atmosphere (Domagal-Goldman et al. 2011). Previous studies have predicted DMS mixing ratios of ∼ 10 -7 -10 -6 for Earth-like planets and super-Earths orbiting low-activity M dwarfs for plausible biomass estimates (Domagal-Goldman et al. 2011; Seager et al. 2013). However, K2-18 b has been described as a moderately active M3 dwarf (Benneke et al. 2019b), and it may not be quiescent in the extreme-ultraviolet (dos Santos et al. 2020). A high Lyman alpha flux could lead to DMS depletion due to interaction with atomic O produced from CO 2 photolysis (Seager et al. 2013). Reaction of DMS with atomic H is significantly slower than with O (see Atkinson et al. 2004; Zhang et al. 2005), and the OH abundance would be limited by H 2 O condensation in K2-18 b. \nIf the DMS abundance on K2-18 b is indeed confirmed by future observations to be greater than ∼ 10 -6 , that result could require very high biological production rates in the ocean and/or new theoretical developments in our understanding of DMS chemistry (including potential abiotic chemistry) in planets such as K2-18 b. We also note that while our retrievals included a wide array of molecules with strong spectral signatures in the observed wavelength range, future theoretical studies and retrievals could consider an even more expanded set, particularly as accurate cross-section data become available. In particular, other hydrocarbon molecules have similar C-H bands in the 3-3.5 µ m range to that of DMS. Therefore, besides our canonical model, we have explored retrievals with other hydrocarbons such as C 2 H 2 , C 2 H 6 , CH 3 OH, HC 3 N, and hazes with optical properties from He et al. (2023), which have been predicted to be relevant for Hycean atmospheres (e.g. Tsai et al. 2021; Madhusudhan et al. 2023), but we found no definitive evidence for their presence. \nBesides DMS, the presence of life could potentially also contribute to the strong chemical disequilibrium indicated by the retrieved atmospheric composition of K218 b. For instance, methanogenic bacteria in Earth's oceans are known to be a significant contributor to the atmospheric CH 4 budget. It is possible that similar biotic sources may to some extent contribute to the ob- \nserved CH 4 abundance in K2-18 b, if indeed life exists on the planet. \nOverall, our findings demonstrate the feasibility of detecting a biosignature molecule in the atmosphere of a habitable-zone sub-Neptune with JWST. This also provides a valuable case study for a framework for biosignature assessment in exoplanets (e.g. Catling et al. 2018; Meadows et al. 2022). The potential inference of DMS in K2-18 b provides a pathway toward the possible detection of life on an exoplanet with JWST and other current and upcoming large observational facilities. The next steps would involve both (a) more theoretical investigations to understand the possible atmospheric and interior processes at play and (b) more observations to verify the present findings and potentially discover other chemical species.", '4.4. Resolving the Missing Methane Problem': "Our strong detection of CH 4 at 5 σ resolves one of the longest-standing conundrums in exoplanet science - 'The Missing Methane Problem' (Stevenson et al. 2010; Madhusudhan & Seager 2011). Low-temperature molecules such as CH 4 and NH 3 are common in the solar system and are seen in the atmospheres of the giant planets (Karkoschka 1998; Encrenaz 2022; Atreya et al. 2018). These molecules are expected to be prominent carriers of carbon and nitrogen in H 2 -rich atmospheres at temperatures below ∼ 600 K, with H 2 O being the dominant oxygen carrier (Burrows & Sharp 1999; Lodders & Fegley 2002). However, no robust detection of CH 4 or NH 3 has been made in any exoplanetary atmosphere with temperatures below ∼ 800 K, despite atmospheric observations made for several such exoplanets with HST and Spitzer at wavelengths sensitive to these molecules, e.g. GJ 436 b (Stevenson et al. 2010; Knutson et al. 2014), GJ 3470 b (Benneke et al. 2019b), K2-18 b (Benneke et al. 2019a), all in the ∼ 300-800 K range (but see Blain et al. 2021). \nAtmospheres of temperate sub-Neptunes are expected to exhibit distinct nonequilibrium chemistry, just as in solar system planets. Several processes can cause chemical disequilibrium, including photochemistry, vertical mixing, and volcanic outgassing (e.g. Yung 1999). However, even strong nonequilibrium chemical mechanisms have difficulty explaining the missing CH 4 in temperate exoplanetary atmospheres (Line et al. 2011; Moses et al. 2013). This missing methane problem has, therefore, remained one of the central puzzles in the area of exoplanetary atmospheres in the pre-JWST era. The present detection of CH 4 in K2-18 b, therefore, demonstrates the detectability of CH 4 and potentially other hydrocarbons with JWST and opens a new era of atmo- \nspheric characterisation of temperate exoplanets in general. The detection suggests that other sub-Neptunes and giant exoplanets with H 2 -rich atmospheres at similar temperatures may also be conducive for detecting CH 4 . Such planets therefore represent important targets for homogeneous studies of carbon chemistry in exoplanetary atmospheres, enabling comparative studies with solar system giant planets.", '4.5. Future Directions': 'Our results demonstrate the potential of candidate Hycean worlds as optimal targets in the search for life on exoplanets. These findings motivate further observations and theoretical work to characterise in detail the atmospheric and potential surface conditions of K218 b and other candidate Hycean worlds (Madhusudhan et al. 2021). Several upcoming JWST observations of K2-18 b will be able to verify the present findings, in particular more observations with NIRSpec G395H (JWST GO 2372) and MIRI LRS (5-10 µ m) (JWST GO 2722). While the former program can confirm the present findings with higher precision, the latter can specifically confirm the presence of DMS which is expected to have a strong spectral feature around 7 µ m (e.g. Figure 7 of Seager et al. 2013). Such observations are also motivated for a number of other promising candidate Hycean worlds orbiting nearby M dwarfs that are even more favorable to observations than K2-18 b (Madhusudhan et al. 2021). \nOverall, the present results pave the way to a new era of atmospheric characterisation of habitable planets and biosignature detection with JWST. The observations also motivate a wide range of theoretical studies to understand in detail the physical, chemical, and biological conditions on Hycean worlds. Our findings present a first step toward the spectroscopic identification of life beyond the solar system and the assessment of our place in the Universe. \nAcknowledgements: This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope as part of Cycle 1 GO Program 2722 (PI: N. Madhusudhan). We thank NASA, ESA, CSA, STScI, everyone whose efforts have contributed to the JWST, and the exoplanet science community for the thriving current state of the field. This work is supported by research grants to N.M. from the UK Research and Innovation (UKRI) Frontier Grant (EP/X025179/1), the MERAC Foundation, Switzerland, and the UK Science and Technology Facilities \nCouncil (STFC) Center for Doctoral Training (CDT) in Data Intensive Science at the University of Cambridge (STFC grant No. ST/P006787/1). N.M. and M.H. acknowledge support from STFC and the MERAC Foundation toward the doctoral studies of M.H. N.M. thanks Tony Roman, Elena Manjavacas, Nestor Espinoza and Sara Kendrew at STScI for their help with planning our JWST observations. J.M. acknowledges support from JWST-GO-02722, which was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127. We thank the anonymous referees for their valuable comments on the manuscript. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the NASA Astrophysics Data System and the Python packages NUMPY , SCIPY , and MATPLOTLIB . \nThis work was performed using resources provided by the Cambridge Service for Data Driven Discovery operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1), and DiRAC funding from STFC (www.dirac.ac.uk). \nAuthor Contributions: N.M. conceived, planned and led the project. N.M. led the JWST proposal with contributions from S.C., S.S., A.P. and J.M. N.M. and S.S. planned the JWST observations. N.M., M.H. and S.S. conducted the data reduction and analyses. N.M. and S.C. conducted the atmospheric retrievals. N.M., J.M. and A.P. conducted the theoretical interpretation. N.M. led the writing of the manuscript with contributions and comments from all authors. \nData Availability: Some/all of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via doi:10.17909/3ds1-8z15. The transmission spectra of K2-18 b reported in this work are available on the Open Science Framework, doi:10.17605/OSF.IO/36DJH. \nFacilities: JWST (NIRISS and NIRSpec) \nAPPENDIX \nFigure 6. HST and JWST observations of K2-18 b. The black points show prior observations of K2-18 b obtained with HST WFC3 in the 1.1-1.7 µ m range, with data from eight transits, presented by Benneke et al. (2019a) using data from HST GO 13665 and GO 14682 programs (PI: B. Benneke). The orange and dark-red points show our JWST NIRISS and NIRSpec observations from one transit each. The data is binned for visual clarity to R ≈ 25 and R ≈ 55, respectively, as shown in Fig. 3. The dark-blue line denotes the median retrieved spectrum (one-offset case), while medium- and lighter-blue regions denote the 1 σ and 2 σ contours, respectively. The yellow points correspond to the median spectrum binned to match the JWST observations. Our JWST NIRISS spectrum is in agreement with the HST WFC3 spectrum for most of the common wavelength range except for two data points toward the blue end of the WFC3 band. \n<!-- image -->', 'A. COMPARISON WITH HST WFC3 OBSERVATIONS': "As discussed above, K2-18 b has been previously observed in the 1.1-1.7 µ m range with HST WFC3 (Tsiaras et al. 2019; Benneke et al. 2019a), with these observations leading to inferences of H 2 O being present in the planet's terminator atmosphere (Tsiaras et al. 2019; Benneke et al. 2019a; Madhusudhan et al. 2020) with a high upper limit on CH 4 (Madhusudhan et al. 2020). The spectrum has also been suggested to be explained by CH 4 or a combination of CH 4 and H 2 O, instead of H 2 O alone, due to the degeneracy between the two molecules in the WFC3 band (Blain et al. 2021). Fig. 6 overlays these prior observations presented by Benneke et al. (2019a) with our new JWST spectrum. It can be seen that the new NIRISS data presented in this work is in good agreement with the prior HST WFC3 data in general, with the notable exception of two data points in the blue end of the WFC3 band, which show a 2-3 σ deviation from the NIRISS observations and corresponding spectral fit. The two deviant WFC3 points are inconsistent with a CH 4 absorption peak in the retrieved spectral fit. As such, it is possible that prior inferences of H 2 O over CH 4 were affected by these two points.", 'B. COMPARISON WITH RETRIEVALS ASSUMING NO CLOUDS': 'As discussed in section 3.5 our retrievals with the canonical model provide nominal constraints on the properties of possible clouds/hazes. Here we show a retrieved spectral fit to our JWST transmission spectrum using a model with no clouds/hazes (Fig. 7). This is applied to the one-offset case discussed in section 3, with all other factors being the same as for the canonical retrieval with clouds/hazes included. While the Bayesian evidence is higher for the canonical model, as discussed in section 3.5, the no clouds/hazes model provides a reasonable fit (Fig. 7) to most of the observed spectrum. It differs from the canonical model fit (Fig. 3) only toward the blue end of the NIRISS spectrum. The retrieved abundances in the present case are also consistent with the canonical case within the 1 σ uncertainties. Therefore, further observations in the optical are needed to more robustly constrain the presence and properties of clouds/hazes in the atmosphere of K2-18 b.', 'C. BAYESIAN PRIORS FOR ATMOSPHERIC RETRIEVAL': 'Table 4 shows the Bayesian prior probability distributions used in the retrievals presented in this work. All but the last six parameters correspond to the canonical model described in section 3.1. δ Nirspec denotes the linear offset applied to the NIRSpec G395H data in the 1-offset case in parts-per-million, while δ NRS1 and δ NRS2 denote the separate offsets applied to observations from the NIRSpec NRS1 and NRS2 detectors, respectively, in parts-per-million. T phot , T het \nFigure 7. Retrieved spectral fit of the JWST transmission spectrum of K2-18b using a model without clouds/hazes in the one-offset case. Details of the figure are the same as for Fig. 3 except that the retrieval does not include clouds/hazes in the model. The retrieval produces a comparable fit to the cloudy case (Fig. 3) across most of the wavelength range, differing mostly towards the bluer end of the NIRISS band, as discussed in Appendix B. The NIRSpec spectrum is vertically offset by -30 ppm, corresponding to the median retrieved offset. \n<!-- image --> \nand f het are the photospheric temperature, starspot/faculae (i.e. heterogeneity) temperature, and coverage fraction, respectively, and are used in the unocculted stellar heterogeneities modelling discussed in section 3.6. U ( X,Y ) denotes a uniform probability distribution between X and Y, while N ( µ, σ 2 ) denotes a normal distribution of mean µ and variance σ 2 . For any additional chemical species considered, such as the hydrocarbons described in section 4.3, we use the same uniform mixing ratio priors as those shown in table 4.', 'REFERENCES': "Abel, M., Frommhold, L., Li, X., & Hunt, K. L. C. 2011, J. Phys. Chem. A, 115, 6805, doi: 10.1021/jp109441f Albert, L., Lafreni`ere, D., Doyon, R., et al. 2023, PASP, 135, 075001, doi: 10.1088/1538-3873/acd7a3 Alderson, L., Wakeford, H. R., Alam, M. 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2024MNRAS.534.1597S

Relic galaxies the oldest ultracompact massive galaxies UCMGs contain almost exclusively pristine stars formed during an intense star formation SF burst at high redshift. As such they allow us to study in detail the early mechanism of galaxy assembly in the Universe. Using the largest catalogue of spectroscopically confirmed UCMGs for which a degree of relicness DoR had been estimated the INSPIRE catalogue we investigate whether or not relics prefer dense environments. The objective of this study is to determine if the DoR which measures how extreme the SF history was and the surrounding environment are correlated. In order to achieve this goal we employ the AMICO galaxy cluster catalogue to compute the probability for a galaxy to be a member of a cluster and measure the local density around each UCMG using machine learningbased photometric redshifts. We find that UCMGs can reside both in clusters and in the field but objects with very low DoR inlineformulatexmath idTM0001 notationLaTeXlt 0.3texmathinlineformula i.e. a relatively extended SF history prefer underdense environments. We additionally report a correlation between the DoR and the distance from the cluster centre more extreme relics when located in clusters tend to occupy the more central regions of them. We finally outline potential evolution scenarios for UCMGs at different DoR to reconcile their presence in both clusters and field environments.

2024-10-01T00:00:00Z

['arXiv:2409.12288', '2024MNRAS.534.1597S', '10.1093/mnras/stae2185', '2024arXiv240912288S', '2024MNRAS.tmp.2152S', '10.48550/arXiv.2409.12288']

['Astrophysics - Astrophysics of Galaxies']

INSPIRE INvestigating Stellar Population In RElics VII. The local environment of ultracompact massive galaxies

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

https://arxiv.org/pdf/2409.12288.pdf

{'No Header': ', 1-12 (2024)', 'INSPIRE: INvestigating Stellar Population In RElics - VII. The local environment of ultra-compact massive galaxies': "Diana Scognamiglio 1 ★ , Chiara Spiniello 2 , 3 , Mario Radovich 4 , Crescenzo Tortora 3 , Nicola R. Napolitano 5 \nJohanna Hartke 15 , 16 , Ignacio Martín-Navarro 13 , 14 , Claudia Pulsoni \n, Rui Li 6 , Matteo Maturi 7 , Michalina Maksymowicz-Maciata 2 , Michele Cappellari 2 , Magda Arnaboldi 8 , Davide Bevacqua 9 , 10 , Lodovico Coccato 8 , Giuseppe D'Ago 11 , 3 , Hai-Cheng Feng 12 , Anna Ferré-Mateu 13 , 14 17 \n- 1 Jet Propulsion Laboratory, California Institute of Technology, 4800, Oak Grove Drive - Pasadena, CA 91109, USA\n- 2 Sub-Dep. of Astrophysics, Dep. of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom\n- 3 INAF - Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131, Naples, Italy\n- 4 INAF - Osservatorio astronomico di Padova, Vicolo Osservatorio 5, I-35122 Padova, Italy\n- 5 Department of Physics 'E. Pancini' University of Naples Federico II C.U. di Monte Sant'Angelo Via Cintia ed. 6, 80126 Naples, Italy\n- 6 School of Physics, Zhengzhou University, Zhengzhou, 450001, China\n- 7 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany\n- 8 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748, Garching, Germany\n- 9 INAF - Osservatorio Astronomico di Brera, via Brera 28, 20121 Milano, Italy\n- 10 DiSAT, Universitá degli Studi dell'Insubria, via Valleggio 11, I-22100 Como, Italy\n- 11 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom\n- 12 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, Yunnan, People's Republic of China\n- 13 Instituto de Astrofísica de Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain\n- 14 Departamento de Astrofisica, Universidad de La Laguna, E-38200, La Laguna, Tenerife, Spain\n- 15 Finnish Centre for Astronomy with ESO (FINCA), FI-20014 University of Turku, Finland\n- 16 Tuorla Observatory, Department of Physics and Astronomy, FI-20014 University of Turku, Finland\n- 17 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': "Relic galaxies, the oldest ultra-compact massive galaxies (UCMGs), contain almost exclusively 'pristine' stars formed during an intense star formation (SF) burst at high redshift. As such, they allow us to study in detail the early mechanism of galaxy assembly in the Universe. Using the largest catalogue of spectroscopically confirmed UCMGs for which a degree of relicness (DoR) had been estimated, the INSPIRE catalogue, we investigate whether or not relics prefer dense environments. The objective of this study is to determine if the DoR, which measures how extreme the SF history was, and the surrounding environment are correlated. In order to achieve this goal, we employ the AMICO galaxy cluster catalogue to compute the probability for a galaxy to be a member of a cluster, and measure the local density around each UCMG using machine learning-based photometric redshifts. We find that UCMGs can reside both in clusters and in the field, but objects with very low DoR ( < 0 . 3, i.e., a relatively extended SF history) prefer under-dense environments. We additionally report a correlation between the DoR and the distance from the cluster centre: more extreme relics, when located in clusters, tend to occupy the more central regions of them. We finally outline potential evolution scenarios for UCMGs at different DoR to reconcile their presence in both clusters and field environments. \nKey words: Galaxies: evolution - Galaxies: formation - Galaxies: elliptical and lenticular, cD - Galaxies: stellar content Galaxies: star formation", '1 INTRODUCTION': "In the Λ -Cold Dark Matter ( Λ CDM) formation scenario, the formation of massive early-type galaxies (ETGs) is consistent with a two-stage formation model (Oser et al. 2010; Naab et al. 2014). In the first stage (at 𝑧 > 2), the central 'bulk' of mass is formed via \nan intense and very fast starburst that quickly ends leaving an ultracompact quiescent galaxy, known as red nugget (Damjanov et al. 2009, 2015a). In a subsequent accretion phase, which is much more extended in time, mergers and gas inflows cause a dramatic size growth but only minor mass change (Daddi et al. 2005; Trujillo et al. 2007; van Dokkum 2008). Although the accreted material is preferentially assembled on the outskirts of a red nugget, it nevertheless contaminates, along the line-of-sight, the 'in-situ', pristine compo- \n, \nnent that encodes the information about high𝑧 baryonic processes, thus affecting its spatial and orbital distributions. \nThe stochastic nature of merging processes suggests that a nonnegligible number of red nuggets at low𝑧 should exist, having slipped through cosmic time without interacting with other systems, thus not changing their stellar populations. These very old, red and ultracompact nearby systems are called relic galaxies (Trujillo et al. 2009, 2014; Ferré-Mateu et al. 2017), as they still bear the memory of the early conditions in which they formed. Therefore, relics offer unprecedented insights into the high𝑧 processes shaping galaxy formation and mass assembly with high precision, comparable to the study of nearby galaxies. Moreover, the number density of relics and its time evolution strongly depends on the physical processes shaping the size and mass evolution of galaxies, e.g. major and minor galaxy mergers and their relative importance, adiabatic expansion driven by stellar mass loss and/or strong feedback (Quilis & Trujillo 2013; Furlong et al. 2015; Wellons et al. 2016; Flores-Freitas et al. 2022; Moura et al. 2024). Hence, finding and precisely counting relics in redshift bins is a very valuable way to constrain the physical scenarios driving the formation and size-evolution of massive ETGs. \nIt is widely acknowledged that the properties of a galaxy are significantly influenced by its surrounding environment. The past merger history should manifest in the size of ETGs, with high-density environments favouring rapid growth through dry merging (Nipoti et al. 2009; van Dokkum et al. 2010). However, studies have yielded mixed results regarding the correlation between ETG size and environment, both at intermediate redshifts and in the local Universe (Kaviraj et al. 2011; Huertas-Company et al. 2013; Cappellari 2013; Hou & Wang 2016). When restricting to UCMGs, without age and star formation histories (SFHs) distinction (i.e., relics and non-relics) the situation is unclear, with some works finding a higher number density in clusters than in the field (Poggianti et al. 2013; Stringer et al. 2015). However, this might be attributed to the fact that the majority of these studies have focused on massive objects which are expected to be more common in denser environments at any size. For instance, Tortora et al. (2020) have performed a statistical analysis of the local environment of photometrically selected ultra-compact ( R e < 1 . 5 kpc) and massive ( 𝑀 ★ > 8 × 10 10 𝑀 ⊙ ) galaxies compared to normal-sized galaxies of similar stellar masses and colours. They have shown that the number density of UCMGs is higher in clusters only because the parent population they are derived from, i.e., red and massive ETGs, are more frequently found in these dense environments. This is also consistent to what is reported in Damjanov et al. (2015b). However, Tortora et al. (2020) have also found that the fraction of UCMGs, calculated with respect to the total parent population in the field is slightly higher compared to that in clusters (see right panel of fig. 1 in Tortora et al. 2020). \nRelic galaxies, i.e., the oldest UCMGs, containing almost exclusively stars formed during the first phase of the formation scenario, intuitively could be expected to be found in low-density environments with less hot gas in the intra-cluster medium (ICM). However, they have been observed also in clusters both in the local Universe (Ferré-Mateu et al. 2017, hereafter FM17) and up to 𝑧 ∼ 0 . 7 (Siudek et al. 2023). Hydro-dynamical simulations have reached very similar conclusions, with relics being identified in clusters and field environments (Peralta de Arriba et al. 2016; Flores-Freitas et al. 2022; Kimmig et al. 2023; Moura et al. 2024). Moreover, both FloresFreitas et al. (2022) and Moura et al. (2024), analysing relics in the Illustris TNG50 simulation, have concluded that at 𝑧 = 0 they are closely connected to the environment in which their progenitors evolved. In particular, the progenitors of relics have been found to live in consistently higher density environments already at 𝑧 ≥ 2, while \nyounger UCMGs residing in clusters were brought to them at a later cosmic time. However, Kimmig et al. (2023), analysing 36 quenched galaxies of stellar mass larger than 3 × 10 10 M ⊙ at 𝑧 = 3 . 42 from the Magneticum Pathfinder simulations, reached an opposite conclusion. They have found that these objects do not inhabit the densest nodes of the cosmic web, but rather sit in local under-densities. \nFromanobservation point of view, FM17 have hinted for a possible correlation between the environment and structural, kinematics and stellar population parameters. Among the three local massive relics analysed, one was found in the field, one in a small group, and one in a cluster. Interestingly, the structural, dynamical, and stellar population properties (size, mass, and SFH) seem to be more extreme for the relic in the centre of a large cluster (NGC 1277), intermediate for the relic living in the outskirts of a small group of galaxies (PGC 032873), and less extreme for the one in isolation (Mrk 1216). This result is however based on only three objects. Now, leveraging the large dataset built by INSPIRE (Spiniello et al. 2021a), we can extend this investigation, also pushing the redshift boundaries outside the local Universe. \nUnderstanding whether systematic differences exist between UCMGs with different SFHs and living in different environments is fundamental to shed light on their origin and evolution. This is the primary objective of this study, which is the seventh of the INSPIRE series. In particular, we aim to determine the potential correlation between the local environment of UCMGs and the ' degree of relicness ' (DoR), qualitatively introduced in FM17 and quantified in Spiniello et al. (2024), hereafter INSPIRE DR3. We define the environment based on local galaxy density through two distinct methodologies. Firstly, a cluster search is conducted utilising the Adaptive Matched Identifier of Clustered Objects (AMICO; Bellagamba et al. 2018; Maturi et al. 2019). Secondly, the Galaxy morphoto-Z with neural Networks (GaZNets; Li et al. 2022) is employed for precise determination of photometric redshifts and galaxy identification. Subsequent analysis explores the correlation between the environment and the DoR. \nThe paper is organised as follows. In section 2, we begin by presenting the data used, starting with a brief summary of the INSPIRE Survey and its dataset, followed by a description of the two catalogues employed to identify clusters, the AMICO galaxy cluster sample and the GaZNets catalogue. In section 3, the main analysis to measure the density of the local environment for the UMCGs is outlined. Section 4 presents our main findings, delving into the characterisation of the local environment and its correlation with the DoR. We discuss the results in Section 5, also trying to relate them to possible formation scenarios. Finally, Section 6 provides a summary and conclusion for the paper.", '2 DATA': "Thanks to data collected as part of the ESO Large Observational program(ID:1104.B-0370,October2019-March2023,PI:C.Spiniello), the INSPIRE project (Spiniello et al. 2021a) has built the first catalogue of spectroscopically confirmed relics outside the local Universe (0 . 1 < 𝑧 < 0 . 5), characterising their kinematics (D'Ago et al. 2023), stellar populations (Spiniello et al. 2021b, 2024), and lowmass end of the Initial Mass Function (IMF; Martín-Navarro et al. 2023; Maksymowicz-Maciata et al. 2024, hereafter MM24). \nHere, we use the final INSPIRE catalogue, presented in INSPIRE DR3.It comprises 52 UCMGs that were originally identified from the Kilo Degree Survey (KiDS; Kuijken 2011) DR3 footprint (de Jong et al. 2017) via a dedicated campaign (Tortora et al. 2016, 2018; \nScognamiglio et al. 2020). Among these, 38 have been confirmed as relics, as they have formed more than 75% of their stellar mass during the first phase of the formation scenario (at 𝑧 > 2). Moreover, as introduced earlier, for each of the 52 UCMGs a DoR has been computed. This is a dimensionless parameter, ranging from 0 to 1 and defined as \nDoR = GLYPH<20> 𝑓 𝑀 ★ tBB = 3 + 0 . 5Gyr 𝑡 75 + 0 . 7Gyr + ( 𝑡 Uni -𝑡 fin ) 𝑡 Uni GLYPH<21> × 1 3 , (1) \nwhere 𝑓 𝑀 ★ tBB = 3 is the fraction of stellar mass formed by 𝑧 = 2, 𝑡 75 is the cosmic time at which a galaxy has assembled 75% of its mass, 𝑡 fin is the final assembly time, when 100% of the stellar mass is in place. Finally, 𝑡 Uni is the age of the Universe at the redshift of the object. Essentially, a higher DoR indicates an earlier and more rapid mass assembly, with the most extreme relics, that have formed the entire totality of their stellar masses at 𝑧 > 2, approaching a value of 1 1 . Conversely, UCMGs with a DoR of 0 have likely undergone a very prolonged star formation (SF), and have just stopped forming stars. \nThe INSPIRE UCMGs span a wide range of DoR, from 0.06 to 0.83, although having very similar sizes, 0 . 5 ≤ R e ≤ 1 . 7 kpc, stellar masses, 0 . 64 × 10 11 ≤ 𝑀 ★ ≤ 2 . 71 × 10 11 𝑀 ⊙ , and colours, 1 . 8 ≤ ( 𝑔 -𝑖 ) ≤ 2 . 3. The DoR has allowed to split the 52 INSPIRE UCMGs in three main families: extreme relics (DoR > 0 . 7), these that have formed the totality of their stellar mass by 𝑧 = 2, relics (0 . 34 ≤ DoR ≤ 0 . 7) which had formed at least 75% of their stellar mass by 𝑧 = 2, and non-relics (DoR < 0 . 34) characterised by a more extended SFH. \nFrom a stellar populations point of view, by definition, the DoR correlates with the integrated stellar age. A strong correlation is also found with stellar metallicities and a mild one with the [Mg/Fe]: objects with a higher DoR have overall larger [M/H] and slightly larger [Mg/Fe] (see INSPIRE DR3). Moreover, it appears that the low-mass end of the IMF slope also correlates with the epoch of the SF (Martín-Navarro et al. 2023, MM24). Finally, relics have systematically larger velocity dispersion values than non-relics of similar stellar mass, both normal-sized and ultra-compact ( INSPIRE DR3; MM24).", '2.1 The AMICO galaxy cluster catalogue': 'The Adaptive Matched Identifier of Clustered Objects (AMICO; Bellagamba et al. 2018, 2019; Maturi et al. 2019) is an algorithm for the detection of galaxy clusters in photometric surveys, based on the Optimal Filtering technique. It allows to maximise the signalto-noise ratio (SNR) of the clusters (Maturi et al. 2005) taking into account the luminosity, spatial distribution and photometric redshifts of galaxies. Briefly, AMICO searches for cluster candidates by convolving the 3D galaxy distribution with a redshift-dependent filter, which is defined as the ratio of a cluster signal, modelled with an analytical recipe, and a noise model derived directly from the data. Bayesian photo𝑧 (BPZ; Benítez 2000), estimated from a templatefitting method, are used here. AMICO thus creates a 3D amplitude map where the candidate clusters are identified as peaks through an iterative approach designed to minimise the blending between nearby objects. The angular positions, redshift, signal amplitude, measuring the cluster galaxy abundance, and the SNR are retrieved for each cluster candidate. The mass of the cluster, based on weak \nlensing (WL) scaling relations, is derived too. Finally, the algorithm provides a probabilistic membership association of galaxies to clusters by exploiting the probability redshift distribution of each galaxy (provided by BPZ redshifts) and the model used for the cluster detection. The cluster model is described by a luminosity function and a radial density profile (Bellagamba et al. 2018), and observationally derived from the galaxy population of clusters detected through the Sunyaev-Zeldovich (SZ) effect (Hennig et al. 2017). \nThe AMICO catalogue based on KiDS DR3 data (de Jong et al. 2017) was presented in Maturi et al. (2019): it covers an area of 414 deg 2 , and comprises 7988 candidate galaxy clusters at 0 . 1 < 𝑧 < 0 . 8. It has been successfully used both to derive the population properties of galaxies in the identified clusters (Radovich et al. 2020; Puddu et al. 2021), as well as for WL (Bellagamba et al. 2019; Ingoglia et al. 2022), and cosmological analyses (Giocoli et al. 2021; Lesci et al. 2022a,b; Romanello et al. 2023). \nHere, we use the newest catalogue derived, applying the same algorithm on the KiDS DR4 (Kuijken et al. 2019). This catalogue, detailed in Maturi et al. (in preparation), spans a total area of 1006 deg 2 , which, after masking, translates to an effective area of 840 deg 2 . It includes 22 614 candidate galaxy clusters within the photometric redshift range of 0 . 1 < 𝑧 < 0 . 8, detected down to a SNR > 3 . 5. The catalogue has an average purity of approximately 80% across the entire redshift range. However, it is worth noting that purity strongly depends on the detection SNR. We refer the readers to Maturi et al. (in preparation) for a more quantitative description of the catalogue.', '2.2 The GaZNets catalogue': 'The Galaxy morphoto-Z with neural Networks (GaZNets), introduced in Li et al. (2022), is a deep learning (DL) tool that combines both images and multi-band photometry measurements for the accurate determination of photometric redshifts. What makes this tool distinctive is its integration of conventional machine learning (ML) regression tools with DL techniques. GaZNets has been already successfully applied to a sample of galaxies from the KiDS DR4 (Kuijken et al. 2019), and tested against other ML based catalogues and classification algorithms (Khramtsov et al. 2019). \nIn this work, we use as input the reference network developed by Li et al. (2022), GaZNet-1, which makes use of a combination of KiDS 𝑟 -band images and the KiDS+VIKING 9-bands catalogue ( 𝑢𝑔𝑟𝑖𝑍𝑌𝐽𝐻𝐾𝑠 , Wright et al. 2018). Of the 65.9 million sources in the original catalogue, roughly 40 millions have a measurement in all the bands, a high-precision photometric redshift ( 𝑧 phot ), with uncertainty Δ 𝑧 = 0 . 038 ( 1 + 𝑧 ) , and a corresponding probability of being classified as quasar, galaxy, or star. We use here the results obtained in Feng et al. (submitted), who classified ∼ 27 . 3 millions sources with 𝑟 -band magnitude ≤ 23 from the KiDS DR5 database in quasars, galaxies, or stars using a multi-modal neural network. In particular, from the classification catalogue, we pre-select only ∼ 7 . 9 million objects that have a very high probability to being a galaxy, 𝑃 gal ≥ 0 . 9, in the redshift range 0 . 1 < 𝑧 < 0 . 5. This threshold for the 𝑃 gal maximizes the completeness while minimizing the contamination. Indeed, using ∼ 20 000 galaxies with a spectroscopic match, Li et al. (2022) have shown that, for both 𝑃 gal ≥ 0 . 5 and 𝑃 gal ≥ 0 . 9 probabilities, 99% of the galaxies are classified correctly. Instead, increasing 𝑃 gal to 0.99, only the 78% of the galaxies will be classified as such. Moreover, considering 𝑃 gal ≥ 0 . 5, 4.6% of the quasi-stellar objects (QSOs) are classified as galaxies. Indeed, using 𝑃 gal ≥ 0 . 5 we would retrieve ∼ 8 . 5 millions of galaxies, hence hinting at a larger contamination. We nevertheless caution the reader that because the spectroscopic sample \nFigure 1. Distribution of the redshift difference between each INSPIRE UCMGs and the AMICO clusters at which they might be associated. The central value ( 𝜇 ) and standard deviation ( 𝜎 ) of the Gaussian distribution fitted to the histogram are shown in the legend. \n<!-- image --> \nis brighter, it is easier to distinguish between quasars, galaxies, and stars.', '3 ANALYSIS': 'In this section, we conduct an in-depth analysis of the local environment for the INSPIRE UCMGs from two perspectives. Firstly, we investigate their potential association with galaxy cluster candidates by cross-referencing INSPIRE catalogue data with the AMICO catalogue. We focus on evaluating the probability of UCMGs being members of these clusters and its possible correlation with the DoR. Secondly, we measure the local density around each INSPIRE UCMGbyconductingacross-match with the GaZNets catalogue and counting galaxies with compatible redshifts. Through this analysis, we aim to determine whether a statistically significant over-density exists, thereby indicating a cluster environment, and its correlation with the DoR.', '3.1 Cluster membership from AMICO': "For each cluster candidate, AMICO provides a list of galaxies with the probability of being a cluster member ( 𝑃 cluster ). The 𝑃 cluster is computed through a cluster model, which assumes a luminosity function and a radial density profile. We note that the probability is distance-dependent in the sense that galaxies spatially more distant from the cluster centre will, by construction, have a lower probability. We cross-match the list of galaxies that could be a member of one of the AMICO clusters with the 52 INSPIRE UCMGs, finding that 45 out of the 52 INSPIRE UCMGs have 𝑃 cluster > 0. However, only 9 of them have a probability of being members greater than or equal to 0.5 (i.e., 𝑃 cluster ≥ 0 . 5), which means that their probability to be in a cluster is larger than their probability to be in the field. We will denote these as 'safe' detections for the remainder of the paper. \nIn Fig. 1, we show the distribution of the redshift difference, Δ 𝑧 = ( 𝑧 UCMG -𝑧 cluster )/( 1 + 𝑧 UCMG ) , between each of the 45 UCMGs with a match and the corresponding AMICO cluster. The redshifts exhibit close proximity, with a mean Δ 𝑧 = 0 . 0011 and a standard deviation 𝜎 = 0 . 0492, as reported in the legend, although there is one object \nFigure 2. Probability to belong to a cluster according to AMICO against the DoR for the 45 INSPIRE UCMGs with a match in AMICO, colour-coded by the purity. The seven objects without a match are visualised as black crosses at 𝑃 cluster = 0. When restricting to 'safe' detections (galaxies with 𝑃 cluster ≥ 0 . 5, see text for more details), a linear relation (grey dashed line) is found, with higher probabilities for objects with larger DoR. \n<!-- image --> \nwith | Δ 𝑧 | > 0 . 1 (J0844+0148, 𝑧 UCMG = 0 . 2837, 𝑧 cluster = 0 . 4413). We stress that the clusters' redshifts are based on the photometric redshifts of their members. \nThe results of the cross-matching between the AMICO cluster catalogue and INSPIRE galaxies are listed in the second block of columns of Table 1, where we list, for the 45 objects with a match, the redshift of the AMICO cluster compatible with that of the UCMG, the probability of the object to be a member of that cluster, the SNR of the detection, the level of the purity ( 𝒫 ), and the virial mass of the cluster ( 𝑀 200 ). The seven objects that lack a match in AMICO are still listed in the table but without a numerical value for these quantities. In fact, for these, the cluster finding algorithm does not find a suitable association to any of the detected cluster candidates. This suggests that they might be field galaxies or reside in groups of galaxies that are too small to be detected by AMICO. \nThe 45 INSPIRE galaxies matched with the AMICO cluster candidates exhibit a broad range in DoR, as visible from Fig. 2, where we colour-coded the data points by the purity of the cluster, 𝒫 . In the same figure, the remaining 7 UCMGs without a match are illustrated as black crosses at 𝑃 cluster = 0. A linear relation, showed as a grey dashed line, emerges when we restrict to 'safe' detections, i.e., limiting to points with 𝑃 cluster ≥ 0 . 5, corresponding to a 50% probability of belonging to the cluster identified by the algorithm. \nIn summary, according to the analysis carried out from AMICO, 9 objects are members of a cluster of galaxies ( 𝑃 cluster ≥ 0 . 5), while 7 are most likely in the field with a high degree of confidence as they do not have a match. The remaining 36 objects exhibit a wide range of 𝑃 cluster values, but all lower than 0.5, hence preventing a safe classification of the local environment. Considering the grouping from INSPIRE DR3, all 9 extreme relics (DoR > 0 . 7) have a match with an AMICO cluster. Of the 7 UCMGs lacking a match in AMICO, 6 are relics (0 . 34 ≤ DoR < 0 . 7) and 1 is a non-relic (DoR < 0 . 34). There are 2 extreme relics, 4 relics, and 3 non-relics with 𝑃 cluster ≥ 0 . 5. Henceforth, there appears to be no clear environmental preference for UCMGs with varying degree of relicness but that a linear correlation exists between DoR and 𝑃 cluster for the 'safe' detections ( 𝑃 cluster ≥ 0 . 5). Furthermore, we notice that UCMGs with \na very extended SFH (DoR < 0 . 3) all show rather low probabilities of being in a cluster.", '3.2 Over-densities identification from the GaZNets catalogue': "In this section, we describe the cross-match between the INSPIRE catalogue and the GaZNets galaxy catalogue described in section 2.2. We select, for each of the 52 UCMGs, all and only galaxies (objects with 𝑃 gal ≥ 0 . 9 in the catalogue by Feng et al., submitted) having redshift compatible to that of the INSPIRE objects and 𝑚 𝑟 < 22. In particular, for the redshift, we compute Δ 𝑧 = ( 𝑧 UCMG -𝑧 phot )/( 1 + 𝑧 UCMG ) and retrieve objects with | Δ 𝑧 | ≤ 0 . 03. To justify our choices in magnitude and redshift range, we use spectroscopic data from the Dark Energy Spectroscopic Instrument (DESI) Survey (Levi et al. 2019; DESI Collaboration et al. 2023). We cross-match the catalogue with the GaZNets one and check the precision of the ML photometric redshifts against the spectroscopic ones. For objects brighter than 𝑚 𝑟 < 22, the uncertainty in redshift estimation is of the order of 0.03, while it increases for fainter objects. Furthermore, we also point the reader to fig. 3 in Li et al. (2022), where the accuracy of the GaZNets ML photometric redshifts is estimated on the training sample of 20 000 galaxies used to test the performances of the ML models. We restrict our analysis to galaxies with 𝑚 𝑟 < 22, corresponding to detecting galaxies with 𝑀 ★ +5 at 𝑧 = 0 . 1 and 𝑀 ★ +2 at 𝑧 = 0 . 4 for the redshifts of the UCMGs.Foreachgalaxy in INSPIRE meeting this criterion and having | Δ 𝑧 | ≤ 0 . 03, we generate a density map within a 10 Mpc radius. This is accomplished using KDEpy 2 library that performs a Kernel Density Estimation (KDE) on both 1D and 2D data. Specifically, we employ an Epanechnikov kernel (Epanechnikov 1969) with a bandwidth of 0.2 Mpc. We extract the peak of the density within a radius of 1.5 Mpc ( ∼ 𝑟 200 for a cluster with 𝑀 200 cluster ∼ 10 14 -10 14 . 5 𝑀 ⊙ ), to check for the presence of nearby over-densities (next nearest neighbour, Σ NNB ). We then select 50 random regions all located from 5 to 9 Mpc from the UCMG and all with radius of 1.5 Mpc. We compute the mean and standard deviation of this distribution ( Σ BKG ± 𝜎 Σ BKG ), using it as a background estimation. Practically speaking, with this procedure, we are estimating the significance of an over-density in the proximity of each UCMG by the quantity \n𝛿 Σ = Σ NNB -Σ BKG 𝜎 Σ BKG . (2) \nTo determine the threshold in 𝛿 Σ for which we expect that an overdensity is significant, we repeat the procedure, while randomising the galaxy positions in the field. The threshold is then defined as the average value of 𝛿 Σ calculated in these randomized fields, which is 𝛿 Σ , min = 3. For one galaxy, J1026+0033, it has not been possible to compute the value of 𝛿 Σ , min . This object lies in proximity to a luminous star, which has been masked in the catalogues, causing a non-homogeneous mapping of the proximity of the UCMGs. \nFigure 3 shows the over-density ( 𝛿 Σ ) values measured for each UCMG against their DoR. The dashed grey lines represent the 𝛿 Σ threshold for an over-density to be significant. The data points are colour-coded by the 𝑃 cluster and those without a match in AMICO are visualised as black crosses. \nWe observe a similar behaviour to that described in the previous section (section 2.1, Fig. 2). Nearly all UCMGs with extended SFHs (DoR < 0 . 3) prefer under-dense environments. However, the scatter \nFigure 3. Over-density as a function of the DoR for the 51 INSPIRE UCMG for which a 𝛿 Σ , min could be computed. The dashed grey lines represent the threshold 𝛿 Σ , min for an over-density to be significant. The data points are colour-coded by their probability to reside in an AMICO cluster and the seven objects without a match in AMICO are visualised as black crosses. \n<!-- image --> \nsuddenly intensifies at intermediate DoR, where UCMGs reach values of 𝛿 Σ as high as 17.06 and as low as 0 . 54. The data points in the figure are colour-coded by the probability to reside in an AMICO cluster. All objects with 𝑃 cluster ≥ 0 . 5, i.e., 'safe' detection, pass the 𝛿 Σ threshold (grey dashed line). This results serves as a confirmation of the consistency between the two methods. \nInterestingly, the vast majority of galaxies with DoR ≥ 0 . 5 are positioned above the 𝛿 Σ , suggesting, once again, that relics might slightly prefer over-dense environments, such as clusters of galaxies. Specifically, almost all (8 out of 9) extreme relics and two-thirds of relics with intermediate DoR values are above the threshold (with the maximum scatter), while roughly 50% of the non-relics fall below the significance threshold. Only one out of the 14 non-relics has a very high value of 𝛿 Σ (J1402+0117, 𝛿 Σ = 17 . 06 3 ). \nHence, according to GaZNets, 34 INSPIRE UCMGs (65% of the sample) reside in over-densities ( 𝛿 Σ ≥ 3). Of these, 8 (out of 9) are extreme relics, 8 are non-relics (out of 14), and only three (out of 9) have DoR < 0 . 3.", '4 RESULTS: CHARACTERISATION OF THE LOCAL ENVIRONMENT': "In this section, we present the results of the analysis performed in the paper, focusing on the relation between the DoR and the environments occupied by the UCMGs. This topic was already introduced in previous papers (e.g., Poggianti et al. 2013; Stringer et al. 2015; Damjanov et al. 2015b; Tortora et al. 2020), which, however, lack any distinction between the ancient massive relics of the early Universe from relatively younger but equally compact and massive systems, that have undergone a much more extended SF. In the local Universe, FM17, based on three confirmed relics, hinted for a correlation between the kinematics, structural, and stellar population parameters of three extreme relics and their local environment. The great step \nTable 1. Classification of the local environment for the 52 INSPIRE UCMGs. Nine objects are in an over-dense region (top rows) with high degree of confidence, 17 definitively in under-dense environments (middle rows). For the remaining, only a tentative environment classification is provided. Within each environment group (horizontal lines), galaxies are ordered in descending order of DoR. We list the INSPIRE ID, the DoR and, the redshifts of the UCMGs ( 𝑧 UCMG ) in the first vertical block (from INSPIRE DR3). In the second block, we list quantities derived from AMICO: redshift of the cluster ( 𝑧 cluster ), the probability for the UCMG to belonging to that cluster ( 𝑃 cluster ), the signal-to-noise ratio (SNR) of detection, its purity ( 𝒫 ), the cluster's virial mass in 𝑀 ⊙ ( 𝑀 200 cluster ), and the logarithmic distance of the UCMG from the cluster centre in kpc (log 𝐷 A ). Finally, in the third block, we list the over-density value ( 𝛿 Σ ), the logarithmic distance of each UCMGfrom its centre in kpc (log 𝐷 GZ ), both derived from GaZNets, and lastly, the environment classification. Rows corresponding to objects without a match in AMICO or GaZNets are listed with a -. \n∗ Tentative environment classification (see the text for more details). \n+ The distances have been calculated from the nearest density peak, which is, however, not significant compared to the background level. \nFigure 4. Summary of number of INSPIRE UCMGs for each environment (outer circles), and DoR family (inner circles). \n<!-- image --> \nforward brought by INSPIRE is the possibility to investigate on this matter with a much larger number statistics and covering a wider range in DoR. Hence, we have checked whether an environmental dependency exists for the stellar masses and sizes (Scognamiglio et al. 2020), velocity dispersion values (D'Ago et al. 2023), and stellar parameters (IMF slope, metallicity, [Mg/Fe]; MM24). Surprisingly, no statistically significant correlation was found, indicating that the larger INSPIRE sample does not support the idea hinted at in FM17. Velocity dispersion and stellar population parameters, although varying as a function of DoR, do not depend upon the density of the local environment in which a UCMG resides. \nTable 1 provides a summary of the characterisation of the local environments for the 52 INSPIRE UCMGs. The first (second) horizontal block of the table lists objects that are residing in an over(under-)dense region with high degree of confidence, having very high (low) values of both 𝑃 cluster and 𝛿 Σ . Among the 26 UCMGs whose environment can be confidently determined (50% of the sample), 9 are situated in clusters (labelled as 'C') and 17 in the field (labelled as 'F'). These galaxies exhibit a diverse range of DoR values in both environments. Notably, we confirm the absence of UCMGs with DoR < 0 . 3 in cluster environments. Out of the remaining 26 INSPIRE objects, 11 are likely to be cluster members, while 15 might inhabit either field environments or small galaxy groups, although a definitive determination remains uncertain. Specifically, the objects marked as 'C ∗ ' in Table 1 exhibit a significant 𝛿 Σ value, exceeding the threshold by more than 1 𝜎 , yet they have an AMICO 𝑃 cluster value below 0.5. This implies that they reside within an over-dense region. Their low AMICO probability could stem from different factors, such as the UCMGs being situated relatively far from the cluster centre 4 , or due to potential inaccuracies in the photometric redshift estimation of the cluster. Lastly, systems denoted with an 'F ∗ ' have a 𝛿 Σ slightly above the threshold (but less than 1 𝜎 away), along with a low 𝑃 cluster value, likely reside in the field or in the outskirts of a small galaxy group, with moderate density. The wedge plot in Fig. 4 provides a graphical visualisation of the numbers of UCMGs based on the environment classification (outer circle) and on the three families defined in INSPIRE DR3 according to the DoR (inner circle). \nFigure 5. KiDS 𝑔𝑟𝑖 cutouts (top panels) and KDE density maps (bottom panels) of size 2 × 2 Mpc 2 for the objects with highest (top row) and lowest (bottom row) DoR in the INSPIRE sample, for which a definitive environment characterisation has been obtained. The cutouts also display DoR, 𝑃 A , and 𝛿 Σ values. The subscripts 'A' is used to denote the 𝑃 cluster from 'AMICO', for brevity and clarity in the images. In the KDE density maps, the yellow star indicates the position of the UCMG, while the green diamond identifies the cluster centre, if any. The circular aperture of 1.5 Mpc used to identify the nearest over-density is also drawn on the map in black. \n<!-- image --> \nFigure 5 shows 4 of the most extreme INSPIRE objects (two highest and two lowest DoR), for which a definitive estimate of the local environment could be obtained (i.e., first two blocks of Table 1). The two objects on the top row are two extreme relics with DoR > 0 . 7, one of them are in a cluster one in the field. The bottom row depicts instead two objects with the lowest DoR, all in the field according to our classification. For each galaxy, the top panel illustrates the 𝑔𝑟𝑖 colour-combined image from the KiDS survey while the bottom shows the KDE density map, both with size of 2 × 2 Mpc 2 and \nFigure 6. The same as in the previous figure, but for four INSPIRE UCMGs with intermediate DoR residing in different environments (cluster on the left, field on the right). We display two 'safe' classifications and two tentative ones. Symbols and scales are as in the previous figure. \n<!-- image --> \nFigure 7. KDE density of 2 × 2 Mpc 2 for five objects in over-dense environments (top row) and five objects in under-dense environment (bottom row), ordered by their 𝛿 Σ value. Symbols as in the previous figures. \n<!-- image --> \nAX [Mpc] \ncentred on the UCMG, which is plotted as a yellow star in the maps. For objects in clusters, we also highlight the position of the centre of the cluster candidate as a green diamond. Figure 6 shows instead four objects with very similar DoR (intermediate ∼ 0 . 4) but residing in different environments. The two top systems live in an over-density, while the two bottom ones inhabit under-dense environments. In all panels, the ID, DoR, 𝛿 Σ , and 𝑃 cluster (reported as 𝑃 A , for brevity and clarity in the images) values are reported. Finally, Fig. 7 shows the KDE maps for additional 10 objects, 5 in over-dense and 5 in under-dense environments to provide a general idea of the variety of \nthe maps for the INSPIRE sample. Objects in the same column (top and bottom) have very comparable DoR values. Moving from left to right within the same row, the 𝛿 Σ values decrease.", '4.1 Correlation between the DoR and the UCMG distance to the cluster centre': "In Table 1, we include the logarithmic values of the distance of an UCMG from both a cluster centre and from an over-density, in kpc. These values are visualized in Fig. 8, where they are plotted against \nthe DoR. In this figure, the dark red points represent AMICO data, while the blue points denote GaZNets data. Galaxies confidently identified as belonging to a cluster or an over-density are labelled with 'C', whereas those considered tentative members are marked with 'C ∗ ' (shaded circles). For both methods, a similar correlation is found: the higher the DoR, the closer to the cluster centre/over-density the UCMGs is located. Indeed, performing a linear fit between the aforementioned distances and the DoR values, a comparably negative correlation is evident in both cases: \nlog 10 𝐷 AMICO = 3 . 02 - ( 0 . 7 × DoR ) log 10 𝐷 GaZNets = 3 . 13 - ( 0 . 6 × DoR ) (3) \nTo assess the significance of the correlation between the DoR and the log 10 𝐷 for the 'safe' detections, we perform a bootstrap analysis. We randomize the DoR and log 10 𝐷 values independently for 49 iterations and include the original data as the 50th iteration. The mean slope from these bootstrapped datasets is 𝜇 = 0 . 01 ± 0 . 46 for AMICO and 𝜇 = -0 . 04 ± 0 . 43 for GaZNets. In contrast, the slopes from the original data are 0.7 for AMICO and 0.6 for GaZNets. These significant differences indicate that the observed correlations are not due to random fluctuations, affirming the robustness and significance of the relationship between DoR and log 10 𝐷 . \nIt is worth noting that the physical significance of the distance from the centre varies significantly depending on whether one considers a very massive cluster of galaxies, which can extend up to much larger distances or small groups, that are instead generally much smaller in size. Nevertheless, from Table 1 it is clear that the cluster candidates identified by AMICO do not exhibit a wide range in total mass, spanning from 4 . 1 × 10 12 𝑀 ⊙ to 2 . 6 × 10 14 𝑀 ⊙ , and with a mean and standard deviation of ( 2 . 6 ± 0 . 1 ) × 10 13 𝑀 ⊙ . Finally, we also note that the distance-DoR relation shows a much larger scatter and a shallower slope when considering all the UCMGs that are tentatively residing in a cluster (i.e., C ∗ , shaded dark red points in Fig. 8). However, as already stressed, the quantities derived by AMICO are based on the detection algorithm that depends on less-precise photometric redshifts and on the analytical cluster model based on a luminosity function and a radial density profile. Furthermore, sometimes a system could be associated (through the membership probability) to more than one cluster. Hence, the estimate of the log 𝐷 for AMICO, especially for tentative classification, could be biased or influenced by unaccounted factors. The over-density values from GaZNets are based on more accurate ML redshifts and simply trace the number of galaxies in the proximity of each UCMG, without assuming any model or light distribution.", '5 DISCUSSION': "From this study it has emerged that UCMGs are present in both over- and under-dense regions, in line with what previously reported (Damjanov et al. 2015b; Tortora et al. 2020). We further found that there is no correlation between their DoR and the density of the local environment. Our observational findings are in perfect agreement by results from simulations (Moura et al. 2024), which have shown that both relics and younger UCMGs exhibit no distinct preference towards either high- or low-density environments at 𝑧 = 0. Instead, they are distributed across a range of densities, as evidenced by their presence in various environments. \nIn this section we focus on possible evolutionary scenarios that are able to explain the presence of UCMGs with all DoR both in over- and under-dense regions and possibly, the lack of objects with DoR < 0 . 3 \nFigure 8. Correlation between DoR and distances from a cluster candidate or an over-density, for AMICO (dark red points) and GaZNets (blue points), labelled with 'C' in Table 1. A linear fit is shown as dashed line for both cases. Lighter points show the UCMGs labelled as 'C ∗ ' in Table 1, which are not included in the fit as the classification of the environment is tentative. \n<!-- image --> \nin cluster environment. We start by focusing on the most extreme relics, hypothesising on how they could avoid any interaction with other cluster members or very quickly stop forming stars in the field. We then focus on UCMGs with a very extended SFH, which seem to prefer under-dense environments. Finally, we stress that a possible third scenario might exist for UCMGs with low and intermediate DoR. Indeed, the observed near-by UCMGs might also be galaxies that had a three phase formation and evolution scenario. After the size growth phase they might have gone through a subsequent phase of re-compaction due to stripping and gas removal (Kapferer et al. 2009; Peluso et al. 2022; Göller et al. 2023), and are therefore observed to be ultra-compact today. This is highly unlikely for extreme relics, where the entire totality of the stellar population is almost as old as the Universe but it cannot be excluded for the remaining cases.", '5.1 Extreme relics in clusters and in the field': "Among the 9 extreme relics (DoR > 0 . 7) which have formed all their stellar mass at high𝑧 , two are in an over-dense environment and one is in an under-dense region (J1438-0127, DoR = 0 . 78) with a high degree of confidence. For the remaining 6 systems, 3 are likely to reside in clusters or groups and 3 more likely to be in the field. Moreover, when extreme relics are in a cluster, they tend to be closer to its centre than UCMGs with lower DoR. \nWhile it is quite intuitive to understand how a high𝑧 isolated, field red nugget can evolve passively and undisturbed through cosmic time without interacting with any other system, it is difficult to explain how it can survive untouched for many Gyr in the central regions of a cluster/group of galaxies. A plausible explanation is that extreme relics have a very deep potential well and incredibly high density, which make them less susceptible to interactions with other galaxies (see also Poggianti et al. 2013). This scenario is also supported by the fact that extreme relics have a larger stellar velocity dispersion (up to ∼ 400 km s -1 at 𝑀 ★ ∼ 10 11 𝑀 ⊙ ) than non-relics, both younger UCMGs and normal-sized ETGs, of similar stellar mass (Ferré-Mateu et al. 2017; Spiniello et al. 2021b, 2024; Grèbol-Tomàs et al. 2023). As found in Peralta de Arriba et al. (2016), the high-velocity dispersion values and the hot ICM can prevent the \ngrowth of an accreted stellar envelope through mergers. Furthermore, a dense environment could help in preventing a continuous SF. Indeed, stripping of the cold gas surrounding a galaxy might happen by the hot, dense diffuse intra-cluster gas, causing SF to cease (Ma et al. 2008; Bekki 2013; Roediger et al. 2014; Steinhauser et al. 2016; Foltz et al. 2018). Finally, the so-called 'galaxy harassment', which is the combined effect of gravitational interactions between galaxies (Merritt 1983; Bialas et al. 2015) and their interaction with the potential well of the cluster as a whole (Byrd & Valtonen 1990), has often been proposed to explain the formation of red sequence galaxies (e.g. Boselli & Gavazzi 2014). So, if the UCMGs entered in the cluster very early-on in time, or formed with it, they stopped forming stars and then evolved passively thereafter. \nOur findings align well with the theoretical scenario described in Moura et al. (2024). The authors have shown that, although in the local Universe relics and younger compact galaxies can be found in all environments, at 𝑧 > 2 the number of relics in clusters is higher when compared to younger UCMGs at the same mass range (see also Kimmig et al. 2023). Hence red nuggets that will be progenitors of relics were preferentially in an over-density at high𝑧 . On the contrary, ultra-compact non-relics in a cluster entered in this dense environment only at later stages. \nIn low-density environments, instead, red nuggets are less likely to merge and increase the 'ex-situ' fraction of stars. Therefore we expect to find these ultra-compact systems still as such in more isolated environment (Peralta de Arriba et al. 2016; Kimmig et al. 2023; Moura et al. 2024). However, if the red nuggets are surrounded by a gas reservoir, they might keep forming stars, although at a very low-rate until they consume it. Hence, extreme relics in the field might be the local descendants of red nuggets have been formed in gas-poor under-dense regions of the Universe at early epochs. This might be the case of J1438-0127, the only INSPIRE extreme relic (DoR = 0 . 78) in the field, and Mrk1216, the only near-by extreme relic definitively the field.", '5.2 Why do UCMGs with more extended SFHs prefer the field?': 'At the other extreme of the distribution, we have found no objects with DoR ≤ 0 . 3 definitively located in an over-dense region. Looking at the tentative classifications (marked with an asterisk (*) in Table 1), only one out of ten UCMGs with such lower DoR could be in a cluster. Among the remaining objects, 6 are definitively in under-dense regions and 3 are tentatively in the field. These objects, although passive and relatively old ( ∼ 4 -6 Gyr) in an integrated sense, are characterised by a much more extended SFH and they are still forming a small percentage of their stellar mass today or stopped very recently, similar to those originally found in Trujillo et al. (2009) and Ferré-Mateu et al. (2012). The preference for UCMGs with extended SFHs to live in the field could be explained by the fact that in these under-dense environment, major (and especially dry) mergers are rare (e.g., de Ravel et al. 2009; Fakhouri & Ma 2009; Darg et al. 2010; Lin et al. 2010; Kampczyk et al. 2012; Ellison et al. 2013). Major mergers can indeed trigger intense bursts of SF, thus shortening the overall SFH of galaxies (Mihos & Hernquist 1996; Gabor et al. 2010; Man & Belli 2018). In contrast, these red nuggets may have had a large reservoir of surrounding gas, which is gradually turned into stars over a much longer timescale. If this scenario is true, then one should expect to find an anti-correlation between the HI column density and the DoR, which is however still too challenging to measure with the current radio telescopes.', '6 SUMMARY AND CONCLUSIONS': "In this seventh paper of the INSPIRE survey, we have investigated whether a correlation exists between the DoR and the local environment for the 52 UCMGs in the sample, as originally hinted at in FM17. We have started with cross-matching the INSPIRE catalogue with a catalogue of cluster candidates from the KiDS survey, obtained using the AMICO algorithm (Bellagamba et al. 2018, 2019; Maturi et al. 2019). Out of the 52 UCMGs, 45 are associated with a cluster as potential members, with a wide range of probabilities, SNR of the cluster detection, and cluster purity values. \nWe have estimated the local density and its significance with respect to the background around each galaxy in INSPIRE . To this aim, we have utilised machine-learning-based photometric redshift from the GaZNets-1 catalogue (Li et al. 2022) and classification into quasars, galaxies, and stars from Feng et. al (submitted). This allowed us to identify galaxies with 𝑚 𝑟 < 22 situated within 1.5 Mpc of each INSPIRE UCMG, with a redshift difference of Δ 𝑧 < 0 . 03. We have then created a density map for each UCMG and estimated its significance over the background local density, obtained selecting 50 random region located around the UCMGs but at larger distances. Wehavefinally assigned, on the basis of the analysis described above, two most likely environments (cluster or field) to each system in the INSPIRE sample. \nThese are the main results from the analysis presented in this paper: \n- · UCMGs can be found in all kind of environments, from dense clusters of galaxies to under-dense field environments. Taking the classifications in Table 1 at face value, we have found 20 UCMGs in a cluster and 32 in the field. Of these systems, 9 are in a cluster and 18 are in the field with high confidence environment classifications, while the remaining are only tentative 5 .\n- · There is no clear correlation between the DoR and the probability for a UCMG to reside in a cluster for the entire sample. However, when restricting the analysis to the 'safe' detections (i.e., 𝑃 cluster ≥ 0 . 5) a linear relation emerges: the most extreme relics exhibit the highest 𝑃 cluster values, as illustrated in Fig. 2.\n- · The density of the local environment where the UCMGs reside does not correlate with the DoR. However, UCMGs with an extended SFH (DoR < 0 . 3) tend to prefer less dense environments, as shown in Fig. 3. In fact, none of the objects with DoR ≤ 0 . 3 reside in a dense environment with high confidence. Only one could tentatively be in a group/cluster (J1154-0016).\n- · A correlation is found between the DoR and the distance from the cluster centre log 10 𝐷 , for both techniques. Indeed, Fig. 8 illustrates that the higher the DoR, the closer the systems are to the centre of the over-density. Additionally, the distances are comparable for the two methods (2 . 6 < log 10 𝐷 < 3 . 2 kpc), at least for the high confidence classifications. \nIn conclusion, our findings suggest that while a weak dependency on DoR exists, relics can be found across diverse environments with different local densities, consistent with both previous observational studies (FM17; Siudek et al. 2023) and hydro-dynamical simulations (Peralta de Arriba et al. 2016; Flores-Freitas et al. 2022; Moura et al. 2024). However, younger UCMGs, with a DoR < 0 . 3 and thus characterised by a more extended SFH, are preferentially found in under-dense environments and reside almost exclusively in the field. \n- 5 Tortora et al. (2020) have found that the fraction of UCMGs in the field is slightly higher compared to that in clusters (see the right panel of their fig. 1). We confirm this result and remind the reader that the INSPIRE sample has been originally drawn from the same sample used in Tortora et al. (2020). \nWe argue that these results are justified by the premise that if a red nugget has formed in an over-density at high redshift (or moved in quite early-on in cosmic time), the hot and dense ICM within the cluster gravitational potential has exerted high pressure and removed all the gas, hence quenching its SF activity (Peralta de Arriba et al. 2016; Boselli et al. 2022). In clusters, extreme relics did not interact with other members, because they possess very deep potential wells, also indicated by their large stellar velocity dispersion values, and incredibly high densities. \nConversely, in under-dense environments, red nuggets at high𝑧 that do not interact via mergers (growing in size) either keep forming stars at very low-rate consuming their gas envelope and becoming non-relics UCMGs, or they do not have any gas surrounding them and hence evolve passively and undisturbed without companions to merge with, nor gas reservoir to form new stars. \nFinally, we highlight that a fraction of the non-relics INSPIRE UCMGs in a cluster might also have been originated by stripping phenomena on an originally larger massive system, that have caused a compaction phase at later cosmic epochs (Dekel & Burkert 2014; van Dokkum et al. 2015). \nOnly by further extending the number of fully characterised and spectroscopically confirmed UCMGs, studying their stellar populations in great detail, we will be able to quantify the number densities of the different types of ultra-compact objects and study their evolution with redshifts. Current and up-coming surveys like Euclid (Laureijs et al. 2011), Vera C. Rubin Observatory's Legacy Survey of Space and Time ( Rubin -LSST; Ivezić et al. 2019 , Multi-Object Spectroscopic Telescope (4MOST; de Jong et al. 2019), and WEAVE at William Herschel Telescope (Jin et al. 2023) will be transformational in extending the number statistics and pushing the redshift boundaries between the near-by and the high𝑧 Universe.", 'DATA AVAILABILITY': 'The INSPIRE data used in this paper are publicly available via the ESO Phase 3 Archive Science Portal under the collection INSPIRE ( https://archive.eso.org/scienceportal/ home?data\\_collection=INSPIRE , https:https://doi.eso. org/10.18727/archive/36 ).', 'ACKNOWLEDGEMENTS': 'The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004), © 2024. All rights reserved. DS is supported by JPL, which is operated under a contract by Caltech for NASA. CS and CT acknowledge funding from the INAF PRIN-INAF 2020 program 1.05.01.85.11. GD acknowledges support by UKRI-STFC grants: ST/T003081/1 and ST/X001857/1.', 'REFERENCES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author.'}

2024A&A...689A...8C

Context. Ultra hot Jupiters gas giants with TSUBeqSUB gt 2000 K are intriguing exoplanets due to the extreme physics and chemistry present in their atmospheres. Their torrid daysides can be characterised using groundbased highresolution emission spectroscopy. Aims. We search for signatures of neutral and singly ionised iron Fe I and Fe II respectively in the dayside of the ultra hot Jupiter WASP76 b as these species were detected via transmission spectroscopy in this exoplanet. Furthermore we aim to confirm the existence of a thermal inversion layer which has been reported in previous studies and attempt to constrain its properties. Methods. We observed WASP76 b on four epochs with ESPRESSO at the VLT at orbital phases shortly before and after the secondary transit when the dayside is in view. We present the first analysis of highresolution optical emission spectra for this exoplanet. We compare the data to synthetic templates created with petitRADTRANS using crosscorrelation function techniques. Results. We detect a blueshifted 4.7 0.3 km sSUP1SUP Fe I emission signature on the dayside of WASP76 b at 6.0. The signal is detected independently both before and after the eclipse and it is blueshifted in both cases. The presence of iron emission features confirms the existence of a thermal inversion layer. Fe II was not detected possibly because this species is located in the upper layers of the atmosphere which are more optically thin. Thus the Fe II signature on the dayside of WASP76 b is too weak to be detected with emission spectroscopy. Conclusions. We propose that the blueshifted Fe I signature is created by material rising from the hot spot to the upper layers of the atmosphere and discuss possible scenarios related to the position of the hotspot. This work unveils some of the dynamic processes ongoing on the dayside of the ultra hot Jupiter WASP76 b through the analysis of the Fe I signature from its atmosphere and complements previous knowledge obtained from transmission studies. It also highlights the ability of ESPRESSO to probe the dayside of this class of exoplanets. Based on Guaranteed Time Observations collected at the European Southern Observatory under ESO programmes 1104.C0350U and 110.24CD.004 by the ESPRESSO Consortium.

2024-09-01T00:00:00Z

['2024arXiv240913519C', '2024A&A...689A...8C', '10.1051/0004-6361/202449935', 'arXiv:2409.13519', '10.48550/arXiv.2409.13519']

['methods: observational', 'techniques: spectroscopic', 'planets and satellites: atmospheres', 'planets and satellites: gaseous planets', 'planets and satellites: individual: WASP-76b', 'Astrophysics - Earth and Planetary Astrophysics']

ESPRESSO reveals blueshifted neutral iron emission lines on the dayside of WASP76 b

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

https://arxiv.org/pdf/2409.13519.pdf

{'ESPRESSO reveals blueshifted neutral iron emission lines on the dayside of WASP-76 b ⋆': "A. R. Costa Silva 1 , 2 , 3 , O. D. S. Demangeon 1 , 2 , N. C. Santos 1 , 2 , D. Ehrenreich 3 , 4 , C. Lovis 3 , H. Chakraborty 3 , M. Lendl 3 , F. Pepe 3 , S. Cristiani 5 , R. Rebolo 6 , 7 , 8 , M. R. Zapatero-Osorio 9 , V. Adibekyan 1 , 2 , Y. Alibert 10 , R. Allart 11 , 3 ⋆⋆ , C. Allende Prieto 6 , 7 , T. Azevedo Silva 1 , 2 , F. Borsa 12 , V. Bourrier 3 , E. Cristo 1 , 2 , P. Di Marcantonio 5 , E. Esparza-Borges 6 , 7 , P. Figueira 3 , 1 , J. I. González Hernández 6 , 7 , E. Herrero-Cisneros 13 , G. Lo Curto 14 , C. J. A. P. Martins 1 , 15 , A. Mehner 14 , N. J. Nunes 16 , E. Palle 6 , 7 , S. Pelletier 3 , J. V. Seidel 14 , A. M. Silva 1 , 2 , S. G. Sousa 1 , A. Sozzetti 17 , M. Steiner 3 , A. Suárez Mascareño 6 , 7 , and S. Udry 3 \n- 1 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal\n- 2 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal\n- 3 Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland\n- 4 Centre Vie dans l'Univers, Faculté des sciences, Université de Genève, Genève 4, Switzerland\n- 5 INAF -Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143, Trieste, Italy\n- 6 Instituto de Astrofísica de Canarias, c / Vía Láctea s / n, 38205 La Laguna, Tenerife, Spain\n- 7 Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain\n- 8 Consejo Superior de Investigaciones Cientícas, Spain\n- 9 Centro de Astrobiología, CSIC-INTA, Camino Bajo del Castillo s / n, E-28692 Villanueva de la Cañada, Madrid, Spain\n- 10 Physics Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland\n- 11 Département de Physique, Institut Trottier de Recherche sur les Exoplanètes, Université de Montréal, Montréal, Québec, H3T 1J4, Canada\n- 12 INAF -Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy\n- 13 Centro de Astrobiología, CSIC-INTA, Crta. Ajalvir km 4, E-28850 Torrejón de Ardoz, Madrid, Spain\n- 14 European Southern Observatory, Alonso de Córdova 3107, Vitacura, Región Metropolitana, Chile\n- 15 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal\n- 16 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, PT1749-016 Lisboa, Portugal\n- 17 INAF -Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, Italy \nReceived 11 March 2024; accepted 9 July 2024", 'ABSTRACT': 'Context. Ultra hot Jupiters (gas giants with Teq > 2000 K) are intriguing exoplanets due to the extreme physics and chemistry present in their atmospheres. Their torrid daysides can be characterised using ground-based high-resolution emission spectroscopy. \nAims. We search for signatures of neutral and singly ionised iron (Fe i and Fe ii , respectively) in the dayside of the ultra hot Jupiter WASP-76b, as these species were detected via transmission spectroscopy in this exoplanet. Furthermore, we aim to confirm the existence of a thermal inversion layer, which has been reported in previous studies, and attempt to constrain its properties. \nMethods. We observed WASP-76 b on four epochs with ESPRESSO at the VLT, at orbital phases shortly before and after the secondary transit, when the dayside is in view. We present the first analysis of high-resolution optical emission spectra for this exoplanet. We compare the data to synthetic templates created with petitRADTRANS , using cross-correlation function techniques. \nResults. Wedetect a blueshifted ( -4.7 ± 0.3 km / s) Fe i emission signature on the dayside of WASP-76 b at 6.0 σ . The signal is detected independently both before and after the eclipse, and it is blueshifted in both cases. The presence of iron emission features confirms the existence of a thermal inversion layer. Fe ii was not detected, possibly because this species is located in the upper layers of the atmosphere, which are more optically thin. Thus the Fe ii signature on the dayside of WASP-76 b is too weak to be detected with emission spectroscopy. \nConclusions. We propose that the blueshifted Fe i signature is created by material rising from the hot spot to the upper layers of the atmosphere, and discuss possible scenarios related to the position of the hotspot. This work unveils some of the dynamic processes ongoing on the dayside of the ultra hot Jupiter WASP-76 b through the analysis of the Fe i signature from its atmosphere, and complements previous knowledge obtained from transmission studies. It also highlights the ability of ESPRESSO to probe the dayside of this class of exoplanets. \nKey words. Methods: observational - Techniques: spectroscopic - Planets and satellites: atmospheres - Planets and satellites: gaseous planets - Planets and satellites: individual: WASP-76 b', '1. Introduction': "In recent years, the field of exoplanet research has partially shifted its focus from the detection of other worlds to their detailed characterisation. The study of exoplanet atmospheres provides extensive knowledge that helps us constrain the formation and evolution models of exoplanetary systems (e.g. Madhusudhan 2019), and it is also considered a crucial step in the search for extraterrestrial life (e.g. Schwieterman et al. 2018; Meadows et al. 2018). \nUltra hot Jupiters (UHJs) are an interesting class of exoplanets to further characterise, due to their extreme equilibrium temperatures (Teq > 2000 K) and even hotter dayside temperatures. It has been predicted that thermal inversion layers exist on the dayside of these scorched planets (e.g. Lothringer et al. 2018; Lothringer & Barman 2019), that is a hotter layer of atmosphere on top of a colder layer, where chemical species produce emission features. Recent observational studies have confirmed this by probing the emission spectra of exoplanets during orbital phases close to the secondary transit (also known as eclipse or occultation), and detecting such features (e.g. Evans et al. 2017; Yan et al. 2020; Borsa et al. 2022; Changeat et al. 2022). \nHowever, it is still unclear what species are absorbing the stellar irradiation at high altitudes and creating the inversion. For some time, titanium oxide (TiO) and vanadium oxide (VO) were assumed to be the opacity sources responsible for this feature (Hubeny et al. 2003; Fortney et al. 2008). Though more recently, Lothringer et al. (2018) have shown that neutral iron (Fe i ) and other atomic metals are also capable of creating an inversion in the atmospheres of UHJs, without the need for TiO or VO. Fe i detections have been reported in multiple UHJs, both in the terminators and on the dayside, through transmission spectroscopy (e.g. Hoeijmakers et al. 2018; Ehrenreich et al. 2020; Bourrier et al. 2020; Borsa et al. 2021) and emission spectroscopy (e.g. Pino et al. 2020; Scandariato et al. 2023), respectively. \nDetecting singly ionised iron (Fe ii ) is also of interest, as it could shed some light on the e ff ect that magnetic fields have on the atmosphere of UHJs (e.g. Perna et al. 2010), and inform modelling works regarding the ionisation (and recombination) fraction of iron. Fe ii has been detected in several hot giants via transmission studies (e.g. Hoeijmakers et al. 2019; Borsa et al. 2021; Prinoth et al. 2022; Bello-Arufe et al. 2022), however, it has only been detected once in emission observations of a UHJ. Borsa et al. (2021) reported Fe ii in the post-eclipse observation of KELT-20 b / MASCARA-2b, but follow-up studies have not been able to confirm the detection (Yan et al. 2022; Kasper et al. 2023; Petz et al. 2024). Further non-detections have been reported for KELT-9 b (Pino et al. 2020; Ridden-Harper et al. 2023), WASP-33 b (Cont et al. 2022), and WASP-121 b (Hoeijmakers et al. 2024). \nThe target of our study is WASP-76 b (West et al. 2016), an UHJ orbiting an F7-star (V = 9.52 mag) on a 1.81-day period, at a separation of 0.033 AU (the parameters of WASP-76 b and its host star are listed in Table 1). The equilibrium temperature of this exoplanet is Teq ∼ 2228 K (Ehrenreich et al. 2020), but as it is tidally locked, the dayside can reach temperatures up to Tdayside ∼ 2693 K (Garhart et al. 2020). At a radius of 1.83 RJup and a mass of 0.92 MJup, WASP-76 b is a benchmark UHJ that has been frequently investigated in recent years. Several atomic, ionised and molecular species have been detected in the atmo- \nre of WASP-76 b, which are summarised in Table 2. Furthermore, this exoplanet has been the target of phase curve observations (Tsiaras et al. 2018; Garhart et al. 2020; May et al. 2021; Fu et al. 2021; Demangeon et al. 2024) and global circulation models (GCMs) have been developed to explain the observed signatures and better understand the mechanics of this atmosphere (e.g. May et al. 2021; Savel et al. 2022; Wardenier et al. 2021, 2023; Schneider et al. 2022; Beltz et al. 2022b,a, 2023; Sainsbury-Martinez et al. 2023; Demangeon et al. 2024). \nIn particular, the detection of Fe i by Ehrenreich et al. (2020) motivates our work. The authors first reported an asymmetrical Fe i detection in the terminators of WASP-76 b, when analysing high-resolution transmission spectra from ESPRESSO 1 (Pepe et al. 2021). This finding was supported by Kesseli & Snellen (2021); Kesseli et al. (2022), and Pelletier et al. (2023), who independently recovered the asymmetrical signature. To explain why Fe i is observed on the evening limb but not on the morning limb, Ehrenreich et al. (2020) suggest that this species must condense as it crosses the colder nightside, and thus should exist in gaseous form on the dayside of WASP-76 b. Furthermore, Pelletier et al. (2023) also report a tentative detection of Fe ii in transmission studies of the atmosphere. Thus we aim to determine whether iron, in both the neutral and ionised forms, is present on the dayside of this planet via emission spectroscopy. \nTo the extent of our knowledge, only one high-resolution emission study has been published for this target so far. Yan et al. (2023) analysed data from CRIRES + 2 (Dorn et al. 2023) and reported the detection of CO and weak H2O emission features. The CO signature was slightly redshifted (2.1 + 0 . 8 -0 . 7 km / s), whereas the H2O signal had a small blueshift ( -5 + 12 -10 km / s, though we note the large uncertainty). These emission lines confirmed the existence of a thermal inversion on the dayside of WASP-76 b, as had been suggested from previous low-resolution spaceborne observations (Edwards et al. 2020; May et al. 2021). \nHigh-resolution spectroscopy has been proven fundamental in the endeavour of characterising atmospheres, as it allows us to distinguish a forest of individual lines in the spectra. Additionally, due to the di ff erent Doppler shifts of the stellar and planetary signal over time, we are often able to disentangle the stellar contribution from the planetary features in high-resolution data (Snellen et al. 2010). In some cases, the line strength is large enough to allow for the analysis of lines individually (Wyttenbach et al. 2015; Allart et al. 2018; Seidel et al. 2019; Allart et al. 2023). However, for weaker spectral features, we can instead employ a cross-correlation function (CCF) to harvest the signal of multiple lines at once, instead of individually (e.g. Snellen et al. 2010; Brogi et al. 2012; Birkby et al. 2013; Hoeijmakers et al. 2015; Allart et al. 2017; Ehrenreich et al. 2020; Azevedo Silva et al. 2022; Prinoth et al. 2023). Since all the planetary lines are shifted by the same radial velocity (RV) value, the robustness of the CCF is enhanced, and so is the significance of the detections. \nIn this work, we present the first analysis of high-resolution emission spectra at visible wavelengths for WASP-76 b obtained with ESPRESSO at the Very Large Telescope (VLT). The main goal is to probe the existence of Fe i and Fe ii in the exoplanet's dayside, which could be contributing to the inverted pressuretemperature (P-T) profile. The thermal inversion layer has been previously confirmed in Yan et al. (2023) from H2O emission \nTable 1. Properties of the planet WASP-76 b and its stellar host. \nlines, but observing the emission features of iron can help further constrain the thermal structure of the planet's dayside. \nThis paper is structured as follows: in Sect. 2, we describe the details of our observations; in Sect. 3, we present the data reduction, processing, and analysis steps for computing the CCFs; in Sect. 4, we report the results of the analysis; we compare our findings to the literature in Sect. 5 and propose an atmospheric scenario that could link our results with those from the literature; finally, we summarise and conclude in Sect. 6.", '2.1. ESPRESSO spectroscopic data': "The WASP-76 system was observed on four di ff erent epochs, two of them covering phases before the planet's secondary transit (2021 September 9 and 2022 October 14) and two after (2021 September 2 and 2022 October 18). The data were obtained with the ESPRESSO spectrograph (Pepe et al. 2021) installed at the VLT, at Cerro Paranal, as part of the programmes 1104.C0350(U) and 110.24CD.004 of the Guaranteed Time Observation. The ESPRESSO mode was set to HR21 (1 unit telescope, 2 × 1 binning), with resolving power of R ∼ 140 000. Fibre A was pointed to our target whereas Fibre B was pointing towards the sky to allow for sky subtraction. Each spectrum covers the visible wavelength range from ∼ 3800 Å to ∼ 7880 Å. \nThis resulted in a total of 148 high-resolution spectra, evenly split between the pre- and post-eclipse observations (74 spectra probing each phase range). The observations lasted between 3.5 h and 3.75 h for each epoch, with the individual exposure time set to 300 seconds. The left panel of Fig. 1 illustrates the orbital phases covered by the observations. We label each epoch \nas data sets I, II, III, and IV, in order of increasing orbital phase at the start of observations. The right panels of this figure show the variation of airmass in each epoch, as well as the signal-to-noise ratio (S / N) of each data set (taken from order 106 of ESPRESSO, at wavelength ∼ 5800 Å). The observation logs are summarised in Table 3.", '2.2. EulerCam photometric data': "We searched for potential photometric variability around the time of the ESPRESSO observations to rule out the presence of strong active regions on the star that could contaminate the retrieved emission spectrum. For this, we observed WASP-76 with EulerCam (Lendl et al. 2012), a 4k × 4k CCD detector installed at the Cassegrain focus of the 1.2-m Leonhard Euler Telescope at ESO's La Silla Observatory. We monitored the star every third night in two di ff erent monitoring campaigns: i) from 2021 June 20 to 2022 January 20, and ii) from 2022 October 1 to 2023 January 10. Each monitoring observation involved taking a sequence of six images in five di ff erent filters, with exposure times indicated in parenthesis: JohnsonB (60 s), JohnsonV (30 s), Sloanr ' (20 s), Sloani ' (30 s) and Sloanz ' (40 s). \nThe raw full-frame EulerCam images for each night of observation are corrected for over-scan, bias and flat-field using the standard reduction pipeline (Lendl et al. 2012). The aperture photometry is performed using circular apertures with radii ranging from 16 to 80 pixels, placed on the target star and three bright stars in the field of view. To mitigate the changing position of stars on the detector, the placement of apertures was performed using their astrometric solution (Lang et al. 2010). The optimal aperture for each night is found by minimising the \n<!-- image --> \nFig. 1. Details of the observations of WASP-76 b with ESPRESSO. Left: Orbital diagram of WASP-76 b showing the epochs and phases ( ϕ ) during which the system was observed. The curved arrow indicates increasing orbital phase for the planet. Right: Variation of signal-to-noise ratio around 5800 Å ( top ) and airmass ( bottom ) for each epoch observed. See more details in Table 3. \n<!-- image --> \nphotometric scatter of the consecutive images. The light curves from JohnsonB and Sloanz ' were removed as they su ff ered from strong systematics. The normalised di ff erential light curves for JohnsonV , Sloanr ' , and Sloani ' are shown in Fig. 2. \nWASP-76 appears to be photometrically quiet around the ESPRESSO eclipse observations, with upper limits on the flux variability from the light curves being 4.95 mmag ( V ), 7.17 mmag ( r ' ), and 5.55 mmag ( i ' ). We note that our observations cannot resolve the two components of the WASP-76 binary system (Wöllert & Brandner 2015). The companion star, WASP76 B, is likely a late-G or early K-type dwarf (Ehrenreich et al. 2020), and it lies at a separation of ∼ 0.44 '' , which corresponds to 53.0 ± 8.8 AU (Ginski et al. 2016; Ngo et al. 2016; Bohn et al. 2020). Though the stars are unresolved, our analysis indicates no photometric activity in general, which would suggest both components A and B are quiet.", "3.1. Reducing ESPRESSO data and extracting planet's spectra": "The raw data were reduced with the ESPRESSO Data Reduction System (DRS, version 3.0.0, Pepe et al. 2021). We proceeded to analyse the S1D sky-subtracted spectra produced by this pipeline, which is in the rest frame of the barycenter of the Solar System. In S1D, all orders of the spectrograph have been merged into a single 1D spectrum for each exposure. Our analysis follows a similar procedure to previous works, and its seven steps are detailed below: \n- -Remove telluric contamination. We removed the telluric contamination using Molecfit (version 4.2, Smette et al. 2015; Kausch et al. 2015) with the ESPRESSO settings. In some regions of the spectra, the telluric features are completely saturated, which makes it impossible to apply a correction. In the following wavelength \nranges (in air), we could not achieve a satisfactory correction and thus they are masked at later stages when calculating the CCF: [5867.56 -6005.55] Å, [6270.23 -6344.15] Å, [6439.08 -6606.96] Å, [6858.15 -7417.40] Å, [7586.03 -7751.12] Å. \n- -Normalise spectra. \nThe continuum contribution was removed from the spectra with RASSINE (Cretignier et al. 2020), via the S-BART Python package (Silva et al. 2022), which optimises the process for ESPRESSO data. This step removes the interference patterns that have been reported to a ff ect the continuum of ESPRESSOobservations (commonly referred to as wiggles). \n- -Fit for the systemic velocity. \nWe found that there was a discrepancy in the systemic velocity (vsys) values presented in the literature. The discovery paper reports this value as -1.0733 ± 0.0002 km / s (West et al. 2016); SIMBAD gives -1.152 ± 0.0033 km / s, from the Gaia Data Release 2 (Soubiran et al. 2018); and the analysis of Ehrenreich et al. (2020) states di ff erent values for each epoch observed: -1.162, -1.167, and -1.171 km / s (with typical uncertainty of the order of 0.002 km / s). This value warrants attention as it is an important parameter to accurately shift the spectra between the di ff erent rest frames. Thus we chose to perform a simple least-squares fit of a Keplerian to the RV values calculated by the pipeline, setting vsys as the only free parameter (the remaining orbital elements were set to those reported in Table 1). We obtained four values of vsys: -1.2113, -1.2134, -1.2064, -1.2100 km / s, for epochs I, II, III, and IV, respectively. The typical uncertainty obs, though the real uncertainty on vsys is expected to be much greater. In the following steps, the data of each epoch was processed \n- tained from this fit is of the order of 0.0003 km / using the corresponding vsys retrieved here.\n- -Create stellar template.\n- All of the spectra were shifted to the stellar rest frame and the median spectrum was computed for each epoch inde- \nFig. 2. Photometric monitoring of WASP76: EulerCam light curves in JohnsonV (top), Sloanr ' (middle), and Sloani ' (bottom) filters, unbinned (red) and binned per day (blue). WASP-76 can be considered photometrically quiet around the time of the ESPRESSO observations (vertical dashed lines). \n<!-- image --> \npendently. The median spectra contain only the stellar lines, which are aligned at the same position in the star's rest frame, increasing their signal. The planetary lines are very faint and their position on the spectra changes significantly over time (the planet's RV changes by ∼ 2.4 km / s between the start of two consecutive exposures). Thus their contribution is diluted when computing the median template in the stellar rest frame, averaging out to values comparable to the noise. \n- -Extract planet's spectra. \nTo obtain spectra that contain only the planetary features, we interpolate all spectra to a common wavelength grid and subtract the stellar median template from every spectrum of the corresponding epoch, in the stellar rest frame. \n- -Compute cross-correlation.\n- Wecompare each exposure of the planetary spectra with syn(Mollière et al. 2019, 2020; Alei et al. 2022, see next section), by computing \nthetic models created with petitRADTRANS a non-weighted CCF according to: \nCCF(RV) = X i simi (RV) (1) \nwhere si is each data point in the planet's spectrum, and mi (RV) is each data point of the model shifted by a given RV lag. At this stage, each spectrum produces one CCF curve, with velocities ranging from -300 to 300 km / s, with a step of 1 km / s. \n- -Co-add CCFs in the planet rest frame. \nThe last step is to shift the individual CCFs to the planetary rest frame, sum them, and assess if there is a detection. The planetary signal has a small amplitude in the individual \nCCFs, but given that the CCF peaks are expected to align when working in the planet's rest frame, then the summation can provide detections at a higher confidence level. We shift the CCFs according to the Keplerian motion of the planet, computed using the parameters shown in Table 1. Our analysis is twofold: firstly, we analyse the co-added CCF resulting from each independent epoch (we combine the 39 CCFs of epoch I into one, the 35 CCFs of epoch II into one, and so on); secondly, we construct a co-added CCF without the separation of epochs, so we combine the 148 CCFs into one. A detection of emission lines will manifest itself as a positive peak at (or close to) RV = 0 km / s, since we are analysing it in the planet's rest frame.", '3.2. Synthetic models for CCF': 'The synthetic models to which we compared the observations were produced with the Python package petitRADTRANS (version 2.7.7) (Mollière et al. 2019, 2020; Alei et al. 2022), which can calculate both transmission and emission spectra of exoplanets. We chose the high-resolution mode ("lbl", λ/ ∆ λ = 10 6 ) to better match our observations, and created separate emission templates with the spectral lines of Fe i and Fe ii (opacities were contributed to petitRADTRANS by K. Molaverdikhani 3 , calculated from the line lists of R. Kurucz 4 ). We set the planetary parameters to those of WASP-76 b, assuming a hydrogen-helium \nFig. 3. Details of models created for the CCF computation. Left: Pressure-temperature (P-T) profile assumed for the dayside of WASP-76 b to create the templates of Fe i and Fe ii , hereafter PT01 (see Sect. 3.2), based on the GCM work of Wardenier et al. (2021, 2023). Right: petitRADTRANS models for Fe i ( top ) and Fe ii ( bottom ), assuming the thermal profile PT01. The shaded regions are excluded from the CCF calculation due to telluric residuals on the empirical spectra. \n<!-- image --> \nTable 2. Literature reports of atom, ion, and molecule detections in the atmosphere of WASP-76 b. \nReferences. (1) Azevedo Silva et al. (2022); (2) Pelletier et al. (2023); (3) Tabernero et al. (2021); (4) Casasayas-Barris et al. (2021); (5) Deibert et al. (2021); (6) Kesseli et al. (2022); (7) Deibert et al. (2023); (8) Ehrenreich et al. (2020); (9) Kesseli & Snellen (2021); (10) Lampón et al. (2023); (11) Seidel et al. (2019), (12) Žák et al. (2019), (13) Seidel et al. (2021), (14) Kawauchi et al. (2022), (15) Fu et al. (2021), (16) Yan et al. (2023), (17) Sánchez-López et al. (2022), (18) Tsiaras et al. (2018), (19) Fisher & Heng (2018), (20) Changeat et al. (2022), (21) Landman et al. (2021), (22) Edwards et al. (2020). \natmosphere with iron as the only trace species. We set both the Fe i and Fe ii abundances to be the solar abundance of Fe i (Lodders 2020), keeping this value constant in every layer of the \natmosphere. The P-T profile was based on the GCM work of Wardenier et al. (2021, 2023) (shown in Fig. 3). We define a two-point model, where the deep and outer atmospheres are represented by isotherms at 1700 K and 3500 K, respectively. The inversion layer is described by a gradient between 1 -100 mbar that connects the two isotherms. At a later stage, we diverge from this model and define new atmospheric profiles to evaluate the e ff ect it has on the CCF signature. \nThe package petitRADTRANS provides spectra in units of spectral flux density (erg cm -2 s -1 Hz -1). We transform them into contrast models (Fplanet / Fstar) by multiplying by the area of the planet disk and dividing by the flux of the star (modelled as a blackbody with Teq = 6329 K. Figure 3 displays the models of Fe i and Fe ii used in the CCF calculations. The models are interpolated to the same wavelength grid as the observations when computing the CCFs.', '4. Results': "The resulting CCFs for neutral and ionised iron are presented in Fig. 4. In the top panels, the CCFs have been co-added in the planet's rest frame for each epoch, calculating the RV shift with the Keplerian solution defined using the parameters presented in Table 1. The CCFs were then converted into a S / N scale by calculating the standard deviation of the baseline (between [ -115, -35] km / s and [35, 115] km / s) and dividing everything by this measurement. Furthermore, we fit a simple 1D Gaussian curve locally around zero to retrieve the RV value of the signature. In the middle panels, we show the CCF resulting from adding all curves from the four epochs in the planet's rest frame (normalised by the baseline after the summation), also fitted with a Gaussian curve. Lastly, in the bottom panels, we produce the Kp - vsys plot. For this, we add all 148 CCFs in the planet's rest frame, but for each row of the plot, we compute the Keplerian solution assuming a di ff erent value for the semi-amplitude velocity, Kp (all other parameters remain constant, see Table 1). If the theoretical orbital parameters are correct and no atmospheric dynamics are detectable in the exoplanet, we should expect a detection at the intersection of Kp = 196.52 km / s and vsys ∼-1.2 km / s (traced by the dashed lines in the figure). \nTable 3. Observation logs. \nNotes. ∗ Values taken from order 106 of ESPRESSO, at ∼ 5800Å. \nTable 4. Results from the Gaussian fits to the CCFs.", '4.1. Detection of blueshifted Fe I in emission': "Wereport a 3.1 σ detection of Fe i in epoch I, and tentative detections in epochs II, III, and IV, with significances of 1.8 σ , 2.4 σ , and 2.4 σ , respectively. The S / Nof these detections is rather low, and other peaks are visible in the co-added CCFs. Some of these peaks lie slightly above what could be considered the noise level, at values distant from the zero-velocity point. The CCF process itself can introduce artefacts resembling detection peaks when lines from the template randomly match with the empirical features of other species scattered throughout the planet's spectra (Borsato et al. 2023). However, in this case, it seems that the noise in the continuum is mostly dominated by red noise. \nThe weakest detection of the individual epochs comes from epoch II, which is puzzling at first because these observations had the best seeing conditions of the four epochs observed, and the spectra have the best S / N. However, this was the only epoch observed with UT2 of the VLT, whereas UT1 was used for the other epochs. UT2 tends to be more a ff ected by the interference pattern created in ESPRESSO (wiggles), therefore the correction of this e ff ect might have left more residuals compared to the other epochs, in turn leading to a less significant detection. Prinoth et al. (2023) reported similar quality issues between data from UT1 and UT2 (see their Appendix A). \nWhen combining all epochs, the neutral iron detection is much clearer, at a S / N of 6.0 σ (see Fig. 4). Table 4 summarises the significance of our findings. Lastly, in the Kp - vsys plot (Fig. A.1), Fe i is detected at the expected Kp value, with a blueshifted vsys, and no other strong peaks are found in the explored parameter space. The detection of emission lines confirms the existence of a thermal inversion layer in the dayside of WASP-76 b. The inversion had been previously hinted at by Edwards et al. (2020); May et al. (2021); Fu et al. (2021), and confirmed by Yan et al. (2023) using near-infrared CRIRES + emission spectroscopy. Our data support their findings. \nFurthermore, we report that our Fe i detection is blueshifted with respect to the planetary rest frame. The CCF for the four epochs produces a peak at an RV of -4.7 ± 0.3 km / s (full width at half maximum, FWHM, of 10.6 km / s). For the co-added CCFs of each epoch, the Gaussian fits are centred at -4.8 ± 0.3, -8.0 ± 0.9, -4.5 ± 0.5, and -1.3 ± 0.8 km / s, with FWHM values \nof 6.3, 13.5, 8.8, and 14.8 km / s, respectively for epochs I, II, III, and IV (see Fig. 4). We note that there is some scatter in the RV shift observed from epoch to epoch, though it is unclear if the di ff erences are caused by the low S / N of the planetary signature or if they indicate a physical variation of the atmosphere. Moreover, our uncertainties are likely underestimated as they represent only the nominal error of the Gaussian fit. It is di ffi cult to properly account for the uncertainties created by the correlated noise of the CCF. Notwithstanding, all the observations presented have clear blueshifted peaks that are identified consistently across the epochs. There seems to be a drift of ∼ 3 km / s between the observations of 2021 (II and III) and 2022 (I and IV) for the same phases. However, it is not possible to draw any significant conclusions at this point. In Fig. 5, we present the combined CCFs by phase range, for pre- or post-eclipse. The observations after the eclipse reveal a smaller shift, -3.3 ± 0.5 km / s (FWHM = 11.9 km / s) than those obtained before the eclipse, -6.0 ± 0.4 km / s (FWHM = 9.4 km / s). The observations of epochs I and II (pre-eclipse) only overlap partially in phase coverage, so this could indicate a gradual change as the planet rotates. However, even with epochs covering very similar phase ranges, such as epochs III and IV, the RV peaks of their CCFs are discrepant at the 5 σ level. The Gaussian fits on epochs I (pre) and III (post) show less broadening than for II (pre) and IV (post), so we consider the fits of I and III to be more significant than their counterparts at the same phase. This implies a consistent blueshift in the emission signal from both the east and west dayside hemispheres of WASP-76 b of about -4.6 km / s (see Fig. 4). Overall, further observations would be useful to confirm if these velocity discrepancies are the result of physical processes, and to shed more light on what could be causing them. \nWe considered if adopting an eccentric orbit, instead of a fully circular orbit, would alter the results and eliminate the blueshift. Given the age of the system (1.8 Gyr, Ehrenreich et al. 2020), we expect the orbit to have circularised (see favourable arguments for this orbital solution in Ehrenreich et al. 2020, Methods, but see also Valente & Correia 2022). Thus we assumed e = 0 in the calculations described so far. However, a slightly eccentric orbit might perhaps explain this signature. Furthermore, Savel et al. (2022) reported that allowing for a small eccentric- \nFig. 4. Results of the CCF analysis for Fe i ( left , detection) and Fe ii ( right , non-detection). Top: Summed CCFs in the planet rest frame, separated by epoch, with 1D Gaussian fits (black lines). Fe i is tentatively detected at a 2 -3 σ level in each epoch and all epochs show a blueshifted signal. Middle: Summed CCFs of the four epochs in the planet rest frame, with no epoch separation. Fe i shows a 6.0 σ detection, as traced by the 1D Gaussian fit (black line). Bottom: Kp - vsys plot of the four epochs combined (see Fig. A.1 for the individual epochs). The black dashed lines indicate the expected position of the signal. The location of the strongest signal is pinpointed by the blue dashes, for the case of Fe i . \n<!-- image --> \nity of 0.01 was a necessary adjustment to reproduce the transit signature of Fe i in WASP-76b (Ehrenreich et al. 2020), combined with high-altitude, optically thick clouds of Fe i , Al2O3, and Mg2SiO4. So we investigated the possibility of WASP-76 b having an eccentric orbit. \nThe phase curve of WASP-76 b was recently observed with CHEOPS 5 (Demangeon et al. 2024). These authors placed an upper limit on the eccentricity of e = 0.0067. We set the eccentricity to this upper limit, computed the CCFs with the newly shifted \nspectra, and compared the two cases. In Fig. 6, we present the coadded CCFs for the cases of e = 0 and e = 0.0067. The change in RV of the CCF peaks of individual epochs is between 0 km / s and 3 km / s (top panel), with the peak in epochs I and II being less blueshifted compared to the zero-eccentricity case, and epochs III and IV being more blueshifted. When all CCFs are co-added (bottom panel), the change is negligible. As the blueshift remains present in all cases, we rule out the possibility that this signature is due to unaccounted-for eccentricity. In Sect. 5, we discuss further possible origins for the blueshifted Fe i .", '4.2. Non-detection of Fe II in emission': 'Wedid not detect the presence of Fe ii , even when adding the 148 CCFs from all epochs (see right panels of Fig. 4). Fe ii has been detected in several UHJs via transmission spectroscopy, thus it is expected to be present on the daysides of these planets. However, it is expected to be more abundant at high altitudes, where Fe i is ionised by the hotter temperatures. These atmospheric layers are more challenging to probe with emission spectroscopy due to being more optically thin. Thus it is not surprising we could not detect Fe ii emission on WASP-76 b, even if we expect it to be present.', '4.3. Constraining P-T profile': 'Once we had confirmed the presence of Fe i in emission, we investigated how changes in the P-T profiles would a ff ect the CCF peak. Thus we defined ten di ff erent P-T profiles, computed the synthetic template with petitRADTRANS , and calculated the cross-correlation for all epochs. Starting from the initial P-T profile (PT01, see Fig. 3 and Sect. 3.2, Wardenier et al. 2021, 2023), we modified either the temperature of the lower layer of the atmosphere, the temperature of the upper layer, the pressure boundaries of the inversion layer, or a combination of these. The characteristics of each P-T profile are shown in the bottom left panel of Fig. 7 and listed in Table B.1. The corresponding synthetic templates that were utilised for the CCF can be found in Fig. B.1. Despite the fact the Fe i opacities in petitRADTRANS were only calculated for temperature values up to 4000 K, we decided to go above this value in some of the P-T profiles. This means that the opacities used by petitRADTRANS when T > 4000 K are the same ones as for T = 4000 K. However, the radiative source function and the atmospheric scale height are not constant for temperature values that extend beyond the pre-defined P-T grid, thus they still hold some useful information. \nWe co-added the 148 CCFs corresponding to each P-T template (top panel of Fig. 7) and fitted the resulting peaks with Gaussian curves. The Fe i detection can be found in all tested profiles except for PT02, which corresponds to the profile with the inversion temperature located deep in the atmosphere. As expected, the significance of the detection changes in the di ff erent cases, with the weakest detection stemming from PT05, another profile that assumes the inversion temperature deeper in the atmosphere, though not as deep as for P0T2. \nModifying the temperature of the lower or upper layers seems to have a limited impact on the significance, as the values only increase or decrease slightly compared to the base model PT01. The case that produces the strongest detections is the one where the temperature of the lower layer was reduced to Tinner = 1200 K. \nFig. 5. Summed CCF curves of the epochs before the eclipse ( top ) and after ( bottom ). The Fe i signal is more blueshifted in phases before the eclipse, and becomes less blueshifted after. \n<!-- image --> \nA further visual investigation of the models revealed that many lines were saturated. This is likely due to the cut-o ff temperature defined for the isotherm at lower pressures. Strong lines that reach the blackbody curve defined by this isotherm thus have the same brightness temperature and appear saturated. In principle, if the lines were not saturated, the significance of the Fe i detection would change, but it would not alter our ultimate scientific conclusion that neutral iron is detected on WASP-76 b.', '5.1. Blueshifted Fe I emission': "The Fe i emission signature we report on the dayside of WASP76 b is blueshifted by ∼ -4.7 km / s. Similar blueshifted signals have been identified in recent works, such as in CO and H2O in WASP-77Ab (Line et al. 2021), and H2O in our target, WASP76 b (Yan et al. 2023). However, the mechanism behind them has not been investigated so far. \nThe Doppler shift observed here cannot be explained by an eccentricity of the orbit, which is very close to zero (see Sect. 4.1). It also does not trace solely the day-to-night wind proposed in recent transmission studies (Seidel et al. 2019, 2021; Ehrenreich et al. 2020), which would appear redshifted in dayside observations (in the planet rest frame). Moreover, this wind is seen at the atmospheric limbs with transit spectroscopy, whereas our observations are most sensitive to integrated dayside emission. It seems to require that additional components be added to the atmospheric dynamics scenario, to ensure that the final diskintegrated signature is blueshifted. In this section, we propose a simple scenario of atmospheric circulation that could explain our observations, and we illustrate the proposed dynamics of the planet's atmosphere in Fig. 8. \nOur interpretation is that material on the dayside of WASP76 b is moving towards the observer, with similar magnitude on both the east and west hemispheres. The day-to-night heat redistribution pattern proposed by GCMs for UHJ in the presence of drag (e.g. Wardenier et al. 2021) would have an RV component close to zero at orbital phases close to the eclipse, when the dayside disk is almost perpendicular to the observer's perspective. \nIn emission spectroscopy, observations are more sensitive to the hottest region of the dayside, in the vicinity of the substellar \n<!-- image --> \nFig. 6. CCFs of Fe i calculated for the cases of e = 0 and e = 0.0067 (CHEOPS upper limit, Demangeon et al. 2024). The CCFs in the top panel are separated by epoch, and the di ff erences in the RV peaks between these eccentricity scenarios are between 0 and 3 km / s. In the bottom panel, all epochs are co-added, and the di ff erence between the two peaks is negligible. Assuming a slightly eccentric orbit, rather than a circular orbit, does not eliminate the blueshifted signature seen for Fe i . \n<!-- image --> \npoint. As such, we could be detecting Fe i atoms that are rising in the atmosphere, in a radial motion from the inner to the outer atmospheric layers. This displacement is possibly generated by the hotspot at or close to the substellar point. A series of works by Seidel et al. (2019, 2020, 2021) has shown that vertical upwards winds in the upper atmosphere of UHJs, including WASP-76b, were the likely cause for the broadening observed in the Na I doublet, being of the order of 20 km / s. Moreover, if these hotter parcels of atmosphere are transported from the dayside to the nightside of the planet, via the day-to-night wind or the super-rotating equatorial jet, the atoms would be detectable at the terminator via transmission spectroscopy, as was reported by Ehrenreich et al. (2020); Kesseli & Snellen (2021); Kesseli et al. (2022); Pelletier et al. (2023). Furthermore, if these atoms condense when reaching the nightside, due to the cooler temperatures, it could explain the glory e ff ect reported by Demangeon et al. (2024) from the light curve analysis. \nThis could mean that the hotspot is generating the upward displacement of a substantial amount of Fe i atoms. SainsburyMartinez et al. (2023) investigated GCMs models that included \nvertical transfer of heated material in the outer atmosphere of UHJs. They concluded that the di ff erences between the day and nightside temperatures ultimately lead to transport in the upward direction on the dayside, whereas the nightside sees a downward motion. However, these authors assumed a P-T profile that combined an adiabat for the deep atmosphere and an isotherm for the outer layers, with no inversion layer present. Had a thermal inversion been included, we do not know to what extent it would alter their findings. \nAs mentioned in the previous section, there is some doubt as to whether the changes in RV shift from one epoch to another are real, or if they are a product of the small S / N residual contamination from activity, or a combination of these, for example. How (or if) the magnitude of the shift varies can help us locate the hotspot of WASP-76 b. If we consider that the planetary signal is more blueshifted before the eclipse than after (see Fig. 5), that could mean the hotspot is shifted to the west. As exemplified in Fig. 8, a westward shift would create a stronger blueshift signature before the occultation. Whereas if we analyse the CCFs for each epoch separately, one can argue that the peaks on epochs II and IV have a less constrained fit than those of epochs I and III, thus we should only consider the two latter results in our discussion. In this case, the blueshift has the same magnitude across both phases, which can point to the lack of an o ff set for the hotspot. With a hotspot that is located at or close to the substellar point, observations before and after the occultation would reveal a mirrored wind structure and thus create a similar atmospheric RV shift overall. May et al. (2021) have reported a negligible o ff set of the hotspot for WASP-76 b, though this conclusion was drawn from Spitzer data which is potentially probing a di ff erent altitude in the atmosphere. Beltz et al. (2022b) show that the hotspot o ff set can be reduced as a result of applying more sophisticated active magnetic drag treatments over more approximate ones. This goes to show the importance of studying planetary magnetic fields and their impact on atmospheric circulation. On the other hand, Wardenier et al. (2021) and Savel et al. (2022) required a hotspot o ff set in their GCMs in order to reproduce the observational findings of transmission spectroscopy. More observations of the dayside of WASP-76 b with higher S / N and better time resolution are necessary to constrain the position of the hotspot. \nTo further dive into the atmosphere of WASP-76 b, it would be necessary to develop a retrieval framework, such as those presented in Brogi & Line (2019); Seidel et al. (2020, 2021); Pelletier et al. (2021); Gandhi et al. (2022), adapted to emission spectroscopy. Such a task is outside the scope of this paper. GCM studies are also an e ffi cient tool to unravel the underlying atmospheric phenomena at play in the atmospheres of UHJs. Wardenier et al. (2021, 2023) have developed a 3D model for WASP-76b, but have produced only the transmission spectra for comparison with the already published transit data (Ehrenreich et al. 2020). Producing GCM to delve into the scorching dayside of this planet is, likewise, not the goal of this observational paper, though we strongly encourage this e ff ort.", '5.2. Lack of Fe II detection': 'Fe ii has only been detected in the dayside of one other UHJ, KELT-20b / MASCARA-2b (Borsa et al. 2022), and only in the post-eclipse data. Despite this, follow-up studies were not able to find its signature on the same planet (Yan et al. 2022; Kasper et al. 2023; Petz et al. 2024). A non-detection has also been reported for the dayside of KELT-9 b by Pino et al. (2020) and Ridden-Harper et al. (2023). Cont et al. (2022) did not detect \nFig. 7. Analysis of the dependence of the iron detection on the model used to compute the CCF. Top: Four-epoch co-added CCFs for Fe i computed with di ff erent templates that correspond to each P-T profile (bottom left panel). Bottom left: P-T profiles (see also Fig. B.1 and Table B.1). Bottom right: Maximum S / N value of Gaussian fit to co-added CCFs of four epochs, computed with the Fe i template corresponding to each P-T profile. \n<!-- image --> \nFe ii on the emission spectra of WASP-33 b, and their injectionrecovery tests concluded that it would not be detectable in their data. For the case of WASP-121 b, a similar planet to WASP76 b, it was observed with ESPRESSO in eight di ff erent epochs, producing about two times as much spectra than we analysed in this work (Hoeijmakers et al. 2024), but Fe ii remained undetected in the dayside of this UHJ. \nFe i ionises at lower pressures and higher temperatures in the atmosphere, decreasing its abundance and increasing that of Fe ii . However, the upper layers of the atmosphere are more optically thin. This is the reason why transmission spectroscopy is a better method to explore them and has been a more successful technique at finding Fe ii . To probe Fe ii at these pressures with emission spectroscopy, it would require a greater abundance of this ion, or a larger number of strong lines available to probe it. Our Fe ii template in the optical regime contains ∼ 1200 lines, which is one order of magnitude smaller than the ∼ 11700 lines present in the Fe i template, a species that is more abundant further down in the atmosphere, and thus more amenable to be detected in thermal emission. The Fe ii lines become stronger for UHJs with higher equilibrium temperatures, which might make this ion traceable. The aforementioned works investigated planets hotter than WASP-76 b, and did not report detections. With \nWASP-76b sitting on the colder edge of the UHJ temperature range, it is expected that we cannot prove the existence of Fe ii on its dayside with CCF techniques. \nThe lack of confident detections or non-detections of Fe ii does not allow for any meaningful conclusions regarding a population trend. We stress that it would be of great interest to trace the ionised state of iron as we expect it to be a ff ected by planetary magnetic fields.', '6. Summary and conclusions': 'Weobserved the dayside of the ultra hot Jupiter WASP-76 b with ESPRESSO on four di ff erent epochs. We collected a total of 148 high-resolution emission spectra. Half of these were obtained just before the planet\'s secondary transit (phases 0.34 - 0.47), and the other half right after (0.54 - 0.62), providing insight into both the east and west hemispheres. This is the first emission spectroscopy study carried out for WASP-76 b at visible wavelengths. We also present monitoring data of WASP-76 from EulerCam, which shows that the host star (and its binary companion) are both quiet stars, with little photometric variation. \nOur main goal was to detect Fe i and Fe ii in emission on the dayside of WASP-76 b. We used the CCF method to compare \nFig. 8. Geometry of the WASP-76 b system. The planet is shown from a polar perspective, at di ff erent phases of its orbit, with the inflated dayside in yellow and the nightside in dark blue. The orange circle is the hotspot, for which we consider two possible locations: (A) at the substellar point (no o ff set) or (B) with a westward o ff set; to compare the possible atmospheric dynamics of each case. The arrows represent the motion of winds, and whether they appear blueshifted or redshifted to an observer on Earth. The dashed black line separates the hemisphere visible to the observer from the non-visible one. A proposed scenario of atmospheric circulation that could lead to the results of this work is presented in the text (see Sect. 5.1 for discussion). \n<!-- image --> \nthe observational data with synthetic models of these chemical species computed with petitRADTRANS . Furthermore, detecting emission features confirms the existence of an inverted atmospheric profile. We then investigated how the pressuretemperature profile impacted the emission signature, by computing synthetic templates based on varying P-T profiles and comparing the resulting CCFs. Our results are summarised as follows: \n- 1. We detect a blueshifted signature (-4.7 ± 0.3 km / s) of Fe i in nearly all epochs, with a detection significance of 6.0 σ from the co-added CCF of the four epochs.\n- 2. We confirm the existence of a thermal inversion layer, which follows from the fact that emission features can only be present if the dayside has an inverted structure.\n- 3. We report a non-detection of Fe ii . Due to the hot temperature of this planet, we expect this ion to exist in the outer atmosphere. However, the non-detection could be due Fe ii being more abundant in the outer atmosphere, which is optically thinner and thus harder to probe with emission spectroscopy. Follow-up studies are required to confirm it its presence.\n- 4. We discuss possible atmospheric scenarios to explain the blueshifted signature. We propose that material is being radially ejected from the hotspot, rising in the atmosphere, and proceeding to the cooler nightside of the planet. We strongly encourage the development of GCM studies that could reproduce this feature.\n- 5. Based on the change of RV shift measured in individual epochs, we are unable to constrain whether the hotspot is located at the substellar point or if it is o ff set. Further observations are required to disentangle the two scenarios.\n- 6. The Fe i signal strength changes when we compare the CCFs resulting from di ff erent assumptions of P-T profile. The \nstrongest significance is attributed to a profile where the temperature of the upper atmosphere is 3500 K and the lower atmosphere is 1200 K, with the inversion located between 1 -100 mbar. \nIn recent years, emission spectroscopy has often been seen as ill-favoured compared to transmission spectroscopy due to the glaring di ff erence in signal strength. However, it is the best spectroscopic avenue to probe the atmospheric structure and composition of exoplanets\' dayside, thus being a crucial technique for exoplanet research. In the lead-up to the next generation of ground-based spectrographs, such as ANDES at the Extremely Large Telescope (Marconi et al. 2022), we provide an example of how useful ground-based high-resolution instruments are for characterising exoplanets. We highlight, in particular, the capability of ESPRESSO to probe the dayside of ultra hot Jupiters, in order to constrain their chemical composition and probe the dynamics of atmospheric circulation. \nAcknowledgements. We thank the anonymous referee for their comments which helped improve the manuscript. The authors acknowledge the ESPRESSO project team for its e ff ort and dedication in building the ESPRESSO instrument. This work was supported by the Fundação para a Ciência e Tecnologia (FCT) and POCH / FSE through the research grants UIDB / 04434 / 2020 and UIDP / 04434 / 2020, and in the framework of the project 2022.04048.PTDC (Phi in the Sky, DOI 10.54499 / 2022.04048.PTDC). This work was co-funded by the European Union (ERC, FIERCE, 101052347). This project has received funding from the European Research Council (ERC) under the European Union\'s Horizon 2020 research and innovation programme (project S pice D une , grant agreement No 947634). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. The authors acknowledge the financial support of the Swiss National Science Foundation (SNSF). This work has also been carried out within the framework of the National Centre of Competence in Research (NCCR) PlanetS supported by the SNSF un- \nthe grants 51NF40\\_182901 and 51NF40\\_205606. A.R.C.S. acknowledges support from FCT through the fellowship 2021.07856.BD. O.D.S.D. is supported in the form of a work contract (DL 57 / 2016 / CP1364 / CT0004) funded by national funds through FCT. C.L. and F.P. would like to acknowledge the SNSF for supporting research with ESPRESSO through the SNSF grants nr. 140649, 152721, 166227, 184618 and 215190. The ESPRESSO Instrument Project was partially funded through SNSF\'s FLARE Programme for large infrastructures. H.C. and M.L. acknowledge the support of the SNCF under grant number PCEFP2\\_194576. M.R.Z.O. and E.H-C. acknowledge financial support from the Agencia Estatal de Investigación (AEI / 10.13039 / 501100011033) of the Ministerio de Ciencia e Innovación and the ERDF "A way of making Europe" through project PID2022-137241NB-C42. E.H-C. acknowledges support from grant PRE2020-094770 under project PID2019-109522GB-C51 funded by the Spanish Ministry of Science and Innovation / State Agency of Research, MCIN / AEI / 10.13039 / 501100011033, and by ERDF, "A way of making Europe". A.S.M. acknowledges financial support from the Government of the Canary Islands project ProID2020010129. A.S.M., J.I.G.H., and R.R. acknowledge financial support from the Spanish Ministry of Science and Innovation (MICINN) project PID2020-117493GB-I00. S.G.S. acknowledges the support from FCT through Investigador FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC) (DOI: 10.54499 / CEECIND / 00826 / 2018 / CP1548 / CT0002)). S.C. acknowledges financial support from the Italian Ministry of Education, University, and Research with PRIN 201278X4FL and the "Progetti Premiali" funding scheme. C.J.A.P.M. acknowledges FCT and POCH / FSE (EC) support through Investigador FCT Contract 2021.01214.CEECIND / CP1658 / CT0001. R.A. is a Trottier Postdoctoral Fellow and acknowledges support from the Trottier Family Foundation. This work was supported in part through a grant from the Fonds de Recherche du Québec - Nature et Technologies (FRQNT). This work was funded by the Institut Trottier de Recherche sur les Exoplanètes (iREx). T.A.S. acknowledges support from FCT through the fellowship PD / BD / 150416 / 2019. E.P. acknowledges financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN / AEI / 10.13039 / 501100011033 and the ERDF \'A way of making Europe\' through project PID2021-125627OBC32, and from the Centre of Excellence \'Severo Ochoa\' award to the Instituto de Astrofisica de Canarias. A.M.S. acknowledges support from FCT through the fellowship 2020.05387.BD. M.S. acknowledges financial support from the SNSF for project 200021\\_200726. E.E-B. acknowledges financial support from the European Union and the State Agency of Investigation of the Spanish Ministry of Science and Innovation (MICINN) under the grant PRE2020-093107 of the Pre-Doc Program for the Training of Doctors (FPI-SO) through FSE funds.', 'References': 'Alei, E., Konrad, B. S., Angerhausen, D., et al. 2022, A&A, 665, A106 Allart, R., Bourrier, V., Lovis, C., et al. 2018, Science, 362, 1384 Allart, R., Lemée-Joliecoeur, P. B., Jaziri, A. 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J., Ivanov, V. D., & Skarka, M. 2019, AJ, 158, 120\n- Wardenier, J. P., Parmentier, V., Lee, E. K. H., Line, M. R., & Gharib-Nezhad, E. 2021, MNRAS, 506, 1258\n- Wardenier, J. P., Parmentier, V., Line, M. R., & Lee, E. K. H. 2023, MNRAS, 525, 4942\n- West, R. G., Hellier, C., Almenara, J. M., et al. 2016, A&A, 585, A126 Wöllert, M. & Brandner, W. 2015, A&A, 579, A129\n- Wyttenbach, A., Ehrenreich, D., Lovis, C., Udry, S., & Pepe, F. 2015, A&A, 577, A62\n- Yan, F., Nortmann, L., Reiners, A., et al. 2023, A&A, 672, A107\n- Yan, F., Pallé, E., Reiners, A., et al. 2020, A&A, 640, L5\n- Yan, F., Reiners, A., Pallé, E., et al. 2022, A&A, 659, A7', 'Appendix A: K p - v sys plots.': 'Fig. A.1. Kp - vsys plots of Fe i and Fe ii for each epoch. The dashed black lines mark the expected position of the planetary signal. Top: Fe i . The signal is detected in all epochs, though at di ff erent amplitudes. Bottom: Fe ii . No detections are found (see Sect. 5.2 for discussion). \n<!-- image -->', 'Appendix B: P-T profile exploration': 'Table B.1. Details of the P-T profiles used to create templates in petitRADTRANS . \nNotes. Pressure (P) and temperature (T) values are given for the lower and upper boundaries of the thermal inversion layer. \nP (bar) \nFig. B.1. petitRADTRANS templates (right) corresponding to the P-T profiles (left), which were utilised for computing the Fe i CCFs. We note that the bottom four models are presented on a larger y-scale than the first six. See Table B.1 for more details on the atmospheric profiles. \n<!-- image -->'}

2024arXiv240908438B

We present the Wavefront Sensor units of the Gravity Plus Adaptive Optics GPAO system which will equip all 8m class telescopes of the VLTI and is an instrumental part of the GRAVITY project. It includes two modules for each Wavefront Sensor unit a Natural Guide Star sensor with highorder 40x40 ShackHartmann and a Laser Guide Star 30x30 sensor. The stateoftheart AO correction will considerably improve the performance for interferometry in particular highcontrast observations for NGS observations and allsky coverage with LGS which will be implemented for the first time on VLTI instruments. In the following we give an overview of the Wavefront Sensor units system after completion of their integration and characterization.

2024-09-01T00:00:00Z

['arXiv:2409.08438', '10.48550/arXiv.2409.08438', '2024arXiv240908438B']

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

GRAVITY Wavefront Sensors HighContrast Laser Guide Star Adaptive Optics systems for the VLTI

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.08438.pdf

{'GRAVITY+ Wavefront Sensors: High-Contrast, Laser Guide Star, Adaptive Optics systems for the VLTI': "Guillaume Bourdarot a , Frank Eisenhauer a,b , S¸enol Yazıcı a , Helmut Feuchtgruber a , Jean-Baptiste Le Bouquin d , Michael Hartl a , Christian Rau a , Jonas Graf a , Nikhil More a , Ekkehard Wieprecht a , Frank Haussmann a , Felix Widmann a , Dieter Lutz a , Reinhard Genzel a,l , Frederic Gont'e c , Sylvain Oberti c , Johann Kolb c , Julien Woillez c , Henri Bonnet c , Daniel Schuppe a , Amit Brara a , Johannes Hartwig a , Armin Goldbrunner a , Christoph Furchtsam a , Franz Soller a , Stefan Czempiel a , Johann Eibl a , David Huber a , Sinem Uysal a , Irmgard Treffler a , Hakan Ozdemir a , Vishaal Gopinath a,b , Pierre Bourget c , Anthony Berdeu h , Stefan Gillessen a , Thomas Ott a , Philippe Berio f , Olivier Boebion f , Florentin Millour f , Roderick Dembet h , Cl'emence ' Edouard h , Tiago Gomes k , Taro Shimizu a , Antonia Drescher a , Maximilian Fabricius a , Jinyi Shangguan a , St'ephane Lagarde f , Sylvie Robbe-Dubois f , Fatm'e Allouche f , Hugo Nowacki d , Denis Defr'ere e , Paulo J. V. Garcia k , Sebastian Hoenig j , Laura Kreidberg g , Thibaut Paumard h , and Christian Straubmeier i \na Max-Planck-Institut fur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching bei Munchen, Germany \nb Department of Physics, TUM School of Natural Sciences, Technical University of Munich, 85748 Garching, Germany \nc European Southern Observatory, Karl-Schwarzschild-Str. 2, Garching bei Munchen, Germany d Universit'e Grenoble Alpes, 621 Av. Centrale, 38400 Saint-Martin-d'H'eres, France e Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium f Univ. Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, Laboratoire Lagrange, France g Max-Planck-Institut fur Astronomie, Konigstuhl 17, 69117 Heidelberg, Germany h LESIA, Observatoire de Paris, Universit'e PSL, Sorbonne Universit'e, Universit'e Paris Cit'e, CNRS, 5 place Jules Janssen, 92195 Meudon, France \ni 1st Institute of Physics, Universitat zu Koln, Zulpicher Straße 77, 50937 Koln, Germany j University of Southampton, University Road, Southampton, SO17 1BJ, United Kingdom k Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Frias, 4200-465 Porto, Portugal and CENTRA - Centro de Astrof'ısica e Gravita¸c˜ao, IST, Universidade de Lisboa, 1049-001 Lisboa, Portugal \nl Departments of Physics and Astronomy, University of California, Berkeley, USA", 'ABSTRACT': 'We present the Wavefront Sensor units of the Gravity Plus Adaptive Optics (GPAO) system, which will equip all 8m class telescopes of the VLTI and is an instrumental part of the GRAVITY+ project. It includes two modules for each Wavefront Sensor unit: a Natural Guide Star sensor with high-order 40x40 Shack-Hartmann and a Laser Guide Star 30x30 sensor. The state-of-the-art AO correction will considerably improve the performance for interferometry, in particular high-contrast observations for NGS observations and all-sky coverage with LGS, which will be implemented for the first time on VLTI instruments. In the following, we give an overview of the Wavefront Sensor units system after completion of their integration and characterization. \nKeywords: interferometry, adaptive optics, high-contrast, laser guide star, GRAVITY, VLTI \nCorrespondence to: Guillaume Bourdarot (bourdarot at mpe.mpg.de)', '1. INTRODUCTION': "The GRAVITY+ project is a major upgrade of the GRAVITY instrument and of the VLTI facility, and will improve by orders of magnitude of sensitivity, contrast and sky coverage of GRAVITY [1] and VLTI instruments as a whole [2]. At the heart of the project is the implementation of Gravity Plus Adaptive Optics (GPAO). Early on in the development of interferometry, AO was identified as a crucial system: unlike small telescopes which are limited by the magnitude of fringe tracking, the limiting magnitude for large telescopes is set by the AO [3]. On large telescopes, for a correction set by the size of an atmosphere turbulent cell, a high-order correction is needed and the limiting magnitude is independent of the telescope diameter. This results in a discrepancy between the fringe-tracking magnitude (in the infrared) which increases with telescope diameter, and the AO limiting magnitude (in the visible) which remains constant, translating in a constraint on the magnitude and the color limit on large telescopes. For example, for an AO system with the same subaperture size as NAOMI, the typical color limit on the UTs is R - K = + 0mag, as estimated in [4]. GPAO will lift the trade-off between high-order correction and limiting magnitude by implementing both a high-order Natural Guide Star (NGS) optimized for high-contrast on bright objects and a LGS mode on faint objects. By doing so, GRAVITY+ achieves a long-term goal of implementing AO and Laser Guide Stars (LGS) for the VLTI, as envisioned from its initial design [5]. \n/dieresis.ts1 \nFigure 1. Predicted Strehl ratio of the GRAVITY+ adaptive optics for NGS (blue dashed) and LGS modes (red dashed). The performances are adapted from ERIS, taking into account the additional throughpout losses of the UT Coud'e train. MACAO on-sky performances are shown in the lower left corner (gray dots). \n<!-- image --> \nThe GPAO NGS mode is designed for a limiting magnitude R=14 (Strehl=0.5), taking into account the losses of the Coud'e train, and a Strehl 0.8 for bright objects. The LGS mode will offer a limiting magnitude R=19, set by the magnitude of the tip/tilt star. The top-level requirement as well as the design of the GRAVITY+ WFS rely largely to the ones of ERIS [6], which was adapted to the UT Coud'e focus. The GPAO system will replace the MACAO curvature sensor [7,8] in operation at VLTI for more than 20 years now. For faint objects, GPAO will increase by more than 5 magnitude the limiting magnitude of the AO star with the LGS compared to MACAO, and by a factor 2 - 2.5 the Strehl ratio for bright objects with NGS. In addition, GPAO will remain compatible with the existing CIAO system, the 9x9 infrared sensor commissioned in 2016 with GRAVITY [9], especially with a LGS+CIAO mode for objects for which no bright visible star is available. \nFigure 2. a) High-redshift quasars: the LGS combined with the G-Wide upgrade will allow to increase the sample of currently 5 quasars to several dozens at redshift z=1-3. b) Galactic Center: the LGS+CIAO mode together with G-Faint upgrade will allow to go fainter than Kmag=21.0 for the imaging of the faint stars around the Galactic Center [10], c) Exoplanets: the NGS 40x40 will allow the observations of exoplanets at high-contrast 10 -6 and short separation < 100mas (adapted from [11]). \n<!-- image --> \n/dieresis.ts1", '2.1 Supermassive Black Holes at Cosmic Noon': 'The study of our local Universe indicates that there is a tight correlation between the evolution of galaxies and the mass of the Supermassive Black Holes (SMBH) which resides at their center. These objects are thought to co-evolve, despite the SMBH only contributes only to a small fraction of the total mass of the galaxy. In order to constrain this evolution, it is crucial to constrain the properties of SMBH across cosmic age. Yet, our understanding is limited by our knowledge of SMBHs and in particular their mass, which relies on scaling relations using AGN reverbation mapping. Constraining the properties of SMBHs at high-redshift, while these objects are at the early stage of their formation, provides fundamental clues on how the black hole and its host galaxy evolve together. GRAVITY+, using the off-axis tracking G-Wide [12], has demonstrated the direct measurement of SMBH dynamical mass at redshift z=2 through spatially resolved observations [13] of the Broad Line Region (BLR). Following the upgrade of the Beam Compressor Delay Lines (BCCDL, [14]), these observations have been recently expanded to 5 additional quasars. The GPAO upgrade, with the implementation of the LGS mode working together with G-Wide, will open up this sample to dozens of samples at redshift z=1 to 3, allowing for the first time to build a statistical sample of the SMBH mass at Cosmic Noon, during the peak of SMBH growth and star formation.', '2.2 Direct Characterization of Exoplanets': 'The direct characterization of planets using GRAVITY the dual-field mode allows for the characterization of exoplanet atmosphere together with an astrometry accuracy of 50-100 µ as. The use of this technique has led to ground-breaking results, such as the first observation of an exoplanet with interferometry [15], the first direct characterization of a planet detected with radial velocity (RV) [16], or the direct discovery of a GAIA and RV inferred planet [17]. Yet, these results were obtained with a moderate AO correction with a typical Strehl of 0.2 using MACAO. Based on GPAO, the high-order correction with the NGS 40x40 will allow a dramatic improvement of the AO correction, translating both in better flux injection and reduced speckle noise. The improved AO correction will also allow the implementation of high-contrast wavefront control techniques, in particular to dig a dark hole at the location of the science fiber [11,18]. Together, these improvements will allow to push the contrast and inner-working angle of high-contrast observation to 10 -6 at short separation < 100mas for the study of young planet in thermal emission. These improvements open up the characterization of GAIA inferred planets, which will give access to the bulk of the population of young gas giant planets at 1 - 3 AU [19]. Finally, the GPAO system will serve as a workhorse for all existing VLTI instruments and future high-contrast visitor instruments at VLTI [20].', '2.3 Galactic Center': "The study of the Galactic Center has allowed precision tests of Einstein's theory of General Relativity, and delivered the strongest evidence that SgrA* is a Schwarschild-Kerr black hole. The next step is the measurement of the spin of the black hole [21], which can be done by means of the Lense-Thiring precession of the orbit of surrounding stars. This requires the detection of a faint star on a close orbit around SgrA*. The imaging of the Galactic Center is currently reaching a limiting magnitude of about ∼ 19 mag [10]. This detection limit is set by the background flux of the stars around SgrA*, with a 1 σ average background noise of 21 . 0 ± 0 . 2 mag in 3.25h integration time. In the case of the Galactic Center, this background originates from the bright surrounding stars, which can hardly be filtered out by the phase-referencing given the number of stars, which signal can hardly be modeled. The CIAO+LGS will be a key gain in sensitivity, which will improve by a factor 2 the Strehl ratio from current S=25% to S=50% (Figure 1). The improvement associated to the AO will be twofold: a direct gain of the injected flux by a factor 2, and the reduction of the noise by a factor 1.5 approximately by reducing the background flux (proportional to 1-S). Together with the reduction of the noise caused by the metrology laser [22], and the upgrade of the Fringe-Tracker [23], the total improvement of the performances should lead to a 3 σ detection limit in 3.25h of 22 mag in the Galactic Center.", '3. SYSTEM': "The GPAO system is located in the Coud'e Train of the UT and is composed of the following sub-systems (Figure 3): \n- · Wavefront Sensors : Shack-Hartmann 40x40 in NGS and 30x30 in LGS, located in the Coud'e focus.\n- · Deformable Mirror : an ALPAO 43x43 with 1.3k actuators located on M8, with an offload of tip/tilt to M2 guiding.\n- · Real Time Calculator (RTC) based on SPARTA upgrade, located in the inner ring of the UTs\n- · Laser Guide Star : is a high-power (20W) Raman fibre laser at sodium wavelength 589.158nm coupled to a telescope launcher located on the UT center piece. This subsystem is the same as the LGS on the Extremely Large Telescope. \nFigure 3. Overview of the Gravity Plus Adaptive Optics subsystems. \n<!-- image -->", '3.1 Wavefront Sensors': "The WFS are composed of a NGS and a LGS module based on Shack-Hartmann sensors. The NGS and LGS design was adapted from the ERIS AO [6,24] from a Cassegrain to a Coud'e focus. The choice of Shack-Hartmann over a Pyramid sensor was driven by the need for linearity over a large range and the need of operational experience, which is lacking for the LGS with a Pyramid. In the WFS unit, each module is mounted on a motorized XY-stage and can patrol the full Coud'e field-of-view (FOV) of 2 arcmin. In operation: \n- · NGS mode is a High-Order (HO) 40x40 subapertures (with 6x6 pixels by subaperture) with a FOV of 2.5 arcsecond per subaperture.\n- · LGS mode is a 30x30 subapertures (with 8x8 pixels by subaperture, FOV of 5 arcsec) working with the Low-Order (LO) 4x4 on the NGS module for the tip/tilt star. The HO-LO switch allows to use on the same NGS camera. \nThe observing modes of GPAO will be the following (see [25]): \nTable 1. Observing modes of the Gravity Plus Adaptive Optics System. \nThe WFS are located after M8 (Deformable Mirror), M9 dichroic beam splitter which separates the visible light (AO) from the infrared light sent to VLTI, and the telecentric lens for the patrol field. GPAO-WFS modules are placed below the Star Separator (STS), at the same position as the MACAO modules. The opto-mechanical design and the positioning in the mechanical frame of the UT Coud'e focus can be found in [26] for a detailed description. \nTable 2. Design of the Natural Guide Star and Laser Guide Star sensors.", '3.2.1 NGS Optical Design': "The L1 lens of the NGS is located close to the Coud'e focus and images the exit pupil of the telecentric lens to the ADC. The lens L2 (located just before the ADC) and L3 (right after the ADC) creates a f/20 beam on a focal plane at the Pupil Steering Mirror (PSM) and the Technical CCD (TCCD). After the PSM, the diaphragm provides an adjustable field stop. After the filter wheel, the lens L4 re-images the pupil, which can be de-rotated by the K-mirror, on the lenslet of the Hi-Lo translation stage. The HO (40x40) and LO (4x4) are mounted on the Hi-Lo stage which allows to select the mode of observations of the NGS. The pupil is finally re-imaged by a relay optics onto the e2v CCD220 chip. The optical layout of the NGS can be found on Figure 4 and Figure 5.", '3.2.2 NGS Subsystems': "NXYT Translation stage The full NGS module is mounted on a XY-translation stage, which allows to cover the 2.0 arcmin patrol-field. The micrometer precision of the stage allows for fine adjustement in the field, as L1 is located close to the Coud'e focus (pixel scale 0.549 as/mm at the Coud'e focus). In tracking mode, the NXYT allows to follow a given trajectory within the patrol field. \nAtmospheric Dispersion Compensator (ADC) The ADC consists of a pair of counter-rotating Amici prisms, based on combination of S-FPM2 and Schott FK5 glasses. The latter is replacing the S-TIM8 in the ERIS design. The ADC is located in a pupil plane of the NGS, between L2 and L3 lenses. It is optimized for a spectral range of 600-1000nm, and is designed for a maximum Zenith angle of 70 · assuming the environment data in terms of height, latitude, humidity of Paranal observatory. \nTechnical Camera (TCCD) The NGS module includes a technical acquisition camera, based on a Prosilica GT 2050 camera. The camera is placed after a dichroic beam-splitter (DCR) located between L3 and the PSM, which reflects the blue part of the spectrum 400-570 nm ( R > 95%) to the TCCD, while transmiting the beam necessary for the OCAM (589-1000 nm, T > 99%). The TCCD is equipped with a lens (NPV) which can be slid in and out of the beam to image the pupil or the field respectively. \nPupil Steering Mirror (PSM) and Field Diaphragm (NDIA) The PSM is located in a focus plane with f/20.0, and allows to stabilize the pupil on the micro-lens. This device is based on a piezo-actuator PI S-334 and allows for good repeatability and travel range. The PSM is located just upstream from a motorized iris which can be used as an adjustable field diaphram (NDIA). The diaphram can be adjusted from 0 to a maximum 8 arcsec approximately, which allows to control the FOV of the subaperture and adapt it to the seeing conditions. \nK-mirror derotator (NROT) The baseline is to operate the WFS without a K-mirror using a software derotation based on a synthetic model of the system, as implemented in []. This model was updated to take into account magnification and anamorphoses, which are known to be important in the CIAO system. Nonetheless, as part of the risk mitagation strategy during the design of GPAO, a derotator was implemented. The K-mirror is made of a FK5 glass, which provide low chromatic dispersion. The dimensions of the K-mirror are such that they allow removing this element withouth changing the overall optical path length. Alternatively, this element can also be used for calibration purpose to generate field rotation. \nFigure 4. Left : Natural Guide Star wavefront-sensor module Right: Laser Guide Star sensor. \n<!-- image --> \nFigure 5. Optical Layout of the NGS and the LGS modules. \n<!-- image --> \nCalibration Unit (NCAL) The WFS also embedds a calibration unit, which consists in a halogen lamp. The lamp is connected to a multimode fiber, which can be placed in or out of the beam, at the position of the UT Coud'e focus just above L1. The calibration unit can therefore be used to generate a point-like source in the field. This calibration mode is used for the daily characterization of the WFS and for monitoring its basic parameters (Detector gain, internal Wavefront Error, Pixel Scale, Shack-Hartmann field-of-view), see 4.1.", '3.3.1 LGS Optical Design': 'The LGS beam is reflected by the dichroic located below the telecentric lens towards the LGS arm. The LGS focus is thus located between the dichroic and the Periscope Folding Mirror (PFM). The beam is then converted from a f/47 to a f/12 beam by the lenses L1 and L2, in order to reduce the focusing range from 1.7m (longitudinal shift at 80km for a f/47 beam) to 120mm. Between L1 and L2, the K-mirror is located close to a pupil plane and allows for field derotation. The focus is ajusted by a translation stage (LFOC), on which the downstream WFS optics are mounted (from L3 to the detector). Like in the NGS, the field is re-imaged on the PSM with an adjustable field stop located in its vicinity. Finally, the pupil is re-imaged by L3 on the 30x30 lenslet array, which is directly mounted in front of the CCD chip (unlike the NGS, the lenslet in the LGS does not have to move). The optical layout of the LGS is shown on Figure 4 and 5.', '3.3.2 LGS Subsystems': "XY Translation stage (LXYT) and Focus Stage (LFOC) The LGS module is mounted on a XYTranslation stage, which allows to cover the patrol field in the same way as the NGS. In addition, the LGS payload comprises a focus stage, which allows to adjust for a laser guide star ranging from 80km to infinity. The LFOC can be placed in tracking mode, which automatically update the position based on the value of the focus from the truth sensor and the altitude of the telescope. \nLGS actuators The control strategy and the actuators in the LGS module are similar to the NGS, which comprises a pupil steering mirror (LPSM), a field-diaphragm (LDIA), a K-mirror (LROT) and a calibration unit (LCAL) at the position of the LGS Coud'e focus. The actuators and these components are the same as the NGS, with the optics adapted to the spectral range of the LGS (569-609nm). \nTable 3. Specifications of the Wavefront Sensor units.", '4.1 Functional tests & Performance': 'The functional tests of the WFS were validated in Europe, both at the unit level (WFS, Garching) and at System level (Nice, see [27]). This sequence was automated through VLT templates (see 4.3) delivered with the instrument, and can be run as a daily health-check. All four WFS units were characterized, and the results were \nFigure 6. Real-Time display in LGS mode with the 30x30 LGS wavefront sensor (left), the NGS wavefront sensor 4x4 Tip-Tilt sensor (center) and the commands applied to the DM (right). In NGS-VIS mode with 40x40 High-Order (not shown), only the NGS sensor is used and the 4x4 is automatically replaced with a 40x40 lenslet. \n<!-- image --> \nvalidated during the Preliminary Acceptance Europe (PAE) at ESO in May 2024. The Wavefront Sensor units have been shipped to Chile in May and June 2024, are being commissioned at Paranal in Summer 2024. \nThe performance of the WFS was validated on the NGS closing the AO loop while controlling up to 1000 modes, without losses of performance (Figure 7). The integral gain K i of the NGS reaches a plateau of performance between 0.2 and 0.7. Both aspect leaves margin for the baseline mode, with 500 modes corrected and an integral gain K i = 0 . 5. For the LGS, similar performances were obtained, with an optimal gain between K i = 0.2 and 0.5 and no performance losses up to 500 modes. The baseline mode for the LGS is currently 400 modes corrected and an integral gain of 0.3. These values are being confirmed on-sky with the commissioning of the GPAO modules at the telescopes as these lines are being written.', '4.2 OCAM2 camera': 'The NGS and LGS modules both rely on the OCAM2 camera from First Light Imaging. The cameras are based on an e2v EMCCD which can be operated full-frame up to 2 kHz with a multiplication gain up to 1000. The read-out noise of the cameras was characterized in the lab and is equal to 0.3e/s on average, for a multiplication gain of 800 and running at 1kHz. The multiplication gain of the camera is known to drift per octant over a period of a few month. Therefore the multiplication gain is monitored over time using the internal calibration unit and corrected through the voltages applied to each octant, in order to keep the homogeneity of the gain on the chip.', '4.3 Python Templates in VLT2022': "The WFS modules were integrated and delivered with templates developed in Python within the VLT2022 framework. These templates are used to perform the characterization tests and the daily health check of the modules. These templates can all run independently of SPARTA (e.g. commissioning purposes, maintenance, etc.), and are complimentary to the templates at system-level presented in [27]. \n<!-- image --> \nFigure 7. Left : Strehl ratio as a function of the number of controlled modes. The AO system shows no decrease of performances up to a number of controled mode of 1000, and shows a good agreement between the measured Interaction Matrix (IM) and the pseudo-Interaction Matrix (PSIM). Right: View of one GRAVITY+ WFS unit installed at the Coud'e focus of UT1 in Paranal. \n<!-- image -->", '4.4 Integration': 'At the time of writing this proceeding, the WFS modules have been shipped and are being integrated in Paranal. The integration and alignment of the WFS modules in Garching were performed using a so-called telescope simulator, which allows to generate a telecentric beam, with either point-source in a field plane with an adjustable pupil stop, or a point in a pupil plane with an adjustable field stop. This source is aligned on a reference target attached to the tower structure, and which reproduces the tilt and centering position of the UT beam. The telescope simulator also includes a sighting telescope coaligned with the reference target, and serves as a reference axis for alignment. In the UTs, a similar set-up is reproduced with a sighting telescope in the STS, aligned on the azimuth axis of the telescope and using the Nasmyth beacon of the UTs.', '5. CONCLUSION': 'The four GPAO units have been assembled and characterized in Europe, and shipped to Paranal in June 2024. The implementation of state-of-the-art AO at the VLTI will allow for the first time high-contrast observations with the NGS mode, and will be followed by the commissioning in 2025 of the LGS guide stars with global sky coverage, and with a significant increase of operability during average seeing conditions compared to MACAO. It completes a long-term vision initiated during the design of the VLTI, and breaks a fundamental sensitivity limitation of interferometry, entering the era of high-contrast observations and all-sky coverage with interferometry.', 'ACKNOWLEDGMENTS': "GRAVITY+ is a consortium composed of German (MPE, MPIA, University of Cologne), French (CNRS-INSU: LESIA, Paris, IPAG, Grenoble, Lagrange, Nice, CRAL, Lyon), British (University of Southampton), Belgian (KU Leuven), and Portugese (CAUP) institutes, built in close collaboration with ESO. DD has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG - 866070). TP has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 101004719.", 'REFERENCES': "- [1] GRAVITY Collaboration, Abuter, R., Accardo, M., Amorim, A., Anugu, N., ' Avila, G., Azouaoui, N., Benisty, M., Berger, J. P., Blind, N., Bonnet, H., Bourget, P., and et al., 'First light for GRAVITY: Phase referencing optical interferometry for the Very Large Telescope Interferometer,' A&A 602 , A94 (June 2017).\n- [2] Gravity+ Collaboration, Abuter, R., Alarcon, P., Allouche, F., Amorim, A., Bailet, C., Bedigan, H., Berdeu, A., Berger, J. P., Berio, P., Bigioli, A., Blaho, R., Boebion, O., and et al., 'The GRAVITY+ Project: Towards All-sky, Faint-Science, High-Contrast Near-Infrared Interferometry at the VLTI,' The Messenger 189 , 17-22 (Dec. 2022).\n- [3] Eisenhauer, F., Monnier, J. D., and Pfuhl, O., 'Advances in Optical/Infrared Interferometry,' ARA&A 61 , 237-285 (Aug. 2023).\n- [4] Woillez, J., Abad, J. A., Abuter, R., Aller Carpentier, E., Alonso, J., Andolfato, L., Barriga, P., Berger, J. P., Beuzit, J. L., Bonnet, H., Bourdarot, and et al., 'NAOMI: the adaptive optics system of the Auxiliary Telescopes of the VLTI,' A&A 629 , A41 (Sept. 2019).\n- [5] Beckers, J. M., 'Planning the VLT interferometer.,' The Messenger 60 , 1-9 (June 1990).\n- [6] Davies, R., Absil, O., Agapito, G., Agudo Berbel, A., Baruffolo, A., Biliotti, V., Black, M., Bonaglia, M., Bonse, M., Briguglio, R., and et al., 'The Enhanced Resolution Imager and Spectrograph for the VLT,' A&A 674 , A207 (June 2023).\n- [7] Roddier, F., 'Curvature sensing and compensation: a new concept in adaptive optics,' Applied Optics 27 , 1223-1225 (Apr. 1988).\n- [8] Arsenault, R., Alonso, J., Bonnet, H., Brynnel, J., Delabre, B., Donaldson, R., Dupuy, C., and et al., 'MACAO-VLTI: An Adaptive Optics system for the ESO VLT interferometer,' in [ Adaptive Optical System Technologies II ], Wizinowich, P. L. and Bonaccini, D., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 4839 , 174-185 (Feb. 2003).\n- [9] Scheithauer, S., Brandner, W., Deen, C., Adler, T., Bonnet, H., Bourget, P., and et al., 'CIAO: wavefront sensors for GRAVITY,' in [ Adaptive Optics Systems V ], Marchetti, E., Close, L. M., and V'eran, J.-P., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9909 , 99092L (July 2016).\n- [10] GRAVITY Collaboration, Abuter, R., Aimar, N., Amorim, A., Arras, P., Baubock, M., Berger, J. P., Bonnet, H., Brandner, W., Bourdarot, G., Cardoso, V., and et al., 'Deep images of the Galactic center with GRAVITY,' A&A 657 , A82 (Jan. 2022).\n- [11] Pourr'e, N., Winterhalder, T. O., Le Bouquin, J. B., Lacour, S., Bidot, A., Nowak, M., Maire, A. L., Mouillet, D., Babusiaux, C., Woillez, J., Abuter, R., Amorim, A., Asensio-Torres, R., Balmer, W. O., Benisty, M., and et al., 'High contrast at short separation with VLTI/GRAVITY: Bringing Gaia companions to light,' A&A 686 , A258 (June 2024).\n- [12] GRAVITY+ Collaboration, Abuter, R., Allouche, F., Amorim, A., Bailet, C., Baubock, M., Berger, J. P., Berio, P., Bigioli, A., Boebion, O., Bolzer, M. L., Bonnet, H., Bourdarot, G., and et al., 'First light for GRAVITY Wide. Large separation fringe tracking for the Very Large Telescope Interferometer,' A&A 665 , A75 (Sept. 2022).\n- [13] Abuter, R., Allouche, F., Amorim, A., Bailet, C., Berdeu, A., Berger, J. P., Berio, P., Bigioli, A., Boebion, O., Bolzer, M. L., Bonnet, H., Bourdarot, G., Bourget, P., and et al., 'A dynamical measure of the black hole mass in a quasar 11 billion years ago,' Nature 627 , 281-285 (Mar. 2024).\n- [14] Fabricius, M., Woillez, J., Abuter, R., Bourdarot, G., and Bourget, P., 'GRAVITY+ Wide: Towards hundreds of z ∼ 2 AGN, larger throughput and improved vibrational control,' in [ Optical and Infrared Interferometry and Imaging ], Sallum, S. and Sanchez-Bermudez, J., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 13095 (2024).\n- [15] GRAVITY Collaboration, Lacour, S., Nowak, M., Wang, J., Pfuhl, O., Eisenhauer, F., Abuter, R., Amorim, A., Anugu, N., Benisty, M., Berger, J. P., Beust, H., Blind, N., Bonnefoy, M., Bonnet, H., Bourget, P., and et al., 'First direct detection of an exoplanet by optical interferometry. Astrometry and K-band spectroscopy of HR 8799 e,' A&A 623 , L11 (Mar. 2019).\n- [16] Nowak, M., Lacour, S., Lagrange, A. M., Rubini, P., Wang, J., Stolker, T., Abuter, R., Amorim, A., Asensio-Torres, R., Baubock, M., Benisty, M., and et al., 'Direct confirmation of the radial-velocity planet β Pictoris c,' A&A 642 , L2 (Oct. 2020)."}

2024ApJ...975..179C

We present 0.22 resolution CO21 observations of the circumnuclear gas disk in the local compact galaxy NGC 384 with the Atacama Large Millimetersubmillimeter Array ALMA. While the majority of the disk displays regular rotation with projected velocities rising to 370 km sSUP1SUP the inner 0.5 exhibits a kinematic twist. We develop warped disk gasdynamical models to account for this twist fit those models to the ALMA data cube and find a stellar masstolight ratio in the H band of ML SUBHSUB 1.34 0.01 1 statistical 0.02 systematic M SUBSUBL SUBSUB and a supermassive black hole BH mass M SUBBHSUB of M SUBBHSUB inlineformula mmlmath overflowscrollmmlmommlmommlmo stretchyfalsemmlmommlmsubsupmmlmn7.26mmlmnmmlmrowmmlmommlmommlmn0.48mmlmnmmlmrowmmlmrowmmlmommlmommlmn0.43mmlmnmmlmrowmmlmsubsupmmlmo stretchyfalsemmlmommlmn1mmlmnmmlmimmlmimmlmspace width0.25emmmlmspacemmlmistatisticalmmlmimmlmsubsupmmlmo stretchyfalsemmlmommlmrowmmlmommlmommlmn1.00mmlmnmmlmrowmmlmrowmmlmommlmommlmn0.55mmlmnmmlmrowmmlmsubsupmmlmo stretchyfalsemmlmommlmisystematicmmlmimmlmo stretchyfalsemmlmommlmo stretchyfalsemmlmommlmommlmommlmsupmmlmn10mmlmnmmlmn8mmlmnmmlmsupmmlmspace width0.25emmmlmspacemmlmsubmmlmiMmmlmimmlmommlmommlmsubmmlmath inlineformula. In contrast to most previous dynamical M SUBBHSUB measurements in local compact galaxies which typically found overmassive BHs compared to the local BH massbulge luminosity and BH massbulge mass relations NGC 384 lies within the scatter of those scaling relations. NGC 384 and other local compact galaxies are likely relics of z 2 red nuggets and overmassive BHs in these relics indicate BH growth may conclude before the host galaxy stars have finished assembly. Our NGC 384 results may challenge this evolutionary picture suggesting there may be increased scatter in the scaling relations than previously thought. However this scatter could be inflated by systematic differences between stellar and gasdynamical measurement methods motivating direct comparisons between the methods for NGC 384 and the other compact galaxies in the sample.

2024-11-01T00:00:00Z

['arXiv:2409.08812', '10.3847/1538-4357/ad7bb0', '2024ApJ...975..179C', '2024arXiv240908812C', '10.48550/arXiv.2409.08812']

['Supermassive black holes', 'Molecular gas', 'Millimeter astronomy', 'Submillimeter astronomy', 'Galaxy kinematics', 'Early-type galaxies', 'Astronomy data modeling', 'Galaxy circumnuclear disk', 'Scaling relations', 'Extragalactic astronomy', '1663', '1073', '1061', '1647', '602', '429', '1859', '581', '2031', '506', 'Astrophysics - Astrophysics of Galaxies']

Modeling ALMA Observations of the Warped Molecular Gas Disk in the Red Nugget Relic Galaxy NGC 384

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

https://arxiv.org/pdf/2409.08812.pdf

{'Modeling ALMA Observations of the Warped Molecular Gas Disk in the Red Nugget Relic Galaxy NGC 384': 'Jonathan H. Cohn, 1, 2 Maeve Curliss, 1 Jonelle L. Walsh, 1 Kyle M. Kabasares, 3, 4, 5 Benjamin D. Boizelle, 6, 1 Aaron J. Barth, 7 Karl Gebhardt, 8 Kayhan Gultekin, 9 David A. Buote, 7 Jeremy Darling, 10 Andrew J. Baker, 11, 12 and Luis C. Ho 13 \n1 George P. and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics & Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA \n- 2 Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA\n- 3 Ames Research Center, National Aeronautics and Space Administration, Moffett Field, CA 94035, USA\n- 4 Bay Area Environmental Research Institute, Ames Research Center, Moffett Field, CA 94035, USA \nDepartment of Physics and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine, CA, 92697-4575, USA \n5 \n6 Department of Physics and Astronomy, N284 ESC, Brigham Young University, Provo, UT 84602, USA \n- 7 Department of Physics and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine, CA 92697-4575, USA\n- 8 Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA \n9 \nDepartment of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109, USA \n10 Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, 389 UCB, Boulder, CO 80309-0389, USA \n- 11 Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road Piscataway, NJ 08854-8019, USA\n- 12 Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Road, Bellville 7535, South Africa\n- 13 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China; Department of Astronomy, School of Physics, Peking University, Beijing 100871, China', 'ABSTRACT': "We present 0 . '' 22-resolution CO(2 -1) observations of the circumnuclear gas disk in the local compact galaxy NGC 384 with the Atacama Large Millimeter/submillimeter Array (ALMA). While the majority of the disk displays regular rotation with projected velocities rising to 370 km s -1 , the inner ∼ 0 . '' 5 exhibits a kinematic twist. We develop warped disk gas-dynamical models to account for this twist, fit those models to the ALMA data cube, and find a stellar mass-to-light ratio in the H -band of M/L H = 1 . 34 ± 0 . 01 [1 σ statistical] ± 0 . 02 [systematic] M ⊙ /L ⊙ and a supermassive black hole (BH) mass ( M BH ) of M BH = (7 . 26 +0 . 43 -0 . 48 [1 σ statistical] +0 . 55 -1 . 00 [systematic]) × 10 8 M ⊙ . In contrast to most previous dynamical M BH measurements in local compact galaxies, which typically found over-massive BHs compared to the local BH mass -bulge luminosity and BH mass -bulge mass relations, NGC 384 lies within the scatter of those scaling relations. NGC 384 and other local compact galaxies are likely relics of z ∼ 2 red nuggets, and over-massive BHs in these relics indicate BH growth may conclude before the host galaxy stars have finished assembly. Our NGC 384 results may challenge this evolutionary picture, suggesting there may be increased scatter in the scaling relations than previously thought. However, this scatter could be inflated by systematic differences between stellar- and gasdynamical measurement methods, motivating direct comparisons between the methods for NGC 384 and the other compact galaxies in the sample.", '1. INTRODUCTION': "Dynamical supermassive black hole (BH) detections have been made in over 100 nearby galaxies, with stellar dynamics accounting for the vast majority of these measurements (Saglia et al. 2016). In recent years, the \nCorresponding author: Jonathan H. Cohn jonathan.cohn@dartmouth.edu \nadvent of the Atacama Large Millimeter/sub-millimeter Array (ALMA), with its improved angular resolution and sensitivity compared to previous mm/sub-mm observatories, has led to a significant increase in molecular gas-dynamical BH mass ( M BH ) measurements in the literature (e.g., Barth et al. 2016; Davis et al. 2017; Boizelle et al. 2019, 2021; Nagai et al. 2019; North et al. 2019; Smith et al. 2021; Cohn et al. 2021, 2023; Kabasares et al. 2022; Nguyen et al. 2022; Ruffa et al. 2023; Kabasares & Cohn et al. 2024; Dominiak et al. 2024a,b). The cold \nmolecular gas detected with ALMA is a more reliable tracer of the gravitational potential around BHs than warm H 2 molecular gas or ionized gas, which often display more turbulent motion (e.g., van der Marel & van den Bosch 1998; Barth et al. 2001; Wilman et al. 2005; Neumayer et al. 2007; Seth et al. 2010; Walsh et al. 2010, 2013; Scharwachter et al. 2013). \nVia these dynamical measurements, BH masses have been found to correlate with large-scale properties of their host galaxies, including stellar velocity dispersion ( σ ⋆ ), bulge mass ( M bul ), and bulge luminosity ( L bul ; e.g., Kormendy & Richstone 1995; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Marconi & Hunt 2003; Gultekin et al. 2009; Kormendy & Ho 2013), indicating BHs and their hosts co-evolve. However, this BH -host galaxy co-evolution is not well understood, in part because the galaxies in which dynamical M BH measurements have been made are not fully representative of the population of galaxies in the Universe. Furthermore, dynamical M BH measurements are not generally possible beyond the local Universe, complicating attempts to study the cosmic evolution of the BH scaling relations. \nHere, we study the circumnuclear molecular gas disk in NGC 384 with ALMA. NGC 384 is a member of a sample of local, massive, and compact early-type galaxies (ETGs) found through the Hobby-Eberly Telescope Massive Galaxy Survey (HETMGS; van den Bosch et al. 2015). Yıldırım et al. (2017) studied the sample's stellar kinematics and photometric properties. These galaxies have large stellar velocity dispersions (Yıldırım et al. 2017), indicating they likely have large BHs ( M BH ≲ 6 × 10 9 M ⊙ ; Kormendy & Ho 2013; Saglia et al. 2016), populating the poorly-sampled high-mass end of the scaling relations. They are also massive (stellar masses M ⋆ ∼ 5 . 5 × 10 10 -3 . 8 × 10 11 M ⊙ ) and compact (effective radii r e ∼ 0 . 7 -3 . 1 kpc), falling on the redshift ( z ) ∼ 2 size -mass relation, despite lying at distances within ∼ 100 Mpc, i.e., at z ∼ 0 (Yıldırım et al. 2017). These systems are fast rotators with cuspy surface brightness profiles, flattened, disk-like shapes, and no evidence in their stellar orbital distributions for major mergers since z ∼ 2 (Yıldırım et al. 2017). They exhibit uniform, old stellar ages ( ≳ 10 Gyr) over several effective radii (Mart'ın-Navarro et al. 2015b; Ferr'e-Mateu et al. 2017; Yıldırım et al. 2017) and elevated fractions of red globular clusters (Beasley et al. 2018; Kang & Lee 2021). Individual objects in the sample have also been shown to have highly concentrated dark matter halos (Buote & Barth 2018, 2019). \nThese local compact galaxies are very different from the brightest cluster galaxies and giant ETGs in the local Universe that typically host the highest-mass BHs. The latter, more typical massive ETGs are commonly thought to evolve from quiescent red nugget galaxies observed at z ∼ 2, growing through accretion and minor/intermediate dry mergers without significant BH growth (e.g., van Dokkum et al. 2010; Hilz et al. 2013). \nIn contrast, local compact galaxies like NGC 384 are likely passively evolved relics of the red nugget galaxies (Trujillo et al. 2014; Yıldırım et al. 2017). Thus, if the passively evolved relics of red nugget galaxies tend to host over-massive BHs, the BHs of common massive local ellipticals may have finished growing by z ∼ 2 (e.g., Ferr'e-Mateu et al. 2015). \nAside from these relic galaxies, most of our knowledge of BH -host galaxy co-evolution at high z stems from single epoch Active Galactic Nucleus (AGN) measurements (e.g., Izumi et al. 2019; Pensabene et al. 2020; Larson et al. 2023; Maiolino et al. 2023; Bogd'an et al. 2024). In particular, recent single epoch M BH determinations from AGN have pointed to BHs that are overmassive at high z compared to the local scaling relations (Pacucci et al. 2023). However, there is a systematic factor of two uncertainty in the local reverberation mapping M BH measurements to which single epoch measurements are anchored (Shen et al. 2023), and single epoch BH masses can be further overestimated by ∼ 0.3 dex when the diversity of quasar properties is not accounted for (Fonseca Alvarez et al. 2020). As such, complementary methods to study highz BH growth are required. The local compact relic galaxy sample thus presents a remarkable alternative view into the history of the BH scaling relations. \nBlack hole mass measurements already exist for five galaxies in this sample, including three using stellar dynamics and two using molecular gas dynamics from ALMA. The measurements from stellar dynamics (in NGC 1277, NGC 1271, and Mrk 1216; van den Bosch et al. 2012; Emsellem 2013; Yıldırım et al. 2015; Walsh et al. 2015, 2016, 2017; Graham et al. 2016a; Krajnovi'c et al. 2018) find BH masses consistent with the M BH -σ ⋆ relation but over-massive compared to the M BH -L bul and M BH -M bul relations by an order of magnitude, even when conservatively using the total stellar luminosities and total stellar masses of the galaxies, rather than their uncertain bulge values (e.g., Graham et al. 2016b; Savorgnan & Graham 2016). Scharwachter et al. (2016) observed molecular gas in NGC 1277 with the IRAM Plateau de Bure Interferometer, finding an M BH consistent with stellar-dynamical studies, albeit with significant uncertainties due to limited angular resolution. At much higher angular resolution, one ALMA measurement (PGC 11179; Cohn et al. 2023) echoed the behavior of the over-massive stellar-dynamical results, but the other (UGC 2698; Cohn et al. 2021) was consistent with all three relations. However, UGC 2698 may have undergone some stellar growth since z ∼ 2 (Yıldırım et al. 2017), evolving it toward the local BH scaling relations. \nAlthough these results could be explained by greater than expected intrinsic scatter in the scaling relations rather than systematically different growth histories, the apparent scatter may be inflated due to a possible systematic offset between stellar- and molecular gas- \ndynamical measurement methods (Cohn et al. 2021). Thus, to decipher whether the local compact galaxies truly display evidence for BHs finishing their growth before their host galaxy finishes assembling stars, we require more BH mass measurements in the sample and direct comparisons between stellar- and gas-dynamical results for individual objects. \nIn this work, we measure the BH mass in NGC 384 with molecular gas dynamics, accounting for a kinematic twist in the gas disk. We adopt an angular diameter distance of 55 Mpc to NGC 384, where 267 pc spans 1 '' , using a ΛCDM cosmology with H 0 = 73 km s -1 Mpc -1 , Ω M = 0 . 31, and Ω Λ = 0 . 69. We use the Virgo + Great Attractor + Shapley Supercluster infall model (Mould et al. 2000) for the Hubble Flow distance in the NASA/IPAC Extragalactic Database 10 . We note that North et al. (2019) made an M BH measurement in NGC 383, which is in the same group as NGC 384, and assumed a distance of 66.6 Mpc to NGC 383 in the process. The BH mass in our models scales linearly with the assumed distance to NGC 384. \nThe composition of the paper is presented below. In § 2, we discuss the Hubble Space Telescope (HST) and ALMA observations of NGC 384. In § 3, we describe our dynamical model, warped disk methodology, and parameter optimization. We present our model results in § 4. In § 5, we discuss our resolution of the BH gravitational sphere of influence (SOI), compare the BH in NGC384 to the scaling relations, and discuss the impact of our results on our understanding of the scaling relations and BH -host galaxy growth. Finally, we present our conclusions in § 6.", '2. OBSERVATIONS': "Confident M BH measurements with molecular gas arise from observations resolving circumnuclear gas extending within or near to the BH SOI. Additionally, the host galaxy's stellar light profile must be characterized to measure the contribution of stars to the gravitational potential on small scales. We therefore obtain HST Wide Field Camera 3 (WFC3) observations, as described in § 2.1. We detail our ALMA observations in § 2.2.", '2.1. HST Observations': 'Program GO-13050 (PI: van den Bosch) observed NGC 384 with HST WFC3 in the IR/F160W ( H -band) and UVIS/F814W ( I -band) filters on 2013 June 6. 1 The H -band observations included three dithered full-', '10 https://ned.ipac.caltech.edu/': "array exposures, with four dithered short sub-array exposures. These sub-array exposures avoid saturating the nucleus while better sampling the point spread function (PSF). These images were processed in Yıldırım et al. (2017) using the calwf3 pipeline and distortioncorrected, cleaned, and combined using AstroDrizzle (Gonzaga et al. 2012). Ultimately, the H -band image has an exposure time of 1354.5 s, a pixel scale of 0 . '' 06 pixel -1 , and a field of view (FOV) of 2 . ' 7 × 2 . ' 6. \nThree dithered full-array exposures were also taken in the I -band. The final I -band image, which has an exposure time of 482.0 s, is drizzled to the same scale as the H -band image and degraded to match the H -band image's resolution, so we can construct I -H maps to characterize the dust disk in NGC 384. These images are shown in Figure 1. \nDust is not clearly visible in the H -band image, but the I -band image shows a ∼ 4 . '' 2-diameter circumnuclear dust disk or ring-like structure. The median I -H color is 1.7 mag, measured just beyond the disk region. The maximum color excess ∆( I -H ), which we find ∼ 0 . '' 7 to the southwest of the nucleus, is ∼ 0.6 mag. We use Vega-relative magnitudes throughout this work.", '2.1.1. Galaxy Surface Brightness Models': "In order to parameterize the stellar surface brightness profile of NGC 384, we fit two-dimensional (2D) Multi-Gaussian Expansions (MGEs) to the H -band image. MGEs are capable of accurately reproducing the stellar surface brightness profiles of ETGs (Emsellem et al. 1994; Cappellari 2002). We complete an initial fit with a 2D regularized MGE (Cappellari 2002) and use that result as the initial guess for a 2D MGE fit in GALFIT (Peng et al. 2010). Gaussian position angles and centers are constrained to be identical for all components in GALFIT . The PSF, which comes from Tiny Tim (Krist & Hook 2004) and is drizzled and dithered identically to the image, is accounted for in the fit. \nFirst, we fit an MGE to the H -band image, masking out foreground stars, other galaxies, and artifacts in the image. This MGE is referred to as the 'original MGE' throughout this paper. The original MGE has ten components, with projected dispersions σ ' ranging from 0 . '' 042 to 51 . '' 459, projected axis ratios q ' between 0.581 and 1.000, and a PA of 136 . 732 · east of north. The MGE is a good fit, with typical residuals of ≲ 5%. \nNext, we follow Cohn et al. (2021) and Cohn et al. (2023) in building an MGE with the dustiest portions of the H -band image masked out. We start with a conservative color cut of I -H = 2 . 05 and fit an MGE to the resultant masked image in GALFIT . Then, we expand the mask to cover pixels with the greatest residuals near the nucleus and fit an MGE using this expanded mask. We repeat this process iteratively until the residuals near the nucleus are ≲ 5%. The final mask and resultant MGE are called the 'dust mask' and the 'dust-masked MGE' throughout this paper. We list the best-fit parameters \nFigure 1. ( Left ) Contours of the HST F160W ( H -band) image (black) with contours of the fiducial model's dust-masked MGE overlaid (red). ( Middle, top ) Central 5 '' of the HST H -band image. ( Right, top ) Central 5 '' of the HST F814W ( I -band) image, which shows a dust disk to the southwest of the nucleus. ( Middle, bottom ) Central 5 '' of the HST H -band contours (black) and the dust-masked MGE (red), with gray shading showing the region that was masked during the MGE fit. The asymmetry in the H -band contours in the dust-masked region points to dust attenuation, and the lack of asymmetry in the red contours indicates the dust-masked MGE accounts for this attenuation. ( Right, bottom ) ALMA CO(2 -1) emission (blue) with HST I -H contours (black) overlaid, showing that the CO(2 -1) emission and dust disk are co-spatial and approximately the same size ( ∼ 4 . '' 2 in diameter). The ellipse in the lower left of the panel corresponds to the ALMA synthesized beam. \n<!-- image --> \nof the dust-masked MGE in Table 1 and compare the dust-masked MGE and H -band image in Figure 1. In Figure 2, we show the surface brightness profiles calculated along the minor axis within the inner ∼ 3 '' for the H -band image, the original MGE, and the dust-masked MGE. The inner ∼ 0 . '' 2 of the galaxy appear mostly dust free. \nFinally, we approximate a dust-corrected MGE by adjusting the H -band image before fitting in GALFIT . This method is described in more detail in Cohn et al. (2023), but we summarize the process here. First, we follow Viaene et al. (2017) and Boizelle et al. (2019) in assuming that the galaxy is oblate axysimmetric and that the dust lies in a thin disk in the galaxy inclined at the same angle as both the gas and the galaxy's stellar component. We use an inclination of 57.3 · from initial flat disk dynamical model results (see § 4.2) to deproject the MGE, then calculate the fraction of light that originates behind versus in front of the dust disk at each pixel, taking the light originating behind the disk to be obscured with a simple screen extinction. The model color excess at each pixel is calculated as a function of intrinsic dust extinction, A V , using Equations 1 and 2 from Boizelle et al. (2019). To convert from A V to A H and A I , we take the standard Galactic R V = 3 . 1 extinction curve \n(Rieke & Lebofsky 1985). Following Cohn et al. (2023), we do not attempt a full pixel-by-pixel correction. The median A H inside the dust masked region is 0.4 mag, with a small standard deviation of 0.1 mag, and we correct the H -band image within the dust masked region by this median value. \nWe then use GALFIT to fit a 2D Nuker model (Faber et al. 1997) to the central 5 '' × 5 '' of the dust-masked H -band image, again accounting for the PSF. The Nuker model consists of a double power-law with an inner slope γ , outer slope β , sharpness of transition α , and break radius between the slopes r b . The fit converges to α = 0 . 43, β = 2 . 86, γ = 0 . 00, and r b = 0 . 91 '' ( ∼ 243 pc). We re-fit this Nuker model to the 0.4 mag-adjusted H -band image in GALFIT , holding β and r b fixed, resulting in γ = 0 . 02 and α = 0 . 44. Next, we replace the dust-masked region of the H -band image with the corresponding pixel values of the best-fit Nuker model, creating a dust-corrected image with a light distribution that varies smoothly. We fit a final MGE to this dust-corrected image in GALFIT and refer to this resultant MGE as the 'dust-corrected MGE' throughout this paper. The dust-corrected MGE consists of ten Gaussians, with σ ' ranging between 0 . '' 047 and 45 . '' 015, q ' between 0.581 and 1.000, a PA of 136.218 · east of north, \nand residuals ≲ 3%. The major- and minor-axis surface brightness profiles of the dust-corrected MGE closely track those of the dust-masked MGE and are thus not included in Figure 2. The MGEs we construct in this work follow a similar profile to the MGE presented for NGC 384 in Yıldırım et al. (2017). Following Cohn et al. (2021) and Cohn et al. (2023), we use the dust-masked MGE in the fiducial model and use the other two MGEs to assess the impact of our treatment of dust on the inferred M BH . \nTable 1. Dust-masked MGE parameters \nNote -MGE parameters fit to the dust-masked HST H -band image of NGC 384 with GALFIT . The MGE component is listed in Column (1). The central surface brightness of each component, given in Column (2), is calculated with an absolute Solar H -band magnitude of 3.37 mag (Willmer 2018) and a Galactic extinction (Schlafly & Finkbeiner 2011) toward NGC 384 of A H = 0 . 032 mag. The projected dispersion along the major axis and the axis ratio for each component are given in Columns (3) and (4), respectively. Projected quantities are listed with primes. All components have a PA of 136 . 212 · east of north.", '2.2. ALMA Observations': "We obtained ALMA band 6 observations of NGC 384 on 2016 October 13 through Program 2016.1.01010.S (PI: Walsh) in Cycle 4. The observations consisted of a single pointing with one spectral window centered at 227.325 GHz, the redshifted frequency of the 230.538 GHz 12 CO(2 -1) emission line, along with two spectral windows centered on continuum with average frequencies of 229.288 GHz and 243.015 GHz. The observations were taken in the C40 -6 configuration with minimum \nMinor axis \nFigure 2. Central surface brightness profiles as a function of projected radius along the minor axis. Profiles are shown for the H -band image (black circles), original MGE (blue triangles), and fiducial dust-masked MGE (pink pluses). As shown in Figure 1, the dust is located to the southwest of the nucleus along the minor axis. The impact of the dust is visible as the difference between the surface brightness profiles in this figure at radii of ∼ 0 . '' 6 -1 '' . \n<!-- image --> \n(maximum) baselines of 16.7 m (3100 m) and a total onsource exposure time of 27.3 minutes. Although dust is detected in the spectral windows centered on continuum, this work focuses on the CO emission. \nWe processed the data with Common Astronomy Software Applications (CASA) version 4.7.2, employing a TCLEAN deconvolution with Briggs weighting ( r = 0 . 5; Briggs 1995). We used the line-free channels to perform uv -plane continuum subtraction. The resultant synthesized beam full width at half maximum (FWHM) is 0 . '' 30 (0 . '' 16) along the major (minor) axis, for a geometric mean of 0 . '' 22, or 59.0 pc. The beam PA is 16 . 17 · east of north. Fluxes were calibrated using the ALMA standard quasar J2253 + 1608 for flux calibration, leading to a 10% uncertainty in the absolute flux calibration at this frequency (Fomalont et al. 2014). \nUltimately, the data cube has 124 frequency channels with widths of 15.07 MHz, corresponding to ∼ 19.88 km s -1 at the redshifted CO(2 -1) frequency. We detect CO emission in channels 43 through 78, corresponding to recessional velocities of cz = 3880 . 0 -4575 . 7 km s -1 . The data cube's pixel scale is 0 . '' 04 pixel -1 . Emissionfree regions have root-mean-square (rms) noise at the 0.3 mJy beam -1 channel -1 level.", '2.2.1. Properties of the CO( 2 -1 ) Emission': "Spatially resolved zeroth (integrated CO(2 -1) emission), first (projected line-of-sight velocity, v los ), and second (projected line-of-sight velocity dispersion, σ los ) moment maps of the ALMA observations of NGC 384 are displayed in Figure 3. During construction, we interactively mask pixels without discernible CO emission, and the final maps are shown within the elliptical fitting region used in dynamical modeling (chosen to encompass almost all the emission in the disk, while minimizing the inclusion of noise; see § 3). To calculate the uncertainty in the first moment, we perform a 1000-iteration Monte Carlo simulation. In each iteration, we perturb the data cube, drawing each pixel from a Gaussian centered on the observed pixel value with a width given by the rms noise calculated in the given channel. Then, we construct the first moment map from that new data cube. The standard deviation of the resultant 1000 moment maps is taken as the moment map uncertainty in each pixel. \nThe CO emission is co-spatial with the NGC 384 dust disk, as seen in Figure 1. The emission also displays a bright knot near the center and a dearth of flux both northwest and southeast of the knot. The first moment map shows that the northwest side of the disk is redshifted and the southeast side is blueshifted, with lineof-sight velocities peaking at ∼ ± 350 km s -1 . There is also a kinematic twist in the first moment map, which we account for in the modeling in § 3.1. The second moment map peaks at 201 km s -1 , northeast of the disk center. At the disk center, the velocity dispersion reaches ∼ 120 km s -1 , and it drops to ∼ 20 km s -1 at a projected radius of ∼ 1 . '' 5. \nWe extract flux densities along the major axis of the data cube with an extraction width equal to the geometric mean of the ALMA beam to construct positionvelocity diagrams (PVDs). For the major axis, we use an angle of 132 . 8 · , measured east of north to the blueshifted side of the disk, corresponding to the best-fit flat disk model disk PA (see § 4.2). Due to the kinematic twist near the nucleus, the central region of the PVD primarily contains features that are along the minor kinematic axis. The observed PVD is shown in Figure 4. \nThe total CO(2 -1) flux is 7 . 20 ± 0 . 06 (stat) ± 0 . 72 (sys) Jy km s -1 , with systematic uncertainties stemming from the flux calibration. We estimate the CO(2 -1) luminosity, L ' CO(2 -1) , from the observed flux following Carilli & Walter (2013). In order to convert this value to a CO(1 -0) luminosity ( L ' CO(1 -0) ), we assume R 21 ≡ L ' CO(2 -1) /L ' CO(1 -0) = 0 . 7 (Lavezzi et al. 1999). We then convert this luminosity to an H 2 mass, using the conversion factor α CO ≡ M H 2 /L ' CO(1 -0) = 3 . 1 M ⊙ pc -2 (K km s -1 ) -1 (Sandstrom et al. 2013), and a total gas mass using the helium-to-hydrogen mass ratio f HE = 0 . 36, such that M gas = M H 2 (1 + f HE ). We find a total gas mass of (7 . 91 ± 0 . 07 [stat] ± 0 . 79 [sys]) × 10 7 M ⊙ , which is on the same order of magnitude as the gas masses found in \nother ETGs with CO emission (e.g., Boizelle et al. 2017; Ruffa et al. 2019, 2023). We likely underestimate the total systematic uncertainties in the gas mass, as we used an α CO calibrated for spiral galaxy disks, which may be different from ETG disks.", '3. DYNAMICAL MODELING': "We use a molecular gas-dynamical modeling code developed and tested in previous work (Cohn et al. 2021, 2023). 2 Following the example of our past work (e.g., Barth et al. 2016; Boizelle et al. 2019; Cohn et al. 2021, 2023; Kabasares et al. 2022), we assume the molecular gas follows circular orbits in a thin disk. We calculate the circular velocity ( v c ) relative to the systematic velocity ( v sys ) as a function of disk radius, based on the enclosed BH and stellar mass. We treat the gas mass as negligible, although this assumption is tested in § 4.2. To calculate the enclosed stellar mass, we multiply the stellar light distribution by the stellar mass-to-light ratio ( M/L H ) and deproject the dust-masked MGE, assuming an oblate axisymmetric shape with inclination angle ( i ) matching that of the flat gas disk. We generate the circular velocities on a grid oversampled relative to the ALMA data cube by a factor of s = 6 and converted to v los using the gas disk i and position angle (Γ). \nWe build intrinsic line profiles along the observed ALMA data cube's frequency axis, with a channel spacing of 15.07 MHz. The line profiles are assumed to be Gaussian, centered on v los at each subsampled point, with a width ( σ turb ) that we take to be constant ( σ 0 ) throughout the disk. The line profiles are weighted by the intrinsic CO flux map, which we estimate with a 10-iteration deconvolution of the zeroth moment map from the ALMA beam using the lucy task (Richardson 1972; Lucy 1974) from scikit-image (van der Walt et al. 2014). The flux in each pixel is divided evenly among the s × s subsampled pixels. A scale factor of order unity, f 0 , accounts for normalization mismatches between the observed and modeled line profiles. Model line profiles are then summed back to the original pixel scale of the ALMA data cube, and each frequency slice of the model is convolved with the ALMA synthesized beam. The model and data cubes are then down-sampled in bins of 4 × 4 spatial pixels to mitigate correlated noise (Barth et al. 2016). \nFinally, the model and data are compared directly within an elliptical fitting region that includes nearly all of the CO emission in each channel while omitting excess noise. The fitting region contains 43 velocity channels, corresponding to | v los -v sys | ≲ 430 km s -1 , with a projected semi-major axis of R fit = 1 . '' 5, an axis ratio q ell = 0 . 54, and a position angle Γ ell = 132 . 8 · east of \nFigure 3. Zeroth ( top row ), first ( middle row ), and second ( bottom row ) moment maps of NGC 384 built from the ALMA data ( left column ) and best-fit fiducial model ( middle column ) within the fiducial elliptical fitting region. The uncertainty in the first moment map ( upper right ) and first moment map residual (data -model) normalized by the uncertainty ( lower right ) are also displayed. The moment maps are constructed on the original ALMA pixel scale of 0 . '' 04 pixel -1 and linearly mapped to their respective scale bars, with each moment's data and model using the same scale. These maps are not used in the fit, as models are fit directly to the data cube ( § 3). The first moment map shows that the CO disk displays a kinematic twist in the inner ∼ 0 . '' 5. \n<!-- image --> \nnorth. Ultimately, in the fiducial model (see § 3.1.2 and § 4), the fitting region contains 6493 data points with 6481 degrees of freedom. 3 \nWe adopt a likelihood of L ∝ exp( -χ 2 / 2), taking χ 2 = ∑ j (( d j -m j ) 2 /σ 2 j ), where d j are the downsampled data points, m j the down-sampled model points, and σ j the noise in channel j , calculated as the standard deviation of an emission-free area of the downsampled data. We optimize the free parameters [ M BH , M/L H , i , Γ, v sys , σ 0 , the BH location ( x 0 , y 0 ), and f 0 ] with dynesty (Speagle 2020), a nested sampling code. Flat priors are sampled for each parameter uniformly in linear space, except for M BH , which is sampled uniformly in logarithmic space. The 68% and 99 . 7% confidence intervals of the parameter posterior distributions \n1 \nFigure 4. PVDs extracted from the observed data cube (top) and best-fit fiducial model cube (bottom) for NGC 384. The best-fit systemic velocity (see Table 2) has been subtracted from the line-of-sight velocities displayed here. The extraction path for the PVDs is along the disk major axis (PA = 132 . 8 · east of north, based on the flat disk model in § 4.2), with a width set by the geometric mean of the synthesized beam (0 . '' 22). Both panels are linearly mapped according to the color bar. As a result of the kinematic twist, the central region of the PVD highlights features of the minor kinematic axis. Our dynamical models are fit directly to the data cube, so these PVDs are not used in the fit. \n<!-- image --> \nare reported as 1 σ and 3 σ uncertainties. See Cohn et al. (2021) for more model details.", '3.1. Warped Disk': "As shown in the first moment map in Figure 3, the inner ∼ 0 . '' 5 of the disk appear to display a twist. We find that a flat disk model must be oversimplified, as it is incapable of reproducing this kinematic twist. Therefore, we implement multiple methods to account for the twist, allowing i and Γ to vary with radius. The resultant model retains circular orbits for the gas, but adjacent orbits are no longer required to exist in the same plane, thus creating a warped disk. \n3.1.1. Tilted Ring Model \nWe first implement a set of variable inclinations and position angles to model the warped disk, preserving the thin disk assumption, following a similar approach to the concentric tilted ring model developed in Boizelle et al. (2019). Here, we choose an integer number N of radial nodes within the fitting region at which Γ n and i n are free parameters. We create continuous Γ( r ) and i ( r ) profiles by interpolating Γ and i linearly between these nodes, then extending i ( r ) and Γ( r ) from the innermost node to r = 0, and from the outermost node to R fit . In order to calculate the initial disk radii r , initial Γ and i are required, so we use the best-fit Γ and i from a flat disk model (see § 4.2). A new warped disk radius R grid is then calculated from i ( r ) and Γ( r ), and the rest of the model is generated as described in § 3 on this grid. This approach avoids potential interpolation concerns when applying tilted-ring models to more moderately warped disks. \nGiven the relatively low signal-to-noise ratio of the NGC 384 observations compared to the data in Boizelle et al. (2019), we test the simplest case, using two nodes evenly spaced over the fiducial fitting ellipse. With a free i and Γ at each of these two nodes, this model includes four new free parameters, which are fit simultaneously with the other model parameters. We test the exact choice of node location in § 4.2 below, finding that results for the model with two nodes are robust and consistent, although the tilted ring Γ is incapable of replicating the central twist in the data.", '3.1.2. Parameterized Warped Disk': "The tilted ring model constructed for NGC 384 is stable in the case with two nodes, but a tilted ring model with only two nodes is incapable of producing the rapid Γ warp seen at the center of the first moment map (see Figure 3). We attempt to construct tilted ring models with N ≥ 3 to account for this warp, but due to the low signal-to-noise of the data and the increasing number of free parameters, these models do not exhibit consistent behavior or provide robust parameter solutions. As such, we develop a new warped disk model using a functional form to allow greater radial changes in the Γ and i parameters, without drastically increasing the number of free parameters in the model. \nFirst, we test a model allowing i and Γ to vary linearly across the entire disk, which we refer to as the 'linear twist' model. In this model, there are two free i and two free Γ parameters, one each at the disk center and one each at the disk edge. As with the tilted ring model, calculating the disk radii r requires an initial i and Γ, for which we use the best-fit flat disk model's i and Γ. The resultant linear i ( r ) and Γ( r ) functions are used to construct a new grid of disk radii R , on which we build the rest of the model, as detailed in § 3. As with the tilted ring model above, this model includes four new free parameters. As discussed in § 4.2, i is well-described with a linear fit, but we find that a linear Γ is unable to \naccount for the relatively flat Γ at large radii combined with the strong central twist. \nAs such, we adopt an exponential function, Γ( r ) = Γ 0 +Γ 1 exp( -r/r Γ ), to allow Γ to warp more strongly at the center. However, we continue to keep the inclination linear in this model, with i 0 free at the disk center and i 1 free at the disk edge. There are five free parameters in this model: i 0 , i 1 , Γ 0 , Γ 1 , and r Γ . Comparison of this parameterized warped disk modeling method showed very similar results for NGC 384 to the tilted ring modeling method presented by Boizelle et al. (2019), with agreement typically within a couple km s -1 and only reaching 10 -20 km s -1 in the innermost couple beam areas. Finally, we also test a parameterized warped disk model that uses an exponential function for both inclination and position angle. This more complex model returns an i ( r ) that is consistent with a linear function, so we move forward with the parameterization using an exponential Γ( r ) and linear i ( r ) in our fiducial model.", '4. MODELING RESULTS': 'Below, we discuss the results of our dynamical modeling of the disk in NGC 384. The fiducial warped disk model, with i and Γ parameterized as linear and exponential functions, respectively, captures the kinematic twist seen in the moment map and yields a significantly improved χ 2 and reduced χ 2 ( χ 2 ν ) over the flat disk, tilted ring, and linear twist models.', '4.1. Fiducial Model Results': "The best-fit parameters for the fiducial model of NGC 384 are listed in Table 2 and posterior distributions of the free parameters are displayed in Figure 5. From the best-fit model, we construct moment maps and PVDs, which are shown on the original ALMA 0 . '' 04 pixel -1 scale in Figures 3 and 4, respectively. The uncertainty and residual maps of the first moment are also also shown in Figure 3. In Appendix A, we compare the fiducial model and observed line profiles on the downsampled pixel scale over the full fitting ellipse. The parameterized warped disk i and Γ profiles are shown in Figure 6, reflecting strong constraints on a linear twist in the inclination and a significant central twist in the position angle. \nWe find M BH = (7 . 26 +0 . 43 -0 . 48 [ +1 . 36 -1 . 74 ]) × 10 8 M ⊙ (1 σ [3 σ ] uncertainties), with χ 2 = 7511 . 1 and χ 2 ν = 1 . 159. We also determine M/L H = 1 . 34 ± 0 . 01[ ± 0 . 03] M ⊙ /L ⊙ . Using a ∼ 12 -13 . 5 Gyr stellar age and a metallicity ∼ 0.05 dex above solar, which match the age and metallicity derived for NGC 384 in Yıldırım et al. (2017), simple stellar population models (Vazdekis et al. 2010) suggest a similar, albeit slightly lower, M/L H ∼ 1 . 2 M ⊙ /L ⊙ for a Kroupa (2001) initial mass function (IMF). Our results are thus consistent with more bottom-heavy IMFs seen at the centers of massive ETGs (e.g., Mart'ın-Navarro et al. 2015a,b; La Barbera et al. 2019; Mehrgan et al. 2024). \nTable 2. Modeling Results \nNote -Best-fit fiducial model results. Free parameters are listed in column (1). Median values of each parameter's posterior distribution are shown in column (2). Statistical 1 σ and 3 σ uncertainties are given in columns (3) and (4), respectively. Prior ranges are shown in column (5). The position angle Γ 0 is measured east of north to the major axis on the blueshifted side of the disk, while the exponential coefficient Γ 1 is measured such that positive values represent a counterclockwise shift. The ( x 0 , y 0 ) coordinates of the BH are given relative to RA = 01 h 07 m 25 . 0181 s and Dec = +32 · 17 ' 33 . '' 798 (J2000), the maximum of the continuum emission. Positive x 0 values correspond to shifts eastward and positive y 0 values to shifts northward.", '4.2. Systematic Uncertainties': "To accurately characterize the uncertainty in the BH mass, we must also account for the choices we made when constructing our models and how they affect M BH (e.g., Boizelle et al. 2019; Cohn et al. 2021, 2023; Kabasares et al. 2022). Below, we describe the impact of various modeling assumptions on M BH and M/L H and characterize the systematic uncertainties implied by their departures from the fiducial model. The overall systematic uncertainties on M BH and M/L H include terms (summed in quadrature) for all alternative models that we judge to provide acceptable fits to the data. To decide whether a given model is 'acceptable' in this context, we primarily consider the Bayesian Information Criterion (BIC) = k log( N ) -2 log( L ), where k = the number of free parameters, N = the number of data points, and L = the maximum likelihood. We exclude models that have ∆BIC ≥ +10 (where ∆BIC = test model BIC -fiducial model BIC), indicating the fiducial model is significantly favored (Liddle 2007). Therefore, the models that we include in our systematic uncertainty \nFigure 5. One-dimensional (1D; top edge) and 2D posteriors of fiducial model parameters for NGC 384. The 2D panels display 1 σ , 2 σ , and 3 σ contours. In 1D panels, black lines correspond to the posterior median and blue dashed lines to 3 σ confidence intervals. Parameter median and 3 σ values are also labeled above each 1D panel. The BH mass is M BH = (7 . 26 +0 . 43 -0 . 48 [1 σ ] +1 . 36 -1 . 74 [3 σ ]) × 10 8 M ⊙ . Our best-fit fiducial model is built with the posterior medians and produces a reduced χ 2 ν = 1 . 159. \n<!-- image --> \n/circledot \n/circledot \ncalculations are reasonable models that sufficiently reproduce the data. For example, we test a model with M BH fixed to 0, which produces a 7% increase to M/L H from the fiducial model. However, this model is ruled out with ∆BIC = +66 . 4 compared to the fiducial model, so we exclude it from our systematic uncertainty calculations. \nWarped disk. As discussed in § 3.1, in addition to the fiducial warped disk model, we test a tilted ring model to account for the kinematic twist in the CO disk. Using the tilted ring model with two i and Γ nodes evenly spaced over the fitting region at r = 0 . '' 5 and r = 1 '' , we find i 0 = 54 . 4 · , i 1 = 57 . 8 · , Γ 0 = 122 . 6 · , and Γ 1 = 133 . 5. As shown in Figure 6, this i profile is consistent with the fiducial warped disk model. The tilted ring Γ( r ) is also consistent with the fiducial model for r ≥ 0 . '' 5, disagreeing only at the innermost radii where the tilted ring Γ is held constant. In this model, M BH decreases 47.0% and M/L H increases 3.9% from the fiducial values, with an increased χ 2 ν of 1.178. We test a variety of node locations for the rings, ranging the inner node from 0 . '' 15 -0 . '' 6 and the outer node from 0 . '' 9 -1 '' . In all cases, \nthe model produces i and Γ profiles consistent with the tilted ring model with evenly spaced nodes. \nAdditionally, we consider a linear twist model, the warped disk model in which both i and Γ are linearly interpolated from the disk center all the way to the disk edge. The resultant i profile is consistent with the fiducial warped disk model, while the Γ profile is relatively flat, dominated by the Γ at larger radii and failing to account for the changing Γ at the center (see Figure 6). The M BH in this model increases 20.3% and M/L H decreases 1.8% from the fiducial model, and χ 2 ν increases to 1.169. \nWe also test a flat disk model, with a single i and Γ value across the full disk, ignoring the observed kinematic twist. This flat disk model has i = 57 . 3 · and Γ = 132 . 8 · , consistent with the fiducial warped disk model results for radii ≳ 200 pc ( ≳ 0 . '' 75). The flat disk model yields a 73.2% lower M BH relative to the fiducial model, a 6.3% increase in M/L H , and a much worse χ 2 ν = 1 . 195. \nThe fiducial model is strongly preferred over all of the above models, with ∆BIC = +112 for the tilted ring model, ∆BIC = +53 for the linear twist model, \nFigure 6. Inclination ( i ; left) and position angle (Γ; right) profiles for the best-fit fiducial warped disk (blue solid line), tilted ring (red dashed line), linear twist (green dash-dotted line), and flat disk (black dotted line) models, plotted as a function of the input radius r over the extent of the disk. Shaded regions correspond to 1 σ confidence intervals. The i ( r ) for the fiducial warped disk model is consistent with the i ( r ) profiles of the tilted ring and the linear twist models, while Γ( r ) for the fiducial warped disk model shows a significant, well-constrained twist at the center. At radii ≳ 200 pc ( ≳ 0 . '' 75, outside of the central twist), the fiducial warped disk Γ( r ) agrees with the flat disk model. \n<!-- image --> \nand ∆BIC = +211 for the flat disk model. Thus, we exclude all of these other disk structure models from our systematic uncertainty calculations as they are very strongly disfavored. \nDust extinction. We adopt the dust-masked MGE from Section 2.1.1 when fitting the fiducial dynamical model. Here, we test the use of the original MGE, which ignores the presence of dust, and the dust-corrected MGE, which assumes A H = 0 . 4 mag to the southwest of the nucleus. The dust-corrected MGE produces a positive shift to the inferred BH mass, with M BH = 7 . 33 × 10 8 M ⊙ (1.0% larger than the fiducial M BH ), and a consistent M/L H of 1.34 M ⊙ /L ⊙ . This model has ∆BIC = -7 . 2 compared to the fiducial model, indicating a better fit to the data. The original MGE produces a negative shift to M BH , with M BH = 7 . 04 × 10 8 M ⊙ (3.0% lower than the fiducial M BH ) and M/L H =1.37 M ⊙ /L ⊙ . However, the original MGE model is a significantly worse fit than the fiducial model, with ∆BIC = +44 . 2, so we exclude it from the systematic uncertainty calculations. \nRadial motion. The kinematic twist near the center of the first moment map in NGC 384 could also reflect radial flows of the molecular gas. Following Cohn et al. (2021) and Cohn et al. (2023), we employ two toy models to determine whether the data favor any radial motion. The first model introduces a radially constant \nvelocity term ( v rad ), which is projected into the line of sight and summed with v los . We find v rad = 13 . 3 ± 8 . 4 km s -1 (3 σ uncertainties), consistent with a low-velocity outflow. M BH decreases by 3.1% and M/L H decreases by 0.9%. The v rad model produces the best fit of all of our models, with χ 2 ν = 1 . 156, for a ∆BIC compared to the fiducial model of -13 . 2. \nGas mass. Here, we test a model accounting for the molecular gas mass. The radial CO surface brightness is measured within elliptical annuli on the zeroth moment map and used to calculate projected mass surface densities. We then calculate circular velocities due to the gas mass ( v c, gas ) in the galaxy midplane by integrating following Binney & Tremaine (2008), assuming a thin disk. The resultant v c, gas is summed in quadrature with v c due to stars and the BH. This model produces a 1.1% increase in M BH and 0.8% decrease in M/L H , fully consistent with the fiducial model's M BH and M/L H within uncertainties. The maximum disk v c, gas is ∼ 33 km s -1 , ∼ 8% of the maximum circular velocity due to stars. This model has χ 2 ν = 1 . 158, with ∆BIC = -6 . 5. \nIn the second model, we use the dimensionless, radially varying term κ (Jeter et al. 2019), which is multiplied by v c , projected onto the line-of-sight, and summed with v los . This model also favors a small outflow, with κ = 0 . 03 ± 0 . 02 (3 σ uncertainties). From this best-fit κ , the median radial velocity in the disk is 9.0 km s -1 , consistent with the best-fit v rad found above. The bestfit M BH decreases by 2.9% from the fiducial model and M/L H decreases by 0.9%, with ∆BIC compared to the fiducial model of -12 . 0. \nTurbulent velocity dispersion. In addition to the radially constant σ turb used in our fiducial model, we test an exponential σ turb ( R ) = σ 0 + σ 1 exp( -R/R σ ) as a function of the warped disk radius. This model converges to σ 0 = 10 . 1 km s -1 (consistent with the fiducial model), R σ = 22 . 6 pc (0 . '' 085), and σ 1 = 221 . 4 km s -1 , although the 3 σ confidence interval on σ 1 extends over the full 0 -500 km s -1 prior range. In this model, M BH decreases by 6.1% and M/L H increases by 0.4%. However, ∆BIC = +10 . 9 compared to the fiducial model, indicating the exponential σ turb is significantly disfavored, so we exclude it from the systematic uncertainty calculations. \nOversampling factor. It is possible for both ionized (e.g., Barth et al. 2001) and molecular (e.g., Boizelle et al. 2019) gas-dynamical M BH measurements to depend on the pixel oversampling factor s . In addition to the fiducial s = 6, we test s =1, 2, 3, 4, and 8. The greatest positive (negative) change to M BH is +0 . 5% ( -10 . 0%) with s = 8 ( s = 1). The greatest positive (negative) change to M/L H is +0 . 4% ( < -0 . 1%) with s = 1 ( s = 8). In all of these models, the ∆BIC compared to the fiducial model is small, ranging from -2 . 9 ( s = 2) to +2 . 1 ( s = 1). \nIntrinsic flux map. Here, we examine how the number of Lucy-Richardson deconvolution iterations on the CO flux map affect our results. The fiducial model utilizes \nten iterations, and we also test five and fifteen. The fiveiteration model produces a 0.3% increase in M BH and a 0.1% increase in M/L H . The 15-iteration model produces a 1.2% decrease in M BH and an identical M/L H . The five-iteration model has ∆BIC = -5 . 4 compared to the fiducial model and the fifteen-iteration model has ∆BIC = +9 . 1, just within our threshold to include the model in the systematic uncertainty calculations. \nDown-sampling factor. In the fiducial model, pixels are down-sampled in groups of 4 × 4 to mitigate correlated noise. The synthesized ALMA beam size is 0 . '' 161 × 0 . '' 302 (4.0 × 7.6 pixels), and it has a position angle of 16 . 17 · east of north, such that the major axis of the beam is mostly aligned with the observed y -axis. Therefore, we test a down-sampling factor of 4 × 8 spatial pixels. We find that M BH decreases by 5.0% and M/L H increases by 1.2%. However, the change in pixel binning means this test and the fiducial model have different data, and a direct BIC comparison between these models is not possible. Nevertheless, this model is a significantly worse fit than the fiducial model, with χ 2 ν = 1 . 188. This is worse than the χ 2 ν of almost every model that we have excluded by BIC comparisons, so we also exclude this model from the final systematic uncertainty calculations. \nFitting ellipse. Here, we vary the semi-major axis of the fitting ellipse, testing R fit = 1 . '' 3 and 1 . '' 7. Fitting regions larger than 1 . '' 7 would include many noisy pixels outside of the CO disk. The model with R fit = 1 . '' 3 produces a 6 . 0% increase in M BH and a 1.2% increase in M/L H . In contrast, the R fit = 1 . '' 7 model finds M BH increases by 4.5% and M/L H decreases by 1.0%. As changing R fit means including or excluding different data, direct BIC comparisons are again not possible. Both models are somewhat worse fits than the fiducial model, with χ 2 ν = 1 . 163 for R fit = 1 . '' 3, and χ 2 ν = 1 . 164 for R fit = 1 . '' 7. However, these χ 2 ν values are comparable to those of models included by BIC comparisons, so we include them in our systematic uncertainty calculations. \nFinal error budget . We calculate the positive and negative systematic (sys) uncertainties on M BH and M/L H by summing the respective changes in quadrature from all of the tested models with acceptable BIC and χ 2 ν values. The greatest positive and negative shifts to M BH are +4 . 5% and -10 . 0%, and come from the R fit = 1 . '' 7 model and oversampling s = 1 model, respectively. The greatest positive and negative shifts to M/L H are +1 . 2% and -1 . 5% from the R fit = 1 . '' 3 and 1 . '' 7 models, respectively. Thus, the BH mass is M BH = (7 . 26 +0 . 43 -0 . 48 [stat, 1 σ ] +1 . 36 -1 . 74 [stat, 3 σ ] +0 . 55 -1 . 00 [sys]) × 10 8 M ⊙ , and the M/L H is M/L H = 1 . 34 ± 0 . 01 [stat, 1 σ ] ± 0 . 03 [stat, 3 σ ] ± 0 . 02 [sys] M ⊙ /L ⊙ .", '5. DISCUSSION': 'Our work, which reports the first dynamical M BH measurement for NGC 384, means the number of molecular gas-dynamical M BH determinations for lo- \ncal compact galaxies now equals the number of stellardynamical measurements in the sample. We discuss the BH SOI in § 5.1, compare NGC 384 and the other relic galaxies with dynamical M BH measurements to the local BH -host galaxy scaling relations in § 5.2, and discuss the significance of our results with respect to the co-evolution of BHs and galaxies in § 5.3.', '5.1. The BH Sphere of Influence and Comparisons to Literature': "The BH SOI, defined here as the radius at which the enclosed stellar mass is equal to M BH , is r SOI = 0 . '' 12 (31 pc) for our fiducial model. If we instead estimate the BH SOI as r g = GM BH /σ 2 ⋆ and take σ ⋆ = 221 km s -1 (Yıldırım et al. 2017), we find r g = 0 . '' 24 (64 pc). To quantify how well the data resolve the BH SOI, we compare the SOI to the geometric mean of the ALMA beam ( θ FWHM = 0 . '' 22) via ξ = 2 r SOI /θ FWHM (Rusli et al. 2013). Using r SOI and r g , we find ξ = 1.1 and 2.2, respectively. These results are comparable to many other ALMA dynamical M BH measurements, which often have ξ ∼ 1 -2 (e.g., Barth et al. 2016; Onishi et al. 2017; Davis et al. 2017, 2018; Smith et al. 2019, 2021; Nguyen et al. 2020; Cohn et al. 2021, 2023; Kabasares et al. 2022; Ruffa et al. 2023; Kabasares & Cohn et al. 2024). \nIn NGC 384, the 1 σ statistical uncertainties in M BH are at the ∼ 6 -7% level, while the statistical uncertainties in M BH for UGC 2698 and PGC 11179 are at the 2 -3% level (Cohn et al. 2021, 2023). The total systematic uncertainties on M BH in NGC 384 remain larger at the level of ∼ 8 -14%. These results continue to demonstrate that accounting for the systematic uncertainties in molecular gas-dynamical modeling is critical for accurately characterizing the confidence in the measured M BH .", '5.2. Comparing the Compact Relic Galaxies to the Local BH Scaling Relations': 'In Figure 7, we compare our measurement of M BH in NGC 384 to the BH scaling relations. For the M BH -σ ⋆ relation, we use the stellar velocity dispersion of NGC 384 measured within a circular aperture at the galaxy half-light radius, σ ⋆ = 221 ± 6 km s -1 (Yıldırım et al. 2017). For the M BH -L bul relation, we calculate the total H -band luminosity from the dust-masked MGE, finding L H = 6 . 13 × 10 10 L ⊙ , then convert to L K using an absolute H -band ( K -band) Solar magnitude of 3.37 (3.27) mag (Willmer 2018) and H -K = 0 . 2 mag from SSP models (Vazdekis et al. 2010). We find L K = 6 . 72 × 10 10 L ⊙ . To calculate the total stellar mass, we multiply L H by the best-fit fiducial M/L H , finding M ⋆ = 8 . 22 × 10 10 M ⊙ . The other local compact galaxies with stellar-dynamical (NGC 1271, NGC 1277, and Mrk 1216; Walsh et al. 2015, 2016, 2017) and molecular gas-dynamical (UGC 2698 and PGC 11179; Cohn et al. 2021, 2023) M BH measurements are also shown in \nFigure 7, using the host properties listed in Cohn et al. (2021) and Cohn et al. (2023). Given debates on the bulge properties of the local compact galaxies (Graham et al. 2016b; Savorgnan & Graham 2016), we use the total galaxy luminosity and stellar mass as upper bounds on bulge values when displaying these objects in Figure 7. Total uncertainties are calculated for all of the BH masses in the figure by summing the systematic and the 1 σ statistical uncertainties in quadrature. Unlike the majority of the local compact galaxies, NGC 384 lies within the scatter of all three scaling relations. \nCompared to the M BH -L bul relation of Kormendy & Ho (2013) and the M BH -M bul and M BH -σ ⋆ relations of Saglia et al. (2016), the M BH we measure for NGC 384 is a factor of 2.2, 2.2, and 1.9 × above the expected value, respectively. This result places the measurement within the upper end of the scatter of each relation, in contrast to the five previously measured BH masses in the local compact galaxy sample, which lie an average of 6.3, 7.4, and 2.0 × above the expected scaling relation values for the M BH -L bul , M BH -M bul , and M BH -σ ⋆ relations, respectively. \nNext, we quantify how offset the BH masses in the local compact galaxy sample as a whole are from the local scaling relations using a Monte Carlo simulation. For each of the six objects with M BH measurements to date, we draw one value from a normal distribution centered on the scaling relation-predicted M BH with a width equal to the relation intrinsic scatter, and one from a distribution centered on the measured M BH with a width given by the total measurement uncertainties. In each iteration, there are thus six scaling relation BH masses and six dynamical BH masses. We repeat this process 10,000 times and find that the median M BH from the six dynamical BH masses is above the median M BH from the six scaling relation-predicted BH masses 100.0%, 99.9%, and 91.5% of the time (over the 10,000 iterations), for the M BH -L bul (Kormendy & Ho 2013), M BH -M bul , and M BH -σ ⋆ (Saglia et al. 2016) relations, respectively. Moreover, the median offset between the two distributions is 2 . 1 × 10 9 M ⊙ , 2 . 0 × 10 9 M ⊙ , and 1 . 2 × 10 9 M ⊙ for the M BH -L bul , M BH -M bul , and M BH -σ ⋆ relations, respectively. \nThese results still hold when we consider only the three ALMA measurements, in which case the median of the dynamically measured distribution is above the scaling relation-predicted distribution for M BH -L bul , M BH -M bul , and M BH -σ ⋆ in 97.2%, 95.2%, and 88.2% of the iterations, respectively. We caution that these comparisons are preliminary, as only six out of the sample of fifteen local compact relic galaxies have dynamical M BH measurements so far. \nComparing to the Zhu et al. (2021) BH mass -core mass scaling relation, and using the total stellar mass as the core mass, our M BH for NGC 384 is a factor of 1.02 above the predicted M BH , consistent with the relation. Likewise, the other five dynamical M BH measure- \nments in the relic sample are consistent with the Zhu et al. (2021) relation within its scatter. This relation is constructed for classical bulges and the cores of elliptical galaxies, which are likely descendants of red nugget galaxies at z ∼ 2.', '5.3. BH and Host Galaxy Co-Evolution': "NGC 384 is a member of a sample of compact galaxies that are local analogs and likely relics of z ∼ 2 red nuggets (Yıldırım et al. 2017). It is a fast rotator with a disky shape, uniform old ( ∼ 10 Gyr) stellar population, super-solar stellar metallicity, and an elevated central surface density that falls off steeply at large radii (Yıldırım et al. 2017). NGC 384 also has a red population of globular clusters (Kang & Lee 2021) and a small effective radius (1.5 kpc; Yıldırım et al. 2017) for its stellar mass (8 . 22 × 10 10 M ⊙ ), consistent with the masssize relation at z ∼ 2 (van der Wel et al. 2014). These properties, typical of the local compact galaxy sample, are also consistent with z ∼ 2 red nuggets and the cores of giant ellipticals (e.g., Trujillo et al. 2014; Ferr'e-Mateu et al. 2015, 2017; Mart'ın-Navarro et al. 2015b; Yıldırım et al. 2017), which tend to host the most massive BHs in the local Universe. \nFurthermore, z ∼ 2 red nuggets are thought to seed the cores of massive local ellipticals whose growth through dry mergers increases bulge stellar mass and luminosity without significantly feeding the BH (e.g., Naab et al. 2009; Oser et al. 2010; van Dokkum et al. 2010). NGC 384 and the other local compact galaxies are thus likely passively evolved relics of z ∼ 2 red nuggets that failed to undergo such mergers (Yıldırım et al. 2017). Such relics of red nuggets have also been observed in cosmological simulations (Wellons et al. 2016). Although some of the local compact galaxies are isolated, most are located in group or cluster environments (Yıldırım et al. 2017), and NGC 384 is a member of Arp 331 (also known as the Pisces Cloud), a galaxy group in Pisces (de Vaucouleurs et al. 1976). Nevertheless, the local compact galaxies have regular isophotes and no evidence of tidal interactions, indicating they likely have not formed through the stripping of outer layers. \nFinding over-massive BHs in most or all passively evolved relics of z ∼ 2 red nuggets would suggest that BHs in massive ETGs tend to finish growing prior to the stars in the galaxy outskirts. Most red nugget galaxies would then have to undergo sufficiently many dry mergers between z ∼ 2 and the present day for their stellar masses and luminosities to catch up to the local M BH -L bul and M BH -M bul relations. We note that the sample's closer agreement to M BH -σ ⋆ despite positive offsets from M BH -L bul and M BH -M bul is unsurprising due to the fact that the Faber-Jackson relation levels off for power-law ETGs (Lauer et al. 2007). Moreover, Matt et al. (2023) showed there is a difference in the predicted number density of high-mass BHs ( M BH > 10 9 M ⊙ ) at 1 ≲ z ≲ 3 based on whether they assume M BH -M bul \nFigure 7. The M BH -σ ⋆ (left), M BH -L bul (middle), and M BH -M bul (right) relationships (Kormendy & Ho 2013; McConnell & Ma 2013; Lasker et al. 2014; van den Bosch 2016; Saglia et al. 2016; Savorgnan et al. 2016), with shaded regions indicating their intrinsic scatter. The local compact relic galaxies with ALMA-based molecular gas-dynamical M BH determinations (Cohn et al. 2021, 2023), including NGC 384 (this work), are shown with blue diamonds. Red squares indicate stellar-dynamical M BH measurements from adaptive optics-assisted integral field spectroscopy (Walsh et al. 2015, 2016, 2017). The relic galaxies are plotted with their total luminosities and masses on the M BH -L bul and M BH -M bul relations, making them upper limits on bulge values. PGC 11179, Mrk 1216, NGC 1271, and NGC 1277 are positive outliers from M BH -L bul and M BH -M bul , but NGC 384 and UGC 2698 are consistent within the intrinsic scatter of all three relations. \n<!-- image --> \nversus M BH -σ ⋆ . The fact that the local red nugget relic sample is more closely aligned with M BH -σ ⋆ may indicate that the local M BH -σ ⋆ relation is a more accurate representation of high-mass BHs at 1 ≲ z ≲ 3 than the local M BH -M bul relation. \nAlthough previous work found evidence for overmassive BHs in relic galaxies (e.g., Ferr'e-Mateu et al. 2015; Walsh et al. 2015, 2016, 2017; Cohn et al. 2023), the local compact galaxies NGC 384 and UGC 2698 challenge this interpretation due to their consistency with the local BH scaling relations. However, UGC 2698 may be a less pristine relic (Yıldırım et al. 2017) and thus could represent an intermediate step between the z ∼ 2 and z ∼ 0 BH scaling relations (Cohn et al. 2021). NGC 384, on the other hand, is a more typical local compact galaxy, showing no indication of any substantial growth since z ∼ 2 (Yıldırım et al. 2017). Therefore, our gas-dynamical result could suggest that not all BHs in z ∼ 2 red nuggets were over-massive compared to the local scaling relations. \nIn this case, the properties of the local compact galaxies could be explained by much greater intrinsic scatter at the high-mass end of the scaling relations than has previously been estimated. However, the apparent scatter could also be inflated by systematic differences in stellar- and gas-dynamical measurement methods. Such systematics remain poorly understood, as there are currently only a handful of objects of any kind with M BH measurements from both stellar- and ALMA-based gas-dynamics. Nevertheless, in three of those cases, the stellar-dynamical measurement is a factor of ∼ 2 × larger than the gas-dynamical measurement (Krajnovi'c et al. 2009; Rusli et al. 2011; Schulze & Gebhardt 2011; Rusli et al. 2013; Barth et al. 2016; Davis et \nal. 2017; Boizelle et al. 2019; Smith et al. 2019; Dominiak et al. 2024b; Waters et al. 2024). Stellar-dynamical M BH measurements for NGC 384, UGC 2698, and PGC 11179 may result in higher M BH values that are more significant outliers from the scaling relations. We note that there are five galaxies with dust disks in the relic sample that are excellent targets for additional molecular gas-dynamical M BH measurements but as-yet have no ALMA observations. Directly cross-checking these ALMA-based measurements with stellar dynamics, as well as obtaining stellar- and molecular gas-dynamical M BH measurements for additional local compact galaxies, is crucial for accurately characterizing the scatter of the scaling relations and determining what implications the sample has for BH -host galaxy co-evolution.", '6. CONCLUSIONS': "We have observed CO(2 -1) emission from the circumnuclear gas disk in the local compact relic galaxy NGC 384 at 0 . '' 22 resolution with ALMA. We measure spatially resolved kinematics of the gas disk, identifying a significant kinematic twist in the first moment map. Therefore, we fit a dynamical model accounting for this twist, such that the disk inclination varies linearly with radius and the position angle varies exponentially with radius. We test a variety of additional dynamical models, finding M BH = (7 . 26 +0 . 43 -0 . 48 [1 σ stat] +0 . 55 -1 . 00 [sys]) × 10 8 M ⊙ and stellar M/L H = 1 . 34 ± 0 . 01 [1 σ stat] ± 0 . 02 [sys] M ⊙ /L ⊙ . The BH SOI is resolved by the data ( ξ = 2 r SOI /θ FWHM ∼ 1 -2), and we find that the total systematic uncertainties are a factor of ∼ 2 × larger than the statistical uncertainties, indicating systematics are vital to consider for molecular gas-dynamical M BH measurements. Obtaining higher signal-to-noise CO imag- \ndetailed gas-dynamical modeling, likely shrinking measurement uncertainties. We find NGC 384 lies within the upper end of the scatter of all three BH -host galaxy scaling relations. \nNGC 384 is a likely relic of a z ∼ 2 red nugget, with an evolutionary history distinct from those of typical local massive ETGs (Yıldırım et al. 2017). However, its location relative to the BH scaling relations is different from the over-massive BHs found in other local compact relic galaxies studied thus far (Walsh et al. 2015, 2016, 2017), including one (PGC 11179) measured with molecular gas (Cohn et al. 2023). Unlike UGC 2698, which is also consistent with all three scaling relations within their scatter (Cohn et al. 2021), NGC 384 shows no evidence of any mergers or growth since z ∼ 2. These properties call into question previous evolutionary interpretations of the over-massive BHs in local compact relic galaxies, which had suggested those BHs might have grown prior to the growth of stars in the galaxy outskirts. \nOur result may instead be evidence that there is greater intrinsic scatter than previously thought, due to a diversity of growth histories in the high-mass end of the scaling relations. Another alternative is that z ∼ 2 galaxies may follow a much steeper M BH -L bul relation than locally. However, the dearth of direct comparisons between molecular gas-dynamical M BH measurements and stellar-dynamical determinations calls into question whether this scatter may result partly from a systematic offset between the two methods. Obtaining stellar-dynamical M BH measurements to compare to the molecular gas dynamics, as well as making M BH measurements for the remainder of the local compact relic galaxies, is required to determine whether the sample truly contains over-massive BHs.", 'ACKNOWLEDGEMENTS': "J.H.C. and J.L.W. were supported in part by NSF grant AST-1814799 and AST-2206219. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.01010.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by \nESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The observations are associated with program #13050. The HST data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute. The specific observations analyzed can be accessed via DOI: 10.17909/emkbrz64. Portions of this research were conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing. This work used arXiv.org and NASA's Astrophysics Data System for bibliographic information. A.J.Bak. acknowledges support from the Radcliffe Institute for Advanced Study at Harvard University. L.C.H. was supported by the National Science Foundation of China (11991052, 12011540375, 12233001), the National Key R&D Program of China (2022YFF0503401), and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06). \nJ.H.C. would like to thank Silvana Delgado Andrade, Kate Pitchford, Taylor Hutchison, Ryan Hickox, Quinn Casey, and Emmanuel Durodola for their support and helpful discussions. We thank the anonymous referee for their valuable comments, which improved our manuscript.", 'Facilities: ALMA, HST (WFC3)': "Software: CASA (v4.7.2; McMullin et al. 2007; CASA Team et al. 2022), dynesty (Speagle 2020, DOI), GALFIT (Peng et al. 2010), kinemetry (Krajnovi'c et al. 2006), mgefit (Cappellari 2002), Tiny Tim (Krist & Hook 2004), AstroDrizzle (Gonzaga et al. 2012; Hack et al. 2012), scikit-image (van der Walt et al. 2014), ASTROPY(Astropy Collaboration et al. 2013, 2018, 2022), MATPLOTLIB (Hunter 2007), NUMPY (van der Walt et al. 2011; Harris et al. 2020), SCIPY (Virtanen et al. 2020).", 'A. LINE PROFILES': 'The model is fit to the data cube on the down-sampled pixel scale within the fitting ellipse, as discussed in § 3, with χ 2 calculated from the model and observed line profiles at each pixel. The fiducial model and data cube line profiles for every down-sampled pixel in the ellipse are plotted in Figure 8.', 'REFERENCES': 'Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167. doi:10.3847/1538-4357/ac7c74 \nAstropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123. doi:10.3847/1538-3881/aabc4f \nFigure 8. Observed (black) and best-fit fiducial model (blue) line profiles in the fitting ellipse, on the down-sampled pixel scale, with noise per channel shown in gray. The BH location is shown with a black star. The x-axis is labeled with the velocity channel ranges for each panel, and the y-axis is labeled with the minimum and maximum flux shown in each panel. The physical extent of the fitting region is also indicated at the corners of the x and y axes. This figure indicates the model is a good fit to the observations across the gas disk. \n<!-- image --> \n- \n- \n- \nTrujillo, I., Ferr\'e-Mateu, A., Balcells, M., et al. 2014, ApJL, 780, L20 \nvan den Bosch, R. C. E., Gebhardt, K., Gultekin, K., et al. 2012, Nature, 491, 729 \nvan den Bosch, R. C. E., Gebhardt, K., Gultekin, K., et al. 2015, ApJS, 218, 10 \nvan den Bosch, R. C. E. 2016, ApJ, 831, 134 \nvan der Marel, R. P. & van den Bosch, F. C. 1998, AJ, 116, 2220. doi:10.1086/300593 \nvan der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, \nComputing in Science and Engineering, 13, 22 \nvan der Walt, S., Schonberger, J., Nunez-Iglesias, J, et al. \n2014, PeerJ, 2, e453 \nvan der Wel, A., Franx, M., van Dokkum, P. G., et al. 2014, ApJ, 788, 28. doi:10.1088/0004-637X/788/1/28 \nvan Dokkum, P. G., Whitaker, K. E., Brammer, G., et al. 2010, ApJ, 709, 1018 \nVazdekis, A., S\'anchez-Bl\'azquez, P., Falc\'on-Barroso, J., et al. 2010, MNRAS, 404, 1639 \nViaene, S., Sarzi, M., Baes, M., et al. 2017, MNRAS, 472, 1286. doi:10.1093/mnras/stx1781 \nVirtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261. doi:10.1038/s41592-019-0686-2 \nWalsh, J. L., Barth, A. J., & Sarzi, M. 2010, ApJ, 721, 762 Walsh, J. L., Barth, A. J., Ho, L. C, & Sarzi, M. 2013, ApJ, \n770, 86 Walsh, J. L., van den Bosch, R. C. E., Gebhardt, K., et al. 2015, ApJ, 808, 183. doi:10.1088/0004-637X/808/2/18 Walsh, J. L., van den Bosch, R. C. E., Gebhardt, K., et al. 2016, ApJ, 817, 2 Walsh, J. L., van den Bosch, R. C. E., Gebhardt, K., et al. 2017, ApJ, 835, 208 Waters, T. K., Gultekin, K., Gebhardt, K., et al. 2024, arXiv:2406.14623 Wellons, S., Torrey, P., Ma, C.-P., et al. 2016, MNRAS, 456, 1030 Willmer, C. N. A. 2018, ApJS, 236, 47 Wilman, R. J., Edge, A. C., & Johnstone, R. M. 2005, MNRAS, 359, 755. doi:10.1111/j.1365-2966.2005.08956.x Yıldırım, A., van den Bosch, R. C. E., van de Ven, G., et al. 2015, MNRAS, 452, 1792. doi:10.1093/mnras/stv1381 Yıldırım, A., van den Bosch, R. E. C., van den Ven, G., et al. 2017, MNRAS, 468, L4216 Zhu, P., Ho, L. C., & Gao, H. 2021, ApJ, 907, 6. \ndoi:10.3847/1538-4357/abcaa1'}

2024A&A...690A.251C

Context. Some giant radio galaxies selected at Xrays with active galactic nuclei AGN show signs of a restarted nuclear activity old lobes plus a nuclear young radio source probed by gigahertz peaked spectra. The study of these sources gives us insights into the AGN activity history. More specifically the kinematics and properties of the outflows can be used as a tool to describe the activity of the source. Aims. One object in this peculiar class is Mrk 1498 a giant lowfrequency double radio source that shows extended emission in O III. This emission is likely related to the history of the nuclear activity of the galaxy. We investigate whether this bubblelike emission might trace an outflow from either present or past AGN activity. Methods. Using a mediumresolution spectroscopy R 10 000 available with MEGARAGTC we derived kinematics and fluxes of the ionised gas from modelling the O III and H features. We identified three kinematic components and mapped their kinematics and flux. Results. All the components show an overall blue to red velocity pattern with similar peaktopeak velocities but a different velocity dispersion. At a galactocentric distance of 2.3 kpc we found a blob with a velocity up to 100 km sSUP1SUP and a high velocity dispersion 170 km sSUP1SUP that is spatially coincident with the direction of the radio jet. The observed O IIIH line ratio indicates possible ionisation from AGN or shocks nearly everywhere. The clumpy structure visibile in HST images at kiloparsec scales show the lowest values of logO IIIH lt 1 which is likely not related to the photoionisation by the AGN. Conclusions. Taking optical and radio activity into account we propose a scenario of two different ionised gas features over the radio AGN lifecycle of Mrk 1498. The radio emission suggests at least two main radio activity episodes an old episode at megaparsec scales formed during a time span of 100 Myr and a new episode from the core gt 2000 yr ago. At optical wavelengths we observe clumps and a blob that are likely associated with fossil outflow. The latter is likely powered by past episodes of the flickering AGN activity that may have occurred between the two main radio phases.

2024-10-01T00:00:00Z

['arXiv:2409.07534', '10.1051/0004-6361/202450045', '2024A&A...690A.251C', '2024arXiv240907534C', '10.48550/arXiv.2409.07534']

['ISM: jets and outflows', 'galaxies: active', 'galaxies: kinematics and dynamics', 'galaxies: nuclei', 'Astrophysics - Astrophysics of Galaxies']

Clues of the restarting active galactic nucleus activity of Mrk 1498 from GTCMEGARA integral field spectroscopy data

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

https://arxiv.org/pdf/2409.07534.pdf

{'Clues of the restarting active galactic nucleus activity of Mrk 1498 from GTC/MEGARA integral field spectroscopy data': 'S. Cazzoli 1 , L. Hernández-García 2 , 3 , I. Márquez 1 , J. Masegosa 1 , G. Bruni 4 , F. Panessa 4 , and L. Bassani 5 \n- 1 IAA-CSIC - IAA - Instituto de Astrofísica de Andalucía (CSIC), Apdo. 3004, 18008, Granada, Spain e-mail: sara@iaa.es\n- 2 Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Providencia, Santiago, Chile\n- 3 Instituto de Física y Astronomía, Universidad de Valparaíso, Av. Gran Bretaña 1111, Playa Ancha, Chile\n- 4 INAF - Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100, I-00133 Roma, Italy\n- 5 Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna. \nReceived 20 March 2024; accepted 23 August 2024', 'ABSTRACT': 'Context. Some giant radio galaxies selected at X-rays with active galactic nuclei (AGN) show signs of a restarted nuclear activity (old lobes plus a nuclear young radio source probed by giga-hertz peaked spectra). The study of these sources gives us insights into the AGN activity history. More specifically, the kinematics and properties of the outflows can be used as a tool to describe the activity of the source. \nAims. One object in this peculiar class is Mrk 1498, a giant low-frequency double radio source that shows extended emission in [O III]. This emission is likely related to the history of the nuclear activity of the galaxy. We investigate whether this bubble-like emission might trace an outflow from either present or past AGN activity. \nMethods. Using a medium-resolution spectroscopy (R ∼ 10 000) available with MEGARA / GTC, we derived kinematics and fluxes of the ionised gas from modelling the [O III] and H β features. We identified three kinematic components and mapped their kinematics and flux. \nResults. All the components show an overall blue to red velocity pattern, with similar peak-to-peak velocities but a di ff erent velocity dispersion. At a galactocentric distance of ∼ 2.3 kpc, we found a blob with a velocity up to 100 km s -1 , and a high velocity dispersion ( ∼ 170 km s -1 ) that is spatially coincident with the direction of the radio jet. The observed [O III] / H β line ratio indicates possible ionisation from AGN or shocks nearly everywhere. The clumpy structure visibile in HST images at kiloparsec scales show the lowest values of log[O III] / H β ( < 1), which is likely not related to the photoionisation by the AGN. \nConclusions. Taking optical and radio activity into account, we propose a scenario of two di ff erent ionised gas features over the radio AGN lifecycle of Mrk 1498. The radio emission suggests at least two main radio activity episodes: an old episode at megaparsec scales (formed during a time span of ∼ 100 Myr), and a new episode from the core ( > 2000 yr ago). At optical wavelengths, we observe clumps and a blob that are likely associated with fossil outflow. The latter is likely powered by past episodes of the flickering AGN activity that may have occurred between the two main radio phases. \nKey words. galaxies: active, galaxies: ISM, galaxies: kinematics and dynamics, techniques: spectroscopic.', '1. Introduction': "Active galactic nuclei (AGN) with radio jets have been observed in phases of activity and dormancy. After a phase of activity ( ∼ 10 8 yr; Schawinski et al. 2015), the fuelling of the radio jets stops and the AGN enters a dormant phase. In some cases ( ∼ 15 %), a new phase of restarted activity can be detected even before the older phase has faded (Kukreti et al. 2023). \nGiant radio galaxies (i.e. radio galaxies with a linear extent above 0.7 Mpc; Garofalo 2022) with old lobes that represent a historical record of past activity are perfect laboratories for studying intermittent activity and galaxy evolution. They hold huge reservoirs of energy that can be observed even when it is not fed by the jets (Dabhade et al. 2020). Hence, the same dataset includes di ff erent phases of nuclear activity from the same nucleus. Bassani et al. (2016) performed radio and X-ray studies of the combined INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) and Swift AGN populations from the 3CR catalogue, which includes 67 sources, the linear sizes of 22 % of which exceed 0.7 Mpc. These are thus classified as giant radio galaxies. Bruni et al. (2019) presented a detailed \nradio study of 13 of these sources and remarkably, 8 sources (61 %) showed signs of restarting activity in a gigahertz-peaked spectrum (GPS). \nThe nuclear activity of galaxies is also modulated by the onset of large-scale outflows. Hence, a complementary view to radio activity that can be used to date the AGN life cycle is to search for evidence for AGN-driven outflows. These latter have been detected through spectroscopy (lines with blue / red wings or large widths; e.g. Cazzoli et al. 2014, 2018, 2020, 2022; Maiolino et al. 2017) and imaging (spatially extended or bubble-like emission; Masegosa et al. 2011; Hermosa Muñoz et al. 2024). The outflow properties (e.g. kinematics, lifetime, and energy) can be used to investigate the history of AGN activity. The outflow properties themselves sensitively depend on the history of the AGN energy injection (e.g. the crossing timescale of the outflow), that is, they are sensitive either to faded or restarted AGN phases (Zubovas & Nardini 2020). It is still not well understood whether the e ff ect of feedback evolves with the AGN lifecycle. \nInterestingly, the optical morphology of giant radio galaxies reveals some potential candidates for di ff use emission that might arise from AGN-driven outflows and hence might be related to the restarted or fading radio activity. In particular, it was shown that the radio activity can also drive outflows (jet-driven outflows; see e.g. Tadhunter 2007, Hardcastle & Croston 2020; Morganti et al. 2021; Cazzoli et al. 2022; Cresci et al. 2023; Speranza et al. 2024). \nThe nearby AGN Mrk1498 (also called WN 1626 + 5153, 1626 + 518 or Swift J1628.1 + 514; z ∼ 0.055) is a giant radio galaxy and of the best candidates for studying the relation between faded and restarted AGN activity and di ff use gaseous emission. It is a Fanaro ff -Riley II radio source showing a classical double-lobed structure (projected separation of 1.1 Mpc) and an unresolved nucleus found by Rottgering et al. (1996) in the Westerbork radio telescope data, more specifically, in data from the Westerbork Northern Sky Survey (WENSS) at 325 MHz. Deep LOFAR images show no connection between the nucleus and the radio lobes (Bruni et al. 2021). At optical wavelengths, Mrk1498 has a prominent broad-line region, as suggested by the very broad H α component (FWHM ∼ 6500 km s -1 ), which resulted in a Sy1.9 classification (Winter et al. 2010). Moreover, Mrk 1498 shows remarkable extended features that are especially evident in Hubble Space Telescope ( HST ) [O III] images (Keel et al. 2015; Hernández-García et al. 2019). These features are relatively [OIII] bright bubble-like structures that extend out to ∼ 10 '' ( ∼ 10 kpc) to the north-east and south-west. They are not aligned with the 1.4 GHz radio jet emission (from the NRAO VLA Sky Survey source catalogue; Keel et al. 2012) or with the soft X-ray emission from the narrow-line region (NLR), as suggested for type 1 AGNs (Gómez-Guijarro et al. 2017). This misalignment and the distance of the bubble-like di ff use [O III] emission rule out any possible relation with the NLR, making its origin intriguing. \nTaking into account the multi-wavelength properties of Mrk1498, previous studies suggested two main possible scenarios to explain the origin of the di ff use bubble-like emission: It might be the aftermath of a merging event or an AGN-driven outflow. The latter is more likely due to the morphology of the features. \nUsing optical integral field spectroscopy (IFS) observations with the Gemini Multiple-Object Spectrometer (GMOS), Keel et al. (2017) demonstrated that a circumnuclear [O III] ring follows the main rotation of the galaxy. These authors also suggested that the clouds in the extended emission are most likely ionised by either a faded AGN or the aftermath of a past merger episode (see also Sartori et al. 2016). \nIn this paper, we use the spectral (R ∼ 10 000) capability of MEGARA / GTC optical IFS observations to study the [O III]bright emission in Mrk 1498. Our goal is to obtain clues about the AGN activity in Mrk 1498 by trying to connect the AGN lifecycle to feedback by searching for outflow signatures. We investigate the multiple distinct kinematic components and explore line flux ratios to constrain the dominant ionisation mechanisms. Furthermore, we make use of the radio properties of Mrk 1498 to link the AGN restarting and fading activity to the observed optical properties. \nThis paper is organised as follows. In Section 2, the data and observations are presented, as is the data reduction. In Section 3, we describe the spectroscopic analysis based on line modelling and map generations. Section 4 highlights the main observational results, which are discussed in Section 5. The \nTable 1. General properties of Mrk 1498 \nNotes. - 'Distance' and 'Scale' are from the Local Group. 'Morphology': Hubble classification. ' i ' is the inclination angle defined as the angle between the line of sight and the polar axis of the galaxy. It is determined from the axis ratio of the isophote in the B band using a correction for intrinsic thickness based on the morphological type. 'PAjet' is measured from radio images at di ff erent spatial scales. Specifically, with images generated from NVSS (arcminute scale) and VLBA (arcsecond scale) data obtained at frequencies of 1.4 and 4.8 GHz, respectively. \nmain conclusions are presented in Section 6. \nAll images and spectral maps are oriented following the standard criterion, that is, north is up, and east is to the left. Throughout the whole paper, angular dimensions are converted into physical distances using the scale distance from the Local Group (1188 pc / '' ; see Table 1).", '2. Observations and data reduction': "Observations were carried out on 2019 July 27 with MEGARA (Multi-Espectrógrafo en GTC de Alta Resolución para Astronomía; Carrasco et al. 2018; Gil de Paz et al. 2018; Gil de Paz 2020) the integral field unit (IFU) at the 10.4 m Gran Telescopio Canarias (GTC). The MEGARA IFU consists of 567 fibres (100 µ m in core size) arranged on a square microlens array that projects a field of 12 '' . 5 × 11 '' . 3 on the sky. Each microlens is a hexagon inscribed in a circle with a diameter of 0 '' . 62 projected on the sky. A total of 56 ancillary fibres (organised in bundles of eight fibres), located at a distance of ∼ 2.0 arcmin from the centre of the IFU field of view (FoV) deliver simultaneous sky observations. \nWe used the MR-R volume-phased holographic (VPH; i.e. VPH521-MR) covering the 4963 - 5445 Å spectral range with a spectral resolution of R ∼ 12 000. The linear dispersion was ∼ 0.122 Å / pixel, hence ∼ 7.3 km s -1 at the corresponding wavelength of [O III] λ 5007. We obtained six exposures of 1360 s each to facilitate cosmic-ray removal. During the observations of Mrk 1498, the dimm-seeing was 1 '' and the airmass was 1.15. \nMEGARA raw data were reduced with the data reduction package provided by Universidad Complutense de Madrid ( megara drp 1 ; version 0.8; Pascual et al. 2020; Pascual et al. 2019) following the MEGARA cookbook 2 . The pipeline allowed us to perform the following steps: sky and bias subtraction, flat-field correction, spectrum tracing and extraction, correction for fibres and pixel transmission, and wavelength and flux calibration (see e.g. Cazzoli et al. 2020, 2022). \nWe applied a regularization grid to obtain square spaxels 3 with a size of 0 '' . 4. The final cube has dimensions of 33 × 30 × 4300, which is equivalent to a total of 990 spectra in the datacube. \nThe point spread function (PSF) of the MEGARA datacube can be described as a Mo ff at function (Mo ff at 1969). In order to avoid any possible PSF contamination in the kinematic measurements, we conservatively considered as the 'nuclear region' a circular area with a radius equal to the width at 10 % intensity of the PSF radial profile. This corresponds to 1 '' . 2 (in radius), as measured from the 2D profile brightness distribution of the standard star. This area is marked (with a circle) in the images and spectral maps from the MEGARA cube and also in the HST image in Fig. 1 The continuum emission from GTC / MEGARA observations (see Fig. 1) covers nearly the entire extension of the di ff use [O III]-bright emission. \nIn each spectrum (i.e. on a spaxel-by-spaxel basis), the e ff ect of instrumental dispersion ( σ INS, i.e. ∼ 0.3 Å) was corrected for by subtracting it in quadrature from the observed line dispersion ( σ obs): σ line = q σ 2 obs -σ 2 INS .", '3. Data analysis': "The spectral coverage of VPH521-MR is limited to ∼ 500 Å (Sect. 2) and the observed spectra lack the stellar features required for an optimal modelling of the stellar continuum. Hence, we did not apply any procedure to subtract the underlying stellar light. The stellar contribution to the Balmer lines in the nuclear spectrum of Mrk 1498 is low (see Fig. 4 in Hernández-García et al. 2019). \nThe spectra in the data cube were visually inspected in order to search for line asymmetries, such as broad wings or double peaks. The latter are present at nearly any distance from the nucleus 4 in the H β and [OIII] emission lines. Hence, we modelled the spectral lines with multiple Gaussian functions (up to three per line; see below). Figure 2 shows examples of the line modelling. \nAll the spectra in the cube were shifted to the rest-frame wavelength using the value of the redshift of Mrk1498 ( z = 0.055625; Table 1). For the line fitting, we then applied a Levenberg-Marquardt least-squares fitting routine under the Interactive Data Analysis (IDL), using mpfitexpr by Markwardt (2009). Following Osterbrock & Ferland (2006), we imposed the intensity ratios of the [O III] λ 4959,5007 lines to be 2.99. Moreover, H β and [O III] were constrained to share the same kinematics (line width and shifts). \nWe excluded from the fitting procedure spectra with a signalto-noise ratio (S / N) per pixel in [O III] λ 5007 lower than 3. This excluded 31 % of the total number of spaxels. \nThe [O III] λ 5007 lines are more intense than H β (see e.g. Fig. 2). In 27 % of the total number of spaxels, H β was not detected, and we only modelled the [O III] lines. These spaxels are generally \nFig. 1. HST sharp-divided image obtained from the data of the Advanced Camera for Surveys using the FR551N filter. It was obtained following the receipt in Márquez et al. (2003), i.e. by dividing the original image, I, by a filtered version of it. The image is displayed with a zoomed-in view that matches the FoV of our MEGARA observations (Sect. 2). The cross is the photometric centre. The size of the MEGARA PSF is indicated in the bottom right part of the figure. The white bar in the upper left corner represents 2 kpc (1 '' . 7). The green, pink, and orange circles mark the spatial location at which we extracted the line profiles (and the corresponding modelling) shown in Fig. 2. The yellow box marks the GEMINI / GMOS field of view from previous IFS observations (Keel et al. 2017). \n<!-- image --> \nlocated at relatively large distances from the nucleus (i.e. > 4 '' . 0). \nWe identified up to three kinematic components to account for double peaks and / or broad wings in the line profiles. To perform the fitting and in order to prevent overfitting, we followed the approach proposed by Cazzoli et al. (2018), which was successfully applied to optical spectra of active galaxies both from long-slit (Hernández-García et al. 2019; Hermosa Muñoz et al. 2020) and IFS (Cazzoli et al. 2020, 2022). Briefly, after fitting all emission lines with one Gaussian component, we evaluated the amplitude of the residuals below the [O III] λ 5007 emission line using the ε line parameter. This parameter is defined as the ratio of the standard deviation of the residuals below the emission lines and that of the line-free continuum. If ε line was greater than 2.5, a further component was added. This evaluation was then repeated after we tested the modelling with two components, and a tertiary component was added if needed. Three Gaussian per line is the largest meaningful number of components that ensures a trade-o ff between the recovery of emission line properties, the S / N, and the goodness of the fit. \nFinally, all the fitting were inspected by eye to ensure a satisfactory line modelling. Figure 2 shows the results of the line fitting for three regions for which three, two, and one Gaussian component were required, respectively. \nOur final modelling of H β did not require a broad component in the nuclear region to confirm the type 1.9 AGN classification (Table 1) of the active nucleus in Mrk 1498. \nFig. 2. Examples of emission line spectra (black) and their modelling at di ff erent galactocentric distances (R; see the top labels and symbols in Fig. 1). Both [O III] λ 5007 (top) and H β features are displayed at the corresponding observed wavelengths (i.e. no correction for redshift; see Sect. 3). For each panel, we show the modelled line profile (red line) and the components (with di ff erent colours). Specifically, Gaussian curves indicate the primary (green), secondary (blue), and tertiary (pink) components we used to model the profiles. Residuals from the fit are shown in the small lower panels, in which vertical yellow lines mark the wavelength range we considered to calculate the ε line (for the [O III] λ 5007 line only; ee Sect. 3 for details). The insets in the upper panels indicate the kinematic values of the components (the emission lines were tied to share the same kinematics). \n<!-- image --> \nThe three components can clearly be distinguished based on their spatial distribution and widths. Specifically, the primary component is detected in near all the FoV, with line widths that are generally larger than 40 km s -1 (with values up to 250 km s -1 ). This provides a good fit of the line wings. The other two components are mostly observed in the north-south direction and have narrower widths, σ< 60 km s -1 (secondary) and σ< 80 km s -1 (tertiary; see Figure 3). \nWe received the following information from the Gaussian functions we used to model each emission line and component: central wavelength, width, and flux intensity, along with their respective fitting errors. These are the 1σ parameter uncertainties weighted for the reduced χ 2 of the fit (see the mpfitexpr documentation). These are lower than 0.5 Å on average for the central wavelengths and widths. \nIn order to map the velocity dispersion of the gas, the intrinsic line widths were computed after we removed the instrumental profile inferred from the sky lines (Sect. 2). The flux maps were obtained directly from the measurements of line intensity on a spaxel-by-spaxel basis. \nThe line maps are shown in Figures 3 and 4. H β and [O III] share the same kinematics, as mentioned above.", '4. Main observational results': "The ionised gas probed by the [O III] emission lines is mostly distributed in the north-south direction. This is in contrast to the orientation of the elliptical host galaxy, PA = 102 · (Vennik et al. 2000), which was calculated based on B and R photometry obtained with CAFOS, which is mounted at the Calar Alto 2.2m telescope (spatial resolution of 0 '' . 4 / pixel). As mentioned in \nTable 2. Main kinematic properties of the di ff erent components of Mrk1498 \nNotes. - PA is the position angle of the kinematic major axis; ∆ V is the observed peak-to-peak semi-amplitude of the velocity field; σ c ( σ avg) is the central velocity dispersion within (excluding) the nuclear region (see Sect. 4.1). The quoted uncertainties for the velocity dispersion measurements are one standard deviation. \nSect. 3, the primary (broadest) component is the most extended and nearly covers the entire MEGARA FoV. \nThe ionised gas probed by the two narrower components (secondary and tertiary) is found to have a similar extension (up to ∼ 6 '' . 0, i.e. 7.1 kpc). The main di ff erence is that part of the secondary component emission is found to be spatially coincident with the direction of the radio jet towards the north-east (PAjet = 45 · ; Table 2; Fig. 3, second row). \nFor all the three components, the faint emission probed by H β has the same morphology as [O III], but at a lower S / N. \nWe summarise below the main results for the three components we used to model the emission lines. We focus on kinematics (Sect. 4.1) and flux (Sect. 4.2). In Table 2 we indicate the main kinematic properties. \nMARK 1498 \nï \nComp: comp2 \nFig. 3. [OIII] λ 5007 velocity field (km s -1 ) and velocity dispersion (km s -1 ) maps and flux intensity maps (erg s -1 cm -2 ) for the [O III] λ 5007 and H β emission lines for the primary component, the broadest of the detected components (Sect. 3). The cross marks the position of the photometric centre. The solid black line indicates the orientation of the radio jet (PAjet = 45 · ; Table 1). The dashed lines indicate the PA we used to extract the diagrams shown in Fig. 5. The maps are spatially smoothed (boxcar average) with a smoothing window of two spaxels (i.e. 0 '' . 8). The size of the MEGARA PSF is indicated with a circle (as in Fig.1 and in all the maps presented in this paper). \n<!-- image --> \nMARK 1498 \nï \nBPTs \nFig. 4. Maps of the standard BPT line ratio log [O III] / H β for the di ff erent components. The maps are smoothed as in Fig. 3. The symbols are the same as in Fig. 3. \n<!-- image -->", '4.1. Kinematics': "The velocity fields for the three components show the same overall behaviour with positive (negative) velocities to the north-west (south-east). The components share a position angle that is similar to that of the maximum velocity gradient and amplitudes (see the dashed lines in all the spectral maps and Table 2). \nThe main di ff erence of the velocity maps is that the broadest (primary) component shows a blob at a distance of 1 '' . 8 (2.2 kpc) towards the south-west in the direction of the radio jet. This blob is observed with positive velocities ( ∼ 75 km s -1 ). At the same spatial position, the σ -map shows values up to ∼ 175 km s -1 , which is higher than what is seen elsewhere (96 ± 24 km s -1 ; \nFig. 5. Position-velocity (P-V) curves of the di ff erent components (labelled at the top) that probe the ionised gas in Mrk 1498. The curves were obtained considering a 0 '' . 8 width pseudo-slit aligned with the major rotation axis (i.e. 148 · ; Table 2). There is a small o ff set between the kinematic centre of the components (filled green circles), i.e. < 0 '' . 8. The radius was therefore calculated as the distance from the photometric centre (continuum peak). For comparison, the pink symbols indicate the P-V curves from Keel et al. (2012) with PA = 149 · . \n<!-- image --> \nσ avg in Table 2). \nThe primary component is about three times broader overall than the secondary and tertiary (excluding the nuclear PSF region). The width ratio of the two narrow components is 0.6 on average (the secondary component is narrowest). \nThe σ -map of the secondary component is rather homogeneous, with an average velocity dispersion in the central region ( σ c) similar to that measured elsewhere (Table 2). Only small deviations exist to the south-east and west of the nucleus at ∼ 40 km s -1 . These deviations lack a counterparts in the velocity or flux intensity maps. \nThe σ -maps of the primary and tertiary components show a similar behaviour. Specifically, they are both enhanced in the same directions: in the direction of the radio jet, and a direction that is almost perpendicular to it (PA ∼ -70 · ). At these locations, the velocity dispersion is about 160 - 220 km s -1 and 60 - 80 km s -1 for the primary and tertiary components, respectively. \nNone of the velocity dispersion maps is therefore as expected in the case of a regular rotating disc (see e.g. Flores et al. 2006). More specifically, the velocity dispersion maps are not centrally peaked. They are either rather flat (secondary component) or have some level of distortions (primary and tertiary components). This suggests the presence of ionised gas discs with different levels of perturbations (see Sect. 5.2).", '4.2. Emission line fluxes and line ratios': "We describe the general behaviour of the flux-intensity of the [OIII] line observed at higher S / N with respect to H β below (as mentioned earlier). \nThe emission in the inner R ∼ 2 '' is not centrally peaked, but elongated (north-south) for all three components. The elongation is strongest for the secondary component. The eccentricity of the ellipses contains 30 % of the flux peak and is 0.5 (Fig. 3, middle row, third panel). The eccentricity is 0.6 and 0.8 for the primary and tertiary components, respectively. \nThis elongated morphological feature might be related to unresolved nuclear motions that cannot be distinguished at the spatial resolution of MEGARA (see Sect. 5.1). \nUp to large galactocentric distances ( ∼ 6 kpc), the evident features are gas clumps that are also visible in the HST image \n(see Fig. 1). These clump-like features show log[O III] λ 5007 / H β lower than 1.0 (0.90 ± 0.08, on average) for the primary (most extended) component, as shown in Fig. 4 (left). Elsewhere, the line ratio is 1.1 ± 0.09 (see Fig. 4). \nThese clumps do not produce perturbations of the overall velocity dispersion maps or velocity field (see Fig. 3).", '5. Discussion': 'In Sect. 5.1, we compared the results from current MEGARA IFS data and previous works. The discussion in Sections 5.2 and 5.3 is dedicated to kinematic and flux features, respectively, and to their possible connection with a putative outflow. In Sect. 5.4 we complete the considerations of the radio jet activity periods (from previous works) and the optical IFS feature analysed here. Figure 5 presents the position-velocity plots along the major kinematic axis (see Table 2) for all three components. Figure 6 is a schematic illustration of the observed features of Mrk 1498 from previous multi-wavelength studies and the current work.', '5.1. Comparison with previous works based on optical data': "The study of the optical properties of both the AGN and the ionised gas emission in Mrk 1498 is mostly based on slit spectroscopy with a low spectral resolution (Keel et al. 2012, 2015; Hernández-García et al. 2019), except for the unique IFS-based work by Keel et al. (2017). The latter was based on data from the Gemini Multiple-Object Spectrometer (GMOS) that covered a region of 3 '' . 5 × 5 '' . 0 and had a image quality of 0 '' . 4 in FWHM at a resolution of 0.5 Å per pixel. The GEMINI data set allowed us to map the rings close to the nuclear region that is observed in the Hubble image (Fig. 6), but not the whole extended emission on kiloparsec scales that is covered by our MEGARA observation (Figures 1 and 6). \nThe nature of the AGN in the nucleus of Mrk 1498 has been debated in the last years (see Hernández-García et al. 2019). At optical wavelengths, the gas surrounding the nucleus is ionised by the AGN radiation field, as indicated by the results from the analysis of slit-spectroscopy data from the Lick and San Pedro Martir observatories by Keel et al. (2012) and Hernández-García et al. (2019) at a spectral resolution of 4 and 2.5 Å per pixel, \nrespectively. This AGN-ionisation scenario is mostly supported by means of standard Baldwin, Phillips & Terlevich (BPT) diagrams (Baldwin et al. 1981). These are empirically derived diagrams based on optical emission line ratios (selected to be una ff ected by reddening) that help us to distinguish di ff erent ionising mechanisms. \nWith the current MEGARA data that only cover H β and [O III] lines (see Sect. 3), we cannot use the BPT diagrams to properly study the possible ionisation mechanism of the di ff erent components found in our analysis. The values of log ([O III] / H β ) still only provide indications for an investigation of the dominant ionisation mechanism. Without the clumpy features mentioned in Sect. 4.2, the typical values of log([O III] / H β ) are > 0.9 (Fig. 4). Considering these typical values and the dividing lines that distinguish ionization from AGN and star formation by Kau ff mann et al. (2003) and Kewley et al. (2006), for the line ratios for the two narrow components suggest that ionisation from AGN is the dominant mechanism. For the primary component, the line ratios are not higher than the dividing lines above, with log([O III] / H β ) ∼ 0.9-1.1. It is therefore essential to measure other standard line ratios to determine the dominant ionisation mechanism. \nThe clumpy structures characterised by log ([O III] / H β ) < 1 are discussed in Sect. 5.3. \nAll the works based on optical slit spectroscopy indicate that the ionised gas in Mrk 1498 has a kinematics that is dominated by rotation, with extreme velocities of up to 300 km s -1 , as measured at large scales (up to R ∼ 25 kpc). In addition to the rotation component, previous works indicated multiple components and distortions in the velocity field (Keel et al. 2012, 2015). \nAs we mentioned in Sect. 4.1, the kinematics of all the three components shows a clear velocity gradient in the north-south direction. From the maps of all the components, we extracted the kinematic values in a 0 '' . 8 pseudo-slit along the maximum velocity gradient 5 . The PA is 148 · . The position-velocity plots in Fig. 5 were obtained from the current MEGARA IFS data in order to compare our kinematic results with those from previous optical spectroscopy (Keel et al. 2012). For all the three components, the velocity curves (P-V curves, Fig. 5) do not reach the plateau within the spatial scales sampled by MEGARA and show some irregularities at R > 3 kpc. For the secondary component, some irregularities are already evident at negative velocities at R > 2 kpc. These could be due to local non-ordered or non-circular motions at large galactocentric distances produced by instabilities from the previous merger episode or non-circular large-scale motions. The velocity amplitude and P-V curve (Table 2 and Fig. 5) from current MEGARA maps agree with those obtained by Keel et al. (2012) along the slit location with PA of 149 · (their Fig. 12). Nevertheless, the measurements do not agree with the value of the velocity amplitude ( ± 700 km s -1 ) reported by Keel et al. (2017) using GMOS IFS-data (their Fig. 9). We find no indication of high velocities like this at the corresponding spatial locations. The same authors also reported H β and [O III] line profiles are narrow with red wings in the south, and about 2 '' . 0 east of the nucleus. As mentioned in Sect. 3, narrow profiles are rare in MGARA data. The MEGARA spectral resolution allows us to reveal and model complex line profiles on spaxel-by-spaxels basis that are mostly characterised by double peaks and broad \nwings (see Fig. 2). \nAs mentioned in Sect. 1, the optical HST -images of the [O III] emission analysed by Keel et al. (2012, 2015) show two remarkable features. On the one hand, they show multiple circumnuclear rings of ionised gas at radii of 0 '' . 5 - 1 '' . 6 (0.6 - 1.8 kpc). On the other hand, relatively bright bubble-like structures that extend out to ∼ 10 '' ( ∼ 10 kpc) to the north-east and south-west are present. For MEGARA data, all the three components show an elongated flux distribution in the innermost region (up to R ∼ 3 '' ; Fig. 3 and Sect. 4.2). This could be related to the ring-like morphology seen in HST data but seen at lower spatial resolution with MEGARA-IFS. Similarly, for the large scale emission, two bubble-like features departing from the nucleus are well visible also in MEGARA data, their clumpiness is better seen in the log[O III] / H β maps (Fig. 4).", '5.2. Kinematic features': "The three velocity components have similar peak-to-peak velocities overall, but di ff erent average and central values of the velocity dispersion. Based on their spatially resolved behaviour, none of the components would trace a regular rotating disc according to the classification by Flores et al. (2006). Following this classification, the kinematics maps (velocity and velocity dispersion) of the primary and tertiary components might both be classified as 'perturbed discs': Their velocity maps are fairly regular, but both deviate from the case of an ideal rotating disc. Although the primary and tertiary components share the same disc classification, we remark that the disc traced by the tertiary component is the more perturbed of the two, with a more disturbed velocity dispersion map. The gas probed by the secondary component follows the main rotational pattern with the lowest values of the velocity dispersion; the main features of its velocity dispersion maps suggest complex motions, as discussed below. \nThe primary component shows a higher velocity dispersion ( ∼ 170 km s -1 ) than average in the σ -map ( ∼ 96 km s -1 ; Table 2) that is mostly spatially extended along the radio axis. At the same spatial location, but only towards the south-west, lies a blob with a velocity up to 100 km s -1 . The velocity is not extreme, but is observed in the direction of the minimum velocity gradient (i.e. north-east / south-west) and in the direction of the jet propagation. \nThe highσ values of the primary component could be associated with shocks and induced turbulence due the passage of the jet through the plane of the galaxy. This is consistent with the elongation of the core of the radio jet in the north-east / south-west direction detected by Hernández-García et al. (2019). Moreover, taking the position angle of the jet into account (PAjet = 45 · , Table 1), we propose that the jet is inclined with respect to the galaxy plane because we also observe perturbations in the tertiary component in two di ff erent directions (in the direction of the jet and nearly perpendicular to it in a sort of boomerang-like shape). \nSignatures of episodic AGN and jet activity have been found in other Seyfert galaxies (e.g. NGC 2639; Rao et al. 2023). We explored a scenario in which the blob at positive velocity seen towards the south-west would be caused by either present or past nuclear activity. For this purpose, we integrated in the nuclear region (i.e. within the PSF size; circles in all figures) the \nobserved [O III] fluxes for the primary component ( ∼ 5 × 10 41 erg / s) and derived the corresponding luminosity. Using the [OIII] luminosity, we calculated a kinematically determined outflow size following Kim et al. (2023). The size of the putative outflow is ∼ 2 kpc, similar to the distance of the blob from the nucleus ( ∼ 2.3 kpc). This similar spatial likeness might indicate that the current AGN activity launched the blob. \nHowever, taking the typical velocity at line peak of the blob into account ( ∼ 100 km s -1 ; Fig. 3), we should detect a velocity dispersion of more than 350 km s -1 , that is, much higher than observed in our MEGARA data. If the scaling between the maximum outflow velocity (V max ) and AGN bolometric luminosity 6 (L bol ) holds (Table 1 and Fig. 2 in Fiore et al. 2017), the blob should move at V max ∼ 500 km s -1 . \nIt is commonly thought that the AGN phenomenon is a phase in the lifecycle of a galaxy in which the nuclear activity can be reactivated 10-100 times during its lifetime, with typical timescales of ∼ 0.5 Myr (Schawinski et al. 2015). This timescale is much shorter than the dynamical time of blob ( ∼ 20 Myr), calculated as in Pereira-Santaella et al. (2016). This indicates that the last episode of AGN activity, the restarted one, is probably not the power mechanism that launched the blob in the putative outflow. \nTaking into account all this evidence, we propose a scenario in which the blob might be associated with a putative outflow powered by a previous more luminous AGN episode. In this scenario, the outflow would have quietly expanded throughout the galaxy with low levels of turbulence. This result is consistent with the one reported in Hernández-García et al. (2019), who found that the currently active nucleus is able to photoionise the gas in its surrounding, but not the large-scale emission on kiloparsec scales. \nBased on the observed kinematics, an alternative scenario is that the young jet has created a blob at its present position and was not pushed out from the nucleus. However, this is di ffi cult to investigate without the knowledge of the other BPT line ratios (e.g. [N II] / H α ), and hence test shocks models. \nTo summarise, a young jet is present in the nucleus of Mrk 1498, as shown by a previous analysis of radio data (Bruni et al. 2019). However, the blob is probably not associated with the recent (restarted) activity, but with the already-faded previous nuclear activity.", '5.3. Flux features (clumps)': 'Similarly as in Sect. 5.2 for the blob, we explored a scenario in which the clumps are indications of either present or previous nuclear activity. \nThe total extension of the clumpy structure is ∼ 6 kpc in radius, and the values of the velocity dispersion of the gas clumps are between 20 and 110 km s -1 , without a clear dependence on the distance from the nucleus. The velocities are about 50220 km s -1 . This implies that the dynamical times required for the gas to reach its current locations are 10-40 Myr (see also Sect. 5.2). The dynamical time of clumps is much longer than the typically time of an AGN duty cycle, indicating that the last \nepisode of AGN activity, the restarted one, is probably not the power mechanism that launched the clumps. \nWhen we assume that the size of the putative outflow is ∼ 2 kpc (Sect. 5.2) and that the clumps are seen up to ∼ 6 kpc, these clumps are likely associated with an AGN-driven outflow from a previous more luminous AGN episode or were launched by the actual restarting activity with an additional energy injection by the radio jet. \nIf the clumps were launched by the faded AGN activity, the putative outflow would be classified as fossil. These outflows persist for an order of magnitude longer than the AGN episode driving it (see Zubovas & Nardini 2020). Other similar cases in the literature are PDS 456 and IRAS F11119 + 3257 (Zubovas & Nardini 2020). \nMrk1498 shows a GPS in the centre (Hernández-García et al. 2019; Bruni et al. 2019). If the outflow were associated with the current restarting young radio activity, the kinematics would be more extreme (large [O III] widths) and the ionised gas morphology would be more disturbed, as proposed by Kukreti et al. (2023). \nAn alternative scenario is that the clumps are simply H II regions that follow the main rotation pattern of the ionised gas in the galaxy due to their low values of log[O III] / H β . \nTo summarise, similarly to the case described in Sect. 5.2, the crossing timescale of the clumps is longer than what is expected for a typical AGN active episode, which suggests that the largescale outflow is unrelated to the current (restarted) AGN phase. The di ff erent kinematics, extension, and morphology of the blob (see Sect. 5.2) and the clumps suggest that the features are associated with di ff erent flickering AGN episodes in the past that are unrelated to the current young radio phase in the nucleus.', '5.4. Considerations about the jet activity periods': 'The two distinct radio activity periods deduced for this source allow us to consider the jet activity period and its possible feedback on the host galaxy. The megaparsec-scale lobes are typically formed in a time span of ∼ 100 Myr (e.g. Orrù et al. 2010), indicating that the jet (thus the AGN) remained active for a similar amount of time. The new radio phase found in the core, spotted through the presence of a peaked radio spectrum and confirmed by VLBI observations at parsec-scale resolution, typically has an age of a few thousand years and a linear size of ∼ 1 kpc (O\'Dea & Saikia 2021). The recent release of the Very Large Array Sky Survey (VLASS) survey images at 3 GHz (3" resolution; Lacy et al. 2020) allows us to place a further constraint on the size of the recently launched jet. The source core shows a projected linear size of ∼ 0 \'\' . 6 (deconvolved), corresponding to ∼ 600 pc at the redshift of the source, in agreement with the expected one for young radio sources. The reactivation of the jet should thus have occurred some > 600pc / c ∼ 2000 years ago. The apparent discontinuity of ∼ 50 kpc between this new phase and the inner edge of the radio emission that is connected with the lobes (Bruni et al. 2021) is thus most probably due to the quiescent period between the two jet activity periods, which could have lasted > 50kpc / c ∼ 200kyr. The blob discussed in this work lies in the gap region between the previous and ongoing radio phases, where the jet that formed the megaparsec-scale lobes has deposited energy for some million years. \nThe AGN flickering may imply changes in the accretion state or AGN power. These are not taken into account in the current discussion because the current data set prevents us from developing \nFig. 6. Cartoon illustrating the di ff erent features of Mrk 1498 from previous multi-wavelength studies and the current work (see text for details). The MEGARA and GMOS fields of view are also indicated. The yellow circle marks the nucleus. \n<!-- image --> \na customised and detailed modelling of the full AGN history as was done in Zubovas et al. (2022).', '6. Conclusions': "On the basis of optical MEGARA IFS data, we have studied the properties of the ionised gas component in the Seyfert 1.9 Mrk1498 after a multiple component fitting, using the H β and [OIII] emission lines as tracers. \nThe conclusions of this study are summarised below. \n- 1. Multiple component fitting. For the first time, we were able based on R ∼ 10,000 optical IFS MEGARA data to disentangle three kinematic components and map their properties. This represents an improvement with respect to all the previous works that were based on optical spectroscopic data, which only studied one component with only weak detections of broad line-wings.\n- 2. Kinematics. All the three components show an overall blue-to-red velocity gradient, with similar peak-to-peak velocities, but a di ff erent velocity dispersion in the centre and on average. The visual inspection of the maps of two components highlights high velocity dispersion values in the direction of the radio jet. We kinematically classified the ionised gas probed by primary and tertiary components as 'perturbed discs' (Flores et al. 2006). The disc traced by the tertiary component is more perturbed than that of the primary component. The secondary component is likely associated with flux clumps in a disc with low levels of turbulence.\n- 3. Ionisation mechanisms. There is no clear trend of the [OIII] / H β ratios on the distance overall. However, our \nMEGARAmaps clearly reveal clumps with line ratios lower than unity in the direction of the maximum velocity gradient. The observed line ratios indicate possible ionisation from AGN or shocks everywhere else. \n- 4. Kinematic features. Towards the south-west at a galactocentric distance of ∼ 2.3 kpc, we observe a blob with a velocity up to 100 km s -1 and a high velocity dispersion ( ∼ 170 km s -1 ) that is spatially coincident with the direction of the radio jet. Taking into account its kinematic, dynamical time ( ∼ 20 Myr), and spatial location, we proposed a scenario in which the blob is powered by a previous more luminous AGN episode with possible additional energy input from the radio jet. We suggest that the jet is likely inclined with respect to the galaxy plane, as we also observe a high velocity dispersion in the tertiary component in two di ff erent directions (in the direction of the jet and nearly perpendicular to it in a sort of boomerang-like shape).\n- 5. The clumpy structure. The dynamical time of the clumps (10-40 Myr) is much longer that the typically time of the AGN duty cycle (0.5 Myr). The clumps are detected up to 6 kpc, which is much farther than the expected radius of a putative outflow launched from the current restarting activity. We proposed two possible scenarios in which the clumps are either an indication of the faded AGN activity (a fossil outflow) or are simply HII regions that follow the main rotation pattern of the ionised gas in the galaxy due to their low values of log[O III] / H β .\n- 6. Radio jet activity periods. Radio lobes on megaparsec scales similar to those found in Mrk 1498 typically form in a time span of ∼ 100 Myr, whereas the new radio phase found in the core typically has an age of a few thousand years. The reactivation of the jet should thus have occurred ∼ 2000 years ago. The apparent discontinuity is thus most probably due to a quiescent period between the two jet activity periods. \nIn summary, based on the optical and radio activity, we propose that two di ff erent ionised gas features are observed over the radio AGN lifecycle of Mrk 1498. On the one hand, we observe at radio frequencies old radio lobes (from the faded AGN) and a young GPS nucleus (from the current restarted AGN). On the other hand, we observe clumps and a blob at optical wavelengths that are likely associated with the past AGN activity (not the current activity) and are likely fossil-outflows related to two different episodes of the flickering AGN activity. When we take the time gap among radio activities and the jet-size evolution into account, the blob discussed in this work could have been launched between the previous and the ongoing radio phases. \nAcknowledgements. SC, IM, and JM acknowledge financial support from the Severo Ochoa grant CEX2021-001131-S funded by MCIN / AEI / 10.13039 / 501100011033. These authors are also supported by the Spanish Ministry of Science, Innovation y University (MCIU) under grants PID2019-106027GB-C41 and PID2022-140871NB-C21. \nLHG acknowledges financial support from FONDECYT Iniciacion 11241477, and ANID Millennium Science Initiative ICN12\\_009. \nGB acknowledges financial support from the Bando Ricerca Fondamentale INAF 2023, for the project: The GRACE project: high-energy giant radio galaxies and their duty cycle . GB acknowledges financial support for the GRACE project, selected via the Open Space Innovation Platform ( https://ideas.esa.int ) as a Co-Sponsored Research Agreement and carried out under the Discovery programme of, and funded by, the European Space Agency (agreement No. 4000142106 / 23 / NL / MGu / my). \nFP acknowledges financial support from the Bando Ricerca Fondamentale INAF 2023: Exploring the origin of radio emission in Radio Quiet AGN. \nThis research has made use of the NASA / IPAC Extragalactic Database \n(NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We acknowledge the usage of the HyperLeda database ( http://leda.univ-lyon1.fr ).", 'References': "Baldwin, J. A., Phillips, M. 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2024arXiv240910113Z

In this work we consider the offdiagonal coupling between two supersymmetric SYK models which preserves both supersymmetry and solvability. We found that the interaction terms of the N2 supersymmetric SYK have a holographic interpretation as a possible supersymmetric traversable wormhole. First we introduce the coupling in the trivial Homologic N1 SYK model as a simplified example. Similar couplings can be implied in N2 chiral SYK model with BPS states. We propose a special form of N4 SYK by introducing supermultiplets and which also naturally include the coupling terms. The holographic picture of N4 SYK does not have an eternal solution in the low energy limit. And the effective actions are studied in both thermal limit and low energy limit. We also investigate the SYKlike thermal field double states of the supersymmetric SYK and the transmission amplitude between singleside N2 models in Lorentz time. Additionally the multiside N24 OTOCs are also studied.

2024-09-01T00:00:00Z

['2024arXiv240910113Z', 'arXiv:2409.10113', '10.48550/arXiv.2409.10113']

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

Offdiagonal coupling of supersymmetric SYK model

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10113.pdf

{'Off-diagonal coupling of supersymmetric SYK model': 'Chenhao Zhang a and Wenhe Cai ∗ a \na Department of Physics, Shanghai University, Shanghai, 200444, China', 'Abstract': 'In this work, we consider the off-diagonal coupling between two supersymmetric SYK models, which preserves both supersymmetry and solvability. We found that the interaction terms of the N=2 supersymmetric SYK have a holographic interpretation as a possible supersymmetric traversable wormhole. First we introduce the coupling in the trivial Homologic N=1 SYK model as a simplified example. Similar couplings can be implied in N=2 chiral SYK model with BPS states. We propose a special form of N=4 SYK by introducing supermultiplets, and which also naturally include the coupling terms. The holographic picture of N=4 SYK does not have an eternal solution in the low energy limit. And the effective actions are studied in both thermal limit and low energy limit. We also investigate the SYK-like thermal field double states of the supersymmetric SYK and the transmission amplitude between single-side N=2 models in Lorentz time. Additionally, the multi-side N=2,4 OTOCs are also studied.', '1 Introduction': 'Recently, supersymmetric holographic wormholes have gained attention. However, these holographic wormholes still lack a simplified and solvable model. In previous works[1, 2, 3, 4], SYK model is introduced as a Gaussian random Majorana fermionic model with good holographic properties, which is dual to a NAdS spacetime. The emergent IR conformal symmetry is also demonstrated in the description of NAdS2/NCFT1 duality[5, 6]. Additionally, the supersymmetric SYK model originates from this research on the cold horizon[7, 8]. Supersymmetric SYK(SSYK) models are supersymmetric extensions of N=0 SYK model with dynamical bosons[9, 10, 11]. They also exhibit an emergent supersymmetric conformal symmetry, which is dual to NAdS spacetime with additional N=1 and N=2 supersymmetry and Grassmann variables[12, 13, 14]. These models have also been extensively studied[15, 16, 17, 18] and are characterized by valuable BPS states[19]. For instance, in double scaling limit[20, 21, 22], both non-supersymmetric and supersymmetric wormholes featuring statistical chords have been introduced[23, 24, 25]. However, the intricate structures of these entangled systems remain unexamined within the context of an adequate holographic model. \nIn this paper, we introduce a first-order interaction into the supersymmetric SYK model, which maintains both supersymmetry and solvability. This model is similar to the approach used in the N=0 coupled SYK model(MQ)[26, 27]. And we explore its low energy effective action[28, 29], thermal phase structure and causality. Previous work has proposed a simple supersymmetrization of JT gravity, exhibiting promising holographic properties[30, 31]. Notably, the supersymmetric JT gravity closely resembles the N=0 model. Furthermore, we extend this to supergravity with N=4 supersymmetry, which also exhibits duality due to the unique form of our N=4 SYK model. \nThe main goal of this paper is to examine the supersymmetric interaction term in both the N=2 and a streamlined N=4 SYK model. These models are formed using a special form of chiral supermultiplet combining two N=2 SYK models and have a natural representation of spinor. Currently, supersymmetric SYK models are under extensive investigation, focusing on aspects such as thermal effective actions, phase structures, and out-of-time correlators. However, they do not involve the detailed study of thermal phase interaction parts as wormholes. In this context, we aim to delve deeper into the causal properties involving excitation transmission and build upon prior research[32, 33, 34, 35, 36, 37], further enrich the understanding of supersymmetric extension. \nThe second goal is to study the N=4 model. Unlike higher-dimensional N=1 and N=2 models[38, 39], various N=4 SYK models have been proposed[40, 41, 42]. The N=4 model includes chiral, vector, and tensor supermultiplets. In this paper, we utilize the 1D supermultiplet. The superfields here differ significantly from the bosonic case. Because of its unique structure, the holographic properties of this N=4 SYK model can be inferred from the N=2 model in superspace, although direct study presents challenges. We introduce interaction terms as in the \nN=1 and N=2 models and examine black hole-wormhole structures. \nThis paper is organized as follows. In Section II, we briefly review the N=1 and N=2 supersymmetric SYK models, and propose a simple form of N=4 SYK model. We then introduce the first-order interaction into these models. In Section III, we begin with the holographic duality of NAdS spacetime and the SYK model, and further examine the supersymmetric NAdS and the supersymmetric SYK model with interaction terms. The holographic properties of N=4 model are also studied. In Section IV, we propose the thermal field double states in supersymmetric SYK model. Section V verifies the causality of the N=2 wormhole through excitation transmission and out-of-time correlation (OTOC). Section VI is the conclusion and discussion.', '2.1 A brief review of N=1 and N=2 SYK': "The N=1 SYK model has been the most widely studied attempt at a supersymmetric SYK model. Subsequent research has revealed that it possesses fewer physical interactions. Nevertheless, the N=1 SYK model still remains a valuable case study for exploring supersymmetry, despite its limited significance in supersymmetric physics. Its thermal properties exhibit similarities to those of the N=2 model. Furthermore, N=1 SYK theory can be formulated using independent supercharges, analogous to the N=0 model \nQ = i q -1 2 ∑ i 1 i 2 ...i q C i 1 i 2 ...i q ψ i 1 ψ i 2 ...ψ i q . (1) \nψ i s represent Majorana fermions on sites ranging from 1 ...N . In the initial definition, the interaction fermions q are assigned a value of 3. The corresponding coupling C ijk is an N × N × N Gaussian random antisymmetric tensor, specified by a constant J with units of energy. The random average of the coupling C exhibits the same form as in the N=0 SYK \nC i 1 i 2 ...i q = 0 , C 2 i 1 i 2 ...i q = 2 J N 2 . (2) \nThe super charge of N=1 model closely resembles that of the N=0 SYK. N=1 Hamiltonian is the square of supercharge \nH = Q 2 = E 0 + ∑ 1 ⩽ i<j<k ⩽ N J ijkl ψ i ψ j ψ k ψ l . (3) \nIt could be considered as a special form of N=0 SYK, and \nE 0 = ∑ 1 ⩽ i<j<k ⩽ N C 2 ijk , \nJ ijkl = -1 8 ∑ a C aij C kla . (4) \nHere, E 0 is a disorder averaged constant, while J ijkl constitutes a random matrix but not Gaussian independent. Supercharge is constructed from Majorana fermions and should adhere to fermionic anticommutation relations. For simplicity, an auxiliary bosonic field can be introduced \n{ Q,ψ i } = Qψ i = i ∑ 1 ⩽ j<k ⩽ N C ijk ψ j ψ k ≡ b i Qb i = Hψ i = ∂ τ ψ i . (5) \nThe Lagrangian consists of original fermionic and auxiliary bosonic field operators, as well as the supercharge relations \nL = ∑ i 1 2 ψ i ∂ τ ψ i -1 2 b i b i + i ∑ 1 ⩽ j<k ⩽ N C ijk b i ψ j ψ k . (6) \nFor simplicity, we can define a superfield within the corresponding superspace. This allows us to constrain the bosons and fermions, and to describe the supersymmetric theory without sacrificing generality. \nΨ ( τ, θ ) = ψ ( τ ) + θb ( τ ) , (7) \nTherefore, we introduce the super covariant derivative \nD θ ≡ ∂ θ + θ∂ τ , \nD 2 θ = ∂ τ . (8) \nWe can define a superfield bilinear correlation function by combining the correlation functions of f-f, f-b, b-f and b-b components \nG ( τ, θ ; τ ' , θ ' ) = ⟨ Ψ ( τ, θ ) Ψ ( τ ' , θ ' ) ⟩ = 〈 ( ψ ( τ ) + θb ( τ )) ( ψ ( τ ' ) + θ ' b ( τ ' ))〉 (9) \n≡ G ψψ ( τ, τ ' ) + √ 2 θG bψ ( τ, τ ' ) -√ 2 θ ' G ψb ( τ, τ ' ) +2 θθ ' G bb ( τ, τ ' ) . (10) \nThen, we return to the original SSYK model and rewrite the effective action into superspace \nS EFF = ∫ dθdτ ( -1 2 Ψ i D θ Ψ i ) + J 3 N 2 ∫ dθ 1 dτ 1 dθ 2 dτ 2 ( Ψ i Ψ i ) 3 . (11) \nNote that this action is diagonalized into the N=0 SYK form using superfields and covariant derivatives. Additionally, the saddle point provides describing equations of motion, incorporating disorder averaging \nD θ G ( τ, θ ; τ '' , θ '' ) + ∫ dτ ' dθ ' G ( τ ' , θ ' ; τ '' , θ '' ) ( J G ( τ ' , θ ' ; τ '' , θ '' ) 2 ) = ( θ -θ '' ) δ ( τ -τ '' ) . (12) \nN=2 supersymmetric SYK can be generated from the symmetry breaking of N=1 theory. We reorganize the square of supercharges into a pair of identical, conjugated supercharges with SU(2) \nQ = i ∑ 1 ⩽ i<j<k ⩽ N C ijk ψ i ψ j ψ k , ¯ Q = i ∑ 1 ⩽ i<j<k ⩽ N ¯ C ijk ¯ ψ i ¯ ψ j ¯ ψ k . (13) \nC and ¯ C represent an independent Gaussian random coefficient and the fields are replaced by a pair of conjugated complex fields. Notice that fermions here only anticommute with fields in conjugated components, where the N=1 supersymmetry breaks by \n{ ψ i , ¯ ψ j } = δ i j , { ψ i , ψ j } = 0 , { ¯ ψ i , ¯ ψ j } = 0 . (14) \nThe N=2 Hamiltonian is determined by the anticommutator of the conjugate supercharges \nQ 2 = 0 , ¯ Q 2 = 0 , H = { Q, ¯ Q } = | C | 2 + ∑ ijkl J kl ij ψ i ψ j ¯ ψ k ¯ ψ l . (15) \nWe can also average the random Gaussian coefficient C using an energy-dependent parameter J \nC ijk ¯ C ijk = 2 J N 2 , (16) \nIt is a random parameter, independent of Gaussian distribution. Analogous to the N=1 model, auxiliary bosonic fields are introduced for both chiral fermions through the commutators of conjugate supercharges \n{ Q,ψ i } = 0 , { ¯ Q, ¯ ψ i } = 0 , b i ≡ i { ¯ Q,ψ i } = ∑ 1 ⩽ j<k ⩽ N ¯ C ijk ¯ ψ j ¯ ψ k , ¯ b i ≡ i { Q, ¯ ψ i } = ∑ 1 ⩽ j<k ⩽ N C ijk ψ j ψ k . (17) \nThe covariant derivative operators are expressed in terms of conjugate supersymmetric components. \nD ≡ ∂ θ + ¯ θ∂ τ , \n¯ D ≡ ∂ ¯ θ + θ∂ τ . (18) \nIn N=2 algebra, various approaches are permitted for structuring the superfield. Additionally, it is convenient to opt for a method known as the chiral superfield, which satisfies \n¯ DΨ i = 0 , D ¯ Ψ i = 0 . (19) \nAnd we obtain the effective actions \nS eff = 1 2 ∫ dθd ¯ θdτ ¯ Ψ i Ψ i + ∫ dθd ¯ θdτ ¯ Ψ i 1 ¯ Ψ i 2 ¯ Ψ i 3 〈 ¯ C i 1 i 2 i 3 C j 1 j 2 j 3 〉 Ψ j 1 Ψ j 2 Ψ j 3 \n, ¯ S eff = 1 2 ∫ d ¯ θdθdτΨ i ¯ Ψ i + ∫ d ¯ θdθdτΨ j 1 Ψ j 2 Ψ j 3 〈 C j 1 j 2 j 3 ¯ C i 1 i 2 i 3 〉 ¯ Ψ i 1 ¯ Ψ i 2 ¯ Ψ i 3 . (20) \nThen, we can define creation and annihilation operators using covariant derivatives, and subsequently construct the bi-linear Fock space from the superfield \nΨ i ( τ, θ, ¯ θ ) = ψ i ( τ + θ ¯ θ ) + θb i , ¯ Ψ i ( τ, θ, ¯ θ ) = ¯ ψ i ( τ + ¯ θθ ) + ¯ θ ¯ b i . (21) \nNotice that the superfields adhere to anticommute relations, and they possess conjugated chiral properties. Additionally, we can define the bi-chiral correlation function with superfields \nG ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) = 〈 ¯ Ψ ( τ, θ, ¯ θ ) Ψ ( τ ' , θ ' , ¯ θ ' )〉 ≡ G ψψ ( τ -θ ¯ θ, τ ' + θ ' ¯ θ ' ) + √ 2 ¯ θG bψ ( τ, τ ' + θ ' ¯ θ ' ) -√ 2 θ ' G ψb ( τ -θ ¯ θ, τ ' ) +2 ¯ θθ ' G bb ( τ, τ ' ) . ¯ G ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) = 〈 Ψ ( τ, θ, ¯ θ ) ¯ Ψ ( τ ' , θ ' , ¯ θ ' )〉 ≡ ¯ G ψψ ( τ + θ ¯ θ, τ ' -θ ' ¯ θ ' ) + √ 2 θ ¯ G bψ ( τ, τ ' -θ ' ¯ θ ' ) -√ 2 ¯ θ ' ¯ G ψb ( τ + θ ¯ θ, τ ' ) +2 θ ¯ θ ¯ G bb ( τ, τ ) . (22) \nWe can also formulate the coupled Schwinger-Dyson equation of chiral-anti chiral, and describe the N=2 SYK system using the conjugate components \nD θ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) + ∫ dτ ' dθ ' d ¯ θ ' G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) ( J G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) q ) = ( ¯ θ -¯ θ '' ) δ ( τ -τ '' -θ ¯ θ + θ '' ¯ θ '' ) , (23) \nD ¯ θ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) + ∫ dτ ' dθ ' d ¯ θ ' ¯ G ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) ( J ¯ G ( τ ' , θ ' , ¯ θ ' ; τ '' , θ '' , ¯ θ '' ) q ) = ( θ -θ '' ) δ ( τ -τ '' -θ ¯ θ + θ '' ¯ θ '' ) . (24) \nIn order to understand the supersymmetric SYK models more precisely, we can also consider the supercharges with a chord diagram. As shown in Figure 1(a), we use the blue line on the left-hand side (LHS) to represent a single SYK system with coupling C ijk (which is known as \nFigure 1: Interpretation of the chord diagram as the supersymmetric SYK model (a) N=1 SYK model (b)N=2 SYK model with chirals \n<!-- image --> \na supercharge). The black line on the right-hand side (RHS) represents the same SYK system that is to be interacted with. We suppose the interaction, represented by chords that start on the LHS and end on the RHS. The pink area includes the interactions between a single fermion ψ i on the LHS and the entire SYK system on the RHS, as well as the classical SYK coupling C ijk inside the RHS. We define this pink region as the auxiliary boson b i . It is easy to prove that the dimension of fermions is equal to the dimension of those auxiliary bosons.As depicted in Figure 1(b), the N=2 supersymmetry forbids the interaction inside a single chiral. When we define the blue line as the chiral supercharge with ψ i , the red line is defined as anti-chiral with ¯ ψ i . The green line will represent the same anti-chiral supercharge that is to be interacted with, while the black line represents the anti-chiral supercharge. We also define the pink region as the chiral boson b i and the yellow region as the chiral boson ¯ b i .", 'N=4 Bilinear SYK': "The N=4 SYK model was derived from considerations of a cold horizon. The operator Φ represents a chiral multiplet that exhibits both bosonic and fermionic dependencies (which is different from the chiral supermultiplet in [40, 41]), and it is characterized by Gaussian randomness. \nL = ∫ d 2 θΩ αβγ Φ α Φ β Φ γ + h.c, (25) \nthe coefficients Ω αβγ as a SYK-like model are also Gaussian random. \n⟨ Ω αβγ ⟩ = 0 , 〈 Ω 2 αβγ 〉 = J. (26) \nNumerous models exhibit a Hamiltonian form. We opt for the bilinear scalar supermultiplet approach. Here we define \nL = ∑ i,α [ ∂ τ ¯ ϕ i ∂ τ ϕ i + ¯ ψ α i ∂ τ ψ i α + ¯ F i F i ] +3 C ijk ( F i ϕ j ϕ k + ϵ αβ ψ i α ψ j β ϕ k ) + h.c. (27) \nParameter α and β correspond to fermionic indices 1 and 2. And ϵ 11 = ϵ 22 = 0, and ϵ 12 = ϵ 21 = 1. We commence our construction using fermions within the framework of N=2 supersymmetry \n{ ψ i α , ¯ ψ β j } = δ β α δ i j . (28) \nNow we consider two identical N=2 SYK models in Fig. (1) b, which is remarked by index 1 and index 2. Since we have introduced an additional dimension as N=2 supercharges with SU(2), an additional U(1) symmetry can emerge between index 1 and index 2 \nQ α = i ∑ 1 ⩽ i<j<k ⩽ N C βγ ijk ψ ' i α ψ j ' β ψ ' k γ , ¯ Q β = i ∑ 1 ⩽ i<j<k ⩽ N ¯ C ijk αγ ¯ ψ ' α i ¯ ψ ' β j ¯ ψ ' γ k . (29) \nSupercharges are essentially Gaussian random, and the coefficient C also involves multiple indices α , β and γ , which vanish with the random coefficient. However, these indices can disappear through transformations, and we will see that it has less physical meaning and would be vanished in integrals. For simplicity, we choose α = β = γ in single supercharge and making the supercharges identical. Thus the indices α and β associated with supercharges at positions 1 and 2 exhibit rotational symmetry. And the auxiliary boson F is pertinent to higher-order cross terms involving supercharges with identical indices. \nb i = { Q α , ¯ ψ α i } , ¯ b i = { ¯ Q α , ψ i α } . (30) \nWe can also generate the bosonic operator b , which has been involved in both N=1 and N=2 supersymmetry. \nH = ψ α ψ β { ¯ Q α , ψ α } 2 ψ α ψ β + ψ α ψ β { ¯ Q α , ψ α } ψ α { ¯ Q α , ψ α } ψ β + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + T { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } . (31) \nHere we use the operator T to exchange of the sort ordering of chiral fermionic pair, which vanishes when we ignore the α -β coupling in supercharges. To establish the N=4 SYK theory, the superfields are defined by \nΦ i = ϕ i + θ α ψ i α + θ 2 F i = ( ψ ' i α + θ α b i ) ( ψ ' i β + θ β b i ) = ψ ' i α ψ ' i β + θ α b i ψ ' i β -θ β b i ψ ' i α + θ α θ β b i b i \n¯ Φ i = ϕ i + ¯ θ α ¯ ψ α i + ¯ θ 2 ¯ F i = ( ¯ ψ ' α i + ¯ θ α ¯ b i ) ( ¯ ψ ' β i + ¯ θ β ¯ b i ) = ¯ ψ ' α i ¯ ψ ' β i + ¯ θ α ¯ b i ¯ ψ ' β i -¯ θ β ¯ b ¯ ψ ' α i + ¯ θ α ¯ θ β ¯ b i ¯ b i , (32) \nHere we have defined the bosonic operators ϕ and F , as well as the fermionic operators ψ α and ψ β , all of which exhibit multiple chiral aspects within the Lagrangian \nϕ \nϕ \ni = ϵ αβ ψ ' i α ψ ' i β , ψ i α = ϵ β α b i ψ ' i β , ψ i β = ϵ α β b i ψ ' i α , F i = b i b i , ¯ i = ϵ αβ ¯ ψ ' α i ¯ ψ ' β i , ¯ ψ α i = ϵ α β ¯ b i ¯ ψ ' β i , ¯ ψ β i = ϵ β α ¯ b i ¯ ψ ' α i , ¯ F i = ¯ b i ¯ b i . (33) \nHere we use ψ i α and ¯ ψ α i to represent fermions in two identical N=2 SYK models. We can also decouple the N=4 theory back to N=2. Given that the superfield theory can also be incorporated into the N=4 theory, the covariant derivative is defined as \nD α = ∂ ∂θ α + ¯ θ α ∂ ∂τ , ¯ D β = ∂ ∂ ¯ θ β + θ β ∂ ∂τ , D 2 θ ≡ 1 4 ϵ αβ D α D β = D 1 D 2 , D 2 ¯ θ ≡ 1 ϵ αβ ¯ D α ¯ D β = ¯ D 1 ¯ D 2 . (34) \n4 \nSince we have introduced the auxiliary parameters and superfields, the action of the N=4 SYK model can be explicitly expressed as \nS = ∫ dτdθ 1 dθ 2 d ¯ θ 1 d ¯ θ 2 ( ¯ Φ i Φ i ) + ∫ dτdθ 1 dθ 2 C ijk Φ i Φ j Φ k + ∫ dτd ¯ θ 1 d ¯ θ 2 ¯ C ijk ¯ Φ i ¯ Φ j ¯ Φ k = ∫ dτ [ ˙ ¯ ϕ i ˙ ϕ i + ¯ ψ α i ˙ ψ i α + ¯ F i F i +3 C ijk ( F i ϕ j ϕ k + ϵ αβ ψ i α ψ j β ϕ k ) + h.c ] . (35) \nThis action can also be constrained by the equation of motion \nD \n2 θ G ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) + ∫ dτ ' d 2 θ ' G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) ( J G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) 2 ) = ( ¯ θ -¯ θ '' ) ( ¯ θ -¯ θ '' ) δ ( τ -τ '' ) . \n(36) \nD \n¯ 2 θ ¯ G ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) + ∫ dτ ' d 2 ¯ θ ' ¯ G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) ( J ¯ G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) 2 ) = ( θ -θ '' ) ( θ -θ '' ) δ ( τ -τ '' ) . \n(37) \nThe N=4 correlation function is represented by \nG ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 ) = 〈 ¯ Φ ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 ) Φ ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 )〉 \n- -√ 2 ¯ θ ' α G ψαϕ ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 ) + ¯ θ 1 ¯ θ 2 θ 1 θ 2 G FF ( τ, τ ' )\n- = G ϕϕ ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 ) + √ 2 θ α G ϕψα ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' α ¯ θ ' α )\n- +2 ¯ θ α θ ' β G ψαψβ ( τ, τ ' ) + ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' β ¯ θ ' β )\n- + θ 1 θ 2 G ϕF ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' ) + ¯ θ 1 ¯ θ 2 G Fϕ ( τ, τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 )\n- + √ 2 ¯ θ α θ 1 θ 2 G ψαF ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' ) + √ 2 ¯ θ 1 ¯ θ 2 θ α G Fψα ( τ, τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' β ¯ θ ' β ) , (38) \n¯ \nG \n( \nτ, θ \n, \n¯ \nθ \n, θ \n, \n¯ \nθ \n; \nτ \n' \n, θ \n' \n, \n¯ \nθ \n' \n, θ \n' \n2 \n1 \n1 \n2 \n2 \n, \n¯ \nθ \n' \n2 \n= \n) \n〈 \nΦ \n( \nτ, θ \n, \n¯ \nθ \n, θ \n, \n¯ \nθ \n; \nτ \n' \n, θ \n' \n, \n¯ \nθ \n' \n, θ \n' \n2 \n1 \n1 \n2 \n2 \n, \n¯ \nθ \n' \n2 \n¯ \n) \nΦ \n( \nτ, θ \n, \n¯ \nθ \n, θ \n, \n¯ \nθ \n; \nτ \n' \n, θ \n' \n, \n¯ \nθ \n' \n, θ \n' \n2 \n, \n- = ¯ G ϕϕ ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 ) + √ 2 ¯ θ α ¯ G ϕψα ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' α ¯ θ ' α ) \n1 \n- -√ 2 θ ' α ¯ G ψαϕ ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 ) + ¯ θ 1 ¯ θ 2 θ 1 θ 2 ¯ G FF ( τ, τ ' )\n- +2 θ α ¯ θ ' β ¯ G ψαψβ ( τ, τ ' ) + ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' β ¯ θ ' β )\n- + ¯ θ 1 ¯ θ 2 ¯ G ϕF ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' ) + ¯ θ 1 ¯ θ 2 ¯ G Fϕ ( τ, τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 )\n- + √ 2 θ α ¯ θ 1 ¯ θ 2 ¯ G ψαF ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' ) + √ 2 θ 1 θ 2 ¯ θ α ¯ G Fψα ( τ, τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' β ¯ θ ' β ) . (39) \nSome of these components would vanish, and we will discuss this problem later.", '2.2 Introduce the coupling': "For N=1 SYK, we can propose a similar first-order interaction term to that used in MQ theory in superspace for consistent. The coupled theory encompasses two separate N=1 SYK models, along with the interaction term \nH int = iµ∂ θ ∑ j ( Ψ j L Ψ j R -Ψ j R Ψ j L ) . (40) \nHere, we have introduced the super derivative to constrain the theory using a delta function in a single-sided superposition. Subsequently, we obtain the action that includes the interaction \n1 \n2 \n2 \n¯ \nθ \n' \n2 \n)〉 \nterm \nS eff N = -log Pf ( D half integer ) + log det ( D integer ( iw n ) 2 ) + ∑ A,B = L,R 1 2 ∫ dτdτ ' [ Σ ψψ,AB ( τ, τ ' ) G ψψ,AB ( τ, τ ' \n) + Σ ψb,AB ( τ, τ ' ) G ψb,AB ( τ, τ ' ) + Σ bψ,AB ( τ, τ ' ) G bψ,AB ( τ, τ ' ) + Σ bb,AB ( τ, τ ' ) G bb,AB ( τ, τ ' ) -( δ AB +(1 -δ AB ) ( -1) ( q -1) / 2 ) J ( G q -1 ψψ,AB ( τ, τ ' ) G bb,AB ( τ, τ ' ) -( q -1) 2 G q -2 ψψ,AB ( τ, τ ' ) G A ψb,AB ( τ, τ ' ) G S bψ,AB ( τ, τ ' ) -( q -1) 2 G q -2 ψψ,AB ( τ, τ ' ) G S ψb,AB ( τ, τ ' ) G A bψ,AB ( τ, τ ' ) )] . (41) \nWe can further utilize the variation in effective action to derive the coupled equation of motion \nD θ G AB ( τ, τ ' ) -∑ C ( -iµϵ AC ∂ θ G AB ( τ, τ ' ) -∫ dτ '' dθ '' Σ AC ( τ, τ '' ) G BC ( τ, τ '' ) ) = δ AB ( θ -θ ' ) δ ( τ -τ ' ) . (42) \nIn the generalization of the N=2 supersymmetric model, the interaction Hamiltonian represents the correlation between chiral and anti-chiral supercharges. The interactions between single-sides should vanish. When considering the two-part interaction term \nH int = iµ∂ θ ∑ j ( Ψ j L ¯ Ψ j R -Ψ j R ¯ Ψ j L ) \n¯ H int = iµ∂ ¯ θ ∑ ( ¯ Ψ j L Ψ j R -¯ Ψ j R Ψ j L ) . \n, j (43) \nThe corresponding N=2 chiral and anti-chiral actions, when combined with the interaction term, lead to the equations of motion \nD θ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ θ G AB ( Z, Z ' ) -∫ dτ '' dθ '' Σ AC ( Z, Z '' ) G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) , \nD ¯ θ ¯ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ ¯ θ ¯ G AB ( Z, Z ' ) -∫ dτ '' d ¯ θ '' ¯ Σ AC ( Z, Z '' ) ¯ G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) . (44) \nWe assume that there is symmetry between the chiral and anti-chiral Green's functions \nG ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) = ¯ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) . (45) \nWe can write down the SD equations. Notably, the N=1 equations are similar to the N=2 \nequations in terms of components \nΣ ψψ,AB ( τ ) = ( q -1) ( -1) ( q -1) / 2 JG q -2 ψψ,AB ( τ ) G bb,AB ( τ ) -( q -1) ( q -2) ( -1) ( q -1) / 2 JG q -3 ψψ,AB ( τ ) , \nG A ψb,AB ( τ ) G S bψ,AB ( τ ) -( q -1) ( q -2) ( -1) ( q -1) / 2 JG q -2 ψψ,AB ( τ ) G S ψb,AB ( τ ) G A bψ,AB ( τ ) \nΣ bb,AB ( τ ) = JG ψψ,AB ( τ ) \nΣ S bψ,AB ( τ ) = -( q -1) JG A ψb,AB ( τ ) G q -2 ψψ,AB ( τ ) \nΣ A bψ,AB ( τ ) = -( -1) ( q -1) / 2 ( q -1) JG S ψb,AB ( τ ) G q -2 ψψ,AB ( τ ) \nΣ S ψb,AB ( τ ) = ( q -1) JG A bψ,AB ( τ ) G ψψ,AB ( τ ) \n2 q , 2 , 2 , 2 , \n( q -1) \nΣ A ψb,AB ( τ ) = ( -1) ( q -1) / 2 2 JG S bψ,AB ( τ ) G ψψ,AB ( τ ) . (46) \nFactor ( -1) ( q -1) / 2 originates from the fermionic sign. In the N=2 theory, we need to adjust the ψ time variable as follows: τ → τ -θ ¯ θ and τ ' → τ ' + θ ' ¯ θ ' . \nAdditionally, it is noteworthy that the components of Green's functions G ψb,AB and G bψ,AB are neither symmetric nor anti-symmetric. Nevertheless, these components can be regularized through the exchange of the Grassmann variables θ and ¯ θ \nG ψb,AB θ → θ ¯ θG ψb,AB , \n¯ θG bψ,AB → G bψ,AB . (47) \n2 , \nIt is easy to check \n-θG ψb,AB ¯ θG bψ,AB →-¯ θθG ψb,AB G bψ,AB , (48) \nwhere we have defined the component -θ ¯ θG ψb,AB is anti-symmetric, whereas the component G bψ,AB is symmetric. Alternatively, we can also define the component G ψb,AB as symmetric and G bψ,AB is anti-symmetric without affecting the physical and numerical results, and this is due to the symmetry between these two components. \nSimilarly, we can apply the same process to make the component G bψ,AB anti-symmetric and make G ψb,AB to symmetric \nG bψ,AB θ → θ ¯ θG bψ,AB , \n¯ θG ψb,AB → G ψb,AB . \nAnother important observation is that these components must satisfy the periodic boundary condition due to the diagonal term, and their low energy ansatz is precisely zero. This allows us to separate them into symmetric and anti-symmetric components and apply the Matsubara method without considering additional subdivisions. The multiple combination of the components \nFigure 2: Supersymmetric Green's function in components fixed J = 3 , µ = 0 . 3 , T = 0 . 01 , q = 5 (a) Single-side fermionic Green's function G ψψ,AA , (b) Single-side bosonic Green's function G bb,AA (c) multi-side Anti-symmetric fermion-bosonic Correlation function G A ψb,LR . (d) multiside symmetric fermion-bosonic Correlation function G S ψb,LR \n<!-- image --> \nθ ¯ θG ψb,AB G bψ,AB can be defined as antisymmetric, similar to the combination θ ¯ θG ψψ,AB G bb,AB . And we can still split the Green function terms into anti-symmetric parts G A ψb,AB and symmetric parts G S ψb,AB , with a similar division for the anti-symmetric G A bψ,AB and symmetric G S bψ,AB . \nThe numerical results in Figure.(2) show that supersymmetric Green's function in components fixed J = 3 , µ = 0 . 1 , T = 0 . 01. In figure.(2)a, the fermionic Green's function behaves like N=0 SYK, but no longer symmetry of Euclidean time τ . Its Retarded side ( τ > 0) decays more faster than its advanced side ( τ < 0). This is very close to the supersymmetric SYK with fermionic chemical potential results in [18]. However, this is different from cSYK[43] and Maldacena-Qi[26]. This model characterized by supersymmetric correlation is constrained on \nboundary, implying that only non-vanishing fermion-bosonic and boson-fermionic correlations exist. And it cannot undergo renormalization similar to that described in [18]. In Figure.(2)b, the bosonic component approximates a delta function, which is an inherent consequence of the original supersymmetric SYK model. We also plot the fermion-bosonic Correlation function in both anti-symmetric figure.(2)c and symmetric figure.(2)d components. Upon including the coupling term, these components are no longer regarded as negligible in previous studies. In Figure.(2)c, the anti-symmetric fermion-bosonic correlation function closely resembles the fermionic correlator G ψψ . Meanwhile, the symmetric fermion-bosonic correlation function in Figure (2)d decreases gradually and approaches zero as we integrate over the periodic time β = 1 /T . The model also exhibits a sudden decay around the origin, time 0, which may be influenced by the delta function present in the bosonic Green's function. One can verify that setting the components G ψψ,LR , G bb,LR , G ψb,AA equal to zero yields a stable solution. Additionally, other Green's functions, such as the boson-fermion conjugate and the left-right conjugate, can be derived from the components discussed in Figure (2). \nAnd the solutions of Green's function in frequency space are \nG ψψ,AB ( ω ) = Det ( A ψψ,AB ( D half integer )) Det ( D half integer ) \nG S ψb,AB ( ω ) = -Det A S ψb,AB ( D integer ) Det ( D integer ) . \n, G bb,AB ( ω ) = -Det ( A bb,AB ( D integer )) Det ( D integer ) , G A ψb,AB ( ω ) = Det ( A A ψb,AB ( D half integer ) ) Det ( D half integer ) , G S bψ,AB ( ω ) = -Det ( A S bψ,AB ( D integer ) ) Det ( D integer ) , G A bψ,AB ( ω ) = Det ( A A bψ,AB ( D half integer ) ) Det ( D half integer ) , ( ) (49) \nIn this work, we introduce auxiliary fermions with bosonic integral frequency and bosons with fermionic half-integral frequency, which should statistically equal zero. The matrices D half integer and D integer represent the Jacobian matrix with symmetric and anti-symmetric properties, respectively, and the coefficient -1 serves as a bosonic factor in this context. We can also absorb this factor and inverse the ς \nD half integer = -iw n -Σ LL,ψψ -Σ LR,ψψ -Σ A LL,ψb -iµ -Σ A LR,ψb -Σ RL,ψψ -iw n -Σ RR,ψψ iµ -Σ A RL,ψb -Σ A RR,ψb -Σ A LL,bψ iµ -Σ A LR,bψ -1 -0 -0 -iµ -Σ A RL,bψ -Σ A RR,bψ -0 -1 -0 , \nD integer = -iw n -0 -0 -Σ S LL,ψb iµ -Σ S LR,ψb -0 -iw n -0 -iµ -Σ S RL,ψb -Σ S RR,ψb -Σ S LL,bψ -iµ -Σ S LR,bψ -1 -Σ LL,bb -Σ LR,bb iµ -Σ S RL,bψ -Σ S RR,bψ -Σ RL,bb -1 -Σ RR,bb . \nZero elements in a matrix represent the possible values for Σ . We assume that the bosonic Σ s are zero, indicating that they also exhibit zero values at half-integer frequencies. In contrast, the fermionic Σ exhibit zero values at integer frequencies. \nAs demonstrated in the equations, these algebraic cofactors of the matrix originate from the Jacobi method applied to Schwinger-Dyson equations. \nThe energy can be evaluated in thermal limit \nE N = ∫ d ¯ θ [ 2 q -1 D G LL + iµ ( 1 -2 q -1 ) G LR ] τ → 0 + + h.c. (50) \nWe can precisely calculate these values within the frequency branch \nE N = 2 q -1 T ∑ half integral ( Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) + µ ( q -3 2 ) Im [ G A LR,ψb ( iw n ) + G A LR,bψ ( iw n ) ] ) + 2 q -1 T ∑ integral ( Σ LL,bb ( iw n ) G LL,bb ( iw n ) + µ ( q -3 2 ) Im [ G S LR,ψb ( iw n ) + G S LR,bψ ( iw n ) ] ) . (51) \nAdditionally, the free energy is derived from the saddle point effective action \nF N = -T log Z N = T S eff N = -T 2 log (2) + ∑ half integral ( log D half integral ( iw n ) ( iw n ) 2 + 2 q -2 q Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) ) + 2 q -2 q Σ A LR,ψb ( iw n ) G A LR,ψb ( iw n ) + Σ A LR,ψb ( iw n ) G A LR,bψ ( iw n ) ] -∑ integral ( log D integral ( iw n ) ( iw n ) 2 -2 q -2 q Σ LL,bb ( iw n ) G LL,bb ( iw n ) -2 q -2 q Σ S LR,ψb ( iw n ) G S LR,ψb ( iw n ) -Σ S LR,ψb ( iw n ) G S LR,bψ ( iw n ) ) . (52) \nIn this section, we have renormalized the determinant terms \n∑ half integral (log ( iw n )) = ln 2 . \nWe have introduced iw n to eliminate the physical influence of auxiliary fermions in D integer . As there are no natural fermions with integral frequencies, the integral part of (log ( iw n )) does not contribute any additional ln 2. \n<!-- image --> \nFigure 3: Free energy and energy of N=2 Supersymmetric SYK fixed J = 3 , q = 5 (a)Free energy under condition µ = 0 . 3 , 0 . 5 , 0 . 8, (b) Energy under condition µ = 0 . 3 , 0 . 5 , 0 . 8, \n<!-- image --> \nThe free energy as a function of temperature is plotted in Figure.(3)a. We cool the system from its initial maximum temperature and heat it again. We consider the number of coupling q = 5 in Eq. (1), which means the theory should correspond to a non-supersymmetrc q=4 SYK. We observe a phase transition in Figure.(3)a. The free energy observed is analogous to the Hawking-Page transition in standard MQ and cSYK models. However, after introducing the supersymmetric coupling, the bosonic and multistate contributions to the free energy term gradually transform the 'wormhole-phase' into a state different from the so-called eternal wormhole. Furthermore, the phase transition is also influenced by this effect(as Figure.(3)a). The energy as a function of inverse temperature β is plotted in Figure.(3)b. Similarly, we can observe a comparable ensemble result regarding the energy behaviors. For T < T c , there are two distinct solution, and consist the phase transition. We can also obtain the smooth curve outside the transition region. \nThe second-order transition ends at a critical point ( T c = -1 . 15 , µ c = 0 . 7). We first cool the system from maximal temperature and then heat it back to its original temperature. We obtain that the phase transition exists after we introduce the super-coupling. The wormhole phase in low energy will also influenced by additional bosonic contribution, and the free energy are no longer constant when we involve the coupling. \nFor N=4 model, we can include the first-order interaction term in the Hamiltonian and while maintaining the supersymmetry and ensuring that the model remains solvable \nH int = ∂ 2 θ iµ ( Φ i L ¯ Φ iR + Φ i R ¯ Φ iL ) = iµF i L ¯ ϕ iR + iµF i R ¯ ϕ iL , ¯ H int = ∂ 2 ¯ iµ ( ¯ Φ iL Φ i + ¯ Φ iR Φ i ) = iµ ¯ ϕ iL F i + iµ ¯ ϕ iR F i . (53) \nθ R L R L \nG 3 = ( ΦΦ ) 3 = ( FF ) ( ϕϕ ) ( ϕϕ ) -( ϕϕ ) ( ψ α ψ α ) ( ψ β ψ β ) + ( ϕϕ ) ( ψ α ψ β ) ( ψ β ψ α ) + 2 ( ϕϕ ) ( Fϕ ) ( ϕF ) -( ψ α ϕ ) ( ϕψ α ) ( ψ β ψ β ) -( ψ β ϕ ) ( ϕψ β ) ( ψ α ψ α ) + ( ψ β ϕ ) ( ϕψ α ) ( ψ α ψ β ) + ( ψ α ϕ ) ( ϕψ β ) ( ψ β ψ α ) . (54) \nThen we can write the N=4 action, including the specific interaction term between the fields \nS eff N = -1 2 log Pf ( D half integer ) + log det ( D integer ) + ∑ A,B = L,R 1 2 ∫ dτdτ ' [ Σ ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' \n- ) + Σ ψ 2 ψ 2 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) + Σ FF,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) + Σ ϕϕ,AB ( τ, τ ' ) G ϕϕ,AB ( τ, τ ' ) + Σ Fϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) + Σ ϕF,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) -J ( ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) -G q -1 ϕϕ,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) -( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) )] . (55) \nand \nD 2 θ G AB ( Z, Z ' ) -∑ ( -iµϵ AC ∂ 2 θ G AB ( Z, Z ' ) -∫ dτ '' dθ '' 1 dθ '' 2 Σ AC ( Z, Z '' ) G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) , \nC D . \n2 ¯ θ ¯ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ 2 ¯ θ ¯ G AB ( Z, Z ' ) -∫ dτ '' d ¯ θ '' 1 ¯ θ '' 2 ¯ Σ AC ( Z, Z '' ) ¯ G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) (56) \nFor each component, we have determined the solution through variations in the associated variables \nΣ ψ 1 ψ 1 ,AB ( τ ) = J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) , \nΣ ψ 2 ψ 2 ,AB ( τ ) = J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) , \nΣ ϕϕ,AB ( τ ) = -( q -1) ( q -2) JG q -3 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) +( q -1) JG q -2 ϕϕ,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) +( q -1) ( q -2) JG q -3 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) , \nΣ FF,AB ( τ ) = -JG q -1 ϕϕ,AB ( τ, τ ' ) , \nΣ ϕF,AB ( τ ) = -J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) , \nΣ Fϕ,AB ( τ ) = -J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) . (57) \nSimilarly, we can write down the saddle point Green's functions in frequency space \nG ψ 1 ψ 1 ,AB ( ω ) = Det ( A ψ 1 ψ 1 ,AB ( D half integer )) Det ( D half integer ) \nG ψ 2 ψ 2 ,AB ( ω ) = Det ( A ψ 2 ψ 2 ,AB ( D half integer )) Det ( D half integer ) \nG ϕϕ,AB ( ω ) = -Det ( A ϕϕ,AB ( D integer )) Det ( D integer ) \n, , , \nG FF,AB ( ω ) = -Det ( A FF,AB ( D integer )) Det ( D integer ) , G ϕF,AB ( ω ) = -Det ( A ϕF,AB ( D integer )) Det ( D integer ) , (58) \nG Fϕ,AB ( ω ) = -Det ( A Fϕ,AB ( D integer )) Det ( D integer ) . \nCorresponding Jacobi matrix is \nD integer = ω 2 n -Σ LL,ϕϕ -Σ LR,ϕϕ -Σ LL,ϕF iµ -Σ LR,ϕF -Σ RL,ϕϕ ω 2 n -Σ RR,ϕϕ iµ -Σ RL,ϕF -Σ RR,ψF -Σ LL,Fϕ iµ -Σ LR,Fϕ 1 -Σ LL,FF -Σ LR,FF iµ -Σ RL,Fϕ -Σ RR,Fϕ -Σ RL,FF 1 -Σ RR,FF , \nD half integer = -iw n -Σ LL,ψ 1 ψ 1 -Σ LL,ψ 1 ψ 2 -Σ LR,ψ 1 ψ 1 -Σ LR,ψ 1 ψ 2 -Σ LL,ψ 2 ψ 1 -iw n -Σ LL,ψ 2 ψ 2 -Σ LR,ψ 2 ψ 1 -Σ LR,ψ 2 ψ 2 -Σ RL,ψ 1 ψ 1 -Σ RL,ψ 1 ψ 2 -iw n -Σ RR,ψ 1 ψ 1 -Σ RR,ψ 1 ψ 2 -Σ RL,ψ 2 ψ 1 -Σ RL,ψ 2 ψ 2 -Σ RR,ψ 2 ψ 1 -iw n -Σ RR,ψ 2 ψ 2 . \nIn Figure.(4), we have plotted the Green's function in N=4 SYK model. In Figure.(4)a, we consider the fermionic Green's function, and we can see that it is very similar to the Green's function in the N=2 SYK model and includes a fermionic chemical potential. In addition, we also consider the bosonic Green's function, which is related to the supersymmetric sector of the theory. In Figure.(4)c, the Green's function of auxiliary bosons approaches the delta function,exhibiting behavior similar to that of bosonic N=2 Green's functions. Furthermore, we can also generate it from the Grassmann algebra. Unlike the N=1 and N=2 models, the N=4 SYK model also contains original bosons that have no direct counterparts in lower-dimensional Grassmann theories. In Figure.(4)b these bosons decay over time. Additionally, we numerically obtain the multiside correlators, which are constrained by supersymmetry within the context of G ϕF,LR and its conjugate. In Figure.(4)d, the multiside correlators have both real and imaginary parts, which influence the phase constant of transformations. The real part in Figure.(4)c behaves like the bosonic component in Figure.(4)b, while the imagine part is also influenced by the fermionic Green' function in Figure.(4)a. One can also make the super-coupling constant between left and right side iµ to be real, and the real side and imagine parts may interact or transform under certain conditions. The components G ϕϕ,LR , G FF,LR , G ϕF,AA , G Fϕ,AA , G ψαψβ,AB , G ψαϕ,AB , G ϕ,ABψα that equal to 0 are stable solutions, while G ψαF,AB , G Fψα,AB should be integrated out in the analysis. \n) \n( \nG \n0.4 \n0.2 \n0 \n-0.2 \n-0.4 \n-0.6 \n-0.8 \n-1 \n-0.5 \n-0.4 \n-0.3 \n-0.2 \n-0.1 \n0 \n0.1 \n0.2 \n0.3 \n0.4 \n0.5 \n/ \n(a) \nFigure 4: Supersymmetric Green's function in components fixed J = 3 , q = 5 (a) Single-side fermionic Green's function G ψψ,AA fixed µ = 0 . 4 i, T = 0 . 01, (b) Single-side bosonic Green's function G ϕϕ,AA fixed µ = 0 . 4 , T = 0 . 01 (c) Single-side auxiliary bosonic Green's function G FF,AA fixed µ = 0 . 4 , T = 0 . 01 (d) multi-side Correlator between different bosons in real Re [ G ϕF,LR ] and imagine Im [ G ϕF,LR ] \n<!-- image --> \n0.2 \n0.1 \n0 \n-0.1 \n-0.2 \n-0.3 \n-0.5 \n-0.4 \n-0.3 \n-0.2 \n-0.1 \n0 \n0.1 \n0.2 \n0.3 \n0.4 \n0.5 \n/ \n(d) \nThe energy can also be calculated as \nE N = ∫ d ¯ θ 1 d ¯ θ 2 [ 2 q -1 D 2 G LL + iµ ( 1 -2 q -1 ) G LR ] τ → 0 + + h.c = 2 q -1 T ∑ half integral ( Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n )) + 2 q -1 T ∑ integral ( Σ LL,ϕϕ ( iw n ) G LL,ϕϕ ( iw n ) + Σ LL,FF ( iw n ) G LL,FF ( iw n ) + µRe ( q -3 2 ) ( G LL,ϕF ( iw n ) + G LL,Fϕ ( iw n )) )] + h.c. (59) \n) \n( \nG \n0.5 \n0.4 \n0.3 \n0.2 \n0.1 \n0 \n-0.1 \n-0.2 \n-0.3 \n-0.4 \n-0.5 \n-0.5 \n-0.4 \n-0.3 \n-0.2 \n-0.1 \n0 \n0.1 \n0.2 \n0.3 \n0.4 \n0.5 \n/ \n(b) \n0.3 \nRe[G \nIm[G \nF \nF \n( \n)] \n( \n)] \nHere we use real or imagine component of the correlation function, depending on how we define the interaction parameter µ and make the energy real. We obtain the free energy from saddle point solutions \nF N = -T log Z N = T S eff N = -T 2 log (2) + ∑ half integral ( log D half integral ( iw n ) ( iw n ) 2 + 4 q -4 q Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) ) -∑ integral ( log D integral ( iw n ) ( iw n ) 2 -2 q -2 q Σ LL,ϕϕ ( iw n ) G LL,ϕϕ ( iw n ) -2 q -2 q Σ LL,FF ( iw n G LL,FF ( iw n ) ) -2 q -2 q Σ LL,ϕF ( iw n ) G LL,ϕF ( iw n ) -Σ LL,Fϕ ( iw n ) G LL,Fϕ ( iw n ) )] . (60) \nIn summary, we consider the cross term in N=4 theory, which is characterized by the component G ϕϕ , the N=4 theory turns to bosonic model. We have studied the free energy and energy as a function of temperature, and there still exists two identical phases. However, the wormhole-black hole picture is altered due to the contribution of G ϕϕ . Because it is mainly a bosonic model, the free energy returns to zero when the temperature is large. And the thermal structure is no longer a eternal solution via the variation of inverse temperature β . We will give a further discussion in the context of the low-energy effective action.", '3.1 Low energy effective action': "The holographic picture of N=0 is relevant to the emergent conformal symmetry in the low energy limit. In previous works, the N=1 supersymmetric algebra has been shown to contain an extra supersymmetric reparametrization. And we can define sets of those reparametrization transformations using a more general coordinates \nτ → τ ' ( τ, θ ) , θ → θ ' ( τ, θ ) . (61) \nUnder a more general reparameterization, the correlation function with certain dimensions should be invariant under coordinate transformation rescaling \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = Ber ( τ ' 1 , θ ' 1 ; τ 1 , θ 1 ) 1 q Ber ( τ ' 2 , θ ' 2 ; τ 2 , θ 2 ) 1 q G ( τ ' 1 , θ ' 1 ; τ ' 2 , θ ' 2 ) , (62) \nwhere the derivative terms in superspace are with Berezinian \nBer ( τ ' , θ ' ; τ, θ ) ≡ Ber ( ∂ τ τ ' ∂ τ θ ' ∂ θ τ ' ∂ θ θ ' ) . (63) \nIn low energy limit, the corresponding Berezinian can be simplified to Grassmann super Jacobian derivative factor, and this simplification gives rise to a nearly super conformal symmetry, with conformal dimension 1/q \nBer ( τ ' , θ ' ; τ, θ ) = D θ θ ' . (64) \nThe correlation functions exhibit a super conformal symmetry in the IR limit which is similar to single side SSYK, and this symmetry is accompanied by the preservation of covariant derivative transformations \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = ( D θ 1 θ ' 1 ) 1 q ( D θ 2 θ ' 2 ) 1 q G ( τ ' 1 , θ ' 1 ; τ ' 2 , θ ' 2 ) . (65) \nwhich is invariant under the super transformation \nτ ' = τ + ϵ + θη, θ ' = θ + η. (66) \nHere ϵ represents an arbitrary bosonic translation, and η is a Grassmann variable. Additionally, we perform a special reparameterization of the SDiff N=1 supersymmetry \nτ → τ ' = f ( τ ) , θ → θ ' = √ ∂ τ f ( τ ) θ. (67) \nWe briefly review the discussion in [9]. Furthermore, a convenient generator of transformations with super conformal symmetry is a reparameterization of inversion \nτ → τ ' = -1 τ , θ → θ ' = θ τ . (68) \nThis algebra is invariant under the transformations of the OSp(1,2) group and can be represented by fractional linear transformations involving three bosonic generators and two fermionic generators \nτ \n= \naτ \n+ \nαθ \n+ \nb \ncτ \n+ \nγθ \n+ \nd \n, \nθ ' = βτ + eθ + δ cτ + γθ + d , (69) \nwhere we have introduced the coefficients to represent certain physical quantities \n( βτ + eθ + δ ) ( e + θβ ) + ( aτ + αθ + b ) ( -γ + θc ) -( cτ + γθ + d ) ( -α + θa ) = 0 , (70) \ni.e. \n' \neβ -aγ + αc = 0 , \ne 2 + βδ +2 αγ + bc -ad = 0 , eδ -γb + αd = 0 . (71) \nIn low energy limit, this generator comprises SL(2,R), and this formalism leads to an N=0 theory with Schwarzian-like properties. This result can be extended to a similar process involving N=0 considerations, thereby deriving a super Schwarzian action. The non-supersymmetric Schwarzian action possesses a reparameterized global conformal symmetry and can be formulated in terms of derivatives, constituting a distinct functional form \n∂ τ 1 ∂ τ 2 log τ ' 1 -τ ' 2 τ 1 -τ 2 = ∂ τ 1 τ ' 1 ∂ τ 2 τ ' 2 ( τ ' 1 -τ ' 2 ) 2 -1 ( τ ' 1 -τ ' 2 ) 2 , (72) \nIn the limit τ 1 → τ 2 the theory gives the Schwarzian action \nS [ f ( τ ) , τ ] = f ''' f ' -3 2 ( f '' f ' ) 2 . (73) \nIn super space extension, since the N=1 SYK has extended the fermion field as a super composite fermionic field, we can substitute the derivative operators with their super-covariant forms \nD τ 1 D τ 2 log τ ' 1 -τ ' 2 τ 1 -τ 2 = D τ 1 τ ' 1 D τ 2 τ ' 2 ( τ ' 1 -τ ' 2 ) 2 -1 ( τ ' 1 -τ ' 2 ) 2 . (74) \n(74) leads to the N=1 super Schwarzian form \nS [ τ ' , θ ' ; τ, θ ] = D 4 θ ' Dθ ' -2 D 3 θ ' D 2 θ ' ( D 4 θ ' ) 2 , (75) \nand which can be written in components \nS [ τ ' , θ ' ; τ, θ ] = S f ( τ ' , θ ' ; τ, θ ) + θS b ( τ ' , θ ' ; τ, θ ) . (76) \nBy analyzing this formula, we can integrate out the Grassmann variable, thereby simplifying the N=1 super Schwarzian action to a bosonic form. \nS A = -Nα S ∫ dτdθ S [ τ ' , θ ' ; τ, θ ] = -Nα S ∫ dτ S b ( τ ' , θ ' ; τ, θ ) . (77) \nα S is a constant derived from the low energy symmetry breaking term and the two point function. Additionally, we can utilize an appropriate reparameterization like N=0 SYK, which simplifies the bosonic component to a function of the Schwarzian derivative supplemented by the fermionic variable η \nS A = -Nα S 2 ∫ dτ S ( tanh h A 2 , τ ) + ηη ''' +3 η ' η '' -S ( tanh h A 2 , τ ) ηη ' . (78) \nIn the IR limit, superconformal symmetry in SSYK allows us to construct a correlation operator between two decoupled models through super conformal reparameterizations h in singleside SSYK. The correlation function in low energy limit admits solutions with ansatz \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = b | τ 1 -τ 2 -θ 1 θ 2 | 2 ∆ , (79) \nwhere the conformal dimension is restricted by ∆ = 1 /q .In addition, we can introduce the coupling action by hyperbolic reparameterizations \nS int = µ 2 ∫ dτdθ [ bD θ ( θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ + θη ( τ )) -h R ( τ + θη ( τ )) -( θ ' ) L ( θ ' ) R 2 ] 1 q . (80) \nThanks to the mathematical relations in SSYK superspace and the terms we have included, the interaction terms in low energy preserve super conformal symmetry and behave similarly to an N=0 theory, except that correlations and derivative have been replaced by super covariant counterparts. Moreover, the correlation form reverts to a non-supersymmetric case when the superspace prolongation vanishes. \nAdditionally, we can introduce an infinitesimal N=2 reparameterization involving two conjugate Grassmann variables ¯ η , η , as well as one bosonic variable h \nτ ' → h ( τ + θ ¯ η ( τ ) + ¯ θη ( τ ) ) , θ ' → exp( ia ( τ )) √ ∂ τ h ( τ ) [ θ + η ( τ ) ( τ + θ ¯ θ )] , ¯ θ ' → exp( -ia ( τ )) √ ∂ τ f ( τ ) [ ¯ θ + ¯ η ( τ ) ( τ -θ ¯ θ )] . (81) \nThis relationship leads to N=2 super Schwarzian action with the low-energy symmetry breaking. \nS A = -Nα S ∫ dτdθd ¯ θ S [ τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ] . (82) \nα S is also a constant that arises from four-point and higher-order modifications. The low energy N=2 Schwarzian action in thermal bosonic sector should exhibit an additional reparameterization U(1) symmetry and denoted by \nS b = Nα S ∫ dτ ( -Sch ( tanh h A 2 , τ ) +( ∂ τ a ) 2 ) . (83) \nCompared to N=2 IR superconformal limit, low energy interaction terms can be classified into chiral and antichiral components, and we can reparameterize the function using a similar thermal form as in the N=1 case \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = b ∣ ∣ τ 1 -τ 2 -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 +2 θ 1 ¯ θ 2 ∣ ∣ 2 ∆ , \nb \n¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ∣ ∣ τ 1 -τ 2 + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -2 θ 1 ¯ θ 2 ∣ ∣ 2 ∆ , S int = µ ∫ dτdθd ¯ θ bD ¯ θ ( ¯ θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ,θ ' , ¯ θ ' ) -h R ( τ,θ ' , ¯ θ ' ) -( θ ' ) L ( ¯ θ ' ) L -( θ ' ) R ( ¯ θ ' ) R +2( θ ' ) L ( ¯ θ ' ) R 2 ∆ , ¯ S int = µ ∫ dτdθd ¯ θ bD θ ( θ ' ) L D ¯ θ ( ¯ θ ' ) R cosh 2 h L ( τ,θ ' , ¯ θ ' ) -h R ( τ,θ ' , ¯ θ ' ) +( θ ' ) L ( ¯ θ ' ) L +( θ ' ) R ( ¯ θ ' ) R -2( θ ' ) L ( ¯ θ ' ) R 2 ∆ . (84) \nThe total action is given by a chiral combination \nS int,total = S int + ¯ S int . (85) \nAs demonstrated in [9], a similar N=2 super transformation involves one bosonic variable ϵ and two fermionic Grassmann variables, namely η and ¯ η \nτ → τ ' = τ + ϵ + θ ¯ η + ¯ θη, θ → θ ' = θ + η, ¯ θ → ¯ θ ' = ¯ θ + ¯ η. (86) \nIn the IR limit, chiral and anti-chiral correlation functions exhibit superconformal symmetry respectively. The N=2 coordinate transformations in the low energy limit are also constrained by the supercovariant derivative and the chiral and anti chiral Jacobians \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D ¯ θ 1 ¯ θ ' 1 ) 1 q ( D θ 2 θ ' 2 ) 1 q G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) , ¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D θ 1 θ ' 1 ) 1 q ( D ¯ θ 2 ¯ θ ' 2 ) 1 q ¯ G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) . (87) \nFor example, we can also introduce chiral-anti chiral coordinates \nτ ± = τ ± θ ¯ θ. (88) \nNote the reparameterization of inversion can also yield a global conformal symmetry \nτ → τ ' = -1 τ , θ → θ ' = θ τ , ¯ θ → ¯ θ ' = ¯ θ τ , τ ± → τ ' ± = -1 τ ± . (89) \nA global superconformal group can be expressed in terms of chiral and anti chiral linear transformations \nτ ' + = aτ + + αθ + b \nτ ' -= ¯ aτ -+ ¯ αθ + b ¯ \nθ ' = βτ + + eθ + δ \n¯ θ ' = βτ -+ ¯ eθ + δ ¯ . \ncτ + + γθ + d , ¯ ¯ cτ -+ ¯ γθ + d , cτ + + γθ + d , ¯ ¯ ¯ cτ -+ ¯ γθ + d (90) \nWhich has an additional restriction defined by coordinates \nτ ' + -τ ' -= 2 θ ' ¯ θ ' , (91) \ni.e. \na ¯ c -¯ ac = 2 β ¯ β, \nb ¯ d -¯ bd = 2 δ ¯ δ, \nα ¯ c -¯ aγ = 2 e ¯ β, \na ¯ γ -c ¯ α = 2 β ¯ e, \nα ¯ d -¯ bγ = 2 e ¯ δ, \nb ¯ γ -¯ αd = 2 δ ¯ e, \na ¯ d + b ¯ c -¯ ad -¯ bc = 2 β ¯ δ +2 δ ¯ β, a ¯ d -α ¯ γ -b ¯ c +¯ ad -¯ αγ -¯ bc = 2 β ¯ δ +2 e ¯ e -2 δ ¯ β. (92) \nCompared to N=0 and N=1, explicit symmetry breaking in low energy limit also leads to a N=2 super Schwarzian action in superderivative form \nS [ τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ] = ∂ τ ¯ D ¯ θ ' ¯ D ¯ θ ' -∂ τ Dθ ' Dθ ' -2 ∂ τ θ ' ∂ τ ¯ θ ' ( ¯ D ¯ θ ' ) ( Dθ ' ) = S f ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + θS θ ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + ¯ θS ¯ θ ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + θ ¯ θS b ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) . (93) \nFor the N=4 theory, we begin with the ladder diagram in the N=2 case. The propagator involving two identical superfields should correspond to a four-point contribution, which is known as solvable ladder diagrams in SYK model. In low energy limit, we consider 0st order kernel \nF 0 ( τ 1 , τ 2 ; τ 3 , τ 4 ) = -G ( τ 1 , τ 3 ) G ( τ 2 , τ 4 ) + G ( τ 1 , τ 4 ) G ( τ 2 , τ 3 ) . \nit could return to two fermionic superfields, and we have obtain the correlation function for the N=4 SYK model \nG ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' 1 , ¯ θ ' 1 , θ ' 2 , ¯ θ ' 2 ) = -b ∣ ∣ τ -τ ' -θ 1 ¯ θ 1 -θ ' 1 ¯ θ ' 1 +2 θ 1 ¯ θ ' 1 ∣ ∣ 2 ∆ ∣ ∣ τ -τ ' -θ 2 ¯ θ 2 -θ ' 2 ¯ θ ' 2 +2 θ 2 ¯ θ ' 2 ∣ ∣ 2 ∆ + b ∣ ∣ τ -τ ' -θ 1 ¯ θ 2 -θ ' 1 ¯ θ ' 2 +2 θ 1 ¯ θ ' 2 ∣ ∣ 2 ∆ ∣ ∣ τ -τ ' -θ 2 ¯ θ 1 -θ ' 2 ¯ θ ' 1 +2 θ 2 ¯ θ ' 1 ∣ ∣ 2 ∆ . (94) \nThe conformal limit still holds for a single propagator but not for kernels. The second term arises from the general reparametrization. And the formal result reverts to a copy of N=2 theories when we trivially reparametrize the theory (they should also be integrated and lead to different physics). \nWe can derive a simple form of the N=4 reparametrization \nθ ' 1 → exp( ia 1 ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ 1 + η 1 ( τ + θ 1 ¯ θ 1 )] , \n¯ θ ' 1 → exp( -ia 1 ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ 1 + ¯ η 1 ( τ -θ 1 ¯ θ 1 )] , θ ' 2 → exp( ia 2 ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ 2 + η 2 ( τ + θ 2 ¯ θ 2 )] , ¯ θ ' 2 → exp( -ia 2 ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ 2 + ¯ η 2 ( τ -θ 2 ¯ θ 2 )] , h 1 = tanh τ + θ 1 ¯ θ 1 + θ 1 ¯ η 1 + ¯ θ 1 η 1 + θ 2 ¯ θ 2 + θ 2 ¯ η 2 + ¯ θ 2 η 2 2 , h 2 = tanh τ -θ 1 ¯ θ 1 + θ 1 ¯ η 1 + ¯ θ 1 η 1 + θ 2 ¯ θ 2 + θ 2 ¯ η 2 + ¯ θ 2 η 2 2 , (95) \nhere we use h 1 and h 2 to denote the reparameterized initial time and final time respectively, and the U(1) variables a 1 and a 2 are also allowed. \nFurthermore, we consider the effective reparameterized interaction action under a special constraint \nS \nint = µ ∫ dτdθd ¯ θ bD ¯ θ 2 ( ¯ θ ' 1 ¯ θ ' 2 ) L D θ 2 ( θ ' 1 θ ' 2 ) R cosh 2 ( h L -h R -( θ ' 1 ) L ( ¯ θ ' 1 ) L -( θ ' 1 ) R ( ¯ θ ' 1 ) R +2 ( θ ' 1 ) L ( ¯ θ ' 1 ) R )( h L -h R -( θ ' 2 ) L ( ¯ θ ' 2 ) L -( θ ' 2 ) R ( ¯ θ ' 2 ) R +2 ( θ ' 2 ) L ( ¯ θ ' 2 ) R ) 2 -bD ¯ θ 2 ( ¯ θ ' 1 ¯ θ ' 2 ) L D θ 2 ( θ ' 1 θ ' 2 ) R cosh 2 ( h L -h R -( θ ' 1 ) L ( ¯ θ ' 2 ) L -( θ ' 1 ) R ( ¯ θ ' 2 ) R +2 ( θ ' 1 ) L ( ¯ θ ' 2 ) R )( h L -h R -( θ ' 2 ) L ( ¯ θ ' 1 ) L -( θ ' 2 ) R ( ¯ θ ' 1 ) R +2 ( θ ' 2 ) L ( ¯ θ ' 1 ) R ) 2 . (96) \nThe N=4 model allows four independent Grassmann variables. Here, we consider an infinitesimal transformation. And for simplicity, we assume that there is no dependence between θ ' 1 and θ 2 , it is easy to check \nD θ 1 θ ' 1 D θ 2 θ ' 2 = D 2 θ ( θ ' 1 θ ' 2 ) , θ ' → θ ' 1 θ ' 2 . (97) \nWe consider an arbitrary physical function F , which should be invariant under a coordinate transformation \nD 2 θ F ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) = D 2 θ τ ' ∂ τ ' F + D 2 θ θ ' 2 ∂ ' 2 θ F + D 2 ¯ θ ¯ θ ' 2 ∂ ' 2 ¯ θ F = D 2 θ θ 2 D ' 2 θ F + D 2 ¯ θ ¯ θ 2 D ' 2 ¯ θ F + ( D 2 θ τ ' -¯ θ ' 1 ∂ θ ' 2 + ¯ θ ' 2 ∂ θ ' 1 -¯ θ ' 1 ¯ θ ' 2 ∂ τ ' ) ∂ τ ' F. (98) \nOne interesting special condition in mathematical manipulations is the process of reducing the higher order terms \nD 2 θ τ ' = ( ¯ θ ' 1 ∂ θ ' 2 -¯ θ ' 2 ∂ θ ' 1 + ¯ θ ' 1 ¯ θ ' 2 ∂ τ ' ) D 2 θ θ ' 2 , D ¯ θ α = D 2 θ ¯ θ α = D 2 θ ¯ θ 2 = 0 . (99) \nThis restriction also leads to a transformation invariant, which exhibits similarities to a certain aspect of super conformal theories. Furthermore, we will see that this condition is highly beneficial in discovering a holographic theory within the context of gravitational physics. \nWe can also consider an analogous approach utilizing Berezin integrals, which is equivalent in certain respects to the previously discussed methods \nBer ( ∂ τ τ ' ∂ τ θ ' 2 ∂ θ 2 τ ' ∂ θ 2 θ 2 ) = Ber ( ∂ τ τ ' ∂ τ θ ' 2 D 2 θ τ ' -( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ τ ) ∂ τ τ ' D 2 θ θ ' 2 -( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ τ ) ∂ τ θ ' 2 ) = Ber ( ∂ τ τ ' ∂ τ θ ' 2 D 2 θ τ ' D 2 θ θ ' 2 ) = ( D 2 θ θ ' 2 ) -1 Ber ( ∂ τ τ ' ∂ τ θ ' 2 ¯ θ ' 1 ∂ θ ' 2 -¯ θ ' 2 ∂ θ ' 1 + ¯ θ ' 1 ¯ θ ' 2 ∂ τ ' 1 ) = ¯ D 2 θ ¯ θ ' 2 , (100) \nwhere we have assumed that the infinitesimal translation exhibits symmetry between the two Grassmann variables θ 1 and θ 2 \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D 2 ¯ θ 1 ¯ θ 2 ' 1 ) 1 q ( D 2 θ 2 θ 2 ' 2 ) 1 q G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) , ¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D 2 θ 2 θ 2 ' 1 ) 1 q ( D 2 ¯ θ 1 ¯ θ 2 ' 2 ) 1 q ¯ G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) . (101) \nTerms with two identical Grassmann variables and derivatives of τ will break the usual super conformal limit of reparametrization in N=4 theory. But an important feature is that, when one of the Grassmann variables is ignored, the N=4 theory should reduce to the N=2 theory. Furthermore, we can construct the N=4 effective action based on the N=2 theory. The aforementioned approach of constructing the N=4 effective action from N=2 theory can also be applied \nto the gravitational sector later. The behavior of the infinitesimal translation in the context of N=4 theory should exhibit some similarities to that in the N=2 theory, but with some notable differences, as follow \nθ 1 → ξ -iϵρ, θ 2 → ξ -iϵρ, ¯ θ 1 → ¯ ξ -iϵ ¯ ρ, ¯ θ 2 → ¯ ξ -iϵ ¯ ρ. (102) \nSince we have the first-order restriction specified in the equation, we can proceed to write down the infinitesimal generators ρ and ¯ ρ in a manner similar to that in the N=2 theory \nρ = -ξ ' ¯ ρ = -¯ ξ ' . (103) \nThen we can characterize the effective action as super Schwarzian \nSchw ( x, ξ, ¯ ξ ; τ, θ, ¯ θ ) = ∂ τ ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) + ( D ¯ θ 1 ¯ ξ ) ∂ τ ( D ¯ θ 2 ¯ ξ ) ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) + ∂ τ ( D θ 1 ξ ) ( D θ 2 ξ ) ( D θ 1 ξ ) ( D θ 2 ξ ) + ( D θ 1 ξ ) ∂ τ ( D θ 2 ξ ) ( D θ 1 ξ ) ( D θ 2 ξ ) -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) = ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 + ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) . (104) \nThis effective action arises from two identical N=2 SYK models, includes a last term that arises from the first-order crossing term. It can be verified that if we eliminate one of the Grassmann variables, the theory will reduce to the N=2 theory. Furthermore, this result indicates that the effective action is mainly Schwarzian in the absence of super space, and the additional cross term would break the emergent super conformal symmetry. \nD 2 θ ξ 2 = ( ∂ ∂θ 1 + ¯ θ 1 ∂ ∂τ ) ξ ( ∂ ∂θ 2 + ¯ θ 2 ∂ ∂τ ) ξ = ∂ 2 θ ξ 2 + D θ 1 ξ ∂ ∂θ 2 ξ + ∂ ∂θ 1 ξD θ 2 ξ + ¯ θ 1 ¯ θ 2 ∂ ∂τ ∂ ∂τ ξ 2 . (105) \nCompared to the mean field methods and SD equations we have studied in the previous section, we can also derive the IR Green's function for the N=2 SYK model. Firstly, when we neglect the higher-order terms and concentrate on the low-energy theory, the effective propagators in the low-energy limit can be written as \nG ψψ ( τ ) ∼ sgn ( τ ) c ψ ( τ ) , G bb ( τ ) ∼ -δ ( τ ) c b ( τ ) ∼ 1 c b ( τ ) . (106) \nwhere the functions c are the conformal factors in propagates that and are determined by the conformal dimension, which is related to the equation G ψψ ( τ ) = -∂ τ G bb ( τ ) \nThe N=4 theory could be defined as a specific theory \nG ψψ ( τ ) ∼ 1 2 sgn ( τ ) c ψ ( τ ) , G FF ( τ ) ∼ δ ( τ ) c F ( τ ) ∼ 1 c ' F ( τ ) , G ϕϕ ( τ ) ∼ 1 2 | τ | c ϕ ( τ ) . (107) \nwith G ϕϕ ( τ ) = ∂ τ G ψψ ( τ ) = ∂ 2 τ G FF ( τ ). However, the component G ϕϕ is weird. Since the absolute value of the time order in the conformal factor should be less than 1, the Green's function Σ ϕϕ is not a convergent function in large τ and β limit(compared to the other component). According to the [39], the components with certain dimension will dominate the bare propagator. Considering the Replica trick, it is also a periodic function. \nThen, we can further qualitatively analyze the thermal entropy using the mean field ansatz, when considering the zero-temperature limit \nS ≡ 1 log ( Trρ n \nexp( -S ) ∼ 1 Z 2 ∫ DϕDψDF exp( -S eff (Σ ϕϕ ( τ ))) \n1 -n ) (108) \nSince the partition function conclude a first-order dependence of logarithm, it will not converge to a certain constant due to the contribution of Σ ϕϕ , and it has a dramatic change around the periodic short time τ = β/ 2(where β is large enough).", '3.2 Gravitational action on boundary': "In this subsection, we consider the holographic duality of the supersymmetric SYK model. First, we briefly review the low energy SYK model with N=0,1,2 supersymmetry and their holography duality, which can be described as JT gravity \nS = -1 16 πG [∫ M d 2 x √ gϕ ( R +2) + 2 ∫ ∂M du √ hK ] , (109) \nwhich is contained in a global AdS 2 metric \nds 2 = dt 2 + dz 2 z 2 . (110) \nWe can also apply a finite curve reparameterization to the boundary \n1 ϵ 2 = t ' 2 + z ' 2 z 2 (111) \nThis form leads to a Schwarzian form with NAdS boundary condition as ϵ approaches zero \nS [ t ( u )] = ∫ duϕ r ( u ) Sch ( t, u ) . (112) \nWe can also consider N=1 super JT gravity as in [12] \nS = -1 16 πG [ i ∫ d 2 zd 2 θEΦ ( R + --2) + 2 ∫ ∂M dudθΦK ] , (113) \nHere we use φ to represent supersymmetric dilatons [44, 45], and E is the supervielbein to harmonize the spinor and scalar fields. In [12], it also gives a special supercomformal gauge condition \ndu 2 +2 θdθdu 4 ϵ 2 = dz ξ E 1 ξ dz π E ¯ 1 π . (114) \nThe superconformal gauge condition is the most important and widely researched among those relevant to quantum gravity and supersymmetric field theories \n( E + θ E 1 θ E + z E 1 z ) = ( e Σ 2 + θe Σ D θ e -Σ 2 -θe Σ -e Σ D θ e -Σ 2 e Σ ) . \nwith supersymmetrization condition. In order to obtain the effective action, the vielbein on both dimensions has to satisfy \nDz = θDθ, z = t + iϵ ( Dξ ) 2 , . \nDt ( u, θ ) = ξ ( u, θ ) Dξ ( u, θ ) \nHere the supersymmetric gravity will contribute an additional spinor component. To convert these spinor terms back to the scalar superspace, we introduce the supervielbein. And we follow the notation E denotes vielbein components relevant to inverse density, which enables us to \nK = T A D T n A T A T A , (115) \nwhere T is a tangent vector along the boundary, normalized the orthogonal vector n to T A n A = 0. \nK = 4 ϵ 2 S [ t, ξ ; u, θ ] . (116) \nThe covariant derivative D T contains contributions from the super derivative and spinor terms, as expressed in the equation D T n A = Dn A + DΩ . Furthermore, the boundary action exhibits properties similar to those of the Schwarzian derivative \nS bdy = ∫ dudθΦ r ( u, θ ) S [ t, ξ ; u, θ ] . (117) \nAnd we can also introduce sets of interaction operators on both left and right boundary \nS int = g ∑ i ∫ dudθO i L ( u, θ ) O i R ( u, θ ) . (118) \nO is a set of N operators with super conformal dimension ∆ . The dimension of g is given by [energy] 2 ∆ -1 [46, 47] \nS int = g ∑ i ∫ dudθd ¯ θ ( ¯ O i L ( u, θ, ¯ θ ) O i R ( u, θ, ¯ θ ) + O i L ( u, θ, ¯ θ ) ¯ O i R ( u, θ, ¯ θ )) , (119) \nwhere O are sets of N operators with super conformal dimension ∆ , and they can be obtained by supersymmetrizing N=0 double traces. g has the dimension as [energy] 2 ∆ -1 . Then we can construct the duality of interaction part of SSYK \n〈 O ( t 1 P ) O ( t 2 P )〉 = ∣ ∣ t 1 P -t 2 P -θ 1 P θ 2 P ∣ ∣ -2 ∆ . (120) \nAfter applying the thermal form reparametrization \nh = tanh ( τ + θη ) 2 , \nθ ' = ( ∂ τ tanh ( ( τ + θη ) 2 )) 1 2 ( θ + η + 1 2 η∂ τ η ) . (121) \nWe can also write the N=2 JT gravity from [13] \nS = -1 16 πG [∫ d 2 zd 2 θEΦ ( R +2) + ∫ d 2 zd 2 ¯ θE ¯ Φ ( ¯ R +2 ) +2 ∫ ∂M dudθd ¯ θ ( Φ + ¯ Φ ) K ] , (122) \nwith the boundary external supercurvature \n∫ ∂M dudθd ¯ θ K = ∫ ∂M dudθK + ∫ ∂M dud ¯ θ ¯ K, K = T A D T n A T A T A , ¯ K = T A ¯ D T n A T A T A . (123) \nA special N=2 supersymmetric gauge constraint is also introduced through a natural process of supersymmetrization \ndu 2 +2 ¯ θdθdu +2 θd ¯ θdu +2 θ ¯ θdθd ¯ θ 4 ϵ 2 = dz ξ E l ξ dz π E ¯ l π . (124) \nHere we use the chiral density and the supervielbeins to rewrite the supergravity in super space. \nThen we have the reparameterization, involving a super Schwarzian effective action and exhibiting holographic duality with the SSYK model \nDz = ¯ θDθ, \n¯ ¯ ¯ \nDz = θ D θ, z = t + iϵ ( Dξ ) ( ¯ D ¯ ξ ) , Dt ( u, θ, ¯ θ ) = ¯ ξ ( u, θ, ¯ θ ) Dξ ( u, θ, ¯ θ ) , ¯ Dt ( u, θ, ¯ θ ) = ξ ( u, θ, ¯ θ ) ¯ D ¯ ξ ( u, θ, ¯ θ ) . \n(125) \nThe extrinsic supercurvature is very similar to the theory with N=1 supersymmetry. \nK = -4 ϵ 2 ξ '' Dξ -ξ ' ( Dξ ' ) ( Dξ ) 2 + ( ¯ D ¯ ξ ' ) ξ ' ( Dξ ) ( ¯ D ¯ ξ ) , K = -4 ϵ 2 ¯ ξ '' ¯ Dξ -¯ ξ ' ( ¯ D ¯ ξ ' ) ( ¯ D ¯ ξ ) 2 + ( Dξ ' ) ¯ ξ ' ( ¯ D ¯ ξ ) ( Dξ ) . (126) \nWe can also rewrite the boundary action in terms of the super Schwarzian \nS bdy = ∫ dudθd ¯ θ ( Φ r + ¯ Φ r ) S [ t, ξ, ¯ ξ ; u, θ, ¯ θ ] . (127) \nWe can also built the N=2 correlation function between boundaries. Since fermions with N=2 supersymmetric interact with conjugate chiral particles, we have restricted the sets of operators O and ¯ O based on their chirality. Likewise, we can utilize the thermal reparametrization solutions \nh 1 = tanh τ + θ ¯ θ + θ ¯ η + ¯ θη 2 , \nh 2 = tanh τ -θ ¯ θ + θ ¯ η + ¯ θη 2 , \nθ ' → exp( ia ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ + η ( τ + θ ¯ θ )] , \n¯ θ ' → exp( -ia ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ + ¯ η ( τ -θ ¯ θ )] , \nS int = g 2 2∆ ∫ dτdθd ¯ θ D θ ( θ ' ) L D ¯ θ ( ¯ θ ' ) R cosh 2 h L ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -h R ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -( θ ' ) L ( ¯ θ ' ) R + ( ¯ θ ' ) L ( θ ' ) R 2 1 q , \n¯ S int = g 2 2∆ ∫ dτdθd ¯ θ D ¯ θ ( ¯ θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -h R ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -( θ ' ) L ( ¯ θ ' ) R + ( ¯ θ ' ) L ( θ ' ) R 2 1 q . (128) \nHere we use h 1 and h 2 to denote the reparameterized initial time and final time. \nWe also want to derive the effective action of N=4 holographic theory in the low temperature limit. First, we consider the special form of N=4 super Schwarzian as discussed in the context of the N=4 low energy limit. \nSchw ( x, ξ, ¯ ξ ; τ, θ, ¯ θ ) = ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 + ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) . (129) \nThis action is not globally conformal or superconformal, unlike the N=0 and N=1,2 SYK models which exhibit special reparameterization symmetries. Unlike the 2D analogues of the SYK model, this gravitational action is also restricted to 1+1d. Since there exists only one time scale, higher order terms with cross terms between the Grassmann variables emerge. These cross terms are not invariant under conformal transformations. A special case is when SL(2,R) emerges in the low energy limit of N=0 SYK, whereas SU (1 , 1 | 1) symmetry does not emerge under N=2 conditions. \nHowever, it can still be derived from N=2 SYK model. Since the infinitesimal translation group is similar to that in the N=2 SYK model, we can propose a generalization based on the N=2 NAdS spacetime. Similarly, in the N=4 case, we expect this N=4 NAdS spacetime to exhibit properties analogous to those of the N=4 SYK model. To extend to N=4 spacetime, we introduce supersymmetry into the relevant Grassmann variable components, building upon the N=2 NAdS spacetime framework. When we eliminate one Grassmann variable α by imposing certain constraints or symmetries \n∂ θ α ∂ ¯ θ α ( θ α ¯ θ α L ) . \nand the theory will return to N=2. \nWe can easily adopt an ansatz for the specific form of the metric within the possible context of N=4 JT gravity \nS JT = -1 16 πG N (∫ M d 2 zd 4 θ E -1 Φ ( R -2) + ∫ M d 2 zd 4 ¯ θ ¯ E -1 ¯ Φ ( ¯ R -2 ) +2 ∫ M dzd 2 θd 2 ¯ θ ( Φ b + ¯ Φ b ) K ) . (130) \nThe parameters in this equation should also be constrained with certain conditions, which means we should restrict the N=4 theory back to N=2 when it involves first-order of the other Grassmann variable. This external curvature term can be written in Grassmann components \n∫ M dzd 2 θd 2 ¯ θ K = ∫ M dzd 2 θK + ∫ M dzd 2 ¯ θ ¯ K. (131) \nSince we want to obtain the formula of the effective action, it is natural to define the external curvature in terms of Grassmann components, which are relevant to or derived from the N=2 theory \nK = T A ¯ D T 1 n A 1 ¯ D T 2 n A 2 T 2 = T A ¯ D 2 T n A T 2 , ¯ K = T A D T 1 n A 1 D T 2 n A 2 T 2 = T A D 2 T n A T 2 . (132) \nHere we use ¯ D T 1 , ¯ D T 2 , D T 1 , D T 2 to express the scalar-spinor covariant derivatives with different Grassmann variables. And these derivatives are performed with a super connection term Ω ξ \nD 2 T n A = D 2 T n A + ( D 2 T z ξ Ω ξ ) n A , \n¯ D 2 T n A = ¯ D 2 T n A + ( ¯ D 2 T z ξ Ω ξ ) n A . (133) \nWe have preliminarily proposed the formalism of the curvature. It is easy to check that this curvature includes N=2 curvature with a single Grassmann component θ or ¯ θ . \nWe can also write the boundary effective action up to 2nd order \n( du 2 +2 ¯ θ 1 dθ 1 du +2 θ 1 d ¯ θ 1 du +2 θ 1 ¯ θ 1 dθ 1 d ¯ θ 1 ) ( du 2 +2 ¯ θ 2 dθ 2 du +2 θ 2 d ¯ θ 2 du +2 θ 2 ¯ θ 2 dθ 2 d ¯ θ 2 ) 4 ϵ 2 = dz ξ E l ξ dz π E ¯ l π . (134) \nCompared to N=2 theory, this equation is generated from the N=4 supercovariant derivative. Parameter E and ¯ E as chiral density leads to supervielbein, which unifies the spinor and scalar components, which also contain N=2 theories. \nIn order to develop the holographic theory of field theory, the boundary reparameterization should satisfy certain constraints \nDz = ( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ z ) D 2 θ 2 \nθ , ¯ Dz = ( θ 1 ∂ ¯ θ 2 -θ 2 ∂ ¯ θ 1 + θ 1 θ 2 ∂ z ) D 2 ¯ θ ¯ θ 2 , z = t + iϵ ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) , \nDt ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) = ( ¯ ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ θ 2 -¯ ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ θ 1 + ¯ ξ 2 ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ z ) D 2 θ ξ 2 , ¯ Dt ( u, θ 1 , θ 2 , , ¯ θ 1 , ¯ θ 2 ) = ( ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ ¯ θ 2 -ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ ¯ θ 1 + ξ 2 ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ z ) D 2 ¯ θ ¯ ξ 2 . (135) \nAn important signature here is that this action should contain the N=2 gravity which is dual to the N=2 SYK model. If we eliminate terms with the Grassmann variable θ 1 and ¯ θ 1 or θ 2 and ¯ θ 2 \n∂ θ α ∂ ¯ θ α ( θ α ¯ θ α S ) , \nThe action now simplifies to N=2 action, which contains a bosonic part that relates to the nonsupersymmetric Schwarzian action. We can easily verify this in terms of the effective action. However, the method for verifying the N=4 construction is incomplete. Naturally, we ignore terms of first order, which do not appear in the N=2 model. Now, we proceed to examine this further. Notice that a stronger condition is required for the full consistency of the theory \n¯ D 2 ¯ θ ¯ ξ 2 = ¯ D ¯ θ ¯ ξ ¯ D ¯ θ ¯ ξ, D 2 θ ξ 2 = D θ ξD θ ξ, (136) \nwhen our reparametrization ξ has only first-order dependence on τ . The single reparametrization ξ should involve a Grassmann variable. Assuming the relevant equation holds, we can then proceed to consider the curvature tensor, which can be expressed in its component form \nK = -2 ϵ 2 D 2 θ ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) , \n¯ K = -2 ϵ 2 ¯ D 2 ¯ θ ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 -∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) . (137) \nOne can easily verify that the curvature exhibits features characteristic of N=2 curvature. \nD 2 θ ξ 2 = ( ∂ ∂θ 1 + ¯ θ 1 ∂ ∂τ ) ξ ( ∂ ∂θ 2 + ¯ θ 2 ∂ ∂τ ) ξ = ∂ 2 θ ξ 2 + D θ 1 ξ ∂ ∂θ 2 ξ + ∂ ∂θ 1 ξD θ 2 ξ + ¯ θ 1 ¯ θ 2 ∂ ∂τ ∂ ∂τ ξ 2 . (138) \nThe correlation term is written in ansatz \n〈 ¯ O ( t 1 P ) O ( t 2 P )〉 = 〈 O ( t 1 P ) ¯ O ( t 2 P )〉 = ∣ ∣ t 1 P -t 2 P -θ 1 α ¯ θ 2 α -¯ θ 1 α θ 2 α -θ 1 β ¯ θ 2 β -¯ θ 1 β θ 2 β ∣ ∣ -2 ∆ . (139) \nFor both N=2 and N=4 theories, the super reparametrization differs significantly from that of N=1 theory.", '4 Thermofield double state': "The thermofield double in the Majorana SYK model is defined as the entangled state that relates the original system to its time-reversed counterpart at a finite temperature \n| TFD β ⟩ = 1 √ Z β ∑ n exp( -βE n / 2) | n ⟩ 1 ⊗| ˆ n ⟩ 2 . (140) \nwhere β = 1 /T . Since N=1 SSYK supercharge has the same structure as Majorana fermions except for parity has changed from odd to even, and we can propose a projection from N=0 to N=1 by changing q to -q. The Hilbert space remains the in its original form to directly represent the supercharge, and it indirectly relates to the system's Hamiltonian. \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -βQ 2 n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 . (141) \nThe interaction term is proportional to Majorana fermions, which act as supercharges in eq.(6) and eq.(9). \nΨ i = ( I + θQ ) ψ i = ( I + θQ ) ( C † i + C i ) , (142) \nwhich exhibits anticommutative relations and behaves like fermions \nH int = iµ∂ θ ∑ i Ψ L i Ψ R i = iµ ∑ i ( QC L † i C R i -QC R † i C L i ) . (143) \nThen, within the N=1 framework, we can explicitly define the eigenstates of the superspace operator \nH | Ψ r n ⟩ = Q 2 | Ψ r n ⟩ = E n | Ψ r n ⟩ , \nH † ∣ ∣ ∣ Ψ l m 〉 = Q 2 † | Ψ r n ⟩ = E ∗ m ∣ ∣ ∣ Ψ l m 〉 , (144) \nwith \n∑ n | Ψ l n ⟩ ⟨ Ψ r n | = I, ⟨ Ψ l n | Ψ r m ⟩ = δ nm . (145) \nGround states of the interaction part are also generated from the N=0 MQ model \n| I ⟩ = ∏ i ( | 1 ⟩ L,i | 0 ⟩ R,i + i | 0 ⟩ L,i | 1 ⟩ R,i ) , (146) \nand \nH † int ∣ ∣ ∣ Ψ l 0 〉 = -µNQ 0 ∣ ∣ ∣ Ψ l 0 〉 , H int | Ψ r 0 ⟩ = -µNQ 0 | Ψ r 0 ⟩ . (147) \nThe TFD β =0 state shall be defined using an anti-unitary transformation, as described in[37, 48] \n| TFD 0 (0) ⟩ = 1 2 N/ 2 ∑ q ∑ n q | n q ⟩ 1 ⊗ ( e -iηπΓ 4 e -iq ( π 2 ) ) P | n q ⟩ 2 , Γ = ( -1) -q + N 2 , P -1 C i P = ηP † i , ⟨ ˆ n -q ' | C R i | ˆ m -q ⟩ = -e -iπ ( -q + N 2 ) ⟨ m q | C L i | n q ' ⟩ , ⟨ ˆ n -q ' | C R † i | ˆ m -q ⟩ = -e -iπ ( -q ' + N 2 ) ⟨ m q | C L † i | n q ' ⟩ . (148) \nWe now define the states and parity P in terms of supercharges. These supercharges in the equation act like Hamiltonian in N=0 model and give a constant without changing the states in Fock space. Since the energy diagram of the N=1 SYK model can be projected onto N=0 model[49, 50, 51, 52], the N=1 SYK model could be considered as a special form of the N=0 SYK model, and is also referred to as 'fake superspace'. The ground states of N=1 SYK model are isomorphic to the trivial homology of supercharges but not to cohomology. Furthermore, we know that the ground state of the N=1 SYK model is not a BPS state. However, we will see that the BPS ground states of the N=2 and N=4 SYK models have an infinite number of projections onto supercharges or onto the N=0 SYK model. \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1 ( iµ ∑ i ( QC L † i C R i -QC R † i C L i )) | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q + N 2 ) 〈 n q ' ∣ ∣ ∣ QC L † i ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ C R i | ˆ m -q ⟩ -iµe iπ ( -q ' + N 2 ) e 2 α 〈 ˆ n -q ' ∣ ∣ ∣ QC R † i ∣ ∣ ∣ n ' q ''' 〉〈 n q ' ∣ ∣ ∣ C R i | m q ⟩ ) , (149) \ngives \n⟨ H int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i [ -iµ 〈 n q ' ∣ ∣ ∣ QC L † i | m q ⟩ ⟨ m q | C L i ∣ ∣ ∣ n q ' 〉 + iµ ⟨ m q | QC L † i ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ C L i | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ QC L † i | m q ⟩ ⟨ m q | C L i ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 QC L † i QC L i | n q ⟩ 1 = Const. (150) \nWe can utilize the same process in the N=2,4 SYK model from Appendix A, which also yield similar results. \nAdditionally, we can calculate the Witten index to confirm the presence of fermionic ground states. \nI ( r ) = Tr [ ( -1) F e 2 πirQ R ] . \nIn the N=2 model, the Witten index is redefined to reflect specific properties or conditions relevant to the model \nI ( r ) = e iπN ( r q -1 2 ) ( 1 -e 2 πir q ) N = ( 2 sin ( πr q )) N . \nThe Witten index is very similar for N=4 SYK, which also incorporates the anti-commutation within the Hamiltonian. The N=4 ground states enforce one of the anti-commutators of the supercharges to be zero, thereby providing give \nI ( r ) = e iπN ( 2 rq -r 2 q 2 -1 2 ) ( 1 -e 2 πi ( 2 rq -r 2 ) q 2 ) N = ( 2 sin π ( 2 rq -r 2 ) q 2 ) N . \nThis quantity also vanishes when r = 0. Here R-charge is defined as Q R = 1 q Q", '5.1 Real time correlation and transmission amplitude': "In both N=1 and N=2 Supersymmetric SYK models, the Lorentz Wightman correlation function can be derived from the properties of superfields \nG > AB ( t 1 , t 2 ) = -i G > AB ( it 1 , it + 2 ) = -i lim G > AB ( it 1 + ϵ, it 2 -ϵ ) \nG < AB ( t 1 , t 2 ) = -i G < AB ( it + 1 , it -2 ) = -i lim G < AB ( it 1 -ϵ, it 2 + ϵ ) \n-ϵ →-0 , ϵ → +0 , \nG R AB ( t 1 , t 2 ) = ϑ ( t 1 -t 2 ) ( G > AB ( t 1 , t 2 ) -G < AB ( t 1 , t 2 ) ) , G A AB ( t 1 , t 2 ) = ϑ ( t 2 -t 1 ) ( G > AB ( t 1 , t 2 ) -G < AB ( t 1 , t 2 ) ) . (151) \nThe Schwinger-Dyson equation in Lorentzian time version is defined as \niD t 1 G > AB ( t 1 , t 2 ) = µϵ AC ∂ θ G > CB ( t 1 , t 2 )+ ∫ dtdθ ( Σ R AC ( t 1 , t ) G > CB ( t, t 2 ) + Σ > AC ( t 1 , t ) G A CB ( t, t 2 ) ) , \niD t 1 G R AB ( t 1 , t 2 ) -µϵ AC ∂ θ G R CB ( t 1 , t 2 ) -∫ dtdθ ( Σ R AC ( t 1 , t ) G R CB ( t, t 2 ) + Σ A AC ( t 1 , t ) G A CB ( t, t 2 ) ) , \n= δ AB ( ¯ θ 1 -¯ θ 2 ) δ ( t 1 -t 2 ) , (152) \nThe corresponding equation of motion \nΣ > AB ( t 1 , t 2 ) = J G > AB ( t 1 , t 2 ) G > AB ( t 1 , t 2 ) Σ R AB ( t 1 , t 2 ) = ϑ ( t 1 -t 2 ) ( Σ > AB ( t 1 , t 2 ) -Σ < AB ( t 2 , t 1 ) ) , (153) \nand written in components \nΣ > ψψ,AB ( t ) = -( q -1) ( -1) ( q -1) / 2 JG >q -2 ψψ,AB ( t ) G > bb,AB ( t ) + ( q -1) 2 ( q -2) ( -1) ( q -1) / 2 JG >q -3 ψψ,AB ( t ) , G >A ψb,AB ( t ) G >S bψ,AB ( t ) + ( q -1) 2 ( q -2) ( -1) ( q -1) / 2 JG >q -2 ψψ,AB ( t ) G >S ψb,AB ( t ) G >A bψ,AB ( t ) , Σ > bb,AB ( t ) = -JG >q ψψ,AB ( t ) , Σ >S bψ,AB ( t ) = ( q -1) 2 JG >A ψb,AB ( t ) G >q -2 ψψ,AB ( t ) , Σ >A bψ,AB ( t ) = ( -1) ( q -1) / 2 ( q -1) 2 JG >S ψb,AB ( t ) G >q -2 ψψ,AB ( t ) , Σ >S ψb,AB ( τ ) = ( q -1) 2 JG >A bψ,AB ( τ ) G > ψψ,AB ( τ ) , Σ >A ψb,AB ( t ) = -( -1) ( q -1) / 2 ( q -1) 2 JG >S bψ,AB ( τ ) G > ψψ,AB ( τ ) . (154) \nWe can rewrite the retarded component using both bosonic and fermionic partitions \nG > ψψ,AB ( ω ) = G R ψψ,AB ( ω ) -( G R ψψ,AB ( ω ) ) ∗ 1 + exp ( -βω ) , G > bb,AB ( ω ) = G R bb,AB ( ω ) -( G R bb,AB ( ω ) ) ∗ -1 + exp ( -βω ) , G >S bψ,AB ( ω ) = G RS bψ,AB ( ω ) -( G RS bψ,AB ( ω ) ) ∗ -1 + exp ( -βω ) , \n∗ \nG >A ψb,AB ( ω ) = G RA ψb,AB ( ω ) -( G RA ψb,AB ( ω ) ) 1 + exp ( -βω ) , G >A bψ,AB ( ω ) = G RA bψ,AB ( ω ) -( G RA bψ,AB ( ω ) ) ∗ 1 + exp ( -βω ) , G >S ψb,AB ( ω ) = G RS ψb,AB ( ω ) -( G RS ψb,AB ( ω ) ) ∗ -1 + exp ( -βω ) . (155) \nIn this context, the bosonic and fermionic factors are incorporated into the contour integrations, rendering the traditional Matsubara methods obsolete. With the frequency being discrete, we can now consider the original bosonic and fermionic factors in real time \nD Lorentz = w n -Σ LL,ψψ -Σ LR,ψψ -Σ A LL,ψb -iµ -Σ A LR,ψb -Σ RL,ψψ w n -Σ RR,ψψ iµ -Σ A RL,ψb -Σ A RR,ψb -Σ A LL,bψ iµ -Σ A LR,bψ -1 -0 -0 -iµ -Σ A RL,bψ -Σ A RR,bψ -0 -1 -0 , D Lorentz inverse = w n -0 -0 -Σ S LL,ψb iµ -Σ S LR,ψb -0 w n -0 -iµ -Σ S RL,ψb -Σ S RR,ψb -Σ S LL,bψ -iµ -Σ S LR,bψ -1 -Σ LL,bb -Σ LR,bb iµ -Σ S RL,bψ -Σ S RR,bψ -Σ RL,bb -1 -Σ RR,bb , G ψψ,AB ( τ ) = Det ( A ψψ,AB ( D Lorentz )) Det ( D Lorentz ) , G bb,AB ( τ ) = -Det ( A bb,AB ( D Lorentz inverse )) Det ( D Lorentz inverse ) , G A ψb,AB ( τ ) = Det ( A A ψb,AB ( D Lorentz ) ) Det ( D Lorentz ) , G S bψ,AB ( τ ) = -Det ( A S bψ,AB ( D Lorentz inverse ) ) Det ( D Lorentz inverse ) , G S ψb,AB ( τ ) = Det ( A S ψb,AB ( D Lorentz inverse ) ) Det ( D Lorentz inverse ) , G A bψ,AB ( τ ) = -Det ( A A bψ,AB ( D Lorentz ) ) . \nDet ( D Lorentz ) (156) \nHere we naturally ignore the excitation at initial time because the numerical approach around point 0 could lead to a divergence in the bosonic Retarded correlator. This implies that we must select the Lorentz time such that t ∈ ( -t max , -1) ∪ (1 , t max ).Since the frequency-dependent Green's functions encompass all the information within this system, omitting the zero point will \nG \nG \n0 \n-0.05 \n-0.1 \n2.5 \n2 \n1.5 \n1 \n0.5 \n0 \n-0.5 \n-1 \n-1.5 \n-2 \n-800 \n0.1 \n0.05 \n80 \n90 \n100 \n110 \n120 \n130 \n140 \n150 \n160 \nt \n(c) \n30 \n40 \n50 \n60 \n70 \n80 \n90 \n100 \n110 \nt \nFigure 5: Package solution of Real-time Green's function fixed J = 3 , q = 5 , mu = 0 . 4 , T = 0 . 01 (a) 1st package (b) 2nd package (c) 3rd package (d) 4th package \n<!-- image --> \nnot affect the system's properties, similar to how the Matsubara method was applied in previous works. We have the wave package solution in numerical limit λ max = 100000 and long-time cut off at t max = 40000 has been plotted in Figure.(5). This package fermionic solution gradually returns to 0, allowing us to employ the numerical long-time cutoff technique. One notable feature is that the bosonic component does not return to 0 in the long-time limit. However, it approaches a harmonic solution and becomes analytical, which we consider reliable when the number of oscillations is sufficiently large. \nWe have plotted the 1-5 packages of the solution in Figure.(5). When we introduce an excitation at the initial Lorentz time, this signal information will periodically recover. Additionally, the excitation decays after recovery, exhibiting behavior similar to the dispersion of packages. The first revival excitation in Figure.(5)a occurs around Lorentz time t 1 = 58, and the nu- \n-600 \n-400 \n-200 \n0 \n200 \n400 \n600 \n800 \nt \n(a) \nRe[G \nRe[G \n,AA \n> \nbb,AA \n> \nbb,AA \nIm[G \n> \nIm[G \nRe[G \n> \n> \nb ,LR \nIm[G \n(t)] \n(t)] \n(t)] \n,AA \n(t)] \nb,LR \n(t)] \n(t)] \nRe[G \nRe[G \n,AA \n> \nbb,AA \n> \nbb,AA \nIm[G \n> \nIm[G \nRe[G \n> \n> \nb ,LR \nIm[G \n(t)] \n(t)] \n(t)] \n,AA \n(t)] \nb,LR \n(t)] \n(t)] \n> \n> \nG \n0.1 \n0.05 \n0 \n-0.05 \n-0.1 \nRe[G \nRe[G \n,AA \n> \nbb,AA \n> \nbb,AA \nIm[G \n> \nIm[G \nRe[G \n> \n> \nb ,LR \nIm[G \n(t)] \n(t)] \n(t)] \n,AA \n(t)] \nb,LR \n(t)] \n(t)] \n> \nmerical Green's function results between peaks are | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 219 and | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 212. The second revival excitation occurs around Lorentz time t 2 = 119 in Figure.(5)b, and we have | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 115 and | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 110. The third revival excitation occurs around Lorentz time t 3 = 178, and | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 080, | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 081. There are also several othe revival excitation t 4 = 234 , t 5 = 292 , t 6 = 349 from Figure.(5)a, and the peaks of the Green's functions decays with time. \nFor N=4 SYK model, the real-time processes do not function effectively. We introduce a long-time cutoff following the periodic condition from 0 ∼ β . However, the bosonic ansatz G ϕϕ ∼ 0 . 5 | t | exhibits long-time divergence, rendering this saddle method unreliable directly. Consequently, we will further approximate this problem using the fully diagonal method in the next subsection.", '5.2 Out-of-time-ordered correlator': 'In this subsection, we investigate the causal and chaotic behaviors using the exact diagonal method and out-of-time-ordered correlator, which allows us to examine these characteristics in a finite system. Here, we primarily consider the correlation between different sectors with a non-zero Hamiltonian. \nFirst, as in the previous section, we replace the fermions in the supercharges with creation and annihilation operators \nψ i = C † i + C i . \nThese operators is represented by 2 × 2 Pauli matrices. For the case of SYK model with N fermions on single side, we can represent the theory using a 4 N × 4 N Fock matrix. \nHere we define the regularized out-of-time-ordered correlator. In order to compare the results with previous work[53, 54], we can shift the multi-side OTOCs by 1-F(t) and invert the phase factors \nF ( t ) = 1 -∑ ij Z ( β ) Tr ( ρW ( t ) ρV (0) ρW ( t ) ρV (0)) Tr ( ρ 2 Wρ 2 W ) Tr ( ρ 2 V ρ 2 V ) , (157) \nwhere the partition function Z ( β ) = Tr (exp ( -βH )) and the density of the finite temperature propagator ρ = exp( -βH/ 4). Here the operators W and V comprise the fermions. And W ( t ) represents the time-evolution of W . \nIn N=1 theory, we can then define the fermionic operator \nW ( t ) = exp ( iH † t )( C L † i + C L i ) exp( -iHt ) , V (0) = C R † j + C R j . (158) \nand it can be treated as a special form of N=0 theory. \n<!-- image --> \nFigure 6: Averaged early-time OTOCs with 60 implement µ = 0 . 15 (a) multi-side OTOC between N=2 algebra ψ L and ¯ b R , (b) multi-side OTOC between N=4 algebra ϕ L and F R , \n<!-- image --> \nAnalyzing the off-diagonal term in the Hamiltonian, we thus define the effective N=2 operators \nW ( t ) = exp ( iH † t )( C L † i + C L i ) exp( -iHt ) , V (0) = ¯ b R j = { Q R , ¯ C R † j + ¯ C R j } = { Q R , ¯ c R † j +¯ c R j } . (159) \nHere, we can simplify the chiral representation by the components C L i and ¯ C R † i or alternatively C L † i and ¯ C R i . \nAnd for N=4 theory, the non-zero terms in Hamiltonian are \nW ( t ) = exp ( iH † t ) ( ϕ L i ) exp( -iHt ) = exp ( iH † t )( c † + c ) αL i ( c † + c ) βL i exp( -iHt ) , V (0) = ¯ F Rj = ¯ b αR j ¯ b βR j = { Q, ¯ c † +¯ c } αL j { Q, ¯ c † +¯ c } βR j . (160) \nHere we have plotted the early-time behaviors of the OTOCs in of N=2 and N=4 theory with 3 identical fermions in single-side supercharge. As shown in Figure.6, we calculate the average of 60 implementations for OTOCs. Our result shows that the early-time OTOCs with larger coupling J and inverse temperature β decay decay much slower, as indicated by the slopes. It is also suggested that a smaller Lyapunov exponent should be implied with larger coupling and lower temperature.', '6 Conclusion': "In this work, we have investigated the interaction terms between two identical supersymmetric SYK models. \nWe first consider a special form of N=4 SYK model with a supermultiplet. This model is generated from two identical N=2 SYK models and is constrained by particle conservation, represented by Grassmann integration, which introduces additional auxiliary bosons. Then, we introduce first-order interaction terms in the N=1,2,4 SYK models using superfields, without breaking the boundary supersymmetry and solvability. In the thermal limit, we evaluate Green's function outside of super space. Subsequently, we analyze the free energy and energy. For N=1,2 theories, there also exists a Hawking-Page-like phase transition, which is also found in the N=0 SYK. However, unlike the N=0,1,2 SYK models, the N=4 SYK is primarily a bosonic model, and its phase structure does not involve a wormhole-black hole transition. \nThe N=1,2 effective action in the low energy limit is reparameterized as superconformal and returns to the super Schwarzian. It should go back to the usual SYK when we ignore the supersymmetric part. The N=4 effective theory is also generated from two identical N=2 theories and is superconformal under certain conditions. In this work, the holographic picture of supersymmetric SYK is also discussed. We consider the supersymmetric JT gravity, which is a natural form of the supersymmetrized nAdS. One notable feature is that the great contribution of component ϕ in the IR limit makes the supersymmetric black holes in N=4 undergo an evaporation process. \nWe also elaborated on these coupled N=1,2,4 SYKs regarding the entanglement property with thermal field double. By definition, these supersymmetric TFD states can be generated from the N=0 case. We have verified that the models with interaction are maximally entangled, and this property is not compromised after supersymmetrizing the SYK. Additionally, we verify the causality of the possible wormholes. The real-time revival dynamics are studied in N=2 theory, and we also calculate the exact diagonals of the N=2,4 theories and study the out-oftime-ordered correlator.", 'Acknowledgements': 'This work is supported by NSFC China(Grants No.12275166, No.11805117 and No.11875184)', 'Appendix A: Calculation of N=2,4 thermalfield double state': "N=2 TFD states are similar to N=1 and N=0 states. However, since we have introduced two independent supercharges but only one Hamiltonian constraint, the ground states are no longer parallel vacuum states but depend on the chiral anticommutation of supercharges. But we can still define maximal entangled states and the TFD states. \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -β { Q, ¯ Q } n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 , (A.1) \nwhile states that exhibit a double nature are relevant to chiral supercharges that possess maximal entanglement \n| n ⟩ 1 = ( | Q ⟩ 1 ∣ ∣ ¯ Q 〉 1 ) n = (∣ ∣ ψ + i ¯ ψ 〉 1 ∣ ∣ ψ -i ¯ ψ 〉 1 ) n , \n| ˆ n ⟩ 2 = ( e -iηπΓ 4 e i ( q +¯ q ) ( ϕ -π 2 ) ) P (∣ ∣ ψ + i ¯ ψ 〉 2 ∣ ∣ ψ -i ¯ ψ 〉 2 ) , Γ = ( -1) -q -¯ q + N 2 , (A.2) \nThe equations satisfy the anti-commute relations and we can build the corresponding Hilbert space to a specific type of entanglement \n{ ψ i + i ¯ ψ i , ψ j + i ¯ ψ j } = i ( δ i j + δ j i ) , { ψ i -i ¯ ψ i , ψ j -i ¯ ψ j } = -i ( δ i j + δ j i ) { ψ i -i ¯ ψ i , ψ j + i ¯ ψ j } = i ( δ i j -δ j i ) = 0 , { ψ i + i ¯ ψ i , ψ j -i ¯ ψ j } = -i ( δ i j -δ j i ) = 0 . (A.3) \nAlgebra can be written using creation and annihilation operators \nψ i = c i † + c i ¯ ψ i = ¯ c i † +¯ c i C † = c i † + i ¯ c i , C = c i + i ¯ c † i , ¯ C = c i † -i ¯ c i , ¯ C † = c i -i ¯ c † i , (A.4) \nAnd we can further simplify the algebra with the N=2 anti-commute relation \nψ i = c i † ¯ ψ i = c i C † = (1 + i ) c i † , C = 0 , ¯ C † = 0 , ¯ C = (1 -i ) c i , (A.5) \nwhich could be generated from N=1 symmetry breaking. This is a simple form of supersymmetric algebra, and we can also utilize the chiral conjugate by deleting C † and ¯ C . \nrecalling \nΨ i = ψ i + θ { ¯ Q,ψ i } , ¯ Ψ i = ¯ ψ i + ¯ θ { Q, ¯ ψ i } . (A.7) \nWe can rewrite the interaction terms with complex chiral fields \nH int = iµ∂ θ ( Ψ Li ¯ Ψ R i -Ψ Ri ¯ Ψ L i ) , \nand \n¯ H int = iµ∂ ¯ θ ( ¯ Ψ L i Ψ Ri -¯ Ψ R i Ψ Li ) , (A.8) \nIn N=2 superspace, the fermion algebra involves the anti-commutation relations between the creation and annihilation operators D or ¯ D \nH int = iµe -iϕ ∂ θ ( ψ i + θ { ¯ Q,ψ i }) L ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) R -iµe iϕ ∂ θ ( ψ i + θ { ¯ Q,ψ i }) R ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) L , = iµe -iϕ { ¯ Q,ψ i } L ¯ ψ R i -iµe iϕ { ¯ Q,ψ i } R ¯ ψ L i = iµe -iϕ { ¯ Q, ( c i † + c i )} L (¯ c i † +¯ c i ) R -iµe iϕ { ¯ Q, ( c i † + c i )} R (¯ c i † +¯ c i ) L = iµe -iϕ { ¯ Q, ( C i † + ¯ C i † + C i + ¯ C i ) 2 } L ( C i † -¯ C i † + C i -¯ C i 2 i ) R -iµe iϕ { ¯ Q, ( C i † + ¯ C i † + C i + ¯ C i ) 2 } R ( C i † -¯ C i † + C i -¯ C i 2 i ) L ¯ H int = iµe -iϕ ∂ ¯ θ ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) L ( ψ i + θ { ¯ Q,ψ i }) R -iµe iϕ ∂ ¯ θ ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) R ( ψ i + θ { ¯ Q,ψ i }) L . = iµe -iϕ { Q, ¯ ψ i } L ψ iR -iµ { Q, ¯ ψ i } R ψ iL = iµe -iϕ { Q, (¯ c i † +¯ c i ) } L ( c i † + c i ) R -iµe iϕ { Q, (¯ c i † +¯ c i ) } R ( c i † + c i ) L = iµe -iϕ { Q, C i † -¯ C i † + C i -¯ C i 2 i } L ( ( C i † + ¯ C i † + C i + ¯ C i ) 2 ) R -iµe iϕ { Q, C i † -¯ C i † + C i -¯ C i 2 i } R ( ( C i † + ¯ C i † + C i + ¯ C i ) 2 ) L (A.9) \n〈 ˆ n -q ' ∣ ∣ ∣ C R i ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q -¯ q + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ C L i ∣ ∣ ∣ n q ' 〉 , 〈 ˆ n -q ' ∣ ∣ ∣ ¯ C R i ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q -¯ q + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ ¯ C L i ∣ ∣ ∣ n q ' 〉 , P -1 C i P = ηP † i . (A.6) \nAnd we can still simplify them into an algebraic form \nH int = iµe -iϕ { ¯ Q, C i † 2 } L ( C i 2 i ) R -iµe -iϕ { ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) R -iµe iϕ { ¯ Q, C i † 2 } R ( C i 2 i ) L + iµe iϕ { ¯ Q, ¯ C i † 2 } R ( ¯ C i 2 i ) L ¯ H int = iµe -iϕ { Q, C i † 2 i } L ( C i 2 ) R -iµe -iϕ { Q, ¯ C i † 2 i } L ( ¯ C i 2 ) R -iµe iϕ { Q, C i † 2 i } R ( C i 2 ) L + iµe iϕ { Q, ¯ C i † 2 i } R ( ¯ C i 2 ) L (A.10) \nSince N=2 Hamiltonian eigenstates are created by applying the appropriate creation and annihilation operators \nH ∣ ∣ Ψ r m , ¯ Ψ r n 〉 = { Q, ¯ Q }∣ ∣ Ψ r m , ¯ Ψ r n 〉 = E mn ∣ ∣ Ψ r m , ¯ Ψ r n 〉 , H † ∣ ∣ ∣ Ψ l m , ¯ Ψ l n 〉 = { Q, ¯ Q } † ∣ ∣ Ψ r m , ¯ Ψ r n 〉 = E ∗ mn ∣ ∣ ∣ Ψ l m , ¯ Ψ l n 〉 , (A.11) \nwith \n∑ n,k ∣ ∣ ∣ Ψ l n , ¯ Ψ l k 〉 〈 Ψ r n , ¯ Ψ r k ∣ ∣ = I, 〈 Ψ l n , ¯ Ψ l k ∣ ∣ ∣ Ψ r m , ¯ Ψ r l 〉 = δ nm δ kl . (A.12) \nAnd we can further propose the ground states \n∣ ∣ Ψ 0 , ¯ Ψ 0 〉 = N ∏ j ( 1 √ 2 | 1 ⟩ L,j | 0 ⟩ R,j + i √ 2 | 0 ⟩ L,j | 1 ⟩ R,j ) = N ∏ j ∑ abcd ( 1 √ 2 | a ± 1 , a ⟩ L,j | b, b ⟩ R,j + i √ 2 | c, c ⟩ L,j | d ± 1 , d ⟩ R,j ) , (A.13) \nwith \nH † int ∣ ∣ ∣ Ψ l 0 , ¯ Ψ l 0 〉 = -µN ( 1 + θ ¯ Q 0 ) ( 1 + ¯ θQ 0 ) ∣ ∣ ∣ Ψ l 0 , ¯ Ψ l 0 〉 , H int ∣ ∣ Ψ r 0 , ¯ Ψ r 0 〉 = -µN ( 1 + θ ¯ Q 0 ) ( 1 + ¯ θQ 0 ) ∣ ∣ Ψ r 0 , ¯ Ψ r 0 〉 . (A.14) \nThe Q 0 is a constant which is relevant to the ground state supercharges. And we use abcd to represent the excited states with abcd energy levels in each supercharge Q L , ¯ Q L and Q R , ¯ Q R . We denote Q = ¯ Q in the N=2 theory in vacuum, while the excitation of Q is higher than ¯ Q by one, these are the first excited states, vice versa. \nWe have imposed the entangled states in the N=2 TFD \n| m q ⟩ 1 ⊗| ¯ m -q ⟩ 2 = N ∏ j ∑ q ∑ kl ( | k q ± m q , k q ⟩ 1 ,j | l -q ± m -q , l -q ⟩ 2 ,j ) (A.15) \nIndicate that \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1 ( iµ ∑ i ( e -iϕ { ¯ Q, C i † 2 } L ( C i 2 i ) R -iµe -iϕ { ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) R -iµe iϕ { ¯ Q, C i † 2 } R ( C i 2 i ) L + iµe iϕ { ¯ Q, ¯ C i † 2 } R ( ¯ C i 2 i ) L )) | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q -¯ q + N 2 ) 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i 2 i ) R | ˆ m -q ⟩ -iµe iπ ( -q ' -¯ q ' + N 2 ) e iϕ 〈 ˆ n -q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } R ∣ ∣ ∣ n ' q ''' 〉 〈 n q ' ∣ ∣ ∣ ( C i 2 i ) L | m q ⟩ + iµe iπ ( -q ' -¯ q ' + N 2 ) e iϕ 〈 ˆ n -q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } R ∣ ∣ ∣ n ' q ''' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i 2 i ) L | m q ⟩ ) (A.16) \nand \n⟨ \nH int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i {-iµe -iϕ 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L | m q ⟩ ⟨ m q | ( C i 2 i ) L ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L | m q ⟩ ⟨ m q | ( ¯ C i 2 i ) L ∣ ∣ ∣ n q ' 〉 + iµe iϕ ⟨ m q | { ¯ Q, C i † 2 } L ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i 2 i ) L | m q ⟩ -iµe iϕ ⟨ m q | { ¯ Q, ¯ C i † 2 } L ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i 2 i ) L | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L | m q ⟩ ⟨ m q | ( C i 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L | m q ⟩ ⟨ m q | ( ¯ C i 2 i ) L ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 { ¯ Q, C i † 2 } L ( C i 2 i ) L -{ ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) L | n q ⟩ 1 = Const. \n(A.17) \nThe Hamiltonian of N=4 SYK model is \nH = ϵ αβ ψ α ψ β { ¯ Q α , ψ α } 2 ϵ αβ ψ α ψ β + ϵ αβ ψ α ψ β { ¯ Q α , ψ α } ψ α { ¯ Q α , ψ α } ψ β ϵ αβ + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + T { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c (A.18) \nTherefore, the N=4 supersymmetric SYK thermal field double is defined as \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -β { ¯ Q α , Q α } { ¯ Q β , Q β } n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 , (A.19) \nThe Hilbert space is also contains the eigenvectors of the supercharges \n| n ⟩ 1 = ( | Q ⟩ α 1 ∣ ∣ ¯ Q 〉 α 1 | Q ⟩ β 1 ∣ ∣ ¯ Q 〉 β 1 ) n ∼ ( ∣ ∣ ψ + i ¯ ψ 〉 α 1 ∣ ∣ ψ -i ¯ ψ 〉 α 1 ∣ ∣ ψ + i ¯ ψ 〉 β 1 ∣ ∣ ψ -i ¯ ψ 〉 β 1 ) n , | ˆ n ⟩ 2 = ( e -iηπΓ 4 e i ( q α +¯ q α + q β +¯ q β )( ϕ -π 2 ) ) P ( ∣ ∣ ψ + i ¯ ψ 〉 α 2 ∣ ∣ ψ -i ¯ ψ 〉 α 2 ∣ ∣ ψ + i ¯ ψ 〉 β 2 ∣ ∣ ψ -i ¯ ψ 〉 β 2 ) , Γ = ( -1) -q α -¯ q α -q β -¯ q β + N 2 , (A.20) \nHere we have defined sets of new eigenvectors, which are also the eigenvectors of Hamiltonian. It is straightforward to verify that these new eigenvectors satisfy the properties of being eigenvectors of the Hamiltonian \n{ ψ i α + i ¯ ψ αi , ψ j β + i ¯ ψ βj } = i ( δ i j + δ j i ) δ αβ , { ψ i α -i ¯ ψ αi , ψ j β -i ¯ ψ βj } = -i ( δ i j + δ j i ) δ αβ { ψ i α -i ¯ ψ αi , ψ j β + i ¯ ψ βj } = i ( δ i j -δ j i ) δ αβ = 0 , { ψ i α + i ¯ ψ αi , ψ j β -i ¯ ψ βj } = -i ( δ i j -δ j i ) δ αβ = 0 . (A.21) \nWe can also rewrite these operators in the Hilbert space \nψ i α = c i † α + c i α ¯ ψ αi = ¯ c † αi +¯ c αi C i α † = c i † α + i ¯ c αi , C αi = c i α + i ¯ c † αi , ¯ C i † α = c i † α -i ¯ c αi , ¯ C i α = c i α -i ¯ c † αi , (A.22) \nwith \n〈 ˆ n -q ' ∣ ∣ ∣ C R αi ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ C L αi ∣ ∣ ∣ n q ' 〉 , 〈 ˆ n -q ' ∣ ∣ ∣ ¯ C R αi ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ ¯ C L αi ∣ ∣ ∣ n q ' 〉 , P -1 C i P = ηP † i . (A.23) \nWe can further introduce the interaction Hamiltonian into the superspace \nΦ = ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β } ¯ Φ = ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β } (A.24) \nThe interaction Hamiltonian is defined \nH int = ∂ 2 θ ( Φ Li ¯ Φ R i + Φ Ri ¯ Φ L i ) ¯ H int = ∂ 2 ¯ θ ( ¯ Φ L i Φ Ri + ¯ Φ R i Φ Li ) , (A.25) \nand \nH int = e -iϕ ∂ 2 θ ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R + e iϕ ∂ 2 θ ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) L , = iµe -iϕ ( { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L ( ¯ ψ α ¯ ψ β ) R + iµe iϕ ( { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R ( ¯ ψ α ¯ ψ β ) L = iµe -iϕ ({ ¯ Q α , ( c i † α + c i α )}{ ¯ Q β , ( c i † β + c i β )}) L (( ¯ c † αi +¯ c αi )( ¯ c † βi +¯ c βi )) R + iµe iϕ ({ ¯ Q α , ( c i † α + c i α )}{ ¯ Q β , ( c i † β + c i β )}) R (( ¯ c † αi +¯ c αi )( ¯ c † βi +¯ c βi )) L = iµe -iϕ ({ ¯ Q α , ( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )}{ ¯ Q β , ( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )}) L (( C i † α -¯ C i † α + C i α -¯ C i α 2 i )( C i † β -¯ C i † β + C i β -¯ C i β 2 i )) R + iµe iϕ ({ ¯ Q α , ( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )}{ ¯ Q β , ( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )}) R (( C i † α -¯ C i † α + C i α -¯ C i α 2 i )( C i † β -¯ C i † β + C i β -¯ C i β 2 i )) L (A.26) \nAnd \n¯ H int = e -iϕ ∂ 2 θ ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) L ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R + e iϕ ∂ 2 θ ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L , = iµe -iϕ ( { Q α , ¯ ψ α } α { Q β , ¯ ψ β }) L ( ψ α ψ β ) R + iµe iϕ ( { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R ( ψ α ψ β ) L = iµe -iϕ ({ ¯ Q α , ( ¯ c † αi +¯ c αi )}{ ¯ Q β , ( ¯ c † βi +¯ c βi )}) L (( c i † α + c i α )( c i † β + c i β )) R + iµe iϕ ({ ¯ Q α , ( ¯ c † αi +¯ c αi )}{ ¯ Q β , ( ¯ c † βi +¯ c βi )}) R (( c i † α + c i α )( c i † β + c i β )) L = iµe -iϕ ({ ¯ Q α , ( C i † α -¯ C i † α + C i α -¯ C i α 2 i )}{ ¯ Q β , ( C i † β -¯ C i † β + C i β -¯ C i β 2 i )}) L (( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )) R + iµe iϕ ({ ¯ Q α , ( C i † α -¯ C i † α + C i α -¯ C i α 2 i )}{ ¯ Q β , ( C i † β -¯ C i † β + C i β -¯ C i β 2 i )}) R (( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )) L (A.27) \nIn this section, we aim to investigate whether the interaction Hamiltonian affects the supersymmetry thermal field double states, and we should concentrate on the diagonal terms within the Hilbert space \nH int = iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) R + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) R + iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( C i α 2 i C i β 2 i ) L + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( ¯ C i α 2 i ¯ C i β 2 i ) L -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( C i α 2 i ¯ C i β 2 i ) L -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( ¯ C i α 2 i C i β 2 i ) L (A.28) \n¯ H int = iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , C i † β 2 i }) L ( C i α 2 C i β 2 ) R + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) L ( ¯ C i α 2 ¯ C i β 2 ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) L ( C i α 2 ¯ C i β 2 ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , C i † β 2 i }) L ( ¯ C i α 2 C i β 2 ) R + iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , C i † β 2 i }) R ( C i α 2 C i β 2 ) L + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) R ( ¯ C i α 2 ¯ C i β 2 ) L -iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β ¯ C i † β 2 i }) R ( C i α 2 ¯ C i β 2 ) L -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , C i † β 2 i }) R ( ¯ C i α 2 C i β 2 ) L (A.29) \nThen the turning terms give \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1 iµ ∑ i e -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) R + iµe iϕ ({ ¯ Q α C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( C i α 2 i C i β 2 i ) L + iµe iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( ¯ C i α 2 i ¯ C i β 2 i ) L -iµe iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( C i α 2 i ¯ C i β 2 i ) L -iµe iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( ¯ C i α 2 i C i β 2 i ) L | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) R | ˆ m -q ⟩ + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) R | ˆ m -q ⟩ + iµe iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) L | ˆ m -q ⟩ + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) L | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) L | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) L | ˆ m -q ⟩ (A.30) \n⟨ H int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i [ -iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 -iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i ¯ C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i ¯ C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 -iµe iϕ ⟨ m q | ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) L | m q ⟩ -iµe iϕ ⟨ m q | ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) L | m q ⟩ + iµe -iϕ ⟨ m q | ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) L | m q ⟩ + iµe -iϕ ⟨ m q | ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) L | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i ¯ C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i ¯ C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) L + ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) L | n q ⟩ 1 -({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) L | n q ⟩ 1 -({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) L | n q ⟩ 1 = Const. (A.31) 54 \nwhere \n∑ n,q ∑ i ⟨ n q | { ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }( C i α 2 i C i β 2 i ) | n q ⟩ = -1 4 ∑ n,q N ( Q n q + Q n q +1 ) 2 (A.32) \nWe can easily verify that the chiral-conjugate component satisfies 〈 ¯ H int 〉 = Const by calculating its value directly from the definitions.", 'References': "- [1] J. Maldacena, and D. Stanford, 'Remarks on the sachdev-ye-kitaev model' Physical Review D 94.10 (2016): 106002.\n- [2] S. Sachdev and J.W. Ye, 'Gapless spin-fluid ground state in a random quantum Heisenberg magnet',Phys. Rev. Lett. 70, 3339342 (1993).\n- [3] S. Sachdev, 'Bekenstein-hawking entropy and strange metals',Phys. Rev. X 5, 041025 (2015).\n- [4] A. Kitaev, 'A simple model of quantum holography',(2015), KITP Strings Seminar and Entanglement Program.\n- [5] J. Maldacena, D. Stanford, and Z. Yang, 'Conformal symmetry and its breaking in twodimensional nearly anti-de Sitter space' Progress of Theoretical and Experimental Physics 2016.12 (2016): 12C104.\n- [6] J. Maldacena. 'Eternal black holes in anti-de Sitter.' Journal of High Energy Physics 2003.04 (2003): 021.\n- [7] D.Anninos, T. Anous, and F. Denef. 'Disordered quivers and cold horizons.' Journal of High Energy Physics 2016.12 (2016): 1-23.\n- [8] J. Maldacena, and L. Susskind. 'Cool horizons for entangled black holes.' Fortschritte der Physik 61.9 (2013): 781-811.\n- [9] W. Fu, D. Gaiotto, J. Maldacena, S. Sachdev. 'Supersymmetric sachdev-ye-kitaev models.' Physical Review D 95.2 (2017): 026009.\n- [10] J. Murugan, D. Stanford, and E. Witten. 'More on Supersymmetric and 2d Analogs of the SYK Model.' Journal of High Energy Physics 2017.8 (2017): 1-99.\n- [11] E. Marcus, and S. Vandoren. 'A new class of SYK-like models with maximal chaos.' Journal of High Energy Physics 2019.1 (2019): 1-26. \n- [12] S. Forste, and I. Golla. 'Nearly AdS2 SUGRA and the super-Schwarzian.' Physics Letters B 771 (2017): 157-161.\n- [13] S. Forste, J. Kames-King, and M. Wiesner. 'Towards the holographic dual of N = 2 SYK.' Journal of High Energy Physics 2018.3 (2018): 1-21.\n- [14] S. Forste, A. Gerhardus, and J. Kames-King. 'Supersymmetric black holes and the SJT/nSCFT1 correspondence.' Journal of High Energy Physics 2021.1 (2021): 1-44.\n- [15] C. Peng, M. Spradlin, and A. Volovich. 'Correlators in the N = 2 supersymmetric SYK model.' Journal of High Energy Physics 2017.10 (2017): 1-28.\n- [16] C. Peng, and S. Stanojevic. 'Soft modes in N = 2 SYK model.' Journal of High Energy Physics 2021.1 (2021): 1-42.\n- [17] K. Bulycheva. ' N = 2 SYK model in the superspace formalism.' Journal of High Energy Physics 2018.4 (2018): 1-45.\n- [18] M. Heydeman, GJ. Turiaci, and W. Zhao. 'Phases of N = 2 Sachdev-Ye-Kitaev models.' Journal of High Energy Physics 2023.1 (2023): 1-53.\n- [19] M. Heydeman, LV. Iliesiu, GJ. Turiaci, W. Zhao. 'The statistical mechanics of near-BPS black holes.' Journal of Physics A: Mathematical and Theoretical 55.1 (2021): 014004.\n- [20] M. Berkooz, V. Narovlansky, and H. Raj. 'Complex Sachdev-Ye-Kitaev model in the double scaling limit.' Journal of High Energy Physics 2021.2 (2021): 1-62.\n- [21] M. Berkooz, M. Isachenkov, V. Narovlansky, G. Torrents. 'Towards a full solution of the large N double-scaled SYK model.' Journal of High Energy Physics 2019.3 (2019): 1-72.\n- [22] M. Berkooz, N. Brukner, V. Narovlansky, A. Raz. 'The double scaled limit of supersymmetric SYK models.' Journal of High Energy Physics 2020.12 (2020): 1-64.\n- [23] HW. Lin, and D. Stanford. 'A symmetry algebra in double-scaled SYK.' SciPost Physics 15.6 (2023): 234.\n- [24] J. Boruch, HW. Lin, and C. Yan. 'Exploring supersymmetric wormholes in N = 2 SYK with chords.' Journal of High Energy Physics 2023.12 (2023): 1-48.\n- [25] HW. Lin. 'The bulk Hilbert space of double scaled SYK.' Journal of High Energy Physics 2022.11 (2022): 1-33.\n- [26] J. Maldacena, and X. Qi, 'Eternal traversable wormhole', arXiv:1804.00491 (2018).\n- [27] X. Qi and P. Zhang,'The coupled SYK model at finite temperature.' Journal of High Energy Physics 2020.5 (2020): 1-14. \n- [42] A. Biggs, J. Maldacena, and V. Narovlansky. 'A supersymmetric SYK model with a curious low energy behavior.' arxiv preprint arXiv:2309.08818 (2023).\n- [43] S. Cao, Y.C. Rui, and X.H. Ge, 'Thermodynamic phase structure of complex SachdevYe-Kitaev model and charged black hole in deformed JT gravity', arXiv preprint arXiv:2103.16270 (2021).\n- [44] AH, Chamseddine, 'Superstrings in arbitrary dimensions.' Physics Letters B 258.1-2 (1991): 97-103.\n- [45] D. Grumiller, P. van Nieuwenhuizen. 'Holographic counterterms from local supersymmetry without boundary conditions.' Physics Letters B 682.4-5 (2010): 462-465.\n- [46] P. Gao, DL Jafferis, AC. Wall. 'Traversable wormholes via a double trace deformation.' Journal of High Energy Physics 2017.12 (2017): 1-25.\n- [47] O. Aharony, M. Berkooz, B. Katz. 'Non-local effects of multi-trace deformations in the AdS/CFT correspondence.' Journal of High Energy Physics 2005.10 (2005): 097.\n- [48] W. Cai, S. Cao, XH. Ge, M. Matsumoto, SJ Sin. 'Non-Hermitian quantum system generated from two coupled Sachdev-Ye-Kitaev models.' Physical Review D 106.10 (2022): 106010.\n- [49] W. Cai, XH, Ge, and GH Yang. 'Diffusion in higher dimensional SYK model with complex fermions.' Journal of High Energy Physics 2018.1 (2018).\n- [50] W. Cai, XH, Ge. 'Superconducting gap ratio from strange metal phase in the absence of quasiparticles.' Communications in Theoretical Physics 73.2 (2021): 025701.\n- [51] C, Zhang, and W. Cai. ' T ¯ T deformation on non-Hermitian two coupled SYK model.' arxiv preprint arXiv:2312.03433 (2023).\n- [52] H, Yuan, XH, Ge, KY Kim . 'Pole-skipping in two-dimensional de Sitter spacetime and double-scaled SYK model' arxiv preprint arXiv:2408.12330 (2024).\n- [53] S. He, P. Hang, C. Lau, Z.Y. Xian, L. Zhao, 'Quantum chaos, scrambling and operator growth in T ¯ T deformed SYK models', Journal of High Energy Physics 2022.12 (2022): 1-32.\n- [54] S, Cao, and XH, Ge. 'Excitation transmission through a non-Hermitian traversable wormhole.' Physical Review D 110.4 (2024): 046022."}

2024arXiv240910317M

The Serkowski relation is the cornerstone of studies of starlight polarization as a function of wavelength. Although empirical its extensive use since its inception to describe polarization induced by interstellar dust has elevated the relation to the status of an indisputable law serving as the benchmark for validating interstellar dust grain models. We revisit the effects of the 3D structure of the interstellar medium ISM on the wavelength dependence of interstellar polarization. We use analytical models to show how the wavelength dependence of both the polarization fraction and direction is affected by the presence of multiple clouds along the line of sight LOS accounting for recent developments in dust distribution modelling and utilizing an expanded archive of stellar polarization measurements. We highlight concrete examples of stars whose polarization profiles are severely affected by LOS variations of the dust grain and magnetic field properties and we provide a recipe to accurately fit multiple cloud Serkowski models to such cases. We present for the first time compelling observational evidence that the 3D structure of the magnetized ISM often results to the violation of the Serkowski relation. We show that 3D effects impact interstellar cloud parameters derived from Serkowski fits. In particular the dust size distribution in single cloud sightlines may differ from analyses that ignore 3D effects with important implications for dust modelling in the Galaxy. Our results suggest that multiband stellar polarization measurements offer an independent probe of the LOS variations of the magnetic field constituting a valuable new tool for the 3D cartography of the ISM. We caution that unless 3D effects are explicitly accounted for a poor fit to the Serkowski relation does not by itself constitute conclusive evidence that a star is intrinsically polarized.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.10317', 'arXiv:2409.10317', '2024arXiv240910317M']

['Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']

3D ISM structure challenges the Serkowski relation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10317.pdf

{'3D ISM structure challenges the Serkowski relation': 'N. Mandarakas 1 , 2, ⋆ , K. Tassis 1 , 2 , and R. Skalidis 3 \n- 1 Department of Physics, University of Crete, Vasilika Bouton, 70013 Heraklion, Greece\n- 2 Institute of Astrophysics, Foundation for Research and Technology - Hellas, 100 Nikolaou Plastira, Vassilika Vouton, 70013 Heraklion, Greece\n- 3 Owens Valley Radio Observatory, California Institute of Technology, MC 249-17, Pasadena, CA 91125, USA \nReceived -; accepted -', 'ABSTRACT': 'Context. The Serkowski relation is the cornerstone of studies of starlight polarizations as a function of wavelength. Although empirical, its extensive use since its inception by Serkowski et al. (1975) to describe polarization induced by interstellar dust has elevated the relation to the status of an indisputable \'law": this is the benchmark against which models of interstellar dust grains are validated. Aims. We revisit the e ff ects of the three - dimensional structure of the interstellar medium (ISM) on the wavelength dependence of \ninterstellar polarization. \nMethods. We use analytical models to show how the wavelength dependence of both the polarization fraction and direction is a ff ected by the presence of multiple clouds along the line of sight (LOS). We account for recent developments in dust distribution modelling and we utilize an expanded archive of stellar polarization measurements to establish the e ff ect of multiple clouds along the LOS. We highlight concrete examples of stars whose polarization profiles are severely a ff ected by LOS variations of the dust grain and magnetic field properties, and we provide a recipe to accurately fit multiple cloud Serkowski models to such cases. \nResults. We present, for the first time, compelling observational evidence that the three - dimensional structure of the magnetized ISM often results to the violation of the Serkowski relation. We show that three - dimensional e ff ects impact interstellar cloud parameters derived from Serkowski fits. In particular, the dust size distribution in single - cloud sightlines may di ff er from analyses that ignore 3D e ff ects, with important implications for dust modelling in the Galaxy. \nConclusions. Our results suggest that multiwavelength stellar polarization measurements o ff er an independent probe of the LOS variations of the magnetic field orientation, and thus constitute a potentially valuable new tool for the 3D cartography of the interstellar medium. Finally, we caution that, unless 3D e ff ects are explicitly accounted for, a poor fit to the Serkowski relation does not, by itself, constitute conclusive evidence that a star is intrinsically polarized. \nKey words. ISM, polarization, etc', '1. Introduction': 'Dust is ubiquitous in the interstellar medium (ISM) of our Galaxy and plays a significant role in many astrophysical processes. The thermal (gray-body) emission of interstellar dust spans many frequencies (100 - 1000 GHz), making dust a powerful tracer of the morphology of ISM structures as well as a diagnostic of their physical properties, such as temperature (Hildebrand 1983; André et al. 2010). At the same time dust emission poses a challenge for cosmological experiments because it veils the cosmic microwave background signal over a wide range of frequencies (Planck Collaboration et al. 2016). Due to the multifaceted role of interstellar dust in astrophysics and cosmology, a detailed understanding of its nature is imperative. \nThe interaction of aspherical dust grains with the magnetic field permeating the ISM leads to their alignment, resulting in the polarization of light, which can be observed from ultraviolet to sub-mm wavelengths (Andersson et al. 2015). Short-wavelength (UV, optical, near-IR) polarization is induced by dichroic extinction of the starlight (which usually starts out unpolarized) by aligned dust grains. The same grains emit long-wavelength (farIR, and sub-mm) polarized thermal radiation. Dust polarization provides one of the most reliable and widely employed methods for tracing the properties of the ISM magnetic fields (e.g., Ska- \nlidis & Tassis 2021; Skalidis et al. 2021, 2022; Pattle et al. 2023; Pelgrims et al. 2023, 2024). \nDust-induced stellar polarization follows a complex empirical relation with wavelength, known as the \'Serkowski relation" (Serkowski et al. 1975). It expresses the polarization fraction P as a function of wavelength λ through the formula \nP ( λ ) = Pmax · exp GLYPH<18> -K · ln 2 λ max GLYPH<19> , (1) \nλ \nwith the use of three free parameters: Pmax the maximum polarization fraction observed at λ max , and K that quantifies the spread of the profile. The polarization angle (also referred to as electric vector position angle, EVPA, denoted here as θ ) is implicitly assumed to be constant with wavelength. \nSince its discovery by Serkowski et al. (1975), this relation has been extensively studied and confirmed by a number of authors (Wilking et al. 1980; Whittet et al. 1992, 2001; Bagnulo et al. 2017; Cikota et al. 2018). It is so well established that it is often dubbed the \'Serkowski law". It is even a standard practice in the literature to use it to extrapolate the polarization fraction in some wavelength from measurements made at another wavelength (Bijas et al. 2024; Neha et al. 2024; Ginski et al. 2024). Most importantly, the Serkowski relation with its complicated shape provides stringent constraints for dust grain models (Guillet et al. 2018; Hensley & Draine 2021; Draine 2024), while in emis- \nsion, the degree of polarization is relatively flat across frequencies (Hensley & Draine 2023). \nThe Serkowski relation applies for single-cloud lines of sight (LOSs), and to date, whenever it is applied, it is implicitly assumed that the observed (integrated) P ( λ ) signal is induced by a single dominant polarization screen (cloud) along the LOS. Reality, however, is more complicated. G aia has opened up the third dimension of the sky (depth, Gaia Collaboration et al. 2018), and since then it has become clear that in the majority of the LOSs there are more than one clouds with comparable contributions in the total dust column (e.g., Green et al. 2019; Edenhofer et al. 2024). The variations of the magnetic field along the LOS are imprinted on the individual stellar polarization measurements, allowing the tomographic mapping of the ISM magnetic field (Pelgrims et al. 2024). \nThe impact of multiple polarizing clouds along the LOS on stellar polarization has been discussed in the literature, though not extensively. Most of the focus has been on the e ff ect on the EVPA, with less attention given to the polarization fraction spectral profile. During the early stages of interstellar polarization (ISP) studies, when the notion of a wavelength-independent EVPA emerged from observational evidence, Treanor (1963) briefly mentioned the possibility of a non-constant EVPA in scenarios involving multiple clouds with di ff erent magnetic field alignment and / or di ff erent grain sizes. Coyne & Gehrels (1966) attributed the observed variability in EVPA with wavelength among their targets to the potential presence of multiple clouds along the LOS. Martin (1974) was the first to construct mathematical models to describe such scenarios, involving multiple clouds with di ff ering properties along the LOS. Some authors focused on the impact of multiple dust components in the LOS on the parameters derived from Eq. 1. For example, Codina-Landaberry & Magalhaes (1976) explored the behavior of the K parameter in a medium with changing grain alignment, while Clarke & Al-Roubaie (1984) investigated how the presence of two clouds along the LOS would influence the parameters K and λ max with simplistic models, finding no firm relationship between their models and contemporary observations. \nWhile modern researchers occasionally reference these earlier works most of the focus remains on the e ff ect of a two-cloud scenario on the EVPA (e.g., McMillan & Tapia 1977; Messinger et al. 1997; Clarke 2010; Whittet 2015; Bagnulo et al. 2017; Whittet 2022). Little attention is given to the broader consequences of such scenarios, in the P -λ profile 1 or the derived dust parameters. \nIn this article, we revisit the e ff ects of the three-dimensional structure of the ISM on the wavelength dependence of ISP, accounting for recent developments in dust distribution modelling and utilizing an expanded archive of polarimetric measurements. We use analytical models to show how both the P -λ and θ -λ profiles are expected to behave in scenarios with multiple clouds along the LOS. For the first time, we present compelling observational evidence to demonstrate these e ff ects and disentangle the polarizing components within the LOS. Ultimately, we demonstrate that the LOS variations of dust and magnetic field properties a ff ect the Serkowski relation in two ways: 1) they challenge the universality of the relation, because Eq. 1 can be a poor fit to the data when 3D e ff ects are significant; and 2) even when Eq. 1 fits the data well, the best-fit parameters may not be representative of dust physics. Conversely, deviations of stellar polarization spectral profiles could potentially be utilized as a diagnostic of the 3D complexity of the magnetized ISM.', '2. Theoretical expectations': 'In the case of multiple dust clouds along the LOS, initially unpolarized star light is a ff ected sequentially by all of them, until it reaches the observer. In the limit of low polarization the normalized Stokes parameters q = Q / I , u = U / I of the transversing light, are additive (e.g. Patat et al. 2010). Therefore the observed q , u in a given wavelength of an intrinsically unpolarized source lying behind NC number of clouds will be \nqobs = NC X i = 1 qi (2) \nand similarily for u , with qi being the q induced by the i -th dust cloud in the LOS. \nIn order to explore this e ff ect, we created analytical models for the cases of 1 and 2 dust clouds along the LOS, for di ff erent parameters of the clouds. Given that the Serkowski relation is an intrinsic property of individual ISM clouds, variations in grain sizes would be manifested by variations in λ max , which is proportional to the average size of aligned grains (Draine 2024). Variations in the plane-of-sky (POS) morphology of the magnetic field are traceable through the polarization angle θ . Variations in λ max and in θ between two or more clouds along the LOS significantly a ff ect the cumulative (observed) polarization signal. For our analytical models we used Eq.1 to calculated Pi ( λ ) for 0 . 1 µ m ≤ λ ≤ 1 . 2 µ m , selecting typical values for Pmax and λ max and fixing K = 1 . 68 · λ max (Wilking et al. 1980). We also assigned a θ i value (constant with wavelength) representing the average POS magnetic field direction within the cloud. Thus, for each cloud i we have qi ( λ ) = Pi ( λ ) · cos 2 θ i and ui ( λ ) = Pi ( λ ) · sin 2 θ i . For the combined model, we calculated qobs ( λ ), uobs ( λ ) from Eq. 2 and finally ˆ P ( λ ), ˆ θ ( λ ) as \nˆ P ( λ ) = q q 2 obs ( λ ) + u 2 obs ( λ ) (3a) \nˆ θ ( λ ) = 1 2 · tan -1 uobs ( λ ) qobs ( λ ) ! , (3b) \nwhere we used the 2 - argument arctan function. \nIn Fig. 1 we present some examples of P -λ and θ -λ profiles resulting from the presence of one or two clouds in the LOS, as explored by our analytical models. The simple case of a single dust cloud (Serkowski relation - Eq.1) is shown in the left panel of Fig. 1 for reference. Three cases with 2 clouds in the LOS are shown in the middle left, middle right, and right panels of Fig. 1. Specifically, the middle left panel demonstrates an example where both the P -λ and θ -λ curves resemble the Serkowski relation, although the profiles are constructed by two components. The middle right panel demonstrates an example where the P -λ curve resembles the Serkowski relation, yet the θ -λ profile deviates from constancy. The right panel demonstrates an example where both the P -λ and the θ -λ profiles deviate from the Serkowski curves. \nAssuming a di ff erence of a factor of two in λ max between our two hypothetical clouds yields a bimodal P ( λ ), which is significantly di ff erent than the shape of a typical Serkowski curve (Fig. 1 upper right panel). Although the di ff erence appears to be significant, it would be challenging to identify it in practice, because multi-band polarization measurements are usually limited to a narrow range of frequencies. \nModest LOS variations in λ max can result in even more complications. In our assumed case of two clouds, a 40% di ff erence \nFig. 1: Demonstration of the e ff ect of 3D structure in the Serkowski realtion, using analytical models for P -λ and θ -λ curves. In each case, the solid black curve corresponds to the combined model and the dashed red and dotted blue lines corresponds to individual clouds. Left column : The simple case of 1 cloud (Eq. 1). Middle left column : An example with 2 Clouds in the LOS, where both the combined P -λ , and θ -λ seem to follow the Serkowski relation. Middle right column : An example with 2 Clouds in the LOS, where the combined P -λ seems to follow the Serkowski relation, but θ -λ not. Right column : An example with 2 Clouds in the LOS where both P -λ and θ -λ profiles deviate from Serkowski expectations. The parameters for each model are: Left : Pmax = 1%, λ max = 0 . 6 µ m , θ = 30 o . Middle left : Pmax 1 = 1 . 5%, λ max 1 = 0 . 55 µ m , θ 1 = 30 o , Pmax 2 = 0 . 7%, λ max 2 = 0 . 5 µ m , θ 2 = 35 o . Middle right : Pmax 1 = 0 . 7%, λ max 1 = 0 . 5 µ m , θ 1 = 0 o , Pmax 2 = 0 . 8%, λ max 2 = 0 . 7 µ m , θ 2 = 60 o . Right : Pmax 1 = 1 . 3%, λ max 1 = 0 . 4 µ m , θ 1 = 0 o , Pmax 2 = 1 . 4%, λ max 2 = 0 . 8 µ m , θ 2 = 75 o . For all clouds we have used K = 1 . 68 · λ max (Wilking et al. 1980). \n<!-- image --> \nin λ max between the two clouds leads to nearly identical to the Serkowski curve profiles (middle right panel in Fig. 1), although the cumulative signal by construction consists of two underlying profiles. In this case, erroneously assuming a single cloud along the LOS would lead to inaccurate constraints for the parameters of the Serkowski relation, and hence for the grain properties. In cases where the λ max and θ parameters are similar between clouds (middle left panel in Fig. 1), the resulting profiles of both polarization and EVPA are indistinguishable from the single cloud case within observational uncertainties, given the current accuracy of optical polarimeters. Significant di ff erences in the EVPAs of two clouds lying along the LOS, such as the one in the employed models which is close to 80 degrees, can be revealed through the variation of the EVPA as a function of λ (bottom middle right and right panels in Fig. 1). However, obtaining a su ffi cient number of data points that would allow us to probe such variations is challenging 2 , and for this reason these data are sparse. \nThese analytic calculations pose a challenge in the interpretation of P ( λ ) observations: while the Serkowski relation would still hold for individual clouds, the majority of the results extracted from multi-band polarization is likely contaminated by the 3D structure of the ISM, leading to inaccurate dust-grain constraints. The upside of this problem is that it opens up new possibilities for studying the magnetized ISM in 3D.', '3.1. Polarimetric data selection - Statistical sample': 'A recent agglomeration of multi-band optical polarization data (Panopoulou et al. 2023), which is the largest to date, allows us, for the first time, to study in a statistical fashion the impact of the 3D ISM structure of the Galaxy on the Serkowski relation, and the consequences for magnetic field and dust physics studies. In order to fit multiple-cloud models to polarization data it is optimal to use polarization measurements across a broad optical wavelength range. However, for most stars in the literature, measurements exist only in a single or in few bands. Even so, it is possible to evaluate the validity of the Serkowski relation in a statistical manner, by studying an ensemble of sources. We designed such an experiment to test whether stars in single-cloud LOSs are better fit by the Serkowski relation than stars in multiple-cloud LOSs. \nFor acquisition of archival optical polarization measurements for the statistical comparison, we explored Tables 4, 6 of the compiled polarization catalog of Panopoulou et al. (2023), which provide polarimetry and distances from G aia EDR3 (Bailer-Jones et al. 2021) respectively for ∼ 42 , 000 sources. We excluded stars with distance greater than 1.25 kpc, as this is the limit of our dust map (see Sect. 3.2). As distance, we take the average of the values (\'r\\_med\\_geo\',\'r\\_med\\_photogeo\') provided in the catalog. We excluded measurements in filters with no well-defined e ff ective wavelength ( λ e f f ), therefore, the excluded filters are \'0", \'20", \'23", \'51" which correspond to \'No filter or unclear", \'weightedmean", 0.735-0.804 µ m , 0.5-0.85 µ m , respectively in the catalog. For targets with multiple measurements in the same filter we used the weighted mean and standard deviation of the q measurements, according to \nq mean = P ( qi · w i ) P w i (4) \nq std = P w i ( xi -¯ x ) 2 GLYPH<16> N -1 N GLYPH<17> · P w i , (5) \nwith w i = 1 /σ 2 qi , and σ qi the uncertainty of the measurement in q , and similarly for u . We excluded measurements with debiased P /σ P < 2. Additionally, we did not consider measurements with polarization uncertainty σ P < 0 . 1% because they are likely underestimated 3 , as the most reliable polarization standards to date (Blinov et al. 2023) are only accurate down to 0 . 1%. After all the cuts, we selected targets that have surviving measurements in at least three bands. We note that we did not impose any cut for intrinsically polarized stars (although the Panopoulou et al. (2023) catalog provides this information), as these targets may have been flagged as intrinsically polarized simply because they do not follow the Serkowski relation. However, the final sample does not contain any star that is flagged as intrinsically polarized. After all the cuts, the final sample contains 223 sources with measurements in at least three optical bands.', '3.2. Extinction': 'We collected extinction data for our target stars from the Edenhofer et al. (2024) dust map. We choose this map, as it provides good angular resolution (14\'), as well as, parsec-scale distance resolution. The map is available in two versions, one that extends out to 1.25 kpc from the Sun, and one that extends up to 2 kpc, but uses less data for the dust profile reconstruction. We used the 1.25 kpc version, as it should be more accurate, and we did not want to introduce any additional sources of noise. Extinction values in the Edenhofer et al. (2024) map are expressed in the units used by Zhang et al. (2023), the work upon which the dust map was based. We converted the extinction values to represent the extinction in the V band ( AV ) based on the conversion table provided by Zhang et al. (2023). \nFor each of our targets, we examined the "mean" (Edenhofer et al. 2024) cumulative and di ff erential AV vs distance ( D ) profiles by eye, in order to identify the number of significant steps ( NC ) in the profile, which presumably correspond to distinct dust clouds. For targets where more than 3 clouds were identified, or the dust profile was very complex, we set NC > 3. The number of target stars in each subgroup with di ff erent number of identified clouds along the LOS are #( NC = 1) = 49, #( NC = 2) = 74, #( NC = 3) = 70, #( NC > 3) = 30. The targets that correspond to each NC are distributed over di ff erent positions on the sky (Fig. 2). \nWe considered the possibility of using a more objective metric to characterize the number of dust clouds along each LOS, as each step in the dust profiles is typically not equally prominent, and in many cases, numerous small steps contribute to the cumulative extinction. Therefore, we explored a method similar to that used by Panopoulou & Lenz (2020), who identified the number of clouds in H i data by detecting peaks in the di ff erential H i column density profile and then normalizing each peak by the height of the tallest peak among them. In our case, the normalized number of clouds in each LOS would be N norm C = P A i V A max V , with A i V the extinction of a given peak in the LOS, and A max V the extinction of the highest peak in the di ff erential AV vs D profile. However, this was not useful in our study for the following reasons. 1) N norm C \nis ambiguous. For example, N norm C = 2 could signify two equally prominent peaks, or one prominent peak plus two peaks half its height, or even one prominent peak plus three peaks a third of its height, and so on. This lack of clarity reduces its utility. 2) The profiles in the Edenhofer et al. (2024) map are sometimes irregular, with many minor peaks that may not represent real features but instead result from sampling noise. Additionally, peaks can appear very close together, making it di ffi cult to distinguish between separate clouds and variations within the same cloud. Although we considered smoothing the profiles to address these issues, we found that the number and relative height of the resulting peaks were highly sensitive to the parameters chosen for smoothing, introducing further ambiguity. 3) Even when a peak appears higher than another, this does not necessarily correlate with increased polarization. The dust-induced polarization of aligned grains depends largely on factors beyond peak height, such as dust density, grain alignment e ffi ciency, magnetic field coherence, and the inclination of the magnetic field. \nIn our attempts to devise an objective method for peak counting, we found that the resulting numbers were often ambiguous or lacked physical meaning. Consequently, we opted for a visual approach. Although this method has its uncertainties and likely introduces some contamination into the subsamples, it is selfconsistent and allows statistical patterns to emerge in a dataset as large as ours. Fig. 3 illustrates examples of dust profiles that we assigned to di ff erent subsamples.', '3.3. Fitting the Serkowski relation': "In order to evaluate the performance of the Serkowski relation in our statistical sample, we used an extremely powerful, if unconventional in Serkowski studies, technique: we fit our data in the Stokes parameters space q -u , exploiting its power to simultaneously trace changes in the profiles of both the degree of polarization and EVPA. Traditionally, the EVPA is not taken into account while fitting the Serkowski formula, thus part of the information imprinted in the polarization signal is ignored. Although the variability of the EVPA with wavelength has been hypothesized to be connected to the existence of multiple clouds (e.g. McMillan & Tapia 1977; Whittet 2015), q -u fitting has never been employed to trace the various components (clouds) contributing to the P -λ and θ -λ profiles. \nAs a first step we converted Eq. 1 to the q -u space, with the introduction of θ in the model as an additional free parameter. Therefore, our model becomes \nˆ q = Pmax · exp GLYPH<18> -K · ln 2 λ max λ GLYPH<19> · cos(2 θ ) , (6a) \nˆ u = Pmax · exp GLYPH<18> -K · ln 2 λ max λ GLYPH<19> · sin(2 θ ) . (6b) \nThe quantity we wished to minimize for the fits is \nm = Lq + Lu + Lq Lu + Lu Lq (7) \nwith \nLq = 1 N λ · X qi -ˆ qi σ qi ! 2 , (8a) \nLu = 1 N λ · X ui -ˆ ui σ ui ! 2 , (8b) \n<!-- image --> \nAmax \nFig. 2: Positions of targets depending on their NC . Maps are in Galactic coordinates, in mollweide projection, centered in (0,0). Parallels are drawn every 30 o . Meridians are drawn every 60 o . Longitude increases towards the right-hand side. Colors correspond to the average λ max of each area. Marked areas with 'A' through 'G' contain stars that are located behind a single as well as multiple clouds. See discussion in Sect. 4.3.Fig. 3: Dust profiles for four example targets belonging in four di ff erent subsamples. From left to right: NC = 1, NC = 2, NC = 3, NC > 3. Top row: Cumulative AV vs D . Bottom row: Di ff erential AV vs D . \n<!-- image --> \nwhere N λ the number of di ff erent bands where we had measurements for each target, qi , ui the measurement in a given band, ˆ qi , ˆ ui our model as introduced in Eq. 6a, 6b, and σ qi , σ ui the uncertainties of the measurements in q and u respectively. The terms Lq and Lu are the reduced χ 2 metric for q and u respectively. The terms Lq / Lu and Lu / Lq were introduced to ensure a balanced contribution from Lq and Lu . For example, if the objective was \nto minimize just Lq + Lu , the optimal value V could be achieved with Lq = V and Lu = 0, Lu = V and Lq = 0, or any combination in between. By introducing Lq / Lu and Lu / Lq , we penalize the model when it favors solutions where Lq = 0 or Lu = 0, thus promoting a more balanced fit. \nFor the modelling procedure we used the O ptuna package in python (Akiba et al. 2019). O ptuna is an automatic hyperpa- \nrameter optimization software framework, originally developed for machine learning hyperparameter-tuning purposes. However, it can also address general optimization problems through the use of a Bayesian optimization framework. We found it highly e ff ective in multi-parameter optimization with few data points and many parameters, such as the case of the q -u modelling, and more e ffi cient in finding the best solution than traditional algorithms such as Markov-Chain Monte-Carlo (MCMC). We used the default options of O ptuna with n\\_trials = 5000 for our fits. Moreover, we adopted the following priors. For Pmax we initially assumed a rather loose Gaussian prior distribution with a mean of 3% and standard deviation of 3%, and enforced it to be positive. For λ max we assumed a gaussian prior distribution with the mean 0.6 µ m and standard deviation 0.2 µ m , and ensured it was always positive. For K we imposed a Gaussian prior distribution with mean 1 . 68 · λ max and standard deviation 0 . 13 · λ max (Wilking et al. 1980), and ensured it was always positive. For θ we assumed a Gaussian prior distribution with mean the average of the polarization values of measurements in individual bands (derived from the average q and u as in Eq. 4 and converted to θ using Eq. 3b) and 10 o standard deviation . \nWe evaluated the quality of our fits by measuring the χ 2 per degree of freedom (DoF): \nχ 2 ν ≡ χ 2 DoF = P GLYPH<18> qi -ˆ qi σ q i GLYPH<19> 2 + P GLYPH<18> ui -ˆ ui σ u i GLYPH<19> 2 2 · N -ν , (9) \nλ \nwhere ν = 4 is the number of free parameters in the model ( Pmax , λ max , K , θ ). We used 2 · N λ as the number of points in Eq. 9 as we accounted for the measurements in both q and u to perform the simultaneous fits. We note that we cannot use m directly (Eq. 7) to evaluate the fits, as m is not a normalized metric, and thus cannot be used for comparing fits to each other.", '3.4. Fitting multiple clouds': 'In order to fit multiple clouds (as in the examples of Sect. 4.2) we used again the q -u fitting technique, following the same logic discussed in Sect. 3.3. The multiple cloud model is derived from Equations 2 and 6, by adding up the q and u components of the individual clouds. \nWhen trying to determine the correct parameters for a model with two or more clouds, the number of parameters is increased (four parameters per cloud), resulting in a substantial number of possible combinations and a vast parameter space to explore. In such situations, identifying the optimal combination of parameters that minimize the cost function can be challenging. To address this, we employed the following procedure. Initially, instead of directly using O ptuna , we utilized an MCMC algorithm to rapidly explore the parameter space. Given that the two clouds are interchangeable, we imposed the condition λ max 1 > λ max 2 to focus the algorithm on a single solution. Subsequently, we visually examined the corner plots of the parameters generated by the MCMC algorithm to establish reasonable constraints for the O ptuna algorithm, thereby significantly narrowing the search space. Finally, we ran the fit using O ptuna , imposing Gaussian priors on the parameters, based on the mean and the spread of the parameters identified by the MCMC algorithm. We chose O ptuna to derive the final set of parameters for the model, as it directly provides the optimal set of parameters by minimizing Eq. 7. The fits are evaluated with χ 2 (Eq. 9). \nν', '4.1. Statistical sample': 'We grouped the sample of sources found in Sect. 3.1 into subsamples depending on the number of clouds ( NC ) lying in their LOS, as derived from the 3D extinction map. We obtained four main subsamples: targets in LOSs with one dominant cloud, NC = 1; with two clouds, NC = 2; three clouds NC = 3; and with more than three clouds, NC > 3. For this sample we fit the traditional Serkowski curve (Eq. 1) for all targets, regardless of NC , to evaluate the capability of the Serkowski relation to fit polarimetric data for subsamples with di ff erent NC . We evaluated the performance of the Serkowski relation in all of the targets as described in Sect. 3.3. We constructed the cumulative density functions (CDFs) of χ 2 ν for the di ff erent subsamples (Fig. 4), and we applied the Kolmogorov-Smirnov (KS) test to identify any statistical discrepancies between the χ 2 ν distributions of the NC = 1 subset and the other three subsets. \nWe found that the χ 2 ν distribution of the NC = 1 subset is significantly di ff erent from those of subsets with NC = 2 or NC = 3. On the contrary, the χ 2 ν distributions of the NC = 1 subset and of the NC > 3 subset are not discrepant. The p -values of the KS tests for all comparisons are quoted in Table 1, along with the median χ 2 ν for the fits of each subsample. Despite the sparse nature of the data (measurements in only ∼ 3 . 5 di ff erent filters on average per target) the sample is su ffi ciently large to allow the statistical distinction of LOSs with two or three clouds from those with only a single cloud. The deterioration of the Serkowski fits in the multi-cloud LOSs is mainly driven by the behaviour of the EVPA with wavelength, and is made clear thanks to our q -u space fits. In contrast, if we perform the fits only on P -λ , it is not possible to discriminate between LOSs with and without multiple clouds with current data. When the structure in the LOS is too complicated ( NC > 3), there is no statistical di ff erence in the polarization profiles compared to the single cloud case. This is not surprising, as with an increasing number of clouds the variations induced by individual screens tend to average out. Even deviations in the EVPA, which should be more prominent, would become smeared and di ffi cult to detect, especially with limited data points across the spectrum. \nTable 1: p - values of the KS test between the subsample of targets with NC = 1 and other subsamples. See Sect. 4.1 for details.', '4.2. Two revealing examples': 'To demonstrate observationally the impact of the LOS variations of λ max and EVPA on the Serkowski relation, we employed archival spectropolarimetric data (Bagnulo et al. 2017). We identified two revealing cases, where the multi-wavelength polarization data show clear evidence of the LOS e ff ects. \nThe LOS toward the main sequence star HD93222 with spectral type O7 represents a prominent example of what the P -λ , and θ -λ profiles look like when 3D e ff ects are significant. Firstly, we \n0.8 \n0.8 \n0.6 \n0. \n(a) \nFig. 4: CDFs of χ 2 for di ff erent subsamples as described in each label. \n<!-- image --> \n(c) \n(b) \n(d) \nν \nobserve that the P -λ deviates from the typical Serkowski curve (Fig. 5 middle panel). Fitting the P -λ data with the Serkowski relation requires unphysical values for both λ max , and K (Table 2), which still yield a merely adequate fit. In addition, θ varies smoothly with wavelength λ , which is unexpected when the polarization is dominated by a single cloud with a well defined mean magnetic field. Thus, both P and EVPA show an atypical behaviour with λ for this star. Using 3D dust extinction maps (Edenhofer et al. 2024), we verified the existence of multiple clouds along the LOS toward the HD93222 star. The clouds are identified as abrupt increments in extinction as a function of distance (left panel in Fig. 5). The average extinction profile shows three prominent clouds located at around 0.25, 1.35 and 1.8 kpc. The target star, whose estimated distance is 2.44 kpc (Bailer-Jones et al. 2021), lies beyond these clouds, hence its polarization signal carries information from all of them. We fit the data with models with di ff erent number of clouds. Although the extinction profile exhibits three distinct steps, the data can be described equally well with both a 2- and a 3-cloud model. This may indicate that two out of three clouds have similar properties, and their constructive profile resembles that of one cloud (such as the case of \nthe middle left panel of Fig. 1). We proceed with the simplest case of a 2-cloud model. \nAs a second example, we show results for the star HD163181, which is an intermediate-size luminous supergiant of spectral class O9.5 (Fig. 6). In this case, the observed degree of polarization follows the Serkowski formula, but the EVPA varies with wavelength. The 3D extinction profile has two significant steps, hence indicating the existence of more than one dominant clouds along this LOS. Assuming the standard Serkowski curve with a single dominant cloud can yield good fits to the data in the P -λ space (Bagnulo et al. 2017), but it fails to explain the observed variability of the EVPA; we verified this by fitting a one cloud model in the q -u plane (dashed blue line in Fig. 6). However, when we considered two clouds along this LOS, as the 3D extinction map suggest, we were able to capture the observed variability of both P and θ with wavelength (red dashed dotted line in Fig. 6 middle panel). \n0.6 \n0.8 \n0.6 \nNc = 2 \nNc > 1 \nNc = 3 \n8 \n8 \nTable 2: Fitted parameters for the stars HD163181 and HD93222 in Sect. 4.2. \n<!-- image --> \n<!-- image --> \nFig. 5: Example of a case where both the P -λ and the θ -λ profiles deviate from Serkowski expectations. We are able to fit both profiles by introducing a 2-Cloud model. Data from Bagnulo et al. (2017) for HD93222. Left : Extinction vs distance profile taken from the 2kpc version of the map by Edenhofer et al. (2024). The vertical red line corresponds to the distance of the source. Middle : Data in the P -λ space, together with three fitted models: Solid black line: Fit using Eq. 1 in P -λ space only. Dashed blue line: Best-fit 1-Cloud model in the q -u space (Eq. 6). Dashed-dotted red line: Best-fit 2-Cloud model in the q -u space (Eq. 6). Dotted orange lines correspond to the two individual components of the 2-Cloud model. Right : Data in θ -λ space together with the three models and the individual components of the 2-Cloud model. Parameters of the fits are quoted in Table 2. \n<!-- image --> \nFig. 6: Example of a case where the P -λ profile follows the Serkowski relation, but the EVPA does not. By employing a model with two components in the LOS, it is possible to fit both P and θ . Data from Bagnulo et al. (2017) for HD163181. Panels and lines similar to Fig. 5. Parameters of the fits are quoted in Table 2 \n<!-- image --> \n.', '4.3. On the parameters of the Serkowski formula': 'Here we demonstrate how the derived parameters of the Serkowski relation can be incorrect, when erroneously assuming a single cloud in the LOS. We explored how 3D e ff ects a ff ect the distributions of λ max , and Pmax that one obtains through the classical fitting of the Serkowski relation in the P -λ space; K \nlinearly correlates with λ max (Wilking et al. 1980), and thus can be omitted from this analysis. We fit the data only in P -λ space (Eq. 1) by minimizing \n1 N λ · X Pi -ˆ Pi σ pi ! 2 , (10) \nFig. 8: Normalized distributions of Pmax for subsamples with di ff erent number of clouds in the LOS. See Sect. 4.3 for details. The following bins are omitted for clarity. For NC = 2: bin at Pmax / AV ≃ 23 with a height of 0.0135. For NC = 3: bin at Pmax / AV ≃ 13 with a height of 0.0145. \n<!-- image --> \nFig. 7: Normalized distributions of λ max for subsamples with di ff erent number of clouds in the LOS. See Sect. 4.3 for details. \n<!-- image --> \nassuming a one-cloud model, and imposing the same priors as discussed in Sect. 3.3. Since the e ff ect of the 3D dust structure is more prominent in θ -λ space, fits in P -λ were comparably good for all four subsamples. We then explored the normalized distributions of the derived parameters for all subsamples (Fig. 7 for λ max , Fig. 8 for Pmax / AV ). \nPast studies of the Serkowski parameters show that the distribution of λ max is singly-peaked, with an average value close to 0.5 µ m(Martin & Whittet 1990). In contrast, we find that for NC = 1 (single-cloud sightlines) the intrinsic distribution of λ max is bimodal, with peaks at approximately 0.28 and 0.62 µ m. On the other hand, all distributions with NC > 1 (multiple-cloud sightlines) are unimodal and peak at around 0.55 µ m, which is close to the value that is considered as the Galactic average. By performing Monte Carlo (MC) simulations we found that the unimodality in the λ max distribution, which is observed for NC > 1, can be reproduced with multiple-cloud models with values of λ max randomly drawn from the NC = 1 distribution. The intrinsic Galactic distribution of λ max may thus be bimodal, and the currently accepted value for the Galactic average of λ max ( ∼ 0 . 5 µ m) is likely contaminated by LOS e ff ects. Notably, the larger spread of the distribution of λ max for the NC = 2 case is not surprising. Clarke & Al-Roubaie (1984) had theorized that in the case of two clouds in the LOS, the acquired λ max from the Serkowski fit is di ff erent from the average λ max of the two clouds when the two clouds have di ff erent parameters. Furthermore, it is worth considering the following. Most areas containing stars behind a single cloud also contain stars behind multiple clouds. We calculated the average λ max for targets in distinct areas on the sky (Fig. 2). In most cases, the average λ max di ff ers significantly between subsets (see Table 3) located in the same area. This suggests that using the λ max of a star from an area with NC > 1 as a characteristic value for the LOS dust properties could be misleading. \nThe distribution of Pmax / AV is also a ff ected by the number of clouds along the LOS. We can immediately see this by assuming, without loss of generality, that two clouds lie along a LOS. The polarization induced by the two clouds can either (partially or totally) add up, or (partially or totally) cancel out, depending on the di ff erence between the EVPAs each would impart on its own.When the EVPA di ff erence is close to 0 o the polarization adds up, while when the di ff erence is close to 90 o it cancels out. \nThis is exactly what we observe in the distributions of Pmax / AV (Fig. 8). The Pmax / AV distribution for NC = 1 is free from LOS e ff ects, hence it should resemble the "ground truth". This distribution peaks at Pmax / AV ≃ 3 -3 . 5% / mag, with only few points with Pmax / AV > 4 . 5% / mag. On the other hand, the NC = 2 distribution peaks in a lower value of Pmax / AV ≃ 2% / mag, with its tail extending to higher values of Pmax / AV than in the NC = 1 case. The shape of the NC = 2 distribution can be understood by the LOS integration e ff ects acting constructively (generating the extended tail), or destructively (generating the lower Pmax / AV peak). The distribution for NC = 3 displays a similar shape, though it is smoother, as the addition of a third cloud facilitates producing polarization closer to the average observed in the NC = 1 case. In the NC > 3 distribution, we generally observe lower values for Pmax / AV compared to the NC > 1 distribution. This is expected, as with more dust components in the LOS, variations in dust and magnetic field properties are more probable, leading to increased depolarization. Overall, both the λ max , and the Pmax / AV are significantly a ff ected by the 3D structure of the ISM, which, if ignored, can lead to erroneous Serkowski best-fit parameters. \nOur results suggest that fitting a single Serkowski relation is invalid in LOSs with multiple clouds. There can be cases where a single Serkowski relation yields good fits, even when 3D e ff ects are important (Sect. 4.2). However, in these cases the best-fit parameters of a single Serkowski curve are misleading, and not representative of the dust grain physics.', '5.1. Implications for dust modeling': 'The findings presented in this study have significant implications for dust modeling. Dust models, and grain alignment theories must satisfy the constraints obtained from Serkowski fits (Martin &Whittet 1990; Andersson et al. 2015; Draine 2024). However, our results indicate that the Serkowski parameters derived from fitting the observed polarization data in complex interstellar environments may not accurately represent the underlying dust properties, leading to erroneous conclusions about grain size dis- \nTable 3: Mean λ max values in µ m and their error on the mean for di ff erent regions in the sky and di ff erent subsamples. Regions with N / A in the error of the mean are cases where there is only one star in the specific region for the subsample. \ntributions and composition. The results from Sect. 4.3 suggest that the intrinsic distribution of λ max in the clouds of our Galaxy may actually be bimodal (Fig. 7). This finding contradicts past studies (Martin & Whittet 1990), which likely stem from (incorrectly) fitting the Serkowski formula in regions with multiple clouds. Although, our employed sample is the largest to-date, the statistics are still limited, and more data are required to further explore the intrinsic distribution of λ max in our Galaxy. \nWe propose the following two options for constraining the populations of dust in di ff erent ISM environments: 1) fit the Serkowski formula only to targets behind a single cloud, or 2) decompose the cumulative polarization signal into its individual components for targets behind multiple clouds to derive the characteristics of each. Moreover, di ff erent regions of the Galaxy (e.g., the disk vs. the halo, or varying distances from the Galactic center) may exhibit distinct dust physics that we are currently missing due to the misapplication of the Serkowski relation. By studying the polarization of targets behind multiple clouds and decomposing the signal to derive individual parameters, it may become possible to uncover dust clouds with diverse and potentially novel properties in di ff erent parts of the Galaxy.', '5.2. Implications for the identification of intrinsically polarized stars': 'It is customary to iterpret deviations from the Serkowski relation in the P ( λ ) behavior of point sources as indicative of intrinsic polarization mechanisms, such as the presence of circumstellar disks (e.g. Topasna et al. 2023). However, the wavelength dependence of polarization in such cases is known to lack a typical pattern (Bastien 2015). In contrast, we have demonstrated that fitting simple two-cloud models can account for deviations from the Serkowski formula quite e ff ectively (Sect. 4.2). This suggests that deviations from the Serkowski curve alone are insu ffi cient to indicate the presence of an intrinsic polarization component. In order to validate that a star not following the Serkowski formula is indeed intrinsically polarized, polarization data could be combined with other evidence, such as detection of significant variability, or comparison with neighboring stars.', '5.3. The possibility of 3D tomography with the Serkowski relation': 'Fitting multiple-cloud models to multiwavelength (optical to nearinfrared) polarization data paves the way to constraining the LOS variations of the plane-of-the-sky magnetic field morphology (tomography). If the number of clouds in the LOS is known, for instance from 3D extinction maps (e.g., Edenhofer et al. 2024; Green et al. 2019), it is possible to extract the parameters of each individual cloud using the techniques discussed in Sect. 3.4. However, the following considerations should be kept in mind. \nFirstly, it is possible that two or more clouds in the LOS have similar parameters. In such cases, the polarization data may be adequately fit with fewer clouds than what the 3D extinction maps suggest. For example, if two clouds along a LOS have comparable λ max and θ , their polarization contributions will combine, resulting in a profile resembling that of a single cloud but with an increased degree of polarization. This might explain our target star HD93222 (Sect. 4.2), where the extinction profile displays at least three distinct steps, yet the data are su ffi ciently well-fit by a 2-Cloud model. Another possibility is that the third cloud has a low abundance of aligned grains, because of, for example, a magnetic field with direction along the LOS. \nAnother strategy one could follow to identify cases of clouds with comparable properties is to obtain multi-band or, ideally, spectropolarimetric measurements of multiple stars at di ff erent distances along the same LOS. The polarization signal would vary with distance, as each star would only be influenced by the foreground dust column. Thus, stars lying behind the same clouds would show similar polarization trends, while distinct changes would only occur for stars behind di ff erent clouds. Following such methods could complement the BISP-1 algorithm (Pelgrims et al. 2023), which employs single-band optical polarization data with parallaxes from Gaia, for performing ISM tomography along many LOSs (Pelgrims et al. 2024) in the context of optopolarimetric surveys, such as the P asiphae survey (Tassis et al. 2018). While the method described by Pelgrims et al. (2023) can determine the number of clouds along a LOS and their distances, it does not fully capture the dust physics within each cloud. Multi-band polarimetric measurements of a few stars at varying distances along the LOS can provide insights into the dust physics within each cloud. Thus, multiwavelength polarization data promises to provide an independent method for performing ISM magnetic field tomography, which could usher in the era of high-precision 3D magnetized ISM cartography.', '6. Summary and Conclusions': 'In this paper, we showcase the e ff ect of the 3D structure of the ISM on the well-known and widely-used Serkowski relation. Our findings are summarized as follows. \n- 1. We revisited theoretically how the Serkowski formula could become invalid in LOSs with many dust components. We showed that the variations of both the degree of polarization and EVPA with wavelength can be severely a ff ected in some stars. In other stars, the polarization profile may misleadingly appear to follow the Serkowski relation, despite the contribution from multiple dust clouds. In such cases, the imprint of the 3D structure can be mainly observed in the θ -λ profile.\n- 2. As a test case, we selected two di ff erent targets that appear to be behind multiple dust clouds, based on their extinction \nprofiles, and for which archival spectropolarimetric data exist. We demonstrated that the single-cloud model, implicitly assumed for Serkowski fits, fails in these cases, while we successfully fitted 2-cloud models. \n- 3. We proposed the q -u fitting technique as the most appropriate approach to fit the Serkowski relation. This method allows for tracing both P and θ as functions of wavelength simultaneously, enabling fitting for more than one clouds along the LOS.\n- 4. We used a sample of 223 stars behind di ff erent number of clouds with polarimetric measurements in three or more bands, to fit and evaluate statistically the performance of the Serkowski formula. We found that the Serkowski formula is not a good model for LOSs with two or more clouds, while it fits well the data in the cases of one or more than three clouds in the LOS. The latter is an outcome of averaging.\n- 5. Our analysis of the fitted parameters of the 223 stars suggests that the intrinsic distribution of the λ max parameter in our Galaxy may be bimodal, something that was veiled until now, due to treating sightlines with many clouds as a single dust component.\n- 6. A poor Serkowski fit by itself does not constitute conclusive evidence of intrinsic polarization in a source.\n- 7. Fitting the Serkowski formula for stars behind multiple dust clouds leads to an incorrect estimation of the fitted parameters, and, thus, of the underlying dust physics.\n- 8. Our results hint towards the possibility of performing magnetic tomography using multiband polarization data in combination with G aia distances. \nIn conclusion, the Serkowski relation should be cautiously applied when the ISM 3D structure is complex. \nAcknowledgements. We thank V. Pavlidou for very helpful discussions. NM and KT were supported by the European Research Council (ERC) under grant agreements No. 7712821. This work was supported by NSF grant AST-2109127. N.M. was funded by the European Union ERC-2022-STG - BOOTES - 101076343. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.', 'References': "Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. 2019, arXiv e-prints, arXiv:1907.10902 \nAndersson, B. G., Lazarian, A., & Vaillancourt, J. 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R., Benetti, S., et al. 2010, A&A, 510, A108 \nPattle, K., Fissel, L., Tahani, M., Liu, T., & Ntormousi, E. 2023, in Astronomical Society of the Pacific Conference Series, Vol. 534, Protostars and Planets VII, ed. S. Inutsuka, Y. Aikawa, T. Muto, K. Tomida, & M. Tamura, 193 Pelgrims, V., Mandarakas, N., Skalidis, R., et al. 2024, A&A, 684, A162 Pelgrims, V., Panopoulou, G. V., Tassis, K., et al. 2023, A&A, 670, A164 Planck Collaboration, Adam, R., Ade, P. A. R., et al. 2016, A&A, 594, A10 Serkowski, K., Mathewson, D. S., & Ford, V. L. 1975, ApJ, 196, 261 Skalidis, R., Sternberg, J., Beattie, J. R., Pavlidou, V., & Tassis, K. 2021, A&A, 656, A118 \nSkalidis, R. & Tassis, K. 2021, A&A, 647, A186 \nSkalidis, R., Tassis, K., Panopoulou, G. V., et al. 2022, A&A, 665, A77 \nTassis, K., Ramaprakash, A. N., Readhead, A. C. S., et al. 2018, arXiv e-prints, \narXiv:1810.05652 \nTopasna, G. A., Mateja, F. M., & Kaltcheva, N. T. 2023, PASJ, 75, 269 \nTreanor, P. J. 1963, AJ, 68, 185 \nWhittet, D. C. B. 2015, ApJ, 811, 110 \nWhittet, D. C. B. 2022, Dust in the Galactic Environment (Third Edition) \nWhittet, D. C. B., Gerakines, P. A., Hough, J. H., & Shenoy, S. S. 2001, ApJ, 547, \n872 \nWhittet, D. C. B., Martin, P. G., Hough, J. H., et al. 1992, ApJ, 386, 562 Wilking, B. A., Lebofsky, M. J., Martin, P. G., Rieke, G. H., & Kemp, J. C. 1980, ApJ, 235, 905 \nZhang, X., Green, G. M., & Rix, H.-W. 2023, MNRAS, 524, 1855"}

2024PASJ..tmp...89X

We present an initial analysis of the XRay Imaging and Spectroscopy Mission XRISM firstlight observation of the supernova remnant SNR N 132D in the Large Magellanic Cloud. The Resolve microcalorimeter has obtained the first highresolution spectrum in the 1.610 keV band which contains Kshell emission lines of Si S Ar Ca and Fe. We find that the Si and S lines are relatively narrow with a broadening represented by a Gaussianlike velocity dispersion of inlineformulatexmath idTM0001 notationLaTeXsigma v sim 450texmathinlineformula km sinlineformulatexmath idTM0002 notationLaTeX1texmathinlineformula. However the Fe Heinlineformulatexmath idTM0003 notationLaTeXalphatexmathinlineformula lines are substantially broadened with inlineformulatexmath idTM0004 notationLaTeXsigma v sim 1670texmathinlineformula km sinlineformulatexmath idTM0005 notationLaTeX1texmathinlineformula. This broadening can be explained by a combination of the thermal Doppler effect due to the high ion temperature and the kinematic Doppler effect due to the SNR expansion. Assuming that the Fe Heinlineformulatexmath idTM0006 notationLaTeXalphatexmathinlineformula emission originates predominantly from the supernova ejecta we estimate the reverse shock velocity at the time when the bulk of the Fe ejecta were shock heated to be inlineformulatexmath idTM0007 notationLaTeX1000 lesssim Vrm rstexmathinlineformula km sinlineformulatexmath idTM0008 notationLaTeX1texmathinlineformula inlineformulatexmath idTM0009 notationLaTeXlesssim 3300texmathinlineformula in the observer frame. We also find that Fe Lyinlineformulatexmath idTM0010 notationLaTeXalphatexmathinlineformula emission is redshifted with a bulk velocity of inlineformulatexmath idTM0011 notationLaTeXsim 890texmathinlineformula km sinlineformulatexmath idTM0012 notationLaTeX1texmathinlineformula substantially larger than the radial velocity of the local interstellar medium surrounding N 132D. These results demonstrate that highresolution Xray spectroscopy is capable of providing constraints on the evolutionary stage geometry and velocity distribution of SNRs.

2024-10-01T00:00:00Z

['2024PASJ..tmp...90A', '2024PASJ..tmp...89A', '10.48550/arXiv.2408.14301', 'arXiv:2408.14301', '10.1093/pasj/psae080', '2024arXiv240814301X', '2024PASJ..tmp...89X']

['Astrophysics - High Energy Astrophysical Phenomena']

The XRISM firstlight observation Velocity structure and thermal properties of the supernova remnant N 132D

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

https://arxiv.org/pdf/2408.14301.pdf

{'The XRISM First Light Observation: Velocity Structure and Thermal Properties of the Supernova Remnant N132D': 'XRISM Collaboration 1 , Marc Audard 2 , Hisamitsu Awaki 3 , Ralf Ballhausen 4,5,6 , Aya Bamba 7 , Ehud Behar 8 , Rozenn Boissay-Malaquin 9,5,6 , Laura Brenneman 10 , Gregory V. Brown 11 , Lia Corrales 12 , Elisa Costantini 13 , Renata Cumbee 5 , Maria Diaz-Trigo 14 , Chris Done 15 , Tadayasu Dotani 16 , Ken Ebisawa 16 , Megan Eckart 11 , Dominique Eckert 2 , Teruaki Enoto 17 , Satoshi Eguchi 18 , Yuichiro Ezoe 19 , Adam Foster 10 , Ryuichi Fujimoto 16 , Yutaka Fujita 19 , Yasushi Fukazawa 20 , Kotaro Fukushima 16 , Akihiro Furuzawa 21 , Luigi Gallo 22 , Javier A. Garcia 5,23 , Liyi Gu 13 , Matteo Guainazzi 24 , Kouichi Hagino 7 , Kenji Hamaguchi 9,5,6 , Isamu Hatsukade 25 , Katsuhiro Hayashi 16 , Takayuki Hayashi 9,5,6 , Natalie Hell 11 , Edmund Hodges-Kluck 5 , Ann Hornschemeier 5 , Yuto Ichinohe 26 , Manabu Ishida 16 , Kumi Ishikawa 19 , Yoshitaka Ishisaki 19 , Jelle Kaastra 13,27 , Timothy Kallman 5 , Erin Kara 28 , Satoru Katsuda 29 , Yoshiaki Kanemaru 16 , Richard Kelley 5 , Caroline Kilbourne 5 , Shunji Kitamoto 30 , Shogo Kobayashi 31 , Takayoshi Kohmura 32 , Aya Kubota 33 , Maurice Leutenegger 5 , Michael Loewenstein 4,5,6 , Yoshitomo Maeda 16 , Maxim Markevitch 5 , Hironori Matsumoto 34 , Kyoko Matsushita 31 , Dan McCammon 35 , Brian McNamara 36 , Franc¸ois Mernier 4,5,6 , Eric D. Miller 28 , Jon M. Miller 12 , Ikuyuki Mitsuishi 37 , Misaki Mizumoto 38 , Tsunefumi Mizuno 39 , Koji Mori 25 , Koji Mukai 9,5,6 , Hiroshi Murakami 40 , Richard Mushotzky 4 , Hiroshi Nakajima 41 , Kazuhiro Nakazawa 37 , Jan-Uwe Ness 42 , Kumiko Nobukawa 43 , Masayoshi Nobukawa 44 , Hirofumi Noda 45 , Hirokazu Odaka 34 , Shoji Ogawa 16 , Anna Ogorzalek 4,5,6 , Takashi Okajima 5 , Naomi Ota 46 , Stephane Paltani 2 , Robert Petre 5 , Paul Plucinsky 10 , Frederick Scott Porter 5 , Katja Pottschmidt 9,5,6 , Kosuke Sato 29 , Toshiki Sato 47 , Makoto Sawada 30 , Hiromi Seta 19 , Megumi Shidatsu 3 , Aurora Simionescu 13 , Randall Smith 10 , Hiromasa Suzuki 16 , Andrew Szymkowiak 48 , Hiromitsu Takahashi 20 , Mai Takeo 29 , Toru Tamagawa 26 , Keisuke Tamura 9,5,6 , Takaaki Tanaka 49 , Atsushi Tanimoto 50 , Makoto Tashiro 29,16 , Yukikatsu Terada 29,16 , Yuichi Terashima 3 , Yohko Tsuboi 51 , Masahiro Tsujimoto 16 , Hiroshi Tsunemi 34 , Takeshi G. Tsuru 17 , Hiroyuki Uchida 17 , Nagomi Uchida 16 , Yuusuke Uchida 32 , Hideki Uchiyama 52 , Yoshihiro Ueda 53 , Shinichiro Uno 54 , Jacco Vink 55 , Shin Watanabe 16 , Brian J. Williams 5 , Satoshi Yamada 56 , Shinya Yamada 30 , Hiroya Yamaguchi 16 , Kazutaka Yamaoka 37 , Noriko Yamasaki 16 , Makoto Yamauchi 25 , Shigeo Yamauchi 46 , Tahir Yaqoob 9,5,6 , Tomokage', 'Yoneyama 51 , Tessei Yoshida 16 , Mihoko Yukita 57,5 , Irina Zhuravleva 58 , Manan Agarwal 55 and Yuken Ohshiro 7,16': "- 1 Corresponding Authors: Hiroya Yamaguchi, Hiromasa Suzuki, Frederick Scott Porter, Caroline Kilbourne, Michael Loewenstein, Jacco Vink, and Yuken Ohshiro\n- 2 Department of Astronomy, University of Geneva, Versoix CH-1290, Switzerland\n- 3 Department of Physics, Ehime University, Ehime 790-8577, Japan\n- 4 Department of Astronomy, University of Maryland, College Park, MD 20742, USA\n- 5 NASA / Goddard Space Flight Center, Greenbelt, MD 20771, USA\n- 6 Center for Research and Exploration in Space Science and Technology, NASA / GSFC (CRESST II), Greenbelt, MD 20771, USA\n- 7 Department of Physics, University of Tokyo, Tokyo 113-0033, Japan\n- 8 Department of Physics, Technion, Technion City, Haifa 3200003, Israel\n- 9 Center for Space Science and Technology, University of Maryland, Baltimore County (UMBC), Baltimore, MD 21250, USA\n- 10 Center for Astrophysics - Harvard-Smithsonian, MA 02138, USA\n- 11 Lawrence Livermore National Laboratory, CA 94550, USA\n- 12 Department of Astronomy, University of Michigan, MI 48109, USA\n- 13 SRON Netherlands Institute for Space Research, Leiden, The Netherlands\n- 14 ESO, Karl-Schwarzschild-Strasse 2, 85748, Garching bei Munchen, Germany\n- 15 Centre for Extragalactic Astronomy, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\n- 16 Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), Kanagawa 252-5210, Japan\n- 17 Department of Physics, Kyoto University, Kyoto 606-8502, Japan\n- 18 Department of Economics, Kumamoto Gakuen University, Kumamoto 862-8680, Japan\n- 19 Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan\n- 20 Department of Physics, Hiroshima University, Hiroshima 739-8526, Japan\n- 21 Department of Physics, Fujita Health University, Aichi 470-1192, Japan\n- 22 Department of Astronomy and Physics, Saint Mary's University, Nova Scotia B3H 3C3, Canada\n- 23 Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA\n- 24 European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), 2200 AG, Noordwijk, The Netherlands\n- 25 Faculty of Engineering, University of Miyazaki, Miyazaki 889-2192, Japan\n- 26 RIKEN Nishina Center, Saitama 351-0198, Japan\n- 27 Leiden Observatory, University of Leiden, P.O. Box 9513, NL-2300 RA, Leiden, The Netherlands\n- 28 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, MA 02139, USA\n- 29 Department of Physics, Saitama University, Saitama 338-8570, Japan\n- 30 Department of Physics, Rikkyo University, Tokyo 171-8501, Japan\n- 31 Faculty of Physics, Tokyo University of Science, Tokyo 162-8601, Japan\n- 32 Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan\n- 33 Department of Electronic Information Systems, Shibaura Institute of Technology, Saitama 337-8570, Japan\n- 34 Department of Earth and Space Science, Osaka University, Osaka 560-0043, Japan\n- 35 Department of Physics, University of Wisconsin, WI 53706, USA\n- 36 Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada\n- 37 Department of Physics, Nagoya University, Aichi 464-8602, Japan \n- 38 Science Research Education Unit, University of Teacher Education Fukuoka, Fukuoka 811-4192, Japan\n- 39 Hiroshima Astrophysical Science Center, Hiroshima University, Hiroshima 739-8526, Japan\n- 40 Department of Data Science, Tohoku Gakuin University, Miyagi 984-8588\n- 41 College of Science and Engineering, Kanto Gakuin University, Kanagawa 236-8501, Japan\n- 42 European Space Agency(ESA), European Space Astronomy Centre (ESAC), E-28692 Madrid, Spain\n- 43 Department of Science, Faculty of Science and Engineering, KINDAI University, Osaka 577-8502, JAPAN\n- 44 Department of Teacher Training and School Education, Nara University of Education, Nara 630-8528, Japan\n- 45 Astronomical Institute, Tohoku University, Miyagi 980-8578, Japan\n- 46 Department of Physics, Nara Women's University, Nara 630-8506, Japan\n- 47 School of Science and Technology, Meiji University, Kanagawa, 214-8571, Japan\n- 48 Yale Center for Astronomy and Astrophysics, Yale University, CT 06520-8121, USA\n- 49 Department of Physics, Konan University, Hyogo 658-8501, Japan\n- 50 Graduate School of Science and Engineering, Kagoshima University, Kagoshima, 890-8580, Japan\n- 51 Department of Physics, Chuo University, Tokyo 112-8551, Japan\n- 52 Faculty of Education, Shizuoka University, Shizuoka 422-8529, Japan\n- 53 Department of Astronomy, Kyoto University, Kyoto 606-8502, Japan\n- 54 Nihon Fukushi University, Shizuoka 422-8529, Japan\n- 55\n- Anton Pannekoek Institute, the University of Amsterdam, Postbus 942491090 GE Amsterdam, The Netherlands\n- 56 RIKEN Cluster for Pioneering Research, Saitama 351-0198, Japan\n- 57 Johns Hopkins University, MD 21218, USA\n- 58 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA\n- ∗ E-mail: yamaguchi.hiroya@jaxa.jp \nReceived 2024 June 28; Accepted 2024 August 20", 'Abstract': 'We present an initial analysis of the X-Ray Imaging and Spectroscopy Mission (XRISM) firstlight observation of the supernova remnant (SNR) N132D in the Large Magellanic Cloud. The Resolve microcalorimeter has obtained the first high-resolution spectrum in the 1.6-10 keV band, which contains K-shell emission lines of Si, S, Ar, Ca, and Fe. We find that the Si and S lines are relatively narrow, with a broadening represented by a Gaussian-like velocity dispersion of σ v ∼ 450 kms -1 . The Fe He α lines are, on the other hand, substantially broadened with σ v ∼ 1670 kms -1 . This broadening can be explained by a combination of the thermal Doppler effect due to the high ion temperature and the kinematic Doppler effect due to the SNR expansion. Assuming that the Fe He α emission originates predominantly from the supernova ejecta, we estimate the reverse shock velocity at the time when the bulk of the Fe ejecta were shock heated to be -1000 < ∼ V rs [kms -1 ] < ∼ 3300 (in the observer frame). We also find that Fe Ly α emission is redshifted with a bulk velocity of ∼ 890 kms -1 , substantially larger than the radial velocity of the local interstellar medium surrounding N132D. These results demonstrate that high-resolution X-ray spectroscopy is capable of providing constraints on the evolutionary stage, geometry, and velocity distribution of SNRs. \nKey words: ISM: individual objects (N132D) - ISM: supernova remnants - X-rays: ISM', '1 Introduction': "consistent with the age derived from the kinematic reconstruction of Law et al. (2020). \nSupernova remnants (SNRs) play a key role in the process of feedback within galaxies. They inject large amounts of kinetic energy into the interstellar medium (ISM), driving shock waves of thousands of kilometers per second, which heat interstellar gas and dust. Their ejecta enrich the ISM with freshly synthesized heavy elements, contributing to the chemical evolution of galaxies. Observations of SNRs provide key information about these processes, through their elemental abundances, morphology, velocity distribution, and thermal properties of the shock-heated materials, providing insight into the pre-explosion evolution and explosion mechanism of their progenitors. \nThe SNR N132D, located in the bar of the Large Magellanic Cloud (LMC), is one of the most studied objects of its class at virtually every wavelength: e.g., radio (Dickel & Milne 1995; Sano et al. 2020), infrared (Williams et al. 2006; Rho et al. 2023), and gamma-ray (Acero et al. 2016; Ackermann et al. 2016; H.E.S.S. Collaboration 2021). Its periphery has an elliptical shape of ∼ 110 '' × 80 '' along the major and minor axes, respectively, which corresponds to ∼ 27 pc × ∼ 19 pc at the distance to the LMC of 50 kpc (Pietrzy'nski et al. 2013). N132D is thought to be in a transitional phase from young to middle-aged (e.g., Bamba et al. 2018). Therefore, both swept-up ISM and ejecta contribute to its electromagnetic spectrum. \nFirst identified as an SNR in the radio band (Westerlund & Mathewson 1966), N132D was categorized as an 'oxygen-rich' SNR by optical observations (e.g., Danziger & Dennefeld 1976; Lasker 1978; Lasker 1980; Dopita & Tuohy 1984; Sutherland & Dopita 1995; Morse et al. 1995). Observations with the Hubble Space Telescope (HST) revealed strong emission of the C/Ne-burning products (i.e., O, Ne, Mg) with little emission of the O-burning products (i.e., Si, S), leading to an interpretation of a Type Ib supernova origin (Blair et al. 2000). A survey of the optical emitting ejecta in the [O III ] band revealed that the O-rich ejecta form a toroidal structure with a diameter of ∼ 9 pc, inclined at an angle of 28 deg to the line of sight (Law et al. 2020), consistent with earlier work of Vogt & Dopita (2011). Assuming homologous expansion from the center of the O-rich ejecta, Law et al. (2020) derived an average expansion velocity of 1745 km s -1 . It is worth noting that a toroidal structure of fast-moving ejecta is found in other core-collapse SNRs, such as Cas A (e.g., Milisavljevic & Fesen 2013), suggesting that the explosion of their progenitors was similarly asymmetric. More recently, the proper motion of the ballistic O-rich ejecta has been measured using multiple epochs of HST data, implying an age of 2770 ± 500 yr (Banovetz et al. 2023), \nThe first X-ray detection of N132D was made using the Einstein Observatory (Long & Helfand 1979; Mathewson et al. 1983), where the luminosity in the 0.5-3.0 keV band was reported to be ∼ 4 × 10 37 ergs s -1 . Hughes (1987) investigated its morphology in detail, and suggested that the SNR had evolved within a cavity in the ISM, likely formed by the pre-explosion stellar wind activity of the progenitor. Using ASCA, Hughes et al. (1998) found that the elemental abundances measured for the entire SNR are consistent with the mean LMC values (Russell & Dopita 1992), suggesting that the X-ray emission is dominated by the swept-up ISM. Later observations using XMM-Newton (Behar et al. 2001) and Chandra (Borkowski et al. 2007) revealed a complex object with ejecta emission mostly coming from the central portion of the remnant, where the X-ray emitting O-rich ejecta have a similar large-scale distribution as the optical emitting ejecta. Furthermore, these observations revealed that the Fe K emission, first detected by BeppoSAX (Favata et al. 1997), exhibits a centrally concentrated morphology, suggesting an ejecta origin. A more recent study by Sharda et al. (2020), who used the Chandra data, revealed that the Fe K emission is distributed throughout the interior of the southern half of the remnant. Suzaku and NuSTAR revealed the presence of Fe Ly α emission in addition to the Fe He α emission (Bamba et al. 2018), indicating that the Fe ejecta are hot ( kT e > ∼ 5 keV) and highly ionized. \nHigh-resolution X-ray spectroscopy of this SNR has been limited to the soft X-ray band ( < 2 keV). Using the crystal spectrometer on board the Einstein Observatory, Hwang et al. (1993) measured the intensities of the forbidden and resonance lines of Ne IX , which were partially resolved in its spectrum. The Reflection Grating Spectrometer (RGS) on board XMM-Newton resolved soft X-ray lines from K-shell ions of C, N, O, Ne, Mg, Si, as well as L-shell ions of S, Ar, Ca and Fe (Behar et al. 2001; Suzuki et al. 2020). In 2016, the Hitomi (ASTRO-H) mission observed N132D, using an X-ray microcalorimeter that could resolve the K-shell lines in the 2-10 keV band. However, it detected only 17 photons in the Fe K band, due to the extremely short exposure ( ∼ 3 . 7 ks) caused by attitude control loss during the observation (Hitomi Collaboration 2018). \nIn this paper, we report on the first high-resolution spectroscopy of N132D with high photon statistics in energies above 2 keV, enabled by the 'first-light' observation 1 of the X-ray Imaging and Spectroscopy Mission (XRISM) \n(Tashiro et al. 2020). This paper focuses on the velocity structure and thermal properties of the hot plasma that can be constrained by analyzing the well-resolved, strong thermal emission lines detected in the spectrum of the entire SNR. The results provide insights into the geometry and dynamical evolution of the SNR, which in turn constrain the nature of the progenitor and its environment. Other investigations, including a search for weak emission features and spatially-resolved spectroscopy, are left for future work. \nThis paper is organized as follows. Section 2 describes the details of the observations and data reduction. In Section 3, we present data analysis. The implication of the results are discussed in Section 4. Finally, we conclude this study in Section 5. Given that this is one of the first papers on the scientific outcomes of the XRISM mission, we provide detailed descriptions about the gain calibration and background model construction in Appendices. The errors quoted in the text and table and error bars given in the figures represent the 1 σ confidence level, unless otherwise stated.", '2 Observation and Data Reduction': "XRISM was launched aboard the H-IIA Launch Vehicle No. 47 on September 7th, 2023 from JAXA's Tanegashima Space Center. The spacecraft was placed into an approximately circular orbit with an inclination of ∼ 31 deg and altitude of ∼ 575 km. The XRISM scientific payload is comprised of two co-aligned instruments, Resolve (Ishisaki et al. 2022) and Xtend (Mori et al. 2022), located at the focal plane of two X-ray Mirror Assemblies (XMAs) with the same design. The former is an X-ray microcalorimeter, enabling non-dispersive high-resolution spectroscopy in the X-ray band. The latter is a traditional X-ray CCD detector, with a wide field of view of 38 ' × 38 ' . In this paper, we focus on spectroscopy of the Resolve data and use the Xtend data for imaging analysis only. \nThe observations of N132D were conducted twice in early December 2023 during its commissioning phase: the first starting at 22:01 UT on December 3 until 00:01 UT on 2023 December 7, and the second starting at 09:53 UT on December 9 until 03:46 UT on December 11. The nominal aim point for both observations was (RA, Dec) = (81. · 25849, -69. · 64122). The radial velocity component (toward N132D) of the Earth's orbital motion with respect to the Sun was about -2 . 7 km s -1 at the time of the observations. \nThe data were reduced utilizing the pre-release Build 7 XRISM software and calibration database (CALDB) libraries, representing updates of their Hitomi predeces- \nsors 2 . The data were reprocessed and screened by the automated XRISM processing pipeline version 03.00.011.008 that started on April 16, 2024.", '2.1 Resolve': "The Resolve observations of N132D were made through a ∼ 250µ mthick beryllium window (Midooka et al. 2020) in the closed aperture door, limiting the bandpass to energies above ∼ 1.6 keV. \nThe Resolve detector gain and energy assignment require correction of the time dependent gain on-orbit (see Appendix 1 for details). A set of 55 Fe radioactive sources on the filter wheel is periodically rotated into the aperture of the instrument to measure the gain. For the observations of N132D, the gain fiducials were measured every orbit for ∼ 30 minutes during earth occultation yielding 500-600 counts in the Mn K α line complex. In total, 72 gain fiducial measurements were conducted. Photon energy is assigned to each event using the fiducial gain curves based on the ground calibration and the standard nonlinear energy scale interpolation method (Porter et al. 2016). As a result, we achieved an energy resolution of 4.43 eV (FWHM) and an energy scale error of 0.04 eV for the N132D observation, confirmed with the fit to the Mn K α spectrum from the 55 Fe radioactive sources (see Appendix 1). On-orbit measurements using Cr and Cu fluorescent sources and Si instrumental lines give array composite systematic uncertainties in the energy scale of < 0 . 2 eV in the 5.4-8.0 keV band and at most 1.3 eV at the low energy end around 1.75 keV. \nEvent-based screening is applied based on that adopted for Hitomi data, with an updated energy upper limit to the pulse-shape validity (SLOPE DIFFER) cut (PI > 22000, 0.5 eV channels), and including the postpipeline RISE TIME and frame event coincidence screening (Kilbourne et al. 2018). GTI-filtering is applied to exclude periods of the Earth eclipse and sunlit Earth's limb, SAA passages, and times within 4300 s from the beginning of a recycling of the 50-mK cooler. The resulting combined effective exposure left after the event screening is 194 ks. We use only Grade (ITYPE) 0 (High-resolution primary) events for the subsequent analysis. \nA redistribution matrix file (RMF) was generated by the r slmkrmf task using the cleaned event file and CALDB based on ground measurements. 3 The following line-spread function components are considered: the Gaussian core, \nexponential tail to low energy, escape peaks, and Si fluorescence. An auxiliary response file (ARF) was generated by the x aarfgen task assuming a point-like source at the aim point as an input. We also generated another ARF using a Chandra image in the 0.5-5.0 keV band as an input sky image. These are consistent to better than 4% across the spectral fitting bandpass. In fact, we confirm no significant difference in results of the following analysis.", '2.2 Xtend': 'The Xtend instrument was operating in the full-window mode during the first observation and in the 1/8-window mode in the second observation, partly for calibration purposes. After the standard event screening, we obtained effective exposures of 123 ks and 71 ks for the first and second observations, respectively. At the time of the observations of N132D, one of the charge-injection rows overlapped with the aim point when the instrument was operating in the full-window mode (Figure 1a) 4 . Therefore, we use only the second observation data for imaging analysis. Figure 1b shows a three-color image of N132D obtained from the second observation: red, green, and blue correspond to 0.3-0.5 keV, 0.5-1.75 keV, and 1.75-10 keV, respectively.', '3 Analysis': 'Figure 2 shows the Resolve spectrum in the 1.6-10 keV band extracted from the entire field of view of 3 × 3 arcmin 2 or 35 pixels. The K-shell emission lines of Si, S, Ar, Ca, and Fe are detected with high significance. The presence of the Fe Ly α emission, first suggested in the Suzaku study (Bamba et al. 2018), is also confirmed. A remarkable difference is found between the He-like emission lines of the intermediate mass elements (IMEs) and Fe. The former are relatively narrow so the forbidden and resonance lines are well resolved, whereas the latter is significantly broadened. This is not an instrumental effect, since the spectral resolution (measured in eV) is approximately constant across the 1.6-10 keV band. It is therefore indicative that the IME and Fe emission originate from different plasma components. \nIn the following subsections, we first look at two narrow band spectra containing the K-shell emission of S ( § 3.1) and Fe ( § 3.2) separately to investigate the broadening and shift of these line complexes. We then analyze the fullband spectrum with physically realistic models of collisionally ionized plasma ( § 3.3). We apply the optimal binning \nthe spectral fits by the appropriate factor (the non-Ls fraction). The other spectral parameters are unaffected by this issue. \nmethod of Kaastra & Bleeker (2016) to the spectrum using the ftgrouppha task in FTOOLS. 5 A spectrum of non X-ray background (NXB) is constructed using the method described in Appendix 2 and taken into account in the following analysis. The spectral fitting is performed based on the C -statistic (Cash 1979), using the XSPEC software version 12.14.0 (Arnaud 1996).', '3.1 Sulfur K band': 'Figure 3 shows the spectrum around the He-like S emission. There are no spectral features due to the NXB in this band. To measure the centroid energy and width of the observed emission lines, we fit the spectrum with Gaussian functions and a bremsstrahlung continuum. The results are given in Table 1, where the rest-frame energies of the identified lines are also listed. We find that both the forbidden ( f ) and resonance ( r ) lines of S XV, whose centroid energies are constrained more stringently than the others, are slightly ( ∼ 2 eV) redshifted with respect to their rest frame energies. If this shift is purely due to the bulk motion of the plasma, the corresponding velocity is v ∼ 250 km s -1 . Given the systematic uncertainty in the gain calibration in this energy band (up to 1.3 eV: Appendix 1), this value is fully consistent with the heliocentric radial velocity of the interstellar gas surrounding N132D, 275 ± 4 km s -1 (Vogt & Dopita 2011). The 1 σ -width is ∼ 4 eV for these lines, corresponding to a velocity dispersion of σ v ∼ 500 km s -1 . \nInterestingly, we detect an unusual emission feature at 2405 eV at a ∼ 4 σ confidence level. If real, the most plausible origin would be highn transitions of Si XIII ( n ∼ 9 → 1), suggesting that charge exchange interactions are taking place. This interpretation is not unrealistic, given the presence of the dense molecular clouds at the periphery of N132D (Williams et al. 2006; Sano et al. 2020). A more quantitative analysis of the possible charge exchange emission will be presented in a future paper (XRISM Collaboration, in preparation).', '3.2 Iron K band': "Figure 4 shows the spectrum in the Fe K band. The broadened emission at ∼ 6.7 keV predominantly originates from the n = 2 → 1 transitions of He-like Fe, but could also have contributions from multiple features produced by lower-ionization states of Fe. The narrower emission feature detected at ∼ 6.95 keV is a mixture of two emission lines, Fe XXVI Ly α 1 and Ly α 2 . In addition, we find a \n<!-- image --> \nFig. 1. (a) Xtend image of N132D obtained with the full-window mode observation conducted from December 3 to 7, where the color corresponds to intensity. The 'gaps' are due to charge injection rows. (b) Xtend image obtained with the 1/8-window mode observation conducted from December 9 to 11. Red, green, and blue correspond to 0.3-0.5 keV, 0.5-1.75 keV, and 1.75-10 keV, respectively. \n<!-- image --> \nFig. 2. (a) Resolve spectrum of the SNR N132D in the 1.6-10 keV band. The red line represents the best-fit Model B spectrum (whose parameters are given in Table 3). The NXB contribution is also taken into account; the narrow features at 7.5 keV and 9.7 keV are Ni-K α and Au L α lines in the NXB, respectively. (b) Same as panel (a) but magnified in the 1.7-2.7-keV band. \n<!-- image --> \nTable 1. Emission lines detected in the S K band (2.3-2.6 keV). \n- ∗ Theoretical Rest-frame energies.\n- † Centroid of two lines. \n‡ Details of this emission feature are discussed in a separate paper (XRISM Collaboration, in preparation). \nFig. 3. The Resolve spectrum in the 2.3-2.6 keV band, where the S XV emission is prominent. Red and green are Gaussian functions and the bremsstrahlung continuum component of an ad hoc model, respectively. The NXB contribution is taken into account but is below the displayed flux level. \n<!-- image --> \nbroad feature around 7.9 keV, possibly a mixture of Fe XXV n =3 → 1 (He β ) emission and Ni XXVII n =2 → 1 (He α ) emission. We note that the contributions of the NXB, indicated by the blue line in Figure 4, are not negligible in this energy band. The narrow lines found at 7.5 keV and 8.0 keV are due to fluorescence of neutral Ni and Cu, respectively, well reproduced by our NXB model. \nSimilar to the previous subsection, we fit the spectrum with a phenomenological model consisting of Gaussian functions and a bremsstrahlung continuum, obtaining the results given in Table 2. The line parameters are consistent with the previous measurement by Bamba et al. (2018). We also note that the Fe He α centroid is constrained more stringently than (but consistent with) the Suzaku measurement (6656 ± 9 eV: Yamaguchi et al. 2014a), and significantly more tightly than the XMM-Newton measurement (6685 +15 -14 eV: Maggi et al. 2016). We then replace the 6.95keV Gaussian with two zgauss components (whose spectral parameters are the source-frame line energy, redshift, width, and flux) to be able to constrain the bulk velocity shift and broadening of a single emission line. We fix the \nFig. 4. The Resolve spectrum in the Fe K band. Red and green are the Gaussian functions and bremsstrahlung continuum components of an ad hoc model, respectively. Blue indicates the NXB spectrum. \n<!-- image --> \nTable 2. Emission line complexes in the Fe K band (6-9 keV). \nline energies to the theoretical rest-frame energies of the Ly α 1 and Ly α 2 lines (6973 eV and 6952 eV, respectively) and the Ly α 1 /Ly α 2 flux ratio to 2 (i.e., statistical weight ratio between the excited states), leaving the redshift and line width as free parameters. This analysis gives a redshift z =2 . 98 +1 . 02 -1 . 05 × 10 -3 and width σ E =17 . 4 +8 . 6 -11 . 9 eV, corresponding to a bulk velocity of v = 894 +306 -315 kms -1 and velocity broadening of σ v =749 +370 -512 kms -1 . The bulk velocity is substantially larger than that obtained from the S K band spectrum (and thus larger than the radial velocity of the LMC ISM).", '3.3 Fullband spectrum': "We now analyze the Resolve spectrum in 1.6-10 keV, the energy band with significant signal from the source. The \nTable 3. Best-fit parameters of the spectral fit in the 1.6-10 keV band. \nanalysis in the previous subsections has revealed that different bulk velocities are required to explain the energy shift observed in the S He α and Fe Ly α emission lines. Therefore, we start the spectral modeling with two components of a bvrnei model in XSPEC, which reproduces velocity-broadened emission from an optically-thin thermal plasma in non-equilibrium ionization (NEI), either ionizing or recombining. The free parameters are the electron temperature ( kT e ), ionization timescale ( τ = n e t , where n e and t are the electron density and time elapsed after the abrupt change of temperature, respectively), redshift ( z ), velocity dispersion ( σ v ), normalization, and the abundances of Si, S, Ar, Ca, and Fe. The abundance of each element is tied between the two components, whereas the other parameters (i.e., kT e , τ , z , σ v ) are left independent between the two. The Ni abundance is linked to the Fe abundance, and the initial plasma temperature ( kT init ) is fixed to 0.01 keV (which is equivalent to the assumption that the plasma is ionizing) for both components. The foreground absorption is not taken into account in our analysis, because its effect is negligible in this energy band due to the low column density to this SNR ( N H < ∼ 10 21 cm -2 ) (Dickey & Lockman 1990; Suzuki et al. 2020). This model yields the best-fit spectrum given in Figure 5a/5b. Although the overall spectrum is well reproduced by this model ( C -stat/dof = 2446.4/2233), it fails to reproduce the observed flux of the Fe Ly α emission. We find that the inferred \nelectron temperatures, kT e ∼ 0 . 83 keV and ∼ 2.3 keV (for the lowT e and highT e components, respectively), are too low to produce a sufficient fraction of H-like Fe ions, if the plasma is ionizing or in equilibrium. \nWe thus modify the kT init value of one of the bvrnei components to 30keV for introducing a recombining plasma (hereafter Model A), similar to the approach taken by Bamba et al. (2018). The best-fit model spectrum and parameters are given in Figure 5c/5d and the 'Model A' column of Table 3. We find that the flux of both Fe He α complex and Ly α emission are successfully reproduced by the highT e component. However, the velocity dispersion of this component ( σ v =1700 +150 -140 kms -1 ), which is mainly determined from the width of the Fe He α complex, is significantly larger than that obtained from the Gaussian modeling of the Fe Ly α emission ( σ v = 749 +370 -512 kms -1 ). In fact, the model does not reproduce well the profile of the Ly α line (Figure 5d). \nThis result leads us to another hypothesis: that different plasma components contribute to the Fe K band spectrum, so that the He α line complex is dominated by a plasma with larger line broadening and the Ly α emission by another plasma with moderate broadening. To confirm this possibility, we introduce a third bvrnei component (hereafter very-highT e component), assuming that all three components are ionizing plasma (hereafter Model B). For the very-highT e component, the redshift and veloc- \n0.1 \nFig. 5. (a) The Resolve spectrum in the 1.6-10 keV band, fitted with the two-component ionizing plasma model (red). The contributions of the lowT e and highT e components are indicated as magenta and green, respectively. Gray indicates the NXB spectrum. (b) Magnified spectrum in the Fe K band, showing that the Fe Ly α mission is not reproduced by the model. (c) and (d) Same as panels (a) and (b), respectively, but fitted with Model A. (e) and (f) Same as panels (a) and (b), respectively, but fitted with Model B. The very-highT e component is now added, which is indicated as blue. \n<!-- image --> \nity dispersion are fixed to the values constrained by the zgauss modeling for the Ly α emission (i.e., z =2 . 98 × 10 -3 and σ v =750 kms -1 ). Since the electron temperature and ionization age of this component are not well constrained with this complex model, we fix these parameters to kT e = 10keV and τ =1 × 10 13 cm -3 s, and expect that the majority of H-like Fe ions are associated with this component The best-fit model spectrum and parameters are given in Figure 5e/5f and the 'Model B' column of Table 3. This model yields a slightly better fit than Model A, especially around the Fe Ly α emission. We confirm that the line broadening ( σ v ) is significantly larger in the highT e component (that reproduces the Fe He α complex) than in the very-highT e component (that reproduces the Fe Ly α emission). The redshift values are also different between the two components. \nIn our spectral analysis, the elemental abundances have been tied among the two or three plasma components, since our models are incapable of constraining them independently (i.e., if the abundances of each component are fitted independently, constrained error ranges of several parameters become extremely large). The abundances of Si and S are determined mainly by the lowT e component (the magenta curve in Figure 5). Therefore, the actual abundances of Si and S are highly uncertain for the other components. Similarly, the Fe abundance is well constrained only for the highT e component (the green curve in Figure 5). Also notable is that the predicted continuum level of the veryhighT e component (the blue curve in Figure 5) is negligibly low compared to the observed continuum level in the whole energy band. Because of this, the observed spectrum can be successfully modeled even if the Fe abundance of this component is set to an extremely high value (e.g., > 10,000 solar). This implies that the very-highT e component could be pure-metal, such as in a deep layer of the supernova ejecta.", '3.4 Narrow band image': 'The result based on Model B in the previous subsection implies that the Fe He α and Ly α emissions originate from different plasma components. If this hypothesis is true, different spatial distributions of these emissions should be expected. We thus generate narrow band images of the Fe He α and Ly α emission as well as the S He α emission to search for morpological differences. The results are shown in Figure 6; the top and bottom panels are the Resolve and Xtend images, respectively. The distribution of the Fe Ly α emission is localized near the SNR center, whereas the Fe He α emission is more widely distributed. This difference is seen by both instruments, although the low statistics in \nTable 4. Summary of the measured bulk velocity and velocity dispersion. \nthe emission lines makes it difficult to determine their true morphology. The morphological difference between the S and Fe emission is also confirmed, consistent with previous studies (Behar et al. 2001; Borkowski et al. 2007; Sharda et al. 2020).', '4 Discussion': 'We have performed line-resolved spectroscopy of the thermal emission from N132D, using the XRISM first-light data. The Resolve spectrum in the 1.6-10 keV band can be modeled by two or three components of NEI plasmas with different electron temperatures. The K-shell emission lines of Si and S are characterized by an ionizing plasma with the electron temperature of ∼ 0.8 keV. The spectrum in the Fe K band (6-9 keV) is, on the other hand, reproduced by either a one-component recombining plasma (Model A) or two-component ionizing plasmas (Model B) with higher temperature. Although the goodness of the fit is comparable between the two models, the different spatial distributions between the Fe He α and Ly α emission, revealed by our imaging analysis (Figure 6), favors Model B as the more likely scenario. The energy shift and broadening of the thermal emission lines have been investigated to constrain the velocity structure of each component. Table 4 summarizes the results based on Model B, where v bulk is the heliocentric radial velocities corrected to the solar system barycentric standard of rest. We discuss the interpretation in the following subsections.', '4.1 Radial velocity': 'We have shown that all the detected emission lines from the IMEs (Si and S) and Fe are redshifted with respect to their rest frame energies. The bulk velocity measured from the IME lines and Fe He α complex are consistent with the radial velocity of the interstellar gas surrounding N132D, 275 ± 4 km s -1 (Vogt & Dopita 2011). On the other hand, the Fe Ly α emission indicates a larger velocity of ∼ 900 km s -1 . Notably, the Hitomi SXS study of this SNR indicated a similarly high bulk velocity of ∼ 1080 km s -1 (Hitomi Collaboration 2018), but this estimate was ob- \nFig. 6. Top: Raw photon count images of the Resolve in (a) 2.4-2.5 keV, (b) 6.5-6.8 keV, and (c) 6.92-6.97 keV, corresponding to the S He α , Fe He α , and Fe Ly α emission, respectively. The overplotted contours are the Xtend image in the 0.5-1.75 keV band. Bottom: Smoothed photon count image of the Xtend in (d) 2.3-2.6 keV, (e) 6.5-6.8 keV, and (f) 6.85-7.05 keV, respectively. \n<!-- image --> \n0.060 \ntained using only 17 photons detected in the Fe He α band (not in the Ly α band). Our measurement of the Fe He α bulk velocity, 249 +96 -117 kms -1 , is lower than the mean value of the Hitomi measurement, but still within its 90% confidence interval of 330-1780 km s -1 . \nThe large redshift observed in the Fe Ly α emission implies that this emission originates from the Fe-rich ejecta with a highly asymmetric velocity distribution. The ejecta scenario is also supported from the high electron temperature that can be achieved by a high-velocity shock. The morphology of the Fe Ly α emission is centrally concentrated (Figure 6), suggesting that this hot Fe ejecta component is present only at the far side of the SNR. Theoretically, an ionization timescale of τ > ∼ 10 12 cm -3 s is required to produce a sufficient fraction of H-like Fe in an ionizing plasma. Therefore, from the estimated SNR age of 2770 yr (Banovetz et al. 2023), the electron density of this component is estimated to be n e = τ/t age > ∼ 11 cm -3 . Such a high density implies that the Fe ejecta form a compact knot. In fact, the normalization of the very-highT e component given in Table 3 corresponds to the emitting volume of ∼ 7 × 10 55 ( n e / 11cm -3 ) -2 cm 3 , less than 0.1% of the total SNR volume ( ∼ 10 59 cm 3 ). \nA highly asymmetric distribution of Fe ejecta has been \nobserved in other core-collapse SNRs, such as Cas A (DeLaney et al. 2010; Hwang & Laming 2012) and G350.10.3 (Borkowski et al. 2020; Tsuchioka et al. 2021). Recent NuSTAR observations of Cas A and SN1987A also revealed that the velocity distribution of radioactive 44 Ti, produced in the same nuclear processes that produce 56 Ni (parent nucleus of Fe), is asymmetric toward one side (Grefenstette et al. 2014; Boggs et al. 2015), similar to what we observe in N132D. Such asymmetry could have been produced by a supernova explosion involving asymmetric effects, such as a standing accretion shock instability (Blondin et al. 2003; Janka et al. 2016), or an asymmetric interaction between the SNR ejecta and ambient medium. For N132D the latter scenario is not unlikely, since the SNR is known to be interacting with dense molecular clouds (Sano et al. 2020).', '4.2 Origin of the line broadening': "One remarkable finding from our high-resolution spectroscopy is that the Fe He α lines are substantially broadened, whereas the K-shell emission lines from Si and S are only slightly broadened (Table 4). There are two plausible causes for the line broadening: (1) thermal Doppler broad- \nning due to high ion temperature or (2) a variation of bulk motion along the line of sight. Here we simply assume σ v = √ σ 2 th + σ 2 kin , where σ th and σ kin indicate the broadening due to (1) and (2), respectively. Both depend on the shock velocity, or more precisely, the upstream bulk velocity in the shock-rest frame, v u, sh . In collisionless shocks, which are generally formed in SNRs, temperature equilibration among different species is not necessarily achieved at the immediate downstream region. In the most extreme case, the relation between the shock velocity and downstream temperature, derived from the Rankine-Hugoniot equations, hold independently among different species i as \nkT i = 3 16 m i v 2 u, sh , (1) \nwhere T i and m i are the temperature and mass, respectively (e.g., Vink et al. 2015). Although the process called 'collisionless electron heating' slightly modifies the temperatures from those predicted by Equation 1 (e.g., Ghavamian et al. 2007; Yamaguchi et al. 2014b), the effect is not essential in the situations we discuss below. The different species then slowly equilibrate to a common temperature via Coulomb collisions in further downstream regions, the timescale of which is discussed later. The thermal Doppler broadening is given as \nσ th = √ kT i m i . (2) \nTherefore, σ th =( √ 3 / 4) · v u, sh is expected when Equation 1 is strictly achieved. On the other hand, the downstream bulk velocity in the shock-rest frame is expected as v d, sh = (1 / 4) · v u, sh , assuming a compression ratio of 4. Therefore, the velocity in the observer frame is calculated as \nv d, obs = V s + v d, sh = V s + 1 4 v u, sh , (3) \nwhere V s is the shock velocity in the observer frame. The line profile depends on the velocity distribution and is not always Gaussian-like. As described in Appendix 3, we can approximate as σ kin ≈| v d, obs | / 2, when the distributions of the plasma density and velocity are spherically symmetric (e.g., expanding shell or sphere). \nThe previous Chandra study revealed that the K-shell emission from Si and S forms the outermost shell (e.g., Borkowski et al. 2007; Sharda et al. 2020), suggesting that the emission predominantly originates from the ISM shocked by the SNR blast wave (although the abundances slightly larger than the mean LMC values of Dopita et al. (2019) may imply a small contribution of ejecta to the emission). Figure 7 shows the thermal equilibration processes due to the Coulomb collisions in the postshock plasma, where various shock velocities (that determine the initial temperatures) and the ISM abundances measured \nFig. 7. Thermal equilibration process among different species via Coulomb collisions in post-shock plasma initially shocked by the blast wave propagating into the ISM. Black, red, and blue indicate temperatures of protons, sulfur, and iron, respectively. From bottom (thick curves) to top (thin curves), the upstream bulk velocities (i.e., blast wave velocity in the observer frame) of 500, 1500, and 3000 km s -1 are assumed. The elemental abundances measured by Suzuki et al. (2020) are assumed. The yellow region indicates the range of τ constrained for the lowT e component. \n<!-- image --> \nby Suzuki et al. (2020) are assumed. The temporal evolution of kT p , kT S , and kT Fe are given as a function of τ . It is clearly indicated that the ion temperatures are equilibrated with the proton temperature at the timescale constrained for the lowT e component (i.e., τ = (0.5-2) × 10 12 cm -3 s: Table 3). Therefore, the thermal Doppler broadening of the S He α lines in the ISM component is calculated to be \nσ th = √ kT S m S ≈ √ kT p m S = √ 3 16 m p m S · V bw , (4) \nwhere V bw (= -v u, sh ) is the blast wave velocity. The line broadening due to the bulk expansion is also obtained as \nσ kin ∼| v d, obs | / 2 = ( V bw -1 4 V bw ) · 1 2 = 3 8 V bw , (5) \nwhich is much larger than σ th . Therefore, we obtain σ v = √ σ 2 th + σ 2 kin ≈ (3 / 8) · V bw ≈ 450kms -1 , and thus V bw ≈ 1200kms -1 . \nUsing the proper motion detected with Chandra, Plucinsky et al. (2024) has measured the blast wave velocity at the bright southern rim to be 1709 ± 386 kms -1 , the mean of which is 1.4 times higher than our measurement. We note that this method directly measures the current shock velocity, whereas the Doppler broadening reflects the shock velocity in the past. It is, therefore, surprising that a larger velocity is obtained from the former. This discrepancy can be explained if the shocked ISM forms a toroidal or ellipsoidal structure with a somewhat low inclination angle as illustrated in Figure 8, rather than a spherically symmetric shell. This interpretation is supported by the fact that the line broadening is well charac- \nelocity in the early evolutionary stage of an SNR. \nAn alternative, more likely possibility is that the SN ejecta contributes significantly to the Fe He α emission, as suggested by some previous work (e.g., Sharda et al. 2020). In shocked ejecta, especially when it consists purely of heavy elements, the expected thermal evolution properties are distinct from the shocked ISM. This is because there are few or no protons or helium nuclei to first equilibrate with, and thus the heavy elements equilibrate with free electrons released from the heavy element atoms themselves. This is quantitatively shown in Figure 9, where the temporal evolution of kT Fe in a pure-Fe plasma is calculated for two cases with different assumptions for the initial conditions: (a) no collisionless electron heating taking place at the reverse shock, or (b) kT e /kT Fe = 0 . 1 is achieved immediately behind the reverse shock due to the efficient collisionless electron heating. Unlike the LMC abundance case (Figure 7), the Fe ions remain at their initial temperature until τ ∼ 10 12 cm -3 s, if the shock velocity is high enough ( > ∼ 3000 km s -1 ). This conclusion is not affected by the efficiency of the collisionless electron heating. Therefore, the thermal Doppler broadening is nonnegligible in this case. \nIn general, the reverse shock in an SNR initially moves outward, and then reverses direction moving inward after a few hundred to thousand years, depending on the ambient density structure and other properties (e.g., Truelove & McKee 1999). The upstream fluid velocity in the shockrest frame, which determines the heating properties, is expressed as \nv u, sh = R rs t -V rs , (6) \nwhere R rs is the reverse shock radius (and thus R rs /t is the free expansion velocity of the outermost unshocked ejecta) and V rs (= dR rs /dt ) is the reverse shock velocity in the observer frame (e.g., Vink et al. 2022). Since Equations 1 and 3 hold in the shock-rest frame, \nkT Fe = 3 16 m Fe ( R rs t -V rs ) 2 (7) \nand \nv ej , obs = 1 4 R rs t + 3 4 V rs (8) \nare obtained, where v ej , obs is the bulk velocity of the shocked ejecta in the observer frame. Assuming that the current Fe temperature is still comparable to the immediate postshock temperature (i.e., Eq. 7), we obtain the thermal Doppler broadening to be \nσ th = √ 3 4 ( R rs t -V rs ) . (9) \nFrom Equations 8 and 9 and an assumption of σ kin ∼ | v ej , obs | / 2, the total broadening is obtained as \nX-ray emitting torus \nObserver \nFig. 8. Top view of the interpreted geometry and velocity structure of the shocked ISM. The blueshifted and redshifted regions are shown in blue and red, respectively. The primary plane of the torus is indicated by the dotted line, but its inclination angle is arbitrary, as it is not constrained in this work. \n<!-- image --> \nterized by a Gaussian profile, because a shell-like geometry with a spherically symmetric velocity distribution leads to a top-flat line profile (see Appendix 3). \nAs mentioned in Section 1, optical observations of N132D revealed a toroidal geometry of the O-rich ejecta (Lasker 1980; Vogt & Dopita 2011) with an inclination angle of ∼ 28 deg (Law et al. 2020). The radius of this torus is ∼ 4.5 pc, smaller than the projected forward shock radius of ∼ 10 pc. Vogt & Dopita (2011) suggested that the Xray emitting ISM shell also forms a toroidal structure and that the tori of the ISM and optical emitting ejecta are aligned and thus physically associated with each other. If the three-dimensional geometry of the SNR is indeed toruslike, the progenitor of N132D must have exploded within a dense CSM with a disk-like geometry. Such CSM distribution is expected if the progenitor is in a binary system, because the pre-explosion stellar wind forms a dense CSM disk on the equatorial plane (e.g., Smith 2017). A similar scenario is suggested for SN 1987A (e.g., Podsiadlowski 2017), where a dense CSM ring is observed in various wavelengths (e.g., Ravi et al. 2024 and references therein). \nThe plasma component responsible for the Fe He α emission has a higher electron temperature and larger line broadening than that responsible for the S He α emission (Table 3). If the Fe He α emission originates from the swept-up ISM, this component should have been shockheated when the blast wave velocity was much higher than it is today. Similar to the previous estimate, we derive V bw ≈ 4450kms -1 in this case, not unreasonable as a shock \nFigure 10 shows the relation between V rs and v u, sh , with which σ v = 1670 +160 -170 kms -1 is expected (derived using Equations 6 and 10). The contributions of the thermal Doppler broadening (red) and the broadening due to the SNR expansion (blue) are also indicated. As expected, the former contribution is more significant when v u, sh is higher, which is expected for an inward moving reverse shock. We find that the observed broadening can be explained when -1000 < ∼ V rs [kms -1 ] < ∼ 3300, indicating that the Fe-rich ejecta in this SNR were shock-heated when the reverse shock was around the turnaround radius. This result is reasonable because it is theoretically expected that the reverse shock remains at a standstill for thousands of years in a middle-aged SNR (e.g., Micelotta et al. 2016). If \n<!-- image --> \n4 \nFig. 9. Thermal equilibration between Fe temperature (blue) and electron temperature (green) via Coulomb collisions in shocked ejecta with pure-Fe composition. From bottom (thick curves) to top (thin curves), the upstream bulk velocities of 1000, 3000, and 5000 km s -1 are assumed. Panels (a) and (b) assume the initial temperature ratio ( kT e /kT Fe ) of m e /m Fe (i.e., no collisionless electron heating at the reverse shock) and 0.1 (i.e., efficient collisionless electron heating), respectively. More details about the calculations are described in Ohshiro et al., in preparation. The yellow region indicates the range of τ constrained for the highT e component. \n<!-- image --> \nσ v = √ 13 64 ( R rs t ) 2 -18 64 R rs t V rs + 21 64 V 2 rs . (10) \nFig. 10. Relation between the reverse shock velocity in observer frame ( V rs ) and the upstream bulk velocity in shock-rest frame ( v u , sh = R rs /t -V rs ) that satisfies σ v = √ σ 2 th + σ 2 kin = 1670 kms -1 (black curve). The observed line width is indicated as the green region with its statistical uncertainty. The red and blue curves indicate the contributions of σ th and σ kin , respectively, for given V rs . The dotted curves correspond to the upper and lower limits of the observed line width. \n<!-- image --> \nthe reverse shock velocity is exactly 0 km s -1 in observer frame, v u, sh = R rs /t ≈ 3700 kms -1 is expected. With this upstream velocity, Fe temperature is expected to remain > ∼ 1 MeV up to τ ∼ 10 12 cm -3 s (Figure 9). Therefore, a significant fraction of the observed line broadening can naturally be attributed to the thermal Doppler broadening. \nAlthough our analysis has successfully provided an estimate for the evolutionary stage of the reverse shock and the thermal properties of the shocked ejecta, it should be emphasized that our analytical model given above is oversimplified. For instance, the geometry of the shocked ejecta must be more complex than we assume, and thus the approximation of the Gaussian-like line profile is likely too simplistic. It is also possible that Fe ions in the shocked ejecta are moderately equilibrated with electrons, and thus the actual thermal Doppler broadening could be slightly smaller. In fact, the Fe Ly α emission shows a narrower width than the Fe He α complex (Table 4), which can be explained if the plasma responsible for the Fe Ly α emission is more equilibrated so that the ion temperature becomes lower. We should also note that, if other heavy elements (e.g., Si and S) are also involved in the thermal evolution of the ejecta, the Fe temperature drops more quickly than is predicted in Figure 9 due to the Coulomb interactions among the heavy elements. If the initial Fe temperature is substantially higher than the current temperature, the required upstream velocity becomes higher than our estimates in Figure 10. To constrain the physical quantities more precisely, detailed calculations based on hydrodynamical simulations need to be employed.", '5 Conclusions': 'We have presented analysis of the XRISM first-light observation data of N132D, the X-ray brighest SNR in the LMC. The excellent performance of the X-ray microcalorimeter Resolve has enabled us to perform high resolution spectroscopy of this SNR in the 1.6-10-keV band, for the first time. Our analysis has revealed that the K-shell emission lines of Si and S, whose origin is thought to be the swept-up ISM, are mildly broadened with σ v ∼ 450 km s -1 . Under an assumption of a nearly symmetrically expanding ISM shell, the radial component of the blast wave velocity is estimated to be ∼ 1200 km s -1 , lower than the recent proper motion measurement with Chandra. On the other hand, the Fe H α complex emission is substantially broadened with σ v ∼ 1670 km s -1 . If this emission originates from the ejecta, the observed line width can be explained through a combination of the thermal Doppler broadening due to the high ion temperature in the non-equilibrium plasma and kinematic Doppler effect due to the expansion of the shocked ejecta. We have also provided an estimate for the evolutionary stage of the SNR, putting a constraint on the reverse shock velocity (in observer frame) to be -1000 < ∼ V rs [kms -1 ] < ∼ 3300, the value at the time when the bulk of the Fe ejecta were shock-heated. If V rs ≈ 0 km s -1 , which is reasonably expected from the theoretical point of view, the current Fe temperature is estimated to be > ∼ 1 MeV. The redshift observed in the Si, S, and Fe He α lines is equivalent to the radial velocity of the ISM surrounding N132D ( ∼ 250 km s -1 ), whereas that observed in Fe Ly α indicates a substantially larger radial velocity of 890 km s -1 , about 600 km s -1 greater than the radial velocity of the local ISM in the LMC. Also our imaging analysis revealed that the Fe Ly α emission is concentrated around the SNR center. These results suggest that the Fe Ly α emission originates from hot Fe ejecta that are present only at the far side of the SNR, distinct from the component responsible for the Fe He α emission. \nThis work represents the first paper to be published by the XRISM Collaboration. The results presented here are uniquely obtained using the spectral power of the Resolve combined with the imaging of Xtend, offering only a modest glimpse of the power of this observatory that will revolutionize our understanding of the Universe.', 'Acknowledgments': 'The XRISM team acknowledges the hundreds, likely thousands, of scientists and engineers in Japan, the United States, Europe, and Canada who contributed to not only this mission, but to all predecessors that came before. This mission is a testament to the long-standing collaborations between the countries and \nspace agencies involved. The authors deeply thank Prof. Kiyoshi Hayashida, who passed away on October 2, 2021, for his significant contribution to the project and whole X-ray astronomy. HY is thankful to Dr. Anne Decourchelle for her helpful comments on this manuscript and to Dr. Daniel Patnaude for discussion about interpretation of the observational results. We also thank the anonymous referee for an insightful and constructive review that improved this work. \nThis work was supported by JSPS KAKENHI grant numbers JP22H00158, JP22H01268, JP22K03624, JP23H04899, JP21K13963, JP24K00638, JP24K17105, JP21K13958, JP21H01095, JP23K20850, JP24H00253, JP21K03615, JP24K00677, JP20K14491, JP23H00151, JP19K21884, JP20H01947, JP20KK0071, JP23K20239, JP24K00672, JP24K17104, JP24K17093, JP20K04009, JP21H04493, JP20H01946, JP23K13154, JP19K14762, JP20H05857, JP23K03459, and JP22KJ1047, and NASA grant numbers 80NSSC20K0733, 80NSSC18K0978, 80NSSC20K0883, 80NSSC20K0737, 80NSSC24K0678, 80NSSC18K1684, and 80NNSC22K1922. LC acknowledges support from NSF award 2205918. CD acknowledges support from STFC through grant ST/T000244/1. LG acknowledges financial support from Canadian Space Agency grant 18XARMSTMA. AT and the present research are in part supported by the Kagoshima University postdoctoral research program (KU-DREAM). SY acknowledges support by the RIKEN SPDR Program. IZ acknowledges partial support from the Alfred P. Sloan Foundation through the Sloan Research Fellowship. Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The material is based upon work supported by NASA under award number 80GSFC21M0002. This work was supported by the JSPS Core-to-Core Program, JPJSCCA20220002. The material is based on work supported by the Strategic Research Center of Saitama University.', 'References': 'Acero, F., Ackermann, M., Ajello, M., et al. 2016, ApJS, 224, 8 Ackermann, M., Albert, A., Atwood, W. B., et al. 2016, A&A, 586, A71 \nArnaud, K. A. 1996, Astronomical Data Analysis Software and Systems V, 101, 17 \nBamba, A., Ohira, Y., Yamazaki, R., et al. 2018, ApJ, 854, 71 Banovetz, J., et al. 2023, ApJ, 948, 33 \nBehar, E., et al. 2001, A&A, 365, 242 \n- Blair, W. P., Morse, J. A., Raymond, J. C., et al. 2000, ApJ, 537, 667\n- Blondin, J. M., Mezzacappa, A., & DeMarino, C. 2003, ApJ, 584, 971\n- Boggs, S. E., Harrison, F. A., Miyasaka, H., et al. 2015, Science, 348, 670 \nBorkowski, K.J., Hendrick, S.P., & Reynolds, S.P. 2007, ApJ, 671, 45 \nBorkowski, K. J., Miltich, W., & Reynolds, S. P. 2020, ApJL, 905, L19 \nCash, W. 1979, ApJ, 228, 939 \nDanziger, I. J. & Dennefeld, M. 1976, ApJ, 207, 394 \nSmith, N. 2017, Handbook of Supernovae, 403 Sutherland, R.S. & Dopita, M.A. 1995, ApJ, 439, 365 Suzuki, H., Yamaguchi, H., Ishida, M., et al. 2020, ApJ, 900, 39 Tashiro, M., Maejima, H., Toda, K., et al. 2020, Proc. SPIE, 11444, 1144422 Truelove, J. K. & McKee, C. F. 1999, ApJS, 120, 299 Tsuchioka, T., Uchiyama, Y., Higurashi, R., et al. 2021, ApJ, 912, 131 Vink, J., Broersen, S., Bykov, A., et al. 2015, A&A, 579, A13 Vink, J., Patnaude, D. J., & Castro, D. 2022, ApJ, 929, 57. doi:10.3847/1538-4357/ac590f Vogt, F. & Dopita, M. A. 2011, Ap&SS, 331, 521 Westerlund, B.E. & Mathewson, D.S. 1966, MNRAS, 131, 371 Williams, B.J., et al. 2006, ApJ, 652, 33 Yamaguchi, H., Badenes, C., Petre, R., et al. 2014, ApJL, 785, L27 Yamaguchi, H., Eriksen, K. A., Badenes, C., et al. 2014, ApJ, 780, 136.', 'Appendix 1 Resolve Gain calibration': 'The Resolve detector gain and energy assignment requires correction of the time dependent gain on-orbit. The Resolve calorimeter detectors are thermal detectors and they thus have both a bolometric and transient response. The bolometric response reflects the thermal radiation environment within the Resolve instrument and impacts the gain of the transient response to X-rays. In addition, the detector gain is impacted by the detector heat sink temperature which is regulated to better than 1 µ K on orbit, and the temperature of the amplifier and control electronics that are affected by spacecraft orientation. To compensate for these effects, the Resolve detector gain is tracked as a function of time using on-board calibration sources. The time dependent energy scale is then reconstructed as a function of time by interpolating a family of temperature dependent gain curves measured during ground calibration (Porter et al. 2016). To measure the gain, a set of 55 Fe radioactive sources on the filter wheel are periodically rotated into the aperture of the instrument. The Mn K α X-rays from the radioactive sources are fit using an empirically measured core line-shape (Hoelzer et al. 1997) for each fiducial interval. For N132D, gain fiducials were measured every orbit for ≈ 30 minutes during earth occultation yielding 500-600 counts in the Mn K α line complex giving a statistical uncertainty in the energy scale of between 0.15-0.2 eV at 5.9 keV for each fiducial interval. N132D was observed early in the commissioning phase for the XRISM mission, before the final time dependent methodology was adopted. In later observations, the time dependent gain fiducial measurements were optimized since far fewer fiducial measurements are needed to track the detector gain. The N132D observations dis- \nFig. 12. Resolve calibration pixel spectrum of Mn K α X-rays from a pencil beam 55 Fe radioactive sources that continuously illuminates the calibration pixel. Two data sets are shown, one during the same fiducial intervals used to track the gain of the main array, and the other just during the observation, not including the fiducial intervals. In both cases, the energy scale reconstruction is the same as that used for the main array and the comparison demonstrates the efficacy of the energy scale reconstruction for this observation. Fits using a Gaussian instrumental function yield energy resolutions of 4.42 and 4.39 eV (FWHM), and energy scale errors of 0.04 and 0.11 eV at 6 keV respectively. \n<!-- image --> \nFig. 11. The Resolve spectrum of Mn K α X-rays from 55 Fe radioactive sources on the filter wheel. The spectrum is a composite of the pixels in the main array and measured during fiducial intervals once per orbit during earth occultation. A fit using the standard Gaussian instrumental function yields an energy resolution of 4.43 eV (FWHM), and an energy scale error of 0.04 eV after energy scale reconstruction. \n<!-- image --> \nd here included 72 gain fiducial measurements, far in excess of what is needed to track the gain with equivalent precision. Later observations of equivalent length would have included only 15 fiducial measurements. \nIn order to assess the energy scale reconstruction and the detector energy resolution during the observation, we perform several additional checks using high-resolution primary (Hp) grade events. The first is to fit the fiducial Mn K α complex per-pixel and as an array composite after energy scale reconstruction. Figure 11 shows the array composite Mn K α complex, the underlying natural line shape (Hoelzer et al. 1997), and a fit using a Gaussian instrumental function. The fit gives an energy resolution of 4.43 eV (FWHM), and an energy scale error of 0.04 eV, both of which are within 0.1 eV of the standard on-orbit performance. Additionally, we use a calibration pixel to verify the performance of the instrument during the main observation outside of the gain fiducials. Resolve includes a standard pixel, identical to the main array, but just outside the field of view. The calibration pixel is continuously illuminated with a pencil beam 55 Fe radioactive source allowing the detector gain and energy resolution to be continuously tracked. Using the calibration pixel, we compare the energy scale reconstruction and energy resolution during the same fiducial intervals as the main array and also just during the observation but using the same energy scale reconstruction method as the main array. These two data sets are shown in Figure 12. Fits using the standard Gaussian instrumental function yield an energy resolution of 4.42 eV (FWHM) and an energy scale error of 0.04 eV during the fiducial intervals, and a resolution of 4.39 eV \n(FWHM) and an energy scale error of 0.11 eV during the N132D observations. On-orbit measurements using Cr and Cu fluorescent sources and a Si instrumental line, give array composite systematic uncertainties in the energy scale of < 0 . 2 eV in the 5.4-8.0 keV band and at most 1.3 eV at the low energy edge of the band at 1.75 keV. On-orbit energy scale measurements and analysis are on-going and we expect that these uncertainties will be reduced in the future with a goal of < 0 . 1 eV across the Resolve bandpass.', 'Appendix 2 Resolve Non X-ray Background': "We developed a model for the Resolve instrumental background (or non-X-ray background, NXB) based on ∼ 7 month data accumulated during periods of Earth eclipse, supplemented by the Hitomi SXS NXB (Kilbourne et al. 2018) and 'blank-sky' data. Starting with the Resolve eclipse data, we applied standard screening, identical to that applied to the on-source data, and produced a spectrum composed by aggregating data from periods of different cut-off rigidity according to the weighting found in the N132D observations. The NXB level is < 10 -3 s -1 keV -1 for the entire array over the energy range of interest. \nWe developed a model for the NXB continuum from this early Resolve NXB data set, but we did not fit the instrumental lines due to distortions and shifts introduced by the initial per-pixel energy scales of the many eclipse segments. Instead, we turned to the Hitomi SXS NXB, deter- \nmined the strengths of the detected instrument lines, and compared these to the line strengths in 226 ks of Resolve blank-sky data with better aligned pixel energy scales than the preliminary Resolve NXB data set. \nThe amplitude of the Au L α 1 line is consistent between the two data sets, but the Mn K α line is significantly weaker in the Resolve data, as expected from ground data. This feature is the result of scattered X-rays from the collimated 55 Fe source pointed at the dedicated calibration pixel, the design of which was modified for Resolve. We determined that the Mn K α line in the Resolve blank-sky data is consistent with that determined in a high-statistics ground measurement, adjusted for the half-life of 55 Fe. The statistics of the other lines were not adequate to inform the model, thus we used the SXS line strengths for all the instrumental lines except Mn K α . We approximated the following 12 lines by Gaussians: Al K α , Au M α 1 , Mn K α 1 , Mn K α 2 , Ni K α 1 , Ni K α 2 , Cu K α 1 , Cu K α 2 , Au L α 1 , Au L α 2 , Au L β 1 , and Au L β 2 . Although better line profiles are known for most of these lines, the statistics of the observation do not warrant a more accurate specification in the model. We separately specified the K α doublets to capture the widths of these profiles, and fixed their normalizations at 2:1. When applying the NXB model to the data of N132D, we adjusted the normalizations of the Mn K α , Ni K α , and Au L α 1 lines, so that their intensities match the observation.", 'Appendix 3 Line Profile for Symmetrically Expanding Shell and Sphere': 'In our spectral analysis, a Gaussian velocity broadening (implemented in the bvrnei model in XSPEC) is assumed to fit the broadened emission lines. However, broadening due to the SNR expansion depends on the radial velocity distribution of the shocked plasma, being not necessarily Gaussian-like. Here we investigate line profiles expected for symmetrically expanding shell and sphere, and compare them with the Gaussian profiles. \nFirst we consider a spherically symmetric shell with a thickness of R out / 12, where R out is the outermost radius (and thus the inner radius is 11 R out / 12), typically expected for the swept-up ISM of SNRs in the Sedov phase. The expansion velocity at the radius r is assumed to be v = v out · ( r/R out ), where v out is the velocity at the radius R out . We also assume uniform plasma density within the shell. Figure 13a shows the expected profiles of a single emission line at 2.45 keV (corresponding to the S He α emission) for v out =450 kms -1 (red), 800 km s -1 (blue), and 1275 kms -1 (green), compared with a velocity-broadened Gaussian with σ v = 450 kms -1 (black). The spectra are \nconvolved with the Resolve RMF and ARF generated in Section 2. We find that a similar line width is expected in the expanding shell model with v out =800 kms -1 and in the Gaussian with σ v =450 kms -1 . We then generate mock Resolve spectra for various v out values ranging from 100 to 2000 km s -1 and fit them with a Gaussian model. Figure 13b shows the relation between the best-fit σ v values and input v out , approximated by a linear function of σ v ≈ 0 . 55 v out . Note that the expanding shell model predicts a characteristic top-flat profile. For this reason, a Gaussian function is not a good approximation, especially when the outermost expansion velocity is high. \nNext we consider a uniform density sphere. Again, the expansion velocity is assumed to be proportional to the radius, v = v out · ( r/R out ). Figure 13c shows the expected profiles of a single emission line at 6.65 keV (corresponding to the Fe He α emission) for v out =1000 kms -1 (red), 2000 kms -1 (blue), and 3000 km s -1 (green), compared with a velocity-broadened Gaussian with σ v =1000 kms -1 (black). Among the three, the second profile gives the best approximation to the Gaussian. Figure 13d shows the relation between σ v and v out for the uniform sphere model, derived similarly to Figure 13b. We find that the relation is approximated by a linear function of σ v ≈ 0 . 45 v out . \nWe make similar investigations for a spherically symmetric shell with different thickness of R out / 12 < l shell < R out , and find that the relation of σ v ≈ α · v out is obtained with 0 . 45 < ∼ α < ∼ 0 . 55. We also find that their is no substantial dependence on the photon energy in this relation. Considering the complexity in actual velocity distribution, we simply assume σ kin ≈ 0 . 5 v out in Section 4.2. \nFig. 13. (a) Profiles of a single emission line at 2.45 keV, expected for the expanding shell model with v out = 450 kms -1 (red), 800 km s -1 (blue), and 1275 kms -1 (green), compared with a velocity-broadened Gaussian with σ v =450 kms -1 (black). (b) The relation between σ v and v out for the expanding shell model. See text for details. (c) Profiles of a single emission line at 6.65 keV, expected for the expanding uniform sphere model with v out =1000 kms -1 (red), 2000 km s -1 (blue), and 3000 km s -1 (green), compared with a velocity-broadened Gaussian with σ v =1000 kms -1 (black). (d) Same as Panel (b), but for the uniform sphere model. The spectra in Panel (a) and (c) are convolved with the Resolve RMF and ARF. \n<!-- image --> \n10'}

2024arXiv240911175L

Vision foundation models which have demonstrated significant potential in multimedia applications are often underutilized in the natural sciences. This is primarily due to mismatches between the nature of domainspecific scientific data and the typical training data used for foundation models leading to distribution shift. Scientific images can have unique structure and characteristics and researchers often face the challenge of optimizing model performance with only a few hundred or thousand labeled examples. Furthermore the choice of vision foundation model can be nontrivial as each model exhibits unique strengths and limitations influenced by differences in architecture training procedures and the datasets used for training. In this work we evaluate the application of various vision foundation models to images from optical and radio astronomy. Our results show that use of certain foundation models improves the classification accuracy of optical galaxy images compared to conventional supervised training. Similarly we achieve equivalent or better performance in object detection tasks with radio images. However their performance in classifying radio galaxy images is generally poor and often inferior to traditional supervised training results. These findings suggest that selecting suitable vision foundation models for astrophysics applications requires careful consideration of the model characteristics and alignment with the specific requirements of the downstream tasks.

2024-09-01T00:00:00Z

['arXiv:2409.11175', '2024arXiv240911175L', '10.48550/arXiv.2409.11175']

['Astrophysics - Instrumentation and Methods for Astrophysics']

Vision foundation models can they be applied to astrophysics data

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11175.pdf

{'Erica Lastufka': 'Department of Computer Science University of Geneva Geneva, Switzerland 1211 erica.lastufka@unige.ch', 'Vitaliy Kinakh': 'University of Geneva Geneva, Switzerland 1211 vitaliy.kinakh@unige.ch', 'Mariia Drozdova': 'University of Geneva Geneva, Switzerland 1211 mariia.drozdova@unige.ch', 'Davide Piras': 'University of Geneva Geneva, Switzerland 1211 davide.piras@unige.ch', 'Svyatoslav Voloshynovskiy': 'University of Geneva Geneva, Switzerland 1211 Svyatoslav.Voloshynovskyy@unige.ch', 'Abstract': 'Vision foundation models, which have demonstrated significant potential in multimedia applications, are often underutilized in the natural sciences. This is primarily due to mismatches between the nature of domain-specific scientific data and the typical training data used for foundation models, leading to distribution shift. Scientific images can have unique structure and characteristics, and researchers often face the challenge of optimizing model performance with only a few hundred or thousand labeled examples. Furthermore, the choice of vision foundation model can be non-trivial, as each model exhibits unique strengths and limitations, influenced by differences in architecture, training procedures, and the datasets used for training. In this work, we evaluate the application of various vision foundation models to images from optical and radio astronomy. Our results show that use of certain foundation models improves the classification accuracy of optical galaxy images compared to conventional supervised training. Similarly, we achieve equivalent or better performance in object detection tasks with radio images. However, their performance in classifying radio galaxy images is generally poor and often inferior to traditional supervised training results. These findings suggest that selecting suitable vision foundation models for astrophysics applications requires careful consideration of the model characteristics and alignment with the specific requirements of the downstream tasks.', '1 Introduction': "In recent years, foundation models have enabled significant advancements in both language and image processing. These large models, trained on vast amounts of data encompassing multiple domains, serve as a cornerstone for numerous everyday applications. Foundation models are designed either for representation learning, where the goal is to capture essential characteristics of the data, or for generative purposes, where the model attempts to produce new data samples similar to the \ntraining distribution. In this work, we focus on the potential of the learned representations from vision foundation models applied to astrophysical images. \nA variety of architectural innovations and training methodologies have been employed to capture complex patterns in large datasets of natural images, some scraped from the internet and others, like ImageNet [DDS + 09], more curated. For many computer vision applications, this provides a robust starting point for further adaptation via fine-tuning for specialized downstream tasks (DSTs). However, the diversity and complexity inherent in foundation models pose substantial hurdles for scientists in fields such as physics, astronomy, and biology, who wish to apply these tools for domain-specific inquiries. \nThe objectives and data used for scientific DSTs are neither known nor used during the training of foundation models. Foundation models are typically trained in a self-supervised or weakly supervised way. Thus, in accordance to the information bottleneck principle [TPB00], it is not obvious whether these models are able to retain features related to the DST, namely sufficient statistics, in their learned representations. If, as suggested, only DST-related information should be kept in the model's representations (and the rest filtered out), it is difficult to determine what, if any, relevant information a foundation model trained on ImageNet might have for astronomy. Furthermore, since there is likely a misalignment between the training data distribution and the DST data distribution, this can result in the problem of distribution shift . \nIn addition to theoretical concerns, practitioners face the challenge of finding optimal fine-tuning techniques . Given that labeled data available for scientific applications is typically limited, identifying the appropriate training objective, data augmentations, model architecture, and training hyperparameters is essential to enhance the model's performance. Firstly, there is the challenge of selecting an appropriate model, which is both critical and non-trivial, requiring a deep understanding of the model's compatibility with the DST data distribution. Secondly, performance variation across different tasks and datasets can be perplexing. Some models excel with minimal labeled data, while others might require substantial label-rich datasets to perform effectively. This performance disparity is often not linear and depends on the nature and volume of the training examples provided for the DST. Thirdly, the feasibility of deploying these models is constrained by available computational capabilities, which varies dramatically based on the model's scale -- from small to giant models. Finally, all of this investigation requires time and substantial effort from domain experts in both the topic of interest and in machine learning best practices. \nThe goal of our paper is to address this first challenge by investigating the differences between foundation models and the implications for obtaining optimal task-specific performance. Our experiments used images from both optical and radio astronomy, and we examined classification and detection tasks. By applying various foundation models to these datasets and DSTs, this paper seeks to bridge the gap between advanced machine learning techniques and practical scientific research.", '2 Use of Foundation Models in Astrophysics': "Figure 1 illustrates the differences between natural images that comprise common training datasets like ImageNet and images from optical and radio astronomy. Unlike natural images, astrophysics images tend to have the following properties: \nSparseness : most images consist of several objects that occupy a small fraction of the total image. Histograms of radio and optical images demonstrate how many pixels have very low brightness. \nNoise : systematic noise is present in the images (this is especially notable for the radio images). \nHigh dynamic range : the brightness of the objects in the image can span several orders of magnitude, so traditional normalization is difficult. The ideal combination of filters or scaling for image display is often determined by research goals (e.g., emphasizing diffuse emission over compact sources). \nArtefacts : instrumental effects or residuals from image reconstruction can form spurious structures of different scales in the images. \nBecause of these fundamental differences, it is commonly assumed in the academic community that vision models trained on natural images are inadequate for performing tasks in astronomy, especially radio astronomy, where images are mathematically reconstructed from sparse samples measured in the Fourier plane. \nFigure 1: Random samples from each of the following datasets: ImageNet-1K (top row), GalaxyMNIST (second row), Radio Galaxy Zoo (RGZ, third row), MeerKAT MGCLS (bottom row). GalaxyMNIST and RGZ are labeled according to morphology class, while MGCLS is labeled by the number of compact sources present. White boxes indicate object bounding boxes used for source detection, when applicable. The rightmost two columns display sample images in their raw data format, unscaled, with an accompanying histogram. GalaxyMNIST combines data from the Dark Energy Camera's r , g , and z channels, while radio images are single-channel continuum images reconstructed from Fourier space, which can therefore include negative values. \n<!-- image --> \nWhile there are many extremely specialized DSTs in both optical and radio astronomy, often taking advantage of the many wavelength channels available, some more common tasks involving a single or small number of channels are listed in Table 1. Machine learning can offer a number of applicable techniques, and pretrained foundation models can be of use in most cases.Table 1: Common tasks in astrophysics and their machine learning analogues.", '2.1 Data': "In this work we evaluated galaxy morphology classification and source detection, two tasks common to both optical and radio astronomy. Details of the datasets used to perform these tasks are in Table 2, and sample images, along with histograms of their un-normalized original images, are in Figure 1. \nClassification datasets are images with a single galaxy centered in the cutout. GalaxyMNIST (GMNIST, [WLG + 22]) is a balanced dataset of four categories: smooth and round (SR), smooth and cigar-shaped (SC), edge-on-disk (E), and unbarred spiral (U). Radio Galaxy Zoo (RGZ) is unbalanced, with labels determined according to number of distinct radio components (C) and number of intensity \npeaks (P) in each source [WGA + 24]. Possible combinations are: 1C 1P, 1C 2P, 1C 3P, 2C 2P, 2C 3P, and 3C 3P, a total of 6 classes. Labeling was done by citizen scientists who performed inspection by varying the relative intensity of continuum radio emission and infrared observations. It is not always visually obvious from the normalized PNG images if there is a difference between a cutouts containing the same number of bright peaks - for example, one containing a single galaxy with two peaks, and another two different components with one peak each. Image pre-processing was by necessity done differently than for natural images, with techniques such as below-threshold zero replacement or channel-specific normalization. Full details are given in their documentation, and for classification both datasets are used as-is, with no additional pre-processing performed. \nTable 2: The datasets used in this study. \nSource detection datasets also consist of cutouts from wide-field images; they may have one to six galaxies in the central area, as in RGZ, or have tens of galaxies in a single cutout, as in MGCLS. Examples of bounding boxes around sources are shown in Figure 1. MGCLS labels are consistent in that the boxes only designate compact sources, and not other examples of extended emission or larger sources that might also be present in the images. Because RGZ's labels also contain extended sources, bounding boxes can much larger and filled with a large amount of noise.", '2.2 Foundation Models': "Foundation models exhibit significant variability in terms of their architectures, training, and performance characteristics. These differences arise from factors such as the size and nature of the training datasets, the number of parameters they incorporate, and their underlying architectural frameworks. Moreover, these models leverage a carefully curated set of data augmentation techniques and diverse pretraining strategies, each tailored to minimize a specific loss function. Table 3 lists the foundation models investigated in this study. With the exception of DINOv2, all were pretrained on ImageNet-1k's 1.2 million training images. \nTable 3: The foundation models used in this study. \nMAE . The Masked AutoEncoder (MAE, [HCX + 21]) uses masked image modeling (MIM) pretraining, reconstructing masked image patches from only a few visible patches. Unlike NLP inspired approaches like BeIT [BDPW21], which discretize visual tokens through an autoencoder, MAE directly processes the visible image patches through an encoder. The resulting output is then combined with mask tokens to reconstruct the original image using a decoder. Reconstruction error is used as the objective training function. \nDINO . DINOv2 [ODM + 23] improves upon its predecessor, DINOv1 [CTM + 21], by utilizing a larger and more curated dataset, LVD-142M. DINOv2 integrates the DINO cross-entropy loss with \nthe MIM objective employed in iBOT [ZWW + 22]. DINOv2 benefits from the Sinkhorn-Knopp batch normalization technique used in SwAV [CMM + 20]. The model employs a teacher-student training paradigm, where a teacher network computes the feature representation of global views of an image, while a student network computes the feature representation of local views, a series of smaller crops. The model is optimized to train the student to replicate the teacher's output, and the teacher is periodically updated using an exponential moving average (EMA) of the student's parameters. \nMSN . Masked Siamese Networks (MSN, [ACM + 22]) combine MIM with Siamese networks to avoid pixel-level and token-level reconstructions. MSN also uses teacher-student training, with the student network computing the feature representation of a partially masked image. \nResNet . Residual Networks are a staple of computer vision, introduced by in [HZRS16]. They are convolutional neural networks (CNNs) that use skip connections to bypass one or more layers, allowing the network to maintain accuracy over very deep networks. ResNet18 consists of 8 blocks, while ResNet50 has 16. We use the pre-trained weights available through torchvision [mc16], which were obtained by training in a supervised fashion for classification of ImageNet-1k using cross-entropy loss. A series of very specific data augmentations, including TrivialAugment [MH21], random erasing, mixup and cutmix, helped increase top-1 accuracy relative to the original augmentation scheme.", '2.3 UMAP Illustration': "Figure 2: GMNIST (upper row) and RGZ (lower row) UMAP for features extracted using DINOv2, MSN, MAE, and Resnet50. \n<!-- image --> \nWe provide empirical evidence that foundation models learn different representations of the same data. Extracting features from each foundation model and dataset, performing PCA and dimensionality reduction using the first 10 components with UMAP illustrates the latent space (Figure 2). We chose UMAP over T-SNE as UMAP better preserves global structure, which is important when local structure in the images might be dominated by noise. For the unbalanced RGZ dataset, a random sample with balanced classes is displayed. \nAlthough UMAP attempts to distill the relationships between thousand-dimensional vectors into two dimensions, it can still reveal clusters where images with similar embeddings reside. This can be deceptive because the model may have not learned relevant embeddings; visualizing the associated class labels can offer insight. \nThe UMAP representation of the DINO and MSN feature spaces for GMNIST is quite similar. Classes SR and U, representing smooth-and-round and unbarred spiral galaxies, are in distinct clusters, although MSN features show more overlap between the two than DINO's. The majority of SR and U galaxy images are separated by ResNet50 as well, although small clusters of these find themselves in other areas. Galaxies that appear predominantly elliptical - smooth-and-cigar-shaped (SC) and edge-on-disk (E) - share similar latent space embeddings; clustering is most distinct for DINOv2. \nLatent space grouping according to class is far less evident with the RGZ data. The best visible separation is seen in ResNet50. The ResNet model we used was trained with the goal of image classification, as opposed to DINO and MSN, which seek to reconcile local and global image characteristics, and MAE, whose objective is image reconstruction. \nThrough visualization of the latent space, we see that our chosen foundation models retain more representations that are class-relevant for GMNIST than RGZ. This is not to say that the learned representations are completely unrelated to the information contained in RGZ, simply that the most important features (according to PCA) are not strongly correlated with image class.", '2.4 Downstream Task Study Methodology': "For image classification , we used the models 'out-of-the-box' by attaching a single-layer linear classifier head to the pre-trained foundation model backbone. Only the classifier was allowed to train, so the backbone is considered frozen in that the weights and biases of the backbone stay fixed. Full fine-tuning of the system by also allowing the backbone parameters to update is known to improve performance, but corresponding compute time is over a hundred times more. As we are investigating model selection, we do not compare with full fine-tuning or hyperparameter optimization. We do compare results with fully supervised classifiers; in this case, both the backbone network used as well as the classifier head were allowed to train. Except for supervised training, all tasks were able to be performed on a NVIDIA GeForce RTX laptop GPU. Training and evaluation took between 10 seconds (10% of training data) to 45 seconds (all training data) per run. Supervised training took from 30 minutes (10% of training data) to 4 hours (all training data) on two NVIDIA TITAN Xp GPUs. \nSource detection was done in a similar fashion, using Faster-RCNN [RHGS17]. The implementation of Faster-RCNN is slightly different for Vision Transformers and CNNs. For a Vision Transformer backbone, a simple feature pyramid network (FPN) based on only the output of the last large-stride feature map of the backbone is used [LMGH22]. The Faster-RCNN implementation using ResNet also uses a FPN, but one that is heirarchical, acting on feature maps from different convolutional blocks in the ResNet rather than just the last one. \nAfter the backbone and FPN, object detection is done by applying a sliding window of the region proposal network (RPN) that predicts whether an object is present or not. These proposals are pooled (RoI pooling) and finally bounding boxes and classes are predicted. We considered the FPN, RPN and RoI layers to comprise the detection head in this situation, as these components are what is attached on top of the backbone network. \nWe again considered the case of a frozen backbone and trainable detection head and compare this to networks trained in a fully supervised fashion. For this source detection task, we also compared with fine-tuning the backbone, as compute time was on the same order of magnitude whether the backbone was frozen or fine-tuned (unlike for classification). Training time on two NVIDIA TITAN Xp GPUs was between approximately 6 to 10 hours depending on the architecture and whether the backbone was frozen or fine-tuned. Training from scratch took 5 to 11 hours.", '3 Galaxy Morphology Classification': "Galaxy morphology classification is a common task in both optical and radio astronomy, and large labeled datasets have been produced thanks to Galaxy Zoo projects [WLG + 22, WGA + 24]. Previous deep learning approaches in optical classification have relied heavily on CNNs trained from scratch (e.g. [DT18, MKEB21, PPOB23, UF24]). A few works have fine-tuned ImageNet-trained models, such as [HRJLW22]'s DenseNet121 and [KPP21]'s EfficientNetB5. Some works have investigated combining the advantages of ViT's attention mechanism with CNNs, such as [WLD + 24], [Dag23] and \n[CXD + 24]. However, application of pure ViTs has been limited and resulted in lower performance than CNNs [LLH + 22], although [KSI23] reports fine-tuning to be better than training from scratch. Radio galaxy classification work has been more limited due to lack of large datasets and a well-defined labeling scheme, and so far only employs CNNs; [BVPG21] and [NGW + 23] provide comprehensive reviews of the topic. Due to the difficult nature of the data, traditional ML techniques, feature engineering, and finding augmentations specific to radio images, are also active fields of study, while some seek to enhance training datasets with synthetic images using generative deep learning. \nWe demonstrate the galaxy morphology classification task with both optical and radio datasets labeled courtesy of the Galaxy Zoo project. Twenty percent of RGZ images contain two labeled galaxies, and four percent contain more than two; these images were all excluded from the classification dataset, reducing the total number of samples to 4692 training images and 1189 test images. Therefore the training data available for RGZ was just over half that available for GMNIST, which was split with 80% of its 10K images in the training set. The images in both datasets were already normalized, so the only data transformation we performed was to scale the images up to the same size (224 × 224 pixels) from their native resolutions (GMNIST: 64 × 64, RGZ: 132 × 132). \nHyperparameters for the classifier training were similar for both datasets, using a learning rate of 0.0005 and a batch size of 64. Classifiers were trained for 100 epochs on GMNIST and 200 on RGZ. Class weights inversely proportional to the number of samples present in the training dataset were used in the cross-entropy loss function used to classify RGZ. To illustrate the performance differences in low- and high-label regimes, the number of samples used for training was varied at 10%, 30%, 50% and 100%, while the test set was kept the same. For comparison, we trained a ViT-Base model with patch size 16 from scratch in a fully-supervised manner, as well as ResNets with both 50 and 18 layers. The best-performing supervised model in both cases was ResNet50. \nFigure 3: F1 scores for optical (left) and radio (right) galaxy morphology classification, as a function of percentage of training labels. Error bars show the maximum and minimum scores out of three different runs. \n<!-- image --> \nFigure 3 shows classification F1 score as a function of training label percentage. While results differ according to dataset, some trends are in common. Performance generally improved when the number of training labels increased, MAE was the worst performer overall, and the smaller ResNet18 did not classify as well as ResNet50. \nOn GMNIST, it was notable that the use of a foundation model improved classification relative to fully supervised training by up to 15%. In fact, the only configuration where starting from a foundation model was outperformed by supervised training is MAE with all available training labels. As supervised training takes hours rather than the tens of seconds required to train a linear classifier layer on frozen, there is a clear benefit to using a foundation model for this particular dataset. Even starting from a model pre-trained on natural images, one observes large increases in performance and speed. \nDINOv2 and MSN both performed better than the ResNets, although ResNet50 improved the most when trained on 100% of the data. One might explain the superior performance of DINO and MSN by \nthe number of parameters of their ViT-Base architectures, were it not for MAE which is also ViT-Base. Unlike DINO and MSN, which use student-teacher paradigms to reconcile partially masked or locally cropped views of images, MAE's objective is to reconstruct an entire image from a few patches. We speculate that MAE retains too much information that is irrelevant to classification; because large portions of GMNIST images consist of empty space, this might overly contribute to the learned representations. \nClassification was more difficult with Radio Galaxy Zoo, with F1 scores of less than 0.6 even when using all available training labels. This could be because more of the image is dominated by noise, and less by the galaxy to be classified. The labeling schematic could be another reason for low scores, as discussed in Section 2.1. Only MSN out-performed supervised training, and only for a low percentage of training labels. \nThe poor performance on radio galaxy classification is not consistent with results from previous works such as [SSW + 24] and [LBT + 24], which have shown much higher F1 scores (up to 0.94 for DINOv2) achieved on the MiraBest dataset [PS23] using pre-trained foundation models. MiraBest is designed for binary classification in the Fanarhoff-Riley scheme, which could be roughly described as a single Gaussian component with one peak versus a single component with two peaks. \nChallenges with the RGZ dataset may stem from the more subtle characteristics that distinguish the classes. Figure 4 (left side) shows that classification for galaxies with two radio components and either two or three bright peaks is especially poor. Re-labeling the images by the number of bright peaks resulted in F1 scores of up to 0.74 (seen on the right side of Figure 4), with all models but MAE out-performing a supervised baseline. Conversely, when re-labeling the images by the number of distinct radio components, the maximum F1 score achieved was 0.63, from MSN with all labels. \nFigure 4: Left: F1 score per class for classification on RGZ. Models can confidently identify single Gaussian sources but struggle with multi-peaked or multi-component sources. Right: Label scheme is re-defined to the number of bright peaks (top) or number of individual radio components (bottom). Higher F1 scores in the top chart show that vision models are better at distinguishing bright peaks in images rather than flux islands defining radio components. \n<!-- image --> \nIt seems that foundation models struggle to identify distinct radio sources, especially when they involve multiple emission peaks. Radio flux islands, containing one or more peaks, are usually identified through analytic source-finding algorithms via connected-component labeling, starting by selecting regions of pixels above a certain flux threshold. This threshold is usually up to 10 times the background RMS, and the brightest pixels it contains can be orders of magnitude greater. This information of relative flux is lost after compression and normalization of the images. \nUnlike GMNIST, RGZ images are reconstructions which contain noise with strong inter-pixel correlations on the scale of the synthesized beam. These statistical differences between ImageNet and radio images suggest that a well-performing model may require knowledge of a very specific set of features; a true case of distribution shift. While the out-of-the-box application of foundation models to GMNIST images performed well, it is not sufficient for classification on RGZ where it offers little to no advantage over supervised training.", '4 Source detection': 'Source detection in both optical and radio images is largely performed analytically. Work using deep learning to replace the time-consuming analytic pipelines has been done by [WWR + 19, FOD + 20, BAC + 19, RMS + 23], mainly through application of Faster- or Mask-RCNN. [SMF + 23] provides an overview of both CNN and ViT-based methods, including YOLO. More recent results based on simulated radio data include [TBDZ + 23] and [CSS + 24], which both surpass the capabilities of analytic source detection. \nSource detection was performed on radio continuum datasets RGZ, which mostly contains single galaxies labeled by morphology, and MGCLS, where a single image contains from 10-100 individual compact sources. MGCLS images may also contain larger sources which are not labeled. \nTable 4: Source detection results \nAs was done for classification, images were upscaled from their native resolution. For RGZ, scaling to 224 × 224 meant that the average single object size became 55 × 55 pixels, while the 25th percentile were 24 × 24 pixels. MGCLS crops were doubled in size from 256 × 256. This was done so that the average compact source occupied 16 × 16 pixels, and the 25th percentile 14 × 14 pixels. The size of the sources relative to the transformer patch size is important; sources unable to occupy most of a patch were difficult to detect. This was proven empirically - performance on detection with MGCLS increased by more than 10% when the image size doubled. \nWe chose to use Faster-RCNN to compare performance on object detection. Our first experiment kept the backbones frozen, allowing only the detection head layers (FPN, RPN and ROI) to train. Due to differences in architecture this resulted in a different number of trainable parameters: 21M for ViT-Det, 17.8M for Resnet50 with FPN, and 11M for Resnet18. Because of the structure of the FPN for ViT-Det, for this task the DINO backbone used was DINOv1, which had available pre-trained weights for the architecture ViT-Base with 16x16 patch size (DINOv2 uses 14x14). In the second experiment we allowed the weights and biases of the backbone networks to update as well (fine-tuning). \nWe kept the default training data augmentations: a randomly applied blur, contrast adjustment, and color jitter. Different learning rates, varied in order to ensure a smooth decay of the training loss, were required depending on the dataset and network. Training was for 100 epochs, with a batch size of 16. Table 4 reports the mean average precision at an intersection-over-union (IOU) threshold of 50%, known as the mAP@50. \nResults showed that MSN and DINO are good foundation models for this task. They outperformed training from scratch, even when the backbone was frozen, except for MSN on MGCLS. Although the ResNet models did not outperform training from scratch when used as a frozen backbone, they \nequaled or exceeded that performance when fine-tuned. Detection on galaxies in the RGZ dataset appears to be a task particularly suited to ResNets, while Vision Transformers do very well at detecting compact sources in MGCLS. Generally, detection was very good on MGCLS for networks larger than ResNet18. Possible reasons for the higher mAP@50 include the single category (compact source) and the tens of sources present in the images compared to the 1-6 sources in RGZ. Additionally, bounding boxes in RGZ can be quite large, including a lot of noise and making them harder to predict accurately. \nThese results show that transfer learning, starting with the pre-trained weights of a foundation model, is a good practice, as performance from even a frozen backbone was not worse than training from scratch, and fine-tuning improved it. It is important to ensure that properties of the architecture, such as patch size in the case of ViTs, are complementary to the data being used for the downstream task.', '5 Conclusions': 'Our results demonstrate that state-of-the-art vision foundation models can immediately perform classification of optical galaxies and source detection of radio galaxies with 72 - 88% precision. In the case of optical galaxy classification, all foundation models out-performed supervised training. F1 scores of close to 0.85 for 800 labeled training images, 10% of the available total, show that foundation models can be used to great effect for initial investigations into using machine learning for scientific tasks. Of the models tested, Vision Transformers DINOv2 and MSN performed the best, along with CNN ResNet50. MAE, despite its success on natural images, is optimized for reconstruction loss and failed to encode characteristics which are highly relevant for galaxy classification. \nShared characteristics of the model training distribution and the images of GMNIST might have contributed to good classification performance. Even though GMNIST images contain a lot of empty space, a good portion of the image is occupied by the galaxy of interest. If stars and other objects are present in the background, they are small relative to the galaxy, even if the galaxy is seen edge-on. On the other hand, classification of radio galaxies was challenging, as even the main radio source was small and the images were dominated by noise and patterns from reconstruction. In this case, supervised training outperformed all foundation models except MSN. \nFor a dataset like RGZ with a clear distribution shift from training to down-stream task data, applicability of foundation models is limited. The models did retain some relevant information about the data - in this case, mainly the information about bright peaks - and designing the data augmentations or label schemes to take advantage of that could improve results. \nThe need for compatibility between the dataset, down-stream task, and foundation model appears in the source detection example as well. Once images were scaled so that the Vision Transformer patch size was smaller than the average radio source, source detection results were encouraging. High mAP@50 scores showed that foundation models larger than ResNet 18 could identify radio sources, especially when the backbone was allowed to fine-tune. However, improvements due to fine-tuning were inconsistent and (in the case of MSN) much smaller than might be expected. This situation might improve given more optimal methods of fine-tuning. \nEven in cases where results are good, such as source detection, there remains a large gap in performance between tasks done on scientific images and tasks done on natural images. State-of-the-art classification on thousand-class ImageNet-1k has a top-1 accuracy of 92%, while the best object detection on the standard COCO dataset has a mAP@50 of 0.73, which may seem low in comparison to our results but is the average metric across all 90 classes. Closing the gap in performance is necessary to achieve scientifically meaningful results, but it will have to be done with far less computing and human resources than are available to large companies. Techniques which can operate with very few labeled training examples and by training a small number of network parameters will be especially valuable for applying vision foundation models to scientific images. In this respect, the semi-supervised learning proposed in [VTK + 20] is a very interesting option for further investigation. \nIn this study, we considered very simple way of fine-tuning. Perhaps a more powerful fine-tuning method, based on the simultaneous use of labeled and unlabeled data, might enhance the results, as suggested in [VTK + 20]. This would be particularly exciting for astrophysics, where unlabeled data is plentiful. In future work, we will examine the potential of such semi-supervised learning framework for fine-tuning.', 'Data availability': 'GalaxyMNIST ( https://github.com/mwalmsley/galaxy\\_mnist ) and MGCLS ( https:// doi.org/10.48479/7epd-w356 ) are public datasets. Radio Galaxy Zoo ( https://radio. galaxyzoo.org/ ) will soon have its first data release, and the dataset used here is available upon reasonable request. The code and instructions to reproduce these experiments is available at https://github.com/elastufka/fm4astro .'}

2024EPJC...84..994R

Dark matter an important portion of compact objects can influence different phenomena in neutron stars. The spontaneous scalarization in the scalartensor gravity has been proposed for neutron stars. Here we investigate the spontaneous scalarization in dark matter admixed neutron stars. Applying the dark matter equations of state we calculate the structure of scalarized neutron stars containing dark matter. The dark matter equations of state are based on observational data from the rotational curves of galaxies and the fermionic selfinteracting dark matter. Our results verify that the spontaneous scalarization is affected by the dark matter pressure in neutron stars. Depending on the central density of scalarized dark matter admixed neutron stars the dark matter pressure alters the central scalar field. The increase of dark matter pressure in lowdensity scalarized stars amplifies the central scalar field. However the pressure of dark matter in highdensity scalarized stars suppresses the central scalar field. Our calculations confirm that the stars in the merger event GW170817 and in the lowmass Xray binary 4U 182030 can be scalarized dark matter admixed neutron stars.

2024-10-01T00:00:00Z

['2024arXiv240907328R', 'arXiv:2409.07328', '10.48550/arXiv.2409.07328', '2024EPJC...84..994R', '10.1140/epjc/s10052-024-13361-w']

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

Effects of dark matter on the spontaneous scalarization in neutron stars

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

https://arxiv.org/pdf/2409.07328.pdf

{'Effects of Dark Matter on the Spontaneous Scalarization in Neutron Stars': 'Fahimeh Rahimi 1 , 2 and Zeinab Rezaei 1 , 2 ∗ \n1 Department of Physics, School of Science, \nShiraz University, Shiraz 71454, Iran. \n2 Biruni Observatory, School of Science, \nShiraz University, Shiraz 71454, Iran.', 'Abstract': 'Dark matter, an important portion of compact objects, can influence different phenomena in neutron stars. The spontaneous scalarization in the scalar-tensor gravity has been proposed for neutron stars. Here, we investigate the spontaneous scalarization in dark matter admixed neutron stars. Applying the dark matter equations of state, we calculate the structure of scalarized neutron stars containing dark matter. The dark matter equations of state are based on observational data from the rotational curves of galaxies and the fermionic self-interacting dark matter. Our results verify that the spontaneous scalarization is affected by the dark matter pressure in neutron stars. Depending on the central density of scalarized dark matter admixed neutron stars, the dark matter pressure alters the central scalar field. The increase of dark matter pressure in low-density scalarized stars amplifies the central scalar field. However, the pressure of dark matter in highdensity scalarized stars suppresses the central scalar field. Our calculations confirm that the stars in the merger event GW170817 and in the low-mass X-ray binary 4U 1820-30 can be scalarized dark matter admixed neutron stars.', 'I. INTRODUCTION': 'Dark matter (DM) which can be captured by relativistic objects affects the various astrophysical phenomena in these stars. Scattering of the DM with nucleons and hyperons in neutron stars (NSs) concerns the transferring of large momentum which leads to the influence of the nucleon structure, the strong interactions, and the momentum dependence of the hadronic form factors on the DM capture rate [1, 2]. The effective capture of DM by NSs alters the cooling process in these stars via the DM impact on the neutrino emissivity resulting in the enhancement of the star cooling and internal relaxation rates [3]. Electroweak multiplet DM is effectively captured in NSs with a large elastic scattering cross-section leading to the possibility of the temperature observation of the NSs [4]. DM-nucleon scattering cross section enhances by the NSs, giving rise to detecting of DM through NS spectroscopy [5]. \nThe sensitivity of NS heating to different properties of DM can be employed to explore the DMimpacts on the relativistic stars [6-12]. Heating of NSs which takes place via the energy transfer by the inelastic DM [6, 7] and leptophilic DM [8] may be detected using infrared telescopes. Capturing the muonphilic DM in NSs is possible due to the existence of stable muons and degenerate electrons in these stars leading to the NS heating up kinetically and via the annihilations [9]. DM scattering and annihilation in the nuclear pasta of NS crust result in the heating of the NS showing the importance of the NS crust as a thermal detector of DM [10]. GeV Dirac fermion DM may also be captured by the NSs causing the heating of NSs via the deposited kinetic energy [11]. Exploring the pseudoscalar-mediated DM through the DM-induced heating of NSs is more favorable compared to the direct mediators and DM searches [12]. \nAstrophysical observations confirm the existence of DM admixed compact stars as well as the significance of DM effects on these objects [13-15]. GW170817 and GW190425 merger events can be regarded as relativistic stars containing both nuclear and bosonic self interacting DM [13]. Relativistic mean-field models [14] and general relativistic model based on the self-interacting scalar field [15] show that the compact component in the GW190814 merger event is a massive NS admixed with DM particles. Properties of DM admixed relativistic stars have been vastly investigated in the literature. The interplay between DM and baryonic matter causes significant effects on NSs [16-19]. Energy injection from DM self-annihilations \ncan cause quark matter droplet nucleation in DM admixed NSs (DMANSs) [16]. Fermionic DM interacting with the nucleons via the Higgs portal grows the rate of DMANS cooling [17]. Thermodynamic properties of symmetric nuclear matter, pure neutron matter, and NS matter are affected by DM in NSs [18]. DM which interacts with the hadronic and quark matter via the exchange of Higgs boson in hybrid NSs changes the discontinuity on the energy density, the star minimum mass, and the mass-radius relation of hybrid stars admixed with DM [19]. DM also modifies the NS equation of state (EoS) [20-23]. Asymmetric bosonic DM as a core in NSs leads to the effective softening of the EoS while it causes the stiffening of the EoS when considered as a DM halo [20]. DM interactions in DMANSs result in the softening of the EoS and the decrease of star maximum mass, radius, and tidal deformability [21-23]. Moreover, the curvature of NSs and their binding energy [24], the gravitational wave emitted from NSs [25], and the distribution of DM as a dense dark core or an extended dark halo in NSs [26] are influenced by the properties of DM in these stars. In consequence, the DMANSs can be considered as probes for constraining the DM characteristics [27, 28]. NSs give constraints on the DM particle mass as well as the DM fractions in the star [27]. High energy neutrino from NSs presents the bounds on the long-lived DM mediators [28]. \nSpontaneous scalarization of NSs which is a tachyonic instability as a result of the nonminimal coupling of scalar field and curvature can arise in the scalar-tensor gravity. This theory of gravity predicts different properties for the NSs compared to the ones in general relativity [29-36]. NSs in scalar-tensor gravity have maximum compactness lower than the compactness of the general relativistic stars [29]. The x-ray pulse profile from the hotspots on the NS surface can be influenced by the star scalar field and deviates from the one in the general relativity [30-32]. The characteristics of NS in the scalar-tensor gravity as well as its scalarization depend on the mutual interplay between magnetic and scalar fields of NSs [33]. NS spontaneous scalarization alters the NS magnetic deformation and the emitted gravitational waves from these stars [34]. Scalar tensor gravity which leads to the scalarized NSs influences the iron line from accreting NSs [35]. For oscillating NSs, scalar-tensor gravity results in the modes in its spectrum which are different from the ones in general relativity [36]. In addition, the observational data related to NSs are used to constrain the parameters of the scalar tensor gravity [37-39]. Applying the data from the NS-black hole gravitational wave events and using the Bayesian inference, the scalar-tensor gravity can be \nconstrained [37]. The properties of spontaneous scalarization in scalar-tensor gravity have been constrained via the data related to the mass and radius of NSs [38, 39]. \nMassive scalar-tensor gravity [40-42], scalar Gauss-Bonnet gravity [43, 44], tensor multiscalar theories of gravity [45-47], multi-scalar Gauss-Bonnet gravity [48], and degenerate higher-order scalar-tensor theories [49] have been considered to explore the scalarized NSs. In scalar-tensor gravity with a massive scalar field, NS maximum mass is higher than the one in general relativity [40]. Considering the self-interacting massive scalar field, the mass of the scalar field and its self-interaction affect the properties of scalarized NSs [41, 42]. In extended Gauss-Bonnet scalar-tensor theories, the scalarization of NSs is due to the curvature of the spacetime rather than the NS matter [43]. Scalar Gauss-Bonnet gravity verifies that scalarized NSs can be formed from the core collapse of a nonscalarized star [44]. Tensor multi-scalar theories of gravity forecast the spontaneous scalarization in the new classes of relativistic stars called topological [45] and non-topological [46] NSs. The spontaneous scalarization of non-topological NSs can also take place in multi-scalar GaussBonnet gravity [48]. Non-uniqueness of scalarized NSs and two solutions for scalarized stars have been reported in tensor multi-scalar theories of gravity [47]. In degenerate higher-order scalar-tensor theories, the mass and radius of NSs are obtained with higher values compared to the ones in general relativity [49]. \nRegarding the above discussions on the DM in relativistic stars and the properties of NSs in scalar-tensor gravity, it seems that the DM can also affect the spontaneous scalarization in NSs. In the present work, we investigate how the DM alters the NSs in scalar-tensor gravity and its spontaneous scalarization. In Section II, the EoSs for the DM which are employed in this paper are introduced. Section III belongs to the model description of DMANSs in the scalar-tensor gravity. In Section IV, we discuss the structural properties of scalarized NSs that contain the DM. Section V concerns the Summary and Conclusions.', 'II. DARK MATTER EQUATIONS OF STATE': 'In this paper, we explore the impacts of the DM pressure on the scalarized NSs. To express the DM EoS, we utilize two models which describe the DM pressure. In the first model, the DM EoS is given by the observational data from the rotational curves of galaxies [50]. The pseudo-isothermal model results in the mass density profile which has regularity \nat the origin. The velocity profile, geometric potentials, and gravitational potential give the EoS in the pseudo-isothermal density profile with the following form [50], \nP DM 1 ( ρ DM ) = 8 p g π 2 -8 [ π 2 8 -arctan √ ρ g ρ DM -1 √ ρ g ρ DM -1 -1 2 ( arctan √ ρ g ρ DM -1 ) 2 ] , (1) \npg \npg \npg \n= \n0.1 \n= \n0.5 \n= \n0.8 \nwith the density, ρ DM , and the pressure, P DM 1 , of DM. Besides, ρ g and p g are the free parameters denoting the central density and pressure of galaxies. For the DM in NSs, the free parameters, ρ g and p g , are of the order of the central density and the pressure of NSs, respectively [51]. Figure 1 (Left) presents the DM pressure in the first model, assuming the value ρ g = 0 . 2 × 10 16 g/cm 3 and different values of p g and also the value ρ 0 = 1 . 66 × 10 14 g/cm 3 . The increase in p g leads to the stiffening of the DM EoS. \n2.0 \n1.5 \n1.0 \n0.5 \n0.0 \n0 \n2 \n4 \n6 \n8 \n10 \n12 \n14 \n/ \nρ \nFIG. 1: Left: Dark matter equation of state in the first model considering ρ g = 0 . 2 × 10 16 g/cm 3 and different values of p g . In this figure and all following figures, p g is in units of 10 35 dyn/cm 2 . Right: Dark matter equation of state in the second model with the mass m = 1 GeV and different values of the interaction between particles, m I . We have assumed ρ 0 = 1 . 66 × 10 14 g/cm 3 . \n<!-- image --> \nρ \nDM \n0 \nIn the second model, we assume the DM as an interacting Fermi gas at zero temperature containing N particles with mass m and spin 1 / 2. The internal energy per particle is related to the one-body term, E 1 , and the interaction two-body term, E 2 , as follows, \nE tot = E 1 + E 2 , (2) \nwith \n0 \nρ \n/ \nDM1 \nP \nE 1 = m 4 c 5 2 π 2 /planckover2pi1 3 1 n DM ∑ i =+ , -1 8 { x ( i ) F √ 1 + x ( i ) F 2 (1 + 2 x ( i ) F 2 ) -sinh -1 ( x ( i ) F ) } , (3) \nin which n DM denotes the total number density of DM particles and x ( i ) F = /planckover2pi1 k ( i ) F mc with k ( i ) F which is the Fermi momentum of a DM particle with spin projection i . For the interaction two-body energy, we apply the form considered in Ref. [52], \nE 2 = u n DM , (4) \nwith the interaction energy density of the particles, u , supposing the spin-independent interaction one. Considering the lowest order approximation, the interaction energy density is given by [52], \nu = n DM 2 m 2 I , (5) \nwith the energy scale of the interaction between DM particles, m I . The first law of thermodynamics, \nP DM 2 = n 2 DM ( ∂E tot ∂n DM ) , (6) \nresults in the pressure of DM in the second model. Figure 1 (Right) shows the DM pressure of the second model with the mass m = 1 GeV and different values of the interaction between particles, m I . The DM EoS is more stiffer with the lower values of m I . Therefore, the decrease of m I which corresponds to the growth of the interaction between DM particles results in the stiffening of the EoS.', 'III. DARK MATTER ADMIXED NEUTRON STARS IN SCALAR TENSOR GRAVITY': 'Spherical symmetric static DMANS can be modeled with a spacetime line element in the scalar-tensor theory in Einstein frame, \nds 2 = -N ( r ) 2 dt 2 + A ( r ) 2 dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (7) \nwith the metric functions N ( r ) and A ( r ) = [1 -2 m ( r ) /r ] -1 / 2 and the mass profile m ( r ). Considering the action of scalar-tensor theory, the field equations lead to five differential equations [53], \ndm dr = 4 πr 2 a 4 ˜ /epsilon1 + r 2 ( r -2 m ) ( dφ dr ) 2 , (8) \nd ln N dr = 4 πr 2 a 4 ˜ p r -2 m + r 2 ( dφ dr ) 2 + m r ( r -2 m ) , (9) \nd 2 φ dr 2 = 4 πra 4 r -2 m [ α (˜ /epsilon1 -3˜ p ) + r (˜ /epsilon1 -˜ p ) dφ dr ] -2( r -m ) r ( r -2 m ) dφ dr , (10) \nd ˜ p dr = -(˜ /epsilon1 + ˜ p ) [ 4 πr 2 a 4 ˜ p r -2 m + r 2 ( dφ dr ) 2 + m r ( r -2 m ) + α dφ dr ] , (11) \ndm b dr = 4 πr 2 a 3 ˜ ρ √ 1 -2 m r . (12) \nHere, φ shows the scalar field and a ( φ ) is the coupling function with the form a ( φ ) = e 1 2 β ( φ -φ 0 ) 2 in which β is the coupling constant, m b denotes the baryonic mass, α ( φ ) = dlna ( φ ) dφ and φ 0 = 0. In addition, the total energy density, ˜ /epsilon1 , and total pressure, ˜ p , are presented by the energy density and pressure of visible ( V ) and dark ( D ) sectors, \n˜ /epsilon1 ( r ) = ε V ( r ) + ε D ( r ) , (13) \n˜ p ( r ) = p V ( r ) + p D ( r ) . (14) \nIn Eq. (14), p V demonstrates the EoS of visible matter. In this work, to describe the visible sector, we consider a system of dense NS matter and apply the piecewise polytropic expansion constrained by the observational data of GW170817 and the data of six lowmass X-ray binaries (LMXB) with thermonuclear bursts or the symmetry energy of the nuclear interaction [54]. For this NS EoS, the polytropic form P = Kρ Γ with four pressure parameters { P 1 , P 2 , P 3 , P 4 } is parameterized. These four parameters are pressure at the densities of { 1 , 1 . 85 , 3 . 7 , 7 . 4 } ρ sat with the saturation density ρ sat = 2 . 7 × 10 14 g cm -3 [55]. Joint analysis results in the constraints on the pressure parameters. Nuclear constraints, gravitational wave data and the LMXB sources with thermonuclear bursts, LMXB source data and the current bounds of M TOV , and LMXB sources with thermonuclear burst mainly determine the values of P 1 , P 2 , P 3 , and P 4 , respectively. These values are P 1 = 3 . 9 × 10 33 , P 2 = 1 . 4 × 10 34 , P 3 = 2 . 0 × 10 35 , and P 4 = 1 . 4 × 10 36 in units of dyn cm -2 . Figure 2 shows this EoS of NS matter. \nFIG. 2: Neutron star matter equation of state constrained by the observational data [54]. \n<!-- image --> \nBesides, p D in Eq. (14), denotes the DM pressure which is chosen from Eq. (1) and Eq. (6), in the first and second model, respectively. We solve the equations employing the boundary conditions, \nm (0) = m b (0) = 0 , lim r →∞ N ( r ) = 1 , φ (0) = φ c , lim r →∞ φ ( r ) = 0 , dφ dr (0) = 0 , ˜ p (0) = p c , ˜ p ( R s ) = 0 . (15) \nHere, R s shows the radius of the star and c denotes the center of the star. Starting with the appropriate boundary condition ( φ (0) = φ c ) at the center of star and the iteration on φ c with the condition [53, 56], \nφ s + 2 ψ s √ ˙ ν 2 s +4 ψ 2 s arctanh [ √ ˙ ν 2 s +4 ψ 2 s ˙ ν s +2 /R s ] = 0 , (16) \nwe solve the differential equations. In the above equation, s presents the surface of the star and ψ s = ( dφ/dr ) s together with ˙ ν s = 2( d ln N/dr ) | s = R s ψ 2 s +2 m s / [ R s ( R s -2 m s )], and the ADM mass, M ADM , \nM ADM = R 2 s ˙ ν s 2 ( 1 -2 m s R s ) 1 2 exp [ -˙ ν s √ ˙ ν 2 s +4 ψ 2 s arctanh ( √ ˙ ν 2 s +4 ψ 2 s ˙ ν s +2 /R s )] , (17) \nand the scalar charge, ω , \nω = -2 M ADM ψ s / ˙ ν s . (18) \nIn this paper, the ADM mass of DMANSs in the scalar-tensor gravity is reported.', 'A. Mass versus the central density of dark matter admixed neutron star': 'Our results for the star mass at different central densities in the cases of NSs and DMANSs have been given in Figures 3 and 4. For NSs with the value of β = -4 . 5, the results of the general theory of relativity (GR) and scalar-tensor theory (STT) are the same. Therefore, the stars with no DM and higher values of β can not be scalarized and no scalarization takes place. This is while for the DMANSs in both DM EoS models, the results of STT deviate from the GR ones even with β = -4 . 5 and these stars can be scalarized. Hence, DM facilitates the star scalarization allowing the existence of scalarized stars with high values of the coupling constant. Considering the lower values of β , i.e. β = -5 or β = -6, the deviation of STT branches from GR ones in DMANSs starts from the higher values of central densities compared to NSs. Moreover, the DMANSs can be scalarized even up to high densities, unlike the NSs. Figures 3 and 4 show that the maximum mass of DMANSs is smaller than the NS one, due to the softening of the EoS by DM. This is in agreement with the results reported in [21-23]. \nWe find from Figure 3 that in the first model for DM pressure, the mass is higher with stiffer DM EoS (larger p g ), for both STT and GR. It should be noted that the lower values of p g , e.g. 0 . 1 × 10 35 dyn/cm 2 , which correspond to softer DM EoSs lead to more reduction of the star mass. Thus the DMANSs with smaller p g are more different from NSs in mass compared to stiffer DM EoSs. Considering the lower values of the coupling constant, the deviation of STT from GR in DMANS is more significant. The stars that have different behaviors in STT and GR are scalarized DMANSs. The stiffening of the DM EoS affects this deviation. In fact, with larger p g and stiffer DM EoS, the STT and GR branches separate at lower densities. The amount of STT deviation increases by growing p g . This means that the higher pressure of DM amplifies the star scalarization. The rate of mass growth versus the density is larger when the stars are scalarized. Figure 3 also verifies that most massive stars are the ones in STT with lower values of β and stiffer DM EoS. \nFigure 4 confirms that in the second model, the decrease in m I and the DM with higher interaction between particles give rise to more massive stars. The increase of mass through the reduction from m I = 1 GeV to m I = 0 . 3 GeV is more remarkable compared to the \none from m I = 300 GeV to m I = 1 GeV . The effects of DM pressure on the star mass are notable when the stars are scalarized. The most massive stars are the scalarized ones with lower coupling constant and stronger interaction between DM particles. DMANSs with higher interaction between particles become scalarized at lower densities. In the case of smaller m I and higher interaction in which the EoS is stiffer, the scalarization of the star is larger. The mass of scalarized stars grows with density more rapidly compared to GR ones. In scalarized stars, the interaction between DM particles enlarges the mass growth with the density. \nFIG. 3: Mass as a function of the central density, ˜ ρ c , for neutron star (NS) and dark matter admixed neutron star (DMANS) in the first model of DM EoS considering the scalar-tensor theory (STT) with different values of the coupling constant, β . The results of the general theory of relativity (GR) are also presented. \n<!-- image -->', 'B. Mass-radius relation of dark matter admixed neutron star': 'In Figures 5 and 6, we have shown the mass versus the radius for the NSs and DMANSs in \n✂ \nFIG. 4: Same as Figure 3 but for the second model of DM EoS. \n<!-- image --> \nSTT and GR. The observational constraints on the NS mass and radius related to EXO 1745248 [57], 4U 1820-30 [58], GW170817 [59, 60], and PSR J0030+0451 [61] are also presented. The mass-radius relation of NSs in STT with β = -4 . 5 coincides with the result of GR, as expected because of the absence of scalarization in NSs considering the higher values of β . However, for the massive scalarized DMANSs with β = -4 . 5, the mass-radius relation deviates from GR one. Thus, the DM affects this relation in STT and GR, differently. The mass-radius relation of NSs in GR and STT satisfies the constraints from EXO 1745-248 and GW170817. The scalarization of NSs leads to the larger masses and radii of stars. \nFigure 5 confirms that in the first model, the level of DM EoS stiffening alters the massradius relation for both GR and STT cases. The increase in DM pressure causes a larger radius for both nonscalarized and scalarized stars. For all DMANSs in both gravities, the range of star radius is more extended considering stiffer DM EoS, i.e. larger p g . Besides, high DM pressure predicts the existence of stars with the same mass but with two sizes. It is clear from Figure 5 that the minimum mass of stars grows by increasing the pressure of DM. \nThe deviation of scalarized stars from nonscalarized ones in the mass-radius relation depends on the value of p g . In DMANSs with smaller DM pressure, the scalarization results in a larger radius for massive stars, similar to NSs . This is while with higher DM pressure, for the stars with lower masses, the scalarized stars are smaller than GR ones, and for massive stars, the scalarization leads to larger sizes. With lower values of the coupling constant, the scalarized DMANSs can be more massive and larger. The case of β = -6 along with p g = 0 . 8 × 10 35 dyn/cm 2 leads to scalarized stars which satisfy the constraints related to the merger event GW170817. Comparison of NSs with DMANSs in the first model of DM EoS approves that the DM extends the range of star radius allowing both smaller and larger DMANSs compared to NSs. \nFigure 6 verifies that the interaction between DM particles raises the radius of stars in STT as well as GR. In most stars, the scalarization of stars also makes the stars larger. Scalarized DMANSs with lower values of the coupling constant and higher interactions between DM particles can have larger masses and radii. Considering higher values of β , the branches of nonscalarized and scalarized stars match for massive stars. However, for β = -6, the massive stars are scalarized and the results of STT deviate from GR ones. With lower values of the coupling constant and higher interactions between DM particles, two scalarized stars can exist with the same radius but different masses. Our calculations verify that the scalarized stars with β = -6 and high interactions between DM particles, i.e. m I = 0 . 3 GeV , are in agreement with the neutron star in the low-mass X-ray binary 4U 1820-30. Accordingly, the scalarized DMANSs in two models of DM EoS are consistent with the given observational data on the assumption that the coupling constant is lower and the DM pressure is higher. Figure 6 also confirms that the DMANSs in the second model of DM EoS are smaller in size compared to NSs.', 'C. Central scalar field of dark matter admixed neutron star': 'In this part, we study the scalarization of NSs and DMANSs by exploring the scalar field at the center of the star. Figures 7 and 8 present the variation of the central scalar field versus the central density for NSs and DMANSs in two models of DM EoS. For NSs in STT with β = -4 . 5, the central scalar field equals to zero at different densities and no scalarization appears. This is while φ c of DMANSs with β = -4 . 5 can be nonzero and \nFIG. 5: Mass versus the radius for NS and DMANS in the first model of DM EoS in GR and STT with different values of the coupling constant, β . Observational constraints on the mass and radius of NS are also given. The constraints are related to EXO 1745-248, 4U 1820-30, GW170817, and PSR J0030+0451. For more details, see the text. \n<!-- image --> \nscalarization takes place. It means that the DM induces the scalar field and scalarization in stars even with higher values of the coupling constant. For the stars, at a special value of the density, the scalarization begins and the scalar field rises from zero. This critical density decreases by the reduction of β for both NSs and DMANSs. In the cases of β = -5 and β = -6, the NS scalar field in high-density stars becomes zero, unlike the DMANSs that are scalarized and experience the nonzero scalar field even up to high densities. We can find from Figures 7 and 8 that the maximum value of the central scalar field for NSs at each coupling constant is smaller than the one in DMANSs. Therefore, the DM intensifies the scalar field in the center of DMANSs. \nFigure 7 verifies that in the first model of DM EoS, the central scalar field increases by growing the density so that with the coupling constant β = -6, the stars with the highest central densities are scalarized. Considering the stars with larger coupling constant, \nFIG. 6: Same as Figure 5 but for the second model of DM EoS. \n<!-- image --> \nthe central scalar field reaches a maximum value and afterward, it reduces by increasing the density. At the lower values of the density in the scalarized stars, the central scalar field grows as p g increases. Therefore, the DM with higher pressure intensifies the central scalar field. This is while with a large enough coupling constant and at high density in the scalarized stars, the scalar field drops by p g . This means that in high-density scalarized stars, the stiffer the DM EoS is, the smaller the value of the central scalar field becomes. In the case of β = -6, the effects of the DM pressure are more significant at lower values of the density in the scalarized stars. The values of the critical density, ˜ ρ crit , for NS and DMANS are given in Tables I and II. The critical density of scalarization in NSs is lower than the one related to DMANSs. It means that in the presence of DM, the star should be more dense to be scalarized. The critical density decreases by p g . The higher pressure of DM results in the easier enhancement of the scalar field and the star scalarization. Table I also shows that the coupling constant also alters the critical density so that it grows by increasing β . Figure 7 shows that the maximum value of φ c gets larger by increasing the DM pressure. Besides, the density corresponding to the maximum scalar field reduces with p g . In DMANSs with \nβ = -4 . 5, at some densities, the scalar field becomes zero and the scalarization terminates. With higher values of the DM pressure, the density corresponding to zero scalar fields is smaller. Consequently, the DM pressure destroys the star scalarization. \nFigure 8 confirms that in the second model of DM EoS, the central scalar field rises from zero and gets to a maximum. Afterwards, φ c reduces by increasing the density. At the lower values of β , the scalar field remains nonzero to high densities, and the stars with high central densities are scalarized. However, with the coupling constant β = -4 . 5, φ c vanishes at high densities. DM EoS also alters the central scalar field. In scalarized DMANSs with low central densities, φ c increases by the reduction of m I . Therefore, the interaction between DM particles amplifies the scalar field in these stars. Nevertheless, in high-density scalarized stars, the stiffer DM EoS results in a decrease in the scalar field. This behavior of DM pressure in the second model of DM EoS at low and high densities is similar to the one in the first model. Considering the DMANSs in the second model of DM EoS, the critical density decreases by the DM pressure. The higher interaction between DM particles gives rise to a nonzero central scalar field and the scalarization at lower densities. Table II presents the critical densities in the second model of DM EoS. The decrease of m I from 1 GeV to 0 . 3 GeV has more significant effects on the critical density. The density corresponding to the maximum central scalar field decreases by m I reduction. Noting the scalarized DMANSs with β = -4 . 5, the scalar field reaches zero at a special density and the stars become nonscalarized once more. The density at which the scalarization disappears decreases by increasing the DM pressure. This reduction means that the DM pressure and the interaction between DM particles assist in the suppression of scalarization. Our results emphasize that the range of the scalarization shrinks by decreasing the value of m I . From Figure 8 we find that the maximum value of the scalar field is not importantly affected by m I . For two models of DM pressure, the stars are scalarized in larger ranges of the central density when the coupling constant is lower.', 'D. Scalar charge in dark matter admixed neutron star': 'In Figures 9 and 10, we have plotted the scalar charge, ω , of NSs and DMANSs applying two models of DM EoS. Considering the case of β = -4 . 5, the scalar charge of NSs is equal to zero, while the DM in DMANSs results in the appearance of a star scalar charge. With \nFIG. 7: The values of the scalar field at the center of the star, φ c , versus the central density, ˜ ρ c , for NS and DMANS in the first model of DM EoS considering different values of the coupling constant, β . \n<!-- image --> \nTABLE I: Critical density, ˜ ρ crit /ρ 0 , at which the scalarization takes place for NS and DMANS in the first model of DM EoS with different values of the coupling constant, β . \nthe values of β = -5 and β = -6 for the coupling constant, the star compactness which is needed to grow the scalar charge from zero is lower in DMANSs. Hence, the DM assists in the increase of the scalar charge even in the stars with lower compactness. However, the \nFIG. 8: Same as Figure 7 but for the second model of DM EoS. \n<!-- image --> \nTABLE II: Same as Table I but for the second model of DM EoS. \nmaximum scalar charge of DMANSs is lower than the one of the NSs. We find from Figure 9 that in the first model of DM, the scalar charge grows versus the star compactness. With a higher coupling constant, the scalar charge can decrease and vanish as the compactness rises. At the lower values of the compactness, the scalar charge is larger in stars with higher values of p g and stiffer DM EoSs. For more compact stars, ω reduces as DM pressure increases. With a special value of M/R , the scalar charge gets a maximum value which this value of \nω increases by p g . The corresponding compactness to the maximum scalar charge reduces with the DM pressure. In stars with larger p g and lower values of the coupling constant, the highest values of the scalar charge can be reached. In each case, at a critical value of the compactness, the scalar charge grows from zero. The critical compactness decreases by p g . This indicates that with higher DM pressure, even the stars with lower compactness can experience the scalar charge. \nThe scalar charge of DMANSs in the second model of DM EoS is presented in Figure 10. Almost in all stars, the scalar charge increases by decreasing m I . The interaction between DM particles leads to a larger scalar charge in DMANSs. The critical compactness at which the scalar charge grows from zero has smaller values when the interaction between DM particles and therefore the DM pressure increases. In two models of DM EoS, the critical compactness of the star grows by increasing the coupling constant. In addition, the range of the star compactness that results in the nonzero scalar charge of the stars is more extended at lower values of β . \nω \nFIG. 9: Scalar charge, ω , as a function of the compactness, M/R, for NS and DMANS in the first model of DM EoS considering different values of the coupling constant, β . \n<!-- image --> \nω \nFIG. 10: Same as Figure 9 but for the second model of DM EoS. \n<!-- image -->', 'E. Critical density of scalarization in dark matter admixed neutron star': 'In the last part of this paper, we sum up the results related to the critical density of scalarization, ˜ ρ crit . Figure 11 belongs to the critical density of scalarization of the NSs and DMANSs in two models of the DM pressure. Considering both NSs and DMANSs with DM EoSs in two models, ˜ ρ crit increases as the coupling constant grows. Hence, the stars with larger values of β ought to have more central densities to be scalarized. The critical density in NSs increases by the coupling constant very smoothly. In the first model of DM EoS, the critical density reduces by increasing p g and the DM pressure as discussed above and presented in Figure 7 and Table I. Considering the first model, the influence of the DM EoS on the critical density is more important at lower values of β . Moreover, the critical density of the second model of DM EoS becomes smaller as m I decreases and the interaction between DM particles grows, see also Figure 8 and Table II. However, in the second model, the pressure of DM alters the critical density more significantly when the coupling constant is larger. \n˜ \nFIG. 11: The critical density of scalarization, ˜ ρ crit , versus the coupling constant, β , for NS and DMANS in the first (Left) and second (Right) model of dark matter pressure. The parameters of DM models are the same as Figure 1. \n<!-- image --> \n<!-- image --> \n˜', 'V. SUMMARY AND CONCLUDING REMARKS': 'In this paper, we have studied the scalarization of dark matter admixed neutron stars in scalar-tensor gravity. The dark matter equation of state based on the observational data from the rotational curves of galaxies (first model) and the equation of state of the fermionic self-interacting dark matter (second model) have been applied to describe the dark sector in neutron stars. With higher-pressure dark matter, the scalarization of neutron stars is more notable. The influence of dark matter pressure on the central scalar field depends on the central density. In low-density neutron stars, the central scalar field increases by the dark matter pressure, while it decreases by the dark matter pressure in high-density scalarized neutron stars. The scalarized stars containing dark matter with higher pressure are the most massive ones. In the first model, the effects of scalarization on the radius of neutron stars depend on the stiffening of the dark matter equation of state. However, in the second model, the scalarized stars are larger than the nonscalarized ones. Our results in the first model show that the stars in the merger event GW170817 can be scalarized dark matter admixed neutron stars. Besides, the scalarized stars in the second model satisfy the constraints of the neutron star in the low-mass X-ray binary 4U 1820-30.', 'Acknowledgments': 'The authors wish to thank the Shiraz University Research Council. \n- [1] N. F. Bell, G. Busoni, T. F. Motta, et al., Phys. Rev. Lett. 127 , 111803 (2021); Phys. Rev. Lett. 129 , 239902 (2022).\n- [2] F. Anzuini, N. F. Bell, G. Busoni, et al., J. Cosmol. Astropart. Phys. 11 , 056 (2021).\n- [3] A. Kumar, H. C. Das, and S. K. Patra, Mon. Not. R. Astron. Soc. 513 , 1820 (2022).\n- [4] M. Fujiwara, K. Hamaguchi, N. Nagata, and J. Zheng, Phys. Rev. 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2022AcAau.190..323B

If humanity is ever to consider substantial longterm colonization of Mars the resources needed are going to be extensive. For a longterm human presence on Mars to be established serious thought would need to be given to terraforming the planet. One major requirement for such terraforming is having the protection of a planetary magnetic field which Mars currently does not have. The Earths magnetosphere helps protect the planet from the potential sterilizing effects of cosmic rays and also helps retain the atmosphere which would otherwise by stripped by large solar storms as they pass over the planet. Mars does have small patches of remnant surface magnetic field but these are localized in the southern hemisphere and are not of sufficient size or magnitude to protect the planet or a colony. P In this article we explore comprehensively for the first time the practical and engineering challenges that affect the feasibility of creating an artificial magnetic field capable of encompassing Mars. This includes the concerns that define the design where to locate the magnetic field generator and possible construction strategies. The rationale here is not to justify the need for a planetary magnetosphere but to put figures on the practicalities so as to be able to weigh the pros and cons of the different engineering approaches. P The optimum solution proposed is completely novel although inspired by natural situations and fusion plasma techniques. The solution with the lowest power assembly and mass is to create an artificial charged particle ring similar in form to a radiation belt around the planet possibly formed by ejecting matter from one of the moons of Mars in a fashion similar to that which forms the JupiterIo plasma torus but using electromagnetic and plasma waves to drive a net current in the rings that results in an overall magnetic field. P With a new era of space exploration underway this is the time to start thinking about these new and bold future concepts and to begin filling strategic knowledge gaps. Furthermore the principles explored here are also applicable to smaller scale objects like manned spacecraft space stations or moon bases which would benefit from the creation of protective minimagnetospheres.

2022-01-01T00:00:00Z

['10.1016/j.actaastro.2021.09.023', '10.48550/arXiv.2111.06887', '2022AcAau.190..323B', '2021arXiv211106887B', 'arXiv:2111.06887']

['Mars', 'Exploration', 'Terraforming', 'Magnetosphere', 'Artificial', 'Plasma', 'Magnetic field', 'Physics - Space Physics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Popular Physics']

How to create an artificial magnetosphere for Mars

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

https://arxiv.org/pdf/2111.06887.pdf

{'No Header': '3', 'How to create an artificial magnetosphere for Mars': 'R.A. Bamford 1 , B.J. Kellett 1 , J. Green 2 , C. Dong 3 , V. Airapetian 4 , R.Bingham 1 , 5 . \n1 \nRAL Space, STFC, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK. 2 NASA Headquarters, Washington DC, U.S. \nDepartment of Astrophysical Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544, U.S. 4 NASA Goddard Space Flight Center (GSFC), 8800 Greenbelt Rd, Greenbelt, MD 20771, U.S. NASA ARC-SST, U.S. 5 SUPRA, Dept of Physics, Uni of Strathclyde, Glasgow, Scotland, U.K.', 'Abstract': "If humanity is ever to consider substantial, long-term colonization of Mars, the resources needed are going to be extensive. For a long-term human presence on Mars to be established, serious thought would need to be given to terraforming the planet. One major requirement for such terraforming is having the protection of a planetary magnetic field - which Mars currently does not have. The Earth's magnetosphere helps protect the planet from the potential sterilizing effects of cosmic rays and also helps retain the atmosphere, which would otherwise by stripped by large solar storms as they pass over the planet. Mars does have small patches of remnant surface magnetic field, but these are localized in the southern hemisphere and are not of sufficient size or magnitude to protect the planet or a colony. \nIn this article we explore comprehensively for the first time, the practical and engineering challenges that affect the feasibility of creating an artificial magnetic field capable of encompassing Mars. This includes the concerns that define the design, where to locate the magnetic field generator and possible construction strategies. The rationale here is not to justify the need for a planetary magnetosphere but to put figures on the practicalities so as to be able to weigh the pros and cons of the different engineering approaches. \nThe optimum solution proposed is completely novel, although inspired by natural situations and fusion plasma techniques. The solution with the lowest power, assembly and mass is to create an artificial charged particle ring (similar in form to a 'radiation belt'), around the planet possibly formed by ejecting matter from one of the moons of Mars (in fashion similar to that that forms the Io-Jupiter plasma torus), but using electromagnetic and plasma waves to drive a net current in the ring(s) that results in an overall magnetic field. \nWith a new era of space exploration underway, this is the time to start thinking about these new and bold future concepts and to begin filling strategic knowledge gaps. Furthermore, the principles explored here are also applicable to smaller scale objects like manned spacecraft, space stations or moon bases, which would benefit from the creation of protective mini-magnetospheres. \nKeywords: Mars, Exploration, Terraforming, Magnetosphere, Artificial, Plasma, Magnetic field.", '1.1 Why do we need a magnetosphere for Mars': "The Earth's magnetic field that originates within the iron core from a dynamo process, encompasses the planet and extends out into the near space environment (see Figure 1). The magnetic field helps to reduce the radiation reaching the surface by re-directing and shielding large numbers of energetic solar particles that would otherwise create a radiation hazard to life. Another important benefit of the Earth's magnetic field is that it inhibits the loss of atmospheric molecules from pick-up by the solar wind during large solar superstorms [1][2][3]. Increasing Mars atmospheric pressure has been proposed as one of the primary requirements in terraforming Mars, along with warming and altering the atmospheric composition (e.g. [4][5][6][7][8][9][10][11]). The aim is to achieve a stable ecosystem or 'ecopoiesis'[12][13][14]. But recent studies suggest that these efforts would be undone by a combination of processes driven by extreme ultraviolet light and solar wind from the Sun, removing atmospheric gases from the upper atmosphere to space [15][16][17][18]. Refs. [19][21][20] show that the presence of a strong intrinsic global magnetic field substantially decreases the loss of molecular ions and alters atmospheric conditions. \nIn contrast smaller, sub-global magnetic fields offer a mixed benefit. The evidence from observations and simulations of the patches of crustal magnetic field that naturally occur already on Mars show that the presence of these anomalies can aid ion loss as much as they might hinder at other times depending upon the orientation of the field and interplanetary environment(e.g.[22][23][24]). \nMars is about half the size of the Earth and has a much lower atmosphere density. This therefore makes atmospheric losses much more significant. Terraforming activities designed to build up the atmospheric pressure and alter its composition on Mars will not want this effort to be undone by the first significant solar superstorm to reach the planet. One of the first goals of terraforming will be to increase the atmospheric pressure above the Armstrong Limit (6.3 kPa), a threshold that removes the requirements of having to wear a full-body pressure suit, although oxygen will still be needed [25]. Below the Armstrong atmospheric pressure limit, water in the lungs, eyes and saliva spontaneously \nboils [26]. Changing the atmospheric pressure can be expected to have wide ranging consequences to many aspects of living and working on Mars including amongst others to weather patterns, dust storms and transportation to name but a few. Primarily though, a global magnetic field generated magnetosphere, Mars could weather the worst of the atmospheric stripping effects of large solar events and help protect from radiation particles. \nThe past several years has seen an increase in the number of serious scientific investigations of many diverse aspects related to manned exploration of Mars and colonization. These include potential missions, interplanetary vehicles, Mars transportation vehicles, habitats but also socioeconomic concerns (e.g. [29],[30][31][32][33][34][35][36][37][38][39]). This indicates that the technology is becoming closer to achievable and affordable. \nIn this article we shall consider potential technological approaches to create an artificial magnetic field to protect Mars. We will not discuss the value or likelihood of humanity colonizing Mars, nor consider the relative merits or performance of magnetospheres, whether they are generated by magnetic fields or otherwise. Nor shall we present an analysis of the possible changes to Mars atmosphere with and without a planetary magnetic field. Such atmospheric modelling requires dedicated articles and will depend on the choice of location of the magnetic field source - for instance below or entirely above the planet's atmosphere. \nWhat will be presented are multiple options for technology approaches and locations for the magnetic field generating infrastructure along with their pros and cons. The assumption is made here that there is a desire to create a magnetic field similar to that of a natural magnetized planet like the Earth and then follow how this could be done from a purely fundamental perspective. This issue, of creating an artificial structure at unprecedented scale, has not been considered in a peer-reviewed journal before. The calculations of power, resources and other relevant parameters are all deliberately made only to first order, as higher precision figures would be meaningless without a comparable level of precision for the engineering. This can be undertaken later. However, before any more detailed engineering design can be proposed there must first be an evaluation of the benefits and limitations of the different approaches and a choice of principle made. The aim here therefore is to discuss and compare the methods and to finally propose a novel solution. \nThe technological options we will consider include: re-starting the planet's iron core, using solid state permanent magnets in either continuous loop or a series of discrete magnets, the use of solid state superconductors or a plasma current loop similar to a current driven plasma torus of an artificial plasmasphere. We shall also consider some of the factors concerning the source location of these generated magnetic fields. Within this analysis, we shall outline the issues and concerns that define the \nFigure 1: An artistic impression of magnetosphere around Mars formed by a magnetic dipole field from an artificial ring or loop of electrical current circulating around the planet. The approximate point at which the pressures balance is the stand-off distance R S . \n<!-- image --> \ndesign such as general mass and electrical current needs. Specific timescales and logistics of installation will not be considered here, as it is anticipated that terraforming Mars will be a worldwide and multicentury endeavor and the potential for paradigm-changing developments would radically alter these. The one exception is the assumption of the development of successful nuclear fusion reactors [27] as an efficient energy generation option. Nuclear fusion is already an extensive international scientific and engineering program that is ever closer to being achieved [27]. Fusion power is a likely necessary enabler for considering substantive colonization and terraforming in general. Fusion based propulsion has been proposed as an important development for human planetary exploration [28], although at this time a successful economic fusion reactor has yet to be developed.", '2 Power needed to create a planetary sized magnetic field': "The primary technical challenge in creating a magnetosphere for Mars is not the strength of the magnetic field but the vast size of the magnetic field needed to encompass an object as large as a whole planet. \nTo first order, for a magnetic field in the path of a flowing plasma (like the solar wind), a stand-off will occur approximately where the magnetic field pressure P B balances the pressure of the solar wind plasma, P sw 1 . For planetary magnetospheres, this distance R s is sometimes called the ChapmanFerraro distance in reference to those that first proposed it [40]. \nR 6 S ∼ B 2 o 2 µ o P sw .R 6 0 (1) \nHere R 0 is the radius of the magnetic field generating structure (iron core, or current/magnetic loop). \nFor Earth, the magnetopause distance ranges from about 5 to 15 Earth radii depending upon conditions. For Mars, without an intrinsic magnetic field, an induced magnetosphere is created close to the planet or in the ionosphere [44]. For Mars this has been observed to mean that during solar superstorms there is a considerable loss of atmospheric molecules [1][41][42]. The creation of an artificial magnetic field will help limit this [19]. We can assume a minimum requirement for an artificial magnetic field for Mars will need to be such that, even during largest solar events, the R S never goes into the planet's ionosphere. \nThe solar wind pressure is usually in the range ∼ 1 to 10 nPa (1 -10 × 10 -9 Nm -2 ) at Earth making the magnetic field intensity (BS) necessary to balance the solar wind ram pressure of ∼ 50-200 nT. This is using magnetic field pressure P B = B 2 S / 2 µ 0 , where µ 0 is the permeability of free-space. At the orbit of Mars this would be ∼ 40-150nT. \nFrom this (as observational data from spacecraft that confirm [44]) we can see that the intensity of magnetic field at the magnetopause needed to disrupt the solar wind, is not very much less than the field of a typical fridge magnet at ∼ 5mT [45]. However, to cover an area at least the size of Mars would mean a ∼ 100nT magnetic field over a minimum of 37 million square kilometres (the radius of Mars is R M ∼ 3400km + ∼ 100km atmosphere). The total energy stored in such a magnetic field is of the order of 10 17 J. This does not include the energy needed to ramp the magnetic field up, which will be affected by the inductive properties of the media in and around the magnetic field generating \nFigure 2: Approaches to creating a magnetic field. The options for the different approaches to creating a current loop are; (A) molten iron core dynamo, (B) solid superconducting current loop or permanent magnets, (C) a chain of discrete coupled current or magnetic sources and (D) a current driven plasma torus. \n<!-- image --> \nstructure. This provides an absolute minimum for the energy needed as that stored in the minimal field once operational. \nThis amount of energy can be compared to the world's total electricity consumption in 2020, which amounted to approximately 583.9 exajoules (10 18 joules) of energy or 553 Quadrillion BTU [46]. For this reason there would be a strong need to have working fusion reactors as an efficient, compact power source as a necessary precursor for embarking on the creation of an artificial magnetosphere for Mars and a general enabler for economic permanent colonization. Nuclear fusion as a power source has the highest energy density - making it compact and reducing the mass needed, which is an important consideration for transportation to Mars. The energy density of fusion fuels, at about 10 12 Jg -1 of deuterium, is orders of magnitude higher than nuclear fission (next nearest) at 10 10 Jg -1 for U 235 and without the radioactive hazard concerns. Nuclear fission also has an energy density millions of times higher than solar power (at 1 AU), or chemical sources like fossil fuel, biofuel, or batteries at less than 10 4 Jg -1 [47].", '3 Approaches and Locations': "To form an artificial magnetosphere a magnetic field needs to be created artificially. There are several ways this might be done in principle. Figure 2 shows the options for the different approaches to creating a suitable current loop. The options are: \n- (A) dynamo circulation of a molten planetary core,\n- (B) a continuous solid superconducting current loop or loop of permanent solid-state magnets,\n- (C) a chain of discrete coupled current sources made of a controlled beam of charged particles forming an electrical current, and \nFigure 3: Options for locating a source of magnetic field; (a) from the core of the planet, (b) the surface of Mars, (c) in orbit of Mars (Low Mars Orbit (LMO) or areostationary orbit), (d) the orbit of Phobos, (e) the orbit of Deimos, and (f) the Mars-Sun L1 Lagrange point. \n<!-- image --> \n- (D) a plasma torus of positively and negatively charged particles with artificial current drive forming a resultant current loop of a solenoid. \nAt some distance from the source of the magnetic field, these solutions will all be indistinguishable from each other. The options for locating the source of magnetic field include in and on the planetary surface and a variety of locations in Martian orbit. \nIn the companion Figure 3 are shown the options for locating the source of magnetic field in \n- (a) the planet's core,\n- (b) on the surface,\n- (c)-(f) a variety of locations in Mars orbit, and\n- (g) at the Sun-Mars L1 point upstream of the planet (first proposed by J. Green) [21]. \nWe shall now consider the factors and practicalities of these options in turn, starting with what would be required to re-start Mars' iron core with the intention of activating a dynamo in the same manner as the Earth's core generates its magnetic field.", "3.1 Restarting Mars' iron core": "The Earth's magnetic field originates from dynamo effects in the outer iron core kept molten in part due to radioactive material and pressure [48], [49]. Earth's iron core is approximately 3,500 km in radius (by coincidence very similar to the size of the entire planet of Mars [50].) Mars' iron core is \nsmall at approximately 1800 km (see [51],[52], [53] for recent reviews). Recent observations by the NASA InSight mission have suggested that the core is in a liquid state [54]. Therefore, there exists the possibility that one could kick-start Mars' iron core back into an active magnetic dynamo, possibly using nuclear material or electrical (inductive) heating. The amount of energy needed, ignoring all other issues such as drilling, would be of the order of 10 26 Joules to raise the ∼ 2 × 10 16 m 3 volume of the iron (II) sulphide [55] core by 1000-2000K (assuming a mean core specific heat of ∼ 800Jkg -1 C -1 and density of 8 × 10 3 kg m -3 ). Since a 1 megaton Hydrogen bomb has an equivalent energy release of 10 9 kg of TNT, which is an amount of energy equal to 4 × 10 15 J [56], the amount of energy alone needed to melt Mars' iron core is the equivalent of 10 11 , 1 megaton H-bombs. This ignores any other issues with the placement and timing of explosions and any subsequent heat losses. However once molten, the retained energy would provide the heat-engine and reduce the viscosity of the core material that would allow the Coriolis forces to create the convection currents that could self-excite the magnetic dynamo into producing a resultant magnetic field. Although the compositions and pressures of Earth and Mars cores may be very different, using a simple proportionate core and planet size argument between Earth and Mars, a restarted core circulation would provide an estimated magnetic field intensity of approximately 10 -4 T at the surface of Mars. \nEven so it is not known how long such a magnetic field would last, given that it is believed early Mars did possess a magnetic field that it lost [51]. In addition, Mars' mantle surrounding the core, is about half the thickness of Earth's mantle making the heat flow and potential uncertainty over the tectonic stability at the surface more significant with a restarted core with convective motions. \nThe volume involved is vast and it is highly unlikely that this option would be feasible. It is also unnecessary since there are other potentially much easier alternative ways of creating an artificial magnetic field that may be more practical. So let us now consider the physics behind solid magnets and current loops as potential sources of magnetic field for creating a magnetosphere [57].", '3.2 Solenoid loop': 'If R 0 is the major radius of the current loop or coil with a total current of I (Amps) where I = NI N , where N is the number of current loops or sub-coils each carrying a current of I N , the following approximations can be applied (in SI units), for a solenoid, the field at the center of the coil is: \nB o glyph[similarequal] µ 0 I 2 R 0 (2) \nClearly a permanent magnet approach does not involve currents but the expression for the decrease \nof the magnetic field intensity with distance will follow the same equation below. At a distance R glyph[greatermuch] R 0 (in any direction) the decrease in magnetic field intensity B ( R ) from a dipole is; \nB ( R ) glyph[similarequal] B 0 ( R 0 R ) 3 (3) \nHere R 0 is the major radius of the magnet or current loop. Magnetic field configurations with higher order poles decrease with distance more rapidly closer to the source. Far from the magnetic field source, all magnetic field configurations reduce to a dipole minimum form because of ∇· B = 0. We assume for simplicity that the magnetic field within the solenoid is uniform. (In fact with a wide major radius the magnetic field could decrease within the loop depending upon the configuration of neighboring coils and minor radius of current loop.) \nWhat Equation 4 shows us is that the magnetic field decrease with distance is not just ∝ 1 /R 3 but ∝ ( R 0 /R ) 3 , and just ∝ B 0 , meaning R 0 is a much more valuable parameter than B 0 and the optimum stand-off distance is achieved with the largest possible coil major radius, R 0 [57]. Furthermore, the gradient of the decrease is less 2 . \nTo protect the entire planet, the magnetic field must also cover all latitudes. This means the current loop would be required to either have a wide minor radius (panel (a) in Figure 4) or to be made up of narrow minor radius coils in a Helmholtz (or similar) configuration (for instance) as in panel (b) of Figure 4. This would produce a continuous magnetic dipole field indistinguishable at a distance from that of a natural field. The plane of the coils does not necessarily have to be that of the ecliptic. \nThe configurations illustrated in Figure 4 show the target area of Mars as seen from the perspective of the Sun and the solar wind and current loops and resultant total magnetic field located surrounding and close to the planet. The same consideration for off-major axis plane magnetic field characteristics is needed regardless of location or orientation. \nAs well as the greater range of wider major diameter, low field coils compared to a smaller high field coils, another potential advantage is that it provides for a potentially safer magnetic field intensities for humans and instrumentation for those working and living close to the source of the magnetic field. It reduces the risk of strong gradients in magnetic field intensity - potentially a more significant hazard than a constant high field. Strong gradients in magnetic field and rapid changes in magnetic field in time, (such as experienced by a spacecraft passing the field coils) could induce large electric fields and currents in instrumentation and humans alike. For a planetary wide artificial field, the ability to work \nFigure 4: To cover the entire area of the planet would require either (a) A wide minor radius, b, current loop (which can be made up of N current loops) or (b) a set of more discrete coils/current loops in Helmholtz (or similar) configuration. \n<!-- image --> \non and around the structure will be very important. Having a magnetic field intensity of the same order as that on the surface of the Earth would clearly not pose a hazard. Most of this argument will also hold for a permanent magnet in the same form of a loop. Once outside the material creating the magnetic field, there is no physical difference. The draw back of a wide diameter current loop is the scale of the structure. This will be discussed later.', '4 Coil locations options: surface, orbit or L1': "Some optimizations between resources in construction and maintenance with operational needs would determine the optimum location. Low Mars Orbit (LMO) would have the advantage of being the easiest to reach although LMO is subject to drag from the planet's tenuous outer atmosphere. The atmosphere of Mars is significantly thinner than that of Earth, with a surface pressure of just 1% of the Earth's. However, because of the much smaller size and mass of Mars, the two atmospheres have a similar scale height. So, objects in LMO would still require periodically boosting in order to avoid orbital decay and re-entry. Here we want to consider some of the specifics. \nTaking the example of Earth's equatorial surface magnetic field strength of B 0 = 31 , 100nT and radius of Earth R E = R 0 = 6 , 400km and using equation 2 above, requires a current I = 3 × 10 8 Amps or 0.3 GAmps. However, for Mars the size of the planet is a lot less ( R M =3,400km) so that the magnetic field need not be that of the surface of Earth. \nThe necessary criteria are: \n- 1. The magnetic field needs to be of sufficient intensity to create an artificial magnetopause.\n- 2. The size of the protected zone of the magnetosphere needs to encompass the entire planet. \n- 3. The magnetosphere must not enter the planet's atmosphere during extreme space weather events (thought to be the major source of atmospheric loss). \nDetermining the parameter ranges for dynamic pressure balance, if the normal solar wind [58] ram pressure is taken as ranging between 3 and 10nPa, a severe solar storm flux of 10 10 protons/cm 2 sec at > 50MeV would correspond to around 8000nPa (if treated as ρv 2 / 2 (deflected flow) or 16000nPa if treated as ρv 2 (absorbed flow)). For normal solar wind conditions, a field of 90-150nT is sufficient to achieve stand-off balance (see equation 2). As has already been mentioned, the magnetic field at the magnetopause of the Earth is of the order of ∼ 100nT. If we choose to have the magnetopause stand-off distance to be a similar number of planetary radii as the Earth's, then R S ∼ 10 R M , where R M = 3 , 400km is the radius of Mars. This is relatively arbitrary but allows for plenty of margin in the solar storm extremes. \nThe question then becomes where to locate the solenoid loop, bearing in mind the analysis above. For simplicity here we shall illustrate using single generic 'current loops' while acknowledging the concerns discussed in the previous section. The options shown in Figures 3 are (A) the iron core (covered already), and current loops locations - (B) on the planetary surface, (C) to (F) in orbit, and (G) located at L1. On the surface of Mars, the current needed would be > 5GAmps producing a surface magnetic field of ∼ 10 -4 T - approximately twice the magnetic field at the surface of the Earth and well within the safe limit for any humans to live with. However, there may well be issues about living and working around the required structures on the surface. Spatial and temporal gradients in magnetic field induce electric fields and are a greater hazard than constant magnitude of the magnetic field. This may be a problem if we consider some of the practical issues with managing a large single loop structure. The facility needs to be able to handle the energy dissipated if the coil is quenched (assuming superconductive coil). The quenched current could be directed to a reversed induction non-superconducting conduit, such as a copper jacket around the coil. Another option is to have a series of connections rather than a single continuous loop, where there are joints or release points. This may have mechanical or energy efficiency drawbacks, but is probably still necessary to allow for maintenance. Some of the issues of living and working near the structure would be avoided if the structure were in orbit. This also allows for a larger R 0 and correspondingly lower B 0 .", '5 Solenoid locations in space': "Placing a solenoid in orbit requires a stable orbit. Mars has a very uneven gravitational pull due to large asymmetries in mass, making station keeping particularly problematic, especially for LMO. This would require considerable delta-V to maintain the orbit [60]. \nFigure 5: The options for locations of a current loop in orbit include Low Mars Orbit, the orbits of the Moons (Phobos and Deimos) and the Mars equivalent of geostationary, areostationary orbit. If we fix the location of the magnetopause at 10 R M with a field of 100nT then the magnetic field intensities created must follow the curve shown and the wider the loop the lower the field, but the larger the structure needed. \n<!-- image --> \nIf a solenoid were to be placed in an areostationary orbit for Mars (the equivalent of Earth's geostationary) of a R 0 = 6 R M then the much wider loop radius drastically reduces the magnetic field intensity needed to reach ∼ 100nT at R S =10 R M of 460nT and consequently requires more than an order of magnitude less current at < 0 . 2GAmps. This is illustrated in Figure 5. During extreme superstorms the magnetopause will move from 10 R M to closer to less than 3 R M (assuming the storm values given previously in Section 2 and reference [21]). We assume that this is still sufficient to still protect the planet 3 . It would be sensible to build in the ability to increase this margin by increasing the current in the coils when needed, and therefore increase the magnitude of the magnetic field for the relatively brief time of the duration of an extreme solar event - which can last from hours to a few days. \nFigure 4 shows the possible locations for the coil (b) on Mars' surface, (c) in an unspecified LMO, or (d) at the orbit of moon Phobos, (e) in areostationary orbit, (f) in the orbit of moon Deimos, and (g) smaller diameter coil at the Sun-Mars Lagrange L1 point. Creating an artificial magnetosphere at the L1 point [21], makes use of the gravity balance point between Mars and the Sun to aid station-keeping of the hardware. The Mars L1 is 333 R M away and the question is then whether Mars would reside within the magnetotail created by the coil. This would depend upon the size of the magnetosphere allowing for the balance of pressures radially from the Sun and those following the Parker Spiral [61] which is more acute at Mars at ∼ 20 · ± 10 · . The benefit of L1 location is the coil can potentially be smaller and the magnetic field larger without concern for the magnetic field intensity being a hazard \nto colonization and operation on the planet due to the distance. However, magnetotails are known to undulate and move around with the variations of the solar wind parameters. This may make maintaining the planet in a 'safe zone' problematic, especially given the distance of the the Mars L1 point from Mars. An analysis of the performance of a magnetic field located at Mars L1 is presented in ref. [21]. \nStation keeping at the Mars L1 point is more challenging than that for Earth. Mars L1 is a very shallow gravitational 'island' and Mars' orbit around the Sun is significantly non-circular so the SunMars L1 point will move by a considerable distance during the 687 days Martian year. Nevertheless, a high field, narrow diameter multiple coil system at Mars L1 would offer a significantly lower mass option with greater safety margins that would more than make up for the extra delta-V needed. \nIf the current ring were to be made of solid-state materials, either superconductors or permanent magnets, and assuming the same minor cross section, then the mass budget would be a factor of 6 greater at the areostationary orbit compared to that for the surface. This brings us to the issue of mass.", '6.1 Permanent magnets and superconductors': 'Many of the difficult technological issues relate to the need to create a magnetic field using either permanent magnets or from an electromagnet (solenoid) that only acts like a magnet when an electric current is passing through it. As described above, the factors for either solution are essentially the same, because we are considering such modest field strengths. The major issue is the trade-off between magnetic field intensity at the source and diameter of the source structure. \nIt is reasonable to expect that, by the time that it becomes practical to terraform Mars, considerable effort and development will have been made that will have led to superconducting materials that operate at much higher temperatures even than liquid nitrogen (current technology) since the current research in the field is heading in this direction [62][63]. \nIf the current carrying conductors are to be solid state, their material will need to have a very low resistivity so as to reduce the energy being lost in the forms of heat within the conductors. High temperature superconductors, that essentially have no electrical resistance would be the ideal material. \nCreating permanent magnetic structures of the scale required would be very challenging. Current ferromagnetic and ferrimagnetic materials on Earth are based on iron and rare-earth elements and these are relatively high density so have high resultant mass. There is potentially an alternative that \nwould avoid or reduce this issue - ferromagnetism has been demonstrated in a lithium gas cooled to less than one kelvin [69]. \nComparing this with superconducting solenoid technology currently these are either pure metals and metalloids that experience a sudden decrease in resistance with low temperature, or are metallic compounds or alloys, using elemental vanadium, technetium, and niobium. At this time, superconductors need to be cooled to at least liquid nitrogen temperatures to exhibit superconductivity. However, it is fair to assume that future superconductors will operate at higher temperatures than this, given the way the technology is progressing [70].', '6.2 Hollow solenoids': 'Due to the size of the structures being discussed here we are inevitably considering hollow loop structures. This radically effects the mass budget. The hollow solenoid coils could be made lattices such as is shown in Figure. 6 which in turn could be constructed of high-performance light weight superconducting (ideally) materials such as nanotubes. Carbon nanotubes (CNT) [64] exists today although they are not superconducting. Any such structure would have to hold the required current density without melting if there is finite resistivity. The electrical conductivity of CNT is ∼ 10 6 -10 7 S/m and 10 8 S/m for pure graphene [65]. If we take the solenoid current calculations from above section, which range between ∼ 0.2-0.5 GAmps (depending upon location) then this would require power of ∼ 10 8 to 10 11 Wusing CNT. While there has yet to be built a fusion-based power station a 2 GW Demonstration Power Plant, known as Demo, is expected to demonstrate large-scale production of electrical power on a continual basis by around ∼ 2040 [27]. This is in the middle of the power range needed. To avoid melting the CNT which would start to occur at about ∼ 2600K [66], using a thermal conductivity of ∼ 2600 -4000 W/mK [67] then the total area of CNT needed would be of the order of 10 11 m requiring a mass of ∼ 10 7 kg assuming a ∼ 100nm depth of density of ∼ 1.4g/cm -3[66]. While this is not an astronomical quantity of mass and power, the required engineering infrastructures to mechanically support and power such a current loop would need to accommodate contingencies, and backups so a more realistic logistical burden estimate is likely to be far greater. \nOne encouraging recent development is that of high-quality ultra-thin superconducting films [68]. This technology could be a game changer for using magnetic fields in space as it has low mass and seems robust enough to deal with launch stress and thermal environment of space. In an era of terraforming Mars, development of this type of material could be a significant driver for such innovations. \nMany of the raw materials could potentially be found on Mars itself - they will obviously need to be refined and processed (using significant energy resources) and then the material will need to \nFigure 6: Solenoid coils could be made from hollow lattice structures to minimize mass. These structure in turn could be comprised of lightweight materials such as nanotube materials. \n<!-- image --> \nbe raised into the required orbit. They will also need to be powered or cooled (or both) in order to operate. The gravity well of Mars being 38% that of Earth will make raising the material to orbit somewhat easier but nevertheless this would require significant fuel, and this makes mass a critical consideration. The exact amount of mass needed to be used in orbit clearly depends very much on the technology used. Some of the options are considered in following sections. \nFor practical operations, each coil would need to be divided into two adjacent but distinctly separate windings so that any fault arising in one of them would not affect the other. This would also allow for down time and servicing.', '7 Placing structures in orbit': "For any orbital structure there is the problem of raising the materials into orbit (even assuming that the raw materials come from Mars). Figure 7 shows the relative gravity well for Earth, Mars and the Moon [71]. As can be seen Mars' lower gravity makes lifting either materials or manufactured structures into space considerably easier than on Earth. \nThe escape velocity for Mars is less than half that of the Earth at 5 m/s. The lower air density on Mars will also make this easier. Mars' moons, Phobos and Deimos are too small to be visible on the figure. Phobos and Deimos offer an obvious resource for the materials needed to construct an artificial magnetic field source in orbit. However, for the size of structures discussed above the moons would not be of sufficient mass. To determine the suitability of these and Mars for resources requires detailed knowledge of the mineral resources and abundances. \nIt is likely the relative Martian abundances of the resources needed to construct an artificial \nDistance from centre of planet (km) \n<!-- image --> \nFigure 7: A comparison of the relative gravity wells between Earth, Mars and the Moon. The gravitational potential is expressed in units of kilometers and is the equivalent distance needed to raise a 1 kg mass in a uniform 1 g (9.8ms -2) ) gravity field. Adapted from Crawford (2015) [71] \nmagnetic field structure will play a large part in the type of technology employed. There is no point considering planetary scale structures that would need rare and exotic minerals in large quantities. Currently, the mineral resources of Mars are not well known or quantified. Iron is likely to be plentiful, particularly in the region around Olympus Mons and the other three giant shield volcanoes in the same region [72]. The mineral composition of Mars' moons is even less well known. However, Phobos and Deimos are believed to be small captured asteroids at 22.2 km and 12.6 km in diameter with total masses of and 1 . 1 × 10 16 kg and 1 . 0 × 10 15 kg respectfully [73]. However, their example suggests materials from captured asteroids may be an option.", '8 Plasma torus': "There is one final alternative to creating a large scale, space-based high current loop that does not require creating a physical structure the dimensions of Mars, that is superconducting, but will not melt. This is to use a plasma torus with a resultant ring current necessary to create a resultant magnetic field. This is illustrated in Figure 8. The concept would be similar to having an open particle accelerator like the Large Hadron Collider (LHC) at CERN [74] but in space and without the goal of accelerating particles. Enrico Fermi first suggested putting a particle accelerator in Earth \nFigure 8: The principle of a plasma torus with current drive that produces a resultant magnetic field. Charged particles are directed between a series of space stations that guide the particles to form the current loop. \n<!-- image --> \norbit to deal with the ever-increasing dimensions needed to reach the ever increasing energy of particles needed for fundamental physics research [75]. The problem with doing this for particle physics is that while the vacuum of space is conductively cold, it is radiatively very hot. This, however, would be an advantage to the plasma torus as the radiative energy could be used to produce the ionization. \nIn the ultra-high vacuum of space charged particles (electrons and ions) could be accelerated and 'beamed' between discrete space stations to create a resultant toroidal current loop. Multiple space stations would be needed to turn the particle streams into a loop and to ensure that the electrical circuit could be completed, to prevent the space stations continuously becoming electrically charged. The stations would introduce the curvature as well as accelerate the particles and maintain the cohesiveness of particle bunches. (This is similar to the action of the particle 'kickers' in terrestrial particle accelerator facilities [76].) \nThe major advantage of this approach is the reduced mass needed rather than any planet encompassing structure solid structure. Plasma structures such as radiation belts naturally occur around planets like the Earth. In these cases, the co-rotating ions and electrons are formed as a result of the rotation of the planet and complex interactions of its natural magnetic field. Here we do the opposite, artificially driving a current in a plasma torus to create a resultant magnetic field. The current can be carried by a current loop where the charge carriers are distributed over an area A, where the total current could be made from a a sum of N beamlets of current I c , where: \nI = NI n = nAvQ (4) \nHere A is the cross-sectional area of the plasma torus and n is the number of charge carriers per unit volume (or charge density), Q is the electronic charge of the particles and v is the drift velocity of the charge carriers. \nFor current drive [77], the toroidal momentum is transmitted to either the plasma electrons or ions. The velocity of the particles in the toroidal direction must exceed their thermal velocity. The electrons need to be given sufficient momentum so that the high inertia ions cannot create a charge balance that cancels the electron current. The ions too could be given an artificial velocity in the opposite direction to enhance the current. The current would then be I = AQ ( n i v i -n e v e ) where n i and n e are the number density of ions and electrons respectively and v i , v e their directed velocities. \nThe principles of current drive in laboratory fusion plasma confinement devices, such as tokamaks, are well established. Clearly for this unique application new technology would need to be applied but for a review of the underlying principles of current drive in plasmas see [77]. There are a number of methods which utilize particle beams or radio frequency waves in any of several frequency regimes. Traveling waves may be induced in the plasma that accelerate the particles via phased arrays or coil arrays and use could be made of plasma waves or electromagnetic waves which exist in nature. The types of wave are varied, the optimum waves could be resonant with lower hybrid waves, Alfv'en, ion cyclotron or electron cyclotron resonant or use more sophisticated modern 'wakefield' approaches [79]. Further discussion of which approach is most suitable would be better suited to a stand-alone article. \nIn general terms in this case, the magnetic field at the outer surface of the plasma loop B b , with minor radius of the plasma torus of b, is approximately, \nB b glyph[similarequal] µ 0 I 2 πb (5) \nIf we assume the magnetic field within the solenoid is uniform then B b = B 0 as used previously. \nFor a plasma torus solenoid by equating equations (2) and (6) with B 0 ∼ B ( b ) this provides b ∼ R 0 / 3 then b ∼ 2000 km the area A = πb 2 ∼ 10 13 m 2 and if we assume Q = 1 and the charge carriers are electrons then current density j = nv = 5 × 10 -5 Amp m -2 for the plasma torus. The attraction of this system, as shown by equation (6), is the 1 /R scaling of the resulting toroidal magnetic field, but the disadvantages would include the continuous power required to drive the current. Having a plasma current not in a solid conductor but exposed in space solves the mass problem but comes with its own technical challenges which we will now explore. \nOver 70 years of laboratory magnetic confinement fusion research have greatly increased our knowl- \nedge of how to confine and control plasma tori [82]. While the criteria for fusion in the laboratory are more exacting than required here, the problems of scale and location do present their own challenges. The stability of a current driven in a laboratory plasma torus is aided by imposing a guiding magnetic field in the same direction as the plasma current (toroidal or torus major axis). This plasma current results in the creation of the magnetic field in the poloidal direction (minor radius cross section). Close to the plasma column, this poloidal component of field aids confinement, using the 'pinch' effect that helps inhibit (though not eliminate) the radial expansion of the plasma particles. Different types of laboratory devices have different relationships between these three basic components of plasma current, toroidal and poloidal magnetic field [78]. Unlike in a natural environment, in an artificial system these are all variables available to be optimized. Here the aim is to maintain the integrity of the plasma current loop sufficiently to result in an overall dipole magnetic field and maintain it against the processes such as diffusion, instabilities, particle pick up and shielding. There are several ways to do this. \nOne way to prevent particles of the solar wind plasma from forming a responsive shell that cancels out the net magnetic field produced by the particle beam current, is to increase the velocity of the particles forming the current given by equation (4) so that their velocity is very much greater than the thermal velocity, v glyph[greatermuch] v th . Particles (generally electrons) like this in tokamaks are called 'runaways' [80], [81] because they are essentially unmagnetized and too energetic to be effected by the bulk plasma flow and hence pass straight through it. In a similar way, highly energetic protons from some solar eruptions can cross the solar system, passing through all the regions of the Earth's magnetosphere and reaching the ground [83]. For this application creating such a directed beam would be an advantage as it would maintain the current and mean that the plasma particles carrying the current would not be easily 'picked-up' or disrupted by the external plasma flows from the space plasma environment, such as effects due to solar storms. To do this the velocity in equation (4) would need to be such that v e ∼ 10 v th ∼ 100keV or ∼ 0 . 1 c where c is the speed of light. \nHowever, at runaway velocities, the plasma torus would be difficult to align and it would thus be difficult maintain a loop, forming a radiation hazard to any transiting spacecraft e.g., [84]. Therefore, an alternatively higher density, lower velocity scheme could be used. Here, since 1 ampere = 6 . 2 × 10 18 particles per square meter per second, if we pick a velocity of 1 ms -1 (non-hazardous) the required density n = 3 × 10 14 charged particles per m 3 this density is so low as to not be a concern to spacecraft. Here the problem will be a greater diffusion of the particles leaking away. To mitigate this, the stations would need to create a linked magnetic geometry or toroidal guide field. This could be done with an array of poloidal solenoids arranged in a toroidal loop. In laboratory plasmas, this \nconfiguration is historically known as a 'Polytron' [85] or 'bumpy torus' [86]. In such a configuration, the superposition of high-ripple toroidal magnetic fields acts as a guide field for the toroidal plasma discharge. \nOnce the magnetic field is created, the full force of the interplanetary solar plasma should not directly reach the current loop, as it will be inside the magnetosphere created. The evidence from natural magnetospheres is that even the non-current driven, natural plasma 'torus-like' structures such as radiation belts and plasmaspheres persist and are not regularly eroded by pick-up processes during storms, at least not significantly enough to be 'blown away'. Sufficient density of higher Z materials will help retain the plasma. At times additional mass loading of the current loop may be needed to ensure dominance of the artificial current during extreme solar events. \nThe closest natural example in space is the Io plasma torus inside Jupiter's magnetosphere [87]. Eruptive material fromIo produces a plasma torus that envelops the entirety of a moon's orbital corridor and has been observed to remain roughly constant, even though the volcanic activity emitting the mostly heavy ions of SO 2 , i.e., S + , S ++ , O + , O ++ on Io is presumably sporadic [88]. Undoubtedly the situation for Io has major differences - the presence of the large, rapidly rotating magnetic field from Jupiter being just one factor [87]. \nThe most efficient source of plasma particles to form the torus is to directly evaporate them off one of the moons of Mars. This would locate the rings at their orbits. This is shown in Figure 9 for Phobos. Using the natural moons of Mars, Phobos (or Deimos), as 'plasma generators' to 'seed' the ring and building this plasma up to form a ring current over many months or years would minimize the resources needed. Phobos is 2.76 Mars radii above the center of Mars (about 6,000 km above the surface) and orbits in 7h and 39m whereas Deimos is in a higher orbit at 6.92 R M from the centre and with a 30.3 h orbital period. Deimos would provide a large surface area to host multiple nuclear power generators and related infrastructure. Both moons orbit above the equator of Mars and Mars is tilted by a very similar angle to that of the Earth at about 25 degrees to the orbital plane of Mars around the Sun. \nThe moons of Mars are so small (Phobos' diameter is 22.2 km and Deimos is 12.6 km) that they have virtually zero escape velocity so, if there was a 'plasma generator' such as an ablation or gas release system (powered by the already referenced nuclear reactors), the plasma would easily escape from the surface. Many elements are easily ionized when exposed to solar radiation even at 1.3AU, creating equal numbers of electrons and ions of a plasma. This would naturally start to form a plasma ring as the moon moves in its orbit. After 30.3 hours (for Deimos) the moon would return to its original orbital position, and the plasma ring would start to be reinforced. Over time a permanent \nFigure 9: Plasma released from the moon of Mars (Phobos or Deimos) (Phobos shown in insert (a)) and artificially accelerated (insert(b)) forming a plasma torus with current drive that produces a resultant magnetic field. Unlike natural systems the particles can be accelerated above the Dricer limit, to form runaway beam relatively unaffected by plasma environment and pickup processes but will need 'kicker' stations to ensure a closed loop is formed. The poloidal magnetic field shown created by current aids the plasma column confinement as well as forming the resultant magnetic field. If the plasma beam is mass loaded with sufficient density to allow for diffusion losses it still would take 10 11 Earth years to exhaust the material of Phobos this way. \n<!-- image --> \nplasma ring could be established. Phobos being bigger, with more surface area, and closer to Mars, it returns to the starting point 4 × quicker. This process would then be similar to what happens with Io, where volcanic eruptions generate the Io plasma torus inside Jupiter's magnetosphere [87]. The Io-Jupiter interaction forms a complex current system accelerating particles into the poles. In the Io-Jupiter case there is an extremely strong pre-existing, natural magnetic field and very rapid rotation. This, unfortunately, will not be the same for Mars. \nHow quickly the plasma will drift away before Deimos or Phobos are able to get back to re-supply it at each location on the next orbit, will depend upon many factors as discussed above. The experience for Io-Jupiter is that the heavy ions stay in place longer. For Io, the torus material is composed of volcanically released sulfur, oxygen, sodium, and chlorine which then get ionized [89]. The composition of Phobos and Deimos is believed to be phyllosilicates or silicate minerals [90]. For the purposes of creating an electrical current any ions that are easily ionized are suitable. The electrons carry the current, but the opposite drift of the ions helps 'anchor' the electrons to associate with the ions via charge separation electric fields. A guide toroidal field would help prevent the loss as would the current driven in the plasma loop. Once the loop is formed the current drive in the torus would result in a 'pinch' effect, with a magnetic field or minor radius oriented (compressional) magnetic field component that would help inhibit outward diffusion [91]. \nEven though Mars' moons are not large, there would be no risk of running out of raw material for the ring itself. Using Phobos, it would take ∼ 10 11 Earth years to completely exhaust the available material. This is taking the total mass of Phobos of ∼ 10 16 kg, creating a current ring of dimensions b=2000km and a R 0 = 2 . 8 R M and a current of ∼ 7 × 10 7 Amps that requires ∼ 5 × 10 5 charge carries per m 3 and therefore a mass of only 15kg per loop if we take a representative atomic mass of 22. This calculation is based on a composition assumed to be dominated by silicon and oxygen for Phobos, that is only singly ionized, and assuming a complete replenishment each orbit. Even with these rough numbers there is plenty of leeway. \nAlthough a plasma torus would need to be actively driven, requiring significant levels of power, the infrastructure would be far less than any solid conductor approach as already discussed above and it offers by far the most credible approach of those we have considered here.", '9 Summary and conclusions': "If Mars is ever to be a long-term abode for human life, it will possibly need the protection of an artificially created magnetic magnetosphere of planetary dimensions. Earth's magnetosphere helps protect the planet from the potential sterilizing effects of cosmic rays and helps retain the atmosphere \nfrom significant stripping during large solar superstorms as they pass over the planet. Here we have shown some simple calculations exploring the basic physics and engineering of what would be needed practically to create a planet sized artificial magnetic field similar to Earth's. Clearly the resources needed would be vast. The purpose here has not been to examine the performance of a magnetic based magnetosphere at Mars, nor to justify the need for magnetic shield. Rather the intention here is to quantifiably explore the practical ways this might be done if humanity chose to do so and to make some estimate of the resources that would be involved. This is done for the first time in a scientific journal. This has deliberately been done to one significant figure precision as each approach presented would need a separate article to detail the level of technology development needed to justify more exacting figures. However, this first brush does allow a comparison of approaches and exploration of ideas. \nNo individual solution comes without vast technical challenges, many of which go beyond what can be described here. The primary challenge is not the intensity of magnetic field needed but the size of the required spatial dimensions. Evidence from Earth's magnetosphere is that the magnitude of the magnetic field intensity to hold back the solar wind is about ∼ 100nT. However, to protect the whole of Mars this would need to be a continuous field over an absolute minimum area of ∼ 10 9 km 2 (the surface area of Mars assuming a 100km atmosphere). To allow for such a magnetosphere to persist during the interaction with the solar wind under all conditions, this would need to be very much larger. \nOf the options considered here it is unlikely that restarting Mars' core will ever be viable option. The problem is not just the minimum of 10 11 , 1 Megaton hydrogen bombs needed to be distributed about the iron core to melt it, but the uncertainty that the dynamo would even restart if this was done or how long any circulation would continue, as it is currently uncertain why Mars' dynamo stopped in the first place - assuming Mars did once have a natural magnetic field arising from its core, like Earth. \nSolenoid loops are the next option and there are a variety of potential locations and technologies. In terms of location these range from on the planet's surface, to stable orbits and co-incident with Mars' moons. With an artificial system the magnitude of the magnetic field at the source and the size of the structure creating it are available to be traded. What we have shown is the advantage that a wide radius ( R 0 ) solenoid loop provides not only in terms of lower magnetic field requirements at the coil surface (which would be safer to work and live around) but that the reduction in the rate of decrease of magnetic field with distance is less by R 3 0 making it much more efficient at covering a wider area than a small radius loop with higher field. A wide diameter current loop requires a larger \nphysical structure to be built in space, however. \nThe magnetic field generating structures could be made of superconducting materials or permeant magnets both of which minimize operating power but have the disadvantage of being heavy and made from rare minerals. Alternatively, carbon nanotubes offer a potentially lighter conducting structure but are fragile and have a finite resistivity requiring continuous power and therefore power losses to overcome. \nWe have shown that the currents required are between ∼ 0.2-0.5 GAmps in one or many solenoid loops. Whilst the power requirement will depend upon the material used, it can be estimated to be between 0.1-100 GW which is between less than one but up to 50 typical 2GW power stations. While not trivial this is not unimaginably large, especially if controlled nuclear fusion has been successfully developed as an efficient energy source in the future. \nOne final approach to reduce the mass burden is to use a beamed plasma current rather than any form of solid conductor. In this scenario the current traverses the vacuum of space. To do this, the charged particles making up the current need to be accelerated to velocities where the interaction with the surround plasma environment is not sufficient to disrupt the loop of current. Exceeding the runaway limit would mean the particles would not be prone to 'pick-up' from solar storms. Once the magnetic field is established, the plasma current channel would reside in the relative protection of its own magnetosphere. Evidence from natural planetary observations is that even non-current driven, non-runaway plasma torii around planets (like the radiation belts) are not totally eroded by solar wind particle pick-up, though losses are inevitable. It would undoubtably be necessary to direct and replenish the plasma current via a series of aligned space stations. However, depending upon its size and location, such a relativistic particle beam could be a potential radiation hazard to transiting spacecraft. So, an alternative would be to 'mass load' the plasma loop by accelerating the particles to a lower non-hazardous velocity but with overwhelming much larger number density. This could be done by evaporating matter from Phoebe or Deimos, ionizing it and using electromagnetic current drive techniques to accelerate the resulting charged particles. The closest natural phenomena to this without the current drive, is the plasma torus created in Io's orbit around Jupiter. What Io's example shows is that a high Z plasma loop around a planet can form and persist (although in Io's case the enormous magnetic field of Jupiter amongst other factors will help confinement). We have shown that,for Mars, the mass needed would not substantially erode the moons at approximately ∼ 15kg per orbit per loop. Using higher Z ions to form the torus will aid retention. \nIn conclusion, as anticipated the resources needed to create a planetary sized magnetic field are non-trivial and there is much further research to be done. What has been presented here are some \nunique solutions for the approaches required to create an artificial planetary sized magnetic field. \nWhilst the ideas presented here are at the scale of a planet like Mars, the principles are equally applicable to smaller scale unmagnetised objects like manned spacecraft, space stations or moon bases, creating protective 'mini-magnetospheres'. \nWith a new era of space exploration now underway, this is the time to start thinking about these new and bold future concepts. As has been proposed by the recent White Paper for NASA's planetary decadal survey Interdisciplinary Research in Terraforming Mars: State of the Profession and Programmatics [92], there is a need to close strategic knowledge gaps and begin to further these concepts and others, in order to work toward a solution that will make colonizing Mars by humans an eventual reality.", 'Acknowledgements': "This paper is dedicated to the memory of John Bradford, in thanks for many fruitful discussions. The authors would like to thank the UK Science Technology Facilities Council and RAL Space in-house research for their support and NASA. Thanks to Lasers and Plasmas Group at Instituto de Plasmas e Fus˜ao Nuclear at T'ecnico Lisboa (IST), Portugal.", 'References': "- [1] B. M. Jakosky, et al., MAVEN observations of the response of Mars to an interplanetary coronal mass ejection, Science, 350, no. M6261 (2015).\n- [2] C. Dong, M. Jin, M. 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2024A&A...691A..96S

We present a study of the white dwarf WD cooling sequence CS in the globular cluster GC Omega Centauri or NGC 5139 hereafter Cen the primary goal of a dedicated Hubble Space Telescope HST programme. We find that the peak at the termination of the WD CS is located at mSUBF606WSUB 30.1 0.2 equivalent to V 31. The brighter part of Cens WD CS is consistent with the presence of massive Hecore WDs in agreement with previous HST analyses with ultraviolet and blue filters. Comparative analyses of the WD luminosity function LF and theoretical counterparts show that a singleage population for the cluster is compatible with the data. However an analysis of only the WD LF cannot entirely exclude the possibility of an age range due to uncertainties in the presentday WD mass function with a star formation history potentially spanning up to 5 billion years predominantly comprising stars about 13 Gyr old with a minority potentially as young as 8 Gyr. This underscores the need for global spectroscopic and photometric investigations that simultaneously include both the WD populations and the previous evolutionary phases in order to fully understand the clusters diverse chemical compositions and ages.

2024-11-01T00:00:00Z

['2024arXiv240904533S', '10.1051/0004-6361/202451288', '2024A&A...691A..96S', '10.48550/arXiv.2409.04533', 'arXiv:2409.04533']

['white dwarfs', 'globular clusters: individual: NGC 5139', 'Astrophysics - Solar and Stellar Astrophysics']

The HST Large Programme on Centauri VII. The white dwarf cooling sequence

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

https://arxiv.org/pdf/2409.04533.pdf

{'The HST Large Programme on ω Centauri - VII. The white dwarf cooling sequence': 'M.Scalco 1 , 2, ⋆ , M. Salaris 3 , 4 , L. Bedin 2 , M. Griggio 1 , 2 , 5 , A. Bellini 5 , M. Libralato 2 , D. Nardiello 6 , 2 , E. Vesperini 7 , J. Anderson 5 , P. Bergeron 8 , A. Burgasser 9 , and D. Apai 10 , 11 \n- 1 Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Via Giuseppe Saragat 1, Ferrara I-44122, Italy\n- 2 Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Padova, Vicolo dell\'Osservatorio 5, Padova I-35122, Italy\n- 3 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\n- 4 Istituto Nazionale di Astrofisica, Osservatorio Astronomico d\'Abruzzo, Via Mentore Maggini, Teramo I-64100, Italy\n- 5 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n- 6 Dipartimento di Fisica e Astronomia "Galileo Galilei", Università di Padova, Vicolo dell\'Osservatorio 3, Padova I-35122, Italy\n- 7 Department of Astronomy, Indiana University, Swain West, 727 E. 3rd Street, Bloomington, IN 47405, USA\n- 8 Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Quebec H3C 3J7, Canada\n- 9 Department of Astronomy & Astrophysics, University of California, San Diego, La Jolla, California 92093, USA\n- 10 Department of Astronomy and Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA\n- 11 Lunar and Planetary Laboratory, The University of Arizona, 1629 E. University Blvd., Tucson, AZ 85721, USA', 'ABSTRACT': "We present a study of the white dwarf (WD) cooling sequence (CS) in the globular cluster (GC) Omega Centauri (or NGC 5139, hereafter ω Cen), the primary goal of a dedicated Hubble Space Telescope (HST) programme. Our analysis has revealed that the peak at the termination of the WD CS is located at m F606W = 30 . 1 ± 0 . 2 (equivalent to V ∼ 31). The brighter part of ω Centauri's WD CS is consistent with the presence of massive He-core WDs, in agreement with previous HST analyses with ultraviolet and blue filters. Comparative analyses of the WD luminosity function (LF) with theoretical counterparts have shown that a single-age population for the cluster is compatible with the data. However, an analysis of just the WD LF cannot entirely exclude the possibility of an age range, due to uncertainties in the present-day WD mass function, with a star formation history potentially spanning up to 5 billion years, predominantly comprising stars about 13 Gyr old, and with just a minority potentially as young as 8 Gyr. This underscores the need for global spectroscopic and photometric investigations that include simultaneously the WD populations together with the previous evolutionary phases to fully understand the cluster's diverse chemical compositions and ages. \nKey words. white dwarfs - globular clusters: individual: NGC 5139", '1. Introduction': 'The white dwarf cooling sequence of globular clusters lies in one of the faintest and least-explored regions of the colourmagnitude diagram (CMD). Deep imaging with HST has reached the peak of the WD number distribution at the faint end of the CS in four GCs: NGC 6397 (Anderson et al. 2008a), M4 (Bedin et al. 2009), 47 Tucanae (Kalirai et al. 2012), and NGC6752 (Bedin et al. 2019, 2023). Each one of these GCs hosts multiple stellar populations (mPOPs) characterized by small mean spreads in initial helium abundances (Milone et al. 2018), and their WD CSs align with predictions for singlepopulation (WD population) systems (Richer et al. 2013; Campos et al. 2016). \nOmega Centauri is a moderately low-reddening GC which is relatively close ( ∼ 5 kpc) to the Sun, making it an ideal target for an e ffi cient study of its faint WD population. It is also one of the most extreme cases of a GC hosting mPOPs; in fact, spectroscopy has disclosed that its stars display a range of [Fe / H], and a very complex pattern of light-element anti-correlations at each [Fe / H] (Marino et al. 2012), while optical CMDs have revealed two main groups of stars with large di ff erences in their \ninitial helium content(Y ∼ 0.40 for the He-rich component; King et al. 2012), as evidenced by its split main sequence (MS) in optical filters (Bedin et al. 2004). The large range of initial chemical abundances also manifests itself through UV-based CMDs, that display a main sequence with at least 15 sub-populations (Bellini et al. 2017b). \nRegarding the WD population, the upper part of the CS in ω Cen exhibits a bifurcation into two sequences, as reported by Bellini et al. (2013): a blue CS populated by standard CO core WDs, and a red CS populated by low-mass WDs with both CO and (mainly) He cores (the presence of a sizable population of He-core WDs was already disclosed by Monelli et al. 2005 and Castellani et al. 2007). \nThe prevailing hypothesis suggests that the blue WD CS is populated by the final stages of evolution of the He-normal stars of ω Cen, while the red WD CS is populated by the final stages of evolution of the He-rich stars. Observing the complete WD CS of this cluster will allow us to investigate the e ff ect of a chemically complex system like ω Cen on the termination of the WD CS (that is an age indicator used in stellar population studies), and set independent constraints on the cluster age spread. \nThis study represents the seventh paper in a series dedicated to the exploitation of an HST multicycle large programme cen- \ntred on ω Cen. The previous publications focused on the analysis of the parallel fields collected as a part of the programme: \n- -Milone et al. (2017, hereafter Paper I) focused on the analysis of the mPOPs of ω Cen at very faint magnitudes.\n- -Bellini et al. (2018b, hereafter Paper II) analysed the internal kinematics of these mPOPs.\n- -Libralato et al. (2018, hereafter Paper III) presented the absolute proper motion estimate for ω Cen.\n- -Scalco et al. (2021, hereafter Paper IV) released the astrophotometric catalogue for two parallel fields of the programme.\n- -Gerasimov et al. (2022, hereafter Paper V) presented a set of stellar models designed to investigate low-mass stars and brown dwarfs in ω Cen.\n- -Finally, Scalco et al. (2024, hereafter Paper VI) presented a comprehensive analysis of the radial distribution of the mPOPs of ω Cen across an extensive part of the cluster. \nThe focus of this study is the WD CS of the cluster and its LF in the primary deep field of this HST large programme. \nThe paper is organized as follows: Section 2 outlines the observations, while section 3 gives details about the data reduction. Section 4 o ff ers a concise overview of the artificial star tests (ASTs) we have performed. The selection criteria employed to construct the cluster CMD are detailed in Section 5. The decontamination of the cluster sample using proper motions (PMs) is discussed in Section 6. Section 7 presents the empirical WD LF derived from the data, and Section 8 discusses the theoretical interpretation of the WD LF. Finally, Section 9 provides a brief summary of the results.', '2. Observations': 'This study is based on images collected with the Wide-Field Channel (WFC) of the Advanced Camera for Surveys (ACS) onboard HST under the multicycle large programme The end of the White Dwarf Cooling Sequences of Omega Centauri (programme IDs: GO-14118 and GO-14662, PI: Bedin). Data were collected over a total of of 132 HST orbits across six epochs, spanning a temporal baseline of ∼ 3 years (from 2015 to 2018). In each epoch, we took 24 deep exposures (of ∼ 1100-1200 s), in the F606W filter and 20 in the F814W filter, resulting in a total of 216 exposures taken with the F606W filter and 180 exposures with the F814W filter. The data set also includes 12 F606W and 10 F814W short exposures (of ∼ 40-50 s). A finding chart of the studied field at two scales, and its position relative to the cluster centre is shown in Fig. 1.', '3. Data reduction': "The data reduction closely follows the procedure outlined in Bedin et al. (2019, 2023), and in other papers of this series (see Paper II; Paper IV; Paper VI). For a detailed description of the reduction process, we refer readers to those papers. \nWe initially conducted a first-pass analysis, wherein fluxes and positions for relatively bright, unsaturated stars were extracted from each image using the FORTRAN routine hst1pass (see Anderson 2022). Each image was analyzed independently to generate a tailored e ff ective Point Spread Function (ePSF), allowing for adjustments to accommodate spatial and temporal variations relative to the library ePSFs provided by Anderson (2006). The tailoring of ePSFs was performed using the method introduced by Anderson & Bedin (2017) for the ultraviolet and \nvisible (UVIS) channel of the Wide Field Camera 3 (WFC3), later extended to ACS / WFC by Bellini et al. (2018a). Both positions and fluxes were corrected for the geometric distortion of the detector following the methods outlined by Anderson (2006). We then created a common, distortion-free reference frame, based on cluster members, to which all individual images were linked by using a six-parameter linear transformation. \nWith the tailored ePSF and transformations obtained during the first-pass analysis, we conducted a second-pass analysis employing the FORTRAN code KS2 introduced by Sabbi et al. (2016); Bellini et al. (2017a) (also see Anderson et al. 2008b; Paper II; Paper III; Paper IV; Nardiello et al. 2018 for a comprehensive description of the software). The KS2 code iteratively identifies, models, and subtracts point sources from the image, initially targeting the brightest sources and progressively addressing fainter sources within the subtraction residuals. This iterative process determines stars' positions and fluxes, alongside crucial diagnostic parameters such as the local sky noise (rmsSKY) and the RADXS parameter (introduced in Bedin et al. 2009), which assesses the resemblance of the source flux distribution to that of the ePSF. For a detailed description of these parameters, we direct readers to Paper II; Paper IV, and Nardiello et al. 2018. \nPhotometry was calibrated to the ACS / WFC Vega-mag system using the procedure outlined in Bedin et al. (2005), employing encircled energy and zero points available from STScI 1 . \nIn panel (a) of Fig. 2, we present a preliminary m F606W versus m F606W -m F814W colour-magnitude diagram (CMD) for sources satisfying | RADXS |< 0.1 in both F606W and F814W filters. This CMDisused exclusively to define the fiducial line of the WD CS (see Section 4). For the subsequent analysis, a CMD obtained with di ff erent selection criteria is employed.", '4. Artificial stars': "The ASTs were conducted following the guidelines outlined by Bedin et al. (2019, 2023). In summary, a fiducial line was established by hand along the bulk of the observed WDs in the preliminary CMD presented in panel (a) of Fig. 2, extending down to where they appeared to stop, and then extrapolated to fainter magnitudes. This fiducial line is shown in magenta in panel (a) of Fig. 2. \nUsing KS2 , 10 5 ASs were introduced along this fiducial line, distributed uniformly with a m F606W magnitude between 24 and 32, and a homogeneous spatial distribution across the field of view. The methodology outlined in (Bedin et al. 2009, Section 2.3) was followed to correct for input-output systematic errors in both real and artificial magnitudes. \nTo determine if an inserted star was successfully recovered, criteria were set such that if an AS was not detected within 0.753 magnitudes ( ∼ -2.5log 2) in both filters and within 1 pixel from the inserted position in both x and y detector coordinates, it was considered unrecovered. Panel (b) of Fig. 2 shows the inserted artificial sources (in magenta) and those that were successfully recovered (grey dots). \nThe combined information from the panels of Fig. 2 was used to bound the region used for counting the WDs of ω Cen. This region was defined by hand-drawn green lines, striking a balance between encompassing observed WDs with significant photometric scatter and excluding the majority of field objects. \n<!-- image --> \n<!-- image --> \nFig. 1. (Left:) A 1 · × 1 · infrared image from the Digital Sky Survey 2 centered on ω Cen . The image is aligned with North up and East toward the left. The region indicated in magenta is the field of view covered by our HST deep field from programmes GO-14118 and GO-14662. (Middle:) Three-chromatic stacked image of the entire ACS / WFC primary field ( ∼ 5 ' × 5 ' ). (Right:) A zoom-in on a dark sub-region (of ∼ 1 ' × 1 ' ) of the ACS / WFC primary field (indicated by the green box in the center panel), at a scale where individual pixels are visible. \n<!-- image --> \nFig. 2. (a) Preliminary m F606W versus m F606W -m F814W CMD of sources within the studied field. Only those having | RADXS |< 0.1 in both F606W and F814W filters are shown. The magenta line denotes the fiducial line for the WD CS, while the green lines bound the region in the CMD used for counting the WDs. (b) On the same scale, the CMD for ASs is presented. The magenta and green lines from panel (a) are retained. The green lines define an area generous enough to encompass both the majority of observed real WDs in panel (a) and the ASs introduced along the WD fiducial line, which were recovered despite exhibiting substantial photometric errors. \n<!-- image -->", '5. Color-magnitude diagram and selections': 'Following the approach by Bedin et al. (2019, 2023), in Fig. 3, we show the impact of our progressive selection criteria on ASs, and then apply the same criteria to real sources. In each panel, the top-right corner is labelled either with an (a) for ASs or with an (r) for real sources. The aim of these selections is to strike a balance that maximizes the inclusion of well-measured WD members of ω Cen while minimizing the inclusion of spurious or poorly measured detections. It is important to note that in the \nsubsequent analysis, we exclusively utilize the long exposures, as they provide the necessary depth to study the WD CS. \nThe light blue and dark blue shaded areas represent the 5 σ and 3 σ regions, respectively, where σ denotes the background noise as measured by Bedin et al. (2019) in both filters. For our analysis, we only consider sources above the 3 σ limits as significantly detected. \nPanel (a1) displays all artificial sources as inserted (in magenta) and as recovered (blue dots). In panel (a2), sources are limited to those within areas observed in at least ∼ 40% of the F814W and F606W images, resulting in a significant reduction of the field of view used for the investigation. \nPanel (a3) further restricts sources to regions where the rmsSKY matches the noise within empty sky patches, indicating areas suitable for detecting faint objects. Panel (a4) excludes all ASs recovered outside the region enclosed within the two green lines shown in panel (a3). \nIn panel (c1), instead of a CMD, the magnitude versus completeness curve is presented (black line). We also display the "good" completeness cg in blue, which represents the completeness calculated within the "good" regions based on the rmsSKY value (see Bedin et al. 2008, for details). This panel indicates that sources passing these selections are 50% complete down to m F606W = 29.45 and 25% complete at m F606W = 30.04. \nPanel (a5) displays the result after applying the RADXS parameter to reject non-stellar objects. Panel (c2) shows the completeness curves after the final RADXS selection. \nThe same selections applied to ASs are then applied to observed real sources in the top panels of Fig. 3. Panel (r5) shows the final sample for ω Cen\'s WD candidates. \nPanel (s1) displays all stars that meet the selection criteria, including those outside the area defined by the two thin green lines in panel (a3). The stars outside this region will be utilized to model field contamination within the WD region in the CMD. The fiducial line defined in Section 4 (and shown in magenta) well represents the mean observed CMD location for WD CS of ω Cen. \nPanel (s2) shows real stars and ASs that meet all selection criteria: ASs -as selected in panel (a5)- are shown in orange, while real sources -as selected in panel (r5)- are shown in blue. This comparison suggests that the majority of the observed real WD CS might not extend to magnitudes as faint as those of the \nrecovered ASs. This could indicate that the peak of the WD CS LF of ω Cen has potentially been reached and surpassed.', '6. Proper motion decontamination': "We also investigated whether the PMs can help in disentangling field objects found within the WD region of ω Cen's CMD and cluster members. To do this, we evaluated PMs following the approach by Bedin et al. (2023) i.e. combining the bulk of the first three epochs ( ∼ 2016.1) to obtain averaged positions for the sources, which are then compared to their averaged positions as measured in the last three epochs of the data ( ∼ 2018.1). In the following analysis, we will consider only sources shown in green in panel (s1) of Fig. 3 but for which it was possible to estimate the PM. \nThe PM analysis is presented in Fig. 4. We colour-code in blue stars surviving the WD selection defined by the two thin green lines, and in orange all other stars. \nIn panels (a), (b) and (c) we present the vector-point diagrams (VPDs) of the sources in our sample. Since PMs are calculated relative to the cluster's overall motion, the cluster members' distribution in the VPD is centred at (0,0). Panels (d), (f) and (g) display CMDs. Panel (e) from the left shows the observed one-dimensional PM ( µ R, obtained by summing in quadrature the PM in the two directions) plotted against m F606W. Among brighter stars, we observe a narrow distribution in µ R for ω Cen's members, predominantly clustered well below µ R < 1 . 5 mas yr -1 , while a broader tail extends towards higher µ R, peaking between 2 < µ R < 10 mas yr -1 , indicative of field objects. However, as we approach a magnitude m F606W ∼ 28, the random errors in positional measurements, compounded over the first and last three epochs, become significant for fainter stars. By m F606W ∼ 28 . 5, it becomes notably challenging to disentangle cluster members and field objects. We defined a PM selection consistent with the PM errors at di ff erent magnitudes represented by a red line. Stars satisfying this selection are shown as filled circles, while stars not passing this selection are indicated with crosses. The subsequent panels display the VPD and CMD for sources positioned to the left or right of the red-line criterion, maintaining the blue colour code for WD candidates. The PMselection e ff ectively separates outliers and objects with large PMs. However, it struggles to distinguish between WDs and field objects fainter than m F606W ∼ 28 . 5. This challenge arises mainly due to the minimal separation between field and cluster members, which is much smaller than measurement errors at magnitudes fainter than m F606W ∼ 28 . 5. \nSince PMs are not e ff ective in distinguishing between WDs and field objects for faint sources, we decided not to use PMs in our analysis.", '7. The corrected WD CS LF': 'We applied a correction to the observed WD CS LF to mitigate the e ff ects of field contamination by unresolved blue galaxies, following the method outlined by Bedin et al. (2023). The process is presented in Fig. 5 and described as follows. In panel (a), we show the CMD of sources identified in panel (s1) of Fig. 3. Using the two lines defined in Fig. 3, we establish on the CMD what we term the WD-region . Additionally, we define two other regions with the same colour width at each magnitude of the WD-region, one at bluer colours and the other at redder colours, referred to as Blueand Red-regions , respectively. These regions are symmetrically o ff set from the WD-region by a fixed colour interval at each magnitude. \nWe then counted the sources within each of these three regions and generated in panel (b) the LFs for each one of them, with error bars representing statistical Poisson errors. The number of contaminants within the CMD WD region is determined as the average of the number of objects observed in the Blueand Red-regions at various magnitudes. This model is shown in magenta, with corresponding errors estimated through linear propagation of Poisson noise. \nIn panel (c), we compare the observed LF with the resulting WD LF corrected for the field-contamination model. The fieldcorrected LF is obtained by subtracting the field model from the observed LF, with errors propagated linearly. \nFinally, in panel (d), we present the completeness-corrected and field-corrected WD CS LF for ω Cen. The errors on this LF were corrected for completeness using a simple approximation involving linear propagation of the errors. \nThe WD CS LF exhibits a peak at an estimated magnitude of m F606W = 30 . 1 ± 0 . 2, followed by a rapid decline leading to zero. Despite the low completeness at these faint magnitudes, which falls well below the commonly accepted limit of 50%, we can consider this result valid as the peak exceeds the 3 σ threshold (dark grey area) that we have set as the lower limit for valid measurements.', '8. Theoretical interpretation of the WD LF': "As briefly summarized in the introduction, ω Cen hosts a very complex population of stars, composed of several subpopulations that show themselves as multiple sequences in CMDs displayed in appropriately chosen filter combinations. These multiple sequences are originated by the range of initial chemical composition - and possibly age- of the cluster's stars. \nSpectroscopic studies have determined a range of iron abundances ∼ -2 . 2 < [Fe / H] < ∼ -0.6 (see, e.g. Johnson & Pilachowski 2010; Marino et al. 2011; Nitschai et al. 2024), with the main component characterized by [Fe / H] ∼ -1.7, and the canonical globular cluster light-element abundance anticorrelations (e.g. the O-Na anticorrelation) present at all [Fe / H]. In addition, there is a sizable component of stars with [Fe / H] ∼ -1.3 and an enhanced helium mass fraction, i.e. Y ∼ 0.36-0.40 (see, e.g., Piotto et al. 2005; King et al. 2012). From the point of view of the stellar ages, the situation is much less clear. For example, Sollima et al. (2005) found an age spread ∆ t ≤ 2 Gyr from their analysis of the CMD of the cluster subgiant branch (SGB) region, a result in agreement with the analysis by Calamida et al. (2009) based on deep near-infrared photometry, and Tailo et al. (2016) derived a negligible ∆ t from the joint analysis of the HB and SGB. On the other hand, the studies by Villanova et al. (2007) and Villanova et al. (2014) of the SGB provided ∆ t equal to several Gyr. \nHere we have investigated theoretically the observed WD CS and the LF of Fig. 5 in light of this complex chemical abundance (and possibly age) distribution, by employing sets of WD cooling tracks and isochrones. To model the CO-core WDs from progenitors not belonging to the very He-enhanced Y ∼ 0 . 4 cluster subpopulation we have employed the BaSTI-IAC WD tracks by Salaris et al. (2022) with hydrogen envelopes and metal-poor progenitors, calculated with the Cassisi et al. (2007) electron conduction opacities, complemented by the Cummings et al. (2018) initial-final mass relation and progenitor lifetimes from the α -enhanced Pietrinferni et al. (2021) models to calculate the corresponding WD isochrones. For the WDs produced by the Y ∼ 0.4 subpopulation, given the lack of He-core WD tracks and progenitor models with Y ∼ 0.40 in the BaSTI-IAC database, we \nFig. 3. Progression of cumulative selections employed to derive a sample of well-measured WDs along ω Cen's WD CS. The sequence progresses from left to right, with top panels representing real stars and bottom panels representing ASs. The number of real stars after each selection is reported in the bottom left of the panels. In panel (a1), a magenta line indicates where ASs were introduced. Panels (c) display the resulting completeness, depicted as (c1) without and (c2) with the inclusion of the selection on RADXS, a highly e ff ective parameter for identifying wellmeasured point sources. Panel (s1) illustrates the impact of RADXS selection on stars both within and outside the defined WD region, enclosed by two green lines (see text for details). Lastly, in panel (s2), a direct comparison between real and artificial stars reveals no distinct decline in the number of artificial stars below m F606W ∼ 30 . 1. \n<!-- image --> \nemployed the models by Althaus et al. (2017) for Y = 0.40 and a metal mass fraction Z = 0.001, close to the value appropriate for the He-rich stars in the cluster (characterized by [Fe / H] ≃ -1.3). Althaus et al. (2017) calculations follow the evolution from the main sequence to the WD stage of models with masses from 0.6 M ⊙ to 2 M ⊙ and Y = 0.40, and provide therefore also a theoretical initial-final mass relation and progenitor lifetimes for these WDs, enabling us to calculate WD isochrones for the Y ∼ 0.4 cluster subpopulation. According to these calculations, MS stars with initial masses up to 0.65 M ⊙ produce He-core WDs, while more massive objects produce CO-core WDs. At ages around 10-13 Gyrs WD isochrones calculated from these models predict along the bright CS He-core WDs with masses equal to 0.440.46 M ⊙ , consistent with the results by Bellini et al. (2013). A couple of points must be noted. The first one is that Althaus et al. (2017) calculations (both progenitors and WDs) do not employ the same physics inputs of the BaSTI-IAC, although some of the main inputs are in common (for example the equation of state for the model WD cores). The second point is that Althaus et al. \n(2017) calculate progenitor models only up to 2 M ⊙ , that produce a CO-core WD model with mass equal to 0.81 M ⊙ . More massive WDs are therefore missing from the isochrones calculated with Althaus et al. (2017) models. \nFigure 6 compares three 12 Gyr isochrones and the corresponding LFs (the result of this comparison does not depend on the chosen age, at least for ages older than a few Gyr), after applying the distance modulus and extinctions assumed for ω Cen. Throughout our analysis we use ( m -M )0 = 13.67 determined by Baumgardt & Vasiliev (2021) as an average of Gaia Early Data Release 3 parallax and kinematic distances, HST kinematic distance, and 26 literature estimates (listed in their Table B.1) based on pulsating variables, eclipsing binaries, CMD fitting, and tip of the red giant branch. The adopted value agrees well with the eclipsing binary distance by Thompson et al. (2001). \nFor the reddening we employed E ( B -V ) = 0.12 (King et al. 2012), close to the average value E ( B -V ) = 0.11 from multiple sources recommended by Lub (2002), and consistent with the mean value determined by Calamida et al. (2005) using Ström- \nFig. 4. (a)-(b)-(c) Vector point diagrams for the samples shown in the corresponding panels below. (d) CMD of the sources in panel (s1) of Fig. 3. In all panels, sources within the WD region (between the two green lines) are represented in blue, while all other sources are shown in orange. The light-grey and dark-grey shaded area indicate the 5 σ and 3 σ limit, respectively, of significant detection for the sources of interest. (e) Onedimensional PM, µ R, as a function of the m F606W magnitude. Bright stars (down to m F606W ∼ 28) exhibit a µ R distribution with a tight dispersion ( < 1 . 5 mas yr -1 ) along with a tail displaying a much broader dispersion. We arbitrarily define two regions, indicated by the red line: one enclosing the bulk of the µ R values at di ff erent magnitudes (indicated by filled circles), and the other containing objects with larger µ R (indicated by crosses). (f)-(g) CMDs for the stars within and beyond the red line defined in panel (e). Neither of the two CMDs solely consists of members or field objects (see text for details). \n<!-- image --> \nren photometry. Using this value of E ( B -V ) we have applied extinction corrections to the F606W and F814W filters dependent on the model T e ff calculated as in Bedin et al. (2005). \nThese LFs (with the same bin size of the observed cluster LF) and all other theoretical LFs discussed later have been calculated using a power-law mass function for the progenitors with a Salpeter exponent x = -2 . 3 and include with a Monte Carlo \ntechnique the photometric error law for the ω Cen CS derived from the AS analysis, as described in Bedin et al. (2023). Two isochrones have been calculated for CO-core WDs from Henormal progenitors with [Fe / H] = -1 . 90 and -0 . 90, respectively, while the third isochrone has been calculated for the progeny of the Y ∼ 0.40 population. This isochrone includes He-core mod- \nFig. 5. (a) The CMD is partitioned into three regions along the WD CS. The azure-shaded region indicates sources that pass the selection on the blue side of the WDs, while the red-shaded region represents those on the redder side. (b) WD CS LF, displaying histograms depicting the number of sources per magnitude interval for observed stars within the WD region and for stars in the two shaded regions. The magenta histogram represents our model for field distribution. (c) The WD CS LF derived by subtracting the field model from the observed WD CS LF. (d) The observed field-corrected WD CS LF, adjusted for completeness, is depicted in black. Grey-shaded areas indicate the 5- and 3σ levels, where below 3σ findings and completeness become unreliable. Errors were linearly propagated and then corrected for completeness. \n<!-- image --> \ndown to m F606W ∼ 28.5, and CO-core models at fainter magnitudes. \nThe isochrones for the two He-normal populations almost completely overlap, and are very similar at their faint end, the age indicator of the population. This is a consequence of the fact that progenitors' metallicity variations a ff ect the WD cooling speed mainly due to the variation of the 22 Ne abundance in the core that a ff ects the energy released by 22 Ne di ff usion and distillation (see, e.g. Deloye & Bildsten 2002; Althaus et al. 2010; Blouin & Daligault 2021; Salaris et al. 2022, 2024, and references therein); however, at the low metallicities of globular clusters, these e ff ects are negligible because of the low abundance of 22 Ne in the WD cores. Moreover, the faint end of the isochrone, where the colours turn to the blue, is populated by the more massive WDs (with decreasing radii), coming from progenitors whose lifetime is much shorter than the isochrone age. As a consequence - remembering that at each point along a WD isochrone the sum of the cooling age of the WD evolving at that point and its progenitor lifetime is constant and equal to the isochrone age- the corresponding magnitudes are set by the cooling times of the WD progeny, and are una ff ected by \nsmall changes in the progenitor lifetimes due to variations in their metallicity. The 'hook' at the blue end of the isochrones is caused by the most massive objects (masses ∼ 1 -1 . 1 M ⊙ that have cooled down faster at this isochrone age, due to an earlier onset of crystallization and phase separation, and are slightly fainter compared to the less massive WDs. \nThe isochrone for the He-rich population is redder for most of its magnitude extension, because it is populated by lower mass objects, and eventually approaches the other two isochrones when the evolving masses become similar. The magnitude of the bottom end of this isochrone is slightly fainter, and the colour extension shorter, this latter e ff ect is due to the lack of models with mass above 0.81 M ⊙ (the other two isochrones include WD models up to 1.1 M ⊙ ). \nDespite these di ff erences, once photometric errors are included, the LFs calculated from the three isochrones are almost equivalent, especially when considering the errors in the observed star counts of the observed LF. \nAs mentioned in the Introduction, Bellini et al. (2013) nearUV and B HST photometry of the bright part of the cluster CS has demonstrated that there are two parallel sequences. Ac- \nFig. 7. Cluster CS (blue filled circles) compared to a 12 Gyr synthetic CMD of the cluster WD population (orange filled circles). The red line is used to define the sample of stars used in our analysis. See text for details. \n<!-- image --> \nFig. 6. Upper panel: Two 12 Gyr WD isochrones for the progeny of Henormal cluster subpopulations with [Fe / H] = -1.9 and -0.9 (blue solid and orange dotted lines, respectively), and a 12 Gyr WD isochrone for the progeny of the Y = 0.4 cluster subpopulation (green dashed line). See text for details. Lower panel: LFs calculated from the three isochrones in the upper panel (same line styles and colours). The number of stars in each LF is the same, approximately equal to the total number of objects in the cluster LF. \n<!-- image --> \ncording to their analysis -which also made use of Cassisi et al. (2009) study of the extreme blue part of the cluster HB- the redder sequence is populated essentially by He-core WDs with mass around 0.45 M ⊙ , the progeny of the He-rich sub-population, while the blue sequence hosts the standard CO-core WDs with mass around 0.55 M ⊙ expected to populate the bright CS of globular clusters. \nThe bright part of our optical CS however does not show a split sequence, due to the small sample of objects and especially the low sensitivity of the (F606W -F814W) colour to T e ff at high temperatures, as shown by the following test. \nFigure 7 displays a qualitative comparison between the cluster cooling sequence and a 12 Gyr (a representative age) synthetic CS made of two components (we used for the synthetic CS the cluster distance modulus and extinction discussed before). The main component, which includes 89% of the synthetic population, represents the WDs produced by all cluster sub-populations other than the Y ∼ 0.4 one. Given the result of the isochrone comparison in Fig. 6, we employed a single WD isochrone for a representative [Fe / H] = -1.7 to produce this WD population. The remaining 11% of the synthetic CS has been calculated using the isochrone calculated with Althaus et al. (2017) Y = 0.4 progenitors and WDs. These percentages come from the analysis of the radial distribution of the cluster sub-populations presented in Paper VI, taking into account that the observed field is located at a radial distance from the centre equal to 2.6 times the half-light radius. \nThe magnitudes along the synthetic CS (which include the photometric errors as determined from the artificial star analysis) have been calculated as described in Bedin et al. (2023). Here we also account for the e ff ect of completeness (we display only objects that pass the completeness test), considering how \nthe completeness fraction varies with m F606W according to the AS analysis. \nThe number of synthetic objects is 1498, the same as on the observed CS, defined in this comparison as the sequence of cluster's stars to the left of the straight line in the figure, with colours larger than -0.5 mag and F606W magnitudes lower than 30.5. \nThe synthetic sequence appears in good overall qualitative agreement with the observations, in terms of shape and colour spread at a given magnitude, apart from the brighter magnitudes below m F606W ∼ 26, where the synthetic CS appears to be slightly bluer than the observations. The synthetic population confirms that with this observed CMD we do not expect to see a welldefined bimodal CS even at the brightest magnitudes where the photometric error is small. \nBefore trying to interpret the cluster LF in terms of age and age spread of the various subpopulations, we compare it in Fig. 8 with three theoretical LFs for 8, 10 and 13 Gyr respectively, calculated for He-normal populations and [Fe / H] = -1.71. The theoretical LFs have been normalized to have the same total number of objects as in the empirical LF for m F606W < 28.0. As shown by Fig. 6, a LF calculated using a single initial chemical composition for the progenitors is appropriate to represent the whole range of chemical compositions of the cluster subpopulations (including the Y ∼ 0.40 subpopulation). \nThe width of the peak at the faint end of the LF encompasses the theoretical LFs for ages between 8 and 13 Gyr, but neither a single age nor a combination of ages between 8 and 13 Gyr can match the observed star counts between m F606W of 29.5 and 30.3. This is very likely an indication that the mass distribution of the WDs along the cooling sequence predicted by our choice of a Salpeter progenitor mass function is inappropriate. (see, e.g., the discussion in Bedin et al. 2023), due to the cluster dynamical evolution. A more top-heavy mass function would increase the relative number of massive WDs and therefore increase the number counts across the region of the peak of the LF. \nFig. 8. Cluster LF (filled circles with error bars) compared to theoretical LFs for ages equal to 13 (blue solid line), 10 (orange dotted line) and 8 Gyr (green dashed line). See text for details. \n<!-- image --> \nFigure 9 indeed shows how an exponent x = -1.6 for the progenitor mass function produces theoretical LFs (normalized as described before) that better match the star counts in the region of the peak of the LF, for various selected combinations of ages. The maximum age range compatible with the observations is 5 Gyr, with 8 Gyr as the minimum age to match the rising branch of the peak region of the LF, while an age of 13 Gyr is required to match the faint, descending branch of the peak region. A single-age population for the cluster provides a good fit to the data, also taking into account that the local maximum of the LF centred at m F606W = 29 . 3 (not matched by the theoretical LF) is very likely a statistical fluctuation; when considering the error bars, the star counts in this magnitude bin are di ff erent from the values at the two adjacent bins by less than 2 σ . \nThat said, an age range among the cluster population cannot be ruled out, so long as the large majority of stars (at least ∼ 80%) in the cluster have an age of 13 Gyr, as shown by the three selected examples in Fig. 9. A high fraction of stars with an age of 13 Gyr is required to match the two points at m F606W = 30.1 and 30.3, while the fraction of stars with ages as low as 8 Gyr has to be small, less than ∼ 10%, otherwise the ratio between the star counts around m F606W = 29 and the number of stars around m F606W = 30 becomes too high compared to the observed LF. The quality of the matches between theory and observations in Fig. 9 is very similar for all four representative cases, and adhoc tweaks of the mass function can improve the match in each of them. Therefore an age range between 8 and 13 Gyr among ω Cen stars cannot be ruled out from the analysis of the WD CS, but if stars younger than 13 Gyr do exist, they should not make up more than ∼ 20% of the total, with less than 10% can having an age as young as 8 Gyr.", '9. Summary': "We have presented our study of the complete WD CS in ω Cen, the primary objective of the HST GO-14118 + 14662 program. We have produced the CMD and corresponding completeness- \nFig. 9. Cluster LF compared to theoretical LFs calculated with an exponent x = -1 . 6 for the WD progenitor mass function, and various age combinations. Panel a : single age 13 Gyr old population; Panel b : 12% of the population 8 Gyr old, 88% is 13'Gyr old; Panel c) : 4% of the population is 8 Gyr, 16% is 10 Gyr old, and 80% is 13 Gyr old; Panel d) : 8% of the population is 8 Gyr old, 8% is 12 Gyr old, and 84% is 13 Gyr old. \n<!-- image --> \norrected LF of the WD CS, which displays a peak at the termination of the CS, located at a magnitude mF 606 W = 30 . 1 ± 0 . 2. \nWe have created a synthetic WD CS for ω Cen, consisting of a main component made of CO-core objects progeny of the Henormal cluster subpopulations (89% of the WDs) and a component made of CO-core and He-core objects produced by the Herich cluster subpopulation. This synthetic CS aligns well with observational data, except at brighter magnitudes where the synthetic sequence appears slightly bluer. \nOur analysis has shown that the chemical complexity of the cluster stellar population has a minor impact on the theoretical interpretation of the observed LF. For a fixed age, LFs calculated for CO-core WDs with varying progenitors' initial chemical compositions, and for He-core WDs produced by the heliumrich cluster subpopulation are very similar, particularly when considering the observational errors. \nWe found that a single-age population can match overall the observed LF, but an age range cannot be entirely ruled out using just the WD LF, given the uncertainties in the present-day WD mass function. LF comparisons suggest that ω Cen's star formation history could potentially span an age range up to ∼ 5 Gyr, however, the majority of stars (at least 80%) must be approximately 13 Gyr old, and only a small fraction (less than 10%) could potentially have ages be as young as 8 Gyr. \nFurther studies, analysing simultaneously the WD LF together with both spectroscopic and photometric data of the previous evolutionary phases, are essential to fully understand the formation and evolution of this extreme globular cluster. In particular, we note that observations of these very same fields are approved and scheduled with JWST under program GO-5110 (PI: Bedin). \nAcknowledgements. Michele Scalco and Luigi Rolly Bedin acknowledge support by MIUR under the PRIN-2017 programme #2017Z2HSMF, and by INAF under the PRIN-2019 programme #10-Bedin."}

2024A&A...691A.320C

Context. Understanding the morphology of galaxies is a critical aspect of astrophysics research providing insight into the formation evolution and physical properties of these vast cosmic structures. Various observational and computational methods have been developed to quantify galaxy morphology and with the advent of large galaxy simulations the need for automated and effective classification methods has become increasingly important. Aims. This paper investigates the use of principal component analysis PCA as an interpretable dimensionality reduction algorithm for galaxy morphology using the IllustrisTNG cosmological simulation dataset with the aim of developing a generative model for galaxies. Methods. We first generate a dataset of 2D images and 3D cubes of galaxies from the IllustrisTNG simulation focusing on the mass metallicity and stellar age distribution of each galaxy. PCA is then applied to this data transforming it into a lowerdimensional image space where closeness of data points corresponds to morphological similarity. Results. We find that PCA can effectively capture the key morphological features of galaxies with a significant proportion of the variance in the data being explained by a small number of components. With our method we achieve a dimensionality reduction by a factor of 200 for 2D images and 3650 for 3D cubes at a reconstruction accuracy below 5. Conclusions. Our results illustrate the potential of PCA in compressing large cosmological simulations into an interpretable generative model for galaxies that can easily be used in various downstreaming tasks such as galaxy classification and analysis.

2024-11-01T00:00:00Z

['10.48550/arXiv.2409.10346', '2024A&A...691A.320C', 'arXiv:2409.10346', '10.1051/0004-6361/202451262', '2024arXiv240910346C']

['methods: statistical', 'techniques: image processing', 'astronomical databases: miscellaneous', 'galaxies: evolution', 'galaxies: kinematics and dynamics', 'galaxies: structure', 'Astrophysics - Astrophysics of Galaxies']

MEGS Morphological Evaluation of Galactic Structure Principal component analysis as a galaxy morphology model

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

https://arxiv.org/pdf/2409.10346.pdf

{'Principal Component Analysis as a galaxy morphology model': 'U. Çakır 1 , 2 and T. Buck 1 , 2 \n- 1 Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 205, D-69120 Heidelberg e-mail: ufuk.cakir@stud.uni-heidelberg.de\n- 2 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany \ne-mail: tobias.buck@iwr.uni-heidelberg.de \nReceived Month, XXXX; accepted Month Day, XXXX', 'ABSTRACT': 'Context. Understanding the morphology of galaxies is a critical aspect of astrophysics research, providing insight into the formation, evolution, and physical properties of these vast cosmic structures. Various observational and computational methods have been developed to quantify galaxy morphology, and with the advent of large galaxy simulations, the need for automated and e ff ective classification methods has become increasingly important. \nAims. This paper investigates the use of Principal Component Analysis (PCA) as an interpretable dimensionality reduction algorithm for galaxy morphology using the IllustrisTNG cosmological simulation dataset with the aim of developing a generative model for galaxies. \nMethods. We first generate a dataset of 2D images and 3D cubes of galaxies from the IllustrisTNG simulation, focusing on the mass, metallicity, and stellar age distribution of each galaxy. PCA is then applied to this data, transforming it into a lower-dimensional image space, where closeness of data points corresponds to morphological similarity. \nResults. We find that PCA can e ff ectively capture the key morphological features of galaxies, with a significant proportion of the variance in the data being explained by a small number of components. With our method we achieve a dimensionality reduction by a factor of ∼ 200 for 2D images and ∼ 3650 for 3D cubes at a reconstruction accuracy below five percent. \nConclusions. Our results illustrate the potential of PCA in compressing large cosmological simulations into an interpretable generative model for galaxies that can easily be used in various downstreaming tasks such as galaxy classification and analysis. \nKey words. Galaxies: structure - Galaxies: fundamental parameters - Galaxies: stellar content - Methods: data analysis - Methods: statistical - Techniques: image processing', '1. Introduction': 'Galaxy morphology, the study of the structural characteristics and visual appearance of galaxies, has been a significant research topic in astrophysics for many years. Early classification schemes, such as the Hubble sequence (Hubble 1926), provided a visual framework for classifying galaxies based on their visual features. Historically, galaxy morphology has also been used to identify di ff erent structural components of a galaxy through an analysis of its light profile. This process is known as structural decomposition and was first presented in an analysis by de Vaucouleurs (1958). However, these classification methods often struggled to capture the complexity and diversity of galaxy morphologies and were unable to scale with the increasing size and quality of astronomical datasets. \nFundamentally, galaxy morphology is the phenomenological realization of the underlying fundamental di ff erences in physical properties. The visual appearance of galaxies is heavily influenced by the joint distribution of stellar mass, metallicity, and age due to the nonlinear dependence of stellar luminosity on these fundamental parameters (e.g. Kuiper 1938). This further implies that di ff erent galaxy components trace di ff erent formation episodes of galaxies. In general, galaxy light can be modeled parametrically as either a single component (usually with a \nSérsic 1963 profile), or with two or more components (e.g. Cook et al. 2019). However, in the current era of large-scale cosmological simulations, a vast amount of data are available, allowing us to investigate galaxies and their appearance in exceptional detail. \nHydrodynamical simulations of galaxy formation, such as the large-volume simulations of the IllustrisTNG suite (Pillepich et al. 2017) or the EAGLE suite (McAlpine et al. 2016) or zoomin models such as the AURIGA (Grand et al. 2017), FIRE (Hopkins et al. 2018), VINTERGATAN (Agertz et al. 2021) or NIHAO (Wang et al. 2015; Buck et al. 2020) simulations generate rich, multidimensional datasets that include various physical properties of galaxies. These datasets provide an excellent resource for studying galaxy morphology in a more rigorous and quantitative manner. In combination with modern methods of machine learning, these simulations enable us to build an interpretable generative model for galaxy morphology (e.g. Lanusse et al. 2021). \nFast and accurate models for galaxy morphology are not only used to classify galaxies along the Hubble (1926) sequence but more recently have gained a lot of attention due to the ESA EUCLID mission. Reliable and accurate models for galaxy images are needed to achieve one of the mission\'s science goals to measure the weak lensing signal (e.g. Euclid Collaboration et al. 2022). \nFig. 1. Galaxy images (upper panels) and image cubes (lower panels) in our dataset after the preprocessing steps as explained in Section 2.2. Each row contains one sample galaxy in three stellar maps - metallicity, age and mass from left to right. The images are normalized in the range [0 , 1] and bright pixels correspond to high values. Note that in the stellar age maps, dark pixels correspond to young stars. The three dimensional plots have been created using the plotly python package. \n<!-- image --> \nIn this work, we set out to study principal component analysis (PCA), a relatively simple, yet powerful, and interpretable model for galaxy morphology. PCA has been widely used in various fields for dimensionality reduction and feature extraction (Jolli ff e & Cadima 2016). In Turk & Pentland (1991) PCA was used to decompose face images into the subspace spanned by image eigenvectors called "eigenfaces", such that each face can be represented by a weighted sum of the eigenfaces. The term eigengalaxy first appeared in De La Calleja & Fuentes (2004), where they used PCA for galaxy classification and termed the basis vectors of the lower-dimensional space as eigengalaxies. \nIn the context of galaxy morphology, Uzeirbegovic et al. (2020) recently used PCA to transform a set of SDSS galaxy images into a space where "closeness" corresponds to visual similarity, allowing for a quantitative measure of morphological likeliness. \nHere, we expand upon their findings and explore how we can use PCA to jointly model the distribution of mass, metallicity, and stellar age in two and three dimensions. This approach specifically aims to condense a high-dimensional data distribution from galaxy simulations into a lower-dimensional space that preserves the morphological features of galaxies and allows for interpretable analysis. Our final model is able to jointly model the distribution of stellar mass, metallicity, and stellar age. In combination with a simple stellar population model to convert these parameters into stellar spectra, it is therefore ideally suited as an interpretable physics model to reconstruct the fundamental joint distribution of these properties from broad-band galaxy images. \nThis paper is structured as follows. In Section 2 we detail our methods. Especially, Section 2.2 discusses the methodol- \nogy used for generating the dataset for our analysis by extracting galaxy data from simulations. Section 2.3 outlines the application of PCA to the image data and the resulting lowerdimensional image space (Section 2.3.1). Our resulting model is presented in Section 3 and its accuracy thoroughly tested in Section 3.3. In Section 3.4 we highlight potential applications of our PCA galaxy model and conclude in Section 4 with a summary and outlook of our results. \nFinally, we publicly release all of our code to reproduce the results of this manuscript via GitHub 1 and refer to Appendix C and Fig. B.1 for an overview of our code and file structure. Our final dataset is publicly available on Zenodo. 2', '2. Methods': 'This section outlines the methodology used to generate the dataset and build the morphology model used in this research.', '2.1. Motivation': 'In this paper we set out to explore the possibilities of PCA to condense a high-dimensional data distribution from galaxy simulations into a lower-dimensional space that preserves key morphological features of galaxies and allows to be employed as a flexible, generative model for galaxy morphology informed by high-resolution state-of-the-art simulations. Our final method is able to project the high-dimensional joint distribution of stellar \nFig. 2. Vectorization in the two dimensional case: This reshaping process converts each galaxy image into a point within a high-dimensional vector space. The first 64 2 values correspond to the values of the pixels in the metallicity map, followed by the values of the other maps. This allows for a unique vector-based representation of each galaxy as I ∈ R d , where d is the dimension (equation 1). \n<!-- image --> \nmass, metallicity, and stellar age into a lower dimensional representation whch can easily be used for downstream generative modelling of galaxies. For example, in combination with a simple stellar population models to convert mass, metallicity and stellar age into stellar spectra, it is therefore ideally suited as an interpretable, generative physics model to reconstruct the fundamental joint distribution of these properties from broad-band galaxy images. Hence, our approach here serves the purpose of tackling the inverse problem of reconstructing physical parameters from observed data.', '2.2. From Simulation to Dataset: Galaxy Image Generation': 'We extracted galaxy data from the IllustrisTNG simulation suite (Nelson et al. 2017; Pillepich et al. 2017; Springel et al. 2017), specifically using the TNG100-1 simulation, which models galaxy formation in the cosmological context and follows 1820 3 resolution elements from redshift 127 to 0. From this dataset we choose snapshot number 99 since it corresponds to redshift z = 0 and extract the galaxies for our later analysis as follows: \n- 1. Selection We choose galaxies within a specified mass range of 10 9 . 5 M ⊙ / h < M ⋆ < 10 13 M ⊙ / h , excluding galaxies with SubhaloFlag = 0, since those are not thought to be of cosmological origin. The final selection yields a total number of N = 11 960 galaxies in the dataset.\n- 2. Rotation : To achieve uniform orientation and meaningful comparisons throughout the data set, we calculated the eigenvectors of the moment of inertia tensor to perform a face-on rotation in the x -y plane and obtain a set of rotated coordinates. Using these coordinates, a temporary image is generated, and a principal component analysis with two components is applied. The first principal component represents a vector that maximizes the variance of the projected particle coordinates(see Uzeirbegovic et al. 2020). From this component, we determine the tilt angle and further rotate the galaxy to align its semimajor axis with the x-axis. This alignment procedure ensures consistent orientation among elliptical and barred spiral galaxies, thereby enabling reliable comparisons.\n- 3. Image Rendering : To accurately capture the visual appearance of the galaxies and reflect the physical, spatial resolution of the simulation, we used the smoothinglength parameter to generate two- and three-dimensional images. Specifically, we render images in the fields of particle mass, mass-weighted metallicity , and \nmass-weighted stellar age 3 This rendering process is performed using the render module from the SWIFTsimIO library (Borrow & Borrisov 2020). We choose an image resolution of 64 × 64 and 64 × 64 × 64 pixels for the two- and three-dimensional case, and the image section to be rendered spans five stellar half-mass radii (5 R half) in each direction. The resolution was chosen to correspond to the native resolution of large observational integral field spectroscopic datasets such as e.g. MANGA (75% of MANGA cubes have less than 64x64 spaxels Bundy et al. 2015) or SAMI (50x50 spaxels Allen et al. 2015) for which we have in mind that our model will be most useful. \n- 4. Normalization and Clipping : Normalization involves rescaling the pixel values to [0 , 1], which standardizes the intensity levels across the images, allowing comparisons between di ff erent galaxy images. Clipping is employed to remove noise by setting a threshold for pixel values, i.e. we discard values that deviate significantly from the majority of pixels. We find that clipping values below the 25 th percentile works best. This helps to highlight important features while reducing the influence of irrelevant variations which mainly arise from shot noise from the limited particle count in individual pixels originating from the limited resolution of the underlying simulations. \nThe normalization and clipping steps are crucial for our subsequent analysis. By eliminating noise and standardizing the pixel values, we facilitate the identification and extraction of meaningful information about galaxy properties, such as morphological characteristics, stellar age distribution, and metallicity patterns. This is also needed for better PCA decomposition by removing the nuisance variance originating from random noise. \nWe iterate over all selected galaxies and computed both twoand three-dimensional images for the three specified fields. The resulting images are then saved in the HDF5 file format, encompassing the entire dataset. In Fig. 1 we show three representative examples of galaxies (face-on spiral, elliptical, and triaxial) in both 2D (upper panels) and 3D (lower panels). In each subfigure we show all three maps (metallicity, age, and mass from left to right). In addition to 2D images and 3D cubes, we store the SubhaloID (i.e the unique identifier of the galaxy in the simulations) and stellar mass values M ⋆ for each galaxy in the attributes subgroup. It is important to note that in the case of IllustrisTNG, we exclude particles with nega- \nve GFM\\_StellarFormationTime , because such particles correspond to gas cells in a wind phase and are not considered relevant for the analysis. The final data set used in this work is further described in detail in Çakır & Buck (2023) and can be found publicly available here: https: // github.com / ufuk- cakir / GAMMA', '2.3. Principal Component Analysis on Images': 'Principal Component Analysis (PCA) is a powerful mathematical technique commonly used in data analysis and dimensionality reduction. In simple terms, PCA finds a new set of basis vectors ( principal components ) that maximizes the variance of the projected data in each direction. The principal components are ordered by their contribution to the total variance of the data, so the first principal component is the direction of maximal variance. Then, the dimensionality of the data space can be reduced by keeping only the top n principal components and projecting all data points onto these principal components to find a lower dimensional representation. \nIn the context of image analysis, PCA o ff ers a rather simple yet powerful approach to extracting not only meaningful features from images and understanding the underlying structures in an unsupervised manner, but also to performing this in a fast and interpretable way. \nIn order to perform PCA, we first need to construct a proper data matrix. Each image in our dataset is represented as a two-[ three] dimensional array of size 64 × 64 [ 64 × 64 × 64] with values ranging from 0 to 1. For each galaxy, the images have been calculated on three di ff erent maps (as explained in Section 2.2). These images can be reshaped into a single row vector (see Fig. 2), transforming each galaxy into a point within a high-dimensional vector space, denoted as I ∈ R d , where the dimensionality d is equal to the total number of pixels in the three combined maps. This results in a dimension of \nd 2 D = 3 · 64 2 = 12 288 d 3 D = 3 · 64 3 = 786 432 (1) \nin the two and three dimensional case. Note that the order in which the three di ff erent fields are flattened is not significant as long as it remains consistent for all the galaxies. This consistency is crucial because we focus on examining the pairwise relationship between each pixel in the high-dimensional space. We finally concatenate all flattened galaxy row vectors to form our data matrix \nD = I 1 I 2 . . . I N ∈ R N × d (2) \nwith N = 11 960 (number of galaxies in the dataset) and d = d 2 D , d 3 D .', '2.3.1. Dimensionality Reduction': 'Our goal is to reduce the dimensionality of the original image space while retaining as much information as possible. PCA addresses this by projecting the data onto a new set of orthogonal axes that maximize the variance of the projected data. \nWe perform PCA on our data matrix D ∈ R N × d using the scikit-learn (Pedregosa et al. 2011) python package. The PCA method implements the following steps in a computationally efficient way: \nFig. 3. Cumulative sum of explained variance ratio for up to 400 eigengalaxies (left): We observe that one needs much more eigengalaxies in three dimensions to achieve the same explained variance as in two dimensions. This Figure shows that in order to achieve more than 90% EVR, you need around 60 (215) eigengalaxies in two (three) dimensions. \n<!-- image --> \n- 1. Centering : Subtract the mean galaxy image to obtain the column-centered data matrix D ∗ i j : \nD ∗ i j = D i j -µ j (3) \nwhere µ ∈ R d is the column-wise mean of the data matrix D The mean images used to center the data are shown in Fig. A.1. \n- 2. Calculate SVD : Singular Value Decomposition (SVD) decomposes the centered data matrix into a product of three matrices', 'D ∗ = U Σ V T': 'where U ∈ R N × N , V ∈ R d × d are unitary matrices, whose columns represent the eigenvectors of D ∗ D ∗ T and D ∗ T D ∗ respectively. The rectangular matrix Σ ∈ R N × d contains the singular values of D ∗ , which are the square roots of the eigenvalues of D ∗ T D ∗ \nThe eigenvectors correspond to the principal components of our data and are ordered by their eigenvalues, which represent the amount of variance explained by each component. The eigenvectors λ ∈ R d represent new orthogonal basis vectors that span our lower-dimensional image space. These eigenvectors are called eigengalaxies , since they can be interpreted as images. \nWe project our high-dimensional data matrix D ∗ onto this subspace spanned by the top n eigengalaxies λ n ∈ R n × d to obtain a lower-dimensional representation.', 'S = D ∗ λ T n': '(4) \nwhere S ∈ R N × n contains the coordinates of N galaxy samples in the n -dimensional eigengalaxy subspace, which we also refer to as the PCA scores . This representation maintains most of the variance in the data while also reducing the dimension from d to n . \nIn order to transform the scores back to the original image space, we simply calculate the data matrix ˆ D containing \nFig. 4. Each column shows the lower dimensional representation using n eigengalaxies. To highlight how close we get, the original images are shown at the bottom. One can see that higher order eigengalaxies account for more detailed small scale structures, however only 16 eigengalaxies are enough to fit the overall morphology of the galaxy. \n<!-- image --> \nthe PCA reconstructed galaxy images as the weighted sum of n -eigengalaxies: \nˆ D = S λ n + µ (5)', '3. Results': 'The PCA-based morphology model provides a quantitative analysis of galaxy morphology by examining the positions of galaxies in the lower dimensional space to study morphological similarities and to model the joint distribution of mass, metallicity \nand stellar age. In this section we present our final PCA decomposition of galaxies in 2D and 3D and evaluate its performance via several quantitative metrics.', '3.1. Dimensionality Reduction and Explained Variance': 'An important quantity to evaluate how good the PCA model preserves information is the explained variance, which refers to the amount of variance of the data that is captured by each principal component, i.e., eigengalaxy. The Explained Variance Ratio (EVR) for each eigengalaxy is the proportion of the total variance of the data set that is explained by that component. Since the eigengalaxies are orthogonal, each additional eigengalaxiy accounts for variance not explained by the eigengalaxies before. \nWe are free to choose the total number n of eigengalaxies we want to keep for the lower-dimensional image space representation (Section 2.3.1). We calculate the cumulative sum of the EVR of each eigengalaxy in Fig. 3 for up to 400 components and find that, e.g. 60 (215) eigengalaxies account for about 90% of the total variance in 2D (3D), which means that most of the information from the dataset can be retained even after significant dimensionality reduction (e.g. a reduction by a factor of 205 (3641)). \nPCA gets better with increasing number of components, however one needs to decide for the trade-o ff between reconstruction accuracy and dimensionality reduction. This is showcased in Fig. 4 which contains the lower-dimensional representation of a sample galaxy (eq. 5) using di ff erent numbers of eigengalaxies. From top to bottom, we show the reconstructed galaxy image using 16 to 512 eigengalaxies while the bottom row shows the original image for comparison. Already with 16 eigengalaxies the main features of the galaxy are well reconstructed, especially the spiral-like structure of stellar age. In general this figure clearly shows that with increasing number of eigengalaxies used for the reconstruction the image accuracy or the finer image details become more and more well approximated. We note that between 256 and 512 eigengalaxies we mostly observe that the noise level in the image is better approximated. \nThe choice of how many eigengalaxies to keep is heavily dependent on the use case, however for further analysis in this paper we use 60 (215) eigengalaxies in two (three) dimensions, since those account for 90% of explained variance as we have shown in Fig. 3. The resulting 60 eigengalaxies in the 2D case are shown in Fig. 5 and the corresponding first 60 eigengalaxies of the 3D case are shown in the appendix in Fig. C.1. We observe that the structures get much more complex as we go up to higher orders up to the scale where we observe pure rotational noise, especially visible in the case of stellar age eigenmaps. This is explained by the fact that the eigengalaxies are ordered by explained variance, and the lower orders account for more global structures. By rotational noise, we mainly refer to the fact, that galaxies show spiral arm structure and other patterns with rotational symmetry. Since PCA is a linear dimensionality reduction, it is not able to faithfully capture these features. Hence higher order eigengalaxies are needed to account for these details. This also points towards a limitation of our model and warrants to explore geometric and equivariant models in future work. \nInterestingly, the physical correlation between mass, metallicity, and age is incorporated by the fact that the stellar age eigengalaxies are almost inverted to those of metallicity and mass, since young stars have on average higher metallicity and are located in overdense regions, i.e. the spiral arms. Note that you even observe eigengalaxies with explicit spiral arms (e.g eigengalaxy 25 in Fig. 5). Thus we conclude that decomposing \nFig. 5. First 60 Eigengalaxies from PCA in 2D: Red corresponds to high positive values, and gray corresponds to low negative values. We show the three di ff erent eigengalaxies for metallicity, stellar age, and mass from top to bottom. \n<!-- image --> \n(c) Masses \nArticle number, page 6 of 13 \nthe physical appearance of galaxies into eigengalaxies indeed results in an interpretable lower-dimensional representation.', '3.2. Reconstruction Accuracy': 'With our final set of 60 eigengalaxies for the 2D case, we calculate the lower dimensional representation of all galaxies in our dataset. In the following we denote by Ioriginal the vector containing the pixel values of the original images and by Irec the vector containing the pixel values of the reconstructed image. In our case I ∈ R d is a d dimensional vector of the image space, where d is given by eq. 1 for the 2D (3D) case. With this, we define the reconstruction error (RE) as the di ff erence in pixel values to measure the discrepancy between the PCA representation vector ˆ I and the original vector I \nRE = P d k = 1 ( Ik -ˆ Ik ) 2 P d k = 1 Ik (6) \nwhere ˆ Ik and Ik represent the k -th elements (pixel) of the reconstructed and original vectors (images), respectively. We calculate the reconstruction error as the di ff erence in the pixel values between the original and PCA representations separately in the three di ff erent maps. An example reconstruction for both the 2D (left panel) and 3D (right panel) case together with the residual image defined as Ioriginal -Irec is shown in Fig. 6. Qualitatively, the reconstruction in all three maps is quite well approximating the original image. We find that especially the stellar age map is approximated very well using only 60 components. \nMore quantitatively, we calculate the reconstruction error as defined via eq. 6 for all galaxies in our sample and show the distribution of reconstruction errors in the top panel of Fig. 7 for both the 2D case (blue histograms) and the 3D case (red histograms). We find that 90% of all images have a reconstruction error less than 0 . 022 in 2D (60 eigengalaxies) and less than 0 . 027 in 3D (215 eigengalaxies), highlighting that our PCA model results in accurately reconstructed images. \nSimilarly, the bottom panel of Fig. 7 shows how the reconstruction error scales with the number of eigengalaxies used. For the 2D case we find that even with as few as ∼ 15 eigengalaxies 90% of all images have a reconstruction error below 0 . 05. In the 3D case however, we need ∼ 60 eigengalaxies to achieve the same reconstruction error below 0 . 05 which is not surprising as the dimensionality of the problem is much larger. This nicely mirrors the results from the explained variance and shows that indeed only a small number of eigengalaxies are needed to accurately describe galaxy morphology.', '3.3. Interpretability of the Model': 'One key advantage of PCA over more advanced machine learning methods such as neural networks is its relatively straightforward interpretability. Since PCA is a linear decomposition of the original image space, each individual component can be interpreted as an image of a very particular morphology. To visualize how eigengalaxies account for di ff erent morphology, we plot the 10 eigengalaxies with the highest absolute score values used for reconstruction in Fig. 8. We show here two example cases of the face-on spiral already shown in Fig. 4 (upper panels in Fig. 8) and the triaxial galaxy from the bottom panels in the 2D example from Fig. 1 (lower panels in Fig. 8). \nThe two columns on the left of Fig. 8 show the original and reconstructed images, while the next ten panels show the contributing eigengalaxies in order of decreasing absolute score. \nThe first thing to note is that vastly di ff erent eigengalaxies contribute to the reconstruction between a spiral and a triaxial galaxy (only 3 out of 10 eigengalaxies are common, though with very di ff erent weights, c.f. 6th vs. 3rd eigengalaxy in the spiral / triaxial case, which have opposite signs), as we would expect from their fundamentally di ff erent morphology. For the spiral galaxy, we find eigengalaxies with a lot of power on the diagonal edges of the image, which is needed to reconstruct the spiral arm feature. For the triaxial galaxy, on the other hand, we find that the eigengalaxies with nearly point symmetry or barlike features contribute most to the reconstruction. \nOur analysis of Fig. 8, although quite qualitative in its nature, clearly shows the potential to describe galaxy morphology through a PCA decomposition. Fitting galaxies with our PCAmodel will allow one to easily study general morphological trends among di ff erent galaxies via the PCA scores of the contributing eigengalaxies (which are all the same for each galaxy by construction). For example, it is easy to select and quantify how bar-like, spiral-like, or centrally concentrated the mass, age, or metallicity distribution of a galaxy is. This further leads to one main application of describing galaxy morphology via PCA, its ability to do galaxy classification and perform a similarity search, which we describe in more depth in the next subsection.', '3.4. Application': 'In this subsection, we highlight two potential applications of our model: i) a similarity search among galaxies to find morphologically similar galaxies ii) outlier detection via a UMAP projection of the PCA components.', '3.4.1. Morphological Similarity Search': 'In order to test whether the lower-dimensional image space encodes morphological information, we can select a sample galaxy and search for its nearest neighbors in the PCA eigenspace using the Euclidean distance: \nd ( ˆ s , si ) = ∥ ˆ s -si ∥ 2 (7) \nwhere ˆ s , si ∈ S (eq. 4) are the scores of the sample and the remaining galaxies. In Fig. 9 we illustrate the five nearest neighbors for the two di ff erent sample galaxies already discussed in the previous Fig. 8. Indeed, this figure shows clearly that there exists a relationship between Euclidean proximity in the PCA eigenspace and the shared morphological characteristics of galaxies. The five nearest neighbors for the spiral galaxy (left panels of Fig. 9) all generally show a pronounced two-arm spiral galaxy where the strongest morphological similarity can be seen in the age maps (middle row). The right-hand side panels of Fig. 9 show that indeed all neighbors of the triaxial galaxy in the PCA eigenspace are also triaxial and share the same strong bar-like feature in mass and metallicity maps. \nSimilarly, the PCA scores can further be used to perform clustering analysis (e.g. using gaussian mixture models or kmeans) to define galaxy types in an unsupervised fashion or to perform an outlier detection to potentially find some interesting galaxies with morphological particularities.', '3.4.2. Outlier Detection': 'To visualize the space in two dimensions, we apply UMAP (Uniform Manifold Approximation and Projection), a dimensionality \nFig. 6. PCA reconstruction in two and three dimensions: The left most column in each panel shows a sample galaxy in the three maps (metallicity, stellar age, and mass from top to bottom) followed by the low-dimensional PCA representation with 60 (215) eigengalaxies in two (three) dimensions. The rightmost column shows the residual image ( Ioriginal -Irec ). You can clearly see that the complex spiral structure in the Stellar Age map is approximated very well in the two-dimensional case while at the same time at n = 60 we smooth the noise in the image. Compare also with Fig. 4 for what happens in the larger n. \n<!-- image --> \nreduction and clustering algorithm developed by McInnes et al. (2020), on the lower-dimensional PCA scores. \nFigure 10 shows the UMAP embedding of the PCA scores of the two-dimensional galaxy images. UMAP separates a cluster of galaxies in the dataset that deviate significantly from the rest, as shown on the left in Figure 10. These galaxies happen to have no stars in the outer regions and therefore are reconstructed using a significantly di ff erent linear combination of eigengalaxies. \nBy employing a threshold on the UMAP coordinates we can filter out around 120 galaxies. We rerun the analysis and fit UMAP on the PCA scores calculated on the filtered data. The resulting plot is shown on the right side of Figure 10. The lower left quadrant of Figure 10 mainly features spherical galaxies with little structure. In the top right section, galaxies with a prominent bar structure are aligned in a strip, whereas in the bottom right, galaxies characterized by a dominant diagonal feature form a separate cluster. Interestingly, the upper middle region of Figure 10 is dominated by galaxies of higher mass. An interactive version of this graph can be accessed using an online dashboard 4 , where one can hover over each distinct point to see the corresponding galaxy image.', '4. Summary & Conclusions': 'In this study, we have explored the application of Principal Component Analysis (PCA) to analyze galaxy images in both twodimensional and three-dimensional cases, jointly modeling the mass, metallicity and age distributions. Our analysis has led to several findings and insights into the potential of using PCA \nto characterize galaxy morphology and perform morphological analysis. \n- 1. We have demonstrated that PCA can be e ff ectively used to reconstruct galaxy images using a relatively small number of eigengalaxies (Fig. 6. The reconstruction accuracy was quantified using the reconstruction error (eq. 6), which measures the di ff erence in pixel values between the original and PCA-reconstructed images. Our results (Fig. 7) indicated that even with a modest number of 60 (215) eigengalaxies in 2D ( 3D ), the PCA model achieved accurate reconstructions, with 90% of the images having reconstruction errors below 0 . 022 ( 0 . 027 ). This points to the potential of PCA for e ffi ciently representing complex galaxy morphology.\n- 2. We discussed the interpretability of the PCA model. Unlike more complex machine learning methods, PCA provides a straightforward interpretation of its components. Each eigengalaxy can be considered an image representing a specific morphological feature. Visualizing the top contributing eigengalaxies for di ff erent types of galaxies revealed their distinctive contributions in Fig. 8 and showcased the potential of PCA for classifying galaxies based on their morphology.\n- 3. In terms of applications, we demonstrate the utility of PCA for performing morphological similarity searches in Fig. 9. By measuring Euclidean distances in the PCA eigenspace, we identified morphologically similar galaxies to a given sample galaxy. This suggests that PCA captures meaningful features of galaxy morphology, allowing for e ffi cient similarity analysis and clustering.\n- 4. We showcase outlier detection using UMAP as an application of our PCA model. By inspecting a UMAP projection \nFig. 7. Reconstruction Error between original images and their PCA projection. Top panel: Reconstruction error for fixed dimensionality reduction onto the 60 (215) dimensional space in two ( three ) dimensions. The dashed line represents the 90% quantile. We observe that 90% of all images have a reconstruction error less than 0 . 022 ( 0 . 027 ). Bottom panel: 90th percentile of the reconstruction error as a function of the number of eigengalaxies used for reconstruction. The reconstruction error is a strong function of eigengalaxies, and already 15 (60) eigengalaxies lead to a reconstruction error better than 5% in 2D (3D). \n<!-- image --> \nof the PCA scores, we are able to filter out particular galaxies that do have no stars in the outskirts. \nIn conclusion, this study highlights the potential of PCA as a powerful tool for analyzing and characterizing galaxy morphology and constructing a flexible, generative model for galaxy morphologies. Its ability to e ffi ciently represent images, interpret its components, and facilitate morphological similarity searches makes it a valuable approach in the field of astrophysics. Our proposed PCA representation can be used to tackle the inverse problem of reconstructing physical parameters from observed data and in the case of 3D eigengalaxies it could even be used to tackle the inverse problem of deprojecting observed galaxies and reconstructing their physics parameters in 3D at the same time. Future work could involve refining the PCA model, e.g. using non-linear PCA, exploring its applications in other astronomical datasets, and developing hybrid approaches that combine PCA with other machine learning techniques for even more comprehensive analyses of galaxy images. \nAcknowledgements. This project was made possible by funding from the Carl Zeiss Stiftung.', 'References': 'Agertz, O., Renaud, F., Feltzing, S., et al. 2021, Monthly Notices of the Royal Astronomical Society, 503, 5826 \n- Allen, J. T., Croom, S. M., Konstantopoulos, I. S., et al. 2015, MNRAS, 446, 1567\n- Borrow, J. & Borrisov, A. 2020, Journal of Open Source Software, 5, 2430 \nBuck, T., Obreja, A., Macciò, A. V., et al. 2020, MNRAS, 491, 3461 Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7 \n- Çakır, U. & Buck, T. 2023, arXiv e-prints, arXiv:2312.06016\n- Cook, R. H. W., Cortese, L., Catinella, B., & Robotham, A. 2019, MNRAS, 490, 4060\n- De La Calleja, J. & Fuentes, O. 2004, Monthly Notices of the Royal Astronomical Society, 349, 87\n- de Vaucouleurs, G. 1958, ApJ, 128, 465\n- Euclid Collaboration, Bretonnière, H., Huertas-Company, M., et al. 2022, A&A, 657, A90\n- Grand, R. J. J., Gómez, F. A., Marinacci, F., et al. 2017, Monthly Notices of the Royal Astronomical Society, 467, 179\n- Hopkins, P. F., Wetzel, A., Kereš, D., et al. 2018, Monthly Notices of the Royal Astronomical Society, 480, 800 \nHubble, E. P. 1926, ApJ, 64, 321 \n- Jolli ff e, I. T. & Cadima, J. 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374, 20150202 Kuiper, G. P. 1938, ApJ, 88, 472\n- Lanusse, F., Mandelbaum, R., Ravanbakhsh, S., et al. 2021, MNRAS, 504, 5543 McAlpine, S., Helly, J., Schaller, M., et al. 2016, Astronomy and Computing, 15, 72\n- McInnes, L., Healy, J., & Melville, J. 2020, UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction\n- Nelson, D., Pillepich, A., Springel, V., et al. 2017, Monthly Notices of the Royal Astronomical Society, 475, 624\n- Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825\n- Pillepich, A., Nelson, D., Hernquist, L., et al. 2017, Monthly Notices of the Royal Astronomical Society, 475, 648\n- Sérsic, J. L. 1963, Boletin de la Asociacion Argentina de Astronomia La Plata Argentina, 6, 41\n- Springel, V., Pakmor, R., Pillepich, A., et al. 2017, Monthly Notices of the Royal Astronomical Society, 475, 676\n- Turk, M. & Pentland, A. 1991, Journal of Cognitive Neuroscience, 3, 71 Uzeirbegovic, E., Geach, J. E., & Kaviraj, S. 2020, Monthly Notices of the Royal Astronomical Society, 498, 4021 \nWang, L., Dutton, A. A., Stinson, G. S., et al. 2015, MNRAS, 454, 83 \nFig. 8. Decomposition of two PCA reconstructed samples: The PCA reconstruction (left) is calculated as the weighted some over 60 eigengalaxies, however here we display the top 10 eigengalaxies (right), chosen based on their absolute weight values (top). On the bottom we show the corresponding index of the eigengalaxy.. This showcases the di ff erent combination of eigengalaxies used to reconstruct galaxies with di ff erent morphological features, e.g., spiral or bar shaped \n<!-- image --> \nFig. 9. Nearest Neighbours in eigenspace: given a sample galaxy (left) we find the five nearest neighbors in euclidean distance in the lower dimensional space (right) for each a spiral (a) and bar shaped (b) representative galaxy sample. \n<!-- image --> \n0 \n1 \n<!-- image --> \nFig. 10. Outlier detection: UMAP projection with n\\_components = 2, n\\_neighbors = 5, min\\_dist = 0.001 calculated (a) on the PCA components of the 2D images to find outliers (b) after filtering outlier galaxy images by choosing galaxies with UMAP1 < 2. We show some of the outliers at the top left of (a). The color map corresponds to the masses of the galaxies in log 10 ( M / M ⊙ ). An interactive version of the plot on the right can be accessed using an online dashboard referenced in the GitHub repository of this work (see Appendix C) \n<!-- image --> \nFig. A.1. Columnwise mean of the Datamatrix as used in equation 3 \n<!-- image --> \nFig. B.1. HDF5File Structure, where each blue node represents a HDF5 group and red nodes are datasets. The data file contains general galaxy parameters under the / Galaxies / Attributes group such as the SubhaloID in the simulation or the total mass of stars particles M ⋆ . The images of particles in the di ff erent fields (i.e metallicity , mass , stellar age ) are calculated in two and three dimensions and saved under the respective subgroup. \n<!-- image -->', 'Appendix A: Column-wise Mean of the Data Matrix': 'The column-wise mean of the data matrix is computed within the PCA implementation of the sklearn library. This mean is visualized in Figure A.1.', 'Appendix B: Data Structure': 'The dataset is stored as a HDF5 file, with the data structure shown in Figure B.1. The dataset is structured into hierarchical groups and datasets. At the top level, general galaxy parameters such as the SubhaloID from the simulation and the total stellar mass ( M ⋆ ) are stored within the / Galaxies / Attributes group. These parameters provide essential metadata for each galaxy in the simulation. Additionally, the file contains images of particle properties across di ff erent fields, including metallicity , mass , and stellar age . These images are calculated in both two and three dimensions and are saved under their respective subgroups. This structure allows for e ffi cient access and storage of large-scale simulation data, facilitating analysis across multiple dimensions and properties.', 'Appendix C: Code and Data Availability': "The source code for the morphology model developed in this study, named MEGS , is made available under an open source license and is publicly hosted on GitHub. \nTo facilitate a wider community's usage and contributions, we have ensured that the repository is well-documented. The repository includes comprehensive documentation hosted on ReadTheDocs that provides an overview of the project, installation instructions, and a guide on how to use the software. \nThe MEGS repository, which contains code, additional documentation, and interactive dashboards, can be found at the following URL: https://github.com/ufuk-cakir/MEGS \nThe cleaned data set with outliers removed is publicly available on Zenodo under the URL: https://zenodo.org/ record/8375344 \nU. Çakır and T. Buck: MEGS: Morphological Evaluation of Galactic Structure \n<!-- image --> \n(c) Masses \nFig. C.1. First 60 Eigengalaxies from PCA in 3D: For better visualization we show the images in the x-z direction"}

2024arXiv240909924L

In a binary merger of two subclusters with comparable masses a pair of merger shocks are typically generated often manifesting as double radio relics. Using cosmological hydrodynamic simulations we identify major merger events with mass ratio mathcalM1mathcalM2lesssim4 and impact parameter brrm vir1lesssim1 where rrm vir1 is the virial radius of the larger subcluster. We analyze merger shock surfaces approximately 1 Gyr after the pericenter passage focusing on their morphology and the distribution of the Mach number Ms of their constituent shock zones. The shock surfaces exhibit an elongated shape with a minortomajor axis ratio of sim0.60.9 and cover the area of sim520 of the enclosed sphere. The area ratio of the two shock surfaces roughly scales with mathcalM1mathcalM2 typically positioning the larger shock ahead of the smaller subcluster. The axis connecting the two subclusters generally does not pass through the centers of the shock surfaces due to the nonzero impact parameter and the turbulent flows around them. The distribution of Ms of shock zones on each surface can be approximated by a lognormal function peaking at Msrmpeakapprox24.5 and extending up to sim10. The surfaceareaweighted and Xrayemissivityweighted average Mach numbers are comparable with langleMsranglermareaapprox2.34.4 and langleMsrangleXapprox24. In contrast the cosmicrayenergyfluxweighted average Mach numbers are higher with langleMsranglermCRapprox35. This discrepancy aligns with the differences between Mach numbers derived from Xray and radio observations of radio relic shocks. On the other hand we find that mostly langleMsrangleXgtrsim2 for simulated merger shocks although shocks with Mrm Xraylesssim2 are often reported in observations.

2024-09-01T00:00:00Z

['2024arXiv240909924L', 'arXiv:2409.09924', '10.48550/arXiv.2409.09924']

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

Morphology and Mach Number Distribution of Merger Shock Surfaces in Merging Galaxy Clusters

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09924.pdf

{'Morphology and Mach Number Distribution of Merger Shock Surfaces in Merging Galaxy Clusters': '<!-- image --> \n1 Department of Physics, College of Natural Sciences, UNIST, Ulsan 44919, Korea \n2 Department of Earth Sciences, Pusan National University, Busan 46241, Korea \n(Received; Revised; Accepted) \nSubmitted to The Astrophysical Journal', 'ABSTRACT': 'In a binary merger of two subclusters with comparable masses, a pair of merger shocks are typically generated, often manifesting as double radio relics. Using cosmological hydrodynamic simulations, we identify major merger events with mass ratio 1 < M 1 / M 2 ≲ 4 and impact parameter b/r vir , 1 ≲ 1, where r vir , 1 is the virial radius of the heavier subcluster. We analyze merger shock surfaces approximately 1 Gyr after the pericenter passage, focusing on their morphology and the distribution of the Mach number, M s , of their constituent shock zones. The shock surfaces exhibit an elongated shape with a minor-to-major axis ratio of ∼ 0 . 6 -0 . 9 and cover the area of ∼ 5 -20% of the enclosed sphere. The area ratio of the two shock surfaces roughly scales as A ss , 2 /A ss , 1 ∝ M 1 / M 2 , typically positioning the larger shock ahead of the lighter subcluster. The axis connecting the two subclusters generally does not pass through the centers of the shock surfaces, due to the nonzero impact parameter and ambient turbulent flows. The distribution of M s of shock zones on each surface can be approximated by a lognormal function, peaking at M s, peak ≈ 2 -4 . 5 and extending up to ∼ 10. The surface-area-weighted and X-ray-emissivity-weighted average Mach numbers are comparable, with ⟨ M s ⟩ area ≈ 2 . 3 -4 . 4 and ⟨ M s ⟩ X ≈ 2 -4. In contrast, the cosmic-ray-energy-flux-weighted average Mach numbers are higher with ⟨ M s ⟩ CR ≈ 3 -5. This discrepancy aligns with the differences between Mach numbers derived from X-ray and radio observations of radio relic shocks. On the other hand, we find that mostly ⟨ M s ⟩ X ≳ 2 for simulated merger shocks, although shocks with M X -ray ≲ 2 are often reported in observations. \nKeywords: galaxies: clusters: general - methods: numerical - shock waves', '1. INTRODUCTION': "During the hierarchical formation of the large-scale structure (LSS) of the universe, galaxy clusters grow through the successive mergers of small subclusters and groups of galaxies. Such mergers are known to induce shock waves and turbulent flow motions in the intracluster medium (ICM). Energy dissipation via shocks and turbulence plays key roles not only in heating the ICM but also in amplifying magnetic fields and acceler- \nCorresponding author: Dongsu Ryu \ndsryu@unist.ac.kr \nCorresponding author: Hyesung Kang \nhskang@pusan.ac.kr \nating cosmic rays (CRs) (e.g., Sarazin 2002; Ryu et al. 2003, 2008; Brunetti & Jones 2014; Porter et al. 2015; Roh et al. 2019; Wittor et al. 2021). \nIn particular, binary mergers between two subclumps of comparable masses with small impact parameters are classified as major mergers and induce a pair of so-called 'merger shocks' after the pericenter passage of dark matter (DM) cores (e.g., Markevitch & Vikhlinin 2007; Ha et al. 2018). The overall picture of merger shocks has been extensively studied through numerical simulations. In simulations of idealized binary mergers with subclusters in hydrostatic equilibrium, it was shown that merger shocks exhibit different properties depending on the merger parameters, such as the mass ratio, impact parameter, viewing angle, etc (e.g., Gabici & Blasi 2003; \nSpringel & Farrar 2007; van Weeren et al. 2011a; ZuHone 2011; Molnar & Broadhurst 2017). \nThe properties of merger shocks were also described in the context of the LSS formation (e.g., Paul et al. 2011; Schmidt et al. 2017; Ha et al. 2018; glyph[suppress]Lokas 2023; Lee et al. 2024; Finner et al. 2024). For instance, analyzing merging clusters found in cosmological hydrodynamic simulations, Ha et al. (2018) described quantities, such as the time variations of the shock speed, v s , the shock Mach number, M s , and the energy dissipation at shock interfaces, in the inherently turbulent ICM 1 . While a variety of mergers with diverse merger parameters are present in these simulations, Ha et al. (2018) concentrated on the cases of nearly head-on collision with the mass ratio of M 1 / M 2 ≈ 2, replicating the characteristics of the well-studied Sausage relic in the cluster CIZA J2242.8+5301 (e.g., van Weeren et al. 2010; Hoang et al. 2017). As illustrated in Figure 1 of Ha et al. (2018), as two subclusters approach and gas clumps are compressed, 'equatorial shocks' first appear in the equatorial plane, propagating toward the direction perpendicular to the merger axis. After the DM core passage, two 'axial shocks' launch along the merger axis in front of subclusters (see also Figure 1 of van Weeren et al. 2011a). \nMerger shocks often manifest as double radio relics in the outskirts of merging clusters (e.g., van Weeren et al. 2011a; Hoang et al. 2018; Golovich et al. 2019). With observations of around a hundred cases to date, it is widely accepted that radio relics exhibit diffuse synchrotron radiation emitted by CR electrons accelerated at merger shocks (see Feretti et al. 2012; Bruggen et al. 2012; Brunetti & Jones 2014; van Weeren et al. 2019, for reviews). Although it has been customary to associate a merger shock with a single Mach number, considering the curved, or often irregular, morphology of radio relics and the turbulent nature of the ICM, the correct picture should be that a merger shock surface is composed of many 'shock zones' characterized by different M s (e.g., Ha et al. 2018; Roh et al. 2019; Botteon et al. 2020; Wittor et al. 2021). Thus, the Mach number estimated from radio or X-ray observations should represent mean values, ⟨ M s ⟩ radio and ⟨ M s ⟩ X -ray , for an ensemble of shock zones associated with each merger shock surface (e.g., Hong et al. 2015; Rajpurohit et al. 2020; DominguezFernandez et al. 2021). \nIn radio observations, radio relics frequently appear as arc-shaped patches at a distance of d s ∼ 1 -2 Mpc \nfrom the cluster center, usually around the virial radius, r vir . The Mach numbers of radio relic shocks are estimated using either the 'injection spectral index', α inj (or α sh ), of the radio spectrum immediately behind the shock in the outer edge of observed relics, i.e., M radio ≈ [(3 + 2 α inj ) / (2 α inj -1)] 1 / 2 , or the 'integrated spectral index', α int , of the volume-integrated spectrum, i.e., M radio ≈ [( α int + 1) / ( α int -1)] 1 / 2 (e.g., Kang 2011). Radio observations indicate typically M radio ≈ 2 -5 (van Weeren et al. 2019). On the other hand, in X-ray observations, merger shocks are detected as discontinuities in the surface brightness or temperature profiles that can be used to estimate the Mach number, M X -ray (e.g., Markevitch et al. 2002; Ogrean et al. 2013; Dasadia et al. 2016; Sanders et al. 2022). These shocks are typically found at d s ≳ 1 Mpc and mostly weak with M X -ray ∼ 1 . 2 -4 (e.g., Eckert et al. 2016; Urdampilleta et al. 2018). \nFor many observed radio relics, the Mach numbers derived from radio and X-ray observations do not coincide, exhibiting the so-called Mach number discrepancy (e.g., Wittor et al. 2021). In most cases, Mach numbers inferred from X-ray observations are lower than those inferred from radio observations, i.e., M X -ray ≲ M radio (e.g., Akamatsu & Kawahara 2013; van Weeren et al. 2016). For instance, in the so-called Toothbrush relic in 1RXS J0603.3, M radio ≈ 3 . 3 with α sh ≈ 0 . 7 and M radio ≈ 3 . 78 with α int ≈ 1 . 15 (Rajpurohit et al. 2018), whereas M X -ray ≲ 2 (Itahana et al. 2015; van Weeren et al. 2016). However, in some relics, the opposite is found; e.g., for a radio relic in Abell 521, M radio ≈ 2 . 27 and M X -ray ≈ 3 . 4 are reported (Giacintucci et al. 2006). \nAs noted above, a merger shock is a surface comprising many shock zones with varying Mach numbers. Previous studies have suggested that radio observations tend to pick up regions with high CR acceleration and, consequently, higher M s , while X-ray observations preferentially reveal regions with higher densities and thus higher X-ray emissivity, which are typically associated with lower M s (e.g., Hong et al. 2015; Ha et al. 2018; Roh et al. 2019; Botteon et al. 2020). This proposition could explain why M radio ≳ M X -ray . However, complex flows surrounding merger shocks in the turbulent ICM could lead to the opposite case. These highlight the needs to understand the detailed properties of merger shocks and the dynamics of surrounding flows and to clarify the nature of observed radio relics. \nIn this work, we study the morphological characteristics of merger shock surfaces and the Mach number distribution of shock zones in major binary merging clusters with different mass ratios and impact parameters using cosmological hydrodynamic simulation data. To \nachieve this, we perform a set of structure formation simulations, identify synthetic merging clusters formed through major binary mergers, and track the time evolution of merger events such as the pericenter passage and the development of merger shocks. Using specific criteria, we try to isolate the surfaces of merger shocks from shocks generated by turbulence in the background ICM and the infall of the warm-hot intergalactic medium (WHIM) along filaments. We then analyze the shape, size, and orientation of the shock surfaces, and examine the probability distribution function (PDF) of M s for shock zones on these surfaces. Additionally, we estimate the average Mach numbers weighted by shock surface area, X-ray emissivity, and CR energy flux, and discuss the implications for observations of radio relics. Figure 1. \nThe paper is organized as follows: In Section 2, we outline the numerical setup, including the generation of synthetic merging clusters and the method for identifying merger shock surfaces. Section 3 presents the results, discussing the morphological characteristics of merger shock surfaces and the Mach number distribution of shock zones. In Section 4, we compare the average Mach numbers of merger shocks with those observed in radio relic shocks. Finally, a brief summary is provided in Section 5.", '2.1. Simulation Setup': 'We generate samples of merging clusters and merger shocks from simulation data for the LSS formation of the universe. Simulations are performed using the particlemesh/Eulerian cosmological hydrodynamic code described in Ryu et al. (1993). We adopt the standard ΛCDM cosmology model with the following parameters: baryon density Ω b = 0.046, dark matter density Ω DM = 0.234, cosmological constant Ω Λ = 0.72, Hubble parameter h ≡ H 0 /(100 km s -1 Mpc -1 ) = 0.7, primordial spectral index n = 0.96, and rms density fluctuation σ 8 = 0.82, complying with WMAP 9 data (Hinshaw et al. 2013). A cubic box of the comoving size of 100 h -1 Mpc with periodic boundaries is employed and divided into 2048 3 uniform grid zones. This results in a uniform spatial grid resolution of ∆ l = 48.8 h -1 kpc. \nIn our simulations, non-gravitational processes, such as radiative cooling and galaxy formation feedback, are not included, as their effects on the dynamical evolution of the ICM on megaparsec scales are expected to be insignificant. Magnetic fields are also not included; given the expected magnetic field strength of ≲ 1 µ G at cluster outskirts (e.g., Ryu et al. 2008; Vazza et al. 2014), the magnetic energy would be an order of magnitude smaller than the kinetic energy, and hence magnetic fields are \n𝑑𝑑 ≈ 𝑟𝑟 \n(a) pre-merger approach \nFigure 1 illustrates the schematic picture of a binary major merger, showing our definitions of the impact pa- \n<!-- image --> \n(b) after DM pericenter passage \nFigure 1. Schematic picture of a binary merger with a nonzero impact parameter b during (a) the pre-merger approach and (b) far after the pericenter passage. Here, r 200 is the radius within which the mean total density of the cluster attains 200 times the critical density, ρ c , of the universe. The subscripts 1 and 2 denote heavier and lighter subclusters, respectively. A pair of merger shocks develop after the pericenter passage, and the one in front of subcluster 1 is labeled as relic shock 1 (blue), while the other in front of subcluster 2 is labeled as relic shock 2 (magenta). \nunlikely to significantly affect the properties of merger shocks, such as their morphology and the Mach number distribution at the shock surface. On the other hand, on microscopic scales, the behavior of magnetic fields, such as amplification and dissipation through kinetic instabilities and wave-particle interactions, should play a crucial role in particle acceleration and subsequent synchrotron emission behind the shock (e.g., Kang et al. 2019; Ha et al. 2021, 2022), but such processes are beyond the scope of this study.', '2.2. Merging Cluster Sample': "From seven different sets of simulations, we identify clusters with an X-ray-weighted temperature T X ≳ 2 keV at redshift z = 0. Here, T X is calculated by weighting the gas temperature, T , with the bremsstrahlung emission, ε ff ∝ T 1 / 2 ρ 2 gas , across the cluster volume (see Kang et al. (1994) and Hong et al. (2015) for details), where ρ gas is the gas density. Among them, we find more than 20 clusters that have gone through major binary mergers. We construct a sample of 12 merging clusters satisfying the following three criteria: (1) mass ratio 1 < M 1 / M 2 ≲ 4; (2) impact parameter b ≲ r vir , 1 ; (3) T X ≳ 2 KeV at the redshift of z relic ≲ 0 . 5. Hereafter, we denote heavier and lighter subclusters with the subscripts 1 and 2, respectively; r vir , 1 is the virial radius of the heavier subcluster, and z relic is the redshift at ∼ 1 Gyr after the pericenter passage, the optimal time for merger shock surfaces to be observed as radio relics (see Ha et al. 2018). \n+ \n𝑟𝑟 \nTable 1. Merging Galaxy Cluster Sample \nrameter and two relic shocks. In panel (a), which depicts a pre-merger phase, two subclusters are about to touch with a distance of d ≈ r 200 , 1 + r 200 , 2 . Here, r 200 is the radius within which the mean total (gas plus DM) density of the cluster attains 200 times the critical density, ρ c , of the universe. The impact parameter, b , is defined as the distance perpendicular to the paths of the subcluster centers. Typically, this corresponds to ∼ 1 . 5 -2 Gyr before the pericenter passage. Panel (b) depicts a stage following the pericenter passage, showcasing two merger shocks, relic shock 1 in front of subcluster 1 and relic shock 2 in front of subcluster 2. We use the term 'relic shock' because, as noted in the introduction, merger shocks are often observed as radio relics. Given that most binary radio relics in major mergers are observed in the post-pericenter passage phase (Golovich et al. 2019), our analysis focuses on the stage shown in panel (b). \nTable 1 lists the 12 merging clusters in our sample. Here, r vir ≈ 1 . 36 r 200 (e.g., Roncarelli et al. 2006; Reiprich et al. 2013), and M 1 and M 2 are the total (gas plus DM) mass contained within the subcluster's virial radius at the epoch of d ≈ r 200 , 1 + r 200 , 2 . The masses and temperatures of the merging clusters in this \n<!-- image --> \nFigure 2. 3D view of shock zones in the comoving volume of (4 . 88 Mpc) 3 that encompasses Cluster 3 (see Table 1). Color displays the Mach number of shock zones. (a) All shock zones within the volume. (b) Shock zones satisfying the density/temperature and distance criteria (see the main text). Here, only the shock zones within ⟨ d s ⟩ ± 1 σ d s are included, where ⟨ d s ⟩ is the mean distance of shock zones from the X-ray center of the cluster, and σ d s is its standard deviation. The images are captured at z relic . \n<!-- image --> \nstudy are somewhat lower than typical values for observed merging clusters (see Section 4) due to the limited computational volume in our simulations. Therefore, we present results primarily for normalized dimensionless quantities, assuming these are approximately scaleindependent.", '2.3. Shock Zones and Merger Shock Surfaces': 'In merged clusters, we identify shocks using the algorithm introduced in Ryu et al. (2003). Along the axis directions, grid zones are tagged as shock zones if they satisfy the following three conditions: (1) ∇ · v < 0, i.e., locally converging flow; (2) ∆ T × ∆ ρ > 0, i.e., the same sign of gas temperature and density gradients; and (3) | ∆log T | > 0 . 11, i.e., the temperature jump larger than that of Mach number M s = 1 . 3. Then, M s is calculated by solving the Rankine-Hugoniot temperature jump, T 2 /T 1 = (5 M 2 s -1)( M 2 s + 3) / (16 M 2 s ). Here, the subscripts 1 and 2 represent the preshock and postshock quantities, respectively. After applying these procedures for all three axis directions, the Mach number of a shock zone is defined as the maximum value of the three Mach numbers along the x , y and z axes, M s = max( M s,x , M s,y , M s,z ). Many grid zones are identified to contain weak shocks, but weak shocks are not energetically important. Thus, we only consider the shock zones with M s ≥ 1 . 5. \nIn the ICM, shocks can be induced by dynamical activities other than mergers of subclusters, such as turbulent flow motions and the infall of the WHIM along filaments. To isolate shock zones that belong to the merger-driven shock surfaces, we employ the following steps. In the first step, considering merger shocks typically appear \nFigure 3. Merging process in Cluster 10 with M 1 / M 2 ≃ 2 . 96 and b/r vir , 1 ≃ 0 . 3 (see Table 1). From top to bottom, 2D slice images of the gas density, ρ gas / ⟨ ρ gas ⟩ , the DM density, ρ DM / ⟨ ρ DM ⟩ , and the gas temperature, T , in the comoving area of (8 h -1 Mpc) 2 around the cluster are shown. Here, ⟨ ρ gas ⟩ and ⟨ ρ DM ⟩ are the mean gas and DM densities of the universe, and T is in units of Kelvin. The pericenter passage occurs at z ≈ 0 . 1, while a pair of merger shocks would appear as radio relics at z ≲ 0 . 05. Relic shock 2 in front of subcluster 2 is larger than relic shock 1 in front of subcluster 1. The black dot at z = 0 draws the circle of the virial radius, r vir . \n<!-- image --> \naround the virial radius (see the next section), we select shock zones satisfying (1) -1 . 4 ≤ log 10 ρ 1 / ⟨ ρ ⟩ ≤ 0 . 3 and (2) -1 . 0 ≤ log 10 T 1 / ⟨ T ⟩ ≤ 0 . 1. Here, ρ 1 and T 1 represent the preshock density and temperature, respectively, and ⟨ ρ ⟩ and ⟨ T ⟩ are the mean values calculated within the virial radius of each cluster. The upper bounds for ρ 1 and T 1 help eliminate turbulence-induced shocks in the dense and hot ICM inside the virial radius, while the lower bounds are imposed to exclude shocks induced by the infall of the WHIM along filaments in cluster outskirts. These criteria have been empirically determined by examining the merger shocks at z relic . In the second step, using the selected shock zones, we estimate the mean distance ⟨ d s ⟩ of each relic shock surface from the cluster center and its standard deviation, σ d s . Throughout the paper, the cluster center is defined as the peak of X-ray emission within the merged cluster. Only those within ⟨ d s ⟩ ± 1 σ d s are chosen as parts of the relic shock surface. We then iteratively update ⟨ d s ⟩ and σ d s and refine the list of constituent shock zones to achieve the best result. \nFigure 2 illustrates three-dimensional (3D) views of the shock surfaces in Cluster 3 at z relic ≈ 0 . 1 (see Table 1). The left panel displays all shock zones within the volume, whereas the right panel shows the shock zones after the refinement according to the above criteria. Below we examine the properties of refined merger shock surfaces.', '3.1. Morphological Properties of Merger Shock Surfaces': "Mergers of subclusters take place during the hierarchical formation of the LSS of the universe; the resulting merger shocks, which develop in the inherently turbulent ICM, usually have complex morphology. Figure 3 displays two-dimensional (2D) slices for the gas density (top panels), DM density (middle panels), and gas temperature (bottom panels) in Cluster 10 (see Table 1). At z = 0 . 18, two prominent clumps approach each other with M 1 / M 2 ≃ 2 . 96 and b/r vir , 1 ≃ 0 . 3. As these clumps close in with a relative velocity of about 10 3 km s -1 , shocks develop along the merger axis and \n(a) \n(b) \nFigure 4(b) plots the mean distance of merger shocks from the cluster center, ⟨ d s ⟩ , as a function of the virial radius of the merged cluster, r vir , at z relic . For our sample clusters with r vir ≈ 1 . 0 -2 . 3 Mpc (proper distance), ⟨ d s ⟩ /r vir ranges ∼ 0 . 6 -1 . 2 with an average value of ∼ 0 . 9. This indicates that at z relic , ∼ 1 Gyr after the pericenter passage, although there is a significant scatter, merger shocks are typically found around the virial radius, consistent with observations of radio relic locations (see the introduction). It is worth noting that Zhang et al. (2019) proposed a 'habitable zone' at distances ≳ r 500 , or ≳ 0 . 5 r vir , beyond which moderately strong shocks appear. Here, r 500 is the radius within which the mean total density is 500 times ρ c . Although their simulation setup and merger parameters differ from ours, the merger shocks in Figure 4(b) are indeed in the habitable zone, aligning with their suggestion. Additionally, the figure shows that there is almost no or only weak dependence of ⟨ d s ⟩ /r vir on r vir , implying that on average, ⟨ d s ⟩ tends to be larger in more massive clusters with larger r vir . \n<!-- image --> \n𝑣𝑣𝑣𝑣𝑣𝑣 \nFigure 4. (a) 3D view of merger shocks at z relic for four representative clusters. Color displays the Mach number of shock zones. The comoving volume of the boxes ranges approximately (4 -5 Mpc) 3 ; the box size varies to optimize the visualization of shock surfaces. Merger shocks for all 12 sample clusters in Table 1 are provided in Figure A1. (b) Mean distance, ⟨ d s ⟩ , between the merger shock surface and the X-ray center of the merged cluster, normalized to the virial radius of each cluster, r vir , versus r vir for 24 merger shocks in Figure A1. Here, r vir is given as the proper distance. The vertical error bars represent the ± 1 σ d s deviation. The horizontal dashed line indicates the average value of ⟨ d s ⟩ /r vir ≈ 0 . 9. Blue dots and lines represent relic shock 1, while red dots and lines represent relic shock 2, the same colors as in Figure 1. The same color scheme is used in subsequent figures. \n= 2.2, = 0.46 \n= 3.08, \n= 0.64 \nbecome visible at z = 0 . 14. The epoch near the pericenter passage occurs around z = 0 . 1. Following this, two merger shocks are observed propagating outward over z = 0 . 1 -0 . 0. The shock surfaces and the turbulent nature of the ICM are most clearly depicted in the temperature images. On the other hand, the gas density images reveal multiple minor mergers and secondary infalls that contribute to turbulent flow motions in the ICM. Although not shown here, the shock speed, v s , and also M s initially increase with time and then tend to converge, though significant fluctuations are observed (see Figure 5 of Ha et al. 2018). \nFigure 4(a) presents 3D views of merger shocks at z relic for four of our sample clusters (Clusters 1, 3, 6, and 11), highlighting pairs of shock surfaces moving in opposite directions. Merger shocks for all 12 sample clusters are shown in Figure A1. In Clusters 6 and 11 with relatively large M 1 / M 2 , a larger shock is in front of the lighter dark matter clump (relic shock 2), while a smaller shock appears in front of the heavier dark matter clump (relic shock 1). Additionally, the figure illustrates the asymmetry, caused by nonzero impact parameters and the turbulent ICM. \n𝑟𝑟 \n(Mpc) \nThe left panel of Figure 5 schematically depicts the geometry of merger shock surfaces as elliptical sections \n0.4 \n/ 𝑟𝑟 𝑣𝑣𝑣𝑣𝑣𝑣 , 1 𝑏𝑏 Figure 5. Left Panel: Schematic picture showing the geometry of relic shocks in a binary merger far after the pericenter passage. The shock surfaces are approximated as ellipses with major axis a ss and minor axis b ss . A ss denotes the surface area of shocks and is calculated as A ss ≈ 1 . 19(∆ l ) 2 N sh , where N sh is the number of shock zones associated with a particular shock surface. Right Panel: (a) Axial ratio of merger shock surfaces, b ss /a ss , for the 24 merger shocks in Figure A1. (b) Ratio of the area of merger shock surfaces, A ss , to the area of the shock spheres of radius ⟨ d s ⟩ , A sphere = 4 π ⟨ d s ⟩ 2 . (c)-(d) Ratio of A ss , 2 /A ss , 1 versus the mass ratio of two subclusters, M 1 / M 2 , and the normalized impact parameter, b/r vir , 1 , for 12 sample clusters. The horizontal dashed lines in (a)-(b) denote the average values for the 24 merger shocks. The solid line in panel (c) draws y = x . \n<!-- image --> \nof spherical shells at distance ⟨ d s ⟩ , described by their semi-major axis a ss and semi-minor axis b ss . The major axis ranges 2 a ss ≈ 1 . 2 -3 . 6 Mpc, while the minor axis ranges 2 b ss ≈ 0 . 7 -2 . 8 Mpc, in terms of the proper distance. As shown in Figure 5(a), the aspect ratio is generally b ss /a ss ≳ 0 . 6 with an average of ∼ 0 . 77, except for an outlier with b ss /a ss ≈ 0 . 4 observed for relic shock 1 in Cluster 5. So, while most merger shock surfaces normally appear moderately elongated, highly elongated, stripe-like surfaces can also form depending on the merger history of clusters as well as on the flow dynamics in the turbulent ICM. \nWe define the 'shock sphere' as a sphere with radius ⟨ d s ⟩ and surface area A sphere = 4 π ⟨ d s ⟩ 2 . For merger shock surfaces, the area is calculated as A ss ≈ 1 . 19(∆ l ) 2 N sh , where N sh is the number of shock zones associated with a given merger shock. The factor of 1.19 accounts for the mean projected area within a 3D zone with random shock normal orientations. In Figure 5(b), the normalized area of merger shock surfaces, A ss /A sphere , is plotted. For most relic shocks, A ss /A sphere ranges ∼ 0 . 05 -0 . 2, with the exception of relic shock 2 in Cluster 8, which has A ss /A sphere ≈ 0 . 36. The average value is A ss /A sphere ∼ 0 . 15, indicating that \nmerger shock surfaces typically occupy a small fraction of the surface area of shock spheres. \nThe morphological properties of merger shock surfaces can vary depending on merger parameters. In idealized binary mergers with subclusters in hydrostatic equilibrium, it was shown that the mass ratio influences the relative size of two merger shock surfaces, while the impact parameter affects the degree of asymmetry around the merger axis (e.g., van Weeren et al. 2011a; Molnar & Broadhurst 2017). Figure 5(c) and (d) show the ratio of the surface areas of two merger shocks, A ss , 2 /A ss , 1 , as a function of merger parameters, M 1 / M 2 and b/r vir , 1 , for our sample clusters. The ratio A ss , 2 /A ss , 1 tends to increase with increasing M 1 / M 2 , consistent with the findings of previous studies on idealized binary mergers mentioned above. In contrast, while there is considerable scatter, no clear dependence of A ss , 2 /A ss , 1 on b/r vir , 1 is found.", '3.2. Mach Numbers as a Function of Angular Distance from the Merger Axis': "As an effort to describe the morphological characteristics of relic shocks in the turbulent ICM, we examine the distribution of M s in merger shock surfaces as a function of the position angle, θ , which is defined as the angle \nFigure 6. (a) 3D view of merger shocks, superimposed on 3D volume-rendered images of the DM density, ρ DM / ⟨ ρ DM ⟩ , for the four clusters shown in Figure 4. The color of merger shocks displays the 'position angle' of shock surfaces, θ , which is the angle between the axis passing through the two DM density peaks (indicated by black line segments) and the line connecting the midpoint of these peaks to shock zones. The images are at z relic as in Figure 4, but with different viewing angles. The corresponding images for all 12 sample clusters are provided in Figure A2. (b) Average Mach number of shock zones associated with relic shocks 1 (blue) and 2 (red) as a function of θ with bins of ∆ θ = 5 · for the four sample clusters. Error bars represent ± 1 σ deviations of M s . Similar plots for all 12 sample clusters are shown in Figure A3. \n<!-- image --> \nlog \n𝐷𝐷𝐷𝐷 \n𝜃𝜃 \n(°) \n𝜌𝜌 \nbetween the axis connecting two DM density peaks and the line extending from the midpoint of these peaks to the given shock zone. Figure 6(a) depicts the 3D distribution of θ for merger shocks, on top of the DM density distribution, for the four clusters shown in Figure 4(a). The corresponding images for all 12 sample clusters are available in Figure A2. Figure 6(b) plot the average and standard deviation of M s in bins of ∆ θ as a function of θ for shock zones that belong to relic shocks 1 (blue) and 2 (red) in the four sample clusters. Similar plots for all 12 sample clusters are provided in Figure A3. \nIn the idealized scenario of a spherical bow shock propagating through a uniform medium, the Mach number distribution is expected to vary as ⟨ M s ⟩ ∝ cos θ (see, e.g., Petrinec & Russell 1997, for further discussion). Russell et al. (2022) investigated the angular dependence of the Mach number for a merger shock in Abell 2146 using X-ray data, finding that it roughly follows the cosine behavior. In contrast, our study examines the angular dependence of the Mach number in 3D. As shown in Figures 6(b) and A3, while some merger shocks, such as relic shock 2 of Cluster 3, exhibit behaviors close to the idealized pattern, many do not follow this trend. Specifically, some shocks exhibit peaks at large nonzero θ values, rather than around θ ∼ 0 · , with a roughly symmetrical decline on either side of the peak (e.g., relic shock 1 of Cluster 11). Others display significant fluctuations in the Mach number distribution with no distinct peak. As a matter of fact, in most cases, the axis connecting DM peaks does not go through the center of \nshock surfaces. This is partly due to nonzero impact parameters in our sample clusters and also due to the fact that merger shock surfaces are disturbed by turbulent flow motions and infalling clumps in the ICM as they propagate from the core to outskirts. Consequently, the center of the shock surface fluctuates and becomes misaligned with the axis connecting the DM peaks. For example, for relic shock 1 of Cluster 1, the shock surface is significantly tilted, so there is no shock zone for θ ≲ 5 · . We point out that Cluster 1 has experienced a major merger at z = 0 . 22, followed by a minor merger with a mass ratio of ∼ 6 in the propagation direction of relic shock 1 at z = 0 . 18. And the shock properties are analyzed at z relic = 0 . 14. In this relic shock, the shock surface morphology is affected by the late minor merger (see the shock on the right hand side of Cluster 1 in Figure 4(a)) and offset from the axis connecting DM peaks.", '3.3. Mach Number Distribution on Merger Shock Surfaces': "The PDFs for M s of shock zones across merger shock surfaces are shown in Figure 7(a) for the four representative clusters. The corresponding PDFs for all 12 sample clusters are provided in Figure A4. In general, the PDFs exhibit positively skewed distributions. We fit the PDFs to a log-normal function, with the fittings shown as gray lines in the figures. The log-normal fits peak around M s, peak ≈ 2 -4 . 5, with a high Mach number tail extending up to M s ∼ 10. This tail is primarily due to \nFigure 7. (a) PDF of the Mach number of shock zones associated with relic shocks 1 (blue) and 2 (red) in the four clusters shown in Figure 4. The vertical dashed lines denote the shock-surface-area-weighted average Mach number, ⟨ M s ⟩ area , while the gray solid lines show the log-normal fitting. The corresponding PDFs for all 12 sample clusters are provided in Figure A4. (b) Standard deviation normalized to the mean value, σ M s / ⟨ M s ⟩ area , as a function of ⟨ M s ⟩ area for the PDFs of 24 merger shocks in 12 sample clusters. The subscript 'area' is omitted for simplicity. The horizontal dashed line represents the average value of σ M s / ⟨ M s ⟩ ≈ 0 . 37. (c) σ ln M s /µ ln M s versus µ ln M s , where µ ln M s and σ ln M s are the mean and standard deviation for the fitted log-normal distribution of the Mach number. \n<!-- image --> \nintermittent accretions of the WHIM, as merger shocks are often attached to infalling filaments. These Mach number distributions agree with the findings of previous numerical studies (e.g., Botteon et al. 2020; Wittor et al. 2021). \nFor the further analysis of their statistical characteristics, we calculate (1) the mean and standard deviation of M s , ⟨ M s ⟩ area and σ M s , using the actual PDFs (blue and red lines in Figures 7(a) and A4, and (2) the mean and standard deviation, µ ln M s and σ ln M s , using the fitted log-normal distributions (gray lines). The subscript 'area' is added to ⟨ M s ⟩ area to specify that it is actually the shock-surface-area-weighted average Mach number. Figures 7(b) and (c) demonstrate that σ M s / ⟨ M s ⟩ area and σ ln M s /µ ln M s are largely independent of ⟨ M s ⟩ area and µ ln M s , with average values of σ M s / ⟨ M s ⟩ area ≈ 0 . 37 and σ ln M s /µ ln M s ≈ 0 . 33, respectively. This implies that the PDFs of M s are more or less universal, although there is scatter owing to diverse merger histories of clusters, including the infall of small clumps, as well as turbulent flow motions in the ICM.", '3.4. Average Mach Numbers of Merger Shock Surfaces': "In addition to M s, peak and ⟨ M s ⟩ area , the Mach number distribution of shock zones on merge shock surfaces can be characterized by other representative values, such as the average Mach numbers weighted with observable quantities. Specifically, we examine the Xray-emissivity-weighted average Mach number, ⟨ M s ⟩ X , and the CR-energy-flux-weighted average Mach number, ⟨ M s ⟩ CR . Here, ⟨ M s ⟩ X is obtained by weighing \nM s with the bremsstrahlung emission, ε ff , over shock zones on the given shock surface. For ⟨ M s ⟩ CR , M s is weighted by the CR energy flux, f CR = η e ( M s ) · f ϕ , where f ϕ = (1 / 2) ρ 1 v s 3 represents the shock kinetic energy flux. The CR electron acceleration efficiency is approximated as η e ≈ 0 . 01 η , where the CR proton acceleration efficiency ranges η ( M s ) ≈ 3 . 6 × 10 -3 -0 . 01 for 2 . 25 ≲ M s ≲ 5 . 0 (Ryu et al. 2019). In this study, we regard ⟨ M s ⟩ CR as a proxy for radio-emissivity-weighted average Mach number, ⟨ M s ⟩ radio . \nIn general, M s, peak tends to be slightly smaller than ⟨ M s ⟩ area ≈ 2 . 3 -4 . 4, as expected with the log-normal shape of the Mach number PDFs, as shown in Figure 8(a). The figure also illustrates that ⟨ M s ⟩ X ∼ 2 -4 is comparable to ⟨ M s ⟩ area . On the other hand, ⟨ M s ⟩ CR ∼ 3 -5 is greater than both ⟨ M s ⟩ area and ⟨ M s ⟩ X . This trend reflects our CR acceleration model, η ( M s ), which predicts higher efficiencies for higher M s , as noted in the introduction. These results are consistent with those of previous studies. For example, using simulations with a cosmological magnetohydrodynamic code, ENZO, Wittor et al. (2021) reported average Mach numbers of ⟨ M s ⟩ X -ray ≈ 1 -4 and ⟨ M s ⟩ radio ≈ 3 -5 for merger shocks; in their work, ⟨ M s ⟩ X -ray and ⟨ M s ⟩ radio were estimated mimicking observations, including projection effects and synchrotron modeling. In summary, M s, peak ≲ ⟨ M s ⟩ area ≈ ⟨ M s ⟩ X ≲ ⟨ M s ⟩ CR , indicating that while ⟨ M s ⟩ X provides a reasonable estimate of ⟨ M s ⟩ area , ⟨ M s ⟩ CR is typically higher. However, there are exceptions; in relic shock 1 of Cluster 1, where a secondary minor merger has disturbed the shock surface, ⟨ M s ⟩ CR is \nFigure 8. (a) Peak of the Mach number PDF, M s, peak (magenta), X-ray-emissivity-weighted average Mach number, ⟨ M s ⟩ X (red), and CR-energy-flux-weighted average Mach number, ⟨ M s ⟩ CR (blue), versus shock-surface-area-weighted average Mach number, ⟨ M s ⟩ area , for 24 merger shocks. (b) ⟨ M s ⟩ X (red) and ⟨ M s ⟩ CR (blue) as a function of the normalized distance of merger shocks from the cluster center, ⟨ d s ⟩ /r vir . \n<!-- image --> \nclose to ⟨ M s ⟩ area , while ⟨ M s ⟩ X is smaller (see the points with ⟨ M s ⟩ area = 4 . 4 in the figure). \nFigure 8(b) presents ⟨ M s ⟩ X and ⟨ M s ⟩ CR , both of which are linked to observable quantities, as a function of the normalized mean distance of merger shocks from the cluster center, ⟨ d s ⟩ /r vir . They generally tend to increase with ⟨ d s ⟩ /r vir . This trend is partly due to the fact that, on average, the gas density decreases and the shock speed increases outwardly in merged clusters (Ha et al. 2018). There is substantial scatter, which would be again attributed to the merger histories of clusters and the influence of turbulent flow motions in the ICM. \nFigure 9. ⟨ M s ⟩ X (top panels) and ⟨ M s ⟩ CR (bottom panels) versus mass ratio M 1 / M 2 (left panels) and impact parameter b/r vir , 1 (right panels) for 24 merger shocks in 12 sample clusters. \n<!-- image --> \nThe Mach number distribution of merger shocks, and hence the representative Mach numbers, could depend on merger parameters, similar to the morphological properties discussed in Section 3.1. Figure 9 shows ⟨ M s ⟩ X and ⟨ M s ⟩ CR as functions of M 1 / M 2 and b/r vir , 1 . The average Mach numbers tend to decrease with increasing M 1 / M 2, although there is relatively large scatter, but the dependence on b/r vir , 1 appears to be only marginal. In Ha et al. (2018), where cases with b ≈ 0 and M 1 / M 2 ≈ 2 were examined, relic shock 1 ahead of the more massive subcluster tends to be stronger with higher Mach numbers than relic shock 2, albeit with large fluctuations (see their Figure 5(a)). However, this tendency is not obvious in our merger shock samples from clusters with nonzero impact factors. As can be seen in Figures 9(a) and (b), in only 5 of 12 clusters, ⟨ M s ⟩ X for relic shock 1 is higher than for relic shock 2. Similarly, in Figures 9(c) and (d), in only 6 of 12 clusters, ⟨ M s ⟩ CR for relic shock 1 exceeds that for relic shock 2. Given the complex interplay of various factors such as merger parameters, merger history, and ICM turbulence, however, we argue that a larger cluster sample would be necessary to establish a more robust conclusion on the relative strengths of M s between relic shock 1 and 2. \nTable 2. Properties of Observed Merger shocks in Galaxy Clusters Hosting Double Radio Relics \nNote - Column1: Name of galaxy clusters with double radio relics. Column2: Redshift of cluster. Column3: X-ray temperature of cluster. Column4: Projected distance of merge shock from the cluster center. Column5: Integrated radio spectral index. Column6: Radio Mach number calculated with α int . Column7: X-ray Mach number derived from temperature jump. Column8: X-ray Mach number derived from density jump. The # symbol indicates that M radio is calculated using α int given in the reference. The * symbol means that M radio can not be estimated, because α int < 1. The † symbol indicates that ⟨ kT X ⟩ is estimated using the L X -T relation in Pratt et al. (2009). For cases where shocks are not detected in X-rays or X-ray Mach numbers are not reported, n/a is indicated. \nReferences -Abell 3376: Akamatsu & Kawahara (2013), George et al. (2015), Urdampilleta et al. (2018); Abell 3667: Akamatsu & Kawahara (2013), Sarazin et al. (2016), Storm et al. (2018), de Gasperin et al. (2022); Abell 3365: Duchesne et al. (2021a), Urdampilleta et al. (2021); ZwCl 0008.8+5215: van Weeren et al. (2011b), Kierdorf et al. (2017), Di Gennaro et al. (2019); Abell 3186: Nesci & Norci (1997), Duchesne et al. (2021a); Abell 1240: Barrena et al. (2009), Hoang et al. (2018), Sarkar et al. (2024); Abell 2345: George et al. (2017), Stuardi et al. (2021); 8C 0212+703: Hoang et al. (2021), Tumer et al. (2023); CIZA J2242.8+5301: Akamatsu & Kawahara (2013), Akamatsu & Kawahara (2013), Di Gennaro et al. (2018), Loi et al. (2020); Abell 2146: Russell et al. (2011), Russell et al. (2012), Hoang et al. (2019); RXC J1314.4-2515: Valtchanov et al. (2002), Venturi et al. (2007), George et al. (2017), Stuardi et al. (2019), Botteon et al. (2020); ZwCl 2341.1+0000 : Zhang et al. (2021), Stuardi et al. (2022); SPT-CL J2032-5627: Bulbul et al. (2019) Duchesne et al. (2021b) ; PSZ1 G096.89+24.17: Finner et al. (2021), Jones et al. (2021); PSZ1 G108.18-11.53: de Gasperin et al. (2015); PSZ2 G233.68+36.14: Ghirardini et al. (2021); MACS J1752.0+4440: van Weeren et al. (2012), Finner et al. (2021); ZwCl 1447.2+2619: Lee et al. (2022); PLCK G287.0+32.9: Bagchi et al. (2011), George et al. (2017); El Gordo: Menanteau et al. (2012), Botteon et al. (2020), Stuardi et al. (2022) \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 10. (a) Comparison of ⟨ M s ⟩ CR (black circles) of simulated merger shocks in our sample clusters with M radio (red squares) of observed radio relics in Table 2, plotted as a function of T X . (b) Comparison of ⟨ M s ⟩ X (black circles) of simulated merger shocks with M X -ray (red and blue squares) of observed radio relics, plotted as a function of T X . (c) and (d) Plots similar to (a) and (b), but shown as a function of the radio relic's distance from the cluster center, d s . Here, for simulated radio relics, d s is the mean 3D distance of the shock surface, ⟨ d s ⟩ . In contrast, for observed radio relics, d s refers to the distance 'projected' onto the sky, listed in Table 2. The vertical lines denotes the errors in observation. \n<!-- image -->", '4. COMPARISON WITH OBSERVATIONS': "In this section, we attempt to compare the properties of merger shocks reproduced in our simulations with those inferred from observed double radio relics. So far, about a hundred clusters hosting radio relics have been identified, and of those, approximately two dozen have double relics (see, e.g., van Weeren et al. 2019). Table 2 provides a summary of the observational data and the Mach numbers inferred from radio and X-ray observations for the double radio relics reported in the literature (see Stuardi et al. 2022, and the references listed below the table). Here, M radio is the radio Mach number calculated using the integrated radio spectral index, α int , while the X-ray Mach number, M X -ray , is estimated from either the temperature jump, T jump , or the density jump, ρ jump , across the relic shock. And d s is the distance from the cluster center, projected onto the sky. \nFigure 11. ⟨ M s ⟩ CR versus ⟨ M s ⟩ X (black circles) for simulated relic shocks in our sample clusters, and M radio versus M X -ray ( T jump ) (red square) for observed radio relics in Table 2. The vertical and horizontal lines denotes the errors in observation. The tilted line draws x = y . \n<!-- image --> \nIn Figure 10, we plot ⟨ M s ⟩ CR and ⟨ M s ⟩ X as functions of the X-ray temperature, T X , and the average distance from the center, ⟨ d s ⟩ , for our 24 simulated merger shocks, along with M radio and M X -ray as functions of T X and the 'projected' distance, d s , for the observed radio relics listed in Table 2. We compare ⟨ M s ⟩ CR with M radio , assuming that ⟨ M s ⟩ CR serves as a proxy of radio-emissivity-weighted average Mach number, since radio synchrotron emissions are produced by CR electrons accelerated via diffusive shock acceleration (DSA). Similarly, we compare ⟨ M s ⟩ X with M X -ray , noting that ⟨ M s ⟩ X is derived from 3D shock properties in our simulations, whereas M X -ray is estimated from the observed X-ray surface brightness map projected onto the sky. \nA couple of points are notable. First, Figures 10(a) and (b) show that T X for simulated merger shocks (filled black circles) is, on average, lower than T X for observed radio relics (filled red and blue squares). This discrepancy arises because the merged clusters in our simulations have relatively low T X ≲ 4 keV due to the limited volume of our simulation box, as pointed in Section 2.2. In contract, X-ray and radio observations preferentially pick up clusters with heavier mass and higher T X . Hence, we present results in dimensionless quantities that would be roughly scale-independent, such as M s and ⟨ d ⟩ /r vir . Second, Figures 10(c) and (d) indicate that ⟨ d s ⟩ for simulated merger shocks is, on average, larger than d s for observed radio relics. This should be partly because the distances projected onto \nthe sky are less than the true 3D distances for observed radio relics. We point out that ⟨ d s ⟩ ∼ r vir for simulated merger shocks, although there is considerable scatter, as shown in Figure 4(b); radio relics are also observed around r vir , as discussed in the introduction. \nGiven the differences between simulated and observed quantities, the comparison should proceed with caution. Nevertheless, we point out the following: (1) The simulations predict ⟨ M s ⟩ CR ∼ 3 -5 for merger shocks, whereas the observed relics exhibit M radio ∼ 2 -5. Considering the limitations inherent in our simulations, this level of agreement would be considered reasonable. On the other hand, it was argued that the critical Mach number for electron preacceleration and subsequent injection to DSA is M crit ≈ 2 . 3 (e.g., Kang et al. 2019; Ha et al. 2021, 2022). Several observed relics have low M radio close to ∼ 2; however, these relics likely have regions with M s ≳ M crit where electrons DSA may occur. (2) In contrast, ⟨ M s ⟩ X ∼ 2 -4 for simulated merger shocks looks greater than M X -ray ∼ 1 . 2 -4 for observed relics in Figure 10. This discrepancy may partly arise from projection effects in X-ray observations (Hong et al. 2015; Hoang et al. 2018). Moreover, Wittor et al. (2021) demonstrated that M X -ray derived from the X-ray surface brightness map projected onto the sky plane can vary significantly depending on the relic's orientation. \nIn Figure 11 we show the relation between the radio and X-ray Mach numbers for both simulated merger shocks and observed radio relics. In both samples, M radio ≳ M X -ray , confirming the earlier interpretation in Section 3.4 that electron acceleration is more efficient in portions of shock surfaces with higher M s , and consequently, M radio tends to be more heavily weighted by those higher M s regions. However, there are exceptions, such as Abell 3667 (NW) and ZwCl 0008.8+5215 (W), where M radio ≲ M X -ray ; these cases require more careful interpretation of radio and X-ray Mach numbers.", '5. SUMMARY': 'A major merger between two subclusters of comparable masses is expected to produce a pair of megaparsecscale bow shocks that propagate into the outskirts of the merged cluster over gigayears following pericenter passage (e.g., van Weeren et al. 2011a). When projected onto the sky, these merger shocks often appear as double radio relics in radio observations or as temperature and surface brightness discontinuities in X-ray observations (see Table 2). \nIn this paper, we analyzed the data from a set of cosmological hydrodynamic simulations to investigate the properties of such shocks induced by major binary mergers. We focused on 12 merging clusters with mass ra- \ntio 1 ≤M 1 / M 2 ≲ 4 and normalized impact parameter b/r , 1 ≲ 1, resulting in a total of 24 merger shocks examined at the optimal redshift for radio relic observation, z relic (see Table 1). Relic shock 1 forms in front of the heavier subcluster 1, while relic shock 2 forms in front of the lighter subcluster 2. We isolated the shock zones associated with the shock surfaces, distinguishing them from background ICM shocks produced by turbulence and infall, based on the criteria outlined in Section 2.3. \nWe inspected the morphological characteristics of these merger shock surfaces, including their shape and area. We then quantified the Mach number distribution across the shock surfaces. Since merger shocks consist of numerous zones with varying Mach numbers, we defined representative average Mach numbers. We explored how these representative Mach numbers manifest in simulated merger shocks and compared them with the Mach numbers inferred in X-ray and radio observations. \nOur main results are summarized as follows: \n- 1. In our sample, merger shock surfaces are located at ⟨ d s ⟩ /r vir ≈ 0 . 6 -1 . 2 from the X-ray center of the clusters at z relic (see Figure 4). The shock surfaces can be approximated as elliptical sections of spherical shells, with an axial ratio of b ss /a ss ≳ 0 . 6. Each shock surface covers approximately ∼ 5 -20 % of the surface area of the shock sphere with r sphere = ⟨ d s ⟩ . The surface area ratio of relic shock 2 to relic shock 1 scales roughly with the mass ratio of subclusters as A ss , 2 /A ss , 1 ∝ M 1 / M 2 (see Figure 5).\n- 2. Due to nonzero impact parameters and turbulent ICM flows, merger shock surfaces become distorted, with their centers offset from the axis connecting the two DM density peaks (see Figures 6(a) and A2). As a result, the distribution of the average Mach number of shock zones, as a function of the position angle θ between the axis connecting the two DM density peaks and the line extending from the mid-point of these peaks to the shock zone, deviates from the typical cos θ pattern. Instead, these distributions often peak at fairly large θ , or exhibit significant fluctuations without a distinct peak (see Figures 6(b) and A3).\n- 3. The PDFs for M s of shock zones across merger shock surfaces are positively skewed and well-fitted by a log-normal function (see Figures 7(a) and A4). The fitted log-normal distributions peak around M s, peak ≈ 2 -4 . 5 with tails extending up to M s ∼ 10. The ratio of the standard deviation to the mean of the PDFs is approximately σ M s / ⟨ M s ⟩ area ≈ 0 . 37 (see Figures 7(b)). Here, the mean of the PDF is the shock-surface-areaweighted average Mach number, ⟨ M s ⟩ area .\n- 4. The area-weighted average Mach numbers, ⟨ M s ⟩ area ≈ 2 . 3 -4 . 4 are comparable to the X-ray- \nvity-weighted average Mach numbers, ⟨ M s ⟩ X ≈ 2 -4. In contrast, the CR-energy-flux-weighted average Mach numbers are higher with ⟨ M s ⟩ CR ≈ 3 -5 (see Figure 8). This discrepancy aligns with observations of radio relics, where generally the Mach numbers inferred from radio data are greater than those derived from Xray data, M radio ≳ M X -ray . \n- 5. In our simulated merger shocks, both ⟨ M s ⟩ X and ⟨ M s ⟩ CR exhibit a weak, decreasing trend with increasing M 1 / M 2 , albeit with significant scatter. On the contrary, the dependence of these Mach numbers on b/r vir , 1 appears to be only marginal (see Figure 9).\n- 6. For simulated merger shocks, we find ⟨ M s ⟩ X ≳ 2, while X-ray observations frequently report shocks with M X -ray ≲ 2 (see Figure 11). As suggested by several previous papers (e.g., Hong et al. 2015; Wittor et al. 2021), this discrepancy may be partly attributed to projection effects in X-ray observations, which can complicate accurate estimation of the true Mach numbers. \nIn this paper, we focused on major binary mergers with 1 ≤M 1 / M 2 ≲ 4 and b/r , 1 ≲ 1, which are expected to present a relatively simple merger scenario. How- \ndue to the complex interplay of factors such as merger history and turbulence, in addition to merger parameters, the structure of merger shocks turns out to be fairly complicated. Our aim was to characterize the properties of shock surfaces, yet a larger sample of simulated merging clusters appears necessary to establish more definitive relationships, such as the dependence of average Mach numbers on merger parameters. \nFinally, some observed radio relics appear to have formed under merger conditions different from those considered here and also may result from multiple merger events. We leave investigations of these cases for future work. \nThe authors would like to thank the anonymous referee for constructive comments and suggestions. This work was supported by the National Research Foundation (NRF) of Korea through grants 2020R1A2C2102800, 2023R1A2C1003131, RS-202200197685, and 2023K2A9A1A01099513. We thank Dr. J.-H. Ha for discussions.', 'REFERENCES': "Akamatsu, H., & Kawahara, H. 2013, PASJ, 65, 16, doi: 10.1093/pasj/65.1.16 \nBagchi, J., Sirothia, S. 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In this appendix, we provide figures for all 12 sample clusters for comprehensive reference. \nFigure A1. Same as in Figure 4(a) for all 12 sample clusters. \n<!-- image --> \nFigure A2. Same as in Figure 6(a) for all 12 sample clusters. \n<!-- image --> \nlog \n𝐷𝐷𝐷𝐷 \n𝜌𝜌 \n𝜃𝜃 \n(°) \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝜃𝜃 \n(°) \n𝜃𝜃 \n(°) \nFigure A3. Same as in Figure 6(b) for all 12 sample clusters. \n<!-- image --> \n𝜃𝜃 \n(°) \n𝜃𝜃 \n(°) \n𝜃𝜃 \n(°) \n𝜃𝜃 \n(°) \n) \n𝑠𝑠 \n𝑀𝑀 \n( \n𝑃𝑃𝑃𝑃 \n𝑃𝑃 \n) \n𝑠𝑠 \n𝑀𝑀 \n( \n𝑃𝑃𝑃𝑃 \n𝑃𝑃 \n) \n𝑠𝑠 \n𝑀𝑀 \n( \n𝑃𝑃𝑃𝑃 \n𝑃𝑃 \n) \n𝑠𝑠 \n𝑀𝑀 \n( \n𝑃𝑃𝑃𝑃 \n𝑃𝑃 \nFigure A4. Same as in Figure 7(a) for all 12 sample clusters. \n<!-- image --> \n0.12 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀 \n𝑠𝑠 \n𝑀𝑀'}

2024arXiv240911347T

After the main sequence phase stars more massive than 2.5 Modot rapidly evolve through the Hertzsprung gap as yellow giants and supergiants YSG before settling into the red giant branch. Identifying YSG in nearby galaxies is crucial for pinpointing progenitors of luminous red novae LRNe astrophysical transients attributed to stellar mergers. In the era of extensive transient surveys like the Vera Rubin Observatorys LSST this approach offers a new way to predict and select common envelope transients. This study investigates potential progenitors and precursors of LRNe by analysing Hubble Space Telescope HST photometry of stellar populations in galaxies within 20 Mpc to identify YSG candidates. Additionally we use ZTF and MeerLICHTBlackGEM to identify possible precursors preparing for future observations by the LSST. We compiled a sample of 369 galaxies with HST exposures in the F475W F555W F606W and F814W filters. We identified YSG candidates using MESA stellar evolution tracks and statistical analysis of colormagnitude diagrams CMDs. Our sample includes 154494 YSG candidates with masses between 3 and 20 Modot and is affected by various contaminants such as foreground stars and extinguished mainsequence stars. After excluding foreground stars using Gaia proper motions contamination is estimated at 1 from foreground stars and 20 from extinction affecting mainsequence stars. Combining our YSG candidates with timedomain catalogs yielded several interesting candidates. Notably we identified 12 LRN precursor candidates for which followup is encouraged. We highlight the importance of monitoring future transients that match YSG candidates to avoid missing potential LRNe and other rare transients. LSST will be a game changer in the search for LRN progenitors and precursors discovering over 300000 new YSG and 100 precursors within 20 Mpc.

2024-09-01T00:00:00Z

['2024arXiv240911347T', '10.48550/arXiv.2409.11347', 'arXiv:2409.11347']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Hertzsprung gap stars in nearby galaxies and the Quest for Luminous Red Novae Progenitors

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11347.pdf

{'Hertzsprung gap stars in nearby galaxies and the Quest for Luminous Red Novae Progenitors ⋆': "Hugo Tranin 1 , 2 , 3 , Nadejda Blagorodnova 1 , 2 , 3 , Viraj Karambelkar 4 , Paul J. Groot 5 , 6 , 7 , 8 , Steven Bloemen 5 , Paul M. \nVreeswijk 5 , Daniëlle L.A. Pieterse 5 , and Jan van Roestel 9 \n- 1 Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c. Martí i Franquès, 1, 08028, Barcelona, Spain e-mail: hugo.tranin@irap.omp.eu\n- 2 Departament de Física Quàntica i Astrofísica (FQA), Universitat de Barcelona (UB), c. Martí i Franquès, 1, 08028, Barcelona, Spain\n- 3 Institut d'Estudis Espacials de Catalunya (IEEC), c. Gran Capità, 2-4, 08034, Barcelona, Spain\n- 4 Cahill Center for Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA\n- 5 Department of Astrophysics / IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands\n- 6 Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch, 7701, South Africa\n- 7 South African Astronomical Observatory, P.O. Box 9, Observatory, 7935, South Africa\n- 8 The Inter-University Institute for Data Intensive Astronomy, University of Cape Town, Private Bag X3, Rondebosch, 7701, South Africa\n- 9 Anton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, 1090 GE Amsterdam, The Netherlands", 'ABSTRACT': "Context. After the main sequence phase, stars more massive than 2.5 M ⊙ rapidly evolve through the Hertzsprung gap as yellow giants and supergiants (YSG), before settling into the red giant branch. Identifying Hertzsprung gap stars in nearby galaxies is crucial for pinpointing progenitors of luminous red novae (LRNe) - astrophysical transients attributed to stellar mergers. In the era of extensive transient surveys like the Vera Rubin Observatory's Legacy Survey of Space and Time (LSST), this approach o ff ers a new way to predict and select common envelope transients. \nAims. This study investigates potential progenitors and precursors of LRNe by analysing Hubble Space Telescope (HST) photometry of stellar populations in galaxies within ∼ 20 Mpc to identify YSG candidates. Additionally, we use the Zwicky Transient Facility (ZTF) and MeerLICHT / BlackGEM to identify possible precursors, preparing for future observations by the LSST. \nMethods. We compiled a sample of 369 galaxies with HST exposures in the F 475 W , F 555 W , F 606 W , and F 814 W filters. We identified YSG candidates using MESA stellar evolution tracks and statistical analysis of colour-magnitude diagrams (CMDs). \nResults. Our sample includes 154,494 YSG candidates with masses between 3 M ⊙ and 20 M ⊙ and is a ff ected by various contaminants, notably foreground stars and extinguished main-sequence stars. After excluding foreground stars using Gaia proper motions, contamination is estimated at 1% from foreground stars (based on TRILEGAL simulations) and ∼ 20% from extinction a ff ecting mainsequence stars. Combining our YSG candidates with time-domain catalogues yielded several interesting candidates. In particular, we identified twelve LRN precursor candidates for which followup is encouraged. \nConclusions. We highlight the importance of monitoring future transients that match YSG candidates to avoid missing potential LRNe and other rare transients. LSST will be a game changer in the search for LRN progenitors and precursors, discovering over 300,000 new YSG candidates and 100 LRN precursors within 20 Mpc. \nKey words. Hertzsprung-Russell and colour-magnitude diagrams - novae - catalogues - Vera Rubin", '1. Introduction': 'The termination of hydrogen burning in a stellar core marks the end of a star\'s main sequence (MS) phase. As the core contracts and temperatures rise, hydrogen shell burning starts, causing the star to expand rapidly. This expansion leads to the star becoming progressively larger and cooler, transitioning into the yellow giant (YG) and supergiant (YSG) phase. However, this evolutionary stage is brief and di ffi cult to observe, creating a "gap" between the MS and the Red Giant Branch (RGB) in the Hertzsprung-Russell (HR) diagram, known as the Hertzsprung gap. Early surveys of YSGs in the Milky Way, Magellanic Clouds, and Andromeda revealed significant discrepancies be- \ntween observations and stellar evolution models, particularly in estimating the duration of this phase (Massey et al. 2000; Drout et al. 2009; Neugent et al. 2010; Drout et al. 2012). \nAdding further complexity, most massive stars are found in binary or multiple systems, with 70% of O-type stars and 50% of B-type stars having companions (e.g., Moe & Di Stefano 2017 and references therein). The rapid expansion of YSGs in such systems often triggers interactions like Case B mass transfer and substantial mass loss (Marchant & Bodensteiner 2023). This interaction makes the YSG phase particularly valuable for studying binary evolution. In cases where a YSG is part of a binary system, unstable mass transfer can lead to a common envelope phase (Paczynski 1976). The transfer of angular momentum of the binary into the envelope can result in its partial (or total) ejection, producing a rare astrophysical transient known as a Luminous Red Nova (LRN). While YSGs have been linked to other \nrare transients, such as core-collapse supernovae (SNe) (Smartt 2015) and failed SNe (Georgy 2012; Neustadt et al. 2021), they are most notably identified as progenitors of LRNe (MacLeod et al. 2017; Blagorodnova et al. 2017, 2021; Cai et al. 2022). \nLRNe are optical and infrared transients with luminosities between those of novae and supernovae, evolving over several weeks to months. Their peak brightness correlates with the progenitor\'s mass (Kochanek et al. 2014; Blagorodnova et al. 2021), enabling us to investigate binary evolution across a wide range of stellar masses, from low-mass to massive stars. Unlike optical LRNe, infrared LRNe may result from common envelope ejections in stars more evolved than YSGs (MacLeod et al. 2022). These stars experience significant mass loss, causing them to become obscured in optical wavelengths while remaining detectable in the infrared. The most well-studied LRN to date, V1309 Sco, was discovered in the Milky Way. Extensive photometric data up to seven years before the onset of the transient revealed its progenitor, a contact binary system with a quickly decaying period, possibly evolving to the merger of its components (Tylenda et al. 2011). Interestingly, the light curve of V1309 Sco exhibited a slow, steady brightening beginning approximately five years prior to the merger. Similar precursor emissions have been observed in extragalactic LRNe (Kankare et al. 2015; Blagorodnova et al. 2017, 2020; Pastorello et al. 2019, 2021a), though their greater distances and faintness have made it di ffi cult to achieve equally detailed sampling. It remains an open question whether this precursor phase reflects continuous or episodic mass loss in the years leading up to the major outburst, potentially signaling the early stages of a binary system merger. Long-term observations of LRN progenitor stars are therefore crucial for testing their binary origins and gaining valuable insights into pre-merger mass loss episodes in these systems. \nThe primary challenge in detecting precursors is their modest absolute magnitude, which means most extragalactic LRNe are only identified during their brightest outbursts. The lack of deep archival data further complicates e ff orts to detect precursor brightening. To overcome this, a comprehensive catalog of potential progenitors in nearby galaxies is essential. A large and deep survey is mandatory to achieve such a goal. While early discoveries of LRNe were limited to the Milky Way and M31 (e.g., V4332 Sgr and V838 Mon, Martini et al. 1999; Tylenda & Soker 2006; M31 LRN-2015, Williams et al. 2015; Blagorodnova et al. 2020), the advent of large synoptic surveys, such as ATLAS (Tonry et al. 2018), the Zwicky Transient Facility (ZTF; Bellm et al. 2019), MeerLICHT (Bloemen et al. 2016), and BlackGEM (Groot et al. 2019; Groot et al. 2024), has paved the way for the systematic detection of LRNe throughout the Local Group and beyond (e.g. Karambelkar et al. 2023). The upcoming Vera C. Rubin Observatory, with its Legacy Survey of Space and Time (LSST), promises to revolutionize transient astronomy, providing an unprecedented opportunity to detect faint LRNe and LRN precursors across vast distances. With LSST expected to generate 10 million transient alerts daily, the primary challenge will be identifying LRN precursors amidst this vast data unless a targeted strategy is developed to detect them. \nIn this study, we present a strategy for identifying LRN precursors by cataloging YSG candidates within 20 Mpc, leveraging archival Hubble Space Telescope (HST) photometry. Our goal is to identify potential precursor candidates by crossmatching future transient alerts with this catalog, o ff ering a new approach to studying the processes driving LRN precursor brightening and triggering mechanisms. An initial study provided a similar census of YGs and YSGs in the Milky Way (Ad- \ndison et al. 2022). Using Gaia DR2 and EDR3 (Gaia Collaboration et al. 2018, 2021) sources, they modelled the distribution of a sample of Milky Way stars in the colour-magnitude ( MG vs. BP -RP ) diagram. Statistical modelling allowed them to select the Hertzsprung gap as the least-populated region of the diagram. As a result, they identified 21 candidates exhibiting signs of a steady increase in brightness. Most of these candidates showed Balmer lines in emission and an infrared excess in their spectral energy distributions, reinforcing the likelihood of an accreting system with dust production. Interestingly, one of them turned out to be the precursor of a type-I X-ray burst (Reig et al. 2022). \nWhile the census of Galactic YSG candidates is crucial, the rate of LRNe is comparable to that of core-collapse supernovae (Karambelkar et al. 2023), suggesting that the largest population of LRN precursors may be found beyond our Galaxy. To address this, we apply a similar methodology to catalog extragalactic YSGs using archival HST photometry of galaxies within 20 Mpc. If a future transient alert matches an object in this catalog, it could be flagged as a potential precursor and prioritized for follow-up observations. This approach o ff ers a practical strategy to sift through the vast number of LSST alerts and conduct the first in-depth study of the processes driving LRN precursor brightening. \nThe paper is structured as follows: in Section 2 we define the data and methods used to select the samples and remove contaminants. Section 3 presents a statistical description of the selected sample and the di ff erent sources of contamination. We crossmatched this catalogue to LRN progenitors from the literature, to the Transient Name Server 1 (TNS) and to time-domain surveys in Section 4, showcasing some individual precursor candidates. These results are discussed in Section 5 and compared to previous searches for YSGs and LRN progenitors. We conclude and give some guidance on how to use this catalogue in Section 6.', '2.1. Selection criteria': 'To identify progenitors of transients produced by massive stars, we need a YSG sample complete down to 8 M ⊙ . This completeness is key for future population studies. According to MESA (Modules for Experiments in Stellar Astrophysics, Paxton et al. 2011) stellar models, an 8 M ⊙ star crossing the Hertzsprung gap has an absolute magnitude of MF 814 W ∼ -5. HST can detect such a star up to 20 Mpc away with a 1-hour exposure ( mlim , F 814 W ∼ 27 . 5). Shorter typical exposures (e.g., 600s) only detect YSGs down to 16 M ⊙ at this distance. Thus, our sample is limited to galaxies within 20 Mpc with deep HST exposures in several bands, allowing us to conduct a census of stars in the Local Universe up to the Virgo cluster, and ensuring a minimum completeness limit of 16 M ⊙ to cover the Hertzsprung gap. \nTo get a complete sample of nearby galaxies, we use the HECATE catalogue of galaxies (Heraklion Extragalactic CATaloguE, Kovlakas et al. 2021), based on the HyperLEDA database (Paturel et al. 2003) but supplemented with robust distance estimates, galaxy mass, metallicity and star formation rate. The base sample, containing 1883 galaxies closer than 20 Mpc, is crossmatched with HST observations available on the Mikulski Archive for Space Telescopes (MAST), to find galaxies with overlapping deep ( t exp > 300 s) exposures in two optical filters 2 . \nFig. 1. Distribution of some properties of the sample of 575 HST-observed galaxies. (Top) Semi-major-axis a and distance from HECATE. (Bottom) Metallicity and Galactic extinction (see text for details). \n<!-- image --> \nOnly galaxies having at least 2% of their area covered by HST are considered, discarding de facto the Magellanic Clouds 3 . The list of selected filters is detailed in Table 1, together with the number of galaxies selected for each filter. We select the F 475 W , F 555 W , F 606 W and F 814 W filters because they have responses close to the Johnson-Cousins V ( F 475 W , F 555 W , F 606 W ) and I ( F 814 W ) filters, allowing us to study V against V -I colourmagnitude diagrams (CMD). The resulting HST-observed sample contains 575 galaxies. The distributions of distance and angular sizes are shown in the top panel of Figure 1.', '2.2. HST data retrieval': "To retrieve HST sources, we use two databases comprising thirdparty catalogues in which source extraction has already been performed. The third release of the Hubble Source Catalog (HSCv3, Whitmore et al. 2016) includes Advanced Camera for Surveys (ACS) and Wide-Field Camera 3 (WFC3) data that were public as of 2017 October 1. As such, it contains data for 449 of our galaxies observed before this date. HSCv3 data were retrieved using MAST queries with the CasJobs API 4 . For 118 additional galaxies, whose observation requirements are met only after 2017, we retrieve their source catalogues from MAST. At the time of the writing, eight galaxies could not be retrieved in \nFig. 2. Density of HST sources under study for the galaxy M31, as retrieved through HSCv3 and MAST databases. The background image is from the Digitized Sky Survey 2 (DSS2, Lasker et al. 1996). \n<!-- image --> \nthese databases, due to proprietary exposures or missing source catalogues. As an example, the southern region of M31, which has sparse coverage in HSCv3, is included in the MAST retrieval. This leads to uniform coverage of the inner galaxy, as shown in Figure 2). \nThe MAST third party catalogues were produced at the time of the data reduction of the corresponding observations, using either DAOPHOT (Stetson 1987), Sextractor (Bertin & Arnouts 1996), or the Hubble Advanced Products pipeline (Tran et al. 2020). For both types of catalogues, magnitudes are retrieved in the large aperture photometry ( MagAper2 ). We made sure that MAST catalogues are reliable and have well calibrated photometry, by comparing their sky coverage and their CMD with HSCv3 data for two test galaxies (NGC 45 and ESO 209-9). In all tested filters, the bias and standard deviation between HSCv3 and MAST photometry is < 0 . 05 mag and < 0 . 2 mag, respectively. Once downloaded, all catalogues of a given galaxy are merged and crossmatched to obtain a single 'master' catalogue, using astropy v5.3.3 (Astropy Collaboration et al. 2013) 5 . To discard compact star clusters, cosmic rays and sources a ff ected by confusion, we discarded sources with concentration indexes (CI) < 0 . 85 or > 1 . 5 or magnitude errors > 0 . 1.", '2.3.1. Stellar evolution tracks': "To analyse the CMD of galaxies, a physical reference is needed, such as isochrones or stellar evolution tracks in the observer's frame. However, the shape of an observed CMD heavily depends on intrinsic and extrinsic parameters. Intrinsic parameters include the galaxy star formation history and its metallicity, while extrinsic parameters correspond to measurement uncertainties and the extinction along the line of sight, commonly modelled with the extinction parameter A λ . Using the synthetic \nTable 1. Summary of HST Filters and Galaxy Selection. \nNotes. HST filters used in this study. The last column gives the number of galaxies selected in the corresponding filters. \nTable 2. MIST parameter grid. \nphotometry of stellar evolution tracks provided by MIST (MESA Isochrones & Stellar Tracks 6 , Dotter 2016; Choi et al. 2016), we generated tracks for a grid of zero-age main sequence (ZAMS) stellar masses, extinctions, and metallicities, summarised in Table 2. The ranges of extinctions and metallicities correspond to the ranges of estimated Galactic extinctions and metallicities of galaxies in our sample, and the mass range encompasses most of known LRNe progenitors (e.g. MacLeod et al. 2022). The upper limit of the mass range does not restrict the YSG selection; instead, it marks the point beyond which stellar masses become poorly estimated, as the Hertzsprung gap can no longer be consistently defined as a continuous post-MS phase. It is important to note that our selection will contain YSG regardless of their stellar evolution phase, either post-MS or post-RSG. While some YSG have been identified as post-RSG in recent studies (e.g. Humphreys et al. 2023), we cannot distinguish between them in our catalog. However, post-RSG YSG typically show only a fast, low-amplitude variability, therefore we expect a stable behaviour in future LSST observations. \nTo estimate extinctions along the line-of-sight for each HECATE galaxy, we use the 2D dust map of Delchambre et al. (2023) released as part of Gaia DR3. Metallicities and galaxy masses are provided in HECATE for only 15% (276) and 62% (1159) of our base sample, respectively. To estimate metallicities, we therefore relied on the absolute B magnitude - oxygen abundance relations of Pilyugin et al. (2004) (Equations 12 and 15 for spiral and irregular galaxies, respectively 7 ): 12 + [ O / H ] = min(5 . 8 -0 . 139 B abs , 6 . 93 -0 . 079 B abs). Gas-phase oxygen abundances were converted to metallicity using the relation Z = [ M / H ] = 12 + [ O / H ] -8 . 69, where 8.69 is the solar oxygen abundance given in Asplund et al. (2009). The resulting distributions of Galactic extinction and metallicity are shown in Figure 1 (bottom panel). In the following, all CMDs are in the Vega magnitude system, aperture-corrected (using the table 8 rec- \nFig. 3. An example of CMD analysis. Extinction-corrected CMD of IC 1613 sources with MIST tracks for a metalllicity Z = -1.2 and an extinction of AV = 0.2. Hertzsprung gap candidates are selected above blue lines. The dotted contour shows the Gaussian mixture model representing the data. \n<!-- image --> \nommended by the HSCv3 documentation) and de-reddened of Galactic extinction.", '2.3.2. HR gap definition': 'The Hertzsprung gap of each galaxy is identified using the MIST tracks at di ff erent masses. Specifically, we define this gap as the time interval between two points, the post-MS bright turno ff (the point when the V -band luminosity first reaches 95% of its postMSbright turno ff value), and the pre-RGB faint turno ff , as illustrated in Figure 3 by the left-hand and right-hand solid blue lines. To account for the internal extinction of the galaxy and retrieve extinguished YSG candidates, we artificially applied a 0.4 mag extinction to the red-side selection cut, using a relative visibility value of R ( V ) = 3 . 1. Although this additional extinction is arbitrary, it ensures the retrieval of most extinguished candidates, as discussed in Section 2.4. YSG candidates are searched between these lines and above the MIST track of a reference ZAMS star mass, which ranges from 3 to 16 M ⊙ depending on the galaxy. To ensure the completeness of the selected samples at this reference mass, it is chosen to be the smallest mass for which the faint end of the Hertzsprung gap is well-observed, i.e., significantly brighter than the sensitivity of HST exposures. Consequently, we exclude 102 galaxies where the available HST data are too shallow to accurately detect YSGs with masses of 16 M ⊙ or below.', '2.3.3. Statistical modelling': 'Due to the significant uncertainties in the estimate of distance, metallicity, Galactic extinction, HST photometry of faint sources, and stellar evolution models, a mismatch between the true locus of the observed Hertzsprung gap and its MIST estimate is inevitable for some galaxies. In particular, some galaxies have part of their MS and / or RGB intersecting the selected area of the CMD. To exclude these densely populated regions, we adopt a similar approach as the YSG selection method of Addison et al. (2022). The CMD of each galaxy is statistically modelled as a mixture of gaussian components (Gaussian Mixture, GM). The number of components is determined based on the logarithm of the number of sources, balancing the need to avoid overfitting in sparse CMDs while allowing for the fitting of com- \nplex structures in dense CMDs. To better retrieve the tails of the distribution (i.e. CMD features such as the RGB), the faintest and most populated regions of dense CMDs, as estimated by Kernel Density Estimation (KDE), were artificially depopulated by a factor of up to 20. This step only removes sources in faint regions and not the YSG selection area. \nTo exclude the dense regions of the CMD, sources located within the GM contour of a reference likelihood are discarded. This contour intersects the Hertzsprung gap of the reference MIST track, with the likelihood value optimised through visual inspection. The resulting contour for IC 1613 is shown by the dotted line in Figure 3, where the reference MIST track corresponds to a 3M ⊙ star. The red giant branch is well excluded from the YSG selection area. A representative subset of the CMDs analysed in this study is shown in Appendix A. This statistical modelling requires a minimum of 100 sources to provide a reliable CMD match. Therefore, we discarded 96 galaxies, leaving 369 galaxies in our sample.', '2.4.1. Removing known contaminants': "Di ff erent types of contaminants may be included at this stage of the selection. First, the line of sight chance alignment of Milky Way stars, notably yellow dwarfs and white dwarfs, with the galaxy under consideration can cause them to appear within the Hertzsprung gap of its CMD. Likewise, background objects such as quasars can also appear in this locus. Second, the extinction a ff ecting main-sequence stars inside each galaxy can displace them from the MS to the Hertzsprung gap in the observed CMD. Additionally, objects other than YSG also naturally appear in the Hertzsprung gap, such as cepheids and other variables within the instability strip, luminous blue variables (LBV), or unresolved globular clusters (e.g. Kraft 1966; Justham et al. 2014; Mora et al. 2007). \nAs an initial cleaning process, we crossmatched between our sample and the Simbad 9 and Gaia DR3 (Gaia Collaboration et al. 2023b) catalogues using a matching radius of 0.5'. Simbad quasars, globular clusters, and classical cepheids were discarded. Additional quasars were found and excluded using the Milliquas (Flesch 2021) and Gaia Extragalactic (Gaia Collaboration et al. 2023a) catalogues. However, we flagged known LBVs, YSGs and red supergiants (RSG) for further consideration within the sample. Sources exhibiting significant Gaia proper motion (PM / PMerr > 4) are identified as Galactic and discarded. Although other types of contaminants cannot be directly excluded, their presence can be quantified statistically.", '2.4.2. Quantifying foreground contamination': "To estimate the line of sight foreground contamination towards each galaxy, we utilised the TRILEGAL (TRIdimensional modeL of thE GALaxy, Girardi et al. 2005) simulation, which models the Milky Way's stellar content, incorporating the Sun's position and various Galactic components. It includes the thin disc, thick disc, halo, and bulge, along with their star formation histories. This simulation, notably used by Dal Tio et al. (2022) to simulate the Milky Way as observed by the upcoming LSST survey, and accessible through an API from NOIRLab 10 , simulates the photometry and stellar parameters of the observed \nFig. 4. Illustration of the estimation of foreground contamination rate using the TRILEGAL Milky Way simulations. The CMD shows TRILEGAL sources overlapping with the footprint of M83 HSC sources. Highlighted in red are the YSG candidates identified within this sample. TRILEGAL sources falling within the black polygon, circled in red, are identified as foreground contaminants. They represent 3.1% of the M83 YSG candidates. \n<!-- image --> \nGalactic population along designated lines of sight. Patches of 0.25 deg 2 were queried for each galaxy in our sample. Given that LSST's coverage is mostly restricted to the Southern Hemisphere (Dec < 0), some galaxies lie outside the simulated regions. For cases where LSST covers the symmetric position relative to the Sun-Galactic poles plane, we used this position instead, assuming similar results due to the symmetric nature of TRILEGAL components. This planar symmetry was verified using test coordinates through the TRILEGAL web interface 11 . For the 30% of galaxies (106 in total) still uncovered, we utilised the TRILEGAL web interface with the same parameters as Dal Tio et al. (2022). Notably, the halo profile in the web interface di ff ers from the LSST simulation, employing a r -1 / 4 profile versus Dal Tio et al.'s r -1 / 2 . 5 profile. We adjusted for this by approximating the profile with r -1 / 4 , Ω= 0.0025 M ⊙ pc -3 and rh = 5700 pc, based on comparable star counts and magnitude distributions. \nTRILEGAL sources matching the footprint of the selected HST exposures and exhibiting synthetic HST photometry within the specific Hertzsprung gap selection region are identified as contaminants. Figure 4 illustrates this method for M83, showing the CMD of TRILEGAL sources overlapping with HST exposures and overlaying the observed YSG candidates. The results of this foreground contamination analysis are presented in Section 3.2.", '2.4.3. Extinguished main-sequence stars': 'Although we account for the Galactic extinction along the line of sight for each galaxy using the appropriate MIST model, intrinsic extinction inside each galaxy is not modelled at this stage. Consequently, a significant fraction of Hertzsprung gap sources may actually be extinguished MS stars. To quantify this, we assigned to each candidate a probability to be an extinguished main-sequence star. Assuming a normal distribution for extinction, we measured extinction scatter by evaluating the standard deviation of AV in a thin region of the CMD, chosen to be the red giant branch, as illustrated in Figure 5. This standard devi- \nFig. 6. Distribution of the standard deviations of extinction for our galaxy sample. \n<!-- image --> \n<!-- image --> \nFig. 5. (Top) Density plot of the RGB of IC 1613 in the (Magnitude relative AV ) plane. The x-axis zeropoint corresponds to the mean of the distribution. The top curve shows the KDE of AV values, with a standard deviation of ± 0.21 (dotted lines). (Bottom) Resulting probability map to be an extinguished MS star in the selection region of YSG sources in the IC 1613 galaxy. \n<!-- image --> \nation is used to model the additional extinction in the selection region, resulting in a probability map for each CMD (Figure 5, bottom panel). This probability is naturally highest at the blue edge of the selection region, which separates the main-sequence region from the Hertzsprung gap. By averaging these probabilities on a per-galaxy basis, we can determine the contamination rate due to extinction. The distribution of AV scatters estimated for all selected galaxies is shown in Figure 6. Besides providing contamination probabilities, it also suggests that the 0.4 mag extinction applied to the red edge of the selection region is sufficient to recover most of the extinguished YSG (Section 2.3.2). As a result, some RSGs are included in the sample, as discussed in Section 5.1 for the case of M31. However, obscured YSGs and RSGs have also been identified as progenitors of certain infrared transients thought to share similarities with LRNe (Jencson et al. 2019).', '3.1. Selected candidates': 'Our analysis identified a total of 154,494 YSG candidates before cleaning, a portion of which being listed in Table 3. The cumulative distribution of their distances is depicted in Figure 7, indicating that about 25% of these candidates are located in galaxies at approximately 0.8 Mpc. These candidates are actually located in M31 (40,566 candidates) and M33 (11,603), the two galaxies with most candidates. The three next galaxies with the most YSG candidates are M101 (8,885), NGC 300 (7,525), and NGC 253 (4,000). \nTable 3. List of YSG candidates.Notes. Column 2: host galaxy. Columns 3-5: HST coordinates (deg) and magnitude (mag). Columns 6-7: MIST inferred stellar parameters. Column 8: Probability to be an extinguished MS star. The complete list will be made available in electronic form. \nAs a result of the CMD analysis, we identified YSG candidates in 353 galaxies. The HST data provenance, available blue / green filter, coverage, and the parameters used for fitting the CMD as described in Section 2.3 are provided for each galaxy in Table 4. This table also details the number of candidates for each galaxy and the estimated contribution from di ff erent contaminants. \nYSG candidates exhibit apparent magnitudes in the F 814 W band ranging from 16 to 25, with a peak at approximately 22.2 (Figure 8). Single-star ZAMS stellar masses, luminosities, and temperatures for these candidates were estimated by interpolation using the MIST stellar evolutionary tracks within the Hertzsprung gap. The distribution of stellar masses, shown in Figure 8, appears bimodal: a lower-mass peak, around approximately 5 M ⊙ , primarily corresponds to candidates in M31, M33 and NGC 253, and a higher-mass peak, around approximately 10 M ⊙ , corresponds to more distant candidates. This bimodality is further illustrated in Figure 9, showing the tight correlation between interpolated stellar masses and F 814 W absolute magnitude. \nTable 4. Galaxies having more than 400 YSG candidates, sorted by distance. \nNotes. Column 2: distance to the galaxy. Columns 3-6: information on their HST dataset. Columns 7-10: parameters used to fit their CMD. Column 11: number of resulting candidates. Columns 12-13: contamination estimates. The complete list of galaxies having YSG candidates will be made available in electronic form. \nFig. 7. Cumulative distribution of distances for YSG candidates before and after cleaning, and estimated number of extinguished mainsequence stars. \n<!-- image -->', '3.2. Contamination fraction': 'To evaluate the contamination fraction from intrinsic extinction and foreground sources, we considered various types of contaminants. Foreground contamination was quantified using the TRILEGAL star counts. According to the method detailed in Section 2.4, approximately 5% of the sources in our YSG sample are expected to be foreground contaminants (including sources having high Gaia proper motion). The foreground contamination mainly a ff ects galaxies at low Galactic latitudes ( | b | < 25º) and 75% of the galaxies have less than 10% of their YSG candidates being foreground, as shown in Figure 10 (right panel). As illustrated in this Figure, galaxies closer to the Galactic plane or toward the Galactic center exhibit higher contamination fractions. After using Gaia proper motions to clean the sample, removing 6,623 sources (4.3% of the sample), we therefore estimate the remaining foreground contamination to be ≲ 1%. \nAdditionally, the probability of each YSG candidate to be an extinguished main-sequence star was computed using the mod- \nFig. 8. Distributions of observed F 814 W magnitude and stellar masses of YSG candidates. \n<!-- image --> \nFig. 9. MIST inferred ZAMS stellar masses of YSG candidates as a function of their absolute F 814 W magnitude. \n<!-- image --> \nelled intrinsic extinction (Section 2.4). The mean probability for all sources in our YSG catalogue is 0.25, with the fraction of contaminants remaining constant across all distances (Figure 7). As expected, this probability is significantly influenced by the temperature of the source, and the filter used in the CMD analysis, as depicted in Figure 11. \nFinally, contaminants such as quasars, Cepheids and luminous blue variables were identified and removed using Gaia and Simbad databases. These known contaminants represent a minority of the sample across all distances (Figure 7). These sources are typically brighter in both observed and absolute magnitudes, as shown in Figure 12. The clean sample of YSG contains 146,502 sources.', '4.1. Completeness for known LRN progenitors': 'Many extragalactic LRNe discovered during the last decade had pre-outburst images taken several years before the transient (e.g. MacLeod et al. 2022 and references therein). For eight of them, listed in Table 5, their host galaxy is in our sample, allowing us to assess the rate of retrieved progenitors in our sample of YSG candidates. Three out of eight progenitors are missing in our sample. For AT2015dl, the transient is located outside the \n<!-- image --> \n2 \nFig. 10. (Left) Foreground contamination rate as a function of Galactic latitude, using the TRILEGAL Milky Way simulations. The colour encodes the Galactic longitude. To maintain readability, only galaxies with a minimum of 100 YSG candidates are included. (Right) Empirical cumulative distribution function (ECDF) of the foreground contamination rate. 75% of galaxies have less than 10% of their YSG candidates being foreground contaminants.Fig. 11. Probability of YSG candidates to be an extinguished MS star as a function of their MIST temperature. Three example galaxies are shown, illustrating the impact of the HST blue / green filter used to analyse the CMD. \n<!-- image --> \nFig. 12. Fraction of identified contaminants using Simbad and Gaia catalogues, as a function of observed and absolute F 814 W magntiude. \n<!-- image --> \nTable 5. Retrieval status of known LRN progenitors in the YSG sample. \nNotes. LRN progenitors identified in the literature, and their retrieval status in our YSG sample. References: [1] MacLeod et al. 2017, [2] Pastorello et al. 2021a, [3] Pastorello et al. 2021b, [4] Blagorodnova et al. 2017, [5] Blagorodnova et al. 2021, [6] Cai et al. 2022 [7] Pastorello et al. 2023, [8] Smith et al. 2016a. \nTable 6. YSG candidates matching TNS objects. \nNotes. The last column shows the type according to TNS with eventual corrections (V*: variable star). Objects are sorted by TNS name. \nfootprint of the HST observations. For NGC4490-OT, although the progenitor was measured and studied by Smith et al. (2016b), its apparent magnitude of m F606W = 23 . 58 ± 0.24 was too faint to be retrieved as an HSCv3 source (the faintest retrieved sources in a 30 arcsec radius circle around the progenitor position were 23.2 mag). For AT2019zhd, the progenitor had a low luminosity ( M F555W = 0 . 21 ± 0 . 14, Pastorello et al. 2021a), placing it in the most populated region of the Hertzsprung gap. Consequently, it could not be selected after the Gaussian mixture cut. Moreover, its proximity to the edge of the field of view prevents this source from being retrieved as a MAST-catalogued source, and its magnitude error would either way exclude it from our sample. Considering these results, our sample appears to represent a relatively comprehensive selection of LRN progenitors, with a ∼ 70% completeness level in regions covered by HST.', '4.2. Progenitors of past and ongoing transients: crossmatch to TNS': "Our YSG sample can be used to find possible progenitors for past transients and further analyse them. To this end, we crossmatched our YSG candidates to the TNS public objects as of 2024 August 31st, with a match radius of 0.6'. The resulting \nFig. 13. HSTcutouts of the 12 candidates resulting from our TNS crossmatch. Images are 10' side-to-side, with a circle of 1' radius pinpointing the location of the YSG source. Adapted from HLA colour composites. \n<!-- image --> \nlist of 24 transients is detailed in Table 6. About half of these transients are already classified and some are well-studied in the literature (e.g. SN2016bau: Aryan et al. 2021. SN2024ggi: Pessi et al. 2024; Jacobson-Galán et al. 2024). For those, our HST matches may provide useful pre-outburst luminosity levels. The TNS report of AT2019ejn mentions multiple discoveries of this transient, suggesting it may be an LBV. For three other transients (AT2020aaqy, AT2021tmm and AT2021ytf), the F 814 W magnitude is brighter than the TNS discovery magnitude, making them likely to also be variable stars. For nine other transients, the type was not straightforwardly identified in TNS and they deserve a detailed analysis of their light curves and HST progenitors. Therefore, we inspected HST colour images for every candidate, using the Hubble Legacy Archive (HLA) website 12 . Cutouts of 10 arcsec side-to-side are displayed in Figure 13 and the position of the transient is marked by a circle. \nTo inspect the temporal evolution of the candidates, we obtained forced photometry for the ZTF public data using the online service ZTF Forced Photometry Service (ZFPS) (Masci et al. 2023). For every candidate, we retrieved all the data starting from the beginning of the public survey in March 2018 (around MJD 58178). We generally used the di ff erence imaging flux obtained during the first year of operations (or a period with low residual flux) as a baseline to calibrate the di ff erence imaging magnitudes. We applied the quality cuts recommended in the documentation and we imposed forcediffimchisq (Reduced chi-square in PSF-fit) < 1 . 3. Data points with S / N > 3 were considered as detections and the upper limits are reported with a 5 σ threshold. To increase the S / N for these faint objects, we binned the fluxes using a 15-day bin size. The resulting light curves are shown in Figure 14. In the following, we report the results of a detailed analysis aimed at identifying potential LRN precursors. \nAT2018mmb : this transient is located at 0.55' from a faint ( mF 555 W = 22.78, mF 814 W = 21.68) YSG in our sample, inside a \nFig. 14. ZTF forced photometry light curves of the 9 candidates resulting from our TNS cross-match. The blue areas indicate the period that was used to set the baseline flux at the location of these transients. \n<!-- image --> \nstellar cluster. However, it is in a crowded region, surrounded by many sources that may cause source confusion (CI = 1.4), and a significantly brighter source ( mF 555 W = 19.07, mF 814 W = 19.38) is located at just 0.37' from the transient. The position of this bright source in the CMD is close the locus of the MS. Although this neighbour is the most likely progenitor of the transient, its recent brightening in 2024 (reaching an absolute magnitude of -9) makes it an interesting candidate for followup. \nAT2019krl : this transient matches a YSG candidate with a separation below 0.1' and a stellar mass ∼ 10 . 3 M ⊙ . Image inspection shows that the YSG is devoid of any bright source in its immediate vicinity. AT2019krl was spectroscopically classified as SN IIn or LBV outburst by The Astronomer's Telegram, No. 12913. However, the absolute magnitude peak at -8.5 is several magnitudes fainter than most SN IIn. It was detected by HST at an absolute magnitude of -6, translating to a ZAMS stellar mass of ∼ 10 M ⊙ , less massive than most LBV. Still, their measured H α line velocity ( ∼ 2000 km / s) rules out a LRN-related mechanism. \nAT2020adbp : this sources matches a YSG in M31 observed by HST in January 2023, with mF 814 W = 18.7. This observation occurred after the transient report on TNS, explaining the bright HST magnitude. There is no pre-brightening image available in the HST archive, which precludes the study of the progenitor. Source confusion is unlikely for this source having CI = 1.1 in both F 814 W and F 475 W filters. Followup of this source is encouraged. \nAT2021ahsv : this transient matches a YSG candidate located at 0.19' with M ∗ ∼ 12 M ⊙ . It is brighter than every close-by source. Standing in the middle of the Hertzsprung gap in the CMD, this candidate progenitor shows an interesting light curve \nwith a slow brightening. Followup of this source is encouraged, as it has showed a steady brightening since 2020. \nAT2022llt : this transient matches a > 20 M ⊙ YSG with a separation of 0.06'. Although ZTF forced photometry does not reveal any long-term behavior, the ATLAS light curve shows a slow brightening of the source in recent years (Figure 16, topmiddle panel). Image inspection shows that source confusion is unlikely for this source (also supported by its CI = 1.05). Followup of this source is encouraged. \nAT2023azz : this transient is located at 0.51' from a ∼ 9 . 6 M ⊙ YSG candidate in M101, according to its TNS coordinates. The ZTF coordinates, taking advantage of a large number of epochs, are even more precise and point to a separation of 0.2'. Image inspection reveals a faint, red source close to a brighter one (at 0.75'). However, confusion may a ff ect the photometry of this source, given the average CI of 1.4. Besides, its location on the CMD makes it also compatible with a RSG. Followup of this object is still encouraged. \nAT2023wgz : this transient has a massive ( > 20 M ⊙ ) YSG counterpart at a 0.58' separation. Its photometry may be moderately a ff ected by source confusion, with a CI of 1.4 in both F 814 W and F 555 W filters. Otherwise, it is not surrounded by any source of similar or greater luminosity. Its light curve shows only a few datapoints, with no detection since 2021. \nAT2023wot : this source matches a ∼ 4 M ⊙ YSG in M31 with a 0.58' separation. Although it is on the faint end of our M31 selection, image inspection shows that it is not surrounded by any source of similar or greater luminosity, ruling out source confusion (CI = 1.13). The Astronomer's Telegram, No. 16319 suggest it to be a nova (Hornoch et al. 2023), which is in agreement with the absolute magnitude at peak of the forced photometry light \ncurve ( ∼ -5 . 3, Figure 14). Such a rapid outburst also supports a nova-like behavior. \nAT2024ikg : this recent transient is found to match a bright ( mF 555 W = 21.73), massive ( ∼ 16 M ⊙ ) YSG candidate located at 0.33' in M101. Querying the HSCv3 detailed catalogue reveals that it was observed in Jan 2003 and Oct 2013, the only repeated filter being F 814 W and F 435 W . The source brightened from 22.06 to 21.65 ( F 814 W ) and 22.62 to 21.35 (F435W) in this interval. Inspection of the image reveals a bright RSG located just 0.25' away from the YSG, but at fainter magnitude ( mF 555 W = 23.05). Besides, the CI in both F 606 W and F 814 W is measured between 0.95 and 1.2 in both epochs, making source confusion unlikely. Followup of this object is encouraged.", '4.3. Other precursor candidates': "LRN precursors are expected to rise by only few magnitudes in several years (e.g. Blagorodnova et al. 2020). In this context, they may not meet the criteria to be reported to TNS (criteria may vary depending on the collaboration) until long after their detectability has begun. In order to retrieve them, we create a Lasair 13 (Smith et al. 2019) watchlist to match our YSG catalogue to all ZTF transient alerts with a matching radius of 0.6', and a Lasair filter to keep only objects with at least 2 positive detections ( ncandgp ≥ 2). \nTo exclude variable stars, we inspected the ZTF light curve of each of the resulting 67 sources using the AlerCE broker (Förster et al. 2021), discarding all sources showing magnitudes in the past at levels brighter or similar to present levels. We used the ATLAS forced photometry tool 14 on di ff erence images to query candidate precursors and confirm the brightening trend on longer timescales. At this stage, we obtained nine precursor candidates consistently brightening over the last few years, including four objects previously analysed in Section 4.2 (AT2020adbp: ZTF20abhtvor, AT2021ahsv: ZTF20abbetli, AT2023azz: ZTF23aaazair, AT2024ikg: ZTF24aalfiak). \nTo obtain photometry for Southern candidates, we crossmatched YSG candidates to MeerLICHT and BlackGEM transients having at least 2 detections, at reasonable signal-tonoise ratio (S / N > 6 . 5) and high probability to be real (rather than bogus) class\\_real > 0 . 8. The matching radius used was 1'. For BlackGEM and MeerLICHT, we obtained 10 and 33 matches, respectively. The large majority of them are variable stars with no clear brightening trend. For three of them, however, a brightening is identified and confirmed with ATLAS forced photometry: MLT28037995, in NGC 300, MLT15613547, in NGC 55, and MLT17180523 in NGC 253, actually corresponding to AT2022llt. The complete list of twelve (ZTF + MeerLICHT + BlackGEM) precursor candidates is detailed in Table 7. Their HST cutouts are presented in Figure 15 and their multi-survey light curves are shown in Figure 16. We fitted a slope to each light curve, in order to quantify the brightening trend. Slope values range between 2 . 1 × 10 -4 and 1 . 2 × 10 -3 mag day -1 , with an average error of 1 . 3 × 10 -4 mag day -1 . In comparison, the brightening of the precursors of M31-LRN2015 and M101 OT2015-1 were respectively of 3 mags over 2 years and 1.5 mags over 6 years (Blagorodnova et al. 2017, 2020), corresponding to slopes of ∼ 4 × 10 -3 mag day -1 and ∼ 7 × 10 -4 mag day -1 , respectively. \nFig. 15. HST cutouts of precursor candidates. Images are 10' side-toside with a 1' radius circle centered on the HST position. \n<!-- image --> \nTable 7. Precursor candidates identified in this study. \nNotes. Precursor candidates (sorted by galaxy name) identified in ZTF, MeerLICHT, or BlackGEM. The last column gives the slope of the bestfit linear trend to the ATLAS light curve.", '5.1. Comparison of our YSG candidates to spectroscopic samples': "The literature provides a comprehensive representation of the brightest stars in M31 and M33, as well as their nature.To assess the validity of our method and further investigate the completeness and reliability of our YSG catalogue, we compare our sample to the spectroscopic YSG and RSG samples in M31 and M33 from Drout et al. (2009), Drout et al. (2012), Gordon et al. (2016), and Massey et al. (2016). In M31, within the CMD re- \nFig. 16. ATLAS forced photometry light curves of the best 12 precursor candidates in ZTF, MeerLICHT and BlackGEM. ATLAS photometry is represented by orange and cyan circles ( o and c bands), ZTF photometry by red and green diamonds ( r and g bands) and MeerLICHT / BlackGEM photometry by black and brown triangles ( q and i bands). ATLAS light curves were rebinned to 60-day bins, other surveys to 15-day bins. \n<!-- image --> \ngion used to select YSGs, Drout et al. (2009) identified 120 probable YSGs and 2772 foreground dwarfs. This identification was achieved by comparing the radial velocity obtained from their spectra with the expected radial velocity at each position in M31, considering its peculiar velocity and rotation curve. Similarly, in M33, they identified 135 YSGs and 781 dwarfs (Drout et al. 2012). In a redder and less luminous region of the CMD, they identified 204 probable RSGs and 204 foreground dwarfs. The studies by Gordon et al. (2016) and Massey et al. (2016) refined these samples by identifying spectral types and members through their spectroscopic campaigns, including the extensive Local Group Galaxy Survey (LGGS, > 1800 spectra). \nConversely, we used Gaia proper motions to eliminate foreground contaminants. To evaluate the e ff ectiveness of this method, we crossmatched the samples from Drout (2009, 2012) and Gordon (2016) with Gaia DR3. For a minority of classified dwarfs, proper motion data is unavailable, constituting an anomaly. These stars are removed from the sample. The distribution of proper motion S / N is shown in Figure 17, categorised \nby the type of star they identified. The vast majority (98.5%) of foreground sources are correctly eliminated using Gaia 's proper motions. Additionally, 96.9% of the eliminated sources are indeed foreground stars. Gaia proper motions prove to be an e ff ective method for detecting foreground stars, comparable to spectroscopy or radial velocity methods. \nComparing the samples from Drout (2009, 2012) and Gordon (2016) to our sample, we obtain 931 associations (separation less than 0.6') for classified stars. The foreground contamination in this magnitude range is substantial, with approximately 89% of selected stars being foreground stars. After our cleaning, 96% of them are eliminated, and only 4% of M31's YSGs are removed. Therefore, the completeness of our sample is preserved, leaving only 23% residual contamination in this bright sample (typically V < 19). At fainter magnitudes, contamination is less significant. In M33, our method performs equally good. The precleaning contamination is 56% (31 out of 56), and none of these 56 stars remain after filtering out high proper motions. \nRSGs represent another type of contaminant in our sample. This is a direct consequence of the selected regions in the CMD, which include a portion of the RGB to avoid missing YSGs affected by extinction. In literature samples, the selection region for RSGs is at a lower magnitude than for YSGs in the CMD, complicating the estimation of the fraction of RSG contaminants. At V < 18 . 5, however, our selection includes 44 YSGs and 14 RSGs, resulting in a contamination rate of 24%. \nIn addition to using proper motions, variability is an independent criterion that helps distinguishing a YSG from a foreground star. For example, massive YSGs have been known to vary erratically compared to low-mass dwarfs. The bottom panel of Figure 17 shows the distribution of the Gaia -measured dGmag , the di ff erence between brightest and faintest G -band magnitude observed during the course of the survey. Interestingly, stars classified as foreground but with a low PM exhibit variability on average 6 times greater than those with proper motion data (average dGmag = 0.2 instead of 0.033, Figure 17). This suggests a misclassification of these stars, which may be actual members of M31. To test this hypothesis, we model the distribution of dGmag as a sum of two distributions: one for high-PM sources identified as foreground, and one for low-PM sources identified as YSGs and RSGs. We can then estimate the contribution of YSGs and RSGs within low-PM foreground-classified stars. The fraction of these candidate members is 68 ± 11%. This suggests that the residual contamination within low-PM stars with V < 18 . 5 is actually ∼ 8% and not 23%. Furthermore, if our sample were to be used to identify progenitors of variable or transient events, the expected value of dGmag would be relatively large, further reducing the probability of association with a foreground source. On the other hand, although we expect most background AGN to be excluded in our sample selection (Section 2.4), some unresolved AGN undetected in X-rays may still produce some alerts in future deep transient surveys. \nFinally, we can quantify the completeness of our YSG catalogue in di ff erent magnitude bins. Using for reference the samples of YSG candidates in M31 from Drout et al. (2009) and Gordon et al. (2016) (see the corresponding CMD in Figure 18), and applying the same proper motion cut as done in our study, we obtain the completeness estimates shown in Figure 19. While the brightest stars are not always detected in third-party HST catalogues (and surveys other than HST are well-adapted to recover those), the fraction of recovered YSGs among recovered HST sources consistently remains around 60%. The lost fraction is primarily due to sources with high CI or located in the dottedline rectangle shown in Figure 18. The latter occurs because our conservative threshold, designed to exclude the main-sequence and some extinguished main-sequence stars, also omits YSGs in the bluer part of the CMD.", '5.2. Impact of photometric uncertainties and source confusion': "Source confusion is commonplace in crowded fields and can significantly bias the photometry of HST sources, leading to incorrect star classifications. This happens when two stars, such as a main-sequence star and a red giant, are in close angular proximity. Their combined light can be mistaken for a single star, such as a YSG, due to the blended colour and luminosity. This e ff ect is another source of contamination a ff ecting our YSG catalogue. \nOne method to quantify source confusion is by using the concentration index (CI), defined as the di ff erence in magnitude between two apertures, typically a smaller aperture of 0.05' and a larger aperture of 0.15' (for ACS and WFC3 / UVIS), normalised \n<!-- image --> \nFig. 17. (Top) Distribution of the Gaia DR3 proper motion signal-tonoise for M31 and M33 stars classified in the literature. (Bottom) Distribution of the variability indicator dGmag for M31 stars. The stars classified as foreground but with a low proper motion are shown in blue. \n<!-- image --> \nFig. 18. CMD of M31 stars classified in the literature, using the photometry of Massey et al. 2016. Smaller markers without contour are stars present in HST footprint but not in HST catalogues, smaller markers with grey contour are stars in HST catalogues but not in our YSG selection, and larger markers with black contour are YSG selected candidates. The dotted lines highlight a region encompassing most of HSTdetected YSG that are missed in our sample. \n<!-- image --> \nsuch that its distribution peaks at 1. Objects with CI values around 1.0 are likely to be stars, while those with significantly higher CI values are likely to be extended sources or the confusion of several stars. Figure 20 presents the distribution of YSG candidates in the CI mF 814 W plane. At bright magnitudes, the CI is close to 1, showing that sources are point-like and source \nFig. 19. Fraction of M31 YSG candidates from the literature recovered as HST sources and HST-selected YSG candidates, as a function of magnitude. \n<!-- image --> \nconfusion is not significant. The faint sources are more numerous and thus denser on the sky, especially in distant galaxies. Therefore, they are more a ff ected by source confusion and can have CI values close to 2. Such sources are currently excluded from our sample, which is restricted to CI values < 1 . 5, thereby reducing its completeness. A future release of the catalog, incorporating DOLPHOT photometry (Dolphin 2016), is anticipated to address this issue more e ff ectively. \nSimilarly, photometric uncertainties can bias the selection of YSGs in HST observations. These uncertainties arise from various sources, such as photon noise, background subtraction errors, and instrumental e ff ects. When dealing with YSG candidates, identified based on their position in the CMD, even small errors in photometry can shift stars into or out of the YSG region. For example, in the NGC 4455 galaxy, an edge-on spiral galaxy located 7.3 Mpc away, observations were conducted using the F 814 W filter for 1030 seconds. These parameters are close to the median values of our sample of galaxies with YSG candidates. When the colour-magnitude diagram is regenerated using magnitude values and errors modelled as a normal distribution, 9% of YSG candidates shift out of the selection region. Conversely, a similar number of previously unselected stars are shifted into the YSG selection region. Likewise, the variability of YSG can cause them to move out of the selection region; however, this variability primarily occurs over short timescales and within specific regions of the HR diagram, such as the instability strip (e.g., Evans 1993). \nIn this context, our conservative selection region helps to maintain a cleaner sample, although it may exclude some legitimate YSGs. Other sources of errors, such as uncertainties in stellar models, errors in extinction, distance, or metallicity of the host, may bias the selection region to a more significant extent.", '5.3. Search for LRN Progenitors and Precursors with LSST': "In a recent paper, Strotjohann et al. (2024) explore the potential for identifying progenitor stars of core-collapse supernovae using data from ground-based wide-field surveys such as ZTF and LSST. Due to challenges like crowding and atmospheric blurring, identifying these progenitor stars in pre-explosion images is di ffi cult. Instead, the study suggests combining numerous preand post-supernova images to detect the disappearance of progenitor stars. As a proof of concept, the authors implemented this approach using ZTF data. Despite analysing hundreds of images \nFig. 20. HST photometric concentration index of YSG candidates as a function of their F 814 W magnitude. \n<!-- image --> \nand achieving limiting magnitudes of approximately 23 in the g and r bands, no progenitor stars or long-lived outbursts were detected for 29 supernovae within a redshift of z ≤ 0.01. The sensitivity limits achieved were several magnitudes less than those in previously detected progenitors. \nConversely, the study projects that LSST, over its 10-year survey, could detect around 50 red supergiant progenitors and several yellow and blue supergiants. It estimates that progenitors of Type Ic supernovae would be detectable if they are brighter than -4.0 magnitudes in the LSST i band, respectively. Given their similar spectral type (A5) compared to YSGs, we assume a similar performance for YSG detection: we therefore expect that 77% of our YSG sample will be redetected by LSST, for sources having MF 814 W < -4 (or M ∗ > 6 M ⊙ , Figure 9). This will provide exquisite variability constraints for sources of r < 23, allowing us to identify new LRN precursors, given LSST repeated observations and the long timescales of these events. Such early precursor discoveries will be essential to schedule precursor spectroscopy, allowing us to probe the line profiles, outflow structure and density (and therefore mass loss), ionization structure, and possible shocks in the system (e.g. Pejcha et al. 2016; Molnar et al. 2017; Blagorodnova et al. 2021). \nFurthermore, LSST can expand the progenitor sample for the 1235 galaxies within 20 Mpc and at Dec < 30º that are not covered by deep multiband HST exposures. In particular, it will cover more than 1200 massive galaxies ( MB < -14 to select starforming hosts), where LRNe are most likely 15 , and and whose YSGs comprise 96% of our sample. Assuming the same distance distribution of YSG candidates as the one in this study, this represents a 130% increase of our M ∗ > 6 M ⊙ sample. It is important to note that some progenitors detected only a few years before the LRN are likely already undergoing mass transfer and can be considered precursors (this is the case of e.g. AT2021blu, imaged both 15 years and 2 years prior to the transient, showing a 1-mag brightening between these two epochs Pastorello et al. 2023). Ongoing large-scale, deep surveys such as Euclid (Laureijs et al. 2011) are also of significant interest for mapping stellar populations and identifying potential transient progenitors in nearby galaxies (Bonanos et al. 2024). \nThe rate of LRNe in the luminosity range -16 ≤ Mr ≤ -11 mag was recently constrained to 7 . 8 + 6 . 5 -3 . 7 × 10 -5 Mpc -3 yr -1 (Karambelkar et al. 2023), comparable to the core-collapse SN rate (Perley et al. 2020). However, LRNe are about three magnitudes fainter, and Karambelkar et al. (2023) suggest a luminosity function in the form dN / dL ∝ L -2 . 5 . At magnitude r < 18 . 5, we thus expect LSST to detect about 50 LRNe (i.e. 1.7% of the 3300 expected bright core-collapse SNe, Strotjohann et al. 2024) over its 10-year survey. \nStrotjohann et al. (2024) also postulate how LSST will detect more than a thousand pre-supernova outbursts, depending on their brightness and duration. In the case of LRNe precursors, a system brightening from -1 to -4 mag in absolute r -band magnitude (similar to the precursor of M31-LRN2015) would be detected in a single LSST exposure up to a distance of 5 Mpc (not taking into account the e ff ects of the luminous background from the galaxy). This distance limit becomes 20 Mpc when using the final survey LSST r -band sensitivity. Considering the host galaxy's brightness, late-time precursors or those from massive stars, with Mr < -6 (Blagorodnova et al. 2020, Figures 14, 16) would have a 6% probability of detection by LSST (Strotjohann et al. 2024). If half LRNe have such a precursor, given their volumetric rate, one can expect about 100 new precursor detections in the era of LSST. The rate, luminosity function, and timing of LRN precursors will be measurable using this large dataset. This might contribute to revealing their intrinsic mechanisms. \nOverall, the cadence of LSST, its multi-band coverage, along with the depth of the survey, allows for the detection of progenitors, years-long faint precursors and variability patterns that precede LRN events. Systematic cataloging and data mining techniques will be crucial in identifying these specific observational signatures within LSST vast datasets. An approach based on the systematic use of archival data and the prediction of future variability using various light-curve analysis techniques is under study (Tranin et al., in prep.).", '6. Conclusion': 'In this study, we use HST imaging of nearby galaxies to find possible LRN progenitors and precursors, making it possible to predict their outburst and rapidly identify new transients matching the position of a candidate YSG. We retrieve the catalogues of HST sources and their photometry for 369 galaxies with distances closer than 20 Mpc, using di ff erent public databases. After building colour magnitude diagrams for each galaxy, we select the Hertzsprung gap stars using MIST stellar evolution tracks, coupled with a statistical representation of the CMD. Foreground contaminants were mostly removed using Gaia proper motions, and the rest of foreground contaminants was quantified using the TRILEGAL simulations of the Milky Way stellar content. The previous spectroscopic identification of foreground stars within the Hertzsprung gap of M31 and M33 showed excellent agreement with our method. Additionally, we constrained the number of contaminants resulting from internal extinction to less than 20% of the sample and quantified this for each source. The use of MIST stellar evolution tracks and a meticulous filtering process to exclude contaminants proved crucial in accurately identifying candidates. \nOur study identified 146,502 yellow supergiant candidates in 353 galaxies, a significant increase over previous research. The resulting sample of candidates was cross-matched with the TNSand the ongoing surveys ZTF, BlackGEM and MeerLICHT. Candidates exhibiting outstanding variability were identified and analysed. In particular, we identified 12 precursor candidates \nbased on their consistent brightening over the past few years. Their spectroscopic followup and identification will be the subject of an upcoming work. \nThe YSG catalogue resulting from this study will be released at the time of publication of this article, together with the pipeline. The Python scripts used to retrieve and analyse HST data are made publicly available at the following address: https://github.com/htranin/LRNsearch . The insights gained from this catalogue can inform future models of stellar evolution and enhance our ability to predict and study rare transient events. This work advances our capabilities to understand yellow supergiants and their role as progenitors and precursors to luminous red novae, filling a gap in previous studies. LSST will be a game changer in the quest for LRNe and their progenitors and precursors: we estimate that the 10-year survey will more than double the number of detected extragalactic YSGs within 20 Mpc, and provide excellent variability constraints for sources of magnitude r < 23. Notably, based on the rate of previous extragalactic LRNe, we expect about 100 LRN precursors to be discovered over the course of LSST, and about 50 bright r < 18 . 5 LRNe. Future research should focus on continuous monitoring of brightening YSG candidates to capture and analyse transient episodes as they occur. We emphasize the importance of closely monitoring these future transients having YSG progenitor, to ensure the identification and study of luminous red novae events and other rare transients. \nAcknowledgements. H. T. and N. B. acknowledge to be funded by the European Union (ERC, CET-3PO, 101042610). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. PJG is supported by NRF SARChI grant 111692. We thank Zeljko Ivezic and the anonymous referee for their valuable comments, which significantly enhanced the quality of this paper. We thank Rick White and Bernie Shiao for their assistance in retrieving the HSCv3 data. We acknowledge the extensive use of the MAST database to conduct this study. Based on observations with the MeerLICHT telescope. MeerLICHT is built and run by a consortium consisting of Radboud University, the University of Cape Town, the South African Astronomical Observatory, the University of Oxford, the University of Manchester and the University of Amsterdam. MeerLICHT is hosted by SAAO. Based on observations with the BlackGEM telescopes. BlackGEM is built and run by a consortium consisting of Radboud University, the Netherlands Research School for Astronomy (NOVA), and KU Leuven with additional support from Armagh Observatory and Planetarium, Durham University, Hamburg Observatory, Hebrew University, Las Cumbres Observatory, Tel Aviv University, Texas Tech University, Technical University of Denmark, University of California Davis, the University of Barcelona, the University of Manchester, University of Potsdam, the University of Valparaiso, the University of Warwick, and Weizmann Institute of science. BlackGEM is hosted and supported by ESO.', 'References': 'Addison, H., Blagorodnova, N., Groot, P. 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Tranin et al.: Quest for Hertzsprung gap stars and LRNe precursors': 'Figure A.1 presents a representative subset of CMDs analysed in this study. The selection area of YSG candidates is shown by solid blue lines. \n<!-- image --> \n<!-- image --> \nIC 2574,Z=-0.9 Av= 0.2,ACS \n<!-- image --> \n<!-- image --> \nM66,Z=-0.6 Av= 0.2,WFC3 \nNGC 1569,Z=-0.9 Av= 0.8, ACS \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nM 33,Z=-1.2 Av= 0.6, ACS \n<!-- image --> \n<!-- image --> \nWFPC2 \n<!-- image --> \nUGCA 193 \nZ=-0.9 Av= 0.0,ACS \nFig. A.1. Representative sample of CMDs analysed in this study. The blue MIST track corresponds to the reference stellar mass at which YSG observation completeness is ensured. \n<!-- image -->', 'Appendix A: Example of CMDs': 'M 81,Z=-0.3 Av= 0.2,ACS \n<!-- image --> \nNGC 4144,Z=-0.6 Av= 0.0,ACS \n<!-- image --> \n<!-- image -->'}

2024arXiv240907625B

The properties of metric perturbations are determined in the context of an expanding Universe governed by a modified theory of gravity with a nonminimal coupling between curvature and matter. We analyse the dynamics of the 6 components of a general helicity decomposition of the metric and stressenergy perturbations consisting of scalar vector and tensor sectors. The tensor polarisations are shown to still propagate luminally in agreement with recent data from gravitational interferometry experiments while their magnitude decays with an additional factor sourced by the nonminimal coupling. We show that the production of these modes is associated with a modified quadrupole formula at leading order. The vector perturbations still exhibit no radiative behaviour although their temporal evolution is found to be modified with spatial dependence remaining unaffected. We establish that the scalar perturbations can no longer be treated as identical. We investigate the scalar sector by writing the modified model as an equivalent twofield scalartensor theory and find the same scalar degrees of freedom as in previous literature. The different sectors are paired with the corresponding polarisation modes which can be observationally measured by their effects on the relative motion of test particles thus providing the possibility of testing the modified theory and constraining its parameters.

2024-09-01T00:00:00Z

['2024arXiv240907625B', 'arXiv:2409.07625', '10.48550/arXiv.2409.07625']

['General Relativity and Quantum Cosmology']

Gravitational wave polarisations in nonminimally coupled gravity

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07625.pdf

{'Gravitational wave polarisations in nonminimally coupled gravity': "∗ † \nMiguel Barroso Varela and Orfeu Bertolami Departamento de F'ısica e Astronomia, Faculdade de Ciˆencias, \nUniversidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal \n(Dated: December 18, 2024) \nThe properties of metric perturbations are determined in the context of an expanding Universe governed by a modified theory of gravity with a non-minimal coupling between curvature and matter. We analyse the dynamics of the 6 components of a general helicity decomposition of the metric and stress-energy perturbations, consisting of scalar, vector and tensor sectors. The tensor polarisations are shown to still propagate luminally, in agreement with recent data from gravitational interferometry experiments, while their magnitude decays with an additional factor sourced by the nonminimal coupling. We show that the production of these modes is associated with a modified quadrupole formula at leading order. The vector perturbations still exhibit no radiative behaviour, although their temporal evolution is found to be modified, with spatial dependence remaining unaffected. We establish that the scalar perturbations can no longer be treated as identical. We investigate the scalar sector by writing the modified model as an equivalent two-field scalar-tensor theory and find the same scalar degrees of freedom as in previous literature. The different sectors are paired with the corresponding polarisation modes, which can be observationally measured by their effects on the relative motion of test particles, thus providing the possibility of testing the modified theory and constraining its parameters.", 'I. INTRODUCTION': "Since their prediction in the early 1900s, gravitational waves (GWs) were seen as a topic of theoretical interest with no hope of any observational detection, mostly due to their significantly small effect on most objects. This all changed with the first detection of GWs by the LIGO experiment in 2015 [1]. Since their experimental confirmation, GWs are now taken as one of the leading opportunities for probing distant events with gravitational effects far beyond any of those that currently take place in our Solar System [2, 3]. This has led to a surge in interest in their theoretical properties in General Relativity (GR) and modified theories, as current and future experiments may provide the means to measure these and thus shed light on a putative more encompassing theory of gravity [3]. \nAs established in their original prediction, GWs in GR are expected to be present in the form of two massless tensor polarisations that propagate luminally through the vacuum. However, modified gravity theories often lead to the presence of additional polarisations, as well as different propagation properties for the tensor modes [4]. For instance, higher-order effective field theories of gravity have been shown to predict deviations from luminal propagation of the tensor modes in the vicinity of Schwarzschild [5] and Reissner-Nordstrom [6] black holes, along with additional deviations in the cosmological context [7]. Many of these properties have been constrained observationally in Earth-based experiments, which have provided us with an upper bound on the mass of the \ngraviton [8], along with lower and upper bounds on the speed of these waves [9]. These measurements can be used to tightly constrain parameters of modified theories or to even rule them out, which is a complex task in other contexts due to the sensible nature of gravitational effects at local scales [10, 11]. Although measuring relatively small modifications to GW properties around exotic backgrounds such as black hole spacetimes seems unlikely in the near future, the cosmological large-scale dynamics may provide promising opportunities to analyse modifications to GR in the proximity of our planet [12]. \nBesides analysing alterations to the already predicted tensor modes, the presence of additional polarisations of GWs in the context of modified gravity has been extensively researched in past literature [4, 10, 11, 13, 14]. This follows from the modification of the dynamics of the scalar and vector modes one considers when decomposing the metric perturbations into their different helicity sectors [15, 16]. The investigation of the properties of these additional modes can be achieved by an explicit perturbative approach or by applying the Newman-Penrose tetrad formalism [17], as seen, for example, in Refs. [18, 19]. In a closer parallel to the work presented here, it has been shown that minimally coupled f ( R ) theories predict the existence of scalar gravitational waves [20-22], which served as motivation for deeper research into this topic. \nIn this work, we consider the properties of metric perturbations in a cosmological expanding background governed by a modified theory of gravity with non-minimal coupling (NMC) of matter and curvature [23]. This advances the work conducted in Ref. [18], where only constant curvature backgrounds were considered as an initial probe for the properties of these waves in the modified regime. This kind of NMC theory has also been exten- \nsively researched in the context of mimicking dark matter profiles [24, 25], analysing the modified theory with solar system constraints [26-28], sourcing cosmological inflation in the early Universe [29-31] and the creation of large-scale structure [32]. More recently, it has been shown that we may solve the open problem of the 'Hubble tension' [33] by considering the late-time effects of the NMC on the model-dependent evolution of cosmic microwave background data used for indirect measurements of the Hubble parameter, while simultaneously providing an observationally compatible mechanism to source the accelerated expansion of the Universe [34]. This further motivates the analysis of the propagation of NMC gravitational waves in the context of an expanding Universe, as it allows for testing of the mathematical consistency of the theory at these scales, as well as providing the means for additional observational tests of the modifications to GR [10, 12]. \nFor simplicity of the analysis carried out here, whenever considering an expanding Universe we use comoving time η , which we relate to cosmic time as dt = a ( t ) dη , and write the background FLRW metric in Cartesian coordinates as \nds 2 = a 2 ( η )( -dη 2 + dx 2 + dy 2 + dz 2 ) (1) \nor more concisely as ¯ g µν = a 2 ( η ) η µν . We define the Hubble parameter by H = ˙ a/a = a ' /a 2 as usual, with dots representing derivatives with respect to cosmic time t and primes representing derivatives with respect to comoving time η . We shall mostly use the 'comoving Hubble parameter' H = a ' /a = aH as this is useful for our analysis, although one can find instances in the literature where H and ˙ H are used for easier interpretation in the context of the expanding Universe. Additionally, we define the full metric as \ng µν = ¯ g µν + a 2 ( η ) h µν = a 2 ( η )( η µν + h µν ) , (2) \nwhich will simplify the form of the equations and the determination of the propagation speed of the perturbations in case they exhibit radiative behaviour. As done in this equation, we will write background quantities with a bar as ¯ ρ whenever the distinction is necessary, while often omitting this when a quantity is already multiplied by another that is explicitly of perturbative first-order. We use the ( -, + , + , +) signature and set c = 8 πG = 1 for simplicity. \nWe have organised this paper as follows. We present the nonminimally coupled model, the relevant field equations and their linearly perturbed form in Section II. The helicity decomposition of the metric and stress-energy tensors, along with their associated polarisations and observable effects on the relative motion of test particles, are described in Section III. This is followed by the analyses of the dynamics of the perturbations in GR and in the nonminimally coupled theory, which are included in Sections IV and V respectively. We present an alternative method to determine the polarisation spectrum \nbased on an equivalent scalar-tensor theory formulation of the NMC model in Section VI. We conclude the work in Section VII, where we discuss the obtained results along with possible extensions of our considerations.", 'A. Action and field equations': "The nonminimally coupled f ( R ) model can be written in action form as [23] \nS = ∫ dx 4 √ -g [ 1 2 f 1 ( R ) + [1 + f 2 ( R )] L m ] , (3) \nwhere f 1 , 2 ( R ) are functions of the scalar curvature R , g is the metric determinant and L m is the Lagrangian density for matter fields [23]. The dynamics of General Relativity can be recovered by setting f 1 = R and f 2 = 0. The inclusion of a cosmological constant can be considered by choosing f 1 = R -2Λ. By varying the action with respect to the metric g µν we obtain the field equations [23] \n( F 1 +2 F 2 L m ) G µν =(1 + f 2 ) T µν +∆ µν ( F 1 +2 F 2 L m ) + 1 2 g µν ( f 1 -F 1 R -2 F 2 R L m ) , (4) \nwhere we have defined ∆ µν ≡ ∇ µ ∇ ν -g µν □ and F i ≡ df i /dR . As can be seen from the presented equations, the choice of the Lagrangian density is non-trivial, as it directly affects the resulting dynamics via the nonminimal coupling, while only appearing in the form of the related stress-energy tensor in General Relativity [35, 36]. With this in mind, throughout this work, we follow the arguments discussed in Refs. [36, 37] and take the Lagrangian density to be L m = -ρ , with ρ denoting the energy density. \nApplying the Bianchi identities ∇ µ G µν = 0 to Eq. (4) leads to the non-conservation law [23] \n∇ µ T µν = F 2 1 + f 2 ( g µν L m -T µν ) ∇ µ R, (5) \n̸ \nwhich reduces to the usual stress-energy tensor conservation equation when we set f 2 = 0. This modification of the conservation equation follows directly from the nonminimal coupling of matter and curvature, as seen by its direct dependence on f 2 and its independence from the minimally coupled part of the theory described by f 1 . In this work, as we focus on the effects of the NMC model independently of the minimally coupled f ( R ) model, we set f 1 = R and single out the remaining effects with f 2 = 0. \nAlthough the form of f 2 will be kept abstract throughout the discussion presented in this work, a useful choice \nis to consider a general power-law expansion in R . Indeed, writing this as \n1 + f 2 ( R ) = ∞ ∑ n = -∞ ( R R n ) n , (6) \nwhere R n can be thought of as setting the scale for which the effects of each term in the series become considerable, we can capture the behaviour of the NMC theory at all orders of the curvature, with positive exponent terms dominating in the early Universe and negative exponent terms coming into play at late-times [34]. This means that we expect the evolution of the perturbations considered in this work to be highly epoch-dependent and for the modifications from GR to be strongest at significantly early and late times of the Universe's expansion.", 'B. Background cosmology in NMC model': "As the behaviour of perturbations will be highly dependent on the background dynamics, it is important to review how the expanding nature of the Universe is modified in the NMC theory. This analysis was originally conducted in Ref. [38], with additional results presented in Refs. [34, 39]. By considering the background metric in comoving coordinates as presented in Eq. (1), we can use the previously discussed field Eqs. (4) to arrive at the modified Friedmann equation \nH 2 = 1 6 F [ 2(1 + f 2 ) a 2 ρ -6 H F ' -a 2 f 1 + a 2 FR ] . (7) \nThe spatial components of the field equations yield the modified Raychaudhuri equation \nH 2 +2 H ' = -1 2 F [ 2 F '' +3 H F ' + a 2 f 1 -a 2 FR +2(1 + f 2 ) a 2 p ] (8) \nand the modified conservation equation of the perfect fluid stress-energy tensor T µ ν = diag( -ρ, p, p, p ), with ρ \nand p representing the energy density and pressure respectively, leads to the usual result \nρ ' +3 H ( ρ + p ) = 0 , (9) \nwhere the choice of L m = -ρ causes the modifications in Eq. (5) to vanish. This greatly simplifies the background cosmological evolution, as we can consider the usual dynamics of the energy density of all kinds of matter with respect to the scale factor ρ ∝ a -3(1+ ω ) , which depends only on the equation of state parameter ω = p/ρ ( ω = 0 for non-relativistic matter, ω = 1 / 3 for radiation). \nAt the background level, we thus expect deviations from the standard ΛCDM model, which is already enough to induce modifications on the propagation of GWs, whose properties are dependent on the dynamics of the expansive nature of the Universe. Based on the research conducted in Refs. [24, 34, 39], we know that observational data from galaxy rotation curves and direct measurements of the Hubble and deceleration parameters can be explained by the NMC theory with the inclusion of negative exponent terms in the power series form of f 2 and without need for the inclusion of a cosmological constant. This means that to a reasonable approximation, one can take the background to evolve according to simpler observationally motivated models such as ΛCDM, especially if the numerical evolution of the perturbations proves too computationally expensive to perform in parallel to the simulation of the FLRW background dynamics in the NMC theory.", 'C. Perturbed NMC equations': "Considering the general perturbations to the FLRW background metric, for which we maintain the convention introduced at the start of this paper, and the matter content, the NMC-modified field Eqs. (4) yield \n( F 1 ,R δR +2 F 2 ,R L m δR +2 F 2 δ L m ) R µν +( F 1 +2 F 2 L m ) δR µν -1 2 g µν F 1 δR -1 2 a 2 h µν f 1 -[ δ ( ∇ µ ∇ ν ) -a 2 h µν □ -g µν δ ( □ ) ] ( F 1 +2 F 2 L m ) -[ ∇ µ ∇ ν -g µν □ ] ( F 1 ,R δR +2 F 2 ,R L m δR +2 F 2 δ L m ) = (1 + f 2 ) δT µν + F 2 T µν δR, (10) \nwhere we have written F i,R = dF i /dR , with the notation F ' i = F i,R R ' still referring to differentiation with respect to η . Given our previous choice of f 1 = R , from now on we shall set F 1 = 1. The linear perturbation of the covariant derivatives is due to the perturbation of the \nChristoffel symbols \nδ Γ ρ µν = 1 2 ¯ g ργ [ ∇ µ ( a 2 h γν ) + ∇ ν ( a 2 h γµ ) -∇ γ ( a 2 h µν )] (11) \nδ ( ∇ µ ∇ ν ) φ = δ Γ ρ µν ∂ ρ φ (12) \nfor any scalar quantity φ . For reasons that will become clear later, it is convenient to write the perturbed field equations in the form \nFδR µ ν + R ν µ δF -1 2 δ µ ν δR +[ δ µ ν □ -∇ µ ν ] δF +[ δ µ ν δ ( □ ) -δ ( ∇ µ ∇ ν )] F = (1 + f 2 ) δ µ ν + F 2 T µ ν δR, (13) \nwhere F = F 1 + 2 F 2 L m = 1 + 2 F 2 L m and so δF = 2( L m F 2 ,R δR + F 2 δ L m ). Additionally, this shows how the explicit dependence on the Lagrangian density is particularly relevant in the context of perturbation theory, as the matter content itself is perturbed, thus leading to a more complex interplay between the evolution of the small changes in the metric and the stress-energy tensor.", 'A. Metric perturbations': "We aim here to analyse general perturbations to an FLRW background metric. The spatial homogeneity of the system means that all background quantities will be functions of comoving time η only. This is especially important for R = R ( η ), as the background modified theory functions f i and their derivatives will be evaluated at this curvature. Most of the assumptions considered in Ref. [18] are no longer valid, such as ∂ µ F i = ∂ µ L m = 0, as these are now evolving quantities in spacetime. This means that the direct analytic conclusions drawn in that work are no longer possible to reach directly, forcing us to explicitly decompose the gravitational waves into 6 general modes and analyse them individually [16]. Nevertheless, when considering the evolution of the perturbations, we may choose to take the sub-horizon (or highfrequency) approximation, in which the background dynamics may be taken to be static in comparison to the rapidly-varying metric perturbations. We write the general helicity decomposition of h µν as [16] \nh ηη =2 ϕ h ηi = β i + ∂ i γ h ij = 1 a h TT ij + 1 3 Θ δ ij + ∂ i ε j + ∂ j ε i + ( ∂ i ∂ j -1 3 δ ij ∇ 2 ) λ, (14) \nwhere we assume that h µν → 0 as we move out to spatial infinity, as one would not expect to feel gravitational effects from an infinitely distant event. We see that we have 6 components from the symmetric h TT ij tensor mode, 3 components for each of the vectors ε i and β i , and 4 scalars ϕ, γ, Θ , λ . This gives a total of 16 modes. Due to the invariance of a ( η ) under spatial rotations, the 4 scalars indeed have helicity 0, the vectors have helicity ± 1 and the tensor has helicity ± 2 [15]. Additionally, we impose the constraints \n∂ i β i = 0 ∂ i ε i = 0 ∂ i h TT ij = 0 δ ij h TT ij = 0 , (15) \nwhich give 1/1/3/1 individual equations respectively, totalling 16-6=10 independent variables, consistent with a 4 × 4 symmetric tensor. Note also that we have written spatial indices in equations like ∂ i ε i = 0 with no indication of being raised or lowered, as these are interpreted as 'helicity' indices [15]. These defined quantities are gauge-dependent, allowing for further simplification [15]. \nWe thus define the gauge-invariant quantities \nΦ ≡ -ϕ + 1 a ∂ η [ a ( γ -1 2 ∂ η λ )] Ψ ≡ 1 6 [ -Θ -∇ 2 λ + H ( γ -1 2 ∂ η λ )] Ξ i ≡ β i -∂ η ε i , (16) \nwhich yields 6 gauge-invariant functions: 1 for Φ, 1 for Ψ, 3 for Ξ i and 6 for h TT ij , minus 3 for ∂ i h TT ij = 0, minus 1 for δ ij h TT ij = 0 and minus 1 for ∂ i Ξ i = 0. We thus have 6 physical and 4 gauge degrees of freedom [16]. The physical quantities are associated with different modes in the relative motion between two test particles [13] and so introduce detectable modifications to the theory. The remaining gauge freedom can be used to choose the conformal Newtonian gauge [15], for which \nλ = γ = β i = 0 (17) \nand so the gauge-invariant quantities can be written as \nΦ ≡ -ϕ Ψ ≡ -1 6 Θ Ξ i ≡ -∂ η ε i , (18) \nwhich allows the line element to be written as \nds 2 = a 2 ( η ) [ -(1 + 2Φ) dη 2 + { (1 -2Ψ) δ ij + ∂ i ε j + ∂ j ε i + 1 a h TT ij } dx i dx j ] . (19) \nThe scalar sector resembles the Newtonian limit of General Relativity, aiding in the interpretation of the effects of the scalar perturbations Ψ and Φ [15].", 'B. Stress-energy tensor perturbations': "Apart from decomposing the metric perturbations, we may also decompose the stress-energy tensor perturbations into their different helicity components [15]. It is convenient to use δT µ ν , whose components can be written as \nδT η η = -δρ δT i η = S i + ∂ i S δT i j = δpδ i j +Σ i j , (20) \nwhere we have defined the anisotropic stress tensor Σ i j . This can in turn be decomposed into scalar, vector and tensor parts as \nΣ ij = ( ∂ i ∂ j -1 3 δ ij ∇ 2 ) σ +( ∂ i σ j + ∂ j σ i ) + σ TT ij , (21) \nwith σ TT ij being traceless ∂ i σ TT ij = 0 and transverse in the sense that δ ij σ TT ij = 0. Specifically, in the case of a perfect fluid, such as the one considered in the context of the FLRW metric, the perturbations to the stress-energy tensor can be written as [15] \nδT η η = -δρ δT i η = -( ρ + p ) v i δT i j = δpδ i j = c 2 s δρδ i j , (22) \nwhere we have used the definition of the speed of sound in a perfect fluid δp = c 2 s δρ ( c 2 s = 0 for non-relativistic matter and c 2 s = 1 / 3 for radiation). Additionally, we have defined the peculiar velocity of particles due to the perturbations, which we can further decompose into its transverse and longitudinal parts as \nv i = V i + ∂ i v, (23) \nwhere V is transverse ( ∂ i V i = 0). Comparing with Eq. (20), we identify S = -( ρ + p ) v , S i = -( ρ + p ) V i and \nΣ i j = 0. This means that the tensor part of the stressenergy perturbations vanishes for a perfect fluid. \nDue to T µ ν also being a tensor, we expect it to transform under the same gauge transformations as discussed in the previous section. This motivates us to define gauge-invariant stress-energy perturbations in terms of the already defined quantities. These are [15] \nδρ ∗ = δρ -3 H ( ρ + p )( v + γ ) v ∗ = v + 1 2 λ ' V i ∗ = V i + β i , (24) \nwhich in our gauge ( λ = γ = β i = 0) simply give \nδρ ∗ = δρ -3 H ( ρ + p ) v (25) \nwith v and V i being gauge-invariant.", 'C. Effects of polarisations on test particles': "The presence of GWs can be inferred from the relative 'stretching/squeezing' of intervals in spacetime. Specifically, this has been used in the LIGO experiment to obtain the first concrete detection of gravitational waves passing through our planet [1]. This observation focused on the more relevant tensorial modes predicted by perturbations to vacuum backgrounds in GR. However, all polarisations discussed here present their respective effects on spacetime intervals and can be associated with different types of effects. To see this, we can analyse perturbations to the geodesic deviation equation due to the metric fluctuations [40] \nD 2 X µ dτ 2 ≡ u α ∇ α ( u β ∇ β X µ ) = R µ αβσ (1) u α u β X σ , (26) \nwhere u α is the four-velocity of the observer, X µ is the displacement of two objects travelling along infinitesimally separated geodesics and the superscript '(1)' refers to taking only first-order contributions to the Riemann tensor. By assuming a non-relativistic observer, we can set u α ≈ ( a -1 , 0 , 0 , 0) and thus obtain \n∂ 2 X i ∂η 2 = -a -2 R i ηjη (1) X j , (27) \nwhere we have used comoving time η to obtain a simpler relation with our previous calculations. Here we have also taken measurements to be made in the local inertial frame (LIF) of the observer, thus simplifying the lefthand side of the equation to a standard partial derivative [40]. Comoving time is particularly useful for this, as our definition of the full metric in Eq. (2) allowed for perturbations to be directly applied to a 'Minkowski' background, as one would observe in its LIF. This allows us to determine the effects of metric perturbations on slowly moving test particles with the use of the perturbed components of R i ηjη . Aligning the propagation of \nany waves with the z-axis, we can separate the Riemann tensor into different modes as [13, 41] \nR i ηjη (1) = P 4 + P 6 P 5 P 2 P 5 -P 4 + P 6 P 3 P 2 P 3 P 1 ij , (28) \nwhere we have labelled the 6 independent polarisation modes as P i . In order of increasing i , these describe the longitudinal, vector-x, vector-y, +, × and breathing modes [13], which are shown in Figure 1. By calculating the respective linearised Riemann tensor components for the chosen gauge-invariant metric perturbations, we obtain \nP 1 = ∂ 2 z Φ+Ψ '' + H (Ψ ' +Φ ' ) P 2 = 1 2 ( H ∂ z Ξ 1 + ∂ z Ξ ' 1 ) P 3 = 1 2 ( H ∂ z Ξ 2 + ∂ z Ξ ' 2 ) P 4 = 1 2 a ( H ' h + + H h ' + -h '' + ) P 5 = 1 2 a ( H ' h × + H h ' × -h '' × ) P 6 = H (Ψ ' +Φ ' ) + Ψ '' , (29) \nwhere the expanding nature of the Universe is present through the comoving Hubble parameter H and its derivative. The association of the different helicity perturbations with the different polarisation modes serves as yet another reason why the chosen decomposition is useful for observational testing. By considering waves varying much faster than the expansion timescale, we obtain the simpler form \nP 1 = ∂ 2 z Φ+Ψ '' P 2 = 1 2 ∂ z Ξ ' 1 P 3 = 1 2 ∂ z Ξ ' 2 P 4 = -1 2 a h '' + P 5 = -1 2 a h '' × P 6 = Ψ '' , (30) \nwhich provides a clear connection between the investigated metric perturbations and the relative motion of test particles. Any detection (or non-detection) of modified behaviour following from the analysis of the NMC theory could provide means to further restrict the theory's parameters or even test for its presence [10, 11, 42, 43].", 'A. Stress-energy tensor perturbations': "As discussed in Section II, the conservation equation in the NMC theory is modified to give \n∇ µ T µ ν = F 2 1 + f 2 ( δ µ ν L m -T µ ν ) ∂ µ R, (62) \nwhere we take L m = -ρ . Perturbing this equation to linear order gives \nδ ( ∇ µ T µ ν ) = F 2 1 + f 2 [( δ µ ν δ L m -δT µ ν ) ∂ µ R + ( δ µ ν L m -T µ ν ) ∂ µ ( δR )] + δ ( F 2 1 + f 2 ) ( δ µ ν L m -T µ ν ) ∂ µ R, (63) \nwhere the left-hand side is unaltered from GR. Noting that in FLRW all background quantities depend only on η , we see that to zeroth order \n( δ µ ν L m -T µ ν ) ∂ µ R = ( δ η ν L m -T η ν ) ∂ η R = δ η ν ( -ρ + ρ ) ∂ η R = 0 , (64) \nwhere we have used our chosen form of L m = -ρ and the ηη component of the diagonal background stress-energy tensor T η η = -ρ . We thus see that the final term in the perturbed conservation equation vanishes for all values of the free index ν . Similarly, by analysing the ν = η \ncomponent we see that \nδ ( ∇ µ T µ η ) = F 2 1 + f 2 [ ( δ µ η δ L m -δT µ η ) ∂ µ R +( δ µ η L m -T µ η ) ∂ µ ( δR ) ] = F 2 1 + f 2 [ ( δ L m -δT η η ) ∂ η R +( L m -T η η ) ∂ η ( δR ) ] = 0 , (65) \nas δT η η = δ L m = -δρ . This shows that the ν = η component of the conservation equation is unaltered by the NMC theory and is thus identical to Eq. (32). \nFor ν = i we have \nδ ( ∇ µ T µ i ) = F 2 1 + f 2 [ -δT η i ∂ η R +( δ µ i L m -T µ i ) ∂ µ ( δR )] = F 2 1 + f 2 [ -δT η i ∂ η R +( L m -p ) ∂ i ( δR )] = -F 2 1 + f 2 [ v i ∂ η R + ∂ i ( δR )] ( ρ + p ) , \n(66) \nwhere we again use v i = V i + ∂ i v and so \nδ ( ∇ µ T µ i ) = -F 2 1 + f 2 ( ρ + p ) V i ∂ η R -∂ i [ F 2 1 + f 2 ( ρ + p )( v∂ η R + δR ) ] . (67) \nHere we have used the fact that f 2 , ρ and p only depend on η at background level. Similarly to the left-hand side of this equation, we have 2 different terms with different helicities. This means we again have 2 separate equations for each spatial component of the conservation equation. The first of these, associated with the vector sector, is \n4 H ( ρ + p ) V i +[( ρ + p ) V i ] ' = -F 2 1 + f 2 R ' ( ρ + p ) V i = -ln (1 + f 2 ) ' ( ρ + p ) V i ⇒ V i = V ( s ) i ( z ) (1 + f 2 )( ρ + p ) a 4 , (68) \nwhere we see that as in GR we can separate the spatial dependence of the vector matter perturbations (here again written with a superscript ( s )) from their temporal evolution, with the latter decaying with an additional factor of (1 + f 2 ) due to the nonminimal coupling. \nFor the scalar sector we have \nc 2 s δρ +( ρ + p )Φ + [( ρ + p ) v ] ' = -[ 4 H +ln(1 + f 2 ) ' ] ( ρ + p ) v -F 2 1 + f 2 ( ρ + p ) δR, (69) \nwhere again we find a contribution of a (1 + f 2 ) factor from the NMC on the evolution of v , along with additional coupling to the scalar metric perturbations via the purely scalar-dependent δR . This is explicitly given by \nδR = 2 a 2 [ ∇ 2 (2Ψ -Φ) -3 H (3Ψ + Φ) ' -6( H 2 + H ' )Φ -3Ψ '' ] , (70) \nshowing the added complexity introduced by the NMC theory to the coupling between the scalar quantities δρ , v , Φ and Ψ, which is not present in minimally coupled f ( R ) theories, as these present no alterations to the stressenergy conservation equation [20, 23].", 'B. Scalar perturbations': "The scalar sector describes the effects of the Ψ and Φ perturbations. The non-vanishing components of the perturbed Einstein tensor are \nδG η η = 6 a 2 H 2 Φ+ 2 a 2 (3 H Ψ ' -∇ 2 Ψ) (40) \nδG i η = 2 a 2 ( H ∂ i Φ+ ∂ i Ψ ' ) (41) \nδG i j = 1 a 2 [ δ i j ( 2( H 2 +2 H ' )Φ + 2 H (Φ ' +2Ψ ' ) + 2Ψ '' ) + ( δ i j ∇ 2 -∂ i ∂ j ) (Φ -Ψ) ] (42) \nwith the respective scalar stress-energy perturbations \nδT η η = -δρ δT i η = -( ρ + p ) ∂ i v δT i j = δpδ i j = c 2 s δρδ i j . (43) \nThe ( i, j ) equations can be split into a δ i j term and a ∂ i ∂ j term. These two kinds of terms behave as different helicity components and can thus be separated into independent equations [15]: \n2( H 2 +2 H ' )Φ + 2 H (Φ ' +2Ψ ' ) + 2Ψ '' = a 2 c 2 s δρ (44) \nand \nΦ -Ψ = 0 ⇒ Φ = Ψ , (45) \nrespectively, where we see that the two scalar perturbations are exactly equal due to the chosen convention in their initial definition in terms of the original perturbation variables. This equality is a direct consequence of the vanishing of σ in the anisotropic stress tensor Σ i j in Eq. (21) when assuming a perfect fluid [15]. \nWe can now analyse the ( i, η ) components of the perturbed field equations with Φ = Ψ. These give \nH ∂ i Φ+ ∂ i Φ ' = -a 2 2 ( ρ + p ) ∂ i v ⇒H Φ+Φ ' = -a 2 2 ( ρ + p ) v, (46) \nwhich we can then apply to the ( η, η ) equation \n2 a 2 ( 3 H 2 Φ+3 H Φ ' -∇ 2 Φ ) = 6 a 2 H ( H Φ+Φ ' ) -2 a 2 ∇ 2 Φ = -3 H ( ρ + p ) v -2 a 2 ∇ 2 Φ = -δρ (47) \nor in the more illuminating form \n∇ 2 Φ = a 2 2 ( δρ -3 H ( ρ + p ) v ) = a 2 2 δρ ∗ . (48) \nThis equation is written fully in terms of gauge-invariant quantities and so is valid in any gauge. It is analogous to the classical Newtonian potential equation ∇ 2 Φ = 4 πGρ [15], here written as ρ/ 2 due to the chosen convention 8 πG = 1 . \nConsidering the δ i j component of the ( i, j ) equation together with the relation Φ = Ψ, we get \n2( H 2 +2 H ' )Φ + 6 H Φ ' +2Φ '' = a 2 c 2 s δρ, (49) \nwhich we combine with the ( i, η ) equation to obtain a gauge-invariant form \nΦ '' +3(1 + c 2 s ) H Φ ' + ( 2 H ' +(1 + 3 c 2 s ) H 2 ) Φ = a 2 2 c 2 s ( δρ -3 H ( ρ + p ) v ) = a 2 2 c 2 s δρ ∗ . (50) \nBy observing that the right-hand side of this equation matches that of Eq. (48) with an additional factor of c 2 s , we obtain the full independent 'master' equation for Φ \nΦ '' -c 2 s ∇ 2 Φ+3(1+ c 2 s ) H Φ ' + ( 2 H ' +(1 + 3 c 2 s ) H 2 ) Φ = 0 , (51) \n̸ \nwhere we see that Φ obeys a wave-like equation with propagation speed c 2 s . The Φ ' term is a friction term that indicates that the wave has the form (Φ a 3(1+ c 2 s ) / 2 ) instead of simply Φ. The physical perturbation Φ thus propagates as a wave for c 2 s = 0, but also decays with the expansion of the Universe as \n̸ \nΦ = Ψ ∝ a -3 2 (1+ c 2 s ) ∼ √ ¯ ρ ( c 2 s = 0) . (52) \n̸ \nDuring the radiation-dominated epoch, we see that the perturbations oscillate while decaying at the same rate as the square root of the background matter density, to which they are inherently connected. After solving for the evolution of Φ we may then determine the behaviour of δρ and v from the previous field and conservation equations [15]. Even though we have found a wave-like equation for the scalar perturbations, this radiative behaviour is only present for c 2 s = 0, meaning that in a matter-dominated epoch, such as the recent past of our Universe, this wave-like evolution of scalar perturbations would not be present, with these instead having a time evolution given by the decoupled Eq. (49) with c 2 s δρ = 0. This equation further simplifies during matter domination, as H 2 + 2 H ' = 0 and c 2 s = 0 during this epoch, leading to the temporal equation for the scalar perturbation Φ '' +3 H Φ ' = 0, which has a solution that is constant in comoving time and one obeying Φ ∝ η -5 ∝ a -5 / 2 [15]. \nAdditionally, by combining the perturbed field and conservation equations for the scalar sector, we may determine the evolution of the matter perturbations δρ in more detail. This is particularly simple in the sub-Hubble limit, with the time scale of perturbations being much \nsmaller than that of the expanding Universe. In this limit, combining the conservation of the stress-energy tensor with the ( η, η ) component of the field equations gives \nδρ '' -c 2 s ∇ 2 δρ = 0 , (53) \nwhich shows how the matter fluctuations also evolve with wave-like behaviour, as one would expect from their intrinsic relation to the scalar metric perturbations. The propagation speed is given by the speed of sound in the corresponding type of perfect fluid, meaning that these matter perturbations propagate at the same speed as the scalar potential Φ (or equivalently Ψ), while also not exhibiting wave-like properties in a matter-dominated background ( c 2 s = 0). \nAlthough we find the scalar perturbations obey wavelike equations in the presence of matter, the same is no longer true in vacuum ( ρ = δρ = 0). In this case, the metric perturbations have a spatial profile given by the Laplace equation ∇ 2 Φ = 0 and time behaviour given by the unsourced version of Eq. (49). This is expected, as GR only predicts the existence of 2 radiative degrees of freedom [16], which will be discussed in the context of the tensor sector below.", 'C. Vector perturbations': "As we consider Ξ i = Ξ i ( η, z ), the condition ∂ i Ξ i = 0 implies Ξ 3 = 0. As in the GR analysis, the fully spatial components of the perturbed field equations have no vector contribution from the stress-energy terms and thus give us the temporal evolution of the vector sector as \nF Ξ ' i -2( F 2 ρ ) ' Ξ i +2 H F Ξ i =[ F Ξ i ] ' +2 H F Ξ i = 0 ⇒ Ξ i = R i ( z ) a 2 F , (84) \nwhich shows that the vector perturbations evolve in comoving time η similarly to how they did in GR, with an additional decay due to the nonminimal coupling term in F . \nThe ( η, i ) components again determine the spatial evolution of the perturbations \n1 2 a 2 F ∇ 2 Ξ i = (1 + f 2 ) δT η i + F 2 ¯ T η i δR = (1 + f 2 )( ρ + p ) V i ⇒∇ 2 R i = 2 V ( s ) i ( z ) , \n(85) \nwhere in the final step we have removed the η -dependence from the equations determined above and from the conservation equations. This is precisely the same spatial equation as in GR, leading us to the conclusion that the spatial dependence of the vector perturbations seems to be unaltered by the NMC theory. Additionally, the separation of the η and spatial evolution means that we have no radiative behaviour and thus the NMC theory does not predict wave-like evolution for the vector terms in the metric perturbation. The modification of the temporal evolution of the vector sector could provide an opportunity to observationally test the presence of a nonminimal coupling in the gravitational theory [10, 42]. However, this would involve obtaining observations at a considerable variety of redshifts and throughout a wide variety of systems in order to draw any solid conclusions.", 'D. Tensor perturbations': "The tensor sector of the field equations is made up of the traceless-transverse h TT ij , which we have normalised as h ij = h TT ij /a for simplicity of the final equations. These have the same form as in the standard GR derivation [40], with h TT xx = -h TT yy = h + and h TT yx = h TT xy = h × . The non-zero Einstein tensor perturbations are \nδG x x = -δG y y = -1 2 a 2 [ ( H 2 + H ' ) h + + □ η h + ] (59) \nδG x y = δG y x = -1 2 a 2 [ ( H 2 + H ' ) h × + □ η h × ] , (60) \nwhere we have defined □ η = -∂ 2 η + ∇ 2 . As discussed previously, for a perfect fluid there are no anisotropic perturbations and so the tensor sector has δT µ ν = 0, leading to the final equations \n□ η h × / + = -( H 2 + H ' ) h × / + = -a 2 R 6 h × / + = -a '' a h × / + , (61) \nwhich represent a wave-like propagation of the tensor modes with luminal speed c 2 gw = 1. Note that we can think of the right-hand side of the wave equations as an 'effective mass' term with m 2 gw = -( H 2 + H ' ). The sign of this quantity depends on the Ricci scalar R . This curvature is 0 during a radiation-dominated epoch ( a ( η ) ∝ η ) and > 0 in a matter-dominated one ( a ( η ) ∝ η 2 ). Thus it seems that in these stages of the evolution of the Universe we could have a negative graviton mass. However, as discussed in Ref. [7], the presence of such a term merely indicates an effective mass, with a vanishing actual mass of the graviton. This distinction is also relevant when considering the causal structure of these waves, where one can typically take the sub-horizon approximation ∂ 2 z h ∼ k 2 h >> ( a '' /a ) h and retrieve the same luminal speed discussed above [7].", 'V. PERTURBATION DYNAMICS IN NONMINIMALLY COUPLING MODEL': 'For simplicity, when analysing GWs in the modified theory we assume all perturbations to be only functions of η and z , as that should not affect any possible radiative behaviour, which can always be aligned with the \nz -axis without loss of generality. With this in mind, any second-order spatial derivative terms in scalar quantities like δR choose no preferred direction and thus one can replace ∂ 2 z for ∇ 2 when extending to more general coordinate dependence. For each sector, we find that the corresponding equations are analogous to their GR counterparts, allowing for a simple reconciliation with general dependence on all spatial coordinates. \nSome analysis of the scalar sector perturbations has been conducted in Ref. [32] in the context of cosmological perturbation theory and the formation of the large-scale structure of the Universe. However, in that work, only approximate behaviour was obtained due to neglecting any time derivatives in comparison with spatial derivatives. While we will sometimes invoke the sub-Hubble regime, we do not make this quasi-static approximation, as we aim to investigate the presence of dynamical gravitational wave-like behaviour in the NMC theory. Another notable difference is the inclusion of vector and tensor sector perturbations, leading to a more complete picture of the evolution of metric fluctuations in this modified theory.', 'B. Tensor perturbations': "The tensor sector of the field equations is relatively simple due to both the zeroth and first-order stressenergy tensor having no tensorial helicity terms. As with the perturbations in GR, the equation for the × term comes from the ( x, y ) component of the field equations, while for the + term an identical equation follows from the ( x, x ) = -( y, y ) components. The evolution of the tensor perturbations is described by \n(1 -2 F 2 ρ ) □ η h + / × +2( F 2 ρ ) ' h ' + / × = [ 2 H ( F 2 ρ ) ' -( H 2 + H ' )(1 -2 F 2 ρ ) ] h + / × , (71) \nwhere we note that the □ η operator indicates a luminal propagation speed c 2 tensor = 1. This equation clearly reduces to the GR equivalent when we set f 2 = 0. However, we now also have a friction term due to the nonminimal coupling term ( F 2 ρ ) ' . To simplify this, we may use the fact that \n(1 -2 F 2 ρ ) h '' -2( F 2 ρ ) ' h ' = Fh '' + F ' h ' = √ F ( √ Fh '' +2 F ' 2 √ F h ' ) = √ F ( √ Fh '' +2 F ' 2 √ F h ' +( √ F ) '' h -( √ F ) '' h ) = √ F ( √ Fh ) '' -√ F ( √ F ) '' h (72) \nto write the equation as \n□ η ( ˜ h + / × ) = -( H F ' F + F '' 2 F -F ' 2 4 F 2 +( H 2 + H ' ) ) ˜ h + / × , (73) \nwhere we have used \n( √ F ) '' √ F = F '' 2 F -F ' 2 4 F 2 (74) \nand defined ˜ h + / × = √ Fh + / × . We thus see that the gravitational wave amplitude will be scaled by a η -dependent factor of F -1 / 2 . For late-time modifications with f 2 = ( R/R n ) n ( n < 0), as used in the context of removing the Hubble tension [34] or mimicking dark matter profiles [24], we see that F 2 ρ < 0 and so F = 1 -2 F 2 ρ > 1, meaning that gravitational waves decay faster as F increases into the future and f 2 dominates. If we take the background to be slowly evolving in comparison to the waves and introduce a stress-energy source, we obtain the modified gravitational wave equation \n□ h TT ij = -16 πG 1 + f 2 F δT ij , (75) \nwhere we have temporarily restored the gravitational constant. This differs from the GR result by a factor of 1+ f 2 F stemming from the non-minimal coupling. We will later identify this as a parameter (Σ = 1+ f 2 F ) that will also modify the weak lensing predictions from the scalar sector perturbations, which may also be thought of as a rescaling of the gravitational constant ˜ G = G Σ [32].", '1. Generation of gravitational waves in NMC theory': "The production of GWs in GR can be analysed by solving the unmodified version of wave Eq. (75) using the Green's function of the □ operator. For the remainder of this section, we shall simplify the equations by writing δT ij as T ij and consider perturbations varying faster than the background, allowing us to approximate t ≈ η and a ( t ) ≈ 1. By applying the GR conservation equation ∂ µ T µν = 0, one can relate the time derivative of the temporal components of the stress-energy tensor to the spatial derivatives of its spatial components. This leads to the well known quadrupole formula \nh ij ( t, ⃗x ) = 2 G r I ij ( t r , r ) = 2 G r d 2 dt 2 ∫ d 3 x ' x ' i x ' j T 00 , (76) \nwhich describes the waves generated by a source at a distance r away and at retarded time t r = t -r . This formula indicates that in GR we do not expect monopoles and dipoles to produce gravitational radiation [40]. \nHowever, in the NMC theory, there are two significant changes. These are the modified dynamical value of the gravitational constant ˜ G and the non-conservation of the stress-energy tensor given in Eq. (5). This means that there are additional steps we must take to derive the form of the waves h ij from the matter content that originates them. We start by applying the Green's function [40] \nh ij ( t, ⃗x ) = 4 G ∫ dt ' ∫ d 3 x ' δ ( t ' -t r ) | ⃗x -⃗x ' | ( 1 + f 2 F ) T ij ≈ 4 G r ∫ d 3 x ' [( 1 + f 2 F ) T ij ] ( t r , ⃗x ' ) , (77) \nwhere we have assumed a source at a distance | ⃗x | = r away with ⃗x ≫ ⃗x ' . To simplify this equation, we assume that the background R ( t, ⃗x ) ≈ R ( t ) such that we may move the scaling of the modified constant outside the spatial integral. This means we now have \nh ij ( t, ⃗x ) = 4 G r ( 1 + f 2 F )∣ ∣ ∣ ∣ t = t r ∫ d 3 x ' T ij ( t r , ⃗x ' ) = 4 ˜ G ( t r ) r ∫ d 3 x ' T ij ( t r , ⃗x ' ) , (78) \nwhich is what one would expect by considering the rescaled gravitational constant as a 'true' constant. We then integrate by parts and neglect boundary terms as we \nexpect no contributions at infinity, while using the conservation equation to substitute T ij terms by T 00 terms. If the conservation equation was unaltered with respect to GR, as is the case in minimally coupled f ( R ) theories, then the quadrupole formula for the spin-2 sector would be recovered as \nh ij ( t, ⃗x ) = 2 ˜ G r I ij ( t r , ⃗x ) , (79) \nwhere in that case we would have ˜ G = G/F 1 . However, the conservation equation in the NMC theory gives \n∂ k T kj = -∂ t T 0 j + F 2 1 + f 2 ( g µj L m -T µj ) ∂ µ R = -∂ t T 0 j + F 2 1 + f 2 ( g 0 j L m -T 0 j ) ˙ R = -1 1 + f 2 ∂ t [(1 + f 2 ) T 0 j ] , (80) \nwhere we have again taken R ≈ R ( t ) and taken the background metric to be flat such that g 0 j = 0. We thus have \nh ij ( t, ⃗x ) = -4 ˜ G r ∫ d 3 x ' x ' i ∂ k T kj = 4 ˜ G r (1 + f 2 ) ∫ d 3 x ' x ' i ∂ t [ (1 + f 2 ) T 0 j ] . (81) \nFollowing the same steps we develop this expression as \nh ij ( t, ⃗x ) = -2 ˜ G r (1 + f 2 ) ∫ d 3 x ' x ' i x ' j ∂ t [ (1 + f 2 ) ∂ k T 0 k ] = 2 ˜ G r (1 + f 2 ) ∫ d 3 x ' x ' i x ' j ∂ t [ (1 + f 2 ) ∂ t T 00 ] = 2 ˜ G r [ I ij + F 2 1 + f 2 ˙ R ˙ I ij ] , (82) \nwhere we have used the conservation equation again to simplify ∂ k T k 0 as \n∂ k T k 0 = -∂ t T 00 + F 2 1 + f 2 ( g 00 L m -T 00 ) ˙ R = -∂ t T 00 + F 2 1 + f 2 ( ρ -T 00 ) ˙ R = -∂ t T 00 , (83) \nas we have chosen the matter Lagrangian to be L m = -ρ = -T 00 . We note that the ˙ R term should only be considered when its dynamics are comparable to those of I ij , as it should otherwise be ignored under the subhorizon approximation. Therefore, under the stated assumptions, we find that the production of gravitational waves is still quadrupolar, obeying a similar equation to the one in GR with a rescaled gravitational constant ˜ G and an additional term due to the modified conservation law, which follows from the nonminimal coupling between matter and curvature. As expected, by setting f 1 = R and f 2 = F 2 = 0, we recover the GR quadrupole formula, \nthus ensuring a smooth reconciliation with GR in the weak modification regime. However, as discussed later in this work, the NMC theory admits scalar gravitational wave degrees of freedom, as previously discussed in Ref. [18]. This additional degree of freedom predicts gravitational radiation from all multipoles, down to monopoles and dipoles, as in most alternative gravitational theories [44]. Nevertheless, showing that the traceless-transverse modes see no effects from monopoles and dipoles at leading order provides clarity on another fundamental property of what we still expect to be the dominant modes of gravitational radiation in the NMC theory.", 'D. Scalar perturbations': "When considering scalar perturbations in GR, we found the equality Ψ = Φ from the purely spatial compo- \nnts of the field equations, leading to a greatly simplified analysis of the remaining components. Analysing the same purely spatial components again reveals the possibility of separating these into δ i j and ∂ i ∂ j terms, as expected from the helicity decomposition. For simplicity, we show these in the high-frequency limit, which gives \nδ i j [ 2(1 + 2 F 2 p )Ψ '' +(1 + 2 F 2 p ) ∇ 2 (Φ -Ψ) -(1 + 2 F 2 p -F ) □ η Ψ+ □ η δF -a 2 (1 + f 2 ) δp ] -∂ i ∂ j [ F (Φ -Ψ) + δF ] = 0 , (86) \nwhere we have not expanded δF into its full dependence on scalar sector quantities for simplicity and again assumed a perfect fluid with p = c 2 s ρ . By setting F = 1, F 2 = 0 and δF = 0, we recover the high-frequency GR result, as expected. There is a strong dependence on the quantity F p ≡ 1 + 2 F 2 p , which interestingly would be precisely the value of F if we had defined the matter Lagrangian in terms of pressure ( L m = p ). We can again set both of these terms equal to zero independently, with the latter giving \nΨ -Φ = δ ln F = δF F = -2 F 2 δρ -2 ρF 2 ,R δR F , (87) \nwhich shows that the NMC, present through F , breaks the equality between the different scalar metric perturbations [32, 45]. This significantly complicates the remaining equations, as we are unable to replace all Ψdependence with Φ-dependence like in GR. Additionally, the ∇∇ δF terms in the perturbed field equations lead to first and second-order derivatives of the density fluctuations δρ , present due to the explicit presence of the matter Lagrangian density in F . However, by rewriting the previous relation as δF = (Ψ -Φ) F and inserting this into the perturbed field Eqs. (13), we may remove most of the complexity from the equations, with stress-energy perturbations now only present in δT µ ν , analogously to GR. \nThe remaining non-trivial equations are obtained from the ( η, η ) component, the ( i, η ) component and the δ i j term in the purely spatial components of the field equations. The choice of only η and z -dependence means that only the i = 3 component of ( i, η ) is non-trivial, with the remaining ( i, η ) components yielding the same equation in the general case and thus not causing any loss of generality. The δ i j term is the same for all i = j components, as expected. This means we are left with 3 non-trivial field equations, along with 2 stress-energy conservation equations for the scalar sector. This would be consistent with the 5 degrees of freedom from Φ , Ψ , v, δρ and δp . However, as we have assumed a perfect fluid with δp = c 2 s δρ , we are only left with 4 independent degrees of freedom and expect one of the equations to be obtained from the others, as is the case with the equations in GR [15]. \nThe ( i, η ) components of the field equations can again be written as ∂ i ( · · · ) = 0, which together with the condition of vanishing perturbations at infinity leads us to \nset ( · · · ) = 0. This can be written as \nF [ H (Φ + Ψ) + (Φ + Ψ) ' ] + F ' (2Φ -Ψ) = -a 2 (1 + f 2 )( ρ + p ) v, (88) \nwhich reduces to the GR result obtained previously when f 2 = 0, as expected. The ( η, η ) component gives \n-a 2 (1 + f 2 ) δρ = -F ∇ 2 (Φ + Ψ) + 3 F ' Ψ ' +3 H F ' Φ +3 F ( H ' -H 2 )(Ψ -Φ) +3 H ( F [ H (Φ + Ψ) + (Φ + Ψ) ' ] + F ' (2Φ -Ψ)) , (89) \nwhere the last term can be simplified with the ( i, η ) equation to give \nF ∇ 2 (Φ + Ψ) -3 F ' (Ψ ' + H Φ) + 3 F ( H 2 -H ' )(Ψ -Φ) = a 2 (1 + f 2 ) δρ ∗ , (90) \nwith δρ ∗ = δρ -3 H ( ρ + p ) v still gauge-invariant in the NMC theory. This is the modified form of the 'Poissonlike' Eq. (48), with additional metric terms proportional to F ' and an added factor of (1+ f 2 ) multiplying the matter content of the equation due to the nonminimal coupling. The final term on the left-hand side is proportional to F , which does not vanish in GR ( F = 1). However, in that scenario, we know that Ψ = Φ and the term will still be removed, thus ensuring a reconciliation with GR. In the sub-horizon regime, where we take the perturbations to vary much faster than the rate expansion of the Universe ( ∂ z ∼ k >> H ), this equation simplifies to \n∇ 2 (Φ WL ) ≡ ∇ 2 (Φ + Ψ) = 1 + f 2 F a 2 δρ ∗ ≡ Σ( a, k ) a 2 δρ ∗ , (91) \nwhich matches the expression found in Ref. [32]. Note that we have taken all background quantities (such as F ) to evolve with the expansion of the Universe as F ' ∼ H F and thus neglect them when compared to derivatives of perturbative quantities. Similarly to what was done in the tensor sector, we have again defined Σ in analogy with the formalism proposed in Ref. [46], where the importance of this parameter in the context of weak lensing observations was discussed. This definition comes from the relationship between the so-called 'weak lensing potential' Φ WL and the density perturbations. As previously pointed out, this effect can be encapsulated as a rescaling of the gravitational constant [32]. \nAll scalar equations presented so far have had explicit dependence on the perturbed matter content. To remove this, we may recognise that to first-order the stressenergy tensor for a perfect fluid obeys T z z + c 2 s T η η = 0, meaning that doing the same to the corresponding components of the perturbed field equations would remove all matter perturbation content. As before, we choose \nthe high-frequency limit for simplicity, which gives \n0 = [ 3(1 + c 2 s ) -F (2 + 3 c 2 s ) ] Ψ '' + [ F (2 + c 2 s ) -2(1 + c 2 s ) ] ∇ 2 Ψ + F Φ '' + [ 1 + c 2 s -F (1 + 2 c 2 s ) ] ∇ 2 Φ . (92) \nThis is the NMC-modified version of the scalar 'wave' Eq. (51). The main difference between these is the dependence on both scalars Φ and Ψ, which are no longer equivalent. It looks like the combination of two wavelike equations for these scalars. Unlike in GR, the assumption of a matter-dominated Universe with c 2 s = 0 no longer fully removes the ∂ 2 z part of the equation. This is due to the presence of the NMC terms proportional to F 2 ρ . The complexity of the coupled equations means we cannot find concrete analytic solutions even under the high-frequency approximation. Nevertheless, one could consider the weak modification regime, which allows for an additional linear treatment of the NMC effects. However, this would only hold in specific circumstances and is thus left as a discussion in Appendix A.", '1. Detection of scalar polarisations': "Even without an analytical solution for the separate metric perturbations, the analysis conducted here provides a possible method for distinguishing between the standard theory and the f ( R ) modified theory. This follows from the initial discussion on the scalar sector in the NMC theory, where we determined that the presence of the nonminimal coupling breaks the equality Ψ = Φ with the introduction of a δ ln F term. When introducing the scalar polarisation components in the tidal tensor ( P 1 and P 6 ), we found that in the high-frequency (subhorizon) limit these depend on the metric perturbations as \nP 1 = ∂ 2 z Φ+Ψ '' ∼ -k 2 Φ -ω 2 Ψ P 6 = Ψ '' ∼ -ω 2 Ψ , (93) \nwhere we note the mixed functional dependence in P 1 . We have also assumed a monochromatic plane wave decomposition of the scalar functions for simplicity in the final step, although one could of course consider more complex Fourier space forms with a mixed frequency spectrum in general. These terms correspond to longitudinal and breathing modes respectively and lead to distinct behaviour from the tensorial + and × modes [16]. In GR, only the massless spin-2 modes are expected to be present, while some modified theories of gravity predict the existence of the aforementioned scalar modes [10, 42]. Particularly, this was already established in the context of minimally and nonminimally coupled f ( R ) theories [18, 20, 22]. This means that the detection of such polarisations would serve as a direct test of modified gravity, although their detection alone would not necessarily distinguish between different modified theories without further determination of their properties. With the above \nresults, we can go further in our predictions, as we not only find the presence of breathing and scalar modes, but we also obtain a clear distinction between their properties, as we no longer have equivalent temporal and spatial metric perturbations in the scalar sector. If we detect both scalar and longitudinal gravitational wave polarisations and are able to compare their effects, one could theoretically use these observations to isolate the effects of the Ψ perturbation present in both P 1 and P 6 and consequently determine the Φ contribution to P 1 . If these are found to be equivalent, then the nonminimally coupled theory could be ruled out, while evidence of a 'decoherence' of Φ and Ψ would indicate the possible presence of the NMC. Of course, this would not necessarily provide definitive proof of its validity over other alternative theories, but it would nevertheless serve as a remarkable step toward obtaining a more general gravitational theory. \nDetecting additional polarisations of GWs would require careful synchronization of multiple interferometers, as we would need data from different orientations to be able to disentangle the 6 possible independent polarisations [47, 48]. As discussed in Refs. [42, 49], the responses of ground-based laser interferometers to breathing and longitudinal modes are completely degenerate, meaning that these can not be distinguished by present experiments. This follows from the equal pattern functions obtained for these polarisations in the context of laser interferometer arrays [48]. The emergence of space-based GW detectors, such as LISA [50], leads to the possibility of probing wider ranges of frequencies with minimal interference from the atmosphere [22]. This means that the distinction and independent analysis of the breathing and longitudinal modes, along with the consequent comparison of the two scalar metric perturbations, could become feasible in the coming years [47, 51-53]. Additionally, there are many recent developments on the detection methods for high-frequency gravitational waves (HFGWs) beyond the range of interferometry experiments like LIGO and LISA. These HFGWs can be generated in the very early Universe (see, for instance, Ref. [54]) or high-energy astrophysical events [12, 55], therefore providing a direct connection to extreme conditions in which modified gravity would be indispensable. They would also be in an optimal frequency range for an electromagnetic response [56-58], whose observation may be used to detect and separate the six polarisations, as discussed in Ref. [59], where it was also found that there is a good complementarity between the proposed electromagnetic response data analysis and existing ground-based telescopes. \nAnother notable avenue for observational tests of modified gravity at the other end of the GW frequency spectrum is the analysis of pulsar timing arrays (PTAs) [48]. This method utilises the highly regular electromagnetic pulsations of distant rotating neutron stars, known as pulsars, to detect low-frequency GWs from the cosmological stochastic background or isolated strong-field events like those caused by super-massive black holes [60]. As \nlight is emitted from the pulsars and received on Earth, both its start and end points are met with the effects of passing gravitational waves from different points in the Universe, leading to path differences that may be observed as measurable alterations to the otherwise periodic pulsation [61]. Each Earth-Pulsar system then serves as an effective interferometer arm of cosmological scale, allowing for the detection of GWs with frequencies ( ∼ nHz) well below those detectable by LIGO ( ∼ 100Hz) or LISA ( ∼ mHz) [60]. Of course, the large distances to these objects mean that we expect a combination of different gravitational waves to affect the observed result. This stochastic gravitational wave background (SGWB) is a complex composition of different wave polarisations, especially if one considers a general metric theory with 6 possible wave modes. Thankfully, this has been thoroughly researched in past literature, where it has been found that the correlation between data from arrays of many different pulsars can be used to decompose this background into its respective polarisations [61-64]. With the constantly increasing number of pulsars under permanent observation and improvements in the respective data analysis, this method provides yet another chance to individually detect the longitudinal and breathing modes, thus allowing for the separation of the effects from Ψ and Φ [65]. This is especially relevant due to the possible early origin of parts of the SGWB, which could provide a window to epochs with some of the more intense NMC presence. Additionally, the longitudinal nature of the P 1 polarisation leads to an increased effect on the irregularity of the pulsar radiation, with sensitivities up to two orders of magnitude greater than those of tensor modes [61].", 'E. Perturbation dynamics in models with f 2 ( R ) = ( R n /R ) n': "The discussion so far has focused on an agnostic form of f 2 ( R ), chosen to keep all arguments as general as possible. However, it is useful to analyse the dynamics resulting from the modified perturbation equations. For this, we choose an inverse power law form given by f 2 = ( R n /R ) n , where n is a positive integer. This choice ensures the decoupling of curvature and matter for large values of R , which from a cosmological point of view corresponds to the early-time Universe, while allowing for the emergence of modified behaviour near the present. Notably, this kind of model has been shown to mimic the effects of dark matter in galaxy rotation curves [24], yield the observed accelerated expansion of the Universe [39] and resolve the Hubble tension [34]. In the latter case, it was found that the NMC model can approximately recreate the late-time behaviour expansion of the ΛCDM model with H 0 ∼ 73 . 2 km/s/Mpc. This means that for small redshifts we can take the evolution of the background quantities from the results of that work, which greatly simplifies the generation of visualisations for the \nworked example shown here. \nIn the remainder of this section, we will focus on models with n = 4 and n = 10, as these were shown to capture different kinds of characteristics of the NMC model in Ref. [34]. The most important function to calculate is F = 1 + 2 F 2 L m , which for these models is given by \nF = 1 + 2 n ( R n R ) n ρ R ≥ 1 , (94) \nwhere we can consider suitable values for the NMC 'characteristic scale' constants R n from the results on the Hubble tension presented in Ref. [34], namely R 4 = 4 . 1 × 10 4 and R 10 = 4 . 4 × 10 4 in the chosen unit convention ( c = 1).", '1. Vector perturbations': 'As discussed earlier, the vector perturbations have a temporal evolution given by \nΞ i ∝ 1 a 2 F = (1 + z r ) 2 F ( z r ) , (95) \nwhere we have defined the redshift as a ∼ (1 + z r ) -1 in order to avoid confusion with the spatial coordinate z used before. Due to the inverse nature of f 2 , we have F 2 < 0, such that F > 1, thus leading to an overall faster temporal decrease of the perturbations due to the expansion of the Universe, as shown in the left panel in Figure 2. Note that the n = 10 model behaves approximately like GR up until z r ≈ 1 . 75, and presents larger perturbation values than n = 4 for all redshifts, with both models predicting a similar ∼ 20% decrease in the present vector perturbation magnitude when compared to GR.', '2. Tensor perturbations': 'The tensor perturbations not only decay with the expansion of the Universe, but also oscillate with their own characteristic frequency, which is redshifted similarly to what is observed for frequencies of light originating in the far past and detected on Earth ( ω obs = ω i a ( t i ) a ( t obs ) ). However, this decreasing frequency is not affected by the NMC model, instead following from the general expanding features of spacetime in the chosen coordinate system. Similarly, we will factor out the decay over distance ( ∼ 1 /r ), an effect present in both GR and in the NMC model, and regard this as due to the dissipation of the waves as they spread over larger distances. With this in mind, we neglect both of these behaviours and focus on the remaining characteristics of the evolution of the amplitude of these waves. From the previously derived equations, we see that even in GR the amplitude of the tensor perturbations will decay as a -1 ∼ (1 + z r ), which has been factored out from the start in the definition of \nFIG. 2. Time evolution of the amplitude of vector (top) and tensor (bottom) metric perturbations in GR and in the NMC modified theory with f 2 = ( R n /R ) n . Here the time dependence is shown in terms of the corresponding redshift z r , with any temporal dependence present in both GR and the modified theories being factored out for easier comparison, as pointed out in the y-axis label. \n<!-- image --> \nthe perturbed metric. However, in the modified model we obtain oscillatory behaviour for the combination √ Fh , which means that the perturbation h will have an additional temporal decrease given by F -1 / 2 . Putting all of this together, we see that \nh t ∝ 1 a √ F = 1 + z r √ F ( z r ) , (96) \nwhere h t denotes the amplitude of the observed tensorial perturbations. The results for the chosen example models are shown in the right panel of Figure 2. Again both models are in agreement with GR for larger redshifts ( z r ⪆ 3), with the n = 10 model maintaining an approximate GR-like behaviour until z r ≈ 1 . 75, after which we see that both models predict a ∼ 10% decrease in the presently observed gravitational wave amplitudes due to the expansion of the Universe in the NMC model.', '3. Scalar perturbations': "The scalar perturbations present more complex equations than the remaining sectors, as seen by the lack of decoupling between the equations for the scalar metric fluctuations Φ and Ψ discussed in the sections above. In fact, even when considering a specific example such as the previously chosen model, we are still unable to obtain analytical solutions for the evolution of these quantities. However, one particular parameter can be predicted in the sub-Hubble limit of Eq. (91), where we defined the 'weak lensing parameter' (Σ) in terms of the relation between the weak lensing potential Φ WL ≡ Φ + Ψ and the density perturbations, similarly to what has been done in past works on cosmological parameters (see Ref. [46] for a review). Distinctly from the vector and tensor sectors, the evolution of this quantity depends not only on F and therefore F 2 ρ , but also on the original NMC function f 2 ( R ). Again applying the generated simulation data from previous work on the Hubble tension [34], we can make predictions on the late-time behaviour of this parameter for different NMC models, as shown in Figure 3. \nFIG. 3. The evolution of the 'weak lensing parameter' Σ in terms of redshift z r in the NMC modified theory with n = 4 and n = 10. The GR prediction is shown for comparison. \n<!-- image --> \nWe find that Σ approaches its GR value (Σ GR = 1) for large redshifts, as expected. In the n = 10 model, this parameter departs from unity later than in the n = 4 model, due to the higher suppression of f 2 with n = 10 for large curvatures. Additionally, Σ consistently decreases almost until the present ( z r = 0), reaching a minimum value of Σ ≈ 0 . 9 before increasing to ∼ 0 . 925, with the turning point occurring around z r = 0 . 2. The n = 4 model is similar in many aspects, such as the initial decrease and consequent turning point. However, there are some notable differences, namely the earlier departure from GR and the earlier turning point around z r = 0 . 7. This latter distinction is of particular importance, as the increase of Σ starts early enough for its value to pass the GR prediction around the same time that it starts increasing \nin the n = 10 model, reaching an increase of ∼ 10% at present. These are noteworthy results, not only because they provide us with predictions for weak lensing properties of the modified theory which could potentially be observationally tested, but also because they offer a clear distinction between the lower-power n = 4 model and its higher-power n = 10 counterpart, based on their effect on weak lensing at lower redshifts. \nRecent attempts to constrain the present value of Σ in the context of testing modified gravity theories typically obtain values around 1, with 1 σ regions in the ∼ [0 . 9 , 1 . 2] range at best [66], which places both of the NMC models' predictions within error of observational constraints [66-69]. However, in Ref. [70] it was found from the Dark Energy Survey (DES) year 3 results that the present value of the weak lensing parameter, analysed by its deviation from GR, i.e. Σ 0 ≡ Σ( z r = 0) -1, can be constrained to Σ 0 = 0 . 6 ± 0 . 4 from DES alone and to Σ 0 = 0 . 04 ± 0 . 05 from DES combined with external data, both of which show a greater tension with the prediction from the n = 10 NMC model than with that of the n = 4 model. Similar conclusions were reached in the Planck experiment's 2018 results [71]. This is particularly relevant when considering that these constraints were obtained when taking a parametrisation for Σ of the form \nΣ par ( z r ) = 1 + Σ 0 Ω Λ ( z r ) Ω Λ ( z r = 0) , (97) \nwhere Ω Λ represents the dark energy density, which does not need to be constant in general. Although this is not equivalent to the predictions of the NMC theories considered here, taking the deviations of Σ from its GR value to be caused by dark energy is in some ways analogous to what happens in late-time NMC models with dependence on inverse powers of R , which have been shown to yield the same accelerated expansion with no need for the presence of dark energy [34, 39]. Improvements to the accuracy of observational constraints on growth parameters in modified gravity could provide crucial evidence to confirm or rule out the presence of modified theories such as the NMC theory considered in this work.", 'VI. GRAVITATIONAL WAVES IN SCALAR-TENSOR THEORY REPRESENTATION': "An alternative method to determine the spectrum of polarisations expected in an f ( R ) NMC gravity model is to look at an equivalent scalar-tensor formulation of the same theory. In fact, it has been shown in Ref. [72] that a two-field scalar-tensor model provides a suitable choice. This is given in the form of a Jordan-Brans-Dicke theory with a potential \nS = ∫ d 4 x √ -g [Ω R -V ( γ, Ω) + 2 (1 + f 2 ( γ )) L m ] , (98) \nwhere γ and Ω are scalar fields and we define \nV ( γ, Ω) = γ Ω -f 1 ( γ ) = γ (Ω -1) (99) \nas the potential. Note that we have already assumed the form of f 1 ( x ) = x as our work focuses on the NMC theory. Varying the action with respect to the fields yields γ = R , along with \nΩ = F 1 ( γ ) + 2 F 2 ( γ ) L m = 1 + 2 F 2 ( γ ) L m , (100) \nwhile varying with respect to the 'physical' metric gives the field equations \nΩ ( R µν -1 2 g µν R ) =( ∇ µ ∇ ν -g µν □ )Ω -1 2 g µν V ( γ, Ω) +[1 + f 2 ( γ )] T µν , (101) \nwhich reduce to the result shown in Section II upon substitution of γ and Ω from the other equations. From now on we will identify γ = R , while keeping Ω as an independent field, as its dependence on the matter Lagrangian L m does not allow us to solve for Ω = Ω( γ ). Taking the trace of the field equations provides a useful relation \n3 □ Ω+2 V ( R, Ω) -R = 3 □ Ω+ R (Ω -2) = (1+ f 2 ) T, (102) \nwhich shows that Ω obeys a Klein-Gordon equation. Perturbing the trace equation around some slowly evolving background curvature R 0 ≈ 0 and field Ω 0 , we obtain \n3 □ δ Ω+(Ω 0 -2) δR = (1 + f 2 ) δT + F 2 TδR, (103) \nwhile the field equations now read \nΩ 0 δG µν =( ∇ µ ∇ ν -g µν □ ) δ Ω -1 2 η µν (Ω 0 -1) δR + F 2 T µν δR +(1 + f 2 ) δT µν , (104) \nwhere we have written the metric perturbations over an approximately flat background as g µν = η µν + h µν . We then choose a gauge in which [18] \n∂ µ ( h µν -1 2 η µν h -η µν δ Ω Ω 0 ) = 0 , (105) \nwhich leads to \nδR µν = ∂ µ ∂ ν δ ˜ Ω -1 2 □ h µν δR = □ δ ˜ Ω -1 2 □ h, (106) \nwhere we have defined δ ˜ Ω = δ Ω / Ω 0 for simplicity. Note that this is equivalent to the quantity δ (ln F ) = δF/F 0 presented in the previous section, where we found that this is the source of the inequality between the two scalar metric perturbations Φ and Ψ. The field equations are then \n□ [ Ω 0 h µν +(2Ω 0 F 2 T µν +(Ω 0 -2) η µν ) δ ˜ Ω -( F 2 T µν + 1 2 η µν ) h ] = -2(1 + f 2 ) δT µν , (107) \nwhile the trace equation gives \n(Ω 0 F 2 T +2Ω 0 -4) □ δ ˜ Ω+ 1 2 (Ω 0 -2 -F 2 T ) □ h = -(1 + f 2 ) δT, (108) \nwhere moving background quantities inside of the derivative operators is justified when considering a slowly evolving background, which may be taken to be approximately constant in comparison to the rapidly evolving perturbations. Note that by considering only the tensor sector, i.e. by neglecting the scalar degree of freedom and the trace of the perturbations, we get the same result as in Section V B, with a modified gravitational constant ˜ G = (1 + f 2 ) G/ Ω 0 = (1 + f 2 ) G/F , as expected. Even when assuming the more complex case of the presence of background matter content, it is clear that the NMC theory introduces at least one additional radiative degree of freedom in the form of δ ˜ Ω, which is associated with the non-zero trace of the perturbations as we would expect for a scalar degree of freedom. \nIndeed, when considering the simpler case where no background matter is present ( ¯ T µν = 0 and Ω 0 = 1), while still allowing for matter perturbations to occur due to the interplay of curvature and matter in the NMC model, the field equations give \n□ ˜ h µν ≡ □ [ h µν -1 2 η µν h -η µν δ Ω ] = -2(1 + f 2 ) δT µν , (109) \nwhere the redefined perturbation ˜ h µν is transverse ( ∂ µ ˜ h µν = 0) due to the previously chosen gauge. In the absence of sources, we thus have a wave equation for the quantity ˜ h µν , which is initially sourced by the matter perturbations at the origin of the GW signal, while the trace equation directly shows the relation between δ Ω and h as □ δ Ω = -1 4 □ h . This means that we may solve this equation with the same traceless-transverse solution used in GR. We then invert the definition of ˜ h µν by using h = -˜ h -4 δ Ω to find the physical metric perturbations \nh µν = ˜ h µν -1 2 η µν ˜ h -η µν δ Ω = ˜ h µν -η µν δ Ω , (110) \nwhere we have taken ˜ h = 0 due to the traceless property taken for the solution. The physical metric perturbation is then given by \nh µν = δ Ω 0 0 0 0 h + -δ Ω h × 0 0 h × -h + -δ Ω 0 0 0 0 -δ Ω µν , (111) \nwhich indicates the presence of luminally propagating breathing and longitudinal polarisation modes induced by the scalar perturbation δ Ω along the standard + and × polarisations present in GR [16, 44]. Note that in terms of the scalar perturbations discussed in Section V, this would mean Ψ = -Φ = δ Ω / 2, leading to the exact relation we found from using a gauge-invariant formalism \nΨ -Φ = δ Ω = δF . We also see that this perturbation obeys h = -4Ω, which agrees with the trace equation in vacuum. \nHowever, it is important to keep in mind that the NMC contribution to δ Ω follows from two distinct terms, namely F 2 δ L m and F 2 ,R L m δR , meaning that the existence of these waves would depend on the presence of background matter to allow for their propagation. When such waves pass through cosmic voids ( L m = δ L m = 0), as will inevitably be the case in a Universe where dark energy is a purely gravitational phenomenon, both of these contributions are negligible or even zero. The specific case of NMC GW perturbations over dark energy-like fluids has been analysed in Ref. [18], but we do not consider these in this work as the NMC theory has no need for dark energy to generate the accelerated expansion of the Universe [34, 39]. Of course, the prior reasoning only applies if we consider a purely NMC theory with no modification to the minimal sector through f 1 , as minimally coupled f ( R ) theories introduce scalar gravitational radiation polarisations even in vacuum [44], as these do not vanish by taking L m = δ L m = 0. This means that, although the non-detection of scalar polarisations in future experiments could rule out the validity of minimally coupled f ( R ) theories, the same cannot be said about their NMC counterparts, as to leading order the scalar polarisations may have been 'washed out' by the passage of the GWs through cosmic voids.", 'VII. CONCLUSIONS': "In this work, we have analysed the properties of metric and matter perturbations in the context of an expanding Universe in a nonminimally coupled theory of gravity. These same perturbations have been thoroughly studied in the context of GR[15, 40], establishing the evolution of small deviations in homogeneous FLRW spacetimes and matter backgrounds and their role in the formation of large-scale structure in the Universe. The latter studies allow for the prediction of the propagation properties of (tensorial) gravitational waves, which radiate at luminal speeds and decay with the expansion of the Universe, as one would expect [7, 15]. GR also predicts the presence of 2 scalar and 2 vectorial degrees of freedom. While the latter do not exhibit any radiative behaviour, evolving independently with respect to time and spatial coordinates, the former behave as one equivalent term with different epoch-dependent properties [15]. The corresponding scalar sector term obeys a wave-like equation during the radiation-dominated epoch of the Universe, propagating at the speed of sound in that medium ( c 2 s = 1 / 3). However, during matter domination ( c 2 s = 0) the time and spatial dependencies separate, leading to a decaying evolution with no oscillatory behaviour. This means that the only radiative behaviour one would expect to detect in a GR-ruled Universe would come from the usual tensorial degrees of freedom, typically separated into + and \n× modes [40]. \nWhen considering the corresponding field equations in the NMC theory, several alterations arise [18]. Although the tensorial modes still propagate luminally, their effective mass is altered and the gravitational waves decay with an additional factor of F -1 / 2 while oscillating. These changes are in line with the analogous result presented in Ref. [18], where the additional time dependence was not obtained due to the assumption of simple constant curvature backgrounds. In late-time conditions, we expect negative exponent terms in a full power series expansion of f 2 to dominate [34, 39], thus leading to a faster decrease in the wave magnitude as it propagates. We then analysed the production of these waves, which under our assumptions maintain their quadrupolar nature at leading order, while monopole or dipole sources would only induce the scalar polarisations [44]. \nThe evolution of vector perturbations is mostly unaltered, with spatial and temporal dependencies separating as in GR, except for the latter being modified by a factor of F -1 , which accelerates their attenuation over time. Even though we cannot classify this sector as containing gravitational waves, any effects stemming from these kinds of metric fluctuations would be affected by the nonminimally coupled theory, thus providing means for future testing of the presence of such modifications to GR [10, 42]. \nThe scalar sector has the most non-trivial changes, mostly due to its inherent connection with the scalar density perturbations, which directly enter the field and conservation equations via the nonminimal coupling and due to our choice of Lagrangian density L m = -ρ [23, 35]. This is particularly relevant when considering the relation between the scalar metric perturbations Ψ and Φ, which are no longer simply equivalent as in GR, but rather related by a more complex coupled equation sourced by the variation in F = F 1 + 2 F 2 L m [32]. This quantity depends on the matter density and the Ricci scalar, which are affected by the perturbations to the stress-energy content and metric components respectively. However, the same equation can be used to remove all density perturbation derivative terms from the field equations, thus allowing for a better comparison between the GR and NMC dynamics. We derived the modified 'Poisson-like' equation, where the Laplacian term is now related to the gauge-invariant density perturbations via an 'effective gravitational constant' ˜ G = G (1+ f 2 ) /F which is unique to the NMC theory [32]. We analysed the effect of this modification on the weak lensing parameter (Σ) in examples where f 2 = ( R n /R ) n , with the n = 4 , 10 models chosen from their previously determined behaviour in Ref. [34]. We found our predictions to be in good agreement with recent observational constraints. \nA particularity of the modified theory considered in this work is the aforementioned inequality of the scalar metric fluctuations Φ and Ψ, as initially introduced in Ref. [32] in the context of large-scale structure formation. Although this adds considerable complexity to the ana- \ntical determination of properties of the scalar modes, it provides us with a general prediction which could be used in tests of the validity of the NMC theory. We have outlined how this hinges on the detection and distinction of the associated breathing and longitudinal modes, which depend on future space-based gravitational wave interferometry experiments, as the two scalar polarisations produce degenerate responses in ground-based laser interferometer experiments [42, 49]. With the predicted creation of future experiments such as LISA, the status of modified theories like the one addressed in this work could thus be put to the test [42]. Recent interest in the detection of lower and higher frequency signals of the SGWB spectrum could also bring about additional tests of modified gravity theories. Among some of the possibilities discussed in this work are the analysis of PTA data ( ∼ nHz) [60] and the proposed adaptation of microwave cavity experiments ( ∼ MHz) [56]. This is a fundamental step in the analysis of the NMC f ( R ) theory, which has already been shown to fit several observational effects (see, for example, Refs. [24, 34, 39]), while only now providing predictions that can be tested without being led by previous results, as any consistent theory must do. \nWe then analysed an alternative method to determine the polarisation spectrum in the NMC theory by making use of the representation of this model as a two-field scalar-tensor theory [72]. We find that the scalar field associated with the nonminimal coupling induces an additional degree of freedom that mixes curvature and matter perturbations. The scalar degree of freedom is associated with the trace of the metric perturbations and presents itself in the form of breathing and longitudinal polarisation modes propagating luminally, in agreement with what has been found in f ( R ) theories [44] and in other work on the NMC theory [18]. However, we discussed the particularities of the propagation of these scalar modes through empty regions such as cosmic voids [73], in which the absence of background matter to perturb would remove any NMC effects, in particular the scalar polarisations. This only applies to pure NMC models with no minimal f ( R ) component, meaning that a future lack of detection of scalar GW polarisations from distant sources could provide insight into an experimental distinction between mixed (MC+NMC) and pure nonminimally coupled f ( R ) theories. \nExtensions of the work described here are varied, with perhaps the clearest direction being performing the analysis of the scalar sector under a general regime. Particularly, research on the evolution of the density perturbations has been conducted in Ref. [32], where the sub-horizon regime was assumed for simplification of the coupled equations. Once a clearer picture of the evolution of scalar quantities is obtained, a logical extension would be to apply those results, together with those presented here, to the prediction of observable primordial imprints of the different perturbation sectors in cosmological data, such as the CMB power spectrum [74]. Additionally, applying the Newman-Penrose tetrad formalism to the \nperturbed NMC theory in a cosmological context could provide further conclusions on the predicted polarisations and their properties, as done in Refs. [18, 19].", 'ACKNOWLEDGMENTS': 'The work of one of us (O.B.) is partially supported by FCT (Funda¸c˜ao para a Ciˆencia e Tecnologia, Portugal) through the project CERN/FIS-PAR/0027/2021, with DOI identifier 10.54499/CERN/FIS-PAR/0027/2021.', 'Appendix A: Scalar perturbations in the weak NMC regime': "Decoupling the scalar field equations analytically in an abstract NMC theory with an unspecified f 2 ( R ) is extremely convoluted. Employing a numerical method could be necessary to extract more quantitative results in general scenarios [15], which with the results of this work would simply consist of evolving the coupled differential equations with a robust numerical integrator with sensible initial conditions and a set chosen NMC theory parameters. With this in mind, some results on a concrete example were presented in Section V E. Nevertheless, some additional conclusions on the scalar sector can be reached in the weak regime, in which we take all modifications to be treated perturbatively to linear order in a separate expansion to that of the metric perturbations. This assumption is not expected to hold under general conditions, as can be seen by considering the hypothetical complete power expansion form of the nonminimal coupling function f 2 discussed in Section II. The behaviour of negative and positive exponents of the curvature scalar is such that we expect the strength of the NMC to be dominant in spacetime regions with significantly small (late-time Universe) [34, 39] or large (earlytime Universe) [29-31] curvatures respectively. However, intermediary situations, such as those expected around the creation of the cosmic microwave background, could provide regions in which the NMC might be treated in the weak regime, as discussed in Ref. [34]. The following perturbative analysis, although not general, is therefore still a relevant example to consider when describing some fundamental points of the evolution of the Universe. \nIn order to simplify the distinction between expansion orders, we introduce a new factor χ multiplying f 2 . As discussed in Refs. [5, 6], this method allows us to analyse any modifications to the equations under the light of their \nGRcounterpart, as they are already first-order terms and thus can be evaluated with the zeroth-order equations in mind. This is particularly useful, as the scalar perturbations Ψ and Φ are equivalent in GR. More specifically, in the perturbative regime we may write \nΨ -Φ = δF F ≈ δF = -2 χ ( ρF 2 ,R δR + F 2 δρ ) = O ( χ ) , (A1) \nwhich may be introduced into the field Eqs. (13). To decouple the equations, we need to remove the δρ dependence, which can be done by using the ( η, η ) component of the GR perturbed field equations, as the δρ term above is already O ( χ ). By taking a Fourier wave form for each scalar sector function with ∇ 2 ∼ -k 2 and the explicit expression for δR given previously, we can rearrange the equation above as \nΨ = Φ + χq (Φ , Φ ' , Φ '' ) (A2) \nor equivalently \nΦ = Ψ -χq (Ψ , Ψ ' , Ψ '' ) , (A3) \nwhere we have introduced the function q (Φ , Φ ' , Φ '' ) for simplicity. This can be applied to the coupled master Eq. (92) by writing Ψ ' = Φ ' + χ ˜ q (Φ , Φ ' , Φ '' , Φ (3) ) and equivalently for Ψ '' , hence removing all Ψ dependence, while now obtaining fourth-order derivative terms in the decoupled master equation. \nHowever, any third or fourth-order derivatives are necessarily O ( χ ) as the zeroth-order master equation only included derivatives of up to second order. We may thus apply Eq.(51) to lower any higher-order derivatives and write the master equation in the form [7] \n∂ 2 η Φ+ β 1 ( η ) ∂ η Φ+ β 0 ( η, k 2 )Φ + α 0 ( η )Φ = 0 , (A4) \nfrom which we can read off the propagation speed c 2 Φ = β 0 /k 2 , with equivalent reasoning leading to analogous equations for Ψ. This is clear when disregarding the friction term, which simply leads to a modified decay of the wave, after which one may take a wave-like time dependence of the form e iωη and solve the equation in the largek limit with ω 2 = β 0 + α 0 . This implies the previously asserted propagation speed c 2 Φ = lim k →∞ ω 2 k 2 = β 0 /k 2 , as the α 0 /k 2 term is negligible in this limit. \nAs expected, these speeds are of the form c 2 Φ / Ψ = c 2 s + χ ∆ c 2 Φ / Ψ , where these are given by \n̸ \n- χ\n- c 2 Φ = c 2 s -a 2 [ -2 H 2 ρF 2 ,R +2 ( H 2 + H ' ) ρF 2 ,R -6 H 2 c 2 s ρF 2 ,R -12 ( H 2 + H ' ) c 2 s ρF 2 ,R +36 H 2 c 4 s ρF 2 ,R -18 ( H 2 + H ' ) c 4 s ρF 2 ,R -2 H 2 F 2 +2 ( H 2 + H ' ) F 2 -18 H 2 c 2 s F 2 +6 ( H 2 + H ' ) c 2 s F 2 -a 2 ρF 2 +2 a 2 c 2 s ρF 2 +3 a 2 c 4 s ρF 2 + +2 H F 2 ,R ρ ' -18 H c 4 s F 2 ,R ρ ' +2 H ρF ' 2 ,R -18 H c 4 s ρF ' 2 ,R -4 ρ ' F ' 2 ,R +12 c 2 s ρ ' F ' 2 ,R +2 H F 2 ' +6 H c 2 s F ' 2 -2 F 2 ,R ρ '' +6 c 2 s F 2 ,R ρ '' -2 ρF '' 2 ,R +6 c 2 s ρF '' 2 ,R -2 F '' 2 ] (A5)\n- c 2 Ψ = c 2 s -χ a 2 [ -6 H 2 F 2 +2 ( H 2 + H ' ) F 2 +42 H 2 c 2 s F 2 -18 ( H 2 + H ' ) c 2 s F 2 -a 2 ρF 2\n- +2 a 2 c 2 s ρF 2 +3 a 2 c 4 s ρF 2 -6 H 2 ρF 2 ,R +2 ( H 2 + H ' ) ρF 2 ,R +54 H 2 c 2 s ρF 2 ,R\n- -12 ( H 2 + H ' ) c 2 s ρF 2 ,R -108 H 2 c 4 s ρF 2 ,R +54 ( H 2 + H ' ) c 4 s ρF 2 ,R (A6)\n- -2 H F 2 ,R ρ ' +18 H c 4 s F 2 ,R ρ ' -2 H F ' 2 -6 H c 2 s F ' 2 -2 H ρF ' 2 ,R\n- +18 H c 4 s ρF ' 2 ,R +4 ρ ' F ' 2 ,R -12 c 2 s ρ ' F ' 2 ,R +2 F 2 ,R ρ ''\n- -6 c 2 s F 2 ,R ρ '' +2 F '' 2 +2 ρF 2 ,R '' -6 c 2 s ρF '' 2 ,R ] . \nNot only do these speeds match at zeroth order, but they still depend on the speed of sound from the perfect fluid considered for the FLRW background. Although in an RD Universe ( c 2 s = 1 / 3) the weak NMC corrections would be relatively small in comparison to the GR speed, this is no longer the case in the more recent MD Universe ( c 2 s = 0), where the radiative behaviour would be fully due to the NMC corrections, even when considering the weak regime. This means that we could in principle use the detection of this behaviour of scalar polarisations of \nGWsvia their effects on test particles to test the presence of a nonminimal coupling in the gravitational theory. We should note that we have ignored the possibility of a dark energy-dominated (ΛD) Universe, as the source of the currently observed accelerated expansion has been shown to be accounted for in the NMC theory with no need for the presence of a cosmological constant [34, 39]. The difference between the scalar perturbation speeds in the NMC theory ( c 2 Ψ = c 2 Φ ) could provide additional avenues of testing [14]. \n- [1] B. P. Abbott et al. 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2024PASA...41...91A

Results from some recent numerical works including ours lend credence to the thesis that the ambient environment that is the magnitude of external pressure affects the starforming ability of clouds and filaments. In continuation with our series of papers on this subject we explore this thesis further by developing new hydrodynamic simulations of accreting filaments confined by external pressures in the range inlineformula texmath 104 7 texmath inlineformula K cminlineformula texmath 3 texmath inlineformula. Our principal findings are i irrespective of linemass filamentfragmentation generally yields spheroidal cores. The initially subcritical filaments in low to intermediate external pressure environments form broad cores suggesting that weakly selfgravitating filaments must fragment via the collect and collapse mode to form broad cores. Transcritical filaments by contrast become susceptible to the Jeanstype instability and form pinched cores ii the ambient environment bears upon the physical properties of filaments including their FWHMinlineformula texmath fil texmath inlineformula. Only the filaments initially suffused with subsonic turbulence in SolarNeighbourhoodlike environments however have FWHMinlineformula texmath fil texmath inlineformula inlineformula texmath sim texmath inlineformula 0.1 pc. In high pressure environs such filaments not only have much smaller widths but also become severely eviscerated. On the contrary filaments suffused with initially supersonic turbulence are typically broader iii the quasioscillatory nature of velocity gradients must be ubiquitous along filament lengths and its magnitude generally increases with increasing pressure. The periodicity of the velocity gradients approximately matches the fragmentation lengthscale of filaments iv oscillatory features of the radial component of the velocity gradient are a unreliable proxy for detecting signatures of accretion onto filaments and v filaments at either extreme of external pressure are inefficient at cycling gas into the dense phase which could reconcile the corresponding inefficiency of starformation in such environments.

2024-11-01T00:00:00Z

['10.1017/pasa.2024.86', '2024PASA...41...91A', 'arXiv:2409.09431', '2024arXiv240909431A', '10.48550/arXiv.2409.09431']

['ISM: clouds physical data and processes: gravitation', 'hydrodynamics stars: formation', 'Astrophysics - Astrophysics of Galaxies']

On the formation of cores in accreting filaments and the impact of ambient environment on it

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09431.pdf

{'On the formation of cores in accreting filaments and the impact of ambient environment on it': 'S. V. Anathpindika \n1 \n<!-- image --> \n1 \n, \n2 \n∗ \nand Di Francesco, J \n<!-- image --> \nIndian Institute of Technology, Kharagpur, West Bengal, India \n2 University Observatory Munchen, Schneirstrasse 1, 81679, Munchen, Germany \n3 National Research Council of Canada, Herzberg, Astronomy & Astrophysics Research Centre, 5071 West Saanich Road, Victoria (BC), \nCanada V9E 2E7', 'Abstract': 'Results from some recent numerical works, including ours, lend credence to the thesis that the ambient environment, i.e., the magnitude of external pressure, affects the star-forming ability of clouds and filaments. In continuation with our series of papers on this subject, we explore this thesis further by developing new hydrodynamic simulations of accreting filaments confined by external pressures in the range 10 4 -7 K cm -3 . Our principal findings are (i) irrespective of linemass, filament-fragmentation generally yields spheroidal cores. The initially sub-critical filaments in low to intermediate external pressure environments form broad cores suggesting that weakly self-gravitating filaments must fragment via the collect - and- collapse mode to form broad cores. Transcritical filaments, by contrast, become susceptible to the Jeans-type instability and form pinched cores; (ii) the ambient environment bears upon the physical properties of filaments including their FWHM fil . Only the filaments initially suffused with subsonic turbulence in Solar-Neighbourhood-like environments, however, have FWHM fil ∼ 0.1 pc . In high pressure environs such filaments not only have much smaller widths, but also become severely eviscerated. On the contrary, filaments suffused with initially supersonic turbulence are typically broader; (iii) the quasi-oscillatory nature of velocity gradients must be ubiquitous along filament lengths and its magnitude generally increases with increasing pressure. The periodicity of the velocity gradients approximately matches the fragmentation lengthscale of filaments; (iv) oscillatory features of the radial component of the velocity gradient are a unreliable proxy for detecting signatures of accretion onto filaments; and (v) filaments at either extreme of external pressure are inefficient at cycling gas into the dense phase which could reconcile the corresponding inefficiency of star-formation in such environments. \nKeywords: ISM : Clouds Physical data and Processes : gravitation, hydrodynamics Stars : formation', '1 INTRODUCTION': "Bulk of the dense mass in molecular clouds (i.e., gas with visual extinction, A v /greaterorsimilar 7), appears filamentary with prestellar cores typically aligned along filament lengths projected on the plane of sky (e.g., Konyves et al. 2015, 2020). So the ubiquity of filaments in the interstellar medium ( ISM ) and their importance in the formation of stars is hardly in doubt. Herschel offered a wide dynamic range of observations in various far-infrared bands which revealed in unprecedented detail the filamentary organisation of gas in molecular clouds (Andr'e et al. 2010, Arzoumanian et al. 2011, Men'schikov et al. 2010). These observations also revealed that filaments seldom occur in isolated environments, and certainly do not resemble the static idealised cylinder often assumed in analytic models (e.g., Stod'olkiewicz 1963, Ostriker \n1964, Nagasawa 1987). On the contrary, accretion from the ambient environment appears to play a key role in determining the vital characteristics of filaments (e.g., Schneider et al. 2010, Kirk et al. 2013, Beuther et al. 2015, Gong et al. 2021). \nDetailed surveys of the filaments in clouds of the Solar-Neighbourhood (i.e., within a radius of ∼ 500 pc ), and those farther away suggest that filaments exhibit a wide range of physical properties. For example, the Herschel Gould Belt Survey showed that filaments in these clouds have a characteristic width (i.e., the FWHM fil ) of ∼ 0.1 pc (e.g., Andr'e et al. 2010, Arzoumanian et al. 2011, 2019), with linemasses in the range 5 - 17 M /circledot pc -1 , peak densities of ∼ 3 - 9 × 10 21 cm -2 and average dust-based temperatures between 14 - 16 K (e.g., Arzoumanian et al. 2019, Orkisz et al. 2019, Suri et al. 2019). Moreover, observations with dense gas \ntracers like H 13 CO + have enabled detection of thinner constituent fibres within filaments (e.g., Hacar et al. 2018). \nIn contrast, Galactic surveys extending beyond the Solar-Neighbourhood, like the HiGAL survey for example, revealed longer filaments expanding over a few parsecs with much higher linemasses that typically range between several tens of M /circledot pc -1 to a few hundred M /circledot pc -1 . These filaments are also generally much warmer than the typical Herschel Solar-Neighbourhood filaments (e.g., Schisano et al. 2020). Similarly, a number of IRDCs 1 also appear elongated and filament-like on the plane of sky. Many IRDCs are located at distances of a few kpc and typically have column densities upward of a few times 10 21 cm -2 , and linemasses /greaterorsimilar 10 2 M /circledot pc -1 (Peretto et al. 2014, Henshaw et al. 2014, Rathborne et al. 2016). Detailed analyses of their column density structures show further that several indeed have filamentary or hub-filamentary morphologies (e.g., Busquet et al. 2016, Henshaw et al. 2017, and the review by Hacar et al. 2022). Detailed analyses of the cloud G0.253+0.016 , also known as the Brick , located in the Galactic CMZ revealed filaments with FWHM fil ∼ 0.17 ± 0.08 pc (Federrath et al. 2016). \nA number of recent numerical Simulations explore filament formation at different spatial scales (e.g., Hennebelle 2013, Federrath et al. 2016, Chen et al. 2016, Inoue et al. 2018, Abe et al. 2021, Federrath et al. 2021). These simulations and others such as those by Smith et al. (2014) and Moeckel & Burkert (2015) show that filaments are a byproduct of the interaction between gas flows in a turbulent environment - the so -called paradigm of turbulence-driven filament -formation ( TDFF ) . In this paradigm, filaments and the cores within them often appear to form almost simultaneously (e.g., Gong & Ostriker 2011, Gom'ez & V'azquez-Semadeni 2014). Numerical Simulations like those by Federrath (2016), for example, for typical conditions in the Galactic arms show filament width in the range 0.05 - 0.15 pc irrespective of the star-formation history. Interestingly, these widths are also consistent with the sonic scale, i.e., the lengthscale on which turbulence transitions from the supersonic to the subsonic regime (see also, Federrath et al. 2021). \nIn other works about evolution of individual filaments Gritschneder et al. (2017), for example, showed that a small density perturbation on an initially subcritical filament amplifies over time and eventually fragments its natal filament. In this genre of simulations individual filaments fragment due to the growth of local density perturbations along their length - the so called geometrical fragmentation of filaments . Heigl et al. (2018), similarly, showed that an initially sub-critical filament becomes susceptible to the sausage-type instability and forms a broad core, i.e., a core that is bigger than the width of its natal filament. An initially supercritical filament, on the other hand, becomes susceptible to the Jeans-type compressional instability to form \npinched cores, i.e., cores of size smaller than the width of its natal filament. These observations are consistent with the analytic findings by Pon et al. (2011) who showed that local perturbations along the axis of a filament amplify faster than its global contraction timescale. \nPrior to this paper we numerically explored the impact of ambient external pressure ( P ext ) on filament evolution (i.e., Anathpindika & Di Francesco 2020, 2021 - hereafter referred to as Papers I and II, respectively). Those papers were motivated by the suggestions that ambient environmental conditions some how affect physical properties of gas, including its ability to form stars. This seems to be true in our own Galactic disk (e.g., Rathborne et al. 2014b, Rice et al. 2016), as well as in the disks of some other nearby galaxies (e.g., Hughes et al. 2010, Hughes et al. 2013, Heyer & Dame 2015). Arguably our thesis about the ambient environment is limited by the assumption that it is equivalent to only the external pressure. In reality, however, the environment could also mean the extended diffuse gas of which an observed filament is just a part. Nevertheless, in this series of papers we are exploring the thesis that ambient environment must affect the evolution of individual filaments just as it also affects evolution of clouds (e.g., Anathpindika et al. 2018). \nWhile filaments discussed in Paper I were accreting, those in Paper II were non-accreting. Naturally, P ext in Paper I was a combination of the thermal pressure of the gas being accreted by the filament and the ram-pressure due to its in-flow velocity. So P ext there was equivalently characterised by the Mach number of the inflowing gas (typically between 0.8-24), and the ratio of the temperature of the inflowing gas to the initial temperature of the gas within the filament (typically between 10-15). In Paper II, however, P ext was purely thermal and was determined only by the temperature of the medium confining the filament. Simulations discussed therein were generated for different choices of the ratio of the temperature of the external medium to the initial temperature of the gas within the filament (typically between 4-30). \nOur principal findings from Papers I & II are : (i) The ambient pressure does in fact bear upon the morphology of filament fragmentation and the cores spawned by it; (ii) Sub-critical filaments in low-pressure environments ( P ext /k B /lessorsimilar 10 4 K cm -3 ) formed cores via the collect-and-collapse mode. Also, these cores were not only bigger than their natal filaments, but the core formation timescale is also comparable to or greater than the e folding timescale that is typically on the order of a few freefall times. In other words, core formation in this instance is relatively slow; (iii) At higher pressures more comparable to that in the Solar-Neighbourhood (i.e., between a few times 10 4 K cm -3 to a few times 10 5 K cm -3 ), however, an initially sub-critical filament contracts to acquire a centrally peaked density profile and forms cores via the Jeans-type compressional instability . These cores are pinched and form on a timescale comparable to or smaller than the freefall \ntime; and (iv) At still higher pressures typically upward of ∼ 10 6 K cm -3 , however, the filaments rupture, i.e., they become severely eviscerated (i.e., losing substantial fraction of their mass) before eventually breaking up into disjointed fragments.", '1.1 Context of this paper vis - a - vis Papers I & II': "Expanding on our work discussed in those papers, we now develop simulations of accreting filaments with a relatively wide range of accretion rates. Crucial questions that we explore here are : (a) Do accreting filaments also become unstable to the sausage-type instability like filaments that either accrete too little mass (Paper I), or none at all (Paper II)? (b) Are the observed fluctuations in velocity gradients ubiquitous and if so, are they indeed cospatial with fluctuations in the density field ? and (c) Is the observation in Paper I of a slight increase in the FWHM fil of filaments with increasing external pressure ( P ext ) inconsistent with the analytic prediction of Fischera & Martin (2012) ? \nThe latter investigation of a correlation between the filament width and the external pressure is necessary because the apparent universality of filament width has been called into question by some observational findings. For example, larger filament widths, typically between ∼ 0.26 pc and ∼ 0.34 pc , were reported in more distant ridges (e.g., Hennemann et al. 2012), and in the Galactic plane filaments (e.g., Schisano et al. 2014) using Herschel . Furthermore, Panopoulou et al. (2014) also observed filaments with larger FWHM fil s in 13 CO emission. Thinner elongated structures with FWHM fil s /lessorsimilar 0.05 pc , on the other hand, have been identified among sub-filaments and fibres using interferometric observations of dense molecular tracers (e.g., Fern'andez-L'opez et al. 2014, Hacar et al. 2018, Dhabal et al. 2018). \nIndeed, Panopoulou et al. (2022) argued that filaments farther away must appear wider due to their poorer resolution. This suggestion no doubt contradicts much of the observational evidence in contemporary literature which shows that ambient environment, i.e., external pressure bears upon the physical properties of the density structure in the ISM. Interestingly, however, convergence tests by Andr'e et al. (2022) reinforce the conclusion by Arzoumanian et al. (2011, 2019) about the existence of a physical lengthscale on the order of ∼ 0.1 pc , at least for the filaments in the Solar neighbourhood. Results by Andr'e et al. (2022) therefore underscore the need for a theoretical framework to reconcile this lengthscale. In view of the suggestion that filament widths roughly correspond to the sonic length (e.g., Federrath 2016), it is plausible that the underlying variations in the local sonic length, or equivalently, in the local Jeans length of the filaments (Anathpindika & Freundlich 2015) generate the observed variation in filament widths. \nWe also explore in this paper the variation of the fraction of putative star-forming gas as a function of \nexternal pressure to understand the impact of ambient environment on the efficiency of star formation. The paper is organised as follows - the numerical method and the initial conditions for the simulations are presented in § 2. The results from our numerical Simulations are then presented and discussed in § 3 and § 4, respectively. We conclude in § 5.", '2 NUMERICAL METHOD AND SET UP': "Numerical Simulations discussed in this work were developed using the SPH code SEREN (Hubber et al. 2011), the features of which were described in Papers I & II. As in Papers I & II, in this work also we model a section of a typical filament as a cylinder of gas having initially uniform density, the length and radius of which are, respectively, L fil = 1 pc and r fil = 0.2 pc for different choices of the initial linemass, f cyl 2 , listed in column 7 of Table 1. Besides these, the filament is also characterised by the initial gas temperature, T gas , listed in column 10 of Table 1. With the critical linemass, M l crit , known at the temperature T gas , the initial mass of the filament listed in column 4 of Table 1 can be readily calculated. \nNote that these are only fiduciary choices of the respective physical parameters meant to represent an early phase during the evolutionary cycle of typical filaments in nearby clouds, and in the more dense elongated IRDCs . We prefer to commence our simulations with a uniform density configuration so that the impact of external pressure on the final density profile of model filaments can be qualitatively and quantitatively studied. Gas within this cylinder is assumed to have the usual molecular composition of the Solar-Neighbourhood. This cylindrical distribution of gas is allowed to accrete mass radially. \nThe set-up of the cylinder and the envelope of gas to be accreted by it is confined by a jacket of particles representing the intercloud medium ( ICM ), which is then placed in a periodic-box having dimensions (0.66,0.66,1.08) pc . We employ ordinary sink particles so that particles exceeding the minimum resolvable density ( ∼ 10 -15 g cm -3 ) are replaced with sinks. This choice of density threshold is good enough to represent typical protostellar objects i.e., the adiabatic contractional phase of a core that spans a density range between 10 -13 g cm -3 and 10 -18 g cm -3 during which it becomes optically thick (e.g., Larson 1973; Low & Lynden-Bell 1976). We note, however, that collating statistical properties of sink particles is not our extant interest. The truncated nature of our model filaments is immaterial because the length or radius of a filament has no bearing on its evolutionary sequence. For it is the density which determines the evolutionary timescale and the fragmentation lengthscale of a filament (e.g., Nagasawa 1987). \n<!-- image --> \nFigure 1.: Cartoon showing the schematic set-up of the test filament and the envelope of gas to be accreted by it. Shown on the left hand is a cross-section of this system; r fil and r env here denote the initial radius of the filament and the radius of the envelope of gas to be accreted, respectively. \n<!-- image --> \nTable 1: Physical parameters for simulations discussed below. \n- b Pressure due to the externally confining intercloud medium ( ICM ). \n- e Mass of individual SPH particle\n- f Initial linemass relative to the critical linemass at temperature T gas (see text). \nThe simulations discussed in this work have an initial average smoothing length h avg = 0.01 pc so that the spatial resolution is 0.02 pc . Following Eqn. (2) in Paper II and with the present choice of filament dimensions, the number of gas particles in the computational domain is N gas ∼ 245000, while that of particles representing the externally confining intercloud medium ( ICM ) is N ICM ∼ 455000. The mass of individual SPH particle, M SPH , for each simulation has also been listed in Table 1. By comparison, the Jeans mass at the minimum achievable temperature (i.e., ∼ 5 K ), and the highest resolvable density (i.e., ∼ 10 -15 g cm -3 ) in these simulations is ∼ 0 . 01 M /circledot , meaning that the Jeans mass is well resolved and satisfies the criterion defined by Hubber et al. (2006) which is the SPH equivalent of the Truelove criterion of minimum resolution in adaptive mesh codes. Even in the worst case scenario, i.e., in Cases 5, 10 & 13 the minimum resolvable mass is comparable to the Jeans mass meaning that we are unlikely to see fictitious fragmentation. Simulations in this work were developed with periodic gravity. The cartoon in Fig. 1 shows a schematic representation of this set-up. \nM acc is the mass of the gas available to be accreted by the filament which is the third free parameter in this set-up besides the initial temperature of the gas in the filament, T gas , and the initial filament linemass, f cyl . This is because we assume that the gas being accreted is transonic so that its Mach number, M inf = 1, and so its inflowing velocity, V inf = a inf , its sound-speed. The model filament is placed in a silo of gas to be accreted by it, the thickness of which is quantified in terms of the average SPH smoothing length and given in multiples of an integer parameter, η ; here η = 2 across all simulations. Thus the rate of accretion, ˙ M acc = M acc · V inf ( r fil + ηh avg ) . The entire set-up is arranged such that there is initially approximate pressure equilibrium across the interface between the filament and the envelope of gas to be accreted. We assume no density contrast between gas in the filament and that to be accreted, or between the gas and the ICM . For any density contrast will naturally create an additional pressure gradient which will further affect filament evolution. We wish to avoid any such possibility. \nThe model filament here is initially superposed by sub-sonic turbulence (initial Mach number M int =0.5) with a power spectrum ∼ k -3 . The choice of a relatively steep power-spectrum is consistent with that for the diffuse optically thin gas at intermediate Mach numbers (e.g., Stanimirovi'c et al. 1999; Stanimirovi'c & Lazarian 2001). The injected turbulence comprises of a natural mixture of the solenoidal and compressional modes. Thus the filament is supported against self-gravity by a combination of thermal pressure and pressure due to the turbulence. The expression for initial pressure balance is, \nP int = ( P therm int + P turb int ) ≡ P ext = -( P inf + P therm ext ); (1) \nwhere P int , P ext , and P inf ( ≡ ρ gas V 2 inf ) are, respectively, the internal pressure, the pressure due to the ICM , and the ram pressure due to the accretion of inflowing gas experienced by the filament; and ρ gas is the average initial density of the gas accreted by the filament. P therm int and P turb int are, respectively, the thermal component of internal pressure and the component of pressure due to internal turbulence. Finally, P therm ext is the thermal component of pressure exerted by the inflowing gas. The -ve sign on the right hand side of Eqn. (1) indicates that this pressure is directed radially inward. \nRearranging Eqn. (1) yields, \nT ICM = ( P int + P inf + P therm ext ) n ICM . (2) \nWith M inf = 1 by choice, the initial gas temperature, T gas , temperature of the gas being accreted, T acc , the initial linemass, f cyl , and the mass accreted, M acc , are the only free parameters while the ICM temperature is calculated according to Eqn. (2). These physical quantities together contribute to the net external pressure denoted as P ext and listed in column 3 of Table 1. To test the convergence of results, we repeated this ensemble of simulations with five different random number seeds used to generate the initial turbulent velocity field that was initially overlaid on the filaments in the respective simulations. So we developed 65 simulations in all. \nThe motivation for this toy-model of accreting filaments has been discussed in the prequel to this paper. While the Mach number of the inflowing gas, M inf , could be crucial, the process of filament assembly is itself a subject of debate. It is unclear if turbulent flows directly assemble filaments (as in e.g., Federrath 2016), or if interacting turbulent gas-flows assemble sheets that subsequently fragment to form density structure. Recently, Rezaei Kh. et al. (2022), for example, suggested that the California MC is sheet-like. Such sheet-like clouds could appear filamentary when projected edge-on on the plane of the sky. Alternately, they could fragment to generate filaments, a mechanism completely different from filaments forming in a box of turbulent gas. So we do not introduce here another variable in the form of M inf . Listed in Table 1 are our choices of various parameters that allow us to span a range of external pressures between a few times 10 4 K cm -3 to a few times 10 7 K cm -3 . \nFor context, the ambient pressure is the lowest in the farthest regions of the Galactic disk and it is upward of ∼ 10 7 K cm -3 towards the Galactic Central Molecular Zone ( CMZ ). Observational works by (e.g., Goodman et al. 1993; Caselli et al. 2002; Pineda et al. 2010 & Hacar et al. 2013) show that relatively dense sub-structure, i.e., clumps & cores, in the Solar-Neighbourhood are thermally supported while turbulent non-thermal motions dominate in the transition region between the dense clumps/cores and the surrounding ICM . For our own MW galaxy the average mid-plane pressure is estimated to be ( P ext /k B ) [ K cm -3 ] ∼ 3 × 10 4 , and includes contributions in roughly equal measures from the diffuse thermal pressure, magnetic field pressure, non-thermal pressure and pressure due to cosmic rays (e.g., Boulares & Cox 1990; Slavin & Cox 1993; Oey & Garc'ıa - Seguera 2004). The pressure is of course higher in active star-forming regions (e.g., Henshaw et al. 2014). The co-existence of a warm tenuous ICM with the cold dense clumps/cores has led authors to conjecture that MC s themselves constitute a multi-phase medium (e.g., McKee 1995 and other references therein). The process of segregation of the respective phases in a purely hydrodynamic calculation was demonstrated in an earlier contribution (Anathpindika 2015). \nEstimates of the average external pressure do not reflect the possible effects of turbulence induced variations in magnetic field strengths. Seta & Federrath (2022), for instance, show that turbulent dynamo, i.e., the generation/amplification of magnetic field in the interstellar medium as the gas is randomly stretched by a turbulent velocity field, can affect the strength of the magnetic field. Studying the impact of such localised enhancement of magnetic field on filament-evolution is, however, beyond the scope of this work as we assume an isotropic external pressure that is a combination of only the thermal and the ram-pressure of the inflowing gas. \nFor the more violent environment such as towards the Galactic CMZ , Federrath et al. (2016), for instance, derive 3D turbulent sonic Mach number of 11 ± 3 for gas in the Brick . This Mach number, though comparable to that for typical molecular clouds in the Solar-Neighbourhood, corresponds to extreme pressure towards the CMZ . The assumed T acc in Table 1 (Cases 8 - 10) is consistent with the observationally inferred gas temperature towards the CMZ (e.g., Rodr'ıguezFern'andez et al. 2001, Mills & Morris 2013, Ao et al. 2013, Rathborne et al. 2014a). Ao et al. , for example, confirmed warm dense gas at temperatures ∼ 100 K through observations of formaldehyde while Rodr'ıguezFern'andez et al. reported hot dense gas (temperature ∼ 400 - 600 K , and density /lessorsimilar 10 6 cm -3 ) in gas envelopes around CMZ clouds through observations of rotational transitions of H 2 . Similarly, an extremely hot gas component (temperature ∼ 700 K ) has also been inferred towards SgrB2, one of the most massive and densest clouds in the CMZ that is known to be forming massive stars, through observations of absorption lines corresponding to higher NH 3 transitions (e.g., Ceccarelli \net al. 2002, Wilson et al. 2006). \nThe values of external pressure P ext so obtained for simulations 1 - 5 are consistent with those inferred observationally for clouds in the outer regions of the Galactic disk and in the Solar-Neighbourhood (e.g., Murphy & Myers 1985, Maddalena et al. 1986, Bally 1987, Loren 1989, Tatematsu et al. 1993). The warm gas in Cases 6 & 7 is representative of the ambience in an active local star-forming region like the Orion MC -complex. Previous observations also show that typical rates of accretion by filaments vary between a few 10s to a few 100s M /circledot Myr -1 pc -1 (e.g., Kirk et al. 2013, Palmeirim et al. 2013, Schisano et al. 2014, Bonne et al. 2020, Gong et al. 2021). So the accretion rates assumed here are consistent with such inferred values. \nFor the sake of convenience, we classify the simulations discussed in this work into three types on the basis of the mass accreted by the respective filaments : Type I (Cases 1, 3, 6, & 8) , where the initially sub-critical filaments were allowed to accrete gas but still remained sub-critical (i.e., f cyl < 1); Type II (Cases 2, 4, 7, & 9 ) , where the initially sub-critical filaments were allowed to accrete enough gas such that they became super-critical (i.e., f cyl > 1); and Type III (Cases 5 & 10) , the initially supercritical filaments, that of course remain supercritical. For reference, Table 2 summarises the nomenclature adopted for discussion of the simulations in the rest of this paper. Observe that the various cases discussed in this work have been listed in Tables 1 & 2 in the order of increasing pressure for each type. We repeat one simulation for each filament type with supersonic initial turbulence, i.e., M int =7. These are listed as respectively, Cases 11, 12 & 13 in Tables 1 & 2. \nFinally, given the universality of the paradigm of structure formation via interaction between turbulent flows, the physics of filament formation must essentially remain the same across environments. So the model of an accreting filament assumed in this work, though simplified, is robust.", '2.1 Limitations': 'The numerical set-up described above and indeed, in the sequel to this paper, is simplified because with some notable exceptions like the Musca filament (e.g., Hacar et al. 2016), isolated filaments are seldom observed in the field. Hub-filamentary structure, i.e., converging network of filaments are often reported in typical star-forming clouds (e.g., Peretto et al. 2021; Kumar et al. 2022). We have, however, preferred the relatively simple set-up of a singular cylinder here because our primary interest is to study the process of filament-fragmentation and not how individual filaments and/or bunch of filaments are assembled in MCs. This latter objective is itself the subject of study of a subsequent paper. \nWhile a number of studies demonstrate the importance of magnetic field in the evolution of the ISM (e.g., \nTable 2: Classification of simulations listed in Table 1 on the basis of f cyl . \nPadoan & Nordlund 2011; Krumbolz & Federrath 2019), the set-up here is purely hydrodynamic. The general consensus of these works, and indeed in other literature, is that the presence of magnetic field dampens the rate of star-formation although the actual impact is fairly modest and that the presence of magnetic field typically reduces the star-formation rate by only a factor of 2-3. In his analytic work, Nagasawa (1987) further showed that the magnetic field slowed the growth of the gravitational instability in a self-gravitating cylinder without affecting the wavelengths of unstable modes. \nEvidently, the absence of magnetic field in this work is therefore unlikely to significantly alter the final outcome (except perhaps the fragmentation timescale) of the simulations discussed herein. However, we appreciate that magnetic fields in a turbulent medium could enhance filament-formation, though probably reduce the number of clumps, cores & stars. Conversely, by stabilising MCs against self-gravity magnetic fields could possibly induce formation of more massive stars and thus enhance the impact of stellar feedback (Hennebelle & Inutsuka 2019), leading to a higher external pressure, but still within the range of external pressure explored in this work.', '3.1 GENERAL EVOLUTIONARY FEATURES OF THE FILAMENTS': 'We have seen in our earlier works that filaments evolve through a series of radial contractions, i.e., circular contractions in the direction perpendicular to the filament axis, followed by fragmentation along the axial direction, irrespective of whether they are accreting (as in Paper I where filaments accreted so little gas that linemass did not vary significantly), or non-accreting (Paper II). As with the filaments in those works, filaments in this work are also allowed to cool dynamically so that the resulting loss of pressure \nFigure 2 shows how C evolves over time (top panel), and the external pressure (central panel), and how n gas varies with radius at the terminal epoch of the simulation (bottom panel). Plots in the top & the central panels show the mean value of C across simulations developed with 5 different random number seeds. The density profiles being mutually similar across simulations with different random number seeds, we show the profiles from only one set. The temporal evolution of the filament radius can be easily inferred from that of C as shown on the top panel of Fig. 2. Also evident from this panel is the fact that with the exception of the filaments in Cases 11-13, the concentration parameter decreases with increasing external pressure. This latter aspect, which essentially means that filaments (again with the exception of those in Cases 11-13), must become thinner with increasing external pressure, is more clearly visible from the plots in the central panel of Fig. 2. Naturally, the filaments in Cases 11-13 also have a bigger central radius, r flat , as can be seen from the corresponding plots on the lower panel of Fig. 2. Evidently, the initially supersonic turbulence in the filaments in Cases 11-13 causes them to be puffed up. They thus have a bigger central radius. This observed variation in the inner radius is also likely to manifest itself in a similar trend for the filament width ( FWHM fil ), a point that we will revisit later in § 3.2.3. \n<!-- image --> \n1 \n0 \nFigure 2.: Top-panel: Temporal evolution of the concentration parameter. Central-panel: The concentration parameter as a function of the external pressure. Data points on both these plots represent the value of C at different epochs of filament evolution. The continuous line here represents the locus of mean values of C for each simulation. Lower-panel: Radial density profiles of the filaments in respective simulations at the terminal epoch of their evolution. \nequilibrium triggers radial oscillations. The temporal evolution of filaments in this work is briefly described in Appendix A and the rendered density images in Figs. A1 - A3 show the evolutionary sequence of the filaments in some of the simulations developed in this work. \nThe sub-critical Type I filaments evolve on rela- \ntively long timescales, i.e., on the order of a few free-fall times and comparable to the e folding timescale, while the super-critical Type II & III filaments evolve more rapidly, on timescales comparable to, or shorter than, the freefall time. As was seen in Papers I & II as well as in the semi-analytic work by Fiege & Pudritz (2002), for example, the filaments remain pressure truncated throughout the courses of their evolution. Like the latter authors, we calculate the concentration parameter, C , defined as log 10 ( r fil ( t ) r 0 ( t ) ) , where r fil ( t ) is the outer radius of the filament at the epoch t , and the natural radial scale factor, \nr 0 ( t ) = a eff ( t ) √ 4 πGρ c ( t ) , (3) \nwhere a eff and ρ c are the effective sound speed and the central density calculated at an epoch t , respectively; r 0 ( t ) defines the effective core radius of the filament (Fiege & Pudritz 2002). Although analogous to the truncation parameter defined for the King models of globular clusters (e.g., Binney & Tremaine 1987), the tidal radius in its definition is replaced here with the outer filament radius, r fil . The concentration parameter thus specifies the radius of truncation of a filament and is therefore a useful proxy for the filament width. \nFinally, the fact that these filaments evolve to have truncated density profiles irrespective of the external pressure is evident from the plot on the bottom panel. Note that these density profiles have been made after eliminating the SPH particles representing the ICM . The radial density distributions within filaments are largely \n<!-- image --> \nFigure 3.: Temporal variation of the radial component of the Divergence of the velocity field within the filament. \n<!-- image --> \nconsistent with the Plummer profile given as, \nρ ( r ) = ρ c [1 + ( r/r flat ) 2 ] p 2 , (4) \nwhere r flat is the spatial extent of the inner flat portion of the density distribution, while the exponent p varies between 2 and 4 for the magnitudes of external pressure considered here. Indeed, some of these density profiles also exhibit a distinct knee so that the profiles are shallower in the outer regions than closer to the centre, and the exponent p also varies over the course of evolution of individual filaments. Interestingly, however, the exponent, p , becomes steeper with increasing external pressure. As previously noted, with the exception of the filaments in Cases 11-13, the central radius, r flat , decreases with increasing external pressure, which is consistent with the trend between the concentration parameter, C , and the external pressure. In fact, r flat becomes vanishingly small for external pressures upward of /greaterorsimilar 10 6 K cm -3 . Rendered density images in the Appendix (Figs. A1-A3) below illustrate the impact of ambient environment on the morphology of filament evolution. Collectively these images show, filaments, irrespective of their linemass, must become eviscerated in high pressure environs (as we shall also see in the following subsection). In low-pressure and Solar-like environs, however, weakly self-gravitating filaments must evolve relatively slowly and form broad cores via the Collect and Collapse mode. \nThe leftand right-hand panels of Fig. 3 show the temporal variation of the radial component of the divergence of the velocity field ( ∇· V ) r within the model filament in Cases 3 ( Type I filament) and 10 ( Type III filament), simulations representative of filament evolution in lowand high-pressure environments, respectively. ( ∇· V ) r is essentially the component of ( ∇· V ) in the radial direction within the filament. Bear in mind that negative values of ( ∇· V ) r denote inward gas-motion, whereas positive values signify gas moving radially outward. For Case 3, the gradual inward contraction of the accreting filament over the course of its evolution is evident from the steadily decreasing ( ∇· V ) r towards the filament axis. Only towards the \nFigure 4.: Temporal variation of the fraction of mass having density /greaterorsimilar 10 18 g cm -3 in different simulations. Individual data points in this plot represent the mean value of M frac at each epoch. See text for further explanation. \n<!-- image --> \nterminal epoch of the simulation when a core begins to collapse do we see a relatively strong inwardly directed velocity gradient. As with the plot in the bottom panel of Fig. 2, the plots shown on either panel of Fig. 3 are also from a simulation out of the ensemble developed with 5 different random number seeds for each case. \nFor Case 10, by contrast, a radially inward travelling compressional disturbance is readily visible in the plot on the right-hand panel of Fig. 3. Note, however, that this disturbance is not a shock wave because the gas that is accreted is transonic ( M inf =1). Indeed, the filament experiences ram-pressure due to this inflowing gas. The accreting filament contracts rapidly which causes pressure to build-up within it. The filament then relaxes, as can be seen from the temporal variation of ( ∇· V ) r in this plot. As the filament in either case evolves through a series of contractions and relaxations, some pockets of gas move inward while others move radially outward which is also significant from an observational perspective. This is because depending on the evolutionary stage of filament, pockets of gas at', 'P ext /k B = 1.21x10 7 K cm -3 (Case 10)': 'Figure 5 (left) shows the temporal variation of the dense gas fraction based on column density for all simulations. Here, the dense gas fraction steadily increases to between 10% - 20% in filaments experiencing external pressure similar to that in the Solar -Neighbourhood (i.e., typically in the range of a few times 10 4 K cm -3 -a few times 10 5 K cm -3 ). For higher external pressures, i.e., upward of ∼ 10 6 K cm -3 , there is a rapid decline in dense gas fraction over time. This behaviour occurs irrespective of the linemass, because such filaments buckle (i.e., experience rapid distortion of their cylindrical geometry) as perturbations on their surfaces amplify rapidly. Indeed, this rapid decline in dense gas fraction could also help reconcile the inefficient nature of star \n<!-- image --> \n<!-- image --> \np \nFigure 5.: Left panel : Temporal variation of the dense gas fraction in different simulations. Right panel : NPDF of molecular Hydrogen in filaments at their respective terminal epochs. As in Fig. 4 the lightly shaded region about each characteristic in either plot represents the variation of the respective quantities over the realisations developed with 5 random number seeds for different choices of the external pressure. \ndifferent locations within it could show signatures of radial expansion or contraction. \nIn the current set of simulations, the inwardly travelling compressional disturbance sets the filament boundary, or equivalently the filament width, unlike in the recent study by Priestley & Whitworth (2021), for instance, where the filament boundary is set by the location of the accretion shock. A decrease in filament width with increasing external pressure is clearly evident from Fig. 3 which also re-emphasises the observation made earlier regarding the central panel of Fig. 2 that shows the variation of the concentration parameter, C , as a function of the external pressure. This inference about possibly thinner filaments at higher external pressures, at least for filaments suffused with subsonic turbulence, though consistent with analytic predictions (e.g., Fischera & Martin 2012), seems at first to be inconsistent with the results we reported in Papers I & II. This inconsistency, as previously noted, is largely because the filament linemasses in those papers barely varied over a factor of a few in the course of their evolution. Naturally then, the self-gravity of those filaments did not significantly change, as they do here.', '3.2.1 DENSE GAS FRACTION': 'The dense gas fraction is a useful proxy to estimate the efficiency of star formation in molecular clouds (see, e.g., Lada et al. 2009). Here we invoke the same definitions used in an earlier work (i.e., Anathpindika et al. 2017) to quantify the fraction of gas in a filament cycled into the dense phase. Recall that we had then used two physical parameters, viz., the fraction of gas having volume density upward of typically ∼ 10 18 g cm -3 and that having column density upward of ∼ 10 21 cm -2 to quantify the fractions of putative star-forming gas. We note that the column density is calculated by taking a \nprojection of these filament in the plane orthogonal to the filament-axis. \nFollowing the definitions of dense gas, Fig. 4 shows the temporal variation of dense gas fraction based on volume density, M frac , in our simulations. As with the plots on the top & the central panels of Fig. 2, individual data points on this plot represent the mean of M frac calculated over the ensemble of realisations developed with 5 random number seeds for each choice of the external pressure, P ext . The lightly shaded region about each characteristic joining these data points represents the variation of M frac about its mean value. As can be seen from this plot, M frac increases steadily irrespective of whether the filament is of Type I or Type II . In general, however, the rate of cycling gas in to the dense phase is slower in Type I filaments than in the Type II filaments. Finally, while the fractional mass, M frac , builds up gradually even for the initially super-critical Type III filament in a Solar-type environment (Case 5), it increases rapidly in a high-pressure environment (e.g., Cases 10 & 13). Type III filaments in general evolve relatively quickly. Evidently, higher external pressure facilitates cycling gas to higher volume densities. We will next explore if such high density filaments are also conducive to star formation. \nFigure 6.: Axial component of velocity gradient for filaments at different epochs of their evolution for different choices of external pressure. Given the identical nature of these plots and in the interests of brevity, we show here only a few cases such that they span the entire range of P ext & linemasses explored in this work. As with the plot on the lower panel of Fig. 2, these plots are also made from one of the simulations developed with 5 different random number seeds for each P ext . \n<!-- image --> \nFigure 7.: Same as in Fig. 6 but now showing the radial component of velocity gradient. \n<!-- image --> \nformation in high-pressure environments. \nThis observed buckling of filaments in high-pressure environments is qualitatively similar to the evolution of shock-confined slabs assembled by gas-flows converging at super-sonic velocities. Such slabs are susceptible to the Shell Instability that manifests itself through rapid amplification of density perturbations on the slab surface due to the transfer of momentum between density crests and troughs, causing the shocked slab to buckle. This buckling motion of the slab (i.e., the \nfilament in present simulations) cycles dense gas within it into the rarefied phase (e.g., Anathpindika et al. 2017). \nIn some of our earlier works we showed that while a higher external pressure may effectively cycle gas to higher densities, the fraction of putative star-forming gas actually decreases with increasing external pressure (Anathpindika et al. 2017, 2018). Now since a higher inflow velocity, V inf , essentially means a larger external pressure, this implies that filaments with a higher rate of gas inflow buckle analogously to the behaviour of shocked \nslabs. So, while accretion enhances the filament mass, even trans-sonic accretion leading to pressures upward of ∼ 10 6 K cm -3 is likely to destabilise the filament and render inefficient the transfer of gas into the dense phase. \nFinally, the filament in Case 5 was initially supercritical and yet the dense gas fraction there declines sharply on a relatively short timescale with external pressure comparable to that in the Solar Neighbourhood, i.e., P ext /k B ∼ 10 5 K cm -3 . In this case, however, the filament contracts on a relatively short timescale and the formation of cores within it appears to be roughly simultaneous. As can be seen in the top left-hand panel of Fig. A3 the filament in this case contracts to form a thin dense spine which explains the low dense gas fraction. Interestingly, this case is the only one with relatively low external pressure in which the dense gas fraction decreases rapidly despite the steady contraction of the filament. In other cases (viz., Cases 6-10) where a similar decline in the dense gas fraction is also observed, the filaments buckled, segregating pockets of dense gas within them which causes the dense gas fraction based on column density to fall. These observed general trends in the temporal variation of the dense gas fraction based on volume density, and on the column density hold true even if M int /greatermuch 1 for the respective filaments (as in cases 11, 12 & 13.) We revisit this point in § 4. \nFinally, Fig. 5 (right) shows the column density probability distribution function, i.e., the NPDF of the ten cases at the terminal epochs of their respective simulations. NPDF , the probability ( p N ) that the column density lies between N and N + dN , is a useful diagnostic to determine the star-forming ability of MC s (e.g., Federrath & Klessen 2012, Kainulainen et al. 2009, Schneider et al. 2013, Stutz & Kainulainen 2015). NPDF s for MCs exhibit a density peak before turning over into the higher and lower density ends of the gas distribution. For typical star-forming regions, in fact, the NPDFs exhibit a well-defined power-law extension with a negative slope towards higher column densities (e.g., Goodman et al. 2009, Kainulainen et al. , 2013). That only a few filaments discussed here had begun forming stars (represented by sink particles) at the time respective simulations were terminated, is evident from the shapes of these NPDFs . Since we do not follow the evolution of individual cores that form in our filaments, it is unsurprising that these NPDFs do not exhibit power law extensions into higher column densities. Nevertheless, the NPDFs derived here are qualitatively similar to those reported for typical filamentary Herschel clouds.', '3.2.2 QUASI-PERIODIC OSCILLATIONS OF THE VELOCITY GRADIENT': "Periodicity in density & velocity structure along filaments has been known for quite some time now (e.g., Schneider & Elmegreen 1979; Dutrey et al. 1991). These suggestions have been confirmed further by several recent observations (e.g., Henshaw et al. 2020; Chen et al. 2020). While the determination of the amplitude of velocity fluctuations may be subject to instrumental \nsensitivity and uncertainties in estimating chemical abundances, the appearance of fluctuations themselves seems to be ubiquitous. So our aim in this work is merely to explore the impact of P ext on the periodicity & on the relative change in the amplitude of such fluctuations in the velocity field. \nFigure 6 shows the axial component of the velocity gradient, i.e., ( ∇ V ) a , along the lengths of the filaments in our simulations at different epochs of their evolution. We calculate the velocity gradients at different locations along the length of the filament first by determining the velocity peaks at these locations, i.e., in the plane of the filament axis, followed by calculating the difference in these velocity peaks over the distance between the locations. By velocity peaks , we mean here maxima in the velocities of gas particles in a slice taken at each location along the filament-axis and may be treated as proxies for the observationally determined line of sight velocities towards filaments. Gradients calculated in this manner are then resolved into their respective components in the axial direction along the filament-axis, i.e., ( ∇ V ) a , and in the radial direction orthogonal to the axis, i.e., ( ∇ V ) r . \nFour crucial inferences can be drawn from these plots (i) ( ∇ V ) a increases with increasing external pressure, or equivalently, with increasing velocity of the gas being accreted ( V inf ); (ii) ( ∇ V ) a observed here is typically 2-3 times higher than those observed in Paper II for non-accreting filaments. Interestingly, the accreting gas does not damp out these quasi-oscillatory features of ( ∇ V ) a . In fact, given that the trigger and subsequent amplification of these velocity fluctuations is qualitatively similar to the typical features of the shell instability (e.g., Vishniac 1983), it is likely that inflows with higher Mach numbers ( M inf ) will generate even larger velocity gradients, (iii) the ambient environment seems to have little impact on the periodicity of the fluctuations in the velocity-field, although not all of them grow, and (iv) the initially supersonic gas in the filament (i.e., Cases 11, 12 & 13) does not affect the peak amplitude of ( ∇ V ) a as it appears consistent with that observed in other simulations with comparable external pressure, but suffused with initially subsonic gas. Nevertheless, the initially supersonic gas seems to excite many more unstable modes as is evident from the higher periodicity of fluctuations in the velocity-field. \nPlots in Fig. 7 are similar to those in Fig. 6 but now show the radial component of the velocity gradient, i.e., ( ∇ V ) r . As with the axial component seen in the magnitude of ( ∇ V ) r also increases with increasing external pressure, i.e., with a higher inflow velocity ( V inf ), though it is generally comparable to the axial component and like it, also exhibits oscillatory features. Recall that such oscillations in ( ∇ V ) r were also seen in the case of non-accreting filaments discussed in Paper II which suggests that oscillatory features in ( ∇ V ) r appear irrespective of whether a filament accretes gas. While we observe a higher magnitude of ( ∇ V ) r for a higher external pressure, ( ∇ V ) r peaks correlate more strongly with the external pressure than with the inflow velocity, \nFigure 8.: Similar to plots in Figs. 6 & 7, but now shown for only the terminal epoch of the filaments in respective simulations and superposed with the location of density peaks along the filament-length. \n<!-- image --> \nas reflected by Spearman rank coefficients ( S n ) of 0.69 and 0.48, respectively. This behaviour suggests that ( ∇ V ) r is more sensitive to the external pressure than it is to the inflow velocity. Inclusion of the filaments with initially supersonic gas in Cases 11-13 does not significantly alter the ( ∇ V ) r -P ext correlation. \nRecently, Chen et al. (2020) also reported similar oscillatory features in the radial direction along filaments in the NGC 1333 region. They could not, however, conclusively identify any signatures of accretion onto the observed filaments. While this lack of evidence could \nbe due to the limited sensitivity of their data, results presented here together with our earlier results in Paper II, however, suggest that the detection of oscillatory features in ( ∇ V ) r towards filaments is not a good proxy for detecting signatures of accretion. Indeed, such features can also appear in non-accreting filaments, a conclusion that is reinforced by the analytic work of Gehman et al. (1996), as well as by the observation here that ( ∇ V ) r peaks are more strongly correlated with P ext . \nFigure 8 shows the radial component ( ∇ V ) a with the density peaks overlaid on it. Continuous and dashed lines in these plots represent respectively the velocity gradient, and the density peaks in each plot. Evidently, the fluctuations of ( ∇ V ) a roughly correlate with the location of density enhancements along the length of the filaments. That these two quantities are indeed correlated is also reflected by the Pearson's Normalised cross-correlation which varies between ∼ 0.75 - 0.87 for these respective plots. In other words, the density and velocity peaks seem co-spatial, though not all perturbations condense out to form cores. The correlation between the density and velocity peaks is also visible in the filaments with the initially supersonic gas (i.e., Cases 11, 12 & 13). Perturbations in all our simulations are seeded purely by white-noise and the typical separation between them is, \nλ sep = 22 . 1 a eff (4 πGρ c ) 1 2 (5) \n(Nagasawa 1987); where a eff is the effective sound speed and ρ c is the mean central density of the filament. \nFor the typical values of physical parameters observed in these simulations, λ sep /lessorsimilar 0.1 pc , which in fact, is also comparable to the separation calculated according to the Gehman et al. analysis. The fact that only some perturbations condense out is especially true for the Type I filaments (i.e., plots corresponding to Cases 1, 3, 6, 8 & 11), and for the Type II filament in Case 12 with initially supersonic gas. Growth of other perturbations in these cases is stymied as they do not acquire sufficient mass which is the essence of the collect-and-collapse mode. Evidently, occurrence of quasi-oscillatory features in the respective velocity gradients is merely indicative of the presence of spatial perturbations along the length of the filaments.", '3.2.3 EXTERNAL PRESSURE AND THE FILAMENT FULL WIDTH AT HALF MAXIMUM (FWHM fil )': 'Figure 9 (left) shows the temporal variation of the FWHM fil of the filament for the set of simulations discussed above. Estimation of the FWHM fil has been described in detail in Paper I. Briefly, we calculate the FWHM fil by measuring the width of the lognormal distribution fitted to the column density across the cross-section of the filament at various locations along its length. If σ r is the dispersion of the corresponding estimates of the filament width, i.e., the average dispersion of these estimated filament widths, then FWHM fil = 2( √ 2 ln 2) σ r . While the FWHM fil s for the Type I filaments in Cases 1 and 3 where the external pressure \nFigure 9 (right) further illustrates that filament width progressively diminishes with increasing external pressure with the exception of the filaments in Cases 11-13. This correlation for the first 10 Cases is consistent with the analytic prediction by Fischera & Martin (2012). Evidently, the filament width is unlikely to be uniform across different environments. Thus it appears that only the filaments with subsonic gas in Solar -Neighbourhood exhibit widths on the order of ∼ 0.1 pc . Similarly, filaments with with subsonic gas in more hostile environments are likely to thinner, but also severely eviscerated. Conversely, those with supersonic gas are likely to be puffed up. \n<!-- image --> \nFigure 9.: Left - panel: Temporal variation of the FWHM fil . As in the Figs. 2, 4 & 5 the lightly shaded region about each characteristic in either plot represents the variation of the FWHM fil over the simulations developed with 5 random number seeds for different choices of the external pressure. Right - panel: FWHM fil as a function of external pressure. Multiple data points for a given external pressure were generated by calculating the FWHM fil at different epochs of filament evolution. The thick black line on this plot represents the locus of points corresponding to the mean FWHM fil for each simulation. As before, individual data points here represent the mean value of FWHM fil across the simulations developed with 5 random number seeds for different P ext . \n<!-- image --> \nis also low are relatively large, filaments suffused with initially supersonic gas in Cases 11-13 are the broadest in this ensemble of simulations and can be seen as the three outliers in this figure. It will be recalled from the plots in Fig. 2 that these are also the filaments with the biggest central radius, r flat , and with a relatively large concentration parameter, C . In the remaining cases, however, we observe that the FWHM fil almost always progressively diminishes over time.', '3.3 OTHER CORRELATIONS': 'The facts that accreting filaments contract radially to acquire a centrally peaked density profile and the accreted gas replenishes turbulence within them motivate these correlations. Figure 10 shows correlations between the column density of filaments, their linemass and the velocity dispersion (Virial parameter). As with similar plots discussed above, data points marked in various panels represent the magnitude of the corresponding physical quantity calculated at different stages of filament evolution. The top panel reveals that the filament linemass is rather poorly correlated with its column density (Spearman rank coefficient, S n =0.08) even when \nthe entire range of external pressure is considered. At relatively low external pressures (i.e., typically a few times 10 4 K cm -3 ; e.g., Cases 1-5), however, this correlation is even worse, and indeed barely visible. The central panel of Fig. 10 shows the correlation between the velocity dispersion ( σ gas ) and mean linemass from our simulations (Spearman rank coefficient, S n =0.11). As with the previous correlation, however, this correlation also appears to be rather weak for lower external pressures of typically only a few times 10 4 K cm -3 . \nFinally, the lower panel of Fig. 10 shows that the Virial parameter ( α vir ) weakly anticorrelates (Spearman rank coefficient, S n = -0.08) with the fractional linemass ( f cyl ) calculated at the mean temperature of the filament at that epoch. The Virial parameter, α vir = 2 σ 2 tot GM l , where σ 2 tot = σ 2 gas + a 2 gas , and a gas is the sound speed for the filament. Admittedly, we do not observe any meaningful correlation between these respective physical quantities. The vertical dashed line on this plot separates the thermally sub-critical filaments from those that are super-critical. Observe that most of our filaments irrespective of whether they are sub-critical/super-critical are virially bound ( α vir < 2), though a few of each are virially unbound. This observation is true even for the filaments initially suffused with supersonic turbulence, i.e., those in Cases 11-13. Admittedly, the boundedness of a filament must have little bearing on its ability to fragment and form putative star-forming cores. We observe that filaments in low pressure environments are typically thermally sub-critical, but virially bound. By contrast, filaments in Solar-type and in extreme environments such as in Cases 10 & 13 generally become super-critical, but can sometimes become virially unbound. \nRespective plots in Fig. 11 illustrate that the col- \nFigure 11.: Similar to the plots in Figs. 9 & 10, but now showing respectively the column density, and the velocity dispersion as a function of the external pressure. \n<!-- image --> \n<!-- image --> \n8 \numn density (left panel), and the velocity dispersion (right panel) both increase with external pressure, a result consistent with the analytic prediction by Fischera & Martin (2012). The velocity dispersion in the current simulations is the result of a combination of momentum injected by the accreting mass as well as the radial contraction. In fact, as with the respective components of ( ∇ V ), the velocity dispersion in the present simulations is also typically a factor of 2-3 higher than that reported in Papers I & II. Accretion of mass is also why we observe a much clearer trend of increasing column density with increasing external pressure. In Paper I where filaments remained relatively starved of mass, however, the dependence of column density on external pressure was weak even in high-pressure environments.', '4 DISCUSSION': 'We observe that accreting filaments evolve through a series of radial contractions and expansions, i.e., a series of inward and outward gas-motion in a circular direction perpendicular to the filament-axis, irrespective of their linemass and external pressure. Filaments in this work become radially unstable due to a combination of dynamical cooling, turbulent dissipation and accretion of gas that increases filament linemass. Irrespective of the external pressure, the filaments develop truncated density profiles. Simulations in this work show that the interplay between turbulence, thermal pressure & self-gravity can generate shallow density profiles. This behaviour is also consistent with the findings of the analytic work by Gehman et al. (1996) as well as some of our earlier numerical work (Anathpindika & Freundlich 2015). \nThese results show that filaments in low-pressure environments (i.e., P ext k B /lessorsimilar 10 4 Kcm -3 ) and those with initially supersonic gas must have a bigger inner radius (and by extension, a higher concentration parameter C ), and develop a shallow density profile with the Plummer exponent p ∼ 2. With the exception of the filaments in Cases 11 - 13 that were superposed with initially supersonic gas, filaments become thinner and \nhave smaller inner radius (and by extension, a lower concentration parameter C ), with increasing pressure. This latter behaviour, which is also consistent with the results from the semi-analytic work by Fiege & Pudritz (2002), can be seen from the plots shown on various panels of Fig. 2. Indeed, the broad filament in Case 13 is consistent with the findings of Federrath et al. (2016) about filaments in the Brick . \nWe note that filament-evolution in all our simulations, including that in Cases 11 - 13, is gravity dominated. Filaments in the latter, i.e., in Cases 11 - 13, are especially interesting because their evolutionary cycle is remarkably different from the turbulence dominated fray and fragment mode observed by Clarke et al. (2017) in those that are initially suffused with supersonic turbulence. In purely isothermal simulations, Clarke et al. observed the formation of fibrous sub-structure within filaments irrespective of whether they were dominated by the compressional or the solenoidal mode of turbulence. We note, however, that unlike the pressure-confined filaments in this work, the accreting filaments modelled by Clarke et al. were essentially placed in vacuum. Furthermore, ˙ M acc assumed in the simulations discussed here is a factor of 6 to 11 smaller than that assumed by Clarke et al. meaning that the turbulence within our filaments is not sufficiently replenished over the course of their evolution. \nVarious rendered images in Figs. A1 - A3 show that accreting filaments in low-pressure environments evolve on relatively long timescales which are comparable to, or greater than the so-called e-folding ( e fold ) timescale. The so-called geometrical mode of filament fragmentation identified by Gritschneder et al. (2017) and Heigl et al. (2018), which manifests itself through the growth of density perturbations, can be seen in the Type I (i.e., the initially sub-critical filaments that remain so even after accreting gas, f cyl < 1) filaments of Cases 1, 3, 6, 8 & 11. This kind of behaviour is also observed in the Type II filament in Case 12 that was initially suffused with supersonic gas. \n5 \nFigure 10.: Similar to the plots in Fig. 9, but now showing respectively the column density, velocity dispersion, and the Virial parameter are shown on the top, central & bottom panels. \n<!-- image --> \nα \nThe filament in these respective simulations becomes unstable to the sausage-type instability (also sometimes referred to as the deformation instability; Nagasawa 1987), so that broad condensations (i.e., mean aspect ratios in the range 1.05 - 1.13 for those in Cases 1, 3, 6, 8, 11 & 12 respectively) begin to appear along the length of the filaments in respective cases. Some of these condensations merge, while a few survive and become massive enough to collapse (i.e., the collect-and-collapse mode). We also observe that the evolutionary timescale of the filament becomes significantly smaller with increasing external pressure. Thus the filaments in Cases \n6, 8 & 12, where the external pressure is in the range of a few times 10 5 K cm -3 and a few times 10 6 K cm -3 , respectively, evolve on less than half the timescale of the filaments in Cases 1 & 3 where P ext /k B ∼ 10 4 K cm -3 . \nGas accreted by filaments in Cases 6 & 8 is warm, a situation typically envisaged towards the Galactic CMZ ). The Type I filaments in these respective cases also fragment via the collect-and-collapse mode, but much faster than in Cases 1 & 2 where P ext is at least 2 orders of magnitude lower. Also, the filament becomes severely ablated in such hostile environs (as in Case 8, for instance). In these two cases the thermal pressure due to the accreted warm gas initially provides buoyancy until it gradually cools and so, the contractional wave driven by the incoming gas is not strong enough to enhance the central density sufficiently so that compressional Jeans-type fragmentation can ensue. In other words, the collect-and-collapse mode of fragmentation must generally dominate weakly self-gravitating filaments. \nThis observation could help us reconcile the wellseparated broad cores in the Musca filament (e.g., Hacar et al. 2016) that look like pearls on a string. Collectively, these findings suggest that sub-critical filaments are more likely to form cores via the collect-and-collapse mode in low to intermediate pressure environments like that in the Solar-Neighbourhood. In any case, star-formation in such filaments is likely inefficient, irrespective of the external pressure. It is therefore clear that irrespective of the linemass, the external pressure does in fact bear upon the evolution of accreting filaments, their density distributions, their inner filament radii and the morphologies of cores that they spawn. It is also important to note that the filaments in Cases 11 - 13 initially suffused with supersonic turbulence also evolve in a manner similar to those in the first 10 Cases where they were suffused with initially subsonic turbulence. The only difference is that the initially turbulent filaments are fluffier than the latter. It remains to be seen how the ambient environment affects the filament accretion, and subsequently, its linemass. We propose to investigate this problem in a more dynamic set-up where filaments are allowed to assemble self-consistently.', '4.1 STAR FORMATION EFFICIENCY': 'There is no dearth of literature showing considerable variation in star formation efficiency across molecular clouds (e.g., Lee et al. 2016; Vutisalchavakul 2016 ). The suggestion that external pressure modulates the efficiency (i.e., the fraction of gas in a volume converted to stars) with which gas is converted into stars has been gaining traction in recent years (e.g., Hughes et al. 2010, 2013; Meidt et al. 2013). Recent numerical work on this subject shows that clouds in high-pressure environments (i.e., P ext /k B /greaterorsimilar 10 6 K cm -3 ) are inefficient at cycling gas into the dense, putative star-forming phase (e.g., Anathpindika et al. (2017; 2018). Equivalently, a relatively poor efficiency of cycling gas into the dense phase could reconcile the observed inefficiency of star-formation in such environments. \nPlots in Fig. 4 show that the dense gas fraction by volume ( M frac ) rises during the contractional phase of filaments before decreasing rapidly during the later phases of evolution when they expand, and that the fraction of gas cycled to higher volume densities increases with increasing external pressure. Crucially, however, the cycling of an increasing fraction of gas to higher volume densities does not necessarily mean a higher dense gas fraction by column density, as is evident from the plot on the left-hand panel of Fig. 5. This apparently contradictory situation arises because the rapid amplification of perturbations on the surfaces of filaments, especially in high-pressure environments (i.e., pressures upward of /greaterorsimilar 10 6 K cm -3 ) causes them to buckle, and sometimes even rupture (i.e., actually broken in to several pieces), which quickly diminishes the dense gas fraction. In the process, filaments may even become severely eviscerated. For example, the filament in Case 9 loses about 40% of its mass by the time the simulation was terminated. The lost mass of course remains in the computational domain and becomes part of the diffuse medium surrounding the filament. The dense gas fraction by column density in Cases 11 & 12 barely reaches a maximum of ∼ 25% before eventually falling off to ∼ 10% at the time of termination of the respective simulations; Case 13 is even worse in regard as the dense gas fraction just about reaches ∼ 12% before tapering off by the time calculations were terminated. \nThe column density PDFs (i.e., NPDFs ) for filaments shown in Fig. 5 (right) are broadly consistent with those found for filamentary Herschel clouds (e.g., Kainulainen et al. 2009; Stutz & Kainulainen 2015). They are also consistent with the semi-analytic calculations of Myers (2017) who deduced similar NPDFs for elongated clouds modelled either as a 2-D truncated Plummer cylinder, a truncated prolate spheroid, or a stretched truncated prolate spheroid. It can be seen from this plot that the NPDFs of filaments experiencing external pressure typically upward of ∼ 10 6 K cm -3 decline rapidly at higher column densities. Similarly, the NPDFs of filaments experiencing lower pressures (typically /lessorsimilar 10 4 K cm -3 ) exhibit long power-law tails at lower column densities ( /lessorsimilar 10 21 cm -2 ). This latter observation suggests that a considerable fraction of gas in such filaments must remain diffuse. Hence, they are likely to be inefficient at forming stars, a result consistent with the observational findings of e.g., Goodman et al. (2009), in respect of the diffuse clouds (i.e., column densities /lessorsimilar 10 21 cm -2 ). \nFurthermore, the NPDFs for filaments experiencing intermediate range of pressures (between typically /greaterorsimilar 10 5 -6 K cm -3 ) peak at column densities /greaterorsimilar 10 21 cm -2 , but exhibit only relatively short power-law tails at low column densities. Such filaments are therefore likely most efficient at forming stars. Interestingly, however, the NPDFs for the Type I & II filaments in Cases 11 & 12 respectively, though qualitatively similar to the others at comparable external pressure, are clipped at the high density end. Also, the peaks of these respective NPDFs are shifted towards lower \ndensities. This observation is not altogether surprising since turbulence inhibits the process of cycling gas into the dense phase. \nHaving lost a substantial fraction of its mass over the course of its evolution, the truncated NPDFs at higher densities for the Type III filaments in Cases 10 & 13 again highlight the fact that only a relatively small fraction of its mass is ever cycled to column densities /greaterorsimilar 10 21 cm -2 . This observation reinforces the fact that star-formation must be inefficient in high-pressure environments. While it is clear that the ambient environment modulates the star-forming ability of clouds, it is also interesting to note that Kainulainen et al. (2013) argued that variations in the strength of the compressional component of turbulence affect the fraction of the putative star-forming gas, as reflected by the slopes of NPDFs towards the high density regime.', '4.2 QUASI-PERIODIC OSCILLATIONS': "Schneider & Elmegreen (1979) and Dutrey et al. (1991) were among the first to report the existence of periodic structure along filaments. Density fluctuations in the ISM are seeded by the interplay between various dynamic instabilities and putative star-forming regions are assembled in regions of converging gas-flows (see review by, e.g., Dobbs & Baba 2014). In an earlier semi-analytic work, Gehman et al. (1996) showed that fluctuations in the density and velocity fields could be generated due to the propagation of waves and instabilities and that there is an approximate correspondence between the two fields, unlike other works (e.g., Nakamura et al. 1991), which suggested magnetohydrodynamic motions like Alfv'en waves as the possible sources of these observed fluctuations in the density/velocity-fields. \nGehman et al. particularly showed that pressure-confined filaments are susceptible to a class of wave-solutions that propagate along their axes, and in the direction orthogonal to their axes. They further showed that these waves have a characteristic fragmentation lengthscale, which in the case of purely logatropic filaments depends on external pressure and varied as P 1 / 2 ext . Their models suggest that the density and the velocity fields are approximately cospatial. They also predict a converging velocity-field towards density peaks, but a non-convergent velocity-field in regions of stable modes (i.e., those that do not grow). \nRecent observations of the ISM by Henshaw et al. (2020), for example, show that fluctuations in the velocity-field are ubiquitous. Analysis of observational data spanning a wide variety of Galactic environments by these authors further showed that the periodicity of velocity structure was often similar to that of the underlying density structure. They therefore conjectured that fluctuations in the density and velocity-fields were likely the key to understanding the origin of gas-flows that assemble star-forming regions in the ISM . On smaller scales, Chen et al. (2020) similarly observed remarkably coherent velocity fluctuations in the NGC \n1333 region where they found that the magnitude of the radial component of the velocity gradient decreased towards the spines of observed filaments and inferred them as signatures of ongoing accretion by filaments. Chen et al. , however, did not detect any strong evidence to suggest large scale accretion by the observed filaments. \nSimilar velocity gradients, typically on the order of a few km s -1 pc -1 , were also reported in the Serpens South star-forming region (e.g., Kirk et al. 2013; Fern'andez-Lop'ez et al. 2014), and the Serpens Main star-forming region (e.g., Lee et al. 2014). Fluctuations in the velocity field were also reported in the Brick and the Sgr B2 region in the Galactic CMZ . Given the extreme ambient environment in the CMZ , peaks of the fluctuations in the centroid velocities are typically an order of magnitude higher than that for clouds in the local neighbourhood (e.g., Henshaw et al. 2016; 2019). In fact, Henshaw et al. (2016) also observed that the fluctuations in the velocity field were approximately co-spatial with the massive ( ∼ 10 4 M /circledot ) clouds in the region which led them to suggest that the observed velocity fluctuations were likely triggered by the gravitational instability. \nStill more recently, Wallace et al. (2022) observed velocity gradients on the order of a few km s -1 towards the Sgr E region which itself is believed to be at the turbulent intersection of the dust lane associated with the Galactic bar and the CMZ and may also be entering the CMZ . Evidently, the ambient environment must affect the peaks of observed velocity fluctuations. Observations of filaments at different length scales show some evidence of larger axial components of the velocity gradient on small spatial scales, with perhaps ( ∇ V ) a scaling as the inverse of filament length. See for e.g., the review by Hacar et al. (2022). \nVarious panels of Figs. 6 and 7 show that quasiperiodic oscillations in the velocity gradient along the axis of the filament, and in the direction orthogonal to it are ubiquitous in the sense that such features are visible irrespective of the magnitude of external pressure and filament linemass. Our findings here are therefore consistent with similar observational inferences suggesting that velocity and density fluctuations are likely ubiquitous as they are visible across a wide range of spatial scales, even up to a few parsecs (e.g., Henshaw et al. 2020). Evidently, the peaks of these oscillations increase with increasing external pressure which is consistent with observational inferences. Various panels of Fig. 8 also suggest that these oscillations overlap with the spatial separation of fragments that form in the respective filaments. This observation is reinforced by the fact that Pearson's Normalised correlation coefficients for these plots is closer to unity. Noticeably, however, not all the fragments condense to form cores that collapse. Taken together, Figs. 6 - 8 along with our observations of similar oscillatory features in the case of non-accreting filaments examined in Paper II, make it clear that such fluctuations in the velocity-field can be triggered even by purely hydrodynamic instabilities.", '4.3 FILAMENT WIDTH': "The idea that filaments, at least those in the nearby molecular clouds, have a universal width, i.e., FWHM fil ∼ 0 . 1 pc (e.g., Arzoumanian et al. 2011, 2013; Andre et al. 2016), has recently come under increasing scrutiny (e.g., Panopoulou et al. 2014, 2022). Convergence tests by Andr'e et al. (2022) for filaments in the Solar neighbourhood, however, show that the conclusion about the FWHM fil being ∼ 0.1 pc is robust. The authors therefore call for a theoretical framework to reconcile this observed lengthscale. That filaments in this work exhibit FWHM fil variations over the course of their temporal evolution is readily clear from the plots shown in Fig. 9 (left). Filaments suffused with initially supersonic gas, however, have the largest FWHM fil and are the three outliers in this figure. Furthermore, the decrease in filament width with increasing external pressure, with the exception of the filaments in Cases 11-13, is also evident from Fig. 9 (right). Indeed, thinner filaments with a FWHM fil significantly smaller than ∼ 0 . 1 pc have also been observed elsewhere in the Galaxy, as noted in § 1 above. See also a more recent review by Hacar et al. (2022). \nIn their semi-analytic work, Gehman et al. (1996) showed that turbulence-supported logatropic filaments are generally wider and exhibit shallower density profiles in comparison with isothermal filaments. So the observations in Cases 11 - 13 here are consistent with the analytic predictions by Gehman et al. More recently, Priestley & Whitworth (2021) showed numerically that filaments having widths on the order of ∼ 0.1 pc can be generated if the gas being accreted is initially mildly supersonic and has a Mach number, M inf /lessorsimilar 3. They further argue that accretional flows with higher M inf generate thinner filaments, and that the width of a filament is determined by the location of the accretion shock within a filament. \nIndeed, Fig. 3 shows the filament width in these simulations is associated with the location of the inwardly propagating compressional disturbance. Filaments in low-pressure environments (i.e., similar to the Solar -Neighbourhood) typically have FWHM fil ∼ 0.1 pc . For higher external pressures, however, the filaments (with the exception of those in Cases 11-13) become thinner, a result consistent with the conclusion drawn by Priestley & Whitworth (2021). Despite M inf being a dimensionless empirical physical quantity, a given Mach number could correspond to a range of external pressures depending on the temperature of the inflowing gas. Even in this work, a transonic inflow (i.e., M inf = 1 ), generates P ext in the range 10 4 -10 7 K cm -3 . So we believe that the external pressure is a better proxy for the ambient environment. Our conclusion here about P ext having a significant bearing upon the FWHM fil of filaments is therefore robust. Our results suggest that the filament width typically signifies the lengthscale on which it acquires a state of dynamic equilibrium.", '4.4 OTHER FILAMENT PROPERTIES': 'Surveys of filaments across a number of clouds in the local Solar-Neighbourhood have shown a direct correlation between the velocity dispersion of gas in filaments, σ gas , their linemass, and the column density (e.g., Arzoumanian et al. 2013). While σ gas may indeed represent the strength of the underlying turbulent velocity field, it also includes contribution due to systemic gas-flows as in a contracting cylinder or in regions of localised collapse. In general, however, σ gas is a good proxy for turbulence, at least before individual density perturbations collapse. \nSuch correlations support the thesis that the gas accreted by filaments is a source of turbulence within them, while the accreted mass simultaneously increases their column density. Indeed, Fig. 10 (top & central panel) shows that such a correlation, albeit a very weak one, is visible in our own simulations, especially those where the respective filaments experience intermediate to high large external pressure, i.e., P ext /k B /greaterorsimilar 10 5 K cm -3 . At lower pressures, however, there is no meaningful correlation. A similar trend between σ gas and the linemass is also seen in the plot in Fig. 10 (central panel). \nFigure 10 (lower) shows that the Virial parameter weakly anticorrelates with the linemass irrespective of whether the filaments are sub-critical or super-critical. This anticorrelation is not as strong as that observed by Arzoumanian et al. (2013) for their sample of filaments. We note, however, that most of the filaments in our present work are virially bound. This is also true of the filaments in Cases 11 - 13 where they were initially suffused with supersonic turbulence, though over the course of their evolution they do sometimes become unbound. Generally speaking, the correlations observed in this work are rather weak which could be due to the idealised nature of our set-up. \nFinally, Fig. 11 shows the correlations between the column density, N H 2 , and external pressure, and σ gas and external pressure. The former correlation is, however, somewhat weaker than that predicted analytically by Fischera & Martin (2012). For the latter, we suggest a higher accretional velocity injects stronger turbulence that manifests itself in the form of a higher velocity dispersion. This latter result corroborates recent observational findings that show a larger coefficient of the size-line width relation towards high-pressure environments (e.g., Rice et al. 2016). That the pressure due to turbulence, P turb ( ≡ ρ gas σ 2 gas ), within the filament correlates tightly with the external pressure, P ext , can indeed be seen from the plot in Fig. 12.', '5 CONCLUSIONS': 'The principal conclusions of this paper are as follows - \n- 1. The magnitude of external pressure definitely bears on the evolution of filaments and on the morphology of cores that spawn within them, irrespective of their linemass. Thus, sub-critical filaments in low to \nFigure 12.: Like the earlier plots in Figs. 9, 10 & 11 but now showing the turbulent pressure, P turb , as a function of the external pressure. The black continuous line represents as before the locus of points corresponding to the mean P turb for each simulation. \n<!-- image --> \n- intermediate pressure environments evolve slowly to form broad cores via the collect-and-collapse mode.\n- 2. With the exception of the filaments suffused with initially supersonic gas that are generally wider, there is a clear trend of decreasing filament width with increasing pressure and filaments in Solar Neighbourhood-like environments have widths typically on the order of ∼ 0.1 pc .\n- 3. Filaments at either extremes of pressure, i.e., towards the Galactic centre in the Central Molecular Zone where P ext /k B /greaterorsimilar 10 7 K cm -3 , or towards the fringes of the Galactic disk where P ext /k B /lessorsimilar 10 4 K cm -3 , are inefficient at cycling gas into higher densities and thus forming stars.\n- 4. We observe quasi-periodic oscillatory features of the velocity gradient in all our simulations and peaks of their fluctuations generally increase with increasing pressure. Fluctuations in the density and the velocity-fields are also roughly cospatial. We suggest that oscillatory features of ( ∇ V ) r are a poor proxy for detecting signals of accretion on to filaments. In fact, we observe that ( ∇ V ) r correlates more strongly with the external pressure than with the inflow velocity of gas being accreted by the filament.\n- 5. Filaments initially suffused with supersonic turbulence have physical properties generally similar to those suffused with subsonic turbulence, except that the former are much less efficient at cycling gas into the dense phase as reflected by a lower dense gas fraction, but have a bigger inner radius and by extension, a bigger FWHM fil .\n- 6. Filaments have higher column densities and higher velocity dispersion (also reflected by a higher pressure due to turbulence within the filament) with increasing external pressure, showing us that external pressure modulates properties of gas even on the scale of individual filaments. We also observe that the filaments modelled here are largely virially bound irrespective of their linemass. Filaments in a \nSolar-type and in extreme environment as in Cases 10 & 13 can, however, sometimes appear to be unbound.', 'Acknowledgements': 'This project was initiated with funds made available under the From GMCs to stars project ( GMCS /000304/2014), funded by the Department of Science & Technology, India. Simulations discussed in this work were developed using supercomputing facilities made available by Digital Research Alliance of Canada (https://alliancecan.ca) and Compute Canada Calcul Canada (www.computecanada.ca). The authors gratefully acknowledge useful discussions with Adam Ginsburg and Cara Battersby. The authors are grateful to anonymous referees for some critical comments that made the original manuscript much clearer.', 'Data Availability Statement': 'No valid data repositories exist as the data generated by numerical Simulations discussed in this work are too big to be shared. Instead we discuss in detail the numerical methods and the initial conditions used to generate these data sets. The initial conditions file and the script of the numerical code can be made available to bonafide researchers on reasonable request.', 'A Filament evolution': 'The Type I (i.e., f cyl < 1, even after accreting gas) filament in a low pressure environment evolves over a longer timescale and forms cores through the collect and collapse 3 mode described in Papers I & II. This behaviour is also visible from the panels specifically corresponding to Cases 1 and 3 (and also in Case 11) in Figs. A1 and A2 (and in Fig. A3, lower panel) , respectively. Interestingly, there is also evidence of mergers between some fragments, as reported by Inutsuka & Miyama (1997). On the contrary, Type II (i.e., f cyl > 1, after accreting gas) filaments, with the exception of that in Case 12, form cores via Jeans - type fragmentation. This mode of fragmentation can be readily seen in the panels specifically corresponding to Cases 2, 4 & 6 in Figs. A1-A3 respectively. \nIn contrast, the continually super-critical ( Type III ) filaments evolve rapidly to form spines, i.e., thin filamentary structures with centrally peaked density profiles, with the exception of that in Case 13. While fragmentation of the filament into distinct pinched cores is visible in Case 5 (e.g., in the picture on the top left hand panel of Fig. A3), no such fragmentation and subsequent formation of cores is visible in the filament in Case 10 (e.g., in the top right hand panel of Fig. A3), whereas the filament in Case 13 is significantly wider than that in Cases 5 & 10. At any rate the filaments in Cases 10 & 13 appear to be composed of a number of dense strands similar to fibres. There is, however, no evidence of core formation in these fibres or in the filament itself. \nRendered density images on lower panels of Fig. A3 similarly show the terminal epoch of the filament in respectively Cases 6, 11, 12 & 13. Observe that as with the \nfilaments superposed with initially subsonic turbulence (i.e., in the first 10 cases), even those with supersonic turbulence (i.e., Cases 11, 12 & 13) evolve via radial contraction followed by axial fragmentation. In other words, the evolution of these latter filaments is still gravity-dominated. This is because as the initial turbulence dissipates, the critical linemass decreases and the filament is readily overwhelmed by an inwardly propagating compressional wave. It means, the inflowing transonic gas in this instance does not sufficiently replenish the turbulence. Thus the choice of the internal Mach number ( M int ) makes no qualitative difference to the evolutionary cycle of filaments which is not particularly surprising given its inefficient replenishment by the inflowing gas. \nRespective images corresponding to Cases 6 & 11 on the lower panel of Fig. A3 make an interesting comparison as the filaments in either case are not only of the Type I kind, but the external pressure is also comparable. Evidently, the turbulence supported filament in Case 11 evolves on a timescale almost twice as long as that in Case 6. More importantly, the cores in the former simulation (i.e., Case 11) are broader than the natal filament and appear to have formed via the Collect and Collapse mode . On the contrary, the filament in Case 6 appears to have spawned cores via the Jeans - type fragmentation. \nThe Type II filament in Case 12 where the external pressure is comparable to that in Cases 6 & 11, also evolves like that in the Case 11 and appears to have formed cores via the Collect and Collapse mode . This is unlike the other Type II filaments in Cases 2, 4 & 7 where cores form via the Jeans-type mode of fragmentation. The filaments in Cases 2 & 4 are shown in Figs. A1 & A2. These observed differences in filament evolution suggest that weakly self-gravitating filaments must fragment via the Collect and Collapse mode. The continually super-critical Type III filament in Case 13 evolves similarly to the one in Case 10, though it is fluffier and is also more profusely striated than the one in Case 10.', 'REFERENCES': "Abe, D., Inoue, T., Inutsuka, S & Matsumoto, T., 2021, ApJ, 916, 83 Anathpindika, S & Freundlich, J., 2015, PASA, 32, 07 Anathpindika, A., Burkert, A & Kuiper, R., 2017, MNRAS, 466, 4633 Anathpindika, A., Burkert, A & Kuiper, R., 2018, MNRAS, 474, 1277 Anathpindika, S. V & Di Francesco, J., 2021, MNRAS, 502, 564A (Paper I) Anathpindika, S. V & Di Francesco, J., 2022, MNRAS, 513, 1275 (Paper II) Andr'e, Ph., Men'schikov, A., Bontemps, S., Konyves, V., Motte, F., Schneider, N et al. , 2010, A&A, 518, L102 Andr'e, Ph., Konyves, V., Roy, A & Arzoumanian, D., 2016, IAU XXIV General Assembly 2015, Proceedings of the IAU, Vol 29B, 2016, 708 Andr'e, Ph., Palmeirim, P & Arzoumanian, D., 2022, A&A, 667, L1 Ao, Y., Henkel, C., Menten, K. M., Requena - Torres, M. A., Stanke, T et al. , 2013, A&A, 550, A135 Arzoumanian, D., Andr 'e, Ph., Peretto, N & K onyves, V., \n2013, A&A, 553, A119 \nArzoumanian, D., Andr'e, Ph., Konyves, V., Palmeirim, P., Roy, A., Schneider, N., Benedettini, M et al. , 2019, A&A, 621, A42 \nFigure A1.: Rendered density images showing the evolutionary sequence of the filaments in respectively Cases 1 ( Upper panel ), and 3 ( Lower panel ). \n<!-- image --> \nBally J., 1987, ApJ, 312, L45 \nBeuther, H., Ragan, S. E., Johnston, K., Henning, Th., Hacar, A & Kainulainen, J., 2015, A&A, 584, A67 \nBinney J. & Tremaine S., 1987, Galactic Dynamics. Princeton Univ. Press, Princeton \nBonne, L., Bontemps, S., Schneider, N., Clarke, S., Arzoumainan, D., Fukui, Y., Tachihara, K et al. , 2020, A&A, 644, A27 \nBoulares, A. & Cox, D. P. 1990, ApJ, 365, 544 \nBusquet, Gemma, Estalella, Robert, Palau, Aina, Liu, HauyuBaobab, Zhang, Quizhou et al. , 2016, ApJ, 819, 139 \nCaselli, P., Benson, P. J., Myers, P. 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2024arXiv240904583P

Subparsec supermassive black hole SMBH binaries are expected to be common in active galactic nuclei AGN as a result of the hierarchical buildup of galaxies via mergers. While direct evidence for these compact binaries is lacking a few hundred candidates have been identified most based on the apparent periodicities of their optical lightcurves. Since these signatures can be mimicked by AGN rednoise additional evidence is needed to confirm their binary nature. Recurring selflensing flares SLF occurring whenever the two BHs are aligned with the line of sight within their Einstein radii have been suggested as additional binary signatures. Furthermore in many cases lensing flares are also predicted to contain a dip whenever the lensed SMBHs shadow is comparable in angular size to the binarys Einstein radius. This feature would unambiguously confirm binaries and additionally identify SMBH shadows that are spatially unresolvable by highresolution VLBI. Here we estimate the number of quasars for which these dips may be detectable by LSST by extrapolating the quasar luminosity function to faint magnitudes and assuming that SMBH binaries are randomly oriented and have massratios following those in the Illustris simulations. Under plausible assumptions about quasar lifetimes binary fractions and Eddington ratios we expect tens of thousands of detectable flares of which several dozen contain measurable dips.

2024-09-01T00:00:00Z

['2024arXiv240904583P', '10.48550/arXiv.2409.04583', 'arXiv:2409.04583']

['Astrophysics - High Energy Astrophysical Phenomena']

Selflensing flares from black hole binaries IV the number of detectable shadows

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.04583.pdf

{'Self-lensing flares from black hole binaries IV: the number of detectable shadows': "Kevin Park 1 , ∗ Chengcheng Xin 2 , Jordy Davelaar 3 , 4 , 5 , 2 , 6 , and Zolt´an Haiman 2 , 1 \n1 Department of Physics, Columbia University, New York, NY 10027, USA \n2 Department of Astronomy, Columbia University, New York, NY 10027, USA \n3 Department of Astrophysical Sciences, Peyton Hall, \nPrinceton University, Princeton, NJ 08544, USA \n4 NASA Hubble Fellowship Program, Einstein Fellow \n5 Center for Computational Astrophysics, Flatiron Institute, \n162 Fifth Avenue, New York, NY 10010, USA and \n6 Astrophysics Laboratory, Columbia University, 550 W 120th St, New York, NY 10027, USA \nSub-parsec supermassive black hole (SMBH) binaries are expected to be common in active galactic nuclei (AGN), as a result of the hierarchical build-up of galaxies via mergers. While direct evidence for these compact binaries is lacking, a few hundred candidates have been identified, most based on the apparent periodicities of their optical light-curves. Since these signatures can be mimicked by AGN red-noise, additional evidence is needed to confirm their binary nature. Recurring self-lensing flares (SLF), occurring whenever the two BHs are aligned with the line of sight within their Einstein radii, have been suggested as additional binary signatures. Furthermore, in many cases, lensing flares are also predicted to contain a 'dip', whenever the lensed SMBH's shadow is comparable in angular size to the binary's Einstein radius. This feature would unambiguously confirm binaries and additionally identify SMBH shadows that are spatially unresolvable by high-resolution VLBI. Here we estimate the number of quasars for which these dips may be detectable by LSST, by extrapolating the quasar luminosity function to faint magnitudes, and assuming that SMBH binaries are randomly oriented and have mass-ratios following those in the Illustris simulations. Under plausible assumptions about quasar lifetimes, binary fractions, and Eddington ratios, we expect tens of thousands of detectable flares, of which several dozen contain measurable dips.", 'I. INTRODUCTION': "Supermassive black holes (SMBHs) with masses between M ≈ 10 6 -10 9 M ⊙ are present in the nuclei of most nearby galaxies [1]. In hierarchical cosmologies, galaxies grow by frequent mergers, which deliver the nuclear SMBHs [2, 3], along with significant quantities of gas [4], to the central regions of the post-merger galaxy. The natural conclusion is that after the two SMBH's separation decreases, SMBH binaries (SMBHBs) should form frequently in galactic nuclei [5]. Hydrodynamical simulations have shown that electromagnetic (EM) emission from these SMBHBs, provided they are surrounded by circumbinary gas, should be detectable starting well before the merger, and should persist all the way to the merger [6-9]. \nThese compact SMBHBs are a fundamental ingredient of galaxy formation and are also prime targets to be observed in gravitational waves (GWs) by LISA [10, 11], and by pulsar timing arrays (PTAs). Indeed, PTAs have recently discovered a stochastic GW background (GWB) in the nHz bands, which is consistent with the cosmological population of coalescing SMBHBs [12-15]. Combining the EM and GW signals from the same source or even the same or overlapping populations of SMBHs would open windows to especially novel science, including astrophysics, cosmology, particle physics, and the nature of gravity [16]. \nThese so-called 'multimessenger' opportunities have stimulated strong interest in finding wider SMBH binaries in EM data. Approximately 300 SMBH binary candidates have been identified among bright active galactic nuclei (AGN) in large time-domain optical surveys [1719] based on their apparent periodicities, and a handful of additional SMBH binary candidates have been identified serendipitously, or through other tentative signatures involving double-peaked or offset emission lines, or spatial structures of radio jets and lobes [20, 21]. These candidates remain controversial because of the lack of a 'smoking gun' binary signature, and in the cases of the periodic candidates, stochastic red-noise AGN variability can mimic periodicities for a few periods [22]. \nApotential SMBHB signature that in some cases could help lift this degeneracy is a 'self-lensing flare' (SLF). If the two SMBHs are aligned within the line-of-sight to within the system's Einstein radius, then whenever one of the SMBHs passes behind the other, its emission will be strongly magnified. These lensing flares occur once or twice per orbit (depending on whether one or both BHs are active). Depending on the binary masses and separations, the flares can last from hours to weeks [23-26]. An AGN identified in the Kepler catalog, dubbed Spikey, has a light-curve consistent with relativistic Doppler modulation from an eccentric binary, with a narrow ∼ 10% spike at the expected orbital phase whose symmetric shape is well fit by a microlensing model [27]. \nIn a fraction of these self-lensing binaries, the size of the SMBH's shadow ( ∼ 5 times its gravitational radius) is commensurate with the (angular) Einstein radius, and an additional feature is imprinted on the light-curves, in \nthe shape of a 'dip' near the peak of each flare. This feature could unambiguously confirm binaries and additionally identify SMBH shadows that are spatially unresolvable by high-resolution VLBI, as was done by the Event Horizon Telescope Collaboration [28, hereafter Paper I]. Toy models for the BH emission show that the precise shapes and sizes of these dips depend on the binary system's parameters [29, Paper II], and recent hydrodynamical simulations find that these 'dip' features exist even in the strongly distorted and fluctuating circumbinary gas, and can be recovered via phase-folding in the face of stochastic noise [30, III]. [31] perform a similar study of the ray-traced emission from SMBH binaries, but using a boosted binary metric and following the binary's inspiral with 3.5PN equations of motion. They recover the self-lensing flares and dips closely matching those in Paper I. \nSelf-lensing flares require nearly edge-on viewing angles, and a natural question is how rare these lensed configurations are. The Vera Rubin Observatory's LSST [32] is expected to contain between 20-100 million bright quasars [33], and combined with its high cadence (with photometric points every few days), is an ideal dataset in which to search for rare sources with periodically recurring flares. Kelley et al. [34] have recently used SMBH populations from the Illustris simulation [35], combined with toy models for binary emission and lensing, and concluded that LSST could detect several hundred selflensed SMBHs with flares lasting for 30 days or longer. \nIn this paper, we follow up on the above studies, to assess the number of self-lensing flares which additionally have detectable 'dips' in their light-curves. In principle, these dips require more stringent alignment, but their incidence rate rises steeply for shorter-duration flares. Here we estimate the number of quasars for which these dips may be detectable by LSST, by extrapolating the quasar luminosity function to faint magnitudes, and assuming that SMBH binaries are randomly oriented and have mass ratios following those in the Illustris simulations. Under plausible assumptions about quasar lifetimes, binary fractions, and Eddington ratios, we predict tens of thousands of detectable flares with durations down to ten days, of which several dozen contain measurable dips. \nThe rest of this paper is organized as follows. In § II, we describe our methodology, including models for the SMBH binary populations, their emission and lensing, the criteria for detectable flares and dips, and the impact of finite source sizes. In § IV, we present our results, in terms of the number of flares and dips detectable in LSST as a function of binary parameters and observational thresholds (magnitude and flare duration). In § IV, we summarize our main results and their implications.", 'II. METHODS': 'In this section, we calculate the number of detectable self-lensing dips in LSST and its dependence on several \nFigure 1. The number of quasars N QSO ( M,z ) in the redshift and mass range z ∈ [0 , 6] and M/ M ⊙ ∈ [10 5 , 10 9 . 5 ]. Integrating over the full mass and redshift ranges gives 20 million, 44 million, 100 million quasars above the assumed LSST magnitude detection limits of m i = 24 , 25 , 26, in agreement with the results of [33]. \n<!-- image --> \nbinary parameters. Results from [36] demonstrate that the evolution of quasars can be reproduced in a model in which they are activated in galaxy mergers. As mentioned above, these galaxy mergers are expected to deliver the two SMBHs to the new galactic nucleus where they form a bound binary [5]. Given these results, our main assumption is that galaxy mergers are responsible for both quasar activity and for producing SMBH binaries [37]. First, we find the expected number of quasars ( N QSO ) in LSST using the quasar luminosity function ( § II A). We then modify N QSO to find the number of binary quasars as a function of the binary orbital period, using estimates of the quasar lifetime at each orbital period ( § II B) allowing the overall binary fraction to be a free parameter. We find the mass ratio distribution for these binaries in different mass and redshift bins using the Illustris simulations ( § II C) and we compute the probability of a detectable self-lensing dip given binary inclination ( § II D) as a function of the mass, mass ratio and orbital period. Finally, we compute similar probabilities for self-lensing flares in the point-source limit ( § II E) and accounting for the finite sizes of the emitting regions ( § II F).', 'A. Number of quasars and binaries': "We follow the calculations in [33] to obtain the number of quasars above LSST's detection threshold, based on the extrapolated quasar luminosity function [QLF; 38]. In Figure 1, we present the number of quasars in LSST's 20,000 deg 2 survey, N QSO , for three magnitude limits, m i = 24, 25 and 26, where m i = 24 corresponds to the \nsingle-exposure magnitude limit of LSST in the i band, and m i = 26 corresponds to the co-added magnitude limit over the whole survey [39]. The BH mass and i -band magnitudes are related using Eq. 1: \nm i = 24+2 . 5 log [ ( f Edd 0 . 3 ) -1 ( M 3 · 10 6 M ⊙ ) -1 ( d L ( z ) d L ( z = 2) ) 2 (1) \nwhere f Edd is defined by the bolometric quasar luminosity L = f Edd L Edd and L Edd is the Eddington luminosity for total mass M . Initially, we assume f Edd = 0 . 3 , but in § IV below we will discuss the dependence of our results to varying f Edd . Quasars in the hatched area are discarded since their magnitudes are below m i = 26. \nUnder the assumption that all quasars are triggered by mergers and are associated with SMBH binaries [40, 41], and that the bright quasar phase lasts for a typical lifetime of τ Q (=say 10 8 years), the number of quasars powered by SMBH binaries with τ m years left to merger scale linearly with the fraction τ m /τ Q , \nN ( M,z ) ≡ ( τ m τ q ) f bin N QSO ( M,z ) . (2) \nN QSO ( M,z ) is the number of quasars shown in Figure 1, which assumes that all quasars are associated with binaries. In this analysis, we vary the fraction of quasars that are SMBH binaries, f bin , from 0.2 to 1. We use residence times τ m = τ res ( q, M, T orb , z ) appropriate to GWor gas-driven binary inspirals, depending on the mass M and orbital period T orb of the binary (see next section). By setting a maximum period T max , N ( M,z ) represents the number of binaries with an orbital period of T max or less, equivalent to the number of binaries with τ m years left to merger.", 'B. Residence time vs. orbital period': "We obtain the residence time of a binary quasar at a given orbital period (or separation) τ res as follows. For short periods, the orbital decay is primarily GW-driven, given by the quadrupole formula [42]: \nτ GW = 1 . 11 × 10 7 q -1 s M -5 / 3 7 ( T orb yr ) 8 / 3 ( 1 + z 4 ) yr , (3) \nwhere q s ≡ 4 q/ (1 + q ) 2 is the symmetric mass ratio, q ≡ M 2 /M 1 ≤ 1 is the mass ratio, M 7 ≡ M/ (10 7 M ⊙ ) is the total mass in units of 10 7 solar masses, and T orb is the orbital period in units of years. \nFor long periods, we assume that the circumbinary gas dominates the binary's inspiral, and adopt the simple scaling from [43]. This assumes that the gas-driven inspiral timescale is proportional to the mass accretion timescale, roughly in agreement with the results of hydrodynamical simulations [44-49]. First, the mass accretion timescale is referenced in units of the Eddington \n] \nFigure 2. The solid blue curve shows the fiducial model for the residence time τ res ( q, M, T orb , z ) of binaries with total mass M = 10 7 M ⊙ , mass ratio q = 1 and redshift z = 1 as a function of observed orbital period T orb . Below the threshold given in Eq. 5 the binaries evolve via GW emission and above the threshold the binaries' inspiral timescale is assumed to be driven by circumbinary gas and increase linearly with T orb . In § IV we also present results for a more conservative model where a constant maximum lifetime of a/ ˙ a gas = 1 . 7 × 10 7 years is imposed (Eq.6), shown by the solid orange curve. Purely GW-driven inspiral, the dashed blue curve, is shown for reference. \n<!-- image --> \nrate: \n˙ m Edd = 4 πGmm p ησ T c = 2 . 26 · 10 -2 × ( η 0 . 1 ) -1 × ( m 10 6 M ⊙ ) M ⊙ yr -1 , (4) \nwhere G is the gravitational constant, m is the mass of the accreting BH, m p is the proton mass, c is the speed of light, σ T is the Thomson scattering cross section and η is the radiative efficiency, assumed to be 0 . 1. Assuming an Eddington accretion rate ˙ m = ˙ m Edd , then \nm ˙ m ∼ 4 . 4 × 10 7 yr , (5) \nand we follow [43] and adopt the residence time at binary separation a \nτ gas = a ˙ a gas = 1 2 . 68 m ˙ m ∼ 1 . 7 × 10 7 yr , (6) \nwhich is constant (independent of a ). At large separations for which the GW-inspiral timescale exceeds this threshold, we assume as our fiducial model that the binary gas-driven residence time increases linearly with the orbital period, as shown by the solid blue curve in Figure 2. As an alternative model, the residence time is fixed at a maximum of 1 . 7 × 10 7 years, shown by the solid orange curve. The illustrations in Figure 2 are for a \nFigure 3. The left panel shows the distribution of the mass ratios of SMBH binaries in the Illustris simulations. In the middle and right panels, we illustrate the redshift and total mass distributions of Illustris binaries, in the four different mass-ratio ranges shown by colors in the left panel. Most binaries are symmetric in mass ( q > 0 . 1) and have relatively low redshift ( z < 3) and total mass ( M < 10 7 M ⊙ ). \n<!-- image --> \nbinary with mass M = 10 7 M ⊙ , redshift z = 1 and mass ratio q = 1. Finally, we constrain the maximum observed orbital period to T max = 5 years, given that for LSST's survey of 10 years, we want to observe a few recurrences of the dip.", 'C. Mass-ratio distributions': "The GW-driven inspiral timescale in Eq. 3 depends on the binary's mass ratio q , which is not constrained by the QLF or other observations. To estimate the mass-ratio distribution we instead use Illustris-3 [35], a cosmological hydrodynamical simulation that self-consistently follows the evolution and mergers of galaxies and their central SMBHs. Illustris provides the merger tree of their SMBH mergers, and we use this to compute to distribution of q as a function of total mass M and redshift z , shown in Figure 3. Illustris embeds MBHs of seed mass ∼ 10 5 M ⊙ which accrete and evolve dynamically. Due to the limitations to spatially resolve closely separated low-mass binaries ( M < 10 6 M ⊙ ), we initially implement a conservative mass cut of 10 6 M ⊙ of Illustris BH binaries and extrapolate the count distribution in ( M,z ) down to the binary total mass of 10 5 M ⊙ . We then divide the Illustris binaries by their mass ratios into 8 logarithmic bins between 10 -4 ≤ q ≤ 1. Figure 3 depicts the fraction of Illustris binaries in each of these 8 bins (left panel). The colors blue, yellow, green, and red each correspond to the binaries with mass ratios in the ranges of [10 -1 , 1] , [10 -2 , 10 -1 ] , [10 -3 , 10 -2 ] , [10 -4 , 10 -3 ]. Using Illustris, we calculate the probabilities that a BH binary with given M and z in one of 50 bins has a mass ratio q in one of the 8 logarithmic bins-see the middle and right panel of Figure 3. Using these mass ratio distributions we \ncan evaluate the GW-driven inspiral timescale for binary quasars in the QLF.", 'D. Self-lensing dips': "As mentioned in § I, for SMBHBs with nearly edgeon orbital planes, periodic self-lensing flares occur as a result of gravitational lensing. Compact binaries close to merger, with an orbital period of 5 years or less in our fiducial model (Table I), are expected to have four or more self-lensing flares detectable within a full ten-year LSST survey. Additionally, GR ray tracing simulations of [28] reveal that self-lensing flares have observable dips, caused by the black hole shadow. In this work, we estimate the number of LSST binaries with detectable selflensing dips for different binary parameters. For this, we recap Eqs. 1-3 of [28], which give an analytical expression for the probability that a binary system with given binary mass ratio q , total mass M , orbital period T orb and redshift z has a detectable self-lensing dip. \nFirst, assuming a circular binary with nearly edge-on binary inclination, the expected phase spacing between the two peaks before and after the dip is the ratio of the diameter of the BH shadow and the circumference of the orbit: \n∆ ϕ = d shadow 2 πa orb , (7) \nwhere a orb is the orbital radius, d shadow = 2 √ 27 GM source /c 2 is the BH shadow diameter of the source, and M source = qM/ (1 + q ) is the mass of the lensed BH (assumed here to be the lower-mass secondary). Expressing the orbital radius in terms of \nTable I. Parameters in our fiducial model (top row) and their ranges considered (bottom row). f bin is the fraction of quasars associated with binaries, f Edd is their Eddington ratio, T max is the maximum orbital period of interest, τ Q is the average total bright quasar lifetime, τ res is the residence time, i.e. the duration a binary quasar spends at each orbital period (shown in Fig. 2), and τ f, min is the minimum required lensing flare duration (not imposed in the fiducial model). \nbinary parameters via a orb = ( GMT 2 orb / 4 π 2 ) 1 / 3 gives \n∆ ϕ = 5 . 63 qM 2 / 3 (1 + q ) T 2 / 3 orb radians , (8) \nwhere G = c = 1 units were used. T orb has units R g /c , where R g = GM source /c 2 is the gravitational radius of M source . The dip in the self-lensing flare can be observed for a range of binary inclinations, ∆ i . When ∆ i is smaller than the angular size of the BH shadow on the sky, i.e. when Eq. 9 is satisfied, the focal point of the lens will be eclipsed by the BH shadow, causing dips at the local maxima of the flares. The inclination window for which this dip is visible is given by \n∆ i ≤ sin -1 ( π ∆ ϕ ) ≡ ∆ i dip . (9) \nAssuming that BH binary orbital inclinations are randomly distributed on a unit sphere, the minimum angular separation between the BHs on the sky occurs when the two BHs are at either end of the semi-minor axes of the orbit's projected ellipse. Then, for a binary chosen at random, the probability that it has an observable self-lensing dip during its orbit is given by \nP dip = ∆ i dip ( q, T orb , M, z ) / 90 · , (10) \ni.e. the probability that ∆ i , which can range from 0 · to 90 · , satisfies Eq. 9. The number of detectable selflensing dips is then N dips = P dip × N ( M,z ), integrated appropriately over the distribution of all binary parameters of mass ratio, observed orbital period, total mass and redshift.", 'E. Self-lensing flares - point source limit': 'We wish to compare our calculations with [34], who have calculated the number of detectable self-lensing flares, irrespective of whether dips from the BH shadow \nare measurable, based on the abundance of binaries in Illustris. To do this, we construct a self-lensing probability similar to the self-lensing dip probability in Eq. 10. We adopt the point-source (PS) magnification [50] \nM PS = u 2 +2 u √ u 2 +4 , (11) \nwhere u = Re( u 1 + u 2 ) is the projected separation in units of the Einstein radius and \nu j = [ ac 2 (cos 2 ϕ j +sin 2 i sin 2 ϕ i ) 4 G ( M -m j ) cos i sin ϕ j ] 1 / 2 , (12) \nwhere i is again the binary inclination relative to the line of sight and ϕ j is the orbital phase of each BH. At the peak of the self-lensing flare ( ϕ 2 = π/ 2), we assume the secondary BH to be the source, and find the maximum orbital inclination ∆ i PS in which a 10% magnification occurs, i.e. M PS ( i ) > 1 . 1. The corresponding self-lensing probability is \nP PS = ∆ i PS ( q, T orb , M, z ) / 90 · (13) \nand the number of self-lensing flares in the point-source limit is then N PS = P PS × N ( M,z ), again integrated appropriately over the distribution of all binary parameters.', 'F. Self-lensing flares - finite source': "According to Figure 2 of [51], for binaries of total mass ≲ 10 8 M ⊙ and orbital periods of a few years, it is necessary to incorporate the accretion disc size of the secondary because the angular size of the accretion disc becomes comparable to the Einstein radius for lower total masses. We adopt their model to account for finitesource effects, based on a multi-color accretion disc extending from the innermost stable circular orbit (ISCO) r ISCO = 6 GM source /c 2 , to the secondary's tidal truncation radius r tidal = 0 . 27 q 0 . 3 a [52] The temperature profile of the disc is given by \nσT 4 ( r ) = 3 GM source ˙ M source 8 πr 3 [ 1 -( r ISCO r ) 1 / 2 ] , (14) \nwhere r is the distance from the central SMBH and ˙ M source is the accretion rate of the lensed BH, for which we assume an Eddington accretion rate. For r > r tidal and r < r ISCO , we set T ( r ) = 0 . The resulting flux from the disc is given by \nF ν ( r ) = πB ν [ T ( r )] , (15) \nwhere B ν is the Planck function. We calculate the lensing magnification by evaluating \nM FS ν = ∫ 2 π 0 ∫ ∞ 0 F ν ( u ' , v ' ) M PS ( u ' ) u ' du ' dv ' ∫ 2 π 0 ∫ ∞ 0 F ν ( u ' , v ' ) u ' du ' dv ' , (16) \nwhere M PS ( u ' ) is given in Eq. 11, evaluated in the lenscentered polar coordinates ( u, v ): \nr 2 ∗ = ( u 2 0 + u 2 -2 u 0 u cos( v -v 0 )) r 2 E (17) \nr = r ∗ √ cos 2 ϕ + sin 2 ϕ cos 2 ( π/ 2 -J ) (18) \nsin ϕ = u sin v -u 0 sin v 0 √ ( u sin v -u 0 sin v 0 ) 2 +( u cos v -u 0 cos v 0 ) 2 . (19) \nHere ( u 0 , v 0 ) is the position of the secondary, and J is the inclination of the source disc relative to the line of sight. We define ∆ i FS as the maximum orbital inclination such that M FS ν ( i ) > 1 . 1 at alignment and the probability that a binary has a detectable self-lensing flare in the finite source limit is \nP FS = ∆ i FS ( q, T orb , M, z, ν, J ) / 90 · . (20) \nThe number of self-lensing flares in the finite-source limit is N FS = P FS × N ( M,z ), integrated over binary parameters. \nIn this paper, we consider the representative observed wavelengths near the center of the six LSST filters u, g, r, i, z, y at 380, 476, 622, 755, 870, 1015 nm, which are chosen by taking the average of the FWHM transmission points of each filter [39]. For example, for a binary at redshift z ∈ [0 , 6], the rest-frame wavelength of 380 / (1+ z ) nm will contribute flux to the u-band, and the rest-frame wavelength is used to evaluate the wavelengthdependent flux in Eq. 15. \nAbsorption by hydrogen clouds in the intergalactic medium could attenuate quasar light depending on the observing wavelength and the quasar redshift. However, according to [53], for observed wavelengths of ν ∼ 400 nm, the mean cosmic transmission is close to 1 for z ≲ 3, where the majority of quasars are. For longer observed wavelengths of ν > 700 nm, the mean cosmic transmission is close to 1 for z ≲ 5, which means intergalactic absorption is mostly negligible for our purposes. In all calculations, we fix the source-disc inclination at a representative value of J = π/ 4 for simplicity.", 'A. Self-lensing dips and their mass- and redshift-dependence': 'We first present the number of detectable dips, for total SMBH binary masses and redshifts in the ranges of M/ M ⊙ ∈ [10 5 , 10 9 . 5 ] and z ∈ [0 , 6]. These mass- and redshift-distributions in the fiducial model are shown in Figure 4, and are also presented numerically in Tables II and III. Overall, we find 41-60 detectable dips, depending \nFigure 4. The distribution of the number of detectable selflensing dips in the M -z plane. The contours for N dips are in units of per unit redshift per unit log mass bin, where the unit bin sizes are ∆ z = 0 . 12 and ∆log 10 M = 0 . 08. Three LSST sensitivity limits, covering the range from a single exposure to the fully co-added survey detection threshold, are shown for reference. Most detectable dips are from quasars at 1 . 5 ≲ z ≲ 3 with total (binary) SMBH masses of 10 6 M ⊙ ≲ M ≲ 10 8 M ⊙ . \n<!-- image --> \nTable II. Dependence of self-lensing dips on binary mass, integrated over all redshifts z ∈ [0 , 6] and mass ratios q ∈ [10 -4 , 1]. This result also assumes the fiducial parameters in Table I. \non the magnitude threshold. Table II shows that ∼ 90% of these binaries with detectable dips have masses between 10 6 M ⊙ to 10 8 M ⊙ . Qualitatively, while the distribution of quasars (Fig. 1) is concentrated at low masses (below 10 6 M ⊙ ), the self-lensing probability in Eq. 10 increases with mass, resulting in the distribution clustering in the intermediate mass regime, see Table II. \nTable III shows that approximately 80% of these binaries are between redshifts z = 1 -3, which is also expected considering that the lensing probability depends only weakly on redshift and the QLF peaks in this range. \nFurthermore, the LSST limits on apparent magnitude, m i < 24 (or, optimistically, m i = 26) constrain the M -z parameter space, which is visualized in the contours of Figure 1. Therefore, we present all of our results below for three different LSST magnitude limits, m i = 24, 25 and 26, covering the range from a single exposure limit to \nTable III. Redshift-dependence of self-lensing dips, integrated over all masses M/M ⊙ ∈ [10 5 , 10 9 . 5 ] and mass ratios q ∈ [10 -4 , 1]. The conditions for f bin and lensing duration were applied as for the total mass dependence in Table I. \nTable IV. Mass-ratio dependence of self-lensing dips, integrated over all redshifts z ∈ [0 , 6] and total masses M ∈ [10 5 , 10 9 . 5 ] M ⊙ . This result also assumes the fiducial parameters in Table I. \nthe fully co-added survey detection threshold. Figure 4 also shows these three magnitude cuts (blue curves), with quasars below m i = 26 are discarded as too faint to be detected (hatched region).', 'B. Mass-ratio dependence': 'There are no ab initio constraints on the Illustris mass ratios, but in practice we find all of them to be within q = 10 -4 -1, with almost all ( > 98%) binaries in the mass-ratio range of 0.1 to 1, as shown in Table IV. This follows from the mass ratio distribution of SMBH binaries from Illustris-the left panel of Figure 3. We note that these data from Illustris are quite uncertain, especially since all of the SMBH binary merger physics is sub-grid. In particular, mergers are assumed to take place instantly when their separation (modeled semi-analytically) decreases below a certain smoothing length. In reality, there can be delays and disruptions by a third SMBH, which can change the mass-ratio distribution [54], and in turn the number of self-lensing dips in each q -bin. We leave a full exploration of this dependence to future work. \nTable V. Number of detectable dips varying with model parameters and LSST limits: f bin is the assumed fraction of quasars that are binaries, f Edd is the Eddington ratio, T max is the maximum allowed orbital period, and τ Q is the typical bright quasar lifetime. The first row corresponds to our fiducial model, while each subsequent row varies one of the parameters, shown in bold. Various maximum allowed lensing durations τ f, min were additionally considered and are shown separately in Table VI.', 'C. Dependence on model parameters': 'We next examine how the number of detectable dips depends on the orbital period and other model parameters. The dependence on T max comes from the probability of an observable self-lensing dip being proportional to ∆ ϕ ∝ T -2 / 3 orb , and the residence time as a function of period τ res ∝ T 8 / 3 orb at shorter orbital periods and τ res ∝ T orb at longer orbital periods. Doubling the fiducial maximum period to 10 years would yield an increase in the number of self-lensing dips by a factor between 4 and 1.26. This is consistent with the increase by a factor of ∼ 1 . 5 as shown in Table V when T max = 10 years. \nOur results also depend on the assumed Eddington ratio f Edd , average quasar lifetime τ Q and the fraction of quasars that are binaries, f bin , as presented in Table V. \nAccording to Eq. 1, changing f Edd from 0 . 3 to 0 . 1 would increase the total mass of our BHs by a factor of 3. The total mass dependence in our model also comes from the self-lensing dip probability ∝ M 4 / 3 , and the residence time, which is either ∝ M -5 / 3 for GW-driven inspiral or independent of total mass ( ∝ M 0 ) for gas-driven inspiral. Therefore, we expect decreasing f Edd from 0 . 3 to 0 . 1 to increase the number of detectable dips by a factor between 0.7 and 4.3, which is consistent with our results, where we find a factor of ∼ 2 increase. \nFinally, our results are linearly proportional to τ -1 Q and f bin , which causes the number of dips to vary within a few orders of magnitude. As mentioned, initially we assume that all quasars are binaries ( f bin = 1). In reality, the fraction of quasars that are binaries is a function of \nFigure 5. Distribution of self-lensing flares in the pointsource approximation, without requiring dips. Blue curves show LSST sensitivity limits, as usual, and the black contour lines show cutoffs resulting from requiring that lensing durations exceed 30, 20, and 10 days for our typical binaries with q = 1 , T orb = 5 years. The gray contour lines correspond to the same minimum lensing durations but for binaries with q = 0 . 1, which are the typical binary parameters when requiring a minimum lensing duration of 30 days. Changing the mass ratio from q = 1 to 0 . 1 increases M lens by a factor of ∼ 2 and lower total masses are needed to produce the same lensing duration contour. We further demonstrate the change in the mass-ratio distribution before and after the lensing requirement in Figure 6. \n<!-- image --> \nbinary parameters M and z . In Table V, we list the numbers for varying f bin . The numbers are also sensitive to the model for the residence time τ res . We find that if we conservatively set a maximum residence time of 1 . 7 × 10 7 years (orange curve in Figure 2), then there are only 5 , 6 , 7 self-lensing dips for the limits m i = 24 , 25 , 26. We also note that the expected phase spacing of the dip, Eq. 7, is valid only for circular orbits and would not hold for non-circular orbits.', 'D. Self-lensing flares and comparison to K21': "Although our main focus in this paper is on the number of detectable dips, we next discuss the number of detectable lensing flares, without considering whether or not a measurable dip may be present. As mentioned in the Introduction, the number of detectable flares has been previously estimated by [34]. We therefore make some adjustments to our fiducial model, to mimic the assumptions in K21 as closely as we can, and then compare our results to theirs in detail. The results, analogous to the number of dips, are shown in Figure 5. \nTo make a direct comparison with the results of [34], we need to apply similar constraints on our SMBH population from the QLF, in terms of lensing amplifica- \nTable VI. The number of detectable self-lensing dips and flares as a function of the required minimum lensing duration τ f , min and LSST magnitude limit. The point-source (PS) approximation is assumed for the flares. We show results for various LSST sensitivity limits. Likewise, the first row shows the results in our fiducial model, with the parameters that are varied shown in bold in subsequent rows. The number of detectable flares is a steep function of the minimum required flare duration. Figures 5 and 8 (right panel) visualize the effects of requiring these lensing durations. \ntion, sensitivity limits, lensing durations and orbital periods. We therefore adopt K21's sensitivity limit of 3 × 10 -30 erg s -1 cm -2 Hz -1 , which corresponds to m i = 25 . 2. We also require that self-lensing durations exceed 30 days (10 intra-flare data points assuming three-day LSST cadence) using the Einstein radius at alignment \nr E = √ 2 ar s cos I orb , (21) \nwhere r s = 2 GM lens /c 2 is the Schwarzschild radius of the lens, M lens = M/ (1 + q ). The approximation for the lensing duration is given by \nτ f = T orb π sin -1 ( r E a ) , (22) \nwhere T orb is the observed orbital period. Finally, we consider only binaries with an observed orbital period of T max ≤ 5 years. Evaluating the number of detectable self-lensing flares in our BH population gives 737 detectable self-lensing flares, compared to their 450 +29 -40 flares, where the superscript and subscript denote the interquartile ranges for their expected number. See Table VII for a detailed summary of the two models, where we compare the parameter ranges of the BH populations, the SLF requirements, and the medians of various binary parameters of the SLFs. \nIn both models, it is notable that as a result of requiring a minimum lensing duration of 30 days, the median mass of the binaries with SLFs are above ∼ 10 8 M ⊙ , significantly higher than the typical mass of both parent SMBH populations, 10 6 -7 M ⊙ . This occurs because the lensing duration increases with total mass. Furthermore, the lensing duration is longer when, for a given total mass M , the mass of the primary M lens = M/ (1 + q ) is larger. As a result, both models select relatively asymmetric mass ratios ( q = 10 -2 ∼ 10 -1 ) from the parent SMBH population, which is predominantly symmetric ( q = 1). \nFigure 7. Peak magnification for a binary with q = 1 , M = 10 8 M ⊙ , z = 2 , T orb = 5 years. In the finite-source approximation, peak magnifications in the r-band (622 nm) and y-band (1015 nm) are shown as a function of orbital inclination in degrees. In this sample binary, the largest orbital inclination which admits a 10% flare is ∆ i FS1 / ∆ i PS = 1 . 24 times larger for the finite-source flare in the r-band compared to pointsource flare. This occurs for most of the binaries in our BH population. Between the 622 nm flare and the 1015 nm flare, there is only a ∆ i FS2 / ∆ i FS1 = 1 . 05 factor difference, which accounts for the mild increase in the number of detectable flares for longer wavelengths. For reference, for this binary ∆ i dip = 0 . 1 · , which is 50 -70 times smaller than ∆ i PS or ∆ i FS . \n<!-- image --> \nFigure 6. The mass-ratio distribution of dips and flares in the point-source approximation, where a minimum lensing duration of 0 days or 30 days is required. The two blue N dips distributions correspond to the mass ratio distributions of the 51 and 9 dips given in the m i = 25 column of Table VI. The two red N flares distributions correspond to the 97118 and 737 flares of the same m i = 25 column in Table VI. \n<!-- image --> \nIncreasing the orbital period increases the lensing duration and so binaries with orbital periods of several years tend to exhibit detectable self-lensing flares. Finally, the redshift dependence of the various lensing requirements is weak and the median redshift is mostly unchanged relative to the parent SMBH binary population. \nWhile K21 requires a minimum flare duration of 30 days, equivalent to 10 intra-flare data points, it could be advantageous to include the much more numerous shorter flares (e.g. 10 or 20 days for 3 or 6 intra-flare data points) and search for additional data in higher-cadence surveys or perform targeted high-cadence follow-up observations. We find that there is a steep dependence on the lensing duration, where ∼ 10 4 -5 flares are reduced to just 734737 after requiring the lensing duration of 30 days. We show the dependence of the number of flares on their duration in Table VI. We fix all other parameters to their fiducial values (first row of Table I). In Figure 5, we further show the cut on the SLF binaries due to minimum duration requirements for two different mass ratios, q = 1 (black curves) and q = 0 . 1 (gray curves). \nFinally, we demonstrate the mass-ratio distribution of the self-lensing flares and dips in our model in Figure 6. The mass-ratio distributions of flares and dips without requiring a minimum lensing duration are shown in red and blue. We assume m i = 25 and the other fiducial parameters given in Table I. In this case, most flares and dips have q = 1, following the mass-ratio distribution of Illustris binaries, previously shown in Figure 3. The mass-ratio distributions of flares and dips after requiring a minimum 30-day lensing duration are shown in dark red and dark blue. As a result of the minimum lensing- \nduration requirements, the peak of the distribution shifts to q = 0 . 1, while the qualitative trend of N dips is relatively unchanged because there is only 1 dip for q < 0 . 1.", 'E. Self-lensing flares - finite source sizes': "To predict the number of detectable self-lensing flares in the previous section, we used the analytic methods in § II E, which assume that the background (lensed) SMBH is a point source. However, in some cases, the size of the source, assuming the emission arises from a minidisk modeled as a standard α -disk, becomes comparable to the Einstein radius, and must be taken into account [51]. \nTo see how a finite source size impacts our results, we re-compute the number of detectable lensing flares using the equations from § II F. Overall, we find a ∼ 25% increase in the number of detectable self-lensing flares relative to the point-source approximation, across the various wavelengths and LSST sensitivity limits, which we summarize in Table VIII. \nTable VII. Comparisons between our model vs. Kelley et al. [34]. The top part of the table compares our assumed parent SMBH populations, where we rely on the QLFs and use Illustris only for mass ratios. In contrast, [34] self-consistently evolve ∼ 10 4 SMBH binaries extracted from Illustris using a semi-analytical model and a re-sampling scheme. Our QLF-based population extends to 10 times lower masses and has roughly 30 times more BHs. We also fix the Eddington ratio to 0.3, whereas [34]'s binaries have varying Eddington ratios typically at ∼ 0 . 01. For a binary with given total mass M and redshift z , our binaries are ∼ 30 times brighter than K21's binaries, which could account for why we predict 2 -10 times more detectable flares for minimum lensing durations of 10 -30 days. In the middle part , we compare the requirements imposed on BH binaries for detectable self-lensing flares. The only significant difference is in the minimum magnification, where we require 10%, whereas [34] require a minimum 5% magnification in addition to the inverse of signal-to-noise ratio SNR , which is dependent on their damped random walk model for intrinsic AGN variability. Finally, in the bottom part , we compare the median parameters of binaries that exhibit detectable self-lensing flares and also the final number of self-lensing flares. Despite our different approaches, the predicted number of flares are comparable (737 vs. 450). \nTable VIII. The number of detectable self-lensing flares in the finite-source approximation for different required minimum lensing durations τ f , min in different LSST filters/wavelengths. The three right-most columns show results for various assumed detection limits. \nThis result is somewhat counter-intuitive, since naively, one would expect point sources to be more highly magnified. To understand why there are more detectable flares from finite-sized sources, we consider ∆ i PS and ∆ i FS , which are defined by the largest orbital inclination that produces a self-lensing flare of 10%. These are proportional to the probabilities P PS , P FS that a binary exhibits a detectable self-lensing flare. We visualize the differences between ∆ i PS and ∆ i FS in Figure 7. This figure shows that for orbital inclinations close to zero, the point-source magnification at alignment is much greater than the finite-source magnification, but the opposite happens for larger orbital inclinations. For the typical binaries that constitute most of our BH population with q ∼ 1 , M/M ⊙ ∈ [10 6 , 10 8 ] , z ∈ [1 , 3] , T orb ∼ 5 years, we find that ∆ i FS / ∆ i PS = 1 . 1 -1 . 3, which accounts for the difference between the number of point-source flares and the number of finite-source flares. Notably, for less-common binaries with q < 0 . 1 or M > 10 9 M ⊙ , we find that the point-source and finite-source magnifications converge, i.e. ∆ i PS ∼ ∆ i FS , and these sources do not affect the overall difference in the number of flares. \nIn Figure 7, we also compare the peak magnifications between different wavelengths as a function of orbital inclination. As expected, shorter wavelengths dominate at orbital inclinations close to zero but fall off at longer wavelengths. Thus, longer wavelengths lead to a wider \n<!-- image --> \n<!-- image --> \nFigure 8. Illustration of going from the parent SMBH population to the detectable lensing dips. Left panel : Mass- and redshiftdistribution of all quasars (same as Fig. 1). For the LSST sensitivity limits of m i = 24 , 25 , 26, we find 20 million, 44 million, and 100 million quasars. Middle panel : Number of binaries with an orbital period of 5 years or less. For the LSST sensitivity limits of m i = 24 , 25 , 26, we find 7 million, 17 million, and 45 million binaries. Right panel : Number of detectable self-lensing dips in binaries (same as Fig. 4). For the LSST sensitivity limits of m i = 24 , 25 , 26, we find 41, 51, and 60 binaries in our fiducial model. The visualizations of the lensing duration cutoffs are shown for q = 1 (black) and q = 0 . 1 (gray). \n<!-- image --> \nrange of orbital inclinations that permit a detectable flare. Between the chosen LSST filter wavelengths, we typically find that discrepancies between ∆ i FS are a few degrees or less, which amount to few % or less differences in the self-lensing probabilities P PS , P FS and the number of detectable self-lensing flares. \nThe steep dependence on the minimum lensing duration which occurred for the point-source flares also occurs in the finite-source case. Furthermore, for both pointlike and finite sources, the number of detectable flares depends sensitively on the LSST sensitivity limits when no lensing duration was required, but when a minimum lensing duration of 30 days is required, the dependence on the LSST sensitivity limits diminishes. \nFinally, we note that in general, in addition to the finite radial extent of the 'minidisks' around the individual SMBHs, one must take into account the finite thickness of these minidisks, as well as the circumbinary gas. This raises the concern that a thick circumbinary disk for a nearly edge-on binary may obscure the lensing phenomena discussed in our paper. However, ray-tracing the emission through a hydrodynamical simulation [30] has shown that the circumbinary disk in the foreground typically does not obscure the self-lensing flare and dip. This is because lensing itself allows us to see into the cavity along bent photon paths.", 'IV. SUMMARY AND CONCLUSIONS': "In this paper, we estimated the number of binary quasars that are sufficiently edge-on to display detectable self-lensing flares and dips. Our approach relies on the assumption that the same galaxy mergers produce SMBH \nbinaries and activate bright quasars. Combined with further assumptions about quasar lifetimes, binary fractions, and Eddington ratios, as well as the mass-ratio distributions of binaries extracted from the Illustris simulations, we computed the number of detectable flares, as well as the number of flares with detectable 'dips' due to the background SMBH's shadow, in LSST's expected catalog of tens of millions of AGN light-curves. \nWe recover earlier results by K21 and find that several hundred lensing flares may be detected by LSST. However, this is based a requiring a minimum flare duration of 30 days. We demonstrated a steep dependence of this number on the minimum required flare duration and found that if much shorter flares (say 10 days) were recoverable, then there would be many more (tens of thousands) of these. \nOur main novel result is that we estimate the number of self-lensing dips from SMBHBs. Under plausible assumptions, we find that several dozen of these should be present and detectable by LSST. 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2024arXiv240906770W

The 408 MHz Haslam map is widely used as a lowfrequency anchor for the intensity and morphology of Galactic synchrotron emission. Multifrequency multiexperiment fits show evidence of spatial variation and curvature in the synchrotron frequency spectrum but there are also poorlyunderstood gain factors between experiments. We perform a Bayesian model comparison across a range of scenarios using fits that include recent spectroscopic observations at sim 1GHz by MeerKAT. A large uncorrected gain factor of about 60 in the Haslam data is strongly preferred partly undermining its use as a reference template.

2024-09-01T00:00:00Z

['arXiv:2409.06770', '10.48550/arXiv.2409.06770', '2024arXiv240906770W']

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

Bayesian evidence for uncorrected gain factors in Galactic synchrotron template maps

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06770.pdf

{'Bayesian evidence for uncorrected gain factors in Galactic synchrotron template maps': 'Michael J. Wilensky , 1 , 2 , † , ∥ ★ Melis O. Irfan , 3 , 4 Philip Bull , 2 , 4 \n1 Department of Physics and Trottier Space Institute, McGill University, 3600 University Street, Montreal, QC H3A 2T8, Canada \n2 \nJodrell Bank Centre for Astrophysics, University of Manchester, Manchester, M13 9PL, United Kingdom \n- 3 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK\n- 4 Department of Physics and Astronomy, University of Western Cape, Cape Town 7535, South Africa\n- † CITA National Fellow \n∥ TSI Postdoctoral Fellow \n12 September 2024', 'ABSTRACT': 'The 408 MHz Haslam map is widely used as a low-frequency anchor for the intensity and morphology of Galactic synchrotron emission. Multi-frequency, multi-experiment fits show evidence of spatial variation and curvature in the synchrotron frequency spectrum, but there are also poorly-understood gain factors between experiments. We perform a Bayesian model comparison across a range of scenarios, using fits that include recent spectroscopic observations at ∼ 1 GHz by MeerKAT. A large uncorrected gain factor of about 60% in the Haslam data is strongly preferred, partly undermining its use as a reference template. \nKey words: methods: data analysis - methods: statistical - (cosmology:) diffuse radiation', '1 INTRODUCTION': "Synchrotron emission from our own Galaxy is a pernicious contaminant of cosmological surveys at radio and microwave wavelengths, particularly those making total intensity (and polarisation) maps of the Cosmic Microwave Background and 21cm line emission. Fortunately, Galactic synchrotron emission is expected to derive from free electron populations with power-law energy distributions, leading to a spectral energy distribution (SED) along each line of sight that closely approximates a power-law in frequency, 𝑆 𝜈 ∝ ( 𝜈 / 𝜈 ref ) 𝛼 . Observed values of the power-law spectral index tend to reside in the range -1 . 2 ≲ 𝛼 ≲ -0 . 5, with some variation as a function of Galactic latitude (Planck Collaboration 2016c). This permits a simple model of synchrotron foreground emission to be constructed and subtracted from radio and microwave data, revealing the cosmological maps of interest up to some corrections for signal loss and residual contamination depending on the foreground removal method that is used. \nAn important and long-standing ingredient of many foreground removal approaches is the Haslam 408 MHz all-sky survey (Haslam et al. 1982). The Haslam map anchors the overall amplitude of the synchrotron power-law model in each direction on the sky. Assuming a power-law SED and a suitable model of the spectral index variation across the sky, the synchrotron intensity can then be predicted across a wide range of frequencies. Multiple sky models, such as the Planck Sky Model (Delabrouille et al. 2013) and Python Sky Model (Thorne et al. 2017), rely on the Haslam data as a proxy for all-sky synchrotron emission amplitude in this way. Additionally, emission models such as the Global Sky Model (Zheng et al. 2017), which use empirical data across a wide range of frequencies, still rely on \nthe Haslam data to provide a low frequency, high angular resolution (56 arcmin) synchrotron emission estimate. Similarly, Bayesian fits to empirical data, such as the Commander fits to the WMAP and Planck data (Planck Collaboration 2016a), use the Haslam data as a prior for synchrotron emission amplitude, with a normalization factor that changes this amplitude across their frequency range. SEDs constructed from multiple experiments across frequencies from tens of MHz to tens of GHz have detected a small degree of curvature in the power-law along some lines of sight (Platania et al. 1998; de Oliveira-Costa et al. 2008; Kogut 2012; Mozdzen et al. 2019; Irfan et al. 2022), and a frequency-dependent correction to the spectral index of the form 𝛼 → 𝛼 + 𝐶 log ( 𝜈 / 𝜈 ref ) has been suggested in Kogut (2012). This hints at energy losses through inverse Compton scattering for the relativistic charged particles responsible for synchrotron emission within our Galactic magnetic field (Strong et al. 2011), and also has implications for the accuracy of the synchrotron model at frequencies significantly higher and lower than 408 MHz. \nThere are known flaws in the Haslam map however. For example, the original map contains unsubtracted (or imperfectly subtracted) point sources, and striping caused by imperfect filtering of correlated noise in the time-ordered data. Many of these issues were addressed in a reprocessing of the original data (Remazeilles et al. 2015), but some hints of residual systematic effects remain. For instance, Planck Collaboration (2016a) note that the synchrotron emission amplitude at 408 MHz requires correcting for use at higher frequencies if the spectral index is to be treated as constant over those frequencies, while Planck Collaboration (2016c) note that a single spectral index value across all frequency ranges can't be used to fit empirical data without generating 'large and spurious gain corrections at 408 MHz.' However, Remazeilles et al. (2015) point out that there is a large possible source of calibration uncertainty associated with the Haslam data in the form of an unknown relationship between the angular scale \nof the map and the brightness temperature scale, due to power outside of the primary beam. This beam efficiency factor can be upwards of 30% for a radio experiment (Planck Collaboration 2014; Du et al. 2016). An important question is whether such issues could have led to incorrect inferences in some of the (many) studies that use the Haslam map as an anchor, e.g. regarding the presence of curvature in the synchrotron SED, or over-/under-subtraction of foreground contamination. \nIn this paper, we use the tools of Bayesian model comparison and new spectroscopic radio observations at ∼ 1 GHz to assess how reasonable gain biases may restrict the ability of sub-GHz data alone to constrain the synchrotron spectral index curvature. We restrict ourselves to considering an overall gain factor for each survey as the only systematic effect in the experiments. \nIn Wilensky et al. (2023), a Bayesian jackknife framework called Chiborg was introduced, wherein an analyst proposes a discrete selection of hypotheses to be compared. Formally, the hypotheses represent hierarchical models (Gelman et al. 2021) with common hyperparameters. In each hypothesis, hyperparameters are adjusted to represent the possibility that a subset of the data are somehow biased. The marginal likelihood, often called the evidence , is then calculated for each hypothesis, and these are normalized to create a posterior probability mass over them. For small data sets, or with carefully chosen hyperparameters, one can perform an exhaustive search over all of the possible bias configurations and determine which are most likely. A null hypothesis is always available where none of the data are biased. \nWe slightly depart from the exact formalism in Wilensky et al. (2023), proposing that the data in question are susceptible to multiplicative gain biases, rather than additive offsets. Since the Chiborg framework compares hypotheses by their marginal likelihoods and the underlying model for the data prohibits analytic marginalization, we use the nested sampler polychord (Handley et al. 2015a,b) to simultaneously calculate the evidence for each hypothesis and constrain free parameters. The evidence calculation allows us to determine which experiments, if any, are likely have uncorrected gain calibration errors, while the parameter constraints allow us to estimate their size. \nWhile it has long been known that the Haslam gain errors might deviate from the usual estimate of 10% uncertainty (e.g. Remazeilles et al. 2015; Monsalve et al. 2021), in this paper we show that there is strong statistical evidence that the gain errors may be as high as 60% on degree scales in some regions. This has potentially serious implications for analyses that rely on the Haslam map, such as joint CMB foreground component separation (Planck Collaboration 2014, 2016a,c; BeyondPlanck Collaboration et al. 2023), and foreground removal strategies in 21-cm experiments (Eastwood et al. 2019; Pagano et al. 2024).", '2.1 Synchrotron SED data and models': 'For this analysis we use the same frequency range and regions on the sky as probed in Irfan et al. (2022): the OVRO Long Wavelength Array (OVRO-LWA) 73MHz data, the Haslam 408 MHz data, and the MeerKLASS 971-1075 MHz data within 1 . 8 · regions centered at (RA, Dec) = (161 · , 2.4 · ), (164 · , 2.7 · ) and (167 · , 3.5 · ). In galactic coordinates, ( 𝑙 , 𝑏 ), these are (246.5 · , 50.7 · ), (249.4 · , 53.1 · ) and (252.1 · , 55.8 · ). We denote these regions as fields 0, 1 and 2, respectively. The data have all been smoothed to a common resolution of \n1 . 8 · with Gaussian beams. Both the OVRO-LWA (Eastwood et al. 2018) and MeerKLASS (Santos et al. 2017) experiments are recent (within the last decade) 21cm intensity mapping endeavors; the former is interferometric, while the latter is used in single-dish mode. For the Haslam values, we use the destriped but not desourced map from Remazeilles et al. (2015), since neither the MeerKLASS nor OVRO-LWA measurements have been desourced (Irfan et al. 2022). We use an aperture photometry method to extract SED data points at each frequency from the available maps. This means each SED data point at each frequency is an average over the pixels in a given field in a way that leaves us insensitive to the absolute zero-level of the maps, but sensitive to multiplicative factors. \nFollowing Irfan et al. (2022), we assume as a null hypothesis that the data follow a curved power law, \n𝑆 ( 𝜈 ) = 𝑆 ( 𝜈 0 ) GLYPH<18> 𝜈 𝜈 0 GLYPH<19> 𝛼 0 + 𝑐 ln ( 𝜈 / 𝜈 0 ) , (1) \nwhere 𝜈 is the frequency, 𝜈 0 is a reference frequency, 𝛼 0 is the spectral index at the reference frequency, and 𝑐 is the curvature. Each experiment quotes a fractional calibration uncertainty. OVRO-LWA quotes between a 5% and 10% gain calibration error based on the flux measurement of 11 known calibrators, including Cygnus A, Cassiopeia A, and Virgo A. The Haslam gain calibration error is often quoted as 10%, which relates to the level of scatter seen when correlating this data set with an absolutely calibrated map at 404 MHz (Pauliny-Toth &Shakeshaft 1962). MeerKLASS calibrate their data with respect to a system temperature model which consists of Galactic and terrestrial contributions; if this calibration were perfect the difference between the data and the model would simply be Gaussian distributed noise centred around zero. As they observe data-model residuals of 0.3K for a 16K system temperature model, they quote their calibration error as good to 2%. The gain uncertainties we choose to carry forward for OVRO-LWA, Haslam, and MeerKLASS are 5%, 10% and 2%, respectively. Each experiment also quotes a thermal noise uncertainty that is very small compared to the calibration uncertainty. Nonetheless, we model the data as \nd = g · s + n , (2) \nwhere the 𝑖 th component of each vector refers to the 𝑖 th measured frequency, d is the data, g are the gains, s is the underlying frequency spectrum, n is the noise, and · represents an elementwise product. \nWe assume that the noise is Gaussian and uncorrelated, \nn ∼ N ( 0 , N ) , (3) \nwhere N is a diagonal matrix containing the quoted thermal variance of each experiment. Conditional on knowing the gains and underlying power law, the data are then specified as \nd | s , g ∼ N ( g · s , N ) . (4) \nIn the absence of prior information about the gain amplitudes other than their quoted uncertainties, we assume, under the null hypothesis H 0 , an uncorrelated Gaussian prior centered on 1, with diagonal covariance G that holds the squares of the quoted fractional errors, \ng | G , H 0 ∼ N ( 1 , G ) . (5) \nWe can marginalize over this prior to obtain \nd | s , N , G , H 0 ∼ N GLYPH<16> s , N + ss 𝑇 · G GLYPH<17> , (6) \nwhere s 𝑇 is the transpose of the vector s . We then propose 31 alternate hypotheses. Each hypothesis in the full set of 32 can be thought of in terms of a 5-bit string. One of these bits corresponds to the curvature, \ni.e. in each hypothesis we either specify that the spectral curvature is exactly 0, or we allow it to range from -0 . 3 to 0 with a flat prior. The other four bits correspond to gain offsets. There is one bit each for OVRO-LWA and Haslam having a possible gain bias. For the MeerKLASS data, we split into a lower and upper subband. The division point was chosen by eye at 1007.34 MHz, roughly 2/5 the way through the full MeerKLASS band. This point was chosen since there is a visually apparent discontinuity in the SED at this point that is consistent in all three fields. We propose one overall gain bias for each subband. Denoting B 𝑗 as the set of hypotheses where the 𝑗 th suband has a non-unity gain factor, i.e. the 𝑗 th gain bit is \'active,\' the corresponding bias 𝜀 𝑗 is assumed to have a prior distribution \n𝜀 𝑗 |H ∈ B 𝑗 ∼ N ( 0 , 0 . 25 ) . (7) \nOtherwise, we assume that bias is exactly 0. \nThis choice of prior width is chosen to balance two concerns. First, for a useful test, hypotheses must be distinguishable in the sense that one must be able to reliably (i.e. on average) tell the difference between data generated from one model versus the other (formally explored in Wilensky et al. (2023) in terms of mutual information ). As a pedagogical example, consider a coin flip experiment where two coins are each flipped 10 times and their results are compared. It is essentially impossible to reliably tell the difference between an experiment where the probability of heads for both coins is exactly 50% and an experiment where the probability of heads might vary independently between 45 and 55% between the two coins. There simply is not enough information to reliably resolve the difference between these two very similar models in such a sparse data set even when results are extreme (e.g. 1 heads and 9 tails for both coins). On the other hand comparing the model of two perfectly fair coin tossing processes to one where each heads probability might vary independently all the way from 0 to 1 can produce a reliable decision rule where extreme results strongly favor the \'potentially unfair coins\' hypothesis and balanced results (heads and tails in roughly even proportions) strongly favor the \'fair coins\' hypothesis, with a small region of ambiguity between these two possibilities. In analogy to this example, it is important to choose gain bias priors that are sufficiently broad to produce this distinguishability, with the understanding that favoring the null hypothesis (gain bias exactly 0) may also be desirable in situations involving small gain biases with no statistically meaningful difference of interpretation. However, the second concern is that choosing an excessively broad prior implies a significant belief in unreasonable (or even unphysical) situations, which is clearly undesirable. \nThe gain hierarchy is constructed so that \ng | 𝜺 ∼ N ( 1 + 𝜺 , G ) , (8) \nwhere 𝜺 is a vector of all the gain biases (tiled appropriately to match the corresponding frequencies for each experiment) and 1 is a vector of all ones. This implies that \nd | s , 𝜺 , N , G , H ∈ B ∼ N GLYPH<16> s · ( 1 + 𝜺 ) , N + ss 𝑇 · G GLYPH<17> , (9) \nwhere B is the set of all hypotheses where any bias exists. \nIn all hypotheses, we adopt a broad, flat prior on the 𝑆 ( 𝜈 0 ) and 𝛼 0 parameters, using 𝜈 0 = 73 MHz as the reference. The prior for 𝑆 ( 𝜈 0 ) extends from half the reported OVRO-LWA SED value to twice its value. The 𝛼 0 prior extends from -1 . 8 to 0. \nWe examine these hypotheses using Bayesian model comparison, wherein we calculate the posterior probability of the hypotheses, 𝑃 (H| d ) , and analyze the data in light of what is most likely. This offers a formal way of comparing models that also naturally accounts for overfitting (Kass & Raftery 1995; Jaynes 2003; Mackay 2003), \nthough it is not without complications (Gelman et al. 2021). By Bayes\' theorem, \n𝑃 (H| d ) = 𝑃 ( d |H) 𝑃 (H) " H 𝑃 ( d |H) 𝑃 (H) (10) \nThe term 𝑃 ( d |H) is the marginal likelihood, or evidence, of the hypothesis, H . Representing the model parameters in any given hypothesis as 𝜽 , the evidence is given by \n𝑃 ( d |H) = ∫ d 𝜽 𝑃 ( d | 𝜽 , H) 𝑃 ( 𝜽 |H) , (11) \nwhere 𝑃 ( 𝜽 |H) is the prior for the model parameters in the hypothesis H . The term 𝑃 (H) represents the prior probability of a given hypothesis. This allows one to disfavor different hypotheses in the selection process if they represent unlikely scenarios according to a particular state of knowledge.', '2.2 Numerical implementation': "The nonlinearity of the model in the power law parameters leads to an analytically difficult evidence integral (Equation 11). We instead use the nested sampler polychord (Handley et al. 2015a,b) to simultaneously compute the evidence of each hypothesis and constrain the remaining degrees of freedom. For the sake of reproducibility, we state our choice of polychord parameter settings. polychord 's evidence estimation is affected by the number of live points during sampling, 𝑁 live , which should be scaled according to the dimensionality of the problem, 𝑁 dim . Different hypotheses have different dimensionality. We use 𝑁 live = 50 𝑁 dim . The polychord parameter 𝑁 repeats controls how many slice sampling steps are performed to generate a new sample, which affects the correlation length of the chain. For reliable evidences, it is recommended to use 𝑁 repeats ∼ 5 𝑁 dim . Wefollowed this recommendation and found the inference to be very similar between several independent chains.", '3 HYPOTHESIS COMPARISON': "In order to form a posterior probability over the discrete hypotheses, we must assign a prior to each hypothesis. The simplest option is to assign equal prior probability to each hypothesis, shown in Figure 1, along with 1 𝜎 Monte Carlo error estimates. Since polychord estimates the logarithm of the Bayesian evidence, the errors are larger for stronger hypotheses. With a flat prior, the posterior is dominated by a few hypotheses in each field, concentrated on 8 hypotheses where Haslam suffers from a gain bias in every one, and curvature is disallowed. Field 1 demonstrates mild confusion about whether the OVRO-LWA datum has a gain bias. At least one biased MeerKLASS band is implicated in all dominant hypotheses. Hypotheses with nonzero curvature are strongly disfavored. \nWhether (or not) models with and without curvature can be distinguished over this frequency range is an important concern. To answer this question, we simulated a small number of data sets with randomly determined 𝛼 0 , 𝑆 ( 𝜈 0 ) , and gain biases, all from distributions much more concentrated than the Chiborg priors. In half the simulated data sets, we set the curvature to 0, and in the other half we assigned a curvature randomly, concentrated on the value -0 . 09. We ran half the simulations with a range of anchor frequencies significantly higher than the data (3, 10, 30, and 100 GHz) while holding other model parameters fixed so that we could investigate the effect of having high-frequency anchors. We then used the same priors as we did on the actual data, and evaluated the marginal posterior probability of there being nonzero curvature, expressed as an odds ratio. To \nFigure 1. Posterior probability over the discrete hypotheses for each field, including Monte Carlo estimation errors. Points within a gray band assume a particular hypothesis. Each corresponding label on the horizontal axis indicates which gain biases are active (Haslam, H; LWA, L; MeerKAT sub-bands 1 and 2, M1 and M2), and whether curvature is allowed (C) in the hypothesis. \n<!-- image --> \nbriefly summarize the results, we find that Chiborg 's ability to determine the presence of curvature in the simulations (unsurprisingly) increases rapidly if we add one more anchor at a significantly higher frequency. Using only MeerKAT as an anchor and only small gain biases, it is somewhat more difficult to distinguish between cases with and without curvature in simulation. However, we remark that the simulations are highly conservative, and that marginal odds against curvature are on the order of 30 in each field. This suggests that the actual data are unlikely to be represented by the highly conservative simulations, and that an anomaly is likely present in the data. \nSince in the alternate hypotheses we propose rather large gain biases, one could assign a small prior probability that this occurs in any given experiment to express skepticism, and then assume that separate experiments commit this type of error randomly. For probabilities less than 0.5, this prior disfavors hypotheses at a rate that is exponential in the number of nonzero biases. We find that one must express decisive skepticism in order to make significant changes compared to Figure 1. For instance, assigning a (very low) prior probability of a bias existing of 1% keeps the mass concentrated on the same 8 hypotheses, but the emphasis shifts towards a more balanced weighting.", '4 FITS AND PARAMETER CONSTRAINTS': "A fully Bayesian estimate of the parameters is formed by marginalizing over the hypotheses. Denoting the full set of parameters as 𝜽 , \n𝑃 ( 𝜽 | d ) = ∑︁ H 𝑃 ( 𝜽 , H| d ) = ∑︁ H 𝑃 ( 𝜽 |H , d ) 𝑃 (H| d ) (12) \nwhere 𝑃 (H| d ) is plotted in Figure 1. The full posterior over the parameters is therefore a mixture over the various hypotheses, and results in weakly multimodal parameter estimates in some instances. Field 0 is almost completely dominated by one hypothesis, which incidentally is the most extreme violation of the null hypothesis. We \nshow an updated fit to the SED data in Field 0 in Figure 2 using just this hypothesis for the parameter estimates. In this hypothesis, the gain bias estimates are rather large for all of the experiments. The OVRO-LWA, Haslam, MeerKAT lower subband, and MeerKAT upper subband gain bias estimates for this hypothesis are shown in the first row of Table 1. Note the slight difference in bias estimate for the two MeerKAT subbands, which helps smooth out the apparent discontinuity in the uncorrected data. \nIn Figure 2, we show the SED data calibrated by each experiment independently, along with 'corrected' data, which is the original data divided by the posterior mean of the gain bias for each point. Wethen show the 1 𝜎 posterior quantile and median for the power law model values under the dominant (dashed with grey shade) and null (dotted with grey shade) hypothesis. At each frequency with data, we add a colored 1 𝜎 error bar to represent the remaining uncertainty in the gain realization and thermal noise assuming these particular parameter values. In other words, we have regressed for the mean of g , given by 1 + 𝜺 , but not g itself. The quadrature sum of these two error quantiles roughly corresponds to the 1 𝜎 quantile of the full posterior predictive distribution of the corrected data. \nThe posterior width of the power law model is much larger when we introduce uncertainty in the gain bias parameters. While the null hypothesis expresses relatively extreme certainty about the power law model, particularly at MeerKAT frequencies, over half of the data lie outside the 1 𝜎 quantile indicated by the gain uncertainties, which dominate the posterior predictive uncertainty for the null hypothesis. For the dominant hypothesis, slightly less than 1/3 of the data lie outside the 1 𝜎 quantile of the best fit predictive distribution, which is roughly within expectation in the instance that the chosen parameter values are correct. The model is clearly not perfect, as there appear to be correlated errors in the MeerKLASS data, i.e. the corrected data seem to consistently over-/under-shoot the model over frequency scales of roughly 20 MHz. A more complete analysis might propose a non-diagonal G parameter and regress for the correlation length. \nWe show a corner plot of the posterior samples for Field 0 in Figure 3. The power law parameters exhibit a strong correlation with \nFigure 2. Data and fits in Field 0 using the null hypothesis and dominant hypothesis reported by Chiborg . The 'corrected data' are the data divided by the best fit gain. The 'best fit predictive' is the 1 𝜎 interval for the predictive distribution of the data conditional on the best fit gain bias and model parameters. The grey shades show the 1 𝜎 quantile for the posterior distribution of the predicted power law. The inset shows the MeerKAT data in more detail. \n<!-- image --> \nTable 1. Summary of posterior parameter constraints for MAP hypothesis in each field. \none another, as well as with the OVRO-LWA gain bias. The reference flux is also slightly correlated with Haslam's gain bias. Additionally, the two MeerKAT subbands have very tightly correlated gain biases. Other parameter pairings exhibit mild or no degeneracy. Despite these apparent degeneracies, which were to be expected in such a model, the bulk of the posterior mass still lies rather far away from possibilities where the experiments are unbiased. This reinforces the notion that the data have gain biases. For brevity, we only show PPDs and corner plots for Field 0. Other fields were similar, though with much less extreme gain biases for MeerKAT, and still relatively strong gain adjustments for OVRO-LWA and Haslam. \nTable 1 shows parameter estimates using, for the sake of simplicity, only the maximum a posteriori (MAP) hypothesis for each field. We find somewhat steeper spectral indices compared to Irfan et al. (2022), however this is to be expected since spectral curvature is fixed at zero. Both OVRO-LWA and Haslam are estimated to have significant positive gain biases that are outside the original quoted gain errors for the experiments, however the OVRO-LWA gain bias has only ∼ 2 𝜎 significance in Fields 0 and 1. \nOf particular noteworthiness is that the Commander synchrotron maps (Planck Collaboration 2016b), which have their own gain factor accounting, also adjust the 408 MHz SED downward in these fields, \nFigure 3. Corner plot for the MAP hypothesis in Field 0. Dashed lines show the MAP parameter value (computed over the full joint posterior). \n<!-- image --> \nas visible in Figure 11 of Irfan et al. (2022). For MeerKAT, Field 0 is estimated to have a somewhat strong gain bias, but rather weaker ones for Fields 1 and 2. While roughly within the original quoted gain error for Fields 1 and 2, at face value this can be interpreted as a statistically significant detection of a small but systematic gain bias that affects the entirety of one of the two subbands in either case. \nApotential issue with this interpretation is that there is significant variation between the different fields. Depending on how the data were calibrated, this variation may be inexplicable as an experimental gain. The MeerKAT data were calibrated as described in Wang et al. (2021), and while it is not implausible to have field-dependent calibration errors, alternatives exist for spatial variation such as unmodeled sources and RFI. Given these alternatives, it is hard to decide the source of these errors without additional data analysis. Similarly, direction-dependent calibration in Eastwood et al. (2018) and other aspects of the calibration routine (such as errors in the beam model) could plausibly give rise to field-to-field variation in the OVRO-LWA gain bias, though these effects are folded into the original 5% quoted error. We remark that detection of the gain biases in the OVRO-LWA data is less significant in the sense that the constraints are much broader and models without an OVRO-LWA gain bias are less disfavored than (e.g.) models where MeerKAT has no gain biases. In contrast to the other experiments, the inferred Haslam gain bias is stable across the three fields, removing the need to examine small-scale direction-dependent sources of error assuming they do not somehow align by coincidence.", '4.1 Robustness test': 'To test the robustness of these inferences, we performed the same exercise twice again, but with either the lower or higher MeerKAT subband entirely removed from the analysis. With the substantial reduction in data volume, we expect that there should be more confusion among the hypotheses along with broader parameter constraints, but that strongly disfavored hypotheses should remain strongly disfavored. In other words, the broad conclusions should be similar, but less sharply supported. The posterior probability over the hypotheses for each test run are shown in Figures 4 and 5 for including only the lower and upper subbands, respectively. We again find that the posterior is strongly concentrated on hypotheses where Haslam has a gain bias and curvature is disallowed. There are apparent differences between inferences regarding the OVRO-LWA and MeerKAT data, which we discuss in more detail. \nFocusing on the low subband test, Field 2 has no significant preference for the configuration of gain biases among the OVRO-LWA and MeerKAT data. Comparing to Figure 1, we see that the MAP hypothesis is one where the low subband has no bias, though there is roughly equal probability in hypotheses where the low subband is biased if we marginalize over all possibilities. This ambiguity regarding the low subband is therefore present even when we consider the high subband data simultaneously. The ambiguity among OVROLWAhypotheses is not present in Figure 1, i.e. it is a new feature that emerges when the high subband is excluded. Meanwhile, Field 0 is relatively unsupportive of a gain bias in the lower MeerKAT subband when the higher one is excluded, and vice-versa in Field 1. Again referring to Figure 1, we see that a gain bias is only supported in', 'Low subband only': 'Figure 5. Same as Figure 1, but only using the higher of the two MeerKAT subbands. \n<!-- image --> \nH \nHypothesis (Haslam biased, nonzero curvature, LWA biased, MK1 biased, MK2 biased) \nFigure 4. Same as Figure 1, but only using the lower of the two MeerKAT subbands.', 'High subband only': '<!-- image --> \nH \nHypothesis (Haslam biased, nonzero curvature, LWA biased, MK1 biased, MK2 biased) \nthe lower subband in Field 0 when one is also included in the higher subband, though there is a small chance that only the higher subband has a gain bias. It is therefore perhaps understandable that a gain bias in the lower subband is somewhat unsupported when the higher subband is entirely excluded from the analysis. The balance of low subband gain bias hypotheses in Field 1 is symmetric between Fig- \nure 1 and Figure 4. Similar to Field 2, both Field 0 and 1 are relatively ambiguous about whether the OVRO-LWA datum has a gain bias. \nThe inference using only the higher MeerKAT subband is nearly the opposite. There is a dominant hypothesis in Field 2 wherein all available data have gain biases. This is essentially also reflected in Figure 1, where there is only one competitive hypothesis excluding \nany of the data used in the higher subband test, and it requires the use of the lower subband data (which was unavailable in this test by construction). Field 0 is in strong support of a MeerKAT gain bias, consistent with Figure 1 where all relevant hypotheses include a gain bias in the upper MeerKAT subband. Field 1 tends more towards excluding a gain bias in the upper MeerKAT subband. Once again consulting Figure 1, the MAP hypothesis involves an absence of a gain bias in the upper MeerKAT subband, though the marginal probability of a gain bias in this subband is roughly evenly split, demonstrating a discrepancy between this inference and the complete-data inference. The probability mass is again roughly equally distributed for and against an OVRO-LWA gain bias in Fields 0 and 1. \nSummarizing, examining the MeerKAT subbands separately roughly preserves marginal inferences about the presence of MeerKAT gain biases, but introduces an ambiguity about the presence of an OVRO-LWA gain bias in most cases. The combination of the subbands therefore seems important for making inferences about the presence of an OVRO-LWA gain bias. We suspect this is because the broader spectrum of measurements gives the combined MeerKATdataastronger lever arm over the spectral index parameter, which is degenerate with the reference flux amplitude and OVROLWA gain bias parameters. Importantly, the separate examination preserves the conclusion that Haslam presents a gain bias and there is no curvature. In further support, the Haslam gain bias posteriors are centered at similar values as when all MeerKAT data are used, but the constraints are broader due to the presence of less data (not shown).', '4.2 Discussion': "Where might such a large calibration error arise in the Haslam map? With limited data, we can only speculate. An analysis conducted with several degree-scale absolutely calibrated maps of large sections of the sky, rather than just a few square degrees, could prove extremely useful in this regard. \nIn any case, we mention one plausible source of calibration error that could give rise to an effect of this size. The conversion from brightness temperature to antenna temperature is scale-dependent, as discussed in Jonas et al. (1998). In particular, the solid angle of the source of interest must be taken into account when converting to brightness temperature. To account for this when studying diffuse emission, the data are often placed on the 'full-beam' scale, meaning the antenna temperature must be scaled by the ratio of the 'main beam' solid angle to the 'full beam' solid angle. The full beam scale usually does not refer literally to an integral out to the horizon, but rather out to some cutoff such as a few main beam widths (Haslam et al. 1974; Reich 1982; Jonas et al. 1998). Since the full beam solid angle is greater than the main beam solid angle, the full beam temperature is generally smaller than the main beam temperature. If the map were not put on the full beam scale, or if the full beam scale were significantly underestimated, then one might observe the size (and sign) of gain bias we claim here. \nHaslam et al. (1974) claims only a 5% difference in the two scales (i.e. 95% of the beam power is in the main beam). This claim is based on 10 · sweeps surrounding Cas A (5 · in either direction). Compared to other instruments, where a 30% difference is more typical, this is a very low conversion factor that usually requires careful design and underillumination to acquire (see §4.4 of Remazeilles et al. (2015) and citations therein, as well as Dickinson et al. (2019)). It therefore seems unlikely that this 5% figure is accurate, which makes this a plausible explanation for why we might see a gain-like effect in the Haslam map as large as we do.", '4.3 Comparison to other results': 'These results uniformly favor the hypothesis that the Haslam map is too bright to be consistent with the other data, and that there is unlikely to be any spectral curvature in these fields once this is corrected for. This result is seemingly inconsistent with Monsalve et al. (2021), where globally scaling the Haslam map up by 21% and subtracting off a small 4.1 Kelvin additive offset improved agreement with absolutely calibrated EDGES data. Furthermore, allowing for spectral curvature had a similar effect on Haslam in that analysis (i.e. it produced agreement even without gain correction), whereas in our analysis we observe no such degeneracy. We argue that these analyses cannot be directly compared in a simple way due to position-dependent complications in the Haslam map and a general incongruity between the analysis in Monsalve et al. (2021) and ours, although this discrepancy does need to be resolved to understand the nature of any correction that would need to be applied to the Haslam map. \nThe Haslam map is constructed from four different surveys made with different telescopes throughout the 60s and 70s (Haslam et al. 1982; Remazeilles et al. 2015). This makes it seem unlikely that a single rescaling over the entire sky should fix calibration errors that may be arising from different physical sources specific to particular surveys. For example, in the simple case where there is a single rescaling procedure required for each independent survey, regions of overlap will require different rescalings to regions of non-overlap. Since we are analyzing a tiny region of the sky compared to the more global analysis of Monsalve et al. (2021), it is unsurprising that these results may end up different (despite the fact that we observe consistent gain corrections for Haslam in each of our small, closely spaced fields). Moreover, given the nature of the Haslam map, it may be desirable to have a way of surgically searching for systematic calibration errors in a position-dependent way, such as with Chiborg . This will be particularly relevant for foreground modeling methods such as in Pagano et al. (2024) that partition the sky. \nIn some fields on the sky, the synchrotron component can contribute a meaningful amount of brightness at the ∼ 30 GHz spinning dust peak (Dickinson et al. 2019; Rennie et al. 2022; Harper et al. 2024). As an example of how an analysis like this one could affect inferences about spinning dust, we take our best fit models under the MAP hypothesis (i.e. gain corrected models) for each field and extrapolate them to 30 GHz. We then compare this to an extrapolation with no curvature or gain corrections. The gain corrections tend to produce shallower spectral indices, resulting in a flux density ratio between the MAP model and no-correction model of 2.6, 1.6, and 2.0 for Fields 0, 1, and 2, respectively. In other words, supposing these data were used as low frequency anchors for inference about spinning dust emission at 30 GHz, there would be significantly different results without accounting for these statistically identified gain biases.', '5 CONCLUSIONS': 'Using a joint Bayesian analysis of data from three radio experiments (OVRO-LWA, the Haslam survey, and MeerKLASS), we found significant evidence for uncorrected gain factors in each experiment, with Haslam in particular showing a consistently large bias. The amount of MeerKAT data used notably affects whether we infer a gain bias in the OVRO-LWA data, which we suspect is related to the increased leverage over the spectral index that is obtained when combining all MeerKAT data. The presence and strength of these \ncalibration errors appear somewhat field-dependent, except in the case of the Haslam map. An in-depth analysis of how the data were reduced for each experiment could show whether this is a sensible result in each case, but initial examination of the calibration methods of each experiment does not exclude this possibility. \nWe emphasize that our model assumed that the only possible systematic effect was a gain factor, however one could imagine different types of systematic effect masquerading as large gain errors in this model, e.g. an additive bias from strong radio frequency interference. In any case, calibration errors or other departures from the astrophysical model such as these can cause significant discrepancies in the inferred spectral properties of synchrotron emission. Errors in the spectral properties have direct relevance to inferences regarding the underlying astrophysical processes (e.g. the presence of curvature or lack thereof). \nThese discrepancies are also deleterious for cosmology experiments such as 21-cm intensity mapping surveys, which rely on foreground modeling for accurate calibration solutions. Even if an accurate calibration can be obtained through other means, such as in situ calibration, an incorrect foreground model could cause structure in the foregrounds to be absorbed by parameters related to the cosmological signal unless appropriate precautions are taken, such as jointly modeling the foregrounds with sufficiently broad a priori uncertainty (Anstey et al. 2021; Pagano et al. 2024). Worse yet, we find several experiments spanning at least one order of magnitude in frequency need to be combined in order to root out these gain biases. The possibility of such biases also makes spectral curvature and higher order spectral structure, formally speaking, difficult to assess without a very broad range of frequencies. Applying this Chiborg -based analysis to a selection of compatible maps (e.g. with appropriate angular resolution) formed from spectroscopic measurements spanning a large range of frequencies would allow for the formulation of a radio sky model complete with formal calibration uncertainties derived jointly among the contributing experiments. Such a sky model would help safeguard against inferential pitfalls in cosmology and astrophysics experiments that depend on accurate sky models.', 'ACKNOWLEDGEMENTS': "We acknowledge Clive Dickinson for very helpful discussions. This result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 948764; MW, PB). MW was funded by a CITA National Fellowship. MI acknowledges support from the South African Radio Astronomy Observatory and National Research Foundation (Grant No. 84156). PB acknowledges support from STFC Grant ST/X002624/1.", 'DATA AVAILABILITY': 'The OVRO-LWA and (destriped) Haslam diffuse radio maps are available in the NASA-hosted LAMBDA archive. 1 Access to the raw MeerKAT data used in the analysis is public (for access information please contact archive@ska.ac.za). Data products are available on request. The python code we wrote for our analysis is available at https://github.com/mwilensky768/sed\\_jackknife .', 'REFERENCES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author.'}

2023PhRvX..13a1048A

We report on the population properties of compact binary mergers inferred from gravitationalwave observations of these systems during the first three LIGOVirgo observing runs. The GravitationalWave Transient Catalog 3 GWTC3 contains signals consistent with three classes of binary mergers binary black hole binary neutron star and neutron starblack hole mergers. We infer the binary neutron star merger rate to be between 10 and 1700 GpcSUP3SUP yrSUP1SUP and the neutron starblack hole merger rate to be between 7.8 and 140 GpcSUP3SUP yrSUP1SUP assuming a constant rate density in the comoving frame and taking the union of 90 credible intervals for methods used in this work. We infer the binary black hole merger rate allowing for evolution with redshift to be between 17.9 and 44 GpcSUP3SUP yrSUP1SUP at a fiducial redshift z 0.2 . The rate of binary black hole mergers is observed to increase with redshift at a rate proportional to 1 z SUPSUP with 2. 9SUB1.8SUBSUP1.7SUP for z 1 . Using both binary neutron star and neutron starblack hole binaries we obtain a broad relatively flat neutron star mass distribution extending from 1.2SUB0.2SUBSUP0.1SUP to 2.0SUB0.3SUBSUP0.3SUPMSUBSUB. We confidently determine that the merger rate as a function of mass sharply declines after the expected maximum neutron star mass but cannot yet confirm or rule out the existence of a lower mass gap between neutron stars and black holes. We also find the binary black hole mass distribution has localized over and underdensities relative to a powerlaw distribution with peaks emerging at chirp masses of 8.3SUB0.5SUBSUP0.3SUP and 27.9SUB1.8SUBSUP1.9SUPMSUBSUB. While we continue to find that the mass distribution of a binarys more massive component strongly decreases as a function of primary mass we observe no evidence of a strongly suppressed merger rate above approximately 60 MSUBSUB which would indicate the presence of a upper mass gap. Observed black hole spins are small with half of spin magnitudes below SUBiSUB0.25 . While the majority of spins are preferentially aligned with the orbital angular momentum we infer evidence of antialigned spins among the binary population. We observe an increase in spin magnitude for systems with more unequalmass ratio. We also observe evidence of misalignment of spins relative to the orbital angular momentum.

2023-01-01T00:00:00Z

['2023PhRvX..13a1048A', '10.1103/PhysRevX.13.011048', 'arXiv:2111.03634', '10.48550/arXiv.2111.03634', '2021arXiv211103634T']

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

Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC3

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

https://arxiv.org/pdf/2111.03634.pdf

{'No Header': ',', 'The population of merging compact binaries inferred using gravitational waves through GWTC-3': "R. Abbott, 1 T. D. Abbott, 2 F. Acernese, 3, 4 K. Ackley, 5 C. Adams, 6 N. Adhikari, 7 R. X. Adhikari, 1 V. B. Adya, 8 C. Affeldt, 9, 10 D. Agarwal, 11 M. Agathos, 12, 13 K. Agatsuma, 14 N. Aggarwal, 15 O. D. Aguiar, 16 L. Aiello, 17 A. Ain, 18 P. Ajith, 19 T. Akutsu, 20, 21 P. F. de Alarc'on, 22 S. Akcay, 13, 23 S. Albanesi, 24 A. Allocca, 25, 4 P. A. Altin, 8 A. Amato, 26 C. Anand, 5 S. Anand, 1 A. Ananyeva, 1 S. B. Anderson, 1 W. G. Anderson, 7 M. Ando, 27, 28 T. Andrade, 29 N. Andres, 30 T. Andri'c, 31 S. V. Angelova, 32 S. Ansoldi, 33, 34 J. M. Antelis, 35 S. Antier, 36 F. Antonini, 17 S. Appert, 1 Koji Arai, 1 Koya Arai, 37 Y. Arai, 37 S. Araki, 38 A. Araya, 39 M. C. Araya, 1 J. S. Areeda, 40 M. Ar'ene, 36 N. Aritomi, 27 N. Arnaud, 41, 42 M. Arogeti, 43 S. M. Aronson, 2 K. G. Arun, 44 H. Asada, 45 Y. Asali, 46 G. Ashton, 5 Y. Aso, 47, 48 M. Assiduo, 49, 50 S. M. Aston, 6 P. Astone, 51 F. Aubin, 30 C. Austin, 2 S. Babak, 36 F. Badaracco, 52 M. K. M. Bader, 53 C. Badger, 54 S. Bae, 55 Y. Bae, 56 A. M. Baer, 57 S. Bagnasco, 24 Y. Bai, 1 L. Baiotti, 58 J. Baird, 36 R. Bajpai, 59 M. Ball, 60 G. Ballardin, 42 S. W. Ballmer, 61 A. Balsamo, 57 G. Baltus, 62 S. Banagiri, 63 D. Bankar, 11 J. C. Barayoga, 1 C. Barbieri, 64, 65, 66 B. C. Barish, 1 D. Barker, 67 P. Barneo, 29 F. Barone, 68, 4 B. Barr, 69 L. Barsotti, 70 M. Barsuglia, 36 D. Barta, 71 J. Bartlett, 67 M. A. Barton, 69, 20 I. Bartos, 72 R. Bassiri, 73 A. Basti, 74, 18 M. Bawaj, 75, 76 J. C. Bayley, 69 A. C. Baylor, 7 M. Bazzan, 77, 78 B. B'ecsy, 79 V. M. Bedakihale, 80 M. Bejger, 81 I. Belahcene, 41 V. Benedetto, 82 D. Beniwal, 83 T. F. Bennett, 84 J. D. Bentley, 14 M. BenYaala, 32 F. Bergamin, 9, 10 B. K. Berger, 73 S. Bernuzzi, 13 C. P. L. Berry, 15, 69 D. Bersanetti, 85 A. Bertolini, 53 J. Betzwieser, 6 D. Beveridge, 86 R. Bhandare, 87 U. Bhardwaj, 88, 53 D. Bhattacharjee, 89 S. Bhaumik, 72 I. A. Bilenko, 90 G. Billingsley, 1 S. Bini, 91, 92 R. Birney, 93 O. Birnholtz, 94 S. Biscans, 1, 70 M. Bischi, 49, 50 S. Biscoveanu, 70 A. Bisht, 9, 10 B. Biswas, 11 M. Bitossi, 42, 18 M.-A. Bizouard, 95 J. K. Blackburn, 1 C. D. Blair, 86, 6 D. G. Blair, 86 R. M. Blair, 67 F. Bobba, 96, 97 N. Bode, 9, 10 M. Boer, 95 G. Bogaert, 95 M. Boldrini, 98, 51 L. D. Bonavena, 77 F. Bondu, 99 E. Bonilla, 73 R. Bonnand, 30 P. Booker, 9, 10 B. A. Boom, 53 R. Bork, 1 V. Boschi, 18 N. Bose, 100 S. Bose, 11 V. Bossilkov, 86 V. Boudart, 62 Y. Bouffanais, 77, 78 A. Bozzi, 42 C. Bradaschia, 18 P. R. Brady, 7 A. Bramley, 6 A. Branch, 6 M. Branchesi, 31, 101 J. Brandt, 43 J. E. Brau, 60 M. Breschi, 13 T. Briant, 102 J. H. Briggs, 69 A. Brillet, 95 M. Brinkmann, 9, 10 P. Brockill, 7 A. F. Brooks, 1 J. Brooks, 42 D. D. Brown, 83 S. Brunett, 1 G. Bruno, 52 R. Bruntz, 57 J. Bryant, 14 T. Bulik, 103 H. J. Bulten, 53 A. Buonanno, 104, 105 R. Buscicchio, 14 D. Buskulic, 30 C. Buy, 106 R. L. Byer, 73 L. Cadonati, 43 G. Cagnoli, 26 C. Cahillane, 67 J. Calder'on Bustillo, 107, 108 J. D. Callaghan, 69 T. A. Callister, 109, 110 E. Calloni, 25, 4 J. Cameron, 86 J. B. Camp, 111 M. Canepa, 112, 85 S. Canevarolo, 113 M. Cannavacciuolo, 96 K. C. Cannon, 114 H. Cao, 83 Z. Cao, 115 E. Capocasa, 20 E. Capote, 61 G. Carapella, 96, 97 F. Carbognani, 42 J. B. Carlin, 116 M. F. Carney, 15 M. Carpinelli, 117, 118, 42 G. Carrillo, 60 G. Carullo, 74, 18 T. L. Carver, 17 J. Casanueva Diaz, 42 C. Casentini, 119, 120 G. Castaldi, 121 S. Caudill, 53, 113 M. Cavagli'a, 89 F. Cavalier, 41 R. Cavalieri, 42 M. Ceasar, 122 G. Cella, 18 P. Cerd'a-Dur'an, 123 E. Cesarini, 120 W. Chaibi, 95 K. Chakravarti, 11 S. Chalathadka Subrahmanya, 124 E. Champion, 125 C.-H. Chan, 126 C. Chan, 114 C. L. Chan, 108 K. Chan, 108 M. Chan, 127 K. Chandra, 100 P. Chanial, 42 S. Chao, 126 C. E. A. Chapman-Bird, 69 P. Charlton, 128 E. A. Chase, 15 E. Chassande-Mottin, 36 C. Chatterjee, 86 Debarati Chatterjee, 11 Deep Chatterjee, 7 M. Chaturvedi, 87 S. Chaty, 36 K. Chatziioannou, 1 C. Chen, 129, 130 H. Y. Chen, 70 J. Chen, 126 K. Chen, 131 X. Chen, 86 Y.-B. Chen, 132 Y.-R. Chen, 133 Z. Chen, 17 H. Cheng, 72 C. K. Cheong, 108 H. Y. Cheung, 108 H. Y. Chia, 72 F. Chiadini, 134, 97 C-Y. Chiang, 135 G. Chiarini, 78 R. Chierici, 136 A. Chincarini, 85 M. L. Chiofalo, 74, 18 A. Chiummo, 42 G. Cho, 137 H. S. Cho, 138 R. K. Choudhary, 86 S. Choudhary, 11 N. Christensen, 95 H. Chu, 131 Q. Chu, 86 Y-K. Chu, 135 S. Chua, 8 K. W. Chung, 54 G. Ciani, 77, 78 P. Ciecielag, 81 M. Cie'slar, 81 M. Cifaldi, 119, 120 A. A. Ciobanu, 83 R. Ciolfi, 139, 78 F. Cipriano, 95 A. Cirone, 112, 85 F. Clara, 67 E. N. Clark, 140 J. A. Clark, 1, 43 L. Clarke, 141 P. Clearwater, 142 S. Clesse, 143 F. Cleva, 95 E. Coccia, 31, 101 E. Codazzo, 31 P.-F. Cohadon, 102 D. E. Cohen, 41 L. Cohen, 2 M. Colleoni, 144 C. G. Collette, 145 A. Colombo, 64 M. Colpi, 64, 65 C. M. Compton, 67 M. Constancio Jr., 16 L. Conti, 78 S. J. Cooper, 14 P. Corban, 6 T. R. Corbitt, 2 I. Cordero-Carri'on, 146 S. Corezzi, 76, 75 K. R. Corley, 46 N. Cornish, 79 D. Corre, 41 A. Corsi, 147 S. Cortese, 42 C. A. Costa, 16 R. Cotesta, 105 M. W. Coughlin, 63 J.-P. Coulon, 95 S. T. Countryman, 46 B. Cousins, 148 P. Couvares, 1 D. M. Coward, 86 M. J. Cowart, 6 D. C. Coyne, 1 R. Coyne, 149 J. D. E. Creighton, 7 T. D. Creighton, 150 A. W. Criswell, 63 M. Croquette, 102 S. G. Crowder, 151 J. R. Cudell, 62 T. J. Cullen, 2 A. Cumming, 69 R. Cummings, 69 L. Cunningham, 69 E. Cuoco, 42, 152, 18 M. Curyglyph[suppress]lo, 103 P. Dabadie, 26 T. Dal Canton, 41 S. Dall'Osso, 31 G. D'alya, 153 A. Dana, 73 L. M. DaneshgaranBajastani, 84 B. D'Angelo, 112, 85 B. Danila, 154 S. Danilishin, 155, 53 S. D'Antonio, 120 K. Danzmann, 9, 10 C. Darsow-Fromm, 124 A. Dasgupta, 80 L. E. H. Datrier, 69 S. Datta, 11 V. Dattilo, 42 I. Dave, 87 M. Davier, 41 G. S. Davies, 156 D. Davis, 1 M. C. Davis, 122 E. J. Daw, 157 \nR. Dean, 122 D. DeBra, 73 M. Deenadayalan, 11 J. Degallaix, 158 M. De Laurentis, 25, 4 S. Del'eglise, 102 V. Del Favero, 125 F. De Lillo, 52 N. De Lillo, 69 W. Del Pozzo, 74, 18 L. M. DeMarchi, 15 F. De Matteis, 119, 120 V. D'Emilio, 17 N. Demos, 70 T. Dent, 107 A. Depasse, 52 R. De Pietri, 159, 160 R. De Rosa, 25, 4 C. De Rossi, 42 R. DeSalvo, 121 R. De Simone, 134 S. Dhurandhar, 11 M. C. D'ıaz, 150 M. Diaz-Ortiz Jr., 72 N. A. Didio, 61 T. Dietrich, 105, 53 L. Di Fiore, 4 C. Di Fronzo, 14 C. Di Giorgio, 96, 97 F. Di Giovanni, 123 M. Di Giovanni, 31 T. Di Girolamo, 25, 4 A. Di Lieto, 74, 18 B. Ding, 145 S. Di Pace, 98, 51 I. Di Palma, 98, 51 F. Di Renzo, 74, 18 A. K. Divakarla, 72 A. Dmitriev, 14 Z. Doctor, 60 L. D'Onofrio, 25, 4 F. Donovan, 70 K. L. Dooley, 17 S. Doravari, 11 I. Dorrington, 17 M. Drago, 98, 51 J. C. Driggers, 67 Y. Drori, 1 J.-G. Ducoin, 41 P. Dupej, 69 O. Durante, 96, 97 D. D'Urso, 117, 118 P.-A. Duverne, 41 S. E. Dwyer, 67 C. Eassa, 67 P. J. Easter, 5 M. Ebersold, 161 T. Eckhardt, 124 G. Eddolls, 69 B. Edelman, 60 T. B. Edo, 1 O. Edy, 156 A. Effler, 6 S. Eguchi, 127 J. Eichholz, 8 S. S. Eikenberry, 72 M. Eisenmann, 30 R. A. Eisenstein, 70 A. Ejlli, 17 E. Engelby, 40 Y. Enomoto, 27 L. Errico, 25, 4 R. C. Essick, 162 H. Estell'es, 144 D. Estevez, 163 Z. Etienne, 164 T. Etzel, 1 M. Evans, 70 T. M. Evans, 6 B. E. Ewing, 148 V. Fafone, 119, 120, 31 H. Fair, 61 S. Fairhurst, 17 A. M. Farah, 162 S. Farinon, 85 B. Farr, 60 W. M. Farr, 109, 110 N. W. Farrow, 5 E. J. Fauchon-Jones, 17 G. Favaro, 77 M. Favata, 165 M. Fays, 62 M. Fazio, 166 J. Feicht, 1 M. M. Fejer, 73 E. Fenyvesi, 71, 167 D. L. Ferguson, 168 A. Fernandez-Galiana, 70 I. Ferrante, 74, 18 T. A. Ferreira, 16 F. Fidecaro, 74, 18 P. Figura, 103 I. Fiori, 42 M. Fishbach, 15 R. P. Fisher, 57 R. Fittipaldi, 169, 97 V. Fiumara, 170, 97 R. Flaminio, 30, 20 E. Floden, 63 H. Fong, 114 J. A. Font, 123, 171 B. Fornal, 172 P. W. F. Forsyth, 8 A. Franke, 124 S. Frasca, 98, 51 F. Frasconi, 18 C. Frederick, 173 J. P. Freed, 35 Z. Frei, 153 A. Freise, 174 R. Frey, 60 P. Fritschel, 70 V. V. Frolov, 6 G. G. Fronz'e, 24 Y. Fujii, 175 Y. Fujikawa, 176 M. Fukunaga, 37 M. Fukushima, 21 P. Fulda, 72 M. Fyffe, 6 H. A. Gabbard, 69 B. U. Gadre, 105 J. R. Gair, 105 J. Gais, 108 S. Galaudage, 5 R. Gamba, 13 D. Ganapathy, 70 A. Ganguly, 19 D. Gao, 177 S. G. Gaonkar, 11 B. Garaventa, 85, 112 F. Garc'ıa, 36 C. Garc'ıa-N'u˜nez, 93 C. Garc'ıa-Quir'os, 144 F. Garufi, 25, 4 B. Gateley, 67 S. Gaudio, 35 V. Gayathri, 72 G.-G. Ge, 177 G. Gemme, 85 A. Gennai, 18 J. George, 87 R. N. George, 168 O. Gerberding, 124 L. Gergely, 154 P. Gewecke, 124 S. Ghonge, 43 Abhirup Ghosh, 105 Archisman Ghosh, 178 Shaon Ghosh, 7, 165 Shrobana Ghosh, 17 B. Giacomazzo, 64, 65, 66 L. Giacoppo, 98, 51 J. A. Giaime, 2, 6 K. D. Giardina, 6 D. R. Gibson, 93 C. Gier, 32 M. Giesler, 179 P. Giri, 18, 74 F. Gissi, 82 J. Glanzer, 2 A. E. Gleckl, 40 P. Godwin, 148 J. Golomb, 1 E. Goetz, 180 R. Goetz, 72 N. Gohlke, 9, 10 B. Goncharov, 5, 31 G. Gonz'alez, 2 A. Gopakumar, 181 M. Gosselin, 42 R. Gouaty, 30 D. W. Gould, 8 B. Grace, 8 A. Grado, 182, 4 M. Granata, 158 V. Granata, 96 A. Grant, 69 S. Gras, 70 P. Grassia, 1 C. Gray, 67 R. Gray, 69 G. Greco, 75 A. C. Green, 72 R. Green, 17 A. M. Gretarsson, 35 E. M. Gretarsson, 35 D. Griffith, 1 W. Griffiths, 17 H. L. Griggs, 43 G. Grignani, 76, 75 A. Grimaldi, 91, 92 S. J. Grimm, 31, 101 H. Grote, 17 S. Grunewald, 105 P. Gruning, 41 D. Guerra, 123 G. M. Guidi, 49, 50 A. R. Guimaraes, 2 G. Guix'e, 29 H. K. Gulati, 80 H.-K. Guo, 172 Y. Guo, 53 Anchal Gupta, 1 Anuradha Gupta, 183 P. Gupta, 53, 113 E. K. Gustafson, 1 R. Gustafson, 184 F. Guzman, 185 S. Ha, 186 L. Haegel, 36 A. Hagiwara, 37, 187 S. Haino, 135 O. Halim, 34, 188 E. D. Hall, 70 E. Z. Hamilton, 161 G. Hammond, 69 W.-B. Han, 189 M. Haney, 161 J. Hanks, 67 C. Hanna, 148 M. D. Hannam, 17 O. Hannuksela, 113, 53 H. Hansen, 67 T. J. Hansen, 35 J. Hanson, 6 T. Harder, 95 T. Hardwick, 2 K. Haris, 53, 113 J. Harms, 31, 101 G. M. Harry, 190 I. W. Harry, 156 D. Hartwig, 124 K. Hasegawa, 37 B. Haskell, 81 R. K. Hasskew, 6 C.-J. Haster, 70 K. Hattori, 191 K. Haughian, 69 H. Hayakawa, 192 K. Hayama, 127 F. J. Hayes, 69 J. Healy, 125 A. Heidmann, 102 A. Heidt, 9, 10 M. C. Heintze, 6 J. Heinze, 9, 10 J. Heinzel, 193 H. Heitmann, 95 F. Hellman, 194 P. Hello, 41 A. F. Helmling-Cornell, 60 G. Hemming, 42 M. Hendry, 69 I. S. Heng, 69 E. Hennes, 53 J. Hennig, 195 M. H. Hennig, 195 A. G. Hernandez, 84 F. Hernandez Vivanco, 5 M. Heurs, 9, 10 S. Hild, 155, 53 P. Hill, 32 Y. Himemoto, 196 A. S. Hines, 185 Y. Hiranuma, 197 N. Hirata, 20 E. Hirose, 37 S. Hochheim, 9, 10 D. Hofman, 158 J. N. Hohmann, 124 D. G. Holcomb, 122 N. A. Holland, 8 I. J. Hollows, 157 Z. J. Holmes, 83 K. Holt, 6 D. E. Holz, 162 Z. Hong, 198 P. Hopkins, 17 J. Hough, 69 S. Hourihane, 132 E. J. Howell, 86 C. G. Hoy, 17 D. Hoyland, 14 A. Hreibi, 9, 10 B-H. Hsieh, 37 Y. Hsu, 126 G-Z. Huang, 198 H-Y. Huang, 135 P. Huang, 177 Y-C. Huang, 133 Y.-J. Huang, 135 Y. Huang, 70 M. T. Hubner, 5 A. D. Huddart, 141 B. Hughey, 35 D. C. Y. Hui, 199 V. Hui, 30 S. Husa, 144 S. H. Huttner, 69 R. Huxford, 148 T. Huynh-Dinh, 6 S. Ide, 200 B. Idzkowski, 103 A. Iess, 119, 120 B. Ikenoue, 21 S. Imam, 198 K. Inayoshi, 201 C. Ingram, 83 Y. Inoue, 131 K. Ioka, 202 M. Isi, 70 K. Isleif, 124 K. Ito, 203 Y. Itoh, 204, 205 B. R. Iyer, 19 K. Izumi, 206 V. JaberianHamedan, 86 T. Jacqmin, 102 S. J. Jadhav, 207 S. P. Jadhav, 11 A. L. James, 17 A. Z. Jan, 125 K. Jani, 208 J. Janquart, 113, 53 K. Janssens, 209, 95 N. N. Janthalur, 207 P. Jaranowski, 210 D. Jariwala, 72 R. Jaume, 144 A. C. Jenkins, 54 K. Jenner, 83 C. Jeon, 211 M. Jeunon, 63 W. Jia, 70 H.-B. Jin, 212, 213 G. R. Johns, 57 A. W. Jones, 86 D. I. Jones, 214 J. D. Jones, 67 P. Jones, 14 R. Jones, 69 R. J. G. Jonker, 53 L. Ju, 86 P. Jung, 56 k. Jung, 186 J. Junker, 9, 10 V. Juste, 163 K. Kaihotsu, 203 T. Kajita, 215 M. Kakizaki, 191 C. V. Kalaghatgi, 17, 113 V. Kalogera, 15 B. Kamai, 1 M. Kamiizumi, 192 N. Kanda, 204, 205 S. Kandhasamy, 11 G. Kang, 216 J. B. Kanner, 1 Y. Kao, 126 S. J. Kapadia, 19 D. P. Kapasi, 8 S. Karat, 1 C. Karathanasis, 217 S. Karki, 89 R. Kashyap, 148 M. Kasprzack, 1 W. Kastaun, 9, 10 S. Katsanevas, 42 E. Katsavounidis, 70 W. Katzman, 6 T. Kaur, 86 K. Kawabe, 67 K. Kawaguchi, 37 \nN. Kawai, 218 T. Kawasaki, 27 F. K'ef'elian, 95 D. Keitel, 144 J. S. Key, 219 S. Khadka, 73 F. Y. Khalili, 90 S. Khan, 17 E. A. Khazanov, 220 N. Khetan, 31, 101 M. Khursheed, 87 N. Kijbunchoo, 8 C. Kim, 221 J. C. Kim, 222 J. Kim, 223 K. Kim, 224 W. S. Kim, 225 Y.-M. Kim, 226 C. Kimball, 15 N. Kimura, 187 M. Kinley-Hanlon, 69 R. Kirchhoff, 9, 10 J. S. Kissel, 67 N. Kita, 27 H. Kitazawa, 203 L. Kleybolte, 124 S. Klimenko, 72 A. M. Knee, 180 T. D. Knowles, 164 E. Knyazev, 70 P. Koch, 9, 10 G. Koekoek, 53, 155 Y. Kojima, 227 K. Kokeyama, 228 S. Koley, 31 P. Kolitsidou, 17 M. Kolstein, 217 K. Komori, 70, 27 V. Kondrashov, 1 A. K. H. Kong, 229 A. Kontos, 230 N. Koper, 9, 10 M. Korobko, 124 K. Kotake, 127 M. Kovalam, 86 D. B. Kozak, 1 C. Kozakai, 47 R. Kozu, 192 V. Kringel, 9, 10 N. V. Krishnendu, 9, 10 A. Kr'olak, 231, 232 G. Kuehn, 9, 10 F. Kuei, 126 P. Kuijer, 53 S. Kulkarni, 183 A. Kumar, 207 P. Kumar, 179 Rahul Kumar, 67 Rakesh Kumar, 80 J. Kume, 28 K. Kuns, 70 C. Kuo, 131 H-S. Kuo, 198 Y. Kuromiya, 203 S. Kuroyanagi, 233, 234 K. Kusayanagi, 218 S. Kuwahara, 114 K. Kwak, 186 P. Lagabbe, 30 D. Laghi, 74, 18 E. Lalande, 235 T. L. Lam, 108 A. Lamberts, 95, 236 M. Landry, 67 P. Landry, 237 B. B. Lane, 70 R. N. Lang, 70 J. Lange, 168 B. Lantz, 73 I. La Rosa, 30 A. Lartaux-Vollard, 41 P. D. Lasky, 5 M. Laxen, 6 A. Lazzarini, 1 C. Lazzaro, 77, 78 P. Leaci, 98, 51 S. Leavey, 9, 10 Y. K. Lecoeuche, 180 H. K. Lee, 238 H. M. Lee, 137 H. W. Lee, 222 J. Lee, 137 K. Lee, 239 R. Lee, 133 J. Lehmann, 9, 10 A. Lemaˆıtre, 240 M. Leonardi, 20 N. Leroy, 41 N. Letendre, 30 C. Levesque, 235 Y. Levin, 5 J. N. Leviton, 184 K. Leyde, 36 A. K. Y. Li, 1 B. Li, 126 J. Li, 15 K. L. Li, 241 T. G. F. Li, 108 X. Li, 132 C-Y. Lin, 242 F-K. Lin, 135 F-L. Lin, 198 H. L. Lin, 131 L. C.-C. Lin, 186 F. Linde, 243, 53 S. D. Linker, 84 J. N. Linley, 69 T. B. Littenberg, 244 G. C. Liu, 129 J. Liu, 9, 10 K. Liu, 126 X. Liu, 7 F. Llamas, 150 M. Llorens-Monteagudo, 123 R. K. L. Lo, 1 A. Lockwood, 245 M. Loh, 40 L. T. London, 70 A. Longo, 246, 247 D. Lopez, 161 M. Lopez Portilla, 113 M. Lorenzini, 119, 120 V. Loriette, 248 M. Lormand, 6 G. Losurdo, 18 T. P. Lott, 43 J. D. Lough, 9, 10 C. O. Lousto, 125 G. Lovelace, 40 J. F. Lucaccioni, 173 H. Luck, 9, 10 D. Lumaca, 119, 120 A. P. Lundgren, 156 L.-W. Luo, 135 J. E. Lynam, 57 R. Macas, 156 M. MacInnis, 70 D. M. Macleod, 17 I. A. O. MacMillan, 1 A. Macquet, 95 I. Maga˜na Hernandez, 7 C. Magazz'u, 18 R. M. Magee, 1 R. Maggiore, 14 M. Magnozzi, 85, 112 S. Mahesh, 164 E. Majorana, 98, 51 C. Makarem, 1 I. Maksimovic, 248 S. Maliakal, 1 A. Malik, 87 N. Man, 95 V. Mandic, 63 V. Mangano, 98, 51 J. L. Mango, 249 G. L. Mansell, 67, 70 M. Manske, 7 M. Mantovani, 42 M. Mapelli, 77, 78 F. Marchesoni, 250, 75, 251 M. Marchio, 20 F. Marion, 30 Z. Mark, 132 S. M'arka, 46 Z. M'arka, 46 C. Markakis, 12 A. S. Markosyan, 73 A. Markowitz, 1 E. Maros, 1 A. Marquina, 146 S. Marsat, 36 F. Martelli, 49, 50 I. W. Martin, 69 R. M. Martin, 165 M. Martinez, 217 V. A. Martinez, 72 V. Martinez, 26 K. Martinovic, 54 D. V. Martynov, 14 E. J. Marx, 70 H. Masalehdan, 124 K. Mason, 70 E. Massera, 157 A. Masserot, 30 T. J. Massinger, 70 M. Masso-Reid, 69 S. Mastrogiovanni, 36 A. Matas, 105 M. Mateu-Lucena, 144 F. Matichard, 1, 70 M. Matiushechkina, 9, 10 N. Mavalvala, 70 J. J. McCann, 86 R. McCarthy, 67 D. E. McClelland, 8 P. K. McClincy, 148 S. McCormick, 6 L. McCuller, 70 G. I. McGhee, 69 S. C. McGuire, 252 C. McIsaac, 156 J. McIver, 180 T. McRae, 8 S. T. McWilliams, 164 D. Meacher, 7 M. Mehmet, 9, 10 A. K. Mehta, 105 Q. Meijer, 113 A. Melatos, 116 D. A. Melchor, 40 G. Mendell, 67 A. Menendez-Vazquez, 217 C. S. Menoni, 166 R. A. Mercer, 7 L. Mereni, 158 K. Merfeld, 60 E. L. Merilh, 6 J. D. Merritt, 60 M. Merzougui, 95 S. Meshkov, 1, ∗ C. Messenger, 69 C. Messick, 168 P. M. Meyers, 116 F. Meylahn, 9, 10 A. Mhaske, 11 A. Miani, 91, 92 H. Miao, 14 I. Michaloliakos, 72 C. Michel, 158 Y. Michimura, 27 H. Middleton, 116 L. Milano, 25 A. L. Miller, 52 A. Miller, 84 B. Miller, 88, 53 S. Miller, 1 M. Millhouse, 116 J. C. Mills, 17 E. Milotti, 188, 34 O. Minazzoli, 95, 253 Y. Minenkov, 120 N. Mio, 254 Ll. M. Mir, 217 M. Miravet-Ten'es, 123 C. Mishra, 255 T. Mishra, 72 T. Mistry, 157 S. Mitra, 11 V. P. Mitrofanov, 90 G. Mitselmakher, 72 R. Mittleman, 70 O. Miyakawa, 192 A. Miyamoto, 204 Y. Miyazaki, 27 K. Miyo, 192 S. Miyoki, 192 Geoffrey Mo, 70 L. M. Modafferi, 22 E. Moguel, 173 K. Mogushi, 89 S. R. P. Mohapatra, 70 S. R. Mohite, 7 I. Molina, 40 M. Molina-Ruiz, 194 M. Mondin, 84 M. Montani, 49, 50 C. J. Moore, 14 D. Moraru, 67 F. Morawski, 81 A. More, 11 C. Moreno, 35 G. Moreno, 67 Y. Mori, 203 S. Morisaki, 7 Y. Moriwaki, 191 G. Morr'as, 256 B. Mours, 163 C. M. Mow-Lowry, 14, 174 S. Mozzon, 156 F. Muciaccia, 98, 51 Arunava Mukherjee, 257 D. Mukherjee, 148 Soma Mukherjee, 150 Subroto Mukherjee, 80 Suvodip Mukherjee, 88 N. Mukund, 9, 10 A. Mullavey, 6 J. Munch, 83 E. A. Mu˜niz, 61 P. G. Murray, 69 R. Musenich, 85, 112 S. Muusse, 83 S. L. Nadji, 9, 10 K. Nagano, 206 S. Nagano, 258 A. Nagar, 24, 259 K. Nakamura, 20 H. Nakano, 260 M. Nakano, 37 R. Nakashima, 218 Y. Nakayama, 203 V. Napolano, 42 I. Nardecchia, 119, 120 T. Narikawa, 37 L. Naticchioni, 51 B. Nayak, 84 R. K. Nayak, 261 R. Negishi, 197 B. F. Neil, 86 J. Neilson, 82, 97 G. Nelemans, 262 T. J. N. Nelson, 6 M. Nery, 9, 10 P. Neubauer, 173 A. Neunzert, 219 K. Y. Ng, 70 S. W. S. Ng, 83 C. Nguyen, 36 P. Nguyen, 60 T. Nguyen, 70 L. Nguyen Quynh, 263 W.-T. Ni, 212, 177, 133 S. A. Nichols, 2 A. Nishizawa, 28 S. Nissanke, 88, 53 E. Nitoglia, 136 F. Nocera, 42 M. Norman, 17 C. North, 17 S. Nozaki, 191 J. F. Nu˜no Siles, 256 L. K. Nuttall, 156 J. Oberling, 67 B. D. O'Brien, 72 Y. Obuchi, 21 J. O'Dell, 141 E. Oelker, 69 W. Ogaki, 37 G. Oganesyan, 31, 101 J. J. Oh, 225 K. Oh, 199 S. H. Oh, 225 M. Ohashi, 192 N. Ohishi, 47 M. Ohkawa, 176 F. Ohme, 9, 10 H. Ohta, 114 M. A. Okada, 16 Y. Okutani, 200 K. Okutomi, 192 C. Olivetto, 42 K. Oohara, 197 C. Ooi, 27 R. Oram, 6 B. O'Reilly, 6 R. G. Ormiston, 63 N. D. Ormsby, 57 L. F. Ortega, 72 R. O'Shaughnessy, 125 E. O'Shea, 179 S. Oshino, 192 \nS. Ossokine, 105 C. Osthelder, 1 S. Otabe, 218 D. J. Ottaway, 83 H. Overmier, 6 A. E. Pace, 148 G. Pagano, 74, 18 M. A. Page, 86 G. Pagliaroli, 31, 101 A. Pai, 100 S. A. Pai, 87 J. R. Palamos, 60 O. Palashov, 220 C. Palomba, 51 H. Pan, 126 K. Pan, 133, 229 P. K. Panda, 207 H. Pang, 131 P. T. H. Pang, 53, 113 C. Pankow, 15 F. Pannarale, 98, 51 B. C. Pant, 87 F. H. Panther, 86 F. Paoletti, 18 A. Paoli, 42 A. Paolone, 51, 264 A. Parisi, 129 H. Park, 7 J. Park, 265 W. Parker, 6, 252 D. Pascucci, 53 A. Pasqualetti, 42 R. Passaquieti, 74, 18 D. Passuello, 18 M. Patel, 57 M. Pathak, 83 B. Patricelli, 42, 18 A. S. Patron, 2 S. Paul, 60 E. Payne, 5 M. Pedraza, 1 M. Pegoraro, 78 A. Pele, 6 F. E. Pe˜na Arellano, 192 S. Penn, 266 A. Perego, 91, 92 A. Pereira, 26 T. Pereira, 267 C. J. Perez, 67 C. P'erigois, 30 C. C. Perkins, 72 A. Perreca, 91, 92 S. Perri'es, 136 J. Petermann, 124 D. Petterson, 1 H. P. Pfeiffer, 105 K. A. Pham, 63 K. S. Phukon, 53, 243 O. J. Piccinni, 51 M. Pichot, 95 M. Piendibene, 74, 18 F. Piergiovanni, 49, 50 L. Pierini, 98, 51 V. Pierro, 82, 97 G. Pillant, 42 M. Pillas, 41 F. Pilo, 18 L. Pinard, 158 I. M. Pinto, 82, 97, 268 M. Pinto, 42 B. Piotrzkowski, 7 K. Piotrzkowski, 52 M. Pirello, 67 M. D. Pitkin, 269 E. Placidi, 98, 51 L. Planas, 144 W. Plastino, 246, 247 C. Pluchar, 140 R. Poggiani, 74, 18 E. Polini, 30 D. Y. T. Pong, 108 S. Ponrathnam, 11 P. Popolizio, 42 E. K. Porter, 36 R. Poulton, 42 J. Powell, 142 M. Pracchia, 30 T. Pradier, 163 A. K. Prajapati, 80 K. Prasai, 73 R. Prasanna, 207 G. Pratten, 14 M. Principe, 82, 268, 97 G. A. Prodi, 270, 92 L. Prokhorov, 14 P. Prosposito, 119, 120 L. Prudenzi, 105 A. Puecher, 53, 113 M. Punturo, 75 F. Puosi, 18, 74 P. Puppo, 51 M. Purrer, 105 H. Qi, 17 V. Quetschke, 150 R. Quitzow-James, 89 F. J. Raab, 67 G. Raaijmakers, 88, 53 H. Radkins, 67 N. Radulesco, 95 P. Raffai, 153 S. X. Rail, 235 S. Raja, 87 C. Rajan, 87 K. E. Ramirez, 6 T. D. Ramirez, 40 A. Ramos-Buades, 105 J. Rana, 148 P. Rapagnani, 98, 51 U. D. Rapol, 271 A. Ray, 7 V. Raymond, 17 N. Raza, 180 M. Razzano, 74, 18 J. Read, 40 L. A. Rees, 190 T. Regimbau, 30 L. Rei, 85 S. Reid, 32 S. W. Reid, 57 D. H. Reitze, 1, 72 P. Relton, 17 A. Renzini, 1 P. Rettegno, 272, 24 A. Reza, 53 M. Rezac, 40 F. Ricci, 98, 51 D. Richards, 141 J. W. Richardson, 1 L. Richardson, 185 G. Riemenschneider, 272, 24 K. Riles, 184 S. Rinaldi, 18, 74 K. Rink, 180 M. Rizzo, 15 N. A. Robertson, 1, 69 R. Robie, 1 F. Robinet, 41 A. Rocchi, 120 S. Rodriguez, 40 L. Rolland, 30 J. G. Rollins, 1 M. Romanelli, 99 R. Romano, 3, 4 C. L. Romel, 67 A. Romero-Rodr'ıguez, 217 I. M. Romero-Shaw, 5 J. H. Romie, 6 S. Ronchini, 31, 101 L. Rosa, 4, 25 C. A. Rose, 7 D. Rosi'nska, 103 M. P. Ross, 245 S. Rowan, 69 S. J. Rowlinson, 14 S. Roy, 113 Santosh Roy, 11 Soumen Roy, 273 D. Rozza, 117, 118 P. Ruggi, 42 K. Ryan, 67 S. Sachdev, 148 T. Sadecki, 67 J. Sadiq, 107 N. Sago, 274 S. Saito, 21 Y. Saito, 192 K. Sakai, 275 Y. Sakai, 197 M. Sakellariadou, 54 Y. Sakuno, 127 O. S. Salafia, 66, 65, 64 L. Salconi, 42 M. Saleem, 63 F. Salemi, 91, 92 A. Samajdar, 53, 113 E. J. Sanchez, 1 J. H. Sanchez, 40 L. E. Sanchez, 1 N. Sanchis-Gual, 276 J. R. Sanders, 277 A. Sanuy, 29 T. R. Saravanan, 11 N. Sarin, 5 B. Sassolas, 158 H. Satari, 86 B. S. Sathyaprakash, 148, 17 S. Sato, 278 T. Sato, 176 O. Sauter, 72 R. L. Savage, 67 T. Sawada, 204 D. Sawant, 100 H. L. Sawant, 11 S. Sayah, 158 D. Schaetzl, 1 M. Scheel, 132 J. Scheuer, 15 M. Schiworski, 83 P. Schmidt, 14 S. Schmidt, 113 R. Schnabel, 124 M. Schneewind, 9, 10 R. M. S. Schofield, 60 A. Schonbeck, 124 B. W. Schulte, 9, 10 B. F. Schutz, 17, 9, 10 E. Schwartz, 17 J. Scott, 69 S. M. Scott, 8 M. Seglar-Arroyo, 30 T. Sekiguchi, 28 Y. Sekiguchi, 279 D. Sellers, 6 A. S. Sengupta, 273 D. Sentenac, 42 E. G. Seo, 108 V. Sequino, 25, 4 A. Sergeev, 220 Y. Setyawati, 113 T. Shaffer, 67 M. S. Shahriar, 15 B. Shams, 172 L. Shao, 201 A. Sharma, 31, 101 P. Sharma, 87 P. Shawhan, 104 N. S. Shcheblanov, 240 S. Shibagaki, 127 M. Shikauchi, 114 R. Shimizu, 21 T. Shimoda, 27 K. Shimode, 192 H. Shinkai, 280 T. Shishido, 48 A. Shoda, 20 D. H. Shoemaker, 70 D. M. Shoemaker, 168 S. ShyamSundar, 87 M. Sieniawska, 103 D. Sigg, 67 L. P. Singer, 111 D. Singh, 148 N. Singh, 103 A. Singha, 155, 53 A. M. Sintes, 144 V. Sipala, 117, 118 V. Skliris, 17 B. J. J. Slagmolen, 8 T. J. Slaven-Blair, 86 J. Smetana, 14 J. R. Smith, 40 R. J. E. Smith, 5 J. Soldateschi, 281, 282, 50 S. N. Somala, 283 K. Somiya, 218 E. J. Son, 225 K. Soni, 11 S. Soni, 2 V. Sordini, 136 F. Sorrentino, 85 N. Sorrentino, 74, 18 H. Sotani, 284 R. Soulard, 95 T. Souradeep, 271, 11 E. Sowell, 147 V. Spagnuolo, 155, 53 A. P. Spencer, 69 M. Spera, 77, 78 R. Srinivasan, 95 A. K. Srivastava, 80 V. Srivastava, 61 K. Staats, 15 C. Stachie, 95 D. A. Steer, 36 J. Steinhoff, 105 J. Steinlechner, 155, 53 S. Steinlechner, 155, 53 S. P. Stevenson, 142 D. J. Stops, 14 M. Stover, 173 K. A. Strain, 69 L. C. Strang, 116 G. Stratta, 285, 50 A. Strunk, 67 R. Sturani, 267 A. L. Stuver, 122 S. Sudhagar, 11 V. Sudhir, 70 R. Sugimoto, 286, 206 H. G. Suh, 7 A. G. Sullivan, 46 T. Z. Summerscales, 287 H. Sun, 86 L. Sun, 8 S. Sunil, 80 A. Sur, 81 J. Suresh, 114, 37 P. J. Sutton, 17 Takamasa Suzuki, 176 Toshikazu Suzuki, 37 B. L. Swinkels, 53 M. J. Szczepa'nczyk, 72 P. Szewczyk, 103 M. Tacca, 53 H. Tagoshi, 37 S. C. Tait, 69 H. Takahashi, 288 R. Takahashi, 20 A. Takamori, 39 S. Takano, 27 H. Takeda, 27 M. Takeda, 204 C. J. Talbot, 32 C. Talbot, 1 H. Tanaka, 289 Kazuyuki Tanaka, 204 Kenta Tanaka, 289 Taiki Tanaka, 37 Takahiro Tanaka, 274 A. J. Tanasijczuk, 52 S. Tanioka, 20, 48 D. B. Tanner, 72 D. Tao, 1 L. Tao, 72 E. N. Tapia San Mart'ın, 53, 20 C. Taranto, 119 J. D. Tasson, 193 S. Telada, 290 R. Tenorio, 144 J. E. Terhune, 122 L. Terkowski, 124 M. P. Thirugnanasambandam, 11 L. Thomas, 14 M. Thomas, 6 P. Thomas, 67 J. E. Thompson, 17 S. R. Thondapu, 87 K. A. Thorne, 6 E. Thrane, 5 Shubhanshu Tiwari, 161 Srishti Tiwari, 11 V. Tiwari, 17 A. M. Toivonen, 63 K. Toland, 69 A. E. Tolley, 156 T. Tomaru, 20 Y. Tomigami, 204 T. Tomura, 192 M. Tonelli, 74, 18 A. Torres-Forn'e, 123 C. I. Torrie, 1 I. Tosta e Melo, 117, 118 D. Toyra, 8 A. Trapananti, 250, 75 F. Travasso, 75, 250 G. Traylor, 6 M. Trevor, 104 M. C. Tringali, 42 A. Tripathee, 184 L. Troiano, 291, 97 A. Trovato, 36 \nL. Trozzo, 4, 192 R. J. Trudeau, 1 D. S. Tsai, 126 D. Tsai, 126 K. W. Tsang, 53, 292, 113 T. Tsang, 293 J-S. Tsao, 198 M. Tse, 70 R. Tso, 132 K. Tsubono, 27 S. Tsuchida, 204 L. Tsukada, 114 D. Tsuna, 114 T. Tsutsui, 114 T. Tsuzuki, 21 K. Turbang, 294, 209 M. Turconi, 95 D. Tuyenbayev, 204 A. S. Ubhi, 14 N. Uchikata, 37 T. Uchiyama, 192 R. P. Udall, 1 A. Ueda, 187 T. Uehara, 295, 296 K. Ueno, 114 G. Ueshima, 297 C. S. Unnikrishnan, 181 F. Uraguchi, 21 A. L. Urban, 2 T. Ushiba, 192 A. Utina, 155, 53 H. Vahlbruch, 9, 10 G. Vajente, 1 A. Vajpeyi, 5 G. Valdes, 185 M. Valentini, 91, 92 V. Valsan, 7 N. van Bakel, 53 M. van Beuzekom, 53 J. F. J. van den Brand, 155, 298, 53 C. Van Den Broeck, 113, 53 D. C. Vander-Hyde, 61 L. van der Schaaf, 53 J. V. van Heijningen, 52 J. Vanosky, 1 M. H. P. M. van Putten, 299 N. van Remortel, 209 M. Vardaro, 243, 53 A. F. Vargas, 116 V. Varma, 179 M. Vas'uth, 71 A. Vecchio, 14 G. Vedovato, 78 J. Veitch, 69 P. J. Veitch, 83 J. Venneberg, 9, 10 G. Venugopalan, 1 D. Verkindt, 30 P. Verma, 232 Y. Verma, 87 D. Veske, 46 F. Vetrano, 49 A. Vicer'e, 49, 50 S. Vidyant, 61 A. D. Viets, 249 A. Vijaykumar, 19 V. Villa-Ortega, 107 J.-Y. Vinet, 95 A. Virtuoso, 188, 34 S. Vitale, 70 T. Vo, 61 H. Vocca, 76, 75 E. R. G. von Reis, 67 J. S. A. von Wrangel, 9, 10 C. Vorvick, 67 S. P. Vyatchanin, 90 L. E. Wade, 173 M. Wade, 173 K. J. Wagner, 125 R. C. Walet, 53 M. Walker, 57 G. S. Wallace, 32 L. Wallace, 1 S. Walsh, 7 J. Wang, 177 J. Z. Wang, 184 W. H. Wang, 150 R. L. Ward, 8 J. Warner, 67 M. Was, 30 T. Washimi, 20 N. Y. Washington, 1 J. Watchi, 145 B. Weaver, 67 S. A. Webster, 69 M. Weinert, 9, 10 A. J. Weinstein, 1 R. Weiss, 70 C. M. Weller, 245 F. Wellmann, 9, 10 L. Wen, 86 P. Weßels, 9, 10 K. Wette, 8 J. T. Whelan, 125 D. D. White, 40 B. F. Whiting, 72 C. Whittle, 70 D. Wilken, 9, 10 D. Williams, 69 M. J. Williams, 69 A. R. Williamson, 156 J. L. Willis, 1 B. Willke, 9, 10 D. J. Wilson, 140 W. Winkler, 9, 10 C. C. Wipf, 1 T. Wlodarczyk, 105 G. Woan, 69 J. Woehler, 9, 10 J. K. Wofford, 125 I. C. F. Wong, 108 C. Wu, 133 D. S. Wu, 9, 10 H. Wu, 133 S. Wu, 133 D. M. Wysocki, 7 L. Xiao, 1 W-R. Xu, 198 T. Yamada, 289 H. Yamamoto, 1 Kazuhiro Yamamoto, 191 Kohei Yamamoto, 289 T. Yamamoto, 192 K. Yamashita, 203 R. Yamazaki, 200 F. W. Yang, 172 L. Yang, 166 Y. Yang, 300 Yang Yang, 72 Z. Yang, 63 M. J. Yap, 8 D. W. Yeeles, 17 A. B. Yelikar, 125 M. Ying, 126 K. Yokogawa, 203 J. Yokoyama, 28, 27 T. Yokozawa, 192 J. Yoo, 179 T. Yoshioka, 203 Hang Yu, 132 Haocun Yu, 70 H. Yuzurihara, 37 A. Zadro˙zny, 232 M. Zanolin, 35 S. Zeidler, 301 T. Zelenova, 42 J.-P. Zendri, 78 M. Zevin, 162 M. Zhan, 177 H. Zhang, 198 J. Zhang, 86 L. Zhang, 1 T. Zhang, 14 Y. Zhang, 185 C. Zhao, 86 G. Zhao, 145 Y. Zhao, 20 Yue Zhao, 172 Y. Zheng, 89 R. Zhou, 194 Z. Zhou, 15 5 115 168 125 1, 70 1 \nX. J. Zhu, Z.-H. Zhu, A. B. Zimmerman, Y. Zlochower, M. E. Zucker, and J. Zweizig 1 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA 2 Louisiana State University, Baton Rouge, LA 70803, USA 3 Dipartimento di Farmacia, Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 4 INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy 5 OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia 6 LIGO Livingston Observatory, Livingston, LA 70754, USA 7 University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 8 OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia 9 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany 10 Leibniz Universitat Hannover, D-30167 Hannover, Germany 11 Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India 12 University of Cambridge, Cambridge CB2 1TN, United Kingdom 13 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena, D-07743 Jena, Germany 14 University of Birmingham, Birmingham B15 2TT, United Kingdom 15 Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA 16 Instituto Nacional de Pesquisas Espaciais, 12227-010 S˜ao Jos'e dos Campos, S˜ao Paulo, Brazil 17 Gravity Exploration Institute, Cardiff University, Cardiff CF24 3AA, United Kingdom 18 INFN, Sezione di Pisa, I-56127 Pisa, Italy 19 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India 20 Gravitational Wave Science Project, National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan 21 Advanced Technology Center, National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan 22 Universitat de les Illes Balears, IAC3-IEEC, E-07122 Palma de Mallorca, Spain 23 University College Dublin, Dublin 4, Ireland 24 INFN Sezione di Torino, I-10125 Torino, Italy 25 Universit'a di Napoli 'Federico II', Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy 26 Universit'e de Lyon, Universit'e Claude Bernard Lyon 1, CNRS, Institut Lumi'ere Mati'ere, F-69622 Villeurbanne, France 27 Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 28 Research Center for the Early Universe (RESCEU), The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 29 Institut de Ci'encies del Cosmos (ICCUB), Universitat de Barcelona, \nC/ Mart'ı i Franqu'es 1, Barcelona, 08028, Spain \n38 \n47 \n55 \n81 \n84 \n66 \n30 Laboratoire d'Annecy de Physique des Particules (LAPP), Univ. Grenoble Alpes, \nUniversit'e Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy, France \n31 Gran Sasso Science Institute (GSSI), I-67100 L'Aquila, Italy \n32 SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom \n33 Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit'a di Udine, I-33100 Udine, Italy \n34 INFN, Sezione di Trieste, I-34127 Trieste, Italy \n35 Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA \n36 Universit'e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France \n37 Institute for Cosmic Ray Research (ICRR), KAGRA Observatory, \nThe University of Tokyo, Kashiwa City, Chiba 277-8582, Japan \nAccelerator Laboratory, High Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan \n39 \nEarthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0032, Japan \n40 California State University Fullerton, Fullerton, CA 92831, USA \n41 \nUniversit'e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France \n- 42 European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy \nSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA \n43 \n44 Chennai Mathematical Institute, Chennai 603103, India \n45 Department of Mathematics and Physics, Gravitational Wave Science Project, \nHirosaki University, Hirosaki City, Aomori 036-8561, Japan \n46 Columbia University, New York, NY 10027, USA \nKamioka Branch, National Astronomical Observatory of Japan (NAOJ), Kamioka-cho, Hida City, Gifu 506-1205, Japan \n48 \nThe Graduate University for Advanced Studies (SOKENDAI), Mitaka City, Tokyo 181-8588, Japan \n49 Universit'a degli Studi di Urbino 'Carlo Bo', I-61029 Urbino, Italy \n50 INFN, Sezione di Firenze, I-50019 Sesto Fiorentino, Firenze, Italy \n51 INFN, Sezione di Roma, I-00185 Roma, Italy \n52 Universit'e catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium \n53 Nikhef, Science Park 105, 1098 XG Amsterdam, Netherlands \n54 King's College London, University of London, London WC2R 2LS, United Kingdom \nKorea Institute of Science and Technology Information (KISTI), Yuseong-gu, Daejeon 34141, Republic of Korea \n56 \nNational Institute for Mathematical Sciences, Yuseong-gu, Daejeon 34047, Republic of Korea \n57 Christopher Newport University, Newport News, VA 23606, USA \n58 International College, Osaka University, Toyonaka City, Osaka 560-0043, Japan \n59 School of High Energy Accelerator Science, The Graduate University for \nAdvanced Studies (SOKENDAI), Tsukuba City, Ibaraki 305-0801, Japan \n60 University of Oregon, Eugene, OR 97403, USA \n61 \nSyracuse University, Syracuse, NY 13244, USA \n62 Universit'e de Li'ege, B-4000 Li'ege, Belgium \n63 University of Minnesota, Minneapolis, MN 55455, USA \n64 Universit'a degli Studi di Milano-Bicocca, I-20126 Milano, Italy \n65 INFN, Sezione di Milano-Bicocca, I-20126 Milano, Italy \nINAF, Osservatorio Astronomico di Brera sede di Merate, I-23807 Merate, Lecco, Italy \n67 LIGO Hanford Observatory, Richland, WA 99352, USA \n68 \nDipartimento di Medicina, Chirurgia e Odontoiatria 'Scuola Medica Salernitana', \nUniversit'a di Salerno, I-84081 Baronissi, Salerno, Italy \n69 SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom \n70 LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA \n71 Wigner RCP, RMKI, H-1121 Budapest, Konkoly Thege Mikl'os 'ut 29-33, Hungary \n72 University of Florida, Gainesville, FL 32611, USA \n73 Stanford University, Stanford, CA 94305, USA \n74 Universit'a di Pisa, I-56127 Pisa, Italy \n75 INFN, Sezione di Perugia, I-06123 Perugia, Italy \n76 Universit'a di Perugia, I-06123 Perugia, Italy \n77 Universit'a di Padova, Dipartimento di Fisica e Astronomia, I-35131 Padova, Italy \n78 INFN, Sezione di Padova, I-35131 Padova, Italy \n79 Montana State University, Bozeman, MT 59717, USA \n80 Institute for Plasma Research, Bhat, Gandhinagar 382428, India \nNicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 00-716, Warsaw, Poland \n82 Dipartimento di Ingegneria, Universit'a del Sannio, I-82100 Benevento, Italy \n83 OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia \nCalifornia State University, Los Angeles, 5151 State University Dr, Los Angeles, CA 90032, USA \n85 \nINFN, Sezione di Genova, I-16146 Genova, Italy \n86 OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia \n87 RRCAT, Indore, Madhya Pradesh 452013, India \n88 \nGRAPPA, Anton Pannekoek Institute for Astronomy and Institute for High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands 89 Missouri University of Science and Technology, Rolla, MO 65409, USA 90 Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia 91 Universit'a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, Italy 92 INFN, Trento Institute for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy 93 SUPA, University of the West of Scotland, Paisley PA1 2BE, United Kingdom 94 Bar-Ilan University, Ramat Gan, 5290002, Israel 95 Artemis, Universit'e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, F-06304 Nice, France 96 Dipartimento di Fisica 'E.R. Caianiello', Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 97 INFN, Sezione di Napoli, Gruppo Collegato di Salerno, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy 98 Universit'a di Roma 'La Sapienza', I-00185 Roma, Italy 99 Univ Rennes, CNRS, Institut FOTON - UMR6082, F-3500 Rennes, France 100 Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India 101 INFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy 102 Laboratoire Kastler Brossel, Sorbonne Universit'e, CNRS, ENS-Universit'e PSL, Coll'ege de France, F-75005 Paris, France 103 Astronomical Observatory Warsaw University, 00-478 Warsaw, Poland 104 University of Maryland, College Park, MD 20742, USA 105 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany 106 L2IT, Laboratoire des 2 Infinis - Toulouse, Universit'e de Toulouse, CNRS/IN2P3, UPS, F-31062 Toulouse Cedex 9, France 107 IGFAE, Campus Sur, Universidade de Santiago de Compostela, 15782 Spain 108 The Chinese University of Hong Kong, Shatin, NT, Hong Kong 109 Stony Brook University, Stony Brook, NY 11794, USA 110 Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA 111 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 112 Dipartimento di Fisica, Universit'a degli Studi di Genova, I-16146 Genova, Italy 113 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands 114 RESCEU, University of Tokyo, Tokyo, 113-0033, Japan. 115 Department of Astronomy, Beijing Normal University, Beijing 100875, China 116 OzGrav, University of Melbourne, Parkville, Victoria 3010, Australia 117 Universit'a degli Studi di Sassari, I-07100 Sassari, Italy 118 INFN, Laboratori Nazionali del Sud, I-95125 Catania, Italy 119 Universit'a di Roma Tor Vergata, I-00133 Roma, Italy 120 INFN, Sezione di Roma Tor Vergata, I-00133 Roma, Italy 121 University of Sannio at Benevento, I-82100 Benevento, Italy and INFN, Sezione di Napoli, I-80100 Napoli, Italy 122 Villanova University, 800 Lancaster Ave, Villanova, PA 19085, USA 123 Departamento de Astronom'ıa y Astrof'ısica, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain 124 Universitat Hamburg, D-22761 Hamburg, Germany 125 Rochester Institute of Technology, Rochester, NY 14623, USA 126 National Tsing Hua University, Hsinchu City, 30013 Taiwan, Republic of China 127 Department of Applied Physics, Fukuoka University, Jonan, Fukuoka City, Fukuoka 814-0180, Japan 128 OzGrav, Charles Sturt University, Wagga Wagga, New South Wales 2678, Australia 129 Department of Physics, Tamkang University, Danshui Dist., New Taipei City 25137, Taiwan 130 Department of Physics and Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan 131 Department of Physics, Center for High Energy and High Field Physics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan 132 CaRT, California Institute of Technology, Pasadena, CA 91125, USA 133 Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan 134 Dipartimento di Ingegneria Industriale (DIIN), Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 135 Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan 136 Universit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France 137 Seoul National University, Seoul 08826, Republic of Korea 138 Pusan National University, Busan 46241, Republic of Korea 139 INAF, Osservatorio Astronomico di Padova, I-35122 Padova, Italy 140 University of Arizona, Tucson, AZ 85721, USA 141 \nRutherford Appleton Laboratory, Didcot OX11 0DE, United Kingdom \n142 OzGrav, Swinburne University of Technology, Hawthorn VIC 3122, Australia \n143 Universit'e libre de Bruxelles, Avenue Franklin Roosevelt 50 - 1050 Bruxelles, Belgium 144 Universitat de les Illes Balears, IAC3-IEEC, E-07122 Palma de Mallorca, Spain 145 Universit'e Libre de Bruxelles, Brussels 1050, Belgium 146 Departamento de Matem'aticas, Universitat de Val'encia, E-46100 Burjassot, Val'encia, Spain 147 Texas Tech University, Lubbock, TX 79409, USA 148 The Pennsylvania State University, University Park, PA 16802, USA 149 University of Rhode Island, Kingston, RI 02881, USA 150 The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA 151 Bellevue College, Bellevue, WA 98007, USA 152 Scuola Normale Superiore, Piazza dei Cavalieri, 7 - 56126 Pisa, Italy 153 MTA-ELTE Astrophysics Research Group, Institute of Physics, Eotvos University, Budapest 1117, Hungary 154 University of Szeged, D'om t'er 9, Szeged 6720, Hungary 155 Maastricht University, P.O. Box 616, 6200 MD Maastricht, Netherlands 156 University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom 157 The University of Sheffield, Sheffield S10 2TN, United Kingdom 158 Universit'e Lyon, Universit'e Claude Bernard Lyon 1, CNRS, Laboratoire des Mat'eriaux Avanc'es (LMA), IP2I Lyon / IN2P3, UMR 5822, F-69622 Villeurbanne, France 159 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universit'a di Parma, I-43124 Parma, Italy 160 INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, I-43124 Parma, Italy 161 Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland 162 University of Chicago, Chicago, IL 60637, USA 163 Universit'e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France 164 West Virginia University, Morgantown, WV 26506, USA 165 Montclair State University, Montclair, NJ 07043, USA 166 Colorado State University, Fort Collins, CO 80523, USA 167 Institute for Nuclear Research, Hungarian Academy of Sciences, Bem t'er 18/c, H-4026 Debrecen, Hungary 168 Department of Physics, University of Texas, Austin, TX 78712, USA 169 CNR-SPIN, c/o Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 170 Scuola di Ingegneria, Universit'a della Basilicata, I-85100 Potenza, Italy 171 Observatori Astron'omic, Universitat de Val'encia, E-46980 Paterna, Val'encia, Spain 172 The University of Utah, Salt Lake City, UT 84112, USA 173 Kenyon College, Gambier, OH 43022, USA 174 Vrije Universiteit Amsterdam, 1081 HV, Amsterdam, Netherlands 175 Department of Astronomy, The University of Tokyo, Mitaka City, Tokyo 181-8588, Japan 176 Faculty of Engineering, Niigata University, Nishi-ku, Niigata City, Niigata 950-2181, Japan 177 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology (APM), Chinese Academy of Sciences, Xiao Hong Shan, Wuhan 430071, China 178 Universiteit Gent, B-9000 Gent, Belgium 179 Cornell University, Ithaca, NY 14850, USA 180 University of British Columbia, Vancouver, BC V6T 1Z4, Canada 181 Tata Institute of Fundamental Research, Mumbai 400005, India 182 INAF, Osservatorio Astronomico di Capodimonte, I-80131 Napoli, Italy 183 The University of Mississippi, University, MS 38677, USA 184 University of Michigan, Ann Arbor, MI 48109, USA 185 Texas A&M University, College Station, TX 77843, USA 186 Department of Physics, Ulsan National Institute of Science and Technology (UNIST), Ulju-gun, Ulsan 44919, Republic of Korea 187 Applied Research Laboratory, High Energy Accelerator Research Organization (KEK), Tsukuba City, Ibaraki 305-0801, Japan 188 Dipartimento di Fisica, Universit'a di Trieste, I-34127 Trieste, Italy 189 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China 190 American University, Washington, D.C. 20016, USA 191 Faculty of Science, University of Toyama, Toyama City, Toyama 930-8555, Japan 192 Institute for Cosmic Ray Research (ICRR), KAGRA Observatory, The University of Tokyo, Kamioka-cho, Hida City, Gifu 506-1205, Japan 193 Carleton College, Northfield, MN 55057, USA 194 University of California, Berkeley, CA 94720, USA 195 Maastricht University, 6200 MD, Maastricht, Netherlands \n196 College of Industrial Technology, Nihon University, Narashino City, Chiba 275-8575, Japan \n197 Graduate School of Science and Technology, Niigata University, Nishi-ku, Niigata City, Niigata 950-2181, Japan \n198 Department of Physics, National Taiwan Normal University, sec. 4, Taipei 116, Taiwan \n199 Astronomy & Space Science, Chungnam National University, \nYuseong-gu, Daejeon 34134, Republic of Korea, Republic of Korea 200 Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara City, Kanagawa 252-5258, Japan 201 Kavli Institute for Astronomy and Astrophysics, Peking University, Haidian District, Beijing 100871, China 202 Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan 203 Graduate School of Science and Engineering, University of Toyama, Toyama City, Toyama 930-8555, Japan 204 Department of Physics, Graduate School of Science, Osaka City University, Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan 205 Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, Sumiyoshi-ku, Osaka City, Osaka 558-8585, Japan 206 Institute of Space and Astronautical Science (JAXA), Chuo-ku, Sagamihara City, Kanagawa 252-0222, Japan 207 Directorate of Construction, Services & Estate Management, Mumbai 400094, India 208 Vanderbilt University, Nashville, TN 37235, USA 209 Universiteit Antwerpen, Prinsstraat 13, 2000 Antwerpen, Belgium 210 University of Biaglyph[suppress]lystok, 15-424 Biaglyph[suppress]lystok, Poland 211 Department of Physics, Ewha Womans University, Seodaemun-gu, Seoul 03760, Republic of Korea 212 National Astronomical Observatories, Chinese Academic of Sciences, Chaoyang District, Beijing, China 213 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Chaoyang District, Beijing, China 214 University of Southampton, Southampton SO17 1BJ, United Kingdom 215 Institute for Cosmic Ray Research (ICRR), The University of Tokyo, Kashiwa City, Chiba 277-8582, Japan 216 Chung-Ang University, Seoul 06974, Republic of Korea 217 Institut de F'ısica d'Altes Energies (IFAE), Barcelona Institute of Science and Technology, and ICREA, E-08193 Barcelona, Spain 218 Graduate School of Science, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan 219 University of Washington Bothell, Bothell, WA 98011, USA 220 Institute of Applied Physics, Nizhny Novgorod, 603950, Russia 221 Ewha Womans University, Seoul 03760, Republic of Korea 222 Inje University Gimhae, South Gyeongsang 50834, Republic of Korea 223 Department of Physics, Myongji University, Yongin 17058, Republic of Korea 224 Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea 225 National Institute for Mathematical Sciences, Daejeon 34047, Republic of Korea 226 Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea 227 Department of Physical Science, Hiroshima University, Higashihiroshima City, Hiroshima 903-0213, Japan 228 School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, UK 229 Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan 230 Bard College, 30 Campus Rd, Annandale-On-Hudson, NY 12504, USA 231 Institute of Mathematics, Polish Academy of Sciences, 00656 Warsaw, Poland 232 National Center for Nuclear Research, 05-400 ' Swierk-Otwock, Poland 233 Instituto de Fisica Teorica, 28049 Madrid, Spain 234 Department of Physics, Nagoya University, Chikusa-ku, Nagoya, Aichi 464-8602, Japan 235 Universit'e de Montr'eal/Polytechnique, Montreal, Quebec H3T 1J4, Canada 236 Laboratoire Lagrange, Universit'e Cˆote d'Azur, Observatoire Cˆote d'Azur, CNRS, F-06304 Nice, France 237 Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada 238 Department of Physics, Hanyang University, Seoul 04763, Republic of Korea 239 Sungkyunkwan University, Seoul 03063, Republic of Korea 240 NAVIER, ' Ecole des Ponts, Univ Gustave Eiffel, CNRS, Marne-la-Vall'ee, France 241 Department of Physics, National Cheng Kung University, Tainan City 701, Taiwan 242 National Center for High-performance computing, National Applied Research Laboratories, Hsinchu Science Park, Hsinchu City 30076, Taiwan 243 Institute for High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands 244 NASA Marshall Space Flight Center, Huntsville, AL 35811, USA 245 University of Washington, Seattle, WA 98195, USA 246 Dipartimento di Matematica e Fisica, Universit'a degli Studi Roma Tre, I-00146 Roma, Italy 247 INFN, Sezione di Roma Tre, I-00146 Roma, Italy 248 ESPCI, CNRS, F-75005 Paris, France 249 Concordia University Wisconsin, Mequon, WI 53097, USA 250 Universit'a di Camerino, Dipartimento di Fisica, I-62032 Camerino, Italy \n251 \nSchool of Physics Science and Engineering, Tongji University, Shanghai 200092, China 252 Southern University and A&M College, Baton Rouge, LA 70813, USA 253 Centre Scientifique de Monaco, 8 quai Antoine Ier, MC-98000, Monaco 254 Institute for Photon Science and Technology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan 255 Indian Institute of Technology Madras, Chennai 600036, India 256 Instituto de Fisica Teorica, Universidad Aut'onoma de Madrid, 28049 Madrid, Spain 257 Saha Institute of Nuclear Physics, Bidhannagar, West Bengal 700064, India 258 The Applied Electromagnetic Research Institute, National Institute of Information and Communications Technology (NICT), Koganei City, Tokyo 184-8795, Japan 259 Institut des Hautes Etudes Scientifiques, F-91440 Bures-sur-Yvette, France 260 Faculty of Law, Ryukoku University, Fushimi-ku, Kyoto City, Kyoto 612-8577, Japan 261 Indian Institute of Science Education and Research, Kolkata, Mohanpur, West Bengal 741252, India 262 Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, Netherlands 263 Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA 264 Consiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi, Piazzale Aldo Moro 5, I-00185 Roma, Italy 265 Korea Astronomy and Space Science Institute (KASI), Yuseong-gu, Daejeon 34055, Republic of Korea 266 Hobart and William Smith Colleges, Geneva, NY 14456, USA 267 International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal RN 59078-970, Brazil 268 Museo Storico della Fisica e Centro Studi e Ricerche 'Enrico Fermi', I-00184 Roma, Italy 269 Lancaster University, Lancaster LA1 4YW, United Kingdom 270 Universit'a di Trento, Dipartimento di Matematica, I-38123 Povo, Trento, Italy 271 Indian Institute of Science Education and Research, Pune, Maharashtra 411008, India 272 Dipartimento di Fisica, Universit'a degli Studi di Torino, I-10125 Torino, Italy 273 Indian Institute of Technology, Palaj, Gandhinagar, Gujarat 382355, India 274 Department of Physics, Kyoto University, Sakyou-ku, Kyoto City, Kyoto 606-8502, Japan 275 Department of Electronic Control Engineering, National Institute of Technology, Nagaoka College, Nagaoka City, Niigata 940-8532, Japan 276 Departamento de Matem'atica da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications, Campus de Santiago, 3810-183 Aveiro, Portugal 277 Marquette University, 11420 W. Clybourn St., Milwaukee, WI 53233, USA 278 Graduate School of Science and Engineering, Hosei University, Koganei City, Tokyo 184-8584, Japan 279 Faculty of Science, Toho University, Funabashi City, Chiba 274-8510, Japan 280 Faculty of Information Science and Technology, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan 281 Universit'a di Firenze, Sesto Fiorentino I-50019, Italy 282 INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 283 Indian Institute of Technology Hyderabad, Sangareddy, Khandi, Telangana 502285, India 284 iTHEMS (Interdisciplinary Theoretical and Mathematical Sciences Program), The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan 285 INAF, Osservatorio di Astrofisica e Scienza dello Spazio, I-40129 Bologna, Italy 286 Department of Space and Astronautical Science, The Graduate University for Advanced Studies (SOKENDAI), Sagamihara City, Kanagawa 252-5210, Japan 287 Andrews University, Berrien Springs, MI 49104, USA 288 Research Center for Space Science, Advanced Research Laboratories, Tokyo City University, Setagaya, Tokyo 158-0082, Japan 289 Institute for Cosmic Ray Research (ICRR), Research Center for Cosmic Neutrinos (RCCN), The University of Tokyo, Kashiwa City, Chiba 277-8582, Japan 290 National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology, Tsukuba City, Ibaraki 305-8568, Japan 291 Dipartimento di Scienze Aziendali - Management and Innovation Systems (DISA-MIS), Universit'a di Salerno, I-84084 Fisciano, Salerno, Italy 292 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands 293 Faculty of Science, Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 294 Vrije Universiteit Brussel, Boulevard de la Plaine 2, 1050 Ixelles, Belgium 295 Department of Communications Engineering, National Defense Academy of Japan, Yokosuka City, Kanagawa 239-8686, Japan 296 Department of Physics, University of Florida, Gainesville, FL 32611, USA 297 Department of Information and Management Systems Engineering, Nagaoka University of Technology, Nagaoka City, Niigata 940-2188, Japan 298 Vrije Universiteit Amsterdam, 1081 HV Amsterdam, Netherlands 299 Department of Physics and Astronomy, Sejong University, Gwangjin-gu, Seoul 143-747, Republic of Korea 300 \nDepartment of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan \nWe report on the population properties of 76 compact binary mergers detected with gravitational waves below a false alarm rate of 1 per year through the cumulative Gravitational Wave Transient Catalog 3 (GWTC-3). The catalog contains three classes of binary mergers: binary black hole (BBH), binary neutron star (BNS), and neutron star-black hole (NSBH) mergers. We infer the BNS merger rate to be between 10 Gpc -3 yr -1 and 1700 Gpc -3 yr -1 and the NSBH merger rate to be between 7.8 Gpc -3 yr -1 and 140 Gpc -3 yr -1 , assuming a constant rate density versus comoving volume and taking the union of 90% credible intervals for methods used in this work. Accounting for the BBH merger rate to evolve with redshift, we find the BBH merger rate to be between 17.9 Gpc -3 yr -1 and 44 Gpc -3 yr -1 at a fiducial redshift ( z = 0 . 2). Using both binary neutron star and neutron star-black hole binaries, we obtain a broad, relatively flat neutron star mass distribution extending from 1 . 2 +0 . 1 -0 . 2 M glyph[circledot] to 2 . 0 +0 . 3 -0 . 3 M glyph[circledot] . We can confidently identify a rapid decrease in merger rate versus component mass between neutron star-like masses and black-hole-like masses, but there is no evidence that the merger rate increases again before 10 M glyph[circledot] . We also find the binary black hole mass distribution has localized over- and under-densities relative to a power law distribution, with peaks emerging at chirp masses of 8 . 3 +0 . 3 -0 . 5 M glyph[circledot] and 27 . 9 +1 . 9 -1 . 8 M glyph[circledot] . While we continue to find the mass distribution of a binary's more massive component strongly decreases as a function of primary mass, we observe no evidence of a strongly suppressed merger rate above ≈ 60 M glyph[circledot] , which would highlight the presence of a upper mass gap. The rate of BBH mergers is observed to increase with redshift at a rate proportional to (1 + z ) κ with κ = 2 . 9 +1 . 7 -1 . 8 for z glyph[lessorsimilar] 1. Observed black hole spins are small, with half of spin magnitudes below χ i ≈ 0 . 25. We observe evidence of negative aligned spins in the population, and an increase in spin magnitude for systems with more unequal mass ratio. We also observe evidence of misalignment of spins relative to the orbital angular momentum.", 'I. INTRODUCTION': 'We analyze the population properties of black holes (BHs) and neutron stars (NSs) in compact binary systems using data through the end of the third observing run of LIGO-Virgo (O3). Gravitational Wave Transient Catalog 3 (GWTC-3) [1] combines observations from the first three observing runs (O1, O2 [2] and O3 [1, 3, 4]) of the Advanced LIGO [5] and Advanced Virgo [6] gravitational-wave observatories. Counting only events with false alarm rate (FAR) of < 0 . 25 yr -1 , we have two binary neutron star (BNS) events, two neutron star-black hole (NSBH) events and 63 confident binary black hole (BBH) events. Considering the BBH population only, as in Section V, we lower the detection threshold to count events with FAR of < 1 yr -1 , resulting in 69 confident BBH events. We distinguish between NSs and BHs using prior information about the maximum NS mass, obtained from constraints on the densematter equation of state [7-9]. We use the observed population of events to infer the properties of the astrophysical BNS, NSBH and BBH populations. In particular, we infer the mass and spin distributions of the NS and BH populations, the overall merger rate, and investigate their cosmological evolution. \nThe population includes a number of exceptional events, notably the discovery in O3 data of two NSBH binaries: GW200105 162426 and GW200115 042309 [10]. In these two systems, the primary mass m 1 is larger than the maximum mass allowed by the NS equation of state, \nand the secondary mass m 2 is consistent with known NS masses. Here and throughout the paper, the primary mass m 1 refers to the larger of the two component masses in the binary, while the secondary mass m 2 refers to the smaller of the two. The inclusion of NSBH events enables the first joint analysis of the full BNS-NSBH-BBH population, including identification of sub-populations of binaries and any mass gaps between them. We also perform an analysis of the NS population properties using both BNS and NSBH systems. \nThe increased number of BBH observations allows for a more detailed investigation of the mass and spin distributions of BH. Overall, our new observations and results are consistent with the expectations about the mass and spin distribution of BBHs derived with our previous observations through GWTC-2 [11], which capture broad features on larger parameter scales than those emphasized in this study, and which we henceforth denote coarse-grained features . We demonstrate the use of nonparametric or broadly modeled methods to characterize the BBH distribution and use these to identify structure in the mass distribution. Another feature of our sample is the accumulation of more BBHs with preferentially negatively aligned spins relative to their orbital angular momentum (e.g., GW191109 010717, GW200225 060421), albeit at a significance that could occur by chance in our large catalog. Finally, the larger sample size allows for more detailed investigations of correlations between black hole masses and spins. \nIn this work, we adopt a high-purity set of candidate events whose selection biases we understand. Our chosen FAR threshold both ensures a sufficiently pure sample for the analyses performed in this work, particularly of the four binaries containing NS in our sample. How- \never, due to the higher observed rate of BBH mergers, even at a less stringent threshold of < 1 yr -1 the relative proportion of background events remains below 10% for analyses of BBH; we therefore adopt this less stringent threshold for analyses of solely the BBH population. Both sensitivity thresholds omit several candidates of moderate significance identified in recent work, including candidates identified by our own search [1, 3], which have required the probability of an event being of astrophysical origin, p astro > 0 . 5 [12]. For example, our chosen FAR threshold excludes some of the most massive events identified in GWTC-3 [1] (e.g., GW190403 051519 and GW200220 061928). We briefly discuss these events, and those identified by other groups, in the context of our reconstructed populations. \nThe remainder of this paper is organized as follows. In Section II we summarize observations we reported through O3, then highlight our key conclusions about them obtained in this study. In Section III we describe the hierarchical method used to fit population models to data, and to validate their results. In Section IV we describe analyses for the whole compact binary population, including both BHs and NSs. In Section V we describe our results for binaries containing one or more NSs. In Section VI and VII we describe our results for BBH masses and spins respectively. In Section VIII we discuss results obtained with other searches or selection criteria, comparing to the populations identified in this work. In Section IX we discuss the astrophysical interpretation of our observations and population inferences. In Section X we comment on prospects for future searches for the stochastic background of gravitational radiation from all compact binary mergers on our past light cone during the next observing run. We conclude in Section XI with the significance of our results. In our Appendices, we provide the details of how we estimate sensitivity to compact binary mergers (Appendix A), a comprehensive description of the population models used in this work (Appendix B), methods we used to validate our study \nagainst prominent sources of systematic error (Appendix C), and additional details ofthe BBH results (Appendix D). In Appendix E, we provide revised posterior distributions for all events used in this work, each reassessed using information obtained from an estimate for the full population.', 'II. SUMMARY OF OBSERVATIONS AND RESULTS': "Atotal of 90 compact binary coalescences (CBCs) have been detected in the first three observing runs [1]. The threshold used in GWTC-3 requires a probability of astrophysical origin of at least 50%. For the population analysis presented here, it is preferable to work with a different threshold to ensure lower contamination from signals of non-astrophysical origin, and to reduce the model dependence in assessing probabilities of astrophysical origin. Consequently, for the majority of analyses presented in this paper, we require a FAR of < 0 . 25 yr -1 in at least one of the search analyses in GWTC-3. This threshold limits the number of events to 67; at this threshold, we expect approximately one event not to be of astrophysical origin. For BBH focused analyses, we loosen the threshold to a FAR < 1 yr -1 due to the higher observed rate of BBH mergers, giving 76 events with available parameter estimates, of which approximately 4.6 are expected to be non-astrophysical. This significantly expands the number of observations since GWTC-2, which included 50 events, of which 47 had FAR of < 1 yr -1 and were used in our previous population analysis [11]. Table I shows selected properties of all events used to infer the astrophysical population of binary mergers in the Universe. The table contains all events with FAR < 1 yr -1 , with less significant events having FAR between 1 yr -1 and 0 . 25 yr -1 which are excluded from all but the BBH analyses clearly identified. Henceforth, we abbreviate candidate names by omitting the last six digits when unambiguous. \n× \n× \n- \nTABLE I: A table of GW events that meet the criteria for inclusion in this work. Events are separated by a horizontal line into sections of FAR min < 0 . 25 yr -1 and 1 yr -1 ≥ FAR min ≥ 0 . 25 yr -1 (lower), where FAR min is the smallest FAR reported over all pipelines. Within these sections, events are listed by the date they were detected. Columns provide the FAR, p astro (from the pipeline with the smallest FAR), and previously-reported estimates of selected parameters. These previously-reported parameters may adopt different priors than our work and do not precisely correspond to our inputs; see Section III for details. The low-significance event GW190531 is not included, lacking parameter inferences. \n× \nFig. 1 shows the properties of the new observations included in this analysis [1]. The shaded regions show two-dimensional marginal distributions for individual events. For reference, the black contours show expected two-dimensional marginal distribution for observed BBH events deduced in our previous analysis of GWTC-2 (the Powerlaw+peak model from [11]). In these plots and henceforth, we define q = m 2 /m 1 and chirp mass \nM = ( m 1 m 2 ) 3 / 5 / ( m 1 + m 2 ) 1 / 5 . (1) \nThe dimensionless spin of each black hole is denoted χ i = S i /m 2 i and the effective inspiral spin parameter [26] \nχ eff = ( m 1 χ 1 + m 2 χ 2 ) · ˆ L m 1 + m 2 , (2) \nwhere ˆ L is the instantaneous orbital angular momentum direction. Finally, z is the redshift of the event, inferred from the measured luminosity distance using H 0 = 67 . 9km s -1 Mpc -1 and Ω m = 0 . 3065 [27]. From these plots, we make several observations that motivate the investigations and results presented in the remainder of the paper. \nNeutron star-black hole binaries. The two NSBH binary observations GW200105 and GW200115 [10] are apparent in Fig. 1 as two of the lowest-mass new sources. Prior to O3, gravitational wave and Galactic observations had not identified any NSBH binaries [10]. We now know that these objects exist and merge, occupying a previously unexplored region in the mass and merger rate parameter space. NSBHs form a distinct population from the BNS and most BBHs, motivating the detailed multi-component analyses pursued in Sec. IV. For the first time, we are able to present rates for BNS, NSBH and BBH inferred jointly from an analysis of all observations. The NSBH merger rate is substantially larger than the BBH merger rate. As a result, our joint analyses produce a marginal mass distribution p ( m 1 ) which differs substantially from our previous work, and from \nanalyses in this work based solely on BBHs: the NSBH merger rate overwhelms the BBH rate at low mass. \nLower Mass Gap. We identify a relative dearth of observations of binaries with component masses between 3 M glyph[circledot] and 5 M glyph[circledot] . This underabundance is visible in the spectrum of observed primary masses plotted in Fig. 1. Gravitational wave and Galactic observations through O3a were consistent with a mass gap for compact objects between the heaviest NSs and the least massive BHs [2831]. The gap was thought to extend from roughly 3 M glyph[circledot] to 5 M glyph[circledot] , potentially due to the physics of core-collapse supernova explosions [32-36]. Both Galactic and gravitational wave observations made contemporaneously with O3 challenge this assumption [23, 37, 38]. Most notably, the secondary in GW190814 sits just above the maximum mass that the dense-matter equation of state is expected to support [23]. The primary of GW200115 [1, 10] may also lie above the maximum NS mass but below 5 M glyph[circledot] . Due to considerable uncertainty in their mass ratio, several binaries' secondaries may also hail from this gap region between 3 M glyph[circledot] and 5 M glyph[circledot] . We investigate the prospect of a mass gap in Sec. IV C, treating all compact objects equivalently. \nNS mass distribution. The observation of NSBH binaries enables a detailed study of the observed mass distribution of NSs, combining results from both BNSs and NSBHs. We discuss this in detail in Sec. V, comparing source classifications informed by the NS equation of state (EOS) as well as the inferred location of the lower mass gap. The inferred NS mass distribution, albeit based upon a limited sample of observations, does not exhibit a peak at 1 . 35 M glyph[circledot] ; in contrast, radio observations of Galactic BNS favor such a peak [39-41]. We investigate the impact of outliers in the mass distribution in Sec. VC, particularly GW190814 whose secondary mass lies above the otherwise inferred NS mass range. \nAdditional substructure in the BBH mass distribution. The observed masses of BBH binaries are clumped. This is most visible on the central panel in \n<!-- image --> \n/circledot \nFIG. 1. New observations since Gravitational Wave Transient Catalog 2 (GWTC-2). The measured properties of new CBC candidates announced since GWTC-2 with FAR < 1 / yr and reported parameters (blue shaded regions), compared to the expected population of detected BBHs (black contours) as inferred from past analysis of GWTC-2 with the same FAR threshold [25]. The left hand plot shows the inferred primary mass m 1 and mass ratio q ; the center plot shows the effective spin χ eff and chirp mass M and the right plot shows redshift z and primary mass. The least-massive sources in this sample include NSBH events GW200105 and GW200115.FIG. 2. Illustrating substructure in the chirp mass distribution for BBH (with FAR < 1 yr -1 , excluding GW190814, as in Sec. VI). Top The individual-event observations versus chirp mass (grey) and an inferred distribution of the observed chirp mass distribution (black solid) using an adaptive kernel density estimator [42, 43]. The kernel bandwidth is optimized for the local event density and a 90% confidence interval (black dashed) is obtained by bootstrapping [44]. Bottom The solid curve is the predicted chirp mass distribution obtained using the flexible mixture model framework (FM); see Sec. III for details. The distribution shows three clusters at low masses and a lack of mergers in the chirp-mass range 10 -12 M glyph[circledot] . \n<!-- image --> \nFig. 1, where overdensities in the chirp mass distribution from 8 to 10 M glyph[circledot] and around 30 M glyph[circledot] are visible. In Fig. 2, we show the one-dimensional chirp mass distribution for BBH events. The top panel shows the observations for individual events, overlaid with the observed distribution. The observations cluster in chirp mass, with about one-eighth of observed events having chirp masses within 8-10 . 5 M glyph[circledot] . Compared to chirp mass accuracy for these events glyph[lessorsimilar] 1 M glyph[circledot] , this region is well-separated from the next most massive binaries in chirp mass. There is also a significant overdensity at M ≈ 30 M glyph[circledot] and a weaker feature at 15 M glyph[circledot] . These features were previously identified using only GWTC-2 [45-48]. In the bottom panel of Fig. 2, we show the inferred astrophysical distribution of chirp mass, as recovered by the same Flexible mixtures (FM) approach that first identified these modulations [47, 49]. The features in the observed distribution are mirrored in the astrophysical one. In Section VI we show that these features are robustly identified by several independent analyses, and demonstrate that the observed structure in the mass distribution is highly significant. Since strong features correlated with chirp mass, but independent of mass ratio, are a priori astrophysically unlikely, these significant overdensities suggest the two-dimensional marginal distribution of the BBH population should also have significant substructure and localized overdensities. We explore this in detail in Sec. VI B \nBBH Rate evolution with redshift . We find that the merger rate density increases with redshift. The right plot in Figure 1 shows the distribution of events as a function of redshift. While there is a clear evolution of the observed mass distribution with redshift, this arises from the detectors' greater sensitivity to higher mass systems. Consequently, from Fig. 1 alone, we are not able to draw inferences about the evolution of the population or merger rate with redshift. We explore these issues in detail in Sec. VI D, where we show that there is no evidence for the evolution of the mass distribution with redshift. However, the merger rate density does increase with redshift. Modeling the rate as ∝ (1 + z ) κ , we find that κ = 2 . 9 +1 . 7 -1 . 8 . Our analysis strongly disfavors the possibility that the merger rate does not evolve with redshift. \nLow BBH spins. The BBH detections exhibit effective inspiral spins concentrated about χ eff ≈ 0, with the highest inferred spins below 0 . 6. The spread is consistent with expectations from GWTC-2. The events include individual candidates that probably have negative effective inspiral spin, consistent with our previous conclusion that the spin distribution contains events with χ eff < 0.", 'A. Data and event selection': "We consider candidate events identified by our search analyses for compact binary mergers using archival \ndata, comprising results from the GstLAL [50-52], PyCBC [53-58], and MBTA [59] analyses using templatebased matched filtering techniques, and the cWB [60, 61] analysis using an excess-energy search that does not assume a physically parameterized signal model. Details of these analyses and the configurations used for O3 data are given in [1, 3, 4]. Out of the thousands of candidates produced, only a small minority correspond to astrophysical merger signals, most being caused by instrumental noise. While methods are emerging for performing a joint population analysis including both signal and noise events [12, 62-64], here we largely follow a simple procedure [11, 25] of imposing a significance threshold to identify events for our population analysis and implicitly treating all events passing the threshold as true signals. The choice of threshold will then limit the expected level of noise contamination. \nThe analyses calculate a ranking statistic for all candidate events, which is used as the basis for estimating the events' FARs. The ranking statistic allows for sources over a broad parameter space of binary component masses and spins to be detected, without making strong assumptions on the form of the source distribution (except in the case of PyCBC-BBH , specialized for comparable-mass BBH mergers). The analyses additionally calculate an estimate of the probability of astrophysical (signal) origin, p astro , using analysis-specific assumptions on the form of the signal distribution (detailed in [1, 3]). Since, in this work, we explore a range of different assumptions and models for the binary merger population, we define our event set by imposing a threshold on FAR values, rather than on p astro [11]. \nOur searches and event validation techniques for gravitational wave transients have so far identified 76 candidates with FAR below 1 yr -1 in LIGO and Virgo data through O3. Table I presents these events. In our analysis here, we remove candidates with probable instrumental origin (e.g., 200219 201407 [1]). Assuming our analyses produce noise triggers independently, we expect ∑ k R T k glyph[similarequal] 4 . 6 false events in our sample, where R is the false alarm rate and T k is an estimate of the time examined by the k th search. For the population studies presented here, the event list can be further restricted by additional FAR thresholds to identify a high-purity list of candidates and to assess the stability of our results to changes in threshold. The choice of FAR threshold to achieve a given level of noise contamination will depend on the number of significant event candidates (and hence, likely signals) considered for an analysis. The most prominent difference concerns analyses for binaries with one or more NS components, in Sections IV and V, as opposed to analyses which only consider BBH systems, in Sections VI and VII. While our data set contains many tens of confidently detected BBH mergers, there is only a handful of comparably significant BNS or NSBH events. This leads us to impose a more stringent threshold of FAR < 0 . 25 yr -1 for all analyses considering NS systems. \nBecause population reconstruction requires careful understanding of search selection biases, we do not include candidates identified by independent analyses [65-71] of the publicly released LIGO and Virgo data [72, 73]. Previous studies [63, 64] suggest that our results are unlikely to change significantly with the inclusion of these events. We similarly omit any triggers produced from our focused IMBH or eccentric binary searches [74], as we have not assessed their sensitivity to the full mass range investigated here using the consistent framework adopted for our primary results. These searches also did not yield any additional significant detections. Future analyses may be able to include events from multiple independent catalogs with a unified framework for calculating event significance independently of specific search methods [75, 76]. \nParameter estimation results for each candidate event [4] were obtained using the lalinference [77], RIFT [78, 79], or Bilby [80, 81] analyses. The parameter estimation analyses use Bayesian sampling methods to produce fair draws from the posterior distribution function of the source parameters, conditioned on the data and a given model for the signal and noise [82]. Unless otherwise noted, we use previouslypublished samples for each event through GWTC-2.1 [2-4]. For GW200105 and GW200115 [10], we use the inferences reported in GWTC-3 [1]. For previouslyreported events through GWTC-2, we adopt the same parameter and event choices reported in our previous population study [11]. For O1 events, we use published samples which equally weight analyses with SEOBNRv3 [83, 84] and IMRPhenomPv2 [85] waveforms, and for new events reported in the GWTC-2 update [4], we use published samples with higher order modes, selected by equally weighting all available higher-order mode analyses ( PrecessingIMRPHM ). The higher-mode analyses associated with GWTC-2 do not include calibration uncertainty. Regarding new events presented in GWTC-2.1, we use the fiducial analysis reported in that work (unless otherwise noted) comprised of merged posterior samples equally drawn from SEOBNRv4PHM [86, 87] and IMRPhenomXPHM [88]. Both models implement precession and include beyond-quadrupole radiation for asymptotically quasicircular orbits. For O3b events newly reported in GWTC-3, we use the publicly released C01:Mixed samples from [1], which equally weigh two analyses with the models SEOBNRv4PHM [87] and IMRPhenomXPHM [88]. These samples lack the impact of calibration error on the SEOBNRv4PHM analyses for GW200316, GW200129, and GW200112. A more complete description of the parameter estimation methods and waveform models used can be found in Section V of [4]. To avoid ambiguity where multiple versions of these samples exist, our input posterior samples adopt the D 2 L prior on luminosity distance D L and have reference spins specified at 20 Hz. In the case of the BNS events GW170817 and GW190425 and the NSBH events GW200105 and GW200115, two versions of the samples are available: one \nthat assumes component spins χ 1 , 2 < 0 . 05 for putative NS, and a less restrictive but event-dependent bound otherwise (e.g., χ 1 , 2 < 0 . 99 for GW200105 and GW200115). We use the latter (high-spin) samples here. \nThe transfer function between the observed strain and astrophysical strain is subject to a systematic calibration uncertainty. Our parameter inferences incorporate our best estimates of calibration uncertainty, as reported in previous work. Since calibration uncertainty has been incorporated independently for each event, we have implicitly assumed any consistent systematic bias applied to all events is small; we estimate less than 0.54% (1.74%) effect for LIGO (Virgo) respectively [89, 90]. For O3a, the amplitude uncertainty was glyph[lessorsimilar] 3% [91]. Because we assume the secular calibration error is much smaller than the calibration error envelope applied when analyzing individual events, we do not incorporate this calibration uncertainty into our estimates of network sensitivity. In O3, this calibration uncertainty implies glyph[lessorsimilar] 10% systematic uncertainty in the sensitive spacetime volume and the inferred merger rate, which is subdominant to our uncertainties from Poisson counting error for most source classes and mass regions. \nEach foreground event in O3 has been rigorously validated [1]. Out of the 108 triggers examined in O3 (including events not included in final search results for this or our companion papers), only 4 were rejected due to the presence of instrumental noise artifacts. The number of vetoed events is comparable to or less than the expected number of false events for our fiducial analysis threshold, and far smaller than the number of events examined in this study.", 'B. Population analysis framework': 'To infer the parameters describing population models, we adopt a hierarchical Bayesian approach, in which we marginalize over the uncertainty in our estimate of individual event parameters; see, e.g., [92-94]. Given a set of data, { d i } , from N det gravitational-wave detections, we model the total number of events as an inhomogeneous Poisson process, giving the likelihood of the data given population parameters Λ as [92, 93, 95] \nL ( { d } , N det | Λ , N exp ) ∝ N N det e -N exp N det ∏ i =1 ∫ L ( d i | θ ) π ( θ | Λ) dθ. (3) \nHere, N exp is the expected number of detections over the full duration of an observation period for the population model Λ, N = N exp /ξ (Λ) is the expected number of mergers over the observation period, with ξ (Λ) the fraction of mergers that are detectable for a population with parameters Λ. The term L ( d i | θ ) is the individual event likelihood for the i th event in our data set that is described by a set of parameters θ . The conditional prior π ( θ | Λ) governs the population distribution \non event parameters θ (e.g., the masses, spins, and redshifts) given a specific population model and set of hyperparameters Λ to describe the model. Constraining the population hyper-parameters describing the distribution of gravitational-wave signals according to different models is one of the primary goals of this paper. A notable simplification results if a log-uniform prior is imposed on N ≡ N exp /ξ (Λ), the total number of events (detectable or not): one can then marginalize Eq. (3) over N to obtain [92, 94, 96] \nL ( { d }| Λ) ∝ N det ∏ i =1 ∫ L ( d i | θ ) π ( θ | Λ) dθ ξ (Λ) . (4) \nTo evaluate the single-event likelihood L ( d i | θ ), we use posterior samples that are obtained using some default prior π ∅ ( θ ). In this case, we can calculate the integrals over the likelihood with importance sampling over the discrete samples where we denote weighted averages over posterior samples as 〈 . . . 〉 . Equation (4), for example, becomes \nL ( { d }| Λ) ∝ N det ∏ i =1 1 ξ (Λ) 〈 π ( θ | Λ) π ∅ ( θ ) 〉 , (5) \nwhere the factor of π ∅ ( θ ) serves to divide out the prior used for initial parameter estimation. The likelihoods are implemented in a variety of software including GWPopulation [97, 98], PopModels [99], Sodapop [100], and Vamana [49]. Each code evaluates one of the likelihoods described above for population models, building a posterior with one of the emcee , dynesty , or stan packages [101-104]. Appendix A describes how we estimate search sensitivity using synthetic sources. \nWe note that the above likelihood formulation includes uncertainty due to the finite number of samples θ per event used in the Monte Carlo integration (see, e.g., [105, 106]). For details of how we alter the likelihood to mitigate this source of uncertainty see Appendix B. In this paper, we refer to both the astrophysical distribution of a parameter - the version as it appears in nature - and the observed distribution of a parameter - what appears in our detectors due to selection effects. The posterior population distribution for a given model represents our best guess for the astrophysical distribution of some source parameter θ , averaged over the posterior for population parameters Λ. \np Λ ( θ ) = ∫ π ( θ | Λ) p (Λ |{ d } ) d Λ . (6) \nThe subscript Λ indicates that we have marginalized over population parameters. Meanwhile, the posterior predictive distribution refers to the population-averaged distribution of source parameters θ conditioned on detection .', 'C. Population models used in this work': 'In this section, we briefly summarize some of the tools and ingredients we use to generate phenomenological models π ( θ | Λ) in this work. Appendix B provides a comprehensive description of the population models used in this work, including their functional form and prior assumptions.', '1. Parametric mass models': 'Neutron star mass models : In the analyses that focus exclusively on the NS-containing events, we model the NS mass distribution as either a power law or a Gaussian with sharp minimum and maximum mass cutoffs. The latter shape is inspired by the Galactic double NS mass distribution [39-41]. In both models, which we call Power and Peak respectively, we assume that the components of BNSs are drawn independently from the common NS mass distribution. For NSBHs, we assume a uniform BH mass distribution and random pairing with NSs. \nFiducial population mass and redshift analysis : In the fiducial power law plus peak ( Power Law + Peak (PP)) model [96, 107], the mass-redshift distribution (per unit comoving volume and observer time) was assumed to be of the form p ( m 1 , q, z ) ∝ q β p ( m 1 )(1 + z ) κ -1 , with p ( m 1 ) a mixture model containing two components: a power law with some slope and limits; and a Gaussian with some mean and variance. [In practice, this model as applied to GWTC-2 also usually included additional smoothing parameters for the upper and lower limit of the power law.] The merger rate normalization is chosen such that the source-frame merger rate per comoving volume at redshift z is given by \nR ( z ) = dN dV c dt ( z ) = R 0 (1 + z ) κ , (7) \nwhere R 0 is the local merger rate density at z = 0 and κ is a free parameter governing the evolution of R ( z ) with higher redshift. The corresponding redshift distribution of BBHs (per unit redshift interval) is [96] \np ( z | κ ) ∝ 1 1 + z dV c dz (1 + z ) κ , (8) \nwhere the leading factor of (1+ z ) -1 converts time increments from the source frame to the detector frame. Past analyses generally fixed the redshift distribution of binaries, assuming a source-frame merger rate that is constant and uniform in comoving volume; this choice corresponds to κ = 0. Our previous population studies [11, 25] additionally considered an evolving merger rate with variable κ . \nPower Law + Dip + Break model (PDB) : To fit the distribution of BH and NS masses, we use a parameterized model described in [108] and [109], consisting of a \nbroken power law with a notch filter. The variable depth of this notch filter allows for a dearth of events between two potential subpopulations at low and high mass. It also uses a low-pass filter at high masses to allow for a potential tapering of the mass distribution at high BH masses. The component mass distribution is then \np ( m | λ ) = n ( m | M gap low , M gap high , A ) × l ( m | m max , η ) × m α 1 if m<M gap high m α 2 if m>M gap high 0 if m>m max or m<m min . (9) \nHere, l ( m | m max , η ) is the low pass filter with powerlaw η applied at mass m max , n ( m | M gap low , M gap high , A ) is the notch filter with depth A applied between M gap low and M gap high . In this model, the primary and secondary masses are fit by the same parameters and are related by a pairing function [110, 111]. Two pairing functions are considered. The first is random pairing: primary and secondary masses take independent values so long as m 2 < m 1 . This model takes the form \np ( m 1 , m 2 | Λ) ∝ p ( m = m 1 | Λ) p ( m = m 2 | Λ) × Θ( m 2 < m 1 ) , (10) \nwhere Θ is the Heaviside step function that enforces primary masses are greater than secondary masses and Λ is the full set of eight hyperparameters. The second is a power-law-in-mass-ratio pairing function, as in [111]. The full mass distribution in the power-law-in-mass-ratio model is thus described by \np ( m 1 , m 2 | Λ) ∝ p ( m = m 1 | Λ) p ( m = m 2 | Λ) × q β Θ( m 2 < m 1 ) . (11) \nUnless otherwise stated, the results from the random pairing model are presented in this work.', '2. Spin models': "Fiducial population spin analyses : Compact binary spins may be parameterized in several different ways. In addition to the dimensionless spin magnitudes χ i ( i ∈ { 1 , 2 } ) and the polar tilt angles θ i between each spin vector and a binary's orbital angular momentum [112], we often appeal to the effective spin parameters χ eff and χ p . The effective inspiral spin χ eff characterizes a mass-averaged spin angular momentum in the direction parallel to the binaries orbital angular momentum. The effective precessing spin χ p , meanwhile, corresponds approximately to the degree of in-plane spin, and phenomenologically parametrizes the rate of relativistic precession of the orbital plane [113]: \nχ p = max [ χ 1 sin θ 1 , ( 3 + 4 q 4 + 3 q ) qχ 2 sin θ 2 ] . (12) \nWe leverage these two descriptions to explore the nature of BBH spins in two complementary ways. First, we use the Default spin model [114] to directly measure the distribution of BBH component spin magnitudes and tilts. We model component spin magnitudes as being independently and identically drawn from a Beta distribution [113], with \np ( χ i | α χ , β χ ) ∝ χ α -1 i (1 -χ i ) β -1 . (13) \nValues of the shape parameters α χ and β χ are restricted to α χ > 1 and β χ > 1 to ensure a non-singular component spin distribution. We describe component spin tilts, in turn, via a mixture between two sub-populations, one with isotropically oriented tilts and another with tilts preferentially concentrated about θ i = 0 [114]: \np (cos θ i | ζ, σ t ) = 1 2 (1 -ζ ) + ζ N [ -1 , 1] (cos θ i ; 1 , σ t ) . (14) \nHere, N [ -1 , 1] (cos θ i ; 1 , σ t ) is a normal distribution truncated to the interval -1 ≤ cos θ i ≤ 1, centered at 1 with a standard deviation σ t . The mixing parameter ζ governs the relative fraction of systems drawn from each sub-population. The form of Eq. (14) is motivated by a desire to capture the behavior of BBHs originating from both dynamical and isolated evolution channels, which are expected to yield preferentially isotropic and aligned spin orientations, respectively. Perfect spin-orbit alignment across the BBH population would correspond to ζ = 1 or σ t = 0, which our prior analysis on GWTC2 ruled out at high confidence [11]. This default spin model is characterized by two parameters characterizing the spin magnitude distribution (e.g., α, β ) and two parameters characterizing the spin misalignment mixture model (i.e., ξ, σ t ). In part because this parameterization approaches isotropy in two independent limits ( σ t = ∞ or ζ = 0), it assigns high prior weight to nearly-isotropic spin distributions. \nGaussian spin model : Our second approach is to instead seek to measure the distribution of effective spin parameters χ eff and χ p . In this case, we phenomenologically model the joint χ eff -χ p distribution as a bivariate Gaussian [115, 116]: \np ( χ eff , χ p | µ eff , σ eff , µ p , σ p , r ) ∝ N ( χ eff , χ p | µ , Σ ) , (15) \ncentered at µ = ( µ eff , µ p ) and with a covariance matrix \nΣ = ( σ 2 eff rσ eff σ p rσ eff σ p σ 2 p ) . (16) \nEquation (15) is truncated to the intervals -1 ≤ χ eff ≤ 1 and 0 ≤ χ p ≤ 1 over which the effective spin parameters are defined. This second model has five parameters for spin: two mean values and three parameters describing the covariance. \nMulti source model (MS): MS models all source categories in a mixture model, with one subpopulation for BNS, NSBH, and BBH. The BBH subpopulation follows the MultiSpin model introduced in [11]. This model features a power law continuum q β m α 1 , plus a peak modeled as a bivariate Gaussian in m 1 , m 2 . Consequently, the mass distribution is similar to the PP model. However, the spin distribution in the power law and Gaussian subpopulations are independent, as are the primary and secondary spins, with each of the four scenarios following the Default spin model, with ζ ≡ 1. \nNew to MS are two additional bivariate Gaussian subpopulations, characterizing BNS and NSBH mergers. The BH component of NSBH follow an independent Gaussian mass distribution. As with BBH, these BH follow an independent Default spin model with ζ = 1. All three types of NS (two in BNS and one in NSBH) are assumed to follow the same Gaussian mass distribution. Each type of NS follows an independent Default spin model, except here the spin magnitudes are scaled down to χ max = 0 . 05, and ζ ≡ 0 since tilts are not well measured.", '4. Nonparametric models': 'Power Law + Spline model (PS): The PS model parameterizes perturbations to a simpler phenomenological primary mass model, that is modeled as a cubic spline function. \np PS ( m 1 | Λ , { f i } ) ∝ p ( m 1 | Λ)exp[ f ( m 1 |{ f i } )] . (17) \nHere, f ( m 1 |{ f i } ) is the perturbation function interpolated from a set of n knots, fixed uniformly in log m 1 space, and with heights { f i } [45]. In this work, we choose as a base model a truncated power law [11, 117] with a low mass taper, similar to our fiducial model but lacking a Gaussian peak in p ( m 1 ). This model has all the parameters of the truncated model in mass and spin, as well as an additional parameter that characterizes the low mass tapering and n more describing the heights of the cubic spline knots. \nFlexible mixtures model (FM): Vamana, the FM model, characterizes the population as a mixture model, summing over individually separable components describing the distribution of chirp mass, mass ratio, and χ i,z [49]. Each component is composed of a Gaussian to model the chirp mass, another Gaussian to model the aligned-spin component, and a power law to model the mass ratio distribution. The weights follow a uniform prior and are proposed using a Dirichlet distribution. We choose eleven components. This choice maximizes the marginal likelihood; however, our results are robust against selecting different numbers of components. \nBinned Gaussian process model (BGP): We also model the two-dimensional mass distribution as a binned \nGaussian Process based on methods outlined in [118, 119]. In this approach, while the redshift and spin distribution are fixed (here, to uniform in comoving volume and isotropic and uniform in magnitude, respectively), we assume the merger rate over distinct mass bins is related via a Gaussian process that correlates the merger rates of neighboring bins. We use conventional techniques provided by pymc3 [120] to explore the hyperparameters of the Gaussian process, in particular its covariance, to optimally reproduce our data.', 'IV. BINARY MERGER POPULATION ACROSS ALL MASSES': 'In this Section, we jointly analyze the masses of all events in Table I for several reasons. First, it allows for the inclusion of all events regardless of their inferred source type. This eliminates issues of ambiguity in source classification for a number of events in O3. Second, it makes possible the detection and characterization of additional features such as a lower mass gap between the lowest-mass objects (likely though not necessarily NS) and the more massive BH populations [108], or multiple subpopulations [119]. Third, it facilitates a selfconsistent calculation of merger rates in different regions of the mass spectrum without explicitly counting the number of events in each category [12, 121]. Last, it naturally produces an overall rate of compact binary coalescences that does not require combining rates produced by disjoint models which may have differing systematics. We choose a detection threshold of FAR < 0 . 25 yr -1 which ensures even sub-populations and features driven by a few events are not contaminated by our background. \nWhen searching for features in the population of compact binary coalescences, we want to draw robust conclusions, stable to different choices of model and approach. We, therefore, fit three independent population models, described in Sec. III. The PDB model uses a parametrized dip in the mass distribution to characterize modulations of a simple broken power law at low mass. The BGP model is a nonparametric method allowing considerable flexibility in the mass distribution, constrained only weakly by certain smoothness priors. The BGP and PDB models assume an isotropically-oriented, uniformin-magnitude spin distribution for simplicity. For most merging binaries and particularly those with component masses below 10 M glyph[circledot] , spin effects have a sub-dominant impact on our sensitivity and thus on our inferences about the compact binary merger rate distribution versus mass, as shown in Appendix C 1. The MS model uses a multicomponent mixture model, treating the mass, rate, and spin parameters of each component almost entirely independently. However, to be directly comparable to the BBH-only analyses presented in Sec. VI, our MS analysis omits the outlier event GW190814. To ensure consistent estimates of spin selection effects, the MS analysis presented here only employs O3 events; however, in Ap- \npendix C, we have demonstrated that our many analyses produce comparable results when including or excluding pre-O3 results.', 'A. Merger Rates': "Models spanning all source classifications allow us to self-consistently measure the merger rates for all detected CBCs, both overall and subdivided into astrophysically interesting mass ranges, assuming they are independent of redshift. Moreover, because events can be classified into each category using mass limits with relatively high confidence, this approach also provides our fiducial BNS, NSBH, and BBH merger rates. Specifically, taking NS masses to lie between 1 and 2 . 5 M glyph[circledot] and BH masses to be between 2 . 5 and 100 M glyph[circledot] and taking the lowest 5% and highest 95% credible interval out of all three models, we infer merger rates between 10 Gpc -3 yr -1 - 1700 Gpc -3 yr -1 for BNS, 7.8 Gpc -3 yr -1 - 140 Gpc -3 yr -1 for NSBH, and 16 Gpc -3 yr -1 - 61 Gpc -3 yr -1 for BBH. Our choice of 2 . 5 M glyph[circledot] as a boundary between BH and NS, albeit different than the nominal threshold of 3 M glyph[circledot] adopted in GWTC-3, is consistent with our subsequent classification, based both on EOS and merger rate,. Table II provides the rate estimates obtained for the three models used in this section and, in addition, shows rates for events in the mass gap, as discussed in detail in Section IV C. \nFor most categories, our merger rate estimates are consistent with previously published estimates. For example, following GWTC-2 we inferred a binary black hole merger rate to be 23 . 9 +14 . 9 -8 . 6 Gpc -3 yr -1 . Our knowledge of the coarse-grained mass spectrum has not significantly evolved since our previous analysis, and we find the inferred BBH rate is consistent with the previously reported rate, which also omitted GW190814. \nWe previously reported a BNS merger rate of 320 +490 -240 Gpc -3 yr -1 [21]. With data from GWTC-3, in addition to inferring the BNS merger rate by fitting various population models, we also make an estimate by fixing the mass, spin, and redshfit distributions under simple assumptions. For this rate estimate, we assume the masses of NSs in merging binaries are uniformly distributed between 1 M glyph[circledot] and 2 . 5 M glyph[circledot] and the merger rate is constant in comoving volume out to a redshift of z = 0 . 15. We also model component spin magnitudes distributed uniformly below 0.4, consistent with assumptions made in [3]. Under these assumptions, we infer a BNS merger rate of 105 . 5 +190 . 2 -83 . 9 Gpc -3 yr -1 . \nFor BNS, the inferred merger rate depends on the presumed mass distribution. With few observations to pin down their behavior at low mass, the three approaches adopted in this Section arrive at different compact binary mass distributions between 1 M glyph[circledot] and 2 . 5 M glyph[circledot] . Because the merger rate in this region scales ∝ 〈 V T 〉 -1 glyph[similarequal] 〈M 15 / 6 〉 -1 (where V T denotes the sensitive 4-volume for a specific binary and the angled brackets denote averag- \nbjects less than 2 . 5 M glyph[circledot] , the upper boundary used in this Section for NS masses), the three methods used in this Section arrive at merger rates within each others' uncertainty but with medians differing by factors of up to approximately ten. \nFor NSBH we previously inferred a merger rate of 45 +75 -33 Gpc -3 yr -1 assuming the observed NSBH are representative of the population or 130 +112 -69 Gpc -3 yr -1 assuming a broad NSBH population [10]. In this work, each of our joint analyses recovers and adopts different mass spectra, producing a broadly consistent rate (between 7.8 Gpc -3 yr -1 and 140 Gpc -3 yr -1 , including sytematics). Combined, our results for the NSBH and BNS merger rates highlight the important role of modeling systematics when drawing inferences about populations with few confident members. \nTo further highlight the impact of model systematics on inferred merger rates, in Table II, we present our deduced merger rates across the mass space using all three models presented in this section. For simplicity, we label mass bins with NS and BH based solely on a boundary at 2 . 5 M glyph[circledot] . We also provide a rate for events in the mass gap between 2 . 5 and 5 M glyph[circledot] , in a binary with either NS or BH. The bin intervals here are chosen for ease of use to roughly capture features in the mass spectrum but do not reflect our methods for event classification nor our inference on features such as the maximum NS mass or edges of any potential mass gaps. \nThe models used in this Section do not model the redshift evolution of the merger rate, and instead report a constant in comoving volume merger rate density [i.e. κ = 0 in Eq. (7)]. For most of the mass intervals considered, our surveys to date extend to only modest redshift, so rate evolution versus redshift can be safely neglected. However, for high-mass binary black holes, our network has cosmologically significant reach, over which the merger rate may evolve. Furthermore, as discussed in Sec. II, we observe structure in the mass distribution for black hole binaries. Therefore, in Sec. VI we provide a more detailed description of BBH merger rates, incorporating both redshift and mass dependence.", 'B. Identifying sub-populations of CBCs': "As discussed in Sec. II, electromagnetic observations had previously suggested a mass gap between black holes and NSs. On the one hand, astrophysical EOS inferences limit nonrotating NS masses to be below the TolmanOppenheimer-Volkoff (TOV) mass, M max , TOV ∼ 2 . 2 -2 . 5 M glyph[circledot] [8, 9, 122-124], and studies of GW170817's remnant limit them to glyph[lessorsimilar] 2 . 3 M glyph[circledot] [125-130]. On the other hand, until recently [23, 37, 38] black holes had not been observed below ∼ 5 M glyph[circledot] . The sparsity of observations between ∼ 2 . 5 M glyph[circledot] and ∼ 5 M glyph[circledot] suggested a potential lower mass gap [28-31]. \nFigure 3 shows the two-dimensional merger rate versus component masses for the three models used in this Sec- \nTABLE II. Merger rates in Gpc -3 yr -1 for the various mass bins, assuming merger rates per unit comoving volume are redshiftindependent. BNS, NSBH and BBH regions are based solely upon component masses, with the split between NS and BH taken to be 2 . 5 M glyph[circledot] . We also provide rates for binaries with one component in the purported mass gap between 2 . 5 M glyph[circledot] and 5 M glyph[circledot] . For all but the last row, merger rates are quoted at the 90% credible interval. For the last row, we provide the union of 90% credible intervals for the preceding three rows, as our most conservative realistic estimate of the merger rate for each class accounting for model systematics. The PDB (pair) model is distinct from the other three models due to its use of a pairing function [111] and is therefore excluded from the union of credible intervals in the final row. In Sec. VI we estimate the merger rate for BBH alone, accounting for variation in merger rate versus redshift. \ntion, as well as the results of FM model applied to BBH. This representation emphasizes the importance of asymmetric binaries to the overall merger rate d R / d m 1 for masses between 1 M glyph[circledot] and 10 M glyph[circledot] . The inferred merger rates further illustrate a falloff in event rate at masses above the BNS scale, with additional peaks associated with both unequal mass binaries consistent with NSBH systems as well as approximately equal mass BBH binaries. The rate of events with at least one component between 2 . 5-5 M glyph[circledot] (i.e. in the purported mass gap) is constrained to be lower than the rate of BNS-like events, but is consistent with the rate of BBH-like events. As further emphasis, Fig. 4 shows the merger rate versus mass for all binaries and also restricting to binaries with q glyph[similarequal] 1 (e.g., the diagonal bins in the BGP model). The rate for approximately equal mass binaries is significantly lower. In other words, because asymmetric mergers like NSBH occur at a much higher rate than BBH but a much lower rate than BNS, in a joint analysis they significantly impact the marginal merger rate d R / d m 1 at the lowest masses. \nThis result highlights another feature: the compact binary population has (at least) three dominant populations: BNS-like systems; significantly asymmetric binaries with small m 2 , comparable to the typical masses of NSs (i.e., including NSBHs as well as GW190814); and the main BBH population with q preferentially more symmetric than 1 / 4 (i.e., including GW190412 but not GW190814). \nFor binaries containing lower mass gap-scale objects, our inferences about the merger rate and its dependence on mass are consistent despite considerable modeling uncertainty. For binaries containing objects between 2 . 5 M glyph[circledot] and 5 M glyph[circledot] and having massive BH-scale primaries ( > 5 M glyph[circledot] ), the mass distribution and merger rate is informed by a few events (GW190814 in particular), thus subject to considerable uncertainty in the inferred component mass distributions. Likewise, for binaries con- \ntaining objects between 2 . 5 M glyph[circledot] and 5 M glyph[circledot] and having NS-scale companions, the merger rate is marginally informed by a few events that may not be associated with this region (i.e., GW200115), exacerbating uncertainty in the inferred NS and BH mass distributions. Providing multiple results for these two source classes explores our systematic uncertainty. The models presented in this section are subject to different sources of systematic uncertainty. For example, MS employs a Gaussian distribution to model components in BNSs, whereas PDB uses a single power law with a sharp turn on at low masses to model all objects below the inferred lower edge of the mass gap. These differences result in considerably different BNS rates due to the limited number of detections in the NS mass range. In particular, differences in pairing function shift the rate inference and add statistical uncertainty in the BNS region. MS and PDB (pair) all assume independent pairing of component masses; PDB (ind) models the pairing of component masses as a power law in mass ratio; and BGP uses a piecewise-constant gaussian process over both component masses. We can therefore directly compare PDB (ind) and PDB (pair) to understand the impact of assuming independent pairing. Independent pairing implies an equal number of equal mass and assymmetric mass mergers, while there have been relatively few unequal mass observations. Thus, a large fraction of PDB (ind)'s assumed population has gone undeteced, resulting in low overall rate. On the other hand, PDB (pair) finds more support for equal mass binaries than asymmetric binaries and produces a higher rate.", 'C. Characterizing suppressed merger rates between NS and BH Masses': 'Figures 3 and 4 show a reduction in the rate above NS masses. It was shown using GWTC-2 that the merger \nFigure 5 shows the differential rate as a function of component mass inferred from all three models. The PDB model infers the location of this drop-off to occur at M gap low = 2 . 1 +0 . 8 -0 . 6 M glyph[circledot] , as shown by black vertical lines. While the other models do not explicitly infer the location of the drop-off, they do clearly show a reduction in the rate at a similar location. The prominence of this \n<!-- image --> \n/circledot \n/circledot \nFIG. 3. Rate density versus component masses for different models inferred from events with FAR < 0 . 25 yr -1 , illustrating consistency on large, coarse-grained scales, but some disagreement and systematics in areas with few observed events. Top left panel : Rate density computed with the FM model assuming no redshift evolution, for binary black holes only. Modulations along lines of constant chirp mass are apparent. Top right panel : Rate density inferred with the BGP model using all compact objects. This model can reproduce observations with localized regions of relatively enhanced rate density. In the binary black hole region, some regions of enhanced density are commensurate with the FM result. Bottom left panel : Rate density inferred with MS. For mergers involving NS, this model reproduces observations with broad distributions, consistent with smoothing the BGP result. For mergers involving typical BH, this model strongly favors equal-mass mergers. Bottom right panel : Rate density inferred with PDB. This model is also consistent with smoothing the FM result, producing features similar to MS, albeit with less structure in the mass ratio distribution for BBH, and by construction lacking a peak near 30 M glyph[circledot] . \nrate between 3 M glyph[circledot] and 7 M glyph[circledot] is suppressed relative to an unbroken power-law extending from higher masses [11]. With additional observations, as well as models and sensitivity estimates that span the full mass range of CBCs, we can now produce a comprehensive perspective on merger rates versus mass throughout the lowmass interval 1-10 M glyph[circledot] . In so doing, we find a dropoff in merger rates above NS-scale masses. As a result, in the detection-weighted population, objects with NSscale mass components are well-separated from objects with BH-scale masses. However, we are unable to confidently infer an absence or presence of a subsequent rise in merger rates from lower mass gap masses. The pur- \nported lower mass gap [28-31] between the NS and black hole populations would produce such a rise, such that the mass gap produces an extended local minimum in the merger rate versus mass. We therefore neither find evidence for nor rule out the existence of a two-sided lower mass gap. \nFIG. 4. Impact of asymmetric binaries on the primary mass distribution, illustrating how the depth and extent of a mass gap in any one-dimensional distribution depends on the choice of slicing or marginalization over the remaining dimension. Differential merger rate as a function of primary mass for the BGP model when considering only the diagonal q glyph[similarequal] 1 bins in Fig. 3 i.e. m 1 d N d m 1 d q d V c d t ∣ ∣ ∣ q glyph[similarequal] 1 and the population of compact binaries across all mass bins. The rate for approximately equal mass binaries is significantly lower highlighting the contribution of asymmetric mergers like NSBHs to the marginal distribution over primary mass. The plot uses the BGP population model inferred from events passing a FAR threshold of < 0 . 25yr -1 . Solid curves represent the median rate densities and shaded areas denote 90% credible regions. \n<!-- image --> \nFIG. 5. Differential merger rate as a function of component mass for the PDB, MS, and BGP model. Three independent methods with different modeling assumptions agree on the merger rate versus mass, while illustrating the importance of modeling systematics on the overall rate for objects with NS-scale masses. Shaded areas denote 90% credible regions, while verrtical black lines denote the median ( solid ) and 90% credible intervals ( dashed ) of the lower boundary of the mass gap, M gap low , in the PDB model rate dropoff location. \n<!-- image --> \ndrop-off can be characterized by comparing the rate of mergers with both masses below 2 . 5 M glyph[circledot] (BNS) to that of mergers with at least one component mass between 2 . 5 and 5 M glyph[circledot] (in the mass gap). For this comparison, we find that the differential merger rate of systems with at least one component in the mass gap is one to two orders of magnitude lower than the BNS rate. Thus, even in the absence of any prior knowledge of the difference between NSs and BHs, the gravitational-wave data suggest two distinct populations of compact objects. This is consistent with results initially found for GWTC-1 by [108] and for GWTC-2 by [109]. \nA subsequent rise in the mass distribution above the putative mass gap is less clearly discernible. The PDB model explicitly parametrizes the mass gap with both low and high-mass transitions M gap low/high and a gap depth A (where A = 0 corresponds to no gap and A = 1 to a lower mass gap containing no events). While the posterior on A peaks around 0.77, i.e. corresponding to a relatively empty mass gap, it has broad support between 0 and 1, indicating an inability to unambiguously differentiate between the presence or absence of a lower mass gap. Additionally, the Bayes factor for a model with no gap ( A = 0) or a completely empty gap ( A = 1), relative to the parametrized model, are 0.073 and 1.4, respectively. This lack of clear preference indicates an inability to resolve the absence or existence of a clear gap-like feature in this part of the mass spectrum. \nA subsequent rise in the mass distribution at M gap high is also less clear to discern. The models infer mass distributions with similar support for both a mildly pronounced gap and a flat transition above M gap high . Both of these are consistent with the finding in [11] of a deviation from a single power law below primary masses of ∼ 7 M glyph[circledot] . \nWe find that if a lower mass gap does exist, it may not be totally empty. While the merger rates show a fall-off above around 2 . 5 M glyph[circledot] in Fig. 5, the rate does not fall to zero. Furthermore, the component masses of 6 events have at least 5% posterior support between M gap low and M gap high when using a population-informed prior [109]. GW190814 stands out as having considerable support for its secondary being within the mass gap or below the dropoff in the rate at M gap low : P ( m 2 , GW190814 ∈ [ M gap low , M gap high ]) = 0 . 76. This event has a mass ratio q = 0 . 112 +0 . 008 -0 . 009 [23], hinting at either a potential subpopulation of lowq , lowm 2 BBHs, or a handful of NSBHs with high NS masses. The former possibility is examined in Sec. VI E, and the latter is discussed in Sec. V C. For both the NSBH systems, there is a ∼ 10% probability of the secondary lying in the mass gap and, for GW200115 the primary has a 70% probability of m 1 < M gap high . Finally, GW190924 021846, which is the BBH event with the lowest total mass, we find roughly equal support for the secondary being either within ( m 2 < M gap high ) or above ( m 2 < M gap high ) the mass gap. \nThe inferred depth of the gap does depend heavily on the assumed pairing function: a model in which objects \nare randomly paired with other objects regardless of mass ratio predicts a more prominent gap than one with a power-law-in-mass-ratio pairing function as in Eq. (11). Similarly, a change of the pairing function will impact the classification of various components as below, in or above the mass gap. Consequently, we do not rely on this methodology for event classification in Sec. V, and instead use EOS-informed limits on the maximum allowed NS mass, and perform leave-one-out analyses with respect to known subpopulations. The lower mass gaprelated results stated here are obtained using a random pairing model. \nThough we report on our analysis with FAR < 0 . 25 yr -1 , to assess the stability of our results to threshold choices we have repeated our analyses using all events with previously reported parameter inferences below 1 yr -1 (i.e., excluding GW190531). Even though such an analysis includes all five candidate NSBH, our key conclusions remain largely unchanged: the derived merger rates versus mass are consistent with the error bars shown in Figs. 4, and 5, and the merger rates reported in Table II are consistent. In particular, we draw similar conclusions about the merger rate between 2 M glyph[circledot] and 10 M glyph[circledot] : suppressed but likely filled, without evidence for or against a true two-sided mass gap.', 'V. MASS DISTRIBUTION OF NEUTRON STARS IN BINARIES': 'In this section, we characterize the astrophysical population of NSs using data from the gravitational wave events that are likely to contain at least one NS. Because of the paucity of low-mass compact binary mergers observed to date, and the difficulty in ascertaining the presence of a NS in these systems, modeling the NS population observed in gravitational waves has been challenging. In our previous population analysis through Gravitational Wave Transient Catalog 2 (GWTC-2) [11], the rate density of BNS and NSBH mergers was estimated, but the shape of the mass distribution of the NSs in these compact binaries was not inferred. The BNS events GW170817 and GW190425 were included in a joint study of the Galactic and gravitational wave populations of BNSs in [131], which linked the two observed populations via a bimodal birth mass distribution. The confident BNS and NSBH detections made to date were analyzed in a study of the gravitational wave population in [100], which found the observed NS masses to be consistent with a uniform distribution. \nWe begin by classifying the observed low-mass compact binaries as BNSs, NSBHs or BBHs. The classifications are based on a comparison of their component masses with an EOS-informed estimate of the maximum NS mass, and are corroborated against the location of the lower mass gap between NSs and BHs as inferred in the previous section. Then, adopting these source classifications as definite and considering the BNS and NSBH \ndetections below a FAR threshold of 0 . 25 yr -1 , we infer the shape of the NS mass distribution in compact binaries. In contrast to Sec. IV, we do not attempt to determine the overall rate of such mergers, nor do we attempt to infer the mass distribution of BHs in coalescing NSBH systems. Our analysis makes a comparison with the observed Galactic population of NSs, and we additionally investigate the impact on the population of the event GW190814, a lower mass-gap merger whose secondary may possibly be a NS, but is more likely a low-mass BH.', 'A. Events containing NSs': "The gravitational-wave signal of a compact binary merger involving a NS differs from that of a BBH due to matter effects in the waveform, most notably the phasing of the gravitational waveform during the inspiral due to the tidal deformation of the NS [132]. Since none of the observations in O3b [1] yield an informative measurement of tidal deformability, the gravitational-wave data do not identify which sources contain a NS. Nonetheless, we can establish whether their components are consistent with NSs by comparing their masses to the maximum NS mass, M max , following the method described in [7]. \nThe precise value of M max is unknown because of uncertainty in the NS EOS. Mass measurements for the heaviest known pulsars [133, 134] set a lower bound of ∼ 2 M glyph[circledot] on M max , while basic causality considerations imply that M max glyph[lessorsimilar] 3 M glyph[circledot] [135, 136]. While individual nuclear theory models for the EOS can produce maximum masses as large as ∼ 3 M glyph[circledot] , astrophysical inferences of the EOS generally predict that the maximum mass of a nonrotating NS, the TOV mass M max , TOV , is between 2 . 2 M glyph[circledot] and 2 . 5 M glyph[circledot] [9, 122-124, 137]. Similarly, studies of GW170817's merger remnant suggest that M max , TOV glyph[lessorsimilar] 2 . 3 M glyph[circledot] [125-130]. Rapid rotation can sustain a maximum mass up to ∼ 20% larger than M max , TOV [138]. However, the astrophysical processes that form compact binaries may prevent the EOS-supported M max from being realized in the population. \nWe can therefore identify objects as NS candidates based on their mass using estimates of M max , as long as we assume a clean separation between the NS and BH mass spectra. Of the events with FAR less than 0 . 25 yr -1 , five have at least one component mass with support below 3 M glyph[circledot] , making them potentially consistent with a BNS or NSBH merger. These events are listed in Table III, and their component mass posteriors are compared to two estimates of M max in Fig. 6. \nFor each of these observed low-mass events, we calculate in Table III the probability that at least one of the component masses is less than the maximum NS mass, marginalizing over statistical uncertainties and assuming a uniform component mass prior. We adopt a threshold probability of 50% for classification as a NS. Our fiducial maximum NS mass estimate is taken to be M max , TOV from the EOS inference of [9], which is based on pulsar \ntiming, gravitational wave and x-ray observations of NSs. That study finds M max , TOV = 2 . 21 +0 . 31 -0 . 21 M glyph[circledot] , and the corresponding posterior distribution is shown for comparison in Fig. 6. Four of the FAR < 0 . 25 yr -1 events have P ( m < M max , TOV ) > 0 . 5 for at least one component, and we deem them either BNSs (if m 1 < M max , TOV ) or NSBHs (if only m 2 < M max , TOV ). The fifth event, GW190814, has P ( m < M max , TOV ) = 0 . 06 and is therefore classified as a BBH. These source classifications do not change if, instead of M max , TOV , we compare against the rotating NS maximum mass, M max ( χ ), as calculated from an empirical relation involving the TOV mass and the component spin χ [139]. This allows for the possibility that one or more of the low-mass components is rapidly rotating. \nWe draw similar conclusions about each event if we interpret the sharp decrease in merger rate near 2 . 5 M glyph[circledot] seen in the PDB analysis as the separation between NS and BH mass ranges. (This interpretation does not imply that M max , TOV and M gap low need to agree: M gap low could be below M max , TOV if the heaviest NSs the EOS can support are not realized in nature, or M gap low could be above M max , TOV if the lower mass gap occurs within the BH mass spectrum.) Following [109], we compare the component mass measurements against the inferred M gap low parameter from the PDB model, as shown in Fig. 6, and list the probabilities P ( m < M gap low ) in Table III. The same four events are consistent with BNSs or NSBHs. \nFig. 6 also plots the component mass posteriors for two FAR < 1 yr -1 events from Table I that may contain NSs, if astrophysical in origin. In particular, GW190426 and GW190917 have masses consistent with NSBH systems [3, 4]. This classification is confirmed by the P ( m < M max , TOV ) and P ( m < M gap low ) probabilities calculated for them in Table III.", 'B. Mass distribution': "Using the FAR < 0 . 25 yr -1 events classified as BNSs or NSBHs in Table III, we infer the mass distribution of NSs in merging compact binaries. We adopt the Power and Peak parametric mass models described in Sec. III and implement a selection function based on a semianalytic approximation of the integrated network sensitivity V T , fixing the redshift evolution of the population and ignoring spins when estimating the detection fraction. The population hyper-parameters are sampled from uniform prior distributions, subject to the condition m min ≤ µ ≤ m max in the Peak model, except that we assume that the maximum mass in the NS population, m max , does not exceed M max , TOV . This is consistent with our use of the nonrotating maximum NS mass to classify the events, and amounts to an assumption that the NSs observed via inspiral gravitational waves are not rotationally supported. In practice, this means imposing a prior proportional to the cumulative distribution function of M max , TOV , as shown in the inset of Fig. 7 and \nFIG. 6. Masses for events with at least one candidate neutron star. Upper panel: one-dimensional posterior distributions for the masses of the candidate NSs, as compared to estimates of the maximum NS mass based on the dense-matter EOS [9] ( M max , TOV ) and on the inferred location of the lower mass gap in Sec. IV's PDB analysis ( M gap low ). Primary components are shown dash-dotted. GW190814's secondary component lies above both estimates of the maximum NS mass. Lower panel: two-dimensional 50% (shaded) and 90% (unshaded) credible regions for the binary masses of each candidate NS merger. The marginal events GW190426 and GW190917 are shown dotted. The 90% credible intervals of the maximum NS mass posterior inferred from the EOS and from the lower mass gap location are also plotted. GW190814 occupies a distinct region of the m 1 -m 2 plane compared to the events deemed BNSs or NSBHs. \n<!-- image --> \n/circledot \ndetailed in Appendix B 1. \nThe inferred mass distributions for these two models are plotted in Fig. 7. The posterior population distribution for the Power model has α = -2 . 1 +5 . 2 -6 . 9 , consistent with a uniform mass distribution, although the median distribution is a decreasing function of mass. The power-law hyper-parameter is most strongly constrained relative to the flat α ∈ [ -12 , 4] prior on the low end. The two bumps in the 90% credible interval visible in Fig. 7 correspond respectively to the minimum and maximum mass cutoffs of the population model realizations with α < 0 and α > 0. The median inferred Peak distribution is relatively flat, and the peak width and location are almost entirely unconstrained relative to the prior: σ = 1 . 1 +0 . 8 -0 . 8 M glyph[circledot] and µ = 1 . 5 +0 . 4 -0 . 3 M glyph[circledot] for a uniform σ ∈ [0 . 01 , 2 . 00] M glyph[circledot] and µ ∈ [1 , 3] M glyph[circledot] prior subject to m min ≤ µ ≤ m max . Thus, the gravitational wave observations to date do not support a NS mass distribution with a pronounced single peak. This contrasts with \nTABLE III. Classifications for low-mass events from Table I. The probability that a component is compatible with a NS is measured by the fraction of its mass posterior lying below an estimate [9] of the maximum nonrotating NS mass, M max , TOV , marginalized over statistical uncertainties. We adopt a 50% threshold for classification as a NS, assuming a clean separation between NS and BH mass spectra. Probabilities are reported relative to a uniform prior on the component mass. They refer to the secondary component of all events except GW170817 and GW190425, in which case the secondary is securely below the maximum NS mass and the probability for the primary is given. The probabilities are similar and the classifications are unchanged when the component masses are compared to M gap low , the location of the lower mass gap between NSs and BHs inferred from Sec. IV's PDB analysis of the FAR < 0 . 25 yr -1 events. \nthe Galactic BNS subpopulation, whose mass distribution is sharply peaked around 1 . 35 M glyph[circledot] [39, 40, 140], as shown for comparison in Fig. 7. The mass distribution of NSs observed in gravitational waves is broader and has greater support for high-mass NSs. This latter point is also true compared to the Galactic NS population as a whole, whose mass distribution has a double-peaked shape [141-143]. \nThe minimum NS mass in the gravitational wave population is inferred to be 1 . 2 +0 . 1 -0 . 2 M glyph[circledot] and 1 . 1 +0 . 2 -0 . 1 M glyph[circledot] in the Power and Peak models, respectively. The lower bound on m min is a prior boundary motivated by the sensitivity model, as the gravitational-wave searches target sources above 1 M glyph[circledot] . The maximum mass in the population is found to be 2 . 0 +0 . 3 -0 . 3 M glyph[circledot] for the Power model and 2 . 0 +0 . 2 -0 . 2 M glyph[circledot] for the Peak model, relative to the EOS-informed m max prior. These values are consistent with the maximum mass inferred from the Galactic NS population, 2 . 2 +0 . 8 -0 . 2 M glyph[circledot] [142], as can be seen in the inset of Fig. 7. The maximum mass is the bestconstrained hyper-parameter in the population models. Its upper bound is more tightly constrained than the Galactic m max in Fig. 7 as a result of the imposed m max ≤ M max , TOV prior, which begins tapering above 2 M glyph[circledot] , and the strong selection bias of gravitational-wave observations towards heavier masses, which renders the non-observation of heavier NSs informative. Nonetheless, the statistical uncertainty in m max remains large, and it is expected that approximately 50 BNS detections will be needed before the maximum mass in the NS population can be measured to within 0 . 1 M glyph[circledot] [144]. \nThe m max value inferred from gravitational waves is also as large as M max , TOV within statistical uncertainties. This would not be the case if, for instance, the astrophysical processes that form coalescing compact binaries prevented 2 M glyph[circledot] NSs from pairing with other compact objects. Such a scenario is compatible with the EOSinformed m max prior that we impose. However, we find \nthere is no evidence that the NS mass spectrum observed with gravitational waves is limited by the astrophysical formation channel: NSs as heavy as can be supported by the EOS can end up in merging compact binaries. \nMoreover, we infer a consistent maximum mass if we adopt a uniform m max prior instead of the EOS-informed one. This relaxes the assumption that the observed NS masses must be below the nonrotating maximum mass, and accounts for the possibility that rapid rotation may cause a NS's mass to exceed M TOV . Specifically, we find m max = 2 . 1 +0 . 8 -0 . 4 M glyph[circledot] in the Power model and 2 . 0 +0 . 8 -0 . 2 M glyph[circledot] in the Peak model. The upper error bar on m max extends to much higher values in this case because it is no longer subject to the tapering EOS-informed prior, which has little support above 2 . 5 M glyph[circledot] . We also obtain consistent results if we expand the event list to include the two marginal NSBH detections listed in Table III, as described in Appendix C 2.", 'C. Outlier events': "The mass-based event classification carried out above deemed GW190814 to be a BBH merger on the basis of the maximum NS mass the EOS can support. We now further demonstrate that it is an outlier from the population of BNSs and NSBHs observed with gravitational waves. \nIf we dispense with its M max , TOV -based classification, and include GW190814 as a NSBH in the population analysis, the inferred maximum mass is shifted up to 2 . 8 +0 . 2 -0 . 2 M glyph[circledot] in the Power model and 2 . 7 +0 . 3 -0 . 2 M glyph[circledot] in the Peak model (cf. m max = 2 . 1 +0 . 8 -0 . 4 M glyph[circledot] in the Power model and m max = 2 . 0 +0 . 8 -0 . 2 M glyph[circledot] in the Peak model without GW190814). These values are obtained relative to a uniform m max prior, since we are no longer consistently enforcing m ≤ M max , TOV ; all results in this subsection refer to this prior. The m max posterior has support up \nFIG. 8. Comparison between GW190814's secondary component and the largest secondary mass in the observed BNS and NSBH population. The Peak model is fit to the population including (respectively, excluding) GW190814. The predicted distribution of the largest secondary mass, max 5 ( m 2 ), observed after five detections-two BNSs and three NSBHsis shown in orange (blue). The shaded region represents the 90% credible interval of the posterior distribution for the mass of GW190814's secondary component. GW190814's m 2 is a 0 . 2%-level outlier from the rest of the observed population of NS secondaries. \n<!-- image --> \n/circledot \nFIG. 7. Inferred neutron star mass distribution. The median mass distribution (solid) and 90% credible interval (shading) inferred for the Power (respectively, Peak ) population model is shown in blue (orange), as compared to the mass distribution of NSs in Galactic BNSs [41] (dot-dashed black) and the mass distribution of all Galactic NSs [142] (solid black). The inferred gravitational-wave population has a greater prevalence of high-mass NSs. The inset shows the posterior distribution for the maximum mass in the NS population for both models, as compared to the Galactic m max . The EOS-informed m max prior, which is proportional to the cumulative distribution function of M max , TOV , is also shown in the inset (dashed). It enforces m ≤ M max , TOV using the maximum Tolman-Oppenheimer-Volkoff mass estimate from [9]. The maximum mass in the gravitational-wave population is as large as M max , TOV within statistical uncertainties. \nto 3 M glyph[circledot] , where the prior truncates and the models' fixed BH mass distribution begins. The inferred NS mass distributions with GW190814 are similar, but flatter and broader, than those depicted in Fig. 7. \nTo test whether GW190814 hails from the same population as GW170817, GW190425, GW200105 and GW200115, we examine the Peak model's posterior predictive distribution for secondary masses with and without GW190814 in the event list. Figure 8 compares GW190814's measured m 2 = 2 . 59 +0 . 08 -0 . 09 M glyph[circledot] against the prediction for the largest observed secondary mass, max 5 ( m 2 ), after two BNS observations and three NSBH observations. That is, we draw two pairs of masses from the posterior predictive distribution for BNSs and three secondary masses from the posterior predictive distribution for NSBHs, take the largest of the five secondaries, and build up the plotted distributions by performing this procedure repeatedly. The probability of observing a secondary mass at least as large as the mean of GW190814's \nm 2 in the population is only 0 . 2% according to the Peak model fit that excludes GW190814. (We characterize GW190814's m 2 by its mean, since it is measured so precisely.) The equivalent probability relative to the Peak model fit that includes GW190814 is 3 . 3%; we expect a rigorous, fully self-consistent calculation of a p-value to lie between these two numbers [145]. Hence, GW190814's secondary component is an outlier from the secondaries in BNS and NSBH systems. In the next section, we also establish GW190814 as an outlier from the BBH population observed in gravitational waves, corroborating our previous analysis [11]. These findings reinforce that it represents a distinct subpopulation of merging compact binaries. \n<!-- image --> \n/circledot", 'VI. MASS DISTRIBUTION OF BLACK HOLES IN BINARIES': 'We find two key new conclusions about the black hole mass distribution using the GWTC-3 dataset to infer a population: that the mass distribution has a substructure, reflected in clustering of detected events, and that observations are consistent with a continuous, monoton- \nically decreasing mass distribution at masses > 50 M glyph[circledot] , providing inconclusive evidence for an upper mass gap. Adopting previous coarse-grained models, we find conclusions consistent with our analysis of GWTC-2 [25]. For the purposes of this section, given our large BBH sample, we adopt a FAR threshold of 1 yr -1 , but we do not include the previously identified outliers GW190917 (a NSBH) and GW190814 (an extreme mass ratio binary) in the BBH population unless otherwise noted. Additionally, unlike the redshift-independent results described in Sec. IV, the new analyses described in this section all account for a redshift-dependent BBH merger rate according to Eq. (7). Specifically, in this Section we present results for the PP model to broadly characterize the mass spectrum and corroborate results found in GWTC-2, as well as the the cubic spline power law perturbation PS model and the binned Gaussian process BGP, as both can capture smaller-scale features in the mass distribution. All three models are described in detail in Section III C. We report on the same BGP analysis as performed in Sec. IV, with FAR < 0 . 25 yr -1 and without allowing for redshift dependence; by contrast, the PS, PP, and FM models allow for redshift dependence and use FAR < 1 yr -1 . Table IV summarizes our results for the overall BBH merger rate, as well as merger rates over restricted mass intervals.', 'A. Broad features of the mass spectrum': "The events from GWTC-3 are broadly consistent with the previously identified population [11]. Figure 9 compares some of the expectations from our previous analysis of GWTC-2 BBHs with the comprehensive sample of GWTC-3 BBH events. The panels compare the observed and expected fractions of all events detected below a threshold in primary mass m 1 , effective inspiral spin χ eff , or source redshift. The panels also show the Wilson score interval [146], a frequentist estimate of the uncertainty in the cumulative distribution F , which is approximately ± 1 . 68 √ F (1 -F ) /N obs when F is significantly different from 0 or 1. \nAll the cumulative distributions in Figure 9 are broadly consistent with our prior expectations based on coarsegrained models used in our previous work. For this reason, we begin by presenting the inferred coarse-grained mass distribution of black hole binaries, making use of the PP model [11] which best fitted the population from GWTC-2. \nFigure 10 shows our inference on the astrophysical primary mass (left) and mass ratio (right) distributions, using the fiducial mass model, compared to what was previously found in GWTC-2 (black). We find a power-law slope for the primary mass, α = 3 . 5 +0 . 6 -0 . 56 , supplemented by a Gaussian peak at 34 +2 . 6 -4 . 0 M glyph[circledot] . On the upper end, the mass of the 99th percentile, m 99% , is found to be 44 +9 . 2 -5 . 1 M glyph[circledot] . The mass ratio distribution is modelled as a power law q β q with β q = 1 . 1 +1 . 7 -1 . 3 \nIn contrast to our GWTC-2 population fit, the inferred mass spectrum decays more rapidly; the m 99% is considerably lower than 60 +14 -13 M glyph[circledot] , as was found with GWTC-2. These results are expected, given that the new observations in GWTC-3 contain a greater fraction of lower mass systems (see, e.g., Fig. 1). The fraction of BBH mergers with primary masses within the Gaussian component of the fiducial model is found to be λ = 0 . 038 +0 . 058 -0 . 026 (0 . 1 +0 . 14 -0 . 071 in GWTC-2), but still rules out zero. This result further highlights that the fraction of higher mass binaries has decreased in GWTC-3. Both the mean and the standard deviation of the Gaussian component are consistent with previous inferences. Furthermore, the inferred mass ratio distribution is less peaked towards equal mass binaries ( β q = 1 . 1 +1 . 7 -1 . 3 ) compared to GWTC-2 ( β q = 1 . 3 +2 . 4 -1 . 5 ), a result driven by the discovery of binaries with support for substantially unequal masses (see, e.g., Fig. 9). \nWe previously used several other phenomenological models to interpret sources in GWTC-2. Using this broader suite of models, we draw similar conclusions to those presented above: the mass distribution is inconsistent with the single power law and has a feature at ∼ 35-40 M glyph[circledot] . The peak's location is also well-separated from the largest black holes predicted by the other component: the over-density and maximum mass are still not associated. The odd ratios discriminating between these models are modest, of order one in three to one in ten. Despite for presentation purposes adopting the PP for illustrating consistency with GWTC-2 results, we cannot decisively differentiate between a peak near 35 M glyph[circledot] versus a more generic transition towards a lower merger rate at higher mass; see Appendix D 1 for details. \nIn Table IV, we provide BBH merger rates for the full population, as well as split based upon the primary mass at m 1 < 20 M glyph[circledot] , m 1 ∈ [20 , 50] M glyph[circledot] and m 1 > 50 M glyph[circledot] to capture the broad features of the mass spectrum: the high rate at low masses, a peak around 35-45 M glyph[circledot] and the falling merger rate at high masses.", 'B. Mass distribution has substructure': "With new discoveries in O3, we are now confident the mass distribution has substructure, with localized peaks in the component mass distribution. For example, we find overdensities in the merger rate ( > 100 . 0% credibilty) as a function of primary mass, when compared to a power law, at m 1 = 10 +0 . 29 -0 . 59 M glyph[circledot] and m 1 = 35 +1 . 7 -2 . 9 M glyph[circledot] . At best, we have modest confidence (less than 10:1 odds) in additional structure. These signs of substructure were identified in O3a [47] and are corroborated by consistent observations in O3b. \nWe arrive at these conclusions through multiple independent analyses. Each of these model agnostic approaches attempts to reconstruct the mass distributions with minimal constraints imposed. Specifically, we employ a flexible mixture model (introduced in Section III C \nFIG. 9. The empirical cumulative density function ˆ F = ∑ k P k ( x ) /N of observed binary parameter distributions (derived from the single-event cumulative distributions P k ( x ) for each parameter x ) are shown in blue for primary mass (left), effective inspiral spin (center), and redshift (right). All binaries used in this study with FAR < 1 / 4yr are included, and each is analyzed using our fiducial noninformative prior. For comparison, the gray bands show the expected observed distributions, based on our previous analysis of GWTC-2 BBH. Solid lines show the medians, while the shading indicates a 90% credible interval on the empirical cumulative estimate and selection-weighted reconstructed population, respectively. GW190814 is excluded from this analysis. \n<!-- image --> \n<!-- image --> \n/circledot \nFIG. 10. The astrophysical BBH primary mass (left) and mass ratio (right) distributions for the fiducial PP model, showing the differential merger rate as a function of primary mass or mass ratio. The solid blue curve shows the posterior population distribution (PPD) with the shaded region showing the 90% credible interval. The black solid and dashed lines show the PPD and 90% credible interval from analyzing GWTC-2 as reported in [11]. The vertical gray band in the primary mass plot shows 90% credible intervals on the location of the mean of the Gaussian peak for the fiducial model.TABLE IV. Merger rates in Gpc -3 yr -1 for BBH binaries, quoted at the 90% credible interval, for the PP model and for three non-parametric models ( Binned Gaussian process , Flexible mixtures , Power Law + Spline ). Rates are given for three ranges of primary mass, m 1 as well as for the entire BBH population. Despite differences in methods, the results are consistent among the models. BGP assumes a non-evolving merger rate in redshift. The merger rate for PP, FM, and PS is quoted at a redshift value of 0.2, the value where the relative error in merger rate is smallest. \nand labelled FM in tables and figures), a cubic spline power law perturbation (PS), and a binned Gaussian process (BGP). Figure 11 shows the inferred rate d R /dm 1 as a function of primary mass for each of the non-parametric models. There is a clear presence of structure beyond an unbroken single power law found when using these more flexible models, with a global maximum of the merger rate at larger masses at around 10 M glyph[circledot] followed by a fall off to lower rates. Modulating this extended decline, the PS, FM and even BGP show indications of additional structure. As the BGP likely cannot resolve small-scale features, we assess these features' details and significance with the remaining two models. \nFigure 12 shows the results of the spline perturbation model, where 1000 posterior draws of the spline function f ( m 1 ) are illustrated, where exp f ( m 1 ) modulates an underlying power-law distribution. The inferred perturbation f ( m 1 ) strongly disfavors zero at both the 10 M glyph[circledot] and 35 M glyph[circledot] peak, finding f ≤ 0 at 0 . 216%, and < 0 . 0325% credibilities respectively. Additionally for the drop in merger rate at 14 M glyph[circledot] , the PS model finds f ≤ 0 at 96 . 1% credibility.", 'C. Inconclusive evidence for upper mass gap': 'Stellar evolution models predict a lack of black holes with masses from 50 +20 -10 M glyph[circledot] to ∼ 120 M glyph[circledot] due to pair production instability [147-153]. The high-mass event GW190521 could have a component lying within this mass gap [22, 154]. Other analyses of this event with independent parameter inferences have argued this event could have both components outside this gap [155-157]. We define a gap as a rapid decline in merger rate at some cutoff mass, followed by a rapid rise in the distribution at a significantly higher mass. Repeating similar analyses with the full O3 data set, we find no evidence for such a gap. Following [155], we extend our PP mass model to allow for masses > 100 M glyph[circledot] , and to include a zero-rate mass interval, parameterized with the lower edge and width of the gap. With this extended model, we find minimal posterior support for the gap to start at < 75 M glyph[circledot] (3 . 1% credibility). When it does, the gap width is constrained to be < 35 M glyph[circledot] . The majority of the posterior support has the gap start above 75 M glyph[circledot] , consistent with the inferred maximum mass cutoff from the PP model without a gap. This allows both component masses of the most massive BBH in the catalog, GW190521, to fall below the cutoff, leaving no observations with masses larger than the start of the gap. We are not able to determine whether or not the mass distribution exhibits a rise again at higher masses. We find a slight preference (ln B = 0 . 06) for the PP model without a gap over one with the gap included, thus we report inconclusive evidence for a zero-rate upper mass gap. Inconclusive support for a zero-rate gap challenges classical conclusions for the pair-instability mass gap. The pair-instability mass gap could start higher than theory expects, or the high-mass binaries in our catalog could \nbe formed in a way that avoids pair-instability.', 'D. Evolution of rate with redshift': "The observation of BBH mergers offers us the means of not only measuring the local merger rate per comoving volume but also the evolution of this merger rate as we look back towards larger redshifts z . Given the limited range of redshift to which our searches are sensitive, we parametrize the merger rate per comoving volume as a simple power law, with R ( z ) ∝ (1 + z ) κ [96]. \nIn our previous study [11], the redshift distribution was weakly constrained, exhibiting a preference for a rate that increased with redshift but still consistent with a nonevolving merger rate. Here, in addition to new events observed in O3b, we leverage updated pipelines and our improved sensitivity models to update our inference of κ . As discussed further in Appendix C 5, these sensitivity model refinements indicate a lower search sensitivity to high-redshift BBH mergers than previously concluded. We now confidently claim to see evolution of the BBH merger rate with redshift in our population with a FAR < 1 yr -1 , inferring that κ > 0 at 99.6% credibility. While the exact distribution of κ does depend on the chosen mass model, we can rule out a redshift-independent merger rate at similar credible levels when adopting any of the parameterized mass distribution models used in [11]. \nFigure 13 shows the marginal posterior on κ given GWTC-3 in blue, obtained while using the PP and Default mass and spin models. The dashed distribution, meanwhile, shows the previously published measurement of κ obtained with GWTC-2. In Fig. 13 we also show our corresponding constraints on R ( z ) itself as a function of redshift. The dark blue line traces our median estimate on R ( z ) at each redshift, while the dark and light shaded regions show central 90% and 50% credible bounds. Our best measurement of the BBH merger rate occurs at z ≈ 0 . 2, at which R ( z = 0 . 2) = 19-42Gpc -3 yr -1 . For comparison, the dashed black line in Fig. 13 is proportional to the Madau-Dickinson star formation rate model [158], whose evolution at low redshift corresponds to κ SFR = 2 . 7. While the rate evolution remains consistent with the Madau-Dickinson star formation rate model, it is not expected for these two rates to agree completely due to the time delays from star formation to merger [159-167]. \nIn most plausible formation scenarios (e.g., if BBHs arise from stellar progenitors), we do not expect R ( z ) to continue growing with arbitrarily high z . Instead, we anticipate that R ( z ) will reach a maximum beyond which it turns over and falls to zero. Even in cases where the peak redshift z p at which R ( z ) is maximized lies beyond the LIGO-Virgo detection horizon, a sufficiently tight upper limit on the stochastic gravitational-wave background due to distant compact binary mergers [168-170] can be leveraged to bound z p from above, potentially yielding a \nFIG. 11. The differential merger rate for the primary mass predicted using three non-parametric models compared to the fiducial PP model. Solid curves are the medians and the colored bands are the 90% credible intervals. These models offer increased flexibility compared to phenomenological models in predicting the population. The PS applies a perturbation to the primary mass in a modified version of our fiducial PP model that does not include the Gaussian peak. FM models the chirp mass, mass ratio, and aligned spin distribution as a weighted sum of mixture components. Both of these models incorporate a single parameter redshift evolution of the merger rate [Eq. (7)]. The BGP models the two-dimensional mass distribution as a binned Gaussian Process which is piecewise constant in log m i , illustrating the same analysis as presented in Sec. IV with FAR < 0 . 25 yr -1 . All three models infer a local maximum in the merger rate at around 10 M glyph[circledot] and 35 M glyph[circledot] . \n<!-- image --> \nFIG. 12. The cubic spline function, f ( m 1 ), describing the perturbations to an underlying power law inferred with the PS model. The thin grey lines show 1000 draws from the posterior while the black lines show the knot locations (vertical) and the 90% credible region of the posterior. The dashed blue lines mark the 90% credible bounds of the Gaussian priors (centered on zero) imposed on each knot's height. The shaded region covers any masses less than the 95th percentile of the marginal posterior distribution on m min . Because the low mass region of the mass distribution is cut off and there are no observations there, the posterior in this region resembles the prior of the cubic spline function. \n<!-- image --> \n/circledot \njoint measurement of κ and z p [171]. As demonstrated in [172], our current instruments are not yet sensitive enough to enable a meaningful joint constraint on κ and z p , even with the inclusion of new events in GWTC-3. \nAs heavy BBHs are primarily believed to arise from low-metallicity stellar progenitors [173-175], one might wonder if more massive BBHs are observed at systemat- \nically higher redshifts than less massive systems. Moreover, any metallicity dependence in the physics of stars, such as the maximum black hole mass imposed by pair instability supernovae (PISN) [148, 152, 176], could yield redshift-dependent features in the black hole mass distribution [177, 178]. Such a redshift dependence would confound efforts to leverage the PISN mass gap as a probe of cosmology. Previous investigations [179] demonstrated using GWTC-2 that redshift dependence of the maximum BBH mass would be required to fit the observations if the BBH mass distribution has a sharp upper cutoff. However, if the distribution decays smoothly at high masses, for example as a power-law, the data are consistent with no redshift dependence of the cutoff location. \nWe revisit this question using the latest BBH detections among GWTC-3, finding that these conclusions remain unchanged. Specifically, by modelling the highmass tail of the distribution with a separate power-law index, we find no evidence that the distribution is redshift dependent, suggesting that the high-mass structure in the BBH mass distribution remains consistent across redshift.", 'E. Outliers in the BBH Population': 'While we inferred the population of most BBH and binaries involving NS, some systems (particularly with significantly asymmetric masses) lie at the boundary between these categories [23, 145]. So far, we have simply excluded these events from our BBH analysis. To demonstrate this choice is internally self-consistent and \nFIG. 14. The posterior distribution on the minimum mass truncation hyper-parameter, m min , inferred with the PP model. The posteriors are shown both including and excluding the two BBH mergers containing low mass secondaries, GW190814 and GW190917. The cutoff at m min = 2 M glyph[circledot] corresponds to the lower bound of the prior distribution. The inclusion of either of these two events significantly impacts the distribution. The shaded regions indicate the 90% credible interval on the m 2 posterior distribution for the two outlier events, GW190814 (purple) and GW190917 (grey). \n<!-- image --> \nFIG. 13. Constraints on the evolution of the BBH merger rate with redshift. Top : Posterior on the power-law index κ governing the BBH rate evolution, which is presumed to take the form R ( z ) ∝ (1 + z ) κ . The blue histogram shows our latest constraints using GWTC-3 ( κ = 2 . 9 +1 . 7 -1 . 8 ), while the dashed distribution shows our previous constraints under GWTC-2. Bottom : Central 50% (dark blue) and 90% (light blue) credible bounds on the BBH merger rate R ( z ). The dashed line, for reference, is proportional to the rate of cosmic star formation [158]; we infer that R ( z ) remains consistent with evolution tracing star formation. \n<!-- image --> \nwell-motivated, we show that these events are outliers from our recovered BBH population. Specifically, we repeat the population analysis using the PP model, highlighting the extent to which the population changes when including these events. \nFor a population consisting of all potential BBH events in O3, including GW190917 and GW190814, the mass distribution must extend to lower masses. In Fig. 14 we plot the recovered distribution for the minimum BH mass, m min , that characterizes the primary mass scale above which black holes follow the parameterized power law distribution. The minimum mass is m min = 2 . 3 +0 . 27 -0 . 23 M glyph[circledot] , with an extremely sharp turn-on of δ m = 0 . 39 +1 . 3 -0 . 36 M glyph[circledot] . By contrast, if we remove the two low-mass events, we find a minimum BH mass of \n<!-- image --> \n/circledot \nm min = 5 . 0 +0 . 86 -1 . 7 M glyph[circledot] , which is consistent with a mass gap, and a broader turn-on of δ m = 4 . 9 +3 . 4 -3 . 2 M glyph[circledot] . It is the secondary masses, m 2 of these events that are in tension with the remainder of the population, as demonstrated in Fig. 14 where the secondary masses are shown by the shaded regions. A single minimum mass is imposed upon all BH, therefore the secondary masses of low-mass or asymmetric binaries have the strongest impact on our inference of m min . \nThese analyses imply two key results about the compact binary population. First, the binary black hole population excluding highly asymmetric systems such as GW190814 is well-defined, and the analyses carried out in this section are well-suited to characterizing the bulk of the BBH population. Second, the existence of GW190814 implies the existence of a subpopulation of highly asymmetric binaries, disconnected from the BBH population but potentially connected to the recently-identified population of NSBH.', 'VII. SPIN DISTRIBUTION OF BLACK HOLES IN BINARIES': 'Compared to our previous work [11], we find two key new conclusions for black hole spins: that the spin distribution broadens above 30 M glyph[circledot] , and that the mass ratio and spin are correlated. Adopting previous coarsegrained models, we find consistent conclusions as our analysis of GWTC-2; notably, we still conclude that a fraction of events probably have negative χ eff . \nThe component spins of binary black holes may offer vital clues as to the evolutionary pathways that produce merging BBHs [180-187]. The magnitudes of \nBBH spins are expected to be influenced by the nature of angular momentum transport in stellar progenitors [173, 188, 189], processes like tides [181, 190, 191] and mass transfer that operate in binaries, and the environment in which the binary itself is formed. Their directions, meanwhile, may tell us about the physical processes by which binaries are most often constructed; we expect BBHs born from isolated stellar evolution to possess spins preferentially aligned with their orbital angular momenta, while binaries that are dynamically assembled in dense environments are predicted to exhibit isotropically oriented spins [183, 185]. \nFigure 15 illustrates our constraints on the component spin magnitudes (left) and spin tilts (right) of BBHs under the Default spin model. Using GWTC-3, we make similar conclusions regarding the spin magnitude distribution as made previously with GWTC-2. In particular, spin magnitudes appear concentrated below χ i glyph[lessorsimilar] 0 . 4, with a possible tail extending towards large or maximal values. Our understanding of the spin tilt distribution, in contrast, has evolved with the addition of new BBHs in GWTC-3. As in GWTC-2, we again exclude the case of perfect spin-orbit alignment (corresponding to ζ = 1 and σ t = 0). With GWTC-3, however, we more strongly favor a broad or isotropic distribution of spin tilts. This shift is seen in the right-hand side of Fig. 15: whereas the cos θ distribution inferred from GWTC-2 was consistent with tilts concentrated preferentially around cos θ = 1, evidence for this concentration is now diminished, with O3b results preferring a flatter distribution across cos θ . \nFigure 16 illustrates our updated constraints on the χ eff and χ p distributions under the Gaussian spin model. As above, our previous results obtained with GWTC-2 are shown in blue, while black curves show our updated measurements with O3b. Measurement of the χ eff distribution with GWTC-2 suggested an effective inspiral spin distribution of non-vanishing width centered at χ eff ≈ 0 . 05, while the χ p distribution appeared incompatible with a narrow distribution at χ p = 0, bolstering the conclusion above that the BBH population exhibits a range of non-vanishing spin-tilt misalignment angles. These conclusions are further strengthened when updating our analysis with GWTC-3. We again infer a χ eff distribution compatible with small but non-vanishing spins, with a mean centered at 0 . 06 +0 . 04 -0 . 05 . Our updated constraints on the effective precessing spin distribution reaffirm the need for non-vanishing χ p among the BBH population. The χ p measurements made previously with GWTC-2 were consistent with both a broad underlying distribution or a very narrow distribution centered around χ p ≈ 0 . 3; this latter possibility is the source of the apparent jaggedness seen in the GWTC-2 result. We draw similar conclusions with GWTC-3, finding that χ p measurements can be explained either by a broad distribution centered at χ p = 0, or a narrow distribution centered at χ p ≈ 0 . 2. If we include GW190814 in our sample (which is otherwise excluded by default from our BBH analyses) support for this second mode is dimin- \nro-centered χ p distribution with standard deviation 0 . 16 +0 . 15 -0 . 08 . \nIn addition to the distributions of effective inspiral spins and component spins χ 1 and χ 2 associated with the more and less massive components of BBHs, respectively, we also explore the distributions of the more and less rapidly spinning components among the BBH population [192]. For a given binary, we define χ A = max | χ | ( χ 1 , χ 2 ) and χ B = min | χ | ( χ 1 , χ 2 ) as the component spins with the larger and smaller magnitudes, respectively. As discussed in Sec. IX, some models for stellar evolution and explosion predict that isolated black holes are born effectively non-rotating and that binary black hole systems primarily acquire spin through tidal spin-up of the secondary component by the first-born (non-spinning) black hole. If this is the case, then we expect to observe a non-vanishing distribution of χ A but a distribution of χ B concentrated at or near zero. Figure 17 shows the resulting distributions of these spinsorted magnitudes χ A (blue) and χ B (green), as implied by the Default model constraints on component spin magnitudes and tilt angles. Light and dark shaded regions show 50% and 90% credible bounds on each parameter, while the dark lines trace the expectation value of p ( χ ) as a function of spin-sorted χ . The χ A distribution, by definition, is concentrated at larger values than the peak seen in Fig. 15 (at χ ≈ 0 . 2). Across the BBH population, these more rapidly spinning components exhibit a distribution that likely peaks near χ A ≈ 0 . 4, with 1st and 99th percentiles at 0 . 07 +0 . 05 -0 . 03 and 0 . 8 +0 . 08 -0 . 08 , respectively. Less rapidly spinning components, meanwhile, are centered at or below χ B glyph[lessorsimilar] 0 . 2, with 99% of values occurring below 0 . 54 +0 . 09 -0 . 08 . \nOne significant question explored in our previous study [11] was the degree to which BBHs exhibit extreme spinorbit misalignment, with tilt angles exceeding θ ≥ 90 · and thus negative effective inspiral spins. Such steeply tilted spins are unlikely for BBH formation from isolated stellar progenitors [193], and hence would serve as a strong indicator of dynamical interaction during BBH evolution. Our GWTC-2 study [11] interpreted the results of the Default and Gaussian spin analyses as indicating the presence of extremely misaligned spins. As seen in Fig. 15, the component spin-tilt distribution is non-vanishing below cos θ = 0. Similarly, in Fig. 16 the χ eff distribution has significant support at χ eff < 0. To check whether this requirement for negative χ eff was a true feature of the data or an extrapolation of the Gaussian population model (which assumes the existence of extended tails), we extended the Gaussian model to truncate the effective inspiral spin on the range χ eff , min ≤ χ eff ≤ 1 (rather than -1 ≤ χ eff ≤ 1) and hierarchically measured the lower truncation bound χ eff , min . We found χ eff , min < 0 at 99 . 1% credibility, concluding that the data required the presence of negative effective inspiral spins. We obtain consistent results if we perform an identical check with GWTC-3; Fig. 18 illustrates our updated posterior on χ eff , min , now inferred to be negative \n<!-- image --> \nFIG. 15. The distributions of component spin magnitudes χ ( left ) and spin-orbit misalignment angles θ ( right ) among binary black hole mergers, inferred using the Default component spin model described further in Sect. B 2 a; e.g., both spin magnitudes are drawn from the same distribution. In each figure, solid black lines denote the median and central 90% credible bounds inferred on p ( χ ) and p (cos θ ) using GWTC-3. The light grey traces show individual draws from our posterior distribution on the Default model parameters, while the blue traces show our previously published results obtained using GWTC-2. As with GWTC-2, in GWTC-3 we conclude that the spin magnitude distribution peaks near χ i ≈ 0 . 2, with a tail extending towards larger values. Meanwhile, we now more strongly favor isotropy, obtaining a broad cos θ i distribution that may peak at alignment (cos θ i = 1) but that is otherwise largely uniform across all cos θ . \n<!-- image --> \n<!-- image --> \nFIG. 16. Left panel : Inferred distribution of χ eff for our latest full analysis in black. For comparison, the blue distribution and interval shows our inferences derived from GWTC2. Right panel : Corresponding result for χ p . While both panels in this figure are derived using the Gaussian spin model, we find similar conclusions with the other spin models used to analyze GWTC-2. \n<!-- image --> \nat 99 . 7% credibility. \nThis interpretation was challenged in [194] and [195], which argued that no evidence for extreme spin misalignment exists if BBH spin models are expanded to allow the existence of a secondary subpopulation with vanishingly small spins. Other avenues of investigation are also in tension with the identification of extreme spin-orbit misalignment. When the χ eff distribution is allowed to correlate with other BBH parameters, like the binary mass ratio (see Sec. VII B), evidence for negative χ eff values diminishes [196]. Motivated by the concerns raised in [194] and [195], we repeat our inference of χ eff , min but under an expanded model that allows for a narrow subpopulation of BBH events with extremely small effective \ninspiral spins: \np ( χ eff | µ eff , σ eff , χ eff , min ) = ζ bulk N [ χ eff , min , 1] ( χ eff | µ eff , σ eff ) +(1 -ζ bulk ) N [ -1 , 1] ( χ eff | 0 , 0 . 01) . \n(18) \nHere, ζ bulk is the fraction of BBHs in the wide bulk population, truncated above χ eff , min , while (1 -ζ bulk ) is the fraction of events residing in the vanishing spin subpopulation, which formally extends from -1 to 1. When repeating our inference of χ eff , min under this expanded model, our data still prefer a negative χ eff , min but with lower significance. As seen in Fig. 18, we now infer that χ eff , min < 0 at 92 . 5% credibility. This expanded model allows us to additionally investigate evidence for the exis- \nFIG. 17. Distribution of magnitudes of the most ( χ A ) and least ( χ B ) rapid component spin among BBHs in GWTC3. Traces show individual draws from our posterior on the spin population under the Default model, while dark curves bound 90% credible bounds on p ( χ A ) and p ( χ B ). \n<!-- image --> \nof a sub-population of BBHs with vanishingly small spins. GWTC-3 prefers but does not require such a subpopulation to exist. We measure ζ bulk = 0 . 54 +0 . 36 -0 . 26 , with ζ bulk > 0 . 2 at 99% credibility, but also find that our posterior remains consistent with ζ bulk = 1.', 'A. Spin distribution consistent as mass increases': 'Our previous analysis adopted the same spin distribution at all masses. The spins of low-mass binaries dominate the reconstructed spin distribution. However, the binaries with the most extreme values of spins have heavier masses: observations GW170729, GW190517, GW190519, GW190620, GW190706, GW190805, and GW191109, constitute 70% of the binaries with moderate to high spins. This preponderance of massive binaries with large spin suggests a one-size-fits-all approach might not fully capture how well we can predict black hole spins given their mass ; conversely, this preponderance can also reflect the increased impact on our search sensitivity at the highest masses. Too, astrophysical formation scenarios often predict correlations between mass and spin, both from isolated and dynamical formation [173, 197]. Using the FM model for the aligned spin components we reconstruct the trend of | s z | versus mass. Figure 19 shows the aligned spin magnitude distribution versus binary chirp mass. At low masses, the aligned spin is consistent with (and well constrained to be close to) zero, (i.e., maximum aligned spin magnitude averaged over chirp masses 30 M glyph[circledot] or less is 0 . 38 at 90% credibility). At heavier masses, the aligned spin is still consistent with zero, albeit with larger dispersion (i.e., maximum aligned spin magnitude averaged over chirp masses 30 M glyph[circledot] or more is 0 . 5 at 90% credibility). This trend is qualitatively consistent with the relative proportion of \nFIG. 18. Cumulative probabilities on the minimum truncation bound on the χ eff distribution as inferred using GWTC-2 and GWTC-3. When modeling the effective inspiral spin distribution as a Gaussian truncated on χ eff , min ≤ χ eff ≤ 1, we inferred using GWTC-2 that χ min , eff < 0 at 99 . 1% credibility, and hence that the data support the existence of BBH mergers with negative effective inspiral spins. Using GWTC3, this same analysis more strongly infers that χ min , eff < 0, now at 99 . 7% credibility. As discussed further below, evidence for negative effective inspiral spins is diminished under an expanded model that allows for a subset of BBHs to possess vanishing effective inspiral spins. When instead modeling the χ eff distribution as a mixture between a broad Gaussian and a narrow Gaussian sub-population centered at χ eff = 0 (e.g., the second consistent with zero spin), we infer χ min , eff < 0 at 92 . 5% credibility. \n<!-- image --> \nevents versus chirp mass: very few observations have high chirp masses, providing relatively little leverage to constrain spins. At high chirp masses, the spin distribution is poorly constrained by only a handful of measurements, closer to our broad prior assumptions, in contrast to the better-constrained distribution at low mass. We have no evidence to support or refute a trend of aligned spin with chirp mass. \nFigure 19 suggests aligned spin magnitude remains constrained to be close to zero independently of the most well-identified peaks in the mass distribution, contrary to what would be expected from hierarchical formation scenarios for these peaks [110, 197-201].', 'B. High spin correlates with asymmetric binaries': "BBHs may exhibit an anti-correlation between their mass ratios and spins, such that binaries with q ∼ 1 favor effective inspiral spin parameters near zero, while binaries with more unequal mass ratios exhibit preferentially positive χ eff values [196]. To evaluate the degree to which q and χ eff are (or are not) correlated, following prior work [196] we adopt a Gaussian model for the χ eff distribution with a mean and standard deviation that are \nFIG. 19. The dependence of aligned spin magnitude on the chirp mass. The light/dark shaded regions are the aligned spin magnitude at a credibility 90%/50%. The distribution is consistent with small values for lower chirp mass binaries, however, the spin magnitude is less tightly constrained for chirp masses of 30 M glyph[circledot] and higher. \n<!-- image --> \nFIG. 20. Posterior constraints on the mean (top) and standard deviation (bottom) of the χ eff distribution as a function of mass ratio q . At 97 . 5% credibility, we find that the mean of the χ eff shifts towards larger values for more unequal mass systems. The grey region in the lower panel shows the area artificially excluded by our prior on the parameters σ 0 and β ; see Eq. (20). \n<!-- image --> \nFIG. 21. Posteriors on the mass ratios and effective inspiral spins of BBHs in GWTC-3, reweighted to a populationinformed prior allowing for a correlation between q and χ eff . We infer that the mean of the BBH χ eff distribution shifts towards larger values with decreasing mass ratios. Accordingly, reweighted events shift considerably, such that events with q ∼ 1 contract about χ eff ≈ 0 while events with q < 1 shift towards larger effective inspiral spins. \n<!-- image --> \nallowed to evolve with q : \np ( χ eff | q ) ∝ exp [ -( χ eff -µ ( q )) 2 2 σ 2 ( q ) ] , (19) \nwith \nµ ( q ) = µ 0 + α ( q -1) (20a) \nlog 10 σ ( q ) = log 10 σ 0 + β ( q -1) . (20b) \nThe new hyperparameters α and β measure the extent to which the location or width of the χ eff distribution changes as a function of mass-ratio. \nWe repeat hierarchical inference of the BBH population, adopting the fiducial model for the primary mass and redshift distribution. At 97 . 5% credibility, we constrain α < 0, indicating that more unequal-mass binaries preferentially possess larger, more positive χ eff . Figure 20 illustrates our constraints on the mean and standard deviation of the χ eff distribution as a function of mass ratio. Each light trace represents a single sample from our hyperposterior, and the solid black lines denote the median values and central 90% bounds on µ ( q ) and σ ( q ) at a given value of q . If we adopt these hierarchical results as a new, population-informed prior, Fig. 21 shows the resulting reweighted posteriors for the BBHs among GWTC-3. Each filled contour bounds the central 90% region for a given event in the q -χ eff plane, while black points mark events' one-dimensional median q and χ eff measurements.", 'VIII. COMPARISON WITH OTHER GW CATALOGS': 'In this paper, we have presented population inferences based upon events identified by the LIGO Scientific, Virgo and KAGRA Collaborations in data taken by the Advanced LIGO and Advanced Virgo instruments during their first three observing runs [1, 2]. We have imposed a FAR threshold of < 0 . 25 yr -1 across all analyses incorporating NS binaries and a lower threshold FAR < 1 yr -1 for BBH analyses. This excludes several events which pass the threshold of p astro > 0 . 5 for inclusion in GWTC-3 . In addition, a number of analyses of the public GW data from O1, O2 and O3a [24, 66-69, 71] have identified additional candidate binary merger events. In the remainder of the paper, we have restricted the primary analysis to events included in GWTC-3. The overriding reason for this is that differences in the analysis methods prevent a detailed evaluation of search sensitivity, as described in Section III, which is critical to interpreting the population. In this Section, we investigate the consistency of the remaining GWTC-3 events and additional non-GWTC events with the population models inferred in this paper. \nFor concreteness, when referring to results reported by external groups, we include all events identified as GWs in their catalogs. In O1, there is one additional event, GW151216 identified in [68]. The additional events from O2 are GW170304, GW170425, and GW170403 which are identified in [69, 71], and GW170121, GW170202, and GW170727 which were also then independently found in open gravitational-wave catalog two (2-OGC) [67]. In O3, we include 16 additional events. These include GW190916 200658 and GW190926 050336 which were originally identified in open gravitational-wave catalog three (3-OGC) [24] and independently identified in Gravitational Wave Transient Catalog 2.1 (GWTC-2.1) [3]; GW190403 051519, GW190426 190642 and GW190514 065416 which are included in GWTC-2.1 but have a FAR below our < 1 yr -1 \nthreshold; GW191113 071753, \nGW191126 115259, \nGW191204 110529, \nGW191219 163120, \nGW200208 222617, \nGW200210 092254, \nGW200220 061928, \nGW200220 124850, \nGW200306 093714, \nGW200308 173609, \nGW200322 091133 which are included in GWTC-3 but again have a FAR below our < 1 yr - 1 threshold. \nIn Figure 22, we show the additional gravitational wave events which were not included in the sample used in this paper. The additional events are broadly consistent with the population presented here although several events lie at the boundaries of the identified population. Specifically, two of the events have aligned spins that lie outside the inferred population. These are GW151216 with a mean χ eff = 0 . 82 and GW170403 with a mean χ eff = -0 . 58. The analysis in [71] used a prior which is constant in χ eff which is significantly different from the uniform in spin-magnitude prior used in the GWTC pa- \ns. A re-analysis of GW151216 and GW170403 [202] leads to inferred χ eff distributions which are more consistent with the population inferred here. Specifically, this gives χ eff = 0 . 5 +0 . 2 -0 . 5 for GW151216 and χ eff = -0 . 2 +0 . 4 -0 . 3 for GW170403. In addition, the sub-threshold events from GWTC-3 extend the distribution to both higher masses and higher mass ratios. However, only low significance events currently populate these regions. Additional observations in future runs will allow us to determine whether these low significance events are more likely spurious, or were the first hints of a broader population in the mass space. \nWith regard to events potentially containing NS, GWTC-3 contains several candidates that do not satisfy our FAR < 1 / yr threshold but do have m 2 potentially consistent with NS masses, namely GW191219 and GW200210. Both events are inferred to have highly asymmetric masses and could possibly be an indication of additional NSBH sources, or asymmetric BBH similar to GW190814. Further observations in future runs will again allow us to investigate these interesting regions of the binary parameter space in greater detail.', '1. Mass distribution': 'The statistical distribution of BH source properties such as their mass, spin and redshift can be used to probe the astrophysics of BH binary formation and evolution [94, 110, 113, 173, 180, 184, 199, 201, 203-214]. The analysis performed in Sec. VI has identified structures in the mass distribution of BBHs that go beyond a standard power-law model and can help to shed light on formation processes. These features were previously identified in [47], but we are now more confident that they are statistically significant (see Sec. VI). \nThe underlying mass distribution of BBHs inferred in this paper peaks at a primary mass ∼ 10 M glyph[circledot] , withthe majority of BBHs having a primary BHs with a mass lower than this value (e.g., see Fig. 11). Formation in globular clusters has been long recognized as an important channel for merging BBHs [215-223]. In this scenario, BBHs are assembled during three body dynamical interactions in a low metallicity environment. The resulting BH mass distribution is generally predicted to peak at > 10 M glyph[circledot] . Three recent studies of globular cluster formation find that the BBH merger rate is severely suppressed where we observe a peak: one study [159] finds that the BBH merger rate is severely suppressed below about m glyph[similarequal] 13 M glyph[circledot] with a corresponding realistic merger rate at this mass value of ∼ 0 . 5 Gpc -3 yr -1 M -1 glyph[circledot] (see their Fig. 2); another recent study [224] finds similar results, with a peak in their mass distribution at about m glyph[similarequal] 1520 M glyph[circledot] (see their Fig. 5); a third analysis [217] finds the \n<!-- image --> \n<!-- image --> \nFIG. 22. The measured properties of the BBH candidates not included in the population study presented in this paper (shaded regions), compared to the inferred population from the PP model presented in Section VI A (black contours). These include both events which fall below our FAR threshold as well as events identified by other groups. The events are color coded based upon the search which first identified them: catalogs from O1 and O2 [68, 69, 71] in red, events from open gravitational-wave catalog three (3-OGC) (which incorporates events in O1-O3a) [24] in green, and from GWTC-3 with FAR > 1 yr -1 threshold in blue. \n<!-- image --> \npeak at m glyph[similarequal] 20 M glyph[circledot] . Taking these results at face value, the inferred high merger rate of sources with glyph[lessorsimilar] 10 M glyph[circledot] may suggest that globular clusters contribute subdominantly to the detected population. Dynamical formation in young clusters is also disfavored to explain the whole BH population at m glyph[similarequal] 10 M glyph[circledot] because lighter BHs are ejected by supernova kicks and do not participate to the dynamical evolution of the cluster [225-227]. \nGalactic nuclei can produce a BBH population with a much wider mass spectrum than both young and globular clusters [228-234]. Because of their high metallicities and escape velocities, nuclear star clusters can form and retain a significant number of lighter BHs, which can then pair and merge. BBH formation near an AGN disk can produce a significant population of BBH mergers with a wide mass spectrum [167, 197, 201, 235, 236]. In such scenarios, the observed low mass overdensities without counterparts in spin could be reflections of supernova physics; by contrast, in these hierarchical formation models no evident mechanism can impart them without a corresponding signature in spin. If the BBHs are formed near an AGN disk, this process might select heavier BHs, hardening the BBH mass function and driving the peak of the mass distribution towards values higher than observed [237]. \nIsolated binary evolution models often predict a peak near m glyph[similarequal] 10 M glyph[circledot] [173, 238-241]. Recent population models find component masses of merging BBHs that peak at 8-10 M glyph[circledot] and come from ∼ 20-30 M glyph[circledot] progenitors [173, 175, 240]. The overall merger rate normalization of the peak remains, however, poorly constrained. Moreover, the peak of the mass distribution can shift significantly depending on the adopted supernova, natal kick, mass transfer, and wind prescriptions, and star formation history of the Universe [32, 166, 175, 227, 239, 242, 243]. \nThe analysis in Section VI suggests two additional peaks in the mass distribution at m ∼ 17 M glyph[circledot] and at \n35 M glyph[circledot] . The three most significant mass peaks are therefore separated by roughly a factor of two from each other [47]. Assuming these peaks exist, an explanation consistent with our constraints on BH spins is that they originate either from the initial BH mass function, or that they are produced by different populations formed by separate physical processes or formation channels. \nThe other feature of the inferred BH mass distribution that was shown in our analysis is the apparent lack of truncation at m ∼ 40 M glyph[circledot] , which confirms our results based on GWTC-1 and GWTC-2 [2, 25]. A mass gap between approximately 50 +20 -10 M glyph[circledot] and ∼ 120 M glyph[circledot] is predicted by stellar evolution models as the result of the pair instability process in the cores of massive stars [147153]. However, due to our limited knowledge of the evolution of massive stars, the formation of BHs heavier than ∼ 40 M glyph[circledot] from stellar collapse cannot be fully excluded [153, 173, 244-247]. The location of the mass gap is sensitive to the uncertain 12 C( α, γ ) 16 O reaction rate, which governs the production of oxygen at the expense of carbon [152, 246, 248]. Moreover, BHs formed from progenitor stars at low metallicities ( Z/Z glyph[circledot] glyph[lessorsimilar] 0 . 1) might avoid all together the mass limit imposed by pairinstability [178, 244, 247]. The lack of a sharp truncation at high masses might indicate a dynamical process, such as the hierarchical merger of BHs [110, 154, 198200, 224, 229, 234, 249-253] or stars [254-257] in dense clusters or in the gaseous disk surrounding a massive BH [213]. In a hierarchical scenario we would expect the more massive BHs to also have the larger spins [199, 200]. While we do observe such a mass-spin correlation above m ∼ 40 M glyph[circledot] (Fig. 19), the binaries with a signature that χ eff is not zero all prefer χ eff > 0 (see Table IV), while hierarchical formation in dynamical environments would lead to isotropically oriented spins. BHs can also increase their birth mass beyond the pair-instability mass gap through the efficient accretion of gas from a stel- \nn or from a surrounding gaseous disk [258262]. Highly coherent accretion on one of the BHs could also explain the negative correlation between χ eff and q shown in Fig. 20, although accretion in gas-free scenarios should be highly super-Eddington in order to impart significant spin [258, 262]. Alternatively, primordial BHs can have masses above the pair-instability mass threshold, although this most likely requires efficient accretion before the reionization epoch in order not to violate current constraints [263].', '2. Redshift distribution': 'In Section VI we showed that the BBH merger rate increases with redshift, as (1 + z ) κ , with κ ∼ 3. Although error bars are large, current data prefer a model in which the merger rate evolves steeply with redshift and at a rate that is consistent with the growth in star formation. For binary formation in the field, the predicted value of κ is sensitive to the assumed efficiency of common envelope ejection: values between κ = 0 . 2 and 2 . 5 are all possible, although relatively small values κ ∼ 1 are preferred [160, 166, 175, 227, 264]. Delay times in the field are also dependent upon stability of mass transfer, e.g., [265]. Similarly, κ glyph[lessorsimilar] 2 is often found in models of BBHs formation in open and young clusters [225, 266]. Dynamical formation in globular clusters predicts κ glyph[lessorsimilar] 2 [164, 267], e.g., [159] find κ = 1 . 6 +0 . 4 -0 . 6 , and show that the most important parameter affecting the value of κ in the globular cluster scenario is the initial cluster half-mass density, ρ h , while uncertainties in other model parameters (e.g., natal kicks, black hole masses, metallicity) have a small effect. Only models in which globular clusters are formed with a high half-mass density, ρ h > 10 5 M glyph[circledot] pc -3 , lead to κ glyph[greaterorsimilar] 2 [159]. While uncertainties are large, improved constraints on the merger rate evolution have the potential to unveil important information about the physics of massive binaries [96, 173, 239, 264] and the initial conditions of clusters across cosmic time [164, 208, 267].', '3. Spin distribution': 'We observe evidence that the spin distribution both requires spin-orbit misalignment and also includes events with anti-aligned spins. BBHs with a large spin-orbit misalignment can be formed in dynamical environments such as globular, nuclear, and young star clusters, or active galactic nuclei [185, 198, 229, 230]. In these systems, two single BHs are paired together during a three body interaction and/or undergo a number of subsequent dynamical interactions before merging. Their spins have directions that are therefore uncorrelated with each other and with the orbital angular momentum of the binary, leading to an isotropic spin-orbit alignment [183, 185]. The evolution of BH spins in AGN disks depends on several uncertain factors, such as the importance of accre- \ntion and dynamical encounters, the initial spin orientation, and the efficiency of migration [201, 268]. If radial migration of BHs is inefficient, the distribution of χ eff skews toward higher values because scattering encounters that randomize spin directions become less frequent. On the other hand, efficient migration would imply more frequent dynamical encounters, producing a χ eff distribution centered around zero [268]. However, the dispersion of the χ eff distribution also increases characteristcally with mass, as with other hierarchical formation scenarios [110, 197, 201]. \nFormation from field binaries is thought to produce components with preferentially aligned spins [181, 183, 185]. Such an alignment, however, is not certain. In fact, all population models of isolated binaries customarily start with the stellar progenitor spins initially perfectly aligned with the orbital angular momentum of the binary. This assumption is made due to simplicity and partly because tidal interactions are thought to quickly remove any spin-orbit misalignment prior to BH formation. However, the observational evidence of close massive binaries with highly inclined spin axes suggests that close massive binaries can form with misaligned spins and that tides might not in all cases be able to realign the spins [269-272]. Moreover, a large spin-orbit misalignment can be produced if a binary is the inner component of a triple system [267, 273-276], where the tertiary component can be either a star, another BH, or even a massive BH [231, 277-279]. In this scenario, the secular gravitational interaction of the binary with an external companion can randomize the orbital plane of the binary. The complex precessional dynamics of the BH spins in triple systems also changes the spin orientation and lead to a distribution of χ eff peaked near zero, although with a marginal preference for aligned spins [273, 276]. A spinorbit misalignment can also be produced in field binaries by a stable episode of mass-transfer prior to the formation of the BHs [280] or by asymmetric mass and neutrino emission during core-collapse [193, 281], although these processes are unlikely to produce a large misalignment for a significant fraction of the population[186, 212, 282]. Weconclude that the presence of systems with misaligned spins is not in contradiction with a scenario in which the majority, if not all, BBHs form in the field of galaxies. On the other hand, the fact that the χ eff distribution is not symmetric around zero, if confirmed, can be used to rule out a model in which all BBHs are formed through dynamical encounters in star clusters [194, 204]. \nCorroborating our previous conclusion based on GWTC-2, we find that the BH population is typically described by small spins. Predictions for BH spin magnitudes vary depending on the assumptions about stellar winds and their metallicity dependence, tides, and are particularly sensitive to the efficiency of angular momentum transport within the progenitor star [188, 189]. If the stellar core remains strongly coupled to the outer envelope during the stellar expansion off the main sequence, then a significant amount of spin can be carried from the \ncore to the envelope. In this case, a BH formed from stellar collapse may be born with nearly zero spins. For formation in isolated binaries, this implies that the firstborn BH will essentially be a Schwarzschild black hole. The second-born BH can still form with significant spin as tidal interactions may realign and increase the spin of its stellar progenitor in between the two supernova explosions [283, 284]. If the binary undergoes chemically homogeneous evolution [285, 286], its components may both be tidally spun up to near break-up velocity, and keep this rotation rate throughout main sequence evolution, evolving into BHs with large and aligned spins. Black holes that form during the QCD phase transition in the early Universe will all have essentially zero natal spins [287, 288]. However, a significant spin can be attained through subsequent gas accretion [263]. Finally, if BBHs are formed or migrate within the accretion disk of a supermassive BH, they can accrete from the surrounding gaseous environment and spin up [268]. \nWe observe neither evidence for nor against an increase in spin magnitude for systems with higher masses [213, 289, 290] and more unequal mass ratios [196]. Current stellar evolution models suggest that larger BH masses should correlate with smaller spins because larger BHs originate from more massive stars which undergo more extensive mass loss, carrying away most of the angular momentum and producing BHs with small spins [173, 181, 284]. A consequence of this should be either a decrease in spin magnitude (and χ eff ) with mass above ∼ 20 M glyph[circledot] or no correlation, where the predicted trend depends on the specific stellar evolution models adopted [173, 181]. Predictions remain uncertain and are strongly dependent on modeling assumptions about the angular momentum transport within the star, spin dissipation during the supernova and the treatment of binary interaction prior to BH formation. If future observations identify such a trend, then an increase in spin magnitude with mass might suggest a hierarchical formation scenario. However, as mentioned above, this scenario seems currently at odds with the fact that binaries with more unequal mass ratios and massive components exhibit preferentially positive χ eff .', 'B. Implications for neutron stars': 'One result from gravitational wave (GW) observations is tension with the strong preference for 1 . 35 M glyph[circledot] mass objects which has been recovered in galactic BNS [41]. Instead, conservatively assuming all objects below the maximum neutron star mass are neutron stars, our unmodeled analysis of the lowest-mass compact objects is consistent with a broad unimodal Gaussian, allowing for highly asymmetric binaries. Our analysis of all individual low-mass (assumed NS) objects suggests a wide NS mass distribution, without the bimodal structure seen in the galactic NS population. The GW-observed population of low-mass mergers is still small. If this tension \npersists, however, several avenues exist to explain a discrepancy between Galactic and GW observations, including but not limited to additional formation channels for GW systems; strong observational selection effects, like those used to explain GW190425 [21, 291, 292] and the smaller body in GW200105; or the prospect that BHs form below the maximum NS mass. \nOur conclusions about the compact object mass spectrum in general and the mass spectrum of NS in particular will have substantial impact on the understanding of the stellar explosions that generate such compact objects [32, 293, 294] and the binary interactions that carry these objects towards merger, assuming a stellar origin for low-mass binary mergers. \nOur analyses show no evidence for or against the presence of a mass distribution feature closely corresponding to the maximum neutron star mass. Rather, the shape of the neutron star mass distribution, the existence of GW190814, and our results for the mass distribution for compact objects between 3 M glyph[circledot] and 7 M glyph[circledot] may instead suggest a continuous mass spectrum, albeit strongly suppressed above the masses of known NS. \nFortunately, the comparatively high prevalence of objects close to the maximum neutron star mass suggests that we will likely observe several objects near this region in the future, potentially providing several avenues to connect features in the NS mass distribution to fundamental nuclear physics. Our analysis of NS in merging binaries alone alone suggests the NS mass distribution extends to the maximum NS mass M TOV expected from the EOS. \nOur analyses are also consistent with both symmetric ( q glyph[similarequal] 0 . 8) and significantly asymmetric ( q < 0 . 8) binaries containing NS in BNS, and modestly ( q ∈ [0 . 5 , 0 . 8]) to highly ( q < 0 . 5) asymmetric binaries in NSBH. Compared to equal-mass mergers [295, 296], modestly asymmetric NS mergers (with either NS or BH counterparts) are potentially strong candidates for multimessenger counterparts [297], since an asymmetric merger can eject more mass [298], produce a larger remnant disk [299, 300], and potentially produce significant associated gamma ray burst emission [301-303]. For BNS, our analyses are consistent with a significant fraction of highly asymmetric events. For NSBH, the discovery of GW200105 and GW200115 demonstrate the existence of asymmetric binaries containing NS with a range of mass ratios. Based on these events, our inferences about the low-mass compact object distribution suggests that EMbright NSBH mergers could occur at a significant fraction of the overall NSBH rate. Generally, a broad mass ratio distribution suggests modestly more favorable prospects for electromagnetic follow-up observations. Conversely, a broad mass ratio distribution complicates simple efforts to interpret existing GW observations which were developed under the assumption that low-mass binary mergers are very frequently of comparable mass [304]. \nFinally, our analyses here leave GW190814 as an outlier both from BBH systems and from systems that con- \ntain a likely NS. Neither component of this binary has exceptional masses; for example, the secondary component could easily be produced from conventional supernova engines [32]. However, based on the merger rates versus mass identified in our study, this system (and the larger sample of high-mass-ratio binaries available at a lower threshold) may require a different formation pathway [230, 305, 306].', 'X. THE GW BACKGROUND FROM BINARY MERGERS': "The observation of binaries with masses in the NSBH range allows us to provisionally complete a census of the different classes of compact binaries that contribute to an astrophysical gravitational-wave background, assuming our existing surveys are sensitive to all relevant sources (i.e., not accounting for frequent subsolar mass mergers). We have previously predicted the contributions of BBH and BNS mergers to the gravitational-wave background, based on the compact binary population observed in GWTC-2 [172]. In Fig. 23, we update this forecast with our latest knowledge of the BBH and BNS population and the newly measured rate of NSBH mergers. \nThe shaded bands on the left side of Fig. 23 shows estimates of and uncertainties on the dimensionless energydensity spectra \nΩ( f ) = 1 ρ c d ρ dln f (21) \nof gravitational waves radiated by each class of compact binary. In Eq. (21), dρ is the gravitational-wave energy density per logarithmic frequency interval d ln f , while ρ c is the critical energy density required to close the Universe. We adopt the same model for the merger history of compact binaries used previously [172, 307], assuming that compact binary formation rate traces a metallicity-weighted star formation rate model [308-310] with a p ( t d ) ∝ t -1 d distribution of time delays t d between binary formation and merger. Time delays are restricted to 20 Myr ≤ t d ≤ 13 . 5 Gyr for BNS and NSBH mergers and 50 Myr ≤ t d ≤ 13 . 5 Gyr for BBHs [175, 311], with binary formation restricted to redshifts below z max = 10. The birth rate of BBH progenitors is further weighted by the fraction of star formation at metallicities Z < 0 . 1 Z glyph[circledot] [174, 312]. \nWithin Fig. 23, the stochastic energy-density due to BBHs has been marginalized over our uncertainty on both the local merger rate and mass distribution of the BBH population, as measured using the PP mass model. At 25 Hz, we estimate the energy-density due to BBHs to be Ω BBH (25 Hz) = 5 . 0 +1 . 4 -1 . 8 × 10 -10 . To estimate the contribution due to BNS systems, we adopt the simple rate measurement presented in Section IV A under a fixed mass distribution, and correspondingly assume a uniform distribution of neutron star masses between 1 \nand 2 . 5 M glyph[circledot] , giving Ω BNS (25 Hz) = 0 . 6 +1 . 7 -0 . 5 × 10 -10 . The contribution due to NSBH systems, meanwhile, is estimated using the BGP rate reported in Table II. For simplicity, we again assume a uniform distribution of neutron star masses between 1 and 2 . 5 M glyph[circledot] and a logarithmically uniform distribution of black hole masses between 5 and 50 M glyph[circledot] among NSBH mergers. Under these assumptions, we find Ω NSBH (25 Hz) = 0 . 9 +2 . 2 -0 . 7 × 10 -10 . \nThe blue band in the right side of Fig. 23 denotes the our estimate of the total gravitational-wave background due to the superposition of these three source classes; we expect a total energy-density of Ω(25 Hz) = 6 . 9 +3 . 0 -2 . 1 × 10 -10 . For comparison, the solid black curve marks our present sensitivity to the gravitational-wave background [172, 313]. Although our estimate for the background amplitude lies well below current limits, it may be accessible with future detectors, such as the planned 'A+' LIGO configuration.", 'XI. CONCLUSIONS': "The third LIGO-Virgo gravitational wave transient catalog (GWTC-3) [1] has increased our census of the population of compact mergers by nearly a factor of two, compared to our analysis of the first half of O3 [11]. We simultaneously employ all observations with FAR < 0 . 25 yr -1 to infer the merger rate versus both component masses across the observed mass spectrum. For NS, we find a broad mass distribution, extending up to 2 . 0 +0 . 3 -0 . 3 M glyph[circledot] , in contrast to the unimodal mass distribution observed for Galactic BNS. We find the BBH mass distribution is nonuniform, with overdensities at BH masses of 10 M glyph[circledot] and 35 M glyph[circledot] . These overdensities may reflect the astrophysics associated with generating coalescing binaries, potentially reflecting properties of stellar physics or astrophysical environments. These features may assist future applications of gravitational wave astronomy. As an example, these sharp features could be redshift-independent and, if so, used as standard candles for cosmology [314, 315]. We find the BH mass distribution exhibits an interval between 2.2 M glyph[circledot] and 6.1 M glyph[circledot] where merger rates are suppressed, which could be consistent with past X-ray observations suggesting a mass gap [28-31]. Our analysis lacks sufficient sensitivity to probe the structure of the mass distribution at the highest masses m 1 > 70 M glyph[circledot] in detail; however, so far, we find no evidence for or against an upper mass gap. \nWe find that observed BH spins are typically small (half less than 0 . 25). We still conclude that at least some of these spins exhibit substantial spin-orbit misalignment. We corroborate a correlation between BBH effective aligned inspiral spins and mass ratio. \nUsing parametric models to infer the distribution of BBH merger rate with redshift, we find the BBH merger rate likely increases with redshift; we cannot yet assess more complex models where the shape or extent of the mass distribution changes with redshift. \nFIG. 23. Forecast of astrophysical gravitational-wave background due to binary mergers following O3. ( Left ): The individual contributions expected from BNS, NSBH and BBH mergers. While uncertainties on the energy-density due to BNS and NSBH are due to Poisson uncertainties in their merger rates, our forecast for the stochastic background due to BBHs additionally includes systematic uncertainties associated with their imperfectly known mass distribution. ( Right ): Estimate of the total gravitational-wave background (blue), as well as our experimental current sensitivity (solid black) [172, 313]. For comparison, we additionally show the expected sensitivities of the LIGO-Virgo network at design sensitivity, as well as that of LIGO's anticipated 'A+' configuraton. \n<!-- image --> \nAnalyses presented in our previous work [11] and in a companion paper [316] employ coarse-grained models for the BBH population, smoothing over some of the subtle features identified above. We find that these coarsegrained models draw similar conclusions on current data as our previous studies; see Sec. VI A. Applications that focus on large-scale features of the mass distribution (e.g., the stochastic background, as described in Sec. X) only require these coarse-grained results. Nonetheless, the mass distribution remains a critical source of systematic uncertainty in any merger rate integrated over any mass interval, particularly in mass intervals with few observations. We specifically find the BNS and NSBH merger rates exhibit considerable uncertainty in the mass distribution, with relative merger rate errors within (and between) models far in excess of the expected statistical The Poisson error associated with the count of these events. These systematics propagate directly into our most conservative estimates for their merger rates. \nThe next GW survey could have a BNS detection range increased by approximately 15-40% [317]. Even without allowing for increased merger rates at higher redshift, the next survey should identify roughly 3 times more events of each class then used in this study, including several new events from the BNS and BHNS category. We continuously revise our assessment of future observing prospects [317].", 'ACKNOWLEDGMENTS': "This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. The authors also \ngratefully acknowledge the support of the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Netherlands Organization for Scientific Research (NWO), for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigaci'on (AEI), the Spanish Ministerio de Ciencia e Innovaci'on and Ministerio de Universidades, the Conselleria de Fons Europeus, Universitat i Cultura and the Direcci'o General de Pol'ıtica Universitaria i Recerca del Govern de les Illes Balears, the Conselleria d'Innovaci'o, Universitats, Ci'encia i Societat Digital de la Generalitat Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the National Science Centre of Poland and the European Union - European Regional Development Fund; Foundation for Polish Science (FNP), the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Social Funds (ESF), the European Regional Development Funds (ERDF), the Royal \nSociety, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the French Lyon Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique (FRS-FNRS), Actions de Recherche Concert'ees (ARC) and Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO), Belgium, the Paris ˆ Ile-de-France Region, the National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Korea, the Natural Science and Engineering Research Council Canada, Canadian Foundation for Innovation (CFI), the Brazilian Ministry of Science, Technology, and Innovations, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan, the United States Department of Energy, and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, INFN and CNRS for provision of computational resources. This work was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, JSPS Grant-inAid for Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Coreto-Core Program A. Advanced Research Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133 and 20H05639 , JSPS Grant-in-Aid for Transformative Research Areas (A) 20A203: JP20H05854, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National Research Foundation (NRF) and Computing Infrastructure Project of KISTI-GSDC in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the Ministry of Science and Technology (MoST) in Taiwan under grants including AS-CDA105-M06, Advanced Technology Center (ATC) of NAOJ, Mechanical Engineering Center of KEK. We would like to thank all of the essential workers who put their health at risk during the COVID-19 pandemic, without whom we would not have been able to complete this work.", 'Appendix A: Sensitivity estimation': "A key ingredient in Eqs. (3) and (4) is the detection fraction ξ (Λ), which estimates the fraction of systems that we expect to successfully detect from some prior volume that extends past our detector's reach. The detection fraction quantifies selection biases, and so it is critical to accurately characterize. For a population described by parameters Λ, the detection fraction is \nξ (Λ) = ∫ P det ( θ ) π ( θ | Λ)d θ. (A1) \nHere, P det ( θ ) is the detection probability: the probability that an event with parameters θ would be detected by a \nparticular search. The detection probability depends on the angular/sky position and orientation of the source binary, and crucially for our purposes, on the masses and redshift of a system, and, to a lesser degree, on the spins. \nGiven the non-ideal nature of the detector data, the variation in network sensitivity over time, and the complexity of both the signal waveforms and the search pipelines, an accurate estimate of P det ( θ ) and ξ (Λ) requires empirical methods, specifically the use of a large suite of simulated signals added to the data: injections. For analyses that focus on the BBH subpopulation in Section VI, we simulate compact binary signals from a reference BBH population and record which ones are successfully detected by the PyCBC , GstLAL or MBTA search pipelines. We omit the cWB search from our volume estimate, since at present any detection of a binary merger was corroborated by a detection in the remaining pipelines. In addition, we also simulate compact binary signals from reference BNS, NSBH and IMBH populations. These injections include binaries with component masses in the range 1-600 M glyph[circledot] , have spins that are isotropic in orientation and are uniform in comoving volume. Spins are drawn from a distribution that is uniform in the dimensionless spin magnitude up to a maximum of χ max = 0 . 998 for black holes and χ max = 0 . 4 for neutron stars. To control computational costs, the expected network signal to noise ratio (SNR) of each injection is computed using representative detector power spectral densitys (PSDs) for O3. Injections with expected SNR below 6 are assumed not to be detected, and are thus removed from the set analyzed by the search pipelines. A thorough description of the injections and their underlying probability distribution is available in [318]. These injections are then combined into a single dataset as a mixture model [1, 318] in order to assess sensitivity across the entire parameter space and subpopulations. Our analyses in Sec. IV make use of these injections to estimate sensitivity. \nUnlike previous synthetic simulation sets used in our population analysis following GWTC-2 [11], the injections used here model spins that are isotropically distributed in orientation and hence allow for orbital precession. Further, the maximum spin magnitude we assume for NS components, 0.4, is significantly larger than for previous injection sets used to estimate BNS merger rates [21]. That said, our injections have an effective χ eff distribution that is narrow and centered at 0 while analyses using BNS populations with small NS spins inherently have χ eff ≈ 0. Because the merger rate depends on spins primarily through the system's χ eff , the specific assumptions made about the spin distribution at low mass have modest impact on the inferred low-mass merger rate. \nFollowing [94, 95, 105, 319], the point estimate for Eq. (A1) is calculated using a Monte Carlo integral over found injections: \nˆ ξ (Λ) = 1 N inj N found ∑ j =1 π ( θ j | Λ) p draw ( θ j ) , (A2) \nwhere N inj is the total number of injections, N found are the injections that are successfully detected, and p draw is the probability distribution from which the injections are drawn. When using this approach to estimate sensitivity, we marginalize over the uncertainty in ˆ ξ (Λ) and ensure that the effective number of found injections remaining after population re-weighting is sufficiently high ( N eff > 4 N det ) following [105]. We also compute (and some analyses like MS employ) semi-analytic approximations to the integrated network sensitivity V T ( θ, κ ) = ∫ d t d z d V c / d z/ (1 + z ) 〈 P det ( θ, z ) 〉 (1 + z ) κ for fiducial choices of κ , appropriate to characterize sensitivity to a population with a fixed redshift evolution. \nFor the O3 observing period, we characterize the found injections as those recovered with a FAR below the corresponding thresholds used in population analyses described in this paper (1 per year and 1 per 4 years) in either PyCBC , GstLAL or MBTA . For the O1 and O2 observing periods, we supplement the O3 pipeline injections with mock injections drawn from the same distribution p draw above. For the mock injections, we calculate P det ( m 1 , m 2 , z, χ 1 ,z , χ 2 ,z ) according to the semianalytic approximation used in our analysis of GWTC2 [25], based on a network signal-to-noise ratio threshold ρ = 10 and representative strain noise power spectral densities estimated from data recorded during the O1 and O2 observing runs. We combine O1, O2 and O3 injection sets ensuring a constant rate of injections across the total observing time [320].", 'Appendix B: Population Model Details': "In this section we provide details about the lowdimensional parameterized population models described \nabove in Section III. Each subsection includes a table with a summary of the parameters for that model and the prior distribution used for each parameter. The prior distributions are indicated using abbreviations: for example, U(0 , 1) translates to uniform on the interval (0 , 1), LU(10 -6 , 10 5 ) translates to log-uniform on the interval 10 -6 , 10 5 , and N(0 , 1) translates to a Gaussian distribution with mean 0 and standard deviation of 1. \nUsing Monte Carlo summations over samples from each event's posterior distribution to approximate the integral in the likelihood (Eq. 4) results in statistical error in the likelihood estimates [105, 106]. In order to avoid including relics from to unconverged Monte Carlo integrals in the posterior distribution, we introduce a data-dependent constraint on the prior, determined by the number of effective samples used in the Monte Carlo integral. We define the effective number of samples as \nN eff = ( ∑ i w i ) 2 ∑ i w 2 i , (B1) \nwhere w i is the weight for the i th event in the Monte Carlo integral. \nFor the Truncated , Power Law + Peak , Power Law + Spline , Power law + dip + break , Default , Gaussian , and power-law population models, we only assign nonzero likelihoods to points in parameter space with an effective sample size of at least the number of observed events in our event list. This is similar to the convergence constraints we enforce when computing sensitivity (see A) [105].", 'a. Truncated mass model': 'Truncated mass model serves as the primary component for some of our mass models. The primary mass distribution for this model follows a power-law with spectral index α , and with a sharp cut-off at the lower end m min and the upper end of the distribution m max : \nπ ( m 1 | α, m min , m max ) ∝ { m -α 1 m min < m 1 < m max 0 otherwise , (B2) \nMeanwhile, the mass ratio q ≡ m 2 /m 1 follows a power-law distribution with spectral index β q \nπ ( q | β q , m min , m 1 ) ∝ { q β q m min < m 2 < m 1 0 otherwise . (B3) \nThe parameters for this model are summarized in Table V. For this model, as well as further mass models where a prior on the total merger rate is not specified, the rate prior is proportional to 1 /R , or equivalently to 1 /N in the notation of Eq.3-4. \nTABLE V. Summary of Truncated model parameters. \nTABLE VI. Summary of Power Law + Peak model parameters.', 'b. Power Law + Peak mass model': "This is equivalent to Model C from [25]. The primary mass distribution is a truncated powerlaw, with the addition of tapering at the lower mass end of the distribution and a Gaussian component: \nπ ( m 1 | λ peak , α, m min , δ m , m max , µ m , σ m ) = [ (1 -λ peak ) P ( m 1 | -α, m max ) + λ peak G ( m 1 | µ m , σ m ) ] S ( m 1 | m min , δ m ) . (B4) \nHere, P ( m 1 | -α, m max ) is a normalized power-law distribution with spectral index -α and high-mass cut-off m max . Meanwhile, G ( m 1 | µ m , σ m ) is a normalized Gaussian distribution with mean µ m and width σ m . The parameter λ peak is a mixing fraction determining the relative prevalence of mergers in P and G . Finally, S ( m 1 , m min , δ m ) is a smoothing function, which rises from 0 to 1 over the interval ( m min , m min + δ m ): \nwith \nS ( m | m min , δ m ) = 0 ( m<m min ) [ f ( m -m min , δ m ) + 1] -1 ( m min ≤ m<m min + δ m ) 1 ( m ≥ m min + δ m ) , (B5) \nf ( m ' , δ m ) = exp ( δ m m ' + δ m m ' -δ m ) . (B6) \nThe conditional mass ratio distribution in this model also includes the smoothing term: \nπ ( q | β, m 1 , m min , δ m ) ∝ q β q S ( qm 1 | m min , δ m ) . (B7) \nThe parameters for this model are summarized in Table VI.", 'c. Power Law + Spline mass model': 'The Power Law + Spline mass model explicitly applies a perturbation to a modified version of the fiducial Power Law + Peak model that does not include the Gaussian peak [45]. Let p ( m 1 | α, m min , m max , δ m ) be the modified Power Law + Peak model without the Gaussian, then primary mass distribution for the Power Law + Spline model is given as: \nTABLE VII. Summary of Power Law + Spline model parameters. \nTABLE VIII. Summary of Flexible mixtures model parameters. All rates are in Gpc -3 yr -1 . \np PS ( m 1 | α, m min , m max , δ m , { f i } ) = k p ( m 1 | α, m min , m max , δ m ) exp( f ( m 1 |{ f i } )) . (B8) \nAbove, k is a normalization factor found by numerically integrating p PS over the range of allowed primary masses, f ( m 1 |{ f i } ) is the perturbation function we model with cubic splines, and { f i } are the heights of the n knots from which f is interpolated. The n knot locations are fixed, spaced linearly in log m 1 space from 2-100 M glyph[circledot] . We additionally restrict the perturbations to converge to the underlying distribution at the boundary nodes by fixing both f 0 and f n -1 to be 0. We chose n = 20 to be the optimal number of knots for this analysis following the same procedure in [45], which adds a total of 18 additional parameters describing the perturbations to the underlying model. In addition to the primary mass, the conditional mass ratio distribution follows the same form as the Power Law + Peak model defined in Eq. (B7). For each mass distribution inference with the Power Law + Spline model, we simultaneously fit the spin distribution with the Default model and the redshift evolution of the merger rate with the Power Law evolution model. The parameters and chosen prior distributions for the Power Law + Spline model are summarized in Table VII.', 'd. Flexible mixtures model': 'The Flexible mixtures model, Vamana, predicts the population using a sum of weighted components. Each component is composed of a Gaussian, another Gaussian and a power-law to model the chirp mass, the aligned spins and the mass ratio respectively. The model is defined as \np ( M , q, s 1 z , s 2 z | λ ) = N ∑ i =1 w i G ( M| µ M i , σ M i ) G ( s 1 z | µ sz i , σ sz i ) σ M i ) G ( s 2 z | µ sz i , σ sz i ) P ( q | α q i , q min i , 1) , (B9) \nwhere G is the normal distribution and P is the truncated power-law. For the presented analysis we use N = 11 components. This choice maximises the marginal likelihood, however, the predicted population is robust for a wide range of N . For detailed description of this model see [49]. Flexible mixtures model uses a power-law to model the redshift evolution of the merger rate, as described in subsection B 3. The merger rate has a uniform-in-log distributed prior; the prior distributions for parameters in Eq. B9 are summarized in Table VIII. \nTABLE IX. Summary of Binned Gaussian process model parameters.', 'e. Binned Gaussian process model': "The Binned Gaussian process models the rate densities, m 1 m 2 d R i d m 1 d m 2 = n i , as a binned Gaussian Process where the index i denotes a particular bin in the two-dimensional log m 1 -log m 2 parameter space [118, 119]. The bin edges in the analysis presented in the paper are located at [1 , 2 , 2 . 5 , 3 , 4 , 5 , 6 . 5 , 8 , 10 , 15 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 100] M glyph[circledot] with the assumption that m 2 ≤ m 1 . The probabilistic model for the logarithm of the rate density in each bin is defined as \nlog n i ∼ N( µ, Σ) , (B10) \nwhere µ is the mean of the Gaussian process and Σ is the covariance matrix that correlates the bins. Each element of the covariance matrix Σ is generated using a squared-exponential kernel k ( x, x ') which is defined as \nk ( x, x ' ) = σ 2 exp ( -( x -x ' ) 2 2 l 2 ) . (B11) \nFor the specific analysis here we take x, x ' to be the bin centers in log m . The parameter σ models the amplitude of the covariances while l is a parameter that defines the length scales over which bins are correlated. The prior distribution chosen here for the length scale is a log-normal distribution with a mean that is the average between the minimum bin spacing \n∆ min ≡ min m 1 ,m 2 ∆log m (B12) \nand the maximum bin spacing \n∆ max ≡ max m 1 ,m 2 ∆log m (B13) \nwith a standard deviation of (∆ max -∆ min ) 4 . This constrains (at '2σ ' in the prior) the correlation length for the GP to lie between 'one bin' and 'all the bins.' For our analyses presented in the paper, the mean and standard deviation are -0 . 085 and 0 . 93 respectively. The Binned Gaussian process model assumes a redshift distribution such that the overall merger rate of compact binaries is uniform-in-comoving volume. The spin distributions for each component are isotropic in direction and uniform in the spin magnitude with a maximum spin of 0 . 998 for BHs and 0 . 4 for NSs; the prior distribution for the relevant parameters in Equations B10 and B11 is summarized in Table IX. \nf. Power law + dip + break model \nThe Power law + dip + break model explicitly searches for separation in masses between two subpopulations by employing a broken power law with a dip at the location of the power law break. As described in [108] and [109], the dip is modeled by a notch filter with depth A , which is fit along with the other model parameters in order to determine the existence and depth of a potential mass gap. No gap corresponds to A = 0, whereas A = 1 corresponds to precisely zero merger rate over some interval. Power law + dip + break also employs a low-pass filter at high masses to allow for a tapering of the mass spectrum, which has the effect of a smooth second break to the power law. \nThe PDB model assumes a merger rate that is uniform in comoving volume. It also assumes a spin distribution with isotropically oriented component spins and uniform component spin magnitudes. \nTABLE X. Summary of Power law + dip + break model parameters. The first entries describe the mass distribution parameters, and the last two entries describe the spin distribution parameters. \nThe joint mass distribution in this model has the form: \np ( m 1 , m 2 ) p ( m 1 ) p ( m 2 )( m 2 /m 1 ) β , \nglyph[lscript] ( m ) = 1 + ( m/m max ) η . \n∝ (B14) p ( m ) ∝ p pl ( m ) n ( m ) glyph[lscript] ( m ) , (B15) n ( m ) = 1 -A (1 + ( M gap low m ) η low )(1 + ( M gap high m ) η high ) , and (B16) 1 (B17) \nwhere p pl ( m ) is a broken power law with exponents α 1 between m min and M gap low and α 2 between M gap low and m max . The parameters for this model are summarized in Table X.", 'g. Neutron star mass models': 'The mass models adopted for the BNS and NSBH events in Sec. V assume a basic mass distribution that is common to all NSs, with random pairing into compact binaries. The basic mass distribution is taken to be either a power law or, inspired by the shape of the Galactic BNS mass distribution [39-41], a Gaussian. The BH mass distribution is fixed to be uniform between 3 and 60 M glyph[circledot] . The NS mass distribution analysis assumes definite source classifications for the events. Thus, the joint mass distribution takes the form \np ( m 1 , m 2 ) ∝ { p ( m 1 ) p ( m 2 ) if BNS U (3 M glyph[circledot] , 60 M glyph[circledot] ) p ( m 2 ) if NSBH , (B18) \nwith p ( m ) either a power law with exponent α , minimum mass m min and maximum mass m max , or a Gaussian with a peak of width σ at µ , plus sharp minimum and maximum mass cutoffs m min , m max . We call these models Power and Peak , respectively. Their hyper-parameters, and the choices for their prior distributions, are listed in Table XI. We additionally impose the constraint m min ≤ µ ≤ m max on the Peak model. Besides the flat m max prior described in the table, for the analyses excluding GW190814 we use a prior proportional to the cumulative distribution function of M max , TOV , i.e., p ( m max ) ∝ ∫ ∞ m max dM max , TOV p ( M max , TOV ). This enforces our expectation that the NS masses in the gravitational-wave population should not exceed M max , TOV . \nTABLE XI. Summary of Power and Peak NS mass model parameters. \nTABLE XII. Summary of Default spin parameters.', '2. Details of spin population models': "a. Default spin model \nThis model was introduced in [25]. Following [113], the dimensionless spin magnitude distribution is taken to be a Beta distribution, \nπ ( χ 1 , 2 | α χ , β χ ) = Beta( α χ , β χ ) , (B19) \nwhere α χ and β χ are the standard shape parameters that determine the distribution's mean and variance. The Beta distribution is convenient because it is bounded on (0,1). The distributions for χ 1 and χ 2 are assumed to be the same. Following [114], we define z i = cos θ i as the cosine of the tilt angle between component spin and a binary's orbital angular momentum, and assume that z is distributed as a mixture of two populations: \nπ ( z | ζ, σ t ) = ζ G t ( z | σ t ) + (1 -ζ ) I ( z ) . (B20) \nHere, I ( z ) is an isotropic distribution, while G t ( z | σ t ) is a truncated two-dimensional Gaussian, peaking at z = 0 (perfect alignment) with width σ t . The mixing parameter ζ controls the relative fraction of mergers drawn from the isotropic distribution and Gaussian subpopulations. The isotropic subpopulation is intended to accommodate dynamically assembled binaries, while G t is a model for field mergers. The parameters for this model and their priors are summarized in Table XII. Additional constraints to the priors on µ χ and σ 2 χ are applied by setting α χ , β χ > 1.", 'b. Gaussian spin model': 'In addition to the distribution of component spin magnitudes and tilts, we explore the distribution of the effective inspiral spin parameter χ eff and the effective precession spin parameter χ p . In particular, we wish to measure the mean and variance of each parameter, and so model the joint distribution of χ eff and χ p as a bivariate Gaussian: \nπ ( χ eff , χ p | µ eff , σ eff , µ p , σ p , ρ ) ∝ G ( χ eff , χ p | µµµ, ΣΣΣ) . (B21) \nThe mean of this distribution is µµµ = ( µ eff , µ p ), and its covariance matrix is \nΣΣΣ = ( σ 2 eff ρσ eff σ p ρσ eff σ p σ 2 p ) . (B22) \nThe population parameters governing this model and their corresponding priors are shown in Table XIII. Equation (B21) is truncated to the physically allowed range of each effective spin parameter, with χ eff ∈ ( -1 , 1) and χ p ∈ (0 , 1). All results in the main text using the Gaussian model are obtained while simultaneously fitting for the \nTABLE XIII. Summary of Gaussian spin parameters. The χ eff , min and ζ parameters appear only in variants of the Gaussian model, as discussed below. \nBBH mass distribution, assuming the Power Law + Peak model, and the evolving redshift distribution model in Appendix B3 below. \nTwo variants of this model are additionally discussed in Sect. VII. In the first, Eq. (B21) is modified such that the effective inspiral spin parameter is truncated not on the interval ( -1 , 1), but on ( χ eff , min , 1), where χ eff , min is another parameter to be inferred by the data. The second variant, inspired by [194] and [195] and defined in Eq. (18), alternatively treats the χ eff distribution as a mixture between a bulk component with a variable mean and width and a narrow zero-spin component centered on χ eff = 0. In this second variant, we measure only the marginal χ eff distribution, implicitly assuming that the remaining spin degree of freedom are distributed uniformly and isotropically. As χ eff is the primary spin measurable, we do not expect this implicit prior to have a strong effect.', '3. Redshift Evolution Model': 'The power-law redshift evolution model parameterizes the merger rate density per comoving volume and source time as [96] \nR ( z ) = R 0 (1 + z ) κ , (B23) \nwhere R 0 denotes the merger rate density at z = 0. This implies that the redshift distribution is \nd N d z = C d V c d z (1 + z ) κ -1 , (B24) \nwhere d V c / d z is the differential comoving volume, and C is related to R 0 by \nR 0 = C d V c d z ) [∫ z max 0 d V c d z (1 + z ) κ -1 ] -1 . (B25) \nWe adopt z max = 2 . 3 as this is a conservative upper bound on the redshift at which we could detect BBH systems during O3, for both detection thresholds used in this work. We employ a uniform prior on κ centered at κ = 0. We take a sufficiently wide prior so that the likelihood is entirely within the prior range, κ ∈ ( -6 , 6).', 'a. Multi source model': 'The Multi source model, introduced in [321], extends the MultiSpin BBH model introduced in [11] to include additional subpopulations for BNS and NSBH systems. Each subpopulation (two for BBH, one for BNS, and one for NSBH) is assumed to have an independent rate parameter. \nThe BBH subpopulation is itself a mixture of two subpopulations, i. a power law mass distribution m -α 1 q β truncated to a range [ m min , BBH , m max , BBH ] which is inferred from the data, and ii. a Gaussian in ( m 1 , m 2 ) with independent mean and standard deviation parameters µ m 1 , BBH , µ m 2 , BBH , σ m 1 , BBH , σ m 2 , BBH . Both subpopulations, and both binary components within them follow independent Default spin models, with ζ ≡ 1. \nTwo more bivariate Gaussians in m 1 , m 2 are used to model BNS and NSBH. The BH in NSBH follow a Gaussian mass distribution, with free parameters µ m, NS BH , σ m, NS BH . As with BBH, these BH follow an independent Default \nspin model with ζ ≡ 1. All three types of NS (two in BNS and one in NSBH) follow the same Gaussian mass distribution, with free parameters µ m, NS , σ m, NS , m max , NS (the minimum mass is assumed to be 1 M glyph[circledot] ). Each type of NS follows an independent Default spin model. To stay within astrophysically plausible spins, the magnitude distributions are scaled down to χ max = 0 . 05. Since NS spin tilts are not well measured, we set ζ ≡ 0, assuming they are isotropic, which has the effect of not wasting any samples from parameter estimation. \nIn addition to any mass cutoffs mentioned above, all BHs component masses are assumed to lie on the range [2 , 100] M glyph[circledot] , with those in NSBHs further restricted to [2 , 50] M glyph[circledot] due to our limited injections. \nPriors for all parameters are given in Table XIV. \nTABLE XIV. Summary of Multi source model parameters. All rates are in Gpc -3 yr -1 , and all masses in M glyph[circledot] . Rate, mass, and spin hyperparameters are separated by horizontal lines.', 'Appendix C: Validation studies': 'We employ several methods to validate our calculations, notably including comparing results from multiple independent analyses; reproducing previous work through O3a [11]; assessing the sensitivity of our results to threshold choices (changing from 1 yr -1 to 0 . 2 yr -1 for BBH; or from 0 . 25 yr -1 to 1 yr -1 for analyses containing NS); and performing posterior predictive checks as in our analysis of GWTC-2 [11]. Though these specific technical checks will not be described here, some of these checks can be reproduced with the data release associated with this paper. \nBelow, we describe additional validation studies we have performed to assess whether our results for merger rates are sensitive to the choice of threshold; waveform systematics; or updates to our sensitivity model.', '1. Effects of the Spin Distribution on Merger Rates Across All Masses': 'In principle, the mass, spin, and redshift distributions of binaries should be fit simultaneously in order to avoid systematic biases in inferred distribution parameters caused by correlations between measurements of these intrinsic parameters [106, 322-328]. However, fixing one or more of these distributions to a realistic form typically introduces biases that have little impact on the parameters of interest. We therefore seek to determine if our choice to fix the spin distribution for the PDB and BGP models has introduced any significant biases in our inference of the mass distribution and overall merger rate. \nWe compare the PDB analysis presented in Section IV with an analysis that utilizes the same mass and redshift distribution but fits for the spin distribution rather than fixing it to one that assumes isotropic and uniformly distributed component spins. For this, we apply the Default [114] spin model described in Section III C 2 and Appendix B2a. The resulting fit is compared to the fiducial analysis in Figure 24. \nWe find some differences between the fixed-spin and fit-spin analyses. Firstly, the hyperposterior for the fitspin analysis is broader than that of the fit-spin analysis, presumably due to an increase in free parameters. Second, some hyperparameters exhibit a slight shift. The most notable shifts are in the rate and upper gap edge parameters. The shift in the rate is to be expected because the fit to the Default model favors lower spinning objects. Since the detectors are slightly less sensitive to low-spin objects, more support for those objects implies a higher astrophysical rate. Nonetheless, all hyperposterior differences are well within statistical uncertainty, so we conclude that both the fixed-spin and fit-spin cases are acceptable, and use the fixed-spin case for our fiducial results for simplicity. \nIn a preprint version of this paper, the rates reported \nby the PDB model were higher than in this version of the paper. This was due to an incorrext approximation of detector senitivity in the region of highly spinning low mass objects. This approximation has been removed in the current version, and the effect has been to lower the BNS and NSBH rates by nearly 1 σ , lowering the overall rate by a similar amount.', '2. NS mass distribution including marginal events': 'If we loosen the FAR threshold to < 1 yr -1 so as to include the marginal events GW190917 and GW190426, and repeat the analysis of Sec. V B, the inferred NS mass distribution is virtually unchanged. This can be seen in Fig. 25, which compares the posterior population distributions inferred with and without the marginal events. Traces from the posterior population distribution with respect to the original FAR threshold are also shown. This alternative analysis strongly suggests that substantial uncertainties in the merger rate versus mass dominate our error budget; the handful of observations made to date is not sufficient to overcome the strong impact of our highly uncertain model priors. Moreover, the masses of the NS secondaries in the marginal events are poorly constrained relative to those in GW170817, GW190425, GW200105 and GW200115, such that the FAR < 0 . 25 yr -1 events continue to drive the inference.', '3. Merger rates including subthreshold triggers': "In the main text, our merger rates were calculated after adopting a fixed significance threshold to identify confident events, then fitting population model families to the recovered events' posteriors. By design, such an approach depends on the threshold. Here we employ an alternative threshold-free method of rate estimation which lacks potential biases from an arbitrary choice of significance threshold [12]. \nWe extend methods from GWTC-2.1 [3], also applied to the discovery of GW200105 and GW200115 [329], to estimate the event rate from the full set of triggers (including subthreshold triggers) from a specific binary merger search: here, GstLAL [50-52]. In doing so, we allow for population distributions that fit our observations and account for still-considerable uncertainty in the mass distribution, rather than adopting a fixed population model with fixed model hyper-parameters. Compared to previous publications, the results presented in this section update the BBH merger rates presented in GWTC-2.1 by including O3b events [1]. We also update the NSBH rate quoted in [329] by incorporating all O3 triggers, rather than as previously truncating to the first 9 months of O3. \nWe use a multi-component mixture model [121] to construct the posterior of astrophysical counts of CBC events by assuming that foreground and background events are \ngap \ngap \nFIG. 24. Corner plot of inferred PDB mass and rate hyperparameters under an analysis that fixes the spin distribution ( blue ) and simultanously fits the spin distribution using the Default model ( orange ). The fit-spin hyperposterior is slightly shifted and widened when compared to the fixed-spin case, but all changes are within statistical uncertainties. \n<!-- image --> \n/circledot \nlow \n/circledot \nhigh \n/circledot \nR \nindependent Poisson processes. We then estimate the space-time volume sensitivity of the pipeline using simulated events which are re-scaled to an astrophysical population model [319]. We then compute the rates as the ratio of the counts to V T . In order to marginalize over population hyper-parameters we compute several V T 's, each corresponding to a population hyper-parameter sample drawn from the inferred hyper-posterior for the astro- \nphysical population model. Finally, we integrate over the count posterior obtained for each of these samples with an appropriate weight, effectively marginalizing over the \nFIG. 25. Inferred neutron star mass distribution with and without the marginal events GW190426 and GW190917. Top panel: Median and 90% confidence region of the inferred NS mass distribution for the Power model, using the event list at a FAR threshold of 0 . 25 yr -1 (blue) and 1 yr -1 (orange). Traces from the posterior population distribution with respect to the stricter FAR threshold are plotted in grey. Bottom panel: Same as the top panel but for the Peak model. The inclusion of the marginal events has a negligible impact on the inferred mass distribution. \n<!-- image --> \n/circledot \npopulation hyper-parameters: \np ( R | glyph[vector]x ) = ∫ p ( R | glyph[vector] Λ , glyph[vector]x ) p (Λ |{ d } )d glyph[vector] Λ = ∫ V T p ( N | glyph[vector]x ) p ( V T | glyph[vector] Λ) p ( glyph[vector] Λ |{ d } )d( V T )d glyph[vector] Λ = ∑ i,j V T ij × p ( N ij | glyph[vector]x ) , (C1) \nwhere glyph[vector]x is the complete set of triggers (including subthreshold triggers) and { d } is the set of data from gravitational-wave detections used in population model inference, as in IIIB. The astrophysical count posterior is given by p ( N | glyph[vector]x ), where N = R × V T ; we evaluate by sampling via N ij = R × V T ij where V T ij is the i'th V T sample drawn from p ( V T | glyph[vector] Λ j ) for the j'th hyperparameter sample glyph[vector] Λ j drawn from the inferred hyperposterior p ( glyph[vector] Λ |{ d } ). 1 Following [121], we take the dis- \ntion p ( V T | glyph[vector] Λ j ) to be \np ( V T | glyph[vector] Λ j ) = 1 V T √ 2 πσ 2 exp [ -[ln V T -ln 〈 V T 〉 ( glyph[vector] Λ j )] 2 2 σ 2 ] , (C2) \nwhere 〈 V T ( glyph[vector] Λ j 〉 ) is calculated by re-weighting simulated sources to an astrophysical population with hyperparameter glyph[vector] Λ j , and σ is the quadrature sum of a calibration error of 10% [91] and Monte-Carlo uncertainty. \nUsing hyper-parameter samples from the posterior inferred using the PP model with data through the end of O3, as in Section VI and imposing a Jeffreys prior ∝ N -1 / 2 on the astrophysical counts, we compute a BBH merger rate of 24 . 81-63 . 58 Gpc -3 yr -1 . A similar calculation for the BGP model, again with a Jeffreys prior ∝ N -1 / 2 imposed on the astrophysical counts, yields an NSBH merger rate of 14 . 57-187 . 96 Gpc -3 yr -1 , which is consistent with 11-140 Gpc -3 yr -1 , the joint inference for the NSBH merger rate presented in the main text. We also compute a BNS merger rate using a fixed population of BNS's, distributed uniformly in component masses that lie with in 1 to 2.5 M glyph[circledot] . This uses the same multicomponent mixture model [121] as described above, with the only difference being, that instead of marginalizing over population hyperparameters like with the BBH and NSBH merger rates, for BNS, we use a fixed population. Hence, it updates the BNS merger rate reported in GWTC-2.1 [3] by including all of O1 through O3 instead of truncating at O3a. We report a BNS merger rate of 28 . 76-462 . 23 Gpc -3 yr -1 which is consistent with the GWTC 2.1 Rate as well as the other BNS rates quoted in this paper that were computed from only highsignificance triggers.", '4. Effect of Waveform Systematics on Population': "All O3b BBH events analyzed in this paper have source properties inferred using two different waveform models: SEOBNRv4PHM [87] and IMRPhenomXPHM [88], both of which include effects of higher-order multipole moments and spin precession. The posterior distribution for each event is then checked for consistency between waveform models before use in our analyses. \nThe event GW200129 is the highest SNR event exhibiting notable inconsistencies between the source properties inferred with the two waveform models. The event analysis using IMRPhenomXPHM infers much more support for unequal masses and precessing spins relative to the analysis using SEOBNRv4PHM. See [1] for an extended discussion of these systematic differences. \nTo test if the inferred BBH spin population depends on the waveform model chosen for this event, we repeat our O3 population inference using the PP model for three different choices of waveform model for GW200129: IMRPhenomXPHM, SEOBNRv4PHM, and a mix of the two. \nFIG. 26. Inferred differential merger rate as a function of the cosine of the tilt angle ( t i ), where i indexes the body of the binary. We demonstrate that the differences in the posterior distribution for GW200129's spin parameters have a minimal effect on the inferred spin tilt population. The population is inferred using posterior distributions for GW200129 using the IMRPhenomXPHM waveform model (orange), SEOBNRv4PHM model (blue), and a mixture of both (green). Dashed lines are 90% credible intervals. \n<!-- image --> \nAs shown in Fig. 26, the inferred spin population is not significantly affected by changes in the waveform model for this event.", '5. Impact of Sensitivity on Redshift Evolution Inference': 'As noted in Sec. II, one change in the sensitivity estimation procedure between this work and our previous study of GWTC-2 [11] is the use of injections that account for the effect of precession and as well as updates to our detection pipelines as detailed in [1]. Since precession was not included in the injections used in [11], the full spin distribution could not be reweighted to calculate the sensitivity via Equation A2, and thus, for the purposes of sensitivity estimation, an approximation was made that S x,y ∈ ( -0 . 5 , 0 . 5). Since we now use precessing injections, we do the reweighting procedure including the full spin distribution as a function of Λ. To test if this difference in our sensitivity estimation procedure is responsible for the change in the inferred redshift evolution, we repeat the population analysis reported in Sec. VI, using our updated sensitivity model, but only including events analyzed in the GWTC-2 populations study [11]. From this analysis, we infer κ > 0 at 97.6% credibility, as opposed to the 85% credibility reported in [11], indicating a much stronger preference for a merger rate increasing with redshift. We conclude that the differences between our current results for the evolution of the BBH merger rate and those reported in [11] are due to improvements to our sensitivity model rather than the presence of the additional events in GWTC-3. \nIn Fig. 27 we compare the redshift dependence of our current sensitivity model to that of the sensitivity model used in [11]. To make this comparison, we reweight the injections used in [11] to the same spin distribution assumed in that study, and assuming a fiducial PP and powerlaw model for the mass and redshift distributions, respectively. We reweight the current injections to this same mass and redshift distribution, but reweight them to the median inferred spin distribution in [11], to mimic a astrophysically-realistic population. Both injection sets only cover the observing times of the O3a observing run. Taking the ratio of the corresponding sensitvities, we find our sensitivity has increased for low redshift events and decreased for high redshift events, relative to the sensitivity model used in [11]. We expect to see an increase in sensitivity between [11] and our current analysis due to updates to the detection pipelines. The relative decrease in sensitivity at higher redshifts indicates a bias in the previous sensitivity estimate, implying that the BBH merger rate at high redshift was underestimated in [11]. Accounting for the shift in sensitivity as a function of redshift causes a relative decrease in local BBH merger rate and a relative increase in high-redshift BBH merger rate, leading to a higher inferred value for κ . \nOne possible explanation for the shift in sensitivity is that the use of precession in the injections for sensitivity estimation caused a non-trivial change in the inferred sensitive hypervolume, given that we do observe precession in the BBH population. Our current detection pipelines use template banks that include only alignedspin components; this can result in up to tens of percent reduced sensitivity to a population of BBHs with spin precession, depending on the degree of precession possible [330-332]. The farthest precessing sources, which, due to their distances, correspond to FARs closest to the detection threshold, are therefore the most susceptible to dropping below the detection threshold with our current pipelines, causing us to see a decrease in sensitivity to a population of BBHs with precession relative to a strictly non-precessing population. \nAdditionally, both the use of population-informed reweighting of the spin distribution to calculate sensitivity to a population and the incorporation of additional detection pipelines may have contributed to a more accurate estimate of our sensitivity across parameter space.', '1. Analyses from GWTC-2': 'We report updated Bayes factor comparisons for these various models in Table XV, showing that the Broken Powerlaw + Peak model is slightly preferred over our fiducial Power Law + Peak model. We highlight the key differences between the model priors for GWTC-2 \nFIG. 27. Comparison of our current BBH merger sensitivity estimate in the O3a observing run ( V T new ) to that used in [11] ( V T old ) as a function of redshift, for events with chirp masses between 20 M glyph[circledot] and 50 M glyph[circledot] . Our current sensitivity model differs from what was used in [11] in two important ways: we use updated detection pipelines relative to those used in [11] and we use injections which include spin precession. Note the relative increase (decrease) in sensitivity at low (high) redshift. Computed by reweighting injections to a fiducial population for each of the two injection sets. \n<!-- image --> \nTABLE XV. Bayes factors for each of the previously used phenomenological mass models relative to the model with highest marginal likelihood, Power Law + Peak . The previous results from GWTC-2 are shown in the second column with the updated catalog results in the third column. \ncompared to GWTC-3: the prior on β q was changed from U( -4, 12) to U( -2, 7), and the main population results now include an evolving redshift model where the prior on κ is changed from 0 to U( -10, 10). \nIn addition to these analyses, we used a variation of the MultiPeak model to study the feature in the mass distribution at ∼ 10 M glyph[circledot] . In GWTC-2 the prior on the mean of the peaks were U(20, 50) and U(50, 100) for the lower and upper mass peaks respectively. We modified these priors to be U(5, 20) and U(20, 100). This updated MultiPeak model is most preferred model with a log 10 B of 1 . 0 compared to the Power Law + Peak model. This further supports our findings of the peaklike feature at ∼ 10 M glyph[circledot] in the mass distribution.', '2. Comprehensive BBH merger rates': 'In Table IV, we evaluate BBH merger rates over targeted mass subsets of the whole BBH space, using models specifically targeted to reproduce new features of the binary black hole mass distribution. For broader context, Table XVI also provides the corresponding merger rates in these intervals from all the models presented in this work.', 'Appendix E: Population-weighted posteriors': "With an increasing number of events, we can use the distribution of the population of compact binaries to inform our priors for parameter estimation. By reweighting the initial analysis of compact binaries with the population distribution we can obtain posterior distribution for the events in GWTC-3 with population-informed priors. Using our population analysis with models Power Law + Peak and Flexible mixtures we provide population-weighted posteriors (Fig. 28) for m 1 , q and χ eff for the BBHs population (69 events). \nSome of our analyses will show apparent changes in the inferences about the mass ratio. These seeminglysubstantive changes reflect the relatively weak constraints provided by the fiducial parameter inferences used as input and shown in black. Specifically, several low-amplitude or low-mass events have extremely weak constraints on mass ratios, with posterior support extending to q < 0 . 4. This extended feature reflects the prior distribution on component masses, conditioned on modest constraints on chirp mass. To be concrete, using the corresponding prior distributions for these events, conditioned on a suitable chirp mass interval, we often find a posterior distribution with comparable support for q = 1 (i.e., the Savage-Dickey estimate of the Bayes factor for unequal mass would be nearly unity). \nExamining Fig. 28 in light of this caveat, we find that our population models and the fiducial model agree. For these events the population reweightings, as expected, strongly favor symmetric component masses (e.g., GW190503 185404, GW190720 000836, GW191127 050227). For a few binaries, however, the two population reweightings disagree. The most notable example is GW190513 205428, where the Flexible mixtures model pulls the posterior distribution to more symmetric component masses and a lower primary mass. Both population models also pull the majority of the posteriors closer to χ eff ∼ 0. However, given the Flexible mixtures models spins as dependent on chirp mass the events with higher mass and higher spin do not drawn to χ eff ∼ 0 as strongly as the Power Law + Peak model (e.g., GW191109 010717) and in some cases the Flexible mixtures reweighted posterior move to higher χ eff values (e.g., GW190706 222641). \n<!-- image --> \n<!-- image --> \nTABLE XVI. Merger rates in Gpc -3 yr -1 for black hole binaries, quoted at the 90% credible interval. Rates are given for three ranges of primary mass, m 1 as well as for the entire population. The PDB, MS, and BGP merger rates are derived assuming the merger rate does not increase with redshift, using a threshold FAR < 0 . 25 yr -1 (Sec. IV). For FM, PS, and PP, merger rates are reported at z = 0 . 2, estimated using a threshold FAR < 1 yr -1 (Sec. VI). The merged rates reported in the merged row are the union of the preceding three rows, which all account for distance-dependent merger rate and adopt a consistent threshold. 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2024JHEP...11..070A

We investigate the retarded Greens function and the greybody factor in asymptotically AdS black holes. Using the connection coefficients of the Heun equation expressed in terms of the NekrasovShatashvili NS free energy of an SU2 supersymmetric gauge theory with four fundamental hypermultiplets we derive asymptotic expansions for both the retarded Greens function and the greybody factor in the small horizon limit. Furthermore we compute the corrections to these asymptotic expansions resulting from the resummation procedure of the instanton part of the NS function.

2024-11-01T00:00:00Z

['2024arXiv240907370B', 'arXiv:2409.07370', '10.1007/JHEP11(2024)070', '2024JHEP...11..070A', '10.48550/arXiv.2409.07370']

['Black Holes', 'AdS-CFT Correspondence', 'Supersymmetric Gauge Theory', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']

The effect of resummation on retarded Greens function and greybody factor in AdS black holes

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

https://arxiv.org/pdf/2409.07370.pdf

{'Julián Barragán Amado a Shankhadeep Chakrabortty, b Arpit Maurya b': "- a Grupo de Física Matemática, Campo Grande Edifício C6, Lisboa 1749-016, Portugal\n- b Department of Physics, Indian Institute of Technology Ropar, Rupnagar, Punjab, India 140001. \nE-mail: \njjamado@fc.ul.pt , s.chakrabortty@iitrpr.ac.in , \narpit.20phz0009@iitrpr.ac.in \nAbstract: We investigate the retarded Green's function and the greybody factor in asymptotically AdS black holes. Using the connection coefficients of the Heun equation, expressed in terms of the Nekrasov-Shatashvili (NS) free energy of an SU (2) supersymmetric gauge theory with four fundamental hypermultiplets, we derive asymptotic expansions for both the retarded Green's function and the greybody factor in the small horizon limit. Furthermore, we compute the corrections to these asymptotic expansions resulting from the resummation procedure of the instanton part of the NS function.", '1 Introduction': "A recent resurgence of black hole perturbation theory using the techniques originally developed for the Nekrasov Shatashvili phase of the Ω -background in 4D supersymmetric gauge theory [1-5] has opened up a significant possibility for achieving the exact analytical form of various physical observables including holographic thermal correlator, quasinormal modes, greybody factors and many more [6-16]. After the separation of variables, the equation of motion for the scalar mode in black hole perturbation theory results in linear second-order ordinary differential equations that can be represented in the form of Heun equations by appropriate redefinition of variables [17]. Heun's equation is known for its generality as a second-order linear differential equation with four regular singular points [18]. These singularities on the Riemann sphere characterize the equation as a generalization of the \nhypergeometric equation [19]. The solution to Heun's equation, also known as the Heun function near a singular point, can be expanded in terms of the solutions around other singularities. The connection coefficients specify the explicit form of such expansion and are computed in terms of the Nekrasov-Shatashvili partition function [20]. \nThe connection coefficients and the underlying connection formula are best understood in the context of 2d Liouville CFT [21]. According to the 2D-4D duality of the Alday-Gaiotto-Tachikawa (AGT) correspondence, the BPS sector of four-dimensional N = 2 , SU (2) gauge theory in Ω is dual to the Liouville CFT on the Riemann sphere [4]. This duality implies that the four-point function of the Liouville CFT corresponds to the NekrasovShatashvili partition function in N = 2 , SU (2) gauge theory [22]. In particular, the duality offers a fascinating connection between the semiclassical limit (large central charge limit) on the parameters of Liouville CFT and the Nekrasov-Shatashvili limit on the parameters of the Ω background in the SU (2) gauge theory [5]. The BPZ equation governing the five-point correlator in Liouville theory provides an avenue for series solutions through the use of conformal blocks. The interplay of conformal blocks in different OPE channels (s, t, and u channels) establishes connections among the solutions to the BPZ equation. In the limit of a large central charge (semi-classical limit), the BPZ equation transforms into Heun's equation, a second-order linear differential equation with four regular singularities where these singular points signifies the insertion of primary operators in the theory. By the virtue of crossing symmetry, the relation among the solutions of the semi-classical BPZ equation leads to a connection formula among the solutions of Heun's equation [21]. \nVery recently, the authors of [23] established an exact analytical expression for the holographic retarded Green's function for AdS 5 Schwarzschild black hole, dual to the correlator in a thermal CFT living in the R 1 × S 3 boundary of the black hole spacetime. They have used a connection formula between the Heun functions associated with the incoming mode expansion of a scalar field near the horizon and those near the boundary of the black hole spacetime. This connection formula nicely captures the response function and source terms present in the boundary expansion of the scalar mode. Finally, by following the prescription of Lorentzian Green's function of black hole spacetime [24], they obtained the retarded Green's function by taking the ratio of source to response function. Further, exact retarded Green's function for a thermal CFT with chemical potential and angular momenta on R 1 × S 3 and also for a thermal CFT living in R 1 × H 3 are constructed in [25] and [26] respectively. \nThe study of greybody factors in asymptotically flat spacetimes has been reviewed using the connection coefficients of the confluent Heun equation, resulting in asymptotic expansions proportional to the area of the black hole horizon in the low-energy limit [7, 9]. Nevertheless, in the case of asymptotically AdS spcetimes, the role of the connection coefficients becomes more elusive due to the nature of the boundary at spatial infinity. Harmark and collaborators [27] computed the greybody factors for static and spherically symmetric spacetime black holes in d -dimensions by splitting up the spacetime in three regions, such that the radial differential equation can be solved analytically, and then matched across the regions. Building on those considerations, [28] and [29] have investigated the greybody \nfactors for rotating and non-asymptotically flat black holes. \nIn this work, we study the retarded Green's function and the greybody factor for Kerr-AdS 5 black hole and Reissner-Nordström-AdS 5 black hole. In doing so, we analyze the uncharged and charged scalar mode perturbation in the respective black hole spacetimes. We show how the Klein-Gordon equation corresponding to the radial mode of scalar field takes the form of Heun's differential equation by successive applications of Möbius transformation followed by a s-homotopic transformation. We find the analytic form of two independent solutions of the Heun's equation at each of the two regular singular points corresponding to the black hole horizon and the boundary of the black hole spacetime. We explicitly show the emergence of the connection formula and the connection coefficients and describe the role they play in the subsequent study of the retarded Green's function and greybody factor. As expected, the Nekrasov partition function plays a crucial role in determining the connection coefficients. Most interestingly, we show how the singularity appearing from the pole structure of the Nekrasov partition function is reincarnated in to the singularity of the pole structure of a , the vacuum expectation value of the scalar in the vector hypermultiplets in the supersymmetric gauge theory, obtained by solving the Matone relation. Within the small black hole approximation, we use a resummation technique to cure the singularity structure in the a order by order in radius of black hole horizon. This exercise surprisingly accommodates a series correction terms in a that can be nicely presented in terms of the generating function of the Catalan numbers. Moreover, the presence of such correction terms in a closed form sigificantly simplify the final analytical results of Green's function and greybody factor. \nThe plan of the paper is the following: In section 2 we discuss the explicit construction of the radial mode of the scalar field in Kerr-AdS 5 black hole spacetime. In section 3 we describe the radial mode of a charged scalar field in Reissner-Nordström-AdS 5 black hole spacetime. Section 4 is dedicated to Heun's equation, the solutions to Heun's equation near regular singular point and the connection formula. In section 5 and in section 6 we discuss the computtaions of retarded Green's function and Greybody factor for Kerr-AdS 5 and Reissner-Nordström-AdS 5 black holes respetively. Finally, we conclude in section 7.", '2 Scalar perturbations in Kerr-AdS 5': "In order to set the backdrop of a specific example of scalar perturbation theory in AdS black hole, let us review the five dimensional KerrAdS 5 black hole as presented in [30]. The explicit form of the corresponding metric takes the form, \nds 2 = -∆ r ρ 2 ( dt -ˆ a 1 sin 2 θ Ξ 1 dϕ 1 -ˆ a 2 cos 2 θ Ξ 2 dϕ 2 ) 2 + ∆ θ sin 2 θ ρ 2 ( ˆ a 1 dt -r 2 +ˆ a 2 1 Ξ 1 dϕ 1 ) 2 + 1 + r 2 /L 2 r 2 ρ 2 ( ˆ a 1 ˆ a 2 dt -ˆ a 2 ( r 2 +ˆ a 2 1 ) sin 2 θ Ξ 1 dϕ 1 -ˆ a 1 ( r 2 +ˆ a 2 2 ) cos 2 θ Ξ 2 dϕ 2 ) 2 + ∆ θ cos 2 θ ρ 2 ( ˆ a 2 dt -r 2 +ˆ a 2 2 Ξ 2 dϕ 2 ) 2 + ρ 2 ∆ r dr 2 + ρ 2 ∆ θ dθ 2 , (2.1) \nwhere \n∆ r = 1 r 2 ( r 2 +ˆ a 2 1 ) ( r 2 +ˆ a 2 2 ) ( 1 + r 2 L 2 ) -2 M = 1 L 2 r 2 ( r 2 -r 2 0 ) ( r 2 -r 2 -) ( r 2 -r 2 + ) , ∆ θ = 1 -ˆ a 2 1 L 2 cos 2 θ -ˆ a 2 2 L 2 sin 2 θ, ρ 2 = r 2 +ˆ a 2 1 cos 2 θ +ˆ a 2 2 sin 2 θ, Ξ 1 = 1 -ˆ a 2 1 L 2 , Ξ 2 = 1 -ˆ a 2 2 L 2 (2.2) \nHere L stands for AdS radius and M , ˆ a 1 and ˆ a 2 are the real parameters, related to the ADM mass ( M ) and angular momenta ( J 1 , J 2 ) respectively [31, 32] \nM = πM (2Ξ 1 +2Ξ 2 -Ξ 1 Ξ 2 ) 4Ξ 2 1 Ξ 2 2 , J 1 = πM ˆ a 1 2Ξ 2 1 Ξ 2 , J 2 = πM ˆ a 2 2Ξ 1 Ξ 2 2 . (2.3) \nWithin a physically sensible range of parameters described as M > 0 , and ˆ a 2 1 , ˆ a 2 2 < 1 , ∆ r = 0 allows two real roots r + and r -signifying the inner and outer horizons of the black hole respectively and one purely imaginary root r 0 that satisfies the following relation [31], \n-r 2 0 = L 2 +ˆ a 2 1 +ˆ a 2 2 + r 2 -+ r 2 + . (2.4) \nFor the purposes of this article, we will see the radial variable, or rather r 2 , as a generic complex number. It will be interesting for us to treat all three roots of ∆ r ,e.g., r 2 + , r 2 -and r 2 0 as Killing horizons. Actually, in the complexified version of the metric (2.1), in all three hypersurfaces defined by r = r 0 , r -and r + we have that each of the Killing fields \nξ k = ∂ ∂t +Ω 1 ( r k ) ∂ ∂ϕ +Ω 2 ( r k ) ∂ ∂ψ , k = 0 , -, + , (2.5) \nbecomes null [33]. The temperatures and angular velocities at each horizon are given by \n̸ \n̸ \nT k = r 2 k ∆ ' r ( r k ) 4 π ( r 2 k +ˆ a 2 1 )( r 2 k +ˆ a 2 2 ) = r k 2 πL 2 ( r 2 k -r 2 i )( r 2 k -r 2 j ) ( r 2 k +ˆ a 2 1 )( r 2 k +ˆ a 2 2 ) , i = j = k, Ω k, 1 = ˆ a 1 Ξ 1 r 2 k +ˆ a 2 1 , Ω k, 2 = ˆ a 2 Ξ 2 r 2 k +ˆ a 2 2 . (2.6) \nWithin the physically sensible range of parameters, T + is positive, T -is negative and T 0 is purely imaginary.", '2.1 The Klein-Gordon equation': "The Klein-Gordon (KG) equation for a scalar of mass µ , in the background (2.1), is determined by \n1 √ -g ∂ µ ( √ -gg µν ∂ ν ) Φ -µ 2 Φ = 0 . (2.7) \nWe consider the following ansatz \nΦ( t, r, θ, ϕ 1 , ϕ 2 ) = e -iωt + im 1 ϕ 1 + im 2 ϕ 2 R ( r ) S ( θ ) , (2.8) \nwhere ω is the frequency of the mode, and m 1 , m 2 ∈ Z are the azimuthal quantum numbers. With the suitable choice of ansatz (2.8),KG equation (2.7) decouples into two second-order ordinary differential equations of the form \n[ 1 r d dr ( r ∆ r d dr ) + ( r 2 +ˆ a 2 1 ) 2 ( r 2 +ˆ a 2 2 ) 2 r 4 ∆ r ( ω -m 1 ˆ a 1 Ξ 1 r 2 +ˆ a 2 1 -m 2 ˆ a 2 Ξ 2 r 2 +ˆ a 2 2 ) 2 -1 r 2 (ˆ a 1 ˆ a 2 ω -ˆ a 2 Ξ 1 m 1 -ˆ a 1 Ξ 2 m 2 ) 2 -λ ℓ -µ 2 r 2 ] R ( r ) = 0 , (2.9a) \n[ 1 sin θ cos θ d dθ ( sin θ cos θ ∆ θ d dθ ) -ω 2 L 2 -m 2 1 Ξ 1 sin 2 θ -m 2 2 Ξ 2 cos 2 θ + Ξ 1 Ξ 2 ∆ θ ( ωL + m 1 ˆ a 1 L + m 2 ˆ a 2 L ) 2 -µ 2 ( ˆ a 2 1 cos 2 θ +ˆ a 2 2 sin 2 θ ) + λ ℓ ] S ( θ ) = 0 , (2.9b) \nwhere λ ℓ is the separation constant. We shall revisit the separation constant and describe its explicit form in section 5. For computational convenience, we introduce a dimensionless radial coordinate \n˜ r = r L , ˜ r k = r k L , k = { 0 , -, + } (2.10) \nand as a result ∆ ˜ r takes the form as ∆ ˜ r = ∆ r L 2 . If we scale the remaining parameters as \n˜ ω = Lω, ˜ a i = ˆ a i L , ˜ µ = Lµ, (2.11) \nwe obtain the dimensionless ODEs, which do not depend on the AdS radius #1 . In terms of the tilde notation, the radial equation (2.9a) takes the following form: \n{ d 2 d ˜ r 2 + ( 1 ˜ r + ∆ ' ˜ r ∆ ˜ r ) d d ˜ r + (˜ r 2 +˜ a 2 1 ) 2 (˜ r 2 +˜ a 2 2 ) 2 ˜ r 4 ∆ 2 ˜ r ( ˜ ω -m 1 ˜ a 1 Ξ 1 ˜ r 2 +˜ a 2 1 -m 2 ˜ a 2 Ξ 2 ˜ r 2 +˜ a 2 2 ) 2 -1 ˜ r 2 ∆ ˜ r (˜ a 1 ˜ a 2 ˜ ω -˜ a 2 Ξ 1 m 1 -˜ a 1 Ξ 2 m 2 ) 2 -λ ℓ ∆ ˜ r -˜ µ 2 ˜ r 2 ∆ ˜ r } R (˜ r ) = 0 , (2.13a) \nwhere \n∆ ˜ r = ( ˜ r 2 -˜ r 2 0 ) ( ˜ r 2 -˜ r 2 -) ( ˜ r 2 -˜ r 2 + ) ˜ r 2 (2.13b) \nEquation (2.13) possesses four regular singular points in ˜ r 2 variable, located at the roots of ∆ ˜ r and the point at infinity. The characteristic exponents of the Frobenius solutions near to each finite singularity are given by \nβ ± k = ± 1 2 θ k , k = { + , -, 0 } (2.14) \nT k = ˜ T k L , Ω k, 1 = ˜ Ω k, 1 L , Ω k, 2 = ˜ Ω k, 2 L . (2.12) \nwhere \nθ k = i 2 π ( ˜ ω -m 1 ˜ Ω k, 1 -m 2 ˜ Ω k, 2 ˜ T k ) , (2.15) \nand for ˜ r = ∞ , we have \nβ ± ∞ = 1 2 (2 ± θ ∞ ) , θ ∞ = √ 4 + ˜ µ 2 := ∆ -2 . (2.16) \nIt turns out that θ + is the variation of the entropy δS of the black hole as it absorbs a quantum of frequency and angular momenta at the outer horizon. Furthermore, θ ∞ can be expressed in terms of the conformal dimension ∆ of the scalar operator dual to a scalar field in the bulk with mass ˜ µ , such that ∆(∆ -4) = ˜ µ 2 [34]. To bring the radial equation (2.13) to the canonical Heun form, we perform a change of variables followed by a s-homotopic transformation #2 , \nz = ˜ r 2 -˜ r 2 -˜ r 2 -˜ r 2 0 , R ( z ) = z β --( z 0 -z ) β -+ (1 -z ) β + ∞ f ( z ) , (2.17) \nwhere \nz 0 = ˜ r 2 + -˜ r 2 -˜ r 2 + -˜ r 2 0 . (2.18) \nThe equation for f ( z ) is \nd 2 f dz 2 + [ 1 -θ -z + 1 -θ + z -z 0 + ∆ -1 z -1 ] df dz + ( κ 1 κ 2 z ( z -1) -z 0 ( z 0 -1) K 0 z ( z -z 0 )( z -1) ) f ( z ) = 0 , (2.19) \nwhere \nκ 1 = 1 2 ( θ -+ θ + -∆ -θ 0 ) , κ 2 = 1 2 ( θ -+ θ + -∆+ θ 0 ) , (2.20a) \n4 z 0 ( z 0 -1) K 0 = -( λ ℓ +∆(∆ -4)˜ r 2 --˜ ω 2 ) ˜ r 2 + -˜ r 2 0 -( z 0 -1) [ ( θ -+ θ + -1) 2 -θ 2 0 -1 ] -z 0 [ 2( θ + -1)(1 -∆) + (∆ -2) 2 -2 ] , (2.20b) \nwhere K 0 is called the accessory parameter. We show the explicit form of angular differential equation in appendix B.", '3 Scalar perturbations in Reissner-Nordström-AdS 5': 'The line element of the Reissner-Nordström-AdS 5 (RN-AdS 5 ) black hole is \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 3 (3.1) \nwhere d Ω 2 3 is the metric of the unit three-sphere and the blackening function f ( r ) reads \nf ( r ) = 1 -M r 2 + Q 2 r 4 + r 2 L 2 (3.2) \nwhere M , Q and L are related to the black hole ADM mass, charge, and the radius of AdS respectively. The function f ( r ) is a polynomial of degree six with only even powers of r , such that the roots of f ( r ) can be expressed as \nf ( r ) = ( r 2 -r 2 0 ) ( r 2 -r 2 1 ) ( r 2 -r 2 2 ) L 2 r 4 . (3.3) \nSince we are interested in space-time configurations that possess black hole horizons, the outer horizon corresponds to the largest positive real root r 2 = r + , while the inner horizon is defined as r 1 = r -satisfying r -≤ r + , and r 0 is purely imaginary for generic values of the mass and charge. Furthermore, one can write mass parameter in terms of the the radius of the outer horizon as \nM = r 2 + + Q 2 r 2 + + r 4 + L 2 , (3.4) \nwhich implies \nr 2 -= L 2 2 -1 -r 2 + L 2 + √ ( 1 + r 2 + L 2 ) 2 + 4 Q 2 L 2 r 2 + r 2 0 = L 2 2 -1 -r 2 + L 2 -√ ( 1 + r 2 + L 2 ) 2 + 4 Q 2 L 2 r 2 + (3.5) \nThe temperature at each horizon is \n̸ \n̸ \nT k = 1 4 π df ( r ) dr ∣ ∣ ∣ ∣ r = r k = 1 2 πL 2 ( r 2 k -r 2 i ) ( r 2 k -r 2 j ) r 3 k , i = j = k (3.6) \nThe electromagnetic potential of the charged black hole can be written as \nA µ dx µ = ( -√ 3 2 Q r 2 + C ) dt, (3.7) \nand for a vanishing potential at spatial infinity, we set C = 0 .', '3.1 The Klein-Gordon equation': "In the charged black hole background (3.1), the Klein-Gordon equation for a massive charged scalar field perturbation reads as \n1 √ -g D µ ( √ -gg µν D ν ) Φ -µ 2 Φ = 0 , (3.8) \nwith D µ = ∂ µ -ieA µ , and e and µ being the charge and the mass of the field, respectively. Equation (3.8) can be decomposed into two second-order ODEs by implementing the following ansatz, \nΦ( t, r, θ, ϕ, ψ ) = e -iωt Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) R ( r ) (3.9) \nThe angular part reduces to the spherical harmonic function on the three-sphere Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) , which satisfies the eigenvalue equation \n∆ Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) = -ℓ ( ℓ +2) Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) , (3.10) \nwhere ℓ is the angular momentum quantum number, and m 1 and m 2 are integers associated with the magnetic quantum numbers. By means of (3.10), the differential equation for the radial function R ( r ) takes the following form \n 1 r d dr ( r 3 f ( r ) d dr ) + r 2 f ( r ) ( ω -√ 3 2 eQ r 2 ) 2 -µ 2 r 2 -ℓ ( ℓ +2) R ( r ) = 0 (3.11) \nFor numerical convenience, we define \n˜ r = r L , ˜ r k = r k L , f (˜ r ) = f ( r ) , k = { 0 , -, + } ˜ Q = Q L 2 , ˜ ω = Lω, ˜ µ = Lµ, ˜ e = √ 3 2 Le, (3.12) \nso that equation (3.11) will take a dimensionless form: \n d 2 d ˜ r 2 + ( 3 ˜ r + f ' (˜ r ) f (˜ r ) ) d d ˜ r + 1 f (˜ r ) 1 f (˜ r ) ( ˜ ω -˜ e ˜ Q ˜ r 2 ) 2 -ℓ ( ℓ +2) ˜ r 2 -˜ µ 2 R (˜ r ) = 0 , (3.13a) \nwhere \nf (˜ r ) = ( ˜ r 2 -˜ r 2 0 ) ( ˜ r 2 -˜ r 2 -) ( ˜ r 2 -˜ r 2 + ) ˜ r 4 . (3.13b) \nEquation (3.13) possesses four regular singular points in ˜ r 2 variable, located at the roots of f (˜ r ) and the point at infinity. The characteristic exponents of the Frobenius solutions near to each finite singularity are given by \nβ ± k = ± 1 2 θ k , k = { + , -, 0 } (3.14) \nwhere \nθ k = i 2 π ˜ T k ( ˜ ω -˜ e ˜ Q ˜ r 2 k ) , (3.15) \nand for ˜ r = ∞ , we have \nβ ± ∞ = 1 2 (2 ± θ ∞ ) , θ ∞ = √ 4 + ˜ µ 2 := ∆ -2 . (3.16) \nSince our aim is to write the equation (3.13) in the canonical Heun form, we introduce a Möbius transformation followed by a s-homotopic transformation \nz = ˜ r 2 -˜ r 2 -˜ r 2 -˜ r 2 0 , R ( z ) = z β --( z 0 -z ) β -+ (1 -z ) β + ∞ f ( z ) , (3.17) \nwhere \nz 0 = ˜ r 2 + -˜ r 2 -˜ r 2 + -˜ r 2 0 , (3.18) \nthat leads to an equation for f ( z ) \nd 2 f dz 2 + [ 1 -θ -z + 1 -θ + z -z 0 + ∆ -1 z -1 ] df dz + ( κ 1 κ 2 z ( z -1) -z 0 ( z 0 -1) K 0 z ( z -z 0 )( z -1) ) f ( z ) = 0 , (3.19) \nwith \nκ 1 = 1 2 ( θ -+ θ + -∆ -θ 0 ) , κ 2 = 1 2 ( θ -+ θ + -∆+ θ 0 ) , (3.20a) \n4 z 0 ( z 0 -1) K 0 = -ℓ ( ℓ +2) + ∆(∆ -4)˜ r 2 --˜ ω 2 ˜ r 2 + -˜ r 2 0 -( z 0 -1) [ ( θ -+ θ + -1) 2 -θ 2 0 -1 ] -z 0 [ 2( θ + -1)(1 -∆) + (∆ -2) 2 -2 ] (3.20b) \nIt is important to note that while the radial equation (3.19) shares a similar formal structure with the radial equation (2.19), they are not identical. The characteristic exponents in each case depend on the details of the background. The accessory parameter K 0 includes the separation constant introduced to decouple the angular and radial components, which, in the case of RN-AdS 5 reduces to the eigenvalues associated to the scalar spherical harmonics on the three-sphere (3.10). Conversely, the angular eigenvalue problem in Kerr-AdS 5 is more complex and it is tipically solved perturbatively in specific limits, e.g., see [35, 36] for asymptotic expansions of λ in the near-equally rotating limit.", '4 Heun functions and connection coefficients': "In this section we show the explicit construction of the normal form of Heun's equation. We elaborate upon the solution of the Heun's equation near regular singular points and derive the connection formula which we frequently use in the computations of retarded Green's function and greybody factor in the subsequent sections. \nThe canonical Heun's differential equation reads \n( d 2 dz 2 + ( γ z + δ z -1 + ϵ z -t ) d dz + αβz -q z ( z -t )( z -1) ) f ( z ) = 0 , (4.1a) \nwhere the coefficients satisfy the condition \nα + β +1 = γ + δ + ϵ, (4.1b) \nand the complex modulus of t is small, | t | ≪ 1 . Via the following transformation \nf ( z ) = z -γ ( t -z ) -ϵ (1 -z ) -δ ψ ( z ) , (4.2) \none can bring (4.1a) into its normal form \n( ∂ 2 z + 1 4 -a 2 0 z 2 + 1 4 -a 2 t ( z -t ) 2 + 1 4 -a 2 1 ( z -1) 2 -1 2 -a 2 0 -a 2 t -a 2 1 + a 2 ∞ + u z ( z -1) + u z ( z -t ) ) ψ ( z ) = 0 , (4.3) \nwith the identification \na 0 = 1 -γ 2 , a t = 1 -ϵ 2 , a 1 = 1 -δ 2 , a ∞ = α -β 2 , u = γϵ -2 q +2 tαβ -t ( γ + δ ) ϵ 2( t -1) , (4.4) \nwhich can be written in terms of a i 's and u as \nα = 1 -a 0 -a 1 -a t + a ∞ , β = 1 -a 0 -a 1 -a t -a ∞ , γ = 1 -2 a 0 , δ = 1 -2 a 1 , ϵ = 1 -2 a t , q = 1 2 + ( a 2 0 + a 2 t + a 2 1 -a 2 ∞ ) t -a t -a 1 t + a 0 (2 a t -1 + t (2 a 1 -1)) + (1 -t ) u. (4.5) \nBy comparing with the radial ODEs (2.19) and (3.19), one can recognize that \nt = z 0 (4.6a) \na 0 = ± θ -2 , a t = ± θ + 2 , a 1 = ± θ ∞ 2 , a ∞ = ± θ 0 2 , (4.6b) \nu = -z 0 K 0 + 1 2 θ -(1 -θ + ) -1 2 ∆(1 -θ + ) -(1 -θ + ) (∆ -1) 2 ( z 0 -1) . (4.6c) \nNote that the explicit form of the parameters, θ k , θ ∞ , z 0 and K 0 depends on the individual details of the corresponding black hole solutions such as the Kerr-AdS 5 and the ReissnerNordström-AdS 5 black holes. \nIn [9, 20, 21], the authors provided explicit expressions for the local expansions of the Heun functions and their connection coefficients in terms of the Nekrasov partition functions. These concepts have been applied to study quasi-normal modes, tidal Love numbers and greybody factors in different space-times [10-14], post-Newtonian (PN) dynamics in the two-body problem [15, 37], as well as the computation of retarded Green's function in asymptotically AdS space-times [23, 25, 26]. \nFollowing [13, 20], two linearly independent solutions around z = t are given by \nf ( t ) -( z ) =HeunG ( t t -1 , q -tαβ 1 -t , α, β, ϵ, δ, z -t 1 -t ) f ( t ) + ( z ) =( t -z ) 1 -ϵ HeunG ( t t -1 , q -αβt 1 -t -( ϵ -1) ( γ + δt t -1 ) , α -ϵ +1 , β -ϵ +1 , 2 -ϵ, δ, z -t 1 -t ) , (4.7a) \nwhile the ones around z = 1 can be written as \nf (1) -( z ) = ( z -t 1 -t ) -α HeunG ( t, q + α ( δ -β ) , α, δ + γ -β, δ, γ, t 1 -z t -z ) f (1) + ( z ) =(1 -z ) 1 -δ ( z -t 1 -t ) -α -1+ δ HeunG ( t, q -( δ -1) γt -( β -1)( α -δ +1) , -β + γ +1 , α -δ +1 , 2 -δ, γ, t 1 -z t -z ) . (4.7b) \nIn addition, the connection formula between solutions near to z = t and near to z = 1 is given by \nt -1 2 + a 0 ∓ a t (1 -t ) -1 2 + a 1 e ∓ 1 2 ∂ a t F ( t ) f ( t ) ( z ) = \n∑ σ = ± M ± σ ( a t , a ; a 0 ) M ( -σ )+ ( a, a 1 ; a ∞ ) t σa e -σ 2 ∂ a F ( t ) ) (1 -t ) -1 2 + a t e iπ ( -a 1 + a t ) e -1 2 ∂ a 1 F ( t ) f (1) + ( z ) \n± ( ∑ σ = ± M ± σ ( a t , a ; a 0 ) M ( -σ ) -( a, a 1 ; a ∞ ) t σa e -σ 2 ∂ a F ( t ) ) (1 -t ) -1 2 + a t e iπ ( a 1 + a t ) e 1 2 ∂ a 1 F ( t ) f (1) -( z )+ ( , \n(4.8) \nwhere F ( t ) is the instanton part of the NS free energy and the connection coefficients M 's are defined as \nM θθ ' ( a 1 , a 2 ; a 3 ) = Γ( -2 θ ' a 2 ) Γ (1 + 2 θa 1 ) Γ ( 1 2 + θa 1 -θ ' a 2 + a 3 ) Γ ( 1 2 + θa 1 -θ ' a 2 -a 3 ) . (4.9) \nBy combining (4.3) with (4.7) we construct the solution to the radial equations (2.19) and (3.19) at the regular singular point z = z 0 (or equivalently z = t ) as \nf ( z ) = C z 0 -f ( z 0 ) -( z ) + C z 0 + f ( z 0 ) + ( z ) . (4.10) \nConsequently, the asymptotic behavior of the radial solution R ( z ) at z = z 0 leads to \nR ( z ) ≃ C z 0 -( z 0 -z ) -θ + / 2 + C z 0 + ( z 0 -z ) θ + / 2 (4.11) \nfor which an incoming wave at the outer horizon z = z 0 (˜ r = ˜ r + ) requires that C z 0 + = 0 . Then, the remaining radial solution according to the boundary condition will be given by \nR ( z ) = C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f ( z 0 ) -( z ) . (4.12) \nNow we can relate the solution around z = z 0 to the local solutions around z = 1 through the appropriate choice of the connection coefficients (4.8). Namely, we have \nf ( z 0 ) -( z ) = ( M --( a t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M --( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a 1 + a t ) e 1 2 ( ∂ a 1 F ( z 0 ) -∂ a t F ( z 0 ) ) f (1) -( z ) + ( M --( a t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a t -a 1 ) e -1 2 ( ∂ a 1 F ( z 0 )+ ∂ a t F ( z 0 ) ) f (1) + ( z ) . (4.13) \nBy introducing (4.13) into (4.12), we obtain the radial solution at the horizon z = z 0 in terms of two local solutions at z = 1 as follows \nR ( z ) = C 1 -C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f (1) -( z ) + C 1+ C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f (1) + ( z ) , (4.14) \nwhere f (1) ± ( z ) are given in (4.7b) and C 1 , ± read \nC 1 -= ( M --( a t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M --( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a 1 + a t ) e 1 2 ( ∂ a 1 F ( z 0 ) -∂ a t F ( z 0 ) ) , (4.15a) \nC 1+ = ( M --( a t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a t -a 1 ) e -1 2 ( ∂ a 1 F ( z 0 )+ ∂ a t F ( z 0 ) ) . (4.15b) \nThe asymptotic behavior of the radial solution (4.14) corresponding to spatial infinity ˜ r → ∞ ( z → 1) \nR ( r ) ≃ C 1 -C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 ∆ ˜ r -∆ + C 1+ C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 (4 -∆) ˜ r ∆ -4 (4.16) \nwhere for ∆ ≥ 4 , the first term converges while the second one diverges at ˜ r → ∞ , and thus the asymptotic solutions correspond to normalizable and non-normalizable solutions, respectively. Furthermore, the retarded Green's function defined as the ratio of the response to the source yields \nG ret (˜ ω, λ ) = ( ˜ r 2 --˜ r 2 0 ) ∆ -2 C 1 -C 1+ = e ∂ a 1 F ( z 0 )+ iπ 2 a 1 ( ˜ r 2 --˜ r 2 0 ) ∆ -2 Γ(2 a 1 ) Γ( -2 a 1 ) Σ 1 -Σ 1+ , (4.17a) \nwhere \n1 \nΣ 1 -= e -2 ∂ a F ( z 0 ) Γ(1 -2 a )Γ( -2 a ) Γ ( 1 2 -a -a 0 -a t ) Γ ( 1 2 -a + a 0 -a t ) Γ ( 1 2 -a + a 1 -a ∞ ) Γ ( 1 2 -a + a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) a + e 1 2 ∂ a F ( z 0 ) Γ(1 + 2 a )Γ(2 a ) Γ ( 1 2 + a -a 0 -a t ) Γ ( 1 2 + a + a 0 -a t ) Γ ( 1 2 + a + a 1 -a ∞ ) Γ ( 1 2 + a + a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) -a (4.17b) \n-1 \nΣ 1+ = e 2 ∂ a F ( z 0 ) Γ(1 -2 a )Γ( -2 a ) Γ ( 1 2 -a -a 0 -a t ) Γ ( 1 2 -a + a 0 -a t ) Γ ( 1 2 -a -a 1 -a ∞ ) Γ ( 1 2 -a -a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) a + e 1 2 ∂ a F ( z 0 ) Γ(1 + 2 a )Γ(2 a ) Γ ( 1 2 + a -a 0 -a t ) Γ ( 1 2 + a + a 0 -a t ) Γ ( 1 2 + a -a 1 -a ∞ ) Γ ( 1 2 + a -a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) -a . (4.17c) \nwhere the radial dictionary for a i , i = { 0 , t, 1 , ∞} is defined in (4.6b), and we will choose the one with the negative sign and we have verified that the final result is independent of this choice of sign. Expression (4.17a) is related to the propagator derived in [23, 25, 26] under the change a 1 →-a 1 , as a result of taking into account the asymptotic analysis of the radial function R ( z ) , instead of ψ ( z ) .", '5 Small Kerr-AdS 5 black holes': "In order to study the retarded Green's function and the greybody factor in asymptotically AdS 5 black holes, we will focus on the small radius limit, while considering equal rotation parameters ˜ a 1 = ˜ a 2 = ˜ a . For this special case the angular equation (2.9b) reduces to a hypergeometric differential equation with the angular eigenvalue given by \nλ ℓ = ( 1 -˜ a 2 ) [ ℓ ( ℓ +2) -2˜ a ˜ ω ( m 1 + m 2 ) -˜ a 2 ( m 1 + m 2 ) 2 ] +˜ a 2 ( ˜ ω 2 +∆(∆ -4) ) . (5.1) \nas it was investigated in [38]. Furthermore, we can define a critical rotation parameter as the maximal rotation parameter at extremality ( T + = 0) \n˜ a c = ˜ r + √ 1 + 2˜ r 2 + , (5.2) \nand we require ˜ a ≤ ˜ a c , in order to guarantee a regular outer horizon. We then parameterize the accessible values for the rotation parameter as ˜ a = α ˜ a c , where α ≤ 1 is a dimensionless extremality parameter for fixed ˜ r + > 0 . In the small ˜ r + limit, the indicial coefficients of the radial equation (2.15) and (2.16), as well as the conformal modulus (2.18) can be written as \nθ -= -i ( m 1 + m 2 ) α 1 -α 2 + iα 2 ( 1 + α 2 ) ˜ ω ˜ r + 1 -α 2 -i ( m 1 + m 2 ) α ( 1 + α 2 ) ( 1 -3 α 2 ) ˜ r 2 + 2 (1 -α 2 ) + iα 2 ( 3 + 5 α 2 -7 α 4 -5 α 6 ) ˜ ω ˜ r 3 + 2 (1 -α 2 ) + i ( m 1 + m 2 ) α ( 5 + 36 α 2 +42 α 4 -52 α 6 -35 α 8 ) ˜ r 4 + 8 (1 -α 2 ) + O ( ˜ r 5 + ) , \n(5.3a) \nθ + = -i ( m 1 + m 2 ) α 1 -α 2 + i ( 1 + α 2 ) ˜ ω ˜ r + 1 -α 2 + i ( m 1 + m 2 ) α ( 1 + α 2 ) ˜ r 2 + 1 -α 2 -2 i ˜ ω ˜ r 3 + 1 -α 2 -i ( m 1 + m 2 ) ( 3 -2 α 2 ) ˜ r 4 + 2 (1 -α 2 ) + O ( ˜ r 5 + ) , (5.3b) \nθ 0 = ˜ ω +( m 1 + m 2 ) α ˜ r + -3 2 ( 1 + α 2 ) 2 ˜ ω ˜ r 2 + -1 2 ( m 1 + m 2 ) α ( 3 + 10 α 2 +5 α 4 ) ˜ r 3 + + 1 8 ( 23 + 28 α 2 +70 α 4 +100 α 6 +35 α 8 ) ˜ ω ˜ r 4 + + O ( ˜ r 5 + ) , (5.3c) \nθ ∞ = ∆ -2 , (5.3d) \nz 0 = ( 1 -α 2 ) ˜ r 2 + -2 ( 1 -α 2 ) ( 1 + 2 α 2 +3 α 4 + α 6 ) ˜ r 4 + + O ( ˜ r 6 + ) , (5.3e) \nand the radial dictionary is defined as follows \na 0 = ± θ -2 , a t = ± θ + 2 , a 1 = ± θ ∞ 2 , a ∞ = ± θ 0 2 , t = z 0 . (5.4) \nIn addition, the Matone relation associates the accessory parameter of the Heun equation with the vacuum expectation value a of the gauge theory [39] \nu = -1 4 -a 2 + a 2 0 + a 2 t + t∂ t F ( t ) , (5.5) \nwhere F ( t ) is the instanton part of the NS free energy defined in (A16). Then, assuming an expansion for small ˜ r + of the form \na = ∞ ∑ n =0 b n ˜ r n + (5.6) \none can compute the coefficients of (5.6) recursively. The small ˜ r + expansion for a reads \na = 1 2 ( ℓ +1) -( 1 + α 2 ) 2 ( 3 ℓ ( ℓ +2) -∆(∆ -4) + 3˜ ω 2 ) 8 ( ℓ +1) ˜ r 2 + -( m 1 + m 2 ) α ( 1 + α 2 ) 2 ( 6 ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) ˜ ω 4 ℓ ( ℓ +1)( ℓ +2) ˜ r 3 + + O ( ˜ r 4 + ) (5.7) \nIn addition, the derivatives of the NS free energy in the case of the equally rotating KerrAdS 5 black hole are \n∂ a F inst = -1 2 ( ℓ +1) ( 1 -α 4 ) ˜ r 2 + + ( m 1 + m 2 )( ℓ +1) α ( 1 + α 2 ) 2 ˜ ω ( (∆ -2) 2 -˜ ω 2 ) ℓ 2 ( ℓ +2) 2 ˜ r 3 + + O ( ˜ r 4 + ) (5.8a) \n∂ a 1 F inst = 1 2 ( 1 -α 4 ) (∆ -2) ˜ r 2 + + ( m 1 + m 2 ) α ( 1 + α 2 ) 2 (∆ -2)˜ ω ℓ ( ℓ +2) ˜ r 3 + + O ( ˜ r 4 + ) (5.8b) \n∂ a t F inst = -i ( m 1 + m 2 ) α ( 1 + α 2 ) ( ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) 2 ℓ ( ℓ +2) ˜ r 2 + \n+ i (1 + α 2 ) ( ( 1 + α 2 ) ( ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) 2 ℓ ( ℓ +2) + ( m 1 + m 2 ) 2 α ℓ ( ℓ +2) ) ˜ ω ˜ r 3 + + O ( ˜ r 4 + ) (5.8c) \nInterestingly, a and ∂ a F inst are independent of the sign choice in (5.4), while ∂ a 1 F inst and ∂ a t F inst will get a global minus sign depending on the choice. Moreover, the poles located at a = ± n/ 2 for n ∈ N in the analytic expansion of the NS free energy (A16) are \ntranslated into poles for the values of the angular momentum quantum number of the form { ℓ, ℓ -1 , ℓ -2 , . . . } and will appear in the higher order terms of the asymptotic expansions for (5.7) and (5.8). It has been pointed out that the resummation procedure can eliminate the existing poles in the instanton partition function [40-42]. Bearing this in mind, one of the authors computed the low-energy absorption cross section of a Reissner-Nordström black hole in rainbow gravity by first applying the resummation procedure to the instanton contributions to the vacuum expectation value a , and then extending this method to the derivatives of the free energy [7]. \nBy inspecting the structure of a , one can recognize that up to a numerical factor, it coincides with the monodromy around two singular points in the Riemann-Hilbert map between the four-punctured Riemann sphere and Fuchsian systems [36]. Therefore, one can attempt an ansatz for the s-wave case ( ℓ = m 1 = m 2 = 0) of the form \na ( ℓ = 0) = 1 2 -ν 0 ˜ r 2 + + O ( ˜ r 3 + ) (5.9) \nand replace into the Matone relation (5.5). The left-hand side is given by equation (4.6c), with the radial accessory parameter K 0 taken from (2.20b), which for small ˜ r + yields \nlhs = -1 2 -(1 + α 2 ) 2 [ (1 -α 2 ) + ∆(∆ -4) 2 -˜ ω 2 2(1 -α 2 ) 2 + α 2 (4 + α 2 )˜ ω 2 2(1 -α 2 ) 2 ] ˜ r 2 + + O ( ˜ r 4 + ) (5.10) \nwhile the right hand side organizes as follows \nrhs = -1 2 + ν 0 ˜ r 2 + -( 1 + α 2 ) 2 (1 + α 4 )˜ ω 2 4(1 -α 2 ) 2 ˜ r 2 + + 1 8 ( 1 -α 4 ) ( ˜ ω 2 -(∆ -2) 2 ) ˜ r 2 + -2 ν 0 x ( 1 + x +2 x +5 x 3 +14 x 4 + . . . ) ˜ r 2 + + O ( ˜ r 4 + ) , (5.11) \nwhere \n1 + x +2 x 2 +5 x 3 +14 x 4 + . . . = 1 -√ 1 -4 x 2 x , (5.12a) \nx = ( 1 + α 2 ) 4 ˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) 2 6 ν 2 0 . (5.12b) \nThe terms inside the parenthesis proportional to ν 0 in equation (5.11) are associated with the sequence of Catalan numbers. Consequently, one can introduce the generating function for these numbers to compute the first correction, ν 0 . Hence, equating (5.10) and (5.11) gives a quadratic equation for ν 0 up to the order O ( ˜ r 2 + ) : \n1 8 ( 1 + α 2 ) 2 ( ∆(∆ -4) -3˜ ω 2 ) +8 ν 0 √ 1 + (1 + α 2 ) 4 ˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) 2 4 ν 2 0 ˜ r 2 + + O ( ˜ r 4 + ) = 0 . (5.13) \nIt turns out that ν 0 in terms of the black hole parameters has a surprisingly simple form \nν 0 = ± 1 8 ( 1 + α 2 ) 2 √ (3˜ ω 2 -∆(∆ -4)) 2 -4˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) , (5.14) \nand x introduced in (5.12b) takes the following form, \nx = ˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) (3˜ ω 2 -∆(∆ -4)) 2 -4˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) . (5.15) \nNotice that the series expansion for the vacuum expectation value (5.9), as well as the generating function of the Catalan numbers make sense if ˜ r 2 + < ν 0 ≤ 1 and | x | < 1 / 4 for given ˜ ω , α and ∆ . \nBy the same token, one can apply the resummation procedure to the derivatives of the NS free energy as follows \n∂ a F inst = -( 2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . ) + O ( ˜ r 2 + ) (5.16a) \n∂ a 1 F inst = 1 2 (∆ -2) [ ( 1 -α 4 ) + ( 1 + α 2 ) 4 ˜ ω 2 4 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ] ˜ r 2 + + O ( ˜ r 4 + ) (5.16b) \n∂ a t F inst = -i ( 1 + α 2 ) 2 ( ˜ ω 2 -(∆ -2) 2 ) ˜ ω 8 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ˜ r + + O ( ˜ r 3 + ) (5.16c) \nwhere ν 0 and x are defined in (5.14) and (5.15), respectively. We have observed that higher order terms in z 0 from the analytic expansion of the instanton part of the NS free energy contribute to the lower order terms and consequently we get a series in x as the correction terms nicely re-summed in the form of the corresponding generating functions #3 \n∂ a F inst := ∂ a F inst (0) + O ( ˜ r 2 + ) = log 4 -2 log ( 1 + √ 1 -4 x ) + O ( ˜ r 2 + ) (5.18a) \n∂ a 1 F inst := ∂ a 1 F inst (2) ˜ r 2 + + O ( ˜ r 4 + ) √ (5.18b) \n= 1 2 (∆ -2) [ ( 1 -α 4 ) + ( 1 + α 2 ) 4 ˜ ω 2 4 ν 0 1 -1 -4 x 2 x ] ˜ r 2 + + O ( ˜ r 4 + ) \n∂ a t F inst := ∂ a t F inst (1) ˜ r + + O ( ˜ r 3 + ) = -i ( 1 + α 2 ) 2 ( ˜ ω 2 -(∆ -2) 2 ) ˜ ω 8 ν 0 1 -√ 1 -4 x 2 x ˜ r + + O ( ˜ r 3 + ) (5.18c)", "5.1 Retarded Green's function": "In the previous section, we have computed the correction to a ( ℓ = 0) , as well as the derivatives of the instanton part of the NS free energy, which appear in the connection coefficients of the solutions of the Heun equation (4.8). Now we aim to derive an asymptotic \n1 + x +2 x 2 +5 x 3 +14 x 4 + . . . = 1 -√ 1 -4 x 2 x , (5.17a) \n2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . = log 4 -2 log ( 1 + √ 1 -4 x ) (5.17b) \nexpansion for the s-wave retarded Green's function in the equal angular momenta limit. This will be achieved by substituting (5.14), (5.15), and (5.18) into (4.17a) and expanding for small ˜ r + \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ∂ a 1 F inst (2) +(∆ -2) ( 1 + 2 α 2 (1 + α 2 ) ) -4(∆ -2) ν 0 ((∆ -2) 2 -˜ ω 2 ) + 2 π sin π ∆ cos π ˜ ω -cos π ∆ e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 + 3 4 ( 1 + α 2 ) 2 ˜ ω ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) )] ˜ r 2 + } + O ( ˜ r 3 + ) , (5.19) \nwhere ψ (0) ( z ) corresponds to the digamma function, ∂ a F inst (0) and ∂ a 1 F inst (2) are given by (5.18a) and (5.18b), and refer to the coefficients of the series expansion in ˜ r + for the derivatives of the instanton part of the NS free energy with respect to a and a 1 , respectively. Note that expression (5.19) assumes that ∆ is not an integer (see [43] regarding the case when ∆ is an integer). Then, the asymptotic expansion for the retarded Green's function is given by \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + 1 4 [ 3 ( 1 + α 2 ) 2 ˜ ω × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) -( 1 + α 2 ) 2 ( 2(∆ -2) ((∆ -2) 2 -˜ ω 2 ) + π sin π ∆ cos π ∆ -cos π ˜ ω ) √ (∆(∆ -4) -3˜ ω 2 ) 2 +2(∆ -2) ( 3 + 4 α 2 +3 α 4 ) ] ˜ r 2 + } + O ( ˜ r 3 + ) . (5.20)", '5.2 Greybody factor': "The computation of the greybody factor in asymptotically AdS spacetimes is a bit more subtle than in asymptotically flat or dS spacetimes. Due to the nature of the boundary condition at spatial infinity, the radiation produced at the horizon can travel all the way to the spatial infinity to be reflected back towards the black hole. As a result, the thermal equilibrium of black holes in AdS is ensured by this infinite mechanism [27]. At the level of the scattering problem, the required boundary conditions cannot be satisfied since one cannot identify an outgoing wave solution at infinity. Nevertheless, using an approximation scheme the greybody factors of static and spherically symmetric, and rotating black holes in asymptotically AdS have been studied in [27, 28], respectively. By considering different regions of the spacetime, the radial equation simplifies and analytical solutions can be found. These solutions can be matched in an overlapping region, resulting in a consistent definition for the fluxes at the horizon and at spatial infinity. \nIn addition, [29] has performed the numerical computation of the absorption cross section in a similar fashion way. The asymptotic analysis at infinity remains the same as [27, 28], while in the near-horizon region, the radial solutions are given in terms of the Heun functions instead of the Hypergeometric functions. Our approach follows both ideas: an approximate radial equation in the far-region and Heun functions at the horizon. However, we will introduce the exact connection coefficients rather than the ratio of the Wronskians as [29]. Finally, we will define the conserved fluxes as \nF = 1 2 i ( R ∗ ˜ r ∆ ˜ r dR d ˜ r -R ˜ r ∆ ˜ r dR ∗ d ˜ r ) , (5.21) \nso that the greybody factor is the ratio between the flux at the horizon and the flux coming in from infinity \nγ ( ℓ ) = F hor F ( ∞ ) in (5.22) \nWe consider radial equation in the far region approximation ˜ r ≫ 1 , such that the radial equation (2.13) reduces to \nR '' + 5 ˜ r R ' -[ ∆(∆ -4) ˜ r 2 + ℓ ( ℓ +2) -˜ ω 2 ˜ r 4 ] R = 0 , (5.23) \nthen, we introduce a new radial coordinate \nu = ˜ ω ˜ r (5.24) \nand consider the limit u ≪ ˜ ω . As a result, equation (5.23) yields \nR '' -3 u R ' + ( 1 -ℓ ( ℓ +2) ˜ ω 2 -∆(∆ -4) u 2 ) R = 0 , (5.25) \nwhose solution is given by the linear combination of the Bessel functions, J ν ( z ) and Y ν ( z ) , of the form \nR ( u ) = C 1 u 2 J ∆ -2 ( √ 1 -ℓ ( ℓ +2) ˜ ω 2 u ) + C 2 u 2 Y ∆ -2 ( √ 1 -ℓ ( ℓ +2) ˜ ω 2 u ) , (5.26) \nwhich can be written, more conveniently, in terms of the Hankel functions \nR ( u ) = 1 2 ( C 1 -iC 2 ) u 2 H (1) ∆ -2 ( u ) + 1 2 ( C 1 + iC 2 ) u 2 H (2) ∆ -2 ( u ) , (5.27) \nsince their asymptotic structures describe the incoming and outgoing part of the wave function. Namely, the H (1) ∆ -2 ( u ) is associated with the incoming part of the solution, while H (2) ∆ -2 ( u ) controls the outgoing part. In the limit ˜ r →∞ the radial solution for the s-wave case behaves like \nR ( r ) ≃ ( C 1 A J + C 2 A Y ) ˜ r -∆ + C 2 B Y ˜ r ∆ -4 , (5.28) \nwhere \nA J = 4 Γ(∆ -1) ( ˜ ω 2 ) ∆ , A Y = -4 π cos π (∆ -2)Γ(2 -∆) ( ˜ ω 2 ) ∆ , B Y = -4 π Γ(∆ -2) ( ˜ ω 2 ) 4 -∆ . (5.29) \nBy comparing the asymptotic behavior of the solutions (5.28) with (4.16), we obtain \nC 1 = A 1 -A J C 1 --A 1+ A Y A J B Y C 1+ , C 2 = A 1+ B Y C 1+ , (5.30) \nwhere C 1 ± are given in (4.15) and \nA 1 -= C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 ∆ , A 1+ = C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 (4 -∆) . (5.31) \nFurthermore, by inspecting the radial solution (5.27), we have \nC in ≡ 1 2 ( C 1 -iC 2 ) , C out ≡ 1 2 ( C 1 + iC 2 ) , (5.32) \nsuch that one can reproduce the (incoming) outcoming coefficients in [29]. Therefore, the asymptotic incoming radial solution with quantum numbers ( ℓ = m 1 = m 2 = 0) reads \nR ( ∞ ) in (˜ r ) = C in ( ˜ ω ˜ r ) 2 H (1) ∆ -2 ( ˜ ω ˜ r ) , (5.33) \nsuch that after substituting (5.33) into equation (5.21), we get the incoming flux at spatial infinity \nF ( ∞ ) in = -2 | C in | 2 ˜ ω 4 π (5.34) \nOn the other hand, the solution at the horizon is given by (4.12), and the associated flux at the horizon is \nF hor = -2 ∆ | C z 0 -| 2 ˜ ω ˜ r 3 + ( 1 + ˜ r 2 + ) ∆ / 2 ( 1 + (1 + 2˜ r 2 + ) α 2 ) 2 √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + 1 + ( 3 + 2(1 + ˜ r 2 + ) α 2 ) ˜ r 2 + + √ 1 + ˜ r 2 + √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + ∆ (5.35) \nThe greybody factor given as the ratio of the flux at the horizon and the incoming flux at spatial infinity reads \nγ ( ℓ =0) = F hor F ( ∞ ) in = | C z 0 -| 2 | C in | 2 2 ∆ -1 π ˜ r 3 + ˜ ω 3 ( 1 + ˜ r 2 + ) ∆ / 2 ( 1 + (1 + 2˜ r 2 + ) α 2 ) 2 √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + 1 + ( 3 + 2(1 + ˜ r 2 + ) α 2 ) ˜ r 2 + + √ 1 + ˜ r 2 + √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + ∆ (5.36) \nSince we are interested in the contribution of the resummation technique for the greybody factor in the small ˜ r + limit, we will derive asymptotic expansions without and with the resummation procedure. For the former we replace (5.7) and (5.8) into (5.36), while keeping \nthe expansions up to second order O ( ˜ r 2 + ) to avoid the pole at ℓ = 0 . Bearing this in mind, the greybody factor reads \nγ (0) = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + 2 2∆+6 π ˜ ω 2∆+5 sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 ( 3˜ ω 2 -∆(∆ -4) ) 4 ((3˜ ω 2 -∆(∆ -4)) 2 -˜ ω 2 (˜ ω 2 -(∆ -2) 2 )) 2 1 χ + O ( ˜ r 4 + ) (5.37) \nwhere \nχ = 4 2∆ Γ ( 1 2 (∆ -˜ ω -2) ) 2 Γ ( 1 2 (∆ + ˜ ω -2) ) 2 ˜ ω 8 +4 4 Γ ( 1 2 (2 -∆ -˜ ω ) ) 2 Γ ( 1 2 (2 -∆+˜ ω ) ) 2 ˜ ω 4∆ + 4 ∆+4 π 2 ˜ ω 2∆+4 cos 2 π ∆ (cos π ∆ -cos π ˜ ω ) ((∆ -2) 2 -˜ ω 2 ) . (5.38) \nOn the other hand, for the latter case we substitute (5.14), (5.15), and (5.16) into (5.36), which implies \nγ (0) = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + 2 2∆+6 π ˜ ω 2∆+5 sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 1 χ + O ( ˜ r 4 + ) (5.39) \nWe observe that the highlighted factor in (5.37) is corrected by a factor (highlighted in blue) containing all the contributions given by the derivative of the instanton part of the free energy at zeroth order, ∂ a F inst (0) . Furthermore, one can replace it by its generating function (5.18a), to obtain \ne ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 = 1 , (5.40) \nwhich reduces the greybody factor (5.39) to \nγ (0) = 2 2∆+6 π ˜ ω 2∆+5 ˜ A sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 χ + O ( ˜ r 4 + ) , (5.41) \nwhere ∆ is not integer, and ˜ A = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + is related to the area of the fivedimensional Kerr-AdS black hole with equal angular momenta in the small-radius limit. Note that in the non-rotating limit α → 0 , equation (5.41) gives the greybody factor for massive scalar fields in small Schwarzschild-AdS 5 black holes. \nIn Fig. 1, we present the greybody factor for small Kerr-AdS 5 black holes with equal rotation parameters and ( ℓ = m 1 = m 2 = 0) modes, as calculated using formula (5.41). The analysis is performed for different values of the extremality parameter α = { 0 , 1 / 3 , 2 / 3 , 9 / 10 , 99 / 100 } , while keeping fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 , and varying the frequency ˜ ω . It is worth mentioning that the greybody spectrum increases with α for fixed ˜ ω , which contrasts with previous results for massless scalar fields in rotating cohomogeneity -1 BH space-times [28]. As observed in [27], the large-amplitude oscillations in the greybody spectrum are consistent with the spacing between the normal modes frequencies in pure \nfive-dimensional AdS spacetime. In addition, we observed an intriguing dynamics in the spectrum at small ˜ ω . As we increase the conformal dimension ∆ , two peaks appear and start to move closer, merging and then separating again, as illustrated in Fig. 2. \nIn Fig. 3, we compare the greybody factor calculated with and without the resummation procedure. The solid curves, representing the asymptotic formula with resummation (5.41), display a compressed spectrum compared to the dashed curves, which correspond to the expression without resummation (5.37). \nFigure 1 . Greybody factor (5.41) as a function of ˜ ω for different values of α , and fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 . \n<!-- image -->", '6 Small Reissner-Nordström-AdS 5 black holes': "We now turn to study the retarded Green's function and the greybody factor of a RN-AdS 5 black hole in the small-radius limit. To this matter, it is convenient to express the charge ˜ Q in terms of ˜ r + . Specifically, the temperature at the outer horizon defined as \n˜ T + = 1 2 π [ 1 ˜ r + -˜ Q 2 ˜ r 5 + +2˜ r + ] , (6.1) \nvanishes at extremality, indicating that the maximal charge is given by \n˜ Q c = ˜ r 2 + √ 1 + 2˜ r 2 + , (6.2) \nsuch that Q ≤ Q c must be satisfied to ensure a regular outer horizon. We then parametrize the accessible charge values as \n˜ Q = q ˜ Q c = q ˜ r 2 + √ 1 + 2˜ r 2 + , 0 ≤ q ≤ 1 , (6.3) \nFigure 2 . Greybody factor as a function of ˜ ω for fixed values of α and ˜ r + , while varying ∆ . \n<!-- image --> \nFigure 3 . Comparison between the greybody factors with (solid lines) and without resummation (dashed lines) for different values of α and fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 . \n<!-- image --> \nwhere q is an extremality parameter, which describes the Schwarzschild-AdS 5 black-hole solution for q = 0 and the extremal RN-AdS 5 black hole for q = 1 . Plugging (6.3) into (3.15) and expanding for small ˜ r + yields \nθ -= i ( q ˜ ω -˜ e ) q 2 ˜ r + 1 -q 2 -i ( ˜ e -3 q 2 ˜ e -q ( 1 -5 q 2 ) ˜ ω ) q 2 ˜ r 3 + 2 (1 -q 2 ) + O ( ˜ r 5 + ) , (6.4a) \nθ + = i (˜ ω -q ˜ e ) ˜ r + 1 -q 2 -i (2˜ ω -q ˜ e ) ˜ r 3 + 1 -q 2 + O ( ˜ r 5 + ) , (6.4b) \nθ 0 = ˜ ω -( 3 2 ( 1 + q 2 ) ˜ ω -q ˜ e ) ˜ r 2 + + 1 8 (( 23 + 26 q 2 +35 q 4 ) ˜ ω -4 ( 3 + 5 q 2 ) q ˜ e ) ˜ r 4 + + O ( ˜ r 6 + ) , (6.4c) \nθ ∞ = ∆ -2 , (6.4d) \nwhile the conformal modulus (3.18) reads \nz 0 = ( 1 -q 2 ) ˜ r 2 + -2 ( 1 -q 4 ) ˜ r 4 + + O ( ˜ r 6 + ) (6.4e) \nNext, one can solve Matone's relation (5.5) to obtain an expansion for the vacuum expectation value a in the form \na = 1 2 ( ℓ +1) -( ( 1 + q 2 ) ( 3˜ ω 2 -∆(∆ -4) + 3 ℓ ( ℓ +2) ) 8( ℓ +1) -q ˜ e ˜ ω 2( ℓ +1) ) ˜ r 2 + + O ( ˜ r 4 + ) (6.5) \nanalogous to (5.7). In addition, the derivatives of the NS free energy are \n∂ a F inst = -1 2 ( 1 -q 2 ) ( ℓ +1) ˜ r 2 + + O ( ˜ r 4 + ) (6.6a) \n∂ a 1 F inst = 1 2 ( 1 -q 2 ) (∆ -2) ˜ r 2 + + O ( ˜ r 4 + ) (6.6b) \n∂ a t F inst = i (˜ ω -q ˜ e ) ( (∆ -2) 2 + ℓ ( ℓ +2) -˜ ω 2 ) ˜ r 3 + 2 ℓ ( ℓ +2) + O ( ˜ r 5 + ) (6.6c) \nDue to the presence of a pole structure in the angular momentum quantum number ℓ within the expansions (6.5) and (6.6), it is necessary to sum the contributions from all orders in z 0 , as demonstrated in [44]. Specifically, we limit our analysis to the s-wave case ( ℓ = 0) , following the same procedure introduced for the equal angular momenta Kerr-AdS 5 black hole in section 5. In other words, we consider an ansatz of the form \na ( ℓ = 0) = 1 2 -ν 0 ˜ r 2 + + O ( ˜ r 4 + ) (6.7) \nand replace into the Matone's relation (5.5). The left-hand side is given by equation (4.6c), with the radial accessory parameter K 0 taken from (3.20b), which for small ˜ r + yields \nlhs = -1 2 -1 4 [ ∆(∆ -4) -6 q 2 (1 -q 2 ) -˜ ω 2 -(3˜ ω -4 q ˜ e ) q 2 ˜ ω -( 1 + q 2 ) q 2 ˜ e 2 -2 ( 1 -q 6 ) (1 -q 2 ) 2 ] ˜ r 2 + + O ( ˜ r 4 + ) (6.8) \nwhile the right hand side organizes as follows \nrhs = -1 2 + ν 0 ˜ r 2 + -1 8 ( 1 -q 2 ) ( (∆ -2) 2 -˜ ω 2 ) ˜ r 2 + -( (1 + q 6 )˜ ω 2 -2(1 + q 4 ) q ˜ e ˜ ω +(1 + q 2 ) q 2 ˜ e 2 ) ˜ r 2 + 4 (1 -q 2 ) 2 -2 ν 0 x ( 1 + x +2 x +5 x 3 +14 x 4 + . . . ) ˜ r 2 + + O ( ˜ r 4 + ) , (6.9) \nwhere \nx = ( (∆ -2) 2 -˜ ω 2 ) ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 2 6 ν 2 0 , (6.10) \nand the terms inside the parenthesis in equation (6.9) are associated with the sequence of Catalan numbers (5.12a), as presented in the previous section. Equating (6.8) and (6.9) leads to an equation for ν 0 that can be solved to yield \nν 0 = ± 1 8 √ √ √ √ √ 4 ( (∆ -2) 2 -˜ ω 2 ) ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) + (( 1 + q 2 ) ( 3˜ ω 2 -∆(∆ -4) ) -4 q ˜ e ˜ ω ) 2 , (6.11) \nwhich in the limit q → 0 gives the correction to Schwarzschild-AdS 5 black-hole solution. Furthermore, x reduces to \nx = 1 ( 4 + ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 2 ((∆ -2) 2 -˜ ω 2 ) ( (1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 ) ) ) . (6.12) \nAnalogously, the derivatives of the NS free energy are \n∂ a F inst = -( 2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . ) + O ( ˜ r 2 + ) (6.13a) \n∂ a 1 F inst = 1 2 (∆ -2) [ ( 1 -q 2 ) + ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 4 ν 0 (6.13b) \n× ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ] ˜ r 2 + + O ( ˜ r 4 + ) \n∂ a t F inst = -i (˜ ω -q ˜ e ) ( (∆ -2) 2 -˜ ω 2 ) 8 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ˜ r + + O ( ˜ r 3 + ) (6.13c) \nwhere ν 0 and x are defined in (6.11) and (6.12), respectively. By means of (5.17), one can substitute the series expansions in (6.13) by their generating functions as follows \n∂ a F inst := ∂ a F inst (0) + O ( ˜ r 2 + ) = log 4 -2 log ( 1 + √ 1 -4 x ) + O ( ˜ r 2 + ) (6.14a) \n∂ a 1 F inst := ∂ a 1 F inst (2) ˜ r 2 + + O ( ˜ r 4 + ) \n= 1 2 (∆ -2) [ ( 1 -q 2 ) + ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 4 ν 0 1 -√ 1 -4 x 2 x ] ˜ r 2 + + O ( ˜ r 4 + ) (6.14b) \n∂ a t F inst := ∂ a t F inst (1) ˜ r + + O ( ˜ r 3 + ) = -i (˜ ω -q ˜ e ) ( (∆ -2) 2 -˜ ω 2 ) 8 ν 0 1 -√ 1 -4 x 2 x ˜ r + + O ( ˜ r 3 + ) (6.14c) \nInterestingly, the corrections to the derivatives of F inst in the case of small RN-AdS 5 BHs match the order of ˜ r + in the corrections found in the case of small Kerr-AdS 5 BHs with equal rotation (5.18). Furthermore, the most singular terms at each given order of ∂ a F inst are responsible for the appearance of the branch cut after the resummation, as we have seen in the coefficient ∂ a F inst (0) , which contains a logarithm.", "6.1 Retarded Green's function": "The retarded Green's function, defined as the ratio of the response to the source, is given by (4.17), where the radial dictionary (4.6b) is realized by (6.4). The new elements in the connection coefficients, such as a ( ℓ = 0) and the derivatives of F inst are obtained from (6.7) and (6.14). Expanding for small ˜ r + , we derive \nG \nret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ∂ a 1 F inst (2) -( e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 -1 4 ( 3(1 + q 2 )˜ ω -2 q ˜ e ) ) ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) ) -( e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 + 1 4 ( 3(1 + q 2 )˜ ω -2 q ˜ e ) ) ( ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) +(∆ -2) ( 1 + 2 q 2 ) -8 ν 0 (∆ -2) ((∆ -2) 2 -˜ ω 2 ) e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) ] ˜ r 2 + } + O ( ˜ r 3 + ) , (6.15) \nwhich can be further simplified, yielding \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + 1 4 [ 6 ( 1 + q 2 ) (∆ -2) -4(∆ -2) √ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ((∆ -2) 2 -˜ ω 2 ) -1 2 ( 4 q ˜ e -6(1 + q 2 )˜ ω + √ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ) × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) ) + 1 2 ( 4 q ˜ e -6(1 + q 2 )˜ ω -√ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ) × ( ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) ] ˜ r 2 + } + O ( ˜ r 3 + ) . (6.16)", '6.2 Greybody factor': 'The computation of the greybody factor for the RN-AdS 5 black hole follows section 5.2. In the far-region approximation, the radial equation (3.13) simplifies to (5.23), then one can consider that the incoming flux at spatial infinity is \nF ( ∞ ) in = -2 | C in | 2 ˜ ω 4 π , (6.17) \nwhile the flux at the horizon corresponds to the incoming radial solution (4.12), expressed in terms of the RN-AdS 5 black hole parameters: \nF hor = -2 ∆ | C z 0 -| 2 ( ˜ ω -q ˜ e √ 1 + 2˜ r 2 + ) ˜ r 3 + √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + 1 + 3˜ r 2 + + √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + ∆ (6.18) \nTherefore, the greybody factor for the s-wave reads \nγ ( ℓ =0) = F hor F ( ∞ ) in = | C z 0 -| 2 | C in | 2 2 ∆ -1 π ˜ r 3 + ˜ ω 4 ( ˜ ω -q ˜ e √ 1 + 2˜ r 2 + ) √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + 1 + 3˜ r 2 + + √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + ∆ , (6.19) \nwhere C in is defined in (5.32). By replacing (6.5) and (6.6) into (6.19), while retaining terms up to second order O ( ˜ r 2 + ) to avoid the pole at ℓ = 0 , the greybody factor in the small ˜ r + limit becomes: \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 ( 1 + ((∆ -2) 2 -˜ ω 2 ) ( (1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 ) ) ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 2 ) 1 χ + O ( ˜ r 4 + ) (6.20) \nwhere \nχ = 4 2∆ Γ ( 1 2 (∆ -˜ ω -2) ) 2 Γ ( 1 2 (∆ + ˜ ω -2) ) 2 ˜ ω 8 +4 4 Γ ( 1 2 (2 -∆ -˜ ω ) ) 2 Γ ( 1 2 (2 -∆+˜ ω ) ) 2 ˜ ω 4∆ + 4 ∆+4 π 2 ˜ ω 2∆+4 cos 2 π ∆ (cos π ∆ -cos π ˜ ω ) ((∆ -2) 2 -˜ ω 2 ) , (6.21) \nNevertheless, the resummation technique gives a contribution to the asymptotic expansion, determined by (6.11) and (6.14), \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 1 χ + O ( ˜ r 4 + ) (6.22) \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 χ + O ( ˜ r 4 + ) (6.23) \nIn Fig. 4, we present the greybody factor for small RN-AdS 5 black holes with ℓ = 0 modes, as calculated using formula (6.23). The analysis is performed for different values of \nthe extremality parameter q = { 0 , 1 / 3 , 2 / 3 , 9 / 10 , 99 / 100 } , while keeping fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 , and varying the frequency ˜ ω . Interestingly, our results show that the greybody spectrum decreases as q increases, which contrasts our previous results in Fig. 1. Although α and q play the role of extremality parameters, their physical nature is different: the former is associated with the angular momentum of the black hole, while the latter corresponds to the charge of the black hole. It is interesting to note that in the γ (0) vs ˜ ω plot for Kerr-AdS 5 , γ (0) increases with α for a fixed ˜ ω . On the other hand, for RN-AdS 5 black hole due the presence of a factor (˜ ω -q ˜ e ) , the greybody factor γ (0) decreases with q for a fixed value of ˜ ω . We see this distinctive behavior by comparing the plot 1 with plot 4. In addition, for 0 < ˜ ω < q ˜ e , the greybody factor is negative, as shown in the panel on the upper-left corner of figure 4, pointing out a superradiant frequency window. The first negative peak increases (in absolute value) as we increase q , contrary to the behavior seen for ˜ ω > q ˜ e , where the spectrum shrinks. \nIn Fig. 5, we compare the greybody factor calculated with and without the resummation procedure. The solid curves, representing the asymptotic formula with resummation (6.23), show a compressed spectrum compared to the dashed curves, which describe the expression without the resummation (6.20). \nFigure 4 . Greybody factor (6.23) as a function of ˜ ω for different values of q , and fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 . \n<!-- image -->', '7 Discussion': "In this paper, we have investigated scalar perturbations in asymptotically AdS sapcetimes, focusing on Kerr-AdS 5 and Reissner-Nordström-AdS 5 black holes. In both cases, the radial ODE can be transformed into a Heun equation -a second-order ODE with four regular singular points. Recently, Heun solutions and their connection coefficients have been computed in terms of Virasoro conformal blocks and, via AGT correspondence, can be expressed in term of the Nekrasov-Shatashvili partition function of an SU (2) supersymmetric gauge theory with four fundamental hypermultiplets [21]. By means of these tools, we have computed the retarded Green's functions and the greybody factor. \nFigure 5 . Comparison between the greybody factors with resummation (solid lines) and without resummation (dashed lines) for different values of q , and fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 . \n<!-- image --> \nInterestingly, these functions introduce new elements, such as the vacuum expectation value of the scalar in the vector hypermultiplet a , and the derivatives of the NS function with respect to a and the masses of the gauge theory, ∂ a F inst and ∂ a i F inst , as seen in (4.15). We observed that by solving Matone's relation for a at small ˜ r + , the poles in the analytic expansion of the NS function (A16), located at a = ± n/ 2 for n ∈ N , correspond to poles in the angular momentum quantum number ℓ for the series expansion of a . Consequently, these poles appear in the expansions of the derivatives of F inst . Bearing this in mind, we perform the computation of the retarded Green's function and the greybody factor for the pole at ℓ = 0 using two strategies: one based on keeping the expansions up to order O ( ˜ r 2 + ) to avoid the pole and one based on curing the pole by considering contributions from all orders in the analytic expansion of the NS function [7]. It is worth mentioning that these features are present in both backgrounds, making the computation analogous, and we restrict our analysis for small ˜ r + black holes, including an extremality parameter. \nAt order O ( ˜ r 2 + ) , the quantities a , ∂ a F inst , and ∂ a i F inst remain finite in the case ℓ = 0 , as shown in (5.8), such that by substituting these into (4.17) and expanding for small ˜ r + , we derive an asymptotic expression for G ret in this limit. For the Kerr-AdS 5 with equal rotation, the result is given by (C1), while for RN-AdS 5 , G ret takes the form (C2). \nIn asymptotically AdS spacetimes, the computation of the greybody factor as the ratio of the flux at the horizon to the incoming flux from infinity is more subtle, due to the asymptotic behavior of the solutions at infinity (4.16), which prevents us identifying the ingoing and outgoing wave behavior. However, using a far-region approximation introduced in [27], we can transform the radial equation into a Bessel's diferrential equation (5.23), where the asymptotic form of the solutions reproduces the ingoing and outgoing waves. For the flux at the horizon, we will use the Heun functions and their connections coefficients. Regarding the computation without the resummation procedure, the greybody factors for small Kerr-AdS 5 and RN-AdS 5 are given in (5.37) and (6.20), respectively. \nOn the other hand, the pole structure in the analytic expansion of the instanton part of \nthe NS free energy suggests that at each order, the most singular term provides a contribution that can be resummed. For instance, we have shown that at ℓ = 0 , the first correction ν 0 to a ( ℓ = 0) is determined by the generating function of the Catalan numbers [45]. It turns out that the derivatives of F inst also receives corrections from the most singular terms at each order. For instance, the leading singularities from ∂ a F inst sum up into a logarithm at order O ( ˜ r 0 + ) . Then, we use them into the formulae (4.17) and (5.22). \nThe asymptotic expansions for the retarded Green's functions in both spacetimes are corrected in a non-trivial manner, involving terms like ν 0 , ∂ a 1 F inst (2) and e ∂aF inst (0) -x e ∂aF inst (0) + x . Surprisingly, these corrections simplify our results (5.20) and (6.16) in comparison with the asymptotic expansions in Appendix C. \nFor the s-wave greybody factor, the effect of resummation simplifies the asymptotic expansions, as well as it leads to a greybody spectrum γ (0) (˜ ω ) that is compressed in comparison with the spectrum without the resummation procedure, as shown in Fig. 3 and 5. Furthermore, we observe in Fig. 1 that the greybody factor of Kerr-AdS 5 with equal rotation increases when the rotation parameter α increases, contrary to previous results [28]. In Fig. 2 an interesting dynamics show up at small frequencies, as we vary the conformal dimension, two peaks emerge and move closer possibly merge and the separate again. \nFor RN-AdS 5 black holes, the spectrum decreases as we increase the extremality parameter q . For 0 < ˜ ω < q ˜ e , the greybody factor is negative, corresponding to a superradiant instability (see Fig. 4). Finally, it will be interesting to incorporate the higher order corrections in a and see the effect in physical observables. It is also worth to explore the effect of resummation in AdS hyperbolic black hole.", 'Acknowledgments': 'We thank Atanu Bhatta for useful discussion and collaboration at the initial stage of the work. JBA acknowledges financial support from the Fundação para a Ciência e a Tecnologia (FCT) - Portugal through the research project 10.54499/2022.03702.PTDC (GENIDE). SC acknowledges ANRF grant CRG/2022/006165. AM would like to thank the Council of Scientific and Industrial Research (CSIR), Government of India, for the financial support through a research fellowship (File No.: 09/1005(0034)/2020-EMR-I).', 'A Nekrasov-Shatashvili function with N f = 4 fundamental hypermultiplets': "Here, we review the SU (2) Nekrasov-Shatashvili function with N f = 4 fundamental hypermultiplets. Nekrasov considered a deformed lagrangian of 4 d , N = 4 gauge theory by introducing two deformation parameters ϵ 1 , 2 [3, 46]. These parameters parameterize the SO (4) rotation of the spacetime R 4 . As a result, the translation symmetry is broken. Nekrasov partition function depends on the coupling parameter τ , VEV ' a ' of the adjoint scalars in the vector multiplets, and hypermultiplet masses m . It consists of three parts, \nnamely, the classical, the one-loop and the instanton part: \nZ ( τ, a, m ; ϵ 1 , 2 ) = Z classical Z 1-loop Z inst (A1) \nIts key characteristic is that it provides the prepotential of the theory in the limit ϵ 1 , 2 → 0 and coincides with the prepotential as determined by the Seiberg-Witten curve. Here, we focus on instanton part of Nekrasov partition function for SU (2) gauge theory which is obtained from the U (2) partition function by dividing it with U (1) -factor [4, 9]. The U (2) partition function is expressed in terms of the combinatorial formula which we review below. Let us denote a partition (Young Tableau) by \nY = ( y 1 , y 2 , ... ) (A2) \nwhere y i is the height of the i -th column and y i = 0 , when i is larger than the width of the tableau. Its transpose is given by \nY t = ( y t 1 , y t 2 , ... ) (A3) \nWe write a vector of Young Tableau as \nY = ( Y 1 , Y 2 ) (A4) \nFor a given Young diagram Y , we denote the arm length and the leg length of a box ' s ' with respect to the diagram Y as \nA Y ( i, j ) = y j -i, L Y ( i, j ) = y t i -j (A5) \nwhere ( i, j ) denotes the coordinates of the box ' s '. We do not restrict ' s ' to be in Y . A Y and L Y can be negative if the box ' s ' lies outside Y [20]. \nThe instanton part of Nekrasov partition function is given by the summation over the Young tableaux, whose summand is the product of factors corresponding to the field content of the Lagrangian. \nZ U (2) inst ( t,⃗a, m ; ϵ 1 , 2 ) = ∑ ⃗ Y t | ⃗ Y | z vec ( ⃗a, ⃗ Y ) z hyp ( ⃗a, ⃗ Y , m ) , (A6) \nwhere ⃗a = ( a 1 , a 2 ) is the VEV of the scalar in the vector multiplet, | ⃗ Y | = | Y 1 | + | Y 2 | denotes the total number of boxes (i.e. instanton number) in both Y 1 and Y 2 . The instanton counting parameter t is given by \nt = e 2 πiτ (A7) \nwhere τ is related with the gauge coupling constant by \nτ = θ 2 π + i 4 π g 2 Y M (A8) \nThe hypermultiplet and vector contributions are given by [47, 48] \nz hyp ( ⃗a, ⃗ Y , m ) = ∏ I =1 , 2 ∏ s ∈ Y I [ a I + m + ϵ 1 ( i -1 2 ) + ϵ 2 ( j -1 2 )] , z vec ( ⃗a, ⃗ Y ) = 2 ∏ I,J =1 ∏ s ∈ Y I 1 a I -a J -ϵ 1 L Y I ( s ) + ϵ 2 ( A Y I ( s ) + 1) × ∏ t ∈ Y J 1 a I -a J + ϵ 1 ( L Y I ( t ) + 1) -ϵ 2 A Y J ( t ) (A9) \nIn our work, we always consider a 1 = -a 2 = a . Furthermore, let us denote with m 1 , m 2 , m 3 and m 4 the masses of the four hypermultiplets. Additionally, we introduce the gauge parameters a 0 , a t , a 1 and a ∞ which are related to the masses m i of the hypermultiplets via \nm 1 = a t + a 0 , m = a a , m = a + a , m = a a . \n2 t -0 3 1 ∞ 4 1 -∞ (A10) \nOn the other hand, the U (1) -partition function for N f = 4 is given by \nZ N f =4 U (1) = (1 -t ) 2( a 1 + ϵ 2 )( a t + ϵ 2 ) /ϵ 1 ϵ 2 (A11) \nwhere ϵ = ϵ 1 + ϵ 2 . Now, we define the SU (2) partition function which is given by \nZ SU (2) inst ( t,⃗a, m 1 , m 2 , m 3 , m 4 ; ϵ 1 , 2 ) = Z -1 U (1) ( t, m 1 , m 2 , m 3 ; ϵ 1 , 2 ) Z U (2) inst ( t,⃗a, m 1 , m 2 , m 3 , m 4 ; ϵ 1 , 2 ) (A12) \nMoreover, to work in Nekrasov-Shatashvili (NS) limit, we consider ϵ 2 → 0 while keeping ϵ 1 fixed (we set ϵ 1 = 1 ). Then, the instanton part of NS free energy is defined as \nF ( N f =4) inst ( a, m i , t ) = lim ϵ 2 → 0 ( ϵ 2 log Z SU (2) inst ( a, m i , t ; ϵ 2 ) ) (A13) \nThis can be rewritten as [9, 20, 23] \nF (4) inst ( a, m i , t ) = lim ϵ 2 → 0 ϵ 2 log [ (1 -t ) -2 ϵ -1 2 ( a 1 + 1 2 )( a t + 1 2 ) ∑ ⃗ Y t | ⃗ Y | z vec ( ⃗a, ⃗ Y ) z hyp ( ⃗a, ⃗ Y , m i ) ] (A14) \nAn interesting property of NS-free energy is that it is a convergent series in t . Hence, it can be written as \nF (4) inst ( a, m i , t ) = ∞ ∑ n ≥ 1 c n ( a, m 1 , m 2 , m 3 , m 4 ) t n (A15) \nThe expansion up to the order t 2 is given below; \nF (4) inst ( a ; m i ; t ) = [ 1 8 ( 1 -4 a 2 ) -1 2 ( m 1 m 2 + m 3 m 4 ) -m 1 m 2 m 3 m 4 2 ( a -1 2 ) + m 1 m 2 m 3 m 4 2 ( a + 1 2 ) ] t + [ 1 128 ( 9 -26 a 2 ) -( m 1 + m 2 ) 2 64 -( m 3 + m 4 ) 2 64 -7 32 ( m 1 m 2 + m 3 m 4 ) -( 1 -4 m 2 1 ) ( 1 -4 m 2 2 ) ( 1 -4 m 2 3 ) ( 1 -4 m 2 4 ) 2048 ( a -1) + ( 1 -4 m 2 1 ) ( 1 -4 m 2 2 ) ( 1 -4 m 2 3 ) ( 1 -4 m 2 4 ) 2048 ( a +1) -m 2 1 m 2 2 ( m 2 3 + m 2 4 ) + m 2 3 m 2 4 ( m 2 1 + m 2 2 ) +4 m 1 m 2 m 3 m 4 (1 -m 1 m 2 m 3 m 4 ) 16 ( a -1 2 ) + m 2 1 m 2 2 ( m 2 3 + m 2 4 ) + m 2 3 m 2 4 ( m 2 1 + m 2 2 ) +4 m 1 m 2 m 3 m 4 (1 -m 1 m 2 m 3 m 4 ) 16 ( a + 1 2 ) + m 2 1 m 2 2 m 2 3 m 2 4 16 ( a -1 2 ) 3 -m 2 1 m 2 2 m 2 3 m 2 4 16 ( a + 1 2 ) 3 ] t 2 + O ( t 3 ) \n(A16)", 'B Angular Equation': 'By two consecutive transformations χ = sin 2 θ , and u = χ/ ( χ -χ 0 ) , with χ 0 = (1 -˜ a 2 1 ) / (˜ a 2 2 -˜ a 2 1 ) , we can take the four singular points of (2.9b) to be located at \nu = 0 , u = 1 , u = u 0 = ˜ a 2 1 -˜ a 2 2 1 -˜ a 2 2 , u = ∞ , (B1) \nand the indicial exponents are \nα ± 0 = ± m 1 2 , α ± 1 = 1 2 (2 ± θ ∞ ) , α ± u 0 = ± m 2 2 , α ± ∞ = ± 1 2 ς, (B2) \nwith \nθ ∞ = ∆ -2 , ς = (˜ ω + m 1 ˜ a 1 + m 2 ˜ a 2 ) . (B3) \nBy introducing the following transformation \nS ( u ) = u α + 0 ( u 0 -u ) α + u 0 (1 -u ) α + 1 y ( u ) , (B4) \nwe bring the angular equation to the canonical Heun form \nd 2 y du 2 + ( 1 + m 1 u + 1 + m 2 u -u 0 + ∆ -1 u -1 ) dy du + ( q 1 q 2 u ( u -1) -u 0 ( u 0 -1) Q 0 u ( u -1)( u -u 0 ) ) y ( u ) = 0 , (B5) \nwhere q 1 , q 2 and the accessory parameter Q 0 are definend as \nq 1 = 1 2 ( m 1 + m 2 +∆ -ς ) , q 2 = 1 2 ( m 1 + m 2 +∆+ ς ) , (B6a) \n4 u 0 ( u 0 -1) Q 0 = ( ˜ ω 2 +˜ a 2 1 ∆(∆ -4) -λ ) 1 -˜ a 2 2 -u 0 [ ( m 2 + θ ∞ +1) 2 -m 2 2 -1 ] -( u 0 -1) [ ( m 1 + m 2 +1) 2 -ς 2 -1 ] . (B6b)', "C Retarded Green's function": "For completeness, we provide the asymptotic expansions of the s-wave retarded Green's functions for both the Kerr-AdS 5 black hole with equal rotation and the RN-AdS 5 black hole in the small ˜ r + limit, without applying the resummation procedure.", 'Kerr-AdS 5 :': 'G \nret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ 3 4 ˜ ω ( 1 + α 2 ) 2 × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) ) -( 1 + α 2 ) 2 ( 3˜ ω 2 -∆(∆ -4) ) ( ( 3˜ ω 2 -∆(∆ -4) ) 2 -˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) ) 8((3˜ ω 2 -∆(∆ -4)) 2 + ˜ ω 2 ((∆ -2) 2 -˜ ω 2 )) 2 π sin π ∆ (cos π ∆ -cos π ˜ ω ) + (∆ -2) ( 2(1 + α 4 )(∆ -3)(∆ -1) -α 2 ( 3˜ ω 2 -∆(∆ -4) ) -3(1 + α 4 )˜ ω 2 ) ((∆ -2) 2 -˜ ω 2 ) +2 α 2 (∆ -2) ] ˜ r 2 + } + O ( ˜ r 3 + ) \n(C1)', 'RN-AdS 5 :': 'G ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ( 3 ( 1 + q 2 ) ˜ ω -2 q ˜ e ) 2 ( ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) ) + (∆ -2) ( (1 + q 2 )(2(∆ -2) 2 -3˜ ω 2 ) -(1 + q 2 )(3˜ ω +2) + 2 q ˜ e (˜ ω +1) ) ((∆ -2) 2 -˜ ω 2 ) + ( ( (1 + q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 8 -( 3 ( 1 + q 2 ) ˜ ω -2 q ˜ e ) 4 + 1 4 ( ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) (˜ ω 2 -(∆ -2) 2 )((1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 )) -1 ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) ) ) ( ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -4(∆ -2) ((∆ -2) 2 -˜ ω 2 ) ) ] ˜ r 2 + } + O ( ˜ r 3 + ) \n(C2)', 'References': "- [1] N. 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2024ApJ...973L..14D

We present cosmological constraints from the sample of Type Ia supernovae SNe Ia discovered and measured during the full 5 yr of the Dark Energy Survey DES SN program. In contrast to most previous cosmological samples in which SNe are classified based on their spectra we classify the DES SNe using a machine learning algorithm applied to their light curves in four photometric bands. Spectroscopic redshifts are acquired from a dedicated followup survey of the host galaxies. After accounting for the likelihood of each SN being an SN Ia we find 1635 DES SNe in the redshift range 0.10 lt z lt 1.13 that pass quality selection criteria sufficient to constrain cosmological parameters. This quintuples the number of highquality z gt 0.5 SNe compared to the previous leading compilation of Pantheon and results in the tightest cosmological constraints achieved by any SN data set to date. To derive cosmological constraints we combine the DES SN data with a highquality external lowredshift sample consisting of 194 SNe Ia spanning 0.025 lt z lt 0.10. Using SN data alone and including systematic uncertainties we find SUBMSUB 0.352 0.017 in flat CDM. SN data alone now require acceleration q SUB0SUB lt 0 in CDM with over 5 confidence. We find inlineformula inlineformula in flat wCDM. For flat w SUB0SUB w SUB a SUBCDM we find inlineformula inlineformula consistent with a constant equation of state to within 2. Including Planck cosmic microwave background Sloan Digital Sky Survey baryon acoustic oscillation and DES 3 2pt data gives SUBMSUB w 0.321 0.007 0.941 0.026. In all cases dark energy is consistent with a cosmological constant to within 2. Systematic errors on cosmological parameters are subdominant compared to statistical errors these results thus pave the way for future photometrically classified SN analyses.

2024-09-01T00:00:00Z

['arXiv:2401.02929', '10.3847/2041-8213/ad6f9f', '2024ApJ...973L..14A', '2024arXiv240102929D', '2024ApJ...973L..14D', '2024arXiv240102929T', '10.48550/arXiv.2401.02929']

['Cosmology', 'Type Ia supernovae', 'Dark energy', 'Dark matter', '343', '1728', '351', '353', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

The Dark Energy Survey Cosmology Results with 1500 New Highredshift Type Ia Supernovae Using the Full 5 yr Data Set

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https://arxiv.org/pdf/2401.02929.pdf

{'The Dark Energy Survey: Cosmology Results With ∼ 1500 New High-redshift Type Ia Supernovae Using The Full 5-year Dataset': "DES Collaboration: T. M. C. Abbott, 1 M. Acevedo, 2 M. Aguena, 3 A. Alarcon, 4 S. Allam, 5 O. Alves, 6 A. Amon, 7 F. Andrade-Oliveira, 6 J. Annis, 5 P. Armstrong, 8 J. Asorey, 9 S. Avila, 10 D. Bacon, 11 B. A. Bassett, 12, 13 K. Bechtol, 14 P. H. Bernardinelli, 15 G. M. Bernstein, 16 E. Bertin, 17, 18 J. Blazek, 19 S. Bocquet, 20 D. Brooks, 21 D. Brout, 22 E. Buckley-Geer, 23, 5 D. L. Burke, 24, 25 H. Camacho, 26, 3 R. Camilleri, 27 A. Campos, 28 A. Carnero Rosell, 29, 3, 30 D. Carollo, 31 A. Carr, 27 J. Carretero, 10 F. J. Castander, 32, 33 R. Cawthon, 34 C. Chang, 23, 35 R. Chen, 2 A. Choi, 36 C. Conselice, 37, 38 M. Costanzi, 39, 31, 40 L. N. da Costa, 3 M. Crocce, 32, 33 T. M. Davis, 27 D. L. DePoy, 41 S. Desai, 42 H. T. Diehl, 5 M. Dixon, 43 S. Dodelson, 28, 44 P. Doel, 21 C. Doux, 16, 45 A. Drlica-Wagner, 23, 5, 35 J. Elvin-Poole, 46 S. Everett, 47 I. Ferrero, 48 A. Fert'e, 25 B. Flaugher, 5 R. J. Foley, 49 P. Fosalba, 32, 33 D. Friedel, 50 J. Frieman, 5, 35 C. Frohmaier, 51 L. Galbany, 32, 33 J. Garc'ıa-Bellido, 52 M. Gatti, 16 E. Gaztanaga, 11, 32, 33 G. Giannini, 10, 35 K. Glazebrook, 43 O. Graur, 11 D. Gruen, 20 R. A. Gruendl, 50, 53 G. Gutierrez, 5 W. G. Hartley, 54 K. Herner, 5 S. R. Hinton, 27 D. L. Hollowood, 55 K. Honscheid, 56, 57 D. Huterer, 6 B. Jain, 16 D. J. James, 58, 59 N. Jeffrey, 21 E. Kasai, 60, 12 L. Kelsey, 11 S. Kent, 5, 35 R. Kessler, 23, 35 A. G. Kim, 61 R. P. Kirshner, 62, 63 E. Kovacs, 4 K. Kuehn, 64, 65 O. Lahav, 21 J. Lee, 16 S. Lee, 47 G. F. Lewis, 66 T. S. Li, 67 C. Lidman, 68, 8 H. Lin, 5 U. Malik, 8 J. L. Marshall, 41 P. Martini, 56, 69 J. Mena-Fern'andez, 70 F. Menanteau, 50, 53 R. Miquel, 71, 10 J. J. Mohr, 20, 72 J. Mould, 43 J. Muir, 73 A. Moller, 43 E. Neilsen, 5 R. C. Nichol, 74 P. Nugent, 61 R. L. C. Ogando, 75 A. Palmese, 28 Y.-C. Pan, 76 M. Paterno, 5 W. J. Percival, 46, 73 M. E. S. Pereira, 77 A. Pieres, 3, 75 A. A. Plazas Malag'on, 24, 25 B. Popovic, 2 A. Porredon, 78 J. Prat, 35 H. Qu, 16 M. Raveri, 79 M. Rodr'ıguez-Monroy, 80 A. K. Romer, 81 A. Roodman, 24, 25 B. Rose, 2, 82 M. Sako, 16 E. Sanchez, 83 D. Sanchez Cid, 83 M. Schubnell, 6 D. Scolnic, 2 I. Sevilla-Noarbe, 83 P. Shah, 21 J. Allyn. Smith, 84 M. Smith, 85 M. Soares-Santos, 86 E. Suchyta, 87 M. Sullivan, 51 N. Suntzeff, 41 M. E. C. Swanson, 50 B. O. S'anchez, 88 G. Tarle, 6 G. Taylor, 8 D. Thomas, 11 C. To, 56 M. Toy, 51 M. A. Troxel, 2 B. E. Tucker, 8 D. L. Tucker, 5 S. A. Uddin, 89 M. Vincenzi, 2 A. R. Walker, 1 N. Weaverdyck, 6, 61 R. H. Wechsler, 90, 24, 25 J. Weller, 72, 91 W. Wester, 5 P. Wiseman, 51 M. Yamamoto, 2 F. Yuan, 8 B. Zhang, 8 and Y. Zhang 1 \n1 Cerro Tololo Inter-American Observatory, NSF's National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile \n2 Department of Physics, Duke University Durham, NC 27708, USA \n14 \n3 Laborat'orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 4 Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA 5 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA 6 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA 7 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 8 The Research School of Astronomy and Astrophysics, Australian National University, ACT 2601, Australia 9 Departamento de F'ısica Te'orica and IPARCOS, Universidad Complutense de Madrid, 28040 Madrid, Spain 10 Institut de F'ısica d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain 11 Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK 12 South African Astronomical Observatory, P.O.Box 9, Observatory 7935, South Africa 13 Mathematics Department, University of Cape Town, South Africa Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706-1390 15 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA 16 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA 17 CNRS, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France 18 Sorbonne Universit'es, UPMC Univ Paris 06, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France 19 Department of Physics, Northeastern University, Boston, MA 02115, USA 20 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universitat, Scheinerstr. 1, 81679 Munich, Germany 21 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 22 Department of Astronomy and Department of Physics, Boston University Boston, MA 02140, USA 23 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA \n24 25 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA \n26 Instituto de F'ısica Te'orica, Universidade Estadual Paulista, S˜ao Paulo, Brazil \n27 School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia", 'The Dark Energy Survey Collaboration': "28 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA \n29 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain 30 Universidad de La Laguna, Dpto. Astrof'ısica, E-38206 La Laguna, Tenerife, Spain 31 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy 32 Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain 33 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain 34 Physics Department, William Jewell College, Liberty, MO, 64068 35 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA 36 NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA 37 Jodrell Bank Center for Astrophysics, School of Physics & Astronomy, University of Manchester, Oxford Rd, Manchester, M139PL, UK 38 University of Nottingham, School of Physics and Astronomy, Nottingham NG7 2RD, UK 39 Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italy 40 Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy 41 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA 42 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 43 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Victoria 3122, Australia 44 NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA 45 Universit'e Grenoble Alpes, CNRS, LPSC-IN2P3, 38000 Grenoble, France 46 Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada 47 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA 48 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway 49 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA 50 Center for Astrophysical Surveys, National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA 51 School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK 52 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 53 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA 54 Department of Astronomy, University of Geneva, ch. d' ' Ecogia 16, CH-1290 Versoix, Switzerland 55 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA 56 Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA 57 Department of Physics, The Ohio State University, Columbus, OH 43210, USA 58 ASTRAVEO LLC, PO Box 1668, MA 01931, USA 59 Applied Materials Inc., 35 Dory Road, Gloucester, MA 01930, USA 60 Department of Physics, University of Namibia, 340 Mandume Ndemufayo Avenue, Pionierspark, Windhoek, Namibia 61 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA 62 TMT International Observatory, 100 West Walnut Street, Pasadena CA 91124 63 California Institute of Technology, 1200 East California Boulevard, Pasadena CA 91125 64 Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia 65 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA 66 Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia 67 Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto ON, M5S 3H4, Canada 68 Centre for Gravitational Astrophysics, College of Science, The Australian National University, ACT 2601, Australia 69 Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA 70 LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France 71 Instituci'o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain 72 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany 73 Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada 74 School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK 75 Observat'orio Nacional, Rua Gal. Jos'e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 76 Graduate Institute of Astronomy, National Central University, 300 Jhongda Road, 32001 Jhongli, Taiwan 77 Hamburger Sternwarte, Universitat Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany 78 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute, 44780 Bochum, Germany 79 Department of Physics, University of Genova and INFN, Via Dodecaneso 33, 16146, Genova, Italy 80 Laboratoire de physique des 2 infinis Ir'ene Joliot-Curie, CNRS Universit'e Paris-Saclay, Bˆat. 100, F-91405 Orsay Cedex, France 81 Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK 82 Department of Physics, Baylor University, One Bear Place #97316, Waco, TX 76798-7316, USA 83 Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas (CIEMAT), Madrid, Spain \n84 Austin Peay State University, Dept. Physics, Engineering and Astronomy, P.O. Box 4608 Clarksville, TN 37044, USA 85 Physics Department, Lancaster University, Lancaster, LA1 4YB, UK \n86 University of Zurich, Physics Institute, Winterthurerstrasse 190/Building 36, 8057 Zurich, Switzerland \n87 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 \n88 Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France \n89 Centre for Space Studies, American Public University System, 111 W. Congress Street, Charles Town, WV 25414, USA 90 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA", 'ABSTRACT': 'We present cosmological constraints from the sample of Type Ia supernovae (SN Ia) discovered and measured during the full five years of the Dark Energy Survey (DES) Supernova Program. In contrast to most previous cosmological samples, in which supernovae are classified based on their spectra, we classify the DES supernovae using a machine learning algorithm applied to their light curves in four photometric bands. Spectroscopic redshifts are acquired from a dedicated follow-up survey of the host galaxies. After accounting for the likelihood of each supernova being a SN Ia, we find 1635 DES SNe in the redshift-range 0 . 10 < z < 1 . 13 that pass quality selection criteria sufficient to constrain cosmological parameters. This quintuples the number of high-quality z > 0 . 5 SNe compared to the previous leading compilation of Pantheon+, and results in the tightest cosmological constraints achieved by any supernova data set to date. To derive cosmological constraints we combine the DES supernova data with a high-quality external low-redshift sample consisting of 194 SNe Ia spanning 0 . 025 < z < 0 . 10. Using supernova data alone and including systematic uncertainties we find Ω M = 0 . 352 ± 0 . 017 in flat-ΛCDM. Supernova data alone now require acceleration ( q 0 < 0 in ΛCDM) with over 5 σ confidence. We find (Ω M , w ) = (0 . 264 +0 . 074 -0 . 096 , -0 . 80 +0 . 14 -0 . 16 ) in flatw CDM. For flatw 0 w a CDM, we find (Ω M , w 0 , w a ) = (0 . 495 +0 . 033 -0 . 043 , -0 . 36 +0 . 36 -0 . 30 , -8 . 8 +3 . 7 -4 . 5 ), consistent with a constant equation of state to within ∼ 2 σ . Including Planck CMB, SDSS BAO, and DES 3 × 2-point data gives (Ω M , w ) = (0 . 321 ± 0 . 007 , -0 . 941 ± 0 . 026). In all cases dark energy is consistent with a cosmological constant to within ∼ 2 σ . Systematic errors on cosmological parameters are subdominant compared to statistical errors; these results thus pave the way for future photometrically classified supernova analyses. \nKeywords: supernovae, cosmology, dark energy', '1. INTRODUCTION': "The standard cosmological model posits that the energy density of the Universe is dominated by dark components that have not been detected in terrestrial experiments and thus do not appear in the standard model of particle physics. Known as cold dark matter and dark energy, their study represents an opportunity to deepen our understanding of fundamental physics. \nThe Dark Energy Survey (DES) was conceived to characterize the properties of dark matter and dark energy with unprecedented precision and accuracy through four primary observational probes (The Dark Energy Survey Collaboration 2005; Bernstein et al. 2012; Dark Energy Survey Collaboration 2016; Lahav et al. 2020). One of these four probes is the Hubble diagram (redshift-distance relation) for Type Ia supernovae (SNe Ia), which act as standardizable candles (Rust 1974; Pskovskii 1977; Phillips et al. 1999) to constrain the history of the cosmic expansion rate. To imple- \nment this probe, the DES SN survey was designed to provide the largest, most homogeneous sample of highredshift supernovae ever discovered. The two papers that first presented evidence for the accelerated expansion of the universe (Riess et al. 1998; Perlmutter et al. 1999) used a total of 52 high-redshift supernovae with sparsely sampled light-curve measurements in one or two optical passbands. Building on two decades of subsequent improvements in SN surveys and analysis, we present here the cosmological constraints using the full 5-year DES SN dataset, consisting of well-sampled, precisely calibrated light curves for 1635 new high-redshift supernovae observed in four bands g, r, i, z . \nFor the last decade, SN Ia cosmology constraints have largely come from combining data from many surveys. The recent Pantheon+ analysis (Scolnic et al. 2022; Brout et al. 2022a) combined three separate midz samples (0 . 1 < z < 1 . 0), 11 different lowz samples ( z < 0 . 1), and four separate highz samples ( z > 1 . 0), \neach with different photometric systems and selection functions (Gilliland et al. 1999; Hicken et al. 2009; Riess et al. 2001, 2004, 2007; Sullivan et al. 2011; Hicken et al. 2012; Suzuki et al. 2012; Ganeshalingam et al. 2013; Betoule et al. 2014; Krisciunas et al. 2017; Foley et al. 2017; Riess et al. 2018; Sako et al. 2018; Brout et al. 2019b; Smith et al. 2020a). The DES sample, which rivals in number the entirety of Pantheon+, does not have the low-redshift ( z < 0 . 1) coverage to completely remove the need for external lowz samples, but at higher redshift enables us to replace a heterogeneous mix of samples with a homogeneous sample of high quality, wellcalibrated light curves. \nA key aim of the DES analysis was to minimize systematic (relative to statistical) errors to enable a robust analysis. Vincenzi et al. (2024) shows that our error budget is dominated by statistical uncertainty, in contrast to most SN cosmology analyses of the last decade, for which the systematic uncertainties equalled or exceeded the statistical uncertainties (Betoule et al. 2014; Scolnic et al. 2018; Dark Energy Survey Collaboration 2019). We also highlight that the most critical sources of systematics are those related to the lack of a homogeneous and well calibrated lowz sample. \nAs the DES sample enables a SN Ia measurement of cosmological parameters that is largely independent of previous SN cosmology analyses, we have been careful to 'blind' our analysis (see Sec. 2.3). The analysis work described in Vincenzi et al. (2024), which stops just short of constraining cosmological parameters, was shared widely with the DES collaboration, evaluated, and approved before unblinding. Unblinding standards included multiple validation checks with simulations and full accounting and explanation of the error budget. No elements of the analysis were changed after unblinding. \nIn this paper we review the analysis of the complete DES SN dataset (as detailed in many supporting papers; see Fig. 1) and present the cosmological results. An important advance on most previous analyses is that we use a photometrically classified rather than spectroscopically classified sample (Moller & de Boissi'ere 2020; Qu et al. 2021), and implement advanced techniques to classify SN Ia and incorporate classification probabilities in the cosmological parameter estimation (Kunz et al. 2007, 2012; Hlozek et al. 2012). While this advance increases the complexity of the analysis, in this work and previous papers (Vincenzi et al. 2023; Moller et al. 2022) we show that the impact of non-SN Ia contamination due to photometric misclassification is well below the statistical uncertainty on cosmological parameters, and this constitutes one of the key results of our analysis.", 'Data:': '- -Calibration (Burke et al. 2018, Brout et al. 2022, Rykoff et al. 2023)\n- -SN photometry (Brout et al. 2019, Sanchez et al. 2024)\n- -SN spectroscopy (Smith et al. 2020a)\n- -DCR and chrom (Lasker et al. 2018, Lee&Acevedo et al. 2023)\n- -Host galaxy redshifts and properties (Lidman et al. 2020, Carr et\n- al. 2021, Wiseman et al. 2020/2021, Kelsey et al. 2023)', 'Simulations:': '-Survey selection effects (Kessler et al. 2019a, Vincenzi et al. 2020) -SN Ia intrinsic and dust properties (Brout&Scolnic 2021, Popovic et al. 2021a/b, Wiseman et al. 2022) and rates (Wiseman et al. 2021) -Contamination (Vincenzi et al. 2019/2020, Kessler et al. 2019b)', 'Analysis:': "Pipeline and Overview \n( Hinton et al. 2020, Vincenzi et al. 2024) \n- - Light-curve fitting (Taylor et al. 2023)\n- - SN classification (Möller & de Boissière 2020, Qu et al. 2021,\n- Vincenzi et al. 2021, Moller et al. 2022)\n- - 'BEAMS' and bias corrections (Kessler & Scolnic 2017), unbinning\n- the SN Hubble diagram (Brout et al. 2020, Kessler et al. 2023)\n- - Effects of host galaxy mismatch (Qu et al. 2023)\n- - Cosmological contour validation (Armstrong et al. 2023)", 'Cosmological results: DES Collaboration 2024': "Testing non-standard cosmological models (Camilleri et al. 2024) \nFigure 1. Overview of supporting papers for DES-SN5YR cosmological results. \nCombining our DES data with a low-redshift sample (see Sec. 2), we fit the Hubble diagram to test the standard cosmological model as well as multiple common extensions including spatial curvature, non-vacuum dark energy, and dark energy with an evolving equation of state parameter. In Camilleri et al. (in prep. 2024) we present fits to more exotic models. \nThe structure of the paper is as follows. We begin in Sec. 2 by describing the dataset, its acquisition, reduction, calibration, and light-curve fitting. We summarize the models we test in Sec. 3 before presenting the results in Sec. 4; our discussion and conclusions follow in Sec. 5 and Sec. 6. The details of our data release, which includes the code needed to reproduce our results, appear in S'anchez (in prep. 2024).", '2.1. DES and Low-redshift SNe': "Our primary dataset is the full five years of DES SNe, which we combine with a historical set of nearby supernovae from CfA3 (Hicken et al. 2009), CfA4 (Hicken et al. 2012), CSP (Krisciunas et al. 2017, DR3) and the \nFigure 2. All DES light curves, showing observed magnitudes in g , r , i , and z bands (left to right respectively) normalized by the maximum brightness of each light curve, and with the time-axis de-redshifted to the rest-frame. Each light curve has been arbitrarily offset by their redshift, with higher-redshift objects higher on the plot (as labeled on vertical axis). Lines show best-fit SALT3 light-curve fits. The g -band and r -band light curves are not used above z ∼ 0 . 4 and z ∼ 0 . 85 respectively because that corresponds to the redshifts at which the lower-wavelength limit of the SALT3 model (3500 ˚ A in the rest frame) passes out of their observed wavelength ranges. \n<!-- image --> \nFoundation SN sample (Foley et al. 2017). We refer to the combined DES plus historical dataset as DESSN5YR . \nThe DES supernova program was carried out over five seasons, August to February from 2013-2018, during which we observed ten ∼ 3 deg 2 fields with approximately weekly cadence in four bands ( g, r, i, z ). Eight of the fields were observed to 5 σ depth of ∼ 23 . 5 mag in all four bands (shallow fields) and two to a deeper limit of ∼ 24 . 5 mag (deep fields). See Flaugher et al. (2015) for a summary of the Dark Energy Camera, Smith et al. (2020a) for a summary of the supernova program, and Diehl et al. (2016, 2018) for observational details. \nThe DES SNe were discovered via difference imaging (Kessler et al. 2015) based on the method of Alard & Lupton (1998). DES images are calibrated following the Forward Global Calibration Method (FGCM; Burke et al. 2018; Sevilla-Noarbe et al. 2021; Rykoff 2023), and both DES and lowz samples are recalibrated as part of the SuperCal-Fragilistic cross calibration effort described in Brout et al. (2022b). SN fluxes are determined using scene modeling photometry (Brout et al. 2019b); we include corrections from spectral energy distribution variations (Burke et al. 2018; Lasker et al. 2019) and from differential chromatic refraction and wavelengthdependent seeing (Lee & Acevedo et al. 2023). We esti- \nFigure 3. Histogram showing the redshift distribution of the DES-SN5YR sample, with new DES SNe in blue and our lowz sample in red. For comparison the distribution of redshifts in the existing Pantheon+ sample is shown in grey (Brout et al. 2022a), which also includes the DES SNe from the DES-SN3YR analysis (blue dashed line). The five-year DES sample contains ∼ 4 × more supernovae above z ∼ 0 . 4 than the Pantheon+ compilation. \n<!-- image --> \nmate the overall accuracy of our calibrated photometry to be ≲ 5 mmag. Host galaxies are assigned following the directional light radius (DLR) method (Sullivan et al. 2006; Gupta et al. 2016; Qu et al. 2023), and host galaxy properties are determined as described by Kelsey et al. (2023) based on Fioc & Rocca-Volmerange (1999) using deep coadded images by Wiseman et al. (2020). Host galaxy spectroscopic redshifts are obtained primarily within the OzDES programme (Yuan et al. 2015; Childress et al. 2017; Lidman et al. 2020). The final data release of photometry of ∼ 20 , 000 candidates, redshifts of hosts, and host galaxy properties is presented in S'anchez (in prep. 2024). \nWe apply strict quality cuts to this sample of candidates to select our final high-quality sample for the Hubble diagram. The same quality cuts were applied to both the lowz sample and the DES supernovae. First, we require a spectroscopic redshift of the host galaxy, good light-curve coverage (at least two detections with SNR > 5 in two different bands), and a well converged light-curve fit using the SALT3 model 1 (Kenworthy et al. 2021; Taylor et al. 2023); this reduces the DES sample size to 3621. Additional requirements include light-curve parameters (stretch and colour) within normal range for SNe Ia, a well-constrained time of peak brightness (uncertainty less than 2 days), good SALT3 fit-probability, and valid distance-bias correction from our simulation (see Table 4 of Vincenzi et al. 2024, for \nmore detail). Our final Hubble-diagram sample includes 1635 supernovae, of which 1499 have a machine-learning probability of being a Type Ia greater than 50% (see Sec. 2.2). Note that we do not perform a cut on this machine-learning probability, rather we use it in the BEAMS formalism that produces our Hubble diagram and to weight the SN distance uncertainties in the fits to the final Hubble diagram (Kessler et al. 2023). The set of all DES light curves is visualised in Fig. 2. \nSince we focus on minimizing potential systematic errors, we only use the best-calibrated, most homogeneous sample of lowz SNe Ia. To reduce the impact of peculiar velocity uncertainties we remove SNe with z < 0 . 025. We furthermore combine only a subset of the available low-redshift samples: CfA3&4, CSP, and Foundation SNe, which are the four largest lowz samples with the most well-understood photometric calibration. Our lowz sample thus totals 194 SNe with z < 0 . 1; this can be compared to Pantheon+, for which the lowz sample was almost four times larger (741 SNe at z < 0 . 1). We have thus exchanged the statistical constraining power of more lowz SNe for better control of systematics. The redshift distribution of our sample compared to the compilation of historical samples in Pantheon+ is shown in Fig. 3. To conclude, the final DES-SN5YR sample includes 1635 DES SNe and 194 lowz external SNe, for a total of 1829 SNe.", '2.2. From light curves to Hubble diagram': "A critical step in the cosmology analysis is to convert each supernova's light curve (magnitude vs time in multiple bands; see examples in Fig. 2) to a single calibrated number representing its standardized magnitude and estimated distance modulus. \nTo achieve this, we use the SALT3 light-curve fitting model as presented in Kenworthy et al. (2021); Taylor et al. (2023) and retrained in Vincenzi et al. (2024) to determine the light-curve fit parameters, amplitude of the SN flux ( x 0 ), stretch ( x 1 ), and color ( c ). These fitted parameters are used to estimate the distance modulus, µ ≡ m -M , using an adaptation of the Tripp equation (Tripp 1998) that includes a correction for observed correlations between SN Ia luminosity and host properties, γG host = ± γ/ 2. Here γ is the size of the step and G host is the property of the host galaxy that is used to determine the step (i.e. mass or color); the sign is + if G host is above the step or -if below. This correction has historically been described as a 'mass step' but we also consider the possibility that it is a 'color step' (see Sec. 2.2 of Vincenzi et al. 2024), \nµ obs ,i = m x,i + αx 1 ,i -βc i + γG host ,i -M -∆ µ bias ,i , (1) \nFigure 4. Hubble diagram of DES-SN5YR. We show both the single SN events and the redshift-binned SN distance moduli. Redshift bins are adjusted so that each bin has the same number of SNe ( ∼ 50). The 1635 new DES supernovae are in blue, and in the upper panel they are shaded by their probability of being a Type Ia; most outliers are likely contaminants (pale blue). The inset shows the number of SNe as a function of redshift (same z -range as the main plot). The lower panel shows the difference between the data and the best fit Flatw CDM model from DES-SN5YR alone (third result in Table 2), and overplots three other best fit cosmological models - Flat-ΛCDM model from DES-SN5YR alone (magenta line, first result in Table 2), Flatw 0 w a CDM model from DES-SN5YR alone (green line, fourth result in Table 2), and Planck 2020 Flat-ΛCDM model without SN data (dashed line, Ω Planck M =0 . 317 ± 0 . 008). \n<!-- image --> \nwhere m x = -2 . 5 log 10 ( x 0 ). 2 The constants α , β , and γ are global parameters determined from the likelihood analysis of all the SNe on the Hubble diagram, while the terms subscripted by i refer to parameters of individual SNe. We find α = 0 . 161 ± 0 . 001, β = 3 . 12 ± 0 . 03, and γ = 0 . 038 ± 0 . 007. We marginalize over the absolute magnitude M (see Sec. 3). The final term in Eq. 1 accounts for selection effects, Malmquist bias, and light curve fitting bias. \nThe nuisance parameters and ∆ µ bias ,i term in Eq. 1 are determined using the 'BEAMS with Bias Corrections' (BBC) framework (Kessler & Scolnic 2017). In particular, bias corrections ∆ µ bias ,i are estimated from a large simulation of our sample. The simulation models the rest-frame SN Ia spectral energy distribution (SED) at all phases, SN correlations with host-galaxy \nproperties, SED reddening through an expanding universe, broadband griz fluxes, and instrumental noise (see Fig. 1 in Kessler et al. 2019a). Using Eq. 1 there remains intrinsic scatter of ∼ 0 . 1 mag in Hubble residuals. Following the numerous recent studies on understanding and modelling SN Ia dust extinction and progenitors (Wiseman et al. 2021, 2022; Duarte et al. 2022; Dixon et al. 2022; Chen et al. 2022; Meldorf et al. 2023), we model this residual scatter using the dust-based model from Brout & Scolnic (2021) [BS21]; Popovic et al. (2023a). In contrast to previously used models in K13, the BS21 model accurately models the Hubble residual bias and scatter as a function of the fitted SALT2 color (see Fig. 5 in Vincenzi et al. (2024), and Fig. 6 in Brout & Scolnic (2021)). Due to uncertainties in the fitted dust parameters (Popovic et al. 2023a), this intrinsic scatter model remains the largest source of systematic uncertainty from the simulation. \nAs we do not spectroscopically classify the SNe and thus expect contamination from core-collapse (CC) supernovae, we perform machine learning light-curve clas- \nsification on the sample following Vincenzi et al. (2023); Moller et al. (2022). We implement two advanced machine learning classifiers, SuperNNova (Moller & de Boissi'ere 2020) and SCONE (Qu et al. 2021) and use state-of-the-art simulations to model contamination (estimated to be ∼ 6 . 5%, see Table 10 and Sec. 7.1.5 of Vincenzi et al. 2024). Classifiers are trained using corecollapse and peculiar SN Ia simulations based on Vincenzi et al. (2021) and using state-of-the-art SED templates by Vincenzi et al. (2019); Kessler et al. (2019b). These DES simulations are the first to robustly reproduce the contamination observed in the Hubble residuals (Vincenzi et al. 2021; Vincenzi et al. 2024, Table 10). \nFor each SN, the trained classifiers assign a probability of being a Type Ia, and these probabilities are included within the BEAMS framework to marginalize over corecollapse contamination and produce the final Hubble Diagram (Kunz et al. 2012; Hlozek et al. 2012). The final DES-SN5YR Hubble diagram is shown in Fig. 4 and includes 1829 SNe. \nAs discussed in Kessler et al. (2023); Vincenzi et al. (2024), the probability that each supernova is a Type Ia ( P Ia ) is incorporated in the BBC fit and used to calculate a BEAMS probability, P B(Ia) (see Eq. 9 in Kessler et al. 2023). BEAMS probabilities are used to inflate distance uncertainties of likely contaminants by a factor ∝ 1 / √ P B(Ia) (see Eq. 10 in Vincenzi et al. 2024). Therefore, the released Hubble diagram data includes distance bias corrections and inflated distance uncertainties (see App. A), enabling users to fit the Hubble diagram without applying additional corrections. With this BEAMS uncertainty weight, we find 75 SNe with distance modulus uncertainties σ µ,i, final > 1 mag and 1331 SNe with σ µ,i, final < 0 . 2 mag. 3 \nVincenzi et al. (2024) stops short of performing cosmological constraints but provides the corrected distance moduli µ along with their uncertainties σ µ , redshifts for each SN, and a statistical+systematic covariance matrix C , which we describe further in Sec. 3. \nArmstrong et al. (2023) presents validation of the cosmological contours produced by our pipeline. Validation that our analysis pipeline is insensitive to the cosmological model assumed in our bias correction simulation appears in Camilleri et al. (in prep. 2024).", '2.3. Unblinding criteria': "Throughout our analysis, cosmological parameters estimated from real data were blinded. We validate our entire pipeline on detailed catalogue-level simulations and examine the cosmological parameters estimated from simulations to test that the input cosmology is recovered. In addition to the many tests described in Vincenzi et al. (2024), the final unblinding criteria that our data passed were: \n- · Accuracy of simulations: Reduced χ 2 between the distribution of data and simulations across a variety of observables (redshift, SALT3 parameters and goodness of the fit, maximum signal-tonoise ratio at peak, host stellar mass) is required to be between 0.7 and 3.0 (see Vincenzi et al. 2024, Fig. 3-4).\n- · Pipeline validation using DES simulations: Demonstrate that our pipeline recovers the input cosmology. We produce 25 data-size simulated samples (statistically independent) assuming a Flat-ΛCDM universe with best-fit Planck value of Ω M and analyze them the same way as real data. We fit each Hubble diagram assuming a Flatw CDM model with a Planck prior and find a mean bias of w -w true ≃ 0 . 001 ± 0 . 020, where w is the mean value of the marginalized posterior of the dark energy equation of state parameter over the 25 samples, and w true = -1 is the model value of that parameter input to the simulation.\n- · Validation of contours: ensuring that our uncertainty limits accurately represent the likelihood of the models (Armstrong et al. 2023).\n- · Independence of reference cosmology: ensuring that our results are sufficiently independent of cosmological assumptions that enter our bias correction simulations (Camilleri et al. in prep. 2024).\n- 2.4. Combining SN with other cosmological probes \nWecombine the DES-SN5YR cosmological constraints with measurements from other complementary cosmological probes. In particular, we use: \n- · Cosmic Microwave Background (CMB) measurements of the temperature and polarisation power spectra (TTTEEE) presented by the Planck Collaboration (2020). We use the Python implementation of Planck's 2015 Plik lite (Prince & Dunkley 2019).\n- · Weak lensing and galaxy clustering measurements from the DES3 × 2pt year-3 magnitude-limited \nTable 1. Variations on the standard cosmological model that are tested in this paper, their Friedmann Equations, and the free parameters in the fit. \n(MagLim) lens sample; 3 × 2-point refers to the simultaneous fit of three 2-point correlation functions, namely galaxy-galaxy, galaxy-lensing, and lensing-lensing correlations (Dark Energy Survey Collaboration 2022, 2023). \n- · Baryon acoustic oscillation (BAO) measurements as presented in the extended Baryon Oscillation Spectroscopic Survey paper (eBOSS; Dawson et al. 2016; Alam et al. 2021), which adds the BAO results from SDSS-IV (Blanton et al. 2017) to earlier SDSS BAO data. Specifically, we use 'BAO' to refer to the BAO-only measurements from the Main Galaxy Sample (Ross et al. 2015), BOSS (SDSS-III Alam et al. 2017), eBOSS LRG (Bautista et al. 2021), eBOSS ELG (de Mattia et al. 2021), eBOSS QSO (Hou et al. 2021), and eBOSS Lya (du Mas des Bourboux et al. 2020). \nWhen combining these data we run simultaneous MCMC fits of the relevant data vectors. We present three combinations: the simplest CMB-dependent combination CMB+SN, a CMB-independent combination BAO+3 × 2pt+SN, and a combination of them all.", '3. MODELS AND THEORY': 'We present cosmological results for the standard cosmological model - flat space with cold dark matter and a cosmological constant (Flat-ΛCDM) - and some basic extensions, such as relaxing the assumption of spatial flatness (ΛCDM), allowing for constant equation of state parameter ( w ) of dark energy (Flatw CDM), and including a linear parameterisation for time-varying dark energy (Flatw 0 w a CDM) in which the equation of state parameter is given by w = w 0 + w a (1 -a ) (Chevallier & Polarski 2001; Linder 2003). \nTo calculate the theoretical distance as a function of redshift we begin with the comoving distance, \nR 0 χ (¯ z ) = c H 0 ∫ ¯ z 0 dz E ( z ) , (2) \nwhere ¯ z is the redshift due to the expansion of the Universe, E ( z ) ≡ H ( z ) /H 0 is the normalized redshift- \nendent expansion rate and is given for each cosmological model by the expression in Table 1, R 0 = c/ ( H 0 √ | Ω K | ) is the scale factor with dimensions of distance (where subscript 0 indicates its value at the present day), and Ω K ≡ 1 -Ω M -Ω Λ is the curvature term. The dimensionless scale factor ( a ≡ R/R 0 ) at the time of emission for an object with cosmological redshift ¯ z is a = 1 / (1 + ¯ z ). The luminosity distance is given by, \nD L ( z obs , ¯ z ) = (1 + z obs ) R 0 S k ( χ (¯ z )) , (3) \nwhere z obs is the observed redshift, and the curvature is captured by S k ( χ ) = sin χ , χ , and sinh χ for closed (Ω K < 0), flat (Ω K = 0), and open (Ω K > 0) universes respectively. 4 \nTo compare data (Eq. 1) to theory we calculate the theoretical distance modulus, which is dependent on the set of cosmological parameters we are interested in (Θ, given in the right column of Table 1), \nµ ( z, Θ) = 5 log 10 ( D L ( z, Θ) / 1 Mpc) + 25 . (4) \nWe compute the difference between data and theory for every i th supernova, ∆ µ i = µ obs ,i -µ ( z i , Θ), and find the minimum of \nχ 2 = ∆ µ i C -1 ij ∆ µ T j , (5) \nwhere C -1 is the inverse covariance matrix (including both statistical and systematic errors) of the ∆ µ vector (see Sec. 3.6 of Vincenzi et al. 2024). \nThe uncertainty covariance matrix includes a diagonal statistical term (discussed Sec. 2.2) and a systematic term. The systematic covariance matrix is built following the approach in Conley et al. (2011) and accounts for systematics such as calibration, intrinsic scatter, and redshift corrections (see Table 6 of Vincenzi et al. 2024). Each element of the covariance matrix expresses the covariance between two of the SNe in the sample. The covariance matrix has dimensions of the number of supernovae N SNe × N SNe and we follow the formalism introduced by Brout et al. (2021) and Kessler et al. (2023). \nFinally, the absolute magnitude of SNe Ia ( M ) and the H 0 parameter (which appears in the luminosity distance) are completely degenerate and therefore they are combined in the single parameter M = M + 5 log 10 ( c/H 0 ). All of our cosmology results are marginalized over this term. Therefore, the value of H 0 has no impact on the fitting of our cosmological results, and we do not constrain H 0 . While M has no impact on cosmology fitting, a precise value is needed to simulate bias corrections. The M uncertainty is below 0.01, resulting in a negligible impact on bias corrections (Brout et al. 2022a; Camilleri et al. in prep. 2024).', '4. RESULTS': 'With the new DES high-redshift supernova sample we can put strong constraints on cosmological models. Of particular interest is whether dark energy is consistent with a cosmological constant or whether its density and/or equation of state parameter varies over the wide redshift range of our sample. The results of our cosmological fits are outlined in this section and summarized in Table 2, and their implications are explored in Sec. 5. \nWe estimate cosmological constraints using Markov Chain Monte Carlo (MCMC) methods as implemented in the CosmoSIS framework (Zuntz et al. 2015), the samplers emcee for best fits (Foreman-Mackey et al. 2013), and PolyChord for tension metrics (Handley et al. 2015), 5 except for fits that include BAO+3 × 2pt, which are calculated using PolyChord for both best fit and tensions. 6 For all fits we present the median of the marginalized posterior and cumulative 68.27% confidence intervals. The chains and code (with the flexibility to test other statistical choices) are publicly available (see Appendix A). Figs. 5, 6, 7 and 8 all present the joint probability contours for 68.3% and 95.5%.', '4.1. Constraints on Cosmological Parameters': 'Figure 5. Constraints on matter density in the FlatΛCDM model from DES-SN5YR only (cyan), DES-SN5YR combined with CMB constraints from Planck Collaboration (2020) (blue), and DES-SN5YR combined with BAO+3 × 2pt (orange), and all probes combined (DESSN5YR+BAO+3 × 2pt and CMB constraints, red). CMB constraints only and BAO+3 × 2pt constraints alone are also shown for comparison (dashed and dotted-dashed respectively). \n<!-- image -->', '4.1.1. FlatΛ CDM': 'For the simplest parameterization, Flat-ΛCDM, Ω M is the only free parameter. We show the probability density function (PDF) of this constraint for DES-SN5YR in Fig. 5; we measure a value of Ω M =0 . 352 ± 0 . 017. We also show the probability distribution of the Planck Collaboration (2020) measurement of Ω Planck M =0 . 317 ± 0 . 008. These are approximately 7 2 σ apart, but not in significant tension as discussed in Sec 4.2. \nCombining DES-SN5YR with Planck CMB gives Ω M =0 . 338 +0 . 016 -0 . 014 , while combining with BAO+3 × 2pt gives Ω M =0 . 330 +0 . 011 -0 . 010 . Combining all three gives Ω M =0 . 315 ± 0 . 007. Interestingly, the combination of all data sets (red in Fig. 5) gives a lower Ω M than any of the other combinations. The reason can be seen in Fig. 6, where all constraints cross the Flat Universe line to the upper left of any individual best fit. \n<!-- image --> \nM \nFigure 6. Constraints for ΛCDM model (non-zero curvature allowed) from the DES-SN5YR dataset only (cyan), from DES-SN5YR combined with BAO+3 × 2pt (orange), from DES-SN5YR combined with CMB measurements (blue), and from all these combined (red). For comparison, we also present cosmological constraints from Planck Collaboration (2020) only (black dashed).Figure 7. Same as Fig. 6 but for the Flat w CDM model. The horizontal dotted line marks the equation of state values for a cosmological constant, i.e. w = -1. \n<!-- image -->', '4.1.2. Λ CDM': 'Fitting DES-SN5YR to the ΛCDM model, we find (Ω M , Ω Λ )=(0 . 291 +0 . 063 -0 . 065 , 0 . 55 ± 0 . 17), consistent with a flat universe (Ω K =0 . 16 ± 0 . 16); see Fig. 6. Combining DES-SN5YR with BAO+3 × 2pt is also consistent with a flat Universe, with uncertainties on Ω K reduced to ∼ ± 0 . 034, while the combination with Planck gives Ω K =0 . 010 ± 0 . 005. The combination of all three gives Ω K =0 . 002 +0 . 004 -0 . 003 .', '4.1.3. Flatw CDM': "Fitting DES-SN5YR to the Flatw CDM model, we measure (Ω M , w ) = (0 . 264 +0 . 074 -0 . 096 , -0 . 80 +0 . 14 -0 . 16 ); see Fig. 7. This is consistent with a cosmological constant (within 2 σ ), although our data favors a w -value that is slightly larger than -1. \nThe w -Ω M contours from SN alone are highly nonGaussian with a curved 'banana'-shaped degeneracy. The best fit value for w or Ω M is thus an insufficient summary of the SN information, as a small shift along the degeneracy direction can result in large shifts in the best-fit values. To address this issue, in Camilleri et al. (in prep. 2024) we introduce a new parameter, Q H ( z ) ≡ -a/ ( aH 2 0 ) ≡ q ( H/H 0 ) 2 . This combination of the deceleration parameter q and the Friedmann equation H/H 0 follows the curve of the degeneracy in the w -Ω M plane. Therefore, measuring Q H ( z ) summarizes the supernova information in a single, almost degeneracy-free value. 8 One has to choose the redshift at which one quotes Q H ( z ), to best match the angle of the degeneracy for the redshift range of the sample. We find Q H ( z = 0 . 2) = -0 . 340 ± 0 . 032 using DES-SN5YR only (see Camilleri et al. in prep. 2024). This Q H value can be used to roughly approximate the DES-SN5YR results and characterize the constraining power without the need for a full fit to the Hubble diagram. \nThe degeneracy in the w -Ω M plane is broken by combining SNe with external probes. Combining with Planck, we measure (Ω M , w ) = (0 . 337 +0 . 013 -0 . 011 , -0 . 955 +0 . 032 -0 . 037 ), again within 2 σ of a cosmological constant. Planck alone provides only a loose constraint on the equation of state parameter of dark energy, w Planck = -1 . 51 +0 . 27 -0 . 18 ; combining with DESSN5YR reduces the uncertainty significantly due to the different degeneracy direction, demonstrating the combined constraining power of these two complementary probes. \nCombining DES-SN5YR with BAO+3 × 2pt we find w = -0 . 922 +0 . 035 -0 . 037 , slightly over 2 σ from the cosmologi- \nFigure 8. Same as Fig. 6 but for the Flatw 0 w a CDM model. The dashed crosshairs mark the equation of state values for a cosmological constant, i.e. ( w 0 , w a ) = ( -1 , 0). The residuals between the DES-SN5YR best fit Flatw 0 w a CDM w.r.t. the Flatw CDM model are presented in Fig. 4. \n<!-- image --> \nl constant. This data combination demonstrates that these late-universe probes alone provide constraints that are consistent with - and of comparable constraining power to - the combination of SN and CMB data. The full combination of all data sets gives w = -0 . 941 ± 0 . 026.", '4.1.4. Flatw 0 w a CDM': "Fitting DES-SN5YR alone to the Flatw 0 w a CDM model gives an equation of state that is slightly over 2 σ from a cosmological constant, marginally preferring a time-varying dark energy (Ω M , w 0 , w a ) =(0 . 495 +0 . 033 -0 . 043 , -0 . 36 +0 . 36 -0 . 30 , -8 . 8 +3 . 7 -4 . 5 ); see Fig. 8. \nCombining DES-SN5YR and the CMB, we find (Ω M , w 0 , w a ) =(0 . 325 +0 . 016 -0 . 012 , -0 . 73 ± 0 . 11, -1 . 17 +0 . 55 -0 . 62 ), which again deviates slightly from the cosmological constant. The same trend is seen when combining with BAO+3 × 2pt and with all data combined. The negative w a means that the dark energy equation of state parameter is increasing with time (sometimes referred to as a 'thawing' model). \nGoodness of fit and tension \n4.2.1. χ per degree of freedom \n4.2. 2 \nTo assess whether our best fits are good fits we calculate the χ 2 per degree of freedom for all our dataset and model combinations; see the last column of Table 2. The χ 2 we use for this test is the maximum likelihood of the entire parameter space, not the marginalized best fit for each parameter. \nThe number of degrees of freedom is the number of data points minus the number of parameters that are common to all datasets (i.e., the cosmological parameters of interest). The number of data points added by the CMB, BAO, and 3 × 2pt is respectively 615, 8, and 471. Due to our treatment of contamination (by inflating the uncertainties of SNe with a low P Ia ), we approximate the effective number of data points in the DES-SN5YR sample by ∑ P B(Ia) = 1735 (rather than the total number of data points, 1829). \nDES-SN5Y \nDES-SN5Y \nDES-SN5Y \nDES-SN5Y \nTable 2. Results for four different cosmological models, sorted into sections for different combinations of observational constraints. These are the medians of the marginalized posterior with 68.27% integrated uncertainties ('cumulative' option in ChainConsumer). For each fit we also show the χ 2 per degree of freedom as a measure of the goodness of fit. \nIdeally, a good fit should have χ 2 / d.o.f. ∼ 1 . 0. The slightly low χ 2 / d.o.f. for the DES-SN5YR data arises because ∑ P B(Ia) only approximates the number of degrees of freedom, and the same behaviour is also seen in simulations.", '4.2.2. Suspiciousness': "Suspiciousness, S , (Handley & Lemos 2019) is closely related to the Bayes ratio, R , 9 and can be used to assess whether different datasets are consistent. However, while the Bayes ratio has been shown to be priordependent (Handley & Lemos 2019), with wider prior widths boosting the confidence, Suspiciousness is prior independent. Therefore, Suspiciousness is ideal for cases such as ours where we have chosen deliberately wide and uninformative priors (Lemos et al. 2021, Sec. 4.2). Trotta (2008) suggests ln S < -5 is 'strong' tension, -5 < ln S < -2 . 5 is 'moderate' tension, and ln S > -2 . 5 indicates the datasets are in agreement. \nWe determine ln S using the ANESTHETIC software (Handley 2019), which produces an ensemble of realizations used to estimate sample variance. Results are \nDES-SN5YR vs CMB \nFigure 9. Measurements of Suspiciousness (∆ ln( S )) between the DES-SN5YR and Planck 2020 datasets for the four models constrained in this paper. Further left indicates higher tension where the shaded regions reflect 'moderate' (yellow) evidence of tension according to Trotta (2008). The values and uncertainties represent the mean and standard deviation of realizations estimating sample variance using the ANESTHETIC software. \n<!-- image --> \nquoted using the mean of the ensemble, with the error bars reflecting the standard deviation. \nIn Fig. 9 we plot the Suspiciousness values for the DES-SN5YR data vs Planck 2020 and vs BAO+3 × 2pt data. We find no indication of tension using any of the four models investigated in this paper. \nFigure 10. Bayesian Evidence difference relative to FlatΛCDM (∆(ln BE )). We present the results for the four different models tested in this analysis and for the three combination of datasets used (DES only in cyan, DES+Planck in blue, DES+BAO+3 × 2pt in orange). An increase (decrease) in ∆(ln BE ) indicates that a model is disfavoured (favoured) compared to Flat-ΛCDM. \n<!-- image -->", '4.3. Model Selection': "Finally, we use Bayesian Evidence to test whether the extra parameters in the more complex models we test are warranted, given the data. In Fig. 10, we present the difference in the logarithm of the Bayesian Evidence, ∆(ln BE ), relative to Flat-ΛCDM for the four different models tested in this analysis and for the three combinations of datasets used in Fig. 10. \nTo evaluate the strength of evidence when comparing Flat-ΛCDM with more complex models, we again use Jeffreys' scale. This empirical scale suggests that ∆(ln BE ) > 2 . 5 (and < -2 . 5) is moderate evidence against (in support of) the more complex model, whereas ∆(ln BE ) > 5 (and < -5) is strong evidence against (in support of) the more complex model (for a review of model selection in cosmology see Trotta 2008). We note that none of the datasets considered in this analysis strongly favours cosmological models beyond Flat-ΛCDM. The priors that we choose for model comparison are w ∈ ( -1 . 5 , -0 . 5), w a ∈ ( -10 , 10) and Ω K ∈ ( -0 . 5 , 0 . 5). We consider these priors (which determine the penalty for more complex models) to be reasonable in terms of general considerations, such as avoiding universes that are younger than generally accepted stellar ages (see Section 5.1.3). Although our chains have been run on uninformative priors, the Bayesian Evidence from those chains may be adjusted for these harmonized priors as described in Appendix 3.", '5.1.1. Is the expansion of the Universe accelerating?': 'Twenty five years ago Riess et al. (1998) found 99.5%99.9% (2 . 8 σ to 3 . 9 σ ) evidence for an accelerating Universe, by considering the deceleration parameter q ≡ ( a a )˙ a -2 and integrating over the likelihood that q 0 < 0. Importantly they note that since q 0 is measured at the present day but the data span a wide range of redshifts, q 0 can only be measured within the context of a model, either cosmographic or physically motivated. They used the ΛCDM model, in which q 0 = Ω M / 2 -Ω Λ . \nDoing the same with DES-SN5YR data gives 99.99998% confidence (5 . 2 σ ) that q 0 < 0 in ΛCDM, or a 2 × 10 -7 chance that the expansion of the Universe is not accelerating. As noted in Section 4.1.3, our confidence is even higher that the universe was accelerating at z ∼ 0 . 2. When we further assume flatness, the confidence in an accelerating Universe is overwhelming (no measurable likelihood for a decelerating Universe) and we find q 0 = -0 . 530 +0 . 018 -0 . 017 . For more fits of q 0 using a cosmographic approach see Camilleri et al. (in prep. 2024).', '5.1.2. Is dark energy a cosmological constant?': "As seen in Sec 4.1, a cosmological constant is a good fit to our data, but not the best fit. Our best fit equation of state parameter is slightly (more than 1 σ ) higher than the cosmological constant value of w = -1 (both for SNe alone and in combination with Planck or BAO+3 × 2pt). Our result agrees with the recent result from the UNION3 compilation analyzed with the UNITY framework (Rubin et al. 2023) (which appeared while this paper was under internal review). The Pantheon+ result (Brout et al. 2022a) is within 1 σ of w = -1, but also on the high side ( w = -0 . 90 ± 0 . 14). \nFurthermore, our analysis slightly prefers a timevarying dark energy equation of state parameter when we fit for w ( a ) such that the equation of state parameter increases with time (again for all data combinations), known as a 'thawing' model. Model selection, however, is inconclusive. \nThe constraints on time-varying w are enabled by the wide redshift range of the DES-SN5YR sample. Our analysis as described in Vincenzi et al. (2024) gives us confidence that systematic uncertainties in this data are below the level of our statistical precision. Nevertheless, it is important to recognize that (a) the lowz sample is the one for which we have the least systematic control and (b) the very high-redshift SNe are the ones for which bias-corrections are large ( > 0 . 1 mag) and more uncertain (e.g., accurate estimation of spectroscopic redshift efficiency is more challenging as we go to higher red- \nshifts), and for which the uncertainties on the rest-frame UV part of the SN Ia spectral energy distribution have more impact on SN distances estimations (see also Brout et al. 2022a). \nTo test whether our fits are dominated by any particular redshift range we ran cosmological fits (a) removing lowz data (i.e., DES SNe alone) and (b) removing highz data (i.e., removing ∼ 80 SNe at z > 0 . 85, for which we use only two bands; see Fig. 2). Most of the cosmological results obtained with the subsamples are consistent with the results found for the full sample. However, we found that removing the lowz sample shifts the contours in the Flatw CDM slightly down, which would make the combined fits more consistent with w = -1. The Flatw 0 w a CDM results are stable to sub-sample selection. See Appendix C for details. \nWe showed in Vincenzi et al. (2024) that systematic uncertainties are sub-dominant to the statistical uncertainties in our sample. Nevertheless, in the future a new low-redshift sample (see Sec. 5.3) would help alleviate any remaining doubt about calibration and systematics in the existing lowz sample, and an even higher-redshift supernova survey would help alleviate any modelling concerns by minimizing selection effects even at z ∼ 1.", '5.1.3. How old is the Universe?': "One of the issues that the discovery of dark energy solved is the age of the Universe ( t 0 ) problem - globular cluster age estimates, in combination with high estimates of H 0 , were inconsistent with models that were not accelerating (VandenBerg et al. 1996; Gratton et al. 1997; Chaboyer et al. 1998). \nOur results, which favor a dark energy equation of state parameter slightly higher than w = -1 would imply that the age is slightly younger than the age found in a Universe where dark energy is a cosmological constant (for the same values of H 0 and present dark energy density). \nTo calculate the Universe's age, one needs a value of H 0 in addition to the best fit cosmological model. Since we do not constrain H 0 in this analysis, we present our measurement of the combination H 0 t 0 . In other words, we give t 0 in units of the Hubble time t H ≡ 1 /H 0 . 10 Our best-fit DES-SN5YR result in Flat-ΛCDM would have an age of (0 . 921 ± 0 . 013) t H . This is ∼ 3% younger than Planck ( t Planck age = (0 . 950 ± 0 . 007) t H ), corresponding to an age difference of approximately -0 . 4 Gyr. Our best fit Flatw 0 w a CDM model gives an age (0 . 86 ± 0 . 02) t H , about 9% younger than the Flat-ΛCDM Planck result, \ncorresponding to an age difference of approximately -1 . 3 Gyr. Such a young age is unlikely given the age of the oldest globular clusters (Valcin et al. 2020; Cimatti & Moresco 2023; Ying et al. 2023). In the future, this information could be used as a prior to limit the feasible range of time-varying dark energy.", '5.1.4. Does our best fit resolve the Hubble tension?': 'As pointed out in Planck Collaboration (2020, their Sec. 5.4), the only basic extensions to the base FlatΛCDM model that resolve the H 0 tension are those in which the dark energy equation of state is allowed to vary away from w = -1. In the w CDM model a phantom equation of state parameter of w ∼ -1 . 5 would help resolve the tension (Di Valentino et al. 2021, their Sec. 5.1), and it is clear from Fig. 7 that CMB alone actually prefers w < -1. In this model, Planck alone does not constrain H 0 very tightly, and they refrain from quoting a value, (see Table 5 of Planck Collaboration (2020)), but lower w correlates with higher H 0 . However, the DES-SN5YR data shows a slight tendency for w > -1, essentially ruling out this solution within w CDM.', '5.2. Comparison with DES-SN3YR and Pantheon+': "It is informative to compare the results of the previous DES-SN3YR analysis (Dark Energy Survey Collaboration 2019; Brout et al. 2019a) with the results of the DES-SN5YR analysis presented in this work. The DES-SN3YR analysis included 207 spectroscopically confirmed SNe Ia from DES and 127 low-redshift SNe from CfA and CSP samples (see also Fig. 3). A fraction of those events is in common between both analyses (55 from lowz external samples and 146 DES SNe). 11 \nHowever, the DES-SN3YR analysis differs from the analysis presented here in many aspects. The SN Ia intrinsic scatter modelling has been significantly improved (from 'G10' and constant σ int floor, to the more sophisticated modelling of intrinsic scatter introduced by Brout & Scolnic 2021; Popovic et al. 2023a), the BBC software has been updated (from BBC '5D' and a binned approach, to BBC '4D' and an unbinned approach), the x 1 -M ⋆ correlations have been incorporated into simulations (following the work by Smith et al. \nFigure 11. Comparison between Hubble residuals for the DES-SN3YR and DES-SN5YR analyses w.r.t. the best fit Flatw CDM for the DES-SN5YR analysis. Hubble residuals are binned in redshift and we present the weighted mean and standard deviation of the mean in each redshift bin. The redshift range covered by the lowz sample is highlighted and shown with thick dotted lines. The two DES samples are consistent with each other. Note the DES-SN3YR analysis only includes spectroscopically confirmed SNe whereas the DES sample in the DES-SN5YR analysis consists entirely of photometrically identified SNe Ia and extends to higherz . \n<!-- image --> \n2020b; Popovic et al. 2021), and the light-curve fitting model has been updated from the SALT2 model to the SALT3 model (see Taylor et al. 2023, for a comparison between SALT2 and SALT3 using the DES-SN3YR sample). Finally, the DES-SN3YR analysis did not require machine-learning classification and the implementation of the BEAMS approach because it is a sample of spectroscopically selected SNe Ia. We compare the final SN distances in Fig. 11 and find consistent results (differences in binned distances are on average 0.02 mag, even in the redshift ranges where contamination is expected to be high). The cosmological results from DES-SN3YR and DES-SN5YR are consistent within uncertainties (when assuming Flat-ΛCDM, Ω M are 0 . 331 ± 0 . 038 and 0 . 352 ± 0 . 017 for DES-SN3YR and DES-SN5YR respectively, while when assuming Flatw CDM and including CMB priors, w are -0 . 978 ± 0 . 059 and -0 . 955 +0 . 032 -0 . 037 ). \nThe other main dataset we can compare to is Pantheon+, which contains a significant amount of independent data (all the highz data). The DES sample is on average much higher redshift than the Pantheon+ sample (see Fig. 3), with over a quarter of the DES-SN5YR sample being at high enough redshift ( z ≳ 0 . 64) to probe the likely decelerating 12 period of the Universe (compared to 6% in Pantheon+). We show a comparison of the contours in Fig. 12. We find very similar constrain- \nFigure 12. Constraints in Flatw CDM from the DESSN5YR sample, the Pantheon+ sample (with and without CMB priors), and the Amalgame sample. The constraining power of the DES-SN5YR and Pantheon+ samples is comparable and consistent, despite Pantheon+ being a spectroscopic SN Ia sample combining 17 different surveys. The 'Amalgame' sample includes the SDSS and PS1 photometric SN samples ( > 1700 intermediate-redshift and high-redshift SNe), however it does not include a lowz anchoring sample (hence the larger contours). DES-SN5YR and Pantheon+ are also combined with CMB constraints (for both we use the Planck lite Python implementation presented by Prince &Dunkley 2019). The horizontal dotted line marks the equation of state values for a cosmological constant. \n<!-- image --> \ng power between Pantheon+ and DES-SN5YR, and the DES-SN5YR value of w is within 1 σ of Pantheon+ (Brout et al. 2022a). These analyses are not fully independent as a fraction of the lowz sample is shared. However, all of the highz dataset is independent, and DES is a photometric sample while Pantheon+ is fully spectroscopic. The constraints on w are similar between DES and Pantheon+ as DES highz has better precision per SN than Pantheon+ and has significantly higher statistical power at z > 0 . 4 (see Fig. 3), but Pantheon+ used 2 × more low-redshift SNe (which we do not include in order to be able to better control systematic uncertainties).", '5.3. DES and Next Generation Supernova Samples': "This analysis has shown that moving from a spectroscopically confirmed sample as done in Dark Energy Survey Collaboration (2019) to a photometric sample can increase the sample size of well-measured supernovae \nsignificantly (from 207 DES SNe Ia in DES-SN3YR to > 1600 in DES-5YR), consistent with an analysis of Pan-STARRS SNe in Jones et al. (2018). This improvement arises because photometric classification alleviates the bottleneck of limited spectroscopic resources. The improvement will increase for future surveys as more candidates are discovered, but the available time for spectroscopy does not increase commensurately. Importantly, the work of Vincenzi et al. (2024) shows that systematic uncertainties due to photometric classification are not limiting. Instead, the 'conventional' systematics of calibration and modeling the intrinsic scatter remain the most significant challenges. \nThere is potential for further increase of the statistical power of the DES sample if one moves to using SNe in which a host galaxy spectroscopic redshift was not acquired and instead relies on photometric redshifts of the SNe and the galaxy. This path was explored by Chen et al. (2022) for a subset of DES SNe, namely ones that occur in redMaGiC galaxies, and has been explored as well for SuperNova Legacy Survey (SNLS, Ruhlmann-Kleider et al. 2022) and the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) in Mitra et al. (2023). These analyses show that the use of photoz s do not introduce systematic uncertainties to a scale similar to the statistical uncertainties. This potential is highlighted by the ≈ 2400 SNe Ia identified without host galaxy spectroscopic redshift in DES that could be used for this type of analysis (Moller & the DES Collaboration in prep. 2024). \nThe DES supernova survey was supported by the 6year OzDES survey on the Anglo-Australian Telescope (described in Lidman et al. 2020), which took multifibre observations of host galaxies to acquire redshifts of host galaxies of SNe. The total investment of this program was 100 nights, and for roughly 75% of the targeted host galaxies a spectroscopic redshift has been secured. This program was fortuitous as the cameras for OzDES and DECam have a nearly identical field-ofview. Enormous resources would be needed to reproduce this joint program for LSST, which will find millions of SNe across 18,000 square degrees (Ivezi'c et al. 2019; S'anchez et al. 2022) (compared to the 27 square degrees of DES SNe). Surveys such as 4MOST will follow-up tens of thousands of these (Swann et al. 2019), but the full wealth of transient information may benefit from an entirely photometric approach. \nAs statistical precision continues to improve thanks to the increased number of supernovae, a main theme for systematic analysis is second-order relations between different systematics. Typically, systematics are treated independently when building the covariance matrix. We \nhave implemented a method to account for calibration systematics along with light-curve model systematics together, but this is currently the only joint exercise. This type of work will grow in importance. For example, while photometric classification does not directly cause a large increase in the error budget, it hinders the ability to constrain the intrinsic scatter model preferred by the data. Potentially, if LSST and other surveys such as those enabled by the Nancy Grace Roman Space Telescope have enough supernovae (Rose et al. 2021), the dataset can enable a forward modeling approach such as the Approximate Bayesian Computation method introduced in Jennings et al. (2016) and worked on in Armstrong et al. (in prep), which could vary all systematics, nuisance, and cosmological parameters at the same time to compare against the data. \nFurthermore, as discussed in Section 5.1.2, modeling of the lowz sample remains a source of systematic uncertainty. This sample comes from a multitude of surveys, even though we have removed many of the older inhomogeneous sources compared to analyses like Pantheon+. In the near future, we expect additions from Zwicky Transient Factory (Smith et al. in prep. 2024), Young Supernova Experiment (Jones et al. 2021; Aleo et al. 2023), and Dark Energy Bedrock All-sky Supernova Survey (DEBASS, PI: Brout) to improve lowz constraints of the SN Hubble Diagram, given their improved calibration and better understood selection function. 13 DEBASS will be particularly fruitful as it is a low-redshift sample taken with DECam, so a single instrument and calibration catalog will be used for the full sample of DEBASS+DES, similar to the singleinstrument PS1 sample in Jones et al. (2019). Using simulations, we estimate that quadrupling the size of our lowz sample (from ∼ 200 to ∼ 800 SNe expected from this next generation of lowz SN surveys) could enable a reduction of uncertainties on w by ∼ 30 per cent (for a Flat w CDM model, using SN data alone). \nLastly, we note that while LSST and Roman may help improve a number of these issues, the first data release is still > 3 years away. We encourage work with the DES-SN sample as presented here, combined with other samples. Popovic et al. (2023b) recently showed the ability to combine separate photometric samples (PS1 and SDSS) into the Amalgame sample (also shown in Fig. 12, and a similar analysis can be done by combining DES with these. It is reasonable to expect that with new low-redshift samples, and combination of high-redshift \nphotometric samples, a sample with > 5000 likely SNe Ia can be compiled in the very near future.", '6. CONCLUSIONS': "The DES Supernova survey stands as a groundbreaking milestone in SN cosmology. With a single survey, we effectively tripled the number of observed SNe Ia at z > 0 . 2 and quintupled the number beyond z > 0 . 5. Here we present the unblinded cosmological results, and in companion papers make public the calibrated light curves and Hubble diagram from the full sample of DES Type Ia supernovae (S'anchez in prep. 2024; Vincenzi et al. 2024). \nAfter combining the 1635 DES SNe (of which 1499 have a probability > 0 . 5 of being a SN Ia) with 194 existing lowz SNe Ia, we present final cosmological results for four variants on ΛCDM cosmology, as summarized in Table 2. \nThe standard Flat-ΛCDM cosmological model is a good fit to our data. When fitting DES-SN5YR alone and allowing for a time-varying dark energy we do see a slight preference for a dark energy equation of state that becomes greater (closer to zero) with time ( w a < 0) but this is only at the ∼ 2 σ level, and Bayesian Evidence ratios do not strongly prefer the Flatw 0 w a CDM cosmology. \nWe compare cosmological results from each of our models to results from the CMB analysis of Planck Collaboration (2020). There are some differences in the best fit values but in each case we find consistency to within 2 σ and a Suspiciousness statistic that indicates agreement among the datasets. \nCritically, the DES-SN5YR analysis shown here demonstrates that contamination due to SN classification and host-galaxy matching is not a limiting systematic for SN cosmology; this opens the path for a new era of cosmological measurements using SN samples that are not limited by live spectroscopic follow-up of SNe. Instead, our analysis shows the SN community that there are other factors that will be crucial for the success of future SN experiments: a high-quality low-redshift sample, a robust UV and NIR extension of light-curve fitting models, excellent control of selection effects across the entire redshift range, and improvement in our understanding of SN Ia intrinsic scatter properties and the role played by interstellar dust. \nFuture work will conclude the Dark Energy Survey by combining these supernova results with the other three pillars of DES cosmology, namely baryon acoustic oscillations, galaxy clustering, and weak lensing.", 'ACKNOWLEDGMENTS': "We acknowledge the following former collaborators, who have contributed directly to this work - Ricard Casas, Pete Challis, Michael Childress, Ricardo Covarrubias, Chris D'Andrea, Alex Filippenko, David Finley, John Fisher, Francisco Forster, Daniel Goldstein, Santiago Gonz'alez-Gait'an, Ravi Gupta, Mario Hamuy, Steve Kuhlmann, James Lasker, Marisa March, John Marriner, Eric Morganson, Jennifer Mosher, Elizabeth Swann, Rollin Thomas, and Rachel Wolf. \nT.M.D., A.C., R.C., S.H., acknowledge the support of an Australian Research Council Australian Laureate Fellowship (FL180100168) funded by the Australian Government, and A.M. is supported by the ARC Discovery Early Career Researcher Award (DECRA) project number DE230100055. M.S., H.Q., and J.L are supported by DOE grant DE-FOA-0002424 and NSF grant AST-2108094. R.K. is supported by DOE grant DE-SC0009924. M.V. was partly supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51546.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. L.K. thanks the UKRI Future Leaders Fellowship for support through the grant MR/T01881X/1. L.G. acknowledges financial support from the Spanish Ministerio de Ciencia e Innovaci'on (MCIN), the Agencia Estatal de Investigaci'on (AEI) 10.13039/501100011033, and the European Social Fund (ESF) 'Investing in your future' under the 2019 Ram'on y Cajal program RYC2019-027683-I and the PID2020-115253GAI00 HOSTFLOWS project, from Centro Superior de Investigaciones Cient'ıficas (CSIC) under the PIE project 20215AT016, and the program Unidad de Excelencia Mar'ıa de Maeztu CEX2020001058-M, and from the Departament de Recerca i Universitats de la Generalitat de Catalunya through the 2021-SGR-01270 grant. R.J.F. and D.S. were supported in part by NASA grant 14-WPS14-0048. The UCSC team is supported in part by NASA grants NNG16PJ34G and NNG17PX03C issued through the Roman Science Investigation Teams Program; NSF grants AST1518052 and AST-1815935; NASA through grant No. AR-14296 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; and fellowships from the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation to R.J.F. We acknowledge the University of Chicago's Research Computing Center for their support of this work. \nFunding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda¸c˜ao Carlos Chagas Filho de Amparo 'a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico and the Minist'erio da Ciˆencia, Tecnologia e Inova¸c˜ao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. \nThe Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energ'eticas, Medioambientales y Tecnol'ogicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci'encies de l'Espai (IEEC/CSIC), the Institut de F'ısica d'Altes Energies, Lawrence \nBerkeley National Laboratory, the Ludwig-Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's NOIRLab, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium. \nBased in part on observations at Cerro Tololo Inter-American Observatory at NSF's NOIRLab (NOIRLab Prop. ID 2012B0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. Based in part on data acquired at the Anglo-Australian Telescope. We acknowledge the traditional custodians of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present. Parts of this research were supported by the Australian Research Council, through project numbers CE110001020, FL180100168 and DE230100055. Based in part on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci'on y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on (Argentina), Minist'erio da Ciˆencia, Tecnologia, Inova¸c˜oes e Comunica¸c˜oes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). This includes data from programs (GN-2015B-Q-10, GN2016B-LP-10, GN-2017B-LP-10, GS-2013B-Q-45, GS-2015B-Q-7, GS-2016B-LP-10, GS-2016B-Q-41, and GS-2017B-LP-10; PI Foley). Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration (PIs Foley, Kirshner, and Nugent). The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. This paper includes results based on data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile (PI Foley), and the Southern African Large Telescope (SALT) (PIs M. Smith & E. Kasai). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian \ncommunity. We are most fortunate to have the opportunity to conduct observations from this mountain. \nThe DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171. The DES participants from Spanish institutions are partially supported by MICINN under grants ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-20160597, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya. Research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We acknowledge support from the Brazilian Instituto Nacional de Ciˆencia e Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2). \nThis research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC0205CH11231 using NERSC award HEP-ERCAP0023923. This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. \nFacilities: CTIO:4m, AAT, Gemini:Gillett (GMOSN), Gemini:South (GMOS-S), Keck:I (LRIS), Keck:II (DEIMOS), Magellan:Baade (IMACS), Magellan:Clay (LDSS3, MagE), SALT \nSoftware: numpy (Harris et al. 2020), astropy (Astropy Collaboration 2013, 2018), matplotlib (Hunter 2007), pandas (Pandas development team 2020), scipy (Virtanen et al. 2020), SNANA (Kessler et al. 2009), Pippin (Hinton & Brout 2020), ChainConsumer (Hinton 2016), Source Extractor (Bertin & Arnouts 1996), MINUIT (James & Roos 1975), SuperNNova (Moller & de Boissi'ere 2020), SCONE (Qu et al. 2021).", 'A. DATA RELEASE AND HOW TO USE THE DES-SN5YR DATA': "Here we explain where to find the data and software necessary to reproduce our analysis. Many of the codes we use are already public (detailed below). The key data, code, and tutorials are available on Github at https: //github.com/des-science/DES-SN5YR. \nThe DES-SN5YR analysis was run using the pippin pipeline framework (Hinton & Brout 2020) 14 that orchestrated SNANA codes for simulations, light curve fitting, BBC, and covariance matrix computation ( SNANA , Kessler et al. 2009), 15 and also integrated photometric classification from Moller & de Boissi'ere (2020) 16 and Qu et al. (2021). 17 Additional analyses codes that run outside the main pipeline include Scene Model Photometry (Brout et al. 2019b), fit \nto measure the SN population of stretch and color (Popovic et al. 2023a), 18 SALT3 training (Kenworthy et al. 2021), 19 and CosmoSIS to fit for cosmological parameters (Zuntz et al. 2015). 20 \nWe release the pippin input files necessary to (i) generate and fit all the simulations used in the analysis (both the large 'biasCor' simulations to calculate bias corrections, and the DES-SN5YR-like simulated samples to validate the analysis); (ii) reproduce the full cosmological analysis, from light-curve fitting to photometric classification, distance estimates and cosmological fitting. Auxiliary files are also available within the SNANA library. 21 \nThe various (intermediate and final) outputs of our analysis pipeline are also provided. This includes (i) light-curve fitted parameters, (ii) light-curve classification results, (iii) the final Hubble diagram and associated uncertainties covariance matrices, and (iv) the cosmology chains.", 'B. PRIORS': "Table 3 lists the prior ranges for our MCMC chains. The priors related to external data sets align with the priors in the original papers. We adapted the prior ranges to enclose the majority of the high likelihood region as appropriate for each data set and model combination. Data-set specific priors are listed in the footnote to the table. \nBayesian Evidence calculations depend on the choice of prior; larger prior ranges used on the same data and likelihoods lead to lower evidences, sometimes referred to as the complex model penalty . Therefore in model comparison using evidence calculations, we took care to choose consistent prior ranges that do not unduly inflate this penalty. Bayes' Theorem states, \np ( M | D ) = p ( D | M ) p ( M ) p ( D ) ∝ p ( D | M ) , (B1) \nwhere D is the data and M is the model, and the proportionality to the Bayesian Evidence p ( D | M ) follows from assuming no prior preference for any model. Writing the model parameters as ⃗ θ we can then write, \np ( D | M ) = ∫ p ( D, ⃗ θ | M ) d N θ = ∫ p ( D | ⃗ θ, M ) p ( ⃗ θ ) d N θ = p ( ⃗ θ ) ∫ p ( D | ⃗ θ, M ) d N θ , (B2) \nwhere the last step assumes a constant prior for each of the N parameters θ i of model M , that fully encompasses the support of the likelihood function (this is true to a very good approximation for the models that are tested here). Making explicit the dependence of the Bayesian Evidence on the model prior by writing p ( D | M ) = BE ( ⃗ θ ), the evidence may then be adjusted for a change in prior volume without recomputing the chains as follows : \nln BE ( ⃗ θ 2 ) = ln BE ( ⃗ θ 1 ) + ln p ( ⃗ θ 1 ) -ln p ( ⃗ θ 2 ) , (B3) \nwhere using ( θ i, min , θ i, max ) for the prior range for each parameter, \np ( ⃗ θ ) = N ∏ i =1 1 θ i, max -θ i, min . (B4)", 'C. TESTS ON SUBSETS OF OUR DATA': 'The large redshift range of the DES-SN5YR sample provides a strong lever arm on the measurement of any time variation of dark energy. We therefore check for potential peculiarities at the extremes of our redshift range that are driving the fit toward non-cosmological-constant values. \nIn Fig. 13 and Table 4, we show the change to the Flat-ΛCDM, Flatw CDM and Flatw 0 w a CDM fits using DES-SN alone (no Lowz external samples) and when using the full DES-SN5YR sample but excluding the highest redshift SNe ( z > 0 . 85, the 5 per cent highest redshift events in our DES SN sample). We show, for example, that in FlatΛCDM excluding the Lowz sample lowers the best fit value to Ω noLow -z M =0 . 330 ± 0 . 024 (∆Ω M = -0 . 022), which closer agreement with the CMB value of Ω Planck M =0 . 317 ± 0 . 008. Similarly, excluding high redshift SNe lowers the best fit value to Ω noHigh -z M =0 . 342 ± 0 . 017 (∆Ω M = -0 . 010). However, it is important to quantify the significance of the observed shifts. \nTable 3. Priors § \n<!-- image --> \nFigure 13. Constraints for the full DES-SN5YR dataset (cyan), when excluding lowz SNe ( z < 0 . 1, grey dashed line), and when excluding highz SNe ( z > 0 . 85, brown dotted-dash line). In Flatw CDM (left) the contours shift primarily along the degeneracy line (and in opposite directions for the lowz and highz cuts), but also slightly perpendicular to the degeneracy direction. In combination with the CMB prior this pushes the result closer to w = -1 in the no-lowz case. The Flatw 0 w a CDM model (right) best fit sees no significant shifts with sub-sample selection. \n<!-- image --> \nThe cosmological contours using the full DES-SN5YR sample, the DES-SN5YR sample without Lowz , and the DES-SN5YR sample without Highz cannot be directly compared as if they were three independent measurements (the three datasets used have large overlaps). Therefore, in order to examine the significance of the observed shifts, we generate 100 independent realizations of the DES-SN5YR Hubble diagram applying the Cholesky Decomposition to the full DES-SN5YR data vector of 1829 SNe, and the associated 1829 × 1829 statistical and systematic covariance matrix. For each independent realization, we fit the cosmological parameters with and without the Lowz and Highz samples and estimate the standard deviation ( σ ) of the estimated ∆Ω M (or ∆ w and/or ∆ w 0 and ∆ w a when fitting for Flatw CDM and Flatw 0 w a CDM). Using this approach, we measure a σ (∆Ω M ) of 0.02 and 0.005 when fitting for Flat-ΛCDM and excluding Lowz and Highz SNe respectively, and we conclude that the ∆Ω M observed on the real data are significant at the 1 . 1 σ and 2 . 1 σ respectively. \nIn Flatw CDM, excluding Lowz gives a best fit w = -1 . 34 ± 0 . 32 (∆ w =0 . 54) and excluding Highz gives a best fit w = -0 . 66 ± 0 . 11 (∆ w = -0 . 14). Using our 100 realizations with systematics, we estimate that the significance of the shifts is 2 . 3 σ and -2 . 2 σ , respectively. \nWe perform the same test incorporating a CMB-like prior. Estimating the best-fit Flatw CDM from our SN subsamples combined with the full CMB likelihood from Planck Collaboration (2020) is computationally expensive and practically unfeasible for data and 100 simulations. For this reason, we use an approximation of a CMB-like prior that uses the R -parameter (defined, e.g., in Komatsu et al. 2009, see Eq. 69) from Planck Collaboration (2020). This CMB-prior approximation is incorporated in the fast minimization cosmological fitting program wfit , available in SNANA. When combining SNe and the approximated CMB prior and fitting for Flatw CDM, we find that the shifts observed in w are not statistically significant (less than 2 σ ). \nWe make similar tests for Flatw 0 w a CDM model. The main results are consistent for the different redshift cuts, with the central value varying less than the Flatw CDM case despite (or because of) the extra flexibility of Flatw 0 w a CDM. If not statistical fluctuations, the observed shifts in w when removing either low or highz SNe would be expected if the Flatw CDM model is inadequate and cannot simultaneously fit the both low and high redshift range in our data; but it is also what you expect if there is some kind of systematic error in the lowz or highz data. Future independent data sets (both supernovae and other measures of expansion such as Baryon Acoustic Oscillations) are essential to \nTable 4. Results using DES data alone (excluding Lowz below z < 0 . 1) and DES-SN5YR without highz SNe ( z < 0 . 85). Shift significance: the significance of shifts in either Ω M (when fitting for FlatΛCDM model) or w (when testing Flat w CDM) is estimated from 100 simulations. \ndetermine which is the better explanation. 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2024arXiv240910497G

In this work we employ the methods of archaeoastronomy to analyze the orientation possibly astronomical of numerous groups of chullpa funerary towers mainly from the 12th to 16th centuries located in the Lauca River valley of the central Bolivian highlands. Despite their great historical relevance both regarding the beliefs and funerary customs of the local populations and the characteristics of the landscape in the highlands little is known about the relationship of these mortuary monuments with the sky. Several authors from chroniclers of the colonial era to more modern explorers indicate that the tomb towers of these regions are oriented in such a way that important parts of their structure in general the entrances of the chullpas point towards the sunrise on the eastern horizon in order to be imbued with the first rays of the Sun. However the sunrise changes its location noticeably at different times of the year. Given the lack of written information or other forms of original documentation in order to affirm the use of a systematic orientation it is necessary to measure a statistically significant number of monuments. We present here the results of the analysis of the precise spatial orientation of the entrances of 80 towers measured in situ during field work in the Lauca River valley. We find that except for a few all the buildings have the openings axes oriented towards the east and within the solar range between the extreme azimuths of the annual movement of the Sun as it crosses the local horizon with a notable concentration of entrances that point slightly towards the north of due east. Our work is the first systematic study of the orientations of the chullpa towers of the Lauca River and can provide crucial information to understand the evolution and scope of the chullpa phenomenon in the Bolivian highlands and in the entire surrounding region.

2024-09-01T00:00:00Z

['arXiv:2409.10497', '2024arXiv240910497G', '10.48550/arXiv.2409.10497']

['Physics - History and Philosophy of Physics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics']

Torres funerarias chullpa en el valle del ro Lauca un primer anlisis arqueoastronmico

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.10497.pdf

{'Alejandro Gangui 1': "Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Argentina. \nCONICET - Universidad de Buenos Aires, Instituto de Astronomía y Física del Espacio (IAFE), Argentina \nRecibido: xx/xx/xx; aceptado: xx/xx/xx \nEn este trabajo empleamos los métodos de la arqueoastronomía para analizar la orientación, posiblemente astronómica, de grupos numerosos de torres funerarias chullpa , principalmente de los siglos XII al XVI, ubicadas en el valle del río Lauca del altiplano central boliviano. A pesar de su gran relevancia histórica, tanto en las creencias y costumbres funerarias de las poblaciones locales como en la constitución del paisaje en el altiplano, poco se sabe sobre la relación de estos monumentos mortuorios con el cielo. Varios autores, desde cronistas de la época colonial hasta exploradores más modernos, indican que las torres-tumba de estas regiones se orientan de forma tal, que partes importantes de su estructura (en general, los vanos o entradas de las chullpas) apuntan hacia el levante en el horizonte oriental, de manera de impregnarse de los primeros rayos del Sol. Sin embargo, el orto solar cambia su ubicación de manera notable en diferentes épocas del año. Dada la falta de información escrita u otra forma de documentación original, para poder afirmar el uso de una orientación sistemática es preciso efectuar la medida de un número estadísticamente significativo de monumentos. Presentamos aquí los resultados del análisis de la orientación espacial precisa de los vanos de 80 torres medidas in situ durante un trabajo de campo en el valle del río Lauca. Hallamos que, excepto unas pocas, todas las construcciones poseen los ejes de los vanos orientados hacia oriente y dentro del rango solar, entre los acimuts extremos del movimiento anual del Sol al cruzar el horizonte local, con una concentración notable de entradas que apuntan levemente hacia el norte del este. El presente constituye el primer estudio sistemático sobre las orientaciones de las chullpas del río Lauca y puede proporcionar información crucial para comprender la evolución y el alcance del fenómeno chullpario en el altiplano boliviano y en toda la región circundante. \nPalabras clave: orientación de torres funerarias, arqueoastronomía, cultura aymara, astronomía y sociedad. \nIn this work we employ the methods of archaeoastronomy to analyze the orientation, possibly astronomical, of numerous groups of chullpa funerary towers, mainly from the 12th to 16th centuries, located in the Lauca River valley of the central Bolivian highlands. Despite their great historical relevance, both regarding the beliefs and funerary customs of the local populations and the characteristics of the landscape in the highlands, little is known about the relationship of these mortuary monuments with the sky. Several authors, from chroniclers of the colonial era to more modern explorers, indicate that the tomb towers of these regions are oriented in such a way that important parts of their structure (in general, the entrances of the chullpas) point towards the sunrise on the eastern horizon, in order to be imbued with the first rays of the Sun. However, the sunrise changes its location noticeably at different times of the year. Given the lack of written information or other forms of original documentation, in order to affirm the use of a systematic orientation, it is necessary to measure a statistically significant number of monuments. We present here the results of the analysis of the precise spatial orientation of the entrances of 80 towers measured in situ during field work in the Lauca River valley. We find that, except for a few, all the buildings have the openings' axes oriented towards the east and within the solar range, between the extreme azimuths of the annual movement of the Sun as it crosses the local horizon, with a notable concentration of entrances that point slightly towards the north of due east. Our work is the first systematic study of the orientations of the chullpa towers of the Lauca River and can provide crucial information to understand the evolution and scope of the chullpa phenomenon in the Bolivian highlands and in the entire surrounding region. \nKeywords: funerary tower orientation, archaeoastronomy, aymara culture, astronomy and society.", 'I. INTRODUCCIÓN': 'La observación del cielo ha ocupado la atención de nuestros antepasados desde épocas muy tempranas. Al carecer de instrumentos sofisticados seguían el movimiento de los cuerpos celestes a simple vista. \nReconocían los momentos singulares de los astros y la repetición cíclica de las estaciones del año. Determinaban, entre otros, las posiciones de salida y puesta del Sol en los solsticios, las de la Luna en los lunasticios, y los ortos y ocasos de las estrellas más prominentes. En reiteradas ocasiones levantaban \nestructuras, a veces monumentales, alineadas con esas direcciones o elegían como emplazamiento de sus lugares sagrados y tumbas, aquellos que se encontraban en un sitio singular de forma que alguno de los fenómenos descritos con anterioridad se produjese sobre una montaña sagrada o en algún otro referente topográfico importante. La relación entre paisaje celeste y paisaje terrestre, es decir, el Paisaje con mayúsculas, ha sido siempre mucho más íntima de lo que hoy en día podría parecer. La importancia de este hecho se refleja también en la reciente aprobación por parte de la UNESCO de la iniciativa "astronomía y patrimonio mundial" que trata de identificar y proteger aquellos lugares, o aquellos elementos intangibles de nuestra cultura, donde la astronomía haya jugado un papel fundamental (Ruggles y Cotte 2011). \nSabemos que el paisaje, en su definición más general, incluyendo aspectos terrestres y celestes, jugó un papel relevante en la localización y la orientación de edificios en diversas culturas del pasado. Esto sucedió en latitudes muy distantes, por ejemplo, en el Egipto antiguo (Belmonte y Shaltout 2009), como así también en regiones más cercanas, siendo el imperio Inca un caso paradigmático (Bauer y Dearborn 1995). En el caso de las organizaciones sociales conocidas históricamente como Confederaciones o "Señoríos" aymara de los siglos XII al XVI, es poco lo que sabemos sobre los aspectos prácticos y cultuales de su astronomía. Estudiar la distribución y orientación de los monumentos chullpa (como expresiones arquitectónicas de costumbres mortuorias), tan relevantes en la cosmovisión andina (Nielsen 2008), nos permitirá indagar de forma novedosa sobre el posible influjo de la cultura Tiwanaku en el culto y en la construcción de tumbas y torres sepulcrales posteriores y que perduraron incluso durante las dominaciones inca y luego colonial hispánica. \nEn los últimos años se han multiplicado los estudios arqueoastronómicos sobre monumentos rituales y funerarios en varios sitios de relevancia arqueológica. Tal es el caso de los monumentos saharianos de piedra seca que representan importantes marcadores culturales (Gauthier 2009) y cuya distribución se ajusta muy bien con la visibilidad de objetos celestes particulares. Se ha verificado que, en el Sahara central, el patrón de orientaciones sigue con alta probabilidad la dirección de la Luna o del Sol nacientes (Gauthier 2015) y, en el caso del Sahara occidental, estudios estadísticos de varios centenares de monumentos muestran un patrón general de orientaciones hacia el horizonte oriental, principalmente agrupadas ligeramente al sur de la posición de salida más meridional de la Luna, el lunasticio mayor sur (Rodríguez-Antón et al. 2023). También en el Cuerno de África, la región oriental de ese continente donde el mar Rojo se conecta con el océano Índico, estudios recientes (Cornax Gómez et al. 2022) sugieren que una gran muestra de montículos de piedras ( cairns ), estelas y entierros antiguos en el campo de túmulos de Heis (Xiis, en Somalilandia) podrían estar orientados hacia objetivos astronómicos. Por ejemplo, estos investigadores llegan a resultados que muestran una alta concentración de orientaciones \nhacia el lunasticio mayor norte, donde la Luna, cuando alcanza su declinación máxima, cruza el horizonte. Dada la importancia de la Luna llena para esos pueblos nómades del desierto, la salida más extrema de las Lunas llenas de invierno a lo largo de los años, definida como la Luna llena antes y después del solsticio de invierno boreal, coincidiría con el patrón que fue encontrado en su estudio.', 'II. LA CULTURA AYMARA: CONTEXTO HISTÓRICO': 'Luego del colapso del imperio y la cultura Tiwanaku, entre los años 1000 y 1100 d.C., aparecen en la región altiplánica boliviano-peruana varios señoríos o etnias que se disputan el territorio. Se trata de grupos aymara-parlantes ("jaqi aru", lengua de la gente o aymara) que irrumpen desde el sudoeste después de una época marcada por una extrema sequía que alteró severamente el sistema agrario manejado por el antiguo imperio (Kolata 1993). Estos reinos o señoríos se desarrollaron en el altiplano hasta aproximadamente el año 1450, cuando los incas invadieron la región. \nLa conquista inca de Carangas, región del Collao (o Collasuyo, así llamado por los incas) que nos ocupará en el presente trabajo, según detallan Gisbert y colaboradores (Gisbert et al. 1994; 1996), comenzó con Pachacuti y se completó durante el reinado de Tupac Inca Yupanqui. Según la crónica del Padre Bernabé Cobo en el siglo XVII, Tupac Yupanqui logró sorprender a Collas y a otras etnias (los Pacajes) en la contienda. El Inca "se encaminó al Collao detrás de las sierras de Vilcanota, y vino a salir a Chungará, tomando por las espaldas al ejército de los Collas". De acuerdo con esta cita el Inca penetró en el Collao por la zona del volcán Sajama, el río Lauca y el lago Chungará (Michel López 2021). En poco tiempo, la mayoría de los señoríos aymara fueron conquistados por los incas, ya sea por pactos de sumisión o por pérdida de batallas y la correspondiente anexión forzosa. \nEn el ámbito arquitectónico lo más característico de todos estos grupos aymara son sus construcciones fortificadas y sus majestuosos monumentos mortuorios. Las primeras, llamadas pucaras en quechua, fueron emplazadas en lo alto de los cerros y en lugares estratégicos para la defensa. Los segundos eran mausoleos o construcciones funerarias que, como ya señalamos, recibieron el nombre de chullpas, torres de variados tamaños, con planta cuadrada o circular y cubiertas con una bóveda por avance, en muchos casos con decoraciones muy vistosas, que se encuentran en grupos formando extensas necrópolis (Gisbert et al. 1996: 7). A las chullpas se las encuentra construidas en adobe, piedra cortada o labrada. Las más antiguas estudiadas datan, en promedio, aproximadamente del año 900 de nuestra era (Pärssinen 1993). En particular, para las torres de la región de Pacajes, este autor da una fecha radiocarbónica que delimita aproximadamente su construcción entre 1450 y 1652. Esto señala que existen en el altiplano chullpas contemporáneas a la ocupación inca y a la llegada de los españoles, como se esperaba, \npues, por ejemplo, las chullpas monumentales de Sillustani (en Perú) muestran una factura similar a la característica constructiva incaica (Gisbert et al. 1996). \nHay todavía amplia discusión sobre el verdadero significado o utilidad de las chullpas en el paisaje altiplánico. Las investigaciones ya han dejado de lado la idea de que las torrecillas eran exclusivamente sepulcros de elite de personajes prominentes ( mallkus o señores de hombres y territorios, cabezas de linaje), considerándolas más como verdaderas encarnaciones monumentales del ancestro mismo que se desea honrar. Y en tal capacidad las chullpas serían responsables de hacer lo que hacen los ancestros, es decir, "proteger los campos y los rebaños, y promover su fertilidad; proteger la cosecha; traer prosperidad a sus descendientes y proporcionarles comida, agua y otros bienes (almacenados); representar al grupo ante extraños; defender la comunidad y su territorio; luchar contra sus enemigos; inspirar decisiones políticas", y demás acciones fundamentales para la comunidad (Nielsen 2008). Además, hay evidencias de que, antes y después de la dominación inca, las chullpas eran empleadas como marcadores de límites territoriales (Fig. 1) o, de alguna manera, como construcciones que señalaban las tierras controladas por diferentes familias, linajes u organizaciones comarcales ( ayllus ) (Hyslop 1977). \nFIG. 1: Dibujo de una chullpa como mojón de delimitación territorial inca en Nueva crónica y buen gobierno , de Guaman Poma de Ayala [1615]. \n<!-- image -->', 'III. CHULLPAS: TORRES FUNERARIAS DEL ALTIPLANO': 'Podemos imaginar la sorpresa de los primeros exploradores europeos al encontrarse con estas vistosas construcciones mortuorias torriformes distribuidas en la extensa llanura altiplánica. Uno de los primeros cronistas, el soldado y explorador Pedro Cieza de León, en La crónica del Perú (1553), relata que "tienen estos indios distintos ritos en hacer las sepulturas, porque en la provincia de Collao (como relataré en su lugar [cap. C]), las hacen en las heredades, por su orden, tan grandes como torres, unas más y otras menos, y algunas hechas de buena labor, con piedras excelentes, y tienen sus puertas que salen al nacimiento del sol, y junto a ellas (como también diré) acostumbran a hacer sus sacrificios y quemar algunas cosas, y rociar aquellos lugares con sangre de corderos o de otros animales" (Cieza, [1553, cap. LXIII] 1984: 266). Y más adelante agrega algunos detalles sobre la ubicación y las características constructivas de las chullpas: "por las vegas y llanos cerca de los pueblos estaban las sepulturas destos indios, hechas como pequeñas torres de cuatro esquinas, unas de piedra sola y otras de piedra y tierra, algunas anchas y otras angostas; en fin, como tenían la posibilidad o eran las personas que lo edificaban. Los chapiteles, algunos estaban cubiertos con paja; otros, con unas losas grandes; y parecióme que tenían las puertas estas sepulturas hacia la parte de levante" (Cieza, [1553, cap. C] 1984: 357). \nPocos años más tarde, en 1571, el cronista y encomendero Juan Polo de Ondegardo no solo describe en detalle los enterramientos de los indios, sino sus costumbres y la pervivencia de las torres chullpa (los "sepulcros de sus mayores"), las que, pese a su prohibición, seguían en uso durante la colonia castellana: "Es cosa común entre indios desenterrar secretamente los defuntos de las iglesias, o ciminterios, para enterrarlos en las Huacas, o cérros, o pampas, o en sepulturas antiguas, o en su casa, o en la del mesmo defunto, para dalles de comer y bever en sus tiempos. Y entonces beven ellos, y baylan y cantan juntando sus deudos y allegados para esto" (Polo de Ondegardo [1571] 1916: 194). \nEs así que, en 1574, para evitar los continuos enterramientos en los chullpares, el virrey Toledo expide una Ordenanza que dictamina "que cada juez en su distrito haga que todas las sepulturas de torres que están en bóvedas en las montañas, e sierras, se derruequen e haga hacer un hoyo grande donde se pongan revueltos los huesos de todos los difuntos que murieron en su gentilidad" (Gisbert et al. 1994: 437). Como sabemos, estas órdenes se ejecutaron solo en parte, a juzgar por la gran cantidad de chullpares que sobrevivieron a nuestros días. Incluso varias décadas después de esta Ordenanza, las chullpas seguían atrayendo la atención de los cronistas (Fig. 2). Es el caso del ya mencionado Padre Cobo, quien, en su Historia del Nuevo Mundo , de 1653, escribía: "Hacíanlas por las vegas, dehesas y despoblados, unas cerca y otras lejos de sus pueblos. Todas eran en forma \nde torrecillas, las menores de un estado [unos 195 cm] de alto, poco más o menos, al talle de nuestras chimeneas, algo más capaces, y las mayores de cuatro a seis estados de alto. Todas tienen las puertas al oriente, y tan bajas y estrechas como bocas de horno, que no se entra en ellas sino pecho por tierra." (Cobo, [Lib. MV, cap. 18] 1964 II: 271-273). \nFIG. 2: Entierro entre Collas, al fondo se ve una chullpa. Dibujo de Nueva crónica y buen gobierno , de Guaman Poma de Ayala [1615]. \n<!-- image -->', 'IV. ORIENTACIÓN DE LAS CHULLPAS: ¿ASTRONOMÍA O TOPOGRAFÍA?': 'Un elemento que se destaca de la narración de estos cronistas tempranos es la orientación de los vanos de entrada de las chullpas hacia el oriente: hacia el "nacimiento del sol" o "la parte de levante" según Cieza de León, o con "las puertas al oriente" según el Padre Bernabé Cobo. Esta orientación que podríamos llamar astronómica es coherente con la idea de que los cuerpos de los difuntos, guardados por la eternidad en el interior de las torrecillas, recibieran los primeros rayos del Sol de cada día, impregnándose de la energía revigorizante del astro. \nSin embargo, el paisaje andino para la cultura aymara estaba poblado de elementos sagrados y su cosmovisión incluía tres planos o "Pachas" (superior, terrestre e interior) que se conectaban a través de sitios especiales, lugares míticos de origen de donde había salido el primer ancestro, llamados pacarinas o huacas (Harris y Bouysse-Cassagne 1988: 246). Por lo tanto, es de esperar que la orientación topográfica, hacia sitios especiales del paisaje terrestre, también haya dejado su rastro en las chullpas. \nComo narra Cieza de León: "cuentan estos indios que tuvieron en los tiempos pasados por cosa cierta que las ánimas que salían de los cuerpos iban a un gran lago, donde su vana creencia les hacía entender haber sido su principio" (Cieza [1553, cap. XCVII] 1984: 349). Así, lagos (como el Titicaca en el actual límite entre Bolivia y Perú) o incluso lagunas prominentes, estaban muchas veces dotados de poderes de recreación o revitalización. La misma veneración se daba con los volcanes y montes nevados que se destacan en el paisaje del altiplano andino, asociados con el culto a los antepasados. Como resultado, es natural que en el trabajo de campo de años subsiguientes se hayan encontrado chullpares orientados hacia los lagos, hacia los cerros tutelares, llamados apus (e.g., Reinhard 1983), o hacia los montes prominentes donde residen los antepasados, conocidos también como "abuelos" ( achachilas ) o gentiles (Harris y BouysseCassagne 1988: 249). \nComo lo señala Gil García (2002), las primeras observaciones planimétricas sistemáticas comenzaron a arrojar resultados en los que las torres chullpa no orientaban sus vanos hacia levante. Por ejemplo, Squier ([1877] 1974: 190-192) observó tempranamente que las chullpas de Acora (en Puno, Perú) se orientaban hacia el norte, en dirección al lago Titicaca. Años más tarde, también Ryden (1947: 343) señaló que las chullpas de la isla de Taquiri, en el Titicaca del lado boliviano, miraban todas hacia las orillas del lago y hacia el horizonte de las altas cumbres. Posteriormente, varios otros exploradores que recorrieron estas regiones del altiplano encontraron grandes necrópolis de chullpas con orientaciones topográficas diferentes, y la evidencia muchas veces mostraba que los vanos de las torres no se relacionaban con el surgir del Sol (ver Gil García 2002: 226 por más ejemplos). \nPor lo que sabemos, entonces, la orientación de estos monumentos, lejos de ser aleatoria, parece seguir ciertos patrones claros: los vanos de las chullpas se orientarían hacia lagos, cursos de agua, montes o elevaciones particulares o, como comentan varios de los primeros cronistas, hacia la salida del Sol en el oriente. Como ya mencionamos, esta última es una característica de las poblaciones andinas, cuyas construcciones en general se orientaban en dirección del Sol naciente. Recordemos que para estos pueblos la orientación era muy relevante y estaba revestida de un valor simbólico (BouysseCassagne 1987: 75). El este, o el oriente en general, era considerado la orientación de la vida y la fertilidad. El oeste (o el poniente), por su parte, estaba asociado con la muerte y la escasez. En particular, hasta hace no mucho tiempo, predominaban las casas de los pastores con las puertas orientadas hacia el levante, mientras que era en la dirección opuesta donde se depositaba la basura. Trabajos etnográficos de los últimos años mostraron que muchos rituales se estructuraban de acuerdo con la importancia de estas orientaciones espaciales. \nVemos entonces que la orientación de las torres chullpa fue un aspecto importante de la cultura aymara (Fig. 3). Sin embargo, el "grado de precisión" de las afirmaciones de los cronistas y exploradores posteriores no fue el principal foco de atención. Sabemos que estos \npueblos no tenían forma de conocer la ubicación de los puntos cardinales más que por la posición del Sol. Pero sabemos también que el Sol, por ejemplo, en la latitud del río Lauca (donde realizaron trabajos Gisbert y colaboradores, y donde efectuamos nuestras mediciones), o en otras zonas cercanas, varía su posición en el horizonte al amanecer en un ángulo de unos 50° (entre los acimuts 65° y 115° aproximadamente) si uno lo observa entre un solsticio y el otro (entre aproximadamente el 21 de junio y el 21 de diciembre). Los arqueoastrónomos se interesan por medir, al menos de manera estadística, las orientaciones precisas de estas construcciones pues, aun a la distancia de siglos, dicen mucho sobre el culto solar de los pueblos y sobre sus actividades en diferentes momentos de su año (agrícola, religioso, etc.). Este es sin duda un tema sumamente interesante, no solo desde la temática astronómica, y es por ese motivo que hemos llevado adelante una misión de campo en una región cultural específica, Carangas, y hemos medido las orientaciones precisas de unas 80 chullpas en el ámbito del río Lauca. \nFIG. 3: Dos torres decoradas ubicadas al sur del río Lauca. Una de ellas muestra una cruz aspada y dentada que ocupa toda la cara frontal con colores rojo y blanco, la otra decorada con un friso formado por rombos dentados de color rojo, verde, negro y blanco. La imagen fue tomada por la mañana y muestra que los vanos se abren hacia el oriente y no miran a lagunas o montes nevados prominentes. Foto del autor. \n<!-- image -->', 'V. LAS CHULLPAS DEL RÍO LAUCA': 'La región de Carangas era uno de los señoríos de habla aymara que ocupaba la parte occidental del altiplano boliviano al oeste del lago Poopó y el río Desaguadero. Según Rivière (1982), los Carangas también dominaban gran parte de la Cordillera Occidental y territorios ubicados en el desierto de Atacama. Un corte transversal de su territorio incluyendo los enclaves de la costa y de los valles muestra para el señorío Carangas una altura media de 3800 msnm, con unos 6542 m de altura para el nevado de Sajama, volcán extinto que constituye su punto más \nalto, y alturas que oscilan entre 600 y 2300 m para las regiones del valle, como Lluta en la costa del Pacífico y Tiquipaya en Cochabamba (Gisbert et al. 1996: 3). \nLos conjuntos de chullpas del presente estudio se concentran en medio del territorio de los Carangas, entre los sectores regidos por los nevados de Sajama y Tata Sabaya, este último al norte del salar de Coipasa, y se ubican en ambas márgenes del río Lauca (departamento de Oruro, a lo largo del piedemonte de la Cordillera Occidental) (Fig. 4). Este río nace en los lagos Chungará y Cotacotani en la provincia de Parinacota, en el actual territorio de Chile, luego ingresa en Bolivia cerca del hito fronterizo de Macaya a una altura de 3860 msnm. Esta región se caracteriza por tener un importante conjunto de chullpas, muchas de las cuales están decoradas con vistosas figuras geométricas. El Lauca penetra por una quebrada entre formaciones cordilleranas que sobrepasan los 4500 m de altura. A ambos lados del río están las lagunas Macaya y Sacabaya, al norte y al sur, respectivamente, dispuestas simétricamente respecto al Lauca "como dos grandes ojos", en un contexto antropomorfo de esa región (Gisbert et al. 1996: 22). \nEstas chullpas están construidas en su mayoría con adobes planos denominados tepes que, en estado fresco, van conformando un entramado en los muros a la manera de la urdimbre textil. Las chullpas decoradas muestran diseños configurados geométricamente y presentan composiciones a base de figuras como el cuadrado, que girados y combinados componen formas típicamente andinas. Las más vistosas muestran listones, ajedrezados y rombos segmentados en colores rojo, blanco, negro y verde que asemejan el arte textil andino (Montero Mariscal et al. 2009: 3). \nFIG. 4: Mapa de la región de Carangas con la localización de los sitios visitados en el valle del río Lauca, entre las lagunas de Macaya (al norte) y Sacabaya (al sur). Muchas de las torres se hallan en grupos densos por lo cual no es simple individualizar cada una de ellas. El mapa incluye la ubicación precisa de las 80 chullpas georreferenciadas. Imagen sobre un mapa cortesía de Google Earth. \n<!-- image -->', 'VI. MEDICIONES Y MÉTODOS DE ANÁLISIS': "Obtuvimos las medidas de orientación de los vanos de las torres con brújulas de alta precisión. Los valores de la declinación magnética para distintos sitios de la \nregión explorada oscilan entre 8°10' y 8°36' oeste (NOAA). La precisión de nuestras medidas de acimut magnético es de aproximadamente 0.5°, por lo que la diferencia en declinación magnética a lo largo del valle del Lauca entra adecuadamente dentro de nuestro error. Como una corroboración adicional, en muchos casos, especialmente con las torres más grandes y regulares, se verificaron las orientaciones medidas con imágenes fotosatelitales. \nEn la Fig. 5 mostramos el diagrama de orientación para las chullpas analizadas. Los valores de los acimuts consignados son los medidos para los vanos de las torres, e incluyen la corrección por declinación magnética en cada sitio particular (NOAA). Las líneas diagonales del gráfico señalan los acimuts correspondientes -en el cuadrante oriental- a los valores extremos para el Sol (acimuts de 65.3° y 115.1° -líneas continuas-, equivalente a los solsticios de invierno y verano australes, respectivamente) y para la Luna (acimuts: 59.1° y 120.8° -líneas rayadas-, equivalente a la posición de los lunasticios mayores). Recordemos brevemente que, en arqueoastronomía, con el término lunasticio mayor (norte o sur) nos referimos a las declinaciones extremas que alcanza la Luna en su movimiento a lo largo del horizonte cuando sale o se pone (ubicadas más de 5° más allá del rango de acimuts barrido por el Sol entre los solsticios). \nComo vemos, casi la totalidad de las torres (75 de un total de 80 medidas) tienen los vanos orientados hacia la salida del Sol en algún momento del año, con un patrón que muestra una cierta preferencia por épocas otoñales e invernales (con acimuts hacia el norte del este geográfico), y una notoria ausencia de orientaciones cercanas al orto solar durante fechas próximas al solsticio de verano austral. Esta es una prueba de la intencionalidad astronómica que se mantuvo a lo largo de varias generaciones de comunidades aymara de esta región del señorío preincaico de Carangas. \nFIG. 5: Diagrama de orientación para los vanos de las torres funerarias aymara de la región del río Lauca. Una clara tendencia a levante predomina en toda la región estudiada, con unas pocas chullpas alineadas levemente hacia el norte y fuera del rango solar. \n<!-- image -->", 'VII. LA ORIENTACIÓN DE LAS CHULLPAS Y EL PAISAJE': 'La región cubierta en nuestro trabajo de campo es extensa (unos 300 kilómetros cuadrados, con una extensión de aproximadamente 40 km de norte a sur a lo largo del piedemonte cordillerano) y, aparte de varios montes nevados -el Sajama en la lejanía hacia el norte, los volcanes Guallatiri y Arintica del lado chileno, el Puquintica en la frontera y otros menos prominentes y más cercanosy lagunas -por ejemplo, las ya mencionadas lagunas Macaya y Sacabaya-, no hay otros accidentes geográficos significativos que pudieran servir de referente topográfico para las construcciones analizadas. Sin embargo, como podemos ver en la Fig. 5, las ochenta chullpas medidas, sin excepción, están orientadas en un rango bastante estrecho de acimuts centrado en una dirección que apunta algo hacia el norte del punto cardinal este. \nEs difícil concebir un medio por el que se hubiese podido alcanzar tal uniformidad en las orientaciones de chullpas muy lejanas a no ser que pensemos en el cielo. Sin duda nos hallamos ante un caso ejemplar de orientación astronómica en esta región de altiplano central boliviano. \nComo ya señalamos, no quedan registros de los pueblos aymara que construyeron estas torres funerarias, para cuya sociedad era tan importante la memoria visual al carecer de escritura. Solo quedan las crónicas posteriores de la época colonial castellana sobre la posible orientación de los vanos. Sin embargo, el peso estadístico del patrón de orientaciones hallado parece ser suficiente para sugerir que las orientaciones se guiaban por el Sol naciente. \nEn tiempos del Intermedio Tardío (años 1200-1438), cuando dominaban los señoríos aymara previo al apogeo de los incas, en los diferentes ayllus de esta región del valle del Lauca, los pobladores capaces habrían estado ocupados en las labores de la tierra desde finales de la primavera y durante los meses de verano, en la producción y almacenamiento de los víveres, realizando principalmente tareas agrícolas que permitirían la futura subsistencia de la comunidad. Una vez que las cosechas hubiesen terminado, el grano ya almacenado y la tierra ya preparada para el nuevo año, la mano de obra de la comunidad habría estado disponible para embarcarse en proyectos de construcción de las necrópolis tan vistosas que, luego de más de 600 años, han llegado a nuestros días. Por ello entendemos que los vanos de las torres, si efectivamente se orientaban en la dirección de la salida del Sol en el día en que empezaba la construcción de cada monumento, deberían tener acimuts mayoritariamente equivalentes a los del orto solar en meses otoñales e invernales. Además, deberíamos hallar pocas chullpas con orientaciones cercanas al naciente del Sol en el solsticio de verano. Y esto es efectivamente lo que muestran nuestras mediciones. \nPor otra parte, nuestro diagrama de la Fig. 5 no presenta acumulación de orientaciones en la vecindad de los valores extremos para la Luna en los lunasticios \nmayores norte o sur. Esto parecería indicar que, si bien como señala Cieza de León en La crónica del Perú , los aymara "tienen en cuenta el tiempo, y conocieron algunos movimientos así del Sol como de la Luna" (Cieza, [1553] 1984: 444), no hay indicios, al menos en la región de Carangas que hemos explorado, de que el orto o el ocaso de la Luna fuesen relevantes a la hora de orientar sus torres. Por último, debemos considerar la posible relevancia de aspectos climáticos propios de la región en donde se ubican las chullpas. En nuestro trabajo de campo hemos podido verificar que la mayoría de las torres medidas presentaba un fuerte grado de erosión y caída de material principalmente en su cara trasera, aquella que en general apunta hacia la cordillera y que resulta constantemente azotada por los fuertes vientos fríos y húmedos provenientes de la montaña (Fig. 6). \nFIG. 6: Espalda fuertemente erosionada de una chullpa decorada con cuadrados rojizos en la cercanía de la laguna e isla de Sacabaya. Como sucede a menudo, el frente y el vano permanecen en buen estado, pero la bóveda y la parte trasera están completamente derrumbadas. Foto del autor. \n<!-- image --> \nSiendo esta una característica del clima propia de la región no sería extraño que los vanos de las chullpas fueran ex profeso ubicados a sotavento, preservando de esta manera los frentes de las torres donde en general se plasmaban los decorados más vistosos y representativos de las comunidades responsables de la construcción y manutención de los monumentos. Y no sería este el primer caso estudiado en donde el clima influye en la orientación de edificios de culto: en un contexto completamente diferente, el de las iglesias coloniales de una isla de Canarias, ya hemos comprobado que la orientación de las puertas de entrada es tal que evita enfrentarse a los vientos locales, en este caso, los alisios provenientes del norte (Gangui et al. 2016). Esta alternativa, sin embargo, requiere llevar a cabo un análisis más profundo de las condiciones climáticas en tiempos pasados, en el valle del Lauca y en diversos sitios del altiplano central con chullpares, y excede el alcance de nuestro trabajo.', 'VIII. PERSPECTIVAS FUTURAS': 'Los pueblos aymara seguían el curso de los astros y ciertos grupos de estrellas cumplían un papel fundamental en su calendario agrícola. El mes de junio, \npor ejemplo, estaba marcado por varias fiestas, especialmente la de la cosecha de la papa, y el orto helíaco de las Pléyades, vistoso cúmulo estelar ubicado en la constelación de Tauro, señalaba el momento de dicha festividad. Además, según el manuscrito de Huarochirí, las Pléyades (las Cabrillas ) tenían una función divinatoria y definían la suerte de la futura cosecha: "cuando las Cabrillas aparecen de gran tamaño, dicen: \'Este año vamos a tener maduración excelente de los frutos\', pero cuando se presentan muy pequeñitas, dicen: \'Vamos a sufrir\'" (Arguedas 1975, Cap. XXIX: 125; Molinié-Fioravanti 1985). En otras palabras, la tierra solo podía ser fértil si recibía la fuerza vital de las Cabrillas ( catachila en aymara) y se lograba establecer un lazo virtuoso entre el mundo de arriba y el plano terrestre (Bouysse-Cassagne 1987: 264). Con tal presencia del movimiento de los astros enraizada en su cultura práctica y simbólica, era natural preguntarse, como lo hemos hecho aquí, si los singulares mausoleos aymara que hemos estudiado no reflejarían también una herencia astronómica en algún aspecto de su construcción. \nEn nuestro trabajo hemos puesto a prueba la hipótesis de la orientación solar de los vanos de un grupo numeroso de torres mortuorias ubicadas en una zona acotada del valle del río Lauca en el altiplano central de Bolivia. Hemos podido comprobar que la totalidad de las chullpas medidas se orienta con acimuts que se acomodan en el cuadrante oriental y, entre estas, prácticamente la totalidad (75 de las 80 medidas, es decir el 94% del total) orientan sus vanos dentro del rango solar (Fig. 5). Este resultado es el esperado a partir de los escritos y relatos de los primeros cronistas europeos, pero se distancia de los resultados de Pärssinen y colaboradores en la provincia Pacajes, por ejemplo, donde los investigadores hallaron que muchas chullpas estaban orientadas directamente hacia cerros y volcanes prominentes (que representan sitios sagrados en la memoria colectiva), o de los resultados de Kesseli y Pärssinen (2005) en Qiwaya, a orillas del Lago Titicaca, donde hallaron 20 chullpas, todas con orientaciones muy diferentes respecto de las del Lauca, con vanos dirigidos hacia el oeste, hacia el sur y el sudeste (Fig. 7). \nFIG. 7: El paisaje que rodea a los chullpares incluye nevados prominentes -sitios de veneración ancestral- como el volcán Sajama de esta imagen. Sin embargo, en nuestras mediciones \n<!-- image --> \nno hay indicios de que los vanos -o incluso las espaldas- de las torres miren hacia estas montañas. Foto del autor. \nSin embargo, el patrón específico de las orientaciones que hemos hallado aquí, donde se evidencia una alta proporción de entradas que se dirigen hacia el norte del este, con muy pocas chullpas que miran hacia el sur del este, no es por el momento simple de explicar. Como fue discutido en secciones anteriores, este resultado podría estar relacionado con aspectos de estacionalidad: simplemente, las chullpas, orientadas hacia Sol naciente, habrían sido construidas durante la parte del año en la cual las comunidades no se hallaban abocadas a los trabajos de la tierra. \nPero nuestro resultado podría deberse también a otras cuestiones, que aún debemos explorar. Hemos visto que muchos chullpares se ubican a los pies de sierras de mediana altura. Esto hace que se vean rodeados por un horizonte irregular y a veces montañoso, sobre todo en la parte trasera de las torres (en la dirección contraria hacia donde se abre el vano de las chullpas), donde el horizonte que hemos registrado en varios sitios excede los 10° de altura. Esto en general no sucede con los horizontes ubicados frente a los vanos, los cuales, como hemos visto, tienden a orientarse hacia el levante y opuestos a la cordillera. Estos últimos horizontes son bajos, y en muchas ocasiones incluso toman valores negativos debido a que las chullpas se ubican sobre pendientes y colinas (Fig. 7). En cualquier caso, sabemos que un análisis completo de nuestros datos -los que hemos reportado aquí y otros que estamos actualmente analizando- requiere tomar en cuenta la altura angular de los horizontes que rodean a las torres, pues un perfil orográfico elevado, especialmente detrás de las chullpas, sin duda cambiaría, por ejemplo, la fecha en la que el Sol en el horizonte podría alinearse con su eje. \nAsí, el trabajo que aún debemos realizar tiene en cuenta no solo los acimuts de los vanos sino también las alturas de los horizontes circundantes adecuadamente corregidos por refracción atmosférica. Este análisis, actualmente en progreso, nos permitirá combinar medidas locales de acimut y altura angular para obtener la declinación astronómica, coordenada ecuatorial que tiene en cuenta cómo afectan tanto la topografía local como la ubicación geográfica a la visibilidad de los objetos celestes (ver, por ejemplo, el análisis desarrollado en Muratore et al. 2023). El valor de esta coordenada estimado para una dada torre, una vez comparado con la declinación del Sol, por ejemplo (que fija aproximadamente un par de días en el año, o solo uno en el caso de los solsticios), nos permitirá verificar, entre otras cosas, si esa torre está o no orientada en una dirección que coincide con el orto solar en meses invernales, y evaluar el peso estadístico de estos resultados. Este análisis estadístico nos proveerá la distribución del número aproximado de torres por cada valor de declinación astronómica posible. Con ese dato podremos entonces verificar si la acumulación de orientaciones en el diagrama de acimuts de la Fig. 5 deja su marca también en un gráfico de declinaciones, que es, en fin de cuentas, aquello que nos señala la \nposible influencia astronómica -ya sea debida al movimiento del Sol o, eventualmente, a una posible orientación estelar- en la orientación de las chullpas estudiadas. \nComo ya mencionamos, este análisis, al igual que una investigación más exhaustiva de los aspectos culturales relacionados con la construcción de chullpas durante la dominación del señorío Carangas, está actualmente en desarrollo y será reportado en otra ocasión.', 'AGRADECIMIENTOS': 'El autor agradece a sus colaboradores Juan Carlos Jemio y Fernando Arispe por iluminadoras discusiones en estos temas y por su apoyo y gentileza durante el trabajo de campo. Este trabajo ha sido financiado parcialmente por CONICET y la Universidad de Buenos Aires.', 'REFERENCIAS': '- J.M. Arguedas (Trad.). 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2024arXiv240904386H

The distribution of different types of atmospheres and surfaces on rocky planets is one of the major questions in exoplanet astronomy but there are currently no published unambiguous detections of atmospheres on any rocky exoplanets. The MIRI instrument on JWST can measure thermal emission from tidally locked rocky exoplanets orbiting small cool stars. This emission is a function of their surface and atmospheric properties potentially allowing the detection of atmospheres. One technique is to measure dayside emission to search for lower thermal emission than expected for a blackbody planet due to atmospheric absorption features. Another technique is to measure phase curves of thermal emission to search for nightside emission due to atmospheric heat redistribution. In this work we compare strategies for detecting atmospheres on rocky exoplanets using these techniques. We simulate secondary eclipse and phase curve observations in the MIRI F1500W and F1280W filters for a range of surfaces and atmospheres on thirty exoplanets selected for their F1500W signaltonoise ratio. Our results show that secondary eclipse observations are highly degenerate between surfaces and atmospheres given the wide range of potential surface albedos. We also show that thick atmospheres can support emission consistent with a blackbody planet in these filters. These two results make it difficult to unambiguously detect or rule out atmospheres using their photometric dayside emission except in a subset of CO2dominated atmospheres. We suggest that an F1500W phase curve could instead be observed for a similar sample of planets allowing the unambiguous detection of atmospheres by nightside emission.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.04386', 'arXiv:2409.04386', '2024arXiv240904386H']

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

Reliable Detections of Atmospheres on Rocky Exoplanets with Photometric JWST Phase Curves

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.04386.pdf

{'Reliable Detections of Atmospheres on Rocky Exoplanets with Photometric JWST Phase Curves': "Mark Hammond , 1 Claire Marie Guimond , 1 Tim Lichtenberg , 2 Harrison Nicholls , 1 Chloe Fisher , 3 Rafael Luque , 4, 5 Tobias G. Meier , 1 Jake Taylor , 3 Quentin Changeat , 2, 6 Lisa Dang , 7 Hamish C. F. C. Hay , 8 Oliver Herbort , 9 and Johanna Teske 10 \n1 \nAtmospheric, Oceanic, and Planetary Physics, Department of Physics, University of Oxford, Parks Rd, Oxford OX1 3PU, UK \n2 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands \n3 Astrophysics, Department of Physics, University of Oxford, Parks Rd, Oxford OX1 3PU, UK \n4 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA \n5 NHFP Sagan Fellow \n6 Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, UK \n7 Trottier Institute for Research on Exoplanets and D'epartement de Physique, Universit'e de Montr'eal, 1375 Ave Th'er'ese-Lavoie-Roux, Montr'eal, QC, H2V 0B3, Canada \n8 Department of Earth Sciences, University of Oxford, S Parks Rd, Oxford OX1 3AN, UK \n9 Department for Astrophysics, University of Vienna, Turkenschanzstrasse 17, A-1180 Vienna, Austria \n10 Earth and Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road, NW, Washington, DC 20015, USA", 'ABSTRACT': 'The prevalence of atmospheres on rocky planets is one of the major questions in exoplanet astronomy, but there are currently no published unambiguous detections of atmospheres on any rocky exoplanets. The MIRI instrument on JWST can measure thermal emission from tidally locked rocky exoplanets orbiting small, cool stars. This emission is a function of their surface and atmospheric properties, potentially allowing detections of atmospheres. One way to find atmospheres is to search for lower day-side emission than would be expected for a black body planet. Another technique is to measure phase curves of thermal emission to search for night-side emission due to atmospheric heat redistribution. Here, we compare strategies for detecting atmospheres on rocky exoplanets. We simulate secondary eclipse and phase curve observations in the MIRI F1500W and F1280W filters, for a range of surfaces (providing our open access albedo data) and atmospheres on thirty exoplanets selected for their F1500W signal-to-noise ratio. We show that secondary eclipse observations are more degenerate between surfaces and atmospheres than suggested in previous work, and that thick atmospheres can support emission consistent with a black body planet in these filters. These results make it difficult to unambiguously detect or rule out atmospheres using their photometric day-side emission alone. We suggest that an F1500W phase curve could instead be observed for a similar sample of planets. While phase curves are time-consuming and their instrumental systematics can be challenging, we suggest that they allow the only unambiguous detections of atmospheres by night-side thermal emission.', '1. INTRODUCTION': "The distribution of atmospheres on rocky planets is one of the major unanswered questions of exoplanet science, with wide-ranging implications for the formation pathway, climate variability, and ultimately surface habitability of potentially Earth-like worlds (Wordsworth & Kreidberg 2022; Lichtenberg & Miguel 2025). So far, there are no rocky exoplanets with unambiguously detected atmospheres. The James Webb Space Telescope (JWST) can measure thermal emission from rocky planets orbiting smaller, cooler stars, which may contain indications of the presence or absence of atmospheres. The signal-to-noise ratio for these observations is highest in \nthe mid-infrared, around 10 to 15 µ m, which overlaps with a CO 2 absorption feature. Observations with the F1500W and F1280W photometric filters of the MIRI instrument on JWST (Rieke et al. 2015) (and possibly the MIRI/LRS instrument for hotter planets) therefore have the ability to detect departures from the thermal emission due to a black body planet. These departures could be caused by changes to albedo, surface spectral features, atmospheric spectral features, or atmospheric heat redistribution. These features can have degenerate effects on thermal emission, especially in the bandpasses of individual photometric filters. \nUsing such observations, STScI is currently implementing a survey of rocky M-dwarf exoplanets with 500 \nhours of JWST Director's Discretionary Time (DDT), informed by the report from the Working Group on Strategic Exoplanet Initiatives with HST and JWST (Redfield et al. 2024). Redfield et al. (2024) proposed a programme to observe 15-20 rocky planets with 2 to 15 secondary eclipses each using the MIRI F1500W filter. Its aim is to detect emission lower than expected for a bare-rock planet without an atmosphere, which is proposed to be a signature of atmospheric heat redistribution (Koll et al. 2019; Koll 2022). Detecting atmospheres in this way is proposed as a test of the 'cosmic shoreline' hypothesis relating to the prevalence of atmospheres on rocky exoplanets (Zahnle & Catling 2017). The purpose of this paper is to simulate potential surfaces and atmospheres on a range of the most observable rocky exoplanets, to identify degeneracies and to identify unambiguous signatures of atmospheres. We aim to use these results to inform the planning and analysis of observations of rocky planets with JWST. \nThe use of MIRI photometric filters to detect signatures of atmospheres on rocky exoplanets was first demonstrated in detail by Deming et al. (2009), which identified the significantly lower emission at 15 µ m that could be produced by CO 2 absorption, as well as a correspondingly higher emission at 11 . 3 µ m. We follow in particular the work of Hu et al. (2012), which identified the variety of surface albedos and identifiable spectral features possible for a range of exoplanet surfaces. \nMansfield et al. (2019) simulated the thermal emission observed by the JWST MIRI/LRS instrument for the surface types considered in Hu et al. (2012), showing how the observed emission depends on the surface type. Mansfield et al. (2019) also showed how bare-rock planets can emit more strongly from their day-sides at longer wavelengths than would be expected from their equilibrium temperatures, due to the relatively higher emissivity of the modelled surfaces at longer wavelengths. They suggested that their results ruled out day-side bare-rock emission below a certain value, so could be used as a method of detecting atmospheres on rocky planets. Koll et al. (2019) compared strategies using MIRI/LRS eclipses and phase curves to detect signs of atmospheric heat redistribution on rocky planets, and concluded that eclipses were a more efficient way to do this, primarily using the results of Mansfield et al. (2019) to discard the possibility of false positives caused by high-albedo bare-rock planets. \nLustig-Yaeger et al. (2019) simulated similar emission spectra for the TRAPPIST-1 system using both barerock and atmospheric models, demonstrating the large potential effect of atmospheric CO 2 absorption on the observed emission around 15 µ m. Ih et al. (2023) anal- \nsed the observed emission from TRAPPIST-1 b in the MIRI F1500W filter using similar models, showing the possibility of deriving additional information by combining it with observations in the F1280W filter. \nThis previous work has been used to interpret observations of rocky exoplanets, such as the secondary eclipse depths of TRAPPIST-1 b and c with the F1500W filter presented in Greene et al. (2023) and Zieba et al. (2023). The observation of TRAPPIST-1 b in Greene et al. (2023) was consistent with the emission from a barerock planet with zero Bond albedo. The emission from TRAPPIST-1 c in Zieba et al. (2023) was lower than that expected for a black body, and could be consistent with a bare-rock surface with a non-zero Bond albedo, or an atmosphere with a weak CO 2 absorption feature. This analysis in Zieba et al. (2023) demonstrated the degeneracy inherent in interpreting observations with a single photometric point. Day-side emission in a single bandpass can be modified by the planetary Bond albedo or atmospheric heat redistribution, or by surface and atmospheric spectral features. Other eclipse observations of rocky planets with JWST have also shown day-side emission consistent with a black body (Mansfield et al. 2024; Xue et al. 2024). \nDucrot et al. (2023) used observations on TRAPPIST1 b in two photometric filters to break some degeneracies inherent in single-filter observations. They combined F1500W and F1280W observations to reveal weaker emission in the F1280W bandpass than the F1500W bandpass published previously in Greene et al. (2023). In the suite of models fitted, this combined dataset was consistent with a high-albedo surface or a pure-CO 2 atmosphere with an inversion. \nKreidberg et al. (2019) showed an alternative approach, measuring an entire phase curve of thermal emission in the 4.5 µ m bandpass of the Spitzer Space Telescope over the orbit of the rocky planet LHS 3844 b. This measurement found a day-side emission consistent with a Bond albedo less than 0.2, but also a nightside emission consistent with zero. This was evidence supporting the lack of a substantial atmosphere, which would have transported heat to the night-side and likely produced a non-zero emission there (although degeneracies with an atmospheric radiating level and planetary albedo still apply). This type of method was previously demonstrated by Seager & Deming (2009), which laid out the possibility of detecting an atmosphere by finding deviations from the thermal phase curve expected for a bare-rock planet. \nDemory et al. (2016) measured a similar thermal phase curve with the same instrument for the 'lava planet' 55 Cancri e, and detected non-zero night-side emission \nthat was suggested to be consistent with an atmosphere transporting heat to the night-side, although a conservative reanalysis suggested that zero night-side emission could not be ruled out (Mercier et al. 2022). Zhang et al. (2024) measured a phase curve of the rocky planet GJ 367b with the MIRI LRS instrument on JWST. Both the day-side and night-side emission from this planet were consistent with the emission expected from a barerock with zero albedo, with no heat redistributed to the night-side. These observations show the opportunity for phase curves to break the degeneracy inherent in the day-side emission. \nIn this study, we simulate secondary eclipse observations and phase curve observations for a suite of barerock and atmospheric models, to explore their degeneracies and to suggest optimal observing strategies for unambiguous detections of atmospheres. We describe our bare-rock model and atmospheric model in Section 2, and then present simulated observations from both in Section 3. We discuss how our results can inform future observing strategies and analysis in Sections 4 and 5, and explore the differences in our methodologies that lead to different conclusions from Mansfield et al. (2019) and Koll et al. (2019), as we favour phase curve observations over eclipse observations.", '2.1. Converting Reflectance Data to Planetary Albedo': "We use reflectance data from the RELAB database (Milliken et al. 2021), and follow the process described in Hu et al. (2012) and Hapke (2012) to derive the reflectances and albedos that we need for the surface model. For each surface listed in Table 1, we combine a bidirectional visible reflectance spectrum with a biconical near- and mid-infrared reflectance spectrum, scaling the latter to align with the former following the RELAB manual (Milliken et al. 2021) (except for the 'Frankenspectra' data which have already been combined). All the bidirectional reflectance data were measured with an incidence angle i = 30 · and an emission angle e = 0 · , giving a phase angle g = 30 · . RELAB provides the wavelength-dependent reflectance factor denoted REFF in Hapke (2012), which we convert to the bidirectional reflectance r dd ( λ, i, e, g ): \nr dd ( λ, i, e, g ) = µ 0 π REFF( λ, i, e, g ) (1) \nwhere µ 0 = cos( i ). We then convert r dd ( λ, i, e, g ) to the single scattering albedo w ( λ ) following Hapke (2012) 1 : \nr dd ( λ, i, e, g ) = w ( λ ) 4 π µ 0 µ 0 + µ H ( µ 0 ) H ( µ ) , (2) \nwhere µ = cos( e ). We use the two-stream approximation for the H functions as described in Hapke (2012): \nH ( x ) ≃ 1 + 2 x 1 + 2 γ ( λ ) x , (3) \nwhere γ ( λ ) = √ 1 -w ( λ ). We then convert the derived w ( λ ) to the directional-hemispheric reflectance r dh ( λ ): \nr dh ( λ ) = 1 -γ ( λ ) 1 + 2 γ ( λ ) µ 0 , (4) \nwhere µ 0 is the incidence angle for the 'direction' in the directional-hemispheric reflectance. In our barerock model this is the local solar zenith angle; r dh ( λ ) then tells us the fraction of incoming stellar radiation scattered in all directions by the surface. \nWe also derive the spherical reflectance r s ( λ ), again following Hapke (2012) and using the approximation: \nr s ≃ 1 -γ ( λ ) 1 + γ ( λ ) ( 1 -1 3 γ ( λ ) 1 + γ ( λ ) ) , (5) \nwhich then defines the emissivity of the surface as ϵ ( λ ) = 1 -r s ( λ ). Figure 1 shows the spherical reflectance r s ( λ ) for all our modelled surfaces. Several simplifying assumptions have been made in deriving the spherical and directional-hemispheric reflectances from the raw reflectance data from the RELAB database, so these quantities may be different for real planetary surfaces. \nThe spherical reflectance r s ( λ ) can be thought of as the wavelength-dependent equivalent of the Bond albedo. Table 1 lists the Bond albedo for each surface, which is the fraction of incident irradiance scattered in all directions. We derive this by integrating the incoming stellar spectrum on the planetary day-side, multiplied by the directional-hemispheric reflectance r dh ( λ ), over the day-side and over all wavelengths. We use TRAPPIST-1 to calculate the Bond albedo values in Table 1. The values will vary depending on the stellar spectrum, which is taken into account when simulating the other planets in our sample. The Bond albedos in Table 1 are generally higher than the fractional decrease in F1500W emission seen in Figure 2, because \nTable 1. The surface types used in the bare-rock model described in Section 2.2. The single-scattering albedo w for each surface, and a script to convert it to different albedos, is available at https://doi.org/10.5281/zenodo.13691959. The horizontal lines denote the conceptual groups of 'Extrusive Igneous Rocks', 'Other Terrestrial Igneous Rocks', and 'Extraterrestrial Rocks and Other Minerals' in which these surfaces are plotted in Figure 1. The 'RELAB Data' column lists the identifiers for the sample datasets in the RELAB database. The Bond albedo values A B are calculated for the stellar spectrum of TRAPPIST-1. † Lagain et al. (2022). \nthe surfaces generally have higher emissivity at longer wavelengths (Mansfield et al. 2019).", '2.2. Bare-Rock Planet Model': 'We use a numerical model of a bare-rock planet with a wavelength-dependent albedo to simulate the surface temperature and resulting thermal emission from our list of targets. We define a 10x10 grid of surface points on the day-side, and calculate their temperatures by balancing the total downward flux F d and the upward flux F u at each grid point on the surface: \nF d ( λ ) = ∫ [1 -r dh ( λ )] F S ( λ ) dλ, (6) \nF u ( λ ) = ∫ ϵ ( λ ) B ( λ, T surf ) dλ, (7) \nwhere F S ( λ ) is the incoming stellar spectrum, and B ( λ, T ) is the Planck function for temperature T . These are integrated over all modelled wavelengths and balanced, giving ∫ F d ( λ ) dλ = ∫ F u ( λ ) dλ at each grid point. We use the stellar spectrum of the member of the SPHINX model grid with the most similar parameters to the modelled star (Iyer et al. 2023), scaled to match the instellation of the effective temperature of the modelled star. We numerically balance the upward and downward fluxes to solve for the local surface temperature T surf at each day-side grid point for a particular surface type, planet, star, and solar zenith angle. Finally, we integrate the emitted flux seen by an observer from each point on the 10x10 planetary surface grid to derive the planetary spectrum.', '2.3. Planetary Atmosphere Model': 'We use AGNI 2 (Nicholls et al. 2024), a 1D radiativeconvective atmosphere model, to simulate the thermal emission from a range of atmospheres on our list of targets. The radiative transfer is implemented under the two-stream and correlated-k approximations through the widely used model SOCRATES (Edwards & Slingo 1996; Manners et al. 2017; Amundsen et al. 2017). We use gas opacity data from the DACE database 3 (Grimm et al. 2021) for the H 2 O and CO 2 atmospheres we simulate here (shown in Figure 1). Collision-induced absorption and Rayleigh scattering are included. As above, we use the SPHINX grid of stellar spectra for the incoming stellar flux (Iyer et al. 2023). AGNI couples the radiative transfer to a convective model using mixing length theory (Joyce & Tayar 2023; Lincowski et al. \n2018). The temperature structure of the atmosphere is found by solving for the state that conserves energy fluxes through each model level, subject to the boundary conditions. The effective temperature is set to zero, meaning that the planet is in radiative equilibrium with the star and is not undergoing secular cooling. A full description of the methods in AGNI can be found in Nicholls et al. (2024). \nTable 2 shows the suite of atmospheric models that we analyse in this paper. The heat redistribution factor scales the instellation applied to the model, and is a simple representation of heat redistribution from the day-side to the night-side by an atmosphere. This is a highly simplified representation of the real process of dynamical heat transport to the night-side, which would be better modelled by parameterised advection Lincowski et al. (2024). As the purpose of the redistribution in our model is just to produce a non-zero brightness temperature on the night-side, we assess that our simple method is sufficient for our purposes and leave more realistic modelling to future work.', '2.4. Simulating MIRI Emission Observations': "We simulate a collection of planets based on the preliminary 'Targets Under Consideration' (TUC) list 4 planned for the survey programme proposed by Redfield et al. (2024). We added TRAPPIST-1 b and TRAPPIST-1 c to this list for comparison with existing observations, ranked the list by MIRI F1500W SNR, and retained the top 30 targets by this metric. The omitted targets span a similar range of equilibrium temperatures as the top 30 targets, but have greater observational uncertainty. \nWe convert the planetary spectra of the bare-rock and atmosphere models to observations in the MIRI 1500W and 1280W filters, following the method in Luger et al. (2017). We use the estimated eclipse SNR from Luger et al. (2017) to estimate the uncertainty of eclipse measurements (so our plots show 1 σ error bars). We calculate later the uncertainty on each point in a simulated phase curve observed with either of these filters, following the method of Luger et al. (2017) 5 . We found that these simple estimates underestimated the uncertainty on real measurements (e.g., Greene et al. 2023; Zieba et al. 2023), so we conservatively scaled up our estimated errors on both the simulated eclipses and phase curves by 50% to match the real uncertainty. \nTable 2. The atmospheric properties used in the suite of AGNI atmospheric models described in Section 2.3. The bracketed pair of values defining the modelled heat redistribution determines the fraction of the incidence irradiation assigned to each of the two columns defining the day-side and night-side. Therefore, [1, 0] corresponds to no heat redistribution, [0.5, 0.5] to full redistribution, and [0.75, 0.25] to the partial redistribution used to calculate the phase curves in Section 3.4. \nWe simulate phase curves by modelling the day-side and night-side emission due to two separate 1D atmospheric models, with 25% of the day-side heating redistributed to the night-side. We define the heat redistribution in this way (rather than the f parameter defined in Hansen 2008), in order to refer directly to the heating applied to each of the 1D models. We set the resulting emission from the day-side and night-side columns as the maximum and minimum of a sinusoidal phase curve with its peak at zero phase (secondary eclipse). We do not add a phase offset to this simulated phase curve, despite the prevalence of such offsets for gaseous tidally locked exoplanets (Parmentier & Crossfield 2017). An offset due to atmospheric dynamics would provide very strong evidence for the presence of an atmosphere, but many 3D simulations of rocky exoplanet atmospheres of this type do not predict significant phase curve offsets (Kane et al. 2021; Hammond & Lewis 2021; Turbet et al. 2022). We therefore note that a phase curve offset is another possible source of evidence for an atmosphere, but we do not rely on it for our conclusions.", '2.5. Choice of surfaces': "We consider 'bare-rock' endmember materials which are conservatively plausible from a geological perspective, as well as those which, despite not likely dominating most planetary surfaces, result in high A B and would therefore be the most dangerous false positives for atmospheres. The surfaces of real planets, including relatively geologically-inactive ones, are macroscopic mixtures-even Mercury has volatile-rich deposits at its poles (Rodriguez et al. 2023, and references therein)nevertheless, mixing between endmembers would not affect the spread in bare-rock A B , being what we are in- \nterested in. With the exceptions of MgSO 4 (magnesium sulfate), FeS 2 (pyrite), and Fe 2 O 3 (hematite), we focus on natural rather than synthetic samples to capture mineralogical variability in the field (e.g., non-stoichiometric components). This list, in Table 1, is not exhaustive for two reasons: first, models cannot predict the detailed surface geology of an exoplanet from observable bulk properties, so our best efforts can only approximate a plausible spread; second, we found that certain surface compositions which may be thermodynamically and geologically plausible could not be included due to data availability constraints. \nThe data availability is primarily limited by (i) which samples are available on the RELAB database (Milliken et al. 2021), and (ii) the necessity for a sample to have both bidirectional reflectance spectra in the visible and near-infrared, as well as biconical reflectance spectra at longer wavelengths. Both datasets are needed to cover the range of stellar spectra and planetary thermal emission that we model. We accounted for grain size variations, as described below, but could not include the effects of surface temperature on observed reflectance spectra (which may be non-monotonic; Bott et al. 2023). \nPartial melting of a silicate mantle produces an igneous crust. Over most of modern Earth (its oceanic crust), this rock takes the particular form of mid-ocean ridge basalt. If many rocky exoplanets experience or have experienced mantle partial melting as a way to lose heat during mantle convection (e.g., Kite et al. 2009), then analogous rock types may be expected in high abundance on bare-rock planet surfaces. However, the bulk silicate compositions of exoplanets need not be the same as rocky planets in the solar system, which in itself would lead to a diversity in oceanic crust-analogues, \neven before divergent thermal histories of planets are invoked (Guimond et al. 2024). To capture this diversity, we include a wide variety of terrestrial igneous rock samples from alkaline and sub-alkaline magma series (the well-known progressions from mafic to felsic compositions upon melting), many being reported in Nair & Mathew (2017) and with detailed geologic settings given there; as well as ultramafic samples with available measurements (lherzolite, harzburgite). These samples thus qualitatively enact some of the expected variability in exoplanet mantle melt fractions, oxygen fugacities, and bulk refractory compositions. We do not exclude felsic volcanic rocks (e.g., rhyolite, phonolite) from this listalthough Earth's granitic continental crust formed in the presence of liquid water, anhydrous fractional melting can also produce felsic rocks and not necessarily in very low volumes (Shellnutt 2013). \nWe also consider materials that appear less abundantly across the planetary surfaces of the inner solar system, yet remain feasible for unknown exoplanets. In particular, the loss of an early atmosphereor prolonged but weak outgassing (see Foley 2024)may remain imprinted in surface mineralogy, via hightemperature gas-rock reactions that proceed geologically quickly and in the absence of water (e.g., Zolotov 2007; Cutler et al. 2020; Filiberto et al. 2020; Teffeteller et al. 2022). Sulfate minerals could be produced by reactions between Mg-silicates and SO 2 gas, demonstrated experimentally and proposed for Venus and other planets (e.g., Renggli et al. 2019; Berger et al. 2019; Rimmer et al. 2021; Byrne et al. 2024). A variety of chemical weathering paths produce hematite (e.g., Fegley et al. 1995). Hematite may also result from escape of an early envelope-oxygen left behind would oxidise FeO in the silicate, as estimated in Kite & Schaefer (2021). Conversely, enough H 2 in the early envelope (with respect to the availability of O) could instead imprint reducing conditions upon the surface, stabilising metals and sulfides (Kite & Schaefer 2021; Schlichting & Young 2021). Such metals and sulfides are also common in enstatite meteorites in our solar system. Planets with reducing surfaces may additionally form through incomplete coremantle segregation during magma ocean crystallisation (Lichtenberg 2021) or accretion of sulfide-rich building blocks in certain systems (Jorge et al. 2022). \nLastly, we include rock samples of extraterrestrial origin and of other genetic backgrounds; namely, lunar samples, martian meteorites, quickly-cooled basaltic glass, and samples representing ejecta. Again, these samples can represent only a slice of possibilities for real planet surfaces. Volcanic ash, for instance, can exhibit very high albedo (Jones et al. 2007), so its possible pres- \nence on a particularly volcanic planet presents a large false positive risk for an atmosphere (E. S. Kite, personal communication). We include one sample of basalt tuff (lithified volcanic ash), but finer-grained and differentlycomposed samples were unavailable. A famous example of a high albedo surface which we do include is lunar anorthosite rock, in this case formed by its staying afloat upon a crystallising magma ocean. Such primitive flotation crusts of lower-density, bright felsic rocks are only observed in the modern solar system on subplanet-size bodies (see Frossard et al. 2019), but we do not strictly rule out the possibility on TUC planets, which has not strictly been ruled out for primordial Earth (Harrison 2009). Not all flotation crusts are so reflective, however-graphite is another material of lowenough density to float on a magma ocean, and its presence in Mercury's crust at weight-percent levels may be part of the cause for low albedos there (Peplowski et al. 2016; Keppler & Golabek 2019). \nMicrometeorite impacts on silicate crust produce small amounts of iron metal, or, again, graphite, and in this way decrease the single scattering albedo of airless surfaces in the solar system and presumably outside of it (Cassidy & Hapke 1975; Lyu et al. 2024). However, space weathering, which groups this and multiple other processes, remains not well understood outside of the Moon and affects different materials differently (e.g., Gaffey 2010; Moroz et al. 2014; Domingue et al. 2014; Pieters & Noble 2016; Dukes et al. 2016; Kaluna et al. 2017; Yumoto et al. 2024); space weathering's effects on albedo should be investigated in future work systematically upon acquiring the necessary data. On real planets, weathering competes with resurfacing processes to determine the 'freshness' of the surface, which will also not be spatially homogeneous. The net effects of these processes might be ground-truthed via disk-integrated spectra of airless bodies in the solar system (Madden & Kaltenegger 2018), although we were unable to use such remote sensing observations for the data availability reasons mentioned above. \nAs surface albedos depend strongly on regolith grain size (e.g., Maturilli et al. 2014), we ensured that a variety of grain sizes are included in Table 1, from very fine dust ( < 1 µ m) to outcrop. In fact, Table 1 indicates that grain size has as big an effect as composition in controlling A B . For comparison, lunar regolith has a mean particle size of ≲ 10-100 µ m, whilst Mercury's is finer (Gundlach & Blum 2013; Domingue et al. 2016). Grain sizes of extrasolar bare-rocks are not known a priori . Nevertheless, dusty regoliths are plausible across the TUC insofar as the velocity of impactors increases with increasing surface gravity, implying greater comminution and smaller \nparticle sizes (Cintala 1992; Gundlach & Blum 2013)though such impacts could also affect albedo in the other direction if they generate more nanophase metals and glass. Hence we include feldspar mineral dust as the highest-albedo endmember material, representing a tiny grain size and being already a light-coloured mineral. Ultimately, whilst the processes that sculpt the surfaces of airless rocky exoplanets can be informed to some degree by analogy to solar system bodies, little is understood about how these and other processes might operate on planets with higher gravity, different mineralogy, and in different radiative and dynamical environments. \nWe used GGchem (Woitke et al. 2018) to test the thermodynamic stability of each surface material (as an equilibrium condensate) under surface pressures of a generously-low 10 nbar and temperatures corresponding to the range in TUC equilibrium temperatures, for various assumptions about bulk element abundances similar to Byrne et al. (2024). The only notable materials not stable under all temperatures were sulfates, which decompose above ∼ 650-750 K depending on bulk composition. No carbonates, ices, or phyllosilicates were found to be stable above ∼ 300 K at 10 nbar (i.e., most of the TUC), hence their exclusion from this particular study. Future work should test whether mineral stabilities are strongly affected by other factors in the space environment, such as proton bombardment from the stellar wind (e.g., McCord et al. 2001; Dukes et al. 2016). Nevertheless, whilst we chose a very low, Io-like atmospheric pressure to be conservative, the stability of phyllosilicates shifts to higher temperatures at slightly less-conservative atmospheric pressures (e.g., they appear at 400 K at 0.01 mbar). Previous theoretical work has shown that phyllosilicates can be important even at 900 K for planetary surfaces at higher pressures than investigated here (Herbort et al. 2020), and that OH absorbs onto silicate grains in protoplanetary disk conditions at 700 K (Thi et al. 2020). Phyllosilicate formation does not require water ice; OH could be supplied instead from organic matter, for example (Hirakawa et al. 2021). These points suggest that completely ruling out any material on unknown exoplanet surfaces will be difficult. Indeed, even the rocky-ness of the TUC is not definitely known, as more than half of the targets do not have both mass and radius measured, and eight planets out of those that do are too under-dense to be consistent with silicate-iron compositions (Unterborn et al. 2023).", '3.1. Examples of MIRI F1500W and F1280 Emission': 'Figure 2 shows the simulated eclipse depth and 1 σ uncertainty in the MIRI F1500W and F1280W filters for \nTRAPPIST-1 c ( T eq = (341 . 9 ± 6 . 6) K, T eff = (2559 ± 50) K, m Ks = (10 . 296 ± 0 . 023) (Gillon et al. 2017)) and GJ 486 b ( T eq = (706 ± 20) K, T eff = (3291 ± 75) K, m Ks = (6 . 362 ± 0 . 018) (Caballero et al. 2022)), for the suite of surface and atmosphere models in Tables 1 and 2 (except the models with redistributions of 25% and 75%). We assume that five eclipses are observed and averaged, with each eclipse observation having a baseline of four eclipse durations outside the eclipse itself (consistent with the photometric observations of TRAPPIST-1 b and c in Greene et al. (2023) and Zieba et al. (2023)). We highlight these two planets because JWST observations of their thermal emission have been published (Zieba et al. 2023; Mansfield et al. 2024), and they span the range of equilibrium temperatures and signalto-noise ratios of our suite of modelled planets. The differences in the fractional emission ranges between our modelled planets are mostly due to their different signalto-noise ratios, and their different equilibrium temperatures (which scales the relative size of their emission in the F1500W and F1280W filters, for the reasons described in Mansfield et al. 2019). \nWe simulate five eclipses because this is consistent with the five and four eclipses observed in Greene et al. (2023) and Zieba et al. (2023). A uniform precision could be achieved for all the modelled planets with fewer eclipses for the targets with higher SNR, or more eclipses for the targets with lower SNR (Redfield et al. (2024) suggests a range of 2 to 15 eclipses over the proposed sample). However, this is irrelevant to our main point as we suggest that the day-side emission is highly degenerate between atmospheres and surfaces regardless of observational precision. \nThe emission values are normalised by the value of the emission from a black-body planet with no heat redistribution, as calculated by our bare-rock model, to highlight deviations from this value due to heat redistribution, atmospheric absorption, and surface albedo and emissivity. Figure 2 shows several important properties of the emission in the MIRI F1500W and F1280W bandpasses from these models. Firstly, the bare-rock emission in both cases spans a wide range of values due to the wide range of Bond albedo values shown in Figure 1, overlapping with most of the atmospheric models. The emission from the atmospheric models also spans a wide range because of the variety of atmospheric structures, opacities in these bandpasses, and redistribution factors. Some of the atmospheres with H 2 O have emission features in these filters, showing how a thick atmosphere can still produce emission consistent with that expected from a black body, instead of the lower emission expected due to atmospheric absorption or redistribution. \n) \n1 \ng \n2 \nm \nc \n( \ny \nt \ni \nc \na \np \nO \nFigure 1. A summary of the input data for the bare-rock and atmospheric models, alongside the thermal emission for the typical temperatures of star and planets in this paper. The top three panels show the spherical reflectances r S ( λ ) for each of the planetary surfaces in Table 1. The single-scattering albedo w for each surface, and a script to convert it to different albedos, is available at https://doi.org/10.5281/zenodo.13691959. The fourth panel shows the opacity of H 2 O and CO 2 at 500K and 1 bar from DACE (Grimm et al. 2021). The fifth panel shows the throughputs of the F1280W and F1500W filters, as well as the black-body emission from a star at 2500 K, a planet at 1000 K, and a planet at 500K, each normalised to their maximum values \n<!-- image --> \n.', 'TRAPPIST-1 c': 'Figure 2. Simulated day-side emission - the average eclipse depth of five eclipses - in the F1500W and F1280W filters for TRAPPIST-1 c and GJ 486 b, for all the surfaces in Table 1, and all the atmospheres in Table 2. The error bars show the 1 σ uncertainty on the eclipse depth, as is typically plotted for this type of measurement (Greene et al. 2023; Zieba et al. 2023). They are shown without heat redistribution here, which we explore in all subsequent figures. Each individual eclipse is assumed to have a baseline of four eclipse durations outside the eclipse. TRAPPIST-1 c represents the cooler planets in our sample, and GJ 486 represents the hotter planets in our sample. The average relative emission in these bandpasses is higher for GJ 486 b than for TRAPPIST-1 c, for the reasons discussed in Section 3.2. \n<!-- image --> \nGJ 486 b \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \nAlbite (dust) \nL \nMagnesium sulfate \nelative to Black Body \n-side Emission R \nDay \nF1500W Black Body \nF1280W Black Body \neccia \nRhyolite \nTholeiitic basalt \nHarzbur \ngite \nDiorite \nLherzolite \nAlk \naline basalt (small) \no \nMars basaltic sher \nBasalt glass \nF1500W \nF1280W \nNorite \nrachyte \nT \nrachybasalt \nAlk \nO \nm \n2 \nC \np \n1 \np \n, \nr \nb \na \n1 \n2 \nC \nO \nm \np \np \n3 \n1 \n0 \n, \nr \nb \na \n1 \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \n2 \nC \nO \nm \np \np \n6 \n1 \n0 \n, \nr \nb \n2 \nC \na \n1 \nO \nm \np \n, \np \n1 \nr \nb \na \n1 \nH \np \n2 \nO \nm \np \n, \n1 \nr \nb \na \n1 \nH \np \n3 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \na \n1 \n2 \nC \nO \nm \np \np \n3 \n1 \n0 \n, \nr \nb \na \n1 \nH \np \n6 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \na \n1 \n2 \nC \nO \nm \np \np \n6 \n1 \n0 \n, \nr \nb \na \n1 \nH \np \n2 \nO \nm \np \n, \n1 \nr \nb \na \n1 \nH \np \n3 \n2 \nC \nO \nm \np \n, \np \n1 \nr \nb \na \n0 \n2 \nC \nO \nm \np \np \n3 \n1 \n0 \n, \nr \nb \n1 \na \n0 \n1 \nO \n2 \nm \np \n0 \n, \n1 \nr \nb \na \n1 \n2 \nC \nO \nm \np \np \n6 \n1 \n0 \n, \nr \nb \na \n0 \n1 \nO \n2 \nm \nH \np \n6 \np \n0 \n, \n1 \nr \nb \na \n1 \n2 \nC \nO \nm \np \n, \np \n1 \nr \nb \na \n0 \nH \np \nF1500W \nF1280W \nO \n2 \nm \np \n, \n1 \nr \nb \na \n0 \nH \np \n3 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \n1 \na \n0 \n1 \n2 \nC \nO \nm \np \np \n3 \n1 \n0 \n, \nr \nb \n1 \na \n0 \n1 \nH \np \n6 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \na \n0 \n1 \n2 \nC \nO \nm \np \np \n6 \n1 \n0 \n, \nr \nb \na \n0 \n1 \nH \np \nF1500W \nF1280W \nO \n2 \nm \np \n, \n1 \nr \nb \na \n0 \nH \np \n3 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \n1 \na \n0 \n1 \nH \np \n6 \n2 \nO \nm \np \n0 \n, \n1 \nr \nb \na \n0 \n1 \nge) \naline basalt (lar \nunar anorthosite \nMars br \nGranite \nGabbr \nAndesite \nBasalt tuff \nPyrite \nT \ngottite \nunar mar \nPhonolite \nephrite \nHematite \nL \ne basalt \nT \nelative to Black Body \n-side Emission R \nDay \nelative to Black Body \n-side Emission R \nDay \n1.2 \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 \nThis can in general be caused by greenhouse warming of the surface and subsequent observation of the surface (or near-surface) emission through a spectral window, or by the formation of a thermal inversion by strong shortwave opacity in the atmosphere. If either of these effects are sufficiently strong, the emission at a particular wavelength could be greater than the expected black-body value. \nSecondly, Figure 2 shows how the difference in emission between the F1500W and F1280W filters is essentially the same for all the bare-rock surfaces due to their similar emissivities over this range (Ih et al. 2023). However, the relative emission between these filters varies for the atmospheric models, due to differences in atmospheric opacity. The largest differences are due to the significant difference in CO 2 opacity between the F1500W and F1280W bandpasses. Some (but not all) of the atmospheric models can be distinguished from bare-rock surfaces by this method (Ih et al. 2023). \nOur models are a limited representation of all the possible surfaces and atmospheres on such a planet. Different surfaces or atmospheres could have higher albedos or emission features, or stronger differences in their F1500W and F1280W emission. Despite our wider selection of surface types than previous studies, the Bond albedos of most types of igneous extrusive rock are relatively low, resulting in relatively high day-side emission, consistent with previous work (Hu et al. 2012; Mansfield et al. 2019; Lyu et al. 2024). The surfaces with small grain sizes, or the more unusual surfaces like magnesium sulfate, provide the highest Bond albedo values. With that said, regolith grain sizes of a few tens of microns are not unusual in the solar system and can result in surprisingly high Bond albedos even for common basalt. We do not attempt to provide a prior likelihood on the prevalence of different surface types; understanding the likelihood of high-albedo surfaces will therefore be crucial to make progress in this area.', '3.2. Detecting Atmospheres with Day-side F1500W Emission': "Figure 3 shows the range of emission values from each planet in our suite, for the bare-rock model, the atmospheric model without heat redistribution, and the atmospheric model with full global heat redistribution. For example, the left-most points show the highest and lowest values for the emission for the bare-rock model of TRAPPIST-1 c. They are joined with a shaded line to show that the models of emission span the region between these two extremes; most of the ranges are covered continuously by the variety of models in each category. \nThis plot is intended to show how an observation of the F1500W emission can be consistent with several potential surfaces or atmospheres. The possibility of bare-rock surfaces with high albedos (see also the 'feldspathic' and 'granitoid' surfaces modelled in Zieba et al. 2023) means that it is very difficult to conclude that a low day-side emission in the F1500W is unambiguously due to an atmosphere. Figure 3 shows that only two of the atmospheres we model with zero heat redistribution can be distinguished from the range of potential surfaces (comparing the lower end of the red and blue ranges for each planet). These are the atmospheres with 1 ppm CO 2 (which provides a strong absorption feature in the F1500W bandpass, without significantly heating the atmosphere) for GJ 806 b and LTT 1445 A b (which both have high signal-to-noise ratios). \nHeat redistribution to the night-side lowers the emission from the atmospheric models (as would an increased atmospheric albedo due to clouds). Many of the CO 2 -dominated atmospheres can be distinguished from barerock surfaces for the planets with the higher signal-tonoise ratios. However, many of the modelled planets are approaching the limit of a detectable eclipse at the low values of fractional emission around 0.25 and below in Figure 3. \nThere are plausible reasons to expect lower albedo values for bare-rock surfaces in general. The lower end of the range of possible surface emission values is driven by high-albedo surfaces types in the 'Extraterrestrial Rocks and Other Minerals' category in Table 1. For TRAPPIST-1 c, Figure 2 shows that the range of F1500W emission values due to the 'Extrusive Igneous Rocks' and 'Other Terrestrial Igneous Rocks' extends down to ∼ 0.45 rather than ∼ 0.20 for the whole sample of surfaces. \nSimilarly, we do not model the effect of space weathering on the albedo and emissivity of the bare-rock surfaces. This could plausibly decrease the albedo of these surfaces, making them more readily distinguished from CO 2 -dominated atmospheres (Hu et al. 2012). However, we suggest this does not allow unambiguous detections of atmospheres. As mentioned above, the extent of darkening due to nanophase metals at a given time reflects a competition between space weathering and (poorlyconstrained) resurfacing processes acting against weathering. Whilst bare-rocks in the solar system frequently appear weathered and darker, these bodies are all small and relatively geologically inactive. The geophysical heat engine would persist for longer on more massive bodies (Stevenson 2003; Kite et al. 2009), promoting frequent reworking of the crust, irrespective of atmosphere retention. Thus, even if we might expect (for ex- \nFigure 3. Comparing the ranges of possible F1500W day-side emission (for five observed eclipses) from the bare-rock model, the atmospheric model with zero heat redistribution, and the atmospheric model with full heat redistribution. Each observation is modelled as the average of five secondary eclipses, with a baseline of four eclipse durations outside the eclipse. Each plotted range compiles the relevant models from the suites like those in Figure 2; for example, the left-most range shows the range of possible F1500W emission values for TRAPPIST-1 c due to all the different surface types in Table 1. The planets are ordered by equilibrium temperature. TRAPPIST-1 c and GJ 486 b are highlighted in bold, as they are plotted in more detail in Figure 2. The bare-rock surfaces can only be distinguished from the atmospheres with zero heat redistribution on GJ 806 b and LTT 1445 A b (that is, the uncertainties on their lower limits do not overlap). The bare-rock surfaces can be distinguished from some of the atmospheres with full heat redistribution on more planets, for a subset of atmospheres with both full heat redistribution and a low CO 2 abundance. \n<!-- image --> \nample) high-albedo lunar anorthosite to be weathered and become darker (Yamamoto et al. 2018), we cannot guarantee that this is the case for an unknown planet. For the purposes of understanding the plausible range of emission, we do not model space weathering because it would simply move the F1500W emission of each surface upwards toward the black body value. As our range of surfaces already spans this range of emission, modelling space weathering would not add to the possible range of F1500W emission from our bare-rock models. \nThe ordering of the planets by equilibrium temperature in Figure 3 shows how the emission from barerock surfaces in the F1500W bandpass (as a fraction of the black body value) rises with planetary equilibrium temperature, most strongly at higher equilibrium temperatures. Mansfield et al. (2019) showed how this is due to the peak of the Planck function moving to shorter wavelengths for hotter planets, where the emissivity of the bare-rock surfaces is relatively lower (see Figure 1), which makes the emission at 15 µ m relatively higher. This is a fractional effect on the change in emission from a black-body, so has a larger absolute effect on the lower end of the emission range than the higher end. \nHowever, the minimum of the emission from the atmospheric models also rises with equilibrium temperature. This is because exactly the same process occurs for the emissivity of a CO 2 -dominated atmosphere (the atmospheres with the lowest F1500W emission are all CO 2 -dominated). Figure 1 shows how CO 2 has low opacity from 5 to 10 µ m; cool planets emit freely in this range and so their emission at 15 µ m is relatively low. However, the peak of the Planck function for hotter planets is lower, from 2 to 5 µ m, where the average CO 2 opacity is higher. This decreases the relative emissivity in this region, which increases the relative emission at 15 µ m. Therefore, the increase in F1500W emission from bare-rock surfaces at higher equilibrium temperature that was identified in Mansfield et al. (2019) does not aid so much in distinguishing them from CO 2 -dominated atmospheres, because exactly the same increase in F1500W emission at higher equilibrium temperatures occurs for CO 2 -dominated atmospheres. \nSo far, we have focused on the possibility of false positive detections of atmospheres, when high-albedo surfaces are observed. It would also be possible to derive a false negative detection of an atmosphere, because the upper range of the emission from the modelled atmo- \nspheres extends up to the expected value for a black body bare-rock planet, and even above this value in many cases. Therefore, a measurement of emission that is consistent with the black body value does not necessarily demonstrate the absence of an atmosphere.", '3.3. Detecting Atmospheres with Day-side F1500W and F1280W Emission': 'Figure 4 shows the difference in emission between the F1500W and F1280W filters for each of our modelled planets, normalised by the F1500W emission from a black body. This difference is almost constant for the bare-rock surfaces, but can vary for the modelled atmospheres, generally due to the CO 2 absorption feature in the F1500W bandpass. \nOur modelling in Figure 4 reaches a similar conclusion for TRAPPIST-1b - only a subset of atmospheres have a detectable difference in the relative emission in these two filters. Some other planets with a higher signal-to-noise ratio have more readily detectable differences in emission. Figure 2 shows that these differences are strongest for a subset of the CO 2 atmospheres (1 bar with 10 3 ppm CO 2 , and 1 bar with 1 ppm CO 2 ). For the highest concentrations of CO 2 , the atmospheric opacity increases in the F1280W bandpass as well as the F1500W bandpass, so the difference in emission between them decreases. This metric does not generally detect H 2 O-dominated atmospheres, as its opacity is similar in both filters (Figure 1). In all cases, the results are similar whether the atmospheres have heat redistribution or not, because the emission in the bandpasses of both filters changes by a similar fraction when heat is redistributed. \nTherefore, while there are some atmospheric compositions that would be detectable from the difference between these two filters, these compositions are a limited set of the total possibilities. Moreover, observing the emission in two filters requires observing twice as many eclipses as for one filter. For example, measuring five eclipses of TRAPPIST-1 b, each with a duration of the eclipse itself plus an additional out-of-eclipse baseline of four eclipse durations, takes 15.1 hours. Each separate observation also requires approximately one additional hour for telescope slewing, target acquisition, and detector stabilisation 6 . Measuring five eclipses for two filters then takes 40.2 hours in total, which is similar to the time of 37.3 hours for a full orbital phase curve observation.', '3.4. Detecting Atmospheres with F1500W Phase Curves': "This section shows simulations of the day-side and night-side emission retrieved from phase curves observed in the F1500W filter. The maximum of the phase curve is the same physical quantity as the day-side emission measured from the simulated secondary eclipses above, so is also affected by the degeneracy between surface Bond albedo, and atmospheric properties. However, the minimum of the phase curve is generally immune to these degeneracies, if it is consistent with non-zero night-side emission, as the emission from the night-side of a bare-rock planet will be negligible. A measurement consistent with zero night-side emission would be degenerate between a bare-rock surface and an atmosphere with weak heat redistribution or weak emission in the observed bandpass. Planets hotter than the sample we consider may have magma oceans whose currents could transport some heat in the absence of atmospheres, but this transport is expected to be inefficient and so would not significantly affect the surface temperature and resulting nightside thermal emission (Kite et al. 2016; Meier et al. 2023; Lai et al. 2024). \nWe simulate phase curves containing two eclipses, with a duration 20% longer than a single orbital period. Their maximum and minimum are calculated by a 1D model for each side of the planet, with stellar heating of 75% and 25% of the total. We then add observational noise and fit a model of a sinusoidal phase curve with a free amplitude, mean, phase offset, and stellar baseline using Salvatier et al. (2016). These atmospheres all assume that 25% of the stellar heating on the day-side is redistributed to the night-side; atmospheres with less redistribution would be less detectable by their nightside emission. This fraction is unknown for exoplanets, and depends on a range of atmospheric properties like surface pressure and composition; our choice of 25% is to demonstrate that an atmosphere with enough heat redistribution can be unambiguously detected. The value of 25% is consistent with the order of magnitude of the heat redistribution in the 3D atmospheric simulations and scaling theory in Koll (2022). It is also consistent with the outgoing longwave radiation on the night-side plotted for a variety of simulations of TRAPPIST-1 e in Turbet et al. (2022). If we modelled planets with full heat redistribution (as we do for the eclipse-only observations above), their night-side emission would be even more detectable. \nFigure 5 shows the range of phase curve maxima (essentially the same as the day-side emission shown in Figure 3), and the range of phase curve minima, for atmospheres 2, 5, 8, and 11 with 25% heat redistribution \nFigure 4. Comparing the ranges of the differences in day-side emission between the F1500W and F1280W filters from the bare-rock model, the atmospheric model with zero heat redistribution, and the atmospheric model with full heat redistribution. Each observation is modelled as the average of five secondary eclipses in each of the two filters, with a baseline of four eclipse durations outside the eclipse. More of the atmospheres can be detected than in Figure 3 by this metric, which makes use of the CO 2 spectral feature in the F1500W bandpass, which is weaker in the F1280W bandpass. Only the atmospheres with 1 ppm CO 2 can be reliably detected by this metric, because for higher abundances of CO 2 its opacity saturates the F1280W bandpass as well as the F1500W bandpass (see Figure 1). The atmospheres with H 2 O cannot be detected by this metric, because the opacity of H 2 O is similar in both bandpasses (Figure 1), resulting in similar emission in both filters (Figure 2). The atmospheres with full heat redistribution are not much more detectable by this metric than the atmospheres with zero redistribution. This is because any redistribution reduces the emission in both the F1500W and F1280W filters by a roughly equal fraction. \n<!-- image --> \nin Table 2. The emission from the 'null hypothesis' of a bare-rock night-side is also shown for each planet, to determine which night-sides could be observationally distinguished from a bare-rock planet. Figure 5 shows that almost all of these simulated atmospheres produce enough night-side emission to be distinguished from a bare-rock planet. \nA phase curve observation also provides a measurement of day-side emission as well as night-side emission, with comparable precision to that shown in Figure 3. This information can be complementary to night-side information; for example, many of the simulated observations in Figure 5 with detectable night-side emission have detectably low day-side emission in Figure 3, providing corroborating evidence for an atmosphere. \nSome atmospheres in Figure 5 produce too little nightside emission to be distinguished from the bare-rock null hypothesis. These are the atmospheres with low amounts of CO 2 , which emit from low pressures but have little greenhouse warming. While this may produce too little emission to be detected on the night-side, the day-side emission will be correspondingly low (see the lowest day-side emission value for each planet in Figure 5). This provides the same evidence for atmospheric \nabsorption on the day-side as would be provided by observing secondary eclipses only. Observing a phase curve with a minimum flux consistent with zero should not be taken as proof of the lack of an atmosphere, as thin atmospheres may redistribute too little heat. Powell et al. (2024) shows that optically thick night-side clouds may produce the same effect, suppressing detectable nightside emission from a thick atmosphere. \nMost of the atmospheres we simulate produce nightside emission that can be distinguished from a bare-rock planet, using an observation of a single phase curve. Any atmosphere that is thick enough to redistribute a nonnegligible fraction of the day-side heating to the nightside will be detectable in this way. An atmosphere detected by a phase curve will also be better characterised than one detected by secondary eclipses only, with a constraint on day-night heat transport and a potential phase curve offset.", '4.1. Comparison to previous work': "We reach some different conclusions to previous studies on this topic. In this section, we compare our methodology and results to Mansfield et al. (2019), \nFigure 5. The ranges of modelled (and then fitted) F1500W phase curve maxima and minima, for the range of atmospheres in Table 1, compared to the zero flux minimum expected on the night-side of a bare-rock planet. Each observation is modelled as a phase curve over one orbit. The plotted points are the true value of the maxima and minima of the phase curves, and the error bars are the 1 σ uncertainty of the values fitted with Salvatier et al. (2016). As we did for the eclipse-only observations, the error bars are conservatively scaled up by 50% to ensure they are at least as large as they would be for real observations. With a heat redistribution of 25% of the day-side energy to the night-side, almost every planet has detectable night-side emission. Those that do not are atmospheres with low CO 2 abundances on planets with low signal-to-noise ratios; these also have detectably weak emission from their day-side as a result. We do not include the day-side emission for bare-rock surfaces as these are shown in Figure 3 with comparable precision to that achieved by a phase curve. Moreover, our aim in the current figure is to identify unambiguous information by distinguishing non-zero night-side emission due to atmospheric heat redistribution. \n<!-- image --> \nWhittaker et al. (2022), Koll et al. (2019), and LustigYaeger et al. (2019) to determine the causes of these differences. \nMansfield et al. (2019) presents 'a new method to detect an atmosphere on a synchronously rotating rocky exoplanet around a K/M dwarf, by using thermal emission during secondary eclipse to infer a high day-side albedo that could only be explained by bright clouds'. They used a similar methodology to Hu et al. (2012) to simulate the broadband MIRI/LRS emission from eight surface types, to conclude that 'a high albedo could be unambiguously interpreted as a signal of an atmosphere for planets with substellar temperatures of T sub =410-1250 K'. This significantly differs from our conclusion in Section 3.2, where we suggest that highalbedo bare-rock planets could produce day-side thermal emission that is much lower than a black body planet, and therefore highly degenerate with an atmosphere. \nWe have determined that the source of this discrepancy is that Mansfield et al. (2019) calculated the surface temperatures of each bare-rock surface using the geometric albedo values derived in Hu et al. (2012) (M. Mansfield, private communication). The geometric albedo is the ratio of planetary flux at zero phase angle \n(from the day-side, for a tidally locked planet) to the flux from a Lambert disk (Seager 2010); it is an observational quantity and does not control planetary energy balance or equilibrium temperature. The Bond albedo determines the fraction of incoming total stellar energy scattered into space in all directions, so is the quantity that controls planetary energy balance and equilibrium temperature as described in Section 2.2 (note that we use its wavelength-dependent equivalent r dh ( λ ) there). \nThe geometric albedo is 50% lower than the Bond albedo for the surfaces modelled here and in Mansfield et al. (2019), because they are assumed Lambertian (Seager 2010). Using the geometric albedo therefore results in significantly higher bare-rock surface temperatures in Mansfield et al. (2019) than for the equivalent surfaces in our Section 3.2 with temperatures calculated using the Bond albedo. The difference in our methods can be seen by comparing the spherical reflectance (which gives the Bond albedo when integrated) for 'Lunar Anorthosite' in our Figure 1, to the 'Feldspathic' albedo value in Figure 3 of Mansfield et al. (2019). Both use the same original RELAB data (LR-CMP-224), but our albedo values are approximately 50% higher. This \ncan be inspected in more detail using the datasets at https://doi.org/10.5281/zenodo.13691959. \nThis is why we are much more pessimistic about the prospects of detecting atmospheres with eclipses only, as our modelled bare-rock surfaces produce approximately 50% weaker emission than the equivalents in Mansfield et al. (2019). Another minor difference is that we consider a wider range of surface types than Mansfield et al. (2019), which broadens the range of possible surface emission values, increasing the degeneracy. We note that Whittaker et al. (2022) applies the same methodology as Mansfield et al. (2019), using the geometric albedo to calculate temperatures of bare-rock surfaces. \nKoll et al. (2019) also tackled this issue, comparing the relative merits of phase curves, spectroscopy, and eclipse photometry, for detecting atmospheres on rocky planets. Koll et al. (2019) concluded that 'infrared photometry of secondary eclipses could quickly identify 'candidate' atmospheres, by searching for rocky planets with atmospheres thick enough that atmospheric heat transport noticeably reduces their day-side thermal emission compared to that of a bare-rock [ . . . ] Candidate atmospheres can be further validated via follow-up spectroscopy or phase curves'. They favour secondary eclipses over phase curves for detecting candidate atmospheres due to the shorter observational time required. They suggest that the false positive bare-rock scenario is unlikely primarily by reference to Mansfield et al. (2019), supported by discussion of low-albedo bare-rocks in the Solar System and on Kreidberg et al. (2019). \nOur conclusions are similar in general to those of Koll et al. (2019): we also find that eclipse photometry could identify signatures of candidate atmospheres, and that these must then be followed up with more detailed measurements. Our point of difference is that we do not discard the bare-rock false positive case following Mansfield et al. (2019), because our use of the Bond albedo rather than the geometric albedo results in much lower thermal emission from bare-rock planets. We therefore suggest that one may as well proceed to the detailed phase-curve 'follow-up' as the secondary eclipse measurements are too degenerate to contain useful information. This is ultimately a qualitative point and that the conclusion of Koll et al. (2019) may still apply if a strategy of initially identifying candidates for later follow-up is preferable. \nOur final comparison to previous work is with LustigYaeger et al. (2019), which focuses mainly on observational signatures of different atmospheres, but also simulated the emission signatures of bare-rock planets. Their Figure 1 shows the secondary eclipse spectrum of TRAPPIST-1 b for a variety of bare-rock surface types. As Lustig-Yaeger et al. (2019) makes clear, these spectra \nare calculated assuming zero Bond albedo so all have the same planetary equilibrium temperature. This results in them emitting similarly to a black body at 15 µ m, unlike our simulations where many surfaces have much lower emission due to their high Bond albedo and lower temperatures. The main conclusion of Lustig-Yaeger et al. (2019) from these results is that 'an airless rock would likely have significantly lower spectral variation than atmospheric features'. This is consistent with our results, although we do not focus on the specifics of the spectral variation of the emission beyond the differences in the F1500W and F1280W filters in Section 3.3.", '4.2. False Positives and False Negatives': 'Detecting atmospheres by eclipse photometry alone depends on our understanding of the link between atmospheric thickness, global heat redistribution, and atmospheric emission in the observed filter. This link is generally suggested to be that thicker atmospheres on tidally locked planets redistribute more heat (Koll 2022), which then emit more weakly from their day-sides. This is suggested to contrast with the higher emission from bare-rock surfaces, as igneous rocks generally have lower Bond albedos (see Table 1) and relatively higher emissivity in the F1280W and F1500W MIRI filters (Mansfield et al. 2019). \nWhile all these statements are physically reasonable, we suggest our results demonstrate plausible false positive and false negative detections of atmospheres by eclipse observations. Firstly, the wide range of surface types we consider have a range of albedos, with the most reflective having Bond albedos over 0.5. While lowalbedo igneous rocks are probably more likely surfaces (especially if they are darkened by space weathering; Hu et al. 2012), it is not possible to entirely rule out known or unknown high-albedo surfaces. Measuring low emission in the F1500W filter from a high-albedo bare-rock planet could therefore provide a false positive detection of an atmosphere. \nSecondly, while thick atmospheres are generally likely to redistribute heat and emit more weakly in the F1500W bandpass, we suggest there are plausible effects that could counteract this. Koll (2022) demonstrates a very compelling scaling relation between atmospheric thickness and heat redistribution in an idealised 3D atmospheric General Circulation Model (GCM), but there are many other surface and atmospheric properties that could affect heat transport such as land-mass distribution, topography, condensation, or clouds (Lewis et al. 2018; Sergeev et al. 2020). \nMoreover, even with strong heat redistribution, an atmosphere may still emit like a bare-rock in a particular \nbandpass. The greenhouse effect could warm the surface beyond its equilibrium temperature, which could then emit at least as strongly as a bare-rock at wavelengths with low atmospheric opacity. This is why some of the atmospheres in Figure 2 emit more strongly than a black body surface would. Secondly, an atmosphere may form a thermal inversion if, for example, its visible and thermal opacities scale differently with pressure (Piette et al. 2023; Zilinskas et al. 2023). It could then produce an emission feature in the F1500W bandpass if it has high opacity in that spectral region. \nBoth of these effects could counteract the overall cooling effect of heat redistribution, raising the emission in a particular bandpass back up to the value expected for a black body. An observation of such an atmosphere in that bandpass could therefore provide a false negative conclusion that there is no atmosphere. Unlike the false positives and false negatives for single-filter observations of secondary eclipses, we suggest that night-side emission can provide an unambiguous detection of an atmosphere.', '4.3. Issues with Phase Curves': "While we suggest that phase curves can resolve the degeneracy between the day-side emission of bare-rock planets and atmospheres, there would still be challenges in making and interpreting observations of night-side emission. \nInstrumental systematics are a key issue for the processing of MIRI observations, particularly a distinctive exponential ramp at the start of an observation (e.g., Zieba et al. 2023; Kempton et al. 2023; Bell et al. 2024). August et al. (2024) concluded that these systematics are more problematic than previously expected when analysing MIRI filter eclipse observations of rocky exoplanets. Repeated eclipse observations and phase curve observations have different advantages and disadvantages when handling these systematics. Repeated eclipse observations require a relatively simple model to be fitted, which should not be degenerate with the shape of an exponential ramp or linear trend. However, every eclipse will have its own systematic shape, requiring a new set of parameters to be fitted each time (Zieba et al. 2023). Observing a phase curve with at least two eclipses provides a single periodic measurement which can separate an exponential systematic ramp from the periodic planetary signal more effectively than separated eclipse observations (Hammond et al. 2024). However, the shape of the phase curve can be degenerate with long-period systematics, with the strongest effects on the inferred night-side emission furthest from the anchoring effect of the eclipses (Kempton et al. 2023). \nThere is an important potential false positive for the detection of an atmosphere using a phase curve. Internal heating processes could lead to a non-zero surface geothermal heat flux at all longitudes, warming the night-side even in the absence of an atmosphere. For internal heating via the decay of radioactive isotopes in an Earth-like concentration, we expect an associated heating rate at the surface on the order of a few tens to a hundred mWm -2 , which is insufficient to be detected in the near- to mid-infrared spectrum with JWST (Meier et al. 2021). Tidal dissipation in the interior adds to this flux; the tidal contribution to surface heating can be estimated from the orbital parameters of the system (which would be refined with a high-precision phase curve), with some assumptions about the interior structure and rheology (e.g., Barr et al. 2018; Hay & Matsuyama 2019; Bolmont et al. 2020; Farhat et al. 2024). \nThe observed globally-averaged surface heat flux on Io (essentially the tidal heat flux here) is a few W m -2 , still undetectable-though note that localised volcanoes could contribute disproportionally to the observed thermal emission. In this context it is irrelevant whether the heat generated in the interior is mostly transported at the surface by conduction through the planet's crust, like Earth, or by the advection of hot magma like Io. The planned DDT programme aims to improve constraints on the eccentricity of the target planets, which should improve the precision of the upper limit on their tidal heating. \nThe greatest issue with phase curves is the observational time that they require, being comparable to the orbital period. It may be possible to reduce this time by observing partial phase curves - for example, starting before an eclipse and finishing after the night-side is observed. However, we suggest that the issues with instrumental systematics encountered for even a full phase curve with two anchoring eclipses in Kempton et al. (2023) imply that a partial phase curve will be impractical for constraining night-side emission.", '4.4. Observing Strategy': "We suggest three general strategies to spend 500 hours with JWST searching for atmospheres rocky exoplanets: \n- 1. Observing secondary eclipses with the F1500W filter only (Redfield et al. 2024); given 500 hours, five eclipses could be observed of each of the top 20 rocky planets sorted by signal-to-noise ratio in the F1500W bandpass. We suggest that these observations would be very susceptible to false positives mistaking high-albedo bare-rock surfaces for atmospheres, or to false negatives mistaking atmospheres with high emission for bare-rock surfaces. \n- 2. Observing secondary eclipses with the F1500W and F1280W filters (Ih et al. 2023); given 500 hours, five eclipses could be observed of each of the top 10 rocky planets sorted by signal-to-noise ratio in the F1500W bandpass. We suggest that these observations would be less degenerate than the first option, confidently identifying atmospheres with an intermediate amount of CO 2 . However, many types of atmosphere would still be degenerate with bare-rock surfaces, and observing eclipses in two filters would be more time-consuming.\n- 3. Observing phase curves with the F1500W filter; given 500 hours, 1.2 orbits (including two eclipses) could be observed for each of 10 rocky planets selected for shorter orbital periods, spanning the range of equilibrium temperatures we model. An example sample would be LHS 1140 c (period 3.8 days), LTT 1445 A c (3.1 days), SPECULOOS-3 b (0.7 days), GJ 3929 b (2.6 days), GJ 1132 b (1.6 days), LHS 475 b (2.0 days), GJ 486 b (1.5 days), LHS 3844 b (0.5 days), GJ 1252 b (0.5 days), and TOI-4527.01 (0.4 days). We suggest that this is the best method for unambiguous detections of atmospheres, as it does not rely on a particular atmospheric composition. Crucially, it avoids the degeneracy between atmospheres and high-albedo surfaces. Any atmosphere that transports enough heat to the night-side would be detectable in this way. \nWe propose the third strategy as a way to provide unambiguous detections of atmospheres. As discussed above, this is a different conclusion about the optimal strategy to Koll et al. (2019), which proposes eclipseonly photometry (the first strategy in our list) as a method (Mansfield et al. 2019) to identify candidate atmospheres for subsequent follow-up with phase curve or spectroscopic observations. The primary reason for this difference is our use of the Bond albedo instead of the geometric albedo to model the temperature of bare-rock planets, resulting in a much stronger degeneracy than identified in Mansfield et al. (2019). \nIt could be beneficial to follow up observations of phase curves in the F1500W filter with observations of phase curves or secondary eclipses in the F1280W filter, if there is weak global emission in the F1500W filter. If this were caused by a CO 2 absorption feature, the F1280W filter could detect stronger global emission as described in Section 3. Another modification could be to additionally use observations from 5 to 12 µ mwith MIRI LRS, as these may be more optimal than the F1500W and F1280W filters for hotter planets emitting more at \nthese shorter wavelengths. Mansfield et al. (2024) presented eclipse observations of GJ 486 b with this instrument, deriving a precise measurement of its brightness temperature but not detecting any clear spectral features. \nThe three separate strategies described above could also be combined. For example, observing just two F1500W eclipses of each of the top 20 targets might identify targets with day-side emission significantly below the black body value, although with relatively large uncertainty. As we argue above that reducing the uncertainty on the day-side emission would very rarely produce a conclusive detection of an atmosphere, the planets with the lowest day-side emission could then be followed up with an observation of a phase curve. We stress again that it is perfectly plausible that the day-side atmosphere of a planet could emit at its black body temperature in the F1500W filter, while also emitting significantly from its night-side. For example, the TRAPPIST-1 c simulation in Figure 5 has dayside emission consistent with its black-body value; for weaker heat redistribution, more planets would also have day-side emission consistent with the black-body value (see the zero redistribution cases in Figure 3). Other processes like thermal inversions or strong greenhouse warming could further increase the day-side emission for atmospheric compositions that we do not model. We reiterate, therefore, that targets for atmospheric detections should not be ruled out on the basis of black body day-side emission. \nThis strategy would be similar to the proposal of Koll et al. (2019) to use 'one to two eclipses with JWST [ . . . ] confirmed by follow-up transit spectroscopy, eclipse spectroscopy, or thermal phase curves'. We suggest that the difficulty of identifying spectral features from transit and eclipse spectroscopy of temperate rocky exoplanets (Lim et al. 2023; Lustig-Yaeger et al. 2023; Wachiraphan et al. 2024) promotes the use of phase curves for these follow-up measurements. Ducrot et al. (2023) also suggests that a combination of broadband emission spectra with phase curve can provide robust detections of atmospheres on rocky planets, as they demonstrate that eclipse observations of TRAPPIST-1b in two photometric filters are not enough to rule out an atmospheric or a bare-rock scenario.", '5. CONCLUSIONS': "We have presented simulations of JWST MIRI observations in the F1500W and F1280W filters, for a range of surfaces and atmospheres on a selection of the most readily observable rocky exoplanets. These have shown that the emission in the F1500W filter is highly degen- \nerate between the surfaces and atmospheres. We suggest that it is more difficult than previously suggested to detect an atmosphere with observations in this filter alone. This is due to the increased range of possible emission values from bare-rock atmospheres that we model, and the similar scaling with temperature of bare-rock emission and emission from CO 2 -dominated atmospheres. We suggest that no observation in this filter alone can unambiguously determine the presence or absence of an atmosphere, being prone to false positives or false negatives. \nWe also modelled an observational strategy using the difference in emission in the F1500W and F1280W filters, which is roughly uniform for the bare-rock surfaces, but can vary for different atmospheric compositions. A subset of atmospheres have detectably different emission in these bandpasses; these are generally atmospheres with enough CO 2 to create significant opacity in the F1500W bandpass, but not so much that there is also significant opacity in the F1280W bandpass. This technique would only be able to identify this subset of atmospheres with the correct spectral properties, with other types still indistinguishable from bare-rock surfaces with high albedo, and would also take longer than single-filter measurements. \nWe then simulated the detectability of night-side emission from phase curve observations in the F1500W filter. This showed that if 25% of the heating on the day-side is redistributed to the night-side, almost every one of the simulated atmospheres would have detectable emission from its night-side given an observation of one full phase curve. We chose this fraction of 25% as an example as this quantity is unknown for rocky planets in general; atmospheres with more or less redistribution would be more or less detectable. We suggest that finding nightside emission would be an unambiguous detection of an atmosphere. This technique would work for any atmosphere redistributing enough of its day-side heating regardless of composition. We suggest that a phase curve can provide a model-independent detection of an atmosphere, relying on the night-side emission that is simply a parameter of the fitted time-series model. Conversely, day-side emission will always be a model-dependent way to search for atmospheres, relying on complex expectations about which surface types could be feasible false positives, and on the emission simulated by more complex atmospheric models. \nDespite their theoretical ability to provide an unambiguous atmospheric detection, phase curves have limitations in reality. They are time-consuming for planets with long orbital periods, and are not necessarily easy to successfully execute due to issues with constraining \ninstrumental systematics discussed in Section 4. Nightside emission due to tidal dissipation could also provide a false positive detection of an atmosphere from a phase curve; we suggest above that this is unlikely to be detectable for these planets, but it must be considered when interpreting a phase curve. \nOur conclusion that observations of eclipses are of very limited use for detecting atmospheres differs from Mansfield et al. (2019) and Koll et al. (2019), which both proposed MIRI eclipses as a method to detect atmospheres on rocky planets (as candidates, in the case of Koll et al. (2019)). Section 4 identified that the primary cause of this difference was our use of the Bond albedo to calculate the temperature of bare-rock planets, rather than the use of the geometric albedo in Mansfield et al. (2019). This produced an erroneously high eclipse depth for the bare-rock surfaces in Mansfield et al. (2019), which was inherited by Koll et al. (2019) in its discussion of the bare-rock false positive. We also expanded on the range of surface types modelled in Hu et al. (2012) and Mansfield et al. (2019), which resulted in a stronger atmosphere-surface degeneracy (together with the corrected albedo). This motivated our argument that night-side emission is the only method by which an atmosphere could be unambiguously detected. \nA strategy focused on phase curves would need to target planets with shorter orbital periods, which might restrict the distribution of the observed planets and the resulting statistical power of a test of the 'cosmic shoreline' hypothesis. However, Redfield et al. (2024) state that the goal of the proposed 500 hour survey is to definitively identify which planets have atmospheres. We suggest that an observing strategy focused on observing phase curves of ∼ 10 planets would allow the unambiguous detections of atmospheres needed to meet this goal. This would still not be a trivial exercise given the effects of instrumental systematics on measurements of nightside emission. Future studies could investigate the sample sizes needed for statistical tests of hypotheses about the distribution of atmospheres on rocky planets. We suggest that any statistical tests, no matter how sophisticated, ultimately depend on unambiguous information content in each observation which could only be provided by phase curve observations. \nM.H. is supported by Christ Church, University of Oxford. C.M.G. is supported by the Science and Technology Facilities Council [grant number ST/W000903/1]. T.L. acknowledges support by the Branco Weiss Foundation, the Alfred P. Sloan Foundation (AEThER project, G202114194), and NASA's Nexus for Exoplanet System Science research coordination network (Alien Earths project, 80NSSC21K0593). H.N. was supported by the Clarendon Fund and the MT Scholarship Trust. C.F. acknowledges financial support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement no. 805445. Support for this work was provided by NASA through the NASA Hubble Fellowship grant #HST-HF2-51559.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. T.G.M. was supported by the SNSF Postdoc Mobility Grant P500PT 211044 \nThis research utilises spectra acquired by Raymond E. Arvidson, Melinda D. Dyar, Bethany L. Ehlmann, William H. Farrand, George Mathew, Jack Mustard, Carle M. Pieters, Hiroshi Takeda, and the Planetary Geosciences Lab (PSI) with the NASA RELAB facility at Brown University. \nThe authors thank Edwin Kite, Eliza Kempton, and Jacob Bean for valuable feedback on an early draft of this manuscript.", 'REFERENCES': "Amundsen, D. S., Tremblin, P., Manners, J., Baraffe, I., & Mayne, N. J. 2017, Astronomy & Astrophysics, 598, A97 August, P. C., Buchhave, L. A., Diamond-Lowe, H., et al. 2024, arXiv preprint arXiv:2410.11048 \nBarr, A. C., Dobos, V., & Kiss, L. L. 2018, Astronomy & \nAstrophysics, 613, A37, \ndoi: 10.1051/0004-6361/201731992 \nBell, T. J., Crouzet, N., Cubillos, P. E., et al. 2024, arXiv preprint arXiv:2401.13027 \nBerger, G., Cathala, A., Fabre, S., et al. 2019, Icarus, 329, 8, doi: 10.1016/j.icarus.2019.03.033 \nBolmont, E., Breton, S. N., Tobie, G., et al. 2020, \narXiv:2010.04587 [astro-ph]. \nhttps://arxiv.org/abs/2010.04587 \nBott, N., Brunetto, R., Doressoundiram, A., et al. 2023, Minerals, 13, 250, doi: 10.3390/min13020250 \nByrne, X., Shorttle, O., Jordan, S., & Rimmer, P. B. 2024, Monthly Notices of the Royal Astronomical Society, 527, 10748, doi: 10.1093/mnras/stad3914 \n' Alvarez, E., Brady, M., et al. 2022, \nCaballero, J., Gonz'alezAstronomy & Astrophysics, 665, A120 \nCassidy, W., & Hapke, B. 1975, Icarus, 25, 371, \ndoi: 10.1016/0019-1035(75)90002-0 \nCintala, M. J. 1992, Journal of Geophysical Research: \nPlanets, 97, 947, doi: 10.1029/91JE02207 \nCutler, K. S., Filiberto, J., Treiman, A. H., & Trang, D. \n2020, The Planetary Science Journal, 1, 21, \ndoi: 10.3847/PSJ/ab8faf \nDeming, D., Seager, S., Winn, J., et al. 2009, Publications of the Astronomical Society of the Pacific, 121, 952 Demory, B.-O., Gillon, M., de Wit, J., et al. 2016, Nature, \n532, 207"}

2024ApJ...973L..11L

The MIRI Exoplanets Orbiting White dwarfs survey is a cycle 2 JWST program to search for exoplanets around dozens of nearby white dwarfs via infrared excess and direct imaging. In this Letter we present the detection of midinfrared excess at 18 and 21 m toward the bright V 11.4 metalpolluted white dwarf WD 0310688. The source of the IR excess is almost certainly within the system the probability of background contamination is lt0.1. While the IR excess could be due to an unprecedentedly small and cold debris disk it is best explained by a inlineformula mmlmath overflowscrollmmlmsubsupmmlmrowmmlmn3.0mmlmnmmlmrowmmlmrowmmlmommlmommlmn1.9mmlmnmmlmrowmmlmrowmmlmommlmommlmn5.5mmlmnmmlmrowmmlmsubsupmmlmath inlineformula M SUBJupSUB cold 248inlineformula mmlmath overflowscrollmmlmsubsupmmlmrowmmlmrowmmlmrowmmlmommlmommlmn61mmlmnmmlmrowmmlmrowmmlmommlmommlmn84mmlmnmmlmrowmmlmsubsupmmlmath inlineformula K giant planet orbiting the white dwarf within the forbidden zone the region where planets are expected to be destroyed during the stars red giant phase. We constrain the source of the IR excess to an orbital separation of 0.12 au marking the first discovery of a white dwarf planet candidate within this range of separations. WD 0310688 is a young remnant of an A or late Btype star and at just 10.4 pc it is now the closest white dwarf with a known planet candidate. Future JWST observations could distinguish the two scenarios by either detecting or ruling out spectral features indicative of a planet atmosphere.

2024-09-01T00:00:00Z

['10.3847/2041-8213/ad74ed', '2024arXiv240816813L', '2024ApJ...973L..11L', '10.48550/arXiv.2408.16813', 'arXiv:2408.16813']

['Infrared excess', 'Extrasolar gaseous giant planets', 'White dwarf stars', 'Debris disks', 'Exoplanet migration', '788', '509', '1799', '363', '2205', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics']

The MIRI Exoplanets Orbiting White dwarfs MEOW Survey Midinfrared Excess Reveals a Giant Planet Candidate around a Nearby White Dwarf

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

https://arxiv.org/pdf/2408.16813.pdf

{'The MIRI Exoplanets Orbiting White Dwarfs (MEOW) Survey: Mid-Infrared Excess Reveals a Giant Planet Candidate around a Nearby White Dwarf': "<!-- image --> \n1 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA \nDepartment of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA \n2 02139, USA \n- 3 Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD 4350, Australia \n4 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8W 2Y2, Canada \n5 Johns Hopkins APL, 11100 Johns Hopkins Rd, Laurel, MD 20723, USA \n6 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706, USA \n7 Department of Astronomy, University of Texas at Austin, Austin, TX, USA \n8 Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden \n9 AURA for the European Space Agency (ESA), ESA Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, Maryland 21218, USA \n10 Gemini Observatory/NSF's NOIRLab, 670 N. A'ohoku Place, Hilo, Hawaii, 96720, USA \n11 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands \n- 12 Aerospace Engineering, TU Delft, Building 62 Kluyverweg 1, 2629 HS Delft, The Netherlands \nEarth and Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road, NW, Washington, DC 20015, USA \n14 \nThe Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA \n15 Las Campanas Observatory, Carnegie Institution for Science Colina el Pino, Casilla 601 La Serena, Chile", 'ABSTRACT': "The MIRI Exoplanets Orbiting White dwarfs (MEOW) Survey is a cycle 2 JWST program to search for exoplanets around dozens of nearby white dwarfs via infrared excess and direct imaging. In this paper, we present the detection of mid-infrared excess at 18 and 21 microns towards the bright ( V = 11.4) metal-polluted white dwarf WD 0310-688. The source of the IR excess is almost certainly within the system; the probability of background contamination is < 0 . 1%. While the IR excess could be due to an unprecedentedly small and cold debris disk, it is best explained by a 3 . 0 +5 . 5 -1 . 9 M Jup cold (248 +84 -61 K) giant planet orbiting the white dwarf within the forbidden zone (the region where planets are expected to be destroyed during the star's red giant phase). We constrain the source of the IR excess to an orbital separation of 0.1-2 AU, marking the first discovery of a white dwarf planet candidate within this range of separations. WD 0310-688 is a young remnant of an A or late B-type star, and at just 10.4 pc it is now the closest white dwarf with a known planet candidate. Future JWST observations could distinguish the two scenarios by either detecting or ruling out spectral features indicative of a planet atmosphere. \nKeywords: Infrared excess, Extrasolar gas giants planets, White dwarfs, Debris disks, Exoplanet migration \nCorresponding author: Mary Anne Limbach \nmlimbach@umich.edu \n∗ NHFP Sagan Fellow \n† NASA Hubble Science Fellow \n13", '1. INTRODUCTION': "White dwarfs represent the final evolutionary stage for most stars, including our Sun. Despite this common endpoint, the fate of their planetary systems remains poorly understood. Detection of exoplanets around white dwarfs have been challenging using traditional methods suitable for main-sequence stars (Endl & Williams 2018; Zhu & Dong 2021). To date, only a few white dwarf exoplanets have been confirmed. \nDue to the limited number of known white dwarf planets, we lack a comprehensive understanding of planetary evolution during the white dwarf phase. Increased detections could facilitate demographic studies and detailed characterization of such systems (Debes et al. 2005; Veras 2021; Ledda et al. 2023; Poulsen et al. 2024). This knowledge could help us ascertain whether planets can endure the death of their host stars and remain in orbit around the resultant white dwarf, as well as identify the conditions under which planets might be disrupted or destroyed during the red giant phase (Debes & Sigurdsson 2002; Zuckerman et al. 2010). Furthermore, a deeper understanding of white dwarf planetary systems could reveal whether life can arise around dead stars (Agol 2011; Fossati et al. 2012; Barnes & Heller 2013; Loeb & Maoz 2013; Kozakis et al. 2018, 2020; Becker et al. 2023; Zhan et al. 2024). \nThe near-featureless infrared spectrum of white dwarfs is a valuable characteristic for exoplanet searches using the detection of infrared (IR) excess, which can indicate the presence of sources colder than the white dwarf within the system such as debris disks, late-type stellar companions, brown dwarfs, or exoplanets (Kilic et al. 2005; Becklin et al. 2005; Farihi et al. 2008b; Barber et al. 2012; Stevenson 2020; Limbach et al. 2022). The IR excess technique enabled the discovery of the first brown dwarf-white dwarf system (Zuckerman & Becklin 1987; Cunningham et al. 2022) and has since been employed to identify numerous similar systems (e.g., Girven et al. 2011; Xu et al. 2015; Rebassa-Mansergas et al. 2019; Hogg et al. 2020; Lai et al. 2021). Although the Spitzer Space Telescope was used in attempts to detect exoplanet-induced IR excess in white dwarf systems (Werner et al. 2004; Mullally et al. 2007; Farihi et al. 2008a; Kilic et al. 2010), no candidates were identified. \nFortunately, with the advent of the JWST, our ability to detect exoplanets around white dwarfs has drastically improved. JWST has multiple programs designed to detect new white dwarf exoplanets (Mullally et al. 2021; Limbach et al. 2023; Poulsen et al. 2023). These search programs have resulted in the detection of multiple candidates (Mullally et al. 2024b, and this work ) and follow-up observations to confirm planet candidates are \nunderway (Venner et al. 2023; Cheng et al. 2024; Mullally et al. 2024a). JWST's capabilities also extend to characterizing known white dwarf exoplanets (MacDonald et al. 2021; Vanderburg et al. 2021; Limbach et al. 2024; Blackman et al. 2024). \nDespite these impressive advancements, no known exoplanets (confirmed or candidate) exist around white dwarfs at mid-separations (0.1-2 AU). Depending on the mass of the star, exoplanets beyond ∼ 2 AU are expected to survive the final stages of stellar evolution intact (Nordhaus & Spiegel 2013). Discovering planets near or inward of this anticipated boundary illuminates our understanding of post-main-sequence exoplanet evolution and migration. The recent discovery of a volatile-rich gaseous debris disk from an evaporating giant planet (Gansicke et al. 2019) and the discovery of a gas giant exoplanet orbiting at only 0.02 AU (Vanderburg et al. 2020) suggest that massive planets may find their way into tight orbits around white dwarfs, and the discovery of dusty disks (generally hotter than giant planets but colder than the white dwarf) around about 2-3% of white dwarfs indicates that large rocky bodies may be torn apart by gravitational tides, producing a closely orbiting hot dust disk. The only other known white dwarf exoplanets and planet candidates orbit beyond the 2 AU boundary where survival is expected (Thorsett et al. 1993; Sigurdsson et al. 2003; Luhman et al. 2011; Blackman et al. 2021; Mullally et al. 2024b). \nThe detection of exoplanets around metal-polluted white dwarfs presents a unique opportunity to explore the intricate dynamics between stellar remnants and planetary systems. Metal-rich white dwarfs are characterized by the presence of heavier elements in their atmospheres. This is surprising because metals should rapidly settle (on timescales as fast as days for hydrogen envelope white dwarfs Kilic et al. 2006; Koester & Wilken 2006; Jura et al. 2007; Koester 2009) below the white dwarf's surface due its strong gravitational fields, leaving only lightweight elements like hydrogen and helium in the atmosphere. Therefore, the presence of metals in white dwarfs atmosphere indicates that they are likely undergoing recent or active accretion. Studies in recent decades have concluded that the source of this accretion (also referred to as pollution) is usually minor planets, comets and asteroids from the progenitor system that survived red giant evolution (Alcock et al. 1986; Jura 2003; Klein et al. 2010; Dufour et al. 2010; Debes et al. 2012; Veras et al. 2024). They likely moved close to the white dwarf due to gravitational interactions with other massive bodies, such as giant planets, in the system. Small bodies, once destabilized, can migrate inward and eventually cross the white dwarf's Roche \nlimit, where they are tidally disrupted into dust and gas. This debris, rich in metals, then accretes onto the white dwarf, polluting its atmosphere. The chemical fingerprints of these accreted materials closely mirror the composition of terrestrial planets, such as Earth, providing strong evidence for this model (Melis et al. 2011; Xu et al. 2019; Trierweiler et al. 2023). \nDiscovering white dwarf systems with planets and disks is crucial to validating the current theories on white dwarf accretion and planetary survival post-main sequence. In this paper, we present the detection of infrared excess at < 2 AU around the metal-polluted white dwarf WD 0310-688. Our observations, conducted as part of the MIRI Exoplanets Orbiting White dwarfs (MEOW) JWST survey program, are detailed in Section 2. Section 3 covers our analysis of the MEOW data, focusing on the detection of infrared excess in the WD 0310-688 system. We use our MEOW data, modeling and archival data to constrain the source of infrared excess and its physical parameters. In Section 4, we discuss the implications of this detection and outline how future observations could confirm the source as an exoplanet. Finally, we summarize our findings in Section 5.", '2. OBSERVATIONS': "The MEOW survey is a JWST Cycle 2 Survey program (analogous to HST 'Snapshot' programs) to collect three-band MIRI imaging of ∼ 20 nearby white dwarfs in search of exoplanets. Utilizing a combination of infrared excess and direct imaging techniques, the MEOW survey is capable of detecting white dwarf exoplanets at all separations, from the Roche limit to the edge of the field of view ( ∼ 1000 AU; depending on the exact distance to the system). MEOW is sensitive to extremely cold planets, with the ability to detect exoplanets with masses as low as that of Saturn and temperatures down to 120 K for the youngest, nearest systems, and planets with temperatures as low as 175 K, equivalent to a 2 M Jup at 3 Gyr, out to a distance of 16 pc. \nThe MEOW survey is volume limited ( < 17 pc). It includes solitary white dwarfs and white dwarfs with faint companions (main sequence stars later than M4V or binary white dwarfs). White dwarfs with bright main sequence companions (e.g., Sirius B, Procyon B, 40 Eridani B, etc) are excluded as a different imaging configuration would be optimal for detecting exoplanets in those systems. \nImaging is taken in three MIRI bands: F770W, F1800W and F2100W (7.7, 18.0 and 21.0 µ m, respectively). For the F770W filter, our total exposure time is 55.5 sec (5 groups/int, 1 int/exp), which is sufficient to achieve an SNR > 200 on all targets in our sample while \nstill avoiding saturation on the nearest white dwarfs. The SNR in 7.7 µ m band is dominated by photon noise from the target star; however, our absolute photometric precision is limited to 2% by the absolute flux calibration of the MIRI imager 1 (Gordon et al. 2022). For the F1800W and F2100W filters, our exposure times are 277.5 sec and 710.4 sec, respectively (12 groups/int, 2 or 5 int/exp), which results in an SNR between 10 and 125 on all targets in our sample. Depending on the target brightness, the noise in the 18 and 21 µ m bands can be dominated by either the thermal background noise or the 2% absolute photometric precision. \nFor all imaging, we used the fast readout mode and a four point cycling dither (starting point 1). We chose a 4-point cycling dither to correct for bad pixels and remove background noise. We used the full MIRI imaging array, which has a 74' × 113' useable field of view, and a detector plate scale of 0.11'/pixel (Bouchet et al. 2015). \nThe observations of WD 0310-688 began Sep 19, 2023 22:50:31 UT and ended at Sep 19, 2023 23:53:06, about an hour later 2 . The false color image constructed from the MIRI imaging in the three bands is shown in Figure 1. \nSome initial findings from the MEOW survey are included in this Letter as they relate to the infrared excess detected around WD 0310-688. However, the majority of the survey's results and analysis will be detailed in a future manuscript. This Letter primarily focuses on reporting the discovery of a planet candidate around WD 0310-688.", '3.1. The White Dwarf WD 0310-688': 'WD 0310-688 is a solitary white dwarf located at a distance of 10.4 pc. With V and G -band magnitudes of 11.4 mag, it is the 5 th brightest white dwarf at visible wavelengths and the single brightest solitary white dwarf in the sky. 3', '3.1.1. Stellar Parameters': "The spectroscopic (B'edard et al. 2017) and photometric (Gentile Fusillo et al. 2021) determinations of WD 0310-688's atmospheric parameters differ slightly, so we \n- 1 jwst-docs.stsci.edu/jwst-data-calibration-considerations/ jwst-data-absolute-flux-calibration\n- 2 Of note, this star was the very first JWST survey observation ever conducted, for any survey program.\n- 3 The five brightest white dwarfs in the optical are Sirius B ( V = 8 . 4), 40 Eri B ( V = 9 . 5), Procyon B ( V = 10 . 9), CD-38 10980 ( V = 11 . 0), and WD 0310-688 ( V = 11 . 4) (Holberg et al. 2008; Gentile Fusillo et al. 2021). \nFigure 1. False color image of WD 0310-688 (bluish-white star in the center) using three JWST/MIRI bands: F770W - blue, F1800W - green, and F2100W - red. Other sources in the field include distant galaxies, background stars, and potentially a few point sources that could be wide-orbit exoplanets bound to WD 0310-688. Although not the main focus of this paper, we note the very faint source 80 AU above the white dwarf, which is marginally detected at 18 and 21 µm , is consistent with a T eff ∼ 140 K planet slightly less massive than Jupiter, however, at this separation the false positive rate is reasonably high. Confirming the any resolved sources as bound exoplanets would necessitate follow-up observations for common proper motion at a later epoch. \n<!-- image --> \ncalculate the ages based on both sets of parameters. This white dwarf is in a regime where a small difference in its mass can significantly affect its main-sequence lifetime. We calculated the white dwarf parameters using wdwarfdate , and the resulting parameters are listed in Table 1. These parameters suggest that the host is a remnant of an A or late B star. We base our calculations on photometric white dwarf parameters because the spectroscopic parameters yield a slightly low white dwarf flux that is in tension with Gaia photometry, thus favoring the photometric solution. The photometric parameters give a white dwarf mass of 0 . 659 ± 0 . 012 M ⊙ and cooling age of 194 ± 13 Myr.", '3.1.2. Metal Lines & Accretion': "WD 0310-688 is classified in the literature as a hydrogen-atmosphere (DA) white dwarf with no metal lines, and has been observed as part of ground based observation campaigns (Koester et al. 2009). However, \nTable 1. WD 0310-688 Spectroscopic & Photometric Derived Parameters \na \nSpectroscopic Teff and log(g) from B'edard et al. 2017; Photometric Teff and log(g) from Gentile Fusillo et al. 2021. \nb \nin a previously-unpublished archival UV HST/STIS E140M+E230M spectrum of this white dwarf (Gaensicke et al. 2015), which we retrieved from the HASP database (Debes et al. 2024), we find O, Si, Mg and Fe absorption in the spectrum near the photospheric radial velocity of WD 0310-688 (see Figure 2). This suggests accretion of metal-rich material on the white dwarf, and \nFigure 2. In a reanalysis of an archival HST/STIS spectrum of WD 0310-688, we detect atomic lines of four metals in the white dwarf's photosphere. This suggests accretion of metal-rich material on the white dwarf, and we reclassify it as a DAZ due to the presence of the metal lines. WD 0310-688 was previously classified as a DA spectral type. \n<!-- image --> \nwe reclassify it as a DAZ. We also note the presence of the Si II 1265 ˚ A line indicating that the UV absorption lines are not from the interstellar medium (Koester et al. 2014) and confirming that there is pollution in the white dwarfs photopshere. Accretion has implication for the disk and exoplanet hypotheses, which we discuss more in Section 4, so we will further constrain this accretion here. \nWe construct a model of WD 0310-688 using an updated version of the model atmosphere code described by Blouin et al. (2018a,b) and references therein. The model atmosphere code has been modified to include all relevant spectral lines and molecular bands up to 30 µ m as described by Limbach et al. (2022). \nWe calculate a log N(Si)/N(H) = -8.1 ± 0.2 from the STIS spectrum by fitting three Si II lines (at 1260.42, 1264.73 and 1265.02 ˚ A). For this white dwarf with a temperature of 15,865 K and log(g) of 8.076, the diffusion timescale of Si is 1.2 × 10 5 seconds (1.4 days; Dufour et al. 2017). The short diffusion timescales for this white dwarf are due to its high temperature, indicating that it must have been accreting in 2016 during the STIS observations. The thickness of the superficial convection zone is 10 -16 . 6 of the total mass (Dufour et al. 2017). By dividing the mass of Si mixed in the convection zone by the sinking timescale, we obtain a steady-state accretion rate of Si of 5.3 × 10 4 g/s. Relative to Si, we find that the other metal abundances, listed in Table 2, are consistent with the bulk Earth (All'egre et al. 1995) once accounting for diffusion timescales. For Fe, the detection is marginal so it is not given in the table, but if we assume a log Fe/Si that corresponds to bulk Earth, we get a synthetic spectrum consistent with the HST spectrum. \nAssuming the accreted material has a bulk Earth composition and based on the steady-state accretion rate of Si, the total accretion rate is 3.3 × 10 5 g/s. This is among the lowest accretion rates observed in polluted white dwarfs (Blouin & Xu 2022). A summary of all the white dwarf's parameters from this work and the literature is provided in Table 2.", '3.2.1. Resolved Sources in the Image': "Prior to conducting a detailed analysis, we visually inspected the color image for resolved sources that could be bound companions to the white dwarf. The purpose of this brief search is to note the presence of possible wide-orbit companions that could serve as perturbers (via planet-planet scattering; see section 4), which is relevant to the IR excess detection presented within this manuscript. \nWe identify several resolved sources in Figure 1 that exhibit colors and fluxes consistent with exoplanets. One of these is relatively close, about 80 AU from the white dwarf. This source (which is not the primary focus of this manuscript), is only detected with a signalto-noise of a few, but is consistent with a ∼ 140 K planet (which given the system's age, would likely be subJovian in mass; Linder et al. 2019). However, at this angular separation the false positive rate is reasonably high in our survey. There are also many other sources at larger angular separations from the white dwarf with photometry consistent with warmer, more massive exoplanet models. We expect that most or all of these point sources at large angular separations are background objects, but additional epochs of MIRI imaging could show whether any share common proper motion with the white dwarf, indicating they are bound exoplanets. This \nTable 2. Summary of WD 0310-688 ParametersRefs: 1. Gaia Collaboration 2020, 2. Napiwotzki et al. 2020, 3. this work , 4. Gentile Fusillo et al. 2021, 5. Koen et al. 2010, 6. Cutri et al. 2003 \ninformation will be crucial for understanding exoplanet occurrence rates around white dwarfs and the systems' dynamical histories. However, we defer detailed consideration of the resolved point sources around WD 0310688 until follow-up observations are conducted.", '3.2.2. Photometry and IR Excess': 'In this subsection, we describe how we (1) reduce the data using a custom pipeline, (2) perform aperture photometry and report the measured flux values, (3) quantify the measurement errors, (4) compute the infrared excess from the photometry, and (5) provide checks to verify that the measured infrared excess is real and not attributable to any emission from the white dwarf itself.', 'Step 1: Data Reduction': 'The MIRI imagery reduced automatically by the JWST pipeline and available in MAST suffers from a nonuniform flatfield subtraction at the reddest wavelengths. To address this, we reprocessed the data using a custom software package, MEOW , that is available on \nGitHub 4 . Version 1.0 of MEOW was used for this reduction. We used JWST pipeline version 1.15.1 and CRDS jwst 1225.pmap. The background subtraction code is based on a STScI JWebbinar demo 5 that produces well flat-fielded Stage 2 data with a custom background subtraction using the multiple dithers on each source.', 'Step 2: Aperture Photometry': "Using the reprocessed Stage 2 data, we then conducted aperture photometry on the white dwarfs in the MEOW dataset. To conduct aperture photometry we used the Python photutils package and the aperture correction values provided in the JWST CRDS 6 and color corrections table provided by the STScI HelpDesk 7 . For the aperture sizes, we used the values corresponding to the full array (as we read out the full MIRI subarray) and the 80% energy encircled aperture sizes (the largest apertures with correction values available) on the target PSF. \nThe resulting measured flux values are given in Table 3. In each band we measure the flux of the white dwarf at each of the four dithers separately and then report the white dwarf's median flux value, using the scatter in the flux measurements divided by the square-root of the number of dithers to compute the error of the measured flux value.", 'Step 3: Error Sources': "The errors in the infrared excess measurement of white dwarfs in the MEOW sample arise from two primary sources: (1) the precision (signal-to-noise) of the measurement, which is primarily limited by photon noise (listed in the middle column in Table 3 for WD 0310688), and (2) the accuracy with which our white dwarf models predict the star's flux. For the MEOW data, the limiting factor can either be photon noise or \nTable 3. Measured WD flux and IR excesses. \nthe accuracy of the JWST photometry and our models, depending on the bandpass and the brightness of the white dwarf. Notably, WD 0310-688, one of the brightest white dwarfs, is consistently limited by model/photometric accuracy across all bands.", 'Step 4: Infrared Excess': 'We compute the infrared excess, defined as the measured flux of the white dwarf relative to the expected (modeled) flux values. We build a model for each white dwarf using the same modeling process described in Section 3.1 for WD 0310-688. This analysis and IR excess calculation was conducted for all white dwarfs in the MEOW sample. Figure 3 illustrates the infrared excess (or deficit) divided by the precision of our measurement for a subset of the white dwarfs in the MEOW sample. Only white dwarfs without a large ( > 100 MG) magnetic field and those amenable to aperture photometry 8 are included in this plot. The white dwarfs included in the plot are WD 0839-327, WD 0840136, WD 1309+853, WD 1820+609, WD 1756+827, WD 1019+637, WD 0752-676, WD 0821-669 and WD0310-688. This measurement precision accounts for both error sources (1) and (2). In this plot, we take error (2) to be 2%, the absolute photometric precision of MIRI as discussed in Section 2. \nIn Figure 3, the green lines representing white dwarfs without notable IR excess are systematically low by 12%. To address this, we apply a correction factor to the measured IR excesses where we have a sufficiently large set of white dwarfs with a signal-to-noise ratio greater than 50 that can be used for calibration. At 7.7 µ m, almost all white dwarfs meet this criterion and at 18 µ m only a couple of the brightest white dwarfs in our sample do. The derived correction factors are: 1.14% at 7.7 µ m and 0.1% at 18 µ m. At 21 µ m, there are no comparatively bright white dwarfs that provide a good reference, so we do not apply an additional correction factor at 21 µ m. \nThe IR excesses for WD0310-688 (orange line), including the correction factors, are -0.9%, 16.6%, and 21.3% at 7.7 µ m, 18 µ m, and 21 µ m bands, respectively (as shown in Figure 5 and listed in Table 3). If the source of the excess is a planet, the infrared excess corresponds to the flux ratio of the planet to the star ( F p /F ∗ ). We calculate the significance of the IR excess detected in the WD 0310-688 system to be 8.3 σ at 18 µ m and 10.7 σ at 21 µ m bands. No significant excess is detected at 7.7 µ m. \nThe purple line in Figure 3 indicates a marginal detection of IR excess for WD 1309+853. This white dwarf \nFigure 3. The measured IR excess or deficit divided by the precision of our measurement (y-axis) in all three spectral bands (x-axis) for a subset of MEOW white dwarfs. The orange line represents WD 0310-688, which shows a highly significant IR excess. The purple line corresponds to another white dwarf with an excess possibly due to an exoplanet but detected with less significance than WD 0310-688. The green lines indicate no significant excess, aligning generally with our models, although they are systematically low by 1-2% at 7.7 µ m and 21 µ m. The white dwarf that is notably low at 21 µ m is likely affected by a nearby bright, red galaxy contaminating the background flux measurement. \n<!-- image --> \nhas a relatively weak magnetic field of 5 MG, which could potentially account for the discrepancies between the photometry and the model. None of the other white dwarfs presented in this plot are known to have magnetic fields. A full analysis and discussion of the IR excesses within the MEOW sample is deferred to a later manuscript.', 'Step 5: Verifying the Result': "To further validate the robustness of the IR excess measurement for WD 0310-688, we conducted two additional tests. First, we compared the fluxes using the Stage 3 data available in MAST for our target, finding the flux values agreed with our custom reduction results to within 2% in all bands (some deviation is expected due to non-uniform flatfield in the Stage 3 data). \nSecond, we generated a range of white dwarf models consistent with the ± 1 σ uncertainties derived from the optical photometry and spectroscopy described in section 3.1. Our analysis revealed that varying the white dwarf model does not produce an IR excess that increases with wavelength, as observed with MIRI. Instead, adjusting the white dwarf parameters results in a near-uniform shift in the IR excess across all MIRI wave- \nngths, owing to the fact that MIRI photometry samples the Rayleigh-Jeans tail of the white dwarf's SED.", '3.2.3. PSF Subtraction': "In this section, we use PSF subtraction to constrain the maximum orbital separation of the IR excess source and determine the background source false positive probability. In an attempt to directly image the source emitting the infrared excess, we conducted a PSF subtraction using our custom flat-field corrected 21 µm Stage 2 data. We used four images of WD 0310-688 for the science frames (see Figure 4) and four reference images from the white dwarf WD 0839-327 (G = 11.82 mag), the white dwarf most similar to WD 0310688 in our MEOW sample, as our reference PSF. We used the python package VIP (Gomez Gonzalez et al. 2017) to crop the images to a size of 100px (1 arcsec), correct NaN values, and align the PSFs to a common center. We then scaled the reference PSF to the science PSF and performed a PSF subtraction (results are shown in Figure 4, center column). We find no notable residuals in the subtraction suggesting the source of infrared excess is unresolved from the star. \nWe then injected companions with the same flux as the detected IR excess at various separations from the target. We find we are able to detect the injected companions after PSF subtraction with a confidence of > 4 σ when the companion is injected at separations of > 2 AU, and therefore we constrain the source to be within 2 AU of the host. \nThe PSF subtraction constrains the maximum separation of the infrared excess source to < 2 AU, or < 0 . 19'. Using this maximum separation, we calculated the probability that our infrared excess is due to a background object rather than something in the WD 0310-688 system. The MIRI field of view is 112.6' × 73.5', which corresponds to a false positive rate of 1/73000 if there was one false positive in the FOV and we observed only one white dwarf. However, MEOW images typically contain four sources within the field of view that are consistent with exoplanet SEDs, and the MEOW survey has observed 17 targets. Assuming the four planet-like sources per field are false positives (though some may actually be planets), we estimate the probability that the IR excess we have detected is a background source to be 1/73000 times 4 FPs/field and 17 targets, giving a false positive probability of < 0.1%. We note that none of the MEOW targets with reliable photometry (no contamination from nearby sources) exhibit an IR excess, except for those with known magnetic fields. We also note that the WD 0310-688 field is less dense than most of the others within the MEOW survey, making this es- \nimate likely conservative. Due to the very low false positive probability, we conclude that the IR excess is almost certainly associated with the white dwarf system.", '3.2.4. Blackbody Retrieval': 'Having ruled out background sources as a plausible explanation for the infrared excess, we conclude that the IR excesses are likely due to a cold dust disk or an exoplanet. We now explore models that can explain the measured excess. \nBecause we only have three photometric data points, the data does not justify a model more complex than a blackbody. The advantage of a blackbody model is that it is agnostic as to whether the source is a disk or an exoplanet, as both are, to first order, blackbodies in the mid infrared. \nWe conducted a blackbody fit using the measured infrared excess. In Figure 5, we plot the measured excess alongside the temperature and size constraints from our blackbody fit. The fit to the photometric measurements yield a T = 248 +84 -61 K blackbody with an emitting area of 1.03 +0 . 67 -0 . 35 R Jup . \nWe note that the size of the emitting body is conspicuously close to that of a giant planet, but we do not disregard the possibility that a cold dust disk could also have an emitting area coincidentally similar to that of a gas giant planet. However, this would constitute a very unusual cold dust disk, as discussed in Section 4. We attempted to run more complex atmospheric retrievals on the data, but found that this resulted in no additional meaningful constraints on the source.', '3.2.5. Minimum Orbital Separation': "We calculate the minimum orbital separation, a , of the source using the fitted blackbody temperature as the equilibrium temperature, T eq . This calculation assumes that the measured brightness temperature is equal to the blackbody temperature, which is a reasonable assumption in the mid infrared, and that there is perfect heat redistribution. The orbital separation is given by \na = R ∗ (1 -α ) 1 2 ( T eff T eq ) 2 (1) \nwhere R ∗ is the radius of the star, α is the albedo of the planet, and T eff is the effective temperature of the white dwarf. This results in a minimum orbital separation of 0 . 18 +0 . 14 -0 . 08 AU (with α = 0 . 4; a typical infrared albedo for a highly reflective gas giant). The 1 σ lower bound gives a minimum orbital separation of 0.10 AU. Orbital separations < 0.1 AU would result in higher brightness temperatures that are inconsistent with the fit. This constraint assumes a circular orbit. However, if the planet \nFigure 4. Left: WD 0310-688 at 21 µm in each of the four dithers, Center: PSF Subtraction at each of the four dithers. Note that the cosmic ray in the first dither and the negative source at a ∼ 30 pixel separation is in the reference star image, not the science image. Right: PSF subtraction after a companion that is 21% the brightness of the star was injected at 2.3 AU. This injected source is retrieved with a statistical significance of 7.0 σ (at each dither the significance of the detections are 3.2 σ , 4.2 σ , 3.7 σ and 2.7 σ ). Injected sources are successfully retrieved at > 4 σ when injected at separations of > 2 AU and therefore we constrain the detected IR excess source to be within 2 AU of the host. \n<!-- image --> \nFigure 5. Measured wavelength-dependent infrared excess (i.e., the planet-to-star flux ratio, if the source of the infrared excess is a planet) from three photometric bands collected with JWST MIRI (F770W, F1800W and F2100W). The infrared excess is consistent with a blackbody of temperature 248 +84 -61 K and an emitting area of 1.03 +0 . 67 -0 . 35 R Jup . Using the derived system age, the planet's temperature, and planet cooling curves, we estimate a planet mass of 3 . 0 +5 . 5 -1 . 9 M Jup . \n<!-- image --> \nhas a highly eccentric orbit (and is migrating), it may pass much closer to the white dwarf, becoming significantly hotter during parts of its orbit. If the source is a optically thin disk (e.g., zodi-like) rather than a planet, the same orbital constraint (0 . 18 +0 . 14 -0 . 08 AU) holds. The exception would be if the source is a planet and exhibits an exceptionally high albedo (e.g., α > 0 . 4), which could potentially allow for slightly closer separations.", '3.2.6. Mass Estimation': "If the source is an exoplanet, we can derive a mass estimate using exoplanet evolutionary models. Using the age of the system and the planet's temperature, along with the 1 σ uncertainties in these two parameters, we constrain the planet's mass to 3 . 0 +5 . 5 -1 . 9 M Jup (Marley et al. 2021), assuming no heat comes from irradiation. If the planet is out at 1-2 AU, the stellar irradiation has essentially no effect on the inferred mass because it would be dominated by the internal heat of the planet, so at those distances the evolutionary masses apply, as previous studies suggest modest irradiation does not change the cooling very much (Fortney et al. 2007). \nHowever, at closer separations, irradiation from the white dwarf could change the evolutionary inferred mass. If the planet orbits close to the white dwarf ( ∼ 0.1 AU), its flux must be dominated by re-radiated stellar radiation. In this scenario, the mass could be much lower. We can still place a lower limit on the mass based on the fact that we need the planet to be sufficiently large in radius. We estimate the minimum mass based on the minimum planet radius from our blackbody fit within the 1 σ constraint, which is 0.68 R Jup . Using the Linder et al. (2019) models, at an age of 1.1 Gyr, we estimate a minimum mass of 0.2 M Jup (slightly less massive than Saturn). However, we note that empirically, there are Jupiter-sized objects with masses as low as a few Earth masses, such as the Kepler 51 super puffs (Masuda 2014; Libby-Roberts et al. 2020).", '3.3. Other Constraints on Companions': '3.3.1. Astrometry \nIf WD 0310-688 is indeed orbited by a giant planet, we can place further constraints on its properties using astrometry. The most precise existing astrometric observations for this target are from the Gaia mission (Gaia Collaboration et al. 2016); however the Gaia epoch astrometry is not yet available, precluding any straightforward search for companions. Nevertheless, it is possible to extract some limits on companion properties from the published Gaia DR3 astrometric solution. \nThe Gaia Renormalized Unit Weight Error (RUWE) parameterizes the significance excess noise of the astro- \nmetric data. Conventionally, a RUWE above > 1.4 is interpreted as significant evidence in favour of astrometric variability (Lindegren 2018). In Gaia DR3 WD 0310688 has a RUWE of only 1.13, which suggests there is no strong evidence for orbital motion in the underlying epoch astrometry. \nWe then attempt to convert the RUWE into a set of upper limits on companion masses. We follow the method of (Belokurov et al. 2020; Korol et al. 2022) to invert the RUWE into the normalized astrometric perturbation, δθ . As the observed RUWE is not significant, we employ RUWE < 1 . 4 as a conservative limit on the astrometric noise, which then gives an assumed limit of δθ < 0 . 19 mas. For orbital periods shorter than or approximately equal to the 1038 d observational baseline of Gaia DR3 (Lindegren et al. 2021), we can assume that this approximates the root mean square of the astrometric reflex amplitude of the theorised orbit. This allows us to assume the conventional astrometric amplitude relation: \nα = m M ∗ a D , (2) \nwhere α is the astrometric amplitude in arcseconds, m is the companion mass, M ∗ is the stellar mass, a is the semi-major axis in AU, and D is the distance in parsecs (Sozzetti 2005). As M ∗ and D are known for WD 0310688, we can derive an upper limit on allowed companion masses as a function of semi-major axis within ⪅ 2 AU. \nMore stringent constraints can be made on companions with wider orbits using Hipparcos-Gaia astrometry (Brandt 2018; Kervella et al. 2019). WD 0310688 was one of only 20 white dwarfs bright enough to be successfully observed with Hipparcos (HIP 14754; Vauclair et al. 1997), and when combined with the more recent Gaia astrometry, the 25 year time baseline allows for strong constraints on longer-period planets. In the Hipparcos-Gaia Catalog of Accelerations (HGCA; Brandt 2021) there is no significant evidence for an astrometric acceleration; the difference between the Gaia proper motion and the averaged HipparcosGaia proper motion in right ascension and declination is ∆ µ = ( -0 . 017 ± 0 . 075, -0 . 012 ± 0 . 080 mas yr -1 ) respectively, equivalent to ∆ v = ( -0 . 8 ± 3 . 7, -0 . 6 ± 3 . 9 m s -1 ) in physical units (Venner et al. 2021, equations 9, 10). We can thus set a strict 3 σ upper limit of < 11 m s -1 on the Hipparcos-Gaia tangential velocity anomaly. \nWe plot the constraints on companion mass as a function of semi-major axis afforded by the astrometric nondetections in Figure 6. Between ≈ 1 -2 AU, the low RUWE allows us to exclude planets more massive than ≳ 3 M J . For wider separations, we use the method of Kervella et al. (2019) to calculate mass limits from \nFigure 6. Constraints on the planets orbiting WD 0310-688 from astrometry. We overplot the planetary parameters implied bu the infrared excess, assuming that the upper limit in projected separation translates to semi-major axis. The absence of a large Gaia DR3 RUWE helps to exclude the more distant, massive edge of the parameter space, but within a < 1 AU the sensitivity of astrometry is low. At larger distances (2 . 5 -10 AU), planets with as low mass as 1 M J can be ruled ruled out due to the absence of a significant Hipparcos-Gaia acceleration. \n<!-- image --> \nthe Hipparcos-Gaia astrometry, from which we find that planets with masses as low as 1 M J can be excluded between 2 . 5 -10 AU. These constraints agree well with and marginally improve upon the non-detection of planets at projected separations above > 2 AU in the MIRI imaging. However, we cannot independently confirm or reject close-separation planets with the existing astrometric data. Our existing astrometric constraints have low sensitivity to planetary-mass companions within a < 1 AU. \nIf the MIRI infrared excess is indeed caused by a giant planet within 2 AU of WD 0310-688, it may be possible to detect the reflex orbital motion in the epoch astrometry that will be released in Gaia DR4. (Sanderson et al. 2022) predict that Gaia astrometry will lead to the detection of 8 ± 2 giant planets around white dwarfs; however, they have assumed that essentially none of these planets will be found within < 2 AU as a result of destruction during stellar evolution.', '3.3.2. Doppler Monitoring': 'In December 2023, we observed WD 0310-688 using the Magellan Carnegie Planet Finder Spectrograph (PFS; Crane et al. 2006, 2008, 2010) to attempt Doppler monitoring for the planet candidate. Although WD 0310-688 is metal-polluted, only the Hydrogen Balmer lines were detectable in the PFS spectrum due to the low level of pollution. Using these lines, we achieved a \nprecision of about 400 m/s over 15 minutes, or 200 m/s per night with 1-hour monitoring. If orbiting at the minimum separation of 0.1 AU, the planet must have a mass near or below the lower end of the self-luminous mass range (e.g., < 1.1 M Jup , as determined in section 3.2.6), or it would be hotter than 248 K. At this separation and mass, we would expect an RV semi-amplitude of 120 m/s. Based on this initial observation, we conclude that detecting the planet via RV monitoring would require an intensive observation program, even if the planet is located in the most favorable part of the allowed parameter space. For the majority of the parameter space, this method is not feasible. Therefore, Doppler monitoring is not a practical method for confirming this planet. However, RV confirmation may be feasible for white dwarf planet candidates with more metal lines in the visible spectrum or with shorter orbital periods (Rogers et al. 2024).', '3.3.3. Other Mid-IR Photometry': 'We searched for archival data that could aid in illuminating the source of the infrared excess. We find that archival Spitzer data show no detectable IR excess. The absence of a Spitzer 8 µm (IRAC band 4) detection of infrared excess (Mullally et al. 2007) is consistent with our non detection of IR excess in the MIRI 7.7 µm band. In the WISE band 3 (12 µ m) we measure an IR excess of 16 ± 14%. Although this excess is consistent with our MIRI detection, it is also consistent with no excess at 1.1 σ , so it provides little additional constraint on the system. In the WISE band 4 observations, the archival measurements do not provide sufficient sensitivity to produce any meaningful constraint on this source.', '4. DISCUSSION': 'We have detected infrared excess emission towards WD 0310-688, and we have shown that the emission is almost certainly coming from the white dwarf system. We can identify two likely explanations: a giant planet, or a cold debris disk. Here, we consider the two hypotheses, give arguments in each of their favor, and discuss the implications of each being correct.', '4.1. Giant Planet': "The first option we consider is a giant planet. It would have a radius similar to that of Jupiter, a mass of 3 . 0 +5 . 5 -1 . 9 M Jup , and an orbital separation between 0.1 and 2 AU. The arguments in favor of this hypothesis are that \n- · Planets of this mass are expected to exist around white dwarfs if they survive the star's red giant phase, particularly since A-type stars have higher \nrates of giant planet occurrence (Reffert et al. 2015). \n- · A wide range of objects, from gas giant planets as small as Saturn to late M-dwarfs, have emitting areas similar to Jupiter, whereas debris disks can vary greatly in size. The radius inferred from a blackbody fit corresponds to the emitting area of a gas giant, which is suggestive. \nThe arguments against this hypothesis are that \n- · The emitting source is close to the white dwarf, which is expected to be rare. We know giant planets orbiting main sequence stars that are close enough to transit are rare (van Sluijs & Van Eylen 2018; Robert et al. 2024). Given our small sample size, one detection does not necessarily indicate a common phenomenon, but finding something so unusual so quickly is surprising. Nevertheless, the short-period giant planet WD 1856 b exists (Vanderburg et al. 2020), and this planet candidate could be a younger analog of that system's planet. Exoplanet astronomy has a history of discovering rare objects sooner than their actual occurrence rates would suggest (Gansicke et al. 2006; Charbonneau et al. 2009; Vanderburg et al. 2015; Gillon et al. 2016).\n- · Modeling the star's evolution (see Appendix A) suggests that the planet is orbiting in a forbidden location (Nordhaus & Spiegel 2013), making it difficult to explain its existence. One explanation that has previously been invoked to explain short and intermediate period orbits around white dwarfs is common envelope evolution, in particular, scenarios involving additional energy sources beyond the planet's gravitational energy (Lagos et al. 2021; Chamandy et al. 2021; Merlov et al. 2021) or scenarios where the planet is engulfed at the end of the AGB phase, and mass loss facilitates the ejection of the envelope (Yamaguchi et al. 2024). However, we argue that a common envelope scenario is unlikely to be a good explanation for an intermediate-separation planet around WD 0310-688. It is already energetically difficult to make common envelope evolution work for WD 1856b. For this planet, which is at a much wider orbital separation then WD 1856 b, this explanation would be even more challenging because amount of orbital energy available to eject the envelope is significantly less. Even greater amounts of additional energy would be required to prevent \nthe planet from crashing into the core (Lagos et al. 2021; Chamandy et al. 2021; Merlov et al. 2021). \nAn alternative explanation for the presence of an intermediate period planet around WD 0310-688 is that the planet is currently undergoing tidal migration. However, in order for tidal migration to take place, the planet's eccentricity must be excited to very high values by additional massive objects in the system. There are no binary companions to this star that could drive tidal migration, however we observe numerous planet candidates at or beyond 80 AU distant in the MEOW images of WD 0310-688. Despite a high contamination rate from background AGN in our survey, if one of these sources is indeed a planet bound to the white dwarf, it could plausibly serve as a Kozai perturber, where the distant companion induces oscillations in the inclination and eccentricity of a planet's orbit, potentially leading to high eccentricity and subsequent tidal migration (Mu˜noz & Petrovich 2020; Stephan et al. 2021). Planetplanet scattering is another plausible mechanism that could be responsible for the planet's current position (Debes et al. 2012; Veras et al. 2016; Maldonado et al. 2021, 2022). Although close-in companions ( < 80 AU) are absent, it is also plausible that an object may have been ejected from this system after interacting with our planet candidate.", '4.2. Debris Disk': 'Another possibility we consider is a disk. Converting an IR excess into precise debris disk parameters is challenging, but we can approach this in a couple of ways. First, we can assume the disk is optically thin and/or puffed up, similar to exozodiacal light. Under this assumption, and using the temperature of 248 K from our blackbody fit for the dust, the semimajor axis would be 0.11 AU, assuming an albedo of α = 0 . 06, similar to that of interplanetary dust in our solar system (Yang & Ishiguro 2015). The effective emitting area of the disk would be approximately the same as the surface area of Jupiter. \nAlternatively, we can assume an optically thick, thin, flat disk (Jura 2003). Using MCMC to fit the IR excess with the Jura disk model, we obtained the following parameters. In this case, the disk is a very skinny ring with inner radius of 171 R WD , outer radius 185 R WD , and is inclined nearly edge-on (see Figure 7). \nThe arguments in favor of this hypothesis are that: \n- · Dust disks are common around white dwarfs, and although we have not previously observed one at such a cold temperature (Farihi et al. 2010), this \nis likely due to the lack of sensitivity before the advent of JWST. It is often more likely to encounter an unusual manifestation of a common phenomenon, like a debris disk, than a typical presentation of an uncommon phenomenon, such as a planet in an unlikely orbital configuration forbidden zone. This aligns with the principle of considering the most probable explanations before exploring less likely alternatives. \n- · The white dwarf is polluted as discussed in section 3.1.2, and a debris disk is the most probable source of the pollution (although we note that pollution does not preclude the presence of a close-in planets Gansicke et al. 2019). Infrared observations indicate white dwarfs with observed debris disks are heavily polluted, typically with total accretion rate > 3 × 10 8 g/s (Farihi et al. 2009), much higher than the accretion rate we measure for WD 0310688.', 'The arguments against this hypothesis': "- · In order to explain the IR emission with a disk model, we must finely tune the disk's geometry to give a surface area similar to Jupiter. In particular, best-fit parameters for an optically thick disk seem a bit contrived, with an extremely thin ring (only 7% of the disk's radial extent) and a nearly edge-on inclination. However, other disks have been found with similarly thin-ring geometries (Ballering et al. 2022), and there is no reason that such disks could not be precisely aligned with our line of sight, so such a seemingly unusual geometry cannot be ruled out. \nFigure 7. Posterior of the disk inclination from the Jura disk model fit (upper left) and a schematic of the disk viewed at 88.21 · (bottom) and face-on (upper right) for comparison. \n<!-- image --> \n- · The location of the dust in the optically thin case is well outside the Roche Limit, which would be contrary to our normal picture of white dwarf disks and pollution (although see the discovery of transiting systems outside the Roche radius for the rocky material that typically makes up dusty debris disks Vanderbosch et al. 2020; Farihi et al. 2022).", '4.3. Implications of the Discovery': 'Regardless of the underlying cause, this IR excess is expanding our understanding of white dwarf planetary systems.', '4.3.1. If it is a Planet': 'Confirming this source as an exoplanet would be particularly exciting for several reasons. It represents the closest planet candidate around a white dwarf discovered to date, at a distance of 10.4 parsecs. It is the first planet candidate identified using the infrared excess detection technique. It is the first white dwarf planet candidate discovered at an intermediate separation (0.12 AU). If confirmed, it would be one of the coldest worlds for which direct spectral atmospheric characterization is possible. \nFurthermore, the host is a remnant of an A or late B star, and the planet would offer critical demographic constraints on this population, which is challenging to study by other means. With few exceptions, planets around A- and B-type main sequence stars are generally accessible only through high-contrast imaging. Searches for such planets have yielded several detections (e.g. Lagrange et al. 2010; Janson et al. 2021a), but are typically limited to planets at relatively large semi-major axes (tens or hundreds of au; Nielsen et al. 2019; Vigan et al. 2021; Janson et al. 2021b). Past the main sequence, in the red giant phase, the stars have expanded and cooled sufficiently to be better suitable for RV studies. Such studies have also yielded several planet candidates in the relevant stellar mass range (e.g. Johnson et al. 2007; Reffert et al. 2015), though in this case the detection range is limited to small separations in the range of typically a few au, and the actual estimation of stellar mass is subject to large uncertainties (e.g. Lloyd 2011; Johnson et al. 2013). As we have seen above, white dwarf imaging with JWST can cover planets over the whole separation space from the Roche limit (via infrared excess) up to hundreds of au (via resolved imaging) simultaneously, providing a much more complete demographic overview for the stellar remnant phase of massive stars. Comparisons with the main sequence and red giant demographics can also provide the first constraints on how \nplanetary systems are affected through the late stages of stellar evolution.', '4.3.2. If it is a disk:': 'Only three WD disks have previous Spitzer/MIPS measurements: G29-38 (Reach et al. 2005), GD 56 and GD 133 (Jura et al. 2007). Few distant and cold disk candidates have been identified around hot, bright white dwarfs with Spitzer at 24 µm . A notable instance is the central star of the Helix Nebula (Su et al. 2007). However, their interpretation as cold dust disks is complicated by dusty outflows associated with their immediate progenitors and binarity (Chu et al. 2011; Clayton et al. 2014). If this source is actually a disk rather than a planet, it would be perhaps unusual compared with other known white dwarf disks and could significantly expand our understanding of how planetary debris is accreted onto polluted white dwarf stars (Chen et al. 2020). \nIf confirmed as a debris disk, this system, alongside PG 1225-079 (Farihi et al. 2010), would represent one of the coldest debris disks ever detected around a white dwarf. These disks may signify distinct evolutionary stages-either at the nascent phase, where the disk is just beginning to form, or nearing the terminal phase, where the disk is predominantly dissipated. Bonsor et al. (2017) predicted that the JWST would uncover numerous small debris disks, suggesting that such findings may be more common than previously anticipated. \nFinally, considering our estimated accretion rate, if this source is indeed a debris disk, it would represent the white dwarf system with the lowest known accretion rate associated with a detectable disk.', '4.4. How to distinguish the two scenarios': "The existing data on this system does not provide enough information to definitely differentiate between the planet and disk hypotheses. We surmise that the easiest way to differentiate between the two will be to obtain a MIRI MRS spectrum of the blended SED. Even in the case of a cloudy planet, several hours of MRS time enables the detection of atmospheric features in an exoplanet's atmosphere that would provide concrete evidence of a planet. Conversely, a lack of detection of planetary atmospheric features consistent with an exoplanet atmosphere would provide strong evidence that the source of infrared excess is instead a cold dust disk. If it is indeed a debris disk, very likely it will have 10 micron silicate feature, which has been seen in white dwarfs disks with mid-infrared data (Jura et al. 2009; Swan et al. 2024).", '5. CONCLUSIONS': "In this paper, we described the detection of infrared excess around the white dwarf WD 0310-688 and found that: \n- · The IR excess is best fit by a blackbody with a temperature of 248 +84 -61 K and an emitting area of 1.03 +0 . 67 -0 . 35 R Jup .\n- · The source of the IR excess is constrained to be within 0.1-2 AU, corresponding to orbital periods of 14 days to 3.4 years.\n- · If the emitting source is an exoplanet, its mass is constrained to 3 . 0 +5 . 5 -1 . 9 M Jup , though it could be as low as 0.2 M Jup if the planet's heat comes from irradiation. \nThe detection of a source, whether it is a planet or a cold dust disk, at this separation is unprecedented. It is one of only a handful of known white dwarf planets or planet candidates. It represents the nearest planet candidate around a white dwarf discovered to date, at a distance of 10.4 parsecs. It is the first planet candidate ever identified using the infrared excess detection technique. This technique is well suited for the detection of white dwarf exoplanets, and may reveal many more candidates during JWST's lifetime. \nIt is the first white dwarf planet candidate discovered at an intermediate separation (0.1-2 AU). This is particularly significant because planets at this separation are expected to have been destroyed during the star's red giant phase. Therefore, if confirmed, the planet must have migrated to its current location after the star evolved into a white dwarf. Confirming and further study of this planet will be crucial for understanding post-main sequence planetary evolution and the fate of planets as their host stars die. \nWe determine that follow-up observations with JWST MRS will enable us to confirm the source of the IR excess as a planet and rule out the disk hypotheses. If confirmed, it will be one of the coldest worlds for which direct spectral atmospheric characterization is possible. Subsequent spectroscopic observations could allow for comparative studies with the coldest directly imaged free-floating planetary-mass worlds (such as the mid Ydwarf WISE 0855-0714 (Luhman et al. 2024), which is comparable in mass, age, and temperature).", 'ACKNOWLEDGEMENT': 'We would like to thank Misty Cracraft, Karl Gordon and the JWST Helpdesk for support with MIRI calibration files and absolute flux calibrations. This research has made use of the SIMBAD database, operated at \nCDS, Strasbourg, France (Wenger et al. 2000). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper leverages data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. This work is based [in part] on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #4403. \nFacilities: JWST, HST, Gaia, Hipparcos, Magellan, Spitzer, WISE . All the JWST data used in this paper can be found in MAST: 10.17909/cjkx-kp07. \nSoftware: astro.py (Astropy Collaboration et al. 2013, 2018, 2022), numpy.py (van der Walt et al. 2011), wdwarfdate (Kiman et al. 2022), POSEIDON (MacDonald & Madhusudhan 2017; MacDonald 2023), MESA version v7503; Dotter 2016, Choi et al. 2016, (Paxton et al. 2011, 2013, 2015), SAOImage DS9 (Smithsonian Astrophysical Observatory 2000), VIP (Gomez Gonzalez et al. 2017), corner.py (Foreman-Mackey 2016) and ChatGPT was utilized to improve wording at the sentence level and assist with coding inquires; Last accessed in August 2024.', "A. THE IMPLICATIONS OF STELLAR EVOLUTION ON A PLANET'S ORBIT": 'Figure 8. Evolution of the stellar radius as a function of age for a 1.9 M ⊙ star, simulated using MESA Isochrones & Stellar Tracks ( MESA version v7503). The shaded pink region represents the range of possible orbital separations of the planetary companion (0.1-2 AU). The dotted lines mark the bounds of this range. The maximum radius reached by the star at the tip of the red giant branch (2.1 AU) exceeds even the most generous estimate of the orbital separation, indicating that the planetary companion likely falls into the forbidden zone. \n<!-- image --> \nWe can draw insights as to whether the planetary companion falls into the forbidden zone by investigating the current orbital separation in comparison to the size of the host when it was at the tip of the red giant branch (RGB). To this end, we simulated the evolution of a 1.9 M ⊙ star using MESA Isochrones & Stellar Tracks 9 ( MESA version v7503; Dotter \n2016, Choi et al. 2016, Paxton et al. 2011, 2013, 2015). We investigated a range of initial metallicities, determining that the orbital separation of the planet was within the maximum RGB radius in all cases. In Figure 8, we illustrate the change in stellar size as a function of age. The estimated bounds of the orbital separation of the companion, 0.1-2 AU, are shown as dotted lines and the intermediary range between these extremes is highlighted in pink. Note that even the most generous orbital separation estimate is less than the maximum RGB size of the host (2.1 AU).', 'REFERENCES': "Bonsor, A., Farihi, J., Wyatt, M. C., & van Lieshout, R. 2017, MNRAS, 468, 154, doi: 10.1093/mnras/stx425 \nBouchet, P., Garc'ıa-Mar'ın, M., Lagage, P. O., et al. 2015, \nPASP, 127, 612, doi: 10.1086/682254 \nBrandt, T. D. 2018, ApJS, 239, 31, \ndoi: 10.3847/1538-4365/aaec06 \n- -. 2021, ApJS, 254, 42, doi: 10.3847/1538-4365/abf93c \nChamandy, L., Blackman, E. G., Nordhaus, J., & Wilson, \nE. 2021, MNRAS, 502, L110, \ndoi: 10.1093/mnrasl/slab017 \nCharbonneau, D., Berta, Z. K., Irwin, J., et al. 2009, \nNature, 462, 891, doi: 10.1038/nature08679 \nChen, C. H., Su, K. Y. L., & Xu, S. 2020, Nature Astronomy, 4, 328, doi: 10.1038/s41550-020-1067-6 \n- Cheng, S., Schlaufman, K., & Caiazzo, I. 2024, A Giant Planet Candidate Orbiting a Young, Massive White Dwarf, JWST Proposal. Cycle 3, ID. #6410\n- Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102 \nChu, Y.-H., Su, K. Y. L., Bilikova, J., et al. 2011, AJ, 142, 75, doi: 10.1088/0004-6256/142/3/75 \nClayton, G. C., De Marco, O., Nordhaus, J., et al. 2014, AJ, 147, 142, doi: 10.1088/0004-6256/147/6/142 \nCrane, J. D., Shectman, S. A., & Butler, R. P. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Ground-based and Airborne Instrumentation for Astronomy, ed. I. S. McLean & M. Iye, 626931, doi: 10.1117/12.672339 \nCrane, J. D., Shectman, S. A., Butler, R. P., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 773553, doi: 10.1117/12.857792 \nCrane, J. D., Shectman, S. A., Butler, R. P., Thompson, I. B., & Burley, G. S. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Ground-based and Airborne Instrumentation for Astronomy II, ed. I. S. McLean & M. M. Casali, 701479, doi: 10.1117/12.789637 \nCunningham, T., Wheatley, P. J., Tremblay, P.-E., et al. 2022, Nature, 602, 219, doi: 10.1038/s41586-021-04300-w"}

2024arXiv240907411S

The Parker Solar Probe PSP spacecraft has transited the innermost regions of the zodiacal cloud and detects impacts to the spacecraft body via its electric field instrument. Multiple dust populations have been proposed to explain the PSP dust impact rates. PSPs unique orbit allows us to identify a region where the impact rates are likely dominated by alphameteoroids small zodiacal grains on approximately circular bound orbits. From the distribution of voltage signals generated by dust impacts to PSP in this region we find the cumulative mass index for grains with radii of sim0.61.4 mum masses of 3times1015 to 3times1014 kg to be alpha 1.1 pm 0.3 from 0.10.25 Rodot. alpha increases toward the Sun with even smaller fragments generated closer to the Sun. The derived size distribution is steeper than previously estimated and in contrast to expectations we find most of the dust mass resides in the smallest fragments and not in large grains inside 0.15 au. As the innermost regions of the zodiacal cloud are likely collisionally evolved these results place new constraints how the solar systems zodiacal cloud and by extension astrophysical debris disks are partitioned in mass.

2024-09-01T00:00:00Z

['2024arXiv240907411S', '10.48550/arXiv.2409.07411', 'arXiv:2409.07411']

['Astrophysics - Earth and Planetary Astrophysics', 'Physics - Space Physics']

Size distribution of small grains in the inner zodiacal cloud

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07411.pdf

{'Size distribution of small grains in the inner zodiacal cloud': "J. R. Szalay , 1 P. Pokorn'y , 2, 3 and D. M. Malaspina 4, 5 \n1 Department of Astrophysical Sciences, Princeton University, 171 Broadmead St., Princeton, NJ 08540, USA 2 Astrophysics Science Divison, NASA Goddard Spaceflight Center, Greenbelt, MD, 20771, USA 3 Department of Physics, The Catholic University of America, Washington, DC, 20064, USA 4 Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, Boulder, CO, USA 5 Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO, USA", 'ABSTRACT': "The Parker Solar Probe (PSP) spacecraft has transited the inner-most regions of the zodiacal cloud and detects impacts to the spacecraft body via its electric field instrument. Multiple dust populations have been proposed to explain the PSP dust impact rates. PSP's unique orbit allows us to identify a region where the impact rates are likely dominated by α -meteoroids, small zodiacal grains on approximately circular, bound orbits. From the distribution of voltage signals generated by dust impacts to PSP in this region, we find the cumulative mass index for grains with radii of ∼ 0.6-1.4 µ m (masses of 3 × 10 -15 to 3 × 10 -14 kg) to be α = 1 . 1 ± 0 . 3 from 0.1-0.25 R ⊙ . α increases toward the Sun, with even smaller fragments generated closer to the Sun. The derived size distribution is steeper than previously estimated, and in contrast to expectations we find most of the dust mass resides in the smallest fragments and not in large grains inside 0.15 au. As the inner-most regions of the zodiacal cloud are likely collisionally evolved, these results place new constraints how the solar system's zodiacal cloud and by extension astrophysical debris disks are partitioned in mass.", '1. INTRODUCTION': "Our solar system's zodiacal cloud comprises multiple dust populations. Of the interplanetary populations, these are typically categorized as α -meteoroids 1 : gravitationally bound grains on elliptic trajectories and β -meteoroids: grains so small the outward force of solar radiation overcomes the inward pull of the Sun's gravity such that they are pushed out of the solar system on hyperbolic trajectories (Zook & Berg 1975; Wehry & Mann 1999). These dust populations are continuously evolving and exchanging material due to dynamical evolution from gravity, solar radiation, and electromagnetic forces, as well as disruption from collisions, sublimation, \nCorresponding author: Jamey Szalay \njszalay@princeton.edu \n- 1 There are two definitions of α -meteoroids in the literature (Sommer 2023): 1) a dynamical subset of gravitationally bound grains with very large eccentricities and 2) all grains gravitationally bound to the Sun. We use the latter, broad definition for α -meteoroids. \nand rotation (Mann et al. 2004). This cloud also provides an important analog to study exozodiacal systems. \nA key quantity that directly affects β -meteoroid production and zodiacal erosion is how collisional fragment particles are distributed in mass, which has not been well constrained near the Sun. Collisional products are often assumed to follow a power-law mass distribution (e.g. Gault & Wedekind 1969; Dohnanyi 1969), such that the immediate fragmented differential mass distribution is f ( m ) ∝ m -( η +1) . These collisional products can then undergo mass-dependent transport and erosion, such that the instantaneous differential mass distribution at any location in the heliosphere has the form \nf ( m ) ∝ m -( α +1) (1) \nwhere α is the cumulative mass index. In this study, we assume the cumulative fragmentation index η and the cumulative mass index α are equal, such η = α . Note, α as the variable used to represent mass index does not have any relation to the term ' α -meteoroid', where we use ' α ' in both cases for historical consistency. Changes in α directly affect how much material the zodiacal cloud \nretains or sheds through its production of β -meteoroids. Hence, an accurate determination of α reveals the effect of collisional fragmentation on zodiacal evolution. \nThere have been a number of both observational and ground-based experimental constraints on α . Groundbased collision experiments found values of α = 0 . 5 -1 . 0 for collisionally generated material (e.g. Krivov et al. 2000, and refs. therein). The size distribution of lunar ejecta in the radius range of 0 . 3 -10 µ m was consistent with α = 0 . 9 (Hor'anyi et al. 2015), while initial Solar Orbiter (SolO) results (Zaslavsky et al. 2021) found values of α = 0 . 3 -0 . 4 for β -meteoroids with radii of ∼ 100-200 µ m. The zodiacal cloud in the outer solar system beyond Saturn's orbit as measured by New Horizons' Student Dust Counter observations (Piquette et al. 2019) was found to be consistent with α = 0 . 5 -1 . 2 by comparing with a dynamical model of the outer zodiacal grain distribution (Poppe 2016). More recent collisional modeling in the inner solar system concluded α = 1 . 11 ± 0 . 03 for grains with radii > 100 µ m (Pokorn'y et al. 2024). From modeling the near-Sun zodiacal cloud, α = 0 . 5 -0 . 6 was expected at a heliocentric distance of 0.1 au for micron-sized grains (Ishimoto & Mann 1998). \nWhile unequipped with a dedicated dust detector, Parker Solar Probe (PSP) spacecraft (Fox et al. 2016) registers dust impacts primarily via voltage measurements with the FIELDS instrument (Bale et al. 2016). There has been an extensive history of measuring dust impacts from spacecraft equipped with electric field instruments, for example: Voyager 2 (Gurnett et al. 1983), Vega (Laakso et al. 1989), DS-1 (Tsurutani et al. 2003; Tsurutani et al. 2004), Wind (Malaspina et al. 2014; Kellogg et al. 2016; Malaspina et al. 2016), MAVEN (Andersson et al. 2015), Cassini (Ye et al. 2014), STEREO (Zaslavsky et al. 2012; Malaspina et al. 2015) and MMS (Vaverka et al. 2018, 2019). \nDuring its first three orbits, impact rates observed by PSP were consistent with fluxes of dominantly highspeed, submicron-sized β -meteoroids leaving the solar system on escaping orbits (Szalay et al. 2020; Page et al. 2020; Malaspina et al. 2020). Grains smaller than β -meteoroids, often termed nanograins, are less susceptible to radiation pressure and their dynamics are dominated by electromagnetic forces; these nanograins were not found to be dominantly responsible for the impact rates during PSP's 2nd orbit (Mann & Czechowski 2020). A two-component model consisting of both α -meteoroids and β -meteoroids was subsequently used to explain the first six orbits of impact rate data (Szalay et al. 2021), which was able to well-reproduce the majority of the impact rates observed. Additionally, a PSP dust database has been published which is continually \nupdated as more data is taken throughout the PSP mission (Malaspina et al. 2023). \nIn this study, we will calculate the cumulative impact amplitude distributions for all PSP-derived dust impact rates (Malaspina et al. 2023). Comparing to the existing two-component model, we identify a region in a subset of PSP orbits dominated by α -meteoroids in the heliocentric distance range of 0 . 1 -0 . 25 au and fit the impact amplitude distributions to a power-law. This will allow us to quantitatively constrain the size distribution of small, bound α -meteoroids in the near-Sun environment.", '2. PSP DUST IMPACT RATES FROM ORBITS 1-17': "Figure 1 shows the dust impact rates (Malaspina et al. 2023) to PSP as a function of position and time from orbits 1-17. There are six orbital groupings for the spacecraft shown: 1-3, 4-5, 6-7, 8-9, 10-16, & 17. There are a few notable features in the impact rates: all orbits exhibit a peak pre-perihelion, with lower impact rates observed at perihelion, followed often by a post-perihelion peak that is lower in total rate than the pre-perihelion peak. A two-component model consisting of prograde α -meteoroids and β -meteoroids was used to successfully model PSP impact rates from orbits 1-6 (Szalay et al. 2021). Note, this model incorporates the updated density profile of the zodiacal cloud derived from PSP imaging (Stenborg et al. 2020). Figure 2a shows an example of this two-component model fit to the impact rates from orbit 13 as a representative example. The model predicts a symmetric pre- and post-perihelion impact rate peak from the α -meteoroid population (green line) and an asymmetric profile for the β -meteoroid population. Due to the spacecraft velocity addition with the solely outbound, prograde β -meteoroids, the impact rate from β -meteoroids is expected to exhibit a strong preperihelion peak and significantly depleted impact rates just after perihelion due to PSP 'catching up' to the β -meteoroids and having much smaller relative impact speeds and corresponding impact rates. \nThe two-component model successfully fits the majority of impact rate structure observed by PSP, with the exception of the post-perihelion peak magnitude for orbits 4-5. During these orbits, which exhibited a postperihelion impact rate peak that could not be explained by the two-component model, it was proposed that a third component of collisionally produced β -meteoroids from the Geminids stream, termed a β -stream, could be responsible for the additional enhancement (Szalay et al. 2021). It was shown that PSP would have intersected the trajectories of β -meteoroids produced during collisions between the Geminids and zodiacal cloud in orbits \n2 \n1 \n0 \nFigure 1. Impact rates for PSP's first 17 orbits. (a) Top-down view in Ecliptic J2000 coordinates, where rates within each orbital group are averaged. The inner-most location of the Helios dust measurements (Grun et al. 1980) outside 0.3 au is also indicated. (b, c) Time-series of each orbit, where the colorbar in (b) is shared with that of (a), as a function days from perihelion. \n<!-- image --> \n4-5. Subsequent directional analysis was also supportive of a β -stream hypothesis (Pusack et al. 2021). Modeling efforts on the overall Geminids stream found the bound α -meteoroids within the Geminids could not be responsible for the additional rate enhancement (Cukier & Szalay 2023), indirectly supporting the β -stream hypothesis. However, without dedicated dust instrumentation it is difficult to definitively conclude the origins of the post-perihelion enhancement during orbits 4-5. \nAside from orbits 4-5, the model is able to wellreproduce the magnitude and location of the postperihelion enhancement. This region, just after perihelion, is the primary focus of this study as it is expected to be dominated by impacts from α -meteoroids, with minimal contributions from β -meteoroids. Figures 2b & c show the impact fraction of α -meteoroids (green) and β -meteoroids (purple) throughout orbit 13. As shown here, the region just after perihelion to ∼ 5 days postperihelion has less than a few percent contribution from β -meteoroids. Therefore, due to PSP's highly eccentric orbit, this region provides a unique opportunity to isolate the α -meteoroid population and analyze its properties. We focus on this region throughout the remainder of this study and derive the size distribution power-law index for α -meteoroids here. Additionally, we focus on orbits 10-16 as they provide seven nearly identical trajectories to sample this region and provide a statistical dataset with which to study α -meteoroids in the nearSun zodiacal cloud.", '3. DETERMINING THE SIZE DISTRIBUTION': "Here, we outline a method to derive the cumulative mass index α from the distribution of impact voltages to PSP. Dust impacts are measured by PSP as a potential change ∆ V ∝ Q imp (Collette et al. 2014; Shen et al. 2023). In order to relate the impact charge distribution to the impact mass distribution, we assume the differential charge distribution has the form f ( Q ) ∝ Q -( α Q +1) , where α Q is the cumulative charge index. In the Appendix, we derive a relation between α Q and α that allows us to estimate the mass index from the distribution of measured impact voltages as similarly done for radar observations of meteors with a speed-dependent radar amplitude (Pokorn'y & Brown 2016). \nTo ensure PSP is being impacted by a single population, we focus on times where the spacecraft resides between 0.1 to 0.25 au (22-54 R ⊙ ) post-perihelion, as the two-component model predicts α -meteoroids contribute > 95% of the total impacts throughout this region for orbits 10-16. . We note interstellar grains could also be impacting PSP in this region (Sterken et al. 2012; Strub et al. 2019), where due to PSP's orbit geometry these grains would be most readily detected during the post-perihelion arc far from the Sun. However, given the ability of the two-component model to reasonably reproduce the location and amplitude of the second peak during orbits 10-16 and that much of the interstellar dust grains are dynamically prevented from transiting very near to the Sun due to radiation pressure, we do not expect this source to appreciably contribute to PSP's impact rates in this region. \nOrbit 13 \nFigure 3a shows the ensemble of cumulative impactamplitude distributions measured by PSP for each 8hour segment within the selected portions of orbits 1016. Each voltage range is displayed at the center of a 50 mV bin, such that the first voltage point in the shaded region corresponds to 0-50 mV, the second point to 50-100 mV, and so forth. Due to the dust impact identification method, impacts with amplitudes below 50 mV (the first point in each curve) are more challeng- \n<!-- image --> \n40 \n40 \nFigure 2. (a) Example impact rate from orbit 13, representative of orbits 10-16, along with the modeled abundances of α -meteoroids and β -meteoroids from our two-component fit (Szalay et al. 2021). (b) Fractional contribution of α -meteoroids and β -meteoroids to total modeled impact rate. (c) Same information as panel (b) shown in the ecliptic J2000 x-y plane with example α -meteoroid & β -meteoroid trajectories. \nR (R \n<!-- image --> \nS \n) \nFigure 3. (a) Cumulative impact rate distribution as a function of impact amplitude for orbits 10-16, with color indicating the power-law fit index above 50 mV for each 8-hr impact rate interval considered. b) Ratio of cumulative impact rates with their corresponding power-law fits.(c) Cumulative charge indices α Q as a function of heliocentric distance, with average values and standard deviation each 5 R ⊙ and the histogram on the right shows the distribution of α Q values. (d) Impact rates as a function of heliocentric distance. Colors for (c) & (d) show orbit number \ning to detect (Malaspina et al. 2023) and we are likely not identifying all impacts below this amplitude, hence the break in the slope of the distribution at this amplitude as shown in Figure 3a. Additionally, this measurement technique saturates above 950 mV , therefore we can probe the size-distribution index within a single decade in mass. We fit the cumulative amplitude distribution above 50 mV to a power-law, where the colors of the various curves in Figure 3a shows the cumulative \ncharge index α Q . Figure 3b shows these same cumulative impact rate curves divided by their corresponding power-law fits. As shown here, the cumulative voltage distributions are highly consistent with power-laws. In Figure 3c, we show the fitted charge indices as a function of heliocentric distance, along with average values in 5 R ⊙ bins. Figure 3d shows the total impact rates for each orbit, where Figures 3c & d are colored by orbit number. \nWe previously estimated the minimum detectable bound α -meteoroid grain radius to be approximately 0.6 µ m for a threshold of 50 mV (Szalay et al. 2021). With the additional uncertainty discussed in the Appendix, this lower limit could be down to ∼ 0.5 µ m, however, grains as small as 0.5 µ m in radius are expected to have β ≈ 0 . 5 and would be unbound. Therefore, we retain our lower limit estimate to be 0.6 µ m As the PSP impact voltages used here have a dynamic range of 50 -950 mV, we are able to assess the mass distribution within a decade in mass from the minimum detectable size such that the derived indices are relevant for grains with radii of approximately 0 . 6 -1 . 4 µ m. \nAs derived in the Appendix, α = α Q (1 + δa ) relates the cumulative charge index α Q to the cumulative mass index α . Figure 4a shows δa as a function of heliocentric distance, where this term is most enhanced for closer heliocentric distances. We use these δa ranges to correct the derived cumulative charge indices α Q from Figure 3c. Considering both the upper and lower limits of δa , Figures 4b & 4c show the range of corrected mass indices. For each 5 R ⊙ heliocentric bin, we determine the full range of possible α values from these two panels and display it in panel (d) with the average value of these ranges shown with black circles. The grey circles overlaid on these show the uncorrected ranges of α Q from Figure 3c. While this mass-dependent correction is important to consider, we find it does not have a substantial effect on the dependence of α on α Q . \nThe mass indices shown in Figure 3d, represent our best estimate of α correcting for mass-dependent biases discussed. Throughout this region, from the full uncertainty ranges in Figure 3d, we find α = 1 . 1 ± 0 . 3. Additionally, we find the mass index is higher near the Sun, exhibiting an enhancement in α inside ∼ 0.15 au systematically above 1.0.", '4. IMPLICATIONS FOR COLLISIONAL PRODUCTION': "Having found α = 1 . 1 ± 0 . 3 in the 0.1-0.25 au range, we now focus on the implications of these mass indices in the near-Sun zodiacal dust cloud. For a collisionally produced population, the mass index determines the \nmaximum size grain that can be produced as a fragment. We investigate how mass is distributed, including the maximum fragment size, as a function of α . For an upper size cutoff m c for fragments, we normalize the distribution such that the total number of particles with mass greater than or equal to m c is 1. Therefore, the differential distribution function is \nf ( m ) = α m -α c -m -α 1 m -( α +1) (2) \nwhere the number of particles between m a and m b is given by \nN = ∫ m b m a f ( m ) dm = m -α a -m -α b m -α c -m -α 1 (3) \nFor a parent particle of mass m 1 , whose minimum fragment mass is m 0 , its collisional fragments must have a total mass of m 1 , hence, \nm 1 = ∫ m 1 m 0 mf ( m ) dm = α m -α c -m -α 1 ∫ m 1 m 0 m -α dm (4) \nTherefore, the mass of the largest fragment m c is given by \n̸ \nm c = ( ( α -1) m 1 αm 1 -α 0 -m 1 -α 1 ) 1 α if α = 1 , m 1 1+ln( m 1 /m 0 ) if α = 1 (5) \n̸ \nFor a more intuitive interpretation, these relations can be approximated by assuming m 0 ≪ m 1 and expanding to zeroth order in m 0 /m 1 when α = 1, \nm c ≈ m 0 ( α -1 α m 1 m 0 ) 1 α if α > 1 , m 1 (1 -α ) 1 α if α < 1 (6) \nAs shown in Equations 5 and 6, for α > 1 the maximum fragment cutoff mass is most strongly governed by m 0 and weakly depends on the ratio m 1 /m 0 , while for α ≤ 1 this cutoff mass is entirely governed by the original grain size m 1 . For α = 1 it is an intermediate case mostly governed by m 1 and weakly depending on m 1 /m 0 . This also means for steeper distributions with α > 1, most of the material is in small grains near the lower end of the mass range, while shallower distributions with α ≤ 1 have most of the material in large grains toward the upper end of the mass range. Figure 5 shows a visualization of these two regimes. \nFigure 6 provides a visualization of these relations. It shows the maximum fragment size r c corresponding to grains with mass m c as a function of assumed smallest fragment size r 0 from 10 nm to 0.3 µ m corresponding to mass m 0 for different values of α . To determine radius \nda \n1 \n- \nda \n1 \n- \n25 \n30 \n35 \n40 \n45 \n50 R \nR (au) \nFigure 4. (a) Correction factor δa for the mass-dependent portion of the impact speed. (b) & (c) Corrected cumulative mass indices α = α Q (1 + δa ) corresponding to the minimum and maximum δa values from (a). (d) Comprehensive estimate of α accounting for mass-dependent biases, with the uncorrected α Q ranges shown with grey dots. \n<!-- image --> \nR (au) \nFigure 5. Distribution of fragmentation products depending on the cumulative mass index. Fragments with α > 1 have most of the mast distributed into small grains, while for α < 1 the majority of mass is distributed into the few largest fragments. \n<!-- image --> \nFragment of 100 um grain \n) \nm \nu \n( \ns \nu \ni \nd \na \nr \nt \nn \ne \nm \ng \na \nr \nf \nm \nu \nm \ni \nx \na \nM \nFigure 6. Maximum fragment cutoff size as a function of minimum fragment size and cumulative mass index α . The shaded region shows the average cumulative mass indices found to be consistent with the near-Sun environment. \n<!-- image --> \nSmallest fragment radius (nm) \nfor a given mass, we assume grains are 'old cometary' origin according to Wilck & Mann (1996), with grain mass densities in the range of 2.0-3.3 g cc -1 for grain radii of 10 nm to 20 µ m. For α < 1 . 0, the dependence of maximum fragment size on r 0 is weak, as most of the mass is partitioned amongst the largest grains. For α > 1 . 0, most of the mass is partitioned to the smallest grains, therefore the smallest fragment size plays a larger role on determining the maximum fragment size. For the mean value of α = 1 . 1, r c = 21 to 30 µ m for r 0 = 10 nm to 0.3 µ m respectively. While we cannot empirically constrain r 0 in this setup, we investigate a lower limit much smaller than that of β -meteoroids. The derived mass-indices in this study, calculated from the slope of the measured voltage distributions, is insensitive to the choice of r 0 . Hence, we include the discussion of r 0 for \nintuitive context and visualization of the consequences of different values of the mass index. \nTo provide an intuitive understanding of how these mass indices affect the near-Sun dust distribution, we visualize the size distribution of collisionally produced grains with r 0 = 50 nm and r 1 = 100 µ m in Figure 7. This value of r 1 is chosen due to the fact that at 1 au, the majority of mass in the zodiacal cloud was found to be in ∼ 100 µ m radius grains (Grun et al. 1985), hence this size grain is likely to be the parent of much of the zodiacal collisional products. Figure 7 shows the size distribution for α = 0 . 6 -2 . 0 with these assumptions. \nWhile the slopes of the different values of α are shown in the left portion of this figure, the schematic on the right visualizes how mass is distributed into collisional fragments. As previously discussed, most of the mass for α < 1 . 0 is distributed into large grains and therefore collisional products in this regime have a few 'large' collisional products that carry the majority of the fragmented material. However, in the regime for α > 1 . 0, as found for the near-Sun environment, most of the mass gets partitioned into very small grains, while still allowing for a few larger grain fragments. With the exception of the few largest grains, the overwhelming majority of collisional mass is partitioned into the very lowest end near m 0 , which for these size ranges would all become β -meteoroids to be ejected from the solar system. \nTo compare to expectations, Figure 8a shows the cumulative flux of meteoroids F ( m ) with masses > m at 1 au (Grun et al. 1985). The cumulative mass index of this distribution can be calculated assuming for sufficiently small ranges in mass that the cumulative mass distribution F ( m ) ∝ m -α , such that \nα = -m F ( m ) d F ( m ) d m (7) \nand shown in Figure 8b. Figure 8b shows our derived mass indices for meteoroids from 0.1 to 0.25 au of α = 1 . 1 ± 0 . 3 with the approximate mass/size range detectable by PSP denoted by the width of the grey rectangle. The corresponding mass index at 1 au is α = 0 . 35 -0 . 4 (Grun et al. 1985), therefore the mass index very near the Sun is significantly larger than its corresponding 1 au value. We also show the modeled expectation of the zodiacal flux and mass index at 0.1 au (Ishimoto & Mann 1998) with the dashed lines in Figure 8, where within the size range PSP can probe, the expected size indices at 0.1 au were expected to be α = 0 . 5 -0 . 6 \nBoth the previous expected mass indices at 0.1 and 1.0 au were below α = 1, therefore, they were both in the regime in which most of the collisional products \n) \nm \n( \nf \nFigure 7. Size distribution of collisional fragments. (left) Differential mass distributions, with α = 1 . 1 ± 0 . 3 highlighted in black and the largest fragment size marked with the vertical dashed lines. (right) Visual representation of fragments, with the single largest fragment in the center and the size of the largest size fragment denoted below each. The fragments that are 1/2 and 1/4 of the largest fragment size are shown in shades of grey, with the number of circles proportional their size distribution. Red indicates where the majority of fragment mass resides, where the largest grains carry the mass for α < 1 and smallest grains carry the mass for α > 1. \n<!-- image --> \nwere still large grains that would remain bound. This is in contrast to the values found here of α = 1 . 3 ± 0 . 2 inside 0.15 au, where effectively all collisional products immediately become β -meteoroids to be expelled from the solar system. This discrepancy may be due to the particularly intense and high speed impact environment near the Sun. \nA similar technique was applied to analyze the distribution of dust impacts to SolO (Zaslavsky et al. 2021), which found values of α = 0 . 3 -0 . 4. However, the index derived from SolO impacts was likely for unbound β -meteoroids with radii of ∼ 100-200 nm, compared to the index calculated here for bound α -meteoroids of radii ∼ 0.6-1.4 µ m. Additionally, the mass index derived from SolO observations was quoted as a lower bound that may be sensitive to the distribution of β -meteoroid impact speeds which could have an appreciate spread in speed at SolO's location (Zaslavsky et al. 2021). Crosscomparing PSP and SolO impactor distributions for similar populations, particularly if β -meteoroids could be isolated in both, would provide an exciting opportunity to evaluate the evolution of the size distribution as a function of heliocentric distances. \nThe population of β -meteoroids was specifically excluded from the calculation of mass indices in this study to focus solely on α -meteoroids which should have a smaller spread in impact speed. If the power-law index found here holds down to β -meteoroids, future modeling efforts to understand how a size distribution of β -meteoroids produced very near the Sun could be com- \nd to all existing and future measurements of β -meteoroids from PSP, SolO, STEREO, WIND, and/or any other spacecraft with the capability to detect β -meteoroids throughout the inner heliosphere (Juh'asz & Hor'anyi 2013; O'Brien et al. 2018; Poppe & Lee 2020, 2022).", '5. DISCUSSION AND CONCLUSIONS': "Leveraging PSP's eccentric orbit, we identify a region near the Sun from 0.1-0.25 au where PSP is effectively only observing bound α -meteoroids due to the very low relative impact speeds and corresponding fluxes from β -meteoroids in this region. From these observations, we derive the cumulative power-law mass index to be α = 1 . 1 ± 0 . 3 for grains with radii in the approximate range of ∼ 0.6-1.4 µ m. These results quantitatively demonstrate the meteoroid size distribution near the Sun differs substantially from the distribution at 1 au. In contrast to expectations, we find most of the collisional fragments produced in the near-Sun environment inside 0.15 au are overwhelmingly partitioned into submicron β -meteoroids, which are effectively 'blown out' of the solar system via the Sun's radiation pressure, or even smaller grains that could become electromagnetically coupled to the solar magnetic field (Czechowski & Mann 2012). The mass index also increases for decreasing heliocentric distance, suggesting even smaller fragments are produced closer to the Sun. The discrepancy between the lower expected mass index and higher value found here could be due to the intense fragmentation process occurring in the very near-Sun environment, \n) \n1 \n- \ns \n2 \n- \nm \n( \nx \nu \nl \nF", 'Radius ( μ m)': "Figure 8. Comparison with size distribution at 1 au (Grun et al. 1985) derived from a large number of observations, as well as the modeled distribution at 0.1 au (Ishimoto & Mann 1998). The grey boxes shows the values of α found here for grains with radii in the approximate range of ∼ 0.6-1.4 µ m detectable by PSP. \n<!-- image --> \nMass (kg) \nwhere impact speeds between grains are likely the highest in the solar system and may provide significantly different fragmentation products compared to lower-speed impacts farther from the Sun. The canonical mass flux at 1 au from Grun et al. (1985) is often scaled by r -0 . 5 to estimate the dust flux at different heliocentric distances; these findings demonstrate that the 1 au fluxes cannot be simply scaled to estimate the near-Sun dust distribution. \nFor this analysis, we assume collisions are driving the size distribution near the Sun. Furthermore, to interpret the mass indices derived here we assume collisional fragmentation produces only dust grains as fragmentation products and consider the production of impact vapor/plasma negligible compared to the mass of the fragments. While modeling efforts have been investigated on grain destruction and the role of vaporization (e.g. Borkowski & Dwek 1995), we elected not to include more complex origin mechanisms in order to interpret the derived mass-indices with minimal assumptions, which future analyses could explore. \nIn addition to collisions, sublimation is certainly also operating on grains that transit very near the Sun, where grains are expected to strongly sublimate within ∼ 10 R ⊙ (Mann et al. 2004). However, the location of peak production of β -meteoroids from a collision \nmodel was found to be consistent with the derived β -meteoroid fluxes to PSP, such that collisions were inferred to be the driving phenomenon for dust grain populations observed by PSP (Szalay et al. 2021). Additionally, a comprehensive dynamical meteoroid model which includes fragmentation and the production of β -meteoroids was found to well-match the PSP observations of β -meteoroid fluxes in the inner heliosphere (Pokorn'y et al. 2024). \nAdditionally, we assume that transport effects have not significantly modified the collisionally produced size distribution from their sources such that the collisional fragmentation index η = α . We anticipate most of the collisions occur below 0.1 au (Szalay et al. 2021), such that the small α -meteoroids we measure outside this distance are likely to have been produced at similar radial distances and may not have been significantly evolved from their original source mass distributions. Future efforts could investigate the effects of transport, specifically how the mass-dependent Poynting-Robertson drag would further modify collisional mass distributions. \nRotational disruption due to radiative torque from sunlight on irregular grains has also been proposed to play an important role for dust evolution within our solar system and in astrophysical systems (Misconi 1993; Silsbee & Draine 2016; Hoang et al. 2019). While grains \nreleased from comets have been found to be imparted with rotation due to gas drag ( ˇ Capek 2014), the extent to which grains experience radiative torques has not been observationally constrained in the zodiacal cloud. While we cannot assess the extent to which rotational disruption plays a role in the zodiacal dust evolution, the size distribution found here provides constraints for any model that would predict the mass partitioning of rotationally disrupted grains and their relative contribution to the overall zodiacal dust population. \nParticles originating from short-period comets have been shown to be resilient to collisions with the zodiacal cloud and have higher fragmentation strength than commonly assumed (Pokorn'y et al. 2024). This translates to longer collisional lifetimes, the ability of larger particles to migrate closer to the Sun, and a smaller fraction of interplanetary dust particles lost to collisions. The higher particle collisional strength might explain the discrepancy in the value of α between our work and existing models (Grun et al. 1985; Ishimoto & Mann 1998), as shown in Figure 8. More particles are able to migrate into the very inner heliosphere to then be detected by PSP without being collisionally fragmented into nano-dust particles that are not detectable by PSP instruments. Our results might provide a crucial missing piece of evidence for determining the crushing laws in the zodiacal cloud and for models of collisional evolution of interplanetary dust clouds (Pokorn'y et al. 2024; Poppe 2016; Poppe et al. 2019). \nWe also note that a fundamental dust grain unit size on the order of 10's of µ mwas found from observations of cometary dust from 67/C-G, interpreted to be the remnants of the early accretion processes (Hornung et al. 2023). If there are grains that represent more 'fundamental units', they could be more resilient to fragmentation than their aggregates. The Grun et al. (1985) curve in Figure 8 shows a steep decline in the value of α once the particle radius drops below ∼ 100 µ m, perhaps suggestive that grains of this size are more resilient to fragmentation. While Hornung et al. (2023) report their findings using the in-situ experiment on 69P/CG, the Grun et al. (1985) model is based on spacecraft observations from HEOS2, Pioneers 8 and 9, and from lunar microcraters, all at 1 au. Interplanetary dust particles experience complex space weathering during their lifetime which influences their chemical composition as well as their size-frequency distribution. While the discrepancy is apparent, how grain fragments relate to the original makeup of their parent grains cannot be directly determined from this analysis, yet provides an interesting topic for further investigation. \nThese results can also be applicable to exozodiacal disks. The grain size distribution, as well as the local stellar conditions at exozodiacal systems, controls the total accretion rate of dust onto exoplanets (Arras et al. 2022). The relative partitioning into α -meteoroids and β -meteoroids in exozodiacal disks plays an important role in their collisional evolution and dictates how remote measurements from these disks are interpreted (Krivov et al. 2000). For example, it was found that β -meteoroid fluxes within the debris disk of β Pictoris could be so large they significantly contribute to the fragmentation of α -meteoroids, hence they may modify the size distribution of their parent α -meteoroids and govern the size dominating in overall cross-section that would be remotely detectable (Krivov et al. 2000). β -meteoroids were also found to play an important role in the luminosities of debris disk halos, where an updated size-frequency distribution could aid in the interpretation of optical and near-IR observations of these halos (Thebault et al. 2023). \nSummarizing our results, \n- · The cumulative mass index for grains with radii of ∼ 0.6-1.4 µ m (masses of 3 × 10 -15 to 3 × 10 -14 kg) is found to be α = 1 . 1 ± 0 . 3 from 0.1-0.25 R ⊙\n- · The cumulative mass index increases toward the Sun, with even smaller collisional fragments generated closer to the Sun.\n- · The derived size distribution is steeper than previously estimated, and in contrast to expectations we find most of the dust mass resides in the smallest fragments inside 0.15 au and not in large grains.\n- · These results place new constraints how the solar system's zodiacal cloud and by extension astrophysical debris disks are partitioned in mass. \nIn conclusion, PSP observations provide a unique platform to observe the inner-most regions of our zodiacal dust disk. While PSP does not have a dedicated dust instrument, impacts detected by its electric field instrument can be used to quantitatively constrain the multiple populations in the near-Sun dust environment. These observations are taken in the nearest region to the Sun ever explored in-situ by spacecraft and provide a critical window into the complex evolution of our zodiacal cloud. \nAcknowledgements. We thank the many Parker Solar Probe team members that enabled these observations. We thank B. Draine, M. Horanyi, B. Hensley, & K. Silsbee for helpful discussions on grain fragmentation, \nD. McComas & G. Livadiotis for helpful discussions related to the power-law analyses, and D. P. Morgan & S. P. Childress for graphical guidance. We also thank the two anonymous reviewers for their helpful comments during review. We acknowledge NASA Parker Solar Probe Guest Investigator grant 80NSSC21K1764. P.P. was additionally supported by NASA Solar System \nWorkings award number 80NSSC21K0153, a cooperative agreement 80GSFC21M0002, and NASA's Planetary Science Division Research Program, through ISFM work packages EIMM and Planetary Geodesy at NASA Goddard Space Flight Center. We used the 'managua' & 'bukavu' colormaps (Crameri et al. 2020) for Figures 3 & 4.", 'A. CONSTRAINING THE UNCERTAINTY IN ESTIMATING THE MASS INDEX': "cumulative impact charge index by \nWhen a high-speed dust grain with mass m imp and speed v imp impacts a surface (PSP) it produces an impact charge Q imp ∝ m a imp v b imp , where the exponents a , b and proportionality constant are determined empirically based on the surface properties (Auer 2001). Typical spacecraft surface materials have a mass exponent of 1 (Collette et al. 2014), and we adopt this value here. . For the speed dependence, we assume the impact speed has a functional form of v imp = ¯ v imp m c imp , such that there is a separable component of the impact speed that depends on mass and ¯ v imp has no mass dependence. Combining the empirically determined mass exponent with a value of 1.0 with the additional mass dependence from the impact speed, the mass dependent exponent of the impact charge is a = 1 + δa , where δa = bc \n. The velocity index b has been empirically determined for a variety of spacecraft materials. MLI (multilayer thermal insulation) covers much of the spacecraft and is the most likely target material to be hit by dust and produce impact signals for bound grains post perihelion that would impact the spacecraft body instead of the heat shield (Szalay et al. 2020). We therefore assuming an exponent of b = 4 . 7 for MLI (Collette et al. 2014). \nWith such a relation between Q and m , the cumulative charge index is given by α Q = α/a = α/ (1 + δa ). Therefore, the cumulative mass index is related to the \nα = α Q (1 + δa ) (A1) \nTo calculate the mass-dependent component of the impact speed, δa , we investigate the dependence of the dust impact speed to PSP. For motion in a two-dimensional orbital plane, a body in a bound orbit about the Sun has an orbital velocity such that \nv = √ µ (1 -β ) ( 2 r -1 a ) φ = √ µ (1 -β ) a (1 -e 2 ) \nr = ± √ µ (1 -β ) ( 2 r -1 a -a (1 -e 2 ) r 2 \nv r v ) \nwhere µ = GM ⊙ = 1 . 327 × 10 20 m 3 s -2 is the standard solar gravitational parameter, β is the mass-dependent ratio of the radiation pressure and gravitational forces from the Sun (Zook & Berg 1975; Burns et al. 1979; Wilck & Mann 1996), r is the heliocentric distance, e is the eccentricity, and a is the semi-major axis. These equations also apply to PSP with β = 0. \nTo reduce complexity in the following calculation, we assume each impacting grain is on a Keplerian orbit with a semi-major axis equal to the heliocentric distance of PSP at any location, such that a d = r , and that grains are in the same orbital plane as PSP. With these assumptions, the dust impact speed v imp = | ⃗v d -⃗v PSP | to PSP is \nv imp = √ √ √ √ µ [ 2 r -1 a + 1 -β r -2 √ (1 -β ) a (1 -e 2 )(1 -e 2 d ) r 3 ± 2 e d √ (1 -β ) r ( 2 r -1 a -a (1 -e 2 ) r 2 ) ] (A2) \nwhere the mass dependence is from β ( m ), e d is the eccentricity of the dust, a = 0 . 41 au & e = 0 . 85 are the semi-major axis and eccentricity of PSP during orbits \n10-16. The ± has a positive value when PSP and the dust grain have the opposite sign for their radial velocity components (e.g. post-perihelion outbound PSP \nDistance = 0.11 au \nFigure 9. (a) Percent change in the impact amplitude due to the finite cadence of 8 hr impact rate intervals and due to the possible eccentricity of the grains. For each effect, six curves are shown corresponding to β = 0 -0 . 5. (b) Power-law envelopes to the family of normalized mass-dependent impact charge curves for heliocentric distances of 0.11 au and 0.24 au. \n<!-- image --> \nand inbound dust grains), and negative when the radial velocity components have the same sign. This impact speed relation will allow us to constrain the uncertainties of our estimates for α discussed at the end of this section. \nWith the analytic expression for the impact speed to PSP given in Eq. A2 we can estimate the uncertainty in determining the mass index due to three effects that all vary as a function of heliocentric distance. First, all impact rates are calculated in 8-hour intervals, which introduces a small uncertainty on the impact speeds. Figure 9a shows the percent change in impact charge amplitude calculated via (1+∆ v/v ) 4 . 7 due to this effect, where v is the impact speed to PSP from a grain on a circular, Keplerian orbit in the center of an 8-hr interval for each distance and ∆ v is the maximum expected difference between v and the impact speeds within the entire 8-hr interval. For the region investigated, this introduces an impact amplitude uncertainty of < 10% and therefore we neglect this effect. \nThe next effect we investigate is possible uncertainty in impact speed due to grains having a non-zero eccentricity. Zodiacal grains very near the Sun are expected to have low eccentricities due to circularization from Poynting-Robertson drag and we consider the case for grains that have eccentricities of up to 0.1 (Pokorn'y et al. 2024). Figure 9a shows the percent change in impact charge amplitude due to such non-circular impactors and we find this introduces an uncertainty in the impact charge amplitude of up to a factor of 2 which translates to a factor of 3 √ 2 ≈ 1 . 3 in radius. As grains on circular orbits would have the slowest impact speed compared to eccentric grains with the same other orbital \nparameters, this uncertainty means we may be overestimating the impactor grain size by a factor of ∼ 1.3. \nBoth of these uncertainties do not introduce any massdependent bias and therefore would not affect the determination of the mass index, they would only affect the estimate of the relevant size range to which the mass index is valid. The third source of uncertainty we consider is due to the impact speed having a mass dependence, which does introduce a mass-dependent bias and propagates to our estimates of α = α Q (1 + δa ). \nTo quantify δa , we calculate the impact speed to PSP given in Eq. A2 as a function of mass, where the mass dependence is solely from β ( m ). We use the β ( m ) relation corresponding to 'old cometary' grains according to Wilck & Mann (1996). For the size range considered, β has values near 0.5 on the lowest mass end and 0.2 on the upper mass end. Figure 9b shows two sets of normalized v 4 . 7 imp curves corresponding to the heliocentric distance boundaries of 0.11 au and 0.24 au considered here. Multiple curves for each family are shown for different values of eccentricity between 0 and 0.1 and using the + or -sign in Eq. A2, with the curves corresponding to a minus sign as the top two of each family and those corresponding to a plus sign as the bottom two. We then determine the slope of power-laws that provide an envelope to these family of curves for each heliocentric distance. In the examples given in Figure 9b, we find power-law functional forms with δa = -0.15 to 0 provide an envelope for the family of curves corresponding to r = 0 . 11 au and δa = 0.01 to 0.06 for r = 0 . 24 au. We calculate these envelopes over the entire range of heliocentric distances considered in this study to determine the range of δa minimum and maximum values.", 'REFERENCES': "- Pusack, A., Malaspina, D. M., Szalay, J. R., et al. 2021, PSJ Shen, M. M., Sternovsky, Z., & Malaspina, D. M. 2023, Journal of Geophysical Research: Space Physics, 128, doi: 10.1029/2022ja030981\n- Silsbee, K., & Draine, B. T. 2016, The Astrophysical Journal, 818, 133, doi: 10.3847/0004-637x/818/2/133\n- Sommer, M. 2023, Planetary and Space Science, 236, 105751, doi: 10.1016/j.pss.2023.105751\n- Stenborg, G., Howard, R., Hess, P., & Gallagher, B. 2020, Astronomy and Astrophysics,\n- doi: 10.1051/0004-6361/202039284\n- Sterken, V. J., Altobelli, N., Kempf, S., et al. 2012, A&A, 538, A102, doi: 10.1051/0004-6361/201117119\n- Strub, P., Sterken, V. J., Soja, R., et al. 2019, Astronomy and Astrophysics, 621, A54\n- Szalay, J. R., Pokorn'y, P., Bale, S. D., et al. 2020, The Astrophysical Journal Supplement Series, 246, 27, doi: 10.3847/1538-4365/ab50c1\n- Szalay, J. R., Pokorn'y, P., Malaspina, D. M., et al. 2021, The Planetary Science Journal, 2, 185, \ndoi: 10.3847/psj/abf928 \n- Thebault, P., Olofsson, J., & Kral, Q. 2023, A&A, 674, A51, doi: 10.1051/0004-6361/202345995\n- Tsurutani, B. T., Clay, D. R., Zhang, L. D., et al. 2003, Geophysical Research Letters, 30, 2134"}

2024arXiv240911613M

The tumultuous effects of ultraviolet photons that source cosmic reionization the subsequent compression and shockheating of lowdensity regions and the modulation of baryons in shallow potential wells induced by the passage of ionization fronts collectively introduce perturbations to the evolution of the intergalactic medium in the postreionization era. These enduring fluctuations persist deep into the postreionization era casting a challenge upon precision cosmology endeavors targeting tracers in this cosmic era. Simultaneously these relics from reionization also present a unique opportunity to glean insights into the astrophysics that govern the epoch of reionization. In this work we propose a first study of the crosscorrelation of lya forest and 21 cm intensity mapping accounting for the repercussions of inhomogeneous reionization in the postreionization era. We investigate the ability of SKA times DESIlike SKA times MUSTlike and PUMA times MUSTlike instrumental setups to achieve a high signaltonoise ratio SNR in the redshift range 3.5 leq z leq 4. Moreover we assess how alterations in integration time survey area and reionization scenarios impact the SNR. Furthermore we forecast the crosscorrelations potential to constrain cosmological parameters under varying assumptions considering or disregarding reionization relics marginalizing over reionization astrophysics and assuming perfect knowledge of reionization. Notably our findings underscore the remarkable capability of a futuristic PUMA times MUSTlike setup with a modest 100hour integration time over a 100 sq. deg. survey to constrain the ionization efficiency error to sigmazeta 3.42 .

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.11613', 'arXiv:2409.11613', '2024arXiv240911613M']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Reionization relics in the crosscorrelation between the Lyalpha forest and 21 cm intensity mapping in the postreionization era

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https://arxiv.org/pdf/2409.11613.pdf

{'Reionization relics in the cross-correlation between the Ly 𝛼 forest and 21 cm intensity mapping in the post-reionization era': 'Paulo Montero-Camacho, 1 , 2 ★ Catalina Morales-Gutiérrez, 3 , 4 Yao Zhang, 2 Heyang Long, 5 , 6 and Yi Mao 2 \n- 1 Department of Mathematics and Theory, Peng Cheng Laboratory, Shenzhen, Guangdong 518066, China\n- 2 Department of Astronomy, Tsinghua University, Beijing 100084, China\n- 3 Department of Physics, University of Costa Rica, 11501 San José, Costa Rica.\n- 4 Space Research Center (CINESPA), University of Costa Rica, 11501 San José, Costa Rica.\n- 5 Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA\n- 6 Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': "The tumultuous effects of ultraviolet photons that source cosmic reionization, the subsequent compression and shock-heating of low-density regions, and the modulation of baryons in shallow potential wells induced by the passage of ionization fronts, collectively introduce perturbations to the evolution of the intergalactic medium in the post-reionization era. These enduring fluctuations persist deep into the post-reionization era, casting a challenge upon precision cosmology endeavors targeting tracers in this cosmic era. Simultaneously, these relics from reionization also present a unique opportunity to glean insights into the astrophysics that govern the epoch of reionization. In this work, we propose a first study of the cross-correlation of Ly 𝛼 forest and 21 cm intensity mapping, accounting for the repercussions of inhomogeneous reionization in the post-reionization era. We investigate the ability of SKA × DESI-like, SKA × MUST-like, and PUMA × MUST-like instrumental setups to achieve a high signal-to-noise ratio (SNR) in the redshift range 3 . 5 ≤ 𝑧 ≤ 4. Moreover, we assess how alterations in integration time, survey area, and reionization scenarios impact the SNR. Furthermore, we forecast the cross-correlation's potential to constrain cosmological parameters under varying assumptions: considering or disregarding reionization relics, marginalizing over reionization astrophysics, and assuming perfect knowledge of reionization. Notably, our findings underscore the remarkable capability of a futuristic PUMA × MUST-like setup, with a modest 100-hour integration time over a 100 sq. deg. survey, to constrain the ionization efficiency error to 𝜎 𝜁 = 3 . 42. \nKey words: intergalactic medium - dark ages, reionization, first stars", '1 INTRODUCTION': "Cosmic reionization, the transformative phase during which our Universe shifted from predominantly neutral to highly ionized, is estimated to have occurred approximately around redshift 𝑧 ∼ 8 (e.g. Planck Collaboration et al. 2020). While the overarching mechanisms driving cosmic reionization are broadly understood (Mesinger 2016, 2019), the lack of direct observations introduces significant uncertainties, particularly regarding the timeline of reionization (Keating et al. 2020; Gnedin 2022; Jin et al. 2023; Roth et al. 2023). With the advent of the James Webb Space Telescope (JWST; Gardner et al. 2006) our knowledge of the sources of ultraviolet photons responsible for reionization is likely to increase dramatically. Moreover, reionization unfolds as a markedly inhomogeneous inside-out process (Lee et al. 2008; Choudhury et al. 2009) wherein denser regions undergo ionization first since sources of ultraviolet photons are expected to predominantly be situated within these dense environments. \nThe post-reionization era is a treasure trove of cosmological information. The relatively high redshifts characteristic of this pe- \nriod not only facilitate the exploration of dark matter candidates (Palanque-Delabrouille et al. 2020; Puchwein et al. 2022) but also offer a unique opportunity to study cosmic evolution and structure formation (Wyithe & Loeb 2009; Visbal et al. 2009) before the onset of nonlinearities imposes significant constraints on the observational landscape. \nThe Lyman𝛼 (Ly 𝛼 ) forest, the absorption features observed in the spectra of background quasars, stands as a pivotal tool for exploring the Universe in the post-reionization era. Its applications extend to the study of H /i.pc and He /i.pc/i.pc reionization (Cen et al. 2009; Upton Sanderbeck & Bird 2020; Montero-Camacho & Mao 2020), the investigation of the matter power spectrum on scales beyond the reach of galaxy surveys (Weinberg et al. 2003), constrain the evolution of the Universe (du Mas des Bourboux et al. 2020), and the inference of cosmological parameters (Chabanier et al. 2019). Moreover, the forest also offers a unique window into the impact of neutrino masses (Yèche et al. 2017). However, obtaining reliable cosmological measurements of the Ly 𝛼 forest at high redshifts proves challenging due to the sparse sampling of quasars (Yèche et al. 2020; Montero-Camacho & Mao \n2021; Chaussidon et al. 2023), particularly regarding potential measurements of the 3D flux power spectrum 1 . \nIntensity mapping (IM) involves a trade-off of sacrificing angular resolution to concentrate on the integrated emission from unresolved sources. Analogous to the Ly 𝛼 forest, IM utilizing the 21 cm hyperfine transition of hydrogen presents a versatile probe capable of delving into various aspects of cosmology in the post-reionization era. It can probe H /i.pc reionization (Long et al. 2023), explore early universe cosmological parameters (e.g. primordial non-gaussianity), and constrain the evolution of the universe and structure formation (Castorina & White 2019; Bull et al. 2015; Wyithe & Loeb 2009; Visbal et al. 2009). Additionally, it can investigate the nature of dark matter (Carucci et al. 2015) and more (Square Kilometre Array Cosmology Science Working Group et al. 2020; Cosmic Visions 21 cm Collaboration et al. 2018). \nNevertheless, the full potential of this observable is impeded by foregrounds, such as galactic synchrotron emission, which surpass the amplitude of the cosmological signal by several orders of magnitude (Wolz et al. 2015). Despite dedicated efforts to mitigate foreground effects (see e.g. Zuo et al. 2023; Diao et al. 2024), a conservative perspective suggests that confirming the cosmic origin of the signal might necessitate cross-correlation with another probe 2 . This cross-correlation typically considers high𝑧 galaxies (Furlanetto & Lidz 2007; La Plante et al. 2023; Hutter et al. 2023) as the additional tracer. \nThe intense heating of the intergalactic medium (IGM) during cosmic reionization triggers a substantial increase in the IGM temperature, reaching a few times 10 4 K. As ionization fronts propagate, inducing a rise in temperature, low-density regions experience shocks that both heat and compress the gas. These shocks, which originate in denser regions due to the increase in Jeans mass with temperature, will push gas in minivoids to higher adiabats compared to gas in denser environments (Hirata 2018) - such as minihalos. This highentropy mean-density gas is then responsible for the long-lasting impact of reionization in the Ly 𝛼 forest (Montero-Camacho et al. 2019; Montero-Camacho & Mao 2020), essentially constituting the long-lasting memory of reionization at 𝑧 ∼ 2 in the forest. Simultaneously, reionization exerts influence over the baryon abundance within a given halo (Long et al. 2022). The thermal kick resulting from the passage of an ionization front expels some baryons from the halo, particularly for halos with shallow potential wells. This modulation is responsible for long-lasting reionization relics in 21 cm IM, persisting up to 𝑧 ∼ 3 (Long et al. 2023). \nTherefore, both the Ly 𝛼 forest and H /i.pc 21 cm intensity mapping exhibit broad-spectrum contamination originating from the imprints of cosmic reionization. This contamination is especially pronounced at large scales and high redshifts in the post-reionization era, where the lingering effects of reionization are more pronounced due to less time to dissipate the additional injected energy (Montero-Camacho & Mao 2020). Managing this systematic is not merely a challenge essential for obtaining cosmological information free from bias in \nthe post-reionization era, but it also represents a novel opportunity to gain insights into the intricate processes governing the reionization history of the Universe (Montero-Camacho & Mao 2021). \nThe observational programs of these two probes are at different developmental stages. The Ly 𝛼 forest has achieved commendable SNR, e.g. eBOSS (Castorina & White 2019; du Mas des Bourboux et al. 2020) and DESI early (Ramírez-Pérez et al. 2023; Gordon et al. 2023; Ravoux et al. 2023) and year-1 results (DESI Collaboration et al. 2024). In stark contrast, the 2-point functions of the cosmological 21 cm signal at higher redshifts ( 𝑧 > 2) are currently constrained only by upper limits (see e.g., Munshi et al. 2023; Abdurashidova et al. 2022; Mertens et al. 2020). Despite this, because of the inherent challenges in measuring the 21 cm auto-power spectrum, there is considerable interest in the cross-correlation of these two distinct tracers as a promising avenue for cosmological studies. \nThe Ly 𝛼 forest × H/i.pc 21 cm IM cross-correlation was initially proposed in Guha Sarkar et al. (2011) as a robust and independent probe of the post-reionization IGM. Carucci et al. (2017) underscored the reduced sensitivity of this cross-correlation to foreground contamination, emphasizing its potential to untangle degeneracies inherent in modeling parameters. Furthermore, the Ly 𝛼 × 21 cm crosscorrelation exhibits promising prospects for constraining dark energy (Dash & Guha Sarkar 2021) and 𝑓 ( 𝑟 ) gravity models, particularly when coupled with cross-correlations involving cosmic microwave background (CMB) lensing (Dash et al. 2023). Moreover, this observable has also emerged as a potential avenue for constraining the nature of dark matter (Sarkar et al. 2019). \nRegarding the observability of this cosmological probe, the Owens Valley Widefield Array (OWFA) (Ali & Bharadwaj 2014; Bharadwaj et al. 2015) and a spectroscopic instrument such as BOSS/eBOSS were anticipated to achieve a robust 6 𝜎 detection at 𝑧 = 3 . 35 with 200 hours of integration (Sarkar et al. 2018). Meanwhile, the Square Kilometre Array Phaew 1 Mid-frequency (SKA1-Mid) in conjunction with an eBOSS-like survey could reach a peak SNR of 15 at 𝑧 = 2 . 5 (Guha Sarkar & Datta 2015). \nHowever, the existing studies/forecasts have yet to incorporate the enduring impact of inhomogeneous reionization on the postreionization IGM. In this work, we delve into the repercussions of these reionization remnants in the Ly 𝛼 forest × H/i.pc 21 cm IM crosscorrelation within the redshift range 3 . 5 ≤ 𝑧 ≤ 4. We specifically target this range because of the heightened abundance of Ly 𝛼 spectra compared to higher redshifts and the pronounced strength of reionization relics in the forest during this period (Montero-Camacho & Mao2020). Nevertheless, it is crucial to note that the impact of reionization imprints will remain significant for this cross-correlation in the broader redshift range of 3 ≲ 𝑧 < 6. \nThe rest of this work is organized as follows. In section 2, we outline our model for capturing the impact of reionization in the postreionization IGM. We present the simulations required to compute the reionization relics in section 3. The instrumental configurations considered in this study are introduced in section 4. Section 5 presents our findings regarding the effectiveness of the instrumental setups in detecting the Ly 𝛼 × 21 cm cross-correlation. Additionally, we consider deviations in survey design and strategy, providing insights into possible gains. In section 6, we demonstrate the importance of accounting for this novel effect through three distinct Fisher forecasts. Finally, we summarized our findings in 7. \nThroughout this work, we use ℎ = 0 . 6774, Ω 𝑏 ℎ 2 = 0 . 0223, Ω 𝑐 ℎ 2 = 0 . 1188, 𝐴 𝑠 = 2 . 148 × 10 -9 , and 𝑛 𝑠 = 0 . 9667. In agreement with the ' TT + TE + EE + lowP + lensing + ext ' cosmology from Planck Collaboration et al. (2016). Furthermore, we use the values of 𝑏 H/i.pc and Ω H/i.pc reported in Crighton et al. (2015). In con- \nSD parameters for the forest from Arinyo-i-Prats et al. (2015).", '2 MODELING THE CROSS-CORRELATION OF Ly 𝛼 AND 21 CM IM WITH REIONIZATION RELICS': "In this work, we consider the inclusion - at first order - of the memory of reionization, that is the long-lasting impact of inhomogeneous reionization, in both Ly 𝛼 flux fluctuations and H /i.pc 21 cm fluctuations. We examine the repercussions of this inclusion when embedded into the traditional cross-correlation of the Ly 𝛼 forest and 21 cm intensity mapping. Mathematically, \n𝑃 21 ,𝐹 ( 𝒌 , 𝑧 ) = 𝑃 Fid . 21 ,𝐹 ( 𝒌 , 𝑧 ) + 𝑃 Mem . 𝐹 ( 𝒌 , 𝑧 ) + 𝑃 Mem . 21 ( 𝒌 , 𝑧 ) , (1) \nwhere the first term on the right-hand side corresponds to the conventional cross-correlation of the Ly 𝛼 forest and 21cm IM without reionization relics. The second term accounts for the memory of reionization present in the Ly 𝛼 forest, originating in the underdense regions. The final term represents the memory of reionization in H /i.pc 21 cm IM, i.e. the memory of reionization sourced by overdense regions. The three terms are given by (Carucci et al. 2017; MonteroCamacho et al. 2019; Long et al. 2023) \n𝑃 Fid . 21 ,𝐹 = 𝑏 𝐹 ( 𝑧 ) 𝑏 H/i.pc ( 𝑧 ) [ 1 + 𝛽 𝐹 ( 𝑧 ) 𝜇 2 ] [ 1 + 𝛽 H/i.pc ( 𝑧 ) 𝜇 2 ] 𝑃 𝑚 ( 𝑘, 𝑧 ) , \n𝑃 Mem . 21 = 𝑏 𝐹 ( 𝑧 ) [ 1 + 𝛽 𝐹 ( 𝑧 ) 𝜇 2 ] 𝑃 𝑚, Ξ ( 𝑘, 𝑧 ) , \n(2) 𝑃 Mem . 𝐹 = 𝑏 H/i.pc ( 𝑧 ) [ 1 + 𝛽 H/i.pc ( 𝑧 ) 𝜇 2 ] 𝑏 Γ ( 𝑧 ) 𝑃 𝑚,𝜓 ( 𝑘, 𝑧 ) , (3) (4) \nwhere 𝑏 𝐹 and 𝛽 𝐹 are the usual flux bias and redshift space distortion parameter, respectively - and similarly for 21 cm quantities. Moreover, 𝜇 is the angle with respect to the line of sight, 𝑏 Γ is the radiation bias defined in Arinyo-i-Prats et al. (2015); Hirata (2018), and 𝑃 𝑚 is the linear matter power spectrum. \nFurthermore, 𝑃 𝑚,𝜓 and 𝑃 𝑚, Ξ are the cross-power spectrum of matter and change of transparency of the IGM due to the impact of reionization, and the cross-power spectrum of matter and change of neutral hydrogen density induced by the passage of reionization fronts, respectively. These cross-power spectra are defined as follows (Montero-Camacho et al. 2019; Long et al. 2023) \n𝑃 𝑚,𝜓 ( 𝑘, 𝑧 obs ) = -∫ d 𝑧 𝜕𝜓 ( 𝑧, 𝑧 obs ) 𝜕𝑧 𝑃 𝑚,𝑥 HI ( 𝑘, 𝑧 ) 𝐷 ( 𝑧 obs ) 𝐷 ( 𝑧 ) , (5) 𝑃 𝑚, Ξ ( 𝑘, 𝑧 obs ) = -∫ d 𝑧 𝜕 Ξ ( 𝑧, 𝑧 obs ) 𝜕𝑧 𝑃 𝑚,𝑥 HI ( 𝑘, 𝑧 ) 𝐷 ( 𝑧 obs ) 𝐷 ( 𝑧 ) , (6) \nwhere the integration covers the epoch of reionization, 𝑃 𝑚,𝑥 HI is our proxy for the correlation of matter and ionized bubble spatial distribution, which accounts for the patchy nature of reionization. 𝐷 is the growth factor, while 𝜓 denotes the transparency of the IGM, and Ξ represents the modulation of the neutral hydrogen due to the passage of ionization fronts. They are given by \n𝜓 ( 𝑧 re , 𝑧 obs | 𝑧 re ) = Δ ln 𝜏 1 = ln GLYPH<20> 𝜏 1 ( 𝑧 re , 𝑧 obs ) 𝜏 1 ( 𝑧 re , 𝑧 obs ) GLYPH<21> , (7) \nΞ ( 𝑧 re , 𝑧 obs | 𝑧 re ) = Δ ln 𝜌 HI = ln GLYPH<20> 𝜌 HI ( 𝑧 re , 𝑧 obs ) 𝜌 HI ( 𝑧 re , 𝑧 obs ) GLYPH<21> , (8) \nwhere 3 . 5 ≤ 𝑧 obs ≤ 4, 𝑧 re is the local redshift of reionization, and 𝑧 re serves as a reference redshift of reionization. The parameter 𝜏 1 is the optical depth that must be assigned in simulations to a patch of gas with mean density and temperature 𝑇 = 10 4 K in order for the mean transmitted flux to match observations, i.e. 𝜏 1 is a normalization factor that guarantees that an optical depth cube return sensible results by matching to the observations of Kim et al. (2007). The transparency \nis defined as a relative measure, benchmarked against a fiducial scenario with 𝑧 re = 8. Conversely, Ξ characterizes the response of halos with shallow potential wells to the passage of ionization fronts, modeling the perturbations in neutral hydrogen density induced by the response relative to those that occur in a fiducial local reionization scenario. We plot some of the ingredients of the reionization relics in Figure B1 for reference. \nSimilarly, the Ly 𝛼 flux and the 21 cm power spectra are respectively given by \n𝑃 3D 𝐹 ( 𝒌 , 𝑧 ) = 𝑏 2 𝐹 ( 1 + 𝛽 𝐹 𝜇 2 ) 2 𝑃 𝑚 + 2 𝑏 𝐹 𝑏 Γ ( 1 + 𝛽 𝐹 𝜇 2 ) 𝑃 𝑚,𝜓 (9) 𝑃 21 ( 𝒌 , 𝑧 ) = 𝑏 2 H/i.pc ( 1 + 𝛽 H/i.pc 𝜇 2 ) 2 𝑃 𝑚 + 2 𝑏 H/i.pc ( 1 + 𝛽 H/i.pc 𝜇 2 ) 𝑃 𝑚, Ξ (10) \nwhere we have neglected terms with higher order in Ξ and 𝜓 . Likewise, we disregard non-linear corrections, justified by our focus on large scales. \nEven though we extract 𝑏 𝐹 and 𝛽 𝐹 from Arinyo-i-Prats et al. (2015), the maximum redshift in their tables is 3. Hence, for our 𝑧 > 3 calculations, we include a redshift evolution factor [( 1 + 𝑧 )/( 1 + 𝑧 𝑝 )] 3 . 55 (Palanque-Delabrouille et al. 2013b) with pivot 𝑧 𝑝 = 3 to account for the evolution of the flux bias and redshift space distortion parameter (see Eq.(10) and surrounding text of Montero-Camacho et al. 2024b). \nWhile both Eq. (5, 6) emerge from the IGM's response to inhomogeneous reionization, they represent distinct response mechanisms. Eq. (7) covers the response to the local reionization process by underdense gas (Hirata 2018). Physically, it originates in minivoids, as underdense gas undergoes ultraviolet heating, shock heating, and compression during the reionization process. In contrast, Eq. (8) captures the response to local reionization by denser regions, predominantly arising from minihalos. Essentially, this is the modulating effect on the number of baryons allowed inside a given halo following the thermal kick delivered by an ionization front (Long et al. 2022). Note that both mechanisms differ in their origin on the small scales but are influenced equivalently by the patchy nature of reionization - parametrized here as 𝑃 𝑚,𝑥 HI and present in both Eq. (5, 6). \nTo facilitate comparison between the different components of Eq. (1), we have chosen to utilize the spherically-averaged power spectrum 3 , i.e. \n𝑃 Sph . 21 ,𝐹 ( 𝑘, 𝑧 ) = 1 2 𝜋 ∫ 2 𝜋 0 𝑑𝜙 ∫ 1 0 𝑑𝜇𝑃 21 ,𝐹 ( 𝒌 , 𝑧 ) , (11) \nwhere the normalization factor already accounts for the extra factor of 2 due to the 𝜇 -symmetry. \nWeplot Eq. (11) in Figure 1 at a few selected redshifts. The impact of reionization is more pronounced at higher redshifts, introducing a competition between the 21 cm effect, arising from the modulation of baryons due to reionization, and the Ly 𝛼 forest effect, originating in underdense regions. The former aims to diminish the signal's amplitude while the latter enhances it. Furthermore, the more substantial (in absolute value) impact of the Ly 𝛼 sourced term compared to the H /i.pc 21 cm term is anticipated due to its greater influence on voids, evident in the bimodal temperature-density relation (see Figure 3 in Montero-Camacho & Mao 2020 versus bayron modulation displayed in Figure 8 of Long et al. 2023). While our focus is on the 3 . 5 < 𝑧 < 4 . redshift range, we expect this qualitative trend to \n<!-- image --> \n<!-- image --> \nFigure 1. The magnitude of the spherically-averaged cross-power spectrum for the Lyman-alpha forest × H/i.pc 21 cm IM cross-correlation is illustrated, encompassing its components, including the imprint of reionization in both probes. The blue dash-dotted line corresponds to the conventional signal without any relics from cosmic reionization. The purple dotted and green dashed curves represent the memory of reionization in the Lyman𝛼 forest and 21 cm power, respectively. We highlight that the memory of reionization in the 21 cm signal is negative, leading to a competition in which the Ly 𝛼 forest, which is positive, dominates. The influence of reionization becomes more pronounced at higher redshifts and larger scales. \n<!-- image --> \npersist at higher redshifts especially given that the impact of selfshielding in minihalos will also enhance the Ly 𝛼 power spectrum at large scales (Park et al. 2023). Regarding the wavenumber trend, the larger deviation at large scales compared to the conventional term is explained by the coupling to the reionization bubble scale in the cross-correlation of the matter and neutral hydrogen fraction field, 𝑃 𝑚,𝑥 H/i.pc .", '3 SIMULATIONS': 'To accurately model the reionization process, it is imperative to probe under the Jeans length prior to the passage of an ionization front ( ∼ 100 kpc). Failure to do so would result in the loss of the ability to track the response by the small-scale structure due to the significant wiping out of these structures caused by the increasing Jeans scale throughout reionization (Hirata 2018). Besides, reionization occurs in an inhomogeneous way. Thus, to secure enough statistical power, there is a need to simulate large comoving volumes (a few hundred Mpc) with substantial ionized bubbles each with a radius of a few Mpc. However, the dynamical range required for these considerations is too large to accomplish with reasonable computational resources using a single simulation. \nWeadopt the methodology introduced by Montero-Camacho et al. (2019); Long et al. (2023) and employ a hybrid approach to compute Eq. (5,6). This approach relies on seminumerical simulations using 21/c.pc/m.pcFAST (Mesinger et al. 2011; Murray et al. 2020), which track the patchy nature of reionization within a 400 Mpc box. In addition, we leverage high-mass resolution small-box ( 𝐿 = 1152 ℎ -1 ckpc) simulations run using a modified version of G/a.pc/d.pc/g.pc/e.pc/t.pc-2 (Springel 2005; Hirata 2018) to monitor the response to ionization fronts in both dense (crucial for H /i.pc 21 cm) and underdense (Ly 𝛼 forest) regions. This hybrid simulation strategy allows us to capture the impact of reionization across a range of scales and redshifts with minimal trade-offs. Naturally, this hybrid methodology will not be sensitive \nto higher-order correlations between the reionization field and the small-scale density field (Montero-Camacho et al. 2019). \nThe small-box simulations include adiabatic expansion (including Hubble expansion), shock heating, Compton heating, and cooling for neutral gas (accounting for residual ionization). In addition, we incorporate the long-lasting impact of X-ray preheating in the post-reionization IGM (Montero-Camacho et al. 2024b). For ionized gas, we also include Compton cooling, He /i.pc/i.pc cooling, recombination cooling, photoionization heating, and free-free cooling. These simulations have been described - and tested - in detail in Hirata (2018); Long et al. (2022). \nTo surpass the limitation imposed by the size of our 21/c.pc/m.pcFAST boxes, we utilize a simple liner bias to estimate the effect of reionization at 𝑘 ≲ 0 . 06 Mpc -1 as follows \n𝑃 𝑚,𝑋 ( 𝑘, 𝑧 ) = 𝑃 𝑚,𝑋 ( 𝑘 cut ≈ 0 . 06 , 𝑧 ) 𝑃 𝑚 ( 𝑘 cut , 𝑧 ) 𝑃 𝑚 ( 𝑘, 𝑧 ) , (12) \nwhere 𝑋 = { 𝜓, Ξ } . This approximation should be sufficient for smaller scales than the ionized bubble scale. Note that this restriction is due to the box side of our large-box simulations and it is a self-imposed restriction based on available computational resources.', '4 TELESCOPES': "To measure the cross-correlation between 21 cm and the Ly 𝛼 forest, we require a radio interferometer and a spectroscopic telescope with some non-negligible overlap in sky coverage and redshift. Unfortunately, our choice for estimating the SNR is not straightforward due to the timeline of the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2022) versus the timeline of the Square Kilometer array (SKA; Braun et al. 2019). While DESI may potentially extend its operational life as DESI-II, allowing for direct operational overlap with SKA, the exact timeline overlap remains uncertain. \nConcurrently, upcoming spectroscopic instruments such as \nz \n=3.94 \nMegaMapper (Schlegel et al. 2019), the MUltiplexed Survey Telescope (MUST; Zhang et al. 2024), and MaunaKea Spectroscopic Explorer (MSE; Percival et al. 2019), categorized as Stage V spectroscopic instruments, are anticipated to be operational around the time when 21 cm radio interferometers come online. However, we underscore that the detailed designs of these instruments are currently in the developmental phase. Navigating the evolving landscape of these instrument timelines and designs will be vital to optimizing an observational program capable of measuring the cross-correlation between 21 cm and the Ly 𝛼 forest. \nGiven these 'instrumental' constraints, we have opted to explore several distinct instrumentation scenarios. \n- · SKA1-Low × DESI-like spectroscopic telescope (referred to as SKA in relevant figures).\n- · SKA1-Low × DESI-MUST-like hybrid (referred to as MUST).\n- · PUMA-like × DESI-MUST-like hybrid (referred to as PUMA). \nAcross all scenarios, we maintain a redshift coverage of 3 . 5 ≤ 𝑧 ≤ 4 and consider two sky coverage overlaps - 100 and 1000 square degrees. Likewise, we vary the integration times, considering both 100 and 1000 hours, which will mainly affect the radio observations as described below. \nOur base reference scenario involves the SKA1-Low × DESI-like pair, with 100 hours of integration time and a 100-square-degree overlap between the instruments. This reference scenario assumes a Planck-like reionization timeline, which may be underestimating the impact of reionization since Planck uses a hyperbolic tangent as a sigmoid to model reionization that proves a poor fit of astrophysical constraints on the timeline of reionization (see Figure 4 of MonteroCamacho et al. 2024a).", '4.1 SKA1-Low': "SKA1-Low, our baseline 21 cm instrument, operates as an aperture array radio interferometer. We consider only the dense core of the array, which is most sensitive to the 21 cm power spectrum. Do note that the outer stations in the array are crucial for calibration and foreground removal purposes. The SKA1-Low, upon completion, offers versatility in supporting various observing modes. Previous works have often focused on observing mode 1, characterized by a dense core comprising 224 stations of 40 meters in diameter, each containing 256 dipole antennas (Square Kilometre Array Cosmology Science Working Group et al. 2020; Zhao et al. 2022). These core stations are distributed over a diameter of approximately one kilometer. \nNevertheless, as pointed out in SKA1-LOW Configuration - Constraints & Performance Analysis §5.4.3, power spectrum measurements are better served by 'substations' with 10 m of diameter. Here we chose to consider only the observing mode 4, as detailed in Table 2 of SKA1-LOW Configuration - Constraints & Performance Analysis. This mode, supported by the instrument's correlator, boasts six times more correlatable elements (substations) at the expense of a reduced bandwidth (8.4 MHz compared to 300 for mode 1). \nTo quantify the unique baselines covered by the dense core, we utilize the number density of baselines 𝑛 𝑏 from Villaescusa-Navarro et al. (2015) and normalize it to recover the total amount of distinct baselines covered by the dense core, as outlined in §2 of Long et al. (2023). \nThe system temperature - the sum of the sky temperature, the \nground reflections, and the temperature of the receiver - is given by \n𝑇 sys ( 𝜈 ) = 60 GLYPH<18> 300 MHz 𝜈 GLYPH<19> 2 . 55 × 1 . 1 + 40 [K] , (13) \nwhile the effective collecting area per station 𝐴 𝑒 is given by (Square Kilometre Array Cosmology Science Working Group et al. 2020) \n𝐴 𝑒 ( 𝜈 ) = 𝐴 𝑒, crit × ( GLYPH<0> 𝜈 crit 𝜈 GLYPH<1> 2 , 𝜈 > 𝜈 crit 1 , 𝜈 ≤ 𝜈 crit , (14) \nwhere 𝜈 crit = 110 MHz and 𝐴 𝑒, crit is the collecting area in m 2 for the 256 dipole antennas of 3.2 m 2 each for observing mode 1. Thus, we approximate the effective area for observing mode 4 by decreasing Eq. (14) by a factor of 6. \nMoreover, the field of view is given by \nFoV ( 𝜆 ) = 𝜆 √ 0 . 7 𝐷 phys ! 2 , (15) \nwhere 𝐷 phys is the diameter of the correlatable element and 𝜆 is the observing wavelength.", '4.2 PUMA': 'ThePacked Ultra-wideband Mapping Array design (PUMA; Cosmic Visions 21 cm Collaboration et al. 2018; Slosar et al. 2019) currently - consists of 32000 antennas distributed in a hexagonal-close packing in a compact circle of roughly 1.5 km diameter. Each antenna has a diameter of 6 meters. We take the baseline number density 𝑛 b directly from Appendix D of Cosmic Visions 21 cm Collaboration et al. (2018). \nFor PUMA, the system temperature is given by \n𝑇 sys ( 𝜈 ) = 25 GLYPH<18> 400 MHz 𝜈 GLYPH<19> 2 . 75 + 2 . 7 + 300 9 + 50 0 . 81 [K] . (16) \nThe effective collecting area per antenna is simply the effective area of the antenna dish, i.e. 𝜋 ( √ 0 . 7 𝐷 phys / 2 ) 2 . Simultaneously, the field of view is given by Eq. (15) but with 𝐷 phys = 6 m. We set the bandwidth to 8.4 MHz (same as SKA1-Low observing mode 4). \nWeconsider PUMA as an optimistic/futuristic 21 cm telescope for our purposes, essentially our best benchmark.', '4.3 DESI-like': 'We assume a DESI-like spectrograph as our baseline Ly 𝛼 forest instrument, which is likely a conservative estimate given the projected timeline of SKA. To gauge the performance of the final - 5 years - DESI data, we use the Quasar Luminosity Function (QLF) to quantify the expected number of Ly 𝛼 quasars observable with DESI (Palanque-Delabrouille et al. 2013a; Yèche et al. 2020) and the spectrograph performance of DESI - see Eq. (21). Detailed information about the DESI instrument is available in DESI Collaboration et al. (2022), the QSO and Ly 𝛼 QSO target selection can be found in Chaussidon et al. (2023), and the spectroscopic pipeline is described in Guy et al. (2023).', '4.4 DESI-MUST-like hybrid': "Werefine our initially conservative DESI-like scenario by enhancing its performance to align more closely with the standards expected for Stage V spectroscopic instruments. Pragmatically, this improvement involves adjusting the aliasing term in the covariance computation, \neffectively augmenting the effective density of lines of sight beyond what is anticipated with our original DESI specs. In summary, we implement an optimistic factor of 3 reduction in both the second and third terms of Eq. (21). Given that MUST will likely be the World's first Stage V spectroscopic instrument 4 , we decided to refer to this scenario as MUST.", '5 RESULTS': 'In the redshift range of interest, the cross-correlation of Ly 𝛼 × 21 cm IMpresents a pragmatically more accessible measurement compared to the auto-correlation analyses of both the Ly 𝛼 forest or of the H /i.pc 21 cm field. This disparity arises from distinct challenges associated with each observable. For the Ly 𝛼 forest, the QLF peaks around 𝑧 ∼ 2, resulting in a substantial reduction in the number density of quasars at 𝑧 ∼ 4. Conversely, for 21 cm intensity mapping, the signal strength is significantly weaker than the foregrounds by several orders of magnitude. Notably, these challenges are uncorrelated, rendering the cross-correlation less susceptible to their impacts (Carucci et al. 2017). Since this is the first study of the effects of the relics from reionization in this observable, we will not consider foreground contamination nor several Ly 𝛼 forest systematics, like continuum fitting (Sun et al. 2023), spectra with broad absorption lines (Filbert et al. 2023), UV clustering (Long & Hirata 2023), spectra with damped Lyman𝛼 systems (Wang et al. 2022), and several other astrophysical and instrumental systematics. Note that this simpler approach is often used in other studies of this cross-correlation (e.g. Guha Sarkar et al. 2011; Sarkar et al. 2019; Carucci et al. 2017) at other redshifts.', '5.1 Impact on SNR': "Under these considerations, we can compute the errors on the observables following Carucci et al. (2017); Villaescusa-Navarro et al. (2015) \n𝜎 2 [ 𝑃 21 ( 𝒌 , 𝑧 )] = GLYPH<16> 𝑃 T 21 ( 𝒌 , 𝑧 ) GLYPH<17> 2 , (17) \n𝜎 2 [ 𝑃 𝐹 ( 𝒌 , 𝑧 )] = GLYPH<16> 𝑃 T 𝐹 ( 𝒌 , 𝑧 ) GLYPH<17> 2 and (18) \n𝜎 2 [ 𝑃 21 ,𝐹 ( 𝒌 , 𝑧 )] = 1 2 GLYPH<16> 𝑃 2 21 ,𝐹 + 𝜎 [ 𝑃 21 ] 𝜎 [ 𝑃 𝐹 ] GLYPH<17> . (19) \nFor 21 cm, 𝑃 Tot . has contributions from cosmic variance and thermal noise, i.e. \n𝑃 T 21 ( 𝒌 , 𝑧 ) = 𝑃 H/i.pc + 𝑇 2 sys ( 𝑧 ) 𝜒 2 ( 𝑧 ) 𝜆 ( 𝑧 ) 1 + 𝑧 𝐻 ( 𝑧 ) GLYPH<18> 𝜆 2 ( 𝑧 ) 𝐴 𝑒 GLYPH<19> 2 GLYPH<18> 𝑆 area FoV ( 𝑧 ) GLYPH<19> × GLYPH<18> 1 𝑁 pol 𝑡 int 𝑛 b ( 𝑢 = 𝑘 ⊥ 𝜒 ( 𝑧 )/ 2 𝜋 ) GLYPH<19> , (20) \nwhere 𝜒 is the comoving distance, 𝐻 is the Hubble parameter, and 𝜆 = 21cm /( 1 + 𝑧 ) . Moreover, 𝑇 sys stands for the system temperature of the radio interferometer, 𝑁 pol = 2 is the number of polarizations, 𝐴 𝑒 the effective area of the correlatable unit (dishes or stations), 𝑆 area is the survey area, FoV is the field of view, 𝑛 b quantifies the number density of baselines in the 𝑢𝑣 -plane as a function of wavenumber, and 𝑡 int corresponds to the integration time. The shot noise is subdominant for the redshift range of interest here (Castorina & Villaescusa-Navarro 2017), thus we do not include it in Eq. (20). \nIn contrast, for the Ly 𝛼 forest we have (McDonald & Eisenstein 2007; Font-Ribera et al. 2014; Montero-Camacho & Mao 2021) \n𝑃 T 𝐹 ( 𝒌 , 𝑧 ) = 𝑃 3D 𝐹 ( 𝒌 , 𝑧 ) + 𝑃 1D 𝐹 ( 𝑘 ∥ , 𝑧 ) 𝑃 2D 𝑤 ( 𝑧 ) + 𝑃 eff 𝑁 ( 𝑧 ) . (21) \nThe first term corresponds to cosmic variance, i.e. it comes from ⟨ 𝛿 𝐹 𝛿 ∗ 𝐹 ⟩ , while the second term represents the aliasing noise due to the sparse sampling of quasars. The third term describes the effective noise due to the spectrograph performance. Note that at high redshifts ( 𝑧 ≳ 3 . 2), the aliasing term tends to dominate Eq. (21) since the limited number of quasars becomes more restrictive (see Figure 6 of Montero-Camacho & Mao 2021). Naturally, this trend reduces the expected signal-to-noise ratio (SNR) of Ly 𝛼 forest surveys attempting a measurement of the 3D flux power spectrum at high redshifts. \nGiven Eq. (17, 18, or 19), the SNR for 𝑃 𝑖 can be written as \nSNR 2 𝑖 = 𝑁 𝑘 𝑃 2 𝑖 𝜎 2 [ 𝑃 𝑖 ] = 𝑉 Survey 𝑘 3 𝜖𝑑𝜇 4 𝜋 2 𝑃 2 𝑖 𝜎 2 [ 𝑃 𝑖 ] , (22) \nwhere 𝜖 = 𝑑𝑘 / 𝑘 and 𝑉 Survey is the - overlap - volume between the radio interferometer and the spectrograph. We consider three redshift bins centered 5 at 𝑧 c = [ 3 . 61 , 3 . 77 , 3 . 94 ] with widths defined by the bandwidth of the radio interferometer, i.e. Δ 𝜈 = 8 MHz. Consequently, our survey volume for a redshift bin centered at 𝑧 c is given by \n𝑉 Survey ( 𝑧 c ) = 4 𝜋 3 𝑓 sky [ 𝜒 ( 𝑧 max ) 3 -𝜒 ( 𝑧 min ) 3 )] (23) \nwith 𝜒 as the comoving distance. Note that 𝑓 sky , the sky coverage, depends on the area of overlap. \nIn what follows, we focus on the spherical-averaged power spectrum, i.e. Eq. (11) and analogs, to compute the appropriate SNR for Ly 𝛼 × 21 cm, Ly 𝛼 forest, and 21 cm intensity mapping. \nWe illustrate the observability of the cross-correlation compared to the auto-correlation of Ly 𝛼 and 21 cm in Figure 2 for our base scenario (SKA1-Low × DESI-like, 𝑡 int = 100 h, 𝐴 sky = 100 sq. deg. and Planck's reionization timeline). We highlight that all of the curves here include the long-lasting relics from cosmic reionization, i.e. Eqs. (5, 6). Unsurprisingly, the SNR for the Ly 𝛼 forest is less than unity because of the drop in available line-of-sight at these redshifts and because of the survey volume. In principle, it is necessary to observe more quasars to reduce the mean separation between forests to make the 3D flux power spectrum observation feasible. Although Figure 2 demonstrates that the 21 cm signal has a larger SNR, this is only true in the absence of foregrounds, which will severely bury the auto-power spectrum. In contrast, the cross-correlation is more robust against foreground contamination (Furlanetto & Lidz 2007; Sarkar et al. 2018; Zhou et al. 2021) and it is already competitive with the 21 cm auto-correlation with the baseline instrumental setup. \nFigure 2 provides a compelling rationale for the significance of the Ly 𝛼 × 21 cm cross-correlation, particularly in the presence of foregrounds. Detection in the cross-correlation can guarantee the cosmological nature of the 21 cm signal. Besides, the SNR is similar to the auto-correlation at the ionized bubble scale (and for our first redshift bin). This is consistent with the findings of Carucci et al. (2017) at a lower redshift ( 𝑧 = 2 . 4). However, we underscore that in the absence of reionization relics, the cross-correlation will not have a similar SNR to that of the auto 21 cm SNR in the lowest redshift bin considered here. The boost provided by the memory of reionization is crucial to be competitive, yet it is not sufficient at higher redshift \nFigure 3 showcases the dependence of the SNR for different integration times and telescope pairs. Unsurprisingly, there is a consistent enhancement in SNR across all redshifts and configurations. In particular, the SKA1-Low configurations benefit the most from this strategy, manifesting improvements of ≈ 2 . 6 in overall SNR (see Table 1). Intriguingly, the PUMA-like × DESI-MUST-like hybrid exhibits negligible improvement at large scales. \n<!-- image --> \n<!-- image --> \nFigure 2. The SNR for the baseline configuration of the Ly 𝛼 forest (purple squares), 21 cm intensity mapping (green pentagons), and Ly 𝛼 × 21 cm (orange diamonds) is presented as a function of wavenumber in our baseline scenario (see text for details). The three panels correspond to different redshift bins. The cross-correlation is, naturally, the more pragmatic measurement since the 21 cm signal would be severely impacted by foregrounds. Notably, the SNR for the forest is somewhat impeded by the chosen survey volume, although the drop in the number density of quasars leads to difficulties at high redshifts regardless of sensible choices for survey volume. The gray-shaded region corresponds to SNR ≤ 0. \n<!-- image --> \nbins because of a significant rise in aliasing noise 𝑃 2D 𝑤 in Eq. (21) - at those redshifts. \nFurthermore, akin to the findings in Carucci et al. (2017), the cross-correlation can surpass the SNR of the 21 cm auto-correlation at smaller scales. However, this trend is obscured at high redshifts due to the poor sampling of quasars. \nHaving established the interest in the cross-correlation of the Ly 𝛼 forest and 21 cm IM at the redshift range of interest (3 . 5 ≤ 𝑧 ≤ 4), we now ponder the impact of the survey strategy.", '5.2 Importance of survey design': "Our main objective here is to underscore the importance of a nonnegligible overlap between spectroscopic instruments and radio interferometers. Given that SKA1-Low is under development in the southern hemisphere and DESI - also likely DESI II - is in the northern hemisphere, we investigate the potential advantages and insights that could be derived from this cross-correlation with diverse instrumental configurations. \nIn order to reduce the noise in the radio interferometer, a widely employed strategy is to consider an increase in the integration time since the thermal noise is inversely proportional to 𝑡 int . Hence, we explore the impact of a tenfold increase in 𝑡 int across various instrumental configurations - specifically SKA1-Low × DESI-like, SKA1-Low × DESI-MUST-like hybrid, and PUMA-like × DESIMUST-like hybrid. \nWe assume that the Ly 𝛼 forest observations will be completed regardless of the integration time; nevertheless, we reward the longer observation time with a factor of two reduction in the spectrograph effective noise 𝑃 eff 𝑁 . In other words, this implies that the extended observation time allows for more exposures. Note that we do not modify the aliasing error - second term in Eq. (21) that significantly dominates the error budget - even when considering additional observation time. \nFrom Eq. (20), the thermal noise of the radio interferometer is inversely proportional to the total integration time. The reason Figure 3 shows a larger impact for the pairs using SKA1-Low is because the radio telescope dominates the error budget for those setups, i.e. it is larger than the cosmic variance contribution and 𝜎 [ 𝑃 H/i.pc ] > 𝜎 [ 𝑃 𝐹 ] . This is also why the change from DESI-like to DESI-MUST-hybrid does not result in a major improvement and largely conserves the increase in SNR that was present in the 100-hour baseline scenario. In contrast, for the PUMA × MUSTsetup, the Ly 𝛼 forest survey now dominates the error, gains are then possible at larger wavenumbers given the large drop at large scales (as can be seen in Figure 2). We also highlight that the number density of baselines also restricts the shape of the SNR at these scales, which is why the SKA1-Low × MUST can be unexpectedly competitive with PUMA × MUST's SNR at some wavenumbers for a 1000 hours of integration time. \nIt is unclear whether major projects like SKA or a Stage V spectroscopic instrument could allocate 1000 hours/many exposures to observe the same overlapping area, given the promising science programs they aim to address. Nevertheless, it is evident that with such a time investment the community could expedite the progress in measurements of the cross-correlation by a decade, achieving observations competitive with what may be available with Stage II 21 cm radio telescopes - at least at large scales. We acknowledge, however, that this is a significant time commitment. \nNaively, one might expect that expanding the overlapping area would perhaps be a more straightforward enterprise, especially for spectroscopic instruments located in the southern hemisphere like \n<!-- image --> \n<!-- image --> \nFigure 3. SNR for the Ly 𝛼 forest × 21 cm intensity mapping accounting for variations in the integration time of the radio telescopes. We assume that the Ly 𝛼 forest survey is also completed during that integration time and reward the additional integration time (see text for details). Shown are the SNR for: PUMA × MUST 100 hours (blue circles) and 1000 hours (cyan squares); SKA1-Low × DESI 100 hours (green pentagons) and 1000 hours (orange octagons); SKA1-Low × MUST 100 hours (purple crosses) and 1000 hours (magenta diamonds). Increasing the total integration time allows for a better measurement in all configurations. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Similar to Figure 3, but modifying the survey area. Notably, increasing the survey area does little to improve the SNR for the configurations with SKA1-Low since the error is dominated by the interferometric instrument and the thermal noise increases with survey area - see Eq. (20). \n<!-- image --> \nthe 4-metre Multi-Object Spectroscopic Telescope (4MOST; Richard et al. 2019). However, it is important to note that increasing the survey area produces a larger thermal noise - see Eq. (20). Conversely, the number of modes 𝑁 𝑘 is directly proportional to the survey area. Thus, as long as the thermal noise is manageable, an increase in the survey area will improve SNR due to the additional modes. For the PUMA setup, where Ly 𝛼 dominates the error budget at large scales, one could anticipate that the increase in modes may give rise to an enhancement in SNR up to a large 𝑘 value. In contrast, the impact of this choice is expected to generate limited gains for the \nSKA and MUST configurations due to the dominance of the 21 cm error, particularly with a tenfold increase in thermal noise. \nWe confirm these expectations in Figure 4 by augmenting the area of overlap by a factor of ten from the baseline scenario (from 100 to 1000 square degrees). \nThe interpretation of this finding is convoluted. Initially, it may be more appealing to allocate 100 hours of a telescope's time to an observation spanning different pointings. Unfortunately, this approach does not lead to a significant enhancement in observations for the late 2020s configurations. In terms of the overall SNR, aggregated across \nthe 𝑘 -bins, there is only a very modest increase of approximately 1 . 05 compared to the baseline scenario. \nMoreover, if radio telescopes happen to be located in the global south and spectroscopic instruments keep the trend of being located in the northern hemisphere, then the overlap may be small to begin with and hence preference would be given to securing a small overlap in the survey footprints. To our knowledge the possible location of PUMA is not decided yet, consequently, our results advocate for the consideration of overlap with Stage V spectroscopic instruments. Furthermore, our results also showcase that a larger overlap can produce significant gains for a PUMA cross-correlation at large scales. For PUMA, the overall SNR augments by a factor of 1 . 7 when enlarging the survey area by a factor of 10. Besides, this enhancement is limited by the assumption of Stage V instrument performance. In principle, PUMA may instead share the skies with potential Stage VI spectroscopic instruments. \nWhile we have only scratched the surface regarding the implications of survey strategy for the Ly 𝛼 × H/i.pc 21 cm cross-correlation at 3 . 5 ≤ 𝑧 ≤ 4, we deem further exploration of the instrumental setup beyond the scope of this work. This study focuses on demonstrating the impact of cosmic reionization in the cross-correlation during the post-reionization era. Therefore, it would be remiss of us not to address the dependence of the signal on the uncertain timeline of reionization (Montero-Camacho & Mao 2021; Jin et al. 2023).", '5.3 Dependence on reionization history': "The timeline of reionization remains uncertain, although abundant direct detection of the ionizing sources is becoming possible thanks to the James Webb Space Telescope (JWST) (Gardner et al. 2006). In fact, JWST has already started to revolutionize our understanding of galaxy formation at the epoch of reionization (EoR) redshifts (e.g. Bradley et al. 2022; Donnan et al. 2023; Atek et al. 2023; Adams et al. 2023). Nevertheless, even though perfect knowledge of the reionization timeline would help to model the EoR, it would still be non-trivial to translate these constraints into a single mapping of reionization astrophysics. \nConsequently, it is intriguing to ponder the impact of different reionization scenarios in our SNR. We limit our study of reionization scenarios to three distinct reionization timelines. Our fiducial case aligns with Planck's timeline of reionization 𝑧 mid = 𝑧 ( 𝑥 H/i.pc = 0 . 5 ) ≈ 7 . 7, besides its duration, defined as Δ 𝑧 = 𝑧 ( 𝑥 H/i.pc = 0 . 10 ) -𝑧 ( 𝑥 H/i.pc = 0 . 90 ) is 3 . 83. Furthermore, we complement this model with scenarios representing later and earlier reionization scenarios, having midpoints at 𝑧 mid = 6 . 88 and 8 . 41, and durations of Δ 𝑧 = 4 . 15 and 3 . 59, respectively. Note that these additional models are roughly consistent with the 1 𝜎 error reported by Planck Collaboration et al. (2020) but we caution that the best fit from Planck is not able to reproduce astrophysical constraints in a satisfactory way - see Figure 4 of Montero-Camacho et al. (2024a) - due to the use of the hyperbolic tangent reionization model (Lewis 2008). A more adequate choice would be to consider the range of reionization profiles allowed in the Gompertz model of reionization. Its best fit to CMB data, in conjunction with astrophysical constraints, indicates a midpoint of reionization at ≈ 7 (Montero-Camacho et al. 2024a), more closely aligned with our late reionization scenario. \nWe obtain our three reionization models by modifying the ionization efficiency 𝜁 in 21/c.pc/m.pcFAST. This parameter governs the overall timeline of reionization, although there are degeneracies (Pober et al. 2014; Park et al. 2019; Montero-Camacho & Mao 2021). Physically, it quantifies the ability of photons to escape their parent galaxies and reach the intergalactic medium (Dayal & Ferrara 2018). \nTable 1. Summary of the total SNR at 𝑧 = 3 . 61 for the different scenarios considered in this work. The Fiducial scenario corresponds to 100 hours, 100 square degrees of survey area, and a Planck-like reionization scenario. We highlight that increasing the survey area by tenfold results in a marginal increase to the SNR for the baseline instrumental setup (see §5.2). \nIn Figure 5, we illustrate the effect of including the memory of reionization in 𝑃 21 ,𝐹 compared to neglecting its existence. Naturally, the influence of the remnants of reionization is more pronounced in the highest redshift bin and for the delayed reionization scenario. As time progresses, gas dissipates the additional injected energy during reionization, resulting in a lesser effect in the Ly 𝛼 forest. Meanwhile, the modulation of the baryons in shallow potential wells becomes more subdued due to the growth of affected galaxies, as a result, the significance of the memory of reionization in H /i.pc 21 cm gradually diminishes as well. \nAs expected, the primary gains in SNR occur at wavenumbers associated with the reionization-bubble scales, and they are minor at small scales. Note that the maximum importance overlaps with the best window in terms of SNR, as shown in Figure 2. Hence, the crosscorrelation of Ly 𝛼 and 21 cm intensity mapping holds significant promise for shedding light on the astrophysics of reionization and it could be used to indirectly constrain the timeline of reionization through its impact in the post-reionization intergalactic medium. Future work will investigate what could be gained from such efforts (see §6 for a preliminary forecast). \nFurthermore, Figure 5 demonstrates that the impact of inhomogeneous reionization on the post-reionization IGM will increase the SNR for all timelines of reionization and redshifts considered here. In fact, the enhancement can reach 30% at its peak. However, we underscore that there is a fundamental competition in 𝑃 21 ,𝐹 between the response of underdense regions (forest) and that of denser regions (intensity mapping) to the reionization process. It is plausible that the overall effect would be a reduction of the signal-to-noise at redshifts close to the tail end of reionization. Fortunately, further scrutiny of this hypothesis will not require high-mass resolution simulations like the ones used throughout this work since the role of the small-scale structure should be subdued at these redshifts. Nevertheless, the feasibility of such a measurement is at best uncertain given the current capabilities of spectroscopic instruments and the sparse density of quasars at such high redshifts. \nWe summarize our findings regarding observation strategy and dependence with reionization history in terms of signal-to-noise ratio in Table 1. \nFigure 6. Forecast for the ability of SKA1-Low × DESI-like to constrain 𝜎 8 and 𝑛 𝑠 using 100 sq. degrees and 100 hours of integration time. Shown are the confidence ellipses - 1 𝜎 and 2 𝜎 - for different strategies where we consider the case of no reionization (I), i.e. the conventional signal; impact of reionization with known reionization (II), and impact of reionization with uncertain reionization (III). Notably this configuration, which uses only three redshift bins, is not very powerful compared to other cosmological programs expected to be operational by the end of the 2020s; however, it still showcases how reionization relics will jeopardize our ability to constrain cosmology from this observable. \n<!-- image --> \n<!-- image --> \nFigure 5. Impact of reionization history in the signal-to-noise ratio of the cross-correlation assuming the SKA1-Low × DESI baseline configuration, i.e. 𝑡 int = 100 h and 𝑆 survey = 100 deg 2 . Shown are the SNR for a late reionization (purple squares), a Planck-like reionization (orange diamonds), and an early reionization scenario (green pentagons). The behavior at small 𝑘 is driven by Eq. (12). \n<!-- image -->", '6 A SIMPLE FORECAST': "Having explored the potential gains in SNR due to different strategies and instrumental setups, we now turn our attention to quantifying the potential cosmological gains using the Fisher Matrix formalism (Heavens 2009). \nSince no other work has looked at the cross-correlation of Ly 𝛼 × 21 cm at this redshift range, we will forecast based on three different scenarios: (i) Conventional signal, i.e. no memory of reionization, (ii) memory of reionization with perfect reionization knowledge, and (iii) marginalizing over the memory of reionization due to uncertainty regarding its timeline/modeling. Hence, we can illustrate the potential advantage gained by the reionization relics but simultaneously demonstrate the dangers of ignoring this effect. Furthermore, note that our strategies do not include any sort of priors from other datasets 6 , and hence future forecasts are likely to improve in these projections. \nThe Fisher matrix is given by \n𝐹 𝛼𝛽 = 𝑧 -bins ∑︁ 𝑖 𝑘 -bins ∑︁ 𝑗 𝜎 -2 𝑧 𝑖 𝑘 𝑗 [ 𝑃 21 ,𝐹 ] 𝜕𝑃 21 ,𝐹 𝜕𝜃 𝛼 ( 𝑧 𝑖 , 𝑘 𝑗 ) 𝜕𝑃 21 ,𝐹 𝜕𝜃 𝛽 ( 𝑧 𝑖 , 𝑘 𝑗 ) , (24) \nwhere we have assumed uncorrelated errors between the different bins and that the posterior distribution can be reasonably welldescribed by a Gaussian. For simplicity, we consider only two cosmological parameters to forecast 𝜃 𝑖 = { 𝜎 8 , 𝑛 𝑠 } in (i) and (ii) . For (iii) , the conservative point of view on the progress of the field, we add the ionization efficiency 𝜁 to 𝜽 . \nThe uncertainty in Eq. (24) is obtained from Eq. (19). Besides, We use the same three redshift bins centered at 3.61, 3.77, and 3.94 used throughout the rest of this paper. For the wavenumbers, we consider twenty logarithmic bins from 0.01 to 1 Mpc -1 . \n<!-- image --> \nn \nTo compute the derivatives in Eq. (24), we run additional simulations that cover deviations of 3% in 𝜎 8 and 𝑛 𝑠 around their fiducial values of 0.8159 and 0.9667, respectively. Meanwhile, for 𝜁 we also consider 3% variation around 𝜁 = 30 but we check the convergence of our choice for the ionization efficiency in Appendix A. Furthermore, we illustrate some of the tendencies for different properties of this simulation suite in Appendix B. \nWe plot the results of the SKA × DESI-like (PUMA × MUSTlike) forecast in Figure 6 (Figure 7) for the three different strategies \nTable 2. Projected errors for the Fisher forecast of §6. Shown are the 1 𝜎 errors obtained for two instrumental configurations (SKA1-Low × DESI-like and PUMA × MUST-like) and for three different scenarios that correspond to no reionization (I), perfect knowledge of reionization/no marginalization over EoR astrophysics (II), and marginalization over ionization efficiency (III), respectively. Both instrumental setups have 100 hours of integration time and 100 sq. deg. of sky coverage. \nPUMA × MUST \n8 \nFigure 7. Similar to Figure 6, but for PUMA × MUST-like. In contrast to Figure 6, this setup places competitive constraints on the cosmological parameters. Therefore, our results emphasize the importance of considering the impact of reionization to prevent miscalculations of errors and the potential introduction of biases. Note that the combination with other datasets and the use of more than three redshift bins will significantly enhance the real constraining power expected from these instruments. \n<!-- image --> \nconsidered in this work. The confidence ellipses correspond to the 1 and 2 𝜎 contours for the 𝑛 𝑠 × 𝜎 8 plane. In addition, we tabulate the projected errors in Table 2. \nBoth instrumental configurations exhibit a trend of greater ability to constrain the tilt of the primordial power spectrum when the impact of reionization is accounted for and known but when marginalization over the ionization efficiency is required the constraint power for the tilt degrades for the PUMA × MUST-like configuration. This trend of better 𝑛 𝑠 constraints was also present in the forecast for Ly 𝛼 forest done by Montero-Camacho & Mao (2021). As seen in their Figures 9 and 10, which include the memory of reionization in the 3D flux power spectrum from the Ly 𝛼 forest, the tilt plays a more important role than the amplitude. We attribute this to the increase (or decrease) of faint galaxies that would happen by modifying the value of 𝑛 𝑠 while a similar increase in 𝜎 8 will affect the environment more uniformly, hence influencing the ionizing sources to a lesser degree than the tilt. Note that reionization is driven by faint galaxies \nin our 21/c.pc/m.pcFAST simulations. The larger significance of the impact of the tilt is also shown in Figure B1 where the impact of changing 𝑛 𝑠 and 𝜎 8 in the reionization history is illustrated. \nIn the absence of reionization relics, the cross-correlation - without any external information from other cosmological probes - seems to constrain 𝜎 8 more than 𝑛 𝑠 . Interestingly, this is the opposite behavior of the results for H /i.pc 21 cm intensity mapping auto power spectrum albeit with 𝐴 𝑠 instead (see the 1 𝜎 error in Table 4 of Long et al. 2023). Disregarding the difference of 𝜎 8 with 𝐴 𝑠 , this trend could be due to the use of Planck priors in the auto-correlation, the additional parameters (like Ω 's that will likely be degenerate with 𝐴 𝑠 and 𝜎 8 ), or perhaps due to the anti-correlated nature of the crosscorrelation since the presence of extra structure would inevitably lead to more absorption in the Ly 𝛼 forest. This extra absorption or lack of could play a significant role in the range where the SNR is not dominated by the 21 cm error. \nThe Ly 𝛼 × 21 cm cross-correlation will eventually become a competitive probe of the astrophysics that governs cosmic reionization. However, the constraining power for 100 hours of integration time is too weak to offer real insights into the reionization process. This research direction is likely to become promising with a more ambitious cross-correlation program, say 1000 hours in a SKA × MUST-like setup. Unexpectedly, even a conservative 100-hour cross-correlation survey will be highly competitive once PUMA (Cosmic Visions 21 cm Collaboration et al. 2018) starts taking data. For reference, 100 hours of integration time would greatly improve on the projected error for a full DESI (5 years) constraint using the 3D flux power spectrum ( 𝜎 𝜁 = 11 . 6, Montero-Camacho & Mao 2021) by roughly a factor of 3 . Its constraining power would be of similar strength to that obtained by demoting the optical depth to reionization to derivedparameter using symbolic regression and CMB data in conjunction with astrophysical data (Montero-Camacho et al. 2024a).", '6.1 Constraint on the timeline of reionization': "Here, we use the results of our PUMA × MUST-like Strategy III forecast - marginalize over reionization astrophysics - to constrain the timeline of reionization, assuming our Fiducial model correctly represents the Universe's neutral hydrogen fraction. In essence, we assume 𝜁 = 30 and we construct the range of allowed 𝑥 H/i.pc based on our forecasted error 𝜎 𝜁 . \nFor clarity, we decided to only consider the PUMA configuration, which is significantly stronger than the SKA1-Low × DESI-like constraint (the error is smaller by almost a factor of 10). This stronger constraining power also justifies our choice of only considering 𝜁 due to the very small errors for 𝜎 8 and 𝑛 𝑠 expected from this instrumental configuration. \nWe plot our findings in Figure 8 along with the current state of direct and indirect observations (i.e. optical depth constraint from the cosmic microwave background). \nWe emphasize that the width of our inferred constraint is the true value added by our forecast. Our fiducial value, based on the default in 21/c.pc/m.pcFAST/v.pc3 and aligned with Planck's inferred reionization history(Planck Collaboration et al. 2020), provides a reasonable description of the Universe's reionization timeline. However, recent advancements in the more reliable quasar damping wing observations (see e.g. Ďurovčíková et al. 2024; Greig et al. 2024) suggest a slight tension with both our fiducial model and the Planck constraint. Notably, these developments also support the Gompertz reionization scenario (Montero-Camacho et al. 2024a), which successfully fits both CMB data and astrophysical constraints on reionization. Future \nFigure 8. Inferred constraint on the reionization history by our PUMA × MUST setup using 100 hours of integration and assuming a 100 sq. deg. of overlap. Also shown are current observational constraints on the timeline of reionization including dark pixel fraction (McGreer et al. 2015; Jin et al. 2023), high-redshift galaxies through their clustering, luminosity evolution (LF) and equivalent width (EW) (Ouchi et al. 2010; Sobacchi & Mesinger 2015; Mesinger et al. 2015; Mason et al. 2018; Hoag et al. 2019; Mason et al. 2019; Morales et al. 2021), and high𝑧 quasars damping wings (DW) (Greig et al. 2022, 2024; Spina et al. 2024; Ďurovčíková et al. 2024). We also include the Planck indirect constraints on the timeline of reionization (Planck Collaboration et al. 2020), which relies on a hyperbolic tangent to parameterize the reionization process. In addition, we include the best fit using Gompertz reionization (Montero-Camacho et al. 2024a), which uses Planck data but does not rely on the hyperbolic tangent parametrization and provides a better fit to astrophysical observations. Note that we have anchored the constraint around the fiducial model, thus the key feature is the width of the constraint, rather than its exact location. \n<!-- image --> \nwork should consider a fiducial model that better aligns with the Gompertz timeline.", '7 SUMMARY': 'The cross-correlation between the Ly 𝛼 forest and 21 cm intensity mapping in the post-reionization era is a promising cosmological probe of the relatively high-redshift intergalactic medium. In particular, it can probe smaller scales before they become fully non-linear (compared to that of traditional galaxy surveys) and it is pragmatically an easier measurement than the auto-correlation of any of those fields (at 𝑧 ∼ 4) due to foregrounds or available quasars line of sight. However, just as the Ly 𝛼 forest (Montero-Camacho et al. 2019) and 21 cm intensity mapping (Long et al. 2023) are sensitive to relics from cosmic reionization, their cross-correlation will also be biased unless appropriate care is taken to handle this broadband effect. \nRegardless of the impact of reionization, our results demonstrate the importance of overlap between radio interferometers and Stage V spectroscopic instruments. Since some of these telescopes are currently in the early planning/design stage, we underscore the significance of guaranteeing a small degree of overlap between the instruments. As shown in Figure 4, even 100 square degrees of overlap could lead to a detection and consequently, it would enhance observational programs aimed at the post-reionzation era. \nFurthermore, we found that the gain in signal-to-noise is small for increased survey area in instrumental setups that use SKA1Low. This trend is caused by the increase in the SKA1-Low thermal noise. Nonetheless, the PUMA × MUST setup does exhibit significant gains, particularly at large scales with increased survey area, \nsince the error is not dominated by thermal noise. In contrast, a longer integration time leads to considerably better measurements across the board, particularly for SKA configurations. Thus, we conclude that our baseline scenario, which we consider our cheap option with 100 hours of integration time and 100 square degrees of overlap, is a wellsuited observational setup although the measurements can likely be improved by further adjusting the survey strategy. For instance, a SKA1-Low × Stage V spectroscopic instrument is likely to perform comparatively to a Stage II radio interferometer at large scales with 1000 hours of integration time (see Figure 3). We emphasize that this result highlights the importance of progressing further than Stage V spectroscopic instruments in the 2040s. \nThe inclusion of reionization relics in the cross-correlation increases the strength of the signal, particularly at high𝑧 and large scales. Interestingly, late reionization produces an enhancement of up to 30% over the predicted level without reionization imprints at 𝑧 ≈ 4. Separating this novel effect from the cosmological information - for instance using physics-inspired templates (Montero-Camacho et al. 2023) - will allow for unbiased inference of cosmological parameters and it would unseal an original methodology to investigate the astrophysics of reionization. Future work will focus on mitigation strategies for this observable. \nWe have demonstrated the expected impact of reionization in the inference of cosmological parameters using a Fisher forecast. Cosmological parameters would be biased if one neglects the reionization relics and there would also be a significant underestimation of the error. However, if our knowledge of the reionization timeline improves significantly, the inclusion of the memory of reionization can result in a stronger constraint (as was the case for our PUMA × MUST forecast in Figure 7). Furthermore, the cross-correlation of the Ly 𝛼 forest and H /i.pc 21 cm IM will be a promising probe of reionization astrophysics in the next decades. \nOur findings should also be interpreted as a cautionary tale - but simultaneously an exciting opportunity - to other post-reionization era high𝑧 ( 𝑧 ≳ 3) tracers, like Ly 𝛼 emission (e.g. Croft et al. 2016; Renard et al. 2021, 2024) and the CO rotational transitions (Breysse et al. 2022; Chung et al. 2022). Future work will assess the impact of reionization relics in other cosmological tracers of the post-reionization era.', 'ACKNOWLEDGEMENTS': "We are grateful to Chris Hirata for his helpful suggestions and comments. This work was supported by The Major Key Project of PCL. We acknowledge the Tsinghua Astrophysics High-Performance Computing platform at Tsinghua University and PCL's Cloud Brain for providing computational and data storage resources that have contributed to the research results reported within this paper. This work made extensive use of the NASA Astrophysics DataSystem and the following open-source python libraries/packages: matplotlib (Hunter 2007), numpy (Harris et al. 2020), and scipy (Virtanen et al. 2020).", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding authors.', 'APPENDIX A: DEPENDENCE OF FORECAST ON 𝛿𝜁': 'The ionization efficiency 𝜁 controls the timing of reionization in 21/c.pc/m.pcFAST by parametrizing the difficulty for an ionizing photon to escape into the IGM. Consequently, this parameter is not as well constrained or studied as cosmological parameters like 𝜎 8 or 𝑛 𝑠 . Here we confirm that the step size of this uncertain parameter does not have a significant impact on our results. We follow Facchinetti et al. (2023) strategy of checking the impact on the forecasted error instead of the performance of the numerical derivative. \nFigure A1 highlights the small difference between the forecasted 𝜁 errors and justifies our choice of 3% variation in ionization efficiency for our Fisher forecast.', 'APPENDIX B: TRENDS FOR THE FISHER SIMULATIONS': 'Here we show some reionization-related properties of the simulations used in §6 to clarify their hierarchy. \nFor instance, it is clear that 𝑛 𝑠 will be the most constrained parameter by the inclusion of reionization relics since it dominates most of the different metrics in Figure B1. In contrast, a weaker constraint in ionization efficiency is also expected, particularly since ionization efficiency does not affect the first term in Eq. (1). \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \n<!-- image --> \nFigure A1. Convergence of the ionization efficiency error with step size. Shown are the forecasted errors for a variation of 3% (blue pentagons), 10% (purple diamonds), and 20% (green triangles) of the fiducial 𝜁 . Given the small difference between the forecasted errors, we opt for the 3% variation in ionization efficiency. \n<!-- image --> \nFigure B1. Properties of the simulations used in the Fisher forecast of section 6. (Top left) Reionization history. (Top right) Cross-correlation of matter and neutral hydrogen field as a function of redshift and evaluated at a 𝑘 representative of the ionized bubble scale. (Middle left) Relative transparency of the IGM 𝜓 as a function of local reionization redshift, Eq. (5). (Middle right) Relative neutral hydrogen density Ξ , Eq. (6) (Bottom left) The cross-power spectrum of matter and transparency of the model divided by that of the fiducial simulation, i.e. Eq. (7) divided by the reference scenario. (Bottom right) Ratio of the cross-power spectrum of matter and relative neutral hydrogen density, i.e. Eq. (8) normalized by the reference scenario. \n<!-- image -->'}

2024SPIE13092E..1WS

One of the primary science goals of the Habitable Worlds Observatory HWO as defined by the Astro2020 decadal survey is the imaging of the first Earthlike planet around a Sunlike star. A key technology gap towards reaching this goal are the development of ultralownoise photon counting detectors capable of measuring the incredibly low count rates coming from these planets which are at contrasts of 1 10SUP10SUP. Superconducting energyresolving detectors ERDs are a promising technology for this purpose as despite their technological challenges needing to be cooled below their superconducting transition temperature lt 1K they have essentially zero read noise dark current or clockinduced charge and can get the wavelength of each incident photon without the use of additional throughputreducing filters or gratings that spread light over many pixels. The use of these detectors on HWO will not only impact the science of the mission by decreasing the required exposure times for exoEarth detection and characterization but also in a wavefront sensing and control context when used for starlight suppression to generate a dark zone. We show simulated results using both an EMCCD and an ERD to dig a dark zone demonstrating that ERDs can achieve the same final contrast as an EMCCD in about half of the total time. We also perform a simple case study using an exposure time calculator tool called the Error Budget Software EBS to determine the required integration times to detect water for HWO targets of interest using both EMCCDs and ERDs. This shows that once a dark zone is achieved using an ERD can decrease these exposure times by factors of 1.52 depending on the specific host star properties.

2024-08-01T00:00:00Z

['arXiv:2409.05987', '10.1117/12.3020603', '10.48550/arXiv.2409.05987', '2024arXiv240905987S', '2024SPIE13092E..1WS']

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

Simulated performance of energyresolving detectors towards exoplanet imaging with the Habitable Worlds Observatory

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

https://arxiv.org/pdf/2409.05987.pdf

{'Simulated performance of energy-resolving detectors towards exoplanet imaging with the Habitable Worlds Observatory': "Sarah Steiger a , Laurent Pueyo a , Emiel H. Por a , Pin Chen b , R'emi Soummer a , Raphael Pourcelot a , Iva Laginja c , and Vanessa P. Bailey b \na Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA b Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA 91109, USA \nc LESIA, Observatoire de Paris, Universit'e PSL, Sorbonne Universit'e, Universit'e Paris Cit'e", 'ABSTRACT': "One of the primary science goals of the Habitable Worlds Observatory (HWO) as defined by the Astro2020 decadal survey is the imaging of the first Earth-like planet around a Sun-like star. A key technology gap towards reaching this goal are the development of ultra-low-noise photon counting detectors capable of measuring the incredibly low count rates coming from these planets which are at contrasts of ∼ 1 × 10 -10 . Superconducting energyresolving detectors (ERDs) are a promising technology for this purpose as, despite their technological challenges, needing to be cooled below their superconducting transition temperature ( < 1K), they have essentially zero read noise, dark current, or clock-induced charge, and can get the wavelength of each incident photon without the use of additional throughput-reducing filters or gratings that spread light over many pixels. The use of these detectors on HWO will not only impact the science of the mission by decreasing the required exposure times for exo-Earth detection and characterization, but also in a wavefront sensing and control context when used for starlight suppression to generate a dark zone. We show simulated results using both an EMCCD and an ERD to 'dig a dark zone' demonstrating that ERDs can achieve the same final contrast as an EMCCD in about half of the total time. We also perform a simple case study using an exposure time calculator tool called the Error Budget Software (EBS) to determine the required integration times to detect water for HWO targets of interest using both EMCCDs and ERDs. This shows that once a dark zone is achieved, using an ERD can decrease these exposure times by factors of 1.5-2 depending on the specific host star properties. \nKeywords: exoplanets, energy-resolving detectors, detectors, Habitable Worlds Observatory, HWO", '1. INTRODUCTION': "The Astro2020 decadal survey recommends the development of a large IROUV space telescope, now known as the Habitable Worlds Observatory (HWO), with the ambitious goal of imaging ∼ 100 star systems and talking spectra of ∼ 25 rocky planets to search for biosignatures that could be indicative of life. This will require the telescope and coronagraph system to disentangle planetary signals that are tens of billions of times fainter than the stars they orbit with planetary count rates on the order of 1-50 photons/hour/ µ m around even the nearest Sun-like stars ( ∼ 10 pc). 1 Due to these incredibly low count rates, a key technology gap that has been highlighted for HWO are low-noise, photon-counting detectors that will be able to make these measurements with the required signal-to-noise ratio (SNR) in reasonable exposure times. \nSuperconducting, energy-resolving detector technologies (ERDs) are of particular interest for this application as, unlike their semiconducting counterparts (e.g. CCDs, EMCCDs, or CMOS detectors), they have no read noise or dark current. They are also 'radiation hard', where radiation sources such as cosmic rays have no lasting impact on the detector array and can easily be identified and removed from further analysis. Perhaps most importantly, many are also inherently energy-resolving: spectra can be directly obtained without a spectrograph, which typically reduces throughput and spreads out light over many pixels. \nS.S. e-mail: ssteiger@stsci.edu \nFigure 1. HWO EAC1-like optical model used for all WFS&C simulations. \n<!-- image --> \nTwo of these superconducting ERDs that have risen to the forefront for astronomical applications are microwave kinetic inductance detectors (MKIDs) and transition edge sensors (TESs). For an MKID, each pixel is a superconducting microwave resonator which measures incident photons through the breaking of Cooper pairs (the charge carriers in the superconductor) which changes the inductance of the material through an effect called the kinetic inductance effect. This causes a shift in the resonant frequency of the pixel where the magnitude of this shift is proportional to the amount of Cooper pairs broken or, equivalently, the energy of the incident photon. 2 Since each pixel can be designed with its own unique resonant frequency, MKIDs have the benefit of easy multiplexability with large-format IROUV instruments in use for astronomy for over a decade. 3-5 TESs are microcalorimeters where a superconducting film is biased near its superconducting transition temperature allowing for small changes in temperature (caused by a photon event) to create a large resistive signal where the size of this signal is proportional to the photon's energy. 6 While TES arrays are not inherently multiplexable, many schemes have been tested to generate large format arrays such as coupling to superconducting quantum interference devices or, more recently, using kinetic inductance current sensors. 7, 8 TESs have also achieved better resolving powers in the optical ( R ∼ 90) 9 than MKIDs ( R ∼ 50) 10 though fundamentally both MKIDs and TESs can achieve the R ∼ 140 needed for bio-signature detection with incremental development from the current state-of-the-art. For this work, we will assume that the energy-resolving performance of these detectors has reached these sufficient levels and will be agnostic to either technology since we are focused on their similar high-level noise properties and ability to discern the energy of incident photons with no dispersive optics. \nInitial trade studies for HWO have already shown that the use of ERDs can increase exo-Earth yields by up to 30% over state-of-the-art semiconductor detectors as well as get spectra of 8 × as many planets by virtue of constantly collecting spectra. 11 These studies show that while superconducting detectors add complexity to the mission by needing to be cooled to cryogenic temperatures ( < 1 K), their potential benefits could not only outweigh these complexities, but even potentially enable the science goals of the mission. \nIn addition to the increase in exo-Earth candidate yield and incidental spectra when using an ERD, in recent years it has become clear that for HWO to achieve its science mission goals, the whole observatory system (from the telescope, to the coronagraph, to the detector) will need to be optimized together to maximize science output. In this work we will explore how ERDs can be integrated with a coronagraphic system and improve the performance of 'digging a dark zone' in a wavefront sensing and control (WFS&C) loop to reduce overheads and allow more mission time to be dedicated to performing science. \nWe start by outlining our optical model and simulation scheme in Section 2, including the detector implementations and assumptions for both EMCCDs and an ERD. We then present the results of the efficiency of using an ERD or EMCCD to dig a dark zone in Section 3. We end in Section 4 with a brief case study using an exposure time calculator tool called the Error Budget Software (EBS) to further demonstrate the increased efficiency of using an ERD for science exposures after a dark zone has been achieved. \nnm \nFigure 2. Input piston, and tip/tilt (PTT) aberrations (surface, in nm) to the primary mirror in the optical model to achieve a starting contrast of ∼ 1 × 10 -6 . \n<!-- image -->", '2.1 Telescope Model': "The optical model used for this work was generated using the same infrastructure as the optical model for the High-contrast Imager for Complex Aperture Telescopes (HiCAT) testbed 12-14 at the Space Telescope Science Institute (STScI). To match the notional exploratory analytic case 1 (EAC1) design for HWO, this existing optical model was updated with the parameters found in Table 1 and illustrated in Figure 1. \nFor this study, an apodized-pupil Lyot coronagraph (APLC) design was chosen for starlight suppression. Specifically, we used the 2-Hex design from Nickson et. al (2022) 15 with 0.2% tolerance to Lyot Stop misalignment, see their Figure 9. We also adopted the same focal-plane mask (FPM) and Lyot stop specifications as is found in that paper. In the presence of no additional aberrations, this design has a mean contrast of 3 . 6 × 10 -11 from 3 . 5 -12 λ/D . \nIn addition to the optical model, we also made use of HiCAT's existing simulator to perform our WFS&C experiments. This simulator uses the open-source Control and Automation for Testbeds Kit 2 (catkit2) 16 as its backend, which allows for high-fidelity simulations of all of the mechanical components of a high-contrast imaging system. Relevant to this work, this includes realistic camera modules, and deformable mirror (DM) control channels. \nIn order to simulate a more realistic observing scenario, we introduced piston and tip/tilt (PTT) aberrations onto the segmented primary mirror (see Figure 2) to achieve a starting contrast of 1 . 2 × 10 -6 which was the starting place from which all WFS&C experiments were run.", '2.2 WFS&C Strategy': "In order to image faint Earth-like planets at contrasts of 10 -10 , diffracted starlight in the focal plane caused by non-common path errors and telescope imperfections needs to be corrected for and removed in the region of interest. This process is typically called 'digging a dark zone' and uses DMs to sense and remove unwanted stellar contamination at the location of potential planets. \nTo dig our dark zone, we used electric field conjugation (EFC) 20 with pairwise probing. In this experiment we used four sets of single actuator probes with wavefront error amplitudes of 10 nm, where each probe set contains a positive and negative probe for a total of 8 probe images per EFC iteration - for more details, see Soummer et. al. (2024). 14 \nTable 1. Optical model parameters.", '2.3 Stellar Flux Calibration': 'Count rates were used to mimic the observing of a star with a V-band magnitude of 8 and were calibrated using the exoscene ∗ library, assuming an optimistic total system throughput of 50%. These count rates were also cross-checked using the EXOSIMS 21 package and found to agree to within a factor of 2. A magnitude 8 star was chosen to reflect the dimmer stars found in the NASA Exoplanet Exploration Program (ExEP) mission star list for the Habitable Worlds Observatory 22 and a typical V-band magnitudes as found in the the Habitable Worlds Observatory preliminary input catalog (HPIC). 23', '2.4 Detector Implementations': "We made use of the EMCCDDetect † package for simulating EMCCD images given a 'perfect' input wavefront which is based on work by Nemati 2020b. 24 This package adequately handles the main sources of noise in the EMCCD including clock-induced charge (CIC), dark current ( ζ ), and read noise (RN). \nWe assume three different detector scenarios whose properties are summarized in Table 1. These reflect the nominal HWO detector as recommended by the LUVOIR and HabEx reports (HWO EMCCD), a detector that has the same properties as the detector in the coronagraphic instrument on the Nancy Grace Roman Space Telescope at its beginning-of-life performance (Roman EMCCD BOL), and a noiseless ERD (ERD). \nIn this study we additionally assume an optimistic quantum efficiency (QE) of 1.0 for the ERD and 0.9 for the HWO EMCCD. We chose to ignore the effects of cosmic rays though think this is an interesting path for future work due to the fundamentally different ways that cosmic rays appear in and ERD compared to an EMCCD. Finally, Poisson noise is added to each image using the HCIPy 25 large poisson utility. \nFigure 3. Final contrast achieved as a function of exposure time per probe image (bottom axis) and total time per EFC iteration (top axis) for an HWO EMCCD (orange), a Roman EMCCD (pink), and an ERD (purple). \n<!-- image -->", '2.4.1 Modeling of an Integral Field Unit (IFU)': "Following the procedure outlined in Nemati et. al. 2020a, 26 to model an IFU a sampling of 4 pixels per lenslet is chosen. When calculating a broadband WFS&C solution, only a small number of spectral elements ( N spec ) over the bandpass is desired to maximize throughput and reduce necessary computation time. In general, \nN spec = R · BW, (1) \nwhere R is the spectral resolution of the IFU and BW is the instrument bandwidth. It then follows that the number of pixels light would be spread over ( N p ) is \nN p = N spec · l samp , (2) \nwhere l samp = 4 is the lenslet sampling. If a dedicated IFU with an R = 30 and BW = 10% is flown on HWO to be used primarily for WFS&C, then N spec = 3 and the factor of 4 to account for the IFU lenslet sampling would be the only multiplicative factor on the detector noise that needs to be considered. This resolution however is too small to detect key bio-signatures and so a second higher resolution 'science-grade' IFU would be needed. \nIf instead a science grade IFU with an R = 140 is used for WFS&C then the detector noise contribution will be much higher. In this case N spec = 140 · 0 . 1 = 14 with the light being spread out over N pix = 14 · 4 = 56 pixels. Rebinning this to the desired spectral resolution ( R = 30) to maximize throughput would mean that the pixel noise contribution per spectral element is 140 30 · 4 ∼ 19 pixels. In this work, we consider this science-grade IFU ( R = 140) case where the detector noise is multiplied by a factor of 19. It is important to note that the ERD does not have these considerations since it can effectively be thought of as an IFU with a variable R and does all wavelength binning in post-processing.", '3. RESULTS': "In this work, we explored four WFS&C strategies and their impacts on the overhead to dig a dark zone and begin taking science observations of exo-Earth candidates. They are (1) using an imaging EMCCD (i.e., no dispersive optics to get wavelength information), (2a) Using an HWO-like EMCCD as part of a science-grade IFU, (2b) using a Roman-like EMCCD as part of a science-grade IFU, and (3) using an ERD. \nFigure 4. Left: Contrast images applying the final DM solution to the ideal (noiseless) optical model for an ERD (top) and EMCCD (bottom). Right: Final camera images at that same base contrast showing the number of counts present in the dark zone of the final images for the ERD (top) and EMCCD (bottom). These images are for t exp = 5 minutes. \n<!-- image --> \nIn case (1) there is an obvious overhead increase due to the need to take exposures at multiple wavelengths to calculate a broadband control solution. For example, consider needing t exp = 1000 seconds of exposure time per spectral bin to acquire enough signal on the detector (at contrasts of 10 -10 ) with N λ = 3 spectral bins over the bandpass. If pairwise probing is used to sense the electric field then it is reasonable to apply between 2-4 sets of probes (pessimistically let us set N p = 8 probes total) per wavelength bin. This results in a total exposure time T = t exp · N λ · N p = 1000 · 3 · 8 = 24000 seconds (6.6 hours) spent sensing the electric field for only a single EFC iteration. \nUsing an ERD or IFU, one broadband image per probe would be used instead which, assuming a flat spectrum over the bandpass, results in needing the same exposure time per probe ( t exp = 1000 s) but with the added benefit of getting all wavelengths simultaneously. This means that only one set of probe images is needed and so T = t exp · N p = 1000 · 8 = 8000 seconds (2.2 hours) per iteration which is a factor of N λ less than the imaging case (see also Table 3). It is worth mentioning that for this application an ERD would only require a modest energy resolution of R ∼ E ∆ E = 600nm 20nm = 30 to resolve 3 spectral bins in 10% bandpass at 600 nm for which both MKIDs and TESs have already surpassed. 9, 10 \nCases (2a), (2b), and (3) are all sensitive to wavelength information and so are instead differentiated by how the detector noise impacts the required exposure times needed to properly sense the electric field or, equivalently, how well the electric field can be sensed and corrected for a given fixed exposure time. In order to test these cases, we ran WFS&C simulations using the detector models described in Section 2.4 and compared the final dark zone contrast as a function of exposure time per image ( t exp ) for an HWO-like EMCCD, Roman-like EMCCD, and ERD. The results are found in Figure 3. Here the final contrast was determined by taking the final DM solution for each scenario, applying it to the optical model, and then calculating the mean contrast in the dark zone using an entirely noiseless detector model. A noiseless detector model is used here since the final images using the noisy detector models contain between 0-2 counts per pixel for all cases. Therefore, using the normal method for determining contrast for these noisy models (taking the mean of all the values in the dark zone) would not be appropriate. At these count rates, using this method would result in the pixels containing no photons artificially deflating the measured contrast - see Figure 4. \nFigure 5. Contrast vs. iteration for the HWO EMCCD (orange) and ERD (purple) with the camera images at select iterations inset. This data is for t exp = 2 minutes. \n<!-- image --> \nTable 2. Time to DZ ContrastTable 3. Time needed to reach a given dark zone (DZ) contrast for different detector WFS&C scenarios. The numbers are all normalized to the ERD with the multiplicative factors denoting how much longer each scenario would take as compared to an ERD. The 'true' times to reach the given contrast are found in the subscripts though it should be noted that these 'true' values will be highly sensitive to particular choices in simulated observing parameters such as stellar magnitude or instrument throughput. \nIn Figure 3, it can be seen that at short exposure times, the ERD outperforms both EMCCD scenarios by about a factor of 2 until the point at which the exposure time is long enough that the detector noise threshold is surpassed and the two cases converge. Figure 5 shows the contrast vs. EFC iteration for the t exp = 2 minutes case with insets showing the coronagraphic camera images for select iterations. At the fifth iteration where the two curves start to diverge, it can be seen that the additional noise in the EMCCD image is beginning to be on par with the speckle signal likely leading to less efficient sensing of the electric field at these count rates. For a summary of how these scenarios effect the total overall time to achieve a given dark zone contrast, see Table 3.", '4. ETC CASE STUDY: DETECTOR NOISE': 'Clock induced charge (CIC), dark current ( ζ ), and read noise (RN) are the three main sources of detector noise that are modeled by yield codes and exposure time calculators (ETCs) to determine the time-to-SNR for an observation, i.e. how long one needs to observe a target to reach the desired SNR for detection or characterization. As part of the coronagraph technology roadmap (CTR) group funded by the ExEP Office, an ETC was developed to study the effects of stability, wavefront error, and wavefront sensing and control systems \non the expected exposure times for exo-Earth candidate detection and characterization called the Error Budget Software (EBS) ‡ . EBS is fundamentally a wrapper for another ETC and yield code called EXOSIMS 21 which it uses to perform all of its backend calculations. \nHere we used EBS to perform a simple case study of how detector noise impacts the time to reach SNR=5 for water detection (R=140) for 5 fiducial HWO target stars as identified by the ExEP Mission Star List for the Habitable Worlds Observatory. 22 A lenslet sampling of 2 was used meaning that the detector noise contribution is coming from spreading the light over 4 pixels per spectral element. The results are found in Figures 6 and 7. The vertical lines denote the detector noises that correspond to the HWO EMCCD (black), the Roman EMCCD beginning-of-life (BOL) performance (dark gray) and the expected Roman end-of-life (EOL) performance (light gray). Figure 7 is a zoom-in of Figure 6 with a linear scaling applied to better differentiate the exposure time differences around the detector properties of interest. \nDepending on spectral type, the increase in exposure times required to achieve the same SNR detection moving from an ERD to the HWO EMCCD ranges between factors of 1.5-2.0.', '5. DISCUSSION AND FUTURE WORK': "This work has shown that using an ERD for digging a dark zone could cut the overhead needed to do so by up to a factor of 2 over even a low-noise EMCCD such as the ones budgeted by the LUVOIR and HabEx reports for HWO. Over a mission lifetime, this translates into potentially thousands of hours that could be spent on additional exo-Earth characterization observations (which has been shown in this work and others to also take less time using an ERD) or executing other general astrophysics programs. \nIn order to fully quantify the effects of using an ERD in a WFS&C context beyond what is presented in this work, more detailed detector models and simulations will be required. First, this work assumed that for all EFC iterations, t exp remains constant. In reality, shorter exposures would be required at the beginning of the digging process and get progressively longer as the signal in the dark zone diminishes and so the effect of having adaptive exposure times on contrast is an obvious next step. The parameters used in the EFC (which were fixed for all experiments), such as probe amplitude, and conditioning number, could also be optimized for better overall contrast performance. Additionally, more simulations should be run to more finely sample the range of exposure times and achieve more accurate DZ digging factors as are found in Table 3. For example, since only a finite amount of exposure times were sampled, if a given contrast was just barely not reached for a given t exp , then the time to DZ contrast would need to be calculated using the next highest t exp studied (say t exp = 5 min instead of 2 min) whereas, in reality, an exposure time in between those two would yield the most optimal time to a given contrast for that WFS&C scenario. \nMore accurate ERD models should also be used since adapting identically 0 noise for these technologies is neither accurate nor realistic. Adopting values of 0 here has traditionally been an acceptable approximation since the noise for an ERD is significantly lower than that of their semiconducting counterparts, but at these very low photon count rates getting these details correct is important. This study has also been agnostic as to which energy-resolving detector technology is chosen, but adapting more realistic TES or MKID specific detector models should be explored since the output format of ERDs is fundamentally different than semiconductor-based detectors. For example, ERDs are truly photon counting and so the outputs are not integrated images, but photon lists where each photon is tagged with its pixel location, energy, and arrival time that can be operated on entirely in post-processing. 27 Such models will additionally allow the exploration of using photon statistical post-processing techniques in the science 'exposures' such as those presented in Steiger et. al. (2021). 28 \nThe choice of using an ERD for a future exo-Earth imaging flagship will have wide-reaching impacts on every aspect of the observatory from digging the dark zone, to collecting science data, to post-processing. For this reason, when looking at trade studies involving these detectors, all of these aspects should be taken into consideration to fully quantify their potential benefits. \nime (hours) \nIntegration T \nDetector Noise (CIC + + RN pixel 1 s 1 ) \n<!-- image --> \nFigure 6. Integration times required to reach a signal-to-noise ratio of 5 for H 2 O detection ( λ =1000 nm, R=140) for 5 fiducial HWO target stars, one F spectral type (HIP 32439, top), two G spectral types (HIP 77052 and HIP 79672, middle), and two K spectral types (HIP 26779 and HIP 113283, bottom). In each plot, the solid colored lines represent the inner habitable zones and the dot-dashed colored lines represent the outer habitable zones. The black vertical dot-dashed lines denote the detector noise corresponding to the HWO EMCCD, the grey vertical dashed lines denote the end-of-life Roman EMCCD detector parameters, and the grey dotted line represents the beginning-of-life Roman EMCCD detector. 19 An energy-resolving detector with 0 detector noise would be off the left-hand side of this plot. \nRequired Integration Time (hr, SNR=5.0) \nFigure 7. Same as Figure 6, but zoomed in and put on a linear scale. A solid black line denoting an ERD was also added at 0 detector noise. \n<!-- image -->", 'ACKNOWLEDGMENTS': "S.S. is supported by an STScI Postdoctoral Fellowship. E.H.P. was supported in part by the NASA Hubble Fellowship grant #HST-HF2-51467.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. I.L. acknowledges partial support from a postdoctoral fellowship issued by the Centre National d'Etudes Spatiales (CNES) in France. \nThe HiCAT testbed has been developed over the past 10 years and benefitted from the work of an extended collaboration of over 50 people. This work was supported in part by the National Aeronautics and Space Administration under Grant 80NSSC19K0120 issued through the Strategic Astrophysics Technology/Technology Demonstration for Exo-planet Missions Program (SAT-TDEM; PI: R. Soummer), and under Grant 80NSSC22K0372 issued through the Astrophysics Research and Analysis Program (APRA; PI: L. Pueyo). \nThis research made use of HCIPy, an open-source object-oriented framework written in Python for performing end-to-end simulations of high-contrast imaging instruments (Por et al. 2018).", 'REFERENCES': "- [1] Villanueva, G. L., Smith, M. D., Protopapa, S., Faggi, S., and Mandell, A. M., 'Planetary Spectrum Generator: An accurate online radiative transfer suite for atmospheres, comets, small bodies and exoplanets,' JQSRT 217 , 86-104 (Sept. 2018).\n- [2] Day, P. K., LeDuc, H. G., Mazin, B. A., Vayonakis, A., and Zmuidzinas, J., 'A broadband superconducting detector suitable for use in large arrays,' Nature 425 , 817-821 (Oct. 2003).\n- [3] Mazin, B. A., Meeker, S. R., Strader, M. J., Szypryt, P., Marsden, D., van Eyken, J. C., Duggan, G. E., Walter, A. B., Ulbricht, G., Johnson, M., Bumble, B., O'Brien, K., and Stoughton, C., 'ARCONS: A 2024 Pixel Optical through Near-IR Cryogenic Imaging Spectrophotometer,' PASP 125 , 1348 (Nov. 2013).\n- [4] Meeker, S. R., Mazin, B. A., Walter, A. B., Strader, P., Fruitwala, N., Bockstiegel, C., Szypryt, P., Ulbricht, G., Coiffard, G., Bumble, B., Cancelo, G., Zmuda, T., Treptow, K., Wilcer, N., Collura, G., Dodkins, R., Lipartito, I., Zobrist, N., Bottom, M., Shelton, J. C., Mawet, D., van Eyken, J. C., Vasisht, G., and Serabyn, E., 'DARKNESS: A Microwave Kinetic Inductance Detector Integral Field Spectrograph for High-contrast Astronomy,' PASP 130 , 065001 (June 2018).\n- [5] Walter, A. B., Fruitwala, N., Steiger, S., Bailey, John I., I., Zobrist, N., Swimmer, N., Lipartito, I., Smith, J. P., Meeker, S. R., Bockstiegel, C., Coiffard, G., Dodkins, R., Szypryt, P., Davis, K. K., Daal, M., Bumble, B., Collura, G., Guyon, O., Lozi, J., Vievard, S., Jovanovic, N., Martinache, F., Currie, T., and Mazin, B. A., 'The MKID Exoplanet Camera for Subaru SCExAO,' PASP 132 , 125005 (Dec. 2020).\n- [6] 'Transition-Edge Sensors,' in [ Cryogenic Particle Detection ], Enss, C., ed., 99 , 63 (2005).\n- [7] Kiviranta, M., Seppa, H., van der Kuur, J., and de Korte, P., 'SQUID-based readout schemes for microcalorimeter arrays,' in [ Low Temperature Detectors ], Porter, F. S., McCammon, D., Galeazzi, M., and Stahle, C. K., eds., American Institute of Physics Conference Series 605 , 295-300, AIP (Feb. 2002).\n- [8] Szypryt, P., Bennett, D. A., Fogarty Florang, I., Fowler, J. W., Giachero, A., Hummatov, R., Lita, A. E., Mates, J. A. B., Nam, S. W., O'Neil, G. C., Swetz, D. S., Ullom, J. N., Vissers, M. R., Wheeler, J., and Gao, J., 'Kinetic inductance current sensor for visible to near-infrared wavelength transition-edge sensor readout,' arXiv e-prints , arXiv:2405.15017 (May 2024).\n- [9] 'Ultra-high efficiency noiseless quantum sensors for hwo and qis..' https://techport.nasa.gov/view/ 146757 . Accessed: 2024-06-08.\n- [10] de Visser, P. J., de Rooij, S. A. H., Murugesan, V., Thoen, D. J., and Baselmans, J. J. A., 'Phonon-TrappingEnhanced Energy Resolution in Superconducting Single-Photon Detectors,' Physical Review Applied 16 , 034051 (Sept. 2021).\n- [11] Howe, A. R., Stark, C. C., and Sadleir, J. E., 'The Scientific Impact of a Noiseless Energy-Resolving Detector for a Future Exoplanet-Imaging Mission,' arXiv e-prints , arXiv:2405.08883 (May 2024).\n- [12] Moriarty, C., Brooks, K., Soummer, R., Perrin, M., Comeau, T., Brady, G., Gontrum, R., and Petrone, P., 'High-contrast imager for complex aperture telescopes (HiCAT): 6. software control infrastructure and calibration,' in [ Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave ], Lystrup, M., MacEwen, H. A., Fazio, G. G., Batalha, N., Siegler, N., and Tong, E. C., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10698 , 1069853 (Aug. 2018).\n- [13] Soummer, R., Por, E. H., Pourcelot, R., Redmond, S., Laginja, I., Will, S. D., Perrin, M. D., Pueyo, L., Sahoo, A., Petrone, P., Brooks, K. J., Fox, R., Klein, A., Nickson, B., Comeau, T., Ferrari, M., Gontrum, R., Hagopian, J., Leboulleux, L., Leongomez, D., Lugten, J., Mugnier, L. M., N'Diaye, M., Nguyen, M., Noss, J., Sauvage, J.-F., Scott, N., Sivaramakrishnan, A., Subedi, H. B., and Weinstock, S., 'High-contrast imager for complex aperture telescopes (HiCAT): 8. Dark zone demonstration with simultaneous closedloop low-order wavefront sensing and control,' in [ Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave ], Coyle, L. E., Matsuura, S., and Perrin, M. D., eds., Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series 12180 , 1218026 (Aug. 2022).\n- [14] Soummer, R., Pourcelot, R., Por, E., Steiger, S., Laginja, I., Buralli, B., Pueyo, L., Nguyen, M., Nickson, B., Sahoo, A., and the extended HiCAT team, 'High-contrast imager for complex aperture telescopes (hicat): 11. system-level static and dynamic demonstration of the apodized pupil lyot coronagraph with a segmented aperture.,' Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 13092 (2024)."}

2024A&A...692A..46D

Context. Theoretical descriptions of convective overshooting in stellar interiors often rely on a basic onedimensional parameterization of the flow called the filling factor for convection. Several different definitions of the filling factor have been developed for this purpose based on 1 the percentage of the volume 2 the mass flux and 3 the convective flux that moves through the boundary. Aims. We examine these definitions of the filling factor with the goal of establishing their ability to explain differences between 2D and 3D global simulations of stellar interiors that include fully compressible hydrodynamics and realistic microphysics for stars. Methods. We study convection and overshooting in pairs of identical twodimensional 2D and threedimensional 3D global simulations of stars produced with MUSIC a fully compressible timeimplicit hydrodynamics code. We examine pairs of simulations for 1 a 3 MSUBSUB red giant star near the first dredgeup point 2 a 1 MSUBSUB premainsequence star with a large convection zone 3 the current sun and 4 a 20 MSUBSUB mainsequence star with a large convective core. Results. Our calculations of the filling factor based on the volume percentage and the mass flux indicate asymmetrical convection near the surface for each star with an outer convection zone. However near the convective boundary convective flows achieve inwardoutward symmetry for each star that we study for 2D and 3D simulations these filling factors are indistinguishable. A filling factor based on the convective flux is contaminated by boundarylayerlike flows making a theoretical interpretation difficult. We present two possible new alternatives to these frequently used definitions of a filling factor which instead compare flows at two different radial points. The first alternative is the penetration parameter of Anders et al. 2022 ApJ 926 169. The second alternative is a new statistic that we call the plume interaction parameter. We demonstrate that both of these parameters captures systematic differences between 2D and 3D simulations around the convective boundary.

2024-12-01T00:00:00Z

['arXiv:2409.09815', '2024A&A...692A..46D', '10.48550/arXiv.2409.09815', '2024arXiv240909815D', '10.1051/0004-6361/202451814']

['convection', 'hydrodynamics', 'stars: evolution', 'stars: interiors', 'Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'Physics - Computational Physics']

The shape of convection in 2D and 3D global simulations of stellar interiors

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

https://arxiv.org/pdf/2409.09815.pdf

{'The shape of convection in 2D and 3D global simulations of stellar interiors': 'M.-G. Dethero 2 , J. Pratt 1 , 2 , 3 , D.G. Vlaykov 3 , I. Bara ff e 3 , 4 , T. Guillet 3 , T. Go ff rey 5 , A. Le Saux 6 , and A. Morison 3 \n- 1 Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA e-mail: pratt34@llnl.gov\n- 2 Department of Physics and Astronomy, Georgia State University, Atlanta GA 30303, USA\n- 3 Astrophysics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, EX4 4QL Exeter, United Kingdom\n- 4 École Normale Supérieure de Lyon, CRAL (UMR CNRS 5574), Université de Lyon 1, 69007 Lyon, France\n- 5 Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom\n- 6 Laboratoire de Meteorologie Dynamique (IPSL), Sorbonne University, CNRS, Ecole Polytechnique, Ecole Normale Superieure, Paris, France \nSeptember 17, 2024', 'ABSTRACT': 'Context. Theoretical descriptions of convective overshooting in stellar interiors often rely on a basic one-dimensional parameterization of the flow called the filling factor for convection. Several di ff erent definitions of the filling factor have been developed for this purpose, based on: (1) the percentage of the volume, (2) the mass flux, and (3) the convective flux that moves through the boundary. \nAims. We examine these definitions of the filling factor with the goal of establishing their ability to explain di ff erences between 2D and 3D global simulations of stellar interiors that include fully compressible hydrodynamics and realistic microphysics for stars. Methods. We study convection and overshooting in pairs of identical two- and three-dimensional global simulations of stars produced with MUSIC , a fully compressible, time-implicit hydrodynamics code. We examine pairs of simulations for (1) a 3 M ⊙ red giant star near the first dredge-up point, (2) a 1 M ⊙ pre-main-sequence star with a large convection zone, (3) the current sun, and (4) a 20 M ⊙ main-sequence star with a large convective core. \nResults. Our calculations of the filling factor based on the volume percentage and the mass flux indicate asymmetrical convection near the surface for each star with an outer convection zone. However, near the convective boundary, convective flows achieve inwardoutward symmetry for each star that we study; for 2D and 3D simulations, these filling factors are indistinguishable. A filling factor based on the convective flux is contaminated by boundary-layer-like flows, making a theoretical interpretation di ffi cult. We present two possible new alternatives to these frequently used definitions of a filling factor, which instead compare flows at two di ff erent radial points. The first alternative is the penetration parameter of Anders et al. (2022). The second alternative is a new statistic that we call the plume interaction parameter . We demonstrate that both of these parameters captures systematic di ff erences between 2D and 3D simulations around the convective boundary. \nKey words. Methods: numerical - Convection - Stars: interiors - Stars: evolution', '1. Introduction': "Predicting the evolution of stars requires a prescription for the amount of mixing caused by convection at di ff erent values of the internal radius of the star. This mixing has been widely linked to the concept of a filling factor for the convective inflows (e.g. Schmitt et al. 1984; Stein & Nordlund 1989; Cattaneo et al. 1989; Toomre et al. 1990; Cattaneo et al. 1991; Zahn 1991; Canuto & Dubovikov 1998; Brummell et al. 2002; Rempel 2004; Browning et al. 2004; Rogers et al. 2006; Käpylä et al. 2017; Cai 2020). Conceptually, a filling factor quantifies, based on fluid motion or heat, how much of the star is moving inward at a given radius. Several early works on stellar convection (e.g. Schmitt et al. 1984; Stein & Nordlund 1989; Hurlburt et al. 1984) predict that the large stratification present in stellar convection zones should produce convective motions that have a pronounced asymmetry between inflows and outflows, corresponding to a filling factor less than one-half. Indeed observations of the solar surface find strong, coherent inflows surrounded by weaker, more di ff use outflows (e.g. as discussed in Nordlund et al. 2009). Where present, such an asymmetry would \nalter the shear interaction between opposing flow structures, and thus change the amount of fluid mixing. The filling factor has thus been viewed as a fundamental measure of how the stellar convection zone is structured. \nEvaluating the filling factor at the boundary of convective instability provides a measure of how much fluid overshoots the convection zone and enters the radiative zone. This provides a link between the structure of convective flows in the convection zone and the depth of convective overshooting or penetration. Several works (Zahn 1991; Brummell et al. 2002) suggest that a filling factor could play a more significant role in determining the overshooting depth than the radial velocity at the convective boundary. The convective velocities typically di ff er between 2D and 3D, and, as noted in these works, the convective flow structures are visually di ff erent; the filling factor has been discussed as a plausible source of di ff erence between two-dimensional and three-dimensional stellar simulations (e.g. Schmitt et al. 1984; Cai 2020; Pratt et al. 2020). \nThe question of how to best define a filling factor remains open because it depends on how to accurately define the edges of structures in a convective flow. Without loss of generality, we call \nsuch structures 'plumes' in this work. We will also refer to convective flows moving inward toward the core as 'inflows' and those moving outward toward the surface as 'outflows.' Many works (e.g. Schmitt et al. 1984; Canuto & Dubovikov 1998; Brummell et al. 2002; Andrássy 2015) have defined the filling factor to be the fractional area occupied by inflows at a given radius interior to the star. These works thus define the edges of convective plumes using the radial velocity. We call this definition the volume-percentage filling factor. In contrast, Rempel (2004) defines a filling factor as the fraction of the volume at a given radius with an inward mass flux; we call this the mass-flux filling factor. Zahn (1991) defines a plume based on the convective flux, and uses the fraction of the convective flux carried by inflows to define a filling factor; we call this the convective-flux filling factor. In a similar vein, Anders et al. (2022) propose a 'penetration parameter,' based on the change in the convective flux between the convection zone and the overshooting zone, as a predictive one-dimensional quantity. \nThese definitions of the filling factor have been evaluated in ideal box-in-a-star type simulations, which often use a moderate stratification and an ideal gas equation of state. However, they have not been examined in the kinds of global stellar simulations that we perform in this work. Our simulations are performed in a spherical shell that includes a large portion of the stellar radius and uses a stratification, temperature gradient, equation of state, and opacity tables that are extracted directly from stellar structure models accurate to the current state-of-the-art in stellar modeling. Our simulations also solve the equations for fully compressible convection; no additional assumptions are made that could impact the asymmetry of the convective flow. 1 We evaluate each definition of a filling factor for suitability to use in theoretical models of convection and overshooting based on two criteria, namely whether they: (1) are di ff erent for di ff erent stellar models and for 2D and 3D simulations, and (2) are correlated with the measured overshooting depth in the simulations. \nThis work is structured as follows. In Section 2, we will discuss the global fluid simulations of stars that we use to study the filling factor. In Section 3 we describe observed di ff erences between 2D and 3D simulations of stars. In Section 4 we describe how the volume-percentage, mass-flux, and convectiveflux filling factors are calculated and present results from our simulations. We also evaluate calculations of the Anders penetration parameter. In Section 5 we present statistics based on the width and numbers of inflows and outflows. We use these diagnostics to build a new nondimensional parameter, which we call the plume interaction parameter, and we show that this parameter is systematically di ff erent in 2D and 3D simulations for the stars we examine. In Section 6, we discuss the implications of our results for future stellar evolution calculations.", '2. Simulations': 'We produce simulations of stellar convection in four di ff erent stellar structures. One star that we examine is a 3 M ⊙ star that is ascending the red giant branch, produced with the open-source Modules for Experiments in Stellar Astrophysics (MESA) code (Paxton et al. 2010); this star has a large outer convection zone because it is near the first dredge-up point. The second star that \nweexamine is a one M ⊙ pre-main sequence star, called the young sun, produced with the Lyon stellar evolution code (Bara ff e & El Eid 1991; Bara ff e et al. 1997, 1998). This model has been described and examined in Pratt et al. (2016, 2017, 2020); we refer to that earlier work for additional details of the young sun, beyond the brief summary of relevant points given here. The third star is a one M ⊙ main-sequence star with a moderate outer convective layer, similar to the current sun, produced with the Lyon stellar evolution code. The fourth star is a 20 M ⊙ star at the zeroage main sequence (ZAMS) with a large convective core, produced with the Lyon stellar evolution code. These stars represent the di ff erent stages of evolution where convection occurs: pre-main sequence, main sequence, and evolved stars. They also represent di ff erent types of convection zones: outer convective envelopes that range between deep and shallow, as well as convective cores. Visualizations of the radial velocities in each of these stars are provided in Fig. 1. \nIn these simulations, each star is assumed to have a homogeneous chemical composition. We perform pairs of 2D and 3D Implicit Large Eddy Simulations (ILES) (Grinstein et al. 2007; Ritos et al. 2018; Margolin 2019) of these stars using the MUltidimensional Stellar Implicit Code ( MUSIC ). In these pairs, the simulation volume and grid for the 3D simulation are identical to the 2D simulation except for the dimensionality, as in Pratt et al. (2020). Our simulations in this work only take convection into account; the possibility of studying additional physical e ff ects such as rotation, a tachocline, chemical mixing, and magnetic fields is omitted from the current study. \nWe use the MUSIC code to solve the inviscid compressible hydrodynamic equations for density ρ , momentum ρ u , and internal energy ρ e : \n∂ ∂ t ρ = -∇ · ( ρ u ) , (1) \n∂ ∂ t ( ρ u ) = -∇ · ( ρ u ⊗ u ) - ∇ p + ρ g , (2) \n∂ ∂ t ( ρ e ) = -∇ · ( ρ e u ) -p ∇ · u + ∇ · ( χ ∇ T ) , (3) \nusing a second-order finite volume method, a MUSCL method (Van Leer 1997; Thornber et al. 2008) of interpolation on a staggered grid, and a van Leer flux limiter (as described in Van Leer 1974; Roe 1986; LeVeque et al. 2006). For 2D simulations, the finite volume method assumes azimuthal symmetry. Time integration in the MUSIC code is fully implicit and uses a Jacobian free Newton-Krylov (JFNK) solver (Knoll & Keyes 2004) with physics-based preconditioning (Viallet et al. 2016; Newman & Knoll 2013; Mousseau et al. 2000; Chen et al. 2014; Holod et al. 2021). The MUSIC code uses an adaptive time step, which is constrained identically for each pair of 2D and 3D simulations. MUSIC simulations use the same tabulated equation of state and opacity that are used by the 1D stellar evolution code that produced the stellar structure. In Eq. (2), g is the gravitational acceleration, a spherically symmetric vector consistent with that used in the stellar evolution calculation. It is not evolved for any of the simulations with an outer convective envelope; the convective core simulation did allow the gravity to be recalculated, but changes in this gravity term were small (see Bara ff e et al. (2023)). \nWe study the pairs of MUSIC simulations in 2D and 3D described in Table 1. In all of our MUSIC simulations, the compressible hydrodynamic equations (1)-(3) are solved in a spherical shell using spherical coordinates: radius r , and angular variables θ and ϕ (in 3D). As we noted in Pratt et al. (2020), grid spacing is particularly important in determining the physics in ILES, \nFig. 1. Visualizations of radial velocity in 2D simulations (from left to right) bg2D , wm2D , dcs2D , and cc2D . Outward flows are indicated in pink, while inward flows are in blue; the zero point in velocity is black. The maximum and minimum values of the color scale are defined by a radial velocity magnitude near the maximum for each simulation: bg2D ( ± 6 . 5 km / s), wm2D ( ± 2 . 9 km / s), dcs2D ( ± 0 . 63 km / s), cc2D ( ± 1 . 5 km / s). \n<!-- image --> \nTable 1. Parameters for compressible hydrodynamic simulations performed with MUSIC . Two di ff erent pairs of young sun simulations are included: (1) the wide2D and wide3D pair have lower radial resolution but the 3D simulation uses a large angle along the equator, while (2) wm2D and wm3D use a smaller equatorial angle but significantly higher radial resolution. The pair of red giant simulations is bg2D and bg3D ; the pair of current sun simulations is dcs2D and dcs3D ; the pair of convective core simulations is cc2D and cc3D . The simulation name, dimensionality, evolutionary state, and stellar mass M in units of the solar mass M ⊙ are provided. The inner radius R i of the spherical shell, the radius of the convective boundary R CB determined by the Schwarzschild criterion, and the outer radius R o of the spherical shell for the simulation are given in units of the total radius of each star, R , as a triplet. The angular extent of the simulation in the polar and equatorial directions is given as ( θ, ϕ ), and the grid spacing in both angular directions is ( ∆ θ, ∆ ϕ ). The average global convective turnover time τ conv is provided as well as its standard deviation in time, and the total time span for each simulation is given in units of the convective turnover time. \nbecause it is directly related to the e ff ective numerical viscosity. In Table 1, the inner and outer radius of the spherical shell for each simulation is noted, and the radial and angular grid spacings are specified. The simulation volume and grid in the r -θ plane are identical for each pair of simulations. To obtain accurate statistics on overshooting, both radial resolution and a longtime series of data are critical. For that reason, it is ideal for us to examine convection in spherical shells. Herwig et al. (2023) have recently described a dipole mode that fills the entire convective core in their simulation of a 25 M ⊙ main-sequence star; that mode is clearly not possible in either of the simulations of the main-sequence star that we examine here. Nevertheless, the 2D and 3D simulations of this star both equally neglect the possibility of such a dipole mode dominating the flow. We leave further study of dipole modes in convective cores to future work. \nTo allow a clear comparison of the convective turnover time between 2D and 3D simulations, we define this fundamental parameter as in Pratt et al. (2016): \nτ conv( t ) = Z CZ d V Hp | v | . Z CZ d V (4) \nIn this expression, Hp is the pressure scale height, and the magnitude of velocity in the denominator is calculated in either 2D or 3D, depending on the simulation. The integration covers the convection zone (CZ) and is volume-weighted using volume element d V . The instantaneous value of τ conv in eq. (4) is averaged over time to produce the values in Table 1. \nIn Table 1, we introduce the quantity Hp , CB / ∆ r . This ratio shows how many grid spaces resolve each pressure scale height at the convective boundary. In this work, we will often refer to the convective boundary; this is the boundary of the region of \nconvective instability calculated by the Schwarzschild criterion, which does not evolve during the simulations in Table 1. Our highest resolution simulation pair is wm2D / wm3D . Simulation wm3D has a grid of r × θ × ϕ = 1312 × 1024 × 64. Our simulations have su ffi cient radial resolution to produce a characteristic radial profile for velocity in 2D. \nAll simulations in this work are ILES, and convergence is not expected in the absolute sense that direct numerical simulations (DNS) converge. A universal shape of the velocity profiles can be observed with su ffi ciently small grid spacing, and the increase in the velocities becomes less as the grid spacing is progressively decreased (see also the discussion of ILES and convergence in Andrassy, R. et al. 2024). A study of the effect of grid spacing was examined systematically for the young sun in Pratt et al. (2016), and for the current sun in Vlaykov et al. (2022). The main sequence core convection simulation was studied in Bara ff e et al. (2023). Similar results are obtained for the red giant simulations. Because of the complications presented by ILES for convergence, we find the use of resolution criteria based on Hp , CB / ∆ r to be more useful for convective overshooting than the traditional DNS-style convergence studies. Such resolution criteria allow for clear comparisons between simulations of di ff erent stars that use di ff erent grids. \nAll data studied here are produced during steady-state convection, a period where the time-averaged value of the total kinetic energy is well-defined and not evolving in time. Each 3D simulation includes at least 3 convective turnover times of data taken during steady-state convection; for each 2D simulation, the time span is more than 30 τ conv . The uncertainties in the calculation of the average convective turnover time have a larger standard deviation for the 2D simulation than the 3D simulation; this is clearly impacted by the longer time series of data available for the 2D simulations (e.g. also observed in Pratt et al. 2020). \nWe examine simulations with two variations on energy boundary conditions that maintain the original radial profiles of density and temperature of the 1D stellar evolution model. For the wide2D / wide3D and wm2D / wm3D simulations, which include the full stellar radius up to the photosphere, the surface radiates energy with the local surface temperature. In this case, the energy flux varies as σ T 4 s , where σ is the Stefan-Boltzmann constant and T s ( θ, t ) is the temperature along the surface. This boundary condition can only be e ff ectively used when the nearsurface layers are included in the simulation volume and the temperature gradient near the surface is su ffi ciently resolved; otherwise, it results in artificially high cooling rates. The young sun simulations have more than one grid space per pressure scale height in this region. For the bg2D / bg3D , cc2D / cc3D , and dcs2D / dcs3D simulations, which do not include the full stellar radius, we hold the energy flux and luminosity constant on the outer radial boundary, at values established by the stellar structure. For an examination of how these boundary conditions a ff ect the dynamics, we refer to Pratt et al. (2016) and Vlaykov et al. (2022). \nIn all of the simulations studied in this work, we use the luminosity profiles accurate to the stellar structure models; we do not employ the tactic of luminosity boosting to shorten the convective turnover time and bring the thermal time-scale of the star closer to the convective turnover time. Luminosity boosting leads to a substantial reduction in computational costs and makes reaching thermal equilibrium feasible when a large enough boost factor is used. However, it can also distort the original background stratification of the star. Even if this is avoided, as discussed in Bara ff e et al. (2021), luminosity boosting increases the overshooting depth, the local heating in the overshooting layer, \nand the shape of the spectrum of waves excited (Lecoanet et al. 2019; Le Saux et al. 2022). Our choice to use luminosities that have not been artificially boosted is motivated by the need to measure an overshooting depth for a specific star. The simulations that we study in this work are far from thermal equilibrium. However the temperature gradient is not evolving during the course of these simulations, and the statistics that we produce are meaningful. \nFor the velocity, we impose non-penetrative and stress-free boundary conditions in the radial directions for all simulations. The energy flux and luminosity are held constant at the inner radial boundary, at the value of the energy flux at that radius in the one-dimensional stellar evolution calculation. On the inner radial boundary of the spherical shell, we impose a constant radial derivative on the density for all simulations. At the outer radial boundary, we apply di ff erent boundary conditions that suit the local derivatives in density best. For simulations bg2D / bg3D , wide2D / wide3D , and wm2D / wm3D this is a hydrostatic equilibrium boundary condition on the density that maintains hydrostatic equilibrium by assuming constant internal energy and constant radial acceleration due to gravity in the boundary cells (Grimm-Strele et al. 2015). For simulations dcs2D / dcs3D and cc2D / cc3D the outer radial boundary has a constant radial derivative imposed on the density. For simulations bg2D / bg3D , wide2d / wide3d , wm2D / wm3D , and dcs2D / dcs3D , we impose periodicity on all physical quantities at the boundaries in θ and ϕ . For simulation cc2D the angular direction is treated with reflective boundary conditions; for simulation cc3D both angular directions are treated with periodic boundary conditions.', '3. Differences between 2D and 3D stellar simulations': "For each of the pairs of simulations that we study, the rootmean-square (RMS) radial velocity profile in the 2D simulation is larger than in the 3D simulation (see for example Fig. 2). This is a common result for stellar convection also noted in Muthsam et al. (1995) and Pratt et al. (2020); the extent to which the 2D velocities are larger appears to be dependent on the stellar model examined. In addition, the radial velocities in 2D and 3D simulations look di ff erent; 3D simulations appear to be 'rougher,' i.e. more small-scale structure is present. This is particularly visible at points where inflows and outflows interact (see Fig. 3). \nThe di ff erences between 2D and 3D simulations also reach beyond velocity amplitudes into the structure of the flow. We examine the radial profile of the local enstrophy in the ϕ -direction, defined as the square of the ϕ component of the vorticity ωϕ = ∇× u | ϕ . The local ϕ enstrophy is larger in 2D than in 3D, with the largest di ff erences occurring at, or near, the convective boundary (see Fig. 4). These plots reveal a di ff erent shape and structuring in the flow that occurs near convective boundaries based on the dimensionality. \nWe calculate the overshooting depth ℓ ov by fitting the distribution of overshooting plumes calculated using the vertical kinetic energy flux with a generalized extreme value distribution, as described in Pratt et al. (2017). We adopt the location parameter from this fit as the overshooting depth ℓ ov . The values of ℓ ov are provided in Table 2 for all of our simulations. For the simulation pairs wm2d / wm3D , bg2D / bg3D , and cc2D / cc3D these numbers are extremely close. For the simulation pair dcs2D / dcs3D , the 2D simulation has somewhat deeper overshooting, while for the simulation pair wide2D / wide3D , the 3D simulation has somewhat deeper overshooting. Given the limited amount of \nFig. 2. The radial profile of RMS radial velocity v r , RMS for 2D and 3D simulations of the 3 M ⊙ red giant star. The lines indicate a time average, taken over the entire simulation time, of the horizontally averaged radial profile. The shaded regions represent one standard deviation above and below the time-averaged line. The radial position of the convective boundary, calculated by the Schwarzschild criterion, is indicated by a vertical black line. The interior radial coordinate of the star r is normalized by the star's radius R . \n<!-- image --> \ndata for the 3D simulations, it is not clear whether these di ff erences between the 2D and 3D simulations are statistically significant.", '4.1. Definition of the filling factor based on volume percentage': "We define a volume-percentage filling factor to be the fractional volume occupied by either the inflows σ vp , in or the outflows σ vp , out : \nσ vp , in = V in ( r , t ) V in ( r , t ) + V out ( r , t ) (5) \nout \nσ vp , out = V ( r , t ) V in ( r , t ) + V out ( r , t ) (6) \nHere V in indicates the total volume of grid cells at a given radius that has an inward velocity, while V out indicates the total volume of grid cells that have an outward velocity. The natural consequence is that the sum of the filling factors of inflows and outflows must be one σ vp , in + σ vp , out = 1. Because of this relation, we will generally use the notation σ vp to indicate the filling factor for the plumes moving toward the convective boundary, dropping the 'in' and 'out' labels. Conceptually, the volumepercentage filling factor is equating the situation where there are many small plumes with an equivalent single large inflow and single large outflow. Many works (Schmitt et al. 1984; Brummell et al. 2002; Andrássy 2015; Käpylä 2023) have used this \nkind of definition for a filling factor. Some have chosen to evaluate the area at the cell surfaces rather than the volume of the cell to calculate this filling fact. These two alternatives converge toward the same value as the cell size is decreased; using the volume to calculate this filling factor is convenient because MUSIC is a finite-volume code. In the work of Zahn (1991), the implication is that the filling factor is a single number, independent of radius. However other authors, including Schmitt et al. (1984); Cai (2020); Canuto & Dubovikov (1998); Käpylä (2023) formulate a filling factor that is a function of the radial depth in the star; doing so allows us to examine ideas about non-local convection throughout the stellar radius. Based on ideas acquired from early simulations of solar convection, Canuto & Dubovikov (1998) expect a highly asymmetric convection pattern with fast, concentrated inflows and slow, broad outflows. Based on observational data at the solar surface and the idea of a stratified star, they thus assert that the filling factor for inflows is always less than half: σ vp , in < 1 / 2. \nA calculation of σ vp in our simulation pair bg2D / bg3D is shown in Fig. 5(a). Both of these simulations evidence a σ vp close to a third near the surface of the star, in rough agreement with observations of solar surface convection (e.g. Nordlund et al. 2009). The similarity between our simulations and observations of the solar surface is striking, given that the resolution of near-surface dynamics is challenging for simulations of the stellar interior. As we examine σ vp deeper in the convective envelope, we find that it grows to approximately one-half, indicating that the convection becomes highly symmetric at the point that plumes are overshooting the bottom of the convection zone. This result is interesting when considered in conjunction with the kinetic energy flux (see Fig. 5(b)). The kinetic energy flux in all of our simulations is negative in the upper and middle parts of the convection zone. It becomes positive near the convective boundary, a result not seen in early works (e.g. Hurlburt et al. 1984). These combined results indicate that flows in the convection zone are more complex than the simple picture of thin, fast-moving inward plumes. In the highly stratified deep interior, inward moving plumes can be both faster and wider than plumes in the near-surface layers. \nWe calculate the value of the volume-percentage filling factor for plumes moving toward the convective boundary in each simulation. We then extract the value at the convective boundary, σ vp , CB , for each of our simulations; these can be found in Table 2. Across all simulations, the mean value of σ vp , CB is 0.499 and the median value is 0.50. For each simulation, the value of 0.5 is within one standard deviation of the time-averaged value. We find no clear di ff erence between the values of σ vp , CB calculated from 2D and 3D simulations. The distributions of these values, calculated at di ff erent points in time during steady state convection, strongly overlap; for some simulations, the time-averaged 2D value is slightly larger, and for others, the time-averaged 3D value is slightly larger. Given the wide range in overshooting depths that we calculate for the four di ff erent stars that we examine in this work, we find no clear correlation between the volume-percentage filling factor at the convective boundary and the overshooting length (see Fig. 6). \nFig. 3. Acomparison of radial velocity in the 3 M ⊙ red giant star from (left) 2D simulation bg2D , and (right) 3D simulation bg3D . The visualization is zoomed in on a small region inside the convection zone to emphasize the di ff erences in the shape of convection between 2D and 3D. Outward flows are indicated in pink, while inward flows are in blue; the zero point in velocity is black. The maximum and minimum values of the color scale are defined by a radial velocity magnitude near the maximum for each simulation: bg2D ( ± 6 . 5 km / s), bg3D ( ± 2 . 9 km / s). \n<!-- image --> \nTable 2. Calculated quantities related to the filling factor for convection. The table includes: the overshooting depth ℓ ov in units of the pressure scale height at the convective boundary, the volume-percentage filling factor σ vp , CB , the mass-flux filling factor σ mf , CB , and the incompressible convective-flux filling factor f z , CB . The subscript CB indicates that the quantity is evaluated at the convective boundary (CB), as defined by the Schwarzschild criterion. Several additional quantities that are evaluated above or below this convective boundary are also shown. This includes the Anders penetration parameter P A and the plume interaction parameter σ int . It also includes for convective envelopes (convective cores) the average number of inflows (outflows) in the overshooting layer N OL and in the convection zone N CZ , and the average width of inflows (outflows) in the overshooting layer W OL and in the convection zone W CZ . The average widths are displayed in units of the percentage of the simulation volume at the given radius.", '4.2. Definition of the filling factor based on mass flux': 'We define filling factors based on the vertical mass flux F mass = ρv r so that \nσ mf , in = GLYPH<12> GLYPH<12> GLYPH<12> F inflows mass GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> F inflows mass GLYPH<12> GLYPH<12> GLYPH<12> + GLYPH<12> GLYPH<12> GLYPH<12> F outflows mass GLYPH<12> GLYPH<12> GLYPH<12> , (7) \nGLYPH<12> \nGLYPH<12> \nσ mf , out = GLYPH<12> GLYPH<12> F outflows mass GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> F inflows mass GLYPH<12> GLYPH<12> GLYPH<12> + GLYPH<12> GLYPH<12> GLYPH<12> F outflows mass GLYPH<12> GLYPH<12> GLYPH<12> . (8) \nThe use of absolute values in the denominator of these ratios is necessary because, due to mass conservation, the direct \nArticle number, page 6 of 15 \nsum of these fluxes is small at any point in time. We note that σ mf , in + σ mf , out = 1, and we will generally use the notation σ mf to indicate the filling factor for the plumes moving toward the convective boundary. This definition of a mass-flux filling factor represents a mass-weighted rather than volume-weighted version of the filling factor. \nTable 2 includes values of the mass-flux filling factor at the convective boundary, σ mf , CB , for each of our simulations. Like the volume-percentage filling factor, the mass-flux filling factor is close to one-half at this point. Fig. 7 demonstrates that, like the volume-percentage filling factor, there is no clear trend between 2D and 3D results, with strongly overlapping distributions \n) \n( \n<!-- image --> \nFig. 4. Radial profiles of the local ϕ enstrophy for 2D and 3D simulations of the 3 M ⊙ red giant star. The lines indicate a time average, taken over the entire simulation time, of the horizontally averaged radial profile. The shaded region represents one standard deviation above and below the time-averaged line. The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image --> \nfrom 2D and 3D simulations. There is also no clear correlation between the mass-flux flux filling factor and the overshooting length. For the selection of stars that we study in this work, the mass-flux filling factor provides similar information to the volume-percentage filling factor. \nCanuto & Dubovikov (1998) defined a filling factor based purely on the velocity (see eq. (49a) of that work). For the simulations examined in this work, this velocity-based filling factor produces results similar to the mass-flux filling factor.', '4.3. Definition of the filling factor based on convective flux': "The analytical work of Zahn (1991) is based on the radial profile of the convective flux \nF conv ( r , t ) = -cp ( r , t ) ρ ( r , t ) ur ( r , θ, t ) T 1( r , θ, t ) . (9) \nThe bar on the right-hand side of this equation indicates an average over the horizontal directions, here indicated by θ . The temperature fluctuation T 1 is the di ff erence between the local temperature field T and the average temperature at a given radius, angle, and time. We conveniently measure this temperature fluctuation as a deviation from an initial radial profile, because the radial profile of the temperature does not evolve during our simulations. This formulation of the convective flux neglects any changes in the density and specific heat that could be dependent on an angle. \nExamining the convective flux in eq. (9), we find that the difference between this flux in the inflows and outflows is large around the convective boundary, resulting in a characteristic negative peak when the total convective flux is calculated (see Fig. 8). This is consistent with earlier works; both Brun et al. \nFig. 5. (a) Radial profile of the volume-percentage filling factor of the inward moving plumes σ vp , in vs the star's internal radius, in units of the total stellar radius R for simulations bg2D / bg3D . (b) Radial profile of the normalized kinetic energy flux, F k for simulations bg2D / bg3D . The solid and dashed lines indicate time-averaged radial profiles. Shaded areas indicate one standard deviation above and below these averaged lines. A thin vertical line indicates the convective boundary determined by the Schwarzschild criterion. \n<!-- image --> \n(2011) and Browning et al. (2004) define the top of the overshooting layer as the point where the time average of the convective flux first changes sign. They then define the bottom of the overshooting layer as the radius in the radiative zone where the convective flux becomes small. As is clear in the figure, using this definition of an overshooting layer along with our fully compressible simulations would mean defining a sizable portion \ns \nFig. 6. The time-averaged volume-percentage filling factor evaluated at the convective boundary determined by the Schwarzschild criterion, σ vp , CB , vs the overshooting depth in units of the pressure scale height at the convective boundary ℓ ov / Hp , CB for all simulations described in Table 1. Error bars indicate one standard deviation around the timeaveraged value of σ vp , CB . Error bars obtained in the calculation of the overshooting depth ℓ ov are smaller than the size of the data points. \n<!-- image --> \ns \nFig. 7. The time-averaged mass-flux filling factor evaluated at the convective boundary determined by the Schwarzschild criterion σ mf , CB vs the overshooting depth in units of the pressure scale height at the convective boundary ℓ ov / Hp , CB for all simulations described in Table 1. Error bars indicate one standard deviation around the time-averaged value of σ mf , CB . Error bars obtained in the calculation of the overshooting depth ℓ ov are smaller than the size of the data points. \n<!-- image --> \nFig. 8. The radial profile of time-averaged convective flux for the 3 M ⊙ red giant star bg2D , normalized to its maximum magnitude value. The shaded region represents one standard deviation in time, above and below the time-averaged line. The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image --> \nof the convection zone as part of the overshooting layer. This shape of the convective flux is not the result of changes in the temperature gradient of the star; the average temperature profile has not evolved during the course of our simulations. The large extent of the shaded area in the figure also demonstrates that the time-variation in the convective flux is substantial. These characteristics are universally present for the di ff erent kinds of stars we have simulated. \nIn their derivation of a filling factor based on the convective flux, Zahn (1991) defines two functions that are 'horizontal structures' of the fluid h 1( θ, t ) and h 2( θ, t ) such that \nur ( r , θ, t ) = u in r , RMS ( r , t ) h 1( θ, t ) , (10) \nT 1( r , θ, t ) = T in 1 , RMS ( r , t ) h 2( θ, t ) . (11) \nHere T in and u in are the temperature fluctuations and radial velocity only in the volume where the fluid is moving radially inward. We will derive the convective-flux filling factor for the inflows; the corresponding equations for outflows are fully analogous. The RMS operation takes the average in the horizontal direction, i.e. over θ if the simulation is 2D, and over both θ and ϕ if the simulation is 3D. Zahn (1991) assumes that h 1 = h 2 ≡ h . To satisfy this assumption requires that the temperature fluctuation and radial velocity are strongly correlated for inflows, or \nur ( r , θ, t ) u in r , RMS ( r , t ) = T 1( r , θ, t ) T in 1 , RMS ( r , t ) . (12) \nIf an average over a su ffi ciently long time period is taken, then the numerators on both sides of this equation will be zero. However, it is unclear how well this assumption is satisfied at any point in time. As the width of a plume changes with the radius, the ratios in eq. (12) could change as well; this could happen if \nthe inflows are less coherent, or if a convective boundary layer alters the flow near the bottom of the convection zone. The degree to which eq. (12) is satisfied could also be di ff erent for di ff erent stars. \nZahn (1991) then defines the convective-flux filling factor f z as a horizontally averaged function from these horizontal structures \nf z ( r , t ) ≡ h 1( r , θ, t ) h 2( r , θ, t ) ≡ h 2 ( r , θ, t ) . (13) \nSolving eqs. (10) and (11) for h 1( θ, t ) and h 2( θ, t ), we express eq. (13) as \nf z ( r , t ) = ur ( r , θ, t ) T 1( r , θ, t ) u in r , RMS ( r , t ) T in 1 , RMS ( r , t ) . (14) \nThis expression clarifies that the filling factor defined by Zahn (1991) is essentially a horizontal average of the convective flux, normalized by a proxy for the convective flux in the inflows. The formula in eq. (14) is in a convenient form for direct calculations. Combining this expression for f z with Zahn's expression for incompressible convective flux in eq. (9), we represent the convective flux in terms of the convective flux filling factor \nF conv ( r , t ) = -cp ( r , t ) ρ ( r , t ) u in r , RMS ( r , t ) T in 1 , RMS ( r , t ) f z ( t ) . (15) \nWe find that f z ( t ) is highly variable in time, and that a long-time average is required to produce a smooth profile; this is unsurprising because this filling factor is related to F conv , which also requires a long-time average to converge. Our formulation of the convective-flux filling factor is dependent on radius, a departure from the formula written by Zahn (1991), where the horizontal structure functions are not radially dependent. \nA characteristic result for the time-averaged radial profile of the convective-flux filling factor is shown in Fig. 9. Far from the convective boundary, this filling factor is larger and positive. However, approaching the convective boundary, the convectiveflux filling factor becomes small and / or negative. This appears to be related to the negative peak in the convective flux at the convective boundary. The implication is that the complex flow patterns, and thus complex fluxes that occur around this boundary contaminate the convective-flux filling factor. A full description of those flows is beyond the scope of the present work, but will be pursued in the future. For a filling factor, which is conceptualized as a percentage of the flow moving inward, a negative number makes little sense. Nevertheless, we have also documented the time-averaged value of f z at the Schwarzschild boundary for convective instability for all of the simulations examined in this work in Table 2. \nFig. 9. The radial profile of the time-averaged filling factor f z as defined in eq. (14) for 2D and 3D simulations of the 3 M ⊙ red giant star. The lines indicate a time average, taken over the entire simulation time, of the horizontally averaged radial profile. The shaded regions represent one standard deviation above and below the time-averaged line. The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image -->", '4.4. Eliminating compressibility as a source of error for the convective flux filling factor': "One possible source of error in the convective-flux filling factor formulated by Zahn (1991) is that the assumption of incompressibility could be unphysical around the convective boundary of a star; we therefore expand on Zahn's incompressible definition of a filling factor. A definition of the horizontally-averaged convective flux that includes compressibility is \nF conv ( r , t ) = -cp ( r , θ, t ) ρ ( r , θ, t ) ur ( r , θ, t ) T 1( r , θ, t ) . (16) \nWe note that Käpylä et al. (2017) use a similar definition of this flux where the horizontal average does not include the specific heat. In this case, we define four horizontal structures such that \nur ( r , θ, t ) = u in r , RMS ( r , t ) h 1( θ, t ) , (17) \nT 1( r , θ, t ) = T in 1 , RMS ( r , t ) h 2( θ, t ) , (18) \nρ ( r , θ, t ) = ρ in RMS ( r , t ) h 3( θ, t ) , (19) \ncp ( r , θ, t ) = c in p , RMS ( r , t ) h 4( θ, t ) . (20) \nAs in the incompressible case, the superscript indicates that the RMS only includes contributions where the fluid is moving radially inward, the RMS operation takes the average in the horizontal direction, and we will derive the filling factor for inward plumes. We then define a compressible-convective-flux filling factor \nf comp ( t ) = h 1( θ, t ) h 2( θ, t ) h 3( θ, t ) h 4( θ, t ) , (21) \n= cp ( r , θ, t ) ρ ( r , θ, t ) ur ( r , θ, t ) T 1( r , θ, t ) c in p , RMS ( r , t ) ρ in RMS ( r , t ) u in r , RMS ( r , t ) T in 1 , RMS ( r , t ) . (22) \nWe calculate f comp for all of the simulations studied in this work. The absolute di ff erence between f z and f comp is small near the convective boundary in all of our simulations (see Fig. 10). The compressible filling factor f comp is slightly larger near the surface in both 2D and 3D, where compressible e ff ects, e.g. Mach numbers, are expected to be slightly larger; this is typical of all of our simulations. Including the e ff ect of compressibility does not resolve the problematic aspects of the convective-flux filling factor near convective boundaries. \nFig. 10. Radial profile of the absolute di ff erence | f z -f comp | between the incompressible convective-flux filling factor and the compressible convective-flux filling factor for 2D and 3D simulations of the 3 M ⊙ red giant star. The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image -->", '4.5. The Anders penetration parameter': "Although it is not strictly a filling factor, we also examine a nondimensional number formulated by Anders et al. (2022), called the 'penetration parameter.' Similar to Zahn (1991), their motivation is to predict convective overshooting and penetration using a ratio of convective fluxes. It is therefore fitting to examine it here as a comparison to the filling factor in our realistic global simulations of stars. The Anders penetration parameter P A is defined as the ratio of the time-averaged convective flux on either side of the convective boundary, namely \nP A = -ˆ F conv | CZ ˆ F conv | OL . (23) \nIn Anders et al. (2022), ˆ F conv | CZ is described as the timeaveraged convective flux evaluated 'slightly' inside the convection zone where both outflows and inflows exist, and ˆ F conv | OL is the time-averaged convective flux located 'slightly' inside the radiative zone, in a layer where only inflows exist. In global simulations of stars, this language is not specific enough to determine where the convective flux should be evaluated. In addition, there is the complication of the complex boundary-layerlike flows that we find inside the convective boundary. To cal- \nFig. 11. The Anders penetration parameter P A vs the overshooting depth in units of the pressure scale height at the convective boundary determined by the Schwarzschild criterion ℓ ov / Hp , CB for all 2D and 3D simulations described in Table 1. Error bars consider one standard deviation of the convective flux at each of the two radial points that contribute to this number. A logarithmic scale is applied to the vertical axis because this parameter, as formulated, can be larger than one. \n<!-- image --> \nlate the Anders penetration parameter across global simulations of di ff erent stars, we define ˆ F conv | CZ as the maximum timeaveraged value of F conv ( r ) in the convective zone, a point that is una ff ected by boundary layer flows. Similarly, we define ˆ F conv | OL as the minimum time-averaged value of F conv ( r ), which can be found near the convective boundary, in the overshooting layer. Values of P A for each of our simulations, calculated in this way, are shown in Table 2. The Anders penetration parameter has the useful characteristic of producing larger average values in 2D than in 3D for all of our simulation pairs (see Fig. 11). We use the standard deviation in time of ˆ F conv | CZ and ˆ F conv | OL to calculate uncertainties for the penetration parameter; the error bars are large, in most cases overlapping data points from both 2D and 3D simulations. Whether the di ff erences between 2D and 3D simulations that we observe in the Anders penetration parameter can explain di ff erences in the overshooting depth in 2D and 3D, however, remains unclear (see Fig. 11). A longer time period of data may also be required to successfully evaluate P A . \nAnother useful result is that the Anders penetration parameter clearly produces di ff erent values for di ff erent stellar models. For the simulation pair cc2D / cc3D , which has small overshooting lengths, the minimum of the convective flux in the overshooting layer is also a small value; consequently, the value of P A becomes large, exceeding 20 for the 3D simulation. This large value, in addition to the large error bars shown in Fig. 11, is a result of how the ratio is formulated. Using 1 / P A would tend to produce values less than one, as the formulations of filling factors did.", '5.1. The width of inflows': 'The essence of the filling factor is as a diagnostic for quantitatively measuring how symmetric or asymmetric inflows and outflows are. The formulation of a filling factor in this way is linked to the early measurement that convective flows observed on the solar surface are structured into thinner regions with intense inflows and broader regions with slower outflows. We therefore consider this early idea, and we examine directly how plume widths change with radius in our simulations. As a number that sums up the widths of inflows, the filling factor is related to a low-order statistic. By instead examining the widths directly, we can pursue higher-order statistics that may be di ff erent in 2D and 3D convective flows. \nAs with the volume-percentage filling factor, we define a single inflow as a continuous set of cells in the θ direction, at a given radius, that all have a negative radial velocity; similarly, we define an outflow based on positive radial velocity. For our 3D simulations, we perform the same calculation for each angle ϕ in our grid. This allows for the 2D and 3D calculations to be directly compared; otherwise to define a two-dimensional perimeter for a convective plume would require a more involved calculation (e.g. Haller 2015; Balasuriya et al. 2018; Rempel et al. 2023). For each of our simulations, a characteristic profile is produced for the average widths of plumes as a function of radius (see Fig. 12). Two important features emerge in each profile: a maximum average width occurs approximately in the middle of the convection zone, which we call W CZ , and a minimum average width occurs near the bottom of the overshooting layer, which we call W OL . In our 2D red giant simulation, we find that the difference in the width of plumes between these two points is more than 10%. Fig. 12 also demonstrates that this characteristic profile is present for simulations with a su ffi cient resolution; as the grid spacing is decreased, the average width is naturally smaller. We find that there is a wider distribution of average widths in the convection zones in each of the 2D simulations that we study than in their 3D counterparts, which may be due to the larger amount of data generated in 2D. \nThe distribution of plume widths in the convection zone is a fundamental diagnostic pointing toward the multi-scale nature of stellar convection. In comparison, the controlled situation of Rayleigh-Bénard convection produces convection rolls that have roughly the same size, dictated by the size of the experiment. In some thin stellar convection zones, this can also be the case; however, in a large convection zone that is defined by a significant density stratification, inflows can have a range of sizes at any given radius. The stellar simulations we examine in this work have convection zones that represent a wide range of stratifications. Our 20 M ⊙ convective core simulation has a density ratio of ∼ 2 from the top to the bottom of the convection zone, the current sun simulations have a density ratio of ∼ 55, the red giant simulations include a density ratio of ∼ 200, and the density ratio in the young sun simulations is greater than 10 5 . Particularly for the young sun, the convection becomes a truly multi-scale flow, with plumes and convection rolls of di ff erent sizes frequently interacting with each other. This is evidenced by the examination of higher-order statistics for plume widths. The time-averaged radial profile of the skewness of inflow width is generally positive throughout the convection zone in our simulations (see for example Figs. 13 and 14), only dipping briefly into the negative for the 3D simulations, where there is a smaller time-series of data available to average. A normal distribution is defined by a skewness of zero; a positive skewness indicates that the distri- \n<!-- image --> \nFig. 12. (a) Widths of inflowing plumes for 2D and 3D simulations of the young sun, wide2D / 3D . (b) Widths of inflows for three 2D simulations of the red giant that have di ff erent grid sizes as labeled, but are otherwise identical. The shaded region represents one standard deviation above and below the time-averaged line. The radial position of the convective boundarydetermined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image --> \ninflowing plume widths is not symmetrical about the mean value, such that inflowing plumes larger than the average are more prevalent than smaller ones. \nThese figures also include the excess kurtosis, which is zero for a normal distribution. In comparison to a Gaussian distribution, a distribution with positive excess kurtosis indicates a greater prevalence of events in the wings of the distribution. Therefore, the excess kurtosis could give us an indication of the \n<!-- image --> \nFig. 13. Time-averaged profiles of skewness and excess kurtosis of the width of inflows for the red giant simulations (a) in bg2D , and (b) bg3D . The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image --> \nimportance of di ff erent scales in a convection zone that involves many length scales. In the 2D red giant simulation bg2D shown in Fig. 13(a), the excess kurtosis is positive throughout. However, the 3D simulation bg3D shown in Fig. 13(b) has an excess kurtosis that is both positive and negative. We find similar results for the young sun pair wm2D / wm3D ; in Fig. 14(b), the 3D simulation has negative excess kurtosis. This reinforces the idea that examination of the plume width can expose di ff erences between 2D and 3D simulations. \n<!-- image --> \nFig. 14. Time-averaged profiles of skewness and excess kurtosis of the width of inflows for the young sun simulations (a) in wm2D , and (b) wm3D . The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image -->', '5.2. Plume numbers': "The width of inflows can be directly related to their number. The number of inflows does not capture information on asymmetry; however, the number of plumes decreases when inflowing plumes are wider and increases when inflowing plumes are thinner. Some theoretical predictions of overshooting have used the number of plumes (Rieutord & Zahn 1995; Pinçon et al. 2016). In addition to this relationship to the plume width, the number of plumes can be related to two possible paradigms for convection in stellar interiors (Käpylä et al. 2017; Brandenburg 2016; \nSpruit 1996). The first has been described as a 'tree-like' structure, where the number of inflows is dependent on depth. The second has been described as a 'forest-like' structure where the number of inflows is depth-independent. To connect these ideas to overshooting, and the di ff erences between 2D and 3D convection, we examine numbers of plumes in our simulations. \nWe calculate the number of inflows, N in by counting up the continuous regions of inflowing cells at each radius; we then average over time. The number of outflows N out is calculated analogously. In each of our simulations with convective envelopes, we find the largest number of inflowing plumes at the surface. As radius decreases, the number of plumes continually decreases until the convective boundary. The average number of inflows increases again just beyond this convective boundary (see Fig. 15(a)), a signature that could indicate the break-up or dissipation of convective flow structures or the interaction between convection and the waves that populate the radiative region. In the figure, the number of inflowing plumes increases rapidly for radii above r / R = 0 . 7; this could be described as a tree-like structure, with large plumes dominating at the bottom of this region and much smaller convective flow structures dominating at the top. Below r / R = 0 . 7, the number of plumes changes only mildly until the bottom of the convection zone; this region could be described as having a 'forest-like' structure, with convective flows of similar size dominating. Following this analogy, a mildly 'tree-like' structure exists in the overshooting layer, as the number of inflowing plumes increases with the depth into the overshooting layer; to our knowledge, this has not been observed before. \nIn the case of core convection (see Fig. 15(b)) the largest number of plumes occurs at the convective boundary. Deeper in the convection core, the number of plumes decreases and then increases toward the inner radial boundary of the simulation. Because the convective core is comparatively small, convection appears to be entirely 'tree-like' in this case. This structure in the number of plumes looks highly similar to the structure that we observe in shallow outer convection zones, such as the current sun. \nThe picture of tree-like or forest-like convection evokes the question of whether plumes split or merge. That question can only clearly be addressed from a Lagrangian point of view, and so it is beyond the scope of this work.", '5.3. Using plume widths and numbers in a non-dimensional parameter': 'Inspired by our relative success in using the Anders penetration parameter to di ff erentiate between 2D and 3D simulations, we construct non-dimensional numbers from the number of plumes and their widths, using values at two radial points. The characteristic radial profile of plume widths (see Fig. 12) indicates that there is a funneling e ff ect on the plumes as they reach and then pass the convective boundary. We therefore construct a nondimensional parameter to indicate the strength of this funneling e ff ect. Because both the widths and numbers of plumes scale with the number of shear interactions between inflows and outflows, we call this the plume interaction parameter , which we define \nσ int = W OL / W CZ . (24) \nFor each of our simulations, the plume interaction parameter is included in Table 2; it is always less than one for our simulations. Resolution of the complex small-scale flows in the overshooting layer can be a challenge for diagnostics. For simulation \n<!-- image --> \nFig. 15. (a) Number of inflows vs. internal radius for 2D and 3D simulations of the 3 M ⊙ red giant star, and (b) number of outflows vs. internal radius for the 20 M ⊙ main-sequence star with a convective core. The radial position of the convective boundary determined by the Schwarzschild criterion is indicated by a vertical black line. \n<!-- image --> \nwide2D , we find σ int = 0 . 45, while for simulation wm2D , where the young sun is resolved about 4 times better, σ int = 0 . 50. This gives a clear indication that the plume interaction parameter is su ffi ciently resolved in our simulations, so that it changes only a small amount with increased resolution. The value of σ int is also always larger for our 3D simulations than our 2D simulations. This suggests that the plume interaction parameter can encapsulate a general di ff erence between 2D and 3D stellar convection. Using the standard deviation of the plume widths to calculate an uncertainty for the average plume interaction parameter, we \nfind that these uncertainties are smaller than the di ff erences between the 2D and 3D values of the plume interaction parameter in about half of our simulation pairs. Because the plume interaction parameter is based solely on the velocity rather than the convective flux, this measure is also physically distinct from the Anders penetration parameter. \nIn addition to the widths of plumes, the number of plumes can be combined in a ratio to produce a second nondimensional number, N CZ / N OL . As with the plume interaction parameter, this number is meaningfully di ff erent for our 2D and 3D simulation pairs, with the 3D simulation always having a larger value. Although a ratio based on the numbers of plumes does include different information from the widths of plumes, for our simulations they appear to be remarkably similar. Because the plume interaction parameter is more clearly linked to a filling factor, we focus on that diagnostic. Fig. 16 shows how both of our nondimensional ratios relate to the overshooting length in our simulations. No clear trend emerges for these pairs of simulations. However, the stellar models that we selected here are very different; investigation of a more similar set of stellar models could more easily exhibit a trend between the plume interaction parameter and the overshooting length; we are pursuing that study.', '6. Summary and Conclusions': 'Because 2D simulations have higher radial velocities than 3D simulations, it has generally been assumed that 2D simulations have a larger overshooting depth. Examining convection in four di ff erent models of stars, we show that the overshooting depth is often very similar in 2D and 3D simulations. Because 2D and 3D convective flows are visibly di ff erent, one possible explanation has been that a filling factor could explain these results. \nIn this work, we have studied di ff erent definitions of the filling factor for convection in realistic global simulations of stars to understand di ff erences in 2D and 3D convection as well as their link to an overshooting depth. Our calculations of a filling factor based on the volume percentage or based on mass flux result in characteristic profiles. These profiles reveal that, for stars with outer convective envelopes, the inward and outward flows are highly asymmetrical near the stellar surface with a value of about one-third (for the volume-percentage filling factor) or twothirds (for the mass-flux filling factor). However, at the bottom of the convection zone, the filling factor is about one-half, and the convection is nearly perfectly symmetrical. For our convective core simulation, these profiles are similar to those of a small convective envelope; at the convective boundary, there is symmetry between inflows and outflows. This is a significant new result for understanding stellar convection because it indicates that a filling factor calculated either using a volume percentage or the mass flux does not distinguish between 2D and 3D simulations. Nor are these filling factors, or any arguments based on asymmetrical convection, able to predict the overshooting depth for di ff erent stars. \nWe study a filling factor based on the convective flux, as suggested by Zahn (1991). This calculation reveals that boundarylayer-like dynamics, as discussed by Kupka & Muthsam (2017), are significant in realistic global simulations of stars. These flows contaminate the convective flux, distorting the signal near the convective boundary, where it might be used to predict an overshooting length. Connected to the convective flux, we also examine the Anders penetration parameter. This is not a filling factor because it involves two di ff erent radial points rather than a separation of inflows and outflows at a single radial point. However, \n<!-- image --> \ns \nFig. 16. Nondimensional ratios (a) the plume interaction parameter, and (b) the ratio of average numbers of plumes N CZ / N OL vs the overshooting depth in units of the pressure scale height at the convective boundary determined by the Schwarzschild criterion, ℓ ov / Hp , CB , for all simulations studied in this work. A logarithmic scale is applied to the vertical axis. \n<!-- image --> \nwe find that this parameter can distinguish between 2D and 3D stellar convection, so it may be useful in explaining the amount of overshooting in 2D and 3D global simulations of stars, as well as for the box-in-a-star simulations where it was developed. The convective flux in the overshooting layer is particularly intermittent, making this diagnostic converge only slowly. \nWe proceed to examine statistics of the widths of inwardflowing plumes (for convective envelopes) and outward-flowing plumes (for core convection). We find clear statistical di ff erences between 2D and 3D simulations in the standard deviation and the \nkurtosis. We identify a universal shape to the radial profile of the average plume width. We also examine the radial profile of the average number of plumes, which we link to pictures of tree-like and forest-like convection. Based on these profiles, we observe tree-like convection in overshooting layers. We construct a nondimensional number from the ratio of the average radial profile of the inward plume widths at two di ff erent radial points; we call this the plume interaction parameter. We demonstrate that the plume interaction parameter captures di ff erences between 2D and 3D simulations. \nAlthough both the Anders penetration parameter and our plume interaction parameter successfully and reliably indicate di ff erences between 2D and 3D stellar convection, for the set of stars that we examine in this work, they do not clearly correlate with the overshooting length. However, this set of stars was selected because they are very di ff erent from each other: they represent stars of di ff erent sizes, with di ff erent kinds of convection zones, at di ff erent evolutionary phases. A more systematic study of how these parameters change with overshooting depth may successfully demonstrate their use in predicting overshooting; that work is underway. \nAcknowledgements. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint e ff ort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. The research leading to these results is partly supported by the ERC grants 787361-COBOM and by the STFC Consolidated Grant ST / V000721 / 1. This work used the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFCDiRACHPCFacility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST / K000373 / 1 and STFC DiRAC Operations grant ST / K0003259 / 1. DiRAC is part of the National E-Infrastructure. This work also used the University of Exeter local supercomputer ISCA. Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC5207NA27344. LLNL-JRNL-857704.', 'References': 'Anders, E. H., Jermyn, A. S., Lecoanet, D., & Brown, B. P. 2022, The Astrophysical Journal, 926, 169 \n- Andrássy, R. 2015, PhD thesis, Universiteit van Amsterdam\n- Andrassy, R., Leidi, G., Higl, J., et al. 2024, A&A, 683, A97\n- Balasuriya, S., Ouellette, N. T., & Rypina, I. 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J., Rheinhardt, M., Brandenburg, A., et al. 2017, The Astrophysical Journal Letters, 845, L23\n- Knoll, D. A. & Keyes, D. E. 2004, J. Comput. Phys., 193, 357\n- Kupka, F. & Muthsam, H. J. 2017, Living Reviews in Computational Astrophysics, 3, 1\n- Le Saux, A., Guillet, T., Bara ff e, I., et al. 2022, Astronomy & Astrophysics, 660, A51\n- Lecoanet, D., Cantiello, M., Quataert, E., et al. 2019, The Astrophysical Journal Letters, 886, L15\n- LeVeque, R. J., Mihalas, D., Dorfi, E., & Müller, E. 2006, Saas-Fee Advanced Course 27. Lecture Notes 1997 Swiss Society for Astrophysics and Astronomy, Vol. 27, Computational Methods for Astrophysical Fluid Flow (Springer Science & Business Media)\n- Margolin, L. G. 2019, Shock Waves, 29, 27\n- Mousseau, V., Knoll, D., & Rider, W. 2000, Journal of computational physics, 160, 743\n- Muthsam, H., Göb, W., Kupka, F., Liebich, W., & Zöchling, J. 1995, Astronomy and Astrophysics, 293\n- Newman, C. & Knoll, D. A. 2013, SIAM Journal on Scientific Computing, 35, S445\n- Nordlund, Å., Stein, R. F., & Asplund, M. 2009, Living Reviews in Solar Physics, 6, 1\n- Paxton, B., Bildsten, L., Dotter, A., et al. 2010, The Astrophysical Journal Supplement Series, 192, 3\n- Pinçon, C., Belkacem, K., & Goupil, M. 2016, Astronomy & Astrophysics, 588, A122\n- Pratt, J., Bara ff e, I., Go ff rey, T., et al. 2017, Astronomy & Astrophysics, 604, A125\n- Pratt, J., Bara ff e, I., Go ff rey, T., et al. 2020, Astronomy & Astrophysics, 638, A15\n- Pratt, J., Bara ff e, I., Go ff rey, T., et al. 2016, Astronomy & Astrophysics, 593, A121\n- Rempel, E. L., Chian, A. C.-L., de SA Silva, S., et al. 2023, Reviews of Modern Plasma Physics, 7, 32\n- Rempel, M. 2004, The Astrophysical Journal, 607, 1046\n- Rieutord, M. & Zahn, J.-P. 1995, Astronomy and Astrophysics, 296, 127\n- Ritos, K., Kokkinakis, I. W., & Drikakis, D. 2018, Computers & Fluids, 173, 307 Roe, P. 1986, Annu. Rev. Fluid Mech., 18, 337\n- Rogers, T. M., Glatzmaier, G. A., & Jones, C. 2006, The Astrophysical Journal, 653, 765\n- Schmitt, J., Rosner, R., & Bohn, H. 1984, The Astrophysical Journal, 282, 316 Spruit, H. C. 1996, arXiv preprint astro-ph / 9605020\n- Stein, R. & Nordlund, Å. 1989, The Astrophysical Journal, 342, L95\n- Thornber, B., Mosedale, A., Drikakis, D., Youngs, D., & Williams, R. 2008, J. Comput. Phys., 227, 4873\n- Toomre, J., Brummell, N., Cattaneo, F., & Hurlburt, N. E. 1990, Computer Physics Communications, 59, 105\n- Van Leer, B. 1974, J. Comput. Phys., 14, 361\n- Van Leer, B. 1997, Journal of computational physics, 135, 229\n- Viallet, M., Go ff rey, T., Bara ff e, I., et al. 2016, A&A, 586, A153 \nVlaykov, D., Bara \nff \ne, I., Constantino, T., et al. 2022, Monthly Notices of the \nRoyal Astronomical Society, 514, 715 \n- Zahn, J.-P. 1991, Astronomy & Astrophysics, 252, 179'}

2021arXiv210508081T

We investigate the possibility of detecting artificial lights from Proxima bs dark side by computing light curves from the planet and its host star. The two different scenarios we consider are artificial illumination with the same spectrum as commonly used LEDs on Earth and a narrower spectrum which leads to the same proportion of light as the total artificial illumination on Earth. We find that the James Webb Space Telescope JWST will be able to detect LED type artificial lights making up 5 of stellar power with 85 confidence assuming photonlimited precision. In order for JWST to detect the current level of artificial illumination on Earth the spectral band must be 103 times narrower. Our predictions require optimal performance from the NIRSpec instrument and even if not possible with JWST future observatories like LUVOIR might be able to detect this artificial illumination.

2021-05-01T00:00:00Z

['arXiv:2105.08081', '2021arXiv210508081T', '10.48550/arXiv.2105.08081']

['Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Popular Physics']

Detectability of Artificial Lights from Proxima b

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https://arxiv.org/pdf/2105.08081.pdf

{'Detectability of Artificial Lights from Proxima b': 'Elisa Tabor 1 and Abraham Loeb 2 \n- 1 Department of Physics, Stanford University, 452 Lomita Mall, Stanford, CA 94305 \n2 Department of Astronomy, Harvard University, 60 Garden St., Cambridge, MA 02138', 'ABSTRACT': "We investigate the possibility of detecting artificial lights from Proxima b's dark side by computing light curves from the planet and its host star. The two different scenarios we consider are artificial illumination with the same spectrum as commonly used LEDs on Earth, and a narrower spectrum which leads to the same proportion of light as the total artificial illumination on Earth. We find that the James Webb Space Telescope (JWST) will be able to detect LED type artificial lights making up 5% of stellar power with 85% confidence, assuming photon-limited precision. In order for JWST to detect the current level of artificial illumination on Earth, the spectral band must be 10 3 times narrower. Our predictions require optimal performance from the NIRSpec instrument, and even if not possible with JWST, future observatories like LUVOIR might be able to detect this artificial illumination.", '1. INTRODUCTION': "Proxima b is one of the best targets outside our solar system in the search for extraterrestrial life (AngladaEscud'e et al. 2016; Ribas et al. 2016; Turbet et al. 2016; Berdyugina & Kuhn 2019). It resides in the habitable zone of its star, suggesting liquid water could exist at its surface; it is a similar mass to Earth, and it orbits the nearest star to the sun, 4.2 light years away, so we can reasonably expect to characterize it in the near future. The important question is then whether Proxima b can sustain intelligent life and how to go about detecting it (Turbet et al. 2016; Kreidberg & Loeb 2016; Lingam & Loeb 2017). \nSince the discovery of Proxima b five years ago, there have been many searches for life on the planet (AngladaEscud'e et al. 2016; Ribas et al. 2016; Turbet et al. 2016), culminating most recently with a tentative detection of a radio signal from BLC-1 originating from its direction (Breakthrough Listen, submitted 2021). Previous studies involved estimating the water content on the planet, projecting the rate at which Proxima b might lose its atmosphere due to stellar flares, reconstructing the evolution of the radius and luminosity of the planet to find its current rotation rates, and studies of its climate and atmosphere (Turbet et al. 2016; Ribas et al. 2016; Kreidberg & Loeb 2016). A generic aspect of a technological civilization is the production of artificial light. In this \netabor@stanford.edu \naloeb@cfa.harvard.edu \nLetter , we examine the detectability of artificial lights originating from Proxima b. \nOwing to its proximity to the star, Proxima b is likely to be tidally-locked with a permanent dayside and nightside. This exacerbates the need for artificial illumination of the nightside for it to be attractive for technological habitability, as it is otherwise permanently dark. \nThe lightcurves of Proxima b involve several factors, including the radius of the planet ( ∼ 1 . 3 R ⊕ for Proxima b) and host star ( ∼ 0 . 14 R glyph[circledot] for Proxima Centauri), orbital period (11 days), orbital semi-major axis ( ∼ 0.05 AU), albedo ( ∼ 0.1 if analogous to the Moon), and orbital inclination (Anglada-Escud'e et al. 2016). \nTo estimate the inclination of Proxima b's orbit, we use information about Proxima c (Damasso et al. 2020). This planet orbits at 1.5 AU, so it remains outside of Proxima Centauri's habitable zone. Proxima c's large semi-major axis has allowed to find its orbital inclination, and the HARPS and UVES spectrograph obtained an orbital inclination of i = 152 ± 14 degrees (Kervella et al. 2020). Since the variation of orbital inclinations in the solar system is about ± 7%, we use an inclination of i = 2 . 65 ± 0 . 43 radians for Proxima b. \nWe consider the James Webb Space Telescope (JWST) to assess the feasibility of detecting artificial lights on Proxima b (Beichman et al. 2014). JWST will allow to characterize the atmosphere of Proxima B and find out how much energy transport occurs on the planet (Kreidberg & Loeb 2016; Turbet et al. 2016). Here we perform a detailed calculation of the lightcurves from Proxima b and simulate signal-to-noise calculations using the JWST Exposure Time Calculator (ETC). \nThe organization of this Letter is as follows. In section 2 we describe our methods for calculating lightcurves and our error analysis. Our results are described in section 3. Finally, we discuss the implications of these findings in section 4.", '2.1. Proxima b Lightcurves': "We calculate the lightcurves from Proxima b using the Exoplanet Analytic Reflected Lightcurves (EARL) (Haggard & Cowan 2018). The uniform albedo map corresponds to the Y 0 0 spherical harmonic and the flux equation, \nF 0 0 = 1 3 π 3 / 2 (sin w -w cos w ) , (1) \nwhere w is the width of a lune cut out by the great circles defined by the subobserver and substellar points. The value of w ranges from 0 to π . We must also multiply the reflected light by the albedo A , for which we make the more conservative approximation of A = 0 . 1, the albedo for the Moon and for typical rocky bodies (Usui et al. 2013). \nThe lightcurve calculated with EARL only shows the light reflected from Proxima b, so to simulate artificial light in addition to the reflected light, we introduce a free variable to equation (1), F ai . We assume the artificial light illuminates the night side of the planet facing away from its host star, so F ai represents the proportion of artificial light coming from Proxima b's dark side relative to the reflected starlight on the daylight hemisphere. Since the maximum value w can take is π , and F 0 0 ( π ) = π 3 π 3 / 2 , in order to purely consider the dark side, we examine π 3 π 3 / 2 -F 0 0 . Thus, our new flux equation is \nF = A 3 π 3 / 2 (sin w -w cos w )+ F ai 3 π 3 / 2 ( π -sin w + w cos w ) . \nTo obtain the desired flux, we multiply our new F by \nπR 2 4 πd 2 \nwhere R is the radius of Proxima b and d is the distance between Proxima b and its star. The top panel of Figure 1 shows the flux dependence on lune width for different values of F ai . The blue curve represents F ai = 1, which occurs when the night side is fully illuminated to the same brightness as the day side facing Proxima Centauri. The green curve, showing F ai = 0, represents no artificial illumination, in which case the night side remains fully dark. \nWe next change coordinates from lune width to a dependence on time, inclination, and orbital angular frequency using the equations in Appendix C of Haggard & \nFigure 1. The lightcurves from Proxima b calculated using EARL, with three different coefficients F ai representing the percentage of stellar power being illuminated on the dark side of the planet. The blue curve represents F ai = 0 . 1, which equals the value we assume for the albedo. Thus the amount of artificial illumination on the night side is equal to the amount of light reflected from the day side. The green curve, for F ai = 0, represents no artificial illumination, so the night side is fully dark. Top panel : the planet to star ratio depends solely on the lune width. Bottom panel : the ratio depends on time (in days), orbital angular frequency, and inclination. \n<!-- image --> \nCowan (2018). Using the inclination of Proxima c of 2.65 radians and an orbital angular frequency of (2 π/ 11)days, we obtain the curves in the bottom panel of Figure 1. The minimum for the green curve in the bottom panel is greater than 0 since Proxima b doesn't transit, so a nonzero fraction of the dayside is always visible (Jenkins et al. 2019). The flux equations give unitless ratios, so we must multiply the reflected light by the stellar spectrum and the artificial light by the predicted spectrum of the artificial illumination on Proxima b. \nWe consider two possibilities for the artificial spectrum: one with commonly used LEDs, and another with the same F ai as currently used on Earth ( ∼ 10 -4 ). These spectra correspond to a Gaussian distribution centered at 1.2 µ m (the peak of Proxima Centauri's light spectrum) with variance 0.12 µ m and 10 -4 µ m, respectively. We normalized the Gaussians so that when F ai = A , the integral of the LED spectra equals the integral of reflected light from the host star. The variation \nin the width of these Gaussians allows us to examine the impact of more focused lights on their detectability. \nFinally, to calculate the spectral radiance density per unit frequency of Proxiam Centauri and Proxima b, we use the blackbody form, \nB ( ν ) = 2 hν 3 c 2 1 e hν/ ( k B T ) -1 \nwith T = 250K for the planet and T = 3000K for the host star (Kreidberg & Loeb 2016). The final lightcurves are the sum of the spectral radiance densities of the star and of the planet, the reflected light from the star, and the artificial illumination of Proxima b.", '2.2. JWST/NIRSpec': "The JWST Exposure Time Calculator (ETC, available at jwst.etc.stsci.edu) allows us to estimate the feasibility of detecting different values of F ai . Since the peak of the stellar flux is around 1.5 microns, we work with NIRSpec's G140M/F100LP disperser-filter combination, which has wavelength range 0.97-1.84 microns and a spectral resolution of R ∼ 1000. 1 We set up using the spectral energy distribution of a blackbody at 3000K, and renormalize the source flux density to 4.384 in the K-band. The ETC produces a graph of the estimated number of photoelectrons per second for different wavelengths, which we use to calculate the predicted errors. The noise corresponds to the square root of the number of electrons N e , so the noise-to-signal ratio is 1/ √ N e where N e is the number of electrons per second n e times the integration time t i . Integrating the photons over one period (11 days) gives us an estimated error of 1 √ N e = 1 √ n e ∗ t i .", '3. RESULTS': 'Wenow combine our error analysis with our calculated lightcurves to obtain full predictions of the lightcurves detected from Proxima b. Here we show the contrast between the light coming from the system at the start and in the middle of an orbital period. \nThe top panel of Figure 2 shows the results we obtained when the artificial light corresponds to commonly used LEDs on Earth, and the bottom panel shows the curve which leads to the same F ai as currently used on Earth, ∼ 10 -4 .', '4. DISCUSSION AND CONCLUSIONS': "We use a reduced chi-square statistic to analyze the confidence of detecting certain values of F ai , shown in \nFigure 2. Difference between the measured spectrum at phase 0 and phase 0.5 in ppm. The black line involves the sum of the spectral radiance densities of Proxima b and its star and the reflected light from the star, while the red line adds the artificial illumination of Proxima b. The top axis shows the frequency (in 10 14 Hz) while the bottom axis shows the wavelength (in µ m). The uncertainties are based on the photon noise from NIRSpec measurement predictions. Top panel : the results obtained when the artificial light corresponds to commonly used LEDs on Earth. Bottom panel : the curve which leads to the same F ai as currently used on Earth. \n<!-- image --> \nFigure 3. For the LED spectrum, we find that JWST will be able to rule out values of F ai > 5% with 85% confidence, and F ai > 9% with 95% confidence. In order for JWST to detect a coefficient of F ai > 0 . 001% (the current value on Earth),with 85% confidence, the spectrum of artificial illumination must be 10 3 times narrower in frequency. In either case, JWST will thus allow us to narrow down the type of artificial illumination being used. \nProxima b is tidally locked if its orbit has an eccentricity below 0.06, where for reference, the eccentricity of the Earth's orbit is 0.017 (Ribas et al. 2016). If Proxima b has a permanent day and nightside, the civilization might illuminate the nightside using mirrors launched into orbit or placed at strategic points (Korpela et al. 2015). In that case, the lights shining onto the perma- \nFigure 3. Chi-squared analysis of data for standard LEDs with width 0.12 µ m. \n<!-- image --> \nnent nightside should be extremely powerful, and thus more likely to be detected with JWST. \nIn summary, we have simulated lightcurves from Proxima b and compared curves corresponding to the reflected stellar spectrum to curves with artificial lights corresponding to a narrower spectrum such as for LEDs. We have found that JWST will be able to show the existence of artificial illumination for standard LEDs 500 times more powerful than those currently found on \nEarth's, and for artificial illumination of similar magnitude to Earth's for a spectrum 10 3 times narrower in frequency. \nFuture extensions of this work could examine the sensitivity of the Large UV Optical Infrared Surveyor (LUVOIR), a telescope in the works with the goal of being launched in 2035 (The LUVOIR Team 2019). LUVOIR will allow us to confirm the artificial illumination, or lack thereof, with a higher degree of precision. Although the noise floor of JWST is not known, current estimates are at about 10 ppm (Schlawin et al. 2021). Detecting an effect of 1 ppm will require pushing the NIRSpec instrument well beyond its expected performance. Even if JWST is not able to detect artificial illumination on Proxima b, LUVOIR and other such telescopes may have significantly improved performance.", 'ACKNOWLEDGMENTS': 'This work was supported in part by the Stanford Physics Department (for E.T.) and in part by a grant from the Breakthrough Prize Foundation (for A.L.). We thank Laura Kreidberg for insightful comments on an early draft of the paper.', 'REFERENCES': 'Anglada-Escud´e, G., Amado, P. J., Barnes, J., et al. 2016, \nNature, 536, 437, doi: 10.1038/nature19106 \nBeichman, C., Benneke, B., Knutson, H., et al. 2014, PASP, \n126, 1134, doi: 10.1086/679566 \nBerdyugina, S. V., & Kuhn, J. R. 2019, AJ, 158, 246, \ndoi: 10.3847/1538-3881/ab2df3 \nDamasso, M., Del Sordo, F., Anglada-Escud´e, G., et al. 2020, Science Advances, 6, eaax7467, \ndoi: 10.1126/sciadv.aax7467 \nHaggard, H. M., & Cowan, N. B. 2018, EARL: Exoplanet Analytic Reflected Lightcurves package. \nhttp://ascl.net/1805.004 \nJenkins, J. S., Harrington, J., Challener, R. C., et al. 2019, MNRAS, 487, 268, doi: 10.1093/mnras/stz1268 \nKervella, P., Arenou, F., & Schneider, J. 2020, A&A, 635, L14, doi: 10.1051/0004-6361/202037551 \nKorpela, E. J., Sallmen, S. M., & Leystra Greene, D. 2015, ApJ, 809, 139, doi: 10.1088/0004-637X/809/2/139 \nKreidberg, L., & Loeb, A. 2016, ApJL, 832, L12, \ndoi: 10.3847/2041-8205/832/1/L12 \nLingam, M., & Loeb, A. 2017, MNRAS, 470, L82, \ndoi: 10.1093/mnrasl/slx084 \nRibas, I., Bolmont, E., Selsis, F., et al. 2016, A&A, 596, \nA111, doi: 10.1051/0004-6361/201629576 \nSchlawin, E., Leisenring, J., McElwain, M. W., et al. 2021, \nAJ, 161, 115, doi: 10.3847/1538-3881/abd8d4 \nThe LUVOIR Team. 2019, arXiv e-prints, \narXiv:1912.06219. https://arxiv.org/abs/1912.06219 \nTurbet, M., Leconte, J., Selsis, F., et al. 2016, A&A, 596, \nA112, doi: 10.1051/0004-6361/201629577 \nUsui, F., Kasuga, T., Hasegawa, S., et al. 2013, ApJ, 762, 56, doi: 10.1088/0004-637X/762/1/56'}

2024ApJ...974...82W

3C 264 is one of the few FRI radio galaxies with detected TeV emission. It is a lowluminosity active galactic nucleus LLAGN and is generally associated with a radiatively inefficient accretion flow RIAF. Earlier multiwavelength studies suggest that the Xray emission originates from a jet. However the possibility that the RIAF can significantly contribute to the Xrays cannot be ruled out. In particular hard Xray emission 10 keV has never been detected making it challenging to distinguish between Xray models. Here we report a NuSTAR detection up to 25 keV from 3C 264. We also present subpixel deconvolved Chandra images to resolve jet emission down to 0.2 from the center of the unresolved Xray core. Together with a simultaneous Swift observation we have constrained the dominant hard Xray emission to be from its unresolved Xray core presumably in its quiescent state. We found evidence of a cutoff in the energy around 20 keV indicating that at least some of the Xrays from the core can be attributed to the RIAF. The Comptonization model suggests an electron temperature of about 15 keV and an optical depth ranging between 4 and 7 following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs. The detected hard Xray emission is at least an order of magnitude higher than that predicted by synchrotron selfCompton models introduced to explain ray and TeV emission suggesting that the synchrotron electrons might be accelerated to higher energies than previously thought.

2024-10-01T00:00:00Z

['arXiv:2409.05943', '2024ApJ...974...82W', '10.3847/1538-4357/ad6a1a', '10.48550/arXiv.2409.05943', '2024arXiv240905943W']

['High energy astrophysics', 'Low-luminosity active galactic nuclei', 'Active galactic nuclei', 'Radio active galactic nuclei', 'X-ray active galactic nuclei', 'Jets', 'Astrophysical black holes', 'Galaxy jets', 'Accretion', 'X-ray astronomy', 'Relativistic jets', 'Gamma-ray sources', '739', '2033', '16', '2134', '2035', '870', '98', '601', '14', '1810', '1390', '633', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies']

NuSTAR Observation of the TeVdetected Radio Galaxy 3C 264 Core Emission and the Hot Accretion Flow Contribution

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https://arxiv.org/pdf/2409.05943.pdf

{'NuSTAR Observation of the TeV-Detected Radio Galaxy 3C 264: Core Emission and the Hot Accretion Flow Contribution': "Ka-Wah Wong, 1 Colin M. Steiner, 1 Allison M. Blum, 1 Dacheng Lin, 2 Rodrigo Nemmen, 3, 4 Jimmy A. Irwin, 5 and Daniel R. Wik 6 \n1 Department of Physics, SUNY Brockport, Brockport, NY 14420, USA \n2 \nDepartment of Physics, Northeastern University, Boston, MA 02115-5000, USA 3 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA 4 Instituto de Astronomia, Geof'ısica e Ciˆencias Atmosf'ericas, Universidade de S˜ao Paulo, S˜ao Paulo, SP 05508-090, Brazil 5 Department of Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35487, USA 6 Department of Physics & Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA", 'ABSTRACT': "3C 264 is one of the few FRI radio galaxies with detected TeV emission. It is a low-luminosity AGN (LLAGN) and is generally associated with a radiatively inefficient accretion flow (RIAF). Earlier multiwavelength studies suggest that the X-ray emission originates from a jet. However, the possibility that the RIAF can significantly contribute to the X-rays cannot be ruled out. In particular, hard X-ray emission ≳ 10 keV has never been detected, making it challenging to distinguish between X-ray models. Here we report a NuSTAR detection up to 25 keV from 3C 264. We also present subpixel deconvolved Chandra images to resolve jet emission down to ∼ 0 . '' 2 from the center of the unresolved X-ray core. Together with a simultaneous Swift observation, we have constrained the dominant hard X-ray emission to be from its unresolved X-ray core, presumably in its quiescent state. We found evidence of a cutoff in the energy around 20 keV, indicating that at least some of the X-rays from the core can be attributed to the RIAF. The Comptonization model suggests an electron temperature of about 15 keV and an optical depth ranging between 4 and 7, following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs. The detected hard X-ray emission is at least an order of magnitude higher than that predicted by synchrotron self-Compton models introduced to explain γ -ray and TeV emission, suggesting that the synchrotron electrons might be accelerated to higher energies than previously thought.", '1. INTRODUCTION': "Studying active galactic nuclei (AGNs) and their relativistic outflows or jets remains one of the most vibrant areas of research in astrophysics today. It is critical not only for probing the properties of black holes but also for understanding their impact on the formation of larger-scale structures. Some AGNs are identified as very high energy (VHE) TeV sources, indicating that particles within them are accelerated to equivalent energy scales. Therefore, studying TeV-emitting AGNs provides crucial insights into the mechanisms of VHE radiation and acceleration (e.g. Madejski & Sikora 2016). \nAmong the VHE sources discovered, blazars are the most common type of extragalactic sources, with their jets oriented closely to our line of sight, and are thought \nto be responsible for the TeV emission. A number of radio galaxies with misaligned jets have also been identified as TeV sources (Rieger & Levinson 2018; Rani 2019; Rulten 2022). In particular, four of these belong to the FRI type (NGC 1275, 3C 264, M87, and Cen A) and seem to form a subclass of VHE sources 1 . Specifically, FRI radio galaxies typically host low-luminosity AGNs (LLAGNs; Wu et al. 2007), which are believed to accrete in a radiatively inefficient accretion flow (RIAF) or a hot mode (Narayan & Yi 1994; Yuan & Narayan 2014). Studying these TeV-emitting FRI sources will undoubtedly provide a unique scientific perspective on the study of VHE particle acceleration in AGNs, whether it be from a misaligned jet (Abdo et al. 2010) or from the hot accretion flow. \n3C 264 is a FRI radio galaxy located at a redshift of z = 0 . 0217 (Baum et al. 1990). TeV emission was detected with VERITAS in 2018 (Mukherjee 2018). The mass of the supermassive black hole is about 2 . 6 × 10 8 M ⊙ (van den Bosch 2016). The X-ray jet has been resolved with Chandra , alongside an unresolved X-ray core (Perlman et al. 2010, hereafter, core means the central source unresolved by Chandra at the subarcsecond scale). The X-ray emission is dominated by the core with an X-ray luminosity of L 0 . 5 -10 keV = 2 × 10 42 ergs s -1 (Sun et al. 2007). Soft X-ray emission below ∼ 2 keV has been detected from the host galaxy NGC 3862, extending up to 6 '' , using Chandra (Sun et al. 2007). The coronal temperature has been measured to be 0 . 65 +0 . 29 -0 . 09 keV and the measured coronal luminosity is L 0 . 5 -2 keV = 1 . 4 × 10 40 ergs s -1 . The contribution of X-ray emission from LMXBs to the total X-ray emission is estimated to be at most 4% (Sun & Murray 2002). \nDue to its low Eddington ratio of L bol /L Edd ≈ 7 × 10 -5 , 3C 264 is a LLAGN (Donato et al. 2004), with the black hole accreting in the RIAF mode. Donato et al. (2004) and Evans et al. (2006) found significant correlations between the X-ray core luminosity and the radio/optical luminosities, suggesting that at least some of the X-rays come from a jet, although an origin from RIAF cannot be completely ruled out. By measuring the X-ray, optical, and radio spectral slopes, Perlman et al. (2010) argue that the X-ray emission is likely due to the synchrotron process, which is consistent with a jet model. These observations suggest that jets can be associated with RIAF as expected (Yuan & Narayan 2014). However, the contribution of X-rays from the RIAF itself remains a possibility in these observations. For instance, the photon index Γ of the RIAF model, especially at very low accretion rates, can be similar to that of a jet model (Nemmen et al. 2014). Similarly, in another TeV FRI galaxy, Cen A, the source of hard X-ray emission could be attributed to the RIAF, a jet, or a combination of both (Furst et al. 2016). \n3C 264 is distinct from the other three VHE FRI galaxies in a number of aspects. For example, Cen A and NGC 1275 have been detected with fluorescent Fe-K line, indicating that X-ray emission can be dominated by accretion flows or coronae illuminating accretion disks (e.g., Fukazawa et al. 2015), while none was detected from 3C 264 or M87, suggesting that X-rays may be dominated by jets instead (see, e.g., Wong et al. 2017, for the discussion of M87). Additionally, M87 is located in a hot gas rich environment (center of the Virgo Cluster), whereas the host galaxy of 3C 264 is offset from its host galaxy cluster (Perlman et al. 2010). Thus, comparing 3C 264 with other VHE FRI galaxies will provide \ninsights into the impact of the accretion flow/jet and the environment on the nature of the X-ray and TeV emissions. \n3C 264 has been observed from radio to TeV energy (Kagaya et al. 2017; Boccardi et al. 2019; Archer et al. 2020). The most favored model to explain the γ -ray and TeV emission is the jet scenario, with synchrotron emission from a jet plus the inverse Compton emission (synchrotron self-Compton, SSC). The full spectral energy distribution (SED) from radio to γ -ray appears to be well explained by the SSC model. However, prior to our study, hard X-ray emission above 10 keV had not been detected. In these SSC models, the hard X-ray region represents the transition between synchrotron and inverse Compton emissions. Therefore, modeling the full SED, including the hard X-ray spectrum, is crucial to identifying the emission origin. \nIn this paper, we report the detection of hard X-ray emission from 3C 264 above 10 keV, using simultaneous NuSTAR and Swift observations. Additionally, we analyzed archival Chandra data to provide better constraints on the origin of the X-ray emission. We describe the X-ray observations in Section 2. In Section 3, we present the image analysis of NuSTAR data, including deconvolved Chandra images, as well as an analysis of surface brightness and hardness profiles. Section 4 describes the spectral analysis results, focusing on the soft excess discovered with Swift , the overall hard Xray emission from the AGN, and the decomposition the hard X-ray emission from the core and the jet. We discuss the origin of the X-ray emission and compare our results with SSC models in Section 5 and conclude in Section 6. \nWeassume a flat universe with H 0 =70kms -1 Mpc -1 , Ω M =0.3, and Ω Λ =0.7. At the redshift of z = 0 . 0217 (Baum et al. 1990), this gives an angular scale of 0.439 kpc arcsec -1 and a luminosity distance of 94.5 Mpc to 3C 264. Errors are given at 1 σ confidence level unless otherwise specified.", '2. X-RAY OBSERVATIONS': "3C 264 was observed with NuSTAR on 2019 July 14, for 55ks (ObsID: 60501016002). All data were reduced using the HEASoft v6.31.1 and CALDB version 20230530. We reprocessed the data using the nupipeline script of the NuSTAR Data Analysis Software package with standard screening criteria, except that we set the filter mode using the parameter SAAMODE=optimized in order to reduce the slightly elevated background caused by the South Atlantic Anomaly. \nA Swift snapshot was taken during the NuSTAR observation on 2019 July 15 for 1.9 ks (ObsID: 00088885001). \nThe Swift-XRT spectrum was extracted using online tools 2 provided by the UK Swift Science Data Centre (Evans et al. 2009). \nWe also included a deep Chandra observation taken with the ACIS-S detector on 2004 January 24 (ObsID: 4916 for 38 ks). All the data were reprocessed using the Chandra Interactive Analysis of Observations ( CIAO ) software version 4.13 and the Chandra CALDB version 4.9.6. The default subpixel event-repositioning algorithm ' EDSER ' was used. After removing the flares using the CIAO deflare script, the final cleaned exposure time came to 34 ks.", '3.1. Spatial Analysis of NuSTAR Image': "With the NuSTAR observation, hard X-rays above ∼ 10 keV are clearly detected from 3C 264 for the first time. We explored images at different energy bands to determine the highest energy that could be detected. Figure 1 shows the NuSTAR images in 3-7 keV and 1520 keV. Each image was created by combining the observations from the two detectors. Relative astrometry was corrected by matching the centroids of the 3-7 keV emission. Hard X-ray emission in 15-20 keV is clearly detected at 4 σ . In the higher energy band of 20-25 keV, the detection significance of the source is at about 2 σ . No emission is detected at higher energy. \nTo quantify the spatial structure of the soft (3-7 keV) and hard (15-20 keV) X-ray emission, and to test if the emissions are point-like or extended, we fit the images using CIAO 's modeling and fitting package, Sherpa . More specifically, we compared the spatial model with a known bright point source, Cyg X-1, as observed with NuSTAR (ObsID: 10002003001). The data for Cyg X-1 were processed in the same way as 3C 264 with standard screening criteria. We fitted the NuSTAR soft and hard images of Cyg X-1 with a 2D Gaussian model and a 2D β -model to represent the core and the wing of the PSF, respectively. The central coordinates of the models are tied. A constant background model was also included in the fittings. The fitting region is limited to a circular region of radius ∼ 3 . ' 5 centered near the peak of the emission. This model fits the point source, Cyg X-1, very well (Figure 2). We use the best-fit 2D Gaussian model + 2D β -model as a point source template, along with a background model, to fit the images of 3C 264. The relative normalizations and all other parameters, except the central coordinates and the normalization of the 2D Gaussian model, are fixed. \nFigure 2 shows that the point source model fits the images of 3C 264 very well. We conclude that both the soft and hard X-ray emissions are unresolved by NuSTAR . The size of the X-ray jet resolved with Chandra is ≲ 2 '' , which is much smaller than the 18 '' FWHM of the NuSTAR PSF. Thus, both the soft and hard X-ray emissions detected with NuSTAR are consistent with an origin that is similarly compact, presumably coming from the Chandra core and jet regions. \nThe locations of the NuSTAR soft and hard X-ray peaks can be determined with about 2 . '' 5 and 10 . '' 4 of precision at 3 σ for two parameters of interest, respectively. The confidence regions are shown in the righthand panel of Figure 1. Due to the 8 '' absolute astrometric uncertainty of NuSTAR , there is an offset of about 10 '' between the Chandra and NuSTAR soft peak. In the figure, a relative astrometric correction has been made so that the peak of the NuSTAR 3-7 keV aligns with that of the Chandra emission. The locations of the NuSTAR soft and hard X-ray peaks are consistent with each other. Thus, the origin of the hard X-ray emission is consistent with that of the soft emission (i.e., core/jet emission).", '3.2. Chandra Image Deconvolution': "Using a maximum entropy deconvolution algorithm, the X-ray jet has already been resolved with Chandra with the size of the jet of about 2 '' pointing toward the northeast direct (Perlman et al. 2010). \nTo resolve the jet structure down to subarcsecond scale, we applied the Lucy-Richardson deconvolution algorithm to the Chandra subpixel resolution images (Richardson 1972; Lucy 1974). For each energy band used, we first performed a raytracing simulation using the Chandra Ray Tracer (ChaRT) 3 . The Chandra spectrum of 3C 264, fitted as an absorbed power-law model, and the aspect solution file from the Chandra observation were used for these raytracing simulations. For each binning size of the image to be deconvoluted, a simulated event file was created by projecting the rays onto the detector plane using the CIAO simulate psf script and the MARX software 4 . This script also created an image of the PSF. Finally, the CIAO arestore script was used, along with the PSF images, to perform the deconvolution. \nTo test the convergence of the deconvolution, we experimented with various image binning sizes (0.0625, 0.1, 0.125, 0.25, and 1 physical pixel), as well as different numbers of iterations (50, 100, and 150). All of \n<!-- image --> \n<!-- image --> \nFigure 1. Smoothed 3-7keV (left-hand panel) and 15-20 keV (middle panel) NuSTAR images of 3C 264. The circle at the center of the image indicates the 30 '' radius spectral extraction region. The circle on top is the background spectral extraction region with the same radius of 30 '' . Using a larger background region gives essentially the same results. A 0.3-6 keV Chandra subpixel resolution image is shown in the right-hand panel, with one image pixel size = 0 . '' 123. The two circles indicate the 3 σ position errors of the emission peaks measured with NuSTAR in different energy bands. Relative astrometric correction has been made so that the peak of the NuSTAR 3-7 keV aligns with that of the Chandra emission (see the text). The arc feature at about 0 . '' 8 southeast from the center is an artifact of the Chandra PSF (Section 3.2). \n<!-- image --> \n<!-- image --> \nFigure 2. Radial surface brightness profiles in the soft (3-7 keV, left-hand panel) and hard (15-20 keV, right-hand panel) bands. NuSTAR data of 3C 264 are shown in blue circles. The Cyg X-1 data, represented by purple circles with much smaller error bars, are normalized to match the surface brightness model of 3C 264. The green and red lines represent the point source model and the background used to fit the 3C 264 data, respectively. The total emission profiles are shown in orange. The very good fitting indicates that both the soft and hard X-ray emissions are unresolved by NuSTAR . \n<!-- image --> \nthese trials yielded similar results, with the same notable features appearing in each image. As a result, we selected a binning size of 0.125 physical pixels and 100 iterations as our nominal choice. \nThe upper panels in Figure 3 show the high-resolution Chandra subpixel resolution image in 0.3-6 keV and the corresponding deconvolved image. The green region in the upper left-hand panel, created by the CIAO make psf asymmetry region script, indicates the artifact of the PSF 5 . Aside from the artifact, the X-ray emission is elongated in the direction of the jet to the northeast. Most of the emission is confined to an area of approximately 2 '' . The deconvolved image reveals more details of the X-ray jet, showcasing a very narrow linear structure. The PSF artifact shows up in the same region and has been excluded on the image. \nThe jet is clearly one-sided with the core located to the southwest. Most of the X-ray emission originates from the core, which has a radius of ∼ 0 . '' 2. The jet is most luminous within ∼ 0 . '' 8, with the surface brightness decreasing progressively from the core. Beyond ∼ 0 . '' 8, the emission has diminished significantly and the jet appears to fade away at about 2 '' . The general spatial structure has been discussed by Perlman et al. (2010) using the maximum entropy deconvolution algorithm. However, the image deconvolved by the Lucy-Richardson method we used is noticeably sharper. For instance, our deconvolved image shows two X-ray tails beyond ∼ 0 . '' 8, one stronger to the northwest and the weaker one appears to turn west. Our deconvolved image is also sharper than that deconvolved with the Chandra HRC-I image observed in 2018 (Archer et al. 2020). \nAll these features are remarkably similar to the HST and VLA images, with much higher angular resolutions presented in Perlman et al. (2010). This suggests that the deconvolution method is robust and that the jet emission originates from the same source across the radio to X-ray spectrum. \nThe lower three panels in Figure 3 show the deconvolved images in three energy bands: the wide band in 0.5-6 keV (lower left-hand panel), the soft band in 0.5-1 keV (lower middle panel), and the hard band in 1-6 keV (lower right-hand panel). These figures were created for the analysis of surface brightness and hardness ratio profiles that follows. The general structures of the jets across the three bands are similar. However, the soft emission appears to be more extended, while the hard emission is narrower.", '3.3. Surface Brightness and Hardness Ratio Profiles of the Jet': "The upper left-hand panel in Figure 4 displays the background-subtracted, exposure-corrected surface brightness profiles of the core and jet within 4 '' from the core center in three energy bands (wide, soft, and hard). The source regions used are depicted in the lower lefthand panel of Figure 3. An angular width of 50 · is chosen to encompass most of the X-ray emission observed within 0 . '' 8. The position angle of the central jet axis is set at 40 . · 43 from north to east (Meyer et al. 2015). We selected a partial annulus background region with an angular width of 270 · extending from 5 '' to 10 '' in the direction opposite the jet. Vignetting correction has been applied using the corresponding exposure maps. The energy bands have been chosen to optimize the hardness ratio analysis that follows. Errors are estimated using the Gehrels prescription 6 (Gehrels 1986). \nBeyond the unresolved core, approximately 0 . '' 2 in radius, the surface brightness rapidly decreases out to about 0 . '' 8, where the most obvious part of the jet lies. Past this point, the jet becomes quite faint and the surface brightness decreases more gradually up to about 2 . '' 4. Any further out, the emission is consistent with background. \nThe deconvolved images from the soft and hard bands enable us to investigate the rough spectral properties at a spatial resolution higher than that achievable through spectral analysis. The latter is limited by PSF blurring. To accomplish this, we performed a hardness ratio analysis, a technique that has been widely used to characterize various X-ray astrophysical sources (see, e.g., Sarazin et al. 2001; Park et al. 2006; Wong et al. 2008; Thimmappa et al. 2022). \nWe experimented to determine the optimal energy bands for the hardness ratio analysis, and two energy bands were chosen: the soft (0.5-1 keV) and hard (16 keV) bands. The lower limit of 0.5 keV was chosen to minimize uncertainty in the Galactic column density, which more significantly impacts soft photons. Additionally, when fitting the Chandra spectrum within the central 2 '' region, we observed residuals around 0.3 keV. This makes the interpretation of energies lower than ∼ 0.5 keV challenging. The upper limit at 6 keV was set to maximize the signal-to-noise ratio. The energy value of 1 keV, which serves as the midpoint between our selected energy bands, is chosen to ensure that the signal-to-noise ratio in each region is greater than three within a 2 '' radius. \nFigure 3. Upper left-hand panel: 0.3-6 keV Chandra subpixel resolution image. The arc feature highlighted in the green region is an artifact of the Chandra PSF. The emission is elongated in direction of the jet to the northeast. Upper right-hand panel: Deconvolved Chandra image in 0.3-6 keV. Lower panels: Deconvolved Chandra image in the wide band (0.5-6 keV; left-hand panel), soft band (0.5-1 keV; middle panel), and hard band (1-6 keV; right-hand panel). The regions shown on the left are used for the analysis of surface brightness and hardness ratio profiles. In all panels, the color bar units are counts per image pixel, and one image pixel size = 0 . '' 0615. For each deconvolved image, the PSF artifact, located within the box region, has been excluded. \n<!-- image --> \n0.0 \n0.1 \n0.3 \n0.7 \n1.6 \n3.3 \n6.6 \n13.2 \n26.6 \n53.1 105.9 \n<!-- image --> \n0 \n1 \n3 \n7 \n14 \n29 \n59 \n119 \n239 \n478 \n952 \n<!-- image --> \n<!-- image --> \n0.0 \n0.4 \n1.2 \n2.8 \n5.9 \n12.3 \n24.8 \n49.9 \n100.4 200.2 399.1 \n<!-- image --> \n0.0 \n0.5 \n1.4 \n3.3 \n7.1 \n14.6 \n29.6 \n59.4 119.5 \n238.5 475.3 \nThe upper right-hand panel in Figure 4 displays the hardness ratio profile, which is derived by dividing the net photon counts in the soft band by those in the hard band for each radial region, or equivalently dividing the soft surface brightness profile by the hard surface brightness profile. The larger the hardness ratio, the softer the emission. The general trend suggests that the hardness ratio remains flat within ∼ 1 . '' 2 and softens from there to ∼ 2 '' , with an anomalous data point at 0 . '' 7 displaying \nparticularly hard emission. This anomalous data point is due to the abrupt decrease in soft photons in that region (see the upper left-hand panel in Figure 4). The surface brightness profiles change significantly around that radius, which could introduce a large systematic uncertainty in that region. The hardness ratio beyond ∼ 1 . '' 2 seems excessively soft for jet emission, which could be due to soft emission from the galaxy or systematic \n<!-- image --> \n<!-- image --> \n1 \n<!-- image --> \nFigure 4. Upper left-hand panel: Background-subtracted, exposure-corrected surface brightness profiles of the jet in different energy bands. The error bars are at 1 σ confidence level. For clarity, the profile in 0.5-6 keV is multiplied by a factor of 10. Upper right-hand panel: Hardness ratio profile defined as the ratio of the count in the soft energy band to the count in the hard energy band (Park et al. 2006). The larger the hardness ratio, the softer the emission. Errors are calculated using standard error propagation procedures. The red data point, calculated using a large region, represents the average value across the brighter part of the jet. The jet emission appears to be softer beyond 1 . '' 2, but this could be due to soft emission from the galaxy or subject to systematic uncertainties. Bottom left-hand panel: Photon index vs. hardness ratio corresponding to the Chandra observation, assuming a power-law model absorbed by the Galactic column density. Bottom right-hand panel: Similar to the upper right-hand panel, but the hardness ratio has been converted to photon index (thin black and red crosses) using the figure in the bottom left-hand panel. The low photon index at 0 . '' 7 appears to be an outlier (see text). The photon indices determined using the spectral fitting method are also shown (thick gray and brown crosses). \n<!-- image --> \nuncertainties. More data is needed to confirm whether this rising trend and the anomalous point are real. \nUpon closer inspection of the hardness ratio profile, the unresolved core within 0 . '' 2 is significantly harder than the jet region beyond that at 0 . '' 2-0 . '' 4. This does not seem to be due to the systematic error related to the PSF's dependence on energy because a larger PSF at higher energy should only make the outer regions harder, not the core. Spectral fitting also shows that the central core is harder than the jet on average, although the spa- \nion of the spectra is limited by the PSF. In principle pileup can make the central spectrum harder, but we have checked this with a PILEUP model in XSPEC 7 and found that it can bias the photon index (Γ) by at most 0.6%, which is negligible. \n7 The X-ray Spectral Fitting Package ( XSPEC ), https://heasarc. gsfc.nasa.gov/xanadu/xspec/ \nTo convert the hardness ratio into the photon index of the absorbed power-law model, we calculate the ratio of model-predicted rates in the soft and hard bands with different photon indices using XSPEC . Galactic absorption was assumed and the corresponding response files were used. The relationship between the photon index and the hardness ratio is shown in the lower left-hand panel of Figure 4. The converted photon index profile within the brighter part of the jet ( < 1 . '' 2) is shown as black crosses in the lower right-hand panel of Figure 4. The photon index at the < 0 . '' 2 core is Γ = 2 . 16 +0 . 03 -0 . 03 , which is significantly harder than that in the jet region of 0 . '' 2-0 . '' 4, where Γ = 2 . 41 +0 . 07 -0 . 08 . We also measured the average hardness ratio of the jet between 0 . '' 2 and 1 . '' 2, and found that the average photon index of the jet is Γ = 2 . 26 +0 . 05 -0 . 05 (red cross in the right-hand panels of Figure 4). Including the core from 0 '' to 1 . '' 2, the average photon index is Γ = 2 . 19 +0 . 03 -0 . 03 (not shown on the figure for clarity). \nWe also created spectra in the corresponding surface brightness regions and performed fitting to obtain the best-fit photon indices (thick crosses in the lower righthand panel of Figure 4). Since the region sizes are smaller than the PSF, the spectra across regions are mixed. Nevertheless, the general trend of the photon indices obtained by spectral fitting and the magnitude of the uncertainties are consistent with those obtained by the hardness ratio method. In particular, the average spectral index of the jet between 0 . '' 2 and 1 . '' 2 determined by the two methods are almost the same, suggesting that the hardness ratio analysis is reliable. \nNote that the innermost two regions are harder according to the spectral fitting, while only the innermost region appears harder in the hardness ratio analysis. This discrepancy might be due to spectral mixing in the spectral analysis, whereas for the hardness ratio analysis deconvolved images are used. The spectral analysis does not confirm the particularly hard region at 0 . '' 7, which could possibly be attributed to spectral mixing.", '4. SPECTRAL ANALYSIS': "We extracted NuSTAR spectra from the central region of 3C 264 for the observation and created the corresponding response files for the unresolved source (Figure 5, and see the discussion below for further details). The extraction region is circular with a 30 '' radius, which is close to the 29 '' half-power radius, centered at the NuSTAR 3-7 keV peak (Figure 1). The background cir- \ngion with the same radius was chosen to be far enough from the center and also located on the same detector chip as the source. The latter criterion is important to minimize instrumental background variations from chip to chip (Wik et al. 2014a). During the spectral analysis with XSPEC , the two NuSTAR spectra from both detectors were joint-fitted during the spectral analysis with XSPEC . To constrain the softer emission of the spectrum, we also included the Swift data taken during the NuSTAR observation (Section 2). \nThe unresolved NuSTAR X-ray emission is overwhelmingly dominated by the AGN activities, specifically the X-ray core and the jet, with only minimal contributions from the diffuse emission from hot gas in the galaxy, the ICM, or low-mass X-ray binaries (LMXBs). \nTo detect fainter sources unrelated to the AGN activities, we incorporated a Chandra observation taken in 2004 (Section 2; see Wong et al. (2014, 2017) for the procedure to detect and analyze Chandra point sources). Only one source is detected in the NuSTAR extraction region with a luminosity of ∼ 10 40 erg s -1 in 0.510 keV if it is at the distance of 3C 264, which is more than two orders of magnitudes fainter than the emission from 3C 264. Sun & Murray (2002) estimated that the contribution from unresolved low-mass X-ray binaries (LMXBs) is at most 4% using the scaling relation from the LMXB X-ray-to-optical ratios in Sarazin et al. (2001). The corresponding luminosity is approximately 1 . 6 × 10 41 erg s -1 in 0.5-10 keV. To assess the impact of all these point sources, we conservatively incorporated a LMXB model with the aforementioned luminosity (Wong et al. 2017; Wik et al. 2014b). The changes in the best-fit photon index and cutoff energy of the LLAGN are negligible, being at most 0.05% and 0.4%, respectively. Thus, we ignore the contributions from these point sources in the spectral analysis.", '4.1. Soft Excess in the Swift Spectrum': "In the Swift spectrum, there is an evident excess of Xray emission around ∼ 0.7-1 keV beyond what a powerlaw model predicts. This excess can be described by an APEC model absorbed by the Galactic absorption. We fixed the redshift to that of 3C 264 and set the metallicity to 0.8 solar, following the value used by Sun et al. (2007) when fitting the X-ray spectrum of the hot gas of the host galaxy. The best-fit temperature is approximately 0.7-0.8 keV. The unabsorbed X-ray flux between 0.5 and 2 keV is about 3 × 10 -13 erg cm -2 s -1 (see Table 1 below). Assuming the distance to 3C 264, the corresponding X-ray luminosity is about 3 × 10 41 erg s -1 . To determine the significance of this excess and ascertain the necessity of the extra APEC model, we sim- \nFigure 5. X-ray spectra with different models as listed in Table 1. POWERLAW models were used on the left-hand panels while CUTOFFPL models were included on the corresponding right-hand panels. Upper left-hand panel: The absorbed POWERLAW model of the NuSTAR spectra (black and red), taken from within a 30 '' circular region centered on the X-ray peak, was jointly fitted with the Swift spectrum (blue). This approach was used to constrain the overall hard X-ray emission originating from AGN activities (Core+Jet). The dotted lines represent the individual components ( POWERLAW + APEC ) as discussed in the text. Upper right-hand panel: Similar to the upper left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model. Middle lefthand panel: Similar to the upper left-hand panel, but including the Chandra jet spectrum (cyan) modeled with POWERLAW . The flux corrected jet models for the NuSTAR and Swift spectra are shown as dotted lines. The unresolved core is also modeled as a POWERLAW (dashed lines). Middle right-hand panel: Similar to the middle left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model for the unresolved core. Lower left-hand panel: Similar to the middle left-hand panel, but including the Chandra core spectrum (magenta) modeled with POWERLAW . Middle right-hand panel: Similar to the lower left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model for the unresolved core. \n<!-- image -->", 'Wong, Steiner et al.': "Table 1. Best-fit Spectral Results using Phenomenological Models \nTable 2. Best-fit Spectral Results using Comptonization ( compPS ) Model \nulated 1000 spectra using the single POWERLAW model and then compared the likelihood ratio for the single POWERLAW model ralative to the APEC+POWERLAW model (likelihood ratio test lrt, in XSPEC ). We found that 957 of the simulated likelihood ratios are smaller than the observed ratio ( p -value = 4.3%), strongly suggesting that the APEC+POWERLAW model is preferred. \nHowever, such a pronounced excess was not observed in the Chandra data, which was taken 15 yr prior to the NuSTAR and Swift observations. Although Sun et al. (2007) identified hot gas in the host galaxy with a temperature roughly around 0.65 keV from the Chandra observation, the flux of the excess in the Swift data is 20 times higher than the hot gas flux they reported. Thus, it is unlikely that the soft excess detected with Swift originates from the diffuse hot gas of the host galaxy. Instead, it might be attributed to an increase in soft flux related to AGN activities, such as an additional Comptonization component (Dewangan et al. 2007), ionized reflection (Crummy et al. 2006), or complex absorption (Gierli'nski & Done 2004; Done et al. 2007). Alternatively, it could also be attributed to another unresolved transient event, such as an off-nuclear hyperluminous X-ray source (HLX) occurring during the NuSTAR and Swift observations. Fortunately, this excess is soft and sufficiently faint in comparison to the LLAGN and jet spectra, ensuring that it does not significantly influence the spectral fitting results. These results are robustly anchored by the broader energy range of the spectra. When excluding the APEC component from the fitting, the deviations in the best-fit photon index and cutoff energy are less than the statistical uncertainties. Nevertheless, the soft excess is taken into account by the absorbed APEC model, with both temperature and normalization being free parameters tied across all spectra.", '4.2. Overall Hard X-ray Emission from AGN Activities': 'To constrain the overall hard X-ray emission from the AGN activities, which includes the unresolved X-ray core and the jets, we fitted the two 3-30 keV NuSTAR spectra jointly with the 0.3-10 keV Swift spectra. All the spectra were grouped with a minimum of one count per bin and were fitted using the C -statistic in XSPEC . Errors of spectral parameters were determined by using ∆ C = 1 (68% confidence) for one parameter of interest. \nAfter accounting for the soft excess (Section 4.1), we modeled the Swift and NuSTAR spectra using a single POWERLAW model to represent the combined emission from both the core and the jet. To investigate the potential presence of a spectral curvature, which could suggest emission from a hot accretion flow or an X-ray corona proximate to the black hole, we also employed \na cutoff power-law ( CUTOFFPL ) model for the fit. No intrinsic absorption was found in the spectra. Thus, the same Galactic absorption model was applied to all components, which was fixed at the Galactic value of N H = 2 . 45 × 10 20 cm -2 (COLDEN program from the Chandra X-ray Center using the data from Dickey & Lockman (1990); see also, Evans et al. (2006) and Perlman et al. (2010)). All the parameters were tied together across all the spectra, with the exception of the normalizations for each spectrum, to account for the crosscalibration uncertainties between the two NuSTAR and the Swift detectors. The best-fit parameters for the overall AGN emission as well as the core and jet components (Section 4.3 below) are listed in Table 1. \nThe overall spectrum, including the unresolved Chandra core and the jet, can be sufficiently characterized by a single POWERLAW model with a photon index of Γ = 2 . 26 +0 . 07 -0 . 07 , absorbed by the Galactic column density (upper left-hand panel in Figure 5). This is consistent with the value of 2 . 19 +0 . 03 -0 . 03 determined using the hardness ratio method described in Section 3.3. This is also consistent with the value of 2 . 24 ± 0 . 05 measured by Perlman et al. (2010). The overall unabsorbed 20-30 keV flux, which is completely dominated by the central regime of the AGN activities (core and jet), is 1 . 69 +0 . 13 -0 . 13 × 10 -13 erg cm -2 s -1 . This translates into a luminosity of 1 . 81 +0 . 14 -0 . 13 × 10 41 erg s -1 . \nFor the absorbed CUTOFFPL model, the best-fit photon index is Γ = 1 . 93 +0 . 18 -0 . 19 (upper right-hand panel in Figure 5). The cutoff energy is E cutoff = 20 +21 -7 keV with a 3 σ lower limit of 6 keV. The unabsorbed 2030 keV flux of the overall AGN-related activities is 1 . 13 +0 . 21 -0 . 18 × 10 -13 erg cm -2 s -1 , giving a luminosity of 1 . 21 +0 . 22 -0 . 20 × 10 41 erg s -1 . \nIn the soft X-ray band of 0.5-10 keV measured with Swift in 2019, both the POWERLAW and CUTOFFPL models estimate a luminosity of about 2 × 10 42 erg s -1 . This is similar to the luminosity of the unresolved core, 1 . 6 × 10 42 erg s -1 , observed with Chandra in 2004 (Perlman et al. 2010), and the power-law luminosity (core) of 2 . 26 × 10 42 erg s -1 observed with XMM-Newton in 2001 (Donato et al. 2004). These fluxes are about three to four times lower than the flux observed during a 2018 outburst observed with Chandra HRC-I (Archer et al. 2020), and it aligns with its low state observed in 2004 and in the preceding years (Boccardi et al. 2019). These observations suggest that the 3C 264 AGN, as observed by NuSTAR and Swift in 2019, is in a quiescent phase.', '4.2.1. Supporting Evidence of a Cutoff in Spectrum': 'The CUTOFFPL model offers a slightly better fit based on the face values of the C -statistics, but this improve- \nment is not statistically significant. The likelihood ratio test ( lrt in XSPEC ) also does not suggest a statistical preference for the CUTOFFPL model. We have calculated the Akaike information criterion (AIC: Akaike 1974) to distinguish between models. The preferred model for the data is the one that minimizes the AIC (Liddle 2007; Medvedev et al. 2021). Table 1 shows that the CUTOFFPL model has a slightly smaller AIC than the POWERLAW model, suggesting that the CUTOFFPL model is slightly preferred. There is also additional evidence supporting the CUTOFFPL model, which is detailed below. \nFirst, a significant discrepancy is observed between the normalizations of the POWERLAW model for Swift and NuSTAR , despite the Swift observation being conducted simultaneously with the NuSTAR observation. No time variation was detected in the NuSTAR light curve, suggesting that this discrepancy is not attributable to variability. The discrepancy between model normalizations, as measured with NuSTAR and Swift , is approximately 20% for the CUTOFFPL model and 45% for the POWERLAW model, respectively. While a cross-calibration uncertainty of 10-15% between different X-ray observatories is within expected limits, and a 20% difference is on the high end but not alarming (Madsen et al. 2017; Molina et al. 2019; Abdelmaguid et al. 2023), a discrepancy exceeding 40% is inconsistent with the current calibration standards 8 . The discrepancy observed in the single POWERLAW model could potentially be explained by adjusting the normalizations to account for spectral curvature. \nMore specifically, the fitting is biased toward the NuSTAR data because there are more photon counts in the NuSTAR data. When fitting a single POWERLAW model to both the NuSTAR and Swift datasets and tying their photon indices, if there is a cutoff in energy, the POWERLAW slope will try to fit the steeper NuSTAR spectra at higher energy. Therefore, if the normalization of Swift is the same as the NuSTAR normalization, the model will overestimate the softer photons measured by Swift . To compensate for this curvature in the fitting, the Swift normalization needs to be lower than the NuSTAR normalization. \nWe simulated the overall spectra with the absorbed CUTOFFPL + APEC model using 1000 times longer exposures for each detector. The model parameters used are listed in Table 1, except that we set all the normalizations of the CUTOFFPL model to those determined by the NuSTAR FPMA detector. The best-fit models to \nFigure 6. Photon index fitted at various energy bins for different dataset. NuSTAR and Chandra data were fitted with absorbed POWERLAW model. For Swift data, since there is a clear soft excess, an absorbed APEC + POWERLAW model was used. \n<!-- image --> \nthe simulated datasets largely reproduce this normalization discrepancy, with the NuSTAR normalizations being more than 30% higher than that of Swift 9 . \nSecond, we divided the NuSTAR spectra into different energy bands and fit the NuSTAR data separately for each band using a POWERLAW model. For comparison, we also fitted the Chandra and Swift data individually. The spectral slope steepens (i.e., the photon index increases) at higher energies (see Figure 6), indicating the presence of an energy cutoff. \nThird, when decomposing the core and jet spectra with Chandra data (Section 4.3.2), the CUTOFFPL model is statistically favored for the core emission, suggesting that a cutoff is needed at least for the core. \nThese points serve as supporting evidence for the CUTOFFPL model. However, to definitively confirm the presence of an energy cutoff, deeper simultaneous observations spanning a broad energy range (e.g., 0.5-40 keV) would be necessary.', '4.2.2. Comptonization Model with Reflection': 'We replace the phenomenological POWERLAW or the CUTOFFPL models with the compPS model (Poutanen & Svensson 1996), which is a physical model that includes the thermal Comptonization emission from hot \nplasma and reflection from a standard accretion disk. Following Ursini et al. (2015), we set the geometry of the hot plasma to spherical ( GEOM = -4). The disk is assumed to have a multicolor temperature, with the inner disk temperature set to 10 eV (Beckmann et al. 2011; Ursini et al. 2015) and the inclination angle set to i = 50 · (Donato et al. 2004). The free parameters for the compPS model include the hot plasma temperature T e , the Compton parameter y , reflection strength R , and normalizations. \nWhile the reflection strength cannot be precisely constrained, the best-fit value is effectively zero ( R = 4 × 10 -14 ), and the 1σ upper limit is R ≤ 0 . 68. Therefore, since there is no evidence of a reflection hump, we fixed R = 0 in the fitting process. \nThe best-fit parameters for the overall AGN emission as well as the core and jet components (Section 4.3 below) are listed in Table 2. The best-fit hot plasma temperature is T e = 15 +3 -2 keV, and the Compton parameter is y = 0 . 56 +0 . 77 -0 . 09 . The optical depth, τ , is related to the Compton parameter and temperature by y = 4 τ ( kT e /m e c 2 ). Therefore, the optical depth falls within the range of τ = 3-12. Using different geometrical settings for the hot plasma (e.g., slab, cylinder, or hemisphere) gives similar results.', '4.3.1. With Chandra Jet Spectrum': "We further constrained the X-ray emission from the individual components of the unresolved X-ray core and the jet by jointly fitting their corresponding Chandra spectra 10 . We first included the Chandra jet spectrum, extracted from the partial annular regions between 0 . '' 2 and 1 . '' 2, as shown in the lower left-hand panel of Figure 3. This jet spectrum was modeled as an extra absorbed POWERLAW . The power-law indices for the NuSTAR , Swift , and Chandra spectra were tied together. Given that the Chandra PSF size scale is comparable to the jet spectral extraction region, some photon leakage from this region is anticipated. However, for the NuS- \nTAR and Swift spectra, the spectral extraction regions are expected to encompass most of the jet emission. By counting the total number of photons in the deconvolved image within a rectangular region that includes most of the jet's photons and comparing this count to that observed in the Chandra jet spectrum, we estimated that the jet spectrum accounts for 59.6% of the total counts. We corrected for this flux loss by tying the NuSTAR and Chandra normalizations to the ratio of this fraction. The Swift and NuSTAR normalizations were also tied together. Based on the deconvolved image, we assessed that the emission beyond 1 . '' 2 contributes less than 5% of the total emission. However, for the emission inside 0 . '' 2, we lack a precise estimate of the unresolved jet's contribution to the X-ray emission. Thus, we did not apply such corrections in the spectral analysis. \nFor the NuSTAR and Swift spectra, after accounting for the jet emission, the remaining emission predominantly originates from the unresolved Chandra core. Similar to Section 4.2, we modeled this emission with either an absorbed POWERLAW model or an absorbed CUTOFFPL model. Additionally, we kept the normalizations for NuSTAR and Swift separate (untied). \nThe best-fit spectra with all the model components are displayed in the middle panels of Figure 5. The best-fit photon indices for the POWERLAW jet component, when jointly fitted with either the single POWERLAW or the CUTOFFPL models of the core, are 2 . 25 +0 . 04 -0 . 04 and 2 . 26 +0 . 04 -0 . 04 , respectively. For the core, the single POWERLAW model yields a photon index of 2 . 27 +0 . 10 -0 . 10 , which is essentially the same as the jet's index. The CUTOFFPL model for the core has a significantly harder photon index of 1 . 68 +0 . 31 -0 . 35 , and the cutoff energy is determined to be E cutoff = 12 +12 -4 keV. The 2030 keV flux of the core, as measured with NuSTAR , is 1 . 25 +0 . 14 -0 . 13 × 10 -13 erg cm -2 s -1 for the POWERLAW model and 0 . 66 +0 . 19 -0 . 16 × 10 -13 erg cm -2 s -1 for the CUTOFFPL model, accounting for about 74% and 58% of the overall AGN emission, respectively. \nIn the case of the POWERLAW model for the core component, the discrepancy between the NuSTAR and Swift normalizations is larger compared to the analysis presented in Section 4.2. The NuSTAR normalizations are twice as high as those from Swift , and they are inconsistent at a level beyond 90% confidence statistically. For the CUTOFFPL model, however, the NuSTAR normalizations are about 25% higher than the Swift normalization, and they are consistent within 1 σ statistically. This further supports the hypothesis of a necessary cutoff in the power-law model for the core. \nUsing the compPS model discussed in Section 4.2.2 for the core, the best-fit hot plasma temperature is deter- \nmined to be T e = 14 +3 -1 keV, and the Compton parameter is y = 0 . 9 +5 . 0 -0 . 4 . The optical depth is within the range of τ = 4-58.", '4.3.2. With Chandra Jet and Core Spectra': "Finally, we extracted a Chandra core spectrum within a radius of 0 . '' 2. PSF correction was applied when generating the response file. As discussed in Section 4.3.1, the amount of emission originating from the unresolved jet within 0 . '' 2 is unknown, and therefore we did not include any corrections for this. \nWe additionally included the Chandra core spectrum and jointly fitted it with the NuSTAR and Swift spectra. We modeled the core component using either a single POWERLAW or a CUTOFFPL model. Given the potential for variability, the spectral shapes of the Chandra and NuSTAR / Swift observations might differ. To investigate the potential spectral variation, we separately fitted the Chandra and NuSTAR spectra in the overlapping 3-8 keV energy band using an absorbed POWERLAW model. For the Swift spectrum, we extended the fitting range to 0.3-8 keV to increase the statistics and included an APEC model, as described in Section 4.1, to account for the soft excess. The photon indices from all three instruments are consistent within a 1 σ level, indicating no evidence of spectral variation. Consequently, we tied the spectral indices of the core components across all observations during the fitting process. The normalizations for NuSTAR , Swift , and Chandra were kept separate (untied) to account for calibration uncertainties or potential flux variations. \nThe best-fit spectra with all model components are shown in the lower panels of Figure 5. The best-fit photon indices of the jet component, when jointly fitted with the single POWERLAW and the CUTOFFPL core models, are 2 . 26 +0 . 04 -0 . 04 and 2 . 25 +0 . 04 -0 . 04 , respectively. These values are essentially the same as those obtained from the analysis without the Chandra core spectrum. \nFor the core, the single POWERLAW model yields a photon index of 2 . 05 +0 . 04 -0 . 04 , which is significantly harder than the jet's spectrum. The photon index for the CUTOFFPL model is 1 . 93 +0 . 06 -0 . 06 , which is also notably harder. The cutoff energy is determined to be E cutoff = 19 +10 -5 keV. The associated 20-30keV flux measured with NuSTAR is 1 . 65 +0 . 10 -0 . 10 × 10 -13 erg cm -2 s -1 and 0 . 81 +0 . 17 -0 . 15 × 10 -13 erg cm -2 s -1 for the POWERLAW and CUTOFFPL models, respectively. These values account for approximately 98% and 72% of the overall AGN emission, respectively. \nIn the case of both the POWERLAW and CUTOFFPL models for the core component, while the Swift data normalizations are 40-50% lower than those of NuSTAR or \nChandra , they remain statistically consistent at the 90% confidence level. Notably, the NuSTAR and Chandra normalizations are consistent within 1 σ . This supports the assumption that there is no significant spectral variation. \nWe conducted a likelihood ratio test ( lrt ) in XSPEC with 1000 simulations to differentiate between the POWERLAW and CUTOFFPL models for the core component. The test results suggest that the POWERLAW model is rejected at a confidence level of 5.4%, strongly indicating the need for a cutoff in the spectrum. \nThe compPS model for the core, as discussed in section 4.2.2, yields a best-fit hot plasma temperature of T e = 14 . 9 +1 . 7 -1 . 3 keV and a Compton parameter of y = 0 . 61 +0 . 15 -0 . 06 . The optical depth falls within the range of τ = 4-7.", '5. DISCUSSION': "We have detected hard X-ray emission ≳ 10 keV from 3C 264. However, both the soft and hard X-ray emissions are unresolved by NuSTAR . The locations of the soft and hard X-ray peaks are consistent with each other, indicating that the origin of the hard X-ray emission aligns with that of the soft emission, likely from the core and the jet resolved by Chandra and by other optical and radio observations. \nWe have presented the highest spatial resolution Xray Chandra image using the Lucy-Richardson deconvolution method. The X-ray jet can be distinctly resolved down to approximately 0 . '' 2 from the center of the unresolved core. The X-ray morphology bears a remarkable similarity to the radio and optical emissions. The Xray jet is most luminous within about 0 . '' 8 from the core and appears linear, indicating that the jet's momentum is dominant. This region, at or less than 0 . '' 8 from the core, is also where four knobs have been resolved with HST , ranging from 0 . '' 15 to 0 . '' 6 away from the core center (Perlman et al. 2010; Meyer et al. 2015). Beyond that, the jet significantly diminishes in brightness and splits into two directions. The stronger path continues northeast, while the weaker one turns to the east. X-ray emissions from the jet are undetectable beyond ∼ 2 '' . \nThe deconvolved Chandra images in the 0.5-1 keV and 1-6 keV bands have been utilized to create a hardness ratio map, allowing us to investigate the spectral properties at a spatial resolution surpassing that achievable through spectral analysis. The average photon index of the jet, determined by this hardness ratio map within the soft X-ray emission range (0.5-6 keV), is Γ = 2 . 26 +0 . 05 -0 . 05 . This finding is in excellent agreement with the spectral analysis, which yields a Γ ranging between 2.21 and 2.30 (see Table 1). The hardness \nratio map also indicates that the photon index of the unresolved Chandra core (within < 0 . '' 2) is significantly harder than in the adjacent regions of the jet. \nWhen jointly fitting the NuSTAR spectra with the Swift spectrum taken during the NuSTAR observation, the overall AGN spectrum (core+jet) can be adequately modeled with either the POWERLAW or CUTOFFPL models. However, there is evidence supporting the presence of a cutoff energy at E cutoff = 20 +21 -7 keV. Including the resolved Chandra core and jet spectra in the soft Xray, taken 15 yr prior to the NuSTAR and Swift observations, further reinforces the likelihood of a cutoff at E cutoff = 19 +10 -5 keV. The CUTOFFPL model also indicates a significantly harder photon index for the core, aligning with the hardness ratio analysis. Such a cutoff in energy implies that the X-ray emission from the core can be at least partially contributed by the RIAF. Notably, the cutoff energy for 3C 264 is the lowest among other LLAGNs measured in X-ray (Chakraborty et al. 2023; Jana et al. 2023). \nWhen fitted with the Comptonization model ( compPS ), the best-fit electron temperature is approximately T e ≈ 15 keV, and the optical depth is more precisely constrained within the range of τ = 4-7, especially when combined with Chandra data. This aligns excellently with the anticorrelation observed between optical depth and electron temperature in a sample of 16 LLAGNs (Chakraborty et al. 2023). Such an anticorrelation has also been previously noted in more luminous Seyfert galaxies (Tortosa et al. 2018) and in the hard state of black hole binaries (Banerjee et al. 2020), suggesting the universality of coronal properties across different black hole masses and accretion rates. This has important implications for a departure from a fixed disk-corona configuration in radiative balance (see Tortosa et al. 2018; Chakraborty et al. 2023, for further discussions). \nThe overall X-ray spectrum is consistent with the X-ray properties of LLAGNs discussed by Ho (2008). The photon index between 0.5 and 10 keV is approximately 2.1, which is within the range of 1 . 4-2 . 2 of typical LLAGNs. The power-law component shows very little intrinsic absorption. There is no evidence of Fe K α emission or Compton reflection, suggesting either the absence of an optically thick accretion disk or that it is truncated and replaced by an optically thin RIAF. Note that RIAF models tend to predict harder X-ray spectra than jet models, as can be seen in Figure 17 of Nemmen et al. (2014). As such, the measured photon index of 1.93 from the CUTOFFPL core model agrees better with a RIAF origin. Detailed modeling of the multiwavelength \nSED will be needed to confirm this (e.g., Nemmen et al. 2014). \nA similar conclusion regarding the existence of RIAFs has been drawn from studies of the LLAGNs Cen A 11 (Furst et al. 2016), M81 (Young et al. 2018), NGC 7213 (Ursini et al. 2015), NGC 3998 and NGC 4579 (Younes et al. 2019). Although Fe lines have been detected in a number of cases, they are attributed to hot thermal plasma or to broad-line regions (see, e.g., Ursini et al. 2015; Young et al. 2018; Younes et al. 2019). For 3C 264, excess in soft emission below ∼ 2 keV is detected, which can be fitted with a thermal plasma model with a temperature of 0.7-0.8 keV. The origin of the soft excess is not clear. All of these are consistent with the X-ray properties of LLAGNs as described by Ho (2008).", '5.1. Implication of the Low Cutoff Energy': 'The cutoff energy E cutoff ≈ 20 keV of 3C 264 is very low when compared to other AGNs, which have a median E cutoff ∼ 200 keV (Ricci et al. 2017). Lower cutoff values are usually observed at very high Eddington ratios (Ricci et al. 2018). The LLAGN 3C 264 has an Eddington ratio of approximately 7 × 10 -5 . This ratio places it outside the typical correlations observed for AGNs with Eddington ratios higher than about 0.001, such as the photon index-Eddington ratio correlation noted in recent studies (e.g., Chakraborty et al. 2023; Jana et al. 2023). For instance, these cited recent studies do not report higher cutoff energies for LLAGNs with Eddington ratios below 0.001, unlike those expected with higher ratios. Some LLAGNs exhibit very low cutoff energies or coronal temperatures, such as a cutoff energy E cutoff ≈ 80 keV and an electron temperature T e ≈ 30-40 keV in low-accreting Seyferts (Jana et al. 2023) and an electron temperature T e ≲ 10-20 keV in LLAGNs (Chakraborty et al. 2023). Therefore, the notably low cutoff of 3C 264, considering the uncertainties, is not overly surprising. \nThe low cutoff energy suggests a cooler corona or plasma within the hot accretion flow. If the X-ray emission arises from the thermal Comptonization of seed disk photons by this hot plasma, particularly in a scenario where the disk is truncated in a RIAF, Comptonization could effectively cool the hot plasma. The smaller the truncation radius, the more efficient the cooling becomes (Tortosa et al. 2018). Consequently, the low cutoff energy indicates a small truncation radius. However, there is no evidence of a truncated disk in 3C 264, such as reflection features or a UV bump (Boccardi et al. \n2019), and the presence of a truncated disk remains unconfirmed. Alternatively, a reduced scale height of the corona or hot accretion flow could also enhance the Compton cooling rate of the hot plasma (Tortosa et al. 2018). The cutoff might also result from bremsstrahlung emission in the outer parts of the RIAF near the Bondi radius. This extended quiescent emission is similarly proposed to explain the low cutoff energy in the X-ray spectrum of Sgr A ∗ (Yuan et al. 2003; Quataert 2002).', '5.2. Testing the Synchrotron Self-Compton Model': 'To account for the γ -ray and TeV emission from 3C 264, the SED has been modeled using the synchrotron self-Compton (SSC) jetted model (Figure 1 in Kagaya et al. 2017). This model adequately fits the softer X-ray emission (up to ≲ 10 keV) and the γ -ray emission observed during the quiescent state of the core. However, it significantly underpredicts the hard X-ray emission (around ∼ 20 keV) observed with NuSTAR by an order of magnitude; our measured SED at 20 keV is around νf ν = 3-4 × 10 -13 erg cm -2 s -1 while the SSC model predicts a value of ∼ 4 × 10 -14 erg cm -2 s -1 . Furthermore, the predicted power-law index of approximately 3 in 1-10 keV is substantially steeper than the value measured. The large discrepancy is likely due to the phenomenological model used to fit the emission below the hard X-ray band. Similarly, recent SSC jetted models of the low state of the 3C 264 core also underestimated the SED at ∼ 20 keV by more than an order of magnitude, and their predicted power-law slope in the 1-10 keV range of ∼ 3-4 is also significantly steeper than our observed values (Boccardi et al. 2019). It is possible that these model fittings underestimated the maximum energy of the electron population in 3C 264. This contrasts with the findings for another TeV radio galaxy, M87, where the SSC model overpredicts the hard X-ray emission at 40 keV by approximately a factor of three during the quiescent state of the core, and the predicted power-law index is flatter than the measured spectral slope (Wong et al. 2017). NuSTAR plays a crucial role in exploring the transition from synchrotron-dominated to (self) inverse Compton-dominated emission around 10 keV, thereby providing key insights into these VHE processes. \nPresently, our understanding is limited by the statistical uncertainties of our data. Future in-depth observations with NuSTAR , extending detection beyond 30 keV, and simultaneous high angular resolution multiwavelength observations, will offer tighter constraints on accretion models and the VHE emission mechanisms.', '6. CONCLUSION': "For the first time, hard X-ray emission ≳ 10 keV from the TeV radio galaxy 3C 264 has been detected, which is unresolved with NuSTAR . The location of the hard X-ray emission is consistent with that of the soft Xray emission. An excess in soft emission below approximately 2 keV has been detected with Swift , and its origin is unclear. We have generated high-resolution deconvolved Chandra images to more precisely determine the origin of the X-ray emission. These deconvolved Chandra images show that the X-ray jet can be resolved down to approximately 0 . '' 2 from the unresolved Chandra core. The X-ray morphology is similar to that of the radio and optical jet. Furthermore, the X-ray spectrum of the Chandra core is harder than that of the jet. The X-ray spectrum can be adequately modeled with a single power-law model. However, evidence suggests the presence of a cutoff in the energy around 20 keV, which indicates that at least some of the X-ray emission from the core can be attributed to the RIAF. The Comptonization model indicates an electron temperature of about 15 keV and an optical depth ranging between 4 and 7, following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs measured in X-rays. Meanwhile, recent SSC models, which were proposed to explain γ -ray and TeV emission, significantly underestimate the hard Xray emission from the low state of the core at 20 keV by an order of magnitude or more. This suggests that the synchrotron electrons might be accelerated to higher energies than previously thought.", 'ACKNOWLEDGMENTS': "The authors would like to thank the referee and the editors for their comments and suggestions, which have improved the manuscript. This work was supported by the following NASA grants: NuSTAR awards 80NSSC21K1855, 80NSSC20K0052, 80NSSC22K0032, and 80NSSC22K0065, Chandra awards GO7-18071B, G09-20111A, and G02-23092X. C.S. and A.B. were also supported by the SUNY Brockport's Summer Undergraduate Research Program (SURP) and the Physics Department's Richard V. Mancuso Summer Research Award and Donald '80 and Diana '81 Hallenbeck Research Scholars Fund. R.N. was supported by NASA through the Fermi Guest Investigator Program (Cycle 16) and a Bolsa de Produtividade from Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico. D.L. was supported by the NASA ADAP grant 80NSSC22K0218. This paper employs a list of Chandra datasets, obtained by the Chandra X-ray Observatory, which are contained in doi:10.25574/cdc.263."}

2023Natur.623..932L

Planets with radii between that of the Earth and Neptune hereafter referred to as subNeptunes are found in closein orbits around more than half of all Sunlike starsSUP12SUP. However their composition formation and evolution remain poorly understoodSUP3SUP. The study of multiplanetary systems offers an opportunity to investigate the outcomes of planet formation and evolution while controlling for initial conditions and environment. Those in resonance with their orbital periods related by a ratio of small integers are particularly valuable because they imply a system architecture practically unchanged since its birth. Here we present the observations of six transiting planets around the bright nearby star HD 110067. We find that the planets follow a chain of resonant orbits. A dynamical study of the innermost planet triplet allowed the prediction and later confirmation of the orbits of the rest of the planets in the system. The six planets are found to be subNeptunes with radii ranging from 1.94RSUBSUB to 2.85RSUBSUB. Three of the planets have measured masses yielding low bulk densities that suggest the presence of large hydrogendominated atmospheres.

2023-11-01T00:00:00Z

['2023Natur.623..932L', 'arXiv:2311.17775', '2023arXiv231117775L', '10.48550/arXiv.2311.17775', '10.1038/s41586-023-06692-3']

['Astrophysics - Earth and Planetary Astrophysics']

A resonant sextuplet of subNeptunes transiting the bright star HD 110067

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

https://arxiv.org/pdf/2311.17775.pdf

{'A resonant sextuplet of sub-Neptunes transiting the bright star HD 110067': "R. Luque 1 , ∗ , H. P. Osborn 2 , 3 , † , A. Leleu 4 , 2 , † , E. Pall'e 5 , 6 , † , A. Bonfanti 7 , O. Barrag'an 8 , T. G. Wilson 9 , 10 , 11 , C. Broeg 2 , 12 , A. Collier Cameron 9 , M. Lendl 4 , P. F. L. Maxted 13 , Y. Alibert 12 , 2 , D. Gandolfi 14 , J.-B. Delisle 4 , M. J. Hooton 15 , J. A. Egger 2 , G. Nowak 16 , 5 , 6 , M. Lafarga 10 , 11 , D. Rapetti 17 , 18 , J. D. Twicken 17 , 19 , J. C. Morales 20 , 21 , I. Carleo 5 , 22 , J. Orell-Miquel 5 , 6 , V. Adibekyan 23 , 24 , R. Alonso 5 , 6 , A. Alqasim 25 , P. J. Amado 26 , D. R. Anderson 10 , 11 , G. Anglada-Escud'e 20 , 21 , T. Bandy 27 , T. B'arczy 28 , D. Barrado Navascues 29 , S. C. C. Barros 30 , 31 , W. Baumjohann 7 , D. Bayliss 10 , J. L. Bean 1 , M. Beck 4 , T. Beck 2 , W. Benz 2 , 12 , N. Billot 4 , X. Bonfils 32 , L. Borsato 33 , A. W. Boyle 34 , A. Brandeker 35 , E. M. Bryant 25 , 10 , J. Cabrera 36 , S. Carrazco Gaxiola 37 , 38 , 39 , D. Charbonneau 40 , S. Charnoz 41 , D. R. Ciardi 34 , W. D. Cochran 42 , K. A. Collins 40 , I. J. M. Crossfield 43 , Sz. Csizmadia 36 , P. E. Cubillos 22 , 7 , F. Dai 44 , 34 , M. B. Davies 45 , H. J. Deeg 5 , 6 , M. Deleuil 46 , A. Deline 4 , L. Delrez 47 , 48 , O. D. S. Demangeon 30 , 31 , B.-O. Demory 12 , 2 , D. Ehrenreich 4 , 49 , A. Erikson 36 , E. Esparza-Borges 5 , 6 , B. Falk 50 , A. Fortier 2 , 12 , L. Fossati 7 , M. Fridlund 51 , 52 , A. Fukui 53 , 5 , J. Garcia-Mejia 40 , S. Gill 10 , M. Gillon 47 , E. Go ff o 14 , 54 , Y. Gomez Maqueo Chew 37 , M. Gudel 55 , E. W. Guenther 54 , M. N. Gunther 27 , A. P. Hatzes 54 , Ch. Helling 7 , K. M. Hesse 3 , S. B. Howell 17 , S. Hoyer 46 , K. Ikuta 56 , K. G. Isaak 27 , J. M. Jenkins 17 , T. Kagetani 56 , L. L. Kiss 57 , 58 , T. Kodama 53 , J. Korth 59 , K. W. F. Lam 36 , J. Laskar 60 , D. W. Latham 40 , A. Lecavelier des Etangs 61 , J. P. D. Leon 56 , J. H. Livingston 62 , 63 , 64 , D. Magrin 33 , R. A. Matson 65 , E. C. Matthews 66 , C. Mordasini 2 , 12 , M. Mori 56 , M. Moyano 67 , M. Munari 68 , F. Murgas 5 , 6 , N. Narita 53 , 62 , 5 , V. Nascimbeni 33 , G. Olofsson 35 , H. L. M. Osborne 25 , R. Ottensamer 55 , I. Pagano 68 , H.", 'The HD 110067 planetary system': "layer (H and He). The equation of state (EOS) of water is the one of ref.[144], the core EOS is the one of ref.[145], and we use for EOS of ref.[146] for the silicate mantle. The thickness of the gas envelope, which depends on the planetary age, mass, etc., is derived from ref.[147]. Note that the influence of the gas layer on the innermost planet (compression and thermal e ff ect) is not included in our model, as the mass of gas layer for the six planets is small (see below). The planetary Si / Mg / Fe molar ratio in all planets is assumed to be equal to the stellar one. The prior distribution of the mass fractions of the three innermost layers (core, mantle, and water layer) is assumed to be uniform on the simplex - the surface defined by the sum of the three mass fractions equal to one. In addition, the mass fraction of the water layer is assumed to be 50% at most [148, 149], and for the gas mass, we use a uniform log prior. \nThe results from this analysis are shown in Extended Data Fig. 6. Our model shows that the gas mass content of all planets is of the order of 10 -3 M ⊕ to 10 -1 M ⊕ (median value, see Table S5), with the notable exception of HD 110067 e (median value of ∼ 10 -7 M ⊕ using the masses from M ethod I, ∼ 10 -3 M ⊕ using M ethod II). The apparent lack of an atmosphere of planet e (located just outside of planet d, which is the most gas-rich of the system, according to the internal structure models) is puzzling. If confirmed by future better determination of its density, the origin of the peculiar internal structure of planet e will have to be understood in the context of the very fragile architecture of the whole HD 110067 system. On the other hand, the water fraction for all planets is essentially unconstrained, due to the still large uncertainty in the planetary masses. However, according to simulations of combined planetary formation and evolution, independently of the accretion mechanism (planetesimal- or pebble-based) all the planets in the system have masses and radii consistent with a formation beyond the ice line [150-152]. Therefore, it is possible that even though the water content is unconstrained in our model, the planets' cores are rich in volatiles. JWST observations of some atmospheric trace gases (particularly ammonia, methane, and / or methanol) could be used as a proxy for the presence of a deep or shallow surface that could break the degeneracies from internal composition models using bulk density measurements alone [153, 154].", 'Bern, Switzerland': "- 3 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n- 4 Observatoire Astronomique de l'Universit'e de Gen'eve, Chemin Pegasi 51, 1290 Versoix, Switzerland\n- 5 Instituto de Astrofisica de Canarias, Via Lactea s / n, 38200 La Laguna, Tenerife, Spain\n- 6 Departamento de Astrofisica, Universidad de La Laguna, Astrof'ısico Francisco Sanchez s / n, 38206 La", 'Laguna, Tenerife, Spain': '- 9 Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North', 'Haugh, St Andrews KY16 9SS, UK': '- 10 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK\n- 11 Centre for Exoplanets and Habitability, University of Warwick, Coventry, CV4 7AL, UK\n- 12 Center for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland\n- 13 Astrophysics Group, Lennard Jones Building, Keele University, Sta ff ordshire, ST5 5BG, United Kingdom \n- 15 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK \n16 \nInstitute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,', 'Grudzicadzka 5, 87-100 Torun, Poland': '- 17 NASA Ames Research Center, Mo ff ett Field, CA 94035, USA\n- 18 Research Institute for Advanced Computer Science, Universities Space Research Association, Washington,', 'DC 20024, USA': "19 SETI Institute, Mountain View, CA 94043, USA \n- 20 Institut de Ciencies de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s / n, 08193 Bellaterra, Spain\n- 21 Institut d'Estudis Espacials de Catalunya (IEEC), Gran Capit'a 2-4, 08034 Barcelona, Spain", '4 The HD 110067 planetary system': 'Extended Data Fig. 4 Results from the ground-based campaign to detect HD 110067 f . a, ∆ WAIC for each of the constrained period bins when compared to a transit-free model. b,c, Best-fit decorrelated photometry with ( b ) and without ( c ) a transit model. Each light curve from each telescope has been o ff set for clarity. Error bars represent 1 σ uncertainties. \n<!-- image --> \nExtended Data Fig. 5 Results from the two radial velocity analyses to measure the mass of each of the planets in the HD 110067 system . Each histogram represents the posterior density function (pdf) of the radial velocity semi-amplitudes as inferred from M ethod I (red) and M ethod II (blue). The area underneath each histogram is normalized to unity. \n<!-- image --> \nExtended Data Fig. 6 Gas mass fraction of the HD 110067 planets as a function of their equilibrium temperature . We infer two values per planet by assuming the di ff erent planetary masses from our M ethod I (red) and M ethod II (blue) radial velocity analyses. The boxes, orange lines, green triangles, and red stars represent respectively the 25% and 75% percentiles, medians, means, and modes of the posterior distributions. The opacity of the vertical lines is proportional to the posterior distribution. \n<!-- image --> \nExtended Data Table 1 CHEOPS observing log. Filler observations aim to catch transits serendipitously in between time-critical observations with higher priority. Boldface notes indicate that a transit event was detected in the data.', 'References': '- [27] Stassun, K. G. et al. The TESS Input Catalog and Candidate Target List. Astron. J. 156 (3), 102 (2018).\n- [28] Stumpe, M. C. et al. Kepler Presearch Data Conditioning I-Architecture and Algorithms for Error Correction in Kepler Light Curves. Proc. Acad. Sci. Pacific 124 (919), 985 (2012).\n- [29] Stumpe, M. C. et al. Multiscale Systematic Error Correction via WaveletBased Bandsplitting in Kepler Data. Proc. Acad. Sci. Pacific 126 , 100 (2014).\n- [30] Smith, J. C. et al. Kepler Presearch Data Conditioning II - A Bayesian Approach to Systematic Error Correction. Proc. Acad. Sci. Pacific 124 , 1000 (2012).\n- [31] Jenkins, J. M. The Impact of Solar-like Variability on the Detectability of Transiting Terrestrial Planets. Astrophys. J. 575 , 493-505 (2002).\n- [32] Jenkins, J. M. et al. Radziwill, N. M. & Bridger, A. (eds) Transiting planet search in the Kepler pipeline . (eds Radziwill, N. M. & Bridger, A.) Software and Cyberinfrastructure for Astronomy , Vol. 7740 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , 77400D (2010).\n- [33] Jenkins, J. M. et al. Kepler Data Processing Handbook: Transiting Planet Search. Kepler Science Document KSCI-19081-003 (2020).\n- [34] Twicken, J. D. et al. Kepler Data Validation I - Architecture, Diagnostic Tests, and Data Products for Vetting Transiting Planet Candidates. Proc. Acad. Sci. Pacific 130 (6), 064502 (2018).\n- [35] Li, J. et al. Kepler Data Validation II-Transit Model Fitting and Multipleplanet Search. Proc. Acad. Sci. Pacific 131 (996), 024506 (2019).\n- [36] Guerrero, N. M. et al. The TESS Objects of Interest Catalog from the TESS Prime Mission. Astrophys. J. Suppl. 254 (2), 39 (2021).\n- [37] Fausnaugh, M. M., Burke, C. J., Ricker, G. R. & Vanderspek, R. Calibrated Full-frame Images for the TESS Quick Look Pipeline. Res. Not. Amer. Astron. Soc. 4 (12), 251 (2020).\n- [38] Hedges, C. et al. TOI-2076 and TOI-1807: Two Young, Comoving Planetary Systems within 50 pc Identified by TESS that are Ideal Candidates for Further Follow Up. Astron. J. 162 (2), 54 (2021).\n- [39] Osborn, H. et al. Two Warm Neptunes transiting HIP 9618 revealed by TESS &Cheops. Mon. Not. R. Astron. Soc. 523 (2), 3069-3089 (2023).', '14 The HD 110067 planetary system': "- [15] Quirrenbach, A. et al. Evans, C. J., Bryant, J. J. & Motohara, K. (eds) The CARMENES M-dwarf planet survey . (eds Evans, C. J., Bryant, J. J. & Motohara, K.) Ground-based and Airborne Instrumentation for Astronomy VIII , Vol. 11447, 114473C. International Society for Optics and Photonics (SPIE, 2020).\n- [16] Cosentino, R. et al. McLean, I. S., Ramsay, S. K. & Takami, H. (eds) Harps-N: the new planet hunter at TNG . (eds McLean, I. S., Ramsay, S. K. & Takami, H.) Ground-based and Airborne Instrumentation for Astronomy IV , Vol. 8446, 84461V. International Society for Optics and Photonics (SPIE, 2012).\n- [17] Holman, M. J. & Murray, N. W. The Use of Transit Timing to Detect Terrestrial-Mass Extrasolar Planets. Science 307 (5713), 1288-1291 (2005).\n- [18] Fulton, B. J. et al. The California-Kepler Survey. III. A Gap in the Radius Distribution of Small Planets. Astron. J. 154 , 109 (2017).\n- [19] Van Eylen, V. et al. An asteroseismic view of the radius valley: stripped cores, not born rocky. Mon. Not. R. Astron. Soc. 479 , 4786-4795 (2018).\n- [20] Kasting, J. F., Whitmire, D. P. & Reynolds, R. T. Habitable Zones around Main Sequence Stars. Icarus 101 (1), 108-128 (1993).\n- [21] Kopparapu, R. K. et al. Habitable Zones around Main-sequence Stars: Dependence on Planetary Mass. Astrophys. J. 787 , L29 (2014).\n- [22] Izidoro, A. et al. Formation of planetary systems by pebble accretion and migration. Hot super-Earth systems from breaking compact resonant chains. Astron. Astrophys. 650 , A152 (2021).\n- [23] Fabrycky, D. C. et al. Architecture of Kepler's Multi-transiting Systems. II. New Investigations with Twice as Many Candidates. Astrophys. J. 790 (2), 146 (2014).\n- [24] Zeng, L. et al. Growth model interpretation of planet size distribution. Proc. Nat. Acad. Sci. USA 116 (20), 9723-9728 (2019).\n- [25] Kempton, E. M. R. et al. A Framework for Prioritizing the TESS Planetary Candidates Most Amenable to Atmospheric Characterization. Proc. Acad. Sci. Pacific 130 (993), 114401 (2018).\n- [26] Otegi, J. F., Bouchy, F. & Helled, R. Revisited mass-radius relations for exoplanets below 120 M ⊕ . Astron. Astrophys. 634 , A43 (2020).", '0.1.1 TESS photometry': "HD 110067 was observed in sectors S23 and S49 (2020 March 18 to 2020 April 16 and 2022 February 26 to 2022 March 26, respectively) of the Transiting Exoplanet Survey Satellite (TESS) [4]. The star was included on the TESS Candidate Target List (CTL) [27] and therefore the target was observed at 120 s cadence in both sectors. The target images are processed by the TESS Science Processing Operations Center (SPOC) pipeline at NASA Ames [5], which calibrates the pixels, performs simple aperture photometry (SAP), flags poor-quality data, and removes systematic trends to create the so-called 'Presearch Data Conditioning' light curve (PDCSAP) [2830]. Finally, the SPOC pipeline runs a wavelet-based transiting planet search for periodic exoplanets [31-33] which in the case of the S23 data revealed two thresholdcrossing events (TCEs) - i.e. candidate planets - which passed Data Validation checks [34, 35]. Manual vetting of these two candidates resulted in the assignment of two TESS Object of Interest (TOI), namely TOI-1835.01 and TOI-1835.02 [36]. \nThe first TCE was alerted by SPOC as TOI-1835.01 with a period of 5.641 d. Although two of the three transits associated with this ephemeris appeared to be of similar depth and duration, the third did not appear to be associated with a clear transit. However, this could have been due to proximity to a systematic dip caused by a momentum dump. A second TCE, TOI-1835.02, was alerted as a single transit at 1948 . 98 TJD TESS Julian Date (TJD ≡ BJD -2457000), where BJD is the Barycentric Julian Date in units of days, as opposed to the period of 11.107 d proposed by SPOC due to an apparent discrepancy in the transit depth between the two transits purportedly linked by the transiting planet search. On the other hand, a peak in the background flux ruled out the planetary origin of an apparent transit feature at TJD = 1940.5. Both the depth and shape of this feature were strongly dependent on the detrending method used in the lightcurve, which is indicative of its spurious nature. \nHD 110067 was later reobserved by TESS in S49. In order to confirm the TCEs and plan immediate follow-up, we downloaded the TESS Image CAlibrator Full Frame Images (TICA FFI) [37] only a week after downlink for each of the first and then second orbits. We computed a light curve from the 10-minute cadence TICA FFI cut-outs using SAP and clipped regions of high brightness due to the Earth and Moon, as well as parts of the light curve a ff ected by systematics such as momentum dumps. The first orbit alone revealed at least 5 new clear transit features. These all appeared to have varying depths and durations, and none were compatible with the 5.64 d period implied by TOI-1835.01. The second orbit showed three more clear transit events, making a total of eight in S49 and five in S23. A 20-second cadence target pixel file was later made available for HD 110067, which resulted in higher-precision photometry.", '0.1.2 CHEOPS photometry': "The CHaracterising ExOPlanets Satellite (CHEOPS) mission is a European Space Agency small-class mission dedicated to studying bright, nearby exoplanet host stars for the purpose of making high-precision photometric observations of transiting planets [6]. We collected 19 separate visits of HD 110067 with CHEOPS between 2022 April 11 and 2022 May 17 under Guaranteed Time Observing programs ID-048 \nand ID-031. The goal of these observations is (1) to confirm the true orbital period of single- and duo-transiting planet candidates and (2) to improve the planetary radius precision and ephemeris of confirmed planets. This has been done for large planets producing deep eclipses from the ground [47, 48], and for small planets from space [39, 49, 50]. An observing log summarizing the duration of each visit, its average observing e ffi ciency (considering the gaps produced by Earth occultations or passages over the South Atlantic Anomaly along the spacecraft's low-Earth orbit), and photometric precision are presented in Extended Data Table 1. \nTo provide the highest quality photometric precision, we opted to perform custom photometric extraction of the CHEOPS imagettes using point-spread-function (PSF) photometry as implemented by the PIPE package [51, 52]. For bright targets such as HD 110067, light curves generated with PIPE exhibit lower median absolute di ff erences than those generated by the CHEOPS Data Reduction Pipeline [53]. The shorter cadence of the CHEOPS imagettes allows a higher cadence light curve, and PSF detrending is also better at removing trends due to systematic factors and background stars. As various PSF models have already been generated and vary as a function of stellar temperature, we opted to use a PSF generated using the star HD 189733, with a similar spectral type of HD 110067. In order to preserve intervisit flux di ff erences, we normalized the entire CHEOPS data together instead of individually. This revealed clear visit-to-visit flux di ff erences due to stellar rotation with an amplitude larger than that of TESS (as stellar activity is typically more pronounced at bluer bandpasses). The final light curves used in our subsequent analyses are shown in Fig. 1 and Fig. S1 in the Supplementary.", '0.1.3 Ground-based photometric campaign': 'We carried out a campaign on the night of May 23rd, 2022 to attempt to confirm the 41.05-day period orbit of HD 110067 f as predicted by our resonance chain analysis. Photometric observations were taken using 14 telescopes using seven di ff erent filters, which observed from various locations to continuously cover a temporal baseline of more than 11 hours (between 2022-05-23UT22:52:55 and 2022-05-24UT10:01:33). This window is long enough to catch the 5-hour transit expected from 02:52 to 07:12UT. However, no single location was able to cover both ingress and egress. A summary of the observations is shown in Extended Data Table 2. Details from each individual observation are shown below. Extended Data Fig. 4 shows the data and best-fit models as discussed in Sect. 0.3.4.', 'Teide Observatory': "We observed HD 110067 on May 23rd, 2022 using the MuSCAT2 instrument installed at the 1.5-m Telescopio Carlos S'anchez (TCS) located at the Teide Observatory, Spain [54]. The images were taken simultaneously in g , r , i , and zs filters with the telescope heavily defocused and with short exposure times of 3 to 5 sec, depending on the band, to avoid saturation. Relative light curves for each band and instrument of HD 110067 were extracted by aperture photometry using a custom pipeline [55] with optimal aperture radii of 8 . '' 1 to 11 . '' 3 depending on the band. Note \nthat there was a technical problem on the dome of the TCS between BJD-2459723 = 0.488 and 0.526; we discarded the data taken during this period. \nWe also observed HD 110067 on May 23rd, 2022 with one of the 1-m telescopes from Las Cumbres Observatory (LCO) global network located at the Teide Observatory, Spain [56]. The observations were obtained through Director's Discretionary Time program 2022A-005 (PI: Wilson). We collected 181 frames with an exposure time of 20 s, covering 2.5 h, using the 4096 × 4096 pix SINISTRO camera. The images were calibrated by the standard LCO BANZAI pipeline [57]. Di ff erential photometric data were extracted using AstroImageJ (AIJ) [58].", 'Paranal Observatory': "We observed HD 110067 on May 23rd, 2022 using the Next Generation Transit Survey (NGTS) facility located at ESO's Paranal Observatory in Chile [59]. NGTS consists of twelve 20-cm, f / 2.8 telescopes with Andor cameras and red-sensitive (600-900 nm) deep-depletion e2v CCDs. Nine NGTS telescopes observed from 23:14 to 04:35UT, covering a predicted transit ingress of HD 110067 f, and spanning an airmass range of 1.7-2.5. Two telescopes started observing two hours late due to a technical issue. All nine telescopes were defocused to avoid saturating the bright target star during the 10-second exposures. The NGTS camera shutters were not functional and so were kept open during the entire observing block. That caused the stars to streak during the 1.5-second readout sequences but without any apparent detrimental e ff ect on the photometry. Observing without using the shutters is now the standard operation mode of NGTS. We performed standard di ff erential aperture photometry, using large aperture radii of 6.5-8.0 pixels, and carefully selecting comparison stars to avoid those that exhibited variability. The light curve of each telescope was normalized individually and no detrending was performed.", 'F. L. Whipple Observatory': 'We observed HD 110067 on May 24th, 2022 using the Tierras instrument installed at the refurbished 1.3-m telescope located at the F. L. Whipple Observatory atop Mount Hopkins, Arizona, United States. The instrument is designed to regularly achieve a photometric precision of 250 ppm on a time scale of both 10 min and a complete observing season. The design choices that permit this precision include a four-lens focal reducer and field-flattener that increase the field-of-view of the telescope, a custom narrow bandpass filter centered around 863.5 nm to minimize precipitable water vapor errors, and a fully automated mode of operation [60]. A total of 1262 4-second exposures were gathered with Tierras for HD 110067. Astrometric calibrations were done in real-time during data gathering and were stored in WCS headers in the FITS files. The FITS files were then passed through the Tierras image reduction pipeline to perform bias corrections and image stitching (the CCD chip is read out through separate amplifiers). AIJ was used for photometric extraction. These observations were gathered shortly after Tierras started science operations, and the data were not flatfielded since knowledge of the flat-field was incomplete at the time. The RMS of the \n15-minute binned data is 323 ppm. The photometric precision on this target is ultimately limited by scintillation, as the target was observed down to an airmass of 2.37. The observations were mildly a ff ected by cirrus.', "San Pedro M'artir Observatory": "Weobserved HD 110067 on May 24th, 2022 with the 1-m SAINT-EX telescope at the Observatorio Astron'omico Nacional de la Sierra de San Pedro M'artir in Baja California, Mexico [61]. SAINT-EX is equipped with a deep-depleted and back-illuminated Andor IKON CCD and a filter wheel. The observations were defocused and acquired in the 'zcut' filter, a custom filter optimized to reduce the systematic uncertainties in the light curves of red stars due to precipitable water vapor, with an exposure time of 10 s. The data were reduced with AIJ using the standard corrections for bias, flatfielding, and dark current. AIJ was also utilized to do the aperture photometry of the time series, producing the light curves and relevant meta-data. The observations were mildly a ff ected by high-altitude cirrus.", 'Haleakala Observatory': "We observed HD 110067 on May 24th, 2022 using the MuSCAT3 instrument mounted on the 2-m Faulkes Telescope North (FTN) at Haleakala Observatory on Maui, Hawaii, United States [62]. The images were taken simultaneously in g , r , i , and zs filters with the telescope heavily defocused and with short exposure times of 3 to 5 sec, depending on the band, to avoid saturation. Relative light curves for each band and instrument were extracted by aperture photometry using a custom pipeline [63] with optimal aperture radii of 8 . '' 1 to 11 . '' 3 depending on the band. There was a guiding issue on the FTN around BJD-2459723 = 0.795, which caused a large shift of the stellar positions on the detectors; we treated the MuSCAT3 data as two independent datasets separated by that time.", '0.1.4 High-resolution imaging': "As part of our standard process for validating transiting exoplanets, and to assess the possible contamination of bound or unbound companions on the derived planetary radii [64], we observed HD 110067 with near-infrared (NIR) adaptive optics (AO) imaging at Palomar Observatory and with optical speckle imaging at Gemini North. Gaia DR3 is also used to provide additional constraints on the presence of undetected stellar companions and wide companions. No close-in ( ≲ 1 '' ) stellar companions were detected by either the NIR adaptive optics or optical speckle imaging.", 'Palomar Observatory': "The Palomar Observatory observations of HD 110067 were made with the PHARO instrument [65] behind the natural guide star AO system P3K [66] on 2020 Jan 08 in a standard 5-point quincunx dither pattern with steps of 5 '' in the narrow-band Brγ filter. Each dither position was observed three times, o ff set in position from each other by 0.5 '' for a total of 15 frames; with an integration time of 1.4 seconds per frame, respectively for total on-source times of 21 seconds. PHARO has a pixel scale \nof 0 . 025 '' per pixel for a total field of view of ∼ 25 '' . The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every 20 · at separations of integer multiples of the central source's FWHM [67]. The Palomar data have a sensitivity ∆ mag = 2 at 0.1 '' and ∆ mag = 9 at 1 '' ; the final sensitivity curve is shown in Fig. S2 of the Supplementary.", 'Gemini Observatory': "Weobserved HD 110067 with the 'Alopeke speckle imaging camera at Gemini North on 2020 June 10 [68]. We obtained five sets of 1000 frames, each frame having an integration time of 60 ms, obtaining images in each of the instrument's two bands (centered at 562 nm and 832 nm). The observations were reduced using our standard software pipeline [69] and reached a 5 σ sensitivity of ∆ mag = 7 (blue channel) and ∆ mag = 6 . 8 (red channel) at separations of 0.5 '' . The reconstructed speckle images show no evidence of additional nearby point sources. The final sensitivity curve is shown in Fig. S2 of the Supplementary.", 'Gaia Space Observatory': 'In addition to the high-resolution imaging, we have utilized Gaia to identify any wide stellar companions that may be bound members of the system [70, 71]. There are no additional widely separated companions identified by Gaia that have the same distance and proper motion as HD 110067. Additionally, the Gaia DR3 astrometry provides additional information on the possibility of inner companions that may have gone undetected by either Gaia or the high-resolution imaging data. The Gaia Renormalised Unit Weight Error (RUWE) is a metric, similar to a reduced chi-square, where values that are ≲ 1 . 4 indicate that the Gaia astrometric solution is consistent with the star being single whereas RUWE values ≳ 1 . 4 may indicate an astrometric excess noise, possibly caused the presence of an unseen companion [e.g., 72]. HD 110067 has a Gaia EDR3 RUWE value of 0.94 indicating that the astrometric fit is consistent with a single-star model.', 'Calar Alto Observatory': "We observed HD 110067 using the CARMENES instrument [15] installed at the 3.5m telescope of Calar Alto Observatory in Almer'ıa, Spain, between 3 July 2020 and 4 July 2021. We collected 39 high-resolution spectra under the observing programs F20-3.5-011 (PI: Nowak) and H20-3.5-013 (PI: Luque). Radial velocities and additional spectral indicators were derived using raccoon [73] and serval [74]. While the mean internal precision of the template matching serval RVs is 3 . 1 m s -1 , the precision of the cross-correlation method raccoon RVs is 2 . 9 m s -1 , so we used the latter in our analyses.", 'Roque de los Muchachos Observatory': "We observed HD 110067 with the HARPS-N spectrograph mounted at the 3.6 m Telescopio Nazionale Galileo [16] of Roque de los Muchachos observatory in \nLa Palma, Spain, between 30 May 2020 and 4 May 2022. We collected 72 high-resolution spectra under the observing programs CAT19A 162 (PI: Nowak), CAT21A 119 (PI: Nowak) and ITP19 1 (PI: Pall'e) that were used to measure the photospheric properties of the star and precise radial velocities. Radial velocities and additional spectral indicators were derived using an online version of the DRS pipeline [75], the YABI tool, and serval [74]. Both the YABI- and serval -derived radial velocities have a median internal precision of 1 . 0 m s -1 , but we used the YABI ones (based on the cross-correlation method) in our final analyses for consistency with the CARMENES dataset.", '0.2.1 Photospheric parameters and abundances': 'To properly characterize the planetary system around HD 110067, we first conduct a series of analyses to determine the properties of the host star. We derive the stellar spectral parameters by applying the widely used ARES+MOOG tools to our co-added HARPS-N spectra [76-78]. ARES [79, 80] measures the equivalent widths of iron lines in the spectrum that are converted into stellar atmospheric parameters using the MOOG radiative transfer code [81] applied to Kurucz model atmospheres [82]. In Extended Data Table 3 we report the e ff ective temperature T e ff , surface gravity log g , and metallicity [Fe / H], obtained upon convergence of ionization and excitation equilibria within this method. Additionally, we measure the stellar v sin i from the HARPS-N spectra using ZASPE [83]. \nWe further study the photospheric parameters by conducting a classical curveof-growth analysis on our co-added HARPS-N spectrum using our aforementioned spectral parameters in order to obtain [Mg / H] and [Si / H] abundances for HD 110067. Utilizing the ARES+MOOG framework detailed above, we obtain the equivalent widths [80] for these elements, which are converted to abundances assuming local thermodynamic equilibrium [81, 82]. The specific details of this analysis are beyond the scope of this paper and can be found in refs.[84, 85]. We report the stellar abundances in Extended Data Table 3.', '0.2.2 Physical parameters': "Using our spectral parameters and the ATLAS [82, 86] and PHOENIX [87] catalogs, we build spectral energy distributions of HD 110067 that we compare to optical and infrared broadband photometry of the star (see Extended Data Table 3) to derive the stellar angular diameter and e ff ective temperature via the infrared flux method [88]. This is conducted in an MCMC approach [89, 90] within which we convert the angular diameter to the stellar radius using the Gaia EDR3 o ff set-corrected parallax [91] with model uncertainties accounted for using a Bayesian modeling averaging. We report the stellar radius R ⋆ in Extended Data Table 3. \nLast, we complete our stellar characterization by determining the mass and age of HD 110067. We constrain two sets of stellar evolutionary models with help of our derived values for T e ff , log g , and R ⋆ [92]. On the one hand, we \nuse an isochrone placement algorithm [93, 94] and interpolate over pre-computed grids of PARSECv1.2S [95] isochrones. On the other hand, we use the Code Li'egeois d' ' Evolution Stellaire [96] combined with a Levenberg-Marquadt minimization scheme [97] to optimize the best-fitting evolutionary track. The results from the two methods are combined to determine the mass and age of the star that is reported in Extended Data Table 3. \nThe [Fe / H] and age of HD 110067 indicate that this star could belong either to the galactic thick disk stellar population or be an older member of the galactic thin disk. The values of [Mg / H] and [Si / H], being within 1 σ of [Fe / H], show that the star is not enhanced in α -capture elements and are indicative of a typical thin disk chemical composition. We determined the kinematic properties of HD 110067 by using the Gaia EDR3 astrometry to compute the Local Standard of Rest space velocities of this star following ref.[98]. From these velocities, we compute that the probability of kinematic membership in the galactic thin disk is 0 . 9911 ± 0 . 0029. Thus, we conclude that HD 110067 is on the older, more metal-poor end of the distribution of the galactic thin disk stellar population.", '0.3.1 Space-based photometry modeling': 'We performed simultaneous modeling of the space-based photometry. We used the quaternion-detrended TESS data combined with the PLD-detrended data for the missing S23 gaps, and the PIPE -detrended CHEOPS data for the three visits containing transits. We built transit models for the six planets with exoplanet [99]. Due to its nature as a rotating telescope on a near-Earth orbit, even PSF-detrended CHEOPS photometry can include systematic trends. However, these typically correlate with other measurements, for example, roll angle, background, or contaminant flux. In order to not bias the transit model and to better propagate uncertainties on the derived parameters, we performed CHEOPS decorrelation alongside our photometric transit modeling. We first fitted each CHEOPS transit individually alongside multiple possible decorrelation factors, allowing us to assess which decorrelation factors are most useful. This also enabled us to test whether such decorrelation is shared among all CHEOPS visits or individual to a single light curve. From this analysis, we included the following parameters in the linear correlation: position centroids, the second harmonic of the cosine of the roll-angle, cos 2 Φ , the change in telescope temperature, and quadratic trends with the x-y centroids. CHEOPS data have also been known to contain flux trends that vary stochastically as a function of roll angle over shorter frequencies [see e.g. 100]. These are not well removed using simple trigonometric functions, hence we also modeled a flexible spline shared between all visits to model shorter-timescale variation. To incorporate stellar variability, a floating mean and flux trend were also fitted to each CHEOPS visit, as well as an individual jitter term. \nInformative priors were used on limb darkening parameters using the theoretical quadratic limb darkening parameters for TESS [101] and CHEOPS [102], with uncertainties inflated to 0.1 in all cases to guard against systematic o ff sets. The impact parameter and radius ratio are fitted from a broad uniform and log-normal prior, \nrespectively, while the period and mid-transit epoch are fitted using broad normal priors from the transits identified and modeled above. Stellar parameters from Extended Data Table 3 were used as inputs to the model with Gaussian priors. Orbits were assumed circular in all cases which is a good approximation for planetary systems with multiple transiting planets [103-105]. The prior and posterior distributions of each parameter in the model are shown in Table S1 of the Supplementary.', '0.3.2 Properties of the unmatched transits': "Our first modeling of the TESS space-based photometry was able to account for a total of 5 transits of planet b (two in TESS S23 and three in TESS S49) and 4 transits of planet c (two each in TESS S23 and S49). However, this analysis left six 'unmatched' transits in the original TESS light curves. In order to pair the transits, we fitted each transit individually using a purely shape-based transit model agnostic to the orbital period using MonoTools [106]. From this analysis, we then compared each transit in duration-depth space, allowing us to clearly see that both transits from S23 shared unique regions of this parameter space with two more transits seen in S49 (duo-transits), while the two longest-duration transits seen only in S49 were solitary (mono-transits). Extended Data Fig. 1 shows this result. \n. \nWe then modeled both duo- and single-transits using MonoTools fitting. This allows long-period planets to be modeled in a way that the transit model is agnostic of the orbital period with the implied period distributions being manipulated using priors. This technique works for transits with single- or duo-transits. In the case of two transit events separated by a long gap, the planetary transit is fitted leaving the orbital period open, and the implied transit shape is used to calculate the probability for each of the possible period aliases. For single transits, potential orbital period windows are computed. In both cases, the period probability distribution comes from a combination of a simple period prior (longer period planets are geometrically disfavored) [107], an eccentricity prior (eccentric orbits are disfavored in multi-transiting systems) [108], and a stability prior using the orbits of other planets in the system (orbit-crossing is disallowed) [further details in 39, 109]. The resulting marginalized period predictions for planets HD 110067 d, e, f, and g are shown in Fig. S3, with posterior values of 21 . 6 + 2 . 9 -1 . 6 , 29 . 9 + 4 . 6 -3 . 3 , 40 . 1 + 7 . 1 -5 1 , and 47 . 0 ± 8 . 0 days, respectively.", '0.3.3 Continuing the resonant chain': 'In this section, we expand the analyses that led to the prediction of the orbits of planets HD 110067 e, f, and g based on the generalized Laplace resonant configuration of the three inner transiting planets in the system. We assume that all events mentioned in the previous section are transits that belong to planets that continue the resonant chain. \nFor transiting systems, generalized three-body Laplace angles can be estimated in 0th order in eccentricity, defined as Ψ e = 0, from the times of mid-transit and the orbital period of the planets [see e.g., 110]. This estimation di ff ers from the actual generalized three-body Laplace angle proportionally to the eccentricities [eq. 15 of 111]. Interestingly, for known systems with a chain of three-body resonances, all \nΨ e = 0 lie close to an equilibrium of the chain, as seen in Extended Data Fig. 2. The largest distance is ∼ 43 degrees for the inner triplet of K2-138 [112]. For HD 110067, the estimated angle Ψ e = 0 , bcd is also at about ∼ 44 degrees from its theorized 180degree equilibrium. Through the study of transit timing variations over several years, one can get constraints on the underlying generalized three-body Laplace angles. In known cases, one can see that these angles oscillate with amplitudes of a few tens of degrees at most around their equilibrium value, see Fig. 2 of [12] for Kepler-60, Fig. 25 of [13] for TRAPPIST-1. \nAs shown above, the two events at 2646.088 TJD and 1937.851 TJD have fully consistent shapes. Among the probable periods computed with MonoTools (Fig. S3), Pe = 30 . 7931 days is the only one that continues the resonant chain, with Pe / Pd = 1 . 5007, landing inside the common 3:2 MMR (see Fig. S4 in the Supplementary). We compute the observed value of the associated generalized three-body Laplace angle Ψ e = 0 , cde = 169 . 995 deg, which is at only 10 deg from the expected 180-degree equilibrium. We hence predict a period of 30 . 7931 days for planet HD 110067 e if it is in the resonant chain. \nFor the remaining two mono-transits, we try a set of first-order MMRs (2 / 1, 3 / 2, 4 / 3, 5 / 4, 6 / 5) between planet #4 and #5 and the same between planet #5 and #6 (hence 25 combinations). Each of these combinations has to be tested assuming that the transit at TJD = 2641.5778 belongs to the 5th planet, and TJD = 2656.0944 belongs to the 6th planet (case A), and vice-versa (case B). Fortunately, many of these 50 possibilities are excluded by existing data. We end up with 4 possibilities for case A and 9 for case B. As seen in Fig. 2, all known chains of Laplace resonances have either their estimated generalized three-body Laplace angle Ψ e = 0, or their actual generalized three-body Laplace angle Ψ close to an equilibrium of the chain. We will hence favor the configurations that are closest to an equilibrium of the chain. For each case, the distance of each estimated angle to its closest equilibrium ∆Ψ = | Ψ e = 0 -Ψ eq | is given in Extended Data Table 4. The case A2, with Pf / Pe = 4 / 3 and P g/ Pf = 4 / 3, comes out as a favorite, with the three outer generalized three-body Laplace angles at less than 20 deg from the closest equilibrium. In addition, one can note that 4 / 3 MMRs are relatively common in resonant chains (see Fig. S4 in the Supplementary). \nFor completeness, we study the role that the eccentricity of the orbit plays in the prediction. To estimate the generalized three-body Laplace angle \nΨ = l λ 1 -( l + m ) λ 2 + m λ 3 , (1) \nat a given epoch, we estimate the value of the λ j as follow. Transits occur when the true longitude of the planet is equal to l 0 = -π/ 2. At first order in the eccentricity, \nλ 0 = l 0 -2 e sin( l 0 -ϖ ) = -π 2 + 2 e cos( ϖ ) . (2) \nWe then assume the planet to be in a circular, unperturbed orbit to compute the value of its mean longitude at the time of transit t 0, λ 0 = -π/ 2. We hence obtain \nλ ( t ) = -π 2 + ( t -t 0) P 2 π. (3) \nThe error on Ψ made by assuming zero eccentricity is hence, at first order [111]: \n| Ψ -Ψ e = 0 | = | 2 le 1 cos ϖ 1 -2( l + m ) e 2 cos ϖ 2 + 2 me 3 cos ϖ 3 | . (4) \nThis error can thus be substantial (several tens of degrees) if the eccentricities are of the order of several parts per hundred, as it is the case for Kepler-223 [110]. Therefore, we study if a given combination of the eccentricities and longitudes of periastron can make Ψ e = 0 closer to the equilibrium than Ψ actually is, or vice-versa. We check this for the cases presented in Extended Data Table 4. \nEach case sets the orbital period of the planets and their mid-transit time. We estimate the planetary masses using the mass-radius relation from [26]. Then, varying the remaining parameters ki = ei cos ϖ i and hi = ei sin ϖ i , we minimize the cost function \nC = A ( Ψ bcd ) + A ( Ψ cde ) + A ( Ψ e f g ) + A ( Ψ f g h ) (5) \nover 200 years, where A ( Ψ ) = 2 π if Ψ circulates, and A ( Ψ ) is the peak-to-peak amplitude of libration of Ψ otherwise. For each case, 40 MCMC runs are conducted to minimize C , using REBOUND [113] for the N -body integration and samsam [114] for the MCMC. For each run, the ki and hi parameters are randomly initialized in the [ -0 . 05 , 0 . 05] range, which are also their boundaries during the MCMC runs. This allows eccentricities that are comparable to those of Kepler-60 [115] and Kepler-223 [110], which are other known chains for which the inner planets are far enough from the star to not have their eccentricities damped by tides. \nThe best solution of each fit is shown in Extended Data Fig. 3. Case A2 is the only one for which the best solutions consistently have a peak-to-peak amplitude of the generalized three-body Laplace angles below 50 degrees on average across the four angles. In all other cases, we were not able to find values of the ki and kj parameters below 85 deg of amplitude on average, with the exception of case A0 for which an average of ≈ 66 deg was reached. The best solutions found across all MCMC runs for the A2 and A0 cases integrated for 1000 years of evolution are shown in Fig. S5 of the Supplementary. This analysis shows that the A2 case remains the one with the highest potential of being close to an equilibrium, while showing that all other cases cannot have an amplitude of libration smaller than 66 deg on average across their generalized three-body Laplace angles, regardless of the values of the ki and kj parameters. The case A2, with Pf / Pe = 4 / 3 and P g/ Pf = 4 / 3, is hence our prediction for the outer architecture of the HD 110067 system.', 'Recovering the missing cadences of TESS S23 observations': 'Based on the dynamical analysis presented above, the likely orbital periods associated with the two mono-transits observed in TESS S49 are approximately 41.05 and 54.74 days, respectively. According to this prediction, both planets transited their host star during TESS S23 observations, but at a time when the photometry was highly a ff ected by scattered light and sky background contamination. The Earth was a significant source of scattered light at the beginning of both S23 orbits (TJD = 1928.09 and TJD = 1941.83) and the Moon was a significant source of scattered light for a few days after the beginning of the second orbit (between 1942 and 1947 TJD). The cadences a ff ected were flagged by SPOC, thus not leaving enough valid data to derive cotrending basis vectors and missing in the PDCSAP light curve. \nOur custom extraction using the PLD method from Sect. 0.1.1 was able to recover the missing data, showing two mono-transits at 1943.6 and 1944.1 TJD (Fig. 1 and also Fig. S6 in the Supplementary). Using MonoTools , we confirmed that the transits were consistent in duration-depth space with the two mono-transits from TESS S49 and separated by an integer number of orbits that matched the orbital periods predicted in our dynamical analysis for planets f and g. With this data reduction, all six planets in the HD 110067 system have been detected in transit at least twice, allowing a precise orbital period determination if we impose priors based on the hypothesis that all planets are trapped in a chain of first-order MMRs. Additionally, we recovered an additional transit of planet b at the beginning of TESS S23 observations.', 'Modeling of the ground-based photometric campaign': 'Targeted observations of HD 110067 were carried out on the night of May 23rd, 2022 to attempt to confirm the 41.05-day orbit of HD 110067 f as predicted via our resonance chain analysis. In order to reveal whether a transit was present in the combined dataset, we built a combined photometric model using all ground-based observations. In order to remove spurious systematic trends in a way that does not bias any transit fit, we opted to perform simultaneous linear decorrelation of each photometric data set using the various meta-data time series available. In all cases, for example, we included an airmass term in the decorrelation as well as a measure of the FWHM width. We also included two position-centroid terms (for MuSCAT-2 and -3), information on comparison star total counts and FWHM width (for LCO, Tierras, and SAINT-EX), and interpolated color time series derived from the relative shift in flux across bands in the MuSCAT-2 and -3 filters [as used in 49]. In all cases, the metadata were normalized to a time series with µ = 0, σ = 1 and modeled using a single scaling parameter with a normal prior of µ = 0, σ = 0 . 5. Quadratic limb darkening parameters were also constrained using normal priors dictated by theoretical limb darkening parameters as computed for each of the nine passbands using LDTK [116] and with inflated uncertainties following the methodology of the space-based photometric analysis. Each of the four time series observed by MuSCAT-3 was split into two around an observing gap that occurred due to the star passing close to the zenith at 2022-05-24UT06:57. Individual time series were used for each of the nine NGTS \ntelescopes, which were decorrelated independently. The final result is 24 individual photometric time series. An o ff set was also applied to each light curve, as well as a single global slope parameter to include the possibility of stellar activity. \nThe transit parameters were constrained based on those found in a fit of the TESS S49 mono-transit. The predicted period used was 4 / 3 × Pe = 41 . 051 ± 0 . 1 d, with the uncertainty implying a divergence from the perfect integer period of 2 . 4 × 10 -3 -larger than those values found for the inner three planets. We limited the period to 41 . 0 ± 0 . 2 days to ensure a transit fit that could be explored with the temporal baseline of the photometry. Due to the non-continuous nature of the photometry, the probability density function of the observed transit time is likely to be asymmetric and could potentially have multiple minima. Therefore, analyses using classical sampling techniques (Markov Chain Monte Carlo, Hamiltonian Monte Carlo, etc.) may not reveal the full picture. In order to initially test this, we kept all other parameters equal but split the range of periods covered by the time series into 36 bins across the anticipated period range (40 . 8 < P < 41 . 2) and fitted a constrained model for each. This would allow us to see the variation in the goodness-of-the-fit as a function of transit epoch. The transit model was built with exoplanet [99] and optimized using pymc3 [117], specifically with the pymc3-ext sampling which enabled correlated parameters for each time series to be grouped together, speeding up the computation. In order to assess whether or not a transit model was justified over a flat model, we used the Watanabe-Akaike information criterion (WAIC) [118, 119] as implemented in arviz [120]. \nOur results show a preference for a transit at the expected period P = 41 . 04 ± 0 . 01 days, with a ∆ WAIC of 9.5 over a transit-free model, as can be seen in Extended Data Fig. 4. The majority of instruments showed a weak preference for a peak at P ∼ 41 . 05d, with the exception of LCO (which observed no in-transit data) and SAINT-EX (which is the most a ff ected by cirrus). This is only equivalent to moderate evidence for a ∼ 41 . 051d period of HD110067 f. The lower two panels of Extended Data Fig. 4 show that both models (with or without transit) fit reasonably well. This is in part because systematic e ff ects dominate over astrophysical signals for transits with depths below 1000 ppm, especially when the target star is observed at a low airmass. A further peak in WAIC is seen at ∼ 40 . 9 d, but this hypothetical transit is covered only by the initial 1 h of MuSCAT-2 photometry and does not fit our predicted period, hence we consider it spurious. This campaign shows that such transits are at the very limit of what is possible with ground-based observations. However, observations during a more favorable observing season and without technical issues such as the meridian flip of MuSCAT-3 (which unluckily coincided with the expected egress) may have constrained better the presence of a transit.', 'Modeling of the radial velocity data': "We carried out an initial frequency-based exploration of the CARMENES and HARPS-N spectroscopic data sets to see which significant signals are present and those related to stellar activity using periodograms [121]. Figures S7 and S8 in the Supplementary show that the dominant signal in the generalized Lomb-Scargle periodograms of both the radial velocities and main activity indicators (CCF-FWHM, \ndi ff erential line width, Mount Wilon's S-index, H α emission) is attributable to the rotational period of the star, measured photometrically to be approximately 20 days using TESS and CHEOPS data. As it is well known, stellar activity induces spurious radial velocity signals [e.g., 122-125], which should be properly removed to unveil induced Keplerian motions in the star. We followed two independent approaches to model the data and minimize the impact of stellar activity e ff ects on the detection and mass determination of the planets in the system.", 'M ethod I: SNfit and breakpoint algorithm': 'Spurious radial velocity signals induced by stellar activity come from the line shape variations in stellar spectra. Those can be quantified through the full-width-at-halfmaximum (FWHM) and the asymmetry of the cross-correlation function (CCF) computed from the spectra [e.g., 126-128]. Following [129], we first fit Skew Normal (SN) functions to the CCFs available from HARPS-N and CARMENES. An SN function is not only characterized by a location and a scale parameter (which are the counterparts of the mean and standard deviation of a Gaussian), but it has a further free parameter that expresses its skewness (hereafter denoted with γ ). For each observation, through the SN-fit we were able to retrieve the stellar radial velocity ( RV , quantified through the SN median), the FWHMSN, the contrast A , and the asymmetry γ . The errors σ RV of the RV measurements were inferred using a bootstrap approach. Denoting with fCCF the flux of a CCF data point, each point was perturbed by sampling values from a Normal distribution whose standard deviation is equal to p f CCF, since the errors a ff ecting the CCF data points are expected to be Poissonian. \nAfter that, we applied the breakpoint ( bp ) method [130] to both the HARPS-N and CARMENES RV time series. The algorithm has been designed to detect those locations along the RV time series where the correlation changes against the vector [FWHMSN, A , γ ] are statistically significant. The goal is to then detrend the RV time series by applying a piece-wise interpolation to each segment found by the bp algorithm rather than performing an overall correction to the whole time series. In this way we are able to better correct for the contamination of stellar variability as shown by ref.[130] and ref.[131]. Finally, we jointly analyzed the RV time series using the MCMCI code [132], where we switched o ff the interaction with stellar evolutionary models to speed up the computations. We set up the detrending function on each piece-wise stationary segment found by the bp algorithm as a polynomial of the following form \n<!-- image --> \nwhere ( kt , kF , kA , k γ , kR ) is the vector of the polynomial orders whose optimal value has been established by launching several MCMC preliminary runs and selecting that \ncombination which produces the minimum Bayesian Information Criterion (BIC) [133]. \nAfter performing a longer MCMCI run made of 4 independent runs (300 000 steps each), which successfully converged as checked through the Gelman-Rubin test [134], we retrieved the posterior distributions of the system parameters. Their median values along with their error bars at the 1 σ level are reported in Tables S2 and S3 of the Supplementary. The full RV time series and the phase-folded RVs of those planets whose detection is above the 3 σ level (planets d and f) are shown in Fig. S9 of the Supplementary. \nM ethod II: multi -dimensional GP On the other hand, we also perform a multidimensional Gaussian Process (GP) approach to characterize the stellar and planetary signals in our RV time series as in [135, 136]. This approach has been proven useful to disentangle stellar and planetary signals in multi-planet systems [e.g., 137, 138]. We create N -dimensional GP models, including N time-series A i , as \nA 1 = A 1 G ( t ) + B 1 ˙ G ( t ) . . . A N = ANG ( t ) + BN ˙ G ( t ) , (7) \nwhere the variables A 1, B 1, · · · , AN , BN , are free parameters which relate the individual time-series to G ( t ) and ˙ G ( t ). In this approach, G ( t ) is assumed to be a latent (unobserved) variable that represents the projected area of the visible stellar disk that is covered by active regions as a function of time. \nWe model the stellar signal using a GP whose covariance between two times ti and t j is given by \nγ QP , i , j = exp -sin 2 [ π ( ti -t j ) / P GP] 2 λ 2 P -( ti -t j ) 2 2 λ 2 e , (8) \nwhere γ QP , i , j is the Quasi-Periodic (QP) kernel, whose hyperparameters are, P GP, the GP characteristic period, λ p, the inverse of the harmonic complexity, and λ e, the long term evolution timescale. \nWe perform a two-dimensional GP model between the RVs and FWHM. We note that these quantities are equivalent in the HARPS-N and CARMENES data. The multidimensional covariance matrix was created using the kernel given in Eq. (8) and its derivatives [135, 136]. We assume that RVs can be described as A i = AiG ( t ) + Bi ˙ G ( t ), while the FWHM time series is described as A i = AiG ( t ). The planetary signals were included in the model as the mean function of the RV time series. We use N Keplerian signals (where N is the number of planetary signals); each one of them depends on the time of minimum conjunction t 0, orbital period P , and Doppler semi-amplitude, K . All orbits are fixed to be circular, so the eccentricity and angle of periastron are fixed. For the FWHM the mean function was treated as an o ff set, \nnoting that we include a di ff erent o ff set per instrument. We also include a jitter term per time series and per instrument to account for unaccounted systematic errors. \nWe perform MCMC samplings of the parameter space using the code pyaneti [136, 139]. We sample the parameter space with 250 walkers and create the posterior distributions with the last 5000 iterations of converged chains with a thin factor of 10. This leads to posterior distributions of 125 000 points for each sampled parameter. Figure 2 shows the spectroscopic time series resulting from this joint analysis. Median values along with their 1 σ uncertainties are reported in Table S4 of the Supplementary. \nModeling techniques employing GP are particularly subject to overfitting giving their flexibility to reproduce the data [see e.g., 140, 141]. To test the robustness of our GP model, we carried out a cross-validation analysis. We repeat the twodimensional GP model described in this section but applied only to the HARPS-N data. Then, we create a predictive model with the inferred parameters and overlay the CARMENESdata with median o ff sets subtracted (similar to a training / evaluation set for machine learning algorithms). Figure S10 shows this analysis, zoomed in to the 2021 observing campaign. The plot shows that the RV data is in agreement with the predictive model, suggesting that our assumption that the stellar signal imprinted in the RVs can be described with a two-dimensional GP is valid for the time span of our observations. For the CCF FWHM CARMENES data, the correlation with the RV measurements is not as strong as for the HARPS-N one, thus the prediction is less accurate in this case. \nBoth the two methods clearly detect planets HD 110067 d and f, M ethod II also detects planet b, however, the detection levels slightly di ff er. Recalling that the two di ff erent techniques are based on di ff erent RV extraction methodologies and on a different treatment of stellar activity, on the one hand, the slight output tension suggests that the RV data alone do not strongly constrain all the six Keplerian signals. We tested additional stellar mitigation approaches, such as sinusoid-fitting at the stellar rotation period and its harmonics or GP decorrelation as a function of time only, but they all turned unsuccessful at constraining the masses of any of the planets (only the time-dependent GP model could recover the signal of planet f, but with a much larger uncertainty, K f = 2 . 0 ± 1 . 0 m s -1 ). On the other hand, the RV semi-amplitudes inferred from the two methods shown in the manuscript are compatible within ∼ 1.5 σ , with the statistical tension ∆ I -II below ∼ 1 σ for planets b, c, d, and g. Extended Data Fig. 5 displays the pairs of posterior density functions for the RV semi-amplitudes of each planet for comparison. \nFor HD 110067 f, after imposing a Gaussian prior centered around 41.05 days in both M ethod I and M ethod II RV models, we indeed recover a significant RV signal with a detection level of ∼ 3 σ . The planet transits only twice in the TESS data and the ground-based photometric campaign hints at moderate evidence for a planetary transit compatible with this value. Therefore, to secure an independent detection of planet f from spectroscopy, we performed an RV-only analysis imposing a uniform unbounded prior (between 30 to 54 days) to the orbital period. The MCMC converged and detected a clear Keplerian signal with a period Pf , uni = 40 . 2 ± 0 . 2 d. Thus, the RV \ndata independently suggest the presence of a putative planet having a period close to 41.05 d. The tension at the ∼ 4 σ level with the predictions from the resonant chain model and the transit observations prove that the current RV data set cannot fully constrain the entire architecture of the planetary system. \nWefinally checked whether there are further Keplerian signals within the RV time series. In particular, given that both planets e and g were not detected via our previous RV analyses where model-dependent values of the orbital periods were imposed as priors, we investigated the presence of potential planetary signals at di ff erent periods that could be attributed to planet e or g. To this end, we performed an MCMC run, where we modeled planets b and c (which are clearly confirmed by the transit events) along with planets d and f (which are clearly detected also in the RV time series). As a result, we produced the Generalized Lomb Scargle periodograms [121] of the residuals, obtained after subtracting the Keplerian signals of all four planets from the activity-cleaned time series (Fig. S11). The high false alarm probability level of the highest peak in both the HARPS-N (18%) and CARMENES (10%) residuals suggests that there are no signals left in the RV data that could be associated with other planets or a misidentification of the orbital periods of planets e and g.', '0.3.5 Final model': 'We computed a final model of the photometric and spectroscopic data sets of the HD 110067 system. Based on the analyses above, neither the light curves nor the radial velocities are precise enough to constrain the eccentricity of the planets. Assuming circular orbits, the photometry and radial velocity thus only constrain jointly the period and phase of a given planet in the system. However, the transit data dominate the precision of these two quantities (by several orders of magnitude). Therefore, for our final model, we opt to perform an independent analysis of the photometry and radial velocity datasets, where priors inform the planet periods and phases in the radial velocity model based on the posterior distributions of the photometry-only fit. Besides, the large number of free parameters in each of the models makes it computationally expensive to run a joint fit, not to mention the complications for numerical samplers to explore the vast multi-dimensional parameter space. Table 1 shows the most relevant planetary parameters of the system based on the photometric fit from Table S1, the radial velocity fit using M ethod II from Table S4, and the stellar parameters from Extended Data Table 3. A corner plot with the posterior distribution of the fitted transit parameters is shown in Fig. S12. The resulting best-fit models and corresponding credibility bands are presented in Fig. 1 for the TESS and CHEOPS photometry and Fig. 2 for the radial velocities.', '0.3.6 Planetary internal structures': 'Using a Bayesian analysis [142, 143], we computed the possible internal structures of the six planets of the system, using the results provided in Tables 1, Extended Data Table 3 and the planetary masses from M ethod I and M ethod II. The forward model used to compute the likelihood is based on a four-layer structure: a central core (iron and sulfur), a silicate mantle (containing Si, Mg, and Fe), a water layer, and a gas', '34 The HD 110067 planetary system': "- [40] Vanderburg, A. et al. TESS Spots a Compact System of Super-Earths around the Naked-eye Star HR 858. Astrophys. J. 881 (1), L19 (2019).\n- [41] Deming, D. et al. Spitzer Secondary Eclipses of the Dense, Modestlyirradiated, Giant Exoplanet HAT-P-20b Using Pixel-level Decorrelation. Astrophys. 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CARMENES and HARPS-N reduced spectra, together with the derived CCF-based radial velocities and spectral indicators are available in Zenodo (https: // doi.org / 10.5281 / zenodo.8211589). All reduced transit photometry and radial velocity measurements used in this work are also provided in Zenodo (https: // doi.org / 10.5281 / zenodo.8211589).', 'Code Availability': 'We used the following publicly available codes, resources and Python packages to reduce, analyze and interpret our observations of HD 110067: numpy [155], matplotlib [156], astropy [157], lightkurve [44], PIPE [51, 52], AstroImageJ [58], raccoon [73], serval [74], ARES [79, 80], MOOG [81], ZASPE [83], emcee [158], CLES [96], exoplanet [99], MonoTools [106], pymc3 [117], ArviZ [120], GLS [121], MCMCI [132], and pyaneti [136, 139]. We can share the code used in the data reduction or data analysis on request.', 'Acknowledgements': "We acknowledge the use of public TESS data from pipelines at the TESS Science O ffi ce and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. CHEOPS is an ESA mission in partnership with Switzerland with important contributions to the payload and the ground segment from Austria, Belgium, France, Germany, Hungary, Italy, Portugal, Spain, Sweden, and the United Kingdom. The CHEOPS Consortium would like to gratefully acknowledge the support received by all the agencies, o ffi ces, universities, and industries involved. Their flexibility and willingness to explore new approaches were essential to the success of this mission. CARMENES acknowledges financial support from the Agencia Estatal de Investigaci'on of the Ministerio de Ciencia e Innovaci'on MCIN / AEI / 10.13039 / 501100011033 and the ERDF 'A way of making Europe' through projects PID2019-107061GB-C61, PID2019107061GB-C66, PID2021-125627OB-C31, and PID2021-125627OB-C32, from the Centre of Excellence 'Severo Ochoa' award to the Instituto de Astrof'ısica de Canarias (CEX2019-000920-S), from the Centre of Excellence 'Mar'ıa de Maeztu' award to the Institut de Ci'encies de l'Espai (CEX2020-001058-M), and from the Generalitat de Catalunya / CERCA programme. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundaci'on Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This article is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos S'anchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide. This paper is based on observations made with the MuSCAT3 instrument, developed by the Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Observatory. Tierras is supported by grants from the John Templeton Foundation and the Harvard Origins of Life Initiative. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. The NGTS facility is operated by the consortium institutes with support from the UK Science and Technology Facilities Council (STFC) under projects ST / M001962 / 1 and ST / S002642 / 1. Some of the observations presented in this paper were carried out at the Observatorio Astron'omico Nacional on the Sierra de San Pedro M'artir (OAN-SPM), Baja California, M'exico. This work makes use of observations from the Las Cumbres Observatory global telescope network. Some of the observations in this paper made use of the High-Resolution Imaging instrument Alopeke and were obtained under Gemini LLP Proposal Number GN-S-2021A-LP105. Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Alopeke was mounted on the Gemini North telescope of the international Gemini Observatory, a program of NSF OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative \nagreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci'on y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on (Argentina), Minist'erio da Ciˆencia, Tecnologia, Inovac¸ ˜oes e Comunicac¸ ˜oes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). This work was supported by the KESPRINT collaboration, an international consortium devoted to the characterization and research of exoplanets discovered with space-based missions. R.Lu. thanks Prof. Daniel Fabrycky for helpful discussions regarding the orbital dynamics of the HD 110067 system. R.Lu. acknowledges funding from University of La Laguna through the Margarita Salas Fellowship from the Spanish Ministry of Universities ref. UNI / 551 / 2021-May 26, and under the EU Next Generation funds. This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40 182901 and 51NF40 205606. A.C.Ca. and T.G.Wi. acknowledge support from STFC consolidated grant numbers ST / R000824 / 1 and ST / V000861 / 1, and UKSA grant number ST / R003203 / 1. O.Ba. acknowledges that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 865624) M.Le. acknowledges support of the Swiss National Science Foundation under grant number PCEFP2 194576. P.F.L.Ma. acknowledges support from STFC research grant number ST / M001040 / 1. Y.Al. acknowledges support from the Swiss National Science Foundation (SNSF) under grant 200020 192038. D.Ga. gratefully acknowledges financial support from the CRT foundation under Grant No. 2018.2323 'Gaseous or rocky? Unveiling the nature of small worlds'. J.A.Eg. acknowledges support from the Swiss National Science Foundation (SNSF) under grant 200020 192038. G.No. thanks for the research funding from the Ministry of Education and Science programme the Excellence Initiative - Research University conducted at the Centre of Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus University in Torun, Poland. D.Ra. was supported by NASA under award number NNA16BD14C for NASA Academic Mission Services. M.La. acknowledges funding from a UKRI Future Leader Fellowship, grant number MR / S035214 / 1. V.Ad. is supported by FCT through national funds by the following grants UIDB / 04434 / 2020, UIDP / 04434 / 2020, and 2022.06962.PTDC. P.J.Am. acknowledges financial support from the grants CEX2021-001131-S and PID2019-109522GB-C52, both funded by MCIN / AEI / 10.13039 / 501100011033 and by 'ERDF: A way of making Europe'. S.C.C.Ba. acknowledges support from FCT through FCT contracts nr. IF / 01312 / 2014 / CP1215 / CT0004. X.Bo., S.Ch., D.Ga., M.Fr. and J.La. acknowledge their role as ESA-appointed CHEOPS science team members. L.Bo., G.Br., V.Na., I.Pa., G.Pi., R.Ra., G.Sc., V.Si., and T.Zi. acknowledge support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. A.Br. was supported by the SNSA. Contributions at the Mullard Space Science Laboratory by E.M.Br. were supported by STFC through the consolidated grant ST / W001136 / 1. S.C.Ga. acknowledges support from UNAM PAPIIT-IG101321. D.Ch. and J.G-M. thank the sta ff at the F. L. Whipple Observatory for their assistance \nin the refurbishment and maintenance of the 1.3-m telescope. W.D.Co. acknowledges support from NASA grant 80NSSC23K0429. This is University of Texas Center for Planetary Systems Habitability Contribution 0063. K.A.Co. acknowledges support from the TESS mission via subaward s3449 from MIT. H.J.De. acknowledges support from the Spanish Research Agency of the Ministry of Science and Innovation (AEI-MICINN) under grant PID2019-107061GB-C66, DOI: 10.13039 / 501100011033. This project was supported by the CNES. The Belgian participation to CHEOPS has been supported by the Belgian Federal Science Policy O ffi ce (BELSPO) in the framework of the PRODEX Program, and by the University of Li'ege through an ARC grant for Concerted Research Actions financed by the Wallonia-Brussels Federation. L.De. is an F.R.S.-FNRS Postdoctoral Researcher. This work was supported by FCT - Fundac¸˜ao para a Ciˆencia e a Tecnologia through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionalizac˜ao by these grants: UID / FIS / 04434 / 2019, UIDB / 04434 / 2020, UIDP / 04434 / 2020, PTDC / FIS-AST / 32113 / 2017 & POCI-010145-FEDER- 032113, PTDC / FIS-AST / 28953 / 2017 & POCI-01-0145-FEDER028953, PTDC / FIS-AST / 28987 / 2017 & POCI-01-0145-FEDER-028987, O.D.S.De. is supported in the form of work contract (DL 57 / 2016 / CP1364 / CT0004) funded by national funds through FCT. B.-O.De. acknowledges support from the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00046. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (project F our A ces grant agreement No 724427). It has also been carried out in the frame of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation (SNSF). D.Eh. acknowledges financial support from the Swiss National Science Foundation for project 200021 200726. E.EBo. acknowledges financial support from the European Union and the State Agency of Investigation of the Spanish Ministry of Science and Innovation (MICINN) under the grant PRE2020-093107 of the Pre-Doc Program for the Training of Doctors (FPISO) through FSE funds. M.Fr. gratefully acknowledges the support of the Swedish National Space Agency (DNR 65 / 19, 174 / 18). J.G-M. acknowledges support by the National Science Foundation through a Graduate Research Fellowship under grant No. DGE1745303 and by the Ford Foundation through a Ford Foundation Predoctoral Fellowship, administered by the National Academies of Sciences, Engineering, and Medicine. The contributions at the University of Warwick by S.Gi. have been supported by STFC through consolidated grants ST / L000733 / 1 and ST / P000495 / 1. M.Gi. is F.R.S.-FNRS Research Director. Y.G.M.Ch. acknowledges support from UNAMPAPIIT-IG101321. E.Go. acknowledge the support by the Thueringer Ministerium fuer Wirtschaft, Wissenschaft und Digitale Gesellschaft. M.N.Gu. is the ESA CHEOPS Project Scientist and Mission Representative, and as such also responsible for the Guest Observers (GO) Programme. M.N.Gu. does not relay proprietary information between the GO and Guaranteed Time Observation (GTO) Programmes, and does not decide on the definition and target selection of the GTO Programme. A.P.Ha. acknowledges support by DFG grant HA 3279 / 12-1 within the DFG Schwerpunkt \nSPP 1992. Ch.He. acknowledges support from the European Union H2020-MSCAITN-2019 under Grant Agreement no. 860470 (CHAMELEON). S.Ho. gratefully acknowledges CNES funding through the grant 837319. This work is partly supported by JST CREST Grant Number JPMJCR1761. K.G.Is. is the ESA CHEOPS Project Scientist and is responsible for the ESA CHEOPS Guest Observers Programme. She does not participate in, or contribute to, the definition of the Guaranteed Time Programme of the CHEOPS mission through which observations described in this paper have been taken, nor to any aspect of target selection for the programme. J.Ko. gratefully acknowledges the support of the Swedish National Space Agency (SNSA, DNR 2020-00104) and of the Swedish Research Council (VR: Etableringsbidrag 2017-04945). K.W.F.La. was supported by Deutsche Forschungsgemeinschaft grants RA714 / 14-1 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. A.L-E. acknowledges support from the CNES (Centre national d''etudes spatiales, France). This work is partly supported by Astrobiology Center SATELLITE Research project AB022006. This work is partly supported by JSPS KAKENHI Grant Number JP18H05439 and JST CREST Grant Number JPMJCR1761. H.L.M.Os. acknowledges funding support by STFC through a PhD studentship. H.Pa. acknowledges the support by the Spanish Ministry of Science and Innovation with the Ramon y Cajal fellowship number RYC2021-031798-I. This work was also partially supported by a grant from the Simons Foundation (PI Queloz, grant number 327127). S.N.Qu. acknowledges support from the TESS mission via subaward s3449 from MIT. S.N.Qu. acknowledges support from the TESS GI Program under award 80NSSC21K1056 (G03268). L.Sa. acknowledges support from UNAM PAPIIT project IN110122. N.C.Sa. acknowledges funding by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. N.Sc. acknowledges support from the Swiss National Science Foundation (PP00P2-163967 and PP00P2-190080) and NASA under award number 80GSFC21M0002. S.G.So. acknowledge support from FCT through FCT contract nr. CEECIND / 00826 / 2018 and POPH / FSE (EC). Gy.M.Sz. acknowledges the support of the Hungarian National Research, Development and Innovation O ffi ce (NKFIH) grant K-125015, a PRODEX Experiment Agreement No. 4000137122, the Lendulet LP2018-7 / 2021 grant of the Hungarian Academy of Science and the support of the city of Szombathely. A.Tu. acknowledges funding support from the STFC via a PhD studentship. V.V.Ey. acknowledges support by STFC through the consolidated grant ST / W001136 / 1. V.V.Gr. is an F.R.SFNRS Research Associate. J.Ve. acknowledges support from the Swiss National Science Foundation (SNSF) under grant PZ00P2 208945. N.A.Wa. acknowledges UKSA grant ST / R004838 / 1. N.Wa. is partly supported by JSPS KAKENHI Grant Number JP21K20376. \nThe HD 110067 planetary system", 'Authors and A ffi liations': 'Author Contributions R.Lu, H.P.Os, A.Le., E.Pa., A.Bo., O.Ba., and T.G.Wi. conceived the project and contributed notably to the writing of this manuscript. R.Lu. and H.P.Os. led the analysis of the photometric data. A.Le. led the dynamical analysis of the system and developed the method with J.-B.De. to predict the orbits of the planets based on their resonant state within the chain. R.Lu., A.Bo. and O.Ba. led the analysis of the radial velocity data and the stellar activity mitigation. T.G.Wi. led the stellar characterization with the help of V.Ad., S.G.So., A.Bo., V.V.Gr., S.Sa., and W.D.Co. Y.Al. and J.A.Eg. led the analysis of the internal structures, while L.Fo. and A.Bo. performed the atmospheric evolution simulations. D.Ra., J.D.Tw., and J.M.Je. improved the TESS data reduction to recover the missing cadences a ff ected by reflected light and high background. R.Lu., E.Pa., and G.No. planned and obtained the time for the observations with CARMENES and HARPS-N. CARMENES observations were made possible by M.La., J.C.Mo., P.J.Am., A.Qu., and I.Ri. HARPS-N observations were made possible by I.Ca., J.O-M., F.Mu., H.J.De., J.Ko., D.Ga., J.H.Li., W.D.Co., E.W.Gu., V.V.Ey., H.L.M.Os., S.Re., E.Go., F.Da., and K.W.F.La. High-resolution imaging observations from Palomar and Gemini North were made possible by A.W.Bo., D.R.Ci, I.J.M.Cr., S.B.Ho., E.Ma., and J.E.Sc. Ground-based photometric observations to catch the transit of planet f were made possible by the MuSCAT2 (R.Lu., E.Pa., N.Na., J.H.Li., K.Ik., E.E-B., J.O-M., N.Wa., F.Mu., G.No., A.Fu., H.Pa., M.Mo., T.Ka., J.P.Le., and T.Ko.), LCO (T.G.Wi., R.Lu., H.P.Os., E.Pa., A.Le., A.Tu., M.J.Ho., Y.Al., and D.Ga.), NGTS (H.P.Os., S.Gi., D.Ba., D.R.An., M.Mo., A.M.S.Sm., E.M.Br., and S.Ud.), Tierras (J.G-M. and D.Ch.), SAINT-EX (N.Sc., Y.G-M-C., L.Sa., S.C-G., and B.-O.De.), and MuSCAT3 (N.Na., J.H.Li., K.Ik., N.Wa., A.Fu., M.Mo., T.Ka., J.P.Le., and T.Ko.) instruments. The remaining authors provided key contributions to the development of the TESS and CHEOPS mission. All authors read and commented on the manuscript, and helped with its revision. \nCorresponding author Correspondence to Rafael Luque (rluque@uchicago.edu). \nCompeting interests The authors declare no competing interests. \nThe HD 110067 planetary system \nExtended Data Figures and Tables \nExtended Data Fig. 1 Transit duration versus transit depth for all unassigned transits in TESS data . TESS Sector 23 and Sector 49 are shown as di ff erent colors. The numbers above each transit denote the mid-transit time in TJD. Contours represent percentile levels, the innermost one corresponding to the 50th percentile and the outermost to the 99th percentile by increments of 10%. The transit of planet f in PLD photometry is marked with ∗ to indicate that its properties are heavily a ff ected by pre-transit systematic noise. \n<!-- image --> \nExtended Data Fig. 2 Generalized three-body Laplace angles for known systems in resonant chains. Included are the Galilean satellites, Kepler-60 [12, 115], Kepler-80 [159], K2-138 [112], Kepler-223 [110], TRAPPIST-1 [13], and TOI-178 [10]. Measurements belonging to the same system are marked with the same color. The line marks the observed distance to the theorized equilibrium (marked with a circle). The distances are estimated at 0th order in eccentricity [110, 111]. For most systems, a single estimation of the generalized Laplace angle is made, while [110] made an estimation for each Kepler quarter. \n<!-- image --> \nExtended Data Fig. 3 Observed distance from the equilibrium for all the simulated scenarios in which planets f and g continue the resonant chain . The y-axis is converted to the mean peak-to-peak amplitude from the generalized three-body Laplace angle using the following expression: mean( A ( Ψ i )) = C / 4. Case A2 remains the one that has the potential to be the closest to an equilibrium. \n<!-- image -->', 'Extended Data Table 2 Ground-based photometric campaign observing log.': 'Extended Data Table 3 Stellar parameters of HD 110067. Error bars represent 1 σ uncertainties. \nExtended Data Table 4 Distance of the estimated generalized three-body Laplace angle Ψ e = 0 to the closest equilibrium for all period ratios that are not excluded by available observations . Case A assumes that the mono-transit at 2641.5778 TJD belongs to the 5th planet, and 2656.0944 TJD belongs to the 6th planet. Case B assumes the opposite. The flag column ∗ = 1 indicates that the position of the equilibria varies with the masses of the planets, in which case the equilibrium is recomputed using [164], with masses computed using a mass-radius relation for sub-Neptunes [26]. For ∗ = 0, the equilibrium is 180 degrees.', '10 The HD 110067 planetary system': 'Supplementary Information Supplementary Information is available for this paper. \nRights and permissions Reprints and permissions information is available at www.nature.com / reprints.'}

2024AJ....168..204B

We investigate the evolutionary stages of four open clustersBerkeley 39 Collinder 261 NGC 6819 and NGC 7789of ages ranging from 1.6 to 6 Gyr. These clusters have previously been classified into dynamically young and intermediate age groups based on the segregation level of BSS with respect to redgiantbranch stars and mainsequence stars respectively. We identify members of these four clusters using the MLMOC algorithm on Gaia DR3 data. To examine the relative segregation of cluster members of different evolutionary stages we utilize cumulative radial distributions proper motion distributions and spatial distributions in galactocentric coordinates. Our analysis shows that Berkeley 39 and NGC 6819 exhibit moderate signs of populationwise segregation from evolved to lessevolved members. NGC 7789 shows signs of mass segregation only in the cumulative radial distributions. On the other hand Collinder 261 exhibits high segregation of BSS in the cumulative radial distribution while other populations show the same level of segregation.

2024-11-01T00:00:00Z

['arXiv:2409.10186', '2024AJ....168..204B', '10.48550/arXiv.2409.10186', '10.3847/1538-3881/ad7a72', '2024arXiv240910186B']

['Star clusters', 'Open star clusters', 'Stellar populations', 'Dynamical evolution', 'Stellar kinematics', 'Blue straggler stars', '1567', '1160', '1622', '421', '1608', '168', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']

Dynamical Evolution of Four Old Galactic Open Clusters Traced by Their Constituent Stars with Gaia DR3

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

https://arxiv.org/pdf/2409.10186.pdf

{'Dynamical Evolution of Four Old Galactic Open Clusters traced by their constituent stars with Gaia DR3': 'Shanmugha Balan , 1 Khushboo K. Rao , 1, 2 Kaushar Vaidya , 1 Manan Agarwal , 3 and Souradeep Bhattacharya 4 \n1 Department of Physics, Birla Institute of Technology and Science, Pilani, 333031, Rajasthan, India 2 Institute of Astronomy, National Central University, 300 Zhongda Road, Zhongli 32001 Taoyuan, Taiwan 3 Anton Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands \n4 Inter University Centre for Astronomy and Astrophysics, Ganeshkhind, Post Bag 4, Pune 411007, India', 'ABSTRACT': 'We investigate the evolutionary stages of four open clusters, Berkeley 39, Collinder 261, NGC 6819, and NGC 7789, of ages ranging from 1.6 - 6 Gyr. These clusters have previously been classified into dynamically young and intermediate age groups based on the segregation level of BSS with respect to red giant branch stars and main sequence stars, respectively. We identify members of these four clusters using the ML-MOC algorithm on Gaia DR3 data. To examine the relative segregation of cluster members of different evolutionary stages, we utilize cumulative radial distributions, proper motion distributions, and spatial distributions in galactocentric coordinates. Our analysis shows that Berkeley 39 and NGC 6819 exhibit moderate signs of population-wise segregation from evolved to less-evolved members. NGC 7789 shows signs of mass segregation only in the cumulative radial distributions. On the other hand, Collinder 261 exhibits high segregation of BSS in the cumulative radial distribution, while other populations show the same level of segregation.', '1. INTRODUCTION': 'In an old stellar system, internal dynamics is the dominant mechanism for mass loss (Meylan & Heggie 1997). A stellar system is led towards energy equipartition through the effect of short- and long-range gravitational interactions called two-body relaxation, eventually resulting in a steady segregation of cluster members based on their masses (Bianchini et al. 2016). At the same time, external factors like the Galactic potential, tidal shocks, and passage through the spiral arms can also play an equally important role in driving the evolution of stellar systems, like open clusters (OCs) (Baumgardt & Makino 2003). \nThese processes significantly affect commonly observed cluster populations, such as the main sequence stars (MSs), main sequence turnoff stars (MSTOs) and red giant branch stars (RGBs), and particulary affect the evolution of binary stars - leading to the formation of exotic cluster populations (Knigge et al. 2009) such as blue straggler stars (BSS; Stryker 1993; Bailyn 1995). BSS are typically defined as a brighter and bluer population located along the MS above the MSTO point of a cluster CMD. Based on spectroscopic observations and photometric variability of a few BSS in OCs and globular clusters (GCs), they are found to be as much as twice the mass of turnoff stars in clusters (Shara et al. 1997). According to single stellar isochrones, the masses of BSS are 1 to 2.5 times the mass of MSTOs of their host clusters (Vaidya et al. 2022; Panthi et al. 2023). This indicates that BSS are among the most massive populations in a star cluster and are hence subjected to larger gravitational drag (Chandrasekhar 1943), consequently leading to their faster segregation in the cluster center compared to most other populations in a cluster. Using this trait of the BSS, various BSS-based studies have been conducted over the last decade to investigate the dynamical ages of GCs (Ferraro et al. 2012, 2020; Beccari et al. 2013, 2023; Sanna et al. 2014; Dalessandro et al. 2015; Cadelano et al. 2022) and OCs (Bhattacharya et al. 2019; Rain et al. 2020; Vaidya et al. 2020; Rao et al. 2021; Rao et al. 2023; Rao et al. 2023). \nFerraro et al. (2012) investigated the dynamical status of GCs using a double normalized BSS radial distribution and sorted them into three families on the basis of their dynamical age - Family I (dynamically young), Family II (intermediate dynamical age) and Family III (dynamically old). Similar studies have been carried out for OCs using normalized BSS radial distributions using RGBs as a reference population and classified them into different families of \ndynamical ages (Bhattacharya et al. 2019; Rain et al. 2020; Vaidya et al. 2020). Later on, Rao et al. (2021) and Rao et al. (2023) studied 23 OCs having a sizeable BSS population ( N BSS > 10), by analysing the relative sedimentation level of the BSS using the A + rh parameter (Alessandrini et al. 2016), defined as the area between cumulative radial distributions of BSS and a reference population 1 up to half-mass radius of a cluster. They determined the relationships between A + rh and N relax (cluster age/central relaxation time) as well as the physical parameters for 23 OCs. Their comparison of these correlations in OCs with those in GCs indicated that OCs have a similar relationship to GCs, albeit with higher uncertainties due to the small number of OCs. \nAmong the studies focusing on OCs, four OCs - Berkeley 39, Collinder 261, NGC 6819, and NGC 7789 - exhibit various dynamical stages in comparison to BSS with RGBs or MS stars. These OCs have ages ranging from 1.6 to 6 Gyrs and contain BSS ≥ 11. Upon comparing the radial distribution of BSS to RGBs, it has been demonstrated that these OCs are dynamically young (Rain et al. 2020; Vaidya et al. 2020). These studies suggest that the most massive population, the BSS, does not show discernible segregation when compared to the most evolved population, the RGBs, possibly due to the small number of BSS in comparison to RGBs. Additionally, these OCs are classified as intermediate age based on the sedimentation level of BSS in comparison to MSTOs and MS (Rao et al. 2023). According to these studies, it is apparent that when comparing BSS with different mass populations, there are varied outcomes regarding dynamical ages of OCs. Additionally, it is also clear that BSS may not be the most optimal approach for determining the dynamical ages of these four OCs. Thus, our goal is to conduct a comprehensive population-wise study in order to explore how different populations are segregated relative to each other. This analysis would allow us understand the mass segregation and dynamical ages of these OCs. We divide cluster members of each evolutionary stages into distinct populations, such as RGBs and SGBs, MSTOs, MS, binaries, and BSS. To carry out this work, we use the spatial and kinematic data of distinct cluster populations from the Gaia DR3 (Gaia Collaboration et al. 2023), the Gaia -ESO Survey (GES; Bragaglia et al. 2022) and the WIYN Open Cluster Survey (WOCS; Mathieu 2000). \nThe rest of the paper is organized as follows: Section 2 provides the details of the membership determination, differential reddening correction, and the selection of sources for each cluster population in the study. Section 3 describes the methods we use to study mass segregation in these clusters. Section 4 discusses the dynamical state of each cluster based on the properties and observations from the analyses. Finally, Section 5 concludes the paper.', '2. DATA AND MEMBERSHIP IDENTIFICATION': 'The Gaia mission (Gaia Collaboration et al. 2016) created one of the largest catalogues of astrometric data with precise positions, proper motions, parallaxes, and photometry data of more than one billion stars. Gaia has been instrumental in identifying members of OCs (Cantat-Gaudin 2022), their fundamental parameters, and even exotic stellar populations like BSS and yellow stragglers with far more accuracy than ever before (Bhattacharya et al. 2019; Rain et al. 2021; Vaidya et al. 2020).', '2.1. Membership Determination': "We determine members of the four OCs using the ML-MOC algorithm (Agarwal et al. 2021) on Gaia DR3 data (Gaia Collaboration et al. 2023). ML-MOC uses a combination of two unsupervised machine-learning algorithms the K -N earest N eighbours algorithm (Cover & Hart 1967) and G aussian M ixture M odels (McLachlan & Peel 2000) - on Gaia parallaxes and proper motions to determine membership of OCs. The advantage of using ML-MOC is that it uses only proper motions and parallaxes without relying on spatial information and can provide members down to G = 20 mag. Moreover, it does not make any assumptions about the nature or density profile of the cluster. However, ML-MOC sometimes struggles to find all members when the field is heavily crowded. Interestingly, it has been capable of finding many special features of OCs, like tidal tails for many OCs (Bhattacharya et al. 2019, 2022) and the bifurcated BSS sequence in Berkeley 17 (Rao et al. 2023). Bhattacharya et al. (2022) determined the completeness of ML-MOC for Gaia EDR3 data on NGC 2448 by comparing the ML-MOC members with deeper Pan-STARRS1 DR2 imaging data and contamination fraction by comparing ML-MOC members with Gaia -ESO Survey data (GES; Bragaglia et al. 2022). Their analysis suggests that ML-MOC is around 90% complete with a contamination fraction of ∼ 2 . 3% down to G = 19 . 5 mag. Since parallaxes and proper motions do not change from Gaia EDR3 to Gaia DR3, members identified using ML-MOC on Gaia DR3 and their completeness are as good as Gaia EDR3 data. For more details on ML-MOC, readers are referred to Agarwal et al. (2021). \n<!-- image --> \n<!-- image --> \nFigure 1. The reddening map (left panel), observed CMD (middle panel), and differential reddening corrected CMD (right panel) for Collinder 261 OC. \n<!-- image --> \nTo use ML-MOC, generally a search radius of around 1.5 times the tidal radius of the cluster is considered. We download Gaia DR3 data for a search radius of 45 ' for Berkeley 39, 30 ' for both Collinder 261 & NGC 6819, and 60 ' for NGC 7789 from the coordinates of cluster centers. The search radii are chosen while keeping in mind the distance of the cluster and the crowdedness of the field along the direction of the cluster. Using ML-MOC, we identify 867 members in Berkeley 39, 2195 members in Collinder 261, 1365 members in NGC 6819, and 2864 members in NGC 7789. To compare the members obtained with ML-MOC, we use members determined by van Groeningen et al. (2023). They use astrometric parameters - the proper motion and the parallaxes - as well as the isochrone shape to identify cluster members. The study reported 923 members in Berkeley 39, 3392 members in Collinder 261, 2535 members in NGC 6819, and 4652 members in NGC 7789, based on sources identified with membership probabilities > 0 . 4. By using a supervised machine learning method, they are able to identify members down to G = 20 mag, thus obtaining a larger sample with more members. However, this approach did not allow for the identification of exotic populations that deviate from the isochrone, such as the BSS, which are crucial to our study. On the other hand, the ML-MOC algorithm does not rely on isochrones or any other prior information about the cluster. Consequently, it is able to identify exotic stellar populations. \nTo determine the cluster center for our four OCs (Table 1), we use the Mean Shift algorithm (Comaniciu & Meer 2002). The mean shift algorithm is used here because it is a non-parametric algorithm and is well suited for this type of clustered data. Using these cluster centers, we find the radial distances of all the sources from the cluster centers and estimate the radii of the clusters. The distance at which the density of cluster members becomes flat and indiscernible from field stars is considered the cluster radius, R , and listed in Table 1. \nWe utilize radial velocity data from the GES (Bragaglia et al. 2022) data for Berkeley 39, from the WOCS (Mathieu 2000) for NGC 6819 (Milliman et al. 2014) and NGC 7789 (Nine et al. 2020), and from Gaia DR3 (Gaia Collaboration et al. 2023) for Collinder 261. We cross-match our members with these catalogues within 2 '' distance from our members. We threshold the data with 2 σ bounds around the mean radial velocity and remove sources with no information in the dataset. The final fraction of sources found in the catalogues after cross-matching and thresholding is tabulated in Table 3. For NGC 6819 and NGC 7789, although we included the binaries as an additional separate population, we could not match enough members as NGC 6819 was observed only down to G = 16.5 mag (Milliman et al. 2014) and NGC 7789 down to G = 15 mag (Nine et al. 2020).", '2.2. Differential Reddening Correction': 'The varying degrees of extinction across a cluster may lead to inaccurate determinations of fundamental parameters and different populations, particularly MS equal mass binaries. To mitigate this issue, we perform differential reddening correction to the identified members following the method from Rao et al. (2023). First, we select the reddening vector, i.e. R G = A G /E(BP -RP), a quantity corresponding to the direction of the distortion of the red clump on the CMDs. We select R G = 0.789 from Rao et al. (2023), which is best fitted to red clump stars across all four OCs. Next, we create a grid over the MS stars with the top border consisting of the reddening vector shifted to the MSTO point of a cluster and the bottom border containing the maximum G magnitude. The right and left borders simply enclose the MS. Using the kNN algorithm, we find 25 spatially closest sources to each cluster member. Of those 25 sources, \nTable 1. The fundamental parameters of the four OCs: the cluster centers (RA and DEC), ages, metallicities, distances, cluster radii ( R ), core radii ( r c ), the number of BSS ( N BSS ), the number of SGB and RGB stars ( N SGB RGB ), the number of MSTO stars ( N MSTO ) and the total number of member stars identified in this work ( N Total ). \nwe compute the mean color ( < BP -RP > ) and mean magnitude ( < G > ) of those which fall within the MS grid. The number 25 is selected after a few trials to ensure that each cluster member has a sufficient number of neighboring sources within the main sequence grid. We then calculate the shift along the reddening vector required to match this mean point to the plotted isochrone. With this, we calculate excess magnitude and color, called differential extinction and reddening, respectively. We then subtract differential extinction and reddening from observed G mag and BP -RP mag, respectively, to obtain the corrected magnitude and color for all enclosed stars. The reddening maps (left panel), observed CMDs (middle panel), and differential extinction and reddening corrected CMDs (right panel) for Collinder 261 are shown in Figure 1 and for the other OCs in Figure A1. From Figure 1 and A1, we can see a significant reduction in the scatter in the redder part of the observed CMDs and each evolutionary sequence appears sharply defined after the differential reddening correction.', '2.3. Selection of Cluster Populations': 'To explore the dynamical evolution of the four OCs, we primarily use five different populations in each of the clusters, namely the upper and lower MS, MSTOs, SGBs and RGBs, and BSS. We were able to identify a binary track in the clusters NGC 6819 and NGC 7789, so we added an additional binary population for these two OCs. We identify these populations using the methodology described in Rao et al. (2021), as follows: we fitted the PARSEC isochrone (Bressan et al. 2012) for each cluster using distances and metallicties from Rao et al. (2023), while modified the values of ages and A V to fit the differential reddening corrected CMDs. The fundamental parameters, such as ages, distances and metallicities of the four OCs are listed in Table 1. The equal mass binary isochrone is plotted by shifting PARSEC isochrones by G = 0 . 75 mag and is used to exclude unresolved binaries above the cluster MSTO point, while the ZAMS isochrone is used to remove hot subdwarfs located just below the BSS. To identify the MSTOs, first, the MSTO point of the CMDs is determined, and then the CMDs are normalized by setting the MSTO point at (0, 0). The shifted magnitude and color are represented by G ∗ and ( Bp -Rp ) ∗ . Next, a trapezoid is defined with an upper limit at G = 0 . 0 and a lower limit at G ∗ = -1 . 1 for NGC 6819 and NGC 7789 OCs and -0 . 75 for Berkeley 39 and Collinder 261 OCs. The redward limit is set at ( Bp -Rp ) ∗ = 0 . 2 and the blueward limit is defined by the equation G ∗ = -15 × ( Bp -Rp ) ∗ -1. Henceforth, we followed the specific steps from Rao et al. (2021) to identify all the populations in our study. We further divided the MS into an upper main sequence (UMS) and lower main sequence (LMS). After some trials, we selected this division by defining a boundary at 1 mag below the MSTO for each cluster. This allows us to obtain a sufficiently large numerical sample which is relatively complete. Figure 2 shows the CMDs of the four OCs with plotted isochrones, equal mass isochrones, and the identified populations. To identify binaries in NGC 6819 and NGC 7789, we plotted isochrones corresponding to various mass ratios (q) 2 over the CMDs (Li et al. 2020) and find that the visible binary sequence could be well separated when we choose q ≥ 0 . 8 for NGC 6819 and q ≥ 0 . 9 for NGC 7789. \nFigure 2. CMDs with fitted single-star PARSEC isochrones (grey solid line) and equal-mass binary isochrones (magneta dashed line). The BSS are shown as blue squares, the RGBs are shown as red triangles, the MSTOs are shown as black squares, and the upper and lower main sequence stars are shown as green and yellow dots. NGC 6819 and NGC 7789 have an equal mass binary track above the MS, and the selected binaries are shown as light blue dots (see Section 2.3 for their selection criterion). Every cluster member is divided into 5 bins based on their magnitude (G) and the average error in G for each bin is calculated. The center of these bins (dark grey dots), and the average error in G scaled × 100 (light grey vertical lines) are shown in the right side of each plot. \n<!-- image --> \nTherefore, we select binaries as sources redder than the q = 0 . 8 (or q = 0 . 9 isochrone for NGC 7789) and fainter than G ≥ 15 . 65 mag for NGC 6819 (or G ≥ 14 . 75 mag for NGC 7789) down till the faintest magnitude limit of the cluster members ( G ≤ 19 mag for NGC 6819 and G ≤ 18 mag for NGC 7789). We identify 119 binaries in NGC 6819 and 214 in NGC 7789.', '3. INVESTIGATING MASS SEGREGATION': '3.1. Cumulative Radial Distribution \nFigure 3. CRDs of the four clusters plotted as a fraction of the sources in each population against the distance from the cluster center normalized by the core radius. The BSS are shown in deep blue, RGBs in red, MSTOs in black, UMS in green, and LMS in yellow. For NGC 6819 and NGC 7789, the additional binary populations are shown in light blue. \n<!-- image --> \nIn order to check the spatial distributions of each cluster population identified in Section 2.3, we plot their cumulative radial distributions (CRDs). We use the BSS, RGBs, MSTOs, UMS, LMS and binaries for NGC 6819 and NGC 7789 for the four clusters, as shown in Figure 3. The radial distances of sources are expressed in the units of core radii r c of their host clusters, adopted from Rao et al. (2023) and listed in Table 1. \nIn order to check if the different populations in our study follow different distributions, we compare the CRDs of two different populations pair-wise using the two-sample Anderson-Darling test (henceforth, AD test, first described in Scholz & Stephens 1987, also see Section 3.4 in Bhattacharya et al. 2021b for more details). The AD test highlights the uniqueness of a cluster population in comparison with another. A significance level is calculated by comparing the test statistic against a critical value. To reject the null hypothesis (i.e., both samples are drawn from different distributions), the computed significance level must be under 5%. However, for significance levels over 5%, no conclusions can be drawn. The AD test results are listed in Table 2 and discussed in detail in Section 4.', '3.2. Proper Motion Distribution': 'Having compared the cluster populations in position space, the next step is to compare them in the proper motion space. For this, we plot proper motion scatter diagrams for only three populations - BSS, RGBs, and MSTOs. \nTable 2. The significance levels (in %) from the AD test for pairwise comparison for every population in each cluster are listed here. We consider the pair of CRDs compared to come from different populations if the significance level is less than 5% (denoted by italics). The significance values are floored at 0.1% and capped at 25%. \nHowever, we do not include LMS and UMS stars for this analysis, as errors in their proper motions are almost of the order of the range of proper motions observed in their host clusters. \nTo visually distinguish the level of segregation better, we use the 2D Gaussian kernel density estimator from the Python package Scipy (Virtanen et al. 2020). The density and concentration of sources towards the mean proper motion are more evident upon overplotting contours in the scatter plot. We plot six contours corresponding to seven equally spaced density levels and the resulting illustration is shown in Figure 4.', '3.3. Spatial and Kinematic distribution in Galactocentric coordinates': 'To further analyze the cluster dynamics, we plot the positions and velocities of different cluster populations of the four OCs in galactocentric coordinates. We seek to understand the dynamics of the heavier populations in the cluster by analyzing their positions and velocities in a 3D space. For this analysis, we combine the UMS and LMS populations into a single MS population. Since the four OCs are fairly far away, the distances from Bailer-Jones et al. (2021) still suffer from significant line-of-sight elongation. This elongation persists in all three projections of galactocentric coordinates. Therefore, we approximate every source in the cluster to be equidistant from us and take the distance of each source to be the distance to the cluster (from Table 1). Using these distances and the RA and DEC of each source, we calculate the positions in the galactocentric coordinates. As a result, we can recover features along two projections while much of the elongation was restricted to the ( x, y ) plane. Figure 5 shows the resulting scatter plots in galactocentric coordinates ( x, y ), ( y, z ), ( x, z ) for the four OCs. We also integrate the orbit for each cluster 20 Myr forward in time following the method from Bhattacharya et al. (2021a, 2022). For this, we use 6D information positions (RA, DEC), distances, proper motions in RA and DEC, and radial velocities - of the cross-matched sources from Table 3 and MWPotential2014 model from the Python package galpy (Bovy 2015). The grey solid lines and the brown dashed lines in each panel of Figure 5 represent the integrated cluster orbits and line-of-sight of each cluster, respectively. From Table 3, we can see that Collinder 261 does not have enough number of member with radial velocity data from Gaia DR3. Therefore, we exclude this OC from the galactocentric velocity-space analysis and only carry out this analysis for Berkeley 39, NGC 6819, and NGC 7789. We determine the galactocentric velocities of our \nFigure 4. Proper motion distribution with overlaid density contour plots of the four clusters. The BSS are shown in blue, RGBs in red and MSTOs in black. The populations are arranged in decreasing order of mass from top to bottom. \n<!-- image --> \nTable 3. The fraction and number (parentheses) of cross-matched sources for each poulation with radial velocity catalogs (Gaia-ESO Public Spectroscopy Survey data (GES) for Berkeley 39 and WIYN Open Cluster Survey data (WOCS) for NGC 6819 and NGC 7789). \ncross-matched members using the 6D information - positions (RA, DEC), distances, proper motions in RA and DEC, and radial velocities. Figure 6 shows the scatter plots of galactocentric velocities - ( v x , v y ), ( v y , v z ), ( v x , v z ) - of the different populations of Berkeley 39, NGC 6819, and NGC 7789.', '4. DISCUSSION': 'We carried out a comprehensive study to explore the dynamical stages of four OCs - Berkeley 39, Collinder 261, NGC 6819, and NGC 7789 -by utilizing different populations, such as BSS, SGBs and RGBs, MSTOs, UMS, LMS, and binaries, These clusters are all older than 1 Gyr, therefore we anticipate that two-body relaxation to be in effect within these OCs. We provide a detailed discussion on each cluster using the analysis conducted in § 3.', '4.1. Berkeley 39': 'Figure 5. Positions of sources in Berkeley 39, Collinder 261, NGC 6819 and NGC 7789 in galactocentric coordinates, arranged row-wise . Different populations are represented by colored dots as per the inset legend, while the black dotted line represents the line-of-sight to the cluster and the solid line represents the cluster orbital direction. \n<!-- image --> \nFigure 6. Velocities of the sources in Berkeley 39, NGC 6819, NGC 7789 in galactocentric coordinates, arranged columnwise . Different populations are represented by colored dots as per the inset legend. \n<!-- image --> \nFrom the CRDs (Figure 3) and the AD results (Table 2), we can see that the BSS appear to show no discernibly different segregation to the RGBs or the MSTOs. The MSTOs appear to be more segregated than UMS, and they both appear to be more segregated than the LMS. From the proper motion scatter diagram guided by density contours as shown in Figure 4, we observe that RGBs are the most tightly bunched in the center compared to BSS and MSTOs. However, BSS and MSTOs show comparable spread in the proper motion space. We observe a similar trend for different populations of Berkeley 39 in the galactocentric position and velocity space shown in Figure 5 and Figure 6, respectively. Since there are very few BSS compared to other cluster populations, their trend is not visible in these plots. RGBs appear to be close to the central values with a lower spread than the other two prominent populations - MSTOs and MS stars. The bulk of the MSTOs appear to have less spread than MS stars. We also observe a weak trend of large negative v x for some sources. Vaidya et al. (2020) classified Berkeley 39 as a Family I OC because of its flat BSS radial distribution normalized with RGBs of the same magnitude range. However, they speculated that it could be a Family III cluster based on its N relax value. Essam & Selim (2015) also suggests that the cluster is dynamically evolved by determining that its N relax = 78 . 4. Rao et al. (2023) claimed that this cluster is of intermediate dynamical age using its A + rh (0.038) and N relax (52), where they used average massive populations of the cluster as reference populations. We expect Berkeley 39 to show mass segregation due to energy equipartition since it is among \nthe oldest OCs of 6 Gyr age. From this analysis, it is evident that BSS and RGBs, being the most massive and evolved populations, respectively, demonstrate a higher level of segregation. Conversely, the LMS, the lightest population, shows the least segregation. Hence, we infer that Berkeley 39 exhibits signs of mass segregation in line with what is expected for its age.', '4.2. Collinder 261': "Collinder 261 is the cluster with the largest BSS population among the four OCs studied in this work. From the CRDs (Figure 3) of different cluster populations and the AD test (Table 2), it is evident that BSS are more segregated than any the other populations of Collinder 261. Interestingly, the BSS population shows a strange feature, i.e., the central BSS up to 0 . 3 r c appear to show the same segregation as other populations, while almost 80% of the BSS population that lies outside this range is highly segregated. The RGBs appear to be more segregated in the outer regions of the cluster than the lighter populations. However, none of the other populations show any clear sign of segregation, as is also observed in the AD test. In contrast to the results shown in CRDs (Figure 3) where a bulk of the BSS population is centrally concentrated, a large spread is observed in its proper motion distribution (Figure 4). Both the RGBs and the MSTOs show a comparable spread to the BSS. \nRain et al. (2020) found that Collinder 261 has a flat BSS radial distribution. They claim that the N relax value is inconclusive in determining the status of mass segregation. Rao et al. (2023) classified the cluster as having intermediate dynamical age based on its relation of A + rh (0.171), with N relax (46). The findings from the current analysis show that all the populations show the same level of segregation except for the BSS. We speculate that the strange behaviour of the BSS is more likely related to their formation mechanisms. Despite having a significantly large number of members, its compactness parameter (log( r t / r c ) = 1.08 ± 0.05) indicates that it has a loose core, implying that the binaries were segregated in the cluster center before the BSS formation. This behaviour is similar to an unusual case observed in the OC Melotte 66, where the BSS are highly segregated in the cluster center, resembling the most evolved OCs and GCs (Rao et al. 2023), even though the cluster's dynamical and physical parameters are similar to Collinder 261. Therefore, one needs to investigate BSS formation channels for Collinder 26 to investigate the reason behind such a large segregation of BSS. Our findings indicate that Collinder 261 is still in the early phase of mass segregation.", '4.3. NGC 6819': "NGC 6819 is one of the two clusters in this study with a clearly visible binary track. According to Figure 3, we can see that each cluster population is segregated based on its mass, except for BSS. The AD test results in Table 2 are inconclusive in determining segregation among BSS, RGBs, and MSTOs. However, the AD test does indicate that these three populations are segregated compared to UMS and LMS, and UMS is segregated compared to LMS. Moreover, according to the AD test, the binary population exhibits different segregation levels compared to BSS, RGBs, and LMS, but the comparison to MSTOs and UMS is inconclusive. \nThe proper motion distribution of NGC 6819 paints a different picture. In Figure 4, we see that the BSS show the least spread among not just other populations in the cluster but other clusters in the study. The presence of outliers in the distribution of the MSTOs indicates it has a larger spread than the RGBs. In the galactocentric positions plot (Figure 5), we observe that most of the members belonging to the heavier BSS, RGB or MSTO populations appear to be more centrally located than the comparatively lighter binary or MS population, which are scattered across the cluster. However, since we were able to match only a small fraction of the MS or binaries, these results are partially conclusive. Almost all cross-matched BSS appear to be bunched in the center of the galactocentric velocities scatter (Fig 6). However, the RGB population does not appear to be as tightly bunched, showing a comparable spread to the MSTO population. The MS and binaries also show a similar spread as RGBs and MSTOs in the galactocentric velocity distribution. This could be due to the small fraction of MS and binary population having radial velocities (Table 3). \nVaidya et al. (2020) observe a flat radial distribution in NGC 6819, much like Berkeley 39 and classify it as a Family I or a Family II cluster. With A + rh = 0 . 248 and N relax = 49 from Rao et al. (2023), NGC 6819 is classified as a intermediate dynamical age cluster. Karata¸s et al. (2023) observe a negative slope in the mass function and obtain a value of N relax = 108 and claim that the cluster shows mild signs of mass segregation. Zwicker et al. (2023) compared the mass distribution of the binaries in the cluster with single stars and claimed that the cluster is mass segregated. From the current analysis, we report that NGC 6819 shows moderate signs of mass segregation. \nNGC 7789 is the youngest cluster in this study and also has a clearly visible binary track. The AD test results (Table 2) and CRDs indicate that LMS is the least segregated compared to BSS, RGBs, MSTOs, and UMS. However, when compared to binaries, the results are inconclusive. Furthermore, binaries show the least segregation when compared to SGBs and MSTOs. The AD test results for other comparisons remain inconclusive. \nIn the proper motion scatter (Figure 4), we observe that the BSS show the largest spread in the study, far more than both the RGBs and MSTOs. The RGBs and MSTOs show a comparable spread with respect to each other; however, the MSTOs have more outliers than the RGBs. In Figure 5, we observe that sources from all populations are scattered across the cluster, showing no signs of segregation. In Figure 6, we find that some of the MSTOs have a far higher value of v y along the cluster's orbit. The RGBs seem to show an opposite trend, with most of them lying in regions with lower v y and higher v x . The cross-matched BSS population or the small fraction of cross-matched MS sources do not show any such bias. To investigate the reason behind this strange behaviour of MSTOs, we plotted radial velocity distribution of these stars and found that these stars have particularly higher radial velocities. When checked their errors, we did not find any strange high errors in the radial velocities. Interestingly, we find that of the 35 MSTOs which are outliers in the v x and v y plot, 30 are very rapid rotators 3 , 2 are binary members, and 3 are binary unknowns. As we can see from its CMD (Figure 2) this cluster falls among the cluster with extended main-sequence turnoff. It is typical for NGC 7789 to contain a large number of rapid rotators among MSTOs due to its age (1.6 Gyr). As these stars are more massive than 1.3 solar masses, they possess radiative envelopes, hence allowing them to maintain the rapid rate of rotation since their formation. \nWu et al. (2007) estimated the concentration parameter and obtained a relaxation time of 516 . 8 Myr for this cluster. Rao et al. (2023) obtained N relax = 4 . 23 and A + rh = 0 . 106. Both claim that the cluster is of intermediate dynamical age. Based on our analyses, we find that NGC 7789 shows almost no signs of mass segregation and classify it as a dynamically young cluster.", '5. SUMMARY AND CONCLUSIONS': 'In this study, we explored the degree of mass segregation of the four OCs - Berkeley 39, Collinder 261, NGC 6819, and NGC 7789 - by examining the kinematics and spatial properties of their members. We used the membership determination algorithm, ML-MOC, on Gaia DR3 data to identify members of the four OCs. To conduct this study, we also used radial velocity data from large public spectroscopic surveys, such as GES Bragaglia et al. (2022) and WOCS Mathieu (2000), in addition to Gaia positions and proper motions. We classified the members into different populations based on their photometric properties, including BSS, RGBs, MSTOs, UMS, and LMS, and included binaries as an additional population for NGC 6819 and NGC 7789. \nOur investigations revealed that NGC 6819 and Berkeley 39 show moderate signs of mass segregation. On the other hand, both Collinder 261 and NGC 7789 exhibit fewer signs of mass segregation. Nonetheless, both clusters demonstrate peculiar characteristics. The BSS population of Collinder 261 is highly segregated in the cluster center compared to any other population. In OCs, BSS usually forms through binary system evolution, especially mass transfer. Therefore, we speculate that this could be due to the loose core of the cluster, where binaries are segregated in the cluster center prior to the BSS formation. However, further exploration of the BSS population would be needed to confirm their formation and unusually higher segregation in the cluster center. Additionally, NGC 7789 has MSTOs with unusually high galactocentric velocities, particularly in v y . We found that these stars are situated at the higher end of the radial velocity distribution, and 30 out of 35 are rapid rotators ( v sin i > 120 km/s). We speculate that the large radial velocities of these stars could be associated with their rapid rotation. Despite the varying degrees of mass segregation observed in these OCs, none of them are fully evolved. As OCs reside in the galactic plane, it is expected that they eventually disintegrate before reaching a fully mass segregated state. While A + rh and N relax serve as good indicators for dynamical evolution when comparing among a large sample of OCs, individual systems may show deviations from the derived cluster dynamical properties based on A + rh and N relax . This is likely due to higher galactic interactions experienced by OCs as they are generally in the disc. Therefore, a more holistic analysis as carried out here is required for better understanding the dynamical state of OCs. By analyzing the cluster populations separately, we have attempted to understand how they represent the overall mass segregation of these four OCs and thus their dynamical status.', 'ACKNOWLEDGEMENTS': 'We thank the referee for their comments that helped improve the manuscript. KKR acknowledges the financial support from National Science and Technology Council, Taiwan, R.O.C. (NSTC 113-2811-M-008-034). SB is funded by the INSPIRE Faculty Award (DST/INSPIRE/04/2020/002224), Department of Science and Technology, Government of India. This work has made use of the early third data release from the European Space Agency mission Gaia (https: //www.cosmos.esa.int/gaia), Gaia DR3 (Gaia Collaboration et al. 2023), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. This research has made use of the Vizier catalogue access tool, CDS, Strasbourg, France. This research made use of ASTROPY, a PYTHON package for astronomy (Astropy Collaboration et al. 2022), NUMPY (Harris et al. 2020), MATPLOTLIB (Hunter 2007), and SCIPY (Virtanen et al. 2020). This research also made use of the Astrophysics Data System governed by NASA (https://ui.adsabs.harvard.edu).', 'REFERENCES': 'Knigge, C., Leigh, N., & Sills, A. 2009, Nature, 457, 288, doi: 10.1038/nature07635 \n- Li, L., Shao, Z., Li, Z.-Z., et al. 2020, ApJ, 901, 49, doi: 10.3847/1538-4357/abaef3\n- Mathieu, R. D. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 198, Stellar Clusters and Associations: Convection, Rotation, and Dynamos, ed. \nR. Pallavicini, G. Micela, & S. Sciortino, 517 McLachlan, G. J., & Peel, D. 2000, Probability and Statistics - Applied Probability and Statistics Section, Vol. 299, Finite mixture models (New York: Wiley) Meylan, G., & Heggie, D. C. 1997, A&A Rv, 8, 1, doi: 10.1007/s001590050008 \nMilliman, K. E., Mathieu, R. D., Geller, A. M., et al. 2014, VizieR Online Data Catalog: WIYN open cluster study. LX. RV survey of NGC 6819 (Milliman+, 2014), VizieR On-line Data Catalog: J/AJ/148/38. Originally published in: 2014AJ....148...38M, doi: 10.26093/cds/vizier.51480038 \nNine, A. C., Milliman, K. E., Mathieu, R. D., et al. 2020, AJ, 160, 169, doi: 10.3847/1538-3881/abad3b \nPanthi, A., Vaidya, K., Vernekar, N., et al. 2023, Monthly Notices of the Royal Astronomical Society, 527, 8325, doi: 10.1093/mnras/stad3750 \nRain, M. J., Ahumada, J. A., & Carraro, G. 2021, A&A, 650, A67, doi: 10.1051/0004-6361/202040072 \n- Rain, M. J., Carraro, G., Ahumada, J. A., et al. 2020, AJ, 159, 59, doi: 10.3847/1538-3881/ab5f0b \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure A1. Differential Reddening correction for Berkeley 39, NGC 6819 and NGC 7789 (from top to bottom) with the reddening map, observed CMD and the reddening corrected CMD for each cluster (from left to right). \n<!-- image --> \nNGC 6819 \nNGC 7789 \n<!-- image -->'}

2023ApJ...954L...4K

We report on the discovery of two lowluminosity broadline active galactic nuclei AGNs at z gt 5 identified using JWST NIRSpec spectroscopy from the Cosmic Evolution Early Release Science CEERS survey. We detect broad H emission in the spectra of both sources with FWHM of 2060 290 km sSUP1SUP and 1800 200 km sSUP1SUP resulting in virial black hole BH masses that are 12 dex below those of existing samples of luminous quasars at z gt 5. The first source CEERS 2782 at z 5.242 is 23 dex fainter than known quasars at similar redshifts and was previously identified as a candidate lowluminosity AGN based on its morphology and restframe optical spectral energy distribution SED. We measure a BH mass of M SUBBHSUB 1.3 0.4 10SUP7SUP M SUBSUB confirming that this AGN is powered by the least massive BH known in the Universe at the end of cosmic reionization. The second source CEERS 746 at z 5.624 is inferred to be a heavily obscured broadline AGN caught in a transition phase between a dustobscured starburst and an unobscured quasar. We estimate its BH mass to be in the range of M SUBBHSUB 0.94.7 10SUP7SUP M SUBSUB depending on the level of dust obscuration assumed. We perform SED fitting to derive host stellar masses M SUBSUB allowing us to place constraints on the BHgalaxy mass relationship in the lowest mass range yet probed in the early Universe. The M SUBBHSUBM SUBSUB ratio for CEERS 2782 in particular is consistent with or higher than the empirical relationship seen in massive galaxies at z 0. We examine the narrow emission line ratios of both sources and find that their location on the BPT and OHNO diagrams is consistent with model predictions for moderately low metallicity AGNs with ZZ SUBSUB 0.20.4. The spectroscopic identification of lowluminosity broadline AGNs at z gt 5 with M SUBBHSUB 10SUP7SUP M SUBSUB demonstrates the capability of JWST to push BH masses closer to the range predicted for the BH seed population and provides a unique opportunity to study the early stages of BHgalaxy assembly.

2023-09-01T00:00:00Z

['2023ApJ...954L...4K', '10.3847/2041-8213/ace5a0', '2023arXiv230200012K', 'arXiv:2302.00012', '10.48550/arXiv.2302.00012']

['Quasars', 'Supermassive black holes', 'High-redshift galaxies', 'Active galactic nuclei', '1319', '1663', '734', '16', 'Astrophysics - Astrophysics of Galaxies']

Hidden Little Monsters Spectroscopic Identification of Lowmass Broadline AGNs at z gt 5 with CEERS

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

https://arxiv.org/pdf/2302.00012.pdf

{'Hidden Little Monsters: Spectroscopic Identification of Low-Mass, Broad-Line AGN at z > 5 with CEERS': 'DALE D. KOCEVSKI, 1 MASAFUSA ONOUE, 2, 3 , ∗ KOHEI INAYOSHI, 2 JONATHAN R. TRUMP, 4 PABLO ARRABAL HARO, 5 ANDREA GRAZIAN, 6 MARK DICKINSON, 5 STEVEN L. FINKELSTEIN, 7 JEYHAN S. KARTALTEPE, 8 MICHAELA HIRSCHMANN, 9 SEIJI FUJIMOTO, 10, 11, 12 STÉPHANIE JUNEAU, 13 RICARDO O. AMORÍN, 14, 15 MICAELA B. BAGLEY, 16 GUILLERMO BARRO, 17 ERIC F. BELL, 18 LAURA BISIGELLO, 19, 20 ANTONELLO CALABRÒ, 21 NIKKO J. CLERI, 22, 23 M. C. COOPER, 24 XUHENG DING, 3 NORMAN A. GROGIN, 25 LUIS C. HO, 2, 26 AKIO K. INOUE, 27, 28 LINHUA JIANG, 2, 26 BRENDA JONES, 29 ANTON M. KOEKEMOER, 30 WENXIU LI, 2 ZHENGRONG LI, 2 ELIZABETH J. MCGRATH, 1 JUAN MOLINA, 31, 2 CASEY PAPOVICH, 22, 23 \nPABLO G. PÉREZ-GONZÁLEZ, 32 NOR PIRZKAL, 33 STEPHEN M. WILKINS, 34, 35 GUANG YANG, 36, 37 AND L. Y. AARON YUNG 38', 'ABSTRACT': 'We report on the discovery of two low-luminosity, broad-line active galactic nuclei (AGN) at z > 5 identified using JWST NIRSpec spectroscopy from the Cosmic Evolution Early Release Science (CEERS) Survey. We detect broad H α emission in the spectra of both sources, with FWHM of 2038 ± 286 and 1807 ± 207 km s -1 , resulting in virial black hole (BH) masses that are 1-2 dex below that of existing samples of luminous quasars at z > 5. The first source, CEERS 1670 at z = 5 . 242, is 2-3 dex fainter than known quasars at similar redshifts and was previously identified as a candidate low-luminosity AGN based on its morphology and rest-frame optical spectral energy distribution (SED). We measure a BH mass of M BH = 1 . 3 ± 0 . 4 × 10 7 M /circledot , confirming that this AGN is powered by the least-massive BH known in the universe at the end of cosmic reionization. The second source, CEERS 3210 at z = 5 . 624, is inferred to be a heavily obscured, broad-line AGN caught in a transition phase between a dust-obscured starburst and an unobscured quasar. We estimate its BH mass to be in the range of M BH /similarequal 0 . 9 -4 . 7 × 10 7 M /circledot , depending on the level of dust obscuration assumed. We perform SED fitting to derive host stellar masses, M /star , allowing us to place constraints on the BH-galaxy mass relationship in the lowest mass range yet probed in the early universe. The M BH / M /star ratio for CEERS 1670, in particular, is consistent with or higher than the empirical relationship seen in massive galaxies at z = 0. We examine the narrow emission-line ratios of both sources and find that their location on the BPT and OHNO diagrams is consistent with model predictions for moderately low-metallicity AGN with Z / Z /circledot /similarequal 0 . 2 -0 . 4. The spectroscopic identification of low-luminosity, broad-line AGN at z > 5 with M BH /similarequal 10 7 M /circledot demonstrates the capability of JWST to push BH masses closer to the range predicted for the BH seed population and provides a unique opportunity to study the early stages of BH-galaxy assembly. \nKeywords: High-redshift galaxies (734); Quasars (1319); Supermassive black holes (1663)', '1. INTRODUCTION': "With the advent of wide-field quasar surveys such as those carried out by the Sloan Digital Sky Survey (SDSS; Fan et al. 2001; Jiang et al. 2016) and the Panoramic Survey Telescope & Rapid Response System 1 (Pan-STARRS1; Bañados et al. 2016; Mazzucchelli et al. 2017), hundreds of quasars have been discovered and characterized at z > 5 (Inayoshi et al. 2020; Fan et al. 2022), with the most distant found a mere 670 million years after the Big Bang (Wang et al. 2021). The super massive black holes (SMBHs) that power these sources have masses of order ∼ 10 9 M /circledot , raising the question of how such systems were built in such a short amount of cosmic \ntime. Most theories involve Eddington-limited or possibly super-Eddington accretion onto seed BHs that are predicted to form at 10 < z < 30 and have masses that range from ∼ 10 2 M /circledot (so called 'light seeds') to over ∼ 10 5 M /circledot ('heavy seeds') with a continuous distribution (e.g., Inayoshi et al. 2020; Volonteri et al. 2021). The relative contribution of each seed type remains largely unconstrained by observations (Miller et al. 2015; Trump et al. 2015). \nMost quasar surveys, which observe /greaterorsimilar 1 , 000 deg 2 down to ∼ 20 mag, are sensitive to only the most luminous quasar populations ( ∼ 10 47 erg s -1 in bolometric luminosity; L bol). These ultra-rare systems, which formed in biased regions of the early universe, place limited constraints on the BH seed population as they would have undergone sustained episodes of exponential growth, even for the most mas- \npredicted seeds, thereby erasing the imprint of the initial seed mass distribution (e.g., Tanaka & Haiman 2009; Volonteri 2010). A complementary approach is to search for lower-luminosity quasars hosting SMBHs with masses closer to the predicted seed mass range at the earliest epochs possible (Somerville et al. 2008; Valiante et al. 2016; Ricarte & Natarajan 2018; Yung et al. 2021; Li et al. 2022). Several deep optical surveys have attempted to do this by reaching a dex fainter in luminosity (e.g., Willott et al. 2007, 2010a; Matsuoka et al. 2016, 2022; Kim et al. 2018, 2020; Fujimoto et al. 2022); however, these samples are still far more luminous than what is observed in the nearby universe ( L bol ∼ 10 43 -44 erg s -1 ; e.g., Greene & Ho 2007, Liu et al. 2018, 2019), biasing our understanding of early SMBHs toward the most massive and active populations (however, see also Mezcua et al. 2018). \nAdditional constraints on the seed mass distribution can be obtained by comparing the masses of high-redshift SMBHs to that of their host galaxies. In the local universe, well established scaling relationships exist between the mass of SMBHs and the bulge properties of their hosts (e.g., Magorrian et al. 1998; Gebhardt et al. 2000; Ferrarese & Merritt 2000; McConnell & Ma 2013; Sun et al. 2015). However, offsets from this relationship at higher redshift can help constrain models of early BH growth and their co-evolution with galaxies (Hirschmann et al. 2010; Habouzit et al. 2022; Hu et al. 2022). Observational studies have produced mixed results in this regard, with several reporting that SMBHs become increasingly overmassive relative to their hosts with increasing redshift (e.g., Trakhtenbrot & Netzer 2010; Bennert et al. 2011; Park et al. 2015; Shimasaku & Izumi 2019; Ding et al. 2020; Neeleman et al. 2021), while other studies report no evolution in the local scaling relationship (e.g., Willott et al. 2017; Izumi et al. 2019; Suh et al. 2020). Pushing such studies to lower SMBH and host masses at high redshifts is expected to provide additional insight into the earliest seeds. Not only are lower-luminosity AGN more representative of the normal BH population (Habouzit et al. 2022), lower mass hosts have a relatively quiet merger history and so represent a robust 'fossil record' of the initial BH-seed mass distribution (Volonteri et al. 2008; Volonteri & Natarajan 2009). \nJWST is expected to be a game changer on both fronts, allowing for the detection of lower luminosity quasars and the light of their host galaxies out to the epoch of cosmic reionization. Since its launch, JWST has already revealed the host morphologies of X-ray and optically selected AGN out to z ∼ 4 (Kocevski et al. 2022; Ding et al. 2022a), detected the host light of a quasar at z /similarequal 6 for the first time (Ding et al. 2022b), and identified a candidate faint quasar at z /similarequal 7 . 7 (Furtak et al. 2022). Recently, Onoue et al. (2023, hearafter O23) reported a candidate low-luminosity AGN at \nz ∼ 5 by exploiting the first NIRCam images of the CEERS program. This AGN candidate, CEERS-AGN-z5-1, has a compact morphology and shows a rest-frame UV-to-optical spectral energy distribution (SED) that can be well explained by an unobscured quasar with L bol = 2 . 5 ± 0 . 3 × 10 44 erg s -1 and strong Balmer and [O III] emission lines. In addition, Carnall et al. (2023) recently reported the detection of broad H α emission from a quiescent galaxy at z = 4 . 658 using JWST, from which they measure the central SMBH mass of M BH = 10 8 . 7 ± 0 . 1 M /circledot . \nHere we report on the detection of broad H α emission from two z > 5 galaxies, including CEERS-AGN-z5-1, using NIRSpec data obtained as part of the second epoch of CEERS observations. The first source, CEERS 1670 at z = 5 . 242, was identified as a result of targeted follow-up of CEERS-AGNz5-1, while the second source, CEERS 3210 at z =5 . 624, was found serendipitously while inspecting the spectra of galaxies with photometric redshifts of z > 8 in the literature. \nWe show that the SMBHs at the heart of these lowluminosity AGN have masses 1-2 dex lower than existing samples of luminous quasars with BH mass estimates at z > 5. We also examine the emission line ratios of both sources and place constraints on the relationship between SMBH and host mass in the lowest mass range yet probed in the early universe. Our analysis is presented as follows: in Section 2, we describe the near-infrared imaging and spectroscopy used for this study, while in Section 3, we discuss the properties of our sample. In Section 4, we outline our methodology for measuring the emission line properties of our sample. Section 5 describes our results, and the implications of our findings are discussed in Section 6. We use vacuum wavelengths for all emission-line features and, when necessary, the following cosmological parameters are used: H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 7, and Ω m = 0 . 3.", '2. OBSERVATIONS & DATA REDUCTION': "The Cosmic Evolution Early Release Science Survey (CEERS) is an early release science program that covers 100 arcmin 2 of the Extended Groth Strip (EGS) with imaging and spectroscopy using coordinated, overlapping parallel observations by most of the JWST instrument suite (Finkelstein et al., in prep). CEERS is based around a mosaic of 10 NIRCam pointings, with six NIRSpec and eight MIRI pointings observed in parallel. Here we make use of NIRCam pointings 3 and 6 obtained on 21 June 2022, as well as NIRSpec pointing 4, obtained on 21 December 2022. In each NIRCam pointing, data were obtained in the shortwavelength (SW) channel F115W, F150W, and F200W filters, and long-wavelength (LW) channel F277W, F356W, F410M, and F444W filters. The total exposure time for pixels observed in all three dithers was typically 2835 s per filter. \nTable 1. AGN Sample \nNOTE-CEERS 1670 is the same source as CEERS-AGN-z5-1 in O23. \nThe NIRSpec observations were taken with the G140M/F100LP, G235M/F170LP and G395M/F290LP R /similarequal 1000 grating/filter pairs as well as with the R /similarequal 30 -300 prism, providing a complete coverage of the 1 -5 µ m range with both configurations. The observation adopted a threenod pattern, each of the nods consisting of a single integration of 14 groups (1036 s). The coadded spectra have a total exposure time of 3107 s in each spectral configuration. Targets for the microshutter array (MSA) configuration included sources selected using the NIRCam imaging in the field from CEERS epoch one (June 2022), especially prioritizing targets with photometric redshifts of z > 6. Each target was observed using a 'slitlet' aperture of three microshutters, and the design also included empty shutters for background subtraction. The shutter configuration for observations taken with the medium resolution gratings and the prism are identical. \nWe performed an initial reduction of the NIRCam images in all four pointings, using version 1.5.3 of the JWST Calibration Pipeline 1 with some custom modifications. We used the current (15 July 2022) set of NIRCam reference files 2 , though we note that the majority were created preflight, including the flats and photometric calibration references. We describe our reduction steps in greater detail in Finkelstein et al. (2022) and Bagley et al. (2022). Coadding the reduced observations into a single mosaic was performed using the drizzle algorithm with an inverse variance map weighting (Fruchter & Hook 2002; Casertano et al. 2000) via the Resample step in the pipeline. The output mosaics have pixel scales of 0 . '' 03/pixel. \nPhotometry was computed on PSF-matched images using SExtractor (Bertin & Arnouts 1996) v2.25.0 in two-image mode, with an inverse-variance weighted combination of the PSF-matched F277W and F356W images as the detection image. Photometry was measured in all seven of the NIRCam bands observed by CEERS, as well as the F606W, F814W, F105W, F125W, F140W, and F160W HST bands using data obtained by the CANDELS and 3D-HST surveys \n(Grogin et al. 2011; Koekemoer et al. 2011; Brammer et al. 2012; Momcheva et al. 2016). \nThe CEERS NIRSpec observations (Arrabal Haro et al., in prep.) were reduced using version 1.8.5 of the JWST Science Calibration Pipeline with the Calibration Reference Data System (CRDS) mapping 1027, starting from the Level 0 uncalibrated data products ('\\_uncal.fits' files) available on MAST. Custom parameters were used for the jump step at the detector-level calibration for a better treatment of the 'snowballs' 3 produced by high-energy cosmic ray events, and a nodded background subtraction was adopted. \nThe reduced two-dimensional (2D) spectra ('s2d') have a rectified trace with a flat slope. The current version (1.8.5) of the pipeline does not correctly identify source locations in the 2D spectra for one-dimensional (1D) spectra extraction. For the sources presented in this work, the 1D spectra were extracted using custom boxcar apertures centered on the visually identified continuum trace. Any remaining artifacts in the extracted spectra were masked after a detailed visual analysis. The flux uncertainties of the reduced 1D spectra appear to be underestimated by a factor of ∼ 2, as estimated from the normalized median absolute deviation (NMAD) of the flux in line-free regions, and so we rescale the flux uncertainty of each spectrum by a factor equal to the ratio of the line-free NMAD to the median pipeline uncertainty. \nThe current (version 1.8.5) of the NIRSpec MSA data reduction uses a flux calibration that relies on pre-flight knowledge of the instrument, which is known to differ from the post-launch performance (see Figure 20 of Rigby et al. 2022). The pipeline applies a correction for 'slit losses' outside the MSA aperture using a pathloss reference file based on a pre-launch model for point sources that has not yet been fully verified on orbit. This correction may be inaccurate for extended sources or non-default spectral extraction apertures, and indeed by comparing spectroscopic fluxes to NIRCam photometry we find some evidence that further corrections are required (see, e.g., section 3). While this may impact our interpretation of individual line fluxes or luminosities, the relative spectrophotometry of the reduced spectra is measured to be reliable, with line ratios of doublets ([O III] λλ 4960, 5008; Storey & Zeippen 2000) and Balmer lines (Osterbrock 1989) that match physical expectations (see additional discussion in Trump et al. 2022, Arrabal Haro et al., in prep.).", '3. SAMPLE DESCRIPTION': "During the initial inspection of our reduced NIRSpec data, we identified two sources with broad H α emission. Information on these sources, referred to as CEERS 1670 and CEERS 3210, is listed in Table 1. CEERS 1670 was ob- \nFigure 1. JWST NIRCam images of our broad-line AGN sample at z > 5 taken in the short-wavelength (F150W and F200W) and longwavelength (F277W, F356W, and F444W) filters. The RGB images are composed of images in the F150W, F277W, and F444W filters. All images are 2 '' × 2 '' in size. The alignment of the NIRSpec microshutter aperture relative to each source is shown in red overtop the F444W image. \n<!-- image --> \nerved as a result of targeted follow-up of the AGN candidate CEERS-AGN-z5-1 identified by O23. CEERS 3210 was selected for observation as it was previously identified as a candidate massive galaxy at z = 8 . 13 by Labbe et al. (2022) and a potential strong-line emitter at z = 5 . 72 by Pérez-González et al. (2022). NIRCam images of both sources are shown in Figure 1, while their 1D and 2D spectra from the G395M grating are shown in Figure 2. Our derived redshifts, based on the [O III] λλ 4960, 5008 narrow lines, for CEERS 1670 and CEERS 3210 are z = 5 . 242 and z = 5 . 624, respectively. \nNeither source is directly detected in the deep (800 ksec) Chandra X-ray observations of the CEERS field from the AEGIS-XD survey (Nandra et al. 2015). However, the shape of their SEDs, coupled with the existence of broad-line emission in their spectra, suggest both sources host lowluminosity AGN. \nIn Figure 3, we show the NIRCam photometry and NIRSpec prism spectrum of both CEERS 1670 and CEERS 3210. In the case of CEERS 1670, we find the prism spectrum must be scaled by a factor of 2 × to match the NIRCam broadband photometry. This may be due to potential slit losses, as CEERS 1670 sits near the edge of its microshutter slit, the outline of which can be seen in Figure 1. We find no such correction is needed for the CEERS 3210 prism spectrum. \nAs discussed by O23, the broad-band photometry of CEERS 1670 is well reproduced by a continuum model with a single power-law function, with the exception of filters that are affected by strong line emission, namely F277W, F410M, and F444M. A single power-law fit to the other four filters yields the best-fit power-law slope α λ = -1 . 14 ± 0 . 03 ( ≡ d ln F λ / d ln λ ), which is consistent with a typical value for unobscured quasars (e.g., Fan et al. 2001; Vanden Berk et al. \n2001). This power-law model yields the absolute magnitude at rest-frame 1450 Å of M 1450 = -19 . 44 ± 0 . 05 mag. Likewise, the monochromatic luminosity at rest-frame 3000 Å and 5100 Å is L 3000 = (4 . 83 ± 0 . 09) × 10 43 erg s -1 and L 5100 = (4 . 48 ± 0 . 08) × 10 43 erg s -1 , respectively. We find that a low-redshift composite spectrum of quasars (the blue model in Figure 3a) from Vanden Berk et al. (2001, hereafter VB01) scaled to match the photometry can explain the observed spectral shape of CEERS 1670 well. \nThe SED of CEERS 3210 shows more complexity. The source has a blue continuum spectrum with a UV slope of α λ = -3 . 0 ± 0 . 3 at λ obs /similarequal 1 -2 µ m and a very steep continuum spectrum ( α λ = 1 . 8 ± 0 . 2) with strong Balmer and [O III] emission lines at longer wavelengths. This steep spectral slope, coupled with the broad H α emission we detect, suggests that this source is a heavily obscured, broad-line AGN (e.g., Gregg et al. 2002). In Figure 3b, we overlay the composite SED of low-redshift broad-line AGN (VB01) reddened assuming a color excess of E ( B -V ) = 0 . 9 and the extinction law discussed in Calzetti et al. (2000). Note that this model shown with the cyan curve is essentially the same as the QSO2 SED template provided in Polletta et al. (2006). This model traces the observed prism continuum at λ obs /greaterorsimilar 3 µ m well; however, the obscured broad-line AGN model does not explain the blue side of the observed spectrum, requiring additional components at these shorter wavelengths. We discuss more complex SED models, including fits using hybrid galaxy plus AGN models, in Section 6.3.", '4. LINE FITTING ANALYSIS': 'The NIRSpec spectra of CEERS 1670 and CEERS 3210 include several prominent emission lines. The G395M/F290LP spectrum of both sources includes strong \n<!-- image --> \nFigure 2. NIRSpec spectra of sources CEERS 1670 and CEERS 3210 taken in the G395M grating with R ∼ 1000. The 2D spectra are shown above with extraction windows highlighted in red. Grey regions in both the 1D and 2D spectra indicate regions masked due to artifacts identified via visual inspection. The location of several prominent emission lines are noted. \n<!-- image --> \nJy] \nμ \n[ \ny \nt \nsi \nn \ne \nd \nx \nu \nl \nF \na \nFigure 3. The SEDs of the two low-luminosity AGN (CEERS 1670 and CEERS 3210) obtained with the JSWT NIRSpec and NIRCam. Left panel (a): the continuum spectral shape is explained by the composite quasar spectrum of VB01 scaled to match the photometry of CEERS 1670 (blue), and is fitted well with a single power law with an index of α λ = -1 . 14 (dashed). The galaxy SED model with M /star /similarequal 6 . 0 × 10 9 M /circledot is overlaid (red), where the stellar continuum in the F356W filter becomes comparable to the observed F356W flux density. This gives a robust upper bound of the underlying stellar population. Right panel (b): the source has a blue continuum spectrum with a UV slope of α λ < -3 . 0 at λ obs /similarequal 1 -2 µ mand a very steep continuum spectrum ( α λ /similarequal 2 . 0). The redder part can be explained either by a heavily obscured quasar (cyan) or a dusty starburst galaxy (red). As a possible explanation of the blue excess in the spectrum, the unobscured broad-line AGN contribution is added to the dusty starburst galaxy (blue). In the dusty galaxy model, the stellar mass is set to M /star /similarequal 6 × 10 10 M /circledot (see the text in Section 6.3). \n<!-- image --> \nCEERS 1670 (z = 5.242) \nNIRCam \nNIRSpec Prism \nQSO (scaled), SDSS composite \nM \nGalaxy ( \n0.7 \n9 \n= 6 10 \n1 \n5 \nM \n) \n2 \nWavelength [μm] \nH α , H β , and [O III] λλ 4960, 5008 emission, and CEERS 3210 also features a He I λ 5877 . 25 line. Both sources exhibit a weak line near the expected wavelength of the [Fe X] λ 6376 coronal emission line. The G235M/F170LP spectrum of both sources includes the [Ne III] λ 3870 . 86 line, while CEERS 3210 also exhibits the H γ λ 4341 . 69 and auroral [O III] λ 4364 . 44 lines. \nWe measure line fluxes and uncertainties with a Levenberg-Marquardt least-squares method implemented by the mpfit IDL code 4 . We fit isolated lines with single Gaussians and simultaneously fit multiple Gaussians for features in the Balmer line regions. The results of our line fits are shown in Figure 4. \nTo account for potential broad components, we fit the H α line with two Gaussians: one narrow with width σ < 350 km s -1 and one broad with width σ > 350 km s -1 . We also attempted to include additional Gaussian components for the [N II] λλ 6550, 6585 doublet, constraining the line widths and relative line centers to narrow H α , but found that the [N II] lines are not significantly ( > 3 σ ) detected and their inclusion does not improve the χ 2 0 of the fit. We report the 1 σ upper limit for [N II] λ 6585 but do not include it in the fits for broad and narrow H α . \nWe also performed a simultaneous fit of the H β emissionline region with components for narrow H β and the [O III] λλ 4960,5008 doublet. In both systems we tested a \nfit that included an additional broad ( σ > 350 km s -1 ) H β component but found that this component is only marginally ( < 1 σ ) detected and including it increases the χ 2 0 of the fit. We report 1 σ upper limits for putative broad H β emission that assume the same width as the broad H α component applied to the local noise of the H β region. \nFinally, we fit single narrow Gaussians for the [O II] λ 3728 . 48 (the 3727+3729 doublet is blended in the R /similarequal 1000 medium-resolution NIRSpec grating), [Ne III] λ 3870 . 86, H γ λ 4341 . 69, and [O III] λ 4364 . 44. The [Ne III] line is significantly ( > 3 σ ) detected in CEERS 1670 and all the other lines are only marginally ( < 3 σ ) detected.', '5.1. Emission Line Properties': 'Our two AGN are identified from their broad H α emission. As described above, we use a two-component fit with both narrow and broad Gaussian components in which the line centers, widths, and fluxes are free parameters. These broad+narrow fits have significantly lower χ 2 0 than single-Gaussian fits for the H α lines. Both objects have best-fit narrow H α components that are unresolved in the R ∼ 1000 NIRSpec spectra, with narrow-H α widths of σ = 135 ± 9 km s -1 and σ = 131 ± 24 km s -1 for CEERS 1670 and CEERS 3210, respectively. The best-fit broad H α components have σ = 840 ± 120 km s -1 and FWHM = 2060 ± 290 km s -1 for CEERS 1670 and σ = 720 ± 87 km s -1 and FWHM=1800 ± 200 km s -1 for CEERS 3210 (fitting σ and FWHM independently). \n10 \n1 \n0.1 \n0.01 \n0.5 \n4 \n3 \nFigure 4. The rest-frame spectra (black histograms) and associated uncertainty (gray error bars) of both sources in regions with emission-line features. Red lines show the best-fit Gaussians for narrow emission lines and the blue line shows the best-fit broad component for H α , which have a FWHM of 2060 ± 286 and 1802 ± 204 km s -1 for CEERS 1670 and CEERS 3210, respectively. \n<!-- image --> \nIn contrast, the H β emission lines of both objects are best-fit by single narrow Gaussians, with no statistical improvement from including a broad component. Both H β lines appear to be unresolved, with best-fit single-Gaussian widths of σ = 145 ± 17 km s -1 for CEERS 1670 and σ = 108 ± 33 km s -1 for CEERS 3210. We compute upper limits for a potential (undetected) broad H β component by assuming a Gaussian of the same width as the measured H α broad lines with the noise properties of the H β region in the spectrum. In both cases the upper limit for potential H β broad emission is statistically consistent with a broad H α/ H β = 3 . 1 (Osterbrock 1989): CEERS 1670 has a lower limit of broad H α/ H β > 2 . 4 (3 σ ) and CEERS 3210 has a lower limit of H α/ H β > 3 . 0 (3 σ ). In other words, both CEERS 1670 and CEERS 3210 are consistent with (undetected) broad H β \nemission that matches typical Type 1 AGN H α /H β ratios, and the lack of observed broad H β in CEERS 1670 and CEERS 3210 cannot be used to classify them as intrinsic Type 1.5 AGN. \nThe narrow Balmer emission lines imply modest dust attenuation in both objects. CEERS 1670 has a measured narrow-line Balmer decrement of H α/ H β = 3 . 9 ± 0 . 5 and CEERS 1670 has a narrow-line H α/ H β = 5 . 3 ± 2 . 1. We use these Balmer decrements as priors to inform the SED fitting in Section 6.2 and 6.3. \nIntriguingly, both AGN have weak emission-line features that are consistent with marginally-detected [Fe X] λ 6376, as seen in Figures 2 and 4. [Fe X] is a coronal emission line with an ionization potential of 262 eV that is observed in low-mass AGN in the local universe (e.g., Molina et al. \n<!-- image --> \nFigure 5. Left Panel (a): The BPT emission-line diagnostic diagram. The gray contours denote the distribution of local star-forming galaxies and AGN as measured by the SDSS survey (York et al. 2000). Black diamonds denote stacked line ratios of CEERS galaxies at z ∼ 5 . 6, z ∼ 4 . 5, and z ∼ 3 . 3 (Sanders et al. 2023). The black long and short-dashed lines denote the z = 0 and z = 2 . 3 boundary between the starforming and AGN regions of the diagram defined by Kauffmann et al. (2003) and Kewley et al. (2013b), respectively. Right Panel (b): the OHNO diagnostic diagram. Black squares denote line ratios of SMACS ERO galaxies at 5 . 3 < z < 8 . 5 (Trump et al. 2022) and gray contours denote the distribution of z ∼ 0 SDSS galaxies. The dashed line denotes the boundary between star-forming and AGN regions as defined in Backhaus et al. (2022). Colored curves in both panels show MAPPINGS V photoionization models (Kewley et al. 2019). The three color-coded sets of curves and points along those curves correspond to different ionization parameters and metallicities, as indicated by the legends, with three curves for each color corresponding to different gas pressures as described in the text. Both of our z ∼ 5 AGN have narrow-line ratios that are consistent with low metallicity and high ionization, with little difference from the emission-line ratios observed for other populations of high-redshift galaxies. \n<!-- image --> \n2021). The putative [Fe X] emission lines are marginally detected with SNR=2.4 for CEERS 1670 and only SNR=1.5 for CEERS3210. Both lines are best-fit to be slightly redder than the other narrow-line features: if the marginal detections represent genuine emission lines then they may indicate a kinematic offset between the extreme-ionization coronal gas and the narrow-line region. \nFinally, in Figure 5 we plot the narrow emission-line ratios of both sources on the BPT ([O III]/H β versus [N II]/H α ; Baldwin et al. 1981) and OHNO ([Ne III]/[O II] versus [O III]/H β ; Backhaus et al. 2022) line-ratio diagnostics that are commonly used to classify galaxies as dominated by emission from AGN or star formation. The colored curves in Figure 5 indicate MAPPINGS V photoionization models from Kewley et al. (2019), with different colored curves for different ionization (log( Q / [cm s -1 ]) = [7 , 8 , 9] increasing left to right), metallicity along each curve ( Z / Z /circledot = [1 , 0 . 4 , 0 . 2 , 0 . 05] as indicated in the legend), and curves shown for each of three thermal pressures (log( Pk -1 B / [K cm -3 ]) = [7 , 8 , 9]). The MAPPINGS V models use α -enhanced abundances as described in Nicholls et al. (2017), such that low metallicities include enhanced relative abundances of O and Ne (and a lower relative abundance of N). Figure 5 also includes comparison samples of highredshift galaxy line ratios from early JWST spectroscopy: \nstacked CEERS measurements from Sanders et al. (2023) in the BPT and SMACS ERO galaxies from Trump et al. (2022) in the OHNO diagram. \nAt low redshift ( z /lessorsimilar 2), AGN typically have higher [N II]/H α , [O III]/H β , and [Ne III]/[O II] ratios due to harder ionizing radiation from the AGN accretion disk, and lineratio diagnostics shown in Figure 5 can be used to separate AGN from star-forming galaxies. However, high-redshift galaxies show systematic offsets relative to galaxies and AGN at z = 0, with higher ionization and lower metallicity in both AGN and from star-forming H II regions (Shapley et al. 2005; Erb et al. 2006; Liu et al. 2008; Kewley et al. 2013b,a; Sanders et al. 2023). Both CEERS 1670 and CEERS 3210 have high [O III]/H β , low [N II]/H α , and high [Ne III]/[O II] line ratios that are consistent with MAPPINGS V photoionization models for high ionization (log( Q / [cm s -1 ]) /similarequal 8) and moderately low metallicity ( Z / Z /circledot /similarequal 0 . 2 -0 . 4). \nThe AGN line ratios and interstellar medium conditions implied in Figure 5 are virtually indistinguishable from starforming galaxies observed at similar redshifts, since highredshift H II regions have similarly high ionization and low metallicity to these z ∼ 5 AGN narrow-line regions. Photoionization models show that low-metallicity AGN can have similar [O III]/H β and [N II]/H α ratios and lie within or even below the star-forming branch (Groves et al. 2006; \nFeltre et al. 2016). Although low-metallicity AGN are rare in the local universe (e.g., Storchi-Bergmann et al. 1998; Groves et al. 2006), recent simulations that make use of the AGN photoionization models presented in Feltre et al. (2016) predict that high-redshift, low-metallicity AGN should primarily occupy the top portion of the local starforming branch (Hirschmann et al. 2019, 2022), in agreement with our findings. The fact that neither source is Xray detected and that their BPT line ratios are similar to that of star-forming galaxies observed at the same redshift means that their broad-line emission may be one of the few ways to detect these high-redshift low-luminosity AGN. Other possible approaches include diagnostics with high-ionziation and extreme-ionization lines (e.g., He II and [Ne V]; Feltre et al. 2016; Nakajima & Maiolino 2022; Cleri et al. 2023). Preselection with photometric colors may also be useful to select fast-growing BHs with M BH ∼ 10 6 -7 M /circledot in metal-poor environments (Inayoshi et al. 2022b).', '5.2. Virial BH Mass Estimates': 'In this section, we estimate the virial BH masses of the two broad-line AGN assuming that their broad H α emission traces the kinematics of gas in the broad-line-region. The single-epoch BH mass estimation method is best calibrated against the width of the broad H β emission line and the restframe 5100 Å continuum luminosity ( L 5100) using the reverberation mapping technique (e.g., Kaspi et al. 2000). However, since we do not detect a broad H β component in our spectra, we instead employ the BH mass relationship proposed by Greene & Ho (2005, hereafter GH05), which relies entirely on H α emission. This method has been widely used in, for example, BH mass estimates for AGN in dwarf galaxies (e.g., Reines et al. 2013; Baldassare et al. 2015). This recipe is based on empirical correlations between Balmer emission-line luminosities and L 5100 and between the line widths of H β and H α . \nIn terms of the broad H α line width and L 5100, the BH mass formula is expressed as: \nM BH = 5 . 04 × 10 6 M /circledot ( L 5100 10 44 erg s -1 ) 0 . 64 ( FWHMH α 10 3 km s -1 ) 2 . 06 . (1) \nThis equation is based on the formula of Kaspi et al. (2000) for H β with the H β line width substituted with that of H α (Equation 3 of GH05). It is important to note that this equation assumes that the 5100 Å continuum luminosity is dominated by light from the AGN. Alternatively, we can directly apply the virial BH mass recipe of GH05, which is based on the broad H α line width and luminosity: \nM BH = 2 . 0 × 10 6 ( L H α 10 42 erg s -1 ) 0 . 55 ( FWHMH α 10 3 km s -1 ) 2 . 06 M /circledot . (2) \nFirst, we use the line width of the broad H α component detected in our NIRSpec spectroscopy, corrected for the R ∼ 1000 instrumental resolution, and L 5100 derived from the photometric SEDs to estimate the virial BH masses of CEERS 1670. Using Equation 1 results in a BH mass of M BH = 1 . 3 ± 0 . 4 × 10 7 M /circledot , with the Eddington ratio of L bol / L Edd = 0 . 15 ± 0 . 04. We use the bolometric luminosity inferred from L 3000 to be consistent with other z > 5 BH mass estimates in the literature. We apply a bolometric correction of L bol = 5 . 15 L 3000 (Richards et al. 2006) to derive L bol = 2 . 49 ± 0 . 04 × 10 44 erg s -1 . Using instead the H α line width and luminosity, Equation 2 yields M BH = 1 . 1 ± 0 . 3 × 10 7 M /circledot . This value is more systematically uncertain than our first estimate, although consistent within the 1 σ error, owing to potential slit losses (see Section 2). \nThe BH mass estimate for CEERS 3210 is complicated because of its potentially obscured nature. Taking the observed H α luminosity at face value and applying Equation 2, we derive a mass of M BH = 9 . 0 ± 2 . 2 × 10 6 M /circledot . We caution that this value is likely a lower limit since the H α emission is likely affected by dust extinction. If we assume that a dust-reddened AGN continuum dominates the observed rest-optical spectrum with AV = 4 (see Section 6.3), the inferred BH mass could be as high as M BH = 4 . 7 ± 1 . 2 × 10 7 M /circledot . A careful decomposition of the AGN/host components, and, if the AGN is dust-reddened, measurements of AGN continuum luminosity at rest-frame infrared wavelengths (Greene et al. 2014; Kim et al. 2015) are required to better estimate the intrinsic continuum luminosity and subsequently the virial mass for this AGN.', '6.1. The M BH -L bol Distribution': 'The successful spectroscopic identification of two lowluminosity broad-line AGN at z > 5 opens up a new parameter space for high-redshift AGN studies, thanks to the unprecedented infrared sensitivity of JWST and the multiwavelength photometric dataset available in the EGS field. Figure 6 shows the distribution of z /greaterorsimilar 5 AGN in the BH mass - bolometric luminosity plane with the two new lowluminosity AGN shown in red and orange. \nAs is discussed in O23, CEERS 1670 is 2-3 dex fainter than known quasars at z /greaterorsimilar 5 (e.g., Willott et al. 2010b; Trakhtenbrot et al. 2011; Shen et al. 2019; Onoue et al. 2019; Matsuoka et al. 2019; Kato et al. 2020) and more comparable to those of typical nearby AGN (e.g., Liu et al. 2019). The virial BH mass estimate we present above now shows that this low-luminosity AGN is by far the least-massive BH known in the universe at the end of cosmic reionization. The modest Eddington ratio of CEERS 1670 suggests that this AGN has been identified after its rapid accretion mode has \nTable 2. Derived AGN Properties \nNOTE-The BH mass for CEERS 1670 uses L 5100 estimated from the photometric SED and the line width of broad H α (FWHMH α , broad) (Equation 1), while for CEERS 3210 we use FWHMH α , broad and line luminosity of broad H α (Equation 2). The bolometric luminosity is also converted from L H α for CEERS 3210. In the third row, we show the case when CEERS 3210 is heavily dust-reddened with AV = 4. The H α luminosities are reported as observed, with no correction for potential slit losses. \nended, although it is possible the system will experience future bursts of heavy accretion (Li et al. 2022). \nFor CEERS 3210, if we use the observed H α luminosity without an extinction correction, then the BH powering this AGN may be comparably low-mass as CEERS 1670. However, if we assume heavy dust attenuation ( AV = 4), it becomes a BH accreting at a rate above the Eddington limit. In Figure 6, we show our results assuming both no extinction for the H α luminosity and AV = 4 with the bolometric luminosity converted from L 5100 estimated from the H α luminosity. Adopting a more moderate level of dust extinction inferred from the observed Balmer decrement in the NIRSpec spectrum (H α/ H β = 5 . 3; AV = 1 . 9), brings the bolometric luminosity of the source closer to the Eddington value. Thus, CEERS 3210 is likely in its most active mode of accretion and on the way to expelling the material that currently obscures it. Fujimoto et al. (2022) report a dust-reddened AGN at z = 7 . 19, the BH mass of which is estimated to be M BH /lessorsimilar 10 8 M /circledot based on the upper limit of its X-ray luminosity. Although not confirmed, their AGN and CEERS 3210 may be drawn from the same population of high-redshift dust-reddened AGN. We discuss this scenario in greater detail in Section 6.3 below.', '6.2. Constraints on the Host Galaxy Mass of CEERS 1670': 'Figure 3a shows the prism spectrum and NIRCam photometric flux densities of CEERS 1670. As discussed in Section 3, the continuum spectral shape can be explained by the low-redshift composite quasar spectrum of VB01. Since the observed spectrum is dominated by the central AGN contribution, it is challenging to estimate the stellar mass of the host galaxy in a plausible way. O23 conducted the SED fitting analysis for the photometric data using templates of metal-poor galaxies (Inoue 2011). The best-fit model with pure galaxy SEDs, where the quasar contribution is neglected, suggests a case with metallicity Z = 0 . 2 Z /circledot , stellar age 500 Myr, star formation rate (SFR) 3 . 6 M /circledot yr -1 , whose stellar mass is 1 . 8 × 10 9 M /circledot . This value is considered to be an upper bound of the stellar mass among the SED templates \nFigure 6. The BH mass - bolometric luminosity plane. Quasar samples at z ≥ 5 are shown as blue and green symbols and contours, while low redshift AGN are shown in black. CEERS 1670 and CEERS 3210 have BH masses 1-2 dex below that of known high redshift quasars and more comparable to those of typical nearby AGN. \n<!-- image --> \nO23 explored, but the true upper bound depends sensitively on the properties of the assumed stellar population. In the following, we give a robust upper bound of the stellar mass built up in the host galaxy at z /greaterorsimilar 5, assuming the SED model parameters that yield a high mass for the given stellar luminosity. \nOne advantage of focusing on z > 5 galaxies is that the stellar age is limited to the age of the Universe, e.g., t /similarequal 1 Gyr at z = 5 . 7. Although the star formation history (SFH) in the galaxy is unconstrained, the mass-to-light ratio ( M /star / L /star ) in the rest-frame optical and near-infrared band tends to increase with time (e.g., Bell & de Jong 2001); for instance, the M /star / L /star ratio in the B -band can be approximated as ∝ t \nat t ∼ 1 Gyr when a constant star formation rate (or decaying with a delay time) 5 is assumed (Into & Portinari 2013). Therefore, for the purpose of deriving an upper bound of the stellar mass, we adopt a characteristic time of t = 1 Gyr. We use the population synthesis code STARBURST99 version v7.0.1 (Leitherer et al. 1999) to generate stellar SEDs of galaxies. Here, we assume the Kroupa IMF (Kroupa 2001; 0 . 1 -100 M /circledot ), the Padova isochrone models, and constant star formation with a duration of 1 Gyr. We consider two values of stellar metallicity ( Z = Z /circledot and 0 . 2 Z /circledot ) to show the metallicity dependence, while we note that the solarmetallicity case gives a higher upper bound of the stellar mass. We take into account dust attenuation by the extinction law of starburst galaxies (Calzetti et al. 2000). The color excess of the stellar continuum is fixed to E s( B -V ) = 0 . 09, which is calculated from the Balmer decrement of the narrow emission lines we detect in the NIRSpec spectra (see Section 5.1). \nThis model, when scaled to the flux density in the F356W filter, results in a host mass of M /star = 6 . 0 × 10 9 M /circledot . This galaxy SED model is shown in Figure 3a as the red curve. Therefore, we argue that the stellar mass of the host galaxy is limited to M /star < 6 . 0 × 10 9 M /circledot for CEERS 1670 so that the stellar continuum flux density does not exceed the observed continuum level. We note that the upper bound depends significantly on the low-mass end ( m /star, min) of the stellar IMF; for instance, the upper bound is reduced by a factor of ∼ 3 for m /star, min = 1 . 0 M /circledot .', '6.3. The Obscured Nature of CEERS 3210': 'Figure 3b shows the prism spectrum and NIRCam photometric flux densities of CEERS 3210. The red spectral shape with an index of α λ /similarequal 2 . 0 at longer wavelengths can be explained either by a heavily obscured quasar (cyan) or a dusty starburst galaxy (red). Both models require the existence of obscuring material along the line of sight: a typical visual extinction of AV /similarequal 3 . 65 and 4 . 0 for the obscured quasar and dusty galaxy model, respectively. We note that this dusty-galaxy SED is calculated with the same galaxy model as discussed in Section 6.2, but assuming a stellar mass of M /star = 6 × 10 10 M /circledot and a different level of extinction. However, neither of the SED models explains the excess of the observed spectrum at λ obs /lessorsimilar 2 µ m, requiring additional blue components. \nOne possible explanation for the blue component is dust (and electron) scattering, which preserves the spec- \ntral shape of the intrinsic broad-line AGN component (e.g., Zakamska et al. 2005). In fact, obscured quasars at low redshifts ( z < 2 . 5) tend to show optical polarization levels higher than those of unobscured populations (Alexandroff et al. 2018). The fraction of the scattered flux relative to the primary component depends on the covering factor of the scattering medium and our viewing angle. For instance, assuming that 0.6% of the radiation flux of the intrinsic spectrum is scattered to our line of sight (see the Torus model in Polletta et al. 2006), the total SED is consistent with the photometric flux densities. Alternatively, the spectral shape of CEERS 3210 could be explained by the combination of quasar emission at short wavelengths and light from a heavily obscured starburst galaxy dominating at long wavelengths. This combination of AGN+galaxy light is shown as the blue curve in Figure 3b. If this is the case, CEERS 3210 would be caught in a transition stage, moving from a dust-obscured starburst to an unobscured luminous quasar by expelling gas and dust. This model hypothesis is consistent with the dustreddened AGN at z = 7 . 19 reported by Fujimoto et al. (2022), the BH mass of which is similar to that of CEERS 3210. This would make CEERS 3210 a dusty progenitor of the luminous, unobscured quasars observed by ground-based quasar surveys. \nWe can place a constraint on the host galaxy mass of CEERS 3210 following the same arguments used for CEERS 1670. Assuming the light at longer wavelengths is entirely dominated by stellar emission and using a dustobscured ( AV = 4 . 0) version of the stellar population described Section 6.2, we obtain an upper limit on the host mass of M /star /lessorsimilar 6 . 0 × 10 10 M /circledot . It is worth noting that the unobscured galaxy SED is modeled so that it has the highest M /star / L /star ratio, and thus our estimate gives a conservative upper bound. Using the hybrid quasar + dusty galaxy model does not appreciably change this upper limit as the steep spectral slope at λ obs > 3 µ m is dominated by the galaxy component in the second scenario. \nNevertheless, it is difficult to distinguish these two scenarios using the current data. Thus, multi-wavelength follow-up observations such as rest-frame infrared and farinfrared imaging are needed to further constrain the nature of CEERS 3210. We leave a more detailed SED analysis of this source to future work.', '6.4. BH-Galaxy Coevolution at z /similarequal 5': 'The empirical relation between the masses of SMBHs and their host galaxies is considered to be one of the most important outcomes of their mutual evolution over the cosmic timescale (e.g., Kormendy & Ho 2013). To constrain how and when the BH-galaxy correlations were established, the M /star -M BH distribution at the earliest epoch of the universe needs to be unveiled. The apparent location of high- \nFigure 7. The BH mass versus stellar mass relation of CEERS 1670 (red) and CEERS 3210 (orange; AV = 4). Circle symbols show the z > 6 quasar samples compiled by Izumi et al. (2021): brighter ones with M 1450 < -25 mag (blue) and fainter ones with M 1450 > -25 mag (cyan). The gray and green cross symbols are the observational samples in the local universe provided by Kormendy & Ho (2013) and Reines & Volonteri (2015), respectively. The diagonal dashed lines represent M BH / M /star = 0 . 1, 0 . 01, and 10 -3 . \n<!-- image --> \nz quasars and their hosts also gives us more information on the BH growth mechanisms and their seeding processes (Inayoshi et al. 2022a; Hu et al. 2022; Scoggins et al. 2023). \nOur first source, CEERS 1670, is a broad-line AGN with a BH mass of M BH /similarequal 1 . 3 × 10 7 M /circledot hosted in a star-forming galaxy with a stellar mass limited below M /star < 6 . 0 × 10 9 M /circledot . Our second source, CEERS 3210, is inferred to be a heavily obscured broad-line AGN with a BH mass of M BH /similarequal 4 . 7 × 10 7 M /circledot (or 9 . 0 × 10 6 M /circledot unless it is obscured). The host stellar mass is possibly as high as M /star /lessorsimilar 6 . 0 × 10 10 M /circledot in the case of the the hybrid quasar + dusty galaxy model. \nFigure 7 shows the M /star -M BH distribution of z /greaterorsimilar 6 quasars compiled in Izumi et al. (2021) (circle) for which the stellar mass is assumed to be the [C II]-based dynamical mass. CEERS 1670 is located at the left-bottom corner of the plane, which is uniquely separated from the z /greaterorsimilar 6 quasar population already known (e.g., Wang et al. 2013; Venemans et al. 2016; Izumi et al. 2021). The mass ratio of M BH / M /star > 2 . 4 × 10 -3 for CEERS 1670 is consistent with or higher than that expected from the empirical relation seen in massive galaxies at z = 0 (black line Kormendy & Ho 2013), but is overmassive compared to the BH-to-galaxy mass ratio measured for nearby broad-line AGN whose virial BH masses are estimated to be as low as that of CEERS 1670 (Reines & Volonteri 2015). On the other hand, adopting the dust-corrected BH mass and dusty-galaxy SED model, CEERS 3210 is located well below the empirical relation at z /similarequal 0. An important caveat is that the upper bound of the stel- \nar mass can be reduced by a factor of /similarequal 3 -5 with a different stellar population and star formation history (see discussion in Section 6.2). Further follow-up observations would give a better estimate of the stellar mass. The existence of such an overmassive BH, if confirmed, provides us with a unique opportunity to study the early stage of the BH-galaxy assembly.', '6.5. Update of z ∼ 5 AGN Luminosity Function': 'We update the UV luminosity function of z = 5 AGN from O23, based on the spectroscopic redshift of CEERS 1670. We do not include CEERS 3210 in our discussion, because of its unconstrained intrinsic UV luminosity. Following O23, we do not aim to provide statistical constraints on the luminosity function based on our small and incomplete sample, but we rather quantify the serendipity of our discovered low-luminosity AGN at z > 5 in the 34.5 arcmin 2 of the first NIRCam pointings of the CEERS survey. Adopting the spectroscopic redshift of z = 5 . 24 and the redshift interval of ∆ z ± 0 . 5, we derive the AGN number density of Φ =1 . 07 × 10 -5 Mpc -3 mag -1 at the UV magnitude of M 1450 = -19 . 4 mag. The difference from O23 ( Φ =1 . 03 × 10 -5 Mpc -3 mag -1 ) is tiny, because the central redshift of z = 5 . 24 only slightly changes from their work ( z = 5 . 15). The updated luminosity function is presented in Figure 8. \nThe faint end of the z > 5 AGN/quasar luminosity function is a matter of debate, because low-luminosity AGN produce more ionizing photons in a certain cosmic volume than do much rarer luminous AGN, and thus its steepness is critical to infer the relative role in the cosmic reionization with respect to star-forming galaxies (e.g., Giallongo et al. 2015; Onoue et al. 2017; McGreer et al. 2018a; Matsuoka et al. 2018; Giallongo et al. 2019; Finkelstein et al. 2019; Grazian et al. 2020, 2022; Niida et al. 2020; Kim et al. 2020; Kim & Im 2021; Jiang et al. 2022; Finkelstein & Bagley 2022a; Yung et al. 2021). The space density that we infer suggests that low-luminosity AGN such as CEERS 1670 may in fact be common, in agreement with previous studies that have identified candidate faint quasars in relatively small survey areas (e.g., Fujimoto et al. 2022). However, a complete survey of low-luminosity AGN with a well-defined selection function as well as a careful analysis of host galaxy contribution to the UV magnitudes (Bowler et al. 2021; Adams et al. 2022; Harikane et al. 2022) is required to statistically argue the AGN abundance at the faint end, and subsequently the relative contribution of AGN to the cosmic hydrogen/helium reionization.', '7. CONCLUSIONS': 'We make use of JWST NIRSpec spectroscopy from the CEERS Survey to identify two low-luminosity AGN at z > 5 with broad H α emission in their spectra. The first source, CEERS 1670 at z = 5 . 242, has a UV magnitude of M 1450 = \nFigure 8. The AGN luminosity function at z ∼ 5 based on CEERS 1670 (red). The 1σ errors have been derived using the low number count statistics by Gehrels (1986). The binned luminosity function from the literature are shown for AGN (McGreer et al. 2018b; Giallongo et al. 2019; Grazian et al. 2020; Niida et al. 2020) and Lyman break galaxies (Harikane et al. 2022; Bouwens et al. 2021). The short-dashed line represents the parametric luminosity function of Niida et al. (2020) and the long-dashed line is from Finkelstein & Bagley (2022b) without a correction term from a double-power law function (i.e., δ = 0 in their Equation 1). \n<!-- image --> \n-19 . 4 ± 0 . 05, making it 2-3 dex fainter than known quasars at similar redshifts. The source was previously identified as a candidate low-luminosity AGN based on broad-band photometry by O23. We measure a FWHM of 2038 ± 286 km s -1 for the broad H α component, resulting in a BH mass of M BH = 1 . 3 ± 0 . 4 × 10 7 M /circledot , making this the least-massive BH known in the universe at the end of cosmic reionization. \nThe second source, CEERS 3210 at z = 5 . 624, has a blue continuum spectrum at short wavelengths ( λ obs < 3 µ m) and a steep spectral slope at longer wavelengths. The SED shape suggests that this source is a broad-line AGN possibly caught in a transition phase between a dust-obscured starburst and an unobscured quasar. We measure a FWHM of 1807 ± 207 km s -1 for the broad H α component, resulting in a BH mass in the range of M BH /similarequal 0 . 9 -4 . 7 × 10 7 M /circledot , depending on the level of dust obscuration assumed. \nWe derive upper limits on the host mass of each AGN and place constraints on the M /star -M BH relationship in the lowest mass range yet probed in the early universe. We find the host of CEERS 1670 is limited to M /star < 6 . 0 × 10 9 M /circledot , while the host mass of CEERS 3210 can be an order of magnitude larger (6 . 0 × 10 10 M /circledot ) if we assume a visual extinction of AV =4 . 0, as inferred from our SED fitting. The M BH / M /star ratio for CEERS 1670, in particular, is consistent with or higher than the empirical relationship seen in massive galaxies at z = 0, but is overmassive compared to the BH-to-galaxy mass \nratio measured for nearby broad-line AGN whose virial BH masses are estimated to be as low as that of CEERS 1670. \nWe examine the narrow emission-line ratios of both sources and find that their location on the BPT and OHNO diagrams is consistent with model predictions for moderately low-metallicity AGN with Z / Z /circledot /similarequal 0 . 2 -0 . 4. The fact that neither source is X-ray detected and their emission line ratios in the BPT diagram are virtually indistinguishable from star-forming galaxies observed at similar redshifts means that their broad-line emission may be one of the few ways to detect these AGN. Other possible approaches include diagnostics with high-ionziation lines (e.g., He and Ne) (Feltre et al. 2016; Nakajima & Maiolino 2022). Preselection with photometric colors may also be useful to select fast-growing BHs with M BH ∼ 10 6 -7 M /circledot in metal-poor environments (Inayoshi et al. 2022b). \nThe spectroscopic discovery of two low-luminosity, lowmass AGN at z > 5 demonstrates the capabilities of JWST to push the BH mass limit closer to the range predicted for the BH seed population. Future work to uncover these lowluminosity AGN, which are the dominant BH population at high redshift, will be the key to further constraining their abundance and the early growth history of SMBHs and their host galaxies.', '8. ACKNOWLEDGMENTS': "This work is supported by NASA grants JWST-ERS-01345 and JWST-AR-02446 and based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. This work also made use of the Rainbow Cosmological Surveys Database, which is operated by the Centro de Astrobiología (CAB/INTA), partnered with the University of California Observatories at Santa Cruz (UCO/Lick,UCSC). \nWe also acknowledge support from the National Natural Science Foundation of China (12073003, 12150410307, 12003003, 11721303, 11991052, 11950410493), and the China Manned Space Project Nos. CMS-CSST-2021A04 and CMS-CSST-2021-A06. PGP-G acknowledges support from Spanish Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 through grant PGC2018-093499-B-I00. \nAG acknowledges financial contribution from the grant PRIN INAF 2019 (RIC) 1.05.01.85.09: 'New light on the Intergalactic Medium (NewIGM)' and support from PRIN MIUR project 'Black Hole winds and the Baryon Life Cycle of Galaxies: the stone-guest at the galaxy evolution supper', contract 2017-PH3WAT."}

2024ApJ...977...87D

We present a weak lensing analysis of the galaxy cluster A2390 at z 0.23 using second moment shape measurements made in 411 short 60 s exposures. The exposures are obtained in three broadband photometric filters g r and i using WIYNODI. Shape measurement in individual exposures is done using a momentmatching algorithm. Forced measurement is used when the momentmatching algorithm fails to converge at low signaltonoise ratio. The measurements made in individual images are combined using inverse error weighting to obtain accurate shapes for the sources and hence recover shear. We use PhoSim simulations to validate the shear measurements recovered by our pipeline. We find the mass of A2390 is in agreement with previously published results. We also find the Emode maps show filamentary structures consistent with baryonic structures and recover most clustersgroups of galaxies found using optical and Xray data. Thus we demonstrate the feasibility of using weak lensing to map largescale structure of the Universe. We also find the central portion of the cluster has a bimodal mass distribution and the relative orientation of the peaks is similar to Xray. We discuss earlier research on this galaxy cluster and show that a latestage merger accounts for all the observed data.

2024-12-01T00:00:00Z

['2024arXiv240912119D', 'arXiv:2409.12119', '10.48550/arXiv.2409.12119', '10.3847/1538-4357/ad7c4f', '2024ApJ...977...87D']

['Weak gravitational lensing', 'Galaxy clusters', '1797', '584', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

Weak Lensing Analysis of A2390 Using Short Exposures

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

https://arxiv.org/pdf/2409.12119.pdf

{'Weak Lensing analysis of Abell 2390 using short exposures': "A.Dutta , 1 J.R.Peterson, 1 T. Rose, 2 M. Cianfaglione, 3 A. Bonafede, 3 G. Li, 4 and G. Sembroski 1 \n1 Department of Physics and Astronomy Purdue University West Lafayette, Indiana, USA 2 Waterloo Center for Astrophysics University of Waterloo Waterloo, Ontario, Canada 3 Department of Physics and Astronomy University of Bologna Bologna, Italy 4 Purple Mountain Observatory West Bejing Road, Nanjing 210008 People's Republic of China", 'ABSTRACT': 'We present a weak lensing analysis of the galaxy cluster Abell 2390 at z = 0.23 using second moment shape measurements made in 411 short 60s exposures. The exposures are obtained in three broadband photometric filters (g, r, i) using WIYN-ODI. Shape measurement in individual exposures is done using a moment matching algorithm. Forced measurement is used when the moment matching algorithm fails to converge at low signal to noise ratio (SNR). The measurements made in individual images are combined using inverse error weight to obtain accurate shape of sources and hence recover shear. We use PhoSim simulations to validate shear measurements recovered by our pipeline. We find the mass of Abell 2390 is in agreement with previously published results. We also find the E-Mode maps show filamentary structures consistent with baryonic structures and recovers most clusters/groups of galaxies found using Optical and X-Ray data. Thus we demonstrate the feasibility of using Weak Lensing to map large scale structure of the universe. We also find the central portion of the cluster has a bimodal mass distribution and the relative orientation of the peaks are similar to X-Ray. We discuss earlier research on this galaxy cluster and show that a late stage merger accounts for all the observed data. \nKeywords: Weak lensing, Galaxy Clusters', '1. INTRODUCTION': 'Galaxy clusters are the largest bound structures in the Universe containing a few hundred to a few thousand individual galaxies. They play an important role in our understanding of the Universe and hierarchical structure formation (Press & Schechter 1974; Bond et al. 1991; Lacey & Cole 1993; Kravtsov & Borgani 2012). Mass profiles of galaxy clusters are fundamental to understanding structure formation and constraining cosmological models (Holder et al. 2001; Cunha et al. 2009). However, the measurement of this is made challenging by the fact that only about 5% of the total mass of galaxy clusters emits light in the visible spectrum. About 15% of the mass emits light in the form of ionized X-Rays and the remaining 80% is expected to be in the form of dark matter (Fukugita et al. 1998). \nDifferent methods of measuring masses of galaxy clusters exist. X-Ray observations of hot ionized ICM are expected to trace the general mass profile of a cluster. This method has been widely used along with assumptions of hydrostatic \nCorresponding author: Anirban Dutta dutta26@purdue.edu \nequilibrium and/or a mass profile such as NFW (Navarro et al. 1995; Evrard et al. 1996; Reiprich & Bohringer 2002; Ettori et al. 2013) to measure the mass of galaxy clusters. These assumptions limit our ability to make accurate measurements since clusters are always in a dynamical state (Piffaretti & Valdarnini 2008). These include cooling flows and AGN feedback (Fabian 1994; Gitti et al. 2012), mergers with other clusters and smaller infalling groups. This is expected since clusters lie at the major intersections of the filamentary structure of the Universe (van Haarlem & van de Weygaert 1993). Another method of measuring cluster masses is with the velocity dispersion of the cluster members along with the assumption of virial equilibrium (Zwicky 1933, 1937; Smith 1936; Danese et al. 1980). However, a significant majority of the clusters are in a dynamical state where the assumption of virial equilibrium might not hold. The Sunyaev-Zeldovich effect (Sunyaev & Zeldovich 1972, 1970, 1980; Carlstrom et al. 2002) has also been used to measure the mass of galaxy clusters. However, this approach is limited by the high quality of data required, mass-observable scaling relation, and dynamical state of the cluster (Krause et al. 2011; Dietrich et al. 2018). \nWeak lensing provides another independent method to measure mass. In weak lensing, coherent distortions in background galaxies (Tyson et al. 1990; Kaiser et al. 1995; Squires et al. 1996) are used to infer mass. Unlike other methods, weak lensing makes no assumption about the state of the cluster and directly probes mass. It is equally sensitive to both baryonic matter and dark matter(DM), making it an ideal tool to directly probe DM distribution in the Universe. Several excellent weak lensing studies on galaxy clusters (Postman et al. 2012; von der Linden et al. 2014; Fu et al. 2022) including Abell 2390 (Squires et al. 1996; Okabe et al. 2010; Umetsu et al. 2009) have been conducted. These have shown masses inferred from weak lensing are consistent with X-ray mass estimates, especially for cool core clusters. For merging or disturbed clusters, mass distributions inferred from X-Ray and weak lensing are significantly different (Clowe et al. 2006). In such cases, weak lensing measurements of mass profiles are expected to be more accurate. These differences between baryonic and DM interactions during mergers can eventually be used to understand and put constraints on DM cross-section. \nIn weak lensing, measuring the shapes of galaxies accurately is of great importance. Traditionally, weak lensing studies have used shape measurements of galaxies made in the co-added image. However, images are obtained under a wide range of seeing conditions, cloud cover and background brightness. Depending on the science requirements, different types of weighted co-add (Zackay & Ofek 2017) are necessary for optimal measurements. It has been suggested that measuring sources in individual images would lead to better measurements since individual images can be weighted optimally to extract better shape information (Tyson et al. 2008; Jee & Tyson 2011; Mandelbaum 2018). Using measurements made in individual exposures avoids information lost in the process of co-addition and allows for small spatial scale photometry and PSF variation to be taken into account. This allows for better PSF correction which in turn leads to better shear recovery. This idea has been used in more recent weak lensing studies such as Miller et al. (2013), Zuntz et al. (2018). However, measuring sources in individual images is challenging since a vast majority of the sources are extremely faint and undetectable. We use forced measurement, a generalization of forced photometry, proposed by Dutta et al. (2024) to measure such low SNR sources. Forced measurement uses reasonable initial guess values of photon counts, shape and size of the source being measured to produce values of photon counts, shape and size that are statistically closer to the true values. The advantage of forced measurement is that it allows for the measurement of sources with S/N as low as 0.1. This allows us to measure all sources in individual images. \nIn this paper, we perform a weak lensing analysis of the galaxy cluster Abell 2390, a rich and massive galaxy cluster at z=0.23 (Abraham et al. 1996). It is a bright X-ray source and shows several arcs due to strong lensing. Abell 2390 have been extensively studied in X-Ray (Pierre et al. 1996; Allen et al. 2001; Martino et al. 2014; Sonkamble et al. 2015), optical (Abraham et al. 1996; Hutchings et al. 2002), radio (Bacchi et al. 2003; Augusto et al. 2006; Sommer et al. 2016; Savini, F. et al. 2019), weak lensing (Squires et al. 1996; Umetsu et al. 2009; Okabe et al. 2010; von der Linden et al. 2014) and strong lensing (Narasimha & Chitre 1993). In X-Ray studies (Allen et al. 2001) Abell 2390 shows hints that it has not relaxed since its last merger. Radio studies such as (Rose et al. 2024a) and (Sommer et al. 2016) have found evidence of gravitational disturbance and turbulence in Intra Cluster Medium (ICM) respectively. Weak Lesning studies have not reported any signs of merger. The paper is organized as follows. In Section 2, we describe our data and the processing pipeline. We describe strategies for co-adding our data to maximize source detection. We then measure the detected sources using a moment matching algorithm and present a novel Monte Carlo Point Spread Function (PSF) correction scheme. We describe in detail how the forced measurement algorithm is used to measure sources in individual exposures. In Section 3, we use PhoSim (Peterson et al. 2015, 2019, 2020, 2024; Burke et al. 2019) to simulate 60 one-minute exposures to show our pipeline recovers shear accurately. Two different cases with different values of input shear are tested. In Section 4, we present our results and compare them \nwith previous X-Ray and Radio data. We show the mass structures we recover from weak lensing correspond to an excess of light density in most regions. We also present a scheme to perform 3D mass reconstruction. In Section 5, we discuss in detail the previous studies and their results conducted on Abell 2390 using X-Ray, Radio and Optical Data. Finally, in Section 6, we show that a late stage merger scenario provides a reasonable explanation for all observed data.', '2.1. Data Acquisition': "Data for this analysis is obtained using WIYN 3.5m telescope at Kitt Peak in Arizona. The diameter of the primary mirror is 3.5m. For this study, we use the ODI instrument (Harbeck et al. 2014; Harbeck et al. 2018). It has a wide field of view, approximately 40 ' x 48 ' . The focal plane of the instrument is populated with 30 Orthogonal Charge Transfer Arrays CCDs (OTAs) in 5 x 6 configuration and has a pixel scale of 0 . 11 '' pixel -1 . The OTAs are a kind of CCD where charges can be moved orthogonally during exposure to increase image sharpness (Harbeck et al. 2018). However, this feature was not available during our imaging sessions. Each OTA is made of 8 × 8 pixel cells. Each pixel cell is 480 × 496 pixels. In this document, we use chips to refer to pixel cells. We record images in 5 separate wide-band photometric filters u, g, r, i, and z over 6 observing runs from 2017 to 2023. The wide-band photometric filters of ODI are very similar to those used by other survey telescopes such as Sloan Digital Sky Survey (Gunn & Weinberg 1994), PanSTARRS (Kaiser et al. 2002) and DES (The Dark Energy Survey Collaboration 2005). The data obtained in the 2022 observing run was unfortunately lost due to corrupt header and metadata values. The issue is likely linked to the damage caused by the forest fires on Kitt Peak in the Fall of 2022. In total, we lost 90 images, each of 60s exposure, due to this issue. We also note one OTA, specifically OTA12 was unavailable/non-functional in 2023. Over the course of 2017 to 2023 a few CCD segments became non-functional as well. In Table 1 we show the total number of exposures, the PSF size of the co-add, and the depth of the co-add in each filter. The PSF size is defined as the median size of stars obtained using the moment matching algorithm described in Dutta et al. (2024). The depth is defined as the AB magnitude brightness of the finest source detected by SExtractor after the cuts mentioned in Section 2. We also show the number of exposures obtained in each filter in each year. \nRejecting the bad quality/unusable data, seeing conditions ranged from excellent (0 . 6 '' ) to poor (3 '' ). The poorest seeing data was obtained in the 2023 observing run which had a median seeing of 2 '' . The normalized histogram of seeing in all images for the i, r, g, and z band is shown in Figure 1. The median seeing in g, r and i bands is around 1 . 0 '' . We used the default 9-Point dithering pattern to fill the CCD gaps. In this process, the telescope pointing is shifted slightly by a pre-determined amount nine times to fill gaps between CCD and ensure uniform coverage of the field. Exposure for each image is 60s. This exposure time is reached as a compromise between short exposure time required for weak lensing (a few seconds ideally) and the 40-50 sec readout time for ODI detectors (Chang et al. 2012; Peterson et al. 2015). Short exposures are important for weak lensing since it allows one to reject or de-weight periods of bad seeing. Overall, we have 720 minutes of data divided roughly equally in each photometric band. The exposures obtained in 2017, 2020 and 2021 were on grey nights. For 2019 the exposures were obtained on a bright night and hence show significantly high background. In 2023, exposures were obtained on a dark night. We visually examined all images and rejected 36 images where satellite trail or wind shake affected the image significantly. \nTable 1. Observing runs (1) Photometric filter (2) corresponding exposure time, (3) PSF size of coadd. This is defined as the median size of stars when they are measured using the moment matching algorithm described in Dutta et al. (2024) (4) approximate depth of the coadd. The depth is defined as the AB magnitude brightness of the faintest source detected by SExtractor after the cuts mentioned in Section 2. The 5 right columns show the number of exposures in each filter obtained in the corresponding year. The missing years i.e. 2018 and 2022 correspond to time lost due to bad weather and technical issues. All exposures are 60s long", '2.2. Correction of systematics and defects': "All raw data is processed using QuickReduce (Kotulla 2013). QuickReduce, the default WIYN data processing pipeline, corrects for common systematics such as saturation and non-linearity, persistency effect, dark subtraction, flat fielding, fringing, removal of cosmic rays, and photometric calibration. Photometric calibration is done using the Pan-STARRS (Kaiser et al. 2002; Chambers et al. 2016) catalog for g, r, i and z bands while the u band is calibrated using SDSS (Gunn & Weinberg 1994) catalog. WCS match with Gaia (Brown et al. 2016) catalog is performed to obtain the final calibrated images. However, QuickReduce is not able to perfectly correct for systematics due to small but noticeable degradation of CCDs. As mentioned previously, a few CCD segments and one OTA gradually degraded and eventually became non functional over the course of 2017 to 2023. The changing nature of detector properties are hard to correct for. This was found to be true for the QuickReduce pipeline as well. This caused a noticeable amplifier glow, non-Gaussian background, unusually high or low pixel values, and cross-talk in the calibrated images. Below we discuss each of these issues in detail and how we correct for them. \nFigure 1. Normalized histogram of seeing (in arcseconds) for 151 r band, 146 i band, 161 g band and 141 z band images. We excluded the u band since the image quality is extremely poor and only a few stars are detectable in the entire image. The seeing values are the FWHM size of the stars as calculated by QuickReduce. \n<!-- image --> \nDuring visual examination of the calibrated images, it was found some chips had correlated noise. An example of this is shown in Figure 2. Such correlated noise leads to several spurious sources in the resulting co-add when employing SExtractor (Bertin & Arnouts 1996) for source detection. This limits our ability to detect faint sources. We decided to de-weight these chips during the co-add process instead of completely rejecting them. We use a weight scheme similar to Annis et al. (2014) \nW = 100 10 Z -25 ( Sσ b ) 2 (1) \nFigure 2. (a) On the left an example of background from a typical chip is shown. On the right, the background from chips with correlated noise is shown. Correlated noise gives rise to spurious sources, especially at the faint end. (b) An example of crosstalk caused by very bright sources. This image shows a portion of the OTA. QuickReduce is unable to get rid of all cases of cross-talk arising from extremely bright sources. Co-adding reduces them to a non-detectable level. However, some of the severe cases are apparent in the co-add. The vertical line of nan's through the bright source is evident. \n<!-- image --> \n<!-- image --> \n(a) \n<!-- image --> \n(b) \nwhere Z = Zero point of the image, W denotes weight, average seeing in an image is denoted by S , and σ b is the background variance calculated using the k=3 sigma clipping algorithm in astropy (Astropy Collaboration et al. 2013a). This weight scheme de-weights the chips with correlated noise since those chips have higher background variance. We note that we do not completely understand the origin of such noise, but this may simply be a form of pink noise from the electronics (Hirata et al. 2024). \nIt was noticed that some images show significant cross-talk patterns. Due to this, pixel values are significantly higher or lower than the background in certain chips adjacent to bright sources. An example of this is shown in Figure 2b. To correct for the lower pixel values we find the lower of 6 standard deviation smaller than the background median and 0 and reject any pixels below this limit. The higher pixel values are harder to correct for since they appear intermittently and mimic real sources. Co-adding described below was able to get rid of most effects due to cross-talk. Any remaining effects are visually identified in the co-add and those regions are masked during source detection. It was also noticed some chips have background variation significantly higher than expected from Poisson statistics. Most of these chips have imperfections and limit our ability to detect extremely faint sources in the co-add. The problem is severe enough that the weighting scheme and cuts presented above were not able to effectively de-weight these chips. To exclude these chips we perform cuts in the background median vs variance space. We use the median instead of the mean because it was found to be much more stable. A graph showing the distribution of median vs variance of the background for all chips in g, r, and i band is shown in Figure 3. Both background median and variance are obtained \nafter k=3 sigma clipping. We consider the condition \n0 . 9 × Median ≤ Variance ≤ Median + 300 (2) \nwhere variance is the background variance and median is the background median obtained after k=3 sigma clipping. If a chip violates this condition in more than half the images, the chip is flagged problematic and its weight is set to 0 for all images. This condition effectively rejects the problematic chips. \nFigure 3. Median of background photon counts vs variance of background photon counts for all chips in every image in g, r, and i band. Shown in red is the y = x + 300 while the black line is y = x. The yellow line shows y = 0.9x line. If a chip is outside the strip which is defined by the red and yellow lines in over half the images, the chip is rejected. This condition was found to be effective in identifying severely defective chips. \n<!-- image -->", '2.3. Co-adding and Source Detection': "The co-adds were performed for each of the five photometric bands separately using the software SWarp (Bertin et al. 2002). SWarp settings used are listed in Table 2. The i and r band images were co-added further. The color image of the entire field is shown in Figure 4. The i+r coadd image was inspected visually for any additional imperfections. A few regions of defects were visually identified. Most of these defects come from clumps of unusually high pixel values. We do not understand the origins of these pixels but they tend to appear at the chip edges or in chips with pre-existing defects. There is also a small amount of cross-talk noise from a few extremely bright stars in the field. We mask out these regions to limit spurious detection. The total percentage of area masked out was less than 2%. This is the final image used for source detection. We employed SExtractor (Bertin & Arnouts 1996) to detect sources. The SExtractor settings used is shown in Table 2. These settings were found to give us the maximum number of sources without contaminating the sample with too many spurious detections. The details of the number of images co-added and the \nFigure 4. RGB Color image of the full field containing the galaxy cluster Abell 2390. We use the co-added images in z, i and r band. They respectively correspond to R, G and B channels. The field of view is approximately 43 ' x 49 ' . \n<!-- image --> \ndepth and PSF size of the final image are shown in Table 1. The final number of sources detected in the i+r co-added image is 54059 and density is ∼ 31.8 arcminute -2 .", '2.4. Co-add Measurement and Source Classification': "The sources detected in the previous step are measured using the moment matching algorithm developed originally by Kaiser et al. (1995) and later modified by Bernstein & Jarvis (2002). We use an adaptive moment matching algorithm with elliptical Gaussian weights. The shape and size of the weight is the best fit elliptical Gaussian for the source being measured. The weighted second moments are calculated as \nQ ij = ∫ θ i θ j f ( θ i , θ j ) W ( θ i , θ j ) dθ i dθ j (3) \nTable 2. List of SWarp parameters used on the left and list of SExtractor parameters used on the right. \nwhere θ represents generalized co-ordinates, W ( θ i , θ j ) is the weight function and f ( θ i , θ j ) is the cutout of the source. The polarization or ellipticity parameters are then defined as \ne 1 = Q 11 -Q 22 Q 11 + Q 22 e 2 = 2 Q 12 Q 11 + Q 22 (4) \nwhere e 1 represents elongation along x (positive values) and y axes (negative values) and e 2 represents elongation along y = x or y = -x lines. Traditionally the quantities Q 11 , Q 22 and Q 12 is referred as σ 2 xx , σ 2 yy and σ 2 xy respectively. This algorithm was modified to produce the photon counts, centroid, size of the source along with ellipticity. We chose the best-fit elliptical Gaussian as our weight function. For a detailed description of this algorithm see Dutta et al. (2024). \nThe measurements are done in the i+r co-add image. In the first step, the optimal cutout size of each source was determined. To estimate the optimal cutout size, a rough guess of size from SExtractor, S sextractor is made. \nS sextractor = √ X 2 IMAGE + Y 2 IMAGE (5) \nwhere X 2 IMAGE is the isophotal image second order moment in x and similarly Y 2 IMAGE . The second order moments are obtained from the SExtractor output file. This estimated size is set at 25% larger than SExtractor size. The lower limit of estimated size was capped at 4 pixels. Next a square cutout with side length 8 times the estimated size is made to measure sources. It was found that in rare cases the moment matching algorithm fails when it is run on such a cutout. However, the algorithm successfully converges when a smaller cutout is made. This happens typically due to light contamination from other nearby sources, leading to gross mis-estimation of background. For cases where the algorithm initially fails, we reduce the size of the cutout by 2 pixels along the x-axis and 2 pixels in the y-axis symmetrically from each side. This is repeated until the algorithm successfully converges or the cutout is less than or equal to 24x24 pixels. This is the minimum size of cutout needed to measure a source. Using this method we obtain a good estimate of the optimal cutout size. \nNext, we attempt to reduce the effect of light contamination from nearby sources, thus increasing the accuracy of our measurements. First, we calculate the bound flag ( bndflag ) using the previous measurements. Flags are calculated using the method outlined in Section 2.6. If the bound flag is raised, it indicates that there is a significant chance that the background estimation might be affected by nearby sources which in turn affects the flux estimation. To correct for this we use two strategies: \n- 1. The cutout is moved by 3 pixels in a direction 180 ° away from the nearest source\n- 2. During the iterative process of moment matching, the background is fixed. To determine the value of background we select three 4x4 pixel regions. These regions are s/ √ 2 away from the central pixel of the cutout, where s is the side length of the cutout. The three regions selected are 90 ° , 180 ° and 270 ° away from the line joining the \nsource to its nearest source. We also create a 4x4 array of zeros. The array of zeros was to ensure the background value does not wander too far from the truth which is very close to 0 in the co-adds. The median value of these 64 pixels in four 4x4 cutouts is the fixed background value. \nIf the bound flag is not raised, we shift the cutout 3 pixels away from the nearest source and re-measure it using the moment matching algorithm. It was found that the amount of shift is not important as long as the shift is small compared to the size of the cutout. We repeat the same measurement process for the co-add in each of the 5 photometric bands. The flux vs size graph from the i+r co-add band measurements for all sources is shown in Figure 5. The vertical column feature arises from stars which have approximately fixed size (size of the PSF) but vary in magnitude. Having a clean sample of stars to measure PSF from is the first step in accurate PSF estimation. We cross-referenced all sources in our field with the star catalog from Gaia EDR3 (Brown et al. 2021). The criteria used for matching was the difference in WCS coordinates of the source in our catalog and Gaia catalog be less than 0 . 72 '' . The sources that were matched are shown as red points in Figure 5a. This gives us an extremely clean sample of stars to determine PSF. \nWhile stars are useful for determining PSF, they are not useful for weak lensing purposes and only serve to dilute the shear signal. Hence an important aspect of any weak lensing pipeline is accurately rejecting stars. In our case rejecting only the stars cross-matched with Gaia EDR3 is not enough. Gaia EDR3 goes to a depth of magnitude 21 in the g band while our i+r co-added image has a depth of magnitude 26. The mismatch in the depth of the two catalogs means that several fainter stars in our field cannot be cross-matched with the Gaia catalogs. Hence, we visually identified the vertical star strip shown as a black box in Figure 5a. Any sources inside the black box were rejected from weak lensing analysis. We note that this method is not perfect and some fainter stars outside the black bounding box enter our galaxy sample. To reject extremely bright sources that show brighter fatter effect, any sources brighter than magnitude 15 shown by the horizontal yellow line were rejected. We also reject any sources with size ≥ 12 pixels. These most likely arise from severe blending. This value was selected somewhat arbitrarily and changing this limit does not significantly affect our analysis. We also wish to reject sources that have size 3 ϵ PSF below the median PSF value, where ϵ PSF is the variation of PSF in the image. This is shown by the yellow line on the left in Figure 5a. All sources left of this yellow line are rejected. A method to determine ϵ PSF is described in the next section. These are likely spurious sources arising from cosmic rays or defective pixels. After these cuts, we have 45022 sources remaining corresponding to a source density of 26.5 arcminute -2 . \nTo calculate photometric redshift we use the magnitude brightness measured in the co-add of the five photometric bands. We use these as inputs to EAZY (Brammer et al. 2008) to determine photometric redshift. A few random galaxies across our field were selected and the photometric redshift was compared to the spectroscopic redshift obtained from the NASA Extragalactic Database (NED). These were generally found to be in good agreement.", '2.5. PSF correction': "PSF causes dilution in the ellipticity and shear signals and hence needs to be corrected. It has also been shown PSF variation across the image can masquerade as shear signal if not taken into account. PSF correction is done in two steps. First, we interpolate the PSF across the image and then in the second step we correct the shape and ellipticity dilution due to PSF. In any astronomical image, PSF can only be precisely determined only at locations where stars are present. To determine the exact PSF shape at other locations a variety of methods can be found in literature. Van Waerbeke et al. (2002) used second order polynomial to model PSF variation across an image. Other elaborate methods have been proposed by Hoekstra (2004), Berg'e et al. (2011) and Chang et al. (2012). We use an inverse distance weight interpolation, similar to one described by Gentile et al. (2012) because of its simplicity and performance. To find σ 2 xx ( PSF ) we use \nσ 2 xx ( PSF ) = n ∑ i =1 σ 2 xx,i w ( i ) (6) \nwhere σ 2 xx,i is the measured Q 11 of the i-th star obtained from the moment matching method, and n is the number of stars used. The weight w(i) is defined as \nw ( i ) = 1 /d i ∑ i 1 /d i (7) \n<!-- image --> \nFigure 5. (a) Magnitude vs size for sources detected and measured in the i + r co-add. The points shown in red are stars matched with the Gaia EDR3 catalog. All sources inside the black box are rejected from the weak lensing analysis pipeline. So are the sources above the horizontal yellow line (15th magnitude) to account for brighter fatter effect. Sources to the right of the vertical yellow line are rejected to avoid blended sources and artifacts. Sources to the left of the curved yellow line are also rejected since these are 3 standard deviations smaller than the median PSF.(b) The photon counts vs size graph for the i+r co-added image simulated using PhoSim for the first case of γ 1 = 0 . 1. The points in the red box are used for PSF interpolation. The points inside the black box are not used for the shear analysis. The condition for the red box is 3 . 3 < size < 3 . 8 pixels and 500 < counts < 10 5 . For the black box the conditions are 2 . 5 < size < 4 . 5 pixels and 10 2 < counts < 10 6 . \n<!-- image --> \nwhere d i is distance of the i-th star from the source. We found the number of stars selected for PSF correction in the range n=5 to n=25 does not affect our analysis significantly. We selected n = 15. Similar process is followed for interpolating σ 2 xy ( PSF ) and σ 2 yy ( PSF ). This completes PSF interpolation. \nA significant portion ( ∼ 35%)of these sources are measured to have a size smaller than PSF. Clearly, these measurements are un-physical. Traditionally sources smaller than PSF have been rejected (Gruen et al. 2013; Applegate et al. 2014; McCleary et al. 2020). However, these sources contain a large amount of information which would be lost if we reject these sources. The measurement of size being smaller than PSF is simply due to statistical fluctuations. In other words, the size of both PSF and sources have Poisson noise and other systematic effects which give rise to size measurements that are smaller than PSF. We introduce PSF correction using a novel Monte Carlo method. If we assume an elliptical Gaussian profile for both the source and PSF, true σ 2 xx of the source is \nσ 2 xx ( true ) = σ 2 xx ( measured ) ± √ S 4 N (1 + K ) -σ 2 xx ( PSF ) ± ϵ ( xx ) PSF (8) \nwhere σ 2 xx ( true ) is the true value , σ 2 xx ( measured ) is the measured value, σ 2 xx ( PSF ) is the value for PSF and ϵ ( xx ) PSF is the error in PSF value. The expression √ S 4 (1 + K ) /N is the Poisson error in σ 2 xx ( measured ) as shown by Dutta et al. (2024), where S is measured size, N is measured photon counts and K for elliptical Gaussian case is 4 πS 2 B/N . B is the background level. In order to find the error in PSF, ϵ ( xx ) PSF we use the bright stars where the Poisson error is negligible. For the co-adds it was found the stars matched with the Gaia catalog are bright enough to neglect the Poisson error. This is also clear from Figure 5a where the red points show a constant width. This width corresponds to PSF variation due to turbulence and other systematics. We perform a k=3 sigma clip for σ 2 xx ( measured ), σ 2 yy ( measured ) and σ 2 xy ( measured ) to reject any stars with significant blending. Stars brighter than magnitude 15.75 are rejected to avoid brighter fatter effect. This threshold is a factor of 2 lower than the threshold mentioned in the previous section. This is to ensure the star sample is well away from the limits at which brighter fatter becomes important. We also reject stars with bflag or vflag raised. The calculation of flags is presented in Section 2.6. The other flags are not considered since the stars matched with Gaia is fairly bright. Of the remaining stars, we find P 84 -P 16 for σ 2 xx , where P i represents the i-th percentile. This value corresponds to two standard deviations i.e. one standard deviation from the median value on either side. Half of this value is defined as ϵ ( xx, Turbulence ) PSF . This gives only the typical \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 6. (a) The points show the σ 2 xx of stars in the i+r co-added image. The red curve shows the best fit Gaussian obtained with a width half the value of P 84 -P 16 where P i represents the i-th percentile value. The fits are reasonably good though not perfect. (b) and (c) shows similar graphs for σ 2 yy and σ 2 xy . The widths of the Gaussians are 0.37, 0.27 and 0.08 respectively. (d) The points show the σ 2 xx of stars in the i+r co-added image simulated using PhoSim for the first case i.e. γ 1 = 0 . 1. The red curve shows the best fit Gaussian for each case. (e) and (f) show similar graphs for σ 2 yy and σ 2 xy . The width of the Gaussians are 0.18, 0.22 and 0.06 respectively. \n<!-- image --> \nerror across the frame due to turbulence. In Figure 6a, 6b and 6c the histograms of stars used for PSF interpolation are shown. The red curve is a Gaussian centered on the median with a width equal to ϵ ( xx, Turbulence ) PSF . \nTo the turbulence error, we add the Poisson error of the stars used for PSF interpolation. Using the errors presented in (Dutta et al. 2024) and standard error propagation, the PSF error due to Poisson noise is \nϵ 2 ( xx, Poisson ) PSF = n ∑ i =1 w 2 ( i ) ( S 4 ( i ) N ( i ) + 4 πBS 6 ( i ) N 2 ( i ) ) 2 (9) \nwhere S(i) is the size of the i-th star and N(i) is the photon counts of the i-th star. The Poisson and the Turbulence component are added in quadrature to find ϵ ( xx ) PSF . It was found the turbulence component always dominates in the co-adds and is an order of magnitude larger than the Poisson component. \nNow we have determined all the terms on the right-hand side of Equation 8. To find σ 2 ( true ) we perform 30k iteration of the Monte Carlo equation 8. In each iteration, the errors are randomly sampled from a Gaussian distribution with mean 0 and standard deviation being the error value. This includes both Poisson error of both the source and the PSF and uncertainty in PSF arising turbulence, instrument, and optics. In each iteration sampling is done for σ 2 xx , σ 2 yy and σ 2 xy . We only consider the cases where the following conditions are satisfied \n- · σ 2 xx ( true ) > 0\n- · σ 2 yy ( true ) > 0 and\n- · 2 σ 2 xy ( true ) < σ 2 xx ( true ) + σ 2 yy ( true ) \nThe first two conditions ensure the size along the x and y axes individually are positive and hence the overall size is positive. The last condition ensures | e 2 | < 1. We take median values of σ 2 xx , σ 2 yy and σ 2 xy in all the iterations that \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 7. (a) and (b) The plot on the left shows the histogram of all stars after they have been passed through the Monte Carlo PSF correction method in the i+r co-added image. Sigma clipping is necessary to reject the stars on the right tail which are not suitable for calculating PSF. The graph has been truncated x = 3 pixels. A similar graph for galaxies is shown on the right. This graph has been truncated at x = 10 pixels. (c) and (d) On the left histogram of all stars after they have been passed through the Monte Carlo PSF correction method in the i+r co-added images simulated using PhoSim with input γ 1 = 0 . 1 is shown. The plot is truncated at x = 3.5 pixels. Not all stars shown in the histogram are used for PSF correction. Sigma clipping is required to reject the stars on the right tail which are not suitable for calculating PSF. A similar graph for galaxies is shown on the right. This plot is truncated at x = 10 pixels. \n<!-- image --> \nsatisfy these conditions to find σ 2 xx ( true ), σ 2 yy ( true ) and σ 2 xy ( true ). We note the conditions listed above were found to be optimal. Several sets of conditions such as constraining overall size > 0 and constraining | e 1 | < 1 were tried and rejected. Sources in which less than 50 samples satisfy the above conditions are considered to have failed PSF correction. Figure 7a and 7b show size histograms of stars and galaxies after PSF correction has been performed in the i+r coadd.", '2.6. Flags': 'In this section, we describe the different flags used in this paper and the conditions that raise these flags. These flags are primarily used to determine which measurements are likely to be contaminated. For instance, sources that lie close to a chip edge or in chip gaps would need to be rejected. Sources that have a very small spatial separation or are blended would likely yield contaminated measurements as well. Below we list all the flags used and the conditions that trigger the flag. \n- 1. Vertical Flag ( vflag ): Some brighter stars have a vertical array of zeros near the center. This can be seen in Figure 2b. These stars are unsuitable for PSF determination. Let us assume the center of the source is at pixel ( x c , y c ). The vflag is raised if any of the following conditions is satisfied\n- · Calculate the average pixel value in a horizontal 3x1 pixel strip two pixels above y c . In other words the mean value of pixels at ( x c -1 , y c +2), ( x c , y c +2) and ( x c +1 , y c +2) is calculated. The mean value of pixels at ( x c -6 , y c +2) and ( x c +6 , y c +2) is also calculated. If the first quantity is less than either of the second then the flag is raised.\n- · Similar calculation as before except we consider a strip three pixels above y c . We find the mean value of pixels at ( x c -1 , y c +3), ( x c , y c +3) and ( x c +1 , y c +3).The mean value of pixels at ( x c -7 , y c +3) and ( x c +7 , y c +3) is also calculated. If the former quantity is smaller than either of the latter, the flag is raised.\n- 2. Bad Flag ( bflag ): As mentioned before some sources at the edges of CCDs or in chip gaps need to be rejected. We check if the cutout is at least 6 times the expected size of the source in that image. It was found this is the minimum cutout size required to recover flux, shape and ellipticity accurately. The expected size is calculated using \nσ 2 image = σ 2 coadd ( true ) + σ 2 image ( PSF ) (10) \nwhere σ 2 coadd ( true ) is obtained after using the Monte Carlo method described above to solve Equation 8 in the i+r co-add. The interpolated values of the PSF in the individual image is σ 2 image ( PSF ). In this equation σ 2 can be substituted with σ 2 xx , σ 2 yy or σ 2 xy . The expected size is \nS = √ σ 2 image,xx + σ 2 image,yy (11) \nIf the cutout size is larger than 6 times the expected size of the source, then we check for pixel values ≤ 0 inside a square box of side 6 times the size. If found, this flag is raised i.e. set to 1. If the initial cutout size is less than 6 times the size of the source being measured, any pixel value ≤ 0 in the cutout would trigger this flag. This flag is exclusively used for cutouts made in individual frames. \n- 3. Background Flag ( bkgflag ): This flag is used in the i+r co-add. We use the larger of σ 2 xx or σ 2 yy to find the size. Size is defined as S = √ σ 2 xx + σ 2 yy . In this case σ 2 xx and σ 2 yy are the values measured by the moment matching method and not corrected for PSF. We replace the smaller of the two with the larger. We also find the average value of background standard deviation in the full image and call this global standard deviation σ g . Next, assuming an elliptical Gaussian profile, we find the radius at which light from the source becomes equal to the background standard deviation. We denote this as r. \nr = √ S 2 log ( 2 πS 2 σ g N ) (12) \nwhere N is the photon counts and S is the size. We also define r 1 = r/ √ 2. We now define 8 regions. The first four regions are simply a 4x4 pixel square at the corners of a square with a size length 2r. The next four regions are similar except now they are at the corners of a square of side length 2r 1 . We find the median value in each region. We also find the median value of the eight median values and call this local median. If the local median is higher than the global median + 2 times the global standard deviation, the flag is raised. We also count the number of regions whose median value is greater than the local median + 1.5 times the local standard deviation or lower than the local median - 1.5 times the local standard deviation. If more than one region exceeds this threshold, then the flag is raised as well. \n- 4. Bound Flag ( bndflag ): We calculate the r for all sources as described above. We cap the lower value of r at 13 pixels. This number is decided since it is just larger than half the size of the minimum cutout of 26x26 pixels. For every source, the distance of the source centroid from the centroid of all other sources is calculated. If the centroid distance is lower than the sum of the r for each pair considered, the flag is raised. This flag indicates if a significant amount of light from a source is present in the cutout of another source. Fewer than 2% of our sources were found to raise this flag. However, it was also found that this flag was not adequate in some cases because, for most astronomical sources, light falls off slower than a Gaussian.', '2.7. Measurement in Individual Images': 'It has been suggested using measurements made in individual images over co-adds could lead to better measurements (Tyson et al. 2008; Jee & Tyson 2011; Mandelbaum 2018). This is because co-adding causes loss of information and averaging of systematic effects. In theory, measurements in individual images can be better corrected to take systematic into account. Measurements made in images with better seeing and lower background brightness can also be given more weight. Indeed, this idea has been used in some recent weak lensing studies such as Miller et al. (2013), Zuntz et al. (2018). However, the challenge with such a scheme is that the sources will be extremely faint in each individual image. In all studies to date, measurements in individual images are performed only when the source has high enough S/N ( ∼ 10 or more). To get around this limitation, we use forced measurement (Dutta et al. 2024) which is capable of measuring sources with SNR << 1. The details of this algorithm is beyond the scope of this paper, but we describe it here briefly. Forced measurement is a generalization of forced photometry (Lupton et al. 2001; Stoughton et al. 2002; Lang et al. 2016). Using the moment matching algorithm described above, we can measure the shapes and position of the detected sources in the co-add. Next, we can use this measurement as our initial guess and run the moment matching algorithm for a single iteration, as opposed to convergence. The flux measurements obtained from this method are identical to forced photometry and so it stands to reason, that the centroid and shapes found could hold some information. This was found to be true and is the essence of forced measurement. First, the pixel values after background subtraction are truncated in a fashion that makes all pixel values positive. This step is crucial to ensure that measured σ 2 values are not negative. Then we use a method similar to forced photometry, where we use the expected shape and size of the source to perform a single iteration of the moment matching algorithm. The values of flux shapes and sizes thus obtained are corrected to take into account the effect of truncation. The details of this algorithm have been described in Dutta et al. (2024). \nWe decided to perform measurements only in g, i, and r band individual images since almost all sources detected in the co-add can be detected in these individual band co-adds as well. To estimate the PSF in individual exposures, a 80 × 80 pixel cutout at the location of the stars is made and moment matching measurement is done on these cutouts. To discard any stars with defects or blending we perform k=3 sigma clip of all the stars in σ 2 xx , σ 2 yy and σ 2 xy . Bright stars with SNR ≥ 100 and no flags raised are selected for PSF estimation. To avoid brighter fatter effect any star having ≥ 10 6 photon counts is rejected. We also reject any stars in chips where the weight has been set to 0 during co-addition. This produces a very pure sample of stars to determine PSF. Interpolation of PSF is done using Equation 6. \nFor the remaining sources, a square cutout 8 times the size of the source is created with the source at the center. To check whether the source is bright enough for the moment matching algorithm to converge, the approximate SNR is calculated. It has been shown in Dutta et al. (2024) when SNR ≥ 15 convergence is guaranteed. SNR is defined as \nSNR = N √ N +4 AB (13) \nwhere N is the photon counts from the source, A is the area and B is the background. The background is taken from the header and is the global median sky background. An alternate approach to determine if the source is bright enough for moment matching to converge would be to run the measurement algorithm on the cutout to check for convergence. However, this approach suffers from the fact that it would be challenging to differentiate between the case where the source is bright enough for convergence and the algorithm converging on a nearby bright source or defect. Hence, this approach was not explored further. Calculating the SNR before actually measuring the source is not possible. Hence, the SNR calculation is done using the expected size, expected flux and overall background of the image. Expected flux ( N expected ) and expected size ( σ 2 image ) is calculated as \nN expected = N coadd × 〈 N image ( star ) N coadd ( star ) 〉 (14) \nσ 2 image = σ 2 coadd ( true ) + σ 2 image ( PSF ) (15) \nwhere σ 2 coadd ( true ) is obtained after using the Monte Carlo method described above to solve Equation 8 in the i+r co-add. The interpolated values of the PSF in the individual image is σ 2 image ( PSF ). In this equation σ 2 can be substituted with σ 2 xx , σ 2 yy or σ 2 xy . The expected photon counts of a source in an image is N expected , N coadd is the photon counts of the source in the corresponding band co-add and ⟨ N image ( star ) /N coadd ( star ) ⟩ is the ratio of photon \ncounts of stars in the image to photon counts in the co-add. The angle brackets in the above equation indicate the median. To find the median value, only the stars that are used for PSF interpolation at the given location are utilized. This was done to minimize the effect of photometric variation across the image. \nIf SNR is lower than 15, we use forced measurement (Dutta et al. 2024). Forced measurement requires a reasonably good starting guess. We use N expected and σ 2 image from the above equations as guess values for forced measurement. After forced measurement has been performed, we similarly perform Monte Carlo PSF correction as before. A slightly modified Monte Carlo equation shown in Equation 17 is used to take into account that the errors in the flux, shape, and size obtained from forced measurement are significantly tighter than Poisson errors. This arises from the fact that forced measurement needs a reasonable guess to start with and only a single iteration is performed. The error in the measured parameters are calculated as \nϵ σ 2 xx = p σ 2 xx ( N,S,B ) √ S 4 N +4 πS 2 B S 4 N 2 (16) \nwhere N is photon counts i.e N expected , S is size and B is the background value and p σ 2 xx is a function of N expected , S and B . We use the piece wise linear function of p σ 2 xx from Dutta et al. (2024). All successful measurements are passed through the Monte Carlo PSF correction scheme similar to Equation 8. For individual exposures the PSF correction equation for σ 2 xx is \nσ 2 xx ( true ) = σ 2 xx ( measured ) ± p σ 2 xx √ S 4 N (1 + K ) -σ 2 xx ( PSF ) ± ϵ ( xx ) PSF (17) \nwhere σ 2 xx ( measured ) is the measured values in individual exposures by forced measurement. In cases where the moment matching method successfully converges, p σ 2 xx is set to 1. The error in PSF is ϵ ( xx ) PSF . Similar equation is used to determined σ 2 yy ( true ) and σ 2 xy ( true ). We perform 30k iterations and use the same condition used in the co-add PSF correction to select which iterations as considered valid. Sources for which less than 50 samples satisfy the above conditions are considered to have failed PSF correction.', '2.8. Combining Data from Individual Images to get Shear': "The individual frame measurements of each source need to be combined into a single shape and ellipticity measurement. To reject contaminated measurements, we apply some sensible cuts. We reject all measurements where any flag has been raised either in the i+r co-add or individual frames. Sources where the size is over a factor of 2 larger than the i+r co-add size after PSF correction is rejected. If this condition fails, it is a strong indication that the source was not measured accurately. We reject measurements where the flux value is negative or nan. This is because negative or nan values do not make physical sense. Cases, where PSF correction failed in single frames and cases where forced measurement produces nans, are rejected as well. Finally, to ensure defects and noise do not affect our measurements, we calculate the sigma-clipped median and standard deviation of individual frame flux measurements in a filter. Any measurement where the flux is 3 standard deviations away from the median is rejected. This is to ensure defects like 'stripes' shown in Figure 2a do not influence the measurement of faint sources. We also reject images with σ 2 xx ( PSF ) or σ 2 yy ( PSF ) greater than 30 pixel 2 which corresponds to a size of approximately 7.7 pixels i.e. a seeing FWHM of 2 . 6 '' . Weak lensing analysis requires very stringent PSF cuts and images with seeing larger than 1 '' have traditionally been discarded (Gruen et al. 2013; Okabe et al. 2010). Also at approximately 4 '' seeing, point sources start to form donut shapes indicating the telescope is extremely out of focus. We also reject images where the sigma clipped standard deviation in stellar σ 2 xx ( PSF ) or σ 2 yy ( PSF ) is greater than 3 pixels. This corresponds to an unusually large variation of PSF across the field and is indicative of issues with the image. These conditions are not mutually exclusive. \nThe images with better seeing conditions and lower background brightness should be given more weight. Put differently, images which have the lowest Poisson error should be given maximum weight. We use inverse error in size as weight when combining the σ 2 xx , σ 2 yy and σ 2 xy of individual measurements. This was found to be optimal. \nσ 2 xx = ∑ i σ 2 xx,i W ( i ) (18) \nwhere σ 2 xx,i is the PSF corrected measurement in the i-th frame and the weight of the i-th image W ( i ) is \nW ( i ) = ϵ -1 σ 2 xx,i ( true ) ∑ i =1 ϵ -1 σ 2 xx,i ( true ) (19) \nwhere ϵ σ 2 xx ( true ) is the error in σ 2 xx ( true ) . This is equivalent to ϵ σ 2 xx and ϵ ( xx ) PSF added in quadrature. \nϵ σ 2 xx,i ( true ) = √ ϵ 2 σ 2 xx ,i + ϵ 2 ( xx ) PSF,i (20) \nSimilar equation for σ 2 yy and σ 2 xy is used. After combining the individual measurements the error in final σ 2 xx is \nϵ σ 2 xx ( true ) = √ ∑ i =1 ϵ 2 σ 2 xx,i ( true ) W 2 ( i ) (21) \nEllipticities e 1 and e 2 are calculated as \ne 1 = σ 2 xx -σ 2 yy σ 2 xx + σ 2 yy (22) \ne 2 = 2 σ 2 xy σ 2 xx + σ 2 yy (23) \nThe error in the final ellipticity measurement can be calculated using standard error propagation equations. This comes out to \nϵ 2 e 1 = 4 ϵ 2 σ 2 xx ( true ) σ 4 xx ( true ) + σ 4 yy ( true ) ( σ 2 xx ( true ) + σ 2 yy ( true )) 4 (24) \nϵ 2 e 2 = 8 ϵ 2 σ 2 xx ( true ) σ 4 xy ( true ) ( σ 2 xx ( true ) + σ 2 yy ( true )) 4 +4 ϵ 2 σ 2 xy ( true ) 1 ( σ 2 xx ( true ) + σ 2 yy ( true )) 2 (25) \nIn the above derivation we have assumed ϵ σ 2 xx ( true ) = ϵ σ 2 yy ( true ) . This is approximately true as long as the PSF and source is not extremely elliptical. Considering only a small fraction of the sources show extremely high ellipticity, this error equation is reasonably close to the true Poisson value. \n<!-- image --> \nFigure 8. (a) Histograms for the difference in shape when sources are measured from the i+r co-add versus when they are measured in individual images of i and r band images. The individual image measurements are combined using inverse error as weight. The reduction in size by weighting individual frames optimally is clear from this plot. This leads to enhanced shear signal and hence shear recovery. (b) Histograms for the difference in shape when sources are measured from the weighted i+r co-add versus when they are measured in individual simulated images of i and r band images produced using PhoSim for the first case i.e. γ 1 = 0 . 1. The individual image measurements are combined using inverse error as weight. The reduction in size by weighting individual frames optimally is clear from this plot. This leads to enhanced shear signal and hence shear recovery. \n<!-- image --> \nOn comparing the size measured from i+r co-added image to the the combined measurement from individual images we see a significant reduction in size for a large fraction of sources. This is shown in Figure 8a. A smaller size increases \nthe shear signal. In Figure 8b we show the same graph for simulated sources. The simulation is described in Section 3. \nTo recover shear from ellipticities, we adopt the method proposed in GREAT3 challenge (Mandelbaum et al. 2014) \ng t ≈ γ t = e t 2(1 -⟨ e 2 ⟩ ) (26) \nwhere γ t is the tangential ellipticity, e t is the tangential ellipticity of a source with respect to some given central point and ⟨ e 2 ⟩ is the average value of ellipticity squared. Reduced tangential shear is denoted by g t . This equation was slightly modified to take into account the weight of each source. \ng t ≈ γ t = ∑ i =1 e t,i w a,i 2(1 -⟨ e 2 ⟩ ) (27) \nwhere e t,i is the tangential component of ellipticity of i-th source about a given point and the weight, w a,i , the weight of the i-th source is defined as \nw a,i = ϵ -2 e,i ∑ i =1 ϵ -2 e,i (28) \nwhere ϵ e is the error in ellipticity and \nϵ 2 e = ϵ 2 e 1 e 2 1 e 2 1 + e 2 2 + ϵ 2 e 2 e 2 2 e 2 1 + e 2 2 (29) \nwhere ϵ 2 e 1 is error in e 1 and ϵ 2 e 2 is error in e 2 . The tangential ellipticity, e t , defined about a point can be written as \ne t = -e 1 cos (2 ϕ ) -e 2 sin (2 ϕ ) (30) \nwhere ϕ is the angle a line joining the point and the source makes and e 1 and e 2 are the ellipticity components. The cross component of ellipticity is similarly written as \ne c = -e 1 sin (2 ϕ ) + e 2 cos (2 ϕ ) (31) \nWe also define \nThe weights w b,i is defined as \nw b,i = ϵ -1 e,i ∑ i =1 ϵ -1 e,i (33) \nUsing error propagation, the error in shear ϵ 2 γ when using Equation 27 is \nϵ 2 γ = γ 2 ∑ i w 2 a,i ϵ 2 e,i ( ∑ i w a,i e tan,i ) 2 + 2 ∑ i w 2 b,i ϵ 2 e,i 1 -( ∑ i e 2 i w b,i ∑ i w b,i ) 2 (34) \nwhere the summation is performed over all sources being considered, γ is the shear obtained, ϵ e,i is the error in the ellipticity of the i-th sources, e tan,i is the tangential ellipticity component of the i-th sources and e 2 i is the ellipticity of the i-th sources. The uncertainty in shear given by Equation 34 is entirely due to Poisson noise. However, when considering real-world data the error in shear is significantly worse due to systematic errors in telescopes, turbulence of the atmosphere, imperfection of CCD's, and a multitude of other effects. Single frame measurements will be affected the most since most of the sources are extremely faint in the individual images of a co-add. Hence, simulations are performed to understand these additional factors better. \n⟨ e 2 ⟩ = ∑ i =1 e 2 i w b,i (32)", '3.1. Simulation and Co-add Measurements': "In order to test the ability of the pipeline described in Section 2 to recover shear accurately we test it on simulated images produced by PhoSim (Peterson et al. 2015, 2019, 2020, 2024). PhoSim is a ray tracing software that can simulate most of the known physics and hence produce extremely realistic images. We use the WIYN-ODI instrument which has been validated with real data from the telescope and simulate 30 images in i and r filter with 60s exposure time. The seeing and airmass were chosen at random from real i and r band images. We chose to simulate sources fainter than 17th magnitude to prevent saturation. We perform two sets of simulations in two different areas of the sky. This ensures we have different ensemble properties for the two cases. In the first case the galaxies in the PhoSim catalog were sheared by γ 1 = 0 . 1 and in the second case by γ 1 = 0 . 05. We pass the images through the exact co-add and detection method described above. The final co-add size is cropped to 10000x10000 pixels or 18.3'x18.3'. Source density in both cases is approximately 55 sources/arcminute 2 . While on one hand, such a high number density from ground-based images is unlikely, on the other hand, it will help to test the performance in a crowded environment where blending issues pose a significant challenge for weak lensing analysis (Hartlap et al. 2011; Dawson et al. 2015). \nFor each image simulated, a 8 x 8 star grid is also simulated with the exact same seed which ensures all random parameters used in PhoSim are identical. This is done for two reasons. One, the clouds in each individual image is slightly different and hence the ZP will be slightly different. By simulating a star grid with stars of known magnitude we are able to calculate ZP accurately. Second, it helps us get an initial estimate of seeing and background. Our pipeline needs ZP, seeing, and background variance to create the weights for the co-add. The magnitude brightness of stars in this grid ranges from 16 to 23. \nThe simulated images were passed through the same pipeline as above. The PSF size of the i+r co-added image is 3.5 pixels and the error in σ 2 xx , σ 2 yy and σ 2 xy is 0.18, 0.22 and 0.06 respectively. The size and errors are very similar for both sets of simulations since the same values of seeing, airmass, and seed were used for both. While the average size of the PSF is comparable to the real data, the errors are significantly smaller. The flux vs size graph is shown in Figure 5b. Points inside the red square are used for PSF estimation while the sources inside the black box are not used for weak lensing analysis to avoid shear dilution due to stars. The condition for the red box is 3 . 3 < size < 3 . 8 pixels and 500 < counts < 10 5 . For the black box, the conditions are 2 . 5 < size < 4 . 5 pixels and 10 2 < counts < 10 6 . These limits were obtained after careful visual examination. We also reject any sources larger than 12 pixels, which are likely due to severe cases of blending or sources brighter than 10 5 photon counts where brighter-fatter effect becomes important. This is clearly seen when considering the sources just above the red rectangle in Figure 5b. In total, the fraction of sources rejected by these cuts is roughly 1/3, most of them being stars enclosed by the black box in Figure 5b. This is in agreement with the PhoSim input catalog in which approximately 1/3 of all sources are stars. In Figure 7c and 7d we show histograms of star and galaxy size after PSF correction. The histogram of galaxy size shows a significant bump at 0.5 pixels which implies contamination from stars. This could potentially explain the slight underestimation of shear in Figure 9. The histogram of star size along with the best fit Gaussian in red is shown in Figure 6d, 6e, and 6f. \nIt was found a few percent of the sources with extremely small ellipticity errors, primarily large and bright sources, bias the shear measurement when using this scheme described in the previous section. Capping ellipticity error to 1/3 of the median ellipticity error values works well to recover shear and hence was adopted. Using this simple modification, we are accurately able to recover shear as shown in Figure 9. In this figure, we plot the shear measured as we start from a central circular region and increase the radius gradually to include more sources. The x-axis shows the radius of the circular region. On y-axis, we plot the recovered shear. We find the shear recovered from individual image measurements is more accurate than using shape measurements made in the co-add. In the second case, there seems to be a slight bias of ∼ 0.005. It was also found ϵ 2 e i.e. the Poisson component of ellipticity error does not accurately predict error bars. This is expected since the simulations accurately take into effect most known physics while our error bars only take into account the Poisson statistics. We found 0.005 to be a more suitable error bar. This error bar also allows us to ignore the bias since it is comparable to the error bar. This error level is approximately 4 times the Poisson error calculated from the co-add and 10 times the error calculated from combined single frame measurements. These factors were adopted and applied to the Poisson error obtained from WIYN-ODI. \n<!-- image --> \nFigure 9. These plots show the ability of our pipeline to accurately recover shear at various levels in the weak lensing regime. To make these plots, we give PhoSim simulations input shear of γ 1 = 0 . 1 in the first case and γ 2 = 0 . 05 for the second case. It is clear both co-add and individual frame measurements accurately recover shear to within 0.005 of the true shear. In the second case, there is slight bias of approximately 0.005. The error bars shown are just the Poisson component. Clearly, single frames have smaller error bars as expected. However, the error bars fall short because of non-Poisson sources of error. Hence we elect to use error bars of 0.005 for shear measured using individual image measurements which is approximately 10 times the Poisson component. \n<!-- image -->", '4.1. Aperture Mass Maps': "A common method of detecting mass concentrations using weak lensing is aperture mass statistics (Kaiser 1995; Schneider 1996; Gruen et al. 2013). It measures the average tangential orientation of background galaxies around some point convolved with a weight. Aperture mass M ap and the associated error σ M ap are defined as \nM ap = ∑ i w ( | θ | - | θ i | ) w p ( i ) g i,t (35) \nσ M ap = √ ∑ i w 2 ( | θ | - | θ i | ) w p ( i ) (36) \nwhere g i,t is the tangential alignment of the i-th galaxy and w p ( i ) is the weight function derived from Poisson error in ellipticity, and the overall weight function is denoted by w ( | θ | ). θ represents the generalized co-ordinates. Specifically the weight due to inverse Poisson error in ellipticity is \nw p ( i ) = 1 ϵ 2 e ( i ) (37) \nSeveral different weight schemes can be found in literature (Schneider 1996; Schirmer et al. 2004; Gruen et al. 2013). We use Gaussian weight (Gruen et al. 2013) for simplicity. \nw ( | θ | ) = exp( -| θ 2 | / 2 σ 2 w ) when | θ | < 3 σ w 0 when | θ | > 3 σ w or | θ | < 50 pixels (38) \nwhere σ w is the width of the weight function. The size of σ w is usually a few arcminutes. A smaller value brings out details at a finer spatial scale but at the cost of more noise. This is because, for a smaller width, averaging of shear is done in a smaller area using a small number of background galaxies. A larger value of width washes out details of mass distribution but has a lower noise. \nFor Poisson error in ellipticity ϵ e , we apply the error cap at 1/3 the median error value in order to prevent the measurements from being dominated by bright sources with extremely low errors. The significance maps is then defined as M ap /σ M ap . M ap is also called the Aperture Mass maps or E-mode maps. To make the B-mode maps the tangential component of shear is replaced by the cross component. In other words, we simply replace g t with g c in Equation 35. This is defined as \ng c ≈ γ c = e c 2(1 -⟨ e 2 ⟩ ) (39) \nA major issue in making the aperture mass maps is that source density varies significantly over a large field. In our case, the source density varies by a factor of 2 in the 40 ' × 43 ' field of view. This produces mass maps that have different error levels in different regions of the image. The problem gets particularly worse at the edges. Hence, we decide to use an adaptive weight scheme. Simply put, we change σ w depending on the source density. The width is smaller in regions of high source density and conversely, the width is larger when the source density is smaller. The width is changed to get σ M ap varying by at most 10% throughout the field, except at the edges. However, we limit the maximum width to 3000 pixels and the minimum width to 500 pixels. The maximum value of width is reached only at the edges. \nIn Figure 10 we show the comparison of the aperture mass map with the light density maps in the central part of the cluster. To make the light density maps we select all galaxies with photometric redshift between 0.15 to 0.55 where the reliability parameter of EAZY is greater than 0.8. We also reject galaxies where the source was measured in less than 3 bands. This range includes the galaxy cluster Abell 2390 which has been spectroscopically confirmed at z=0.23 (Abraham et al. 1996). We reject any galaxy with size greater than 10 pixels. Then we divide our images into grids of 50 × 50 pixels. For each grid position, we calculate the light from the galaxies in a 5 . 5 ' radius and weigh them by a Gaussian of width 0 . 73 ' . The adaptive E-Mode maps are made by fixing the value of σ M ap within the range 3 . 5 × 10 -4 ± 1 . 75 × 10 -5 and considering the galaxies in the redshift range z = 0.4 to 2.0. We do not consider galaxies with z > 2.0 since in this range a large fraction of the sources represent catastrophic failure of the photo-z code. Once again we only consider galaxies where the reliability score of photometric redshift is greater than 0.8 and flux information is available in at least 3 filters. We also reject sources larger than 10 pixels in size or sources where the bkgflag is raised. These conditions are not mutually exclusive and there is significant overlap. Changing the conditions slightly does not significantly affect our results. The main peak of the E-mode maps line up extremely well with the galaxy density maps which in turn lines up with the central cD galaxy and X-Ray maps Allen et al. (2001). In Figure 11a we show the contours of the Aperture Mass Map overlayed on the light density map of galaxies in the redshift range z = 0.15 to 0.55. The symbols show the location of galaxy clusters and groups obtained from the NASA Extragalactic Database. Figure 12 shows the light density map of galaxies in the redshift range z = 0.35 to 0.75 with contours of Aperture Mass Map overlayed on it. We discuss these maps in detail in Section 6.", '4.2. Mass Maps': "Aperture mass maps are useful for locating peaks and the general structure of mass distribution. However, one of the key objectives of weak lensing studies are measurement of mass. We follow the method of Kaiser & Squires (1993) which was later generalized by Seitz & Schneider (1995) to estimate the mass. κ ( θ ) is given as \nκ ( θ ) ≈ 1 2 π ¯ n N ∑ n =1 W ( θ -θ n , s ) γ n,t θ 2 n (40) \nwhere γ n,t is the tangential shear of the n-th source, ¯ n is the density of sources and \nW ( x, s ) = 1 -( 1 + x 2 2 s 2 ) exp ( -x 2 2 s 2 ) (41) \nwhere x is the distance of the source from the point about which we calculate κ ( θ ) and s is the window width. The weight function W ( x ) in effect suppresses noise at a small radius. If a very small window function is chosen then the image becomes extremely noisy. On the other hand, if a large value of s is chosen then the details of mass distribution are washed out. It was found that s = 200 pixels or 22 '' produces optimal result. To produce the κ map the exact same procedure used to make E-Mode maps is followed, except we now no longer use adaptive values. We also only consider galaxies in a radius of 7 . 33 ' for any given point. The map of κ ( θ ) for the entire field is shown in Figure 11b . The error level in each pixel was determined using error propagation and multiplying the resulting Poisson error with a factor of 10 to take into account systematic/non-Poisson sources of error, as determined from PhoSim simulations. \nTo convert from κ ( θ ) to enclosed mass we use \nκ ( θ ) = Σ Σ crit (42) \nΣ crit = c 2 4 πG D s D l D ls (43) \nwhere Σ is the mass enclosed and \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 10. (a)The contours of E-Mode (i.e. M ap ) maps shown in blue are overlayed on the galaxy light density maps. Contours are drawn at 0.012, 0.024, 0.036 and 0.048. The light density maps are made by adding light from galaxies in the range z = 0 . 15 to 0 . 55 weighted by a Gaussian of width 0 . 73 ' . The main peak matches up perfectly. The smaller peak on the lower left is distinctly separate from the E-Mode peak. The two peaks of the E-Mode map are shown as yellow circles (b) Same as before except the contours are from the convergence map. Contours are drawn at 0.1, 0.15, 0.25 and 0.3. The two peaks of the convergence map are shown as yellow circles (c) In color shown is the X-ray image from Chandra with the convergence map contours overlayed. A slight swirl pattern in the X-Ray can be seen matching the swirl pattern of the convergence contours. (d) The convergence map is shown as background. Contours show the compact component subtracted radio images from LOFAR obtained after careful re-analysis of DR2 data by Cianfaglione et al (in prep). Contours are drawn at 0.0014, 0.0021, 0.0028, 0.0035 Jy/beam which corresponds to 2, 3, 4 and 5 σ significance respectively \n<!-- image --> \nWhere D s is the angular distance to the source, D l is the angular distance to the lens, and D ls is the angular distance measured by an observer at the location of the lens to the source. This is not the same as simply subtracting D s and D l and depends on the curvature of the universe. It should be noted that Σ crit is defined for a single source. However, in weak lensing analysis, one typically has thousands of background sources at various redshifts for a given lens system. In our case, the lens system is Abell 2390. Hence an average value of D s D ls is determined to find Σ crit . \n<!-- image --> \nFigure 11. (a)The contours show the E-Mode (i.e. M ap ) maps constructed from the galaxy sample in the range z=0.4 to 2.0 overlayed on light density map of galaxies in redshift range 0.15 - 0.55. We fixed σ M ap = 3 . 5 × 10 -4 ± 1 . 75 × 10 -5 . Contours are drawn at levels 0.012, 0.024, 0.036 and 0.048. The background is light density map that was smoothed with a Gaussian width of 0 . 73 ' . Four major filamentary structures connecting the galaxy cluster are visible and are shown in green arrows. One to the north, one to the west (right), and one towards the southwest (down right). Another structure to the southeast is also visible. The filament seen from the contours broadly agrees with the filaments seen in the light density maps. The location clusters obtained from NED are shown as symbols. The squares are obtained from X-ray studies while the stars are obtained from optical data. Cross symbols show cluster location obtained using the SZ effect on Planck data. Each symbol and the corresponding literature is discussed in detail in Section 6. (b) Contours of κ ( θ ) is overlayed on the light density maps smoothed with a Gaussian of 0 . 73 ' of galaxies in the range z = 0.15 to 0.55. Contours have been drawn at levels 0.1, 0.2 and 0.3. The κ ( θ ) map has been smoothed with a Gaussian of width 16.5 '' . The bimodal mass distribution of the central region of Abell 2390 can be clearly seen at R.A = 21h:53m:36.8s and Dec = 17 ° :41 ' :43 '' . The other peaks are consistent with the mass concentrations shown by E-Mode maps. We find also increased noise at the edges of κ ( θ ) map. \n<!-- image --> \nThe average value is defined as (Okabe et al. 2010) \n〈 D s D ls 〉 = ∫ D s D ls dp dz dz (44) \nwhere dp/dz is the redshift probability distribution of source galaxies used for weak lensing measurements. We use the redshift value obtained using EAZY photo-z as described in Section 2.4. In our case, this value comes out to 0.66. This leads to a Σ crit value of 4.23 x 10 15 M ⊙ /Mpc 2 which is in agreement with the value of 4.55 x 10 15 M ⊙ /Mpc 2 obtained by Squires et al. (1996). Our value for ⟨ D s /D ls ⟩ is in agreement with the value of 0.69 obtained by Okabe et al. (2010) with images comparable depth to ours. This is remarkable since both Squires et al. (1996) and Okabe et al. (2010) were unable to perform photo-z due to lack of color information. A comparison of mass enclosed in the inner part of Abell 2390 with Squires et al. (1996) is shown in Figure 13. We find an excellent match up to a radius of 0.7 Mpc from the cluster center. The mismatch after that is likely due to mass sheet degeneracy which states that κ ( θ ) can be approximately determined to an additive constant. For a detailed discussion on mass sheet degeneracy see Bradaˇc et al. (2004).", '4.3. 3-D Slices': "The ability to make slices of mass distribution at various redshifts would be the ultimate test of any deep weak lensing analysis. One of the first maps 3-D weak lensing reconstruction was done by Massey et al. (2007). It used high-quality images obtained from HST to measure shear in each redshift bin. However, such reconstructions are extremely rare in literature, especially with ground-based data. One of the challenges in making such a 3-D map is \nFigure 12. The contours show the E-Mode (i.e. M ap ) maps constructed from the galaxy sample in the range z=0.4 to 2.0 overlayed on light density map from galaxies in the redshift range 0.35 to 0.75. Contours are drawn at levels 0.012, 0.024, 0.036 and 0.048. We fixed σ M ap = 3 . 5 × 10 -4 ± 1 . 75 × 10 -5 . The background is light density map that was smoothed with a Gaussian width of 0 . 73 ' . \n<!-- image --> \ngalaxy density. In weak lensing one typically requires a few tens of galaxies arcminute -2 to measure shear. However, when these are divided into redshift bins, the galaxy density is drastically decreased which introduces more noise. Below we describe our approach. \nWe construct an E-Mode map, E 1 with all galaxies in the redshift range z= 0.01 to 2.0. We only consider the sources that satisfy the conditions mentioned in Section 4.1. This map, in an ideal case, traces most cosmic structures from z = 0 to 2.0. We construct a second E-Mode map, E 2 with all galaxies in the range z = 0.5 to 2.0. This E-Mode map boosts the structures in the range z =0.01 to 0.5 since it only contains galaxies background to z = 0.5. This can be written down as \nE 1 = S 1 ( N 2 N 1 + N 2 ) + S 2 (45) \nE 2 = S 1 + S 2 (46) \nwhere S 1 is the cosmic structure is the foreground slice and S 2 is the structure in the background slice. N 1 is the number of sources in redshift range z = 0.01 to 0.5 i.e. the foreground slice and N 2 is the number of sources in redshift range z = 0.5 to 2.0 i.e. background slice. The factor ( N 2 N 1 + N 2 ) signifies that only the population N 2 trace structure S 1 . \nFigure 13. Enclosed mass as a function of radius from the center of the galaxy cluster. The center is located on the BCG at R.A = 21h:53m:36.8s and Dec = 17 ° :41 ' :43 '' . We used Poisson error bars and multiplied them with a factor of 10, as found in PhoSim simulations, to take into account systematic/non-Poisson errors. We find our data matches extremely well Squires et al. (1996) up to radius of 700 kpc. The mismatch beyond that is likely due to mass sheet degeneracy and can be remedied using a small additive constant. \n<!-- image --> \nIt was found significant noise arises from from the fact that the resolution of E-mode maps are different, with E 1 having significantly better resolution than E 2 . This is because the number of galaxies used to construct E 2 is approximately half of the number of galaxies used to construct E 1 . To remedy this we bring all E-Mode maps to the same resolution. We smooth E 1 by a Gaussian of width 4 pixels i.e. 22 '' and E 2 by a Gaussian of width 2 pixels i.e. 11 '' . This was visually determined after careful examination of the two E-Mode images i.e. E 1 and E 2 . \nWhile in theory all pixels of the E-Mode maps matter, we note that in reality, the shear signals are extremely small. Hence only the brightest few pixels in the E-Mode image contain information. It was found that capping the minimum value of E-mode maps at 1 standard deviation from the media of all pixels in the image produces better results. This value in our case comes out to approximately 0.014 for E 1 and 0.018 for E 2 . The E-Mode maps were produced in an adaptive manner constraining σ M ap = 4 . 5 × 10 -4 ± 2 . 25 × 10 -5 . Next, using inversion of Equations 45 and 46 mentioned above we recovered the two slices. The E-Mode contours of both the foreground and background slice are shown in Figure 14 We find the foreground slice contains signal from Abell 2390 while the background slice is completely devoid of signal from Abell 2390. A few of the more prominent sub-structures can be seen in the foreground slice. The background slice is very noisy and the contours do not seem well correlated to light density maps. This is not surprising since after all the cuts mentioned above, E 2 had an extremely low source density of 8 galaxies/arcmin -2 .", '5. DISCUSSION': "Abell 2390 has been extensively studied in X-Ray (Pierre et al. 1996; Allen et al. 2001; Martino et al. 2014; Sonkamble et al. 2015), optical (Abraham et al. 1996; Hutchings et al. 2002), radio (Bacchi et al. 2003; Augusto et al. 2006; Sommer et al. 2016; Savini, F. et al. 2019), weak lensing (Squires et al. 1996; Umetsu et al. 2009; Okabe et al. 2010; \n<!-- image --> \nFigure 14. (a)The adaptive E-Mode contours from the foreground slice (z = 0.01 to 0.6) overlayed on an image showing the light density of galaxies in the redshift range z = 0.15 to 0.55 smoothed is a Gaussian of width 0 . 73 ' . (b) The contours depicted are from the adaptive E-Mode maps of the background slice (z = 0.5 to 2) overlayed on an image showing the light density of galaxies in the redshift range z = 0.35 to 0.75 similarly smoothed. In both cases, contours are placed at levels of 0.02, 0.07, and 0.12. There is significant noise in our maps. The signal from Abell 2390 appears only in the foreground slice as expected. \n<!-- image --> \nvon der Linden et al. 2014) and strong lensing (Narasimha & Chitre 1993). The central continuum source of the BCG shows young, compact, and self-absorbed jets (Edge et al. 1999; Augusto et al. 2006). Accretion towards the central supermassive black hole has also been inferred from CO, CN, SiO, HCN and HCO+ absorption lines against the radio continuum source, which show molecular gas clouds moving in toward the galaxy center at roughly 100 to 300 km/s (Rose et al. 2019, 2024b). In Figure 15 we show a composite image of X-ray (in red), the convergence map (in blue) overlaid on the optical image of Abell 2390. In the following sections, we briefly discuss these previous findings and compare them to our results.", '5.1. X-Ray': "High-resolution X-ray images of Abell 2390 were obtained and analyzed by Allen et al. (2001). ROSAT data for this cluster has been analyzed by Pierre et al. (1996). However, the Chandra data is much more detailed and hence we focus on that. Allen et al. (2001) found that the X-ray profile can be fit well to a NFW profile. A variety of other profiles such as softened isothermal sphere and full isothermal sphere were also found to provide reasonably good fits. They also note that this cluster does not seem to be completely relaxed. Both Allen et al. (2001) and Rose et al. (2024a), find evidence of excess X-ray emission approximately 5 '' South-East of the central peak, which coincides with the BCG. We note here that this excess was found using two independent methods. Allen et al. (2001) used adaptive smoothing to find the excess, while Rose et al. (2024a) subtracted a double beta model. Both methods show the excess emission is approximately 20 kpc (5 '' ) from the central peak. It is hypothesized by Allen et al. (2001) that the cluster has not fully relaxed from the last merger. The model subtracted images of Rose et al. (2024a) are interesting because they seem to show two cores orbiting one another (their figure 6 ). However, there is a chance the excesses and depressions seen in the model subtracted images are a result of AGN activity. Indeed this has been suggested by Sonkamble et al. (2015). Evidence of current AGN activity at kpc scale was found by Augusto et al. (2006), with moderate high radio frequency variability of the radio continuum since 2015 also identified by Rose et al. (2022).", '5.2. Radio Observations': "It has been known for decades that massive merging clusters are likely to host radio emission in the form of radio halos and radio relics (van Weeren et al. 2019). The synchrotron radio emission would be caused by the re-acceleration of cosmic ray electrons by turbulent motions that develop in the ICM during cluster mergers. \nFigure 15. A composite image of Abell 2390. The X-ray obtained using Chandra(Allen et al. 2001) is shown in red and our mass map (convergence) in blue. It is overlayed on the RGB color image of Abell 2390 created using g band coadd for blue, r band coadd for green and i band coadd for red. Optical images have been obtained using WIYN-ODI. At the center of the image, the bright elliptical galaxy is the BCG. The X-Ray shows a swirl-like pattern in the same direction as the DM. \n<!-- image --> \nIn the past years, radio observations have started to reveal radio halos in clusters that are not undergoing major mergers, and that - in some cases - host a cool core (e.g. Bonafede et al. (2014); Sommer et al. (2016); Venturi et al. (2017); Savini, F. et al. (2019); Biava et al. (2024)). These results indicate that these sources might be connected to the occurrence of minor/off-axis mergers, though it remains unclear how minor mergers could initiate continuum emission on megaparsec scales. \nAbell 2390 is one of these clusters, as it hosts both a cool core and signs of minor dynamical disturbances from the X-ray morphological parameters. Diffuse radio emission in Abell 2390 was first discovered and classified as a radio mini halo by Bacchi et al. (2003) and then as a radio halo by Sommer et al. (2016). LOFAR observations revealed the presence of a double radio galaxy with the lobes extending in the east-west axis for ∼ 600 kpc, and Savini, F. et al. (2019) could not distinguish the radio emission from the radio galaxy from a possible contribution from the radio halo, leaving the possibility of a radio halo open. Savini, F. et al. (2019) also noted that radio galaxies of such a size are uncommon at the center of galaxy clusters, as the ICM prevents the expansion of the lobes to such large scales. \nLOFAR observations of Abell 2390 have also been published in the LOFAR Data Release 2 (Botteon et al. 2022), and the authors concluded that most of the emission came from the radio galaxy. \nCianfaglione et al (Master thesis, in prep) have re-analyzed the data from the LOFAR DR2, and subtracted the central AGN using an approach tuned 'ad-hoc' for this cluster. Specifically, they subtracted all the emission on scales smaller than 375 '' , corresponding to 1390 kpc, and re-imaged the data at low resolution (that is 1 ' ) to gain sensitivity towards the extended emission (Figure 10d). They found residual radio emission with a flux density S (144 MHz) = 0 . 16 ± 0 . 03 Jy, corresponding to a monochromatic power P (144 MHz) = 2 . 7 ± 0 . 5 × 10 25 W/Hz. In addition, the emission radial profile follows the exponential profile typically found in radio halos. The radio halo emission extends for 1450 kpc. As the size, power, and radio profile are all in line with those of radio halos in clusters of similar mass (Cuciti et al. 2023), they concluded that the residual emission is actually a radio halo. The presence of a radio halo reinforces the results obtained in this work, as they are in line with the presence of dynamical activity. If the merger is indeed in a late stage, as derived in this work, we expect the radio halo emission to be steep (Brunetti & Jones 2014; Savini, F. et al. 2019; Biava et al. 2024). Sommer et al. (2016) using VLA data estimates the overall spectral index to be 1.6 which is fairly high. \nHigh-resolution radio images of the BCG using ALMA to detect CO reveal an extended tail Rose et al. (2024a). They considered various scenarios, including both outflows and inflows, and came to the conclusion that a recent gravitational disturbance of the central BCG is the most likely explanation of the extended tail. Alcorn et al. (2023) using CFHT/SITELLE finds the BCG has a tail matching the findings of Rose et al. (2024a). All of this points to a fairly disturbed structure in the central region.", '5.3. Lensing and Spectroscopy': "Abraham et al. (1996) made spectroscopic measurements of this galaxy cluster using Canada-France-Hawaii Telescope (CFHT). They found spectroscopic evidence of an in-falling galaxy stream from the NW corner approximately 10 ' from the center of A2390. This can also be seen from the light density maps shown in Figure 11a. This group of galaxies has also been identified using optical cluster finding algorithms, shown as yellow and black stars in Figure 11a. \nA strong lensing study of this cluster was performed by Narasimha & Chitre (1993). Weak lensing studies have also been conducted using data obtained from CFHT Telescope (Squires et al. 1996) Subrau Telescope (Umetsu et al. 2009; Okabe et al. 2010). It was also studied by von der Linden et al. (2014) by using data from both telescopes. These studies, except for von der Linden et al. (2014) do not find evidence of a merger in the galaxy cluster using weak lensing maps. However, we note here that the only two studies with depth comparable to ours are Okabe et al. (2010) and von der Linden et al. (2014). We also note von der Linden et al. (2014) come to the conclusion that evidence of merger activity from the North-West, presumably the same galaxy group discovered by Abraham et al. (1996), is present in this cluster. While our light maps in Figure 11a clearly show this, we are not able to recover this with very high confidence. We detect the peak of this smaller group with somewhat lower confidence. We believe this is a minor and early-stage merger event. In addition to this, we also find evidence of gravitational disturbance in the central region of Abell 2390. Our convergence maps show the DM cores are in the process of merger. Aperture mass maps of von der Linden et al. (2014) look similar to ours, in the sense that they find a bimodal mass distribution at the core. Due to the significantly worse resolution of their map, unfortunately, any further comparison is difficult.", '5.4. Late Merger Hypothesis': "We believe Abell 2390 is a case of extreme late-stage merger as suggested by Allen et al. (2001). It seems to be the most likely explanation of all observed data. During this merger, the hot gas experienced friction when orbiting the cluster while the DM component experienced very little of this friction, if any. This caused the hot gas to lose angular momentum and fall into the central core faster than the DM. Currently, the hot gas is almost merged with the central core since it can only be detected using adaptive smoothing (Allen et al. 2001) or model subtraction (Rose et al. 2024a). The DM core still shows signs of an ongoing merger. In our mass maps, the mass excess is found southeast of the main peak at a separation of approximately 1 ' or 220 kpc. This separation, while in the exact same direction as the X-ray excess, is about 10 times more when compared to the X-ray separation of 5 '' i.e. 20kpc. A slight swirl pattern in the lower left of the X-ray image in Figure 10c is visible and matches up with the swirl seen on the overlaid convergence contours lending support to this hypothesis. \nThe late-stage merger would cause gravitational disturbance primarily in the central regions of the cluster. This is supported by the extended tail of the BCG found by Rose et al. (2024a). The direction of the tail also matches with \nthe expected direction from a late-stage merger. The process of an ongoing merger from the southeast would cause the plume in the BCG to be in the northwest direction. \nThe extended radio emission found by Sommer et al. (2016) and Cianfaglione et al. also supports our hypothesis. The late-stage orbital motion of two heavy DM cores will introduce significant turbulence and ripples in the ICM which leads to the re-energizing of leptons, which in turn leads to a radio halo with a steep spectral index. Sommer et al. (2016) estimated the spectral index to 1.6 which is very steep and consistent with our hypothesis of a late-stage merger.", '6. SMALLER GROUPS AND LARGE SCALE STRUCTURE': 'Weak lensing provides one of the most powerful ways to map large-scale structures of Dark Matter through the Universe (Refregier 2003). This is especially useful in probing regions of lower density such as filamentary structures and smaller galaxy groups since shear depends linearly on mass enclosed. Other methods such as X-rays are ineffective in studying these regions since X-ray brightness depends on density squared (Ettori 2000). Hence X-ray signals from low-density regions cannot be detected. These regions are difficult to detect in Radio wavelengths as well due to lower concentrations of ICM and ultra-relativistic particles in the ICM. Reliable detection of these lower mass structures can be used in conjunction with mass, redshift, and distribution of larger structures, such as galaxy clusters, to better constrain cosmological models (Bernardeau et al. 1997; Jain & Seljak 1997). \nIn Figure 11a we show the light density of the i+r co-add as the background color. We only consider galaxies in the redshift range z = 0.15 to 0.55 and the galaxies that pass all the size, flag, and flux cuts mentioned in Section 4.1. This is the baryonic mass distribution and is roughly expected to follow the large-scale structure. The contours from the aperture mass map are overlaid on this. The symbols show the reported galaxy groups and clusters on NED (NASA Extragalactic Database). The star symbols were obtained using cluster-finding methods that rely on optical data. The yellow star corresponds to the list published by Wen et al. (2012) and later updated by Wen & Han (2015). The black star corresponds to the list published by Rozo et al. (2015). Both of these methods primarily use SDSS data. The green star corresponds to the list published by Gal et al. (2009) using data from the Digitized Second Palomar Observatory Sky Survey. X-ray analysis of this region of the sky is primarily based on XMM-Newton data. The location of smaller groups and clusters based on available X-Ray data is shown in squares. The black square shows the location of groups mentioned in Haines et al. (2018). The yellow square corresponds to locations mentioned in Giles et al. (2022). The red cross is cluster location derived from the SZ effect in Planck data (Ade et al. 2016; Khatri 2016). It is clear in Figure 11a that most of the smaller galaxy groups are successfully detected by our aperture mass map. There are some regions in our map that have high light density but is neither detected by weak lensing nor the different galaxy group detection methods mentioned above. This could arise from a variety of factors. The light maps could be contaminated with light from sources that are not within the desired redshift range. It could also be the case that our images are significantly deeper than any existing data to which the cluster-finding algorithms have been applied. Thus, these structures are visible in our light map only. \nIt has been shown that aperture mass statistics or significance maps are able to trace the cosmic filamentary structures (Jauzac et al. 2012; HyeongHan et al. 2023). We are able to recover a few broad filamentary structures as shown in Figure 11a. We notice 4 major structures in the light density maps. One towards the north, one to the west, one to the southwest, and one to the southeast. All of these are broadly traced by the contours. The contours also trace a structure to the northeast where there seems to be a small galaxy group. We note that neither the aperture mass map nor the convergence map is able to detect the filament northwest of the central region. In Figure 12 we overlay the E-Mode contours on light density maps of galaxies in the redshift range z= 0.35 and 0.75. The light density maps are made using the same method as before.', '7. CONCLUSION': "In this paper we present a weak lensing analysis of the galaxy cluster Abell 2390 using extremely deep images obtained from WIYN-ODI. We introduce a novel method that allows us to obtain shear information from galaxies which are measured to be smaller than the PSF. This allows us to create mass maps with higher source density than previously possible with images of similar depth. We measured shapes in individual exposures using a moment matching algorithm. The forced measurement method was used when SNR was too low for convergence. The aperture mass maps obtained show that we are able to recover most of the smaller galaxy groups, identified using optical and X-ray cluster-finding algorithms. A group of galaxies approximately 10 ' North-West of the cluster was also found and \nappears to be in the process of infalling into the cluster from spectroscopic data. In addition to this, most of the filamentary structures around Abell 2390 were also recovered in the aperture mass maps. Within a radius of ∼ 220 kpc from the BCG, Abell 2390 appears to have a bi-modal mass distribution, with the smaller peak being southeast of the main peak, which is consistent with the X-ray excess found in Chandra data. The separation between the peaks is 220 kpc whereas the separation in X-Ray is 20 kpc. This suggests that Abell 2390 is a case of an extreme late-stage merger, with the hot gas close to the center now being completely relaxed following the most recent merger. However, due to the lack of friction during the infall period, the dark matter is still actively merging with the main DM core. The merger hypothesis is supported by CO radio observation using ALMA which find a tail in the BCG that can be explained by a recent gravitational disturbance. The high spectral index of 1.6 found using VLA data and radio halo found in both VLA and careful re-examination of LOFAR DR2 also support this hypothesis. If true, more such findings in other galaxy clusters along with simulations might help us put upper limits on the cross-section of DM. We find our mass estimate for this galaxy cluster is consistent with previously published results.", '8. ACKNOWLEDGMENTS': 'The authors thank the anonymous referee for the useful comments and suggestions. The authors would like to thank Purdue University for its continued support. We are also very grateful to the WIYN-ODI PPA team, especially Wilson Liu, Nick Smith, Arvind Gopu and all the telescope operators for their help in obtaining excellent quality data. We also thank the Purdue Rosen Center for Advanced Computing (RCAC) for access to computing facilities that have been extensively used in this paper. This data analysis has been done on python and the authors acknowledge the use of astropy(Astropy Collaboration et al. 2013b, 2018, 2022), numpy(Harris et al. 2020), scipy(Virtanen et al. 2020), matplotlib(Hunter 2007) and aplpy (Robitaille & Bressert 2012; Robitaille 2019).This research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. \nFacilities: WIYN-ODI 3.5m,', 'REFERENCES': 'Abraham, R. G., Smecker-Hane, T. A., Hutchings, J. B., et al. 1996, The Astrophysical Journal, 471, 694-719, doi: 10.1086/177999 \nAde, P. A. R., Aghanim, N., Arnaud, M., et al. 2016, Astronomy and Astrophysics, 594, A27, doi: 10.1051/0004-6361/201525823 Alcorn, L. Y., Yee, H. K. C., Drissen, L., et al. 2023, MNRAS, 522, 1521, doi: 10.1093/mnras/stad948 Allen, S., Ettori, S., & Fabian, A. 2001, Monthly Notices of the Royal Astronomical Society, 324, 877-890, doi: 10.1046/j.1365-8711.2001.04318.x Annis, J., Soares-Santos, M., Strauss, M. A., & et al. 2014, The Astrophysical Journal, 794, 120, doi: 10.1088/0004-637x/794/2/120 Applegate, D. E., von der Linden, A., Kelly, P. L., et al. 2014, Monthly Notices of the Royal Astronomical Society, 439, 48-72, doi: 10.1093/mnras/stt2129 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013a, aap, 558, A33, doi: 10.1051/0004-6361/201322068 -. 2013b, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 \nAstropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Astropy Collaboration, Price-Whelan, A. M., Lim, P. 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2024arXiv240912790F

We investigate the oneparticle sector for the fieldtheoretical model of gravitationally induced decoherence for a scalar field in 1 with a special focus on the renormalisation of the oneparticle master equation. In contrast to existing models in the literature where the renormalisation is usually performed after the Markov and rotating wave approximation and often only for certain limits such as the non or ultrarelativistic limit here we apply the renormalisation directly after the oneparticle projection. With this strategy we show that UVdivergent contributions in the oneparticle master equation can be identified with the vacuum contributions in the selfenergy of the scalar field in the effective quantum field theory and depending on the chosen oneparticle projection method its vacuum bubbles while the additional thermal contributions in the selfenergy are all UVfinite. To obtain the renormalised oneparticle master equation we use an onshell renormalisation procedure of the underlying effective QFT. We then apply the Markov and rotating wave approximation specifying a condition under which the Markov approximation can be applied in the case of the ultrarelativistic limit. We compare our results with those available in the literature. This includes an analysis of two different kinds of oneparticle projections a comparison of the application and effects of renormalisation of quantum mechanical and field theoretical models the nonrelativistic and ultrarelativistic limits of the renormalised oneparticle master equations and a comparison with a quantum mechanical toy model for gravitationally induced decoherence in the context of neutrino oscillations.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.12790', 'arXiv:2409.12790', '2024arXiv240912790F']

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

Gravitationally induced decoherence of a scalar field investigating the oneparticle sector and its interplay with renormalisation

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12790.pdf

{'Gravitationally induced decoherence of a scalar field: investigating the one-particle sector and its interplay with renormalisation': 'Max Joseph Fahn ∗ and Kristina Giesel † \nInstitute for Quantum Gravity, Theoretical Physics III, Department of Physics, FAU Erlangen-Nurnberg, Staudtstr. 7, 91058 Erlangen, Germany. \nWe investigate the one-particle sector for the field-theoretical model of gravitationally induced decoherence for a scalar field in [1] with a special focus on the renormalisation of the one-particle master equation. In contrast to existing models in the literature, where the renormalisation is usually performed after the Markov and rotating wave approximation and often only for certain limits such as the non- or ultra-relativistic limit, here we apply the renormalisation directly after the one-particle projection. With this strategy, we show that UV-divergent contributions in the one-particle master equation can be identified with the vacuum contributions in the self-energy of the scalar field in the effective quantum field theory and depending on the chosen one-particle projection method, its vacuum bubbles, while the additional thermal contributions in the self-energy are all UV-finite. To obtain the renormalised one-particle master equation, we use an on-shell renormalisation procedure of the underlying effective QFT. We then apply the Markov and rotating wave approximation, specifying a condition under which the Markov approximation can be applied in the case of the ultra-relativistic limit. We compare our results with those available in the literature. This includes an analysis of two different kinds of one-particle projections, a comparison of the application and effects of renormalisation of quantum mechanical and field theoretical models, the non-relativistic and ultra-relativistic limits of the renormalised one-particle master equations, and a comparison with a quantum mechanical toy model for gravitationally induced decoherence in the context of neutrino oscillations.', 'CONTENTS': 'References \n74', 'I. INTRODUCTION': "Since we expect closed and thus isolated quantum systems to be an idealisation, there is increasing interest in various areas of physics in the investigation of open quantum systems. These are usually modelled by a total system consisting of two parts: the system under consideration and the environment as well as the interaction between the two. By tracing out the degrees of freedom of the environment, the effective dynamics of the system under consideration are obtained, which, in contrast to the closed system, is still affected by the interaction with the environment [2-4]. The effective dynamics of the system under consideration is often formulated in the form of a so-called master equation, of which the Lindblad equation is a special and prominent example [5, 6]. An important question in the context of open quantum systems is, what to choose as the environment and how to model the interaction with the system. Since we assume that all standard matter is at least coupled to gravity, interest in open quantum systems with a gravitational environment has increased in recent years, see for instance [7-13] and the reviews in [14, 15] and references therein. This has been investigated in the context of gravitationally induced decoherence due to a quantum gravitational environment in the framework of linearised gravity in several works in the literature. For instance in [1, 8, 9] a scalar field is considered as the matter field, while in [12, 16] a photon field is considered and in [11] a model for a generic bosonic matter field is investigated. All these works have in common that they start from the underlying field theoretical model and then derive a corresponding master equation for the matter system under consideration. \nIn addition, there are also phenomenological models, see for instance [7, 10, 17-20], that do not necessarily derive a master equation from an underlying action, but rather set up physically motivated ansatze for the form of the dissipator, which encodes the effective influence of the environment in the master equation. The latter are often formulated in the context of quantum mechanics and thus for finitely many degrees of freedom, where a given master equation is often less difficult to work with or even to solve. Furthermore, many of these models already used the Lindblad equation as a starting point, so that the physical properties of the environment are often less accessible compared to the underlying field theoretical models. An interesting question is therefore, on the one hand, how the underlying field theoretical model can be linked to a given quantum-mechanical phenomenological model and, on the other hand, what additional insights the underlying field-theoretical model can provide that are no longer accessible to us in the quantum-mechanical model. In this work we want to address both question in the context of models for gravitationally induced decoherence. \nTo answer these questions, the one-particle sector of field theory needs to be investigated to establish a link to microscopic quantum mechanical models. Furthermore, in order to investigate the connection to existing phenomenological models for gravitationally induced decoherence, one needs to understand more precisely how certain methods such as renormalisation and specific approximations such as the Markov or rotating wave approximation, which are often performed to finally arrive at a Lindblad-type master equation, affect the field-theoretical model respectively its one-particle sector. Some of these questions have already been discussed and answered in the works in [8, 9, 12]. There, the one-particle sector was derived from a field-theoretical model and quantum mechanical master equations were derived for the non-relativistic and ultra-relativistic cases. The new aspect we aim at investigating in this work is the role of renormalisation in this context and its interplay with the further approximations, such as the Markov and rotation wave approximation, that one needs to apply in the derivation of the final master equation. \nTo the best of the authors' knowledge, renormalisation for field-theoretical models of gravitationally induced decoherence has been performed in the existing literature after applying the Markov \nand rotational wave approximation, often at the level of the corresponding one-particle sector in certain limits such as the ultra-relativistic limit [12] or the non-relativistic limit [9]. In contrast in this work we will perform the renormalisation at the level of the effective field theory before we apply any of the above mentioned approximations or limits. This strategy allows us to obtain a more detailed understanding about the UV-divergent contributions in the one-particle master equations. We will follow the methods introduced in [21], where one scalar field was coupled to a second scalar field as an environment and extend those techniques to the case of a gravitational environment. These methods allow us to identify individual contributions in the one-particle master equations with specific Feynman diagrams of the underlying effective field-theoretical model. Since the starting point of the model in this paper is the canonical formulation of a scalar field coupled to linearised gravity, in a first step we introduce non-covariant Feynman rules adapted to the canonical model, following [22, 23], where this was introduced for the case of QED. Interestingly, the connection between the covariant and non-covariant Feynman rules can be used to show that the divergent contributions in the one-particle master equation are involved in the self-energy of the scalar field in the effective field theory. The self-energy can be decomposed into a vacuum and a thermal contribution, the latter vanishing when we consider the zero temperature limit of thermal gravitational waves in the environment. The vacuum contributions can then be renormalised using a standard procedure, and the renormalised one-particle master equation can be obtained. \nEquipped with this result, we can use it and investigate what kind of effect a Markov and rotating wave approximation have and compare it to the existing literature, where these two approximations are mostly done before a renormalisation is preformed. At least for the case of the ultra-relativistic limit, we are able to provide a condition under which the Markov approximation can be applied in the model considered here. For the general case, this is quite a challenging task, since the integrals involved, which have to be analysed for the environmental correlation functions, are quite complicated. In the case of the rotating wave approximation in the existing literature the pre- and post-trace application (see for instance [12, 24-26] for analyses and applications) is discussed and we apply the latter in this work and determine the final one-particle master equation where both the Markov and rotating wave approximation have been applied. This is used together with the intermediate results before the individual approximations in some applications to compare with the existing results in the literature, in particular the work in [9, 11, 12] for field-theoretical models and the quantum mechanical model in [27] in the context of gravitationally induced decoherence in neutrino oscillations, which is based on a quantum mechanical toy model for gravitationally induced decoherence from [28]. For the comparison with the quantum mechanical model, we are especially interested in the extent to which we can relate the application and effect of renormalisation in the field-theoretical model and in the quantum mechanical model and how we can thereby gain new insights into the differences, and similarities respectively as well as the physical properties of these models. \nThe paper is structured as follows: after the introduction in section I, we briefly review the field-theoretical model from [1] in section II whose one-particle projection is derived in section III, where we consider two types of projections, a non-extended and an extended one, which differ in whether each individual operator in the master equation preserves the one-particle sector or whether only the combinations of operators that enter the final master equation must do so. The renormalisaton of the one-particle master equation is discussed in section IV. First we identify those contributions in the one-particle master equations that are UV-divergent in subsection IV A. Subsection IV B and IV C introduce the non-covariant and covariant Feynman rules and discuss their relation as well as show how the UV-divergent contributions can be identified with the vacuum part of the scalar field's self-energy. The renormalisation of the self-energy is discussed in \nsubsection IV D and in subsection IV E we use these results to determine the final form of the renormalised one-particle master equation. Afterwards in section V we apply the Markov and rotating wave approximation that are separately discussed in subsection V A and V B respectively. As possible applications in section VI we discuss in subsection VI A the evolution of the populations of the one-particle master equation before and after renormalisation as well as after in addition the Markov approximation has been applied, see subsections VI A 1, VI A 2 and VI A 3 and compare our results with the ones in [11]. To compare more in detail to the existing results in [8, 9, 12] we consider the non-relativistic and ultra-relativistic limit in subsections VI B and VI C. The comparison with quantum mechanical model from [27] that considers gravitationally induced decoherence in the context of neutrino oscillations can be found in subsection VI D. Finally we summarise and conclude in section VII. In addition, details of the calculations required to obtain the results of each section are provided in the appendix to make the article self-contained.", 'II. REVIEW OF THE UNDERLYING FIELD-THEORETICAL MASTER EQUATION': "As we aim at investigating the one-particle sector of model considered in [1] in this work and compare the results obtained here to results in the existing literature such as in [8, 9, 12], we briefly review the main results from [1] that are taken here as a starting point for the further analysis. In [1] a step-by-step derivation of a second order, time-convolutionless master equation for gravitationally induced decoherence of a scalar field is presented with the aim to compute the effective dynamics of the scalar field evolving in an environment consisting of thermal gravitational waves. The starting point in [1] is the classical action of general relativity coupled to a scalar field, where the mostly plus signature is used for the metric. To be able to apply canonical quantisation later on, general relativity is then formulated in the Hamiltonian (ADM) framework ([29]) formulated in terms of Ashtekar-Barbero variables ([30-33]), which encodes the gravitational degrees of freedom in terms of an SU(2)-connection A i a ( ⃗x, t ) and its canonically conjugate momenta denoted as densitised triads E a i ( ⃗x, t ). Here, the first index a is a spatial one and the second one i an SU(2)-Lie algebra index. The metric can be determined by the densitised triads only up to a rotation and this yields to an additional so-called Gauß constraint next to the Hamiltonian and spatial diffeomorphism constraint already present in the ADM formulation. \nThe reason that [1] works with these elementary variables is that they form the elementary canonical variables in the context Loop Quantum Gravity (LQG, see [34, 35] for an introduction) and hence formulating the classical model in terms of them makes the application of a quantisation using LQG techniques possible, as discussed in [1]. After the inclusion of a boundary term for an asymptotic flat spacetime ([36]), the gravitational sector of the system is linearised around a Minkowski spacetime, where κ = 8 πG N c 4 plays the role of the perturbation parameter and G N denotes Newton's constant. The matter contribution is included in terms of a post-Minkowski approximation [37]. As the linearised model still includes gauge symmetries, suitable Dirac observables are then constructed in perturbation theory using the relational formalism ([38-44]) to extract the physical degrees of freedom. For gravity they correspond to the linearised symmetric transverse traceless components of the connection and the triad fields. These constitute of two degrees of freedom each, equivalent to the two polarisations of gravitational waves and their conjugate momenta which are the linearised excitations one obtains when using the metric and its conjugate momenta as the elementary variable. In addition one can construct Dirac observables for the scalar field and its canonically conjugate momentum for the matter sector. \nAfter a Fock quantisation, a time-convolutionless master equation ([2]) is derived by tracing out the gravitational environment, which is described by a Gibbs state with temperature 1 parameter \nΘ, and in this way treating the scalar field as an open quantum system. For this, the projection operator techniques (see [2, 45, 46]) are employed, for which one splits the Hilbert space into a relevant and an irrelevant part. In the model considered here the scalar field is the relevant and the gravitational environment the irrelevant part. Then the strategy is to solve the Liouville-von Neumann equation of the relevant part perturbatively using an approximated, truncated solution for the irrelevant part. The final master equation in [1] is time-convolutionless and truncated after second order in √ κ . It reads \n∂ ∂t ρ S ( t ) = -i [ H S + κ U + κ H LS , ρ S ( t )] + D [ ρ S ( t )] . (2.1) \nHere, ρ S ( t ) denotes the density matrix describing the state of the scalar field at physical 2 time t . Furthermore, H S is the Hamiltonian of the free scalar field, \nH S := ∫ R 3 d 3 k ω k a † k a k , (2.2) \nwhere a ( † ) k denote the annihilation/creation operator valued distributions for momentum ⃗ k corresponding to the scalar field, that obey the standard commutation relations [ a k , a † l ] = δ 3 ( ⃗ k, ⃗ l ), and ω k = √ ⃗ k 2 + m 2 its dispersion relation. The other terms in (2.1) arise due to the coupling to linearised gravity and represent a self-interaction term U and a Lamb-shift-like term H LS (given in (A.I.2)) as well as a dissipator term D (given in (A.I.3)). The first one arises when the Hamiltonian is expressed in terms of the independent Dirac observables mentioned above and the Lamb-shiftlike term H LS contributes to the unitary evolution of the scalar field, whereas the dissipator leads to to non-unitary evolution encoding effects like decoherence. A similar form for a master equation can be obtained by using the Born approximation instead of the projection operator technique, see section 4 in [1] for more details. \nWhile this master equation predicts the effective dynamics of the scalar field under the influence of a gravitational environment, we still need to deal with infinitely many degrees of freedom which makes investigating the solution of the master equation very challenging. For this reason in this work the master equation for a single scalar particle that follows from the master equation in (2.1) is derived. \nIn general, this master equation, as well as the full field theoretical one from [1], might not be completely positive and usually to check this is a challenging task for the field theoretical models. To obtain a completely positive master equation that allows a probability interpretation of the density matrix for all times larger than the initial time, a common procedure is to invoke several approximations. In this work, we also discuss two frequently applied approximations of this kind in detail, which are the Markov and the rotating wave approximation, and investigate their applicability as well as their interplay with the renormalisation of the model.", 'III. PROJECTION OF THE MASTER EQUATION TO THE ONE-PARTICLE CASE': 'In this section, we discuss how such a master equation for a single scalar particle can be obtained starting with equation (2.1) in the field theory context. When working with field theoretical master equations such a one-particle projection is commonly applied to investigate some features of the master equation, see for instance the works in [9, 11, 12, 21]. There exist however different methods \non how to perform this projection in detail. In [9, 11] the procedure is carried out such that the final one-particle master equation is probability conserving, which requires to neglect some terms that would otherwise be present in a direct projection. In contrast, in [21] a different strategy based on Thermo Field Dynamics (TFD), which is a formulation of Quantum field theory at finite temperature (see [47, 48] and for an introduction [49]), is employed, in which these terms still contribute to the one-particle master equation. Here we follow the method used in [9, 11], but keep all possible terms and investigate their influence on the one-particle master equation. It will turn out that after applying an on-shell renormalisation and Markov approximation, they will not play any role for decoherence, but will remove the remaining contribution of the Lamb-shift-like term to the unitary evolution after the rotating wave approximation has been applied. \nTo obtain the one-particle projection of the master equation in (2.1), we replace the density matrix with the corresponding density matrix for a single particle in momentum representation \nρ 1 ( t ) = ∫ R 3 d 3 u ∫ R 3 d 3 v ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v , (3.1) \nin the master equation and neglect all contributions that project out of the single particle space. In this formulation, ρ ( ⃗u, ⃗v, t ) is the (quantum mechanical) density matrix of a single particle in momentum representation. \nIn the following we will discuss the corresponding individual contributions in (2.1) separately and further will discuss the assumptions used in the model considered here as well as compare them to the existing literature: \nThe first term of the master equation, representing the evolution of a free scalar particle, can be computed immediately to yield \n-i [ H S , ρ 1 ( t )] = -i ∫ R 3 d 3 u ∫ R 3 d 3 v ( ω u -ω v ) ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v . (3.2) \nThe contribution of the second term depends on the structure of the form of the operator U . The detailed expression is given in [1] in equation (3.12), for the analysis here it is however enough to know that U consists of different combinations of always four creation and/or annihilation operators for the scalar field, i.e. \nU = ∫ d 2 k 1 ∫ d 2 k 2 ∫ d 2 k 3 ∫ d 2 k 4 [ ζ 1 a k 1 a k 2 a k 3 a k 4 + ζ 2 a † k 1 a k 2 a k 3 a k 4 + ζ 3 a k 1 a † k 2 a k 3 a k 4 + ... + ζ N a † k 1 a † k 2 a † k 3 a † k 4 ] , (3.3) \nwith the coefficient distributions ζ i = ζ i ( ⃗ k 1 , ⃗ k 2 , ⃗ k 3 , ⃗ k 4 ) that contain delta distributions that relate some of the momenta. For more details, we refer to the definition in equation (3.12) in [1]. When applying normal ordering to this operator, as it is done in [1], then it will not contribute after the one-particle projection: in the summands of U , where the number of creation operators is not equal to the number of annihilation operators, the resulting terms would project out of the one-particle space. In the other summands, there are exactly two creation and two annihilation operators which, when normal ordered, annihilate any one-particle state. In [9, 12] the normal ordering of U is applied differently: in their work the four annihilation and/or creation operators are normal ordered pairwise 3 . In that case contributions of the form : a † k 1 a k 2 : : a † k 3 a k 4 : preserve the one-particle space and thus still contribute after the one-particle projection. To distinguish \nthese two types of operator orderings, we denote the first one, where U is normal ordered, total normal ordering, and the second one partial normal ordering. In this work we consider a totally normal ordered Hamiltonian as in [1]. \nThe third term, the Lamb-shift-like Hamiltonian, as well as the fourth contribution, the dissipator, both contain the same building blocks. To evaluate them, it is sufficient to consider the following three combinations: \nj A r ( ⃗ k, ⃗p ) † j B r ( ⃗ k, ⃗ l ) ρ 1 ( t ) ︸ ︷︷ ︸ ( I ) and ρ 1 ( t ) j A r ( ⃗ k, ⃗p ) † j B r ( ⃗ k, ⃗ l ) ︸ ︷︷ ︸ ( II ) and j B r ( ⃗ k, ⃗ l ) ρ 1 ( t ) j A r ( ⃗ k, ⃗p ) † ︸ ︷︷ ︸ ( III ) , (3.4) \nwhere the j A r ( ⃗ k, ⃗p ) denote individual and different normal-ordered current operators labeled by A ∈ { 1 , 2 , 3 , 4 } that carry a polarisation label r ∈ {±} and two momentum arguments. These current operators are defined in detail starting in equation (A.I.8) and are of the form \nj 1 r ( ⃗ k, ⃗p ) ∝ a † p a k + p j 2 r ( ⃗ k, ⃗p ) ∝ a † -p -k a -p (3.5) \nj 3 r ( ⃗ k, ⃗p ) ∝ a -p a k + p j 4 r ( ⃗ k, ⃗p ) ∝ a † p a † -k -p . (3.6) \nHence they consist of two creation and/or annihilation operators with different momentum labels. At this point arises the question whether we want to enforce trace preservation in the one-particle master equation, which corresponds to probability conservation. In [9, 11] this is done, which results in the exclusion of specific terms from the one-particle master equation. These terms can be identified from the general form of the master equation in (2.1) as we will discuss now: it is evident that when applying the trace the commutator vanishes and one is left with \n∂ ∂t tr { ρ S ( t ) } = tr {D [ ρ S ( t )] } . (3.7) \nInserting the definition of the dissipator given in [1] in equation (4.74) then yields \n∂ ∂t tr { ρ S ( t ) } = κ 2 ∫ d 3 k d 3 p d 3 l (2 π ) 3 ∑ r ∈{±} 4 ∑ A,B =1 R AB ( ⃗ p, ⃗ l ; ⃗ k, t ) · tr { j B r ( ⃗ k, ⃗ l ) ρ S ( t ) j A r ( ⃗ k, ⃗p ) † -1 2 { j A r ( ⃗ k, ⃗p ) † j B r ( ⃗ k, ⃗ l ) , ρ S ( t ) } } , (3.8) \nwhere the R AB are time-dependent coefficients. When the current operators j A r are individually projected onto the one-particle space, due to the cyclicity of the trace all terms in the difference of the two traces are exactly canceled and one obtains a preserved trace of the density matrix, hence probability conservation. This is the approach used for instance in [9, 11]. Another option is to apply the one-particle projection in such a way that each entire term in the master equation has to preserve the one-particle space. This is for instance done in [21], where two scalar fields are considered, one as the system and the other one as the environment. In this case there will remain terms in the one-particle projection of the product of two current operators in the last term of (3.8) that have no counterpart in the first term of (3.8) and thus will not cancel in the difference of the two traces. \nTo keep our analysis as generic as possible, we will include these terms in this work and investigate their effect in the one-particle master equation and denote this one-particle projection the extended one-particle projection. To take them into account in our further calculations, we will introduce a \nfactor δ P in these contributions to be able to switch between the extended one-particle projection ( δ P = 1) and the non-extended one ( δ P = 0). \nThe detailed derivation of the one-particle projection of the master equation following these methods introduced here can be found in appendix A.I. The additional terms that are present in the extended one-particle projection correspond to physical situations in QFT in which in the intermediate steps two particles are created and annihilated afterwards. This also includes the case where the original particle is left invariant and a vacuum bubble is created. The latter case thus requires a renormalisation, which is also carried out in appendix A.I. Following these projection methods, the one-particle master equation for the density matrix in momentum representation is then given in (A.I.47) and reads \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) -κ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u [ C ( ⃗u, ⃗ k, t ) + δ P C P ( ⃗u, ⃗ k, t ) ] + P v ( ⃗ k ) ω v -k ω v [ C ∗ ( ⃗v, ⃗ k, t ) + δ P C ∗ P ( ⃗v, ⃗ k, t ) ] } ρ ( ⃗u, ⃗v, t ) + κ 2 ∫ d 3 k (2 π ) 3 P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω u ω v + k ω v { C ( ⃗u + ⃗ k, ⃗ k, t ) + C ∗ ( ⃗v + ⃗ k, ⃗ k, t ) } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) (3.9) \nwith P u ( ⃗ k ) := P ijln ( ⃗ k ) u i u j u l u n which contains the symmetric transverse traceless (STT-)projector \nP ijln ( ⃗ k ) = 1 2 [ P il ( ⃗ k ) P jn ( ⃗ k ) + P in ( ⃗ k ) P jl ( ⃗ k ) -P ij ( ⃗ k ) P ln ( ⃗ k )] , (3.10) \nthat in turn consists of combinations of the transverse projectors \nP ij ( ⃗ k ) = δ ij -k i k j ⃗ k 2 . (3.11) \nThe presence of this projector is a consequence of the chosen Dirac observables, and thus the physical degrees of freedom of the linearised gravitational field. The coefficients in (3.9) are defined as \nC ( ⃗u, ⃗ k, t ) = ∫ t -t 0 0 dτ Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k -ω u ) τ + N ( k ) e i (Ω k -ω u -k + ω u ) τ } (3.12) \nC P ( ⃗u, ⃗ k, t ) = ∫ t -t 0 0 dτ Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k + ω u ) τ + N ( k ) e i (Ω k -ω u -k -ω u ) τ } , (3.13) \nwhere N ( k ) = 1 e β Ω k -1 is the Bose-Einstein distribution of the gravitational waves in the environment with frequencies Ω k := √ ⃗ k 2 and β = ( k B Θ) -1 , where k B is the Boltzmann constant and Θ the temperature parameter of the Gibbs state that characterises the environment of thermal gravitational waves. The term in the first line of the master equation in (3.9) represents the standard unitary evolution of the free scalar particle. The remaining terms describe the influence of the environment and encode in general different physical processes like energy shifts, dissipation and decoherence. While the expressions in lines two and three only depend on the state ρ ( ⃗u, ⃗v, t ) considered at time t , the last line links this state to other states ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) at time t . This master equation however still has some problematic contributions that possess UV-divergences and hence needs to be UV-renormalised, as will be discussed in the next section.', 'IV. RENORMALISATION OF THE TCL ONE-PARTICLE MASTER EQUATION': 'Upon investigation of the individual contributions in (3.9) it becomes evident that some terms exhibit divergences as will be discussed in detail below. This raises the question of at what stage of the derivation of the master equation the renormalisation procedure should be carried out. In the literature, there are different strategies how to deal with these divergent contributions. For gravitationally induced decoherence, they have often not been computed in detail due to the reason that they are expected to not modify decoherence but only influence the unitary evolution, see for instance the discussion in [8, 11]. In [9, 12] the renormalisation of these contributions has been performed in the end after a Markov and rotating wave approximation have already been applied. \nIn this work we choose the strategy to renormalise the master equation first before applying further approximations or deriving physical implications. It will turn out that effects predicted with a non-renormalised master equation might get modified or even vanish when working with the renormalised version instead. An example of this kind is also discussed in [27], where a quantum mechanical toy model for gravitationally induced decoherence based on the model in [28] is applied in the context of neutrino oscillations. In that case the necessary renormalisation is very trivial compared to the model considered in this work. There, the renormalisation causes the contributions of the Lamb-shift Hamiltonian to cancel exactly. Consequently, all physical implications involving contributions of the Lamb-shift Hamiltonian, as discussed for example in [50], would be absent when working with the renormalised model presented in [27]. \nIn order to carry out the renormalisation, we will first identify the diverging terms. As we will discuss below in more detail, these are in particular the terms in the second and third line of the master equation (3.9) that will also be present in the case where the temperature parameter vanishes, that is for Θ = 0, in which the thermal state merges into a vacuum state. They are of the form ∫ d 3 k 1 | ⃗ k | 3 and thus yield a logarithmic UV-divergence. Once these contributions are identified, we express them in the form of Feynman diagrams of the underlying effective QFT. For this purpose, we follow the strategy in [21], where a master equation for a scalar field with an environment consisting of another scalar field is presented. Here the treatment is extended so that the linearised gravitational field can be included as an environment. We will proceed in five steps: first in subsection IV A we will identify the divergent contributions in the master equation and then present the corresponding Feynman rules following from the underlying effective QFT based on the non-covariant formulation in subsection IV B. Afterwards in subsection IV C we provide a set of equivalent, covariant Feynman rules in terms of which we perform the renormalisation of the divergent contributions in subsection IV D. Finally we discuss the resulting renormalised one-particle master equation in subsection IV E.', 'A. Identification of the UV divergences in the one-particle master equation': 'Starting from the master equation in (3.9), we want to investigate which terms on the righthand side are UV-divergent with respect to the ∫ R 3 d 3 k integration. As the projector P ijln ( ⃗ k ) is independent of the absolute value of ⃗ k , it does not influence the UV behaviour. Then, one can identify four different types of contributions in the integrands after performing the τ -integration and introducing the following sign factors σ, σ 1 , σ 2 ∈ {± 1 } : \n- (a) 1 ω u -k Ω k 1 Ω k + ω u -k + σω u : for large | ⃗ k | , i.e. for | ⃗ k | >> | ⃗u | , m this term becomes 1 | ⃗ k | 3 and thus leads to a logarithmic UV-divergence under the integral.\n- (b) 1 ω u -k Ω k 1 Ω k + ω u -k + σω u ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ): assuming that ρ ( ⃗x, ⃗y, t ) is a proper, normalisable density matrix in position space for which the Fourier transform exists leads to the requirement that ρ ( ⃗u, ⃗v, t ) has to decrease rapidly for large ⃗u, ⃗v . Therefore this expression is UV-finite.\n- (c) N ( k ) 1 ω u -k Ω k 1 Ω k + σ 1 ω u -k + σ 2 ω u : a series expansion of the denominator of N ( k ) yields N ( k ) = 1 β | ⃗ k | + O ( | ⃗ k | 2 ) . Hence, this term tends to zero for large | ⃗ k | and also the combination x n e x -1 decreases rapidly for x →∞ , thus this kind of contribution is UV-finite.\n- (d) N ( k ) 1 ω u -k Ω k 1 Ω k + σ 1 ω u -k + σ 2 ω u ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ): this contribution is a combination of cases (b) and (c) and also UV-finite. \nFrom this analysis follows that the expressions involving N ( k ), that would be absent in the vacuum case and are thus denoted as thermal contributions in the following, are all UV-finite. Some of the vacuum contributions, these are the ones that do not involve N ( k ), lead to UV-divergences which we want to cure by a renormalisation. To achieve this, in the next section we show in a first step that these terms correspond exactly to the self-energy diagrams for the scalar particle in the form of Feynman diagrams.', 'B. Non-covariant Feynman rules and self-energy': "In this section we present the Feynman rules in non-covariant form corresponding to the effective quantum field theory containing a scalar field coupled to linearised gravity in [1], where the latter is considered as the environment, which is the basis for the master equation in (3.9). Then, we rewrite these rules in the next section in a covariant form to be able to follow the strategy of [21], where a suitable renormalisation for a master equation for a scalar fields with a second scalar field as the environment is discussed. Here, we slightly extend these methods in order to apply them to the case where the linearised gravitational field is treated as the environment. The Feynman rules can be constructed from [1]: \n- · The scalar field has the standard propagator , which follows from its quantised mode expansion in (3.4) and (3.5) in [1], which we denote by a solid line and which reads in the mostly plus signature convention: \n= -i k 2 + m 2 -iϵ . (4.1) \nHere, k 2 = -( k 0 ) 2 + ⃗ k 2 . \n- · The propagator of the triad field was derived in [1] in equation (4.46) and is denoted in terms of a curved line: \n= 1 κ P abcd ( ⃗ k ) [ -i k 2 -iϵ +2 πN ( k ) δ ( k 2 ) ] , (4.2) \nwhere the first summand is the vacuum and the second one the thermal part. The existing tensor structure manifests in the form of the tensor structure of the STT-projector defined \nabove in (3.10). When contracted with a quantity that is symmetric in ( ab ) as well as in ( cd ), like it is the case for the interaction vertex introduced below, the STT-projector reduces to \nP abcd ( ⃗ k ) = δ ac δ bd -1 2 δ ab δ cd + 1 2 ⃗ k 4 k a k b k c k d -1 ⃗ k 2 ( 2 δ ac k b k d -1 2 δ ab k c k d -1 2 δ cd k a k b )] . (4.3) \n- · The coupling between the scalar field and linearised gravity is encoded in the interaction part of the total action in [1] that is given as a reformulation of equation (3.17) in that work by: \nS int = -∫ dt ∫ d 3 x H int ( ⃗x, t ) = -∫ dt ∫ d 3 k ⃗ k 2 κ { ⃗ k 2 δ E ab ( ⃗ k, t ) T ab ( -⃗ k, t ) -1 2 T 00 ( ⃗ k, t ) T a a ( -⃗ k, t ) -1 4 T 00 ( ⃗ k, t ) T 00 ( -⃗ k, t ) + T 0 a ( ⃗ k, t ) T 0 b ( -⃗ k, t ) ( δ ab -k a k b 4 ⃗ k 2 )} , (4.4) \nwhere T µν ( ⃗ k, t ) denotes the Fourier transform of the scalar field's energy momentum tensor 4 . The interaction vertex between the scalar and the triad field can then be read off and is related to T ab . Due to the fact that T ab depends on derivatives of the scalar field, the expression for the triad-scalar-field vertex is different depending on the direction of the momenta involved in the diagrams. Considering the Fourier transform of T ab ( ⃗x, t ), where the the scalar fields can be factorised, we find the expression ˜ T ab ( p, q ) ϕ ( p ) ϕ ( q ) with \n˜ T ab ( p, q ) := 1 2 δ ab [ -p 0 q 0 + ⃗ p · ⃗q -m 2 ] -1 2 [ p a q b + p b q a ] . (4.5) \nHence, the triad-scalar-field-vertex is given by \np q = p q = iκ ˜ T ab ( σ p p, σ q q ) , \nwhere σ p is + if particle p is incoming and -if it is outgoing. \n- · The remaining terms in the interaction part of the action (4.4) give rise to an additional second order vertex which cannot be split into first order vertices due to the lack of a suitable intermediate particle in this effective field theory. They have the form \n-κ ∫ d 3 x ∫ d 3 y -1 4 T 00 ( ⃗x ) T 00 ( ⃗y ) + T 0 a ( ⃗x ) T 0 b ( ⃗y ) δ ab -1 4 ( T 00 ( ⃗x ) T a a ( ⃗y ) + T a a ( ⃗x ) T 00 ( ⃗y )) 4 π || ⃗x -⃗y || -κ ∫ d 3 x ∫ d 3 y ∫ d 3 z -1 4 ∂ a ⃗x T 0 a ( ⃗x ) ∂ b ⃗y T 0 b ( ⃗y ) (4 π ) 2 || ⃗x -⃗z |||| ⃗y -⃗z || . (4.6) \nT 00 = T 00 = 1 2 [ π 2 +( ∂ a ϕ ) ( ∂ a ϕ ) + m 2 ϕ 2 ] = ϵ ( ϕ, π ) , T 0 a = T a 0 = -δ ab T 0 b = -δ ab π ∂ b ϕ = -δ ab p b ( ϕ, π ) , T ab = δ ac δ bd T cd = δ ab [ π 2 -( ∂ c ϕ ) ( ∂ c ϕ ) -m 2 ϕ 2 ] +( ∂ a ϕ ) ( ∂ b ϕ ) . \n2 \nThe corresponding symmetrised Feynman rule reads: \nk p q u v = -iκ ⃗ k 2 NI ( p, q, u, v ) \nwith \nNI ( p, q, u, v ) := [ -1 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) + ˜ T 0 a ( p, q ) ˜ T 0 b ( -u, -v ) ( δ ab -k a k b 4 ⃗ k 2 ) -1 4 ˜ T 00 ( p, q ) ˜ T a a ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] , (4.7) \nwith ⃗ k defined using momentum conservation as ⃗ k = ⃗ p + ⃗q = ⃗u + ⃗v . Note that a similar vertex does also appear in QED when quantised in Coulomb gauge and there it represents the Coulomb interaction, see for instance [22, 23]. \n- · As external lines we only have the scalar field in the cases we are interested in here. This follows the standard case for a quantised scalar field, the detailed expressions are however not required for the following discussions. \nEquipped with these Feynman rules, we will now show in the subsequent section that the divergent contributions in the one-particle master equation can be identified exactly with the contribution of the self-energy diagram constructed with the above Feynman rules. In the model considered here this corresponds to the following Feynman diagram: \nu u -k u k (4.8) \nWith the Feynman rules introduced above, the amplitude represented by this diagram and denoted by Π( u 2 ) has the form \nΠ( u 2 ) := ∫ R 4 d 4 k (2 π ) 4 1 κ P abcd ( ⃗ k ) [ -i k 2 -iϵ +2 πN ( k ) δ ( k 2 ) ] [ iκ ˜ T ab ( u, -( u -k )) ] · -i ( u -k ) 2 + m 2 -iϵ [ iκ ˜ T cd ( u -k, -u ) ] = κu a u b u c u d ∫ R 4 d 4 k (2 π ) 4 P abcd ( ⃗ k ) [ -i k 2 -iϵ +2 πN ( k ) δ ( k 2 ) ] 1 ( u -k ) 2 + m 2 -iϵ = κ ∫ R 3 d 3 k 2(2 π ) 3 P u ( ⃗ k ) { (Ω k + ω u -k ) Ω k ω u -k (Ω k + ω u -k + u 0 -iϵ )(Ω k + ω u -k -u 0 -iϵ ) -N (Ω k ) [ 1 Ω k ( u 0 -Ω k + ω u -k -iϵ )( u 0 -Ω k -ω u -k + iϵ ) + 1 Ω k ( u 0 +Ω k + ω u -k -iϵ )( u 0 +Ω k -ω u -k + iϵ ) ]} =Π vac ( u 2 ) + Π Θ ( u 2 ) (4.9) \nIn the first step the definition of ˜ T ab in (4.5) was used and in the second step the k 0 -integration was performed, where for the vacuum part the residue theorem was applied. In the last step, we have defined the vacuum and thermal contribution to the self-energy as \nΠ vac ( u 2 ) := κ ∫ R 3 d 3 k 2(2 π ) 3 P u ( ⃗ k ) { (Ω k + ω u -k ) Ω k ω u -k (Ω k + ω u -k + u 0 -iϵ )(Ω k + ω u -k -u 0 -iϵ ) \nΠ Θ ( u 2 ) := -κ ∫ R 3 d 3 k 2(2 π ) 3 P u ( ⃗ k ) N (Ω k ) 1 Ω k ( u 0 -Ω k + ω u -k -iϵ )( u 0 -Ω k -ω u -k + iϵ \n(4.10) [ ) + 1 Ω k ( u 0 +Ω k + ω u -k -iϵ )( u 0 +Ω k -ω u -k + iϵ ) ]} . (4.11) \nIf we now want to identify contributions in the one-particle master equation with the self-energy, the following subtlety results: a key difference between the master equation in (3.9) and standard quantum field theory is that the latter is constructed for the limit t 0 →-∞ , t →∞ when evaluating scattering amplitudes. To take this into account, we apply the method presented in [21] for situations where there is a finite temporal interval. In this way, we can transform the self-energy diagram into the second line of the right-hand side of the master equation in (3.9): \nΞ( ω u , ⃗u, t 0 , t ) := ∫ t t 0 dτ ∫ R du 0 Π( u 2 ) cos[( u 0 -ω u )( t -τ )] = ∫ t -t 0 0 dτ ∫ R du 0 Π( u 2 ) cos[( u 0 -ω u ) τ ] . (4.12) \nThe standard QFT-limit can be recovered, in which t → ∞ and t 0 → -∞ , and using this the integral over the temporal interval can be rewritten as a δ -distribution as ∫ ∞ 0 dτ cos[( u 0 -ω u ) τ ] = 1 2 ∫ ∞ -∞ dτe -i ( u 0 -ω u ) τ = πδ ( u 0 -ω u ), that will set the external momentum u on-shell. \nAfter evaluating the u 0 integration we obtain for finite times t and t 0 : \nΞ( ω u , ⃗u, t 0 , t ) = κ 2 ∫ R 3 d 3 k (2 π ) 3 P u ( ⃗ k ) πi 2 ω u -k [ C ( ⃗u, ⃗ k, t -t 0 ) + C P ( ⃗u, ⃗ k, t -t 0 ) ] , (4.13) \nwhich indeed, multiplied by a factor 2 i πω u , can be identified with the first term on the right hand side of the second line of the one-particle master equation in (3.9) in the extended one-particle projection. Given this results, it is now also easy to discuss the case of the non-extended one-particle projection: the master equation for this case, i.e. for δ P = 0, can just be obtained by replacing the cosine in (4.12) by 1 2 e -i ( u 0 -ω u ) τ . We find that in the QFT-limit the difference between the extended and non-extended one-particle projection manifests itself in a factor of 2. \nTo obtain the second term in the second line of (3.9), we can follow the same steps and just have to replace ⃗u by ⃗v and take the complex conjugate. \nWith the results in this section we have shown that the UV-divergent terms in the one-particle master equation correspond to the self-energies of the scalar particle. What remains is to discuss the renormalisation of this self-energy. In order to be able to apply the standard procedure for renormalisation in this case, however, the corresponding covariant Feynman rules of the model considered here must first be derived.", 'C. Covariant Feynman rules': "To be able to employ the standard renormalisation technique for the loop associated with the scalar particle's self-energy, we introduce in this section the covariant Feynman rules corresponding to the effective QFT under consideration here. For this, we follow [22, 23], where the procedure is outlined for QED. \nIn a first step we will demonstrate that specific sums of non-covariant Feynman diagrams add up to the corresponding covariant Feynman diagram. For this purpose, we consider the sum of the second order vertex in (4.6) with a second order combination of the non-covariant scalar field-triad vertex, shown below (4.5). As will be derived below, the second order vertex in (4.6) is precisely that term which restores covariance if we work with a fully covariant triad propagator and a covariant vertex. \nWe will restrict our discussions mainly to a Coulomb-scattering type of diagram here which is sufficient for our later applications. At the end of this section we will also briefly discuss the diagram associated with the scalar particle's self-energy. In the case of the Coulomb-scattering type diagram, the above mentioned equivalence in terms of Feynman diagrams reads \n<!-- image --> \nwhere the curly line corresponds to the covariant triad propagator. \nNext, we will present the covariant Feynman rules, then specialise them to the case of the Coulombscattering type diagram to show the above equivalence. The corresponding covariant Feynman rules for the propagators and vertices discussed in the last section are as follows: \n- · The scalar propagator remains unchanged \n= -i k 2 + m 2 -iϵ . (4.14) \n- · The covariant triad propagator becomes \n= 1 κ -i k 2 -iϵ P µνρσ (4.15) \nwith \nP µνρσ := 1 2 [ η µρ η νσ + η µσ η νρ -η µν η ρσ ] . (4.16) \nIn the context of a linearised gravitational environment there is no multi-triad vertex and therefore in the effective QFT considered in this work the triad propagator always couples only to the scalar field-triad-vertex, see also below. The latter is symmetric in ( µν ) as well as in ( ρσ ). This allows us to slightly simplify the projector P µνρσ whenever it occurs in combination with scalar field-triad-vertices and express it as \nP µνρσ := η µρ η νσ -1 2 η µν η ρσ . (4.17) \n- · The covariant vertex is given by \np q = \np q = -iκ 2 ( -2 ˜ T µν ( σ p p, σ q q ) + η µν ˜ T ρ ρ ( σ p p, σ q q )) , \nwhere \n˜ T µν ( p, q ) = 1 2 η µν ( p ρ q ρ -m 2 ) -1 2 ( p µ q ν + p ν q µ ) . (4.18) \nWhenever this is combined with a triad propagator, the second term of the vertex contribution vanishes because we have \nη µν ˜ T α α [ η µρ η νσ + η µσ η νρ -η µν η ρσ ] = 2 η ρσ ˜ T α α -2 η ρσ ˜ T α α = 0 , (4.19) \nwhere we used that η µν η µν = 4. Hence, for processes like Coulomb scattering we can replace the expression for the vertex with \niκ ˜ T µν ( σ p p, σ q q ) . (4.20) \n- · Since the second order vertex was used to obtain a covariant propagator and vertex, there is no analogue of the second order vertex in the covariant case.\n- · The external lines for the scalar field remain unmodified and the ones for the triad field are not important for this work. \nThese covariant Feynman rules are in accordance with the ones 5 presented in [51, 52], where also a scalar field is coupled to a linearised gravitational field, except for the usual differences caused by the choice of different signatures for the metric, as they use the mostly minus signature. Using momentum conservation, i.e. k = p + q = u + v , yields \nk µ ˜ T µ 0 ( p, q ) = k 0 ˜ T 00 ( p, q ) + k a ˜ T a 0 ( p, q ) = 1 2 [ q 0 ( p 2 0 -⃗ p 2 -m 2 ) + p 0 ( q 2 0 -⃗q 2 -m 2 )] , (4.21) \nk µ ˜ T µa ( p, q ) = k 0 ˜ T 0 a ( p, q ) + k b ˜ T ba ( p, q ) = 1 2 [ q a ( p 2 0 -⃗ p 2 -m 2 ) + p a ( q 2 0 -⃗q 2 -m 2 )] . (4.22) \nIf the scalar field is on-shell, which we assume for a moment for the Coulomb-scattering diagram, then the right hand side of both expressions vanishes. With this, one can directly show the equivalence of using the covariant set of Feynman rules for the Coulomb scattering diagrams discussed above of this. We do present this in appendix A.II. \nNote that in perturbation series, the diagram containing the two vertices is of second order in the expansion and hence obtains an additional factor 1 2 compared to the second order vertex diagram. \nAs a next step we would like to show a similar equivalence between the non-covariant and covariant \nFeynman rules for the self-energy diagram, namely the following equivalence in terms of Feynman diagrams \n+ = \nNote that the diverging term actually only contains the first of the two Feynman diagrams on the left hand side. However, the second term is the self-energy contribution which vanishes in the one-particle projection of the master equation, so we can add this diagram as in the one-particle master equation its contribution vanishes for normal ordering. \nThe equivalence for the Feynman diagrams on both sides of this equation is however much more difficult to prove compared to the Coulomb scattering diagram, which is also the case in QED, since the momentum inside the loop is not on-shell, which prevents a similar calculations as done for the Coulomb scattering tree level graph. Given that, to prove this equivalence goes beyond the scope of this work here and we refer here to the fact that the covariant set of Feynman rules can also be derived from the same underlying action using a different approach and gauge, which are then used for instance in [51, 52]. Hence, independently of the derivation, we expect that they describe the same physics. Based on this, it is now possible to specify the expression corresponding to the scalar particle self-energy diagram in covariant form and renormalise it, which will be discussed in detail in the next section. As mentioned at the beginning of this section, such a replacement of non-covariant by covariant Feynman rules along the lines presented here is also employed in QED when quantising in Coulomb gauge, as for instance in [22, 23].", 'D. UV-renormalisation of the self-energy of the scalar particle': 'In terms of the Feynman rules introduced in the previous section, the self-energy diagram for the vacuum propagator, which was defined in (4.10), can be expressed as \nΠ vac ( u 2 ) = ∫ d 4 k (2 π ) 4 [ iκ ˜ T µν ( u, -( u -k )) ] [ iκ ˜ T ρσ ( u -k, -u ) ] 1 κ -i k 2 -iϵ · 1 2 [ η µρ η νσ + η µσ η νρ -η µν η ρσ ] -i ( u -k ) 2 + m 2 -iϵ = κ 2 ∫ d 4 k u 2 k 2 +2 m 2 uk -2 m 4 [( k + u ) 2 -iϵ ][ k 2 + m 2 -iϵ ] . (4.23) \nAs the thermal part Π Θ ( u 2 ), defined in (4.11), is not divergent, as it has been discussed in subsection IVA, we only consider the vacuum part here. For the renormalisation we follow the strategy in [53]. Using dimensional regularisation with d = 4 -ϵ , the STT-projector is slightly modified in d dimensions and reads (see e.g. [52]): \nP µνρσ := 1 2 [ η µρ η νσ + η µσ η νρ -2 2 -ϵ η µν η ρσ ] . (4.24) \nDue to this, the expression for the self-energy diagram slightly changes and becomes (for ϵ ≪ 1): \nΠ vac ( u 2 ) = κ 2 µ ϵ ∫ d d k u 2 k 2 +2 m 2 uk -2 m 4 ( 1 + ϵ 4 ) [( k + u ) 2 -iϵ ][ k 2 + m 2 -iϵ ] , (4.25) \nwhich coincides with the expression derived in [52]. Here, we rescaled κ → κµ ϵ to keep the dimension of κ for any value of d . As later we will encounter also IR-divergences, we introduce a small artificial triad mass λ in the triad propagator that becomes \n1 κ -i k 2 + λ 2 -iϵ P µνρσ . (4.26) \nWith this, the self-energy diagram reads \nΠ vac ( u 2 ) = κ 2 µ ϵ ∫ d d k u 2 k 2 +2 m 2 uk -2 m 4 ( 1 + ϵ 4 ) [( k + u ) 2 + λ 2 -iϵ ][ k 2 + m 2 -iϵ ] , (4.27) \nThis can then be evaluated using the standard methods for dimensional regularisation (see appendix A.III). The result is that the divergent part can be isolated such that one obtains \nΠ vac ( u 2 ) = -2 π 2 κm 2 ϵ ( m 2 + u 2 ) + Π reg vac ( u 2 ) (4.28) \nwith the finite part Π reg vac ( u 2 ). The infinite part then has to be renormalised by introducing a suitable counter term. As the finite part of this counter term can in principle be chosen arbitrarily, Π reg vac ( u 2 ) can still change. In our case, we choose the finite part of the counter term according to the on-shell renormalisation procedure. This then yields for the final renormalised loop Π R vac ( u 2 ): \nΠ R vac ( u 2 ) := Π reg vac ( u 2 ) -Π reg vac ( -m 2 ) -( u 2 + m 2 ) ∂ ∂u 2 Π reg vac ( -m 2 ) . (4.29) \nThis specific form is determined by the on-shell renormalisation scheme that sets the pole of the scalar propagator to m 2 and also fixes its residue according to the following two conditions: \nΠ R vac ( u 2 = -m 2 ) ! = 0 (4.30) \n∂ ∂u 2 Π R vac ( u 2 = -m 2 ) ! = 0 . (4.31) \nIt can readily be seen that the definition in (4.29) satisfies these two conditions. Note that in [52], they apply a similar procedure without fixing the residue of the pole and therefore also not including an artificial triad mass, because for their purposes it is sufficient to fix the pole of the propagator. \nThe consideration above suggests that we have to include the following counter term: \nδ Π( u 2 ) = 2 π 2 κm 2 ϵ ( m 2 + u 2 ) -Π reg vac ( -m 2 ) -( u 2 + m 2 ) ∂ ∂u 2 Π reg vac ( -m 2 ) (4.32) \nsuch that \nΠ( u 2 ) + δ Π( u 2 ) = Π R vac ( u 2 ) + Π Θ ( u 2 ) , (4.33) \nwhere Π Θ ( u 2 ) denotes the finite thermal contribution to the loop defined in (4.11). From (A.III.19) we have for λ → 0: \nΠ reg vac ( -m 2 ) = 0 , (4.34) \nthus \nδ Π( u 2 ) = [ 2 π 2 κm 2 ϵ -∂ ∂u 2 Π reg vac ( -m 2 ) ] ( m 2 + u 2 ) , (4.35) \nwhere the expression in the square brackets only depends on m 2 . In order to implement a suitable counter term, we introduce a renormalised mass m R by m 2 = m 2 R + m 2 R δ m , where m denotes the bare mass we have used so far and δ m a mass counterterm, as well as a renormalised wave function \nφ R = 1 √ Z 2 φ . Then the renormalised scalar field propagator (a Greens function containing twice φ ) reads up to the one-loop contribution: \niG (1) ( u 2 ) = 1 Z 2 -i u 2 + m 2 = -i u 2 + m 2 R + -i u 2 + m 2 R [ -i ( u 2 δ 2 + m 2 R ( δ 2 + δ m ))+Π( u 2 )] -i u 2 + m 2 R + O ( κ 2 ) , (4.36) \nwhere we expanded Z 2 = 1 + δ 2 . From this follows that \n-i ( u 2 δ 2 + m 2 R ( δ 2 + δ m )) ! = δ Π( u 2 ) (4.37) \nand it becomes evident that only the wave function has to be renormalised in the following manner: \nδ m = 0 (4.38) \nδ 2 = i [ 2 π 2 κm 2 R ϵ -∂ ∂u 2 Π reg ( -m 2 R ) ] . \nDue to the counter-term, We have to replace in the old set of Feynman rules m by m R and obtain the following additional Feynman rule of order κ : \n= -iδ 2 ( u 2 + m 2 R ) . (4.39) \nTo simplify notation, we continue to use m , in particular as we have seen that m R = m . The result is therefore an additional counterterm in the Lagrangian which leads in renormalised perturbation theory to an additional interaction of order κ that we have to include when evaluating the loop. Then the former diverging term Π( u 2 ) becomes finite and only Π R vac ( u 2 ) is left. This yields a modification of the right hand side of the master equation. As calculated in appendix A.III.2, the contribution of the renormalised vacuum loop terms to the master equation vanishes: \nΞ R ( ω u , ⃗u, t 0 , t ) = ∫ t t 0 dτ ∫ R du 0 Π R vac ( u 2 ) cos[( u 0 -ω u )( t -τ )] = 0 , (4.40) \nΞ R ( ω v , ⃗v, t 0 , t ) = ∫ t t 0 dτ ∫ R dv 0 Π R vac ( v 2 ) cos[( v 0 -ω v )( t -τ )] = 0 . (4.41) \nTherefore neither the dimensional constant µ , nor the artificial triad mass λ play a role in the physical predictions made with the master equation.', 'E. Renormalised one-particle master equation': "With the renormalisation carried out in the previous subsections, a first renormalised version of the one-particle master equation (3.9), where only the former diverging terms are modified, reads: \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) -κ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u [ C R ( ⃗u, ⃗ k, t ) + δ P C R P ( ⃗u, ⃗ k, t ) ] + P v ( ⃗ k ) ω v -k ω v [( C R ( ⃗v, ⃗ k, t ) ) ∗ + δ P ( C R P ( ⃗v, ⃗ k, t ) ) ∗ ] } ρ ( ⃗u, ⃗v, t ) + κ 2 ∫ d 3 k (2 π ) 3 P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω u ω v + k ω v { C ( ⃗u + ⃗ k, ⃗ k, t ) + C ∗ ( ⃗v + ⃗ k, ⃗ k, t ) } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) (4.42) \nwith \nand \nC ( ⃗u, ⃗ k, t ) = ∫ t -t 0 0 dτ Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k -ω u ) τ + N ( k ) e i (Ω k -ω u -k + ω u ) τ } . (4.45) \nAt the level of the operator equation, the renormalisation removed the Θ-independent terms from the terms in the second and third line of (4.42), hence leaving us with the following dissipator: \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ a,b =1 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 1 Ω k { -( j b r ( ⃗ k, ⃗ l ) ρ S ( t ) j a r ( ⃗ k, ⃗p ) † ) f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. + N ( k ) [ j a r ( ⃗ k, ⃗p ) † , [ j b r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. } . (4.46) \nIf working with the non-extended projection δ P = 0, then there was probability conservation before the renormalisation, i.e. ∫ d 3 u ∂ ∂t ρ ( ⃗u, ⃗u, t ) = 0. Now, due to the vacuum term in (4.45) this probability conservation is destroyed. As the renormalisation is a purely technical procedure that should not change the physics, in particular not basic principles as probability conservation, we also replace C ( ⃗u, ⃗ k, t ) by C R ( ⃗u, ⃗ k, t ) in the last line of the master equation. Another reason for this is that the term in the last line of the master equation is based on the same QFT as the terms in the second and third line, hence they should be renormalised in the same way 6 . The final renormalised one-particle master equation is thus \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) -κ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u [ C R ( ⃗u, ⃗ k, t ) + δ P C R P ( ⃗u, ⃗ k, t ) ] + P v ( ⃗ k ) ω v -k ω v [( C R ( ⃗v, ⃗ k, t ) ) ∗ + δ P ( C R P ( ⃗v, ⃗ k, t ) ) ∗ ] } ρ ( ⃗u, ⃗v, t ) + κ 2 ∫ d 3 k (2 π ) 3 P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω u ω v + k ω v { C R ( ⃗u + ⃗ k, ⃗ k, t ) + ( C R ( ⃗v + ⃗ k, ⃗ k, t ) ) ∗ } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) (4.47) \nand the dissipator at operator level \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ a,b =1 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k { [ j a r ( ⃗ k, ⃗p ) † , [ j b r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. } . (4.48) \nC R ( ⃗u, ⃗ k, t ) = 2 ∫ t -t 0 0 dτ Ω k N ( k ) cos[Ω k τ ] e -i ( ω u -k -ω u ) τ (4.43) \nC R P ( ⃗u, ⃗ k, t ) = 2 ∫ t -t 0 0 dτ Ω k N ( k ) cos[Ω k τ ] e -i ( ω u -k + ω u ) τ (4.44) \nCompared to the expression before renormalisation given in [1] in equation (4.60), the vacuum contribution in the dissipator vanishes. \nIn [27] a quantum mechanical model based on the model in [28] is considered where a system is coupled to an environment of harmonic oscillators. The bath of harmonic oscillators mimics the thermal gravitational waves and the model serves as toy model for gravitationally induced decoherence. In [27] the system was then specified to neutrinos in order to investigate gravitationally induced decoherence in the context of neutrino oscillations. A more detailed discussion on the model from [27] can be found in section VI D. \nThe quantum mechanical master equation in that work consists of a dissipator term and a Lambshift, where the latter contained divergences and is finally removed by a renormalisation in [27]. We are now interested to discuss the similarities and differences of that quantum mechanical renormalisation and the renormalisation of the one-particle master equation presented in this work. In the quantum mechanical case, the interaction Hamiltonian has the following form: \nˆ H I = ˆ H S ⊗ N ∑ i =1 g i ˆ q i (4.49) \nwith the Hamiltonian ˆ H S , describing the neutrino propagation, coupling constants g i and the position operators of the harmonic oscillators in the environment ˆ q i . This provides a toy model for the coupling of the energy momentum tensor and the metric perturbations in the field theoretical model. The form of the coupling in the quantum mechanical model implies that the coefficients Λ of the Lamb-shift and Γ of the dissipator in the final Lindblad equation only depend on the environment and are defined in the following way, see [27] equations (4) and (5): \nΛ( t -t 0 ) := ∫ t -t 0 0 dτ ∫ ∞ 0 dω J ( ω ; Ω , η ) sin( ωτ ) (4.50) \nΓ( t -t 0 ) := 2 ∫ t -t 0 0 dτ ∫ ∞ 0 dω J ( ω ; Ω , η ) cos( ωτ ) coth ( β ℏ ω 2 ) , (4.51) \nwhere a spectral density J ( ω ; Ω , η ) was used which depends on the frequencies of the harmonic oscillators denoted by ω , an effective coupling parameter η and a UV-cutoff frequency Ω. From this form it becomes evident that the Lamb-shift term is independent of the temperature parameter β = 1 k B Θ , where k B denotes Boltzmann's constant. Thus the Lamb-shift contribution only encodes vacuum effects, while the prefactor Γ of the dissipator depends on Θ and yields a non-vanishing contribution for Θ = 0. The renormalisation in this model applied in [27] then removes the Lamb-shift contribution completely, as it depends on the unphysical cutoff frequency Ω, while the prefactor of the dissipator is not altered, as here the dependency on Ω vanishes after the Markov approximation. \nNext, let us discuss to what extent it is possible to connect the renormalisation and its effects of the one-particle master equation presented above with the renormalisation applied in the quantum mechanical toy model. For this purpose first we discuss the two forms of the original full field theoretical master equation derived in [1]. The first form is given in that work in equation (4.52): \n∂ ∂t ρ S ( t ) = -i [ H S + κ U, ρ S ( t )] -κ 2 ∫ t 0 ds ∑ r ∫ R 3 d 3 k { iD ( ⃗ k, t -s ) [ J r ( ⃗ k ) , { J r ( -⃗ k, s -t ) , ρ S ( t ) }] + D 1 ( ⃗ k, t -s ) [ J r ( ⃗ k ) , [ J r ( -⃗ k, s -t ) , ρ S ( t ) ]] } , (4.52) \nwhere D and D 1 are two coefficients that arise from combinations of the environmental correlation functions similar to Λ and Γ in the quantum mechanical model and they read: \nD ( ⃗ k, t -s ) := -sin(Ω k ( t -s )) Ω k (4.53) \nD 1 ( ⃗ k, t -s ) := coth ( β Ω k 2 ) cos(Ω k ( t -s )) Ω k . (4.54) \nThe operators J r ( ⃗ k, t ) and J r ( ⃗ k ) := J r ( ⃗ k, 0) were defined in [1] in equation (3.18) and contain a combination of two creation and/or annihilation operators of the scalar field along with their time evolution. As a first difference to the master equation in [27] it turns out that the term proportional to D in (4.52) cannot be written as a simple commutator, as it is the case with the Lamb-shift contribution in the quantum mechanical toy model. If the system's operator J were to commute with the system Hamiltonian, then this would be possible, and this is the case in the quantum mechanical toy model in [27]. Then this would imply that the Lamb-shift is independent of the temperature parameter Θ and therefore a pure vacuum effect. \nFor the field-theoretical model, a similar form where one has a Lamb-shift contribution and a dissipator is the one given at the beginning of Appendix A.I.1. Here, the coefficients of the Lamb-shift term are S ab and the prefactors of the dissipator are R ab . From their definitions in (A.I.5) and (A.I.6) one can see that in general they have a different form as Λ and Γ in (4.50) and (4.51) above and the Lamb-shift includes vacuum as well as thermal contributions. A similar result is obtained in [2], where quantum electrodynamics is discussed from the point of view of open quantum systems with the standard interaction Hamiltonian of QED. There, the resulting Lamb-shift Hamiltonian is therefore split into a vacuum part, denoted as Lamb-shift and a thermal part, denoted as Stark-shift. \nAs discussed above, we would expect that if the system's operator in the interaction Hamiltonian commutes with the system Hamiltonian, that we can then recover the form of the quantum mechanical model in [27]. Indeed, if we had [ J, H S ] = 0, then the phases e ± iω a ( ⃗ k,⃗p ) t and e ± iω b ( ⃗ k, ⃗ l ) t coming from the time evolution of the J operators would vanish in the definitions of S ab and R ab in (A.I.5) and (A.I.6). This allows the remaining terms to be combined into a form similar to Λ and Γ. In particular the thermal contribution of the Lamb-shift would vanish, as it is the case in the quantum mechanical toy model. \nLet us now compare the renormalisations of the two models: in the quantum mechanical model in [27], the effect of the renormalisation is to remove the Lamb-shift Hamiltonian which only consisted of a vacuum part. The renormalisation applied in the present work removes the vacuum parts in the Lamb-shift Hamiltonian and the dissipator. The thermal part of the Lamb-shift Hamiltonian however remains. From the discussion of the open QED model from [2], one would expect a similar result in a quantum mechanical model where a thermal contribution in the Lamb-shift is present. The dissipator of the quantum mechanical model [27] is left unmodified by the renormalisation, in particular the vacuum contribution is present there. This is in contrast to the procedure here, where the renormalisation removes all vacuum terms, also the ones from the dissipator. In the quantum mechanical toy model, these contributions are however removed at a later stage when the Markov approximation is applied and hence also not present in the final Lindblad equation. \nThis concludes the discussion on the renormalisation of the one-particle master equation. In the next section, we discuss how one can apply specific physical approximations to draw physical implications from the renormalised one-article master equation.", 'V. APPLICATION OF THE MARKOV AND ROTATING WAVE APPROXIMATIONS TO TRANSFORM THE TCL MASTER EQUATION INTO LINDBLAD FORM': 'The renormalised TCL one-particle master equation (4.47) describes the evolution of a single scalar particle in an environment filled with thermal gravitational waves. Since this master equation is not in Lindblad form, we cannot directly conclude that it is completely positive and provides physically meaningful implications based on positive probabilities for all chosen time intervals. For such models, one usually has to investigate case by case whether further assumptions such as the Markov and rotating wave approximation are justified that are usually used to obtain a master equation in Lindblad form. It is often possible to understand from the involved time scales in the system and environment of the open quantum model in which scenarios these approximations are a good choice, see for example [2] for a discussion in quantum optics. For models with finitely many degrees of freedom, there are also results that suggest time scales which allow to judge when the Markov approximation can be applied that are completely determined by the properties of the environment, such as its spectral density as well as the coupling constant, which encodes the strength of the coupling to the system in the interaction Hamiltonian [54]. \nThe derivation of master equations in the context of field-theoretical models with gravity as an environment is less well explored in the literature in comparison and has been presented in the context of gravitationally induced decoherence recently for instance in [1, 8, 9, 11, 12, 16]. While the works in [1, 8, 11, 16] focus on the derivation of a TCL master equation, in [9, 12] a Lindblad equation is used, for which further approximations are employed, among these the Markov approximation and the rotating wave approximation. \nCompared to the above-mentioned open quantum mechanical models for gravitationally induced decoherence, a detailed analysis of the applicability of such approximations is much more challenging and beyond the scope of this article. An important difference to the present work is that in [9, 12] the approximations are applied on the non-renormalised one-particle master equation. Given the results of the last section, we can instead perform the Markov and rotating wave approximations for the renormalised one-particle master equation and investigate whether applying these approximations before or after renormalisation leads to differences in the final one-particle master equation, considering both the extended and non-extended one-particle projection. \nIn this section we consider both approximations separately, in subsection V A we discuss the Markov approximation and in subsection V B the rotating wave approximation. In addition, for the case of an ultra-relativistic limit, we also specify some conditions when the Markov approximation can be used for the model considered here.', 'A. Markov approximation': 'The Markov approximation consists in the assumption that the correlation functions of the environment are strongly peaked around the initial time and decay rapidly. If this is given, the integral ∫ t -t 0 0 dτ over these environmental correlation functions has the main contribution from around their peak. Thus the error obtained when shifting the initial time t 0 → -∞ and therefore the upper integration limit t -t 0 →∞ is negligibly small. As a consequence, the parameters involved in the dissipator of the final Lindblad equation will no longer depend on the temporal coordinate. \nAs discussed above for a field theoretical model, even in the single particle sector, to develop generic criteria for which the Markov approximation can be applied that can easily be checked for a given model, is difficult. For instance the methods developed in [54] strongly rely on the fact that the model is formulated in a quantum mechanical context. Motivated by the physical applications \nin section VI D to ultra-relativistic particles, in particular neutrinos and their oscillations, as a first step, we investigate this special case more in detail in this context and present a condition under which the Markov approximation can be applied for the model under consideration. The details are discussed in appendix A.IV.1 by analysing the individual parts of the master equation and where we show that a suitable condition for the applicability of the Markov approximation in the ultra-relativistic case is the requirement that \nu, v ≫ 1 cβ , (5.1) \nwhere u := | ⃗u | , v := | ⃗v | and c denotes the speed of light. The reason why β and thus the temperature parameter Θ are involved here is because we use a Gibbs state to trace out the environmental and thus gravitational degrees of freedom. The specific time scales then yield that for the application in VI D the correction terms to the Markov approximation are negligible. If another than the ultra-relativistic case is considered, the above condition could be violated, and therefore a more comprehensive analysis is needed to understand when and under what conditions the Markov approximation can be applied, which we envisage for future work. \nBefore applying the Markov approximation to the renormalised one-particle master equation, we briefly discuss the main steps that are involved: the first step is to perform the τ -integration using the identity \n∫ ∞ 0 dτ e -iωτ = ∫ ∞ -∞ dτ Θ( τ ) e -iωτ = 2 π ∫ ∞ -∞ dx Θ( x ) e -2 πiωx = πδ ( ω ) -PV ( i ω ) , (5.2) \nwhere PV denotes the Cauchy principal value, and secondly the evaluation of the ⃗ k -integration which simplifies due to the first step. With this, the affected terms in the master equation can then be split into two classes: one class that consists of contributions involving the delta distribution that will yield a real contribution to the master equation and hence lead to decoherence. Another class that contains terms including the principal value, which result in an imaginary contribution that affects the unitary evolution. The detailed computation for the contributions leading to decoherence can be found in appendix A.IV.2. Those contributions that involve the principal value \nare evaluated in appendix A.IV.3. The final result, given in (A.IV.43), takes the form: \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) + κ 16 πβ { -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arctanh ( u ω u ) -2 ω v v m 4 arctanh ( v ω v ) -ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 2 3 -m 2 u 2 + m 4 ω u u 3 arctanh ( u ω u )] -ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 2 3 -m 2 v 2 + m 4 ω v v 3 arctanh ( v ω v )] } ρ ( ⃗u, ⃗v, t ) + iκ 2(2 π ) 3 1 √ ω u ω v ∫ d 3 k P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω v + k N ( k ) Ω k ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) { PV ( 1 Ω k + ω u -ω u + k ) -PV ( 1 Ω k -ω u + ω u + k ) -PV ( 1 Ω k + ω v -ω v + k ) +PV ( 1 Ω k -ω v + ω v + k ) } -(1 -δ P ) iκ 2(2 π ) 2 lim ϵ → 0 [ u 4 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω u ω u -k k 2 -( ω u -k -ω u ) 2 -v 4 ω v ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω v ω v -k k 2 -( ω v -k -ω v ) 2 ] ρ ( ⃗u, ⃗v, t ) . \n(5.3) \nThe contributions in lines two to four arose from the delta distributions and are real, so they cause decoherence in the evolution of the scalar particle. The remaining terms are imaginary and therefore contribute to the unitary evolution of the density matrix. When working with the extended one-particle projection, then the expressions in the last two lines vanish. The real part in lines two to four remains unaffected by the rotating wave approximation that will be applied in the next subsection, hence it already possesses its final form. \nIn the existing literature, the master equation in the one-particle picture is usually directly specified or derived for the non-relativistic (see e.g. [9] for scalar particles) or the ultra-relativistic case (see e.g. [12] for photons). In these cases, the dissipator has a simpler form and there are no arctanhterms present as it is the case here in the general master equation, where neither the non- nor the ultra-relativistic limits have been applied yet. \nTo further compare with the existing literature, in section VI D we will consider the non- and ultra-relativistic limit of this master equation above and show that the arctanh does not appear in either limit. Thus, it indeed leads to the results obtained in the literature. \nAnother difference to similar work in [9, 12] is that the master equation in the present work has already been renormalised, i.e. all vacuum contributions that are independent of the temperature parameter Θ have been removed in these terms, which arise due to the gravitational influence in (5.3), which in particular contains all vacuum fluctuations of the gravitational field. This can be seen by setting the temperature parameter equal to zero, as then all terms including the gravitational influence vanish.', 'B. Rotating wave approximation': 'After having applied the Markov approximation, the rotating wave approximation is usually a next step in order to cast the master equation into a completely positive Lindblad form. The physical idea behind the approximation is to take into account that detectors only have a finite resolution and cannot resolve arbitrarily fast oscillations, but only measure a coarse-grained result. In the literature, there exist different ways to apply the rotating wave approximation. One possibility is, following the nomenclature in [24], the pre-trace RWA, where the approximation is applied at the level of the interaction Hamiltonian by dropping counter-rotating terms. This is often employed e.g. in quantum optics and leads to the Jaynes-Cummings model, see [55, 56], which is nowadays extensively studied for instance in quantum technology, see [57]. This pre-trace RWA, which is also applicable in closed quantum systems, has been studied from several angles yielding different results in the last years among other things on its higher order corrections and a renormalisation of the resulting series (see [26]) as well as also on the bounds of its applicability (see [25]). From the analysis in [24] it follows that in open quantum systems the second version of the RWA, the post-trace rotating wave approximation which is applied at the level of the master equation after tracing out the environment, yields dynamics which are expected to be closer to the true system dynamics. This analysis in [24] is carried out for quantum mechanical models and we expect that more work is required to extend it to the full field theoretical case. Nevertheless, we take this discussion as a motivation to apply in this work the post-trace RWA, which was also employed in similar analyses, for instance in [12]. This post-trace RWA is implemented by considering the master equation in the interaction picture and then dropping terms that rotate very fast compared to the other ones. The detailed implementation and computation can be found in appendix A.V, here we only state the result: \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) + κ 16 πβ { -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arctanh ( u ω u ) -2 ω v v m 4 arctanh ( v ω v ) -ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 2 3 -m 2 u 2 + m 4 ω u u 3 arctanh ( u ω u )] -ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 2 3 -m 2 v 2 + m 4 ω v v 3 arctanh ( v ω v )] } ρ ( ⃗u, ⃗v, t ) -(1 -δ P ) iκ 2(2 π ) 2 lim ϵ → 0 [ u 4 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω u ω u -k k 2 -( ω u -k -ω u ) 2 -v 4 ω v ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω v ω v -k k 2 -( ω v -k -ω v ) 2 ] ρ ( ⃗u, ⃗v, t ) . (5.4) \nIt can be seen when comparing this result to (5.3) that the effect of the rotating wave approximation is to remove the remaining part of the Lamb-shift in the extended projection. In the non-extended projection, there still survives one term of the Lamb-shift which corresponds to the last two lines in (5.4). Apart from that, the rotating wave approximation causes no further modifications on the master equation. This is due to the fact that all other terms that would be removed by the approximation were already dropped when performing the one-particle projection of the master equation. The general dissipator at the operator level can however be written in Lindblad form \nafter the RWA, see (A.V.45) in the appendix: \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} ∫ R 3 d 3 k (2 π ) 2 δ ( k ) N ( k ) Ω k ( L r ( ⃗ k ) ρL r ( ⃗ k ) † -1 2 { ρ, L r ( ⃗ k ) † L r ( ⃗ k ) } ) (5.5) \nwith Lindblad operators \nL r ( ⃗ k ) := ∫ R 3 d 3 p 1 √ 1 -( k + p cos( θ p )) 2 ω 2 k + p J 2 r ( ⃗ k, ⃗p ) , (5.6) \nwhere θ p denotes the angle between ⃗ k and ⃗ p and J 2 r ( ⃗ k, ⃗p ) = 2 j 1 r ( ⃗ k, ⃗p ) with the latter being defined in (A.I.8). With this, we have derived the final form of the renormalised one-particle master equation after Markov and rotating wave approximation. In the next section, we discuss some applications and investigate some features of the master equation at different intermediate stages before, during and after the applied approximations.', 'VI. APPLICATIONS OF THE ONE-PARTICLE MASTER EQUATION': 'In this section, we discuss some applications of the one-particle master equation derived in the previous sections. We start with analysing the evolution of the populations of the one-particle density matrix with a special focus on the interplay between the renormalisation and Markov and rotating wave approximations in subsection VI A and compare the results to [11] where the evolution of the populations of the non-renormalised TCL master equation is derived. Next we discuss the non-relativistic limit of the one-particle master equation in subsection VI B and compare the results to the ones in [9]. Furthermore, we investigate the ultra-relativistic limit in section VI C, compare it to [12], and discuss the relation to the quantum mechanical model for gravitationally induced decoherence in neutrino oscillations in [27] in section VI D. This further allows to connect to phenomenological models that investigate the influence of gravity on neutrino oscillations, like for instance in [7, 10, 17, 58].', 'A. Evolution of the populations of the one-particle master equation': 'We start by analysing the dynamics of the populations, that is the diagonal elements, in momentum representation predicted by the master equation at different stages in the derivation of the final Lindblad equation. We have chosen this application because it is an example that allows us to discuss and compare the implications that arise depending on the stage of the calculation at which the renormalisation procedure is performed. \nTo investigate the evolution of the populations, we take the different versions of the master equation and compute it for ρ ( ⃗ k, t ) := ρ ( ⃗ k, ⃗ k, t ) before and after the renormalisation as well as after the Markov approximation. As the rotating wave approximation only affects the Lamb-shift Hamiltonian, the dynamics of the populations will not get modified after its application.', '1. Before renormalisation': "The dynamics of the populations in the one-particle master equation (3.9) before renormalisation and further approximations can be obtained by evaluating the master equation for ρ ( ⃗ k, t ) := ρ ( ⃗ k, ⃗ k, t ). In this case, we have no contribution from the unitary dynamics and in the dissipator all \nimaginary parts will vanish 7 and one obtains a dissipator that is purely real. In this subsection we adapt the notation to the one used in [11] in order to better facilitate the comparison with their results: \n˙ ρ ( ⃗ k, t ) = -κ ∫ d 3 k ' (2 π ) 3 P k ( ⃗ k ' -⃗ k ) Ω k ' -k ω k ' ω k · · {[ [ N ( k ' -k ) + 1] sin( χ ( t -t 0 )) χ + N ( k ' -k ) sin( χ ' ( t -t 0 )) χ ' + δ P [ N ( k ' -k ) + 1] sin( η ( t -t 0 )) η + δ P N ( k ' -k ) sin( η ' ( t -t 0 )) η ' ] ρ ( ⃗ k, t ) -[ [ N ( k ' -k ) + 1] sin( χ ' ( t -t 0 )) χ ' + N ( k ' -k ) sin( χ ( t -t 0 )) χ ] ρ ( ⃗ k ' , t ) } (6.1) \nwith ˙ ρ ( ⃗ k, t ) = ∂ t ρ ( ⃗ k, t ) and χ := Ω k ' -k -ω k + ω k ' , χ ' := Ω k ' -k + ω k -ω k ' , η := Ω k ' -k + ω k + ω k ' and η ' := Ω k ' -k -ω k -ω k ' as well as P k ( ⃗ k ' -⃗ k ) = P ijln ( ⃗ k ' -⃗ k ) k i k j k l k n . Using that P ijln ( ⃗ k -⃗ k ' )( ⃗ k -⃗ k ' ) i = 0 as P ijln ( ⃗ k -⃗ k ' ) projects onto the symmetric transverse traceless part and therefore removes the longitudinal part ∝ ⃗ k -⃗ k ' , which can be seen from the definition in (3.10), we can use \nP ijln ( ⃗ k -⃗ k ' ) k i = P ijln ( ⃗ k -⃗ k ' ) k ' i (6.2) \nand hence rewrite \nP k ( ⃗ k ' -⃗ k ) = P ijln ( ⃗ k ' -⃗ k ) k i k j k ' l k ' n . (6.3) \nFrom equation (6.1) one can also once more see the implication of the chosen projection, i.e. whether δ P = 0 or δ P = 1, on the probability conservation, which was discussed below equation (3.8). When working with the non-extended projection δ P = 0, then we have \n∫ R 3 d 3 k ˙ ρ ( ⃗ k, t ) = 0 (6.4) \ndue to symmetry and thus probability in the scalar particle's subsystem is conserved. If working with the extended one-particle projection δ P = 1 instead, it can be seen in equation (6.1) that the terms containing η and η ' lack a symmetric counterpart to be cancelled and hence in that case probability conservation is not given any more, as it was also discussed below equation (3.8). \nIn [11] the dynamics of the population for a master equation of a photon coupled to linearised gravity are discussed. We obtain an agreement with their result if we specialise to a massless scalar particle and choose as the initial time t 0 = 0. In addition we need to consider the non-extended one particle projection (i.e. δ P = 0), in order to adapt to their chosen normal ordering as well as choose the temperature parameter Θ to be zero. The latter corresponds to a vacuum state of the gravitational waves environment. Inserting these assumptions in the evolution of the populations this equation becomes \n˙ ρ ( ⃗ k, t ) = -κ ∫ d 3 k ' (2 π ) 3 P k ( ⃗ k ' -⃗ k ) Ω k ' -k ω k ' ω k [ sin( χt ) χ ρ ( ⃗ k, t ) -sin( χ ' t ) χ ' ρ ( ⃗ k ' , t ) ] , (6.5) \nwhich has a very similar form as the one in [11] for a photon. The only difference arises due to the fact that for the photons in [11] the polarisation vectors couple to the symmetric transverse traceless projector while here for the scalar particles, as they do not carry any polarisation, This role is taken over by the momentum, which is the only direction-dependent quantity that scalar particles possess.", '2. After renormalisation': "As discussed in section IV E, the effect of the renormalisation was that the vacuum part in the one-particle master equations, these are the contributions not involving N ( ⃗ k ), vanishes. At the practical level this can be implemented by replacing everywhere N ( k ' -k ) + 1 → N ( k ' -k ). Then the dynamics of the populations becomes \n˙ ρ ( ⃗ k, t ) = -κ ∫ d 3 k ' (2 π ) 3 P k ( ⃗ k ' -⃗ k ) Ω k ' -k ω k ' ω k · N ( k ' -k ) · {[ sin( χ ( t -t 0 )) χ + sin( χ ' ( t -t 0 )) χ ' + δ P sin( η ( t -t 0 )) η + δ P sin( η ' ( t -t 0 )) η ' ] ρ ( ⃗ k, t ) -[ sin( χ ' ( t -t 0 )) χ ' + sin( χ ( t -t 0 )) χ ] ρ ( ⃗ k ' , t ) } . (6.6) \nWe realise that now all terms depend on N ( k ' -k ). As a consequence, the entire evolution of the populations is trivial, that is ˙ ρ ( ⃗ k, t ) vanishes, if we consider the specific case of a vanishing temperature parameter Θ = 0 yielding directly N ( k ' -k ) = 0 for all k, k ' . The comparison to the non-renormalised master equation shows that the physical properties of the two one-particle master equations are quite different as far as the dynamics of the populations is concerned. For this reason the discussions and physical implications drawn in [11] based on the dynamics of the populations in the non-renormalised equation (6.5) are problematic, as the evolution of the diagonal terms vanishes after renormalisation in the zero temperature limit.", '3. After the Markov approximation': 'Due to the fact that for the diagonal elements the coefficients always enter in the form C + C ∗ , only real terms in the one-particle master equation after the second Markov approximation in (5.3) remain: \n∂ ∂t ρ ( ⃗ k, t ) = ˙ ρ ( ⃗ k, t ) = κ 16 πβ { -20 3 k 2 +4 ω 2 k -4 ω k k m 4 arctanh ( k ω k ) +4 k 2 [ 2 3 -m 2 k 2 + m 4 ω k k 3 arctanh ( k ω k )] } ρ ( ⃗ k, t ) =0 . (6.7) \nThis means that the Markov approximation removes the dynamics of the populations also in the case of non-vanishing temperature and independently of the extended projection δ P . It therefore \nalso restores probability conservation, as it removes all terms from the extended projection in the dissipator. The rotating wave approximation only affects the imaginary parts of the master equation, thus it does not change the evolution of the populations any more. \nThis result that the dynamics of the populations vanishes is also obtained in [9] for a non-relativistic one-particle master equation that was renormalised after the application of Markov and rotating wave approximation, and for the one of a photon after renormalisation and application of the same two approximations in [12].', 'B. Non-relativistic limit': 'In the following, we apply the renormalised one-particle master equation after Markov and rotating wave approximation (5.4) to non-relativistic particles in order to compare the decoherence with the one derived in [9]. \nIn the non-relativistic limit we have u 2 m 2 ≪ 1 and v 2 m 2 ≪ 1 and due to this the one-particle master equation simplifies. In this case we can expand the arctanh as \n-2 m 2 m u 1 √ 1 + u 2 m 2 arctanh u m 1 √ 1 + u 2 m 2 = -2 m 2 m u [ u m -2 3 u 3 m 3 + 8 15 u 5 m 5 + O ( u 6 m 6 )] . (6.8) \nGiven this we find for the contribution from lines two to four in (5.4) which is the part leading to decoherence the following expression: \nκ 16 πβ { -16 15 u 4 + v 4 m 2 -16 15 u 2 v 2 m 2 (1 -3 cos 2 ( γ ) } = -κ 5 πβm 2 [ 1 3 ( u 4 + v 4 ) + u 2 v 2 ( 1 3 -cos 2 ( γ ) )] , (6.9) \nwhere γ is defined as the angle between ⃗u and ⃗v , i.e. ⃗u · ⃗v = uv cos( γ ). We work with the extended projection δ P = 1 here, as a consequence there is no Lamb-shift contribution left and the master equation becomes \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t )( ω u -ω v ) -κ 5 πβm 2 [ 1 3 ( ⃗u 4 + ⃗v 4 ) + ⃗u 2 ⃗v 2 ( 1 3 -cos 2 ( γ ) )] ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( u 2 2 m -v 2 2 m ) -κ 5 πβm 2 [ 1 3 ( ⃗u 4 + ⃗v 4 ) + ⃗u 2 ⃗v 2 ( 1 3 -cos 2 ( γ ) )] ρ ( ⃗u, ⃗v, t ) , (6.10) \nwhere we used in the last step ω u = √ ⃗u 2 + m 2 = m √ u 2 m 2 +1 ≈ m + u 2 2 m and likewise for ω v . The master equation (60) in [9], where a Lindblad equation is used after Markov and rotating wave approximation to also describe a scalar field coupled to a linearised gravitational field, reads in momentum representation 8 \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( u 2 2 m R -v 2 2 m R ) -2 κ 3 βm 2 R [ 1 3 ( ⃗u 4 + ⃗v 4 ) -1 3 ⃗u 2 ⃗v 2 ( 1 + cos 2 ( γ ) ) ] ρ ( ⃗u, ⃗v, t ) . (6.11) \nWhile in that work, they use the same underlying physical system, one of the differences is that there a gauge fixing is used while in this work the elementary physical variables were identified in [1] \nby choosing geometrical clocks with respect to which suitable Dirac observables were constructed. Additionally, the Hamiltonian used in [9] is not completely normal ordered, while here we worked with a completely normal ordered one. A more detailed discussion of these two points can be found in [1]. Furthermore, the renormalisation is carried out in a different manner: in this work it is done before the Markov and rotating wave approximation are applied. In contrast in [9] the final master equation is renormalised after the application of these two approximations and after going into the non-relativistic limit. Their renormalisation procedure involves the introduction of a cutoff Λ ≪ m which is later absorbed in a redefinition of the renormalised mass m → m R , while here we found in equation (4.38) that only the wave function needs to be renormalised, see section IV D. In [9] compared to our result here, there are some additional unitary terms left due to using the non-extended one-particle projection. These contributions are proportional to the UV-cutoff Λ and to u 4 m 2 R , which is why they are dropped in [9] from the final master equation in the non-relativistic limit, even though in the limit Λ → ∞ they would diverge. As our results demonstrate, using the extended projection and a renormalisation before the application of the approximations hence removes the necessity to drop diverging terms by hand. \nAdditional differences between (6.10) and (6.11) are the prefactor in front of the dissipator and the structure inside the square brackets. In these two points the results derived here do not agree with the results in [9]. Particularly regarding the last point, our result however agrees with a similar derivation for photons in [12] where more intermediate steps are provided and where the final structure in the square brackets is the same as in (6.10).', 'C. Ultra-relativistic limit': "In this subsection we apply the one-particle master equation to ultra-relativistic particles. Possible applications are one-particle master equations for photons as discussed in [12] as well as gravitationally induced decoherence in neutrino oscillations as for instance discussed in [27], where a quantum mechanical toy model was used. \nIn the ultra-relativistic limit we have m 2 u 2 ≪ 1 as well as m 2 v 2 ≪ 1. Taking this into account, we neglect all terms of order O ( m 2 u 2 ) and O ( m 2 v 2 ) respectively and higher order contributions. Note, that this also includes terms involving arctanh function because \nm 2 u 2 1 √ 1 + m 2 u 2 arctanh 1 √ 1 + m 2 u 2 = O ( m 2 u 2 ) . \nThis leads to the the following simplification for the decoherence term in (5.4): \nκ 16 πβ { -4 3 ( u 2 + v 2 ) -4 3 [ uv -3( ⃗u · ⃗v ) 2 ] } = -κ 4 πβ [ 1 3 ( u 2 + v 2 ) + uv ( 1 3 -cos 2 ( γ ) )] . (6.12) \nThe remaining computation of the imaginary part in the dissipator can be found in (A.IV.40) in the appendix. Combining all results, the renormalised one-particle master equation in the ultra- \nrelativistic limit can be written in the form \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( u -v ) -κ 4 πβ [ 1 3 ( u 2 + v 2 ) + uv ( 1 3 -cos 2 ( γ ) )] ρ ( ⃗u, ⃗v, t ) -(1 -δ P ) iκu 4 ρ ( ⃗u, ⃗v, t ) 105(2 π ) 2 ω u { 4 [ π 4 15 β 4 u 4 -7 π 2 6 β 2 u 2 -6 ln(1 -e -βu ) βu +4 Li 2 ( e -βu ) β 2 u 2 -6 Li 3 ( e -βu ) β 3 u 3 -6 Li 4 ( e -βu ) β 4 u 4 ] +35 -35 ln ( e βu -1 ) βu -14 u ∫ ∞ u dk N ( k ) k 2 +3 u 3 ∫ ∞ u dk N ( k ) k 4 } +(1 -δ P ) iκv 4 ρ ( ⃗u, ⃗v, t ) 105(2 π ) 2 ω v { 4 [ π 4 15 β 4 v 4 -7 π 2 6 β 2 v 2 -6 ln(1 -e -βv ) βv +4 Li 2 ( e -βv ) β 2 v 2 -6 Li 3 ( e -βv ) β 3 v 3 -6 Li 4 ( e -βv ) β 4 v 4 ] +35 -35 ln ( e βv -1 ) βv -14 v ∫ ∞ v dk N ( k ) k 2 +3 v 3 ∫ ∞ v dk N ( k ) k 4 } , (6.13) \nwhere γ denotes the angle between ⃗u and ⃗v and Li s ( x ) denotes the poly-logarithm function defined in (A.IV.41). In the extended projection, i.e. for δ P = 1, this becomes \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( u -v ) -κ 4 πβ [ 1 3 ( u 2 + v 2 ) + uv ( 1 3 -cos 2 ( γ ) )] ρ ( ⃗u, ⃗v, t ) , (6.14) \nwhich can be rewritten in terms of an operator equation as \n∂ ∂t ˆ ρ ( t ) = -i [ ˆ H, ˆ ρ ( t )] + κ 8 πβ ( δ il δ jm -1 3 δ ij δ lm )[ ˆ p i ˆ p j ˆ p 0 , [ ˆ ρ ( t ) , ˆ p l ˆ p m ˆ p 0 ]] , (6.15) \nwith ˆ p 0 := √ ˆ p n ˆ p n + ξ m 2 1 . In this definition, ξ m 2 is a small regulator that removes the eigenvalue zero from the spectrum of ˆ p 0 , as in that case the operator would not be invertible. For massive particles, this regulator corresponds to the mass squared m 2 which is still present in the ultrarelativistic limit, even though very small compared to the other summand. This is, up to a factor of 2, the same result for decoherence as derived in [12] for gravitationally induced decoherence of photons. This difference of a factor of 2 is already present when comparing the field-theoretical models of [12] and [1]. Note that in [12] the derivation and in particular the application of the approximations is performed without a prior renormalisation of the one-particle master equation, which is done in the end to get rid of the diverging Lamb-shift term. As expected from the analysis in this work, they find a logarithmic divergence in the end. The derivation of the master equation in [12] is very similar to the one in [9], hence we refer for a detailed comparison to the discussion in subsection VI B. The renormalisation in [12] is done after performing the approximations and the ultra-relativistic limit such that the detailed procedure depends on the cutoff frequency Λ and its relation to the photon frequency ω u (in our case the scalar particle's frequency). In the end in [12] the electric and magnetic fields as well as the coupling constant are renormalised.", 'D. Application to neutrino oscillations': "Finally, we want to discuss the relation of the results obtained in this work with the one presented in [27], where gravitationally induced decoherence in neutrino oscillations is investigated based on a quantum mechanical toy model [28] with neutrinos as system of interest and a collection of Harmonic oscillators to model the thermal gravitational waves environment. For this purpose we consider the decoherence of neutrinos predicted by the ultra-relativistic one-particle master equation derived in this work. \nThe connection to the work in [27] is of interest to us in several respects: first due to the quantum mechanical nature of the the toy model used in [27], the coupling parameter encoding the strength for the coupling between system and environment cannot be determined from first principles. Instead a free parameter was introduced, that after introduction of a spectral density was denoted by 9 η , similar to what was done also in [28]. Second, the quantum mechanical model requires the choice of a spectral density to derive the final master equation together with an appropriate cut-off function that regulates the integral over the frequency domain. In [27], four different commonly used cut-off functions were considered and it was shown that the final result of the master equations cannot distinguish between the different choices. For the spectral density, the usual linear dependence on the frequency, which is widely used in the context of quantum optics, was considered. As the one-particle master equation in this work is derived from an underlying field theory model presented in [1] the situation is different here. As the matter, which is a scalar field in the present work, couples to linearised gravity, the coupling constant in the interaction Hamiltonian is naturally build into the model and given by κ = 8 πG N , where G N is Newton's constant. Furthermore, due to the field-theoretical character of the model, it is not necessary to introduce a spectral density by hand, since the interaction Hamiltonian contains an integral over all modes from the beginning. As a third aspect we want to compare the application of the Markov approximation in the quantum mechanical toy model and in the one-particle master equation derived in this work. \nEven though the equation derived here is, taken strictly, only applicable to scalar particles, we still apply it to the case of neutrinos in order to discuss the relation with the results in [27]. That this can be done in this context is due to the reason that the quantum mechanical toy model investigated in [27] treats the neutrinos as plane waves and thus does not take the full spinorial nature of neutrinos into account. \nWe assume that the two momenta ⃗u and ⃗v are approximately parallel to each other in order to have intersection probability and to be able to measure them in a neutrino detector. With this, the one-particle master equation in the ultra-relativistic limit becomes \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( u -v ) -κ 12 πβ ( u -v ) 2 ρ ( ⃗u, ⃗v, t ) . (6.16) \nAs for ultra-relativistic particles we have ˆ H ˆ ρ ≈ u ˆ ρ , we indeed obtain the same form for the master equation as in [27] in the effective mass basis 10 : \n∂ ∂t ˆ ρ ( t ) = -i [ ˆ H, ˆ ρ ( t )] + κ 6 πβ ( ˆ H ˆ ρ ( t ) ˆ H -1 2 { ˆ H 2 , ˆ ρ ( t ) } ) . (6.17) \nThe master equation above can be solved in the energy eigenbasis, where we denote the energy eigenvalues by E i . With respect to this basis we denote the individual elements of ˆ ρ by ρ ij whose \nsolutions read \nρ ij ( t ) = ρ ij (0) e -i ( E i -E j ) t -κ 12 πβ ( E i -E j ) 2 t . (6.18) \nThis result agrees, up to the different prefactor of 2 in front of the decoherence term mentioned in subsection VI C, with the one obtained for the one-particle projection evaluated for motion in one dimension for a photon in [12]. \nWe will now discuss the comparison of the three aspects mentioned above. We start with the comparison of the coupling parameter. Such a comparison can be obtained by comparing the prefactors of the decoherence terms in the model from [27] and here. For the latter we have \nκ 12 πβ = 2 G N k B Θ 3 c 4 → 2 G N k B Θ 3 ℏ 2 c 5 , (6.19) \nwhere we restored the correct units in the last step. Comparing with the decoherence rate in [27], which is 4 η 2 k B Θ ℏ 3 , and introducing the Planck length ℓ P , we find that \nη 2 = ℏ G N 6 c 5 = ℓ 2 P 6 c 2 ≈ 5 · 10 -88 s 2 . (6.20) \nAn estimate for the value of η 2 inspired from field theory was already discussed in the appendix in [27]. The difference in the orders of magnitude compared to the analysis in the appendix in [27] arises due to the numerical prefactors that could not be determined precisely by the analogy analysis and estimate in [27]. Similar results of the coupling strength can be found in [9, 12]. As the one-particle master equation considers the case of a scalar field with a thermal gravitational background, more work is needed in order to develop more sophisticated models for neutrinos or fermions in general to derive a similar master equation for a fermionic system under consideration and for more general environments. In addition the model is based on linearising gravity around a flat Minkowski background, whereas it would be interesting to also consider decoherence models for longer propagation distances and consider master equations based on a model on a cosmological background as the presence of a scale factor could modify the decoherence effect, as analysed for instance in [59-61]. \nCompared to the quantum mechanical model [27], as mentioned above, here it was not necessary to introduce a spectral density, which is a kind of continuum limit for the frequencies of the oscillators in the environment that is typically used in similar quantum mechanical models to avoid Poincare recurrences, see for instance [2]. Furthermore, the cut-off function that needed to be used in the quantum mechanical model to regularise divergent integrals is not required for the one-particle master equation here. Instead the divergent contributions in the one-particle master equation could be linked to Feynman diagrams of a corresponding effective field theory giving a clearer physical interpretation than in the quantum mechanical toy model. With that given, the divergent contributions were treated using a standard renormalisation procedure known from quantum field theory that would be applied also in other situations in quantum field theory where such kind of diagrams play a role. \nNot entirely unrelated to the latter paragraph is the discussion of the application of the Markov approximation in the model considered here and the quantum mechanical toy model in [27]. In the latter, due to its simplicity compared to the one-particle master equation considered here, it was explicitly shown that the environmental correlation functions are strongly peaked around the initial time and decay rapidly after the peak. Such environmental correlation functions depend on both \nthe chosen spectral density and the chosen cut-off function. In this work, however, none of these choices are made, but the corresponding quantities are determined and set from the beginning when formulating the model. In section V A a condition was discussed under which the Markov approximation can be applied for the ultra-relativistic limit. Considering here the application to neutrinos we can discuss whether this condition is satisfied in this application and how it relates to the application of the Markov approximation in the quantum mechanical toy model in [27]. The condition applied in this work, equation (5.1), states that the Markov approximation is justified if 11 \nu, v ≫ k B Θ c . (6.21) \nIn the case of ultra-relativistic neutrinos where we neglect their masses, this is equivalent to \nE u , E v ≫ k B Θ , (6.22) \nwhere E u and E v denote the neutrino energies. Typical neutrino energies investigated in [27] start at energies of 1 GeV ≈ 1 . 6 · 10 -10 J . Given the Boltzmann constant k B ≈ 1 . 4 · 10 -23 J K and the temperature parameter Θ of the thermal gravitational waves of around 1 K used in [27], is the condition for the applicability of the Markov approximation used in this work \n1 . 6 · 10 -10 J ≫ 1 . 4 · 10 -23 J . (6.23) \nBoth sides of the inequality still differ by more than ten orders of magnitude, so the approximation can also be used for neutrinos with lower energies or for higher values of the temperature parameters. Thus, we can conclude given the proof presented in this work, the application of the Markov approximation to the physical scenario used in [27] is not only justified at the level of the quantum mechanical model as shown in [27] but also if one derives that model from the one-particle master equation of the QFT model. \nFinally, compared to the quantum mechanical toy model in [27], the more general one-particle master equation derived here also allows to consider the generalisation to decoherence models with wave packets, whereas in [27] plane waves were considered. This is due to the general form of the one particle density matrix defined in (3.1) as \nρ 1 ( t ) = ∫ R 3 d 3 u ∫ R 3 d 3 v ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v . (6.24) \nWhen choosing suitable initial conditions for ρ ( ⃗u, ⃗v, 0), one can model different descriptions for neutrinos like wave packets or plane waves, where the latter just correspond to delta distributions in this context.", 'VII. CONCLUSIONS': 'In this paper, we investigate the one-particle sector of the field-theoretic model in [1] for gravitationally induced decoherence for a scalar field coupled to linearised gravity. One of the main foci of this work is the renormalisation of the one-particle master equation, which, in contrast to the existing literature [9, 12], is performed before applying approximations such as the Markov or rotating wave approximation. As our results show this strategy provides a physical interpretation \nof the UV-divergent contributions in the one-particle master equation as being the vacuum part of the self-energy of the scalar field. To demonstrate this explicitly several steps are necessary: first the one-particle projection, where we considered two different kinds in this work, the non-extended and extended one. The latter also includes those processes in which in an intermediate steps two particles are created and annihilated afterwards or when the initial particle is left invariant and a vacuum bubble is created. Compared to the non-extended projection the last situation requires and additional renormalisation. While the non-extended projection yields a one-particle master equation with probability conservation, the extended projection does not have that property before any renormalisation, approximations or limits are taken into account. Then for both one-particle projections we identify the UV divergent contributions in the one-particle master equation. It turns out that all thermal parts of the one-particle master equation are UV finite and only the vacuum contributions in those terms which do not include other states than the one under consideration, are the divergent ones. Since in open quantum mechanical models the Lamb-shift Hamiltonians often require renormalisation and their physical interpretation is given, we wanted to address the question of the physical interpretation of the UV-divergent contributions in the one-particle master equation determined from an underlying field-theoretic model. \nFor this purpose, we used the methods introduced in [21] and applied them to an open quantum model with thermal gravitational waves as environment, instead of an environment consisting of a scalar field as in [21]. These methods allow us to identify contributions in the one-particle master equation with certain Feynman diagrams of the effective field theory for the scalar field. Since the model in [1] is based on a canonical quantisation of the master equation in a first step we consider the interaction part of the effective quantum field theory model in its canonical from and introduce the corresponding non-covariant Feynman rules along the lines of [22, 23], where similar methods are used in the framework of QED. Equipped with them, we consider the (non-covariant) self-energy diagram, which can be decomposed into a thermal and a vacuum part, and show that the latter can be identified with the UV-divergent contributions in the one-particle master equation. In order to apply a standard on-shell renormalisation procedure for the self-energy of the scalar field, we relate the non-covariant Feynmann rules to the covariant rules. Interestingly, a sum of two non-covariant Feynman diagrams, expressed by non-covariant propagators, can be combined into one diagram containing the corresponding covariant propagator. In the work in [21] the Feynman rules were directly available in covariant form and therefore the introduction of non-covariant Feynman rules was not necessary there. We then present the final renormalised one-particle master equation. \nAs our results show, the renormalisation leads to a renormalisation of the wave function in the model considered in the present work and the effect of the renormalisation is that all vacuum contributions in the Lamb-shift terms as well as in the dissipator are no longer present in the renormalised one-particle master equation. Comparing our results to the one in [9], where the renormalisation is performed after a Markov and rotating wave approximation have been applied and the non-relativistic limit has been considered, there the renormalisation procedure consists of introducing a cutoff function that is later absorbed in a redefinition of the renormalised mass. In [12] the effective model for the photon is also renormalised after applying the Markov and rotational wave approximation and considering the ultra-relativistic limit, and there the electric and magnetic field as well as the coupling constant are renormalised. Our results also allow a comparison to the renormalisaiton performed in the quantum mechanical model in [27]. While there the Lamb-shift Hamiltonian, that consists in that model only of vacuum contributions, needs to be renormalised and is absent after renormalisation, in the renormalised one-particle master equation in this work, the thermal part of the Lamb-shift Hamiltonian remains. This agrees with the situation one has when QED is treated as an open quantum model [2], where only the vacuum \ncontribution in the Lamb-shift terms is denoted as Lamb-shift and the thermal contribution as Stark shift. From the discussion of the open QED model from [2], one would expect a similar result in a quantum mechanical model where a thermal contribution in the Lamb-shift is present. We also see differences in how renormalisation affects the dissipator. While in the model presented here the renormalisation removes the vacuum contributions from the dissipator, in [27] the dissipator is not changed when the renormalisation is applied and therefore the vacuum contributions remain. However, these are removed as soon as the Markov approximation is applied and therefore do not contribute to the final Lindblad equation in [27]. \nGiven the renormalised one-particle master equation we then discuss the application of the Markov and rotation wave approximation. While a general analysis of the applicability of the Markov approximation is beyond the scope of this article, we can find a condition for the ultra-relativistic case under which the Markov approximation can be applied. For the applications to neutrino oscillations considered in [27], this condition is very mild and does not lead to severe restrictions. This fits with the fact that it could be explicitly shown for the quantum mechanical model in [27] that the Markov approximation can be applied. After the Markov approximation we also applied a post-trace rotatating wave approximation. Here we obtain a difference for the extended and non-extended one-particle projection. While for the extended projection, the rotating wave approximation has the effect that it removes the remaining part of the Lamb-shift term, in the case of the non-extended projection there is still one term left in the Lamb-shift contribution. The dissipator term is not affected by this approximation because those terms that would be potentially affected have been already removed by the one-particle projection. \nAs the first application of the renormalised one-particle master equation we discuss the evolution of the population and compare our results to the one in [11]. We demonstrate that although there exists a non-trivial evolution of the populations before renormalistion, which is consistent with the results obtained in [11] for the effective model of a photon if we choose the non-extended one-particle projection and specialise to the case of a vanishing temperature parameter, after renormalisation the evolution of the populations, that is the diagonal elements of the effective density matrix, becomes trivial. This shows the relevance of a renormalisation when analysing physical effects with the master equation, as the non-trivial dynamics of the populations and hence the effect discussed in [11] is removed by the renormalisation applied here. Hence, we conclude that if renormalisation is taken into account, we have no physically interesting effect in the evolution of the populations. In addition from our results for the evolution of the populations we learn that after applying the Markov approximation to the renormalised one-particle master equation, the extended and non-extended one-particle projection yield the same result. \nIn addition, we analyse the form of the one-particle master equation in the non- and ultrarelativistic limit. In these cases, the dissipator obtains a rather simple form which is similar to the ones in [9] for a non-relativistic scalar particle and in [12] for a photon. While the detailed structure of the dissipator in [9] differs from ours, [12] has the same one we have. \nAnother application we present is the comparison with the model in [27], where a quantum mechanical toy model for gravitationally induced decoherence, based on the model in [28], is investigated in the context of neutrino oscillations. Considering the renormalised one-particle master equation after applying the Markov and rotating wave approximation and in the special case of the ultra-relativistic limit, we obtain the renormalised Lindblad equation derived from the underlying quantum mechanical microscopic model in [27]. This comparison allows to fix the coupling constant in the interaction Hamiltonian that was an open parameter in [27]. As already discussed in [27], \nwhere a first estimate of this coupling constant was given as well as the discussions in [12, 62-65] the value is fairly tiny. Whether more sophisticated models for gravitationally induced decoherence will involve larger values of the coupling parameter is an interesting question to discuss in future research. In addition, in contrast to the quantum mechanical model [27], no spectral density, which can be understood as a kind of continuum limit of the frequencies of the harmonic oscillators in the environment of the model in [27], has to be introduced into the field-theoretical model in this work, but the corresponding integrals are automatically determined from the underlying action of the model. The results obtained here also allow an extension of the model in [27], where the plane wave approach for the neutrinos is used, to models including wave packets, which can be implemented by an appropriate choice of the one-particle density matrix. We plan to consider an analysis of such a model in future work. \nThere are several directions for further extensions and generalisations of the results obtained in this article. First, one can consider field-theoretical models where an operator ordering is chosen for which the self-interaction term of the scalar field does not vanish, as in [9]. In this case, we do not expect the renormalisation procedure to change, since we expect the same identification of non-covariant and covariant Feynman diagrams to hold for the self-energy, with the only difference that the second diagram in the sum of non-covariant diagrams does not vanish, as is the case in the model considered here. Instead of introducing the covariant Feynman rules in order to apply the renormalisation, one can also study if and how the methods from [66] for one-loop renormalisation of QED in Coulomb gauge can be employed to perform the renormalisation directly based on the non-covariant Feynman rules. Furthermore, a more general analysis of the question under which criteria the Markov approximation can be applied would be interesting. In particular, whether it is also possible to formulate conditions that depend only on the properties of the environment, as it was done in [54] for open quantum models with finitely many degrees of freedom. Also a study of the applicability and effect of different versions of the Markov approximation for gravitationally induced decoherence models is an interesting point, as some of them directly yield a Lindblad equation and therefore remove the necessity to apply an additional rotating wave approximation. Such modified Markov approximations is for instance the one discussed in [67, 68]. There, in addition to the density matrix, the operators of the system in the Schrodinger picture, which appear in the double commutator in the master equation, are Taylor expanded around the final time. Another possibility is a generalisation to field theory of the approximation for quantum mechanical models in [69], where one obtains a completely positive master equation by replacing an arithmetic mean of the spectral density by a geometric one. \nAlthough there are quantum field theoretical models for gravitationally induced decoherence for scalar fields [1, 8, 9], photons [12, 16] or generic bosons [11], a model that includes fermions is still missing in the literature. Such a model would also be interesting in connection with the application to neutrino oscillations. Another possible generalisation is to consider quantisations other than Fock quantisations for the one-particle model, see for example [70], where a quantisation inspired by loop quantum gravity was used in the framework of polymerised quantum mechanics to formulate an open scattering model.', 'ACKNOWLEDGMENTS': 'MJF and KG would like to thank Michael Kobler for the valuable discussions at an early stage of the project. MJF would like to thank Renata Ferrero and Roman Kemper for the fruitful discussions on some computational aspects of the project. MJF thanks the Heinrich-Boll foundation for financial support.', 'Appendix A.I ONE-PARTICLE PROJECTION OF THE MASTER EQUATION': 'In this section, we explicitly carry out the projection of the master equation (2.1) on the oneparticle sector. As discussed in the main text, we proceed by inserting the density matrix of a single particle \nρ 1 ( t ) = ∫ d 3 u ∫ d 3 v ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v (A.I.1) \ninto the master equation (2.1), where ⃗u and ⃗v denote labels of the momentum and then by neglecting all contributions that do not preserve the one-particle subspace. The projection of the Hamiltonian H S of the free scalar field and the gravitational self-interaction of the scalar field U have already been discussed in the main text. Here we focus on the remaining terms, the Lamb-shift-like Hamiltonian H LS and the dissipator D . First, we give the detailed expressions for these two contributions of the master equation in subsection A.I.1 and then project them separately on the one-particle space in subsection A.I.2.', 'A.I.1 Individual contributions to the master equation': "In [1] the original dissipator was decomposed into two parts by separating a purely complex contribution entering the master equation analoguously to H S and U in terms of a commutator with ρ S . This term was then defined to be the so-called Lamb-shift Hamiltonian that contributes to the unitary evolution of the density matrix and the remaining terms were identified as a new dissipator term D that has a form similar to the first standard form (see e.g. [2] for the different forms of dissipators). The Lamb-shift Hamiltonian is given in [1] as \nH LS := 1 2 ∫ d 3 k d 3 p d 3 l (2 π ) 6 2 ∑ r ; a,b S ab ( ⃗ p, ⃗ l ; ⃗ k, t ) j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) , (A.I.2) \nand the dissipator as \nD [ ρ S ] := κ 2 ∫ d 3 k d 3 p d 3 l (2 π ) 6 2 ∑ r ; a,b R ab ( ⃗ p, ⃗ l ; ⃗ k, t ) ( j b r ( ⃗ k, ⃗ l ) ρ S ( t ) j a r ( ⃗ k, ⃗p ) † -1 2 { j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) , ρ S ( t ) } ) , (A.I.3) \nwhere S ab and R ab are coefficient functions and the j a r are operators containing combinations of scalar field's creation and annihilation operators respectively. These quantities will be defined below. From the structure of H LS and D follows that three different kinds of terms have to be evaluated for the projection: \nj a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) ρ 1 ( t ) ︸ ︷︷ ︸ ( I ) and ρ 1 ( t ) j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) ︸ ︷︷ ︸ ( II ) and j b r ( ⃗ k, ⃗ l ) ρ 1 ( t ) j a r ( ⃗ k, ⃗p ) † ︸ ︷︷ ︸ ( III ) . (A.I.4) \nThe coefficient functions S and R are defined as \nS ab ( ⃗ p, ⃗ l ; ⃗ k, t ) := 1 2Ω k [ [ N (Ω k ) + 1] { e -i (Ω k + ω b ( ⃗ k, ⃗ l )) t -1 Ω k + ω b ( ⃗ k, ⃗ l ) + e i (Ω k + ω a ( ⃗ k,⃗p )) t -1 Ω k + ω a ( ⃗ k, ⃗p ) } -N (Ω k ) { e i (Ω k -ω b ( ⃗ k, ⃗ l )) t -1 Ω k -ω b ( ⃗ k, ⃗ l ) + e -i (Ω k -ω a ( ⃗ k,⃗p )) t -1 Ω k -ω a ( ⃗ k, ⃗p ) }] (A.I.5) \nand \nR ab ( ⃗ p, ⃗ l ; ⃗ k, t ) := i Ω k [ [ N (Ω k ) + 1] { e -i (Ω k + ω b ( ⃗ k, ⃗ l )) t -1 Ω k + ω b ( ⃗ k, ⃗ l ) -e i (Ω k + ω a ( ⃗ k,⃗p )) t -1 Ω k + ω a ( ⃗ k, ⃗p ) } -N (Ω k ) { e i (Ω k -ω b ( ⃗ k, ⃗ l )) t -1 Ω k -ω b ( ⃗ k, ⃗ l ) -e -i (Ω k -ω a ( ⃗ k,⃗p )) t -1 Ω k -ω a ( ⃗ k, ⃗p ) }] (A.I.6) \nwith the Bose-Einstein distribution \nN (Ω k ) =: N ( k ) = 1 e βk -1 , (A.I.7) \nwhere we use the graviton frequency Ω k = √ ⃗ k 2 =: k and β = 1 k B Θ with the Boltzmann constant k B and a temperature parameter Θ that determines the Gibbs state of the gravitational waves environment. Furthermore we have, defining ω k := √ m 2 + ⃗ k 2 with the scalar field's mass m : \nj 1 r ( ⃗ k, ⃗p ) := a † p a k + p 1 2 √ ω p ω k + p [ p a p b [ P -r ( ⃗ k )] a b ] ω 1 ( ⃗ k, ⃗p ) := ω p -ω k + p (A.I.8) \nj 2 r ( ⃗ k, ⃗p ) := a † -p -k a -p 1 2 √ ω p ω k + p [ p a p b [ P -r ( ⃗ k )] a b ] ω 2 ( ⃗ k, ⃗p ) := ω k + p -ω p (A.I.9) \nj 3 r ( ⃗ k, ⃗p ) := a -p a k + p 1 2 √ ω p ω k + p [ p a p b [ P -r ( ⃗ k )] a b ] ω 3 ( ⃗ k, ⃗p ) := -ω p -ω k + p (A.I.10) \nj 4 r ( ⃗ k, ⃗p ) := a † p a † -k -p 1 2 √ ω p ω k + p [ p a p b [ P -r ( ⃗ k )] a b ] ω 4 ( ⃗ k, ⃗p ) := ω p + ω k + p . (A.I.11) \nHere, a k and a † k are annihilation and creation operator valued distributions acting on the underlying bosonic Fock space for a scalar field in the standard way. [ P ± r ( ⃗ k )] a b are the projectors on the individual transverse modes defined via \n[ P ± ( ⃗ k )] b a := m a ( ± ⃗ k ) m b ( ± ⃗ k ) , (A.I.12) \nwhere { ⃗ k | ⃗ k | , m ( ⃗ k ) , m ( ⃗ k ) } form an orthonormal basis of R 3 , details can be found in [1]. In what follows, we only need the properties of orthonormality of the three basis elements and the fact that the symmetric transverse-traceless projector is \nP abcd ( ⃗ k ) = ∑ r ∈{±} [ P r ( ⃗ k )] a i [ P -r ( ⃗ k )] c j δ ib δ cd = 1 2 [ P ac ( ⃗ k ) P bd ( ⃗ k ) + P ad ( ⃗ k ) P bc ( ⃗ k ) -P ab ( ⃗ k ) P cd ( ⃗ k )] . \n(A.I.13) \nwith the transverse projector \nP ab ( ⃗ k ) = δ ab -k a k b ⃗ k 2 . (A.I.14) \nNow we can proceed to project H LS and D on the one-particle space.", 'A.I.2 Computation of the one-particle projection': 'We start with the evaluation of the three kinds of terms (I)-(III) and consider all possible combinations ( a, b ) that give a one-particle state after application. To keep track of all different combination in the three cases, we do this using a table. Considering only the creation and annihilation operator valued distributions a ( † ) k in the j -operators, we find: \n- (I) j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,1) a † k + p a p a † l a l + k a † u | 0 ⟩ ⟨ 0 | a v = δ 3 ( ⃗ l + ⃗ k -⃗u ) a † k + p a p a † l | 0 ⟩ ⟨ 0 | a v = δ 3 ( ⃗ p + ⃗ k -⃗u ) δ 3 ( ⃗ p -⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,2) a † k + p a p a † -l -k a -l a † u | 0 ⟩ ⟨ 0 | a v = δ 3 ( ⃗ l + ⃗u ) δ 3 ( ⃗ p + ⃗ l + ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,1) a † -p a -p -k a † l a k + l a † u | 0 ⟩ ⟨ 0 | a v = δ 3 ( ⃗ k + ⃗ l -⃗u ) δ 3 ( ⃗ p + ⃗ l + ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,2) a † -p a -p -k a † -l -k a -l a † u | 0 ⟩ ⟨ 0 | a v = δ 3 ( ⃗ l + ⃗u ) δ 3 ( ⃗ p -⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (4,4) δ P a -k -p a p a † l a † -k -l a † u | 0 ⟩ ⟨ 0 | a v = δ P ( a ) see below\n- (II) a † u | 0 ⟩ ⟨ 0 | a v j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l )\n- (1,1) a † u | 0 ⟩ ⟨ 0 | a v a † k + p a p a † l a l + k = δ 3 ( ⃗ p + ⃗ k -⃗v ) δ 3 ( ⃗ p -⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,2) a † u | 0 ⟩ ⟨ 0 | a v a † k + p a p a † -l -k a -l = δ 3 ( ⃗ k + ⃗ p -⃗v ) δ 3 ( ⃗ p + ⃗ l + ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,1) a † u | 0 ⟩ ⟨ 0 | a v a † -p a -p -k a † l a k + l = δ 3 ( ⃗ p + ⃗v ) δ 3 ( ⃗ p + ⃗ l + ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,2) a † u | 0 ⟩ ⟨ 0 | a v a † -p a -p -k a † -l -k a -l = δ 3 ( ⃗ p + ⃗v ) δ 3 ( ⃗ p -⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (4,4) δ P a † u | 0 ⟩ ⟨ 0 | a v a -k -p a p a † l a † -k -l = δ P (b) see below', '(III) j b r ( ⃗ k, ⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v j a r ( ⃗ k, ⃗p ) †': "- (1,1) a † l a l + k a † u | 0 ⟩ ⟨ 0 | a v a † k + p a p = δ 3 ( ⃗ l + ⃗ k -⃗u ) δ 3 ( ⃗ k + ⃗ p -⃗v ) a † u -k | 0 ⟩ ⟨ 0 | a v -k\n- (1,2) a † -l -k a -l a † u | 0 ⟩ ⟨ 0 | a v a † k + p a p = δ 3 ( ⃗ k + ⃗ p -⃗v ) δ 3 ( ⃗u + ⃗ l ) a † u -k | 0 ⟩ ⟨ 0 | a v -k\n- (2,1) a † l a k + l a † u | 0 ⟩ ⟨ 0 | a v a † -p a -p -k = δ 3 ( ⃗ p + ⃗v ) δ 3 ( ⃗ l + ⃗ k -⃗u ) a † u -k | 0 ⟩ ⟨ 0 | a v -k\n- (2,2) a † -l -k a -l a † u | 0 ⟩ ⟨ 0 | a v a † -p a -p -k = δ 3 ( ⃗ p + ⃗v ) δ 3 ( ⃗u + ⃗ l ) a † u -k | 0 ⟩ ⟨ 0 | a v -k \nThe expressions for (a) and (b) are calculated separately because they include the vacuum bubbles mentioned in the main text that require a renormalisation. By applying the commutators and commuting all annihilation operators towards the vacuum state one obtains: \n( a ) = a -k -p a p a † l a † -k -l a † u | 0 ⟩ ⟨ 0 | a v =[ δ 3 ( ⃗ p + ⃗ k + ⃗ l ) δ 3 ξ ( ⃗ p + ⃗ k + ⃗ l ) + δ 3 ( ⃗ p -⃗ l ) δ 3 ξ ( ⃗ p -⃗ l ) + δ 3 ( ⃗ p + ⃗ k + ⃗ l ) δ 3 ( ⃗u + ⃗ k + ⃗ p ) + δ 3 ( ⃗ p -⃗ l ) δ 3 ( ⃗u + ⃗ k + ⃗ p ) + δ 3 ( ⃗ p -⃗u ) δ 3 ( ⃗ p -⃗ l ) + δ 3 ( ⃗ p -⃗u ) δ 3 ( ⃗ k + ⃗ p + ⃗ l )] a † u | 0 ⟩ ⟨ 0 | a v , (A.I.15) \n( b ) = a † u | 0 ⟩ ⟨ 0 | a v a -k -p a p a † l a † -k -l =[ δ 3 ( ⃗ p + ⃗ k + ⃗ l ) δ 3 ξ ( ⃗ p + ⃗ k + ⃗ l ) + δ 3 ( ⃗ p -⃗ l ) δ 3 ξ ( ⃗ p -⃗ l ) + δ 3 ( ⃗ p + ⃗ k + ⃗ l ) δ 3 ( ⃗v + ⃗ k + ⃗ l ) + δ 3 ( ⃗ p -⃗ l ) δ 3 ( ⃗v + ⃗ k + ⃗ l ) + δ 3 ( ⃗v -⃗ l ) δ 3 ( ⃗ p -⃗ l ) + δ 3 ( ⃗v -⃗ l ) δ 3 ( ⃗ k + ⃗ p + ⃗ l )] a † u | 0 ⟩ ⟨ 0 | a v . (A.I.16) \nNote that under the map ( ⃗u, ⃗p, ⃗ l ) ↔ ( ⃗v, ⃗ l, ⃗p ) we can get from (a) to (b) and vice versa. An important remark here is that the first two terms in both expressions contain the square of a Dirac delta distribution. This is a problematic term since, as can be shown, the corresponding integral over this expression still diverges when for the individual delta distributions the regularised version is considered and the regulator is removed after the integration is performed. We will deal with this issue further below in this subsection where we renormalise the density matrix to handle the divergent contributions involved. For now, we replace one of the two delta distributions by a function including a regulator δ 3 ( ⃗ k ) → δ 3 ξ ( ⃗ k ), where the regulator is sent to zero after performing the corresponding integrations. \nIn a next step, we evaluate the entire expressions appearing in the Lamb-shift Hamiltonian \nand the dissipator. For each term, where the delta distributions resolve two integrals, that is for all terms but the diverging ones, we choose to resolve the integrals over the range of the variables ⃗ p and ⃗ l respectively. Then we are left with the integration over the variables ⃗ k , ⃗u and ⃗v as well as with the sum over the polarisation labels r . Considering for V ab which is either R ab or S ab , that were defined above in (A.I.6) and (A.I.5), the three expressions \n(I): ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v (A.I.17) \n(II): ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) a † u | 0 ⟩ ⟨ 0 | a v j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) (A.I.18) \n(III): ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) j a r ( ⃗ k, ⃗p ) † a † u | 0 ⟩ ⟨ 0 | a v j b r ( ⃗ k, ⃗ l ) , (A.I.19) \nthat appear in (A.I.2) and (A.I.3), we obtain: \n- (I) ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,1) V 11 ( ⃗u -⃗ k, ⃗u -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω u ω u -k P u ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,2) V 12 ( ⃗u -⃗ k, -⃗u ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω u ω u -k P u ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,1) V 21 ( -⃗u, ⃗u -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω u ω u -k P u ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,2) V 22 ( -⃗u, -⃗u ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω u ω u -k P u ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (4,4) δ P (a) (see below) \n(II) ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) a † u | 0 ⟩ ⟨ 0 | a v j a r ( ⃗ k, ⃗p ) † j b r ( ⃗ k, ⃗ l ) \n- (1,1) V 11 ( ⃗v -⃗ k,⃗v -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω v ω v -k P v ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (1,2) V 12 ( ⃗v -⃗ k, -⃗v ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω v ω v -k P v ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,1) V 21 ( -⃗v, ⃗v -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω v ω v -k P v ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (2,2) V 22 ( -⃗v, -⃗v ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 ω v ω v -k P v ( ⃗ k ) a † u | 0 ⟩ ⟨ 0 | a v\n- (4,4) δ P (b) (see below) \n(III) ∑ r ∫ d 3 p ∫ d 3 l V ab ( ⃗ p, ⃗ l ; ⃗ k, t ) j a r ( ⃗ k, ⃗p ) † a † u | 0 ⟩ ⟨ 0 | a v j b r ( ⃗ k, ⃗ l ) \n- (1,1) V 11 ( ⃗v -⃗ k, ⃗u -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 √ ω v ω v -k ω u ω u -k P u,v ( ⃗ k ) a † u -k | 0 ⟩ ⟨ 0 | a v -k\n- (1,2) V 12 ( ⃗v -⃗ k, -⃗u ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 √ ω v ω v -k ω u ω u -k P u,v ( ⃗ k ) a † u -k | 0 ⟩ ⟨ 0 | a v -k\n- (2,1) V 11 ( -⃗v, ⃗u -⃗ k ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 √ ω v ω v -k ω u ω u -k P u,v ( ⃗ k ) a † u -k | 0 ⟩ ⟨ 0 | a v -k \n(2,2) V 22 ( -⃗v, -⃗u ; ⃗ k, t ) ρ ( ⃗u, ⃗v ) 1 4 √ ω v ω v -k ω u ω u -k P u,v ( ⃗ k ) a † u -k | 0 ⟩ ⟨ 0 | a v -k , \nwhere we defined 12 \nP u ( ⃗ k ) := ∑ r [ ⃗u · ⃗ m ( -r ⃗ k )] 2 [ ⃗u · ⃗ m ( r ⃗ k )] 2 = 2[ ⃗u · ⃗ m ( -⃗ k )] 2 [ ⃗u · ⃗ m ( ⃗ k )] 2 = P abcd ( ⃗ k ) u a u b u c u d (A.I.20) \nP u,v ( ⃗ k ) := ∑ r [ ⃗u · ⃗ m ( -r ⃗ k )] 2 [ ⃗v · ⃗ m ( r ⃗ k )] 2 = P abcd ( ⃗ k ) u a u b v c v d . (A.I.21) \nNext, we evaluate in more detail the coefficient functions S and R which were defined in (A.I.5) and (A.I.6). Firstly excluding all terms from the extended projection, i.e. the ones arising from the combination (4 , 4), it turns out that the coefficients appearing in ( I ) -( III ) are equal to each other in each group: \nS 11 ( ⃗u -⃗ k, ⃗u -⃗ k ; ⃗ k, t ) = S 12 ( ⃗u -⃗ k, -⃗u ; ⃗ k, t ) = S 21 ( -⃗u, ⃗u -⃗ k ; ⃗ k, t ) = S 22 ( -⃗u, -⃗u ; ⃗ k, t ) =: S ( -) 1 P ( ⃗u, ⃗ k, t ) = 1 Ω k { N ( k ) + 1 Ω k + ω u -k -ω u (cos [(Ω k + ω u -k -ω u ) t ] -1) -N ( k ) Ω k -ω u -k + ω u (cos [(Ω k -ω u -k + ω u ) t ] -1) } , (A.I.22) \nR 11 ( ⃗u -⃗ k, ⃗u -⃗ k ; ⃗ k, t ) = R 12 ( ⃗u -⃗ k, -⃗u ; ⃗ k, t ) = R 21 ( -⃗u, ⃗u -⃗ k ; ⃗ k, t ) = R 22 ( -⃗u, -⃗u ; ⃗ k, t ) =: 2 R ( -) 1 P ( ⃗u, ⃗ k, t ) = 2 Ω k { N ( k ) + 1 Ω k + ω u -k -ω u sin [(Ω k + ω u -k -ω u ) t ] + N ( k ) Ω k -ω u -k + ω u sin [(Ω k -ω u -k + ω u ) t ] } (A.I.23) \nand \nR 11 ( ⃗v -⃗ k, ⃗u -⃗ k ; ⃗ k, t ) = R 12 ( ⃗v -⃗ k, -⃗u ; ⃗ k, t ) = R 21 ( -⃗v, ⃗u -⃗ k ; ⃗ k, t ) = R 22 ( -⃗v, -⃗u ; ⃗ k, t ) =: 2 R (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) = i Ω k { [ N ( k ) + 1] ( e -i (Ω k + ω u -k -ω u ) t -1 Ω k + ω u -k -ω u -e i (Ω k + ω v -k -ω v ) t -1 Ω k + ω v -k -ω v ) -N ( k ) ( e i (Ω k -ω u -k + ω u ) t -1 Ω k -ω u -k + ω u -e -i (Ω k -ω v -k + ω v ) t -1 Ω k -ω v -k + ω v )} . (A.I.24) \nWith these expressions, we can rewrite the equations for the Lamb-shift Hamiltonian and the dissipator that contain the contributions of ( a, b ) ∈ { 1 , 2 } : \n-i [ H { 1 , 2 } LS , ρ 1 ( t )] = -iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u S ( -) 1 P ( ⃗u, ⃗ k, t ) + iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v S ( -) 1 P ( ⃗v, ⃗ k, t ) (A.I.25) \nas well as \nD { 1 , 2 } [ ρ 1 ( t )] = -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u R ( -) 1 P ( ⃗u, ⃗ k, t ) -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v R ( -) 1 P ( ⃗v, ⃗ k, t ) + κ ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u -k | 0 ⟩ ⟨ 0 | a v -k P u,v ( ⃗ k ) √ ω u -k ω u ω v -k ω v R (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) . (A.I.26) \nIt remains to deal with the (4 , 4)-terms that arise when using the extended projection. The four summands from (a) and (b) without the terms containing the ξ -regulator yield \n-i [ H { 4 , 4 } LS , ρ 1 ( t )] = -iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u + k ω u ˜ S (+) 1 P ( ⃗u, ⃗ k, t ) δ P + iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v + k ω v ˜ S (+) 1 P ( ⃗v, ⃗ k, t ) δ P (A.I.27) \nwith \n˜ S (+) 1 P ( ⃗u, ⃗ k, t ) := S 44 ( -⃗u -⃗ k, ⃗u ; ⃗ k, t ) = S 44 ( -⃗u -⃗ k, -⃗u -⃗ k ; ⃗ k, t ) = S 44 ( ⃗u, ⃗u ; ⃗ k, t ) = S 44 ( ⃗u, -⃗u -⃗ k ; ⃗ k, t ) = 1 Ω k { N ( k ) + 1 Ω k + ω u + k + ω u (cos [(Ω k + ω u + k + ω u ) t ] -1) -N ( k ) Ω k -ω u + k -ω u (cos [(Ω k -ω u + k -ω u ) t ] -1) } . (A.I.28) \nAfter a substitution ⃗ k →-⃗ k in the integration we find: \n-i [ H { 4 , 4 } LS , ρ 1 ( t )] = -iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u S (+) 1 P ( ⃗u, ⃗ k, t ) δ P + iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v S (+) 1 P ( ⃗v, ⃗ k, t ) δ P (A.I.29) \nwhere \nS (+) 1 P ( ⃗u, ⃗ k, t ) := ˜ S (+) 1 P ( ⃗u, -⃗ k, t ) . (A.I.30) \nThe same substitution and the definition \nR (+) 1 P ( ⃗u, ⃗ k, t ) := 1 Ω k { N ( k ) + 1 Ω k + ω u -k + ω u sin [(Ω k + ω u -k + ω u ) t ] + N ( k ) Ω k -ω u -k -ω u sin [(Ω k -ω u -k -ω u ) t ] } (A.I.31) \nlead to: \nD { 4 , 4 } [ ρ 1 ] = -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u R (+) 1 P ( ⃗u, ⃗ k, t ) δ P -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v R (+) 1 P ( ⃗v, ⃗ k, t ) δ P . (A.I.32) \nDefining additionally \nS (1) 1 P ( ⃗u, ⃗ k, t ) := S ( -) 1 P ( ⃗u, ⃗ k, t ) + S (+) 1 P ( ⃗u, ⃗ k, t ) δ P (A.I.33) \nR (1) 1 P ( ⃗u, ⃗ k, t ) := R ( -) 1 P ( ⃗u, ⃗ k, t ) + R (+) 1 P ( ⃗u, ⃗ k, t ) δ P (A.I.34) \ngives us the opportunity to rewrite \n-i [ H LS , ρ 1 ( t )] = -iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u S (1) 1 P ( ⃗u, ⃗ k, t ) + iκ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v S (1) 1 P ( ⃗v, ⃗ k, t ) (A.I.35) \nas well as \nD [ ρ 1 ] = -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u R (1) 1 P ( ⃗u, ⃗ k, t ) -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v R (1) 1 P ( ⃗v, ⃗ k, t ) + κ ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u -k | 0 ⟩ ⟨ 0 | a v -k P u,v ( ⃗ k ) √ ω u -k ω u ω v -k ω v R (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) . (A.I.36) \nThis can be summarised as \n-i [ H LS ,ρ 1 ( t )] + D [ ρ 1 ] = = -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P u ( ⃗ k ) ω u -k ω u [ R (1) 1 P ( ⃗u, ⃗ k, t ) + iS (1) 1 P ( ⃗u, ⃗ k, t )] -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u | 0 ⟩ ⟨ 0 | a v P v ( ⃗ k ) ω v -k ω v [ R (1) 1 P ( ⃗v, ⃗ k, t ) -iS (1) 1 P ( ⃗u, ⃗ k, t )] + κ ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ⃗u, ⃗v, t ) a † u -k | 0 ⟩ ⟨ 0 | a v -k P u,v ( ⃗ k ) √ ω u -k ω u ω v -k ω v R (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) . (A.I.37) \nFinally, the terms in the extended projection that contain the ξ -regulator, that is the expressions arising from the first two terms in (A.I.15) and (A.I.16), are analysed. These terms are equal for (a) and (b) and read: \nδ P lim ξ → 0 ∫ d 3 p d 3 l d 3 k d 3 ud 3 v (2 π ) 3 2 a † u | 0 ⟩ ⟨ 0 | a v V 44 ( ⃗ p, ⃗ l ; ⃗ k, t ) P p,l ( ⃗ k ) 4 √ ω p ω k + p ω l ω k + l ρ ( ⃗u, ⃗v, t ) · { δ 3 ( ⃗ p + ⃗ k + ⃗ l ) δ 3 ξ ( ⃗ p + ⃗ k + ⃗ l ) + δ 3 ( ⃗ p -⃗ l ) δ 3 ξ ( ⃗ p -⃗ l ) } = δ P lim ξ → 0 ∫ d 3 p d 3 k (2 π ) 3 2 V 44 ( ⃗ p, ⃗p ; ⃗ k, t ) P p ( ⃗ k ) 2 ω p ω k + p δ 3 ξ ( ⃗ 0) ρ 1 ( t ) , (A.I.38) \nwhere we used that V 44 ( ⃗ p, ⃗p ; ⃗ k, t ) = V 44 ( ⃗ p, ⃗ -k -p ; ⃗ k, t ). Due to the equality of these terms for (a) and (b), they drop out of the Lamb-shift Hamiltonian and are only left in the dissipator term: \nD div [ ρ 1 ] = -δ P κ 4 lim ξ → 0 ∫ d 3 k d 3 p (2 π ) 3 2 R 44 ( ⃗ p, ⃗p ; ⃗ k, t ) P p ( ⃗ k ) 2 ω p ω k + p δ 3 ξ ( ⃗ 0) ρ 1 ( t ) =: Z ( t ) ρ 1 ( t ) . (A.I.39) \nWritten in this form, it becomes evident that they do act as a multiplicative constant and do not modify the state ρ 1 . Therefore they are nothing but vacuum bubbles expressed in QFT language. With the definition of Z ( t ) above the entire master equation for the single particle is given by \n∂ ∂t ρ 1 ( t ) = -i [ H S + H LS , ρ 1 ( t )] + D [ ρ 1 ] + Z ( t ) ρ 1 ( t ) , (A.I.40) \nwhere from now on the terms absorbed in Z ( t ) are dropped from the definition of D [ ρ 1 ]. We can see that the diverging term Z ( t ) can be absorbed by a renormalisation of the density matrix, likewise to a renormalisation of the wave function known from QFT: \nρ 1 ( t ) → ρ ( ren ) 1 ( t ) := exp (∫ t 0 dt ' Z ( t ' ) ) ρ 1 ( t ) . (A.I.41) \nIn terms of the renormalised density matrix, the one-particle master equation then reads: \n∂ ∂t ρ ( ren ) 1 ( t ) = -i ∫ d 3 ud 3 v ρ ( ren ) ( ⃗u, ⃗v, t ) | ⃗u ⟩ ⟨ ⃗v | ( ω u -ω v ) -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ren ) ( ⃗u, ⃗v, t ) | ⃗u ⟩ ⟨ ⃗v | P u ( ⃗ k ) ω u -k ω u [ R (1) 1 P ( ⃗u, ⃗ k, t ) + iS (1) 1 P ( ⃗u, ⃗ k, t )] -κ 2 ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ren ) ( ⃗u, ⃗v, t ) | ⃗u ⟩ ⟨ ⃗v | P v ( ⃗ k ) ω v -k ω v [ R (1) 1 P ( ⃗v, ⃗ k, t ) -iS (1) 1 P ( ⃗v, ⃗ k, t )] + κ ∫ d 3 k d 3 ud 3 v (2 π ) 3 ρ ( ren ) ( ⃗u, ⃗v, t ) | ⃗u -⃗ k ⟩ ⟨ ⃗v -⃗ k | P u,v ( ⃗ k ) √ ω u -k ω u ω v -k ω v R (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) . (A.I.42) \nTo simplify the notation, we drop the label ( ren ) from the density matrix from this point on. The complex combinations R ± iS can be evaluated further and one finds \nR (1) 1 P ( ⃗u, ⃗ k, t ) ± iS (1) 1 P ( ⃗u, ⃗ k, t ) = = ± i Ω k { N ( k ) + 1 Ω k + ω u -k -ω u ( e ∓ i (Ω k + ω u -k -ω u ) t -1 ) -N ( k ) Ω k -ω u -k + ω u ( e ± i (Ω k -ω u -k + ω u ) t -1 ) + δ P N ( k ) + 1 Ω k + ω u -k + ω u ( e ∓ i (Ω k + ω u -k + ω u ) t -1 ) -δ P N ( k ) Ω k -ω u -k -ω u ( e ± i (Ω k -ω u -k -ω u ) t -1 ) } = ∫ t 0 dτ Ω k { [ N ( k ) + 1] e ∓ i (Ω k + ω u -k -ω u ) τ + N ( k ) e ± i (Ω k -ω u -k + ω u ) τ + δ P [ N ( k ) + 1] e ∓ i (Ω k + ω u -k + ω u ) τ + δ P N ( k ) e ± i (Ω k -ω u -k -ω u ) τ } . (A.I.43) \nFor R (2) we have: \nR (2) 1 P ( ⃗u, ⃗v, ⃗ k, t ) = i 2Ω k { [ N ( k ) + 1] ( e -i (Ω k + ω u -k -ω u ) t -1 Ω k + ω u -k -ω u -e i (Ω k + ω v -k -ω v ) t -1 Ω k + ω v -k -ω v ) -N ( k ) ( e i (Ω k -ω u -k + ω u ) t -1 Ω k -ω u -k + ω u -e -i (Ω k -ω v -k + ω v ) t -1 Ω k -ω v -k + ω v )} = ∫ t 0 dτ 2Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k -ω u ) τ + N ( k ) e i (Ω k -ω u -k + ω u ) τ +[ N ( k ) + 1] e i (Ω k + ω v -k -ω v ) τ + N ( k ) e -i (Ω k -ω v -k + ω v ) τ } . (A.I.44) \nDefining then \nC ( ⃗u, ⃗ k, t ) = ∫ t -t 0 0 dτ Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k -ω u ) τ + N ( k ) e i (Ω k -ω u -k + ω u ) τ } (A.I.45) \nC P ( ⃗u, ⃗ k, t ) = ∫ t -t 0 0 dτ Ω k { [ N ( k ) + 1] e -i (Ω k + ω u -k + ω u ) τ + N ( k ) e i (Ω k -ω u -k -ω u ) τ } , (A.I.46) \nwhere we have restored the initial time 13 t 0 that was set to 0 in [1], the one-particle master equation in momentum representation has the form \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) -κ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u [ C ( ⃗u, ⃗ k, t ) + δ P C P ( ⃗u, ⃗ k, t ) ] + P v ( ⃗ k ) ω v -k ω v [ C ∗ ( ⃗v, ⃗ k, t ) + δ P C ∗ P ( ⃗v, ⃗ k, t ) ] } ρ ( ⃗u, ⃗v, t ) + κ 2 ∫ d 3 k (2 π ) 3 P u,v ( ⃗ k ) √ ω u + k ω u ω v + k ω v { C ( ⃗u + ⃗ k, ⃗ k, t ) + C ∗ ( ⃗v + ⃗ k, ⃗ k, t ) } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) . (A.I.47)", 'Appendix A.II EQUIVALENCE OF THE NON-COVARIANT AND COVARIANT FEYNMAN RULES FOR THE COULOMB-LIKE SCATTERING DIAGRAM': 'Here we show that the non-covariant and the covariant Feynman rules introduced in sections IV B and IVC respectively yield the same result for the Coulomb-like scattering diagram \n<!-- image --> \nThe left hand side of the equation is expressed in terms of the non-covariant Feynman rules and the first diagram, which we label A , reads \nA = 1 2 ( -iκ ) 2 ˜ T ab ( p, q ) ˜ T cd ( -u, -v ) -i κ 1 k 2 P abcd ( ⃗ k ) = iκ 2 k 2 ˜ T ab ( p, q ) ˜ T cd ( -u, -v ) [ ( δ ac -k a k c ⃗ k 2 )( δ bd -k b k d ⃗ k 2 ) -1 2 ( δ ab -k a k b ⃗ k 2 )( δ cd -k c k d ⃗ k 2 ) ] = iκ k 2 [ 1 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T b b ( -u, -v ) + 1 4 ⃗ k 4 k a k b k c k d ˜ T ab ( p, q ) ˜ T cd ( -u, -v ) -1 ⃗ k 2 { ˜ T ab ( p, q ) ˜ T a c ( -u, -v ) k b k c -1 4 ˜ T ab ( p, q ) ˜ T c c ( -u, -v ) k a k b -1 4 ˜ T a a ( p, q ) ˜ T cd ( -u, -v ) k c k d } ] , (A.II.1) \nwhere the overall factor of 1 2 arises due to the fact that it is a diagram of second order in the expansion of the Dyson series. Next, we can make use of energy-momentum conservation, as we assumed the scalar particles to be on-shell, which reads, as introduced in the main text in (4.21) and (4.22) for k = p + q = u + v : \n1 \n2 \nk µ ˜ T µ 0 ( p, q ) = k 0 ˜ T 00 ( p, q ) + k a ˜ T a 0 ( p, q ) = 2 [ q 0 ( p 2 0 -⃗ p 2 -m 2 ) + p 0 ( q 2 0 -⃗q 2 -m 2 )] = 0 (A.II.2) k µ ˜ T µa ( p, q ) = k 0 ˜ T 0 a ( p, q ) + k b ˜ T ba ( p, q ) = 1 [ q a ( p 2 0 -⃗ p 2 -m 2 ) + p a ( q 2 0 -⃗q 2 -m 2 )] = 0 . (A.II.3) \nFrom these we find \nk a k b T ab ( p, q ) = -k 0 k b T 0 b ( p, q ) = ( k 0 ) 2 T 00 ( p, q ) \n˜ ˜ ˜ (A.II.4) k a ˜ T ab ( p, q ) = -k 0 ˜ T 0 b ( p, q ) (A.II.5) k a ˜ T 0 a ( p, q ) = -k 0 ˜ T 00 ( p, q ) (A.II.6) \nwhich leads to \nA = iκ k 2 [ 1 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T b b ( -u, -v ) + ( k 0 ) 4 4 ⃗ k 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) -( k 0 ) 2 ⃗ k 2 { ˜ T a 0 ( p, q ) ˜ T a 0 ( -u, -v ) -1 4 ˜ T 00 ( p, q ) ˜ T c c ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) } ] . (A.II.7) \nCombining this with the expression for the second term in the above Feynman diagram for Coulomb-like scattering, which we call B and which reads \nB = -iκ ⃗ k 2 [ -1 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) + ˜ T 0 a ( p, q ) ˜ T 0 b ( -u, -v ) ( δ ab -k a k b 4 ⃗ k 2 ) -1 4 ˜ T 00 ( p, q ) ˜ T a a ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] = -iκ k 2 [ -k 2 4 ⃗ k 2 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) + k 2 ⃗ k 2 ˜ T 0 a ( p, q ) ˜ T a 0 ( -u, -v ) -k 2 ( k 0 ) 2 4 ⃗ k 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) -k 2 4 ⃗ k 2 ˜ T 00 ( p, q ) ˜ T a a ( -u, -v ) -k 2 4 ⃗ k 2 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] , (A.II.8) \none can obtain \nA + B = iκ k 2 [( ( k 0 ) 4 + k 2 ⃗ k 2 + k 2 ( k 0 ) 2 4 ⃗ k 4 ) ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) -( ( k 0 ) 2 + k 2 ⃗ k 2 ) ˜ T 0 a ( p, q ) ˜ T a 0 ( -u, -v ) + 1 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T b b ( -u, -v ) + ( ( k 0 ) 2 + k 2 4 ⃗ k 2 ) ˜ T 00 ( p, q ) ˜ T c c ( -u, -v ) + ( ( k 0 ) 2 + k 2 4 ⃗ k 2 ) ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] = iκ k 2 [ 1 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) -˜ T 0 a ( p, q ) ˜ T a 0 ( -u, -v ) + 1 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) -1 4 ˜ T a a ( p, q ) ˜ T b b ( -u, -v ) + 1 4 ˜ T 00 ( p, q ) ˜ T c c ( -u, -v ) + 1 4 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] . (A.II.9) \nOn the other hand, we obtain with the covariant Feynman rules for the right side of the Coulomblike scattering diagram above 14 , which we name C : \nC \n= 1 2 iκ k 2 ˜ T µν ( p, q ) ˜ T ρσ ( -u, -v ) ( η µρ η νσ -1 2 η µν η ρσ ) = iκ 2 k 2 [ ˜ T µν ( p, q ) ˜ T µν ( -u, -v ) -1 2 ˜ T µ µ ( p, q ) ˜ T ρ ρ ( -u, -v ) ] = iκ 4 k 2 [ 2 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) + 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) + 4 ˜ T 0 a ( p, q ) ˜ T 0 a ( -u, -v ) -˜ T 0 0 ( p, q ) ˜ T 0 0 ( -u, -v ) -˜ T 0 0 ( p, q ) ˜ T c c ( -u, -v ) -˜ T a a ( p, q ) ˜ T 0 0 ( -u, -v ) -˜ T a a ( p, q ) ˜ T b b ( -u, -v ) ] = iκ k 2 [ 1 4 ˜ T 00 ( p, q ) ˜ T 00 ( -u, -v ) ˜ T 0 a ( p, q ) ˜ T 0 a ( -u, -v ) + 1 2 ˜ T ab ( p, q ) ˜ T ab ( -u, -v ) --1 4 ˜ T a a ( p, q ) ˜ T b b ( -u, -v ) + 1 4 ˜ T 00 ( p, q ) ˜ T c c ( -u, -v ) + 1 4 ˜ T a a ( p, q ) ˜ T 00 ( -u, -v ) ] . (A.II.10) \nNote that we use the mostly plus signature of the metric, hence pulling a temporal index results in a sign change. By comparing (A.II.9) and (A.II.10) we can see that they are identical, therefore we indeed have that the non-covariant and the covariant Feynman rules produce the same result for Coulomb-like scattering. \nWhen considering a loop diagram as discussed at the end of section IV C, then the momentum inside the loop is not on-shell. Due to this, in a similar calculation as the one shown in this appendix, there will remain correction terms to the relations in (A.II.4) - (A.II.6) that will prevent one from directly seeing the equivalence. However, as we discuss in the main text, as both sets can be derived from the same underlying Lagrangian, we expect that they yield the same physics.', 'Appendix A.III DETAILED COMPUTATION OF THE UV-RENORMALISATION': 'In this appendix we present the evaluation of the vacuum self-energy diagram from equation (4.27), \nΠ vac ( u 2 ) = κ 2 µ ϵ ∫ d d k u 2 k 2 +2 m 2 uk -2 m 4 ( 1 + ϵ 4 ) [( k + u ) 2 + λ 2 -iϵ ][ k 2 + m 2 -iϵ ] , (A.III.1) \nwhich encodes the self-energy in vacuum of the scalar particle of mass m caused by an internal triad particle of small auxiliary mass λ . For the evaluation, we follow the strategy in [53] for the regularisation of the QED self-energy at one-loop level.', 'A.III.1 Computation of the Loop integral': 'In a first step we use the identity \n1 x -iϵ = i ∫ ∞ 0 dz e -iz ( x -iϵ ) (A.III.2) \nin order to rewrite equation (A.III.1) as \nΠ vac ( u 2 ) = -κ 2 µ ϵ ∫ d d k [ u 2 k 2 +2 m 2 uk -2 m 4 ( 1 + ϵ 4 )] · ∫ ∞ 0 dz 1 ∫ ∞ 0 dz 2 e -iz 1 ( k 2 + m 2 -iϵ ) e -iz 2 (( k + u ) 2 + λ 2 -iϵ ) = -κ 2 µ ϵ ∫ ∞ 0 dz 1 ∫ ∞ 0 dz 2 ∫ d d k [ u 2 k 2 +2 m 2 uk -2 m 4 ( 1 + ϵ 4 )] · e -i ( z 1 + z 2 ) ( k + z 2 u z 1 + z 2 ) 2 -[ ϵ ( z 1 + z 2 )+ iλ 2 z 2 + im 2 z 1 + iu 2 z 2 -i z 2 2 u 2 z 1 + z 2 ] . (A.III.3) \nSubstituting k → k -z 2 u z 1 + z 2 then yields \nΠ vac ( u 2 ) = -κ 2 µ ϵ ∫ ∞ 0 dz 1 ∫ ∞ 0 dz 2 ∫ d d k [ u 2 k 2 -2 u 2 z 2 uk z 1 + z 2 + z 2 2 u 4 ( z 1 + z 2 ) 2 +2 m 2 uk -2 m 2 z 2 u 2 z 1 + z 2 -2 m 4 ( 1 + ϵ 4 )] · e -i ( z 1 + z 2 ) k 2 -[ ϵ ( z 1 + z 2 )+ i ( λ 2 + u 2 ) z 2 + im 2 z 1 -i z 2 2 u 2 z 1 + z 2 ] . (A.III.4) \n= \n- \n= \n- \nϵ \nµ \nϵ \nµ \ndz \nlim \ndz \nlim \nξ \n→ \n0 \nξ \n→ \n0 \n- \nβ \ndβ β e \nDue to symmetry, all terms linear in k vanish and only terms of two different kinds remain for which the k -integrations can be performed directly: \n∫ d d k e -i ( z 1 + z 2 ) k 2 = √ -πi z 1 + z 2 d (A.III.5) ∫ d d k k 2 e -i ( z 1 + z 2 ) k 2 = i d d ( z 1 + z 2 ) ∫ d d k e -i ( z 1 + z 2 ) k 2 = d 2 π √ -πi z 1 + z 2 d +2 . (A.III.6) \nEmploying these, one can rewrite the self-energy as: \nΠ vac ( u 2 ) = -κ 2 µ ϵ ∫ ∞ 0 dz 1 ∫ ∞ 0 dz 2 e -[ ϵ ( z 1 + z 2 )+ i ( λ 2 + u 2 ) z 2 + im 2 z 1 -i z 2 2 u 2 z 1 + z 2 ] · { √ -πi z 1 + z 2 d [ z 2 2 u 4 ( z 1 + z 2 ) 2 -2 m 2 z 2 u 2 z 1 + z 2 -2 m 4 ( 1 + ϵ 4 ) ] + u 2 d 2 π √ -πi z 1 + z 2 d +2 } . (A.III.7) \nTo continue, we use 15 \n1 = lim ξ → 0 ∫ ∞ ξ dβ β δ ( 1 -z 1 + z 2 β ) . (A.III.8) \nBy substituting z 1 → z 1 β and z 2 → z 2 β we then obtain \nΠ vac ( u 2 ) = -κ 2 µ ϵ ∫ ∞ 0 dz 1 ∫ ∞ 0 dz 2 lim ξ → 0 ∫ ∞ ξ dβ δ (1 -( z 1 + z 2 )) β e -β [ ϵ ( z 1 + z 2 )+ i ( λ 2 + u 2 ) z 2 + im 2 z 1 -i z 2 2 u 2 z 1 + z 2 ] · {√ -πi β ( z 1 + z 2 ) d [ z 2 2 u 4 ( z 1 + z 2 ) 2 -2 m 2 z 2 u 2 z 1 + z 2 -2 m 4 ( 1 + ϵ 4 ) ] } \nκ \n2 \nκ \n2 \n∫ \n0 \n∫ \n0 \n1 \n1 \n∫ \nξ \n∫ \nξ \n∞ \n[ \nϵ \n+ \ni \n( \nλ \n2 \n+ \nu \n2 \n) \nz \n+ \nim \n2 \n(1 \n- \nz \n) \n- \niz \n2 \nu \n2 \n] \n· √ -πi β d [ z 2 u 4 -2 m 2 zu 2 -2 m 4 ( 1 + ϵ 4 )] + u 2 d 2 π √ -πi β d +2 \n∞ \n[ \ndβ e \n- \niβ \n- \niϵ \n+( \nλ \n2 \n+ \nu \n2 \n) \nz \n+ \nm \n2 \n(1 \n- \nz \n) \n- \nz \n2 \nu \n2 \n] \n· { -π 2 β ϵ 2 β √ -iπ -ϵ [ z 2 u 4 -2 m 2 zu 2 -2 m 4 ( 1 + ϵ 4 )] \n+ iπ 2 4 -ϵ 2 u 2 √ -iπ -ϵ β ϵ 2 β 2 } , (A.III.9) \n̸ \n+ u 2 d 2 π √ -πi β ( z 1 + z 2 ) d +2 \nwhere in the last step we used d = 4 -ϵ . Next, we perform the β -integration, that is \nlim ξ → 0 ∫ ∞ ξ dβ e -iβa β ϵ 2 β n , (A.III.10) \nwhere we already set ϵ = 0 in the exponential, with a = ( λ 2 + u 2 ) z + m 2 (1 -z ) -z 2 u 2 and n ∈ { 1 , 2 } . To obtain the result, we apply the residue theorem. For a > 0 the contour can be closed by a quarter circle from ∞ to -i ∞ + ξ and a line from -i ∞ + ξ to ξ . With this, the pole at β = 0 can be avoided. The integral then becomes \nlim ξ → 0 ∫ ∞ ξ dβ e -iβa β ϵ 2 β n = -lim ξ → 0 ∫ ∞ 0 dt e -t ( t + iξa ) ϵ 2 -n ( -i ) ϵ 2 -n +1 a -ϵ 2 + n -1 , (A.III.11) \nwhere we substituted t = βa . Expanding the term ( t + iξa ) ϵ 2 -n for small ξ yields \n( t + iξa ) ϵ 2 -n = t ϵ 2 -n + iξa ( ϵ 2 -n ) t ϵ 2 -n -1 + O ( ξ 2 ) . (A.III.12) \nWith the definition of the Gamma function \nΓ( z ) = ∫ ∞ 0 dt t z -1 e -t (A.III.13) \nit then follows that \nlim ξ → 0 ∫ ∞ ξ dβ e -iβa β ϵ 2 β n = ( ia ) n -1 -ϵ 2 [ Γ ( ϵ 2 -n +1 ) + lim ξ → 0 O ( ξ ) ] = ( ia ) n -1 -ϵ 2 Γ ( ϵ 2 -n +1 ) . (A.III.14) \nFor a < 0, the contour is closed by a quarter circle from ∞ to i ∞ + ξ and a line from i ∞ + ξ to ξ . The result turns out to be the same as for the case a > 0. This thus yields \nΠ vac ( u 2 ) = -κ 2 µ ϵ ∫ 1 0 dz { -π 2 √ -iπ -ϵ [ z 2 u 4 -2 m 2 zu 2 -2 m 4 ( 1 + ϵ 4 )] i -ϵ 2 · [( λ 2 + u 2 ) z + m 2 (1 -z ) -z 2 u 2 ] -ϵ 2 Γ ( ϵ 2 ) + iπ 2 ( 2 -ϵ 2 ) u 2 √ -iπ -ϵ i 1 -ϵ 2 [( λ 2 + u 2 ) z + m 2 (1 -z ) -z 2 u 2 ] 1 -ϵ 2 Γ ( ϵ 2 -1 ) } . (A.III.15) \nNext, we expand all terms in ϵ and then perform the z -integration which results in \nΠ vac ( u 2 ) = -π 2 κ ϵ [ 2 m 2 ( m 2 + u 2 ) + λ 2 u 2 ] + π 2 κ 2 u 2 { u 2 [2 m 2 ( m 2 + u 2 )( γ E -2) + λ 2 ( u 2 ( γ E -1) -m 2 )] + m 2 ( m 2 +2 u 2 + λ 2 ) √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 · [ arctanh ( m 2 + u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 ) -arctanh ( m 2 -u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 )] +ln( π )[ u 4 λ 2 +2 m 2 u 2 ( m 2 + u 2 )] + ln ( λ m ) [ m 6 -3 m 2 u 2 λ 2 -m 2 λ 4 ] +ln( λ )[3 u 2 m 4 +2 u 4 ( m 2 + λ 2 )] + ln( m )[ u 2 m 2 (2 u 2 + m 2 )] -ln( µ 2 )[2 u 2 m 2 ( m 2 + u 2 ) + λ 2 u 4 ] } + O ( ϵ ) (A.III.16) \nGiven this, it becomes evident that the pole in ϵ arises from the term -π 2 κ ϵ [ 2 m 2 ( m 2 + u 2 ) + λ 2 u 2 ] , which yields in the limit of vanishing graviton mass -π 2 κ ϵ [ 2 m 2 ( m 2 + u 2 ) ] . A suitable counter term should remove this divergence, which is discussed in the main text, and we are left with the regularised version \nΠ reg vac ( u 2 ) = π 2 κ 2 u 2 { u 2 [2 m 2 ( m 2 + u 2 )( γ E -2) + λ 2 ( u 2 ( γ E -1) -m 2 )] + m 2 ( m 2 +2 u 2 + λ 2 ) √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 · [ arctanh ( m 2 + u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 ) -arctanh ( m 2 -u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 )] +ln( π )[ u 4 λ 2 +2 m 2 u 2 ( m 2 + u 2 )] + ln ( λ m ) [ m 6 -3 m 2 u 2 λ 2 -m 2 λ 4 ] +ln( λ )[3 u 2 m 4 +2 u 4 ( m 2 + λ 2 )] + ln( m )[ u 2 m 2 (2 u 2 + m 2 )] -ln( µ 2 )[2 u 2 m 2 ( m 2 + u 2 ) + λ 2 u 4 ] } . (A.III.17) \nIn order to determine the (arbitrary) finite part of the counter term, we pick the on-shell renormalisation condition. This requires the pole in the propagator to be at u 2 = -m 2 , that is it imposes the condition Π( u 2 = -m 2 ) ! = 0, and the fixing of its residue such that ∂ ∂u 2 Π( u 2 = -m 2 ) ! = 0. Together, these two conditions imply the following formula for the renormalised Π R vac : \nΠ R vac ( u 2 ) := Π reg vac ( u 2 ) -Π reg vac ( -m 2 ) -( u 2 + m 2 ) ∂ ∂u 2 Π reg vac ( -m 2 ) . (A.III.18) \nApplied to (A.III.16), one obtains for the additional terms: \nΠ reg vac ( -m 2 ) = -π 2 κ 2 λ { ( m 2 -λ 2 ) √ λ 2 -4 m 2 [ arctanh ( 2 m 2 -λ 2 λ √ λ 2 -4 m 2 ) +arctanh ( λ √ λ 2 -4 m 2 )] + λ 3 ln ( m λ ) + m 2 λ [ γ E +ln ( πλ 5 µ 2 m 3 )] } . (A.III.19) \nas well as \n∂ ∂u 2 Π reg vac ( -m 2 ) = π 2 κ 2 m 2 { 5 m 4 λ +2 m 2 λ 3 -λ 5 √ λ 2 -4 m 2 [ arctanh ( λ 2 -2 m 2 λ √ λ 2 -4 m 2 ) -arctanh ( λ √ λ 2 -4 m 2 )] +(2 γ E -3) m 4 + m 2 λ 2 ( γ E -2 + ln( π )) + ( m 4 -λ 4 ) ln ( m λ ) +2 m 2 λ 2 ln( λ ) + 2 m 4 ln( mπλ ) -m 2 (2 m 2 + λ 2 ) ln ( µ 2 ) } . (A.III.20) \nIf we had not introduced the small triad mass λ , then this last expression would be divergent, see for instance [52, 53]. Hence we continue to work now with Π R vac ( u 2 ). This concludes the discussion of the renormalisation of the self-energy diagram.', 'A.III.2 Contribution to the master equation': 'In order to see the effect of the renormalisation on the master equation, one has to evaluate the following expression, as discussed at the end of section IV B: \nΞ R ( ω u , ⃗u, t 0 , t ) = ∫ t t 0 dτ ∫ R du 0 Π R vac ( u 2 ) cos[( u 0 -ω u )( t -τ )] . (A.III.21) \nSubstituting t -τ → τ yields \nΞ R ( ω u , ⃗u, t 0 , t ) = ∫ t -t 0 0 dτ ∫ R du 0 Π R vac ( u 2 ) cos[( u 0 -ω u ) τ ] (A.III.22) \nand due to symmetry it holds that \n∫ R du 0 Π R vac ( u 2 ) cos[( u 0 -ω u ) τ ] = ∫ R du 0 Π R vac ( ⃗u 2 -u 2 0 ) [cos( u 0 τ ) cos( ω u τ ) + sin( u 0 τ ) sin( ω u τ )] = cos( ω u τ ) ∫ R du 0 Π R vac ( ⃗u 2 -u 2 0 ) cos( u 0 τ ) . (A.III.23) \nTo solve the integrations, we first consider all terms that depend on u 0 in the form ( u 2 + m 2 ) = ( ω 2 u -u 2 0 ) with ω u = √ ⃗u 2 + m 2 . \nTo evaluate this, we would like to use the distributional integration ∫ R du 0 cos( u 0 τ ) = πδ ( τ ). However, in order for this to be true, we would need to have a Schwartz function paired with the distribution under the τ integration and its integration domain should be R . To have this, we modify the cosine slightly and we will see when evaluating the integration that this modification does not affect the final value. We introduce the Schwartz function S c ( ω u τ ) which coincides on the interval [ ϵ, t -t 0 -ϵ ] with cos( ω u τ ), where ϵ ≪ 1. On the interval [ -ϵ, ϵ ] it is a smooth function constructed in such a way that S c (0) = 1 2 cos(0) = 1 2 , d 2 dτ 2 S c ( ω u τ ) | τ =0 = 1 2 d 2 dτ 2 cos( ω u τ ) | τ =0 = -ω 2 u 2 \nand S c ( ω u τ ) = 0 for τ < ϵ . For [ t -t 0 -ϵ, t -t 0 + ϵ ] it is also constructed as a smooth function from S c ( ω u ( t -t 0 -ϵ )) = cos( t -t 0 -ϵ ) to 0 for S c ( ω u τ ) for τ > t -t 0 + ϵ . The construction is in such a way, that S c ( ω u τ ) is a Schwartz function on R . Then we have: \n∫ t -t 0 0 dτ cos( ω u τ ) ∫ R du 0 ( ω 2 u -u 2 0 ) cos( u 0 τ ) = ∫ R dτ S c ( ω u τ ) [ πδ ( τ ) ω 2 u + d 2 dτ 2 ∫ R du 0 cos( u 0 τ ) ] = ∫ R dτ S c ( ω u τ ) [ πδ ( τ ) ω 2 u + π d 2 dτ 2 δ ( τ ) ] = ∫ R dτ [ πδ ( τ ) ω 2 u S c ( ω u τ ) + πδ ( τ ) d 2 dτ 2 S c ( ω u τ ) ] = π [ ω 2 u 2 -ω 2 u 2 ] = 0 . (A.III.24) \nUsing this 16 , what remains is \nπ 2 κ 2 ∫ t -t 0 0 dτ cos( ω u τ ) ∫ R du 0 cos( u 0 τ ) { ( m 4 -λ 4 ) m 2 + u 2 u 2 ln ( λ m ) + m 2 m 2 +2 u 2 + λ 2 u 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 · · [ arctanh ( m 2 + u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 ) -arctanh ( m 2 -u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 )] + λ ( m 2 -λ 2 ) √ λ 2 -4 m 2 [ arctanh ( 2 m 2 -λ 2 λ √ λ 2 -4 m 2 ) +arctanh ( λ √ λ 2 -4 m 2 ) ]} , (A.III.25) \nwhich is independent of µ . Before explicitly evaluating the integrations, we simplify the integrands by taking the limit λ → 0 where possible, as λ was only introduced as artificial small graviton mass to be able to fix the residuum of the pole in the propagator. As arctanh( i 0) and arctanh( i ∞ ) are finite, the last two lines vanish. Also, lim λ → 0 λ 4 ln( λ ) = 0 and, when expanding the square root in the second line, we find that lim λ → 0 λ arctanh(1 + λ ) = 0 as arctanh( x ) = 1 2 ln(1 + x ) -1 2 ln(1 -x ). \nThis leaves us with \nπ 2 κ 2 ∫ t -t 0 0 dτ cos( ω u τ ) ∫ R du 0 cos( u 0 τ ) { m 4 m 2 + u 2 u 2 ln ( λ m ) + m 2 m 2 +2 u 2 u 2 ( m 2 + u 2 ) · · [ arctanh ( m 2 + u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 ) -arctanh ( m 2 -u 2 m 2 + u 2 ) ]} . (A.III.26) \nFrom the above named relation arctanh( x ) = 1 2 ln(1 + x ) -1 2 ln(1 -x ) follows that \narctanh ( x y ) = 1 2 ln ( x + y y -x ) . (A.III.27) \nThis yields for the last line: \narctanh ( m 2 -u 2 m 2 + u 2 ) = 1 2 ln ( m 2 u 2 ) . (A.III.28) \nFor the line before, we expand the argument in second order for small λ and obtain \narctanh ( m 2 + u 2 -λ 2 √ ( m 2 + u 2 ) 2 -2 λ 2 ( m 2 -u 2 ) + λ 4 ) ≈ arctanh ( 1 -2 λ 2 u 2 ( m 2 + u 2 ) 2 ) . (A.III.29) \nHigher orders will not contribute in the final limit λ → 0. Expressing the arctanh again in terms of logarithms, we obtain \narctanh ( 1 -2 λ 2 u 2 ( m 2 + u 2 ) 2 ) = 1 2 ln ( 2 -2 λ 2 u 2 ( m 2 + u 2 ) 2 ) -1 2 ln ( 2 λ 2 u 2 ( m 2 + u 2 ) 2 ) ≈ 1 2 ln (2) -1 2 ln ( 2 λ 2 u 2 ( m 2 + u 2 ) 2 ) , (A.III.30) \nwhere we neglected terms of order λ 2 and higher. With these simplifications one then finds that \nΞ R ( ω u , ⃗u, t 0 , t ) = π 2 κ 2 ∫ t -t 0 0 dτ cos( ω u τ ) ∫ R du 0 cos( u 0 τ ) { m 4 m 2 + u 2 u 2 ln ( λ m ) -m 2 2 m 2 +2 u 2 u 2 ( m 2 + u 2 ) · · [ ln ( 2 λ 2 u 2 ( m 2 + u 2 ) 2 ) +ln ( m 2 2 u 2 ) ]} = π 2 κ 2 ∫ t -t 0 0 dτ cos( ω u τ ) ∫ R du 0 cos( u 0 τ ) { m 4 m 2 + u 2 u 2 ln ( λ m ) -m 2 2 m 2 +2 u 2 u 2 ( m 2 + u 2 ) · · ln ( λ 2 m 2 ( m 2 + u 2 ) 2 ) } . (A.III.31) \nThe next step is to solve the following two integrations: \n(A) m 4 ln ( λ m )∫ R du 0 cos( u 0 τ ) ω 2 u -u 2 0 ⃗u 2 -u 2 0 (A.III.32) \n(B) -m 2 2 ∫ R du 0 cos( u 0 τ ) m 2 +2 ⃗u 2 -2 u 2 0 ⃗u 2 -u 2 0 ( ω 2 u -u 2 0 ) ln ( λ 2 m 2 ( ω 2 u -u 2 0 ) 2 ) . (A.III.33) \nWe start with (A) : \n∫ R du 0 cos( u 0 τ ) ω 2 u -u 2 0 ⃗u 2 -u 2 0 = -∫ R du 0 e iu 0 τ ω 2 u -u 2 0 ( u 0 -| ⃗u | )( u 0 + | ⃗u | ) . (A.III.34) \nApplying the residue theorem by closing the contour with a semi-circle in the upper half plane leaves us with the contributions of the poles that lie on the contour, hence contribute + 1 2 their residua, yielding \n-∫ R du 0 e iu 0 τ ω 2 u -u 2 0 ( u 0 -| ⃗u | )( u 0 + | ⃗u | ) = -πi [ e i | ⃗u | τ m 2 2 | ⃗u | -e -i | ⃗u | τ m 2 2 | ⃗u | ] = πm 2 | ⃗u | sin( | ⃗u | τ ) . (A.III.35) \nTherefor we have \n(A) = πm 6 | ⃗u | sin( | ⃗u | τ ) ln ( λ m ) . (A.III.36) \nWe proceed with (B) : \n∫ R du 0 cos( u 0 τ ) m 2 +2 ⃗u 2 -2 u 2 0 ⃗u 2 -u 2 0 ( ω 2 u -u 2 0 ) ln ( λ 2 m 2 ( ω 2 u -u 2 0 ) 2 ) = -∫ R du 0 e iu 0 τ m 2 +2 ⃗u 2 -2 u 2 0 ( u 0 -| ⃗u | )( u 0 + | ⃗u | ) ( ω 2 u -u 2 0 ) ln ( λ 2 m 2 ( ω 2 u -u 2 0 ) 2 ) . (A.III.37) \nWe apply the residue theorem once again. The integrand has singularities at u 0 = ±| ⃗u | . Working with the principal value logarithm, i.e. the complex logarithm with branch cut at the negative real axis, the logarithm here does not have any branch cuts given that ω u > 0, hence the equation λ 2 m 2 ( ω 2 u -u 2 0 ) 2 ! = -α with α ∈ R , α > 0 does not have any solution for u 0 ∈ C . Due to the prefactor ( ω 2 u -u 2 0 ), there is no singularity in u 0 = ± ω u . We pick a closed integration contour from -∞ to -| ⃗u | -ϵ , then go in a semi-circle clockwise around the pole to -| ⃗u | + ϵ , continue to | ⃗u | -ϵ , again go around the singularity in a semi-circle clockwise to | ⃗u | + ϵ , continue to + ∞ and close it with a semicircle in the upper half-plane. The closed contour does not contain any singularities, hence its contribution vanishes. Due to the exponential e iu 0 τ , also the semi-circle at infinite radius in the upper half-plane vanishes. Hence only the contributions of the singularities at u 0 = ±| ⃗u | remain. As we went for the closed contour around them clockwise, the singularity contributions have to be evaluated counter-clockwise and added to the closed contour. We start by investigating the one at \nu 0 = + | ⃗u | and replace u 0 = | ⃗u | + ϵe iϕ : \n-lim ϵ → 0 ∫ π 0 dϕ iϵe iϕ e iτϵe iϕ + iτ | ⃗u | m 2 -4 | ⃗u | ϵe iϕ -2 ϵ 2 e 2 iϕ ϵe iϕ (2 | ⃗u | + ϵe iϕ ) ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) · ln ( λ 2 m 2 ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = -i lim ϵ → 0 ∫ π 0 dϕ e iτ | ⃗u | m 2 -4 | ⃗u | ϵe iϕ -2 ϵ 2 e 2 iϕ 2 | ⃗u | + ϵe iϕ ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) ln ( λ 2 m 2 ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = -i lim ϵ → 0 ∫ π 0 dϕ e iτ | ⃗u | m 2 1 2 | ⃗u | m 2 ln ( λ 2 m 2 ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = -i m 4 2 | ⃗u | e iτ | ⃗u | lim ϵ → 0 ∫ π 0 dϕ ln ( λ 2 m 2 ) = -i πm 4 | ⃗u | e iτ | ⃗u | ln ( λ m ) , (A.III.38) \nwhere in the first step we expanded e iτϵe iϕ + iτ | ⃗u | = e iτ | ⃗u | (1 + O ( ϵ )) and neglected all but the zeroth order due to the limit, in the second step we expanded 1 2 | ⃗u | + ϵe iϕ = 1 2 | ⃗u | ( 1 -ϵ 2 | ⃗u | e iϕ + O ( ϵ ) ) and applied the limit to the terms depending on ϵ where possible. In the third step, we expanded \nln ( λ 2 m 2 ( m 2 -2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = ln ( λ 2 m 2 ) + ϵ 4 | ⃗u | m 2 e iϕ + O ( ϵ 2 ) . (A.III.39) \nFor the other singularity at u 0 = -| ⃗u | we find analogously for u 0 = -| ⃗u | + ϵe iϕ : \nlim ϵ → 0 ∫ π dϕ iϵe iϕ e iτϵe iϕ -iτ | ⃗u | m 2 +4 | ⃗u | ϵe iϕ -2 ϵ 2 e 2 iϕ ϵe iϕ ( -2 | ⃗u | + ϵe iϕ ) ( m 2 +2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ \n-0 ) · ln ( λ 2 m 2 ( m 2 +2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = + i lim ϵ → 0 ∫ π 0 dϕ e -iτ | ⃗u | m 2 1 2 | ⃗u | m 2 ln ( λ 2 m 2 ( m 2 +2 | ⃗u | ϵe iϕ -ϵ 2 e 2 iϕ ) 2 ) = i m 4 2 | ⃗u | e -iτ | ⃗u | lim ϵ → 0 ∫ π 0 dϕ ln ( λ 2 m 2 ) = i πm 4 | ⃗u | e -iτ | ⃗u | ln ( λ m ) . (A.III.40) \nCombining these two yields \n(B) = -πm 6 | ⃗u | sin( | ⃗u | τ ) ln ( λ m ) = -(A) . (A.III.41) \nFrom this follows that we have \nΞ R ( ω u , ⃗u, t 0 , t ) = 0 . (A.III.42) \nDue to the way the renormalised quantities entered in the master equation, the result is now independent of the scale µ as well as of the artificial graviton mass λ , whose limit to zero can therefore be taken without problems.', 'Appendix A.IV APPLICATION OF THE MARKOV APPROXIMATION': 'In this appendix, we apply the Markov approximation to the renormalised one-particle master equation (4.47). Applying the formula given in (5.2), we obtain two classes of terms, one class that contains the δ -distributions and the other one containing the Cauchy principal value. Before we evaluate them in subsections A.IV.2 and A.IV.3, we first discuss the applicability of the Markov approximation for ultra-relativistic particles with focus on neutrinos in subsection A.IV.1.', 'A.IV.1 Applicability of the Markov approximation for ultra-relativistic particles': 'In general, the Markov approximation can be applied if the timescales τ B on which the correlation functions decay are much smaller than the timescales τ R on which the state of the system varies (see [2]). The identification of these timescales is however hard without solving the one-particle master equation before the application of the approximation. As the Markov approximation corresponds to sending t 0 → -∞ and hence ∫ t -t 0 0 dτ -→ ∫ ∞ 0 dτ , we will analyse the error one makes when extending the integration domain from t -t 0 to ∞ . If the integrand is strongly peaked around τ = 0, which is usually assumed when deriving Markovian master equations, then the error of the additional contribution should be negligible. For this, we analyse the different parts of the renormalised one-particle master equation in (4.47): \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = -iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) -κ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u [ C R ( ⃗u, ⃗ k, t ) + δ P C R P ( ⃗u, ⃗ k, t ) ] + P v ( ⃗ k ) ω v -k ω v [( C R ( ⃗v, ⃗ k, t ) ) ∗ + δ P ( C R P ( ⃗v, ⃗ k, t ) ) ∗ ] } ρ ( ⃗u, ⃗v, t ) + κ 2 ∫ d 3 k (2 π ) 3 P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω u ω v + k ω v { C R ( ⃗u + ⃗ k, ⃗ k, t ) + ( C R ( ⃗v + ⃗ k, ⃗ k, t ) ) ∗ } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) (A.IV.1) \nwith \nC R ( ⃗u, ⃗ k, t ) = 2 ∫ t -t 0 0 dτ Ω k N ( k ) cos[Ω k τ ] e -i ( ω u -k -ω u ) τ (A.IV.2) \nC R P ( ⃗u, ⃗ k, t ) = 2 ∫ t -t 0 0 dτ Ω k N ( k ) cos[Ω k τ ] e -i ( ω u -k + ω u ) τ . (A.IV.3) \nWe start with the real part of C R in the second line, which will later lead to decoherence. The term we have to take into account is \n-κ 2 ∫ d 3 k (2 π ) 3 P u ( ⃗ k ) ω u -k ω u ∫ t -t 0 0 dτ Ω k N ( k ) [cos((Ω k + ω u -k -ω u ) τ ) + cos((Ω k -ω u -k + ω u ) τ )] = -κ 2(2 π ) 2 ∫ ∞ 0 dk ∫ π 0 dθ u 4 2 sin 5 ( θ ) ω u -k ω u kN ( k ) · [ sin(( k + ω u -k -ω u )( t -t 0 )) k + ω u -k -ω u + sin(( k -ω u -k + ω u )( t -t 0 )) k -ω u -k + ω u ] , (A.IV.4) \nwhere in the second line we went to spherical coordinates with ⃗ k z || ⃗u and performed the τ -integration. The main contribution from the integrand will come from small k , because for k = 0 the denominator tends to zero and N ( k ) decreases rapidly for large k . For the Markov approximation however the behaviour depending on ( t -t 0 ) is important. To extract this, we first substitute \nµ := ω u -k = √ ω 2 u + k 2 -2 uk cos( θ ) (A.IV.5) \nand assume, already adapting the scenario of section VI D where we will apply the one-particle master equation to ultra-relativistic particles, that ω u ≈ u , which then yields \nµ (0) = | k -u | µ ( π ) = k + u dθ = dµ µ uk sin( θ ) (A.IV.6) \nas well as \nsin 2 ( θ ) = 1 -( u 2 + k 2 -µ 2 ) 2 4 u 2 k 2 . (A.IV.7) \nUsing this and defining ∆ t := t -t 0 , equation (A.IV.4) reads \n-u 2 κ 4(2 π ) 2 ∫ ∞ 0 dk N ( k ) ∫ k + u | k -u | dµ [ 1 -( u 2 + k 2 -µ 2 ) 2 4 u 2 k 2 ] 2 [ sin(( k + µ -ω u )∆ t ) k + µ -ω u + sin(( k -µ + ω u )∆ t ) k -µ + ω u ] . (A.IV.8) \nNow we make some assumptions on the involved quantities motivated by the application to ultrarelativistic neutrinos to simplify the integration. In order to continue, we assume that \nu ≫ 1 cβ . (A.IV.9) \nTo resolve the absolute value in the µ -integration, we split the k -integration into two regions, one with k < u and another one with k > u : \n∫ u 0 dk N ( k ) ∫ u + k u -k + ∫ ∞ u dk N ( k ) ∫ u + k k -u . (A.IV.10) \nThe second integral is now negligible compared to the first one, as on the one hand side the dominant contribution of the integrand is around k = 0 where the root of the denominator lies, and on the other hand due to condition (A.IV.9) for k > u ≫ 1 cβ also N ( k ) will strongly damp the integrand in that region. Hence we only continue with the first integral: \n-u 2 κ 4(2 π ) 2 ∫ u 0 dk N ( k ) ∫ k + u u -k dµ [ 1 -( u 2 + k 2 -µ 2 ) 2 4 u 2 k 2 ] 2 [ sin(( k + µ -ω u )∆ t ) k + µ -ω u + sin(( k -µ + ω u )∆ t ) k -µ + ω u ] . (A.IV.11) \nThe µ -integration can be solved and one obtains \n-u 2 κ 4(2 π ) 2 ∫ u 0 dk N ( k ) k 4 (∆ t ) 8 u 4 [ 16 k ∆ t (45 + (∆ t ) 2 ( -6 k 2 +( -27 + 4 k 2 (∆ t ) 2 ) u 2 +(∆ t ) 2 u 4 )) +4 k ∆ t (135 + 2(∆ t ) 2 ( -3 k 2 +( -36 + k 2 (∆ t ) 2 ) u 2 +(∆ t ) 2 u 4 )) cos(2 k ∆ t ) -6(105 + 2(∆ t ) 2 ( -15 k 2 +( -30 + 7 k 2 (∆ t ) 2 ) u 2 +(∆ t ) 2 u 4 )) sin[2 k ∆ t ] ] . (A.IV.12) \nFIG. 1. Decay of the integral I (∆ t ) in (A.IV.12) for different values of u and β (see main text) in percent normalised such that I (10 -11 s ) = 1. \n<!-- image --> \nThis can be integrated numerically. For 17 the three different values of u = 2 . 4 · 10 22 1 s , u = 2 . 4 · 10 24 1 s , u = 2 . 4 · 10 26 1 s and for the two values of β = 10 -11 s, β = 10 -13 s one obtains that the result vanishes rapidly for increasing times ∆ t , see figure 1. \nIt hence is visible that the error made when not only considering time until t -t 0 but until ∞ is negligible, given that its value drops rapidly when increasing t -t 0 . In section VI D typical propagation times of neutrinos through the Earth are used, which are of the order t -t 0 ≈ 10 -2 s . \nIn the general case, in which we do not specify to ultra-relativistic neutrinos, the integration boundaries for the µ -integration in (A.IV.8) would be µ (0) = √ m 2 +( k -u ) 2 as well as µ ( π ) = √ m 2 +( k + u ) 2 . As the temperature parameter Θ characterising the environment is finite, one can always find a parameter ˜ k such that ˜ k ≫ 1 cβ and hence the k -integration can be approximated ∫ ∞ 0 dk ≈ ∫ ˜ k 0 dk , since N ( k ) damps the result strongly on the remaining interval. The general integration of this quantity goes beyond the scope of this work. To facilitate its computation, one can use the specific properties of the considered model similarly to the way it is applied to ultra-relativistic neutrinos in this work. \nNext, we consider the real terms arising from the extended projection in line two of equation (A.IV.1). The same steps for the ultra-relativistic conditions as above yield \n-u 2 κ 4(2 π ) 2 ∫ u 0 dk N ( k ) ∫ k + u u -k dµ [ 1 -( u 2 + k 2 -µ 2 ) 2 4 u 2 k 2 ] 2 [ sin(( k + µ + ω u )∆ t ) k + µ + ω u + sin(( k -µ -ω u )∆ t ) k -µ -ω u ] . (A.IV.13) \nNote that in contrast to the previous case, now the denominator is never zero, hence the contribution of this term is less dominant than the one of the previous terms. Solving the µ -integration and plotting the term with the same parameters as above again shows that the result vanishes rapidly for increasing values of t -t 0 . The absolute value of the integral is however several orders of magnitude below the contribution in (A.IV.12), hence it is negligible compared to the above term. \nFor the imaginary terms in the second line of (A.IV.1), one can perform the same analysis and also obtains a strong decay in t -t 0 for the terms. The analogous arguments (if v ≫ 1 cβ ) hold for the \nterms in the third line of (A.IV.1). For the last line, also the wave function ρ depends on ⃗ k , but as the structure of the involved terms is very similar and the same main arguments can be carried over (roots of the denominator, same main quantities u , v , β and damping due to N ( k )), also the behaviour of this term depending on t -t 0 is similar as above. Hence the Markov approximation is justified here under the above named assumptions of ultra-relativistic particles that fulfil (A.IV.9).', 'A.IV.2 Evaluation of the delta terms': "The main step in the application of the Markov approximation consists in replacing \n∫ t -t 0 0 dτ e -iωτ -→ ∫ ∞ 0 dτ e -iωτ = πδ ( ω ) -PV ( i ω ) , (A.IV.14) \nas argued in detail in the main text. In this subsection, we investigate the terms containing the delta distribution and in the next subsection we evaluate the principal value terms. The delta distribution parts of the terms in lines two and three of (A.IV.1) become \n-πκ 2 ∫ d 3 k (2 π ) 3 { P u ( ⃗ k ) ω u -k ω u N ( k ) Ω k [ δ (Ω k -ω u -k + ω u ) + δ (Ω k + ω u -k -ω u ) + δ P δ (Ω k -ω u -k -ω u ) + δ P δ (Ω k + ω u -k + ω u ) ] + P v ( ⃗ k ) ω v -k ω v N ( k ) Ω k [ δ (Ω k -ω v -k + ω v ) + δ (Ω k + ω v -k -ω v ) + δ P δ (Ω k -ω v -k -ω v ) + δ P δ (Ω k + ω v -k + ω v ) ]} . (A.IV.15) \nTo evaluate the delta distributions, we first have to determine the zeroes of the arguments. As the part for v is exactly the same as the one for u , we focus on the latter one. The four equations to solve then read, expressing the ⃗ k -integration in spherical coordinates ( k, θ, ϕ ) and picking them such that ⃗u || ⃗ k z : \n± k = √ ω 2 u + k 2 -2 uk cos( θ ) -ω u (A.IV.16) \n± k = √ ω 2 u + k 2 -2 uk cos( θ ) + ω u , (A.IV.17) \nwhere u := | ⃗u | . With the limits set by the spherical coordinates, i.e. k ∈ { 0 , ∞} and cos( θ ) ∈ {-1 , 1 } it is evident that the right hand side of the equation in the second line is always positive, hence δ (Ω k + ω u -k + ω u ) = 0. For the positive sign in the second line we find \n( k -ω u ) = √ ω 2 u + k 2 -2 uk cos( θ ) ⇐⇒ kω u = uk cos( θ ) , (A.IV.18) \nwhich is solved for k = 0. There is no other solution, as the remaining equation ω u = √ m 2 + u 2 = u cos( θ ) is never fulfilled for m> 0. Note however that k = 0 is here only a solution of the squared equation and not of the original one 18 in (A.IV.17), hence we also have δ ( -Ω k + ω u -k + ω u ) = 0. \nThis means that here all additional terms arising from the extended projection vanish. For the two equations in the first line we have \n( ± k + ω u ) = √ ω 2 u + k 2 -2 uk cos( θ ) ⇐⇒ ∓ kω u = uk cos( θ ) . (A.IV.19) \nThis is solved by k = 0, which is also a solution to both non-squared equations. Apart from that, there is no other solution as again | ∓ ω u | = | ∓ √ m 2 + u 2 | > u while | u cos( θ ) | ≤ u . From this we obtain \nδ ( ± Ω k -ω u -k + ω u ) = δ ( k ) ∣ ∣ ∣ ± 1 + u cos( θ ) ω u ∣ ∣ ∣ = δ ( k ) 1 ± u cos( θ ) ω u . (A.IV.20) \nApplying this to the original expression in spherical coordinates, we find \n-πκ 2 1 (2 π ) 2 ∫ ∞ 0 dk ∫ π 0 dθ k sin( θ ) { P u ( ⃗ k ) ω u -k ω u N ( k ) [ δ ( k ) 1 + u cos( θ ) ω u + δ ( k ) 1 -u cos( θ ) ω u ] + P v ( ⃗ k ) ω v -k ω v N ( k ) [ δ ( k ) 1 + v cos( θ ) ω v + δ ( k ) 1 -v cos( θ ) ω v ]} , (A.IV.21) \nwhere in the second line we picked different spherical coordinates with ⃗v || ⃗ k z and defined v := | ⃗v | . Evaluation of the delta yields, using \nP u ( k, θ, ϕ ) = 1 2 [ u 2 -( uk cos( θ )) 2 k 2 ] 2 = u 4 2 [ 1 -cos 2 ( θ ) ] 2 = u 4 2 sin 4 ( θ ) = P u ( θ, ϕ ) (A.IV.22) \nand with l'Hospital's limit \nlim k → 0 k N ( k ) = lim k → 0 k e βk -1 = lim k → 0 1 βe βk = 1 β , (A.IV.23) \nthe following result 19 , where we get an additional factor of 1 2 as the point k = 0, where the delta distribution does not vanish, lies at the edge of the integration area: \n-πκ 8 β 1 (2 π ) 2 ∫ π 0 dθ sin 5 ( θ ) { u 4 ω 2 u [ 1 1 + u cos( θ ) ω u + 1 1 -u cos( θ ) ω u ] + v 4 ω 2 v [ 1 1 + v cos( θ ) ω v + 1 1 -v cos( θ ) ω v ]} = πκ 4 β 1 (2 π ) 2 ∫ π 0 dθ sin 5 ( θ ) { u 2 cos 2 ( θ ) -ω 2 u u 2 + v 2 cos 2 ( θ ) -ω 2 v v 2 } = πκ 4 β 1 (2 π ) 2 [ -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arccoth ( ω u u ) -2 ω v v m 4 arccoth ( ω v v ) ] . (A.IV.24) \nNext, we compute the last line of equation (A.IV.1). The terms containing the delta distributions read \nπκ 2 ∫ d 3 k (2 π ) 3 P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω u ω v + k ω v N ( k ) Ω k · { δ (Ω k + ω u -ω u + k ) + δ (Ω k -ω u + ω u + k ) + δ (Ω k + ω v -ω v + k ) + δ (Ω k -ω v + ω v + k ) } ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) . (A.IV.25) \nProceeding the same way as above it follows that \nδ ( ± Ω k + ω u -ω u + k ) = δ ( k ) ∣ ∣ ∣ 1 ∓ u cos( α uk ) ω u ∣ ∣ ∣ = ω u δ ( k ) ω u ∓ u cos( α uk ) , (A.IV.26) \nwhere α uk is the angle between ⃗u and ⃗ k . Due to the presence of the two directions ⃗u and ⃗v in the prefactor, it is not possible to pick ⃗ k z parallel to both ⃗u and ⃗v , as in general ⃗u || ⃗v does not hold. Given that P ijln ( ⃗ k ) = P ijln ( θ, ϕ ), the delta distribution can be applied and one obtains 20 \nπκ 2(2 π ) 3 β ∫ π 0 dθ ∫ 2 π 0 dϕ sin( θ ) P ijln ( θ, ϕ ) u i u j v l v n ω u ω v · { ω 2 u ω 2 u -u 2 cos 2 ( α uk ) + ω 2 v ω 2 v -v 2 cos 2 ( α vk ) } ρ ( ⃗u, ⃗v, t ) . (A.IV.27) \nNow we choose in the first term the spherical coordinates such that ⃗u || ⃗ k z and in the second one ⃗v || ⃗ k z and obtain \nπκ 2(2 π ) 3 β ∫ π 0 dθ ∫ 2 π 0 dϕ sin( θ ) P ijln ( θ, ϕ ) u i u j v l v n ω u ω v { ω 2 u ω 2 u -u 2 cos 2 ( θ ) + ω 2 v ω 2 v -v 2 cos 2 ( θ ) } ρ ( ⃗u, ⃗v, t ) . (A.IV.28) \nThe the only quantity depending on ϕ is P ijln ( θ, ϕ ). The appearing contraction is, assuming ⃗u || ⃗ k z : \nP ijln ( θ, ϕ ) u i u j v l v n = [ ⃗u · ⃗v -( ⃗u · ⃗ k )( ⃗v · ⃗ k ) ⃗ k 2 ] 2 -1 2 [ ⃗u 2 -( ⃗u · ⃗ k ) 2 ⃗ k 2 ][ ⃗v 2 -( ⃗v · ⃗ k ) 2 ⃗ k 2 ] = [ ⃗u · ⃗v -u cos( θ )( ⃗v · ˆ ⃗ k ) ] 2 -1 2 [ u 2 -u 2 cos 2 ( θ ) ] [ v 2 -( ⃗v · ˆ ⃗ k ) 2 ] = ( ⃗u · ⃗v ) 2 + u 2 cos 2 ( θ )( ⃗v · ˆ ⃗ k ) 2 -2 u cos( θ )( ⃗u · ⃗v )( ⃗v · ˆ ⃗ k ) -1 2 u 2 sin 2 ( θ ) [ v 2 -( ⃗v · ˆ ⃗ k ) 2 ] , (A.IV.29) \nwhere the unit vector ˆ ⃗ k is defined as \nˆ ⃗ k = sin( θ ) cos( ϕ ) sin( θ ) sin( ϕ ) cos( θ ) . (A.IV.30) \nFrom this follows that \n∫ 2 π 0 dϕ P ijln ( θ,ϕ ) u i u j v l v n =2 π ( ⃗u · ⃗v ) 2 + u 2 cos 2 ( θ )[2 π cos 2 ( θ ) v 2 z + π sin 2 ( θ )( v 2 x + v 2 y )] -4 πuv z cos 2 ( θ )( ⃗u · ⃗v ) -πu 2 2 sin 2 ( θ )[2 v 2 -2 v 2 3 cos 2 ( θ ) -sin 2 ( θ )( v 2 1 + v 2 2 )] = π [ 2( ⃗u · ⃗v ) 2 -u 2 2 ( v 2 + v 2 z ) ] + π cos 2 ( θ ) [ u 2 ( v 2 + v 2 z ) -4 uv z ( ⃗u · ⃗v ) ] + π cos 4 ( θ ) [ u 2 2 (3 v 2 z -v 2 ) ] = -π u 2 2 ( v 2 -3 v 2 z ) + π cos 2 ( θ ) u 2 ( v 2 -3 v 2 z ) -π cos 4 ( θ ) u 2 2 ( v 2 -3 v 2 z ) = -π 2 u 2 ( v 2 -3 v 2 z ) [ 1 -cos 2 ( θ ) ] 2 = -π 2 u 2 ( v 2 -3 v 2 z ) sin 4 ( θ ) , (A.IV.31) \nwhere we used in the second step that we chose the coordinate system such that ⃗u = u⃗e z , hence ⃗u · ⃗v = uv z . The θ -integration then has the form \n-π 2 u 2 ( v 2 -3 v 2 z ) ∫ π 0 dθ sin 5 ( θ ) ω 2 u ω 2 u -u 2 cos 2 ( θ ) = -πω u ( v 2 -3 v 2 z ) [ 5 3 ω u -ω 3 u u 2 + m 4 u 3 arctanh ( u ω u ) ] . (A.IV.32) \nDue to symmetry, we get the same result for the other term just with the replacement u ↔ v , as for the corresponding terms we can analogously choose ⃗v = v⃗e z . The contribution of the last line of equation (A.IV.1) is therefore 21 : \nκ 16 πβ { ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 5 3 -( 1 + m 2 u 2 ) + m 4 ω u u 3 arctanh ( u ω u )] + ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 5 3 -( 1 + m 2 v 2 ) + m 4 ω v v 3 arctanh ( v ω v )] } . (A.IV.33) \nCollecting all contributions from the delta terms yields \nκ 16 πβ { -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arctanh ( u ω u ) -2 ω v v m 4 arctanh ( v ω v ) -ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 2 3 -m 2 u 2 + m 4 ω u u 3 arctanh ( u ω u )] -ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 2 3 -m 2 v 2 + m 4 ω v v 3 arctanh ( v ω v )] } . (A.IV.34) \nThis is the final form of the real part of the dissipator that causes decoherence. The rotating wave approximation which is carried out as a next step leaves this part of the master equation invariant.", 'A.IV.3 Evaluation of the Cauchy principal value contributions': 'For the Markov approximation, it remains to compute the terms that contain the Cauchy principal value in (A.IV.1) after the approximation (A.IV.14). The terms in line two of (A.IV.1) read \niκ 2 ∫ d 3 k (2 π ) 3 u 4 2 sin 4 ( α uk ) 1 ω u -k ω u N ( k ) Ω k [ PV ( 1 Ω k + ω u -k -ω u ) -PV ( 1 Ω k -ω u -k + ω u ) + δ P PV ( 1 Ω k + ω u -k + ω u ) -δ P PV ( 1 Ω k -ω u -k -ω u ) ] , (A.IV.35) \nwhere α uk is the angle between ⃗u and ⃗ k . It can be seen that the term inside the principal value causes problems for ⃗ k → ⃗ 0. Thus we exclude a small region of radius ϵ around ⃗ k = ⃗ 0, perform the integration and take the limit ϵ → 0 in the end. We then obtain in spherical coordinates 22 with ⃗ k z || ⃗u : \nlim ϵ → 0 iκu 4 4(2 π ) 2 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) 1 ω u -k kN ( k ) [ 1 Ω k + ω u -k -ω u -1 Ω k -ω u -k + ω u + δ P 1 Ω k + ω u -k + ω u -δ P 1 Ω k -ω u -k -ω u ] . (A.IV.36) \nFor δ P = 1 this can be simplified to \nlim ϵ → 0 iκu 4 4(2 π ) 2 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) 1 ω u -k kN ( k ) [ 2 u cos( θ ) ω u -k kω 2 u -ku 2 cos 2 ( θ ) ] = lim ϵ → 0 iκu 5 2(2 π ) 2 ω u [ ∫ ∞ ϵ dk N ( k ) ][ ∫ π 0 dθ cos( θ ) sin 5 ( θ ) ω 2 u -u 2 cos 2 ( θ ) ] ︸ ︷︷ ︸ =0 = lim ϵ → 0 0 = 0 . (A.IV.37) \nWithout the principal value the k -integration would diverge: \n∫ ∞ 0 dk N ( k ) = ∫ ∞ 0 dk 1 e βk -1 = iπ β -lim ϵ → 0 2 β arctanh ( 1 -2 e ϵβ ) . (A.IV.38) \nWithout prior renormalisation, some terms arising due to the additional term present in the nonrenormalised coefficients C ( ⃗u, ⃗ k, t ) in (4.45) compared to the renormalised one would remain here and lead to logarithmic divergences, as expected from the discussion in section IV A. \nFor the non-extended projection, i.e. for δ P = 0, the situation is more complicated. In that case we find, again using spherical coordinates and implementing the principal value by excluding a sphere of radius ϵ around the critical point ⃗ k = ⃗ 0: \nlim ϵ → 0 iκu 4 4(2 π ) 2 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) 1 ω u -k kN ( k ) [ 1 Ω k + ω u -k -ω u -1 Ω k -ω u -k + ω u ] = -lim ϵ → 0 iκu 4 2(2 π ) 2 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω u ω u -k k 2 -( ω u -k -ω u ) 2 . (A.IV.39) \nThe θ -integration leads to a complicated result that can be simplified when considering e.g. the ultra-relativistic limit. Then it yields, where the limit ϵ → 0 can be taken also before the integration: \n-iκu 4 105(2 π ) 2 ω u { -4 ∫ u 0 dk N ( k ) ( k 3 u 4 -7 k u 2 ) + ∫ ∞ u dk N ( k ) ( 35 1 u -14 u k 2 +3 u 3 k 4 ) } = -iκu 4 105(2 π ) 2 ω u { 4 [ π 4 15 β 4 u 4 -7 π 2 6 β 2 u 2 -6 ln(1 -e -βu ) βu +4 Li 2 ( e -βu ) β 2 u 2 -6 Li 3 ( e -βu ) β 3 u 3 -6 Li 4 ( e -βu ) β 4 u 4 ] +35 -35 ln ( e βu -1 ) βu -14 u ∫ ∞ u dk N ( k ) k 2 +3 u 3 ∫ ∞ u dk N ( k ) k 4 } . (A.IV.40) \nHere, Li s ( x ) denotes the poly-logarithm function defined by \nLi s ( x ) := ∞ ∑ n =1 x n n s . (A.IV.41) \nThe remaining two integrations cannot be performed analytically, but they can be solved numerically given a specific temperature Θ and a value for u , and are finite as long as u > 0. For the terms in line three of (A.IV.1) we get the same results when replacing ⃗u → ⃗v and applying complex conjugation. For the terms in line four we obtain: \niκ 2(2 π ) 3 1 √ ω u ω v ∫ d 3 k P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω v + k N ( k ) Ω k ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) { PV ( 1 Ω k + ω u -ω u + k ) -PV ( 1 Ω k -ω u + ω u + k ) -PV ( 1 Ω k + ω v -ω v + k ) +PV ( 1 Ω k -ω v + ω v + k ) } . (A.IV.42) \nWithout further specification of ρ , this cannot be simplified further at this point. \nIn summary, after the second Markov approximation we hence have \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = ) \n-iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) + κ 16 πβ { -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arctanh ( u ω u ) -2 ω v v m 4 arctanh ( v ω v -ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 2 3 -m 2 u 2 + m 4 ω u u 3 arctanh ( u ω u )] -ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 2 3 -m 2 v 2 + m 4 ω v v 3 arctanh ( v ω v )] } ρ ( ⃗u, ⃗v, t ) + iκ 2(2 π ) 3 1 √ ω u ω v ∫ d 3 k P ijln ( ⃗ k ) u i u j v l v n √ ω u + k ω v + k N ( k ) Ω k ρ ( ⃗u + ⃗ k,⃗v + ⃗ k, t ) { PV ( 1 Ω k + ω u -ω u + k ) -PV ( 1 Ω k -ω u + ω u + k ) -PV ( 1 Ω k + ω v -ω v + k ) +PV ( 1 Ω k -ω v + ω v + k ) } -(1 -δ P ) iκ 2(2 π ) 2 lim ϵ → 0 [ u 4 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω u ω u -k k 2 -( ω u -k -ω u ) 2 -v 4 ω v ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω v ω v -k k 2 -( ω v -k -ω v ) 2 ] ρ ( ⃗u, ⃗v, t ) , (A.IV.43) \nwhere the last two lines vanish when working with the extended projection.', 'Appendix A.V APPLICATION OF THE ROTATING WAVE APPROXIMATION (RWA)': "In this appendix the detailed implementation of the rotating wave approximation for the renormalised Markovian master equation under consideration in this work is discussed. We proceed in the standard way by considering the master equation in interaction picture and removing all terms that oscillate fast (see e.g. [2, 12]. For this we start in the field theory and consider the full dissipator from (4.60) in [1]: \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ a,b =1 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 1 Ω k { [ j a r ( ⃗ k, ⃗p ) † , j b r ( ⃗ k, ⃗ l ) ρ S ( t ) ] f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. + N ( k ) [ j a r ( ⃗ k, ⃗p ) † , [ j b r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. } , \n(A.V.1) \nwhere f ( ω ; t ) := ∫ t -t 0 0 dτ e -iωτ and we assume that the Markov approximation has already been applied, thus the former functions f ( ω ; t ) are now distributions f ( ω ) = πδ ( ω ) -iPV ( 1 ω ) independent of time. The different ω a and j r a were defined starting in (A.I.8). Taking into account that the renormalisation removed the terms that are independent of N ( k ), the dissipator becomes (see \n(4.48)): \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ a,b =1 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k { [ j a r ( ⃗ k, ⃗p ) † , [ j b r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω b ( ⃗ k, ⃗ l )) + h.c. } . (A.V.2) \nThe basis for the RWA is the dissipator in interaction picture, which is obtained by substituting \nj a r ( ⃗ k, ⃗p ) -→ j a r ( ⃗ k, ⃗p ) e iω a ( ⃗ k,⃗p ) t . (A.V.3) \nThus we get as time-dependent frequencies that cause oscillations terms of the form \ne ± i [ ω a ( ⃗ k,⃗p ) -ω b ( ⃗ k, ⃗ l )] t . (A.V.4) \nNext we apply the RWA, which means that we discard all the rapidly oscillating terms and only keep those where ω a ( ⃗ k, ⃗p ) = ω b ( ⃗ k, ⃗ l ) holds. For a = 3 and a = 4 this means that only b = 3 and b = 4 survive. However, from the definition of the ω a ( ⃗ k, ⃗p ) follows that ω 1 ( ⃗ k, ⃗p ) = ω 2 ( ⃗ k, -⃗ k -⃗ p ), so also terms of the form a = 1, b = 2 and vice versa will remain. To simplify this, we introduce J A r ( ⃗ k, ⃗p ) and ω A ( ⃗ k, ⃗p ) with A ∈ { 2 , 3 , 4 } , similar as in [12], such that \nJ 2 r ( ⃗ k, ⃗p ) := 2 j 1 r ( ⃗ k, ⃗p ) ω 2 ( ⃗ k, ⃗p ) := ω 1 ( ⃗ k, ⃗p ) = ω 2 ( ⃗ k, -⃗ k -⃗ p ) = ω p -ω k + p (A.V.5) \nJ 3 r ( ⃗ k, ⃗p ) := j 3 r ( ⃗ k, ⃗p ) ω 3 ( ⃗ k, ⃗p ) := ω 3 ( ⃗ k, ⃗p ) = -ω p -ω k + p (A.V.6) \nJ 4 r ( ⃗ k, ⃗p ) := j 4 r ( ⃗ k, ⃗p ) ω 4 ( ⃗ k, ⃗p ) := ω 4 ( ⃗ k, ⃗p ) = ω p + ω k + p . (A.V.7) \nMaking use of the fact that j 1 r ( ⃗ k, ⃗p ) = j 2 r ( ⃗ k, -⃗ k -⃗ p ), we can rewrite the dissipator in terms of a sum over capital letters A,B : \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ A,B =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k { [ J A r ( ⃗ k, ⃗p ) † , [ J B r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω B ( ⃗ k, ⃗ l )) + h.c. } . (A.V.8) \nThe RWA-requirement to keep only the terms where ω A ( ⃗ k, ⃗p ) = ω B ( ⃗ k, ⃗ l ) then keeps the following summands \nA = B = 2 A = B = 3 A = B = 4 (A.V.9) \ncompletely, which were also exactly the same summands that survived the one-particle projection, while for the remaining six summands it yields the following conditions: \nA = 2 , B = 3 : ω p -ω k + p + ω l + ω k + l = 0 (A.V.10) \nA = 2 , B = 4 : ω p -ω k + p -ω l -ω k + l = 0 (A.V.11) \nA = 3 , B = 4 : -ω p -ω k + p -ω l -ω k + l = 0 , (A.V.12) \nand for the other three summands the same conditions with the role of ⃗ p and ⃗ l swapped. For mass m > 0, which implies ω k > 0, the last condition is never fulfilled. For m = 0 there is a solution, namely ⃗ k = ⃗ p = ⃗ l = 0. The first condition reads \nω k + p = ω p + ω l + ω k + l . (A.V.13) \nThis is never fulfilled, as we show in the following (where we define p := || ⃗ p || , l := || ⃗ l || , k := || ⃗ k || and assume 23 m> 0): \nω k + p ≤ √ ( k + p ) 2 + m 2 < √ p 2 + m 2 + √ l 2 + m 2 + √ ( k -l ) 2 + m 2 ≤ ω p + ω l + ω k + l . (A.V.14) \nThe inequality in the middle can be proven by considering the square of both sides (as each summand individually is positive, the direction of the inequality remains unaffected), which yields: \nkp + kl < m 2 + l 2 + √ p 2 + m 2 √ ( k -l ) 2 + m 2 + √ p 2 + m 2 √ l 2 + m 2 + √ l 2 + m 2 √ ( k -l ) 2 + m 2 . (A.V.15) \nWe can now estimate the right hand side downwards as \nm 2 + l 2 + p | k -l | + pl + l | k -l | < m 2 + l 2 + √ p 2 + m 2 √ ( k -l ) 2 + m 2 + √ p 2 + m 2 √ l 2 + m 2 + √ l 2 + m 2 √ ( k -l ) 2 + m 2 , (A.V.16) \nwhich yields \nkp + kl < m 2 + l 2 + p | k -l | + pl + l | k -l | (A.V.17) \nto be proven. Due to the absolute value, we consider two different cases: Let's first assume that k > l . Then the inequality reads \nkp + kl < m 2 + kp + kl , (A.V.18) \nwhich is true as long as m> 0. In the second case, i.e. for l ≥ k , we are left with \nkp + kl < m 2 +2 l 2 +2 pl -pk -kl , (A.V.19) \nor equivalently \n0 < m 2 +2[ l 2 -kl ] + 2[ pl -pk ] . (A.V.20) \nHowever, as we are considering the case l ≥ k , both brackets yield non-negative results and thus the inequality is fulfilled for m > 0. Hence (A.V.10) does also not have any solutions as long as m> 0. For m = 0 we have to solve the following equality: \n| ⃗ k + ⃗ p | -p ! = | ⃗ k + ⃗ l | -l . (A.V.21) \nHowever, we also have \n| ⃗ k + ⃗ p | -p ≤ k ≤ | k -l | + l ≤ | ⃗ k + ⃗ l | -⃗ l , (A.V.22) \nhence in order for equation (A.V.21) to hold, all ≤ signs must become equalities. For the first one this is the case if ⃗ k ∥ ⃗ p and for the last one if ⃗ k ∥ -⃗ l . For the one in the middle, we have to consider two cases on how to resolve | k -l | . If k ≥ l , then we can directly drop the absolute value and the middle ≤ becomes an equality. In case k < l we find | k -l | + l = 2 l -k > 2 k -k = k , so there is no equality. Hence for m = 0 the following solutions exist: \n⃗ k ∥ ⃗ p ∧ ⃗ k ∥ -⃗ l ∧ k ≥ l . (A.V.23) \nIt remains to investigate (A.V.11). Isolating ω p on one side and following the same argumentation as above (squaring the inequality and estimating downwards the rights hand side) we end up with \nkp + kl < k 2 + m 2 + l 2 + l | k -p | + | k -p || k -l | + l | k -l | . (A.V.24) \nHere we have to consider four different cases: \n- · k ≥ p and k ≥ l : We obtain 2 kp < 2 k 2 + m 2 , which has no solution for m> 0 and for m = 0 and k = p we get solutions, thus k ≥ l, k = p, ⃗ k ∥ -⃗ l, ⃗ k ∥ -⃗ p .\n- · k ≥ p and k < l : We obtain 0 < m 2 +2 l 2 -2 pl which has no solution for m ≥ 0.\n- · k < p and k ≥ l : We obtain 0 < m 2 , which has only for m = 0 a solution, thus there k ≥ l, k < p, ⃗ k ∥ -⃗ l, ⃗ k ∥ -⃗ p .\n- · k < p and k < l : We obtain 0 < m 2 +2[( p -k )( l -k ) + l ( l -k )] and thus no solution as every bracket is positive. \nSummarising, for equality (A.V.11) we get again no solution if m> 0 and for m = 0 we have \n⃗ k ∥ -⃗ p ∧ ⃗ k ∥ -⃗ l ∧ p ≥ k ≥ l . (A.V.25) \nSo in total, for a positive mass non of the non-diagonal terms survives the rotating wave approximation. If the mass is zero, the following non-diagonal terms survive: \nA = 2 , B = 3 : ⃗ k ∥ ⃗ p ∧ ⃗ k ∥ -⃗ l ∧ k ≥ l (A.V.26) A = 2 , B = 4 : ⃗ k ∥ -⃗ p ∧ ⃗ k ∥ -⃗ l ∧ p ≥ k ≥ l (A.V.27) A = 3 , B = 4 : ⃗ k = ⃗ l = ⃗ p = 0 (A.V.28) A = 3 , B = 2 : ⃗ k ∥ ⃗ l ∧ ⃗ k ∥ -⃗ p ∧ k ≥ p (A.V.29) A = 4 , B = 2 : ⃗ k ∥ -⃗ l ∧ ⃗ k ∥ -⃗ p ∧ l ≥ k ≥ p (A.V.30) A = 4 , B = 3 : ⃗ k = ⃗ l = ⃗ p = 0 . (A.V.31) \nIf we plug these special cases into the dissipator, which contains a projection of ⃗ l and also of ⃗ p onto the plane perpendicular to ⃗ k , all extra terms containing ⃗ k ∥ ± ⃗ l and ⃗ k ∥ ± ⃗ p vanish. Then only the special solution ⃗ k = ⃗ l = ⃗ p = 0 remains in all six cases (due to polar coordinates and thus also spherical coordinates being non-unique for zero radius, in that case still all directions are possible). However, as this is only one point regarding the radius integrations, it will vanish under the integral. Thus all the extra correction terms vanish 24 and we are left with the dissipator after the RWA in the form \nD [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k · { [ J A r ( ⃗ k, ⃗p ) † , [ J A r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f (Ω k + ω A ( ⃗ k, ⃗ l )) + h.c. }∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) (A.V.32) \n. \nIn order to explicitly write the hermitian conjugate, we split f ( ω ) = f δ ( ω ) + f PV ( ω ), as the two parts behave differently under complex conjugation (the first part is real, the second one purely imaginary) and compute them in the following two subsections.", 'A.V.1 Computation of the delta terms in the RWA': 'The delta terms remains unaffected by the complex conjugation 25 , thus we get, using that the terms in the first line, which are independent of N ( k ), vanished due to the Markov approximation: \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 f δ (Ω k + ω A ( ⃗ k, ⃗ l )) Ω k N ( k ) · { ( J A r ( ⃗ k, ⃗p ) † ρJ A r ( ⃗ k, ⃗ l ) -1 2 { ρ, J A r ( ⃗ k, ⃗ l ) J A r ( ⃗ k, ⃗p ) † } ) + ( J A r ( ⃗ k, ⃗ l ) ρJ A r ( ⃗ k, ⃗p ) † -1 2 { ρ, J A r ( ⃗ k, ⃗p ) † J A r ( ⃗ k, ⃗ l ) } ) }∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) . (A.V.33) \nUsing the following equalities: \nJ 2 r ( ⃗ k, ⃗p ) † = J 2 r ( -⃗ k, ⃗ k + ⃗ p ) ω 2 ( -⃗ k, ⃗p + ⃗ k ) = -ω 2 ( ⃗ k, ⃗p ) (A.V.34) \nJ 3 r ( ⃗ k, ⃗p ) † = J 4 r ( -⃗ k, -⃗ p ) ω 3 ( -⃗ k, -⃗ p ) = -ω 4 ( ⃗ k, ⃗p ) (A.V.35) \nJ 4 r ( ⃗ k, ⃗p ) † = J 3 r ( -⃗ k, -⃗ p ) ω 4 ( -⃗ k, -⃗ p ) = -ω 3 ( ⃗ k, ⃗p ) , (A.V.36) \nwe can simplify the δ -part of the dissipator and obtain \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 f δ (Ω k + ω A ( ⃗ k, ⃗ l )) + f δ (Ω k -ω A ( ⃗ k, ⃗ l )) Ω k N ( k ) · { ( J A r ( ⃗ k, ⃗ l ) ρJ A r ( ⃗ k, ⃗p ) † -1 2 { ρ, J A r ( ⃗ k, ⃗p ) † J A r ( ⃗ k, ⃗ l ) } ) }∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) . (A.V.37) \nAdditionally, all terms involving the extended projection, i.e. all terms where A = 3 or A = 4, vanished due to the Markov approximation, hence the δ -part of the dissipator simplifies even further: \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 f δ (Ω k + ω 2 ( ⃗ k, ⃗ l )) + f δ (Ω k -ω 2 ( ⃗ k, ⃗ l )) Ω k N ( k ) · { ( J 2 r ( ⃗ k, ⃗ l ) ρJ 2 r ( ⃗ k, ⃗p ) † -1 2 { ρ, J 2 r ( ⃗ k, ⃗p ) † J 2 r ( ⃗ k, ⃗ l ) } ) }∣ ∣ ∣ ∣ ∣ ω 2 ( ⃗ k,⃗p )= ω 2 ( ⃗ k, ⃗ l ) (A.V.38) \nAs shown above, f δ ∝ δ ( k ), hence the RWA condition reads \nω 2 ( ⃗ k, ⃗p ) = ω p -ω k + p = 0 = ω l -ω k + l = ω 2 ( ⃗ k, ⃗ l ) (A.V.39) \n. \nand is therefore automatically fulfilled, thus it can be dropped. One can furthermore evaluate the f δ and obtains, using (A.IV.20): \nf δ (Ω k + ω 2 ( ⃗ k, ⃗ l )) + f δ (Ω k -ω 2 ( ⃗ k, ⃗ l )) = πδ ( k ) [ 1 1 + l cos( θ l ) ω l + 1 1 -l cos( θ l ) ω l ] = δ ( k ) 2 π 1 -l 2 cos 2 ( θ l ) ω 2 l , (A.V.40) \nwhere θ l denotes the angle between ⃗ k and ⃗ l and l = | ⃗ l | , k = | ⃗ k | . This yields \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 δ ( k ) Ω k 2 π 1 -l 2 cos 2 ( θ l ) ω 2 l N ( k ) · { ( J 2 r ( ⃗ k, ⃗ l ) ρJ 2 r ( ⃗ k, ⃗p ) † -1 2 { ρ, J 2 r ( ⃗ k, ⃗p ) † J 2 r ( ⃗ k, ⃗ l ) } ) } . (A.V.41) \nThe rotating wave condition further implies \n∂ ∂ | k | ω 2 ( ⃗ k, ⃗p ) = ∂ ∂ | k | ω 2 ( ⃗ k, ⃗ l ) ⇐⇒ k + p cos( θ p ) ω k + p = k + l cos( θ l ) ω k + l . (A.V.42) \nWe therefore can use that \nδ ( k ) 2 π 1 -l 2 cos 2 ( θ l ) ω 2 l = 2 πδ ( k ) 1 1 -( k + l cos( θ l )) 2 ω 2 k + l = 2 πδ ( k ) 1 √ 1 -( k + l cos( θ l )) 2 ω 2 k + l 1 √ 1 -( k + p cos( θ p )) 2 ω 2 k + p (A.V.43) \nand, defining the Lindblad operators \nL r ( ⃗ k ) := ∫ R 3 d 3 p 1 √ 1 -( k + p cos( θ p )) 2 ω 2 k + p J 2 r ( ⃗ k, ⃗p ) , (A.V.44) \nwe can recast the dissipator in Lindblad form: \nD δ [ ρ S ] = κ ∑ r ∈{ + , -} ∫ R 3 d 3 k (2 π ) 2 δ ( k ) N ( k ) Ω k ( L r ( ⃗ k ) ρL r ( ⃗ k ) † -1 2 { ρ, L r ( ⃗ k ) † L r ( ⃗ k ) } ) . (A.V.45) \nAs the rotating wave approximation dropped the same terms as the single-particle projection and led to a condition on the frequencies that is already implemented in δ ( k ), which is present in every term of the δ -part of the dissipator after the Markov approximation, the RWA does not change the form of the δ -part of the dissipator compared to its form after the Markov approximation in (A.IV.34).', 'A.V.2 Computation of the Cauchy principal value terms in the RWA': "The contributions involving the Cauchy principal value, denoted as the PV-part of the dissipator (A.V.32) in the following reads after renormalisation, which removes the terms independent of \nN ( k ): \nD PV [ ρ S ] \n= -κ 2 ∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k · { [ J A r ( ⃗ k, ⃗p ) † , [ J A r ( ⃗ k, ⃗ l ) , ρ S ( t ) ]] f PV (Ω k + ω A ( ⃗ k, ⃗ l )) + h.c. }∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) . (A.V.46) \nAs f ( ω ) is purely imaginary, it switches sign under the hermitian conjugation and we obtain: \n. \nD PV [ ρ S ] = -κ 2 ∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k · { [[ J A r ( ⃗ k, ⃗p ) † , J A r ( ⃗ k, ⃗ l ) ] , ρ S ( t ) ] f PV (Ω k + ω A ( ⃗ k, ⃗ l )) }∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) (A.V.47) \nWe can then rewrite this part of the dissipator as \nD PV [ ρ S ] = -i κ 2 [ V LS , ρ ] (A.V.48) \nwith \nV LS = -∑ r ∈{ + , -} 4 ∑ A =2 ∫ R 3 d 3 k d 3 p d 3 l (2 π ) 3 N ( k ) Ω k PV ( 1 Ω k + ω A ( ⃗ k, ⃗ l ) ) [ J A r ( ⃗ k, ⃗p ) † , J A r ( ⃗ k, ⃗ l ) ] ∣ ∣ ∣ ∣ ∣ ω A ( ⃗ k,⃗p )= ω A ( ⃗ k, ⃗ l ) . (A.V.49) \nNote that the rotating wave approximation hence removed the imaginary terms in the fifth to seventh line of the Markovian master equation in (5.3), while it did not change the other imaginary terms. These were vanishing when working with the extended projection, hence in that case we find V LS = 0. If the non-extended projection is used, only the terms for A = 2 are left and there the RWA condition is already implemented in the one-particle projection, as the case (1 , 1) includes δ ( ⃗ p -⃗ l ), see table (I) and (II) at the beginning of section A.I.2. Hence the RWA does not change anything in the remaining PV-terms. The final one-particle master equation then becomes \n∂ ∂t ρ ( ⃗u, ⃗v, t ) = \n-iρ ( ⃗u, ⃗v, t ) ( ω u -ω v ) + κ 16 πβ { -10 3 ( u 2 + v 2 ) + 2( ω 2 u + ω 2 v ) -2 ω u u m 4 arctanh ( u ω u ) -2 ω v v m 4 arctanh ( v ω v ) -ω u ω v ( v 2 -3 ( ⃗u · ⃗v ) 2 u 2 )[ 2 3 -m 2 u 2 + m 4 ω u u 3 arctanh ( u ω u )] -ω v ω u ( u 2 -3 ( ⃗u · ⃗v ) 2 v 2 )[ 2 3 -m 2 v 2 + m 4 ω v v 3 arctanh ( v ω v )] } ρ ( ⃗u, ⃗v, t ) -(1 -δ P ) iκ 2(2 π ) 2 lim ϵ → 0 [ u 4 ω u ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω u ω u -k k 2 -( ω u -k -ω u ) 2 -v 4 ω v ∫ ∞ ϵ dk ∫ π 0 dθ sin 5 ( θ ) kN ( k ) 1 -ω v ω v -k k 2 -( ω v -k -ω v ) 2 ] ρ ( ⃗u, ⃗v, t ) . (A.V.50) \n- [1] M. J. Fahn, K. 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2023arXiv230602465E

We present an overview of the James Webb Space Telescope JWST Advanced Deep Extragalactic Survey JADES an ambitious program of infrared imaging and spectroscopy in the GOODSS and GOODSN deep fields designed to study galaxy evolution from high redshift to cosmic noon. JADES uses about 770 hours of Cycle 1 guaranteed time largely from the NearInfrared Camera NIRCam and NearInfrared Spectrograph NIRSpec instrument teams. In GOODSS in and around the Hubble Ultra Deep Field and Chandra Deep Field South JADES produces a deep imaging region of 45 arcmin2 with an average of 130 hrs of exposure time spread over 9 NIRCam filters. This is extended at medium depth in GOODSS and GOODSN with NIRCam imaging of 175 arcmin2 with an average exposure time of 20 hrs spread over 810 filters. In both fields we conduct extensive NIRSpec multiobject spectroscopy including 2 deep pointings of 55 hrs exposure time 14 medium pointings of 12 hrs and 15 shallower pointings of 4 hrs targeting over 5000 HST and JWSTdetected faint sources with 5 low medium and highresolution dispersers covering 0.65.3 microns. Finally JADES extends redward via coordinated parallels with the JWST MidInfrared Instrument MIRI featuring 9 arcmin2 with 43 hours of exposure at 7.7 microns and twice that area with 26.5 hours of exposure at 12.8 microns For nearly 30 years the GOODSS and GOODSN fields have been developed as the premier deep fields on the sky JADES is now providing a compelling start on the JWST legacy in these fields.

2023-06-01T00:00:00Z

['arXiv:2306.02465', '2023arXiv230602465E', '10.48550/arXiv.2306.02465']

['Astrophysics - Astrophysics of Galaxies']

Overview of the JWST Advanced Deep Extragalactic Survey JADES

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2306.02465.pdf

{'Overview of the JWST Advanced Deep Extragalactic Survey (JADES)': "Daniel J. Eisenstein, 1 Chris Willott, 2 Stacey Alberts, 3 Santiago Arribas , 4 Nina Bonaventura, 5, 6, 3 Andrew J. Bunker, 7 Alex J. Cameron, 7 Stefano Carniani, 8 Stephane Charlot, 9 Emma Curtis-Lake, 10 Francesco D'Eugenio, 11, 12 Pierre Ferruit, 13 Giovanna Giardino, 14 Kevin Hainline, 3 Ryan Hausen, 15 Peter Jakobsen, 5, 6 Benjamin D. Johnson, 1 Roberto Maiolino, 11, 12, 16 Marcia Rieke, 3 George Rieke, 17 Hans-Walter Rix, 18 Brant Robertson, 19 Daniel P. Stark, 3 Sandro Tacchella, 11, 12 Christina C. Williams, 20 Christopher N. A. Willmer, 3 William M. Baker, 11, 12 Stefi Baum, 21 Rachana Bhatawdekar, 22, 23 Kristan Boyett, 24, 25 Zuyi Chen, 3 Jacopo Chevallard, 7 Chiara Circosta, 26 Mirko Curti, 27, 11, 12 A. Lola Danhaive, 11 Christa DeCoursey, 3 Ryan Endsley, 28 Anna de Graaff, 18 Alan Dressler, 29 Eiichi Egami, 3 Jakob M. Helton, 3 Raphael E. Hviding, 3 Zhiyuan Ji, 3 Gareth C. Jones, 7 Nimisha Kumari, 30 Nora Lutzgendorf, 31 Isaac Laseter, 32 Tobias J. Looser, 11 Jianwei Lyu, 3 Michael V. Maseda, 32 Erica Nelson, 33 Eleonora Parlanti, 8 Michele Perna, 4 D'avid Pusk'as, 11, 12 Tim Rawle, 34 Bruno Rodr'ıguez Del Pino, 4 Lester Sandles, 11, 12 Aayush Saxena, 7, 16 Jan Scholtz, 11, 12 Katherine Sharpe, 1 Irene Shivaei, 3 Maddie S. Silcock, 35 Charlotte Simmonds, 11, 12 Maya Skarbinski, 1 Renske Smit, 36 Meredith Stone, 3 Katherine A. Suess, 19, 37 Fengwu Sun, 3 Mengtao Tang, 3 Michael W. Topping, 3 Hannah Ubler, 11, 12 Natalia C. Villanueva, 1 Imaan E. B. Wallace, 7 Lily Whitler, 3 Joris Witstok, 11, 12 and Charity Woodrum 3 \n1 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA 2 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada \n3 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson AZ 85721 USA \nCentro de Astrobiolog'ıa (CAB), CSIC-INTA, Cra. de Ajalvir Km. 4, 28850- Torrej'on de Ardoz, Madrid, Spain \n5 Cosmic Dawn Center (DAWN), Copenhagen, Denmark \n- 6 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark \nDepartment of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK \n7 \n8 Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy \nSorbonne Universit'e, CNRS, UMR 7095, Institut d'Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France \n9 \nCentre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 \n9AB, UK \n11 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK \n12 Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, UK \n- 13 European Space Agency, European Space Astronomy Centre, Camino Bajo del Castillo s/n, 28692 Villafranca del Castillo, Madrid, Spain \n14 ATG Europe for the European Space Agency, ESTEC, Noordwijk, The Netherlands \n15 \nDepartment of Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 \n16 \nDepartment of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK \n17 Steward Observatory and Dept of Planetary Sciences, University of Arizona 933 N. Cherry Avenue Tucson AZ 85721 USA 18 Max-Planck-Institut fur Astronomie, Konigstuhl 17, D-69117, Heidelberg, Germany \nDepartment of Astronomy and Astrophysics University of California, Santa Cruz, 1156 High Street, Santa Cruz CA 96054 USA \n20 \nNSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719 USA \n21 \nDepartment of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada \n- 22 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Ca˜nada, Madrid, Spain \n23 European Space Agency, ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, NL \n24 School of Physics, University of Melbourne, Parkville 3010, VIC, Australia \n25 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia \n26 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Ca˜nada, Madrid, Spain \n27 European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany \n28 Department of Astronomy, University of Texas, Austin, TX 78712 USA \n29 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101 \nAURA for European Space Agency, Space Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD, 21210 \n31 European Space Agency, Space Telescope Science Institute, Baltimore, Maryland, US \n- 32 Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter St., Madison, WI 53706 USA \n33 Department for Astrophysical and Planetary Science, University of Colorado, Boulder, CO 80309 USA \n10 \n19 \n30 \n4 \n- 34 European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villafranca del Castillo, Madrid, Spain\n- 35 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK \n- 37 \nKavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305 USA", 'ABSTRACT': 'We present an overview of the James Webb Space Telescope (JWST) Advanced Deep Extragalactic Survey (JADES), an ambitious program of infrared imaging and spectroscopy in the GOODS-S and GOODS-N deep fields, designed to study galaxy evolution from high redshift to cosmic noon. JADES uses about 770 hours of Cycle 1 guaranteed time largely from the Near-Infrared Camera (NIRCam) and Near-Infrared Spectrograph (NIRSpec) instrument teams. In GOODS-S, in and around the Hubble Ultra Deep Field and Chandra Deep Field South, JADES produces a deep imaging region of ∼ 45 arcmin 2 with an average of 130 hrs of exposure time spread over 9 NIRCam filters. This is extended at medium depth in GOODS-S and GOODS-N with NIRCam imaging of ∼ 175 arcmin 2 with an average exposure time of 20 hrs spread over 8-10 filters. In both fields, we conduct extensive NIRSpec multi-object spectroscopy, including 2 deep pointings of 55 hrs exposure time, 14 medium pointings of ∼ 12 hrs, and 15 shallower pointings of ∼ 4 hrs, targeting over 5000 HST and JWST-detected faint sources with 5 low, medium, and high-resolution dispersers covering 0.6-5.3 µ m. Finally, JADES extends redward via coordinated parallels with the JWST Mid-Infrared Instrument (MIRI), featuring ∼ 9 arcmin 2 with 43 hours of exposure at 7.7 µ m and twice that area with 2-6.5 hours of exposure at 12.8 µ m. For nearly 30 years, the GOODS-S and GOODS-N fields have been developed as the premier deep fields on the sky; JADES is now providing a compelling start on the JWST legacy in these fields. \nKeywords: early universe - galaxies: evolution - galaxies: high-redshift', '1. INTRODUCTION': 'The James Webb Space Telescope (JWST) is revolutionizing the study of galaxy evolution by giving us unprecedented access to deep, sharp, and nuanced infrared imaging and spectroscopy. Designed to push the redshift frontier and bring the early growth of galaxies into clear focus, the telescope is performing at, or even better than, expectations (Rigby et al. 2023). JWST takes marvelous advantage of the faintness of the zodiacal foregrounds at 2-10 µ m and state-of-the-art infrared detectors to unlock the rest-frame optical at redshifts z > 4, combining large collecting area and diffractionlimited imaging. Exploiting this telescope are ambitious multi-purpose instruments (Gardner et al. 2023) that give us dozens of selectable filters, exquisite slitless, integral-field or multi-object spectroscopy (MOS) and multiple coronagraphs. \nUnraveling the physics of high-redshift galaxies will require the combination of many different kinds of observations. While there are important observations of rare, extreme phenomena, many goals in studying the general population are well served by deep-sky general surveys, as each image contains an unbiased superposition of all epochs of galaxy evolution. Of course, low \nredshifts are best served by wider, shallower data, but great depth is required to identify and characterize highredshift galaxies. \nThe Hubble Deep Field was a dramatic advance in this regard. It boldly unveiled the high-redshift Universe in a single multi-color blank-field image (Williams et al. 1996; Ferguson et al. 2000). Since then, such surveys have been vigorously pursued, utilizing virtually every high-sensitivity narrow-field telescope. This field and that of the Chandra Deep Field South (Giacconi et al. 2002) were broadened out to form the Great Observatories Origins Deep Survey (GOODS Giavalisco et al. 2004), utilizing the new opportunities of the HST Advanced Camera for Surveys (ACS) and Spitzer infrared telescope to partner with deep Chandra X-ray imaging (Luo et al. 2008). Soon after, the Hubble Ultra Deep Field (HUDF) was sited in the heart of GOODS-S (Beckwith et al. 2006). It has since become the standard bearer of deep fields, pushing into the epoch of reionization by leveraging exceptional optical and infrared Hubble Space Telescope imaging, e.g., the UDF09 (Bouwens et al. 2010) and UDF12 programs (Ellis et al. 2013), with tremendous investments across the electromagnetic spectrum from many other imaging and spectroscopic facilities. \nJWST is designed to pursue such surveys; the telescope provides exquisite image quality and depth, but it moves slowly enough that deep fields are operationally favored. Every pointing of practical depth reveals tens of thousands of galaxies, including many at z > 6 where the intergalactic medium (IGM) completely blocks optical light. Further, JWST provides a novel opportunity to conduct detailed faint multi-object spectroscopy beyond 2 µ m, where critical rest-optical lines are shifted at high redshift. Numerous projects have already been approved to start this work, which we expect will be one of the enduring legacies of the telescope (Robertson 2022). \nHere, we describe the JWST Advanced Deep Extragalactic Survey (JADES), a collaboration of the NearInfrared Camera (NIRCam) and Near-Infrared Spectrograph (NIRSpec) Instrument Development Teams. The plans to conduct deep-field imaging and spectroscopy were featured in the original instrument proposals, with the intent to devote a substantial amount of guaranteed time to this topic. In 2015, the teams joined to form a larger and more coordinated project, now called JADES, to focus on the exceptional opportunities of JWST onto the GOODS-S and GOODS-N fields. In doing so, it became possible to carry out a project with fewer compromises: deep and wide enough to support the geometry of the instrument footprints and utilize efficient parallel observations, with robust well-dithered imaging and spectroscopy in many filters and several dispersion modes. \nAt about 770 hours of observing time plus coordinated parallels, JADES is the largest program operating in JWST Cycle 1 and is a very large investment of instrument team guaranteed time. The time was roughly evenly split between the NIRCam and NIRSpec GTO budgets, with a supplementary contribution from the MIRI-US team. By applying the experience of the teams that designed and commissioned the instruments, we aim to provide an exquisite legacy data set for these deep fields. \nIn this overview paper, we describe the scientific motivations and resulting survey design of JADES, providing an overview of how we developed our strategy to maximize performance in both imaging and spectroscopy in the targeted fields. Subsequent papers will describe the imaging data reduction and spectroscopic target selection, as well as the first data release centered on the HUDF (Bunker et al. 2023a; Rieke et al. 2023a).', '2.1. Motivations': "The sensitivity and instrumentation of JWST provide a singular opportunity to study the evolution of galaxies from the earliest epochs ≲ 300 Myr after the Big Bang, through the Epoch of Reionization during the first billion years of cosmic history, and on to Cosmic High Noon where the stellar mass and black hole mass densities of the universe were well-established. Unlike all other previous studies of high-redshift (i.e., z > 3) galaxy populations where only rest-frame ultraviolet spectral properties have been accessible, JWST enables for the first time, via its instruments NIRCam (Rieke et al. 2023b), NIRSpec (Jakobsen et al. 2022; Ferruit et al. 2022), NIRISS (Doyon et al. 2023) and MIRI (Wright et al. 2023), photometry and spectroscopy extending from blueward of the Lyman break to redward of the Balmer/4000 ˚ A break region. In this section we describe how our view of the Universe before the launch of JWST shaped the design of JADES and how early JWST observations support our decisions. \nAt the earliest times in cosmic history (e.g., < 500 Myr), the first abundant population of star-forming galaxies developed. Galaxy formation is a self-regulated process, and the ways in which early galaxies respond to the rapid accretion and cooling of gas greatly affects their bulk properties like luminosity and size. Through perseverance in HST imaging surveys, a handful of galaxies at z ∼ 10 -11 were discovered (Coe et al. 2013; Oesch et al. 2016), enabling a first glance at the primitive galaxy formation process. \nJWST has the sensitivity and the required array of infrared filters to identify galaxies selected in rest-frame ultraviolet (UV) at z > 12, farther than any tentative HST detections. Hundreds of hours of NIRCam multifilter imaging will yield sufficient source counts to measure the UV luminosity function evolution out to z ∼ 10 and beyond. JADES includes medium-band filters in the NIRCam long wave channel that can help identify and distinguish high-z candidates from dusty, strong emission line sources at lower redshifts e.g. Zavala et al. 2023; Fujimoto et al. 2022; Arrabal Haro et al. 2023). The most distant galaxies will inevitably be very faint and these NIRCam discoveries will require very long integrations with the NIRSpec low-resolution prism for redshift confirmation, especially at redshifts z > 10 where the strongest optical lines are redshifted beyond the NIRSpec range (Robertson et al. 2023a; Curtis-Lake et al. 2023). \nExtending redder than Hubble and reaching many magnitudes deeper than ever achieved with Spitzer, \nJWST greatly improves our census of the early Universe, tracing the growth of galaxies at this early epoch. Combining the SFR and stellar masses derived from spectralenergy distributions (SED) spanning the rest-UV and optical, NIRCam and MIRI imaging allow estimates of the stellar birthrate of galaxies out to z ∼ 10 and can deliver our earliest constraints on the efficiency of galaxy formation (e.g. Labbe et al. 2022; Endsley et al. 2022; Tacchella et al. 2023). Stellar masses measured at z ∼ 10 allow us to infer a bulk star formation rate to z ≳ 12 (given the minimum ∼ 100 Myr timescale typically required for the development of strong restframe optical breaks; Whitler et al. 2023a; Dressler et al. 2023). Rest-UV emission lines, such as [CIV](1549 ˚ A), HeII(1640 ˚ A), OIII](1663 ˚ A) and CIII](1909 ˚ A), are accessible to NIRSpec to the highest redshifts to measure the physical properties of nebular gas and infer their sources of ionization (Bunker et al. 2023b; Hsiao et al. 2023; Tang et al. 2023). Our window into early galaxies with JWST becomes dramatically richer at just slightly later times ( z ∼ 8 -9) where NIRSpec can measure the rest-frame optical strong lines (e.g., [OII], [NeIII], H β , [OIII], and even H α at z < 7) and both the Lyman and Balmer/4000 ˚ A breaks (Cameron et al. 2023; Sanders et al. 2023; Reddy et al. 2023a; Tang et al. 2023). NIRCam imaging in medium and wide filters can measure these breaks and strong lines, and given enough filters can differentiate between the two (Endsley et al. 2022; Williams et al. 2023; Withers et al. 2023). \nFrom previous surveys of the cosmic microwave background, quasars and galaxies, we know that these epochs experience the first important contributions of galaxies to the cosmic reionization process (Planck Collaboration et al. 2020; Fan et al. 2022; Robertson et al. 2022). Here, JWSTsimultaneously constrains the evolving rest-frame UV galaxy luminosity density and provides information on the hardness of the ionizing continuum and the escape fraction of Lyman continuum photons (Simmonds et al. 2023; Mascia et al. 2023; Endsley et al. 2022; Donnan et al. 2023; Bouwens et al. 2023). These measurements will result in a more accurate 'balancing of the budget' for cosmic reionization, where we weigh the cosmic ionization rate against the recombination of the intergalactic hydrogen in determining the evolving bulk IGM neutrality. Mapping with spectra the 'Lymanα disappearance', measuring the evolving fraction of UVdropout selected galaxies that show (or not) Lymanα emission, can track how the increased IGM neutrality at earlier times extinguishes observed line emission in progressively more of the sources (Stark et al. 2010; Fontana et al. 2010; Pentericci et al. 2014; Mason et al. 2018; Ouchi et al. 2020; Jones et al. 2023). Correlating the \ntransmission of Lymanα with photometric or spectroscopic measurements of environment can provide further insight into the topology of ionized bubbles during this epoch (Tang et al. 2023; Witstok et al. 2023a; Endsley et al. 2023; Jung et al. 2023; Lu et al. 2023; Whitler et al. 2023b). \nRest-frame optical line spectroscopy at z ∼ 6-9 dramatically extends our knowledge of the chemical enrichment of galaxies and reveals the physical conditions in the warm interstellar medium of early star-forming galaxies. Line excitation diagrams provide insights into the ionization state of the star-forming ISM in these systems; NIRSpec allows to apply these diagnostics in an entirely new redshift regime (e.g., Cameron et al. 2023; Sanders et al. 2023; Reddy et al. 2023a), connecting them with the properties of the exciting stellar populations. Via the combined measures of star formation rate in the rest-UV, stellar mass in the rest-optical, and metallicity from nebular lines we can explore whether the fundamental metallicity and mass-metallicity relations are already in place after only ∼ 1 billion years of cosmic history (Curti et al. 2023; Nakajima et al. 2023). \nJWST is revealing the emergence of morphological structures at z > 2 through superb infrared imaging (Robertson et al. 2023b; Kartaltepe et al. 2023; Ferreira et al. 2022a,b; Jacobs et al. 2023; Huertas-Company et al. 2023; Magnelli et al. 2023; Baker et al. 2023). JWST can resolve these galaxies from the rest-UV to the rest-optical, providing spatially-resolved measures of color gradients and stellar population properties, vastly outperforming HST (e.g. Figure 1). We can distinguish the clumpy UV-bright morphology from the rest-optical light on a galaxy-by-galaxy basis, and thereby constrain the role of large-scale gravitational instability in setting galaxy structures at z ∼ 2 -3. NIRSpec spectroscopy with the medium- or high-resolution gratings will connect these morphological measures to the dynamics of the galaxies, and through measuring outflows further constrain the role of feedback in shaping these maturing galaxies. \nHST has found compact red galaxies at z = 2, but JWST's angular resolution, sensitivity and redder bands are proving revolutionary to explore old stellar populations and their morphology at z > 3 (Carnall et al. 2023; Ji et al. 2023, Suess et al. in prep). With NIRCam imaging, including medium filters, Balmer and D 4000 breaks can be cleanly picked out. Spectroscopy at R = 100 with the NIRSpec prism provides precise redshifts and break strengths, but higher resolution spectroscopy enables more detailed constraints on SFH and abundances at z > 3 -4, an era that prior to JWST was prohibitive or impossible to study, but critical to our understand- \n<!-- image --> \nFigure 1. A grand design spiral at redshift 2 revealed in JADES imaging in GOODS-S. ( left ) The JWST NIRCam image combining F090W, F200W, and F444W filters in the Deep Prime region of program 1180. ( right ) The HST ACS F850LP image, where this star-forming galaxy is all but invisible. \n<!-- image --> \ning of why and how galaxies stop forming stars (Carnall et al. 2023; Nanayakkara et al. 2022). The importance of burstiness in the evolution of galaxies is now becoming clear with several examples of z > 5 galaxies undergoing 'mini-quenching' episodes (Looser et al. 2023b; Strait et al. 2023). Additional evidence for burstiness is now becoming apparent in statistical samples of NIRCam SEDs (Endsley et al. 2023; Dressler et al. 2023). \nDust attenuation and reddening in the rest-frame UV and optical spectra and SEDs of galaxies strongly affect the inferred physical quantities, so it is important to study the underlying dust properties. JWST opens a new window to identify and characterize obscured populations that were completely missed by even the deepest Spitzer and HST surveys. ALMA has revealed that such hidden galaxies likely contribute significantly to the cosmic star formation rate density at 3 < z < 8 (e.g. Williams et al. 2019; Fudamoto et al. 2021; Algera et al. 2023), and early JWST data is supporting this finding (Barrufet et al. 2023). The stellar SEDs and morphologies of dust obscured galaxies at z > 3 can now be characterized in detail for the first time (G'omezGuijarro et al. 2023; Nelson et al. 2022; P'erez-Gonz'alez et al. 2023). JWST is indicating that massive galaxies have non-negligible dust content at high-redshift, pointing to efficient production mechanisms even out to z ∼ 8 (McKinney et al. 2023; Akins et al. 2023). The combina- \nof multi-band NIRCam photometry and multi-line NIRSpec spectroscopy offers the opportunity to tackle these issues through SED-fitting and through line diagnostics such as the Paschen and Balmer decrements and 2200 ˚ A bump (e.g., Witstok et al. 2023b; Shapley et al. 2023; Reddy et al. 2023b; Sandles et al. 2023). \nThe intimate connection between the growth of galaxies and their supermassive black holes can be traced back to the earliest epochs with deep JWST imaging and spectroscopy. Active galactic nuclei (AGN) are being discovered via their pointlike morphology (particularly in the redder bands, e.g. Labbe et al. 2022; Furtak et al. 2022), broad wings of Balmer emission lines (e.g. Figure 2) and highly-ionized narrow lines (Kocevski et al. 2023; Harikane et al. 2023; Maiolino et al. 2023; Scholtz et al. 2023; Larson et al. 2023). In most cases these AGN are not detectable even by very deep Chandra or JVLA imaging, putting JWST at the forefront of the quest for the earliest supermassive black holes. With JADES we obtain deep NIRCam and MIRI imaging plus deep NIRSpec spectrocopy to discover these previously hidden AGN and investigate the evolving relationship between supermassive black holes and their host galaxies. \nThroughout these epochs, the development of the galaxy populations remains tightly connected with the structure formation process in our ΛCDM cosmology \n<!-- image --> \n<!-- image --> \nFigure 2. Example spectra for the z = 4 . 65 galaxy JADES-GS+53.13284-27.80185 (ID=00008083) from the JADES Deep/HST observations. The upper panel shows the low-resolution prism spectrum (total integration time 10 5 seconds). This spectrum reveals many emission lines and high S/N continuum. However, some emission lines are blended or have low equivalent width, motivating the acquisition of grating spectra for most of the galaxies with prism spectra in JADES. The rms uncertainty is shown in the gray shaded spectrum at the bottom of the plot. The lower-left panel shows the JADES Deep imaging data of this galaxy with an overlay of the position of the microshutters during the three nods. The green color of the galaxy indicates [OIII] and H β line emission dominating in the F277W filter. For the angular scale, we remind that the individual microshutters are 0.2 '' by 0.46 '' . The lower-right panel shows the H α spectra obtained with the medium G395M and high G395H resolution gratings. Both these spectra reveal a broad ( σ = 800kms -1 ) emission line from a low-luminosity AGN broad line region (Maiolino et al., in prep.). However, the G395H grating is required to spectrally resolve the narrow line emission from the galaxy itself (observed σ = 65kms -1 , compared to instrumental line spread function σ = 30kms -1 ; de Graaff et al. (in prep.)). \n<!-- image --> \nuniverse. The rates of star formation, stellar population aging, merging, and dynamical and morphological transformation are ultimately manifestations of the growth of dark matter halos. JWST is providing a new context for understanding the connection between galaxy and dark matter structure formation by aiming to discover the earliest galaxies that form in rare peaks of the density field, establishing both the SFR-halo mass and stellar mass-halo mass relations out to z ∼ 10, watching the emergence of dynamically cold galactic structures, and by observing the assembly of the first massive galaxies that form primarily through dissipationless \nmergers. The new spectroscopic capabilities allow us to identify physically-associated galaxies in the early universe rather than just projected overdensities (Kashino et al. 2022; Helton et al. 2023; Morishita et al. 2023) and enable us to distinguish between how central and satellite galaxies evolve further back in time than has previously been possible. The combination of area and depth allow for clustering analyses down to very faint magnitudes on spectroscopically-informed samples with well-constrained redshift selection functions. This combination will also address critical gaps in our knowledge of environmentally-driven galaxy evolution. The key \nepochs of stellar growth and the subsequent quenching in groups and (proto-)clusters likely often occur in a dust-obscured phase (see Alberts & Noble 2022, for a review), necessitating rest-frame near- and mid-infrared observations that are robust against extinction and directly probe obscured activity. In all, JWST allows a more physically complete view of galaxy formation that builds directly from the underlying ΛCDM framework. \nWe stress that most if not all of these science drivers require a substantial survey volume, not just depth. We aim to slice the galaxy samples in a variety of parameters for inter-comparison. Going deep may (slowly) reveal the less luminous galaxies, but we need to gather sufficient samples of the L ∗ and brighter ones as well. Rare phases, such as AGN and extreme starbursts, can be important for the evolutionary story. Large-scale structure is prominent even at high redshifts because galaxies are extremely biased tracers of the underlying density field. On the scale of one NIRCam or NIRSpec MOS pointing, this can cause the fluctuations in the number of objects, particularly those from the most massive halos, to vary substantially (Steinhardt et al. 2021). Larger surveys allow one to measure more accurate luminosity functions, but also to potentially measure the clustering amplitude itself, which bears on the mass of the host halos as well as on possible Mpc-scale environmental drivers in galaxy evolution. \nThe above science cases can all be addressed efficiently through a deep extragalactic survey. Typical high-redshift galaxies are common on the sky but very faint. The scientific exploitation of images is necessarily broad simply because of projection of the line of sight, but this is also true for efficient use of multi-object spectroscopy. Since the advent of the Hubble Deep Field, the community has been focusing its resources onto a small number of deep fields, so that the synergies between different types of data can be best exploited. Our survey follows this same logic.", '2.2. Opportunities of a Combined Imaging & Spectroscopy Program': 'As the combination of imaging and spectroscopy is a key driver of our coordinated parallel strategy, we want to stress that JWST imaging and spectroscopy reinforce each other in numerous critical ways. \nFirst, one has the obvious aspect that imaging and spectroscopy constrain different physical properties of the galaxies, which we seek to combine. \nSecond, having accurate redshifts is important for the interpretation of imaging in terms of luminosities, restframe colors, and proper sizes. For example, the conversion of SEDs to stellar masses and star formation \nhistories can easily be degenerate with redshift uncertainty. Spectroscopy is the gold standard for redshifts, and JWST has sufficient sensitivity and multiplex to provide spectroscopic redshifts for thousands of galaxies all the way to z > 10. Moreover, our program is providing large training samples for the photometric redshift methods that will supply redshifts for the rest of the imaging sample. \nThird, there are technical synergies. NIRCam broadband filters in the rest-frame optical can have substantial contribution from very strong emission lines, as illustrated by the recent JWST results (Cameron et al. 2023; Matthee et al. 2022). NIRSpec spectroscopy is providing the location and fluxes of these lines, enabling us to subtract them to accurately measure the continuum SED. This is important for the estimation of stellar population age distributions. We will do this subtraction directly in thousands of objects, but also measure the trends and variations needed to model the purely photometric samples. \nNIRCam, in turn, is important for NIRSpec MOS to understand its slit losses, background subtraction, and to aid in the interpretation of emission line kinematics. Unlike ground-based slit masks, the NIRSpec micro-shutter array (MSA) provides a fixed grid of slits. Galaxies will fall at various registrations relative to those slits and with a range of sizes. Achieving accurate line flux calibration requires the imaging to provide a model of this. Further, NIRSpec MOS background subtraction requires the subtraction of neighboring shutters; only NIRCam can provide a deep 2-5 µ m probe of contaminating objects in these shutters. Eventually, we expect that the sharp NIRCam images will provide morphological templates for more ambitious extraction of undersampled NIRSpec spectra, going beyond just summing along the spatial direction of a slit. \nFourth, JWST imaging can allow more efficient target selection for NIRSpec MOS spectroscopy of rare populations. While HST can provide Lyman-dropout selection for UV bright targets, the longer wavelength coverage of JWST yields much improved photometric redshifts for redder objects. NIRCam medium-band imaging can isolate objects with strong rest-frame optical line emission that can then be targeted for line profile studies with the NIRSpec gratings. \nFinally, there are more subtle astrophysical synergies. For strong line emitters, the high S/N and the excellent angular resolution of NIRCam provides, via the comparison of different filters, measurements of size and morphology of the line emission relative to the stellar light. NIRCam imaging can reveal color gradients to be correlated with spectral properties. For spatially extended \ngalaxies, one can even connect these to resolved spectral variations along the slits. \nWe note that while the combination of imaging and spectroscopy is critically important, it is not the case that one requires spectroscopy for every imaging object. Rather one intends to use the spectroscopy to build models of the trends, so that one can perform statistical work on the non-spectroscopic sample.', '3.1. Field Selection': 'JADES seeks to combine deep multi-band imaging and spectroscopy in pursuit of the science goals described in § 2. It is designed to bring NIRCam and NIRSpec MOS together on a common region of the sky, while covering substantial areas in two different fields in order to increase the statistical reach for rare objects and sample large-scale structure. \nWe observe two fields in JADES, to avoid concern that the large-scale structure of highly biased tracers and the radiation transport of reionization could make any one field peculiar and limit confidence in any unusual results. Further, we wanted to spread the observing around the year to ease the constraints on scheduling such a large program. Of course, even more fields would better mitigate the concerns about cosmic variance, but this would limit the depth and area of each. \nThe choice of location was driven by the availability of deep pan-chromatic imaging and spectroscopy, as the study of galaxy evolution draws on a wide range of such input. This led us clearly to the GOODS-South and GOODS-North fields, which have received huge investments of telescope time over the past 25 years from essentially every facility that bears on the high-redshift Universe. \nGOODS-South, home of the Chandra Deep Field South and the Hubble Ultra Deep Field as well as very deep ALMA (Walter et al. 2016; Dunlop et al. 2017; Franco et al. 2018; Hatsukade et al. 2018) and JVLA data (Rujopakarn et al. 2016; Alberts et al. 2020), is the preeminent deep field on the sky. We chose this as the primary field for JADES and focused the majority of the observing time there. GOODS-North, home of the Hubble Deep Field and exceptionally deep Chandra data, was chosen as the second field. \nBy placing JADES in the legacy GOODS fields, we seek to augment the rich HST community data products with a comprehensive set of JWST imaging and spectroscopy. The GOODS (Giavalisco et al. 2004), CANDELS (Grogin et al. 2011; Koekemoer et al. 2011), and UDF (Beckwith et al. 2006; Ellis et al. 2013; Illingworth et al. 2013) data have been reduced and released as com- \nponents of the Hubble Legacy Fields (HLF Illingworth et al. 2016; Whitaker et al. 2019a). The HLF reductions provide an excellent matched set of HST images in multiple ACS and WFC3 bands for use with the JADES JWST NIRCam imaging. We also utilize astrometric registration to Gaia performed by G. Brammer (private communication) using the methods of Kokorev et al. (2022) and grizli 1 .', '3.2. Tiers and Geometrical Constraints': "JADES is built as a two-layer wedding cake, with Deep portions of both imaging and spectroscopy, flanked by larger Medium depth regions. Bringing imaging and spectroscopy to bear on the same targets, while making efficient use of coordinated parallel observations, are driving goals of the survey design. Here we begin to describe these considerations. \nThe differing on-sky geometries of the NIRCam and NIRSpec instruments mean that these two cannot be efficiently overlapped with a single pointing of NIRCam. It takes at least a 2x2 mosaic of NIRCam to produce a filled area large enough to cover one NIRSpec MOS pointing. Further, the ability to use two instruments at once is an important opportunity to increase the science return, but the angular separation between the instantaneous fields of the instruments drives one to a large field. We note that, as a consequence of the visibility constraints, neither of the JADES fields allows JWST to return at a 180 · position angle, so as to swap the instrument locations. Instead, we have to construct an adequately sized mosaic, and choose the parallels to maximize the science return. Most of the MIRI parallel data and the NIRSpec MOS parallel data falls on NIRCam imaging, and nearly all of the NIRSpec parallel data falls on the GOODS/CANDELS HST imaging. \nWe placed the Deep portion of JADES in GOODS-S, while the Medium data are in both fields. We considered placing Deep pointings in GOODS-N, but as full support of the NIRSpec MOS footprint requires 4 NIRCam pointings, this would have become overly expensive. \nThe NIRCam data in JADES fall into 4 categories. There are contiguous portions in regular mosaics, of both deep and medium variety; we call these 'Prime'. Other portions occur as parallel exposures to NIRSpec MOS pointings, whose positions are therefore dictated by the location and position angle of the spectroscopy; we call these 'Parallel'. Again, these come in deep and medium variety. \nIn detail, ten of the Medium-depth Prime NIRCam exposures were taken with NIRSpec in parallel, but the structure of JWST coordinated parallel observations is that NIRSpec is formally prime in the planning tool. We refer to NIRCam as prime (and NIRSpec as parallel) despite this, because the exact pointing and exposure times were being dictated by the NIRCam science goals. \nOther NIRCam pointings were taken with MIRI in parallel; these yield Deep and Medium MIRI imaging. \nThe JADES NIRSpec MOS data fall into 3 tiers. One tier comprises two deep pointings, one scheduled early in the program and targeted without JWST imaging, the other scheduled at the end and targeted from the JADES imaging. These are called Deep/HST and Deep/JWST, respectively. Then there are two tiers of medium depth. One named Medium/JWST is targeted from JADES imaging, the other named Medium/HST has target selection prior to JWST imaging and is somewhat shallower. All of the NIRSpec MOS data is taken with NIRCam in parallel. \nThe Medium-depth designs with both NIRCam and NIRSpec MOS are shaped heavily by a desire to take well-dithered data, with at least 6 pixel locations to provide robustness to bad pixels and the undersampled point-spread function, and to use long enough exposures to keep observatory overheads low (these concerns are easily satisfied with the Deep data). Hence, even this flanking data is quite deep in comparison to pre-JWST opportunities. Because of the full use of coordinated parallel observations, we paid particular attention to minimizing the data rate for telemetry, typically utilizing the DEEP8 readout pattern for NIRCam and SLOW readouts for MIRI. This kept the program to about 2 GB/hr of data volume. \nAlthough JADES was designed to be observed in a single year, over-scheduling in Cycle 1 resulted in it being scheduled over 18 months. In particular, the large investment in GOODS-S was spread over two observing seasons. This led to some reoptimization relative to the original design, induced by exact position angles, instrument problems for some observations, and further on-orbit appreciation of science opportunities. We will focus in this paper on the observed program, making only passing mention of the original layout.", '3.3. JADES Footprint': 'The footprint of the JADES survey is shown in Figures 3 for GOODS-S and 4 for GOODS-N. As the geometry is complicated with the various overlapping footprints, we include separate images of each major portion of the survey. We stress that portions of the GOODS-S footprint are provisional: the position angle and exact \nlocation of most of the Medium/JWST NIRSpec and parallel NIRCam data have not been confirmed yet, and observational hiccups may yet occur. \nIn GOODS-S, one can see how the primary Deep portion of the survey, centered on the UDF, and the Medium portion to the west support each other with coordinated parallel observations. For instance, the MIRI parallels are largely partnered with NIRCam imaging, including the first NIRCam Deep Parallel. We also generate NIRSpec Medium/HST parallels that fall back on the UDF, providing high-multiplex spectroscopy for our Deep imaging. Finally, the NIRSpec Medium/JWST and Deep/JWST pointing create NIRCam parallels that fan out to create more imaging area. \nIn GOODS-N, the area of the prime survey is a little smaller, which makes it harder to fully utilize the parallels. We do utilize the NIRCam Medium Prime fields to cover the HDF and provide imaging for the initial NIRSpec Medium/HST spectroscopy as well as targets for the Medium/JWST followup. The parallels from that later spectroscopy produce NIRCam images that cover additional GOODS/CANDELS imaging. Two of the NIRCam prime pointings did not have MIRI or NIRSpec parallels that would fall on NIRCam imaging or GOODS; we opted to use MIRI, to be partnered with HST CANDELS imaging and perhaps future NIRCam data. \nA major constraint on the layout of the survey comes from the intention to be able to conduct NIRSpec followup of NIRCam-selected targets within a single observing window. We judged that 60 days would be the minimum separation of these visits, and therefore that it is needed to perform the NIRCam imaging early in the window and the NIRSpec follow-up late in the window. In GOODS-N, this plan is what has been scheduled. Unfortunately, our GOODS-S observations ended up split over two years, and we are seeking to use more optimal position angles (PA) and observing windows in the second year. \nIn GOODS-S, the year 1 NIRCam Deep Prime imaging was taken at V3 PA = 298.56 · , in early October 2022. The year 1 Medium Prime imaging was taken a week later at V3 PA = 308 · . The first NIRSpec Deep/HST pointing followed at V3 PA = 321 · . When we return in year 2, we will seek to match the Deep Prime and Medium Prime position angles to those of year 1. The position angles of the year 2 follow-up spectroscopy are assigned but might change, and the exact locations will depend on the location of high-priority targets. \nIn GOODS-N, we observed the NIRCam Medium Prime data at PA = 241 · in early February 2023. The \nDec (ICRS) \nFigure 3. Layout of the JADES observations in the GOODS-South field. JADES observations with NIRCam, NIRSpec MOS, and MIRI are shown as colored shaded regions. Higher opacity indicates higher exposure time for NIRCam and MIRI or overlapping MSA pointings for NIRSpec. Dithers and nods smaller than 2 arcseconds are not plotted. For NIRCam, only the SW quadrants are shown for clarity. For NIRSpec, only the active area of the MSA that was used for target placement, excluding regions that lead to truncated prism spectra, is shown. Some of the observations yet to be made in Cycle 2 (NIRSpec Deep/JWST and most Medium/JWST and their associated NIRCam parallels) have positions and orientations that are still to be defined based on scheduling. Outlines of other surveys, including the HST/ACS UDF and CANDELS, are shown with black and olive green curves. The smaller sub-panels show the same information split by instrument for clarity because it can be difficult to see the details when all observations are plotted together. The NIRCam sub-plot in upper-right additionally includes field outlines for the public JWST Cycle 1 NIRCam imaging from the JEMS, FRESCO and NGDEEEP programs. \n<!-- image --> \nJADES GOODS-South Field Layout \nJADES GOODS-North Field Layout \nFigure 4. Layout of the JADES observations in the GOODS-North field. Details as in Figure 3. GOODS-North data collection is nearly complete, so this figure is nearly final. Only the southern-most of the Medium/JWST pointings remains, due to a guide-star acquisition failure; this pointing will be repeated at an unknown location and position angle. The NIRCam sub-plot in the upper-right additionally includes the outlines of the public JWST Cycle 1 NIRCam imaging from the FRESCO program. \n<!-- image --> \nNIRSpec Medium/JWST data then followed in early May, at PA = 150.48 · . Due to a fault with acquiring guide stars, one observation was skipped and had to be redesigned and observed at PA = 132 . 93 · .', '3.4. Other Overlapping JWST Data': "The GOODS fields, UDF, and HDF have of course been observed by other programs in JWST Cycle 1. Here we briefly describe programs whose data overlap JADES. Figures 3 and 4 show the JADES NIRCam footprint with overlays with some of these and other programs. We surely expect this list to grow in future cycles! \nThe First Reionization Epoch Spectroscopic Complete program (FRESCO, PI: Oesch, Program 1895, Oesch et al. 2023) conducted NIRCam F444W slitless spectroscopy (2 hr depth), paired with F182M and F210M medium-band imaging (1 hr each). These 8-pointing mosaics in GOODS-S and GOODS-N overlap heavily with JADES. FRESCO has proven to be highly complementary to JADES, as the strong emission lines are imaged as excesses in JADES photometry and then redshifts are obtained in FRESCO. The additional mediumband imaging, while shallower than JADES, provides more spectral resolution on mid-redshift galaxies. \nThe JWST Extragalactic Medium-band Survey (JEMS; PIs: Williams, Tacchella, & Maseda; Program 1963, Williams et al. 2023) observed one pointing on the UDF in F182M, F210M, F430M, F460M, and F480M, with 4-8 hrs/filter. This dataset heavily overlaps with JADES, providing additional filters with compelling depth. In addition, JEMS conducted a NIRISS parallel in F430M and F480M, about half of which overlaps JADES NIRCam imaging. \nBetween these two, we note that F182M and F210M will be available for a notable portion of JADES. We have co-reduced JEMS and FRESCO NIRCam imaging with JADES, and include these data in JADES photometric catalogs. \nIn addition to FRESCO's wider medium-depth slitless spectroscopy, the Next Generation Deep Extragalactic Exploratory Public survey (NGDEEP, PI: Finkelstein, Program 2079) is producing very deep NIRISS slitless spectroscopy on the UDF (Bagley et al. 2023). We expect that these programs will complement the deeper but targeted NIRSpec MOS spectroscopy. The NGDEEP NIRCam imaging parallel falls off the JADES footprint, to the south-east. \nThere is also other deep MIRI GTO imaging in GOODS-S. Program 1283 (PI: Oestlin) is observing one extremely deep pointing in the UDF, reaching 49 hrs in F560W. The NIRISS parallel from this pointing falls \nat the southern edge of JADES NIRCam imaging. For wider MIRI coverage, program 1207 (PI: G. Rieke) is observing 15 pointings in 8 filters, all overlapping the JADES NIRCam imaging. \nThe push for deep imaging and spectroscopy in GOODS-S will continue in Cycle 2 with an approved GO program 3215 (PI: Eisenstein) that will return to the footprint of the JADES 1210 program to add very deep NIRCam imaging in six medium bands-F162M, F182M, F210M, F250M, F300M, and F335M-to refine the search for z > 15 Lyman dropouts. At the same time, ultra-deep NIRSpec MOS spectroscopy will be obtained on top of the JADES Deep Prime imaging. \nFinally, there are several other NIRSpec programs in the fields. Two examples are program 2674 (PI: Arrabal Haro), which conducted NIRSpec MOS and coordinated NIRCam imaging in GOODS-N that will complement the JADES footprint, and program 2198 (PI: Barrufet), which conducted NIRSpec MOS and NIRCam pre-imaging on two fields in GOODS-S. Other spectroscopic programs are the GTO NIRSpec WIDE MOS survey from programs 1211 and 1212 (PI: Luetzgendorf) and IFU program 1216 (PI: Luetzgendorf). Additional NIRCam pure-parallel imaging overlapping both fields is being obtained by the PANORAMIC program (PI: Williams, Program 2514).", '4. NIRCAM OBSERVATIONS': 'In this section, we detail the JADES NIRCam imaging. We begin by describing the details of the four categories of NIRCam imaging: Deep Prime, Medium Prime, Deep Parallel, and Medium Parallel. We then discuss cross-cutting aspects: filter selection, data quality, and a summary of depths.', '4.1. NIRCam Deep Prime': "In the heart of GOODS-S centered on the UDF, we observe 4 NIRCam fields to image a 4 . 4 ' by 6 . 1 ' rectangular field. A pair of NIRCam pointings is offset by ∼ 61 . 5 '' in V2 so as to cover the NIRCam inter-module gap, and each pointing contains many exposures including offsets in V3 to cover the shortwave chip gap. This pattern is then repeated in the second pair of pointings, offset in V3 leaving only a small overlap with the first pair. We size this offset to the height of the long-wave (LW) footprint, as this is slightly smaller than the shortwave (SW) footprint. We note that the mosaic was laid out for V3 PA of 300 · but observed at 298.56 · , causing a small cosmetic deviation from a rectangular layout. \nWe use the DEEP8 readout pattern with 7 groups, yielding individual exposure times of 1375 seconds. DEEP8 was used to reduce the data volume, but will \n<!-- image --> \nFigure 5. A small portion of the Hubble Ultra Deep Field, comparing HST and JWST imaging. ( left ) The F160W image from HST WFC3, from the Hubble Legacy Field reduction of Whitaker et al. (2019b), with an exposure time of about 65 hrs. This is the single deepest H-band image taken with HST. ( right ) The F150W JWST NIRCam image from the first year of JADES with an exposure time of about 10 hours. The superior depth and image quality of JWST is clear. \n<!-- image --> \nTable 1. Overview of the NIRCam Deep imaging in GOODS-S in different filters, listing the number of separate exposures and the total exposure time per pointing, in ksec. We note that some pointings overlap, doubling the depth, and that many Deep pointings also overlap with Medium, further increasing the depth. \nyield somewhat worse recovery from cosmic rays than a MEDIUM choice would have. In total, each pointing utilizes 114 such exposures, each with a SW and LW filter choice. Because each pointing includes more than 2 days of exposure time, we must split the observations into 3 visits. The first two use 9-point subpixel dithers \n(and no primary dither), each with 5 filter pairs, yielding 45 exposures. These two visits are offset in V3 by ∼ 5 '' to step over the SW chip gap. The third visit uses a mosaic of two pointings, each with 4-point subpixel dithers and 3 filter pairs, so 24 total exposures. The mosaic is designed with a row overlap chosen to result in the same V3 step and with the pointing chosen to result in the same footprint as the first two visits. Due to a mistake in correcting for a small PA shift, the third visit of pointing 2 is offset mildly, 3 '' , from the exact overlap. All filters are included in both 9-point subpixel dither visits and therefore most points in the image are observed by at least 18 different NIRCam pixels (9 if the location falls in the SW chip gap in one of the visits). \nThese observations are all taken with MIRI in coordinated parallel, the results of which will be described in § 6. Because of this, in the first season, we selected the subpixel dither based on F770W stepsizes. Having inspected these results, we concluded that a mildly larger stepsize would be better for background subtraction; we therefore changed to the F1500W dither pattern in year 2. \nWe utilize 9 filters in the Deep Prime pointings: F090W, F115W, F150W, and F200W on the SW arm, and F277W, F335M, F356W, F410M, and F444W on the LW arm. The exposure times per filter are listed in Table 1. It is important to note that about over a \nthird of the total field is covered by two of the pointings, doubling the total exposure times. Because of the importance of high-redshift dropouts, we slant the SW exposure times toward the F115W and F150W filters, while on the LW side we favor the longer filters where the zodiacal background is reducing the sensitivity. A comparison of the imaging we obtained in the NIRCam Deep Prime with that from the HUDF HST imaging is shown in Fig. 5. \nAlthough JADES was designed to be observed in one year, Cycle 1 scheduling constraints caused the program to be split. We opted to observe all 4 pointings in each year. In year 1, all pointings were observed with one of the 9-point dither visits, and two were observed with the 4-point dithers. This will be repeated in year 2, completing the observing. Unfortunately, this segmentation also caused a delay in the spectroscopic followup of the Deep imaging. However, it does create an opportunity to consider year-scale time variation in this nearly 25 arcmin 2 deep field. \nIn total, this part of the JADES program was an investment of 229 hours and resulted in 174 open-shutter hours of data, a utilization of 76%.", '4.2. NIRCam Medium Prime': "To provide shallower flanking coverage and increase the area available for NIRSpec MOS targeting, JADES includes 18 pointings in (mostly) regular mosaics yielding contiguous coverage to medium depth. 7 of these are in GOODS-N and 11 in GOODS-S. Because of the footprint, 8 of the pointings have MIRI in parallel, while 10 have NIRSpec MOS in parallel. In most cases, the pointings are paired with a 62 '' step in V2 so as to fill a long rectangle covering the inter-module gap, with some double coverage. In detail, the step is chosen to match the width of one SW chip, so that the pointings cover the V3-parallel chip gap of their partner. \nCycle 1 scheduling constraints caused half of the Medium Prime time in GOODS-S to be delayed until fall 2023. We opted to observe in 2022 the six pointings of the mosaic that did not overlap the NIRCam Deep Prime field, so that most of the prime area could be observed in fall 2022, including the footprint of the deep MIRI parallels. These pointings also provided NIRSpec MOS parallels on and around the UDF. The GOODS-S mosaic was designed for and observed at v3PA of 308 · . The small PA difference from the Deep Prime mosaic was included to make the program more easily schedulable. \nFor these six pointings, the filters are as in Deep Prime save for omission of F335M. Each of the 8 filters received 6 exposures, falling on 6 different pixels. The \nfilter pair of F115W and F444W received 1159 second exposures using DEEP8 and 6 groups. The other three filter pairings receive 945 second exposures; these use DEEP8 with 5 groups to reduce data volume. Unfortunately, one pointing failed due to shorts in the NIRSpec MSA; this one will be repeated in fall 2023. Another pointing had half of the LW imaging impacted by the shorts; we opted to accept this one and adjust a future pointing to compensate. \nAfter this start, and in view of the ongoing scheduling of the NIRCam parallel observations, we opted to make some minor adjustments. We had originally planned for 19 Medium pointings, but we decided to remove one pointing in GOODS-S and compress certain exposure times in GOODS-N in order to add F335M back into filter set for 11 of the 12 remaining pointings, pairing with a second exposure of F115W. We also had to adjust times to balance the program within its allocation, due to small on-orbit alterations in the parallel observation timing model. The detailed exposure times are presented in Table 2 for GOODS-S and Table 3 for GOODSN. Most exposures use the DEEP8 readout mode, but the shorter ones use MEDIUM8. In GOODS-S, relative to the original grid, we removed one pointing that overlapped the Deep Prime imaging and mildly compressed the time on a second that filled only a small gap between Deep and the rest of Medium. We then took the northern most row and displaced one pointing to compensate for the short-impacted year 1 pointing, and the other pointing to fill more of the gap between Deep Prime and the expected location of future Parallel imaging. In GOODS-N, we decreased the time in the 3 pointings with MIRI parallels; in one case we were unable to include F335M. \nFor the 7 pointings in GOODS-N, we pack four of them tightly in V2 to cover the intermodule gap. The other three have no intermodule coverage, and we further have chosen these to have less exposure time, as the MIRI parallels are more flexible than the NIRSpec MOS parallels. The northern portion of the mosaic is therefore somewhat shallower, but able to cover more of our early NIRSpec Medium/HST data. We decided that it was more important to maintain a 6-fold dither than to insist on a filled footprint, expecting that this field will likely attract larger coverage in future Cycles. The mosaic was designed and observed at v3PA of 241 · . \nRegarding the dither strategy, when operated with MIRI parallels, we use 3 dithers with the INTRAMODULEX pattern and 2-point subdithers. These were based on the MIRI F1800W PSF in GOODS-N and F2100W in GOODS-S, chosen to increase the dither step size for background subtraction. This dither uses small \nTable 2. Overview of the NIRCam GOODS-S Medium imaging in different filters, listing the number of separate exposures and the total exposure time per pointing, in ksec. We note that some pointings overlap, increasing depth. Pointing 4 (observation 22) with MIRI parallels is shorter than the others to balance the time within the allocation; this pointing is substantially overlapped by the Deep Prime mosaic. Pointing 25 of the NS-HST set lost half of the LW exposure time to illumination from a short circuit in NIRSpec. \nTable 3. Overview of the NIRCam GOODS-N Medium imaging in different filters, listing the number of separate exposures and the total exposure time per pointing, in ksec. We note that some pointings overlap, increasing depth. Pointing 3 (observation 3) with MIRI parallels is shorter than the other two to balance the time within the allocation. Pointings 4-7 are with NS-HST, while pointings 8-11 are with NS-JWST. \nsteps at 45-degrees in V2 and V3, stepping over both SW chip gaps in each. Hence most points in the chip gaps receive 4 SW exposures; only small overlaps from the cross in the middle generate only 2. We remind that in most cases, the center of each arm is covered by the other pointing in the pair. \nWhen operated with NIRSpec parallels, the strategy is mildly different. Here, we split the 6 exposures into two sets of 3. Each triplet is a different MSA design with largely independent targets, to be described further in § 5. In each triplet, NIRSpec will execute its 3-step nod along the slits of the MSA. We note that these steps are \nroughly at 45 · on the NIRCam pixel grid. For the next triplet, we step the central pointing purely in the V3 direction by an amount to cover the V2-parallel chip gap. The V3-parallel gap is not covered within this pointing, but usually will be by the partner in the mosaic. \nIn summary, this part of JADES was an investment of 195 hours, 123 in GOODS-S and 72 in GOODS-N, resulting in 131 open-shutter hours (with the formally prime instrument), a utilization of 67%. \nFigure 6. A small portion of JADES NIRCam GOODS-S imaging, combining F090W, F200W, and F444W filters in the Deep Parallel region of program 1210. This image shows the great diversity of galaxies revealed in every JWST image, with a wide variety of colors and morphologies. North is up. \n<!-- image --> \nIn GOODS-S, JADES executes two long NIRSpec MOS pointings ( § 5), each of 200 ks open-shutter spectroscopy. Each of these is used to make a long NIRCam parallel exposure, the location of which depends on the position angle and observing window, which in turn was subject to the programmatic constraints of the NIRSpec targeting. \nIt turned out, fortuitously, that the first of these pointings was at a position angle (V3PA of 321 · ) that caused the NIRCam parallel to fall on top of the deep MIRI parallels produced by the NIRCam Deep Prime program. \nThe second pointing needs to be later in the observing window so that the targets resulting from the analysis of the full NIRCam data set can be used. We are designing this for v3PA of 53 · , which will place the NIRCam parallel north of the Deep Prime field, near the location of the first Medium/JWST parallel. The exact placement will depend on the location of the most interesting high-redshift candidates in the Deep Prime field. \nEach of these two deep fields use the same nine filters as the Deep Prime program. We also use the same DEEP8 readout mode with 7 groups, yielding 1375 second integrations. Each field uses 144 such integrations for a total of 55 hours of open-shutter imaging, indeed mildly deeper than a single Deep Prime pointing (but without the overlaps of the mosaic). The exposure time per filter is shown in Table 1. \nOne limitation of these data is that NIRSpec only employs 9 dither locations, 3 nod locations in each of 3 different slit configurations. We were careful to arrange that each filter is observed at least twice at every location. In detail, the NIRSpec exposures are just over twice as long as the NIRCam integration, so each nod position results in a pair of back-to-back otherwise identical NIRCam integrations, 72 in all. We note that the dither pattern is set by the geometry of the NIRSpec MSA and therefore not tuned to the pixel scale of NIRCam; that said, the steps are not commensurate with the NIRCam pixel scale and so the intra-pixel behavior is sampled. \nAnother limitation is that the 3 MSA configurations in NIRSpec are stepped only a short distance ∼ 0.8 '' , to limit the effects of distortions across the MSA. This means that the data set does not fill the SW chip gaps. On the flip side, the data maximize depth in the region that is covered. An example of the superb quality obtained in this program is shown in Figure 6.", '4.4. NIRCam Medium Parallel': 'JADES executes twelve medium-depth NIRSpec MOS pointings, 4 in GOODS-N and 8 in GOODS-S, to be described in § 5. Each of these produces a NIRCam coordinated parallel observation, with a total of 45 exposures. The exposure times are given in Tables 2 and 3. \nAs these parallel fields were at risk to fall off of the HST GOODS and CANDELS imaging, depending on the final position angles, we opted to include the F070W filter in addition to the nine filters used in the Deep imaging. This improves isolation of z ∼ 5 . 5 Lyman α dropouts. \nAs with the NIRSpec Deep program, these observations also use 9 closely spaced dither pointings, via 3 nod locations in each of 3 slit configurations. The SW chip gaps are not covered. We ensure that each NIRCam filter is observed in at least 2 of the 3 slits and hence at 6 dither locations. 2 of the 3 slits have all ten filters observed; the remaining one is missing F070W, F200W, F335M, and F356W. Therefore, the area covered by all ten filters is reduced by a tiny amount. \nAs the prime spectroscopy will be placed on the NIRCam Prime mosaic, these Medium parallels in both GOODS-S and GOODS-N fall almost entirely outside of the NIRCam Prime footprint, thereby providing a substantial amount of additional area at medium depth. In GOODS-N, we observed at v3PA of 150.48 · and 132.92 · , placing the new imaging on the northeastern portion of the HST GOODS-N field. Because of the location of high-priority spectroscopic targets, these do not form a regular grid, but they are close enough to map a sizable near-contiguous region. By coincidence, the orientation of NIRCam in this Parallel imaging is almost 90 · rotated from the Prime imaging, leaving an obvious pattern for future observations to fill in the gap between the two. Unfortunately, a guide-star acquisition failure caused Observation 8 to be skipped, requiring a replan (Obs 98) that was observed a few weeks later with 18 · of rotation relative to the other three pointings. \nIn GOODS-S, we will spread the spectroscopic pointings over the full Medium mosaic. In the original plan, these would all have been late in the first observing window, but given the scheduling delay, we are opting to spread them out. The first field (observation 1) was observed on January 12-13, 2023, at a V3PA of 56.17 · . The other 7 will be observed in late 2023, likely at a range of position angles. Here, we assume V3 PA of 30 · for display and summary purposes. The exact parallel footprints cannot be specified until the targets are in hand, and may move substantially.', '4.5. Comments on Filter Selection': 'Early on in the design of JADES, we recognized that the strong increase in zodiacal emission longward of 4.5 µ m would increase the background in F444W, such that F410M could be competitive despite its narrower bandpass. We therefore opted to include both filters to increase the resolution of the spectral energy dis- \nribution. This is particularly important because of the strong rest-optical emission lines expected in some high-redshift galaxies, as indicated by Spitzer imaging and now confirmed with early JWST spectroscopy. The spacing of the H α +[NII] and H β +[OIII] complexes is such that only one can fall into F444W at any given redshift, and including F410M means that the comparison will separate the line from continuum emission at most redshifts (with ambiguity if the lines fall on the shoulder of the filter curves). This separation is important for stellar population modeling, as we want to measure the rest-optical continuum color relative to the ultraviolet. \nAs we studied this, we concluded that F335M and F356W offered a similar opportunity and that the likelihood of strong H α or H β +[OIII] emission recommended splitting this exposure time as well. This has been borne out in practice: we have found the abilty to isolate strong emission lines at 3-5 µ mto be very useful and interesting. In addition to the lines themselves, the extra spectral resolution helps to isolate the Balmer jump at high redshift and to measure the rest-optical continuum. \nWhere possible, we observe F277W somewhat longer than F356W because of the extra coverage from F335M. We also chose to slant the exposure time toward making F115W deeper, emphasizing the selection of z > 9 candidates. \nIn summary, we have found the NIRCam coverage in the nine base bands to be highly effective for photometric redshifts. For example, at z ≈ 7, one observes the Lyman α drop in F090W and H β +[OIII] in the longest bands.', '4.6. Data Quality Caveats': "While a detailed description of the data reduction and performance will be left for later papers (Robertson et al., in prep.; Tacchella et al., in prep.), we here describe some issues that we have already seen and that might be of interest to other users. \nOur observing was split into many separate visits rather than long campaigns, and we encountered substantial persistence at the start of some visits, left over from the immediately preceding program. These signals last for several hours and will require detailed modeling. The effect is more severe on the SW chips A3, B3, and B4; the other SW and both LW chips are much milder. \nMost dramatically, in the Deep program, observations 7 and 10, were observed immediately after observations of the bright Trapezium nebula, leaving substantial diffuse emission over portions of SW chips A3, B3, and A4. This is most severe in the F090W filter (the first used), but there are faint traces in the next filter (F115W), 3.5 hours later. \nWe stress that the persistence is not simply coming from bright stars in the previous images, but the change in the diffuse background illumination level from the previous program. This greatly increases the affected area, particularly in A3 and B4. The decay time for A3 is particularly long. Observation 4 in 1181 shows a similar morphology of persistence incurred by a change in the background level relative to the previous program. \nWe also see persistence in A3 in visits following wavefront sensing operations, creating a moderate-size ( ∼ 100 pixel across) hexagonal image. While such sensing is common, the reference star is planned to fall on a consistent part of the chip. We hope that the frequent reoccurence of this signal will allow that area of the chip to be particularly well characterized. 1180 observations 15 and 27 are affected by this. \nWe have also seen a case (1180 observation 18) in which a bright star happened to fall on the NIRCam field during the preceding MIRI observation, even though NIRCam was not being used in parallel. NIRCam is typically left open to the sky when not used. One can see the imprint of the whole MIRI dither pattern, as well as the trail when the telescope slewed away from the field. \nOf course, these regions do also incur persistence from brighter sources in our own observations. For instance, in 1181 observation 2, there is a bright star in the bad portion of A3 that creates a recurrent glow in all 6 dither locations. \nMitigating these persistent signals is particularly vexing when one is using a small-angle dither, as a pixel may never find a blank-sky location away from a larger galaxy. We are therefore increasing our sub-pixel dither selection in the remainder of our observations. However, this is not always possible when observing jointly with NIRSpec. We caution that we have not yet considered the effect of persistence on our photometry, but in the worst regions of these chips, we believe this should be studied. By construction, the dither pattern is repeated between filters, so the persistence from the first filter will affect the next. \nLike many NIRCam observations, we occasionally have noticeable illumination of the SW detector through an off-axis stray-light path, producing the so-called 'wisps'. These affect F200W and F150W most strongly. While these signals are known to modulate in amplitude due to the brightness of stars in the source region on the sky, we have found that the exact morphology of the pattern also varies within our program. We are still analyzing this, but hope that a low-dimensional set of templates will suffice to remove them. \nWe have found it very helpful to visualize the calibrated exposures of a dither sequence in animations, fixed in pixel coordinates so that the true objects move and the detector artifacts stay still. \nWe have found it easier to disentagle persistence from wisps when F090W or F115W are observed first in a visit, so that the persistence has decayed away before the wisp-affected F150W and F200W bands are observed. \nWe now turn to rarer problems. The second half of one pointing (observation 30) of the Medium Prime mosaic in GOODS-S had to be skipped due to an on-board issue unrelated to our program. There was not enough time in the observing window (constrained by the spectroscopic coordinated parallel) to try again. Fortunately, this pointing is at the edge of the mosaic. The first half of this same pointing had its LW data badly contaminated by a glowing short circuit in the NIRSpec MSA. We will repeat this entire pointing in October 2023 to complete the 6-dither coverage. \nOne half of another pointing (observation 25, first 4 dither sequences) was also affected by the NIRSpec glowing short (Rawle et al. 2022). In this case, because the location of the field was favorable to access in a different way in the year 2 observing, we opted not to repeat this imaging location but instead combine the NIRSpec re-do with another pointing. \nIn both of these cases, we found that the LW data were badly affected. The background was roughly doubled, but further there are many patterns of concentric rings, with spacing depending on wavelength, likely due to some diffractive pattern from where the light from NIRSpec has bounced off of the tertiary mirror. As the short circuit is apparently not particularly hot, the effect on SW is much less and we think this data is usable. There are a handful of faintly detected rings in F200W, chips A1, A2, A4, and B4. \nNext, in the 3 deep visits of 1210, we find an enigmatic set of arcsecond-scale blobs near an edge of B3. These are bright in the second visit, but detectable in the other two. They are therefore not due to persistence, and we are confident they are not astrophysical as they do not appear in 1180 images of the same region. The morphology is very different from a wisp. \nSome of our exposures have a plume of what appears to be scattered light in a corner of B4. We hypothesize that this may be due to a bright star striking the chip mask, as the signal changes slightly between the three visits of 1210, which move only at the arcsecond level, suggesting a well-focused source. However, we also see it in observations 27, 28, 29, and 30 of 1180, so it seems that the cause is not particularly rare.", '4.7. NIRCam Imaging Depth': "As shown in Figures 3 and 4, the NIRCam pointings often overlap, so that the survey is deeper than what appears in Table 1-3. Further, the depth varies because of the geometry of these overlaps. To provide a useful summary of the NIRCam program, we divide the footprint into 3 disjoint regions: Deepest, Deep, and Medium. Deepest refers to the area where two or more of the Deep Prime pointings overlap. Deep refers to the remainder of the area of the Deep Prime and Parallel fields. Medium refers to the rest of the area, including both GOODS-S and GOODS-N. In each case, we add up all of the exposure time in each filter, including the contribution of Medium pointings to the Deep regions, along with a corresponding point source depth from the Exposure Time Calculator. These areas and averages are approximate, as we do not yet have the final exposure locations and have not used an exact weight map, which would include the detailed impact of the dithers on the boundaries and the impact of bad pixels. We also remind that the exposure times do vary within the regions; for example, the northern portion of GOODS-N is shallower by a factor of 2-3 in exposure time than the bulk of the Medium region. Nevertheless, these summaries are reasonable averages for forecasts and contextual comparisons. \nWe then convert the representative exposure times to anticipated 10σ depths for background-limited 0 . 2 '' diameter apertures with point source aperture corrections. We do this by using the first pointing of 1286, as a representative and currently non-overlapped Medium pointing. The aperture error is derived from measuring many such apertures in blank regions of the mosaic and computing the rms in bands of exposure time. We note that the Deep survey was observed with longer exposure times and usually lower zodiacal background level, which will make it slightly deeper than the scaling from this 1286 pointing. However, the final mosaic will have mildly lower effective exposure time due to bad pixels and cosmic rays. \nWe remind that the Deep area is all in GOODS-S, while the medium area is split approximately evenly between GOODS-S and GOODS-N. The average depths in GOODS-N is mildly shallower than in GOODS-S, but the two fields are sufficiently similar that we do not separate them for this summary. \nIn Figure 7, we show a visualization of the estimated depth of the JADES, JEMS, and FRESCO NIRCam and JADES MIRI imaging, along with the estimates for HST ACS and WFC3 imaging (Whitaker et al. 2019b) in the HUDF and CANDELS fields that JADES overlaps. We have measured the JEMS and FRESCO depths in the \nFigure 7. Depth versus Wavelength for JADES and other data sets. Black, blue, and red solid squares show the 10σ point source depth for JADES Deepest, Deep, and Medium NIRCam data, using the variation in 0.2 '' blank-sky apertures. Purple and magenta circles show the depth of the JEMS and FRESCO medium-band imaging. Horizontal ranges show the filter widths. The JADES Deep and Medium MIRI depths are shown with grey and pink points on the right. The 0.2 '' aperture is mildly too small for the HST WFC3 image quality, causing these estimates of depth to be too optimistic by 0.1-0.2 mag compared to larger apertures. Comparison is shown (dark and light green) to the depths measured in the same method from the Hubble Legacy Field (Whitaker et al. 2019b) mosaics, separating the single HUDF pointing from the broader CANDELS region. The HUDF ACS (WFC3) footprint is 11 (4.7) square arcminutes, comparable to the area of the Deepest JADES data. The CANDELS area exceeds the JADES Medium area. JADES improves over even the HUDF in area and spatial resolution, and at wavelengths longward of 1.6 µ m, the gains in depth and resolution are immense. \n<!-- image --> \nsame manner as that of JADES, using our own reductions of these data. One sees that the JADES imaging is comparable in depth in the deepest HUDF data, which covered only a single HST pointing. One also sees that the ratio of optical ACS to infrared JADES depth is less favorable in the broader CANDELS region compared to that of the HUDF.", '5. NIRSPEC OBSERVATIONS': "As introduced in § 3, the JADES NIRSpec MOS data fall into 3 tiers: Deep, Medium/JWST, and Medium/HST. All 3 tiers use the low-resolution prism as well as several gratings, with grating spectra available for most of the prism targets. Table 5 provides a summary of disperser configurations and exposure times for each tier. This section presents some of the features common to all tiers before describing each tier separately. \nTable 4. A Summary of Average Exposure Times and Depths in the NIRCam Deep and Medium Surveys. The 6 Deep pointings cover a total footprint of ∼ 45 arcmin 2 , with ∼ 9 of these being double-covered and marked as Deepest. The 31 Medium pointings are anticipated to cover a total additional footprint of ∼ 175 arcmin 2 exclusive of the Deep footprint. Within each, we compute the average exposure time per filter, including the contribution of Medium to the deeper regions. We then present the 10σ 0 . 2 '' diameter aperture depth, including point source aperture correction, for these average exposure times, scaling from observed errors in the first observed pointing of 1286 as described in the text. a The F070W filter is used in only a subset of Medium pointings, covering an area of ∼ 76 arcmin 2 . We quote the average exposure time and depth in this smaller area. b The F335M filter is not used in some Medium pointings, so that this filter covers a Medium footprint of ∼ 134 arcmin 2 . We quote the average exposure time and depth in that smaller region. \nEach of the NIRSpec gratings are used with a matching long-pass filter to prevent overlap of first order spectra by higher orders. For the band 1 G140M grating there are two available filters; F070LP and F100LP. F100LP blocks all light below 1 µ m, thereby ensuring no second order overlap within the nominal spectral range up to 1.8 µ m. F070LP allows through light at > 0 . 7 µ m, so it has the advantage of enabling observation in the range 0.7 to 1.0 µ m. This corresponds to the wavelength of the Lymanα transition at redshifts of 4.8 to 7.2, bridging the epoch of the end of cosmic reionization (Robertson 2022). Therefore we chose to use the F070LP filter for all the JADES G140M spectroscopy. We accept that there will be second order overlap from the sky, increasing the sky background at > 1 . 4 µ m, and from galaxies at redshifts below 7 that have flux in this wavelength region. However, much like the overlapping grating spectra allowed by our MSA configurations described in the following subsection, this increased continuum flux will have little impact on our emission line measurements from the G140M spectra.", '5.1. MSA Configuration Design': "JADES designed its NIRSpec multi-object observations using the tool eMPT (Bonaventura et al. 2023) that provided key features beyond what was found in the baseline tools for MSA design. In particular, eMPT allowed us to: \n- 1. Constrain the NIRSpec pointings to a rigid mosaic of NIRCam fields, once the position angle is specified.\n- 2. Impose a detailed prioritization system for our targets and have complete control over the order in which each class of targets is attempted placed on the MSA at a given pointing.\n- 3. Optimize repeated observations across multiple overlapping MSA designs to maximize exposure time on the highest priority targets.\n- 4. Identify and eliminate beforehand targets having contaminating objects falling within their (nodded) slitlets.\n- 5. Avoid the use of shutters leading to prism spectra truncated by the NIRSpec detector gap or contaminated by the spectra of failed open shutters.\n- 6. Enable overlap of the grating spectra (except for some high priority targets whose grating spectra are protected from overlap) to maximize the gratings multiplexing, while keeping those of the prism distinct.\n- 7. Open additional blank-sky shutters that disperse onto unused detector real estate to support master background subtraction. \nTo do this, eMPT contains the full NIRSpec model of the astrometric distortions and multi-shutter geometry \nand constraints, from which it can accurately predict how given astrometric positions will fall onto shutters, whether those shutters are available to use, and where the resulting spectra will fall on the detectors (and thereby whether they will overlap). The code thereby revealed the detailed outcome of each target, with which one can proceed to accept targets in complex priority orders. After this process determines which shutters were to be opened, the final optimal pointings and matching MSA masks were imported into the standard APT/MPT workflow for further execution. \nThe MSA shutters are on a rigid grid, and the opaque regions between the shutters block enough light from compact sources that one typically chooses to retain only the fraction of targets that are sufficiently well centered in their shutters. Not all shutters function properly with 22 failed open that always disperse light onto the detector and 17.5% of the unvignetted shutters that are permanently closed (Boker et al. 2023). Our MSA masks require a 3-shutter-high slitlet for nodding and background subtraction. Locations where one can open a 3-shutter-high slitlet are therefore limited by this MSA operability leading to the concept of a 'viable slitlet' map. In all the JADES tiers we perform MSA reconfigurations with small (always < 10 and mostly < 1 arcsec) offsets where we attempt to obtain spectra of at least the highest priority targets in multiple configurations. Simulations have shown that these offsets need to be kept small to maximize the overlap of the viable slitlet maps and to avoid the astrometric distortion at the NIRSpec MSA plane that cause some objects to become insufficiently centered. For most tiers of JADES, the main constraint on these offsets is to ensure maximal coverage of the highest priority targets in multiple MSA configurations. \nBecause of these constraints, one needs a very high target density in order to achieve a high multiplex of assigned targets. JADES typically supplies at least 200 targets per square arcminute, yielding about 150 assigned targets on the prism designs, corresponding to an average assignment rate of only 9%. An obvious consequence of this low average rate is that one does not want to serve the rarer higher-value targets in this limited way. We therefore developed a detailed prioritization of the targets, largely by redshift and flux (see § 5.2). The eMPT allows us to assign slits in a greedy order, assigning higher priorities first. While this slightly decreases the total multiplex, it yields much higher assignment rates on high value targets. \nNIRSpec prism spectra are relatively short, allowing multiple columns of non-overlapping spectra. For a given prism MSA design, there is a matching grat- \ning MSA design that is nearly identical. Importantly, we keep nearly all of the same shutters open for the grating configuration so that most of our galaxies will have information at multiple spectral resolutions. The longer traces of the gratings may overlap, as may the zero-order emission, but the emission lines are sparse in these spectra. Emission lines can be associated to their parent object in multiple ways: the location along the three shutter tall 1 . 5 '' slit, the prism spectrum, where the lines appear unoverlapped, and the wavelength ratio of multiple detections. The dispersed continua of the typical faint targets is below the detector noise in the grating spectra and hence the continua of the overlapping spectra do not substantially increase the noise. For the highest priority targets and for the infrequent brighter targets, we do close some shutters to avoid overlap; so a small fraction of objects are observed only with the prism.", '5.2. Target Prioritization': 'Here we summarize the design of the JADES spectroscopic target selection process. The main criteria for placing galaxies into priority classes are their redshifts and fluxes. Redshifts are estimated via photometric redshift algorithms and/or Lyman-break color selection. The highest redshifts are prioritized both because they are rare and because one of the main scientific goals of JADES is to understand the earliest phase of galaxy evolution. Galaxies with higher fluxes (in continuum or predicted emission lines, depending on the category) are prioritized since higher S/N spectra allow a wider range of science investigations. \nThe details of the priority classes depend on the tier (Deep, Medium/JWST or Medium/HST) and whether the targeting is based only on HST imaging or on the JADES imaging. Deeper spectroscopic observations have lower flux limits for similar classes so that the achieved S/N will be similar for the different tiers. \nThe highest priority class contains relatively bright galaxies at the highest redshifts; z > 8 . 5 for Deep and Medium/JWST and z > 5 . 7 for Medium/HST program. After these, we prioritize fainter galaxies at the same redshift and then progressively lower redshift bins, favoring the brighter galaxies. Through this, we aim to build up a statistical sample between redshift 1 . 5 < z < 5 . 7 in the lower priority classes over the tiers, with the shallower tiers contributing to the bright end and deeper tiers contributing to the faint end. In addition to these classes we also include a small fraction of galaxies identified as special in other data, e.g. with ALMA, Chandra, Lymanα emitters selected with the ESO VLT/MUSE instrument. \nTable 5. Summary of the NIRSpec MOS Observations. For each program, we list the number of separate MSA fields, as well as the exposure time per disperser in kiloseconds. Each field consists of 1 to 3 sub-pointings, each with two nearly identical MSA designs: one for the prism and a second for the grating; the latter closes a few shutters to protect certain high-priority spectra from overlap. The quoted times are summed over the sub-pointings, but not all targets can be placed on all sub-pointings. The number of unique targets in each subsurvey is listed; bold values indicate completed observations, whereas other values are estimates at this time. The long-pass filter choices for the gratings are F070LP, F170LP, F290LP, and F290LP, respectively. a Each Medium/HST pointing is split into two distinct MSA locations, separated by the primary dither step across the NIRCam SW chip gaps. As the larger dither causes us to typically observe distinct galaxies, we account each pointing as two fields with one sub-pointing. When the slit registrations are favorable, we do reobserve high-priority targets to double the exposure times on these. b Twelve MSA locations were planned, but only 4 were completed in cycle 1 due to instrument problems. 3 will be redone in cycle 2. c These 5 re-dos were organized to be similar to two Medium/JWST locations, one with 3 sub-positions and one with 2. \nWe also prioritize a few bright ( H AB < 23 . 5) moderate-redshift ( z > 1 . 5) galaxies, enabling the collection of exquisite infrared spectra from objects around cosmic noon. After these, we prioritize in photometric redshift bins, favoring the rarer brighter examples. Galaxies with photometric redshifts below z < 1 . 5 are used as a low priority filler sample; nevertheless, the large number of these targets yields a substantial observed set. Further details on the target prioritization, including variations per tier, are provided in the JADES Deep/HST data release paper (Bunker et al. 2023a). \nThe distribution of spectroscopic redshift and F444W magnitudes are shown in Figure 8. One sees that NIRSpec is recovering redshifts to extremely faint flux levels, particularly in the Deep pointing.', '5.3. MSA Target Acquisition': 'Successful use of NIRSpec MOS depends critically on high-quality astrometry. Astrometric distortions in the target coordinates will tend to perturb targets away from the centers of their shutters, lowering performance. More insidiously, NIRSpec MSA Target Acquisition relies on a few bright compact sources to align the MSA to the desired location on the sky, so astrometric errors on those few objects can cause the entire MSA to be misaligned. \nJADES was fortunate to be able to utilize recent reductions of HST imaging in the GOODS fields that had been aligned to the Gaia DR2 reference frame (G. Bram- \nmer priv. comm., Gaia Collaboration et al. 2016, 2018). This alleviated concerns of distortions or mismatches between faint targets and bright acquisition sources. \nHowever, the GOODS-S and GOODS-N fields, being intentionally placed in regions of very low stellar density, do present a severe deficit of stars suitable for target acquisition. Instead, we had to use the HST imaging to identify compact galaxies, using the longest exposure time for the acquisition image to reach down to 24-27 mag. All the JADES target acquisitions so far were successful, although the increased centroiding error for such extended targets may somewhat have reduced the alignment accuracy of the NIRSpec observations. In later spectroscopy, when NIRCam data was available, we used its imaging to select roughly circular compact galaxies and stars, using the multi-band near-IR photometry to more confidently estimate the fluxes in the NIRSpec CLEAR and F140X target acquisition filters and to enforce isolation criteria.', '5.4. NIRSpec Deep Spectroscopy': 'JADES features two long NIRSpec multi-object observations, each of 200 ksec total exposure time and both located in GOODS-S on the UDF and mostly inside the NIRCam Deep Prime footprint. The first pointing was designed to be observed early and be targeted using HST GOODS and CANDELS imaging, supplemented with other pre-JADES data. The second deep pointing will be observed at the end of the program and will use tar- \nFigure 8. The observed distribution of F444W AB magnitude versus spectroscopic redshift for the Deep/HST pointing (top) and the first Medium/JWST pointing (bottom). The target selection has successfully weighted toward higher redshift galaxies, resulting in a more even redshift distribution. \n<!-- image --> \ngets from the full JADES imaging. We refer to these two pointings as Deep/HST and Deep/JWST, respectively. \nThe NIRSpec Deep/HST observations occurred over three visits between 21 and 25 October 2022 at a V3PA of 321 · . As described above, these observations were intended to be targeted solely on pre-JWST data. Shortly before the scheduled visits, NIRSpec suffered some shorts on the MSA that required us to replan the MSA configurations for these observations. The NIRCam Deep Prime data were taken at the start of October, and an early reduction of the multi-band images and photometry catalogs were available. This enabled us to include some JWST-selected targets, at the expense of lower priority HST-selected targets, in the NIRSpec \nobservations. We also re-prioritised the catalog using the NIRCam data to improve high-redshift photometric redshift accuracy, and to homogenize the selection in the lower priority classes with respect to the Deep/JWST to be based on the F444W filter. We note that the pointings of the observations were not changed, only the choice of which shutters to open. Two of the additional target galaxies with photometric redshifts from the NIRCam imaging (Robertson et al. 2023a) were spectroscopically confirmed in Deep/HST to lie at the highest redshifts known of 12.6 and 13.2 (Curtis-Lake et al. 2023). \nThe NIRSpec Deep spectroscopy utilizes five dispersers: the prism and four gratings: G140M/F070LP, G235M/F170LP, G395M/F290LP, and G395H/F290LP. The prism is observed for 100 ksec, and the gratings for 25 ksec each. \nEach pointing uses slitlets of 3 shutters, with the 3point nod. To provide additional pixel diversity and some dithering in the spectral direction, we design 3 subpointings for each pointing, typically separated by 3-5 shutters in the dispersion direction and 1-2 shutters in the spatial direction such that the optimal common coverage of the highest priority targets in all three dithered pointings is achieved. This results in the target light from a given wavelength for the majority of sources appearing in up to 9 pixel locations. As described earlier, there are actually 2 MSA designs for each sub-pointing (6 in total) because we use a separate configuration for the prism relative to that of the gratings. Each integration uses NRSIRS2 readout with 19 groups, yielding 1400 second apiece. We conduct two integrations per exposure. For the gratings, this means that each nod location is visited only once for 2 consecutive integrations. For the prism, the telescope repeats the nodding 4 times, with 2 integrations per time. \nAs discussed in § 5.1, it is inevitable that not all targets can be placed on all three sub-pointings, even though this is our preference. We accept this and fill in some targets with only 1 or 2 sub-pointings. Using a smaller dither step increases the ability to repeat targets. \nThis part of JADES is an investment of 145 hours and results in 111 prime open-shutter hours, a utilization of 76%. The exceptional line-flux sensitivity achieved as a function of wavelength is shown in Figure 9.', '5.5. NIRSpec Medium/JWST Spectroscopy': 'JADES includes 12 medium-depth pointings that are targeted from JWST imaging, but are otherwise scaled down versions of the Deep pointings. Relative to Deep, the prism is scaled down more than the gratings, reflecting the goal of studying the galaxy spectra in more \ndetail. Exposure times are listed in Table 5. The original plan was 8665 sec in each of the 5 dispersion modes. However, small on-orbit changes in the timing model for parallel observations caused us to make small adjustments in the exposure times to be more efficient with the NIRCam parallels and to fit the program into the allocation. \nAs with Deep, the observing uses 3-point nods with 3-shutter slits at each of 3 sub-pointing locations, for a total of 9 pixel dither locations. Each of these exposures is a single integration, using NRSIRS2 readout with 12 or 14 groups. As for the Deep pointings, some targets can only be placed on one or two sub-pointings, resulting in proportionally lower exposure time. \nFour of the twelve Medium/JWST pointings are placed in GOODS-N and eight are in GOODS-S that has wider JADES NIRCam imaging. Three of the GOODSN pointings were observed between April 30 and May 5, 2023 at V3 PA 150.48 · , covering a large fraction of the NIRCam Medium Prime mosaic. The fourth was delayed by an observatory failure to acquire guide stars and was observed at V3 PA 132.93 · on May 27, 2023. \nUnfortunately, this final JWST/Medium observation in GOODS-N was affected by a short circuit in the NIRSpec MSA for some configurations, which when triggered produced a glow of light that flooded the detectors, ruining the NIRSpec data (Rawle et al. 2022). Half of the configurations were affected by these shorts (two of three prism and one of three grating). The other three configurations did not address the susceptible column of the MSA and therefore did not have the glow. The possible effects on the NIRCam coordinated parallel imaging data are still being investigated at this time. \nFor GOODS-S, four pointings are in the NIRCam Deep Prime mosaic, with the other four on the Medium Prime mosaic. Most of the GOODS-S Medium/JWST was delayed until Cycle 2, save one pointing observed on January 12 and 13, 2023, at V3 PA 56.17 · . The exact pointing positions and orientations of the remaining seven observations are still to be determined. The pointings will be based on the highest priority targets from the NIRCam Deep Prime, NIRCam Medium Prime and first NIRCam Deep parallel observations. \nThis part of JADES is an investment of 222 hours, 77 in GOODS-N and 145 in GOODS-S, resulting in 146 prime open-shutter hours, a utilization of 66%. As before, the line detection sensitivity as a function of wavelength is shown in Figure 9.', '5.6. NIRSpec Medium/HST Spectroscopy': "The final tier is the shallowest and results from 'parallel' observations during ten of the NIRCam Medium \nPrime fields, four in GOODS-N and six in GOODS-S. We remind that our naming convention is following the instrument that is driving our science design; in all coordinated parallels including NIRSpec MOS, NIRSpec is formally the prime instrument. The HST name refers to the fact that in the survey design, the JWST imaging was not yet available and hence the targeting was from HST data. We will describe explicitly the few cases we could use JWST-based targets, due to interruptions in the program. \nAs the imaging program requires a large 7 '' offset to step over the V2-parallel SW chip gap, we opt to split each parallel opportunity into two largely distinct sets of targets. Offsets of this size when combined with the NIRSpec astrometric distortion at the MSA plane would otherwise cause many targets to become poorly registered within their slits. Each pointing in the mosaic was given a small 1 '' freedom of motion to optimize the slit centration of a few highest priority targets. \nFor each of these 20 target sets, we use 4 dispersers, omitting the G395H higher resolution grating. Exposure times are listed in Table 5. GOODS-S was observed first, and here we used NRSIRS2 readout with 17 groups for the prism and 14 groups for the gratings, each observed once at each of 3 nod locations. Based on the first tranche of data in GOODS-S, we replanned GOODS-N so that the prism was observed twice with 14 groups at each pointing, to increase the S/N. \nAs this program results from the parallels of a reasonably tightly packed imaging mosaic, these NIRSpec MOS fields overlap substantially. However, any given object in the footprint of a single MSA pointing will often be poorly centered in its possible shutter, leading to many targets being rejected due to the low expected throughput. Collisions of prism spectral traces block many other targets. Therefore, several returns to a given area can be supported without much duplication. We do allow our targets to be observed twice (and our highest priority targets up to four times), if the slit registration is favorable. \nOur observations of the GOODS-S portion of this mosaic (observations 25-30 of program 1180) were affected by two short circuits in the NIRSpec MSA, similar to those described in the previous section. Some configurations did not use these columns of the MSA and therefore did not have the glow. The NIRCam coordinated parallel imaging data was unaffected in most cases, but the first half of observations 25 and 30 suffer from a short so bright that the illumination even reached NIRCam, as described in § 4.6. \nOnly 4 of the 12 target sets were successfully observed without a bright short glow, and even in these cases \nFigure 9. The unresolved line detection limit as a function of wavelength for two JADES NIRSpec tiers. We assume a well-centered point source and use a 10 σ detection threshold. The blue long lines are the prism, while the orange, green, and cyan lines are the three R = 1000 gratings. The solid lines show the depth in the Deep/HST pointing, while the dashed lines are a representative Medium/JWST pointing. For an unresolved line, the G395H grating has a similar line detection limit to G395M plotted here. \n<!-- image --> \nthere is some persistence that affects the prism exposure in a small portion of the image. These are MSA configurations 2, 4, 6, and 10, which are the second half of observations 25, 26, 27, and 29, respectively. For a 5th target set (configuration 5, the first half of observation 26), the grating exposures are unaffected, but the prism exposures were flooded by the short. We plan to complete this 5th set with a return in October 2023. \nOne of the target sets was not observed due to an unrelated telescope issue (configuration 12, second half of observation 30). As the first half of observation 30 was badly affected by shorts, we will perform a complete repeat of this pointing in October 2023 at the original position angle, albeit with a different set of targets based on NIRCam selection. \nThe other 5 target sets were re-observed on January 27 & 28, 2023, without new NIRCam parallels. Because the new position angle was already going to require a complete re-plan, we opted to collect the 5 single-nod designs into two pointings with smaller dithers, akin to the Medium/JWST program. Observation 134 has two subpointings; observation 135 has three. For these, we use the same NIRCam-based target selection that was used for the first Medium/JWST observation in GOODS-S.", '5.7. Data Quality Caveats': 'The NIRSpec MOS observations have been processed by using a pipeline developed by the ESA NIRSpec Science Operations Team and the NIRSpec GTO Team. The major steps of the data processing are described in (Bunker et al. 2023a), while a detailed description of the pipeline and its performance will be reported in a forthcoming paper (Carniani, in prep.). Here we discuss the main issues encountered during the data processing and analysis phase. \nAs mentioned in the previous sections, program 1180 was affected by short circuits in the NIRSpec MSA that produced bright glows of artificial light, ruining most of the exposures. In particular, 76 out of 132 exposures are ruined by such a bright glow. The other exposures did not suffer from this problem, but we found some persistence for the prism exposures that contaminate the spectra of ∼ 10% targets in the MSA design. In some cases, the signal of the persistence is as high as the sky background emission. \nBy inspecting the count rate maps before background subtraction, we noticed that some shutters dedicated to the targets did not open (Rawle et al. 2022). We excluded these temporarily failed shutters from the data processing workflow. In each pointing, we found, on average, that ∼ 1% of the targets in the MSA masks are affected by this issue. In most cases, only one of the 3-slitlet shutters was unexpectedly closed, but there are the same targets in which two or even all three shutters did not open during the observations. In these cases, the noise of the final products increases as the total exposure time is reduced. \nAlthough the target selection was optimized to adopt the 3-point nod strategy for the background subtraction process, some background shutters were contaminated by either background or foreground source. We have thus exploited either HST or NIRCam images to identify automatically contaminated shutters and excluded them in the background subtraction steps of the pipeline. This increases the noise of the background subtraction image but avoids a possible over-subtraction of background emission that could alter the final spectra of the targets.', '6.1. MIRI Deep Parallel': "The NIRCam Deep Prime program creates very deep MIRI parallels in GOODS-S, totaling 43.1 hrs of openshutter time in each of the four fields. The fields overlap only slightly, so that the deep area is about 9 arcmin 2 . \nThe fields were designed to overlap the NIRCam Medium Parallel mosaic, but it turned out that one of the NIRCam Deep Parallel fields substantially overlaps \nTable 6. Summary of MIRI Observations. The number of pointings, area in arcmin 2 , exposure times per filter, and the final AB magnitude detection threshold per filter for a 10σ point source. For GOODS-N Medium/MIRI and GOODS-S Deep/MIRI, these thresholds are based on or extrapolated from already obtained observations assuming a 0.4 '' aperture and aperturecorrected for both filters. The GOODS-S Medium/MIRI values are projected from GOODS-N Medium/MIRI. Note that the Deep/MIRI pointings partially overlap those of Medium/MIRI, but we do not coadd these exposure times when computing the depth. a One of the GOODS-S pointings is mildly shallower, with 5.6 and 21.5 ks of exposure in F770W and F1280W, respectively. b One of the GOODS-N pointings is mildly shallower, with 6.6 ks of exposure in F1280W. c To date, 61-94 ks have been obtained per pointing of GOODS-S Deep/MIRI. \nFigure 10. RGB (F770W, F4442W, F335M) image of a trio of z ∼ 6 photometric redshift candidates detected in the GOODS-S JADES Deep/MIRI parallel (24-25.6 AB with SNR ∼ 20 -80 in F770W). All three are likely emission line galaxies, with H α in F444W. The leftmost candidate additionally has significant [OIII]5007 emission in the F335M medium band filter, resulting in a more purple color. The F770W probes the rest-frame 1 µ m emission, a powerful constraint on the properties of older stellar populations. \n<!-- image --> \nas well. This allows these MIRI images to provide a very deep look at the high-redshift universe. We note that while the MIRI fields are not on the UDF itself, they do fall near the center of the Chandra Deep Field South. \nWe decided to focus this time to the study of restframe 1-2 µ m imaging of galaxies at redshifts above 3. For this, we selected the F770W filter as the most promising compromise between the rising background to the red and the lever arm relative to the deeper F410M \nand F444W data. An example of robust F770W detections of a trio of z ∼ 6 photometric redshift candidates in the GOODS-S Deep/MIRI footprint can be seen in Figure 10. F770W is somewhat less sensitive in AB magnitude than F560W, but it more than doubles the logarithmic wavelength gap relative to 4.4 µ m, allowing it to better detect power-law deviations in the slope of the near-IR SED. The longer band is also less likely to suffer from rest-optical emission line contamination of the continuum light measurement; H α will enter F560W at 6 . 6 < z < 8 . 4, which might be at the frontier of detectability with these long exposures. \nThe MIRI data must of course follow the NIRCam exposure times and dither pattern. To reduce data volume, we use SLOWR1 readout mode with 57 groups to yield a single integration of 1361 seconds per exposure. Each pointing then has 114 of these exposures, taken at 22 different dither points, for a total of 155.2 ksec. However, on short time scales, the MIRI data is taken cycling through 9 or 4 subpixel dither locations. The resulting 10 σ point source sensitivity is 27.1 AB (Table 6).", '6.2. MIRI Medium Parallel': 'In addition to the deep data, we conduct MIRI parallels with eight of the NIRCam Medium Prime pointings. Five of these are in GOODS-S and three in GOODS-N. \nAs these data are considerably shallower and yet only mildly more area, we opt to focus on the science of intermediate-redshift galaxies ( z ∼ 3 -5), where we can place strong constraints on the stellar emission SED, such as the regime of the contribution from TP-AGB stars, robustly identify the rising continuum associated with AGN, and look for unusual SEDs. To accomplish these goals, we take moderately deep exposures in F770W and then use most of the time in the F1280W, which gains in sensitivity over the WISE W3 band by a factor of ∼ 1000. We note that some but not all of \nthe medium area in GOODS-S is also covered by the far deeper F770W imaging from the Deep parallels. In GOODS-N, 2/3 of the MIRI pointings fall off of the planned NIRCam coverage, but are covered by CANDELS. \nThe exposure times are listed in Table 6. In all pointings, the data set uses 6 dither locations. The dither locations have 3 close pairs with relatively long strides between them, and hence there is a relatively large boundary region that has only 2 or 4 exposures. However, since MIRI is Nyquist sampled, this was considered acceptable. We always use the SLOWR1 readout, so as to reduce the data rate. F770W uses 1 integration per readout; F1280W uses 2 or 3 to avoid saturation on the background. In GOODS-S, where the available exposure times are longer, we observe F770W once per dither location and return to F1280W four times. In GOODS-N, our exposures are shorter and we do not have any coverage from the Deep MIRI parallels; we therefore opt to include two exposures each of F770W and F1280W per dither location.', '6.3. Data Quality Caveats': 'So far, we have not encountered any substantial concerns with the MIRI data acquired. We did find that the subpixel dither pattern used in the first Deep data, based on the F770W PSF size, was smaller than we would have preferred to use for the generation of sky flats. We find this can be mitigated by generating sky flats from roughly contemporaneous exposures over multiple pointings. Nevertheless, we have increased the dither steps in later observations. Moreover, the return to the same Deep field in year 2 will improve the pixel sampling, so we expect this will not be a lasting concern.', '7. PREPARING FOR JADES': "Like many JWST observing programs, the JADES team engaged in substantial preparations for the data set. We were particularly driven by the tight time scale, likely at most 6 weeks, to provide targets for multiobject spectroscopic follow-up from the NIRCam and MIRI imaging. This central goal of the program requires image reduction, mosaicing, source detection, source photometry, photometric redshift generation, target selection, and MSA design to be ready to run in quick order. We also sought to prepare for analysis of the spectroscopy, most obviously for the data reduction, extraction, and spectral analysis, as clearly the spectroscopic results would be desired to feed into the processing of the second field to be observed. \nPart of this preparation was inherent in the needs of the instrument teams to support a wider range of \ncommissioning and early science observations. Most obviously, we rely on the effort to develop exposurelevel reduction software such as NCDhas written by K. Misselt and the pre-processing pipeline developed by the ESA NIRSpec Science Operations Team (Birkmann et al. 2022), which subsequently became the basis for the STScI stage 1 and 2 pipelines. To build and validate these tools pre-launch required creation of detailed codes to simulate instrument data: Guitarra 2 for NIRCam and the NIRSpec Instrument Performance Simulator (IPS; Dorner et al. 2016). \nFor NIRCam, we used Montage 3 to combine the individual exposures into a mosaic. In order to identify outlier pixels (such as cosmic rays that have not been picked up by NCDhas), we first construct a medium-based mosaic, which we then projected this medium-based mosaic back to the individual exposures. We identified and masked outlier pixels that are 3 σ outliers. In a final step, we constructed the mean-based mosaic and fully propagated the errors. \nAn important additional aspect of JADES preparation included optimizing our instrument configurations, integration times, and area to maximize our key science goals (e.g. bottom panel of Figure 4). To this end, we developed JAGUAR, a novel phenomenological model of galaxy evolution out to z ∼ 15 (JAdes extraGalactic Ultradeep Artificial Realizations; Williams et al. 2018) 4 , incorporating known galaxy abundances, flux, color, and morphology relations across redshift. A key utility of JAGUAR includes both mock SEDs and full-resolution spectra, which we generated using BEAGLE (Chevallard & Charlot 2016) based on selfconsistent models of stellar radiation and its transfer through the interstellar and intergalactic medium. Beyond survey design, JAGUAR also enabled simulation of realistic galaxy fields using mock imaging tools like Guitarra, mock NIRSpec spectra and optimizing our MSA design procedures. \nWith these tools, JADES then performed data challenges, simulating mock galaxy fields down to individual ramps and then reducing the data to make high-level products. For NIRCam, this included mosaicing, object detection, photometry, and photometric redshifting. For NIRSpec, mock target lists were used to build MSA designs with the eMPT code (Bonaventura et al. 2023), and simulated spectra (Chevallard et al. 2018) were run through reduction and extraction to develop tools for \nredshift and line flux estimation (Giardino et al. 2019). A key advantage of these data challenges was to define data models for the interfaces between segments of the analysis. They also allowed us to generate test suites to validate each segment. \nJADES conducted Data Challenge 1 (DC1) to initiate this process. The NIRCam component of DC1 concentrated on a single visit of Proposal 1180 (Observation 7, Visit 2), which is part of the NIRCam Deep Prime pointings. The DC1 simulations used 9 ramps with 7 DEEP8 groups for all filters used in the original JADES deep survey design (F090W, F115W, F150W, F200W, F277W, F356W, F410M and F444W). The source catalog used for DC1 was derived solely from the JAGUAR mock catalog, where we assigned random positions in R.A and declination for the selected galaxies but in such a way to maintain the same surface density of galaxies as the JAGUAR parent sample. \nThe NIRSpec component of DC1 was based on the NIRSpec Deep/JWST program. 200 point source galaxies with realistic JAGUAR spectra were simulated in the prism and medium resolution grating dispersers in each of the three Deep/JWST pointings. The data were processed with the NIRSpec IPS Pipeline Software (NIPS; Dorner et al. 2016). The resulting output was simulated, flux-calibrated, combined spectra for 370 galaxies, used to validate our choice of integration times for these faint targets (Giardino et al. 2019). \nFollowing improvements in the code base, we then conducted Data Challenge 2 (DC2) as a set of more comprehensive exercises. DC2 covered about 2/3 of the area of the JADES survey in GOODS-S ( ∼ 80 arcmin 2 ) and included Deep and Medium NIRCam pointings, using the same setup as the planned observations (e.g., exposure time, read out mode, dither positions) as recovered from the APT file. In contrast to DC1, the DC2 simulations included the field-of-view distortions, particularly important to test the ability to make astrometrically correct mosaics in sky coordinates and accounting for pixel sub-sampling when constructing these mosaics. Cosmic-ray hits were also added to the individual ramps. The DC2 sample used a combination of the CANDELS (Grogin et al. 2011; Koekemoer et al. 2011) catalog with S'ersic parameters estimated by van der Wel et al. (2012) and objects from JAGUAR, which also provides S'ersic parameters. The latter enabled including objects beyond the apparent magnitude limit and redshift cutoff of the HST data. In this process, we used the positions and shapes of all observed galaxies and supplemented these with mock galaxies where a fraction of the latter were included until the counts in apparent magnitudes and redshifts were as close as \npossible to the JAGUAR magnitude-redshift distribution. In addition to the galaxy catalog, we also created a separate set of images with stellar sources, that were used to verify the photometric calibration procedure. In both Data Challenges a few objects with abnormal colors and Population III galaxies (Zackrisson et al. 2011) were added to test the efficacy of algorithms being tailored to detect outliers. For DC2, HST fluxes were also calculated (though no HST images created) which were used to estimate the number of low-redshift contaminants in the photometric redshift calculation. For the JAGUAR galaxies in DC2, noise was added to the mock HST fluxes and uncertainties were estimated according to the depth of the available ancillary HST imaging. \nFor NIRSpec, these data challenges not only served to verify and practice ingesting NIRCam-generated and -formated target catalogs and images into the NIRSpec target selection and MSA mask design work flow, but also provided a critical opportunity to augment the eMPT with needed features. In particular, we modified eMPT to be able to point the 'prime' NIRSpec instrument such that the 'secondary' NIRCam instrument achieved its intended elaborate mosaicking of the NIRCam Medium Prime fields described in Sections 4.2 and 5.6, while simultaneously exploiting the ≃ 1 arcsec level permissible deviations from the nominal NIRCam pointing pattern to optimize the parallel NIRSpec exposures such that the largest possible number of the highest priority HST targets were captured by the MSA. This task was further complicated by the peculiar manner in which the roll orientation of the NIRSpec MSA assigned to an observation by STScI does not refer to the center of the MSA, but rather to a reference point defined by median location of all targets contained in the NIRSpec input catalog entered into the APT (Bonaventura et al. 2023). Limiting the impact of this complication over the 6 . 4 ' lever arm between the field centers of the two instruments required an iterative approach in which the NIRSpec input catalog was gradually trimmed down to match the outer envelop of the final NIRSpec footprint. \nThe NIRSpec component of DC2 simulated an approximation of the GOODS-South Medium/HST tier. The NIRCam source scene described above was used to assign spectra and morphologies to the known HST prioritized target catalog. The eMPT was exercised to determine the optimum set of six pairs of pointing locations, within the small tolerance allowed given that in the real Medium/HST a NIRCam mosaic would be made in parallel, that maximized the number of highest priority targets assigned shutters. The eMPT was then run to assign targets to shutters in order of priority class. Spectra were simulated and processed in a similar \nmanner to DC1. One difference is that contaminants that would fall within the target or background shutter were included in the simulation to assess the effects of contamination. \nTo manipulate these Data Challenges and to prepare for the real data, JADES also built visualization tools. To browse the sky, we developed FitsMap (Hausen & Robertson 2022), inspired in part by the Legacy Survey viewer led by D. Lang (Dey et al. 2019). FitsMap allows us to zoom and pan the sky, easily changing between image layers, with overlays from various catalogs that provide pop-up access to the database information. To study the SEDs and photometric redshift outputs, we developed JADESview 5 , which shows image thumbnails, photometric SEDs with best-fit template overlays, and photometric redshift likelihoods versus redshift. \nWe have found these preparations to be invaluable in handling the in-flight data. That said, unsurprisingly the real data have presented additional challenges to which the team (and the community more broadly) must adjust. Our reduction of on-sky data will be described further in Rieke et al. (2023a) and Bunker et al. (2023a), as well as upcoming papers (Alberts et al., in prep.; Carniani et al., in prep.; Robertson et al., in prep; Tacchella et al., in prep.).", '8. CONCLUSIONS': "The JWST Advanced Deep Extragalactic Survey is bringing an ambitious deep imaging and spectroscopic infrared view of the GOODS-S and GOODS-N fields in the first cycle of JWST observing. With JADES, we use 545 hours of open-shutter dual-band NIRCam imaging and 240 open-shutter hours of MIRI imaging to cover about 220 arcmin 2 to very faint flux levels in 12 distinct bands. We then conduct extensive multi-object infrared spectroscopy using 339 open-shutter hours of NIRSpec MOS, observing over 5000 faint targets with both prism and grating dispersers. \nThe resulting JADES imaging and spectra will provide an exquisite sample for the study of galaxy evolution. Already the data set has yielded many candidates at redshifts above 8 (Hainline et al. 2023) and provided spectroscopic confirmation of 5 galaxies at z > 10 (Robertson et al. 2023a; Curtis-Lake et al. 2023; Tacchella et al. 2023; Bunker et al. 2023b). The amount of detail in both imaging and spectroscopy is very impressive and is revealing high-redshift galaxies to be a diverse set, with clear variations in morphology, emission-line ratios, and star-formation histories (e.g. Dressler et al. 2023; Ends- \nley et al. 2023; Looser et al. 2023a). The spectra reveal the imprint of reionization through variations in Lyman α emission (Saxena et al. 2023; Witstok et al. 2023a) and signatures of the Gunn-Peterson damping wing (CurtisLake et al. 2023). \nAt the time of this writing, the GOODS-S data set has been about 40% completed, with half of the imaging and much of the follow-up spectroscopy to happen in the last quarter of 2023. The GOODS-N observing has been successfully completed. \nJADES also provides a useful design example for deep surveys, which we have documented in this paper. We have found great value in the medium-band F335M and F410M imaging and provide examples to achieve high pixel-diversity in both imaging and spectroscopy. We have demonstrated how the multiplex of grating spectroscopy can be increased by allowing these spectra to overlap and using the shorter prism spectra to disambiguate emission lines. We are also confronting a number of challenges in carrying out the survey, such as recovering from lost data in a survey with substantial geometrical constraints and concerns with NIRCam persistence. We expect these will be useful learning experiences as the JWST mission matures. \nAs listed in § 3.4, JADES is one of several extragalactic surveys being carried out in Cycle 1 of the JWST mission. These span a range of depth, areas, filter sets, and fields, and there is a productive complementarity in these choices. JADES is important because of its deep and reasonably wide coverage of the GOODS-S/HUDF and GOODS-N/HDF fields, where there is an awesome amount of multi-wavelength imaging and spectroscopy, and because of its close coordination of JWST imaging and spectroscopy. \nThe first release of JADES data, focusing on year 1 Deep NIRCam imaging and NIRSpec multi-object spectroscopy on the HUDF, is presented in Bunker et al. (2023a) and Rieke et al. (2023a) and available at https://archive.stsci.edu/hlsp/jades and http: //jades.idies.jhu.edu/. Additional releases will follow successively in the coming year, and we post science updates from the survey at the JADES Collaboration website, https://jades-survey.github.io. JADES will provide the foundation for JWST's study of these two premier deep fields, and we look forward to many years of utilization and extension of this data set. \nThe JADES Collaboration thanks the Instrument Development Teams and the instrument teams at the European Space Agency and the Space Telescope Science Institute for the support that made this program possible. We also thank our program coordinators at STScI for their help in planning complicated parallel observations. \nProcessing for the JADES NIRCam data release was performed on the lux cluster at the University of California, Santa Cruz, funded by NSF MRI grant AST 1828315. This research makes use of ESA Datalabs (datalabs.esa.int), an initiative by ESA's Data Science and Archives Division in the Science and Operations Department, Directorate of Science. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk). \nMR, AD, EE, DJE, BDJ, BR, GR, FS, and CNAW acknowledge support from the NIRCam Science Team contract to the University of Arizona, NAS5-02015. DJE is further supported as a Simons Investigator. SAr acknowledges support from Grant PID2021-127718NBI00 funded by the Spanish Ministry of Science and Innovation/State Agency of Research (MICIN/AEI/ 10.13039/501100011033). AJB, AJC, JC, IEBW, AS & GCJ acknowledge funding from the 'FirstGalaxies' Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056). AJC acknowledges funding from the 'FirstGalaxies' Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 789056). ECL acknowledges support of an STFC Webb Fellowship (ST/W001438/1). \nFunding for this research was provided by the Johns Hopkins University, Institute for Data Intensive Engineering and Science (IDIES). The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant no.140. RM, WB, FDE, TJL, JS, LS, and JW acknowledge support by the Science and Technology Facilities Council (STFC) and by the ERC through Advanced Grant 695671 'QUENCH'. RM also acknowledges funding from a research professorship from the Royal Society. JW further acknowledges support from the Fondation MERAC. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. ALD thanks the University of Cambridge Harding Distinguished Postgraduate Scholars Programme and Technology Facilities Council (STFC) Center for Doctoral Training (CDT) in Data intensive science at the University of Cambridge (STFC grant number 2742605) for a PhD studentship. BRP acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation/State Agency of Research (MICIN/AEI/ 10.13039/501100011033) RS acknowledges support from a STFC Ernest Rutherford Fellowship (ST/S004831/1). CWo is supported by the National Science Foundation through the Graduate Research Fellowship Program funded by Grant Award No. DGE-1746060. DP acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. H U gratefully acknowledges support by the Isaac Newton Trust and by the Kavli Foundation through a Newton-Kavli Junior Fellowship. LW acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2137419. MP acknowledges support from the research project PID2021-127718NB-I00 of the Spanish Ministry of Science and Innovation/State Agency of Research (MICIN/AEI/ 10.13039/501100011033), and the Programa Atracci'on de Talento de la Comunidad de Madrid via grant 2018-T2/TIC-11715 MSS acknowledges support by the Science and Technology Facilities Council (STFC) grant ST/V506709/1. REH acknowledges acknowledges support from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1746060. SC acknowledges support by European Union's HE ERC Starting Grant No. 101040227 - WINGS. The research of KB is supported in part by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.", 'REFERENCES': '- -. 2019b, ApJS, 244, 16\n- Whitler, L., Endsley, R., Stark, D. P., et al. 2023a, MNRAS, 519, 157\n- Whitler, L., Stark, D. P., Endsley, R., et al. 2023b, arXiv e-prints, arXiv:2305.16670\n- Williams, C. C., Curtis-Lake, E., Hainline, K. N., et al. 2018, ApJS, 236, 33\n- Williams, C. C., Labbe, I., Spilker, J., et al. 2019, ApJ, 884, 154\n- Williams, C. C., Tacchella, S., Maseda, M. V., et al. 2023, arXiv e-prints, arXiv:2301.09780\n- Williams, R. E., Blacker, B., Dickinson, M., et al. 1996, AJ, 112, 1335\n- Withers, S., Muzzin, A., Ravindranath, S., et al. 2023, arXiv e-prints, arXiv:2304.11181 \nWitstok, J., et al. 2023a, submitted \n- Witstok, J., Shivaei, I., Smit, R., et al. 2023b, arXiv e-prints, arXiv:2302.05468\n- Wright, G. S., Rieke, G. H., Glasse, A., et al. 2023, PASP, 135, 048003 \nZackrisson, E., Rydberg, C.-E., Schaerer, D., Ostlin, G., & Tuli, M. 2011, ApJ, 740, 13 \n- Zavala, J. A., Buat, V., Casey, C. M., et al. 2023, ApJL, 943, L9'}

2023ApJS..265....5H

We conduct a comprehensive study on dropout galaxy candidates at z 916 using the first 90 arcminSUP2SUP James Webb Space Telescope JWST Near Infrared Camera images taken by the early release observations ERO and early release science programs. With the JWST simulation images we find that a number of foreground interlopers are selected with a weak photoz determination SUP2SUP gt 4. We thus carefully apply a secure photoz selection criterion SUP2SUP gt 9 and conventional color criteria with confirmations of the ERO Near Infrared Spectrograph spectroscopic redshifts and obtain a total of 23 dropout galaxies at z 916 including two candidates at zmathrmphot16.250.460.24 and 16.410.550.66 . We perform thorough comparisons of dropout galaxies found in our work with recent JWST studies and conclude that our galaxy sample is reliable enough for statistical analyses. We derive the UV luminosity functions at z 916 and confirm that our UV luminosity functions at z 9 and 12 agree with those determined by other Hubble Space Telescope and JWST studies. The cosmic star formation rate SFR density decreases from z 9 to 12 and perhaps to 16 but the densities at z 1216 are higher than the constant star formation efficiency model. Interestingly there are six bright galaxy candidates at z 1016 with M SUBUVSUB lt 19.5 mag and M SUBSUB 10SUP89SUP M SUBSUB. Because a majority 80 of these galaxies show no signatures of active galactic nuclei in their morphologies the high cosmic SFR densities and the existence of these UVluminous galaxies are explained by the lack of suppression of star formation by the UV background radiation at the prereionization epoch andor an efficient UV radiation production by a topheavy initial mass function with Population IIIlike star formation.

2023-03-01T00:00:00Z

['10.48550/arXiv.2208.01612', '2022arXiv220801612H', 'arXiv:2208.01612', '2023ApJS..265....5H', '10.3847/1538-4365/acaaa9']

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

A Comprehensive Study of Galaxies at z 916 Found in the Early JWST Data Ultraviolet Luminosity Functions and Cosmic Star Formation History at the Prereionization Epoch

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

https://arxiv.org/pdf/2208.01612.pdf

{'A Comprehensive Study on Galaxies at z ∼ 9 -16 Found in the Early JWST Data: UV Luminosity Functions and Cosmic Star-Formation History at the Pre-Reionization Epoch': 'Yuichi Harikane, 1 Masami Ouchi, 2, 1, 3 Masamune Oguri, 4, 5 Yoshiaki Ono, 1 Kimihiko Nakajima, 2 Yuki Isobe, 1, 6 Hiroya Umeda, 1, 6 Ken Mawatari, 2 and Yechi Zhang 1, 7 \n1 Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan 2 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 4 Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 5 Department of Physics, Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 263-8522, Japan 6 Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan \n- 7 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan \n(Accepted for Publication in ApJS)', 'ABSTRACT': 'We conduct a comprehensive study on dropout galaxy candidates at z ∼ 9 -16 using the first 90 arcmin 2 JWST/NIRCam images taken by the early release observations (ERO) and early release science (ERS) programs. With the JWST simulation images, we find that a number of foreground interlopers are selected with a weak photoz determination (∆ χ 2 > 4). We thus carefully apply a secure photoz selection criterion (∆ χ 2 > 9) and conventional color criteria with confirmations of the ERO NIRSpec spectroscopic redshifts, and obtain a total of 23 dropout galaxies at z ∼ 9 -16, including two candidates at z phot = 16 . 25 +0 . 24 -0 . 46 and 16 . 41 +0 . 66 -0 . 55 . We perform thorough comparisons of dropout galaxies found in our work with recent JWST studies, and conclude that our galaxy sample is reliable enough for statistical analyses. We derive the UV luminosity functions at z ∼ 9 -16, and confirm that our UV luminosity functions at z ∼ 9 and 12 agree with those determined by other HST and JWST studies. The cosmic star-formation rate density decreases from z ∼ 9 to 12, and perhaps to 16, but the densities at z ∼ 12 -16 are higher than the constant star formation efficiency model. Interestingly, there are six bright galaxy candidates at z ∼ 10 -16 with M UV < -19 . 5 mag and M ∗ ∼ 10 8 -9 M glyph[circledot] . Because a majority ( ∼ 80%) of these galaxies show no signatures of AGNs in their morphologies, the high cosmic star-formation rate densities and the existence of these UV-luminous galaxies are explained by no suppression of star-formation by the UV background radiation at the pre-reionization epoch and/or an efficient UV radiation production by a top-heavy IMF with Population III-like star formation. \nKeywords: galaxies: formation - galaxies: evolution - galaxies: high-redshift', '1. INTRODUCTION': "One of the most important goals in astronomy today is to understand galaxy formation from their birth stage to current stage (Stark 2016; Dayal & Ferrara 2018; Ouchi et al. 2020; Robertson 2021). To accomplish the goal, observations for present galaxies to first galaxies are key to revealing the entire process of galaxy formation, while observations of early high redshift galaxies, especially \nhari@icrr.u-tokyo.ac.jp \nfirst galaxies, are missing (e.g., Zackrisson et al. 2011; Nakajima & Maiolino 2022). \nOver the past 2-3 decades, large telescopes have driven observational studies of galaxy formation with millions of galaxies at a redshift up to z ∼ 10 since the start of deep imaging observations represented by the legendary Hubble Deep Field project with Hubble Space Telescope (HST; Williams et al. 1996). To date, deepfield imaging observations have reached detection limits of glyph[similarequal] 30 mag in the wavelength range of 0 . 4 -1 . 6 µ m with HST/ACS and WFC3 instruments in the Hubble Ultra Deep Field (HUDF; Beckwith et al. 2006, see \nBouwens et al. 2021 and references therein) with the moderately deep ultra-violet (UV) extension, UVUDF (0 . 2 -0 . 4 µ m; Windhorst et al. 2011; Teplitz et al. 2013). Albeit with shallower detection limits of ∼ 26 -29 mag, HST GOODS, COSMOS, and CANDELS, and the associated parallel-field programs have covered a total area of square degrees in the blank fields (Giavalisco et al. 2004; Scoville et al. 2007; Grogin et al. 2011; Koekemoer et al. 2011). Complementary ground-based observations of Subaru Hyper Suprime-Cam survey have been completed optical imaging covering ∼ 1000 deg 2 with ∼ 26 mag depth (Aihara et al. 2022; see also Harikane et al. 2022b), while the ground-based nearinfrared (1 -2 µ m) and Spitzer Space Telescope imaging (3 -8 µ m) are limited to a total of few deg 2 with the similar or moderate depths of ∼ 25 -26 mag. Beyond deep imaging in blank fields, the HST programs, Hubble Frontier Fields (HFF) and Reionization Lensing Cluster Survey (RELICS), target 6 and 41 massive clusters, respectively, with depths of ∼ 26 -29 mag to study faint high redshift galaxies behind the clusters, exploiting gravitational lensing magnification (Lotz et al. 2017; Coe et al. 2019). These deep imaging data provide more than 4 million photometrically-selected dropout galaxies at z ∼ 4 -10 (Bouwens et al. 2021; Harikane et al. 2022b) and up to z ∼ 13 (Harikane et al. 2022a). Albeit with much small high redshift galaxy samples, spectroscopic observations confirm galaxies up to z = 9 . 1 with ALMA (Hashimoto et al. 2018) and z = 11 . 0 by HST/WFC3 grism and Keck/MOSFIRE spectroscopy (Oesch et al. 2016; Jiang et al. 2021). Star formation in even higher redshift ( z glyph[greaterorsimilar] 14) is discussed based on Balmer break galaxy candidates at z ∼ 6 (Mawatari et al. 2020b). \nWith the galaxy samples photometrically selected in the rest-frame UV wavelengths, a number of studies have derived rest-frame UV luminosity functions reaching up to z ∼ 10 -13. The UV luminosity functions show the redshift evolution from z ∼ 3 to 10 with a decrease of the normalization φ ∗ and an increase of the faint-end slope α , and no evolution of characteristic luminosity L ∗ on the basis of the Schechter function parameterization (Bouwens et al. 2015, 2021; Finkelstein et al. 2015b; Ishigaki et al. 2018). At z ∼ 4 and above, there are claims that the bright-end of the UV luminosity function is explained with the double power-law function, but not with the Schechter function, due to the excessive number of bright galaxies (Bowler et al. 2014, 2020; Ono et al. 2018; Stevans et al. 2018; Harikane et al. 2022b). Such bright galaxy population includes the galaxies with spectroscopic redshifts at z ∼ 10 (Oesch et al. 2016; Jiang et al. 2021) and perhaps galaxy candidates at z ∼ 13 (Harikane et al. 2022a), while it is not clearly concluded \nwith the sufficient statistical accuracy and the spectroscopic confirmations (e.g., Stefanon et al. 2019; RojasRuiz et al. 2020; Bowler et al. 2020). \nOver the cosmological volumes, the redshift evolution of cosmic star formation rate (SFR) density is revealed with the UV luminosity function measurements, and shows a monotonic decrease from z ∼ 3 to z ∼ 10 with a small contribution of dusty starbursts at z glyph[greaterorsimilar] 6 (e.g., Bouwens et al. 2022a; Barrufet et al. 2022). The UV luminosity function measurements provide the physical picture of galaxy formation over the redshift range of z ∼ 0 -10, tying galaxies and dark-matter halos via abundance matching techniques (e.g. Behroozi et al. 2013, 2019; Moster et al. 2013, 2018; Finkelstein et al. 2015a). There is an increasing trend of stellar-to-halo mass ratio towards high redshift for a given halo mass (e.g., Behroozi et al. 2013, 2019; Harikane et al. 2016, 2018), which is consistent with the original idea of the galaxy-formation downsizing picture (Cowie et al. 1988). The galaxy-dark matter halo connection probed by the clustering analysis indicates that the star formation efficiency, defined by the ratio of the SFR to the dark matter accretion rate, SFR/ ˙ M h , is almost constant across the redshift of z ∼ 2 -7 given the dark matter halo mass (Harikane et al. 2018, 2022b), and the constant star formation efficiency model can reproduce the evolutional trend of the cosmic SFR density (e.g., Bouch'e et al. 2010, Mason et al. 2015, Harikane et al. 2018, 2022b, Tacchella et al. 2018). \nThe UV luminosity function measurements, especially at the faint end, are clue to understanding galaxy formation (Yue et al. 2016) as well as cosmic reionization (Robertson 2021), where abundant faint star-forming galaxies are thought to be sources of cosmic reionization. The faint-end ( glyph[greaterorsimilar] -15 mag) UV luminosity function at z ∼ 6 -10 is probed with galaxies behind massive clusters such with the HFF data via gravitational lensing magnification (Atek et al. 2015, 2018; Ishigaki et al. 2015, 2018; Livermore et al. 2017; Laporte et al. 2016; Oesch et al. 2018), while the faint-end slopes and luminosity function turnovers are poorly constrained, due to the limited statistics and lensing magnification systematics (Bouwens et al. 2017, 2022a; Kawamata et al. 2018; Yang et al. 2022). \nHere the James Webb Space Telescope (JWST) was launched in the end of 2021, and just started its operation in the early 2022. The first data sets of JWST were released on July 12, 2022, taken by the early release observations (ERO) whose targets include a massive cluster SMACS J0723.3-7327 (SMACS J0723, z = 0 . 39) and Stephan's Quintet. The ERO imaging data taken with NIRCam (Rieke et al. 2005) are deep enough to \ndetect high redshift galaxies with the depths of ∼ 30 mag, and multi-band data covering glyph[greaterorsimilar] 2 µ m wavelengths allow us to detect galaxies at the previously unreachable redshift range up to z ∼ 20. Rest-frame optical emission at z glyph[greaterorsimilar] 10 is redshifted to the mid infrared bands and can be covered with the Mid-Infrared Instrument (MIRI; Bouchet et al. 2015). The ERO spectroscopic data of NIRSpec (Jakobsen et al. 2022) taken in the multi-object spectroscopy mode confirmed galaxies up to z = 8 . 5 with rest-frame optical lines in the 2 -5 µ m wavelengths. The slit-less spectroscopy of NIRISS (Doyon et al. 2012) supplement spectroscopic redshift determinations in the wavelength range of ∼ 1 -2 µ m. All of these data sets are revolutionizing the galaxy formation studies. The JWST observatory subsequently releases the director's discretionary early release science (ERS) data that include NIRCam, NIRSpec, and NIRISS data taken by the ERS programs of Cosmic Evolution Early Release Science (CEERS; Finkelstein et al. 2017, 2022c) and GLASS James Webb Space Telescope Early Release Science (GLASS; Treu et al. 2022). Further releases will deliver data of Cycle 1 observations that include Public Release IMaging for Extragalactic Research (PRIMER; Dunlop et al. 2021), UNCOVER (Labbe et al. 2021), and COSMOS-Webb (Kartaltepe et al. 2021) once the observations complete. Programs of guaranteed time observations (GTO), such as JWST Advanced Deep Extragalactic Survey (JADES; Bunker et al. 2020) will be also completed in the early years. \nThis is a great development of observational astronomy, presenting the unprecedentedly deep and high quality data covering the infrared band ( > 2 µ m). In fact, after the releases of the ERO and ERS data sets, we find the explosive progresses of galaxy formation studies. The mass models of the ERO target cluster, SMACS J0723, are improved with the NIRCam imaging and NIRSpec spectroscopic data (Mahler et al. 2022; Pascale et al. 2022; Caminha et al. 2022). High redshift galaxies are searched in the ERO SMACS J0723 and ERS CEERS and GLASS fields, and are identified at z ∼ 9 -20 (Naidu et al. 2022b; Castellano et al. 2022; Leethochawalit et al. 2022; Adams et al. 2022; Morishita & Stiavelli 2022; Atek et al. 2022; Donnan et al. 2022; Finkelstein et al. 2022b; Yan et al. 2022). The morphological properties are investigated with the NIRCam images of the ERO SMACS J0723 and the CEERS observations via the comparisons of HST images for galaxies at z ∼ 3 -6 (Ferreira et al. 2022) and the rest-frame optical and near-infrared bands for galaxies at z ∼ 1 -2 (Suess et al. 2022), respectively. The infrared photometric properties of galaxies at z ∼ 1 -2 are studied with the NIRCam and MIRI images of the ERO SMACS J0723 \nobservations in conjunction with the ALMA archival data (Cheng et al. 2022). The ERO NIRSpec observations in SMACS J0723 provide high-quality spectra that allow to identify 10 galaxies at z = 1 . 2 -8 . 5, three of which reside at z = 7 . 7 -8 . 5 (Carnall et al. 2022), and to characterize the inter-stellar medium of the galaxies (Schaerer et al. 2022; Curti et al. 2022). NIRISS spectroscopic data complements the NIRSpec observations and provide spectroscopic sample of z ∼ 1 -8 galaxies (Roberts-Borsani et al. 2022; Wang et al. 2022; Boyett et al. 2022; Marchesini et al. 2022). More JWST results for galaxy formation are being actively reported. \nIn this paper, we present a comprehensive study on high redshift galaxies using the first JWST/NIRCam datasets taken by the ERO and ERS programs. The deep infrared imaging data taken with NIRCam allow us to search for galaxies at z glyph[greaterorsimilar] 9, and to constrain the UV luminosity function and the cosmic SFR density in the universe 600 Myrs after the Big Bang. We will also perform thorough comparisons of galaxies found in our work and recent JWST studies. \nThis paper is organized as follows. Section 2 presents the JWST observational data sets used in this study. Section 3 explains our sample selection and galaxy photometry catalog. In Section 4, we describe the mass model for the lensing cluster. We show our main results of UV luminosity functions and cosmic SFR densities in Section 5, and discuss the physical properties of early galaxies in Section 6. Section 7 summarizes our findings. Throughout this paper, we use the Planck cosmological parameter sets of the TT, TE, EE+lowP+lensing+BAO result (Planck Collaboration et al. 2020): Ω m = 0 . 3111, Ω Λ = 0 . 6899, Ω b = 0 . 0489, h = 0 . 6766, and σ 8 = 0 . 8102. All magnitudes are in the AB system (Oke & Gunn 1983).", '2.1. JWST/NIRCam Data': "We use four JWST NIRCam datasets obtained in the ERO and ERS programs, ERO SMACS J0723, ERO Stephan's Quintet, ERS CEERS, and ERS GLASS (Table 1). The total area is ∼ 90 arcmin 2 . We retrieved law data ( uncal.fits) from the MAST archive and reduced the data using the JWST pipeline version 1.6.3 development version (1.6.3.dev34+g6889f49, almost the same as 1.7.0). We use the Calibration Reference Data System (CRDS) context file of jwst 0995.pmap released in October, whose calibration values were derived using calibration observations of three different standard stars placed in all of the 10 NIRCam detectors. These new flux calibrations were verified using imaging of the globular cluster M92 (Boyer et al. 2022). In addition \nTable 1. Limiting Magnitudes of the JWST Data \nNote -Columns: (1) Field. (2) Effective area in arcmin 2 . (3)-(12) Typical limiting magnitudes which correspond to 5 σ variations in the sky flux measured with a circular aperture of 0 . '' 2-diameter in the deepest region. \nFigure 1. Comparison of magnitudes. Magnitudes measured in the JWST F 150 W -band (left) and F 356 W -band (right) are compared with those in the HST F 160 W -band and Spitzer [3.6]-band, respectively. The measured magnitudes agree well with those in the HST and Spitzer images within ∼ 10%, indicating that the flux is reasonably calibrated. Note that we include a 10% error floor on all measured fluxes to account for possible systematic uncertainties. \n<!-- image --> \n21 \n22 \n23 \n24 \n25 \nF356W (JWST) \nusing iraf tasks geomap and geotran . To check the reliability of the flux calibration, we compare our measured magnitudes in the JWST images with those in the HST and Spitzer images. As shown in Figure 1, the measured fluxes are almost consistent with those in the HST and Spitzer images, indicating that the flux is reasonably calibrated. \nThe limiting magnitudes were measured in 0 . '' 1, 0 . '' 2, and 0 . '' 3-diameter circular apertures by randomly placing apertures in sky areas using Python packages Astropy/photutils . Sky areas were defined as pixels without objects detected by SExtractor . We measured the limiting magnitudes in bins of the weight values to take into account inhomogeneity of the depth. Us- \n25 \n24 \n23 \n22 \n21 \nto the standard reduction, we added some processes to obtain better reduced images as follows. Before the Stage 2 calibration, we subtracted stray light features called 'wisps' by using a script provided by the NIRCam team 1 , and removed striping by using a script provided in the CEERS team (Bagley et al. 2022b) 2 . We ran the SkyMatch step individually on each frame of Stage 2 calibrated data before Stage 3 calibration, following a suggestion by the CEERS team (Bagley et al. 2022b). The images were pixel-aligned with a pixel scale of 0 . '' 015 / pixel, except for ones in the Stephan's Quintet field with a scale of 0 . '' 03 / pixel to reduce image sizes. Because the pipeline-processed images still showed a gradient of the sky background, we further subtracted the sky background using SExtractor (Bertin & Arnouts 1996). The intracluster light around the cluster center of SMACS J0723 is also removed in this process, although more sophisticated processes are sometimes employed (e.g., Livermore et al. 2017; Pagul et al. 2021). Note that our galaxy samples are not severely affected by systematics due to the intracluster light removal, because as shown in Section 3.5, the only candidate selected in the SMACS J0723 field, SM-z12-1, is located in the parallel field. The results of luminosity functions do not change beyond the errors if we remove the datapoint estimated with the SMACS J0723 data. Finally, we corrected for an astrometric offset between each detector and band \n[3.6] (Spitzer) \ning the weight map, we masked some regions around the edge of the detectors whose exposure time is short. We also measured a full-widths at half-maximum (FWHM) of the point spread function (PSF) in each image by selecting stellar objects in the magnitude-FWHM diagram. The measured limiting magnitudes in a 0 . '' 2diameter circular aperture, effective areas, and typical FWHMs of the PSFs are presented in Table 1. Here the effective area is defined as an area that is observed with all available bands before the foreground removal. The effect of the foreground will be taken into account in the completeness estimate (Section 5.1). In the following sections we detail our observational dataset in each field.", '2.1.1. ERO: SMACS J0723': 'A massive galaxy cluster at z = 0 . 39, SMACS J0723, was deeply observed with NIRCam, NIRSpec, MIRISS, and MIRI in the ERO (ERO-2736). The NIRCam images were taken in the six bands of F 090 W , F 150 W , F 200 W , F 277 W , F 356 W , and F 444 W , covering 11 . 0 arcmin 2 . The exposure time in each filter is ∼ 7500 seconds, and the 5 σ limiting magnitude in the F 356 W band is 29.9 mag.', "2.1.2. ERO: Stephan's Quintet": "Stephan's Quintet, a group of five local galaxies, was observed with NIRCam and MIRI in the ERO (ERO2732). The NIRCam images were taken in the six bands of F 090 W , F 150 W , F 200 W , F 277 W , F 356 W , and F 444 W , covering 42 arcmin 2 . We have masked central regions of the field that is affected by the five local galaxies in Stephan's Quintet, resulting in an effective area of 37 . 2 arcmin 2 , corresponding to ∼ 4 NIRCam pointings. The exposure time in each filter is roughly ∼ 1200 seconds, and the 5 σ limiting magnitude in the F 356 W band is 28.9 mag.", '2.1.3. ERS: CEERS': 'A part of the HST/CANDELS Extended Groth Strip (EGS) field is observed with JWST in the Cosmic Evolution Early Release Science (CEERS) survey (ERS-1345; Finkelstein et al. 2017, Finkelstein et al. 2022c). We use four pointing datasets of NIRCam obtained in June 2022 with the seven bands of F 115 W , F 150 W , F 200 W , F 277 W , F 356 W , F 410 M , and F 444 W , covering a total of 33 arcmin 2 . The exposure time in each filter is ∼ 2800 -6200 seconds, and the 5 σ limiting magnitude in the F 356 W band is 29.7 mag. Since the exposure times are not uniform across the four NIRCam pointings, we separately analyze the four pointing data. \nA massive galaxy cluster, Abell 2744, was observed with JWST in the ERS program of Through the Looking GLASS (ERS-1324; Treu et al. 2017, 2022). Deep NIRCam images were taken in June 2022 in a parallel mode of NIRISS observations targeting the center of the cluster, in seven bands F 090 W , F 115 W , F 150 W , F 200 W , F 277 W , F 356 W , and F 444 W , covering 6 . 8 arcmin 2 . The exposure time in each filter is ∼ 5600 -23000 seconds, and the 5 σ limiting magnitude in the F 356 W band is 29.9 mag. The lensing magnification is negligible in the field of the NIRCam observations that is ∼ 5 arcmin away from the cluster center of Abell 2744.', '2.2. JWST/NIRSpec': 'We use publicly available data from the ERO NIRSpec observations targeting the field of the SMACS J0723 cluster (ERO-2736). The NIRSpec observations consist of two pointings with the same Multi Shutter Array (MSA) configuration. NIRSpec observations were carried out by using the disperser-filter combinations of G235M/F170LP and G395M/F290LP, which cover the wavelength range from 1.66 µ m to 5.16 µ m with a spectral resolution of R ∼ 1000. The total exposure time of the two individual pointings is 8,840 seconds for each grating. The NIRSpec observations have taken spectra for a total of 35 objects, three of which, s04590 ( z spec = 8 . 495), s06355 ( z spec = 7 . 664), and s10612 ( z spec = 7 . 659), are securely identified at z > 7 whose dropouts can be covered with the NIRCam F 090 W band. See K. Nakajima et al. in prep. for details of the reduction and analysis of the NIRSpec data.', '2.3. HST/ACS and WFC3': "HST multi-band images are available in the fields of SMACS J0723 and CEERS (EGS). We downloaded HST ACS and WFC3 images in the SMACS J0723 and CEERS fields from the websites of RELICS (Coe et al. 2019) 3 and CEERS 4 , respectively. We found a small offset ( ∼ 0 . '' 2) of the WCS between the JWST and HST images. In this paper we use coordinates of the JWST images.", '3.1. Photometric Catalog': "We construct multi-band source catalogs from the JWST data to select the F 115 W , F 150 W , and F 200 W -dropout galaxies. We use SWarp (Bertin et al. 2002) \nto produce our detection image that is a weighted mean image of the bands redder than the Lyman break in each dropout selection (i.e., F 150 W , F 200 W , F 277 W , F 356 W , F 410 M , and F 444 W for the F 115 W -dropout selection, F 200 W , F 277 W , F 356 W , F 410 M , and F 444 W for the F 150 W -dropout selection, and F 277 W , F 356 W , F 410 M , and F 444 W for the F 200 W -dropout selection). To measure object colors, we match the image PSFs to the F444W-band images whose typical FWHM of the PSF is glyph[similarequal] 0 . '' 16, the largest of the NIRCam multi-band images. \nWe perform source detection and photometry with SExtractor (version 2.5.0; Bertin & Arnouts 1996). We found that the photometry with SExtactor MAG AUTO performs better as a total magnitude than that with photutils isophotal flux , which is one of the outputs of the JWST calibration pipeline, using CEERS simulated images with a mock galaxy catalog created with the Santa Cruz Semi-Analytic Model (Somerville et al. 2021; Yung et al. 2022). 5 We run SExtractor in dual-image mode for each image with its detection image, having the parameter set as follows: DETECT MINAREA = 5, DETECT THRESH = 3 . 0, ANALYSIS THRESH = 3 . 0, DEBLEND NTHRESH = 32, and DEBLEND MINCOUNT = 0 . 005. The total number of the objects detected is ∼ 250,000. We measure the object colors with the MAG APER magnitudes defined in a 0 . '' 3diameter circular aperture in the PSF-matched images. Source detections are evaluated with 0 . '' 1 and/or 0 . '' 2diameter circular apertures in the original (not PSFmatched) images. The total magnitudes are estimated from the 0 . '' 3-diameter aperture magnitudes with the aperture correction. The value of the aperture correction is defined as the difference between the MAG AUTO magnitude and the 0 . '' 3-diameter aperture magnitude in an image of the weighted mean of the PSF-matched images whose wavelengths are longer than the Lyman break (i.e., F 150 W , F 200 W , F 277 W , F 356 W , F 410 M , and F 444 W for the F 115 W -dropout selection, F 200 W , F 277 W , F 356 W , F 410 M , and F 444 W for the F 150 W -dropout selection, and F 277 W , F 356 W , F 410 M , and F 444 W for the F 200 W -dropout selection). Furthermore, we correct for a small offset ( ∼ 0 . 1 mag) between the measurement of MAG AUTO and the true total magnitude due to the wing of the PSF not captured with MAG AUTO (see Sections 2.2 and 2.5.1 in Finkelstein et al. 2022a). We measure this offset by inserting mock galaxy images randomly in the real images, and measure the magnitudes using SExtractor in a similar manner \nas the completeness simulation described later in Section 5.1. To account for systematic uncertainties of the flux measurements (e.g., zero-point correction), we include 10% error floor on all measured fluxes. Finally we correct for the galactic extinction using Schlegel et al. (1998) and Schlafly & Finkbeiner (2011) and make final photometric catalogs.", '3.2. Dropout Selection': "From the photometric catalogs constructed in Section 3.1, we construct z ∼ 9 -16 dropout galaxy catalogs based on the Lyman break color selection technique (e.g., Steidel et al. 1996; Giavalisco 2002). As shown in Figure 2, galaxy candidates at z ∼ 9, 12, and 16 can be selected by the F 115 W , F 150 W , and F 200 W -dropout selections, respectively. \nFirst, to identify secure sources, we select source whose signal-to-noise ratios (S/Ns) in a 0 . '' 2-diameter circular aperture are higher than 5 in the detection images. We also require sources to be detected at > 3 . 5 σ levels in at least two bands redder than the Lyman break. We then select dropout galaxy candidates by using their broadband spectral energy distribution (SED) colors. We adopt the following color criteria: \nF 115 W -dropout ( z ∼ 9): \n( F 115 W -F 150 W > 1 . 0) ∧ (1) \n( F 150 W -F 277 W < 1 . 0) ∧ (2) \n( F 115 W -F 150 W > ( F 150 W -F 277 W ) + 1 . 0) (3) \nF 150 W -dropout ( z ∼ 12): \n( F 150 W -F 200 W > 1 . 0) ∧ (4) \n( F 200 W -F 356 W < 1 . 0) ∧ (5) \n( F 150 W -F 200 W > ( F 200 W -F 356 W ) + 1 . 0) (6) \nF 200 W -dropout ( z ∼ 16): \n( F 200 W -F 277 W > 1 . 0) ∧ (7) \n( F 277 W -F 444 W < 1 . 0) ∧ (8) \n( F 200 W -F 277 W > 1 . 5( F 277 W -F 444 W ) + 1 . 0) (9) \nWe select sources with prominent breaks with the criteria of Equations (1), (4), and (7), and measure the slope of the continuum and remove red interlopers with Equations of (2-3), (5-6), and (8-9). To measure the slope of the continuum, we use the bands that have a large wavelength difference as much as possible, while not biting the Balmer break in the redder band. To remove foreground interlopers, we exclude \nF150W -F277W --Figure 2. Two-color diagrams of F 115 W -F 150 W vs. F 150 W -F 277 W (left), F 150 W -F 200 W vs. F 200 W -F 356 W (center), and F 200 W -F 277 W vs. F 277 W -F 444 W (right) corresponding to the color selections for F 115 W -dropouts at z ∼ 9, F 150 W -dropouts at z ∼ 12, and F 200 W -dropouts at z ∼ 16, respectively. The red squares represent our dropout galaxy candidates that meet the color selection criteria indicated with the red lines. The blue lines denote colors of the dropout galaxy models with UV spectral slopes of β UV = -2 . 3 and -1 . 3 whose redshifts are indicated with the numbers and the blue circles with an interval of ∆ z = 0 . 2. The black dotted, dashed, dot-dashed lines show colors of typical elliptical, Sbc, and irregular galaxies (Coleman et al. 1980) redshifted from z = 0 to 7. The star marks present expected colors of Galactic dwarf stars (Patten et al. 2006; Kirkpatrick et al. 2011). These colors of dwarf stars are estimated by interpolating available flux measurements obtained by ground based telescopes ( J , H , and K -bands) and the Spitzer telescope ([3.6] and [4.5]). \n<!-- image --> \nsources with continuum detections at > 2 σ levels in the 0 . '' 1- or 0 . '' 2-diameter apertures in bands bluer than the Lyman break, i.e., the F 090 W band for the F 115 W -dropouts, F 090 W and F 115 W bands for the F 150 W -dropouts, and F 090 W , F 115 W , and F 150 W bands for the F200W-dropouts. To select reliable candidates, we restrict our dropout selections in fields where the bluer band than the Lyman break is available; the F 115 W -dropout selection is only performed in the GLASS field. We also apply a criterion of SExtractor stellarity parameter, CLASS STAR , of < 0 . 9, to remove stellar contaminants. Finally, we visually inspect images of the selected sources to remove spurious sources or sources affected by nearby bright objects and diffraction spikes of bright stars. We removed about half of the selected objects in this process. We also visually inspect HST images of the selected sources in the SMACS J0723 and CEERS fields to check whether the source is consistent with being a high redshift galaxy, although the HST images are typically ∼ 1 -2 mag shallower than the JWST images in these fields.", '3.3. SED Fitting': 'To further remove low redshift interlopers, we perform galaxy SED fitting with the flexible Bayesian inference code prospector (Johnson et al. 2021), and derive the photometric redshift. Model spectra are derived from Flexible Stellar Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) package with the modules for Experiments in Stellar Astrophysics Isochrones \nand Stellar Tracks (MIST; Choi et al. 2016). The boost of ionizing flux production of massive stars are included in the MIST isochrones (Choi et al. 2017). Here we assume the stellar initial mass function (IMF) determined by Chabrier (2003), the Calzetti et al. (2000) dust extinction law, and the intergalactic medium (IGM) attenuation model by Madau (1995). Note that the choice of the IGM attenuation model does not affect our galaxy selection at z ∼ 9 -16 because the flux bluer than the Ly α break is almost absorbed by the highly neutral IGM at these redshifts regardless of the choice of the IGM attenuation model. The Ly α emission line is also masked considering the high IGM neutral fraction at these redshifts. We adopt a flexible star formation history with five bins that are spaced equally in logarithmic times between 0 Myr and a lookback time that corresponds to z = 30, where the SFR within each bin is constant. We change the redshift, optical depth in the V -band, star formation history, and total stellar mass as free parameters, while fix the metallicity to Z = 0 . 2 Z glyph[circledot] . We assume a continuity prior for the star formation history, and flat priors for other parameters in the range 0 < z < 20, 0 < τ V < 2, and 6 < log( M ∗ /M glyph[circledot] ) < 12. We search for the best-fit model to the observed photometry with the MCMC method by using emcee (Foreman-Mackey et al. 2013). \nBased on the results of the SED fitting, we select objects whose high-redshift solution is more likely than the low-redshift ones, by using ∆ χ 2 , which is defined as a difference between the χ 2 values of the best high- \nhift solution and the lower-redshift solution, ∆ χ 2 = χ 2 ( z low ) -χ 2 ( z high ). Previous studies use a criterion of ∆ χ 2 > 4 . 0, corresponding to a 2 σ level (e.g., Bowler et al. 2020; Harikane et al. 2022a; Donnan et al. 2022; Finkelstein et al. 2022b). However, given the small number of the available JWST bands bluer than the Lyman break and the expected small number density of z > 9 galaxies, it is possible that this criterion is not sufficient to remove low redshift interlopers. To determine the threshold value for ∆ χ 2 , we use the CEERS simulated NIRCam images. In the CEERS simulated images, mock galaxies at z = 0 -10 in Yung et al. (2019, 2022) are inserted using the JWST data simulator Mirage (Hilbert et al. 2019). We measure fluxes of mock galaxies in each band in the same manner as our real dropout galaxy selection (Section 3.1), and conduct the SED fitting using prospector . As shown in Figure 3, at least eight sources at z true ∼ 3 -4 in the simulations have the best photometric redshifts of z phot ∼ 12 -15 and ∆ χ 2 = 4 -9, indicating that the criterion of ∆ χ 2 > 4 is not sufficient to remove low redshift interlopers. Thus in this study, we instead adopt a strict screening criterion of ∆ χ 2 > 9 . 0, which can remove these interlopers. The inclusion of this strict criterion does not introduce a bias with respect to a color of the UV continuum for bright galaxies, because the strength of the break is the most important factor to determine the ∆ χ 2 value. For faint galaxies, about 40% of them at ∼ 29 -30 mag will be missed due to the inclusion of this criterion, and this effect is taken into account in the completeness estimate in Section 5.1.', '3.4. Comparisons with Spectroscopic Redshifts': 'To test the reliability of our galaxy selections and SED fitting, we compare our photometric redshift estimates with the spectroscopic results obtained in the NIRSpec observations (Section 2.2). Since there are currently no z > 9 source spectroscopically confirmed with NIRSpec in the filelds used in this study, we focus on the three galaxies at z > 7, s04590 ( z spec = 8 . 495), s06355 ( z spec = 7 . 664), and s10612 ( z spec = 7 . 659). We measure fluxes of the three galaxies in the NIRCam images and estimate photometric redshifts using prospector , in the same manner as our dropout galaxies. Figures 4 and 5 present results of the SED fitting and comparison with the spectroscopic redshifts. We found that the estimated photometric redshifts agree well with the spectroscopic redshifts within ∼ 2 σ uncertainties, indicating that our SED fitting works well to estimate the redshift from the NIRCam photometry.', '3.5. Final Sample': "Finally we select 13, 8, and 2 dropout galaxy candidates at z ∼ 9, 12, and 16, respectively (Table 2). The photometric redshifts range from z ∼ 8 . 7 to 16 . 4, demonstrating the power of JWST exploring the early universe (Figure 6). Examples of the snapshots and SEDs of selected galaxies at z ∼ 9, 12, and 16 are presented in Figures 7 and 8. These sources show a sharp discontinuity around the Lyman break band, a flat or blue continuum, and non-detection in the bluer bands than the break, all of which are consistent with a high redshift galaxy. Photometric properties of our galaxy candidates are summarizes in Tables 3-5. Note that no objects appear in more than one final dropout sample. \nTo investigate morphological properties of our galaxy candidates, we fit our galaxy candidates with the S'ersic profile using galfit (Peng et al. 2010). We find that some bright candidates, i.e., GL-z9-1, GL-z12-1, CR2z12-1, CR-z16-1, and S5-z16-1, are clearly more extended than the PSF, although GL-z12-1 is compact compared to other candidates, implying potential AGN activity. We stack images of other faint candidates at each redshift, and find that the stacked images also show extended profiles with respect to the PSF. We thus conclude that the stellar contamination is negligible. Details of the morphological properties of our candidates are presented in Ono et al. (2022). \nOne of the highest redshift source candidates in our catalogs is CR2-z16-1 at z = 16 . 25 +0 . 24 -0 . 46 in the CEERS2 field. As discussed later in Section 3.6.6, CR2-z16-1 is firstly identified as a z = 16 . 4 source (ID 93116) in Donnan et al. (2022). As shown in the middle panel of Figure 8, our measured fluxes are almost consistent with those presented in Naidu et al. (2022a) and Finkelstein et al. (2022c), while fluxes in Donnan et al. (2022) are systematically fainter than our measurements, probably because Donnan et al. (2022) assume the PSF for the aperture correction. The colors measured in these three studies (this study, Donnan et al. 2022, Naidu et al. 2022a, and Finkelstein et al. 2022c) consistently show a clear break around F 200 W -band, consistent with a z = 16 . 3 galaxy, although there are some discussions about a possible solution of a dusty line emitter at z ∼ 5 (Zavala et al. 2022; Naidu et al. 2022a). Although the NIRSpec spectroscopy is required to conclude the redshift of CR2-z16-1, we include this source as a F 200 W -dropout galaxy candidate. \nThe other candidate at z ∼ 17 is S5-z16-1 at z = 16 . 41 +0 . 66 -0 . 55 identified in the Stephan's quintet field. Although this source is located in a region whose exposure time is relatively short compared to the central region of the field, our position-dependent estimates of flux uncertainties indicate that the source detection, color, and", 'Interlopers in Simulations': "Figure 3. Examples of mock galaxies whose true redshifts are z true ∼ 3 but selected as F 150 W -dropout galaxies at z phot ∼ 12 -15 identified in the simulated NIRCam images. For each object, the top left panel shows the 1 . '' 5 × 1 . '' 5 snapshots in NIRCam bands with a 3-pixel smoothing whose band names are indicated with the red labels. The bottom left panel presents the SED of the object. The red symbols with error bars are measured magnitudes or 2 σ upper limits, and the blue curve shows the best-fit model. The true and estimated redshifts with 2 σ errors are indicated with the black and blue texts, respectively. The χ 2 value is shown in the right panel as a function of redshift. The black curve is the true SEDs at z ∼ 3, while the blue curve denotes χ 2 values of our SED fitting for our photometric redshift determination. These objects meet the weak photometric redshift criterion of ∆ χ 2 > 4, but do not meet our strict criterion of ∆ χ 2 > 9, where ∆ χ 2 is the χ 2 difference between the best high redshift solution and a lower redshift solution, ∆ χ 2 = χ 2 ( z low ) -χ 2 ( z high ). See texts for details. \n<!-- image --> \nFigure 4. Top: NIRSpec spectrum of s04590 at a spectroscopic redshift of z = 8 . 495. The spectroscopic redshift is confirmed with the H β , H γ , H δ , [Oiii] λλ 5007,4959, [Oiii] λ 4363, [Oii] λ 3727, [Ne iii ] λ 3967, [Ne iii ] λ 3869, and the tentative Ciii] λ 1909 lines. The flux is arbitrary. Bottom left: Optical to near-infrared SED of s04590. The red circles and arrows indicate the measured magnitudes and 2 σ limits, respectively. The filled (open) red symbols denote the measurements and the limits obtained with JWST/NIRCam (HST/ACS and WFC3). The blue curve and the blue redshift label represent the best-fit model SED and the photometric redshift with 2 σ errors derived by our photometric redshift technique, which is compared with the spectroscopic redshift indicated with the black label. The images on this panel show 1 . '' 5 × 1 . '' 5 cutout images of s04590 in the NIRCam bands with a 3-pixel smoothing whose band names are indicated with the red labels. Bottom right: χ 2 as a function of redshift. The blue curve denotes χ 2 values of our SED fitting for our photometric redshift determination. The vertical dashed line indicates the spectroscopic redshift that agrees well with the photometric redshift. \n<!-- image --> \nFigure 5. Same as Figure 4, but for s06355 at z spec = 7 . 664 and s10612 at z spec = 7 . 659. Our estimates of the photometric redshifts agree well with the spectroscopic redshifts. \n<!-- image --> \nFigure 6. Absolute UV magnitude as a function of redshift for galaxies at 6 < z < 16. The red diamonds represent our dropout galaxy candidates selected with the JWST images. The red open circles show HD1 and HD2 previously found by the combination of the images taken with Sptizer and ground-based telescopes (Harikane et al. 2022a). The gray square and circles denote GN-z11 (Oesch et al. 2016; Jiang et al. 2021) and dropout galaxies selected with deep HST images (Bouwens et al. 2015). \n<!-- image --> \nTable 2. Number of Our Dropout Candidates \nnon-detections are robust against the uncertainties. An emission line feature is detected with ALMA in S5-z161 (Fujimoto et al. 2022), which would be interpreted as either [Oiii] 52 µ m at z = 16 . 01 or [Cii]1 58 µ m at z = 4 . 61. We include this possible candidate in the luminosity function calculation, although the luminosity is remarkably high compared to our expectations at this high redshift. \nFigure 7. (Top:) The left panel presents the optical to near-infrared SEDs of the z ∼ 9 dropout galaxy, GL-z9-1. The red circles and arrows show the measured magnitudes and 2 σ upper limits, respectively. The blue curve denotes the best-fit model SED whose redshift and 2 σ errors are presented in the upper left with the blue labels. The gray curve is a significantly worse fit of a low redshift solution. The images on this panel are 1 . '' 5 × 1 . '' 5 cutout images of GL-z9-1 in the NIRCam bands with a 3-pixel smoothing whose band names are indicated with the red labels. The right panel shows χ 2 values of the SED fitting as a function of redshift. (Middle:) Same as the top panels, but for another z ∼ 9 dropout galaxy candidate, GL-z9-2. (Bottom:) Same as the top panels, but for a z ∼ 12 dropout galaxy candidate, GL-z12-1. \n<!-- image --> \nFigure 8. Same as Figure 7, but for z ∼ 12 dropout galaxy candidate, CR2-z12-1 (top), and z ∼ 16 dropout galaxy candidates, CR2-z16-1 (middle) and S5-z16-1 (bottom). The open red symbols denote the measurements and the limits obtained with HST/ACS and WFC3. The orange, pink, and brown open symbols in the middle panel are measurements in Donnan et al. (2022), Naidu et al. (2022a), and Finkelstein et al. (2022c), respectively. \n<!-- image --> \nTable 3. Summary of Our F 115 W -Dropoout Candidates at z ∼ 9 \nNote -(1) Name. (2) Right ascension. (3) Declination. (4) Total magnitude in the F 356 W band with 1 σ errors. (5) F 115 W -F 150 W color with 1 σ errors. (6) F 150 W -F 277 W color with 1 σ errors. (7) Absolute UV magnitude with 1 σ errors. (8) Photometric redshift with 2 σ errors. (9) χ 2 difference between the best high redshift solution and a lower redshift solution, ∆ χ 2 = χ 2 ( z low ) -χ 2 ( z high ). (10) Reference (This: this work, N22: Naidu et al. (2022b), C22: Castellano et al. (2022), D22: Donnan et al. (2022)) and note for a reason why the source is not selected in this study (1: > 2 σ detection in F 090 W , 2: ∆ χ 2 < 9 . 0). \nTable 4 . Summary of Our F 150 W -Dropoout Candidates at z ∼ 12 \nTable 4 continued \nTable 4 (continued) \nTable 4 continued \nJWST Galaxies at z ∼ 9 -17 \nTable 4 (continued) \nNote -(1) Name. (2) Right ascension. (3) Declination. (4) Total magnitude in the F 356 W band with 1 σ errors. (5) F 150 W -F 200 W color with 1 σ errors. (6) F 200 W -F 356 W color with 1 σ errors. (7) Absolute UV magnitude with 1 σ errors. Values of galaxies in the SMACS J0723 field are after the lensing magnification correction with glafic . (8) Photometric redshift with 2 σ errors. (9) χ 2 difference between the best high redshift solution and a lower redshift solution, ∆ χ 2 = χ 2 ( z low ) -χ 2 ( z high ). (10) Reference (This: this work, N22: Naidu et al. (2022b), C22: Castellano et al. (2022), D22: Donnan et al. (2022), F22: Finkelstein et al. (2022b), Y22: Yan et al. (2022)) and note for a reason not selected in this study (1: > 2 σ detection in F 090 W , 2: F 150 W -F 200 W < 1 . 0, 3: ∆ χ 2 < 9 . 0, 4: < 5 σ in the detection image). \nTable 5 . Summary of Our F 200 W -Dropoout Candidates at z ∼ 16 \nNote -(1) Name. (2) Right ascension. (3) Declination. (4) Total magnitude in the F 356 W band with 1 σ errors. (5) F 200 W -F 277 W color with 1 σ errors. (6) F 277 W -F 444 W color with 1 σ errors. (7) Absolute UV magnitude with 1 σ errors. Values of galaxies in the SMACS J0723 field are after the lensing magnification correction with glafic . (8) Photometric redshift with 2 σ errors. (9) χ 2 difference between the best high redshift solution and a lower redshift solution, ∆ χ 2 = χ 2 ( z low ) -χ 2 ( z high ). (10) Reference (This: this work, D22: Donnan et al. (2022)) and note for a reason not selected in this study (1: F 200 W -F 277 W < 1 . 0, 2: ∆ χ 2 < 9 . 0).", '3.6. Comparison with Previous Studies': 'Some other studies identified galaxy candidates at z > 9 using the JWST NIRCam ERO and/or ERS \ndatasets. Here we review these studies and compare their samples with our galaxy samples. Tables 3-5 summarize properties of other possible candidates that were \nselected in other studies but did not meet our selection criteria. These comparisons were conducted on November 20, 2022, and we clarify the version of the paper we compared in the following sections.', '3.6.1. Naidu et al. (2022b)': 'Using the ERS CEERS and GLASS datasets, Naidu et al. (2022b) found two bright galaxy candidates at z ∼ 10 and 12, GLASS-z10 and GLASS-z12, which correspond to GL-z9-1 and GL-z12-1 in our sample, respectively. Their estimates of the photometric redshifts ( z = 10 . 35 +0 . 38 -0 . 51 and z = 12 . 38 +0 . 13 -0 . 27 for GL-z101 and GL-z12-1, respectively, with prospector , from the ApJL published version) are consistent with our estimates ( z = 10 . 49 +0 . 53 -0 . 72 and z = 12 . 28 +0 . 08 -0 . 07 ).', '3.6.2. Castellano et al. (2022)': 'Castellano et al. (2022) identified seven galaxy candidates at z ∼ 9 -12 with the color selection using the ERS GLASS dataset. Among the six candidates from the ApJL published version, three candidates, GHZ1, GHZ2, and GHZ4, are selected in our selection. GHZ1 (GHZ2) is GL-z9-1 (GL-z13-1) in our sample, and their photometric redshift, z = 10 . 53 -10 . 63 ( z = 12 . 11 -12 . 30) is comparable with our estimates. GHZ4 was identified in our selection as GL-z9-2, and their photometric redshift ( z = 9 . 93 -10 . 08) agrees with our estimate ( z = 10 . 46 +0 . 45 -0 . 99 ). The other three candidates, GHZ3, GHZ5, and GHZ6, did not meet our selection criteria, due to a possible detection in the F 090 W -band or ∆ χ 2 < 9, although GHZ3 and GHZ5 were also reported to have low redshift solutions in Castellano et al. (2022).', '3.6.3. Leethochawalit et al. (2022)': 'Leethochawalit et al. (2022) studied photometric properties of galaxies at 7 < z < 9 using the ERS GLASS dataset. We refer to the manuscript version 2 that was submitted to arXiv on October 4. Since their galaxies are identified from the F 090 W -dropout selection, we do not expect significant overlap between their galaxy sample and ours.', '3.6.4. Adams et al. (2022)': 'Adams et al. (2022) identified four galaxy candidates at 9 < z < 12 in the ERO dataset taken in the SMACS J0723 field. We refer to the manuscript version 2 that was submitted to arXiv on August 9. Since three of them are expected to be at z < 10, it is reasonable that they are not identified in our F 150 W -dropout selection ( z ∼ 12). The other source (ID 10234) is estimated to be at z = 11 . 42, around the edge of our redshift selection window (see Figure 11), and not selected in \nour study due to its insufficient F 150 W -F 200 W color and a possible detection in the F 090 W -band.', '3.6.5. Atek et al. (2022)': 'Using the ERO dataset in the SMACS J0723 field, Atek et al. (2022) selected 10 galaxy candidates at 10 < z < 16. We refer to the manuscript version 2 that was submitted to arXiv on October 31. Among them, four candidates, SMACS z12a, SMAC z12b, SMACS z16a, and SMACS z16b, have photometric redshifts of z > 12, and are expected to be overlapped in our galaxy catalogs. However, none of them are selected as high redshift galaxy candidates in our study, due to their insufficient colors or ∆ χ 2 < 9.', '3.6.6. Donnan et al. (2022)': 'Donnan et al. (2022) selected 45 galaxies at z > 8 . 5 using ERO SMACS J0723 and ERS GLASS and CEERS datasets. We refer to the manuscript version 2 that was submitted to arXiv on October 22. Among the 45 galaxies, three galaxies (IDs 1698, 6415, and 17487) are identified in the GLASS dataset, and are also selected in this study as GL-z9-1, GL-z9-4, and GL-z12-1, respectively. Their photometric redshifts ( z = 10 . 45 +0 . 26 -0 . 16 , z = 10 . 79 +0 . 45 -0 . 66 , and z = 12 . 42 +0 . 27 -0 . 21 for IDs 1698, 6415, and 17487, respectively) are consistent with our estimates ( z = 10 . 49 +0 . 53 -0 . 72 , z = 10 . 19 +0 . 63 -0 . 55 , and z = 12 . 28 +0 . 08 -0 . 07 ). The brightest candidate in Donnan et al. (2022) is ID 93316 at z = 16 . 39 +0 . 32 -0 . 22 , which is CR2z16-1 at z = 16 . 25 +0 . 24 -0 . 46 in our catalog. In the version 2, Donnan et al. (2022) newly selected ID 32395 2 at z = 12 . 29 +0 . 91 -0 . 32 , which is also selected in this study as CR2-z12-1 at z = 11 . 63 +0 . 51 -0 . 53 , which was firstly identified in Finkelstein et al. (2022b) Donnan et al. (2022) presented other three candidates at z > 12, but these candidates are not selected in this study due to ∆ χ 2 < 9.', '3.6.7. Finkelstein et al. (2022b)': "One galaxy candidate at z ∼ 14, dubbed Maisie's Galaxy in Finkelstein et al. (2022b), is also selected in this study as CR2-z12-1. We refer to the manuscript version 2 that was submitted to arXiv on September 7. The photometric redshift presented in Finkelstein et al. (2022b) is z = 11 . 8 +0 . 3 -1 . 2 , consistent with our estimate ( z = 11 . 63 +0 . 51 -0 . 53 ).", '3.6.8. Yan et al. (2022)': 'Yan et al. (2022) identified a total of 88 galaxy candidates at z ∼ 11 -20 in the ERO SMACS J0723 field, including 63 F 150 W -dropouts and 15 F 200 W -dropouts, possibly overlapping with our F 150 W and F 200 W -dropout candidates, respectively. We refer to the manuscript version 1 that was submitted to arXiv \nW \n0 \n5 \n1 \nF \n- \nW \n5 \n1 \n1 \nF \n1 \n0 \n3 \n2 \n1 \n0 \n1 \n2 \n1 \n0 \n1 \n2 \nF277W \nF444W \nF150W -F277W --Figure 9. Same as Figure 2, but for evaluating the interlopers of foreground galaxies. The gray curves indicate colors of model galaxies at z = 0 -8 that are produced with PANHIT (Mawatari et al. 2020a). See texts for details of the models. The black arrow indicates a shift of the colors with dust extinction of ∆ E ( B -V ) = +0 . 1. The magenta circle in the right panel is a dusty starburst galaxy at z ∼ 5 that may appear as a z > 15 galaxy discussed in Zavala et al. (2022). Our color selection criteria avoid these low-redshift interlopers at z ∼ 0 -8. \n<!-- image --> \non July 23. Out of 61 and 15 sources in their F 150 W -dropouts and F 200 W -dropouts, we identify 54 and 11 objects in our original photometric catalogs, respectively. However, we cannot identify counterparts of the remaining 11 sources, F150DA-013, F150DA-047, F150DA-057, F150DB-004, F150DB-023, F150DB-056, F150DB-058, F200DB-015, F200DB-109, F200DB-175, and F200DB-181, probably because their SNRs are not sufficient to be identified in this study, the source is severely affected by nearby bright objects, or a WCS offset between Yan et al. (2022) and this study is too large to identify the counterparts. Among the 54 objects identified as F 150 W -dropouts, F150DA-053 at z = 11 . 71 +1 . 56 -0 . 54 is SM-z12-1 at z = 12 . 47 +1 . 19 -0 . 72 in this study. We have checked photometry of the other 53 and 11 objects identified in our original photometric catalogs, but none of them are selected as high redshift candidates in this study, due to their insufficient colors of the break, ∆ χ 2 < 9, and/or an insufficient S/N in the detection image.', '3.6.9. Summary of the Comparisons': 'In summary, we have found that bright candidates reported in previous studies are reproduced in this study, such as GL-z9-1, GL-z12-1, CR2-z12-1, and CR2-z161. However, some of faint candidates reported in other studies are not selected in our selection criteria, because most of these faint candidates are selected by photometric redshifts but with a weak criterion (e.g., ∆ χ 2 > 4) or by relatively weak color selection criteria (e.g., F 150 W -F 200 W > 0 . 5). It is expected that the contamination fraction in such faint candidates is high, given the small ∆ χ 2 values (see discussions in Section 3.3). These comparisons indicate that our selection \ncriteria are conservative enough to remove foreground interlopers while keeping bright and reliable candidates.', '3.7. Contamination': 'We check whether our sample is largely contaminated by foreground interlopers or not. One of the major sources for contamination is low redshift galaxies whose Balmer breaks are redshifted to the wavelength of the Lyman break of our dropout galaxies. To test the effect of such contamination, we make a mock catalog of galaxies with Balmer breaks at z = 0 -8. We first generate model spectra of galaxies by using PANHIT (Mawatari et al. 2020a) assuming a delayedτ star formation history, τ = 0 . 01, 0 . 03, 0 . 1, 0 . 3, and 1 Gyr; stellar age of 0 . 01 -1 . 3 Gyr; and metallicity of Z = 0 . 0001, 0.0004, 0.004, 0.008, 0.02, and 0.05. In Figure 9 we plot model tracks of the z = 0 -8 galaxies with colors of our selected galaxy candidates and our color selection criteria. We find that the z = 0 -8 galaxies with Balmer breaks have a relatively small break color (e.g., F 150 W -F 200 W < 1 . 0 in the F 150 W -dropout selection) or larger break color and red continuum color ( F 150 W -F 200 W > 1 . 0 and F 200 W -F 356 W > 0). Our color selection criteria avoid model tracks of these z = 0 -8 galaxies, and most of our candidates are located far from these model tracks. In the right panel of Figure 9, we also plot a dusty starburst galaxy at z ∼ 5 that may appear as a z > 15 galaxy discussed in Zavala et al. (2022). Such a dusty interloper is also removed from our sample due to the red continuum color ( F 277 W -F 444 W > 1 . 0). \nTo evaluate the effect of contamination from low redshift objects scattering into our selection criteria due \nF115W-dropout \n( \nz \n∼ \n9) \nW \n7 \n7 \n2 \nF \n- \nW \n0 \n0 \n2 \nF \n1 \n0 \n3 \n2 \nF200W-dropout \n( \nz \n∼ \n16) \nZavala+22', 'F115W-Dropout (stacked)': "F150W-Dropout (stacked) \nFigure 10. Stacked images of our F 115 W -dropouts (top) and F 150 W -dropouts (bottom). The size of the images is 1 . '' 5 × 1 . '' 5. There are no positive signals found at the positions of the dropouts in the F 090 W image (top) and the F 090 W and F 115 W images (bottom) whose wavelength ranges (rest-frame < 1216 ˚ A) do not include emission from z ∼ 9 and 12 sources, indicating that our samples are not significantly contaminated by foreground interlopers. \nto the photometric noise at the depth of the observations, we conduct Monte Carlo simulations using the real datasets in the same manner as previous studies (e.g., Bouwens et al. 2015; Ono et al. 2018; Harikane et al. 2022b). We start from multi-band catalogs constructed in the GLASS field whose images are sufficiently deep. We create 100 mock catalogs by perturbing the measured fluxes by adding photometric scatters based on the flux uncertainties in each band in the CEERS and Stephan's Quintet fields whose depths are shallower than the GLASS field. We select high redshift galaxies from the mock catalogs with the same selection criteria as our real selection. In the same manner as Bouwens et al. (2015), we classify sources that are selected but that show detections in the band bluer than the break in the original catalogs as contaminants. Based on these simulations, we find that the contamination rate due to the scatter is < 6% for the F 150 W and F 200 W -dropout selections. The contamination rates for the F 115 W -dropout selection cannot be evaluated with this procedure, because the galaxy selection is conducted only in the GLASS field where we start the simulation. However, the good agreement in z ∼ 9 number densities between our results and previous studies (Section 5.2) indicates that the contamination is not significant in our F 115 W -dropout sample. \nTo further test the contamination, we stack images of the 13 F 115 W -dropout candidates and 8 F 150 W -dropout candidates. If the sample is significantly contaminated by low redshift interlopers, the stacked images should show signals in a band whose wavelength is bluer than the Lyman break. Figure 10 presents stacked images of our F 115 W - and F 150 W -dropout candidates. There are no significant positive signals found in the F 090 W -band and in the F 090 W - and F 115 W -bands for \nthe F 115 W - and F 150 W -dropout candidates, respectively, suggesting that our samples are not significantly contaminated by low redshift interlopers. These tests and comparisons in Section 3.6 indicate that our conservative selection criteria with careful screening of low redshift interlopers provide a reliable sample of z ∼ 9 -16 galaxy candidates, suitable for statistical studies such as luminosity function measurements.", '4. MASS MODEL': "Various mass models for SMACS J0723 are produced by parametric mass modeling algorithms. The RELICS survey (Coe et al. 2019) team provides the mass models 6 developed with the Lenstool (Jullo et al. 2007; Fox et al. 2022) and glafic (Oguri 2010) codes, both of which are constructed using the HST data. A new model with the Lenstool code is constructed with the JWST ERO data in Mahler et al. (2022). Golubchik et al. (2022) recently present a mass model of SMACS J0723 developed by the light-traces-mass ( LTM ; Broadhurst et al. 2005; Zitrin et al. 2009, 2015) approach before the JWST ERO data release, and subsequently Pascale et al. (2022) present the LTM mass modeling with the JWST ERO data. Caminha et al. (2022) also develop the mass model of SMACS J0723 using Lenstool . \nIn this paper, we construct an updated glafic (Oguri 2010, 2021) strong lens mass model of SMACS0723 using the new JWST ERO data. The magnification factors predicted by the updated glafic mass model are compared with those from the other existing mass models to evaluate the lens-model uncertainty. The glafic code performs the so-called parametric lens mass modeling, \nwhere shapes of the mass distributions of the cluster are described by a superposition of a small number of lens mass components with known profile shapes, and parameters characterizing the lens mass components are determined so as to reproduce observed positions of multiple images. \nAs a specific procedure, we largely follow the methodology described in Kawamata et al. (2016). We model the dark matter halo by an elliptical Navarro-FrenkWhite (NFW; Navarro et al. 1997) density profile with an approximation to speed up the calculation of its lensing property (see Oguri 2021). Model parameters associated with the NFW component are the mass, the center, the ellipticity and its position angle, and the concentration parameter. In addition to the main NFW halo, we place an additional NFW component whose center is fixed to a bright cluster member galaxy located at North-West, (R.A., Decl.)=(110 . 7928634, -73 . 4476417). We also fix the concentration parameter of the additional NFW component to c = 10, and fit its mass, ellipticity and position angle only. Cluster member galaxies selected by photometric redshifts from the RELICS HST data (Coe et al. 2019) are modeled by an elliptical pseudo Jaffe profile. In order to reduce the number of parameters, the velocity dispersion σ and the truncation radius r trunc of each cluster member galaxy are assumed to scale with its luminosity (in HST F814W band) as σ ∝ L 1 / 4 and r trunc ∝ L η with their normalization and η being treated as free parameters. In addition we include an external shear to improve the fitting. For multiple image sets without spectroscopic redshifts, we simultaneously fit their redshifts. \nWe search for the best-fitting model by the standard χ 2 minimization, where χ 2 is computed from the differences between observed and model-predicted positions. We assume the positional error of 0 . '' 4, which is a typical positional accuracy achieved by the parametric strong lens mass modeling (Kawamata et al. 2016). The χ 2 is evaluated in the source plane, taking account of the full magnification tensor at each multiple image position (see Appendix 2 of Oguri 2010). Errors on model parameters are derived using the standard Markov Chain Monte Carlo technique. Multiple images are identified iteratively, starting with secure sets of multiple images that are obvious from their colors, morphologies, and redshifts, constructing a preliminary mass model with those sets of multiple images, and searching for new multiple sets with help of the preliminary mass model. In this work, we use conservative 12 sets of multiple images for our strong lens mass modeling, with the total number of multiple images of 38. These multiple image sets are mostly consistent with other work using different \nTable 6. Summary of Magnification Factors Estimated by Various Mass Models \nNote - (1) Name. (2) Spectroscopic or photometric redshift. (3)-(6) Magnification factors estimated by grafic , Mahler et al. (2022), Pascale et al. (2022), and Caminha et al. (2022). \nlens modeling codes (Pascale et al. 2022; Mahler et al. 2022). We adopt spectroscopic redshifts for five sets of multiple images given in the literature (Golubchik et al. 2022; Pascale et al. 2022; Mahler et al. 2022). Our bestfitting model has χ 2 = 28 . 3 for degree of freedom of 32, representing a good fit. The root-mean-square (rms) of differences between observed and predicted multiple image positions is 0 . '' 35. \nWith the updated glafic mass model, we calculate the magnification factors µ of our dropout galaxy candidates and the effective survey volume. Table 6 summarizes the magnification factors of our dropout galaxy candidate and spectroscopically confirmed galaxies at z > 7 calculated by glafic , Mahler et al. (2022), Pascale et al. (2022), and Caminha et al. (2022). We find that the magnification factor calculated by each model agree well typically within ∼ 20%.", '5.1. Sample Completeness': "To derive the rest-frame UV luminosity function, we estimate the completeness of our dropout galaxy selection in the same manner as previous studies (e.g., Ouchi et al. 2009; Ono et al. 2018; Harikane et al. 2022b). We conduct Monte Carlo simulations with real NIRCam images and artificial galaxies mocking high redshift galaxies. The mock high redshift galaxies follow the sizeM UV redshift distribution revealed with the HST legacy data sets for galaxies at z ∼ 0 -10 (Shibuya et al. 2015) that is extrapolated to our redshift ranges, where the sizeM UV distribution is the log-normal distribution. Our initial measurements of sizes for our z > 9 galaxy candidates are consistent with this assumption within the uncertainties. We adopt the S'ersic index n = 1 found in typical galaxies at z ∼ 5 -10 (Ono et al. 2013; Shibuya \net al. 2015) and the flat distribution of the intrinsic ellipticity in the range of 0.0-0.8. Recent studies indicate that morphologies of z ∼ 9 -16 galaxies identified in the JWST datasets are consistent with these assumptions (e.g, Ono et al. 2022). The SEDs of the mock high redshift galaxies uniformly distribute over magnitude and redshift, and have a color distribution agreeing with the M UV -β UV relation observationally determined at z ∼ 8 (Bouwens et al. 2014), where β UV is the UV spectral slope index. The IGM absorption of Inoue et al. (2014) is applied to the SEDs, which produces absorption features in the wavelengths shorter than the Ly α line. We produce 100 artificial objects of the mock high redshift galaxies with IRAF mkobject in each redshift and magnitude bin, and place the artificial objects on the real JWST NIRCam images. With the images, we perform the object detections, photometry, the color selection, and the SED fitting in the same manner as Section 3. In the SMACS J0723 field, we consider the source magnification and multiply lensed images by using the mass model made with glafic described in Section 4. Finally we calculate the selection completeness as a function of magnitude and redshift, C ( m,z ), with the photometric catalogs of the artificial high redshift galaxies. Figure 11 presents examples of the selection completeness thus obtained. Although the average redshifts are z = 10 . 1 for F 115 W -dropouts, z = 13 . 8 for F 150 W -dropouts, and z = 18 . 7 for F 200 W -dropouts, we use the median of photometric redshifts of our selected candidates, z = 9 . 1 for F 115 W -dropouts, z = 12 . 0 for F 150 W -dropouts, and z = 16 . 3 for F 200 W -dropouts as the representative redshifts of our each dropout sample. \nBased on the results of these selection completeness simulations, we estimate the survey volume per unit area as a function of apparent magnitude (Steidel et al. 1999), \nV eff ( m ) = ∫ C ( m,z ) dV ( z ) dz dz, (10) \nwhere dV ( z ) /dz is the differential comoving volume as a function of redshift. The space number density of our galaxy candidates that is corrected for incompleteness is calculated with the following equation: \nψ ( m ) = n ( m ) V eff ( m ) , (11) \nwhere n ( m ) is the surface number density of selected galaxies in an apparent magnitude bin of m . We convert the number density as a function of apparent magnitude, ψ ( m ), into the UV luminosity functions, Φ[ M UV ( m )], which are the number densities of galaxies as a function of rest-frame UV absolute magnitude. Assuming a flat rest-frame UV continuum, we calculate the absolute UV \nFigure 11. Selection completeness for our dropout galaxies. The purple, blue, and green curves show selection completeness for the F 115 W -, F 150 W -, and F 200 W -dropout galaxies whose rest-frame UV ( ∼ 1500 ˚ A) magnitudes are F 200 W = 27 . 0 mag, F 277 W = 27 . 0 mag, and F 356 W = 27 . 0 mag, respectively. Each selection window is smoothed by ∆ z = 1 . 0. \n<!-- image --> \nmagnitudes of galaxies from their apparent magnitudes in the bluest band not affected by the Lyman break, i.e., F 200 W , F 277 W , and F 356 W -bands for F 115 W , F 150 W , and F 200 W -dropout galaxy candidates, respectively. The 1 σ uncertainty is calculated by taking into account the Poisson confidence limit (Gehrels 1986) and the cosmic variance. We estimate the cosmic variance in the number densities using the bias values of z ∼ 7 galaxies obtained in Harikane et al. (2016), following the procedures in Somerville et al. (2004).", '5.2. Results': 'Figures 12 and 14 present our luminosity functions at z ∼ 9, 12, and 16 together with luminosity functions obtained by previous work including the latest JWST studies (Naidu et al. 2022b; Donnan et al. 2022; Finkelstein et al. 2022b; Bouwens et al. 2022b). Our measurements of the luminosity functions are summarized in Table 7. Comparing with the previous measurements of the luminosity functions, we find that our luminosity functions at z ∼ 9 and 12 agree well with those of the previous HST and JWST studies within the uncertainties, as shown in Figure 12. In Figure 14, we compare the luminosity function of our possible candidates at z ∼ 16 newly determined by this study with those available at lower redshifts at z ∼ 14 constrained by JWST. We confirm that these luminosity functions are comparable. \nWe conduct χ 2 minimization fitting of the double power-law and Schechter functions to the luminosity functions that include the measurements at the bright \n<!-- image --> \n<!-- image --> \nUV \nFigure 12. UV luminosity functions at z ∼ 9 (left) and 12 (right). The red diamonds represent the number densities of our galaxy candidates, while the red arrows indicate the 1 σ upper limits. The errors include the cosmic variance (see text). The red solid and dashed lines are our best-fit double power-law and Schechter functions, respectively. In the left panel, the orange circles indicates the luminosity functions at z ∼ 9 obtained in Donnan et al. (2022) using JWST data, and the gray symbols and shades denote the results at z ∼ 9 derived by the previous studies using HST or ground-based telescope data, Oesch et al. (2013, squares), McLeod et al. (2016, right-pointing triangles), Morishita et al. (2018, pentagons), Stefanon et al. (2019, triangles), Bowler et al. (2020, circles), Bouwens et al. (2021, diamonds), Rojas-Ruiz et al. (2020, hexagons), Leethochawalit et al. (2022, down-pointing triangle), Finkelstein et al. (2022a, shade with dotted lines), and Bagley et al. (2022a, shade with dashed lines). In the right panel, the orange circles, the red circle, the orange down-pointing triangle, and the orange squares indicate the number density of galaxies at z ∼ 12, z ∼ 13, z ∼ 10 -13, and z ∼ 12 -13 reported by Donnan et al. (2022), Harikane et al. (2022a), Naidu et al. (2022b), and Bouwens et al. (2022b), respectively. The gray open symbols indicate the luminosity functions at z ∼ 10 obtained by McLeod et al. (2016, diamonds), Oesch et al. (2018, squares), Morishita et al. (2018, pentagons), and Bowler et al. (2020, circle). The green open star mark represents the number density of GN-z11 (Oesch et al. 2016). See Harikane et al. (2022a) for the estimate of the number density and the UV magnitude of GN-z11. Our estimated luminosity functions at z ∼ 9 and 12 agree well with previous HST and JWST results.Figure 13. Same as the right panel of Figure 12, but with the fitting results without the brightest datapoint in Harikane et al. (2022a). \n<!-- image --> \nend in the literature. In the fitting, we use the results of this study, Morishita et al. (2018), Bowler et al. \n(2020), and Bouwens et al. (2021) for the z ∼ 9 luminosity function, the results of this study and Harikane et al. (2022a) for the z ∼ 12 luminosity function assuming that the UV luminosity function does not rapidly change at z ∼ 12 -13, and the result of this study for the z ∼ 16 luminosity function. We show the best-fit functions in Figures 12 and 14, and present the best-fit parameters in Table 8. At z ∼ 9, the χ 2 values of the fitting suggest that the double power-law function explains the luminosity functions ( χ 2 / dof = 2 . 3 / 9) better than the Schechter functions ( χ 2 / dof = 3 . 6 / 10), albeit with the moderately small difference of χ 2 ( ∼ 1 σ ). At z ∼ 12 and 16, we find no significant differences between the double power-law and Schechter functions in the χ 2 values, probably due to the large uncertainties of the measurements. At z ∼ 12, we also fit only the measurements of this study, excluding the brightest datapoint in Harikane et al. (2022a), as shown in Figure 13. The best-fit DPL and Schechter functions are slightly flatter than the fitting results with the datapoint in Harikane et al. (2022a) at the bright end. \nFigure 14. UV luminosity function at z ∼ 16. The red diamond and the arrows represent the number density of our galaxy candidates and the 1 σ upper limits, respectively. For reference, we show the UV luminosity functions at the lower redshifts, z ∼ 14 (Donnan et al. 2022; orange filled circle), z ∼ 14 (Finkelstein et al. 2022b; orange filled square) z ∼ 12 (this study; gray open diamonds), z ∼ 13 (Harikane et al. 2022a; gray open circle), and z ∼ 10 -13 (Naidu et al. 2022b; gray open down-pointing triangle). \n<!-- image --> \nFigure 15 presents the redshift evolution of the luminosity function. We find the continuous decrease of luminosity functions from z ∼ 5 to z ∼ 12. We do not find a significant decrease from z ∼ 12 to 16 beyond the uncertainty. There is a hint of a small evolution from z ∼ 12 to 16, while the small number statistics do not allow us to conclude whether the evolutionary trend changes from z ∼ 5 -12 to 12 -16. \nFigure 16 compares the observed luminosity functions at z ∼ 12 and 16 with those predicted by theoretical models (Dayal et al. 2014, 2019; Yung et al. 2020; Behroozi et al. 2020; Wilkins et al. 2022; Mason et al. 2022). At z ∼ 12, most of the models in Figure 16 explain the observational measurements in the faint magnitude range from -20 to -18 mag, while some models do not reproduce the moderately high number densities of the observational measurements at the bright magnitude of M UV < -20 mag. At z ∼ 16, most of the models cannot reproduce the observed number density of bright galaxies at M UV < -20 mag, except for the FLARES (Lovell et al. 2021; Vijayan et al. 2021; Wilkins et al. 2022) whose prediction at z ∼ 15 agrees with our number density estimate within uncertainties. Similarly, Figure 17 shows the predicted number of bright galaxies at z ∼ 12 -16 with M UV < -20 mag. Figure 17 indicates that the models underpredict the number of galaxies compared to the observation, although the sig- \nFigure 15. Evolution of the UV luminosity functions from z ∼ 4 to z ∼ 16. The yellow, red, and pink diamonds represent our measurements of the luminosity functions at z ∼ 9, 12, and 16, respectively, whereas the red circle is the one obtained by Harikane et al. (2022a) at z ∼ 13. The orange, yellow, green, blue, purple, brown, and gray symbols indicate the luminosity functions at z ∼ 10, 9, 8, 7, 6, 5, and 4, respectively. The circles at z ∼ 4 -7 and 8 -10 are the data taken from Harikane et al. (2022b) and Bowler et al. (2020), respectively. The squares at z ∼ 4 -9 and z ∼ 10 are the data of Bouwens et al. (2021) and Oesch et al. (2018), respectively. The diamond at z ∼ 10 represents the result of McLeod et al. (2016). The lines denote the double powerlaw functions derived by the previous studies for z ∼ 4 -7 (Harikane et al. 2022b) and z ∼ 8 -13 (Bowler et al. 2020). For clarity, we shift the data point of Bowler et al. (2020) at z ∼ 10 by -0 . 2 mag. \n<!-- image --> \nUV \nnificance is small and more data are needed to obtain the conclusion. This difference of the observations and models would suggest that the feedback effects in the models may be too strong to produce abundant bright galaxies, lower dust obscuration in these bright galaxies than the model assumptions, and/or that there exist hidden AGNs that produce radiation comparable with or more than stellar components of the galaxies (e.g., Bowler et al. 2014, 2020; Ono et al. 2018; Stevans et al. 2018; Shibuya et al. 2022; Harikane et al. 2022b; Pacucci et al. 2022; Mason et al. 2022), although there is also a possibility that this difference may be caused by other physical processes, as discussed in Section 6.', '5.3. Cosmic SFR Density': 'We derive the cosmic SFR densities at z ∼ 9, 12, and 16. We integrate the best-fit double power-law functions (Table 8) down to -17 mag, the same limit as previous studies (e.g., Bouwens et al. 2015; Oesch et al. 2018; Harikane et al. 2022a), and obtain the UV luminosity \n<!-- image --> \n<!-- image --> \nUV \nFigure 16. Comparison of the luminosity-function measurements with theoretical predictions and the empirical models at z ∼ 12 (left) and z ∼ 16 (right). The blue lines show the theoretical and empirical models obtained by Dayal et al. (2014, 2019, solid line), Yung et al. (2020, dotted line), Behroozi et al. (2020, dotted-dashed line), Wilkins et al. (2022, double-dotted dashed line), and Mason et al. (2022, dashed line; their no dust model). The red and orange symbols show observational results in the same manner as Figures 12 and 14. The red diamonds and arrows represent the measurements and upper limits obtained by this study. The orange circles, the red circle, the down-pointing orange triangle, and the orange square in the left (right) panel indicate the number densities reported by Donnan et al. (2022), Harikane et al. (2022a), Naidu et al. (2022b), and Bouwens et al. (2022b) (Finkelstein et al. (2022b)), respectively.Figure 17. Theoretical predictions for the number of bright galaxies at z ∼ 12 -16 with M UV < -20 mag detected in our survey area of ∼ 90 arcmin 2 . These numbers are based on the theoretical models of Dayal et al. (2014, 2019), Yung et al. (2020), Behroozi et al. (2020), Wilkins et al. (2022), and Mason et al. (2022). The red horizontal line with the shaded region indicates the number of observed galaxies at z ∼ 12 -16 with M UV < -20 mag ( N obs = 4 ± 2), which is higher than these model predictions. \n<!-- image --> \nTable 7. Obtained Luminosity Function at Each Redshift \nNote -Errors and upper limits are 1 σ . \nTable 8. Fit Parameters for Luminosity Functions \nNote -Errors are 1 σ . \nTable 9. Obtained Cosmic UV Luminosity Density and SFR Density \nNote -Errors are 1 σ . ρ SFR , UV and ρ SFR are SFR densities without and with dust extinction correction, respectively. \ndensities, ρ UV . We correct ρ UV for the dust extinction, following the attenuation-UV slope ( β UV ) relation (Meurer et al. 1999) and β UV -M UV relation at z = 8 in Bouwens et al. (2014). The choice of these assumptions (e.g., using the attenuation-UV slope law in de Barros et al. 2014 instead) does not affect our conclusions because the correction factor is very small ( glyph[lessorsimilar] 0 . 1 dex). We calculate SFRs from UV luminosities, L UV , corrected for dust extinction by the relation, \nSFR ( M glyph[circledot] yr -1 ) = K UV L UV (erg s -1 Hz -1 ) , (12) \nwhere K UV is the conversion factor that depends on the recent star-formation history, metal enrichment history, and the choice of the IMF. Here we apply K UV = 1 . 15 × 10 -28 M glyph[circledot] yr -1 / (erg s -1 Hz -1 ) that is used in Madau & Dickinson (2014). This value of K UV is valid for the Salpeter (1955) IMF, and consistent with the cosmic star-formation history and the evolved stellar metallicity (10 -0 . 15 z Z glyph[circledot] ; Madau & Dickinson 2014) up to z ∼ 10. Table 9 summarizes our measurements of the cosmic UV \nluminosity density, SFR densities without and with dust extinction correction at each redshift. \nFigure 18 presents the cosmic SFR density evolution. In this figure, we show the cosmic SFR density measurements at z ∼ 0 -10 obtained by previous studies, all of which are converted to the calibration of Madau & Dickinson (2014) with the Salpeter (1955) IMF (Equation (12)). We confirm that our SFR density at z ∼ 9 is consistent with the previous measurements. We compare the observational measurements of the SFR densities with the constant star-formation efficiency ( SFR/ ˙ M h ( z ) = const . ) model (Harikane et al. 2022b) together with the extrapolation of the Madau & Dickinson (2014) estimates at z = 0 -8. We find that the cosmic SFR densities significantly decrease from z ∼ 9 to 12. A decrease of the cosmic SFR densities may exist from z ∼ 12 to 16, while the decrease is not larger than the errors. Interestingly, The constant star-formation efficiency model explains the evolution of the cosmic SFR densities up to z ∼ 10 (Harikane et al. 2022b), while \nFigure 18. Cosmic SFR density evolution. The red circles represent the cosmic SFR densities obtained by our study, with the double power-law luminosity functions integrated down to M UV = -17 mag. The black circles indicate the cosmic SFR densities derived by Madau & Dickinson (2014), Finkelstein et al. (2015b), McLeod et al. (2016), Bhatawdekar et al. (2019), and Bouwens et al. (2020). The orange circles are results in Donnan et al. (2022). The blue dashed curve is the best-fit function of the cosmic SFR densities in Harikane et al. (2022b, their Equation 60). In Harikane et al. (2022b), they assume the constant star formation efficiency at z > 10, resulting in the power-law decline with ρ SFR ∝ 10 -0 . 5(1+ z ) . The gray dashed curve shows the best-fit function at z glyph[lessorsimilar] 8 determined by Madau & Dickinson (2014) extrapolated to z > 8. All results are converted to those of the Salpeter (1955) IMF. \n<!-- image --> \nour measurement at z ∼ 12 is higher than the model prediction beyond the uncertainty. Moreover, there is a hint of a high cosmic SFR density at z ∼ 16 above the model, although it is not statistically significant due to the large error. Such higher SFR densities than the constant efficiency model at z ∼ 15 is actually consistent with observations of Balmer break galaxy candidates at z ∼ 6 (Mawatari et al. 2020b).', '6.1. Possible High Cosmic SFR Density at z > 10': 'Our observational measurements suggest that the SFR densities at z ∼ 12 -16 are higher than the constant starformation efficiency model of Harikane et al. (2022b). Although the constant star-formation efficiency model well explains the cosmic SFR densities at z ∼ 0 -10, this model underpredicts those at z ∼ 12 -16. Here we discuss the following three possibilities that explain the observed high SFR densities at z ∼ 12 -16. \n- (A) No star formation suppression at the prereionization epoch. The universe at z ∼ 12 -16 is at the pre-reionization epoch when the IGM is highly neutral (Ouchi et al. 2020; Robertson 2021). At the epoch of reionizaton (EoR; z ∼ 6 -12) and the epoch of post-reionization (postEoR; z glyph[lessorsimilar] 6), galaxies and AGN produce UV radiation by their star-formation and nuclear activity, and produce strong UV background radiation. The UV background radiation heats up Hi gas in low-mass halos of M h glyph[lessorsimilar] 10 8 -9 M glyph[circledot] with negligible Hi self-shielding, suppressing star-formation at the EoR and post-EoR (Barkana & Loeb 2000; Susa & Umemura 2004; Hoeft et al. 2006; Pawlik & Schaye 2009; Mesinger et al. 2009; Sawala et al. 2010; Bland-Hawthorn et al. 2015). Although the halo masses of our galaxies at z ∼ 12 -16 are unknown, the maximum halo mass existing at z ∼ 15 given the survey volume of this study is M h glyph[similarequal] 3 × 10 9 M glyph[circledot] in the structure formation model with the Planck cosmology (Behroozi et al. 2020). In other words, most of halos (with glyph[lessorsimilar] 10 9 M glyph[circledot] ) at z ∼ 15 are not affected by the UV background at the pre-reionization epoch, while the similar halos at z glyph[lessorsimilar] 10 experience the suppression of star formation by the UV background at the EoR and postEoR. To test whether this effect can quantitatively explains the observed SFR densities, we construct a model of the SFR density evolution including the enhancement of the star formation that is free from the suppression by the UV background at the pre-EoR. We use a model in Barkana & Loeb (2000) with a reionization redshift of z reion = 13, and multiply the prediction of the constant star formation efficiency model (Harikane et al. 2022b) by a factor of the star formation rate enhancement due to no suppression by the UV background in Barkana & Loeb (2000). The left panel of Figure 19 presents this hybrid model including the effect of star formation enhancement at pre-EoR, which reproduces the observed SFR densities at z ∼ 12 -16 within uncertainties. This agreement indicates a possibility that the star formation efficiency at z ∼ 12 -16 is higher than those at z glyph[lessorsimilar] 10 due to no suppression of the star formation activity at the pre-EoR.\n- (B) Presence of AGN activity. Another possibility is that a large fraction of the observed UV luminosity densities at z ∼ 12 -16 is produced by AGN, and there is no excessive SFR densities at z ∼ 12 -16 beyond the constant star-formation efficiency model. This is an interesting scenario that \n<!-- image --> \nFigure 19. Possible scenarios to explain the observed SFR densities at z > 10. (Left:) Scenario of no star formation suppression at pre-reionization epoch. At the reionization epoch and after that, star formation in low-mass halos is suppressed by strong UV background radiation, while before the reionization epoch such a suppression of star formation activity does not occur. The upper edge of the blue shaded region indicates the enhancement of the star formation by this effect (Barkana & Loeb 2000), which explains the observed SFR densities (see texts for details). (Right:) Scenario of Pop III star formation. As shown in Figure 20, Pop III stellar populations with a top-heavy IMF produces a significant amount of UV photons at a given SFR, resulting in the overestimates of the SFR densities if we use the canonical UV-SFR conversion factor. The red and orange filled circles at z > 10 . 5 are SFR densities calculated based on the conversion factor for a Pop III stellar population with a top-heavy IMF (the PopIII.1 model in Figure 20), which agree well with the constant star formation efficiency model in Harikane et al. (2022b). The open circles are SFR densities based on the canonical conversion factor. \n<!-- image --> \nmitigates the existence of supermassive black holes (SMBH) at z ∼ 7 (Mortlock et al. 2011, Ba˜nados et al. 2018, Wang et al. 2021) by efficient gas accretion on SMBHs creating AGNs, while a standard gas accretion limited by the Eddington accretion rate does not explain the existence of the SMBHs at z ∼ 7. However, our z ∼ 12 -16 candidates except for GL-z12-1 show extended morphologies (Section 3.5). Thus the fraction of AGN radiation dominated galaxies is as small as ∼ 10% (= 1 / 10) at z ∼ 12 -16. Although the excessive SFR density estimate at z ∼ 16 is unclear due to the small statistics, the one at z ∼ 12 cannot be explained by AGN activity. \n(C) A top-heavy IMF. The third possibility is an overestimate of the SFR density due to a top-heavy IMF possibly with the Population III (Pop III) stellar population. In our estimate of the SFR density, we use the canonical UV luminosityto-SFR conversion factor of K UV = 1 . 15 × 10 -28 M glyph[circledot] yr -1 / (erg s -1 Hz -1 ), which is for the Salpeter (1955) IMF, while K UV depends on starformation history, metallicities, and IMFs (e.g., Madau & Dickinson 2014; Tacchella et al. 2018). Indeed in the early universe, the IMF is expected to be more top-heavy because of a lower metallic- \nty or a higher CMB temperature (e.g., Omukai et al. 2005; Chon et al. 2022), resulting in a higher Jeans mass, especially for Pop III stellar populations (e.g., Hirano et al. 2014, 2015). To test whether this effect can explain the observed densities, we calculate the UV-to-SFR conversion factor, K UV , for different metallicity and IMF assumptions using Yggdrasil (Zackrisson et al. 2011). Figure 20 presents K UV for different metallicities and IMFs as a function of stellar age. We find that Pop III stellar populations with top-heavy IMFs (PopIII.1 and PopIII.2 in Yggdrasil ) produce ∼ 3 -4 times more UV photons than the canonical assumption given the SFR, because nebular continuum emission boosts the UV luminosity as discussed in previous studies (e.g., Zackrisson et al. 2008; Schaerer & de Barros 2009, 2010). This low conversion factor reduces the SFR density estimates at z ∼ 12 -16 as shown in the right panel of Figure 19, resulting in the SFR densities consistent with the constant star formation efficiency model. \nBased on these discussions, we conclude that (A) no star formation suppression at pre-reionization epoch or (C) a top-heavy IMF with a Pop III-like star formation can explain the observed high SFR densities at \nFigure 20. UV luminosity-SFR conversion factor, K UV , for various metallicities as a function of stellar age. The green, cyan, and blue curves show the conversion factor for metallicities of Z = 0 . 02, 0 . 004, 0 . 0004, respectively, calculated with Yggdrasil (Zackrisson et al. 2011) assuming a constant star formation history with a unity gas covering fraction. These factors are values for a UV luminosity at 1500 ˚ A in the Salpeter (1955) IMF in the interval of 0 . 1 -100 M glyph[circledot] . Note that the original outputs from Yggdrasil are for the Kroupa (2001) IMF, and we correct for the IMF difference by multiplying the outputs by 1.49. The solid (PopIII.1) and dashed (PopIII.2) purple curves show the conversion factors for Pop III stellar populations with an extremely top-heavy IMF (50-500 M glyph[circledot] , the Salpeter 1955 slope) and a moderately top-heavy IMF (log-normal with characteristic mass of M c = 10 M glyph[circledot] , dispersion σ = 1 M glyph[circledot] , and wings extending from 1 -500 M glyph[circledot] ). If galaxies at z > 10 are dominated by Pop III stellar populations, the conversion factor is significantly lower than the typically assumed value ( K UV = 1 . 15 × 10 -28 M glyph[circledot] yr -1 / (erg s -1 Hz -1 ); the black line), resulting in the overestimate of the SFR. \n<!-- image --> \nz ∼ 12 -16. These possibilities can be further investigated by follow-up observations with JWST/MIRI covering a longer wavelength than the Balmer break to obtain the robust stellar mass measurements and star formation history, or with JWST/NIRSpec and MIRI spectroscopy to search for signatures of Pop III-like stellar populations and AGN activity.', '6.2. Properties of Luminous Galaxy Candidates': 'In this study, we have found several luminous galaxy candidates at the early epoch of z ∼ 10 -16, when the age of the universe is only ∼ 200 -500 Myrs after the Big Bang. Here we discuss physical properties of these luminous galaxy candidates. \nTable 10. SFRs and Stellar Masses of Luminous Galaxy Candidates with M UV < -19 . 5 mag \nNote -Assuming the Chabrier (2003) IMF and metallicity of Z = 0 . 2 Z glyph[circledot] . The SFR is averaged over the past 50 Myr in the same manner as Tacchella et al. (2022). See Section 3.3 for the details of the SED fitting. \n- † This candidate shows a compact morphology indicative of AGN activity, while the profile is spatially extended more than the PSF (Section 3.5). \nTable 10 summarizes SFRs and stellar masses of six galaxy candidates whose UV magnitudes are brighter than M UV = -19 . 5 mag, constrained by the SED fitting in Section 3.3 assuming the Chabrier (2003) IMF. Our estimates of SFRs and stellar masses agree with previous estimates by Naidu et al. (2022b), Donnan et al. (2022), and Finkelstein et al. (2022b), indicating that these luminous galaxies are very massive with stellar masses as high as M ∗ ∼ (1 -10) × 10 8 M glyph[circledot] at z ∼ 10 -16. While the contributions from AGN radiation to the SEDs may be suspected in one of the objects, GL-z12-1 (see Sections 3.5 and 6.1), at least the rest of the objects (i.e. ∼ 80% of the bright z ∼ 10 -16 galaxies) would be truly stellar massive. Although the NIRCam photometry is limited to < 5 µ m and does not trace the SEDs beyond the Balmer break (4000 ˚ A) corresponding to 5 -7 µ m in the observed frame at z ∼ 10 -16, these stellar mass estimates provide rough lower limits that miss the contribution from old stellar populations beyond the Balmer break, given high specific SFRs of these galaxy candidates, SFR/M ∗ ∼ 10 -8 yr -1 . \nHere is a question how these galaxies with the large stellar masses form at this early epoch of z ∼ 10 -16. To discuss the formation scenario of these massive galaxy candidates, we estimate the stellar-to-halo mass ratio (SHMR) of these galaxies. Using the abundance matching technique in the same manner as Harikane et al. (2016, their Equation (66)), we estimate the halo mass of the most massive halo that can be observed with the survey volume in this study, resulting in 5 × 10 10 M glyph[circledot] and 5 × 10 9 M glyph[circledot] at z ∼ 12 and z ∼ 16, respectively. \nFigure 21. Stellar masses of our galaxy candidates as a function of redshift. The red filled diamonds show the stellar mass estimates for six luminous galaxy candidates with M UV < -19 . 5 mag at z ∼ 10 -16 (Table 10), and open red circles are results for other candidates. The gray shaded region indicates the stellar mass whose number density is below the observational limit, calculated from the cosmic baryon fraction (Ω b / Ω m = 0 . 16) and the maximum halo mass that can be observed with the survey volume of this study. The black and blue curves indicate the stellar masses calculated from the maximum halo mass with M ∗ /M h = 0 . 1 and the maximum M ∗ /M h value at each redshift in Behroozi et al. (2020), respectively. The massive stellar masses ( M ∗ ∼ 10 9 M glyph[circledot] ) of the two z ∼ 16 candidates can be explained by a very high SHMR of M ∗ /M h = 0 . 1, indicating a star formation efficiency as high as ∼ 60%. The other four luminous candidates at z ∼ 10 -13 also show higher stellar masses compared to the predictions from Behroozi et al. (2020). \n<!-- image --> \nFrom the stellar mass estimates discussed above, the SHMRs of z ∼ 12 and z ∼ 16 galaxies are ∼ 0 . 01 and ∼ 0 . 1, respectively. Because the cosmic baryon to dark matter density ratio is Ω b / Ω m = 0 . 16 (Planck Collaboration et al. 2020), the SHMRs of the z ∼ 16 galaxies reach ∼ 60% of the cosmic baryon fraction, as shown in Figure 21. In other words, more than a half of baryon gas in the halos is converted to stars, which is unlikely found in lower-redshift and present-day galaxies whose SHMRs are ∼ 0 . 02 -0 . 03 at maximum (e.g., Harikane et al. 2016; Behroozi et al. 2019). A similar conclusion is obtained from the comparison of the UV luminosity functions (Inayoshi et al. 2022). However, theoretical models predict such efficient star formation at the pre-reionization epoch where 70 -80% of baryon are converted stars (Susa & Umemura 2004) in halos with 10 8 -9 M glyph[circledot] masses in a few hundred Myr, when the UV background radiation is too weak to suppress star forma- \non (A) in Section 6.1). The other four galaxies at z ∼ 11 -14 also show higher stellar masses compared to the predictions from the maximum SHMR in Behroozi et al. (2020), indicating elevated star formation efficiencies, probably due to no suppression of star formation activity at the pre-reionization epoch. Another possibility is that the SFR and the stellar mass of these bright galaxies are overestimated due to the assumption of the IMF and metallicity in the SED fitting, as discussed in Section 6.1 (discussion (C)). Indeed, if we assume that the stellar population of these galaxies are dominated by Pop III with a top-heavy IMF, the SFR and stellar mass are reduced by a factor of ∼ 3 -4, more comparable to the observed SHMRs at lower redshifts. These comparisons, together with the discussions in Section 6.1, indicate that the observed properties of z ∼ 10 -16 galaxies (i.e., high cosmic SFR densities and massive stellar masses) can be explained by either no star formation suppression by UV background radiation at the pre-reionization epoch or a top heavy IMF possibly with a Pop III-like stellar population.', '7. SUMMARY': "In this paper, we have conducted comprehensive analyses for the JWST/NIRCam images taken by the JWST ERO SMACS J0723, Stephan's Quintet, ERS GLASS, and CEERS projects, covering a total of ∼ 88 . 7 arcmin 2 , in conjunction with the supports of the ERO SMACS J0723 NIRSpec spectra. We reduced the NIRCam datasets using the new calibration parameters released in October, based on calibration observations of three different standard stars placed in all of the 10 NIRCam detectors. Our major findings are summarized below: \n- 1. We have selected dropout galaxy candidates at z ∼ 9, z ∼ 12, and z ∼ 16 showing significant continuum breaks in the NIRCam F 115 W , F 150 W , and F 200 W -bands, respectively, by the color criteria, confirming clear non-detections in the band(s) whose wavelength is shorter than the continuum breaks including the F 090 W band (Section 3.2, Figure 2). Because we have found that a weak photoz criterion of ∆ χ 2 > 4 cannot remove a number of foreground interlopers on the bases of the JWST simulation data produced by the CEERS project team (Figure 3), we apply a stringent photoz determination criterion of ∆ χ 2 > 9 with the prospector code for our galaxy selection. We thus identify 13, 8, and 2 dropout galaxy candidates at z ∼ 9, z ∼ 12, and z ∼ 16, respectively (Table 2). We confirm that our photometric redshifts agree well with the spectroscopic redshifts, by applying our photometric redshift tech- \nat z spec ∼ 8 -9 found by the ERO NIRSpec observations (Figures 4 and 5). \n- 2. We have thoroughly compared our dropout galaxy candidates with other high redshift galaxies reported in a number of recent studies in the ERO SMACS J0723 and the ERS GLASS+CEERS NIRCam fields. We have summarized the candidates so far claimed in the literature together with our dropouts in Tables 3-5. For bright galaxy candidates, we find that a reasonable fraction of galaxies are commonly selected in our and previous studies. We confirm that, among all of the candidates, our dropout galaxies show the significant Ly α continuum breaks and flat UV continua with non-detections of continua whose wavelengths are shorter than the break (Figures 7-8), and conclude that we do not miss many reliable candidates in the redshift range of z ∼ 9 -16 in our selection.\n- 3. We have derived the UV luminosity functions at z ∼ 9, 12, and 16 (Figures 12 and 14). The UV luminosity functions at z ∼ 9 and 12 agree with those of previous HST and JWST studies within uncertainties including the cosmic variance, and the UV luminosity function at z ∼ 16 is newly constrained. The double power-law function is preferred to the Schechter function at z ∼ 9, albeit with the moderately small difference of χ 2 .\n- 4. The cosmic SFR densities at z ∼ 9, 12, and 16 are derived by the integration of the best-fit UV luminosity functions (Figure 18). By the comparisons with the previous low-redshift determinations of cosmic SFR densities, we find that the cosmic SFR densities significantly decrease from z ∼ 9 to 12. A decrease of the cosmic SFR densities may exist from z ∼ 12 to 16, while the decrease is not larger than the errors. Our measurements of the cosmic SFR density at z ∼ 12 is higher than predictions from the constant star-formation efficiency model (Harikane et al. 2022b), while the model explains the cosmic star-formation history at z glyph[lessorsimilar] 10. Moreover, there is a hint of a high cosmic SFR density at z ∼ 16 above the model, although it is not statistically significant due to the large error.\n- 5. There are several luminous and massive galaxy candidates with M UV < -19 . 5 mag at the early epoch of z ∼ 10 -16, when the age of the universe is only ∼ 200 -500 Myrs after the Big Bang (Figure 21). We confirm that our stellar mass estimates are comparable with those of the previous studies. Although one of the objects \nmay have contributions of UV radiation from an AGN suggested by their morphologies, a majority ( ∼ 80%) of the galaxies may be truly stellar massive. By the comparisons with the structure formation models that provide the upper limits of the dark-matter halo masses observed in this study, the SHMR of the luminous galaxy candidates at z ∼ 16 is M ∗ /M h ∼ 0 . 1, corresponding to ∼ 60% of the baryon to dark matter density ratio in the Planck cosmology, indicating that most of baryon may be converted to stars, unlike lower-redshift and present-day galaxies with a reasonably small SHMR up to M ∗ /M h glyph[similarequal] 0 . 02 -0 . 03 (e.g., Harikane et al. 2016; Behroozi et al. 2019). The other candidates at z ∼ 10 -13 also have stellar masses more massive than predictions from the maximum SHMR in Behroozi et al. (2020). \n- 6. This study identifies two interesting observational properties of galaxies at z ∼ 10 -16, the cosmic SFR densities higher than the constant starformation efficiency model and the existence of the UV-luminous galaxies with high stellar masses. The possibility of the AGN contribution can be ruled out, because the small fraction of galaxies have compact morphologies suggesting no dominant radiation from the AGN activity. Instead, there are two scenarios that explain the observational properties (Figure 19). One scenario is that the UV background radiation does not suppress the star formation at the pre-reionization epoch unlike at the EoR and post-EoR. Efficient star formation may take place at z ∼ 10 -16, producing the high cosmic SFR densities and the stellar massive galaxies. The other scenario is that a top-heavy IMF possibly with Pop III (or similarly metal poor) stellar populations produces strong UV radiation. The strong UV radiation may result in the overestimates of SFR densities above the constant star-formation efficiency model and of the stellar mass of the luminous galaxies. Further observational and theoretical studies are needed to test these two scenarios.", 'ACKNOWLEDGMENTS': 'We thank the anonymous referee for a careful reading and valuable comments that improved the clarity of the paper. We thank the CEERS team, especially Micaela Bagley and Steven Finkelstein, for providing many useful scripts for the NIRCam data reduction and datasets of the CEERS simulated images, and L. Y. Aaron Yung for providing SEDs of the mock galaxies. We are grateful to Rychard Bouwens, Seiji Fujimoto, Kohei Inayoshi, Akio Inoue, Tadayuki Kodama, Sandro Tacchella, Kenichi Tadaki, and Masayuki Umemura for useful comments and discussions. We thank Pratika Dayal and L. Y. Aaron Yung for sharing their data of the luminosity function. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with programs 2732, 2736, 1324, and 1345. The authors acknowledge the ERO, GLASS, and CEERS teams led by Klaus M. Pontoppidan, Tommaso Treu, and Steven L. Finkelstein, respectively, for developing their observing programs with a zero-exclusive-access period. This publication is based upon work supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and KAKENHI (20H00180, 20H00181, 20H05856, 21K13953, 21H04467, 22H01260) through Japan Society for the Promotion of Science. This work was supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo. \nSoftware: PANHIT (Mawatari et al. 2020a), Prospector (Johnson et al. 2021), SExtractor (Bertin & Arnouts 1996), SWarp (Bertin et al. 2002), Yggdrasil (Zackrisson et al. 2011)'}

2024A&A...690A.351T

Recent measurements of the cosmicray electron plus positron spectrum in several experiments have confirmed the presence of a break at 1 TeV. The origin of the break is still not clearly understood. In this work we explored different possibilities for the origin which include an electron source spectrum with a broken power law a power law with an exponential or superexponential cutoff and the absence of potential nearby cosmicray sources. Based on the observed electron plus positron data from the DAMPE and the H.E.S.S experiments and considering supernova remnants as the main sources of cosmic rays in the Galaxy we find statistical evidence in favor of the scenario with a broken powerlaw source spectrum with the bestfit source parameters obtained as 2.39 for the source spectral index ESUB0SUB 1.6 TeV for the break energy and f 1.59 10SUP48SUP ergs for the amount of supernova kinetic energy injected into cosmicray electrons. This powerlaw break in the spectrum has been predicted for electrons confined inside supernova remnants after acceleration via diffusive shock acceleration process and also indicated by the multiwavelength study of supernova remnants. All of this evidence shows that the observed spectral break provides a strong indication of a direct link between cosmicray electrons and their sources. Our findings further show that electrons must undergo spectral changes while escaping the source region in order to reconcile the difference between the spectral index of electrons observed inside supernova remnants and that obtained from Galactic cosmicray propagation studies.

2024-10-01T00:00:00Z

['10.48550/arXiv.2409.05509', 'arXiv:2409.05509', '2024A&A...690A.351T', '10.1051/0004-6361/202348607', '2024arXiv240905509T']

['acceleration of particles', 'diffusion', 'cosmic rays', 'ISM: supernova remnants', 'Astrophysics - High Energy Astrophysical Phenomena']

Origin of the break in the cosmicray electron plus positron spectrum at 1 TeV

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

https://arxiv.org/pdf/2409.05509.pdf

{'Origin of the break in the cosmic-ray electron plus positron spectrum at ∼ 1 TeV': 'Satyendra Thoudam /star \nDepartment of Physics, Khalifa University, PO Box 127788, Abu Dhabi, United Arab Emirates \nOctober 30, 2024', 'ABSTRACT': 'Recent measurements of the cosmic-ray electron plus positron spectrum in several experiments have confirmed the presence of a break at ∼ 1 TeV. The origin of the break is still not clearly understood. In this work, we explored di ff erent possibilities for the origin, which include an electron source spectrum with a broken power law, a power law with an exponential or super-exponential cuto ff , and the absence of potential nearby cosmic-ray sources. Based on the observed electron plus positron data from the DAMPE and the H.E.S.S experiments, and considering supernova remnants as the main sources of cosmic rays in the Galaxy, we find statistical evidence in favor of the scenario with a broken power-law source spectrum, with the best-fit source parameters obtained as Γ = 2 . 39 for the source spectral index, E 0 ≈ 1 . 6 TeV for the break energy, and f = 1 . 59 × 10 48 ergs for the amount of supernova kinetic energy injected into cosmic-ray electrons. This power-law break in the spectrum has been predicted for electrons confined inside supernova remnants after acceleration via di ff usive shock acceleration process, and also indicated by the multi-wavelength study of supernova remnants. All of this evidence shows that the observed spectral break provides a strong indication of a direct link between cosmic-ray electrons and their sources. Our findings further show that electrons must undergo spectral changes while escaping the source region in order to reconcile the di ff erence between the spectral index of electrons observed inside supernova remnants and that obtained from Galactic cosmic-ray propagation studies. \nKey words. cosmic rays - di ff usion - acceleration of particles - ISM: supernova remnants', '1. Introduction': 'High-energy cosmic-ray (CR) electrons and positrons with energies above ∼ 10 GeV su ff er radiative losses mainly through synchrotron and inverse Compton processes during their propagation through the Galaxy. For a power-law source spectrum of E -Γ as predicted by the di ff usive shock acceleration (DSA) theory (e.g., Drury 1983), radiative losses coupled with an energydependent di ff usive propagation of CRs are expected to produce an equilibrium electron spectrum in the Galaxy that follows a power law E -( Γ+ 1 -ξ ) for stationary sources uniformly distributed throughout the Galactic disk (Ginzburg & Ptuskin 1976; Atoyan et al. 1995), where ξ ∼ (0 . 2 -0 . 35), depending on the index of the CR di ff usion coe ffi cient D ( E ) ∝ E a , which is typically in the range of a ∼ (0 . 3 -0 . 6) in common models of CR propagation in the Galaxy (Jones et al. 2001; Strong et al. 2010). \nIn recent years, unprecedented high-precision measurements of CR electrons plus positrons in several experiments, such as those with the Alpha Magnetic Spectrometer (AMS-02; Aguilar et al. 2019a), Calorimetric Electron Telescope (CALET; Adriani et al. 2017), Dark Matter Particle Explorer (DAMPE; Ambrosi et al. 2017), Fermi Large Area Telescope ( Fermi -LAT; Abdollahi et al. 2017), and the High Energy Stereoscopic System (H.E.S.S.; Aharonian et al. 2008, 2009), have revealed spectral structures that deviate from theoretical expectations. Two prominent features are a gradual hardening above ∼ (30 -50) GeV and a sharp break or steepening at ∼ 1 TeV ( = 10 3 GeV). Evidence for the break was first reported by the H.E.S.S. experiment and later confirmed by the \nDAMPE experiment with its high-resolution measurement covering an energy range of 25 GeV -4 . 6 TeV. The observed spectrum closely follows E -3 and E -4 below and above the break, respectively. \nThe origin of the spectral break is still not clearly understood. However, it can be understood that the break cannot be due to the observed suppression in the positron flux at high energies, as positrons represent only a small fraction of the combined electron and positron flux. The positron fraction increases steadily from about 5% at ∼ 8 GeV to a maximum of about 15% at ∼ (300 -400) GeV, above which it shows a continuous decrease (Aguilar et al. 2019b). At ∼ 1 TeV, positrons make up only about 10% of the combined electron plus positron flux. This indicates that the spectral break has to do with the electron component, which is therefore our main focus in this work. \nThe break is di ffi cult to explain as an e ff ect of the propagation of CR electrons in the Galaxy. In the standard model of CR propagation in the Galaxy, where we sought a steady-state solution, CR electrons are expected to have a spectrum that steepens continuously with energy without any break as discussed above, unless certain special conditions are imposed (Lipari 2019). Features may arise that are related to the sudden drop in the energy loss rate of high-energy electrons as the inverse Compton scattering cross-section changes from the Thomson limit to the Klein-Nishina regime (Schlickeiser & Ruppel 2010). However, this drop is expected to produce an upturn in the spectrum (Evoli et al. 2020), not a steepening, which is what is shown by the measurements. \nThe observed break is most likely caused by an e ff ect related to the source properties of CRs, such as the absence of nearby sources and a cuto ff or break in the source spectrum itself. At \nenergies where radiative losses are significant, electrons cannot travel far distances in the Galaxy, and the space-time volume that contains the sources contributing to the observed CR electrons becomes smaller (see e.g., Lipari 2019). This can lead to only a small number of sources contributing at high energies, and can thus produce a spectrum at the Earth that is exponentially suppressed (Kobayashi et al. 2004). For a typical interstellar medium (ISM) environment and standard CR propagation parameters, electrons of 1 TeV have an average di ff usion length of ∼ 1 kpc in the Galaxy (see Sect. 2 for details). Therefore, if potential CR sources are missing or if they are significantly less numerous in the local ISM within ∼ 1 kpc of Earth, the observed CRelectron flux can be significantly suppressed at TeV energies. Hereafter, we refer to this as the \'missing sources" scenario. \nBased on theoretical grounds and observational evidence, supernova remnants (SNRs) are considered to be the most plausible sources of CRs in the Galaxy (e.g., Ginzburg & Syrovatsky 1961; Drury 1983; Reynolds & Keohane 1999). Theoretically, the DSA process at the supernova shocks can accelerate suprathermal particles of the ISM to very high energies. For electrons, the maximum energy can be limited by di ff erent factors, which can lead to di ff erent shapes of the spectral cuto ff at the highest energies. For instance, for the widely adopted case of Bohm-type di ff usion of particles, the electron spectrum follows the most commonly adopted form, the "exponential cuto ff " exp( -E / E 0), if the maximum energy is limited by the presence of a free-escape boundary ahead of the shock in the upstream region (see Yamazaki et al. 2014 and references therein). This type of spectral cuto ff is widely used in CR propagation studies, for both the leptonic and hadronic species (Kobayashi et al. 2004; Thoudam et al. 2016), as well as in the study of radio and X-ray observations from SNRs (Reynolds & Keohane 1999). If the maximum energy is limited by radiative losses or by the finite age of the SNR, a \'superexponential cuto ff " exp( -E 2 / E 2 0 ) is expected (Kang et al. 2009; Kang 2011). This shape of the spectral cuto ff is primarily expected for cases where the CR di ff usion coe ffi cient around the shock has a strong energy dependence, such as in the case of a Bohm-like di ff usion (Zirakashvili & Aharonian 2007; Blasi 2010). The super-exponential cuto ff is also found to explain better the observed synchrotron spectrum from shell-type SNRs when a more correct calculation of synchrotron emission is implemented, instead of the widely adopted δ -function approximation (Zirakashvili & Aharonian 2007). Recently, this form for the spectral cuto ff was used to study spectral features in the CR electron spectrum (Fang et al. 2018; Evoli et al. 2020). \nAnother spectral feature that is naturally expected to be found for electrons present inside SNRs is a break in the powerlaw spectrum (hereafter, the "broken power-law" scenario). In the DSA process applied to supernova shocks, the majority of the particles escape downstream of the shock after acceleration and remain confined, during which time high-energy electrons suffer radiative losses. At a given age of the remnant, t age, electrons below a certain energy E b, whose energy loss time t loss > t age will maintain the spectrum generated at the shock E -Γ , while those above E b with t loss < t age will follow a steeper spectrum E -( Γ+ 1) (Thoudam & Hörandel 2011; Ohira & Yamazaki 2017). Detailed simulations of time-dependent DSA have confirmed the presence of such a break in the power-law electron spectrum inside SNRs (Kang 2011). They found the break at around 1 TeV when the SNR age is about 1000 yr, while the exponential cuto ff remained at ∼ (40 -50) TeV for a reasonable set of ISM parameters and a downstream magnetic field of ∼ 100 µ G for \nthe remnant. Recently, Morlino & Celli (2021) have shown that generation of this sharp break in the electron spectrum requires a large amplification of the magnetic field in the downstream region of the shock, possibly generated by the combined e ff ect of both CR-related and turbulent instabilities. Also discussed in recent works (Diesing & Caprioli 2019; Cristofari et al. 2021) is the importance of this e ffi cient magnetic field amplification operating until the late stages of the remnant\'s evolution, in order to produce a steeper source index of electrons compared to that of protons, as revealed by Galactic CR propagation studies. \nIn this work, we compared the di ff erent scenarios described above - missing sources, exponential cuto ff , super-exponential cuto ff , and broken power-law - in terms of the quality of fit to the observed CR electron spectrum, and show that the broken power-law scenario best explains the observed break as well as the overall spectrum. \nThis paper is organized as follows. In Sect. 2, we present the CR propagation model used in our study, and in Sect. 3, the di ff erent scenarios for the spectral break are described in detail. Section 4 presents the electron plus positron data that we used to test the di ff erent scenarios, and Sect. 5 describes the spectrum calculation for the di ff erent scenarios and the fitting procedure that we adopted for this work. Section 6 presents the fit results, and Sect. 7 discusses the results in detail.', '2. Propagation of cosmic-ray electrons in the Galaxy': "For a stationary and uniform distribution of CR sources in the Galactic disk, the di ff usive propagation of CR electrons in the Galaxy subject to synchrotron and inverse Compton radiative losses is described as \n∇· ( D ∇ N ) + ∂ ∂ E { b ( E ) N } = -q , (1) \nwhere E is the kinetic energy, N ( r , E ) is the di ff erential number density at a position r ≡ ( r , φ, z ) in cylindrical coordinates (with r representing the radial position, φ the azimuthal, and z the perpendicular position to the Galactic plane), and q ( r , E ) is the source term denoting the electron injection rate per unit volume into the ISM. The di ff usion coe ffi cient is taken as D ( E ) = D 0 β ( E / E 0) a , where D 0 = 1 . 55 × 10 28 cm 2 s -1 is the di ff usion constant, β = v / c is the ratio of the particle velocity v to the velocity of light c , E 0 = 3 GeV, and a = 0 . 54 is the di ff usion index (Thoudam & Hörandel 2013). This form of D ( E ) neglects the break at ∼ 300 GeV reported in some recent studies based on the secondary-to-primary ratios measured by the AMS-02 experiment (Ferronato et al. 2024). b ( E ) is the radiative energy loss rate, which is the sum of synchrotron and inverse Compton losses. Following Schlickeiser & Ruppel (2010), we took \nb ( E ) = 4 3 σ T c ( E m e c 2 ) 2 U B + 4 ∑ i = 1 W i E 2 i E 2 + E 2 i , (2) \nwhere σ T is the Thomson cross-section, m e c 2 is the electron rest-mass energy, and U B = B 2 / 8 π is the energy density of the magnetic field of strength B . We took B = 3 µ G. This value of the magnetic field is the average over the z = ± 5 kpc region below and above the Galactic plane, calculated at the galactocentric radius of 8 . 5 kpc using the relation B ( z ) = B 0 e -| z | / 3kpc µ G adopted in the GALPROP CR propagation code (Strong et al. 2010). We took B 0 = 6 µ G, the local magnetic field value (Beck 2000). A recent detailed study reports a slightly lower \nvalue of the Galactic magnetic field (Unger & Farrar 2024). The summation in Eq. 2 represents the sum of the inverse Compton losses in the Klein-Nishina limit over four di ff erent di ff use photon fields present in the ISM - (1) photons from spectral type B stars, (2) photons from spectral type G-K stars, (3) infrared radiation, and (4) cosmic microwave background, where W i denotes the respective photon energy density, and E i is the critical Klein-Nishina energy above which the Thomson cross-section starts to break down. ( W i , E i) for the four photon fields are taken as (0 . 09 eV / cm 3 , 40 GeV), (0 . 3 eV / cm 3 , 161 GeV), (0 . 4 eV / cm 3 , 4 . 0 × 10 4 GeV), and (0 . 25 eV / cm 3 , 3 . 0 × 10 5 GeV), respectively (Schlickeiser & Ruppel 2010). More details on the relativistic treatment of inverse Compton losses of high-energy electrons can be found in Delahaye et al. (2010). \nAs high-energy electrons cannot travel far distances in the Galaxy because of radiative losses, the e ff ect of di ff usion boundary of the Galaxy on the solution of Eq. (1) can be neglected for energies within our scope of interest, that is, above ∼ 10 GeV. The Green's function of Eq. (1) can be obtained as \nG ( r , r ' , E , E ' ) = 1 8 π 3 / 2 b ( E ) A 3 / 2 exp ( -( r -r ' ) 2 4 A ) (3) \nwhere A ( E , E ' ) = ∫ E ' E D ( u ) b ( u ) du . (4) \nThe CR density at a position r is obtained by convolving the Green's function with the source term q ( r , E ) as \nN ( r , E ) = ∫ ∞ E dE ' ∫ ∞ -∞ d r ' G ( r , r ' , E , E ' ) q ( r ' , E ' ) . (5) \nFor a uniform source distribution in the Galactic disk, and assuming azimuthal symmetry, the source term can be written independent of r and φ as q ( r , E ) = ν Q ( E ) δ ( z ), where ν denotes the frequency of supernova explosions per unit surface area in the Galactic disk, and Q ( E ) is the source spectrum. We took ν = 25 Myr -1 kpc -2 , which corresponds to ∼ 3 supernova explosions per century in the Galaxy (Grenier 2000). Then, the CR density at the position of the Earth, taken to be r = 0, due to all the sources being located beyond a radial distance r 0 in the Galactic disk, is obtained as \nN ( E ) = 2 πν ∫ ∞ E dE ' ∫ ∞ r 0 r ' dr ' ∫ ∞ -∞ dz ' G ( r ' , z ' , E , E ' ) Q ( E ' ) δ ( z ' ) = ν 2 √ π b ( E ) ∫ ∞ Q ( E ' ) √ exp -r 2 0 4 A dE ' . \nDefining r d( E , E ' ) = 2 √ A ( E , E ' ), Eq. (6) can be written as \nE A (6) \nN ( E ) = ν √ π b ( E ) ∫ ∞ E Q ( E ' ) r d exp -r 2 0 r 2 d dE ' . (7) \nr d( E , E ' ) corresponds to the average di ff usion length of electron of energy E ' before it cools down to energy E . For the di ff usion coe ffi cient D ( E ) ∝ E a and energy loss coe ffi cient b ( E ) ∝ E 2 , as expected in the case of synchrotron losses and inverse Compton losses in the Thomson regime, we can write an exact analytical expression of Eq. (4) as \nA ( E , E ' ) = 1 1 -a [ D ( E ) t loss( E ) -D ( E ' ) t loss( E ' ) ] , (8) \nwhere t loss( E ) ∝ 1 / E is the energy loss time for electrons of energy E . For the values of the di ff usion parameters and the Galactic magnetic field strength of B = 3 µ Gused in the present study, the di ff usion length for an electron of 1 TeV before losing half its energy was calculated to be r d ∼ 1 . 16 kpc. In general, for a < 1 and E ' >> E , that is, when the electron has lost a significant fraction of its energy ( E = η E ' , where η << 1), from Eq. (8), we have r d ∝ √ D ( E ) t loss( E ) ∝ E ( a -1) / 2 . For a = 0 . 54, which is the value adopted in this work, r d was found to scale with energy as r d ∝ E -0 . 23 , which shows that high-energy electrons traverse a smaller region of the Galaxy compared to the low-energy ones. This can lead to a significant suppression of the observed electron flux at high energies if potential CR sources are absent in the nearby region, due to the dominant e ff ect of the exponential term in Eq. (7). \nBy setting r 0 = 0 in Eq. (7), the CR electron flux at Earth from a continuous distribution of sources throughout the Galactic disk can be obtained as \nN ( E ) = ν √ π b ( E ) ∫ ∞ E Q ( E ' ) r d dE ' . (9) \nFor a single power-law source spectrum in the form Q ( E ) ∝ E -Γ , Eq. (9) gives a spectrum at Earth that follows N ( E ) ∝ E -Γ -1 + ξ for the case of D ( E ) ∝ E a and b ( E ) ∝ E 2 . The spectrum steepens mainly due to radiative losses, but with a correction factor ξ = (1 -a ) / 2 arising from the di ff usive nature of the propagation. For a = 0 . 54, the correction factor is ξ = 0 . 23.", '3. Different scenarios for the spectral break': 'As mentioned briefly in Sect. 1, the observed spectral break at ∼ 1 TeV is most likely due to an e ff ect related to the sources, such as missing nearby CR sources or an intrinsic cuto ff (or break) in the CR source spectrum. In this section, we discuss the di ff erent possibilities in detail.', '3.1. Missingsources': 'The absence of potential CR sources in the local ISM can lead to a suppression in the CR electron flux at high energies, which may appear as a break or exponential cuto ff in the observed spectrum (Kobayashi et al. 2004; Lipari 2019). The energy above which the suppression occurs depends on the distance of the closest source. To produce a suppression above ∼ 1 TeV, the nearest source must be located around 1 kpc from Earth, as discussed in Sect. 2 based on Eq. (7). \nThe shape of the spectral break in this case is determined by the exp ( -r 2 0 / r 2 d ) term in Eq. (7). Defining r 0 as the di ff usion length of electrons corresponding to the break energy E 0, we can write r 0 = 2 √ D ( E 0) t loss( E 0) ∝ E ( a -1) / 2 0 . For a = 0 . 54, this gave r 0 ∝ E -0 . 23 0 , and together with the r d ∝ E -0 . 23 obtained in Sect. 2, the exponential cuto ff in Eq. (7) could be written in terms of energy as exp [ -( E / E 0) 0 . 46 ] , which is close to exp ( -√ E / E 0 ) . Therefore, the missing sources scenario is expected to exhibit a spectral cuto ff that is harder than the commonly adopted exponential and super-exponential cuto ff s, which are discussed below.', '3.2. Exponentialcutoff': "In the DSA process, if the maximum energy of the particles is limited by free escape from the acceleration region, the spectral \ncuto ff can follow an exponential cuto ff of the form exp ( -E / E 0) (see e.g., Caprioli et al. 2009; Ohira et al. 2010 and references therein). This type of cuto ff is also most commonly used in CR propagation studies (e.g., Kobayashi et al. 2004; Thoudam et al. 2016), as well as in the study of non-thermal radio and X-ray emission from SNRs (Reynolds & Keohane 1999; Bamba et al. 2003). \nIn the test particle approximation, assuming the presence of an escape boundary at a distance L ahead of the shock in the upstream region, the maximum energy E m of the particles that remain confined is given as the condition D s( E m) / u = L , where D s ∝ E β is the di ff usion coe ffi cient of the particles in the accelerator region, and u is the velocity of the shock expanding into a stationary medium. The spectrum at the shock in this case follows (Caprioli et al. 2009; Ohira et al. 2010; Yamazaki et al. 2014), as \nf 0( E ) ∝ exp -q ∫ E 1 1 -exp [ -uL D s( E ' ) ] d log E ' , (10) \n where q = 3 r / ( r -1), with the shock compression ratio r = 4 for strong shocks. At energies E /greatermuch E m, the spectrum reduces to f 0( E ) ∝ exp [ -q ( E / E m) β ] . For particles di ff using in the Bohm regime, β = 1, which gives a cuto ff shape of the form exp ( -E / E 0), where we took E 0 = E m / q .", '3.3. Super-exponentialcutoff': "If the maximum energy of the electrons in the DSA process is limited by radiative losses or by the finite age of the accelerator, the high-energy cuto ff is expected to follow a super-exponential cuto ff exp [ -( E / E 0) 2 ] for the widely adopted Bohm-type diffusion (Zirakashvili & Aharonian 2007; Kang et al. 2009; Blasi 2010). \nThe asymptotic form of the electron spectrum at the shock front in the very high energy regime where the radiative energy losses dominate over acceleration is given as (Zirakashvili & Aharonian 2007) \nf 0( E ) ∝ K 0( E ) exp [ -S 0( E )] , (11) \nwhere, for the same energy dependence of the radiative losses and the di ff usion coe ffi cient in the downstream and upstream regions of the shock, K 0( E ) ∝ √ b ( E ) / E , and S 0( E ) can be approximated as (see also Yamazaki et al. 2014) \nS 0( E ) ≈ ( q u ) 2 ∫ E b ( E ' ) D ( E ' ) E ' 2 dE ' . (12) \nUsing the condition that the maximum energy is determined by equating the acceleration time t acc( E m) ≈ 4 D s( E m) / u 2 (Bell 2013), with the radiative energy loss time t loss( E m) ∝ 1 / E m, Eq. (12) becomes S 0( E ) ≈ q ( E / E m) β + 1 . For Bohm-like di ff usion, the shape of the spectral cuto ff then follows exp [ -( E / E 0) 2 ] , where E 0 = E m / √ q .", '3.4. Brokenpower-law': 'In the standard DSA theory as applied to supernova shock waves, a major fraction of the particles escape downstream of the shock and remain confined within the remnant without crossing the shock again. During the confinement, if high-energy electrons su ff er severe radiative losses, this can lead to a power-law break \nin the cumulative electron spectrum (Thoudam & Hörandel 2011; Kang 2011; Ohira & Yamazaki 2017). However, the presence of such a break requires a large magnetic field amplification in the downstream that continues until the late stages of the SNR evolution (Morlino & Celli 2021). This strong field is also required to produce the di ff erence in the source spectral index between protons and electrons above ∼ 10 GeV that is required in CR propagation studies (Diesing & Caprioli 2019; Cristofari et al. 2021). \nThe temporal evolution of the electron spectrum during the confinement downstream, subject to severe radiative losses and continuous injection into the downstream after acceleration at the shock fronts, can be written as (Kardashev 1962; Thoudam & Hörandel 2011) \nf ( E , t ) ∝ E -( Γ+ 1) [ 1 -(1 -E / E 0) Γ -1 ] , for E < E 0 ∝ E -( Γ+ 1) , for E ≥ E 0 , (13) \nwhere Γ is the spectral index of electrons accelerated at the supernova shocks. Equation (13) shows that the electron spectrum exhibits a break at an energy, E 0 ∝ 1 / t , whose value depends on the age of the remnant, t . It can be easily checked using Eq. (13) that the spectrum below the break follows E -Γ , while above, the spectrum becomes steeper as E -( Γ+ 1) . Interestingly, the electron spectrum measured by the H.E.S.S and the DAMPE experiments also exhibit a similar spectral di ff erence below and above the break energy of 1 TeV.', '4. Measured electron plus positron spectrum': 'Recently, various experiments, such as the AMS-02 (Aguilar et al. 2019a), CALET (Adriani et al. 2017), DAMPE (Ambrosi et al. 2017), Fermi -LAT (Abdollahi et al. 2017) and H.E.S.S. (Aharonian et al. 2008, 2009) instruments, have provided high-quality data on the total electron plus positron spectrum up to energies of a few teraelectronvolts. Their combined spectrum indicates a hardening above ∼ (30 -50) GeV with a power-law index of ∼ 3 . 0 and a steepening above ∼ 1 TeV to an index close to ∼ 4 . 0. While the results from the di ff erent experiments show a similar behavior over a wide energy range, there are systematic di ff erences between the measured fluxes. For this reason, and also because DAMPE covers an energy range (25 GeV -4 . 6 TeV) wide enough to detect the presence of the spectral break at ∼ 1 TeV in a single measurement, we use the DAMPE data to test the di ff erent scenarios for the spectral break described in Sect. 3 together with the H.E.S.S data to increase the statistic at TeV energies.', '5. Calculation of CR electron spectrum and fitting the electron plus positron data': "CR electron spectrum for the di ff erent scenarios can be calculated from Eq. (7) or Eq. (9) by taking their appropriate source spectrum Q ( E ). For the missing sources scenario, Eq. (7) was used by keeping r 0 a model parameter, while for the rest, Eq. (9) was used where r 0 was set to 0. Below are the di ff erent forms of Q ( E ) adopted in our calculation based on the description of the \nTable 1. Model parameters ( f , Γ , E 0 , r 0) and their ranges considered in the fitting procedure. \n- \n- \n- \n- \n- \n- \n- \n- \nNotes. The step sizes used for scanning the parameters are: ∆ f = 0 . 001 ( × 10 49 ergs), ∆Γ = 0 . 01, ∆ E 0 = 2 GeV and ∆ r 0 = 0 . 05 kpc. \ndi ff erent scenarios given in Sect. 3: \nQ ( E ) = kE -Γ , \nkE -Γ 1 + ( E E 0 ) 1 /δ , Broken powerlaw , \nMissing sources = kE -Γ exp ( -E E 0 ) , Exponential cuto ff = kE -Γ exp -E 2 E 2 0 , Super exponential cuto ff = -δ (14) \n where k is a normalization constant that is proportional to the fraction f of the supernova kinetic energy of 10 51 ergs injected into CR electrons, Γ is the source index, and E 0 is the exponential or super-exponential cuto ff . For the broken power-law scenario, the function reproduces the spectral shapes E -Γ below the break energy E 0 and E -( Γ+ 1) above the break as described by Eq. (13). The smoothness parameter δ was fixed at 0 . 05 so as to produce a similar level of sharpness at E 0 as the spectrum given as Eq. (13). \nFor each scenario, the fit to the observed data was performed using three parameters related to the source properties: ( f , Γ , r 0) for the missing sources and ( f , Γ , E 0) for the all other cases. We used the fraction f instead of k for the fit as it directly represents the amount of supernova kinetic energy that is converted into CR electrons. In this study, we calculated f on the basis of energy injected into electrons above 1 GeV. The fit was performed in two steps. First, as f only scales the normalization and does not a ff ect the shape of the final spectrum, we treated it as a nuisance parameter and marginalized it to determine the set of values for ( Γ , r 0) or ( Γ , E 0) that gives the maximum likelihood, L = ∑ f L f , fit to the observed data. The summation over f was carried out in steps of ∆ f = 0 . 02 (in units of 10 49 ergs), and L f = exp ( -χ 2 / 2 ) with \nχ 2 = ∑ i [ I D( E i) -I M( E i)] 2 σ 2 D ( E i) + ∑ i [ I H( E i) -I M( E i)] 2 σ 2 H ( E i) , (15) \nwhere the summations are over the measured energy points E i. ( I D , σ D), and ( I H , σ H) are the measured electron plus positron flux and the associated errors from the DAMPE and the H.E.S.S. experiments, respectively, at their respective E i's. I M( E i) = I e -( E i) + I e + ( E i). As such, I e -( E i) = ( v / 4 π ) N ( E i) is the electron flux predicted by the model (Eqs. 7 or 9), and I e + ( E i) is the observed positron flux from the AMS-02 experiment, which can be described as (Aguilar et al. 2019b) \nIe + ( E i) = E 2 i ˆ E 2 [ K 1 ( ˆ E E 1 ) γ 1 + K 2 ( ˆ E E 2 ) γ 2 exp ( -ˆ E E c )] , (16) \nwhere the first term represents contribution of the di ff use positron flux produced from the collision of CR nuclei with interstellar matter, and the second term represents an additional contribution of positrons, which could be originated from dark matter annihilation or astrophysical sources such as pulsars in the Galaxy. ˆ E = E i + φ , with φ = 1 . 1 GeV, is the solar modulation parameter, K 1 = 6 . 51 × 10 -2 (m 2 sr s GeV) -1 , K 2 = 6 . 80 × 10 -5 (m 2 sr s GeV) -1 , E 1 = 7 . 0 GeV, E 2 = 60 GeV, γ 1 = -4 . 07, γ 2 = -2 . 58, and E c ≈ 813 GeV. In writing Eq. (15), we assumed that the data points are uncorrelated. \nIn the next step, ( Γ , r 0) or ( Γ , E 0) were fixed to the values that maximize L , and we scanned over f in finer steps of ∆ f = 0 . 001 to determine its best-fit value that gives the minimum value of χ 2 . Table 1 lists the model parameters and their ranges used in the fit. The fit was performed for the energy range (20 -3500) GeV. This range excludes the three highest energy data points of the measured spectrum shown in Fig. 1. Although measurement uncertainties are large at these energies, the rising trend in the flux might indicate the presence of one or several strong nearby sources, which we did not include in this study (Kobayashi et al. 2004). Considering SNRs located within 1 kpc of Earth and using a reasonable form of energy-dependent CR escape from the SNRs, Thoudam & Hörandel (2012) showed that nearby sources, in particular the Vela remnant, can be responsible for the steep rise in the observed spectrum at the highest energies. We note that for the energy range considered in this work, the e ff ects of processes such as solar modulation, reacceleration by interstellar magnetic turbulence or by encounters with old SNRs, convection by the Galactic wind, and energy losses due to ionization and bremsstrahlung on the electron spectrum at Earth are expected to be negligible. These processes are important mostly at energies below ∼ 20 GeV. \nIn order to check the e ff ect of the selected energy range on the fit results, we also performed a fit for another energy range of (100 -3500) GeV. The ranges of the model parameters for this fit range are also listed in Table 1.", '6. Results': 'The fit results for the (20 -3500) GeV range are shown in Fig. 1 for the di ff erent scenarios of the spectral break where the bestfit electron plus positron spectrum (thick solid line), the electron component (dashed line) and the positron spectrum (thin solid line) are shown along with the measurements from the DAMPE and the H.E.S.S experiments. The positron spectrum in Fig. 1, given as Eq. 16, is a basic fit to the observed data from AMS-02 (Aguilar et al. 2019b), which is also shown in the figure. \nAll of the scenarios produce a reasonably good fit to the data up to energies close to ∼ 1 TeV, whereas above this energy, \nFig. 1. Fit to the CR electron plus positron data from the DAMPE (Ambrosi et al. 2017) and the H.E.S.S. (Aharonian et al. 2008) experiments for the (20 -3500) GeV fit range for di ff erent scenarios of the spectral beak: missing sources (top left), exponential cuto ff (top right), super-exponential cuto ff (bottom left), and broken power-law (bottom right). Line representation: best-fit electron plus positron spectrum (thick solid line), electron spectrum (dashed line) and positron spectrum (thin solid line). The positron spectrum and its data are from AMS-02 measurements (Aguilar et al. 2019b). The fits exclude the three highest energy data points. The best-fit values of the model parameters are listed in Table 2. \n<!-- image --> \nTable 2. Best-fit values of the model parameters ( f , Γ , E 0 , r 0) and the χ 2 values of the fits. \n- \n- \nNotes. The values listed are for the two di ff erent fit ranges: (20 -3500) GeV and (100 -3500) GeV. The χ 2 calculation includes both the DAMPE and the H.E.S.S. data. \nthe broken power-law and the super-exponential cuto ff scenarios show significantly better fits. The missing source and exponential cuto ff scenarios did not produce spectra which are steep enough to explain the observed steepening above 1 TeV. Figure 1 also demonstrates that the positron fluxes do not account for the observed spectral break. The break is determined by the electron component. \nTo compare the quality of the fits between the di ff erent scenarios for the (20 -3500) GeV fit range, we plotted the best-fit electron plus positron spectra of the di ff erent scenarios from Fig. 1 together into a single plot in Figure 2 (top-left panel). It can be seen that all the scenarios predict an almost similar spectrum at \nlower energies, but above ∼ 1 . 5 TeV, their predictions start to deviate from each other, with the broken power-law (thick red line) and the super-exponential cuto ff (blue line) scenarios explaining the data better than the missing sources (green line) and the exponential cuto ff (black line) scenarios. The positron spectrum and its data shown in Fig. 2 are the same as in Fig. 1. The residuals of the spectral fits are shown in the bottom-left panel, calculated only with respect to the DAMPE data, although both DAMPEandH.E.S.S data were used in the fits. The filled circles in the residual plots show the data points that were not included in the fit, and the dashed lines show the ± 20% residual levels to guide the eye. The right panels of Fig. 2 show similar plots for \nFig. 2. Comparison of the best-fit electron plus positron spectra of the di ff erent scenarios for the two di ff erent fit ranges: (20 -3500) GeV (left) and (100 -3500) GeV (right). Line representation: missing sources (green line), exponential cuto ff (black line), broken power-law (thick red line), and super-exponential cuto ff (blue line). For the (20 -3500) GeV fit, the best-fit electron plus positron spectra are the same as shown in Fig. 1. The top plots show the spectral fits, and the bottom plots show the fit residuals. Residuals are shown only with respect to the DAMPE data, but the χ 2 calculation includes both the DAMPE and the H.E.S.S. data. Filled circles in the residual plots represent data points not included in the fit. The dashed lines represent ± 20% residual levels, shown only for reference. The best-fit values of the model parameters and the χ 2 values are listed in Table 2. All of the data shown and the positron spectrum (thin red line) are the same as in Fig. 1. \n<!-- image --> \nthe (100 -3500) GeV fit range. Above 100 GeV, the individual spectra and the relative di ff erences between them are similar to those observed in the (20 -3500) GeV fit range. \nTable 2 gives the best-fit values of the di ff erent parameters, that is, ( f , Γ , r 0) for the missing sources scenario and ( f , Γ , E 0) for the other cases, along with the χ 2 values of the fits. Based on the χ 2 values, the broken power-law is found to give the best fit of all the scenarios for both the fit ranges. For the (20 -3500) GeV fit, it gives the smallest χ 2 value of 69 for a number of degrees of freedom of nd f = 43. The next best case, which is the superexponential cuto ff , gives χ 2 = 87 for the same nd f . The exponential cuto ff and the missing sources give χ 2 = 114 and 125, respectively. For the 100 -3500 GeV fit, we find similar results, where χ 2 = 57 for the broken power-law compared to χ 2 = 67 for the super-exponential cuto ff , χ 2 = 89 for the exponential cuto ff , and χ 2 = 111 for the missing sources scenario for nd f = 33. For both of the fit ranges, we find that the overall fit quality decreases in the following order of the scenarios: broken power-law, super-exponential cuto ff , exponential cuto ff , and missing sources. This demonstrates the robustness of the fit with respect to the chosen energy range. \nFor the (20 -3500) GeV fit, which is the main result of this work, we find that the broken power-law scenario gives Γ = 2 . 39, E 0 = 1576 GeV, and f = 0 . 159 × 10 49 ergs, whereas the (100 -3500) GeV fit gives a flatter source index of Γ = 2 . 36 together with smaller E 0 and f values of E 0 = 1482 GeV and f = 0 . 142 × 10 49 ergs. This is because the (20 -3500) GeV fit has to accommodate for the observed steeper spectrum below ∼ 100 GeV, which then requires larger values of E 0 and f to explain both the observed spectral break and the flux level at the same time. For the exponential and the super-exponential cuto ff s, we find the best-fit values for the (20 -3500) GeV fit to be ( Γ = 2 . 37, E 0 = 6762 GeV, f = 0 . 154 × 10 49 ergs) and ( Γ = 2 . 39, E 0 = 4588 GeV, f = 0 . 159 × 10 49 ergs). These val- \nso larger than the corresponding values obtained for the (100 -3500) GeV fit. The missing sources scenario gives Γ = 2 . 23, r 0 = 0 . 85 kpc, and f = 0 . 272 × 10 49 ergs for the (20 -3500) GeV fit, compared to the values of Γ = 2 . 08, r 0 = 1 . 3 kpc, and f = 0 . 378 × 10 49 ergs for the (100 -3500) GeV range. This can be explained as: a flatter Γ will require a larger r 0 value (which corresponds to a lower cuto ff energy) to compensate for the increase in the flux at higher energies, due to the flatter spectrum. \nWe note that our best-fit value for the source index value (for instance, Γ = 2 . 39 for the broken power-law scenario) is larger than the typical value of 2 . 0 predicted by the standard theory of DSAfor strong shocks. However, this larger value of Γ is typical for CR propagation studies (see e.g., Ackermann et al. 2012). A similar value of Γ = 2 . 39 for CR electrons was also found in a recent study presented in Evoli et al. (2020). It is possible that this type of steep index is produced by CR modified shocks inside SNRs, as discussed recently in Caprioli et al. (2020), or by the presence of an e ffi cient magnetic field amplification operating until the late stages of SNR evolution, when the shock already becomes weak (Diesing & Caprioli 2019; Cristofari et al. 2021).', '7. Discussions and conclusions': 'We have explored di ff erent scenarios related to CR source properties for the origin of the observed break in the CR electron spectrum at ∼ 1 TeV, and find that the scenario with the broken power-law source spectrum best describes the data with the bestfit values of Γ = 2 . 39, E 0 = 1576 GeV, and f = 0 . 159 × 10 49 ergs. While the widely adopted exponential cuto ff scenario as well as the scenario of the missing nearby sources are statistically disfavored, we find that the super-exponential cuto ff also explains the data reasonably well, although it is statistically less favorable than the broken power-law case. Moreover, multi-wavelength \nobservations of SNRs provide a strong indication favoring the broken power-law scenario for the TeV break, as discussed below. \nDSA applied to supernova shocks in a regular ISM predicts an exponential or super-exponential cuto ff s in the electron spectrum depending on the limiting condition for the maximum energy, as discussed in Sect. 3, but the cuto ff s are expected to be at ∼ (30 -100) TeV (Reynolds 2008), which is much higher than the position of the observed break in the CR electron spectrum. Moreover, combined spectral fitting of radio and X-ray data of several young SNRs have also found a similar range of very high energy cuto ff s for electrons (Reynolds & Keohane 1999), thereby disfavoring the exponential or super-exponential cuto ff s in the source spectrum as a plausible explanation for the observed TeV break. \nOn the other hand, DSA simulations using standard SNR parameters have predicted a break in the power-law spectrum of electrons at GeV -TeV range (where the position of the break depends on the age of the remnant) arising from the radiative losses of electrons during their confinement in the downstream region (Kang 2011). The break energy which is given as the condition, t loss = t age, follows E b ≈ 1 / ( α t age) for a continuous injection of electrons into the downstream (see e.g., Thoudam & Hörandel 2011). For synchrotron losses, α = 2 . 5 × 10 -18 ( B /µ G) 2 GeV -1 s -1 , indicating that our best-fit break energy of E 0 = 1576 GeV ≈ 1 . 6 TeV corresponds to an SNR age of t age ≈ 3 . 3 × 10 4 yr for a conservative magnetic field strength of B = 15 µ G in the downstream. This age is close to the onset of the radiative phase of an SNR, t r = 2 . 7 × 10 4 ( E SN / 10 51 ergs) 0 . 24 ( n H / cm 3 ) -0 . 52 yr ≈ 2 . 7 × 10 4 yr, estimated for a standard supernova explosion energy of E SN = 10 51 ergs and ISM density of n H = 1 H cm -3 (Sturner et al. 1997). It is generally considered that as an SNR enters the radiative phase, the shock slows down significantly and all of the particles confined downstream are released into the ISM. It is worth keeping in mind that our estimate of t r does not take into account the variety of interstellar environments in which supernova explosions can occur in the Galaxy. While thermonuclear or Type Ia supernova explosions generally occur in the regular ISM, core-collapse or Type II supernova explosions can occur in a more complex environment, which can be a low-density cavity formed by powerful winds originated from their massive progenitor stars (O-type stars) or a high-density environment of molecular clouds if the progenitors are of lower mass, such as B-type stars (Weaver et al. 1977; Chevalier 1999). In addition, most supernovae are expected to be found within star clusters where the environment can be even more complex due to the interacting stellar winds (Longmore et al. 2014). For a typical range of n H = (0 . 1 -10) H cm -3 , we estimate the start of the radiative phase to be in the range of t r ≈ (8 × 10 3 -9 × 10 4 ) yr. The SNR age of t age ≈ 3 . 3 × 10 4 yr that we obtained corresponding to our best-fit break energy of E 0 ∼ 1 . 6 TeV falls well within this range. \nStrong observational evidence in favor of the broken powerlaw case comes from the extensive H.E.S.S. measurement of SNR RXJ1713 . 7 -3946, which is one of the most well-studied SNRs across di ff erent wavelengths (Abdalla et al. 2018). Simultaneous fitting of high-quality X-ray and GeV -TeV gamma-ray data shows a clear preference for a broken power-law particle spectrum which, for the leptonic origin, requires a power-law break at E b = 2 . 5 TeV with an index Γ = 1 . 78 (2 . 93) below (above) the break and an exponential cuto ff of E c ∼ 88 TeV for a magnetic field strength of B ∼ 14 µ G. The Γ value below the break obtained by the H.E.S.S measurement is smaller \nthan our best-fit value of Γ = 2 . 39, but the spectral di ff erence of ∆Γ ∼ 1 . 15 found with H.E.S.S. below and above the break is quite close to the ∆Γ ∼ 1 . 0 predicted by the power-law scenario in the present study, which is also the observed di ff erence in the CR electron spectrum. In another study, a fit to the multiwavelength data using a broken power-law electron spectrum with a super-exponential cuto ff gives a slightly di ff erent result: Γ = 2 . 0 below the break, E b = 3 . 5 TeV, E c = 80 TeV, and B = 16 µ G(Zang & Liu 2019). Although the exact origin (leptonic or hadronic) of the non-thermal emission from RX J1713 . 7 -3946 is still under debate, we note that the hadronic scenario requires certain environmental conditions for the SNR to generate a broken power-law source spectrum, for instance, a supernova explosion in a molecular cloud that has already been swept away by a strong wind from the progenitor star (Gabici & Aharonian 2014). Such an environment may not be readily available for the majority of the SNRs in the Galaxy. On the other hand, in the leptonic case, a power-law spectral break is naturally expected, as discussed above in Sect. 3.4. \nAdditional evidence for the broken power-law case can be found in the multi-wavelength studies of several other SNRs. Zang & Liu (2019) studied a sample of seven middle-aged (several thousand years old) shell-type SNRs and finds that the majority of the SNRs in their sample show an electron spectrum with a power-law break at E b ∼ (1 . 0 -3 . 5) TeV and Γ ∼ 1 . 9 -2 . 2 below the break, along with a high-energy super-exponential cuto ff in the range of ∼ (30 -80) TeV and a magnetic field in the range of B ∼ (9 -26) µ G. \nZeng et al. (2019) considered a larger sample of 35 middleaged and moderately old SNRs, and also finds a particle distribution that follows a broken power-law spectrum. They find the break energy to decrease with the SNR age in the range of ∼ 100 TeV -1 GeV over ∼ (10 3 -6 × 10 4 ) yr age range with a spectral index Γ ∼ 1 . 2 -2 . 3 below the break energy. However, they considered a combined population of electrons and protons to fit the multi-wavelength data, and assumed the two population to have the same spectral index and break energy, which may not necessarily be true in all cases (Diesing & Caprioli 2019; Morlino & Celli 2021). These reservations make it di ffi cult infer information from their results that can be directly compared with our findings in this study, where our focus was purely on the electron spectrum. \nOverall, the observation of a spectral break with ∆Γ ∼ 1 . 0 in the electron spectra of several SNRs, such as RX J1713 . 7 -3946 and others as presented in Zang & Liu (2019), and our independent finding of a similar break and similar ∆Γ in the electron source spectrum from the study of CR propagation in the Galaxy, provide a strong indication of a direct link between the observed CR electrons and their sources. However, the Γ values of ∼ (1 . 9 -2 . 2) inferred from the SNRs are found to be relatively smaller than our best-fit value of 2 . 39 obtained from the CR electron data. The fact that the Γ values from the SNRs are quite close to the index predicted by the DSA process indicates that the electrons confined inside the remnants mostly preserve the shape of the spectrum produced at the shocks, except at energies above the break where the spectrum steepens by ∆Γ ∼ 1 . 0 due to radiative cooling. However, what is necessary for the study of CR propagation is the electron spectrum released by SNRs into the ISM. This spectrum is poorly known, and it may be di ff erent from the spectrum produced at the shocks. Our findings strongly indicate that a modification (steepening) of the electron spectrum must be happening while the electrons escape through the source region. Recent studies have shown that CR protons escaping from their sources can generate magnetic fluctuations around \nthe source region due to both resonant and non-resonant streaming instabilities (Schroer et al. 2021, 2022). These fluctuations lead to the formation of low-di ff usivity CR inflated bubbles surrounding the sources where CRs spend a considerable amount of time before finally being released into the ISM. For electrons, while they di ff use through these bubbles, their spectrum can be changed due to radiative losses. This may explain the observed di ff erence in the spectral index of electrons obtained from SNRs and that required by the Galactic propagation study. A similar argument has also been raised to explain the di ff erence in the source index of protons and electrons, as revealed by CR propagation studies considering that protons and electrons can su ff er di ff erent spectral changes during their propagation through these bubbles (Schroer et al. 2021; Cristofari et al. 2021). The presence of such bubbles or cocoons around potential CR sources has also been proposed to predict the additional contribution of secondary CRs from the sources leading to a hardening in the secondary-to-primary ratios at TeV energies (Cowsik & Burch 2010). \nBased on the best-fit results obtained in this work using the observed CR electron spectrum and the evidence reported from the multi-wavelength studies of SNRs, we conclude that if SNRs are the main sources of CRs in the Galaxy, the power-law break in the measured CR electron plus positron spectrum at ∼ 1 TeV is most likely an imprint of the radiative cooling break present in the electron source spectrum. Our findings further show that in order to reconcile the di ff erence between the spectral index of electrons obtained from the study of SNRs and CR propagation, CR electrons must undergo spectral modifications while escaping the source region. A detailed investigation of this topic can help further strengthen the link between SNRs and CRs in the Galaxy. \nAcknowledgements. Fundings from the ADEK grant (AARE19-224), and the Khalifa University ESIG-2023-008 and RIG-S-2023-070 grants are acknowledged.', 'References': 'Abdalla, H., et al. 2018, A&A 612, A6 \nAbdollahi, S. et al. 2017, PRD, 95, 082007 \nAckermann, M., et al. 2012, ApJ, 750, 3 \nAdriani, O. et al. 2017, PRL, 119, 181101 \nAguilar, M. et al. 2019a, PRL, 122, 101101 \nAguilar, M. et al. 2019b, PRL, 122, 041102 \nAharonian, F. A. et al. 2008, PRL, 101, 261104 \nAharonian, F. A., et al. 2009, A&A, 508, 561 \nAmbrosi, G. et al. 2017, Nature, 552, 63 \nAtoyan, A.M., Aharonian, F.A., & Völk, H.J., 1995, Phys. Rev. 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2024arXiv240905391M

Globular clusters offer a powerful way to test the properties of stellar populations and the late stages of lowmass stellar evolution. In this paper we study oscillating giant stars and overtone RR Lyraetype pulsators in the nearest globular cluster M4 with the help of highprecision continuous light curves collected by the Kepler space telescope in the K2 mission. We determine the frequency composition of five RRc stars and model their physical parameters with a grid of linear pulsation models. We are able for the first time to compare seismic masses of RR Lyrae stars directly to the masses of the very similar red horizontal branch stars in the same stellar population independently determined from asteroseismic scaling relations. We find a close match with an average seismic mass of 0.651pm0.028Modot for RR Lyrae stars and 0.657pm0.034Modot for red horizontalbranch stars. While the validity of our RR Lyrae masses still relies on the similarity of neighboring horizontal branch subgroups this result strongly indicates that RRc stars may indeed exhibit highdegree l 8 and 9 nonradial modes and modeling these modes can provide realistic mass estimates. We also determine the He content of the cluster to be Y 0.266pm 0.008 and compare the seismic masses for our sample of RR Lyrae to theoretical mass relations and highlight the limitations of these relations.

2024-09-01T00:00:00Z

['2024arXiv240905391M', 'arXiv:2409.05391', '10.48550/arXiv.2409.05391']

['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - Astrophysics of Galaxies']

Matching seismic masses for RR Lyraetype and oscillating red horizontalbranch stars in M4

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05391.pdf

{'Matching seismic masses for RR Lyrae-type and oscillating red horizontal-branch stars in M4': 'László Molnár 1 , 2 , Henryka Netzel 3 , Madeline Howell 4 , 5 , Csilla Kalup 1 , 2 , and Meridith Joyce 1 , 6 \n- 1 Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences, MTA Centre of Excellence, Konkoly-Thege Miklós út 15-17, H-1121, Budapest, Hungary\n- e-mail: molnar.laszlo@csfk.org; lmolnar@konkoly.hu\n- 2 ELTE Eötvös Loránd University, Institute of Physics and Astronomy, 1117, Pázmány Péter sétány 1 / A, Budapest, Hungary\n- 3 Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland\n- 4 School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia\n- 5 ARC Centre of Excellence for Astrophysics in Three Dimensions (ASTRO-3D), Australia\n- 6 University of Wyoming, 1000 E University Ave, Laramie, WY USA', 'ABSTRACT': 'Globular clusters o ff er a powerful way to test the properties of stellar populations and the late stages of low-mass stellar evolution. In this paper we study oscillating giant stars and overtone RR Lyrae-type pulsators in the nearest globular cluster, M4, with the help of high-precision, continuous light curves collected by the Kepler space telescope in the K2 mission. We determine the frequency composition of five RRc stars and model their physical parameters with a grid of linear pulsation models. We are able, for the first time, to compare seismic masses of RR Lyrae stars directly to the masses of the very similar red horizontal branch stars in the same stellar population, independently determined from asteroseismic scaling relations. We find a close match, with an average seismic mass of 0 . 651 ± 0 . 028 M ⊙ for RR Lyrae stars and 0 . 657 ± 0 . 034 M ⊙ for red horizontal-branch stars. While the validity of our RR Lyrae masses still relies on the similarity of neighboring horizontal branch subgroups, this result strongly indicates that RRc stars may indeed exhibit high-degree, l = 8 and 9 non-radial modes, and modeling these modes can provide realistic mass estimates. We also determine the He content of the cluster to be Y = 0 . 266 ± 0 . 008, and compare the seismic masses for our sample of RR Lyrae to theoretical mass relations and highlight the limitations of these relations. \nKey words. globular star clusters - asteroseismology - RR Lyrae variable stars', '1. Introduction': "Masses of RR Lyrae stars are notoriously di ffi cult to determine. This is a problem, because the masses of Horizontal Branch (HB) stars, and in particular the masses of RR Lyrae stars, located at the intersection of the HB and the instability strip of classical pulsating stars, are crucial pieces of information for stellar evolution theories and models (Catelan 2009). Masses would provide information not only on the mass distribution along the HB, but also on the amount of mass loss during the red giant phase of stellar evolution and on the accuracy of mass estimates based on pulsational envelope models, as well. Thus, many avenues of research have been already explored to determine the masses of RR Lyrae stars. \nBinarity, for example, would be a straightforward way to infer dynamical masses. However, not only do we not know any eclipsing binaries with an RR Lyrae member, it is very di ffi cult to find systems that contain an RR Lyrae star at all (Wade et al. 1999; Kennedy et al. 2014; Hajdu et al. 2015). Since bona fide RR Lyrae stars are considered to be very old, their companions have either evolved into white dwarfs or other stellar remnants already, or must be smaller stars, most likely still evolving along the lower main sequence as K-M dwarfs. As such, the contribution of RR Lyr companions to the total light output of the system is expected to be minimal, precluding the detection via spectral energy distributions or as double-lined spectroscopic binaries. \nThe dearth of eclipsing RR Lyrae stars can also be explained by the evolutionary pathways of these pulsators. These stars moved past the red giant phase without any significant interactions with a companion that could have directed the primary away from evolving onto the HB and towards binary evolution scenarios. Companions thus must be distant to avoid any mass transfer, and a distant companion orbiting a now shrunken horizontal-branch star would make the probability of observing eclipses from the system very low. The only eclipsing binary candidate system turned out to be an impostor, a Binary Evolution Pulsator (Pietrzy'nski et al. 2012; Karczmarek et al. 2017). In this case, mass transfer directed a much lighter primary to rapidly cross the instability strip and to experience RR Lyraelike pulsations. We note that binarity and a limited amount of mass transfer has been proposed to explain metal-rich RR Lyrae stars, but those systems are expected to have wide orbits, too (Bobrick et al. 2024). \nMore indirect methods have shown increased promise, but all have their own caveats. For example, radial velocity searches are hindered by the expected long orbital periods and small signals relative to the much larger pulsation velocities. A recent work proposed new binary candidates, but it also failed to confirm the expected periastron passage of the oldest binary candidate, TU UMa (Wade et al. 1999; Barnes et al. 2021). \nAn alternative way to detect binaries and estimate masses is the light-time e ff ect, but this method has been impeded by the \nprevalence of the Tseraskaya-Blazhko e ff ect 1 , the quasi-periodic modulation of the pulsation amplitude and phase (Blažko 1907; Shapley 1916; Kurtz 2022). Nevertheless, several binary candidates have been identified in the OGLE survey (Hajdu et al. 2015, 2021). And while Hajdu et al. (2021) did not address the masses of the primaries directly, these authors found a trimodal distribution in the mass functions of the secondaries. \nAs we have shown, dynamical methods to determine masses of RR Lyrae stars su ff er from various theoretical and / or practical shortcomings. This limits us to other, more indirect, modeldependent methods. Fundamental physical parameters such as luminosity, T e ff and log g , along with elemental abundances, can be fitted with evolutionary tracks and atmosphere models. A relation to calculate masses from other physical parameters (period, luminosity, T e ff ) was first proposed by van Albada & Baker (1971) and by multiple authors since. However, they come with strong uncertainties in various model parameters that are very loosely constrained by observations, if at all. These include the He content of the models, the approximations chosen for internal processes like convection and overshoot, or various aspects of mass loss along the evolutionary pathway (see, e.g., Marconi et al. 2018; Anders & Pedersen 2023; Joyce & Tayar 2023; Joyce et al. 2024). Nemec et al. (2011) used such relations to calculate masses for the RR Lyrae stars in the Kepler field, for example, but without providing uncertainties for them. \nOne way to limit uncertainties in stellar parameters is to study RR Lyrae stars in globular clusters. There are currently close to three thousand RR Lyrae stars known in 115 Galactic globular clusters (Cruz Reyes et al. 2024). For these stars, parameters like distance and metallicity can be determined accurately for the members, and using that information, the van Albada & Baker (1971) relation was employed in many cases to estimate masses. M3, for example, was studied by multiple authors, and Valcarce & Catelan (2008) determined the mass distribution for the entire HB of the cluster based on evolutionary models, and found the distribution to be potentially bimodal. However, they discuss how that finding can be complicated by limitations present in the stellar tracks. A di ff erent approach was taken by Kumar et al. (2024), who fitted the light curves of the M3 RR Lyrae variables directly, based on a grid of non-linear pulsation models. Direct light curve fitting is a promising technique (see also, Bellinger et al. 2020, and Das et al., in prep), but current one-dimensional non-linear pulsation models still cannot reproduce light curves very accurately, due to limitations in handling convection, for example (Kovács et al. 2023). \nUncertainties in evolutionary and non-linear pulsation models can also be mitigated if we turn towards asteroseismology, and model the frequencies of the oscillatory modes observed in the star (see, e.g., Kjeldsen & Bedding 1995; Aerts 2021). Incorporating asteroseismic constraints has shown great potential recently for stars outside the instability strip. However, classical pulsators like RR Lyrae stars face obstacles in this aspect, too: in these stars, most often only a single mode can be observed, severely limiting our ability to constrain physical parameters from frequencies alone. Masses for a frew doublemode RR Lyrae stars have been calculated from non-linear nonadiabatic hydrodynamical simulations, (Molnár et al. 2015b), but, as mentioned before, the accuracy of non-linear models have been called into question (Smolec & Moskalik 2008). \nMore recently, a new method based on the discovery of lowamplitude signals in overtone RR Lyrae stars has been proposed \n(Netzel et al. 2015b,c). Theoretical predictions by Dziembowski (2016) suggested that these modes may be high-order, ℓ = 8, 9 non-radial modes, which can be calculated from linear pulsation models. Unlike modeling radial modes in the non-linear regime, fitting radial and non-radial modes together using a linear model grid is a technique very similar to the asteroseismology of red giant stars. This method was developed by Netzel &Smolec (2022), and was further explored, incorporating other observables such as T e ff , L and [Fe / H] information into the fit in Netzel et al. (2023). \nBut, as we have shown, we still lack a way to test the validity of these new RR Lyrae masses through direct methods. We therefore searched for alternatives. RR Lyrae stars make up only a small portion of the He-core-burning regime, and stars exist blue- and redward of them on the HB. Indeed, the evolution of HB stars and their positions on the HB are largely defined by the mass of their H-rich envelopes, as well as the mass loss they experience before reaching the HB. At temperatures below the red edge of the instability strip, we find the red horizontal branch (rHB) stars. These stars are very similar to the RR Lyrae pulsators, except for hosting a slightly more extended envelope. Since their envelopes are also convective, these stars show solarlike oscillations instead of pulsations, and these solar-like oscillations can be analysed in a way very similar to that of red giant and red clump stars (Matteuzzi et al. 2023). If rHB and RR Lyrae stars can be observed and modeled within the same population of stars, such as in globular clusters, they can potentially o ff er a way to compare asteroseismic masses determined by independent methods. \nThis opportunity came with the Kepler space telescope. Kepler observed multiple globular clusters during the K2 mission, including the cluster closest to the Sun, Messier 4 (M4). This cluster was bright enough that seismic data was obtainable not only for the bright red giants, but fainter stars, such as the rHB stars in it, as well. Howell et al. (2022) calculated seismic masses for several stars and compared the averages of various evolutionary phases to estimate the integrated mass loss between stages in the cluster. M4 also contains several RR Lyrae stars, including overtone ones that have been observed by Kepler . If non-radial modes can be detected in the overtone stars, their seismic masses could be compared directly to the seismic masses of rHB stars, validating the results. \nIn Section 2, we process the Kepler data of five RRc stars observed in M4 and determine their physical parameters. In Section 3, we analyse the photometry and the frequency content of the RRc stars and fit them with seismic models. In Section 4, we compare the RRc physical parameters to the (updated) parameters of the red giant stars of the cluster, as well as to theoretical mass relations. Finally, we summarize our findings in Section 5.", '2. Data and methods': 'The Kepler space telescope observed M4 during Campaign 2 of the K2 mission in late 2014 for 80 days nearly continuously. The cluster was at the edge of module 12 of the detector; hence M4 was only partially covered by the campaign.', '2.1. K2 photometry': 'A large portion of the cluster was covered with an extended custom pixel aperture. We identified three RRc stars within this aperture: M4 V6, V42 and V61. To extract their light curves, we followed the same di ff erential-image method based on the \nFITSH photometry package of Pál (2012) that we used on M80 (Molnár et al. 2023b), and on other crowded, faint, and / or moving targets (see, e.g., Molnár et al. 2015a, 2018; Kalup et al. 2021). The output for these stars is flux variations relative to the subtracted master frame. We shifted the median variations to the Gaia DR3 G-band average brightnesses (Gaia Collaboration et al. 2021; Riello et al. 2021). \nSince M4 has a large angular diameter, it extends beyond the K2 custom aperture. We located two additional RRc stars farther away from the center (V43 and V76), on the outskirts of the cluster: these were recorded with individual pixel apertures. For these two stars, we used the simple aperture photometry (SAP) light curves provided by the mission. \nRaw K2 photometric data contains systematic signals due to the limited pointing capabilities experienced by the telescope during the mission. It was found in earlier works focusing on RRLyrae stars (Plachy et al. 2019; Bódi et al. 2022; Molnár et al. 2023b) that the best method to separate the instrumental signals from the pulsations is the K2 Systematics Correction (K2SC) method developed by Aigrain et al. (2016). We applied the same corrections here to remove fast systematics. Slow trends were then removed with the algorithm developed by Bódi et al. (2022), fitting a high-order polynomial to the light curve that is optimized for best overlap of the pulsation cycles via phase dispersion minimization. The resulting light curves are shown in Fig. 1. Photometric data is available in Appendix B. \nOne more target, M4 V75, which was identified as a possible RRc star by Yao et al. (1988), is located within the K2 custom aperture as well. This star, however, shows no periodic variation, confirming previous non-detections by Stetson et al. (2014) and Safonova et al. (2016).', '2.2. Observational constraints': 'A detailed asteroseismic analysis requires further constraints on the physical parameters (such as L , T e ff , log g or [Fe / H]) of the stars. Howell et al. (2022) used scaling relations for the red giants in the cluster, which relies on luminosities and e ff ective temperatures. These quantities can be calculated from photometry, but they require accurate corrections for interstellar extinction. Here we used the reddening map and E(B-V) zero point determined specifically for M4 by Hendricks et al. (2012). \nIn contrast, our approach for the RRc stars is akin to peakbagging, where we collect and fit individual oscillation peaks (Appourchaux 2003). However, since we are limited to very few modes, it is still important to constrain our modeling space with classical observables such as allowed luminosity, T e ff and metallicity ranges. \nWe collected the average brightness values of the RRc stars both in Gaia DR3 passbands, and in Johnson passbands from the observations of Stetson et al. (2014) 2 . The Johnson data set is displayed in Fig. 2, with the RRc stars and the stars studied by Howell et al. (2022) highlighted. However, since input parameters for previous RRc model fits were based on Gaia brightnesses, we decided to use those here, as well. \nFirst, we calculated the interstellar extinction in the Gaia passbands with the seismolab software package, which uses the mwdust code and the Bayestar 3D dust map for that purpose (Bódi et al. 2022; Bovy et al. 2016; Green et al. 2019). We then used the distance modulus provided by Baumgardt & Vasiliev \nFig. 1. Corrected K2 light curves of the five RRc stars that were targeted by the mission. \n<!-- image --> \n(2021), µ = 11 . 337 ± 0 . 018 mag, to convert the extinctioncorrected values into absolute magnitudes. \nWe used M G and ( BP -RP )0 to derive luminosity and effective temperature. The e ff ective temperature T e ff was calculated using the relation from eq. 1 from Mucciarelli & Bellazzini (2020), which requires metallicity, [Fe / H], and colour. We employed the relation derived for giant stars and colour ( BP -RP )0. We set the metallicity to [Fe / H] ≈ -1 . 1 ± 0 . 07 (Harris 1996; Marino et al. 2008; MacLean et al. 2018). The luminosity was calculated using the formula: \nL / L ⊙ = 10 ( -0 . 4 ( M bol -( M ⊙ bol )) , (1) \nwhere M bol is a bolometric brightness of a star, and M ⊙ bol is a bolometric brightness of the Sun, set to 4.74 mag. We converted M G to M bol using bolometric correction calculated with the Gaiadr3 Bcg tool by Creevey et al. (2023). The input parameters to calculate bolometric correction are gravity (log g ), metallicity, and e ff ective temperature. We assumed log g = 3. \nMetal, Z , and helium, Y , contents were calculated based on [Fe / H]. First, we transformed [Fe / H] to [M / H] using formula by Salaris et al. (1993): \nFig. 2. Color-magnitude diagram of M4 in Johnson passbands using photometry from Momany et al. (2002) and Mochejska et al. (2002). Stars targeted by our study and by Howell et al. (2022) are highlighted with colored points. The Gaia membership sample is from Vasiliev & Baumgardt (2021). \n<!-- image --> \n[M / H] ≈ [Fe / H] + log GLYPH<16> 0 . 638 · 10 [ α/ Fe] + 0 . 362 GLYPH<17> , (2) \nwhere [ α / Fe] is enhancement by α elements. Then, we transformed [M / H] to Z , using the relation: \n[M / H] = log Z Z ⊙ -log X X ⊙ , (3) \nwhere Z ⊙ and X ⊙ are solar values, which we set to Z ⊙ = 0 . 0134 and X ⊙ = 0 . 7381 (Asplund et al. 2009). \nAssuming that hydrogen content, X , can vary from 0.70 to 0.76, and α -element enhancement can vary from 0 to 0.4, we calculated the modeling range for Z to be 0.00086 - 0.0032.', '3. Photometric results on the RRc stars': 'We used the Period04 software to determine the frequency composition of each star (Lenz & Breger 2005). The detection limit was set to S / N > 4.0, relative to the nearby average noise level in the frequency spectrum. First, we subtracted the pulsation frequency and its harmonics, then searched for any extra signals above or below the pulsation frequency. Folded light curves and frequency spectra are displayed in Fig. 3.', '3.1. Extra modes': 'According to Dziembowski (2016), the signals around period ratios P / P O1 ≈ 0 . 63 and 0.61, relative to the first overtone, correspond to (the first harmonic of) l = 8 and l = 9 modes, respectively. Here we label these as f 63 and f 61 frequencies, although this family of modes is also called f 61 or fX modes collectively. The actual mode frequencies are the subharmonics of these signals at f 61 / 2 and f 63 / 2, but those are usually harder to detect due to geometric cancellation e ff ects. \nWe clearly identified the modes in four stars: V6, V42, V43 and V76. We also see subtle di ff erences between them. In V6, the subharmonic at the true pulsation frequency is the strongest, which is unusual among RRc stars as we expect the harmonic to be stronger. In V42 and V76, we detect multiple peaks and the corresponding harmonics and subharmonics. In V43, the peak appears to be incoherent and can be fitted with three close-by frequencies. Here we observe a very wide structure at the subharmonic that extends below the expected range of the f 61 modes. However, these frequencies are not low enough to be either the fundamental mode or an f 68 mode. \nWhile f 61 modes are prevalent in the RRc population of M4, we did not detect the low-frequency f 68 mode clearly in either of the targets. The latter is a separate group of extra modes, discovered recently in RRc stars, which appear at lower frequencies ( f / f O1 ≈ 0 . 686), and whose origin has not been identified yet (Netzel et al. 2015a; Benk"o & Kovács 2023). The lack of detection is in agreement with the mode abundance results of Netzel et al. (2024). They found that while the f 61 modes are most frequent around [Fe / H] ≈ -1 . 0, the frequency of the f 68 mode drops drastically above -1.3, and the metallicity of M4 is higher than that limit. \nIn contrast to the results above, V61 turned out to be the rather di ff erent from the four other stars. Here we only detect significant extra frequency peaks at the subharmonic range, but not at the expected f 61 range. The two peaks appear at period ratios 0.7335 and 0.7877. The former value may potentially correspond to the fundamental mode, with a period of 0.36168 d: however, that would put the star to [Fe / H] values higher than that of M4 in the Petersen diagram (Szabó et al. 2004; Chen et al. 2023). We investigate the modulation properties of the star in Appendix A.', '3.2. Petersen diagram': 'We present the identified f 61 signals on a Petersen diagram in Fig. 4, showing the period ratios against the longer period (here fixed to the first overtone). We plot the M4 results in red against the points presented by Molnár et al. (2023b). We find that the points line up well with other samples that have moderate metallicities, such as the bulge and NGC 6362. In this regime most frequencies clearly fall onto the main ridges at period ratios P 1 / PX ≈ 0 . 632 and 0.613, corresponding to the ℓ = 8 and 9 modes. We also detect signals in the middle ridge, which Dziembowski (2016) interprets as combination frequencies. Further signals appear below, at P 1 / PX ≈ 0 . 605 which might also be combination peaks. We did not detect any ℓ = 10 peaks which should be at P 1 / PX ≈ 0 . 6 or below, according to Dziembowski (2016).', '3.3. Comparison with other photometric results': 'The K2 observations were analysed before by Wallace et al. (2019a) who searched the cluster for new variables. They de- \nFig. 3. Left: Fourier spectra of the five RRc stars. Here we removed the main pulsation frequency and its harmonics belonging to the first overtone to reveal the low-amplitude extra modes. Modes are labeled with red, and combination frequencies with black. Right: light curves folded with the first overtone period. \n<!-- image --> \ntected higher scatter in V61 than in other RR Lyrae stars, but did not recognize it as modulated. They also detected two stars they classified as millimagnitude RR Lyrae stars (Wallace et al. 2019b). Both of those two stars show only a single periodicity. \nGround-based multicolor photometry was collected for the RRLyrae members (among others) by Stetson et al. (2014). They did not recognize V61 as a modulated star either, although upon reanalysis of their photometry, a side peak is visible. The only other star common with our targets that has a useful amount of time series photometry is V6. There a frequency signal is marginally detectable at the position of the subharmonic ( f 63 / 2). This shows that seismic modeling of RRc stars requires extensive, high-precision photometry to detect the low-amplitude extra modes.', '3.4. RRc pulsation models': 'For each star, we calculated pulsation models to match the observed first-overtone period and period ratio(s). To calculate the theoretical models, we used the envelope pulsation code of \nDziembowski (1977). The input physical parameters required by the code are mass, luminosity, e ff ective temperature, hydrogen and metal abundances. Additionally, this code requires an approximate value of dimensionless frequency for the non-radial mode. We followed the approach of Netzel & Smolec (2022) for individual calculations. Namely, we used the estimates that relate non-radial mode frequencies from the linear fits to sequences in the Petersen diagram (see Equations 4 and 5 in Netzel & Smolec 2022), which were then converted to dimensionless frequencies, σ , using the formula from Eq. 6 in Netzel & Smolec (2022). Furthermore, we set the range of possible starting values of nonradial mode frequencies to σ ± 0 . 1. Consequently, for each model we were able to get the non-radial mode with the highest driving rate, i.e., it is the most strongly trapped in the envelope. \nIn order to find the best matches between periods and period ratios in theoretical models and in observed values for each star, we employed genetic algorithms 3 , which used the pulsation code with di ff erent input physical parameters. The ranges of physical \nTable 1. Input parameter ranges, pulsation periods and period ratios for each RRc star that we used for modeling. PR1 and PR2 are the mode period ratios relative to the first overtone. Mass ranges are left deliberately wide to accommodate realistic values over the grid. \nFig. 4. Petersen diagram of the f 61-type modes. Here we plot signals detected in M4 (in red) over existing literature data. Frequency peaks fall onto the main ridges discovered in the OGLE data. Positions of the f 61 and f 63 ridges are labelled respectively. \n<!-- image --> \nparameters were set based on observational constraints (see Table 1). In the case of mass, we set a wide range of 0 . 5 -0 . 9 M ⊙ to accommodate any realistic mass result. For each star, we executed 100 separate runs. For each run we set the maximum number of iterations to 150, population size of 100, mutation probability of 0.1, elitist ratio to 0.01, crossover probability to 0.5, and parents portion to 0.3. We note that we performed calculations with di ff erent parameters for genetic algorithms beforehand, including the numbers of populations and iterations, to ensure that the results are robust. The values and errors of physical parameters were derived as the means and standard deviations of the results from each of the hundred individual runs. \nCalculated ranges of physical parameters used to constrain the models for each star are summarized in Table 1.', '4. New and revised physical parameters': 'With the RRc physical parameters obtained, we proceeded to compare them to parameters of other RR Lyrae stars, as well as to the average properties of M4. For further comparison, we \nFig. 5. Positions of the seismically determined T e ff and log g values for the five RRc stars in M4, against the spectroscopically observed RRc stars in the T e ff -log g plane, as collected by Molnár et al. (2023b). \n<!-- image --> \nrecalculated the masses for the oscillating rHB stars originally studied in Howell et al. (2022), too.', '4.1. RR Lyrae masses and other physical parameters': 'As listed in Table 2, the masses of four out of the five RRc stars are within 0.06 M ⊙ , between 0.631 and 0.683 M ⊙ with an average uncertainty of ± 0 . 022 M ⊙ . The only outlier is the modulated star, V61, at 0.751 M ⊙ , but for that star our mode identification was more uncertain. However, mass outliers exist among the other evolutionary stages, such as among the rHB stars in the cluster as well. These masses align well with the mass-period distribution of RRc stars published by Netzel et al. (2023). \nWe compared the calculated T e ff and log g values to the distribution of RR Lyrae stars presented by Molnár et al. (2023b). As Fig. 5 shows, the stars belong to the cooler RRc stars, lying between 6500-7000 K, with log g values being very close to 3.0 for all five stars. For chemical composition, we find average bulk abundances of X = 0 . 734 ± 0 . 0075 and Z = 0 . 0019 ± 0 . 0002. These values indicate a He abundance of Y = 0 . 266 ± 0 . 008 for M4. \nThe comparison of the mass ratio of heavy elements, Z , to the observed [Fe / H] index is not straightforward, as it is influenced by the abundance di ff erences of individual elements (see, \nTable 2. Modeled physical parameters of the five RRc stars based on seismic modeling. \ne.g., Hinkel et al. 2022). Nevertheless, we can give an estimate for the correctness of our seismic Z values. Following the relations described in Section 2.2, in reverse order, and using an α -enhancement of [ α / Fe] = 0 . 39 ± 0 . 05 (Marino et al. 2008), we get an average seismic [Fe / H] of -1 . 13 ± 0 . 05 for the five RRc stars, which agrees with the spectroscopic [Fe / H] value of the cluster.', '4.2. Seismic masses for red giants and the average rHB mass': "Pressure modes in solar-like oscillators - such as red giants are characterized by two global seismic parameters; the large frequency separation, ∆ ν , and the frequency of the maximum acoustic power, ν max. These quantities are correlated to stellar properties, which are used to derive seismic mass scaling relations (Ulrich 1986; Brown et al. 1991; Kjeldsen & Bedding 1995). For time-series photometry with short baselines (e.g. the K2 mission), it has been shown that accurate asteroseismic masses can be measured independently of a ∆ ν estimate (e.g. Howell et al. 2022, 2024). This method uses the following scaling relation: \nM M ⊙ ! ≃ ν max ν max , ⊙ ! L L ⊙ ! T e ff T e ff , ⊙ ! -7 / 2 . (4) \nWeuse the red giant sample and their corresponding K2 light curves from Howell et al. (2022) to measure seismic masses using this relation above. They detected solar-like oscillations in 75 red giants across three phases of evolution (RGB, rHB and early AGB), and used the pySYD pipeline (Chontos et al. 2022) to measure the asteroseismic parameters. We adopt the Howell et al. (2022) estimates for T e ff and luminosity, although not their ν max determinations. Here we report updated measurements of ν max and the corresponding uncertainty for each star using a new asteroseismic pipeline, pyMON (Howell et al., in prep). \nThe pyMON pipeline is an adaptation of pySYD (see Chontos et al. 2022 for more details), where the main di ff erence is that ∆ ν is not measured. In both pipelines, the power spectrum is smoothed using an estimate for ∆ ν . pyMON substitutes this measurement with an estimate from a calibrated ∆ ν -ν max scaling relation (Stello et al. 2009; Howell et al. 2022). This is useful for low signal-to-noise data (e.g. our M4 photometry), because it is di ffi cult to measure an accurate ∆ ν . A comparison between the pySYD and pyMON methods shows that the measurement of ν max remains consistent, however there is a reduction by a factor of ∼ 2 in the ν max uncertainty, due to the reduced uncertainty in ∆ ν . Consequently, this results in a decrease in the seismic mass uncertainties by a factor of ∼ 1 . 5. \nWe provide our individual seismic masses in the bottom panel of Figure 6. Note that we have removed the stars identified as mass outliers in Howell et al. (2022), and also limited the RGB sample to below the luminosity bump (hereby known as \nlower RGB or LRGB). This ensured there was no bias to lower masses due to any mass loss that might have occurred. In the top panel of Figure 6, we represent the distribution of our masses as kernel density estimation (KDE) functions for each evolutionary phase. A KDE is calculated by modelling each data point as a Gaussian, adopting the mass as the mean and the uncertainty as the standard deviation. The final KDE is the summation of the Gaussian components, where the area has been normalised to a value of one. \nTo estimate a new average mass for the HB sample, we determine the peak value of the corresponding KDE. This represents the mode mass for this sample, and was measured to be 0 . 655 ± 0 . 007 M ⊙ The uncertainty is calculated as the standard error on the mean. Our average HB mass is consistent with the value found previously in Howell et al. (2022). This was similarly found for the RGB and early AGB samples, where we calculated average masses of 0 . 80 ± 0 . 009 M ⊙ and 0 . 55 ± 0 . 01 M ⊙ respectively 4 . Individual and average masses relative to the calculated luminosities are plotted in Fig. 7. \nThe integrated mass loss rates between evolutionary phases are also consistent with Howell et al. (2022). The mass loss from the lower RGB to the HB decreases slightly to ∆ M RGB -HB = 0 . 145 ± 0 . 011, whereas we find a mass loss from ∆ M HB -AGB = 0 . 105 ± 0 . 013 from the HB to the early AGB. \nWe plot the distributions of the masses along the HRD in Fig. 8. We also include a MIST 5 (MESA Isochrones and Stellar Tracks; Choi et al. 2016) isochrone for comparison. This isochrone has an age of 12.2 Gyr, an average of various literature sources for M4 (Cabrera-Ziri & Conroy 2022). Since MIST does not o ff er α -enhanced isochrones, we use a value of [Fe / H] = -0.9 to emulate the higher [ α / Fe] abundances, based on the scaling described by Joyce et al. (2023). The mass loss prescriptions used by MIST for low-mass stars are η R = 0 . 1 and η B = 0 . 2 for the RGB and AGB phases, following the (Reimers 1975) and Bloecker (1995) schemes, respectively. Colors represent masses: it is clear that the conservative mass loss prescription used in MIST's pre-computed isochrone database does not capture the true amount of mass loss in the cluster. This is most evident in the later evolutionary stages: while the isochrone has masses around 0.82-0.83 M ⊙ in the RGB, this only drops to 0.79 by the time of the HB and early AGB, which is significantly higher than the observed values. A consequence of the overly conservative mass loss prescription is that the isochrone barely reaches the red edge of the RHB region and therefore cannot reproduce the structure of the HB at all. We note that mass loss parameters cannot be adjusted in MIST through the web interface, either. \nFig. 6. a) The mass distributions (calculated as KDEs) for the M4 sample of red giants and RR Lyraes. We have separated the KDEs into evolutionary phases. For the HB sample, we have included a KDE consisting of just the rHB sample (blue), and a KDE which combines both the rHB and RR Lyrae samples (cyan). We note that the area of each KDE is normalised to one, and as such the heights of the distributions do not correspond to sample sizes. b) Individual masses for the red giant and RR Lyrae samples plotted against Gaia magnitude. \n<!-- image --> \nTable 3. Average masses of each evolutionary group. \nOur results indicate that for old, low-mass stellar populations, such as globular clusters, pre-packaged MIST isochrones (and any other isochrone set with similarly conservative mass loss prescriptions) can only reliably reproduce the masses of stars up to the lower RGB, before reaching the red giant branch bump. This highlights the importance of cluster seismology and realistic mass loss estimates, because these studies can provide the first observational constraints on the required level of mass loss through the later stages of low-mass stellar evolution.", '4.3. Comparison of the rHB and RRc masses': 'As we discussed in Section 1, estimating the masses of RR Lyrae stars in a model-independent way is an extremely di ffi cult task. While evolutionary models can be used to constrain HB masses in globular clusters (see, e.g., Gratton et al. 2010; Howell et al. \n2024; Kumar et al. 2024), and masses for RRd stars can be estimated from non-linear models (Molnár et al. 2015a), these approaches still need independent verification. Here we present, for the first time, simultaneous mass estimates based on two independent seismic approaches for stars along the HB in a globular cluster. And although these seismic models and scaling relations could still contain their own model uncertainties, they can now be verified against each other. \nFor M4, we found a combined HB mass of 0 . 655 ± 0 . 007 M ⊙ . If we split this sample into their constituent RR Lyrae and rHB parts, we find average masses of M rHB = 0 . 657 ± 0 . 034 M ⊙ and M RRc = 0 . 651 ± 0 . 028 M ⊙ , respectively. A small non-zero di ff erence is in agreement with the evolutionary predictions that stellar (envelope) masses along the HB decrease towards the blue (Salaris & Cassisi 2006). Therefore, we can conclude that our two seismic mass estimates, the scaling relations for rHB stars, and the linear RRc model grid of Netzel & Smolec (2022), are in agreement. Thus, we may conclude that fitting the f 61 modes o ff ers a reliable way to estimate RR Lyrae star masses, even if it is only available for overtone stars.', '4.4. Comparison with mass relations': 'Relations to calculate masses for RR Lyrae, or more broadly for HB stars from observational constraints, have been given by multiple authors, derived from pulsation and evolutionary \nFig. 7. The distribution of masses with increasing luminosity among the evolutionary groups. Lines indicate the average masses and uncertainties. \n<!-- image --> \nmodels. Here we compare seven prescriptions with our seismic masses. The first widely adopted relation on dependence of the pulsation period on the physical parameters ( M, L, T e ff ) was published by van Albada & Baker (1971). This was later refined and updated by multiple authors, also incorporating a metallicity term (Bono et al. 1997; Jurcsik 1998; Marconi et al. 2003; Di Criscienzo et al. 2004; Marconi et al. 2015). A more general relation between mass, color and metallicity was given for HB stars by Gratton et al. (2010). \nThe accuracy of these prescriptions, however, strongly depend on the modeling choices and uncertainties intrinsic to parameter choices, which are not often explored. Furthermore, many prescriptions are based on T e ff and L values, which have to be converted from observations and have many uncertainties and potential parameter degeneracies themselves. Converting apparent brightness to theoretical luminosity, for example, involves uncertainties in the distance of the object, in the amount of interstellar absorption between the observer and the object, and even in the calculation of the bolometric correction. Similarly, converting observed colors to T e ff involves uncertainties coming from interstellar absorption and the choice of extinction law parameter, as well as from the accuracy of the calibration of the colorT e ff conversion scales themselves. \nWith these limitations in mind, we calculated the RR Lyrae masses using these relations and compared them to our seismic results in two ways. First we used the classical constraints, L \nFig. 8. The seismic HRD of M4. Here we show stars in the log L -T e ff plane for which masses are available. Colors indicate stellar mass. The gap in the HB between RRc and rHB stars is where RRab stars reside, for which no mass estimate is available. The solid line is a MIST isochrone with an age of 12.2 Gyr and an [Fe / H] index of -0.9. \n<!-- image --> \neff \nand T e ff values, computed from the Gaia photometry, which also served as initial values for our seismic model fits. These results are plotted in the left panel of Fig. 9. As the figure shows, the values are generally in the right range, between 0.55 and 0.8 M ⊙ , however, di ff erences from the seismic values can reach 0.1 M ⊙ , and uncertainty ranges can be even wider than that, ranging from ± 0 . 03 to ± 0 . 12 M ⊙ . \nWe then recalculated the masses using the L and T e ff values obtained from the seismic model fits listed in Table 2, and plotted them in the right panel of Fig. 9. In contrast with the results above, these correlate very well with our seismic masses. The matches are not perfect, however, as clear systematic di ff erences are visible. Most relations give results very similar to the original one published by van Albada & Baker (1971), underestimating the masses by an average di ff erence of -0.020 to -0.028 M ⊙ , relative to the seismic masses. The only relation that overestimates the masses is that of Marconi et al. (2015), with an average difference of + 0.032 M ⊙ . These results highlight the true di ff erence between various models and their parameter choices, rather than the uncertainties coming from the observations. The main source of observational uncertainty left in these fits is in the frequencies of the modes we fit. \nGratton et al. (2010) determined a di ff erent mass relation based on the B -V (or V -I ) color indices and [Fe / H] indices \n<!-- image --> \nFig. 9. Comparison of the RRc masses with results from various mass relations based on physical parameters. On the left: masses calculated from classical observational constraints obtained for L and T e ff . Right: masses calculated from the L and T e ff outputs produced by our seismic results listed in Table 2. Seismic masses are shifted by small amounts in both plots to make individual error bars visible. The dashed line is the identity line in both plots. \n<!-- image --> \nFig. 10. Comparison of the seismic HB masses, both for the RRc and rHB stars, with the mass relation published by Gratton et al. (2010). Empty symbols indicate marginal detections. Large grey symbols indicate the averages for both groups. The dashed line is the identity line. \n<!-- image --> \nof HB stars. We calculated the masses of both the RRc and the rHB stars using our dereddened B -V colors. The results are displayed in Fig. 10. This relation results in a larger mass di ff erence (0.037 M ⊙ ) between the two groups than what we found from the seismic data. We suspect that this di ff erence is coming from the fact that the relation is based on zero-age HB stellar models, whereas the stars in M4 are not necessarily on the ZAHB, but rather at evolutionary stages dictated by their common ages and thus common masses.', '5. Conclusions': "Determining the masses of RR Lyrae variables is a notoriously di ffi cult task. While relations based on physical parameters exist, and pulsational masses can, in principle, provide masses for \ndouble-mode stars, these are strongly model-dependent methods that could not be verified by independent mass estimates. Asteroseismology o ff ers a new way to determine stellar masses (Aerts 2021). While seismic masses still depend on stellar envelope models and on the validity of scaling relations, they can be tested against other observational techniques, e.g., dynamical masses in eclipsing binaries. \nGlobular clusters, such as M4, provide an opportunity to probe the late stages of low-mass stellar evolution. Building on the unique capabilities of the Kepler space telescope, we calculated new, more accurate masses for the oscillating giant stars using asteroseismic scaling relations and ν max values determined with the new pyMON code (Howell et al., in prep.). We then analysed the overtone RR Lyrae stars in the cluster and determined their masses using peak-bagging and fitting the frequencies with the seismic model grid of Netzel & Smolec (2022). With these, we were able to relate RR Lyrae seismic masses to seismic masses of red HB stars, and found that they match each other closely. Furthermore, we estimate a He content of Y = 0 . 266 ± 0 . 008 for the cluster from our models. \nWhile our result is not yet a direct measurement of RR Lyrae masses, as it relies on the assumption that RR Lyrae and rHB stars in the cluster have similar masses, we were able to test model-dependent seismic (or pulsational) masses against independent mass estimates for the first time. We find that the average RRc and rHB masses agree within the errors, with the RRc stars having slightly lower masses, as expected from the stucture of the HB. We compared the modeled physical parameters of the RRc stars to various relations between M, L, T e ff and the pulsation periods, and found that while the generally agree, systematical di ff erences do exist. We also show, however, that the usability of these relations is rather limited, as L and T e ff values have to be determined very accurately, otherwise the calculated masses will be very uncertain. \nThis study not only tests RR Lyrae masses for the first time, but it also lends further support to the hypothesis of the f 61 modes being high-order ℓ modes, as proposed by Dziembowski (2016). Therefore, it is now possible to estimate masses for field RRc and RRd stars as well, where enough modes can be detected from high-precision photometry (Netzel & Smolec 2022; Netzel et al. 2023). Furthermore, RRc stars now o ff er a way to measure masses on the HB in globular clusters, even where rHB stars are too faint for asteroseismology. RR Lyrae stars thus open up a new way to study mass loss in more clusters, either with Kepler , or with future instruments, such as the Roman Space Telescope \n(Howell et al. 2024; Molnár et al. 2023a) or the HAYDN telescope project, which is designed specifically to observe dense stellar fields (Miglio et al. 2021). \nAcknowledgements. This research was supported by the 'SeismoLab' KKP137523 Élvonal grant of the Hungarian Research, Development and Innovation O ffi ce (NKFIH). Cs.K. was supported by the ÚNKP-23-3, New National Excellence Program of the Ministry of Culture and Innovation from the source of the National Research, Development and Innovation Fund. This research has received support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 947660). H.N. is funded by the Swiss National Science Foundation (award PCEFP2\\_194638). M.J. gratefully acknowledges funding of MATISSE: Measuring Ages Through Isochrones, Seismology, and Stellar Evolution , awarded through the European Commission's Widening Fellowship. This project has received funding from the European Union's Horizon 2020 research and innovation programme. This research made use of NASA's Astrophysics Data System Bibliographic Services, as well as of the SIMBAD database operated at CDS, Strasbourg, France.", 'References': 'Aerts, C. 2021, Reviews of Modern Physics, 93, 015001 \nAigrain, S., Parviainen, H., & Pope, B. J. S. 2016, MNRAS, 459, 2408 \n- Anders, E. H. & Pedersen, M. G. 2023, Galaxies, 11, 56\n- Antipin, S. V. & Jurcsik, J. 2005, Information Bulletin on Variable Stars, 5632, 1 Appourchaux, T. 2003, Ap&SS, 284, 109\n- Asplund, M., Grevesse, N., Sauval, A. 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We calculated the Fourier-spectra of both the amplitude and phase variations. The former is also shown in Fig. B.1; the spectrum of the phase variation looks very similar. Both show to separate frequency peak, from which we calculated two modulation periods: Pm 1 = 11 . 39 ± 0 . 05 d and Pm 2 = 19 . 63 ± 0 . 25 d.', 'Appendix B: Data tables': 'In Table B.1, we present the reduced K2 light curves for the five RRc stars analysed in this work. The full table will be available online. In Table B.4, we list the updated seismic and physical parameters for the RGB, AGB and rHB stars. In Tables B.2 and B.3 we list the mass estimates we show in Figures 9 and 10. \nTable B.1. K2SC- and trend-corrected photometric data for the five RRc stars. The full table is available in machine-readable format online.Table B.2. Mass values for RR Lyrae and rHB stars based on the mass relation of Gratton et al. (2010), compared to our seismic masses. \nTable B.3. Mass values calculated from the various mass relations, using either the classical observational constraints or the physical parameters from our seismic fits as constraints.Table B.4. The updated global seismic properties and calculated physical param- \neters for the giants in M4. \nTable B.4. continued.'}

2024arXiv240912684C

We present a neural network classification model for detecting the presence of hyperonic degrees of freedom in neutron stars. The models take radii andor tidal deformabilities as input and give the probability for the presence of hyperons in the neutron star composition. Different numbers of observations and different levels of uncertainty in the neutron star properties are tested. The models have been trained on a dataset of wellcalibrated microscopic equations of state of neutron star matter based on a relativistic meanfield formalism. Real data and data generated from a different description of hyperonic matter are used to test the performance of the models.

2024-09-01T00:00:00Z

['arXiv:2409.12684', '2024arXiv240912684C', '10.48550/arXiv.2409.12684']

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

Detecting Hyperons in neutron stars a machine learning approach

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12684.pdf

{'Detecting Hyperons in neutron stars - a machine learning approach': 'Valéria Carvalho, 1, ∗ Márcio Ferreira, 1, † and Constança Providência 1, ‡ \n1 \nCFisUC, Department of Physics, University of Coimbra, P-3004 - 516 Coimbra, Portugal (Dated: September 20, 2024) \nWe present a neural network classification model for detecting the presence of hyperonic degrees of freedom in neutron stars. The models take radii and/or tidal deformabilities as input and give the probability for the presence of hyperons in the neutron star composition. Different numbers of observations and different levels of uncertainty in the neutron star properties are tested. The models have been trained on a dataset of well-calibrated microscopic equations of state of neutron star matter based on a relativistic mean-field formalism. Real data and data generated from a different description of hyperonic matter are used to test the performance of the models.', 'I. INTRODUCTION': "Neutron stars (NSs) are among the densest objects in the Universe, yet their internal properties and composition remain open questions. The equation of state (EOS) of NS matter, dense and asymmetric nuclear matter, is the main research focus in NS physics. In particular, observations of massive NSs provide constraints on the EoS at intermediate and high baryonic densities. In addition, the advent of multi-messenger astrophysics, which combines different sources of information about NS such as gravitational waves (GWs), photons, and neutrinos, has improved our knowledge of NS physics. \nAt moderate and high baryonic densities, constraints on the EOS of NS matter are primarily provided by observations of massive NSs. Key measurements include 1 . 908 ± 0 . 016 M ⊙ for PSR J1614-2230 [1-3], 2 . 01 ± 0 . 04 M ⊙ for PSR J0348-0432 [4], 2 . 08 ± 0 . 07 M ⊙ for PSR J0740+6620 [5], and 2 . 13 ± 0 . 04 M ⊙ for PSR J1810+1714 [6]. The rise of multi-messenger astrophysics has further enriched our understanding of NS physics by combining information from various sources, including GWs, photons, and neutrinos. The detection of compact binary coalescence events, such as GW170817 [7] and GW190425 [8] by the LIGO/Virgo collaboration, has placed additional constraints on the EOS. Moreover, recent mass and radius inferences from the NICER (NS Interior Composition ExploreR) experiment, including PSR J0030+045 [9, 10], and PSR J0740+6620's radius [11-13] have significantly narrowed the range of possible NS matter scenarios. Additionally, observations of PSR J0427-4715 [14, 15], the brightest and closest known millisecond pulsar, provide further constraints on both mass and radius, complementing the previous measurements. Future experiments, with enhanced precision, are expected to further refine our understanding of the EOS and NS properties. Missions such as the enhanced X-ray Timing and Polarimetry mission (eXTP) [16, 17], the Spectroscopic Time-Resolving Observatory for Broadband Energy (STROBE-X) [18], and \nthe Square Kilometer Array (SKA) telescope [19] will be crucial in constraining the different theoretical scenarios for NS matter. \nFrom the theoretical side, the low-density region of the NS EOS is constrained by chiral effective field theory (cEFT) [20-22], while perturbative quantum chromodynamics (pQCD) becomes reliable at high densities [23] (see [24] for a review). Inference of the NS EOS, given a set of theoretical, experimental and observational constraints, is frequently carried out using Bayesian inference frameworks [13, 22, 2533]. Several studies take an agnostic modeling of the EOS, e.g., piecewise polytropes, the speed of sound formalism, spectral approaches, Gaussian processes [25, 30, 34-40]. Although these methods provide valuable insights into the EOS, they do not offer information about the underlying particle composition of NS matter. \nNSs are composed of five distinct layers: the atmosphere, outer crust, inner crust, outer core, and inner core. However, the exact composition of the core remains a subject of ongoing investigation. Several hypotheses have been proposed, suggesting that the core could consist of nucleonic matter, quark matter, hyperons, or even dark matter. One of the most discussed possibilities is the presence of hyperons in the inner core, see for example [41-47]. Due to the extreme densities at the center of an NS and the rapid increase in the chemical potential of the nucleon with increasing density, the formation of strange hadrons, such as hyperons, could become energetically favorable. While the appearance of hyperons seems almost inevitable under such conditions, their presence would soften the EOS, which could lead to a maximum NS mass that contradicts the observed 2 M ⊙ NSs [44, 48-50]. This discrepancy, known as the hyperon puzzle, remains unsolved. \nThe use of machine learning frameworks, and specially feed-forward neural networks (NNs) based methods, as inference tools in high-energy physics has seen a growing surge of interest across various disciplines [51]. In NS physics, NNs have been extensively applied to study the EoS of NSs in several works [52-58]. In addition, Bayesian NNs, which are models capable of uncertainty quantification, have been employed to map NS observations to critical quantities such as the speed of sound squared and the proton fraction \nwithin NS cores [59], as well as to investigate the properties of nuclear matter [60]. Recently, an upgraded method for obtaining the probability distribution of NS M ( R ) relationships was introduced in [61], improving upon earlier works [62-64]. This new approach uses Monte Carlo integration rather than Gaussian input noise to refine predictions, while also incorporating new observational data from NICER and the GW170817 event. Generative modelling has also begun to emerge as a promising alternative for decoding NS characteristics. For instance, the use of normalizing flows to detect phase transitions in NSs through anomaly detection [65], or the use of conditional variational autoencoders as EOS sampler (conditional on an observational set) to gain insights into NS properties [66]. A framework aimed to infer the EOS directly from telescope observations based on normalizing flows model coupled with Hamiltonian Monte Carlo methods was introduced in [67]. Studies focused on GW also play a significant role in constraining the EoS of NS. Machine learning can be used to analyze GW signals from binary NS mergers. For instance, [68] explored the use of the Audio Spectrogram Transformer model, a type of NN architecture inspired by how we process sound, to classify the EoS solely based on GW signals. \nThis work investigates the potential identification of the presence of hyperons within NSs by employing NN classification models to analyze key NS observables. Specifically, we explore three types of input data: mass-radius, mass-tidal deformability, and a combination of both, while also varying the level of input noise. The training dataset consists of relativistic mean-field (RMF) models, which are constrained by minimal conditions such as low-density properties and pure neutron matter, in addition to describing two solar-mass stars using Bayesian inference. To evaluate the models' performance, we first tested the trained models using two theoretically based datasets: one derived from the same dataset used for training, and another generated with a different formalism but also constructed within the RMF framework. Following this, we assessed the models' predictive capabilities for massradius observables, utilizing observational data from our previous study [60]. The observational data were carefully adjusted to match the respective training regions, ensuring consistency between the theoretical and real-world test cases. The paper is organized as follows: Section II introduces the selected family of nuclear models and explains the generation of mock observational datasets. Section III provides a brief overview of NNs. The results are presented and discussed in Section IV, followed by concluding remarks in Section V.", 'II. DATASET': 'To achieve our goal of identifying the potential existence of hyperons within NSs, we first need to construct a comprehensive dataset, which will serve as the foundation for developing our machine learning models. This section outlines the process of creating the different datasets used throughout this study.', 'A. Theoretical concepts of dataset': 'The data used to train the machine learning models is derived from nuclear models based on an RMF theory, where nucleon interactions are mediated by the exchange of scalarisoscalar, vector-isoscalar, and vector-isovector mesons [6974]. To allow a realistic description of nuclear matter, the Lagrangian densities include self-interacting and mixed meson terms. These terms define the density dependence of the EOS and the symmetry energy. We refer to this approach and the corresponding EOS as nonlinear (NL). An alternative would be to consider density dependent meson couplings [75]. In Sec. IV C a set of EOSs obtained within this last formalism in [28] will also be introduced and referred to as DDB EOSs. A detailed description of the EOS used in the present study can be found in [33], where within a Bayesian inference calculation the parameters of the RMF model were constrained. The data set, which is publicly available 1 , contains 17810 EoS of pure nucleonic matter and 18728 EoS of hadronic matter containing both nucleons and hyperons. The Λ and Ξ hyperonmeson couplings are allowed to vary in a range compatible with the properties of hypernuclei as determined in the studies [76-78]. The corresponding mass-radius curves, calculated using the Tolman-Oppenheimer-Volkoff (TOV) equations, are shown in Fig. 1. The Bayesian framework used to generate the two sets of EOSs imposes minimal constraints on various nuclear saturation properties. This approach ensures the reproduction of 2 M ⊙ NSs and maintains consistency with lowdensity pure neutron matter as predicted by N 3 LO calculations in chiral effective field theory (see table I). \nTABLE I: The different constraints imposed via Bayesian inference: the binding energy per nucleon ϵ 0 , incompressibility K 0 , and symmetry energy J sym at the nuclear saturation density n 0 (with 1 σ uncertainties). The pressure of pure neutron matter (PNM) is considered at densities of 0.08, 0.12, and 0.16 fm -3 , obtained from a χ EFT calculation [20]. \nFIG. 1: Mass-radius, left plot, and mass-log(tidal deformability), right plot, curves for nucleonic matter (17810 curves, shown in blue) and hyperonic matter (18728 curves, shown in red). \n<!-- image -->', 'B. Mock datasets generation': 'The goal of this work is to use simulated NS observations, represented by the set X , as inputs to a machine learning model aimed at classifying whether or not an NS contains hyperons, represented by the set Y . The input set X consists of D rows of vectors, denoted by x , while the output set Y consists of D rows of scalars, denoted by y . The value of D defines the size of the data set. To clarify, the output set Y is expressed as Y = { y ( i ) } D i =1 , and the input set X is expressed as X = { x ( i ) } D i =1 , i.e, the EOSs in the data set are characterized by the tuples ( x ( i ) , y ( i ) ) . The output scalar y is binary, where 0 indicates the presence of hyperons and 1 indicates their absence. The input vectors x can take three different forms, depending on the data type ( R , Λ , or R Λ ) or the number of NS considered ( Q = 5 , 10 or 15): \n- 1. The R datasets contain simulated R ( M ) data. The observations along the R ( M ) curve are denoted by pairs ( M R q , R q ) , and the input vectors take the form x R = { ( M R 1 , R 1 ) , ( M R 2 , R 2 ) , ..., ( M R Q , R Q ) } , with dimension of 10, 20, and 30, when considering 5, 10, 15 NS, respectively.\n- 2. The Λ datasets contain simulated Λ( M ) data. The observations along the Λ( M ) curve are denoted by pairs ( M Λ q , Λ q ) , and the input vectors take the form x Λ = { ( M Λ 1 , Λ 1 ) , ( M Λ 2 , Λ 2 ) , ..., ( M Λ Q , Λ Q ) } , with dimension of 10, 20, and 30, when considering 5, 10, 15 NS, respectively.\n- 3. The R Λ datasets contain both simulated R ( M ) and Λ( M ) data. The observations along the R ( M ) and Λ( M ) curves are denoted by pairs ( M R q , R q ) and ( M Λ q , Λ q ) , and the input vectors take the form x R Λ = { ( M R 1 , R 1 , M Λ 1 , Λ 1 ) , ( M R 2 , R 2 , M Λ 2 , Λ 2 ) , ..., \n( M R Q , R Q , M Λ Q , Λ Q ) } , with dimension of 20, 40, and 60, when considering 5, 10, 15 NS, respectively. \nThe use of three types of datasets is aimed to investigate how much information is available in NSs observations, GW observations, and the combination of both when inferring the possible presence of hyperons in NS. To ensure a balanced representation of each composition in the dataset, the larger number of hyperonic EoS was adjusted by selecting 17810 EOS from both nucleonic and hyperonic compositions, resulting in a total of 35620 EOS, meaning a D =35620. The dataset was then randomly divided into a training set containing 90% of the data and a test set comprising the remaining 10%. \nThe statistical procedure for generating the mock data from the M ( R ) curves with distinct input noises follows the following steps, as detailed in [60]. For a given EoS, we first randomly sample Q NS mass values, M 0 q ∼ U [1 , M max ] , where M max is the maximum mass corresponding to the respective TOV curve, and determine the radius from the TOV solution, ( M 0 q , R ( M 0 q )) , for q = 1 , ..., Q . Then, the actual noisy observation values ( M R q , R q ) are obtained by sampling from Gaussian distributions centered at the TOV solution: M R q ∼ N ( M 0 q , σ 2 q,M R ) and R q ∼ N ( R ( M 0 q ) , σ 2 q,R ) , where σ q,R ∼ U [0 , σ R ] and σ q,M R ∼ U [0 , σ M R ] for q = 1 , ..., Q . The values of σ R and σ M R are shown in Table II. These Q pairs ( M R q , R q ) collectively characterize the M ( R ) diagram of a given EoS. Hereafter, for a given EOS, we denote by single observation the corresponding input vector x = [ M R 1 , · · · , M R Q , R 1 , · · · , R Q ] . To incorporate the tidal deformability, we take a similar statistical procedure: we first sample M Λ q ∼ U [1 , M max ] and then Λ q ∼ N (Λ( M Λ q ) , σ 2 Λ ( M Λ q )) , where Λ( M Λ q ) represents the tidal deformability of the star, and the values σ 2 Λ ( M Λ q ) can \nbe found in Table II. The value of σ Λ ( M Λ q ) is defined as σ Λ ( M Λ q ) = constant × ˆ σ ( M Λ q ) , where ˆ σ ( M Λ q ) is computed from the training dataset as the standard deviation of Λ( M ) , we use this value given the broad range of values of the tidal deformability. For datasets containing tidal deformability, a single observation consist of an input structure as x = [ M Λ 1 , · · · , M Λ Q , Λ 1 , · · · , Λ Q ] , while when both massradius and tidal deformability are considered becomes x = [ M R 1 , · · · , M R Q , R 1 , · · · , R Q , M Λ 1 , · · · , M Λ Q , Λ 1 , · · · , Λ Q ] . During training, the logarithm of tidal deformability is used due to its vastly different scale compared to mass and radius. \nFor each EOS, we replicate the above procedure n s times, i.e., the input vector x is resampled n s times. This approach expands the dataset to D = n s × D . For training, each EoS was simulated with 20 mock observations ( n s = 20 ), whereas for testing n s = 1 was used. This setup mimics a real-world scenario where typically only a single mock observation of the "true" EoS is available. \nIndependent datasets were generated with distinct input noise levels for each input size. Therefore, for each dataset type ( R , Λ and R Λ ), we have three possibilities for the input size Q (5, 10 or 15 NS) and three noise levels (see Table II), corresponding to a total of 27 different models. Where for these 27 different models we always have the same D size, which is D = 20 × 32058 for training and D = 1 × 3562 for testing. \nThe three levels of input noise are the following: i) a noiseless case, X 0 ; ii) a small amount of noise measured by σ M R = 0 . 1 M ⊙ , σ R = 0 . 2 km, and σ Λ ( M Λ q ) = 0 . 5ˆ σ ( M Λ q ) , X 1 ; and iii) a larger noise level compatible with present observations, given by σ M R = 0 . 136 M ⊙ , σ R = 0 . 626 km, where the process to obtain this values is described in our previous works [60, 65], and σ Λ ( M Λ q ) = 2ˆ σ ( M Λ q ) . We have chosen to characterize the tidal deformability noise by the standard deviation of the training dataset ˆ σ ( M Λ q ) , e.g., ˆ σ (1 . 4 M ⊙ ) = 130 . 05 . The primary goal of creating these three distinct datasets is to evaluate the model\'s performance under varying noise levels.', 'III. NEURAL NETWORKS': 'The objective of this study is to apply NNs to classify NS observations, distinguishing between those that suggest the presence of hyperons in the NS composition and those that do not. This classification will be performed across datasets with varying input noise levels and sizes, as explained in the previous section. Additionally, we will explore the impact of incorporating tidal deformability into the classification process. In this section, we will delve into the mechanics of NNs and their specific application to our research. \nTABLE II: The parameters used for generating the mock data for each set. The interpolation function ˆ σ ( M Λ q ) was derived from the training dataset and represents the standard deviation for each possible value of mass.', 'A. How it is defined': 'A NN consists of interconnected neurons organized into layers, including input s = 0 , hidden s = 1 , ..., S -1 and output layers, s = S , where S +1 is the total number of layers. The data is composed of a set of ( x ( i ) , y ( i ) ) tuples, i.e., D = { ( x ( i ) , y ( i ) ) } D i =1 , where x ( i ) ∈ R Q and y ( i ) ∈ R K . Each layer is composed of neurons, which are connected to neurons in adjacent layers through weights, denoted as the matrix W . Additionally, each neuron has a bias term, denoted as b , which serves as a threshold. The network output is defined as ˆ y = NN θ ( x ) , which is parameterized by θ = { ( W [ s ] , b [ s ] ) } S s =1 . The computation within each neuron involves multiplying the weights with the corresponding neurons from the previous layer and summing these products for each hidden unit. The bias term is also added to the sum, giving a result of Σ = Wx + b . Subsequently, a non-linear function, known as the activation function, and denoted by ϕ ( · ) , is applied to each neuron within both the hidden layers and the output layer. One of the most commonly used activation functions is the Rectified Linear Unit (ReLU) [81], defined as ϕ ( x ) = max(0 , x ) . This activation function determines the neuron\'s output, essentially dictating how "active" or responsive the neuron becomes. The final output for the n th neuron is therefore expressed as a = ϕ ( Σ ) . For the first hidden layer, the output of the hidden units is given by \n a ( i ) 11 . . . a ( i ) n 1 = ϕ [1] W [1] 11 · · · W [1] 1 q . . . . . . . . . W [1] n 1 · · · W [1] nq x ( i ) 1 . . . x ( i ) q + b [1] 1 . . . b [1] n . (1) \nPassing through all layers, the final result is calculated as \nNN θ ( x ) = ϕ [ S ] ( W [ S ] ϕ [ S -1] ( · · · ϕ [1] ( W [1] x + b [1] ) · · · ) ︸ ︷︷ ︸ a S -1 + b [ S ] ) , (2) \nin matrix form, this becomes \n ˆ y ( i ) 1 . . . ˆ y ( i ) k = ϕ [ S ] W [ S ] 11 · · · W [ S ] 1 h . . . . . . . . . W [ S ] k 1 · · · W [ S ] kh a ( i ) 1 S -1 . . . a ( i ) hS -1 + b [ S ] 1 . . . b [ S ] k . (3) \nNote that in the present work, while the input vector x ( i ) has different dimensions, depending on the dataset type or number of NS being considered (see Table II), e.g., x ( i ) ∈ R 2 Q , for R and Λ and x ( i ) ∈ R 4 Q for R Λ with Q = { 5 , 10 , 15 } , the output dimension is always fixed to one, as we are dealing with a single probability (scalar), i.e., y ( i ) ∈ R 1 , meaning K=1. In a typical multi-class classification problem the final activation function ϕ [ S ] consists of a softmax function [82], however for a binary classification problem is the sigmoid activation function, ϕ ( x ) = 1 / (1 + exp( -x )) , ensuring that the model\'s probability predictions, NN θ ( x ( i ) ) ≡ p , is constrained between 0 and 1.', 'B. How it is trained': "The primary objective during training is to minimize a designated loss function by optimizing the model's parameters, denoted as θ , in order to attain the lowest value of the loss \nθ ∗ = arg min θ L ( θ ) . (4) \nTo achieve this, the back-propagation algorithm is employed, consisting of two main phases: the forward pass and the backward pass. In the forward pass, the model computes predictions and evaluates the loss function, which measures the error between the true output and the model's predictions. For binary classification tasks, this is typically done using the binary cross-entropy loss function [83], defined as \nL ( θ ) = 1 D D ∑ i =1 [ y ( i ) log( NN θ ( x ( i ) )) +(1 -y ( i ) ) log(1 -NN θ ( x ( i ) )) ] . (5) \nThe backward pass calculates derivatives of the loss function with respect to each parameter in the NN. These derivatives are then subtracted from the corresponding parameter values \n∂L ( θ ) ∂ θ = 1 D D ∑ i =1 ∂l ( y ( i ) , NN θ ( x ( i ) )) ∂ θ , (6) \nθ ' = θ -η ∂L ( θ ) ∂ θ , (7) \nwhere η represents the learning rate, a hyperparameter that governs the step size during parameter updates. The derivatives are computed using the chain rule \n∂l ( y ( i ) , NN θ ( x ( i ) )) ∂ θ = ∂l ( y ( i ) , NN θ ( x ( i ) )) ∂ NN θ ( x ( i ) ) ∂ NN θ ( x ( i ) ) ∂ϕ [ S ] ∂ϕ [ S ] ∂ θ . (8) \nWeight updates in NNs are commonly accomplished through minibatch gradient descent. It involves segmenting the dataset into smaller mini-batches, with the batch size determining the number of data points used to update the weights in each iteration. This method offers a strike balance between accurate optimizations and computation efficiency. An epoch is then when all the training data has been used.", 'C. Training procedure for our problem': 'A schematic representation of the problem is present in Fig. 2, where the three different input types, each associated with three distinct noise levels, are represented (see Table II). Additionally, each input type varies by the number of stars used (5, 10, or 15). The output is a scalar value, defined as a probability distribution, p , where we assign the class No Hyperons (NH) as 1 and the class Hyperons (H) as 0. For the \nFIG. 2: Schematic representation of the classification model. Depending on the type of model, the size of the input vectors is 2 Q ( R i and Λ i models) and 4 Q ( R Λ i model), where Q indicates the number of NS used (5, 10 and 15). The output of the model is a scalar value p ∈ [0 , 1] , where the value 0 identifies the hyperons class whereas 1 indicates the absence of hyperons (no hyperons class). \n<!-- image --> \ntraining procedure, a random subset of the training data was reserved for validation, with an 80/20 split for training and validation, respectively. The input data X was standardized. \nThe optimal machine learning model is identified through the tuning of its hyperparameters, including the number of \nneurons, layers, and activation functions. For the hidden layers, we explored ReLU, Softplus, and Sigmoid activation functions, with a Sigmoid activation function for the output layer. For the sets R and Λ , which share the same input size, various NN architectures were tested. A grid search revealed that all input configurations performed equally well with the same architecture, see Table III. \nFor the R Λ sets, a separate grid search was performed to identify the most effective model architectures for each of the three input sizes. The results, presented in Table IV, show that the optimal architecture varies with input size. This suggests that as the number of input features increases \nand the data becomes more complex-incorporating tidal deformability and mass-radius pairs-the model architecture must adapt accordingly. For instance, with an input size of 20 ( Q = 5 stars), the model requires a greater number of layers, likely because the complexity of the input data demands a more sophisticated approach to capture the underlying patterns and interactions effectively than when more information is given for an input size of 40 ( Q = 10 stars). One might wonder why the same architecture is not used for input sizes of 20 in the R , Λ , and R Λ sets. The answer lies in the higher complexity of the R Λ dataset, which combines information from both radius and tidal deformability, creating a more intricate problem for the model to solve. Consequently, the R Λ set requires a distinct architecture to handle this increased complexity, even when the input size is the same. All grid searches were performed using set X 0 , with the assumption that similar behavior will be observed across the other sets. \nDuring training, a linear scheduler was employed to gradually reduce the learning rate from 0.01 to 0.001. The ADAM optimizer [84], enhanced with the AMSgrad improvement [85], was used. The models were trained for 2000 epochs, with the model achieving the lowest validation loss being selected, as early stopping was not implemented. The training process utilized a mini-batch size of 1024. The implementation was carried out using the PyTorch library [86]. \nTABLE III: Parameters for Datasets R and Λ .TABLE IV: Parameters for Datasets R Λ .', 'IV. RESULTS': "After training the NN classification models, we evaluate their performance using several metrics. In this section, we describe these evaluation metrics and present the models' performance on an independent test set that shares the same theoretical framework as the training data. We then test the models on a dataset generated within a different theoretical framework to assess their robustness. Finally, we apply the trained models to real observational data, demonstrating their practical applicability.", 'A. Metrics': "To assess the performance of the probabilistic classifiers models, we convert the output probability estimates into binary classes. We defined that a given sample belongs to the class H if p < 0 . 5 while the sample belongs to the class NH (only nucleons are present) if p ≥ 0 . 5 . The models evaluations rely on key metrics derived from the confusion matrix, represented in Fig. 3. The confusion matrix offers a detailed overview of the model's injected (actual) samples vs. predicted outcomes, including the counts of True Hyperons (TH), True No Hyperons (TNH), False Hyperons (FH), and False No Hyperons (FNH). For instance, the value of TH gives the number of H samples correctly classified, while FNH denotes the number of wrongly classified H samples. These metrics provide a robust framework for analyzing the effectiveness and accuracy of our models. \nThe primary metric employed is accuracy, which represents the proportion of correctly classified instances (both TH and TNH) out of all instances. It is defined as \nAccuracy = TH + TNH TH + TNH + FH + FNH . (9) \nWhile accuracy is useful, it does not always provide a complete picture, especially when dealing with imbalanced datasets. To address this, we also use the F1 score, which is the harmonic mean of precision and recall, also known as True Positive Rate (TPR), where the precision is defined as \nPrecision = TNH TNH + FNH , (10) \nand the recall as \nRecall = TNH TNH + FH . (11) \nFIG. 3: Left: The non-normalized confusion matrices for our specific classification task (detecting the presence of hyperons). The size of the test sets is 1737 H samples and 1825 NH samples. The labels designate the four possible pairs of model input/prediction: True Hyperons (TH), True No Hyperons (TNH), False Hyperons (FH), and False No Hyperons (FNH). Right: Relation between the probability p and the labels TH, TNH, FH and FNN. The threshold has been taken at p = 0 . 5 , which means that Hyperons ( No Hyperons ) corresponds to p < 0 . 5 ( p ≥ 0 . 5) . \n<!-- image --> \nThe F1 score offers a more balanced measure by considering both FH and FNH, and it is particularly effective for imbalanced classes. It is given by \nF1 score = 2 × TNH 2 × TNH + FH + FNH . (12)", 'B. Application to mock data': "The confusion matrices for all models trained are show in Fig. 4, which were computed on the test set. The different models, with architectures defined in Tab. III and IV, are denoted by the nature of their input data, meaning the data to which they were trained and the data to which they will be tested: R i for R ( M ) data, Λ i for Λ( M ) , and R Λ i when both R ( M ) and Λ( M ) data are considered. The index i = 0 , 1 , 2 denotes the three possible noise levels (see Table II). A clear trend emerges from the analyses of the confusion matrices: as noise increases i = 0 to 2 across all input sizes and types, the counts of TH and TNH decrease. This trend indicates that the model's precision diminishes as noise levels rise, which is an expected outcome. The higher number of Actual NH instances observed in the confusion matrices is due to the random class distribution of the test set, which consists of 1737 H samples and 1825 NH samples. This test set distribution will be consistently used for evaluation purposes throughout the analysis. \nFor the R 0 , Λ 0 , and R Λ 0 (noiseless) models, the number of true classifications converges and ceases to improve once the input size reaches Q = 10 NS. In the case of R Λ , this convergence is observed across all noise levels. However, for R and Λ models, there is a noticeable improvement when increasing the number of NSs to 15 under noisy conditions. This suggests that when less information is provided to the model (as in the separate R and Λ inputs), more input NS are necessary to achieve a higher count of true classifications, \nespecially in the presence of noise. Conversely, without noise, the model's precision reaches a plateau at Q = 10 NS. \nA detailed analysis of false predictions shows a substantial increase when comparing X 0 and X 2 , which can be quantified by the ratio X 2 / X 0 , where we are using ( FH ( X 2 ) + FNH ( X 2 )) / ( FH ( X 0 ) + FNH ( X 0 )) . For the R Λ models, this ratio is 9.5, 3.5, and 3.7 from R Λ 0 to R Λ 2 for Q = 5 , Q = 10 , and Q = 15 , respectively. In comparison, the R models exhibit ratios of 9.8, 7.5, and 5.3 for the same values of Q . The Λ models, however, display significantly higher ratios, with false predictions increasing by 29.3, 35.4, and 20.3 for Q = 5 , Q = 10 , and Q = 15 , respectively. These differences highlight varying sensitivities to the number of samples across different sets. The Λ models exhibit a significantly higher increase in false values compared to the R and R Λ models, leading to a more pronounced decline in performance. In contrast, the R Λ models demonstrate greater stability, with less fluctuation in false values and maintaining a more consistent performance across different sample sizes. \nLet us analyze the behaviour of the Accuracy, Eq. (9), over the different datasets. As can be observed in Fig. 5, accuracy increases progressively as the noise level decreases from 2 to 0 (noiseless) across all type models ( R i , Λ i , and R Λ i ). The impact of varying the number of NS in each observation (denoted by Q) is best analyzed in Tab. V, where we also present the F1 score and the Area Under the Curve (AUC) for accuracy as the threshold varies. \nThe F1 score is a consequence of the classes representing the presence (H class) or absence (NH class) of hyperons being equally weighted - implying that the models treat false H and false NH with equal importance. As is typical in binary classification tasks, we have set a probability threshold p thr of 0 . 5 for the binary classification: a sample belongs to the H class if p < 0 . 5 otherwise it belongs to the NH class ( p ≥ 0 . 5 ). To discuss the consequences of choosing a different probabil- \nFIG. 4: Non-normalized Confusion matrices for datasets using R (left), Λ (middle) and R Λ (right) inputs. The size of the test sets is 1737 H samples and 1825 NH samples. Each panel contains three columns for the different number of stars in each observation contains ( Q = 5 , 10 , 15 ) and three rows for the different levels of noise considered X i , i = 0 , 1 , 2 defined in Table II. \n<!-- image --> \nFIG. 5: Accuracy for all dataset types, R (in red), Λ (in blue) and R Λ (in gray) (see Table II). The x-axis indicates the noise size X i , i = 0 , 1 , 2 (see Table II) and the symbols denote the number of NSs of each observation ( Q = 5 , 10 , 15 ). \n<!-- image --> \ny threshold p thr on the performance of the models, we introduce the concept of the AUC of the model's accuracy as the threshold p thr varies. While AUC is commonly associated with the Receiver Operating Characteristic (ROC) curve, as discussed in [87], we use it here to evaluate the accuracy of the model across different thresholds. This approach allows us to measure how well the model performs independently of a particular threshold. An AUC of 1 would indicate perfect classification. \nThe highest accuracy values, see Tab. V, although differing by only 0.001, are observed with Q = 10 for both R 0 and Λ 0 . For R Λ 0 a higher accuracy is obtained with a smaller number of pairs ( Q = 5 ), probably due to the additional informa- \non provided by the combination of mass-radius and masstidal deformability inputs. For the R 1 to R 2 and Λ 1 to Λ 2 models, accuracy increases steadily as the number Q of pairs increases, with more pronounced improvements observed in higher noise scenarios, where each additional pair significantly improves the performance of the model. In contrast, the accuracy for R Λ 1 and R Λ 2 tends to plateau at Q = 10 , indicating that beyond this point, adding more pairs does not improve accuracy further, possibly due to the model reaching its capacity to effectively process the combined data. The F1 score results show a very similar behaviour to the accuracy results, and we can see that they only differ by the third decimal place for some values. The AUC value shows the same behaviour as the Accucay in the majority of cases. This consistency suggests that the results are not very sensitive to the choice of threshold. \nOverall, our models demonstrate strong performance across all evaluated sets, with consistently high metric values, particularly in terms of accuracy and F1 score. Although there is a noticeable decline in precision as noise is introduced - especially when examining the predictions for X 2 , which contains noise levels similar to those found in actual observations - the models still maintain a high level of accuracy. This is particularly evident in the R and R Λ models, where accuracy remains robust despite the added noise. These results are encouraging as they suggest that our models are well-equipped to handle real-world observational data with inherent noise.", 'C. Application to different hadronic models': "Having confirmed the ability of our model to predict the test set, we extend our work to evaluate its performance with additional data sets generated within a different description of nuclear matter. Specifically, we consider the sets DDB from [47] with 19140 EoS with hyperons and 8794 EoS without hyperons. These sets have been calculated within an RMF framework with density dependent couplings [28, 75, 88]. \nTABLE V: Accuracy, F1 score and AUC value across the multiple models, as plotted in Fig. 5. \nTo test the model on this new EoS data, we have created corresponding datasets for each model, incorporating the same levels of noise as defined in Tab. II, following the procedure outlined in section II for the test set with n s = 1 . This resulted in a combined dataset of D = 27934 samples. The results for accuracy are presented in Fig. 6, and Tab VI provides a detailed breakdown of the accuracy, F1 score, and AUC values. The overall accuracy remains consistently above 73%, demonstrating that the model reliably predicts the results. Notably, accuracy increases with the number of observations ( Q ) for most sets, except for a slight dip from Q = 5 to Q = 10 in sets R Λ 1 and R Λ 2 . The higher accuracy observed for the set X 1 compared to X 0 can be explained by the regularizing effect of noise, which helps prevent overfitting and improves the model's ability to generalize to unseen data, i.e. adding a small amount of noise acts as a regularizer, preventing the model from fitting too closely to the specific details of the training data. In the absence of noise, the model may overfit by focusing too much on small patterns, reducing its generalizability. Specifically, the DDB sets predict a larger radius and tidal deformability upper limit for nuclear matter, as well as a smaller lower limit for both quantities in the presence of hyperons, compared to the model's training set, defined as NL. The observed decline in accuracy for R Λ at Q = 10 relative to Q = 5 likely comes from increased discrepancies in the joint probability of R Λ , reflecting formalism differences alongside the uncertainties in set X 1 . For set X 2 , which includes larger uncertainties, the expected trend resumes, with accuracy increasing as the number of observations, Q , grows. Also, we should be aware that maybe some slight overfitting can be happening here given that for Q=10 which now has the smaller performance in the previous section had the biggest. \nLooking at Tab. VI, given the relatively small proportion of NH, it is important to consider not only accuracy but also the F1 score. The F1 score closely mirrors the accuracy trend, reaching values of 0.876, 0.873, and 0.871 for the topperforming sets ( R 0 , R 1 , R Λ 0 ) at Q = 15 , suggesting the model effectively handles class imbalances and difficult-toclassify samples. In addition, the AUC values, which quantify the model's ability to distinguish between classes independently of the classification threshold, range from 0.534 (for \nΛ 2 at Q = 5 ) to 0.867 (for R 0 at Q = 15 ). Most R and R Λ sets achieve AUC values above 0.8 at higher Q levels, indicating strong discriminatory power, particularly when more observations (NSs) are available.FIG. 6: Accuracy for the new dataset, analogous to Fig. 5, showing results for R (red), Λ (blue), and R Λ (grey) (see Table II). The x-axis represents the noise size X i where i = 0 , 1 , 2 (refer to Table II). Symbols on the plot indicate the number of NSs for each observation, corresponding to Q = 5 , 10 , and 15 . \n<!-- image --> \nOverall, the models R and R Λ demonstrated superior performance, while the model struggled the most with Λ 2 , which exhibited lower accuracy, F1-scores, and AUC values. This suggests that the dataset used to train the model contains excessive noise, making it harder for the model to capture patterns effectively. In the case of Λ 2 , the absence of radius information, unlike in R Λ 2 , further complicates the model's ability to learn the underlying data behavior. In summary, while the model performs well across most datasets, there is room for improvement, particularly in handling the Λ datasets with higher noise levels. In the following sections, we will investigate the model's performance using real observational data. \nTABLE VI: Accuracy, F1 score and AUC value across the multiple models, as plotted in Fig. 6 for the new set.", 'D. Application to real data': "As a final test of our classification models, we apply the models that use R ( M ) as input data (denoted as R in Table II) to real NS data. Although the use of tidal deformability data obtained from GW observations would be ideal for constraining the EOS and searching for evidence of hyperons, the current lack of sufficient observational data limits this approach. However, future advances in GW detectors, such as the next generation of ground- and space-based observatories, are expected to play a critical role in this area. Given the model's limited extrapolation capabilities, we carefully selected a subset of NSs observations, listed in Table 1 of [89], ensuring they fell within the model's training region. For the R 0 and R 1 models, we have selected nine observations where the central values of the radius distributions were within ± 0 . 2 km of the training data range, as shown in Fig. 7. \nFIG. 7: Observations taken from Table 1 of [89], for the 68% CI, the gray, blue, and black bands represent the R 2 , R 1 , and R 0 data sets, respectively, and include both solutions with and without hyperons. \n<!-- image --> \nFor R 2 (the trained model with a larger noise value), which better matches the dispersion of the observational data (as illustrated in Fig. 7), we included all observations except M28 . This allowed model R 2 to exhibit an input noise level simi- \nlar to the standard deviation of the mass-radius pairs, providing a more realistic test scenario. To minimize extrapolation given the broad distribution within 1 σ for each observation, we opted to use the mean values of the observations as inputs to the NN models. To generate test data, we employ sampling without replacement to generate 100 samples of the input vector of five NS out of the 9 observations for models R 0 and R 1 and 17 for R 2 . The statistics of the models outputs are shown in Fig. 8, which displays the mean and the more relevant percentiles for the output predictions. This figure is based on Fig. 3 and also considers the probability p = 0 . 5 as the threshold to define the labels Hyperons and No Hyperons . The results indicate that although the model produces predictions across the whole spectrum, the 35 th percentile falls in the No Hyperons range completely across all three models. This implies that 65% of the data across the three sets lies in the No Hyperons region. In addition, the median is completely at p = 1 , suggesting that the observational data are more consistent with NSs lacking hyperons. Note that the confidence of the No Hyperons case is larger for the R 2 model, as can be clearly seen especially for the 12 th percentile, where it is the only model that already falls within the range indicating the absence of hyperons. These results are especially meaningful because R 2 is the model trained with data that closely matches the spread observed in NS observations, thereby reinforcing the reliability of this outcome. \nWe further tested the R 2 model using the model with Q = 10 and Q = 15 , considering the 17 available observations. Due to the limited number of values available for varying the input vectors, we conducted 10 samplings without replacement, resulting in 10 different input vectors. For Q = 15 , all predictions were equal to 1, and for Q = 10 , all predicted class probabilities were p > 0 . 896 . These results further validate our earlier conclusion, especially given the high scores achieved for the metrics with Q = 10 and 15 in the previous section, indicating the reliability of our model's predictions.", 'V. CONCLUSIONS': "We investigated a classification problem aimed at determining the potential presence of hyperons within NSs based \nFIG. 8: Some statistics (mean, median, 35 th , 12 th and 10 th percentiles) for the R 2 , R 1 , and R 0 models represented in the y axis, for Q = 5 . The probability p = 0 . 5 was considered as the threshold to define the labels Hyperons and No Hyperons . \n<!-- image --> \non their macroscopic properties. We investigate the capacity of NNs for the present binary classification problem of detecting the presence of hyperons. The dataset used has been constructed from an RMF approach within a Bayesian framework, incorporating crucial constraints derived from both nuclear matter properties and NS observations. The dataset consists of 17810 EOSs for pure nucleonic matter and 18728 EOSs including hyperons. This approach was chosen to achieve our primary objective: to investigate the presence of hyperons within NSs based on observational data, which requires the use of microscopic models. From this comprehensive dataset, we generated three distinct sets by selecting either 5, 10, or 15 pairs of mass-radius ( R ), mass-tidal deformability ( Λ ), or both ( R Λ ) as input parameters. We varied the input noise to create three different datasets for each input size, as summarized in Table II. These sets represent different levels of noise scatter in the input data. Consequently, a total of 27 NN models were trained, each corresponding to one of these possible combinations. We conducted a grid search to optimize the model architecture specifically for the R 0 , Λ 0 , and R Λ 0 sets. Once the optimal architecture was identified, we evaluated the model's performance using confusion matrices and key metrics such as accuracy and F1 score. To ensure that the model's performance is not biased by a fixed threshold, we also calculated the AUC for accuracy as a function of the threshold. The AUC values demonstrated consistent behavior with the accuracy measured at the standard threshold of 0 . 5 . Our findings reveal that as input noise increases, the model requires a larger input size to maintain high accuracy. In contrast, for datasets without noise, the model quickly \nreaches peak accuracy with a smaller number of input pairs. This suggests that for observational data-roughly equivalent to the sets with X 2 noise levels-larger input sizes are needed to achieve accuracy comparable to the noise-free case. Nevertheless, our model achieves high scores across all evaluated metrics, including in scenarios with the highest input noise. We further tested the model on a different dataset, also generated using an RMF approach but with density-dependent couplings and a different particle composition. The results demonstrated good accuracy across all models, consistently exceeding 73%. For the AUC value and the F1 score, the performance decreased slightly. The F1 score is particularly significant given the highly imbalanced nature of this test set, and has indicated that the model is effectively handling and correctly predicting outcomes despite the data imbalance. While the model performs well across most datasets, there is room for improvement, particularly in handling the Λ datasets with higher noise levels. Finally, we tested our models on real observational data for the R models with Q = 5 , using a subset of observations that lie within the model's training region. The results revealed that the majority of predictions fell within the region without hyperons, suggesting that hyperons may be absent inside NSs. More specifically, 65% of the data across all datasets indicated predictions in the region without hyperons. We also conducted tests with fewer samples for the R 2 set at Q = 10 and Q = 15 , achieving solid results that consistently indicate the absence of hyperons. \nLooking ahead, future observations will enable us to test models that incorporate tidal deformability, potentially providing additional insights into NS compositions. A significant direction for future work involves the application of deep ensembles or the incorporation of BNNs. These approaches would allow us to quantify both epistemic and aleatoric uncertainties, providing a more comprehensive understanding of model reliability. Given the results obtained with different nuclear models it could also be advantageous to perform a more rigorous model optimization, such as using Bayesian search methods, to enhance performance and accuracy. However, this approach can be resource-intensive in terms of both time and memory. Additionally, expanding the model to consider the presence of a quark phase in NS compositions and training and testing it under these new conditions could offer intriguing insights into the physics of dense matter.", 'ACKNOWLEDGMENTS': 'This work was partially supported by national funds from FCT (Fundação para a Ciência e a Tecnologia, I.P, Portugal) under the projects 2022.06460.PTDC with the DOI identifier 10.54499/2022.06460.PTDC, and UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers 10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively. \n- [1] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature 467 , 1081 (2010).\n- [2] E. Fonseca et al. , Astrophys. J. 832 , 167 (2016), arXiv:1603.00545 [astro-ph.HE] .\n- [3] Z. Arzoumanian et al. (NANOGrav), Astrophys. J. Suppl. 235 , 37 (2018), arXiv:1801.01837 [astro-ph.HE] .\n- [4] J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, M. H. van Kerkwijk, M. Kramer, C. Bassa, V. S. 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2024PhRvD.110j4068C

We construct the firstorder perturbative TaubNewmanUntiTamburino TaubNUT black hole solutions in Einstein gravity extended with a cubic curvature invariant. The corrected thermodynamic quantities are then obtained by the standard method and the first law and Smarr relation are satisfied. We also study the perturbative correction to thermodynamics using the ReallSantos RS method and verify that the method is still applicable even though the metrics are no longer asymptotic to Minkowski spacetime. We then apply the RS method to obtain the leading correction to the thermodynamics of the complicated KerrTaubNUT black holes.

2024-11-01T00:00:00Z

['10.1103/PhysRevD.110.104068', '10.48550/arXiv.2409.07692', 'arXiv:2409.07692', '2024PhRvD.110j4068C', '2024arXiv240907692C']

['General relativity', 'alternative theories of gravity', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

TaubNUT black hole in higher derivative gravity and its thermodynamics

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

https://arxiv.org/pdf/2409.07692.pdf

{'Taub-NUT Black Hole in Higher Derivative Gravity and Its Thermodynamics': 'Yu-Qi Chen 1 , Hai-Shan Liu 1 and H. Lu 1 , 2 \n1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China \n2 \nJoint School of National University of Singapore and Tianjin University, International Campus of Tianjin University, Binhai New City, Fuzhou 350207, China', 'ABSTRACT': 'We construct the first-order perturbative Taub-NUT black hole solutions in Einstein gravity extended with a cubic curvature invariant. The corrected thermodynamic quantities are then obtained by the standard method and the first law and Smarr relation are satisfied. We also study the perturbative correction to thermodynamics using the Reall-Santos (RS) method and verify that the method is still applicable even though the metrics are no longer asymptotic to Minkowski spacetime. We then apply the RS method to obtain the leading correction to the thermodynamics of the complicated Kerr-Taub-NUT black holes.', '1 Introduction': "The Taub-NUT metric [1,2] is one of the most notable exact solutions in General Relativity. It is a simple generalization of the Schwarzschild black hole: in addition to the mass parameter m , there is a NUT parameter n that measures the strength of the time bundle over the foliating round S 2 , with the SU (2) isometry group preserved. The introduction of NUT parameter n has the effect that the metric is no longer asymptotic to Minkowski spacetime, since the time bundle does not degenerate at the asymptotic region. The metric is still asymptotic to local Minkowski spacetime in that the curvature tensors fall off fast enough in the asymptotic region. The NUT parameter brings an advantage that the metric becomes absent from a local curvature singularity as the radial coordinate runs from positive infinity to negative infinity. The price is that the time bundle creates a Misner string singularity [3]. This singularity can be avoided by imposing a period condition on the time coordinate. However this is not necessary and geodesic can still be completed even in the presence of the Misner string [4, 5]. This is because the Misner string, as in the case of the Dirac string, is a global singularity and cannot be observed by local geodesic motions. \nNevertheless, owing to the poor understanding of the Misner string, Taub-NUT metric was traditionally studied in the Euclidean signature, as part of the gravitational instanton that was important to understanding quantum gravity [6]. However, recent works showed that the collapsed objects GRO J1655-40 and M87 /star might contain NUT charges [8-10]. As suggested in [11, 12], primordial black holes, as a potential dark matter candidate, could also contain NUT. These results indicate that we should treat the Taub-NUT metric in Lorentz signature more seriously. In Lorentz signature, the Taub-NUT has an event horizon regardless the value of the parameter m , be it positive, negative or zero, with the topology of R 2 /multicloseleft S 2 . \nTreating Taub-NUT as a black hole faces a challenge to fit the well established black hole thermodynamics. Requiring that the number of independent thermodynamic variables be the same as the number of parameters, one expects that the first law takes the form \nδM = TδS +Φ n δQ n . (1) \nHowever, in this differential equation equation, only ( T, S ) are well established with no ambiguity. The mass M , NUT charge Q n and its thermodynamic conjugate Φ n are not obvious to determine. Since the metric is not asymptotic to Minkowski spacetime, the AdM formalism of determining the mass no longer applies. A seemingly natural proposal of treating m as the mass would allow the black hole to have arbitrarily negative mass. There have been several attempts of defining NUT charge and mass in order to get the appropriate differential form of the thermodynamic first law recently [13-30]. Some specific choices of thermodynamics have been \nused in the holographic study successfully [31-34]. And recent results argue that the ill-defined thermodynamics can break weak cosmic supervision conjecture [35], although a more careful examination on the subject is required [36]. In this paper, we adopt the thermodynamics developed in [29] mainly for the following three reasons \n- · The mass, or the energy, of the black hole is always positive.\n- · The Euclidean action gives rise to the Gibbs free energy, G = M -TS -Φ n Q n , analogous to the Kerr and Reissner-Nordstrom black holes.\n- · There exists a smooth n → 0 limit. \nArguably, the last criterium may be the weakest requirement. \nIn the framework of quantum field theory, Einstein theory coupled to conventional matter fields is UV divergent [37] and non-renormalizable. A consistent quantum theory of gravity is still beyond reach. One way to study the higher energy physics at low energy scale is to consider the effective field theory approach. In the premise of preserving the covariance of the gravitational theory, the only choice of the low energy effective action of quantum gravity is adding all possible diffeomorphism invariant curvature terms [38] which is given by \nL = R -2Λ 0 + 1 M 2 p L 4 + 1 M 4 p L 6 + .... (2) \nwhere M p is Planck mass, which is related to the UV behavior and L i denotes the combination of all the i 'th order derivative terms. If we treat the higher-order terms as part of classical action, they will generally introduce additional massive modes including ghost modes. These modes can give rise to new black holes with additional hair [39]. Indeed, some new Taub-NUT solutions were constructed numerically in a few recent papers [40-46]. In the perturbative effective theory approach, on the other hand, there will be no new modes and hence no new hair, but the original Ricci-flat Taub-NUT solution can be modified by the perturbation. For the Taub-NUT black hole that has S 2 symmetry even with the time bundle, the perturbation can be explicitly solved. However, for the complicated Kerr Taub-NUT, an exact solution, even at the perturbative level, is unlikely to be found. This provides a serious challenge to discuss the higher-derivative correction to the black hole thermodynamics. Recently, Reall and Santos developed a technique for evaluating the leading-order correction to black hole thermodynamics without perturbative solutions [47] based on the Euclidean action. The RS method has been very successful in the case of asymptotically flat and AdS black holes. (For the latter more complicated case, see, e.g. [47-49].) However, Taub-NUT or Kerr Taub-NUT metrics are neither asymptotically flat nor AdS. In this paper, we would like to test the RS \nmethod using the simpler Taub-NUT metric and then apply the method directly on the KerrTaub-NUT. \nThe paper is organized as follows: In section 2, we review Taub-NUT black hole and its thermodynamics obtained in [29]. In Section 3, we apply the RS method to obtain the corrected thermodynamics of Taub-NUT black hole. In Section 4, we construct the firstorder perturbative solution of Taub-NUT and adopt the standard Euclidean action method to evaluate the corrected thermodynamics. Then we compare them with the results obtained by the RS method and prove that the RS method is applicable in Taub-NUT. In Section 5, we use the RS method to obtain the leading-order correction of the Kerr Taub-NUT black hole thermodynamics. Finally, we conclude the paper in Section 6. In the appendix, we consider perturbations to other proposals of the Taub-NUT thermodynamics.", '2 Thermodynamics of Taub-NUT black hole': "In this section, we briefly review the thermodynamics of the Taub-NUT black hole in Einstein gravity. As mentioned earlier, there are many proposals for the Taub-NUT black hole thermodynamics. In this paper, we adopt the proposal of [29] for the reasons explained in the introduction. \nWe begin with the Ricci-flat Taub-NUT black hole metric \nds 2 = -h ( dt +2 n cos θdφ ) 2 + dr 2 f +( r 2 + n 2 )( dθ 2 +sin 2 θdφ 2 ) , (3) \nwhere \nf ( r ) = h ( r ) = f 0 ( r ) ≡ r 2 -2 mr -n 2 r 2 + n 2 . (4) \nThe metric has two parameters, m and n . When n = 0, the metric reduces to the Schwarzschild black hole of mass m , which has an event horizon for positive mass m . When n is turned on, the event horizon continue to exist and its location r 0 is given by the largest positive root of \nf . The corresponding entropy and temperature can be assigned as \nT = √ f ' ( r 0 ) h ' ( r 0 ) 4 π = 1 4 πr 0 , S = Area 4 = π ( r 2 0 + n 2 ) . (5) \nHowever, a new feature emerges when n is nonzero: an event horizon exists regardless the value and the sign of m , be it positive, negative or zero. Therefore, as was pointed out in [29], it is no longer sensible to treat m , a quantity unbounded below, as the mass. Instead, based on the generalized Komar integration, the mass is shown to be \nM = m + n 2 r 0 = √ m 2 + n 2 , (6) \nwhich is always positive regardless the value of m . The NUT charge Q n and its thermodynamic conjugate Φ n are given by \nQ n = n r 0 , Φ n = 1 2 n. (7) \nIn particular, the NUT potential is determined from the degenerate Killing vector at the north and south pole, i.e. \n/lscript ± = ∂ φ ∓ 2 n∂ t = ∂ φ ∓ 4Φ n ∂ t . (8) \nThis definition of Φ n becomes more apparent in the Kerr Taub-NUT black hole, as was illustrated in [29]. The thermodynamic quantities satisfy the first law (1). Furthermore, the Euclidean action I , expressed in terms of the above set of thermodynamic quantities, give precisely the Gibbs free energy, namely \nF G = I β = M -TS -Φ n Q n , (9) \nwhich is inline with other well established black holes, e.g. Kerr and RN black holes, whose Euclidean actions all give rise to the Gibbs free energy.", '3 Thermodynamic corrections from the RS method': "In this section, we adopt the RS method [47] to obtain the leading-order higher-derivative correction to the Taub-NUT black hole. The simplest higher derivative extension to Einstein gravity is to consider quadratic invariants of Riemann tensor, including R 2 , R µν R µν and R µνρσ R µνρσ terms. However, in four dimensions, the Gauss-Bonnet combination is a total derivative, and furthermore, there can be the field redefinition g µν → g µν + c 1 R µν + c 2 Rg µν that does not alter the physics perturbatively. Therefore, the nontrivial perturbative extension requires at least the cubic invariants. Using the appropriate field redefinition [50,51], we have \nL 6 = √ -g ( α 1 Riem 3 + α 2 ˜ Riem 3 ) , Riem 3 = R µν ρσ R ρσ αβ R αβ µν , ˜ Riem 3 = R µ ν ρ σ R ν α σ β R α µ β ρ . (10) \n(Here, we do not consider the parity odd combinations that involve one four-index epsilon tensor.) In fact, since the Taub-NUT metric is Ricci flat, the terms involving two Ricci tensor and scalars will naturally drop out also. Note that the third-order Lovelock combination vanishes in four dimensions, it follows that for Ricci-flat metrics, we must have \nRiem 3 = 2 ˜ Riem 3 . (11) \nWe therefore only need to consider the total Lagrangian \nL tot = √ -g ( R + α Riem 3 ) , (12) \nwhere α is the coupling constant of the cubic Riemann tensor invariant. The on-shell Euclidean action is thus given by \nI E = 1 16 π ∫ bulk L tot + I surf . (13) \nThe surface term associated with the leading-order Einstein-Hilbert action is the well-known Gibbon-Hawking-York term. The term associated with the cubic invariant can be complicated, but as we shall see presently, it gives no contributions owing to its fast falloff in the asymptotic region. \nAccording to the RS method, the total Euclidean action, up to and including the linear α order can be evaluated on the background Ricci-flat metric g 0 , namely \nI E = I 0 ( g 0 ) + I 1 ( g 0 ) + O ( α 2 ) , (14) \nwhere I 0 ( g 0 ) is Euclidean action of the Einstein theory and I 1 ( g 0 ) is given by \nI 1 ( g 0 ) = α 16 π ∫ bulk Riem 3 ∣ ∣ ∣ g → g 0 . (15) \nIt is worth noting that for the Taub-NUT metric, we have \nRiem 3 = 96 m 3 -3 mn 2 r 9 + O ( r -10 ) , (16) \nwhich is highly convergent asymptotically. \nAs we have pointed out in the previous section, in the thermodynamic interpretation of [29] we adopted, the Euclidean action is associated with the Gibbs free energy, where the two independent thermodynamic variables are T and Φ n . Therefore, in the RS method (14), these two variables remain their unperturbed values ( T 0 , Φ n 0 ), namely \nT → T 0 = 1 4 π ˜ r 0 + O ( α 2 ) , Φ n → Φ n 0 = ˜ n 2 + O ( α 2 ) . (17) \nFor the later purpose of comparing the results of finding explicit perturbative solutions, we find it useful add tildes on parameters r 0 and n , which can be a priori perturbed by the cubic Riemann invariant. We find \nI 0 ( g 0 ) = ˜ r 2 0 -˜ n 2 4˜ r 0 T 0 , I 1 ( g 0 ) = 5˜ n 2 -7˜ r 2 0 14˜ r 3 0 (˜ r 2 0 + ˜ n 2 ) T 0 . (18) \nThen, the corrected Gibbs free energy is \nG RS ( T 0 , Φ n 0 , α ) = T 0 I E ( T 0 , Φ n 0 , α ) = ˜ r 2 0 -˜ n 2 4˜ r 0 + α (5˜ n 2 -7˜ r 2 0 ) 14˜ r 3 0 (˜ r 2 0 + ˜ n 2 ) . (19) \nNote that we have omitted the ' O ( α 2 )' expression for simplicity. Entropy, NUT charge can be computed by the partial derivative of free energy and mass can be obtained by using a Legendre transformation \nS RS = -( ∂G RS ∂T 0 ) | Φ n , Q n RS = -( ∂G RS ∂ Φ n 0 ) | T , M RS = G RS + T 0 S RS +Φ n 0 Q n RS . (20) \nSpecifically, we have \nQ \nS RS = π (˜ r 2 0 + ˜ n 2 ) -α 6 π (5˜ n +6˜ n ˜ r 0 -7˜ r 0 ) 7˜ r 2 0 (˜ r 2 0 + ˜ n 2 ) 2 , n RS = ˜ n ˜ r 0 -α 24˜ n 7˜ r 0 (˜ r 2 0 + ˜ n 2 ) 2 , M RS = ˜ r 2 0 + ˜ n 2 2˜ r 0 -α 5˜ n 4 +22˜ n 2 ˜ r 2 0 -7˜ r 4 0 7˜ r 3 0 (˜ r 2 0 + ˜ n 2 ) 2 . (21)", '4 Higher derivative correction to Taub-NUT metric': 'As mentioned in Introduction, Taub-NUT spacetime is not asymptotically Minkowskian but only locally flat. It is thus instructive to verify the correction of the cubic Riemann invariant from the more traditional approach. In this section we shall solve the equations of motion and construct the first-order perturbative solution to the Ricci-flat Taub-NUT and use the traditional method to compute the thermodynamic quantities.', '4.1 Perturbative solution': "The equation of motion (EOM) that follows from the theory (12) can be conveniently written as \nP acde R cde b -1 2 g ab L -2 ∇ c ∇ d P acdb = 0 , (22) \nwhere \nP abcd = ∂L ∂R abcd = 1 2 ( g ac g bd -g ad g bc ) + 3 2 α ( R e f a c R bedf -R e f a d R becf ) . (23) \nWe now move on to consider the ansatz (3) of Taub-NUT black hole. We wish to construct perturbative solutions to the first order in α . We therefore consider metric functions in small α expansion that gives \nf ( r ) = f 0 ( r ) + αf 1 ( r ) + O ( α 2 ) , h ( r ) = f 0 ( r ) + αh 1 ( r ) + O ( α 2 ) . (24) \nHere f 0 was given in (4). We substitute the expressions into the EOM and extract the linear differential equations of f 1 and h 1 at the linear α order. In order to simplify these equations, we first utilize the rr component of EOM that yields \nf 1 =( -( r 2 + n 2 ) 7 ( n 4 + n 2 (4 m -5 r ) r -2 mr 3 ) h 1 +( n 2 +2 mr -r 2 )( -24( -6 mn 2 r (9 n 8 -60 n 6 r 2 -42 n 4 r 4 +132 n 2 r 6 -23 r 8 ) + 2 m 3 (27 n 8 r -84 n 6 r 3 -126 n 4 r 5 +108 n 2 r 7 -5 r 9 ) + n 2 ( n 10 +27 n 8 r 2 -126 n 6 r 4 -210 n 4 r 6 +189 n 2 r 8 -9 r 10 ) +3 m 2 (5 n 10 -105 n 8 r 2 +42 n 6 r 4 +294 n 4 r 6 -111 n 2 r 8 +3 r 10 )) + r ( r 2 + n 2 ) 8 h ' 1 )) / ( r ( r 2 + n 2 ) 7 ( -2 mn 2 +3 n 2 r + r 3 )) . (25) \n4 2 2 4 \nThe tt component of EOM then gives a second order differential equation of h 1 : \nh '' 1 +(48(8 m 4 n 2 r 3 (213 n 8 -924 n 6 r 2 +846 n 4 r 4 -188 n 2 r 6 +5 r 8 ) -48 m 2 n 2 r 3 (106 n 10 -841 n 8 r 2 +1176 n 6 r 4 -190 n 4 r 6 -122 n 2 r 8 +15 r 10 ) + 8 n 4 r 3 (23 n 10 -674 n 8 r 2 +1602 n 6 r 4 -168 n 4 r 6 -273 n 2 r 8 +18 r 10 ) + m 3 (15 n 14 +831 n 12 r 2 -15129 n 10 r 4 +36399 n 8 r 6 -18795 n 6 r 8 -459 n 4 r 10 +1013 n 2 r 12 -35 r 14 ) + mn 2 ( n 14 -175 n 12 r 2 +9225 n 10 r 4 -41151 n 8 r 6 +27867 n 6 r 8 +6075 n 4 r 10 -2661 n 2 r 12 +51 r 14 )) + n 2 ( r 2 + n 2 ) 8 (4 r 3 + m ( n 2 -3 r 2 )) h 1 + r ( n 2 + r 2 ) 8 (5 n 2 r 3 + r 5 -m ( n 4 +5 n 2 r 2 )) h ' 1 ) / ( r 2 ( n 2 + r 2 ) 9 ( -2 mn 2 +3 n 2 r + r 3 )) = 0 . (26) \nThe equation can be solved analytically. We thus have \nf 1 = ∆ f 1 + f bdy , h 1 = ∆ h 1 + h bdy , (27) \nwhere \n∆ f 1 = 8 7 r 2 ( r 2 + n 2 ) 7 [ -3 mn 2 r (27 n 8 +2116 n 6 r 2 -8598 n 4 r 4 +6900 n 2 r 6 -973 r 8 ) + m 3 (81 n 8 r +1036 n 6 r 3 -10290 n 4 r 5 +6972 n 2 r 7 -343 r 9 ) + 3 m 2 (9 n 10 +111 n 8 r 2 -5298 n 6 r 4 +10226 n 4 r 6 -2871 n 2 r 8 +63 r 10 ) -n 2 (27 n 10 +669 n 8 r 2 -5478 n 6 r 4 +8390 n 4 r 6 -3861 n 2 r 8 +189 r 10 ) ] . (28) ∆ h 1 = 8 7 r ( r 2 + n 2 ) 7 [ 4 n 4 r (15 n 6 -237 n 4 r 2 +265 n 2 r 4 -27 r 6 ) -12 m 2 (33 n 8 r -219 n 6 r 3 +247 n 4 r 5 -45 n 2 r 7 ) -3 mn 2 (9 n 8 -404 n 6 r 2 +1230 n 4 r 4 -516 n 2 r 6 +17 r 8 ) + m 3 (27 n 8 -476 n 6 r 2 +1050 n 4 r 4 -588 n 2 r 6 +35 r 8 ) ] . (29) \nThe functions f bdy and h bdy are terms associated with the integration constants ( c 1 , c 2 ): \nf bdy = c 1 2 n 4 +( -3 + 4 m ) r 3 +3 n 2 r (2 m +( -2 + r ) r ) ( -1 + ( -3 + 2 m ) n 2 ) r 2 ( n 2 + r 2 ) + c 2 -n 4 +(1 -2 m ) r 3 + n 2 r ( -3( -1 + r ) r + m ( -3 + r 2 )) ( 1 + ( 3 + 2 m ) n 2 ) r 2 ( n 2 + r 2 ) , \nh bdy = -c 1 2 mn + r (3 n ( -2 + r ) + r ( -3 + 2 r )) ( r +(3 -2 m ) n 2 r )( n 2 + r 2 ) + c 2 ( -1 + r )( -mn 2 (1 + r ) + r (3 n 2 + r )) 2 2 2 . \n--(30) 2 2 ( r +(3 -2 m ) n r )( n + r ) (31) \nThe integration constants can be fixed in the following consideration. In the large r region, f 1 and h 1 should vanish. We find \nlim r →∞ f 1 = 0 , lim r →∞ h 1 = 2 c 1 -c 2 -1( -3 + 2 m ) n 2 , → c 2 = 2 c 1 . (32) \nThen, we have \nf 1 = c 1 r + O ( 1 r 2 ) . (33) \n/negationslash \nThe perturbative metric function f 1 should have a [ L ] -4 dimension. It indicates that if c 1 = 0, it would have the following form \nc 1 = 1 ∑ 3 i =0 β i m i n 3 -i . (34) \nwhere β i are some purely numerical constants. Obviously, the metric is divergent when m = 0 and n = 0. Therefore, c 1 must vanish, and our perturbative solution should be f 1 = ∆ f 1 and h 1 = ∆ h 1 .", '4.2 Thermodynamics': "In our perturbative approach, the black hole horizon is no longer located at r 0 but corrected by higher derivative terms. It can be determined by f ( r h ) = 0 = h ( r h ). The horizon is thus shifted by (at the linear α order) \nr h = r 0 + α 27 n 2 -35 r 2 0 7 n 2 r 3 0 +7 r 5 0 . (35) \nThe Hawking temperature can be evaluated by the standard method \nT = √ f ' ( r h ) h ' ( r h ) 4 π = 1 4 πr 0 + α 7 r 2 0 -5 n 2 14 πr 3 0 ( r 2 0 + n 2 ) 2 . (36) \nIn higher derivative gravity, the entropy is not simply given by one quarter of the area of the event horizon. Instead, we need to use the formula obtained by Wald's prescription [52, 53], namely \nS = 1 8 π ∫ d 2 x √ h/epsilon1 ab /epsilon1 cd P abcd , (37) \nwhere h is determinant of the induced metric on the horizon, /epsilon1 ab is the binormal to the horizon with /epsilon1 ab /epsilon1 ab = -2 and P abcd is defined in (23). Specifically, we find the Wald entropy of our linearly perturbed solution is given by \nS = π ( r 2 0 + n 2 ) + α 2 π (7 r 2 0 -15 n 2 ) 7 r 2 0 ( r 2 0 + n 2 ) . (38) \nThe degenerate Killing vectors (8) at the north and south pole remain intact under the perturbation; therefore, the NUT potential remains the same \nΦ n = n 2 = Φ n 0 . (39) \nThe remainder of the thermodynamic variables can be obtained from the free energy associated with the Euclidean action, namely \nG = I β , Q n = -( ∂G ∂ Φ n ) ∣ ∣ ∣ T , M = F + TS +Φ n Q n . (40) \nHere we adopt the brutal-force method and simply substitute the perturbed solution into the Euclidean action, evaluate it up to and including the α order. Note that the surface term associated with the cubic invariant term can be ignored, owing to its fast falloff on the boundary. We find \nG = m 2 + α 5 n 2 -7 r 2 0 7 r 3 0 ( r 2 0 + n 2 ) , (41) \nM = r 2 0 + n 2 2 r 0 -α 10 n 2 7 r 3 0 ( r 2 0 + n 2 ) , Q n = n r 0 -α 10 n 7 r 3 0 ( r 2 0 + n 2 ) . (42) \nIt is straightforward to verify that they do obey the first law (1) up to and including the α order. It is worth noting that there exists a relation between ∆ Q n and ∆ M : \n∆ M = n ∆ Q n = 2Φ n ∆ Q n . (43) \nIt indicates that the mass can be conveniently written as an α -independent expression \nM = m +2Φ n Q n , (44) \neven under the α -order correction.", '4.3 Verification of The RS Method in Taub-NUT': 'We have obtained the α -order correction to the Taub-NUT black hole thermodynamics using two different methods. We now show that they are equivalent. We note that in the RS approach, the thermodynamic quantities are expressed in terms of (˜ r 0 , ˜ n ), such that the thermodynamic variables ( T, Φ) are fixed to T 0 and Φ 0 . We therefore need to perform redefinition on ( r 0 , n ) in the second approach \nr 0 → ˜ r 0 + δ ˜ r 0 , δ ˜ r 0 = -α 2(5˜ n 2 -7˜ r 2 0 ) 7˜ r 0 (˜ r 2 0 + ˜ n 2 ) 2 , n → ˜ n, (45) \nso that the temperature and NUT potential are both unperturbed, namely \nT → T 0 = 1 4 π ˜ r 0 + O ( α 2 ) , Φ n → Φ n 0 + O ( α 2 ) . (46) \nIt is straightforward to verify that the second approach would reproduce the RS thermodynamic quantities precisely. We therefore establish the RS method is indeed applicable in Taub-NUT geometries. \nIt is also of interest to consider thermodynamic system where the entropy is the potential, in which case, the mass and NUT charge are the thermodynamic variables that are unperturbed under the α correction. Starting from the second approach, we need perform a further field redefinition \nr 0 = ˆ r 0 + αδr 0 , n → ˆ n + αδn ; δr 0 = 0 , δn = 10ˆ n 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) , (47) \nand \nunder which the mass and NUT charge are both fixed \nM = ˆ r 2 0 + ˆ n 2 2ˆ r 0 + O ( α 2 ) , Q ˆ n = n ˆ r 0 + O ( α 2 ) . (48) \nThe linear α correction of the remaining thermodynamic quantities are \nT = 1 4 π ˆ r 0 + α (7ˆ r 2 0 -5ˆ n 2 ) 14ˆ r 3 0 (ˆ r 2 0 + ˆ n 2 ) 2 , S = π (ˆ r 2 0 + ˆ n 2 ) + 2 πα (7ˆ r 2 0 -5ˆ n 2 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) , Φ n = ˆ n 2 + 5 α ˆ n 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) . (49) \nWe thus reestablish the identity [47] \n∆ S ∣ ∣ ∣ M,Q n = 2 πα (7ˆ r 2 0 -5ˆ n 2 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) = -∆ I ∣ ∣ ∣ T, Φ n . (50) \nIf we instead start from the RS method, we can get the same result by performing the field redefinition \n˜ r 0 = ˆ r 0 + αδ ˜ r 0 , ˜ n → ˆ n + αδ ˜ n ; δ ˜ r 0 = 2(5ˆ n 2 -7ˆ r 2 0 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) 2 , δ ˜ n = 10ˆ n 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) . (51)', '4.4 Comparing the two approaches': 'In both approaches described in this section, we obtain the correction to the Gibbs free energy, from which we read off the thermodynamic quantities. However, the Gibbs free energy is expressed in terms of the parameters associated with the horizon radius and the NUT parameter n . By itself, we cannot read off the thermodynamic quantities directly. In the RS approach, the thermodynamic variables are assumed to be fixed, without corrections. We can then read off the remaining thermodynamic quantities associated with the differential relation from dG = -SdT -Q n d Φ n . In the approach of constructing explicit perturbative solution, the issue becomes more subtle. Since the explicit solution is given, we can determine the temperature and entropy of the perturbed solution by the standard methods, which leads to Q n d Φ n = -SdT -dG . However, we cannot determine (Φ n , Q n ) simply from Q n d Φ n . We need an independent formula to determine either Φ n or Q n . The key of the thermodynamic approach of [29] is that the NUT potential is determined by (8). Since the degenerate Killing vectors remain the same for our perturbative solution, it follows that Φ n is unperturbed in our second approach. We can then read off the perturbed NUT charge Q n from the perturbed Gibbs free energy. \nThus we see that in the both RS and standard method, the Gibbs free energy can be determined, but one needs to know independently the ( T, Φ n ) in order to read off the complete set of thermodynamic variables. In the RS case, by procedure, we have ( T, Φ n ) = ( T 0 , Φ n 0 ). For the standard method of constructing perturbative solution, T is no longer T 0 , but it can be \nobtained by the standard procedure, and Φ n happens to be Φ n 0 . Thus it is a highly nontrivial exercise that the RS and perturbative solution methods match precisely and it provides an important validation to the thermodynamic proposal of [29]. \nIn the appendix, we consider some other thermodynamic proposals. One can always apply the RS formalism, but its verification using the perturbative solution becomes less convincing, since there is no clear independent calculation of the thermodynamic variables from the perturbative solution.', '5 First-order correction to Kerr-Taub-NUT thermodynamics': 'Having established that the RS method is applicable to the Taub-NUT black hole, we now apply it to the Kerr Taub-NUT black hole, for which the analytic perturbative solution is unlikely to be found. The Ricci-flat rotating Taub-NUT metric is given by \nds 2 = ( r 2 + v 2 )( dr 2 ∆ + du 2 1 -u 2 ) + 1 r 2 + v 2 ((1 -u 2 ) e 2 1 -∆ e 2 2 ) , e 1 = adt -( r 2 + a 2 + n 2 ) dφ, e 2 = dt +(2 nu -a (1 -u 2 )) dφ, ∆ = r 2 -2 mr + a 2 -n 2 , v = n + au, u = cos θ. (52) \nThe metric consists of three integration constants ( m,a,n ). For n = 0, it reduces to the standard Kerr metric. The metric contains a Cauchy horizon r -and an event horizon r + ≥ r -, which are defined by ∆( r ± ) = 0. The thermodynamic quantities based on [29] are: \nT = r 2 + + n 2 -a 2 4 πr + ( r 2 + + n 2 + a 2 ) , Ω = a r 2 + + n 2 + a 2 , Φ n = n 2 , S = π ( r 2 + + n 2 + a 2 ) , Q n = n r + , M = m + n 2 r + , J = Ma. (53) \nAgain, the free energy associated with the Euclidean action is of the Gibbs type: \nG ( T, Φ n , Ω) = M -TS -Φ n Q n -Ω J . (54) \nWe now apply the RS method. We choose the redefinition of integration constant r + → ˜ r 0 , a → ˜ a , n → ˜ n . The NUT parameter is also not redefined as we discussed in the last section. We have \nI 0 ( g 0 ) = ˜ r 2 0 -˜ n 2 +˜ a 2 4˜ r 0 β , I 1 ( g 0 ) = β 14 ˜ r 0 3 (˜ a 2 -2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 3 (˜ a 2 +2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 3 [ ˜ a 12 +6˜ a 10 (˜ n 2 -3˜ r 2 0 ) +(5˜ n 2 -7˜ r 2 0 )(˜ n 2 + ˜ r 2 0 ) 5 -2˜ a 2 (˜ n 2 + ˜ r 2 0 ) 3 (5˜ n 4 +78˜ n 2 ˜ r 2 0 -7˜ r 4 0 ) -˜ a 8 (29˜ n 4 +198˜ n 2 ˜ r 2 0 +25˜ r 4 0 ) + 12˜ a 6 (3˜ n 6 +25˜ n 4 ˜ r 2 0 +25˜ n 2 ˜ r 4 0 +3˜ r 6 0 ) \n+˜ a 4 ( -9˜ n 8 +84˜ n 6 ˜ r 2 0 +450˜ n 4 ˜ r 0 4 +420˜ n 2 ˜ r 0 6 +63˜ r 0 8 ) ] . (55) \nThe Gibbs-type free energy is then simply G = ( I 0 ( g 0 )+ I 1 ( g 0 )) T 0 . we can obtain the perturbed entropy, angular momentum, NUT charge and mass: \nS = -( ∂F ∂T 0 ) | Φ n , Ω , J = -( ∂F ∂ Ω 0 ) | Φ n ,T , Q n = -( ∂F ∂ Φ n 0 ) | T, Ω , M = F + T 0 S +Φ n 0 Q n +Ω 0 J. (56) \nSpecially, the corrections are \n∆ S = α (2 π (˜ a 2 + ˜ n 2 + ˜ r 0 2 )(3˜ a 14 +3˜ a 12 (5˜ n 2 -3 ˜ r 0 2 ) -3(˜ n 2 + ˜ r 0 2 ) 5 (5˜ n 4 +6˜ n 2 ˜ r 0 2 -7 ˜ r 0 4 ) -˜ a 10 (105˜ n 4 +150˜ n 2 ˜ r 0 2 +101˜ r 0 4 ) + ˜ a 8 (195˜ n 6 +469˜ n 4 ˜ r 0 2 -919˜ n 2 ˜ r 0 4 -233 ˜ r 0 6 ) +˜ a 2 (˜ n 2 + ˜ r 0 2 ) 3 (45˜ n 6 +179˜ n 4 ˜ r 0 2 +503˜ n 2 ˜ r 0 4 +49˜ r 0 6 ) -3˜ a 6 (45˜ n 8 +108˜ n 6 ˜ r 0 2 +150˜ n 4 ˜ r 0 4 +284˜ n 2 ˜ r 0 6 +69˜ r 0 8 ) + ˜ a 4 ( -3˜ n 10 -207˜ n 8 ˜ r 0 2 +514˜ n 6 ˜ r 0 4 +1218˜ n 4 ˜ r 0 6 +465˜ n 2 ˜ r 0 8 -35 ˜ r 0 10 ))) / (7 ˜ r 0 2 (˜ a 2 -2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 (˜ a 2 +2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 ) , (57) \n∆ J = -α ((2˜ a (˜ a 2 + ˜ n 2 + ˜ r 0 2 )(9˜ a 12 (˜ n 2 -˜ r 0 2 ) + 2˜ a 10 (5˜ n 4 -126˜ n 2 ˜ r 0 2 -11 ˜ r 0 4 ) +(˜ n 2 + ˜ r 0 2 ) 5 (25˜ n 4 -96˜ n 2 ˜ r 0 2 +7˜ r 0 4 ) + 4˜ a 6 (˜ n 2 + ˜ r 0 2 ) 2 (35˜ n 4 +198˜ n 2 ˜ r 0 2 +19˜ r 0 4 ) -˜ a 4 (˜ n 2 + ˜ r 0 2 ) 2 (25˜ n 6 -407˜ n 4 ˜ r 0 2 -809˜ n 2 ˜ r 0 4 -89 ˜ r 0 6 ) -2˜ a 2 (˜ n 2 + ˜ r 0 2 ) 3 (27˜ n 6 +281˜ n 4 ˜ r 0 2 -23˜ n 2 ˜ r 0 4 -21 ˜ r 0 6 ) -˜ a 8 (105˜ n 6 +473˜ n 4 ˜ r 0 2 +71˜ n 2 ˜ r 0 4 -9 ˜ r 0 6 ))) / (7 ˜ r 0 3 (˜ a 2 -2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 (˜ a 2 +2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 )) , (58) \n∆ Q n = -(4 α ˜ n (9˜ a 14 +10˜ a 12 (˜ n 2 -18 ˜ r 0 2 ) + 6 ˜ r 0 2 (˜ n 2 + ˜ r 0 2 ) 6 +5˜ a 2 (˜ n 2 + ˜ r 0 2 ) 4 (5˜ n 4 -2˜ n 2 ˜ r 0 2 -23 ˜ r 0 4 ) -˜ a 10 (105˜ n 4 +766˜ n 2 ˜ r 0 2 +181˜ r 0 4 ) -2˜ a 4 (˜ n 2 + ˜ r 0 2 ) 2 (27˜ n 6 +406˜ n 4 ˜ r 0 2 +411˜ n 2 ˜ r 0 4 -16 ˜ r 0 6 ) + 2˜ a 8 (70˜ n 6 +675˜ n 4 ˜ r 0 2 +900˜ n 2 ˜ r 0 4 +247˜ r 0 6 ) +˜ a 6 ( -25˜ n 8 +420˜ n 6 ˜ r 0 2 +2130˜ n 4 ˜ r 0 4 +2324˜ n 2 ˜ r 0 6 +639˜ r 0 8 ))) / (7 ˜ r 0 3 (˜ a 2 -2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 (˜ a 2 +2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 ) , (59) \n∆ M = -( α (˜ a 16 +8˜ a 14 (5˜ n 2 -2 ˜ r 0 2 ) + 16˜ a 2 ˜ n 2 (˜ n 2 + ˜ r 0 2 ) 4 (5˜ n 4 -12˜ n 2 ˜ r 0 2 -37 ˜ r 0 4 ) +(˜ n 2 + ˜ r 0 2 ) 6 (5˜ n 4 +22˜ n 2 ˜ r 0 2 -7 ˜ r 0 4 ) -4˜ a 12 (217˜ n 2 ˜ r 0 2 +15˜ r 0 4 ) -32˜ a 10 (10˜ n 6 +76˜ n 4 ˜ r 0 2 +25˜ n 2 ˜ r 0 4 + ˜ r 0 6 ) -4˜ a 4 (˜ n 2 + ˜ r 0 2 ) 2 (50˜ n 8 +711˜ n 6 ˜ r 0 2 +723˜ n 4 ˜ r 0 4 -55˜ n 2 ˜ r 0 6 -21 ˜ r 0 8 ) + 2˜ a 8 (225˜ n 8 +2350˜ n 6 ˜ r 0 2 +3600˜ n 4 ˜ r 0 4 +1338˜ n 2 ˜ r 0 6 +55˜ r 0 8 ) +8˜ a 6 ( -7˜ n 10 +210˜ n 8 ˜ r 0 2 +1030˜ n 6 ˜ r 0 4 +1232˜ n 4 ˜ r 0 6 +441˜ n 2 ˜ r 0 8 +22˜ r 0 10 ))) / (7 ˜ r 0 3 (˜ a 2 -2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 )4(˜ a 2 +2˜ a ˜ n + ˜ n 2 + ˜ r 0 2 ) 4 ) . (60) \nSince in Gibbs-type free energy, the temperature is an independent variable. We can thus take (smooth) extremal limit by simply setting T 0 = 0, which implies that ˜ r 2 0 = ˜ a 2 -˜ n 2 . The results \nbecome significantly simpler, given by \nS = 2 π ˜ a 2 -4 πα (2˜ a 2 +3˜ n 2 ) 7(˜ a 2 -˜ n 2 ) 2 , J = ˜ a 3 √ ˜ a 2 -˜ n 2 -α ˜ a 3 (3˜ a 2 +17˜ n 2 ) 7(˜ a 2 -˜ n 2 ) 3 √ ˜ a 2 -˜ n 2 , Q n = ˜ n √ ˜ a 2 -˜ n 2 + α ˜ n (11˜ a 2 +9˜ n 2 ) 7(˜ n 2 -˜ a 2 ) 3 √ ˜ a 2 -˜ n 2 , M = ˜ a 2 √ ˜ a 2 -˜ n 2 + α ( -˜ a 4 -13˜ a 2 ˜ n 2 -6˜ n 4 ) 7(˜ a 2 -˜ n 2 ) 3 √ ˜ a 2 -˜ n 2 . (61) \nFinally, we consider corrections with the charges, namely ( M,J,Q n ), fixed. This can be achieved by field redefinitions \n˜ r 0 → ˆ r 0 + αδ ˜ r 0 , ˜ n → ˆ n + αδ ˜ n, ˜ a → ˆ a + αδ ˜ a . (62) \nwhere \nδ ˜ r 0 = -2( -3ˆ a 18 -(5ˆ n 2 -7ˆ r 2 0 )(ˆ n 2 + ˆ r 2 0 ) 8 +ˆ a 16 ( -49ˆ n 2 +11ˆ r 2 0 ) + 7ˆ a 2 (ˆ n 2 + ˆ r 2 0 ) 6 (15ˆ n 4 -42ˆ n 2 ˆ r 2 0 +7ˆ r 4 0 ) + 8ˆ a 14 (14ˆ n 4 +111ˆ n 2 ˆ r 2 0 +9ˆ r 4 0 ) -16ˆ a 4 (ˆ n 2 + ˆ r 2 0 ) 4 (17ˆ n 6 +149ˆ n 4 ˆ r 2 0 -71ˆ n 2 ˆ r 4 0 -7ˆ r 6 0 ) + 8ˆ a 12 (25ˆ n 6 +195ˆ n 4 ˆ r 2 0 +63ˆ n 2 ˆ r 4 0 +13ˆ r 6 0 ) -2ˆ a 10 (375ˆ n 8 +3728ˆ n 6 ˆ r 2 0 +4966ˆ n 4 ˆ r 4 0 +1208ˆ n 2 ˆ r 6 0 -21ˆ r 8 0 ) + 24ˆ a 6 (ˆ n 2 + ˆ r 2 0 ) 2 (ˆ n 8 +183ˆ n 6 ˆ r 2 0 +351ˆ n 4 ˆ r 4 0 +109ˆ n 2 ˆ r 6 0 +4ˆ r 8 0 ) + 2ˆ a 8 (319ˆ n 10 +1863ˆ n 8 ˆ r 2 0 +1190ˆ n 6 ˆ r 4 0 -770ˆ n 4 ˆ r 6 0 -405ˆ n 2 ˆ r 8 0 +11ˆ r 10 0 )) / (7ˆ r 0 (ˆ a 2 -2ˆ a ˆ n + ˆ n 2 + ˆ r 2 0 ) 4 (ˆ a 2 +2ˆ a ˆ n + ˆ n 2 + ˆ r 2 0 ) 4 ( -ˆ a 4 +(ˆ n 2 + ˆ r 2 0 ) 2 )) , \n(63) \n- \nδ \n˜ n \n=2ˆ n \n( \nˆ \na \n2 \n(ˆ \n- \nn \n15ˆ a \n2 \n18 \n2 \n0 \n+ ˆ \n6 \n0 \n15ˆ \nr \n1018ˆ \n- \nn \n4 \n) \nr \n+5(ˆ n \n6 \n) \n- \n4 \n0 \nˆ \nr \n628ˆ \nn \n2 \n(55ˆ \n4ˆ \n- \n6 \n0 \nˆ \nr \n10 \n0 \n+451ˆ \nr \n)) \n/ \n(7ˆ \na \n12 \nn \n2 \n4 \n+ ˆ r \n- \n2 \n0 \n) \n+ˆ a \n274ˆ \n(115ˆ \n2244ˆ \n- \nn \n2 \nn \n6 \n6 \n0 \nˆ \nr \n8 \n0 \n353ˆ \nr \n2 \n0 \nr \n(ˆ \na \n9 \nn \nˆ \nr \n2 \n0 \n16 \n(29ˆ n \n2 \n+349ˆ r \n4 \n0 \n+279ˆ \n+1145ˆ \n- \n841ˆ \n) + 2ˆ \n2 \n- \na \n2ˆ \na \nˆ \n8 \nn \n+ ˆ \nr \n8 \n0 \n- \nn \n) \n4 \nˆ \nr \n- \n2 \n0 \nr \n) + 4ˆ \na \n+1201ˆ \n4ˆ \n125ˆ \nn \nn \na \n6 \n10 \n2 \n0 \n+ ˆ \nr \n(ˆ \nn \n2 \nn \n2 \n0 \n4 \n2 \n) + 4ˆ \n(ˆ \nn \n4 \n0 \na \n14 \n2 \n0 \n+ ˆ \nr \n(29ˆ \n4 \n) \n6 \n0 \n+363ˆ \n2 \nˆ \nr \n2 \n0 \n+ ˆ \nr \n+687ˆ \n4 \n) \n(ˆ \na \nn \n) \n2 \n0 \nˆ \nr \n+2ˆ \na \nˆ \n(31ˆ \nn \nr \n8 \nn \n4 \n(41ˆ \n+170ˆ \nn \n6 \n) + 2ˆ \na \n+908ˆ \n+5950ˆ \nn \n+ ˆ \nn \nn \n6 \nˆ \nr \n4 \n0 \n2 \n0 \n+ ˆ \nr \n) \nn \nn \n2 \nˆ \nr \n+187ˆ \n10 \n6 \nˆ \nr \nn \n2 \n0 \n4 \n(295ˆ \n2 \n0 \nn \n+77ˆ \n2 \n0 \nˆ \nr \n8 \n- \nr \n4 \n0 \n) \n- \n701ˆ \nn \n+2332ˆ \n+986ˆ \n+8638ˆ \n4 \n( \n- \nˆ \na \nn \nn \n6 \n0 \nˆ \nr \n+(ˆ \nn \nˆ \nr \n4 \n0 \n+3951ˆ \n2 \n0 \n+ ˆ \nr \n) \nn \n2 \n6 \nˆ \nr \n4 \n0 \n2 \n0 \nˆ \nr \nn \n2 \n)) \n, \nδ ˜ a =2( -ˆ a 17 -2ˆ a 15 (11ˆ n 2 + ˆ r 2 0 ) + ˆ a 13 (38ˆ n 4 +364ˆ n 2 ˆ r 2 0 -2ˆ r 4 0 ) + ˆ a (ˆ n 2 + ˆ r 2 0 ) 6 (45ˆ n 4 -214ˆ n 2 ˆ r 2 0 +21ˆ r 4 0 ) -2ˆ a 3 (ˆ n 2 + ˆ r 2 0 ) 4 (69ˆ n 6 +537ˆ n 4 ˆ r 2 0 -253ˆ n 2 ˆ r 4 0 -49ˆ r 6 0 ) + 2ˆ a 11 (65ˆ n 6 +501ˆ n 4 ˆ r 2 0 +55ˆ n 2 ˆ r 4 0 +3ˆ r 6 0 ) -4ˆ a 9 (95ˆ n 8 +928ˆ n 6 ˆ r 2 0 +1172ˆ n 4 ˆ r 4 0 +228ˆ n 2 ˆ r 6 0 -15ˆ r 8 0 ) +2ˆ a 5 (ˆ n 2 + ˆ r 2 0 ) 2 (21ˆ n 8 +1188ˆ n 6 ˆ r 2 0 +2146ˆ n 4 ˆ r 4 0 +876ˆ n 2 ˆ r 6 0 +89ˆ r 8 0 ) + 2ˆ a 7 (143ˆ n 10 +729ˆ n 8 ˆ r 2 0 +350ˆ n 6 ˆ r 4 0 -70ˆ n 4 ˆ r 6 0 +243ˆ n 2 ˆ r 8 0 +77ˆ r 10 0 )) / (7ˆ r 2 0 (ˆ a 2 + ˆ n 2 + ˆ r 2 0 )(ˆ a 2 -2ˆ a ˆ n + ˆ n 2 + ˆ r 2 0 ) 4 (ˆ a 2 +2ˆ a ˆ n + ˆ n 2 + ˆ r 2 0 ) 4 ) . (65) \nUnder these redefinitions, the mass, NUT charge and angular momentum are fixed at the linear α order, \nM = M 0 + O ( α 2 ) , Q n = Q n 0 + O ( α 2 ) , J = J 0 + O ( α 2 ) . (66) \n+ \n8 \n0 \nˆ \nr \n2 \n2 \n4 \n4 \n4 \n2 \n2 \n8 \n2 \n2 \n( \n2 \n(64) \nWe find that the identity, \nremains intact. \n∆ S ∣ ∣ ∣ M,Q n ,J = -∆ I ∣ ∣ ∣ T, Φ n , Ω , (67)', '6 Conclusion': 'In this paper, we considered Einstein gravity in four dimensions extended with a cubic Riemann curvature invariant and studied how this term modifies perturbatively the thermodynamics of the original Ricci-flat Taub-NUT and Kerr-Taub-NUT metrics. \nThe perturbative solution for the Taub-NUT metric can be obtained straightforwardly owing to its SU (2) isometry. However, the exact higher-order solution for the Kerr-TaubNUT metric is unlikely to be found, even at the linear perturbative order. We thus adopted the RS method to compute the leading-order correction to the thermodynamics. The RS method does not require us to construct explicit perturbative solutions and was well established for asymptotically flat and AdS black holes. However, Taub-NUT or Kerr-Taub-NUT are not asymptotically flat nor AdS, it is therefore useful to verify the RS method using the simpler Taub-NUT solution by explicitly constructing the perturbative solution. Furthermore, there are multiple proposals of thermodynamics for Taub-NUT or Kerr-Taub-NUT metrics in literature. We adopted the proposal of [29] for the three reasons given in the introduction. (We also gave some discussions on a few other proposals in the appendix.) One important advantage of this proposal in [29] is that three quantities ( T, S ) and Φ n , together with the Euclidean action, can be independently obtained even in the approach of finding perturbative solution to the higher-derivative curvature correction. We found that the RS method matched precisely the perturbative solution approach. This then provides a convincing validation of the thermodynamic approach [29]. \nWe then applied the RS method directly on Kerr-Taub-NUT and obtained the leadingorder correction of the cubic curvature invariant to the black hole thermodynamics. The general expressions are rather complicated, but the corrections in the extremal limit reduce significantly. In future directions, one may construct the perturbed Kerr-Taub-NUT solution numerically in higher derivative gravity to verify our results and also the proposal of [29]. This also allows to check whether the near-horizon geometry in the extremal limit becomes irrational as in the case of [54]. Furthermore, our results can be generalized to Taub-NUT AdS spacetime. In asymptotically AdS spacetime, the Reall-Santos method becomes more subtle. In [55] and [56], two different improved RS methods are proposed, which have been successfully applied in many cases [56-58]. We can construct the perturbed Taub-NUT AdS solution and \ncompare the results from the two methods.', '7 Acknowledge': 'We are grateful to Peng-Ju Hu and Liang Ma for the useful discussion. This work is supported in part by NSFC (National Natural Science Foundation of China) Grants No. 12075166, No. 11935009, No. 12375052 and Tianjin University Graduate Liberal Arts and Sciences Innovation Award Program (2023) No. B1- 2023-005.', 'A Other proposals of Taub-NUT thermodynamics': 'There have been different proposals for deciphering the thermodynamics of Taub-NUT black holes in literature. In the main text, we adopt the proposal of [29], which perhaps is the only proposal satisfying all the three criteria listed in the Introduction. Nevertheless, we would like explore in this appendix how some other proposals fare under the cubic invariant perturbation. It should be remarked that the Euclidean action and its correction (18) based on the RS method are the same for all the proposals, but they lead to different thermodynamic quantities in different proposals, depending on the choices of the thermodynamic variables. However, using the perturbative method to verify the RS method may not be possible, if the proposal itself does not tell us how to calculate the thermodynamic variables from the perturbative solutions.', 'A.1 The first version proposed in [13]': 'The first version was proposed in [13], where the thermodynamic quantities are \nM 0 = m, Φ n 0 = 1 8 πn , Q n 0 = -4 πn 3 r 0 . (68) \nThe temperature and entropy are the same as before, since there is no controversy in these two quantities. The free energy associated with the Euclidean action is the Gibbs free energy, with ( T 0 , Φ n 0 ) as its variables. This proposal thus satisfies one of the three criteria listed in Introduction, but it suffers from (1) the mass can be arbitrarily negative and (2) there is no smooth n → 0 limit. Similar to the example we discussed in the main text, the RS method can also be established in this case. The ( T 0 , Φ n 0 ) are fixed with the redefined constants ˜ r 0 \nand ˜ n . The modified NUT charge and mass are \nQ n = -4 π ˜ n 3 ˜ r 0 + α 96 π ˜ n 3 7˜ r 0 (˜ r 2 0 + ˜ n 2 ) 2 , M = ˜ m + α 7˜ r 2 0 -5˜ n 2 7˜ r 3 0 (˜ r 2 0 + ˜ n 2 ) . (69) \nWe then focus on the situation with fixed mass and NUT charge, the redefined constants are given by \n˜ r 0 → ˆ r 0 + αδ ˜ r 0 , ˜ n → ˆ n + αδ ˜ n, δ ˜ r 0 = 6(5ˆ n 4 +6ˆ n 2 ˆ r 2 0 -7ˆ r 4 0 ) 7ˆ r 0 (ˆ r 2 0 + ˆ n 2 ) 2 (ˆ n 2 +3ˆ r 2 0 ) , δ ˜ n = 10ˆ n 7ˆ r 2 0 (3ˆ r 2 0 + ˆ n 2 ) . (70) \nUnder the redefinition, we obtain \nM = ˆ r 2 0 -ˆ n 2 2ˆ r 0 + O ( α 2 ) , Q n = -4 π ˆ n 3 ˆ r 0 + O ( α 2 ) , T = 1 4 π ˆ r 0 + 3( -5ˆ n 4 -6ˆ n 2 ˆ r 2 0 +7ˆ r 4 0 ) 14 π ˆ r 3 0 (ˆ n 2 + ˆ r 2 0 ) 2 (ˆ n 2 +3ˆ r 2 0 ) , Φ n = 1 8 π ˆ n -5 α 28 π ˆ n ˆ r 2 0 (ˆ n 2 +3ˆ r 2 0 ) , S = π (ˆ r 2 0 + ˆ n 2 ) + 2 πα (7ˆ r 2 0 -5ˆ n 2 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) . (71) \nWe also verified the identity \n∆ S ∣ ∣ ∣ M,Q n = 2 πα (7ˆ r 2 0 -5ˆ n 2 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) = -∆ I ∣ ∣ ∣ T, Φ n . (72) \nIn the second approach, the perturbative solution and corresponding Gibbs free energy were already presented in section 3. The temperature and entropy can be determined by the standard method. We cannot make decisions for [13] how the Φ n should be corrected; however, if we assume that it is unmodified by the perturbative solution, then the resulting perturbed thermodynamics is equivalent to the RS method.', 'A.2 The Second Version Proposed in [13]': 'In [13], the authors also proposed a second version. The leading thermodynamic quantities are \nM 0 = m, T 0 = 1 4 πr 0 , S 0 = π ( r 2 0 -n 2 ) , Φ n 0 = -n 2 , Q n 0 = n r 0 . (73) \nNote that the entropy in this version is no longer proportional to the area of the horizon. The free energy associated with the Euclidean action is now Helmholtz type, namely \nF 0 ( T 0 , Φ n 0 ) = M 0 -T 0 S 0 . (74) \nTherefore, the RS method requires that the temperature and NUT charge fixed under the perturbation, and the remaining quantities can be read off as \nS RS = -( ∂F RS ∂T 0 ) ∣ ∣ ∣ Q n , Φ n RS = ( ∂F RS ∂Q n 0 ) ∣ ∣ ∣ T , M RS = F RS + T 0 S RS . (75) \nThis leads to \nS RS = π (˜ r 2 0 -˜ n 2 ) + α 6 π (7˜ r 2 0 -5˜ n 2 ) 7˜ r 2 0 (˜ r 2 0 + ˜ n 2 ) , Φ n RS = -˜ n 2 + α 12 7˜ n (˜ r 2 0 + ˜ n 2 ) 2 , M RS = ˜ m + α 7˜ r 2 0 -5˜ n 2 7˜ r 3 0 (˜ r 2 0 + ˜ n 2 ) . (76) \nIf we would like to consider fixed ( M,Q n ), which requires the field redefinition \n˜ r 0 = ˆ r 0 + αδ ˜ r 0 , ˜ n = ˆ n + αδ ˜ n, δ ˜ r 0 = 2(5ˆ n 2 -7ˆ r 2 0 ) 7ˆ r 0 (ˆ r 4 0 -ˆ n 4 ) , δ ˜ n = 2ˆ n (5ˆ n 2 -7ˆ r 2 0 ) 7ˆ r 2 0 (ˆ r 4 0 -ˆ n 4 ) , . (77) \nWe find that the following relation is satisfied \n∆ S ∣ ∣ ∣ M,Q n = 2 πα (7ˆ r 2 0 -5ˆ n 2 ) 7ˆ r 2 0 (ˆ r 2 0 + ˆ n 2 ) = -∆ I ∣ ∣ ∣ T,Q n . (78) \nIn other words, the NUT charge can be treated literally as a constant, instead of being a thermodynamic variable. \nIn this thermodynamic proposal, the entropy does not satisfy the Wald formalism, making it more or less impossible to determine the correction to the entropy from the perturbative solution. With an additional ambiguity of determining the correction to Q n independently, any ad hoc method to make the results consistent with the RS method will not be likely to be convincing. We therefore are not sure whether the RS method is still applicable in this case.', 'A.3 Multiple Hair Version': 'Recently, a multiple hair interpretation of the Taub-NUT thermodynamics was proposed in [17]. They introduce a new conserved charge J = mn that is analogous to the Kerr black hole. In their prescription, the first law can be written as \ndm = TdS + ωdJ +Φ n dn. (79) \nwith \nω = n r 2 0 + n 2 , Φ n = -2 nr 0 r 2 0 + n 2 . (80) \nThe free energy associated with the Euclidean action then satisfies. \nF 0 = m -T 0 S 0 -ω 0 J 0 -n 2 Φ 0 (81) \nIn this proposal, there are three thermodynamic variables ( T 0 , ω, Φ n ) associated with the free energy, and they are not independent, but parameterized by two parameters ( r 0 , n ) only. Consequently, one cannot read off the conjugate thermodynamic quantities from the free energy, even if the reverse procedure of finding the free energy from the thermodynamic quantities can be achieved. \nThe key to the RS method is to find the higher-order correction to the free energy where the variables are fixed. We then read off the thermodynamic quantities conjugate to the variables from the differential relation. Thus the RS method is inapplicable to this proposal.', 'References': "- [1] A.H. Taub, 'Empty space-times admitting a three parameter group of motions,' Annals Math. 53 , 472-490 (1951) doi:10.2307/1969567\n- [2] E. Newman, L. Tamburino and T. Unti, 'Empty space generalization of the Schwarzschild metric,' J. Math. Phys. 4 , 915 (1963) doi:10.1063/1.1704018\n- [3] C.W. Misner, 'The flatter regions of Newman, Unti and Tamburino's generalized Schwarzschild space,' J. Math. Phys. 4 , 924-938 (1963) doi:10.1063/1.1704019\n- [4] G. Cl'ement, D. Gal'tsov and M. 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2024PhRvD.110j3031D

The Hubble constant inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula is a crucial parameter in cosmology. However various cosmic observations have produced differing posterior values for inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula resulting in what is referred to as the inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula tension. To resolve this discrepancy utilizing other cosmological probes to constrain inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula is advantageous. In the quest to identify dark matter candidates the QCD axion and axionlike particles collectively referred to as axions have become leading contenders. These elusive particles can coalesce into dense structures known as axion stars via BoseEinstein condensation. When these axion stars exceed a critical mass typically through accretion or merging they experience a selfinduced collapse. This process results in short radio bursts assuming a decay constant inlineformulammlmath displayinlinemmlmsubmmlmifmmlmimmlmiammlmimmlmsubmmlmommlmommlmsupmmlmn10mmlmnmmlmn13mmlmnmmlmsupmmlmtext mmlmtextmmlmtext mmlmtextmmlmiGeVmmlmimmlmathinlineformula with the frequency depending on the axion mass and the luminosity determined by both the axion mass and decay constant. Therefore we propose that collapsing axion stars could serve as a novel standard candle to constrain inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula. Even more interesting is that the radio bursts emitted by collapsing axion stars with specific parameters match the characteristics of observed nonrepeating fast radio bursts. Thus fast radio bursts generated by collapsing axion stars have the potential to be used as standard candles to constrain inlineformulammlmath displayinlinemmlmsubmmlmiHmmlmimmlmn0mmlmnmmlmsubmmlmathinlineformula.

2024-11-01T00:00:00Z

['10.48550/arXiv.2409.05120', '2024arXiv240905120D', '10.1103/PhysRevD.110.103031', 'arXiv:2409.05120', '2024PhRvD.110j3031D']

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

Novel standard candle Collapsing axion stars

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

https://arxiv.org/pdf/2409.05120.pdf

{'Novel standard candle: Collapsing axion stars': 'Haoran Di, 1, ∗ Lijing Shao, 2, 3 Zhu Yi, 4, 5 and Shi-Bei Kong 1 \n1 School of Science, East China University of Technology, Nanchang 330013, China 2 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China 3 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China 4 Faculty of Arts and Sciences, Beijing Normal University, Zhuhai 519087, China 5 Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai 519087, China \nThe Hubble constant, H 0 , is a crucial parameter in cosmology. However, various cosmic observations have produced differing posterior values for H 0 , resulting in what is referred to as the H 0 tension. To resolve this discrepancy, utilizing other cosmological probes to constrain H 0 is advantageous. In the quest to identify dark matter candidates, the QCD axion and axionlike particles, collectively referred to as axions, have become leading contenders. These elusive particles can coalesce into dense structures known as axion stars via Bose-Einstein condensation. When these axion stars exceed a critical mass, typically through accretion or merging, they experience a self-induced collapse. This process results in short radio bursts, assuming a decay constant f a ≲ 10 13 GeV, with the frequency depending on the axion mass and the luminosity determined by both the axion mass and decay constant. Therefore, we propose that collapsing axion stars could serve as a novel standard candle to constrain H 0 . Even more interesting is that the radio bursts emitted by collapsing axion stars with specific parameters match the characteristics of observed non-repeating fast radio bursts (FRBs). Thus, FRBs generated by collapsing axion stars have the potential to be used as standard candles to constrain H 0 .', 'I. INTRODUCTION': "The standard cosmological model, known as the Λ cold dark matter (ΛCDM) model, is the simplest framework that aligns well with observational data. Notably, the Planck satellite mission's observations [1] strongly support a basic 6-parameter ΛCDM cosmology. However, cracks are beginning to appear in this model, as evidenced by tensions between different astrophysical observations that rely on the 6-parameter ΛCDM framework. For instance, the standard ΛCDM model, informed by Planck observations of the cosmic microwave background (CMB) power spectra, predicts a lower Hubble constant ( H 0 ) than what is measured locally using the distance ladder method. The H 0 value obtained from Cepheid-calibrated Type Ia supernovae (SNe Ia) is 73 . 04 ± 1 . 04 km s -1 Mpc -1 [2], which is 5 σ higher than 67 . 4 ± 0 . 5 km s -1 Mpc -1 [3] estimated from Planck CMB data within the ΛCDM framework. Numerous efforts have been made to resolve the so-called 'Hubble tension', but none have provided a convincing explanation so far (see Ref. [4] and references therein). This underscores the need to develop new, precise cosmological probes for cross-check. The gravitational-wave (GW) standard siren method [5, 6] is among the most promising approaches to address the Hubble tension. The unique multi-messenger observation, GW170817, provided the first H 0 measurement using the standard siren method, achieving about 14% precision [7], which is still insufficient to resolve the Hubble tension. \nAnother challenge for the ΛCDM model is the unresolved nature of cold dark matter. The QCD axion [8, 9], \nwhich arises from the Peccei-Quinn mechanism [10, 11], is one of the leading dark matter candidates, providing a prominent solution to the strong-CP problem. Additionally, string theory strongly suggests the existence of a wide range of axionlike particles (ALPs) across various mass scales, leading to the concept of the 'axiverse' [12]. For brevity, we will refer to both the QCD axion and ALPs as 'axion' throughout this paper. Axions, which can emerge through various mechanisms [13-17], have the potential to attain extremely high phase space density, leading to the intriguing phenomenon of Bose-Einstein condensation (BEC) [18] due to their bosonic nature. This condensation enables axions to coalesce into gravitationally bound structures known as axion stars [19-22]. In the conventional post-inflationary scenario, where the U (1) PQ symmetry undergoes spontaneous breaking after the inflationary period, axion stars could potentially make up as much as 75% of the dark matter component [23-25]. Moreover, studies of microlensing events from HSC and OGLE data consistently suggest that around 27 +7 -13 % of dark matter may exist in the form of axion stars [26]. The observed anomalies in the orbits of trans-Neptunian objects could be explained by the presence of an axion star captured within the solar system [27]. Axion stars for ALPs with specific parameters may undergo collapse, emitting millisecond-long radio bursts with a peak luminosity of 1 . 60 × 10 42 erg / s, consistent with the characteristics of observed non-repeating fast radio bursts (FRBs) [28]. Other studies have also explored the connection between axion stars and FRBs [29-32]. The astronomical investigation of axion stars can complement laboratory searches for axions. \nThis paper is organized as follows. In section II, we introduce another cosmological probe, FRBs, and discuss their limitations. In section III, we briefly present the \nconcept of the dilute axion star. In section IV, we explore the stimulated decay [28, 33-35] of collapsing axion star and explain why this process is suitable for use as a novel standard candle. In section V, we propose that FRBs generated by collapsing axion stars can serve as potential standard candles. The conclusion is presented in section VI. Throughout this paper, we use natural units with c = ℏ = 1.", 'II. FAST RADIO BURSTS': "FRBs are intense, transient radio signals that last only a few milliseconds [36-38], and both their physical origin and the mechanism behind their emission remain largely enigmatic. To date, about a thousand FRBs have been observed, with a small subset exhibiting repeating patterns [39-42]. It is widely believed that repeating and non-repeating FRBs are produced by different physical processes. Numerous models have been proposed to explain the sources of FRBs (see Refs. [43-49] for reviews). The frequency spectrum of FRBs typically ranges from about 400MHz to 8GHz [46], with total emitted energies between 10 38 and 10 40 erg [38, 50]. While there is a confirmed association between at least some FRBs and magnetars [51, 52], the exact mechanisms that trigger these mysterious events and their radiation processes remain hotly debated. Similar to the diversity seen in supernovae, it is possible that the observed FRB events originate from a mix of different populations or progenitor mechanisms. Despite these uncertainties, FRBs hold great potential as valuable cosmological probes. \nAs FRB photons travel through plasma, they interact with free electrons, causing dispersion across different frequencies. The Dispersion Measure is defined as the integral of the free electron number density along the propagation path, and it is directly proportional to the cosmological distance. This makes Dispersion Measure a useful tool for measuring cosmological distances. If an FRB can be precisely localized to a specific galaxy, its redshift can be inferred from the host galaxy [53-55]. However, accurately determining redshift by identifying the burst's host galaxy has proven difficult, as the contributions to Dispersion Measure from the host galaxy and the inhomogeneities of the intergalactic medium cannot be precisely determined from observations. Of the approximately one thousand of FRBs detected, only about thirty have been precisely localized [56, 57]. Despite the limited number of accurately localized events, FRBs have already been utilized to estimate H 0 [58-61]. However, more data are needed to improve the accuracy of these measurements and determine the most reliable value. For a cosmological probe to be precise, a large dataset is necessary to minimize random errors. Consequently, FRBs have the potential to be excellent cosmological probes if a significantly greater number can be localized.", 'III. DILUTE AXION STARS': "The QCD axion is a pseudoscalar boson with spin 0, characterized by its small mass m ϕ , extremely weak selfinteraction, and very feeble interactions with Standard Model particles. To ensure Lagrangian shift symmetry invariance, the axion potential V ( ϕ ) must exhibit periodic behavior with respect to ϕ : V ( ϕ ) = V ( ϕ + 2 πf a ), where f a is the axion decay constant, representing the energy scale at which the U (1) PQ symmetry spontaneously breaks. The instanton potential [10] is the most commonly used model for the axion potential in phenomenological studies: \nV ( ϕ ) = m 2 ϕ f 2 a [ 1 -cos ( ϕ f a )] = 1 2 m 2 ϕ ϕ 2 + λ 4! ϕ 4 + ..., (1) \nwhere λ = -m 2 ϕ /f 2 a is the attractive self-interaction coupling constant. As bosonic particles, axions can reach very high phase space densities, enabling the formation of BECs. These BECs can give rise to axion stars, which may exist in both dilute and dense forms [62-65]. However, dense axion stars may have lifespans too short to be of significant cosmological relevance as astrophysical objects [19, 66-69]. \nA stable dilute axion star can be described as a system where the attractive self-gravity of the axions is counterbalanced by the repulsive gradient energy. This equilibrium is maintained as long as the star's density remains low enough that self-interactions are negligible. The maximum mass and corresponding minimum radius of a dilute axion star can be expressed as follows [70, 71]: \nM max ∼ 5 . 073 M pl √ | λ | , R min ∼ √ | λ | M pl m ϕ λ c , (2) \nwhere M pl represents the Planck mass and λ c denotes the Compton wavelength of the axion. When the axion star's mass increases and surpasses the maximum mass specified by Eq.(2) due to merger events [72-77] or the accretion of axions from the surrounding environment [78-80], the attractive self-gravity and self-interaction of the axions will exceed the repulsive gradient energy, leading to the collapse of the axion star. As the star begins to collapse, its density increases rapidly. When the collapsing axion star reaches a critical radius, the stimulated decay of the star can be triggered, generating short radio bursts, provided the decay constant f a < 1 . 08 × 10 13 GeV [28]. For larger decay constants, when the radius of an axion star that has exceeded its maximum mass contracts to the axion's Compton wavelength, 2 π/m ϕ , which is greater than the critical radius triggering stimulated decay, the axions begin to annihilate and transition into relativistic states [81]. This leads to a rapid loss of the axion star's energy, a phenomenon known as a 'bosenova' [81-84].", 'IV. A NOVEL STANDARD CANDLE: COLLAPSING AXION STARS': "In the framework of the instanton potential, the general Lagrangian for the axion can be written as follows: \nL = 1 2 ∂ µ ϕ∂ µ ϕ -1 2 m 2 ϕ ϕ 2 -λ 4! ϕ 4 + 1 4 g aγγ ϕF µν ˜ F µν + ..., (3) \nwhere F µν represents the electromagnetic field tensor, and ˜ F µν is its dual tensor, defined as F αβ ϵ µναβ / 2. The axion-photon coupling constant, g aγγ , is defined as g aγγ = αK/ (2 πf a ), where α represents the fine structure constant, and K is a model-dependent constant typically around one. For example, in the standard Kim-ShifmanVainshtein-Zakharov (KSVZ) model [85, 86], K is approximately -1 . 95, while in the Dine-Fishler-SrednickiZhitnitskii (DFSZ) model [87, 88], K is around 0 . 72. For simplicity, we assume K = 1 in the following discussion. For QCD axions, their interaction with gluons establishes a well-known relationship between the decay constant f a and the axion mass [89]: m ϕ ≃ 6 µ eV ( 10 12 GeV /f a ) . By substituting the attractive self-interaction coupling constant λ = -m 2 ϕ /f 2 a and the Compton wavelength of the axion λ c = 2 π/m ϕ into Eq. (2), we obtain the following expressions for the maximum mass and minimum radius of a dilute axion star: \nM max ∼ 5 . 97 × 10 -12 M ⊙ ( m ϕ 10 -5 eV ) -1 ( f a 10 12 GeV ) , (4) \nR min ∼ 2 . 41 × 10 2 km ( m ϕ 10 -5 eV ) -1 ( f a 10 12 GeV ) -1 , (5) \nwhere M ⊙ denotes the solar mass. These equations show that the maximum mass and minimum radius of a dilute axion star depend solely on the axion mass and the decay constant. For more general axion stars, we denote the mass and radius as M AS and R AS , respectively. \nAxions are not completely stable, mainly because of their interaction with the electromagnetic field, as described by the interaction term L int = 1 / 4 g aγγ ϕF µν ˜ F µν . This interaction causes axions, in their rest frame, to spontaneously decay into two photons with the same helicity due to angular momentum conservation. The decay rate is given by [33] \nΓ ϕ = 1 8 π ( 1 2 m ϕ ) 1 2 ∑ λ ' = ± |M ( ϕ → γ ( λ ' ) γ ( λ ' )) | 2 = 1 . 02 × 10 -50 s -1 ( m ϕ 10 -5 eV ) 3 ( 10 12 GeV f a ) 2 , (6) \nwhere λ ' = ± denotes the helicity of the photons, and M ( ϕ → γ ( λ ' ) γ ( λ ' )) represents the transition matrix element determined by the interaction term L int = \n1 / 4 g aγγ ϕF µν ˜ F µν . Since the axion is a boson, the boson enhancement effect must be considered within the axion star. The evolution of the photon number density with a given helicity λ ' = ± within an axion star, driven by axion decays and inverse decays, is described by the Boltzmann equation [33]: \ndn λ ' dt = ∫ dX LIPS |M ( ϕ → γ ( λ ' ) γ ( λ ' )) | 2 ×{ f ϕ ( p )[1 + f λ ' ( k 1 )][1 + f λ ' ( k 2 )] -f λ ' ( k 1 ) f λ ' ( k 2 )[1 + f ϕ ( p )] } , (7) \nwhere f ϕ and f λ ' represent the phase space densities of axions and photons, respectively. The photon number density, n λ ' , is calculated as n λ ' = ∫ d 3 k/ (2 π ) 3 f λ ' . The phase space integration is carried out using the standard Lorentz-invariant measure, which accounts for the momenta of both the axion and the photons. To solve this equation, we assume that the phase space distribution of axions and photons within the dilute axion star is roughly homogeneous and isotropic. Under this assumption, the time derivative of total photon number density can be expressed as follows [34]: \ndn γ dt = 2Γ ϕ n ϕ + 16 π 2 m 3 ϕ v Γ ϕ n ϕ n γ -16 π 2 3 m 3 ϕ ( v + 3 2 ) Γ ϕ n 2 γ , (8) \nwhere n ϕ represents the axion number density, and v is the maximum velocity of axions within the axion star, estimated to be around 1 / (2 R AS m ϕ ) based on the Heisenberg uncertainty principle. In this equation, the first term corresponds to the spontaneous decay of axions. The second term, which is directly proportional to the product of the axion number density and the photon number density, represents the stimulated decay of axions. The final term, proportional to n 2 γ , accounts for the process of inverse decay. Additionally, the photons generated through spontaneous and stimulated decay escape from the axion star at a rate given by Γ e = R -1 AS , which is the inverse of the axion star's radius. \nBy integrating Eq. (8) and accounting for photons that escape from the axion star, we derive the following set of coupled differential equations describing the evolution of the photon and axion numbers within the axion star: \ndN γ dt = 2Γ ϕ N ϕ + 12 π m 3 ϕ vR 3 AS Γ ϕ N ϕ N γ -2 π (2 v +3) m 3 ϕ R 3 AS Γ ϕ N 2 γ -Γ e N γ , (9) \ndN ϕ dt = -Γ ϕ N ϕ -6 π m 3 ϕ vR 3 AS Γ ϕ N ϕ N γ + 4 πv m 3 ϕ R 3 AS Γ ϕ N 2 γ . (10) \nFIG. 1. Luminosities of collapsing axion stars with a decay constant f a = 5 × 10 12 GeV and axion masses of 10 -5 eV, 10 -4 eV, and 10 -3 eV, respectively. The peak luminosities of these bursts are approximately 10 42 erg / s, with the total energy emitted being around 10 39 erg, 10 38 erg, and 10 37 erg, respectively. For a fixed decay constant, the duration of the radio bursts varies depending on the axion mass. \n<!-- image --> \nFor a dilute axion star with a minimum radius, substituting Eq. (5) into Γ e = R -1 AS changes the photon escape rate to \nΓ e ∼ 1 . 24 × 10 3 s -1 ( m ϕ 10 -5 eV ) ( f a 10 12 GeV ) . (11) \nThe photon number N γ ≃ (2Γ ϕ / Γ e ) N ϕ in dilute axion stars with minimum radius, resulting from spontaneous decay, is not sufficient to trigger stimulated decay. However, when the axion star's mass exceeds the critical mass, causing it to collapse until its size becomes smaller than the critical radius \nR cr ≃ 24 π Γ ϕ M max m 3 ϕ ∼ 6 . 67 × 10 -4 km ( 10 -5 eV m ϕ )( 10 12 GeV f a ) , (12) \ni.e., when 12 π Γ ϕ N ϕ N γ / ( m 3 ϕ vR 3 AS ) ≃ 2Γ ϕ N ϕ , stimulated decay is initiated. Once the axion star's radius reaches the critical value, the photon escape rate becomes \nΓ e ≃ m 3 ϕ 24 π Γ ϕ M max ∼ 4 . 50 × 10 8 s -1 ( m ϕ 10 -5 eV ) ( f a 10 12 GeV ) . (13) \nWhen the star's size approaches the Compton wavelength of the axion, 2 π/m ϕ , axions begin to annihilate, transitioning into relativistic states [81]. To induce stimulated radiation rather than generating relativistic axions, the critical radius R cr ≃ 24 π Γ ϕ M max /m 3 ϕ must be larger than the Compton wavelength, 2 π/m ϕ . This condition requires that f -1 a ≳ 10 -13 GeV -1 , as illustrated in Fig. 2 and Fig. 3. \nFIG. 2. Total energy emitted by collapsing axion stars with varying parameters. The total energy depends only on the axion mass and decay constant. With a constant decay constant, the total energy emitted by the collapsing axion star decreases as the axion mass increases. Conversely, with a constant axion mass, the total energy emitted increases as the decay constant rises. The rectangular area corresponds to the luminosity and frequency of currently detected FRBs, with frequency mapped to the axion mass. The gray area indicates where collapsing axion stars would produce relativistic axions rather than radio bursts. \n<!-- image --> \nThe luminosity of the collapsing axion star due to stimulated decay is given by \nL ϕ = 1 2 m ϕ N γ R -1 cr = N γ m 4 ϕ 48 π Γ ϕ M max ∼ 3 . 60 × 10 40 erg / s ( N γ 10 49 ) ( m ϕ 10 -5 eV ) 2 ( f a 10 12 GeV ) , (14) \nwhich is determined solely by the mass of the axion and the decay constant f a . By substituting Eqs. (6), (12) and (13) into Eqs. (9) and (10), and then numerically solving these coupled equations, we can obtain N γ . Substituting this into Eq. (14) allows us to determine the temporal variation of the collapsing axion star's luminosity, as illustrated in Fig. 1. In Fig. 1, we present three numerical solutions for a representative set of parameters with a decay constant f a = 5 × 10 12 GeV and varying axion mass. Once the size of the axion star falls below the critical radius, it will produce a brief radio burst, as shown in Fig. 1. Therefore, a collapsing axion star with an inverse decay constant f -1 a ≳ 10 -13 GeV -1 will trigger stimulated decay. The spectrum of the radio burst will be nearly monochromatic at a frequency of ν ≃ m ϕ / (4 π ) ≈ 1 . 21( m ϕ / 10 -5 eV)GHz, which is a distinctive signal that could potentially be detected by radio telescopes. If a radio signal consistent with the stimulated decay of a collapsing axion star is observed, it can serve as a standard candle due to the strong luminosity of this radio burst, which is determined solely by the axion mass and the decay constant. Thus, collapsing \naxion stars could serve as excellent cosmological probes. \nThe above results are based on the instanton potential of axions. If other potentials are considered, the luminosity and total radiated energy of collapsing axion stars will change. For example, the chiral potential [90] has a corresponding coupling constant for the self-interaction of the axion of -0 . 34 m 2 ϕ /f 2 a [91], which is about one-third of the coupling constant for the instanton potential. This will increase the maximum mass of the dilute axion star and raise the critical radius corresponding to the onset of stimulated decay. The maximum luminosity remains unchanged, while the total energy radiated by the axion star increases. Additionally, the strength of the interaction between axions and photons depends on the selected K values, meaning that different choices for K will affect the luminosity of axion stars. Specifically, the decay rate of an axion is proportional to the square of K , rendering the result independent of the sign of K . For the KSVZ axion model, K is approximately -1 . 95, with its absolute value being about twice that of the value we consider. Consequently, both the decay rate of the axion and the critical radius will increase, resulting in a decrease in the maximum luminosity of the axion star, while the total energy radiated remains unchanged.", 'V. A POTENTIAL STANDARD CANDLE: FAST RADIO BURSTS': 'Interestingly, the radio signals from the stimulated decay of collapsing axion stars may have already been detected. FRBs are bright, transient radio signals lasting just milliseconds, with a frequency spectrum ranging from approximately 400MHz to 8GHz, and a total energy emission typically between 10 38 and 10 40 erg. Fig. 2 shows the variation of the total energy radiated by a collapsing axion star with different decay constants and axion masses. It can be observed from Fig. 2 that the total energy and photon frequency radiated by a collapsing axion star for certain parameters are consistent with some FRBs. For constraints on the axion parameter space from FRBs, refer to Fig. 3. Therefore, the stimulated decay of collapsing dilute axion stars might account for some of the observed non-repeating FRBs. Additionally, in the conventional post-inflationary scenario, axion stars could comprise up to 75% of the dark matter component. Given the significant abundance of axion stars in the Universe, collapsing axion stars may be common enough to serve as sources of FRBs. The rate of collapsing axion stars deserves further dedicated study. The frequency of the observed radio signals is given by \nν obs = ν em 1 + z = m ϕ 4 π (1 + z ) , (15) \nwhere ν em is the emitted frequency of the radio signal, determined by the mass of the axion. Therefore, the \nFIG. 3. Constraints on the axion parameter space. The region above the orange solid line is excluded based on constraints from Planck 2018 data combined with BAO measurements [92]. The green area indicates exclusions derived from white dwarf observations [93]. The latest constraints on the axion parameter space from white dwarf observations can be found in Ref. [94]. The purple line denotes the parameter space for QCD axions. Below the gray line, collapsing axion stars are predicted to emit relativistic axions [81] rather than radio bursts. The blue region corresponds to parameter space that aligns with the luminosity and frequency characteristics of non-repeating FRBs. \n<!-- image --> \nredshift can be expressed as \nz = m ϕ 4 πν obs -1 . (16) \nFig. 3 illustrates the constraints on the axion parameter space based on the hypothesis that some FRBs are explained by collapsing axion stars. This provides a potential direction for experimental axion searches. Once the axion mass is accurately determined through experiments, the redshift of radio bursts can also be precisely measured. The luminosity flux of the collapsing axion star as observed from Earth is given by the formula: \nF ϕ = L ϕ 4 πd 2 L = N γ m 4 ϕ 192 π 2 Γ ϕ M max d 2 L , (17) \nwhere d L represents the luminosity distance. The luminosity distance can then be expressed as \nd L = ( L ϕ 4 πF ϕ ) 1 / 2 = m 2 ϕ 8 π ( N γ 3Γ ϕ M max F ϕ ) 1 / 2 , (18) \nwhere N γ , Γ ϕ and M max depend solely on the axion mass and decay constant. Thus, by measuring the flux and knowing the axion mass and decay constant, we can determine the luminosity distance d L . The luminosity distance, measured in this way, provides a cross-check with the cosmological distance obtained from the Dispersion Measure, which is defined as the integral of the number density of free electrons along the propagation path. This \ncross-check improves the accuracy of the luminosity distance, which in turn can help pinpoint the host galaxy of the FRB. As more FRBs are precisely located, the value of the Hubble constant can be measured with greater accuracy, which may help resolve the Hubble tension. Therefore, FRBs generated by collapsing axion stars have the potential to be used as standard candles and serve as valuable cosmological probes.', 'VI. CONCLUSIONS': "The Hubble constant, H 0 , is a crucial parameter in cosmology, but different observational methods have produced varying estimates, leading to what is known as the H 0 tension. To resolve this discrepancy, it is helpful to employ additional cosmological probes. The GW standard siren method is among the most promising approaches for addressing the Hubble tension. The unique multi-messenger observation event, GW170817, provided the first measurement of H 0 using the standard siren method, achieving about 14% precision, which is not sufficient to resolve the tension. 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2024CQGra..41u7002T

We point out that dark matter and dark energy arise naturally in a recently proposed model of combinatorial quantum gravity. Dark energy is due to the groundstate curvature at finite coupling dark matter arises from allotropy in the discrete structure of spacetime. The stable structure of the spacetime crystal represents the curved background the coexisting metastable allotropes of higher curvature and energy are natural candidates for dark matter. We thus suggest that dark energy and dark matter are two manifestation of quantum gravity.

2024-11-01T00:00:00Z

['2024CQGra..41u7002T', 'arXiv:2409.09385', '10.1088/1361-6382/ad7acf', '2024arXiv240909385T', '10.48550/arXiv.2409.09385']

['dark matter', 'dark energy', 'quantum gravity', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Dark matter and dark energy in combinatorial quantum gravity

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

https://arxiv.org/pdf/2409.09385.pdf

{'Dark matter and dark energy in combinatorial quantum gravity': "C. A. Trugenberger 1 \n1 SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland \nWe point out that dark matter and dark energy arise naturally in a recently proposed model of combinatorial quantum gravity. Dark energy is due to the ground-state curvature at finite coupling, dark matter arises from allotropy in the discrete structure of space-time. The stable structure of the space-time 'crystal' represents the curved background, the coexisting metastable allotropes of higher curvature and energy are natural candidates for dark matter. We thus suggest that dark energy and dark matter are two manifestions of quantum gravity.", 'INTRODUCTION': "Dark matter (for a recent review see [1]) and dark energy (for a recent review see [2]) are the two major puzzles of contemporary cosmology. The mass-energy of the universe is made only by 5% of ordinary baryonic matter, the remainder being 27% dark matter, an unknown form of matter interacting with ordinary matter only through gravity and 68% dark energy, an unknown form of energy causing the measured acceleration of the universe expansion (for a review see [3]). Even after decades of dedicated e ff orts, no proposed theory matches the data. \nThe main candidates for the explanation of dark energy are a cosmological constant, a background negative pressure driving the universe expansion (see [2-4]) or quintessence [6], a dynamical version of the cosmological constant representing a fifth fundamental force. \nThere are two possible avenues to tackle the dark matter problem. One is to posit a new type of particle, with essentially only gravitational interactions, which is the mainstream belief today, the other is to posit a modification of general relativity (GR). In this latter approach, mostly classical models have been proposed, like modified Newtonian dynamics (MOND) [7] or its relativistic generalization, tensor-vectorscalar gravity (TeVeS) [8]. Apart from the dark matter and dark energy problems themselves, however, no other classical alteration of GR is required. What is called for, instead is a theory of quantum gravity. If dark matter is not matter, it is most likely that it is explained by the generalization of GR required to reconcile it with quantum mechanics. A first proposal in this sense is the so-called entropic gravity [9], a model in which space time is considered as an emergent property of entangled qubits. \nIt is the purpose of this paper to show that both dark energy and dark matter are natural consequences of quantum gravity. Based on recent advances in discrete geometry, we recently proposed to formulate general relativity on abstract metric spaces, focusing, specifically on random graphs [1012]. The Hamiltonian governing this model is Ollivier's combinatorial Ricci curvature [13-16], hence the name 'combinatorial quantum gravity'. Two of the characteristic features of this model are that there is no real distinction in the nature of space-time and matter, the two being just two di ff erent phases of the same elementary constituents, and that quantum mechanics in three spatial dimensions emerges simultaneously \nwith gravity, in the spirit of Wheeler's 'it from bit' hypothesis [17-19]. As we now show, both dark energy and dark matter emerge naturally in this theory.", 'HYPERBOLIC HOLOGRAPHY': "The model posits a negative-curvature surface, which can be thought of as a curved holographic screen [20-22], and a universal Newtonian time parameter t which governs its statistical fluctuations, including the Brownian motion of point-like defects. At distances larger than the curvature radius, Brownian motion on a negative-curvature surface decomposes into a dominant ballistic and deterministic component and a random correction [17-19]. The dominant deterministic component is equivalent (up to exponentially small corrections) to the geodesics of a de Sitter surface of positive curvature of the same absolute value, i.e. to the expansionary drag of a positive cosmological constant [2-4] . The radial coordinate of the surface becomes dynamically soldered to universal time and emerges as coordinate time. \nWhen this emergent coordinate time is used to parametrize dynamics, the residual slow, stochastic component of Brownian motion, as seen by co-moving observers, is quantum mechanics in an emergent space with 3 spectral space dimensions and one time dimension. The spectral space dimension 3 is a consequence of the local limit theorem in negative curvature [23] and goes by the name of 'pseudodimension' or 'dimension at infinity' in the mathematical literature [24]. The distances in this 3D emergent space are inherited from the hyperbolic surface and, as a consequence the e ff ective positive curvature e ff ects are reproduced also in 3D (see [19] for a review). \nOverall, the picture of the universe in this model is as follows. At scales larger than the curvature radius of the holographic surface we have an e ff ective (3 + 1)-dimensional de Sitter space with the slow, co-moving dynamics of massive particles governed by quantum mechanics. At scales smaller than the curvature radius, Lorentz invariance and the very concept of coordinate time are lost and the model reduces to di ff usion on an e ff ectively flat 2D surface. At even smaller scales, below the Planck scale, we see the discrete nature of this surface, as we will discuss in the next section. \nIt is important to stress that, as a consequence of the hyperbolic holography, the curvature e ff ects in this 'emergent \nuniverse' reduce to the fundamental curvature e ff ects on the holographic surface. This will the subject of the rest of the paper.", 'EMERGENCE OF 2D GEOMETRY FROM NETWORKS': 'The idea of formulating a discrete version of general relativity goes back a long time. Most attempts are based on simplicial complexes, either causal ones [25] or growing flavoured ones [26, 27]. Simplicial complexes, however are still piece-wise flat chunks of Euclidean space. In [10, 11] we proposed to formulate general relativity on abstract metric spaces, with no reference whatsoever to concepts such as a manifold, not even locally, focusing in particular on random graphs (for a review see [28]). To this end we used a then recently developed combinatorial form of Ricci curvature, the Ollivier curvature [13-16]. A modified version of the Ollivier curvature was also proposed as a quantum version of Ricci curvature in [29, 30]. The Ollivier Ricci curvature, however, is the only combinatorial curvature (to our best knowledge) that has been proven to converge to continuum Ricci curvature for geometric networks defined on manifolds [31, 32]. For another recent attempt to obtain an emergent space-time from purely random structures see [33]. \nThe holographic surface is formulated as a statistical mechanics model on the configuration space of 4-regular, incompressible random graphs (IRG) with N vertices [10, 11]. The partition function Z is \nZ = X IRG e -1 g H , (1) \nwhere g is the dimensionless coupling. Incompressible graphs are those for which short cycles, i.e. triangles, squares and pentagons, do not share more than one edge. This condition is the loop equivalent of the hard-core condition for bosonic point particles. As the hard-core condition prevents the infinite compressibility of Bose gases, the graph incompressibility condition prevents graph crumpling by requiring that loops can \'touch\' but not \'overlap\' on more than one edge. The graph incompressibility condition can be also formulated as an excluded sub-graph condition [11]. The Hamiltonian H is the total Ollivier-Ricci curvature, a discrete combinatorial analogue of the Riemannian Einstein-Hilbert action, \nH = -4 X i ∈ G κ ( i ) = -4 X i ∈ G X j ∼ i κ ( i j ) , (2) \nwhere we denote by j ∼ i the neighbour vertices j to vertex i in graph G , i.e. those connected to i by one edge ( i j ). Here κ ( i j ) is the Ollivier-Ricci curvature of edge ( i j ) [13-16] (see Appendix) and 4 is the degree of the regular graph. \nWhile the Ollivier-Ricci curvature is cumbersome to compute in the generic case, it simplifies substantially on 2D- \nregular incompressible graphs [11], \nκ ( i j ) = Tij 2 D -" 1 -2 + Tij + Sij 2 D # + -" 1 -2 + Tij + Sij + Pij 2 D # + . (3) \nwhere the subscript \' + \' is defined as [ α ] + = max(0 , α ) and Tij , Sij and Pij denote the number of triangles, squares and pentagons supported on egde ( i j ). This gives \nH = H global + H local , H global = 16 N -9 8 T -S -5 8 P ! , H local = X ( i j ) ∈ E 1 GLYPH<16> ( Tij + Sij ) -2 GLYPH<17> + X ( i j ) ∈ E 2 GLYPH<16> ( Tij + Sij + Pij ) -2 GLYPH<17> , (4) \nwhere T , S and P are the total numbers of triangles, squares and pentagons on the graph and E 1 and E 2 are the ensembles of edges for which the respective summands are strictly positive. If we consider edges supporting each no more than two short cycles, the local term can be neglected. \nLet us consider, for simplicity, this case so that we can focus on the global term. This shows that the formation of short cycles is favoured since it minimizes the energy, with triangles being most favoured, followed by squares and lastly pentagons. The incompressibility condition, however, requires that a triangle can share an edge only with a pentagon, not with an another triangle or a square. But an edge supporting two squares contributes less to the energy than an edge supporting a triangle-pentagon couple. As a consequence, squares are favoured and triangles and pentagons can survive at most as rare isolated defects. Numerical simulations fully confirm this simple argument [11]. From now on it will be simpler, thus to consider only bipartite graphs, which have no odd cycles at all. \nRandom regular graphs have essentially no short loops, these being exceedingly rare in large graphs [34]. The ground state of the statistical model, instead is reached when the number of squares takes the maximal value S = N . One should thus expect a phase transition when g is decreased from g = ∞ , where the contribution of the highest-energy random regular graphs is large to g = 0, where only the ground state survives. \nThere is indeed a continuous phase transition between a random phase, with extremely sparse cycles on the graph and a geometric phase, in which the graph defines the tiling of a negative curvature surface (for a review see [19]). The phase transition is due to a condensation of square loops (4-cycles), with triangles and pentagons appearing only as possible rare defects [11], as the dimensionless coupling constant (playing the role of temperature) is decreased. For a finite number of vertices, this is a genus-decreasing transition from an infinite genus surface at the critical point to a torus ground state at g = 0. For an infinite number of vertices, we have a transition from infinite curvature at the critical point to a flat \nEuclidean plane at g = 0. For all intermediate coupling we have a negative-curvature surface with two scales: an ultraviolet scale, below which the random character survives, and an infrared radius of curvature. When the coupling is reduced, both the ultraviolet scale and the curvature decrease, until a uniform flat plane is obtained as the ground state at zero coupling. \nThe ultraviolet scale is the correlation length of the continuous phase transition. As always, below this scale, the disordered phase, in this case the random phase, survives. In this graph transition, however, this has an additional consequence. In the geometric phase, distances scale as √ N while in the disordered phase, where a random graph survives, distances scale as log N [28, 34]. When N becomes very large, the dimensions of a disordered random ball, as viewed from the geometric phase, becomes point-like. Therefore, the surviving domains of random phase at any finite coupling have a natural interpretation as Planck-size matter particles, with a rest energy proportional to their excess curvature with respect to the ground state [17, 18]. It is these point-like objects, which we shall call baryonic matter, that di ff use on the surface as a consequence of statistical fluctuations, as describe above. In this model, there is no real distinction between matter and spacetime, both are made of the same constituents, graph links. \nTo conclude this section, it is interesting to note that most discrete models of gravity based on networks produce negative-curvature ground states. Indeed, this result was found both in growing simplicial complexes approaches [27] and in models based on an alternative combinatorial Ricci curvature [29], although the authors of this latter work chose to interpret this as \'quantum flatness\'.', 'DARK ENERGY': 'Dark energy, in the present model, is a simple consequence of ground state curvature at finite coupling. In the geometric phase, each vertex i can be surrounded by si = 1 , 2 , 3 , 4 squares (we have already treated the si = 0 case in the previous section). The minimum of the free energy at a given coupling g is realized for one of these values, which we shall call ¯ s and which is the most frequent value for this g . Let us consider a uniform configuration with ¯ s at each vertex and all edges of fixed length ℓ . As we discuss in detail in the next section, this is the tiling of a hyperbolic plane with a given curvature dependent on ¯ s . This curvature, by means of the hyperbolic holography described above, plays the role of a cosmological constant [4], i.e. of dark energy. We will discuss the role of deviations from the most frequent value ¯ s in the next section. \nIf we consider the coupling g as a dynamical variable, it is possible to identify 1 /g itself with universal time. In this case, the big bang can be identified with the critical point at which the universe emerges as an infinite-curvature point-like ball of baryonic matter. From there on, the curvature decreases from extremely high values towards zero while more and more mat- \nter is transformed into space-time. Today, we expect the radius of curvature scale to lie between the Planck scale and the smallest scales probed by present accelerators: below this scale one would see 2D physics. The very large curvature near the critical point would be perceived at large distances in the e ff ective de Sitter description as an extremely accelerated expansion, a form of topological inflation when space-time emerges from randomness by a topological re-arrangement of the graph. Of course, one should also take into account the gravitational attraction between the residual matter. The details of such a computation are beyond the scope of the present paper but it is clear that this would slow considerably the perceived expansion acceleration, perhaps even cancelling it completely for a period. But when more and more matter transforms into space-time, so that the residual attraction becomes small, the ground-state curvature takes over again and the perceived expansion will accelerate again, albeit at a much smaller rate, which actually tends to zero as the ground state curvature becomes smaller and smaller at tiny couplings. The ground state curvature is thus a natural explanation of dark energy.', 'DARK MATTER': "Every vertex i in the geometric phase can exist in four possible states, corresponding to the number si = 1 , 2 , 3 , 4 of squares surrounding it, which, as we now show, amounts to a quantized curvature. We have identified network regions with si = 0 with baryonic matter particles, while the value ¯ s realized in the minimum of the free energy at a given coupling defines what we call space and has the lowest curvature, giving rise to dark energy as explained above. But what about vertices with di ff erent values of si ? \nAs always when slowly lowering the coupling in continuous phase transitions, there typically survive domains of higher free energy. These are metastable states, which, depending on their free energy di ff erence to the minimum, may be extremely long-lived. In the present case, these domains interact gravitationally due to their higher curvature. Such domains, characterized by 0 < si < ¯ s are neither baryonic matter nor space but, rather, represent natural candidates for dark matter. Dark matter, in this model, appears like crystal allotropy (for a review see [35]) in the fabric of space-time. \nLet us now explain this in details. Every regular graph with degree larger than 2 has a genus, corresponding to the smallest genus of a surface on which it can be embedded with no edge crossings. The graph becomes thus the 1-skeleton of a 2-cell embedding, for which the graph cycles are homeomorphic to open disks on the surface. The 2-cell embedding is called a 'map'. A combinatorial map can be geometrized by assigning the same fixed geodesic length ℓ to each edge so that it becomes a tiling of the surface, in which the combinatorial cycles become regular polygonal faces of the tiling [36-38]. Here we shall consider infinite graphs in the geometric phase and, for simplicity, we shall focus on the geometric structure \nat large distances, approximating the baryonic matter domains with points. \nSu ffi ciently deep in the geometric phase, each edge of the map supports exactly two regular polygons. Let ki , i = 1 . . . 4, be the number of edges of the 4 polygons surrounding a vertex and let us call the vector k = ( k 1 , k 2 , k 3 , k 4) the vertex type. For simplicity, we shall consider homogeneous maps, for which the edge type is the same for all vertices, so that the geometrization gives rise to semi-regular tilings. Moreover, for homogeneous maps, the tiled surface is a constant curvature surface [36-38]. \nThe combinatorial Ollivier-Ricci curvature of such a homogeneous map is given by (see (4) \nκ comb = -16 N 1 -9 8 T N -S N -5 8 P N ! = -H , (5) \nwhere T , S and P are the total numbers of triangles, squares and pentagons on the graph, and H is its energy. This has to be compared with the geometric curvature of the corresponding semi-regular tiling, given by the angle sum parameter \nα = 4 X i = 1 ki -2 ki . (6) \nIf α < 2 the surface is spherical, i.e. of positive constant curvature, if α = 2 it is the flat Euclidean plane and, finally, if α > 2 it is hyperbolic, i.e. of negative constant curvature [36-38]. \nFor every hyperbolic tiling there is exactly one geodesic length so that the sum of interior angles at each vertex sums exactly to 2 π . To see this, let us use the cosine rule for a hyperbolic n-gon, \nsin GLYPH<18> θ 2 GLYPH<19> = cos GLYPH<16> π n GLYPH<17> cosh GLYPH<16> ℓ 2 R GLYPH<17> , (7) \nto write the total interior angle at a vertex of a tiling as \n4 X i = 1 2 arcsin cos GLYPH<16> π ki GLYPH<17> cosh GLYPH<16> ℓ 2 R GLYPH<17> = 2 π , (8) \nwhere R is the radius of curvature of the Poincar'e disk on which the tiling is constructed. First of all let us note that this is a monotonically decreasing function of ℓ/ R . Second, for ℓ/ R →∞ the total interior angle vanishes. In the opposite limit ℓ/ R → 0, the geometry becomes Euclidean and the interior angle sum becomes larger than 2 π because of the above angle-sum condition α > 2. Therefore, given a specific semiregular tiling, there is exactly one possible parameter ℓ/ R that solves (8). If the length unit ℓ is held fixed, the hyperbolic radius R of the Poincar'e disk varies and, with it, the curvature K = -1 / R 2 . Otherwise, we can describe varying curvature at a fixed radius by letting the length unit ℓ change. In any case, the tiling determines the curvature of the underlying surface. This shows also that, for strictly negative Ollivier curvature, \nFIG. 1. An allotropy region (blue) with vertices surrounded by two squares within an hyperbolic tiling with all vertices surrounded by three squares. The hyperbolic tiling represents space of a given curvature; the blue allotropy region represents a region of dark matter, with a higher absolute curvature and energy. Created by Eryk Kopczy'nski using RogueViz [39]. \n<!-- image --> \nwe cannot take the length scale to zero; this is possible only if the curvature also vanishes. In the infrared, the model is pushed to the completely ordered phase at zero coupling, corresponding to a flat geometry. \nThe geometric phase is realized by the condensation of squares and corresponds, thus to vertex types with at least one ki = 4 and all other ki = 4 or ki ≥ 6 for all vertices. At each coupling, one of the four possible vertex types of the semiregular tiling, with one, two, three of four squares will constitute the minimum of the free energy. When the coupling is decreased from the critical value, the number of squares corresponding to the free-energy minimum will increase, see (5). At each step, one of the ki decreases from a higher value to 4. In order to maintain the total interior angle sum as 2 π , as in (8), the quantity ℓ/ R has to decrease, which means the absolute value of the curvature of the tiled surface decreases, until it vanishes for four squares. \nSuppose now the coupling decreases from the critical value to a lower value. If, at this new value, the free-energy difference between the actual minimum and the previous minimum at higher couplings is su ffi cently small, very long-lived metastable domains of a di ff erent tiling can survive, exactly as in crystal allotropy (for a review see [35]), the paramount example being unstable diamond in an environment which favours graphite as the stable configuration of carbon. These domains have higher absolute negative curvature than the stable ground state, as shown in the example in Fig. 1. In the e ff ective 3D Lorentzian picture arising from holography they have, correspondingly higher positive curvature and are thus natural candidates for dark matter. \nIn this model, dark matter arises thus essentially as 'crystallographic' defects in the fabric of space-time. Of course, \nfor couplings just above the transition values between two allotropes, small domains of lower absolute curvature can form. In the Lorentzian picture, these would correspond to antigravity domains with positive curvature lower than the dominant de Sitter background.", 'APPENDIX: OLLIVIER-RICCI CURVATURE': 'Ricci curvature on manifolds is a measure of how much (infinitesimal) spheres around a point contract (positive Ricci curvature) or expand (negative Ricci curvature) when they are transported along a geodesic with a given tangent vector at the point under consideration. The Ollivier curvature is a discrete version of the same measure. For two vertices i and j it compares the Wasserstein (or earth-mover) distance W GLYPH<16> µ i , µ j GLYPH<17> between the two uniform probability measures µ i , j on the spheres around i and j to the distance d ( i , j ) on the graph and is defined as \nκ ( i , j ) = 1 -W GLYPH<16> µ i , µ j GLYPH<17> d ( i , j ) . (9) \nThe Wasserstein distance between two probability measures µ i and µ j on the graph is defined as \nW GLYPH<16> µ i , µ j GLYPH<17> = inf X i , j ξ ( i , j ) d ( i , j ) , (10) \nwhere the infimum has to be taken over all couplings (or transference plans) ξ ( i , j ) i.e. over all plans on how to transport a unit mass distributed according to µ i around i to the same mass distributed according to µ j around j .', 'DATA AVAILABILITY': 'Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.', 'COMPETING INTERESTS': "The author declares that he has no competing interests. \n- [1] Arbey, A. & Mahmoudi, F. Dark matter and the early universe: a review. Progress in Particle and Nuclear Physics 119 , 103865 (2021).\n- [2] Brax, P. What makes the Universe accelerate? A review on what dark energy could be and how to test it. Rep. Prog. Phys. 81 , 016901 (2018)\n- [3] Peebles, P. J. E. Ratra, B. The cosmological constant and dark energy. Rep. Mod. Phys. 75 , 559-606 (2003). \n- [4] Padmanabhan T. Cosmological constant: the weight of the vacuum. Phys.Rept. 380 , 235 (2003).\n- [5] Ratra, B. Peebles, P. J. E. Cosmological consequences of a rolling homogeneous scalar field. Phys.Rev. D 37 , 3406 (1988).\n- [6] Wetterich, C. Cosmology and the fate of dilatation symmetry. Nucl. Phys. B 302 , 668-696 (1988).\n- [7] Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophysical Journal 270 , 364-370 (1983).\n- [8] Bekenstein, J., Relativistic gravitation theory for the modified Newtonian dynamics paradigm. Phys. Rev. D 70 , 083509 (2004).\n- [9] Verlinde, E. On the Origin of Gravity and the Laws of Newton. JHEP 2011 , 29 (2011).\n- [10] Trugenberger, C., A. Combinatorial quantum gravity: geometry from random bits. JHEP 09 , 045 (2017).\n- [11] Kelly, C. Trugenberger, C. A. & Biancalana, F. Self-Assembly of Geometric Space from Random Graphs. Class. Quant. Gravity 36 , 125012 (2019).\n- [12] Kelly, C. Trugenberger, C. A. & Biancalana, F. Emergence of the circle in a statistical model of random cubic graphs. Class. Quant. Gravity 38 , 075008 (2021).\n- [13] Ollivier, Y. Ricci curvature of metric spaces. C. R. Math. Acad. Sci. Paris 345 , 643-646 (2007).\n- [14] Ollivier, Y. Ricci curvature of Markov chains in metric spaces. J. Funct. Anal. 256 , 810 (2009).\n- [15] Linn Y. Lu S. & Yau, T. Ricci curvature of graphs. Tohoku Math. J. 63 , 605-627 (2011).\n- [16] Jost J. & Liu, S. Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs. Discrete Comp. Geom. 51 , 300-322 (2014).\n- [17] Trugenberger, C., A. Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity. JHEP 04 , 019 (2022).\n- [18] Trugenberger, C., A. Emergent de Sitter space, quantum behaviour and large-scale spectral dimension (3 + 1). JHEP 03 , 186 (2023).\n- [19] Trugenberger, C., A. Combinatorial Quantum Gravity and Emergent 3D Quantum Behaviour. Universe 9 , 499 (2023).\n- [20] t'Hooft G. Dimensional reduction in quantum gravity. arXiv:gr-qc / 9310026\n- [21] Susskind, L. The world as a hologram. J. Math. Phys. 36 , 63776396 (1995).\n- [22] Bousso R. The holographic principle for general backgrounds. Class. Quant. Gravity 17 , 997 (2000).\n- [23] Ledrappier F. & Lim, S. Local limit theorem in negative curvature. arXiv:503.04156\n- [24] Anker J.-P. & Bougerol, P. & Jeulin, T. The infinite Brownian loop on a symmetric space. Rev. Mat. Iberoamericana 18 , 4197 (2002).\n- [25] Ambjorn J. & Gorlich, J. & Jurkiewicz, J. & Loll, R. Nonperturbative quantum gravity. Phys. Rep. 519 , 127-210 (2012).\n- [26] Bianconi G. & Rahmede, C. Network geometry with flavor: from complexity to quantum geometry. Phys. Rev. E 93 , 032315 (2016).\n- [27] Bianconi G. & Rahmede, C. Emergent hyperbolic network geometry. Sci. Rep. 7 , 41974 (2017).\n- [28] Albert R. & Barabasi L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74 , 47 (2002).\n- [29] Klitgaard N. & Loll, R. Introducing quantum Ricci curvature. Phys. Rev. D 97 , 046008 (2018).\n- [30] Klitgaard N. & Loll, R. Implementing quantum Ricci curvature. Phys. Rev. D 97 , 106017 (2018).\n- [31] van der Hoorn P. & Cunnigham, W. J. & Lippner, G. & Trugen-\n- berger, C. A. & Krioukov, D. Ollivier-Ricci curvature convergence in random geometric graphs. Phys. Rev. Res. 3 , 013211 (2021).\n- [32] Kelly C. & Trugenberger, C. A. & Biancalana F. Convergence of combinatorial quantum gravity. Phys. Rev. D 105 , 124002 (2022).\n- [33] Kleftogiannis, I. & Amanatidis, I. Emergent spacetime from purely random structures. arXiv:2210.00963\n- [34] McKayB.D. & Wormald, N. C. & Wysocka B. Short cycles in random regular graphs. The Electronic Journal of Combinatroics 11 , R66 (2004).\n- [35] Bernstein, J. Polymorphism in molecular crystals (Oxford Uni- \nversity Press, 2002). \n- [36] Datta B. & Gupta, S. Semi-regular tilings of the hyperbolic plane. Discrete & Computational Geometry 65 , 531-553 (2019).\n- [37] Maiti A. Quasi-vertex-transitive maps on the plane. Discrete Math. 343 , 7 (2020).\n- [38] Maiti A. Pseudo-homogenoeus tiling of the hyperbolic plane. arXiv:2302.05661\n- [39] Kopczynski E. & Celinska, D. & Ctranct, M. HyperRogue: playing with hyperbolic geometry. Bridges Conference Proceedins (2017)."}

2015MNRAS.446..521S

We introduce the Virgo Consortiums Evolution and Assembly of GaLaxies and their Environments EAGLE project a suite of hydrodynamical simulations that follow the formation of galaxies and supermassive black holes in cosmologically representative volumes of a standard cold dark matter universe. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and active galactic nuclei AGN in which thermal energy is injected into the gas without the need to turn off cooling or decouple hydrodynamical forces allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles we calibrate them to the presentday galaxy stellar mass function and the amplitude of the galaxycentral black hole mass relation also taking galaxy sizes into account. The observed galaxy stellar mass function is reproduced to 0.2 dex over the full resolved mass range 10SUP8SUP lt MSUBSUBMSUBSUB 10SUP11SUP a level of agreement close to that attained by semianalytic models and unprecedented for hydrodynamical simulations. We compare our results to a representative set of lowredshift observables not considered in the calibration and find good agreement with the observed galaxy specific star formation rates passive fractions TullyFisher relation total stellar luminosities of galaxy clusters and column density distributions of intergalactic C IV and O VI. While the massmetallicity relations for gas and stars are consistent with observations for MSUBSUB 10SUP9SUP MSUBSUB MSUBSUB 10SUP10SUP MSUBSUB at intermediate resolution they are insufficiently steep at lower masses. For the reference model the gas fractions and temperatures are too high for clusters of galaxies but for galaxy groups these discrepancies can be resolved by adopting a higher heating temperature in the subgrid prescription for AGN feedback. The EAGLE simulation suite which also includes physics variations and higher resolution zoomedin volumes described elsewhere constitutes a valuable new resource for studies of galaxy formation.

2015-01-01T00:00:00Z

['10.48550/arXiv.1407.7040', '2015MNRAS.446..521S', 'arXiv:1407.7040', '2014arXiv1407.7040S', '10.1093/mnras/stu2058']

['methods: numerical', 'galaxies: evolution', 'galaxies: formation', 'cosmology: theory', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

The EAGLE project simulating the evolution and assembly of galaxies and their environments

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

https://arxiv.org/pdf/1407.7040.pdf

{'The EAGLE project: Simulating the evolution and assembly of galaxies and their environments': 'Joop Schaye, 1 glyph[star] Robert A. Crain, 1 Richard G. Bower, 2 Michelle Furlong, 2 Matthieu Schaller, 2 Tom Theuns, 2 , 3 Claudio Dalla Vecchia, 4 , 5 Carlos S. Frenk, 2 I. G. McCarthy, 6 John C. Helly, 2 Adrian Jenkins, 2 Y. M. Rosas-Guevara, 2 Simon D. M. White, 7 Maarten Baes, 8 C. M. Booth, 1 , 9 Peter Camps, 8 Julio F. Navarro, 10 Yan Qu, 2 Alireza Rahmati, 7 Till Sawala, 2 Peter A. Thomas, 11 2', 'James Trayford': "- 1 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands\n- 2 Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK\n- 3 Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium \n4 \nInstituto de Astrof'ısica de Canarias, C/ V'ıa L'actea s/n,38205 La Laguna, Tenerife, Spain \n- 5 Departamento de Astrof'sica, Universidad de La Laguna, Av. del Astrof'ısico Franciso S'anchez s/n, 38206 La Laguna, Tenerife, Spain\n- 6 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\n- 7 Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany\n- 8 Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium\n- 9 Department of Astronomy & Astrophysics, The University of Chicago, Chicago, IL 60637, USA\n- 10 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada\n- 11 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK \n3 October 2014", 'ABSTRACT': "We introduce the Virgo Consortium's EAGLE project, a suite of hydrodynamical simulations that follow the formation of galaxies and supermassive black holes in cosmologically representative volumes of a standard ΛCDM universe. We discuss the limitations of such simulations in light of their finite resolution and poorly constrained subgrid physics, and how these affect their predictive power. One major improvement is our treatment of feedback from massive stars and AGN in which thermal energy is injected into the gas without the need to turn off cooling or decouple hydrodynamical forces, allowing winds to develop without predetermined speed or mass loading factors. Because the feedback efficiencies cannot be predicted from first principles, we calibrate them to the present-day galaxy stellar mass function and the amplitude of the galaxy-central black hole mass relation, also taking galaxy sizes into account. The observed galaxy stellar mass function is reproduced to < ∼ 0 . 2 dex over the full resolved mass range, 10 8 < M ∗ / M glyph[circledot] < ∼ 10 11 , a level of agreement close to that attained by semi-analytic models, and unprecedented for hydrodynamical simulations. We compare our results to a representative set of low-redshift observables not considered in the calibration, and find good agreement with the observed galaxy specific star formation rates, passive fractions, Tully-Fisher relation, total stellar luminosities of galaxy clusters, and column density distributions of intergalactic C iv and O vi . While the mass-metallicity relations for gas and stars are consistent with observations for M ∗ > ∼ 10 9 M glyph[circledot] ( M ∗ > ∼ 10 10 M glyph[circledot] at intermediate resolution), they are insufficiently steep at lower masses. For the reference model the gas fractions and temperatures are too high for clusters of galaxies, but for galaxy groups these discrepancies can be resolved by adopting a higher heating temperature in the subgrid prescription for AGN feedback. The EAGLE simulation suite, which also includes physics variations and higher-resolution zoomed-in volumes described elsewhere, constitutes a valuable new resource for studies of galaxy formation. \nKey words: cosmology: theory - galaxies: formation - galaxies: evolution", '1 INTRODUCTION': "Cosmological simulations have greatly improved our understanding of the physics of galaxy formation and are widely used to guide the interpretation of observations and the design of new observational campaigns and instruments. Simulations enable astronomers to 'turn the knobs' much as experimental physicists are able to in the laboratory. While such numerical experiments can be valuable even if the simulations fail to reproduce observations, in general our confidence in the conclusions drawn from simulations, and the number of applications they can be used for, increases with the level of agreement between the best-fit model and the observations. \nFor many years the overall agreement between hydrodynamical simulations and observations of galaxies was poor. Most simulations produced galaxy mass functions with the wrong shape and normalisation, the galaxies were too massive and too compact, and the stars formed too early. Star formation in high-mass galaxies was not quenched and the models could not simultaneously reproduce the stellar masses and the thermodynamic properties of the gas in groups and clusters (e.g. Scannapieco et al. 2012 and references therein). \nDriven in part by the failure of hydrodynamical simulations to reproduce key observations, semi-analytic and halo-based models have become the tools of choice for detailed comparisons between galaxy surveys and theory (see Baugh 2006 and Cooray & Sheth 2002 for reviews). Thanks to their flexibility and relatively modest computational expense, these approaches have proven valuable for many purposes. Examples include the interpretation of observations of galaxies within the context of the cold dark matter framework, relating galaxy populations at different redshifts, the creation of mock galaxy catalogues to investigate selection effects or to translate measurements of galaxy clustering into information concerning the occupation of dark matter haloes by galaxies. \nHowever, hydrodynamical simulations have a number of important advantages over these other approaches. The risk that a poor or invalid approximation may lead to overconfidence in an extrapolation, interpretation or application of the model is potentially smaller, because they do not need to make as many simplifying assumptions. Although the subgrid models employed by current hydrodynamical simulations often resemble the ingredients of semi-analytic models, there are important parts of the problem for which subgrid models are no longer required. Since hydrodynamical simulations evolve the dark matter and baryonic components self-consistently, they automatically include the backreaction of the baryons on the collisionless matter, both inside and outside of haloes. The higher resolution description of the baryonic component provided by hydrodynamical simulations also enables one to ask more detailed questions and to compare with many more observables. Cosmological hydrodynamical simulations can be used to model galaxies and the intergalactic medium (IGM) simultaneously, including the interface between the two, which may well be critical to understanding the fuelling and feedback cycles of galaxies. \nThe agreement between hydrodynamical simulations of galaxy formation and observations has improved significantly in recent years. Simulations of the diffuse IGM al- \nready broadly reproduced quasar absorption line observations of the Ly α forest two decades ago (e.g. Cen et al. 1994; Zhang et al. 1995; Hernquist et al. 1996; Theuns et al. 1998; Dav'e et al. 1999). The agreement is sufficiently good that comparisons between theory and observation can be used to measure cosmological and physical parameters (e.g. Croft et al. 1998; Schaye et al. 2000; Viel et al. 2004; McDonald et al. 2005). More recently, simulations that have been reprocessed using radiative transfer of ionizing radiation have succeeded in matching key properties of the high-column density H i absorbers (e.g. Pontzen et al. 2008; Altay et al. 2011; McQuinn et al. 2011; Rahmati et al. 2013a). \nReproducing observations of galaxies and the gas in clusters of galaxies has proven to be more difficult than matching observations of the low-density IGM, but several groups have now independently succeeded in producing disc galaxies with more realistic sizes and masses (e.g. Governato et al. 2004, 2010; Okamoto et al. 2005; Agertz et al. 2011; Guedes et al. 2011; McCarthy et al. 2012; Brook et al. 2012; Stinson et al. 2013; Munshi et al. 2013; Aumer et al. 2013; Hopkins et al. 2013; Vogelsberger et al. 2013, 2014b; Marinacci et al. 2014). For the thermodynamic properties of groups and clusters of galaxies the progress has also been rapid (e.g. Puchwein et al. 2008; McCarthy et al. 2010; Fabjan et al. 2010; Le Brun et al. 2014). The improvement in the realism of the simulated galaxies has been accompanied by better agreement between simulations and observations of the metals in circumgalactic and intergalactic gas (e.g. Stinson et al. 2012; Oppenheimer et al. 2012), which suggests that a more appropriate description of galactic winds may have been responsible for much of the progress. \nIndeed, the key to the increase in the realism of the simulated galaxies has been the use of subgrid models for feedback from star formation that are more effective in generating galactic winds and, at the high-mass end, the inclusion of subgrid models for feedback from active galactic nuclei (AGN). The improvement in the resolution afforded by increases in computing power and code efficiency has also been important, but perhaps mostly because higher resolution has helped to make the implemented feedback more efficient by reducing spurious, numerical radiative losses. Improvements in the numerical techniques to solve the hydrodynamics have also been made (e.g. Price 2008; Springel 2010; Read et al. 2010; Saitoh & Makino 2013; Hopkins 2013) and may even be critical for particular applications (e.g. Agertz et al. 2007; Bauer & Springel 2012), but overall their effect appears to be small compared to reasonable variations in subgrid models for feedback processes (Scannapieco et al. 2012). \nHere we present the EAGLE project 1 , which stands for Evolution and Assembly of GaLaxies and their Environments. EAGLE consists of a suite of cosmological, hydrodynamical simulations of a standard ΛCDM universe. The main models were run in volumes of 25 to 100 comoving Mpc (cMpc) on a side and employ a resolution that is sufficient to marginally resolve the Jeans scales in the warm ( T ∼ 10 4 K) interstellar medium (ISM). The simulations use state-of-the-art numerical techniques and subgrid models for radiative cooling, star formation, stellar mass loss and metal \nenrichment, energy feedback from star formation, gas accretion onto, and mergers of, supermassive black holes (BHs), and AGN feedback. The efficiency of the stellar feedback and the BH accretion were calibrated to broadly match the observed z ∼ 0 galaxy stellar mass function (GSMF) subject to the constraint that the galaxy sizes must also be reasonable, while the efficiency of the AGN feedback was calibrated to the observed relation between stellar mass and BH mass. The goal was to reproduce these observables using, in our opinion, simpler and more natural prescriptions for feedback than used in previous work with similar objectives. \nBy 'simpler' and 'more natural', which are obviously subjective terms, we mean the following. Apart from stellar mass loss, we employ only one type of stellar feedback, which captures the collective effects of processes such as stellar winds, radiation pressure on dust grains, and supernovae. These and other feedback mechanisms are often implemented individually, but we believe they cannot be properly distinguished at the resolution of 10 2 -10 3 pc that is currently typical for simulations that sample a representative volume of the universe. Similarly, we employ only one type of AGN feedback (as opposed to e.g. both a 'radio' and 'quasar' mode). Contrary to most previous work, stellar (and AGN) feedback is injected in thermal form without turning off radiative cooling and without turning off hydrodynamical forces. Hence, galactic winds are generated without specifying a wind direction, velocity, mass loading factor, or metal mass loading factor. We also do not need to boost the BH Bondi-Hoyle accretion rates by an ad-hoc factor. Finally, the amount of feedback energy (and momentum) that is injected per unit stellar mass depends on local gas properties rather than on non-local or non-baryonic properties such as the dark matter velocity dispersion or halo mass. \nThe EAGLE suite includes many simulations that will be presented elsewhere. It includes higher-resolution simulations that zoom into individual galaxies or galaxy groups (e.g. Sawala et al. 2014a). It also includes variations in the numerical techniques (Schaller et al. 2014) and in the subgrid models (Crain et al. 2014) that can be used to test the robustness of the predictions and to isolate the effects of individual processes. \nThis paper is organised as follows. We begin in § 2 with a discussion of the use and pitfalls of cosmological hydrodynamical simulations in light of the critical role played by subgrid processes. We focus in particular on the implications for the interpretation and the predictive power of the simulations, and the role of numerical convergence. In § 3 we describe the simulations and our definition of a galaxy. This section also briefly discusses the numerical techniques and subgrid physics. The subgrid models are discussed in depth in § 4; readers not interested in the details may wish to skip this section. In § 5 we show the results for observables that were considered in the calibration of the subgrid models, namely the z ∼ 0 GSMF, the related relation between stellar mass and halo mass, galaxy sizes, and the relations between BH mass and stellar mass. We also consider the importance of the choice of aperture used to measure stellar masses and investigate both weak and strong convergence (terms that are defined in § 2). In § 6 we present a diverse and representative set of predictions that were not used for the calibration, including specific star formation rates and passive fractions, the Tully-Fisher relation, the mass-metallicity relations, var- \nperties of the intracluster medium, and the column density distributions of intergalactic metals. All results presented here are for z ∼ 0. We defer an investigation of the evolution to Furlong et al. (2014) and other future papers. We summarize and discuss our conclusions in § 7. Finally, our implementation of the hydrodynamics and our method for generating the initial conditions are summarized in Appendices A and B, respectively.", '2 IMPLICATIONS OF THE CRITICAL ROLE OF SUBGRID MODELS FOR FEEDBACK': 'In this section we discuss what, in our view, the consequences of our reliance on subgrid models for feedback are for the predictive power of the simulations ( § 2.1) and for the role of numerical convergence ( § 2.2).', '2.1 The need for calibration': "Because the recent improvement in the match between simulated and observed galaxies can, for the most part, be attributed to the implementation of more effective subgrid models for feedback, the success of the hydrodynamical simulations is subject to two important caveats that are more commonly associated with semi-analytic models. \nFirst, while it is clear that effective feedback is required, the simulations can only provide limited insight into the nature and source of the feedback processes. For example, suppose that the implemented subgrid model for supernovae is too inefficient because, for numerical reasons, too much of the energy is radiated away, too much of the momentum cancels out, or the energy/momentum are coupled to the gas at the wrong scale. If we were unaware of such numerical problems, then we might erroneously conclude that additional feedback processes such as radiation pressure are required. The converse is, of course, also possible: the implemented feedback can also be too efficient, for example because the subgrid model underestimates the actual radiative losses. The risk of misinterpretation is real, because it can be shown that many simulations underestimate the effectiveness of feedback due to excessive radiative losses (e.g. Dalla Vecchia & Schaye 2012), which themselves are caused by a lack of resolution and insufficiently realistic modelling of the ISM. \nSecond, the ab initio predictive power of the simulations is currently limited when it comes to the properties of galaxies. If the efficiency of the feedback processes depends on subgrid prescriptions that may not be good approximations to the outcome of unresolved processes, or if the outcome depends on resolution, then the true efficiencies cannot be predicted from first principles. Note that the use of subgrid models does not in itself remove predictive power. If the physical processes that operate below the resolution limit and their connection with the physical conditions on larger scales are fully understood and can be modelled or observed, then it may be possible to create a subgrid model that is sufficiently realistic to retain full predictive power. However, this is currently not the case for feedback from star formation and AGN. As we shall explain below, this implies that simulations that appeal to a subgrid prescription for the generation of outflows are unable to predict the \nstellar masses of galaxies. Similarly, for galaxies whose evolution is controlled by AGN feedback, such simulations cannot predict the masses of their central BHs. \nTo illustrate this, it is helpful to consider a simple model. Let us assume that galaxy evolution is self-regulated, in the sense that galaxies tend to evolve towards a quasiequilibrium state in which the gas outflow rate balances the difference between the gas inflow rate and the rate at which gas is locked up in stars and BHs. The mean rate of inflow (e.g. in the form of cold streams) evolves with redshift and tracks the accretion rate of dark matter onto haloes, which is determined by the cosmological initial conditions. For simplicity, let us further assume that the outflow rate is large compared to the rate at which the gas is locked up. Although our conclusions do not depend on the validity of this last assumption, it simplifies the arguments because it implies that the outflow rate balances the inflow rate, when averaged over appropriate length and time scales. Note that the observed low efficiency of galaxy formation (see Fig. 8 in § 5.2) suggests that this may actually be a reasonable approximation, particularly for low-mass galaxies. \nThis toy model is obviously incorrect in detail. For example, it ignores the re-accretion of matter ejected by winds, the recycling of stellar mass loss, and the interaction of outflows and inflows. However, recent numerical experiments and analytic models provide some support for the general idea (e.g. Finlator & Dav'e 2008; Schaye et al. 2010; Booth & Schaye 2010; Dav'e et al. 2012; Haas et al. 2013a,b; Feldmann 2013; Dekel et al. 2013; Altay et al. 2013; Lilly et al. 2013; S'anchez Almeida et al. 2014). This idea in itself is certainly not new and follows from the existence of a feedback loop (e.g. White & Frenk 1991), as can be seen as follows. If the inflow rate exceeds the outflow rate, then the gas fraction will increase and this will in turn increase the star formation rate (and/or, on a smaller scale, the BH accretion rate) and hence also the outflow rate. If, on the other hand, the outflow rate exceeds the inflow rate, then the gas fraction will decrease and this will in turn decrease the star formation rate (and/or the BH accretion rate) and hence also the outflow rate. \nIn this self-regulated picture of galaxy evolution the outflow rate is determined by the inflow rate. Hence, the outflow rate is not determined by the efficiency of the implemented feedback. Therefore, if the outflow is driven by feedback from star formation, then the star formation rate will adjust until the outflow rate balances the inflow rate, irrespective of the (nonzero) feedback efficiency. However, the star formation rate for which this balance is achieved, and hence also ultimately the stellar mass, do depend on the efficiency of the implemented feedback. If the true feedback efficiency cannot be predicted, then neither can the stellar mass. Similarly, if the outflow rate is driven by AGN feedback, then the BH accretion rate will adjust until the outflow rate balances the inflow rate (again averaged over appropriate length and time scales). The BH accretion rate, and hence the BH mass, for which this balance is achieved depend on the efficiency of the implemented feedback, which has to be assumed. According to this toy model, which appears to be a reasonable description of the evolution of simulated galaxies, the stellar and BH masses are thus determined by the efficiencies of the (subgrid) implementations for stellar and AGN feedback, respectively. \nThe simulations therefore need to be calibrated to produce the correct stellar and BH masses. Moreover, if the true efficiency varies systematically with the physical conditions on a scale resolved by the simulations, then the implemented subgrid efficiency would also have to be a function of the local physical conditions in order to produce the correct mass functions of galaxies and BHs. \nA similar story applies to the gas fractions of galaxies or, more precisely, for the amount of gas above the assumed star formation threshold, even if the simulations have been calibrated to produce the correct GSMF. We can see this as follows. If the outflow rate is determined by the inflow rate, then it is not determined by the assumed subgrid star formation law. Hence, if we modify the star formation law, 2 then the mean outflow rate should remain unchanged. And if the outflow rate remains unchanged, then so must the star formation rate because for a fixed feedback efficiency the star formation rate will adjust to the rate required for outflows to balance inflows. If the star formation rate is independent of the star formation law, then the galaxies must adjust the amount of star-forming gas that they contain when the star formation law is changed. \nHence, to predict the correct amount of star-forming gas, we need to calibrate the subgrid model for star formation to the observed star formation law. Fortunately, the star formation law is relatively well characterised observationally on the ∼ 10 2 -10 3 pc scales resolved by large-volume simulations, although there are important unanswered questions, e.g. regarding the dependence on metallicity. Ultimately the star formation law must be predicted by simulations and will probably depend on the true efficiency of feedback processes within the ISM, but resolving such processes is not yet possible in simulations of cosmological volumes. \nIt is not obvious how the efficiency of feedback from star formation should be calibrated. We could choose to calibrate to observations of outflow rates relative to star formation rates. However, those outflow rates are highly uncertain and may be affected by AGN feedback. It is also unclear on what scale the outflow rate should be calibrated. In addition, the outflow velocity and the wind mass loading may be individually important. Moreover, unless the interaction of the wind with the circumgalactic medium is modelled correctly and resolved, then obtaining a correct outflow rate on the scale used for the calibration does not necessarily imply that it is also correct for the other scales that matter. \nWe choose to calibrate the feedback efficiency using the observed present-day GSMF, as is also common practice for semi-analytic models. We do this mostly because it is relatively well constrained observationally and because obtaining the correct stellar mass - halo mass relation, and hence the correct GSMF if the cosmological initial conditions are known, is a pre-condition for many applications of cosmological simulations. For example, the physical properties of the circumgalactic medium (CGM) are likely sensitive to the halo mass, but because halo mass is difficult to measure, observations and simulations of the CGM are typically compared for galaxies of the same stellar mass. \nOne may wonder what the point of hydrodynamical \nsimulations (or, indeed, semi-analytic models) is if they cannot predict stellar masses or BH masses. This is a valid question for which there are several answers. One is that the simulations can still make predictions for observables that were not used for the calibration, and we will present such predictions in § 6 and in subsequent papers. However, which observables are unrelated is not always unambiguous. One way to proceed, and an excellent way to learn about the physics of galaxy formation, is to run multiple simulations with varying subgrid models. It is particularly useful to have multiple prescriptions calibrated to the same observables. EAGLE comprises many variations, including several that reproduce the z ∼ 0 GSMF through different means (Crain et al. 2014). \nA second answer is that making good use of simulations of galaxy formation does not necessarily mean making quantitative predictions for observables of the galaxy population. We can use the simulations to gain insight into physical processes, to explore possible scenarios, and to make qualitative predictions. How does gas get into galaxies? What factors control the size of galaxies? What is the origin of scatter in galaxy scaling relations? What is the potential effect of outflows on cosmology using weak gravitational lensing or the Ly α forest? The list of interesting questions is nearly endless. \nA third answer is that cosmological, hydrodynamical simulations can make robust, quantitative predictions for more diffuse components, such as the low-density IGM and perhaps the outer parts of clusters of galaxies. \nA fourth answer is that calibrated simulations can be useful to guide the interpretation and planning of observations, as the use of semi-analytic and halo models has clearly demonstrated. In this respect hydrodynamical simulations can provide more detailed information on both the galaxies and their gaseous environments.", '2.2 Numerical convergence': "The need to calibrate the efficiency of the feedback and the associated limits on the predictive power of the simulations call the role of numerical convergence into question. The conventional point of view is that subgrid models should be designed to yield numerically converged predictions. Convergence is clearly a necessary condition for predictive power. However, we have just concluded that current simulations cannot, in any case, make ab initio predictions for some of the most fundamental observables of the galaxy population. \nWhile it is obvious that we should demand convergence for predictions that are relatively robust to the choice of subgrid model, e.g. the statistics of the Ly α forest, it is less obvious that the same is required for observables that depend strongly and directly on the efficiency of the subgrid feedback. One could argue that, instead, we only need convergence after recalibration of the subgrid model. We will call this 'weak convergence', as opposed to the 'strong convergence' that is obtained if the results do not change with resolution when the model is held fixed. \nIf only weak convergence is required, then the demands placed on the subgrid model are much reduced, which has two advantages: \nFirst, we can take better advantage of increases in resolution. The subgrid scale can now move along with the res- \nolution limit, so we can potentially model the physics more faithfully if we adopt higher resolution. \nA second advantage of demanding only weak convergence is that we do not have to make the sacrifices that are required to improve the strong convergence and that might have undesirable consequences. We will provide three examples of compromises that are commonly made. \nSimulations that sample a representative volume currently lack the resolution and the physics to predict the radiative losses to which outflows are subject within the ISM. Strong convergence can nevertheless be achieved if these losses are somehow removed altogether, for example, by temporarily turning off radiative cooling and calibrating the criterion for switching it back on (e.g. Gerritsen 1997; Stinson et al. 2006). However, it is then unclear for which gas the cooling should be switched off. Only the gas elements into which the subgrid feedback was directly injected? Or also the surrounding gas that is subsequently shock-heated? \nOther ways to circumvent radiative losses in the ISM are to generate the outflow outside the galaxy or to turn off the hydrodynamic interaction between the wind and the ISM (e.g. Springel & Hernquist 2003; Oppenheimer & Dav'e 2006; Oppenheimer et al. 2010; Puchwein & Springel 2013; Vogelsberger et al. 2013, 2014b). This is a valid choice, but one that eliminates the possibility of capturing any aspect of the feedback other than mass loss, such as puffing up of discs, blowing holes, driving turbulence, collimating outflows, ejecting gas clouds, generating small-scale galactic fountains, etc. Furthermore, it necessarily introduces new parameters that control where the outflow is generated and when the hydrodynamics is turned back on. These parameters may directly affect results of interest, including the state of gas around galaxies, and may also re-introduce resolution effects. A potential solution to this problem is to never re-couple and hence to evaluate all wind interactions using a subgrid model, even outside the galaxies, as is done in semi-analytic models. \nHowever, bypassing radiative losses in the ISM is not by itself sufficient to achieve strong convergence. In addition, the feedback must not depend on physical conditions in the ISM since those are unlikely to be converged. Instead, one can make the feedback depend on properties defined by the dark matter, such as its local velocity dispersion or halo mass (e.g. Oppenheimer & Dav'e 2006; Okamoto et al. 2010; Oppenheimer et al. 2010; Puchwein & Springel 2013; Vogelsberger et al. 2013, 2014b), which are generally better converged than the properties of the gas. As was the case for turning off cooling or hydrodynamic forces, this choice makes the simulations less 'hydrodynamical', moving them in the direction of more phenomenological approaches, and it also introduces new problems. How do we treat satellite galaxies given that their subhalo mass and dark matter velocity dispersion are affected by the host halo? Or worse, what about star clusters or tidal dwarf galaxies that are not hosted by dark matter haloes? \nIn practice, however, the distinction between weak and strong convergence is often unclear. One may surmise that keeping the physical model fixed is equivalent to keeping the code and subgrid parameters fixed (apart from the numerical parameters controlling the resolution), but this is not necessarily the case because of the reliance on subgrid prescriptions and the inability to resolve the first generations \nTable 1. The cosmological parameters used for the EAGLE simulations: Ω m , Ω Λ , and Ω b are the average densities of matter, dark energy and baryonic matter in units of the critical density at redshift zero; H 0 is the Hubble parameter, σ 8 is the square root of the linear variance of the matter distribution when smoothed with a top-hat filter of radius 8 h -1 cMpc, n s is the scalar power-law index of the power spectrum of primordial adiabatic perturbations, and Y is the primordial abundance of helium. \nof stars and BHs. For typical subgrid prescriptions, the energy, the mass, and the momentum involved in individual feedback events, and the number or intermittency of feedback events do not all remain fixed when the resolution is changed. Any such changes could affect the efficiency of the feedback. Consider, for example, a star-forming region and assume that feedback energy from young stars is distributed locally at every time step. If the resolution is increased, then the time step and the particle mass will become smaller. If the total star formation rate remains the same, then the feedback energy that is injected per time step will be smaller because of the decrease in the time step. If the gas mass also remains the same, then the temperature increase per time step will be smaller. A lower post-feedback temperature often leads to larger thermal losses. If, instead, the subgrid model specifies the temperature jump (or wind velocity), then the post-feedback temperature will remain the same when the resolution is increased, but the number of heating events will increase because the same amount of feedback energy has to be distributed over lower-mass particles. There is no guarantee that more frequent, lower-energy events drive the same outflows as less frequent, higher-energy events. \nMoreover, for cosmological initial conditions, higher resolution implies resolving smaller haloes, and hence tracing the progenitors of present-day galaxies to higher redshifts. If these progenitors drive winds, then this may impact the subsequent evolution. \nIn § 5.1 we investigate both the weak and strong convergence of our simulations, focusing on the GSMF. We test the weak convergence for a wide variety of predictions in sections 5 and 6.", '3 SIMULATIONS': "EAGLE was run using a modified version of the N -Body Tree-PM smoothed particle hydrodynamics (SPH) code gadget 3, which was last described in Springel (2005). The main modifications are the formulation of SPH, the time stepping and, most importantly, the subgrid physics. \nThe subgrid physics used in EAGLE is based on that developed for OWLS (Schaye et al. 2010), and used also in GIMIC (Crain et al. 2009) and cosmo-OWLS (Le Brun et al. \n2014). We include element-by-element radiative cooling for 11 elements, star formation, stellar mass loss, energy feedback from star formation, gas accretion onto and mergers of supermassive black holes (BHs), and AGN feedback. As we will detail in § 4, we made a number of changes with respect to OWLS. The most important changes concern the implementations of energy feedback from star formation (which is now thermal rather than kinetic), the accretion of gas onto BHs (which now accounts for angular momentum), and the star formation law (which now depends on metallicity). \nIn the simulations presented here the amount of feedback energy that is injected per unit stellar mass decreases with the metallicity and increases with the gas density. It is bounded between one third and three times the energy provided by supernovae and, on average, it is about equal to that amount. The metallicity dependence is motivated by the fact that we expect greater (unresolved) thermal losses when the metallicity exceeds ∼ 10 -1 Z glyph[circledot] , the value for which metal-line cooling becomes important. The density dependence compensates for spurious, numerical radiative losses which, as expected, are still present at our resolution even though they are greatly reduced by the use of the stochastic prescription of Dalla Vecchia & Schaye (2012). The simulations were calibrated against observational data by running a series of high-resolution 12.5 cMpc and intermediate resolution 25 cMpc test runs with somewhat different dependencies on metallicity and particularly density. From the models that predicted reasonable physical sizes for disc galaxies, we selected the one that best fit the z ∼ 0 GSMF. For more details on the subgrid model for energy feedback from star formation we refer the reader to § 4.5. \nAs described in more detail in Appendix A, we make use of the conservative pressure-entropy formulation of SPH derived by Hopkins (2013), the artificial viscosity switch from Cullen & Dehnen (2010), an artificial conduction switch similar to that of Price (2008), the C 2 Wendland (1995) kernel and the time step limiters of Durier & Dalla Vecchia (2012). We will refer to these numerical methods collectively as 'Anarchy'. Anarchy will be described in more detail by Dalla Vecchia (in preparation), who also demonstrates its good performance on standard hydrodynamical tests (see Hu et al. 2014 for tests of a similar set of methods). In Schaller et al. (2014) we will show the relevance of the new hydrodynamical techniques and time stepping scheme for the results of the EAGLE simulations. Although the Anarchy implementation yields dramatic improvements in the performance on some standard hydrodynamical tests as compared to the original implementation of the hydrodynamics in gadget 3, we generally find that the impact on the results of the cosmological simulations is small compared to those resulting from reasonable variations in the subgrid physics (see also Scannapieco et al. 2012). \nThe values of the cosmological parameters used for the EAGLE simulations are taken from the most recent Planck results (Planck Collaboration 2013, Table 9) and are listed in Table 1. A transfer function with these parameters was generated using CAMB (Lewis et al. 2000, version Jan 12). The linear matter power spectrum was generated by multiplying a power-law primordial power spectrum with an index of n s = 0 . 9611 by the square of the dark matter trans- \nTable 2. Box sizes and resolutions of the main EAGLE simulations. From left-to-right the columns show: simulation name suffix; comoving box size; number of dark matter particles (there is initially an equal number of baryonic particles); initial baryonic particle mass; dark matter particle mass; comoving, Plummer-equivalent gravitational softening length; maximum proper softening length. \nfer function evaluated at redshift zero 3 . Particles arranged in a glass-like initial configuration were displaced according to 2nd-order Lagrangian perturbation theory using the method of Jenkins (2010) and the public Gaussian white noise field Panphasia (Jenkins 2013; Jenkins & Booth 2013). The methods used to generate the initial conditions are described in detail in Appendix B. \nTable 2 lists box sizes and resolutions of the main EAGLE simulations. All simulations were run to redshift z = 0. Note that contrary to convention, box sizes, particles masses and gravitational softening lengths are not quoted in units of h -1 . The gravitational softening was kept fixed in comoving units down to z = 2 . 8 and in proper units thereafter. We will refer to simulations with the same mass and spatial resolution as L100N1504 as intermediate resolution runs and to simulations with the same resolution as L025N0752 as high-resolution runs. \nParticle properties were recorded for 29 snapshots between redshifts 20 and 0. In addition, we saved a reduced set of particle properties ('snipshots') at 400 redshifts between 20 and 0. The largest simulation, L100N1504, took about 4.5 M CPU hours to reach z = 0 on a machine with 32 TB of memory, with the EAGLE subgrid physics typically taking less than 25 per cent of the CPU time. \nThe resolution of EAGLE suffices to marginally resolve the Jeans scales in the warm ISM. The Jeans mass and length for a cloud with gas fraction, f g , are, respectively, M J ≈ 1 × 10 7 M glyph[circledot] f 3 / 2 g ( n H / 10 -1 cm -3 ) -1 / 2 ( T/ 10 4 K) 3 / 2 and L J ≈ 2 kpc f 1 / 2 g ( n H / 10 -1 cm -3 ) -1 / 2 ( T/ 10 4 K) 1 / 2 , where n H and T are the total hydrogen number density and the temperature, respectively. These Jeans scales can be compared to the gas particle masses and maximum proper gravitational softening lengths listed in columns 4 and 7 of Table 2. \nSimulations with the same subgrid physics and numerical techniques as used for L100N1504 were carried out for all box sizes (12.5 - 100 cMpc) and particles numbers (188 3 - 1504 3 ). We will refer to this physical model as the reference model and will indicate the corresponding simulations with the prefix 'Ref-' (e.g. Ref-L100N1504). As detailed in § 4, we re-ran the high-resolution simulations with recalibrated parameter values for the subgrid stellar and AGN feedback to improve the match to the observed z ∼ 0 GSMF. We will use the prefix 'Recal-' when referring to the simulations with this alternative set of subgrid parameters \nTable 3. Values of the subgrid parameters that vary between the models presented here. The parameters n H , 0 and n n control, respectively, the characteristic density and the power-law slope of the density dependence of the energy feedback from star formation (see equation 7 in § 4.5.1). The parameter C visc controls the sensitivity of the BH accretion rate to the angular momentum of the gas (see equation 9 in § 4.6.2) and ∆ T AGN is the temperature increase of the gas during AGN feedback (see § 4.6.4). \n(e.g. Recal-L025N0752). Note that in terms of weak convergence, Ref-L100N1504 is more similar to model RecalL025N0752 than to model Ref-L025N0752 (see § 2.2 for a discussion of weak and strong convergence). In addition, we repeated the L050N0752 run with adjusted AGN parameters in order to further improve the agreement with observations for high-mass galaxies. We will refer to this model with the prefix 'AGNdT9'. Table 3 summarizes the values of the four subgrid parameters that vary between the models presented here. Crain et al. (2014) and Schaller et al. (2014) will present the remaining EAGLE simulations, which concern variations in the subgrid physics and the numerical techniques, respectively. Finally, Sawala et al. (2014a) present very high-resolution zoomed simulations of Local Group like systems run with the EAGLE code and a physical model that is nearly identical to the one used for the Ref-L100N1504 model described here. \nFigure 1 illustrates the large dynamic range of EAGLE. It shows the large-scale gas distribution in a thick slice through the z = 0 output of the Ref-L100N1504 run, colour-coded by the gas temperature. The insets zoom in on an individual galaxy. The first zoom shows the gas, but the last zoom shows the stellar light after accounting for dust extinction. This image was created using three monochromatic radiative transfer simulations with the code skirt (Baes et al. 2011) at the effective wavelengths of the Sloan Digital Sky Survey (SDSS) u, g & r filters. Dust extinction is implemented using the metal distribution predicted by the simulations and assuming that 30 per cent of the metal mass is locked up in dust grains. Only material within a spherical aperture with a radius of 30 pkpc is included in the radiative transfer calculation. More examples of skirt images of galaxies are shown in Figure 2, in \nFigure 1. A 100 × 100 × 20 cMpc slice through the Ref-L100N1504 simulation at z = 0. The intensity shows the gas density while the colour encodes the gas temperature using different colour channels for gas with T < 10 4 . 5 K (blue), 10 4 . 5 K < T < 10 5 . 5 K (green), and T > 10 5 . 5 K (red). The insets show regions of 10 cMpc and 60 ckpc on a side and zoom into an individual galaxy with a stellar mass of 3 × 10 10 M glyph[circledot] . The 60 ckpc image shows the stellar light based on monochromatic u, g and r band SDSS filter means and accounting for dust extinction. It was created using the radiative transfer code skirt (Baes et al. 2011). \n<!-- image --> \nthe form of a Hubble sequence. This figure illustrates the wide range of morphologies present in EAGLE. Note that Vogelsberger et al. (2014a) showed a similar figure for their Illustris simulation. In future work we will investigate how morphology correlates with other galaxy properties. More images, as well as videos, can be found on the EAGLE web sites at Leiden, http://eagle.strw.leidenuniv.nl/ , and Durham, http://icc.dur.ac.uk/Eagle/ . \nWe define galaxies as gravitationally bound subhaloes identified by the subfind algorithm (Springel et al. 2001; Dolag et al. 2009). The procedure consists of three main steps. First we find haloes by running the Friends-of-Friends (FoF; Davis et al. 1985) algorithm on the dark matter particles with linking length 0.2 times the mean interparticle \nseparation. Gas and star particles are assigned to the same, if any, FoF halo as their nearest dark matter particles. Second, subfind defines substructure candidates by identifying overdense regions within the FoF halo that are bounded by saddle points in the density distribution. Note that whereas FoF considers only dark matter particles, subfind uses all particle types within the FoF halo. Third, particles that are not gravitationally bound to the substructure are removed and the resulting substructures are referred to as subhaloes. Finally, we merged subhaloes separated by less than the minimum of 3 pkpc and the stellar half-mass radius. This last step removes a very small number of very low-mass subhaloes whose mass is dominated by a single particle such as a supermassive BH. \nFigure 2. Examples of galaxies taken from simulation Ref-L100N1504 illustrating the z = 0 Hubble sequence of galaxy morphologies. The images were created with the radiative transfer code skirt (Baes et al. 2011). They show the stellar light based on monochromatic u, g and r band SDSS filter means and accounting for dust extinction. Each image is 60 ckpc on a side. For disc galaxies both face-on and edge-on projections are shown. Except for the 3rd elliptical from the left, which has a stellar mass of 1 × 10 11 M glyph[circledot] , and the merger in the bottom-left, which has a total stellar mass of 8 × 10 10 M glyph[circledot] , all galaxies shown have stellar masses of 5-6 × 10 10 M glyph[circledot] . \n<!-- image --> \nFor each FoF halo we define the subhalo that contains the particle with the lowest value of the gravitational potential to be the central galaxy while any remaining subhaloes are classified as satellite galaxies. The position of each galaxy is defined to be the location of the particle belonging to the subhalo for which the gravitational potential is minimum. \nThe stellar mass of a galaxy is defined to be the sum of the masses of all star particles that belong to the corresponding subhalo and that are within a 3-D aperture with radius 30 pkpc. Unless stated otherwise, other galaxy properties, such as the star formation rate, metallicity, and half-mass radius, are also computed using only particles within the 3-D aperture. In § 5.1.1 we show that this aperture gives a nearly identical GSMF as the 2-D Petrosian apertures that are frequently used in observational studies. \nWe find the effect of the aperture to be negligible for M ∗ < 10 11 M glyph[circledot] for all galaxy properties that we consider. However, for more massive galaxies the aperture reduces the stellar masses somewhat by cutting out intracluster light. For example, at a stellar mass M ∗ = 10 11 M glyph[circledot] as measured using a 30 pkpc aperture, the median subhalo stellar mass is 0.1 dex higher (see § 5.1.1 for the effect on the GSMF). Without the aperture, metallicities are slightly lower and half-mass radii are slightly larger for M ∗ > 10 11 M glyph[circledot] , but the effect on the star formation rate is negligible.", '4 SUBGRID PHYSICS': 'In this section we provide a thorough description and motivation for the subgrid physics implemented in EAGLE: radiative cooling ( § 4.1), reionisation ( § 4.2), star formation ( § 4.3), stellar mass loss and metal enrichment ( § 4.4), energy feedback from star formation ( § 4.5), and supermassive black holes and AGN feedback ( § 4.6). These subsections can be read separately. Readers who are mainly interested in the results may skip this section.', '4.1 Radiative cooling': 'Radiative cooling and photoheating are implemented element-by-element following Wiersma et al. (2009a), including all 11 elements that they found to be important: H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe. Wiersma et al. (2009a) used cloudy version 4 07.02 (Ferland et al. 1998) to tabulate the rates as a function of density, temperature, and redshift assuming the gas to be in ionisation equilibrium and exposed to the cosmic microwave background (CMB) and the Haardt & Madau (2001) model for the evolving UV/Xray background from galaxies and quasars. By computing \nthe rates element-by-element, we account not only for variations in the metallicity, but also for variations in the relative abundances of the elements. \nWe caution that our assumption of ionisation equilibrium and the neglect of local sources of ionizing radiation may cause us to overestimate the cooling rate in certain situations, e.g. in gas that is cooling rapidly (e.g. Oppenheimer & Schaye 2013b) or that has recently been exposed to radiation from a local AGN (Oppenheimer & Schaye 2013a). \nWe have also chosen to ignore self-shielding, which may cause us to underestimate the cooling rates in dense gas. While we could have accounted for this effect, e.g. using the fitting formula of Rahmati et al. (2013a), we opted against doing so because there are other complicating factors. Selfshielding is only expected to play a role for n H > 10 -2 cm -3 and T < ∼ 10 4 K (e.g. Rahmati et al. 2013a), but at such high densities the radiation from local stellar sources, which we neglect here, is expected to be at least as important as the background radiation (e.g. Schaye 2001; Rahmati et al. 2013b).', '4.2 Reionization': 'Hydrogen reionization is implemented by turning on the time-dependent, spatially-uniform ionizing background from Haardt & Madau (2001). This is done at redshift z = 11 . 5, consistent with the optical depth measurements from Planck Collaboration (2013). At higher redshifts we use net cooling rates for gas exposed to the CMB and the photo-dissociating background obtained by cutting the z = 9 Haardt & Madau (2001) spectrum above 1 Ryd. \nTo account for the boost in the photoheating rates during reionization relative to the optically thin rates assumed here, we inject 2 eV per proton mass. This ensures that the photoionised gas is quickly heated to ∼ 10 4 K. For H this is done instantaneously, but for He ii the extra heat is distributed in redshift with a Gaussian centred on z = 3 . 5 of width σ ( z ) = 0 . 5. Wiersma et al. (2009b) showed that this choice results in broad agreement with the thermal history of the intergalactic gas as measured by Schaye et al. (2000).', '4.3 Star formation': "Star formation is implemented following Schaye & Dalla Vecchia (2008), but with the metallicity-dependent density threshold of Schaye (2004) and a different temperature threshold, as detailed below. Contrary to standard practice, we take the star formation rate to depend on pressure rather than density. As demonstrated by Schaye & Dalla Vecchia (2008), this has two important advantages. First, under the assumption that the gas is self-gravitating, we can rewrite the observed Kennicutt-Schmidt star formation law (Kennicutt 1998), ˙ Σ ∗ = A (Σ g / 1 M glyph[circledot] pc -2 ) n , as a pressure law: \n˙ m ∗ = m g A ( 1 M glyph[circledot] pc -2 ) -n ( γ G f g P ) ( n -1) / 2 , (1) \nwhere m g is the gas particle mass, γ = 5 / 3 is the ratio of specific heats, G is the gravitational constant, f g is the mass fraction in gas (assumed to be unity), and P is the total pressure. Hence, the free parameters A and n are determined by observations of the gas and star formation rate \nsurface densities of galaxies and no tuning is necessary. Second, if we impose an equation of state, P = P eos ( ρ ), then the observed Kennicutt-Schmidt star formation law will still be reproduced without having to change the star formation parameters. In contrast, if star formation is implemented using a volume density rather than a pressure law, then the predicted Kennicutt-Schmidt law will depend on the thickness of the disc and thus on the equation of state of the star forming gas. Hence, in that case the star formation law not only has to be calibrated, it has to be recalibrated if the imposed equation of state is changed. In practice, this is rarely done. \nEquation (1) is implemented stochastically. The probability that a gas particle is converted into a collisionless star particle during a time step ∆ t is min( ˙ m ∗ ∆ t/m g , 1). \nWe use A = 1 . 515 × 10 -4 M glyph[circledot] yr -1 kpc -2 and n = 1 . 4, where we have decreased the amplitude by a factor 1.65 relative to the value used by Kennicutt (1998) because we use a Chabrier rather than a Salpeter stellar initial mass function (IMF). We increase n to 2 for n H > 10 3 cm -3 , because there is some evidence for a steepening at high densities (e.g. Liu et al. 2011; Genzel et al. 2010), but this does not have a significant effect on the results since only ∼ 1% of the stars form at such high densities in our simulations. \nStar formation is observed to occur in cold ( T glyph[lessmuch] 10 4 K), molecular gas. Because simulations of large cosmological volumes, such as ours, lack the resolution and the physics to model the cold, interstellar gas phase, it is appropriate to impose a star formation threshold at the density above which a cold phase is expected to form. In OWLS we used a constant threshold of n ∗ H = 10 -1 cm -3 , which was motivated by theoretical considerations and yields a critical gas surface density ∼ 10 M glyph[circledot] pc -2 (Schaye 2004; Schaye & Dalla Vecchia 2008). The critical volume density, n H = 0 . 1 cm -3 , is also similar to the value used in other work of comparable resolution (e.g. Springel & Hernquist 2003; Vogelsberger et al. 2013). Here we instead use the metallicity-dependent density threshold of Schaye (2004) as implemented in OWLS model 'SFTHRESZ' (eq. 4 of Schaye et al. 2010; equations 19 and 24 of Schaye 2004), \nn ∗ H ( Z ) = 10 -1 cm -3 ( Z 0 . 002 ) -0 . 64 , (2) \nwhere Z is the gas metallicity (i.e. the fraction of the gas mass in elements heavier than helium). In the code the threshold is evaluated as a mass density rather than a total hydrogen number density. To prevent an additional dependence on the hydrogen mass fraction (beyond that implied by equation 2), we convert n H into a mass density assuming the initial hydrogen mass fraction, X = 0 . 752. Because the Schaye (2004) relation diverges at low metallicities, we impose an upper limit of n ∗ H = 10 cm -3 . To prevent star formation in low overdensity gas at very high redshift, we also require the gas density to exceed 57.7 times the cosmic mean, but the results are insensitive to this value. \nThe metallicity dependence accounts for the fact that the transition from a warm, neutral to a cold, molecular phase occurs at lower densities and pressures if the metallicity, and hence also the dust-to-gas ratio, is higher. The phase transition shifts to lower pressures if the metallicity is increased due to the higher formation rate of molecular hydrogen, the increased cooling due to metals and the in- \ncreased shielding by dust (e.g. Schaye 2001, 2004; Pelupessy et al. 2006; Krumholz et al. 2008; Gnedin et al. 2009; Richings et al. 2014). Our metallicity-dependent density threshold causes the critical gas surface density below which the Kennicutt-Schmidt law steepens to decrease with increasing metallicity. \nBecause our simulations do not model the cold gas phase, we impose a temperature floor, T eos ( ρ g ), corresponding to the equation of state P eos ∝ ρ 4 / 3 g , normalised to 5 T eos = 8 × 10 3 K at n H = 10 -1 cm -3 , a temperature that is typical for the warm ISM (e.g. Richings et al. 2014). The slope of 4 / 3 guarantees that the Jeans mass, and the ratio of the Jeans length to the SPH kernel, are independent of the density, which prevents spurious fragmentation due to the finite resolution (Schaye & Dalla Vecchia 2008; Robertson & Kravtsov 2008). Following Dalla Vecchia & Schaye (2012), gas is eligible to form stars if log 10 T < log 10 T eos + 0 . 5 and n H > n ∗ H , where n ∗ H depends on metallicity as specified above. \nBecause of the existence of a temperature floor, the temperature of star forming (i.e. interstellar) gas in the simulation merely reflects the effective pressure imposed on the unresolved, multiphase ISM, which may in reality be dominated by turbulent rather than thermal pressure. If the temperature of this gas needs to be specified, e.g. when computing neutral hydrogen fractions in post-processing, then one should assume a value based on physical considerations rather than use the formal simulation temperatures at face value. \nIn addition to the minimum pressure corresponding to the equation of state with slope 4 / 3, we impose a temperature floor of 8000 K for densities n H > 10 -5 cm -3 in order to prevent very metal-rich particles from cooling to temperatures characteristic of cold, interstellar gas. This constant temperature floor was not used in OWLS and is unimportant for our results. We impose it because we do not wish to include a cold interstellar phase since we do not model all the physical processes that are needed to describe it. We only impose this limit for densities n H > 10 -5 cm -3 , because we should not prevent the existence of cold, adiabatically cooled, intergalactic gas, which our algorithms can model accurately.", '4.4 Stellar mass loss and type Ia supernovae': "Star particles are treated as simple stellar populations (SSPs) with a Chabrier (2003) IMF in the range 0 . 1 -100 M glyph[circledot] . The implementation of stellar mass loss is based on Wiersma et al. (2009b). At each time step 6 and for each stel- \nlar particle, we compute which stellar masses reach the end of the main sequence phase using the metallicity-dependent lifetimes of Portinari et al. (1998). The fraction of the initial particle mass reaching this evolutionary stage is used, together with the initial elemental abundances, to compute the mass of each element that is lost through winds from AGB stars, winds from massive stars, and core collapse supernovae using the nucleosynthetic yields from Marigo (2001) and Portinari et al. (1998). The elements H, He, C, N, O, Ne, Mg, Si, and Fe are tracked individually, while for Ca and S we assume fixed mass ratios relative to Si of 0.094 and 0.605, respectively (Wiersma et al. 2009b). In addition, we compute the mass and energy lost through supernovae of type Ia. \nThe mass lost by star particles is distributed among the neighbouring SPH particles using the SPH kernel, but setting the mass of the gas particles equal to the constant initial value, m g . Each SPH neighbour k that is separated by a distance r k from a star particle with smoothing length h then receives a fraction m g ρ k W ( r k , h ) / Σ i m g ρ i W ( r i , h ) of the mass lost during the time step, where W is the SPH kernel and the sum is over all SPH neighbours. To speed up the calculation, we use only 48 neighbours for stellar mass loss rather than the 58 neighbours used for the SPH. \nIn Wiersma et al. (2009b) and OWLS we used the current gas particle masses rather than the constant, initial gas particle mass when computing the weights. The problem with that approach is that gas particles that are more massive than their neighbours, due to having received more mass lost by stars, carry more weight and therefore become even more massive relative to their neighbours. We found that this runaway process can cause a very small fraction of particles to end up with masses that far exceed the initial particle mass. The fraction of very massive particles is always small, because massive particles are typically also metal rich and relatively quickly converted into star particles. Nevertheless, it is still undesirable to preferentially direct the lost mass to relatively massive gas particles. We therefore removed this bias by using the fixed initial particle mass rather than the current particle mass, effectively taking the dependence on gas particle mass out of the equation for the distribution of stellar mass loss. \nWe also account for the transfer of momentum and energy associated with the transfer of mass from star to gas particles. We refer here to the momentum and energy related to the difference in velocity between the star particle and the receiving gas particles, in addition to that associated with the mass loss process itself (e.g. winds or supernovae). We assume that winds from AGB stars have a velocity of 10 kms -1 (Bergeat & Chevallier 2005). After adjusting the velocities of the receiving gas particles to conserve momentum, energy conservation is achieved by adjusting their entropies. Momentum and energy transfer may, for example, play a role if the differential velocity between the stellar and gas components is similar to or greater than the sound speed of the gas, although we should keep in mind that the change in the mass of a gas particle during a cooling time is typically small. \nhave verified that our results are unaffected by this reduction in the sampling of stellar mass loss from older SSPs. \nAs in Wiersma et al. (2009b), the abundances used to evaluate the radiative cooling rates are computed as the ratio of the mass density of an element to the total gas density, where both are calculated using the SPH formalism. Star particles inherit their parent gas particles' kernel-smoothed abundances 7 and we use those to compute their lifetimes and yields. The use of SPH-smoothed abundances, rather than the mass fractions of the elements stored in each particle, is consistent with the SPH formalism. It helps to alleviate the symptoms of the lack of metal mixing that occurs when metals are fixed to particles. However, as discussed in Wiersma et al. (2009b), it does not solve the problem that SPH may underestimate metal mixing. The implementation of diffusion can be used to increase the mixing (e.g. Greif et al. 2009; Shen et al. 2010), but we have opted not to do this because the effective diffusion coefficients that are appropriate for the ISM and IGM remain unknown. \nThe rate of supernovae of type Ia (SNIa) per unit initial stellar mass is given by, \n˙ N SNIa = ν e -t/τ τ , (3) \nwhere ν is the total number of SNIa per unit initial stellar mass and exp( -t/τ ) /τ is a normalised, empirical delay time distribution function. We set τ = 2 Gyr and ν = 2 × 10 -3 M glyph[circledot] -1 . Figure 3 shows that these choices yield broad agreement with the observed evolution of the SNIa rate density for the intermediate resolution simulations, although the AGNdT9-L050N0752 may overestimate the rate by ∼ 30 per cent for lookback times of 4-7 Gyr. The highresolution model, Recal-L025N0752, is consistent with the observations at all times. \nAt each time step for which the mass loss is evaluated, star particles transfer the mass and energy associated with SNIa ejecta to their neighbours. We use the SNIa yields of the W7 model of Thielemann et al. (2003). Energy feedback from SNIa is implemented identically as for prompt stellar feedback using the stochastic thermal feedback model of Dalla Vecchia & Schaye (2012) summarized in § 4.5, using ∆ T = 10 7 . 5 K and 10 51 erg per SNIa.", '4.5 Energy feedback from star formation': "Stars can inject energy and momentum into the ISM through stellar winds, radiation, and supernovae. These processes are particularly important for massive and hence short-lived stars. If star formation is sufficiently vigorous, the associated feedback can drive large-scale galactic outflows (e.g. Veilleux et al. 2005). \nCosmological, hydrodynamical simulations have traditionally struggled to make stellar feedback as efficient as is required to match observed galaxy masses, sizes, outflow rates and other data. If the energy is injected thermally, it tends to be quickly radiated away rather than to drive a wind (e.g. Katz et al. 1996). This 'overcooling' problem is typically attributed to a lack of numerical resolution. If the \nFigure 3. The evolution of the supernova Ia rate density. Data points show observations from SDSS Stripe 82 (Dilday et al. 2010), SDSS-DR7 (Graur & Maoz 2013), SNLS (Perrett et al. 2012), GOODS (Dahlen et al. 2008), SDF (Graur et al. 2011), and CLASH (Graur et al. 2014), as compiled by Graur et al. (2014). Only data classified by Graur et al. 2014 as the 'most accurate and precise measurements' are shown. The 1 σ error bars account for both statistical and systematic uncertainties. The simulations assume that the rate is a convolution of the star formation rate density with an exponential delay time distribution (eq. 3) with e-folding time τ = 2 Gyr, normalised to yield ν = 2 × 10 -3 M glyph[circledot] -1 supernovae Ia per unit stellar mass when integrated over all time. \n<!-- image --> \nsimulation does not contain dense, cold clouds, then the star formation is not sufficiently clumpy and the feedback energy is distributed too smoothly. Moreover, since in reality cold clouds contain a large fraction of the mass of the ISM, in simulations without a cold interstellar phase the density of the warm, diffuse phase, and hence its cooling rate, is overestimated. \nWhile these factors may well contribute to the problem, Dalla Vecchia & Schaye (2012, see also Dalla Vecchia & Schaye 2008, Creasey et al. 2011 and Keller et al. 2014) argued that the fact that the energy is distributed over too much mass may be a more fundamental issue. For a standard IMF there is ∼ 1 supernova per 100 M glyph[circledot] of SSP mass and, in reality, all the associated mechanical energy is initially deposited in a few solar masses of ejecta, leading to very high initial temperatures (e.g. ∼ 2 × 10 8 K if 10 51 erg is deposited in 10 M glyph[circledot] of gas). In contrast, in SPH simulations that distribute the energy produced by a star particle over its SPH neighbours, the ratio of the heated mass to the mass of the SSP will be much greater than unity. The mismatch in the mass ratio implies that the maximum temperature of the directly heated gas is far lower than in reality, and hence that its radiative cooling time is much too short. Because the mass ratio of SPH to star particles is independent of resolution, to first order this problem is independent of resolution. At second order, higher resolution does help, because the thermal feedback can be effective in generating an outflow if the cooling time is large compared with the sound crossing time across a resolution element, and the latter decreases with increasing resolution (but only as m 1 / 3 g ). \nThus, subgrid models are needed to generate galactic winds in large-volume cosmological simulations. Three types of prescriptions are widely used: injecting energy in kinetic form (e.g. Navarro & White 1993; Springel & Hernquist 2003; Dalla Vecchia & Schaye 2008; Dubois & Teyssier 2008) often in combination with temporarily disabling hydrodynamical forces acting on wind particles (e.g. Springel & Hernquist 2003; Okamoto et al. 2005; Oppenheimer & Dav'e 2006), temporarily turning off radiative cooling (e.g. Gerritsen 1997; Stinson et al. 2006), and explicitly decoupling different thermal phases (also within single particles) (e.g. Marri & White 2003; Scannapieco et al. 2006; Murante et al. 2010; Keller et al. 2014). Here we follow Dalla Vecchia &Schaye (2012, see also Kay et al. 2003) and opt for a different type of solution: stochastic thermal feedback. By making the feedback stochastic, we can control the amount of energy per feedback event even if we fix the mean energy injected per unit mass of stars formed. We specify the temperature jump of gas particles receiving feedback energy, ∆ T , and use the fraction of the total amount of energy from core collapse supernovae per unit stellar mass that is injected on average, f th , to set the probability that an SPH neighbour of a young star particle is heated. We perform this operation only once, when the stellar particle has reached the age 3 × 10 7 yr, which corresponds to the maximum lifetime of stars that explode as core collapse supernovae. \nThe value f th = 1 corresponds to an expectation value for the injected energy of 8 . 73 × 10 15 erg g -1 of stellar mass formed, which corresponds to the energy available from core collapse supernovae for a Chabrier IMF if we assume 10 51 erg per supernova and that stars with mass 6 -100 M glyph[circledot] explode (6 -8 M glyph[circledot] stars explode as electron capture supernovae in models with convective overshoot; e.g. Chiosi et al. 1992). \nIf ∆ T is sufficiently high, then the initial (spurious, numerical) thermal losses will be small and we can control the overall efficiency of the feedback using f th . This freedom is justified, because there will be physical radiative losses in reality that we cannot predict accurately for the ISM. Moreover, because the true radiative losses likely depend on the physical conditions, we may choose to vary f th with the relevant, local properties of the gas. \nBy considering the ratio of the cooling time to the sound crossing time across a resolution element, Dalla Vecchia & Schaye (2012) derive the maximum density for which the thermal feedback can be efficient (their equation 18), \nn H ,t c ∼ 10 cm -3 ( T 10 7 . 5 K ) 3 / 2 ( m g 10 6 M glyph[circledot] ) -1 / 2 , (4) \nwhere T > ∆ T is the temperature after the energy injection and we use ∆ T = 10 7 . 5 K. This expression assumes that the radiative cooling rate is dominated by free-free emission and will thus significantly overestimate the value of n H ,t c when line cooling dominates, i.e. for T glyph[lessmuch] 10 7 K. In our simulations some stars do, in fact, form in gas that far exceeds the critical value n H ,t c , particularly in massive galaxies. Although the density of the gas in which the stars inject their energy will generally be lower than that of the gas from which the star particle formed, since the star particles move relative to the gas during the 3 × 10 7 yr delay between star formation and feedback, this does mean that for stars forming at high gas densities the radiative losses may well exceed those that \nwould occur in a simulation that has the resolution and the physics required to resolve the small-scale structure of the ISM. As we calibrate the total amount of energy that is injected per unit stellar mass to achieve a good match to the observed GSMF, this implies that we may overestimate the required amount of feedback energy. At the high-mass end AGN feedback controls the efficiency of galaxy formation in our simulations. If the radiative losses from stellar feedback are overestimated, then this could potentially cause us to overestimate the required efficiency of AGN feedback. \nThe critical density, n H ,t c , increases with the numerical resolution, but also with the temperature jump, ∆ T . We could therefore reduce the initial thermal losses by increasing ∆ T . However, for a fixed amount of energy per unit stellar mass, i.e. for a fixed value of f th , the probability that a particular star particle generates feedback is inversely proportional to ∆ T . Dalla Vecchia & Schaye (2012) show that, for the case of equal mass particles, the expectation value for the number of heated gas particles per star particle is (their equation 8) \n〈 N heat 〉 ≈ 1 . 3 f th ( ∆ T 10 7 . 5 K ) -1 (5) \nfor our Chabrier IMF and only accounting for supernova energy (assuming that supernovae associated with stars in the range 6-100 M glyph[circledot] each yield 10 51 erg). Hence, using ∆ T glyph[greatermuch] 10 7 . 5 K or f th glyph[lessmuch] 1 would imply that most star particles do not inject any energy from core collapse supernovae into their surroundings, which may lead to poor sampling of the feedback cycle. We therefore keep the temperature jump set to ∆ T = 10 7 . 5 K. Although the stochastic implementation enables efficient thermal feedback without the need to turn off cooling, the thermal losses are unlikely to be converged with numerical resolution for simulations such as EAGLE. Hence, recalibration of f th may be necessary when the resolution is changed.", '4.5.1 Dependence on local gas properties': 'We expect the true thermal losses in the ISM to increase when the metallicity becomes sufficiently high for metal-line cooling to become important. For temperatures of 10 5 K < T < 10 7 K this happens when Z > ∼ 10 -1 Z glyph[circledot] (e.g. Wiersma et al. 2009a). Although the exact dependence on metallicity cannot be predicted without full knowledge of the physical conditions in the ISM, we can capture the expected, qualitative transition from cooling losses dominated by H and He to losses dominated by metals by making f th a function of metallicity, \nf th = f th , min + f th , max -f th , min 1 + ( Z 0 . 1Z glyph[circledot] ) n Z , (6) \nwhere Z glyph[circledot] = 0 . 0127 is the solar metallicity and n Z > 0. Note that f th asymptotes to f th , max and f th , min for Z glyph[lessmuch] 0 . 1Z glyph[circledot] and Z glyph[greatermuch] 0 . 1Z glyph[circledot] , respectively. \nSince metallicity decreases with redshift at fixed stellar mass, this physically motivated metallicity dependence tends to make feedback relatively more efficient at high redshift. As we show in Crain et al. (2014), this leads to good agreement with the observed, present-day GSMF. In fact, Crain et al. (2014) show that using a constant f th = 1 ap- \npears to yield even better agreement with the low-redshift mass function, but we keep the metallicity dependence because it is physically motivated: we do expect larger radiative losses for Z glyph[greatermuch] 0 . 1 Z glyph[circledot] than for Z glyph[lessmuch] 0 . 1 Z glyph[circledot] . If we were only interested in the GSMF, then equation (6) (or f th = 1) would suffice. However, we find that pure metallicity dependence results in galaxies that are too compact, which indicates that the feedback is too inefficient at high gas densities. As discussed above, this is not unexpected given the resolution of our simulations. Indeed, we found that increasing the resolution reduces the problem. \nWe therefore found it desirable to compensate for the excessive initial, thermal losses at high densities by adding a density dependence to f th : \nf th = f th , min + f th , max -f th , min 1 + ( Z 0 . 1Z glyph[circledot] ) n Z ( n H , birth n H , 0 ) -n n , (7) \nwhere n H , birth is the density inherited by the star particle, i.e. the density of its parent gas particle at the time it was converted into a stellar particle. Hence, f th increases with density at fixed metallicity, while still respecting the original asymptotic values. We use n Z = n n = 2 / ln 10. The seemingly unnatural value 2 / ln 10 ≈ 0 . 87 of the exponent is a leftover from an equivalent, but more complicated expression that was originally used in the code. Using the round number 1 instead of 0.87 would have worked equally well. We use n H , 0 = 0 . 67 cm -3 , a value that was chosen after comparing a few test simulations to the observed presentday GSMF and galaxy sizes. The higher resolution simulation Recal-L025N0752 instead uses n H , 0 = 0 . 25 cm -3 and a power-law exponent for the density term of -1 / ln 10 rather than -2 / ln 10 (see Table 3), which we found gives better agreement with the GSMF. Note that a density dependence of f th may also have a physical interpretation. For example, higher mean densities on 10 2 -10 3 pc scales may result in more clustered star formation, which may reduce thermal losses. However, we stress that our primary motivation was to counteract the excessive thermal losses in the high-density ISM that can be attributed to our limited resolution. \nWe use the asymptotic values f th , max = 3 and f th , min = 0 . 3, where the high asymptote f th , max is reached at low metallicity and high density, and vice versa for the low asymptote. As discussed in Crain et al. (2014), where we present variations on the reference model, the choice of the high asymptote is the more important one. Using a value of f th , max greater than unity enables us to reproduce the GSMF down to lower masses. \nValues of f th greater than unity can be motivated on physical grounds by appealing to other sources of energy than supernovae, e.g. stellar winds, radiation pressure, or cosmic rays, or if supernovae yield more energy per unit mass than assumed here (e.g. in case of a top-heavy IMF). However, we believe that a more appropriate motivation is again the need to compensate for the finite numerical resolution. Galaxies containing few star particles tend to have too high stellar fractions (e.g. Haas et al. 2013a), which can be understood as follows. The first generations of stars can only form once the halo is resolved with a sufficient number of particles to sample the high-density gas that is eligible to form stars. We do not have sufficient resolution to resolve the smallest galaxies that are expected to form in the \nreal Universe. Hence, the progenitors of the galaxies in the simulations started forming stars, and hence driving winds, too late. As a consequence, our galaxies start with too high gas fractions and initially form stars too efficiently. As the galaxies grow substantially larger than our resolution limit, this initial error becomes progressively less important. Using a higher value of f th , max counteracts this sampling effect as it makes the feedback from the first generations of stars that form more efficient. \nThe mean and median values of f th that were used for the feedback from the stars present at z = 0 . 1 in Ref-L100N1504 are 1.06 and 0.70, respectively. For RecalL025N0752 these values are 1.07 and 0.93. Hence, averaged over the entire simulation, the total amount of energy is similar to that expected from supernovae alone. A more detailed discussion of the effects of changing the functional form of f th is presented in Crain et al. (2014). In that work we also present models in which f th is constant or depends on halo mass or dark matter velocity dispersion.', '4.6 Black holes and feedback from AGN': "In our simulations feedback from accreting, supermassive black holes (BHs) quenches star formation in massive galaxies, shapes the gas profiles in the inner parts of their host haloes, and regulates the growth of the BHs. \nModels often make a distinction between 'quasar-' and 'radio-mode' BH feedback (e.g. Croton et al. 2006; Bower et al. 2006; Sijacki et al. 2007), where the former occurs when the BH is accreting efficiently and comes in the form of a hot, nuclear wind, while the radio mode operates when the accretion rate is low compared to the Eddington rate and the energy is injected in the form of relativistic jets. Because cosmological simulations lack the resolution to properly distinguish these two feedback modes and because we want to limit the number of feedback channels to the minimum required to match the observations of interest, we choose to implement only a single mode of AGN feedback with a fixed efficiency. The energy is injected thermally at the location of the BH at a rate that is proportional to the gas accretion rate. Our implementation may therefore be closest to the process referred to as quasar-mode feedback. For OWLS we found that this method led to excellent agreement with both optical and detailed X-ray observations of groups and clusters (McCarthy et al. 2010, 2011; Le Brun et al. 2014). \nOur implementation consists of two parts: i) prescriptions for seeding low-mass galaxies with central BHs and for their growth via gas accretion and merging (we neglect any growth by accretion of stars and dark matter); ii) a prescription for the injection of feedback energy. Our method for the growth of BHs is based on the one introduced by Springel et al. (2005a) and modified by Booth & Schaye (2009) and Rosas-Guevara et al. (2013), while our method for AGN feedback is close to the one described in Booth & Schaye (2009). Below we summarize the main ingredients and discuss the changes to the methods that we made for EAGLE.", '4.6.1 BH seeds': 'The BHs ending up in galactic centres may have originated from the direct collapse of (the inner parts of) metal-free \ndwarf galaxies, from the remnants of very massive, metalfree stars, or from runaway collisions of stars and/or stellar mass BHs (see e.g. Kocsis & Loeb 2013 for a recent review). As none of these processes can be resolved in our simulations, we follow Springel et al. (2005a) and place BH seeds at the centre of every halo with total mass greater than 10 10 M glyph[circledot] /h that does not already contain a BH. For this purpose, we regularly run the friends-of-friends (FoF) finder with linking length 0.2 on the dark matter distribution. This is done at times spaced logarithmically in the expansion factor a such that ∆ a = 0 . 005 a . The gas particle with the highest density is converted into a collisionless BH particle with subgrid BH mass m BH = 10 5 M glyph[circledot] /h . The use of a subgrid BH mass is necessary because the seed BH mass is small compared with the particle mass, at least for our default resolution. Calculations of BH properties such as its accretion rate are functions of m BH , whereas gravitational interactions are computed using the BH particle mass. When the subgrid BH mass exceeds the particle mass, it is allowed to stochastically accrete neighbouring SPH particles such that BH particle and subgrid masses grow in step. \nSince the simulations cannot model the dynamical friction acting on BHs with masses < ∼ m g , we force BHs with mass < 100 m g to migrate towards the position of the minimum of the gravitational potential in the halo. At each time step the BH is moved to the location of the particle that has the lowest gravitational potential of all the neighbouring particles whose velocity relative to the BH is smaller than 0 . 25 c s , where c s is the speed of sound, and whose distance is smaller than three gravitational softening lengths. These two conditions prevent BHs in gas poor haloes from jumping to nearby satellites.', '4.6.2 Gas accretion': 'The rate at which BHs accrete gas depends on the mass of the BH, the local density and temperature, the velocity of the BH relative to the ambient gas, and the angular momentum of the gas with respect to the BH. Specifically, the gas accretion rate, ˙ m accr , is given by the minimum of the Eddington rate, \n˙ m Edd = 4 πGm BH m p glyph[epsilon1] r σ T c , (8) \nand \n˙ m accr = ˙ m Bondi × min ( C -1 visc ( c s /V φ ) 3 , 1 ) , (9) \nwhere ˙ m Bondi is the Bondi-Hoyle (1944) rate for spherically symmetric accretion, \n˙ m Bondi = 4 πG 2 m 2 BH ρ ( c 2 s + v 2 ) 3 / 2 . (10) \nHere m p is the proton mass, σ T the Thomson cross section, c the speed of light, glyph[epsilon1] r = 0 . 1 the radiative efficiency of the accretion disc, and v the relative velocity of the BH and the gas. Finally, V φ is the rotation speed of the gas around the BH computed using equation (16) of Rosas-Guevara et al. (2013) and C visc is a free parameter related to the viscosity of the (subgrid) accretion disc. The mass growth rate of the BH is given by \n˙ m BH = (1 -glyph[epsilon1] r ) ˙ m accr . (11) \nThe factor ( c s /V φ ) 3 /C visc by which the Bondi rate is multiplied in equation (9) is equivalent to the ratio of the Bondi and the viscous time scales (see Rosas-Guevara et al. 2013). We set C visc = 2 π for Ref-L100N1504, but increase the value of C visc by a factor 10 3 for the recalibrated highresolution model, Recal-L025N0752, and by a factor 10 2 for AGNdT9-L050N0752 (see Table 3). Since the critical ratio of V φ /c s above which angular momentum is assumed to reduce the accretion rate scales with C -1 / 3 visc , angular momentum is relatively more important in the recalibrated simulations, delaying the onset of quenching by AGN to larger BH masses. As demonstrated by Rosas-Guevara et al. (2013), the results are only weakly dependent on C visc because the ratio of V φ /c s above which the accretion rate is suppressed, which scales as C -1 / 3 visc , is more important than the actual suppression factor, which scales as C visc . \nOur prescription for gas accretion differs from previous work in two respects. First, the Bondi rate is not multiplied by a large, ad-hoc factor, α . Springel et al. (2005a) used α = 100 while OWLS and Rosas-Guevara et al. 2013 used a density dependent factor that asymptoted to unity below the star formation threshold. Although the use of α can be justified if the simulations underestimate the gas density or overestimate the temperature near the Bondi radius, the correct value cannot be predicted by the simulations. We found that at the resolution of EAGLE, we do not need to boost the Bondi-Hoyle rate for the BH growth to become selfregulated. Hence, we were able to reduce the number of free parameters by eliminating α . Second, we use the heuristic correction of Rosas-Guevara et al. (2013) to account for the fact that the accretion rate will be lower for gas with more angular momentum (because the accretion is generally not spherically symmetric as assumed in the Bondi model, but proceeds through an accretion disc).', '4.6.3 BH mergers': 'BHs are merged if they are separated by a distance that is smaller than both the smoothing kernel of the BH, h BH , and three gravitational softening lengths, and if their relative velocity is smaller than the circular velocity at the distance h BH , v rel < √ Gm BH /h BH , where h BH and m BH are, respectively, the smoothing length and subgrid mass of the most massive BH in the pair. The limit on the allowed relative velocity prevents BHs from merging during the initial stages of galaxy mergers.', '4.6.4 AGN feedback': "AGN feedback is implemented thermally and stochastically, in a manner analogous to energy feedback from star formation. The energy injection rate is glyph[epsilon1] f glyph[epsilon1] r ˙ m accr c 2 , where glyph[epsilon1] f = 0 . 15 is the fraction of the radiated energy that is coupled to the ISM. As was the case for the stellar feedback efficiency, f th , the value of glyph[epsilon1] f must be chosen by calibrating to observations, in this case the normalisation of the relation between BH mass and stellar mass. As demonstrated and explained by Booth & Schaye (2010, see also Booth & Schaye 2009), the value of glyph[epsilon1] f only affects the BH masses, which are inversely proportional to glyph[epsilon1] f . In particular, the outflow rate generated by the AGN and hence also the factor by which \nthe star formation is reduced, are highly insensitive to glyph[epsilon1] f provided it is nonzero. This can be explained by self-regulation: the BH accretion rate adjusts until the rate at which energy is injected is sufficient for outflows to balance inflows. \nWe use the same value for the AGN efficiency as in OWLS, glyph[epsilon1] f = 0 . 15 and glyph[epsilon1] r = 0 . 1, which implies that a fraction glyph[epsilon1] f glyph[epsilon1] r = 0 . 015 of the accreted rest mass energy is returned to the local ISM. As was the case for stellar feedback, the required value will depend on the radiative losses in the ISM, which may depend on the resolution and the precise manner in which the energy is injected. We do not implement a dependence on metallicity, because metals are not expected to dominate the radiative losses at the high temperatures associated with AGN feedback. As shown in Figure 10, a constant value of glyph[epsilon1] f = 0 . 15 yields broad agreement with observations of the relation between BH mass and stellar mass. \nEach BH carries a 'reservoir' of feedback energy, E BH . After each time step ∆ t , we add glyph[epsilon1] f glyph[epsilon1] r ˙ m accr c 2 ∆ t to this reservoir. If the BH has stored sufficient energy to heat at least n heat particles of mass m g , then the BH is allowed to stochastically heat each of its SPH neighbours by increasing their temperature by ∆ T AGN . For each neighbour the heating probability is \nP = E BH ∆ glyph[epsilon1] AGN N ngb 〈 m g 〉 , (12) \nwhere ∆ glyph[epsilon1] AGN is the change in internal energy per unit mass corresponding to the temperature increase, ∆ T AGN (we convert the parameter ∆ T AGN into ∆ glyph[epsilon1] AGN assuming a fully ionised gas with primordial composition), N ngb is the number of gas neighbours of the BH and 〈 m g 〉 is their mean mass. We then reduce E BH by the expectation value for the injected energy. We use n heat = 1 and limit the time step of the BHs such that we expect 8 P < 0 . 3 (see § A1.1). \nThe most important parameter for the AGN feedback is the temperature increase ∆ T AGN . Larger values will make individual feedback events more energetic, generally resulting in smaller radiative losses in the ISM. However, larger values will also make the feedback more intermittent. We set ∆ T AGN = 10 8 . 5 K in the L100N1504 reference model, but use 10 9 K for our recalibrated high-resolution model RecalL025N0752 and model AGNdT9-L050N0752 (see Table 3). These temperatures exceed the value of 10 8 Kused in OWLS and the ∆ T = 10 7 . 5 K that we use for stellar feedback. As can be seen from equation (4), the critical density above which the feedback energy is expected to be radiated away increases with the value of ∆ T . Because the density of the ambient gas around the BH tends to increase with resolution, we found that we need to increase ∆ T when increasing the resolution. Similarly, because the gas density around the BH often reaches values that are much higher than is typical for star-forming gas, we require higher temperature jumps for AGN feedback than for stellar feedback. \nFigure 4. The galaxy stellar mass function at z = 0 . 1 for the EAGLE simulations Ref-L100N1504 (blue), AGNdT9-L050N0752 (red), and Recal-L025N0752 (green-blue). The curves switch from solid to dashed at the high-mass end when there are fewer than 10 objects per (0.2 dex) stellar mass bin. At the low-mass end the curves become dotted when the stellar mass falls below that corresponding to 100 baryonic particles. Data points show measurements with 1 σ error bars from the GAMA survey (open circles; z < 0 . 06; Baldry et al. 2012) and from SDSS (filled circles; z ∼ 0 . 07; Li & White 2009). The high-resolution model Recal-L025N0752 is noisier because of its small box size. The intermediate-resolution models slightly underestimate the galaxy number density at the knee of the mass function and slightly overestimate the abundance at M ∗ ∼ 10 8 . 5 M glyph[circledot] . The galaxy number density agrees with the data to < ∼ 0 . 2 dex. \n<!-- image -->", '5 COMPARISON WITH OBSERVABLES CONSIDERED DURING THE CALIBRATION OF THE FEEDBACK': "In this section we will compare the main EAGLE simulations to z ∼ 0 observations of the GSMF, the related stellar mass - halo mass relation, galaxy sizes, and the relation between BH mass and stellar mass. Since these observables were considered during the calibration of the subgrid models for feedback, we cannot consider the EAGLE results reported in this section to be 'predictions'. However, note that we had no control over the slope of the M BH -M ∗ relation and that galaxy sizes were only used to rule out strongly discrepant models (i.e. models without a density dependence of the energy feedback from star formation).", '5.1 The galaxy stellar mass function': "Figure 4 shows the z = 0 . 1 galaxy stellar mass function (GSMF) from EAGLE. The dark blue curve shows RefL100N1504, the green curve shows the high-resolution simulation Recal-L025N0752, and the red curve shows AGNdT9L050N0752. Recall that AGNdT9-L050N0752 employs a higher heating temperature for AGN feedback than the reference model, which makes the feedback more efficient. While this is unimportant for the GSMF, we will see in § 6.4 that it offers a significant improvement for the intracluster \nmedium. At the high-mass end the curves switch from a solid to a dashed line style where there are fewer than 10 objects per (0.2 dex) stellar mass bin. At the low-mass end the curves become dotted when the stellar mass falls below that corresponding to 100 baryonic particles, where sampling effects associated with the limited resolution become important, as can be seen by comparing the intermediateand high-resolution simulations. \nThe GSMF of the high-resolution simulation RecalL025N0752 is noisier because the box size is too small to provide a representative sample. Note that the main problem is not Poisson noise due to the small number of objects per bin, but the small number of large-scale modes that modulate the local number density of galaxies of various masses. Indeed, Fig. 7 shows that the GSMF of Recal-L025N0752 has the same wiggles as that of Ref-L025N0376, which uses the same box size and, apart from the change in resolution, the same initial conditions. The wiggles that are present for Ref-L025N0376 are absent for model Ref-L100N1504, even though these two simulations use identical resolutions and (subgrid) parameter values. This confirms that the wiggles in the GSMF of Recal-L025N0752 are caused by the small size of its simulation volume. We will therefore focus on the larger volume simulations when comparing the simulated and observed GSMFs. \nThe simulation results are compared with observations from the Galaxy And Mass Assembly (GAMA) survey (Baldry et al. 2012; open circles) and from SDSS (Li & White 2009; filled circles). For the intermediate-resolution simulations the galaxy number densities agree with the observations to < ∼ 0 . 2 dex over the full mass range for which the resolution and box size are adequate, i.e. from 2 × 10 8 M glyph[circledot] to over 10 11 M glyph[circledot] (slightly below 10 11 M glyph[circledot] for Recal-L025N0752). The observed shape of the GSMF is thus reproduced well. \nAt fixed number density, the differences in stellar mass between the simulations and observations are smaller than 0.3 dex for Ref-L100N1504 and AGNdT9-L050N0752. Given that even for a fixed IMF, uncertainties in the stellar evolution models used to infer stellar masses are ∼ 0 . 3 dex (e.g. Conroy et al. 2009; Behroozi et al. 2010; Pforr et al. 2012; Mitchell et al. 2013), there is perhaps little point in trying to improve the agreement between the models and the data further. \nThe subgrid models for energy feedback from star formation and for BH accretion have been calibrated to make the simulated GSMF fit the observed one, so the excellent agreement with the data cannot be considered a successful prediction. However, success was by no means guaranteed given that the computational expense of hydrodynamical simulations severely limits the number of test runs that can be performed and, more importantly, because the freedom built into the model is rather limited. For example, while the mass scale above which AGN feedback becomes dominant is sensitive to the parameter C visc of the subgrid model for BH accretion (see equation 9 in § 4.6.2), the efficiency of the AGN feedback was calibrated to the observed relation between BH mass and stellar mass and does not affect the shape of the GSMF (Booth & Schaye 2009, 2010). \nFigure 5 shows that the level of correspondence between the data and EAGLE is close to that attained for semi-analytic models (left panel) and is unprecedented for large, hydrodynamical simulations (right panel). As can be \nseen from the right panel, even though Oppenheimer et al. (2010), Puchwein & Springel (2013), and Illustris (Vogelsberger et al. 2014a; Genel et al. 2014) all adjusted their subgrid feedback models to try to match the data, the fits to the data are substantially less good than for EAGLE. In particular, their models all produce mass functions that are too steep below the 'knee' of the Schechter function and too shallow for larger masses. It is worth noting that each of these three groups implemented the feedback from star formation kinetically, scaled the wind velocity with the velocity dispersion of the dark matter, determined the dependence of the wind mass loading on the dark matter velocity dispersion by assuming a constant wind energy, and temporarily turned off the hydrodynamical forces on wind particles to allow them to escape the galaxies. This contrasts with EAGLE, where the feedback was implemented thermally rather than kinetically, the feedback energy varied with local gas properties, and the hydrodynamical forces were never turned off. \nHence, contrary to the other models shown, EAGLE's subgrid model does not impose any particular wind velocity or mass loading or any dependence on dark matter or halo properties. The injected energy does depend on the local metallicity and gas density, but the relation between the outflow properties and the energy injected at the star formation site is an outcome of the simulation. Crain et al. (2014) will show that while varying the feedback energy with local gas properties is necessary to obtain reasonable galaxy sizes, the z ∼ 0 GSMF is actually also reproduced by the EAGLE model that injects a constant energy per unit stellar mass (equal to the energy from supernovae) without any calibration. \nWhile the excellent fit to the lowz GSMF is encouraging, the success of the model can only be judged by comparing to a wide range of observables and redshifts, particularly those that were not considered during the calibration. We will consider a diverse selection of observables in § 6 and will investigate their evolution in Furlong et al. (2014) and other future papers.", '5.1.1 Effect of the choice of aperture': "For the simulations we chose to define a galaxy's stellar mass as the sum of the mass of the stars that are part of a gravitationally bound subhalo and that are contained within a 3-D aperture of radius 30 proper kiloparsec (see § 3). Figure 6 shows the effect of the choice of aperture for Ref-L100N1504. For M ∗ < 10 11 M glyph[circledot] the results are insensitive to the aperture, provided it is > ∼ 30 pkpc. However, for M ∗ > 10 11 M glyph[circledot] the aperture does become important, with larger apertures giving larger masses. \nThe same is true for the observations, as can be seen by comparing the data from Li & White (2009) with the re-analysis of SDSS data by Bernardi et al. (2013) (open triangles in Fig. 6). Baldry et al. (2012) and Li & White (2009) are in good agreement, but Bernardi et al. (2013) find a much shallower bright-end slope than previous analyses. For M ∗ > 10 11 M glyph[circledot] Bernardi et al. (2013) attribute substantially more mass to galaxies than Li & White (2009) and Baldry et al. (2012). Part of the difference is due to the assumed mass-to-light ratios (even though all studies assume a Chabrier IMF) and the way in which the background \n<!-- image --> \nFigure 5. Comparisons of the GSMF from EAGLE's Ref-L100N1504 with the semi-analytic models of Gonzalez-Perez et al. (2014), Henriques et al. (2013), and Porter et al. (2014) (left panel) and with the large hydrodynamical simulations of Oppenheimer et al. (2010), Puchwein & Springel (2013), the Illustris simulation (Vogelsberger et al. 2014b, data taken from Genel et al. 2014), and the MassiveBlack-II simulation (Khandai et al. 2014) (right panel). All models are for a Chabrier IMF (Gonzalez-Perez et al. 2014 and Khandai et al. 2014 have been converted from Kennicutt and Salpeter IMFs, respectively). The EAGLE curve is dotted when galaxies contain fewer than 100 stellar particles and dashed when there are fewer than 10 galaxies per stellar mass bin. Except for Oppenheimer et al. (2010), all simulations include AGN feedback. Apart from MassiveBlack-II, all models were calibrated to the data (the Galform semi-analytic model of Gonzalez-Perez et al. 2014 was calibrated to fit the K-band galaxy luminosity function). The agreement with the data is relatively good for both EAGLE and the semi-analytic models, but EAGLE fits the data substantially better than the other hydrodynamical simulations do. \n<!-- image --> \nis subtracted (see e.g. Bernardi et al. 2013 and Kravtsov et al. 2014 for discussion). Most of the difference between Li & White (2009) and Bernardi et al. (2013) can probably be attributed to the way in which a galaxy's light is measured. Li & White (2009) integrate the light within a 2-D aperture of size twice the Petrosian radius, defined to be the radius at which the mean local surface brightness is 0.2 times the mean internal surface brightness. Bernardi et al. (2013) on the other hand, estimate the total amount of light by integrating S'ersic plus exponential profile fits. Hence, the Bernardi et al. (2013) mass function potentially includes intracluster light and the discrepancy between different authors is related to the fact that it is unclear where cD galaxies end. Baldry et al. (2012) integrate single S'ersic fits to the light profiles, which we would expect includes less intracluster light than the S'ersic plus exponential fits of Bernardi et al. (2013) but more than the Petrosian apertures of Li & White (2009). However, Bernardi et al. (2013) find that the high-mass end of the Baldry et al. (2012) mass function is affected by their redshift cut ( z < 0 . 06). \nWe believe the Baldry et al. (2012) and Li & White (2009) data to be the most suitable for comparison to our results, since our definition of a galaxy excludes intracluster light. For Li & White (2009) this is confirmed by our finding that a 3-D aperture of 30 pkpc gives nearly identical results to a 2-D Petrosian cut, as can be seen from Figure 6. \nThus, for masses > 10 11 M glyph[circledot] comparisons of the GSMF with observations would benefit from mimicking the particular way in which the mass is estimated for real data. This would, however, have to be done separately for each survey. For our present purposes this is unnecessary, also because \nour simulation volume is in any case too small to study the GSMF at masses glyph[greatermuch] 10 11 M glyph[circledot] .", '5.1.2 Numerical convergence': "The left panel of Figure 7 compares the GSMFs for model Ref-L025N0376, which has the same resolution as the largest EAGLE volume Ref-L100N1504, and the higher-resolution model Ref-L025N0752. The two Ref-L025 simulations use identical subgrid parameters, but the mass and spatial resolution differ by factors of 8 and 2, respectively. In § 2.2 we termed a comparison between models with identical parameters a 'strong convergence test'. Below 10 9 M glyph[circledot] the mass function is substantially flatter in the high-resolution model. However, at M ∗ ∼ 10 9 M glyph[circledot] its GSMF is up to 0.4 dex higher than for the fiducial resolution, leading to disagreement with the data. The largest discrepancy is the stellar mass corresponding to a number density of ∼ 2 × 10 -2 cMpc -3 , which is about an order of magnitude higher than observed. \nThe thin curves in Figure 7 show the strong convergence test of Vogelsberger et al. (2013) using the galaxy formation model that was also used for Illustris. Clearly, the strong convergence is similarly poor. This is somewhat surprising, since Illustris uses a subgrid model for feedback from star formation that was designed to give good strong convergence. In particular, the parameters of the subgrid wind model vary with the velocity dispersion of the dark matter rather than with the properties of the gas and hydrodynamical interactions between the wind and the ISM are not modelled. \nThat the strong convergence is not particularly good \n<!-- image --> \nFigure 7. Strong (left panel) and weak (right panel) tests of the convergence of the GSMF with numerical resolution. Models L025N0752 have a better mass and spatial resolution than L025N0376 by factors of 8 and 2, respectively. The strong convergence test compares models with identical subgrid parameter values, while the weak convergence test compares the original, intermediate-resolution model Ref-L025N0376 with a high-resolution model Recal-L025N0752 for which the parameters of the subgrid models for feedback from star formation and for gas accretion onto BHs were recalibrated in order to reproduce the observed GSMF. For comparison, the thin curves in the left panel show the strong convergence test for the galaxy formation model used for the Illustris simulation as reported by Vogelsberger et al. (2013). The EAGLE curves are dotted where galaxies contain fewer than 100 stellar particles and dashed where there are fewer than 10 galaxies per stellar mass bin. \n<!-- image --> \nFigure 6. The effect of the choice of aperture on the GSMF. Curves show the z = 0 . 1 GSMF from Ref-L100N1504 for different 3-D apertures: radii of 30, 50, and 100 proper kiloparsec, a 2-D Petrosian aperture, and no aperture at all. In all cases only stellar mass bound to a subhalo is considered. The simulation curves are dotted where galaxies contain fewer than 100 stellar particles and dashed where there are fewer than 10 galaxies per stellar mass bin. Data points indicate observations. The Li & White (2009) and Bernardi et al. (2013) data points are both for SDSS, but use Petrosian magnitudes and integrals of S'ersic plus exponential fits, respectively. The Baldry et al. (2012) data points are for the GAMA survey and use integrals of single S'ersic fits. The choice of aperture is important for M ∗ > 10 11 M glyph[circledot] , both for the simulation and the observations. \n<!-- image --> \nfor EAGLE is unsurprising for the reasons discussed in § 2.2 and § 4.5. For M ∗ < 2 × 10 8 M glyph[circledot] galaxies in Ref-L025N0376 contain fewer than 100 star particles, which is insufficient to properly sample the feedback from star formation in the context of EAGLE's subgrid model. Because the feedback can be modelled down to lower masses in Ref-L025N0752, galaxies with M ∗ ∼ 10 9 M glyph[circledot] have had systematically different histories than galaxies of a similar mass in Ref-L025N0376. In addition, higher resolution enables the gas density distribution to be populated by particles up to higher densities, where our fiducial implementation of thermal feedback becomes inefficient (equation 4 in § 4.5). \nIn § 2.2 we argued that hydrodynamical simulations such as EAGLE should recalibrate the efficiency of the subgrid feedback when the resolution is changed substantially. In general, keeping the subgrid parameters fixed does not imply that the physical model remains unchanged, since the energy, mass or intermittency associated with the feedback events changes. Moreover, the efficiency of the feedback cannot, in any case, be predicted from first principles, even if the convergence were perfect. \nRecal-L025N0752 is our recalibrated high-resolution simulation. As detailed in § 4.5.1 and Table 3, the dependence of the feedback energy per unit stellar mass on the gas density is somewhat different between the different resolutions. However, the mean values of f th , which is equal to the expectation value of the amount of injected energy in units of the energy available from core collapse supernovae, are nearly identical: 1.06 at intermediate resolution (for stars formed at z > 0 . 1 in Ref-L100N1504) and 1.07 at high resolution (for stars formed at z > 0 . 1 in Recal-L025N0752). The asymptotic maximum of f th , reached at low metallicity and low gas density, is 3 in both cases. As detailed in \n§ 4.6.2 and Table 3, Recal-L025N0752 also uses a different value for the parameter that controls the importance of angular momentum in suppressing accretion onto BHs, making the accretion rate more sensitive to the angular momentum of the accreting gas. Without this change, AGN feedback would become important at too low masses. Finally, the high-resolution model uses a higher AGN feedback temperature, ∆ T AGN = 10 9 K rather than 10 8 . 5 K, which helps to suppress the increase in the cooling losses that would otherwise occur due to the higher gas densities that are resolved in the higher resolution model. Without this change the AGN feedback would be insufficiently effective. \nThe right panel of Figure 7 shows a 'weak convergence test', i.e. a comparison of the GSMFs of the calibrated intermediate resolution model Ref-L025N0376 and the recalibrated high-resolution model Recal-L025N0752. The two curves show some of the same bumps and wiggles, because the initial conditions used for the two simulations share the same large-scale modes. In the mass range for which galaxies in the intermediate-resolution model are resolved with more than 100 star particles ( M ∗ > 2 × 10 8 M glyph[circledot] ) the difference in the galaxy number density is smaller than 0.2 dex. We conclude that the weak convergence is good.", '5.2 The relation between stellar mass and halo mass': "The GSMF can be thought of as a convolution between the mass function of dark matter haloes and a function describing the galaxy content of the haloes as a function of their mass. The halo mass function can be predicted accurately when the cosmology is known, but the galaxy content of haloes is very sensitive to the baryonic processes involved in the formation of galaxies. As modelling galaxy formation is EAGLE's primary goal, it is of interest to compare the relation between stellar mass and halo mass in the simulations to the relation inferred from observations. Because the subgrid model for feedback was calibrated to fit the z ∼ 0 GSMF, the relation between stellar and halo mass can hardly be considered a prediction. We therefore discuss this relation in this section, even though we did not calibrate the simulations to fit the relation inferred from observations. \nFigure 8 shows the 'galaxy formation efficiency', ( M ∗ /M 200 ) / (Ω b / Ω m ), for central galaxies as a function of either the mass of their host halo (left panel) or their stellar mass (right panel). Here the halo mass, M 200 , is defined as the total mass contained within the virial radius R 200 , defined to be the radius within which the mean internal density is 200 times the critical density, 3 H 2 / 8 πG , centred on the dark matter particle of the corresponding FoF halo with the minimum gravitational potential (see § 3). If the baryon fraction in the halo were equal to the cosmic average of Ω b / Ω m ≈ 0 . 16, then an efficiency of unity would indicate that the stellar mass accounts for all the halo's share of baryons. We focus on central galaxies because the strong tidal stripping to which satellite haloes are subject obscures the underlying relation between galaxy formation efficiency and halo mass. \nThe simulation clearly shows that galaxy formation is most efficient in haloes with mass ∼ 10 12 M glyph[circledot] , as has been found by many others. In fact, it would be more appropriate to say that this is the mass where galaxy formation is 'least \ninefficient' as the efficiency is only ∼ 10% at the peak. The efficiency is sharply peaked at a stellar mass of ∼ 10 10 . 4 M glyph[circledot] , which corresponds to the onset of the knee in the GSMF (Fig. 4). As is the case for most models of galaxy formation, in EAGLE the sharp reduction at lower masses is mostly due to stellar feedback, while the drop off at higher masses can in part be attributed to inefficient cooling, but is mostly caused by AGN feedback. \nAlthough halo masses can be measured observationally, e.g. from gravitational lensing or satellite kinematics, the errors are still relatively large and it is difficult to disentangle central and satellite galaxies. In Figure 8 we therefore compare with results obtained through the abundance matching technique. In its most basic form abundance matching relates central galaxies to haloes by matching the observed GSMF to the halo mass function predicted from a collisionless simulation, assuming that the stellar masses of galaxies increase monotonically with the masses of their host haloes (e.g. Vale & Ostriker 2004). Modern versions allow for scatter and evolution, and assume that the masses of satellite galaxies are set at the last time they were centrals. \nFigure 8 compares EAGLE to the abundance matching results of Behroozi et al. (2013) and Moster et al. (2013). Note that the abundance matching studies assumed the WMAP7 cosmology, whereas we assume the Planck cosmology. For EAGLE we use the total mass of the halo in the hydrodynamical simulation, whereas abundance matching studies use collisionless simulations. Because feedback processes reduce halo masses, we expect M 200 to be biased high by ∼ 10% for the abundance matching results (e.g. Sawala et al. 2013; Cui et al. 2014; Velliscig et al. 2014; Cusworth et al. 2014; Martizzi et al. 2014; Sawala et al. 2014a; Vogelsberger et al. 2014b), but this effect is small compared to the dynamic range shown 9 . Beyond the peak the results become increasingly sensitive to the aperture used to measure the galaxy's light. For example, Kravtsov et al. (2014) show that using the Bernardi et al. (2013) GSMF as input increases the efficiency by ∼ 0 . 5 dex at M 200 = 10 14 M glyph[circledot] relative to the values of Behroozi et al. (2013) and Moster et al. (2013). However, as discussed in § 5.1.1, our use of a fixed 30 pkpc aperture means that comparison to Bernardi et al. (2013) is inappropriate at the high-mass end. In § 6.4 we will show that a more robust comparison with observations of the total stellar content of massive galaxies reveals good agreement with EAGLE. \nThe convergence with resolution is good and the galaxy formation efficiency in EAGLE is very close to that inferred from abundance matching. This was of course to be expected, given the good convergence and the good agreement with the observations for the GSMF. The peak efficiency is 0.1-0.2 dex lower in EAGLE and is reached at a slightly ( ∼ 0 . 2 dex) higher stellar mass, which is consistent with the fact that EAGLE slightly undershoots the observed GSMF at the knee (see Fig. 4). \n<!-- image --> \nFigure 8. The ratio of the stellar to halo mass, relative to the universal baryon fraction, as a function of halo mass (left panel) and stellar mass (right panel) for central galaxies. The simulation curves are dotted where there are fewer than 100 stellar particles per galaxy. Filled circles show individual objects where there are fewer than 10 objects per bin. The shaded regions show the 1 σ scatter in the simulations. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in Ref-L100N1504 for M ∗ > 10 10 M glyph[circledot] . The EAGLE results agree with results inferred from observations through the technique of abundance matching (grey, solid curves; Moster et al. 2013; Behroozi et al. 2013). The small difference between EAGLE and the abundance matching in the location and height of the peak are consistent with EAGLE's small underestimate of the GSMF around the knee (see Fig. 4). \n<!-- image -->", '5.3 Galaxy sizes': "The parameters of the subgrid model for feedback from star formation and AGN were calibrated to observations of the z ∼ 0 GSMF. The parameter that controls the importance of the angular momentum of the gas in suppressing BH accretion was set to a value for which AGN feedback causes the GSMF to turn over at a mass similar to what is observed. As will be shown in Crain et al. (2014), we found that for EAGLE, calibration of the stellar feedback is actually unnecessary to reproduce the GSMF. Fixing the amount of energy injected per unit stellar mass to that available in the form of core collapse supernovae, i.e. f th = 1, works well, as does the physically motivated dependence on the gas metallicity that we use (eq. 6). However, such models produce galaxies that are far too compact because of excessive radiative losses at high gas densities, and we can show analytically that these spurious cooling losses are caused by our limited numerical resolution (see § 4.5). \nWe consider it reassuring that the breakdown of the subgrid model for feedback from star formation at high density is understood and leads to a clear conflict with observations. On the other hand, the fact that such an unrealistic model has no trouble matching the observed GSMF emphasizes the importance of comparing to a wide range of observables. \nTo counteract the numerical radiative losses occurring at high gas densities, we introduced a dependence of the feedback energy from star formation on the gas density, while keeping both the maximum and mean amounts of energy reasonable (see § 4.5.1). Although we could not afford the computational expense of calibrating the models to fit both the z ∼ 0 GSMF and the size distribution in detail, we did reject models that produced galaxies that were far too \nFigure 9. Galaxy size as a function of stellar mass for galaxies at z = 0 . 1 in proper kiloparsec. The coloured curves show the median, projected half-mass radii for the simulations and the shaded regions show the 1 σ scatter. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in Ref-L100N1504 for M ∗ > 10 10 M glyph[circledot] . The simulation curves are dotted below the resolution limit of 600 stellar particles. Where there are fewer than 10 galaxies per bin, individual objects are shown as filled circles. The models are compared with S'ersic half-light radii from SDSS (Shen et al. 2003; the grey, solid line shows the median and the grey dotted lines indicate 1 σ scatter) and GAMA (Baldry et al. 2012; data points with error bars indicate the 1 σ scatter, shown separately for blue and red galaxies). The simulations and Shen et al. (2003) only include late-type galaxies, i.e. a S'ersic index n s < 2 . 5. \n<!-- image --> \nsmall. As a consequence of this strategy, the z ∼ 0 galaxy sizes cannot be regarded as true predictions. \nFigure 9 plots the median value of the half-mass radius, R 50 , i.e. the radius that encloses 50 per cent of the stellar mass in projection, as a function of galaxy stellar mass. The half-mass radii were determined by fitting S'ersic laws to the projected, azimuthally averaged surface density profiles, as in McCarthy et al. (2012). Following Shen et al. (2003), only galaxies with S'ersic index n s < 2 . 5 are included. For Ref-L1001504, 94% of the galaxies with more than 600 star particles have n s < 2 . 5. \nThe high-resolution Recal-L025N0752 agrees very well with the intermediate-resolution models for M ∗ > 10 9 M glyph[circledot] , which corresponds to about 600 star particles for the intermediate-resolution runs. For this mass the median R 50 is about three and a half times the maximum gravitational softening length (see Table 2). Hence, we take the stellar mass 600 m g as the minimum value for which we can measure half-mass radii. We thus require six times more stellar particles to measure sizes than we need to measure mass. \nThe simulations are compared to data from SDSS (Shen et al. 2003) and GAMA (Baldry et al. 2012). Note that the observations fit surface brightness profiles and provide halflight radii rather than half-mass radii, so the comparison with the models is only fair if the stellar mass-to-light ratio does not vary strongly with radius. As mentioned above, Shen et al. (2003) select galaxies with n s < 2 . 5, as we have done here. Baldry et al. (2012) on the other hand present results separately for red and blue galaxies, finding that the latter are ∼ 0 . 2 dex more extended at fixed stellar mass. Shen et al. (2003) use Petrosian apertures, which we expect to yield results similar to the 3-D apertures of 30 pkpc that we use for the simulations (see § 5.1.1). \nFor M ∗ glyph[greatermuch] 10 8 M glyph[circledot] Shen et al. (2003) agree better with the Baldry et al. (2012) results for red galaxies, even though n s < 2 . 5 should pick out more disky and hence bluer galaxies. The differences between the two data sets are indicative of the level of correspondence between independent measurements of observed galaxy sizes. \nFor 10 9 < M ∗ / M glyph[circledot] < 10 10 the simulation results fall in between those of Baldry et al. (2012) for red and blue galaxies. For M ∗ < 10 9 M glyph[circledot] and M ∗ > 10 10 M glyph[circledot] the simulations agree very well with the sizes of blue and red galaxies, respectively. At 10 11 M glyph[circledot] the red sample of Baldry et al. (2012) gives sizes that are about 0.1-0.2 dex larger than found for both the simulations and the data from Shen et al. (2003). This difference may be due to the fact that Shen et al. (2003) use Petrosian sizes, whereas Baldry et al. (2012) do not. Indeed, if we do not impose any 3-D aperture, then the simulation curve follows the results of the red sample nearly exactly for M ∗ > ∼ 10 11 M glyph[circledot] , while the sizes of lower-mass galaxies remain unchanged (not shown). The agreement with Shen et al. (2003) is excellent: the difference with the simulations is ≤ 0 . 1 dex for all models and for the full range of stellar mass. \nFor M ∗ > 10 10 M glyph[circledot] the scatter in the sizes of the simulated galaxies is similar to the observed dispersion, but at lower masses it appears to be smaller. This could be due to a lack of resolution or some other deficiency in the simulations or halo finder, but it could also be due to observational errors or to the fact that we have ignored variations in the stellar mass-to-light ratio and dust extinction.", '5.4 The relation between BH mass and stellar mass': "Figure 10 shows the mass of the central supermassive BH as a function of the galaxy's stellar mass. The simulation results are compared with the compilation of observations from McConnell & Ma (2013). The observed stellar mass was obtained by extrapolating a fit to the mass profile of the bulge inferred from kinematic data. Because the observed galaxies were selected to be early-type, the bulge likely dominates the stellar mass, at least for the massive systems. \nThe three EAGLE simulations give nearly identical results, indicating good convergence. For M ∗ glyph[lessmuch] 10 10 M glyph[circledot] the BH mass asymptotes to 10 5 M glyph[circledot] /h , which is the mass of the seed BHs that are inserted into FoF haloes with mass > 10 10 M glyph[circledot] /h that do not already contain BHs. As can be seen from Fig. 8, a halo mass of 10 10 M glyph[circledot] corresponds to M ∗ ∼ 10 8 M glyph[circledot] . Above M ∗ ∼ 10 10 M glyph[circledot] the relation between BH mass and stellar mass steepens, but it quickly flattens off to a relation that agrees very well with the observations for M ∗ > ∼ 10 11 M glyph[circledot] . The rapid growth of the BHs between M ∗ = 10 10 and 10 11 M glyph[circledot] coincides with the steepening of the GSMF (compare Fig. 4) and the sharp increase in the fraction of galaxies that are passive (right panel of Fig. 11). This is understandable, as the AGN feedback associated with the rapid BH growth quenches star formation. \nThe agreement with the observations is good, although the observed scatter is larger. In terms of the normalisation of the M BH -M ∗ relation the good agreement is perhaps not a surprise. The normalisation is determined by the assumed efficiency of the AGN feedback, glyph[epsilon1] f glyph[epsilon1] r , i.e. the amount of energy that is injected per unit of accreted mass (e.g. Booth & Schaye 2009, 2010). We used the same value ( glyph[epsilon1] f glyph[epsilon1] r = 0 . 015) as was used for OWLS and cosmo-OWLS, which Booth & Schaye (2009) and Le Brun et al. (2014) found to give agreement with the observed M BH -M ∗ relation. Fig. 10 shows that this efficiency also works for EAGLE, even though the mass resolution of EAGLE is nearly two orders of magnitude better than for OWLS and about 3 orders of magnitude better than for cosmo-OWLS. Note, however, that we used higher AGN heating temperatures than the ∆ T AGN = 10 8 K that was used in OWLS (see Table 3). \nIt would clearly be desirable to extend the comparison to observations to lower masses, but in this regime a more careful analysis is required. This is because of the importance of systematic and selection effects for the observations (e.g. Lauer et al. 2007; Schulze & Wisotzki 2011) and because a bulge-to-disc decomposition would be necessary for the simulations since most low-mass galaxies are disky. The same issues likely also affect the comparison of the scatter.", '6 COMPARISON WITH OTHER OBSERVATIONS': 'In this section we will compare the results of EAGLE to a diverse set of lowz observations of galaxies, galaxy clusters, and the IGM. The results reported in this section were not used to calibrate the subgrid models for feedback and can therefore be considered predictions that can be used as independent consistency checks. During the testing phase, we did look at earlier, more basic versions of some of the plots \nFigure 10. The relation between the mass of the central supermassive black hole and the stellar mass of galaxies. The coloured curves show the median relations for the simulations and the shaded regions show the 1 σ scatter. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in RefL100N1504 for M ∗ > 10 10 M glyph[circledot] . Where there are fewer than 10 objects per bin, individual objects are shown as filled circles. Data points with 1 σ error bars show the compilation of observations from McConnell & Ma (2013). The simulations show the total stellar mass (within a 3-D aperture of 30 pkpc), while observations show bulge masses. However, the observed galaxies were selected to be early-type. The simulations agree with the observations, although the observed scatter is larger. \n<!-- image --> \nshown here, so most of the predictions cannot be considered blind. However, we have not adjusted any model parameters to improve the results shown in this section. \nThere are two exceptions to the above statements. First, we plotted the metal column density distributions ( § 6.5) for the first time after the simulations had finished, so this was a truly blind prediction. Second, the discrepancy between the gas fraction in clusters predicted by Ref-L100N1504 and inferred from X-ray observations that will be discussed in § 6.4 was the motivation for running model AGNdT9-L050N0752. This model represents an educated guess in terms of the modifications to the subgrid AGN feedback, because we could only afford to calibrate models using volumes of 25 cMpc on a side, which are too small to contain clusters of galaxies. \nThe observables presented in this section were not selected because the models reproduce them accurately. They were selected because they give a broad overview of the z ∼ 0 EAGLE universe, because we had the tools to compute them, and because we are currently not preparing separate papers on them. Future papers will present more observables as well as results for higher redshifts.', '6.1 Specific star formation rates and passive fractions': "The left panel of Figure 11 shows the specific star formation rate (SSFR), ˙ M ∗ /M ∗ , of actively star-forming galaxies as a function of stellar mass. Here, galaxies are classified to \nbe star-forming if the SSFR > 0 . 01 Gyr -1 , which is indicated by the horizontal, dashed line in the left panel. The higher and lower diagonal lines in the left panel indicate the SSFR corresponding to 10 star-forming gas particles (assuming a gas density of n H = 10 -1 cm -3 , the star formation threshold that we impose at the metallicity Z = 0 . 002) at intermediate and high resolution, respectively. To the left of these curves resolution effects become important, which we indicate by using dotted lines. In particular, the increase in the SSFR at low stellar mass that is clearly visible for the intermediate resolution simulations is a numerical effect: the curves trace lines of constant numbers of star-forming particles. Compared with the intermediate-resolution models, the high-resolution simulation Recal-L025N0752 predicts slightly higher SSFRs. The difference is 0.2 dex at M ∗ = 10 9 M glyph[circledot] and less than 0.1 dex above 10 10 M glyph[circledot] . \nThe models are compared with observations from Bauer et al. (2013), who measured the SSFRs of ∼ 73 , 000 galaxies from the GAMA survey using spectroscopic H α measurements and dust corrections based on Balmer decrements. The intermediate-resolution simulations agree with the data at the high-mass end, but underpredict the SSFR at low masses, reaching a maximum discrepancy of 0.3 - 0.4 dex at 10 9 M glyph[circledot] . The high-resolution model also underpredicts the SSFR, but the discrepancy is less than 0.2 dex. These differences are comparable to the systematic uncertainty in the data. For example, even for a fixed IMF the systematic uncertainty in the stellar mass, which shifts the data parallel to the diagonal lines, is ∼ 0 . 3 dex (Conroy et al. 2009; Behroozi et al. 2010; Pforr et al. 2012; Mitchell et al. 2013) and the systematic error in the star formation rate, which shifts the data vertically, is likely to be at least as large (e.g. Moustakas et al. 2006). The scatter in the simulations is ∼ 50% smaller than observed, but the observed scatter includes measurement and systematic uncertainties. \nThe right panel of Figure 11 shows the fraction of galaxies that are passive as a function of stellar mass. For the simulations we classify galaxies as passive if they have SSFR < 0 . 01 Gyr -1 , but the observational papers use somewhat different and varying criteria. We leave a more precise comparison for future work, e.g. using colours and accounting for dust extinction for the simulated galaxies. At low stellar masses the curves become dashed where there are, on average, fewer than 10 star-forming gas particles in a galaxy with SSFR = 0 . 01 Gyr -1 . These parts of the curves are unreliable and the upturn of the passive fraction at low mass is thus due to the limited resolution of the simulations. This interpretation is confirmed by the fact that the upturn shifts to eight times lower masses if the particle mass is decreased by a factor of eight, switching from the intermediate resolution Ref-L100N1504 to the high-resolution Recal-L025N0752. \nFor M ∗ glyph[greatermuch] 10 9 M glyph[circledot] , where the simulations are close to converged, both the simulations and the observations show a strong increase of the passive fraction with mass, from ∼ 10 per cent at 10 9 M glyph[circledot] to ∼ 90 per cent at 10 11 . 5 M glyph[circledot] . Relative to the data, the simulation curves are shifted towards higher stellar masses by about 0.3 dex. This difference is similar to the systematic uncertainty in the observed stellar masses. We also find shifts of similar magnitudes if we vary the critical SSFR below which simulated galaxies are classified as passive by a factor of two. \nWe conclude that in the regime where the simulations \nFigure 12 shows the relation between the maximum of the rotation curve and stellar mass for disc galaxies, i.e. a close relative of the Tully-Fisher relation (Tully & Fisher 1977). For the simulations we classify galaxies with S'ersic index n s < 2 . 5 as late-type, as we did when considering galaxy sizes ( § 5.3). We use circular velocities ( v c = √ GM ( < r ) /r ) rather than trying to estimate rotation velocities, since the latter become noisy for galaxies that are not resolved with many particles. \n<!-- image --> \nFigure 11. Left panel: Specific star formation rate, ˙ M ∗ /M ∗ , for actively star-forming galaxies as a function of stellar mass at z = 0 . 1. Galaxies are classified as star-forming if their SSFR > 10 -2 Gyr -1 , indicated by the horizontal, dashed line. The coloured curves show simulation medians and the shaded regions show the 1 σ scatter. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in Ref-L100N1504 for M ∗ > 10 10 M glyph[circledot] , all at z = 0 . 1. The higher and lower diagonal lines correspond to 10 starforming gas particles (assuming n H = 0 . 1 cm -3 ) at intermediate and high resolution, respectively. To the left of these lines the curves are dotted to indicate that the results are unreliable due to sampling effects. In particular, the sharp upturns at the lowest masses trace lines of fixed numbers of star-forming gas particles. The data points show observations from GAMA (0 . 05 < z < 0 . 32; Bauer et al. 2013) with the error bars indicating the 1 σ scatter. Right panel: Fraction of passive galaxies, i.e. galaxies with SSFR < 10 -2 Gyr -1 , as a function of stellar mass at z = 0 . 1. In both panels the simulation curves are dotted where they are unreliable due to poor resolution ( < 10 star-forming gas particles) and dashed were there are < 10 objects per bin. Data points show observations from Bauer et al. (2013) and Moustakas et al. (2013). \n<!-- image --> \ncan be trusted, the predicted SSFRs and passive fractions are slightly lower than the observations but agree with them to within the expected (systematic) errors.", '6.2 Tully-Fisher relation': 'The data points with 1 σ error bars correspond to the set of homogenised observations of disc galaxies compiled by Avila-Reese et al. (2008) and the grey line indicates the median. The stellar masses have been reduced by 0.15 dex, which is necessary to convert to a Chabrier IMF (AvilaReese, private communication). In addition, following McCarthy et al. (2012) and Dutton et al. (2011), we applied a small correction to the stellar masses using the expression given in the appendix of Li & White (2009) to improve the consistency with those derived from more accurate five-band SDSS data. \nAll simulations track each other very closely, implying excellent numerical convergence. The simulations are in excellent agreement with the data. Over the mass range 10 9 < ∼ M ∗ / M glyph[circledot] < 10 11 the difference in velocity between the \nmodels and the data compiled by Avila-Reese et al. (2008) is less than 0.03 dex, which is smaller than the 0.1 dex 1 σ error on the fit to the observations. At higher masses, which are only probed by Ref-L100N1504, the difference with the observations increases, reaching 0.12 dex at M ∗ = 10 11 . 3 M glyph[circledot] . However, most of these very massive galaxies do not look disky and would probably not be selected by Avila-Reese et al. (2008). \nNote that we have not attempted to analyse the simulations and the data in the same manner, because this would go beyond the scope of the current study. As mentioned above, we use maximum circular velocities, whereas the observations are based on maximum gas rotation velocities, which may show more scatter if the orbits are not all circular. In addition, the observations probe only the inner parts of the halo, whereas we consider the entire halo. McCarthy et al. (2012) found that for the GIMIC simulations the maximum circular velocities are nearly always reached within two effective radii for M ∗ > ∼ 10 9 . 5 M glyph[circledot] , and should therefore be easily accessible to the observations, but it is possible that for smaller masses the observations underestimate the maximum rotation velocity.', '6.3 Mass-metallicity relations': "The left panel of Figure 13 shows the metallicity of the ISM, which we take to be star-forming gas for the simulations, as a function of stellar mass. For both the intermediate- and the high-resolution models the gas metallicity increases with stellar mass and flattens off for M ∗ > 10 10 M glyph[circledot] . However, the \nFigure 12. The relation between the maximum of the rotation curve and stellar mass, i.e. an analogue of the Tully-Fisher relation, for late-type galaxies at z = 0 . 1. The coloured curves show the medians for the simulations. The curves are dotted below the resolution limit of 100 stellar particles. Where there are fewer than 10 galaxies per bin, individual objects are shown as filled circles. The shaded regions show the 1 σ scatter in the simulations. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in Ref-L100N1504 for M ∗ > 10 10 M glyph[circledot] . The simulation results only include galaxies with S'ersic index n s < 2 . 5 and are based on maximum circular velocities. The data points with 1 σ error bars correspond to the set of homogenised observations of disc galaxies compiled by Avila-Reese et al. (2008) and the grey line indicates the median. The model predictions are in remarkable agreement with the data. \n<!-- image --> \nhigh-resolution simulation, Recal-L025N0752, predicts systematically lower metallicities. For M ∗ > ∼ 10 10 M glyph[circledot] the difference is less than 0.15 dex, but it increases with decreasing mass, reaching a maximum of 0.4 dex at M ∗ ∼ 10 8 . 5 M glyph[circledot] . Because there is no clear mass below which the two resolutions diverge, it is unclear where to put the resolution limit and we therefore have not dotted any part of the curves. \nInterestingly, model Ref-L025N0752 (not shown) yields a mass-metallicity relation that agrees better with RefL100N1504 than the prediction of Recal-L025N0752 does, particularly for M ∗ < 10 9 M glyph[circledot] . The high-resolution run again predicts lower metallicities than the intermediate-resolution version, but the maximum difference is smaller than 0.2 dex. For M ∗ < 10 7 . 5 M glyph[circledot] the metallicity is actually lower at intermediate resolution than at high resolution. Hence, for the mass-metallicity relation the strong convergence is considerably better than one might infer from the comparison of RefL025N0752 and Recal-L025N0752. Recall that the latter was recalibrated to fit the GSMF, which meant the efficiency of feedback had to be increased relative to the reference model, particularly at M ∗ ∼ 10 9 M glyph[circledot] (see Fig. 7). Apparently, the stronger outflows in Recal-L025N0752 reduce the metallicity of the ISM. Thus, the 'strong convergence' is better than the 'weak convergence'. This is possible because in this case the weak convergence test compares simulations that were each calibrated to fit the GSMF, not the mass-metallicity relation. \nThe two sets of observations that are shown in the left panel of Figure 13 are both derived from SDSS data. Tremonti et al. (2004) estimated the metallicity statistically based on theoretical model fits to various strong emission lines, while Zahid et al. (2014) derived metallicities using the R23 strong line method as calibrated by Kobulnicky & Kewley (2004). The two studies do not agree with each other. In particular, while Tremonti et al. (2004) and Zahid et al. (2014) agree at M ∗ ∼ 10 11 M glyph[circledot] , the former find a steeper relation than the latter, resulting in metallicities that are about 0.2 dex lower for 10 9 -10 10 M glyph[circledot] . The difference is due to the uncertain calibration of the emission-line diagnostics. In fact, as shown by Kewley & Ellison (2008), the systematic uncertainty is even larger than suggested by this plot. For example, the empirical calibration of Pilyugin & Thuan (2005) yields a metallicity that is 0.75 dex lower than that of Tremonti et al. (2004) at 10 11 M glyph[circledot] and an almost flat relation with stellar mass, dropping by only 0.2 dex when the stellar mass decreases to 10 9 M glyph[circledot] . Besides the calibration issues, the gas phase abundance likely underestimates the total metallicity of the ISM because a non-negligible fraction of the metals may condense onto dust-grains (e.g. Dwek 1998; Mattsson & Andersen 2012). Finally, the systematic uncertainty in the stellar mass, for a fixed IMF, is about 0.3 dex (e.g. Conroy et al. 2009). \nThe metallicities predicted by the simulations are also subject to significant systematic uncertainties unrelated to the galaxy formation physics. Even for a fixed IMF, the nucleosynthetic yields are uncertain at the factor of two level (e.g. Wiersma et al. 2009b). However, we choose not to simply re-scale the simulation metallicities within this uncertainty because that would make them inconsistent with the radiative cooling rates used during the simulation. \nGiven the large systematic uncertainties in both the normalisation and the shape of the observed massmetallicity relation, and the systematic uncertainties in the yields adopted in the simulations, care needs to be taken when comparing the models and the data. We will nevertheless proceed to make such a comparison. \nThe median mass-metallicity relations predicted by the intermediate-resolution simulations agree with Zahid et al. (2014) to better than 0.2 dex at all masses and to better than 0.1 dex for M ∗ > 10 9 . 5 M glyph[circledot] , but the observed relation is steeper at lower masses. The predicted scatter is larger than observed by Tremonti et al. (2004), particularly for the highest masses. The scatter in the gas metallicity of these massive objects is large in the simulations because they typically contain very few star-forming gas particles. This causes strong sampling effects and large variations in time following AGN outbursts. \nThe median metallicity predicted by the high-resolution model Recal-L025N0752 matches Tremonti et al. (2004) to better than 0.2 dex over the full mass range covered by both the simulation and the observations (10 8 . 5 < M ∗ / M glyph[circledot] < 10 11 ) and to better than 0.1 dex for M ∗ > 10 9 . 2 M glyph[circledot] . Apparently, the increase in the efficiency of energy feedback from star formation that is required to make the GSMF fit the observations (and which was implemented by changing the density dependence of the efficiency, see § 4.5.1), simultaneously decreases the metallicity of the ISM of low-mass galaxies to the values observed by Tremonti et al. (2004). \nThe predicted relations between stellar metallicity and \nFigure 13. The metallicity of the ISM (left panel) and of stars (right panel) as a function of stellar mass. The conversion from the absolute oxygen abundances shown along the left y -axis in the left panel to the metallicities relative to solar shown along the right y -axis assumes 12 + log 10 (O / H) glyph[circledot] = 8 . 69 (Allende Prieto et al. 2001). Note that the two panels show the same range in metallicity. Curves show the median relations for the simulations at z = 0 . 1, where we take ISM to be all star-forming gas, and the shaded regions show the 1 σ scatter. For clarity we only show the scatter in Recal-L025N0752 for M ∗ < 10 10 M glyph[circledot] and in Ref-L100N1504 for M ∗ > 10 10 M glyph[circledot] . Where there are fewer than 10 galaxies per bin, individual objects are shown as filled circles. The high-mass galaxies with very low gas metallicities correspond to objects that are nearly devoid of gas, leading to sampling problems in the simulations. The data points show observations reported by Zahid et al. (2014) and Tremonti et al. (2004) for gas, and by Gallazzi et al. (2005) and Kirby et al. (2013) for stars (converted to solar abundances assuming Z glyph[circledot] = 0 . 0127 and 12 + log 10 (Fe / H) glyph[circledot] = 7 . 52, respectively). The dashed line in the right panel shows the best-fit relation given by Kirby et al. (2013), which includes also lower-mass galaxies than shown here. \n<!-- image --> \nmass are shown in the right panel of Figure 13 and compared with observations from SDSS from Gallazzi et al. (2005) and for dwarf galaxies from Kirby et al. (2013). The trends and differences largely parallel those seen for the gasphase abundances in the left panel. For M ∗ > ∼ 10 9 M glyph[circledot] simulation Recal-L025N0752 is relatively close to the data, but at lower masses all models predict higher metallicities than observed by Kirby et al. (2013). As was the case for the gas metallicity, the (strong) convergence is actually much better than suggested by this figure. For M ∗ > 10 7 . 5 M glyph[circledot] simulation Ref-L025N0752 (not shown) predicts a stellar metallicity that is lower, but within 0.1 dex of the metallicity predicted by Ref-L100N1504. Model AGNdT9-L050N0752 predicts slightly higher metallicities than Ref-L100N1504 for M glyph[greatermuch] 10 10 M glyph[circledot] , which agrees better with the data. \nThe main difference between the conclusions that can be drawn from the gas and stellar metallicities concerns the scatter. While the scatter in the gas phase abundances was overestimated in the simulations, the scatter in the stellar abundances appears to be strongly underestimated. However, it would be surprising for the scatter in the observed stellar metallicity to be so much larger than the observed scatter in the gas phase metallicity, which suggests that the scatter in the observed stellar metallicities may be dominated by errors. Indeed, while the mean relation from the CALIFA integral field survey is close to that of Gallazzi et al. (2005), the scatter is about a factor of two smaller (Gonz'alez Delgado et al. 2014).", '6.4 X-ray observations of the intracluster medium': "In this section we will consider some parameters that are commonly measured from X-ray observations of the intragroup and intracluster gas. The comparison to observations is more like-for-like than in previous sections, because all simulation results are derived by applying observational analysis techniques to virtual X-ray observations of the simulations. Simulation Recal-L025N0752 is not considered here because the simulation box is too small to produce clusters of galaxies. \nThe methods used to generate the plots are identical to those employed for cosmo-OWLS in Le Brun et al. (2014) and we refer the reader to § 2.2 of that paper for details. Briefly, gas density, temperature and metallicity profiles are determined by fitting single temperature, single metallicity 'Astrophysical Plasma Emission Code' (APEC) (Smith et al. 2001) models to synthetic Chandra X-ray spectra in three-dimensional radial bins centred on the minimum of the gravitational potential in the halo. Mass profiles are obtained by fitting the functions proposed by Vikhlinin et al. (2006) to the density and temperature profiles and assuming hydrostatic equilibrium. We then determine the radius within which the mean internal density equals 500 times the critical density, R 500 , hse , and the corresponding spherical overdensity mass, M 500 , hse . We will use the subscript 'hse' to indicate that the quantity has been inferred from virtual observations under the assumption of hydrostatic equilibrium (which holds only approximately, see Le Brun et al. 2014 and references therein). Mean X-ray temperatures and elemental abundances within R 500 , hse are determined by fitting APEC models to a single radial bin. We include all z = 0 \nFigure 14. I -Band luminosity within R 500 , hse as a function of M 500 , hse at z = 0. Black data points represent observations of Sanderson et al. (2013), Gonzalez et al. (2013), and Kravtsov et al. (2014), and the dashed black line represents the SDSS image stacking results of Budzynski et al. (2014). Where necessary, observations were converted to the I -band following Le Brun et al. (2014). The observational studies and the simulations both include contributions from satellites and diffuse intracluster light (ICL). The simulations agree well with the data. \n<!-- image --> \nhaloes with FoF mass > 10 12 . 5 M glyph[circledot] but plot only results for haloes with M 500 , hse > 10 13 M glyph[circledot] for which the correspondence between M 500 and M 500 , hse is good for most objects, except that M 500 , hse is systematically biased low by ∼ 20 per cent (see Fig. B1 of Le Brun et al. 2014). \nFigure 14 shows the (Cousins) I -band luminosity within R 500 , hse as a function of M 500 , hse . Each point corresponds to a single simulated or observed object. The predicted luminosity-mass relation matches the observations very well. As the I -band luminosity is a proxy for stellar mass and the simulations were calibrated to the observed GSMF, this may at first sight not be surprising. However, the high-mass tail of the GSMF was not calibrated to any observations, because the test simulations were too small to contain such rare objects. Moreover, here we plot the total luminosity within R 500 , a radius that exceeds the aperture used for the GSMF by more than an order of magnitude. Hence, the results shown here include contributions from satellites and the intracluster light, both for the observations and simulations. \nFigure 15 shows the gas mass fraction, M gas , 500 , hse /M 500 , hse as a function of mass M 500 , hse . Because the gas mass is derived from the (virtual) X-ray data, it only correctly accounts for gas that has a temperature similar to that of the gas that dominates the X-ray emission. For the reference model the gas mass inferred from X-ray observations, under the assumption of hydrostatic equilibrium, is about 0.2 dex higher than observed, except perhaps for the two most massive objects. \nFigure 15. The z = 0 gas mass fraction within R 500 , hse as a function of M 500 , hse . All quantities are inferred from (virtual) X-ray observations. Black data points represent observations of Vikhlinin et al. (2006), Maughan et al. (2008), Allen et al. (2008), Pratt et al. (2009), Sun et al. (2009), and Lin et al. (2012). The reference model overpredicts the gas fractions. Model AGNdT9L050N0752, which employs a higher heating temperature for AGN feedback, performs well for groups of galaxies, but may also overpredict the gas fraction in higher mass ( > ∼ 10 14 M glyph[circledot] ) clusters. \n<!-- image --> \nLe Brun et al. (2014) have shown that the gas fraction is particularly sensitive to the temperature to which the AGN heat the surrounding gas in our subgrid prescription for AGN feedback. In particular, higher heating temperatures, which correspond to more energetic but less frequent bursts, eject the gas more effectively, yielding lower gas fractions. This was the motivation for running model AGNdT9L050N0752, which uses a heating temperature ∆ T AGN of 10 9 K, compared with 10 8 . 5 K for the reference model. Before running this model, we used a 25 cMpc version to (approximately) recalibrate the BH accretion model so as to maintain the good match with the GSMF, in particular the location of the knee. We could, however, not afford to run multiple 50 cMpc models and could therefore not calibrate to observations of groups of galaxies. \nAs can be seen from Figure 15, contrary to model RefL100N1504, model AGNdT9-L050N0752 does appear to reproduce the observations of group gas fractions. That is, for M 500 , hse < 10 13 . 5 M glyph[circledot] the simulation points agree with an extrapolation of the observations for high-mass systems. There is a strong hint that the gas fraction may again become too high for more massive clusters, although with only 1 object with M 500 , hse > 10 13 . 5 M glyph[circledot] it is hard to judge the significance of this deviation. \nLe Brun et al. (2014) found that the cosmo-OWLS simulations, which use 2 × 1024 3 particles in 400 h -1 cMpc volumes, reproduce these and many other observations of groups and clusters over the full mass range of 10 13 -10 15 M glyph[circledot] for ∆ T AGN = 10 8 K. This may seem surprising \nFigure 16. The soft (0.5-2.0 keV) X-ray luminosity as a function of the X-ray temperature at z = 0. Only points for which M 500 , hse > 10 13 M glyph[circledot] are shown. Black data points represent observations of Horner (2001), Osmond & Ponman (2004), Pratt et al. (2009), and Mehrtens & et al. (2012). The reference model predicts too high X-ray luminosities for clusters above 1 keV, but simulation AGNdT9-L050N0752 is consistent with the data. \n<!-- image --> \ngiven that EAGLE requires higher values of ∆ T AGN . Note, however, that because the particle mass in cosmo-OWLS is more than 3 orders of magnitudes larger than for EAGLE, the energy in individual AGN feedback events in cosmo-OWLS is still much larger than that in AGNdT9L050N0752. \nFigure 16 shows the X-ray luminosity in the 0.5-2.0 keV band as a function of the temperature measured from the (virtual) X-ray data. For the reference model the agreement with the observations is reasonably good at low temperatures (the lack of simulated points with L glyph[lessmuch] 10 42 erg s -1 is due to the fact that we only selected systems with M 500 , hse > 10 13 M glyph[circledot] ), but the predicted luminosity is about a factor of three too high above 1 keV. Model AGNdT9L050N0752 appears to match the data well, but more objects with k B T > 1 keV are needed to better assess the degree of correspondence.", '6.5 Column density distributions of intergalactic metals': 'The galactic outflows that we invoke to reproduce observations of galaxies also disperse heavy elements into the IGM. Furthermore, the winds shock-heat the gas, which may, in turn, change its ionisation balance. Hence, it is interesting to compare the predicted distribution of intergalactic metal ions to the observations. This is a strong test for the model, since the subgrid feedback was only calibrated to match the stellar properties of galaxies. \nFigure 17 compares the predicted column density distribution functions (CDDFs) of C iv (left panel) and O vi \n(right panel) with measurements derived from quasar absorption line observations, mainly from the Hubble Space Telescope (HST). Note that this prediction was completely blind. \nThe CDDF is conventionally defined as the number of absorbers per unit column density, N , and per unit absorption distance, dX . The number of absorbers per unit absorption distance is obtained from the quantity that is actually observed, the number of absorbers per unit redshift, via dX = dz ( H 0 /H ( z ))(1+ z ) 2 . The redshift ranges of the observations vary and are indicated in the legend. All observations are for z < 1 and most for much lower redshift. For clarity we only show the simulation results for our z = 0 . 27 snapshots. However, limiting the comparison to z = 0 . 27 does not affect our conclusions because the evolution is weak. \nFor the simulations we compute ion fractions for each gas particle using cloudy photoionisation models, assuming the gas is in ionisation equilibrium and exposed to the Haardt & Madau (2001) model for the UV/X-ray background from galaxies and quasars. We then obtain the CDDF by projecting the simulation cube onto a 2-D grid and applying SPH interpolation to compute the ion column density in each cell. We use a grid cell size of 10 ckpc, which is sufficiently small to obtain convergence, and have verified that projection effects are negligible by comparing results obtained from simulations using different box sizes. \nObservationally, the CDDF is obtained by decomposing the identified absorption features into Voigt profiles and grouping those into systems using criteria that differ between observers and that are not always well defined. We intend to mimic the observational procedures more closely in future work. From Figure 17 it is clear that the differences between different sets of observations exceed the reported statistical uncertainties, suggesting the presence of significant systematic errors. Particularly for O vi , the analysis of COS spectra by Danforth et al. (2014) yields systematically more absorbers than the earlier analyses of STIS/FUSE/GHRS data by Danforth & Shull (2008), Thom & Chen (2008), and Tripp et al. (2008). \nAs discussed in § 6.3, even for a fixed IMF the nucleosynthetic yields are uncertain at the factor of two level (e.g. Wiersma et al. 2009b). This suggests that we are free to rescale the metal column densities, i.e. to shift the curves in Figure 17 horizontally by up to 0.3 dex. However, doing so would break the self-consistency of the simulations as the metal abundances determine the cooling rates. \nThe simulation predictions generally agree well with the data, falling in between the different sets of observations, both for C iv and O vi . The simulations appear to produce too few ultra-strong absorbers, i.e. systems with column densities ∼ 10 15 cm -2 . However, the frequency of these extremely rare systems is particularly sensitive to systematics and hence requires a more careful comparison. \nFor O vi the difference between Ref-L100N1504 and Recal-L025N0752 is substantial for N O VI ∼ 10 14 cm -2 with the high-resolution model yielding up to a factor of 3 more absorbers. However, this does not lead to any disagreement with the data as all simulations fall in between the different sets of observations. Recall that in low-mass galaxies feedback from star formation is more effective in the recalibrated, high-resolution model Recal-L025N0752 than in the reference model. It is interesting that while this boost in \n<!-- image --> \nFigure 17. The column density distribution functions of C iv (left panel) and O vi (right panel). The coloured curves show the simulation predictions and the data points with 1 σ error bars indicate observations taken with STIS/FUSE (Danforth & Shull 2008), COS (Danforth et al. 2014), STIS/FUSE/GHRS (Cooksey et al. 2010), and STIS (Thom & Chen 2008; Tripp et al. 2008). The redshift ranges of the observations vary and are indicated in the legend. For clarity we only show the simulation results for z = 0 . 27. The predictions are consistent with the data. \n<!-- image --> \nthe feedback efficiency decreases the metallicity of the ISM (Fig. 13), it boosts the abundance of metal ions in the IGM. It is tempting to conclude that the more effective feedback transports more metals from galaxies into the IGM. However, whether this is the case is not clear from the results presented here due to the importance of ionisation corrections. \nIn future work we will compare with high-redshift data and with absorption line observations of the gas around galaxies of known mass. For now, we are encouraged by the fact that a model that was calibrated to the GSMF and galaxy sizes, also yields good agreement with observations of intergalactic metals.', '7 SUMMARY AND DISCUSSION': "We have introduced the EAGLE project, where EAGLE stands for 'Evolution and Assembly of GaLaxies and their Environments'. EAGLE consists of a suite of large, hydrodynamical cosmological simulations. In this introductory paper we have focused on a set of simulations for which the subgrid parameters for feedback were calibrated to match the observed z ∼ 0 galaxy stellar mass function (GSMF), subject to the constraint that the galaxy sizes must also be reasonable. Crain et al. (2014) will present models in which the subgrid physics is varied. \nThe largest EAGLE simulation, Ref-L100N1504, uses nearly 7 billion (2 × 1504 3 ) particles in a 100 cMpc box. This corresponds to an initial baryonic particle mass of 1 . 8 × 10 6 M glyph[circledot] and a force resolution of 0.7 proper kpc (smaller at high redshift), which we refer to as 'intermediate resolution'. The resolution was chosen to marginally resolve the Jeans scales in the warm ( T ∼ 10 4 K) ISM. The highresolution model, Recal-L025N0752, has eight times better \nmass resolution and two times better spatial resolution, thus resolving a galaxy like the Milky Way with ∼ 10 6 particles. \nThe simulations were run with the code gadget 3, but with a modified implementation of SPH, the time stepping, and the subgrid models. The simulations include subgrid prescriptions for (element-by-element) radiative cooling, star formation, stellar evolution and mass loss, energy feedback from star formation, the growth of supermassive BHs, and AGN feedback. The prescription for star formation accounts for the observation that stars form from molecular clouds and that the H i -H 2 transition depends on metallicity. The subgrid model for accretion onto BHs accounts for the fact that angular momentum suppresses the accretion rate. \nThe most critical parts of the model are the implementations of energy feedback from star formation and AGN. We argued that present-day simulations of representative volumes cannot predict the efficiency of the feedback processes from first principles because of their reliance on subgrid models, because of spurious radiative losses due to the limited resolution, and because they lack the resolution and do not include all the physics necessary to model the structure of the ISM. \nWe discussed some of the implications of the inability to predict the efficiency of the feedback from first principles. We argued that current cosmological simulations can predict neither BH nor stellar masses, which implies that the subgrid models for feedback need to be calibrated to observations. Another consequence is that it is difficult to distinguish different physical feedback mechanisms that operate nearly simultaneously, such as winds driven by supernovae and radiation pressure. Furthermore, unless one can demonstrate that the model does not suffer from overcooling due to limited numerical resolution, one cannot conclude that there is a need for a new, physical feedback process just because the implemented feedback is insufficiently effective. \nBecause the spurious radiative losses depend on the resolution, one may have to recalibrate when the resolution is changed. We termed this 'weak convergence' as opposed to the 'strong convergence' that corresponds to the same physical model giving consistent results at different resolutions. However, we argued that most subgrid models for feedback effectively change with resolution even if the subgrid parameters are kept constant. \nThe quest for strong convergence of simulations that lack the resolution to model the ISM has led to significant sacrifices, which generally involve disabling aspects of the hydrodynamics during feedback. Examples include temporarily turning off radiative cooling, temporarily turning off hydrodynamical forces, and making the feedback efficiency dependent on dark matter velocity dispersion rather than on local properties of the gas. However, until the cooling losses can be predicted, even fully converged simulations will be unable to predict stellar and BH masses from first principles. We therefore prefer to minimize the sacrifices and to opt for weak convergence. Nevertheless, we demonstrated that the strong convergence of our model is reasonably good (Fig. 7). \nMotivated by the above considerations, we chose to keep our subgrid models for feedback as simple as possible. We employ only one type of stellar feedback and hence we do not distinguish between stellar winds, radiation pressure, and core collapse supernovae. Similarly, we include only one type of AGN feedback and therefore do not implement separate 'quasar' and 'radio modes'. We find that a more complex approach is not required to match observational data. \nWe implement both feedback from star formation and AGN thermally using the stochastic prescription of Dalla Vecchia & Schaye (2012). By injecting the energy stochastically rather than at every time step, we can specify both the temperature jump of the heated gas and the expectation value for the amount of energy that is injected. This enables us to better mimic the physical conditions associated with observed feedback processes, in particular the high heating temperatures that suppress the initial radiative losses, than would otherwise be possible given the limited resolution of the simulations. The velocities and mass loading factors of galactic winds are thus not imposed, but are an outcome of the simulation. \nThe temperature jump associated with feedback events is chosen to balance the need to minimize both the initial, radiative losses (which are largely numerical) and the time between feedback events (to allow for self-regulation). The probability of heating events then needs to be calibrated by comparing the simulation results for some observable to real data. The subgrid efficiency of the AGN feedback, i.e. the expectation value for the amount of energy that is injected into the ISM per unit of accreted gas mass, is constant and was chosen to match the normalisation of the observed relation between the masses of galaxies and their central supermassive BHs. This parameter is, however, unimportant for observables other than the masses of BHs. The subgrid efficiency of the feedback from star formation, f th , i.e. the expectation value for the amount of energy that is injected into the ISM in units of the energy available from core collapse supernovae, was chosen to reproduce the observed GSMF for M ∗ < 10 10 . 5 M glyph[circledot] , i.e. below the knee of the Schechter function. Finally, the value of the parameter that controls the sensitivity of the BH accretion rate to the angular mo- \nntum of the surrounding gas was adjusted to make the mass function turn over at the onset of the exponential drop of the observed GSMF. \nWe made f th a function of both metallicity and density. We use a physically motivated metallicity dependence with f th dropping when the metallicity is increased from values glyph[lessmuch] 0 . 1 Z glyph[circledot] to glyph[greatermuch] 0 . 1 Z glyph[circledot] . This reduction in the efficiency is meant to capture the increase in radiative losses that is expected when metal-line cooling becomes important, which happens for Z > 0 . 1 Z glyph[circledot] at the temperatures relevant for gas shockheated in galactic winds (e.g. Wiersma et al. 2009a). \nWhile a constant value of f th = 1, or a pure metallicity dependence, each give an excellent fit to the GSMF, they result in galaxies that are far too compact (Crain et al. 2014). This happens because, at the resolution of EAGLE, the stochastic implementation for stellar feedback is still subject to numerical radiative losses at high gas densities, as we demonstrated analytically. To compensate for these spurious losses, we increase f th at high gas densities. However, f th never exceeds 3 and the mean value is smaller than 1.1. \nWe compared EAGLE to a diverse set of observations of the low-redshift Universe, carefully distinguishing between observations that were considered during the calibration (the GSMF and thus also the directly related M ∗ -M 200 relation, galaxy sizes, and the M BH -M ∗ relation) and those that were not. We came to the following conclusions: \n- · The observed GSMF is reproduced over the range 10 8 < M ∗ / M glyph[circledot] < ∼ 10 11 . At fixed mass, the difference in number density relative to the data is < ∼ 0 . 2 dex. At fixed number density, the difference in mass is smaller than 0.3 dex (Fig. 4). Even for a fixed IMF, this discrepancy is comparable to the systematic uncertainty in the observed masses due to stellar evolution alone. This level of agreement with the data is close to that obtained by semi-analytic models and is unprecedented for hydrodynamical simulations (Fig. 5).\n- · Three-dimensional apertures of 30 proper kpc, which we used throughout the paper, give results close to the Petrosian masses that are often used for observations, e.g. by SDSS. For M ∗ > 10 11 M glyph[circledot] larger apertures yield higher masses (Fig. 6).\n- · The stellar mass - halo mass relation for central galaxies is close to that inferred from abundance matching. The efficiency of galaxy formation, M ∗ /M 200 , peaks at the halo mass M 200 ∼ 10 12 M glyph[circledot] and at the stellar mass M ∗ ∼ 10 10 . 4 M glyph[circledot] (Fig. 8).\n- · Disc galaxy sizes are well matched to the observations. Over the full range of stellar mass, 10 8 < M ∗ / M glyph[circledot] < 10 11 . 5 , the median stellar half-mass radii of late-type galaxies agree with the observed half-light radii to within 0.1 dex (Fig. 9).\n- · The median relation between BH mass and stellar mass agrees with the observations, but the scatter in the model is smaller than observed. The simulations predict that galaxies with total stellar masses of 10 9 -10 10 M glyph[circledot] typically host BHs with masses that fall below the extrapolation of the highmass power-law relation (Fig. 10).\n- · The predicted relation between the median specific star formation rate ( ˙ M ∗ /M ∗ ; SSFR) and stellar mass for starforming galaxies, i.e. the 'main sequence of star formation', agrees with the observations to within 0.2 dex over the observed range of 10 9 < M ∗ / M glyph[circledot] < 10 11 at high-resolution and \nto within 0.35 dex at intermediate resolution (Fig. 11, left panel). \n- · The predicted fraction of galaxies that are passive, which we define as SSFR < 10 -2 Gyr -1 for the simulations, increases sharply with stellar mass between 10 10 and 10 11 . 5 M glyph[circledot] , in agreement with the observations (Fig. 11, right panel).\n- · The predicted median relation between the maximum of the rotation curve and stellar mass of late-type galaxies, i.e. a close analogue of the Tully-Fisher relation, agrees with the observations to better than 0.03 dex over the observed mass range of 10 9 < ∼ M ∗ / M glyph[circledot] < 10 11 (Fig. 12).\n- · The relations between ISM metallicity and stellar mass and between stellar metallicity and stellar mass are predicted to flatten with stellar mass. For the gas the predicted median metallicities agree with the observed values to within 0.1 dex for M ∗ > 10 9 . 5 M glyph[circledot] at intermediate resolution and down to the lowest observed mass, M ∗ ∼ 10 8 . 5 M glyph[circledot] , at high resolution. At lower masses the predicted relations are less steep than extrapolations of the observed trends. For the stellar metallicities the discrepancies are larger. For M ∗ > 10 10 M glyph[circledot] all simulations agree with the data to better than 0.2 dex, but the difference increases with decreasing mass. At M ∗ ∼ 10 8 M glyph[circledot] the stellar metallicities in the intermediate- and high-resolution simulations are higher than observed by about 0.7 and 0.3 dex, respectively.\n- · For the mass-metallicity relations the strong convergence is significantly better than the weak convergence, i.e. simulations that keep the subgrid parameters fixed converge better with numerical resolution than simulations for which the feedback is (re)calibrated to the z ∼ 0 GSMF at each resolution. Hence, the increase in the efficiency of the feedback from star formation that was applied at high resolution in order to match the observed GSMF, simultaneously steepens the Z ( M ∗ ) relations, improving the agreement with the data.\n- · A comparison to observations of groups and clusters of galaxies with M 500 , hse > 10 13 M glyph[circledot] , where the subscript 'hse' indicates that the quantity was estimated from virtual observations under the assumption of hydrostatic equilibrium, revealed that:\n- - The predicted relation between the total I -band light within R 500 , hse and M 500 , hse agrees with the data. Note that this includes contributions from satellites and intracluster light (Fig. 14).\n- - The gas mass fractions, M gas , 500 , hse /M 500 , hse , are overestimated by about 0.2 dex in the reference model. For M 500 , hse < 10 13 . 5 M glyph[circledot] this can be remedied by increasing the subgrid AGN heating temperature, as implemented in model AGNdT9-L050N0752. At higher masses this change may be insufficient, although larger simulation volumes are needed to confirm this (Fig. 15).\n- - The reference model predicts soft X-ray luminosities that are about 0.5 dex higher than observed for clusters with spectroscopic temperatures ∼ 1 keV. However, model AGNdT9-L050N0752 is consistent with the observations (Fig. 16).\n- · The column density distributions of intergalactic C iv and O vi are in good agreement with the data, falling in between the results obtained by different surveys (Fig. 17). \nHence, in the resolved mass range, which spans 10 9 < ∼ M ∗ / M glyph[circledot] < ∼ 10 11 for some observables and 10 8 < ∼ M ∗ / M glyph[circledot] < ∼ 10 11 for others, EAGLE agrees with a diverse set of low-redshift observations of galaxies. At the same time, EAGLE reproduces some key observations of intergalactic metals. The only discrepancies found in this work that substantially exceed observational uncertainties concern the gas and stellar metallicities of dwarf galaxies, which are too high, and the predictions of the reference model for X-ray observations of the intracluster medium. The metallicity problem is only substantial at intermediate resolution, so it is possible that it can be resolved simply by increasing the resolution further. We already demonstrated that the problem with groups of galaxies can be remedied by increasing the heating temperature used in the subgrid model for AGN feedback, as implemented in model AGNdT9-L050N0752, without compromising the successes of the reference model. However, larger volumes are needed to judge whether the increase in the heating temperature that was implemented in this model suffices to obtain agreement with the data for massive ( M 500 > ∼ 10 14 M glyph[circledot] ) clusters of galaxies. \nIn future papers we will test many more predictions of EAGLE. Although we will undoubtedly uncover problems, so far we have no reason to believe that the results shown here are unrepresentative. We will show that the success of EAGLE extends to other areas that have in the past proven to be challenging for hydrodynamical simulations, such as the bimodal distribution of galaxies in colour-magnitude diagrams. We will also demonstrate that the relatively good agreement with the data is not limited to low redshift. In addition to further exploring the models that have been presented here, we plan to use the larger suite of physical models presented in Crain et al. (2014) to gain insight into the physical processes underlying the observed phenomena. Finally, we have already begun to carry out higher-resolution resimulations of individual structures (e.g. Sawala et al. 2014a,b) with the code used for EAGLE. \nAlthough the relatively good agreement between EAGLE and the observations, as well as that between other recent, hydrodynamical simulations of representative volumes and the data (e.g. Vogelsberger et al. 2014a), is encouraging, we should keep in mind that we have not attempted to model many of the physical processes that may be important for the formation and evolution of galaxies. For example, EAGLE does not include a cold interstellar gas phase, radiation transport, magnetohydrodynamics, cosmic rays, conduction, or non-equilibrium chemistry, and EAGLE does not distinguish between different forms of energy feedback from star formation and between different forms of AGN feedback. We argued that at present there are good reasons for such omissions, but many of those arguments would no longer apply if the numerical resolution were increased by several orders of magnitude. While it will take some time for simulations of representative volumes to attain the resolution that is required to model the cold ISM, simulations of individual objects can already do much better. Ultimately, simulations should be able to predict the efficiencies of the most important feedback processes and hence to predict, rather than calibrate to, the GSMF. \nWe hope that EAGLE will prove to be a useful resource \nfor the community. 10 The agreement with observations is sufficiently good for the simulations to be used in ways that have so far been reserved for semi-analytic models of galaxy formation. At the same time, because hydrodynamical simulations provide much more detailed 3-D information, make fewer simplifying assumptions, and simultaneously model the galaxies and the IGM, EAGLE enables one to ask many questions that are difficult to investigate with semi-analytic models.", 'ACKNOWLEDGMENTS': "We would like to thank Volker Springel for sharing his codes and for his advice and discussions. We gratefully acknowledge discussions with Jarle Brinchmann, Shy Genel, Justin Read, Debora Sijacki. We are also thankful to Martin Bourne and Laura Sales for their contributions to the initial phase of the project, Amandine Le Brun for her help with the X-ray plotting routines, Peter Draper and Lydia Heck for their help with the computing resources in Durham, and to Wojciech Hellwing for help with computing in Poland. We are also grateful to all the people working on the analysis of the EAGLE simulations and would like to thank the anonymous referee for a constructive report. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. We also gratefully acknowledge PRACE for awarding us access to the resource Curie based in France at Tr'es Grand Centre de Calcul. This work was sponsored by the Dutch National Computing Facilities Foundation (NCF) for the use of supercomputer facilities, with financial support from the Netherlands Organization for Scientific Research (NWO) and by the HPC Infrastructure for Grand Challenges of Science and Engineering Project, co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme and conducted at the Institute for Mathematical and Computational Modelling at University of Warsaw. The research was supported in part by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreements 278594-GasAroundGalaxies, GA 267291 Cosmiway, and 321334 dustygal, the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office ([AP P7/08 CHARM]), the National Science Foundation under Grant No. NSF PHY11-25915, the UK Science and Technology Facilities Council (grant numbers ST/F001166/1 and ST/I000976/1), Rolling and Consolodating Grants to the \n10 We intend to make the simulation output public in due course, starting with galaxy properties, which we will make available using the SQL interface that was also used for the Millennium simulation (Lemson & Virgo Consortium 2006; Springel et al. 2005b). 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L., 1995, ApJL, 453, L57", 'APPENDIX A: HYDRODYNAMICS': "Recently, much effort has been directed at solving a wellknown issue with the standard SPH implementation: multivalued particle pressure and large artificial viscosity causing unphysical surface tension at contact discontinuities (for a detailed description of the problem see, e.g. Agertz et al. 2007). This surface tension impedes the development of hydrodynamical instabilities resulting in poor mixing of gas phases, which could in principle compromise simulations of galaxy formation (e.g. Sijacki et al. 2012; Nelson et al. 2013). Several solutions have been suggested in order to smooth the pressure at contact discontinuities (e.g. Ritchie & Thomas 2001; Price 2008; Read et al. 2010; Saitoh & Makino 2013; Hopkins 2013), and to reduce the artificial viscosity away from shocks (e.g. Morris & Monaghan 1997; Cullen & Dehnen 2010). \nAs described in more detail below, we employ the fully conservative SPH formulation derived by Hopkins (2013), of which the solutions suggested by Ritchie & Thomas (2001), Read et al. (2010) and Saitoh & Makino (2013) are special cases. We use the artificial viscosity switch from Cullen & Dehnen (2010) and a switch for artificial conduction similar to that of Price (2008). We apply the time step limiters of Durier & Dalla Vecchia (2012). \nWe adopt the C 2 Wendland (1995) kernel with N ngb = 58 neighbours. This kernel inhibits particle pairing (Dehnen & Aly 2012) and the number of neighbours was chosen to give an effective resolution that is close to that of the cubic spline kernel with 48 neighbours that was used in OWLS. \nThe methods used here are collectively referred to as 'Anarchy' and will be described in more detail in Dalla Vecchia (in preparation), who also demonstrates its performance on standard hydrodynamical tests. In Schaller et al. (2014) we compare the results of EAGLE cosmological simulations with different hydrodynamics and time stepping schemes. Consistent with previous work (e.g. Scannapieco et al. 2012), we find that our results are generally substantially less sensitive to changes in the hydrodynamical techniques than to reasonable variations in the subgrid physics.", 'A1 SPH': "Following Hopkins (2013), the generalised equation of motion is \nm i d v i d t = -N ∑ j =1 x i x j [ P i y 2 i f ij ∇ i W ij ( h i ) + P j y 2 j f ji ∇ i W ij ( h j ) ] , (A1) \nwhere m i , v i and P i are the particle mass, velocity and pressure, respectively; W ij is the SPH kernel; h i is the SPH smoothing length; f ij is the correction term for variable smoothing lengths (the so-called 'gradh ' term), given by \nf ij = 1 -˜ x j x j ( h i n D ˜ y i ∂y i ∂h i )[ 1 + h i n D ˜ y i ∂ ˜ y i ∂h i ] -1 , (A2) \nwhere n D is the number of spatial dimensions. In the above equations, ˜ x i and its SPH smoothed value, ˜ y i = ∑ j ˜ x j W ij ( h i ), define the particle volume, ˜ V i = ˜ x i / ˜ y i . The particle smoothing length is defined by the relation \n4 π 3 h 3 i = N ngb ˜ V i , (A3) \nwhere N ngb is the number of neighbouring particles. 11 In our implementation we chose ˜ x i = m i and ˜ y i ≡ ρ i = ∑ j m j W ij ( h i ), the SPH particle density. \nThe remaining quantities, x i and y i = ∑ j x j W ij ( h i ), define the 'thermodynamical volume', and can be chosen in order to obtain a smooth representation of the pressure. Since we follow the evolution of the gas pseudo entropy, A ≡ P/ρ γ , the natural choice is then x i = m i A 1 /γ i and y i ≡ ¯ P 1 /γ i = ∑ N j =1 m j A 1 /γ j W ij ( h i ) as suggested by Read et al. (2010). With this definition, the weighted pressure, ¯ P i , is now single-valued and varies smoothly through contact discontinuities. \nIn practice, it is convenient to define a weighted density that can be used in the conversion between thermodynamical quantities (entropy, internal energy, temperature) and that can be predicted for inactive particles. We define the weighted density by writing the entropic function, P = Aρ γ , as follows: \n¯ P i = A i ( 1 A 1 /γ i N ∑ j =1 m j A 1 /γ j W ij ( h i ) ) γ = A i ¯ ρ γ i . (A4) \nNote that this definition of the density is the only one that is consistent with the definition of the pressure (Read et al. 2010). \nThe formulation of the SPH equation in terms of the pressure and entropy thus introduces the notion of a weighted density ¯ ρ i . Despite having the units of a density, this quantity should not be confused with the physical density ρ i = ∑ j m j W ij ( h i ). The weighted density should be thought of as an intermediate quantity required for the calculation of other thermodynamics quantities and for the SPH equation of motion. As a consequence, both densities must be used in the subgrid recipes. If the model requires a density (cooling, enrichment), then we use the physical density ρ i . On the other hand, if the quantity of interest is the pressure or the temperature, then we use the weighted density ¯ ρ i for consistency with the SPH equations. \nFinally, equation (A1) can be written as \nd v i d t = -N ∑ j =1 m j [ A 1 /γ j A 1 /γ i ¯ P i ¯ ρ 2 i f ij ∇ i W ij ( h i ) + A 1 /γ i A 1 /γ j ¯ P j ¯ ρ 2 j f ji ∇ i W ij ( h j ) ] , (A5) \nwhere the gradh terms are (see equation A2): \nf ij = 1 -1 A 1 /γ j ( h i n D ρ i ∂ ¯ P 1 /γ i ∂h i )[ 1 + h i n D ρ i ∂ρ i ∂h i ] -1 . (A6)", 'A1.1 Injection of feedback energy': 'When the equations of SPH are formulated using the pressure and entropy as main variables, particles do not carry a numerical field for their internal energy. This quantity has to be computed as a weighted sum over the particle neighbours in the same way as the density is computed in other formulations of SPH. Energy from feedback events can hence not be implemented by simply increasing the internal energy of the particle by some amount ∆ u . Furthermore, because the weighted density, ¯ ρ i , and the entropic function, A i , of a particle are coupled, a na¨ıve change of A i during energy injection would be incorrect as the corresponding weighted density would also change, making the total thermal energy of the gas (across all particles in the simulation volume) change by an amount different from ∆ u . \nIn Anarchy this problem is partially solved by performing a series of iterations during which A i and ¯ ρ i are changed until the two quantities have converged: \nA i,n +1 = ( γ -1)( u old +∆ u ) ¯ ρ γ -1 i,n , ¯ ρ i,n +1 = ¯ ρ i,n A 1 /γ n -m i W (0) A 1 /γ i,n + m i W (0) A 1 /γ i,n +1 A 1 /γ i,n +1 , (A7) \nwhere m i is the mass of particle i and W is the kernel function. This approximation is valid for reasonable values of ∆ u and is crucial for injecting thermal feedback in the gas phase. \nFor high thermal jumps with more than one particle being heated, as can for example occur for our AGN feedback scheme, the approximation provided by these iterations is not sufficiently accurate to properly conserve energy. We hence limit the amount of energy that can be injected in the gas phase by AGN in a single event by limiting the heating probability to 0 . 3 (effectively limiting the number of particles being heated at the same time in a given neighbourhood) for which tests show that the correct amount of energy is distributed to the gas.', 'A2 Artificial viscosity': "SPH requires artificial viscosity to capture shocks. The artificial viscosity switch has been implemented following Cullen & Dehnen (2010). Their algorithm enables a precise detection of shocks and avoids excessive viscosity in pure shear flows. As in Cullen & Dehnen (2010), particles are assigned individual values of the viscosity coefficient, α v ,i . This is recomputed at every time step n , and if it exceeds the value \nat the previous step, α n v ,i > α n -1 v ,i , the viscosity coefficient is set to min ( α n v ,i , α v , max ). If α n v ,i ≤ α n -1 v ,i , the viscosity coefficient decays towards α n v ,i on a time scale proportional to the particle's sound-crossing time, τ i = h i / (0 . 1 c i ): \nα v ,i = α n v ,i +( α v ,i -α n v ,i ) e -∆ t/τ i , (A8) \nand limiting the minimum allowed value, α v ,i ≥ α v , min ≥ 0. We adopt α v , min = 0 . 05 in order to facilitate particle ordering, and allow the coefficient to range up to α v , max = 2. We found that if the number of neighbours is sufficiently large ( ∼ 10 2 ), the calculation of the velocity divergence in gadget is sufficiently accurate for standard hydrodynamical tests. Therefore, we did not implement any expensive matrix calculation of the velocity divergence (Cullen & Dehnen 2010; Read & Hayfield 2012; Hu et al. 2014).", 'A3 Entropy diffusion': 'SPH is by construction non-diffusive. However, some diffusion mechanism is required during mixing of gas phases in order to mimic thermal conduction. We do not attempt to model physical diffusion; the implemented diffusion is purely numerical. We also do not implement diffusion to solve numerical problems at contact discontinuities; these are solved by the adopted SPH scheme. \nThe thermal energy, u , is diffused according to the following equation (e.g. Monaghan 1997; Price 2008), \nd u i d t = N ∑ j =1 α d ,ij v d ,ij m j ρ ij ( u i -u j ) ∇ i W ij ( h i , h j ) , (A9) \nwhere v d ,ij = max( c i + c j + v ij · r ij /r ij , 0), and the diffusion coefficient, α d ,ij , density and kernel derivative are averages among particle pairs. The purely numerical switch, similar to the one of Price (2008), is triggered by the spatial second derivative of the internal energy \n˙ α d ,i = β h i ∇ 2 i u i √ u i , (A10) \nwhere the growth speed of α d ,i can be tuned through the coefficient β . We adopt β = 0 . 01. With this choice, diffusion is mild and there is no need of any further limiter in the presence of gravity. Finally, the diffusion coefficient evolves with time as \nα d ,i ( t +∆ t ) = α d ,i ( t ) -( α d ,i ( t ) -α d , min τ i -˙ α d ,i ) ∆ t, (A11) \nwhere the decay time scale, τ i , is the same as employed in the artificial viscosity, and α d , min = 0. We set the maximum allowed value to α d , max = 1, but this is unimportant because α d ,i glyph[lessmuch] 1 even for large discontinuities in the internal energy.', 'A4 Time stepping': "The accuracy of the time integration is increased by using a time-step limiter (e.g. Saitoh & Makino 2013). We adopted the solution of Durier & Dalla Vecchia (2012) which ensures that sudden changes in the particle internal energy, e.g. caused by feedback, are promptly captured and propagated to neighbouring particles by shortening their time step and by activating them. We set the maximum ratio of neighbouring particles' time steps to four.", 'APPENDIX B: GENERATION OF THE INITIAL CONDITIONS': 'We have made two types of initial conditions: dark matter only with all particles the same mass, and dark matter with gas. The dark matter with gas simulations are created starting from a corresponding dark matter only simulation so we first describe how the dark matter only initial conditions were made.', 'B1 Building dark matter only initial conditions': "The initial conditions are created in three steps. Firstly, a particle load, representing an unperturbed homogeneous periodic universe in a 3-torus is produced. Secondly, a realisation of a Gaussian random density field with the appropriate linear power spectrum is created over the 3-torus. Thirdly the displacements and velocities, consistent with the pure growing mode of gravitational instability, are calculated from the Gaussian realisation and applied to the particle load producing the initial conditions. \nThe unperturbed particle loads for the dark matter only initial conditions have a glass-like particle distribution produced by applying the method first described in White (1994). This method, available as an option in the gadget-2 code (Springel 2005), was applied, with periodic boundary conditions, to make a 'primitive' cubic glass distribution with 47 3 particles. The particle loads required for each of the EAGLE initial conditions were built by tiling this primitive cubic glass file n times in each of the three principal coordinate directions across a larger cubic 3-torus, giving particle loads with a glass distribution with (47 n ) 3 particles. \nThe dark matter only initial conditions were generated using the ic 2lpt gen code using the method described in Jenkins (2010) to create second order Lagrangian perturbation theory (2lpt) resimulation initial conditions. The ic 2lpt gen code outputs Zeldovich initial conditions plus a '2lpt mass' for each particle. The EAGLE version of gadget 3 is then used to solve a Poisson equation sourced by the 2lpt masses placed at their unperturbed positions. The solution of this Poisson equation yields second-order Lagrangian growing mode displacements and velocities for each particle. Adding these to the Zeldovich displacements and velocities of all the particles produces the final 2lpt initial conditions. The 2lpt masses can then be discarded and the usual equations of motion are solved by integrating the initial conditions forward in time.", 'B2 Choice of phases': "Generating a Gaussian random field requires choosing a set of random phases. For the EAGLE simulations we take these phases from Panphasia which is a public multiscale Gaussian white noise field (Jenkins 2013; Jenkins & Booth 2013). Using Panphasia provides a simple way to publish the linear phases that define the EAGLE volumes. Table B1 lists the 'phase descriptors' which define the location of the phase information of each volume within the much larger Panphasia field (Jenkins 2013). These phase descriptors define the phases on all scales and uniquely determine the phases not only for the simulations published here, but for any possible zoom simulation of any subregion of these volumes, \nTable B1. The phases for the EAGLE simulation volumes are taken from the public multiscale Gaussian white noise field Panphasia (Jenkins 2013). For completeness we publish the phases for all the volumes in the EAGLE series, but note that we have not yet carried out baryonic simulations in boxes greater than 100 cMpc. These periodic cubic volumes have side-lengths given by 6 . 25 × 2 n cMpc, where n is an integer, n = 0 -10. \nand at any resolution (down to sub-Earth mass resolution if needed) in the future. In principle sufficient information is provided in this paper to enable anyone to re-run these simulations, or to resimulate objects identified from the EAGLE database. The information required is provided by the combination of the phase descriptors, the cosmological parameters and the linear matter power spectrum, and for the volumes themselves the details of how the particle load was constructed.", 'B3 Particle indexing': 'To make it possible to trace particles easily between the initial conditions and snapshots, each particle in the initial conditions was given a unique 42-bit integer index. The index was generated by assigning each particle a location on a space-filling Peano-Hilbert curve defined with a resolution of 14 bits per Cartesian coordinate over the simulation volume. The location for each particle was determined from its unperturbed position in the particle load. The particle index therefore encodes a Lagrangian position for the particle. Using a 42-bit index allows the Lagrangian position to be determined to a cubic cell of side length 1/16384 of the box size. This is small compared to the interparticle separations of particles in the initial conditions, which means that each particle has a unique index. The primitive 47 3 glass file and routines to calculate the Peano-Hilbert indices are available at http://eagle.strw.leidenuniv.nl/ .', 'B4 Making the full initial conditions': 'The initial conditions for the hydrodynamical simulations are generated from the dark matter only sets of initial conditions. Each dark matter particle is replaced with a pair of particles consisting of a dark matter particle and gas particle with a combined mass equal to that of the original dark matter particle. The ratio of the gas and dark matter particles is equal to Ω baryon / (Ω matter -Ω baryon ). These particle pairs are positioned so that the centre of mass of the pair corresponds to the position of the original particle in the dark matter only initial conditions. The particle pairs are aligned with the (1,1,1) coordinate direction and the gas \nparticle is positioned in the (1,1,1) direction relative to its corresponding dark matter particle. The magnitude of the displacement between the pair is chosen so that an initial cubic grid with mean density in the dark matter only initial conditions would transform into a body-centred cubic grid with dark matter (gas) particles at the centres of cubic cells made of gas (dark matter) particles. \nFor the hydrodynamical simulations the index of the dark matter particles is taken to be exactly twice that of the corresponding index in the dark matter only initial conditions. The index of the gas particle is chosen to be one more than its corresponding dark matter particle. Thus, all dark matter particles have even indices, and all gas particles odd indices.'}

2024MNRAS.534..948W

The vertical shear instability VSI is widely believed to be effective in driving turbulence in protoplanetary discs PPDs. Prior studies on VSI exclusively exploit the reflecting boundary conditions BCs at the disc surfaces. VSI depends critically on the boundary behaviours of waves at the disc surfaces. We extend earlier studies by performing a comprehensive numerical analysis of VSI with partially reflecting BCs for both the axisymmetric and nonaxisymmetric unstable VSI modes. We find that the growth rates of the unstable modes diminish when the outgoing component of the flow is greater than the incoming one for highorder body modes. When the outgoing wave component dominates the growth of VSI is notably suppressed. We find that the nonaxisymmetric modes are unstable and they grow at a rate that decreases with the azimuthal wavenumber. The different BCs at the lower and upper disc surfaces naturally lead to nonsymmetric modes relative to the disc midplane. The potential implications of our studies for further understanding planetary formation and evolution in PPDs are also briefly discussed.

2024-10-01T00:00:00Z

['2024MNRAS.tmp.2096W', '2024arXiv240907898W', '2024MNRAS.534..948W', '10.48550/arXiv.2409.07898', 'arXiv:2409.07898', '10.1093/mnras/stae2141']

['Astrophysics - Earth and Planetary Astrophysics']

Vertical shear instability with partially reflecting boundary conditions

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

https://arxiv.org/pdf/2409.07898.pdf

{'Vertical Shear Instability with Partially Reflecting Boundary Conditions': 'Yuzi Wu ( 吴 玉 孜 ), 1 , 2 Cong Yu ( 余 聪 ), 1 , 2 ★ Can Cui ( 崔 灿 ) 3 \n1 School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China \n- 2 CSST Science Center for the Guangdong-Hong Kong-Macau Greater Bay Area, Zhuhai 519082, China\n- 3 Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'The vertical shear instability (VSI) is widely believed to be effective in driving turbulence in protoplanetary disks. Prior studies on VSI exclusively exploit the reflecting boundary conditions (BCs) at the disk surfaces. VSI depends critically on the boundary behaviors of waves at the disk surfaces. We extend earlier studies by performing a comprehensive numerical analysis of VSI with partially reflecting BCs for both the axisymmetric and non-axisymmetric unstable VSI modes. We find that the growth rates of the unstable modes diminish when the outgoing component of the flow is greater than the incoming one for high-order body modes. When the outgoing wave component dominates, the growth of VSI is notably suppressed. We find that the non-axisymmetric modes are unstable and they grow at a rate that decreases with the azimuthal wavenumber. The different BCs at the lower and upper disk surfaces naturally lead to non-symmetric modes relative to the disk midplane. The potential implications of our studies for further understanding planetary formation and evolution in protoplanetary disks (PPDs) are also briefly discussed. \nKey words: hydrodynamics - instabilities - protoplanetary discs.', '1 INTRODUCTION': 'Protoplanetary disks serve as the nurseries for planet formation. Turbulence is a key factor driving angular momentum transport within the disks, and further exploration of the physical factors that drive turbulence would enhance our understanding on disk mass and angular momentum transfer, which is of great significance in the processes of planet formation and evolution (Youdin & Lithwick 2007; Lesur et al. 2023). With the development of observational technology, the finer structures of protoplanetary disks are gradually emerging. The high-resolution observations by the Atacama Large Millimeter/submillimeter Array (ALMA) reveal the distribution of dust and gas within protoplanetary disks, as well as clear substructures such as rings and gaps (Teague et al. 2016; Yamaguchi et al. 2024). Concurrently, ALMA has detected a certain level of turbulence within protoplanetary disks, suggesting that turbulence may play a significant role in various stages of planet formation (Manger & Klahr 2018; Latter & Papaloizou 2018; Flaherty et al. 2019, 2024). The investigation into the origin of disk turbulence will provide an indispensable theoretical framework for a more profound comprehension of the formation and evolution processes of exoplanets. \nIn different regions of the protoplanetary disk, the dominant mechanisms for generating turbulence are diverse (Pfeil & Klahr 2019). In the inner disk region, with radii smaller than about 0.3 astronomical units and the outermost disk region, beyond 100 astronomical units, turbulence is primarily driven by magnetorotational instability (MRI). In the intermediate region, however, the weak ionization results in a poor coupling between the magnetic field and gas, suppressing the MRI (Bai 2011; Cui & Bai 2021). Within this "dead \nzone", purely hydrodynamic instabilities become the primary driver of turbulence (Yellin-Bergovoy et al. 2021), such as the VSI (Urpin & Brandenburg 1998; Urpin 2003), zombie vortex instability (ZVI) (Marcus et al. 2015), and convective overstability (COS) (Klahr & Hubbard 2014). The operation of these instabilities depends on specific environmental conditions. VSI, due to its extensive influence in protoplanetary disks, has received widespread attention. \nThe VSI arises in the presence of angular velocity shear 𝜕 Ω / 𝜕𝑧 within protoplanetary disks, serving as an effective source of turbulence. The VSI is similar in form to the Goldreich-Schubert-Fricke instability (GSF), which was initially discovered in slowly rotating stars (Goldreich & Schubert 1967). It does not require strong radial stratification and is induced solely by vertical shear (Urpin 2003). \nFor protoplanetary disks, the dependence of Ω on 𝑧 significantly alters the stability characteristics of the disk. However, fluid elements also experience stabilizing effects due to vertical buoyancy, hindering the growth of VSI. Effective rapid cooling can overcome this stabilizing effect caused by buoyancy. Under rapid cooling with 𝑡 c < Ω -1 K ℎ | 𝑞 |/( 𝛾 -1 ) , VSI will be effective (Lin & Youdin 2015), where 𝑡 c is the cooling timescale, ℎ is the disk aspect ratio, 𝑞 is the power-law index of temperature profile and 𝛾 is the adiabatic index. For cooling times longer than the critical cooling time of VSI, COS modes will emerge, experiencing similar growth rates as GSF (Klahr 2024). Realistic protoplanetary disks have complex density and temperature structures, with thermal relaxation depending on the intricate spatial distribution of local disk stratification. Threedimensional simulations have studied the thermal coupling of gas and dust particles in optically thick and optically thin regions within protoplanetary disks. The decoupling of oscillations of dust and gas particles in the upper atmosphere leads to ineffective thermal relaxation, thereby suppressing VSI turbulence (Pfeil & Klahr 2021). \nThe VSI is classified into surface modes and body modes (Nelson et al. 2013; Lin & Youdin 2015). Surface modes concentrate on the disk surface. Body modes extend vertically within the disk, exhibiting prominent vertical oscillations with radial wavenumbers much larger than vertical wavenumbers. Based on the symmetry of body modes, they can be further classified into breathing modes and corrugation modes (Nelson et al. 2013). \nNelson et al. (2013) studied the nonlinear evolution of VSI through numerical simulations. They showed that, although surface modes grow rapidly, unstable modes are eventually dominated by body modes, especially corrugation modes, which are characterized by a narrow band of large-scale vertical motions (Nelson et al. 2013). This can lead to turbulence with an effective parameter 𝛼 ≈ 10 -4 ∼ 10 -3 , causing non-negligible angular momentum transport (Nelson et al. 2013; Stoll & Kley 2014). \nThe VSI mainly occurs in the "dead zone" with weak magnetic coupling within protoplanetary disks. The magnetic field provides a stabilizing effect through magnetic tension or magnetorotational turbulence, suppressing the VSI. With increasing magnetization, surface modes will disappear before body modes (Cui & Lin 2021; Latter & Kunz 2022). The Ohmic resistivity in non-ideal MHD can overcome magnetic stability, restoring the instability (Cui & Lin 2021). Disk winds could coexist and interact with the VSI, enhancing the accretion rate of disk winds and triggering strong VSI turbulence. Weak ambipolar diffusion strength or enhanced coupling between gas and magnetic fields will also suppress shear instability (Cui & Bai 2020). \nPrevious studies on the VSI adopted a vertically global treatment with the reflecting BCs (e.g. Nelson et al. 2013; Lin & Youdin 2015; Cui & Lin 2021; Svanberg et al. 2022). The effects of the disk\'s vertical extent on the VSI were studied by Barker & Latter (2015). Lin & Youdin (2015) applied free and rigid BCs to the VSI. Free BCs establish a zero Lagrangian pressure perturbation at the boundary, resembling a reflecting BC with a phase difference, as we detailed in Section A. The rigid BC means no vertical velocity perturbation at the boundary, consistent with the one used in Barker & Latter (2015). It is also a reflecting BC, albeit with half-wave loss, as we explained in Section 4.3. This work investigates the VSI with partially reflecting BCs, introducing a new factor worth further exploration to enhance our understanding of the VSI\'s behavior in more realistic astrophysical scenarios. \nThe complex environment of protoplanetary disks at their surfaces can indeed give rise to the outgoing flows. Zhang et al. (2024) considered that full disks and transition disks with small cavities have a superheated atmosphere and cool midplane, with this temperature structure producing an outgoing flow layer at optical depth 𝜏 ∗ < 1 on top of an ingoing flow layer at 𝜏 ∗ ∼ 1. Notably, magnetohydrodynamic (MHD) disk winds are one such phenomenon that is effectively launched and driven from the disk surface, providing a physical foundation for understanding and justifying the existence of outgoing BCs. In addition, the presence of anti-symmetric magnetic field modes in protoplanetary disks results in one-sided characteristics for disk winds, meaning that these winds occur solely on one side of the disk relative to its midplane, which can potentially lead to asymmetry in the outgoing flow (Wang et al. 2024). \nConsequently, it is essential to fill the research gap regarding the behavior of VSI under conditions of partial reflection at the boundaries. We focus on the effect of partially reflecting BCs, considering the axisymmetric case and further including a nonzero azimuthal wavenumber to account for non-axisymmetric VSI effects. \nThe structure of the paper is as follows: In § 2, we present the dynamical equations. In § 3, we introduce the vertical shear of the equilibrium disk model. In § 4, we show the linearized equations for \nboth axisymmetric and non-axisymmetric perturbations and describe the partially reflecting BCs. In § 5, we analyze the growth rates and symmetry of VSI. Finally, we summarize our results in § 6.', '2 BASIC EQUATIONS': 'The continuity, momentum and energy conservation equations of hydrodynamics for a protoplanetary disk are as follows: \n𝜕𝜌 𝜕𝑡 + ∇ · ( 𝜌 𝒖 ) = 0 , (1) \n𝜕 𝒖 𝜕𝑡 + ( 𝒖 · ∇) 𝒖 = -1 𝜌 ∇ 𝑃 - ∇ Φ , (2) \n𝜕𝑃 𝜕𝑡 + ( 𝒖 · ∇) 𝑃 = -𝛾𝑃 ∇ · 𝒖 -Λ , (3) \nwhere 𝜌, 𝒖 , 𝑃 are the density, three-dimensional velocity and pressure, respectively. To study the protoplanetary disk, we adopt the cylindrical coordinates( 𝑟, 𝜙, 𝑧 ). The gravitational potential by the central star with mass 𝑀 is Φ = -𝐺𝑀 /( 𝑟 2 + 𝑧 2 ) 1 / 2 , where 𝐺 is the gravitational constant. In equation (3), 𝛾 is the adiabatic index, and the sink term Λ on the right side represents the radiative cooling.', '3 EQUILIBRIUM DISK MODEL': 'The radial power-law distribution for both density and temperature in the cylindrical coordinate are: \n𝜌 ( 𝑟 ) = 𝜌 0 GLYPH<18> 𝑟 𝑟 0 GLYPH<19> 𝑝 , 𝑇 ( 𝑟 ) = 𝑇 0 GLYPH<18> 𝑟 𝑟 0 GLYPH<19> 𝑞 , (4) \nwhere 𝑝 is the power-law index of the mass density and 𝑞 is the power-low index of the temperature, 𝜌 0 and 𝑇 0 are the density and temperature at the fiducial radius 𝑟 0 , respectively. We adopt the ideal equation of state, 𝑃 = R 𝜇 𝜌𝑇 . Here R is the gas constant and 𝜇 is the mean molecular mass of gas. For a characteristic timescale 𝑡 c, the radiative cooling takes the form: \nΛ = 𝜌 R 𝜇 𝑇 -𝑇 eq 𝑡 c = 𝑃 -𝑐 2 s 𝜌 eq 𝑡 c , (5) \nwhere 𝑇 eq and 𝜌 eq are the equilibrium temperature and density, respectively. In the absence of cooling, the disk will be stable against the Solberg-Hoiland criteria. Sufficiently rapid cooling is an indispensable factor to trigger VSI since it requires rapid thermal relaxation to inhibit buoyancy. \nThe equilibrium state of the disk at radial and vertical directions meet: \n𝜕 Φ 𝜕𝑟 + 1 𝜌 𝜕𝑃 𝜕𝑟 = 𝑟 Ω 2 , 𝜕 Φ 𝜕𝑧 + 1 𝜌 𝜕𝑃 𝜕𝑧 = 0 , (6) \nwhere Ω = Ω ( 𝑟, 𝑧 ) is the equilibrium rotation field. The density and pressure are related by the isothermal sound speed 𝑐 s, which is 𝑐 s ( 𝑟 ) = √︁ 𝑃 / 𝜌 . The density profile is: \n𝜌 ( 𝑟, 𝑧 ) = 𝜌 ( 𝑟 ) exp " 1 𝑐 s 2 𝐺𝑀 √︁ 𝑟 2 + 𝑧 2 -𝐺𝑀 𝑟 !# , (7) \nThe equilibrium angular velocity is \nΩ 2 ( 𝑟, 𝑧 ) = Ω 2 K ( 𝑟 ) " 1 + 𝑞 + ( 𝑝 + 𝑞 ) ℎ 2 -𝑞 𝑟 √︁ 𝑟 2 + 𝑧 2 # , (8) \nwhere Ω K = √︁ 𝐺𝑀 / 𝑟 3 is the Kepler frequency. We assume thin disk models with a small disc aspect ratio ℎ , i.e. ℎ = 𝐻 / 𝑟 ≪ 1, where 𝐻 = 𝑐 𝑠 / Ω 𝐾 is the scale height of the disc. In this case, the density profile in equation (7) is simplified as: \n𝜌 ( 𝑟, 𝑧 ) = 𝜌 ( 𝑟 ) exp GLYPH<18> -𝑧 2 2 𝐻 2 GLYPH<19> . (9) \nRadial variations in temperature inevitably lead to a vertical gradient in the angular velocity, or vertical shear: \n𝜕 Ω 2 𝜕𝑧 = 𝑞𝑧 𝑟 2 Ω 2 K GLYPH<0> 1 + 𝑧 2 / 𝑟 2 GLYPH<1> 3 / 2 . (10) \nThe radial gradient of the angular velocity squared is: \n𝜕 Ω 2 𝜕𝑟 = -3 Ω 2 𝑟 -𝑞𝑧 2 𝑟 3 Ω 2 K GLYPH<0> 1 + 𝑧 2 / 𝑟 2 GLYPH<1> 3 / 2 . (11)', '4 LINEAR PERTURBATION EQUATIONS': 'In this work, the linear perturbations for both the axisymmetric and non-axisymmetric cases with partially reflecting BCs will be discussed. The Eulerian perturbations ( 𝛿𝑢 𝑟 , 𝛿𝑢 𝜙 , 𝛿𝑢 𝑧 , 𝛿𝜌, 𝛿𝑃 ) have space and time dependence of the form exp ( 𝑖𝑘 𝑟 𝑟 + 𝑖𝑚𝜙 -𝑖𝜐𝑡 ) , where 𝑘 𝑟 is the radial wavenumber, 𝑚 is the azimuthal wavenumber and 𝜐 is the complex frequency. The linearized perturbation equations for the velocity, density and pressure perturbations are: \n𝑖 ( 𝜐 -𝑚 Ω ) 𝛿𝜌 𝜌 = 𝑖𝑘 𝑟 𝛿𝑢 𝑟 + 𝛿𝑢 𝑟 𝑟 + 𝑖𝑚 𝑟 𝛿𝑢 𝜙 + 𝜕𝛿𝑢 𝑧 𝜕𝑧 + 𝜕 ln 𝜌 𝜕𝑧 𝛿𝑢 𝑧 , (12) \n𝑖 ( 𝜐 -𝑚 Ω ) 𝛿𝑢 𝑟 = 𝑖𝑘 𝑟 𝛿𝑃 𝜌 -2 Ω 𝛿𝑢 𝜙 , (13) \n𝑖 ( 𝜐 -𝑚 Ω ) 𝛿𝑢 𝜙 = 𝑖𝑚 𝑟 𝛿𝑃 𝜌 + 𝑟 𝜕 Ω 𝜕𝑧 𝛿𝑢 𝑧 + 𝜅 2 2 Ω 𝛿𝑢 𝑟 , (14) \n𝑖 ( 𝜐 -𝑚 Ω ) 𝛿𝑢 𝑧 = 𝑑 𝑑𝑧 GLYPH<18> 𝛿𝑃 𝜌 GLYPH<19> + 𝜕 ln 𝜌 𝜕𝑧 GLYPH<18> 𝛿𝑃 𝜌 -𝑐 s 2 𝛿𝜌 𝜌 GLYPH<19> , (15) \n𝑖 ( 𝜐 -𝑚 Ω ) 𝛿𝑃 𝜌 = 𝑐 s 2 𝛾 GLYPH<18> 𝑖𝑘 𝑟 𝛿𝑢 𝑟 + 𝛿𝑢 𝑟 𝑟 + 𝑖𝑚 𝑟 𝛿𝑢 𝜙 + 𝜕𝛿𝑢 𝑧 𝜕𝑧 GLYPH<19> + 𝑐 s 2 𝜕 ln 𝜌 𝜕𝑧 𝛿𝑢 𝑧 + 1 𝑡 c GLYPH<18> 𝛿𝑃 𝜌 -𝑐 s 2 𝛿𝜌 𝜌 GLYPH<19> , (16) \nwhere 𝜅 is the epicyclic frequency and 𝜅 2 ≡ 4 Ω 2 + 𝑟 𝜕 Ω 2 𝜕𝑟 . Note that our equations retain the curvature terms neglected in Lin & Youdin (2015). The impact caused by the curvature terms is weak. We have ignored the radial gradients of the equilibrium density and pressure to simplify the equation. This type of radially local approximation was initially proposed in McNally & Pessah (2014) and has also been adopted in Lin & Youdin (2015). However, we retain the radial temperature gradient to ensure the VSI operates.', '4.1 Wave Equations for the Axisymmetric VSI': 'For the axisymmetric VSI, we take 𝑚 = 0. In terms of the variables 𝑦 1 ≡ 𝛿𝑢 𝑧 , and 𝑦 2 ≡ 𝛿𝑃 / 𝜌 , Equations (12)-(16) can be cast into two \nfirst-order ordinary differential equations (ODEs) 1 : \n𝑑𝑦 1 𝑑 ˆ 𝑧 = 𝐴 11 𝑦 1 + 𝐴 12 𝑦 2 , 𝑑𝑦 2 𝑑 ˆ 𝑧 = 𝐴 21 𝑦 1 + 𝐴 22 𝑦 2 . (17) \nThe relevant coefficients in the above linear ODEs are \n𝐴 11 = ( 𝜒 + 𝑄 ) ˆ 𝑧 , 𝐴 12 = 𝑖 ˆ 𝜐 ( 𝜒 + 𝑈 ) , (18) \n𝐴 21 = 𝑖 ˆ 𝜐 -ˆ 𝑧 2 ( 𝜒 -1 ) 𝑖 ˆ 𝜐 , 𝐴 22 = - ( 𝜒 -1 ) ˆ 𝑧 , (19) \nwhere 𝜒 = ( 1 -𝑖 ˆ 𝜐𝛽 )/( 1 -𝑖 ˆ 𝜐𝛽𝛾 ) , 𝑄 = 𝑖ℎ𝑞 ˆ 𝑘 𝑟 + 𝑞ℎ 2 , and 𝑈 = ˆ 𝑘 2 𝑟 -𝑖 ˆ 𝑘 𝑟 ℎ . Note that the equations above have been written in a dimensionless form adopting the following rules: 𝑧 = ˆ 𝑧𝐻 , 𝑘 𝑟 = ˆ 𝑘 𝑟 / 𝐻 , 𝜐 = ˆ 𝜐 Ω K , 𝛽 = 𝑡 c Ω K . The eigenvalue ˆ 𝜐 is a complex number. Its real part ˆ 𝜔 is the frequency and the imaginary part ˆ 𝜎 means the growth rate. With our convention, the sound speed 𝑐 𝑠 = 𝐻 Ω 𝐾 is normalized to unity. \nWe are interested in the frequency range in which 𝜐 is small compared to the epicyclic frequency 𝜅 . This low-frequency approximation filters out the acoustic waves and retains the inertial-gravity waves (Lubow & Pringle 1993). With this approximation and rapid cooling 𝛽 = 10 -3 , the dispersion relation (B1) in Appendix B can be written as \nˆ 𝜐 2 ≃ ˆ 𝑘 2 𝑧 + 𝑖 GLYPH<16> 𝑖ℎ𝑞 ˆ 𝑘 𝑟 + 𝑞ℎ 2 + 1 GLYPH<17> ˆ 𝑧 ˆ 𝑘 𝑧 1 + ˆ 𝑘 2 𝑟 -𝑖 ˆ 𝑘 𝑟 ℎ , (20) \nSince ℎ ≪ 1 and ˆ 𝑘 𝑟 ≫ 1 , ˆ 𝑘 𝑧 , the equation shows the dispersion relation for inertial waves (Goodman & Lackner 2009) \n𝜐 2 ≈ Ω 2 𝐾 ˆ 𝑘 2 𝑧 + 𝑖 GLYPH<16> 𝑖ℎ𝑞 ˆ 𝑘 𝑟 + 1 GLYPH<17> ˆ 𝑧 ˆ 𝑘 𝑧 ˆ 𝑘 2 𝑟 ∼ Ω 2 𝐾 ˆ 𝑘 2 𝑧 ˆ 𝑘 2 𝑟 . (21) \nIt is worthwhile to note that the VSI actually corresponds to the inertial wave. The vertical component of the group velocity and phase velocity for inertial waves are always in the same direction.', '4.2 Wave Equations for the Non-axisymmetric VSI': "In the non-axisymmetric case, the azimuthal wavenumber 𝑚 ≠ 0. Westill adopt the low-frequency approximation to simplify our analysis, i.e., the Doppler shifted frequency is much smaller than the epicyclic frequency, | 𝜐 -𝑚 Ω | 2 ≪ 𝜅 2 . The governing ODEs for the non-axisymmetric case read: \n𝑑𝑦 1 𝑑 ˆ 𝑧 = 𝐴 ' 11 𝑦 1 + 𝐴 ' 12 𝑦 2 , 𝑑𝑦 2 𝑑 ˆ 𝑧 = 𝐴 ' 21 𝑦 1 + 𝐴 ' 22 𝑦 2 , (22) \nwhere \n𝐴 ' 11 = GLYPH<0> 𝜒 ' + 𝑄 ' GLYPH<1> ˆ 𝑧 , (23) \n𝐴 ' 12 = 𝑖 ( ˆ 𝜐 -𝑚 ) GLYPH<0> 𝜒 ' + 𝑈 ' GLYPH<1> + 𝑊 2 -ˆ 𝑘 𝑟 𝑚𝑞ℎ 3 2 ˆ 𝑧 2 , (24) \n1 In a more compact matrix form, \n𝑑 𝒀 𝑑 ˆ 𝑧 = 𝑨𝒀 , where 𝑨 = GLYPH<18> 𝐴 11 𝐴 12 𝐴 21 𝐴 22 GLYPH<19> , 𝒀 ≡ ( 𝑦 1 , 𝑦 2 ) 𝑇 = GLYPH<18> 𝑦 1 𝑦 2 GLYPH<19> . \nIt should be stressed that all the wave propagation properties are hidden in the matrix 𝑨 . The two eigenvalues ( 𝑖 ˆ 𝑘 𝑧 1, 𝑖 ˆ 𝑘 𝑧 2) and two eigenvectors ( 𝒓 1, 𝒓 2) of the matrix 𝑨 are important for our analysis of BCs. (Detailed derivation of the eigenvalues and eigenvectors are given in Appendix A. Further explanation can be found in Appendix B.) \n𝐴 ' 21 = 𝑖 ( ˆ 𝜐 -𝑚 ) -ˆ 𝑧 2 ( 𝜒 ' -1 ) 𝑖 ( ˆ 𝜐 -𝑚 ) , 𝐴 ' 22 = -GLYPH<0> 𝜒 ' -1 GLYPH<1> ˆ 𝑧 . (25) \nRelevant variables in the above matrix coefficients are \n𝜒 ' = [ 1 -𝑖 ( ˆ 𝜐 -𝑚 ) 𝛽 ] /[ 1 -𝑖 ( ˆ 𝜐 -𝑚 ) 𝛽𝛾 ] , 𝑈 ' = 𝑈 + 𝑚 2 ℎ 2 , \n𝑄 ' = 𝑄 + ( ˆ 𝜐 -𝑚 ) 𝑚𝑞ℎ 2 / 2 , 𝑊 = 4 𝑖𝑚ℎ 2 -3 𝑚ℎ ˆ 𝑘 𝑟 , \nrespectively. Compared to equation (17), the additional terms in the above equations are due to the non-axisymmetric effect.", '4.3 Boundary Conditions': "We adopt the WKB approximation to specify the BCs. Note that it is a local approximation of the linear second-order ODEs. The WKBapproximation is only exploited locally at the boundary, which provides certain constraints on the global integration. Our global solutions are obtained by numerical integration rather than the WKB approximation. According to Appendix A, the WKB approximation can be understood in terms of the eigenstructures of the coefficient matrix of the wave equations. The variables 𝒀 ≡ ( 𝑦 1 , 𝑦 2 ) 𝑇 can be viewed as the sum of two eigenvectors 𝒓 1 and 𝒓 2 : \n𝒀 = 𝑎 1 𝒓 1 + 𝑎 2 𝒓 2 . (26) \nAs mentioned in Appendix B 2 , we know that the wave group velocity associated with 𝒓 1 is incoming and 𝒓 2 outgoing. The coefficients 𝑎 1 and 𝑎 2 represent the strength of waves in these two directions, respectively. The boundary parameter 𝑅 ≡ 𝑎 2 / 𝑎 1 is determined by 𝑦 1 and 𝑦 2 in the following way (Please refer to Appendix A for the derivation): \n𝑅 = h -𝐴 1 ˆ 𝑧 + 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 i 𝑦 1 -𝐵 1 𝑦 2 h 𝐴 1 ˆ 𝑧 + 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 i 𝑦 1 + 𝐵 1 𝑦 2 . (27) \nThe parameters, 𝐴 1 , 𝐴 2 , 𝐴 3 and 𝐵 1 in equation (27) are explicitly given in Appendix A. Note that equation (27) can be written alternatively as a linear combination of 𝑦 1 and 𝑦 2 : \nGLYPH<18> 𝐴 1 ˆ 𝑧 + 𝑅 -1 𝑅 + 1 𝑖 √︃ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<19> 𝑦 1 + 𝐵 1 𝑦 2 = 0 . (28) \nThe above equation is the actual BC adopted in our numerical calculations. Perfect reflection occurs when 𝑅 = 1, \n𝐴 1 ˆ 𝑧𝑦 1 + 𝐵 1 𝑦 2 = 0 . (29) \nFor a reflection with a half-wave loss, the BC can be expressed as 𝑦 1 = 0 with 𝑅 = -1, i.e. 𝛿𝑢 𝑧 = 0 at ˆ 𝑧 = ± ˆ 𝑧 max. As 𝑅 approaches infinity, the mode solely exhibits an outgoing inertial wave. Purely outgoing boundary conditions are widely used in the studies of Rossby wave instability (Li et al. 2000; Yu & Lai 2013; Huang & Yu 2022). \nTheBCforthenon-axisymmetric case could be similarly obtained, which is: \nGLYPH<18> 𝐴 ' 1 ˆ 𝑧 + 𝑅 -1 𝑅 + 1 𝑖 √︃ 𝐴 3 ' + 𝐴 ' 4 ˆ 𝑧 4 -𝐴 2 ' ˆ 𝑧 2 GLYPH<19> 𝑦 1 + GLYPH<16> 𝐵 1 ' -𝐵 2 ' ˆ 𝑧 2 GLYPH<17> 𝑦 2 = 0 , (30) \nwhere \n𝐵 ' 1 = 2 𝑖 ( ˆ 𝜐 -𝑚 ) GLYPH<0> 𝜒 ' + 𝑈 ' GLYPH<1> + 𝑊 , 𝐵 ' 2 = ˆ 𝑘 𝑟 𝑚𝑞ℎ 3 , (31) \n𝐴 ' 1 = 2 𝜒 ' + 𝑄 ' -1 , (32) \n𝐴 ' 2 = GLYPH<0> 𝑄 ' + 1 GLYPH<1> 2 -2 𝑖 ( ˆ 𝜐 -𝑚 ) 𝐵 ' 2 + 4 GLYPH<0> 𝜒 ' -1 GLYPH<1> GLYPH<0> 𝑄 ' -𝑉 GLYPH<1> , (33) \n𝐴 ' 3 = 4 ( ˆ 𝜐 -𝑚 ) 2 GLYPH<0> 𝜒 ' + 𝑉 GLYPH<1> , 𝐴 ' 4 = 2 𝑖 ( 𝜒 ' -1 ) ˆ 𝜐 -𝑚 𝐵 ' 2 , (34) \nrespectively. The parameter 𝑉 is given by 𝑉 = 𝑈 ' -𝑖𝑊 /[ 2 ( ˆ 𝜐 -𝑚 )] . Following Lubow & Pringle (1993), it should be noted that when applying the low-frequency approximation to non-axisymmetric analyses, a critical requirement exists that the modes must be tightly wrapped, leading to a condition 𝑚 ≪ ˆ 𝑘 𝑟 / ℎ .", '5 NUMERICAL RESULT': 'To investigate the axisymmetric or non-axisymmetric linear perturbations of VSI, we need to solve Equation (17) or Equation (22). The eigenvalues of these equations are closely related to the BCs, which we adopt as Equation (28) or Equation (30). These BCs represent a linear combination of outgoing and incoming wave components of different strengths. The numerical solution of this kind of complex two-point boundary eigenvalue problem can be efficiently obtained using the relaxation method (Press et al. 1992). In this method, we discretize the ODEs into finite difference equations (FDEs) across a mesh that spans the domain of interest. Through iterative trial and error, we progressively determine eigenvalues and eigenfunctions by a method analogous to the Newton -Raphson method. Throughout these iterations, both the ODEs and BCs are satisfied simultaneously. In this study, we employ 5001 uniform mesh points to achieve a high-accuracy numerical solution with an average error of less than 10 -8 . The basic parameters of the disk are set as 𝑞 = -1, 𝑝 = -1 . 5, ℎ = 0 . 05, 𝛾 = 1 . 4, respectively.', '5.1 Axisymmetric VSI unstable modes': 'Usually, VSI modes exhibit odd or even symmetry. Our calculations shown in Fig. 2-3 with symmetric BCs at 𝑧 = -𝑧 max and 𝑧 = 𝑧 max obtain similar results to Nelson et al. (2013). However, there exist highly asymmetric cases, as demonstrated in the bottom right panel of Figure 20 in Nelson et al. (2013). There is still a lack of a clear explanation for the asymmetry. For simplicity, we choose the domain 𝑧 ∈ [ 0 , 𝑧 max ] to avoid such asymmetric surface modes. At the midplane 𝑧 = 0, we set the BCs according to the parity of the eigenfunction. While at 𝑧 = 𝑧 max, we set partially reflecting BCs. \nOur numerical results with the cooling parameter 𝛽 = 10 -3 are shown in Fig. 1, which depicts the variation of growth rate with the frequency for partially reflecting BCs. Each symbol denotes a specific order mode. For a general set of parameters, lower-order modes possess a smaller eigenfrequency | 𝜔 | . In the upper panel, we present the result for smaller radial wavenumber ˆ 𝑘 𝑟 = 10. Only the body modes are observed in this case. The results for larger radial wavenumber ˆ 𝑘 𝑟 = 30 are displayed in the lower panel. In this case, both the body and surface modes show up. The growth rate varies with the frequency in a non-monotonic way. With the gradual increase of frequency, the growth rate first increases with the frequency (i.e., lower-order modes). Once the growth rate reaches a maximum, it would decrease with the frequency (i.e., higher-order modes). The growth rate of the low order body mode is almost not affected by \nFigure 1. Variations of the growth rate with frequency for the radial wavenumber ˆ 𝑘 𝑟 = 10 and ˆ 𝑘 𝑟 = 30, respectively. The cooling parameter is 𝛽 = 10 -3 . The disk respect ratio ℎ = 0 . 05 and maximum disk height 𝑧 max = 5 𝐻 . The boundary condition parameter is set to 𝑅 = 2 (magenta pluses), 𝑅 = 0 . 5 (blue pluses) and 𝑅 = 1 (black pluses), respectively. \n<!-- image --> \nthe BCs. However, for the higher-order body modes, the influence of the BCs becomes more obvious. However, for surface modes, the boundary has a relatively minor effect on the growth rate. As the parameter 𝑅 decreases, the growth rates become larger. On the other hand, as the parameter 𝑅 increases, the growth rates decrease. Our result indicate that the surface of the disk actually acts as a trap of the inertial waves. When the trapping effects become weaker ( 𝑅 becomes larger), the VSI would become weaker as well. \nFig. 2 shows the vertical velocity perturbation, 𝛿𝑢 𝑧 , of the loworder corrugation mode (symmetric with respect to midplane) with growth rate 𝜎 = 0 . 73 ℎ Ω K . In the upper panel, perfectly reflecting BC is set at both the upper and lower boundaries. The resulting eigenfunction is symmetric relative to the midplane. In the lower panel, the perfectly reflecting BC ( 𝑅 = 1) is adopted at the lower boundary 𝑧 = -5 𝐻 and a partially reflecting BC ( 𝑅 = 2) at the upper boundary 𝑧 = 5 𝐻 . The strength of the outgoing wave is twice that of the incoming wave, breaking the symmetry with respect to the midplane. At the upper boundary, where the flow is predominantly outgoing, the vertical velocity perturbations are stronger than those observed at the lower boundary. Owing to its nature as a low-order mode, the growth rate is barely affected by BCs. For higher order modes, such as the breathing mode (anti-symmetric with respect to the midplane) with a frequency | 𝜔 | = 2 . 15 ℎ Ω 𝐾 shown in Fig. 3, the growth rate is clearly affected by the BCs. For the partially reflecting BC with 𝑅 = 2, the growth rate has been suppressed by 0 . 02 ℎ Ω 𝐾 . \nFigure 2. Vertical velocity perturbation 𝛿𝑢 𝑧 of the low order corrugation mode. Top: with same boundary parameter 𝑅 = 1 at 𝑧 = ± 𝑧 max. Bottom: with boundary parameter 𝑅 = 1 on the lower boundary and boundary parameter 𝑅 = 2 on the upper boundary, respectively. \n<!-- image --> \nFigure 3. Same as Figure 2 but for higher order breathing mode. \n<!-- image --> \nMeanwhile, the anti-symmetric configuration originally relative to the midplane can no longer be maintained. \nWecompare three cases of 𝑅 = 0 . 01, 𝑅 = 1, and 𝑅 = 100 in Fig. 4. When the strength of the outgoing wave is much greater than that of the incoming wave, the growth rate is significantly suppressed. Especially, the growth rate of the higher order modes with frequency | 𝜔 | ≳ 5 ℎ Ω K are preferentially damped. An increase in the strength of the outgoing component also results in a shift of the wave frequency \nFigure 4. Growth rate and intrinsic frequency of the unstable modes for ˆ 𝑘 𝑟 = 30andextremeboundaryparameters 𝑅 = 0 . 01(yellow pluses), 𝑅 = 100 (green pluses) and the case of 𝑅 = 1 (black pluses) as a reference. \n<!-- image --> \nFigure 5. Growth rate of unstable modes vs. vertical extent. Blue and yellow pluses symbolize boundary parameters 𝑅 = 1 and 𝑅 = 2 for 𝑧 max = 5 𝐻 , while black and magenta pluses represent the same parameters for 𝑧 max = 8 𝐻 . \n<!-- image --> \nwhere the maximum growth rate is achieved towards the lower frequency. We find that the VSI is quite similar to the Papaloizou-Pringle instability (PPI), which requires a reflecting boundary for PPI to operate (Papaloizou & Pringle 1984). As a comparison, there exists more robust instability, such as Rossby wave instability (Lovelace et al. 1999; Huang & Yu 2022). The reflecting boundary is not necessary for RWI. The fully outgoing boundary conditions (i.e., 𝑅 = ∞ ) are often adopted in the RWI calculations. \nIn Fig. 5, we note that the growth rate varies non-monotonically with the frequency. Below a certain threshold frequency, the growth rate increases with the frequency. Above the threshold, the growth rate decreases with the frequency. The disk vertical extent has an important effect on the threshold frequency and corresponding growth rate. This phenomenon applies not only to reflecting BCs but also to partially reflecting BCs, as shown in Fig. 5. The threshold frequency for 𝑧 max = 8 𝐻 is larger than the threshold for 𝑧 max = 5 𝐻 . A larger vertical extent has a higher growth rate at the threshold frequency as well. Comparing the changes in growth rates due to contrasting BCs across the two vertical extents, the partially reflecting BCs have less effect on the growth rate in the larger one. Even so, as the ver- \nFigure 6. Growth rate and wave frequency of the unstable modes with ˆ 𝑘 𝑟 = 30, disk respect ratio ℎ = 0 . 05, and thermal relaxation timescale 𝛽 = 0 . 001 (black), 𝛽 = 0 . 01 (magenta), 𝛽 = 0 . 02 (blue), 𝛽 = 0 . 04 (yellow) for different boundary parameters 𝑅 = 2 (colored pluses) and 𝑅 = 1 (colored dots). \n<!-- image --> \nFigure 7. Growth rate and frequency of the unstable modes with radial wavenumber ˆ 𝑘 𝑟 = 10, disk aspect ratio ℎ = 0 . 05, cooling time 𝛽 = 10 -3 . We set perfectly reflecting BCs 𝑅 = -1 and azimuthal wavenumbers 𝑚 = 0 (black pluses), 𝑚 = 1 (magenta pluses), 𝑚 = 2 (blue pluses), 𝑚 = 3 (cyan pluses). \n<!-- image --> \nincreases, a very high 𝑅 still significantly suppresses the unstable modes, similar to the findings in Fig. 4. \nSince the cooling is essential for VSI to work, it would be interesting to study how the cooling affects VSI with our new BCs. We compare in Fig. 6 the mode growth rate and frequency for different values of cooling parameter 𝛽 with 𝑅 = 2 (partially reflecting) and 𝑅 = 1 (perfectly reflecting). It shows that, for either partially reflecting or perfectly reflecting BCs, the growth rates increase as 𝛽 varies from 0.04 to 0.001. At sluggish thermal relaxation, the growth of unstable modes is notably reduced. In this case, only the lower order unstable modes can survive. Furthermore, even with slower cooling, VSI is still affected by BCs. The augmentation of outgoing wave leads to a reduction in the growth rate of higher order modes, consistent with the scenario observed when cooling is set at 0.001 (as illustrated in Fig. 1). \n̂ \nFigure 8. unstable modes with different BCs and same azimuthal wavenumber (top), same BC but different azimuthal wavenumbers (bottom). \n<!-- image -->', '5.2 Non-axisymmetric VSI unstable modes': 'In Fig. 7, we show our results of non-axisymmetric VSI. The results for ˆ 𝑘 𝑟 = 10 with both 𝑚 ≠ 0 (non-axisymmetric) and 𝑚 = 0 (axisymmetric) are displayed. The parameter 𝑅 = -1 means perfect reflection with a half-wave loss. With the increase of 𝑚 , there is an overall decrease in the growth rate. Note that the azimuthal wavenumber 𝑚 ≪ ˆ 𝑘 𝑟 / ℎ , thus the low-frequency approximation could be used to deal with the non-axisymmetric perturbations. \nFig. 8 shows non-axisymmetric VSI with larger radial wavenumber for both the body and surface modes. The top panel shows the effect of BCs on non-axisymmetric unstable modes with 𝑚 = 2. Consistent with the trend in the axisymmetric scenario, boundaries characterized by a stronger outgoing component suppress the growth rate of nonaxisymmetric instability as well. The bottom panel elucidates the variation in the growth rate caused by non-axisymmetric effects. In the lower panel, we fix the parameter 𝑅 = 2 at the upper boundary 𝑧 = 5 𝐻 . For instance, at the frequency of | 𝜔 | = 3 . 22 ℎ Ω 𝐾 , the growth rate decreases from 0 . 381 ℎ Ω 𝐾 to 0 . 307 ℎ Ω 𝐾 as the wavenumber 𝑚 increases from 0 to 3. \nComparing the two panels of Fig. 8, the evident changes of unstable modes caused by BCs occur primarily for higher-order body modes, whereas non-axisymmetric effects demonstrate a global impact across the entire frequency range for both surface and body modes. The non-axisymmetric effect is significant and can not be neglected. Though we do not yet have a satisfactory physical explanation for the 𝑚 -dependence of VSI, our calculations indicate the term -3 𝑚ℎ ˆ 𝑘 𝑟 in the variable 𝑊 = 4 𝑖𝑚ℎ 2 -3 𝑚ℎ ˆ 𝑘 𝑟 plays an essential role for this dependence.', '6 CONCLUSION': 'In this paper, we analyze the VSI with partially reflecting BCs for vertically global and radially local protoplanetary disks. The partially reflecting BCs are viewed as a linear combination of the incoming and outgoing wave components. By adjusting the relative magnitudes of the two wave components, we can investigate the response of the VSI to different BCs. \nWe find that the growth rate of VSI is closely related to the BCs, especially for higher-order breathing and corrugation modes whose frequency exceeds the threshold. The growth rate decreases when the boundary parameter, 𝑅 , increases. In other words, the growth rate of the instability diminishes when the outgoing wave component dominates. As the outgoing component increases, the location of the maximum growth rate shifts toward a lower frequency. \nHow waves propagate away from the surface of realistic protoplanetary disks is determined by the atmosphere of the disk (Ogilvie & Lubow 1999). The fraction of the reflected and transmitted wave depends critically on the transition region between the disk and the atmosphere. In this paper, we use a free boundary parameter, 𝑅 , to describe the wave behavior at this disk-atmosphere transition region. Whether the disk atmosphere enhances or suppresses the VSI depends on the characteristics of this transition region. Our current understanding of the physical conditions of protoplanetary disks and their atmosphere is still limited. It would be interesting to perform further studies on wave propagation in a more realistic disk-atmosphere transition region. \nThe vertical velocity perturbation exhibits symmetric or antisymmetric properties in the case of reflective BC at both the upper and lower boundaries (Nelson et al. 2013). It is possible that the upper and lower disk-atmosphere transition regions have different physical properties. In this case, the waves would behave differently at the upper and lower boundaries, and the inherent symmetry (or anti-symmetry) would be broken. \nWe examine the growth rate of VSI with different vertical extents. As the vertical extent increases, the maximum growth rate of VSI increases as well. The finding is consistent with the discoveries made by Barker & Latter (2015). Under partially reflecting BCs, this property remains unchanged. In Lin & Youdin (2015), the free BCs establish a zero Lagrangian pressure perturbation, consistent with a reflecting boundary with a phase difference. Moreover, the rigid BC ( 𝛿𝑢 𝑧 = 0) they used aligns with our calculations with 𝑅 = -1. We have also performed calculations with boundary parameters 𝑅 = 1 (reflecting BC) and 𝑅 = -1. In both of these cases, the growth rates of VSI are found to be consistent. \n. We find that the non-axisymmetric modes are also unstable and they grow at a rate that decreases with the azimuthal wavenumber. In a non-axisymmetric scenario, the BCs favoring the outgoing wave component still have a suppression effect on the VSI. \nMHDdisk winds can potentially initiate outgoing flow in realistic PPDs. The presence of anti-symmetric magnetic field modes leads to one-sided characteristics in disk winds, which can further cause asymmetry in the outgoing flow of the disk. These factors will affect the development of VSI and may significantly influence planet formation and evolution. Realistic protoplanetary disks are magnetized. It would be interesting to take into account the effects of magnetic fields, incorporating our new treatment of BCs to explore the VSI in magnetized protoplanetary disks.', 'ACKNOWLEDGEMENTS': "We are grateful for the anonymous referee's valuable comments and suggestions that improve the manuscript. This work has been supported by the National SKA Program of China (Grant No. 2022SKA 0120101) and the National Key R&D Program of China (No. 2020YFC2201200), the science research grants from the China Manned Space Project (No. CMS-CSST-2021-B09, CMSCSST2021-B12 and CMS-CSST-2021-A10), and opening fund of State Key Laboratory of Lunar and Planetary Sciences (Macau University of Science and Technology) (Macau FDCT Grant No. SKLLPS(MUST)-2021-2023). C.Y. has been supported by the National Natural Science Foundation of China (Grant Nos. 11521303, 11733010, 11873103, and 12373071).", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'REFERENCES': 'Bai X.-N., 2011, ApJ, 739, 51 Barker A. J., Latter H. N., 2015, MNRAS, 450, 21 Cui C., Bai X.-N., 2020, ApJ, 891, 30 Cui C., Bai X.-N., 2021, MNRAS, 507, 1106 Cui C., Lin M.-K., 2021, MNRAS, 505, 2983 Cui C., Tripathi A., Yu C., Lin M.-K., Youdin A., 2024, arXiv e-prints, p. arXiv:2407.02103 Flaherty K., Hughes A. M., Mamajek E. E., Murphy S. J., 2019, ApJ, 872, 92 Flaherty K., et al., 2024, in American Astronomical Society Meeting Abstracts. p. 106.04 Goldreich P., Schubert G., 1967, ApJ, 150, 571 Goodman J., Lackner C., 2009, ApJ, 696, 2054 Huang S., Yu C., 2022, MNRAS, 514, 1733 Klahr H., 2024, arXiv e-prints, p. arXiv:2404.15933 Klahr H., Hubbard A., 2014, ApJ, 788, 21 Latter H. N., Kunz M. 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P., 1992, Numerical recipes in FORTRAN. The art of scientific computing. Press Syndicate of the University of Cambridge Schiff L. I., 1968, Quantum Mechanics, 3rd edn. McGraw-Hill Education Stoll M. H. R., Kley W., 2014, A&A, 572, A77 Svanberg E., Cui C., Latter H. N., 2022, MNRAS, 514, 4581 Teague R., et al., 2016, A&A, 592, A49 Urpin V., 2003, A&A, 404, 397 Urpin V., Brandenburg A., 1998, MNRAS, 294, 399 Wang L., Xu S., Wang Z., Fang M., Goodman J., 2024, ApJ, 972, 142 \nYamaguchi M., et al., 2024, PASJ, 76, 437 \nYellin-Bergovoy R., Umurhan O. M., Heifetz E., 2021, Geophysical and Astrophysical Fluid Dynamics, 115, 674 Youdin A. N., Lithwick Y., 2007, Icarus, 192, 588 Yu C., Lai D., 2013, MNRAS, 429, 2748 Zhang S., Zhu Z., Jiang Y.-F., 2024, ApJ, 968, 29', 'APPENDIX A: DERIVATION OF THE BOUNDARY CONDITIONS': 'In this appendix, we show the derivation of Equation (28). According to the WKB approximation 𝑦 1 , 𝑦 2 ∝ 𝑒 𝑖 ˆ 𝑘 𝑧 ˆ 𝑧 , we can readily know that 𝑑𝑦 1 / 𝑑 ˆ 𝑧 = 𝑖 ˆ 𝑘 𝑧 𝑦 1 , 𝑑𝑦 2 / 𝑑 ˆ 𝑧 = 𝑖 ˆ 𝑘 𝑧 𝑦 2 . Note that ˆ 𝑘 𝑧 here is a local constant associated with the boundaries 3 . Then Eq. (17) becomes \n( 𝐴 11 -𝑖 ˆ 𝑘 𝑧 ) 𝑦 1 + 𝐴 12 𝑦 2 = 0 , 𝐴 21 𝑦 1 + ( 𝐴 22 -𝑖 ˆ 𝑘 𝑧 ) 𝑦 2 = 0 . (A1) \nTo ensure the existence of nontrivial solutions of 𝑦 1 and 𝑦 2 , the matrix 𝑨 must satisfy \nGLYPH<12> GLYPH<12> 𝑨 -𝑖 ˆ 𝑘 𝑧 𝑰 GLYPH<12> GLYPH<12> = GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 𝐴 11 -𝑖 ˆ 𝑘 𝑧 𝐴 12 𝐴 21 𝐴 22 -𝑖 ˆ 𝑘 𝑧 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> = 0 . (A2) \nThe equation (A2) can be rewritten as a quadratic equation for 𝑖 ˆ 𝑘 𝑧 : \nGLYPH<16> 𝑖 ˆ 𝑘 𝑧 GLYPH<17> 2 -𝑖 ˆ 𝑘 𝑧 ( 1 + 𝑄 ) ˆ 𝑧 -Δ = 0 , (A3) \nwhere Δ = -ˆ 𝜐 2 ( 𝜒 + 𝑈 ) + ( 𝜒 -1 ) ˆ 𝑧 2 [ 𝑄 -𝑈 ] . This quadratic equation for 𝑖 ˆ 𝑘 𝑧 can be readily solved. The two roots are \n𝑖 ˆ 𝑘 𝑧 1 = 1 2 GLYPH<20> ( 1 + 𝑄 ) ˆ 𝑧 + 𝑖 √︃ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<21> , (A4) \n𝑖 ˆ 𝑘 𝑧 2 = 1 2 GLYPH<20> ( 1 + 𝑄 ) ˆ 𝑧 -𝑖 √︃ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<21> . (A5) \nNote that there are two eigenvectors associated with the two eigenvalues of the matrix 𝑨 4 . According to the standard algorithm in linear algebra, the two eigenvectors associated with 𝑖 ˆ 𝑘 𝑧 1 and 𝑖 ˆ 𝑘 𝑧 2 can be written as ： \n𝒓 1 = GLYPH<18> 𝐵 1 -𝐴 1 ˆ 𝑧 + 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<19> , 𝒓 2 = GLYPH<18> 𝐵 1 -𝐴 1 ˆ 𝑧 -𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<19> , (A6) \nwhere \n𝐴 1 = 2 𝜒 + 𝑄 -1 , 𝐴 2 = ( 𝑄 + 1 ) 2 + 4 ( 𝜒 -1 ) ( 𝑄 -𝑈 ) , (A7) \n3 We adopt the Wentzel-Kramers-Brillouin (WKB) approximation, which is typically used for a semi-classical calculation in quantum mechanics in which the wave function is recast as an exponential function (Schiff 1968). That is the theme of our treatment for the boundary condition. This method aims to find approximate solutions to linear ODEs with spatially varying coefficients. For the global vertical shear instability analysis, the relevant disk physical variables are supposed to be viewed as wave components with vertical wave number that varies with the vertical positions. But when it comes to the boundaries, we actually focus on the very regions that are spatially local. As a result, the variations of wave number are small and could be approximated as a local constant. In addition to quantum mechanics, such WKB approximations are also widely used in the astrophysical wave dynamics (Li et al. 2000; Cui et al. 2024). \n𝐴 3 = 4ˆ 𝜐 2 ( 𝜒 + 𝑈 ) , 𝐵 1 = 2 𝑖 ˆ 𝜐 ( 𝜒 + 𝑈 ) , (A8) \nrespectively. Note that these two eigenvectors represent waves with different group velocities. \nThe equation (26) can be written explicitly as: \n( 𝑎 1 + 𝑎 2 ) = 𝑦 1 / 𝐵 1 , (A9) \n( 𝑎 1 -𝑎 2 ) = 𝑦 2 + 𝐴 1 ˆ 𝑧 ( 𝑎 1 + 𝑎 2 ) 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 = 𝑦 2 + 𝐴 1 ˆ 𝑧𝑦 1 / 𝐵 1 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 . (A10) \nWith some simple manipulations, we can arrive at: \n𝑅 ≡ 𝑎 2 𝑎 1 = h -𝐴 1 ˆ 𝑧 + 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 i 𝑦 1 -𝐵 1 𝑦 2 h 𝐴 1 ˆ 𝑧 + 𝑖 √︁ 𝐴 3 -𝐴 2 ˆ 𝑧 2 i 𝑦 1 + 𝐵 1 𝑦 2 , (A11) \nwhich could be recast as a linear combination of 𝑦 1 and 𝑦 2 , namely: \nGLYPH<18> 𝐴 1 ˆ 𝑧 + 𝑅 -1 𝑅 + 1 𝑖 √︃ 𝐴 3 -𝐴 2 ˆ 𝑧 2 GLYPH<19> 𝑦 1 + 𝐵 1 𝑦 2 = 0 . (A12) \nThis is exactly Equation (28) in the main text. A larger 𝑅 indicates a greater outgoing component. Its non-axisymmetric counterpart, the Equation (30) could be obtained in a similar manner. \nNote that our BCs are quite general and the free BC (zero Lagrangian pressure perturbation, i.e., Δ 𝑃 = 0) in Lin & Youdin (2015) can be recovered with an appropriate value of 𝑅 free . Numerical calculation shows that the modulus of | 𝑅 free | = 1. The free BC actually means a reflecting boundary with a phase difference between the incoming and outgoing waves.', 'APPENDIX B: DISPERSION RELATION AND IDENTIFICATION OF WAVE PROPAGATION DIRECTION': 'With the WKB approximation, the local dispersion relation is: \n𝑐 4 ˆ 𝜐 4 + 𝑐 3 ˆ 𝜐 3 + 𝑐 2 ˆ 𝜐 2 + 𝑐 1 ˆ 𝜐 + 𝑐 0 = 0 , (B1) \nwhere coefficients 𝑐 0 , 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 are \n𝑐 0 = ˆ 𝑘 2 𝑧 + 𝑖 ( 𝑄 + 1 ) ˆ 𝑧 ˆ 𝑘 𝑧 , 𝑐 1 = 𝑖 𝛽 ( 𝛾 -1 ) ( 𝑄 -𝑈 ) ˆ 𝑧 2 , (B2) \n𝑐 2 = 𝛽 2 𝛾 2 𝑐 0 + 𝑖 𝛽𝛾𝑐 1 -𝑈 -1 , 𝑐 3 = -𝑖 𝛽 ( 𝛾 -1 ) , (B3) \n𝑐 4 = -𝛽 2 𝛾 [ 1 + 𝛾𝑈 ] , (B4) \nrespectively. Note that this dispersion relation is equivalent to the Equation (A3). According to the definition of group velocity 𝑣 𝑔 = 𝜕𝜐 𝜕 ˆ 𝑘 𝑧 , we know that the group velocity is ： \n𝑣 𝑔 = -( ˆ 𝜐 2 𝛽 2 𝛾 2 + 1 ) GLYPH<2> 2 ˆ 𝑘 𝑧 + 𝑖 ( 𝑄 + 1 ) ˆ 𝑧 GLYPH<3> / 𝐷 . (B5) \nwhere 𝐷 = 4 𝑐 4 ˆ 𝜐 3 + 3 𝑐 3 ˆ 𝜐 2 + 2 𝑐 2 ˆ 𝜐 + 𝑐 1 . The inverse function rule is applied during the calculation because ˆ 𝑘 𝑧 depends on 𝜐 and not the other way around. \nNote that there are two roots of the quadratic equation (A3) for ˆ 𝑘 𝑧 . Each root of ˆ 𝑘 𝑧 has a corresponding eigenvector. The specific form of each eigenvector is shown in Equation (A6). It is clear that, for the wave associated with 𝒓 2 , the group velocity 𝑣 𝑔 > 0 (outgoing) at the outer boundary where 𝑧 = 5 𝐻 . For the wave associated with 𝒓 1 , the group velocity 𝑣 𝑔 < 0 (incoming) at outer boundary 𝑧 = 5 𝐻 . The \nwave assignment is general for all eigenfunctions and independent of the chosen parameters. It also holds in general for the case of 𝑚 ≠ 0. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author.'}

2024arXiv240913011S

Massive blackhole binaries will be the loudest sources detectable by LISA. These systems are predicted to form during the hierarchical assembly of cosmic structures and coalesce by interacting with the surrounding environment. The hardening phase of their orbit is driven by either stars or gas and encodes distinctive features into the binary black holes that can potentially be reconstructed with gravitationalwave observations. We present a Bayesian framework to assess the likelihood of massive mergers being hardened by either gaseous or stellar interactions. We use stateoftheart astrophysical models tracking the cosmological evolution of massive blackhole binaries and construct a large number of simulated catalogs of sources detectable by LISA. From these we select a representative catalog and run both parameter estimation assuming a realistic LISA response as well model comparison capturing selection effects. Our results suggest that at least within the context of the adopted models future LISA observations can confidently constrain whether stars or gas are responsible for the binary hardening. We stress that accurate astrophysical modeling of the blackhole spins and the inclusion of subdominant emission modes in the adopted signal might be crucial to avoid systematic biases.

2024-09-01T00:00:00Z

['arXiv:2409.13011', '10.48550/arXiv.2409.13011', '2024arXiv240913011S']

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

Stars or gas Constraining the hardening processes of massive blackhole binaries with LISA

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.13011.pdf

{'Stars or gas? Constraining the hardening processes of massive black-hole binaries with LISA': "Alice Spadaro , Riccardo Buscicchio , David Izquierdo-Villalba , 1, 2 3 3 \n1, 2, ∗ 1, 2, 3 1, 2 \nDavide Gerosa , Antoine Klein , and Geraint Pratten \n1 Dipartimento di Fisica 'G. Occhialini', Universit'a degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy \n2 INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy \n3 Institute for Gravitational Wave Astronomy & School of Physics and \nAstronomy, University of Birmingham, Birmingham, B15 2TT, UK \n(Dated: September 23, 2024) \nMassive black-hole binaries will be the loudest sources detectable by LISA. These systems are predicted to form during the hierarchical assembly of cosmic structures and coalesce by interacting with the surrounding environment. The hardening phase of their orbit is driven by either stars or gas and encodes distinctive features into the binary black holes that can potentially be reconstructed with gravitational-wave observations. We present a Bayesian framework to assess the likelihood of massive mergers being hardened by either gaseous or stellar interactions. We use state-of-theart astrophysical models tracking the cosmological evolution of massive black-hole binaries and construct a large number of simulated catalogs of sources detectable by LISA. From these, we select a representative catalog and run both parameter estimation assuming a realistic LISA response as well model comparison capturing selection effects. Our results suggest that, at least within the context of the adopted models, future LISA observations can confidently constrain whether stars or gas are responsible for the binary hardening. We stress that accurate astrophysical modeling of the black-hole spins and the inclusion of subdominant emission modes in the adopted signal might be crucial to avoid systematic biases.", 'I. INTRODUCTION': "According to the current ΛCDM model of structure formation, galaxies form hierarchically [1]. In this context, the formation of massive black-hole (BH) binaries is a direct consequences of galaxy mergers [2]. Present evidence for massive BH binaries relies on observational signatures of active galactic nuclei, including offset or double-peaked spectral lines and periodic variations in light curves [3, 4]. While electromagnetic observations lack the capability to directly resolve massive BH binaries below the parsec scale, the recent evidence for a nanoHertz stochastic gravitational-wave (GW) signal in pulsar timing array datasets points to the existence of a large cosmic population of high mass ( > 10 7 M ⊙ ) merging BH binaries [5-8]. The detection of individual mergers of 10 4 -10 7 M ⊙ BHs is a cornerstone in the science case of future space-based GW detectors, notably the Laser Interferometer Space Antenna (LISA) [9]. \nThe astrophysical environments surrounding massive BH binaries are crucial in shaping the binary dynamics and promoting the final coalescence [10]. In particular, mergers can only be explained if dissipation mechanisms of astrophysical nature are at play in the early inspiral, before binaries enters their GW-driven regime. Modeling such processes is an active field of research, as different assumptions lead to different predictions for both properties and rates of merging massive BHs. Broadly speaking, hardening mechanisms can be divided into 'gas hardening' \nand 'stellar hardening' [11]. The former, which dominates in gas-rich environments, is driven by the interaction of the massive BH binary with a gaseous circumbinary disk [12, 13]. Conversely, the latter takes place primarily in gas-poor environments, where the binary orbital separation shrinks due to three-body interactions with individual stars [14]. Both gas and stellar hardening affect the merger rate [15, 16] and can potentially be reconstructed with LISA observations [17]. \nWith this in mind, we present a Bayesian framework to compare single detections of massive BH binaries from LISA-simulated catalogs against state-of-theart astrophysical simulations. We use the L-Galaxies semi-analytic model [18] applied to the high-resolution Millenium-II dark-matter (DM) merger trees. We specifically investigate whether it will be possible to infer that putative massive BH binaries observed by LISA have evolved in either gas or stellar environments. We construct realistic LISA catalogs, perform parameter estimation including the full LISA response with the Balrog [19-21] code, and implement a Bayesian model comparison between the two hardening mechanisms while considering selection effects [22]. \nThis paper is organized as follows. In Sec. II, we illustrate the adopted astrophysical models. In Sec. III, we present the construction of our LISA catalogs. In Sec. IV, we describe our parameter-estimation and model-selection strategies. In Sec. V, we present and discuss our findings. Finally, in Sec. VI we summarize our results and outline possible future developments. Throughout this paper, we use units where c = 1.", 'A. Galaxy evolution model': 'We use the L-Galaxies semi-analytical model as presented in Refs. [18, 23, 24]. In particular, the model is built on top of merger trees from the MilleniumII [25] cosmological simulation, which follows the evolution of 2160 3 dark matter particles with a mass of 6 . 885 × 10 6 M ⊙ /h within a comoving box of side 100 Mpc /h , later rescaled [26] to match the Planck cosmology [27] (here h = 10 -2 H 0 Mpc s/km and H 0 is the Hubble constant). \nL-Galaxies associates a fraction of baryonic matter to each newly resolved DM halo in the form of a diffuse, spherical, and quasi-static hot gas atmosphere. As the gas cools down, it settles into a disk [28]; this facilitates episodes of star formation, which in turn results in the assembly of a stellar disk component. Galaxies are also allowed to assemble an overdensity of stars in their central regions (i.e. their bulges). This is triggered by either internal processes, namely non-axisymmetric instabilities that redistribute the stellar matter, or galaxy mergers.', 'B. Black-hole growth': 'Each newly resolved DM halo is initialized with a massive BH seed spanning a mass range from ∼ 10 2 to ∼ 10 5 M ⊙ [24]. The initial dimensionless spin parameter χ is set uniformly in [0 , 0 . 998] [29]. \nThe evolution of masses and spins is influenced by the accretion of surrounding gas into the BHs as well as BH-binary coalescences. In particular, gas accretion is triggered by both galaxy mergers and disk instabilities [3032], and predominantly drives the mass growth [33]. The cold gas available for accretion settles in a reservoir of mass M res around the BH, which is progressively consumed through a sequence of transient accretion disks [23, 34]. As for the BH spin, its value is set by the frequency of accretion events consumed in prograde or retrograde orbits, which in L-Galaxies is linked to the coherence of the bulge kinematics [35, 36]. The contribution of binary coealescences to the BH mass and spin evolution is important at low redshifts ( z < 2), where galaxies have mostly depleted their gas reservoirs through star formation and feedback processes. \nThe post-merger BH masses and spins are computed using fits to numerical-relativity simulations [37, 38]. During the BH evolution, L-Galaxies tracks the evolution of the spin magnitude χ but not the spin direction, which is however important for LISA observations. In this work, we construct this property for each binary component using the gas fraction of the environment f gas = M res / ( M res + M ), where M is the total mass of the BH binary. If f gas > 0 . 5, coalescence occurs in a gas-rich environment and the two spins are assumed to \nFIG. 1. Density distribution of the total mass M and redshift z of massive BH binary mergers. Our fiducial population is shown in gray, where the contour encloses 99% of the estimated source density. The red (blue) distribution corresponds to the sub-population of mergers hardened by gas (stars); contours contain 50%, 90%, and 99% of the estimated source density. Thick contours refer to additional realizations to capture the statistical fluctuations in the underlying astrophysical model. Circles denote the 44 sources from TheCatalog used in our analysis, with colors indicating their respective f gas values. Note the two pairs of overlapping sources, c.f. Fig. 4. \n<!-- image --> \n/circledot \nalign to the binary orbital angular momentum via the Bardeen-Petterson effect [39-41]. If instead f gas < 0 . 5, the environment is gas-poor and the BH spins are assumed to be isotropically distributed [33, 42].', 'C. Binary hardening': 'The dynamical pathway of massive BH binaries can be divided into three phases: pairing, hardening, and GW inspiral [2]. Following a galaxy merger, the pairing phase reduces the BH-BH separation from ∼ kpc to ∼ pc through dynamical friction. This is implemented in LGalaxies as in Ref. [43]. As the binary forms, different physical processes contribute to the shrinking of the binary semi-major axis depending on the gas fraction f gas . For f gas > 0 . 5, gas hardening dominates and the decay of the binary orbit is set by accretion torques from the circumbinary disk. In gas-poor environments with f gas < 0 . 5, instead, the binary evolution is driven by capture and ejection of stars. \nGravitational-radiation reaction becomes dominant at roughly milliparsec scales and drives the system to the merger. For details on the L-Galaxies implementation see Ref. [23]. \nFIG. 2. Distribution of the gas fraction f gas of the environment across the fiducial population. Sources in blue (red) with f gas < 0 . 5 ( f gas > 0 . 5) are classified as mergers occurring in a gas-poor (gas-rich) environment. The two sub-population distributions are jointly normalized to the full population. \n<!-- image -->', 'D. Fiducial population': "The model described in the previous subsections provides the astrophysical population of BH mergers we refer to as 'fiducial', which is depicted in grey in Fig. 1 and presented in Ref. [44]. This population comprises two distinct sub-populations with peaks at f gas ∼ 0 (stellar hardening) and f gas ∼ 1 (gas hardening) parameter (Fig. 2). The latter dominates the overall merger rate by a factor of ∼ 5. Figure 1 shows the respective mass and redshift distributions for each sub-populations. Systems evolving in gaseous environments merge on average at higher redshifts z ≲ 8 and have lower masses M ∼ 10 4 -10 6 M ⊙ compared to binaries hardened by stellar interactions which instead are more prominent at z ≲ 3 and have M ∼ 10 5 -10 9 M ⊙ . Nonetheless, the two populations overlap significantly in the M -z parameter space, making their distinguishability with LISA non trivial.", 'III. MOCK LISA CATALOGS': "We build mock LISA observation catalogs on top of the L-Galaxies simulations. The latter provides the number density of massive BH binary mergers, d 2 N/ d z d V c , per unit redshift z and comoving volume V c . We proceed as follows: \n- (i) We compute the total merger rate of massive BH mergers predicted by the fiducial model [45] \nd N d t obs = ∫ d N d z w ( z ) d z, (1) \nwhere \nw ( z ) = 4 π [ d L (1 + z ) ] 2 d z d V c . (2) \nIn the equations above, d L is the luminosity distance to the source and t obs is the time measured at the detector. \n- (ii) We draw the number of sources N astro according to a Poisson distribution with mean value λ = T × d N/ d t obs where T is the duration of the observing period. Massive BH binary signals in LISA typically last hours to months and gradually accumulate signal-to-noise ratio (SNR) over their inspiral. We conservatively set T = 5yr in the calculation above, which accommodates for signals merging after the nominal duration T LISA = 4yr [9].\n- (iii) We extract N astro sources from the astrophysical population with relative weights given by w i = w ( z i ). Each source is characterized by the binary component masses m 1 , 2 , mass ratio q = m 2 /m 1 , the aligned dimensionless spins χ 1 , 2 , and the redshift z .\n- (iv) From these, we further select sources with a cutoff frequency f cut > 1 × 10 -4 Hz and q > 0 . 01. We set f cut = 5 f ISCO where f ISCO is the detector-frame GW frequency at innermost stable circular orbit. This is to ensure that the dominant mode is in the LISA sensitivity band and that the adopted waveform model is reliable.\n- (v) We use the IMRPhenomXHM [46] waveform approximant which captures the full coalescence of quasi-circular, non-precessing BH binaries. The harmonics h lm are calibrated to numerical relativity and include the ( l, | m | ) = { (2 , 2) , (2 , 1) , (3 , 3) , (3 , 2) , (4 , 4) } multipoles. The implementation of the LISA response to such GW signal in the Balrog code has been presented in Ref. [20].\n- (vi) For each source, we draw its extrinsic parameters by sampling uniformly the time to merger t m ∈ [0 , 5yr], the initial phase ϕ 0 ∈ [0 , 2 π ], the polarization angle ψ ∈ [ -π/ 2 , π/ 2], the ecliptic sine latitude sin β ∈ [ -1 , 1] , the ecliptic longitude λ ∈ [0 , 2 π ], and the cosine inclination cos ι ∈ [ -1 , 1].\n- (vii) The SNR of each source, given its parameters θ , is evaluated using the three noise-orthogonal TDI channels h = { h k ; k = A,E,T } as follows \nSNR 2 = ∑ k 〈 h k ( θ ) ∣ ∣ h k ( θ ) 〉 k . (3) \nWe assume constant and equal-armlength approximation [47]. The inner product is given by \n〈 a | b 〉 k = 2 ∫ f max f min ˜ a ( f ) ˜ b ∗ ( f ) + ˜ a ∗ ( f ) ˜ b ( f ) S k ( f ) d f, (4) \n<!-- image --> \n<!-- image --> \nGas hardening \nFIG. 3. Distribution of massive BH binary mergers across the generated catalogs. The left (gray), middle (blue), and right (red) panel shows the full population of BHs, the sub-population of stellar-hardened sources, and the sub-population of gas-hardened sources, respectively. Light histograms show the astrophysical distributions [steps (i) -(iii) in Sec. III] while dark histograms show the distributions of merger events detectable by LISA during an observation time T = 5yr [steps (iv)-(vii) in Sec. III]. The dashed distributions are a variation of the latter where we include higher-order harmonics and remove the condition on f cut [step (iv)]. The vertical dotted line denotes the selected catalog used to perform the statistical analysis of Sec. IV. \n<!-- image --> \nwhere ˜ a ( f ) denotes the Fourier transform of the time series a ( t ) and S k ( f ) the noise power spectral density of the k -th TDI channel. For the latter, we use the semi-analytical expression of Ref. [48] which models the superposition of LISA stationary instrumental noise and astrophysical confusion noise from unresolved Galactic binaries. We use a low-frequency cut-off f min = 0 . 1 mHz, as set in the ESA definition study report [9], and integrate up to f max = 1Hz, which is well above the maximum frequency of all generated GW signals. We consider sources detectable if they exceed the threshold SNR = 10 [9]. \nWe iterate the above procedure 1000 times, resulting in a distribution of LISA catalogs. The corresponding distributions for the number of merger events is shown in Fig. 3. Specifically, steps (i) -(iii) generate the light-gray histogram in the left panel, depicting the distribution of astrophysical mergers predicted by the fiducial model. The dark-gray histogram is obtained by applying steps (iv) -(vii) to the astrophysical catalogs and shows the distribution of massive BH merger events detectable by LISA assuming the dominant l = 2, | m | = 2 emission mode. The blue (red) histograms in the middle (right) panel show the subpopulations of mergers that evolved via stellar (gas)-hardening. Overall, the LISA catalogs have about one order of magnitude fewer events compared to the astrophysical catalogs, which is due to the detection criteria described above. The observable catalogs predominantly feature massive BH binary mergers occurring in gas-rich environments (46 on average) compared to systems evolving in gas-poor environments (1 on average). The averaged count of detectable sources mildly increases to 50 when we relax the assumption on the mass ratio, allowing q to be as small as 0.001. These sources, and those at even lower mass ratio, border the extreme mass-ratio inspiral regime and will need to be treated differently; we \nleave this to future work. \nThe dashed distributions in all the three panels of Fig. 3 show the detectable events modeled using the additional modes ( l, | m | ) = { (2 , 1) , (3 , 3) , (3 , 2)(4 , 4) } in the waveform approximant and removing the condition on f cut [step (iv)]. We find higher-order modes marginally increase the number of detectable sources (see Ref. [49]). This is more significant for massive binaries with M ≳ 10 8 M ⊙ , where the (2,2) mode frequency falls out of band. As shown in Fig. 3, this preferentially impacts binaries hardened via stellar processes: including higher-order modes in LISA analyses might have important repercussions on the astrophysical interpretation of the data. \nIn the following, we study in detail one specific realization of our simulated LISA catalogs, which we refer to as TheCatalog . This specific catalog is close to the medians of the distributions in Fig. 3 and can thus be taken as representative. The 45 detectable sources of TheCatalog are shown as colored circles in the right panel of Fig. 4 together with the astrophysical population from which they are extracted. The left panel puts these sources into context by showing the common 'waterfall plot', i.e. the averaged SNR in the mass-redshift space [9]. LISA is expected to detect massive BH binary systems in the mass range of M tot ∼ 10 4 -10 7 M ⊙ and out to z ∼ 7, covering an SNR range from ∼ 10 to ∼ 3000. The 13 sources marked with white triangles in Fig. 4 are undetectable by LISA. Of these, 9 sources with M tot > 10 4 M ⊙ merge after the LISA mission nominal duration, thereby limiting their SNR growth, cf. step (ii) in Sec. III. Conversely, the 4 sources with M tot < 10 4 M ⊙ merge within the mission lifetime T LISA but remain with SNR < 10 nonetheless. These undetectable sources are excluded from the analysis presented in Sec. IV. \nStellar hardening \nFIG. 4. Left panel. Contour lines of constant SNR from massive BH binaries ( q = 0 . 5, χ 1 , 2 = 0 . 2) detectable by LISA, adapted from Ref. [9]. The SNR shown on the color scale has been averaged over sky location, polarization and inclination assuming the dominant quadrupole emission mode. Right panel. The gray contour encompasses 99% of the estimated astrophysical BH population simulated with L-Galaxies . Circles indicate the 45 detectable sources of TheCatalog , colored according to their SNR [see step (vii) in Sec. III]. Of note, two pairs of sources overlap in this plot: one, located at z ∼ 4 . 5 with M ∼ 6 × 10 3 M ⊙ , have the same weights w i [i.e. identical source parameters, see step (iii) in Sec. III]; the other pair, located at z ∼ 2 with M ∼ 4 × 10 4 M ⊙ , are characterized by very similar parameter values. White triangles indicate sources with SNRs below the detection threshold. Note that undetectable sources with high masses ( M > 10 4 M ⊙ ) are affected by the cut-off on the time to merger [step (ii) in Sec. III]. The dashed contour lines of constant SNR are the same of the left panel. \n<!-- image --> \n/circledot \n/circledot", 'IV. STATISTICAL INFERENCE': 'We compare a single GW events against distributions of simulated sources using the Bayesian formalism spelled out in Ref. [22]; see also Refs. [50, 51]. Specifically, we quantify the relative degree of consistency between individual detections from TheCatalog and the two subpopulation models of binary hardening: gas (G) and stellar (S). \nGiven the posterior distribution p ( θ | d, U) from each individual GW event obtained with some uninformative priors p ( θ | U), we compute the Bayes factor between models G and S as follows: \nB G / S = B G / U B S / U = ∫ p ( θ | d, U) p ( θ | G) p ( θ | U) d θ ∫ p ( θ | d, U) p ( θ | S) p ( θ | U) d θ . (5) \nThe astrophysical probability densities p ( θ | G) and p ( θ | S) act as new informative priors on the targeted parameters which we reconstruct from the simulated sources using Gaussian kernel density estimates (KDEs) [52]. \nParameter estimation is performed through Balrog on θ = { θ α , θ ζ } where θ α = {M c , δµ, χ 1 , χ 2 , d L } represent the quantities used to perform the astrophysical model selection [Eq. (6)] and θ ζ = { t m , ϕ 0 , ψ, sin β, λ, cos ι } are the additional extrinsic parameters characterizing the \nmodeled signals. In particular, we use the redshifted chirp mass M c and the dimensionless mass difference δµ = ( m 1 -m 2 ) / ( m 1 + m 2 ) as they are the two mass parameters that enter the Post-Newtonian (PN) evolution at the leading- and next-to-leading order, respectively. We run full Bayesian inference on simulated data using the nested sampling algorithm [53] as implemented in Nessai [54]. All the injections are in zero noise and we choose uniform priors p ( θ | d, U) on each parameter over either its entire definition domain or a range that is sufficiently large to enclose the entire posterior. In particular, we classify sources into three SNR intervals: low (10 -100), moderate (100 -1000), and high ( > 1000). For each of these, we estimate prior limits by calibrating against a fiducial source selected at the lower boundary of that range. Therefore, the terms p ( θ α | U), p ( θ ζ | U), p ( θ ζ | G), p ( θ ζ | S) [see step (vi) of Sec. III] are constant, we can factor them out the integrals of Eq. (5) and evaluate the KDEs exclusively on θ α . This approach significantly improves the computational efficiency. Equation (5) thus simplifies to \nB G/S = ∫ p ( θ α | d, U) p ( θ α | G)d θ α ∫ p ( θ α | d, U) p ( θ α | S)d θ α . (6) \nNote that our detection statistics is approximate as GW detectability depends on the data realization (for detailed \nFIG. 5. Logarithmic Bayes factor between the gas- and stellarhardening hypotheses as a function of the source SNR. Circles represent the 44 sources from TheCatalog used in the population analysis, with colors indicating their respective f gas values. Horizontal dashed lines denote the threshold values of the Bayes factor according to the Jeffrey scale. Black error bars refer to the statistical (but not systematic) fluctuations associated to the underlying astrophysical model. \n<!-- image --> \ndiscussions on this point see e.g. Refs. [55, 56]). We verified that all of our posterior samples have SNR > 10 and q > 0 . 01, which ensure consistency with the adopted detection criterion. Furthermore, it is important to note that posterior samples constrained to a parameter space θ α that is not supported by the gas [stellar] probability density, i.e. p ( θ α | G) = 0 [ p ( θ α | S) = 0], result in a formally null [infinite] value for the Bayes factor in Eq. (6). As a concrete example, see the discussion on the spin distributions in Sec.V. \nCrucially, one needs to account for selection effects. The detectability-conditioned Bayes factor is given by [22] \nD G / S = P (det | S) P (det | G) B G / S , (7) \nwhere P (det | S) and P (det | G) are the fraction of sources from the sub-populations S and G, respectively, that can be detected a priori given adopted detection statistic SNR > 10 and q > 0 . 01. Our numerical implementation follows that described in Ref. [22].', 'A. Inference': "We apply the formalism from Sec. IV to the 45 mergers in TheCatalog , which LISA is expected to detect. Our parameter-estimation pipeline successfully recovers 44 sources, with posterior distributions on source parameters θ well confined within physically unbounded priors. \nHowever, for one source with SNR ∼ 13, the stochastic algorithm fails to converge, leading to poor-quality posteriors. Therefore, we choose to exclude this source from the population analysis. Additionally, sources with SNRs in the range 10 -100 frequently exhibit non-linear correlations and multimodalities on θ ζ ; however both are sufficiently decoupled from the reconstructed θ α 's, which are the target parameters of our astrophysical model selection. In the following, we quote parameter estimates at 90% confidence interval. \nAmong the target parameters, M c is the most precisely measured. Specifically, we constrain it with a relative precision ∆ M c / M c of ∼ 10 -6 -10 -4 for sources with moderate and high SNRs. For low SNRs, the relative precision is ∼ 10 -4 -10 -3 . The spin components χ 1 and χ 2 are measured with a relative precision of 0 . 006% -1% at high and moderate SNRs, and 1% to 65% at low SNRs. \nSix systems in the low -SNR range exhibit mild biases on δµ due to the broad posteriors (∆ δµ/δµ ∼ 6% -160% at 90% credible level) and projection effects associated to non-linear parameter correlations. Of these, three systems have posterior distributions that include the injected values only in their 99 . 9% confidence inteval. For the other three sources, the true injected value lies within the 98 . 5% credible interval. Finally, we emphasize that LISA will be able to measure the source luminosity distance with a relative precision ∆ d L /d L of ∼ 0 . 3% -2% for high SNRs, ∼ 1% -45% for moderate SNRs, and ∼ 10% -115% for low SNRs. \nUncertainties related to sky localization and host galaxy identification, which are crucial for multimessenger detections of gas-rich mergers [57], will be addressed in future work.", 'B. Model selection': "We use the posteriors described above to evaluate the detection-conditioned Bayes factors D G / S of Eq. (7). Results are presented in Fig. 5, where sources are ordered by their SNR and colored according to their true astrophysical sub-population. For all the sources we find 'decisive' evidence [58] in favor of the gas (stellar) sub-population model, with log 10 D G / S > 2 (log 10 D G / S < -2). Additionally, no significant correlation is observed between D G / S and the SNR of each source. This is largely due to the highly precise measurement of the target parameters, as described above. Even the wider posterior distributions are well localized within the sub-population parameter space, so increasing the SNR does not lead to improved discrimination between the two astrophysical models. We observe that most of the Bayes factors fall within the range 10 ≲ | log 10 D G / S | ≲ 25, with three exceptions that exhibit lower values. Among theese, two sources with SNR ∼ 2000 and 7 ≲ | log 10 D G / S | ≲ 10 evolved trough a gas-hardening phase ( f gas ∼ 0 . 72). Conversely, the lowest Bayes factor ( | log 10 D G / S | = 5 . 63) is associated with \nFIG. 6. Marginal distribution of the aligned spin component of the primary BH χ 1 (left panel) and the secondary BH χ 2 (right panel) for mergers in gaseous (red) and stellar (blue) environments. Thick histograms indicate our fiducial model; thin histograms indicate additional realizations. Blue dots denote the median value of source s 1 from TheCatalog as well as the two additional sources s 2 , 3 , c.f. Sec. V. The error bars indicate the 90% confidence intervals. The two sub-population distributions are jointly normalized to the full population. \n<!-- image --> \nthe only source in TheCatalog that hardened through stellar interactions ( f gas ∼ 0 . 14). \nWe further test the analysis pipeline by generating ten additional realizations of our astropyhysical model to evaluate the impact of simulation uncertainties on the results. The error bars in Fig. 5 indicate the resulting variability associated with each log-Bayes factor. Even when accounting for these statistical fluctuations, model G is confidently favored for all the G sources, thus validating the previous results. For the one S source, model S remains favored, though to a lesser extent. Crucially, these error bars capture the statistical fluctuations in the underlying astrophysical model and not the systematic effects due to the many assumptions entering the model itself.", 'C. Source parameters': 'Based on these results, we now investigate how the astrophysical properties of massive BH binaries might influence the distinguishability between models G and S. In Fig. 1, we show the 44 analyzed systems from TheCatalog together with the astrophysical sub-population distributions marginalized on the total mass and redshift. Notably, we observe that the gas-hardened sources are predominantly supported by the gas-hardening subpopulation, with most falling within (outside) the 90% contour level of the gas (stellar) distribution. Conversely, the stellar-hardened source, hereafter referred to as s 1 , falls outside the 90% contour level of the stellar-hardening sub-population but is located close to the peak (i.e. \nwithin the 50% contour level) of the gas-hardening subpopulation. This suggest that additional parameters may be impacting its log 10 D G / S value, favoring the stellarhardening model. \nTo investigate further, we analyze the one-dimensional distribution of the aligned component spins χ 1 and χ 2 , see Fig. 6. For the source s 1 considered so far, the primary component spin χ 1 = 0 . 49 +0 . 08 -0 . 08 lies below the 92 nd and 1 . 5 th percentiles of the stellar- and gas-hardening distributions, respectively, thus placing it confidently in the former. Additionally, the secondary component spin χ 2 = 0 . 39 +0 . 13 -0 . 13 is well below the 90 th percentile of the stellar-hardening distribution and lies within only the 0 . 3 rd percentile of the gas-hardening distribution. We find that the recovered δµ parameter does not provide significant support for model S, with the full posterior information falling below the 14 th percentile of the stellarhardening distribution but within the 50 th percentile of the gas-hardening sub-population.', 'D. Importance of the BH spins': 'From the discussion above, we conclude that, at least for the astrophysical models considered here, the component spins are the crucial parameters for inferring the hardening processes of massive BH binaries. We stress that the employed models describe a simplified scenario. Massive BH binaries evolving in gas-rich environments experience accretion torques that might align their component spins with the orbital angular momentum of the binary. The models used here assume that the spin alignment pro- \ncess occurs rapidly, typically within the time-scale of the merger, thus leading to high and positive spin configuration at merger [42] ( χ 1 , 2 ∼ 0 . 98 on average, as indicated by the red distributions in Fig. 6). In contrast, binaries that evolve in gas-poor environments do not experience significant gas accretion and likely enter the GW dominated phase with roughly isotropic spin orientations, as reflected by the blue distributions in Fig. 6. Nevertheless, as pointed out in the literature, the overall picture is more complex due to the internal properties of the disk. Indeed, the effectiveness of disk alignment may be influenced by the gas temperature [40], accretion rate [59], as well as the disk viscosity [60] and its initial angle of misalignment, which can lead to critical disk-breaking configurations [61]. Accurate astrophysical models accounting for these properties are crucial for addressing such scenarios in future study. \nFinally, we extend the analysis from TheCatalog by including two additional sources, s 2 and s 3 , selected from the LISA stellar-hardening distribution shown in Fig. 3. In order to provide a representative sample of LISA observations, we specifically select sources with SNR in the range 100 -1000, which is most densely populated interval for detectable massive BH binaries in LISA (see Fig. 4). This region is populated by sources with M ∼ 10 4 -10 6 M ⊙ at redshift z ∼ 2 -4. Additionally, we select the two sources such that their primary spin components fall in the tails of the stellar-hardening sub-population from Fig. 6. This allow us to further investigate the role of spins in the model comparison. \nFirst, as shown in Fig. 6, the primary component spin χ 1 = -0 . 8 +0 . 1 -0 . 1 for source s 2 is entirely constrained within a region where p ( χ 1 | G) = 0, thus making the KDE evaluation unnecessary, as pointed out in Sec. IV. Therefore, we can definitely favor model S over model G (log 10 D G / S ≪-2) without even considering the influence of the other binary properties. For source s 3 , the analysis indicates a preference for the stellar-hardening scenario with log 10 D G / S = -4 . 65. This case is particularly challenging because the gas-hardening model has also strongly support over the source posterior. Specifically, s 3 has a mass M ∼ 7 × 10 4 M ⊙ and is located at a redshift z ∼ 1 . 6, positioning it at the peak of the M -z distribution for the gas-hardening sub-population (see Fig. 1). Additionally, it has a measured primary spin component of χ 1 = 0 . 98 +0 . 01 -0 . 03 . This value is notably high and exceeds the 99 . 5 th percentile of the stellar-hardening distribution and the 7 th percentile of the gas-hardening distribution. As illustrated in Fig. 6, s 3 is situated in a high-density region for gas-hardened sources but in a relative sparse region for stellar-hardened sources. Furthermore, the recovered value of δµ falls below the 9 th (38 th ) percentile for the stellar (gas) sub-population, providing no further support for model S with respect model G. On the other hand, the secondary component spin strongly favors model S. In fact, the recovered value χ 2 = 0 . 79 +0 . 04 -0 . 01 falls below the 95 th (1 . 2 nd ) percentile for the stellar- (gas-) hardening \nsub-population. \nThis further emphasizes that, at least in the astrophysical setup considered here, component spins play a crucial role in constraining the hardening mechanisms responsible for the final coalescence of massive BH binaries, even in ambiguous but rare cases (less than 1% of the stellar-hardening population sources detectable by LISA). However, simulation-induced fluctuations reduce the logBayes factor of source s 3 to as low as 0.35, indicating only a weak preference for model S. Incorporating spin precession into the population analysis may be crucial to enhance model discrimination and gain further insights into the galactic environments where massive BH binary mergers occur. This is left to future work.', 'VI. CONCLUSIONS': "In this work, we studied the astrophysical origin of merging massive BH binaries detectable by LISA, with a specific focus on the hardening mechanisms that drive the binary toward the final coalescence. \nCombining results from large-scale cosmological simulations and state-of-the-art models for massive BH binary evolution, we constructed a large set of realistic mock catalogs for LISA. Our analysis showed that most detectable sources are expected to evolve in gas-rich environments. We then performed Bayesian inference and model selection on a representative set of LISA detections and quantifies the statistical evidence in favor of either the gas-hardening or stellar-hardening channel. \nOur results demonstrated that component spins might play a crucial role in distinguishing between these two astrophysical sub-populations and LISA will be able to provide clear insight given the expected precision on measured parameters. This is a rather strong conclusion, which might however depend on the specific astrophysical model adopted in this paper. Reversing the argument, our findings imply that accurate modeling of the BH spins, both magnitudes and directions, is crucial to avoid major systematics in LISA's astrophysical inference. \nIndividual source inference with a richer waveform phenomenology -including precession effects, smaller massratios, and potential eccentricity- as well as more complete astrophysical simulations will yield a more realistic estimate of the LISA potential to uncover the astrophysics of massive BHs. For instance, in this paper we find that including higher-order modes in the adopted GW signal preferentially impacts sources that evolved through specific astrophysical formation pathways. While we leave further explorations to future work, we stress that the methodology presented here is general and can be readily applied to more sophisticated models.", 'ACKNOWLEDGMENTS': "Data supporting this paper (including all mock LISA catalogs and posterior samples) are publicly released at Ref. [62]. We thank Matthew Mould, Monica Colpi, Alberto Sesana, Christopher Moore, Nathan Steinle, and Jonathan Gair for discussions. A.S. and D.G. are supported by ERC Starting Grant No. 945155-GWmining, Cariplo Foundation Grant No. 2021-0555, MUR PRIN Grant No. 2022-Z9X4XS. A.S., R.B. D.I.-V., and D.G. are supported by MUR Grant 'Progetto Dipartimenti di Eccellenza 2023-2027' (BiCoQ), and the ICSC National Research Centre funded by NextGenerationEU. R.B. is supported by Italian Space Agency Grant 'Phase \n- [1] S. D. M. White and M. J. Rees, Mon. Not. R. Astron. Soc. 183 , 341 (1978).\n- [2] M. C. Begelman, R. D. Blandford, and M. J. Rees, Nature 287 , 307 (1980).\n- [3] A. 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2024arXiv240912907M

Phosphorus P is an important element for the chemical evolution of galaxies and many kinds of biochemical reactions. Phosphorus is one of the crucial chemical compounds in the formation of life on our planet. In an interstellar medium phosphine PH3 is a crucial biomolecule that plays a major role in understanding the chemistry of phosphorusbearing molecules particularly phosphorus nitride PN and phosphorus monoxide PO in the gas phase or interstellar grains. We present the first confirmed detection of phosphine PH3 in the asymptotic giant branch AGB carbonrich star IRC10216 using the Atacama Large MillimeterSubmillimeter Array ALMA band 6. We detect the J 1000 rotational transition line of PH3 with a signaltonoise ratio SNR of geq3.5sigma. This is the first confirmed detection of phosphine PH3 in the ISM. Based on LTE spectral modelling the column density of PH3 is 3.15pm0.20times1015 cm2 at an excitation temperature of 52pm5 K. The fractional abundance of PH3 with respect to H2 is 8.29pm1.37times108. We also discuss the possible formation pathways of PH3 and we claim that PH3 may be created via the hydrogenation of PH2 on the grain surface of IRC10216.

2024-09-01T00:00:00Z

['2024arXiv240912907M', '10.48550/arXiv.2409.12907', 'arXiv:2409.12907']

['Astrophysics - Astrophysics of Galaxies', 'Physics - Chemical Physics']

Confirmation of interstellar phosphine towards asymptotic giant branch star IRC10216

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.12907.pdf

{'Confirmation of interstellar phosphine towards asymptotic giant branch star IRC + 10216': 'Arijit Manna 1 , Sabyasachi Pal 1,* \n1 Department of Physics and Astronomy, Midnapore City College, Paschim Medinipur, West Bengal, India 721129 \n* Corresponding author. E-mail: sabya.pal@gmail.com \nAbstract. Phosphorus (P) is an important element for the chemical evolution of galaxies and many kinds of biochemical reactions. Phosphorus is one of the crucial chemical compounds in the formation of life on our planet. In an interstellar medium, phosphine (PH3) is a crucial biomolecule that plays a major role in understanding the chemistry of phosphorus-bearing molecules, particularly phosphorus nitride (PN) and phosphorus monoxide (PO), in the gas phase or interstellar grains. We present the first confirmed detection of phosphine (PH3) in the asymptotic giant branch (AGB) carbon-rich star IRC + 10216 using the Atacama Large Millimeter / Submillimeter Array (ALMA) band 6. We detect the J = 10-00 rotational transition line of PH3 with a signal-to-noise ratio (SNR) of ≥ 3.5 σ . This is the first confirmed detection of phosphine (PH3) in the ISM. Based on LTE spectral modelling, the column density of PH3 is (3.15 ± 0.20) × 10 15 cm -2 at an excitation temperature of 52 ± 5 K. The fractional abundance of PH3 with respect to H2 is (8.29 ± 1.37) × 10 -8 . We also discuss the possible formation pathways of PH3 and we claim that PH3 may be created via the hydrogenation of PH2 on the grain surface of IRC + 10216. \nKeywords. ISM: individual objects (IRC + 10216) - ISM: abundances - ISM: kinematics and dynamics - stars: formation - astrochemistry', '1. Introduction': "Phosphorus is one of the rare elements in the interstellar medium (ISM). Phosphorus is the 13 th element in the meteoritic material and the 11 th element in the crust of Earth (Maci'a et al., 1997). In ISM, the prebiotic chemistry of phosphorus (P) has attracted attention in astrochemical communities because P-bearing molecules, such as phosphorus monoxide (PO) and phosphorus nitride (PN), play important roles in the production of phospholipids and nucleic acids (Turner et al., 1990; Fontani et al., 2016). Phospholipids and nucleic acids are important for the formation of life on our planet. P-bearing molecules may play an important role in the production of large complex biomolecules that store genetic information in nucleic acids (Maci'a et al., 1997). P-bearing molecules also play major roles in the synthesis of DNA and RNA (Maci'a et al., 1997). Earlier millimeter and submillimeter wavelength observations indicated the depletion of P-bearing molecules by a factor of ≥ 100 with respect to the cosmic abundance of P ( ∼ 3 × 10 -7 ) in cold and dense parts of the ISM (Turner et al., 1990; Fontani et al., 2016; Lefloch et al., 2016). The depletion of P-bearing molecules suggests that the bulk of P-bearing species is blocked by interstellar grains (Fontani et al., 2016). Recently, \nthe ESA Probe Rosetta and the ALMA demonstrated that P-bearing species came to Earth through comets (Altwegg et al., 2016; Rivilla et al., 2020). Comets carry several biomolecules because they travel between several star-forming regions (Altwegg et al., 2016). In ISM, P-bearing molecules, such as PO, PN, HCP, CCP, and CP, were detected in the envelopes around evolved stars (Gu'elin et al., 1990; Tenenbaum et al., 2007; Halfen et al., 2008; Milam et al., 2008; Tenenbaum et al., 2008; De Beck et al., 2013; Ziurys et al., 2018). \nIn ISM, phosphine (PH3) is a relatively stable oblate symmetric top molecule. The rotational levels of PH3 are given by two quantum numbers ( J , K ), and radiative transitions are allowed within the levels of the K ladder ( ∆ J = 1, ∆ K = 0). The electric dipole moment of PH3 is 0.573 Debye (Davies et al., 1971). SousaSilva et al. (2020) claimed that PH3 acts as a biosignature in the space. Except for our planet, evidence of PH3 is also found in the atmospheres of Saturn and Jupiter with mixing ratios of 2 ppm and 0.6 ppm using the Voyager data (Maci'a et al., 2005). Subsequently, Fletcher et al. (2009) demonstrated the global distribution of PH3 in the atmosphere of Saturn and Jupiter by using Cassini / CIRS observations. Recently, Gapp et al. (2024) also found evidence of PH3 in the atmosphere of Jupiter using Herschel / PACS. At high tem- \nTable 1 : Stellar properties of IRC + 10216. \nures and pressures, PH3 is produced in the deep atmosphere on large gas planets (Bregman et al., 1975; Tarrago et al., 1992). Recently, Greaves et al. (2021) reported the identification of the absorption line of PH3 at a frequency of 266.944 GHz using ALMA and JCMT on the deck of Venus with a mixing ratio of 20 ppb. Several questions exist regarding the detection of PH3 and the chemical models of the atmosphere of Venus. First, Greaves et al. (2021) could not explain the formation of highly abundant PH3 in the atmosphere of Venus by using steady-state and photochemical models. Greaves et al. (2021) showed di ff erent abiotic chemical routes to explain the high abundance of PH3 in the atmosphere of Venus. Subsequently, Villanueva et al. (2021) and several other authors raised questions regarding the spectroscopic data analysis by Greaves et al. (2021). Villanueva et al. (2021) and other authors clearly showed there is no evidance of PH3 in the atmosphere of Venus. Subsequently, Cordiner et al. (2022) also attempted to search for the emission lines of other transitions of PH3 at frequencies of 533 GHz and 1067 GHz using the NASA SOFIA aircraft, but they could not detect any absorption lines of PH3. Therefore, Cordiner et al. (2022) estimated that the upper limit of PH3 in Venus in the altitude range of 75-110 kmis ≤ 0.8 ppb. Earlier, Olsen et al. (2021) also searched for evidence of PH3 towards the atmosphere of Mars, but they did not detect PH3. The upper limit abundance of PH3 towards Mars is ≤ 0.6 ppbv (Olsen et al., 2021). Except for our solar system, the emission line of PH3 was tentatively detected in the envelope of the carbonrich star IRC + 10216 using the IRAM 30 m single-dish telescope (Ag'undez et al., 2008). This detection is tentative because Ag'undez et al. (2008) did not properly \nidentify the emission line of PH3 at 266.944 GHz, owing to the limitation of the spectral resolution ( ∼ 1.25 MHz) of IRAM. Ag'undez et al. (2008) also claimed that IRC + 10216 is one of the sources in the ISM where a very high abundance of PH3 is still present because the PH3 / HCPabundance ratio is similar to the NH3 / HCN in the envelope of this source. \nIRC + 10216 (alternatively CW Leonis) is known as a carbon-rich asymptotic giant branch (AGB) star located at a distance of 130 pc (Menten et al., 2012). This carbon-rich star is near the end of its evolution and is very close to being converted into a protoplanetary nebula (Skinner et al., 1998; Osterbart et al., 2000). The physical properties of IRC + 10216 are presented in Table 1. IRC + 10216 loses mass at a very high rate (2 × 10 -5 M ⊙ yr -1 ) because the object is very close to the end of its AGB lifetime and this source is covered by an extensive circumstellar envelope (CSE) (Crosas & Menten, 1997). This source is ideal for studying complex organic molecular lines because of its carbon-rich environment. Approximately 70 individual molecules have been detected in IRC + 10216, including MgNC, AlNC, NaCN, AlCl, NaCl, KCl, AlF, and carbon-chain molecules (Cernicharo & Guelin, 1987; Cernicharo et al., 2000; Ziurys et al., 2002). \nIn this letter, we present the first confirmation of interstellar phosphine (PH3) towards the carbon-rich star IRC + 10216 using ALMA. For the detection of the emission line of PH3, we used the local thermodynamic equilibrium model (LTE). The observations and data reduction are presented in Section 2.. The result of the detection of the rotational emission line of PH3 is shown in Section 3.. The discussion and conclusion are shown in Section 4. and 5.. \nFigure 1 : Rotational emission line of PH3 with transition J = 10-00 towards IRC + 10216 (left panel). The green lines represent the observed molecular spectrum of IRC + 10216. The black spectra indicate the best-fit LTE model spectra for PH3. The red spectrum is the global Gaussian model. The blue spectra are the Gaussian model corresponding to the SiS ( V = 4) emission line, and the violet spectra are the Gaussian model corresponding to the PH3 emission line. The yellow vertical line indicates the rest frequency position of the PH3. The right panel image shows the corner diagram plot based on MCMC fittings. Corner plots showing the 3D posterior probability distributions of column density in cm -2 , excitation temperature in K, and FWHM in km s -1 . \n<!-- image -->", '2. Observations and data reductions': "Weused the cycle 0 archival data of IRC + 10216, which was observed using a high-resolution ALMA in band 6 (PI: Cernicharo, Jose, ID: 2011.0.00229.S) with a 12 m array. This observation was performed on April 8, 2012, to study the emission lines of HCN with transitions J = 3-2 and J = 8-7 and observation of the dust formation zone in IRC + 10216 with an on-source integration time of 32.76 min. The phase centre of IRC + 10216 observation was ( α, δ )J2000 = 09:47:57.406, + 13:16:43.561. At the time of observation, 3C 279 and 3C 273 were used as bandpass and flux calibrators, respectively. J0854 + 201 was used as the phase calibrator. During the observation period, 16 antennas were used, with a minimum baseline of 15.7 m and a maximum baseline of 384.1 m. Observations were performed in the frequency ranges of 265.05-266.92 GHz, 266.15268.03 GHz, 267.54-269.42 GHz, and 268.05-269.92 GHz with a spectral resolution of 976.56 kHz. \nFor data analysis, we used the Common Astronomy Software Application (CASA 5.4.1) with the ALMA data analysis pipeline (McMullin et al., 2007). We applied the Perley-Butler 2017 flux calibration model for each baseline for flux calibration using the SETJY task (Perley & Butler, 2017). We also used the pipeline tasks HIFA bandpassflag and HIFA flagdata for flag- \nging bad antenna data and channels, which were performed after flux and bandpass calibration. After a preliminary reduction of the data, we split the target data (IRC + 10216) using the MSTRANSFORM task with all the rest frequencies. We also used the UVCONTSUB task to subtract continuum emission from the UV plane of the calibrated data. After data reduction, continuum emission images of IRC + 10216 were created using the TCLEAN task with a HOGBOM deconvolver for line-free channels. Earlier, Ag'undez et al. (2015) discussed the dust continuum emission of IRC + 10216 using the same data. Therefore, we do not discuss dust continuum emissions in this study. We also created spectral data cubes using the TCLEAN task with the SPECMODE = CUBE parameter and Briggs weighting with a robust value of 0.5. We also applied a multiple self-calibration method to improve the RMS of the data cubes. To correct the primary beam pattern, we used the IMPBCOR task.", '3.1 Identification of PH 3 towards IRC + 10216': "To study the rotational emission line of PH3, we only focus on the spectral data cube that was observed between the frequency range of 266.15 GHz and 268.03 GHz because the rest frequency of PH3 ( J = 10-00) \nFigure 2 : Rotational emission line of carbon monoxide (CO) with transition J = 3-2 towards IRC + 10216. The blue line indicates the observed spectra of CO, and the black spectrum is the best-fitting Gaussian model. \n<!-- image --> \nis 266.944 GHz. We extracted the molecular spectra of IRC + 10216 from the spectral data cube to create a 2.3 '' diameter circular region over the line-emitting region of the source, which is larger than the synthesized beam size of the spectral data cube. The synthesized beam size of the spectral data cube is 0.90 '' × 0.48 '' . The systemic velocity ( V LSR) of IRC + 10216 is -26.5 km s -1 (Ag'undez et al., 2015). To identify the rotational emission line of PH3, we used the local thermodynamic equilibrium (LTE) model spectra with the Cologne Database for Molecular Spectroscopy (CDMS) database (Muller et al., 2005). To fit the LTE spectra to the observed spectra of PH3, we used the Markov chain Monte Carlo (MCMC) algorithm in CASSIS (Vastel et al., 2015). The MCMC analysis using the CASSIS is well described in Manna & Pal (2024). The MCMC method initializes by randomly selecting a seed point ( X 0) in the three-dimensional parameter space. Then, it randomly chooses a nearby point ( X 1) based on a variable step size, which is recalculated for each iteration. The χ 2 value of the new state ( X 1) is calculated, and if the ratio p = χ 2 ( X 0) / χ 2 ( X 1) > 1, the new state is accepted. However, even if p < 1, the new state may still be accepted with a certain probability. If the new state is rejected, the original state ( X 0) remains, and another nearby point ( X 1) is randomly selected. By allowing a finite probability of accepting a worse χ 2 value, the algorithm avoids converging directly to a local minimum and instead ensures a more thorough exploration of the entire parameter space. During our MCMC analysis, we employed 1000 walkers, uniformly distributed within the specified parameter ranges, and ran the chains \nfor a burning sequence of 20,000 steps to ensure convergence. The MCMC approach allows us to vary all parameters, including column density, excitation temperature, and full width at half maximum (FWHM) until the best fit is obtained. After the LTE spectral analysis, we detected the emission line of PH3 at a frequency of 266.944 GHz with transition J = 10-00 in the spectra of IRC + 10216. We also observed that the emission line of PH3 is closely associated with the emission line of SiS V = 4 with transition J = 15-14. The rest frequency of the emission line of SiS V = 4 is 266.941 GHz, which was obtained from Ag'undez et al. (2008). Previously, Ag'undez et al. (2008) and Ag'undez et al. (2014) attempted to search the emission line of PH3 ( J = 10-00) using the IRAM 30 m telescope, but the authors did not detect the proper peak of the emission line of PH3 because of the limitation in the resolution of the IRAM. Our detection of the emission line of PH3 using ALMA is the first confirmation of the presence of PH3 in IRC + 10216. After spectral analysis, we see that the emission line of PH3 is non-blended. The upper state energy ( Eu ) and Einstain coe ffi cients ( Aij ) of the identified PH3 transition are 12.81 K and 2.43 × 10 -5 s -1 . The full-width half maximum (FWHM) of the LTE spectra is 5.12 ± 0.62 km s -1 . According to the LTE model, the best-fit column density of PH3 is (3.15 ± 0.20) × 10 15 cm -2 with an excitation temperature of 52 ± 5 K and a source size of 0.90 '' . Our estimated excitation temperature of PH3 is similar to the excitation temperature of another P-bearing molecule, CP, which was estimated by Milam et al. (2008). The excitation temperature of PH3 is relatively low because the AGB star is near the end of its evolution. The peak and integrated intensities of the detected emission line of PH3 are 4.26 ± 0.12 K and 20.46 ± 0.56 K km s -1 respectively. The optical depth of the spectra of PH3 is 5.81 × 10 -2 . The estimated optical depth indicates that the identified rotational emission line of PH3 is optically thin. The LTE-fitted rotational emission line of PH3 towards IRC + 10216 is shown in Figure 1. In addition, we created a corner plot corresponding to the LTE fitting values. The corner plot shows the 3D posterior probability distributions of column density in cm -2 , excitation temperature in K, and FWHM in km s -1 of PH3 towards IRC + 10216, which is shown in the right panel of Figure 1.", '3.2 Estimation of the column density of molecular H 2 towards IRC + 10216': 'To determine the column density of molecular H2 towards IRC + 10216, we used the following equation: \nN(H2) = 2 . 0 × 10 20 × W( 12 CO) K kms -1 , (1) \nFigure 3 : Integrated emission map of PH3 towards IRC + 10216. The contour levels are started at 3 σ and are increased by a factor of √ 2. The blue circle represents the synthesized beam of the integrated emission map. \n<!-- image --> \nwhere W( 12 CO) is the integrated intensity of 12 CO ( J = 3-2) at the corresponding velocity intervals. This equation was taken from Isequilla et al. (2021). To determine the emission line properties of CO, we used the 2016.1.00251.S (PI: Vlemmings Wouter) ALMA data. The analysis of this ALMA data is well described in Siebert et al. (2022). The emission line of CO towards IRC + 10216 is shown in Figure 2. We fitted a Gaussian model over the emission line of CO and estimated that the integrated intensity of the CO line is 190.39 K kms -1 . Using the above equation, the estimated column density of H2 towards IRC + 10216 is (3.80 ± 0.58) × 10 22 cm -2 .', '3.3 Abundance of PH 3 towards IRC + 10216': "To derive the fractional abundance of PH3, we used the column density of PH3 inside the 0.90 '' synthesized beam, which was divided by the column density of H2. The fractional abundance of PH3 with respect to molecular H2 towards IRC + 10216 is (8.29 ± 1.37) × 10 -8 , where the column density of molecular H2 towards IRC + 10216 is (3.80 ± 0.58) × 10 22 cm -2 . Previously, Ag'undez et al. (2008) and Ag'undez et al. (2014) estimated the tentative abundance of PH3 towards IRC + 10216, which varied between ∼ 10 -9 and ∼ 10 -8 . Our estimated fractional abundance of PH3 using ALMA data is similar to those of Ag'undez et al. (2014) but one order of magnitude higher than those of Ag'undez et al. (2008). Previously, Lefloch et al. (2016) attempted to detect the emission line of PH3 at a frequency of 266.944 GHz using the IRAM 30 m telescope towards the solar-type star-forming region L1157, however, they could not successfully detect the emission line of PH3. The upper limit of the abundance of PH3 towards L1157 is ≤ 10 -9 (Lefloch et al., 2016). Recently, Furuya & Shimonishi (2024) also searched the emission line of PH3 ( J = 1000) towards L1544 using the ALMA, but they could \nnot detect PH3. The upper limit column density and abundance of PH3 towards L1544 are ≤ 7.6 × 10 10 cm -2 and ≤ 6.7 × 10 -12 , respectively (Furuya & Shimonishi, 2024). Therefore, we confirm that IRC + 10216 is the only source in the ISM where evidence of PH3 is found.", '3.4 Spatial distribution of PH 3': "We created an integrated emission map of PH3 ( J = 1000) towards IRC + 10216 using the IMMOMENTS task. In the IMMOMENTS task, we used the channel ranges of the spectral data cubes, where the emission lines of PH3 were identified. The integrated emission map of PH3 is shown in Figure 3. The emission map clearly shows that the emission line of PH3 originated from the inner envelope of IRC + 10216. We also fitted a 2D Gaussian over the integrated emission map of PH3 using the IMFIT task. The following equation is used to estimate the emitting region of PH3 \nθ S = q θ 2 50 -θ 2 beam (2) \nIn the above equation, θ 50 = 2 √ A /π indicates the diameter of the circle whose area ( A ) surrounds the 50% line peak of PH3 and θ beam indicates the half-power width of the synthesized beam of the integrated emission map of PH3 (Manna et al., 2023). The size of the emitting region of PH3 is 0.83 '' . The synthesized beam size of the integrated emission map of PH3 is 0.90 '' × 0.48 '' . We observed that the emitting region of PH3 is slightly smaller than the synthesized beam size of the integrated emission map. This indicates that the detected emission line of PH3 is not spatially resolved towards IRC + 10216. Therefore, we cannot draw any conclusions regarding the spatial distribution morphology of PH3 towards IRC + 10216. Higher angular and spatial resolution observations with better spectral resolution are required to understand the chemical morphology of PH3 towards IRC + 10216.", '4.1 Possible formation mechanism of PH 3': "Previous chemical modelling studies have indicated that PH3 is an important biomolecule that plays a major role in the production of other P-bearing molecules using the surface chemistry of the interstellar grains (Charnley & Millar, 1994; Aota & Aikawa, 2012; Nguyen et al., 2020). Earlier, Nguyen et al. (2020) showed that PH3 is created on the grain surface rather than in the gas phase. Charnley & Millar (1994) and Nguyen et al. (2020) proposed a possible formation mechanism for \nFigure 4 : The proposed gas phase and grain surface chemical network for the formation of PH3 and link with other P-bearing molecules. The green box indicates P-bearing molecules which were previously detected in IRC + 10216. \n<!-- image --> \nPH3 via the hydrogenation of atomic P on the grain surface of the highly dense part of star-formation regions and hot molecular cores via the following reactions. \nP + H -→ PH, ∆ = -277 KJ mol -1 (1) \nPH + H -→ PH2, ∆ = -347 KJ mol -1 (2) \nPH2 + H -→ PH3, ∆ = -326 KJ mol -1 (3) \nSince reactions 1-3 are barrierless, PH3 can be produced on the grain surface. At low temperatures ( ∼ 10 K), the hydrogen atom (H) can di ff use and react with other compounds on the grain surface (Hama & Watanabe, 2013). Previously, no P-bearing molecules, including PH3, were observed in the solid state. Turner et al. (2015, 2018, 2019) experimentally observed that solid PH3 is converted into phosphoric acid, diphosphate, and methyl phosphonic acid at low temperatures. Chantzos et al. (2020) also showed that the PH3 is destroyed due to the reaction of C + (C + + PH3 -→ PH3 + + C) and H + (H + + PH3 -→ PH3 + + H). Earlier, Ag'undez et al. (2014) computed the chemical model of PH3 in the IRC + 10216 environment using reaction 3, and they found that the model abundance of PH3 is 1.0 × 10 -8 . Our observed abundance of PH3 towards IRC + 10216 using the ALMA is very close to the modelled abundance of PH3, which was estimated by Ag'undez et al. (2014). This indicates PH3 is formed via hydrogenation of PH2 on the grain surface of IRC + 10216.", '4.2 Chemical link-up between PH 3 and other molecules': "Previously the emission lines of HCP, CP, PO, and PN were detected towards the IRC + 10216 (Matthews et al., \n1987; Gu'elin et al., 1990; Milam et al., 2008). We created a chemical network to understand the prebiotic chemistry of PH3 and the chemical link-up between all detected P-bearing species towards IRC + 10216, as shown in Figure 4. The chemical reactions are taken from Charnley & Millar (1994), Aota & Aikawa (2012), Hama & Watanabe (2013), Nguyen et al. (2020), and UMIST 2012 (McElroy et al., 2013) molecular reaction database. Our chemical network clearly shows that PH acts as a possible precursor of PH3, PN, and PO. Previously, Ag'undez et al. (2014) showed that the gas phase chemistry is not su ffi cient for the formation of PH3. Therefore, there is a high chance of the formation of PH3 via hydrogenation of PH2 on the grain surface of IRC + 10216. A new chemical model using the gas-grain chemistry and quantum chemical studies using the density functional theory (DFT) is needed with proper formation and destruction pathways of PH3 with reaction rates to understand the chemical evolution and proper formation and destruction pathways of PH3 towards IRC + 10216.", '5. Summary and conclusion': 'In this letter, we present the first confirmation of the rotational emission line of PH3 towards the carbon-rich AGB star IRC + 10216 at a frequency of 266.944 GHz using the ALMA band 6. The abundance of PH3 towards IRC + 10216 is (8.29 ± 1.37) × 10 -8 . We discuss the possible formation pathways of PH3 and we claim that PH3 may be formed via the hydrogenation of PH2 on the grain surface of IRC + 10216. The confirmed detection of PH3 indicates that the grain surface chemistry is su ffi cient for the production of other P-bearing molecules because PH3 and PH act as possible precursors of other P-bearing molecules. Our chemical network shows that PH3, PN, and PO are chemically connected to the PH. A detailed spectral line study and chemical modelling are required to understand other Pbearing molecules towards IRC + 10216, which will be carried out in our next follow-up study.', 'Acknowledgement': 'We thank the anonymous referee for the helpful comments that improved the manuscript. A.M. acknowledges the Swami Vivekananda Merit cum Means Scholarship, Government of West Bengal, India, for financial support for this research. The emission spectra of PH3 are available on our github repository. The emission map and chemical network of PH3 within this paper is available from the corresponding author upon reasonable request. This paper makes use of the following \nALMA data: ADS / JAO.ALMA#2011.0.00229.S and 2016.1.00251.S. ALMA is a partnership of ESO, NSF (USA), and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The JAO is operated by ESO, AUI / NRAO, and NAOJ.'}

2024Natur.633...58S

The canonical theory for planet formation in circumstellar disks proposes that planets are grown from initially much smaller seedsSUP15SUP. The longconsidered alternative theory proposes that giant protoplanets can be formed directly from collapsing fragments of vast spiral armsSUP611SUP induced by gravitational instabilitySUP1214SUPif the disk is gravitationally unstable. For this to be possible the disk must be massive compared with the central star a disktostar mass ratio of 110 is widely held as the rough threshold for triggering gravitational instability inciting substantial nonKeplerian dynamics and generating prominent spiral armsSUP1518SUP. Although estimating disk masses has historically been challengingSUP1921SUP the motion of the gas can reveal the presence of gravitational instability through its effect on the diskvelocity structureSUP2224SUP. Here we present kinematic evidence of gravitational instability in the disk around AB Aurigae using deep observations of SUP13SUPCO and CSUP18SUPO line emission with the Atacama Large Millimetersubmillimeter Array ALMA. The observed kinematic signals strongly resemble predictions from simulations and analytic modelling. From quantitative comparisons we infer a disk mass of up to a third of the stellar mass enclosed within 1 to 5 on the sky.

2024-09-01T00:00:00Z

['10.1038/s41586-024-07877-0', '2024arXiv240902196S', '10.48550/arXiv.2409.02196', '2024Natur.633...58S', 'arXiv:2409.02196']

['Astrophysics - Earth and Planetary Astrophysics']

Gravitational instability in a planetforming disk

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

https://arxiv.org/pdf/2409.02196.pdf

{'Gravitational instability in a planet-forming disk': "J. Speedie 1 , R. Dong 1 , 2 , C. Hall 3 , 4 , C. Longarini 5 , 6 , B. Veronesi 7 , T. Paneque-Carreño 8 , 9 , G. Lodato 5 , Y. Tang 10 , R. Teague 11 , J. Hashimoto 12 , 13 , 14 \n- 1 Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada\n- 2 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, People's Republic of China \n- Department of Physics and Astronomy, The University of Georgia, Athens, GA 30602, USA \n- 7 Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis,-Laval, France\n- 8\n- Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, the Netherlands 9 European Southern Observatory, Karl-Schwarzschild-Str 2, 85748 Garching, Germany\n- 10 Academia Sinica, Institute of Astronomy and Astrophysics, 11F of AS/NTU AstronomyMathematics Building, No.1, Sec. 4, Roosevelt Rd., Taipei, Taiwan \n- 12 Astrobiology Center, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 1818588, Japan\n- 13 Subaru Telescope, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan 14 Department of Astronomy, School of Science, Graduate University for Advanced Studies (SOKENDAI), Mitaka, Tokyo 181-8588, Japan \nThe canonical theory for planet formation in circumstellar disks proposes that planets are grown from initially much smaller seeds 1-5 . The long-considered alternative theory proposes that giant protoplanets can be formed directly from collapsing fragments of vast spiral arms 6-11 induced by gravitational instability (GI) 12-14 -if the disk is gravitationally unstable. For this to be possible, the disk must be massive compared to the central star: a disk-to-star mass ratio of 1 / 10 is widely held as the rough threshold for triggering GI, inciting significant non-Keplerian dynamics and generating prominent spiral arms 15-18 . While estimating disk masses has historically been challenging 19-21 , the motion of the gas can reveal the presence of GI through its effect on the disk velocity structure 22-24 . Here we present kinematic evidence of gravitational instability in the disk around AB Aurigae, using deep observations of 13 CO and C 18 O line emission with the Atacama Large Millimeter/submillimeter Array (ALMA). The observed kinematic signals strongly resemble predictions from simulations and analytic modeling. From quantitative comparisons, we infer a disk mass of up to 1 / 3 the stellar mass enclosed within 1 '' to 5 '' on the sky. \nWe targeted the disk around AB Aurigae (AB Aur), a 2 . 5 -4 . 4 Myr old 25-28 Herbig Ae 29 star of intermediate mass ( M ⋆ = 2 . 4 M ⊙ ) 26,27,30 at a distance of 155 . 9 ± 0 . 9 pc 31 . AB Aur is at a relatively late stage of protostellar evolution, classified as a Class II Young Stellar Object 32,33 (YSO). To probe the velocity structure of the disk, we obtained deep ALMA Band 6 observations of molecular emission lines 13 CO ( J = 2 -1 ) and C 18 O ( J = 2 -1 ) with high velocity resolution (channel widths of v chan = 42 m/s and 84 m/s respectively). The observations were taken in two array configurations with baselines ranging 14 to 2 , 216 m, reaching a total on-source integration time of 5.75 hours. Imaging with a Briggs robust value of 0 . 5 provided image cubes with a spatial resolution or beam size of 0 . 237 '' × 0 . 175 '' (beam position angle, PA = 1 . 2 · ) equivalent to 37 × 27 au. We collapse the 3D image cubes into 2D moment maps to expose the velocity-integrated intensity (moment 0), intensity-weighted line-of-sight velocity ( v los , moment 1) and emission line width (moment 2). This collection is shown in Extended Data Figure 1. \nTo reveal the spiral arms in the disk, we apply a high-pass filter 34 (see Methods) to the ALMA 13 CO moment maps (Figure 1bcd). In the filtered line-of-sight velocity (moment 1) map, we observe spiral-shaped disturbances in the gas velocity field throughout the disk (Figure 1b). With the filtered velocity-integrated intensity (moment 0) and line width (moment 2) maps, we visually highlight regions of peak density and temperature (Figure 1cd). Compression and shock-heating are expected to lead to temperature enhancements (and thus localized line broadening) within GI-induced density spirals in self-regulating disks 13,23 . The VLT/SPHERE H -band scattered light image of AB Aur originally presented in Boccaletti et al. (2020) 35 is shown for comparison (Figure 1a). Scattered light comes from the disk surface, probing the distribution of (sub-)micron-sized dust usually well-coupled with the gas. Previous simulations have shown that GIinduced density spirals are prominent in scattered light 16,36 . At least seven spiral structures (S1-S7) have been previously identified in the H -band image 35,37 , though not all occupy the same radial region and some may be branches of adjacent arms 38 . The disk rotates counter-clockwise (the spiral arms are trailing), and the south side is the near side, tilted toward us 38-40 . \nTo provide a qualitative comparison to the ALMA observations, we run 3D smoothed-particle hydrodynamic (SPH) simulations of a gravitationally unstable disk (see Methods). The simulations were postprocessed with radiative transfer and then further processed to have the same viewing angle, sensitivity, spectral and angular resolution as the AB Aur data. To place the disk comfortably within the gravitationally unstable regime ( M disk /M ⋆ ≳ 0 . 1 ), we set the total gas mass to 0 . 3 × the mass of the star. For sustained spiral arms, we set the cooling timescale to 10 × the local dynamical timescale ( β = 10 ). The simulated GI disk shows spiral structures in all three moment maps, resembling those in the AB Aur disk (Extended Data Figures 1 and 2). Overall, the AB Aur disk hosts a global architecture of spiral arms at 100 to 1,000 au scales across all azimuths in multi-wavelength observations tracing different disk components and quantities, strongly indicating ongoing gravitational instability. \nOne characteristic kinematic feature in the AB Aur disk can be found in the isovelocity curve at the systemic velocity v sys in the moment 1 map - Figure 2a shows a sinusoidal pattern at v los = v sys (along the minor axis; white color), more prominent towards the south. This signature, known as a 'minor axis GI wiggle' 22 , has been predicted in hydrodynamic simulations 22,24 and analytic theory 23 as a clear \nkinematic signature of gravitational instability (Figure 2bc). It is one of a global set of GI wiggles in isovelocity curves we observe throughout the AB Aur disk (Extended Data Figure 3). These wiggles are generated by self-gravitating spiral arms, which constitute local minima in the gravitational potential field and induce corresponding oscillations in the gas velocity field. The synthetic moment 1 map of the SPH GI disk simulation shows a minor axis GI wiggle with similar morphology as the observed one (Figure 2c), completely distinct from the linear pattern found in a disk undergoing Keplerian rotation with no radial motions (Figure 2bc insets). \nAmong all GI wiggles, the minor axis GI wiggle has been known and targeted in past studies for its convenience in quantitative analysis 23,24 . Due to projection effects, only the radial and vertical components of the disk velocity field ( v r or v z ) contribute to v los at the systemic velocity traced by this wiggle. In the case of GI-induced velocity perturbations, the v r contribution is expected to dominate 22 . As we show with 2D analytic calculations of gravitationally unstable disks (see Methods), a self-gravitating spiral arm induces radial motion convergent on itself, appearing as a wiggle in the moment 1 map at v sys where the spiral crosses the minor axis (c.f. Extended Data Figure 5). The filtered moment 1 map in Figure 1b displays redand blue-shift patterns corresponding to convergent flows toward spiral S5 (visible in both scattered light and 13 COmoment 0 and 2; Figure 1acd), supporting the interpretation that the GI wiggle along the southern minor axis in Figure 2a is generated by a self-gravitating spiral arm. \nHaving identified evidence of gravitational instability in disk kinematics and in the detections of spirals across multiple tracers and moment maps, we now quantitatively analyze the GI wiggle along the southern minor axis to constrain the disk mass. We extract the 13 CO and C 18 O emission spectra along the southern disk minor axis (Figure 3ab) and detect the wiggle in position-velocity space (hereafter referred to as the 'PV wiggle'), which is a different view of the position-position wiggle in Figure 2a. Slicing the 3D image cubes this way more comprehensively exposes the gas velocity structure and enables us to quantify the perturbation in units of velocity. We measure the emission line centers by performing a quadratic fit to the spectrum in each spatial pixel of the image cube 41 . This method achieves sub-spectral resolution precision on the line center and yields statistically meaningful and robust uncertainties 42 . We find remarkably similar sinusoidal morphology between the PV wiggles in 13 CO and C 18 O emission (Figure 4a). \nTheoretical studies have shown that the dynamical response of a disk to its own self-gravity is sensitive to the disk-to-star mass ratio and the cooling rate 23,24 . Specifically, the amplitude of the induced radial velocity perturbations is proportional to ( M disk /M ⋆ ) 2 and β -1 / 2 (Eqns. 11 & 18 in Methods). This allows us to use the observed minor axis PV wiggle to infer the disk mass once we make assumptions on the disk cooling rates. Following Longarini et al. (2021) 23 , we employ a statistical metric to quantify the 'magnitude' of the minor axis PV wiggle, defined as the standard deviation of the line center velocities over a radial range. Bounded by the inner central cavity and outer edge of recovered C 18 O emission, our radial range spans 1 '' to 5 '' ( 155 to 780 au). We find a magnitude of 37 . 4 ± 2 . 9 m/s for the southern minor axis PV wiggle in 13 CO and 44 . 2 ± 1 . 3 m/s in C 18 O (Figure 4b). For comparison, the gravitationally unstable disk in the SPH simulation has a southern minor axis PV wiggle in 13 CO emission with quantitatively similar amplitude and sinusoidal morphology (Figure 3c), and a magnitude of 39 . 1 ± 1 . 8 m/s (Extended Data Figure 7a). \nQuantifying the minor axis PV wiggle magnitude as above, we perform comparisons against analytic models to identify the combinations of disk mass ( M disk /M ⋆ ) and cooling timescale ( β ) that satisfy the AB Aur observations. A proof of concept of this technique with the SPH simulation is shown in Extended Data Figure 7b. Using the analytic modeling code giggle 1 of Longarini et al. (2021) 23 (Methods), we calculate the minor axis PV wiggle magnitude in gravitationally unstable disk models for 60 × 60 combinations of M disk /M ⋆ and β , letting each vary within the ranges 0 . 0 ≤ M disk /M ⋆ ≤ 0 . 4 and 10 -2 ≤ β ≤ 10 2 . A demonstrative analytic curve for the minor axis PV wiggle from the same model shown in Figure 2b is underlaid in Figure 4a for qualitative comparison. Figure 4c shows the resulting map of 60 × 60 analytic minor axis PV wiggle magnitudes. Overlaying contours in this map at the magnitude values measured for the AB Aur 13 CO and C 18 O southern minor axis PV wiggles, we find a disk mass in the gravitationally unstable regime with 0 . 1 ≲ M disk /M ⋆ ≲ 0 . 3 for a cooling timescale of 0 . 1 < β < 10 . This result is robust to plausible variations in the analytic model parameter choices (Extended Data Figure Figure 8). This disk mass range is broadly consistent with the observed spiral morphology - a lower disk mass may result in a large number of more tightly wound spirals than we observe, and vice versa 12,43 . To demonstrate that the implied cooling timescales are compatible with the constrained disk mass values, Figure 4c also displays ranges of β derived from independent radiative cooling prescriptions (see Methods). \nThe detection of GI in the disk around AB Aur, a Class II YSO 32,33 , demonstrates that gravitational instability can take place during later evolutionary stages. This result, together with previous reports of multiple protoplanet candidates in and amongst spiral arms in the system 35,44-46 (Extended Data Figure 9), provides a direct observational connection between gravitational instability and planet formation. Looking forward, the AB Aur system can be an ideal testbed for understanding how planet formation is facilitated by GI-induced spiral arms - whether by fragmentation into gas clumps enabled by rapid cooling 7-10 ( β ≲ 3 ), or by dust collapse of solids concentrated within spiral arms sustained by slow cooling 47-50 ( β ≳ 5 ). \n2 \nFigure 1: Global spirals in the AB Aur disk. (a) VLT/SPHERE H -band scattered light image of the AB Aur disk (ref. 35 ) tracing spiral structure in (sub-)micron-sized dust grains. The labelled spirals S1-S7 are taken from previous works (ref. 37,38 ). (b) Filtered ALMA 13 CO intensity-weighted mean velocity (moment 1) map, revealing residual gas motion within the bulk flow. The synthesised beam is shown in the bottom left corner as an ellipse. The inset zooms into the region around where S5 crosses the minor axis, highlighting converging flows on the two sides of S5 indicated by arrows. (c) Filtered ALMA 13 CO integrated intensity (moment 0) map, highlighting peaks in the gas density and/or temperature. (d) Filtered ALMA 13 CO emission line width (moment 2) map, showing localized line broadening within the spiral arms. Insets in panels (c,d) zoom into the same region as panel (b) inset, showing enhanced gas density/temperature caused by the radially converging flows around S5. 5 \n<!-- image --> \nFigure 2: Detection of the GI wiggle in the AB Aur disk. (a) ALMA 13 CO intensity-weighted mean velocity (moment 1) map showing line-of-sight velocity ( v los ) of gas in the AB Aur disk. The observations display the 'GI wiggle' along the minor axis (arrow) predicted by Hall et al. (2020) 22 as a clear kinematic signature of gravitational instability. (b) v los map of a gravitationally unstable disk at the inclination and position angle of the AB Aur disk, computed with 2D analytic modeling 23 . Self-gravitating spiral arms crossing the minor axis induce radial motion that appears as a wiggle (arrow). (c) Synthetic ALMA 13 CO moment 1 map of the 3D SPH GI disk simulation, revealing the same GI wiggle signature (arrow). The insets in panels (b) and (c) show corresponding images for Keplerian disks with no radial gas motion, where the isovelocity curve at the systematic velocity appears as a straight line along the minor axis. \n<!-- image --> \nFigure 3: The GI wiggle in position-velocity space, or PV wiggle. Emission spectra (intensity as a function of velocity) extracted along the southern minor axis of the disk, plotted with distance from the star. The line centers are shown as yellow points. The insets at the bottom right of each panel show the corresponding line center map, with black circles delineating 1 '' radial increments. The yellow line along the southern minor axis is the narrow ( 0 . 5 · -wide) wedge-shaped mask within which the spectra and line centers are extracted. In all PV diagram panels, the grey box in the bottom left corner has horizontal width equal to the beam major axis and vertical height equal to the channel width. (a) ALMA observations of the AB Aur disk in 13 CO and (b) in C 18 O. (c,d) Synthetic ALMA 13 COobservations generated from 3D SPH simulations of a gravitationally unstable disk with a disk-to-star mass ratio of 0.3 and a cooling rate described by β = 10 . In (d) , the simulated disk has its velocity structure artificially post-processed to be Keplerian. \n<!-- image --> \nFigure 4: PVwiggle mophology, magnitude, and constraints on the AB Aur disk mass. (a) The ALMA 13 CO and C 18 Oline centers along the southern minor axis from Figure 3ab, after quadratic detrending (Methods). Uncertainties on the line centers are shown by yellow shaded regions. For qualitative comparison, the minor axis PV wiggle of the analytic GI model disk from Figure 2b is shown in the background in light grey. (b) The magnitude of the southern minor axis PV wiggle in AB Aur is measured to be 37 . 4 ± 2 . 9 m/s in 13 CO and 44 . 2 ± 1 . 3 m/s in C 18 O. (c) A map of the minor axis PV wiggle magnitude of 3600 analytic GI model disks, calculated for a 60 × 60 grid of disk-to-star mass ratios and cooling timescales. Each cell in the map represents the minor axis PV wiggle magnitude from a different model. A yellow contour is drawn at each of the AB Aur 13 CO and C 18 O measured magnitude values, and dashed lines represent the quoted uncertainties. White shaded regions denote 1 σ , 2 σ and 3 σ departures from a Keplerian signal in 13 CO (c.f. Fig. 3d). White horizontal bars indicate independently derived β ranges at a selection of M disk /M ⋆ values (Methods). \n<!-- image -->", 'References': "- 1. Salyk, C. et al. 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This accretion rate, taken together with the current age t 0 = 2 . 5 -4 . 4 Myr 3-6 , implies a high 'latent disk mass': M latent disk = ˙ M ( t 0 ) × t 0 = 0 . 25 -0 . 44 M ⊙ , or M latent disk /M star ∼ 0 . 1 -0 . 2 . M latent disk provides an accretion rate-based assessment of disk mass, assuming a constant stellar accretion rate ˙ M and we are observing the system mid-way through the disk's lifetime 7, 8 . This is a conservative estimate as the accretion rate at earlier epochs is likely higher 9 . In millimeter continuum observations, the disk shows a dust ring at ∼ 1 '' and a cavity inside 10 , likely caused by the trapping of millimeter-sized dust at a pressure bump. The dust ring is located inside the main spirals in both the scattered light and gas emission. Late infall from above or below the main disk plane 10-13 is likely encouraging GI by providing a source of mass to maintain a high M disk /M ⋆ value 14,15 . \nALMAobservations. We observed AB Aur with ALMA in April, May and September 2022 under ALMA program ID 2021.1.00690.S (PI: R. Dong). Measurements were taken with the Band 6 receivers 16 in array configurations C-3 (2 execution blocks) and C-6 (6 execution blocks). In total, the 8 execution blocks reached an on-source integration time of 5.75 hours, making this the longest fine-kinematics ( v chan < 100 m/s) program toward a single protoplanetary disk to date. Extended Data Table 1 provides details of the observations. We centered one spectral window (SPW) at the 13 CO J = 2 -1 molecular emission line transition rest frequency (220.3986 GHz), covering a bandwidth of 58.594 MHz with 1920 channels, resulting in the highest achievable spectral resolution of 41.510 m/s after default spectral averaging with N = 2 by Hanning smoothing within the correlator data processor. A second SPW was centered at the C 18 O J = 2 -1 rest frequency (219.5603 GHz) covering the same bandwidth with half as many channels (960 channels; due to sharing a baseband with another SPW), achieving a 83.336 m/s spectral resolution. To enable self-calibration, our correlator setup sampled the continuum in another SPW centered at 233 . 012 GHz with 128 channels each 15.625 MHz in width, obtaining the full available 2.0 GHz bandwidth. Using the continuum data, all execution blocks were aligned to a common phase center in the uv -plane. We performed a series of phase-only self-calibration iterations, and avoided combining by SPW in the first two rounds to remove any potential per-SPW phase offsets. We also carried out one round of amplitude and phase self-calibration. Finally, we applied the phase center realignments and calibration gain tables (that we generated with the continuum data) to the line data. We performed continuum subtraction in the uv -plane using the uvcontsub task. \nAll imaging was performed with the CASA tclean task. We used the multiscale deconvolution algorithm 17 with (Gaussian) deconvolution scales [ 0 . 02 '' , 0 . 1 '' , 0 . 3 '' , 0 . 6 '' , 1 . 0 '' ]. We did not image with a Keplerian mask so as not to restrict our ability to observe non-Keplerian emission. After experimentation with CASA's auto-multithresh masking algorithm 18 , we adopted an imaging strategy similar to PHANGS-ALMA 19 , in which we clean conservatively, with a broad mask ( usemask='pb' \nand pbmask=0.2 ), forcing frequent major cycles 2 . To achieve frequent major cycles we set the maximum number of minor cycle iterations per channel to cycleniter=80 , the minor cycle threshold to max\\_psf\\_sidelobe\\_level=3.0 and minpsffraction=0.5 , and the maximum assigned clean component to gain=0.02 times the peak residual. We adopted a Briggs robust weighting scheme, and generated two sets of image cubes; one with a robust value of 0 . 5 and a second with robust 1 . 5 . The corresponding beam sizes for 13 CO are 237 × 175 mas, 1 . 2 · for robust 0 . 5 and 390 × 274 mas, -1 . 4 · for robust 1 . 5 . We imaged with a FOV out to the primary beam FWHM ( 38 '' ) with 0 . 02 '' pixels ( 9 or 12 pixels per synthesized beam minor or major axis, respectively). We imaged in LSRK velocity channels at 42 m/s for 13 CO and 84 m/s for C 18 O respectively (nearly native channel spacing). The CLEAN threshold was set to 5 × the rms noise measured in 20 line-free channels of the dirty image cube. We applied JvM correction 20,21 and primary beam correction. The rms noise in the resulting 13 CO cubes imaged with robust 0 . 5 and robust 1 . 5 is 2 . 0 mJy/beam and 1 . 2 mJy/beam respectively, and 0 . 6 mJy/beam in the C 18 Ocube imaged with robust 1 . 5 . \nWe used the robust 0 . 5 image cubes for our position-position analysis (moment maps; Figures 1 & 2) and the robust 1 . 5 cubes for our position-velocity analysis (PV diagrams and line centers; Figures 3 & 4). We made the moment 0, 1 and 2 maps using the bettermoments 22,23 methods 'collapse\\_zeroth, 'collapse\\_first', and 'collapse\\_percentiles', respectively. We note that we calculate our 'moment 2' maps as the average of the red- and blue-shifted line widths about the intensity-weighted median line center (i.e., as the average of the wpdVr and wpdVb maps). Mathematically this is a different approach to find the line width than the classic moment 2 approach, though in our case we find the two yield nearly identical outcomes. We applied sigma-clipping at 5 × the rms noise and performed no spectral smoothing. \nGeometric properties. We used the Python package eddy 24 to infer geometric properties of the disk, namely to constrain the disk center x 0 , y 0 , the disk inclination i , the position angle PA , the systemic velocity v sys , and the dynamical stellar mass M ⋆ . We performed an MCMC to fit the C 18 Omoment 1 map (Extended Data Figure 4) with a geometrically thin Keplerian disk rotation profile: \nv 0 = √ GM ⋆ r · sin i · cos ϕ + v sys , (1) \nwhere r is the disk radius, ϕ is the azimuthal angle around the disk, and G is the gravitational constant. Following convention, we fix the inclination to the value found from fitting the continuum, i = 23 . 2 · (ref. 10,25 ), and the distance to 155.9 pc (Gaia DR3 26,27 ). We assumed flat priors for all values and spatially downsampled the rotation map to the beam FWHM prior to the likelihood calculation so that only spatially independent pixels were considered. The calculation of the posterior distributions was run with 128 walkers and an initial burn-in period of 10,000 steps before the posterior distributions were sampled for additional 10,000 steps. The resulting posterior distributions were x 0 = -5 ± 7 mas, y 0 = -17 ± 7 mas, PA = 236 . 7 ± 0 . 3 · , M ⋆ = 2 . 23 ± 0 . 02 M ⊙ , and v sys = 5858 ± 5 m/s, where we report the uncertainties represented by the 16th and 84th percentiles about the median value. The latter three values are consistent with constraints from previous observations 4, 5, 10, 25, 28 . \nHydrodynamic simulations and synthetic ALMA observations. We performed 3D global smoothedparticle hydrodynamic (SPH) simulations with the PHANTOM code 29 using 1 million SPH particles. We assumed a central star mass of 2 . 4 M ⊙ (ref. 4, 5 ), represented by a sink particle 30 with accretion radius set to 60 au. The initial inner and outer disk radii were set to r in , SPH = 80 au and r out , SPH = 500 au, respectively. We set the initial gas mass to 0 . 7 M ⊙ , corresponding to M disk /M ⋆ = 0 . 29 . The surface density profile follows Σ ∝ r -p (where the power-law index p = 1 . 0 ), and the sound speed profile follows c s ∝ r -q (where q = 0 . 25 ). The initial disk aspect ratio was set to H/r = 0 . 05 at 80 au. We set α SPH such that α min ≤ α SPH ≤ α max , with α min = 0 . 001 and α max = 1 . 0 , with the value of α SPH set by the Cullen & Dehnen (2010) 31 switch that increases viscosity only in the case of converging flows. This results in a Shakura-Sunyaev viscosity of α SS ≈ 0 . 01 throughout the disk. \nWe assumed an adiabatic equation of state, with heating from compressional P d V work and shock heating. The disk cools by Gammie cooling 32 (a.k.a. β -cooling) where the cooling timescale is proportional to the local dynamical time by the factor β , such that t cool ( r ) = β Ω -1 ( r ) , where Ω( r ) = ( GM ⋆ /r 3 ) 1 / 2 is the Keplerian frequency. We set β = 10 , a typical value used or found in simulations 33-36 . We let the simulation evolve for five orbital periods of the outermost particle, at which point the disk settles into a state in which the Toomre Q parameter is between 1 and 2 from r in , SPH to 1.1 r out , SPH . \nWe computed the disk thermal structure and 13 CO ( J = 2 -1 ) model line cubes using the Monte Carlo radiative transfer code MCFOST 37,38 . We assumed the 13 CO molecule is in local thermodynamic equilibrium (LTE) with its surroundings and that the dust is in thermal equilibrium with the gas ( T gas = T dust ). We set the 13 CO/H 2 abundance to 7 × 10 -7 (ref. 33,39 ) and we used ≈ 10 7 photon packets to calculate T dust . Voronoi tesselation was performed on 990,972 SPH particles which corresponded to 99% of the mass in the simulation. We set the total dust mass to 1% of the total SPH gas mass and used a dust grain population with 50 logarithmic bins ranging in size from 0 . 1 µ m to 3 . 0 mm. The dust optical properties are computed using Mie theory. The central star was represented as a sphere of radius 2 . 5 R ⊙ radiating isotropically at an effective temperature T eff = 9770 K, set to match AB Aur 40-43 . The disk was given an inclination of 23 . 2 · , a position angle of 236 . 7 · (where PA is measured east of north to the red-shifted major axis), and placed at a distance of 155 . 9 pc, all consistent with the AB Aur system. \nWe used the same PHANTOM simulation to create both the GI and Keplerian model line cubes shown in Figure 2c and 3. We created the Keplerian counterpart with MCFOST, using the flags -no\\_vr and -no\\_vz to force the radial and vertical velocities to be zero, and -vphi\\_Kep to force the azimuthal velocities to be Keplerian. Both 13 CO model line cubes were generated with MCFOST, binned at the observed spectral resolution of 42 m/s, and gridded in the image plane to have 2048 × 2048 pixels of size 0 . 02 '' . We assumed a turbulent velocity of 0 . 05 km/s. \nWe generated synthetic ALMA image cubes from the 13 CO model line cubes using syndisk 3 to match the properties of the observed AB Aur 13 CO image cubes (robust 0 . 5 and 1 . 5 ). In the latter case \nthe model line cube was convolved with a beam of size 0 . 390 '' × 0 . 274 '' and PA -1 . 4 · . Correlated noise was added with an rms of 1 . 2 mJy/beam. The model data were then smoothed with a Hanning spectral response function with a resolution of 42 m/s. Effects associated with interferometric or spatial filtering are not captured by this process, and our synthetic ALMA image cubes are effectively fully-sampled in the uv -plane. The synthetic cubes were collapsed into moment maps following the same procedure as the AB Aur data (Extended Data Figure 1). \nAnalytic modeling. We analytically compute the velocity fields of gravitationally unstable disks using the giggle 4 package developed by Longarini et al. (2021) 35 . Working in 2D polar coordinates ( r , ϕ ), giggle considers a geometrically thin disk with surface density profile Σ 0 ∝ r -p and inclination i , centered on a star of mass M ⋆ . It computes the projected line-of-sight velocity field as \nv los = ( v r sin ϕ + v ϕ cos ϕ ) sin i + v sys , (2) \nwhere v r and v ϕ are the radial and azimuthal components of the disk velocity field. The basic state of the disk (i.e., considering only the gravitational potential contribution from the central star) is assumed to be Keplerian: v r = 0 and v ϕ = v Kep . The scheme of the model is to determine the perturbations in v r and v ϕ generated by gravitational instability by taking into account the additional gravitational contribution from the disk, which is initialized as marginally unstable and imprinted with global spiral density perturbations. The model computes the velocity field under the assumption that the disc is self-regulated. This state is imposed by assuming a balance between heating (by compression and shocks within the spiral arms) and cooling (by radiative processes). As such, the amplitude of the spiral density perturbations A Σ spir / Σ 0 saturated to a finite value proportional to the cooling timescale β : 44,45 \nA Σ spir Σ 0 = χβ -1 / 2 , (3) \nwhere the proportionality factor χ is of order unity 35,45 . The imprinted spiral density perturbation is assumed to be small relative to the background surface density, so that all the relevant quantities (density Σ , gravitational potential Φ , velocities v r and v ϕ , and enthalpy h ) can be written as a linear sum of the basic state and the perturbation: \nX ( r, ϕ ) = X 0 ( r ) + X spir ( r, ϕ ) . (4) \nThe spiral perturbation in density is given the form \nΣ spir ( r, ϕ ) = ℜ [ A Σ spir e j ( mϕ + ψ ( r )) ] , (5) \nwhere j = √ -1 (as we are using i to represent the disk inclination), and m is the azimuthal wavenumber. The 'shape function' ψ ( r ) is described by m and the spiral pitch angle α pitch as: \nψ ( r ) = m tan α pitch log r , (6) \nwhich is related to the radial wavenumber k by dψ/dr = k . The spiral density perturbation necessarily introduces a corresponding perturbation to the gravitational potential: \nΦ spir ( r, ϕ ) = -2 πG | k | Σ spir ( r, ϕ ) . (7) \nThe negative proportionality Φ spir ∝ -Σ spir is the definition of self-gravitating spiral arms. As a result, corresponding perturbations in the azimuthal and radial velocities are driven: \nv r ( r, ϕ ) = ℜ [ A v r ( r ) · e j ( mϕ + ψ ( r )) ] , (8) \nv ϕ ( r, ϕ ) = ℜ [ A v ϕ ( r ) · e j ( mϕ + ψ ( r )) ] + r Ω , (9) \nwhere we note r Ω = v Kep because the angular frequency Ω includes super-Keplerian rotation from the disk mass contribution: \nΩ 2 = GM ⋆ r 3 + 1 r ∂ Φ disk ∂r . (10) \nBy assuming the disk is marginally unstable, and by maintaining the self-regulated state condition, the amplitude of the radial and azimuthal velocity perturbations A v r ( r ) and A v ϕ ( r ) are determined: 35 \nA v r ( r ) = 2 jmχβ -1 / 2 ( M disk ( r ) M ⋆ ) 2 v Kep ( r ) , (11) \nA v ϕ ( r ) = -1 2 jχβ -1 / 2 ( M disk ( r ) M ⋆ ) v Kep ( r ) , (12) \nwhere M disk ( r ) is the disk mass enclosed within radius r . With a surface density profile Σ 0 ( r ) ∝ r -p , then M disk ( r ) ∝ r -p +2 , and the amplitude of the radial perturbation is described by A v r ( r ) ∝ r -2 p +7 / 2 . For p < 7 / 4 , A v r ( r ) is an increasing function of radius. The factor of imaginary number j in Eqn. 11 has important physical consequences: when the real component of A v r ( r ) is taken (Eqn. 8), the radial velocity perturbation is π/ 2 out of phase with the spiral density perturbation (Eqn. 5), and convergent at the locations where Σ spir takes a maximum. Explicitly, \nv r ( r, ϕ ) ∣ ∣ ∣ ∣ ϕ = π/ 2 ∝ -sin ( m π 2 + ψ ( r ) ) , (13) \nΣ spir ( r, ϕ ) ∣ ∣ ∣ ∣ ϕ = π/ 2 ∝ cos ( m π 2 + ψ ( r ) ) . (14) \nFor qualitative visual comparison with the AB Aur moment 1 map in Figure 2a, we compute the projected line-of-sight velocity field of a gravitationally unstable disk with β = 10 and M disk /M ⋆ = 0 . 3 in Figure 2b. We set m = 3 and α pitch = 15 · to approximately match the 13 CO spirals in the AB Aur disk (Figure 1), and assume p = 1 . 0 and χ = 1 . 0 (ref. 45 ). The dominant azimuthal wavenumber is expected to be inversely related to the disk-to-star mass ratio q , roughly obeying m ∼ 1 /q (ref. 45-47 ), so our choice of m = 3 is consistent with M disk /M ⋆ ≈ 0 . 3 . \n̸ \nRevealing global spiral structure. Weobtain the residual moment maps shown in Figure 1 using a variation on the conventional high-pass filtering (a.k.a. unsharp masking) technique. The conventional method is to convolve the image with a Gaussian kernel and subtract the blurred image from the original. It is a common technique to increase the visual contrast of variations in an image and has been used successfully to reveal spiral structure disks (e.g. 6, 48-53 ). Here, we perform the convolution with a radially expanding kernel 5 - that is, with a Gaussian kernel whose FWHM, w , increases with radial distance from the image center (i.e., with disk radius) with a simple power-law dependence: \nw ( r ) = w 0 · ( r/r 0 ) γ , (15) \nwhere w 0 is the kernel width at r 0 = 1 '' . A radially expanding kernel provides a way to highlight variations more evenly throughout the disk, given the spatial scales of the variations -which are expected to track with the local scale height and increase with radius- and the dynamical range of the variations, which fall with radius. After experimentation we adopt w 0 = 0 . 3 '' and γ = 0 . 25 , though we emphasize this is a qualitative choice and the key spiral features, such as their locations, are robust against a variety of choices in kernel parameters. The high-pass filter technique is also flexible to the disk emission surface morphology, and can capture global scale deviations from Keplerian rotation in the background disk. Extended Data Figure 4 compares the residual moment 1 maps in 13 CO and C 18 O obtained after subtracting the axisymmetric geometrically thin Keplerian model (Eqn. 1) vs. after subtracting a blurred version of the moment 1 map made with the expanding kernel filter. The Keplerian residuals (panels c and h) show signs of global scale deviation from Keplerian: the east (west) side is generally blue-shifted (red-shifted), hinting at super-Keplerian rotation, signatures of disk mass contributing to the total mass of the system. While spiral structure is indeed also visible in the Keplerian residuals, the expanding kernel residuals (panels e and j) reveal the underlying spiral structure in a spatially even manner, indicating that the expanding kernel background model (panels d and i) more successfully captures the quasi-local background disk velocity. We note that this background model is non-axisymmetric; it displays excess blueshifted velocity in the southeast quadrant of the disk such that the contour of v los = v sys diverges westward from the minor axis south of the star, possibly indicative of a global disk warp. This is what necessitates a detrending of the line centers to isolate the sinusoidal component of the southern minor axis PV wiggle in Figure 4a (see section 'Measuring the magnitude of AB Aur's minor axis PV wiggle'). Filtered moment maps for the synthetic ALMA observations of the simulated SPH GI disk are shown in Extended Data Figure 2. \nGlobal kinematics of self-gravitating spiral arms. Radially convergent motion (as in Figure 1bcd insets) serves as a kinematic signature for the location of self-gravitating spiral arms at disk azimuths where the radial velocity perturbation contributes sufficiently strongly to the observed velocity field, and thus cannot be a fully unambiguous locator at disk azimuths away from the minor axis. Extended Data Figure 5c and g provide maps of velocity residuals from Keplerian for the 2D analytic GI disk model and the SPH GI disk simulation. The convergent motion toward the spiral spines is visible for a range of azimuths around the minor axis, but becomes progressively less clear moving toward the major axis as the azimuthal velocity -super-Keplerian rotation- contributes progressively more to the line-of-sight. However, high-pass filtering (panel h) captures and removes the background super-Keplerian rotation, leaving a residual map that \nresembles the isolated radial component (panel d). Extended Data Figure 5i-l overlays the locations of 13 CO spirals in the AB Aur disk (from filtered moment 0/2; Figure 1cd) onto the filtered moment 1 maps, in order to illustrate where convergent motion does or does not serve as a locator throughout the disk. Ambiguity occurs around the major axis, which is a location of transition in the sign of v r sin i sin ϕ (first term of Eqn. 2), and when two spirals are not well separated and their motions superimpose. Three of the seven spiral structures in VLT/SPHERE scattered light appear to be spatially associable with those in 13 CO (S1, S5, S7; panel l inset). Offsets in the southeast quadrant of the disk (S2, S3, S4) may be further indication of a disk warp (Extended Data Figure 4di), or other non-trivial phenomena (e.g., vertical density and temperature gradients, projection effects 54 ). \nThe kinematic signatures observed in the present ALMA dataset -probing disk scales ∼ 100 to 1,000 au- are recognizably different from what is expected for planet-driven perturbations. Planetary wakes are dampened and become nearly circular as they propagate away from the planet 55-57 , whereas GI-driven spirals maintain their modest pitch angles with radius and the amplitude of the induced velocity perturbations depends on the enclosed disk mass (Eqns. 11 & 12). In the planetary case, the density and radial velocity perturbations are in phase (their peaks spatially coincide), and the pattern of motion within an arm along a radial cross-section is divergent 58,59 . Overall, the essential characteristic of GI-induced spirals is that they occur globally 33,35 (c.f. Figure 1, Extended Data Figures 2, 3 & 5). In previous datasets probing smaller spatial scales -within the AB Aur disk's central cavity- planetary candidates P1/f1 (ref. 25,48 ), P2/b (ref. 25,43,60-62 ), and f2 (ref. 48 ) are known to be associated with -or driving- spiral arms, as observed in VLT/SPHERE scattered light and/or ALMA 12 CO emission. As shown in Extended Data Figure 9, due to their small separations ( ≲ 0 . 7 '' ), kinematic signatures from these candidates are inaccessible to our ALMA observations. Clump-like signals 'c' and 'd' seen by HST/STIS (ref. 43 ) at wide separations ( ∼ 2 . 75 '' and ∼ 3 . 72 '' respectively) are in locations tentatively suggestive of constituting spiral arm fragments and may warrant further investigation. \nPosition-velocity analysis. Weuse the robust 1 . 5 image cubes for our position velocity analysis to maximize the recovery of emission at large disk radii. Owing to the clear association with a self-gravitating spiral arm (Figure 1bcd insets), we target the wiggle on the southern minor axis. A clear spiral arm in moment 0/2 crossing the northern minor axis is also observed, but at the outer edge of the recovered 13 CO and C 18 O emission ( ∼ 3 '' ; c.f. Extended Data Figure 5kl). We obtain the position-velocity diagrams shown in Figure 3 using eddy 24 to extract spectra from pixels within a 0 . 5 · -wide wedge-shaped mask oriented 90 · clockwise of the red-shifted major axis (shown in Figure 3 insets). Our quantitative analysis of the minor axis PV wiggles is performed with maps of the line centers made using the quadratic method of bettermoments 22,23 , which fits a quadratic curve to the spectrum in each pixel of the cube: \nI ( v ) = a 0 + a 1 ( v -v peak ) + a 2 ( v -v peak ) 2 , (16) \nwhere v peak is the channel of peak intensity in the spectrum. We select this approach over the traditional intensity-weighted mean velocity (moment 1) method specifically for its ability to provide well characterized, statistically meaningful uncertainties on the line center, σ v los (ref. 23 ). The statistical uncertainty on \neach line center is computed as: \nσ v los = √ σ 2 I 8 ( 3 a 2 2 + a 1 2 a 2 4 ) , (17) \nwhere σ I is the rms noise of the intensities (see ref. 23 for a derivation). The quadratic method also has the advantage of being unaffected by sigma-clipping and of automatically distinguishing the front side of the disk from the back side 23 . Prior to the quadratic fitting we spectrally smooth the data with a Savitzky-Golay filter of polynomial order 1 and filter window length of 10 channels ( 420 m/s) in the case of 13 CO and 3 channels ( 252 m/s) in the case of C 18 O. The former was also applied to the two synthetic ALMA 13 CO image cubes generated from the SPH simulations. We extract the values from the resulting line center and line uncertainty maps within the same wedge mask described above. The extracted line center values are shown as yellow points in Figure 3 and the uncertainties are shown as yellow shaded regions in Figure 4a. \nMeasuring the magnitude of the minor axis PV wiggle. Following Longarini et al. (2021) 35 , we measure the 'magnitude' of a minor axis PV wiggle as the standard deviation of the line center values over a radial range. Bounded by the inner central cavity and the outer edge of C 18 O emission, we adopt a radial range of 1 . 0 '' to 5 . 0 '' . We estimate the uncertainty on the magnitude measurement using a resampling procedure: we take 10,000 draws from Gaussian distributions centered on the observed line centers with standard deviation σ v los (Eqn. 17) to create 10,000 instances of the minor axis PV wiggle; we compute their magnitudes; and then report the uncertainty as the standard deviation of those 10,000 magnitude estimates. \nIn addition to the wiggle, the 13 CO and C 18 O emission on the southern disk minor axis also exhibit an underlying monotonic blueward trend with disk radius, seen in Figure 3ab as a subtle downward bend with radius of the line centers, or equivalently in Figure 2a as a westward or clockwise shift in the contour of v los = v sys . We earmark this feature as a possible disk warp (Extended Data Figure 4di), and adopt a least-squares fitting approach to isolate the sinusoidal component of the PV wiggle. This approach yields the background trendline that minimizes the standard deviation of the residuals, thus providing the most conservative estimate for the magnitude of the detrended PV wiggle. We fit a quadratic trendline (Extended Data Figure 6a) as it more closely resembles the high-pass filter background curve than a linear one (Extended Data Figure 6bc). We show the quadratically-detrended PV wiggles in Figure 4a and report their magnitudes in Figure 4b. We find very similar magnitudes for both the 13 CO and C 18 O wiggles, despite C 18 O likely tracing lower optical depths in the AB Aur disk. This empirically substantiates comparisons with the 2D analytic model (next section). \nPerforming the same procedure outlined above on the synthetic 13 CO minor axis PV wiggle of the GI disk in the SPH simulation, we find a wiggle magnitude of 39 . 1 ± 1 . 9 m/s (Extended Data Figure 7). \nConstraining disk mass with quantitative comparisons to analytic models. We perform quantitative comparisons between the observed 13 CO and C 18 Ominor axis PV wiggles and the projected radial velocity component in our analytic model, v r sin i (ref. 35 ). From Eqns. 8 and 11, the projected radial velocity on the \nminor axis ( ϕ = π/ 2 ) is: \nv r ( r, ϕ ) ∣ ∣ ∣ ∣ ϕ = π/ 2 · sin i = -2 mχβ -1 / 2 ( M disk ( r ) M ⋆ ) 2 v Kep ( r ) sin ( m π 2 + ψ ( r ) ) · sin i . (18) \nThis curve reflects the disk mass enclosed within the inner and outer radii of the model, which we set to span the same projected radial range as the observed PV wiggles ( 1 '' to 5 '' ). We compute 3600 of these curves for a 60 × 60 grid of models with (total enclosed) M disk /M ⋆ linearly spaced ∈ [0 . 0 , 0 . 4] and β logarithmically spaced ∈ [10 -2 , 10 2 ] . Again we set m = 3 and α pitch = 15 · to match the AB Aur disk, and assume p = 1 . 0 and χ = 1 . 0 (ref. 45 ). For qualitative comparison, we plot an example analytic minor axis PV wiggle behind the data in Figure 4a; the model has β = 10 and M disk /M ⋆ = 0 . 3 . We show in Extended Data Figure Figure 8 that m = 3 reproduces the observed wiggles better than other choices, and that p = 1 . 5 could also provide a satisfying match, while p = 2 . 0 is too steep. Since the wiggle amplitude is independent of α pitch (Eqn. 11), the magnitude is constant with α pitch when sampled over the same range in phase (not shown). \nWe measure the minor axis PV wiggle magnitude of the 3600 models and present the resulting magnitude map in Figure 4c. By drawing contours in the Figure 4c map at the magnitude values measured for AB Aur ( 37 . 4 ± 2 . 9 m/s in 13 CO and 44 . 2 ± 1 . 3 m/s in C 18 O), we find every combination of M disk /M ⋆ and β that satisfy the observations. Repeating this procedure with our synthetic ALMA observations of the SPH GI disk simulation shown in Figure 3c, we find that this technique successfully recovers the disk mass set in the underlying SPH simulation (Extended Data Figure 7). \nFor independent physical estimates of plausible β values between 1 '' to 5 '' ( 155 to 780 au), we rely on radiative cooling prescriptions 63,64 . From Equation 39 of Zhang & Zhu (2020) 64 , β is a function of r and depends on M disk through the surface density Σ . We assume T = ( ϕL ⋆ 8 π r 2 σ SB ) 1 / 4 , where σ SB is the StefanBoltzmann constant, L ⋆ = 59 L ⊙ is the stellar luminosity of AB Aur 43 , and ϕ = 0 . 02 represents the flaring angle 65 . We use the DSHARP Rosseland mean opacity 66 κ R = κ R ( T, a max ) for a power-law grain size distribution truncated at a max . We set a max to 0 . 1 mm and the dust-to-gas mass ratio to f = 0 . 1% , based on radial drift arguments and lack of (sub-)mm emission at these large radii. We compute a β ( r ) profile for each M disk /M ⋆ ∈ [0 . 0 , 0 . 4] and extract the values at 1 '' and 5 '' . We overlay the resulting β ( M disk /M ⋆ ) ranges as white shaded regions in Extended Data Figure 8 (where the dependence on p arises from the dependence on Σ ), and in Figure 4 as white horizontal bars at a selection of M disk /M ⋆ values. For example, for M disk /M ⋆ = 0 . 2 and p = 1 . 0 , we find β (1 '' ) = 5 . 3 and β (5 '' ) = 3 . 6 × 10 -2 . While knowledge of cooling in disks is very limited, these estimates help to emphasize that not all values of β are equally likely. \nData availability All observational data products presented in this work are available through the CANFAR Data Publication Service at https://doi.org/10.11570/24.0087 . This includes final reduced and calibrated ALMA measurement sets, image cubes and moment maps, and processed SPHERE data. All simulated data products including hydrodynamic simulations and synthetic ALMA data are available at https://doi.org/10.5281/zenodo.11668694 . The raw ALMA data are publicly available via the ALMA archive https://almascience.nrao.edu/aq/ under project ID 2021.1.00690.S. The raw VLT/SPHERE data are publicly available via the ESO Science Archive Facility https://archive. \neso.org/eso/eso\\_archive\\_main.html under programme 0104.C-0157(B). \nCode availability ALMA data reduction and imaging scripts are available at https://jjspeedie. github.io/guide.2021.1.00690.S . The Python packages used in this work are available: bettermoments ( https://github.com/richteague/bettermoments ), eddy ( https://github.com/richteague/ eddy ), giggle v0 ( http://doi.org/10.5281/zenodo.10205110 ), PHANTOM ( https:// github.com/danieljprice/phantom ), MCFOST ( https://github.com/cpinte/mcfost ). \nAcknowledgements We thank our referees for their careful and insightful comments that improved the manuscript. We thank Kaitlin Kratter for enlightening discussions and valuable suggestions. J.S. thanks Ryan Loomis, Sarah Wood and Tristan Ashton at the North American ALMA Science Center (NAASC) for providing science support and technical guidance on the ALMA data as part of a Data Reduction Visit to the NAASC, which was funded by the NAASC. The reduction and imaging of the ALMA data was performed on NAASC computing facilities. J.S. thanks Christophe Pinte, Daniel Price and Josh Calcino for support with MCFOST, Luke Keyte and Francesco Zagaria for discussions on self-calibrating ALMA data, and Chris White for sharing perceptually uniform colormaps. J.S. acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Canada Graduate Scholarships Doctoral (CGS D) program. R.D. acknowledges financial support provided by the Natural Sciences and Engineering Research Council of Canada through a Discovery Grant, as well as the Alfred P. Sloan Foundation through a Sloan Research Fellowship. C.L. and G.L. acknowledge funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement # 823823 (RISE DUSTBUSTERS project). C.L. acknowledges funding from UK Science and Technology research Council (STFC) via the consolidated grant ST/W000997/1. B.V. acknowledges funding from the ERC CoG project PODCAST No 864965. Y.W.T. acknowledges support through NSTC grant 111-2112-M-001-064- and 112-2112-M-001-066-. J.H. was supported by JSPS KAKENHI Grant Numbers 21H00059, 22H01274, 23K03463. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2021.1.00690.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. Based on data products created from observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 0104.C-0157(B). This work has made use of the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/LAM/CESAM (Marseille), OCA/Lagrange (Nice), Observatoire de Paris/LESIA (Paris), and Observatoire de Lyon. This research used the Canadian Advanced Network For Astronomy Research (CANFAR) operated in partnership by the Canadian Astronomy Data Centre and The Digital Research Alliance of Canada with support from the National Research Council of Canada the Canadian Space Agency, CANARIE and the Canadian Foundation for Innovation. \nAuthor Contributions R.D. led the ALMA proposal. J.S. processed the ALMA data. J.H. processed the VLT/SPHERE data. C.H. performed the SPH simulations. J.S. performed the radiative transfer calculations. C.L. and G.L. developed the analytic model. J.S. performed all presented analyses. J.S. and R.D. wrote the manuscript. All co-authors provided input to the ALMA proposal and/or the manuscript. \nCompeting Interests The authors declare that they have no competing financial interests. \nCorrespondence Correspondence and requests for materials should be addressed to: J.S. (email: jspeedie@uvic.ca), R.D. (email: rbdong@uvic.ca).", 'Extended Data': "Extended Data Figure 1: Moment maps: AB Aur observations and GI disk simulations. (ac) Integrated intensity (moment 0), intensity-weighted mean velocity (moment 1), and intensityweighted line width (moment 2) maps for the ALMA 13 CO observations toward AB Aur. Panel (b) appears in the main text as Figure 2a. (d-f) Moment 0, 1, and 2 maps for the synthetic ALMA 13 COobservations of the SPH GI disk simulation. Like the AB Aur observations, the simulated GI disk displays a prominent GI wiggle along the southern minor axis (indicated by white arrows). \n<!-- image --> \nExtended Data Figure 2: Filtered moment maps: AB Aur observations and GI disk simulations. Expanding kernel filter residuals of the maps shown in Extended Data Figure 1, highlighting global spirals and velocity disturbances generated by GI. Panels (a-c) appear in the main text as Figure 1b-d. The minor axis GI wiggle indicated by white arrows in Extended Data Figure 1b and e is shown here as an isovelocity contour at v los = v sys ± v chan in all panels. \n<!-- image --> \nExtended Data Table 1: Details of the ALMA Band 6 observations. \nExtended Data Figure 3: Global GI wiggles in analytic models, SPH simulations, and the AB Aur disk. Isovelocity contours in line-of-sight velocity maps at the velocity values indicated by the colour bar. (a) v los map of the 2D analytic GI disk model (shown in Figure 2b). (b) v los map of the 2D analytic Keplerian disk model (shown in Figure 2b inset). (c) Synthetic ALMA 13 CO moment 1 map for the 3D SPH GI disk simulation (shown in Figure 2c). (d) Synthetic ALMA 13 CO moment 1 map for the 3D SPH Keplerian disk simulation (shown in Figure 2c inset). (e) Observed ALMA 13 CO moment 1 map for the AB Aur disk, imaged with robust 0 . 5 (shown in Figure 2a). (f) Like (e), but imaged with robust 1 . 5 . \n<!-- image --> \nExtended Data Figure 4: Obtaining velocity residuals in the AB Aur disk. (a) ALMA 13 CO moment 1 map, imaged with robust 0 . 5 , as shown in Figure 2a. (b) Background model made with a Keplerian rotation profile, assuming a geometrically thin axisymmetric disk (Eqn. 1). (c) Velocity residuals after subtracting the model in panel (b). Global spiral substructure is visible, but unevenly so. The model does not capture the non-axisymmetric emission surface morphology and/or superKeplerian rotation. (d) Background model made with the expanding kernel filter (Eqn. 15). (e) Velocity residuals after subtracting the model in panel (d), as shown in Figure 1b. (f-j) Like (a-e) but with the ALMA C 18 O moment 1 map, imaged with robust 1 . 5 . \n<!-- image --> \nExtended Data Figure 5: Kinematics of GI-driven spiral arms. (a-d) 2D analytic modeling (Longarini et al. 2021) 35 . (e-h) Synthetic ALMA 13 CO observations of the 3D SPH GI disk simulation. (i-l) ALMA observations of the AB Aur disk. (a) Disk surface density (Eqn. 4 & 5). (b) Line-of-sight velocity (Eqn. 2), as in Figure 2b. (c) Velocity residuals from Keplerian (i.e., subtracting Figure 2b inset). (d) Line-of-sight component of the radial velocity (first term of Eqn. 2). (e) Filtered moment 0. (f) Moment 1. (g) Moment 1 residuals from Keplerian. (h) Filtered moment 1. (i) ALMA 13 CO filtered moment 0. (j) ALMA 13 CO filtered moment 2. (k) ALMA 13 CO filtered moment 1. (l) ALMA C 18 O filtered moment 1 (robust 1 . 5 ). Panel (l) inset overlays the VLT/SPHERE H -band scattered light spirals S1-S7 (ref. 14,67 ) in red, and 13 CO spirals S1-S9 we identify in black. 29 \n<!-- image --> \nExtended Data Figure 6: Methods for isolating the sinusoidal component of the southern minor axis PV wiggle in the AB Aur disk. (a) Detrending the ALMA 13 CO line centers from Figure 3a with linear and quadratic trendlines found by a least-squares fit. (b) Detrending with the expanding kernel high-pass filter, varying the kernel width parameter w 0 and keeping the kernel radial power-law index fixed to γ = 0 . 25 (Equation 15). We find the background trendlines by extracting the velocity values from the high-pass filter background map (e.g. Extended Data Figure 4d) within the same 0 . 5 · -wide wedge-shaped mask as we do for the line centers, positioned along the southern disk minor axis. (c) Like (b), but varying γ and keeping w 0 fixed to w 0 = 0 . 30 '' . The high-pass filter detrending approach converges to the same measured PV wiggle magnitude as the quadratic fit approach. \n<!-- image --> \n<!-- image --> \nExtended Data Figure 7: PV wiggle morphology, magnitude, and disk mass recovery in the SPH GI disk simulation. Like Figure 4, but for the synthetic ALMA observations of the SPH GI disk simulation. (a) The synthetic ALMA 13 CO line centers along the southern minor axis from Figure 3c, after quadratic detrending. Uncertainties on the line centers are shown by yellow shaded regions. The magnitude of this PV wiggle is measured to be 39 . 1 ± 1 . 9 m/s. The analytic model shown in the background for qualitative comparison has the same parameters as the underlying SPH simulation ( M disk /M ⋆ = 0 . 29 and β = 10 ) and its PV wiggle magnitude is 39 . 0 m/s. (b) As in Figure 4c, a map of the minor axis PV wiggle magnitude of 60 × 60 analytic models on a grid of disk-to-star mass ratios and cooling timescales. A contour is drawn at the measured magnitude of the synthetic 13 CO PV wiggle in panel (a), and dashed lines represent the quoted uncertainties. The technique successfully recovers the disk mass set in the SPH simulation. \n<!-- image --> \nExtended Data Figure 8: Comparisons to additional sets of analytic models. Like Figure 4, but varying the azimuthal wavenumber m and surface density power-law index p in the comparison grid of analytic GI model disks. Each upper subpanel shows the quadratically detrended 13 CO and C 18 O line centers (yellow) behind a demonstrative analytic PV wiggle (black) computed with the combination of m and p indicated by the row and column labels (keeping M disk /M ⋆ = 0 . 3 and β = 10 fixed). Each lower subpanel shows the corresponding map of PV wiggle magnitude computed for a 60 × 60 grid of analytic models in M disk /M ⋆ and β , again with the combination of m and p indicated by the row and column labels. The two yellow contours are drawn at the magnitude values measured for the observed AB Aur 13 CO and C 18 O southern minor axis PV wiggles. The white shaded region between two white curves represents plausible β ranges from r = 1 -5 '' . The combination shown in Figure 4c is m = 3 , p = 1 . 0 . 32 \nExtended Data Figure 9: Candidate sites of planet formation. Coloured × 's mark the locations of candidate protoplanets reported in the literature 25,43,48 . A table providing the candidates' coordinates on the sky, estimated masses and the reporting references is available as source data . (a) Filtered ALMA 13 CO moment 0 map, as in Figure 1c. (b) VLT/SPHERE H -band scattered light image (ref. 48 ), as in Figure 1a. The inset zooms into the central 2 '' × 2 '' region to show the spiral structures in different tracers at spatial scales unresolved by the present ALMA observations. The H -band scattered light image is shown after high-pass filtering, and orange contours show the two spirals identified in ALMA 12 CO J = 2 -1 moment 0 (ref. 25 ) at levels from 25 to 50 mJy/beam km/s in increments of 5 mJy/beam km/s. \n<!-- image -->"}

2016arXiv161100036D

DESI Dark Energy Spectroscopic Instrument is a Stage IV groundbased dark energy experiment that will study baryon acoustic oscillations BAO and the growth of structure through redshiftspace distortions with a widearea galaxy and quasar redshift survey. To trace the underlying dark matter distribution spectroscopic targets will be selected in four classes from imaging data. We will measure luminous red galaxies up to z1.0. To probe the Universe out to even higher redshift DESI will target bright O II emission line galaxies up to z1.7. Quasars will be targeted both as direct tracers of the underlying dark matter distribution and at higher redshifts 2.1 lt z lt 3.5 for the Lyalpha forest absorption features in their spectra which will be used to trace the distribution of neutral hydrogen. When moonlight prevents efficient observations of the faint targets of the baseline survey DESI will conduct a magnitudelimited Bright Galaxy Survey comprising approximately 10 million galaxies with a median zapprox 0.2. In total more than 30 million galaxy and quasar redshifts will be obtained to measure the BAO feature and determine the matter power spectrum including redshift space distortions.

2016-10-01T00:00:00Z

['10.48550/arXiv.1611.00036', '2016arXiv161100036D', 'arXiv:1611.00036']

['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Cosmology and Nongalactic Astrophysics']

The DESI Experiment Part I ScienceTargeting and Survey Design

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/1611.00036.pdf

{'The DESI Experiment Part I: Science,Targeting, and Survey Design': "DESI Collaboration: Amir Aghamousa 73 , Jessica Aguilar 76 , Steve Ahlen 85 , Shadab Alam 41 , 59 , Lori E. Allen 81 , Carlos Allende Prieto 64 , James Annis 52 , Stephen Bailey 76 , Christophe Balland 88 , Otger Ballester 57 , Charles Baltay 84 , Lucas Beaufore 45 , Chris Bebek 76 , Timothy C. Beers 39 , Eric F. Bell 28 , Jos Luis Bernal 66 , Robert Besuner 89 , Florian Beutler 62 , Chris Blake 15 , Hannes Bleuler 50 , Michael Blomqvist 2 , Robert Blum 81 , Adam S. Bolton 35 , 81 , Cesar Briceno 18 , David Brooks 33 , Joel R. Brownstein 35 , Elizabeth Buckley-Geer 52 , Angela Burden 9 , Etienne Burtin 12 , Nicolas G. Busca 7 , Robert N. Cahn 76 , Yan-Chuan Cai 59 , Laia Cardiel-Sas 57 , Raymond G. Carlberg 23 , Pierre-Henri Carton 12 , Ricard Casas 56 , Francisco J. Castander 56 , Jorge L. Cervantes-Cota 11 , Todd M. Claybaugh 76 , Madeline Close 14 , Carl T. Coker 26 , Shaun Cole 60 , Johan Comparat 67 , Andrew P. Cooper 60 , M.-C. Cousinou 4 , Martin Crocce 56 , Jean-Gabriel Cuby 2 , Daniel P. Cunningham 1 , Tamara M. Davis 86 , Kyle S. Dawson 35 , Axel de la Macorra 68 , Juan De Vicente 19 , Timoth'ee Delubac 74 , Mark Derwent 26 , Arjun Dey 81 , Govinda Dhungana 44 , Zhejie Ding 31 , Peter Doel 33 , Yutong T. Duan 85 , Anne Ealet 4 , Jerry Edelstein 89 , Sarah Eftekharzadeh 32 , Daniel J. Eisenstein 53 , Ann Elliott 45 , St'ephanie Escoffier 4 , Matthew Evatt 81 , Parker Fagrelius 76 , Xiaohui Fan 90 , Kevin Fanning 48 , Arya Farahi 40 , Jay Farihi 33 , Ginevra Favole 51 , 67 , Yu Feng 47 , Enrique Fernandez 57 , Joseph R. Findlay 32 , Douglas P. Finkbeiner 53 , Michael J. Fitzpatrick 81 , Brenna Flaugher 52 , Samuel Flender 8 , Andreu Font-Ribera 76 , Jaime E. Forero-Romero 22 , Pablo Fosalba 56 , Carlos S. Frenk 60 , Michele Fumagalli 16 , 60 , Boris T. Gaensicke 49 , Giuseppe Gallo 52 , Juan Garcia-Bellido 67 , Enrique Gaztanaga 56 , Nicola Pietro Gentile Fusillo 49 , Terry Gerard 29 , Irena Gershkovich 48 , Tommaso Giannantonio 70 , 78 , Denis Gillet 50 , Guillermo Gonzalez-de-Rivera 54 , Violeta Gonzalez-Perez 62 , Shelby Gott 81 , Or Graur 6 , 38 , 53 , Gaston Gutierrez 52 , Julien Guy 88 , Salman Habib 8 , Henry Heetderks 89 , Ian Heetderks 89 , Katrin Heitmann 8 , Wojciech A. Hellwing 60 , David A. Herrera 81 , Shirley Ho 41 , 47 , 76 , Stephen Holland 76 , Klaus Honscheid 26 , 45 , Eric Huff 26 , Eric Huff 45 , Timothy A. Hutchinson 35 , Dragan Huterer 48 , Ho Seong Hwang 87 , Joseph Maria Illa Laguna 57 , Yuzo Ishikawa 89 , Dianna Jacobs 76 , Niall Jeffrey 33 , Patrick Jelinsky 89 , Elise Jennings 52 , Linhua Jiang 69 , Jorge Jimenez 57 , Jennifer Johnson 26 , Richard Joyce 81 , Eric Jullo 2 , St'ephanie Juneau 12 , 81 , Sami Kama 44 , Armin Karcher 76 , Sonia Karkar 88 , Robert Kehoe 44 , Noble Kennamer 37 , Stephen Kent 52 , Martin Kilbinger 12 , Alex G. Kim 76 , David Kirkby 37 , Theodore Kisner 76 , Ellie Kitanidis 47 , Jean-Paul Kneib 74 , Sergey Koposov 61 , Eve Kovacs 8 , Kazuya Koyama 62 , Anthony Kremin 48 , Richard Kron 52 , Luzius Kronig 50 , Andrea Kueter-Young 34 , Cedric G. Lacey 60 , Robin Lafever 89 , Ofer Lahav 33 , Andrew Lambert 76 , Michael Lampton 89 , Martin Landriau 76 , Dustin Lang 23 , Tod R. Lauer 81 , Jean-Marc Le Goff 12 , Laurent Le Guillou 88 , Auguste Le Van Suu 3 , Jae Hyeon Lee 42 , Su-Jeong Lee 45 , Daniela Leitner 76 , Michael Lesser 90 , Michael E. Levi 76 , Benjamin L'Huillier 73 , Baojiu Li 60 , Ming Liang 81 , Huan Lin 52 , Eric Linder 89 , Sarah R. Loebman 28 , Zarija Luki'c 76 , Jun Ma 72 , Niall MacCrann 13 , 45 , Christophe Magneville 12 , Laleh Makarem 50 , Marc Manera 17 , 33 , Christopher J. Manser 49 , Robert Marshall 81 , Paul Martini 13 , 26 , Richard Massey 16 , Thomas Matheson 81 , Jeremy McCauley 76 , Patrick McDonald 76 , Ian D. McGreer 90 , Aaron Meisner 76 , Nigel Metcalfe 60 , Timothy N. Miller 76 , Ramon Miquel 55 , 57 , John Moustakas 34 , Adam Myers 32 , Milind Naik 76 , Jeffrey A. Newman 30 , Robert C. Nichol 62 , Andrina Nicola 58 , Luiz Nicolati da Costa 75 , 82 , Jundan Nie 72 , Gustavo Niz 21 , Peder Norberg 16 , 60 , Brian Nord 52 , Dara Norman 81 , Peter Nugent 27 , 76 , Thomas O'Brien 26 , Minji Oh 73 , 93 , Knut A. G. Olsen 81 , Cristobal Padilla 57 , Hamsa Padmanabhan 58 , Nikhil Padmanabhan 84 , Nathalie Palanque-Delabrouille 12 , Antonella Palmese 36 , Daniel Pappalardo 26 , Isabelle Pris 2 , Changbom Park 87 , Anna Patej 42 , 90 , John A. Peacock 59 , Hiranya V. Peiris 33 , Xiyan Peng 72 , Will J. Percival 62 , Sandrine Perruchot 3 , Matthew M. Pieri 2 , Richard Pogge 26 , Jennifer E. Pollack 62 , Claire Poppett 89 , Francisco Prada 63 , Abhishek Prakash 30 , Ronald G. Probst 81 , David \nRabinowitz 84 , Anand Raichoor 12 , 74 , Chang Hee Ree 73 , Alexandre Refregier 58 , Xavier Regal 3 , Beth Reid 76 , Kevin Reil 71 , Mehdi Rezaie 31 , Constance M. Rockosi 24 , 92 , Natalie Roe 76 , Samuel Ronayette 3 , Aaron Roodman 71 , Ashley J. Ross 13 , 26 , Nicholas P. Ross 59 , Graziano Rossi 25 , Eduardo Rozo 46 , Vanina Ruhlmann-Kleider 12 , Eli S. Rykoff 71 , Cristiano Sabiu 73 , Lado Samushia 43 , Eusebio Sanchez 19 , Javier Sanchez 37 , David J. Schlegel 76 , Michael Schneider 77 , Michael Schubnell 48 , Aurlia Secroun 4 , Uros Seljak 47 , Hee-Jong Seo 20 , Santiago Serrano 56 , Arman Shafieloo 73 , Huanyuan Shan 74 , Ray Sharples 14 , Michael J. Sholl 5 , William V. Shourt 89 , Joseph H. Silber 76 , David R. Silva 81 , Martin M. Sirk 89 , Anze Slosar 10 , Alex Smith 60 , George F. Smoot 47 , 76 , Debopam Som 2 , Yong-Seon Song 73 , David Sprayberry 81 , Ryan Staten 44 , Andy Stefanik 52 , Gregory Tarle 48 , Suk Sien Tie 26 , Jeremy L. Tinker 38 , Rita Tojeiro 91 , Francisco Valdes 81 , Octavio Valenzuela 65 , Monica Valluri 28 , Mariana Vargas-Magana 68 , Licia Verde 55 , 66 , Alistair R. Walker 81 , Jiali Wang 72 , Yuting Wang 80 , Benjamin A. Weaver 38 , Curtis Weaverdyck 48 , Risa H. Wechsler 71 , 83 , David H. Weinberg 26 , Martin White 47 , Qian Yang 69 , 90 , Christophe Yeche 12 , Tianmeng Zhang 72 , Gong-Bo Zhao 80 , Yi Zheng 73 , Xu Zhou 80 , Zhimin Zhou 80 , Yaling Zhu 89 , Hu Zou 72 , Ying Zu 13 , 79 \n(Affiliations can be found after the references)", 'Abstract': 'DESI (Dark Energy Spectroscopic Instrument) is a Stage IV ground-based dark energy experiment that will study baryon acoustic oscillations (BAO) and the growth of structure through redshift-space distortions with a wide-area galaxy and quasar redshift survey. To trace the underlying dark matter distribution, spectroscopic targets will be selected in four classes from imaging data. We will measure luminous red galaxies up to z = 1 . 0. To probe the Universe out to even higher redshift, DESI will target bright [O II] emission line galaxies up to z = 1 . 7. Quasars will be targeted both as direct tracers of the underlying dark matter distribution and, at higher redshifts (2 . 1 < z < 3 . 5), for the Lyα forest absorption features in their spectra, which will be used to trace the distribution of neutral hydrogen. When moonlight prevents efficient observations of the faint targets of the baseline survey, DESI will conduct a magnitude-limited Bright Galaxy Survey comprising approximately 10 million galaxies with a median z ≈ 0 . 2. In total, more than 30 million galaxy and quasar redshifts will be obtained to measure the BAO feature and determine the matter power spectrum, including redshift space distortions.', '1 Overview': "DESI is a Stage IV ground-based dark energy experiment that will study baryon acoustic oscillations (BAO) and the growth of structure through redshift-space distortions (RSD) with a wide-area galaxy and quasar redshift survey. DESI is the successor to the successful Stage-III BOSS redshift survey and complements imaging surveys such as the Stage-III Dark Energy Survey (DES, operating 2013-2018) and the Stage-IV Large Synoptic Survey Telescope (LSST, planned start early in the next decade). DESI is an important component of the DOE Cosmic Frontier program, meeting the need for a wide-field spectroscopic survey identified in the 2011 'Rocky-III' dark energy community planning report. In addition to providing Stage IV constraints on dark energy, DESI will provide new measurements that can constrain theories of modified gravity and inflation, and that will measure the sum of neutrino masses. \nThe DESI instrument is a robotically-actuated, fiber-fed spectrograph capable of taking up to 5,000 simultaneous spectra over a wavelength range from 360 nm to 980 nm. The fibers feed ten three-arm spectrographs with resolution R = λ/ ∆ λ between 2000 and 5500, depending on wavelength. This powerful instrument will be installed at prime focus on the 4-m Mayall telescope in Kitt Peak, Arizona, along with a new optical corrector, which will provide a three-degree diameter field of view. The DESI collaboration will also deliver a spectroscopic pipeline and data management system to reduce and archive all data for eventual public use. \nThe DESI instrument will be used to conduct a five-year survey designed to cover 14,000 deg 2 . To trace the underlying dark matter distribution, spectroscopic targets will be selected in four classes from imaging data. We will measure luminous red galaxies (LRGs) up to z = 1 . 0, extending the BOSS LRG survey in both redshift and survey area. To probe the Universe out to even higher redshift, DESI will target bright [O II] emission line galaxies (ELGs) up to z = 1 . 7. Quasars will be targeted both as direct tracers of the underlying dark matter distribution and, at higher redshifts (2 . 1 < z < 3 . 5), for the Lyα forest absorption features in their spectra, which will be used to trace the distribution of neutral hydrogen. When moonlight prevents efficient observations of the faint targets of the baseline survey, DESI will conduct a magnitude-limited Bright Galaxy Survey (BGS) comprising approximately 10 million galaxies with a median z ≈ 0 . 2. In total, more than 30 million galaxy and quasar redshifts will be obtained to measure the BAO feature and determine the matter power spectrum, including redshift space distortions. \nIn the following document, we primarily refer to this baseline survey, which would span 14,000 deg 2 . We also calculate numbers for a minimum survey spanning 9,000 deg 2 , which is still sufficient to meet the requirements of a Stage-IV project. \nDESI provides at least an order of magnitude improvement over BOSS both in the comoving volume it probes and the number of galaxies it will map. This will significantly advance our understanding of the expansion history of the Universe, providing more than thirty sub-percentaccuracy distance measurements. Precision on the expansion history of the Universe is a powerful probe of the nature of dark energy. This can be quantified with the Dark Energy Task Force figure of merit (DETF FoM), which measures the combined precision on the dark energy equation of state today, w 0 , and its evolution with redshift w a . DESI galaxy BAO measurements achieve a DETF FoM of 133, more than a factor of three better than the DETF FoM of all Stage-III galaxy BAO measurements combined. The FoM increases to 169 with the inclusion of Lyα forest BAO, and 332 including galaxy broadband power spectrum to k = 0 . 1 h Mpc -1 . DESI clearly satisfies the DETF criteria for a Stage-IV experiment. Moreover, the FoM grows to 704 when the galaxy broadband power spectrum data out to k < 0 . 2 h Mpc -1 are included. \nIn addition, DESI will measure the sum of neutrino masses with an uncertainty of 0.020 eV (for k max < 0 . 2 h Mpc -1 ), sufficient to make the first direct detection of the sum of the neutrino masses \nat 3σ significance and rule out the the inverted mass hierarchy at 99% CL, if the hierarchy is normal and the masses are minimal. DESI will also place significant constraints on theories of modified gravity and of inflation by measuring the spectral index n s and its running with wavenumber, α s . The BGS will enable the best ever measurements of low redshift BAO and RSD, including the use of multiple-tracer methods that exploit galaxy populations with different clustering properties, and it will yield novel tests of modified gravity theories using the velocity fields of cluster infall regions. Because the nearby galaxies of the BGS are too clustered to fill all of the targets, in parallel with the BGS, DESI will conduct a survey of Milky Way stars, that can be used to trace the dark matter halo of the Milky Way and probe the small-scale structure of ΛCDM. \nDESI will provide an unprecedented multi-object spectroscopic capability for the U.S. through an existing NSF telescope facility. Many other science objectives can be addressed with the DESI wide field survey dataset and through bright time and piggy-back observation programs. Much as with SDSS, a rich variety of projects will flow from the legacy data from the DESI survey. \nDESI will overlap with the DES and LSST survey areas, which are primarily in the Southern hemisphere but which will have equatorial and northern ecliptic regions. DESI will be a pathfinder instrument for the massive spectroscopic follow-up required for future large area imaging surveys such as LSST. \nThis portion of the Final Design Report summarizes the DESI scientific goals, the target selection, and survey design. The accompanying instrument portion of the FDR describes the instrument and optical design, integration and test plan, and the data management system. The companion Science Requirements Document provides information that guides the design. The DESI construction management plan is presented in the accompanying Project Execution Plan. Likewise, project cost and schedule are available in appropriate Project Office documents.", '2.1 Introduction': "DESI will explore some of the most fundamental questions in physical science: what is the composition of the Universe at large and what is the nature of space-time? These questions are now open to exploration because of recent discoveries. We summarize here the framework used to express these questions and the parameters used to quantify our understanding. \nThere are several pillars of the cosmological model that are now well established: 1) a period of rapid acceleration inflation or a similar process - occurred in the early Universe, generating the primordial fluctuations, which seeded large scale structures, galaxies and galaxy clusters, which grew during the decelerating, matter dominated era 2) gravitational instabilities produced acoustic oscillations in the plasma, which were imprinted about 400,000 years after this inflation period, when photons decoupled from atoms and produced the Cosmic Microwave Background 3) this was followed by a period of matter domination, when small density fluctuations grew into large-scale structure, 4) comparatively recently, there was a transition to accelerated expansion driven by either a modification to General Relativity or a new form of energy dark energy - not due to any particles known or unknown, and which contributes about 68% of the Universe's energy density, and 5) about 27% of the energy density today is due to matter outside the Standard Model of particle physics dark matter - which is responsible for large-scale structure formation and accounts for galaxy rotation curves and the motions of galaxies in clusters. \nThat the Universe is expanding more and more rapidly was first revealed through measurements of Type Ia supernovae [1, 2], and subsequently confirmed using other techniques. Within General Relativity, accelerated expansion requires ρ + 3 p < 0, where ρ is the total energy density and p is the total pressure of the matter, radiation, and other ingredients. The total equation of state w = p/ρ must be less than -1 / 3 for accelerated expansion. The equation of state need not be a constant; in general it depends on time, or equivalently the scale size of the universe a = 1 / (1 + z ). From now on, we let w denote the equation of state of the dark energy component alone. \nFor ordinary non-relativistic matter, the pressure is negligible compared to the energy due to the rest mass and thus w = 0. For photons and other massless particles, w = 1 / 3. The cosmological constant term is equivalent to dark energy with w = -1. Generally, energy with an equation of state w ( a ) evolves as ρ ( a ) = ρ ( a = 1) F ( a ), where F ( a ) = 1 for a cosmological constant and for a general equation of state w ( a ) is \nF ( a ) ≡ exp [ 3 ∫ 1 a da ' a ' (1 + w ( a ' )) ] . (2.1) \nIt is standard to parameterize the equation of state as \nw ( a ) = w 0 +(1 -a ) w a , (2.2) \nwhich accurately reproduces distances for a wide range of models. \nThe contributions to the energy density of the Universe are conventionally expressed relative to the critical density \nρ crit = 3 H 2 0 8 πG , (2.3) \nwhich would be just sufficient to slow the expansion ultimately to zero in the absence of a dark energy component Ω Λ . We write \nΩ m = ρ m ρ crit . (2.4) \nFigure 2.1: The expansion history of the Universe for different models of dark energy, holding the present-day Hubble constant fixed. The inset shows the spacing between five models with constant w ranging from -0 . 97 to -1 . 03, showing the exquisite precision required to distinguish these. Overlaid are measurements of the distance-redshift relation, translated into errors on lookback time at each redshift. Measurements from current supernovae, binned in redshift, are shown in blue; current BAO measurements from BOSS DR9, WiggleZ, and 6dF are shown in red; projections for DESI are shown in black. DESI measurements have the ability to make very tight constraints on dark energy, although we caution that this figure shows variations in only one cosmological parameter. Full forecasts, such as those presented in § 2.4.3, must marginalize over other cosmological parameters such as Ω m and H 0 . \n<!-- image --> \nWe define Ω r for radiation and Ω DE for dark energy analogously. The curvature term Ω k = -k/H 2 0 is defined so that General Relativity requires \nΩ r +Ω m +Ω k +Ω DE = 1 (2.5) \nfor a Universe with spatial curvature k . The expansion rate of the Universe is given by \nH ( a ) ≡ ˙ a a = H 0 [ Ω r a -4 +Ω m a -3 +Ω k a -2 +Ω DE F ( a ) ] 1 / 2 . (2.6) \nThe contribution from radiation, Ω r is negligible today and inflation predicts that the curvature is zero. The Hubble constant today is H 0 = h × 100 km/s/Mpc ≈ 70 km/s/Mpc. \nglyph[negationslash] \nThe Dark Energy Spectroscopic Instrument (DESI) [3] will provide precise spectroscopic redshifts of more than thirty million objects. From these will come three-dimensional maps of the \nWe have three possible explanations for the accelerating expansion of the Universe: a cosmological constant, equivalent to static dark energy with w = -1; a dynamical dark energy with w ( a ) = -1; or a failure of General Relativity. DESI is designed to address this fundamental question about the nature of the Universe. The challenge of distinguishing the cosmological constant solution from dark energy with w near -1 is displayed in Figure 2.1. \ndistribution of matter covering unprecedented volume. DESI will survey an enormous volume at 0 . 4 < z < 3 . 5 using luminous red galaxies, emission line galaxies, and quasars, producing tight constraints on the large-scale clustering of the Universe. In addition, DESI will perform a Bright Galaxy Survey (BGS) of the z < 0 . 4 Universe, allowing the study of cosmic structure in the darkenergy-dominated epoch with much denser sampling. These data will help establish whether cosmic acceleration is due to a mysterious component of the Universe or a cosmic-scale modification of GR, and will constrain models of primordial inflation. \nDESI will have a dramatic impact on our understanding of dark energy through its primary measurement, that of baryon acoustic oscillations. Waves that propagated in the electron-photonbaryon plasma before recombination imprint a feature at a known comoving physical scale (150 Mpc or 4 . 6 × 10 24 m) in the distribution of separations between pairs of galaxies. Localizing this baryon acoustic oscillation (BAO) feature and comparing its apparent size to the known physical scale provides a measurement of the distance to the galaxy sample and thus the expansion history of the Universe. The BAO measurement was singled out by the Dark Energy Task Force [4] as having the fewest experimental uncertainties among the techniques for measuring dark energy; it simply depends on the galaxy locations, rather than their shapes or brightnesses. DESI's two-point correlation measurements will also detect the anisotropies in galaxy clustering - redshift space distortions (RSD) - due to the peculiar velocities of galaxies generated by density perturbations. This gives a direct measurement of the properties of gravity at each redshift, through its effect on galaxies' motions. \nIn addition to the constraints on dark energy, the galaxy and Lyα flux power spectra will reflect signatures of neutrino mass, scale dependence of the primordial density fluctuations from inflation, and possible indications of modified gravity. To realize the potential of these techniques requires an enormous number of redshifts over a deep, wide volume and thus a substantial investment in a new instrument with capabilities well beyond existing facilities and for which we can utilize a substantial portion of the observing time. \nThe DESI survey will have considerable impact beyond these cosmological highlights on the study of galaxies, quasars, and stars. Spectroscopy is a core tool of astrophysics, and the ability to combine many millions of spectra with modern wide-field, multi-wavelength imaging surveys will yield rich opportunities. While the DESI collaboration includes members planning to work on these topics, we do not discuss these in this design report, as they are not driving requirements. We make one brief exception for the Milky Way Survey ( § 2.6), as it will involve a substantial number of targets that piggyback on the Bright Galaxy Survey, using fibers that have no suitable galaxy available within their patrol radius.", '2.2 Measuring Distances with Baryon Acoustic Oscillations': 'DESI will measure the expansion of the Universe by observing the imprint of baryon acoustic oscillations set down in the first 380,000 years of its existence. This pattern has the same source as the pattern seen in the cosmic microwave background, but DESI will map it as a function of cosmic time, while the CMB can see it only at one instant. The pattern is imprinted on all matter at large scales and can be viewed by observing galaxies of various kinds or by observing the distribution of neutral hydrogen across the cosmos, and shows up as excess correlations at the characteristic distance of the sound horizon at decoupling.', '2.2.1 Theory': "Initial fluctuations in density and pressure provided sources for sound waves that propagated in the photon-electron-baryon plasma of the early Universe (see, for example, [5]). These sound waves propagated with a speed approximately c/ √ 3 until the Universe cooled sufficiently for electrons and ions to recombine to neutral atoms, causing the sound speed to drop dramatically. An excess of matter was left both at the source of the wave and at the surface where these waves terminated. The matter excesses at these locations left their imprint on the large-scale structure of galaxies and hydrogen gas. Before a wave stopped, it traveled a co-moving distance s ≈ 150 Mpc, which can be computed to precision 0.3% from cosmological parameters extremely well measured in CMB. \nViewed transversely, the 150-Mpc ruler subtends an angle θ such that \ns = (1 + z ) D A ( z ) θ = θ ∫ z 0 c dz ' H ( z ' ) (2.7) \nwhere D A ( z ) is the angular-diameter distance to an object at redshift z . The final equality holds only if the curvature is zero. \nWhile the CMB gives us a purely angular correlation function, the characteristic scale is present in the three-dimensional distribution of large-scale structure. Viewed along the line of sight, correlations are enhanced for galaxy pairs separated by ∆ z such that \nc ∆ z H ( z ) ≈ s (2.8) \nThis latter measurement requires a spectroscopic survey to resolve the full three-dimensional density distribution of galaxies. \nThe observation of the peak in the two-point correlation function thus provides a means of measuring both the angular diameter distance, D A ( z ) and the Hubble expansion rate, H ( z ). The ability of the BAO method to directly probe H ( z ) is unique among dark energy probes. This becomes progressively more important at higher redshifts since H measures the instantaneous expansion rate (and through it, the total energy density of the Universe) while D A measures the integrated expansion history. Measuring both improves our ability to distinguish between different cosmological models.", '2.2.2 BAO in Galaxies': 'The best developed application of the BAO technique uses galaxies as tracers of the matter distribution; the BAO feature appears in the two-point correlation function of galaxies, the probability, in excess of random, that two galaxies are separated by a distance r . This has been achieved with high statistical significance in several measurements spanning the redshift range from z = 0 to z = 1. \nFigure 2.2: The angle-averaged correlation functions [left] and power spectra [right], before [top] and after [bottom] reconstruction measured using the BOSS DR11 CMASS galaxy sample [6]. The BAO feature is clearly detected at over 7 σ as a peak in the correlation function and a corresponding set of oscillations in the power spectrum. \n<!-- image --> \nThe highest significance detection ( > 7 σ ) is currently that of the Baryon Oscillation Spectroscopic Survey (BOSS) using the z > 0 . 45 sample [6, 7]. We show representative data in Figure 2.2. These data measured the distance-like quantity D V ( z ) ≡ ((1 + z ) D A ) 2 / 3 ( cz/H ( z )) 1 / 3 to a redshift of 0.57 to 1.0%, the most precise measurement using the BAO technique. The lower redshift z < 0 . 45 sample in BOSS constrained the same combination of distances to 2%. At still lower redshifts, the 6-Degree Field Galaxy Redshift Survey [8] measured the distance to z = 0 . 106 with 4.5% accuracy At a somewhat higher redshift, the WiggleZ galaxy survey measured the distance to a redshift of 0 . 7 to 4% [9]. This combination of these measurements has for the first time enabled mapping the distance-redshift relation purely from BAO measurements. \nMost of these measurements used the galaxy correlation function averaged over the orientation of the pair to the line of sight to measure D V , a combination of D A and H . More recent work has also measured the correlation functions transverse and parallel to the line of sight, allowing one to break the degeneracy between D A and H that exists in purely angle averaged measurements. \nThe current generation of surveys is an excellent proving ground for analysis techniques. For instance, the BOSS experiment compared analyses done in Fourier and in configuration space and used different algorithms for estimating distances from the resulting two-point functions. All these yielded consistent distance measurements, given the statistical precision of the measurements. While the level of consistency is not at the level required by DESI, ongoing surveys provide a clear roadmap for developing and validating improvements to these analysis techniques. The current measurements provide an important validation of our forecasts for DESI presented below. \nThe non-linear evolution of the matter density field broadens the acoustic peak, potentially decreasing the precision on the distance measurement, and causes a small shift in the peak location, thereby biasing the distance. Ref. [10] pointed out that because this broadening is caused by the large-scale velocity flows resulting from gravitational forces, the effect may be substantially reversed by estimating the velocity fields from the large-scale structure map and moving the galaxies back \nto their initial positions. In addition to a notable improvement in the recovered statistical errors, this reconstruction also mitigates the shifts in the distance scale due to nonlinear evolution, with numerical tests showing suppression to below 0 . 1%. Reconstruction was first applied to the SDSSII galaxy survey [11], improving the statistical precision by a factor of 1 . 7. Galaxy samples from more recent SDSS results, DR11, yield similar improvements after reconstruction. See Figure 2.2. As with the other analysis methods, we expect improvements to reconstruction algorithms before the DESI measurements become available. We however choose to be conservative and assume a reconstruction performance similar to what has already been demonstrated with current data.', 'Observational Systematics': "The BAO method is simple in principle - all one requires are the three-dimensional positions of galaxies. The need to preserve the BAO feature along the line of sight sets the requirement on redshift precision. This precision, as stated in the Level 2 Survey Data Set Requirements is σ z / (1 + z ) ∼ 0 . 0005 per galaxy, which is easily within the state-of-the-art and achieved throughout our wavelength range in the spectrograph design. \nThe angular and radial selection functions of the survey can induce systematic uncertainties. The angular selection function is determined by the imaging survey used for targeting, and may be spuriously modulated by photometric calibrations, seeing and extinction variations, and image deblending. All of these effects are intrinsically angular effects and therefore may be separated from the BAO feature, which is a feature in three-dimensional physical space (not isolated to the angular degrees of freedom). A similar separation is possible for systematics in the radial selection function of the survey. The impact of these is therefore expected to be small. In addition, there has been considerable work [12, 13] developing techniques to further mitigate these effects. \nThe ongoing BAO surveys provide the opportunity to identify and quantify observational systematics. DESI will benefit greatly from this work, but it also faces some unique challenges. The most important of these arise from the fiber positioning system and from the forest of sky lines, which impinge on the radial selection function. The limited patrol radius of the fiber positioners causes the highest density regions to be sampled less completely than lower density regions. This particularly affects the observer's line of sight and can skew the anisotropic correlation pattern. High sky brightness at certain wavelengths makes it difficult to find [O II] emission lines, thereby reducing the spectroscopic completeness at specific redshifts. Initial studies have shown that these survey artifacts can influence the measured clustering, but we expect both to be correctable to good accuracy, as the source of the variations can be tracked with high fidelity. Finding the optimal method to achieve the full statistical precision inherent in the data is an ongoing project of the science team.", 'Theoretical Systematics': 'The robustness and accuracy of the BAO method derive from the simplicity of the early Universe and the precision with which we know the speed and time of propagation of sound waves in the primordial plasma. The evolution of density fluctuations in the Universe is very well described by linear perturbation theory and is now exquisitely tested by the recent measurements of temperature fluctuations in the Cosmic Microwave Background radiation by the Planck satellite [14, 15, 16]. The current CMB measurements constrain the size of the BAO standard ruler to 0 . 3%. This uncertainty is folded into our forecasts for DESI. Furthermore, any miscalibrations in the acoustic scale would affect principally the determination of the Hubble constant, not the dark energy constraints [17]. \nThe sound waves travel a comoving distance of 150 Mpc, setting the BAO scale to be much larger \nthan the scale of gravitational collapse even in the present Universe (about 10 Mpc). Analytical calculations, verified by direct numerical simulations, have found the nonlinear evolution of the density field alters the BAO scale by 0 . 3% at the present epoch, and even less at the higher redshifts probed by DESI. \nGalaxy formation may result in an additional shift in the BAO scale due to mismatched weighting of high and low density regions. Initial perturbative and numerical studies [18, 19, 20, 21, 22, 23, 24] also find these shifts to be small, with the most extreme shifts of order 0 . 5%. As mentioned above, density-field reconstruction applied to simulations reduces these shifts to the 0.1% level without the need for further modeling. We expect that further modeling from theory and simulations will allow us to robustly limit these uncertainties to well below the DESI statistical limits. In addition, the DESI target samples are designed to overlap in multiple redshift ranges, allowing empirical tests of the robustness of the BAO measurements to different tracer populations. \nA recently discovered astrophysical effect that could affect the BAO feature arises from the relative velocities of the baryons and the dark matter at the recombination epoch [25, 26]. This modulates the formation of the earliest protogalaxies and potentially could persist to their descendants (some of which would be measured by DESI). This modulation is due to the same pressure forces that create the BAO, and the impact could shift the measured acoustic scale. While this effect is expected to be negligible for the galaxies probed by DESI, the possibility of a systematic bias in the inferred distance scale can not be ruled out on theoretical grounds. Fortunately, [27] demonstrate that this effect would also create a distinctive three-point function signal measurable in DESI that would diagnose any contamination from this effect (also [28]). \nAll of the above strongly argue that the theoretical systematic effects associated with the BAOscale measurements are either intrinsically or correctable to below the 0 . 1% level required by DESI.', '2.2.3 BAO in the Lyα Forest': 'Measuring BAO with galaxies as tracers is a mature method [29, 9]. Such measurements become much more difficult for z glyph[greaterorsimilar] 2 . 0 where galaxy redshifts are harder to get. However, measuring dark energy properties at this high redshift allows us to probe the Universe well before the advent of accelerated expansion. An interesting possibility is that dark energy density does not become completely negligible at high redshift, as predicted by the cosmological constant or other models with w glyph[similarequal] -1, but rather remains at a level predicted by some particle-physics models and detectable by future surveys [30, 31, 32, 33, 34, 35]. Such a component can only be measured or excluded by a technique sensitive to the expansion history at high redshift. \nThe Lyα forest provides the means to measure BAO at redshifts larger than 2. The forest is a collection of absorption features in the spectra of distant quasars blue-ward of the Lyα emission line [36]. These features arise because the light from a quasar is absorbed by neutral hydrogen in the intergalactic medium. Since the quasar light is constantly red-shifting, hydrogen at different redshifts absorbs at different observed wavelengths in the quasar spectrum. The amount of absorption reflects the local density of neutral hydrogen, which in turn traces the dark matter field on sufficiently large scales. Numerical simulations and analytical work show that for plausible scenarios, the Lyα forest is well within the linear biasing regime of scales relevant for BAO [37, 38, 39]. Therefore, measuring three-dimensional correlations in the flux fluctuations of the Lyα forest provides an accurate method for detecting BAO correlations [37, 40, 41, 42] \nUsing the Lyα forest to measure the three-dimensional structure of the Universe became possible with the advent of BOSS, which was the first survey to have a sufficiently high density of quasars to measure correlations on truly cosmological scales. This was done in 2011 [43]. At the beginning of 2013, the first detection of BAO in the Lyα forest was published in a series of papers \n<!-- image --> \n<!-- image --> \nFigure 2.3: Correlation functions of Lyα forest flux fluctuations based on the BOSS DR11 quasars [47], binned in the cosine of the angle to the line of sight, µ ( µ = 1 is along the line of sight, µ = 0 is perpendicular to the line of sight). From left to right, the bins are µ > 0 . 8, 0 . 5 < µ < 0 . 8 and µ < 0 . 5. The points are the measured correlation function, the solid line is the best fit model, while the dashed line is the best fit assuming a fiducial cosmology. These results measure the optimal combination D 0 . 3 A H -0 . 7 to 2%. \n<!-- image --> \n[44, 45, 46]. These were recently updated to the almost complete BOSS sample in [47] (Figure 2.3) yielding a 5 σ detection of the BAO feature. \nThe redshift-space distortions in the Lyα forest are larger than in galaxy-based measurements [43, 37]. Thus the signal-to-noise for the radial modes is considerably higher than for transverse modes. Consequently, in contrast to the galaxy measurements, the Lyα forest BAO measurements measure the Hubble parameter H ( z ) with greater precision than the angular diameter distance D A ( z ). For instance, [47] find that the combination D 0 . 3 A H -0 . 7 is optimally constrained to ∼ 2%.', 'Systematics': 'Inevitably, there will be systematic effects that could distort the Lyα measurements, but these should produce broadband contamination and would not affect our ability to measure an isolated feature in the data, such as the BAO peak. However, unless carefully accounted for, these systematics could contaminate secondary science, such as Lyα broadband power measurements, neutrino masses and warm dark matter constraints. \nAstrophysical contaminants include sources of non-gravitational large scale fluctuations, such as He II reionization and fluctuations in the photo-ionization background [48, 49, 50, 51, 52]. There are also targeting systematics - quasars with significant absorption in the forest region are considerably easier to target, since they are easier to distinguish from stars. As a result, observed Lyα forest regions are not sampling the Universe randomly, but prefer overdense lines of sight. Back-of-theenvelope calculations show that this effect is small, although more work should be done to confirm this 1 . Finally, there are metal contaminations. For example, Si III that tracks the hydrogen fluctuations produces a line that contaminates the Lyα forest flux measurements at separation of 2271 km/s. The cross-correlation between Lyα forest absorption and Si III absorption, if misinterpreted as Lyα -to-Lyα correlations could bias the BAO measurements [43, 47]. Further contamination arises where the metal absorption traces large scale structures at a significantly different redshift. For example C IV traces structure at z = 1 . 7 at wavelengths which probe the Lyα forest at z = 2 . 4 [53]. For BAO measurements these can be reliably corrected by including them as a part of the model. For other uses, such as broadband power spectrum measurements, a combination of nuisance modeling, accurate mock spectra and numerical simulations should remove \nany potential biases associated with these complications. \nPerhaps the most important systematic effects will come from imperfections in the instrument and data reduction. For example, artificial features in the mean transmission at the position of galactic Balmer transitions were noticed in BOSS data [44]. These were tracked down to the imperfect interpolation in calibration vectors when these features were masked in calibration stars. Although such effects are on average calibrated out, they can in principle produce sharp features in correlation at certain pairs of wavelengths that could potentially contaminate the BAO measurements. Other effects include noise calibration and its Poisson nature, imperfect sky subtraction, etc. Fortunately, there are no fundamental obstacles to modeling the listed systematics with a carefully executed pipeline. The sheer amount of data that will be available and the relatively high signal-to-noise of true small scale fluctuations in the forest will allow us to check the data in many different ways and validate the data reduction pipeline.', '2.3 Measuring Growth of Structure with Redshift Space Distortions': 'DESI will observe redshifts, which reflect the velocities due to expansion, but also the peculiar velocities due to gravitational attraction by large scale structure. Peculiar velocities are observable in redshift surveys because they alter the correlations between galaxies along the line of sight, resulting in an anisotropy in the observed clustering. Comparing the expansion history and the growth of large scale structure from redshift space distortions will allow DESI to test General Relativity.', '2.3.1 Theory': 'Galaxies and quasars are point tracers of the underlying cosmic structure. The physics of how they trace the dark matter fluctuations is well understood based on arguments about locality of galaxy formation [54, 55, 56]. On very large scales bias is scale independent and redshift-space distortions are described by linear perturbation theory. Beyond-linear perturbative corrections can be used on intermediate scales before perturbation theory breaks down entirely on small scales [57, 58, 59]. \nThe measurement of the growth of structure relies on redshift-space distortions seen in galaxy surveys. Even though we expect the clustering of galaxies in real space to have no preferred direction, galaxy maps produced by estimating distances from redshifts obtained in spectroscopic surveys reveal an anisotropic galaxy distribution. The anisotropies arise because galaxy redshifts, from which distances are inferred, include components from both the Hubble flow and peculiar velocities driven by the clustering of matter. Measurements of the anisotropies allow constraints to be placed on the rate of growth of clustering [60, 61]. \nOn large scales, the observed large-scale structure is basically described by a small fractional perturbation δ ( x ) = δρ ( x ) /ρ = ( ρ ( x ) -ρ ) /ρ to the uniform density. Ignoring the higher-order contributions, the perturbation in redshift space ( δ s ) is related to the real space perturbation at directional cosine µ between line-of-sight direction and the wave-number k , by the Kaiser relation [62], \nδ s ( k ) = δ ( k )(1 + βµ 2 ) (2.9) \nHere β = f/b , where b is the galaxy bias and f is related to the linear growth function D ( a ) by \nf = d ln D ( a ) d ln a . (2.10) \nIn the linear regime, density perturbations grow proportional to D ( z ) which increases with decreasing z . \nIn GR, D ( z ) is completely specified by the expansion history even in the presence of dark energy; this is no longer generically true in alternative theories of gravity. The behavior of f in GR is given, to a good approximation, by \nf glyph[similarequal] Ω m ( z ) γ , (2.11) \nwhere γ is the growth index, approximately 0.55 in GR, and where Ω m ( z ) is the fraction of the total energy density in the form of matter at redshift z . In alternative gravity theories, a common simple parameterization of the modified growth rate is to alter the growth index γ . [63] demonstrated that a DESI-like survey could constrain γ to 0.04 (7%). More general modifications might involve modifying (in a time- and scale-dependent manner) the potentials that enter the metric. Precise growth measurements over a wide range of redshifts and scales, combined with constraints from overlapping CMB and weak lensing surveys, make large galaxy surveys like DESI excellent probes of gravity (see [64] for a recent review). Here, we focus on scale-independent growth rates for large-scale structure, but the DESI data set will allow more complicated investigations. \nAs an important example of extensions, we highlight the Bright Galaxy Survey, where we will be mapping a smaller volume ( z < 0 . 4) at substantially higher number density and with more diversity of galaxies. This redshift range is crucial because it is when dark energy dominates and any associated modifications of gravity would be expected to be strongest. Getting the best precision out of this limited volume requires spectroscopy to produce a 3-D map of the density field. The BGS will test for modifications of gravity directly via the redshift-distortion method, including the novel methods of using multiple tracers in order to suppress sample variance [65]. But the search can be extended via spectroscopic detection of clusters and groups, along with galaxy halo occupation modeling, to measure the amplitude of clustering by halo abundances [66, 67]. The maps can also be correlated with weak lensing maps (e.g., from DES, LSST, Euclid, or CMB-S4) to measure the amplitude of clustering [68, 69]. Comparing the observed velocity field to the expected velocity field sourced from the lensing matter overdensities enables further tests of modified gravity models of cosmic acceleration [60]. Finally, the more detailed map will allow tests of screening theories on smaller scales [70, 71], in which one considers the response of individual galaxies to the predicted gravitational field. \nIn the Kaiser approximation, the redshift space power spectrum, P s , is given by \nP s ( k ) = ( b + fµ 2 ) 2 P m ( k ) (2.12) \nwhere P m is the linear theory mass power spectrum. In principle, this prescribed anisotropy provides a means of measuring f , and through it the growth of gravitational structures. However, in the above, the measurements of f are degenerate with the amplitude of the matter power spectrum. Therefore the combination f ( z ) σ 8 ( z ) is the actual observable, where the normalization of the power spectrum P ( k ) is proportional to σ 2 8 ( z ) 2 .', '2.3.2 Systematics': "Galaxies are expected to follow the same gravitational potential as the dark matter and hence have the same velocities. The main theoretical systematic uncertainty in RSD is that nonlinear velocity effects extend to rather large scales and give rise to a scale-dependent and angle-dependent clustering signal. It is easy to see these effects in any real redshift survey: one sees elongated features along the line of sight, called the Fingers of God (FoG). The FoG are caused by random velocities inside virialized objects such as clusters, which scatter galaxies along the radial direction in redshift space, even if they have a localized spatial position in real space. This is just an extreme example and other related effects, such as nonlinear infall streaming motions, also cause nonlinear corrections. In addition, RSD measure velocities as sampled at the galaxy positions. One is thus probing not the velocity field, but rather the momentum density field. Galaxies are a biased tracer of the dark matter and this introduces scale dependent effects into RSD statistics even if galaxies are simply a linear tracer of the dark matter. \nThere are a plethora of approaches [72, 73, 74, 57, 58, 59] to modeling redshift space distortions in the literature, and the analyses in Table 2.1 make use of many of them. It has been firmly established that the Kaiser formula is inadequate to recover information faithfully on the quasilinear scales of interest, and so most analyses now adopt some form of perturbative corrections. However, because these corrections depend strongly on the halo bias [75, 76], methods calibrated on purely the dark matter power spectrum are of limited utility. Moreover, the details of the mapping between galaxies and dark matter halos also strongly modify the correlation function, mostly through FoG \nTable 2.1: Compilation of RSD-based fσ 8 measurements from [89]. For the BOSS DR11 galaxy sample we cite the measurement of [85]. Other analyses of DR11 find consistent results [87, 84] \n± \neffects. All of these effects can induce 10% effects on RSD at k ∼ 0 . 1 h /Mpc. Current models of RSD are able to reproduce these nonlinear effects at the percent level for k < 0.05-0.1 h /Mpc. Extending this to smaller scales would increase the power of the DESI RSD survey. This will require us to improve our bias models and the realism of our simulations. \nMost of the observational systematics examined in detail in the SDSS-III BOSS [see 12] primarily affect clustering on the largest scales; currently these are of little concern for RSD measurements, for which the signal comes primarily from the smallest scales included in the measurements. The most important systematic effect is the estimate of a survey's radial selection function [77, 12]. Since the redshift distribution of targets cannot be predicted precisely a priori, it must be measured directly from the observed galaxies' redshift distribution. Doing so removes some cosmological radial modes from the observed galaxy overdensity field, resulting in a bias in the monopole-quadrupole amplitudes at the < 0 . 2 σ level. The ratio of systematic to statistical uncertainty should remain relatively constant with survey area for a given redshift distribution, since the statistical errors on the correlation function and n ( z ) shrink at the same rate.", '2.3.3 Current Status of RSD Measurements': "Redshift-space-distortion measurements have now been performed on a host of surveys, which we summarize in Table 2.1 and show in the left panel of Figure 2.4; taken together, these surveys provide a measure of the growth rate of cosmic structure good to about 3% in the low redshift Universe. Almost all of these measurements of fσ 8 are derived from the anisotropy in the twopoint correlations of the observed galaxy density field. The anisotropic correlation from SDSS-III BOSS DR11 CMASS sample is shown in Figure 2.5. While there have been some analyses directly on the two-dimensional correlation function ξ ( r p , r π ) [e.g., 78, 79, 80, 81], most authors further compress the data into multipoles [e.g., 82, 77, 83, 84, 85] or wedges [86, 87]. Efficient information compression is necessary when the covariance matrix of the observables are estimated from a finite number of mock surveys [88]. \nMost of these measurements assume a flat ΛCDM cosmology to model the redshift-distance relation (see [81] for an exception); dropping this assumption degrades the measurement of fσ 8 . However, the combination of geometric and dynamical constraints available from the analysis of \n<!-- image --> \nFigure 2.4: Left: The data points show the CMASS DR11 measurement of fσ 8 (gold pentagon; [85]) along with similar, low redshift, measurements and 1 σ error bars as presented in Table 2.1. The three stripes show theoretical predictions for different gravity models allowing for uncertainty in the background cosmological parameters, constrained using only the WMAP 7 data [94]. Figure adapted from [89]. Right: Joint constraints in the Ω m -γ plane from BOSS DR11, where γ is the growth index of structure, as defined in Eq. (2.11). Figure taken from [85]. \n<!-- image --> \nFigure 2.5: The two-dimensional correlation function of the BOSS DR11 CMASS galaxies, measured perpendicular (x-axis) and parallel (y-axis) to the line of sight. The BAO ring, distorted by redshift space distortions is clearly visible, as is the characteristic squashing of the correlation function on large scales. \n<!-- image --> \nanisotropic galaxy clustering is quite complementary to isotropic BAO measurements for constraining dark energy. For instance, in the case of SDSS-III BOSS DR11 for a flat w CDM cosmology, the combination of Planck and the BOSS BAO measurements constrain w = -1 . 01 ± 0 . 08 [6], while including the geometric and dynamical information in the quadrupole correlation function (term proportional to µ 2 ) yields w = -0 . 993 ± 0 . 056 [85]. \nConsidering instead tests of gravity given a 'known' expansion history, Figure 2.4 shows that for a flat ΛCDM cosmology in general relativity, the predicted redshift evolution of the observable \nfσ 8 is quite mild in the redshift range that has been studied observationally. These observations can begin to distinguish between gravity models ( f ( R ) and DGP are shown), though there is still substantial uncertainty in the theoretical predictions simply due to uncertainties in both the matter density Ω m and overall matter power spectrum normalization, σ 8 . The right-hand side of Figure 2.4 shows constraints in the Ω m -γ plane from BOSS DR11 [85]. These data yield a 16% constraint on the growth index. DESI will improve on the precision of the growth constraint from all previous measurements by a factor of ∼ 4-10 [95], depending on advances in analysis and theoretical modeling. In addition, it will provide measurements to significantly higher redshifts. \nTwo surveys in particular are pathfinders for DESI targets: WiggleZ [96] analyzed emission line galaxies with bias b near 1, while SDSS-II and SDSS-III BOSS study luminous red galaxies (LRGs) with a bias near 2. WiggleZ included much smaller scales in their RSD analysis, which led to impressive constraints given the number of galaxies in the survey. However, they were not able to generate easily a large N -body simulation volume capable of resolving the halos expected to host emission line galaxies, and so their theoretical modeling is necessarily less well-tested. By comparison, LRGs are hosted by massive halos that can easily be simulated. The perturbation-based model of [83] was carefully calibrated against N -body-based mock-galaxy catalog and included realistic effects like the 'Fingers-of-God' (the elongated structure in the right panel of Figure 2.5). However, because these effects are so strong, their analysis was restricted to relatively large scales. \nOngoing progress in combining the perturbative analytic results with those of N-body simulations should pave way for the increased theoretical prediction accuracy necessary to extract RSD information at small spatial scales", '2.4 Distance, Growth, Dark Energy, and Curvature Constraint Forecasts': "DESI's observational program defined in the Requirements Document and described in this Report specifies the numbers of galaxies and Lyα forest sources and their distribution that will be measured. Using the specified quality of those observations, we can predict the precision with which cosmological parameters will be determined by DESI. Thanks to the unprecedented scope of DESI's spectroscopic measurements, these measurements will take us to a new level - Stage-IV - in cosmological exploration.", '2.4.1 Forecasting Overview': "We use the Fisher matrix formalism to estimate the parameter constraining power of the finished survey, largely following [95]. Our baseline cosmological model is flat ΛCDM. This model is specified by seven parameters, which are listed together with their fiducial values in Table 2.2. Parameter symbols have their conventional meanings. Our standard fiducial parameter values follow the Planck 2013 results, specifically the P+WP+highL+BAO (P from Planck , WP from WMAP , highL from high resolution CMB experiments like ACT and SPT) column of Table 5 of [14]. The difference from the Planck 2015 is negligible for these purposes. In addition to the conventional six parameters of the minimal cosmological model, we also always vary the amount of tensor modes; however this is largely irrelevant because the T/S measurement is completely dominated by Planck and essentially uncorrelated with other parameters. \nIsolating the BAO feature gives the most robust, but also most pessimistic, view of the information that one can recover from galaxy clustering measurements, since BAO can be measured even in the presence of large unknown systematic effects (very generally, these will not change the BAO scale [22]). We quote errors on the transverse and radial BAO scales as errors on D A ( z ) /s and H ( z ) s , respectively, where s is the BAO length scale. For galaxy and quasar clustering, these measurements are correlated at each redshift with a correlation coefficient of 0.4. \nWe also quote errors on an isotropic dilation factor R/s , defined as the error one would measure on a single parameter that rescales radial and transverse directions by equal amounts. In this case, for a small change in R , the corresponding variations in the model values of D A and H are \nD A = ( 1 + δR R fid ) D A, fid (2.13) \nand \nH = ( 1 + δR R fid ) -1 H fid (2.14) \nwhere D A, fid ( z ) and H fid ( z ) are the angular diameter distance and Hubble parameter in a fiducial Universe. An explicit definition of R in terms of the measured H and D A is generally not needed and depends on the experimental scenario. The simplest cases are easy to understand: for a purely transverse measurement (e.g., photometric survey) R = D A , while for a purely radial measurement (e.g., something closer to the Lyα forest, although it is not purely radial) R = H -1 (or R = H -1 H fid D A, fid , if one is concerned about inequivalent units). For intermediate cases like typical galaxy clustering, the appropriate combination of H and D A can always be determined given the covariance matrix between them. For example, it is approximately proportional to D V ( z ) ≡ ((1+ z ) D A ) 2 / 3 ( cz/H ( z )) 1 / 3 in analyses of spherically averaged clustering, such as from 6dF, BOSS, and WiggleZ. \nGoing beyond BAO, we use 'broadband' galaxy power, i.e. measurements of the power spectrum as a function of redshift, wavenumber and angle with respect to the line of sight. This \nTable 2.2: Parameterization of the cosmological model and parameter values for the fiducial model. The seven parameters in the upper part of the table are always free. Parameters in the second half of the table are extensions of the simplest model discussed below. \ntreatment automatically recovers all available information from the two-point clustering, i.e. not just the shape of the isotropic power spectrum, but also redshift-space distortions, Alcock-Paczynski [97], and the BAO information. \nThe broadband Fisher matrix is calculated by combining the inverse variance of the power spectrum P ( k ) of each Fourier mode with the derivative of power in each mode with respect to set of cosmological parameters. We divide the survey into a set of redshift slices and coadd the resulting matrices. The model for the three-dimensional power spectrum of the galaxy or Lyα distribution is \n˜ P ( k, µ, z ) = b ( z ) 2 (1 + β ( z ) µ 2 ) 2 P mass ( k, z ) D ( k, µ, z ) , (2.15) \nwhere µ is the angle of the wavevector to the line of sight, k is the wavenumber, b is the linear bias parameter, β the redshift space distortion parameter and D ( k, µ, z ) is a non-linear correction calibrated from simulations (for the Lyα forest this is given by [98] and for galaxies it is based on the information damping factors of [99]). The Fisher matrix calculation will integrate over all µ and a suitable range of k . The inverse variance of the power spectrum of each mode gets contributions from both the intrinsic sample variance and the shot noise. This results in an effective volume V eff ( ˜ P ) of each redshift slice that is given by V eff ( ˜ P ) = [1 + 1 / ( n ˜ P )] -2 V survey [100]. The value nP represents the ratio of true clustering power to that from shot noise. Alternatively, it can be seen \nas the signal-to-noise ratio per mode (redshift, wavenumber, and orientation slice): if nP > 1 then roughly the signal exceeds the sample variance uncertainty for that mode. \nFor the galaxy survey, we use large-scale broadband power up to some quoted k max . At small scales, k > k max , we continue to use BAO information. We use two simple choices of k max : 0.1 h Mpc -1 and 0.2 h Mpc -1 . These cutoffs are intended to indicate sensitivity of results to the effective scale where information is recovered after making corrections for non-linearity, after marginalization over suitable non-linear bias parameters. It will be a major program of the next decade to figure out exactly how to do this fitting in practice for a high precision survey like DESI; how well we can do this will determine how well we can measure parameters. As discussed in [95], k max ∼ 0 . 1 h Mpc -1 corresponds roughly to the performance of current analyses, while k max ∼ 0 . 2 h Mpc -1 is more of a stretch goal for the DESI era (some improvement over current analysis can be expected simply by going to higher redshift where the non-linear scale is smaller). \nThe redshift-space distortions can effectively constrain two parameter combinations, b ( z ) σ ( z ) and f ( z ) σ ( z ), where σ ( z ) ∝ P 1 / 2 mass ( z, k ) is the RMS normalization of the linear mass density fluctuations as a function of z . In Table 2.3, we quote projected constraints on fσ for different maximum k assumptions e.g., fσ 0 . 1 means the error calculation included information up to k max = 0 . 1 h Mpc -1 . These fractional errors are equivalent to what one usually sees quoted as an error on ' fσ 8 '. The fσ k precision we project for DESI, aggregated over all redshifts, is ∼ 0.74% for k max = 0 . 1 h Mpc -1 , or ∼ 0.38% for k max = 0 . 2 h Mpc -1 .", '2.4.2 Baseline Survey': 'Our baseline assumption for science projections is that DESI runs over an approximately five-year period covering 14,000 deg 2 in area. DESI will target four types of objects: Bright Galaxies (BGS), Luminous Red Galaxies (LRGs), Emission Line Galaxies (ELGs) [101], and quasars. Details on how these objects are targeted can be found in Section 3. In what follows, most calculations are done for this baseline survey. We additionally provide several relevant calculations for the required minimum survey with the same target number densities over 9,000 instead of 14,000 deg 2 in area. \nThe number densities used here, plotted in Figure 2.6, are based on the selection criteria for each object type described in the following chapter. \nWe assume fiducial biases follow constant b ( z ) D ( z ), where D ( z ) is the linear growth factor normalized by D ( z = 0) ≡ 1. For LRGs we use b LRG ( z ) D ( z ) = 1 . 7. For ELGs we use b ELG ( z ) D ( z ) = 0 . 84 [101]. For quasars we use b QSO ( z ) D ( z ) = 1 . 2 (loosely based on [102]). For the BGS, we use b BGS ( z ) D ( z ) = 1 . 34, but the results are insensitive to this value because of the much higher number density in most of the BGS volume. Note that these forms keep the observed clustering amplitude of each individual tracer constant with redshift, in agreement with observations (more detailed references for bias evolution are given below, in sections 3.1, 3.2, 3.3.1, and 3.4.1 for BGS, LRGs, ELGs, and QSOs, respectively). \nThe signal-to-noise for typical BAO-scale modes in redshift space is shown in Figure 2.7, along with the same quantity computed for several other experiments for comparison [95]. \nWe evaluate ¯ nP at k = 0 . 14 h Mpc -1 , µ = 0 . 6, an approximate center-of-weight point for BAO measurements. We chose these values by looking for the point where ¯ nP = 1 corresponded to the optimum in a trade-off between area and number density at fixed total number of objects (specifically, for the full range of parameters covered by DESI LRGs and ELGs). This definition reflects the origin of the idea that ¯ nP = 1 is a special point, but it should be kept in mind that achieving ¯ nP by this definition does leave a survey significantly farther away from the sample variance limit than the traditional definition k = 0 . 2 h Mpc -1 , µ = 0. \nFigure 2.6: DESI number densities, per unit z , per square degree, used in cosmology projections (Table 2.3 and 2.7). \n<!-- image --> \nFigure 2.7: Signal to noise comparison of the DESI galaxy survey against other precursor (Stage II and Stage III) and upcoming (Stage IV) spectroscopic surveys. Shown is ¯ nP ( k = 0 . 14 h Mpc -1 , µ = 0 . 6). The DESI forecasts do not include the Lyα forest contribution. Including this would give an effective ¯ nP ∼ 0 . 3 at z ∼ 2 . 5. Note that the large area covered by DESI provides an advantage reflected in Figure 2.9. \n<!-- image --> \nFigure 2.8: Signal-to-noise ratio per ˚ Aused for DESI quasar spectra (detector noise, not absorption noise), for different g magnitudes, accounting for mean Lyα forest absorption. \n<!-- image --> \nFigure 2.9: The fractional error on the dilation factor, R , as a function of redshift presented in comparable bins for DESI, BOSS, Euclid , WFIRST , HETDEX, and eBOSS. This gives an indicative error on distance measurements to each redshift. The forecasts for a 14,000 deg 2 DESI Bright Galaxy Survey (BGS) are also shown. DESI will provide the best measurements over much of the region and is competitive with space-based missions, which will come later. We use 50 million total galaxies for Euclid, following their Definition Study Report [104], although it has been suggested that this may be optimistic [105]. \n<!-- image --> \nThe spectral signal-to-noise ratio that we use, computed using the bbspecsim code [103], is shown in Figure 2.8.', '2.4.3 Summary of Forecasts': "Table 2.3 lists the basic galaxy and quasar BAO distance measurement projections, and RSD f ( z ) σ 8 ( z ) error projections for two different k max values for our baseline 14K survey. We provide the same set of calculations in Table 2.4 for our threshold 9K survey. Tables 2.5 and 2.6 shows the projections for the Bright Galaxy Survey for 14K and 9K square degrees, respectively. Table 2.7 lists the Lyα forest BAO distance measurement projections, including cross-correlations with quasars in the same redshift range for a z > 1 . 9 Lyα forest survey; Table 2.8 presents the same calculations for the threshold 9K survey. The BAO errors are also shown in Figure 2.9, along with those from other experiments for comparison (see [95] for a description of the other experiments). \nDESI will provide high precision measurements of the Universe's expansion rate over billions of years. Using the Lyα forest technique, coverage will include the early times when the expansion rate was decreasing (when the matter density, not the dark energy density, was controlling the rate). In Figure 2.10 we show how DESI will improve these measurements over those existing today. \nTable 2.9 shows Dark Energy Task Force (DETF) Figures of Merit (FoMs) [4]. For the common normalization convention that we follow, the FoM is simply ( σ w p σ w ' ) -1 where w ( z ) = w p +( a p -a ) w ' and a p is chosen to make the errors on w p and w ' independent. Because the DETF FoM model is defined to include the possibility of curvature, we include curvature projections in Table 2.9. The figure of merit results are reflected in Figure 2.11. \nImportantly, Table 2.9 shows that these surveys exceed the Stage IV FoM threshold. We take this to be a value of 110, based on a 10-fold improvement of the value of 11 from [109]. This \nTable 2.3: Summary of forecasted constraints achievable by DESI, covering 14,000 deg 2 . Indications of signal to noise, nP , are given at two values of k, µ = { 0 . 2 h Mpc -1 , 0 } and { 1 . 4 h Mpc -1 , 0 . 6 } . The fractional error on the normalization of f ( z ) P 1 / 2 ( k, z ) is σ fσ k /fσ k , assuming known shape of the power spectrum and known geometry, using k max = k h Mpc -1 . The dilation factor R is defined to be a parameter rescaling the radial and transverse distances by equal factors.Table 2.4: Like Table 2.3, except with DESI covering only 9,000 deg 2 . \nTable 2.5: Like Table 2.3, except for the DESI Bright Galaxy Survey, covering 14,000 deg 2 . \nTable 2.6: Like Table 2.5, but for a 9,000 deg 2 Bright Galaxy Survey.Table 2.7: z > 1 . 9 Lyα forest quasar survey, over 14000 sq. deg. Parameter errors are in percent relative to the BAO scale, s . \nTable 2.8: Like Table 2.7, except with DESI covering only 9,000 deg 2 . \nFigure 2.10: Expansion rate of the Universe as a function of redshift. In the upper plot, the filled blue circle is the H 0 measurement of [106], the solid black square shows the SDSS BAO measurement of [107], the red square shows the BOSS galaxy BAO measurement of [6], the red circle shows the BOSS Lyα forest BAO measurement of [47], and the red x shows the BOSS Lyα forest BAO-quasar cross-correlation measurement of [108]. The lower plot shows projected DESI points. \n<!-- image --> \nFigure 2.11: The w 0 -w a plane showing projected limits (68%) from DESI using just BAO and using the broadband (BB) power spectrum. Also shown is the limit from BOSS BAO. Planck priors are included in all cases, and DESI includes the BGS and non-redundant part of BOSS. The figure of merit of the surveys is inversely proportional to the areas of the error ellipses. \n<!-- image --> \nTable 2.9: DETF Figures of Merit and uncertainties σ w p and σ Ω k . σ w p is the error on w at the pivot redshift, which also equal to the error on a constant w holding w a = 0. σ Ω k is the error on the curvature of the Universe, Ω k . All DESI lines contain the BGS, and BOSS in the range 0 . 45 < z < 0 . 6 that does not substantially overlap with DESI. All cases include Planck CMB constraints. The pivot point, where w ( a ) has minimal uncertainty is indicated by a p . We note that a FoM of 110 is 10 times the Stage II level of [109], which we take to be the definition of Stage IV. DESI BAO galaxy exceeds this threshold even with a 9,000 square degree survey. \nis the same Stage IV definition that LSST used in their Conceptual Design Report. The 9,000 square degrees DESI survey achieves 121 with galaxies and Lyα forest BAO. We note that these computations include only BAO and CMB, without even the Stage II Supernovae Ia results from [109]. Including DESI galaxy broadband clustering or other dark energy probes boost the Figure of Merit well above 110. \nAs this 9000 square degree survey forecast meets the Stage IV threshold and hence the Mission Need, we have adopted it as the quantitative basis for the Level 1 Science Requirement for the DESI project. We aggregate the BAO performance into three redshift ranges, R in 0 . 0 < z < 1 . 1 and 1 . 1 < z < 1 . 9 and H in 1 . 9 < z < 3 . 7, for the L1 requirements, so as to leave flexibility in the exact redshift distribution of targets. An extensive discussion of how the FOM depends on variation in survey parameters was presented in the DESI Conceptual Design Review. \nThe measurements of fσ 8 from redshift-space distortion provide the means for testing General Relativity. Figure 2.12 shows the rate of growth of structure, f , as a function of the redshift. Forecasted DESI errors, assuming information at k < 0 . 2 h Mpc -1 , are shown on the ΛCDM curve. Alternative gravity models generically predict scale-dependent growth, and here we show theoretical expectations for the f ( R ) modified theory of gravity evaluated at two scales (two values of k ), as well as predictions for the DGP braneworld theory. DESI can clearly distinguish between these models.", 'Galaxy and Quasar Clustering': 'Our treatment of isolated galaxy BAO follows [99], assuming 50% reconstruction, i.e., reduction of the BAO damping scale of [99] by a factor 0.5, except at very low number density, where we degrade reconstruction based on [110]. \nBias uncertainty is modeled by a free parameter in each redshift bin, generally of width ∆ z = 0 . 1, for each type of galaxy. Our results are not sensitive to the redshift bin width [95]. For the broadband signal, we use the same information damping factors from [99] as we use for BAO. This is well-motivated from a theoretical point of view as the non-linear clustering suppresses all linear \n1.0 \nFigure 2.12: Growth of structure, f , as a function of redshift, showing projected DESI measurements and their ability to discriminate against alternative gravity models, f ( R ) (whose scaledependent growth we show evaluated at two different scales) and DGP. The brown (light) error bars at z < 0 . 5 correspond to DESI Bright Galaxy Survey; these are expected to improve when information from the multiple tracers in the BGS is included. Adopted from the Snowmass report on the growth of cosmic structure [64]. \n<!-- image --> \ntheory information, not just BAO [19]. We also include the reconstruction factor (50% reduction in damping length), assuming that reconstruction will recover non-BAO information as well. See [95] for more discussion.', 'Lyα Forest': 'DESI will also probe large-scale structure using the Lyα forest [111, 43], i.e., the Lyα absorption by neutral gas in the intergalactic medium in the spectra of high redshift quasars (it may be possible to do even better at faint magnitudes using Lyman-break galaxies [41]). The distribution of intergalactic gas can be used as a complementary tracer to galaxies of the underlying matter distribution for BAO and broadband power spectrum characteristics. \nThe constraints from the Lyα forest are difficult to predict accurately, because they require careful simulation [112, 113]. The forecasts described below we believe are a conservative assessment. We limit the application of Lyβ forest data to BAO only (see below), and do not include crosscorrelations with quasar density, nor statistics beyond the power spectrum, such as the bispectrum, which are known to be powerful for breaking IGM model degeneracies (e.g., [114]). Finally, we only use the redshift range z = 2 -2 . 7. \nWe model the three dimensional power spectrum of Lyα using Eq. (2.15) and, except as otherwise noted, we use the method of [41] to estimate the errors obtainable by DESI. We use Table I of [37] to model the dependence of b , β , and fitting parameters of D . While these are primarily valid near z ≈ 2 . 25, for BAO the model dependence is not significant. For broadband spectra constraints the bias and damping parameters depend on the amplitude and slope of the linear power spectrum, temperature-density relation [115], and mean level of absorption [116], all of which are varied in our Fisher matrix calculations. To help constrain these parameters, we include the one-dimensional power spectrum, which could be measured from the hundreds of existing high resolution spectra [116, 117]. \nWhile past projections used the rest wavelength range 1041 < λ < 1185 ˚ A (following [111]), for the BAO constraints only, we expand the range to include the Lyβ forest and move slightly closer to the quasar, 985 < λ < 1200 ˚ A, reflecting our increasing confidence that we understand the relevant issues well enough to measure BAO across this range [118]. (The Lyβ forest is the wavelength range ∼ 973 -1026 ˚ A where there is Lyβ absorption on top of the Lyα absorption. This Lyβ absorption corresponds to the same gas we see in the standard Lyα forest and should provide some extra information, but we simply assume it can be mostly removed as a source of noise and the underlying Lyα used to measure BAO to shorter wavelengths in each quasar spectrum.) Gains from this enhancement of effective number density (and cross-correlations with quasars) are substantial because the measurement is quite sparse, i.e., in what for galaxies we would call the shot-noise limited regime. \nThe cross-correlation of quasars with the Lyα forest [119] provides a complementary measurement of BAO at high redshift. We combine the two probes of structure in the same volume as described in [95]. The correlation of Lyα absorption in quasar spectra can also provide other cosmological information, beyond BAO: cosmological parameter constraints from the line of sight power spectrum [111, 120, 121, 122], and from the full shape of the three-dimensional clustering [37]. In the projections below we distinguish between Lyα forest BAO measurements and broadband measurements that include the one-dimensional power spectrum measurement.', '2.5 Cosmology Beyond Dark Energy': 'While the fundamental goal of DESI is the measurement of the expansion rate of the Universe through BAO and RSD, the enormous spectroscopic survey will measure the two-point correlation function and power-spectrum over a broad range of scales and redshifts. These data will open up broader investigations into cosmology and particle physics. \nThe broadband power spectrum will provide tests of inflation through its scale dependence. Inflation can also be tested through the scale dependence of the bias of dark matter halos, which constrains the primordial non-Gaussianity. The power spectrum will also reflect the damping of structure by free-streaming neutrinos and thereby give a measure of the sum of the neutrino masses, and possibly reveal previously unknown nearly massless species.', '2.5.1 Inflation': "The inflationary paradigm is the leading explanation for the origin of the fluctuations of primordial density, which in turn seeded the large-scale structure we observe today. In its simplest formulation, inflation predicts perturbations in the initial distribution that are very nearly scale-independent and Gaussian-distributed about the mean. Inflation has been tested primarily with the CMB observations - starting with COBE measurements on large scales in the early 1990s and continuing with the increasingly precise WMAP and Planck measurements in this millennium. However the CMB temperature measurements are not expected to improve greatly after Planck (though CMB polarization has a lot to offer, in particular in testing for signatures of inflationary gravity waves). Large-scale structure measurements have become increasingly precise thanks to 2dF, SDSS, and WiggleZ. These complement the CMB measurements in temporal and spatial scales. The next frontier for tests of inflation is large-scale structure. DESI's unparalleled three-dimensional picture of the evolution of structure will contribute powerfully.", 'Spectral Index and Its Running': "Inflation predicts that the primordial spectrum of density fluctuations is nearly a power law in wavenumber k . The power law is specified by the spectral index defined as \nn s ( k 0 ) = d ln P d ln k ∣ ∣ ∣ ∣ k 0 (2.16) \nwhere k 0 is some reference scale, typically chosen to be k 0 = 0 . 05 Mpc -1 . A perfect power law would correspond to a constant n s ; in reality, inflation also predicts a small 'running' with wavenumber parameterized with the parameter α = dn s /d ln k , again defined at k 0 . The primordial power spectrum can therefore be written as [123] \nP ( k ) = P ( k 0 )( k/k 0 ) n S ( k 0 )+ 1 2 α ln( k/k 0 ) . (2.17) \nThe exact Harrison-Zel'dovich primordial spectrum has n s = 1, while inflation predicts slight deviations from unity. Ruling out n s = 1 at a significant level of confidence would strengthen the case for inflation [124]. Recent Planck data currently favor n s < 1 at 5 σ ; n s = 0 . 968 ± 0 . 006 [125]. The current limit on running of the spectral index obtained by Planck is dn s /d ln k = -0 . 003 ± 0 . 007 (95% CL). Because it is in the regime of linearity for a wide range of k , the Lyα forest is an excellent complementary probe of α s . \nIn Table. 2.10 we present forecasts on inflationary observables obtained with the Fisher-matrix formalism described in Section 2.4.1, applied to the power spectrum obtained from DESI galaxies, \nTable 2.10: Projected constraints on inflationary observables obtained by DESI. In all cases, we include constraints from the Planck satellite and BAO information from DESI galaxies, quasars and the Lyα forest. We show the result of including information from the broadband galaxy power spectrum ('Gal') out to k max = 0 . 1 and 0 . 2 h Mpc -1 , and from the Lyα forest. The numbers in parentheses show the relative improvement over Planck . Broadband Lyα forest constraints include ∼ 100 existing high resolution spectra to constrain the IGM model. n s constraints assume fixed α s . Both constraints are marginalized over Σ m ν , and the fiducial values are n s = 0 . 963, α s = 0. \nquasars, and Lyα forest, combined with CMB data from the Planck satellite. The table shows strong constraints on n s , and improvements up to a factor of three over Planck alone, under the assumption that there is no significant running in the spectral index. Achieving these constraints will require excellent control of broad-band systematics in the Lyα forest and galaxy analyses. But the effort is worthwhile, as these measurements can have far-reaching implications on our understanding of the very early Universe, as we now describe. \nFor the spectral index, the increased accuracy implies much better constraints on models of inflation. With the DESI+ Planck constraints, excellent constraints on the spectral index will effectively reduce the allowed region in the plane of n s and r , the ratio of tensor to scalar modes, to a vertical line pinned at the measured value of n s . Combining these results with better measurements of the r from the small-scale CMB experiments will lead to much better constraints on inflationary models. Even without the accompanying r measurements, better determination of the spectral index is important: for example, for inflationary potentials V ( φ ) ∝ φ m , where φ is the inflaton field, the spectral index and the total number of e-folds of inflation N are related via 1 -n s = ( m +2) / (2 N ) [126]. Hence, for this class of models the duration of the inflationary phase would be determined by DESI very precisely. \nImplications of the precise measurements of the running of the spectral index α s are even more impressive. In standard single-field slow-rolling inflationary models, the running of the spectral index is of the order O ((1 -n s ) 2 ) ∼ 1 × 10 -3 if n s ∼ 0 . 96. This means that DESI will start to approach the region of expected detection in minimal inflationary models. More importantly, a detection of running larger than the slow-roll prediction would imply either that inflation involves multiple fields, or a breakdown of the slow roll approximation [127], or else that a non-canonical kinetic term is controlling inflationary dynamics [128]. Any detection of the running of the spectral index would represent a significant advance in our understanding of the physics of inflation.", 'Primordial non-Gaussianity': "One of the fundamental predictions of the simplest inflationary models is that the density fluctuations in the early Universe that seeded large-scale structure were nearly Gaussian distributed. A single field slow-roll inflation with canonical kinetic energy and adiabatic vacuum predicts very small amount of non-Gaussianity. A violation of any of these conditions, however, may lead to large non-Gaussianity. A simple, frequently studied model is that of non-Gaussianity of the local type, Φ = φ G + f NL ( φ 2 G -〈 φ 2 G 〉 ), where Φ is the primordial curvature fluctuation and φ G is a Gaussian random field. A detection of nonzero f NL would rule out the simplest model of inflation, while a \nnon-detection at a level of f NL < O (1) would rule out many of its alternatives. \nThe tightest existing upper limits on non-Gaussianity have been obtained from observations of the cosmic microwave background by the Planck experiment[129]. Recently, a number of inflationary models have been proposed which predict a potentially observable level of non-Gaussianity, these include those from fast-roll inflation [130, 131, 132, 133, 134], quasi-single field inflation [135, 136], warm inflation [137, 138], and non-Bunch-Davies or excited initial states [130, 139, 140, 141]. There are also hybrids of multi-field and non-slow-roll models [142, 143, 144], and the inclusion of isocurvature modes in the non-Gaussian correlations [145, 146, 147]. Improved limits on nonGaussianity would rule out some of these models. Conversely, a robust detection of primordial non-Gaussianity would dramatically overturn the simplest model of inflationary cosmology, and provide information that would help us significantly improve our understanding of the nature of physical processes in the early Universe. \nUntil recently, the most powerful methods to place limits on f NL were based on the bispectrum of the CMB. The constraints from CMB data have improved starting from σ ( f NL ) glyph[similarequal] 3000 with COBE [148] to σ ( f NL ) glyph[similarequal] 20 with WMAP [149], to the tight constraint of σ ( f NL ) glyph[similarequal] 5 . 8 with Planck 's first year data [150] and finally to σ ( f NL ) glyph[similarequal] 5 . 0 with the 2015 data from Planck [129]. It is therefore impressive and maybe even surprising that a powerful LSS survey such as DESI can provide comparable but highly complementary constraints to Planck . Moreover, as we now show, DESI and Planck in combination can provide very tight constraints on distinct classes of physically motivated inflationary models. \nPowerful constraints on non-Gaussianity can come from the effect that it has on the clustering of dense regions on very large scales [151]. Essentially, the bias of dark matter halos assumes a unique, scale-dependent form at large spatial scales in the presence of primordial non-Gaussianity of local type \nb ( k ) ≡ b 0 +∆ b ( k ) = b 0 + f NL ( b 0 -1) δ c 3Ω M H 2 0 ag ( a ) T ( k ) c 2 k 2 , (2.18) \nglyph[negationslash] \nwhere b 0 is the usual Gaussian bias (on large scales, where it is constant), f NL is the parameter that indicates departures from Gaussianity (when f NL = 0), δ c ≈ 1 . 686 is the collapse threshold, T ( k ) is the transfer function and g ( a ) is the growth suppression factor. Notice the unique k -2 scale dependence in the presence of primordial non-Gaussianity. Since the bias b ( k ) is readily measured from the correlation function of galaxies or quasars, classes of inflationary models can be tightly constrained. A first application of this method has been presented using the large-scale clustering of quasar and luminous red galaxies (LRG) galaxy data from the Sloan Digital Sky Survey (SDSS) [152]. The result, a non-detection with one sigma error σ ( f NL ) glyph[similarequal] 25, was (at the time) comparable to the CMB constraints from WMAP . DESI will provide constraints competitive, and very complementary, to those from Planck , provided that we have systematics under control [153, 154, 155] \nForecasts for DESI indicate that the 1 σ error on the local model from DESI alone will be σ ( f NL ) glyph[similarequal] 5, and about a factor of two better when combined with the final Planck temperature and polarization data. From the fundamental physics point of view, these constraints are very exciting, as they probe not only primordial non-Gaussianity but are likely to detect the additional non-Gaussian signal due to late-time nonlinear interactions of the photon-baryon fluid with gravity (with f NL late glyph[similarequal] few [156, 157]), and thus provide an additional test of cosmology. \nMore generally, inflationary models predict a range of possibilities for the scaling of the bias ∆ b ∝ k -m . For example, m = 2 for the local model parameterized by f NL as in Eq. (2.18); multifield inflationary models generically produce 0 < m glyph[lessorsimilar] 2, and models with modifications to the initial quantum state can produce an even stronger scaling with m = 3. Because many of these \nFigure 2.13: Constraints on the models of primordial non-Gaussianity with 'running', where the usual parameter f NL is promoted to a power-law function of wavenumber, f NL ( k ) = f ∗ NL ( k/k ∗ ) n f NL . The larger contours show constraints on f ∗ NL and n f NL from a first analysis that was applied to WMAP 7 data [158]. The size of the red dot shows the 68% C.L. forecast on the joint constraint expected from the combination of the DESI and full Planck data sets, based on projections in Ref. [159]. \n<!-- image --> \nmodels therefore leave a strong imprint in the clustering of galaxies and quasars, DESI will be able to strongly constrain whole classes of inflationary models. We show an illustration in Figure 2.13, where we present constraints on the models with 'running' of non-Gaussianity, where the usual parameter f NL now runs with wavenumber, f NL ( k ) = f ∗ NL ( k/k ∗ ) n f NL . The larger contours show constraints on f ∗ NL and n f NL from a first analysis that was applied to WMAP 7 data [158], while the small, red contour shows the 68% C.L. forecast on the joint constraint expected from the combination of the DESI and full Planck data sets, based on projections in Ref. [159]. The latter constraint will shrink the area in the f ∗ NL -n f NL plane by about a factor of 100. \nTo achieve such excellent constraints, the galaxies measured in DESI must have sufficiently large bias, since only for biased tracers is the non-Gaussian scale-dependent clustering revealed. One way to further improve the errors is by combining two tracers of LSS, one with a high bias and one with a low bias. In this case it may possible to cancel sampling variance, which is the dominant source of error on large scales [160, 161], but due to low number density this will have to include an additional tracer of structure, potentially combining with the LSST and DES data. \nMore detailed studies of halo mass distribution of BOSS galaxies, combined with numerical simulations of non-Gaussian models [162] as well as studies of how to mitigate the large-angle systematic errors [163, 164, 155] are needed to provide a better definition of the ultimate reach of DESI for non-Gaussianity studies. However it seems certain that DESI constraints will be at least comparable to the best limits from CMB and that they will provide an excellent temporal and spatial complement to the latter. \nTable 2.11: Constraints on the sum of neutrino masses from DESI forecasts in combination with constraints from the Planck satellite. The experiment combinations are identified as described in the caption of Table 2.10. The last four cases include the information from Planck and DESI BAO measurements. Fiducial values are Σ m ν = 0 . 06 eV, N ν, eff = 3 . 04. Σ m ν constraints assume fixed N ν , while N ν is marginalized over Σ m ν .", '2.5.2 Neutrinos': 'The effects of neutrinos in cosmology are well understood (for a review, see [165]). They decouple from the cosmic plasma when the temperature of the Universe is about 1 MeV, just before electron-positron annihilation. While ultra-relativistic, they behave as extra radiation (albeit not electromagnetically coupled) with a temperature equal to (4 / 11) 1 / 3 of the temperature of the cosmic microwave background. As the Universe expands and cools, they become non-relativistic and ultimately behave as additional dark matter.', 'Neutrino Mass': "The mass of neutrinos has two important effects in the Universe [165]. First, as the neutrinos become non-relativistic after the time of CMB decoupling they contribute to the background evolution in the same way as baryons or dark matter, instead of becoming completely negligible as they would if massless (like photons). This affects anything sensitive to the background expansion rate, e.g., BAO distance measurements. Second, the process of neutrinos becoming non-relativistic imprints a characteristic scale in the power spectra of fluctuations. This is termed the 'free-streaming scale' and is roughly equal to the distance a typical neutrino has traveled while it is relativistic. Fluctuations on smaller scales are suppressed by a non-negligible amount, of the order of a few percent. This allows us to put limits on the neutrino masses. \nFrom neutrino mixing experiments we know the differences of the squares of masses of the neutrino mass eigenstates. The splitting between the two states with similar masses is ∆ m 2 21 = (7 . 50 ± 0 . 20) × 10 -5 eV 2 , while the splitting between the highest and lowest masses squared is ∆ m 2 32 = 2 . 32 +0 . 12 0 . 08 × 10 -3 eV 2 . Two things are not known: the absolute mass scale, and whether the two states close together are more or less massive than the third state. In what is called the normal hierarchy, the close states are less massive. In this configuration, the lowest possible masses in eV are 0, 0.009, and 0.048, so the minimal sum of neutrino masses is 0.057 eV. In the inverted hierarchy, the minimal masses are 0, 0.048, and 0.049 eV, for a total of 0.097 eV. This is shown in Figure 2.14. \nTable 2.11 shows our projected Σ m ν constraints, obtained through Fisher matrix calculations as discussed above and in [95]. \nWith a projected resolution of 0.020 eV, DESI will make a precision measurement of the sum of the neutrino masses independent of the hierarchy and therefore determine the absolute mass scale for neutrinos, a measurement that is otherwise very challenging. Furthermore, if the masses were \nFigure 2.14: The two possible neutrino mass hierarchies. Also shown is what fraction of each mass eigenstate corresponds to a neutrino flavor eigenstate. DESI will be sensitive to the sum of the neutrino masses and possibly to the mass hierarchy. \n<!-- image --> \nminimal and the hierarchy normal, DESI would be able to exclude the inverted hierarchy at 2 σ .", 'Dark Radiation (e.g., sterile neutrinos)': "The other parameter relevant for neutrino physics is the effective number of neutrino species N ν, eff , which parameterizes the energy density attributed to any non-electromagnetically interacting ultrarelativistic species (including e.g. axions) in units of the equivalent of one neutrino species that fully decouples before electron-positron annihilation. Extra radiation shifts the redshift of matter radiation equality and changes the expansion rate during the CMB epoch, although it does not significantly affect the Universe at the epoch probed by DESI. The value for the standard cosmological model is N ν, eff = 3 . 04 3 [166]. The detection of any discrepancy from the expected value would be a truly major result, as it would indicate a sterile neutrino [167], a decaying particle [168], a nonstandard thermal history [169], or perhaps that dark energy does not fade away to ∼ 10 -9 at the time of recombination as expected for the cosmological-constant model [170]. All of these possibilities represent important extensions of the standard cosmological model, and uncovering them would present a major advance of our understanding of the Universe. Our forecasts for this parameter are also shown in Table 2.11. Again we see that the effective number of neutrino species will be measured to ∼ 10% or better, providing strong constraints on the alternative models involving extra sterile neutrinos, axions or partly thermalized species. \nIn Figure 2.15 we show the improvement in the measurement of several fundamental parameters from cosmology and neutrino physics. The standard is taken to be the results from BOSS together with Planck . Displayed is the ratio of the uncertainty from BOSS over the uncertainty from DESI, with Planck always included. \nFigure 2.15: Improvement in the measurements of w p , w ' = w a , Ω k , ∑ m ν the sum of the neutrino masses, n s the spectral index, α s the running of the spectral index, and N ν, eff the number of neutrino-like (relativistic) species. \n<!-- image --> \n7", '2.6 The Milky Way Survey: Near-Field Cosmology from Stellar Spectroscopy': "During conditions unusable for faint galaxy work, DESI will pursue the Bright Galaxy Survey, mapping 10 million galaxies to z ∼ 0 . 4 in pursuit of the clustering analyses, such as from BAO and RSD, as described earlier in this chapter. As detailed in section 3.1, the areal density of these bright galaxies is comparable to the fiber density of DESI. Achieving a high completeness in the face of clustering and Poisson fluctuations requires multiple visits, leading to an excess of fibers compared to targets. Indeed, some fibers will be unable to reach a viable galaxy target even on the first pass, and this fraction increases on subsequent passes. \nBright stars are the natural secondary target, and we expect that any bright galaxy survey with the DESI fiber positioner will produce a very large sample of stars as a by-product. This sample is also of high science interest, leading to the definition of the Milky Way Survey. At 17th magnitude, even a short (8-10 min) DESI exposure measures an excellent spectrum with S/N = 25 per pixel, which will yield the radial velocity to a few km/s precision and the metallicity. We expect the BGS to generate at least 10 million such spectra. Spectroscopy of individual stars provides radial velocity, effective temperature, surface gravity, chemical abundance distribution, and approximate age. The assembly history of the Milky Way is encoded in the spatial distributions, kinematics, and chemical composition of the various distinct Galactic stellar populations. This information can test cosmological predictions for how galaxies like the Milky Way form and evolve on small scales that are difficult or impossible to test elsewhere in the Universe, and provide a critical test of the small-scale predictions of the ΛCDM model. \nThe European Space Agency GAIA satellite has been successfully launched and will provide a catalog of parallaxes, proper motions, and spectrophotometry for a billion point sources down to V ∼ 20 over the whole sky. The satellite's RVS spectrograph will supplement those data with radial velocities for millions of brighter stars, although the flux limit is still under investigation due to higher than expected scattered light. DESI can substantially enhance the science return from GAIA by providing radial velocities and metallicities for stars much fainter than what the GAIA spectrograph can provide. While other projects are planned for spectroscopic follow-up of GAIA stars, DESI's higher multiplex, wide field of view, and extremely rapid reconfiguration give it a clear advantage. \nThe stellar program will put exceptional new constraints on the distribution of dark matter in the Milky Way, a vital measurement that links Galactic science, galaxy formation and cosmology. The Milky Way gravitational potential can be probed via the rotation of the Milky Way beyond 15 kiloparsecs, the motions of newly discovered tidal streams, and the kinematics of bright stars in the distant stellar halo. The uncertainty in the Milky Way mass, density profile, and internal structure currently are critically important systematics in the interpretation of direct and indirect dark matter searches, and the measurements possible with the stellar program will substantially reduce these uncertainties. \nJoint metallicity and velocity distribution functions for stars far beyond the solar neighborhood will reveal the recent assembly history of the outer disk and vastly improve our understanding of the structure and formation of the thick disk. The first-ever deep spectroscopic survey of halo main-sequence turn-off stars to 30 kiloparsecs can be used to reconstruct the history of the Galaxy in its first two billion years and its interaction with other galaxies, shedding new light on enigmatic halo substructures like the Virgo overdensity and Hercules-Aquila cloud. Moreover, a survey of millions of stars will have huge potential for the discovery of kinematically and chemically peculiar stars in as-yet unexplored regions of the Galaxy.", '2.7 Complementarity with Other Surveys': "While DESI's spectroscopic survey will by itself yield incisive results in cosmology, its power is increased when combined with other experiments. DESI's BAO results are directly connected to CMB measurements via its dependence on the acoustic scale, but additional information can be obtained by directly cross-correlating the CMB with the density distribution and redshift space distortions from DESI. Large imaging surveys, including DES and LSST, will provide vast amounts of complementary data, allowing increased precision for both cosmological and neutrino measurements. This combination of imaging and spectroscopic surveys is particularly powerful for distinguishing dark energy from modified gravity models for cosmic acceleration.", '2.7.1 Synergies with Planck and Future CMB Experiments': "The cross-correlation of Planck and potential future CMB experiments, such as Advanced ACTPol and CMB-S4, with DESI enables cosmological measurements not possible with either individually, and opens up new opportunities to constrain fundamental physics, in the properties of dark energy and gravity discussed in 2.4 and the nature of neutrinos and inflation summarized in 2.5. \nOn large scales, cross-correlating CMB temperature fluctuations with the galaxy density field measures the Integrated Sachs-Wolfe effect, probing the time evolution of the gravitational potential and independently constraining dark energy [171]. The combination of CMB lensing and the foreground galaxies or quasars will also improve not only the signal-to-noise of CMB lensing leading to stronger cosmological constraints on the matter content, but also our understanding of the foreground tracers in large-scale structure, as lensing allows a clean measurement of the bias of the foreground sources. \nThe combination of CMB lensing and the RSD measurements from DESI will allow a probe of the two relativistic gravitational potentials independently (see e.g. [60] for an application of this test but for the case of gravitational lensing of background galaxies, not the CMB), testing the GR prediction of their equality over a wide redshift range [172]. CMB lensing and RSD measurements will also provide complementary constraints on the sum and differences of the neutrino masses, that in combination could help infer the neutrino hierarchy. \nDESI will provide highly complementary constraints on inflation to those from Planck and a number of upcoming CMB small scale temperature and polarization experiments. An exciting realization in inflationary theory is that discerning the scale-dependence, or 'shape', of the bispectrum (the 3-point function) could provide a direct insight into the inflationary mechanism, through how non-Gaussianity is generated [173, 174]. CMB 3-point correlation measurements constrain a wide range of primordial bispectrum configurations, while DESI will provide more detailed information about the properties in the squeezed limit, a regime that could provide characteristic information about the underlying mechanism driving inflation e.g. whether it is multi-field, sourced from a non-Bunch Davies vacuum state, or includes non-trivial kinetic terms in the inflationary action. \nCross correlating the galaxy velocity field (inferred from the 3D density distribution) with the CMB will measure the kinetic Sunyaev-Zeldovich (kSZ) effect at the percent level. These measurements provide constraints on more exotic deviations from our standard cosmological models [175, 176, 177]. In addition, these measurements are astrophysically important since the kSZ effect is an unbiased probe of electrons and can be used to inventory the baryons in the Universe [178].", '2.7.2 Synergies of DESI with DES and LSST': 'The massive spectroscopic survey provided by DESI will provide a unique and important complement to direct-imaging science projects currently being planned. We focus here on the Dark \nEnergy Survey (DES) and the Large Synoptic Survey Telescope (LSST), but DESI will complement other future imaging surveys in similar ways. Although both DES and LSST are located in the Southern Hemisphere, their planned surveys have overlap of a few thousand square degrees with the baseline DESI survey. In addition, only some of the cosmological tests described below rely on overlap between the photometric and spectroscopic surveys. \nDESI can provide critical input into photometric redshifts which can help control the systematic uncertainty associated with cosmological measurements from photometric surveys like DES and LSST. For instance, cross correlation of photometric lensing sources with spectroscopic galaxy samples enable the reconstruction of the redshift distribution of the lensing sources [179, 171, 180, 181] providing a critical consistency test on the photometric redshifts used for cosmic shear and/or calibrating the mass of galaxy clusters for cluster abundance tests. Likewise, magnificationbased lensing measurements of spectroscopic sources [182] can provide a consistency test for shape systematics and/or photometric redshift systematics in shear-based calibration of cluster masses. \nJust as importantly, the combination of photometric and spectroscopic surveys is significantly more powerful than either set of surveys alone. An example is the utility of using galaxy-galaxy lensing, in which one uses the lensing of background galaxies by galaxies from the spectroscopic sample to measure the galaxy-mass cross-correlation of the spectroscopic sample. On small scales, this measures the properties of the host dark matter halo, testing galaxy bias models; on larger scales, it can be used to measure the mass-mass auto-correlation and hence the amplitude of structure [68, 183]. Several studies have forecast cosmological constraints from a combination of DES-like and DESI-like experiments [184, 185, 186, 187], and while the range of assumptions and forecasts varies from work to work, there is agreement that the combination of DES and DESI/LSST gives substantial benefits in terms of measured cosmological and non-cosmological parameters. This is particularly true within the context of modified gravity models, where the combination of surveys enables entirely new types of measurements that are ideally suited for addressing such questions. For instance, recent theoretical work suggests that comparing the shear field generated by galaxy clusters to the corresponding galaxy velocity can significantly improve current modified gravity constraints [188]. \nAs an example of improvement in another type of constraint that can be achieved through the combination of DESI with imaging surveys, Figure 2.16 shows the joint constraint on the sum of the neutrino masses in eV against the dark energy density ω DE = Ω DE h 2 obtained by combining anticipated results for DESI BAO with LSST weak lensing. Similarly, Figure 2.17 shows prospective constraints in the Ω m -Ω Λ plane obtained by combining anticipated results for DESI BAO with LSST weak lensing (these forecasts assume the surveys are not overlapping on the sky, although it makes practically no difference [95, 189]). \nFinally, DES and LSST will provide world-leading samples for supernova cosmology. The BAO and SNe Ia methods for measuring the cosmic distance scale are highly complementary: supernovae excel at low redshifts, where the SNe are brighter and where the BAO is more limited by cosmic variance due to the small cosmic volume. The combination of DESI with ground-based supernovae samples spanning from z = 0 to z ≈ 0 . 8 will be a powerful view of the distance-redshift relation and the expansion history of the Universe. While we have focused on Figure of Merits drawn only from BAO and the DESI clustering samples, the inclusion of low to intermediate-redshift supernovae provides a notable improvement to current BAO constraints, as highlighted in numerous papers, such as [6, 190]. Essentially one is using BAO to calibrate the relative distance scale provided by the SNe. The redshift overlap of the two methods provides a further systematic cross-check. The exquisite precision of DESI at z > 0 . 6 will find an excellent partner in the DES and LSST supernova samples. \nDESI will directly support the coming decade of supernova cosmology by providing spectroscopic \nFigure 2.16: Constraint on the sum of the neutrino masses in eV against the dark energy density ω DE = Ω DE h 2 obtained by combining DESI BAO with LSST weak lensing, in each case including Planck CMB constraints. More powerful constraints are obtained when the full power spectrum from DESI is used. See Table 2.11. \n<!-- image --> \nFigure 2.17: Prospective constraints in the Ω m -Ω Λ plane obtained by combining DESI BAO with LSST weak lensing. More powerful constraints are obtained when the full power spectrum from DESI is used. See Table 2.9. \n<!-- image --> \nredshifts for many tens of thousands of SNe host galaxies. This will happen both for the faint galaxy survey out to z ∼ 1, but also with the BGS at z < 0 . 4. Over a 10-year period, a typical ( L ∗ ) galaxy has at least a 1% probability of having a detectable SN Ia. This means that the BGS will contain of order 10 5 supernova host galaxies, and the LRG sample of more massive galaxies could produce a comparable number at higher redshift. While photometric redshifts are planned for the large LSST and DES supernova samples, spectroscopic redshifts allow more precision, particularly at \nlow redshift where the uncertainty in the redshift and resulting luminosity distance overwhelm the intrinsic precision of the standard candle. Samples of many tens of thousands of hosts can only be achieved with multi-object wide-field surveys. We note that with DESI there is no need to wait to select the host galaxies after the explosion: at z < 0 . 2, the BGS will include more than half of all SN Ia host galaxies in the survey footprint. Having a pre-existing redshift will also enable better allocation of follow-up resources for rare transients from surveys such as LSST.', '2.7.3 Synergies of DESI with Euclid/WFIRST': "Euclid is a medium class European Space Agency survey mission designed to measure Dark Energy [104]. Recently, NASA has become a partner, enabling a group of 40 US astronomers to join the international consortium. Euclid will perform a 15,000 deg 2 survey jointly undertaking visible imaging to measure weak lensing and simultaneous near- infrared observations split into sequential imaging (for photometric redshift measurement) and slitless spectroscopy. Two Deep Fields about 2 magnitudes deeper than the wide survey and covering around 20 deg 2 each will be also observed, primarily for calibrations of the wide survey data but also extending the scientific scope of the mission to faint high redshift galaxies, quasars and AGNs. The spectroscopic survey is focused on H α emitting galaxies and is most powerful at high redshifts 1 < z < 2. \nThe timeline for DESI is prior to Euclid (which is scheduled to launch in December 2020 for 6 scheduled years of data collection), but even in the era of Euclid , at redshifts z < 1 the combination of LRGs and ELGs that DESI will observe will remain the world-leading data set for spectroscopically confirmed galaxies with good redshift measurements. At z > 2 the DESI measurements from Lyα will also remain unique. Euclid may surpass DESI in the redshift range 1 < z < 2 provided the slitless spectroscopy is as effective as hoped. DESI could help Euclid clustering measurements by providing important information on the potential confusion of the Euclid slitless spectroscopy in this redshift range. The combination of Euclid space-based weak lensing with the large spectroscopic samples from DESI will be a strong opportunity for galaxygalaxy weak lensing, similar to what was discussed in the DES/LSST context in the previous subsection. DESI's contribution of z < 1 lenses is particularly important in this regard. \nWFIRST-AFTA is an envisaged NASA mission using a 2.4 m diameter primary mirror satellite being designed to perform a 2000 deg 2 near-infrared survey, including a slitless spectroscopic component [105]. The current narrow/deep WFIRST-AFTA concept is highly complementary to the wide/shallow Euclid strategy, and will provide deeper, denser galaxy samples. However, the smaller area covered compared to either Euclid or DESI means that the direct expansion rate and growth rate measurements would be weaker. \nComparisons of the precision of the BAO measurement projected for DESI, Euclid , and WFIRST are shown in Figure 2.9. \nDESI will be highly complementary to the weak lensing surveys to be performed for Euclid and WFIRST-AFTA , providing spectroscopic galaxy samples at the same redshifts as the matter that is causing the lensing, thus enabling many innovative analyses from these combined datasets. DESI will help in the calibration of photometric redshifts - which are essential for these lensing experiments - and aid in investigating systematic issues such as intrinsic alignments. Likewise, Euclid and WFIRST-AFTA will greatly enhance the legacy value of DESI, providing high resolution optical and NIR imaging of all DESI targets, greatly improving the prospects for non-dark energy science, e.g., the morphology-density relationship at z > 1.", '3 Target Selection': 'The DESI survey will measure with high precision the baryon acoustic feature imprinted on the large-scale structure of the Universe, as well as the distortions of galaxy clustering due to redshiftspace effects. To achieve these goals, the survey will make spectroscopic observations of four distinct classes of extragalactic sources - the bright galaxy sample (BGS), luminous red galaxies (LRGs), star-forming emission line galaxies (ELGs), and quasi-stellar objects (QSOs). Each of these categories requires a different set of selection techniques to acquire sufficiently large samples of spectroscopic targets from photometric data. To ensure high efficiency and spectroscopic completeness, we select objects with spectral features expected to produce a reliable redshift determination or a Lyα forest measurement within the DESI wavelength range. \nThe characteristics of our baseline samples for each of these target classes are summarized in Table 3.1. This Table specifies the primary redshift range, the photometric bands for targeting, the projected areal density (in terms of number of targets, number of fibers allocated across all pointings accounting for multiple exposures, and the number of useful redshifts resulting per square degree), as well as the total number of objects in the desired class for which redshifts are expected to be obtained for each of these samples. This table may be compared to Table 1 in the Science Requirements Document (SRD). The SRD considers both a threshold survey of 9,000 deg 2 and a baseline survey of 14,000 deg 2 . Throughout this chapter, we consider only the latter scenario; simulations for reduced focal planes indicate that we would achieve essentially the same sample surface densities as for the baseline scenario, so that sample sizes would simply scale with survey area. In the following sections, we will describe the basis of these numbers in more detail.', 'Summary of Target Samples': "The lowest-redshift sample of DESI targets will be the Bright Galaxy Sample (BGS). These galaxies will be observed during the time when the moon is significantly above the horizon, and the sky is too bright to allow efficient observation of fainter targets. Approximately the 10 million brightest galaxies within the DESI footprint will be observed over the course of the survey, sampling the redshift range 0 . 05 < z < 0 . 4 at high density. This sample alone will be ten times larger than the SDSS-I and SDSS-II 'main sample' of 1 million bright galaxies observed from 1999-2008. \nAbove redshift z = 0 . 4, DESI will observe luminous red galaxies (LRGs). These luminous, \nTable 3.1: Summary of the properties for each DESI target class. The bands listed are for the target selection, where g , r , and z are optical photometry and W 1 and W 2 denote are WISE infrared photometry. The exposure densities are increased over the target densities due to some objects being observed on multiple passes. The number of good redshifts and baseline sample sizes (in millions) are for successful redshifts. \nmassive galaxies have long since ceased star formation and therefore exhibit evolved, red composite spectral energy distributions (SEDs). The BOSS survey targeted these objects to z ≈ 0 . 6 using SDSS gri colors and measured spectroscopic redshifts using the prominent 4000 ˚ A break continuum feature. While DESI will aim to achieve 350 LRGs/deg 2 over 14,000 square degrees, the BOSS sample of 119 LRGs/deg 2 will contribute significantly to our science analyses over the 10,000 deg 2 footprint in which it exists; DESI may extend this low-redshift sample over a larger footprint, but this is not in the current baseline plan. DESI will target LRGs to z ≈ 1 . 0, where they may be most efficiently selected using the prominent 1.6 µ m (rest frame) 'bump,' which causes a strong correlation between optical/near-infrared (NIR) color and redshift in this regime. We will use 3.4 µ m photometry from the space-based Wide-Field Infrared Survey Explorer ( WISE ) to select LRGs efficiently in the redshift range of 0 . 6 < z < 1 . 0. DESI can exploit the 4000 ˚ A break to obtain secure redshifts for LRGs over this full redshift range. \nThe majority of the spectroscopic redshift measurements for DESI will come from ELGs at redshifts 0 . 6 < z < 1 . 6. These galaxies possess high star formation rates, and therefore exhibit strong emission lines from ionized H II regions around massive stars, as well as SEDs with a relatively blue continuum, which allows their selection from optical grz -band photometry. The prominent [O II] λλ 3726 , 29 doublet in ELG spectra consists of a pair of emission lines separated in rest-frame wavelength by 2.783 ˚ A. This wavelength separation of the doublet provides a unique signature, allowing definitive line identification and secure redshift measurements. The goal of the DESI ELG target selection will be to provide a large sample of ELGs with sufficient [O II] line flux to obtain a detection and redshift measurement to z = 1 . 6. \nThe highest-redshift target sample will consist of QSOs. We will measure large-scale structure using QSOs as direct tracers of dark matter in the redshift range 0 . 9 < z < 2 . 1. At higher redshifts, we will utilize the foreground neutral-hydrogen absorption systems that make up the Lyα forest; DESI spectra cover the Lyα transition at λ = 1216 ˚ A for objects at z > 2 . 1. We will use optical photometry combined with WISE infrared photometry in the W1 and W2 bands to select our primary sample of QSOs. QSOs are ∼ 2 mag brighter in the near-infrared at all redshifts compared to stars of similar optical magnitude and color, providing a powerful method for discriminating against contaminating stars. QSOs at z > 2 . 1 used for Lyα forest measurements do not require homogeneous selection on the sky for cosmological measurements, as we do not rely on the clustering of the QSOs themselves. As a result, DESI may exploit optical variability and additional passbands where available to enhance this sample. Those z > 2 . 1 QSOs which are selected via uniform methods across the sky may also be used to enhance clustering measurements. DESI will obtain additional exposures on the confirmed z > 2 . 1 quasars to measure the Lyα forest to the required S/N.", 'Summary of Required Imaging': "All DESI target samples will be selected using optical grz -band photometry from ground-based telescopes and near-infrared photometry from the WISE satellite. The observations assumed in our baseline targeting plan are summarized in Table 3.2. This imaging plan has been developed through a detailed analysis of alternative telescope/instrument combinations. The imaging depths will be at least 24.0, 23.4, 22.5 AB (5 σ for an exponential profile r 3 = 0 . 45 '' ) in g , r , z and 20.0, 19.3 AB (5 σ ) in WISE W1,W2. All sample magnitude limits quoted in this section are total (model-like) magnitudes for the BGS and for LRGs and ELGs, or PSF magnitudes for QSOs. \nThe optical imaging for the DESI targets will be provided from three telescopes at two sites, Cerro Tololo and Kitt Peak. The DECam camera on the Blanco 4-m telescope will provide grz imaging over 9000 deg 2 in the DESI footprint at Dec ≤ +34 deg. The first 6700 deg 2 of this \nTable 3.2: Summary of telescopes being used for targeting. \nfootprint (DECaLS) has been approved as a 64-night NOAO 'Large Survey' program during the period August 2014 through July 2017 and is 40% completed. An 8-night extension of this program (DECaLS+) has been approved for the 2016A semester to obtain another 800 deg 2 in the Northern Galactic Cap. A proposal to observer the remainder of the DESI footprint in the South Galactic Sky will be submitted in future semesters. The Bok 2.3-m telescope is providing gr imaging over the 5000 deg 2 region of the North Galactic Cap (NGC) that lies at Dec ≥ +34 deg with the existing 90Prime camera. The 220 nights necessary for these observations are guaranteed via an MOU with the University of Arizona / Steward Observatory. Observations were taken in Spring 2015 which identified electronics problems in the camera that were corrected in September 2015. The Bok observations re-started in January 2016 and are now 15% complete. The Mayall 4-m telescope will provide z -band imaging over the same NGC footprint using the existing MOSAIC-2 camera upgraded with 4 red-sensitive CCDs. Those observations will be conducted over 220 nights in 2016 and 2017. The Mayall observations began in February 2016 and are now 15% complete. All imaging data are planned to be completed by August 2017, where the Mayall observations must be complete as that telescope will being taken off-line for DESI installation. \nThe WISE satellite is obtaining infrared imaging to sufficient depths for DESI target selection over the full sky. An initial 13-month survey is being supplemented with a 3-year extended mission known as NEOWISE that began 1 December 2013 and will complete in December 2016. The initial WISE survey and first year of NEOWISE data are publicly available, with the final two data releases scheduled for March 2016 and March 2017. \nThe DESI analyses will be performed separately in each of the three regions of the DESI footprint: the NGC at DEC > +34 deg, the NGC at DEC < +34 deg, and the South Galactic Cap (SGC). Based on SDSS-III/BOSS experience with separately-calibrated regions, we expect to analyze these separately and combine the cosmological constraints downstream. The DECam and Bok/MOSAIC coverage will have some overlap (at the DEC ≈ +34 · strip and by targeting specific calibration fields like COSMOS, Bootes, and DEEP-2) in order to tie together calibrations and understand the subtle variations in target selection resulting from differences in filter+telescope response between the two datasets. \nIn the remainder of this Section, we demonstrate that our baseline optical/infrared color selections can select the targets listed in Table 3.1, and summarize the key properties of each sample. The accompanying instrument volume of the FDR details the design of the DESI instrument, which informs a spectral simulator presented in that volume. The spectral simulator aids in the design of the targeting strategy (such as magnitude limits), calculates exposure times, and estimates redshift measurement efficiencies. Given the expected target densities and exposure times, the overall survey strategy is developed in Section 4. Included in the survey strategy is an optimized method to tile the sky that maximizes the area covered and number of target redshifts obtained, while minimizing the overall time required for the survey. The outlines for a strategy for fiber allocation \nFigure 3.1: Surface density of BGS targets as a function of r -band magnitude from a numerical simulation. This mock is calibrated to match low-redshift data from SDSS. \n<!-- image --> \nare given in the accompanying FDR. This strategy leads to the values given in Table 3.1.", '3.1.1 Overview of the Sample': 'The galaxy sample for the BGS will be a flux-limited, r -band selected sample of galaxies. The magnitude limit is determined by the total amount of observing bright time and the exposure times required to achieve our desired redshift efficiency. This target selection is, in essence, a deeper version of the galaxy target selection for the SDSS main galaxy sample (MGS). We explore the properties of the BGS target sample through mock catalogs created from numerical simulations. These mocks have identical properties to the MGS at low redshift, including the luminosity function, color distribution, and clustering properties. At higher redshifts, the mock BGS is calibrated using data from the much smaller areas of the GAMA ( z ∼ 0 . 3) and DEEP2 ( z glyph[lessorsimilar] 1 . 0) surveys.', 'Surface Density': 'In Figure 3.11 we show the surface density of candidate ELGs in our grz selection box (see Figure 3.10) as a function of the r -band magnitude limit. At a depth of r AB ≈ 23 . 4, we achieve our goal of 2400 targets per square degree. As we discuss below, we conservatively estimate that at least 65% of these will be bona fide ELGs in the redshift range 0 . 6 < z < 1 . 6 with a strong enough [O II] emission-line doublet (in tandem with other nebular emission \nFigure 3.11: Surface density of ELGs as a function of limiting r -band magnitude. The solid black line shows the surface density of objects which lie within the target selection box shown in Figure 3.10 as a function of r AB magnitude based on a 35 deg 2 region of DECaLS observed to the final survey depth. For comparison, the dashed line is the set of objects selected from CFHTLS-Deep photometry [200] which has been transformed and degraded to the anticipated depth of DECaLS. The horizontal dashed red line shows our goal density of 2400 targets deg -2 , which is achieved at a depth of r AB glyph[lessorsimilar] 23 . 4. We note that the differences in the two curves is most likely due to the scatter in the transformations between the CFHTLS and DECaLS photometric systems. \n<!-- image --> \nlines available at z glyph[lessorsimilar] 1) to yield a secure redshift. Out of this sample, at most 270,000 ELGs over 500-1,500 deg 2 may be targeted by SDSS-IV/eBOSS, representing a sample that could be used for further validation of DESI targets.', 'Redshift Distribution': 'Figure 3.12 shows the anticipated redshift distribution of our candidate grz -selected sample of ELGs, determined based on those DEEP2/EGS objects which are both selected by our candidate cuts (after transforming to the DECaLS photometric system and degrading to the expected depth of the survey) and exhibit sufficient [O II] flux for DESI redshift measurements to succeed, reweighted to account for DEEP2 target selection rates. 4 \nThe ELG sample is designed to have a product of the number density and the power spectrum, ¯ nP , that exceeds 1 over some scales. This is shown as the dashed blue line in Figure 3.12, which is the surface density for which ¯ nP = 1 when evaluated at wave number k = 0 . 14 h Mpc -1 and orientation relative to the line-of-sight µ = 0 . 6). Below this limit, shot noise will dominate errors in measuring the BAO signal (cf. § 2.4.2). Our candidate ELG selection exceeds the ¯ nP = 1 curve to redshift z ∼ 1 . 3. \nFigure 3.12: Expected redshift distribution of ELG targets based on our analysis of the DEEP2/EGS survey data (see Figure 3.10). The overall normalization of the distribution has been fixed to 1280 ELGs deg -2 (from a targeted sample of 2400 targets deg -2 ) to reflect conservative estimates of the overall efficiencies of fiber assignment, target selection, and redshift measurement. The ELG redshift distribution drops to a level where shot noise dominates errors in BAO measurements (i.e., ¯ nP < 1) only at z glyph[greaterorsimilar] 1 . 3 (dashed blue line). \n<!-- image -->', 'Redshift measurement method': 'The adopted grz color-cuts are designed to maximize the selection of galaxies at z ≈ 1 with significant [O II] emission-line flux. In Figure 3.13 we plot [O II] flux as a function of redshift using the DEEP2/EGS sample. The red curve shows the limiting [O II] flux above which DESI simulations predict we will detect emission lines at > 7 σ , resulting in secure redshifts. Galaxies at redshift z > 1 . 0 will have the [O II] doublet as the only strong spectroscopic feature, while those at lower redshifts will show H β (at z < 0 . 5) and [O III] (at z < 1 . 0).', 'Large-scale-structure bias': 'In order to predict the strength of the BAO feature in galaxy clustering measurements, we must assume a value for the ratio of galaxy clustering to dark matter clustering, commonly referred to as the large-scale structure bias. On large scales this may be approximated as a function of redshift that is independent of scale, b ( z ). We can anticipate that the bias for z > 0 . 6 luminous red galaxies should be at least as large as that of BOSS LRGs, as only the most extreme objects will be able to assemble a large amount of mass and cease star formation by this earlier epoch. We therefore assume a bias of the form b ( z ) = 1 . 7 /D ( z ), where D ( z ) is the growth factor; this matches the value \nFigure 3.8: DESI LRG redshift distribution for our candidate sample from two studies: (black) Photometric redshift distribution for a sample selected using DECam imaging in the COSMOS field, which has full redshift coverage but suffers from high sample variance (as seen from the feature at z ≈ 0 . 77). (blue) Spectroscopic redshift distribution for galaxies selected using SDSS Stripe 82 photometry and assigned the redshift of the object with the nearest color from a BOSS ancillary program. The latter sample has low sample variance, but the high-redshift tail is suppressed by the lack of redshifts at 20 < z SDSS < 20 . 46. Shown in red is the redshift distribution of lowz LRGs, many of them already observed by SDSS-I/II and SDSS-III/BOSS, which will be included in the DESI analysis. \n<!-- image --> \nmeasured by SDSS-I at z = 0 . 34 [193] and by SDSS-III at z = 0 . 57 [203]. We have extrapolated this trend to z = 1 for the DESI LRGs.', 'Target selection efficiency': 'Targets selected as ELGs could fall short in several ways: they could entirely fail to yield a redshift (e.g., if the galaxy is at z glyph[greaterorsimilar] 1 . 63 then no strong emission lines will be detected by DESI); they could prove to be low-redshift galaxies, z < 0 . 6; they could be QSOs instead of galaxies (and hence useful for higher-redshift clustering analyses but likely outside the redshift range of the ELGs); or they could be stars. Based on the DEEP2/EGS sample, we estimate that ∼ 10% of the objects targeted via the baseline selection criteria are expected to be stars, ∼ 5% will be lower-redshift interlopers, and ∼ 5% will be at z glyph[greaterorsimilar] 1 . 6, while contamination from QSOs is expected to be negligible. Combining all these factors, the fraction of ELG targets which are in fact galaxies in the correct redshift range is approximately 80%. Among these objects, about 85% will have a high enough [O II] flux to securely measure a redshift more than 95% of the time (see Figure 3.13). Combining all these factors with the 78% fiber assignment rate expected for an input target density of 2400 targets deg -2 , we obtain an a final density of 1220 ELGs deg -2 .', 'Areas of risk': 'The primary source of risk in our ELG selection is the limitations of the datasets available for developing and assessing selection algorithms. DEEP2 is the only large current survey which resolves the [O II] doublet critical for obtaining secure redshifts at z > 1; however, due to the z > 0 . 75 color cut applied by DEEP2 in three of four survey fields, it can be used to assess the low-redshift tail of the ELG selection in only a limited area, the Extended Groth Strip used for all analyses here. Because of the limited area, the number of DEEP2 ELGs within our color box is relatively small, so both Poisson noise and sample/cosmic variance have a significant effect on our predicted redshift distributions. Furthermore, the \nlack of DEEP2 coverage of [O II] at z > ∼ 1 . 4 means that our assessments of performance in that regime are subject to some amount of uncertainty. Despite these shortcomings, even more assumptions and extrapolations would be necessary with any other existing dataset. The consistency of VVDS and COSMOS results-together with the initial SDSS-IV/eBOSS observations-with the DEEP2-based predictions builds confidence that these uncertainties are not substantial. \nThe second potential source of risk which would cause performance to fall short of our projections is that the redshift success rate for DESI ELGs could not simply be a function of signal-to-noise ratio, but may also depend in more subtle ways upon the instrumental resolution and the intrinsic galaxy velocity dispersions. For example, it would be difficult to directly discriminate between [O II] or another single-line feature at lower redshift for a population of ELGs with unusually large velocity dispersions σ v > 150 km s -1 (though the rarity of low-luminosity objects with extremely high velocity dispersions, as would be implied by a false identification, may allow such cases to be resolved). \nTo conclude, the ELG selection methods used for our baseline plan will yield 2400 targets deg -2 . From these targets, DESI should securely measure redshifts for approximately 1220 ELGs deg -2 in the redshift range 0 . 6 < z < 1 . 6 (see Table 3.1). This sample will enable constraints on cosmological parameters over a broad redshift range centered on z ≈ 1, which can be directly compared to results from the independently observed samples of LRGs at z < 1 and quasars at z > 1.', '3.2.1 Overview of the Sample': 'The lowest-redshift dark-time sample for DESI will come from targeting 350 candidate luminous red galaxies (LRGs) per square degree [192]. These objects are both high in luminosity and red in rest-frame optical wavelengths due to their high stellar mass and lack of ongoing star formation. They exhibit strong clustering and a relatively high large-scale-structure bias, which enhances the amplitude of their power spectrum, and hence the BAO signal ([193], [194], [195]). Because of their strong 4000 ˚ A breaks and their well-behaved red spectral energy distributions, low-redshift LRGs at z < 0 . 6 can be selected efficiently and their redshifts estimated based on SDSS-depth photometry [196]. The BOSS survey has targeted 119 LRGs per deg 2 with z glyph[lessorsimilar] 0 . 6 using SDSS imaging. \nDESI science analyses will incorporate existing BOSS spectroscopic samples (which cover 10,000 deg 2 of the DESI footprint) where available, as well as applying BOSS-like target selection algorithms (in regions not yet covered) to target LRGs at low z . Because the BOSS target selection is well understood and documented in SDSS papers, we will not discuss it further here. Extending the LRG sample to redshifts z > 0 . 6, where the 4000 ˚ A break passes beyond the r band and the optical colors of LRGs overlap with those of red stars, requires different selection techniques, taking advantage of available near-infrared imaging from space. The remainder of this section will focus on the strategy we will use in that domain.', '3.2.2 Selection Technique for z > 0 . 6 LRGs': "The spectral energy distributions of cool stars exhibit a local maximum around a wavelength of 1 . 6 µ m, corresponding to a local minimum in the opacity of H -ions [197]. This feature, commonly referred to as the '1.6 µ m bump', represents the global peak in the flux density ( f ν ) for stellar populations older than about 500 Myr [198], such as those in LRGs. In Figure 3.5 we plot an example LRG template spectrum from [199], illustrating both the strength of this peak and the depth of the 4000 ˚ A break. The lowest-wavelength channel in WISE , the W1 band centered at 3.4 µ m, is nearly optimal for selecting luminous red galaxies; it overlaps the bump at redshift near z = 1, so that higher-redshift LRGs will be bright in WISE photometry but comparatively faint in the optical. As may be seen in Figure 3.6, a simple cut in r -W 1 color can therefore select LRGs effectively, while adding in information on r -z color can help in rejecting non-LRGs. WISE data are particularly well suited for this application, as the survey depth was designed specifically for detection of L ∗ red-sequence galaxies to z = 1; LRGs are generally significantly brighter than this limit. In addition, we currently apply an i SDSS > 19 . 9 cut to emulate rejection of previous BOSS-like targets. \n-13 \nFigure 3.5: Atemplate spectrum based upon observations of the nearby elliptical galaxy NGC 4552, drawn from the work of [199]. The spectrum f ν is plotted as a function of rest-frame wavelength; we overplot the total (telescope + instrument + detector) response curves for DECam grz and WISE W 1 and W 2 imaging at the appropriate rest frame wavelengths for an LRG at z = 0 . 9. The 1.6 micron bump, the key spectral feature that enables our LRG selection method, corresponds to the peak in this spectrum. In the inset, we plot flux f λ over a limited wavelength range in order to illustrate clearly the 4000 ˚ A break and the abundance of spectral absorption features in this vicinity, which will be exploited by DESI to measure redshifts for LRGs. \n<!-- image --> \nFigure 3.6: An optical/near-infrared color-color diagram for galaxies observed by both DECam and WISE in the COSMOS field, where highly accurate 30-band photometric redshifts are available and used to label points the points shown. In this and subsequent figures, r indicates DECam r -band AB magnitude, z indicates DECam z AB , and W1 indicates WISE 3.4 µ m AB magnitude. Galaxies with LRG-like spectral energy distributions also having z > 0 . 6 are indicated by points color-coded according to their redshift, whereas small black points indicate blue galaxies at all redshifts. The dashed lines indicate the borders of our LRG selection box; our baseline sample assumes that objects above and to the right of these lines that also have magnitude z AB < 20 . 46 will be targeted by DESI as high-redshift LRGs. \n<!-- image --> \nFigure 3.7: Surface densities of targeted candidate z > 0 . 6 LRGs as a function of limiting z -band magnitude. We plot here the surface density of objects that lie within the target selection box shown in Figure 3.6 as a function of their z AB magnitude, as determined from DECam data in the 35 square degree DECaLS Early Data Release region. We also indicate our goal density of 350 targets per square degree via the magenta dashed line. Our baseline LRG sample size is attained at a depth of z AB < 20 . 46. At this limit, an average of roughly two spectroscopic measurements per LRG will be required to attain secure redshifts for > 98% of targets. \n<!-- image --> \nWe have tested selection techniques using optical grz catalogs derived from CFHT Legacy Survey [200], SDSS Stripe 82 data, or DECam grz imaging; NIR imaging from WISE ; and redshifts and rest-frame colors derived from DEEP2 spectra [201] or accurate 30-band COSMOS photometric [202] redshifts. A BOSS ancillary program has obtained roughly 10 , 000 redshifts of magnitude z SDSS < 20 LRG candidates selected using SDSS and WISE photometry with somewhat broader color cuts than DESI will likely use, which provide additional tests of our basic techniques.", '3.2.3 Sample Properties': 'The baseline LRG selection cuts for DESI are shown by the solid lines in Figure 3.6. This selection, applied to a sample with a total DECam z -band magnitude limit of z AB = 20 . 46, relies on optical photometry in the r and z bands and infrared photometry in the WISE W1 band. DESI target LRGs will often not be detected in the anticipated g band imaging, but are well above the depth limits in the r , z , and W 1 bands, having r < 23 and W 1 < 19 . 5. \nThis selection is already sufficient to meet all DESI design requirements, though we anticipate further improvements in the future. The major properties of this sample are:', '3.3.1 Overview of the sample': "Emission-line galaxies (ELGs) constitute the largest sample of objects that DESI will observe. The galaxies exhibit strong nebular emission lines originating in the ionized ('H II') regions surrounding short-lived but luminous, massive stars. ELGs are typically late-type spiral and irregular galaxies, although any galaxy actively forming new stars at a sufficiently high rate will qualify as an ELG. Because of their vigorous ongoing star formation, the integrated rest-frame colors of ELGs are dominated by massive stars, and hence will typically be bluer than galaxies with evolved stellar populations such as LRGs. The optical colors of ELGs at a given redshift will also span a larger range than LRGs due to the greater diversity of their star formation histories and dust properties. \nDESI leverages the fact that the cosmic star formation rate was roughly an order of magnitude higher at z ∼ 1 than today, which causes galaxies with strong line-emission to be very common at that epoch [204, 205, 206]. Figure 3.9 shows an example rest-frame spectrum of an ELG, which is characterized by a blue stellar continuum dominated by massive stars, a Balmer break at ∼ 3700 ˚ A (whose strength depends on the age of the stellar population), and numerous nebular emission lines, the most prominent of which are H α λ 6563, H β λ 4861, the higher-order Balmer lines, and the forbidden [O III] λλ 4959 , 5007 and [O II] λλ 3726 , 3729 nebular emission-line doublets. The inset provides a zoomed-in view of the [O II] doublet (assuming an intrinsic line-width of 70 km s -1 ), which the DESI instrument is designed to resolve over the full redshift range, 0 . 6 < z < 1 . 6. By resolving the [O II] doublet, DESI will avoid the ambiguity of lower-resolution spectroscopic observations, which cannot differentiate between this doublet and other single emission lines [207].", '3.3.2 Selection Technique for z > 0 . 6 ELGs': "The DESI/ELG targeting strategy builds upon the success of the DEEP2 galaxy redshift survey, which used cuts in optical color-color space to effectively isolate the population of z glyph[greaterorsimilar] 0 . 7 galaxies for follow-up high-resolution spectroscopy using the Keck/DEIMOS spectrograph [209, 201]. More recently, several SDSS-III/BOSS and SDSS-IV/eBOSS ancillary programs have confirmed that optical color-selection techniques can be used to optimally select bright ELGs at 0 . 6 < z < 1 . 7 [210, 211, 212]. \nFigure 3.9: Example rest-frame spectrum of an ELG showing the blue stellar continuum, the prominent Balmer break, and the numerous strong nebular emission lines. The inset shows a zoomedin view of the [O II] doublet, which DESI is designed to resolve over the full redshift range of interest, 0 . 6 < z < 1 . 6. The figure also shows the portion of the rest-frame spectrum the DECam grz optical filters would sample for such an object at redshift z = 1. \n<!-- image --> \nIn Figure 3.10 we plot the g -r vs r -z color-color diagram for those galaxies with both highly-secure spectroscopic redshifts and well-measured [O II] emission-line strengths from the DEEP2 survey of the Extended Groth Strip (EGS) [201]. The grz photometry of these objects is drawn from CFHTLS-Deep observations of this field [208], transformed and degraded to the anticipated depth of our DECam imaging (see § 3.6.1). As discussed in the next section, we expect to achieve a very high redshift success rate for ELGs with integrated [O II] emission-line strengths in excess of approximately 8 × 10 -17 erg s -1 cm -2 . This integrated [O II] flux corresponds to a limiting star-formation rate of approximately 1 . 5, 5, and 15 M glyph[circledot] yr -1 at z ∼ 0 . 6, 1, and 1 . 6, respectively, which lies below the 'knee' of the star formation rate function of galaxies at these redshifts [213, 214]. \nFigure 3.10 shows that strong [O II]-emitting galaxies at z > 0 . 6 (blue points) are wellisolated from the population of lower-redshift galaxies (pink diamonds) and the stellar locus (grey contours). The separation between galaxies above and below z glyph[similarequal] 0 . 6 occurs due to the spectrum blueward of the Balmer break ( λ rest ∼ 3700 ˚ A; cf. Figure 3.9) shifting into the r -band filter, which rapidly reddens the r -z color. Similarly, at z glyph[greaterorsimilar] 1 . 2 the Balmer break moves into the z -band filter, causing both the g -r and r -z colors to be relatively blue at higher redshifts. The black polygon in Figure 3.10 delineates the target selection box to isolate the population of strong [O II]-emitting ELGs at 0 . 6 < z < 1 . 6. By targeting galaxies in this box to a depth of r AB = 23 . 4, we strike a balance between maximizing the number of z ∼ 1 ELGs with significant [O II] flux while simultaneously minimizing contamination from stars and lower-redshift galaxies. ELGs galaxies with the very bluest colors are not included in the selection box, as their 'flat' spectra exhibit similar colors at all redshifts and are therefore difficult to select in our redshift range. \nFigure 3.10: Optical g -r vs. r -z color-color diagram based on spectroscopy from the DEEP2 Galaxy Redshift Survey, illustrating our preliminary selection for ELGs at z > 0 . 6 with significant [OII] emission-line flux. Although the galaxy photometry is based on deep CFHTLS imaging [208], the colors have been transformed and degraded to the expected depth of the DECaLS imaging. This plot shows that strong [O II]-emitting galaxies at z > 0 . 6 (blue points) are in general well-separated from both the population of lower-redshift galaxies (pink diamonds) and from the locus of stars in this color space (grey contours). The selection box (thick black polygon) selects those galaxies with strong [O II]-emission while minimizing contamination from stars and lower-redshift interlopers. \n<!-- image -->", '3.3.3 Sample Properties': 'The baseline ELG selection criteria for DESI are based on our analysis of the DEEP2/EGS survey data, which targeted galaxies more than half a magnitude fainter and with considerably higher spectroscopic signal-to-noise ratio than DESI. Because of this greater depth, we anticipate that any galaxies with sufficiently strong [O II] flux to yield a redshift with DESI also yielded a successful redshift measurement in DEEP2. We have also cross-verified our selection criteria and redshift distributions for ELGs using data from the 1 . 3 deg 2 COSMOS field [215] and from the 0 . 6 deg 2 VVDS-Deep field [216]; both of these samples give consistent results, within the expected variation due to both sample variance and systematic differences between the samples. Our selection, when applied to imaging with magnitude limits of g AB = 24, r AB = 23 . 4 and z AB = 22 . 5 (i.e., the anticipated depth of DECam Legacy imaging), is sufficient to meet all DESI science requirements (although we do anticipate to refine the sample selection even further). The major properties of this sample are as follows.', 'Large-Scale Structure Bias': 'We estimate the linear clustering bias of our sample of ELGs relative to their dark matter halos using the DEEP2 data. Employing methods similar to those of [217] and [218], we have measured the clustering of ELGs at quasilinear scales of 1 -10 h -1 Mpc in three overlapping redshift bins centered at z = 0 . 87, 1 . 0 and 1 . 2. The observed galaxy clustering is constant within errors at all redshifts, even as the amplitude of matter clustering increases at lower redshift [219]. The observations can thus be described by a galaxy bias which is inversely proportional to the growth factor of dark matter fluctuations. Based on our measurements we adopt b ( z ) = 0 . 84 /D ( z ), where D ( z ) is the growth factor at redshift z ( D ( z ) = 1 today). This increase in the bias with redshift for star-forming galaxies is consistent with other studies of similar objects at z =0.5-2.2 [220, 221, 222]. \nFigure 3.13: [OII] flux as a function of redshift for DEEP2/EGS galaxies. The light blue squares represent all galaxies in the sample, while the dark blue points are those objects targeted as DESI ELGs (see Figure 3.10). DESI will detect emission lines at 7 σ for the bulk of the targeted sample, corresponding to those objects above the 95% efficiency line in red. \n<!-- image -->', '3.4.1 Overview of the sample': 'The highest-redshift coverage of DESI will come from quasars (a.k.a. quasi-stellar objects, or QSOs), extremely luminous extragalactic sources associated with active galactic nuclei. QSOs are fueled by gravitational accretion onto supermassive black holes at the centers of these galaxies. The QSO emission can outshine that of the host galaxy by a large factor. Even in the nearest QSOs, the emitting regions are too small to be resolved, so QSOs will generally appear as point sources in images. These are the brightest population of astrophysical targets with a useful target density at redshifts z > 1 where the population peaks [223, 224]. \nDESI will use QSOs as point tracers of the matter clustering mostly at redshifts lower than 2.1, in addition to using QSOs at higher redshift as backlights for clustering in the Lyα forest. This enlarges the role of QSOs relative to the BOSS project, which only selected QSOs at z > 2 . 15 for use in the Lyα forest, and enhances their role relative to eBOSS where QSOs are used in a similar fashion as in DESI although with lower densities. DESI will select 170 QSOs per deg 2 over its footprint, of which 50 per deg 2 will be at z > 2 . 1 and suitable for the Lyα forest. \nDESI pilot programs, [224] updated in [225], have answered the long-standing uncertainties in the faint end of the QSO luminosity function. The surface density for z > 0 . 9 QSOs derived from these studies is shown in Figure 3.14, along with previous estimates from [226] (25% lower) or from the LSST science book [227, 228] (40% higher). Brighter than magnitude g = 23 . 0 ( r = 23 . 0 respectively), we infer that a complete QSO sample would contain about 185 (200, resp.) QSOs per deg 2 at z < 2 . 1 and about 75 (90, resp.) at z > 2 . 1. DESI will target and obtain redshifts for 120 and 50 QSOs per deg 2 in the redshift ranges z < 2 . 1 \nFigure 3.14: Cumulative surface density of quasars (objects per deg 2 ) as a function of g magnitude for z > 0 . 9, derived from different estimates of the QSO luminosity function. \n<!-- image -->', 'and z > 2 . 1, respectively.': 'Because of their point-like morphologies and with photometric characteristics that mimic faint blue stars in optical wavelengths (Figure 3.16, middle plot), QSO selection is challenging. The photometric selection used by BOSS to target Lyα QSOs at z > 2 . 15 has attained a 42% targeting efficiency (i.e., fraction of targets that prove to have the desired class and be in the desired redshift range), yielding 17 z > 2 . 15 QSOs per deg 2 down to the SDSS photometric limit of g < 22 . 1 [12]. The selection technique for DESI needs to achieve a minimum efficiency of about 65%; unlike for BOSS, however, QSOs at z < 2 . 15 are considered successes. A baseline scheme for QSO selection that achieves our goals for DESI is presented below.', '3.4.2 Selection Technique': "QSOs commonly exhibit hard spectra in the X-ray wavelength regime, bright Lyα emission in the rest-frame UV, and a power-law spectrum behaving as F ν ∝ ν α with α < 0 in the mid-infrared bands [229] (c.f. Figure 3.15). In the mid-optical colors, QSOs at most redshifts are not easily distinguished from the much more numerous stars. Successful selection of a highly-complete and pure QSO sample must make use of either UV or infrared photometry; DESI relies upon optical and infrared photometry for its baseline selection. \nThe QSO target selection is a combination of optical-only and optical+IR selections. The greatest separation from the stellar locus in the optical comes from ugr colors where the 'UV excess' in u -g produces bluer colors than those of most stars (Figure 3.16 left). In the absence of u band in the baseline imaging plan, the bulk of the QSO targets are identified in an optical+IR selection (Figure 3.16 right), where the excess infrared emission from QSOs results in a clear segregation from stars with similar optical fluxes. Stellar SEDs indeed sample the rapidly declining tail of the blackbody spectrum at those wavelengths, where QSOs have a much flatter SED. We defined a color selection to depths r = 23 . 0 with cuts in g -r vs. r -z and in r -Wvs. g -z shown in Figure 3.16, using DECaLS+ WISE photometry from the DR2 data release. We restrain the selection to objects with stellar morphology, to avoid an almost 10-fold contamination by galaxies that otherwise enter our selection region. \nFigure 3.15: QSO spectrum exhibiting the main emission lines used in their identification. \n<!-- image --> \nThe WISE data are available on the whole sky, and are photometered deeper than the public WISE catalogs using the Tractor-forced photometry (see section 3.8). Although WISE and optical data are not synchronous, the color difference between QSOs and stars is so large that QSO variability has minimal effect on the color selection. The WISE satellite has been reactivated, and will improve by a factor of two in signal-to-noise prior to DESI. \nThis baseline QSO target selection was tuned over the Stripe82 region where we led DESI pilot surveys (ancillary programs in BOSS and eBOSS, complemented by MMT observations) in order to build catalogs of spectroscopically identified QSOs at all redshift, which we use as truth tables. These pilot surveys selected highly complete samples of g < 23 or r < 23 QSOs from combined color and variability information (cf. section 3.4.5), using deep SDSS ugriz and WISE near-infrared data sets. Our baseline selection was then tested on an independent region of Stripe82. \n<!-- image --> \n<!-- image --> \nFigure 3.16: Colors in the optical ( ugrz ) or near-infrared (W is a linear combination of WISE W1 and W2 bands) of objects photometrically classified as stars (blue points) or spectroscopically classified as QSOs. Orange contours indicate the locus of tracer QSOs at z < 2 . 1 , red contours of Lyα QSOs at z > 2 . 1 , and red dots are for z > 3 . 5 QSOs. Left panel is based on SDSS photometry, middle and right panels on DECaLS-DR2. Black lines mark the boundaries of the selection regions described in the text. \n<!-- image --> \nWe also investigated an alternative algorithm based on a machine-learning algorithm called Random Forest. We trained it on all 47000 identified QSOs over the DECaLS-DR2 footprint, and used, for the star sample, a selection of 80000 unresolved objects in Stripe82, stripped of known QSOs and sources exhibiting QSO-like variations in their light curve. As for the previous selection, the algorithm relies solely on object colors and is restrained to unresolved sources with r < 23. It selects 97% of the known QSOs recovered by the more traditional color selection, but exhibits a better performance than the latter, in particular at redshifts above 2.1 or faint magnitudes. \nConsidering the completeness of the color cut or of the Random Forest approach as a function of redshift and magnitude, measured over truth regions, and applying it to the QSO luminosity function of [225], both selections result in over 170 QSOs per deg 2 , among which over 40 per deg 2 (55 per deg 2 for the Random Forest) are at z > 2 . 1. The non-QSO targets are stellar contaminants (about 80 per deg 2 in the color-cut selection, and 60 per deg 2 in the Random Forest selection). \nDESI may supplement its high-redshift QSOs with more sophisticated selection algorithms and other supplementary photometry as it becomes available. Time-domain data enable variability selection methods (as described in Section 3.4.5). UV ( u -band) data improve QSO selection, and allow discrimination between low-redshift and high-redshift QSOs. Algorithmically, neural-network based algorithms [230] and an extreme deconvolution method that models the distributions of stars and quasars at the flux limit [231] have been in use by BOSS where they allowed an increase of up to 20% in selection efficiency over traditional selection algorithms [232]. They are also applied, and thus further tested, in eBOSS. A combination of these additional data and algorithms will allow DESI to target QSOs in excess to those currently planned, with a small impact on the overall fiber budget. \nThe main contaminants to a grz + WISE QSO selection are very low-redshift star-forming galaxies with strong PAH emission, currently excluded using a star-galaxy separation based on ground-based optical imaging; a few high-redshift obscured galaxies, which are rare at bright optical magnitudes; and faint stars that artificially drift into the QSO locus because of poor optical photometry.", '3.4.3 Sample Properties': 'Two selections using optical grz and near-infrared data achieved a performance at the level of our goals for the DESI sample. Application of additional data and more sophisticated selection algorithms may be used to boost, in particular, the high-redshift QSO densities. To be conservative, we consider below the color-cut selection as the baseline DESI QSO selection. The major properties of the baseline DESI QSO sample are : \n- · Surface Density: The current grz + WISE color-box selection yields a total of 260 targets per deg 2 to a limit r = 23, of which about 140 per deg 2 are expected to be QSOs with z < 2 . 1 and about 40 per deg 2 are QSOs at z > 2 . 1, similar to the required densities of Table 3.1. Based on the QSO luminosity function of [225], this corresponds to about 60% of all QSOs in this magnitude range. The Random Forest selection increases this rate to 67%, with 55 z > 2 . 1 QSOs per deg 2 . We anticipate that the deeper WISE data expected before the start of DESI will allow us to further increase the completeness and decrease the stellar contamination.\n- · Redshift distribution: The expected redshift distribution of the QSO sample is illustrated in Figure 3.17 as the thick red histogram, which is determined by assuming the QSO \n30 \nFigure 3.17: Expected distribution of QSO redshifts from DESI (thick red histogram) using the targeting efficiency measured for the baseline DECaLS-DR2 selection over truth regions. For comparison, we also show the QSO luminosity function to r < 22 . 5 (blue dashed line) and r < 23 . 0 (red dotted line). \n<!-- image --> \ncompleteness for QSOs brighter than r < 23 measured in the truth region for the colorcut selection. For comparison, we show on the same plot the QSO luminosity function to r < 22 . 5 (blue dashed line) and r < 23 (red dotted line). \n- · Redshift measurement method: The key features contributing to the classification and redshifts of QSOs are the Lyα , CIV, CIII] and MgII emissions (c.f. Figure 3.15). From our experience with BOSS, eBOSS and MMT pilot programs, we estimate that in a single DESI visit we will fail to obtain redshifts for QSO targets about 10% of the time, mostly for objects at g > 22 . 5 [224, 225]. All QSO targets will be observed once early in the survey. Those confirmed to be QSOs at z > 2 . 1 will be re-observed in subsequent passes over the sky in order to obtain higher signal-to-noise spectra of the Lyα .\n- · Large-scale-structure bias: QSO bias has been measured in BOSS via QSO-Lyα crosscorrelation studies to be 3 . 6 at z = 2 . 4 [233], in agreement with previous measurements [234, 235]. For QSOs at lower redshifts, we project a bias of the form b ( z ) = 1 . 2 /D ( z ), where D ( z ) is the growth factor. At z > 2 . 1, clustering information is computed from the transmitted flux in the Lyα forest and not directly from correlations between objects; the flux bias of Lyα absorbers is estimated to be about -0.2 (it is negative because a larger matter density implies a higher absorption and thus a lesser transmitted flux) [236], and is strongly enhanced along the line of sight by redshift-space distortions.\n- · Target selection efficiency: From the first pass of targeting over the sky, we expect to identify 170 QSOs per deg 2 from a sample of 260 targets per deg 2 , for a target selection efficiency (including redshift failures) of 65%. For the subsequent passes, the target selection efficiency will be near 100%, as only objects identified as z > 2 . 1 QSOs will be re-observed. After four passes, the average target selection efficiency is therefore of order 80%.', '3.4.4 Recent and near-term developments for QSO target selection': 'During 2015, we focused on building large truth tables of QSOs against which to test current and improved selection algorithms. We developed comprehensive selections of quasars using the deep and multi-epoch SDSS photometry in the Southern Equatorial region called Stripe 82, where variability selections are notably efficient (cf. Sec. 3.4.5 and [237, 224]). These pilot programs led, in particular, to a sample of 18,000 spectroscopically-confirmed QSOs over 120 deg 2 to an extinction-corrected magnitude g c < 22 . 5, as well as to a smaller but deep sample of 175 deg -2 QSOs to g c < 23 over ∼ 10 deg 2 . They also allowed us to update the QSO luminosity function and make it more robust at faint magnitudes [225]. We are planning further dedicated programs to be run at MMT and AAT to extend the truth tables to r c < 23 as required for DESI. We also applied for a program at MMT to test the current target selection algorithms relying solely upon DECaLS+ WISE data, in a field where the WISE data already have the depth of the final 4-year survey, with the aim of providing the first validation the QSO selection for DESI. \nIn parallel, work has begun on machine-learning algorithms to take better advantage of the imaging data available for DESI. In BOSS, the XDQSO algorithm [231] led to nearly 20% improvement compared to color cuts and we can reasonably expect a similar increase in the yield of the QSO selection for DESI by the Random Forest algorithm that we are focusing on. These developments have started with DECaLS optical data and existing WISE infrared data. They will be iterated as additional depth is acquired on WISE.', '3.4.5 Variability Data Improves Selection of High-Redshift QSOs': "Time-domain photometric measurements can enhance QSO selection. They allow us to exploit the intrinsic variability of QSOs [238] to distinguish them from stars of similar colors. They therefore complement the color-selection techniques presented in Sec. 3.4.2. We have so far used variability information extensively to build truth tables against which to test QSO selection. In a second step, we will use variability to select additional high-redshift Lyα QSOs, for which uniformity of selection across the sky is not required. \nBecause the accretion region around a quasar is highly compact, its luminosity can vary substantially on timescales ranging from days to years, with a pattern distinct from that seen in variable stars. The time variability of astronomical sources can be described using a measure of the amplitude of the observed magnitude variability ∆ m as a function of the time delay ∆ t between two observations. This 'structure function' is modeled as a power law parameterized in terms of A , the mean variation amplitude on a one-year time scale (in the observer's reference frame) and γ , the logarithmic slope of the variation amplitude with respect to time: ∆ m = A (∆ t ) γ . \nWe have tested variability techniques in DESI pilot surveys, both in Stripe 82 [237] that was the subject of repeated SDSS observations totaling about 50 epochs, and elsewhere on the sky, where time-domain information was derived from 5-10 epochs of PTF R -band data. As illustrated in Figure 3.18, the segregation between QSOs and stars is much reduced with poorer data, but variability remains competitive. This technique allowed us to identify 30% more QSOs in the Stripe 82 field than with time-averaged optical photometry only [237], and a combined color and variability selection from CFHT and PTF imaging data in the CFHTLS D3 field allowed us to achieve a record-high surface density of 207 QSOs per deg 2 to g = 23. The gain relative to the baseline QSO targeting with full WISE depth is likely \n<!-- image --> \nFigure 3.18: Left panel: Structure function parameters for 50-epoch gri light curves from SDSS in Stripe 82 (left), where the parameters are amplitude (A) and time duration exponent (Γ). Right panel: Structure function parameters for the 6-epoch R light curves from PTF, where the discriminating power is diminished but still valuable with fewer epochs and filters. \n<!-- image --> \nto be less dramatic, and will be evaluated as those data become available. Even with only four epochs of WISE data (two stacks per year, two years of observations), preliminary tests indicate that variability information from WISE can be used to reduce the contamination of the target sample by about 10%. The full 4-year survey WISE will allow 8-epoch light curves, and further gain over current estimates. \nImaging surveys that could provide useful time-domain information for variability selection include the PTF and follow-on iPTF surveys (in the deepest areas of their footprint), the DES survey or the WISE survey. Variability is not assumed in our baseline targeting plan, but it is expected to be valuable for selecting the Lyα QSOs wherever coverage exists.", '3.5 Calibration Targets': "Target selection is also responsible for providing lists of standard stars for flux calibration, and lists of blank sky locations to be used for modeling the sky. \nMain-sequence F stars will be used as the primary spectrophotometric standard stars. These stars are well-described by stellar atmosphere models, making them ideal targets for spectrophotometric calibration at optical wavelengths. A stellar template of appropriate temperature, surface gravity and metallicity will be derived for each star and used to derive the spectral response including the time-varying atmospheric absorption bands. \nThe selection will be similar to the color-magnitude selection of BOSS to identify lowmetallicity targets through a selection in ( u -g ),( g -r ), ( r -i ), and ( i -z ) colors. The restrictive BOSS selection yields 10 stars per deg 2 ; to obtain a larger number of potential targets using the new grz photometry, DESI will broaden this selection and include higher metallicity standard stars. With Gaia spectrophotometry of F stars that span a range of metallicity, and upcoming data from SDSS-IV/eBOSS in which a broader selection is applied, we plan to evaluate the value of a mix of lower and higher metallicity F stars to serve as flux calibration standards for DESI. Finally, we will perform a cross-calibration of low-metallicity and higher metallicity F-stars during the commissioning stages of DESI, thus providing validation of the standard star selection. \nBlank sky locations will be determined as part of the object detection algorithms applied \nFigure 3.19: The primary imaging surveys that will result in targeting data for the DESI project. The footprint at DEC ≤ +34 · will be covered using the Dark Energy Camera (DECam) on the Blanco 4m telescope at Cerro Tololo Inter-American Observatory. The Dark Energy Camera Legacy Survey (DECaLS, in yellow), the Dark Energy Survey (DES, in orange), and the extended DECaLS in the North Galactic Cap (DECaLS+, in purple on left) are underway. A proposal for the remaining extended DECaLS in the South Galactic Cap (DECaLS+, in purple on right) will be submitted. Imaging of the North Galactic Cap region at DEC ≥ +34 · (cyan) will be covered with the 90Prime camera at the Bok 2.3-m telescope in g -and r -bands (BASS: the Beijing-Arizona Sky Survey) and with the upgraded MOSAIC-3 camera on the Mayall 4m telescope in z -band (MzLS: the MOSAIC z-band Legacy Survey). Both the Bok and Mayall telescopes are located on Kitt Peak National Observatory. \n<!-- image --> \nto the input imaging, ensuring that there are no detectable sources within the fiber diameter in any of the input bands. These will be provided at a density such that every fiber (when possible) will have the option of a blank sky if it isn't otherwise assigned to a science target.", '3.6 Baseline Imaging Datasets': 'The samples described above can be selected given highly-uniform optical imaging data in the g , r , and z bands, as well as all-sky imaging from the WISE satellite. The same imaging data for selected science targets will be used to identify calibration targets (standard stars and sky fibers). A combination of three telescopes will be used to provide the baseline targeting data for DESI: the Blanco 4m telescope at Cerro Tololo, the Bok 90-inch and the Mayall 4m telescope at Kitt Peak. The footprints of the primary surveys using these telescopes that will deliver the targeting data are shown in Figure 3.19 and the next three subsections discuss these surveys and their current status in more detail. The status of the WISE data is presented in § 3.6.4.', '3.6.1 Blanco/DECam Surveys (DEC ≤ 34 · )': "The Dark Energy Camera (DECam) on the Blanco 4m telescope, located at the Cerro Tololo Inter-American Observatory, will provide the optical imaging for targeting over 2/3 of the \nDESI footprint, covering both the North and South Galactic Cap regions at Dec ≤ 34 · . Due to the combination of large field of view and high sensitivity from 400-1000 nm, DECam is the most efficient option for obtaining photometry in the g , r , and z bands. \nDECam can reach the required depths for DESI targets in modest total exposure times of 100, 100 and 200 sec in g , r , z in median conditions. These data reach required 5 σ depths of g =24.0, r =23.4 and z =22.5 for an ELG galaxy with half-light radius of 0.45 arcsec. For a 3-dither observing strategy, accounting for weather loss, DECam is capable of imaging 9000 deg 2 of the DESI footprint to this depth in 81 scheduled nights. These depth estimates have been vetted with grz photometry in the COSMOS field in Spring 2013 (Section 3.3.1). \nA public survey, 'The DECam Legacy Survey of the SDSS Equatorial Sky' (DESI collaborators D. Schlegel and A. Dey are PIs), has been approved to obtain optical imaging to the required depth over 6200 deg 2 . This 'DECaLS' survey has been allocated 64 nights spread out over 3 years (2014A to 2017B semesters) as part of the NOAO Large Surveys program. The survey began in August 2014 and has thus far had 30 scheduled nights and 6 Director's discretionary nights (near full moon), during which 23% of the g + r and 49% of the z imaging has been completed. The current coverage is shown in Figure 3.20. \nThe DECaLS program is making use of other DECam data within the DESI footprint as those data become public. The most significant of these other data sets is from the Dark Energy Survey, which includes a 500 deg 2 contiguous area in the South Galactic Cap. \nFigure 3.20: Left panel: Coverage map of the DECaLS survey through February 2016. The coverage in the g , r and z filters is indicated by the color as blue ( g -only), yellow ( r -only), green ( g + r ), purple ( g + z ), orange ( r + z ) or black ( g + r + z ). Each panel represents one of the 3 passes, where pass 1 is observed in the best weather conditions. \n<!-- image --> \nDECaLS is explicitly not re-imaging that area, and making use of those raw data as the proprietary period expires 12 months after the date of observation. \nDECaLS will cover ≈ 2/3 of the planned DESI footprint at Dec ≤ 34 · . The DECaLS team successfully applied for an 8-night extension (DECaLS+) that will obtain imaging for the remaining 800 deg 2 in the North Galactic Cap. An additional 500 deg 2 in the South Galactic Cap is being observed by the Dark Energy Survey, with those raw data publicly available 12 months after the date of observation. A proposal to observer the remainder of the DESI footprint in the South Galactic Sky will be submitted in future semesters. Data from these programs are treated the same and reprocessed uniformly to ensure consistency for DESI target selection. \nThe DECam data have been reduced to calibrated images at NOAO and catalogs constructed using the Tractor algorithm (see § 3.8). These catalogs have been used for the DESI target selection tests described elsewhere in this chapter.", '3.6.2 Bok/90Prime Survey (DEC ≥ 34 · )': "The NGC footprint at Dec ≥ +34 deg will be observed by the Bok 2.3-m telescope in two optical bands ( g and r ) for DESI targeting. The Bok Telescope, owned and operated by the University of Arizona, is located on Kitt Peak, adjacent to the Mayall Telescope. The 90Prime instrument is a prime focus 8k × 8k CCD imager, with four University of Arizona ITL 4k × 4k CCDs that have been thinned and UV optimized with peak QE of 95% at 4000 ˚ A[239]. These CCDs were installed in 2009 and have been operating routinely since then. 90Prime delivers a 1.12 deg field of view, with 0.45 '' pixels, and 94% filling factor. Typical delivered image quality at the telescope is 1.5 '' . The g and r -band survey over 5000 deg 2 is projected to require 180 nights of scheduled telescope time for average weather. The throughput and performance in these bands were demonstrated with data in September 2013. \nThe BASS survey tiles the sky in three passes, similar to the DECaLS survey strategy. At least one of these passes will be observed in photometric conditions (P1) and seeing conditions better than 1.7 arcsec. \nThe Bok survey (known as the Beijing-Arizona Sky Survey; Zhou Xu and Xiaohui Fan, PIs; see http://batc.bao.ac.cn/BASS ) was awarded 56 nights in Spring 2015 and 100 nights in each of Spring 2016 and 2017. The Bok survey will target 5500 deg 2 in the NGC, including 500 deg 2 of overlap with the region covered by the DECam surveys in order to understand and correct for any systematic biases in the target selection. The existing Bok g -band filter is well-matched to the DECam g -band filter. The existing Bok r -band filter had a significantly different bandpass as compared to the DECam r -band filter, therefore we acquired a new r -band filter from Asahi that was delivered in April 2015. \nThe BASS survey began observations in Spring 2015. A number of instrument control software updates, new flexure maps, and new observing tools were implemented that greatly improve the pointing accuracy, focusing of the telescope, and observing efficiency. 15% of the g -band and 2% of the r -band tiles were observed in that semester. It was discovered that those data suffered from defective electronics in the read-out system that introduced A/D errors, gain variations and non-linearities. Those electronics were replaced in September 2015 followed by a recommissioning of the system in Fall 2015. \nBASS has been scheduled for the 100 darkest nights in the 2016A semester (JanuaryJune), and expects to schedule a comparable number of nights in 2017A. Through February 17, 2016, the survey has completed 10, 10 and 0% of the pass 1, 2, 3 tiles in g -band, and \n14, 13, 5% of the tiles in r -band (see Figure 3.21). The raw and calibrated images will be publicly served through the NOAO Science Archive. These data will be included in the Legacy Survey catalogs beginning with Data Release 3 in 2016.", '3.6.3 Mayall/MOSAIC Survey (DEC ≥ 34 · )': 'The Mayall z -band Legacy Survey (MzLS) will image the DEC ≥ +34 · region of the DESI North Galactic footprint. It will use the MOSAIC-3 camera at the prime focus of the 4meter Mayall telescope at Kitt Peak National Observatory. MzLS will be scheduled for 230 nights during semesters 2016A and 2017A through an agreement between the National Science Foundation and the Department of Energy. 116 of these nights have been scheduled in the 2016A semester, with a survey start on February 2, 2016. The imaging camera has undergone a major upgrade in 2015 to improve its z -band efficiency. The KPNO 4m telescope control system and the imaging camera software have been upgraded for improved operational efficiency. NOAO has purchased a new z -band filter to match the DECam filter bandpass and to thereby minimize any differences between the DECam and MOSAIC z surveys. \nThe MOSAIC-3 camera is a new version of the prime focus imaging system. This upgrade has made use of the dewar from the MOSAIC-2 camera at CTIO and the MOSAIC-1.1 mechanical system and guider from KPNO. Yale University designed and built a new cold plate for the dewar which it populated with four super-thick (00 µ m-thick) fully-depleted 4096 2 pixel CCDs with the same 15-micron pitch. The readout system consists of four DESI controllers, one for each CCD that simultaneously reads the four quadrants of each device. These controllers were modified to synchronize to a single clock. The dewar was delivered to NOAO in September 2015 where it was integrated with the MOSAIC-1.1 mechanical enclosure, shutter, filter wheel and acquisition and guider system. This upgraded camera, christened MOSAIC-3, saw first light in October 2015 and underwent further on-sky commissioning runs in November and December 2015. The z -band efficiency has been measured to be improved by 60% as compared to the MOSAIC-1.1 camera. \nThe MzLS survey tiles the sky in three passes, similar to the DECaLS survey strategy. At least one of these passes will be observed in photometric conditions (P1) and seeing conditions better than 1.3 arcsec. Through March 8, 2016, the survey has completed 23, 19 and 8% of its pass 1, 2 and 3 tiles (see Figure 3.21). \nThe The MOSAIC z -band survey project will be run similarly to the DECaLS survey, with the initial processing being done using the NOAO pipeline and calibration and catalog construction being carried out at LBNL/NERSC. The raw and pipeline-processed images are public as they are available, typically at the end of each lunar cycle, through the NOAO Science Archive. These data will be included in the Legacy Survey catalogs beginning with Data Release 3 in 2016.', '3.6.4 WISE All-Sky Survey': 'Infrared imaging from the Wide-field Infrared Survey Explorer ( WISE ) satellite are critical to the DESI targeting algorithm for LRGs and QSOs. During its primary 7-month mission from through August 2010, WISE conducted an all-sky survey in four bands centered at 3.4, 4.6, 12 and 22 µ m (known as W1, W2, W3 and W4) [240] 99.99% of the sky was imaged at least 8 times, while regions near the ecliptic poles were observed more than 100 \n<!-- image --> \nFigure 3.21: Left panel: Coverage map of the Bok/BASS survey based on data collected though March 6, 2016, and excluding data prior to the electronics fixes in September 2015. The coverage in the g and r filters is indicated by the color as blue ( g -only), yellow ( r -only) or green ( g + r ). Each panel represents one of the 3 passes, where pass 1 is observed in the best weather conditions. Right panel: Coverage map of the MzLS z -band survey based on data collected though March 6, 2016. The coverage is indicated in each of the three passes. \n<!-- image --> \ntimes. Following the primary 4-band mission, WISE continued survey operations in the three shortest bands for 2 months, then the two shortest bands for an additional 4 months for a total of a 13-month mission that completed in September 2011. NASA re-activated the satellite in Fall 2013 and is continuing two-band survey observations for an additional 3 years starting December 1, 2013, as the NEOWISE project. The first NEOWISE data release occurred on March 25, 2015, the second release will be March 23, 2016, and the final release will be in March 2017. \nDESI target selection utilizes the two shortest-wavelength bands at 3.4 (W1) and 4.6 µ m (W2). Photometry in these bands is measured using the the Tractor algorithm (see Section 3.8) measured on the re-stacked WISE and NEOWISE Level 1 imaging that retains the intrinsic resolution of the data and are appropriate for preserving the available signalto-noise [241]. Data Release 1 of the Legacy Survey (DECaLS and WISE ) made use of the initial 13-month data set, reaching 5σ limiting magnitudes of 20.0 and 19.3 AB mag in W1 and W2. Data Release 2 made use of approximately twice as much WISE data with the first year of NEOWISE . The final Legacy Survey catalogs will use the full WISE and NEOWISE data sets, reaching 0.7 mag fainter than the Legacy Survey Data Release 1 or the WISE All-Sky Data Release.', '3.7 Additional Imaging Data': 'Additional imaging data, if available, can supplement the target selection data and may be used, in particular, to improve the selection of the high-redshift Lyα forest QSO sample. This is because the Lyα forest analysis is based on the clustering of absorption systems along the line of sight, and therefore does not require a spatially uniform QSO sample. As a result, the QSO target selection can utilize datasets that may not be uniform (in depth, bandpass, or time sampling) over the DESI footprint. In this section, we summarize the key datasets that may contribute to this effort, if they prove to be available. These data sets are not assumed to be available for our baseline target selection plans, but rather should improve the efficiency of targeting higher-redshift ( z > 2 . 1) QSOs beyond the baseline targeting strategy presented above.', '3.7.1 SDSS': 'The Sloan Digital Sky Survey [242] has obtained multi-band ( ugriz ) photometry (in photometric conditions) over a 10,000 deg 2 extragalactic footprint in the North Galactic and South Galactic Caps. The Northern Cap and four stripes in the Southern Cap were imaged in 1998-2004. The bulk of the Southern Cap was imaged in 2008-2009, and the SDSS camera was then retired from service in December 2009. The median 5 σ magnitude depths for the SDSS ugriz bands are 22.15, 23.13, 22.70, 22.20, and 20.71, respectively, but with substantial variation in depth from seeing. SDSS may provide a reference photometric point for variability selection of high-redshift QSOs, allowing variability over long time baselines to be measured.', '3.7.2 PanSTARRS-1': 'The PanSTARRS-1 (PS1) 3 π survey [243] is a transient-sensitive survey designed to observe 30,000 deg 2 of sky over 12 epochs in each of the five grizy filters. The multi-band photometry generated from the co-added exposures reaches depths that are comparable to SDSS in gr and potentially deeper in iz . These depths would potentially be adequate for the DESI BGS and LRG samples, but not the ELG or QSO samples. The PS1 survey completed observations in 2013. The PS1 time-domain photometry may be useful for enhancing the selection of Lyα QSOs at the brighter magnitudes. The DECaLS survey is currently using a bright star catalog from PS1 to provide initial photometric and astrometric calibration across its footprint. The PS1 co-added imaging and catalogs are not available as of March 2016.', '3.7.3 PTF, iPTF, and ZTF': 'The Palomar Transient Factory (PTF) [244] was a photometric survey designed to find transients via repeated imaging over 20,000 deg 2 in the Northern Hemisphere. In February 2013, the next phase of the program, iPTF (intermediate PTF) began. Both have used the CFH12K camera on the 1.2 m Oschin Telescope at Palomar Observatory, which covers 7.2 deg 2 of sky in a single pointing with a pixel scale of 1.01 arcsec. \nFour years of survey operations have so far yielded a total of 5,000 deg 2 in R -band and 1,000 deg 2 in g -band to useful depths for QSO selection based on variability. LBNL is a partner in the PTF and iPTF collaborations, and DESI has access to these data. \nAn upgraded Zwicky Transient Factory (ZTF) has been funded through an NSF Mid-Scale Innovations Program in Astronomical Sciences. ZTF will utilize the same telescope with a new 46 square-degree imager, beginning operations in 2017. The ZTF survey will cover the entire sky at declinations Dec > -20 deg, including the full DESI footprint. ZTF will operate with a g -band similar to the DECam and Bok g -band, an R -band (MouldR ) that is broader, and potentially an i -band. These data, which will be available to DESI collaboration for the purposes of target selection, are expected to eventually achieve the DESI targeting depths in g and R bands, but likely not before the start of DESI spectroscopic operations. The time sampling of ZTF is planned to be highly non-uniform over the DESI footprint, with different areas of sky covered in different years. Therefore, ZTF is not viable for the baseline DESI target selection, but PTF, iPTF and ZTF may be used to supplement the high-redshift QSO selection for DESI.', '3.7.4 CFHT': "The Canada-France-Hawaii Telescope (CFHT) is a 3.6-m meter telescope on Mauna Kea, Hawaii. CFHT is a joint facility of the National Research Council of Canada, the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The CFHT prime focus imager MegaCam, a very efficient instrument for imaging large areas of sky, consists of 36 2k × 4k e2v CCDs, covering a field of view of 0.97 deg 2 with a pixel scale of 0.185 arcsec per pixel. MegaCam started operations in 2003 and has conducted a number of large imaging surveys, the largest being the CFHT Legacy Survey covering 155 deg 2 . \nThe CFHT community is in discussions with the Euclid consortium and may play a role in providing ugri imaging data over the northern Euclid footprint. However, no plan is currently in place. There is an ongoing u -band survey ('CFHT-Luau: The CFHT Legacy Survey for the u-band all-sky universe'; A. McConnachie and R. Ibata, PIs) aimed at providing imaging over 4000 deg 2 of the high-Galactic-latitude northern sky, approximately split between the North and South Galactic caps. CFHT-Luau will complete in the 2016B semester (with data becoming public 1 year after observation). ( u -g ) color selection is an efficient discriminator between low-redshift and high-redshift QSOs. Hence, the CFHT data may be used to supplement the high-redshift Lyα forest QSO selection in DESI, especially in combination with variability data.", '3.7.5 SCUSS': 'The South Galactic Cap U-band Sky Survey [245] is a survey of 4000 deg 2 in the South Galactic Cap using the 90Prime instrument on the Bok 2.3-m telescope. The survey was a joint project among the Chinese Academy of Sciences, its National Astronomical Observatories unit, and Steward Observatory 5 . The survey was conducted between September 2010 and October 2014 with typical exposure times of 5 minutes per field. The limiting magnitude reached by the data is u ∼ 23 mag (5 σ point source), with some variation due to varying seeing conditions. These data may be used to supplement the high-redshift Lyα forest QSO selection in DESI, especially in combination with variability data.', '3.8 The Tractor Photometry for Target Selection': "The DESI target selection combines photometry from optical imaging and from WISE . DESI Imaging Scientist Dustin Lang has developed the Tractor forward-modeling approach to perform source extraction on pixel-level data [246]. 6 This is a statistically rigorous approach to fitting the differing PSF and pixel sampling of these data, which is particularly important as the optical data have a typical PSF of ≈ 1 arcsec and the WISE PSF is ≈ 6 arcsec. \nThe Tractor takes as input the individual images from multiple exposures in multiple bands, with different seeing in each. A simultaneous fit is performed for sources to the pixel-level data of all images. Thus, if a source is determined to be a point source, it is photometered as a point source in every band and every exposure. If it is found to be a morphologically extended source, then the same light profile is consistently fit in all images. This produces object fluxes and colors that are consistently-measured across the wide-area imaging surveys input to DESI target selection \nFor bright objects that were cleanly detected by WISE alone, we find our pixel-level measurements to be consistent with catalog-level measurements (see Figure 3.22). However, we are also able to measure the fluxes of significantly fainter objects, as well as to study collections of objects that are blended in the WISE imaging but that are resolved in the optical images. Figure 3.23 compares a traditional optical-infrared color-magnitude diagram, based on matching sources between catalogs at different wavelengths, to the results of our WISE forced photometry, which requires no such matching. This demonstrates how The Tractor increases the color-space information available to DESI targeting. \nIn general, The Tractor improves target selection for all DESI classes by allowing information from low signal-to-noise measurements to be utilized. The Tractor is particularly important for QSO targeting. Up to 15% of QSO spectra exhibit broad absorption lines that potentially reduce the measured flux in broadband imaging. High-redshift (Lyα ) QSOs will drop out of some imaging bands completely. Finally, the 5 σ optical limit at the extremes of DESI targeting corresponds to a < 5 σ limit in WISE for QSOs (c.f. Sec. 3.4.2). The Tractor successfully differentiates between the QSOs that are detected in WISE , and the QSOs that in general are not detected (c.f. Figure 3.16), whereas traditional 'catalog-matching' approaches would not be successful. \nTarget selection of LRGs and QSOs for the SDSS-IV/eBOSS, which began observations in July 2014, utilized The Tractor . For eBOSS targets, the Tractor was applied to obtain forced photometry based upon galaxy profiles measured by the SDSS imaging pipeline. Those profiles were convolved with the WISE point-spread function, and then a linear fit was performed on the full set of WISE imaging data. The result was a set of flux estimates for all SDSS objects, constructed so that the sum of flux-weighted profiles best matched the WISE images. DESI will make use of this same fitting approach, using optical images from surveys being conducted with the DECam, Bok and Mayall telescopes (c.f. Sec. 3.6) in place of the SDSS images. \nThe Tractor has already been applied to the DECam survey imaging that will be used as part of DESI target selection. This survey, which is known as DECaLS, attained its second release (DR2) of imaging early in 2016. Tractor catalogs based on this DR2 data are publicly available 7 . DECaLS DR2 comprises all grz imaging conducted with DECam \nFigure 3.22: Forced photometry results from the Tractor code, using information from SDSS detections and light profiles to measure the flux from objects in the WISE images to below the WISE detection limit. Left panel: The results agree for bright objects that are detected in the WISE catalog. The widening locus below W1 ∼ 14 is due to our photometry treating larger objects as truly extended, in contrast to the point-source-only assumptions in the public WISE catalog. Right panel: A demonstration of the increased depth made possible from using the Tractor . By using optical imaging from SDSS to detect objects, photometry is measured for objects that are well below the WISE detection limit. \n<!-- image --> \nFigure 3.23: Forced photometry results from the Tractor code, contrasted with traditional 'catalog-matching'. Left: Color-magnitude diagram from matching SDSS to WISE catalogs. Many objects below the WISE catalog detection limits are lost. Right: Results from forced photometry of the WISE images based on SDSS detections. No matching is required, and objects that would be detected in WISE at only few-sigma significance can readily provide flux measurements. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3.24: An example 'brick' covering 0 . 25 × 0 . 25 deg 2 from the DECaLS survey. From left to right, the panels show the actual grz imaging data, the rendered model based on the Tractor catalog of the region, and the residual map. The Tractor catalog represents an inference-based model of the sky that best fits the observed data. \n<!-- image --> \nprior to June 2015 that lies within the DESI footprint. This includes both imaging conducted specifically for DECaLS and public raw imaging re-extracted using the Tractor . The co-added images and Tractor catalogs are presented in 'bricks' of approximate size 0.25 · × 0 . 25 · (see Figure 3.24) and DECaLS DR2 contains approximately 260 million unique sources spread over 97,554 bricks. \nIn total, DECaLS DR2 contains about 2000 deg 2 of imaging in both g - and r -band and roughly 5300 deg 2 in z -band only. 1800 deg 2 has been observed in all three optical filters. DECaLS is on schedule to observe its projected 6200 deg 2 of imaging over 3 years (c.f. Sec. 3.6.1). Based on formal errors from The Tractor , the median 5 σ point source depths for areas in DECaLS DR2 with full coverage in each band are g = 24 . 65, r = 23 . 61, z = 22 . 84, meeting the depth requirements for DESI target selection. WISE fluxes based on forced photometry using the Tractor are available for all sources extracted as part of DECaLS DR2. \nCatalogs generated by the Tractor will be vetted for DESI target selection using a series of image validation tests. Catalogs of galaxies are expected to be generated in a manner that is model-independent across all bands and that should achieve a 5 σ , extinction-corrected depth of g =24.0, r =23.4 and z =22.5. 90% of the DESI footprint requires full-depth imaging, but 95% (98%) must be within 0.3 (0.6) magnitudes of full-depth. The photometric system produced by the Tractor must be uniform and stable, with < 1% systematic errors (RMS) in g - and r -band, < 2% in z -band, and < 2% from morphological mis-classifications. The z -band image quality must exceed 1.3 '' in at least one pass everywhere in the DESI footprint. The systematic and random errors in astrometry must be less than 30 mas and 90 mas RMS, respectively. In order to facilitate these imaging tests, which are ongoing, The Tractor catalogs will ultimately include source positions, fluxes, shape parameters, and morphological quantities that can be used to discriminate extended sources from point-sources, together with errors on these quantities.", '4.1 Introduction': 'The DESI instrument will make largest spectroscopic survey to date. The design of the survey is optimized by selecting a footprint that is as large as possible from the Mayall telescope while staying clear of the Milky Way. The survey strategy will establish the order in which the observations will be made. The strategy will be modified in detail by atmospheric conditions, but the overall plan will be established to optimize the best science results for both the complete survey and results from intermediate years.', '4.2 Survey Footprint': "The DESI survey footprint is defined to be 14,000 square degrees that can be observed spectroscopically from Kitt Peak. This footprint will be one contiguous region selected from the North Galactic Cap (NGC) and one contiguous region in the South Galactic Cap (SGC). The instrumented area of the focal plane is 7.50 square degrees. 14,000 square degrees can be covered nearly completely with little overlap using 2,000 tiles, where each tile represents one DESI observation. We refer to the full 2000-tile coverage of the footprint as a 'layer'. Five layers with altogether 10,000 tiles covers each coordinate of the footprint with an average of 5.24 fibers. The DESI footprint is formally defined as any position on the sky within 1.605 deg of any of these selected tile centers. \nThe DESI spectroscopic survey will primarily select targets from catalogs derived from imaging with the Blanco/DECam camera, the Bok/90Prime camera, the Mayall/MOSAIC2 camera and the Wide-field Infrared Survey Explorer ( WISE ). Although WISE imaging covers the entire sky, the imaging from DECam, the Bok Telescope, and the Mayall telescope impose an external constraint on the DESI footprint, as targets must be selected from large contiguous regions imaged with the same instruments. The Bok and Mayall will provide targeting in the NGC at Dec > +30 deg. The Blanco will provide targeting in both the NGC and SGC at Dec < +30 deg. An area of approximately 800 sq. deg. in the SGC at Dec > +30 (and -32 < b < -15) is 'orphaned' and excluded from the DESI survey as it would be a small area observed with a different camera. \nThe footprint is constrained, as well, by the need to avoid regions that would require long exposures due to airmass or dust, by weather patterns at Kitt Peak, and by regions of high stellar density. The resulting footprint is shown in Figure 4.1.", '4.3 Field Centers': "We refer to 'tiling' as the process by which field centers are assigned in a manner to cover the footprint with optimal coverage of each coordinate on the sky. The single-layer tiling of the sky mentioned in Section 4.2 is a preliminary solution that is achieved using the icosahedral tiling [247] with 5762 tile centers distributed on the full sphere 8 . This tiling is very-well matched to the DESI focal plane size. The first layer rotates the above tiling solution by 90 deg in RA. This rotation conveniently puts rows of tile centers along lines of approximately constant declination at the north and south boundaries of the DESI survey. Each of additional layers 2 through 5 have an additional rotation of the tile centers by 1.08 \nFigure 4.1: Tile centers for the DESI footprint in an equal-area projection. Declination limits are imposed at -8 . 2 < Dec in the NGC (left), and -18 . 4 < Dec < +30 in the SGC (right). Approximately 1% of tiles have exposure factors larger than 2.5 (shown in blue), but are included to avoid unwanted holes in the footprint. The five layers are shown in separate (but overlapping) colors. The spots indicate the centers of focal plane positions and overlap between layers. The symbols do not represent the size of the area in the sky subtended by the focal plane. Every location inside the footprint is within reach of a fiber, on average, 5.24 times during the survey. \n<!-- image --> \ndeg in RA. This gives large dithers on most of the sky (except at the pole, which is not in the DESI footprint), thus filling the gaps in the focal plane with subsequent visits. Nonuniformity in coverage could artificially introduce structure in the targeting of LSS-tracers; alternative tilings based on the same first layer but with subsequent layers obtained with more disparate rotations will be further studied for possible improvements to the uniformity. \nA descoped instrument has been considered which would conduct the DESI survey over 9000 square degrees rather than 14,000 square degrees. This descoped instrument would populate only six of the 10 wedges on the focal plane with 3000 instead of 5000 fibers. The populated wedges are best arranged in a 'Pacman' format. A different tiling solution is necessary for covering a smaller survey area with the same mean coverage per survey area. First, 240 tile centers are placed on the celestial equator uniformly separated in RA. Stripes of tiles are then placed on lines of constant celestial latitude spaced every 2.765 deg. At each stripe, the number of tiles is reduced by the factor cos(Dec) from the 240 placed on the celestial equator. This results in a tiling solution with similar uniformity and coverage statistics as the baseline survey, with 4% more tiles than would be necessary under the assumption of a simple scaling with focal plane area. \nThe pattern of fiber positioners in the focal plane is shown in Figure 4.2. Combining \nthis with the tiling gives a purely geometric measure of the coverage for each position within the DESI footprint. The distribution of this coverage is shown in Figure 4.3 and in Table 4.1. The average coverage is about 5.1 fibers available per coordinate, with only 3.5% of the footprint having a coverage of less than 3 fibers. The mean relative to the value of 5.24 reported earlier is slightly reduced due to increase of edge effects over the smaller area tested. The edges of the footprint have the least coverage. The results of a similar study for the reduced 'Pacman' focal plane are shown in the right hand panel of Figure 4.3. \n<!-- image --> \nFigure 4.2: Left: Fiber positioner locations for the full DESI instrument. Right: The locations for the reduced instrument 'pacman' configuration. 'Missing' positioner locations are for the guidefocus arrays (square regions) and fiducial markers for the fiber view camera. \n<!-- image --> \nFigure 4.3: Coverage pattern on the sky after five layers over a 4 degree by 4 degree patch. This is shown for a region away from the edges of the footprint. Left: The fully-populated focal plane with 5000 fibers. Right: The reduced focal plane with 3000 fibers. \n<!-- image --> \nTable 4.1: The fraction of the footprint covered by 1, 2, 3, ... 8 fibers. The first row shows the results using all five layers in the baseline survey. The second row shows the results using just the four layers that include LRG and quasar targets. The mean is slightly decreased and the rms slightly increased by edge effects.", '4.4.1 Sequence of Observations': 'The placement of field centers presented in Section 4.3 is designed to cover the footprint in five independent tilings. Given the 1940 hours of scheduled time each year, roughly 20% (390 hours) will occur under conditions when the moon is above the horizon and the remainder under dark conditions. Each year, 20% of the fields in a full independent tiling of 2000 field centers will be observed using the scheduled time in grey conditions. This layer is planned to include only ELG targets because their spectral features are predominantly found at redder wavelengths and redshift success rates are less susceptible to increased sky background from the moon. On the other hand, the darkest 80% of the scheduled time (1550 hours) will be used to observe the QSO and LRG targets at highest priority, leaving the remaining science fibers for ELG targets. \nThere remains additional freedom to determine the order in which the tiles over the four dark time layers are observed. Full simulations of the program will be used to determine the optimal approach. The simulations will factor in seeing, transparency and weather variations for each exposure via Monte Carlo simulations to predict the quality of spectra and the variations in final survey areal coverage. Each exposure will be tuned to a grid of targets parameterized by magnitude and redshift using an exposure time calculator that approximates the sensitivity of the instrument. Weather conditions will be mocked using monthly statistics at Kitt Peak and the results will be used to determine likely redshift success rates over all target classes. A description of these simulations is found in the document DESI-1658. The approach that optimizes intermediate and final cosmology results will be chosen. \nWe provide a baseline strategy in the accompanying document on long-term strategy. This program is assembled without consideration of weather and other variables. The survey is designed to get an early complete sample over 10% of the footprint as early as possible. The survey also provides distinct milestones for data products and cosmological analysis at the end of each year. \nFinally, we will investigate the target strategies, exposure depths, quality of data, and expected spectroscopic completeness during a phase of survey validation. Survey validation will occur during the end of commissioning before the full survey begins. The baseline program for this phase of the project is presented in the accompanying document on survey validation.', '4.4.2 Exposure Times and Margin': "Over five years, DESI is projected to observe 14,000 sq. deg. of the footprint presented in Section 4.2. The exact subset of this footprint to be observed will be contiguous regions in each of the NGC and the SGC that best fit the expected allocation of time. We have simulated the choice of final tile centers and the average exposure times according to an observing schedule of 1940 hours of dark and grey time per year as defined in the Site Alternatives study (DESI-311). The simulation includes a two minute overhead between fields and variations in exposure time for each field due to airmass and Galactic dust extinction. All exposure times are split into two separate exposures (with one minute of read time). This split limits the number of cosmic rays in an individual exposure, and also effectively maximize the S/N in variable sky conditions. The split is not assumed in the baseline for spectroscopic depth and completeness, so the time per field is larger in these simulations than in the baseline design. The accumulated S/N will be measured by the exposure time calculator (see the DESI Performance Studies in the Instrument FDR). We project that 57% of the scheduled time will deliver usable data, where 'usable data' is assumed in conditions when the dome is open and seeing is better than 1.5 arcsec. Although DESI will observe when the seeing is worse than 1.5 arcsec, those data have been ignored in these estimates of survey duration. \nWe simulate the full suite of observations accounting for airmass and Galactic dust extinction by choosing an hour angle for each field that maximizes the overall survey depth while fitting into the allocated time. Exposure times are estimated for each field to produce uniform depth in dust-extinction and atmosphere-extinction corrected spectra. In preliminary estimates, we assume the same dependence of S/N on airmass as was measured with BOSS, and degradation in S/N due to Galactic extinction for the sky-noise-limited case of the faintest targets. In future iterations, we will include a more sophisticated interpretation of redshift success rate for representative targets, thus accounting for the wavelength-dependent S/N estimates of each target class. For the 14,000 sq. deg. footprint observed with 10,000 tiles, we find an average exposure time of 1800 seconds. Scaling this to an observation taken at zenith with no Galactic dust extinction (as shown in the figure in the DESI Performance Studies in the Instrument FDR) produces an equivalent exposure time of 1226 seconds. In other words, each exposure will have a S/N equivalent to a 1226 second exposure taken at zenith, under photometric conditions, median sky brightness and median seeing. As explained in Simulations Section of the Instrument FDR this fiducial exposure time of 1226 seconds allows the 1000 second exposures that are predicted to produce the required redshift success rates for each DESI target class. This projection leaves a 22% margin in exposure time for worse-than-projected weather, throughput performance, instrument downtime, or other factors that could slow the pace of the survey. \nSimilarly, we have estimated the average and fiducial exposure times for the reduced focal plane of the DESI KPP survey. The 'Pacman' tiling of Section 4.2 leads to an average exposure time of 1700 seconds for 10,600 tile centers covering 9,000 sq. deg. Even though the average exposure time is somewhat lower than the 14,000 sq. deg. survey, the fiducial exposure time of 1270 seconds is actually larger because the average field in a 9,000 sq. deg. program lies at lower airmass and lower Galactic extinction than the average field in a 14,000 sq. deg. program. The projected margin for the 9,000 sq. deg. KPP survey is 27%.", '4.5.1 Introduction': 'A portion of DESI operations will be affected by increased sky brightness from the moon, so as to make conditions unsuitable for observing the targets above z = 0 . 6. DESI expects to observe in the darkest 21 nights of the month, but some of those nights are affected in part by moon, adding up to about 440 hours per year of time. Assuming the same average weather statistics used in planning for dark time, we expect 250 hours per year on average of open-dome bright time. During this time, the DESI collaboration will conduct a survey of bright galaxies which will increase performance for the cosmology goals. This Bright Galaxy Survey (BGS) will be the primary bright-time survey program. In addition, the density of fibers in the DESI focal plane will enable a simultaneous survey of Milky Way Stars (The Milky Way Survey; MWS) during bright time. The MWS will target some of the oldest stars in the Galaxy with the goal of understanding the mass distribution, formation and evolution of the Galaxy. We refer to these two combined programs as the Bright Time Survey (BTS).', '4.5.2 Survey Footprint': 'The Bright Time Survey will use the same 14,000 square degree footprint as the dark time project. This will enable the BTS to benefit from the optimization of the dark time footprint for observability. The BGS targets will be selected from the same imaging data as the dark time targets. The MWS will use Gaia photometry and proper motions for target selection. The Gaia survey is all-sky, and so covers the DESI 14,000 square degree footprint.', '4.5.3 Field Centers': 'The BTS will use the same tiling pattern as the DESI Key Project, but with only 3 layers totaling 6000 tiles. There are roughly 1400 galaxies per square degree to r = 20 . 0. With 4500 science fibers per tile, the BTS will place about 27 million fibers, more than the ∼ 20 million BGS targets. However, the presence of clustering and Poisson fluctuations of bright galaxies implies that we must incur these extra layers if we want to achieve a higher completeness. For fiber assignment, BGS targets are divided into two priorities; brighter targets with r < 19 . 5 ( ∼ 800 deg -2 ) receive high priority, while fainter 19 . 5 < r < 20 . 0 ( ∼ 600 deg -2 ) targets receive secondary priority. Preliminary simulations of DESI fiber assignments using this priority scheme yields 3-layer completeness values of 92% for the bright sample and 77% for the faint sample, for an overall fiber completeness of 86%, or roughly 17M targets. \nThere are about 600 stars with effective temperatures higher than 4700 K per square degree to r = 18 at Galactic latitude greater than 40 degrees from the equator. The DESI focal plane is 7.5 square degrees, so at each layer there will be many fibers available for the MWS.', '4.5.4 Observation Strategy': 'Completing 6000 tiles in the 1250 hours of available open-dome time indicates an average time of 12.5 minutes per tile. Survey simulations accounting for the increased exposure time required as a function of airmass and extinction indicate that we would have 400 seconds available for a reference exposure at unit airmass and zero extinction. We are planning for \na 300 second reference exposure, therefore leaving a 33% margin. Our spectral simulations ( § 3.1 and Figure 3.3) indicate that a 5-minute exposure will yield a redshift success of 97% for galaxies down to r = 19 . 5, and 92% for the fainter 19 . 5 < r < 20 . 0 sample. \nFor stars, 5 minute exposures will result in spectra with S/N per Angstrom of 14 at λ > 650 nm for stars of r magnitude 16.5-18, depending on their spectral energy distributions. Spectra of that quality are sufficient to yield radial velocity and chemical abundance information. \nBecause the BTS needs only three layers, it will be possible to combine multiple exposures for fainter objects for higher S/N. For example, it would be possible to re-expose many of the 5% of BGS targets that fail to achieve a redshift in the first two layers. We will perform fiber assignment simulations that combine the MWS and BGS samples to determine the optimal way to assign fibers that accounts for galaxy clustering and the variation in stellar density across the footprint, and which achieves maximum redshift and radial velocity completeness for faint targets. \nWith this basic strategy we expect to obtain spectra of roughly 10 million galaxies in the BGS and 10 million stars in the MWS. More simulations of the BTS are required to determine how to prioritize sky coverage versus completeness to enable early science. The BTS simulations will use the same survey simulation code as the dark time program, adapting it as required to account for scheduling around lunar phase and separation angle between the field and the moon.', 'Acknowledgements': 'This research is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DEAC0205CH1123, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; additional support for DESI is provided by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to the National Optical Astronomy Observatory; the Science and Technologies Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the HeisingSimons Foundation; the National Council of Science and Technology of Mexico, and by the DESI Member Institutions: Aix-Marseille University; Argonne National Laboratory; Barcelona Regional Participation Group; Brookhaven National Laboratory; Boston University; Carnegie Mellon University; CEA-IRFU, Saclay; China Participation Group; Cornell University; Durham University; cole Polytechnique Fdrale de Lausanne; Eidgenssische Technische Hochschule, Zrich; Fermi National Accelerator Laboratory; Granada-Madrid-Tenerife Regional Participation Group; Harvard University; Korea Astronomy and Space Science Institute; Korea Institute for Advanced Study; Institute of Cosmological Sciences, University of Barcelona; Lawrence Berkeley National Laboratory; Laboratoire de Physique Nuclaire et de Hautes Energies; Mexico Regional Participation Group; National Optical Astronomy Observatory; Siena College; SLAC National Accelerator Laboratory; Southern Methodist University; Swinburne University; The Ohio State University; Universidad de los Andes; University of Arizona; University of California, Berkeley; University of California, rvine; University of California, Santa Cruz; University College London; University of Michigan at Ann Arbor; University of Pennsylvania; University of Pittsburgh; University of Portsmouth; University of Queensland; University of Toronto; University of Utah; UK Regional Participation Group; Yale University. 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In: ArXiv e-prints (Dec. 2009). arXiv: 0912.0201 [astro-ph.IM] .\n- [229] D. Stern et al. 'Mid-Infrared Selection of Active Galaxies'. In: Astrophys. J. 631 (Sept. 2005), pp. 163-168. eprint: arXiv:astro-ph/0410523 .", 'Author Institutions': "- 1 2137 Frederick Reines Hall, Irvine, CA 92697, USA\n- 2 Aix Marseille Univ, CNRS, LAM, 13388 Marseille, France\n- 3 Aix Marseille Univ, CNRS, OHP, 04870 Saint-Michel-l'Observatoire, France\n- 4 Aix Marseille Universit'e, CNRS/IN2P3, CPPM UMR 7346, 13288, Marseille, France\n- 5 Alphabet Inc., 1650 Charleston Rd. Mountain View, CA 94043, USA\n- 6 AMNH, Department of Astrophysics, American Museum of Natural History, New York, NY 10024, USA\n- 7 APC, Universit'e Paris Diderot-Paris 7, CNRS/IN2P3, CEA, Observatoire de Paris, 10, rue Alice Domon & Lonie Duquet, Paris, France\n- 8 Argonne National Laboratory, High-Energy Physics Division, 9700 S. Cass Avenue, Argonne, IL 60439, USA\n- 9 Astronomy Department, Yale University, P.O. Box 208101 New Haven, CT 06520-8101, USA\n- 10 Brookhaven National Laboratory, Upton NY 11973, USA\n- 11 Carreterra M'exico-Toluca S/N, La Marquesa, Ocoyoacac, Edo. de M'exico C.P. 52750, M'exico\n- 12 CEA Saclay, IRFU F-91191 Gif-sur-Yvette, France\n- 13 Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA\n- 14 Centre for Advanced Instrumentation, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK\n- 15 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia\n- 16 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK\n- 17 Centre for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, UK\n- 18 Cerro Tololo Inter-American Observatory (CTIO), Colina El Pino s/n, Casilla 603, La Serena, Chile\n- 19 CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain\n- 20 Clippinger Laboratories, Room 333, Ohio University, Athens, OH 45701, USA\n- 21 Departamento de F'ısica, Universidad de Guanajuato - DCI, C.P. 37150, Leon, Guanajuato, M'exico\n- 22 Departamento de F'ısica, Universidad de los Andes, Cra. 1 No. 18A-10, Edificio Ip, Bogot'a, Colombia\n- 23 Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, Canada M5S 3H4\n- 24 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95065, USA\n- 25 Department of Astronomy and Space Science, Sejong University, Seoul 143-747, Republic of Korea\n- 26 Department of Astronomy, The Ohio State University, 4055 McPherson Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA\n- 27 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA\n- 28 Department of Astronomy, University of Michigan, 1085 S. University Avenue, Ann Arbor, MI 481091107, USA\n- 29 Department of Astronomy, Yale University, Steinbach Hall, 52 Hillhouse Avenue, New Haven, CT 06511, USA\n- 30 Department of Physics & Astronomy and Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, Pittsburgh, PA 15260, USA\n- 31 Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA\n- 32 Department of Physics & Astronomy, University of Wyoming, 1000 E. University, Dept. 3905, Laramie, WY 82071, USA\n- 33 Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK \n- 34 Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA\n- 35 Department of Physics and Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA\n- 36 Department of Physics and Astronomy, University College London, 3rd Floor, 132 Hampstead Road, London, NW1 2PS, UK\n- 37 Department of Physics and Astronomy, University of California, 4129 Frederick Reines Hall, Irvine, CA 92697, USA\n- 38 Department of Physics and Center for Cosmology and Particle Physics, New York University, New York, NY 10003, USA\n- 39 Department of Physics and JINA Center for the Evolution of the Elements, University of Notre Dame, Notre Dame, IN 46556, USA\n- 40 Department of Physics and Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA\n- 41 Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA\n- 42 Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA\n- 43 Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA\n- 44 Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, USA\n- 45 Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA\n- 46 Department of Physics, University of Arizona, 1118 E. Fourth Street, PO Box 210081, Tucson, AZ 85721, USA\n- 47 Department of Physics, University of California, Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA 94720-7300, USA\n- 48 Department of Physics, University of Michigan, 450 Church St., Ann Arbor, MI 48109, USA\n- 49 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\n- 50 Ecole Polytechnique F'ed'erale de Lausanne, CH-1015 Lausanne, Switzerland\n- 51 European Space Astronomy Centre (ESAC), 38205 Villanueva de la Ca˜nada, Madrid, Spain\n- 52 Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA\n- 53 Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA\n- 54 HCTLab Research Group, Escuela Politecnica Superior, Universidad Aut'onoma de Madrid, C/Francisco Tomas y Valiente 11, 38049, Spain\n- 55 Instituci'o Catalana de Recerca i Estudis Avan¸cats (ICREA), Pg. de Llu'ıs Companys 23, 08010 Barcelona, Spain\n- 56 Institut de C'ıencies de l'Espai, IEEC-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain\n- 57 Institut de Fisica dAltes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain\n- 58 Institute for Astronomy, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland\n- 59 Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK\n- 60 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\n- 61 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\n- 62 Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK\n- 63 Instituto de Astrofisica de Andaluc'ıa, Glorieta de la Astronom'ıa, s/n, E-18008 Granada, Spain\n- 64 Instituto de Astrof'ısica de Canarias, C/ Va L'actea, s/n, 38205 San Crist'obal de La Laguna, Santa Cruz de Tenerife, Spain\n- 65 Instituto de Astronomia, Universidad Nacional Aut'onoma de M'exico, Apartado Postal 70264, 04510 M'exico D.F., M'exico\n- 66 Instituto de C'ıencias del Cosmoc, (ICCUB) Universidad de Barcelona (IEEC-UB), Mart'ı i Franqu'es \n- 1, E08028 Barcelona\n- 67 Instituto de F'ısica Te'orica (IFT) UAM/CSIC, Universidad Aut'onoma de Madrid, Cantoblanco, E28049, Madrid, Spain\n- 68 Instituto de F'ıisica, Universidad Nacional Aut'onoma de M'exico, Cd. M'exico C.P. 04510\n- 69 Kavli Institute for Astronomy and Astrophysics at Peking University, PKU, 5 Yiheyuan Road, Haidian District, Beijing 100871, P.R. China\n- 70 Kavli Institute for Cosmology, Cambridge, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK\n- 71 Kavli Institute for Particle Astrophysics and Cosmology and SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA\n- 72 Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China\n- 73 Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea\n- 74 Laboratoire dAstrophysique, Ecole Polytechnique F'ed'erale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland\n- 75 Laborat'orio Interinstitucional de e-Astronomia, Rua Gal. Jose Cristino 77, Rio de Janeiro, RJ 20921400, Brazil\n- 76 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA\n- 77 Lawrence Livermore National Laboratory, P.O. Box 808 L-211, Livermore, CA 94551, USA\n- 78 Ludwig-Maximilians University Munich, University Observatory, Scheinerstr. 1, 81679 Munich, Germany\n- 79 McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA\n- 80 National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd. 100012, Beijing, P.R. China\n- 81 National Optical Astronomy Observatory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA\n- 82 Observatorio Nacional, R. Gal. Jose Cristino 77, Rio de Janeiro, RJ 20921-400, Brazil\n- 83 Physics Department, Stanford University, Stanford, CA 93405, USA\n- 84 Physics Department, Yale University, P.O. Box 208120, New Haven, CT 06511, USA\n- 85 Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA\n- 86 School of Mathematics and Physics, University of Queensland, 4101, Australia\n- 87 School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Republic of Korea\n- 88 Sorbonne Universit'es, UPMC Universit Paris 06, Universit'e Paris-Diderot, CNRS-IN2P3 LPNHE 4 Place Jussieu, F-75252, Paris Cedex 05, France\n- 89 Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA\n- 90 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA\n- 91 SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK\n- 92 University of California Observatories, 1156 High Street, Sana Cruz, CA 95065, USA\n- 93 University of Science and Technology, Daejeon 34113, Republic of Korea"}

2024ApJ...976L..12J

On 2024 May 1011 the strongest geomagnetic storm since 2003 November occurred with a peak Dst index of 412 nT. The storm was caused by NOAA active region AR 13664 which was the source of a large number of coronal mass ejections and flares including 12 Xclass flares. Starting from about May 7 AR 13664 showed a steep increase in its size and free magnetic energy along with increased flare activity. In this study we perform 3D magnetic field extrapolations with the NF2 nonlinear forcefree code based on physicsinformed neural networks R. Jarolim et al.. In addition we introduce the computation of the vector potential to achieve divergencefree solutions. We extrapolate vector magnetograms from the Solar Dynamics Observatorys Helioseismic and Magnetic Imager at the full 12 minute cadence from 2024 May 5 0000 to 11 0436 UT in order to understand the ARs magnetic evolution and the large eruptions it produced. A decrease in the calculated relative free magnetic energy can be related to solar flares in 90 of the cases and all considered Xclass flares are reflected by a decrease in the relative free magnetic energy. Regions of enhanced free magnetic energy and depleted magnetic energy between the start and end times of major Xclass flares show spatial alignment with brightness increases in extremeultraviolet observations. We provide a detailed analysis of the X3.9class flare on May 10 where we show that the interaction between separated magnetic domains is directly linked to major flaring events. With this study we provide a comprehensive data set of the magnetic evolution of AR 13664 and make it publicly available for further analysis.

2024-11-01T00:00:00Z

['10.3847/2041-8213/ad8914', '2024ApJ...976L..12J', '2024arXiv240908124J', 'arXiv:2409.08124', '10.48550/arXiv.2409.08124']

['Solar flares', 'Solar activity', 'Solar magnetic fields', 'Solar magnetic reconnection', 'Magnetohydrodynamical simulations', '1496', '1475', '1503', '1504', '1966', 'Astrophysics - Solar and Stellar Astrophysics']

Magnetic Field Evolution of the Solar Active Region 13664

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

https://arxiv.org/pdf/2409.08124.pdf

{'No Header': '3', 'Magnetic Field Evolution of the Solar Active Region 13664': 'Robert Jarolim, 1 Astrid M. Veronig, 2, 3 Stefan Purkhart, 2 Peijin Zhang, 4 and Matthias Rempel 1 \n1 High Altitude Observatory, NSF NCAR, 3080 Center Green Dr., Boulder, CO 80301, USA \n2 University of Graz, Institute of Physics, Universitatsplatz 5, 8010 Graz, Austria \nUniversity of Graz, Kanzelhohe Observatory for Solar and Environmental Research, Kanzelhohe 19, 9521 Treffen am Ossiacher See, Austria \n4 New Jersey Institute of Technology, 323 Dr Martin Luther King Jr Blvd, Newark, NJ 07102, USA', 'ABSTRACT': 'On 2024 May 10/11, the strongest geomagnetic storm since November 2003 has occurred, with a peak Dst index of -412 nT. The storm was caused by NOAA Active Region (AR) 13664, which was the source of a large number of coronal mass ejections and flares, including 12 X-class flares. Starting from about May 7, AR 13664 showed a steep increase in its size and (free) magnetic energy, along with increased flare activity. In this study, we perform 3D magnetic field extrapolations with the NF2 nonlinear-force free code based on physics informed neural networks (Jarolim et al. 2023). In addition, we introduce the computation of the vector potential to achieve divergence-free solutions. We extrapolate vector magnetograms from SDO/HMI at the full 12 minute cadence from 2024 May 5-00:00 to 11-04:36 UT, in order to understand the active regions magnetic evolution and the large eruptions it produced. The computed change in magnetic energy and free magnetic energy shows a clear correspondence to the flaring activity. Regions of free magnetic energy and depleted magnetic energy indicate the flare origin and are in good correspondence with observations in Extreme Ultraviolet. Our results suggest that the modeled solar flares are related to significant topological reconfigurations. We provide a detailed analysis of the X4.0-class flare on May 10, where we show that the interaction between separated magnetic domains is directly linked to major flaring events. With this study, we provide a comprehensive data set of the magnetic evolution of AR 13664 and make it publicly available for further analysis. \nKeywords: Sun: flares, Sun: magnetic fields, methods: numerical', '1. INTRODUCTION': "Solar flares and coronal mass ejections (CMEs) are the most energetic phenomena in our solar system, and can cause severe effects on the space weather at Earth and other solar-system planets. Flares and CMEs are different facets of the same physical process. They are known to result from instabilities in the coronal magnetic field and the impulsive release of vast amounts of energy by magnetic reconnection (Priest & Forbes 2002; Schrijver 2009), which is subsequently converted into kinetic energy of high-energy particles, plasma motions and heating (e.g., Veronig et al. 2005; Fletcher et al. 2011). The magnetic energy that is suddenly released in solar flares (on time scales of minutes to hours) has been previously accumulated and stored (on time scales of days to weeks), through the emergence of magnetic flux from the convection zone to the solar atmosphere and by shearing motions producing strong electric cur- \nrents in the Active Region's (AR) corona (e.g., Forbes et al. 2006; Sun et al. 2012; Wiegelmann et al. 2014). Therefore, it is important to model the coronal magnetic field and to study the 3D topology of ARs that may lead to major flaring (Wiegelmann et al. 2014; Janvier et al. 2015; Kors'os et al. 2024). \nStatistical studies have shown that large flares are preferentially produced by large ARs of high magnetic complexity (e.g., Leka & Barnes 2007; Schrijver 2007). Sammis et al. (2000) found that about 60% of the X-class flares (100% of flares ≥ X4) resulted from ARs of Mount Wilson magnetic class βγδ and a size > 1000 µ hem. However, where the bulk of the magnetic energy is stored, and where and how the energy release in flares is triggered is still a topic of intense research (e.g., Green et al. 2018; Kusano et al. 2020; Gupta et al. 2021). \nDuring 2024 May 2 to 14, NOAA AR 13664 was visible from Earth, and developed to one of the largest and most flare-productive ARs in the recent decades. From May 4 to 7, it grew in size from about 110 to 2700 µ hem (see the overview in Hayakawa et al. 2024), and starting from May 6 it was of magnetic class βγδ . Over its lifetime, AR 13664 produced 12 X-class flare (including an X8.7 flare when it has already rotated behind the Western limb) and 52 M-class flares. Noteworthy, its high activity caused the strongest geomagnetic storm since November 2003, with a peak Dst index of -412 nT on 2024 May 11, around 4 UT. \nThe exceptional flaring activity and the fast evolution of AR 13664 while it was on the Earth-facing hemisphere, makes it an ideal candidate for a detailed investigation of the evolution of its magnetic complexity, the energy storage in the AR and the energy release during the major flares it produced. \nIn this study, we model the ARs magnetic field during its transition across the solar disk and analyze the magnetic topology and magnetic energy build-up/release mechanisms. In Sect. 2, we introduce the used data, magnetic model, and analysis methods. Section 3 summarizes the result of our magnetic field extrapolations and the connection to the flaring activity of AR 13664. We investigate the energy build-up and release processes and specifically focus on the X4.0 flare on 2024 May 10. Finally, we provide a summary and interpretation of the model results (Sect. 4).", '2. METHOD': 'In this study, we primarily examine the magnetic topology and evolution of AR 13664. Non-linear ForceFree (NLFF) extrapolations are frequently applied to obtain a realistic estimate of the coronal magnetic field from photospheric vector magnetograms (e.g., Wiegelmann & Sakurai 2021; Wheatland & Leka 2011; Wiegelmann & Inhester 2010). However, NLFF extrapolations typically require additional pre-processing of observational data and are computationally expensive (Wiegelmann et al. 2006; Wiegelmann & Sakurai 2021). \nPhysics-Informed Neural Networks (PINNs; Raissi et al. 2019) are a novel method for solving partial differential equations and have the ability to smoothly integrate noisy data and incomplete physical models (Karniadakis et al. 2021). In Jarolim et al. (2023), PINNs were introduced for NLFF extrapolations, and demonstrated the ability to provide reliable magnetic field extrapolations in quasi real-time. This specifically enables the efficient computation of extrapolation series at a high temporal resolution. \nA critical aspect of the NLFF methods is the divergence free condition, which is typically part of the optimization method (e.g., Wiegelmann et al. 2012). Here, we build on the PINN implementation from Jarolim et al. (2023) and introduce in addition extrapolations through the vector potential, instead of directly modeling the magnetic field (Sect. 2.2). With this approach we can intrinsically obtain solenoidal magnetic field solutions.', '2.1. Data': 'We use vector magnetograms from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) onboard the Solar Dynamics Observatory (SDO; Pesnell et al. 2012). Specifically, we use Space Weather Active Region Patches (SHARP; Bobra et al. 2014) available with a cadence of 12 min. Here, the vector magnetograms are reprojected to a Cylindrical Equal Area (CEA) grid with B x , B y , B x , which we use as input for our magnetic field extrapolations. For our extrapolations we use the full resolution with a spatial scale of 0.36 Mm per pixel. Note that this resolution is typically not reproduced by our method, due to the disagreement between the force-free assumption and the observation. However, extrapolating the data at full resolution causes no significant increase in computing time. In addition, we use the provided error maps as input to our extrapolation method. \nFor the evaluation, we use observations in Extreme Ultraviolet (EUV) spectral bands from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) onboard SDO. To better visualize flaring activity, we create integrated EUV maps, where we use SDO/AIA observations from a single spectral band at a one minute cadence, and integrate the resulting image stack along the temporal axis. We specifically use the AIA 94 and 131 ˚ A filters, which are most sensitive to the hot flaring plasma, with peaks in the temperature response curves at T ≈ 7 and T ≈ 11 MK, respectively. \nFor our study, we use observations from 2024 May 5-00:00 to 11-04:36 UT. The series contains SDO/HMI data gaps. The only significant data gap occurs between 8-16:36 and 9-00:00, which coincides with several M-class flares and an X1.0 flare (8-21:08). Therefore, these events were not considered in our analysis.', '2.2. Non-linear force-free extrapolations': 'For the Non-Linear Force Free (NLFF) extrapolations of AR 13664, we use the method from Jarolim et al. (2023), which is based on PINNs. Here, a Neural Network is trained to act as a function approximation of the modeled magnetic field ⃗ B . The model optimization is performed by iteratively sampling points from \nFigure 1. Overview of the evolution of NOAA AR 13664 from 2024 May 5-00:00 to 11-04:36 UT. a) Evolution of the vertical distribution of free magnetic energy. b) GOES 1-8 ˚ A soft X-ray flux. c) Observed SDO/HMI radial magnetic field maps, SDO/AIA 131 ˚ A EUV maps, and the modeled current density integrated along the vertical axis. We show the initial flux emergence (snapshot at 2024 May 7 00:00 UT) and the major flare eruptions (indicated by dashed lines). The magenta line indicates the point where the right edge of the SHARP reaches 60 · longitude, beyond which the reliability of the vector magnetograms and the modeled coronal field decreases. The circles in the integrated current density maps indicate points of flux domain interactions. An animation of the full time series is available in the online journal. Left top: B z of the modeled photospheric magnetic field. Right top: EUV observations from SDO/AIA 131 ˚ A EUV. Left bottom: maps of integrated current density. Right bottom: maps of integrated free magnetic energy. The movie shows the re-configuration of magnetic domains (dark lines in current density maps) during strong solar eruptions. \n<!-- image --> \nthe boundary condition and randomly from within the simulation volume (x, y, z). The coordinate points are used as input to the neural network and mapped to the corresponding magnetic field vector (B x , B y , B z ). For the boundary condition the network is optimized to match the observed magnetic field vector, while for the randomly sampled points the residuals of the force-free \nequation \nL ff = ∥ ( ⃗ ∇× ⃗ B ) × ⃗ B ∥ 2 ∥ ⃗ B ∥ 2 + ϵ , (1) \nand the divergence-free equation \nL div = | ⃗ ∇· ⃗ B | 2 , (2) \nare minimized. The derivatives are computed by automatic differentiation and used to construct the partial differential equations. Given that the neural network is fully-differentiable, automatic differentiation can be used to compute smooth derivatives of the model outputs with respect to the input coordinates. As shown in Jarolim et al. (2023), typically it is not possible to achieve a perfect agreement between the observations and the force-free model, and consequently a trade-off between the two conditions is found. This shortcoming is intrinsic to NLFF extrapolations, and can only be mitigated by building on a more complex physical model (e.g., magneto-hydrodynamic modeling) or by additional observational constrains (e.g., chromospheric magnetic field measurements; Jarolim et al. 2024; Fleishman et al. 2019; Yelles Chaouche et al. 2012). \nFor the optimization of the boundary condition, we take the provided error maps into account. Here, we compute the difference between the modeled ( ⃗ B ) and the reference magnetic field ( ⃗ B 0 ) \n⃗ B diff, 0 = abs( ⃗ B -⃗ B 0 ) . (3) \nWeaccount for uncertainties in the measurement by subtracting the error map ⃗ B error and clipping negative values \n⃗ B diff,clipped = max { ⃗ B diff, 0 -⃗ B error , 0 } . (4) \nTherefore, we only optimize values where the modeled field exceeds the error threshold. For the loss of the boundary magnetic field, we compute the vector norm of the clipped difference vector \nL B = ∥ ⃗ B diff,clipped ∥ 2 . (5) \nIn this study, we replace the direct modeling of the magnetic field ⃗ B , with the computation of the vector potential ⃗ A which is derived by taking the curl of ⃗ B \n⃗ B = ∇× ⃗ A. (6) \nFrom the fundamental relation of vector calculus this implies ∇· B = ∇· ( ∇× A ) = 0, which directly leads to solenoidal field solution. Note that this computation is performed with auto-differentiation and that a finite-differences approach will show deviations from a perfectly solenoidal field. \nOmitting the optimization for the divergence-free condition, we obtain the final loss from the force-free and boundary condition \nL = λ ff L ff + λ B L B . (7) \nFor our initial extrapolation we exponentially decay λ B from 1 , 000 to 1 over 50 , 000 iterations. We throughout set λ ff to 0.1.', '2.3. Metrics': "From the resulting magnetic field extrapolations we transfer the neural representation to a grid representation by sampling each point in our simulation volume at a resolution of 0.72 Mm per pixel, which corresponds to a rebinning by a factor of 2 from the original SDO/HMI resolution. In addition to the magnetic field vector ⃗ B , we compute the current density ⃗ J according to Ampere's law \n⃗ J = c 4 π ∇× ⃗ B, (8) \nwhere c corresponds to the speed of light. Here, we use directly automatic differentiation of the neural representation, to obtain smooth derivatives that are independent of the spatial grid resolution. Note that computed derivatives are in units of the normalized input coordinates and are converted to physical units (Mm) for our evaluation. \nFrom the resulting cubes we compute measurements of magnetic energy, free magnetic energy, and the quality metrics. The magnetic energy is defined as \nE = N ∑ i B 2 i 8 π , (9) \nwhere i refers to the i th grid cell and N to the total number of grid cells. The free magnetic energy corresponds to the difference between the magnetic energy of the NLFF field and the potential field \nE free = E FF -E PF . (10) \nHere, we compute the potential field solution using the Green's function approach as proposed by Sakurai (1982). As input for the potential field extrapolation we use the modeled bottom boundary of our NLFF solution (i.e., the adapted boundary). \nFor the visualization of spatial distribution we compute the local magnetic energy density ( E i = B 2 i / 8 π ), and integrate along the vertical or horizontal axis. We compute energy difference maps analogously, by calculating the energy difference per grid cell (∆ E i = E t2 ,i -E t1 ,i ) and integration along the vertical axis. The total energy difference is computed by integrating over the full simulation domain \n∆ E total = N ∑ i ∆ E i . (11) \nTo provide an upper estimate of the released magnetic energy during flares, we compute the total depleted magnetic energy ∆ E -by clipping local energy increases prior to the integration: \n∆ E -= N ∑ i min(0 , ∆ E i ) . (12) \nThis excludes the energy build-up that may be continuously ongoing in the AR also during the flaring time (e.g., due to flux emergence). However, we note that also spatial redistribution may falsely count towards released magnetic energy (e.g., movement of magnetic elements). Therefore, this metric can only provide an upper estimate of released magnetic energy, while more advanced metrics would be needed to discern between the parallel processes. \nWe compute metrics for divergence- and force-freeness to verify the validity of our method. For this we use the normalized divergence \nL div,n ( ⃗ B ) = ∑ i | ⃗ ∇· ⃗ B i | / ∥ ⃗ B i ∥ . (13) \nwhich we compute from the sampled grid based on finite differences. In addition, we compute the current weighted angle between the magnetic field and the current density θ J = sin σ J , with \nσ J ( ⃗ B ) = ( ∑ i ∥ ⃗ J i × ⃗ B i ∥ ∥ ⃗ B i ∥ ) / ∑ i ∥ ⃗ J i ∥ . (14) \nTo estimate the difference from the boundary condition, we compute the deviation above the error map ⃗ B error as \n⃗ B diff,clipped = max { ⃗ B diff, 0 -⃗ B error , 0 } , (15) \nwhere ⃗ B diff, 0 refers to the absolute difference between the modeled and observed magnetogram. From this expression, we compute the vector norm of the clipped difference vector to quantify the deviation from the boundary condition \n∆ B = ∥ ⃗ B diff,clipped ∥ . (16)", '2.4. Squashing factor and twist number': 'We use the squashing factor (Q-factor) Q ( x, y, z ) and twist ( T w ) to characterize the topology of the magnetic field (Titov et al. 2002; Titov 2007). \nThe Q-factor is derived from the differential mapping of the magnetic field lines. For a given magnetic field line connecting two surfaces ( S 1 , S 2 ) with on-surface coordinates ([ x 1 , y 1 ], [ x 2 , y 2 ]), the Jacobian matrix of differential mapping is defined as: \nD 1 2 = ( ∂x 2 ∂x 1 ∂x 2 ∂y 1 ∂y 2 ∂x 1 ∂y 2 ∂y 1 ) ≡ ( a b c d ) (17) \nand the squashing factor (Q-factor) at [ x 1 , y 1 ] can be expressed as: \nQ ( x 1 , y 1 ) = a 2 + b 2 + c 2 + d 2 ∣ ∣ ∣ det D 1 2 ∣ ∣ ∣ . (18) \nThe Q-factor is an invariant along a magnetic field line. A large Q-factor indicates the diverging nature of the local magnetic field. \nThe twist number ( T w ) characterizes the amount of winding along a magnetic field line, expressed as: \nT w = ∫ L ∇× ⃗ B · ⃗ B 4 πB 2 d l, (19) \nwhere the integral range L is a segment of a magnetic field line. \nIn this study, we use FastQSL (Zhang et al. 2022), which utilizes graphics processing units (GPUs) to perform efficient computations of Q and T w for the full 3D volume.', '3. RESULTS': 'We apply our extrapolation method to the full series of SDO/HMI SHARP vector magnetograms from 2024 May 5-00:00 to 11-04:36 UT, with a 12 min cadence. We make use of the series training approach, using the extrapolation result from the previous time step as initial condition for the training with the next boundary condition. With this approach, we only need to perform a single extrapolation from scratch, while the remaining extrapolations of the series can be obtained in realtime (about 6 minutes per extrapolation with 4 NVIDIA A100 GPUs; for details see Jarolim et al. 2023). Due to the HMI data gap from May 8-16:36 to May 9-00:00 UT, we separate the series into two parts, where we continue the series extrapolation from an additional initial extrapolation on May 9-00:00 UT. From our extrapolation results we render the cubes of magnetic field at a spatial sampling of 0.72 Mm per pixel. Note that the model is trained with a resolution of 0.36 Mm per pixel, however we found the reduced sampling sufficient and the decrease in data volume allows for easier access. The full resolution cubes are available through the model snapshots. \nIn Fig. 1 we provide an overview of the ARs evolution. The GOES series in panel b shows the flare related increases in X-ray intensity. Panel c shows the radial magnetic field maps, SDO/AIA 131 ˚ A EUV filtergrams and integrated current density throughout the modeled time sequence (dashed lines in panel b). The magnetic energy release related to flares can be seen from the change in free magnetic energy (panel a). The major flux emergence starts on 2024 May 7, as can be seen from the build-up in free magnetic energy and the magnetic field maps (red circle). In Sect. 3.2 we further discuss the relation of energy depletion and the observed intensity increases in EUV. The white circles in panel c indicates supposed separatrix layers that are related to re-configurations during the solar flares (see Sect. 3.3). \nFigure 2. Overview of the temporal evolution of unsigned magnetic flux, magnetic energy, free magnetic energy, and the ratio between free magnetic energy and magnetic energy ( E free /E ) of AR 13664. Blue lines correspond to the direct quantities, while orange lines refer to the first order time derivatives of those quantities. Blue and orange shaded bars indicate M- and X- class flares, respectively. The bars outline flare start to end times, according to the GOES flare catalog. \n<!-- image -->', '3.1. Magnetic energy evolution': "We compute the integrated quantities of magnetic energy E (Eq. 9), free magnetic energy E free (Eq. 10), and the ratio of free magnetic energy to magnetic energy E free /E . In addition, we compute the total unsigned magnetic flux from the SHARP vector magnetograms. \nFigure 2 shows an overview of the computed time series (blue lines) and the corresponding time derivatives (orange lines). The unsigned magnetic flux shows a continuous increase over the modeled time frame, which is also reflected by the energy quantities. A notable shortterm flare-related change in the energy ratio only occurs for the X4.0 and X5.8 flares. This is most probably attributed to the continuous flux emergence which dominates the global energy evolution. \nFrom the corresponding time derivatives plotted in Fig. 2, we can identify a much better correspondence to the AR's flaring activity. We indicate M- and X-class flares as shaded areas from flare start- to end-time. For all the X-class flares, one can see a distinct depletion of magnetic energy, most prominently for the free magnetic energy and the ratio of free to total magnetic energy E free /E . Note that the trend in the energy change profiles is still positive due to the continuous strong flux emergence. For the X1.1 flare on May 9 17:23 UT we note that the primary drop in magnetic energy occurs after the GOES flare end time. From the SDO/AIA \nEUV images in Movie 1 , an increased emission at the central flare location and across the AR can be observed up to two hours past the designated flare end time, in agreement with the modeled energy decrease. Similar to the magnetic energy release during X-class flares, also a series of M-class flares can result in a strong energy decrease, as can be seen from the flares on May 10 18:30 19:00 UT and on May 9 06:00 UT (black arrows). Also the M-class flares prior to the X1.0 flare on 2024 May 8 led to a noticeable drop in ∆ E total . \nDespite the strong energy depletion derived on May 11 01:00 UT, which is in agreement with the observed X5.8 flare, we caution that the used magnetograms are obtained close to the limb and suffer from increased uncertainties (e.g., spectropolarimetric inversions, azimuth disambiguation) and projection effects.", '3.2. Major solar flares': 'We further investigate the change of magnetic energy during seven flare events in the spatially resolved maps, in order to identify the locations of the stored and the released free magnetic energy within the AR. We primarily consider five X-class flares that occurred between May 8 and May 11, and analyze in addition two M-class flares for context. Note that the X1.0 flare on May 8 01:33 is not related to AR 13664, the X1.0 flare on May 8 21:08 UT occurred during the HMI data gap, and the \nX1.5 flare on May 11 11:15 UT is too close to the solar limb to obtain reliable extrapolations. In Fig. 3 we show maps of EUV emission, time-integrated over the duration of the event, the free magnetic energy prior to the flare eruption, and difference maps in magnetic energy between the post- and pre-flare field. The red contours outline the 10 12 erg/cm 2 threshold of the depleted free magnetic energy E -(Eq. 12). \nThe maps of free magnetic energy highlight regions with increased flaring potential. The increase in free magnetic energy directly links to the major flux emergence in the eastern part of AR 13664, and also relates to the locations of flare occurrence. \nFor all X-class flares we note a good agreement between locations of magnetic energy change and increased EUV emission (see Fig. 3). The X1.0 (May 8 - 04:37 UT), the X2.3 (May 9 - 08:45 UT) and the X5.8 (May 11 - 01:10 UT) flares show additional signatures of flux emergence (red regions) during the major energy decrease (surrounding blue regions). \nFor comparison, we show the same maps also for two of the more than 50 M-class flares that the AR has produced (first and fourth row in Fig. 3). Here, we note a much smaller energy release. Specifically, when comparing the central part of the AR during the M3.1 flare (May 9 - 11:52 UT), which occurs in the western part of the active region, the magnetic energy change mainly shows increases due to emerging flux. In contrast, for the weaker M2.1 flare (May 7 - 20:18 UT) we note regions of energy decrease that align well with the enhanced EUV emission in the central part of the active region. \nIn Table 1, we summarize the total energy difference and the released magnetic energy for the individual Xand M-class flares. Here, we compare the total energy difference (∆ E total ; Eq. 11) and the estimated total depleted magnetic energy (∆ E -; Eq. 12). We use the first extrapolation prior to the flare start and past the flare end, as defined by the GOES catalog, to compute the energy difference. The continuous energy build-up largely dominates the global energy evolution during the flares, which only results in an energy decrease during the two largest flares (X4.0 and X5.8) and the early M2.1 flare. When only considering the depleted magnetic energy ∆ E -, we can provide an upper estimate of the released magnetic energy during the flare. Here, most of the X-class flares reach > 10 32 erg, in agreement with estimates from X-ray observations, where the estimated total flare energy ranges from 2 . 4 · 10 32 to 6 . 0 · 10 32 erg for flares between X1.0 and X2.8 (Woods et al. 2006). The largest flare in the sequence (X5.8) reaches an upper estimate of 2 . 1 · 10 32 erg in depleted magnetic energy. The modeled M3.1 flare is one magnitude lower in energy. \nFigure 3. Major solar flares during 2024 May 7-11 and their magnetic energy distribution. The left column shows for each event the AIA 94 ˚ A EUV map time-integrated over the flare duration. The middle column shows the corresponding maps of free magnetic energy E free , which outline the primary regions where energy can be released. The right column shows the change in magnetic energy, indicating where energy is decreased (blue) or increased (red) over the event duration. The red contours in the integrated EUV maps indicate the 10 12 erg/cm 2 level of energy depletion (∆ E -). For all flare events we note a correspondence between the observed EUV emission and regions of magnetic energy change, both in terms of energy depletion (extended blue regions) and energy increase (red regions). \n<!-- image -->', '3.3. Separatrix layers': 'From the NLFF coronal field extrapolations, we can identify distinct layers that separate different magnetic domains (i.e., quasi-separatrix layers; Demoulin et al. 1996). Figure 4 shows AR 13664 during flaring activity at 2024 May 8 03:00 UT. As can be seen from the EUV maps, the flare loops clearly outline a dark separating layer (see arrow in Fig. 4). We compare this observation to our modeled integrated current density, the squashing factor at a horizontal slice at 5 Mm, and a field line plot. The squashing factor indicates the cross-section of the modeled magnetic domains, where white lines represent the separation layers where field lines strongly diverge (Titov et al. 2002). Similarly, the field line plot shows the magnetic topology of the subframe of the AR. \nTable 1. Energy change during the major solar eruptions. ∆ E total corresponds to the magnetic energy difference between the flare start- and end-time. ∆ E -refers to the integrated negative magnetic energy change over the flare duration, which provides an upper estimate of released magnetic energy during the event. The X1.0 flare on May 8 - 21:08 UT falls into the HMI data gap, and the active region is too close to the solar limb on May 11 - 11:15 UT (X1.5 flare). Therefore, both flares are not considered in this analysis. \nSpecifically, the field line traces show the strong separation between the northern flux rope channel and the southern region. The central part of the AR ( x ≈ 250 Mm, y ≈ 75 Mm) shows highly convoluted domains and multiple highly twisted fields. \nThe integrated current density shows the strongest correspondence to the EUV observation, where we can clearly identify the bright regions as twisted fields in the current density map. In particular, current-free layers show a good agreement both with lines of high squashing factor and the EUV observations. We associate these layers of low current density as a trace of quasi separatrix layers that separate magnetic domains in the active region (c.f. flux-free regions; Janvier et al. 2015). \nThis is particularly relevant for the modeled flares, where we throughout observe a re-configuration in the current maps associated to solar flares. In Movie 1 we show the temporal evolution of the current density map, where we note rapid motion and closing in of currentfree layers prior to major solar eruptions. We further discuss this in Sect. 3.4, where we provide an analysis of the current density evolution of the X4 flare on 2024 May 10. Figure 1c further highlights the features in the current density maps that we associate with flaring activity and the accompanying movie demonstrates the relation to the flare occurrence. \n3.4. X4.0 flare on 10 May 2024 and associated filament eruption \nFigure 4. Example of a supposed separatrix layer and relation to the modeled magnetic field on 2024 May 8-03:00 UT. a) The observed vertical component of the magnetic field with outline of the subregion which is illustrated in panels b)-e). The red arrow indicates the supposed separatrix layer in the SDO/AIA EUV observation in the 131 ˚ A filter during a solar flare event (b). We can identify the separatrix layer as current-free layer in the vertically integrated current density (c). The squashing factor at a height of 5 Mm outlines different magnetic domains (d). Here, the separatrix layer appears as a clear separation line. From the magnetic field line plot we can further identify the different magnetic domains in the solar atmosphere (e). The color coding refers to the local current density j . \n<!-- image --> \nIn this section, we focus on the X4.0 flare associated with a filament eruption that occurred on 2024 May 10 with a GOES start time of 06:27:00 UT and a peak time of 06:54:00 UT. Figure 5 shows an overview of the preflare NLFF magnetic field extrapolation at 05:58:43 UT. \nFigure 5. Magnetic field configuration and parameters prior to the X4.0 solar flare and associated filament eruption, as derived from a pre-flare NLFF coronal magnetic field extrapolation on 2024 May 10-05:58:43 UT. a) Contours of the AIA 1600 ˚ A flare ribbons on top of the pre-flare SHARP LOS magnetogram. The flare contours mark regions that exceed 150, 600, and 2400 ct/s between 06:30 and 07:00 UT. The blue line marks the position of the slice used for panel d. Panel b and c show a zoom-in of the primary flaring region. b) Squashing factor ( Q ) map at the bottom layer of the extrapolation volume ( z = 0 Mm). The red contours are the same as in panel a. c) Overview of selected magnetic field structures, showing the filament channel colored by the local current density, a fan-spine structure in green, and overlying fields connecting both ribbons in gray. d) Twist ( T w ) map along the vertical plane indicated by the blue line in panel a. An animation of the magnetic field line plot is available in the online journal. The series shows the evolution of the magnetic topology from 05:58:43 to 12:58:43 UT, where we note a reconfiguration of the fan-spine structure and a contraction of the central flux rope. \n<!-- image --> \nPanel a shows the cumulative flare ribbons and kernels over the impulsive phase of the flare as observed by the AIA 1600 ˚ A channel. The two main flare ribbons are located in the northeastern part of the AR, with the southern ribbon extending along a narrow region of positive magnetic polarity and the northern ribbon covering a weaker region of negative magnetic polarity. The squashing factor map (panel b) shows that this main flare region was strongly separated from the rest of the AR by a separatrix layer running along the narrow positive polarity region. \nThis configuration resembles the one we already found two days earlier (2024 May 8), shown in Fig. 4. Similarly to that previous event, during the X4.0 flare the southern part of the AR is also involved in the flare, although the main eruption takes place in the strongly separated northeastern part (see the AIA 131 ˚ A time series in Fig. 6). We observe the formation of two separate flare arcades in EUV that both connect to the southern flare ribbon in the narrow positive polarity region. \nPanel c in Fig. 5 shows the magnetic field configuration of the pre-flare NLFFF extrapolation, but only focuses on structures associated with the main, northern part of the event. The NLFFF extrapolation reveals strongly sheared field lines that form the filament channel and run along the polarity inversion line between the two flare ribbons. Gray field lines start from seed sources placed in selected regions within the northern flare ribbon contours derived from AIA 1600 ˚ A maps. They reveal parts of the field structure above the filament that will be stretched by the erupting filament and reconnected beneath it, resulting in the flare ribbons and flare loops spanning between them. Green field lines show a fan-spine structure near the western anchor point of the filament. This structure corresponds to a circular portion in the AIA 1600 ˚ A contours that connects the two flare ribbons in the west and follows the same strong separatrix layer shown in Fig. 5b (c.f., Masson et al. 2009). Some of the green field lines of the fan-spine structure closely follow the filament channel towards its western anchor point, while others extend further towards the northern part of the northern flare ribbons. This suggests that the fan-spine structure was involved in the reconnection process during the filament eruption. \nThe movie accompanying Fig. 5 shows the temporal evolution of field structures started from fixed seed sources for each magnetic field extrapolation between 05:58:43 and 12:58:43 UT on 2024 May 10 ( Movie 2 ). The movie shows a shift of the overlying fields, a northward drift and contraction of the western half of the filament channel, and a decay of the extended connec- \ntion of the fan-spine structure to the northern flare ribbon. Note that the SHARP region moves slightly in relation to the AR during this period and the magnetic field evolves, but the seed sources from which the field lines are drawn are stationary. This means that the movie does not show the evolution of fixed field lines, and some changes can also be due to the relative motion of the magnetic field and the seed sources. \nIn panel d we show the twist in the pre-flare magnetic field along a cross-section through the western part of the filament. The filament channel is part of the negative twist region between Y = 100 to 120 Mm. This configuration shows an anemone type structure, where the small positive flux region (red) is overlaid by the magnetic flux rope structure (blue in panel d). This shows similarity to the mini-filament eruption mechanism proposed in Sterling et al. (2015) and could also be a triggering mechanism in this large-scale filament eruption. \nFig. 6 shows maps of integrated current density and SDO/AIA 131 ˚ A maps at 1 hour cadence. The blue arrow indicates the magnetic flux rope which undergoes a re-configuration during the flare event and is supposedly associated with the observed filament eruption. By comparing the flux rope before (2024 May 10-06:00) and after the eruption (2024 May 10- 08:00), a significant topology change can be observed, where the flux rope shows a larger extent after the flare eruption. The black arrow indicates the supposed separatrix configuration, where magnetic field lines of opposite polarity are compressed (c.f., squashing factor in Fig. 5). The reconfiguration during the solar flare can be clearly identified from the observation on 2024 May 10-09:00 UT, where the magnetic domains are merged, and the northern part of the dark separation layer opens, forming the extended magnetic flux rope.', '4. DISCUSSION': "AR 13664 showed a rapid flux emergence from 2024 May 7 that lead to a series of M- and X-class flares, causing a multitude of X- and M-class flares and associated CMEs and being the source of the largest geomagnetic storm with a peak Dst index of -412 nT of the last two decades. \nIn this study, we provided an in-depth overview of the magnetic topology and evolution of AR 13664, while it was present on the Earth-facing solar hemisphere. We applied NLFF extrapolations to study the 3D magnetic topology and the evolution of the energy build up and flare-associated energy releases from 2024 May 5-00:00 to 11-04:36 UT. All our extrapolation results are publicly available (Section 5) as HDF5 and VTK files. We \nFigure 6. Comparison of the temporal evolution of the modeled vertically integrated current density (left) and the observed SDO/AIA 131 ˚ A filtergrams (right) for the X4.0 flare and filament eruption on 10 May 2024. The blue arrows indicate the reconfiguration of the magnetic flux rope, visible in the current density map. The black arrow highlights the current-free layer which undergoes substantial reconfiguration during the flare and associated filament eruption. \n<!-- image --> \nalso provide the model save states, which are storage efficient ( ∼ 2 MB per file), and codes for extracting and evaluating the data cube 1 . \nThe resulting data set can be used to (1) further study the relation of the solar flares to coronal mass ejections (Murray et al. 2018). (2) Analyze magnetic parameters from the 3D field prior to the flare eruptions (Kors'os et al. 2024; Kusano et al. 2020). (3) Analyze the temporal evolution of the modeled flaring events (e.g., Purkhart et al. 2023). (4) Complement related observations, from both in-situ and remote sensing instruments (e.g., Hayakawa et al. 2024). \nOur extrapolations show a realistic approximation of the observed flaring activity. The derived changes in free magnetic energy shows a direct relation to the solar flares, most prominently for the strong X-class flares and for the occurrence of multiple M-class flares over a short time frame (Sect. 3.1). In contrast, the integrated quantities of magnetic energy and free magnetic energy are subject to permanent energy increase which complicates the direct identification of flare related energy release processes. For the estimate of total released energy we are neglecting regions of energy increase, providing an upper estimate of modeled released magnetic energy. For the total depleted magnetic energy E -, we estimate released magnetic energies in the order of > 10 32 erg for X-class flares, while M-class flares typically show a lower energy difference (order of 10 31 erg). To better understand NLFF modeled magnetic energy release processes in the presence of emerging flux, a comparison to MHD simulations could give further insights (e.g., Chen et al. 2023). \nWe provided a detailed analysis of the X4.0 flare on 2024 May 10-06:27 UT (Sect. 3.4) to demonstrate how our NLFF magnetic field extrapolations can be applied to the interpretation of solar flares and filament eruptions. The magnetic field topology in a pre-flare extrapolation shows a high degree of agreement with AIA 1600 ˚ A flare ribbons and AIA EUV images. They reveal the main event geometry consisting of the filament channel, overlying loops, and a fan-spine structure. The squashing factor map derived from this extrapolation reveals a strong separatrix layer running along the southern flare ribbon, highlighting its additional magnetic connection to the southern part of the AR, which was also active during this flare. This is in agreement with previous studies that compared reconnection signatures with the location of quasi-separatrix layers (e.g., Dud'ık \net al. 2014; Zhao et al. 2014; Dalmasse et al. 2015; Janvier et al. 2015). Our analysis suggest that we can use layers with strong gradients of current density as reference for separatrix layers that outline magnetic domains (Sect. 3.3). In Fig. 6 we show the magnetic re-configuration during the X4.0 flare, where our model suggests the formation of a new current channel and reconfiguration of magnetic domains. The strong current density build-up, close correspondence between the flare ribbons and the quasi-separatrix layer, and the topological reconfiguration, are strong evidence for magnetic reconnection (Aulanier et al. 2012; Janvier et al. 2013, 2015). \nFinally, with this study, we provide further extensions to PINN-based NLFF modeling, by introducing the vector potential for solenoidal simulations. While this assures that the fundamental physical law is satisfied, this leads to increased deviations from the (non-divergence free) boundary condition and requires additional compute resources.", '5. DATA AVAILABILITY': 'All our extrapolation results are publicly available. \n- · Data: https://app.globus.org/file-manager? origin id=4263de78-cfdb-401e-a62b-dae3b935530a& origin path=%2F\n- · Code: https://github.com/RobertJaro/NF2\n- · Documentation for data usage: https://github. com/RobertJaro/NF2/wiki/AR-13664 \nWith this study we provide FastQSL (Zhang et al. 2022) in the NF2 framework (Jarolim et al. 2023). \nThe SDO HMI and AIA data is provided by JSOC (http://jsoc.stanford.edu/).', '6. ACKNOWLEDGMENTS': 'RJ was supported by the NASA Jack-Eddy Fellowship. This material is based upon work supported by the NSF National Center for Atmospheric Research (NCAR), which is a major facility sponsored by the U.S. National Science Foundation under Cooperative Agreement No. 1852977. We would like to acknowledge highperformance computing support from the Derecho system (doi:10.5065/qx9a-pg09) provided by NCAR, sponsored by the National Science Foundation. This research was funded in part by the Austrian Science Fund (FWF) 10.55776/I4555 (AMV, SP). \nThis research has made use of SunPy (Mumford et al. 2020; Barnes et al. 2020), AstroPy (Astropy Collaboration et al. 2013), PyTorch (Paszke et al. 2019) and Paraview (Ahrens et al. 2005). \nFacilities: \nSDO (HMI, AIA). \nSoftware: AstroPy (Astropy Collaboration et al. 2013, 2018), SunPy (Barnes et al. 2020; Mumford et al. \n2020; Glogowski et al. 2019), Pytorch (Paszke et al. 2019), Paraview (Ahrens et al. 2005), NF2 (Jarolim et al. 2023).', 'A. QUALITY': 'We compute quality metrics over the full extrapolation series, to assure the consistency of our results. In Fig. 7 we compare the deviation from the boundary condition, the current-weighted angle between the magnetic field and current density ( θ J ), and the normalized divergence ( L div = | ⃗ ∇· ⃗ B | / ∥ B ∥ ), as well as the arcsine of the θ J angle. The dashed line indicates the approximate start time of increased flux emergence. Note that the divergence is computed from the grid representation based on finite-difference, while the divergence computed based on smooth derivatives is per definition zero (vector potential; < 10 -6 G/Mm). As compared to the divergence estimates in Jarolim et al. (2023) and Jarolim et al. (2024), the normalized divergence is two orders of magnitude lower when the vector potential is applied. The metrics for force-freeness ( θ J and σ J ) show slightly larger values than reported in Jarolim et al. (2023) for the λ ff = 0 . 1 configuration. This is likely caused by the enforced divergence freeness. The decrease in θ J over the time series is related to the flux emergence (i.e., increase in J). As can be seen from the fluctuating behaviour at the end of the sequence, the high projection angle causes significant variations. For this reason, we refrain from performing extrapolations at later points in time. \nWith the use of the vector potential the solutions are per definition divergence-free. However, since the observations are typically not divergence-free (due to e.g., noise, grid scaling, neglected corrugation, errors from inversion and disambiguation method) this results in an additional deviation from the boundary condition as compared to direct modeling of the magnetic field. Our uncertainty estimates further emphasize that observations close to the solar limb suffer from increased errors (see App. B). \nThe modeling of the vector potential has further implications for the computation requirements. The extrapolation from scratch requires about 7 hours on 4 NVIDIA A100 GPUs. For the computation of the series we require ∼ 5 min per time step with the same computational setup. Therefore, the extrapolation of the time series can still be performed in quasi real-time, but requires more computational power and computing time than direct modeling of the magnetic field B ( ≈ 1 hour for an extrapolation from scratch).', 'B. UNCERTAINTY QUANTIFICATION': 'We use ensemble modeling to estimate model dependent uncertainties. For all our model fitting we start from a randomly initialized model and use randomly sampled points during the optimization. From this, we expect that regions that are less constrained by the boundary data (e.g., weak field regions, larger errors in the magnetograms), or where the boundary condition and force-free model are in disagreement, show an increased difference among the individual extrapolations. \nFor our uncertainty estimation we perform five individual extrapolations and compute the standard deviation across the ensemble \nδB = √ √ √ √ ∑ N i ∥ ∥ ∥ ⃗ B i -⃗ B ∥ ∥ ∥ N , (B1) \nwhere ⃗ B refers to the average magnetic field vector per grid cell and N to the number of ensemble runs ( N = 5). For the visualization of uncertainty maps we compute the average uncertainty along the vertical axis. \nThe ensemble modeling requires parallel model training from scratch, therefore we can only provide uncertainty estimates for selected examples. Here, we provide uncertainty estimates for extrapolations on 2024 May 10 06:00 UT prior to the X4.0 flare, and on 2024 May 11 01:00 UT prior to the X5.8 flare. Specifically, the later extrapolation uses observations that are close to the solar limb. In Fig. 8 we show maps of integrated current density of the ensemble runs, the corresponding standard deviation of the current maps, and the uncertainty maps ( δB ). The current density maps show throughout very similar configuration, indicating that our method converges to similar solutions in terms of magnetic topology. This can also be seen from the low standard deviation. \nFigure 7. Metrics for divergence freeness (top) and force-freeness (bottom) over the evolution of the active region. The dashed line indicates the start of major flux emergence (c.f., Fig. 1). The decrease of θ J can be related to the increase in magnetic currents. \n<!-- image --> \nThe uncertainty maps highlight the regions of increased variation, which are primarily located in the southern part and the weak field region between the two polarities. Importantly, the central complex region with mixed polarities shows low uncertainties, which indicates that this region is well constrained by the boundary condition. For both extrapolations we note increased uncertainties close to the boundaries. This further highlights the discrepancy of the assumed potential field boundary conditions. For the second extrapolation, close to the solar limb, we note a drastic increase in uncertainty, particularly towards the solar west. This suggests that extrapolations close to the solar limb should be excluded from further evaluation, particularly for points that exceed 60 · longitude.', 'REFERENCES': 'Ahrens, J., Geveci, B., & Law, C. 2005, The visualization \nhandbook, 717 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 \nAstropy Collaboration, Price-Whelan, A. M., Sip"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Aulanier, G., Janvier, M., & Schmieder, B. 2012, A&A, 543, A110, doi: 10.1051/0004-6361/201219311 \nBarnes, W. T., Bobra, M. G., Christe, S. D., et al. 2020, The Astrophysical Journal, 890, 68 Bobra, M. G., Sun, X., Hoeksema, J. T., et al. 2014, SoPh, 289, 3549, doi: 10.1007/s11207-014-0529-3 Chen, F., Cheung, M. C. M., Rempel, M., & Chintzoglou, G. 2023, ApJ, 949, 118, doi: 10.3847/1538-4357/acc8c5 Dalmasse, K., Chandra, R., Schmieder, B., & Aulanier, G. \n2015, A&A, 574, A37, doi: 10.1051/0004-6361/201323206 \nFigure 8. Uncertainty maps prior to the X4.0 and X5.8 flare. We show the vertically integrated current density of the five ensemble runs and the corresponding standard deviation at the top. The bottom plots show the reference magnetogram and the corresponding uncertainty maps. The stronger projection effects on 2024 May 11-01:00 leads to a substantial increase in uncertainty towards the west limb. \n<!-- image --> \nDemoulin, P., Henoux, J. C., Priest, E. R., & Mandrini, C. H. 1996, A&A, 308, 643 Dud´ık, J., Janvier, M., Aulanier, G., et al. 2014, ApJ, 784, 144, doi: 10.1088/0004-637X/784/2/144 Fleishman, G., Mysh\'yakov, I., Stupishin, A., Loukitcheva, M., & Anfinogentov, S. 2019, ApJ, 870, 101, doi: 10.3847/1538-4357/aaf384 Fletcher, L., Dennis, B. R., Hudson, H. S., et al. 2011, SSRv, 159, 19, doi: 10.1007/s11214-010-9701-8 Forbes, T. G., Linker, J. A., Chen, J., et al. 2006, SSRv, 123, 251, doi: 10.1007/s11214-006-9019-8 Glogowski, K., Bobra, M. G., Choudhary, N., Amezcua, A. B., & Mumford, S. J. 2019, Journal of Open Source Software, 4, 1614, doi: 10.21105/joss.01614 Green, L. M., T¨or¨ok, T., Vrˇsnak, B., Manchester, W., & Veronig, A. 2018, SSRv, 214, 46, doi: 10.1007/s11214-017-0462-5 Gupta, M., Thalmann, J. K., & Veronig, A. M. 2021, A&A, 653, A69, doi: 10.1051/0004-6361/202140591 Hayakawa, H., Ebihara, Y., Mishev, A., et al. 2024, arXiv e-prints, arXiv:2407.07665, doi: 10.48550/arXiv.2407.07665 Janvier, M., Aulanier, G., & D´emoulin, P. 2015, SoPh, 290, 3425, doi: 10.1007/s11207-015-0710-3 Janvier, M., Aulanier, G., Pariat, E., & D´emoulin, P. 2013, \nA&A, 555, A77, doi: 10.1051/0004-6361/201321164 \nJarolim, R., Thalmann, J. K., Veronig, A. M., & Podladchikova, T. 2023, Nature Astronomy, doi: 10.1038/s41550-023-02030-9 Jarolim, R., Tremblay, B., Rempel, M., et al. 2024, ApJL, 963, L21, doi: 10.3847/2041-8213/ad2450 Karniadakis, G. E., Kevrekidis, I. G., Lu, L., et al. 2021, Nature Reviews Physics, 3, 422, doi: 10.1038/s42254-021-00314-5 Kors´os, M. B., Jarolim, R., Erd´elyi, R., et al. 2024, ApJ, 962, 171, doi: 10.3847/1538-4357/ad18bd Kusano, K., Iju, T., Bamba, Y., & Inoue, S. 2020, Science, 369, 587, doi: 10.1126/science.aaz2511 Leka, K. D., & Barnes, G. 2007, ApJ, 656, 1173, doi: 10.1086/510282 Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, SoPh, 275, 17, doi: 10.1007/s11207-011-9776-8 Masson, S., Pariat, E., Aulanier, G., & Schrijver, C. J. 2009, ApJ, 700, 559, doi: 10.1088/0004-637X/700/1/559 Mumford, S. J., Christe, S., Freij, N., et al. 2020, SunPy, v1.1.4, Zenodo, doi: 10.5281/zenodo.3871057 Murray, S. A., Guerra, J. A., Zucca, P., et al. 2018, SoPh, 293, 60, doi: 10.1007/s11207-018-1287-4 Paszke, A., Gross, S., Massa, F., et al. 2019, in Advances in Neural Information Processing Systems 32, ed. H. Wallach, H. Larochelle, A. Beygelzimer, F. d \' Alch´e-Buc, E. Fox, & R. Garnett (Curran Associates, Inc.), 8024-8035'}

2023BAAS...55c.250L

The entire Heliophysics research system hinders discourages proper worklife balance. A core issue stems from the culture of workabovelife associated with mission development and implementation but also the expectations that seem to originate from announcements by agencies. We are calling for worklife balance plans and implementation to be a criterion used for mission downselection.

2023-07-01T00:00:00Z

['2023dsss.rept..250L', '10.48550/arXiv.2306.05444', '2023arXiv230605444L', 'arXiv:2306.05444', '2023BAAS...55c.250L', '10.3847/25c2cfeb.007ec3a4']

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

WorkLife Balance Starts with Proper Deadlines and Exemplary Agencies

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

https://arxiv.org/pdf/2306.05444.pdf

{'Work-Life Balance Starts with Proper Deadlines and Exemplary Agencies': 'Noé Lugaz 1 , Réka M. Winslow 1 , Nada Al-Haddad 1 , Christina O. Lee 2 , Sarah K. Vines 3 , Katharine Reeves 4 , Amir Caspi 5 , Daniel Seaton 5 , Cooper Downs 6 , Lindsay Glesener 7 , Angelos Vourlidas 3 , Camilla Scolini 1 , Tibor Török 6 , Robert Allen 3 , Erika Palmerio 6 \n1 Space Science Center, University of New Hampshire; 2 Space Science Laboratory, University of California - Berkeley, 3 Johns Hopkins University/Applied Physics Laboratory, 4 Center for Astrophysics, Harvard Smithsonian, 5 Southwest Research Institute, 6 Predictive Science Inc., 7 University of Minnesota \nSynopsis : Diversity, equity and inclusion (DEI) programs can only be implemented successfully if proper work-life balance is possible in Heliophysics (and in STEM field in general). One of the core issues stems from the culture of \'work-above-life\' associated with mission concepts, development, and implementation but also the expectations that seem to originate from numerous announcements from NASA (and other agencies). The benefits of work-life balance are well documented; however, the entire system surrounding research in Heliophysics hinders or discourages proper work-life balance. For example, there does not seem to be attention paid by NASA Headquarters (HQ) on the timing of their announcements regarding how it will be perceived by researchers, and how the timing may promote a culture where work trumps personal life. The same is true for remarks by NASA HQ program officers during panels or informal discussions, where seemingly innocuous comments may give a perception that work is expected after "normal\' work hours. In addition, we are calling for work-life balance plans and implementation to be one of the criteria used for down-selection and confirmation of missions (Key Decision Points: KDP-B, KDP-C). \nThe benefits of proper work-life balance are well documented in peer-reviewed publications for more than 30 years (e.g., see Crosby, 1991). They include benefits to the individual, their family but also to the employers themselves, in terms of increased productivity, worker retention, improved recruitment capabilities and increased loyalty, among others (see Sirgy & Lee, 2017). This white paper does not dive into this large body of research but invite the Decadal Survey committee to hear from researchers from the social sciences if they need further arguments. Instead, we detail some of the current organization of research in Heliophysics that is detrimental to work-life balance and offer potential solutions. \nAt the core of work-life balance is the peer pressure and perceived (or real) expectations that individual researchers feel in their work in Heliophysics. Numerous small, seemingly inconsequential announcements and decisions can create an overall culture of \'work first, life second.\' Agencies should carefully consider the ramifications of their announcements, even informal, before releasing them and train program officers for careful considerations of any announcement related to short deadline or late/long work hours, even in \'informal settings\' (for example while running a panel). In addition, the move to partially online/remote-work has made many of existing issues significantly more prominent and created new issues with work-life balance. We discuss a few examples below. We invite agencies to consider the consequences of deadlines and announcements. We propose using the actual implementation of work-life balance plans during a mission Phases A and B as one of the selection criteria for missions during key decision points (KDPs). This would ensure that the culture of \'a successful mission takes a 24/7 work commitment\' doesn\'t continue into the next decades.', 'Example 1: Overall Work Culture': "Many informal discussions between the lead author and graduate students and post-doctoral researchers at COSPAR 2022 centered around questions such as 'can you be successful while taking at least one day off every week?' (where week is the 7-day week) or 'do you have nightmares about your work?'. Discussions within the group of authors of this white paper highlighted that these are common questions and extend to later career stages. Overall, this highlights that issues of work culture and work-life balance within STEM in general and geospace/heliophysics science in particular, are especially dire. This issue is compounded by multiple developments: a) The increase over the past decade of data science jobs and engineering jobs in 'New Space' companies with high pay; b) The decrease in effective (inflation-adjusted) salary for many early-career scientists, associated with salary increases from universities or laboratories that fall below the level of inflation and increases in cost-of-living (especially rent, which has increased beyond inflation). Taken together, early-career scientists are often confronted with the prospect of having spent around a decade in higher education to have a low-paid job with non-existent work-life balance.", 'Solutions:': "- ∞ Review work-life balance as part of Key Decision Points (KDPs) for missions. At a minimum, this would be a review of the work-life balance plan submitted within the mission proposal and its implementation. It may require changes in the scope of the Standard Review Board (SRB) Handbook and the addition of members in the SRB that can assess this plan and its implementation. This could also be in the form of additional reviews if there is staff turnover over a given threshold, to see if this is simply due to staff leaving for more exciting projects or if this is related to burnout or a caustic work culture in the team. Having a high-level requirement within a mission regarding work-life balance that can be tracked (employee turnover, number of vacation days taken) is another venue that could be taken.\n- ∞ Ensure that holidays and vacation time is integrated into the instrument and mission master schedule and that key reviews are not held during or just after major holidays (Thanksgiving, end-of-the year). This could be connected to the work-life balance plan. Heliophysics missions often do not have extremely tight launch constraints, as is the case for Planetary missions (for which, targeting a specific planet, moon or planetary body requires a launch in a tight window). For example, it is possible to launch to L1 more than 3 weeks every month. As such, Heliophysics mission schedule can more easily take into consideration work-life balance. NASA should be willing to pay for the additional costs, if any, associated with a slightly longer schedule.\n- ∞ Integrate not only a DEI Plan but also a training plan for instrument leads, PIs, deputies, and project managers regarding work-life balance as part of the proposal.\n- ∞ Develop ways for a lower 'cost' of entry (in term of work-life balance) into missions for early-career scientists, maybe through the development of SIMPLEX-like concept. While reducing the number of pages for the proposal is tempting, it can have the opposite effect of requiring a more polished proposal unless the risk tolerance of TMC maturity is further decreased for step-1 selections. While the 'cost' in terms of proposal preparation could be similar for a proposal submitted to SIMPLEX as compared to H-FORT, the higher budget for SIMPLEX could reduce the pressure on the PI and Project Manager (PM) to juggle multiple projects.\n- ∞ NASA should also consider offering access to laboratories for concept studies (like for the decadal survey concept studies) after a pre-selection based on science and/or following the SIMPLEX example of funding a Phase A/B so that there is a reward to be selected for a concept study (rather than having to invest more 'unpaid' time for a Phase A concept study). This latter option also improves equity for proposers from institutions that do not have the internal resources to invest in unpaid mission design exercises, or prior mission heritage - allowing the science to be driven by the 'right' institution by partnering with existing mission design houses.", 'Example 2: Work Culture Surrounding Instruments and Missions': "Participation in a mission is often seen as the ultimate achievement in Heliophysics. Early- and mid-career researchers are often welcomed within instrument or mission teams with 'badge of honor/horror' stories, such as 'that day we ran our first EM testing during Christmas,' or 'remember this week of overnighters we pulled before the proposal submission.' Such stories are obviously detrimental to the perception of work requirement and work-life balance in STEM, and what it takes to run a successful mission or build a successful instrument. For early- and mid-career researchers proposing or participating in their first mission, at the Mission of Opportunity or Explorer level, the expectations in terms of proposal preparations are often daunting unless they are being mentored actively by researchers who have had leadership role within missions, or unless they are parts of centers (such as NASA centers or APL) where there is significant level of internal support to develop mission concepts (like the presence of the APL Concurrent Engineering, or ACE, laboratory). Mentoring is often offered through 'patronage,' where researchers that are being mentored to become future instrument leads or part of mission management are chosen because they are at the right institute at the right time. While such a model might be appropriate for instrument leads, where there is an institutional heritage, it is not appropriate for other mission roles, such as PI, deputy PI and Project/Mission Scientist, and science working group lead, who do not require institutional heritage.", 'Example 3: Schedule of Remote Review Panels': "Before COVID, most NASA panels used to be in person in the greater DC area or in San Antonio, TX. Expectations of work in the evening used to make (some) sense when researchers were staying in a hotel away from their family. The same is, however, not true for remote panels where the panelists are logging in from their home. The problem is compounded by the three to six time zones (including Hawaii and Alaska) that the United States spans. Some NASA panels have been starting at 7am PT or running beyond 7pm ET with the expectation that researchers will do \nadditional writing after the 'day' is finished. First, this gives a wrong impression from NASA Headquarters program officers regarding the expectations that NASA has in term of work schedule, since panels are one of the only ways where early and mid-career scientists interact with NASA HQ program officers. Second, this puts exceptional pressure on early and mid-career scientists, especially (but not only) women who are more often in charge of childcare and have a larger share of housework. Participation in panels is voluntary and eligible panelists are given a modest honorarium. However, participation in panels is seen by many early-career scientists as a necessary a) to understand the selection process, b) to build up their resume. While remote participation in panels eliminates travel time and provides more equitable access to panel participation by people who cannot easily travel (e.g., due to family obligations), care must be taken to ensure that remote participation is not more onerous than in-person.", 'Solution:': '- ∞ In general, announcements should be avoided on Friday afternoon or just before a holiday.\n- ∞ Significant deadlines (the white paper one, the 2022 SMEX ones) are often rumored to be forthcoming for months before the actual announcement. Agencies and NASEM should make more use of official announcements of a forthcoming deadline within a particular quarter (say summer 2022 for the white paper, fall 2022 for the SMEX). This would enable early-career researchers or researchers without a developed network to 1) be aware of the deadline and 2) for those without dedicated institutional funding, to prepare more adequately for this critical deadline. In general, the agencies and academies should announce a worst-case (rather than a hope-for) deadline so that they can stick to the deadline as much as possible without constant changes. In general, a longer time between the announcement and the deadline might be advisable, as it gives more time for researchers to prepare and set their own schedule.\n- ∞ We propose that major recurring proposals (Explorers, H-FORT, Strategic capabilities, DRIVE centers, etc.) are given a set of 4-6 yearly and fixed deadlines (for example March 1, May 1, July 1, September 1, November 1) and that if there is a shift in the schedule, it is to the next deadline. As NASA is already able to release the yearly ROSES omnibus on February 14 every year, we are confident that the agencies would be able to stick to these fixed deadlines. All announcements on the schedule should be early. NASA should consider the cost-benefit in increasing the time interval between the final AO publication and due date from 90 days to 120 or beyond.', 'Example 4: The Decadal Survey White Paper Deadline and Other Announcements': 'The white paper deadline was announced on June 24 with a deadline of August 18 (pushed since then to August 24 and to September 7). The initial time period for white paper preparation was under 8 weeks (and under 7 when considering delays in broad announcements to the community) and corresponds to the time period of a) school summer holidays (which last from June 10 to August 22 in Fairfax county, VA for example), b) heavy conference schedule with COSPAR, TESS/SPD, the IAU General Assembly and the SHINE workshop, among others during the 2month time period, c) the release of the Heliophysics SMEX draft AO in June 22, and the deadline for the Heliophysics Flight Opportunities and Technology and Instrument Development for Science in August 31/September 1, d) the time when many researchers do take vacation, especially after two years of restricted travel due to the COVID situation. \nWhile this is just one example, schedules for other announcements have also been hard to understand. For example, the community announcement for the 2022 SMEX and Mission of Opportunity call was sent at 2pm ET on 2021 December 23 rd . As the final AO was originally to be released in June 2022, it is unclear why the initial announcement could not wait until early 2022 but had to instead occur just before a major holiday season. In addition, the final AO initially expected in June 2022 was then changed to August, then changed to September. As of this WP deadline, the SMEX 2022 proposal target date is now planned to be due in December 2022. Forward planning for things like conference attendance (e.g., AGU Fall meeting, which is taking \nplace less than a week after the current deadline), and scientific collaborations for people involved in large mission proposals is impossible when projected deadlines for these proposals are constantly shifting, as has happened with the current round of Heliophysics SMEX call. This also applies to holidays and vacations, for which planning is nearly impossible.', 'References:': 'Crosby, F. J. (1991). Juggling: the unexpected advantages of balancing career and home for women and their families . New York: Free Press. \nSirgy, M., Lee, DJ. Work-Life Balance: an Integrative Review . Applied Research Quality Life 13 , 229-254 (2018). https://doi.org/10.1007/s11482-017-9509-8'}

2023ApJ...951L...8A

We report multiple lines of evidence for a stochastic signal that is correlated among 67 pulsars from the 15 yr pulsar timing data set collected by the North American Nanohertz Observatory for Gravitational Waves. The correlations follow the HellingsDowns pattern expected for a stochastic gravitationalwave background. The presence of such a gravitationalwave background with a powerlaw spectrum is favored over a model with only independent pulsar noises with a Bayes factor in excess of 10SUP14SUP and this same model is favored over an uncorrelated common powerlaw spectrum model with Bayes factors of 2001000 depending on spectral modeling choices. We have built a statistical background distribution for the latter Bayes factors using a method that removes interpulsar correlations from our data set finding p 10SUP3SUP 3 for the observed Bayes factors in the null nocorrelation scenario. A frequentist test statistic built directly as a weighted sum of interpulsar correlations yields p 5 10SUP5SUP to 1.9 10SUP4SUP 3.54. Assuming a fiducial f SUP23SUP characteristic strain spectrum as appropriate for an ensemble of binary supermassive black hole inspirals the strain amplitude is 2.40.60.7times 1015 median 90 credible interval at a reference frequency of 1 yrSUP1SUP. The inferred gravitationalwave background amplitude and spectrum are consistent with astrophysical expectations for a signal from a population of supermassive black hole binaries although more exotic cosmological and astrophysical sources cannot be excluded. The observation of HellingsDowns correlations points to the gravitationalwave origin of this signal.

2023-07-01T00:00:00Z

['10.48550/arXiv.2306.16213', '2023arXiv230616213A', '10.3847/2041-8213/acdac6', 'arXiv:2306.16213', '2023ApJ...951L...8A']

['Gravitational waves', 'Gravitational wave astronomy', 'Millisecond pulsars', 'Radio pulsars', 'Supermassive black holes', '678', '675', '1062', '1353', '1663', 'Astrophysics - High Energy Astrophysical Phenomena', 'General Relativity and Quantum Cosmology']

The NANOGrav 15 yr Data Set Evidence for a Gravitationalwave Background

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

https://arxiv.org/pdf/2306.16213.pdf

{'The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background': "Gabriella Agazie, 1 Akash Anumarlapudi, 1 Anne M. Archibald, 2 Zaven Arzoumanian, 3 Paul T. Baker, 4 Bence B'ecsy, 5 Laura Blecha, 6 Adam Brazier, 7, 8 Paul R. Brook, 9 Sarah Burke-Spolaor, 10, 11 Rand Burnette, 5 Robin Case, 5 Maria Charisi, 12 Shami Chatterjee, 7 Katerina Chatziioannou, 13 Belinda D. Cheeseboro, 10, 11 Siyuan Chen, 14 Tyler Cohen, 15 James M. Cordes, 7 Neil J. Cornish, 16 Fronefield Crawford, 17 H. Thankful Cromartie, 7, ∗ Kathryn Crowter, 18 Curt J. Cutler, 19, 13 Megan E. DeCesar, 20 Dallas DeGan, 5 Paul B. Demorest, 21 Heling Deng, 5 Timothy Dolch, 22, 23 Brendan Drachler, 24, 25 Justin A. Ellis, 26 Elizabeth C. Ferrara, 27, 28, 29 William Fiore, 10, 11 Emmanuel Fonseca, 10, 11 Gabriel E. Freedman, 1 Nate Garver-Daniels, 10, 11 Peter A. Gentile, 10, 11 Kyle A. Gersbach, 12 Joseph Glaser, 10, 11 Deborah C. Good, 30, 31 Kayhan Gultekin, 32 Jeffrey S. Hazboun, 5 Sophie Hourihane, 13 Kristina Islo, 1 Ross J. Jennings, 10, 11, † Aaron D. Johnson, 1, 13 Megan L. Jones, 1 Andrew R. Kaiser, 10, 11 David L. Kaplan, 1 Luke Zoltan Kelley, 33 Matthew Kerr, 34 Joey S. Key, 35 Tonia C. Klein, 1 Nima Laal, 5 Michael T. Lam, 24, 25 William G. Lamb, 12 T. Joseph W. Lazio, 19 Natalia Lewandowska, 36 Tyson B. Littenberg, 37 Tingting Liu, 10, 11 Andrea Lommen, 38 Duncan R. Lorimer, 10, 11 Jing Luo, 39, ‡ Ryan S. Lynch, 40 Chung-Pei Ma, 33, 41 Dustin R. Madison, 42 Margaret A. Mattson, 10, 11 Alexander McEwen, 1 James W. McKee, 43, 44 Maura A. McLaughlin, 10, 11 Natasha McMann, 12 Bradley W. Meyers, 18, 45 Patrick M. Meyers, 13 Chiara M. F. Mingarelli, 31, 30, 46 Andrea Mitridate, 47 Priyamvada Natarajan, 48, 49 Cherry Ng, 50 David J. Nice, 51 Stella Koch Ocker, 7 Ken D. Olum, 52 Timothy T. Pennucci, 53 Benetge B. P. Perera, 54 Polina Petrov, 12 Nihan S. Pol, 12 Henri A. Radovan, 55 Scott M. Ransom, 56 Paul S. Ray, 34 Joseph D. Romano, 57 Shashwat C. Sardesai, 1 Ann Schmiedekamp, 58 Carl Schmiedekamp, 58 Kai Schmitz, 59 Levi Schult, 12 Brent J. Shapiro-Albert, 10, 11, 60 Xavier Siemens, 5, 1 Joseph Simon, 61, § Magdalena S. Siwek, 62 Ingrid H. Stairs, 18 Daniel R. Stinebring, 63 Kevin Stovall, 21 Jerry P. Sun, 5 Abhimanyu Susobhanan, 1 Joseph K. Swiggum, 51, † Jacob Taylor, 5 Stephen R. Taylor, 12 Jacob E. Turner, 10, 11 Caner Unal, 64, 65 Michele Vallisneri, 19, 13 Rutger van Haasteren, 66 Sarah J. Vigeland, 1 Haley M. Wahl, 10, 11 Qiaohong Wang, 12 Caitlin A. Witt, 67, 68 Olivia Young, 24, 25 \nThe NANOGrav Collaboration \n1 Center for Gravitation, Cosmology and Astrophysics, Department of Physics, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA \n2 Newcastle University, NE1 7RU, UK \n3 X-Ray Astrophysics Laboratory, NASA Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA \n- 4 Department of Physics and Astronomy, Widener University, One University Place, Chester, PA 19013, USA \n5 Department of Physics, Oregon State University, Corvallis, OR 97331, USA \n6 Physics Department, University of Florida, Gainesville, FL 32611, USA \n- 7 Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA 8 Cornell Center for Advanced Computing, Cornell University, Ithaca, NY 14853, USA\n- 9 Institute for Gravitational Wave Astronomy and School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK\n- 10 Department of Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV 26506, USA\n- 11 Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA\n- 12 Department of Physics and Astronomy, Vanderbilt University, 2301 Vanderbilt Place, Nashville, TN 37235, USA \n13 \nDivision of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA \n14 \nKavli Institute for Astronomy and Astrophysics, Peking University, Beijing, 100871 China \n- 15 Department of Physics, New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801, USA 16 Department of Physics, Montana State University, Bozeman, MT 59717, USA\n- 17 Department of Physics and Astronomy, Franklin & Marshall College, P.O. Box 3003, Lancaster, PA 17604, USA \nDepartment of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada \n19 \nJet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA \n20 George Mason University, resident at the Naval Research Laboratory, Washington, DC 20375, USA \nCorresponding author: The NANOGrav Collaboration \ncomments@nanograv.org \n18 \n33 \n66 \n43", 'The NANOGrav Collaboration': "- 21 National Radio Astronomy Observatory, 1003 Lopezville Rd., Socorro, NM 87801, USA \n22 \nDepartment of Physics, Hillsdale College, 33 E. College Street, Hillsdale, MI 49242, USA \n23 Eureka Scientific, 2452 Delmer Street, Suite 100, Oakland, CA 94602-3017, USA \n24 School of Physics and Astronomy, Rochester Institute of Technology, Rochester, NY 14623, USA \nLaboratory for Multiwavelength Astrophysics, Rochester Institute of Technology, Rochester, NY 14623, USA \n26 Bionic Health, 800 Park Offices Drive, Research Triangle Park, NC 27709 \n27 Department of Astronomy, University of Maryland, College Park, MD 20742 \n28 Center for Research and Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771 29 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA \n30 Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046, Storrs, CT 06269-3046, USA \n31 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA \n32 Department of Astronomy and Astrophysics, University of Michigan, Ann Arbor, MI 48109, USA \nDepartment of Astronomy, University of California, Berkeley, 501 Campbell Hall #3411, Berkeley, CA 94720, USA \n34 Space Science Division, Naval Research Laboratory, Washington, DC 20375-5352, USA \n35 University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA \nDepartment of Physics, State University of New York at Oswego, Oswego, NY, 13126, USA \n37 NASA Marshall Space Flight Center, Huntsville, AL 35812, USA \n38 Department of Physics and Astronomy, Haverford College, Haverford, PA 19041, USA \n39 Department of Astronomy & Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada 40 Green Bank Observatory, P.O. Box 2, Green Bank, WV 24944, USA \n41 Department of Physics, University of California, Berkeley, CA 94720, USA \n42 Department of Physics, University of the Pacific, 3601 Pacific Avenue, Stockton, CA 95211, USA \nE.A. Milne Centre for Astrophysics, University of Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK \n- 44 Centre of Excellence for Data Science, Artificial Intelligence and Modelling (DAIM), University of Hull, Cottingham Road, Kingston-upon-Hull, HU6 7RX, UK \n45 \nInternational Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia \n46 Department of Physics, Yale University, New Haven, CT 06520, USA \n47 Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany \n48 Department of Astronomy, Yale University, 52 Hillhouse Ave, New Haven, CT 06511 \n49 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA 02138 \n50 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George St., Toronto, ON M5S 3H4, Canada \n51 Department of Physics, Lafayette College, Easton, PA 18042, USA \n52 Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA \n53 \nInstitute of Physics and Astronomy, Eotvos Lor'and University, P'azm'any P. s. 1/A, 1117 Budapest, Hungary \n54 Arecibo Observatory, HC3 Box 53995, Arecibo, PR 00612, USA \n55 Department of Physics, University of Puerto Rico, Mayaguez, PR 00681, USA \n56 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA \n57 Department of Physics, Texas Tech University, Box 41051, Lubbock, TX 79409, USA \n58 Department of Physics, Penn State Abington, Abington, PA 19001, USA \n59 Institute for Theoretical Physics, University of Munster, 48149 Munster, Germany \n60 Giant Army, 915A 17th Ave, Seattle WA 98122 \n61 Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA \n62 Center for Astrophysics, Harvard University, 60 Garden St, Cambridge, MA 02138 \n- 63 Department of Physics and Astronomy, Oberlin College, Oberlin, OH 44074, USA \n64 Department of Physics, Ben-Gurion University of the Negev, Be'er Sheva 84105, Israel \n65 Feza Gursey Institute, Bogazici University, Kandilli, 34684, Istanbul, Turkey \nMax-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut), Callinstrasse 38, D-30167, Hannover, Germany \n- 67 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208 68 Adler Planetarium, 1300 S. DuSable Lake Shore Dr., Chicago, IL 60605, USA", 'ABSTRACT': 'We report multiple lines of evidence for a stochastic signal that is correlated among 67 pulsars from the 15-year pulsar-timing data set collected by the North American Nanohertz Observatory for Gravitational Waves. The correlations follow the Hellings-Downs pattern expected for a stochastic gravitational-wave background. The presence of such a gravitational-wave background with a powerlaw-spectrum is favored over a model with only independent pulsar noises with a Bayes factor in excess \n25 \n36 \nof 10 14 , and this same model is favored over an uncorrelated common power-law-spectrum model with Bayes factors of 200-1000, depending on spectral modeling choices. We have built a statistical background distribution for these latter Bayes factors using a method that removes inter-pulsar correlations from our data set, finding p = 10 -3 (approx. 3 σ ) for the observed Bayes factors in the null no-correlation scenario. A frequentist test statistic built directly as a weighted sum of inter-pulsar correlations yields p = 5 × 10 -5 -1 . 9 × 10 -4 (approx. 3 . 5-4 σ ). Assuming a fiducial f -2 / 3 characteristic-strain spectrum, as appropriate for an ensemble of binary supermassive black-hole inspirals, the strain amplitude is 2 . 4 +0 . 7 -0 . 6 × 10 -15 (median + 90% credible interval) at a reference frequency of 1 yr -1 . The inferred gravitational-wave background amplitude and spectrum are consistent with astrophysical expectations for a signal from a population of supermassive black-hole binaries, although more exotic cosmological and astrophysical sources cannot be excluded. The observation of Hellings-Downs correlations points to the gravitational-wave origin of this signal. \nKeywords: Gravitational waves - Black holes - Pulsars', '1. INTRODUCTION': "Almost a century had to elapse between Einstein's prediction of gravitational waves (GWs, Einstein 1916) and their measurement from a coalescing binary of stellarmass black holes (Abbott et al. 2016). However, their existence had been confirmed in the late 1970s through measurements of the orbital decay of the Hulse-Taylor binary pulsar (Hulse & Taylor 1975; Taylor et al. 1979). Today, pulsars are again at the forefront of the quest to detect GWs, this time from binary systems of central galactic black holes. \nBlack holes with masses of 10 5 -10 10 M ⊙ exist at the center of most galaxies and are closely correlated with the global properties of the host, suggesting a symbiotic evolution (Magorrian et al. 1998; McConnell & Ma 2013). Galaxy mergers are the main drivers of hierarchical structure formation over cosmic time (Blumenthal et al. 1984) and lead to the formation of close massive-black-hole binaries long after the mergers (Begelman et al. 1980; Milosavljevi'c & Merritt 2003). The most massive of these (supermassive black-hole binaries, SMBHBs, with masses 10 8 -10 10 M ⊙ ) emit GWs with slowly evolving frequencies, contributing to a noiselike broadband signal in the nHz range (the GW background, GWB; Rajagopal & Romani 1995; Jaffe & Backer 2003; Wyithe & Loeb 2003; Sesana et al. 2004; McWilliams et al. 2014; Burke-Spolaor et al. 2019). If all contributing SMBHBs evolve purely by loss of circular orbital energy to gravitational radiation, the resultant GWB spectrum is well described by a simple f -2 / 3 characteristic-strain power law (Phinney 2001). \n- ∗ NASA Hubble Fellowship: Einstein Postdoctoral Fellow \nHowever, GWB signals that are not produced by populations of inspiraling black holes may also lie within the nHz band; these include primordial GWs from inflation, scalar-induced GWs, and GW signals from multiple processes arising due to cosmological phase transitions, such as collisions of bubbles of the post-transition vacuum state, sound waves, turbulence, and the decay of any defects such as cosmic strings or domain walls that may have formed (see, e.g., Guzzetti et al. 2016; Caprini & Figueroa 2018; Dom'enech 2021, and references therein). \nThe detection of nHz GWs follows the template outlined by Pirani (1956, 2009), whereby we time the propagation of light to measure modulations in the distance between freely falling reference masses. Estabrook & Wahlquist (1975) derived the GW response of electromagnetic signals traveling between Earth and distant spacecraft, sparking interest in low-frequency GW detection. Sazhin (1978) and Detweiler (1979) described nHz GW detection using Galactic pulsars and (effectively) the solar system barycenter as references, relying on the regularity of pulsar emission and planetary motions to highlight GW effects. The fact that pulsars are such accurate clocks enables precise measurements of their rotational, astrometric, and binary parameters (and more) from the times-of-arrival of their pulses, which are used to develop ever-refining end-to-end timing models . Hellings & Downs (1983) made the crucial suggestion that the correlations between the timeof-arrival perturbations of multiple pulsars could reveal a GW signal buried in pulsar noise; Romani (1989) and Foster & Backer (1990) proposed that a pulsar timing array (PTA) of highly stable millisecond pulsars (Backer et al. 1982) could be used to search for a GWB. Nevertheless, the first multi-pulsar, long-term GWB limits were obtained by analyzing millisecond-pulsar residuals independently, rather than as an array (Stinebring et al. 1990; Kaspi et al. 1994). \n<!-- image --> \n= 13/3 \n<!-- image --> \n<!-- image --> \nFigure 1. Summary of the main Bayesian and optimal-statistic analyses presented in this paper, which establish multiple lines of evidence for the presence of Hellings-Downs correlations in the 15-year NANOGrav data set. Throughout we refer to the 68 . 3%, 95 . 4%, and 99 . 7% regions of distributions as 1 / 2 / 3 σ regions, even in two dimensions. (a): Bayesian 'free-spectrum' analysis, showing posteriors (gray violins) of independent variance parameters for a Hellings-Downs-correlated stochastic process at frequencies i/T , with T the total data set time span. The blue represents the posterior median and 1 / 2 σ posterior bands a for a power-law model; the dashed black line corresponds to a γ = 13 / 3 (SMBHB-like) power-law, plotted with the median posterior amplitude. See § 3 for more details. (b): Posterior probability distribution of GWB amplitude and spectral exponent in a HD power-law model, showing 1 / 2 / 3 σ credible regions. The value γ GWB = 13 / 3 (dashed black line) is included in the 99% credible region. The amplitude is referenced to f ref = 1yr -1 (blue) and 0 . 1 yr -1 (orange). The dashed blue and orange curves in the log 10 A GWB subpanel shows its marginal posterior density for a γ = 13 / 3 model, with f ref = 1yr -1 and f ref = 0 . 1 yr -1 , respectively. See § 3 for more details. (c): Angular-separation-binned inter-pulsar correlations, measured from 2,211 distinct pairings in our 67-pulsar array using the frequentist optimal statistic, assuming maximum-a-posteriori pulsar noise parameters and γ = 13 / 3 common-process amplitude from a Bayesian inference analysis. The bin widths are chosen so that each includes approximately the same number of pulsar pairs, and central bin locations avoid zeros of the Hellings-Downs curve. This binned reconstruction accounts for correlations between pulsar pairs (Romano et al. 2021; Allen & Romano 2022). The dashed black line shows the Hellings-Downs correlation pattern, and the binned points are normalized by the amplitude of the γ = 13 / 3 common process to be on the same scale. Note that we do not employ binning of inter-pulsar correlations in our detection statistics; this panel serves as a visual consistency check only. See § 4 for more frequentist results. (d): Bayesian reconstruction of normalized inter-pulsar correlations, modeled as a cubic spline within a variable-exponent power-law model. The violins plot the marginal posterior densities (plus median and 68% credible values) of the correlations at the knots. The knot positions are fixed, and are chosen on the basis of features of the Hellings-Downs curve (also shown as a dashed black line for reference): they include the maximum and minimum angular separations, the two zero crossings of the Hellings-Downs curve, and the position of minimum correlation. See § 3 for more details. \n<!-- image --> \nFrom a statistical-inference standpoint, the problem of detecting nHz GWs in PTA data is analogous to GW searches with terrestrial and future space-borne experiments, in which the propagation of light between reference masses is modeled with physical and phenomenological descriptions of signal and noise processes. It is distinguished by the irregular observation times, which encourage a time- rather than Fourierdomain formulation, and by noise sources (intrinsic pulsar noise, interstellar-medium-induced radio-frequencydependent fluctuations, and timing-model errors) that are correlated on timescales common to the GWs of interest. This requires the joint estimation of GW signals and noise, which is similar to the kinds of global fitting procedures already used in terrestrial GW experiments, and proposed for space-borne experiments. GW analysts have therefore converged on a Bayesian framework that represents all noise sources as Gaussian processes (van Haasteren et al. 2009; van Haasteren & Vallisneri 2014), and relies on model comparison (i.e., Bayes factors, which are ratios of fully marginalized likelihoods) to define detection (see, e.g., Taylor 2021). This Bayesian approach is nevertheless complemented by null hypothesis testing, using a frequentist detection statistic 1 (the 'optimal statistic' of Anholm et al. 2009; Demorest et al. 2013; Chamberlin et al. 2015) averaged over Bayesian posteriors of the noise parameters (Vigeland et al. 2018). \nThe GWB-rather than GW signals from individually resolved binary systems-is likely to become the first nHz source accessible to PTA observations (Rosado et al. 2015). Because of its stochastic nature, the GWB cannot be identified as a distinctive phase-coherent signal in the way of individual compact-binary-coalescence GWs. Rather, as PTA data sets grow in extent and sensitivity one expects to first observe the GWB as excess low-frequency residual power of consistent amplitude and spectral shape across multiple pulsars (Romano et al. 2021; Pol et al. 2021). An observation following this behavior was reported in 2020 (Arzoumanian et al. 2020, henceforth NG12gwb) for the 12.5-year data set collected by the North American Nanohertz Observatory for Gravitational waves (NANOGrav, McLaughlin 2013; Ransom et al. 2019), and then confirmed (Goncharov et al. 2021a; Chen et al. 2021) by the Parkes Pulsar Timing Array (PPTA, Manchester et al. 2013) and the European Pulsar Timing Array (EPTA, Desvignes et al. 2016), following many years of null results and steadily decreasing upper limits on the GWB am- \nplitude. A combined International Pulsar Timing Array (IPTA, Perera et al. 2019) data release consisting of older data sets from the constituent PTAs also confirmed this observation (Antoniadis et al. 2022). Nevertheless, the finding of excess power cannot be attributed to a GWBorigin merely by the consistency of amplitude and spectral shape, which could arise from intrinsic pulsar processes of similar magnitude (Goncharov et al. 2022; Zic et al. 2022), or from a common systematic noise such as clock errors (Tiburzi et al. 2016). Instead, definitive proof of GW origin is sought by establishing the presence of phase-coherent inter-pulsar correlations with the characteristic spatial pattern derived by Hellings and Downs (1983, henceforth HD): for an isotropic GWB, the correlation between the GW-induced timing delays observed at Earth for any pair of pulsars is a universal, quasi-quadrupolar function of their angular separation in the sky. Even though this correlation pattern is modified if there is anisotropy in the GWB-which may be the case for a GWB generated by a SMBHB population (Mingarelli et al. 2013; Taylor & Gair 2013; Cornish & Sesana 2013; Mingarelli & Sidery 2014; Mingarelli et al. 2017; Roebber & Holder 2017)-the HD template is effective for detecting even anisotropic GWBs in all but the most extreme scenarios (Cornish & Sesana 2013; Cornish & Sampson 2016; Taylor et al. 2020; B'ecsy et al. 2022; Allen 2023). \nIn this letter we present multiple lines of evidence for an excess low-frequency signal with HD correlations in the NANOGrav 15-year data set (Figure 1). Our key results are as follows. The Bayes factor between an HD-correlated, power-law GWB model and a spatially uncorrelated common-spectrum power-law model ranges from 200 to 1,000, depending on modeling choices (Figure 2). The noise-marginalized optimal statistic, which is constructed to be selectively sensitive to HDcorrelated power, achieves a signal-to-noise ratio of ∼ 5 (Figure 3 and Figure 4). We calibrated these detection statistics by removing correlations from the 15-year data set using the phase-shift technique, which removes interpulsar correlations by adding random phase shifts to the Fourier components of the common process (Taylor et al. 2017). We find false-alarm probabilities of p = 10 -3 and p = 5 × 10 -5 for the observed Bayes factor and optimal statistic, respectively (see Figure 3). \nFor our fiducial power-law model ( f -2 / 3 for characteristic strain and f -13 / 3 for timing residuals) and a log-uniform amplitude prior, the Bayesian posterior of GWB amplitude at the customary reference frequency 1 yr -1 is A GWB = 2 . 4 +0 . 7 -0 . 6 × 10 -15 (median with 90% credible interval), which is compatible with current astrophysical estimates for the GWB from SMBHBs (e.g., \nBurke-Spolaor et al. 2019; Agazie et al. 2023a). This corresponds to a total integrated energy density of Ω gw = 9 . 3 +5 . 8 -4 . 0 × 10 -9 or ρ gw = 7 . 7 +4 . 8 -3 . 3 × 10 -17 ergs cm -3 (assuming H 0 = 70 km / s / Mpc) in our sensitive frequency band. For a more general model of the timing-residual power spectral density with variable power-law exponent -γ , we find A GWB = 6 . 4 +4 . 2 -2 . 7 × 10 -15 , and γ = 3 . 2 +0 . 6 -0 . 6 . See panel (b) of Figure 1 for A GWB and γ posteriors. The posterior for γ is consistent with the value of 13 / 3 predicted for a population of SMBHBs evolving by GW emission, although smaller values of γ are preferred; however, the recovered posteriors are consistent with predictions from astrophysical models (see Agazie et al. 2023a). We also note that, unlike our detection statistics (which are calibrated under our modeling assumptions), the estimation of γ is very sensitive to minor details in the data model of a few pulsars. \nThe rest of this paper is organized as follows. We briefly describe our data set and data model in § 2. Our main results are discussed in detail in § 3 and § 4; they are supported by a variety of robustness and validation studies, including a spectral analysis of the excess signal ( § 5.2), a correlation analysis that finds no significant evidence for additional spatially correlated processes ( § 5.3), and cross-validation studies with singletelescope data sets and leave-one-pulsar-out techniques ( § 5.4). In the past two years we have performed an endto-end review of the NANOGrav experiment, to identify and mitigate possible sources of systematic error or data set contamination: our improvements and considerations are partly described in a set of companion papers: on the NANOGrav statistical analysis as implemented in software (Johnson et al. 2023), on the 15-year data set (Agazie et al. 2023b, hereafter NG15), and on pulsar models (Agazie et al. 2023c, hereafter NG15detchar). More companion papers address the possible SMBHB (Agazie et al. 2023a) and cosmological (Afzal et al. 2023) interpretations of our results, with several more GW searches and signal studies in preparation. We look forward to the cross-validation analysis that will become possible with the independent data sets collected by other IPTA members.", '2. THE 15-YEAR DATA SET AND DATA MODEL': "The NANOGrav 15-year data set 2 (NG15) contains observations of 68 pulsars obtained between July 2004 and August 2020 with the Arecibo Observatory \n(Arecibo), the Green Bank Telescope (GBT), and the Very Large Array (VLA), augmenting the 12.5-year data set (Alam et al. 2021a,b) with 2 . 9 years of timing data for the 47 pulsars in the previous data set, and with 21 new pulsars 3 . For this paper we analyze narrowband times of arrival (TOAs), which are computed separately for sub-bands of each receiver, and focus on the 67 pulsars with a timing baseline ≥ 3 years. We adopt the TT(BIPM2019) timescale and the JPL DE440 ephemeris (Park et al. 2021), which improves Jupiter's orbit with ranging and VLBI observations of the Juno spacecraft. Uncertainties in the Jovian orbit impacted NANOGrav's 11-year GWB search (Arzoumanian et al. 2018; Vallisneri et al. 2020), but they are now negligible. \nFor each pulsar, we fit the TOAs to a timing model that includes pulsar spin period, spin period derivative, sky location, proper motion, and parallax. While not all pulsars have measurable parallax and proper motion, we always include these parameters because they induce delays with the same frequencies for all pulsars ( f = 0 . 5 yr -1 for parallax and f = yr -1 plus a linear envelope for proper motion), so there is a risk that a parallax or proper motion signal could be misidentified as a GW signal. Fitting for these parameters in all pulsars reduces our sensitivity to GWs at those frequencies; however, this effect is minimal for GWB searches since these frequencies are much higher than the frequencies at which we expect the GWB to be significant. For binary pulsars, the timing model includes also five orbital elements for binary pulsars and additional non-Keplerian parameters when these improve the fit as determined by an F test. We fit variations in dispersion measure as a piecewise constant 'DMX' function (Arzoumanian et al. 2015; Jones et al. 2017). The individual analysis of each pulsar provides best-fit estimates of the timing residuals δt , of white measurement noise, and of intrinsic red noise, modeled as a power law (Cordes 2013; Lam et al. 2017; Jones et al. 2017). 4 White measurement noise is described by three parameters: a linear scaling of TOA uncertainties ('EFAC'), white noise added to the TOA uncertainties in quadrature ('EQUAD'), and noise common to all sub-bands at the same epoch ('ECORR'), with independent parameters for every receiver/backend combination (see NG15detchar). We summarize white noise by its maximum a posteriori (MAP) covariance C . See App. A for more details of our instruments, observations, and data- \neduction pipeline: a complete discussion of the data set can be found in NG15. \nIn our Bayesian GWB analysis, we model δt as a finite Gaussian process consisting of time-correlated fluctuations that include intrinsic red pulsar noise and (potentially) a GW signal, along with timing-model uncertainties (van Haasteren et al. 2009; van Haasteren & Vallisneri 2014; Taylor 2021). The red noise is modeled with Fourier basis F and amplitudes c (Lentati et al. 2013). All Fourier bases (the columns of F ) are sines and cosines computed on the TOAs with frequencies f i = i/T , where T = 16 . 03 yr is the TOA extent. The timing-model uncertainties are modeled with design-matrix basis M and coefficients ϵ . The singlepulsar log likelihood is then \nln p ( δt | c , ϵ ) = -1 2 [ r T C -1 r +lndet(2 π C ) ] , (1) \nwith \nr = δt -Fc -Mϵ . (2) \nThe prior for the ϵ is taken to be uniform with infinite extent, so the posterior is driven entirely by the likelihood. The set of the { c } for all pulsars take a joint normal prior with zero mean and covariance \n⟨ c ai c bj ⟩ = δ ij ( δ ab φ ai +Φ ab,i ) ; (3) \nhere a, b range over pulsars and i, j over Fourier components; δ ij is Kronecker's delta. The term φ ai describes the spectrum of intrinsic red noise in pulsar a , while Φ ab,i describes processes with common spectrum across all pulsars and (potentially) phase-coherent inter-pulsar correlations. The { c } prior ties together the single-pulsar likelihoods (Equation 1) into a joint posterior, p ( c , ϵ , η | δt ) ∝ p ( δt | c , ϵ ) p ( c , ϵ | η ) p ( η ), where we have dropped subscripts to denote the concatenation of vectors for all pulsars, and where η denotes all the hyperparameters (such as red-noise and GWB power spectrum amplitudes) that determine the covariances. We marginalize over c and ϵ analytically, and use Markov chain Monte Carlo techniques (see App. B) to estimate p ( η | δt ) for different models of the intrinsic red noise and common spectrum. \nThe data-model variants adopted in this paper all share this probabilistic setup, but differ in the structure and parametrization of Φ ab,i . For a model with intrinsic red noise only (henceforth irn ), Φ ab,i = 0; for common-spectrum spatially-uncorrelated red noise ( curn ), Φ ab,i = δ ab Φ CURN ,i ; for an isotropic GWB with Hellings-Downs correlations ( hd ), Φ ab,i = Γ( ξ ab )Φ HD ,i , with Γ the Hellings-Downs function of pulsar angular \nseparations ξ ab \nΓ( ξ ab ) = 3 2 x ln( x ) -1 4 x + 1 2 + 1 2 δ ab , (4) \nx = 1 -cos ξ ab 2 . (5) \nIn NG12gwb we established strong Bayesian evidence for curn over irn ; finding that hd is preferred over curn would point to the GWB origin of the common-spectrum signal. We also investigate other spatial correlation patterns, e.g., monopole or dipole, introduced in § 5.3. \nThroughout this paper, we set the spectral components φ ai of intrinsic pulsar noise (which have units of s 2 , as appropriate for the variance of timing residuals) to a power law, \nφ ai = A 2 a 12 π 2 1 T ( f i f ref ) -γ a f -3 ref , (6) \nintroducing two dimensionless hyperparameters for each pulsar: the intrinsic-noise amplitude A a and spectral index γ a . We use log-uniform and uniform priors, respectively, on these hyperparameters; their bounds are described in App. B. More sophisticated intrinsic-noise models are discussed in § 5.1 and NG15detchar. In models curn γ and hd γ , the common spectra Φ CURN ,i and Φ HD ,i follow Equation 6, \nΦ CURN ,i = A 2 CURN 12 π 2 1 T ( f i f ref ) -γ CURN f -3 ref , (7) \nΦ HD ,i = A 2 HD 12 π 2 1 T ( f i f ref ) -γ HD f -3 ref , (8) \nintroducing hyperparameters A CURN , γ CURN and A HD , γ HD respectively. However, we set γ HD = 13 / 3 for the GWB from a stationary ensemble of inspiraling binaries, and refer to that fiducial model as hd 13 / 3 . For specific 'free spectrum' studies we will instead model the individual Φ CURN ,i or Φ HD ,i elements, and refer to models curn free and hd free . Throughout this article we use frequencies f i = i/T with i = 1-30 for intrinsic noise ( f = 2-59 nHz), covering a frequency range over which pulsar noise transitions from red-noise-dominated to white-noise-dominated. For common-spectrum noise, we limit the frequency range in order to reduce correlations with excess white noise at higher frequencies. Following NG12gwb, we fit a curn γ model enhanced with a power-law break to our data, and limit frequencies to the MAP break frequencies ( i = 1-14 or f = 2-28 nHz; see App. C).", '3. BAYESIAN ANALYSIS': "When fit to the 15-year data set, the curn γ and hd γ models agree on the presence of a loud time-correlated \nFigure 2. Bayes factors between models of correlated red noise in the NANOGrav 15-year data set (see § 5.3 and App. B). All models feature variableγ power laws. curn γ is vastly favored over irn (i.e., we find very strong evidence for common-spectrum excess noise over pulsar intrinsic red-noise alone); hd γ is favored over curn γ (i.e., we find positive evidence for Hellings-Downs correlations in the common-spectrum process); dipole and monopole processes are strongly disfavored with respect to curn γ ; adding correlated processes to hd γ is disfavored. While the interpretation of 'raw' Bayes factors is somewhat subjective, they can be given a statistical significance within the hypothesis-testing framework by computing their background distributions and deriving the p -values of the observed factors, e.g., Figure 3. \n<!-- image --> \nstochastic signal with common amplitude and spectrum across pulsars 5 . The joint A HD -γ HD Bayesian posterior is shown in panel (b) of Figure 1, with 1-D marginal posteriors in the horizontal and vertical subplots. The posterior medians and 5-95% quantiles are A HD = 6 . 4 +4 . 2 -2 . 7 × 10 -15 and γ HD = 3 . 2 +0 . 6 -0 . 6 . The thicker curve in the vertical subplot is the A HD posterior for the hd 13 / 3 model, for which A HD , 13 / 3 = 2 . 4 +0 . 7 -0 . 6 × 10 -15 . These amplitudes are compatible with astrophysical expectations of a GWB from inspiraling SMBHBs (see § 6). The A HD posterior has essentially no support below 10 -15 . \nThe strong A HD -γ HD correlation is an artifact of using the conventional frequency f ref = 1yr -1 in Equation 6, and it largely disappears when f ref is moved to the band of greatest PTA sensitivity; see the dashed contours in panel (b) of Figure 1 for f ref = (10 yr) -1 . The γ HD posterior is in moderate tension with the theoretical universal binary-inspiral value γ HD = 13 / 3, which lies at the 99% credible boundary: smaller values of γ HD could be an indication that astrophysical effects, such as stellar scattering and gas dynamics, play a role in the evolution of SMBHBs emitting GWs in this frequency range (see § 6 and Agazie et al. 2023a). This highlights the importance of measuring this parameter. Furthermore, its estimation is sensitive to details in the modeling of intrinsic red noise and of interstellar-medium timing delays in a few pulsars (see the analysis in § 5.2). Notably, in the 12.5-year data set γ HD = 13 / 3 was recovered at ∼ 1 σ below the median (NG12gwb); this anomaly is reversed in the newer data set. It is likely that more expansive \ndata sets or more sophisticated chromatic noise models, e.g., next generation Gaussian process models such as in § 5.1 (Goncharov et al. 2021b; Chalumeau et al. 2022; Lam et al. 2018), will be needed to infer the presence of possible systematic errors in γ HD . \nOur Bayesian analysis provides evidence that the common-spectrum signal includes Hellings-Downs interpulsar correlations. Specifically, the Bayes factor between the hd γ and curn γ models ranges from 200 (when 14 Fourier frequencies are included in Φ i ) to 1,000 (when 5 frequencies are included, as in NG12gwb). Results are similar for hd 13 / 3 vs. curn 13 / 3 . Figure 2 recapitulates Bayes factors between a variety of models, including some with the alternative spatial-correlation structures discussed in § 5.3. The very peaked A HD posterior in panel (b) of Figure 1, significantly separated from smaller amplitudes, supports the very large Bayes factor between irn and curn γ . The 15-year data set favors hd γ over curn γ , and over models with monopolar or dipolar correlations, and it is inconclusive about, i.e., gives roughly even odds for, the presence of spatially correlated signals in addition to hd γ . \nWe can also regard the hd γ vs. curn γ Bayes factor as a detection statistic in a hypothesis-testing framework, and derive the p -value of the observed Bayes factor with respect to its empirical distribution under the curn γ model. We do so by computing Bayes factors on 5,000 bootstrapped data sets where inter-pulsar spatial correlations are removed by introducing random phase shifts , drawn from a uniform distribution from 0 to 2 π , to the common-process Fourier components (Taylor et al. 2017). This procedure alters inter-pulsar correlations to have a mean of zero, while leaving the amplitudes of intrinsic pulsar noise and CURN unchanged, thus provid- \nFigure 3. Empirical background distribution of hd γ -tocurn γ Bayes factor (left, see § 3) and noise-marginalized optimal statistic (right, see § 4), as computed by the phase-shift technique (Taylor et al. 2017) to remove inter-pulsar correlations. We only compute 5,000 Bayesian phase shifts, compared to 400,000 optimal statistic phase shifts, because of the huge computational resources needed to perform the Bayesian analyses. For the optimal statistic, we also compute the background distribution using 27,000 simulations (orange line) and compare to an analytic calculation (green line). Dotted lines indicate Gaussian-equivalent 2 σ , 3 σ , and 4 σ thresholds. The dashed vertical lines indicate the values of the detection statistics for the unshifted data sets. For the Bayesian analyses, we find p = 10 -3 (approx. 3 σ ); for the optimal statistic analyses, we find p = 5 × 10 -5 -1 . 9 × 10 -4 (approx. 3 . 5-4 σ ). \n<!-- image --> \n- \n2 \n0 \n2 \n4 \n6 \nNoise-Marginalized Mean S/N \ning a way to test the null hypothesis that no inter-pulsar correlations are present. The resulting background distribution of Bayes factors is shown in the left panel of Figure 3-they exceed the observed value in five of the 5,000 phase shifts ( p = 10 -3 ). We also performed sky scramble analyses (Cornish & Sampson 2016), which remove the dependence of inter-pulsar spatial correlations on the angular separations between the pulsars by attributing random sky positions to the pulsars. Sky scrambles generate a background distribution for which inter-pulsar correlations are present in the data but they are independent of the pulsars' angular separations: for this distribution, we find p = 1 . 6 × 10 -3 . A detailed discussion of sky scrambles and the results of these analyses can be found in App. F. \nAs in NG12gwb, we also carried out a minimally modeled Bayesian reconstruction of the inter-pulsar correlation pattern, using spline interpolation over seven splineknot positions. The choice of seven spline-knot positions is based on features of the Hellings-Downs pattern: two correspond to the maximum and minimum angular separations (0 · and 180 · , respectively), two are chosen to be at the theoretical zero crossings of the HellingsDowns pattern (49 . 2 · and 121 . 8 · ), one is at the theoretical minimum (82 . 5 · ), and the final two are between the end points and zero crossings (25 · and 150 · ) to allow additional flexibility in the fit. Panel (d) of Figure 1 shows the marginal 1-D posterior densities at these spline-knot positions for a power-law varied-exponent model. The reconstruction is consistent with the overplotted Hellings-Downs pattern; furthermore, the joint 2-D marginal posterior densities for the knots, not shown \nin panel (d) of Figure 1, at the HD zero-crossings is consistent with (0 , 0) within 1 σ credibility.", '4. OPTIMAL STATISTIC ANALYSIS': "We complement our Bayesian search with a frequentist analysis using the optimal statistic (Anholm et al. 2009; Demorest et al. 2013; Chamberlin et al. 2015), a summary statistic designed to measure correlated excess power in PTA residuals. (Note that there is no accepted definition of 'optimal statistic' in modern statistical usage, but the term has become established in the PTA literature to refer to this specific method, so we use it for this reason.) It is enlightening to describe the optimal statistic as a weighted average of the inter-pulsar correlation coefficients \nρ ab = δt T a P -1 a ˜ Φ ab P -1 b δt b Tr P -1 a ˜ Φ ab P -1 b ˜ Φ ba , (9) \nwhere δt T a are the residuals of pulsar a , and P a = 〈 δt a δt T a 〉 is their total auto-covariance matrix. The cross-covariance matrix ˜ Φ ab encodes the spectrum of the HD-correlated signal, normalized so that Φ ab = A 2 Γ( ξ ab ) ˜ Φ ab (see Pol et al. 2022), and where elements of Φ ab are given by Equation 3. Indeed, the ρ ab have expectation value A 2 Γ( ξ ab ), but their variance σ 2 ab = (Tr P -1 a ˜ Φ ab P -1 b ˜ Φ ba ) -1 + O ( A 4 ) is too large to use them directly as estimators. Thus we assemble the optimal statistic as the variance-weighted, Γ-template-matched average of the ρ ab , \nˆ A 2 = ∑ a>b ρ ab Γ( ξ ab ) /σ 2 ab ∑ a>b Γ 2 ( ξ ab ) /σ 2 ab . (10) \nPhase shifts \nSimulations \nAnalytic background \n2 \ns \n3 \ns \n4 \ns \nFigure 4. Optimal statistic S/N for HD correlations, distributed over curn γ (solid lines) and curn 13 / 3 (dashed lines) noise-parameter posteriors. The vertical lines indicate the mean S/Ns. We find S/Ns of 5 ± 1 and 4 ± 1 for curn γ and curn 13 / 3 , respectively. \n<!-- image --> \nThis equation represents the optimal estimator of the HD amplitude A 2 ; it can also be interpreted as the best-fit ˆ A 2 obtained by least-squares-fitting the ρ ab to the Hellings-Downs model ˆ A 2 Γ( ξ ab ). Because ˆ A 2 is a function of intrinsic-red-noise and common-process hyperparameters through the P a , we use the results of an initial Bayesian-inference run to refer the statistic to MAP hyperparameters, or to marginalize it over their posteriors. As discussed in Vigeland et al. (2018), we obtain more accurate values of the amplitude by this marginalization. \nTo search for inter-pulsar correlations using the optimal statistic, we evaluate the frequency (the p -value) with which an uncorrelated common-spectrum process with parameters estimated from our data set would yield ˆ A 2 greater than we observe. In the absence of a signal, the expectation value of ˆ A 2 is zero, and its distribution is approximately normal. Thus we divide the observed ˆ A 2 by its standard deviation to define a formal signalto-noise ratio \nS / N = ∑ a>b ρ ab Γ( ξ ab ) /σ 2 ab [∑ a>b Γ 2 ( ξ ab ) /σ 2 ab ] 1 / 2 . (11) \nFigure 4 shows the distribution of this S/N over curn γ and curn 13 / 3 noise-parameter posteriors, with S/Ns of 5 ± 1 and 4 ± 1, respectively (means ± standard deviations across noise-parameter posteriors). We use 14 frequency components to model the signal: the dependence on the number of frequency components is very weak. \nBecause the distribution of ˆ A 2 is only approximately normal (Hazboun et al. 2023), the S/N of Equation 11 does not map analytically to a p -value, and it cannot be interpreted as a 'sigma' level. Instead, optimalstatistic p -values can be computed empirically by re- \noving inter-pulsar correlations from the 15-year data set with phase shifts (Taylor et al. 2017). We draw random phase offsets from 0 to 2 π for the common-process Fourier components, which is equivalent to making uniform draws from the background distribution of CURN, and ask how often a random choice of phase offsets produces a HD-correlated signal. The right panel of Figure 3 shows the distribution of noise-marginalized S/N over 400,000 phase shifts. There are 19 phase shifts with noise-marginalized S/N greater than observed, with p = 5 × 10 -5 . We compare the phase-shift distribution with backgrounds obtained by simulation (right panel of Figure 3, orange line) and analytic calculation (green line). For the former, we simulate 27,000 curn γ realizations using MAP hyperparameters from the 15-yr data and compute the optimal-statistic S/N for each; for the latter, we evaluate the generalized χ 2 distribution (Hazboun et al. 2023) with median curn γ hyperparameters. Although neither method includes the marginalization over noise-parameter posteriors, we find good agreement with phase shifts, with p = 1 . 8 × 10 -4 from simulations, and p = 1 . 9 × 10 -4 from the analytic calculation. Finally, we use sky scrambles to compute the p -value for the null hypothesis that inter-pulsar correlations are present, but they have no dependence on the angular separation between the pulsars, for which we find p < 10 -4 (see App. F). \nAveraging the cross-correlations ρ ab in angularseparation bins with equal numbers of pulsar pairs reveals the Hellings-Downs pattern, as shown in panel (c) of Figure 1 for 15 bins. The ρ ab were evaluated with MAP curn 13 / 3 noise parameters. The black dashed curve traces the expected correlations for an HDcorrelated background with the MAP amplitude; the vertical error bars display the expected 1 σ spreads of the binned cross-correlations, accounting for the ⟨ ρ ab ρ cd ⟩ covariances induced by the HD-correlated process (Romano et al. 2021; Allen & Romano 2022). (Neglecting those covariances yields 20-40% smaller spreads. Note that they are not included in p -value estimates because those are calculated under the null hypothesis of no spatially correlated process.) \nAlthough each draw from the noise-parameter posterior would generate a slightly different plot, as would different binnings, the quality of the fit seen in Figure 1 provides a visual indication that the excess lowfrequency power in the 15-year data set harbors HD correlations. The χ 2 for this 15-bin reconstruction with respect to the Hellings-Downs curve is 8 . 1, where we account for ρ ab covariance in constructing the bins, and the covariance between bins in constructing the χ 2 (Allen & Romano 2022). This corresponds to a p -value of 0.75, \ncalculated using simulations based on the hd γ model, or 0.92 if one assumes this value follows a canonical χ 2 with 15 degrees of freedom. These p -values are representative of what we find with different binnings: we find p > 0 . 3 when using eight to 20 bins (assuming a canonical χ 2 distribution).", '5. CHECKS AND VALIDATION': "Prior to analyzing the 15-year data set, we extensively reviewed our data collection and analysis procedures, methods, and tools, in an effort to eliminate contamination from systematic effects and human error. Furthermore, the results presented in § 3 and § 4 are supported by a variety of consistency checks and auxiliary studies. In this section we present those that offer evidence for or against the presence of HD correlations, reveal anomalies, or otherwise highlight features of note in the data: alternative DM modeling ( § 5.1), the spectral content ( § 5.2) and correlation pattern ( § 5.3) of the excessnoise signal, as well as the consistency of our findings across data set 'slices,' pulsars, and telescopes ( § 5.4).", '5.1. Alternative DM models': 'In this paper and in previous GW searches (e.g., NG12gwb), we model fluctuations in the DM using DMX parameters (a piecewise-constant representation, see NG15). Adopting this DM model as the standard makes it easier to directly compare the results here to those in NG12gwb. An alternative model where DM variations are modeled as a Fourier-domain Gaussian process, DMGP, has been used by Antoniadis et al. (2022), Chen et al. (2021), and Goncharov et al. (2021a). The Fourier coefficients follow a power law similar to those of intrinsic and common-spectrum red noise, but their basis vectors include a ν -2 radio-frequency dependence, and the component frequencies f i = i/T range through i = 1-100. Under the DMGP model we also include a deterministic solar-wind model (Hazboun et al. 2022) and the two chromatic events in PSR J1713+0747 reported in Lam et al. (2018) which are modeled as deterministic exponential dips with the chromatic index quantifying the radio-frequency dependence of the dips left as a free parameter. If these chromatic events are not modeled, they raise estimated white noise (Hazboun et al. 2020). A detailed discussion of chromatic noise effects can be found in NG15detchar. \nUsing the DMGP model in place of DMX has minimal effects on nearly all pulsars in the array. Only PSRs J1713+0747 and J1600 -3053 show notable differences in their recovered intrinsic-noise parameters. However, DMGP does affect the parameter estimation of common red noise, as seen in Figure 5, shifting the posterior for \nFigure 5. curn γ posterior distributions using DMGP (red) and DMX (blue) to model DM variations. The dashed line marks γ CURN = 13 / 3. While the posteriors are broadly consistent, DMGP shifts the γ CURN posterior to higher values, making it more consistent with γ CURN = 13 / 3. \n<!-- image --> \nγ to higher values that are more consistent with 13 / 3. Despite this, we still recover HD correlations at the same significance as when we use DMX to model fluctuations in the DM, implying that the evidence reported for the presence of correlations in this work is independent of the choice of DM noise modeling.', '5.2. Spectral analysis': "Adopting power-law spectra for curn and hd is a useful simplification that reduces the number of fit parameters and yields more informative constraints; furthermore, it is expedient to identify hd 13 / 3 with the hypothesis that we are observing the GWB from SMBHBs. Nevertheless, the standard γ = 13 / 3 power law for GW inspirals may be altered by astrophysical processes such as stellar and gas friction in nuclei (see, e.g., Merritt & Milosavljevi'c 2005 for a review), by appreciable eccentricity in SMBHB orbits (Enoki & Nagashima 2007), and by low-number SMBHB statistics (Sesana et al. 2008). hd γ parameter recovery may also be biased if intrinsic pulsar noise is not modeled well by a power law. Indeed, our data show hints of a discrepancy from the idealized hd 13 / 3 model: the γ HD posterior in panel (b) of Figure 1 favors slopes much shallower than 13/3, and the hd γ -tocurn γ Bayes factor drops from 1,000 to 200 when Fourier components at more than five frequencies are included in the model. \nWe examine the spectral content of the 15-year data set using the curn free and hd free models, which are parametrized by the variances of the Fourier components at each frequency. Their marginal posteriors are shown in the left panel of Figure 6, where bin number i corresponds to f i = i/T , with T = 16 . 03 yr the extent of the data set. For the purpose of illustration, we overlay best-fit power laws that thread the posteriors in a way \n√ Figure 6. Left : Posteriors of Fourier component variance Φ i for the curn free (left) and hd free (right) models (see § 2), plotted at their corresponding frequencies f i = i/T with T the 16.03-yr extent of the data set. Excess power is observed in bins 1-8 (somewhat marginally in bin 6); Hellings-Downs-correlated power in bins 1-5 and 8. The dashed line plots the best-fit power law, which has γ ≃ 3 . 2 (as in panel (d) of Figure 1); the fit is pushed to lower γ by bins 1 and 8. The dotted line plots the best-fit power law when γ is fixed to 13/3; it overshoots in bin 1 and undershoots in bin 8. Right : Posteriors of variance Φ 2 in Fourier bin 2 ( f 2 = 3 . 95 nHz) in a curn free + hd free + monopole free + dipole free model, showing evidence of a quasi-monochromatic monopole process (dashed). No monopole or dipole power is observed in all other bins of that joint model, with Φ CURN ,i and Φ HD ,i posteriors consistent with the left panel. \n<!-- image --> \nsimilar to the factorized PTA likelihood of Taylor et al. (2022) and Lamb et al. (2023). \nWe deem excess power, either uncorrelated for curn free or correlated for hd free , to be observed in a bin when the support of the posterior is concentrated away from the lowest amplitudes. No power of either kind is observed above f 8 , consistent with the presence of a floor of white measurement noise. Furthermore, no correlated power is observed in bins 6 and 7, where a power-law model would expect a smooth continuation of the trend of bins 1-5 (cf. the dashed fit of Figure 6): this may explain the drop in the Bayes factor. However, correlated power reappears in bin 8, pushing the fit toward shallower slopes. Indeed, repeating the fit by omitting subsets of the bins suggests that the low recovered γ HD is due mostly to bin 8 and to the lower-than-expected correlated power found in bin 1. Obviously, excluding those bins leads to higher γ HD estimates. \nTo explore deviations from a pure power law that may arise from statistical fluctuations of the astrophysical background or from unmodeled systematics (perhaps related to the timing model), in App. D we relax the normal c k prior (cf. Equation 3) to a multivariate Student's t -distribution that is more accepting of mild outliers. The resulting estimate of γ CURN peaks at a higher value and is broader than in curn γ , with posterior medians and 5-95% quantiles of γ CURN = 3 . 5 +1 . 0 -1 . 0 . \nSimilarly, spectral turnovers due to interactions between SMBHBs and their environments can result in reduced GWB power at lower frequencies, which might explain the slightly lower correlated power in bin 1. We investigate this hypothesis in App. E using the turnover \nspectrum of Sampson et al. (2015). For this curn turnover model, the 15-year data favor a spectral bend below 10 nHz (near f 5 ), but the Bayes factor against the standard hd γ is inconclusive. \nFuture data sets with longer time spans and the comparison of our data set with those of other PTAs should help clarify the astrophysical or systematic origin of these possible spectral features.", '5.3. Alternative correlation patterns': "Sources other than GWs can produce inter-pulsar residual correlations with spatial patterns other than HD. For example, errors in the solar-system ephemerides create time-dependent Roemer delays with dipolar correlations (Roebber 2019; Vallisneri et al. 2020), and errors in the correction of telescope time to an inertial timescale (Hobbs et al. 2012, 2020) create an identical time-dependent delay for all pulsars (i.e., a delay with monopolar correlations). \nGair et al. (2014) showed that, for a pulsar array distributed uniformly across the sky, HD correlations can be decomposed as \nΓ HD ,ab = ∞ ∑ l =0 g l P l (cos ξ ab ) , g 0 = 0 , g 1 = 0 , g l = 3 2 (2 l +1) ( l -2)! ( l +2)! for l ≥ 2 , (12) \nwhere the P l (cos ξ ab ) are Legendre polynomials of order l evaluated at the pulsar angular separation ξ ab . In other words, a HD-correlated signal should have no power at l = 0 or l = 1. \nFigure 7. Multiple-component optimal statistic for a Legendre polynomial basis (Equation 12) with with l max = 5. The violin plots show the distributions of the normalized Legendre coefficients A 2 l = A 2 g l over curn γ noise-parameter posteriors. The black dashed line shows the Legendre spectrum of a pure-HD signal, with the median posterior ˆ A 2 HD . \n<!-- image --> \nWe can perform a frequentist generic correlation search using Legendre polynomials 6 with the multiplecomponent optimal statistic (MCOS; Sardesai & Vigeland 2023)-a generalized statistic that allows multiple correlation patterns to be fit simultaneously to the correlation coefficients ρ ab . Figure 7 shows the constraints on A 2 l = A 2 g l obtained by fitting the correlations ρ ab to this Legendre series using the MCOS and marginalizing over curn γ noise-parameter posteriors. The quadrupolar structure of the data is evident, along with a small but significant monopolar contribution. \nThe same feature from the Legendre decomposition appears if we use the MCOS to search for multiple correlations simultaneously: a multiple regression analysis favors models that contain both significant HD and monopole correlations (see App. G). From simulations of 15-year-like data sets (see App. H.1), we find a p -value of 0.11 (approx. 2 σ ) for observing a monopole at this significance or higher with a pure-HD injection of amplitude similar to what we observe. We also perform a model-checking study to assess whether the observed monopole is consistent with the hd 13 / 3 model (see App. H.2), and we find a p -value of 0.11 for producing an apparent monopole when the signal is purely hd 13 / 3 . Thus, we conclude that it is possible for a HDcorrelated signal to appear to have monopole correlations in an optimal statistic analysis at this significance level. \nIn contrast, Bayesian searches for additional correlations do not find evidence of additional monopole- or dipole-correlated red noise processes: as shown in Figure 2, the Bayes factors for these processes are ∼ 1. We also perform a general Bayesian search for correlations using a curn free + hd free + monopole free + dipole free model, which allows for independent uncorrelated and correlated components at every frequency bin. We note that this analysis is more flexible than the ones described above, which assume a power-law power spectral density. We find no significant dipole-correlated power at any frequency, and we find monopole-correlated power only in the second frequency bin ( f 2 = 3 . 95 nHz); posteriors of variance for that bin are shown in the right panel of Figure 6. \nMotivated by this finding, we perform a search for hd γ + sinusoid , which includes a deterministic sinusoidal delay (applied to all pulsars alike, as appropriate for a monopole) with free frequency, amplitude, and phase. The sinusoid's posteriors match the free-spectral analysis in frequency and amplitude; however, the Bayes factor between hd γ + sinusoid and hd γ calculated using two methods (Hee et al. 2015; Hourihane et al. 2023), is only ∼ 1, so the signal cannot be considered statistically significant. Astrophysically motivated searches for sources that produce sinusoidal or sinusoid-like delays in the residuals, such as an individual SMBHB or perturbations to the local gravitational field induced by fuzzy dark matter (Khmelnitsky & Rubakov 2014), also yield Bayes factors ∼ 1. Thus we conclude that there is some evidence of additional power at 3.95 nHz with monopole correlations; however, the significance in the Bayesian analyses is low, while the optimal-statistic S/N could be produced by a HD-correlated signal. Therefore, we cannot definitively say whether the signal is present, or determine the source. We note that performing an MCOS analysis after subtracting off realizations of a sinusoid using hd γ + sinusoid posteriors reduces the (S / N) monopole ≃ 0 while (S / N) HD remains unchanged, indicating that this single-frequency monopole-correlated signal is likely causing the nonzero monopole signal observed in the MCOS analysis. \nSimilar hints of a monopolar signal (though weaker) were found in the NANOGrav 12.5-year data set, unsurprisingly given that it is a subset of the current data set. To exercise due diligence, we audited the correction of telescope time to GPS time at the Arecibo Observatory and at the Green Bank Telescope, and found nothing that could explain our observations. The subsequent steps in the time-correction pipeline rely on very accurate atomic clocks and are unlikely to introduce considerable systematics (Petit 2022). An important test will \nbe whether this signal persists in future data sets. If this monopolar feature is a truly an astrophysical signal, we would expect it to increase in significance as our data set grows. Comparisons with other PTAs and combined IPTA data sets will also provide crucial insight.", '5.4. Dropout and cross-validation': "The GWB is by its nature a signal affecting all of the pulsars in the PTA, although it may appear more significant in some based on their observing time span, noise properties, and on the particular realization of pulsar and Earth contributions (Speri et al. 2023). One way to assess the significance of the GWB in each pulsar is a Bayesian dropout analysis (Aggarwal et al. 2019; Arzoumanian et al. 2020), which introduces a binary parameter that turns on and off the common signal (or its inter-pulsar correlations) for a single pulsar, leaving all other pulsars unchanged. The Bayes factor associated with this parameter, also referred to as the 'dropout factor,' describes how much each pulsar likes to 'participate' in the common signal. \nFigure 8 plots curn γ vs. irn dropout factors for all 67 pulsars (blue). We find positive dropout factors (i.e., dropout factors > 2) for an uncorrelated common process in twenty pulsars, while only one has a dropout factor < 0 . 5. For comparison, in the NANOGrav 12.5year data set ten pulsars showed positive dropout factors for an uncorrelated common process, while three had negative dropout factors. We also show HD correlations vs. curn γ dropout factors (orange). For these, the uncorrelated common process is always present in all pulsars, but the cross-correlations for all pulsar pairs involving a given pulsar may be dropped from the likelihood. We find positive factors for HD correlations vs. curn γ in seven pulsars, while three are negative. We expect more pulsars to have positive dropout factors for curn γ vs. irn than for Hellings-Downs vs. curn γ because the Bayes factor comparing the first two models is significantly higher than the one comparing the second two models (see Figure 2). Negative dropout factors could be caused by noise fluctuations or they could be an indication that more advanced chromatic noise modeling is necessary (Alam et al. 2021a). They could also be caused by the GWB itself, which induces both correlated and uncorrelated noise in the pulsars (the so-called 'Earth terms' and 'pulsar terms'; Mingarelli & Mingarelli 2018). \nIn addition to Bayes factors, the goodness-of-fit of probabilistic models can be evaluated by assessing their predictive performance (Gelman et al. 2013). Specifically, given that the GWB is correlated across pulsars, we can (partially) predict the timing residuals δt a of \nFigure 8. Support for curn γ (blue) and hd γ correlations (orange) in each pulsar, as measured by a dropout analysis. Dropout factors greater than 1 indicate support for the curn γ or hd γ while those less than 1 show that the pulsar disfavors it. We find significant spread in the dropout factors among pulsars with long observation times, but overall more pulsars favor curn γ participation and hd γ correlations than disfavor them. \n<!-- image --> \n10 \npulsar a from the residuals δt -a of all other pulsars by way of the 'leave-one-out' posterior predictive likelihood (PPL) \np ( δt a | δt -a ) = ∫ d θ a p ( δt a | θ a ) p ( θ a | δt -a ) , (13) \nwhere θ a are all the parameters and hyperparameters that affect pulsar a in a given model. As discussed in Meyers et al. (2023), we compare the predictive performance of curn 13 / 3 and hd 13 / 3 for each pulsar in turn by taking the ratio of the corresponding leave-one-out PPLs. These ratios are closely related to the dropout factors plotted in Figure 8. Multiplying the PPL ratios for all pulsars yields the pseudo Bayes factor (PBF). For the 15-year data set we find PBF 15 yr = 1,400 in favor of hd 13 / 3 over curn 13 / 3 . The PBF does not have a 'betting odds' interpretation, but we obtain a crude estimate of its significance by building its background distribution on 40 curn 13 / 3 simulations with the MAP log 10 A CURN inferred from the 15-year data set. For all simulations except one, the PBF favors the null hypothesis, and log 10 PBF 15 yr is displaced by approx. three standard deviations from the mean log 10 PBF. \nA different sort of cross-validation relies on evaluating the optimal statistic for temporal subsets of the data set, as in Hazboun et al. (2020). In the regime where \nFigure 9. S/N growth as a function of time and number of pulsars. As we move from left to right we add an additional six months of data at each step. New pulsars are added when they accumulate three years of data. The blue violin plot shows the distribution of the optimal statistic S/N over curn γ noise parameters. The dashed orange line shows the number of pulsars used for each time slice. \n<!-- image --> \nthe lowest frequencies of our data are dominated by the GWB, the optimal statistic S/N should grow with the square root of the time span of the data and linearly with the number of pulsars in the array (Siemens et al. 2013); in this regime increasing the number of pulsars is the best way to boost PTA sensitivity to the GWB. To verify that this is indeed the case, we analyze 'slices' of the data set in six-month increments, starting from a six-year data set. Once a new pulsar accumulates three years of data, we add it to the array. We perform a separate Bayesian curn γ analysis for each slice and calculate the Hellings-Downs optimal statistic over the noise-parameter posterior. In Figure 9, we plot the S/N distributions against time span and the number of pulsars. As expected, we observe essentially monotonic growth associated with the increase in the number of pulsars. \nThe signal should also be consistent between timing observations made with Arecibo and GBT. To test this, we analyze the two split-telescope data sets (see App. A); both show evidence of common-spectrum excess noise. Figure 10 shows Arecibo (orange) and GBT (green) curn γ posteriors, which are broadly consistent with each other and with full-data posteriors (blue). Arecibo yields log 10 A = -14 . 02 +0 . 18 -0 . 22 and γ = 2 . 78 +0 . 70 -0 . 64 (medians with 68% credible intervals), while GBT yields log 10 A = -14 . 2 +0 . 15 -0 . 17 and γ = 3 . 37 +0 . 40 -0 . 38 . \nThe split-telescope data sets are significantly less sensitive to spatial correlations than the full data set, because they have fewer pulsars and therefore pulsar pairs (see Figure 12 of App. A). Nevertheless, we can search them for spatial correlations using the optimal statistic. We find a noise-marginalized Hellings-Downs S/N \nFigure 10. curn γ posterior distributions for Arecibo (orange) and GBT (green) split-telescope data sets, and for the full data set (blue). The dashed line marks γ CURN = 13 / 3. The posteriors for the split-telescope data sets are consistent with each other and with the posteriors for the full data set. \n<!-- image --> \nof 2.9 for Arecibo and 3.3 for GBT, consistent with the split-telescope data sets having about half the number of pulsars as the full data set. The S/Ns for Arecibo and GBT are comparable: while telescope sensitivity, observing cadence, and distribution of pulsars all affect GWB sensitivity, the dominant factor is the number of pulsars because the S/N scales linearly with the number of pulsars but only as ∝ ( σ √ c ) -1 /γ , where σ is the residual root-mean-squared, and c is the observing cadence (Siemens et al. 2013). We also note that the distributions of angular separations probed by Arecibo and GBT are similar, although GBT observes more pulsar pairs with large angular separations (see Figure 12).", '6. DISCUSSION': "In this letter we have reported on a search for an isotropic stochastic GWB in the 15-year NANOGrav data set. A previous analysis of the 12.5-year NANOGrav data set found strong evidence for excess low-frequency noise with common spectral properties across the array, but inconclusive evidence for Hellings-Downs inter-pulsar correlations, which would point to the GW origin of the background. By contrast, the 12.5-year data disfavored purely monopolar (clock-error-like) and dipolar (ephemeris-error-like) correlations. Subsequent independent analyses by the PPTA and EPTA collaborations reported results consistent with ours (Goncharov et al. 2021a; Chen et al. 2021), as did the search of a combined data set (Antoniadis et al. 2022)-a syzygy of tantalizing discoveries that portend the rise of low-frequency GW astronomy. \nWe analyzed timing data for 67 pulsars in the 15-year data set (those that span > 3 years), with a total time span of 16.03 years, and more than twice the pulsar pairs than in the 12.5-year data set. The common-spectrum stochastic signal gains even greater significance and is \ndetected in a larger number of pulsars. For the first time, we find compelling evidence of Hellings-Downs interpulsar correlations, using both Bayesian and frequentist detection statistics (see Figure 1), with false-alarm probabilities of p = 10 -3 and p = 5 × 10 -5 -1 . 9 × 10 -4 , respectively (see Figure 3). \nThe significance of Hellings-Downs correlations increases as we increase the number of frequency components in the analysis up to five, indicating that the correlated signal extends over a range of frequencies. A detailed spectral analysis supports a power-law signal, but at least two frequency bins show deviations that may skew the determination of spectral slope (Figure 6). These discrepancies may arise from astrophysical or systematic effects. Furthermore, slope determination changes significantly using an alternative DM model (Figure 5). The study of spatial correlations with the optimal statistic confirms a Hellings-Downs quasiquadrupolar pattern (Figure 7 and panel c of Figure 1), with some indications of an additional monopolar signal confined to a narrow frequency range near 4 nHz. However, the Bayesian evidence for this monopolar signal is inconclusive, and we could not ascribe it to any astrophysical or terrestrial source (e.g., an individual SMBHB or errors in the chain of timing corrections). \nThe GWB is a persistent signal that should increase in significance with number of pulsars and observing time span. This is indeed what we observe by analyzing slices of the data set (see Figure 9). Furthermore, the signal is present in multiple pulsars (Figure 8), and can be found in independent single-telescope data sets (Figure 10). We are preparing a number of other papers searching the 15-year data set for stochastic and deterministic signals, including an all-sky, all-frequency search for GWs from individual circular SMBHBs. This search, together with the same analysis of the 12.5-year data set (Arzoumanian et al. 2023), indicates that the spectrum and correlations we observe cannot be produced by an individual circular SMBHB. \nIf the Hellings-Downs-correlated signal is indeed an astrophysical GWB, its origin remains indeterminate. Among the many possible sources in the PTA frequency band, numerous studies have focused on the unresolved background from a population of close-separation SMBHBs. The SMBHB population is a direct byproduct of hierarchical structure formation, which is driven by galaxy mergers (e.g., Blumenthal et al. 1984). In a post-merger galaxy, the SMBHs sink to the center of the common merger remnant through dynamical interactions with their astrophysical environment, eventually leading to the formation of a binary (Begelman et al. 1980). GW emission from a SMBHB at nHz frequencies \nis quasi-monochromatic because the binaries evolve very slowly. Under the assumption of purely GW-driven binary evolution, the expected characteristic-strain spectrum is ∝ f -2 / 3 (or f -13 / 3 for pulsar-timing residuals). \nThe GWB spectrum may also feature a low-frequency turnover induced by the dynamical interactions of binaries with their astrophysical environment (e.g., with stars or gas, see Armitage & Natarajan 2002; Sesana et al. 2004; Merritt & Milosavljevi'c 2005) or possibly by non-negligible orbital eccentricities persisting to small separations (Enoki & Nagashima 2007). We find little support for a low-frequency turnover in our data (see App. E). \nThe GWB amplitude is determined primarily by SMBH masses and by the occurrence rate of close binaries, which in turn depends on the galaxy merger rate, the occupation fraction of SMBHs, and the binary evolution timescale; population models predict amplitudes ranging over more than an order of magnitude (Rajagopal & Romani 1995; Wyithe & Loeb 2003; Jaffe & Backer 2003; McWilliams et al. 2014; Sesana 2013), under a variety of assumptions. Figure 11 displays a comparison of hd γ parameter posteriors with power-law spectral fits from an observationally constrained semianalytic model of the SMBHB population constructed with the holodeck package (Kelley et al. 2023). This particular set of SMBHB populations assumes purely GW-driven binary evolution, and uses relatively narrow distributions of model parameters based on literature constraints from galaxy-merger observations (see, e.g., Tomczak et al. 2014). While the amplitude recovered in our analysis is consistent with models derived directly from our understanding of SMBH and galaxy evolution, it is toward the upper end of predictions implying a combination of relatively high SMBH masses and binary fractions. A detailed discussion of the GWB from SMBHBs in light of our results is given in Agazie et al. (2023a). \nIn addition to SMBHBs, more exotic cosmological sources such as inflation, cosmic strings, phase transitions, domain walls, and curvature-induced GWs can also produce detectable GWBs in the nHz range (see, e.g., Guzzetti et al. 2016; Caprini & Figueroa 2018, and references therein). Similarities in the spectral shapes of cosmological and astrophysical signals make it challenging to determine the origin of the background from its spectral characterization (Kaiser et al. 2022). The question could be settled by the detection of signals from individual loud SMBHBs or by the observation of spatial anisotropies, since the anisotropies expected from SMBHBs are orders of magnitude larger than those produced by most cosmological sources (Caprini & Figueroa \nFigure 11. Posteriors of hd γ amplitude (for f ref = 1yr -1 ) and spectral slope for the 15-year data set (blue), compared to power-law fits to simulated GWB spectra (red, dashed) from a population of SMBHBs generated by holodeck (Kelley et al. 2023) under the assumption of purely GW-driven binary evolution, and narrowly distributed model parameters based on galaxy merger-observations. We show 1 / 2 / 3 σ regions, and the dashed line indicates γ = 13 / 3. The broad contours confirm that population variance can lead to a significant spread of spectral characteristics. \n<!-- image --> \n2018; Bartolo et al. 2022). We discuss these models in the context of our results in Afzal et al. (2023). \nThe EPTA and Indian Pulsar Timing Array (InPTA; Joshi et al. 2018), PPTA, and Chinese Pulsar Timing Array (CPTA; Lee 2016) collaborations have also recently searched their most recent data for signatures of a gravitational-wave background (Antoniadis et al. 2023; Reardon et al. 2023; Xu et al. 2023), and an upcoming IPTA paper will compare the results of these searches. The IPTA's forthcoming Data Release 3 will combine the NANOGrav 15-year data set with observations from the EPTA, PPTA, and InPTA collaborations, comprising about 80 pulsars with time spans up to 24 years, and offering significantly greater sensitivity to spatial correlations and spectral characteristics than single-PTA data sets. Future PTA observation campaigns will improve our understanding of this signal and of its astrophysical and cosmological interpretation. Longer data sets will tighten spectral constraints on the GWB, clarifying its origin (Pol et al. 2021). Greater numbers of pulsars will allow us to probe anisotropy in the GWB (Pol et al. 2022) and its polarization structure (see, e.g., Arzoumanian et al. 2021, and references therein). The observation of a stochastic signal with spatial correlations in PTA data, suggesting a GWB origin, expands the horizon of GW astronomy with a new Galaxy-scale observatory sensitive to the most massive black-hole systems in the Universe and to exotic cosmological processes.", 'ACKNOWLEDGMENTS': "Author contributions. An alphabetical-order author list was used for this paper in recognition of the fact that a large, decade timescale project such as NANOGrav is necessarily the result of the work of many people. All authors contributed to the activities of the NANOGrav collaboration leading to the work presented here, and reviewed the manuscript, text, and figures prior to the paper's submission. Additional specific contributions to this paper are as follows. G.A., A.A., A.M.A., Z.A., P.T.B., P.R.B., H.T.C., K.C., M.E.D., P.B.D., T.D., E.C.F., W.F., E.F., G.E.F., N.G., P.A.G., J.G., D.C.G., J.S.H., R.J.J., M.L.J., D.L.K., M.K., M.T.L., D.R.L., J.L., R.S.L., A.M., M.A.M., N.M., B.W.M., C.N., D.J.N., T.T.P., B.B.P.P., N.S.P., H.A.R., S.M.R., P.S.R., A.S., C.S., B.J.S., I.H.S., K.S., A.S., J.K.S., and H.M.W. developed the 15-year data set through a combination of observations, arrival time calculations, data checks and refinements, and timing model development and analysis; additional specific contributions to the data set are summarized in NG15. S.R.T. and S.J.V. led the search and coordinated the writing of this paper. P.T.B., J.S.H., P.M.M., N.S.P., J.S., S.R.T., M.V., and S.J.V. wrote the paper, made the figures, and generated the bibliography. K.P.I., N.L., N.S.P., X.S., J.S., J.P.S., S.R.T., and S.J.V. performed analyses and developed new techniques on a preliminary 14-year data set. P.T.B., B.B., B.D.C., S.C., B.D., G.E.F., K.G., A.D.J., A.R.K., L.Z.K., N.L., W.G.L., N.S.P., S.C.S., L.S., X.S., J.S., J.P.S., S.R.T., C.U., M.V., S.J.V., Q.W., and C.A.W. analyzed preliminary versions of the 15-year data set. J.S.H., J.G., N.L., N.S.P., J.P.S., and J.K.S. performed noise analyses on the data set. P.T.B., B.B., R.B., R.C., M.C., N.J.C., D.D., H.D., G.E.F., K.A.G., S.H., A.D.J., N.L., W.G.L., P.M.M., P.P., N.S.P., S.C.S., X.S., J.S., J.P.S., J.T., S.R.T., M.V., S.J.V., and C.A.W. performed the Bayesian and frequentist analyses presented in this paper. K.G., J.S.H., P.M.M., J.D.R., and S.J.V. computed the background distribution of our detection statistics. P.R.B., P.M.M., N.S.P., and M.V. performed simulations that were used to compute false alarm probabilities. A.M.A., P.T.B., B.B., B.D., S.C., J.A.E., G.E.F., J.S.H., A.D.J., A.R.K., N.L., M.T.L., K.D.O., T.T.P., N.S.P., J.S., J.P.S., J.K.S., S.R.T., M.V., and S.J.V. contributed to the development of our code. P.T.B., N.J.C., J.S.H., A.D.J., T.B.L., P.M.M., J.D.R., and M.V. performed an internal code review. K.C., C.C., J.A.E., J.S.H., D.R.M., M.A.Mc., C.M.F.M., K.D.O., N.S.P., S.R.T., M.V., R.v.H., and S.J.V. were part of a NANOGrav Detection Study Group. M.A.M. and P.N. served on the IPTA Detection Committee. K.C., J.S.H., T.J.W.L., P.M.M., N.S.P., J.D.R., X.S., S.R.T., S.J.V., and C.A.W. responded to the Detection Checklist from the IPTA Detection Committee. \nAcknowledgments. The NANOGrav collaboration receives support from National Science Foundation (NSF) Physics Frontiers Center award numbers 1430284 and 2020265, the Gordon and Betty Moore Foundation, NSF AccelNet award number 2114721, an NSERC Discovery Grant, and CIFAR. The Arecibo Observatory is a facility of the NSF operated under cooperative agreement (AST-1744119) by the University of Central Florida (UCF) in alliance with Universidad Ana G. M'endez (UAGM) and Yang Enterprises (YEI), Inc. The Green Bank Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc. The National Radio Astronomy Observatory is a facility of the NSF operated under cooperative agreement by Associated Universities, Inc. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Specifically, it used the Bridges-2 system, which is supported by NSF award number ACI-1928147, at the Pittsburgh Supercomputing Center (PSC). This work was conducted using the Thorny Flat HPC Cluster at West Virginia University (WVU), which is funded in part by National Science Foundation (NSF) Major Research Instrumentation Program (MRI) Award number 1726534, and West Virginia University. This work was also conducted in part using the resources of the Advanced Computing Center for Research and Education (ACCRE) at Vanderbilt University, Nashville, TN. This work was facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington. This research was supported in part through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor. NANOGrav is part of the International Pulsar Timing Array (IPTA); we would like to thank our IPTA colleagues for their feedback on this paper. We thank members of the IPTA Detection Committee for developing the IPTA Detection Checklist. We thank Bruce Allen for useful feedback. We thank Valentina Di Marco and Eric Thrane for uncovering a bug in the sky scramble code. We thank Jolien Creighton and Leo Stein for helpful conversations about background estimation. L.B. acknowledges support from the National Science Foundation under award AST-1909933 and from the Research Corporation for Science Advancement under Cottrell Scholar Award No. 27553. P.R.B. is supported by the Science and Technology Facilities Council, grant number ST/W000946/1. S.B. gratefully acknowledges the support of a Sloan Fellowship, and the support of NSF under award #1815664. The work of R.B., \nR.C., D.D., N.La., X.S., J.P.S., and J.T. is partly supported by the George and Hannah Bolinger Memorial Fund in the College of Science at Oregon State University. M.C., P.P., and S.R.T. acknowledge support from NSF AST-2007993. M.C. and N.S.P. were supported by the Vanderbilt Initiative in Data Intensive Astrophysics (VIDA) Fellowship. K.Ch., A.D.J., and M.V. acknowledge support from the Caltech and Jet Propulsion Laboratory President's and Director's Research and Development Fund. K.Ch. and A.D.J. acknowledge support from the Sloan Foundation. Support for this work was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc./National Radio Astronomy Observatory. Support for H.T.C. is provided by NASA through the NASA Hubble Fellowship Program grant #HST-HF251453.001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. K.Cr. is supported by a UBC Four Year Fellowship (6456). M.E.D. acknowledges support from the Naval Research Laboratory by NASA under contract S-15633Y. T.D. and M.T.L. are supported by an NSF Astronomy and Astrophysics Grant (AAG) award number 2009468. E.C.F. is supported by NASA under award number 80GSFC21M0002. G.E.F., S.C.S., and S.J.V. are supported by NSF award PHY2011772. K.A.G. and S.R.T. acknowledge support from an NSF CAREER award #2146016. The Flatiron Institute is supported by the Simons Foundation. S.H. is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1745301. N.La. acknowledges the support from Larry W. Martin and Joyce B. O'Neill Endowed Fellowship in the College of Science at Oregon State University. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). D.R.L. and M.A.Mc. are supported by NSF #1458952. M.A.Mc. is supported by NSF #2009425. C.M.F.M. was supported in part by the National Science Foundation under Grants No. NSF PHY-1748958 and AST-2106552. A.Mi. is supported by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy - EXC 2121 Quantum Universe - 390833306. P.N. acknowledges support from the BHI, funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. The Dunlap Institute is funded by an endowment established by the David Dunlap family and the University of Toronto. K.D.O. was supported in part by NSF Grant No. 2207267. T.T.P. acknowledges \nsupport from the Extragalactic Astrophysics Research Group at Eotvos Lor'and University, funded by the Eotvos Lor'and Research Network (ELKH), which was used during the development of this research. S.M.R. and I.H.S. are CIFAR Fellows. Portions of this work performed at NRL were supported by ONR 6.1 basic research funding. J.D.R. also acknowledges support from start-up funds from Texas Tech University. J.S. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2202388, and acknowledges previous support by the NSF under award 1847938. C.U. acknowledges support from BGU (Kreitman fellowship), and the Council for Higher Education and Israel Academy of Sciences and Humanities (Excellence fellowship). C.A.W. acknowledges support from CIERA, the Adler Planetarium, and the Brinson Foundation through a CIERA-Adler postdoctoral fellowship. O.Y. is supported by the National Science Foundation \nGraduate Research Fellowship under Grant No. DGE2139292. \nDedication. This paper is dedicated to the memory of Donald Backer: a pioneer in pulsar timing arrays, a term he coined; a discoverer of the first millisecond pulsar; a master developer of pulsar timing instrumentation; a founding member of NANOGrav; and a friend and mentor to many of us. \nFacilities: Arecibo, GBT, VLA \nSoftware: acor , astropy (Astropy Collaboration et al. 2022), ceffyl (Lamb et al. 2023), chainconsumer (Hinton 2016), ENTERPRISE (Ellis et al. 2019), enterprise extensions (Taylor et al. 2018), hasasia (Hazboun et al. 2019), holodeck (Kelley et al. 2023), Jupyter (Kluyver et al. 2016), libstempo (Vallisneri 2020), matplotlib (Hunter 2007), numpy (Harris et al. 2020), PINT (Luo et al. 2021), PTMCMC (Ellis & van Haasteren 2017), scipy (Virtanen et al. 2020), Tempo2 (Hobbs & Edwards 2012)", 'REFERENCES': "Armitage, P. 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Kitagawa (New York, NY: Springer \nNew York), 199-213, doi: 10.1007/978-1-4612-1694-0 15 \nAlam, M. F., Arzoumanian, Z., Baker, P. T., et al. 2021a, \nApJS, 252, 4, doi: 10.3847/1538-4365/abc6a0 \n-. 2021b, ApJS, 252, 5, doi: 10.3847/1538-4365/abc6a1 \nAllen, B. 2023, PhRvD, 107, 043018, \ndoi: 10.1103/PhysRevD.107.043018 \nAllen, B., & Romano, J. D. 2022, arXiv e-prints, \narXiv:2208.07230, doi: 10.48550/arXiv.2208.07230 \nAnholm, M., Ballmer, S., Creighton, J. D. E., Price, L. R., & Siemens, X. 2009, PhRvD, 79, 084030, \ndoi: 10.1103/PhysRevD.79.084030 \nAntoniadis, J., Arzoumanian, Z., Babak, S., et al. 2022, \nMNRAS, 510, 4873, doi: 10.1093/mnras/stab3418 \nAntoniadis, J., et al. 2023, in preparation", 'A. ADDITIONAL DATA SET DETAILS': "The observations included in the NANOGrav 15-year data set were performed between July 2004 and August 2020 with the 305-m Arecibo Observatory (Arecibo), the 100-m Green Bank Telescope (GBT), and, since 2015, the 27 25-m antennae of the Very Large Array (VLA). We used Arecibo to observe the 33 pulsars that lie within its declination range (0 · < δ < +39 · ); GBT to observe the pulsars that lie outside of Arecibo's range, plus J1713+0747 and B1937+21, for a total of 36 pulsars; the VLA to observe the seven pulsars J0437 -4715, J1600 -3053, J1643 -1224, J1713+0747, J1903+0327, J1909 -3744, and B1937+21. Six of these were also observed with Arecibo, GBT, or both; J0437 -4715 was only visible to the VLA. Figure 12 shows the sky locations of the 67 pulsars used for the GWB search (top) and the distribution of angular separations for the pulsar pairs (bottom). \nInitial observations were performed with the ASP (Arecibo) and GASP (GBT) systems, with 64-MHz bandwidth (Demorest 2007). Between 2010 and 2012, we transitioned to the PUPPI (Arecibo) and GUPPI (GBT) systems, with bandwidths up to 800 MHz (DuPlain et al. 2008; Ford et al. 2010). We observe pulsars in two different radio-frequency bands in order to measure pulse dispersion from the interstellar medium: at Arecibo, we use the 1.4 GHz receiver plus either the 430 MHz or 2.1 GHz receiver (and the 327 MHz receiver for early observations of J2317+1439); at GBT, we use the 820 MHz and 1.4 GHz receivers; at the VLA, we use the 1.4 GHz and 3 GHz receivers with the YUPPI system. \nIn § 5.4 we analyze also two split-telescope data sets: 33 pulsars for Arecibo, and 35 for GBT (excluding J0614 -3329, which was observed for less than three years). For the two pulsars timed by both telescopes (J1713+0747 and B1937+21), we partition the timing data between the telescopes and obtain independent timing solutions for each. We do not analyze a VLA-only data set, which would have shorter observation spans and significantly reduced sensitivity.", 'B. BAYESIAN METHODS & DIAGNOSTICS': "The prior probability distributions assumed for all analyses in this paper are listed in Table 1. We use Markov chain Monte Carlo (MCMC) techniques to sample randomly from the joint posterior distribution of our model parameters. Marginal distributions are obtained simply by considering only the parameter of interest \n<!-- image --> \nFigure 12. Top: Sky locations of the 67 pulsars used in the 15-year GWB analysis. Markers indicate which telescopes observed the pulsar. Bottom: Distribution of angular separations probed by the pulsars in the full data set (orange), the Arecibo data set (blue), and the GBT data set (red). Because Arecibo and GBT mostly observed pulsars at different declinations, there are few inter-telescope pairs at small angular separations, resulting in a deficit of pairs for the full data set in the first bin. \n<!-- image --> \nin each sample. To assess convergence of our MCMC runs beyond visual inspection we use the Gelman-Rubin statistic, requiring ˆ R < 1 . 01 for all parameters (Gelman & Rubin 1992; Vehtari et al. 2021). We performed most runs discussed in this paper with the PTMCMC sampler (Ellis & van Haasteren 2017) and postprocessed samples with chainconsumer (Hinton 2016). \nIn NG12gwb we use an analytic approximation for the uncertainty of marginalized-posterior statistics (Wilcox 2012). Here we instead adopt a boostrap approach: we resample the original MCMC samples (with replacement) to generate new sets that act as independent sampling realizations. We then calculate the distributions of the desired summary statistics (e.g., quantiles, marginalized posterior values) over these sets. From these distributions, we determine central values and uncertainties (either medians and 68% confidence intervals, or means and standard deviations). \nTable 1. Prior distributions used in all analyses performed in this paper. \nWe rely on a variety of techniques to perform Bayesian model comparison. The first is thermodynamic integration (e.g., Ogata 1989; Gelman & Meng 1998), which computes Bayesian evidence integrals directly through parallel tempering: we run N β MCMC chains that explore variants of the likelihood function raised to different exponents β , then approximate the evidence for model H as \nln p ( d |H ) = ∫ 1 0 ⟨ ln p ( d | θ ) ⟩ β d β ≈ 1 N β ∑ β ⟨ ln p ( d | θ ) ⟩ β , (B1) \nwhere all likelihoods and posteriors are computed within model H , θ denotes all of the model's parameters, and the expectation ⟨ ln p ( d | θ ) ⟩ β is approximated by MCMC with respect to the posterior p β ( θ | d ) ∝ p ( d | θ, H ) β p ( θ, H ). The inverse temperatures β are spaced geometrically, as is the default in PTMCMC . \nTo compare nested models, which differ by 'freezing' a subset of parameters, we also use the Savage-Dickey density ratio (Dickey 1971): if models H and H 0 differ \nby the fact that (say) θ 0 is frozen to 0 in the latter, then p ( d |H 0 ) /p ( d |H ) = p ( θ 0 = 0 | d, H ) /p ( θ 0 = 0 |H ). \nWhen comparing disjoint models with different likelihoods (e.g., hd versus curn ), we use product-space sampling (Carlin & Chib 1995; Godsill 2001). This method treats model comparison as a parameter estimation problem, where we sample the union of the unique parameters of all models, plus a model-indexing parameter that activates the relevant likelihood function and parameter space of one of the sub-models. Bayes factors are then obtained by counting how often the model index falls in each activation region and taking ratios of those counts. \nIn some situations, it can be difficult to sample a computationally expensive model directly. In these cases, we sample a computationally cheaper approximate distribution and reweight those posterior samples to estimate the posterior for the computationally expensive model (Hourihane et al. 2023). The reweighted posterior can be used in the thermodynamic-integration or Savage-Dickey methods. In addition, the mean of the weights yields the Bayes factor between the expensive \nand approximate models, which may be of direct interest (e.g., hd can be approximated by curn ). We estimate Bayes-factor uncertainties using bootstrapping and, for product-space sampling, with the Markov-model techniques of Cornish & Littenberg (2015) and Heck et al. (2019).", 'C. BROKEN POWER-LAW MODEL': 'As shown in NG12gwb, the simultaneous Bayesian estimation of white measurement noise and of red-noise processes described by power laws biases the recovery of the spectral index of the latter (Lam et al. 2017; Hazboun et al. 2019). Just as in NG12gwb and Antoniadis et al. (2022), we impose a high-frequency cutoff on the red-noise processes. To choose the cutoff frequency, we perform inference on our data with a curn γ model modified so that the common process has power spectral density \nS ( f ) = A 2 12 π 2 ( f f ref ) -γ [ 1 + ( f f break ) 1 /ℓ ] ℓγ f -3 ref ; (C2) \nthen set the cutoff to the MAP f break . Equation C2 is fairly generic, allowing for separate spectral indices at low ( γ ) and high ( δ ) frequencies. The break frequency f break dictates where the broken power law changes spectral index, while ℓ (which we set to 0.1) controls the smoothness of the transition. \nThe marginal posterior for f break , obtained in the factorized-likelihood approximation using the techniques of Lamb et al. (2023), has median and 90% credible region of 3 . 2 +5 . 4 -1 . 2 × 10 -8 Hz, and a MAP value of 2 . 75 × 10 -8 Hz. The latter is close to f 14 = 14 /T in our frequency basis (with T the total span of the data set), so we use 14 frequencies to model common-spectrum noise processes (see § 2 and NG12gwb).', 'D. T -PROCESS SPECTRUM MODEL': "The free-spectrum analysis of our data ( § 5.2 and Figure 6) shows that the frequency bins at f 1 , f 6 , f 7 , and f 8 appear to be in tension with a pure power law, skewing the estimation of γ and reducing the hd 13 / 3 vs. curn 13 / 3 Bayes factor. Assuming that those frequency components reflect unmodeled systematics or strongerthan-expected statistical fluctuations, we can make our inference more robust to such outliers with a 'fuzzy' power-law model that allows the individual Φ i to vary more freely around their expected values. To wit, we introduce the t -process spectrum (TPS) \nΦ TPS ,i = x i Φ powerlaw ,i with x ∼ invgamma( x i ; 1 , 1) , (D3) \nwhere Φ powerlaw ,i follows Equation 6 and x follows the inverse gamma distribution with parameters α = β = 1; the resulting Gaussian mixture yields a Student'st distribution for the Φ TPS ,i . Figure 13 shows curn γ powerlaw posteriors and curn TPS modified power-law posteriors, obtained in the factorized-likelihood approximation (Taylor et al. 2022; Lamb et al. 2023) and compared to curn free bin variances. The TPS model is spread more widely and deviates from the perfect power law at bins f 1 , f 6 , f 7 , and f 8 , as expected. The right panel of Figure 13 shows the joint log 10 A,γ posteriors for curn γ and curn TPS . The latter is more consistent with steeper power laws, and it includes γ = 13 / 3 at 1 σ credibility.", 'E. TURNOVER MODEL': "The final parameterized spectral model that we investigate is motivated by the idea that the dynamics of SMBHBs are influenced by their environments at sub-parsec separations (Armitage & Natarajan 2002; Sesana et al. 2004; Merritt & Milosavljevi'c 2005). These interactions affect binary evolution and the resulting spectrum of the GWB. The process of bringing two SMBHs together after galaxy mergers involves a complex chain of interactions: despite significant theoretical work, the lack of observational constraints makes it difficult to draw any conclusions. PTAs, however, provide a unique opportunity to probe the timescale over which two SMBHs evolve from the merger of their galaxies to a bound binary that produces GW signals in the PTA sensitivity band. \nWhen dynamical interactions dominate orbital evolution, binaries will traverse the GW spectrum more quickly, reducing GW emission compared to a GWdriven inspiral. This kind of behavior is captured by the turnover model (Sampson et al. 2015): \nS ( f ) = A 2 12 π 2 ( f f ref ) -γ [ 1 + ( f 0 f ) κ ] -1 f -3 ref . (E4) \nThis is qualitatively similar to the broken power law discussed earlier, except that here f 0 represents the GW frequency at which typical binary evolution transitions from environmentally dominated (at lower frequencies and wider separations) to GW-dominated (at higher frequencies and smaller separations). The parameter κ controls the shape of the spectrum below f 0 , and depends on the orbital-evolution mechanism. Note that the actual turning point of the spectrum is not at f 0 but at f bend = f 0 × (3 κ/ 4 -1) 1 /κ (NG9gwb). \nApplying this model to our data, we find hints of departures from a pure power law: the transition frequency f 0 lies below 10 nHz with 65% credibility, while the bend frequency lies below 10 nHz with 75% credibility. \n<!-- image --> \nFigure 13. Power-law ( curn γ , blue) and t -process power-law ( curn TPS , orange) spectral posteriors. Left : reconstructed spectra, compared to free-spectral bin-variance posteriors ( curn TPS , violin plots). Right : joint (log 10 A,γ ) posteriors. The 'fuzzy' t -process allows local deviations from a perfect power law, producing wider constraints that are more consistent with γ = 13 / 3 (dashed line). \n<!-- image --> \nNevertheless, Bayesian comparison of this curn turnover model with curn γ reports an inconclusive Bayes factor of 1 . 46 ± 0 . 02 in favor of curn turnover . Furthermore, the estimation of curn turnover parameters is sensitive to DM modeling (see § 5.1). While the spectra are broadly consistent whether we use DMX or DMGP to model DM fluctuations, there are differences in the power at certain frequencies that lead to differences in the turnover parameters. This is discussed in greater detail in Agazie et al. (2023a).", 'F. SKY SCRAMBLES': "In the sky-scramble method (Cornish & Sampson 2016), inter-pulsar correlations are analyzed as if the pulsars occupied random sky positions, with the purpose of creating a background distribution of PTA detection statistics for null-hypothesis testing, in alternative to phase shifts (Taylor et al. 2017; see § 3 and § 4). If a correlated signal is present in the data, phase shifts and sky scrambles actually test different null hypotheses: phase shifts test the hypothesis that no inter-pulsar correlations are present, while sky scrambles assume that interpulsar correlations are present at the level measured in the data, but test the hypothesis that these correlations have no dependence on angular separation. \nAs is the convention in the literature, we require that scrambled overlap reduction functions (ORFs) be independent of each other and of the unscrambled ORF using a match statistic, \n̸ \n¯ M = ∑ a,b = a Γ ab Γ ' ab √ ( ∑ a,b = a Γ ab Γ ab )( ∑ a,b = a Γ ' ab Γ ' ab ) , (F5) \nwhere Γ ab and Γ ' ab are two different ORFs. For the sky scrambles used in our analysis, the scrambled ORFs have ¯ M < 0 . 1 with respect to the unscrambled ORF, and ¯ M < 0 . 17 with each other. We generate 10,000 sky scrambles, owing to the difficulty in obtaining large numbers of scrambled ORFs that satisfy the match threshold; because of limitations of computational resources, we obtain our detection statistics for 5,000 of those ORFs. Figure 14 shows the resulting background distributions for the hd γ -tocurn γ Bayes factor (left panel) and the optimal-statistic S/N (right panel). The Bayes factors exceed the observed value in eight of the 5,000 sky scrambles ( p = 1 . 6 × 10 -3 ), while none of the sky scrambles have noise-marginalized mean S/N greater than observed ( p < 10 -4 ). \nWe note that the null distribution recovered by the sky scrambles is not very sensitive to the choice of match threshold for | ¯ M | ≲ 0 . 2. Figure 15 compares the null distributions when the match threshold for all ORFs with each other and with the unscrambled ORF is set to | ¯ M | < 0 . 17 (blue), | ¯ M | < 0 . 1 (orange), and | ¯ M | < 0 . 08 (green). There is very little difference among the distributions; however, imposing a smaller threshold means that fewer sky scrambles can be used (6,043 with | ¯ M | < 0 . 1 and 1,534 with | ¯ M | < 0 . 08, compared to 10,000 with | ¯ M | < 0 . 17), which limits the precision with which the p -value can be measured. We find no evidence that the recovered null distribution is biased when including sky scrambles with matches up to 0.17. \n̸ \n̸ \nFigure 14. Empirical background distribution of hd γ -tocurn γ Bayes factor (left, see § 3) and noise-marginalized optimal statistic (right, see § 4), as computed in 5,000 sky scrambles, which erases the dependence of inter-pulsar correlations on the angular separation between the pulsars. Dotted lines indicate Gaussian-equivalent 2 σ , 3 σ , and 4 σ thresholds. The dashed vertical lines indicate the values of the detection statistics for the unscrambled data set. We find p = 1 . 6 × 10 -3 (approx. 3 σ ) for the Bayesian analysis, and p < 10 -4 ( > 3 σ ) for the optimal-statistic analysis. \n<!-- image --> \nTable 2. Multiple-correlation optimal statistic best-fit coefficients ˆ A 2 k , S/Ns, and AIC probabilities \nNote -All values were computed for the 15-year data set, assuming a power-law power spectral density using the 14 lowest frequency components. Here ˆ A 2 , S/N, and AIC are marginalized over pulsar noise parameters with fixed γ = 13 / 3. The numbers in parentheses represent the mean least-squares errors for the ˆ A 2 k coefficients and standard deviations over noise-parameter posteriors for S/Ns. We compute p (AIC) with respect to the model with the lowest mean AIC (i.e., HD + monopole).", 'G. MULTIPLE-CORRELATION OPTIMAL STATISTIC': "The multiple-correlation optimal statistic (MCOS; Sardesai & Vigeland 2023) fits the inter-pulsar correlation coefficients ρ ab with a linear model that includes multiple components with different correlation patterns, but with the same spectral shape. The linear-model coefficients are the squared amplitudes of the components. Within such a model, the significance of each component can be quoted as a S/N given by its best-fit coefficient divided by the fit error. Just as for the noise-marginalized optimal statistic (Vigeland et al. 2018), the posterior distribution of pulsar noise parameters induces a distribution of MCOS statistics. \nWe fit the 15-year data with models that include HD, monopole, and dipole-correlated components in various combinations. Table 2 lists the noise-marginalized am- \nplitude estimates and S/N for all models. The goodnessof-fit of the models can be compared using the Akaike Information Criterion (AIC; Akaike 1998): \nAIC = 2 k + χ 2 , (G6) \nwhere k is the number of model parameters and χ 2 is the fit's chi-squared, computed without accounting for GW-induced ρ ab correlations. (This can be thought of as a pseudo-Bayes factor, with the factor of 2 k imposing an Occam penalty.) The relative probability of a model compared to the most-favored model is then given by \np (AIC) = exp [(AIC min -AIC) / 2] , (G7) \nwhere AIC min is the minimum AIC across all models. Table 2 lists the AIC probabilities, computed by averaging the AIC of each model over pulsar noise parameters. The HD-correlated model is preferred among \nFigure 15. Comparison between empirical background distributions for the noise-marginalized optimal statistic, as computed by the sky-scramble technique. We show distributions computed using a match threshold of ¯ M < 0 . 17 (blue), ¯ M < 0 . 1 (orange), and ¯ M < 0 . 08 (green). Dotted lines indicate Gaussian-equivalent 2 σ , 3 σ , and 4 σ thresholds. The dashed vertical lines indicate the values of the detection statistics for the unscrambled data set. We find little difference between the background distributions computed using different match thresholds, modulo the fact that imposing a smaller threshold yields fewer sky scrambles, which limits the precision to which the p -value can be measured. \n<!-- image --> \nthe models with a single correlated process. The models with both HD and monopole correlations are preferred among all models: for a model with HD and monopole correlations, we find S/N of 3 . 4 ± 0 . 8 for HD correlations and 2 . 9 ± 0 . 8 for monopolar correlations, while for a model with HD, monopole, and dipole correlations, we find S/N of 2 . 9 ± 0 . 6 for HD correlations, 2 . 4 ± 0 . 6 for monopole correlations, and 0 . 6 ± 0 . 4 for dipole correlations (means ± standard deviations across noise-parameter posteriors). The statistical significance of these S/Ns can be quantified empirically using simulations of 15-year-like data sets (see App. H.1), which report p -values < 10 -2 and ≃ 4 × 10 -2 for the observed mean HD and monopole statistics across data replications with no spatially correlated injections. \nAs discussed in Sardesai & Vigeland (2023), the optimal statistic and the MCOS are metrics of the apparent spatial correlation pattern of the data, but they have a limited ability to identify its actual source. That is because a real HD signal may also excite the monopole optimal statistic and the monopole component of the MCOS; conversely, a real monopolar signal may also excite the HD optimal statistic and the HD component of the MCOS; and so on. The S/Ns quoted in Table 2 quantify how often we would expect to measure the observed value of the optimal statistic if only uncorrelated noise is present, but they do not describe how often one type of correlated noise would produce a given value of the optimal statistic for a different type of correlation. \n<!-- image --> \nFigure 16. Results of the MCOS analysis, which prefers a model including both HD and monopole correlations. Top : MCOS S/N for HD correlations (solid blue) and monopole correlations (dashed orange), marginalized over curn 13 / 3 noise-parameter posteriors. The vertical lines indicate the mean S/Ns. We find a S/N of 3 . 4 ± 0 . 8 for HD correlations and 2 . 9 ± 0 . 8 for monopole correlations. Bottom : Binned cross-correlations ρ ab (black error bars), computed with MAP noise parameters from a curn 13 / 3 run. The solid blue and dashed orange curves show best-fit HD and HD+monopole correlation patterns, corresponding to ˆ A 2 = 6 . 8 × 10 -30 and to ˆ A 2 HD = 5 . 5 × 10 -30 , ˆ A 2 monopole = 8 × 10 -31 , respectively. The monopolar component accounts for the vertical shift of the cross-correlations with respect to the HD curve. We use the standard version of the optimal statistic that does not include inter-pulsar correlations to compute ρ ab , so the points and errors do not match those shown in panel (c) of Figure 1. \n<!-- image --> \nThis effect can be characterized using simulations (see App. H.1), which report a p -value of 0 . 11 for the observed mean monopole statistic when a HD-correlated signal with the MAP 15-year amplitude is included in the simulated data sets. We conclude that there are some indications of a possible monopole-correlated signal in the data with S/N comparable to but smaller than the S/N for HD correlations; however, from simulations we conclude that it is possible for such a signal to appear in an MCOS analysis if only a HD-correlated stochastic process is present. \nFigure 17. Top: MCOS HD S/N values recovered in the three simulations described in App. H.1, compared to the MCOS HD S/N measured in the real data set (vertical dashed red line), which has p -values of < 10 -2 for simulations i and iii, and 0 . 64 for simulations ii. Bottom: MCOS monopole S/N values recovered in the three simulations, compared to the real-data MCOS monopole S/N (vertical dashed red line), which has p -values of 4 × 10 -2 , 1 . 1 × 10 -1 , and < 10 -2 for simulation i, ii, and iii respectively. \n<!-- image -->", 'H. MULTIPLE-CORRELATION OPTIMAL STATISTIC SIMULATIONS': "In this appendix we obtain the distribution of the MCOS over an ensemble of simulated data sets, with the goal of characterizing the probability that the observed S/Ns could have been produced by pulsar noise alone, or by a GWB with HD correlations. Unlike our Bayesian analysis, the MCOS prefers a model that includes both HD and monopolar components. So we are especially interested in asking how frequently we may expect the observed MCOS monopole if the data contain only the GWB. In Apps. H.1 and H.2 we present two different types of simulations: 'astrophysical,' where we generate synthetic data with MAP noise parameters inferred from the 15-year data set, both with and without the GWB; and 'model checking,' where we create data replications following the hd 13 / 3 posteriors for the real data set. Note that neither simulation attempts to account for the monochromatic character of the putative monopolar signal (see § 5.2).", 'H.1. Astrophysical simulations': "Following Pol et al. (2021), we generate simulated data sets adopting MAP pulsar-noise parameters obtained from the real data independently for each pulsar; these 'noise runs' include an additional power-law process to reduce contamination between the putative GWB and the pulsars' intrinsic red noise (Taylor et al. 2022). We produce 100 realizations each of three different simu- \nFigure 18. Distribution of real-data and replicated MCOS monopole S/Ns. Each point represents a draw η ( k ) from hd 13 / 3 posterior, which is used to simulate δt sim , ( k ) and to compute both S/Ns. The replicated monopole S/N is greater for 11% of the simulations. \n<!-- image --> \nlations: (i) injecting no spatially correlated power-law GWB or excess uncorrelated common-spectrum noise; (ii) injecting a spatially correlated power-law GWB with amplitude 2 . 7 × 10 -15 and spectral index 13 / 3; and (iii) injecting no GWB or common-spectrum noise, but omitting the additional power-law process in the estimation of intrinsic pulsar noise, with the goal of testing how often excess common-spectrum noise is recognized as a spatially correlated GWB. \nWe compute HD + monopole + dipole MCOS S/Ns for all synthetic data sets (see Figure 17). The mean HD S/Ns observed in the real data (see App. G) correspond to p -values of < 10 -2 for simulations (i) and (iii), and 0 . 64 for simulation (ii). The mean monopole S/Ns observed in the real data set correspond to p -values of 4 × 10 -2 , 1 . 1 × 10 -1 , and < 10 -2 for simulations (i), (ii), and (iii) respectively. We conclude that it is unlikely that we would measure HD correlations at the level observed in real data when no correlated signal is present (simulation (i)) or when only uncorrelated common-spectrum red noise is present (simulation (iii)). In addition, the HD S/Ns obtained from a HD-correlated GWBinjection (simulation (ii)) are fully consistent with the S/N observed in real data. By contrast, the observed monopole S/N could have been produced by intrinsic pulsar noise alone, or by a real HD signal.", 'H.2. Model-checking simulations': 'In App. H.1 we have tackled the question of monopole S/N significance using simulations based on real-data MAP estimates η MAP of pulsar-noise and GW parameters. In this appendix we adopt a procedure with \na stronger Bayesian flavor, evaluating the MCOS on a population of data replications created using hd 13 / 3 as a generative model with noise hyperparameters η drawn from the hd 13 / 3 real-data posterior. This can be seen also as a Bayesian model-checking exercise (Gelman et al. 1996, 2013): if we find that the summary statistic of interest (the monopole MCOS) has a much more extreme value in real data than in data replications, we should suspect that the data model (here hd 13 / 3 ) is missing something. \nWe perform the test by drawing 500 parameter vectors { η ( k ) } from the hd 13 / 3 real-data posterior; for each η ( k ) we simulate a data set δt sim , ( k ) ∼ p ( δt | η ( k ) ) and compare MCOS( δt sim , ( k ) ; η ( k ) ) with MCOS( δt ; η ( k ) ). Our notation emphasizes the dependence of the MCOS on the pulsar noise parameters through the P matrices in Equation 9. Figure 18 shows the resulting distribution of monopole S/Ns. The replicated monopole S/N is greater than its observed counterpart for 11% of the draws. Thus, it is plausible that the MCOS could measure the observed monopole S/N in data that contain only a HD-correlated GWB. Conversely, the observed monopole S/N does not by itself suggest that hd 13 / 3 is misspecified.'}

2024A&A...690L..11B

Jupiters icy moon Ganymede has a tenuous exosphere produced by sputtering and possibly sublimation of water ice. To date only atomic hydrogen and oxygen have been directly detected in this exosphere. Here we present observations of Ganymedes COSUB2SUB exosphere obtained with the James Webb Space Telescope. COSUB2SUB gas is observed over different terrain types mainly over those exposed to intense Jovian plasma irradiation as well as over some bright or dark terrains. Despite warm surface temperatures the COSUB2SUB abundance over equatorial subsolar regions is low. COSUB2SUB vapor has the highest abundance over the north polar cap of the leading hemisphere reaching a surface pressure of 1 pbar. From modeling we show that the local enhancement observed near 12 h local time in this region can be explained by the presence of cold traps enabling COSUB2SUB adsorption. However whether the release mechanism in this highlatitude region is sputtering or sublimation remains unclear. The north polar cap of the leading hemisphere also has unique surfaceice properties probably linked to the presence of the large atmospheric COSUB2SUB excess over this region. These COSUB2SUB molecules might have been initially released in the atmosphere after the radiolysis of COSUB2SUB precursors or from the sputtering of COSUB2SUB embedded in the HSUB2SUBO ice bedrock. Dark terrains regiones more widespread on the north versus south polar regions possibly harbor COSUB2SUB precursors. COSUB2SUB molecules would then be redistributed via cold trapping on icerich terrains of the polar cap and be diurnally released and redeposited on these terrains. Ganymedes COSUB2SUB exosphere highlights the complexity of surfaceatmosphere interactions on Jupiters icy Galilean moons.

2024-10-01T00:00:00Z

['arXiv:2409.13364', '2024A&A...690L..11B', '10.1051/0004-6361/202451599', '10.48550/arXiv.2409.13364', '2024arXiv240913364B']

['planets and satellites: atmospheres', 'planets and satellites: composition', 'planets and satellites: individual: Ganymede', 'Astrophysics - Earth and Planetary Astrophysics']

A patchy COSUB2SUB exosphere on Ganymede revealed by the James Webb Space Telescope

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https://arxiv.org/pdf/2409.13364.pdf

{'No Header': 'L etter to the E ditor', 'A patchy CO 2 exosphere on Ganymede revealed by the James Webb Space Telescope': 'Dominique Bockelée-Morvan 1 , Olivier Poch 2 , François Leblanc 3 , Vladimir Zakharov 1 , Emmanuel Lellouch 1 , Eric Quirico 2 , Imke de Pater 4 , 5 , Thierry Fouchet 1 , Pablo Rodriguez-Ovalle 1 , Lorenz Roth 6 , Frédéric Merlin 1 , Stefan Duling 7 , Joachim Saur 7 , Adrien Masson 1 , Patrick Fry 8 , Samantha Trumbo 9 , Michael Brown 10 , Richard Cartwright 11 , Stéphanie Cazaux 12 , Katherine de Kleer 10 , Leigh N. Fletcher 13 , Zachariah Milby 10 , Audrey Moingeon 2 , Alessandro Mura 14 , Glenn S. Orton 15 , Bernard Schmitt 2 , Federico Tosi 14 , and Michael H. Wong 4 \n- 1 LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 92195, Meudon, France, e-mail: Dominique.Bockelee@obspm.fr\n- 2 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France\n- 3 LATMOS / CNRS, Sorbonne Université, UVSQ, Paris, France\n- 4 Department of Astronomy, University of California, 22 Berkeley, CA 94720, USA\n- 5 Department of Earth and Planetary Science, University of California, 22 Berkeley, CA 94720, USA\n- 6 Space and Plasma Physics, KTH Royal Institute of Technology, Stockholm, Sweden\n- 7 Institute of Geophysics and Meteorology, University of Cologne, Albertus Magnus Platz, 50923 Cologne, Germany\n- 8 University of Wisconsin, Madison, WI, 53706, USA\n- 9 Department of Astronomy & Astrophysics, University of California, San Diego, La Jolla, CA 92093, USA\n- 10 Division of Geological and Planetary Sciences, Caltech, Pasadena, CA 91125, USA\n- 11 Johns Hopkins University Applied Physics Laboratory, 11001 Johns Hopkins Rd. Laurel, MD 20723, USA\n- 12 Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands \n13 \nSchool of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK \n- 14 Istituto Nazionale di AstroFisica - Istituto di Astrofisica e Planetologia Spaziali (INAF-IAPS), 00133 Rome, Italy\n- 15 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA \nReceived / Accepted', 'ABSTRACT': "Jupiter's icy moon Ganymede has a tenuous exosphere produced by sputtering and possibly sublimation of water ice. To date, only atomic hydrogen and oxygen have been directly detected in this exosphere. Here, we present observations of Ganymede's CO2 exosphere obtained with the James Webb Space Telescope. CO2 gas is observed over di ff erent terrain types, mainly over those exposed to intense Jovian plasma irradiation, as well as over some bright or dark terrains. Despite warm surface temperatures, the CO2 abundance over equatorial subsolar regions is low. CO2 vapor has the highest abundance over the north polar cap of the leading hemisphere, reaching a surface pressure of 1 pbar. From modeling we show that the local enhancement observed near 12 h local time in this region can be explained by the presence of cold traps enabling CO2 adsorption. However, whether the release mechanism in this high-latitude region is sputtering or sublimation remains unclear. The north polar cap of the leading hemisphere also has unique surface-ice properties, probably linked to the presence of the large atmospheric CO2 excess over this region. These CO2 molecules might have been initially released in the atmosphere after the radiolysis of CO2 precursors, or from the sputtering of CO2 embedded in the H2O ice bedrock. Dark terrains (regiones), more widespread on the north versus south polar regions, possibly harbor CO2 precursors. CO2 molecules would then be redistributed via cold trapping on ice-rich terrains of the polar cap and be diurnally released and redeposited on these terrains. Ganymede's CO2 exosphere highlights the complexity of surface-atmosphere interactions on Jupiter's icy Galilean moons. \nKey words. Planets and satellites: individual: Ganymede, Planets and satellites: atmospheres, Planets and satellites: composition, Infrared: planetary systems", '1. Introduction': "Jupiter's icy satellites, Europa, Ganymede and Callisto, are known to have rarefied atmospheres. The surface composition of these moons is dominated by H2O ice and non-ice components, possibly salts, hydrated minerals and organics, hosting volatiles such as CO2 (Carlson et al. 1996; McCord et al. 1998; Tosi et al. 2024). Sublimation and weathering processes, such as sputtering by charged particles from Jupiter's magnetosphere and micrometeoroids bombardment, lead to the formation of weakly bound \natmospheres composed primarily of H2O, O2, OH, H, O, and CO2 species. Because of strong telluric absorption by Earth's atmosphere, detection of atomic and molecular emissions from icy moon exospheres is di ffi cult from ground-based facilities. Most of our knowledge comes from the detection of auroral O and H emission lines in the atmospheres of the three icy moons (Hall et al. 1998; Cunningham et al. 2015; Barth et al. 1997; Roth et al. 2017a,b; de Kleer et al. 2023), with some constraints obtained on H2O vapor content for Ganymede (Roth et al. 2021). So far, atmospheric CO2 was only detected in the atmosphere of Callisto \n(Carlson 1999; Cartwright et al. 2024). The maximum CO2 column densities do not coincide with the subsolar region, nor the regions with the greatest solid-state CO2 abundance on Callisto's surface, suggesting that CO2 gas may be partly sourced by outgassing from its crust (Cartwright et al. 2024). Characterizing how icy moon exospheres are formed and sustained is pivotal for understanding surface-atmosphere interactions, geomorphological and chemical changes driven by erosion. \nGanymede is the only known moon with an intrinsic magnetic field, resulting in a complex space plasma environment which has been explored by the in-situ flybys of Galileo and Juno spacecraft (e.g., Kivelson et al. 1996; Allegrini et al. 2022; Ebert et al. 2022; Clark et al. 2022). The intrinsic magnetic field directs most of the external Jovian magnetospheric plasma in a way that it primarily interacts with the moon's surface where Ganymede's mini-magnetosphere has open field lines, i.e., around the polar regions (Poppe et al. 2018; Liuzzo et al. 2020; Greathouse et al. 2022). This results in specific surface properties with respect to shielded equatorial latitudes, such as the formation of H2O ice-rich patches at the polar caps (Khurana et al. 2007; Ligier et al. 2019; Stephan et al. 2020; King & Fletcher 2022) with higher amounts of amorphous H2O ice (Ligier et al. 2019; Bockelée-Morvan et al. 2024), radiolytically produced H2O2 (Trumbo et al. 2023), and CO2 possibly trapped in amorphous H2O ice (Bockelée-Morvan et al. 2024). In addition, asymmetries between the north / south polar caps, and leading / trailing hemispheres are observed (Ligier et al. 2019; de Kleer et al. 2021; Trumbo et al. 2023; Bockelée-Morvan et al. 2024). Here we present the first detection of CO2 in the exosphere of Ganymede, achieved using the James Webb Space Telescope (JWST), and we link the observed highly hetereogeneous CO2 exosphere to surface properties and processes. This paper follows the investigation of Ganymede's surface properties from the same JWST data set (Bockelée-Morvan et al. 2024, hereafter Paper I).", "2. JWST observations of Ganymede's exosphere": "Observations undertaken with the NIRSpec / IFU instrument provided spatially resolved (0.1' pixel size, with ∼ 190 pixels across Ganymede's disk) spectra of the leading and trailing sides of Ganymede in the 2.9-5.2 µ m range at high spectral resolution ( R ∼ 3000) (Paper I and Appendix A). Ro-vibrational emission lines of the CO2 ν 3 band at 4.26 µ m were detected within the broad solid-state CO2 absorption band (Figs 1, B.1). We used several data processing techniques to extract the CO2 gas signal and best evaluate the confidence level of the detection for weak signals (Appendices B, C). CO2 column densities were inferred using a non-LTE excitation model (Appendices D, E). The distributions of column densities for the two hemispheres are shown in Fig. 1. Figure 2 presents the dependence on latitude, from the analysis of spectra after averaging pixels over ranges of latitude. \nThe distribution of CO2 gas shows strong regional variations (Fig. 1) and is at odds with expectations that the peak surface location of the exosphere would be at the dawn terminator due to condensation on the surface at night and early morning reevaporation (Stecklo ff et al. 2022). The CO2 exosphere is most prominent over the north polar regions of the leading hemisphere, peaking at 81 · W, 51 · N (12 h local time), with a column density along the line of sight of (1.5 ± 0.11) × 10 18 m -2 corresponding to a pressure at the surface of 1 pbar. The rotational temperature of CO2 measured in this region (107 ± 5 K, Fig. H.1) constrains the gas kinetic temperature in Ganymede's exosphere (Appendix H). A point-spread function (PSF) correc- \nthe CO2 column density map suggests that the decrease poleward of 50 · N is real (Appendix M). At southern latitudes of the leading hemisphere, and on the trailing hemisphere, the CO2 exosphere is on average at least five-times less dense (Fig. 2). Low column densities are measured at or near equatorial latitudes for both hemispheres. The trailing hemisphere displays a north / south asymmetry, with the exosphere extending over a broader range of latitudes in the southern hemisphere. Noticeable in Fig. 1 (see also Fig. O.1) is a CO2 gas enhancement in a large region around (30 · W, 25 · N), encompassing the Tros crater (27 · W, 11 · N). CO2 excess is also present at around 30 · S on the leading side, which corresponds to the position of the expected southern open-closed-field-line boundary (OCFBs, Appendix J). \nExospheric H2O was unsuccessfully searched for in 5.5-7.1 µ m spectro-imaging data acquired with the JWST MIRI / MRS instrument (see Appendix I). Our 3σ upper limit on the H2O column density for the subsolar region of the leading side (Table 1) is about an order of magnitude higher than the minimum of 6 × 10 18 H2O / m 2 required to explain UV HST data of atomic oxygen emission lines (Roth et al. 2021). On the other hand, for the trailing side, our derived upper limit for a 105 K atmosphere is slightly below the lower limit from HST (3.6 × 10 19 H2O / m 2 , Roth et al. 2021). Since HST constrains the H2O / O2 ratio and not directly the H2O abundance, this could imply that the atmosphere is overall more dilute and that both H2O and O2 densities are lower than assumed in Roth et al. (2021). This would contradict recent results that suggested a denser global atmosphere based on plasma measurements (Carnielli et al. 2020a; Waite et al. 2024). Alternatively, a higher atmospheric temperature (e.g., 130 K, Table 1), as might be expected above subsolar regions, increases the JWST upper limit to values consistent with the HST lower limits.", "3. Processes releasing CO 2 in Ganymede's exosphere": "Possible processes releasing CO2 into Ganymede's exosphere include surface ice sublimation and sputtering by energetic particles. We investigated whether these mechanisms, acting either on H2O ice containing CO2 molecules or on pure CO2 ice, could be distinguished from the observed properties of Ganymede's exosphere. For this purpose, we used the Exospheric Global Model Leblanc et al. (EGM, 2017), a multi-species 3D Monte Carlo model that considers sources and sinks (photodestruction, surface sticking, gravitational escape) of such exospheres (Appendix K). The simulations (Appendix L) were designed to explain to first order the CO2 column density peak observed in the northern latitudes of the leading hemisphere, and the dichotomy between the trailing and leading hemispheres. \nA key question to address is the localized character of the atmosphere. Mean surface temperatures, even in the polar regions (100-110 K, Fig. K.1), are much warmer than the expected condensation temperature of pure CO2 (73 K at 1 pbar pressure). Hence, the CO2 atmosphere might have been expected to be more global, as shown by EGM calculations considering mean surface temperatures (Appendix L.1). This indicates that CO2 interacts with the surface material much more strongly than expected in such a simplistic view. Similar conclusions were reached for O2 gas at Ganymede (Waite et al. 2024), but also at Dione and Rhea (Teolis & Waite 2016), based on inconsistencies on O2 column densities between exospheric models and measurements. In those Saturn's moons, the O2 source rates implied by the observations are 50 (Dione) - 300 (Rhea) times less than expected from the known O2 radiolysis yields from \nFig. 1. CO2 in Ganymede's exosphere. Top and bottom rows are for the leading and trailing sides, respectively. The first, second and third columns show: 1) Bond albedo maps derived by de Kleer et al. (2021) from Voyager-Galileo mosaic; 2) Line-of-sight CO2 column density maps inferred from spectral modeling (Appendices B-D); trailing data were smoothed using a 3 × 3 boxcar filter; color scales for the leading and trailing sides di ff er, and indicated above the plots; pixel sizes are 0.1 × 0.1' and the PSF is ∼ 0.19' (FWHM); CO2 maximal emission in the leading hemisphere (based on central contour) is at 81 · W, 51 · N ( ∼ 12h local time); correcting for the line of sight, the maximum vertical column density is at 72 · W, 45 · N (12.6 h local time), Figs. 4A, O.2; 3) CO2 gaseous emission spectra obtained after removing the continuum emission from Ganymede's surface, averaged over latitudes 45-90 · N for leading (top), and 30-60 · S for trailing (bottom); best fit synthetic spectra are shown in cyan, with a fitted rotational temperature of 108 ± 8 K for the leading side, and a fixed rotational temperature of 105 K for the trailing side; the y scale is µ Jy per pixel. The green lines in the maps show the open-closed-field line boundary at the time of the JWST observations (Appendix J, Duling et al. 2022). Ganymede was north of and within the plasma sheet at the time of the leading and trailing sides observations, respectively. The subsolar point at the time of the JWST observations is shown by a red star in the Bond albedo maps. \n<!-- image --> \nTable 1. H2O and CO2 line-of-sight column densities in selected Ganymede's areas. \na Using extracted spectra from either the subsolar region (Solar Zenith Angle SZA < 15 · , average of 8 pixels for MIRI, 27 pixels for NIRSpec) or from the region with large CO2 gas emission (leading, 40-65 · N, 46-100 · W, 7 pixels for MIRI, 15 pixels for NIRSpec). b Using the most intense 10-15 ro-vibrational lines expected in absorption in the 5.7-6.2 µ m spectral range. c Using the most intense 10-15 ro-vibrational lines expected in emission in the 6.2-7.1 µ mspectral range. d 3σ upper limits on H2O line-of-sight column density combining upper limits obtained for the two spectral ranges ( < 6.2 µ mand > 6.2 µ m) (Appendix I). f Upper limits are 3σ . \nion-irradiated pure water ice measured in the laboratory, and surface interactions (adsorption / di ff usion) appear to control the exospheric structure, density, and seasonal variability. We note that for CO2 at Ganymede, the prime evidence for strong surface / atmosphere interactions comes from the non-global character of the atmosphere, rather than the absolute CO2 column densities (which remain di ffi cult to explain, see below). \nTo explain the atmospheric patchiness, Ganymede's surface may have properties increasing the e ff ective binding and desorption energies of O2 and CO2. In addition to surface roughness producing cold traps, surface irradiation (creating defects) and microstructure (enabling di ff usion, and re-adsorption on ad- \njacent grains) could increase e ff ective binding and desorption energies of adsorbates (Yakshinskiy & Madey 2000; Cassidy et al. 2015; Sarantos & Tsavachidis 2020). Such hypotheses were drawn to explain the distribution of alkali gases surrounding Mercury and the Moon. In the EGM simulations, the surface temperature model considers surface roughness, as constrained from JWST / MIRI brightness temperature maps (Paper I), and simulates the presence of local cold spots through a temperature distribution (Appendix K). As shown in Appendix L.1, CO2 diffusion is, to a large extent, controlled by the ability of molecules to condense on cold traps, thereby explaining localized enhancements of the CO2 exosphere at high latitudes. \nFig. 2. Variation of CO2 gas line-of-sight column density with latitude. Blue and red symbols refer to the leading and trailing sides, respectively. Column densities were derived from spectra that have been averaged in latitude bins of 7.5 · (leading, Fig. B.2) and 15 · (trailing). The blue (resp. pink) vertical domains show the latitude range of the open-closed-fieldline boundaries (OCFB) for the leading and trailing sides, respectively, restricted to longitudes of 10-130 · W(leading) and 210-330 · W(trailing). \n<!-- image --> \nFig. 3. Calculated line-of-sight column-density maps of the CO2 exosphere of Ganymede above the leading side from the EGM model (in unit of 10 18 m -2 ). Left: CO2 release associated with H2O sublimation with a CO2 / H2O relative abundance of 5 for an H2O areal ice fraction of 50% at latitudes > 50 · N. Right: sputtering of H2O ice with CO2 / H2O = 0.01 at latitudes > 40 · N; the result was multiplied by 382 to match the observations (Appendices K, L). The green lines display the boundary between open and close field lines (OCFB). The subsolar point is at 2.6 · N, 82 · W. \n<!-- image --> \nFigure 3 shows simulations of Ganymede's CO2 exosphere above the leading hemisphere, assuming that the release of CO2 is induced by the sublimation (left panel), or the sputtering (right panel) of H2O ice containing CO2 molecules. Sublimation of CO2 ice was also investigated (Fig. L.2c, d). In all three cases, the CO2 column density peaks at the right latitude, as long as the source region covers the north polar cap (latitude > 40-50 · N, longitude range 0-180 · ), and follows a diurnal / longitudinal trend with a maximum at ∼ 13.1 to 13.4 h, slightly shifted from the maximum surface temperature (12.5 h, Fig. K.1) and observed CO2 peak (12 h). Sputtering explains the smooth diurnal variation of the CO2 column density better than sublimation (Fig. L.2f). \nIn our models in which CO2 is released through sputtering of H2O ice with 1% CO2 molecules, we had to multiply the sputtered flux from Leblanc et al. (2017) by a factor ∼ 380 to match the observed peak column density. The need to increase the sputtered flux significantly might be related to the approach used to calculate this flux (which follows Leblanc et al. (2017), see Appendix K), which consisted in using the yield definition of Cassidy et al. (2013) and a precipitating Jovian ion flux of 10 6 particles / cm 2 / s , ignoring any sputtered component from electron impact. In fact, Carnielli et al. (2020a) modeled the ion \npopulation in the ionosphere and concluded, based on electron measurements from the Galileo spacecraft, that ionospheric ions could be a significant source of ion precipitation, especially on the leading hemisphere (Carnielli et al. 2020b). Using measurements from the Juno spacecraft, Waite et al. (2024); Vorburger et al. (2024) concluded that low-energy electrons are an important sputtering agent. Another source of uncertainty is sputtering yields for production of CO2 by ion and electron impacts, which are unconstrained because the relevant experiments are sparse. Simulations investigating sputtering on the entire Ganymede's surface (Appendix L.4) show that it might be possible to explain the overall distribution of CO2 exosphere by considering strong regional variations of surface properties. \nAconsequence of the factor of 380 enhancement of the sputtered flux from Leblanc et al. (2017) is that to first order the O2 column density in our model is multiplied by the same factor, bringing it to values ∼ 1.5 × 10 17 O2 / cm 2 , at odds with results from Leblanc et al. (2023) and Roth et al. (2021), based on the atomic O line intensities in the UV. This result goes in the same direction as the ionospheric calculations of Carnielli et al. (2020b) and the post-Juno analyses of Vorburger et al. (2024) and Waite et al. (2024), who advocated for O2 columns ∼ 20 times enhanced compared to previous estimates, but the discrepancy is here much larger, which at face value could be taken as an argument against sputtering being at the origin of the CO2 atmosphere. \nIn our sublimation models in which CO2 gas is released in proportion with the H2O sublimation flux, to reproduce the peak column density requires an unrealistic CO2 abundance relative to water, three orders of magnitude larger than estimated for the surface ( ∼ 1% in mass, Paper I). Hence, this scenario cannot explain the CO2 exospheric excess on the northern polar cap of the leading side. On the other hand, direct sublimation of CO2 ice is a possible mechanism as only a very small amount of surface coverage (3 × 10 -14 , Table L.1) is required to explain the peak column density, albeit with an expected diurnal variation more extreme than observed (Fig. L.2d). Regarding sub-solar regions and considering H2O ice sublimation with an areal H2O abundance of 20% appropriate for the leading side (Ligier et al. 2019), our model predicts a H2O column density of 4.1 × 10 19 m -2 , consistent with the JWST upper limit for Ganymede's leading hemisphere (Table 1), but a factor of 7 above the minimum value derived from HST data for this hemisphere (6 × 10 18 m -2 , Roth et al. 2021). \nIn summary, processes releasing CO2 in Ganymede's exosphere are not well understood. The smooth diurnal variation of the CO2 column density favors sputtering, but explaining the measured column densities with this process requires further model developments taking advantage of the most recent magnetospheric data acquired by the Juno mission. We can anticipate that the interpretation of Callisto's CO2 exosphere (Cartwright et al. 2024), which is one order of magnitude denser than Ganymede's CO2 exosphere , will be similarly challenging.", "4. Linking Ganymede's CO 2 exosphere to surface properties": "The 4.26µ m absorption band of surface CO2 is ubiquitous on Ganymede, and is caused by CO2 under di ff erent physical states. However, the CO2 gas column density does not correlate with the CO2 surface distribution globally (Fig. 4A, B ; see Appendix N and Figs N.1A, C, D, F). Rather, the prominence of the CO2 exosphere on the northern polar cap is associated with other surface properties. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 4. Comparing CO2 exosphere to surface properties on Ganymede's leading hemisphere. A) Vertical CO2 gas column density (unit of 10 18 m -2 , this work, see also Fig. O.2); B) Depth of CO2-solid absorption band (Paper I); C) Central wavelength of CO2-solid absorption band (Paper I); D) Reflectance at 3.65 µ m (Paper I); E) Central wavelength of H2O Fresnel peak (Paper I); F) Relative amplitude of the maximum reflectance between 3.5 and 4 µ m (H2O interband amplitude, Paper I). The north pole of the leading hemisphere possesses the most red-shifted absorption band center of solid CO2, consistent with CO2 trapped in amorphous H2O ice (Paper I). It also has the higher reflectance at 3.65 µ m and H2O interband amplitude, indicative of a higher density of facets in H2O ice for the photons (i.e., smaller grains and / or more internal defects and / or higher micro-roughness / porosity), and the most blue-shifted central wavelength of the H2O Fresnel peak due to a higher proportion of amorphous water ice (Mastrapa et al. 2009). \n<!-- image --> \nAccording to Galileo high-resolution images, Ganymede's polar 'caps' are actually made of discrete patches of optically thick ice, preferentially located on pole-facing slopes (Khurana et al. 2007), likely formed by H2O ice sputtering and subsequent re-deposition on these coldest locations (Khurana et al. 2007). On both hemispheres, Ganymede's north polar regions show spectral properties indicative of H2O ice particles having a higher density of facets for the photons (i.e., smaller grains and / or more internal defects and / or higher microroughness / porosity) causing multiple scattering and a higher proportion of amorphous ice than the south polar regions (Denk et al. 2009; Ligier et al. 2019, ; Paper I) (Fig. 4D,E,F). \nRemarkably, these north / south polar asymmetries in spectral properties are most pronounced on the leading hemisphere. As shown in Fig. 4, the fact that the peak column density of CO2 gas is found over regions where water ice has the highest density of facets, the largest amorphous fraction, and the most redshifted absorption band center of solid CO2, indicative of CO2 trapped in amorphous H2O ice, suggests that all these properties are probably linked. They are co-located poleward of 40 · N, so they are probably specific to the ice-rich patches constituting the so-called 'polar cap'. The CO2 exosphere is maximum over the polar cap, but it extends over all of the northern open-field-lines area. \nDuring the JWST observation of the leading side, the southern hemisphere of Ganymede was facing towards the center of the plasma sheet, where the column density of plasma along Jupiter's magnetospheric field lines is larger than on the northern side of the plasma sheet. On the plasma sheet facing hemisphere, the auroral band of Ganymede is brighter than on the other hemisphere (Saur et al. 2022; Greathouse et al. 2022; Milby et al. 2024). The reason for the auroral asymmetry is not fully understood. It could be due to the larger plasma momentum and resultant larger magnetic stresses on hemispheres facing the \nplasma sheet center and / or asymmetric reconnection processes (e.g., Saur et al. 2022; Milby et al. 2024). The larger auroral brightness requires larger auroral electron fluxes of which the largest fraction will collide with the surface. Additionally, the hemisphere facing the center of the plasma sheet is facing larger fluxes of energetic ions and electrons. Integrating these electron and ion fluxes over a full Jovian synodic rotation period should however lead to similar fluxes on the northern and southern polar regions (Poppe et al. 2018; Liuzzo et al. 2020). The observations of higher density of CO2 gas and of enhanced / specific surface properties on the northern hemisphere of the leading side are thus not consistent with what would be expected from either instantaneous or time-averaged plasma e ff ects. Therefore, the specifics of the north polar regions of the leading hemisphere are likely an inherent property of Ganymede's surface. \nThe north and south polar caps mainly di ff er in the nature of their underlying terrains. Galileo Regio, the largest patch of the darker and more cratered terrains on Ganymede, encompasses much of the leading north polar latitudes, while the leading south polar latitudes have fewer of such dark cratered terrains (Fig. O.3 from Patterson et al. (2010)). The low-albedo material, concentrated in topographic lows by sublimation and mass wasting (Prockter et al. 1998), may be a remnant of Ganymede's formation building blocks and / or may have been deposited by cometlike bodies (Zahnle et al. 1998; Bottke et al. 2013), so it could contain CO2 precursors (organic and inorganic carbon-bearing components), whose radiolysis and / or disaggregation by energetic particles may produce and / or release CO2. As shown by laboratory experiments, the radiolysis of complex organic matter (Gomis & Strazzulla 2005; Raut et al. 2012) or carbonates (Costagliola et al. 2017) in the presence of H2O forms CO2. In addition, the disaggregation of carbonaceous chondrite-like material (Yuen et al. 1984) or the radiolysis of some of their inorganic carbon-bearing components (carbonates and other miner- \nals, Nakamura et al. 2023) could also release or produce CO2. This CO2 production may be specifically enhanced in the northern open-field-lines area of the leading hemisphere because they host the largest extent of dark cratered terrains than the southern ones (Fig. O.3). \nHowever, the peak in CO2 column density is not only over the fraction of Galileo Regio poleward of the OCFB, but over the water ice polar cap (Fig. 4A, D, F ; Fig. N.1B), covering diverse terrain types (Fig. 1). Moreover, the peak of the CO2 vertical column density is at 72 · W, 45 · N (12.6 h local time), on the boundary between Galileo Regio and the bright terrain Xibalba Sulcus (Fig. 1, Fig. O.3). Therefore, if the CO2 is initially produced on the dark terrains, it should migrate and accumulate over the polar cap on the long term, before being diurnally released and redeposited over the polar cap as is possibly observed. This redistribution might also occur if the CO2 is initially produced from other sources, for example by a relatively recent resurfacing event (impact, mass movement) that would have exposed to the surface CO2 or CO2 precursors from the subsurface and / or from the impactor. Notably, several impact craters with bright ejecta are present over the part of Xibalba Sulcus showing maximum CO2 column density (Collins et al. 2014), and the ice bedrock of this relatively recent region has been suspected of containing significant CO2 based on its geomorphology (Moore et al. 1999). If this CO2 exosphere is permanent, geological mass-wasting events (Moore et al. 1999; Pappalardo et al. 2004) and possibly micro-meteoritic gardening, may regularly expose new CO2 or CO2 precursors to the surface, maintaining the CO2 exosphere over the long-term. The release of CO2 gas from on-going volcanic activity seems unlikely given the surface age (0.5-1 Ga with large uncertainties, Zahnle et al. 1998; Showman et al. 2004), but gravity anomalies were identified around this region (Gomez Casajus et al. 2022). \nProduced CO2 may then preferentially co-deposit with H2O and accumulate on high-latitude cold traps, potentially explaining the red-shift of the CO2 absorption band with latitude (Fig. 4C). The maximum CO2 column densities and the H2O ice having the highest density of facets both peak around a longitude at the maximum solar illumination and maximum surface temperature at this latitude (Paper I), suggesting a diurnal process releasing CO2 from the ice in the atmosphere (Fig. 4A, D, F). According to our analyses, sputtering appears to have a temperature dependence that is the most consistent with the observation (Fig. L.2). At mid-day, the maximum temperature of the ice enhances sputtering and thermal stress that may generate more ice facets, resulting in surface micro-roughness or internal cracks, which could further enhance CO2 release (Baragiola 2003, and references therein). Later in the day and night, re-deposition and / or molecular movements induced by energetic ions might fill in these pores or cracks, decreasing the density of facets and trapping the CO2 again.", '5. Summary': "In summary, the north / south polar asymmetry in the distribution of CO2 gas of the leading hemisphere could be explained by the larger extent of dark terrains over the northern polar region, providing a larger initial source of CO2 produced by radiolysis of organic or inorganic precursors. The existence of other initial sources specific to this region (impact, mass movement, cryovolcanism) cannot be excluded, but lack compelling evidence. After its initial production, CO2 may migrate and accumulate on cold traps of the polar cap and be diurnally released and redeposited, explaining the co-location of the northern polar atmo- \nsphere with the H2O and CO2 surface properties. Whether the release mechanism in this high-latitude region is sputtering or sublimation remains unclear. Outside of the open-field-line areas, CO2 gas is located above various terrain types, including the dark terrain Melotte and some other terrains having more (or smaller) grains of H2O ice or H2O-bearing minerals / salts (Fig. N.1B, E). This spatial distribution suggests the existence of several mechanisms producing and releasing CO2. Future investigations of Ganymede from JWST and space missions, together with further models and experiments dedicated to sputtering processes, are needed to unravel the origin of Ganymede's patchy CO2 exosphere. \nAcknowledgements. This work is based on observations made with the NASA / ESA / CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program 1373, which is led by co-PIs Imke de Pater and Thierry Fouchet and has a zero-exclusive-access period. D.B.-M, E.Q., E.L., T.F., and O.P. acknowledge support from the French Agence Nationale de la Recherche (program PRESSE, ANR-21-CE49-0020-01). I.dP and M.H.W. were in part supported by the Space Telescope Science Institute grant nr. JWST-ERS-01373. L.F. was supported by STFC Consolidated Grant reference ST / W00089X / 1; for the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to the Author Accepted Manuscript version arising from this submission. Some of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).", 'References': 'Allegrini, F., Bagenal, F., Ebert, R. W., et al. 2022, Geophys. Res. Lett., 49, e2022GL098682 \n- Baragiola, R. A. 2003, Planetary and Space Science, 51, 953\n- Barth, C. A., Hord, C. W., Stewart, A. I. F., et al. 1997, GRL, 24, 2147\n- Blauer, J. & Nickerson, G. 1973, GA survey of vibrational relaxation rate data for processes important to CO2-N2-H2O infrared plume radiation, Tech. rep., U.S. Geological Survey, technical Report AFRPL-TR-73- 57 Ultrasystems, Inc.\n- Bockelée-Morvan, D., Lellouch, E., Poch, O., et al. 2024, A&A, 681, A27 (Paper I)\n- Bottke, W. F., Vokrouhlický, D., Nesvorný, D., & Moore, J. M. 2013, Icarus, 223, 775\n- Carlson, R., Smythe, W., Baines, K., et al. 1996, Science, 274, 385 \nCarlson, R. W. 1999, Science, 283, 820 \n- Carnielli, G., Galand, M., Leblanc, F., et al. 2020a, Icarus, 343, 113691\n- Carnielli, G., Galand, M., Leblanc, F., et al. 2020b, Icarus, 351, 113918\n- Cartwright, R. J., Villanueva, G. L., Holler, B. J., et al. 2024, Planetary Science Journal, 5, 60\n- Cassidy, T. A., Merkel, A. W., Burger, M. H., et al. 2015, Icarus, 248, 547\n- Cassidy, T. A., Paranicas, C. P., Shirley, J. H., et al. 2013, Planetary and Space Science, 77, 64\n- Clark, G., Kollmann, P., Mauk, B. H., et al. 2022, Geophys. Res. Lett., 49, e2022GL098572\n- Collins, G. C., Patterson, G. W., Head, J. W., et al. 2014, Global geologic map of Ganymede, Tech. Rep. 3237, U.S. Geological Survey, iSSN: 2329-132X Publication Title: Scientific Investigations Map\n- Costagliola, A., Vandenborre, J., Blain, G., et al. 2017, The Journal of Physical Chemistry C, 121, 24548, publisher: American Chemical Society \nCrovisier, J. 1987, A&A Suppl., 68, 223 \n- Cunningham, N. J., Spencer, J. R., Feldman, P. D., et al. 2015, Icarus, 254, 178 Davidsson, B. J. R. & Hosseini, S. 2021, MNRAS, 506, 3421\n- de Kleer, K., Butler, B., de Pater, I., et al. 2021, PSJ, 2, 5\n- de Kleer, K., Milby, Z., Schmidt, C., Camarca, M., & Brown, M. E. 2023, Planetary Science Journal, 4, 37\n- Denk, T., Neukum, G., Khurana, K. K., & Pappalardo, R. T. 2009, in European Planetary Science Congress 2009, 572\n- Duling, S., Saur, J., Clark, G., et al. 2022, GRL, 49, e2022GL101688\n- Ebert, R. W., Fuselier, S. A., Allegrini, F., et al. 2022, Geophys. Res. Lett., 49, e2022GL099775\n- Famá, M., Shi, J., & Baragiola, R. A. 2008, Surface Science, 602, 156\n- Fray, N. & Schmitt, B. 2009, Planetary and Space Science, 57, 2053', 'Appendix A: JWST observations and data reduction': "NIRSpec / IFU observations of the leading and trailing sides of Ganymede were obtained as part of the ERS programme #1373 (PIs I. de Pater, T. Fouchet). These observations, acquired with the G395H / F290LP grating / filter pair, provided spatially resolved imaging spectroscopy in the range 2.86-5.28 µ m over a 3' × 3' field of view with 0.1' × 0.1' spatial elements (310 × 310 kmat Ganymede), and a nominal spectral resolution of R ∼ 2700. The estimated full width at half maximum of the point spread function (PSF) is ∼ 0.19' (Appendix M). Detailed information on these observations is provided in Trumbo et al. (2023) and Paper I, focused on the analysis of solid state spectral features from CO2, H2O, and H2O2. For the data reduction we followed the procedure adopted in Bockelée-Morvan et al. (2024) (Paper I). The updated JWST pipeline version 1.12.5 and context file version jwst \\_1148 . pmap were used. Correction for the 1 / f noise was done as explained in Trumbo et al. (2023) and Paper I. \nERS #1373 comprised also observations of the leading and trailing sides of Ganymede using the Mid-Infrared Instrument / medium resolution spectroscopy (MIRI / MRS), which are described in Paper I. These observations, made with the four IFU channels, provided spatially-resolved unsaturated spectra in the 4.9-11.7 µ m range. Channel 1 (4.9-7.65 µ m) covers the ν 2 vibrational band (and weaker ν 2 + ν 3ν 3 and ν 2 + ν 1ν 1 hot-bands) of H2O in vapor phase from which the H2O content in Ganymede's exosphere can be studied. For Channel 1, the spaxel (aka pixel in main text) size is 0.13' and the spectral resolution is ∼ 3700. The data were re-reduced using most recent JWST pipeline version 1.11.3, and context file jwst \\_1119 . pmap , and processed as in Paper I. \nGanymede spectra are crowded with solar lines. For the study of solid-state features in NIRSpec spectra, the output of the JWST pipeline, calibrated in radiance units (MJy / sr), were divided by the solar spectrum (Hase et al. 2010) at the spectral resolution of NIRSpec, giving data in units of radiance factor I / F (Paper I). To obtain spectra in radiance units and corrected from solar lines, the data in I / F units were multiplied by the solar continuum. In the spectral 4.2-4.3 µ m region where strong ro-vibrational lines of the CO2 ν 3 band are present, solar lines are not numerous and much fainter than in nearby spectral regions. Nevertheless, we payed special attention to solar-line removal as gaseous emission lines from Ganymede are faint. We determined that solar lines present in the 4.4-4.6 µ m range are best removed when applying a correction factor of ∼ 0.87 to the nominal spectral dispersion provided by JWST documentation (i.e., increasing the wavelength-dependent spectral resolution by 1.15). We used this factor (giving R = 3365 at 4.2-4.3 µ m) in subsequent analyses, including for producing synthetic line profiles. \nIn MIRI spectra, the most intense ro-vibrational lines from the H2O ν 2 band are expected between 5.6 and 7.4 µ m(Fig. I.1). In this spectral region, both reflected light and thermal emission from Ganymede's surface contribute to the continuum, especially at the lowest wavelengths where the two components have similar intensities (Paper I). Therefore, spectra were corrected from solar lines by isolating the reflected-light component, and applying the method used for NIRSpec data.", 'Appendix B: Extraction of CO 2 gaseous lines': "The ν 3 bands of CO2 in gaseous form and in solid state lie at the same wavelengths. The solid-state absorption band of CO2 shows strong variations in shape and intensity on the surface of \nFig. B.1. CO2 ν 3 band gas spectra from Ganymede's exosphere. Left (A-C): North polar cap of the leading side of Ganymede (averaged spectra for latitudes > 45 · N); Right (D-F): Southern hemisphere of the trailing side (latitudes 30-60 · S). A, D) Observed spectra showing both the CO2 ν 3 absorption band from CO2 in solid state, and ro-vibrational emission lines of gaseous CO2; B, E) CO2 gaseous emission spectra obtained after removing the continuum emission shown in red in panels A and D (Appendix B). C, F) Residual CO2-gas DIFF spectra obtained by removing the continuum obtained from low-pass filtering (Appendix B). Best fit synthetic spectra are shown in blue, with a fitted rotational temperature of 108 ± 8 K for the leading side, and a fixed rotational temperature of 105 K for the trailing side. In all plots, vertical-axis unit is µ Jy per pixel (1 pixel = 0.1' × 0.1'). \n<!-- image --> \nGanymede (Paper I). Hence, to isolate the weak ro-vibrational emission lines from CO2 gas from the broad absorption band, we developed specific tools, which were tested on synthetic spectra. We restricted the analysis to the 4.220-4.295 µ m range where the strongest CO2 gaseous lines and only weak solar lines are present. \nIn a first step, the solid-state contribution was estimated by applying low-pass filtering with a Butterworth filter. The optimum cuto ff frequency that preserves best the gaseous signatures was determined by applying the method to synthetic spectra combining the Ganymede CO2-solid absorption band and CO2gas fluorescence emission. In a second step, the residual CO2 gas signature (called DIFF) was obtained by subtracting this estimated solid-state signal from the observed spectra. Two examples of residual DIFF spectra are shown in Figs B.1C, F. This method does not allow retrieval of the correct shape of the CO2 gaseous band. We show in Figs B.1B, E (and Fig. 1) CO2 gas spectra from Ganymede displaying the expected ro-vibrational structure of the CO2 ν 3 band for fluorescence emission. They were obtained through several iterations, by computing the envelope of the residual DIFF signal and adding the bottom part of the envelope to the solid-state signal extracted from low-pass filtering. From synthetic spectra processed in the same manner, we found that this third step produced an overestimation of the strength of the CO2 gas signature, especially for faint signals at the limit of noise. Hence, analyses were made on DIFF spectra. \nIn order to evaluate the significance of detection of gas emissions, we computed the Pearson correlation coe ffi cient Cp between the DIFF spectra and a forward model. The forward model consists in a DIFF spectrum computed by applying the same treatment as for the data to a synthetic spectrum obtained by combining the Ganymede average CO2 absorption band observed on the leading side (obtained from low-pass filtering) and a CO2 fluorescence spectrum at 105 K convolved to the \ninstrumental spectral resolution R = 3365. The Pearson correlation coe ffi cient Cp was computed for each individual spaxel on the leading side. Due to the faintness of the CO2 gas emission lines on the trailing side, the trailing data were smoothed using a 3 × 3 boxcar filter. Cp ranges from -0.22 to 0.87 on the leading side, and from -0.11 to 0.50 on the trailing side (left panels of Fig. O.1). Except for the northern regions of the leading Ganymede disk, Cp values do not exceed 0.5. Hence the confidence level of the detection of the CO2 exosphere is rather low for several regions, calling for the use of other detection criteria (cross-correlation technique). \nFig. B.2. DIFF spectra on the leading hemisphere as a function of latitude. Data were averaged over latitude bins of 7.5 · and treated as explained in Appendix B. Fitted fluorescence DIFF spectra are shown in black. The Pearson correlation coe ffi cient for each spectrum is given in the legend. Vertical-axis unit is the flux density per pixel. \n<!-- image -->", 'Appendix C: Cross-correlation technique': 'We used the cross-correlation technique to obtain additional criteria for confirming weak CO2 detections. This method is widely used, e.g., to search for molecular signatures in exoplanet spectra (Snellen et al. 2010; Mâlin et al. 2023). We computed the cross-correlation function (CCF) between the Ganymede DIFF spectra and the forward model over a velocity range (-3000, 3000) km / s (in total 127 spectral resolution elements, aka spectels) using velocity steps δ v spaced by 10 km / s (0.21 spectels). Cross-correlating the forward model with itself, the maximum of the autocorrelation function is obviously at δ v = 0. Because of the periodicity in frequency of the CO2 ro-vibrational lines, which are equally spaced every 0.003 µ m( ∼ 4 spectels), the autocorrelation function and CCFs present strong secondary peaks (reaching 80 % of the maximum for the closest secondary peaks) spaced by the corresponding value in velocity units (Fig. C.1). The criterion used for confident CO2 exosphere detection is that the maximum of the CCF stands close to δ v = 0, namely is shifted by at most one spectel element. \nWe estimated the signal-to-noise ratio (S / NCCF) of the CCF at δ v = 0 to quantitatively measure the confidence level of the detections. For that purpose, we generated synthetic spectra adding random Gaussian noise to the Ganymede average CO2 absorption band observed on the leading side obtained from low-pass filtering. As there is no possibility of estimating the noise level \nFig. C.1. Cross-correlation functions (CCFs). Top: from averaged data at latitudes > 45 · N on the leading hemisphere. Bottom: from averaged data at latitudes 30-60 · S on the trailing hemisphere. The forward model for computing the cross-correlation is a fluorescence CO2 spectrum. In both cases the CCF peaks at δ v = 0, indicating CO2 exosphere detection. The maximum of the CCF is lower for the trailing side due to a fainter CO2 signal. \n<!-- image --> \nfrom the acquired spectra themselves (noise-like features are dominated by residuals in solar lines subtraction), we used the error cube given in the Level 3 hyperspectral data cubes to set the r.m.s., and assumed that it scales as σ / √ n , when n spaxels are averaged. The generated synthetic spectra were then processed as for the Ganymede data, and the resulting DIFF spectra were cross-correlated with the forward model to obtain a cross correlation function CCFnoise for a spectrum containing only noise. We then measured the standard deviation of the CCFnoise curve. Eighty random-noise synthetic spectra were processed in this way, to derive a representative standard deviation σ CCFnoise from the median of the values obtained for each shot. For processed Ganymede spectra, S / NCCF is obtained by dividing the CCF at δ v = 0 by σ CCFnoise. \nThe cross-correlation technique was applied on data averaged over latitude bins (Figs. 2, C.1, C.2). Spectra with a Pearson correlation coe ffi cient Cp > 0.3 all display a CCF with S / NCCF > 5, and a maximum shifted by less than 1 spectel element. From those criteria, the CO2 exosphere is detected with good confidence both in the northern and southern hemispheres of the leading and trailing sides of Ganymede. On the other hand, the S / N (and inferred CO2-gas signal) is low near the equator for both hemispheres, indicating a more tenuous CO2 exosphere in these regions. The decrease of the CO2 signal observed at the most polar latitudes could be related to PSF blurring since spaxels probing extreme polar latitudes are near the limb of Ganymede disk. However, this decrease is still observed after deconvolution with modeled NIRSpec PSFs (Fig. M.1). \nFig. C.2. Study of 4.22-4.295 µ mspectra averaged over latitude bins. A) Pearson correlation coe ffi cient between residual CO2 DIFF spectra and a synthetic DIFF spectrum which uses a CO2 fluorescence synthetic spectrum at T = 105 K as input. B) CO2 line-of-sight column density retrieved from the fit of CO2 DIFF spectra. C) shift, in fraction of spectel, of the maximum of the cross-correlation function (CCF) between CO2 DIFF spectra and the synthetic DIFF spectrum; secure detection is indicated when the shift is close to zero. D) S / N of the cross-correlation function at shift = 0. The S / N is obtained by computing the CCF noise obtained by using as input a simulated noisy spectrum, with the r.m.s deduced from the ERR entry in the Level 3 data cube, and scaled in √ n , where n is the number of averaged spaxels (S / N = CCF / CCF noise at shift = 0). In all plots, blue and red symbols correspond to the leading and trailing sides, respectively. The blue (resp. pink) vertical domains show the latitude range of the open-closed-field line boundaries for the leading and trailing sides, respectively, restricted to longitudes of 10-130 · W(leading) and 210-330 · W(trailing). \n<!-- image -->', 'Appendix D: SMART-EVE excitation/radiative transfer model': "A non-LTE Stochastic Modeling of Atmospheric Radiative Transfer-Exospheric Vibrational Excitation (SMART-EVE) has been developed to calculate the ro-vibrational populations of the (1) H2O ν 2 mode (010) at 6.25 µ m, and (2) CO2 ν 3 mode (001) at 4.25 µ m. SMART-EVE solves the locally defined statistical equilibrium equations (SEEs) for all the energy levels considered and the radiative transfer equations (RTEs) for all the bands connecting these levels. Due to non-linearities arising from radiative transfer and / or collisional coupling, the resulting equation system is solved iteratively using the Accelerated Lambda iteration approach which alternates SEE calculations involving all the energy levels with RTE calculations involving all atmospheric layers. \nThe 1D model of the atmosphere is described by the kinetic gas temperature, assumed vertically uniform, and the gas density which follows hydrostatic equation. The model assumes that the atmosphere is illuminated by the Sun from the top, and by the surface thermal emission and the reflected solar and atmospheric radiance from below. The model parameters are: the column density and kinetic temperature T kin of the atmosphere, the surface temperature Ts and reflectance factor Ref , and the heliocentric distance. Electron-impact excitation of CO2 is not considered, as most likely insignificant (Supplementary Information). \nThe radiative processes considered are spontaneous and stimulated emissions, absorption of the upward thermal flux, incident solar, and reflected solar and atmospheric irradiance from the surface, as well as exchanges between layers. A single collisional process is considered for the vibrational state, its vibration-to-translation (V-T) relaxation / excitation in intermolecular collisions. However, vibrational de-excitation by collisions is insignificant in Ganymede's exosphere (see Appendix G). It is assumed that rotational levels are at LTE at all altitudes with a rotational temperature T rot = T kin. \nThe model is run from the surface up to 100 km, with 1km thick layers. The spectral data are taken from the HITRAN database (Gordon et al. 2022). We considered only (010)-(000) (H2O) and (001-000) (CO2) vibrational transitions, with a total number of lines of 1017 for H2O and 129 for CO2. The solar spectrum was taken from Paper I.", 'Appendix E: Determination of CO 2 column density': "CO2 line-of-sight column densities were derived using a twostep approach. First, column densities were derived under the assumption of fluorescence equilibrium and optically thin lines. In the second step, a correction factor was applied, using prescriptions obtained from the SMART-EVE model described above. \nFor CPU-time considerations, we used the Planetary Spectrum Generator (PSG) (Villanueva et al. 2018) for the first step. Optically thin CO2 ν 3 band fluorescence spectra at high spec- \nFig. D.1. CO2 simulated nadir spectra. Input model parameters are: CO2 column density N (CO2) = 10 18 m -2 , surface temperature Ts = 145 K, gas rotational temperature T rot = 105 K, for I / F values of 0.04 (red) and 0.12 (green). The fluorescence spectrum (black) corresponds to I / F = 0 and Ts = 0 K. Ganymede's exosphere is described by hydrostatic equilibrium. \n<!-- image --> \nral resolution (0.1 cm -1 ) were generated (outputs for cometary atmospheres) and downloaded for a range of rotational temperatures in steps of 1 K (and fixed column density). As done for the forward model described above, they were combined with Ganymedes's solid-state CO2 band and synthetic DIFF spectra were computed. This bank of synthetic spectra was utilized to fit the Ganymede DIFF spectra using the Levenberg-Marquardt algorithm (we used the lim f it Python package), with a normalizing factor as free parameter. The rotational temperature was set as a free parameter for the analysis of the high S / N spectra of the north hemisphere of the leading side (Fig. H.1), and fixed to 105 K elsewere (i.e., equal to the inferred value in leading north hemisphere, Appendix H). The CO2 column density was derived from the inferred normalizing factor. For the uncertainty in the CO2 column density, we used that provided by the lim f it package and derived from the covariance matrix. \nModel simulations for the CO2 ν 3 band (Fig. D.1) show that spectral profiles from Ganymede's exosphere are expected to be less intense than in the assumption of cometary-like fluorescence emission (case Ref = 0, Ts = 0), which neglects absorption of surface reflected solar radiation by CO2 gas, reflection of CO2 gas emission on the surface, and surface thermal radiation. While this latter process is not significant at 4.25 µ m, the other two processes a ff ect the total band intensity of the ν 3 band BA according to: \nBA = BA 0 × (1 -4 . 2 × I / F ) , for N(CO2) = 10 17 m -2 (E.1) \nBA = BA 1 × (1 -3 . 6 × I / F ) , for N(CO2) = 10 18 m -2 (E.2) \nBA = BA 2 × (1 -2 . 1 × I / F ) , for N(CO2) = 10 19 m -2 (E.3) \nwhere I / F is the radiance factor on the surface, and BA 0, BA 1, BA 2 are equal to 4.10 × 10 -8 , 4.27 × 10 -7 , and 3.84 × 10 -6 Wm -2 sr -1 , respectively. BA 0 is consistent with the value of 4.06 × 10 -8 Wm -2 sr -1 retrieved from PSG (Villanueva et al. 2018) for optically thin cometary-like fluorescence emission at 4.95 au from the Sun with N (CO2) = 10 17 m -2 . Eqs E.1-E.3 were obtained from multiple simulations fixing Ts = 145 K and T rot = 105 K, and varying I / F . \nWe applied a correction factor intermediate between Eqs E.1 and E.2 (i.e., slope of -4.0 for the dependence with I / F ) on the CO2 column density inferred assuming fluorescence equilibrium, using radiance factors I / F at ∼ 4.25 µ m measured from JWST (Paper I).", 'Appendix F: Electron impact excitation of CO 2': 'We made estimations for electron-impact excitation of the CO2 ν 3 band using cross-sections from Itikawa (2002). Electron populations were assumed to follow a Maxwellian distribution around a mean temperature. For the total electron density and the temperature, values that explain the highest UV brightnesses (OI 1356 Å) of 1000 R (Waite et al. 2024) measured for Ganymede were used. Specifically, we assumed an electron temperature of 20 eV and a high number density of 2500 cm -3 . CO2 emission from this process is found to be more than two orders of magnitude lower than fluorescence emission. The CO2 ν 3 band could be excited by much cooler electrons, well below 10 eV. However, information on these cold electrons is missing. Cross-sections for electron-impact excitation of the CO2 ν 3 band increase with decreasing electron energy (Itikawa 2002). Using an electron temperature of 1 eV and the same number density, CO2 emission from electron impact excitation is only two times higher than for 20 eV electrons.', 'Appendix G: CO 2 ν 3 -band collisional relaxation': "We have evaluated the role of de-excitation of the CO2 ν 3 band by collisions with H2O, O2, and CO2 versus spontaneous emission. The result is that these processes are not significant in Ganymede's exosphere. The rate for collisional de-excitation of the CO2 ν 3 band via CO2-H2O collisions is 1.2 × 10 -13 cm 3 / s at 120 K (Blauer & Nickerson 1973). The H2O number density is at most 1.4 × 10 10 cm -3 at the surface, derived from hydrostatic equilibrium for a water column density of 5 × 10 20 m -2 (Roth et al. 2021). This gives a collision rate of at most 1.7 × 10 -3 s -1 , which is much lower than the spontaneous emission rate of the ro-vibrational levels (on the order of 400 s -1 ). So the quenching is negligible. \nThe rates for de-excitation of CO2 ν 3 band via collisions with O2 and CO2 are much lower than for CO2-H2O collisions. So these collisional processes are still less significant.", 'Appendix H: Rotational temperature of CO 2': 'For fluorescence emission, the relative intensities of the rovibrational lines of the CO2 ν 3 band are set by the population distribution in the ground vibrational state, described by a Boltzmann distribution at the rotational temperature T rot. Rotational temperatures of CO2 derived on the northern latitudes ( > 30 · N) of the leading hemisphere are shown in Fig. H.1 and are on the order of 105-110 K (see Appendix B for details on how T rot was derived). This is slightly lower than the surface temperature of Ganymede at these latitudes (from 120 to 140 K, Fig. K.1). This rotational temperature possibly reflects the kinetic temperature of the exosphere at low altitudes where collisions with the major gas (H2O or CO2) are still e ffi cient enough to thermalize CO2 molecules. Alternatively, it might reflect the rotational energy of the CO2 molecules when they left the surface, and be representative of the temperature of the surface where CO2 molecules were released. The CO2 molecule has no dipole moment, so radiative rotational decay within the ground vibrational state does not take \nplace. The rotation temperature is expected to increase with residence time in the atmosphere due to radiative decay from the excited vibrational states. However, one should mention that during their residence time in the exosphere (at most 18 h, which is the CO2 lifetime set by electron-impact ionization), CO2 molecules undergo at most 7 fluorescence cycles. Based on fluorescence calculations for cometary atmospheres, CO2 molecules reach a warm fluorescence equilibrium only after about 3500 fluorescence cycles at 5 au from the Sun Crovisier (1987). In summary, the measured T rot should reflect the thermal environment where last thermalizing collisions occurred, or the excitation state of the molecules when they left the surface. \nFig. H.1. CO2 rotation temperature in the north hemisphere of the leading hemisphere. Spaxels within latitude bins of width 7.5 · were averaged. Values were derived from the fitting of DIFF spectra using as model fluorescence emission with T rot as a free parameter (Appendix B). The weighted mean value is T rot = 107 ± 5 K. \n<!-- image -->', 'Appendix I: H 2 O analysis': 'Weanalysed MIRI / MRSChannel-1 spectra obtained by either 1) averaging spaxels around the subsolar point, namely eight spaxels for which the solar zenith angle (SZA) is less than 15 · at the center of the spaxel; 2) averaging seven spaxels covering the region of the leading hemisphere where the CO2 exosphere is prominent. Neither one shows any hint of the presence of water lines (Fig. I.1) and the non-detection of H2O was further confirmed by applying the cross-correlating technique using a forward model of a synthetic spectrum of H2O computed with the SMART-EVE code. \nIn the exosphere of Ganymede, the thermal radiation from the surface competes with the Sun\'s direct radiation for the excitation of the H2O ν 2 band at 6.2 µ m (Paper I). In addition, in nadir viewing, absorption of the radiation from Ganymede\'s surface by the H2O exosphere might compete with ν 2 fluorescence emission, so that the band might be in absorption under certain conditions and a simple fluorescence model would not apply. Therefore, we used the SMART-EVE model described above to derive upper limits on the H2O column density ( N (H2O)). \nRadiance factor values of 0.04 and 0.08 were assumed for the leading and trailing hemispheres, respectively (see Fig. 26 of Paper I). The surface temperature was chosen such that the brightness temperature in the synthetic spectra matches the TBB value at 6.2 µ m measured on the MIRI spectrum. For the "subsolar" spectra (SZA < 15 · ), Ts (and TBB ) are closed to 155 K (leading) \nFig. I.1. Continuum-divided spectra of Ganymede observed with MIRI and synthetic H2O spectra. Spectra for the leading and trailing sides are shown in red and blue, respectively, with the spectrum of the trailing side shifted vertically. Spaxels for which the solar zenith angle is less than 15 · at the center of the spaxel have been averaged. Synthetic spectra are superimposed, with input parameters indicated in the legend (Appendix I) and N (H2O) = 10 20 m -2 . The Ganymede spectra do not show any hint of H2O lines. \n<!-- image --> \nand 160 K (trailing). For the spectrum extracted at the position of the CO2 northern source, Ts is about 140 K. Synthetic spectra for the subsolar region are shown in Fig. I.1. Ro-vibrational lines at λ < 6.2 µ mare expected in absorption whereas emission lines are expected at λ > 6.2 µ m. As a matter of fact, the vibrational temperature of the ν 2 band (mainly controlled by solar IR pumping) is ∼ 158-159 K, i.e. very close to the brightness temperature near 6.4 µ m. The change from absorption to emission regimes is related to the fact that the vibrational temperature is close to the wavelength-dependent brightness temperatures near 6.2 µ m. \nTable 1 presents measured 1σ uncertainties for the H2O band area in the 5.7-6.2 and 6.2-7.1 µ mspectral ranges, considering the 10-15 expected strongest lines (with intensities > 0 . 2 the intensity of the strongest line). From the measured band areas in each wavelength window, we derived a 3σ upper limit for line-of-sight N (H2O), using SMART-EVE model with appropriate parameters. The results were then combined. The final results are given in Table 1 for two values (105 and 130 K) of the H2O rotational temperature.', 'Appendix J: Open-closed-field-line Boundary': "The OCFB location is determined through magnetohydrodynamic modeling of Ganymede's magnetosphere similar to the method described in Duling et al. (2022). Due to the variation of the upstream magnetic field and plasma density at Ganymede's position relative to the Jovian current sheet, the OCFB location can oscillate with an amplitude ranging between 2 to 6 degree latitude during Jupiters's approximative 10-h rotation period (Saur et al. 2015). We modeled the OCFB analogous (Duling et al. 2022) by adapting the upstream conditions to estimates for the times of the JWST observations. During the observation of the leading side, Ganymede was above the center of the current sheet and we used 61 amu / cm 3 and (-11, -66, -79)nT for the upstream plasma mass density and magnetic field respectively. During the trailing side observation Ganymede was at the cen- \nter of the current sheet and we used 100 amu / cm 3 and (-18, -6, -79)nT.", 'Appendix K: CO 2 , H 2 O Ganymede exospheric model': "We simulated the CO2 exosphere using the Exospheric Global Model (EGM), a multi-species Monte Carlo model describing the fate of test particles in a gravitational field, interacting with a surface or an atmosphere and subject to sources of ionization and dissociation. EGM has been extensively used to model the exospheres of H2O and related species (e.g., O2, H) in various objects, in particular Ganymede (Leblanc et al. 2017, 2023). We considered two possible mechanisms of ejection of the CO2 molecules from the surface: i) sputtering, i.e. ejection following bombardment of H2O ice containing CO2 molecules by the incident Jovian energetic ions and electrons; and ii) sublimation of the CO2 molecules from Ganymede surface. We considered the release of CO2 either from the sublimation of pure CO2 ice or from the sublimation of H2O ice containing CO2 molecules. We took into account that CO2 molecules re-impacting cold areas of the surface eventually recondense. The H2O exosphere is also computed. The calculated images from the simulations (e.g., Fig. 3) consider the orbital position of Ganymede around Jupiter at the time of the JWST observations and the viewing geometry of JWST observations (for the observations of the leading hemisphere, sub-observer coordinates were 2 · N, 72 · W). Convolution with a FWHM = 0.185' PSF is applied (Appendix M). Line-ofsight CO2 column densities averaged over latitude bins of 7.5 · or 15 · were computed for comparison with the data shown in Fig. 2. For the study of the CO2 exosphere above the north polar cap of the leading hemisphere, we extracted the longitudinal variation of the CO2 column density for latitudes in the range 42-62 · N. \nSublimation: For a CO2 release associated with the sublimation of water ice, the release rate is in proportion with the H2O sublimation rate (cm -2 s -1 ): \nF (CO2) = fc × q CO2 × 2 . 17 10 32 e -U 0 k B Ts √ Ts . (K.1) \nF (H2O) = q H2O × 2 . 17 10 32 e -U 0 k B Ts √ Ts , (K.2) \nwhere U 0 / kB = 5950 K, and q H2O is the areal surface fraction of H2O. The relative abundance of CO2 in the sublimated gases (in number) is q CO2 / q H2O. The description of F (H2O) follows Leblanc et al. (2023). Ts is the surface temperature. fc is a factor introduced to reproduce the CO2 JWST data. \nFor the sublimation of CO2 ice, the sublimation rate (cm -2 s -1 ) is given by: \nF (CO2) = fc × N tot q CO2 τ 0 √ Ts e -U 1 k B Ts , (K.3) \nwhere N tot = 10 18 cm -2 s K 0 . 5 is determined from a fit of the polynomial relation of CO2-ice vapor pressure with temperature (Fray & Schmitt 2009) and using U 1 / kB = 2860 K (surface binding energy for CO2 on H2O ice, (Sandford & Allamandola 1990)) and τ 0 = 3.45 10 -13 s (Sandford & Allamandola 1990). fc is a factor introduced to reproduce the JWST data. \nSputtering: The ejection of CO2 molecules by sputtering is described by the e ffi ciency by which CO2 molecules are emitted \nfrom a surface when an incident ion or electron impacts the surface with a given energy. We hypothesized that CO2 molecules are trapped in / on H2O ice, so we assumed that the sputtering yield follows the same temperature dependence as for H2O, and used the same definition as for H2O (Cassidy et al. 2013; Leblanc et al. 2023): \nY (CO2) = fc × Y 0 × (1 + Y 00 × e -U 00 k B Ts ) , (K.4) \nwith Y 0 = 1200. In fact, we made the assumption that CO2 molecules are released into the exosphere along with H2, O2, H2O2 and H2O molecules ejected when pure H2O ice is bombarded, therefore U 00 and Y 00 are set to be the same as for the bombardment of pure H2O ice (Famá et al. 2008): U 00 = 0.06 eV (700 K), Y 00 = 220. As for the energy and angular distributions of the CO2 molecules when ejected from the surface, we followed the approach used for sputtered O2 in Leblanc et al. (2017) and assumed a Maxwell-Boltzmann energy distribution at the local surface temperature. Regarding the intensity and spatial distribution of the Jovian ions impacting the surface, we assumed a given ion flux of 10 6 particles / cm 2 / s derived from Cassidy et al. (2013) as in Leblanc et al. (2017), impacting Ganymede's surface only in the open-field-line regions. Electron impacts are not considered. The flux of the CO2 molecules released at a given position on Ganymede's surface is therefore the product of Y (CO2) (Eq. K.4) times q CO2 times the flux of impacting particles. The flux of H2O follows the same equation, using q H2O instead. A multiplying factor fc is introduced in Eq. K.4 with respect to Cassidy et al. (2013) and Leblanc et al. (2023) that is adjusted to reproduce the CO2 column density measured by JWST. This factor is also applied to the flux of H2O sputtered molecules. In our model, sputtering of water ice is assumed to release mainly H2O molecules with a ratio H2O / O2 = 20 (Leblanc et al. 2017; Cassidy et al. 2013). \nSurface adsorption: To determine the fate of a CO2 molecule re-impacting the surface, we define the CO2 residence time at the surface as: \nτ = τ 0 e U 1 k B Ts . (K.5) \nwhere U 1 / kB = 2860 K is the binding energy for CO2 adsorbed on H2O ice, and τ 0 = 3.45 10 -13 s (Sandford & Allamandola 1990). We considered that when the CO2 residence time is longer than the model time step (0.25 s), any particle hitting the surface gets trapped in the surface. We then calculated at each step and for each trapped particle a probability to be re-ejected as being equal to the ratio between the time step of the simulation and the residence time calculated from the surface temperature at the position of the particle. This probability is then compared to a random number between 0 and 1 and if larger than this random number, the particle is re-emitted into the exosphere. We checked that the results are not sensitive to the model time step. \nSurface temperature: In Leblanc et al. (2017, 2023), the Ganymede's surface temperature was calculated using a 1-D heat conduction model. Such description had some limitations, in particular it ignored surface roughness that leads to a distribution of facet temperatures (instead of a single temperature) at a given latitude, longitude and local time. Bockelée-Morvan et al. (2024) (Paper I) showed that matching the JWST / MIRI brightness temperature maps, in particular the low-to-high latitude and the noon-to-dawn / dusk temperature contrasts, requires considering surface roughness e ff ects. In the framework of a model for the distribution of slopes inherited from Hapke (1984), they found that the data could be fit by invoking mean slope angles s \nFig. K.1. Ganymede's surface temperature used in EGM model, representative of the leading hemisphere. Left panel: facet temperature distribution (cumulative probability) of the surface temperatures at 12 h local time for various latitudes indicated in the plot. Right panel: latitude / longitude map of the average surface temperature with the subsolar point being at a latitude of 2 · N (as for JWST observations, Paper I) and longitude of 180 · . \n<!-- image --> \n= 15 · -20 · on the trailing side and 20 · -25 · on the leading side, with some variations depending on the adopted surface albedo model. Here we adopted the following parameters, relevant to the leading side: s = 25 · , Bond albedo = 0.30, thermophysical parameter Θ = 0.3 (i.e. thermal inertia Γ = 22.5 SI units). We used a spatially constant Bond albedo to keep the number of free parameters tractable. Such rough temperature distributions were calculated on a 37 × 48 latitude × local time (or longitude) grid, i.e. with a 5 · latitude and 0.5 h local time step. Figure K.1 (left) shows examples of cumulative facet temperature distributions at noon local time and various latitudes, while the right panel shows the facet-averaged temperature map, where the maximum temperature is at 12.5 h local time. The multiplicity of temperatures at a given latitude / local time enables condensation in regions where it would not be expected without surface roughness. At the equator, the probability to find a surface element at a temperature smaller than 73 K (the theoretical condensation temperature of CO2 at Ganymede atmospheric pressure of 1 pbar) is zero, even in presence of surface roughness. However, the probability of encountering temperatures lower than 73 K increases with latitude, to 1% at + 30 · , 12% at + 60 · and 30% at + 90 · . \nFor describing the temperature of the surface of the trailing side (used for the calculations shown in Appendix L.4), we adopted the following parameters, which fit at best the JWST / MIRI brightness temperature map of this hemisphere: s = 20 · , Bond albedo = 0.20, thermophysical parameter Θ = 0.3. \nWe stress that, as indicated in Paper 1, our thermal model describes roughness purely as a slope e ff ect, and does not account for other more complex e ff ects associated with topography, such as shadowing and self-heating due to scattering and reabsorption of solar and thermal radiation within craters, as done for investigating cold traps for water ice on the Moon (Hayne et al. 2021; Davidsson & Hosseini 2021). Applying such more advanced thermophysical models is left to future investigations.", 'Appendix L: Simulated exospheres from EGM simulations': "Appendix L.1: CO 2 gas spreading \nA question to address is the localized character of Ganymede's CO2 exosphere. Indeed, CO2 does not condense e ffi ciently at \nthe typical Ganymede's surface temperatures, even in the polar regions (100-110 K), and therefore could spread out over the whole illuminated disk, possibly condensing only in the nonilluminated areas. To study the spreading of CO2 molecules, we performed EGM simulations (Appendix K) assuming that CO2 is released by the sublimation of CO2 ice from a small surface area (300 × 300 km) at 52 · N. The calculations were performed using a distribution of facet temperatures for each location, as computed with our thermal model with surface roughness (Appendix K), and, for comparison, using instead the facet-average temperature map (Fig. K.1, right). Figure L.1 shows the vertical CO2 column density in the two cases. It shows that the resulting atmospheric distribution is quite di ff erent for the rough surface. The migration distance is much smaller in this case, showing that the di ff usion of CO2 molecules is substantially controlled by the ability to condense on cold traps. These cold traps are probably the discrete patches of optically thick ice, preferentially located on pole-facing slopes, that constitute the polar cap (Khurana et al. 2007). The limited horizontal spreading of CO2 gas indicates that local column-density maxima are associated with local sources.", 'Appendix L.2: CO 2 exosphere: Leading CO 2 source only': 'The most significant feature of Ganymede\'s CO2 exosphere being the large excess in the northern hemisphere of leading side, our simulations (see Appendix K for model description) were designed to reproduce this feature. Specifically, we calculated line-of-sight column density maps N (CO2), and extracted latitudinal and longitudinal profiles of N (CO2) for comparison with observations. Model simulations were performed assuming q CO2 = 1% in a northern cap extending from latc to 90 · in latitude and ranging from 0 to 180 · Win longitude. Elsewhere, q CO2 was set to 0. The input value q CO2 = 1% was set as it is consistent within a factor of a few with the rough estimation of the CO2 abundance at the surface based on the depth of the CO2-solid 4.3 µ m band (CO2 / H2O ∼ 1% in mass, Paper I). The limiting latitude latc and the correction factor fc to the CO2 flux were adjusted to best reproduce the position in latitude and value of the column-density peak in the latitudinal profile of N (CO2) shown in Fig. 2. \nColumn density maps obtained for sputtering, CO2 release associated with H2O sublimation and sublimation of CO2 ice \nFig. L.1. Vertical CO2 column density using a distribution of temperature at each location (rough surface, left panel) or the mean temperature (right panel). The simulations consider the sublimation of CO2 ice from a 300 × 300 km region at 52 · N, with q CO 2 × fc = 3 × 10 -14 . \n<!-- image --> \nTable L.1. Line-of-sight column densities from EGM simulations. \n- ( a ) q CO2 in a northern cap covering latc -90 · in latitude and ranging from 0 to 180 · Win longitude. Elsewhere, q CO2 is set to 0.\n- ( b ) Correction factor to simulated CO2 flux from the surface to reproduce the CO2 line-of-sight column density peak in the latitude profile (Fig. 2).\n- ( ∗ ) Values are low because q CO2 = 0 for latitudes < latc . \n(Appendix K) are shown in Fig. L.2 (and in the main text Fig. 3 for the first two production mechanisms). EGM latitudinal profiles of line-of-sight column densities are compared to observations in the left panels of Fig. L.2. Longitudinal variations for latitudes in the range 42-62 · N (i.e., encompassing the northern region of the leading hemisphere with CO2 gas enhancement) are compared in the right-hand panels of Fig. L.2. Table L.1 lists model input parameters, and average column densities in the sub-solar region and so-called "Leading CO2 source" defined in Table 1. For CO2 release associated with H2O sublimation (top panels), the latitudinal profile is reproduced for latc = 50 · N and multiplying the flux of CO2 molecules by fc = 260 (as fc × q CO2 = 260 × 0.01 = 2.6, this corresponds to a CO2 / H2O relative abundance of 5 for an H2O areal ice fraction of 50%). Since the H2O sublimation flux is highly dependent of the surface temperature (Eq. K.2) and is therefore strongly favored at low latitudes, latc must be close to the latitude at which the peak of the CO2 column density is observed (Fig. 1). For the CO2-ice sublimation case (middle panels), latc is also close to 50 · N, and fc = 3 × 10 -12 . For the sputtering scenario (bottom panels), a limiting latitude latc = 40 · N best reproduces the latitudinal trend of the column density (a too narrow distribution in latitude is obtained for latc = 50 · N), and the flux of the sputtered CO2 molecules had to be multiplied by fc = 382. \nAs shown in Fig. L.2, the sputtering only scenario (panels e and f) provides a better fit of the spatial distribution of CO2 exosphere than sublimation. Especially, for the sublimation sce- \nnarios, a strong variation with longitude (i.e., local time) is obtained whereas the diurnal variation is flatter and almost consistent with the observations for sputtering. This is essentially due to the di ff erence in the temperature dependence of these mechanisms (Eqs K.1, K.3, K.4). However, the peak of the CO2 line-ofsight column-density distribution for sputtering is more shifted towards the afternoon ( ∼ 13.4 h) than in the sublimation cases ( ∼ 13.1 h) and for the observed peak ( ∼ 12 h).', 'Appendix L.3: H 2 O exosphere': 'EGM simulations of Ganymede\'s H2O exosphere were already performed by Leblanc et al. (2017, 2023), but the used surface temperature model did not consider surface roughness, unlike the present calculations. Figure L.3 displays the H2O column density as seen from JWST for the sublimation (left panel) and sputtering (right panel) cases. The H2O ice areal surface fraction q H2O is set to 20% all over the surface, a value which is consistent with measured water ice abundances on the leading side for latitudes ≤ 30-40 · Ligier et al. (2019). For latitudes of 4050 · N, values of 40-50% would be more appropriate Ligier et al. (2019). As expected the sublimation of H2O follows the surface temperature distribution. A North / South asymmetry can be clearly seen which is driven by the small asymmetry in surface temperature associated with the positive subsolar latitude (see Fig. K.1). A dawn to dusk asymmetry with larger H2O column densities towards the dusk is also present. A similar dawn / dusk \nFig. L.2. CO2 gas line-of-sight column density as a function of latitude (left) and longitude (right) for the leading side. Light blue symbols show model results from EGM and dark blue symbols refer to JWST observations. Top panels (a, b): CO2 release associated to H2O sublimation with surface flux fc × q CO 2 = 2.6 at latitudes > 50 · N and q CO 2 = 0 elsewhere. Middle panels (c, d): CO2 release associated to CO2 ice sublimation with surface flux fc × q CO 2 = 3. × 10 -14 at latitudes > 50 · N and q CO 2 = 0 elsewhere. Bottom panels (e, f): sputtering only with surface abundance q CO 2 = 1% at latitudes > 40 · N, and q CO 2 = 0 elsewhere, and a multiplying factor to surface CO2 flux fc = 382. In the left panels (a, c, e) column densities are averages in latitude bins of 7.5 · . Longitudinal variations shown in right panels (b, d, f) consider latitudes in the range 42-62 · N (i.e., encompassing the northern region of the leading hemisphere with CO2 gas enhancement). Calculated CO2 column density maps for all three cases are shown in the left panels. \n<!-- image --> \nFig. L.3. Calculated line-of-sight column-density maps of the H2O exosphere of Ganymede (in 10 18 m -2 ). The green lines display the boundary between open and close field lines (OCFB). Left panel: sublimation only source. Right panel: sputtering only source with fc = 382. The H2O areal surface fraction is set to q H 2 O = 20%. The subsolar point is at 2.6 · N, 82 · W. \n<!-- image --> \nasymmetry, related to surface thermal inertia, is present for the CO2 simulated exosphere (Fig. 3, left panel). \nThe calculated line-of-sight H2O column densities in the leading subsolar region (solar zenith angle SZA < 15 · ) and so called "Leading CO2 source" are given in Table L.1. They are below the upper limits set by JWST (7-17 × 10 19 m -2 for subsolar, 2 × 10 19 m -2 for CO2 source region, Table 1), summing the contributions from sputtering and sublimation. When consider- \na more appropriate value of q H2O = 40-50% for the northern region of the leading hemisphere, the modelled column density for this region is of the order of the JWST upper limit. The value obtained for the subsolar region (4.1 10 19 m -2 ) is a factor of 7 above the minimum value derived from HST OI data for the leading hemisphere (6 × 10 18 m -2 , Roth et al. (2021)). \nWe stress that calculated H2O sputtered fluxes are under the hypothesis that H2O is the major species released by sputtering \nFig. L.4. Simulation of the entire CO2 exosphere formed by sputtering only. No sputtering is simulated inside the closed-field-line region. Input parameters are given in the text. Panel a: 2D line-of-sight column density map of CO2 (in 10 18 CO2 / m 2 ), Ganymede-leading side. Panel b: same as panel a but for the Ganymede-trailing side. Panel c: latitudinal variation of the longitudinal average column density for the Ganymede-leading side (for bins of 7.5 · in latitude). Panel d: same as for panel c but for the Ganymede-trailing side and bins in latitude of 15 · . Dark blue symbols: JWST observations. Light blue symbols: EGM simulation. Red symbols: JWST observations. Orange symbols: EGM simulation. \n<!-- image --> \nof H2O ice Cassidy et al. (2013). However, the major mass loss from water ice by particle bombardment might not be H2O, as assumed here, but rather via the ejection of O2 and H2 (or at least in significant proportion) according to laboratory experiments Teolis et al. (2017). Hence, H2O sputtered fluxes given in Table L.1 can be considered as upper limits. This reinforces our conclusion that JWST H2O upper limits measured in the north polar cap of the leading hemisphere are fully consistent with expectations.', "Appendix L.4: Simulation of CO 2 sputtering on entire Ganymede's surface": "We also investigated CO2 sputtering over the southern openfield-line region of the leading hemisphere and over the southern and northern open-field-line regions of the trailing hemisphere. This approach allowed us to illustrate how the sputtering ejection rate would need to be changed from a region to another to explain the JWST observations. The surface temperature of the trailing side was calculated using the same approach as for leading side (see parameters of the thermal model in Appendix K). Figure L.4 provides one example of results obtained by using the same q CO2 and fc values for the two hemispheres, the only di ff erence being the location of the OCFBs. The distribution of sputtered CO2 over the whole surface is described according to: \n- -a region (1) of sputtering above a latitude of 50 · N, with a flux of sputtered CO2 molecules determined as described in section I, with fc = 640 and q CO2 = 1%,\n- -a region (2) in the northern hemisphere open field lines region with fc = 80 and q CO2 = 1% (describing e.g. regions in between northern OCFB and > 50 · Npolar cap on the leading hemisphere),\n- -a region (3) in the southern hemisphere open field lines region with fc = 60 and q CO2 = 1%,\n- -a region (4) in close field lines region with fc = 0. \nAssuming the same fc = 640 value for all regions, would be equivalent to adopting q CO2 = 1%, 0.13%, 0.09 %, and 0% for regions (1), (2), (3), (4) respectively. With these assumptions, the shape of the latitudinal distribution observed on the leading side is reproduced but not that of the trailing side (Fig. L.4). Reproducing at least approximately Ganymede's CO2 exosphere with the sputtering mechanism might be possible by adjusting fc or q CO2 parameters geographically over the surface. However, this model cannot explain the exospheric excess observed at equatorial to mid latitudes in the southern regions of the trailing hemisphere (Fig. 1) taking into account that, at the time of the trailing-side observation, Ganymede was inside the plasma sheet of Jupiter's magnetosphere, so southern polar regions were not over-exposed to plasma bombardment with respect to northern regions (Appendix J).", 'Appendix M: PSF deconvolution of CO 2 gas map': "In order to analyse the spatial distribution of the CO2 exosphere on Ganymede's leading side in more detail, we applied a deconvolution procedure. The deconvolution process was done using the AIDA algorithm in classical mode (Hom et al. 2007) that requires science and PSF data files. Reported Full Width at Half Maximum (FWHM) measured at 4.25 µ m from JWST point-source observations range from 0.14 to 0.17' ()Deugenio2023 and are expected to vary with the observational mode (e.g., number of dithers). Unfortunately, no reference star was observed during the observations of Ganymede, so we used the WebbPSF software \nFig. M.1. Top: deconvolved CO2-gas maps (leading hemisphere) obtained with the AIDA algorithm using NIRSpec PSFs calculated with WebbPSF at 3.5 µ m(FWHM = 0.165'), 4.2 µ m(FWHM = 0.185') and 5 µ m(FWHM = 0.205'). Bottom: residuals of the deconvolution for each calculated PSF; plotted are ( O -C ) 2 , where O is the original CO2 column-density map (leading hemisphere, top-right plot), and C is the convolution of the deconvolved O map. Color bars are in unit of 10 18 m -2 . \n<!-- image --> \n( https://www.stsci.edu/jwst/science-planning/ proposal-planning-toolbox/psf-simulation-tool ) which can calculate monochromatic PSFs for NIRSPec in spectroscopic mode. PSFs were generated at various wavelengths to explore how deconvolved images vary with the PSF FWHM. \nWe show in Fig. M.1 the deconvolution of the CO2 gas map above the leading hemisphere for PSFs with FWHMs of 0.165, 0.185 and 0.205' corresponding to WebbPSF outputs at 3.5, 4.2 and 5 microns, respectively. For this purpose, we used the CO2 column density map derived for the full 37 × 43 spaxels IFU frame (i.e., including results outside Ganymede disk). The deconvolved maps were then reconvolved with the PSFs used for the deconvolution. Residuals with respect to the original data are shown in Fig. M.1 (bottom row) for the three assumed PSFs, and do not di ff er much. The deconvolved CO2 gas distribution confirms that the northern CO2 excess is confined in longitude and latitude, and that the decrease of the column density above latitudes of 60 · N is real. The deconvolved map also clearly shows the excess nearby the southern OCFB.", 'Appendix N: Spatial variations of surface CO 2 band depth': 'The CO2 absorption band around 4.26 µ m is widespread over the surface of Ganymede. Globally, the CO2 band depth appears anti-correlated with bond albedo and water ice absorption band depths, with the maximum CO2 band depth on the equatorial regions but much lower values on regions poleward of 30°N (Fig. N.1, Hibbitts et al. (2003), Paper I). The CO2 band center (Fig. 4C) and shape show stronger relationship with surface brightness for both hemispheres, red-shifting and getting narrower / asymmetric as we reach polar latitudes, respec- \nvely. As discussed in Paper I these changes of band position and shape may be due to contributions of CO2 under varying physical states / matrices depending on the latitude: adsorbed on minerals or salts at the equatorial latitudes, and possibly mixed in amorphous water ice at the poles (Fig. 4C). But why is the CO2 band depth weaker at the polar regions, where CO2 gas appears to be released from the surface? Actually, the poles of Callisto also have weaker CO2 band depths and finer-grained water ice than the equatorial regions, just like Ganymede. From Galileo / NIMSdata, this was interpreted as fine-grained ice physically covering and spectrally masking the CO2, although such a masking e ff ect was not demonstrated numerically or experimentally (McCord et al. 1998; Hibbitts et al. 2000; Hibbitts et al. 2003). JWST / NIRSpec data of Ganymede also show some anti-correlation between the spatial distribution of the CO2 band depth and that of the H2O 4.5µ m band depth (Fig. N.1). However, the lower CO2 band depth at the polar regions may not be (only) due to a putative spectral-masking e ff ect, but to other factors. Maybe the surface areal abundance of CO2 is lower within the open-field-line areas because the irradiation releases it more e ffi ciently from its mineral association. As a result, CO2 may be less concentrated over non-ice mineral-rich terrains and only present on bright ice-rich patches (the coldest surfaces), reducing its geographically averaged band depth and making its band shape and position more compatible to CO2 mixed in water ice. Another explanation might also be that the absorption coe ffi -cient of CO2 decreases at the poles because CO2 is in a di ff erent state / matrix and / or at a di ff erent temperatures than at equatorial regions. \nDespite the lack of correlation between the global distribution of CO2 gas column density and solid CO2 band depth, there are some local areas where both are minimum (in the south po-', "Bockelée-Morvan et al.: Ganymede's CO2 exosphere": "Fig. N.1. Comparing CO2 exosphere to H2O and CO2 distribution on Ganymede's surface. Top and bottom rows are for the leading and trailing hemispheres, respectively. A, D: CO2 gas column density maps at a 3 × 3 smoothed resolution (this work) ; B, E: H2O band depth at 4.5 µ m(Paper I); C, F: CO2-solid 4.3µ mband depth (Paper I). In these plots, unlike in Figs 1 and O.1, the same color scales and boxcar smoothing are used for leading and trailing. \n<!-- image --> \nlar regions and in an area at about 260-300 · Wand 30 · S-50 · N, Fig. N.1), and some spaxels at the extreme north of the leading hemisphere show an enhanced CO2 band depth (Fig. 4).", 'A & A proofs: manuscript no. aa-main-revised': "Fig. O.1. CO2 gas distribution above Ganymede surface. First and second rows are for the leading side, at the original and 3 × 3 boxcar smoothed (for higher SNR) resolutions, respectively. Bottom row is for the trailing side at 3 × 3 smoothed resolution. Plots on the first column show the Pearson correlation coe ffi cient between continuum-filtered residual CO2 gas emission and a forward CO2 fluorescence model at 105 K (Fig. H.1). Plots on the second and third column show the line-of-sight CO2 column density and SNR inferred by fitting synthetic CO2 fluorescence spectra (Appendix B). The color scales for the leading and trailing sides are di ff erent, and indicated above the plots. Pixel sizes are 0.1 × 0.1'. The green lines show the open-closed-field line boundary Duling et al. (2022) \n<!-- image -->", 'Leading (unsmoothed)': '<!-- image --> \nFig. O.2. CO2 vertical column density maps of the leading (left) and trailing (right) hemispheres. They were deduced from the line-of-sight column density maps, by multiplying them by the cosine of the angle between local zenith and line of sight directions. For the leading hemisphere, the central contour for the north excess is at about 72 · W(12.6 h local time), 45 · N. Trailing data were smoothed using a 3 × 3 boxcar filter. \n<!-- image --> \nFig. O.3. Geological map of Ganymede (Plate 2 of Patterson et al. (2010)). The green lines show the open-closed-field line boundary at the time of the JWST observations Duling et al. (2022). \n<!-- image -->'}

2024RNAAS...8..222W

The dispersion measures of fast radio bursts have been identified as a powerful tool for testing the zeromass hypothesis of the photon. The classical approach treats the massive photoninduced and plasmainduced time delays as two separate phenomena. Recently Y.B. Wang et al. suggested that the joint influence of the nonzero photon mass and plasma effects should be considered and proposed a revised time delay for massive photons propagating in a plasma medium denoted as inlineformula inlineformula which departures from the classical dispersion relation SUP2SUP. Here we discuss the derivation presented by Y.B. Wang et al. and show that the classical dispersion relation remains valid based on Proca equations.

2024-09-01T00:00:00Z

['2024RNAAS...8..222W', '10.48550/arXiv.2409.04672', 'arXiv:2409.04672', '2024arXiv240904672W', '10.3847/2515-5172/ad7676']

['Radio transient sources', 'Intergalactic medium', 'Particle astrophysics', '2008', '813', '96', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']

The Dispersion Relation of Massive Photons in Plasma A Comment on Bounding the Photon Mass with Ultrawide Bandwidth Pulsar Timing Data and Dedispersed Pulses of Fast Radio Bursts

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https://arxiv.org/pdf/2409.04672.pdf

{"The Dispersion Relation of Massive Photons in Plasma: A Comment on 'Bounding the Photon Mass with Ultrawide Bandwidth Pulsar Timing Data and Dedispersed Pulses of Fast Radio Bursts'": 'BAO WANG 1, 2 AND JUN-JIE WEI 1, 2 \n1 \nPurple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China 2 School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, China', 'ABSTRACT': "The dispersion measures of fast radio bursts have been identified as a powerful tool for testing the zero-mass hypothesis of the photon. The classical approach treats the massive photon-induced and plasma-induced time delays as two separate phenomena. Recently, Wang et al. (2024) suggested that the joint influence of the nonzero photon mass and plasma effects should be considered, and proposed a revised time delay for massive photons propagating in a plasma medium, denoted as ∆ t ' m γ ∝ ν -4 , which departures from the classical dispersion relation ( ∝ ν -2 ). Here we discuss the derivation presented by Wang et al. (2024) and show that the classical dispersion relation remains valid based on Proca equations. \nKeywords: Radio transient sources (2008) - Intergalactic medium (813) - Particle astrophysics (96)", '1. INTRODUCTION': "Since the discovery of fast radio bursts (FRBs), numerous studies have employed the dispersion measures (or equivalently, the frequency-dependent time delays) of FRBs to place upper limits on the rest mass of the photon ( m γ ; Wu et al. 2016; Bonetti et al. 2016, 2017; Shao & Zhang 2017; Xing et al. 2019; Wei & Wu 2020; Wang et al. 2021; Lin et al. 2023; Wang et al. 2023; Ran et al. 2024). In the classical treatment, two separate time delays are considered: the plasma-induced time delay ( ∆ t d ∝ ν -2 ) and the massive photon-induced time delay ( ∆ tm γ ∝ ν -2 ). These two contributions are considered to be directly additive. However, in a recent work, Wang et al. (2024) proposed a revised time delay, denoted as ∆ t ' m γ ∝ ν -4 , which is attributed to the joint influence of the nonzero photon mass and plasma effects. In this comment, we point out the contradictions in the revised time delay proposed by Wang et al. (2024), and derive the dispersion relation of massive photons propagating in plasma based on Proca equations (Proca 1936), thereby proving the validity of the classical time delay ( ∝ ν -2 ).", "2. COMMENT ON TIME DELAY ∆ t ' m γ ∝ ν -4": "In this section, we follow the derivation presented in Wang et al. (2024) to examine the revised time delay. We first define four types of arrival times: \n- (i) The arrival times of massless photons in vacuum and plasma are t 1 and t 2, respectively.\n- (ii) The arrival times of massive photons in vacuum and plasma are t ' 1 = ∫ 1 υ γ g d l and t ' 2 = ∫ 1 υ new g d l , respectively, where υ γ g and υ new g are the group velocities of massive photons in vacuum and plasma, respectively. \nAccording to Wang et al. (2024), the observed time delay between massive photons with different frequencies ( ν low, ν high; ν low <ν high) in a plasma medium is expressed as \n∆ t ' obs = ( t ' 2 , low -t ' 1 , low ) -( t ' 2 , high -t ' 1 , high ) = ∫ ( 1 υ new g , low -1 υ γ g , low ) d l -∫ ( 1 υ new g , high -1 υ γ g , high ) d l ≃ A 1 ( 1 ν 2 low -1 ν 2 high ) DM + A 2 ( 1 ν 4 low -1 ν 4 high ) DM , (1) \nwhere A 1 = e 2 2 π mec , A 2 = e 2 c 3 m 2 γ 4 π meh 2 , and e and me are the charge and mass of an electron, respectively. Here DM = ∫ ne d l is the dispersion measure contribution from the plasma, which \nis defined as the integral of the number density of electrons ne along the line of sight. The second term in the last row of Equation (1) is regarded as the time delay arises from the combined effects of plasma and nonzero photon mass, which has the form of ∝ ν -4 that differs from previous results ( ∝ ν -2 ; Wu et al. 2016; Shao & Zhang 2017). \n̸ \nIt is clear that Equation (1) (see also Equation (3) in Wang et al. 2024) presents a contradiction: when the electron number density ne is equal to zero, the time delay is eliminated. This indicates that there is no dispersion for massive photons in vacuum. Figure 1 illustrates the timelines for calculating the time delays in two scenarios: massless photons and massive photons. In the scenario of massless photons in plasma, the observed time delay between low- and high-frequency photons is ∆ t obs = t 2 , low -t 2 , high = ( t 2 , low -t 1 , low) -( t 2 , high -t 1 , high), where t 1 , low = t 1 , high due to the constancy of the light speed c in vacuum. However, for massive photons, the observed time delay is ∆ t ' obs = t ' 2 , low -t ' 2 , high = ( t ' 2 , low -t ' 1 , low ) -( t ' 2 , high -t ' 1 , high ). The additional subtractions of the arrival times of massive photons in vacuum (i.e., t ' 1 , low and t ' 1 , high ) in Wang et al. (2024) cause the elimination of the ν -2 behavior. If we use the correct formula, then the firstorder delay time term would exhibit a frequency dependence of ν -2 , i.e., \n∆ t ' obs = t ' 2 , low -t ' 2 , high = ∫ ( 1 υ new g , low -1 υ new g , high ) d l ≃ A 1 ( 1 ν 2 low -1 ν 2 high ) ( DM + DM γ ) , (2) \nwhere the group velocity υ new g is expressed as \nυ new g = c ( 1 -m 2 γ c 4 h 2 ν 2 ) 1 / 2 ( 1 -ν 2 p ν 2 ) 1 / 2 ≃ c ( 1 -1 2 ν 2 p ν 2 -1 2 m 2 γ c 4 h 2 ν 2 ) , (3) \nwhere ν p = ( nee 2 /π me ) 1 / 2 is the plasma frequency. Analogous to DM, the 'effective dispersion measure' (DM γ ) arising from massive photons is defined as DM γ ≡ π mec 5 m 2 γ h 2 e 2 ∫ d l (Shao & Zhang 2017). Equation (2) shows that the classical dispersion relation ( ∝ ν -2 ) for massive photons propagating in plasma still holds.", '3. THE DISPERSION RELATION DERIVED FROM PROCA EQUATIONS': 'The classical Maxwell equations are founded upon the zero-mass hypothesis of the photon. Proca (1936) first considered the addition of a mass term and modified the Maxwell equations, thereby establishing the Proca equations. The', 'Massless': 'Figure 1. Timeline schematic for calculating the time delays. The blue and red timelines illustrate the scenarios of massless and massive photons, respectively. The definitions of arrival times can be referred to section 2. \n<!-- image --> \nProca Lagrangian is given as (Jackson 2021) \nL Proca = -1 16 π F αβ F αβ -1 c J α A α + µ 2 8 π A α A α . (4) \nIn this Lagrangian, F αβ is the electromagnetic tensor, A α = ( ϕ, A ) is the 4-vector potential, J α = ( cnee , J ) is the 4-vector current, and µ = m γ c / ¯ h is the parameter related to the photon mass. According to the Proca Lagrangian, the equation of motion can be derived as ∂ β F βα + µ 2 A α =4 π J α / c . Combined with the Bianchi identity ∂ ( µ F αβ ) = 0, the vector form of the Proca equations can be written as \n∇· E = 4 π nee -µ 2 ϕ, (5) \n∇× E = -1 c ∂ B ∂ t , (6) \n∇· B = 0 , (7) \n∇× B = 4 π c J + 1 c ∂ E ∂ t -µ 2 A . (8) \nOnce we establish the Proca equations, the dispersion relation of massive photons traveling in plasma can be derived. We first separate the variables into two parts: an equilibrium part indicated by the subscript 0, and a perturbation part indicated by the subscript 1. In a plasma with B 0 = J 0 = E 0 = A 0 = 0, the linear approximation allows Equation (8) to be converted to \nc ∇× B1 = 4 π J1 + ∂ E1 ∂ t -c µ 2 A1 . (9) \nBy taking the time derivative of this equation, we obtain \nc ∇× ∂ B1 ∂ t = 4 π ∂ J1 ∂ t + ∂ 2 E1 ∂ t 2 -c µ 2 ∂ A1 ∂ t . (10) \nThe subsequent steps are to transform each term into the function of E1 . Taking the curl of Equation (6), we have \n∇× ∂ B1 ∂ t = -c ∇× ( ∇× E1 ) = c ∇ 2 E1 -c ∇ ( ∇· E1 ) = c ∇ 2 E1 + c µ 2 ∇ ϕ 1 , (11) \nwhere the condition ∇· E1 = -µ 2 ϕ 1 from Equation (5) is used. Since the current is generated by electron motion, by combining the expressions J1 = nee v1 and me ∂ v1 /∂ t = -e E1 , the time derivative of the current can be expressed as \n∂ J1 ∂ t = -nee 2 me E1 . (12) \nWealso need the relation between potentials ( ϕ 1 , A1 ) and E1 : \n∂ A1 ∂ t = -c E1 -c ∇ ϕ 1 , (13) \nwhich can be derived from the time derivative of B1 = ∇× A1 and Equation (6). We assumed that the perturbation parts oscillate sinusoidally, i.e., ∝ exp[ i ( kx -ω t )], where k and ω are the wave vector and frequency, respectively. Hence, the time derivative and space gradient can be replaced as ∂/∂ t → -i ω and ∇→ ik . Combining Equations (11)-(13), Equation (10) \ncan be rewritten as \n( ω 2 -c 2 k 2 -ω 2 p -c 2 µ 2 ) E1 = 0 , (14) \nwhere ω p = 2 πν p = (4 π nee 2 / me ) 1 / 2 is the plasma frequency. Requiring the terms in the bracket being equal to zero, one has \nω 2 = c 2 k 2 + ω 2 p + c 2 µ 2 . (15) \nThis is the dispersion relation of massive photons propagating in plasma. Furthermore, the group velocity of massive photons can be calculated by \nυ g = ∂ω ∂ k = c ( 1 -ω 2 p + c 2 µ 2 ω 2 ) 1 / 2 ≃ c ( 1 -1 2 ω 2 p ω 2 -1 2 c 2 µ 2 ω 2 ) . (16) \nOne can see from Equation (16) that the velocity modifications from the plasma and nonzero photon mass effects are independent of each other, with the first-order expansion terms being ∝ ν -2 . This is the natural consequence based on the Proca theory. Therefore, the classical dispersion relation that is adopted for photon mass limits remains valid. \nThis work is supported by the NSFC (grant Nos. 12422307 and 12373053). 1 2', 'REFERENCES': 'Bonetti, L., Ellis, J., Mavromatos, N. E., et al. 2016, Physics \nLetters B, 757, 548, doi: 10.1016/j.physletb.2016.04.035 \n-. 2017, Physics Letters B, 768, 326, \ndoi: 10.1016/j.physletb.2017.03.014 \nJackson, J. D. 2021, Classical electrodynamics (John Wiley & \nSons) \nLin, H.-N., Tang, L., & Zou, R. 2023, MNRAS, 520, 1324, \ndoi: 10.1093/mnras/stad228 \nProca, A. 1936, J. Phys. Radium 7, 7, 347 \nRan, J.-Y., Wang, B., & Wei, J.-J. 2024, Chinese Physics Letters, \n41, 059501, doi: 10.1088/0256-307X/41/5/059501 \nShao, L., & Zhang, B. 2017, PhRvD, 95, 123010, \ndoi: 10.1103/PhysRevD.95.123010 Wang, B., Wei, J.-J., Wu, X.-F., & López-Corredoira, M. 2023, JCAP, 2023, 025, doi: 10.1088/1475-7516/2023/09/025 Wang, H., Miao, X., & Shao, L. 2021, Physics Letters B, 820, 136596, doi: 10.1016/j.physletb.2021.136596 Wang, Y.-B., Zhou, X., Kurban, A., & Wang, F.-Y. 2024, ApJ, 965, 38, doi: 10.3847/1538-4357/ad2f99 Wei, J.-J., & Wu, X.-F. 2020, Research in Astronomy and Astrophysics, 20, 206, doi: 10.1088/1674-4527/20/12/206 Wu, X.-F., Zhang, S.-B., Gao, H., et al. 2016, ApJL, 822, L15, doi: 10.3847/2041-8205/822/1/L15 Xing, N., Gao, H., Wei, J.-J., et al. 2019, ApJL, 882, L13, doi: 10.3847/2041-8213/ab3c5f'}

2024MNRAS.534.3761A

We introduce Astrophysical HybridKinetic simulations with the FLASH code inlineformulatexmath idTM0001 notationLaTeXtt AHKASHtexmathinlineformula a new Hybrid particleincell PIC code developed within the framework of the multiphysics code FLASH. The new code uses a secondorder accurate Boris integrator and a predictorpredictorcorrector algorithm for advancing the Hybridkinetic equations using the constraint transport method to ensure that magnetic fields are divergencefree. The code supports various interpolation schemes between the particles and grid cells with postinterpolation smoothing to reduce finite particle noise. We further implement a inlineformulatexmath idTM0002 notationLaTeXdelta ftexmathinlineformula method to study instabilities in weakly collisional plasmas. The new code is tested on standard physical problems such as the motion of charged particles in uniform and spatially varying magnetic fields the propagation of Alfvn and whistler waves and Landau damping of ion acoustic waves. We test different interpolation kernels and demonstrate the necessity of performing postinterpolation smoothing. We couple the TURBGEN turbulence driving module to the new Hybrid PIC code allowing us to test the code on the highly complex physical problem of the turbulent dynamo. To investigate steadystate turbulence with a fixed sonic Mach number it is important to maintain isothermal plasma conditions. Therefore we introduce a novel cooling method for Hybrid PIC codes and provide tests and calibrations of this method to keep the plasma isothermal. We describe and test the hybrid precision method which significantly reduces by a factor inlineformulatexmath idTM0003 notationLaTeXsim 1.5texmathinlineformula the computational cost without compromising the accuracy of the numerical solutions. Finally we test the parallel scalability of the new code showing excellent scaling up to 10000 cores.

2024-11-01T00:00:00Z

['10.1093/mnras/stae2188', '10.48550/arXiv.2409.12151', '2024arXiv240912151A', 'arXiv:2409.12151', '2024MNRAS.tmp.2132C', '2024MNRAS.534.3761A']

['Astrophysics - Instrumentation and Methods for Astrophysics', 'Astrophysics - Earth and Planetary Astrophysics', 'Physics - Computational Physics']

AHKASH a new Hybrid particleincell code for simulations of astrophysical collisionless plasma

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https://arxiv.org/pdf/2409.12151.pdf

{'AHKASH : a new Hybrid particle-in-cell code for simulations of astrophysical collisionless plasma': 'Radhika Achikanath Chirakkara, \n1 \n★ \n1 \n, \n2 \n<!-- image --> \n<!-- image --> \n1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2 Australian Research Council Centre of Excellence in All Sky Astrophysics (ASTRO3D), Canberra, ACT 2611, Australia \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': "Weintroduce A strophysical H ybridK inetic simulations with the flASH code ( AHKASH ) - a new Hybrid particle-in-cell (PIC) code developed within the framework of the multi-physics code FLASH . The new code uses a second-order accurate Boris integrator and a predictor-predictor-corrector algorithm for advancing the Hybrid-kinetic equations, using the constraint transport method to ensure that magnetic fields are divergence-free. The code supports various interpolation schemes between the particles and grid cells, with post-interpolation smoothing to reduce finite particle noise. We further implement a 𝛿 𝑓 method to study instabilities in weakly collisional plasmas. The new code is tested on standard physical problems such as the motion of charged particles in uniform and spatially varying magnetic fields, the propagation of Alfvén and whistler waves, and Landau damping of ion acoustic waves. We test different interpolation kernels and demonstrate the necessity of performing post-interpolation smoothing. We couple the TurbGen turbulence driving module to the new Hybrid PIC code, allowing us to test the code on the highly complex physical problem of the turbulent dynamo. To investigate steady-state turbulence with a fixed sonic Mach number, it is important to maintain isothermal plasma conditions. Therefore, we introduce a novel cooling method for Hybrid PIC codes and provide tests and calibrations of this method to keep the plasma isothermal. We describe and test the 'hybrid precision' method, which significantly reduces (by a factor ∼ 1 . 5) the computational cost, without compromising the accuracy of the numerical solutions. Finally, we test the parallel scalability of the new code, showing excellent scaling up to 10,000 cores. \nKey words: methods: numerical - plasmas - turbulence", '1 INTRODUCTION': "Manyastrophysical plasmas, such as the solar wind (Verscharen et al. 2019), hot accretion flows in the galactic center (Event Horizon Telescope Collaboration et al. 2019), and the hot intracluster medium of galaxy clusters (Schekochihin & Cowley 2006; Kunz et al. 2022) are weakly collisional or collisionless plasmas, and their evolution is coupled with magnetic fields of dynamically important strength. The collisionality of a plasma can be determined from the ratio of the Coulomb mean free path ( 𝜆 mfp ) to the characteristic length scale of the system ( 𝐿 ). For example, the hot interstellar medium is characterised by 𝜆 mfp / 𝐿 ∼ 10 -3 -10 -2 , which can be suitably modelled as a collisional plasma (Ferrière 2020), where the magnetohydrodynamical (MHD) approximation holds reasonably well. For the solar wind, 𝜆 mfp / 𝐿 ∼ 1, for hot accretion flows, 𝜆 mfp / 𝐿 ∼ 1, and the hot intracluster medium, 𝜆 mfp / 𝐿 ∼ 0 . 1, making these plasmas weakly collisional or collisionless. Such astrophysical systems are interesting laboratories to study the nature of magnetized collisionless and weakly collisional plasma and it is important to develop numerical schemes that can suitably model such systems, which is the aim of this work. \nMagnetohydrodynamic (MHD) codes have been used extensively to model magnetized astrophysical gas (Fryxell et al. 2000; Dubey \nChristoph Federrath, \net al. 2008a; Mignone et al. 2007; Stone et al. 2008; Brandenburg & Dobler 2010; Stone et al. 2020; Pencil Code Collaboration et al. 2021). Assuming gas as a conducting fluid, these codes solve the continuity equation, Navier Stokes equation, induction equation for magnetic field evolution and energy equation for a given equation of state. MHD numerical simulations have become an established method to model astrophysical plasma in the last few decades. However, when the plasma is weakly collisional or collisionless, the continuum limit of the fluid equations breaks down and MHD is no longer valid for such plasmas. In such situations, we resort to a kinetic treatment of the plasma (Kulsrud 2005), which is our approach for this work. \nKinetic codes solve the Vlasov or the Vlasov-Landau equation (Birdsall & Langdon 1991), to study the evolution of the ion distribution functions, along with Maxwell's equations. Numerically this can be achieved in many ways, including solving the six-dimensional (three spatial + three velocities) equations for the distribution function (Ganse et al. 2023) or using a particle-in-cell (PIC) approach to sample the moments of the distribution function while using a threedimensional spatial computational grid (Spitkovsky 2005; Cerutti et al. 2013; Shalaby et al. 2017; Derouillat et al. 2018; Nättilä 2022). Here we focus on the PIC method. The PIC method can simulate the physics on the ion and electron time and length scales, but doing so makes this approach computationally expensive and challenging. Moreover, it can be very difficult to run these simulations for larger physically interesting time and length scales, as the scale separa- \non between physical processes is often large. For example, in the intracluster medium, the dynamic length scale ∼ 10 kpc, the ion Larmor scale ∼ 10 -12 kpc and the electron Larmor scale ∼ 10 -14 kpc, making PIC codes infeasible for certain physical problems. \nAnother numerical complication is modelling turbulence which is prevalent in most astrophysics systems. Turbulence is also present in weakly collisional astrophysical environments like the hot intracluster medium (ICM) of galaxy clusters and solar wind. Turbulence in such environments may be driven by merger events, feedback from active galactic nuclei, or by in-falling galaxies stirring the ICM (Simionescu et al. 2019). Observational signatures of turbulence in such environments have been studied by Hitomi Collaboration et al. (2016); Gatuzz et al. (2022a,b, 2023). Regions of the ICM can also host subsonic or supersonic turbulence of rotational or irrotational nature (Zinger et al. 2018). This is also true of collisionless astrophysical plasma like the solar wind (Howes et al. 2014; Bruno & Carbone 2013; Klein & Howes 2015; Verscharen et al. 2019). \nWe adopt the middle ground approach of solving the Hybridkinetic equations using so-called HybridPIC in our new code AHKASH 1 , which builds on the architecture of the FLASH code (Fryxell et al. 2000; Dubey et al. 2008a,b). In Hybrid PIC, positively charged ions are modelled as particles, which sample the moments of the ion distribution function, while the electrons are modelled as a mass-less fluid 2 (Horowitz et al. 1989; Lipatov 2002; Bagdonat & Motschmann 2002; Gargaté et al. 2007; Müller et al. 2011; Kunz et al. 2014b; Le et al. 2023). We note that we solve the Hybrid-kinetic equations in the non-relativistic limit (the ratio of typical speeds to the speed of light in the hot ICM ≲ 10 -3 ). With this approach, we capture the physics at the ion scales, like the Larmor gyration of ions and the damping of ion acoustic waves, without having to encapsulate electron scale physics, such as Debye oscillations. This also significantly reduces the computational cost of performing these simulations relative to the full PIC approach, while still being more computationally expensive than MHD simulations, but allowing us to model the physics of weakly collisional plasmas. \nThe viscous heating of plasma due to turbulence, and the thermodynamics of weakly collisional plasma are different from collisional MHD plasma (Howes 2010; Kunz et al. 2011; Squire et al. 2023a). It is therefore important to study plasma turbulence and heating of collisionless plasmas like the solar wind, by using PIC methods. For example, recent studies use Hybrid PIC to study the heating of ions and electrons in the solar wind (Arzamasskiy et al. 2019; Squire et al. 2023b). The Hybrid PIC approach also enables the study of magnetic fields in a turbulent, magnetized, weakly collisional plasma (St-Onge & Kunz 2018; Achikanath Chirakkara et al. 2024). While Rincon et al. (2016) solve the Vlasov equation in their numerical experiments, St-Onge & Kunz (2018) and Achikanath Chirakkara et al. (2024) use Hybrid PIC simulations. We note that the numerical experiments described in Achikanath Chirakkara et al. (2024) have been performed using AHKASH . \nTurbulence affects the thermodynamics of the plasma and controls the evolution of magnetic fields. Therefore, to properly understand the evolution of weakly collisional astrophysical systems, it is important to numerically model turbulence. Several kinetic codes have been used to study turbulence in weakly collisional or collisionless plasma \n1 AHKASH is an acronym for Astrophysical Hybrid-Kinetic simulations with the flASH code. AHKASH (pronounced as ¯ak¯aś) means 'the sky' in many Indian languages such as Hindi, Malayalam, and Gujarati. \n(Spitkovsky 2005; Kunz et al. 2014b). Turbulence dissipates energy and heats the plasma which makes it difficult to maintain isothermal conditions, especially for controlled numerical experiments. Here we aim to accurately model the effects of turbulence in collisionless plasma with our code using an Ornstein-Uhlenbeck process (Federrath et al. 2010, 2022), especially to maintain the isothermal nature of plasma via the novel cooling in the presence of plasma heating due to turbulence. \nWeuse the second-order accurate predictor-predictor-corrector integration scheme to update the particles and electromagnetic fields in our code, similar to Kunz et al. (2014b). We use the Boris integrator (Boris 1970) and the constrained transport method to update the magnetic field while keeping it divergence-free (Yee 1966). We also implemented a 𝛿 𝑓 method to enable the study of instabilities in collisonless plasma. We study the effect of using different interpolation kernels, performing smoothing operations and varying the number of particles per grid cell in our numerical simulations. Finally, we have implemented a new plasma cooling method, which can be used to maintain a steady sonic Mach number in driven fully-developed turbulence in weakly collisional plasmas, even when such plasmas experience strong local compression and/or rarefaction. This method is important for studying turbulence in collisionless plasma, particularly in the supersonic regime. \nThe rest of this paper is organised as follows. We describe the Hybrid-kinetic equations and the related numerical methods we use in Sec. 2. We test the accuracy of our new Hybrid PIC code with various standard test problems in Sec. 3. We discuss the interpolation tests in Sec. 4 and describe the ion cooling method we developed alongside numerical tests in Sec. 5. We discuss the 'hybrid precision' method, performance of the code and present scaling tests in Sec. 6. Finally, we summarize our study in Sec. 7.", '2 METHODS': 'In this section, we present the equations and methods used in AHKASH . Sec. 2.1 introduces the equations that govern the motion and coupling of positively-charged ions (particles) with an electron fluid in Hybrid-kinetic plasmas. Sec. 2.2 describes the operations that the code performs in each time step. To track ion trajectories, we use the Boris integrator, described in Sec. 2.3. The particle-grid interpolation and smoothing operations are explained in Sec. 2.4. The 𝛿 𝑓 method is introduced in Sec. 2.5, and the magnetic field update is described in Sec. 2.6. We discuss the predictor-predictor-corrector integration scheme in Sec. 2.7, with the corrections to the interpolated electromagnetic fields explained in Sec. 2.8, and higher-order numerical hyper-resistivity discussed in Sec. 2.9. Finally, we describe the turbulent driving in Sec. 2.10, and the novel cooling method, introduced to overcome plasma heating, in Sec. 2.11. Time-step constraints are discussed in Sec. 2.12.', '2.1 Hybrid-kinetic equations': "The evolution of a weakly collisional plasma is governed by the Vlasov equation, which ensures the conservation of phase-space den- \nsity, \nd 𝑓 i ( 𝑡, r , v ) d 𝑡 = 0 , 𝜕 𝑓 i 𝜕𝑡 + 𝜕 r 𝜕𝑡 · 𝜕 𝑓 i 𝜕 r + 𝜕 v 𝜕𝑡 · 𝜕 𝑓 i 𝜕 v = 0 , 𝜕 𝑓 i 𝜕𝑡 + v · 𝜕 𝑓 i 𝜕 r + 𝑞 i 𝑚 i ( E + v × B ) · 𝜕 𝑓 i 𝜕 v = 0 , (1) \nwhich describes the evolution of the ion distribution function, 𝑓 i ( 𝑡, r , v ) , where 𝑡 is time, r is the position vector, and v is the velocity vector 3 . The electromagnetic force on the ions is described by the Lorentz force, ( 𝑞 i / 𝑚 i )( E + v × B ) , where 𝑞 i and 𝑚 i are the charge and mass of the ions, respectively. E and B are the electric and magnetic fields experienced by the ions. We solve Eq. 1 using the PIC approach, for which we can write the equations of motion for the ions, \nd r 𝑝 d 𝑡 = v 𝑝 , (2) \n𝑝 \nd \nv \n𝑡 \nd \n= \n𝑞 \ni \n𝑚 \ni \nE \n+ \nv \n𝑝 \n× \nB \n, \n(3) \nwhere r 𝑝 and v 𝑝 are the position and velocity of the 𝑝 th particle, respectively, and 𝑝 = 1 , 2 , . . . , 𝑃 , where 𝑃 is the total number of particles. We note that this model can be easily expanded for multiple positively-charged ion species, however, here we only consider protons. \nThe evolution of the electrons is described by the Vlasov-Landau equation (similar to Eq. 1, but now with a collision operator on the right-hand side), \n𝜕 𝑓 e 𝜕𝑡 + v · 𝜕 𝑓 e 𝜕 r + 𝑞 e 𝑚 e ( E + v × B ) · 𝜕 𝑓 e 𝜕 v = 𝐶 [ 𝑓 e ] , (4) \nwhere 𝑓 e ( 𝑡, r , v ) , 𝑞 e, and 𝑚 e, are the electron distribution function, charge, and mass of the electrons, respectively, and 𝐶 [ 𝑓 e ] is the electron collision operator. \nPerforming an expansion of the electron distribution function in powers of the electron-to-ion mass ratio ( 𝑚 e / 𝑚 i ) 1 / 2 , we can simplify the Vlasov-Landau equation (Appendix A1.1 of Rosin et al. (2011)). Up to the lowest orders in this expansion, one finds that the zeroth-order distribution function is Maxwellian as shown in Appendix A1.3 of Rosin et al. (2011). Multiplying the expression obtained from Eq. 4 up to order ( 𝑚 e / 𝑚 i ) 1 / 2 by 𝑚 e 𝑣 , integrating over velocity space and using that the zeroth-order electron distribution function is Maxwellian, as detailed in Appendix A1.5 of Rosin et al. (2011), we obtain the following the Ohm's law for the electric field with resistivity (index 'nores'), \nE nores = -( J e × B ) 𝜌 e + ∇ 𝑝 e 𝜌 e , (5) \nwhere J e, 𝜌 e, and 𝑝 e, are the electron current, electron charge density, and electron pressure, respectively. The electron collision operator is modelled using Ohmic resistivity, which gives us \nE = -( J e × B ) 𝜌 e + 𝑘 B 𝑇 e ∇ 𝑛 e 𝜌 e + 𝜂𝜇 0 J , (6) \nwhere J is the total current and 𝜂 is the magnetic diffusivity. This additional term acts as a sink for magnetic energy. 𝜇 0 is the magnetic permeability constant. We have also used the ideal gas equation of state in Eq. 6 with a constant temperature for the electrons, \n3 Note that 𝜕𝑓 𝜕 r = GLYPH<16> 𝜕𝑓 𝜕𝑥 , 𝜕𝑓 𝜕𝑦 , 𝜕𝑓 𝜕𝑧 GLYPH<17> , and 𝜕𝑓 𝜕 v = GLYPH<16> 𝜕𝑓 𝜕𝑣𝑥 , 𝜕𝑓 𝜕𝑣𝑦 , 𝜕𝑓 𝜕𝑣𝑧 GLYPH<17> . \nGLYPH<0> \nGLYPH<1> \n𝑝 e = 𝑛 e 𝑘 B 𝑇 e, where 𝑛 e, 𝑇 e, and 𝑘 B , are the electron number density, constant electron temperature, and the Boltzmann constant, respectively. The total current is the sum of the ion current ( J i ) and the electron current ( J e), \nJ = J i + J e . (7) \nThe charge density of the ions, 𝜌 i , can be obtained from the zeroth moment of the distribution function and can be written as \n𝜌 i = 𝑞 i ∫ 𝑓 i 𝑑 3 v , (8) \nand assuming quasi-neutrality of the plasma, we have \n𝜌 i + 𝜌 e = 0 , (9) \nwhich implies that the charge density of the ions and electrons is the same, 𝜌 i = -𝜌 e. \nThe first moment of the distribution function corresponds to the mean (or bulk) velocity of the ions, which defines the ion current, \nJ i = 𝑞 i ∫ v 𝑓 i 𝑑 3 v . (10) \nUsing this and 𝜌 e = 𝑞 e 𝑛 e in Eq. 6, the Ohm's law can be re-written as \nE = ( J -J i ) × B 𝜌 i -𝑘 B 𝑇 e 𝑞 e ∇ 𝜌 i 𝜌 i + 𝜇 0 𝜂 J . (11) \nFinally, the evolution of the positively-charged ions and electrons and the Ohm's law (Eq. 11) are coupled to the following Maxwell's equations, \nJ = ∇ × B 𝜇 0 , (12) \nwith the magnetic field calculated from Faraday's law, \n𝜕 B 𝜕𝑡 = -∇ × E , (13) \nand the solenoidal condition for the magnetic field, \n∇ · B = 0 . (14)", '2.2 Hybrid PIC algorithm': 'We solve the Hybrid-kinetic equations described in Sec. 2.1 using the PIC method. Particles are used to sample the moments of the ion distribution function, 𝑓 i . In this section, we go through a simplified description of what the code does in one computational time step, Δ 𝑡 . A schematic diagram of this simplified algorithm is shown in Fig. 1 and the computational loop is described below. \n- (i) Particle evolution: In the presence of electric and magnetic fields, the ions are accelerated by the Lorentz force, 𝑞 i / 𝑚 i ( E nores + v × B ) , where E nores is the electric field experienced by the ions, excluding resistive effects. The resistive term is excluded here to ensure the conservation of momentum of the overall system (see Sec. 4.5.2 of Lipatov (2002) and Bagdonat (2004)). The particle velocities and positions are updated based on this acceleration and velocity, respectively. More details on the particle evolution are provided in Sec. 2.3.\n- (ii) Depositing moments of the distribution function onto the computational grid: The ion charge density and current are obtained from the zeroth and first moments of the distribution function, using Eq. 8 and Eq. 10. These moments are sampled by the particles and deposited onto the grid, using an interpolation method (see Sec. 2.4 and Sec. 2.5).', '(i) Update particle positions and velocities': "Figure 1. Simplified schematic of the Hybrid particle-in-cell (Hybrid PIC) algorithm for one time step Δ 𝑡 (see Sec. 2.2). (i) The particle positions and velocities are updated based on the velocity and acceleration from the Lorentz force, respectively (see Sec. 2.1 and Sec. 2.3). (ii) Following this, the updated charge density, 𝜌 i, and the ion current, J i, are deposited onto the computational grid using an interpolation operation (see Sec. 2.4 and Sec. 2.5). (iii) The charge density and ion current are source terms in the Ohm's law, Eq. 34, which is used to compute the electric field, E , on the grid. (iv) The magnetic field, B , is then calculated using Faraday's law, Eq. 13, on the grid (see Sec. 2.6). Finally, the updated electric and magnetic fields are interpolated from the grid to the particles for the next particle evolution step. \n<!-- image --> \n(iii) Computing the electric field (Ohm's law): The charge density and current on the grid are used in the Ohm's law, Eq. 11. The total current is calculated at this stage using Ampère's law, Eq. 12 (see Sec. 2.1). \n(iv) Computing the magnetic field (Faraday's law): The magnetic field update on the grid is described in more detail in Sec. 2.6. The E and B fields are now interpolated from the grid to the particles for the next cycle. \nOur code uses the predictor-predictor-corrector integration algorithm (Kunz et al. 2014b), which is more sophisticated and involved than the simplified description above, and is second-order accurate, capable of stably propagating Alfvén and whistler waves. We provide details on the full scheme in Sec. 2.7. The time step, Δ 𝑡 , is computed as the minimum of several particle and wave time steps, as discussed in detail in Sec. 2.12.", '2.3 Particle integrator': "To update the particle positions and velocities, we solve the equations of motion, Eq. 2 and 3, using the Boris integrator. The Boris integrator is commonly used in PIC codes for its ability to accurately evolve particle dynamics for a large number of time steps (Boris 1970; Qin et al. 2013). The velocity update in one time step, Δ 𝑡 , can be denoted as v 𝑡 → v 𝑡 + Δ 𝑡 , where v 𝑡 is the velocity at time 𝑡 and v 𝑡 + Δ 𝑡 is the velocity at time 𝑡 + Δ 𝑡 . Then, v 𝑡 + Δ 𝑡 is calculated via \nv -= v 𝑡 + 𝑎 e E 𝑡 + Δ 𝑡 / 2 nores (15) \nv + = v -+ 𝑎𝑏 v ' × e B 𝑡 + Δ 𝑡 / 2 (17) \nv ' = v -+ 𝑎 v -× e B 𝑡 + Δ 𝑡 / 2 (16) \nv 𝑡 + Δ 𝑡 = v + + 𝑎 e E 𝑡 + Δ 𝑡 / 2 nores , (18) \nwhere v -, v ' and v + are intermediate time-step solutions for the velocity. The coefficients 𝑎 and 𝑏 are calculated each time-step and defined as 𝑎 = 𝑞 i /( 2 𝑚 i ) Δ 𝑡 and 𝑏 = 2 /( 1 + 𝑎 2 e B 2 ) . e E 𝑡 + Δ 𝑡 / 2 nores and e B 𝑡 + Δ 𝑡 / 2 are the electric and magnetic fields interpolated to the particles at the intermediate time-step, 𝑡 + Δ 𝑡 / 2 (how these are obtained is described in Sec. 2.7). As shown above, the scheme splits the electric field evolution into two steps. Eq. 15 describes the first electric field acceleration step at the beginning of the scheme. The magnetic field evolution is done in between the electric field updates, in the form of two rotations as shown by Eq. 16 and 17 (see fig. 4 in Boris 1970). The final electric field advancement is done in Eq. 18, and the velocity update is complete. \nThe update of the particle positions, r 𝑡 → r 𝑡 + Δ 𝑡 , is carried out as \nr 𝑡 + Δ 𝑡 = r 𝑡 + Δ 𝑡 2 ( v 𝑡 + v 𝑡 + Δ 𝑡 ) . (19)", '2.4.1 Interpolation': "The PIC method requires particle-to-grid and grid-to-particle interpolations (see Fig. 1). The grid is used to evolve the electromagnetic fields, while the particles represent the ions, which exhibit Lagrangian motion governed by the electric and magnetic fields through the equations of motion, Eq. 2 and 3. \nIn general, the interpolation operation provides us with the interpolated scalar quantity Q at position r , based on a quantity 𝑄 at positions r 𝑙 , via \nQ( r ) = 𝐿 ∑︁ 𝑙 = 1 𝑄 ( r 𝑙 ) 𝑊 ( r -r 𝑙 ) , (20) \nwhere 𝑊 ( r -r 𝑙 ) is the weight (interpolation kernel) function, and \n𝑙 = 1 . . . 𝐿 denotes the index of all 𝐿 particles (cells) within the kernel, for interpolations from the particles (grid cells) to the grid (particles). We describe the nearest-grid-point (NGP), cloud-in-cell (CIC), and the triangular-shaped-cloud (TSC) weight functions in Appendix A. For vector quantities, Eq. (20) is performed componentwise. \nFor grid-to-particle (G → P) interpolations, r 𝑙 = r 𝑔 and r = r 𝑝 in Eq. 20. In this case, 𝑄 ( r 𝑙 ) = 𝑄 ( r 𝑔 ) is the grid quantity to be interpolated to the particles. For example, the electromagnetic fields are interpolated from the centre of the computational grid cells to each particle's position using the same weight function used to perform the P → G interpolations. This is described by Fig. 2b. We note that the default interpolation kernel we use is the CIC weight function. \nFor particle-to-grid (P → G) interpolations, r 𝑙 = r 𝑝 are the particle positions and r = r 𝑔 is the centre of each computational grid cell in Eq. 20. This is shown by Fig. 2a. The quantity 𝑄 ( r 𝑙 ) = 𝑄 ( r 𝑝 ) may be further expressed as a product, 𝑄 ( r 𝑝 ) = 𝑞 𝑝 v 𝑚 𝑝 , where 𝑞 𝑝 and v 𝑝 are the charge and the velocity of the 𝑝 th particle, respectively, and 𝑚 = 0 , 1 , 2 , ... corresponds to the 𝑚 th moment of the ion distribution function. The most common application of P → G interpolations is to compute the charge density, which is the zeroth moment ( 𝑚 = 0), 𝜌 i = Q( r )/ Δ 𝑉 , where Δ 𝑉 = Δ 𝑥 Δ 𝑦 Δ 𝑧 , based on the charge of each particle, 𝑄 ( r 𝑝 ) = 𝑞 𝑝 . Likewise, the first moment ( 𝑚 = 1) represents the ion current, J i = Q( r )/ Δ 𝑉 , by setting 𝑄 ( r 𝑝 ) = 𝑞 𝑝 v 𝑝 . \nThe influence of using different kernels (NGP, CIC, TSC) is quantified in Sec. 4.", '2.4.2 Post-interpolation smoothing to reduce particle noise': 'As described above, we sample the moments of the ion distribution function using averages over a population of particles. However, since one can only use a finite number of particles, the distribution function can only be sampled to a certain accuracy, and therefore finite particle counts always introduce noise in any of the particleto-grid interpolated quantities. This can be particularly problematic for the charge density, as the particle noise may introduce spurious electric fields, due to the fact that the thermo-electric term in Eq. (11) generates electric fields based on gradients in the electron pressure (which is proportional to the charge density). Thus, noise in the charge density may lead to large spurious gradients. To address this issue, we perform a smoothing operation on the grid for the ion charge density and current, every time these moments are deposited from the particles to the grid. \n𝐷 ( 𝑖, 𝑗, 𝑘 ) → 𝐷 ( 𝑖 -1 , 𝑗 , 𝑘 ) 4 + 𝐷 ( 𝑖, 𝑗, 𝑘 ) 2 + 𝐷 ( 𝑖 + 1 , 𝑗 , 𝑘 ) 4 , (21) \nThe smoothing (filtering) operation is defined such that a P → G interpolated quantity, 𝐷 , at cell index ( 𝑖, 𝑗, 𝑘 ) is transformed as \nin the 𝑖 th spatial direction. In case of a 3D simulation, this operation is performed in the 𝑖 th , 𝑗 th and 𝑘 th direction, consecutively. This means that during a single smoothing operation, cell ( 𝑖, 𝑗, 𝑘 ) passes 1/2 of its value to its left and right neighbours, and receives 1/4 of the value of its left and right neighbour, respectively, therefore conserving the overall sum. \nThe smoothing operation (Eq. 21) can be repeated 𝑁 smooth times to achieve a smoother and smoother representation of the interpolated quantity, 𝐷 ( 𝑖, 𝑗, 𝑘 ) . We investigate the appropriateness of the smoothing operation in Sec. 4 below. We find that using 2 smoothing passes ( 𝑁 smooth = 2) avoids excessive smoothing, retaining smallscale detail, and at the same time provides an appropriate reduction of particle noise, at least in applications of turbulent dynamo amplification. We note that the details of required smoothing passes \nmay depend on the specific problem and on the affordable number of particles per cell (with tests provided in Sec. 4).', '2.5 Studying perturbations with Hybrid particle-in-cell simulations': 'The 𝛿 𝑓 method can be used to study perturbations and instabilities in collisionless plasmas (Kunz et al. 2014a,b). In this method, the particles are used to sample the difference between the full distribution function, 𝑓 , and the zeroth-order equilibrium distribution function, 𝑓 0 , \n𝛿 𝑓 = 𝑓 -𝑓 0 . (22) \nFirst, the ion charge density is deposited onto the grid as \n𝜌 i ( r 𝑔 ) = 𝜌 0 ( r 𝑔 ) + ˜ 𝜌 i ( r 𝑔 ) , (23) \nwhere 𝜌 0 ( r 𝑔 ) is the equilibrium charge density, and ˜ 𝜌 i ( r 𝑔 ) is obtained by a P → G interpolation with 𝑄 ( r 𝑝 ) = 𝑞 𝑝 𝑤 𝑝 in Eq. 20, where 𝑤 𝑝 is the factor the particle moments are weighted with in the 𝛿 𝑓 method (Parker & Lee 1993; Denton & Kotschenreuther 1995). The ion current is deposited as \nJ i ( r 𝑔 ) = 𝜌 0 ( r 𝑔 ) 𝑣 0 ( r 𝑔 ) + ˜ J i ( r 𝑔 ) , (24) \nwhere 𝑣 0 ( r 𝑔 ) is the equilibrium mean velocity, obtained from 𝑓 0 . ˜ J i ( r 𝑔 ) is obtained by a P → Ginterpolation with 𝑄 ( r 𝑝 ) = 𝑞 𝑝 v 𝑝 𝑤 𝑝 in Eq. 20. A common choice for the equilibrium distribution function is the Maxwell distribution function, for which 𝜌 0 ( r 𝑔 ) = 𝜌 0 = 𝑞 i 𝑛 0 , where 𝑛 0 is the mean number density, and 𝑣 0 ( r 𝑔 ) = 0. The 𝛿 𝑓 method is useful to study the plasma when 𝛿 𝑓 ≪ 𝑓 . However, as the perturbations grow and 𝛿 𝑓 ∼ 𝑓 , the 𝛿 𝑓 method is no longer suitable to model the plasma, which is usually the case when turbulence is present.', '2.6 Constrained transport and solenoidality of magnetic fields': "To update the magnetic fields we use the constrained transport (CT) method on the three-dimensional computational grid to solve Faraday's law, Eq. 13 (Yee 1966). Fig. 3 shows a schematic of a single grid cell with the location of the ion charge density, ion current, and electric and magnetic fields. The charge and current density are sampled from the particles and deposited at the centre of each grid cell as 𝜌 i ( 𝑖, 𝑗, 𝑘 ) and J i ( 𝑖, 𝑗, 𝑘 ) , respectively. We note that ( 𝑖, 𝑗, 𝑘 ) represents the coordinate of the cell centre denoted as r 𝑔 , which is described in Sec. 2.4.1. \nThe magnetic fields are calculated on the face-center of the grid cell and represent the face-averaged value of the magnetic fields. The 𝑥 -component of the magnetic field is calculated on the face of the grid cell parallel to the 𝑦 -𝑧 plane, 𝐵 𝑥 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 ) . The magnetic field's 𝑦 and 𝑧 -components are defined similarly to the 𝑥 -component. These points are depicted by the grey circles in Fig. 3. \nIn the following, we discretize the electromagnetic fields on the grid, where 𝑖 → 𝑖 + 1 represents 𝑥 → 𝑥 + Δ 𝑥 , where Δ 𝑥 is the grid cell length along the 𝑥 direction, and analogously for 𝑗 and 𝑘 in the 𝑦 and 𝑧 direction, respectively. The electric fields, E , required to update the magnetic fields, B , are calculated on the edge centres of the grid cell using the Ohm's law, Eq. 34. The 𝑥 -component of the electric field is calculated on the edge centre parallel to the 𝑥 -axis, 𝐸 𝑥 ( 𝑖, 𝑗 -1 / 2 , 𝑘 -1 / 2 ) . The 𝑦 and 𝑧 -components of the electric field are defined in analogy to the 𝑥 -component. These are shown by the blue squares on the edge centre of the grid cell in Fig. 3. \nNext, we discretize Faraday's law and update the magnetic fields in \n<!-- image --> \n/uni0394 /u1D465 (a) Particle-to-grid (P → G) interpolation \n/uni0394 \n/u1D465 \n<!-- image --> \n(b) Grid-to-particle (G → P) interpolation \n/uni0394 /u1D466 x x This paper has been typeset from a T E X/L A T E X file prepared by the author. Figure 2. Schematic diagram describing (a) particle-to-grid (P → G) and (b) grid-to-particle (G → P) interpolations with the cloud-in-cell weight function in two-dimensions. The shaded region shows the cloud-in-cell interpolation kernel ( 𝑊 ), which has the same dimensions as the grid cell, centred on the particle position, r 𝑝 . The cell centres, r 𝑔 , are depicted by the cross symbols. The arrows show the direction of the interpolation (a) from the particles to the grid cell centres and (b) from the grid cells to the particle (see Sec. 2.4.1 for further details). \n/uni0394 /u1D466 /u1D44A ( r /u1D45D -r /u1D454 ) time. In one time step, i.e., 𝑡 → 𝑡 + Δ 𝑡 , the evolution of the magnetic field components is given by \nr \n/u1D45D \n/uni0394 /u1D465 r /u1D454 x x 𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 ) = 𝐵 𝑡 𝑥 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 ) + Δ 𝑡 Δ 𝑧 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 + 1 / 2 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 -1 / 2 ) GLYPH<3> -Δ 𝑡 Δ 𝑦 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 -1 / 2 , 𝑗 + 1 / 2 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 -1 / 2 , 𝑗 -1 / 2 , 𝑘 ) GLYPH<3> , (25) \npaper has been typeset from a T \nE \nX/L \nA \nT \nE \nlated at the cell centre is thus \n∇ · B 𝑡 + Δ 𝑡 ( 𝑖, 𝑗, 𝑘 ) = 1 Δ 𝑥 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖 + 1 / 2 , 𝑗 , 𝑘 ) -𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 ) GLYPH<3> + 1 Δ 𝑦 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑦 ( 𝑖, 𝑗 + 1 / 2 , 𝑘 ) -𝐵 𝑡 + Δ 𝑡 𝑦 ( 𝑖, 𝑗 -1 / 2 , 𝑘 ) GLYPH<3> + 1 Δ 𝑧 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 + 1 / 2 ) -𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 -1 / 2 ) GLYPH<3> . (28) \nX file prepared by the author. \n/uni2605 𝐵 𝑡 + Δ 𝑡 𝑦 ( 𝑖, 𝑗 -1 / 2 , 𝑘 ) = 𝐵 𝑡 𝑦 ( 𝑖, 𝑗 -1 / 2 , 𝑘 ) + Δ 𝑡 Δ 𝑥 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 + 1 / 2 , 𝑗 -1 / 2 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 -1 / 2 , 𝑗 -1 / 2 , 𝑘 ) GLYPH<3> -Δ 𝑡 Δ 𝑧 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 -1 / 2 , 𝑘 + 1 / 2 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 -1 / 2 , 𝑘 -1 / 2 ) GLYPH<3> , (26) \n© \n𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 -1 / 2 ) = 𝐵 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 -1 / 2 ) + Δ 𝑡 Δ 𝑦 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 + 1 / 2 , 𝑘 -1 / 2 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 -1 / 2 , 𝑘 -1 / 2 ) GLYPH<3> -Δ 𝑡 Δ 𝑥 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 + 1 / 2 , 𝑗 , 𝑘 -1 / 2 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 -1 / 2 , 𝑗 , 𝑘 -1 / 2 ) GLYPH<3> . (27) \nE-mail: publications@ras.ac.uk (KTS) We also require the cell-centred values of the magnetic fields to interpolate these fields to the particle positions. The cell-centered values of the magnetic field at 𝑡 + Δ 𝑡 are obtained by averaging the face-centred values. \n2024 The Authors The divergence of the magnetic field from the CT method calcu- \nDue to the staggered positions of the electric and magnetic fields in the construction of the stencil, we find ∇ · B 𝑡 + Δ 𝑡 ( 𝑖, 𝑗, 𝑘 ) = 0 when Eqs. 25-27 are plugged into Eq. 28. The construction of the CT method ensures that the analytical expression, ∇ · (∇ × C ) = 0, is satisfied for any vector C , and therefore it can be used to ensure the solenoidality of the magnetic field, to machine precision. \nE-mail: publications@ras.ac.uk (KTS) 2024 The Authors For comparison with the CT method, we also describe the cellcentred finite difference method to update magnetic fields in Appendix B, which we show also ensures that the magnetic fields are divergence-free. Similar to the CT method, the cell-centred finite difference method ensures the solenoidality of magnetic fields (although on a larger stencil, which has exactly twice the linear extent of the CT method) using the construction of the numerical stencil. Our code includes both the CT and the cell-centred finite difference method to update magnetic fields, but we use the CT method as the default.", '2.7 Predictor-predictor-corrector integration scheme': "Hybrid PIC codes use different integration schemes to advance the particle positions, velocities, and electromagnetic fields in time. Examples of these are the predictor-corrector method and the cyclic leapfrog (CL) with the current-advance method (CAM-CL) (Holmstrom 2009; Holmström 2011; Winske et al. 2022). Here we use the predictor-predictor-corrector algorithm (Kunz et al. 2014a), which requires two predictions for the electric and magnetic fields at an intermediate time-step, 𝑡 + Δ 𝑡 / 2. The predictor-predictor-corrector method is second-order accurate and can stably propagate Alfvén \nx \nx \nr \n/u1D454 \n/uni0394 \n/u1D465 \nr \n/uni0394 \n/u1D465 \nr \n/u1D45D \n/u1D45D \n/u1D446 \n/u1D446 \n/uni0394 \n/u1D465 \nx \nr \n( \n/u1D454 \nx \n/uni0394 \n/u1D465 \nx \nr \n/u1D45D \nx \nr \n- \nr \n/u1D454 \n- \n/u1D454 \nREFERENCES Figure 3. Staggered construction of the electric and magnetic fields for the constrained transport method to solve Faraday's law, Eq. 13. The electric fields are depicted by the blue squares on the edge centres and the magnetic fields are shown by the grey circles on the face centres of the grid cell. The charge density and ion current are always deposited on the grid-cell centre, shown by the yellow circle. \n<!-- image --> \nThis paper has been typeset from a T \nAs described in Sec. 2.3, to update the particle velocity, v 𝑡 → v 𝑡 + Δ 𝑡 , we require the electromagnetic fields e E 𝑡 + Δ 𝑡 / 2 and e B 𝑡 + Δ 𝑡 / 2 at the intermediate time-step, 𝑡 + Δ 𝑡 / 2. Furthermore, the constrained transport method also requires the electric field, e E 𝑡 + Δ 𝑡 / 2 , to advance the magnetic fields, B 𝑡 → B 𝑡 + Δ 𝑡 , as discussed in Sec. 2.6. Therefore, we need estimates of e E 𝑡 + Δ 𝑡 / 2 and e B 𝑡 + Δ 𝑡 / 2 . \nand whistler waves, as demonstrated in Sec. 3.2. A complete description of the predictor-predictor-corrector integration scheme is provided in Kunz et al. (2014b). \n(KTS) \ne E 𝑡 + Δ 𝑡 / 2 = 1 2 ( E 𝑡 + E 𝑡 + Δ 𝑡 pred , 1 ) , (29) \nThe electric field is calculated from the Ohm's law, Eq. 34, and is a function of the charge density, ion current and magnetic field, E ( 𝜌 i , J i , B ) . In the first predictor step of the predictor-predictorcorrector algorithm, B 𝑡 + Δ 𝑡 pred , 1 is calculated using E 𝑡 . The first prediction for the electric field, E 𝑡 + Δ 𝑡 pred , 1 , is calculated using 𝜌 𝑡 i , J 𝑡 i and B 𝑡 + Δ 𝑡 pred , 1 . Once E 𝑡 + Δ 𝑡 pred , 1 and B 𝑡 + Δ 𝑡 pred , 1 are estimated, the electromagnetic fields at 𝑡 + Δ 𝑡 / 2 are calculated as \ne B 𝑡 + Δ 𝑡 / 2 = 1 2 ( B 𝑡 + B 𝑡 + Δ 𝑡 pred , 1 ) . (30) \nThesecondprediction for the magnetic field, B 𝑡 + Δ 𝑡 pred , 2 , is calculated using e E 𝑡 + Δ 𝑡 / 2 . We advance the particles from 𝑡 → 𝑡 + Δ 𝑡 using e E 𝑡 + Δ 𝑡 / 2 and e B 𝑡 + Δ 𝑡 / 2 to obtain 𝜌 𝑡 + Δ 𝑡 i and J 𝑡 + Δ 𝑡 i , which we use along with B 𝑡 + Δ 𝑡 pred , 2 to predict E 𝑡 + Δ 𝑡 pred , 2 . Advancing the particles is computationally the most expensive step in the algorithm, but using an additional particle-advancement step helps us better estimate the electromagnetic fields directly from the motion of the particles. The quantities e E 𝑡 + Δ 𝑡 / 2 and e B 𝑡 + Δ 𝑡 / 2 are then re-calculated using E 𝑡 + Δ 𝑡 pred , 2 and B 𝑡 + Δ 𝑡 pred , 2 , \nX/L \nA T E similar to Eq. 29-30, \nX file prepared by the author. \ne E 𝑡 + Δ 𝑡 / 2 = 1 2 ( E 𝑡 + E 𝑡 + Δ 𝑡 pred , 2 ) , (31) \ne B 𝑡 + Δ 𝑡 / 2 = 1 2 ( B 𝑡 + B 𝑡 + Δ 𝑡 pred , 2 ) . (32) \nFinally, B 𝑡 + Δ 𝑡 is calculated using e E 𝑡 + Δ 𝑡 / 2 and the particle positions and velocities are updated using e E 𝑡 + Δ 𝑡 / 2 and e B 𝑡 + Δ 𝑡 / 2 .", '2.8 Corrections for interpolated electromagnetic fields': 'In the absence of thermo-electric and resistive effects, the electric and magnetic fields are perpendicular to each other during their calculation on the computational grid. However, when these fields are interpolated to the particles, this may no longer be true due to interpolation errors. To address this issue, we perform the following correction to the interpolated electric field at the particle positions (Lehe et al. 2009; Kunz et al. 2014b; Achikanath Chirakkara et al. 2024), \nE ∗ nores = E nores + [( E nores · B ) G → P -E nores · B ] B | B | 2 , (33) \nwhere all the terms in the above equation are fields interpolated to the particle positions. ( E nores · B ) G → P is the dot product of E nores and B , calculated on the grid and interpolated to the particle positions. E ∗ nores is the corrected electric field, which the particles experience. \nE \n( \nr \n/u1D45D \nr \n/u1D454 \n) \n) \n/uni0394 \n/u1D466 \n/uni0394 \n/u1D466', '2.9 Hyper-resistivity': "Hyper-resistivity is introduced in Hybrid PIC codes to remove the propagation of grid-scale fluctuations by damping magnetic field noise on grid-cell scales (hence the use of a 4th-order spatial derivative). To do this, we add a hyper-diffusivity ( 𝜂 hyper ) term on the right-hand side of the Ohm's law, Eq. 11, \nE = ( J -J i ) × B 𝜌 i -GLYPH<16> 𝑘 B 𝑇 e 𝑞 e GLYPH<17> ∇ 𝜌 i 𝜌 i + 𝜇 0 𝜂 J -𝜇 0 𝜂 hyper ∇ 2 J . (34) \nThe choice for the resistivity and hyper-resistivity depends on the physical problem and the grid resolution (for more details, see Sec. 2.5.2 in Achikanath Chirakkara et al. 2024).", '2.10 Turbulent driving': 'To drive turbulence, we add a turbulent acceleration field f to Eq. 3, \nd v 𝑝 d 𝑡 = 𝑞 i 𝑚 i GLYPH<0> E nores + v 𝑝 × B GLYPH<1> + f . (35) \nThe turbulent acceleration field f is constructed in Fourier space and evolved by an Ornstein-Uhlenbeck process, using TurbGen (Federrath et al. 2010, 2022). We drive turbulence on large scales in our numerical simulations. This turbulent energy injection is restricted to large scales (usually half the computational domain). This energy then self-consistently cascades to smaller and smaller scales, where it is ultimately dissipated as heat (Frisch 1995; Federrath et al. 2021). Thus, the injection of turbulent energy via driving leads to an increase in the temperature of the ions. \nThe turbulent velocity fluctuations excited via driving can be quantified in terms of the sonic Mach number, \nM = 𝑣 turb 𝑣 th , (36) \nwhere 𝑣 turb is the turbulent speed and 𝑣 th is the thermal speed. As a result of energy dissipation on small scales, the temperature increases in the absence of cooling, and therefore, the thermal speed increases and the Mach number of the plasma decreases. This means that statistically-steady turbulence cannot be maintained in the absence of cooling, which is problematic, because the properties of the turbulence are sensitive to the Mach number (Federrath et al. 2011; Seta & Federrath 2021; Achikanath Chirakkara et al. 2021; Seta & Federrath 2022). This problem is even more significant for turbulence in the supersonic regime ( M > 1), where 𝑣 th increases faster compared to the subsonic regime ( M < 1), as more turbulent energy is dissipated in the supersonic case, due to dissipation in shocks.', '2.11 Plasma cooling': "As explained in the previous section, maintaining isothermal plasma conditions is critical for certain physics applications and controlled numerical experiments. Therefore, to maintain an isothermal plasma, we introduce a cooling method, with a schematic diagram shown in Fig. 4. The goal of this cooling method is to achieve a constant (in space and time) target ion thermal speed, \n𝜎 target = 𝑁 𝑑 𝑘 B 𝑇 target 𝑚 i ! 1 / 2 , (37) \nwhere 𝑇 target is the target ion temperature and 𝑁 𝑑 = 1 , 2 or 3, depending on the dimensionality of the problem. We perform this cooling on a user-controlled timescale, ( Δ 𝑡 ) cool (in units of 𝑡 cool ), which is \ntested and calibrated in Sec. 5. When 𝑇 target = 𝑇 e, cooling ensures that electrons and ions have the same temperature. \nWe begin by noting that the total particle velocity can be decomposed as \nv = ( v blk ) G → P + v th , (38) \ni.e., the sum of the local bulk flow velocity, ( v blk ) G → P , and the particle thermal velocity, v th . To cool the plasma, we first perform P → G interpolations using Eq. 20 with 𝑄 ( r 𝑝 ) = 𝑚 i v 𝑝 and 𝑄 ( r 𝑝 ) = 𝑚 i v 2 𝑝 , where both quantities are vectors with each component representing the 3 spatial coordinate directions, i.e., v 𝑝 = ( 𝑣 𝑝,𝑥 , 𝑣 𝑝,𝑦 , 𝑣 𝑝,𝑧 ) and v 2 𝑝 = ( 𝑣 2 𝑝,𝑥 , 𝑣 2 𝑝,𝑦 , 𝑣 2 𝑝,𝑧 ) , respectively. Following this, the two vector quantities are interpolated back from the computational grid to the particle locations as ( 𝑚 i v ) G → P and ( 𝑚 i v 2 ) G → P , respectively. The ion mass density on the computational grid, 𝜌 𝑚 , is also interpolated from the grid to the particle positions (G → P) as ( 𝜌 𝑚 ) G → P using Eq. 20 with 𝑄 ( r 𝑔 ) = 𝜌 𝑚 . These back-and-forth interpolation operations are performed to obtain the bulk velocities and thermal speed at the particle positions. At each particle position, we then have \n( v blk ) G → P = ( 𝑚 i v ) G → P ( 𝜌 𝑚 ) G → P , (39) \nUsing the above, we can calculate \n( v 2 ) G → P = ( 𝑚 i v 2 ) G → P ( 𝜌 𝑚 ) G → P . (40) \nσ G → P = GLYPH<16> ( v 2 ) G → P - ( v blk ) 2 G → P GLYPH<17> 1 / 2 , (41) \nwhere σ G → P is the thermal speed in the 𝑥 , 𝑦 and 𝑧 directions at the particle locations. \nHaving gathered all this information, we are now ready to perform the cooling step. From Eq. (38), each particle's thermal velocity, v th = v - ( v blk ) G → P , which is re-scaled (corrected) by the proportional mismatch between the measured σ G → P from Eq. (41) and the target thermal speed, then the bulk speed is reattached, such that the total particle velocity after the cooling step is set to \nv → v cooled = 𝜎 target | σ G → P | v th + ( v blk ) G → P = 𝜎 target | σ G → P | ( v - ( v blk ) G → P ) + ( v blk ) G → P . (42) \nThe cooling is performed every ( Δ 𝑡 ) cool , in units of 𝑡 cool , which is defined as \nThis re-scaling adjusts the thermal velocity component of each particle, while keeping the direction of the thermal velocity component at each particle position unchanged (i.e., using the magnitude, | σ G → P | of the three σ G → P components in Eq.(42), to maintain a constant target plasma temperature. \n𝑡 cool = 𝑡 th M , (43) \nusing the thermal crossing time, 𝑡 th = Δ 𝑙 / 𝑣 th , where Δ 𝑙 = min ( Δ 𝑥, Δ 𝑦, Δ 𝑧 ) is the minimum side length of a computational grid cell and 𝑣 th is the thermal speed. Cooling is done after the predictorpredictor-corrector integration scheme (see Sec. 2.7) is complete. This method ensures isothermal plasma conditions locally (on the scale of individual grid cells), and by extension also globally.", '2.12 Time-step constraints': 'To resolve the different physical timescales involved in weakly collisional plasmas, the following time-step constraints must be fulfilled: \n<!-- image --> \n<!-- image --> \nCooling \n-----→ \n(b) \nFigure 4. Schematic diagram of the cooling method. A single grid cell with 3 particles is shown for simplicity. (a) Using interpolation operations, the particle velocity is decomposed into the bulk, ( v blk ) G → P, and the particle thermal velocity, v th, as described in Eq. 38. (b) Then v th = v - ( v blk ) G → P, is rescaled to the target value, 𝜎 target, and ( v blk ) G → P reattached (not shown in the schematic), as described in Eq. 42. This method ensures isothermal conditions locally and globally throughout the plasma. \n<!-- image --> \n(i) Larmor gyration: to resolve the Larmor motion of ions in the presence of a magnetic field, one needs to use a time step that is significantly smaller than the timescale required for a charged particle to perform one gyration about the magnetic field, \n𝑡 Larmor = 2 𝜋𝑚 i 𝑞 i | B | , (44) \nwhere 𝑚 i and 𝑞 i are the mass and charge of the ions, respectively. The required time step to resolve Larmor gyration is problem-dependent, and will be tested in Sec. 3.1. \n(ii) Particle speed: the particle speeds set a time-step constraint determined by the time taken by the fastest moving particle in the computational domain to travel across the grid cell of minimum side length Δ 𝑙 = min ( Δ 𝑥, Δ 𝑦, Δ 𝑧 ) . This is given by \n𝑡 part = Δ 𝑙 | v | max , (45) \n(iii) Alfvén waves: the timescale for the propagation of Alfvén waves across a computational grid with cells of minimum side length Δ 𝑙 is \nwhere | v | max is the speed of the fastest particle within the domain. \n𝑡 Alfvén = Δ 𝑙 𝑣 A , (46) \nwith the Alfvén speed 𝑣 A = | B |/ √ 𝜇 0 𝑚 i 𝑛 i , where 𝑛 i and 𝜇 0 are the number density and the vacuum magnetic permeability, respectively. \n(iv) Whistler waves: the timescale for the propagation of whistler waves across a grid cell is \n𝑡 whistler = Δ 𝑙 𝑣 whistler , (47) \nwhere 𝑣 whistler = 𝜋 | B |/( 𝑞 i 𝜇 0 𝑛 i Δ 𝑙 ) is the speed of whistler waves with wavenumber, 𝑘 = 2 𝜋 / 2 Δ 𝑙 . Note that 𝑡 whistler ∝ Δ 𝑙 2 . \n(v) Ohmic resistivity: the magnetic diffusivity of the plasma, 𝜂 , sets the resistive timescale \n𝑡 𝜂 = ( Δ 𝑙 ) 2 𝜂 . (48) \n(vi) Hyper-resistivity: the higher-order numerical diffusivity sets an additional dissipative time step, the hyper-resistive timescale \n𝑡 𝜂 hyper = ( Δ 𝑙 ) 4 𝜂 hyper , (49) \nwhere 𝜂 hyper is the hyper-diffusivity. \n(vii) Thermal crossing: the thermal crossing timescale is the time taken by a sound wave to cross a grid cell, defined as \n𝑡 th = Δ 𝑙 𝑣 th , (50) \nwhere 𝑣 th is the thermal speed of the plasma. \nThe overall time step is calculated as the minimum of the all above time-step constraints, \nΔ 𝑡 = min { 𝑠 L 𝑡 Larmor , 𝑠 part 𝑡 part , 𝑠 A 𝑡 Alfvén , 𝑠 w 𝑡 whistler , 𝑠 𝜂 𝑡 𝜂 , 𝑠 𝜂 hyper 𝑡 𝜂 hyper , 𝑠 th 𝑡 th } , (51) \nwhere 𝑠 L , 𝑠 part, 𝑠 A , 𝑠 w, 𝑠 𝜂 , 𝑠 𝜂 hyper , and 𝑠 th , are the safety factors for the Larmor, particle speed, Alfvén, whistler, resistive, hyper-resistive, and thermal time steps, respectively. We introduce these safety factors to ensure that each time step is sufficiently resolved. The default value for all the safety factors is 0.1, except for the particle-speed time step, which is 𝑠 part = 0 . 5. Depending on the physical problem, these safety factors can be adjusted.', '3 PARTICLE, WAVE, AND WAVE-PARTICLE INTERACTION TESTS': 'To test the accuracy of our code, we perform the following standard tests and compare our numerical results with the analytical solutions. First, we test the Larmor gyration of charged particles in Sec. 3.1. In Sec. 3.2 we test the propagation of Alfvén and whistler waves. Finally, we test whether the code can capture physical instabilities arising from particle-wave interactions in Sec. 3.3.', '3.1 Charged particle in a uniform magnetic field': "We test our implementation of the Boris integrator by studying the motion of a charged particle in a uniform magnetic field. We set up an external (non-evolving) magnetic field, B = 𝐵 0 ˆ 𝑧 , where 𝐵 0 = 10 -8 Tesla, and give the charged particles an initial velocity perpendicular to the direction of the magnetic field, v = 𝑣 0 ˆ 𝑥 , where 𝑣 0 = 1 . 0 m s -1 . Due to the effect of the Lorentz force, the charged particles will gyrate in a circle in the 𝑥 -𝑦 plane with the Larmor radius, 𝑟 Larmor = 𝑚 i 𝑣 0 / 𝑞 i 𝐵 0 , where 𝑚 i and 𝑞 i are the mass and charge of the particle, respectively. The analytical solution for the evolution of the 𝑥 -coordinate of the particle's position is 𝑥 ( 𝑡 ) = 𝑟 Larmor sin ( Ω i 𝑡 ) , \nFigure 5. The 𝑥 -coordinate of a charged particle undergoing gyrations in the presence of a uniform magnetic field, normalised to the Larmor radius ( 𝑟 Larmor), as a function of time in units of the Larmor time ( 𝑡 Larmor). We plot this for simulations with Δ 𝑡 = 0 . 1 𝑡 Larmor for the Boris integrator and a non-conserving particle integrator. The solid black line depicts the analytical solution. We find that the gyration radius of our numerical solutions does not change with the Boris integration method, while with the non-conserving particle integrator, the gyration radius increases over time. \n<!-- image --> \nwhere Ω i = 𝑞 i 𝐵 0 / 𝑚 i is the gyration frequency of the charged particle. The total kinetic energy of the particle does not change as the magnetic field does not do any work on the particle. \nWe pick the time step, Δ 𝑡 = 0 . 1 𝑡 Larmor , where 𝑡 Larmor = 2 𝜋 / Ω i is the time taken by the charged particle to complete one gyration. Fig. 5 shows the 𝑥 -coordinate of the particle, normalised to 𝑟 Larmor , as a function of time, normalised to the Larmor time ( 𝑡 Larmor , defined in Eq. 44), along with the analytical solution. We also perform this test with a non-conserving cyclic leapfrog particle integrator (Holmstrom 2009). \nWefindthat the amplitude (radius) of the Larmor gyration is stable in the Boris integration method, as the energy is conserved up to the machine precision level in this case. However, with a non-conserving particle integrator (dotted red line), the radius of gyration increases over time, due to the particle gaining energy as a result of integration inaccuracies. Thus, we conclude that using the Boris integrator provides the basic requirement for stably modelling Larmor gyration, and thus we use this integration scheme as the default for all PIC simulations. \nWhile the Boris integrator provides stable Larmor (circular) orbits for any choice of time step Δ 𝑡 , for a uniform magnetic field it may not do so for a spatially varying magnetic field, and instead, the choice of time step may play a role. This is tested next, where we follow the motion of a charged particle in a non-uniform magnetic field, \n𝐵 𝑧 ( 𝑥 ) = 𝐵 min GLYPH<26> 2 -exp GLYPH<20> -GLYPH<16> 𝑥 2 Δ 𝑥 GLYPH<17> 2 GLYPH<21> GLYPH<27> , (52) \nwhere 𝑥 is the 𝑥 -coordinate of the grid cell centre, Δ 𝑥 is the cell length in the x-direction and 𝐵 min is the magnetic field strength at 𝑥 = 0. This profile is shown by the colour map in Fig. 6. The charged particle is given an initial velocity in the 𝑥 -direction, similar to the test in Fig. 5, and we plot the results up to 1 𝑡 Larmor , where 𝑡 Larmor is calculated based on the maximum magnetic field strength, 𝐵 max = 2 𝐵 min . We vary the time step, Δ 𝑡 / 𝑡 Larmor = 10 -3 , 10 -2 and 10 -1 , and show the results in Fig. 6. The near-perfect orbit is obtained for a very small time step-here Δ 𝑡 / 𝑡 Larmor = 10 -3 (dashed line). For a relatively \nFigure 6. Orbit of a charged particle in a spatially varying magnetic field, 𝐵 𝑧 ( 𝑥 ) , described by Eq. 52, using the Boris integrator for varying time steps, Δ 𝑡 / 𝑡 Larmor = 10 -3 , 10 -2 , and 10 -1 . Using a relatively large Δ 𝑡 / 𝑡 Larmor = 10 -1 , we find that the trajectory of the particle deviates significantly from the cases that use Δ 𝑡 / 𝑡 Larmor = 10 -2 and 10 -3 . While the required time step depends on the physical problem, this test shows that it is important to pick a time step that can suitably resolve the trajectory of the particles. \n<!-- image --> \nlarge time step, Δ 𝑡 / 𝑡 Larmor = 10 -1 (blue crosses), we see that the orbit is significantly distorted, with a clear orbital precision. Using Δ 𝑡 / 𝑡 Larmor = 10 -2 (green pluses), the trajectory is fairly close to the optimal orbit. This test demonstrates that it is important to choose a suitable time step to achieve the desired accuracy in the trajectories of the charged particles, especially when the magnetic field is nonuniform, which is almost always the case for PIC applications.", '3.2 Propagation of waves in collisionless plasma': 'By using perturbation analysis on the Hybrid-kinetic equations, one can obtain the dispersion relation for the propagation of circularly polarized waves. Right-handed waves follow the dispersion relation 𝜔 2 R = 𝑘 2 ( 1 + 𝜔 R ) andleft-handed waves follow the dispersion relation 𝜔 2 L = 𝑘 2 ( 1 -𝜔 L ) (Kulsrud 2005). We note that in these equations, the frequency, 𝜔 , is normalised to the ion gyration frequency, Ω i , and the wavenumber, 𝑘 , is normalised to the ion inertial length, 𝑑 i . In the low-wavenumber limit, 𝑘 ≪ 1, both the right- and lefthanded waves follow the Alfvén wave dispersion relation, 𝜔 2 = 𝑘 2 . In the high-wavenumber limit, 𝑘 ≫ 1, the dispersion relation for the right-handed waves is that of whistler waves, 𝜔 R = 𝑘 2 , and for the left-handed waves, it is that of the ion-cyclotron waves, 𝜔 L → 1 asymptotically as 𝑘 →∞ . \nTo test this with our numerical code, we initialise a mean magnetic field, 𝐵 0 , along the 𝑥 -axis and oscillatory magnetic fields along the 𝑦 and 𝑧 -axes, 𝐵 𝑦 = 𝛿𝐵 sin ( 𝑘𝑥 ) and 𝐵 𝑧 = 𝛿𝐵 cos ( 𝑘𝑥 ) , where 𝑘 = 2 𝜋 / 𝐿 𝑥 , 𝛿𝐵 = 10 -3 𝐵 0 and 𝐿 𝑥 is the length of the computational domain in the 𝑥 -direction. Our test simulations have cold ions and electrons, 𝑇 i = 𝑇 e = 0. The Ohmic dissipation and hyper-resistivity have been switched off in these tests. We choose 𝑁 ppc = 100 particles per cell and periodic boundary conditions. We also resolve the Alfvén and whistler wave time steps and use a safety factor of 0.1 for both these time steps (see Sec. 2.12 for details). \nFig. 9 shows the squared amplitude of the density fluctuations as a function of time, normalised to the sound-crossing timescale, 𝑡 𝑐 s = 1 /( 𝑘𝑐 s ) , where 𝑐 s is the sound speed based on 𝑇 e. We set the ratio of the ion to electron temperature, 𝑇 i / 𝑇 e = 0 . 1. We find that the density fluctuations decay exponentially with time, by locating the local maxima of the density fluctuations and fitting an exponential decay \n<!-- image --> \ni \nFigure7. Dispersion relationship of Alfvén, whistler and ion-cyclotron waves, i.e., frequency of the propagating wave, 𝜔 , normalised to the ion gyration frequency, Ω i, for different values of the wavenumber 𝑘 (in units of the ion inertial length, 𝑑 i), for a wide range of wavenumbers, 𝑘𝑑 i = [ 0 . 05 , 9 ] with 𝑘 Δ 𝑥 = 2 𝜋 / 64 (i.e. 64 grid cells per wavelength), where Δ 𝑥 is the grid-cell length in the direction of wave propagation. The red points show the numerical results with our new code, AHKASH . The black dashed curves show the analytical solutions for the left-handed and right-handed waves. In the low-wavenumber regime ( 𝑘𝑑 i ≪ 1), the Alfvén wave dispersion relation ( 𝜔 / Ω i = 𝑘𝑑 i) is followed and at higher wavenumbers ( 𝑘𝑑 i ≫ 1), the whistler waves ( 𝜔 / Ω i = ( 𝑘𝑑 i ) 2 ) and the ion-cyclotron waves ( 𝜔 = Ω i) appear. The measured errors in the frequencies are very small ( ≲ 4%). For the tests with 𝑘𝑑 i = 0 . 1 , 0 . 9 and 9, we vary the grid resolution, 𝑘 Δ 𝑥 = 2 𝜋 / 32 and 2 𝜋 / 128 and find that the numerical solutions are converged with 𝑘 Δ 𝑥 = 2 𝜋 / 32 in different 𝑘𝑑 i regimes. We find that our results are in excellent agreement with the analytical solutions and show stable and accurate propagation of waves.', '3.2.1 Wavelength dependence': 'The initial wave propagates across the computational domain and to accurately extract the frequencies of the waves, we perform a Fourier analysis on the time evolution data. We then use a lognormal fitting function to fit the frequency peaks in the data. The frequency we report is the mean of the lognormal distribution and the standard deviation of the distribution is a measure of its uncertainty. In Fig. 7, we show the frequency of the waves normalised to the ion gyration frequency, 𝜔 / Ω i , for different values of the wavenumber normalised to the ion inertial length, 𝑘𝑑 i = [ 0 . 05 , 9 ] . We also plot the analytical solutions for the dispersion relation and find excellent agreement with the analytical solutions across all 𝑘𝑑 i regimes. This shows that our code can stably and accurately capture the propagation of waves. \nWe pick 𝑘 Δ 𝑥 = 2 𝜋 / 64 in our tests, where Δ 𝑥 is the grid-cell length in the direction of wave propagation and 𝑘 = 2 𝜋 / ℓ , where ℓ is the wavelength. This implies there are 64 cells per wavelength, ℓ = 64 Δ 𝑥 . The tests with 𝑘𝑑 i = 0 . 1 , 0 . 9 and 9 are repeated with varying grid resolution, Δ 𝑥 = ℓ / 32 or 𝑘 Δ 𝑥 = 2 𝜋 / 32 and Δ 𝑥 = ℓ / 128 or 𝑘 Δ 𝑥 = 2 𝜋 / 128. In Fig. 7, we show that the numerical results with 𝑘 Δ 𝑥 = 2 𝜋 / 32 are converged in different 𝑘𝑑 i regimes.', '3.2.2 Grid resolution criteria': 'In this section, we study in more detail the effect of the grid resolution on the wave test. We vary the grid-cell length in the direction of wave propagation, Δ 𝑥 = ℓ / 8 ℓ / 128, where ℓ = 2 𝜋 / 𝑘 is the wavelength, and measure the frequency of wave propagation, 𝜔 , normalised to the ion gyration frequency, Ω i , for the test with 𝑘𝑑 i = 0 . 9. The \nFigure 8. The frequency of the propagating wave, 𝜔 , normalised to the theoretical frequency, 𝜔 theory, for the wave test with 𝑘𝑑 i = 0 . 9, as a function of grid resolution, ℓ / Δ 𝑥 , where ℓ = 2 𝜋 / 𝑘 is the wavelength and Δ 𝑥 is the grid-cell length. For tests with Δ 𝑥 = ℓ / 8 and Δ 𝑥 = ℓ / 16, the analytical solutions are not obtained for the right-handed waves. As the grid resolution is increased Δ 𝑥 ≳ ℓ / 32, the analytical solutions are recovered, and the numerical solutions are converged. \n<!-- image --> \nresults are shown in Fig. 8. We find that with ℓ = ( 32 - 128 ) Δ 𝑥 , the analytical solutions are recovered (to within one sigma) and the numerical solutions are converged with ℓ = 32 Δ 𝑥 . Using a lower grid resolution, the analytical solutions are not obtained, and we also find spurious frequencies start to dominate in the Fourier analysis.', '3.3 Wave-particle interaction': 'Here we test the decay of ion acoustic waves due to Landau damping using the 𝛿 𝑓 method (see Sec. 2.5). Landau damping of ion acoustic waves follows the dispersion relation 𝑑𝑍 ( 𝜁 )/ 𝑑𝜁 = 2 𝑇 i / 𝑇 e, where 𝑍 ( 𝜁 ) is the plasma dispersion function and 𝜁 is the phase velocity, which has a real (oscillatory) and an imaginary (decaying) part, and 𝑇 i and 𝑇 e are the ion and electron temperature, respectively. The decay rate of the wave depends on the ratio of the ion to electron temperature. We will compare our numerical solutions with the solutions of the analytical dispersion relation, 𝑑𝑍 / 𝑑𝜁 = 2 𝑇 i / 𝑇 e, which we solve numerically using the Newton-Raphson method for a given 𝑇 i / 𝑇 e. \nThe length of our computational domain is 𝐿 𝑥 = 16 𝑑 i along the 𝑥 -axis, where 𝑑 i is the ion inertial length. We initialise the velocity of the ions (particles) using a Maxwell-Boltzmann distribution with temperature 𝑇 i and fixed electron temperature 𝑇 e. We initialise the density with the profile 1 + 𝑎 cos ( 𝑘𝑥 ) , where we choose a small perturbation amplitude, 𝑎 = 0 . 01 and the wavenumber of the perturbation, 𝑘 = 2 𝜋 / 𝐿 𝑥 . We used 𝑁 ppc = 10 6 particles per cell and periodic boundary conditions.', '3.3.1 Results': 'Figure 9. Landau damping of ion acoustic waves shown by the decay of the squared amplitude of the density fluctuation as a function of time, normalised to the sound-crossing time, 𝑡 𝑐 s , for a ratio of the ion temperature to the electron temperature of 𝑇 i / 𝑇 e = 0 . 1. The local maxima used for fitting are shown by the red dots. We fit for the Landau damping decay rate, shown by the dashed line and find a decay rate of 𝛾 fit = ( 4 . 5 ± 0 . 2 ) × 10 -2 𝑡 -1 𝑐 s , which shows good agreement with the theoretical value, 𝛾 theory. \n<!-- image --> \nFigure 10. SameasFig. 9, but for 𝑁 ppc = 10 2 , 10 3 , 10 4 , 10 5 and 10 6 particles per cell. For 𝑁 ppc ≳ 10 4 , the Landau damping of ion acoustic waves are resolved, while particle noise dominates below 𝑁 ppc ∼ 10 4 . \n<!-- image --> \nfunction. We find a decay rate of 𝛾 fit = ( 4 . 5 ± 0 . 2 ) × 10 -2 𝑡 -1 𝑐 s . We measure the decay rates between successive data points and the decay rate we report is the mean of all the intervals and the error reported is the standard deviation of the decay rate across all the intervals. We compare this with the solutions of the dispersion relation and find good agreement with the theoretical solution, 𝛾 theory = 0 . 042 𝑡 -1 𝑐 s .', '3.3.2 Particle resolution criterion': 'Here we repeat the Landau damping test with 𝑁 ppc = 10 2 , 10 3 , 10 4 and 10 5 particles per cell. The results are shown in Fig. 10. We find that with low particle count 𝑁 ppc ≲ 10 4 , particle noise dominates, while for higher 𝑁 ppc the Landau damping of ion acoustic waves are resolved.', '4 PARTICLE-GRID INTERPOLATION TESTS': 'After confirming that the code agrees with analytical solutions for standard particle, wave, and particle-wave problems, we perform collisionless turbulent dynamo tests to understand the effect of using different interpolation kernels and smoothing in our numerical simulations. The turbulent dynamo mechanism converts turbulent kinetic energy into magnetic energy, exponentially amplifying magnetic fields (Kazantsev 1968; Moffatt 1978; Ruzmaikin et al. 1988). The properties of the turbulent dynamo are sensitive to the Mach number, nature of turbulence driving and the magnetic and kinetic Reynolds number of the plasma (Schekochihin et al. 2004; Haugen et al. 2004a,b; Federrath et al. 2011; Schober et al. 2012; Seta et al. 2020; Seta & Federrath 2020; Achikanath Chirakkara et al. 2021; Kriel et al. 2022; Gent et al. 2023) and recently numerical studies have extended this to the collisionless regime using Hybrid-kinetic codes (Rincon et al. 2016; St-Onge & Kunz 2018; Achikanath Chirakkara et al. 2024).', '4.1 Turbulence driving setup and initial conditions': 'In all the numerical tests presented in this and the following sections, we drive turbulence with a random acceleration field, f , following an Ornstein-Uhlenbeck process (Eswaran & Pope 1988; Federrath et al. 2010, 2022). The acceleration field is constructed in Fourier space to contain only large-scale modes, 𝑘𝐿 / 2 𝜋 = [ 1 , 3 ] , where 𝐿 is the size of the cubic computational domain. The turbulence driving amplitude is modelled as a parabolic function that peaks at 𝑘𝐿 turb / 2 𝜋 = 2 and is zero at 𝑘𝐿 / 2 𝜋 = 1 and 3, where 𝐿 turb is the peak driving scale of the turbulence. The turbulent velocity dispersion, 𝑣 turb , is controlled by adjusting the amplitude of the driving. We inject purely solenoidal acceleration modes (∇· f = 0 ) (Federrath et al. 2010) into the plasma for these tests. For all the tests in this section, we initialize the velocity of the particles using a Maxwellian distribution with temperature 𝑇 i , which is equal to 𝑇 e. \nAll the numerical tests in this section are performed on a periodic three-dimensional computational domain with 𝑁 3 grid = 128 3 grid cells. We drive the turbulence to produce a steady-state (fullydeveloped) turbulent Mach number of M = 0 . 2. Using the steadystate kinetic energy, we initialise the magnetic field such that the ratio of magnetic to turbulent kinetic energy is ( 𝐸 mag / 𝐸 kin ) init = 10 -8 , and the initial ratio of the Larmor radius to the box size is ( 𝑟 Larmor / 𝐿 ) init = 10 2 (see Sec. 2.5 of Achikanath Chirakkara et al. (2024) for further details). The convergence of the turbulent dynamo simulations with respect to the grid and particle resolution is studied in App. D of Achikanath Chirakkara et al. (2024). The magnetic Reynolds number of the plasma, defined as \nRm = 𝑣 turb 𝐿 turb 𝜂 , (53) \nis set to Rm = 500. The hyper-resistive magnetic Reynolds number is defined as \nRm h = 𝑣 turb ( 𝐿 turb ) 3 𝜂 hyper , (54) \nand is set to Rm h = 10 7 . For all the tests in this section, we use cooling (tested and discussed separately in detail in Sec. 5 below) with a cooling frequency of ( Δ 𝑡 ) cool = 0 . 1 𝑡 cool (see Eq. 43 and Sec. 2.11). \nFigure 11. Particle → grid interpolation tests using collisionless turbulent dynamo simulations. The Mach number, M , is shown in the top panel, and the magnetic-to-kinetic energy ratio, 𝐸 mag / 𝐸 kin, is shown in the bottom panel, as a function of time normalised to the turbulent eddy turnover time, 𝑡 0. For the standard set of numerical parameters, we use the cloud-in-cell (CIC) interpolation kernel with 𝑁 ppc = 100 particles per cell and two smoothing passes, 𝑁 smooth = 2. We find that using nearest-grid-point (NGP) or triangularshaped-cloud (TSC) interpolation functions does not noticeably change the box-averaged solutions for the Mach number or the magnetic energy (provided 𝑁 smooth = 2). We also find that changing the number of particles per cell, 𝑁 ppc = 50 , 200, or 400, does not change the box-averaged solutions. Changing the number of smoothing operations when 𝑁 smooth ≥ 1 also has no significant impact on the box-averaged solutions. However, without any smoothing, 𝑁 smooth = 0 (red dashed line), the growth in magnetic energy is slightly underestimated. \n<!-- image -->', '4.2 Comparison of global quantities in interpolation tests': "We test three interpolation kernels: the nearest-grid-point (NGP), the cloud-in-cell (CIC), and the triangular-shaped-cloud (TSC) interpolation functions, detailed in Sec. 2.4.1 and Appendix A. We also vary the number of smoothing passes, 𝑁 smooth = 0 , 1 , 2, and 4, as discussed in Sec. 2.4.2. Finally, we investigate the influence of different numbers of particles per cell, 𝑁 ppc = 50 , 100 , 200, and 400. \nFig. 11 shows the time evolution (in units of the turbulent turnover time, 𝑡 0 = 𝐿 turb / 𝑣 turb ) of the sonic Mach number ( M ; top panel) and the magnetic-to-kinetic energy ratio ( 𝐸 mag / 𝐸 kin ; bottom panel), for all the interpolation tests considered here. We see that all test simulations yield very similar results, with the sonic Mach number reaching a steady state of M ∼ 0 . 2. We also see that after an initial transient phase of about 2-3 𝑡 0 , the magnetic energy grows exponentially in all tests. These are typical characteristics of small-scale turbulent dynamogrowth during the so-called 'kinematic phase' (Brandenburg & Subramanian 2005; Schekochihin et al. 2004; Federrath 2016; Seta", '& Federrath 2020; Kriel et al. 2022; Achikanath Chirakkara et al. 2024).': 'Comparing the different tests in Fig. 11, we find that all runs are virtually identical in terms of their box-averaged global evolution, except for the simulation that does not use any smoothing ( 𝑁 smooth = 0) after particle → grid interpolations (red dashed line).', '4.3 Spatial structure of interpolated fields': 'Fig. 12 shows the relative charge density at 𝑡 = 10 𝑡 0 for all the interpolation test simulations above. The top left panel shows the test with the standard set of numerical parameters (CIC, 𝑁 smooth = 2, 𝑁 ppc = 100), which serves as the main reference simulation. Comparing the tests in the first row (CIC vs. NGP vs. TSC), we find no significant difference in the spatial distribution of the charge density. While the NGP method on its own is generally inferior compared to the CIC and TSC interpolation methods, the fact that we perform 𝑁 smooth = 2 smoothing passes after these interpolations nullifies any specific differences between NGP, CIC, and TSC. \nFinally, the last row of Fig. 12 shows the effect of varying the number of particles per cell, 𝑁 ppc = 50, 200, 400. While using a higher 𝑁 ppc improves the spatial smoothness of the charge density field, we found no significant impact on the global evolution of the system in Fig. 11. It is worth noting that the test with 𝑁 smooth = 2 and 𝑁 ppc = 400 (bottom right panel) is visually very similar to the case with 𝑁 smooth = 4 and 𝑁 ppc = 100 (middle row, righthand panel). This suggests that a balanced combination of 𝑁 ppc and 𝑁 smooth can lead to comparably good results for a desirably low 𝑁 ppc, as it saves computational resources. However, in general, we suggest that the number of smoothing passes should be kept sufficiently low, in particular in supersonic flows, which would result in excessive smearing of discontinuities (shocks). An optimal setting of 𝑁 smooth and 𝑁 ppc is therefore problem-dependent and requires experimentation and testing in any PIC simulation. \nThe 2nd row of Fig. 12 shows a comparison of 𝑁 smooth = 0 (no post-interpolation smoothing), 𝑁 smooth = 1, and 4. We see that not smoothing after interpolation (middle row, left-hand panel) leaves a very noisy density distribution. This substantially improves already with a single smoothing pass (central panel). Using 4 smoothing passes ( 𝑁 smooth = 4; right panel) leads to a relatively smooth distribution. However, neither 𝑁 smooth = 1 nor 4 are substantially different from the standard 𝑁 smooth = 2 results. While the spatial details of the interpolated fields depend on the number of smoothing passes, we saw in Fig. 11 that the choice of smoothing passes (except for 𝑁 smooth = 0) has no significant impact on the global evolution of the system.', '5 COOLING TESTS': 'In this section, we demonstrate the working of our cooling method for ions described in Sec. 2.11.', '5.1 Cooling vs. no cooling': 'Here we investigate the requirement of plasma cooling to model subsonic and supersonic turbulent flows. We run the following numerical simulations on a triply periodic uniform computational domain with 𝑁 3 grid = 128 3 grid cells and 𝑁 ppc = 100 particles per cell for subsonic, M target = 0 . 2, and supersonic, M target = 2 target Mach numbers, without and with cooling. The turbulence driving used here is described in Sec. 4.1. We note that magnetic fields have not been \nFigure 12. Slice plots of the relative charge density (colour map) at 10 𝑡 0 for the interpolation test simulations shown in Fig. 11. The superimposed vectors and streamlines show the velocity and magnetic field, respectively. For the standard test (top left panel), we use the cloud-in-cell (CIC) interpolation kernel with 2 smoothing passes ( 𝑁 smooth = 2), and 𝑁 ppc = 100 particles per cell. First row: comparison of different interpolation kernels, including the nearest-grid-point (NGP) and triangular-shaped-cloud (TSC) interpolation kernels, with 𝑁 smooth = 2 and 𝑁 ppc = 100. Second row: tests without smoothing ( 𝑁 smooth = 0), with 𝑁 smooth = 1, and 𝑁 smooth = 4, using the CIC interpolation kernel and 𝑁 ppc = 100. Third row: tests with different numbers of particles per cell, 𝑁 ppc = 50 , 200, and 400. These tests show that it is important to smooth the particle-sampled fields with at least 𝑁 smooth = 1 smoothing pass (somewhat better results still, are obtained here with 𝑁 smooth = 2), and to work with a sufficiently large number of particles per cell ( 𝑁 ppc ∼ 100) to limit the effects of particle interpolation noise. \n<!-- image --> \nincluded for all the tests described in this section as they play no role in the cooling functionality.', '5.1.1 Time evolution of global thermodynamic and turbulent quantities': 'The time evolution of the cooling experiments is shown in Fig. 13. The top panel shows the turbulent speed, normalised to the fixed target thermal speed, 𝑣 turb /( 𝑣 th ) target, in our numerical tests as a \nfunction of time. We find that without cooling (red curves), it is difficult to maintain the target turbulent speeds, 𝑣 turb /( 𝑣 th ) target = 0 . 2 in the subsonic regime and 𝑣 turb /( 𝑣 th ) target = 2 in the supersonic regime. After an initial maximum, 𝑣 turb decreases in the tests without cooling (red curves). When cooling is included, we find that the target 𝑣 turb is reached and maintained throughout the simulations. The middle panel of Fig. 13 shows the thermal speed, normalised to the target thermal speed, 𝑣 th /( 𝑣 th ) target. For the tests without cooling, we find that 𝑣 th increases monotonically as turbulence energy dissipates, \nFigure 13. Turbulent speed normalised to the target thermal speed, 𝑣 turb /( 𝑣 th ) target (top panel), thermal speed normalised to the target thermal speed, 𝑣 th /( 𝑣 th ) target (middle panel), and the sonic Mach number, M (bottom panel), as a function of time, normalised to the eddy turn-over time, 𝑡 0, for turbulence driving simulations on 𝑁 3 grid = 128 3 grid cells with 𝑁 ppc = 100 particles per cells with cooling (blue) and without cooling (red), in tests with a target Mach number of 0.2 (dashed lines) and 2 (solid lines), respectively. We find that without cooling it is impossible to maintain the target Mach number. This is primarily the result of heating due to turbulent dissipation and shocks (for the supersonic test), demonstrating the necessity for cooling to model steady-state turbulent flows. \n<!-- image --> \nt \n/ \nt \n0 \nwhile with cooling, 𝑣 th is maintained at its target level. We also find that the thermal speed of the plasma increases faster in the supersonic regime when cooling is not included. For both the tests with cooling in the subsonic and supersonic regime, ( Δ 𝑡 ) cool = 0 . 1 𝑡 cool , where 𝑡 cool is the cooling timescale (more details on the role of ( Δ 𝑡 ) cool below). The ratio of the data in the top and bottom panels is shown in the bottom panel of Fig. 13, i.e., the sonic Mach number, M , which reflects the issues discussed separately for 𝑣 turb and 𝑣 th without cooling. Over time, the plasma becomes increasingly subsonic, which makes studying maintained supersonic flows practically impossible without cooling. However, even for the subsonic regime, turbulence cannot be maintained at a steady Mach number without cooling. This demonstrates the necessity for cooling in applications such as steady-state turbulence.', '5.1.2 Statistical and spatial distribution of the temperature': 'Hereweinvestigate the distribution and spatial structure of the plasma temperature in more detail. Fig. 14 shows the probability density \nFigure 14. Probability density functions (PDFs) of the thermal speed, 𝑣 th, normalised to the target thermal speed, ( 𝑣 th ) target, for simulations with target Mach number, M target = 0 . 2 (dashed lines) and M target = 2 (solid lines), without cooling (red) and with cooling at 𝑡 / 𝑡 0 = 0 . 5 (blue) and time-averaged in 𝑡 / 𝑡 0 = [ 2 , 10 ] (cyan). The tests without cooling show very broad distributions, i.e., not only is the target average temperature overestimated, but the plasma is also spatially at very different temperatures. For the tests with cooling, the target thermal speed is reached and maintained (see time-averaged curves shown in cyan) to within 0.6% accuracy in both the subsonic and supersonic regimes. The spatial fluctuations remain reasonably small with a standard deviation of 0.3% and 4.3% in the subsonic and supersonic regimes, respectively. The higher residual fluctuations in the supersonic regime are due to the presence of shocks. \n<!-- image --> \nfunction (PDF) of the thermal speed, 𝑣 th , normalised to the target thermal speed, ( 𝑣 th ) target. For the subsonic and supersonic test without cooling at 𝑡 = 0 . 5 𝑡 0 , we find 𝑣 th /( 𝑣 th ) target = 1 . 172 ± 0 . 010 and 1 . 189 ± 0 . 117, respectively. While the mean values for both these tests are similar with about 17-19% higher than the target, the spread of the thermal speed is much higher for the supersonic simulation. This can be attributed to more turbulent kinetic energy available to heat the plasma and the presence of shocks dissipating large amounts of energy in the supersonic regime. By contrast, when cooling is included, we find 𝑣 th /( 𝑣 th ) target = 1 . 001 ± 0 . 003 and 1 . 000 ± 0 . 002 for the same subsonic and supersonic test comparison, respectively. For the simulations with cooling, we also show the PDFs averaged between 2 -10 𝑡 0 , for which we find 𝑣 th /( 𝑣 th ) target = 1 . 000 ± 0 . 003 and 1 . 006 ± 0 . 043, respectively, i.e., the target thermal speeds are reasonably well maintained. \nThe respective spatial distribution of the thermal speed is shown in Fig. 15. We see that without cooling, the distribution of the thermal speed is inhomogeneous. In the case of supersonic turbulence, this distribution is significantly more inhomogeneous as more energy is injected into the plasma and dissipation occurs primarily in shocks. Comparing the over- and under-densities in the bottom right-hand panels we see that they are associated with the under- and over-heated regions in the bottom left-hand panels, respectively, i.e., adiabatic heating and cooling occur in relatively over- and under-dense regions, respectively. Once cooling is introduced, the resulting thermal speed distribution is fairly homogeneous. This shows that it is important to cool the plasma locally (not just globally), to obtain a spatially homogeneous isothermal plasma. \n(a) Thermal speed relative to the average thermal speed. \n<!-- image --> \n(b) Plasma density relative to the average density. \n<!-- image --> \nFigure 15. (a) Slice plots of the thermal speed, 𝑣 th, normalised to the mean thermal speed, ⟨ 𝑣 th ⟩ , at 𝑡 / 𝑡 0 = 0 . 5, for the same test simulations as in Fig. 13 (as indicated in the panel labels). We see that without cooling, the thermal speed is inhomogeneous, particularly in the supersonic regime. Cooling homogenises the thermal speed, showing that cooling is required locally to achieve reasonable isothermal conditions throughout the plasma. (b) Same as (a), but for the ion mass density, 𝜌 m, normalised to the mean ion mass density, ⟨ 𝜌 m ⟩ . The fluctuations in the mass density are much smaller in the subsonic tests when compared to the supersonic tests, as expected. Shocked structures with large density gradients are apparent in the supersonic regime, which correlates with regions of higher thermal speeds, showing that shocks dissipate kinetic energy efficiently.', '5.2 Choosing the cooling frequency': 'In order to keep the plasma isothermal, we need to apply the cooling operator at regular time intervals, ( Δ 𝑡 ) cool ∝ 𝑡 cool (see Sec. 2.11). Here we investigate the required cooling frequency, ( Δ 𝑡 ) -1 cool , to maintain approximate isothermal plasma conditions in the subsonic and supersonic regime of turbulence.', '5.2.1 Subsonic tests': 'In Fig. 16 we vary the cooling time for the subsonic turbulence simulations presented above, ( Δ 𝑡 ) cool = 0 . 01, 0 . 1, 1 and 10 𝑡 cool . We perform these simulations with a target M = 𝑣 turb /( 𝑣 th ) target = 0 . 2 and list the results in Tab. 1. The figure legend also lists the cooling timescale in units of the thermal crossing time, 𝑡 cool = 𝑡 th /M = 5 𝑡 th (see Eq. 43). The middle panel of Fig. 16 shows that for larger values of the cooling time, the thermal speeds start increasing, while for lower values of ( Δ 𝑡 ) cool , 𝑣 th is maintained close to the target thermal speed. We find that for the turbulent speed and the Mach number, shown by the top and the bottom panel of Fig. 16, respectively, the numerical solutions converge and the target 𝑣 turb and M are attained with ( Δ 𝑡 ) cool ∼ 0 . 1 𝑡 cool = 0 . 5 𝑡 th .', '5.2.2 Supersonic tests': 'We repeat the same experiment with a supersonic target Mach number, M = 𝑣 turb /( 𝑣 th ) target = 2, and show the results in Fig. 17. We list the results in Tab. 1. We find that for ( Δ 𝑡 ) cool = 1 and 10 𝑡 cool , the plasma heats up very quickly and the target thermal speed is not maintained. Performing the cooling with a higher frequency, by using ( Δ 𝑡 ) cool = 0 . 1 𝑡 cool = 0 . 05 𝑡 th , the target thermal speed is acquired and maintained and the target Mach number is achieved. \nTable 1. List of simulations shown in Fig. 16 and Fig. 17 with the corresponding model name, turbulent speed normalised to the target thermal speed, 𝑣 turb /( 𝑣 th ) target, thermal speed normalised to the target thermal speed, 𝑣 th /( 𝑣 th ) target, and the sonic Mach number, M . These quantities are measured for 𝑡 / 𝑡 0 = 2 -10, where 𝑡 0 is the turbulent turnover time. This shows that the thermal speed and Mach number are converged (to within one sigma) as the cooling frequency is increased, in both the subsonic and the supersonic regimes of turbulence.', '5.2.3 Summary': 'The thermal crossing time, 𝑡 th , is fixed between the subsonic and supersonic tests, as both have been configured to reach the same target thermal speed. Since the Mach number is 10 times higher in the supersonic runs, we need a 10 times higher cooling frequency (in absolute terms) in the supersonic tests, compared to the subsonic tests. However, when expressed in terms of the cooling time, which takes the target Mach number into account (Eq. 43), the required cooling frequency is the same in the supersonic and subsonic regimes. That is, we find that for ( Δ 𝑡 ) cool ∼ 0 . 1 𝑡 cool , i.e., using a safety factor of ∼ 0 . 1, the target thermal speed and Mach number are reasonably achieved and maintained throughout. \n〉 \nth \n〈 \nth \n〉 \n〈 \nFigure 16. Same as Fig. 13, but for subsonic turbulence with target M = 0 . 2 and different cooling times, ( Δ 𝑡 ) cool = 0 . 01, 0 . 1, 1 and 10 𝑡 cool, where 𝑡 cool is the cooling timescale. We see that the target thermal speed is overestimated with ( Δ 𝑡 ) cool = 1 and 10 𝑡 cool. By contrast, the thermal speed is converged and maintained close to the target value for ( Δ 𝑡 ) cool ≲ 0 . 1 𝑡 cool. \n<!-- image -->', '6 CODE OPTIMISATION AND PERFORMANCE': "In this section, we discuss the performance and scalability of our code. We discuss the 'hybrid precision' method introduced in Federrath et al. (2021). We extend and test the method for Hybrid PIC in Sec. 6.1. Parallel scaling tests are provided in Sec. 6.2.", '6.1 Hybrid precision': "Hybrid PIC simulations can be extremely computationally demanding, as they require both high grid-cell counts and high particle counts. As we have seen in Sec. 4, 𝑁 ppc ∼ 100 particles per cell is a standard requirement for sufficient interpolation accuracy. With a grid-cell count of just 100 3 cells, this means that even at such moderate grid resolutions, we very quickly end up with an enormous number of resolution elements 4 . Therefore, it is important to investigate and develop numerical methods that will help reduce \n4 For example, if we wanted to do a Hybrid PIC turbulence simulation with 1000 3 grid cells and 𝑁 ppc = 100, we would have 1000 3 × 100 = 10 11 resolution elements to compute on. This is not only significant in terms of memory requirements, but the compute time per time step is also significantly higher for particle operations compared to MHD grid-cell operations. \nFigure 17. SameasFig.16,but for supersonic turbulence with a target M = 2. As for the subsonic case, a cooling time of ( Δ 𝑡 ) cool ∼ 0 . 1 𝑡 cool, is required to maintain 𝑣 th at its target level and therefore maintain the target Mach number. We also perform a test with ( Δ 𝑡 ) cool = 10 -3 𝑡 cool, to demonstrate the convergence of the thermal speed and the Mach number as the cooling frequency is increased in the supersonic regime and tabulate the results in Tab. 1. \n<!-- image --> \n0 \nthe computational resources required to perform these simulations, without compromising the accuracy of the numerical solutions. \nOne such approach for optimisation is to store and compute solutions on single precision (4 byte) floating-point numbers instead of the standard double precision (8 byte per floating-point number). This reduces the memory footprint of an application by a factor of 2. The time for computations and parallel message passing interface (MPI) communications on single precision numbers is also approximately halved compared to double precision, as long as communication times are dominated by bandwidth and not latency 5 . However, using single precision without safeguards may lead to substantial inaccuracies in the numerical solution 6 . Thus, we will use the 'hybrid \n5 This depends on the size of the problem and the number of cores used. However, for normal production applications, communication is often bandwidth limited, as communication is only required once per time step, and the data-package size scales with the number of particles, noting that practically all applications use 𝑁 ppc ≫ 1, and thus, the total number of particles determines the relevant problem size rather than the number of grid cells. Packing MPI messages and communicating many quantities per particle means that applications using this Hybrid PIC code are usually bandwidth-dominated and therefore benefit from using hybrid precision compared to double precision. \nprecision' approach introduced in Federrath et al. (2021). The basic approach is to store and communicate all quantities that occupy large arrays (such as any quantity that is defined per grid cell or per particle) in single precision arrays. However, computations that require double precision for sufficient accuracy are carried out in double precision by explicit promotion to double precision during those operations. The operations that require double precision include summing up the mass, charge and momentum of all the particles. Additionally, the time step is promoted to double precision throughout the code. All global integral quantities for grid variables, such as electric and magnetic energy, as well as particle variables like the componentwise mean particle velocity and the particle Larmor radius, are also calculated in double precision. As shown in Federrath et al. (2021), this approach cuts down memory storage and compute time requirements by nearly a factor of ∼ 2 overall, while retaining sufficient (near double precision) accuracy for all relevant quantities. \nHere we implement and test this hybrid precision approach for Hybrid PIC, and compare the results of this with pure single precision and pure double precision solutions. To do this, we use the same basic turbulence setup as in the previous two sections, as these represent complex and relevant use cases of the code. Fig. 18 shows the fluctuations in the average particle velocity over time, as a function of the total number of particles, 𝑁 , with single and hybrid precision. The fluctuations in the average particle velocity, 𝜎 (⟨ 𝑣 p ⟩) , are calculated as the standard deviation of the average particle speed, ⟨ 𝑣 p ⟩ , over time ( ≈ 0 -1 𝑡 0 ). The average particle speed and its fluctuations are expected to be zero to within machine precision, as there are no bulk flows in the plasma for the turbulent driving setup. Using single precision, we find that 𝜎 (⟨ 𝑣 p ⟩)/ 10 -6 increases strongly as the number of particles is increased, where 10 -6 is the expected precision level for single precision. This makes using single precision undesirable, as with a higher number of particles the numerical solutions become progressively worse and unreliable. On the other hand, using hybrid precision, we find 𝜎 (⟨ 𝑣 p ⟩)/ 10 -6 ≲ 1, and this value does not change as the number of particles is increased. Thus, the absolute error with hybrid precision remains bound to the level of single precision, while with pure single precision, the errors increase very quickly. \nWenote that while Fig. 18 shows a specific metric for comparison, other metrics show a similar behaviour. Thus, hybrid precision provides a stable and accurate (to within ≲ 10 -6 ) solution, while single precision does not. We also note that pure double precision outperforms hybrid precision in this metric, as expected (the fluctuations in the average particle speed are zero to within double precision, as expected). However, the main goal here is to establish that in the hybrid-precision approach (while naturally only providing an absolute accuracy between that of single and double precision), the accuracy of the solution is stable with hybrid precision, while it quickly deteriorates with pure single precision. \nTo further demonstrate that the relevant physical solution is practically as accurate with hybrid precision as it is with pure double precision, we show the Mach number and the magnetic-to-kinetic energy ratio in tests with double and hybrid precision in Fig. 19. We find excellent agreement between the double and hybrid precision runs. We also checked other quantities, including probability density functions and power spectra, which all show excellent agreement as well (see Appendix C). The following subsection compares the computational cost of hybrid vs. double precision, and quantifies the parallel scaling of the code. \nprecision - the question is whether the quantity of interest is sufficiently accurate in the context of a specific application. \nFigure 18. The time- and particle-averaged variations in the average particle velocity (averaged over all particles and across the 𝑥 , 𝑦 and 𝑧 components), normalised to the expected precision level with single precision ≈ 10 -6 , as a function of the total number of particles, 𝑁 , for numerical tests with single precision (red) and hybrid precision (blue). The error bars are calculated as the standard deviation of the 𝑥 , 𝑦 and 𝑧 components of 𝜎 (⟨ 𝑣 p ⟩) . Using single precision, the variation in the average particle speed increases, becoming worse with increasing number of particles. By contrast, with hybrid precision, the error is bounded, as desired. \n<!-- image --> \nFigure 19. Same as Fig. 11, but for simulations comparing double and hybrid precision. The numerical solutions for the Mach number, M , and the ratio of magnetic energy to kinetic energy, 𝐸 mag / 𝐸 kin, show excellent agreement. \n<!-- image --> \n0 \nFigure 20. Time taken by the code to perform one-time step (averaged over a total of 100 time steps), normalised by the total number of particles (top x-axis) as a function of the number of compute cores for the turbulence simulations described in Sec. 6.1, with double precision (magenta) and hybrid precision (blue). The respective dashed lines represent fits with a constant (ideal weak scaling). Both double and hybrid precision cases show excellent parallel scaling up to ∼ 10 4 compute cores. Moreover, hybrid precision simulations are a factor of ∼ 1 . 5 faster than pure double precision calculations. The error bars are obtained from the standard deviation of sub-intervals of 10 time steps over the total time-step interval. \n<!-- image -->", '6.2 Parallel code scaling': 'Here we test how the performance of our code scales with an increasing number of computational cores for numerical tests with hybrid and double precision, discussed in Sec. 6.1. We show the time per particle per time step as a function of the number of compute cores in Fig. 20. We note that this is a weak scaling test, i.e., the number of particles grows proportionally with the number of compute cores (see top x-axis for the absolute number of particles). Therefore, ideal scaling is indicated by a constant time per particle per time step. We calculate the time by taking an average of over 100 time steps, where the error bars have been estimated as the standard deviation over intervals of 10 time steps in the total time step range. The dashed lines show fits with a constant, i.e., ideal scaling. \nWe find excellent (near ideal) parallel scaling of the code with both double and hybrid precision up to ∼ 10 4 compute cores. Fig. 20 also shows that the hybrid precision simulations are a factor of ≈ 1 . 5 faster than the double precision calculations. The theoretical speed-up could be as high as factor 2, as discussed in Sec. 6.1 7 , but significant care must be taken in promoting relevant single-precision operations to double precision, in order to maintain sufficient overall accuracy of the code (c.f., Fig. 19) Therefore, a reduction of the required compute time by ∼ 33% with hybrid precision compared to pure double precision is highly beneficial for computationallyexpensive Hybrid PIC simulations.', '7 SUMMARY AND CONCLUSIONS': "We introduced AHKASH - a new Hybrid particle-in-cell (PIC) code within the FLASH astrophysics code framework (Fryxell et al. 2000; Dubey et al. 2008a,b) to study weakly collisional plasmas. The Hybrid PIC code includes state-of-the-art numerical methods for time integration of particle trajectories, i.e., the Boris integrator (Boris 1970; Zenitani & Umeda 2018, Sec. 2.3), and the predictor-predictorcorrector integration algorithm (Kunz et al. 2014b, Sec. 2.7). We use the constrained transport (CT) method to ensure that magnetic fields remain divergence-free (Yee 1966, Sec. 2.6). The code supports various grid → particle and particle → grid interpolation schemes, as well as post-interpolation smoothing to reduce finite particle noise (see Sec. 2.4). It further supports the ' 𝛿 𝑓 method' to study instabilities in weakly collisional plasmas (see Sec. 2.5). The code implements a series of time step constraints to ensure that physical plasma time scales are appropriately resolved (see Sec. 2.12). \nWeperform several tests to demonstrate the ability and accuracy of the new code to model standard physical problems, such as the motion of a charged particle in the presence of magnetic fields (Sec. 3.1), the propagation of Alfvén and whistler waves (Sec. 3.2), and Landau damping of ion acoustic waves (Sec. 3.3). \nIn Sec. 4 we compare the quality of various particle → grid interpolation schemes. We find that global quantities are largely independent of the details of the interpolation scheme, provided that at least 1 smoothing pass is performed after the particle deposition step. However, structural details can vary significantly with the interpolation scheme and with the number of particles per cell ( 𝑁 ppc). We suggest that a reasonable compromise between computational feasibility and accuracy is achieved with the cloud-in-cell interpolation and 2 post-deposition smoothing passes for 𝑁 ppc ∼ 100 particles per cell, noting that detailed requirements are case-specific and problemdependent. \nThe new code further supports turbulence driving (Federrath et al. 2010, 2021), modelled with the Ornstein-Uhlenbeck process (Sec. 2.10). To study steady-state turbulence in weakly collisional plasma, we have to maintain isothermal conditions across the computational domain and to do so we introduced a novel cooling method in the code (Sec. 2.11). This allows the modelling of fully developed, steady-state turbulence, including turbulent dynamos in weakly collisional plasmas, whose properties are sensitive to the Mach number of the plasma. In particular, we show that the cooling method is essential for modelling steady-state supersonic turbulence in Hybrid PIC. We demonstrate and quantify the ability of the cooling method to keep the plasma both locally and globally isothermal (Sec. 5). \nSince Hybrid PIC simulations can be computationally demanding, we extend the hybrid precision method introduced in Federrath et al. (2021) for grid-based hydrodynamics to our Hybrid PIC code (Sec. 6.1). This method uses a hybrid approach that promotes critical computations to double precision arithmetic (8 bytes per floatingpoint number), but otherwise stores and communicates all quantities in single precision (4 bytes per floating-point number). This provides a factor of ∼ 1 . 5 speed-up over a pure double precision calculation (c.f. Fig 20), but retains sufficient accuracy and precision throughout the calculations (c.f. Figs.18 and 19). Finally, Fig. 20 shows that the new code exhibits excellent parallel scalability up to 10,000 compute cores. \nWeplan to use AHKASH to study astrophysical problems, especially the physics of the turbulent dynamo in supersonic plasmas and its application to the intracluster medium of galaxy clusters.", 'ACKNOWLEDGEMENTS': 'We thank Matthew W. Kunz for sharing his expertise on developing Hybrid PIC codes and for many useful discussions that have been very beneficial to this work. R. A. C. acknowledges that this work was supported by an NCI HPC-AI Talent Program 2023 Scholarship (project gp08), with computational resources provided by NCI Australia, an NCRIS-enabled capability supported by the Australian Government. C. F. acknowledges funding provided by the Australian Research Council (Discovery Project DP230102280), and the AustraliaGermany Joint Research Cooperation Scheme (UA-DAAD). We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391), the Australian National Computational Infrastructure (grant ek9) and the Pawsey Supercomputing Centre (project pawsey0810) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software, FLASH , was in part developed by the Flash Centre for Computational Science at the University of Chicago and the Department of Physics and Astronomy at the University of Rochester.', 'DATA AVAILABILITY': 'The Hybrid particle-in-cell code AHKASH has been implemented in our private fork of the FLASH code. The code and results from our simulations will be shared on reasonable request to the corresponding author.', 'REFERENCES': 'Ruzmaikin A., Shukurov A., Sokoloff D., 1988, Magnetic Fields of Galaxies. Academic Press, Dordrecht \nSchekochihin A. A., Cowley S. C., 2006, Physics of Plasmas, 13, 056501 Schekochihin A. A., Cowley S. C., Taylor S. F., Maron J. L., McWilliams J. C., 2004, ApJ, 612, 276 \nSchober J., Schleicher D., Federrath C., Klessen R., Banerjee R., 2012, Phys. Rev. E, 85, 026303 \nSeta A., Federrath C., 2020, MNRAS, 499, 2076 \nSeta A., Federrath C., 2021, Physical Review Fluids, 6, 103701 \nSeta A., Federrath C., 2022, MNRAS, 514, 957 \nSeta A., Bushby P. J., Shukurov A., Wood T. S., 2020, Physical Review Fluids, 5, 043702 \nShalaby M., Broderick A. E., Chang P., Pfrommer C., Lamberts A., Puchwein E., 2017, ApJ, 841, 52 \nSimionescu A., et al., 2019, Space Sci. Rev., 215, 24 \nSpitkovsky A., 2005, in Bulik T., Rudak B., Madejski G., eds, American Institute of Physics Conference Series Vol. 801, Astrophysical Sources of High Energy Particles and Radiation. pp 345-350 ( arXiv:astro-ph/0603211 ), doi:10.1063/1.2141897 \nSquire J., Kunz M. W., Arzamasskiy L., Johnston Z., Quataert E., Schekochihin A. A., 2023a, Journal of Plasma Physics, 89, 905890417 \nSquire J., Meyrand R., Kunz M. W., 2023b, ApJ, 957, L30 \nSt-Onge D. A., Kunz M. W., 2018, ApJ, 863, L25 \nStone J. M., Gardiner T. A., Teuben P., Hawley J. F., Simon J. B., 2008, ApJS, 178, 137 \nStone J. M., Tomida K., White C. J., Felker K. G., 2020, ApJS, 249, 4 \nVerscharen D., Klein K. G., Maruca B. A., 2019, Living Reviews in Solar \nPhysics, 16, 5 \nWinske D., Karimabadi H., Le A., Omidi N., Roytershteyn V., Stanier A., 2022, arXiv e-prints, p. arXiv:2204.01676 \nYee K., 1966, IEEE Transactions on Antennas and Propagation, 14, 302 \nZenitani S., Umeda T., 2018, Physics of Plasmas, 25, 112110 \nZinger E., Dekel A., Birnboim Y., Nagai D., Lau E., Kravtsov A. V., 2018, MNRAS, 476, 56', 'APPENDIX A: WEIGHT FUNCTIONS': 'The Cartesian components of r and r 𝑙 are ( 𝑥, 𝑦, 𝑧 ) and ( 𝑥 𝑙 , 𝑦 𝑙 , 𝑧 𝑙 ) , respectively. Using these, we define h = ( ℎ 𝑥 , ℎ 𝑦 , ℎ 𝑧 ) , where ℎ 𝑥 = | 𝑥 -𝑥 𝑙 |/ Δ 𝑥 , ℎ 𝑦 = | 𝑦 -𝑦 𝑙 |/ Δ 𝑦 and ℎ 𝑧 = | 𝑧 -𝑧 𝑙 |/ Δ 𝑧 . Δ 𝑥 , Δ 𝑦 and Δ 𝑧 are the grid-cell size in the 𝑥 , 𝑦 and 𝑧 directions, respectively. \nThe nearest-grid-point weight function, where the particle is assigned completely to its nearest grid cell, is defined as \n𝑊 NGP ( r -r 𝑙 ) = ( 1 , if | r -r 𝑙 | = min (| h |) 0 , otherwise . (A1) \nThe distance between the particle and each grid-cell centre is calculated and the minimum value, min (| h |) , gives us the grid-cell centre closest to the particle. \nIn the cloud-in-cell (CIC) interpolation kernel, a \'cloud" is created by each particle in the shape of a grid cell and the weight function can be described as \n𝑊 CIC ( r -r 𝑙 ) = ( ( 1 -ℎ 𝑥 )( 1 -ℎ 𝑦 )( 1 -ℎ 𝑧 ) , if ℎ 𝑥 , ℎ 𝑦 , ℎ 𝑧 < 1 0 , otherwise . \n(A2) \nThe triangular-shaped-cloud weight function is defined co-ordinate wise as \n𝑊 TSC ( r 𝑖 - ( r 𝑙 ) 𝑖 ) = 1 -2 ℎ 2 𝑖 , if ℎ 𝑖 ≤ 0 . 5 2 ( 1 -ℎ 𝑖 ) 2 , if 0 . 5 < ℎ 𝑖 < 1 0 , otherwise . (A3) \nThe total weight function, 𝑊 TSC ( r -r 𝑙 ) , is then given by \nIn both CIC and TSC interpolation kernels, the size of the weight function is one grid cell ( ∼ Δ 𝑥 ). One can also construct larger stencils for the weight function, however, the computational cost of the interpolation operation increases as the extent of the weight function increases. \n𝑊 TSC ( r -r 𝑙 ) = 𝑊 TSC ( x -x 𝑙 ) 𝑊 TSC ( y -y 𝑙 ) 𝑊 TSC ( z -z 𝑙 ) . (A4)', 'APPENDIX B: CELL-CENTERED FINITE DIFFERENCE METHOD': "Wealsohaveacell-centered approach to update magnetic fields using Faraday's law, Eq. 13. In this method, the electric and magnetic fields are defined and evolved on the cell-centers. This numerical stencil is depicted in Fig. B1. \nIn this method, the magnetic field is updated in time in the following way \n𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖, 𝑗, 𝑘 ) = 𝐵 𝑡 𝑥 ( 𝑖, 𝑗, 𝑘 ) + Δ 𝑡 2 Δ 𝑧 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖, 𝑗, 𝑘 + 1 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖, 𝑗, 𝑘 -1 ) GLYPH<3> -Δ 𝑡 2 Δ 𝑦 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖, 𝑗 + 1 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖, 𝑗 -1 , 𝑘 ) GLYPH<3> , (B1) \n𝐵 \n+ \n𝑡 \n𝑦 \n𝑖, 𝑗, 𝑘 \n= \n𝐵 \n𝑡 \n𝑦 \n𝑖, 𝑗, 𝑘 \n+ Δ 𝑡 2 Δ 𝑥 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 + 1 , 𝑗 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑧 ( 𝑖 -1 , 𝑗 , 𝑘 ) GLYPH<3> Δ 𝑡 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 𝑖, 𝑗, 𝑘 1 e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 𝑖, 𝑗, 𝑘 1 GLYPH<3> , (B2) \n( \n) \n( \n-2 Δ 𝑧 ( + ) -( -) \n𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 ) = 𝐵 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 ) + Δ 𝑡 2 Δ 𝑦 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 + 1 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑥 ( 𝑖, 𝑗 -1 , 𝑘 ) GLYPH<3> -Δ 𝑡 2 Δ 𝑥 GLYPH<2> e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 + 1 , 𝑗 , 𝑘 ) -e 𝐸 𝑡 + Δ 𝑡 / 2 𝑦 ( 𝑖 -1 , 𝑗 , 𝑘 ) GLYPH<3> . (B3) \nThe divergence of the magnetic field in the cell-centered finite difference method can be written as \n∇ · B 𝑡 + Δ 𝑡 ( 𝑖, 𝑗, 𝑘 ) = 1 2 Δ 𝑥 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖 + 1 , 𝑗 , 𝑘 ) -𝐵 𝑡 + Δ 𝑡 𝑥 ( 𝑖 -1 , 𝑗 , 𝑘 ) GLYPH<3> + 1 2 Δ 𝑦 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑦 ( 𝑖, 𝑗 + 1 , 𝑘 ) -𝐵 𝑡 + Δ 𝑡 𝑦 ( 𝑖, 𝑗 -1 , 𝑘 ) GLYPH<3> + 1 2 Δ 𝑧 GLYPH<2> 𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 + 1 ) -𝐵 𝑡 + Δ 𝑡 𝑧 ( 𝑖, 𝑗, 𝑘 -1 ) GLYPH<3> . (B4) \nUsing Eq. B1 - Eq. B3 in Eq. B4, we obtain that ∇ · B 𝑡 + Δ 𝑡 ( 𝑖, 𝑗, 𝑘 ) = 0. This is ensured by the construction of the cell-centered finite difference method similar to the constrained transport method (see Sec. 2.6), albeit on a stencil twice the size. The above calculations show that as long as the construction of a numerical method ensures that the analytical expression, ∇·∇× C = 0, is satisfied for any vector C , it can be used to ensure that magnetic fields are divergence-free.", 'APPENDIX C: HYBRID PRECISION - MAGNETIC SPECTRA AND PROBABILITY DENSITY FUNCTIONS': "Probability density functions (PDFs) and power spectra are key quantities in the study of turbulence and magnetic field amplification \nΔ \n𝑡 \n) \ny \nFigure B1. Cell-centered finite difference method to solve Faraday's law, Eq. 13. The charge density and ion current are deposited on the grid-cell center, depicted by the yellow circle. The electric and magnetic fields are also calculated on the cell-centers. The magnetic field of the central grid cell is updated using the electric field of the neighbouring grid cells as described by Eq. B1 - Eq. B3. \n<!-- image -->", '(Federrath 2013; Federrath & Banerjee 2015; Seta & Federrath 2020,': '2021). In this section, we compare the PDFs and power spectra obtained in the hybrid vs. double precision tests described in Sec. 6.1, with Fig. C1 showing the results. We find excellent agreement between the double and hybrid precision tests for all the examined quantities, further demonstrating that the numerical solutions with hybrid precision are in practice as accurate as those with double precision. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author. \nFigure C1. Probability density functions (PDFs) of the natural logarithm of the mass density normalised to the mean mass density, ln ( 𝜌 m /⟨ 𝜌 m ⟩) (top left), the 𝑥 -component of the velocity normalised to the root mean square velocity, 𝑢 𝑥 / 𝑢 rms (top right), the 𝑥 -component of the magnetic field normalised to the root mean square magnetic field, 𝐵 𝑥 / 𝐵 rms (bottom left), and the power spectra of magnetic energy (bottom right), for the hybrid vs. double precision tests discussed in Sec. 6.1. The PDFs and power spectra are time-averaged in the kinematic regime of the turbulent dynamo ( ≈ 3 -15 𝑡 0). We find excellent agreement between the double and hybrid precision statistics. \n<!-- image -->'}

2024arXiv240905979F

The Epoch of Reionization Spectrometer EoRSpec is an upcoming Line Intensity Mapping LIM instrument designed to study the evolution of the early universe z 3.5 to 8 by probing the redshifted CII 158 mum finestructure line from aggregates of galaxies. The CII emission is an excellent tracer of star formation since it is the dominant cooling line from neutral gas heated by OB star light and thus can be used to probe the reionization of the early Universe due to star formation. EoRSpec will be deployed on PrimeCam a modular directdetection receiver for the 6meter Fred Young Submillimeter Telescope FYST currently under construction by CPI Vertex Antennentechnik GmbH and to be installed near the summit of Cerro Chajnantor in the Atacama Desert. This instrument features an image plane populated with more than 6500 Microwave Kinetic Inductance Detectors MKIDs that are illuminated by a 4lens optical design with a cryogenic scanning FabryPerot Interferometer FPI at the pupil of the optical system. The FPI is designed to provide a spectral resolving power of Rsim100 over the full spectral range of 210420 GHz. EoRSpec will tomographically survey the ECOSMOS and ECDFS fields with a depth of about 4000 hours over a 5 year period. Here we give an update on EoRSpecs final mechanicaloptical design and the current status of fabrication characterization and testing towards first light in 2026.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.05979', '2024arXiv240905979F', 'arXiv:2409.05979']

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

CCAT A status update on the EoRSpec instrument module for PrimeCam

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05979.pdf

{'CCAT: A status update on the EoR-Spec instrument module for Prime-Cam': 'Rodrigo Freundt a , Yaqiong Li b , Doug Henke c , Jason Austermann d , James R. Burgoyne e , Scott Chapman c,e,f , Steve K. Choi g , Cody J. Duell b , Zach Huber b , Michael Niemack b,a , Thomas Nikola h , Lawrence Lin b , Dominik A. Riechers i , Gordon Stacey a , Anna K. Vaskuri d , Eve M. Vavagiakis b,j , Jordan Wheeler d , Bugao Zou k , and the CCAT collaboration 1 \n- a Department of Astronomy, Cornell University, Ithaca, NY 14853, USA. \nb Department of Physics, Cornell University, Ithaca, NY 14853, USA. \n- c NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada.\n- d Quantum Sensors Division, National Institute of Standards and Technology, Boulder, CO 80305, USA.\n- e Department of Physics and Astronomy, University of British Columbia, Vancouver, V6T 1Z1, Canada.\n- f Department of Physics and Atmospheric Science, Dalhousie University, Halifax, NS, B3H 4R2, Canada.\n- g Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA h Cornell Center for Astrophysics and Planetary Sciences, Cornell University, Ithaca, NY 14853, USA.\n- i \nInstitut fur Astrophysik, Universitat zu Koln, Zulpicher Strasse 77, 50937 Koln, Germany. j Department of Physics, Duke University, Durham, NC 27708, USA. k Department of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA.', 'ABSTRACT': "The Epoch of Reionization Spectrometer (EoR-Spec) is an upcoming Line Intensity Mapping (LIM) instrument designed to study the evolution of the early universe (z = 3.5 to 8) by probing the redshifted [CII] 158 µ m finestructure line from aggregates of galaxies. The [CII] emission is an excellent tracer of star formation since it is the dominant cooling line from neutral gas heated by OB star light and thus can be used to probe the reionization of the early Universe due to star formation. EoR-Spec will be deployed on Prime-Cam, a modular direct-detection receiver for the 6-meter Fred Young Submillimeter Telescope (FYST), currently under construction by CPI Vertex Antennentechnik GmbH and to be installed near the summit of Cerro Chajnantor in the Atacama Desert. This instrument features an image plane populated with more than 6500 Microwave Kinetic Inductance Detectors (MKIDs) that are illuminated by a 4-lens optical design with a cryogenic, scanning Fabry-Perot Interferometer (FPI) at the pupil of the optical system. The FPI is designed to provide a spectral resolving power of R ∼ 100 over the full spectral range of 210-420 GHz. EoR-Spec will tomographically survey the E-COSMOS and ECDFS fields with a depth of about 4000 hours over a 5 year period. Here we give an update on EoR-Spec's final mechanical/optical design and the current status of fabrication, characterization and testing towards first light in 2026. \nKeywords: Epoch of Reionization, Line Intensity Mapping, CII, Fabry-Perot Interferometer, MKIDs \nFurther author information: (Send correspondence to R.F.) \nR.F.: E-mail: rgf57@cornell.edu", '1. INTRODUCTION': "An important outcome of the formation of the first stars and galaxies in the early universe is the so-called Epoch of Reionization (EoR), a cosmic time that has been technically difficult to probe due to the intrinsically faint, redshifted emission from these first luminous objects. During the EoR, the primordial hydrogen left after recombination ( z ∼ 1100) underwent a gradual changeover from neutral to fully ionized as the strong radiation fields produced by these first stars and galaxies heated the interstellar and intergalactic mediums. There is solid observational evidence from the absorption spectra of distant quasars and from measurements of the integrated optical depth to reionization in CMB experiments that this process was mostly completed at around redshift z ∼ 6. 1 Cosmological simulations and targeted observations of high-redshift star forming galaxies suggest that the reionization of the universe was neither homogeneous nor instantaneous, but rather a process that evolved in patches, starting at z ∼ 15-20, and whose evolution was possibly correlated with the star formation history of the universe. 2 We are thus interested in the details of the reionization history, such as the characteristic scale of the reionization bubbles, if any, and the statistics of the population of sources that drove this process. From a cosmological viewpoint, building a picture of the EoR by tracking the ionization fraction over cosmic time is of paramount interest, as it might encode information about the expansion of the universe, which can help break tensions in cosmology, as well as shed light on the nature of dark matter. \nWhile galaxy surveys have probed deeper and deeper into the high redshift universe, 3 it is still technically infeasible to survey the whole population of sources over large cosmic volumes. An alternative technique called Line Intensity Mapping (LIM) 4 has been proposed to overcome this difficulty. LIM maps atomic and molecular lines in aggregate without resolving individual line emitters. By using the spectroscopic redshift of specific target lines, LIM can tomographically build three-dimensional intensity maps (spectral cubes) of large cosmological volumes such as the EoR and cosmic noon. In the past decade, LIM has seen wider adoption in the experimental astrophysics and cosmology community with fielded experiments such as CHIME, COMAP, and TIME, among many others. 4 \nThe Epoch of Reionization Spectrometer (EoR-Spec) 5 is an imaging spectrometer designed to obtain tomographic maps of the late EoR ( z ∼ 3 . 5-8 . 0) through LIM of fluctuations in the aggregate clustering signal encoded in the 158 micron fine-structure transition line from ionized carbon. This emission originates primarily ∗ in Photo Dissociation Regions (PDRs) at the surface of dense molecular clouds, and is the dominant cooling mechanism for star formation in these regions. 6 [CII] has been proven to be an excellent tracer of star formation, both in the local and early universe, where it is redshifted to sub-mm wavelengths and thus accessible from the ground with facilities like the Fred Young Submillimeter Telescope (FYST), 7 which our collaboration is building near the top of Cerro Chajnantor at an elevation of 5600-m in the Atacama desert. Prime-Cam 8, 9 is a cryogenic, direct-detection receiver for FYST and populates the focal plane with up to seven modular instruments. The EoR-Spec instrument modules will be dedicated to carry out the CCAT Deep Spectroscopic Survey (DSS), a 4000-hour LIM survey of E-COSMOS and E-CDFS fields. EoR-Specs' angular (sub arc-minute) and spectral resolution ( R ∼ 100) are matched to the expected clustering signal. Thus, a significant detection of [CII] at z ⪅ 6 is expected, 10,11 as well as important limits at higher redshifts. 12 \nIn this paper, we give an update on the final design of EoR-Spec. In section 2, we describe the instrument and present changes made to the optomechanical design, while in section 3 we comment on potential optical systematics. Finally, in sections 4, we report on recent progress on the fabrication and testing of the detector arrays for this instrument.", '2. INSTRUMENT DESIGN': "EoR-Spec features three large-format arrays of Microwave Kinetic Inductance Detectors (MKIDs), with a total of 6528 non-polarized pixels divided into two bands. Given that we expect a brighter signal at the lowest redshift range, we have weighted the number of detector such that two arrays are sensitive to the 210-315 GHz range or \nLow Frequency (LF) band, while the third sub-array is sensitive to the 315-420 GHz frequency range or HighFrequency (HF) band. All three arrays will be lumped-element aluminum MKIDs that are front-side illuminated with an aluminum feedhorn array that efficiently couples radiation into the detector. A series of high-purity silicon lenses reimage the telescope's focal plane into the aperture plane of the feedhorns. \nThe element that gives the spectral response to the camera design described is the scanning Fabry-Perot Interferometer, which is strategically located at the 4 Kelvin stage and aligned with the pupil of the optics, where collimation ( F/ # > 100) is maximized. A FPI consists of two identical, highly reflective mirrors that form a resonant cavity, sometimes also called the etalon, whose resonant frequency (maximum transmission) can be tuned to a certain wavelength range by increasing the spacing between the mirrors systematically while maintaining high parallelism with respect to each other. For EoR-Spec, we are fielding a novel etalon design with Silicon-Substrate-Based (SSB) mirrors that feature etched metamaterial Anti Reflection Coating (ARC) on one side and a evaporated metal mesh on the other surface, as described in 13 and 14. These SSB mirrors form a resonant cavity that essentially behaves as a variable filter, with a spectral response that also depends on the angle of incidence. We operate the SSB etalon in second and third interference orders, corresponding to the LF and HF bands, respectively. \nThe mechanical design of the instrument module (Figure 1) has evolved from Prime-Cam's 280 GHz instrument module design 15 and the Simons Observatory's Large Aperture Telescope Receiver optics tubes. 16 Hence, it inherits the same cryogenic design that consists of rigid aluminum cylinders that mount critical elements along the optical axis (such as lenses, filters and the cold stop), as well as the use of carbon fiber structures to thermally isolate stages at different cryogenic temperatures. EoR-Spec leverages on this general camera designs but adds a fourth silicon lens to the optical train and the FPI at the cold stop. \nDue to the lower cooling power available at 1 K, the cold Lyot stop (and thus the FPI) was relocated to the 4 K temperature stage with minimal additional radiation loading on the detector array. This also makes the FPI easier to mount to the instrument module; by adding an intermediate section to the 4 K tube, the FPI can be easily accessed for troubleshooting during commissioning. \nFigure 1. A semi-exploted view of the Computer-Aided Design (CAD) of the EoR-Spec instrument module. The light beam enters from the left and propagates throughout the 4-element (L1-L4) refractive optics design, painted in red, as well as Low-Pass Edge (LPE) filters at 4K and 1K. L1 and L2 are located at the 4 K stage, as well as the FPI (middle section). L3 and L4 are located at 1 K. The 1 K temperature stage also features a 3-vane baffle design for stray light mitigation, which also works as a radiation shield. The three detector arrays are located behind L4. The area between L1 and L2 is covered with Microwave Metamaterial Absorber (MMA) tiles 16 for further stray light mitigation. From end to end, the length of the instrument module is about 1.2 meters. \n<!-- image -->", '2.1 Cryogenic Optomechanical Design': "The optical design for EoR-Spec originally required the use of biconic lenses 17 to meet the specification of the instrument, including high collimation at the Lyot stop ( F/ # > 100) while maintaining high image quality (Strehl ratio above 0.8 over the full 1 . 3 · field of view). After many iterations, the final design 18 now incorporates only aspheric lenses without sacrificing optical performance. Aspheric lenses are easier and cheaper to fabricate than biconic lenses and do not require clocking alignment. Figure 1 shows the four silicon lenses colored red. Not shown on the image are the metamaterial ARCs that have to be machined on the surfaces of the lenses to avoid ghosting (Section 3). We label these lenses with their numbered position from sky to detectors (L1-L4). \nIn this final optical design, the distance between the final lens in the module and the detector focal plane is significantly shorter in EoR-Spec than in Prime-Cam's other instrument modules, making the design of a shorter carbon fiber truss 15 impossible without increasing thermal conduction from 1 K to 100 mK. One solution was to incorporate L4 to the detector array plate, but this would have increased the heat capacity at the 100 mK stage, where cooling power is limited. Instead, we decided to keep the diameter of L4 small enough to fit inside the carbon fiber truss (Figure 2), which complicates the assembly procedure but keeps the same truss length and thermal leakage as in Prime-Cam's other modules. Other mechanical constraints set by carbon fiber tube diameter required us to keep the physical diameter of L3 no larger 336 mm. \nFigure 2. Left: Detailed view of the 1 K-to-0.1 K interface. Because of the short distance that need to be maintained between L4 and the image plane, the lens cell for L4 was designed such that it fits inside the carbon fiber truss. Right: Close-up view of detector arrays and readout hardware. \n<!-- image -->", '2.2 Readout, Thermometry and FPI Control': "MKIDs allow for frequency-multiplexed readout schemes, where a large number of detectors with unique resonant frequencies can be excited and read out simultaneously with a single transmission line. The resonators are designed to have high quality factors and adequate spacing to avoid collisions in the working bandwidth. \nThe same cold readout design as the 280 GHz module 15 has been used for EoR-Spec but has been scaled down to the number of readout channels required. The LF detector arrays need three networks each (two coaxial cables per network) to read out 1728 MKIDs, while the HF array is planned to double the amount of networks to read out a total of 3072 detectors. Low Noise Amplifiers (LNA) † cooled to 4 K are required at the exit port of each network. A bundle of flexible coaxial cables are routed from the LNAs (depicted also in Figure 2) to Prime-Cam's readout harness where the coaxial cables transition into flexible striplines that are heat sunk at 4 K and 40 K, to finally be digitized and processed by the warm readout electronics inside the instrument cabin. 19 \nThermometers will be located strategically across the instrument module. Thermometry cables are routed in between the 4 K shell and the carbon fiber shell, as they need to be connected to the readout harness from the back of the module. Triaxial and twisted-pair cables are also required for the capacitive sensors and cryogenic stepper motors, respectively, which are used to scan the FPI during the survey.", '3. STRAY LIGHT ANALYSIS': 'Given the major changes (described in Section 2) from the general camera design, we decided to carry out a stray light analysis similar to the ones described in 20-22, but focused on the effects of unwanted reflection from the FPI structures, including the etalon. Figure 3 shows a non-sequential light path in ZEMAX Optic Studio , where light reflected back and forth from the image plane and the FPI mirror form a secondary image. This ghost image is symmetric with respect to the center of the image for any given field angle, and is produced because both the etalon and the feedhorn surface are parallel to each other and the optics is telecentric. ‡ \nFigure 3. Stray light analysis. Left: Non-sequential ray trace of an off-axis field point. The ray trace was filtered so that it shows only unwanted reflections from the image plane that interact with the etalon. An uniform scattering profile was assigned to the image plane for simplicity, while the designed SSB mirror transmission profile was assigned to the dummy surface (orange) at the Lyot stop. Right: Spurious images formed at the image plane from unwanted reflections off optical and mechanical components. Normalized and smoothed with the Point Spread Function (PSF). Point-like ghost image on the right side peaks at -14 dB below the primary image, while diffuse ghost images (from reflections off silicon lenses) are at the -35 dB level. \n<!-- image --> \nTo determine the level of ghosting, a suite of simulations were carried out in CST studio software to understand scattering off the feedhorn array surface. For this, a on-axis Gaussian beam was scanned across a 7 × 7 feedhorn array section, including the highly-reflective metal areas between the feedhorns. The waist of the beam was approximated to fit an f/# of 2.1. The percentage of the total power reflected off this interface and out of the simulation box (i.e., not coupled into any adjacent feedhorn) is shown in Figure 4 for an array of 25 locations. This reflectivities take into account both specular and non-specular reflections. We combined these results, along with the designed transmission profile of the SSB mirrors, to determine that the spurious image is approximately -14 dB below the main image. This is a pessimistic estimate because we only study on-axis reflections (with respect to the feedhorn aperture plane) and we assume that all scattered power reflects specularly rather than considering a proper scattering distribution function. Regardless, the ghosting level is still comparable with the diffraction limit and thus will not affect observations. \nThe same suite of simulations was used to confirm the sections of the instrument module that needed to be coated with absorbers or else they could introduce undesired loading onto the detectors if a fraction of those stray-light paths ends up at 300 K (from a time-reverse perspective). We decided to coat the front of the FPI structure with an microwave absorber material, which could be either a metamaterial tiles like in 16 or cryogenic epoxy loaded with charcoal. \n<!-- image --> \nFigure 4. Scattering analysis for f = 262 . 5 GHz. Left: CST simulation of an on-axis Gaussian beam focused at the aperture plane of the feedhorn array (49 horns), with the beam waist 0 . 84 × λ × N = 2 . 015 mm to approximate a beam with f/# of 2.1. Figure shows a slice of the scattering distribution function. Right : Scattered power (%) for an on-axis Gaussian beam at different offsets across the array. \n<!-- image -->', '4. LOW FREQUENCY DETECTOR ARRAY': "We have started fabricating the first LF detector array for EoR-Spec. Two out of three arrays at the image plane are LF arrays, with 1728 pixels each and band centered at 265 GHz. Each hexagonal array is divided into 3 rhombus-shaped groups of pixels that are read out with a single transmission line. Figure 5 shows the fabricated aluminum box that houses the detector's silicon wafer stack inside. The design of the detector box and the assembly procedure is described in more detail in Ref. 23.", '4.1 Feedhorn Array Design and Fabrication': 'The feedhorn array is based on previous designs of close-packed, simple conical horns like in 24 and 25. The spacing of feedhorns is determined by the pixel pitch of 2.75 mm (1.2 F λ ), optimized from calculations of mapping sensitivity. 8 For an f/ # of 2.1, we have optimized the aperture diameter, flare length and wall thickness for maximum coupling efficiency, spillover efficiency and ease of machining. \nThe feedhorn array was fabricated with a conventional CNC milling approach using a series of custom cutters on a monolithic block of ALCOA QC-10 alloy. The material of choice was based on ease of machining and the fact that the material is stress-relieved. The machining procedure involves first the removal of the bulk of the material with a series of rough cutters, and second the use of custom-made taper-ball-end cutters and reamers that drill the final shape of the feedhorn profile as shown in Figure 5. To avoid cutter failure, sets of ∼ 100 feedhorns were drilled at a time before exchanging cutters, allowing uniform horn surface roughness and hole roundness throughout the process. This number was determined from the fabrication of prototypes (middle pane in Figure 5) that were later sliced to measure the surface with feedhorns with a profilometer.', '4.2 Detector Array Fabrication and Testing': 'The LF detector array shown in Figure 6 has been fabricated by the Quantum Sensors Division at the National Institute of Standards and Technology (NIST). The full silicon wafer stack consists of the detector array, the Waveguide Interface Plate (WIP), a spacer wafer, and a blank wafer, as described in more detail in ref. 23. For the first cryomechanical test, we placed only the detector array and the spacer wafer inside a dark box at cryogenic temperatures, and ran a frequency sweep to locate the resonators (Figure 7), resulting in a preliminary yield of 85%. Figure 6 also shows our current test setup, where the full wafer stack in placed inside the detector array box (with feedhorns included), which is mounted to the milli Kelvin stage of an Adiabatic Demagnetization Refrigerator (ADR). We read out each of the 3 channels with a single coaxial cable by using a cold RF switch. \n<!-- image --> \n<!-- image --> \nFigure 5. LF feedhorn array design and fabrication. Left: Half power bandwidth (HPBW) of the feedhorn design. HPBW ranges from 20 to 35 degrees depending on azimuthal cut and the operating frequency. Middle: Array of 10 × 10 feedhorn fabricated to test the machining process. The sample was sliced through the first and last rows to reveal the feedhorn profile. Right: Full LF feedhorn array under fabrication. We expect that the nanometer-scale oxidation layers formed on the QC-10 alloy from the coolant used during the milling process will not affect the coupling efficiency at the wavelengths of interest. \n<!-- image --> \n<!-- image --> \nFigure 6. Low Frequency MKID array fabrication and testing. Left: Photograph of detector array wafer, where interdigitated capacitors and feedlines are visible. Right : Detector box with the silicon wafer stack was mounted inside an ADR for testing. \n<!-- image --> \nThe next steps planned for the LF array characterization is to carry out LED mapping to pinpoint and assign resonators to individual pixels in the focal plane, as well as characterizing the optical performance of the detectors using a cold load at different tone powers. 26 Based on the results, we will evaluate if collisions and cross-talk can be resolved with capacitor trimming to increase the yield of this array.', '5. CONCLUSIONS AND FUTURE WORK': 'Mapping [CII] in a continuous redshift range with a background-limited imaging spectrometer like EoR-Spec will enable direct mapping of the large-scale distribution of the sources of reionization in the early universe. The final optomechanical design for EoR-Spec has been presented, where a careful analysis of critical components was carried out to maximize instrument performance. Technical drawings for the 4 K and 1 K shells have been produced and parts are now under fabrication. Testing of the first LF detector array will continue, as well as development and fabrication of the HF band array. We expect first light on FYST in 2026.', 'ACKNOWLEDGMENTS': 'The construction of EoR-Spec is supported by NSF grant AST-2009767. The CCAT project, FYST and PrimeCam instrument have been supported by generous contributions from the Fred M. Young, Jr. Charitable Trust, Cornell University, and the Canada Foundation for Innovation and the Provinces of Ontario, Alberta, and British Columbia. The construction of the FYST telescope was supported by the Großgerate-Programm of the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under grant INST 216/733-1 FUGG, as well as funding from Universitat zu Koln, Universitat Bonn and the Max Planck Institut fur Astrophysik, Garching. G.S., T.N., and R.F. acknowledge support in part by NSF grant AST-1910107. \nFigure 7. Preliminary S 21 for each of the three channels of the LF array. A detector yield of 85% was accomplish with only the detector wafer and spacer wafer. This yield is expected to improve in the final configuration, where better grounding is achieved. \n<!-- image -->', 'REFERENCES': "- [1] Robertson, B. E., Ellis, R. S., Furlanetto, S. R., and Dunlop, J. 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2024arXiv240906769B

The 21cm brightnesstemperature field of neutral hydrogen during the Epoch of Reionization and Cosmic Dawn is a rich source of cosmological and astrophysical information primarily due to its significant nonGaussian features. However the complex nonlinear nature of the underlying physical processes makes analytical modelling of this signal challenging. Consequently studies often resort to seminumerical simulations. Traditional analysis methods which rely on a limited set of summary statistics may not adequately capture the nonGaussian content of the data as the most informative statistics are not predetermined. This paper explores the application of machine learning ML to surpass the limitations of summary statistics by leveraging the inherent nonGaussian characteristics of the 21cm signal. We demonstrate that a welltrained neural network can independently reconstruct the hydrogen density spintemperature and neutralfraction fields with crosscoherence values exceeding 0.95 for kmodes below 0.5 Mpc h1 based on a representative simulation at a redshift of z approx 15. To achieve this the neural network utilises the nonGaussian information in brightness temperature images over many scales. We discuss how these reconstructed fields which vary in their sensitivity to model parameters can be employed for parameter inference offering more direct insights into underlying cosmological and astrophysical processes only using limited summary statistics of the brightness temperature field such as its power spectrum.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.06769', '2024arXiv240906769B', 'arXiv:2409.06769']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Inferring the density spintemperature and neutralfraction fields of HI from its 21cm brightness temperature field using machine learning

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.06769.pdf

{'Inferring the density, spin-temperature and neutral-fraction fields of HI from its 21-cm brightness temperature field using machine learning': 'Bohdan Bidenko, 1 , 2 Léon V. E. Koopmans, 1 ★ P. Daniel Meerburg 2 \n- 1 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, NL-9700 AV Groningen, the Netherlands\n- 2 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands \nAccepted XXX. Received YYY; in original form ZZZ', 'ABSTRACT': 'The 21-cm brightness-temperature field of neutral hydrogen during the Epoch of Reionization and Cosmic Dawn is a rich source of cosmological and astrophysical information, primarily due to its significant non-Gaussian features. However, the complex, nonlinear nature of the underlying physical processes makes analytical modelling of this signal challenging. Consequently, studies often resort to semi-numerical simulations. Traditional analysis methods, which rely on a limited set of summary statistics, may not adequately capture the non-Gaussian content of the data, as the most informative statistics are not predetermined. This paper explores the application of machine learning (ML) to surpass the limitations of summary statistics by leveraging the inherent non-Gaussian characteristics of the 21-cm signal. We demonstrate that a well-trained neural network can independently reconstruct the hydrogen density, spin-temperature, and neutral-fraction fields with cross-coherence values exceeding 0.95 for 𝑘 -modes below 0 . 5 Mpc h -1 , based on a representative simulation at a redshift of 𝑧 ≈ 15. To achieve this, the neural network utilises the non-Gaussian information in brightness temperature images over many scales. We discuss how these reconstructed fields, which vary in their sensitivity to model parameters, can be employed for parameter inference, offering more direct insights into underlying cosmological and astrophysical processes only using limited summary statistics of the brightness temperature field, such as its power spectrum. \nKey words: dark ages, reionization, first stars - methods: data analysis - software: machine learning', '1 INTRODUCTION': "Throughout much of the history of the evolution of the universe in its first billion years, its baryonic component consisted mostly of neutral hydrogen. Analysis of physical processes during this period is possible with the help of the 21-cm emission line caused by the spin-flip transition of the electron-pair making up the hydrogen atom. \nThe unique physical information contained in this signal has led to the development of an entire field known as 21-cm cosmology (e.g., Barkana & Loeb 2001; Furlanetto et al. 2006; Pritchard & Loeb 2012; Mesinger 2019). Forecasts for experiments such as the GMRT(Paciga et al. 2013), LOFAR (Patil et al. 2017; Mertens et al. 2020; Ghara et al. 2020), NenuFAR (Mertens et al. 2021; Munshi et al. 2024), LWA (Eastwood et al. 2019), MWA (Barry et al. 2019; Li et al. 2019; Trott et al. 2020), PAPER (Kolopanis et al. 2019), HERA (DeBoer et al. 2017; Abdurashidova et al. 2022), and SKA (Koopmans et al. 2015; Mellema et al. 2015) aim to significantly improve constraints on the models describing the astrophysics of the first ionizing sources as well as cosmological parameters (Liu & Shaw 2020). \nThe traditional approach to studying the 21-cm signal has relied on measuring or constraining fluctuations of the 21-cm brightness temperature, which can be qualified by taking moments of the distribution, of which the most well-studied example is the power spectrum \n- ★ E-mail: koopmans@astro.rug.nl \n(see e.g., Morales & Hewitt 2004). Current attempts to measure the 21-cm power spectrum (see references above) mostly aim to detect a signal from the period when the universe was completely neutral to mostly ionized. This window is referred to as Cosmic Dawn (CD) and the Epoch of Reionization (EoR) and roughly spans 30 ≤ 𝑧 ≤ 5. Below 𝑧 = 5, the 21-cm signal can still be observed inside of halos/galaxies as a biased tracer of structure on large scales, which is the target for current experiments such as CHIME (Amiri et al. 2022), Tianlai (Chen 2012), HIRAX (Crichton et al. 2022) and MEERKAT (Cunnington et al. 2022), who recently claimed detection of the 21cm signal power spectrum at low redshift. \nWhile most effort has been put into detecting the power spectrum (or cross-power with large scale structure in the case of low𝑧 21-cm signal, see e.g. (Pen et al. 2009; Chang et al. 2010; Masui et al. 2013; Anderson et al. 2018; Wolz et al. 2022)), the distribution of 21-cm fluctuations is non-Gaussian and its power spectrum does not capture all information present in the field. As such, relying on the power spectrum alone is sub-optimal in constraining the physics that sources the signal. In this paper, we focus on the late stages of the CD or early stages of the EoR, when the universe is still mostly neutral, but patches of ionized regions start to appear. Several studies have explored the non-Gaussian nature of the signal in this epoch, both using higher-order moments, such as the bi-spectrum (Saxena et al. 2020; Hutter et al. 2020; Watkinson et al. 2021, 2022), as well as the morphology (Chen et al. 2019; Gazagnes et al. 2021; Giri & Mellema 2021; Kapahtia et al. 2021). The bi-spectrum is the most sensitive \nwhen non-Gaussianity is weak, which is not the case during the CD and EoR. \nWhat complicates the application of summary statistics is the lack of a closed analytical formulation of the fluctuations during this epoch (see, for example, McQuinn & D'Aloisio (2018); Qin et al. (2022) for recent attempts). This stands in contrast to the post-EoR epoch and the 21-cm signal from the Dark Ages. Thus, both the modelling and analysis of the signals during the CD and EoR epochs heavily rely on simulations. Consequently, the non-linear nature of the processes governing astrophysics and cosmological evolution, as well as the resulting non-Gaussian signal, are challenging to qualify analytically. Developing a standardised estimator that optimally captures this information is likely impossible. Fortunately, with the advancement of machine learning (ML), it has become possible to infer complex relationships between different datasets, enabling us to qualify data beyond its summary statistics. \nApplying ML techniques to cosmological and astrophysical problems is a rapidly developing field. For example, in the context of 21-cm data, deep-learning-based methodologies were shown to be successful in identifying ionised regions in the presence of challenging astrophysical foreground and instrumental noise contamination (Gagnon-Hartman et al. 2021; Bianco et al. 2021, 2024), in applying sample generation and parameter inference (Schmit & Pritchard 2017; Zhao et al. 2022; List & Lewis 2020) and reconstructing the density fields (Villanueva-Domingo & Villaescusa-Navarro 2021). \nOne can think of the 21-cm field during the CD and EoR as a product of cosmological and astrophysical evolution (Madau & Rees 2004). The key ingredients of the 21-cm brightness signal are the spin temperature 𝑇 𝑆 , the ionization fraction 𝑥 HI and baryonic matter density fluctuations 𝛿 𝑏 . Optimal use of the data for astrophysical and cosmological inference would benefit from separating astrophysical quantities 𝑇 𝑆 and 𝑥 HI fromthecosmological signal 𝛿 𝑏 , and vice versa. The same reasoning applies when considering the astrophysics hidden in the signal. It is known that at the level of the power spectrum, parameters that are associated with cosmology and astrophysics can have strong degeneracies, which can only be partially broken when considering measurements of the power spectrum at several redshifts (see e.g., Greig & Mesinger 2017). Given the non-Gaussian nature of the 21-cm signal during the CD and EoR, it is only natural to explore whether non-Gaussian information is capable of reducing these degeneracies and providing additional information. This is the main motivation behind this paper. \nIn particular, inspired by applications of machine learning aimed at density-field reconstruction (Flöss & Meerburg 2024), we develop a three-dimensional field-to-field reconstruction network that is capable of reconstructing the three fields 𝑇 𝑆 , 𝑥 HI and 𝛿 𝑏 directly from the brightness fluctuations 𝛿𝑇 𝑏 , i.e. the observed field. The network utilizes the full non-Gaussian information in the observed field to reconstruct the other fields. We show that the information in the power spectra of these fields is complementary to the power spectrum of 𝛿𝑇 𝑏 , and if combined, they could lead to improved parameter constraints. In this paper, we consider a simple set-up focused on a single redshift, but considering a range of reionization histories simulated using the semi-numerical public code 21cmFAST (Mesinger et al. 2011; Greig & Mesinger 2015; Park et al. 2019). Performing parameter forecasts and exploring applications of the reconstructed fields will be presented in future publications. As such, this paper serves as a proof-of-principle. \nOur paper is organized as follows. In section 2, we briefly describe the difference between the observable 21-cm signal and the underlying physical fields and motivate using a simple two-parameter Fisher forecast why reconstructing the field could benefit parameter infer- \nence. In section 3, we describe the simulations used for training and testing of the neural network and provide details on its architecture and the training procedure. In section 4, we present the results of the reconstruction and qualify the performance of the network by measuring the cross-spectra. Finally, we discuss limitations and discuss future directions in section 5.", '2 THE 21-CM SIGNAL OF NEUTRAL HYDROGEN': "In this section, we summarise the brightness temperature field's dependence on the neutral hydrogen fraction, its spin temperature, and the baryonic overdensity field. Through a straightforward Fisher information analysis, we demonstrate that the power spectra of these fields provide additional information beyond that contained in the power spectrum of the brightness temperature field alone.", '2.1 The Brightness Temperature Field': "The observed 21-cm brightness temperature is defined by the difference between the spin temperature 𝑇 𝑆 and the Cosmic Microwave Background (CMB) temperature 𝑇 CMB , modulated by the Sobolov optical depth along the line of sight. It can be approximated as follows (Madau & Rees 2004; Furlanetto et al. 2006): \n𝛿𝑇 𝑏 = 𝑇 𝑆 -𝑇 CMB 1 + 𝑧 ( 1 -e -𝜏 𝜈 ) ≈ 27 𝑥 HI ( 1 + 𝛿 𝑏 ) GLYPH<18> Ω 𝑏 ℎ 2 0 . 023 GLYPH<19> GLYPH<18> 0 . 15 Ω 𝑚 ℎ 2 1 + 𝑧 10 GLYPH<19> 1 / 2 × GLYPH<18> 𝑇 𝑆 -𝑇 CMB 𝑇 𝑆 GLYPH<19> GLYPH<20> 𝜕 𝑟 𝑣 𝑟 ( 1 + 𝑧 ) 𝐻 ( 𝑧 ) GLYPH<21> mK , (1) \nHere, 𝑥 HI is the neutral-hydrogen fraction, 𝛿 𝑏 is the fractional overdensity of baryonic matter, and 𝜕 𝑟 𝑣 𝑟 is the velocity gradient along the line of sight. The spin temperature measures the number of hydrogen atoms in the two hyperfine spin-states (parallel or antiparallel). During the initial period after the CMB has formed, the CMB and the gas temperature are tightly coupled, and a balance of collisional and radiative transitions is set, 𝑇 𝑆 = 𝑇 CMB = 𝑇 𝑔 . As the universe expands, the CMB photons decouple from the gas while collisions couple the spin temperature to the gas temperature, which cools adiabatically. This is the moment ( 𝑧 ∼ 200) when the 21-cm brightness can be seen in absorption against the CMB. As the gas dilutes further, collisions become inefficient, and the spin temperature approaches the CMB temperature ( 𝑧 ∼ 50). When the first stars form, Lyman𝛼 photons again couple the spin temperature to the gas, leading to a peak in the brightness temperature (still in absorption) at 𝑧 ∼ 20, until X-rays from the first stars heat the gas during the EoR. The brightness temperature can then be observed in either absorption ( 𝑇 𝑆 < 𝑇 CMB ) or emission ( 𝑇 𝑆 > 𝑇 CMB ). During most of the EoR, 𝑇 𝑆 > 𝑇 CMB , and hence the brightness temperature is positive. This is the regime we will study in this paper. We refer to Pritchard & Loeb (2012) and Mesinger (2019) for a more detailed review of the above processes. \nThe primary focus of nearly all studies has been the summary statistics of the 𝛿𝑇 𝑏 field. When observing the 21-cm signal, the 𝛿𝑇 𝑏 field is the only direct observable. As shown by Eq. (1), however, the brightness temperature is related to three other fields 𝑇 𝑆 , 𝑥 HI and 𝛿 𝑏 . Note that in principle, it also couples to fluctuations in the CMB temperature, but these are typically negligible since Δ 𝑇 CMB / 𝑇 CMB ∼ O( 10 -5 ) . The evolution of the fluctuations in these \nFigure 1. Fisher forecast for 𝜁 , 𝛼 X and 𝑇 vir derived from the power spectrum of the brightness temperature field (blue), the spin temperature field (red), and the ionization field (green). We used cosmic variance limited power spectra in the range 0 . 04 h / Mpc ≤ k ≤ 0 . 6 h / Mpc. The power spectra were obtained from simulations at 𝑧 = 15 obtained with the semi-numerical code 21cmFAST , fixing all other parameters. The derivatives were computed using finite difference derivatives. This simple Fisher analysis shows how different fields respond to reionization parameters and whether the information in these fields is independent. Combining their information would break degeneracies and improve parameter constraints. \n<!-- image --> \nfields is determined by complex cosmological and astrophysical processes. The baryonic matter density perturbations, 𝛿 𝑏 , grow through gravitational interactions but, on small scales, are expected to be affected by astrophysical feedback effects. The neutral fraction of hydrogen (and its fluctuations), 𝑥 HI , is determined by photoionization by early stars and quasars. The spin temperature field, 𝑇 𝑆 , is defined by collisional coupling in dense regions, coupling with CMB radiation in lower kinetic temperature regions, X-ray heating, and Ly𝛼 pumping mechanism (for details see e.g. Barkana & Loeb 2001; Furlanetto et al. 2006; Pritchard & Loeb 2012; Mesinger 2019). \nModelling these effects analytically is challenging (see e.g. McQuinn & D'Aloisio 2018; Qin et al. 2022) and we rely on numerical simulations to make predictions about the brightness temperature field, which relies on 𝑇 𝑆 , 𝛿 𝑏 and 𝑥 HI . Full numerical codes that include ray-tracing are computationally expensive, and to render the number of cosmological boxes necessary for parameter inference, or in this case, for the training of a neural network, is infeasible at the moment. Hence, for that purpose, we use the semi-numerical code 21cmFAST (Mesinger et al. 2011; Park et al. 2019). It starts with cosmological initial conditions rendered using 2LPT using standard cosmological parameters. The code models the formation and evolution of dark-matter halos using the Press-Schechter formalism, incorporates star formation and feedback processes within these halos, and calculates the propagation of ionizing radiation from stars and galaxies. Key parameters are the ionizing efficiency 𝜁 , representing the number of ionizing photons emitted per baryon in collapsed structures and 𝑇 vir , which sets the conditions under which gas can col- \nwithin dark matter halos and 𝛼 X which quantifies how efficiently X-ray photons can heat the surrounding intergalactic medium (IGM) compared to ionizing photons produced by stars and galaxies.", '2.2 Fisher Information': 'In this subsection, we show the information these fields contain on three astrophysical parameters to motivate our effort to reconstruct the fields determining the observed brightness temperature. \nIn Figure 1, we show the one- and two-dimensional marginal constraints determined by performing a simple Fisher forecast (Fisher 1935) derived directly from 21cmFAST simulations using the power spectra of 𝛿𝑇 𝑏 (blue contours), 𝑇 𝑆 (red) and 𝑥 HI (green). Because we focus on astrophysical parameters rather than containing cosmology (as done by Flöss & Meerburg 2024), we can ignore the baryonic density field. The results are obtained for a noiseless cosmic-variancelimited experiment, and the power spectra are measured at redshift 𝑧 = 15 using 256 3 resolution boxes with a side scale of 1024 Mpc/h, ensuring 𝑇 CMB / 𝑇 𝑆 is not so small such that the effect of 𝑇 𝑆 is negligible. \nWenotice that the X-ray spectral index parameter, 𝛼 X , has stronger constraints from the 𝑇 𝑆 power spectrum than from 𝛿𝑇 𝑏 due to stronger 𝑇 𝑆 coupling with the X-ray background (Field 1959; Mesinger et al. 2011). The parameter that defines the ionizing efficiency of UV radiation from early galaxies 𝜁 is constrained stronger by the 𝑥 HI field. The minimal virial temperature parameter log 10 ( 𝑇 min vir ) has similar constraints from 𝑇 𝑆 and 𝛿𝑇 𝑏 . However, due to different types of degeneracies with the 𝛼 X parameter, the constraints could be optimised by combining information from these fields (after properly accounting for their covariance). This simple parameter Fisher forecast shows that if one were to have access to these additional fields, a simple summary statistic, such as their power spectra, could already improve astrophysical parameter constraints. \nHowever, since we do not have direct access to these fields, we explore the potential of a deep learning-based algorithm for the reconstruction of the 𝑇 𝑆 , 𝑥 HI and 𝛿 𝑏 using only the observed 𝛿𝑇 𝑏 field. Note that the information to reconstruct these fields must be derived from the non-Gaussian nature of the brightness temperature. In principle, all information is present in this one observable field, but this information is contained in multiple higher-order moments. If we were able to capture and model that information correctly, the resulting constraints would be more optimal. However, given the challenge of both modelling and measuring these higher-order moments, a direct reconstruction would allow for a relatively straightforward analysis using the power spectra after reconstruction.', '3 METHODOLOGY': 'In this section, we shortly describe the simulations of the brightnesstemperature field and the neural network used to infer the neutralfraction, baryonic-density and spin-temperature fields from the brightness-temperature field.', '3.1 Simulations': "We aim to recover underlying physical fields using supervised machine learning based on the information present in the observable. As a first attempt, we will use simulations without including any observational and instrumental effects. Therefore, the simulations used in this work do not contain any noise, and the information \nFigure 2. Architecture of the neural network used for reconstruction. Image credit: Flöss & Meerburg (2024) \n<!-- image --> \nabout the physical processes is limited only by the quality of the simulations, their size, and the binning effect. We leave that to a future analysis. We simulate the four fields using 21cmFAST . For the training and validation set, we generate simulations with different astrophysical parameters: 𝜁 ∈ { 28 , 32 } , 𝛼 x ∈ { 0 . 935 , 1 . 065 } , and log 10 ( 𝑇 min vir ) ∈ { 4 . 67 , 4 . 69 , 4 . 71 , 4 . 73 } . We chose parameters whose constraints could benefit from the field reconstruction as shown in Figure 1. The chosen range of parameters lies within 1 𝜎 of the parameter constraints forecast for upcoming experiments (e.g. Saxena et al. 2023). The test data is generated with parameter values between the test grid points: 𝜁 = 30, 𝛼 x = 1, and log 10 ( 𝑇 min vir ) = 4 . 7, which corresponds to the 'FAINT GALAXIES' model of Greig & Mesinger (2017). With this set of simulations, we determine to what extent our network can interpolate to models that are not part of the training set. A more elaborate study of reconstruction quality as a function of parameter space is a topic of ongoing investigation. The range considered here should again serve as a proof-of-principle. \nWe chose to study the reconstruction in a regime close to the maximumaverage brightness temperature. Therefore, we simulate all fields at a redshift 𝑧 = 15. Each simulation data cube has a resolution of 256 3 voxels and a physical side size of 1024 Mpc/h. Besides 𝛼 X , 𝑇 vir and 𝜁 , all other astrophysical and cosmological parameters in our simulations were set to the default values of 21cmFAST . All simulations used for training, validation and testing are generated with different randomised initial realisations (seeds).", '3.2 U-net architecture': 'For the reconstruction of the physical fields, we use the neural networkwiththesamearchitecture as was proposed in Flöss & Meerburg (2024), which was inspired by the work of Gagnon-Hartman et al. (2021). The reconstruction algorithm starts with the construction of \nsix additional fields 1 for each input 𝛿𝑇 𝑏 field: \n𝒖 ( 𝒙 ) = ∫ 𝑑 3 𝒌 -𝑖 𝒌 𝑘 2 𝛿𝑇 𝑏 ( 𝒌 ) 𝑒 𝑖 𝒌 · 𝒙 (2) \n𝜕 𝑖 ( 𝒖 ) 𝑗 ( 𝒙 ) = ∫ 𝑑 3 𝒌 𝑘 𝑖 𝑘 𝑗 𝑘 2 𝛿𝑇 𝑏 ( 𝒌 ) 𝑒 𝑖 𝒌 · 𝒙 , (3) \nwhere 𝛿𝑇 𝑏 ( 𝒌 ) is a Fourier transform of the temperature brightness field. It was shown by Flöss & Meerburg (2024) that adding the gradient fields improve reconstruction. \nThe neural network architecture utilised in our study consists of an initial convolutional layer with a 3 × 3 × 3 kernel size, followed by a context block consisting of two additional convolutional layers with the same kernel size. A residual connection is then applied, combining the output of the initial convolution with that of the context block. This process is repeated, with the initial convolutional layer now employing a stride of two, resulting in the down-sampling of feature maps by a factor of two in each spatial dimension. This down-sampling procedure is iterated three more times until the feature maps reach a size of 16 × 16 × 16 cells. Subsequently, the feature maps are upsampled by a factor of two, and the outputs of the residual connections from the down-sampling phase are concatenated, creating skip connections. These concatenated feature maps are then fed into a localisation block, comprising a 3 × 3 × 3 ordinary convolutional layer followed by a 1 × 1 × 1 convolutional layer, with this process repeated until the input size reaches 256 × 256 × 256. Finally, one more convolutional layer of size 1 × 1 × 1 is applied to generate a single feature map representing the reconstructed field. Throughout the architecture, except for the final convolutional layer, each layer consists of convolution, instance normalisation, and leaky ReLU activation functions. \nmK \nFigure 3. Slices of the four fields projected along the third dimension of the data cube. First column: the input brightness temperature field. Second column: corresponding target field. Third column: reconstructed field. Fourth column: the difference between the reconstructed and true field. The first row shows the reconstruction of the spin temperature field 𝑇 𝑆 , the second row shows the reconstruction of the ionization fraction 𝑥 HI, and the third row shows the reconstruction of the baryonic matter fractional overdensity 𝛿 𝑏 . Residuals are very small, indicating high-quality reconstruction for all three target fields. \n<!-- image -->', '3.3 Training': "The U-net is trained independently for each type of the reconstructed physical field due to memory limitations. Thus, we train three independent neural networks in this work. For each network, we use 240 simulations of 𝛿𝑇 𝑏 field data cubes as input dataset and corresponding 𝛿 𝑏 , 𝑇 𝑠 , and 𝑥 HI as output datasets. The training set contains 15 field realisations for each point in parameter space. These simulations are used to optimise neural network parameters using a mean square error loss function: \nL MSE = 1 𝑁 sim 1 256 3 𝑁 sim ∑︁ 𝑖 = 1 GLYPH<0> 𝐹 𝑖, true -𝐹 𝑖, rec GLYPH<1> 2 , (4) \nwhere 𝐹 𝑖, true and 𝐹 𝑖, rec are the true target and reconstructed field, respectively, and the difference is calculated between individual voxel values. The loss function is normalised by the number of voxels and simulations in the set 𝑁 sim . The validation dataset consists of 48 simulations (three per parameter-space point), and the test dataset consists of 90 simulations with one fixed parameter set as discussed in section 3.1. \nFor the spin temperature field, 𝑇 𝑆 , we cap the field's maximum value at the 99.5th percentile. This limit is essential to prevent instability in the learning process that arises when the U-net reconstructs very large localised peaks and affects only a few voxels. This adjustment has a negligible impact on the power spectrum and other statistical properties of the fields. Finally, for training purposes, we \nnormalize fields using the standard deviation of the training set fields and subtract the mean value of the training fields. The true values of the reconstructed fields are recovered through inverse re-scaling and applying a corresponding shift.", '4 RESULTS': 'In this section, we present a representative reconstruction of the neutral-fraction, baryonic-density, and spin-temperature fields from a random brightness-temperature field, which was given as input to the trained U-net. We visually inspect these, analyse their power spectra, and assess their cross-coherence with the true underlying fields. Again, we emphasise that this is proof of concept, and a full exploration of a wider range of redshifts and model parameters is left for future analyses.', '4.1 Visual inspection': "To qualitatively assess the reconstructed fields, we first perform a visual comparison. Representative examples of the network reconstruction are shown in Figure 3. In the first column, we show a slide of the input 𝛿𝑇 𝑏 field along one of the data cube's axes. We note that these are the same for each row but shown for ease of comparison with the other fields. The second column shows the true target fields 𝑇 𝑆 , 𝑥 HI , and 𝛿 𝑏 in the first, second and third rows, respectively. \nThe third column shows the results of the reconstruction, obtained with a U-net for each field. The difference between the true and the reconstructed field is shown in the fourth column. \nRemarkably, there are no strong visual imperfections in residuals across all the simulations used in training, validation and tests. Residuals do not feature apparent structure or inhomogeneities. The 𝑥 HI residuals show some minor localised peaks. These peaks are associated with a small misalignment of the positions of ionised regions between the reconstructed and the true field. However, due to the small-scale nature of these features, they do not introduce significant errors on the scales that forthcoming experiments are expected to probe with reasonable signal-to-noise (i.e., 𝑘 ≤ 0 . 5 h/Mpc).", '4.2 Power spectra of the reconstructed and true fields': 'To quantitatively assess the reconstructed fields, we compute the power spectrum of the reconstructed field and compare it to the power spectrum of the target field. The power spectrum of the binned three-dimensional fields is calculated using the following estimator: \n𝑃 ( 𝑘 𝑖 ) = ⟨| 𝐹 ( 𝒌 )| 2 ⟩ = 1 𝑁 𝑘 𝑖 ∑︁ 𝒌 ∈ 𝑘 𝑖 ± Δ 𝑘 | 𝐹 ( 𝒌 )| 2 , (5) \nwhere 𝐹 ( 𝒌 ) is a Fourier transform of the field 𝐹 . The square magnitude of the Fourier transform is calculated as the spherical layer average in the wavenumber space defined by the radius 𝑘 𝑖 , with a width Δ 𝑘 equal to the fundamental Fourier mode of the data cube. The number of Fourier modes in the spherical layer 𝑁 𝑘 𝑖 is taken into account as a normalization factor. \nWe present the power spectra in Figure 4, averaged over the test dataset. From left to right, the panel shows the statistical properties of the 𝑇 𝑆 field, the 𝑥 HI field, and 𝛿 𝑏 field, respectively. The power spectra of the reconstructed fields (shown in red lines) in all three cases deviate from true target power spectra (blue dashed lines) by less than 10% across the entire range of scales. We also show the residual power spectra (black lines). The residual power is ≤ 7% of the field power spectrum for 𝑘 ≤ 0 . 5 h / Mpc. The 𝑥 HI residual power spectrum increases on small scales (high 𝑘 values) as expected from the localized peaks in the residual. Despite this behaviour of the residual, the deviation of the reconstructed power spectrum is ≤ 2 . 5% in the case of 𝑥 HI field (black dashed line in the bottom panel of Figure 4). Similarly, the reconstructed power spectra of 𝑇 𝑆 and 𝛿 𝑏 deviate ≤ 7%. Overall, the reconstructions are remarkably good.', '4.3 Cross-coherence between the reconstructed and true fields': "While the previous section showed that the U-net reconstruction method successfully recovers the power spectra of the target fields, a more ambitious goal is to recover the target fields on a pixel-to-pixel level. This is possible because the network relies on a pixel-by-pixel comparison of the loss function used in training (Eq. (4)). To further quantify the quality of the reconstructed fields, we compute the cross-coherence between the target (i.e. true) and the reconstructed field. Similar to the power spectra, the coherence is defined as the normalised cross-power spectrum, as follows: \n𝐶 𝑖, 𝑗 ( 𝑘 ) = ⟨ 𝐹 𝑖 ( 𝒌 ) 𝐹 ∗ 𝑗 ( 𝒌 )⟩ 2 ⟨| 𝐹 𝑖 ( 𝒌 )| 2 ⟩⟨| 𝐹 𝑗 ( 𝒌 )| 2 ⟩ = 𝑃 2 𝑖, 𝑗 ( 𝑘 ) 𝑃 𝑖 ( 𝑘 ) 𝑃 𝑗 ( 𝑘 ) . (6) \nHere 𝑃 𝑖, 𝑗 is cross-power spectrum of field 𝑖 and field 𝑗 , 𝑃 𝑖 and 𝑃 𝑗 are auto-power-spectra of fields 𝑖 and 𝑗 , obtained using Eq. (5), respectively. Coherence values of unity correspond to identical target \nand reconstructed fields, and values of zero correspond to completely uncorrelated target and reconstructed fields. \nWe show the coherence between reconstructed and target fields (black lines), as well as the coherence of the target with the input field 𝑇 𝑏 (blue), in Fig. 5. The reconstructed fields in the test dataset have a mean coherence of ≥ 0 . 92 for scales 𝑘 ≤ 0 . 5h / Mpc with the target field. The coherence increases on larger physical scales, which is in agreement with the trend in the power spectrum residual. It is also indicative that non-Gaussian information is being used to reconstruct these scales. The rapid decrease in cross-coherence on smaller scales ( 𝑘 ≥ 0 . 15h / Mpc) between the reconstructed field and target field shows that they are not simply scaled versions of each other. \nTo test that a single point in the reconstructed field depends on a wide range of scales and surrounding points, we examined the network's reconstruction capabilities by applying cuts in Fourier space, thereby removing certain scales from the input field similar to Gagnon-Hartman et al. (2021). As anticipated, the quality of the reconstruction deteriorates, showing sensitivity to the specific scales being excluded. When large scales (small 𝑘 values) are removed, the networks can still reconstruct modes absent in the observed fields, indicating the importance of small, non-Gaussian scales in the reconstruction process. Conversely, removing small scales hampers the reconstruction, as it eliminates critical non-Gaussian information. A comprehensive analysis of the specific scales contributing to the reconstruction will be addressed in future work.", '5 DISCUSSION AND CONCLUSIONS': 'We have presented a deep learning-based (U-net) method that successfully reconstructs all three physical fields (i.e., baryonic density, neutral fraction and spin-temperature) that largely determine the observed 21-cm brightness temperature field of neutral hydrogen during the Epoch of Reionization and Cosmic Dawn. The motivation to build such a network stems from the fact that these fields have different responses to both astrophysical and cosmological parameters, making it possible to independently harvest the information in these fields to improve parameter constraints. The network utilises the non-Gaussian information present in the brightness temperature field. This non-Gaussian information non-trivially couples modes at different scales and the networks use this coupling to reconstruct the target fields. This is most evident from the fact that the observed field 𝑇 𝑏 has poor cross-correlation with the target fields on small scales, while after reconstruction, we find very high cross-correlation on those same scales. Since this is not simply an overall re-scaling of the field, the information is recovered from non-linear relations, i.e. from non-Gaussian information present in the brightness temperature data cube. Concretely, we find the following: \n- · The power spectra of the reconstructed spin temperature 𝑇 𝑆 , neutral hydrogen fraction 𝑥 HI and baryonic matter density perturbation 𝛿 𝑏 field - simulated with the semi-numerical code 21cmFAST -deviate ≤ 7% at 𝑘 = 0 . 5 h/Mpc from the power spectra of the true target field, and even less on larger scales.\n- · The cross-coherence of the reconstructed fields with the true target fields is found to be 𝐶 ≥ 0 . 92 on scales 𝑘 < 0 . 5h / Mpc, much higher than the cross-coherence with the brightness-temperature field. This confirms that the reconstruction extends beyond the similarity of statistical properties but features a high degree of pixel-topixel accuracy of the reconstructed field. \nThe primary conclusion of our proof-of-principle analysis is that \nFigure 4. Top panel: Power spectrum of reconstructed field (red), true target field (blue dashed), and their residual (black). Left: spin temperature field; central: neutral hydrogen fraction; right: baryonic density fluctuations. Bottom panels: fractional deviation of the reconstructed field power spectrum from the target true field power spectrum. \n<!-- image --> \nFigure 5. Cross-correlation (coherence) between the target field from the test dataset brightness temperature fields (blue) and the reconstructed field (black). The left panel is the spin temperature field; the central panel is the hydrogen neutral fraction; the right panel is the density field fluctuations. Curves represent the mean value for the entire test dataset. On large scales, the brightness temperature is coherent with all other fields. On small scales, the brightness temperature becomes more decoherent. After reconstructing the target fields, we obtain significant coherence up to 𝑘 ∼ 0 . 5 Mpc/h. \n<!-- image --> \nthe redshifted 21-cm brightness temperature field retains most of the information regarding the three underlying fields: baryonic matter density, neutral fraction, and spin temperature. These fields can be accurately reconstructed using a neural network that effectively disentangles them from a single input field. \nHowever, several additional steps are necessary to fully evaluate the practical applications of these methods. First, it is crucial to assess whether the inclusion of realistic noise, instrumental effects, and foregrounds impacts the reconstruction quality. Second, performing a Fisher analysis on the reconstructed fields is essential to determine their utility for parameter inference, as demonstrated by Flöss & Meerburg (2024). Third, our current setup focuses on a relatively high redshift of 𝑧 = 15, where the ionized regions are small 2 . Some preliminary tests, however, show that the reconstruction quality diminishes when large ionized patches are present in the observed field. To mitigate this, one possible approach is to complement the 21-cm brightness measurements with other line tracers, such as OIII or CO , which exhibit significant fluxes in ionised regions. Integrating \nthese tracers into the reconstruction process would be interesting to examine. \nIn addition to using the reconstructed fields for parameter constraints, another promising application is their use in crosscorrelation studies. For instance, inhomogeneous (patchy) reionization induces secondary fluctuations in the CMB. It is feasible to reconstruct the associated patchy 𝜏 map, which serves as a proxy for the ionization power spectrum integrated along the line of sight. According to Eq. (1), the 21-cm brightness temperature is sensitive to 𝑥 HI . Cross-correlations between the brightness temperature and the reconstructed patchy 𝜏 signal have been explored in the literature (Meerburg et al. 2013; Roy et al. 2020). However, the line-of-sight mode is typically lost when foregrounds are either avoided or removed, complicating the direct measurement of this cross-correlation. Based on our analysis, it is likely that the reconstructed 𝑥 HI field, even with large-scale cuts to account for foreground removal or avoidance, may be less affected by this issue. Moreover, the reconstructed 𝑥 HI field is expected to have a high coherence with the patchy 𝜏 signal, given that both measure the same underlying field, albeit one integrated along the line of sight. \nWe are currently investigating this and other potentially valuable applications, which we aim to report in future publications.', 'ACKNOWLEDGEMENTS': 'We link to thank Thomas Flöss, and Anchal Saxena for useful discussions and initial collaboration on the project. Furthermore, we like to thank the Center for Information Technology of the University of Groningen for their support and for providing access to the Habrok high-performance computing cluster. BB is supported by the Fundamentals of the Universe research program at the University of Groningen. LVEK acknowledges the financial support from the European Research Council (ERC) under the European Union\'s Horizon 2020 research and innovation programme (Grant agreement No. 884760, "CoDEX").', 'DATA AVAILABILITY': 'Accompanying code is available at https://github.com/ bidenkobd/reconstruction/ . The data underlying this article will be shared on reasonable request to the corresponding author.', 'APPENDIX A: CROSS-SEED TEST': "When training our network, we use randomised seeds (phases) of the initial density field. This ensures that the network does not 'learn' a particular realisation of the fields but instead must learn the correlation structure. To verify that the randomisation of the seeds is sufficient and that the network does not pick up any recurring structure, we computed the correlation coefficients (coherence) of the true fields ( 𝑇 𝑆 , 𝑥 HI , and 𝛿 𝑏 ) generated from different initial seeds and compared these to the correlation coefficients of the reconstructed fields, derived from applying the network to the simulated 𝛿𝑇 𝑏 sourced with randomised seeds. The results are shown in Fig. A1. \nOverall, the correlation coefficient is ≪ 1, as expected, given that these fields have different phases. Any correlation structure should \nFigure A1. Coherence between simulated fields with different initial conditions (black) and corresponding reconstructed fields (red) \n<!-- image --> \nbe due to the chance correlation of the two random fields. The reconstructed fields also exhibit similar behaviour, where the chance correlations almost perfectly match those of the true fields. Only for the ionization field is there a small decrease in correlation after reconstruction on scales 𝑘 > 0 . 3 ℎ Mpc -1 . \nThe cause of this increase is not entirely clear. We have explored several hypotheses, but none adequately explains this behaviour. The effect is extremely small and may be simply a result of the coherence between the true field and the reconstructed fields dropping towards smaller scales. However, the absence of this effect in the other two reconstructed fields is somewhat puzzling. Further investigation is necessary to rule out any issues with the network fully. Because the effect is minor and has a negligible impact on the quality of the reconstruction, we will defer this investigation to future work. construction, we will leave this for future work. \nThis paper has been typeset from a T E X/L A T E X file prepared by the author."}

2024arXiv240909977B

Measurements of the redshift drift the real time variation of the redshift of distance sources are expected in the next couple of decades using next generation facilities such as the ANDES spectrograph at the ELT and the SKAO survey. The unprecedented precision of such observations will demand precise theoretical and numerical modeling of the effect in the standard LambdaCDM cosmology. In this work we use the Gadget4 Nbody code to simulate the redshift drift and its fluctuations in LambdaCDM cosmologies deriving the corresponding power spectra from a simulation with 10243 particles in a 1textrmGpch1 box. Our results provide an estimate for the distribution and amplitude of the fluctuations and the spectra which match previous work in the literature using EinsteinBoltzmann solvers to within an order of magnitude. Our work provides a methodology for performing statistical analysis of the redshift drift effect and deriving its fluctuation power spectra from future large scale surveys.

2024-09-01T00:00:00Z

['2024arXiv240909977B', '10.48550/arXiv.2409.09977', 'arXiv:2409.09977']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Redshift Drift fluctuations from Nbody simulations

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09977.pdf

{'Redshift Drift fluctuations from N -body simulations': 'Pedro Bessa 1 Valerio Marra 2 , 3 , 4 Tiago Castro 3 , 4 , 5 \n1 PPGCosmo, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil \n2 Dep. de Física, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Brazil \n3 INAF - Osservatorio Astronomico di Trieste, 34131 Trieste, Italy \n4 IFPU - Institute for Fundamental Physics of the Universe, 34151, Trieste, Italy \n5 INFN - Sezione di Trieste, I-34100 Trieste, Italy \nE-mail: pedvbessa@gmail.com \nAbstract. Measurements of the redshift drift - the real time variation of the redshift of distance sources - are expected in the next couple of decades using next generation facilities such as the ANDES spectrograph at the ELT and the SKAO survey. The unprecedented precision of such observations will demand precise theoretical and numerical modeling of the effect in the standard ΛCDM cosmology. In this work, we use the Gadget4 N -body code to simulate the redshift drift and its fluctuations in ΛCDM cosmologies, deriving the corresponding power spectra from a simulation with 1024 3 particles in a 1Gpc h -1 box. Our results provide an estimate for the distribution and amplitude of the fluctuations and the spectra, which match previous work in the literature using Einstein-Boltzmann solvers to within an order of magnitude. Our work provides a methodology for performing statistical analysis of the redshift drift effect and deriving its fluctuation power spectra from future large scale surveys. \nKeywords: redshift drift, numerical simulations \nArXiv ePrint: \n1234.56789', '1 Introduction': "As cosmology develops further into its precision era, the theoretical and computational modeling of astrophysical and cosmological phenomena becomes ever more important in order to make proper sense of observations, break degeneracies and probe new regimes. The capabilities of next and current generation surveys allow observational cosmology to probe astrophysical and cosmological phenomena previously thought unreachable, and use these new probes to further test the standard cosmological model. Among these new probes and facilities, with the advent of new, state-of-art spectrographs such as ESPRESSO at the ELT [1, 2], and radio surveys such as SKAO [3, 4], we expect to obtain a first measurement of the time variation of the redshift of far away sources, namely the redshift drift, in the next couple of decades. This first measurement would impose constraints on the cosmological parameters of the ΛCDM and its extensions in a model-independent way in the local Universe from real-time cosmology, an important complementary measurement to distance based and cosmographic measurements. \nThe redshift drift provides a new, powerful probe of cosmological dynamics, since it is a model independent test of the Universe's rate of expansion change, relying purely on metric and kinematic quantities. Its theoretical prediction was first derived by Sandage and McVittie [5, 6] as a general consequence of non-static cosmological models. Its detection and measurement through the observation of the Lymanα forest of distant quasars, was proposed by Loeb [7], in what is now known as the Sandage-Loeb test, although measurements from photometric surveys and strong gravitational lenses have also been investigated [8, 9]. \nWith the possibility of measuring this effect in the near-future, and its use as a test of the cosmological model, research has focused on understanding the evolution and redshift dependency of the redshift drift in general cosmological space-times, in particular perturbed Robertson-Walker metrics [10, 11]; its use as a test of dark energy and alternative cosmological scenarios [12]; and as a probe of cosmic homogeneity [13]. Important work has been done on forecasting the measurement of the effect through observations of the Ly α forest by facilities like the ELT [14-16] and the SKAO [17-19]. \nFrom the theoretical point of view, by extending the redshift drift definition to arbitrary spacetimes, the works [10, 12] have allowed the calculation of perturbative and contaminating effects from cosmological inhomogeneities and local structures on the drift. In particular, the authors in [11] derived the gauge-invariant expression for the redshift drift in a perturbed Robertson-Walker spacetime and its power spectrum, which allows one to interpret future observations and its perturbative effects to first order, in full concordance with the standard cosmological model and its independent late and early time tests. \nIn scenarios where direct observational data is limited, simulations and numerical implementations provide a controlled framework to test hypotheses and explore the effects of various cosmological parameters on a given effect or observable. Modeling of the redshift drift and observational errors through N -body simulations was performed first in [20], and its power spectra was first numerically obtained through the use of Einstein-Boltzmann solvers in [11], and using relativistic numerical codes in [21]. There has been, however, a lack of direct modeling of the effect and its power spectra through N -body simulations in the ΛCDM setting, including peculiar velocity and acceleration effects. \nThis work aims to fill this gap in the literature by using Newtonian N -body simulations and its mapping to relativistic cosmology as a model for a Universe with structure, and deriving the scale and redshift dependency of the drift through its power spectra obtained directly from simulations, providing a first estimate of the effect in the ΛCDM model from N -body simulations, as well as consistency check for future survey observations and data of this effect. \nThe paper layout is as following: In Section 2, we review the mathematical background needed to derive the redshift drift fluctuations in a Robertson-Walker cosmology. In Sections 3 and 4, we review the derivation of the redshift drift fluctuations power spectrum in ΛCDM and its approximations, as well as the map between Newtonian perturbations obtained from the N -body simulations and the theoretical relativistic quantities. In Section 5, we detail our simulation parameters and set-up and the numerical derivation of the power spectra. We also detail our error estimation and modeling. In Section 6, we write our conclusions and discuss the results.", '2 Redshift drift and fluctuations': "We model the redshift drift fluctuations using linear cosmological perturbation theory on a homogeneous and isotropic background, given by Robertson-Walker metric. Since we are interested in the concordance ΛCDM model, we focus on the flat, K = 0 case. First, we obtain its expression in flat RW and then generalize it to arbitrary spacetimes, such that the fluctuations become a particular case where the metric is given by the perturbed RW metric. In a flat Robertson-Walker spacetime, with line element given by \nds 2 = -a 2 ( t )[ dt 2 + δ ij dx i dx j ] , (2.1) \nwhere we denote cosmic time by τ and conformal time by t . Using the subscripts o and s to denote measurements at the observer and source, respectively. The expression for the redshift drift can be easily derived through the cosmic time derivative of the redshift [7] \ndz dτ o = d dτ o ( a o a s ) = 1 dτ o [ a o (1 + H o dτ o ) a s (1 + H s ( a s /a o ) dτ o ) -a o a s ] = H o (1 + z ) -H s . (2.2) \nThe expression for the drift above shows that observing the redshift of a source at different observer times τ gives us a measure of its evolution at current time τ O . From (2.2) it is clear that the redshift drift has a positive value when the Hubble parameter increases monotonically, indicating an accelerating phase of the scale factor. At the background level, this is true for any given observer since we have spacetime homogeneity and isotropy, but once perturbations are introduced, the existence of cosmic structures changes the path of light rays through their gravitational potential and peculiar velocities, such that equation (2.2) has to be modified. \nIn [10], the author derived a generalization of expression (2.2) for the redshift drift in an arbitrary spacetime, relying only on the assumption of a 3 + 1 slicing of the spacetime. For an observer o with 4-velocity u µ and a light ray emitted from source S with tangent 4-vector k µ , the redshift drift is measured by the observer is \ndz dτ o = (1 + z ) [ h -∇ e E E ] o -[ h -∇ e E E ] s , with h := 1 3 θ + σ µν e µ e ν -a µ e µ , (2.3) \nwhere we have the usual kinematic quantities - the expansion, acceleration, and traceless expansion tensor - along the observer's congruence, defined respectively by \nθ = ∇ u u, a µ = ∇ µ θ, σ µν = ∇ ( µ u ν ) -1 3 θh µν , (2.4) \nwhere h µν = g µν + u µ u ν . Expression (2.3), in the case of the metric (2.1) simplifies to (2.2). We will use it to derive the expression for the drift in a Universe with perturbations.", '2.1 Redshift drift in perturbed Robertson-Walker': "We now focus on the case of a Universe with inhomogeneities of the matter and metric fields described by small fluctuations relative to the background metric, modeled by a perturbed Robertson-Walker metric. In the longitudinal gauge, the line element for this metric is given by [22] \nds 2 = -a 2 ( t ) 2 (1 + 2Ψ) dt 2 + a 2 ( t )(1 -2Φ) δ ij dx i dx j , (2.5) \nwhere the potentials Φ and Ψ are the gauge-invariant Bardeen potentials. We also write the observer's 4-velocity in this spacetime as \nu µ = 1 a (1 -Ψ , v i ) , v i ≡ ∂ i v, (2.6) \nwith v the velocity potential, and u µ a global defined congruence of comoving observers. \nFollowing fully gauge-invariant cosmological perturbation theory, in [11], the authors obtained the gauge-invariant expression for the redshift drift (2.3) in the perturbed RobertsonWalker metric, which here we express in terms of the longitudinal gauge used in (2.5) \ndz dτ o = (1 + z ) [ -H + ˙ Ψ+¯ n i ∂ i ( ˙ v + H v ) + ˙ H H (Φ -¯ n i ∂ i v ) + ( ˙ H-H 2 H + ¯ n i ∂ i ) ∫ o s dλ ( ˙ Φ+ ˙ Ψ ) ] , (2.7) \nwhere ¯ n i is the space-like vector of the source direction or spatial propagation. In deriving this expression, we neglected observer-dependent monopole and dipole terms, namely the observer's peculiar velocity and gravitational potential, as we are focused in survey and simulation data averaged over sources in the sky. This observer dependent drift and its impact on measurements of the redshift drift has been studied in [23]. \nThe potential terms in expression (2.7) have been shown to be orders of magnitude smaller than the spatial dependent and velocity terms, particularly in the redshift range z ∈ [0 , 1 . 5] we expect to measure the drift effect [11], and thus we assume that they are subdominant to the other perturbative quantities and do not contribute to the overall fluctuations of the drift. Focusing on terms containing the velocity and its time derivatives, and by defining d ¯ z d ¯ τ o as the background value of the redshift drift given by (2.2), we obtain the expression for the perturbations of the redshift drift \nδ ˙ z := dz dτ o -d ¯ z d ¯ τ o d ¯ z d ¯ τ o ≈ -1 H [ ¯ n i ˙ v i -( ˙ H-H 2 H ) ¯ n i v i ] , (2.8) \nwhere we note again that all observer and potential terms were neglected. In the following sections, we will derive the power spectrum of the signal (2.8) as a function of redshift measured by the observer.", '3 Redshift drift power spectrum': "The redshift drift fluctuations power spectrum can be obtained directly from the primordial metric fluctuations power spectrum through the potential and velocity transfer functions and Einstein's equations, and this has been implemented and checked numerically in [11] through Einstein-Boltzmann solvers. Here we briefly describe how one can obtain the redshift-drift power spectra from the primordial perturbations. \nThe primordial spectrum of curvature perturbations P Ψ generated by primordial fluctuations formed through a standard inflationary mechanism in the early Universe can be written as [22] \nk 3 P Ψ ≡ 2 π 2 ( 3 5 ) 2 P R = (2 π ) 3 δ ( k -k ' ) A ( k k ∗ ) n S -1 , (3.1) \nwhere k ∗ is a pivot scale that characterizes the perturbation scale, A is the primordial amplitude of the Bardeen Potential Ψ fluctuations, and n S is the spectral index defining the scale dependency of the perturbations. P R is the scale-invariant primordial spectrum, which is \nrelated to the Fourier transform of the 2-point correlation function of the primordial fluctuations by 2 π 2 ( 3 5 ) 2 P R = k 3 〈 Ψ in ( k )Ψ ∗ in ( k ' ) 〉 . The measured primordial amplitude, obtained via observation of the CMB is the primordial curvature fluctuation amplitude A S , related to A via [22] \nA = 2 π 2 ( 3 5 ) 2 A S . (3.2) \nThe cosmological parameters A S , n S have been measured and constrained through observations of the CMB by the Planck satellite, and an up-to-date review of their values can be found in [24]. The values for these parameters used throughout this paper can be found in table 1. \nThe value of the Bardeen potential at a given redshift Ψ( z ) is related to the primordial fluctuations Ψ in ( k ) through the Transfer Function T Ψ ( k, z ) and the growth factor D + ( z ), measures of the cosmological structure formation up to redshift z , via [25] \nΨ( k, z ) = T Ψ ( k, z )Ψ ini ( k ) (3.3) = 9 10 D + ( a ) a T ( k )Ψ ini ( k ) . \nThe Transfer Function T ( k ) is characteristized by the evolution of the cosmological model and its parameters and can be obtained either analytically through an ansatz [26], or numerically using Einstein-Boltzmann solvers. Einstein's Equations allows one to use the Transfer Function T ( k ) to obtain the fields v , Φ and Ψ at a given redshift from the primordial power spectrum P R using the relations [27] \nV ( k, z ) = T V ( k, z )Ψ ini ( k ) , (3.4a) \nΦ( k, z ) = T Φ ( k, z )Ψ ini ( k ) , (3.4b) \nT Φ = T Ψ , (3.4c) \nT V = 2 a 3Ω m k H 2 0 ( H T Ψ + ˙ T Ψ ) , (3.4d) \nkT Ψ = ˙ T V + H T V . (3.4e) \nApplying these relations to (2.7) and decomposing observed fields in spherical harmonics Y /lscriptm ( n ) over the celestial sphere with unit vector n , the observed angular power spectrum of the redshift drift fluctuations at redshift z S is given by \nC δ ˙ z /lscript ( z s ) = 2 A π ∫ dk k ( k k ∗ ) n S -1 | F /lscript ( k, z s ) | 2 , (3.5) \nwhere the Kernel F /lscript ( k, z S ) is given by \nF /lscript ( k, z s ) ≈ -1 H [ j /lscript ( kr s ) ( ˙ T ψ + ˙ H H T Ψ ) + j ' /lscript ( kr s ) ( ˙ T V -˙ H-H 2 H T V )] , \nwhere j /lscript are the spherical Bessel functions of order /lscript , and r S is the comoving radial distance from the observer to the source. Here, we neglect the time-varying integral terms, which are small, while keeping the potential terms.", '4 Power spectrum from N -body simulations': "To test the theoretical predictions for the redshift drift power spectra C δ ˙ z /lscript ( z s ) obtained in the previous section, we use the Gadget4 1 N -body cosmological code [28] to perform a Newtonian N -body simulation and a modified version of the class Einstein-Boltzmann solver [29, 30] to validate our results, first used in [11] to obtain numerical results for the power spectra. \nIn the longitudinal gauge the metric potentials in (2.5) and the velocities in (2.6) map directly to the Newtonian gravitational potential ψ and the particles' peculiar velocity vector v i [31], allowing us to treat them as the particle fields in a Newtonian N -body simulation. The theoretical prediction for the perturbed redshift drift field (2.7), obtained using firstorder cosmological perturbation theory, can then be directly mapped to the fields from the simulation with no loss in accuracy, as first shown in [31]. To obtain the power spectra at different redshifts from the Gadget4 simulations then, we follow this correspondence, formalized in the dictionary between relativistic and Newtonian cosmologies first defined in [31]. \nThe angular power spectrum at redshift z of an arbitrary 3D field X is related to its power spectrum through the expressions [25]: \nC X /lscript ( z ) = 2 π ∫ ∞ 0 dkk 2 P X obs ( k, z ) W 2 r,/lscript ( k ) = 4 π ∫ ∞ 0 dk k ∆ 2 ( k, z ) W 2 r,/lscript ( k ) , (4.1) \nW r,/lscript ( k ) = ∫ r 0 d ¯ rW (¯ r ) j ( i ) /lscript ( k ¯ r ) , (4.2) \nwhere W (¯ r ) is an arbitrary window function used to filter a certain range of redshifts or scales, and j ( i ) /lscript is the i -th derivative of the spherical Bessel function j /lscript , which appears in the Fourier transform of direction-dependent fields X ( n ). The dimensional power spectrum is related to the dimensionless one by P X obs ( k, z ) = 2 π 2 k 3 ∆ 2 ( k, z ). \nBy taking the field X to be the redshift drift fluctuation, the observed angular power spectrum of the redshift drift derived from an N -body simulation is given by \nC δ ˙ z /lscript ( z ) = 4 π ∫ ∞ 0 dk k ∆ 2 ( k, z ) W 2 r,/lscript ( k ) (4.3) \n∆( k, z ) = F { -1 H [ -˙ ψ + n · ˙ v + ˙ H H ψ -( ˙ H-H 2 H ) n · v ]} , (4.4) \nwhere we used the mapping between the relativistic Bardeen potential Ψ and the Newtonian potential ψ , and between the relativistic 4-velocity v µ of the observed and the peculiar velocity of particles v = ( v x , v y , v z ). F denotes the 3D Fourier transform of the redshift drift field, and n is the unit vector on the sphere. \nWe further simplify expression (4.3) by neglecting the potential terms, which are at least an order of magnitude smaller than the velocity terms, to arrive at the final expression \nfor the angular redshift drift power spectrum, given by \nC δ ˙ z /lscript ( x ) = 4 π ∫ ∞ 0 dk k ∆ 2 ( k, z ) W 2 r,/lscript ( k ) , (4.5) \nW r,/lscript ( k ) = ∫ r 0 d ¯ r W (¯ r ) j ' /lscript ( k ¯ r ) , (4.6) \n∆( k, z ) = F { -1 H [ n · ˙ v -( ˙ H-H 2 H ) n · v ]} . (4.7) \nFor scales /lscript /greatermuch 1, it is usual to adopt the Limber approximation, which we detail in appendix A. Using this approximation, (4.1) can be further simplified to \nC X /lscript ( x ) = 2 π ∫ ∞ 0 dk k ∆ 2 ( k, z ) W 2 r,/lscript ( k ) ≈ 1 ( /lscript +1 / 2) 3 ∫ d ¯ r ¯ r ∆ 2 ( /lscript +1 / 2 ¯ r , z ) W 2 (¯ r ) . (4.8) \nFor distance modes satisfying /lscript ≳ /lscript c = ¯ r/ ∆¯ r the Limber approximation has been shown to be accurate to smaller than 1% accuracy levels in comparison to the fully integrated power spectra [22, 25]. We shall make use of this approximation as a consistency check when validating the simulations. \nFurthermore, we use a top-hat window function, which filters modes outside a comoving size ∆¯ r . Modes k such that /lscript ≳ k ∆¯ r 2 π ¯ r are integrated out of the angular power spectra. The explicit form of this window function is given by \nW r (¯ r ) = { 1 / ∆ r for r 1 < ¯ r < r 2 , ∆ r = r 2 -r 1 0 otherwise . (4.9) \nOne can also interpret the window function above as selecting fluctuations inside the redshift slice ∆ z ≈ H ( z )∆ r/c centered on the redshift z . \nThis window function, when inserted in equation (4.8), allows one to simplify further the observed angular power spectrum expression in the Limber approximation. By selecting a small distance slice ∆¯ r of the observed cosmic region, we have \nC X /lscript ( z ) ≈ 1 ( /lscript +1 / 2) 3 ∫ d ¯ r ¯ r ∆ 2 ( /lscript +1 / 2 ¯ r , z ) W 2 (¯ r ) ∆¯ r/ ¯ r /lessmuch 1 -----→ /lscript c ( /lscript +1 / 2) 3 ∆ 2 ( /lscript +1 / 2 ¯ r , z ) . (4.10) \nEquation (4.10) allows simple implementation of the the theoretical power spectrum (4.1), and although we use the FFTlog-and-beyond 2 [32] routine to calculate the integrals [33], at smaller scales the Limber approximation provides consistency check for the numerical power spectra. \nWe note that approximations (4.8) and (4.10) are valid for integrals containing no derivatives of the Bessel functions, such as for the potential terms in (3.5). In the final expression (4.5) the Limber approximation is not a good approximation due to the different oscillating behavior of the spherical Bessel function derivatives.", '5 Simulation setup and derived power spectrum': 'To model the observed power spectra of the redshift drift fluctuations (4.5) we perform a Newtonian N -body simulation using the Gadget4 code. Our simulation was run on a \nTable 1 : Parameters used to generate the N -body simulation with Gadget4 and numerical spectra through Einstein-Boltzmann solvers. \n1 Gpc h -1 Box with 1024 3 particles, and initial conditions were generated via the built-in NGEN-ic code at redshift z = 49. Cosmological parameters, particle mass, and gravitational softening /epsilon1 soft can be found in Table 1. We produced three snapshots at the redshifts z = 0 . 5 , 1 , 2. For the window function, we use a redshift slice of ∆ z = 0 . 1 and a top-hat window function, which corresponds to a comoving size of 0 . 1 c/H ( z ) Mpc for the projected field in equation (4.1).', '5.1 Validation': 'We compute the matter power spectra at redshifts z = 0 . 5 , 1 , 2 with the Pylians3 library 3 [34], and test them against the linear and nonlinear ( halofit ) matter power spectra derived from class . The simulation snapshots are read in HDF5 format, and we employ a triangularshaped cloud mass assignment scheme in Pylians3 to generate the density fields. We plot the results in figure 1. The curves obtained from the simulations match the expected behavior. We subtracted the shot-noise contribution P N = V/N from the power spectrum measured from the simulation ˆ P , that is, we show P = ˆ P -V/N . \nTo validate the routines used to compute the angular power spectra in equation (4.5), we compare the results obtained from the simulation with the FFTlog-and-beyond libraryagainst those from the class code (the classgal module). Additionally, we include the Limber approximation as a consistency check for the angular power spectra derived from the simulations. The results of this comparison are presented in figure 2. The agreement is not perfect because classgal computes the angular power spectrum on the lightcone, while we are using the power spectrum of a given snapshot. We checked the importance of non-linearities by feeding the linear power spectrum from class to the FFTlog-and-beyond routine: we found the expected suppression of power at smaller scales and lower redshifts. We estimate the cosmic variance on the angular power spectrum via Var [ C /lscript ] = 2 C 2 /lscript 2 /lscript +1 . \nFigure 1 : Matter power spectra for redshifts z = 0 . 5 , 1 , 2. In the figures we compare two routines against the power spectra obtained from the Gadget4 simulations: the halofit routine and the linear matter power spectrum from CLASS . The vertical red lines mark 2 k fund and k Nyq / 2. \n<!-- image --> \nTo avoid effects due to finite box size and resolution, we restrict our spectra to modes k ∈ [2 k fund , k Nyq / 2] and angular correlations to modes /lscript ∈ [ 2 /lscript fund = 2 k fund r ( z ) (1+ z ) , /lscript Nyq / 2 = k Nyq r/ 2 (1+ z ) ] , above the fundamental frequency k fun = 2 π/ Box Size and below the Nyquist frequency k Nyq = N grid × k fun .', '5.2 Redshift drift': "To generate the 3D array of the redshift drift field, we discard potential terms and follow equation (4.7) to derive the drift expression from the acceleration and velocity fields. Via Pylians3 we obtain the 1024 3 array with the total drift per voxel. We then divide the total redshift drift field by the particle number field, in order to obtain the average redshift drift field. It can happen that there are empty voxels; in this case we set the average redshift drift to zero. The amount of empty voxels is less than 100 in a total of 2 30 voxels. We then subtract the average redshift drift in the snapshot from the redshift drift field and divide by the same average to obtain the redshift drift fluctuations of (2.8). We plot the redshift drift fluctuation distribution histogram in figure 3 for the three redshifts analyzed, which provide a glimpse at the gaussianity and amplitude of the fluctuations. \nFigure 2 : Angular power spectrum of matter for redshifts z = 0 . 5 , 1 , 2. The power spectra are calculated in three different ways: using the CLASSgal code, using the FFTlog routine to obtain the angular spectra from the matter power spectrum of the simulation, and using the Limber approximation for the angular spectra from the simulation. The vertical red lines mark 2 /lscript fund and /lscript Nyq / 2. \n<!-- image --> \nIn obtaining the power spectrum, we make use of the flat-sky approximation, where one assumes the direction vector n on the sphere is approximated by a vector α , where α is a 2D coordinate vector in the plane perpendicular to the observer's reference direction. This is equivalent to choosing the reference axis z in the simulation box as being the observer direction, as well as assuming that the box size corresponds to a sizeable fraction of the celestial sphere. We further detail this approximation in appendix B, while noticing that for big enough box sizes, it provides an accurate estimate of the angular power spectrum [35]. The power spectrum is then calculated using the Pk\\_library from pylians (same routine that we used to compute the matter power spectrum of figure 1). \nThe shot noise for the drift field can be computed following the formalism of [36] such that \nP δ ˙ z N /similarequal V N 〈 δ ˙ z 2 〉 , (5.1) \nthat is, it is the matter power spectrum shot noise multiplied by the variance of the drift \nFigure 3 : Histogram for the redshift drift fluctuations in the simulation box for redshifts z = 0 . 5 , 1 , 2. The distribution is nearly gaussian with standard deviation σ ≈ 2 · 10 -2 for all redshifts. \n<!-- image -->", 'field. We then have': 'P δ ˙ z = ̂ P δ ˙ z -P δ ˙ z N , (5.2) \nwhere we subtracted the shot-noise contribution from the power spectrum measured from the simulation ˆ P . We plot the dimensionless power spectra ∆ 2 ( k, z ) of the redshift drift in figure 4 with and without subtraction of the shot noise. We can see that the shot-noise subtraction correctly removes the excess power at small scales, k ∼ 2 h /Mpc. √ \nWe see that the amplitude of the redshift drift fluctuations is | δ | ∼ ∆ 2 ≈ O (10 -3 ), and that the power spectrum is stronger at higher redshifts, where the cosmic velocity and acceleration fields in (2.8) are known to be larger, since velocity perturbations decay as a function of cosmic time in ΛCDM [22]. This result is in agreement with the results found in [11] and around two orders of magnitude higher than the results found in [20], which also used a Newtonian simulation to obtain an estimate of the fluctuation amplitude. We note that the results in [20] are based on an Einstein-de Sitter background cosmology, such that some of the discrepancies can be attributed to the effect of both the background cosmology and structure formation in ΛCDM, which is known to be slower due to the late time acceleration. \nFigure 4 : Dimensionless redshift drift power spectra at redshifts z = 0 . 5 , 1 , 2 with and without shot noise. The vertical red lines mark 2 k fund and k Nyq / 2. \n<!-- image --> \nTable 2 : Shot noise for the redshift drift power spectra and the expected noise for the redshift drift experiment with SKA1 and SKA2, taken from [18]. The value for z = 2 is extrapolated from the expected noise at z = 1 . 5. \nFor N particles = 1024 3 and the cosmic parameters given in table 1, the values of the shot noise P δ ˙ z N at the analyzed redshifts can be found in table 2. The expected shot noise in the SKA survey experiment aimed at measuring the redshift drift has been estimated in [18], and by assuming that the survey will be able to precisely measure the fluctuations to constrain the background value of the redshift drift, we can compare the results of table 2 to the error due to shot noise predicted for the SKA1 and SKA2 experiments as found in [11, 18], which is expected to observe N ∼ 10 7 galaxies per redshift bin and the noise term proportional to 1 / √ N . Table 2 shows that the shot noise obtained from our work is small compared to the noise predicted from the survey capabilities, such that one does need to take into account \nFigure 5 : Redshift drift angular power spectra for redshifts z = 0 . 5 , 1 , 2 . The green line is the prediction for the angular power spectrum using the implementation in the class cosmology code. The vertical red lines mark 2 /lscript fund and /lscript Nyq / 2. \n<!-- image --> \nthe shot noise from the drift fluctuations when estimating the power spectrum.', '5.3 Comparison with class and discussion': 'The modified class code used to generate the angular power spectrum C δ ˙ z /lscript operates strictly within linear perturbation theory, meaning that nonlinear effects are not included in the redshift drift power spectra. However, since our simulations probe the nonlinear regime, deviations from linearity appear at smaller scales, leading to modifications in the slope of the angular spectrum compared to the linear spectra produced by class . We normalize the spectra derived from simulations, based on (4.5), to match the convention used in [37], enabling a proper comparison between the amplitude of the N -body angular spectra and the predictions from class . This normalization convention is detailed in Appendix C for reference. \nIn figure 5, we plot both the spectra obtained from the simulations and the spectra generated using the class implementation of the redshift drift fluctuations spectra. We find an agreement within an order of magnitude between both spectra at all scales. The discrepancies are attributed to the flat-sky approximation used in computing the redshift drift component at large scales, the non-linearities at small scales, and to the calculation of \nclustering based on the simulation snapshot drift power spectrum rather than on the lightcone, as also highlighted in figure 2. Previous works on the amplitude and angular power spectrum of the redshift drift fluctuations, using both Newtonian N -body simulations [20] and fully relativistic simulations [21], found a difference of more than two orders of magnitude for the amplitude of the fluctuations in relation to the class implementation found in [11]. \nIn the ΛCDM case, our work points to an effect of at most order 10 -3 in relation to the background redshift drift for angular correlations, in agreement with both [11] and [13]. With respect to the order of magnitude and size of the effects, our work converges with the literature, where a precision of ∆ ˙ z/ ˙ z ∼ 10 -3 is needed in order to measure such fluctuations. Nevertheless, even in our idealized simulation setting, we were able to constrain the amplitude and scaling of the power spectrum of the drift fluctuations through observations of the cosmic velocity and acceleration fields, which together with the galaxy number count spectra, as discussed in [11], could help in detecting a statistically significant signal of the drift by constraining contaminating effects due to the peculiar motion of sources.', '6 Conclusions': 'In this paper, we derive the redshift drift power spectrum at different redshifts from an N -body simulation using the Gadget4 code. To obtain the power spectrum from the simulation, we model it using the theoretical prediction found in [11] and numerically calculate it using the velocity and acceleration fields from the Gadget4 snapshots and power spectrum and FFT routines found in the pylians and FFTlog-and-beyond python libraries. We validate our code and simulation using the Einstein-Boltzmann solver class , and once validated, compare our results to the numerical predictions obtained using the same code in [11]. Our work provides a methodology for modeling the redshift drift using N -body simulations and derive one and two point statistics from simulation data. \nOur findings show an agreement with the numerical implementation of the redshift drift spectra using the class code to within an order of magnitude, with a signal that approximately increases from large to intermediary scales for all redshifts and peaks at scales ∼ O (10 2 ), and strictly decreases afterwards, up to scales /lscript ∼ 10 3 . Our results also show a peak of the signal at the same scales as the class implementation and an overall agreement with the scale behavior of previous work on ΛCDM settings such as [11]. The discrepancies are attributed to the flat-sky approximation used in computing the redshift drift component at large scales, the non-linearities at small scales, and to the calculation of clustering based on the simulation snapshot drift power spectrum rather than on the light-cone. Previous works using N -body simulations to model a measurement of the redshift drift and its fluctuations find a disagreement of more than 2 orders of magnitude in relation to our results and the results of [11], although the authors assumed a different cosmological background and energy content. Further work assuming the same background cosmologies, possibly on the past lightcone, is needed in order to properly compare the results from different numerical simulations of the effect. \nOur work agrees with previous literature in that measuring the fluctuations on the drift significantly impacts the observation of the background effect, in particular in measuring the velocity and acceleration fields distribution over the sky. Although of order 10 -3 on the background drift, which is already small, it is within the capabilities of surveys such as SKA [11, 21]. \nAt last, our simulation assumes an idealized dark matter plus Λ only Universe, with structure formation and peculiar motion of sources with no baryonic effects or feedback, and a simulation box size of 1Gpc. The SKA radio telescope will have a sky coverage of 75% and probe redshifts z ∈ [0 , 5] in the radio, HI and 21cm regions, observing baryonic feedback in regimes previously unseen. Larger simulations, including baryonic and hydrodynamic effects in structure formation and the momentum fields, are to be done to prepare for the upcoming observational challenge in measuring the drift with next-generation surveys. Such a task is beyond the current computational capabilities of the current work, and we leave it for future research.', 'Acknowledgements': 'It is a pleasure to acknowledge Enea Di Dio, Cullan Howlett, and Ruth Durrer for their valuable discussions. PB acknowledges financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Fundação de Apoio à Pesquisa do Espírito Santo (FAPES) for his PhD and visiting PhD fellowship. VM thanks CNPq (Brazil) and FAPES (Brazil) for partial financial support. TC is supported by the Agenzia Spaziale Italiana (ASI) under - Euclid-FASE D Attivita\' scientifica per la missione - Accordo attuativo ASI-INAF n. 2018-23-HH.0, by the PRIN 2022 PNRR project "Space-based cosmology with Euclid: the role of High-Performance Computing" (code no. P202259YAF), by the INFN INDARK PD51 grant, and by the FARE MIUR grant \'ClustersXEuclid\' R165SBKTMA. We acknowledge the computing centre of Cineca and the use of the Santos Dumont supercomputer of the National Laboratory of Scientific Computing (LNCC, Brazil). We would also like to acknowledge the use of the computational resources provided by the Sci-Com Lab of the Department of Physics at UFES, which was funded by FAPES and CNPq.', 'A Limber approximation': "The Limber approximation makes use of the fact that the spherical Bessel functions j /lscript ( x ) peak around x = [ /lscript ( /lscript +1)] 1 / 2 ≈ /lscript +1 / 2 [38]. For large /lscript , this peak is even more pronounced, such that one may take j /lscript ( x ) /lscript /greatermuch 1 --→ √ π ( /lscript +1 / 2) δ ( x -/lscript -1 / 2), such that, for a function f ( k ) = ∫ dx ' f ( x ' ) j /lscript ( kx ' ), we have \n∫ ∞ 0 dkf ( k ) = ∫ ∞ 0 ∫ dkdx ' f ( x ' ) j /lscript ( kx ' ) ≈ ∫ ∞ 0 dkf ( /lscript + 1 2 ) . (A.1) \nThis approximation has been used extensively on scales /lscript ≳ O (10) for cosmological angular correlations, including in class [37].", 'B Flat sky approximation': 'In the flat-sky approximation we assume that we are observing a small enough area of the sky such that the curvature of the celestial sphere can be neglected. More precisely, we substitute the spherical coordinates over the whole celestial sphere by \nn = r ( z )( θ, φ ) -→ e z = r ( z )( x, y ) , (B.1) \nwhere e z is the unitary vector in a given coordinate in the simulation box, assumed to be the observer or survey reference direction, and ( x, y ) are 2D coordinates in the plane \nperpendicular to e z . For small scales, this approximation is very accurate, as smaller areas of the sky are less affected by curvature effects. In this approximation, the modes /lscript are conjugate to the coordinates α = ( x, y ), such that they are related by a 2D Fourier transform. See [35] for a comprehensive discussion of the flat-sky approximation.', 'C Units': 'As the Gadget4 code works with internal code units, one needs to convert from internal units to physical units and then map these fields to the physical fields found in equation (4.5) through the conversion factors. We list the conversion factors and units used throughout the paper in the following table, and refer the reader to the Gadget4 documentation at wwwmpa.mpa-garching.mpg.de/gadget4 for a comprehensive list of the conversion factors. \nTable 3 : Fields given in simulation units and physical units, and conversion factor from simulation to physical units. \nFurthermore, the class code outputs the angular power spectra of fields X using the convention \n/lscript ( /lscript +1) 2 π C XX /lscript = ( /lscript ( /lscript +1) 2 π ) 4 π ∫ dk k ∆ 2 /lscript ( k, z ) , (C.1) \nwhere ∆ 2 ( k, z ) is given by the | F /lscript | 2 in expression (3.5).', 'References': "- [1] C.J.A.P. Martins et al., Fundamental physics with ESPRESSO: Constraints on Bekenstein and dark energy models from astrophysical and local probes* , Phys. Rev. D 105 (2022) 123507 [ 2205.13848 ].\n- [2] S. Cristiani et al., Spectrographs and Spectroscopists for the Sandage Test , 2, 2023 [ 2302.04365 ].\n- [3] F. Combes, Science with SKA , in Semaine de l'astrophysique française 2021 , 7, 2021 [ 2107.03915 ].\n- [4] R. Braun, T. Bourke, J.A. Green, E. Keane and J. Wagg, Advancing Astrophysics with the Square Kilometre Array , PoS AASKA14 (2015) 174.\n- [5] A. Sandage, The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes , Astrophys. J. 136 (1962) .\n- [6] G. McVittie, Appendix to the change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes , Astrophys. J. 136 (1962) .\n- [7] A. Loeb, Direct Measurement of Cosmological Parameters from the Cosmic Deceleration of Extragalactic Objects , Astrophys. J. Lett. 499 (1998) L111 [ astro-ph/9802122 ].\n- [8] A.G. Kim, E.V. Linder, J. Edelstein and D. Erskine, Giving Cosmic Redshift Drift a Whirl , Astropart. Phys. 62 (2015) 195 [ 1402.6614 ].\n- [9] O.F. Piattella and L. Giani, Redshift drift of gravitational lensing , Phys. Rev. D 95 (2017) 101301 [ 1703.05142 ]. \n- [10] A. Heinesen, Multipole decomposition of redshift drift - model independent mapping of the expansion history of the Universe , Phys. Rev. D 103 (2021) 023537 [ 2011.10048 ].\n- [11] P. Bessa, R. Durrer and D. Stock, Perturbations of cosmological redshift drift , JCAP 11 (2023) 093 [ 2306.13911 ].\n- [12] A. Heinesen, Redshift drift as a model independent probe of dark energy , Phys. Rev. D 103 (2021) L081302 [ 2102.03774 ].\n- [13] S.M. Koksbang and A. Heinesen, Redshift drift in a universe with structure: Lemaître-Tolman-Bondi structures with arbitrary angle of entry of light , Phys. Rev. D 106 (2022) 043501 [ 2205.11907 ].\n- [14] J. Liske et al., Cosmic dynamics in the era of Extremely Large Telescopes , Mon. Not. Roy. Astron. Soc. 386 (2008) 1192 [ 0802.1532 ].\n- [15] R. Cooke, The ACCELERATION programme: I. Cosmology with the redshift drift , Mon. Not. Roy. Astron. Soc. 492 (2020) 2044 [ 1912.04983 ].\n- [16] ANDES collaboration, Cosmology and fundamental physics with the ELT-ANDES spectrograph , Exper. Astron. 57 (2024) 5 [ 2311.16274 ].\n- [17] C.M.J. Marques, C.J.A.P. Martins and C.S. Alves, Fundamental cosmology from ANDES precision spectroscopy , Mon. Not. Roy. Astron. Soc. 522 (2023) 5973 [ 2305.01446 ].\n- [18] H.-R. Klöckner, D. Obreschkow, C. Martins, A. Raccanelli, D. Champion, A.L. Roy et al., Real time cosmology - A direct measure of the expansion rate of the Universe with the SKA , PoS AASKA14 (2015) 027 [ 1501.03822 ].\n- [19] K. Bolejko, C. Wang and G.F. Lewis, Direct detection of the cosmic expansion: the redshift drift and the flux drift , 1907.04495 .\n- [20] S.M. Koksbang, Redshift drift in a universe with structure. II. Light rays propagated through a Newtonian N-body simulation , Phys. Rev. D 107 (2023) 063544 [ 2303.06900 ].\n- [21] S.M. Koksbang, A. Heinesen and H.J. Macpherson, Redshift drift in a universe with structure. III. Numerical relativity , Phys. Rev. D 110 (2024) 063519 [ 2404.06242 ].\n- [22] R. Durrer, The Cosmic Microwave Background , Cambridge University Press, 2 ed. (2020), 10.1017/9781316471524.\n- [23] T. Inoue, E. Komatsu, W. Aoki, T. Chiba, T. Misawa and T. Usuda, The effect of our local motion on the Sandage-Loeb test of the cosmic expansion , Publ. Astron. Soc. Jap. 72 (2020) 131 [ 1911.01467 ].\n- [24] Planck collaboration, Planck 2018 results. VI. Cosmological parameters , Astron. Astrophys. 641 (2020) A6 [ 1807.06209 ].\n- [25] S. Dodelson and F. Schmidt, Modern Cosmology , Elsevier (2021).\n- [26] D.J. Eisenstein and W. Hu, Power spectra for cold dark matter and its variants , Astrophys. J. 511 (1997) 5 [ astro-ph/9710252 ].\n- [27] C. Bonvin and R. Durrer, What galaxy surveys really measure , Phys. Rev. D 84 (2011) 063505 [ 1105.5280 ].\n- [28] V. Springel, R. Pakmor, O. Zier and M. Reinecke, Simulating cosmic structure formation with the gadget-4 code , Mon. Not. Roy. Astron. Soc. 506 (2021) 2871 [ 2010.03567 ].\n- [29] E. Di Dio, F. Montanari, J. Lesgourgues and R. Durrer, The CLASSgal code for Relativistic Cosmological Large Scale Structure , JCAP 11 (2013) 044 [ 1307.1459 ].\n- [30] D. Blas, J. Lesgourgues and T. Tram, The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes , JCAP 2011 (2011) 034 [ 1104.2933 ]. \n- [31] S.R. Green and R.M. Wald, Newtonian and Relativistic Cosmologies , Phys. Rev. D 85 (2012) 063512 [ 1111.2997 ].\n- [32] X. Fang, E. Krause, T. Eifler and N. MacCrann, Beyond Limber: Efficient computation of angular power spectra for galaxy clustering and weak lensing , JCAP 05 (2020) 010 [ 1911.11947 ].\n- [33] X. Fang, E. Krause, T. Eifler and N. MacCrann, Beyond limber: efficient computation of angular power spectra for galaxy clustering and weak lensing , Journal of Cosmology and Astroparticle Physics 2020 (2020) 010-010.\n- [34] F. Villaescusa-Navarro, Pylians: Python libraries for the analysis of numerical simulations , Astrophysics Source Code Library 1811.008 (2018) .\n- [35] W.L. Matthewson and R. Durrer, The Flat Sky Approximation to Galaxy Number Counts , JCAP 02 (2021) 027 [ 2006.13525 ].\n- [36] C. Howlett, The redshift-space momentum power spectrum - I. Optimal estimation from peculiar velocity surveys , Mon. Not. Roy. Astron. Soc. 487 (2019) 5209 [ 1906.02875 ].\n- [37] E. Di Dio, F. Montanari, J. Lesgourgues and R. Durrer, The CLASSgal code for Relativistic Cosmological Large Scale Structure , J. Cosmol. Astropart. Phys. 2013 (2013) 044.\n- [38] M. LoVerde and N. Afshordi, Extended Limber Approximation , Phys. Rev. D 78 (2008) 123506 [ 0809.5112 ]."}

2024A&A...691A..43W

Context. Stars and planets form in regions of enhanced stellar density subjecting protoplanetary discs to gravitational perturbations from neighbouring stars. Observations in the Taurus starforming region have uncovered evidence of at least three recent stardisc encounters that have truncated discs HVDO Tau RW Aurigae and UX Tau raising questions about the frequency of such events. Aims. We aim to assess the probability of observing truncating stardisc encounters in Taurus. Methods. We generated a physically motivated dynamical model including binaries and a spatialkinematic substructure to follow the historical dynamical evolution of the Taurus starforming region. We used this model to track stardisc encounters and the resulting outer disc truncation over the lifetime of Taurus. Results. A quarter of discs are truncated below 30 au by dynamical encounters but this truncation mostly occurs in binaries over the course of a few orbital periods on a timescale 0.1 Myr. Nonetheless some truncating encounters still occur up to the present age of Taurus. Strongly truncating encounters ejecting 10 percent of the disc mass occur at a rate 10 MyrSUP1SUP sufficient to explain the encounter between HV and DO Tau 0.1 Myr ago. If encounters that eject only 1 percent of the disc mass are responsible for RW Aurigae and UX Tau then they are also expected with encounter rate SUBencSUB 100200 MyrSUP1SUP. However the observed sample of recent encounters is probably incomplete since these examples occurred in systems that are not consistent with a random drawing from the mass function. One more observed example would statistically imply additional physics such as replenishment of the outer disc material. Conclusions. The marginal consistency of the frequency of observed recent stardisc encounters with theoretical expectations underlines the value of future large surveys searching for external structures associated with recent encounters. The outcome of such a survey offers a highly constraining novel probe of protoplanetary disc physics.

2024-11-01T00:00:00Z

['2024arXiv240913021W', '2024A&A...691A..43W', '10.1051/0004-6361/202450842', 'arXiv:2409.13021', '10.48550/arXiv.2409.13021']

['planets and satellites: formation', 'protoplanetary disks', 'binaries: general', 'circumstellar matter', 'stars: formation', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - Solar and Stellar Astrophysics']

Running with the bulls The frequency of stardisc encounters in the Taurus starforming region

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

https://arxiv.org/pdf/2409.13021.pdf

{'The frequency of star-disc encounters in the Taurus star forming region': "Andrew J. Winter, 1, ⋆ Myriam Benisty, 1 , 2 Linling Shuai, 3 , 4 Gaspard Dûchene, 2 , 5 Nicolás Cuello, 2 Rossella Anania, 6 Corentin Cadiou, 7 Isabelle Joncour 2 \n- 1 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, 06300 Nice, France\n- 2 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France\n- 3 Astronomy Department, University of Michigan, Ann Arbor, MI 48109, USA\n- 4 Department of Astronomy, Xiamen University, 1 Zengcuoan West Road, Xiamen, Fujian 361005, China\n- 5 Astronomy Department, University of California Berkeley, Berkeley CA 94720-3411, USA\n- 6 Dipartimento di Fisica 'Aldo Pontremoli', Universita degli Studi di Milano, via Celoria 16, Milano, 20133, Italy\n- 7 Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 43, SE-221 00 Lund, Sweden \nReceived September 15, 1996; accepted March 16, 1997", 'ABSTRACT': 'Context. Stars and planets form in regions of enhanced stellar density, subjecting protoplanetary discs to gravitational perturbations from neighbouring stars. Observations in the Taurus star-forming have uncovered evidence of at least three recent, star-disc encounters that have truncated discs (HV / DO Tau, RW Aurigae, UX Tau), raising questions about the frequency of such events. \nAims. \nWe aim to assess the probability of observing truncating star-disc encounters in Taurus. \nMethods. We generate a physically motivated dynamical model including binaries and spatial-kinematic substructure to follow the historical dynamical evolution and stellar encounters in the Taurus star forming region. We track the star-disc encounters and outer disc radius evolution over the lifetime of Taurus. \nResults. A quarter of discs are truncated below 30 au by dynamical encounters, but this truncation mostly occurs in binaries over the course of a few orbital periods, on a time-scale ≲ 0 . 1 Myr. Nonetheless, some truncating encounters still occur up to the present age of Taurus. Strongly truncating encounters (ejecting ≳ 10 percent of the disc mass) occur at a rate ∼ 10 Myr -1 , su ffi cient to explain the encounter between HV and DO Tau ∼ 0 . 1 Myr ago. If encounters that eject only ∼ 1 percent of the disc mass are responsible for RW Aurigae and UX Tau, then they are also expected with encounter rate Γ enc ∼ 100 -200 Myr -1 . However, the observed sample of recent encounters is probably incomplete, since these examples occurred in systems that are not consistent with random drawing from the mass function. One more observed example would statistically imply additional physics, such as replenishment of the outer disc material. \nConclusions. The marginal consistency of the frequency of observed recent star-disc encounters with theoretical expectations underlines the value of future large surveys searching for external structures associated with recent encounters. The outcome of such a survey o ff ers a highly constraining, novel probe of protoplanetary disc physics. \nKey words. protoplanetary discs - planet formation - star forming regions', '1. Introduction': "Planet formation proceeds in a 'protoplanetary disc' of dust and gas over a time-scale of ∼ 3 Myr (e.g. Haisch et al. 2001). During this time, nascent planetary systems typically inhabit star forming regions with a local stellar density that far exceeds the average in the galactic neighbourhood (e.g. Lada & Lada 2003). Neighbouring stars in these regions can feedback on the planet formation process in a variety of ways. These may include: external irradiation driving thermal winds (Winter & Haworth 2022, and references therein), chemical enrichment (Bastian et al. 2013; Lichtenberg et al. 2016; Parker & Schoettler 2023), late-stage gas in-fall (Dullemond et al. 2019; Ku ff meier et al. 2020, 2021, 2023) and star-disc encounters (Cuello et al. 2019, 2020, 2023). While all of these processes are of great interest for understanding the diversity of the observed exoplanet population, in this work we focus on the latter phenomenon. 'Star-disc encounters' refer to gravitational perturbations expe- \nenced by a protoplanetary disc during the close passage of a neighbouring star. \nVarious phenomena such as stellar accretion outbursts (Pfalzner 2008; Forgan & Rice 2010; Vorobyov et al. 2021; Dong et al. 2022) and spiral arms in protoplanetary discs (e.g. De Rosa & Kalas 2019) and free-floating planets (e.g. Vorobyov et al. 2017) - possibly including enigmatic binary planet-mass objects in the Orion Nebula cluster (Pearson & McCaughrean 2023; Wang et al. 2023; Portegies Zwart & Hochart 2023) may all be feasibly attributed to stellar encounters. There remain alternative explanations; for example, spiral arms may be produced by gravitational instability (e.g. Douglas et al. 2013; Meru et al. 2017; Baehr & Zhu 2021) or companion stars / brown dwarfs / planets (e.g. Dong et al. 2015; Ren et al. 2020). A search for flyby candidates that may have generated spiral arms by Shuai et al. (2022) revealed no evidence that nearby stars recently had a su ffi ciently close approach. However, such an e ff ort is challenging due to proper motion uncertainties and incompleteness of reliable measurements; particularly if a perturber is \nvery low mass or is itself a binary. To assess the role of star-disc encounters for protoplanetary disc evolution, we must estimate the rate of encounters in dynamically evolving star forming regions. \nThe rate of stellar encounters, and their role for planet formation, is dependent on the local stellar density of the star forming region. The stellar mass density ρ ∗ of star forming regions can vary dramatically in the range 1 M ⊙ pc -3 ≲ ρ ∗ ≲ 10 6 M ⊙ pc -3 . The latter limit represents the most extreme stellar densities in the galaxy, such as in the cores of globular clusters that remain bound for a Hubble time (e.g. Krumholz et al. 2019). Intermediate densities ρ ∗ ∼ 10 3 M ⊙ pc -3 are typical in the cores of open clusters that may survive against galactic tides over 100 Myr or Gyr time-scales. However, possibly the most common environments in which stars and planets form are the low density regions that produce loose 'associations', which are globally unbound against galactic tides. The average density in such regions has been assumed to be too low to induce frequent tidal encounters throughout the disc lifetime (e.g. Winter et al. 2018c). \nTaurus is an example of a (globally) low density region, which does however host convincing evidence of at least three recent close star-disc encounters. These cases are RW Aurigae (Cabrit et al. 2006; Dai et al. 2015; Rodriguez et al. 2018), HV and DO Tau (Howard et al. 2013; Winter et al. 2018a) and UX Tau (Zapata et al. 2020; Ménard et al. 2020). Each of these systems exhibits significant external structure which appears to be explained by recent flybys, su ffi ciently close such that they were capable of unbinding disc material. In an unstructured star forming region at the average density of Taurus, the probability of a one-o ff random encounter over the disc lifetime is vanishingly small. This puzzle may appear to be partially resolved for some cases if the system is bound, such that close encounters occur periodically over the binary orbit. However, in isolation this is not a su ffi cient explanation. While repeated encounters in a multiple system can drive spiral arms in the disc (e.g. Alaguero et al. 2024), the disc should also be rapidly truncated such that on subsequent close approaches the gravitational pertubation no longer produces large extended tidal tails (e.g. Ménard et al. 2020). It is therefore necessary to perturb star-disc systems in Taurus stochastically, and do so over its ∼ 1 -3 Myr lifetime. In this manuscript, we aim to answer the question: do we expect su ffi -cient star-disc encounters in Taurus to explain the frequency of observed recent encounters? \nThe role of stellar encounters has been the focus of numerous studies focusing both on the evolution of protoplanetary discs (e.g. Clarke & Pringle 1993; Ostriker 1994; Pfalzner et al. 2005; Winter et al. 2018b) and mature planetary systems (e.g. Spurzem et al. 2009; Shara et al. 2016; Winter et al. 2022; Li et al. 2024). Studies using N-body simulations often perform parameter studies or implement scaling relations to model star forming regions. For example, one approach is to adopt the putative mass-radius (e.g. total stellar mass M c, half-mass radius R c) relationship for star forming regions (e.g. R c ∝ M 1 / 2 c - Adams et al. 2006). However, the nature of this relationship varies depending on the sample / definition of mass and radius (Pfalzner & Govind 2021), and exhibits significant scatter. Mature / massive star clusters also do not appear to follow this relationship (Krumholz et al. 2019), and particularly for low mass / density regions it is challenging to unambiguously define an individual star forming region. In addition, even within a well-defined 'individual' star forming region, internal structure has been shown to have a strong influence on the role of encounters (e.g. Craig & Krumholz 2013; Parker 2023). Assessing encounter rates therefore requires quantitatively matching present day position-velocity structure. Yet it \nis not clear how well the widely adopted method for generating fractal initial conditions, sampling hierarchical boxes (Craig & Krumholz 2013), reproduces the observed spatial and kinematic structure in star forming regions. \nStructure in giant molecular clouds is set by the turbulent fragmentation, from which the power-spectrum and Mach number determine the mass density distribution (e.g. VazquezSemadeni 1994; Padoan et al. 1997). An ideal approach to studying the role of dynamical encounters in young star forming regions is therefore to model the star formation process directly through hydrodynamic simulations. Such simulations have shown that encounters are common during early disc evolution, possibly determining the initial distribution of protoplanetary disc radii (Bate 2018). However, these experiments are computationally expensive, following star formation only for ∼ 0 . 1 Myr time-scales. This makes parameter studies or tailored modelling of individual star forming regions in this way impracticable. \nHere we present a complementary approach to the above works, targeted at generating N-body initial conditions tailored to match young star forming regions. We achieve this by simulating a Taurus-like star forming region, with physically and empirically motivated initial conditions, including binaries. We draw stellar positions and velocities from an empirically constrained power spectrum, reflecting how turbulent energy is distributed across di ff erent scales. Our main aim is to develop a dynamical model to track the history of stellar encounters in the Taurus star forming regions. In doing so, we will assess whether the rate of disc-truncating star-disc encounters in Taurus is su ffi cient to produce the three known examples: HV / DO Tau, RW Aurigae and UX Tau. This goal requires ensuring that our dynamical model closely reproduces the present day spatial and kinematic structure in Taurus. We therefore review these structural properties in Section 2, which we then adopt for benchmarking our dynamical model. We discuss our approach for initialising initial conditions and dynamically evolving the model in Section 3, in which we also draw comparisons with the structure metrics introduced in Section 2. We discuss the rate of truncating encounters over time in our simulation in Section 4, quantifying the degree to which the observed examples of recent star-disc encounters are statistically expected. We summarise our conclusions in Section 5.", '2.1. Aim': 'An empirically motivated dynamical model for Taurus requires accurately quantifying kinematic substructure. If we had arbitrarily accurate and complete data for the 3D positions and velocities of all the stars in Taurus, then it would be trivial to use these data to generate N-body initial conditions to compute the future evolution of Taurus. However, as discussed in Section 2.2, while we have a fairly complete census of stars in terms of their projected spatial distribution, kinematic data is far more limited. Given the e ff ects of multiplicity and extinction, parallax measurements also typically have associated uncertainties that make them impractical for use in setting initial conditions. In addition, we are interested in quantifying the frequency of encounters in the past, while we do not have direct measurements of the early spatial-kinematic stellar configuration in Taurus. For these reasons, we require metrics that characterise kinematic substructure. These metrics both guide our choice of initial conditions and o ff er a benchmark comparison for our models. \nIn the following, we first review the data for the stellar population in Taurus (Section 2.2). We then discuss how we quantify spatial structure in Section 2.3 and kinematic structure in Section 2.4.', '2.2. Data': 'We use the census of Taurus members by Luhman (2023), with astrometric data from Gaia DR3 (Gaia Collaboration et al. 2016, 2023; Babusiaux et al. 2023). This catalogue contains 532 members, with a high degree of completeness for spectral types earlier than M6-M7. When we consider proper motion di ff erences between neighbours (Section 2.4), we restrict the sample to the 271 that have a reliable astrometric solution by the canonical criteria that the Renormalised Unit Weight Error (RUWE 1 ) is smaller than 1 . 4 in Gaia DR3. \nWecannot use the observed 3D positions directly to generate initial conditions for several reasons. Most obviously, the sample for which we have parallax is incomplete (417 / 532), and the typical uncertainties on the parallax correspond to a few pc. This means that small scale structure is not resolvable. We therefore generate our initial conditions parametrically, closely comparing with the observed structure in Taurus. We consider only the projected separations between stars when inferring the spatial and velocity structure of the region.', '2.3. Spatial substructure': "We aim to understand the role of close neighbours on disc evolution. A sensible metric to quantify structure is therefore the normalised pair separation function, which can be defined in two dimensions ˆ Σ pairs( ∆ R ). This is the averaged surface density of neighbours for any given star, normalised to unity when integrated over 2 π ∆ R d ∆ R . This surface density evolves over time, and we will therefore use this metric to ensure that we capture the time at which the dynamical state in our simulation is similar to that in Taurus. \nTwo related but complementary metrics are the one- and twopoint correlation functions, Ψ ( ∆ R ) and ξ ( ∆ R ) respectively. These metrics have been applied by Joncour et al. (2017) and Joncour et al. (2018) to study the structure in Taurus. They are broadly defined as the excess of pairs with a given separation compared to a random, uniformly distributed population of stars in the same area. The one-point correlation Ψ is an excess of nearest pairs, while the two-point correlation represents the excess of all pairs. Of particular relevance for this work, Joncour et al. (2017) applied the one-point correlation function to demonstrate that approximately 40 percent of stars in Taurus are in ultra-wide binaries, with separations in the range 1 . 6 × 10 3 -5 × 10 4 au. We will use this inference to motivate our initial conditions, and the one- and two-point correlation functions to validate our model a posteriori . In particular, from the one-point correlation function Joncour et al. (2017) find a region of 'inhibition' (a smaller number of pairs than expected from random sampling) between ∼ 0 . 1 · and ∼ 0 . 5 · , while up to ∼ 0 . 2 · , Ψ ∝ ∆ R -1 . 5 . For the two-point correlation function, ξ > 1 out to ∆ R ≈ 2 · .", '2.4. Velocity substructure': 'Stellar velocities are inherited from the velocities of the material from which they form, and are thus dependent on the kinematic \nFig. 1. Proper motion di ff erence as a function of angular separation on the plane of the sky for nearest neighbours in the Luhman (2023) sample for Taurus. Red data points are the observed sample, with uncertainties propagated from Gaia uncertainties. The black line shows the the velocity dispersion as a function of separation we adopt for our model, with a Keplerian component that dominates at small separations. The transition between the two power-laws is the break between the binary and individual systems ( ∼ 5 × 10 4 au, comparable to the galactic tidal radius for typical stars). The colour bar shows the corresponding MaxwellBoltzmann distribution, normalised for each separation. See text for details. \n<!-- image --> \nstructure of the parental molecular cloud. This velocity structure in a turbulent medium is driven by interacting waves that generate an energy cascade that is described by an energy spectrum E ( k ). Across a wide range of length scales λ , it can be approximated by a power-law (e.g. Elmegreen & Scalo 2004): \nE ( k ) = P ( k ) k 2 ∼ σ 2 v , k k ∝ k -β ∝ λ β , (1) \nwhere P ( k ) is the (three dimensional) power-spectrum, k is the wavenumber, σ v , k is the characteristic velocity for k . \nA range of β values in the interstellar medium (ISM) have been inferred observationally. These range from the original estimate for the size-linewidth relation for molecular clouds yielding β ≈ 1 . 76 (Larson 1981) to larger β ≈ 2 . 3 (Münch 1958; Heyer & Brunt 2004, for example). For weakly compressible turbulence, the energy spectrum for density and velocity follow the same power-law. Qian et al. (2018) find β ≈ 2 on small scales ≲ 2 pc (suggestive of compressible turbulence) and β ≈ 5 / 3 in the range ∼ 5 -10 pc (suggestive of incompressible turbulence, see also Brunt 2010). Ha et al. (2022) estimated β ≈ 1 . 7 from the stellar population and β ≈ 1 . 8 from H α , both on ∼ 10 pc scales. \nFor our purposes we adopt β = 2, appropriate for supersonic, rapidly cooling turbulence (Burgers 1948). The velocity dispersion relation we adopt is then: \nσ v ( ∆ r ) = σ v , 0 ∆ r ∆ r 0 ! 0 . 5 , (2) \nwhere we choose normalisation constants σ v , 0 = 0 . 6 km s -1 for ∆ r 0 = 1 pc. This yields a velocity dispersion similar to that empirically inferred for the youngest stars on galactic scales in the solar neighbourhood (e.g. Holmberg et al. 2009), while more importantly matching the dispersion in Taurus down to binary length scales. In Figure 1, we show the distribution of proper \nmotion di ff erences for nearest neighbours as a function of their separation in Taurus. We also show the Maxwell-Boltzmann distribution: \ngN D ( ∆ v | σ v ) = 2 ∆ v N D -1 Γ ( N D / 2) 1 4 σ 2 v ! N D / 2 exp -∆ v 2 4 σ 2 v ! (3) \nin two dimensions ( N D = 2) for equation 2 with a Keplerian component appropriate for a star of mass m ∗ = 0 . 5 M ⊙ added in quadrature. This shows that the majority of neighbours follow the expected separation-velocity relation. Neighbours with a much larger relative velocity may have larger physical than projected separations. We will apply the size-velocity relation given by equation 2 to generate stellar velocities, and compare the appropriate Maxwell-Boltzmann distribution to the nearest neighbour relative velocities we generate in our model, both initially and after dynamical evolution.', '3.1. Overview': 'Our approach for simulating a Taurus-like star forming region is to use a physically and empirically motivated set of initial conditions, including binary systems. We then benchmark our model against the spatial-kinematic properties of the present day Taurus to ensure that the simulation reflects the observed dynamical state. With this model, we are able to extract encounters which we can convert into the protoplanetary disc radii evolution using analytic formulae fit to numerical experiments for truncating encounters. \nIn the remainder of this section, we detail this process. First, we discuss our method for generating physically motivated initial conditions, starting with a gas density distribution in Section 3.3, which we convert to a stellar density distribution in Section 3.4. We then discuss implementing empirically motivated binaries in Section 3.6 and the appropriate velocity substructure in Section 3.8. We validate our dynamical model with respect to the observed dynamical state in Taurus in Section 3.9. Finally, we discuss our approach for extracting close stellar encounters from the simulation in Section 3.10 and the resultant disc evolution in Section 3.11.', '3.2. Nbody code': 'Throughout this work, we use NBODY6 ++ (Aarseth 2003; Spurzem 1999; Wang et al. 2015) to integrate the stars and binary systems under gravity for 3 Myr. Given our interest in the short term evolution of a low mass star forming region, we do not include stellar evolution. Nor do we include tidal binary circularisation, or an external potential. To capture the role of several neighbours, we insist on a short time-step factor for the irregular force polynomial 5 × 10 -3 . The parameter adjustment time in N-body units is 10 -4 in code time-units that are 3 . 22 Myr, at which the regularisation parameters are updated. The parameter and initial condition file for our fiducial model are available online.', '3.3. Lognormal gas density field': 'We aim to initially generate a physically motivated gas density distribution, from which to draw our stellar population. Numerical experiments have shown that the mass density ρ of isothermal turbulent flows is well approximated by a lognormal distribution (e.g. Vazquez-Semadeni 1994; Nordlund & Padoan 1999;', 'Gaussian field box selection': "Fig. 2. Illustration of how we select a box from which to generate a lognormal density field from a larger scale, low resolution lognormal density field. We illustrate the underlying density grid by drawing faint black points with density proportional to the local density field. Cyan points indicate our selection of the top 20 density maxima from the density field. The red star indicates the position of the adopted box centre. The dashed red lines indicate the boundary of the new zoom-in box. \n<!-- image --> \nOstriker et al. 1999). In order to generate a lognormal gas distribution, we must first generate a Gaussian density field over a 3D grid. Generating a density field that we can use to produce initial conditions for the N-body simulation is not trivial, since a Gaussian random field may exhibit peaks close to the edges of the grid. These peaks would be the site of high stellar density (or 'NESTS' - Joncour et al. 2018), and may be partially excluded by the grid. Indeed, the centre of the grid can represent a 'void' (underdensity) in the Gaussian field, which would undermine our goal to explore the interactions within and between stars in overdensities. Since we also need a grid that covers a large dynamical range (from the size scale of Taurus, down to the wide-binary scale), it is not practicable to simply choose a subset of the complete Gaussian random field with a very high resolution. It is therefore useful to apply an approach that ensures that we can centre the grid on an overdensity. \nWhile we could attempt to do this by multiplying the density field by a centrally concentrated profile (such as a 3D Gaussian or Plummer profile), this would fundamentally change the nature of the density field and corresponding power spectrum. To avoid this we adopt a two-stage process, inspired by cosmological zoom-in simulations and applying the publicly available genet IC 2 code (Stopyra et al. 2020, 2021). This code generates a zoom-in or splice (Cadiou et al. 2021) self-consistently within a surrounding Gaussian random field by Fourier-space filtering. The net e ff ect is that we are able to generate an initial, coarse Gaussian field, and then zoom in on an overdensity that represents the centre of our model for Taurus. \nOur approach for this is represented graphically in Figure 2. We first generate a Gaussian random field over a 256 3 grid with \nside length 100 pc. We adopt a power spectrum index of -3. We do not quantitatively fit for this index, but select a posteriori between indices -2, -2 . 5, -3 and -4, finding that -3 reproduces a stellar density structure that matches the observed structure. Empirically, this index is comparable to that inferred in Taurus by Brunt (2010). \nFrom the large scale density field, we then locate local maxima over 5 pc length scales using the ndimage.maximum\\_filter algorithm from S cipy (Virtanen et al. 2020). We choose the 20 greatest local maxima, and select the closest to the centre of the grid, ensuring the zoom-in grid will be within the range of the initial grid. We then use genet IC to generate a new Gaussian density field on a grid that is smaller by a factor three (box side length 33 pc), and with 1024 3 grid cells. This defines a field which has a high density centre, and a resolution of 0 . 032 pc, or ∼ 7000 au, down to length scales comparable to wide binary separations. Finally, we enforce a dispersion σ ln ρ = ln GLYPH<16> 1 + 0 . 25 ⟨ v 2 ⟩ / c 2 s GLYPH<17> = 5 on the logarithmic density field. This is approximately the expected dispersion given a sound speed c s = 0 . 2 km s -1 and the root mean square velocity ⟨ v 2 ⟩ 1 / 2 = √ 3 σ v ∼ 5 km s -1 for ∆ r ∼ 30 pc, according to equation 2.", '3.4. Stellar spatial distribution': 'We now wish to draw a stellar spatial distribution from our underlying gas density profile. To do this, we must consider which regions of the cloud are able to rapidly collapse to form stars. We consider the time-scale for this collapse from rest: \nτ ff = 1 . 52 ρ 100 M ⊙ pc -3 ! -1 / 2 Myr . (4) \nTo infer a free-fall time across our grid, we require an absolute density scale, for which we assume a total mass within the box of 10 4 M ⊙ , corresponding to the approximate total gas mass in the Taurus complex (Goldsmith et al. 2008). We then impose a constraint that grid cells must have τ ff < 1 Myr in order to host a star, comparable to the empirical age dispersion Luhman (2023). We assign probabilities to these grid cells p ∗ ∝ ρτ -1 ff ∝ ρ 3 / 2 - i.e. the probability of forming a star is proportional to the quantity of material multiplied by the rate at which that material is expected to collapse. For a selected grid cell, we then o ff set the location of the formed star such that it is placed randomly within the cell. We validate this choice of spatial distribution in our dynamical model a posteriori in Section 3.9. \nFor the the total number of systems, we draw N sys that yields a total number of stars N ∗ that is broadly consistent with the findings of Luhman (2023). This is not trivial, both because there is not a clear detection limit for that sample and because we include binaries stochastically from the initial population of potential primaries, as well as the ultrawide binary fraction inferred by Joncour et al. (2017). Further, it is not straightforward to interpret what fraction of binaries are resolved / detected by Luhman (2023). We perform a two stage process as follows in Section 3.5 and 3.6.', '3.5. Ultrawide pairs': "First of all, we correct the total number of stars for a given fraction of ultrawide binaries F uwb = 2 / 3. We consider 400(1 -F uwb / 2) = 267 stars with masses > 0 . 08 M ⊙ (half of these will end up in ultrawide binaries). We then add the fraction of brown \nFig. 3. Histogram of stellar masses in our dynamical model for Taurus. The solid black shows all stars including binary companions, while the red line is just the primaries and single stars. The black dashed line corresponds to the number that would be expected in each bin from the Kroupa (2001) IMF (equation 5). The bin sizes are similar to those adopted by Luhman (2004) in Figure 13, top panel. \n<!-- image --> \n∗ \n/circledot \ndwarfs we expect below this limit from the initial mass function (IMF). For this, we use the following IMF (e.g. Kroupa 2001): \nξ ( m ∗ ) ∝ m -0 . 3 ∗ 0 . 01 M ⊙ ≤ m ∗ < 0 . 08 M ⊙ m -1 . 3 ∗ 0 . 08 M ⊙ ≤ m ∗ < 1 . 3 M ⊙ m -2 . 3 ∗ 0 . 5 M ⊙ ≤ m ∗ < 1 . 3 M ⊙ m -2 . 7 ∗ 1 . 3 M ⊙ ≤ m ∗ 0 otherwise , (5) \nwith normalisation constants such that ξ is continuous and integrates to unity over all m ∗ . For each of these stars, we then draw an ultrawide companion with total probability F uwb. This yields 660 stars and sub-stellar objects in total, 38 percent of which have masses < 0 . 08 M ⊙ . We then draw the masses of all stars from the equation 5. \nThe ultrawide pair fraction we adopt is somewhat higher than F uwb ≈ 0 . 55, inferred by Joncour et al. (2017); 186 of the 338 stars in their sample are in ultrawide pairs. However, some of these companions will separate during dynamic evolution. We also note that Joncour et al. (2017) found evidence that members of ultrawide pairs are ∼ 15 percent more likely to be themselves in a shorter period binary. However, given that this is a relatively minor correction to the binary fraction, and it is challenging to disentangle dynamical versus primordial origins for these statistics, we do not include this enhancement. While we do not forward model the initial fraction, we validate our choice by comparing the one-point correlation function in our model with that computed by Joncour et al. (2017) in Section 3.9. \nThe separations for the ultrawide pairs is approximately loguniform between 1 . 6 × 10 3 au and 5 × 10 4 au (see Fig. 7 of Joncour et al. 2017), and we therefore draw semi-major axes of these companions similarly. We assume a uniform eccentricity distribution up to 0 . 9, and a random orientation (as described in Section 3.6). \nWe define binaries separately from 'ultrawide pairs', described above and treated as single stars in terms of their mass-function. We consider the entire population of stars (including each member of the ultrawide pairs) to be potential primary stars in a binary system, and add companions to a subset. \nThe binary population we include is empirically motivated, based on the findings of Moe & Di Stefano (2017). In brief, we use the same functional form for the probability of each primary having a companion in each dex of orbital period space, modified in a number of ways. Firstly, we exclude binaries with period P < 10 5 days ( ≲ 30 au); such binaries are close enough to be largely uninfluenced by encounters, and in the context of protoplanetary discs may host circumbinary discs. At longer periods, we remove the exponential taper that Moe & Di Stefano (2017) infers for P > 10 5 . 5 days, instead assuming a constant binary fraction per dex out to P = 10 7 . 7 days, approximately corresponding to the scale at which we transition to ultrawide binaries. The log-uniform distribution is consistent with the observed separations for the ultrawide pairs (see Fig. 7 of Joncour et al. 2017), and for wide binaries in the field (Lépine & Bongiorno 2007). For all binaries, we draw eccentricities uniformly from 0 -0 . 9 (e.g Abt 2006, and review by Duchêne & Kraus 2013). \nOnce we have randomly drawn the periods for the companions of our initial population, we draw the mass-ratio and eccentricity. For the ultrawide binaries, we draw masses from the IMF, as described in Section 3.4. For the regular binaries, we draw mass-ratios q in the range 0 . 1 < q < 1 from a probability density function p ( q ). To construct p , we define two power-law regime with indices γ small for q < 0 . 3 and γ large for q ≥ 0 . 3. Following Moe & Di Stefano (2017), we also include an excess probability of the star having a twin, where a twin is defined as having a mass ratio 1 -∆ twin < q ≤ 1. This is defined in practice by computing the quantity: \np ' twin = F twin R 1 0 . 1 p ' ( ˜ q ) d ˜ q ∆ twin(1 -F twin) 1 -∆ twin < q < 1 0 otherwise , (6) \nwhere \np ' ( q ) = ( q γ small 0 . 1 ≤ q < 0 . 3 0 . 3 γ small 0 . 3 γ large q γ large q ≥ 0 . 3 . (7) \nThen finally the probability density function for q is: \np ( q ) = p ' + p ' twin R 1 0 . 1 p ' ( ˜ q ) + p ' twin ( ˜ q ) d ˜ q . (8) \nSince we only consider P > 10 5 days and largely low mass stars, we have only two regimes for the power-law indices. For P < 10 6 days we have γ small = 0 . 4 and γ large = -0 . 4, and otherwise we have γ small = 0 . 5 and γ large = -1 . 1. We fix F twin = 0 . 1, for ∆ twin = 0 . 05. These values are consistent with observational constraints, although typical uncertainties are high for many of these values (El-Badry et al. 2019). When we generate our initial population, we also exclude any companions that are generated that have masses below our lower IMF limit (0 . 01 M ⊙ ), although in practice this only influences brown dwarf primaries. The overall binary fraction is ∼ 30 percent for binaries with orbital periods > 10 5 days, which is broadly consistent with the field population (e.g. Niu et al. 2021). We show the statistics of our binary and ultrawide pair distribution for the stellar population in Figure 4. \nFinally, we generate the positions and velocities of the companion population by randomly sampling cos i (for inclination i ), the argument of periapsis, longitude of ascending node and true anomaly uniformly over the appropriate ranges. We then compute the appropriate position and velocity vector of the companion with respect to the primary. We thus produce the initial binary population that we evolve dynamically during our simulation.", '3.7. Mass function': 'We have already described our stellar mass drawing procedure for single stars and binaries in Sections 3.5 and 3.6. To validate our mass distribution including the binary population, we show the histogram of stellar masses in Figure 3. The bins used for this histogram are similar to those used by Luhman (2004, top panel of their Figure 13). The number of approximately solar mass stars found by Luhman (2004) is ∼ 40. This number is close to complete for Taurus, and is similar to our drawing from the IMF. More recently, Esplin & Luhman (2019) found a peak around m ∗ ∼ 0 . 15 and ∼ 25 stars with m ∗ ≳ 1 M ⊙ across the whole of Taurus, again comparable to our IMF draw. The mass function for all the stars, including binary companions, is shown as the black histogram in Figure 3. Despite including a di ff erent mass function for companions, the mass function is not greatly altered from the Kroupa (2001) IMF we initially assume for the primary population. We therefore conclude that we have drawn a similar stellar population to that of Taurus.', '3.8. Initial velocities': 'We must assign velocities to the primary stars we have generated, motivated by our discussion in Section 2.4. We tackle this by generating velocities following a Gaussian process, such that velocities of stars that are close to each other are more highly correlated than those of stars at large separations. If ∆ r < ∆ r max, we can write the corresponding kernel or covariance function (e.g. see equation 2.19 of Rasmussen et al. 2006): \nk ( r , r \' ) = k ( ∆ r ) = σ 2 v , max -σ 2 v ( ∆ r ) , (9) \nwhere σ v , max = σ v ( ∆ r max). By this definition, the covariance function is not well defined for large ∆ r > ∆ r max, where k becomes negative. However, we are free to choose an arbitrary large ∆ r max. In this case, we can also rewrite the covariance function: \nk ( ∆ r ) ≈ σ 2 v , max GLYPH<16> 1 - D 2 r GLYPH<17> (10) \nwhere we have defined: \nD r = " 1 -exp -∆ r ∆ r max !# 0 . 5 . (11) \nEquations 9 and 10 are equivalent for large ∆ r max / ∆ r → ∞ , but at large separations equation 10 defines a maximal dispersion between stellar velocities. We choose equation 10 because it yields well defined covariance for any ∆ r . In practice, we anyway choose ∆ r max = 100 pc so that our choice is not important. In order to assign velocities given this kernel function, we define the covariance matrix K = [ kij ] = [ k ( r i , r j )]. We then perform a Cholesky decomposition K = LL T , where L is a lower triangular matrix. We define a vector w α which is a vector with a length \nFig. 4. Histogram showing the distribution of companions at a given semi-major axis (left) eccentricity (middle) and mass-ratio (right) for primary stars with masses m ∗ > 0 . 08 M ⊙ . The red histogram is for those pairs we define as ultrawide pairs, while the blue shows the binaries. \n<!-- image --> \nFig. 5. Data points show the proper motion di ff erence versus angular separation (assuming a distance of 140 pc) for nearest neighbours from the initial conditions in our model. The black line shows the one dimensional velocity dispersion as a function of separation and the colour bar shows the normalised Maxwell-Boltzmann distribution for each separation, as in Figure 1. \n<!-- image --> \ncorresponding to the number of stars for which we assign velocities. We draw each wj α ∼ N (0 , 1) independently for each spatial dimension α = 1 , 2 , 3. We then define the velocity components for the stellar population u α = Lw α . \nIn Figure 5 we show the velocity di ff erence between di ff erent nearest neighbours in our model initial conditions, including binaries. It is clear from Figure 5 that our synthetic stellar population (scatter points) have relative velocities that follow a similar size-velocity relation, σ v ( ∆ r ) as found in Section 2.4 - cf. Figure 1.', '3.9. Model validation': "We validate our model by analysing the structure metrics discussed in Section 2. First, we consider the pairwise separation distribution, as shown in Figure 6. The pair surface density profile ˆ Σ pairs shows an excess at very small projected separations ( ∆ R ≲ 5 × 10 -4 pc), where the Luhman (2023) sample may be missing closer binaries. There is also a small excess around the \nbinary transition at a few 10 -2 pc. This excess is quickly lost as the system evolves, and the spatial structure is in good agreement with the observed population at ∼ 1 Myr. We therefore adopt 1 Myr as the 'present day' in our simulation. \nAt this time, the velocity structure (illustrated in Figure 7) remains similar to the velocity structure we inferred in Taurus in Section 2.4. When comparing Figure 7 to Figure 1, we note that there are some di ff erences in how they are constructed. For example, close binaries are complete in our model, but not for those in Taurus. Indeed, in Taurus the sample is restricted to only stars with Gaia proper motions, with all the biases that implies. Nonetheless, the correlation between velocity and separation remains broadly similar. The transition in nearest neighbour velocity di ff erence from 'binary' to 'field' coincides with the change in the power-law surface density profile in Figure 6 for ∆ r ∼ 3 × 10 -2 pc. \nWe also show the one- and two-point correlation functions in Figure 8. These can be compared directly to Fig. 4 of Joncour et al. (2017). We find a similar functional form for both, again indicating that the structure in our model matches the physical structure of Taurus. We obtain a very similar region of 'inhibition' in which Ψ < 1 in the range of separations 0 . 1 -0 . 5 · . We find a similar power-law for Ψ where ∆ R < 0 . 2 · . We also recover ξ > 1 for ∆ R ≲ 1 · , close to the result of Joncour et al. (2017). \nWe conclude that our model, with initial conditions based on physical and empirical arguments, reproduces the observed dynamical state of Taurus at 1 Myr. The age in our model of 1 Myr is somewhat younger than the 1 -3 Myr typically estimated for Taurus (e.g. Luhman 2023). If Taurus is typically older on average, this might suggest that the stellar distribution was initially more compact than the initial conditions in our simulation; although in this case we would also expect more rapid dynamical evolution. It is also possible that residual gas in the star forming region slows down the dispersal of structure. However, then it would be plausible that discs in the high density regions are replenished by accretion of this residual gas (e.g. Ku ff meier et al. 2020), blurring the lines of what ' t = 0' means for disc evolution. In truth, probably a mixture of these influences, as well as a finite period of star formation, somewhat influence the dynamical evolution of the region. Nonetheless, for the purpose of this work, we are satisfied that the current state of stars and discs in Taurus is well approximated by our simulation at 1 Myr. In this context, we note that a posteriori we find that the frequency of strong encounters does not change rapidly between 1 -3 Myr in our simulation (see Section 4.3). \nFig. 6. The normalised pair surface density ˆ Σ pairs as a function of projected separation ∆ R in parsecs. The red line shows the observed surface density for Taurus from the Luhman (2023) census, which may be incomplete at the smallest separations. The coloured lines show the outcome from our model, at snapshot outputs indicated by the colour bar. The 1 Myr snapshot is also marked by black crosses, at which time the pair distribution function is in good agreement between observations and simulation. \n<!-- image --> \nFig. 7. As in Figure 5, except for our simulation after 1 Myr of evolution. \n<!-- image -->", '3.10. Tracking encounters': "Due partly to our inclusion of binaries, interactions between stars are too complex to be easily identified with a simple criterion 'on the fly' during the simulation. We have therefore chosen to analyse encounter properties by post-processing high frequency outputs. Specifically, we have 9299 snapshots from our simulation over 3 Myr, corresponding to a time-step of 323 years. To accurately extract the correct encounter properties, we then perform the following analysis for each star i : \n- 1. Identify the closest neighbour j ( t ) for each time-step, spatially separated by vector ∆ r i j and with velocity di ff erence ∆ v i j .\n- 2. If the nearest neighbour j to i has a bound companion k which is separated from j by vector ∆ r jk such that | ∆ r jk | < \nFig. 8. One-point ( Ψ , blue points) and two-point ( ξ , grey points) correlation functions computed from our model at 1 Myr. The red line shows the Ψ ∝ ∆ R -1 . 5 relationship for ∆ R < 0 . 2 · and the two dashed black lines enclose the region of inhibition (where Ψ < 1), as inferred by Joncour et al. (2017, see their Figure 4). \n<!-- image --> \n- 0 . 1 | ∆ r i j | , we consider j and k to be a single star with the combined mass and momentum of j and k . In the following we will refer to star j as the binary barycentre.\n- 3. We search for a sign change in the vector ∆ r i j · ∆ v i j from negative (approaching) to positive (receding). We define snapshot l , at time tl , for which | ∆ r i j | is minimal between the two adjacent snapshots ( l , l + 1).\n- 4. We compute the analytic eccentricity e , closest approach distance r p, time of pericentre t p. If the predicted closest approach is not between tl and tl + 1, then this must be a nonhierarchical multiple interaction, occurring on a short timescale ( < 323 years). In this case, we assume that the closest approach is at tl (with closest approach distance given by the separation at this time), although in practice this is rare. \nThis procedure ensures that, despite the finite temporal resolution of our simulations, we are able to resolve the majority of encounters that are relevant to disc truncation. We limit the number of encounters to one per time-step, thus we do not count every closest approach for close binaries which have an orbital period < 323 years. Although we could in principle include multiple encounters per time-step, these encounters anyway quickly truncate the disc on short time-scales. While our method may also not be accurate particularly for chaotic multiple interactions in cases where multiple interactions occur on time-scales less than 323 years, we show in Appendix A that our results are not influenced if we increase the output frequency.", '3.11. Disc truncation and initial radii': "To compute the post-encounter disc radius, we use the analytic functions inferred by Winter et al. (2018c). These functions were established by fitting a scale-free, angle-averaged expression to numerical test particle simulations, depending on the ratio of the closest approach distance r p to the outer disc radius R out, the eccentricity of the encounter e and the mass-ratio of the perturber q . Since these fitting functions were inferred for unbound encounters, we will adopt e = 1 for encounters with e < 1. We expect this to be a reasonable approximation. For example, Man- \nara et al. (2019) fit an analytic functional form to the steady state truncation radius inferred from the numerical results of Artymowicz & Lubow (1994). For an equal mass system on a circular orbit, this estimate implies a truncation radius ∼ R out ≈ 0 . 33 r p, which is the same as the steady state truncation radius implied by our prescription (see Figure 4 of Winter et al. 2018c). In practice, the majority of encounters that strongly truncate the disc at late times also have e ∼ 1. We also consider all discs which experience encounters with r p < 10 au to be 'destroyed', making the assumption that the resultant compact disc may be rapidly accreted or photoevaporated (e.g. Clarke et al. 2001). \nUnless otherwise stated, we initialise all discs with outer radii following (Andrews 2020): \nR out , 0 = 250 m ∗ 1 M ⊙ ! 0 . 9 au . (12) \nWhile this is empirically motivated by observed outer radii as inferred from CO emission lines, this relationship has a large scatter (which may in part be driven by dynamical truncation). The measurements themselves also come with numerous caveats originating from systematic uncertainties in outer radius definition, optical depth and chemistry (see e.g. Miotello et al. 2023). However, with these caveats we will here assume that the relationship is exact. We also do not consider viscous expansion (e.g. Lynden-Bell & Pringle 1974) or wind-driven contraction (e.g. Tabone et al. 2022). The discs therefore only evolve (shrink) as a result of star-disc encounters.", '3.12. Caveats for the physical model': 'There are two main caveats for the dynamical model we present here. The first is that, while we are interested in establishing the rate of strongly disc-truncating interactions at a given system age, this depends on there being a well-defined age of the system at large. More specifically, we implicitly assume that the spread in stellar ages is much smaller than the age of the system itself. This is almost certainly not the case, with the groups in Taurus having ages in the range ∼ 1 -3 Myr (Luhman 2023). However, our e ff orts are still valid in that a star-disc system will only undergo encounters with nearby neighbours, which we generally expect to have very similar ages. None of the existing examples of recent encounters have been suggested to be particularly young, but our conclusions can be reconsidered depending on the inferred ages of post-encounter systems. \nThe other major caveat is that we do not include the selfgravity of the ISM. Without including a fully hydrodynamical model, it is not possible to include this potential in a physical way. Our choice to ignore the gas potential is in a sense a similar assumption to the one discussed above, specifically that local star formation is completed instantly, and the gas is instantaneously dispersed (e.g. by stellar feedback). If instead a large quantity of residual gas remained within the individual star forming regions that comprise the Taurus complex, then groups of stars may remain bound for longer. This may in turn increase the local encounter rate at later times. It is not possible to capture this gas potential without either full hydrodynamics (making matching exactly the structural properties of Taurus impossible), or by developing a scheme for integrating a complex potential that allows several components that co-move with stellar groups (beyond the scope of this work). However, we do find agreement between the dynamical state (i.e. correlation functions) in our model and observations, which would suggest the role of ISM self-gravity on the population dynamics as a whole cannot be dramatic. While \nthis nonetheless remains a shortcoming of this work, we discuss in Section 4.5 that if high density gas alters our results this would have other interesting physical implications.', '4.1. Types of late-stage encounter': 'First, we qualitatively explore the nature of star-disc encounters in our simulation. For this we show some examples of encounters that occur at 1 ± 0 . 2 Myr in our simulation. The location of all these close encounters are shown in the central panel of Figure 9, from which we extract the following categories: \n- a) Stable, long period eccentric binary: Possibly the most straightforward kind of encounter is a very long period binary which remains unperturbed by unbound stars, as shown in the top-left of Figure 9. This kind of encounter can still produce relatively large changes in outer disc radius if the orbital period is su ffi ciently long.\n- b) Chaotic multiple: Shown in the top-middle of Figure 9 is an example of a bound triple system that interacts to eventually eject one of the stars at ∼ 0 . 8 Myr, leaving a tighter binary that can further truncate the disc.\n- c) Perturbed eccentric binary: If a stable binary is perturbed by an encounter with an external (unbound) star, then this may also result in a tightening of the binary. A closer periastron distance can then further truncate the disc. We show an example of this in the top-right of Figure 9 (although in this case, the external star is in fact marginally bound).\n- d) Random unbound encounters: This is the best-studied type of encounter, the rate of which can be understood analytically at a given local stellar density and velocity dispersion. However, in our simulation, we do not find any examples of random encounters. These encounters are more common in denser regions, but in Taurus we find that the majority of stardisc interactions are mediated by wide binary companions. \nWhile this categorisation illuminates the value of physically motivated structure and multiplicity within models for star forming regions, they are not rigid, and types of encounter may blur into each other. Our findings emphasise that unbound encounters and binary interactions cannot be studied separately during the dynamical evolution of a young star forming region.', '4.2. Disc evolution': "We consider how encounters shape the global disc radius distribution in Taurus. This is illustrated in Figure 10, where we show the cumulative distribution of disc outer radii over the course of the simulation. We find that the vast majority of discs are rapidly truncated below their initial radius by encounters, and ∼ 1 / 4 are truncated below 30 au. However, by the present time (1 Myr), dynamical encounters are not changing the overall distribution of disc radii significantly. This is because, while external perturbation does sculpt the outer disc for a large fraction of the population, disc truncation mostly occurs in stable binaries. Observations of discs in binary systems appear broadly consistent with theoretical expectations (Manara et al. 2019; Rota et al. 2022). We conclude that, for a region like Taurus, the role of encounters for the disc population as a whole is largely dominated by early interactions in binaries, rather than an ongoing process of disc truncation. \nFig. 9. In the bottom panel, we show the spatial distribution of stars (mass > 0 . 08 M ⊙ ) at 1 Myr and contemporary disc-truncating encounters in our simulation. We show as coloured star or square symbols the location of all stars that underwent encounters yielding a fractional truncation -∆ R out / R out > 0 . 01 over the age range 1 ± 0 . 2 Myr. Encounters that yielded -∆ R out / R out > 0 . 1 are highlighted with red circles. In the panels along the top we show the separation between specific stars and their stellar neighbours over the first 1 . 5 Myr of the simulation. Each line represents the distance to a single stellar neighbour, linearly interpolated between snapshots, where neighbours that are one of the two closest stars at any given time-step are included (if they pass within 10 4 au). In each plot, we mark the location of a logged 'closest approach' as a cross (inferred analytically from the closest time-step - see Section 3.10). \n<!-- image -->", '4.3. Truncating encounter frequency': 'We now turn to the primary motivation of this work, and ask the question: do we expect su ffi cient close encounters in Taurus to produce the examples of recent dynamical interaction? To answer this question we require two definitions: \n- 1. Which encounters produce significant external structures (e.g. tidal tails)?\n- 2. For how long does this observable external structure persist? \nTo answer these questions, we consider the rate of encounters that yield a fractional truncation -∆ R out / R out above some threshold, in the context of the observed flyby candidates in Taurus.', '4.3.1. HV and DO Tau': 'The huge extended dust bridge between HV and DO Tau (Howard et al. 2013) appears to be the result of a strongly truncating encounter, possibly occurring during the dynamical decay of a quadruple system (Winter et al. 2018a). While the model of Winter et al. (2018a) is probably not a unique scenario for producing the observed structure, we can make some quantitative arguments as to the requirements of such an encounter. Assuming some grain growth occurred within the disc, the mass of material expelled during the encounter is M ex ∼ 10 -4 M ⊙ (Winter et al. 2018a), which is ∼ 10 percent of the current mass M disc of \nFig. 10. Disc outer radius R out evolution for the population of stars with m ∗ > 0 . 08 M ⊙ in our simulation. The colour bar shows the time at which the distribution is measured. The initial outer radius distribution is shown in red. \n<!-- image --> \nthe disc around HV Tau C (Stapelfeldt et al. 2003). If the surface density of the disc Σ ∝ R -1 , for R the cylindrical radius inside R out, this implies a fractional truncation -∆ R out / R out ∼ M ex / M disc ∼ 0 . 1. To reach the present day projected separation of ≳ 10 4 au, the encounter must have occurred ∼ 0 . 1 Myr ago. This would suggest a rate of ∼ 10 encounters per Myr for encounters that result in a fractional truncation -∆ R out / R out ≳ 0 . 1.', '4.3.2. RW Aurigae': 'For RW Aurigae, the best-fitting model explored by Dai et al. (2015) had an initial outer radius of 60 au and a final outer radius in the range ∼ 40 -57 au. This estimate is mostly based on the geometry of the spiral arm as inferred from their simplified radiative transfer. The 12 CO emission detected around RW Aurigae is at least partly optically thick, so it is not possible to reliably infer a total mass that has been ejected in the encounter. The conclusions of Dai et al. (2015) may change if the apparent morphology as seen in CO di ff ers when detailed chemistry, photodissociation or radiative transfer e ff ects are taken into account. If the physical arm is longer and wider than they assume, the authors show that a weaker encounter can be consistent with the observed structure. Dai et al. (2015) also infer a time since closest approach ∼ 600 years from their dynamical model. This conclusion may depend somewhat on the deprojected geometry of the spiral and the system orientation.', '4.3.3. UX Tau': 'In the case of UX Tau, Ménard et al. (2020) focused on two di ff erent flyby simulations, with outer radius R out = 60 au and 90 au, both with r p = 100 au, with a mass-ratio of the perturber q ∼ 0 . 08 -0 . 22. The former radius is close to the observed outer radius (post-encounter). Both simulations produce clear spiral arms and some extended structure, so a severely truncating encounter is not required. There are currently no estimates for the mass of the external structure with respect to the disc mass. From their simulations, Ménard et al. (2020) estimate ∼ 1000 years since closest approach, with some margin for uncertainty in the system geometry as in the case of RW Aurigae.', '4.3.4. Overall rates': "The considerations above lead us to conclude that a small fraction of the disc mass may be capable of producing some of the observed external structures. We therefore explore a range of fractional truncation thresholds -∆ R out / R out > 10 -3 , 10 -2 and 10 -1 . Choosing a threshold smaller than 10 -3 makes little di ff erence to the overall encounter rate. As discussed above, 'weak' encounters with -∆ R out / R out ∼ 10 -2 may be su ffi cient to reproduce the structures observed around RW Aurigae and UX Tau, while the stronger encounter threshold would correspond to systems like HV and DO Tau. While we cannot be confident of the exact -∆ R out / R out that result in RW Aurigae and UX Tau-like systems, our results will motivate future studies quantifying the mass in the external structure surrounding systems that have undergone recent encounters, given that M ex / M disc ∼ -∆ R out / R out. \nThe overall encounter rate is summarised in Figure 11 for these thresholds. We have binned the encounters by the time at which they occur and by their fractional disc truncation. We also show the evolution of individual discs by connected lines between individual encounters, so that the evolution of the outer disc radii in (for example) binary systems is clear. As expected from Section 4.2, the majority of discs are truncated rapidly in binary systems during the first ∼ 0 . 1 Myr. However, there remain examples of individual encounters persisting throughout the course of the simulation. In order to explore the degree to which our results are stochastic, we run two additional versions of our experiment described in Appendix B. \nQuantitatively, we can see that the rate of the strongest encounters -∆ R out / R out > 0 . 1 remains at ∼ 10 Myr -1 in the time range ∼ 1 -3 Myr (see also Appendix B). Thus HV and DO Taulike encounters, observable for ∼ 0 . 1 Myr, are likely (probability ∼ 50 percent) to be found somewhere in Taurus. We conclude that the occurrence of the HV and DO Tau encounter is expected in the context of our dynamical model. \nThe more recent RW Aurigae- and UX Tau-like encounters require a considerably higher encounter rate. The rate of weaker encounters -∆ R out / R out > 10 -2 is Γ enc ∼ 100 Myr -1 , with approximately a factor order unity in stochastic variation (Appendix B). This rate is a factor ∼ 2 -3 larger for -∆ R out / R out > 10 -3 . We can write the probability of observing at least N encounters from a Poisson distribution: \nP obs( N obs ≥ N ) = 1 -N -1 X i = 0 ( Γ enc τ obs) i exp( -Γ enc τ obs) i ! , (13) \nwhere τ obs is the period for which the encounter is observable. Given that we are equally likely to observe the disc at any stage during this period of observability, this implies the average time of observation ⟨ t obs ⟩ = τ obs / 2. We therefore adopt a moderately generous τ obs = 2000 years, a factor ∼ 2 larger than the time since periastron for UX Tau and RW Aurigae. Taking a range of encounters Γ enc = 100 -200 Myr -1 for N obs ≥ 2 (RW Aurigae and UX Tau) yields P obs = 0 . 018 -0 . 062. Therefore, if encounters with -∆ R out / R out ≳ 10 -2 can produce these systems, then the tension with our model are not significant (or very marginally significant at ∼ 2 σ ). If a much more truncating encounter is required, then this tension may become significant. This marginal agreement underlines the importance of future studies quantifying the fraction of mass in the extended structure. \nIn the absence of additional constraints, we conclude that the expected rate of encounters in Taurus is marginally su ffi cient to produce UX Tau and RW Aurigae without appealing to additional physics (see Section 4.5). However, this would not be the \nFig. 11. The global rate of disc truncating encounters for discs around stars with mass m ∗ > 0 . 08 M ⊙ throughout the evolution of our dynamical model. The top panel shows the rate of all truncating encounters that decrease the disc radius by at least 0 . 1 percent (light gray), 1 percent (gray) and 10 percent (black) of the pre-encounter radius. The colour scale of the bottom panel shows the encounter rates binned into both the time of the encounter ( x -axis) and the degree of truncation ( y -axis). Each logged encounter is shown by a pink square, and subsequent encounters for the same disc (if any) are connected by faint pink lines. \n<!-- image --> \ncase if the sample of known recent, truncating star-disc encounters is incomplete. Indeed, there is some reason to suspect that this may be the case, as discussed below.", '4.4. Masses of stars undergoing close encounters': 'We can ask whether truncating encounters occur more or less frequently for high mass stars. We show the distribution of the masses of stars that undergo truncating encounters after 1 Myr in Figure 12. This mass distribution is indistinguishable to the overall mass function ( m ∗ > 0 . 08 M ⊙ ). By contrast, RW Aurigae A and B have masses ∼ 1 . 3 -1 . 4 M ⊙ and ∼ 0 . 7 -0 . 9 M ⊙ respectively (Ghez et al. 1997; Woitas et al. 2001). HV Tau C has a mass ∼ 0 . 5 -1 M ⊙ (Duchêne et al. 2010) and DO Tau ∼ 0 . 3 -0 . 7 M ⊙ (Beckwith et al. 1990; Hartigan et al. 1995). UX Tau A has a mass ∼ 1 . 3 M ⊙ and UX Tau C has mass ∼ 0 . 16 M ⊙ (Kraus & Hillenbrand 2009; Zapata et al. 2020). The origin of the extended structures are the discs around RW Aurigae A, HV Tau C, DO Tau and UX Tau A. The masses of these stars appear to be systematically greater than what would be obtained from randomly sampling from the IMF. Assuming nominal masses of 1 . 4 M ⊙ , 0 . 8 M ⊙ , 0 . 5 M ⊙ and 1 . 3 M ⊙ for RW Aurigae A, HV Tau C, DO Tau UX Tau A respectively yields KS test p -value \n0 . 023 percent when comparing to the stars that undergo encounters in our model. It is quite possible that observations could be biased to detect evidence of encounters involving more massive stars with brighter discs. However, this would imply many more undetected external structures that are evidence of encounters involving low mass stars in Taurus. Following equation 13, if evidence for just one more encounter similar to RW Aurigae or UX Tau were uncovered in Taurus, this would being the 2 σ encounter rate to Γ enc ∼ 680 Myr -1 . Alternatively, this would yield P obs ≈ 10 -3 even for weak encounters with Γ enc ∼ 100 Myr -1 . Finding further examples would introduce significant tension between the frequency of such events and the encounter rates in our model (Section 4.3.4), necessitating additional physics. This highlights the importance of future unbiased surveys to search for evidence of recent star-disc encounters in Taurus. \nWe caveat our findings with the fact that we do not invoke any primordial mass segregation (Zinnecker et al. 1993; Moeckel & Bate 2010; Plunkett et al. 2018). However, mass segregation is not evident in Taurus (Dib & Henning 2019). \n/circledot \nFig. 12. Cumulative distribution function for the masses m ∗ of stars with m ∗ > 0 . 08 M ⊙ in our simulation (black line), compared with those that underwent a truncating encounter -∆ R out / R out > 0 . 01 (red line). The distributions are not significantly di ff erent, with KS test probability p KS = 0 . 87. We also show mass estimates for the stars with discs that are responsible for the observed external structure. We do not show uncertainties in these estimates for clarity, but errors quoted are typically ∼ ± 0 . 3 M ⊙ . The distribution of masses of stars that have been inferred to have experienced recent truncating encounters are significantly different from those in our simulation, with p KS = 0 . 023. \n<!-- image -->', '4.5. Mechanisms to increase late encounter rates': 'Given the apparently high probability that the observed sample of discs with external structure produced during a recent stardisc encounter is incomplete, we consider a number of possibilities that may enhance the frequency of encounters late-on during the dynamical evolution of Taurus.', '4.5.1. (Viscous) re-expansion': 'For discs that experience multiple encounters that are either unbound or in long period binaries, viscous angular momentum transport (e.g. Lynden-Bell & Pringle 1974) or in-fall of material from the ISM (e.g. Padoan et al. 2005; Manara et al. 2018; Ku ff meier et al. 2023; Gupta et al. 2023; Padoan et al. 2024; Winter et al. 2024) may replenish outer disc material such that subsequent encounters will have a stronger influence than under our assumption that disc radius is fixed. This replenishment should be correlated with stellar mass (as suggested by observed stellar accretion rates - Manara et al. 2017), so this would also preferentially enhance encounter rates for high mass stars, consistent with observations (Section 4.4). Conversely, if angular momentum is extracted by magnetohydrodynamic winds (e.g. Bai & Stone 2013) then this could exacerbate the encounter rate problem.', '4.5.2. Self-gravity of the interstellar medium': 'As discussed in Section 3.12, for practical reasons we have ignored ISM self-gravity in our calculations. If included, we would expect this self-gravity to increase the time for which groups of stars remain bound and therefore potentially enhance the frequency of truncating encounters at late times. This in-of-itself \nFig. 13. Critical density at which the ISM is bound against turbulent pressure (black line, left hand side y -axis) and the resultant BondiHoyle-Lyttleton (BHL) accretion rate for a solar mass star (red dashed line, right hand side y -axis). Both are shown as a function of spatial scale λ . \n<!-- image --> \n/circledot \nmay explain any shortcoming in the frequency of encounters in our model compared to observations. However, for this to keep stars bound together, gas must be at approximately the critical density to undergo gravitational collapse. On length scales λ relevant for the Taurus region ( σ v ≫ c s and λ ≪ h , the galactic scale height), this is e ff ectively the Jeans criterion: the gravitational potential must balance the turbulent energy. At such densities, as discussed above, a number of authors have suggested that disc replenishment by Bondi-Hoyle-Lyttleton (BHL) accretion may be substantial (Padoan et al. 2005; Throop & Bally 2008; Ku ff meier et al. 2023; Winter et al. 2024). \nTo test whether we expect stellar dynamics-altering ISM self-gravity to also substantially replenish protoplanetary discs, we derive the critical density ρ c as a function of λ following Winter et al. (2024), shown as the black line in Figure 13. We then estimate the expected typical BHL accretion rate: \n˙ M BHL ∼ 4 π G 2 m 2 ∗ ρ c σ 3 v (14) \nfor a solar mass star m ∗ = 1 M ⊙ , shown as the red dashed line in Figure 13. Here σ v ∝ λ 0 . 5 as we assume when implementing kinematic substructure. For typical regions of size λ ∼ 1 pc (Schmalzl et al. 2010; Joncour et al. 2018), our calculations imply a BHL accretion rate ˙ M BHL ∼ 10 -8 M ⊙ yr -1 , comparable to observed stellar accretion rates (e.g. Manara et al. 2023). Therefore, if local dynamics is influenced by ISM self-gravity, we expect this also to replenish protoplanetary discs as discussed above in Section 4.5.1. \nWe summarise that, while we cannot rule out ISM selfgravity as a possible driver of present day encounters in Taurus, this would have extremely interesting consequences for disc evolution. In particular, it would imply that replenishment from the ISM is indeed common.', '4.5.3. Tidal torques from the disc on the perturber': 'If the disc is su ffi ciently massive and the encounter su ffi ciently close, then it is possible that the tidal dissipation of the orbital energy of the perturber leads to capture and tightening of the bound orbit (Clarke & Pringle 1993; Ostriker 1994; Muñoz et al. 2015). In this case the subsequent encounters may be stronger as \nthe periastron separation decreases. This is a promising way to generate delayed strong encounters in binary / multiple systems where multiple encounters have occurred, particularly given the evidence for several close passages in the RW Aurigae system (Rodriguez et al. 2018). Since the disc mass is a superlinear function of star mass (e.g. Ansdell et al. 2016), this mechanism could also be more e ff ective for perturbations to discs around massive stars. A key question however, is whether discs retain su ffi cient mass to generate orbital decay to the present day; this may be the subject of future hydrodynamic simulations for RW Aurigae and UX Tau.', '4.5.4. Embedded planets or brown dwarfs': 'If brown dwarfs or massive planets form and remain embedded in the protoplanetary discs, then it is possible that during the encounter they undergo large eccentricity perturbations (Heggie & Rasio 1996) or chaotic evolution, perhaps to be captured by the perturbing star (e.g. Fregeau et al. 2004). This would e ff ectively be a late-stage enhancement of the binary fraction, which may be expected to enhance the encounter rate. As for the previous two mechanisms, massive planets appear to be more common around high mass stars (Johnson et al. 2010). While an interesting possibility in the context of spiral arms in discs, given the presence of a stellar-mass perturber in the cases of RW Aurigae and UX Tau this does not appear to be a convincing explanation.', '4.5.5. Spatial variation of the binary fraction': 'In our simulations, we initiated binaries independently of their location. It is plausible that binary formation is more probable in regions of enhanced local density, where enhanced tides can in principle result in fragmentation into multiple systems (Horton et al. 2001). This would enhance the degree to which binaries typically interact with other stars, and may therefore lead to enhanced encounter rates. However, RW Aurigae at least does not appear to be located in a region of high stellar density (e.g. Pfalzner & Govind 2021).', '4.5.6. Prospects': 'While these explanations discussed in this section are not yet necessary based on existing constraints, two future developments may change this conclusion. First, the quantity of mass in the extended structure surrounding RW Aurigae and UX Tau could be a substantial fraction of the disc mass (that may imply | ∆ R out / R out | ≫ 10 -2 ). Second, any new discoveries of extended structure in Taurus originating from a star-disc encounter will considerably increase the required encounter rate to Γ enc ∼ 680 Myr -1 , much greater than the rate for even weak encounters in our model Γ enc ∼ 200 Myr -1 . Should an additional recent encounter be discovered, and given the arguments outlined above, possibly the most promising mechanism for enhancing the encounter rate is the (viscous) re-expansion or ISM replenishment of the disc. This hypothesis should be quantitatively investigated in the event of additional discoveries.', '5. Conclusions': "We have presented an N-body model for the Taurus star forming region which is physically and empirically motivated. Our initial stellar population is generated probabilistically from a turbulent gaseous medium, via a zoom-in on a Gaussian field us- \ning the cosmology simulation tool genet IC (Stopyra et al. 2020). We applied the inferred size-velocity relation in Taurus to generate an initial velocity dispersion, combined with an empirically motivated binary population. Without any additional tuning, this yields a dynamical model with excellent agreement to the observed structure in Taurus by the metrics of the pair separation distribution and separation-velocity correlation. This has allowed us, for the first time, to accurately assess the global frequency of encounters in the Taurus star forming region. \nIn this way we have shown that, like the bull that is its namesake, stars in the Taurus star forming region rarely settle for one close encounter. Instead, star-disc systems act as stellar matadors, often enduring several close approaches with a neighbouring star. The closest approach distances change stochastically over time as binaries are themselves perturbed by low velocity neighbours. High order multiples can also form and break up on time-scales much shorter than the lifetime of Taurus. As a result, strong encounters at the present day can occur in one of four ways. The most common are: during the evolution of a chaotic multiple system; when an eccentric binary is perturbed; or as a result of a close approach in a very long period eccentric binary. Random encounters between single stars are rare, with close encounters in Taurus being mostly mediated through a binary companion. These categories can be blurred, and in some sense to distinguish between binary and unbound encounters is a false dichotomy. Since binaries and nearby low velocity stars influence each other, inferring accurate encounter rates requires that the spatial and kinematic structure of a region is quantitatively taken into account. \nOverall, ∼ 1 / 4 of discs are truncated below 30 au by dynamical encounters. However, the majority of these dynamical truncation events happen in the first few 0 . 1 Myr of the cluster evolution, over the course of a few binary periods. After this time, the role of encounters in sculpting the overall distribution of disc radii is largely finished in a low density star forming region such as Taurus. \nNonetheless, individual strong encounters still occur over the region as a whole, and we consider whether the examples of HV and DO Tau (Howard et al. 2013; Winter et al. 2018a), RW Aurigae (Dai et al. 2015; Rodriguez et al. 2018) and UX Tau (Zapata et al. 2020; Ménard et al. 2020) should be observed in our model. We conclude that events resembling HV and DO Tau can occur at a rate of ∼ 10 Myr -1 and therefore, given that it has remained observable for ∼ 0 . 1 Myr, observing one such event at any given time is expected. If weak encounters that eject only ∼ 0 . 1 -1 percent of the disc mass are responsible for the external structure around RW Aurigae and UX Tau, then they are also expected given the encounter rate Γ enc ∼ 100 -200 Myr -1 in our model. However, we highlight that the systems that have been inferred to have experienced recent truncating encounters do not appear to be consistent with random drawing from the mass function, while those in our model are. This hints that the known sample of discs that have recently experienced truncating encounters is incomplete: just one more observational example would be in strong tension with our model, implying additional physics. \nWe discuss a number of physical mechanisms that should be explored by future work that may yield enhancements in the rate of truncating encounters, such as the tightening of binaries due to star-disc interaction (e.g. Muñoz et al. 2015) or reexpansion / replenishment in the outer regions of protoplanetary discs. We also show that substantial replenishment via BHL accretion must be proceeding if ISM self-gravity, which we neglect \nin this work, is significant for the dynamical evolution of high order multiples across the Taurus complex. \nWe summarise that star-disc encounters are an important probe of disc physics. This work highlights the need for a systematic search for extended structure generated by star-disc encounters in Taurus. \nAcknowledgements. We thank the anonymous referees for their useful comments that helped improve the clarity of this manuscript. This project has received funding from the European Research Council (ERC) under the European Union's Horizon research and innovation programme (grant agreement No. 101002188, project PROTOPLANETS, and grant agreement No. 101042275, project Stellar-MADE). AJW has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No 101104656. LS was supported by NSFC grant Nos. 11890692, 12133008, and 12221003. LS thanks T Fang for support and acknowledges science research grants from NSFC, grant Nos. 11890692, 12133008, and 12221003. This work has made use of data from the European Space Agency (ESA) mission Gaia ( https://www.cosmos.esa.int/gaia ), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium ). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.", 'References': '- Aarseth, S. J. 2003, Gravitational N-Body Simulations Abt, H. A. 2006, ApJ, 651, 1151 Adams, F. C., Proszkow, E. M., Fatuzzo, M., & Myers, P. C. 2006, ApJ, 641, 504 Alaguero, A., Cuello, N., Ménard, F., et al. 2024, V892 Tau: A tidally perturbed circumbinary disc in a triple stellar system Andrews, S. M. 2020, ARA&A, 58, 483 Ansdell, M., Williams, J. 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Lunine, 429', 'Appendix A: Output frequency exploration': "Our approach to computing the encounter frequency has been to post-process high frequency outputs, following the method described in Section 3.10. For our fiducial model, we adopt an output frequency of one every 323 years. While this frequency is su ffi cient to capture even binary encounters if the semi-major axis a ≳ 50 au, it is possible that in some instances binary-single or binary-binary interactions result in miscalculation of the true encounter properties. Here we test if such events may change our results. \nWhile it would be laborious (and challenging) to check every single star's encounter history in our sample for examples where our approach does not capture close encounters, we can more directly test our choice on the disc truncation history. We therefore decrease the output time-step by factor ten (one every 32 years), and repeat the same analysis on the rate of disc truncation as before. The results are summarised in Figure A.1, which can be compared directly to Figure 11. Statistically and qualitatively, our results are very similar. We therefore conclude that our approach for extracting encounters is not substantially altering our conclusions. However, we do notice some small di ff erences in the encounter history that correspond to a small number of individual encounter histories. These di ff erences are not important for our conclusions, but we investigate them as follows. \nWe first identify an example of a system for which we observe di ff erences in the encounter history depending on the output frequency. We show one such encounter history in Figure A.2 for our fiducial model, which is a chaotic triple interaction. The close encounters identified by our algorithm are shown as crosses. Comparison with the high frequency output simulation (Figure A.3) shows that at early times, both encounter histories are identical. They diverge after ∼ 0 . 3 Myr, from which point the dynamics of the systems evolve chaotically to di ff erent endstates. This suggests that the di ff erences in the encounter histories inferred from Figure A.1 compared to Figure 11 are not due to di ff erences in our encounter-extraction algorithm, but di ff erences in the N-body integration itself. While in principle changing the frequency of outputs should not alter the integration, N body 6 ++ performs a number of accountancy operations at the time of output. It is possible that these operations slightly alter other numbers in the code that enter into numerical calculations via, for example, the adjustment of parameters. We demonstrate that altering the parameter adjustment time-step ( DTADJ ) can have a similar influence on the dynamical evolution of chaotic multiples in Figure A.4, where we reduce this time-step by five compared to our fiducial model. \nWe do not here investigate what changes to the output timestep lead to an altered chaotic evolution of high order multiples when using the N body 6 ++ code. By the nature of chaotic interactions, such changes may be tiny (such as machine precision) and in this case no particular solution is obviously more accurate. This is particularly irrelevant physically, since for these cases other processes may also change the dynamical outcome. However, we are satisfied that the statistical distribution of star-disc encounters our model is not dependent on our choice of output time-step, and that our algorithm for extracting encounters is adequate for our purposes.", 'Appendix B: Stochastic encounter history': 'To ensure that we are not dominated in our quantitative conclusions by stochastic variations in the encounter rates, we perform two additional numerical integrations of a Taurus-like re- \ngion. We do this by performing two resampling experiments for the same density field as we adopt for our fiducial model, then adding a new binary population. The outcome comparing the encounter rates across all the simulations is shown in Figure B.1. Figures comparable to Figure 11 are shown in Figures B.2 and B.3. While we are not able to repeat the experiment enough times to gain a full distribution at each time interval, we can estimate a factor ∼ 2 variation in the encounter rates is typical. We conclude that the rate of truncating encounters that we predict is only stochastic to within a factor of order unity. \nFig. A.1. As in Figure 11, but for a factor ten higher output frequency. \n<!-- image --> \nFig. A.2. An example of a chaotic multiple interaction in our fiducial model. We show the separation of two neighbouring stars from a third star as a function of time. Crosses mark the location where the closest encounter is recorded by our post-processing analysis. \n<!-- image --> \nFig. A.3. As in Figure A.2, for the same stars but with a factor ten higher output frequency. \n<!-- image --> \nFig. A.4. As in Figure A.2, but with a factor five smaller time-step between the adjustment of simulation paramaters ( DTADJ parameter in N body 6 ++ ). \n<!-- image --> \nFig. B.1. Stochastic variation of the encounter rate between di ff erent simulations (shown by di ff erent line styles). The colour of the lines refers to the threshold truncation used to define the encounter (as in Figure 11). \n<!-- image --> \nFig. B.2. As in Figure 11 but for a second random drawing of the stellar and binary population. \n<!-- image --> \nFig. B.3. As in Figure B.2 but for a third random drawing of the stellar and binary population. \n<!-- image -->'}

2024arXiv240908499Z

Deep and low massratio contact binaries DLMCBs are believed to be in the final stage of their contact phase potentially leading to the formation of fastrotating single stars such as FK Comtype stars and blue stragglers as well as luminous red novae. These systems serve as an excellent laboratory for studying stellar coalescence and merging processes. Our search for DLMCBs began in 2004 and has since identified a group of such systems. Together with that collected from the literature more than 100 DLMCBs have been detected so far. Half of them have had their periods investigated based on OC curves. Some have shown period increases while others have exhibited period decreases. Among them more than half DLMCBs have cyclic variations suggesting the possibility of the existence of a third body orbiting around the DLMCBs. Furthermore with more data obtained extending the span of the OC curve more cyclic variations could be detected. The high proportion of signs of the presence of third bodies makes them an essential factor to consider when studying the merger of contact binaries.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.08499', '2024arXiv240908499Z', 'arXiv:2409.08499']

['Astrophysics - Solar and Stellar Astrophysics']

Deep and low massratio contact binaries and their third bodies

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.08499.pdf

{'Deep and low mass-ratio contact binaries and their third bodies': 'Zhu Liying, 1 , 2 , 3 Qian Shengbang, 4 Liao Wenping, 1 , 2 , 3 Zhang Jia, 1 , 2 , 3 Shi Xiangdong, 1 , 2 , 3 Li Linjia, 1 , 2 , 3 Meng Fangbin, 1 , 2 , 3 Wang Jiangjiao 1 , 2 , 3 Matekov Azizbek 1 , 2 , 5 \n- 1 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China; zhuly@ynao.ac.cn\n- 2 Key Laboratory of the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China\n- 3 University of Chinese Academy of Sciences, No.1 Yanqihu East Rd, Huairou District, Beijing, P.R.China 101408\n- 4 School of Physics and Astronomy, Yunnan University, Kunming 650091, China\n- 5 Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, 33 Astronomicheskaya str., Tashkent, 100052, Uzbekistan \nAbstract. Deep and low mass-ratio contact binaries (DLMCBs) are believed to be in the final stage of their contact phase, potentially leading to the formation of fastrotating single stars such as FK Com-type stars and blue stragglers, as well as luminous red novae. These systems serve as an excellent laboratory for studying stellar coalescence and merging processes. Our search for DLMCBs began in 2004 and has since identified a group of such systems. Together with that collected from the literature, more than 100 DLMCBs have been detected so far. Half of them have had their periods investigated based on O-C curves. Some have shown period increases, while others have exhibited period decreases. Among them, more than half DLMCBs have cyclic variations, suggesting the possibility of the existence of a third body orbiting around the DLMCBs. Furthermore, with more data obtained extending the span of the O-C curve, more cyclic variations could be detected. The high proportion of signs of the presence of third bodies makes them an essential factor to consider when studying the merger of contact binaries.', '1. Introduction': 'Contact binaries comprise two stars with components intricately connected, sharing a common convective envelope (CCE). This shared envelope leads to the exhibition of EW-type light curves with similar depths in both minima, even when the two components di ff er significantly in mass. Contact binaries can originate from various processes, including the fragmentation of a molecular cloud, the capture of a passing star, and stellar collisions. Over time, a combination of mechanisms such as tidal friction, magnetic braking, Kozai cycles within triple systems, and mass transfer between the components acts to bring the stars into closer proximity, giving rise to contact binaries. This con- \ntinuous evolution may eventually lead to the merger of the two stars. The well-known luminous red nova, V1309 Scorpii, provides confirmed evidence that contact binaries can indeed undergo the merger process. \nA luminous red nova is a stellar explosion characterised by a distinct red colour, and a light curve that fades slowly with resurgent brightness in the infrared. They were not only discovered in the Milky Way galaxy such as V4332 Sgr (Martini et al. 1999), V838 Mon (Munari et al. 2002; Bond et al. 2003) and V1309 Sco (Mason et al. 2010), but also discovered in other Galaxies including the nearest major galaxy M31(Mould et al. 1990) and the elliptical galaxy M85 (Kulkarni et al. 2007). V1309 Sco was discovered in September 2008 (Nakano 2008) and underwent an evolutionary transition from an F-type giant to a late M-type giant (Mason et al. 2010). it presented a unique opportunity for the study of stellar mergers, facilitated by the recorded light curves of its progenitor captured by OGLE. These light curves strongly suggested that V1309 Scorpii originated as a contact binary (Tylenda et al. 2011). Zhu et al. (2016) further revealed that its progenitor was a low-mass ratio (q 0.1) and deep contact binary star (f = 90%) in 2002. The search for and study of deep and low mass-ratio contact binaries (DLMCBs) holds significance in unraveling the destiny of contact binaries and advancing our understanding of stellar coalescence and merging processes. \nThe criterion for binary star merging is defined by the condition that the orbital angular momentum is less than three times the rotational angular momentum (Hut, 1980). Given the orbital constraints imposed by the contact configuration, contact binaries with lower mass ratios tend to approach this merging condition. On the other hand, deep contact implies a thick common envelope (CE), which is inherently unstable. Hence, contact binaries characterized by both deep contact and low mass ratios emerge as promising objects for the study of stellar mergers. Our exploration of deep and low mass-ratio contact binaries (DLMCBs) began in 2004 (Qian & Yang 2004), and we proposed that a contact binary could be categorized as a DLMCB if its mass ratio q ≤ 0 . 25 and the fill-out factor f ≥ 50% (Qian et al., 2005a; 2006b). This classification provides a useful framework for identifying binaries that are conducive to investigating the dynamics of stellar merging.', '2. Candidates selection': 'Thanks to several photometric surveys conducted around the world, a significant number of eclipsing binaries with EW-type light variation have been discovered.Together with spectroscopic survey by the Large Sky Area Multiobject Fiber Spectroscopic Telescope (LAMOST), not only the light curves, but also the stellar atmospheric parameters including the e ff ect temperature T e f f , the gravitational acceleration Log(g), the metallicity [Fe / H] and the radial velocity V r of a large number of contact binaries have been obtained. This provides a valuable database for studying the physical properties of contact binaries. By cross-checking the VSX catalog (the international variable star index, Watson et al. 2006) with LAMOST targets, we have compiled a catalog of EW binaries with all the available parameters and continue to update it with new observations (Qian et al., 2017, 2020). The entire catalogue is available online, with an electronic version accessible through the website (http: // search.vbscn.com / 2020EW.table1.txt). \nBased on the catalog, a new orbital period distribution of contact binaries has been derived with a strong maximum and a very sharp edge at around 0.15 days. The maximum of the distribution is at about 0.31 d, and most EWs are in the orbital period \nrange from 0.285 to 0.345 d (Qian et al., 2020). The period-color (or temperature) relation is a well-known relation for contact binaries (Eggen 1967; Rucinski 1998). To investigate this relation in detail using LAMOST stellar atmospheric parameters, the new relation is presented in Figure 1. As shown in the figure, most EWs are located within the two cyan lines, which represent the boundaries of normal EWs. Green dots represent EWs observed by low-resolution spectra (LRS), while dark dots represent EWs observed by medium-resolution spectra (MRS). Systems near the right boundary usually have longer orbital periods for a given temperature and higher orbital angular momentum. They usually have marginal (or shallow) contact configurations with fillout factors less than 20%. Systems located near the left boundary have shorter orbital periods and are usually deep contact systems. Therefore, the LAMOST data are very useful for selecting targets for detailed follow-up observations and investigations, and more and more DLMCBs will be detected in the future. \nFigure 1. Correlation between the orbital period and e ff ective temperature based on normal EWs observed by LRS and MRS (green and dark dots respectively). Blue dots refer to binaries located above the left boundary of normal EWs, while red dots to systems below the right boundary. Systems near the left border are deep contact binaries. \n<!-- image -->', '3. Recent results': 'Since 2004, our team has published a series of papers to report the discovery of Deep and Low Mass-Ratio Contact Binaries (DLMCBs) (Qian & Yang, 2004; Qian et al., 2005a, 2005b, 2006, 2007, 2011; Zhu et al., 2005, 2011; Yang et al., 2005, 2009, 2013; Liao et al., 2017, 2022; Sarotsakulchai et al., 2018; Meng et al., 2023). In addition to our own discoveries, many DLMCBs have been reported by other researchers, including Pribulla et al., 2009; Zola et al., 2017; Kjurkchieva et al., 2017; ; Li et al., 2022; Christopoulou et al., 2022; Guo et al., 2023 etc. To date, more than 100 DLMCBs have \nbeen discovered, and approximately half of them have had their periods investigated based on the O-C curve. Among these systems, 33 show period increases, 20 show period decreases, and 25 exhibit periodic variations due to the light travel time caused by a third body orbiting around the central DLMCBs. Here we introduce our recent results on two active DLMCBs with confirmed additional body based on spectroscopic observation and the changes of the O-C curve due to the light travel time e ff ect.', '3.1. NY Bootes: An Active DLMCB with a Cool Companion in a Hierarchical Triple System': 'NY Bootes (NY Boo, TIC 309809572) was first discovered as an overcontact binary system by the Northern Sky Variability Survey (Ho ff man et al. 2009). The Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) provided high-precision photometric data for NY Boo in five sectors (sector16, sector23, sector24, sector49, and sector50), with an exposure time of 2 minutes. The phased light curves for all sectors are shown in the upper-left panel of Figure 2. As can be seen from this figure, the light curves of NY Boo exhibit noticeable variations over time, indicating significant magnetic activity in the system. Six low-resolution spectra of NY Boo are available from the LAMOST database, and three infrared spectra were obtained from the SDSS Scientific Archive Server. By using the cross correlation function (CCF) method on the LAMOST and SDSS spectra, we obtained its radial velocity (RV) curve, which is shown in the upper-right panel of Figure 2. A simultaneous analysis of the light curve and RV curve reveals that NY Boo is a contact binary with a mass ratio ( q = 0 . 139) and fillout factor ( f = 73%). It is worth noting that NY Boo is a partial eclipsing binary with an inclination of only 67.4 degrees, and its light curves exhibit significant changes in a short time. This makes it di ffi cult to confirm that it is a DLMCB without the support of RV curve. Most of the DLMCBs detected to date are total eclipsing binaries, as the reliable photometric mass ratio can be determined in this case. LAMOST is a reflecting Schmidt telescope with an e ff ective aperture of 4 m that can simultaneously observe and obtain spectra for close to 4000 targets. With more spectra of EW binaries, more DLMCBs could potentially be discovered in the future. \nIn order to study the orbital period variation, we collected all available timings of NY Boo, together with the others calculated by us, we constructed the O-C curve of NY Boo based on these times of light minimum, which span 23 years. A detailed investigation of its O-C diagram reveals that NY Boo undergoes a long-term period decrease while also experiencing cyclical variations in its orbital period. The corresponding fit curves can be seen in the lower-left panel of Figure 2. The rate of orbital period decrease is dP / dt = -1 . 305( ± 0 . 022) × 10 -6 dayyr -1 . Comparing the orbital period reduction timescale with the thermal timescale of the primary component, we found that NY Boo may be in the fast-shrinking stage. The cyclical variation of the orbital period with an amplitude A = 0 . 00713 day and a period P = 16 . 9 yr could be due to the light-travel time e ff ect of the tertiary companion, which is consistent with the third light features detected from the CCF profiles of the SDSS spectra (lower-right panel of Figure 2). Therefore, NY Boo should be a triple system. Evidence of the existence of the third body was only found in the infrared spectrum, suggesting that it is a late-type star. By combining the third light contribution derived from the CCF fit with the results of the O-C curve fit, we derived the orbital inclination, mass, and semimajor axis of the third body as i 3 = 50, m 3 = 0 . 31 M J , and a 3 = 5 . 82 au, respectively, This indicates that the third body is non-coplanar with the central pair. The combination of \na low mass ratio, high fill-out factor, fast-shrinking, and rapid light variation in the inner pair with a non-coplanar tertiary component makes NY Boo an excellent target for studying the late evolution of contact binaries and stellar mergers (Meng et al., 2023). \nFigure 2. Upper-left panel: The TESS phase-flux diagram of NY Boo that varied with time. Upper-right panel:The RV curves of NY Boo. The filled dots represent data from LAMOST spectra and open dots represent data from SDSS spectra. Lower-left panel: The O-C diagram of NY Boo. The red dots refer to the eclipse times obtained from TESS. Lower-right panel: CCF profile of the LAMOST (upper panel) and SDSS (lower panel) example spectra. The solid black line indicates the CCF profile. The red dashed line represents the double Gaussian fit. The blue and pink dashed lines denote individual Gaussian components. \n<!-- image -->', '3.2. V410 Aur: An Active DLMCB in a quintuple stellar system': "The variability of V410 Aur was first discovered by the Hipparcos satellite, and its radial velocity study was conducted by Rucinski et al.(2003). They determined that V410 Aur is a spectroscopic double-lined binary with a mass ratio of q sp = 0 . 144(13) and third light contribution l 3 = 26(1)%. The light curves of V410 Aur are known to vary with time, indicating the star's activity. In 2005, we modeled the light curves of V410 Aur and concluded that it is a DLMCB (Yang et al. 2005). However, the study of the O-C curve at that time suggested that the orbital period of V410 Aur was continuously increasing. \nIn 2022, we reanalyzed the O-C curve and found that the previous period increase was only part of the cyclic variation (Liao et al., 2022). We analyzed the cyclical variation for the light-travel time e ff ect and determined that the minimum mass of the third \nbody is 1 . 39(0 . 13) M J which is much larger than the spectroscopically inferred value of (0.97 M J ). This indicates that the spectroscopically detected tertiary is actually a single-lined spectroscopic binary with an unseen component. We also determined the maximum orbital semimajor axis of the third body to be 6 . 19(0 . 67) au. Gaia detected a visual companion to V410 Aur at practically the same distance from the Sun, providing further confirmation of its physical bond. These results reveal that V410 Aur contains a single-lined spectroscopic binary with a visual companion in a quintuple stellar system, making it an interesting active merger progenitor in a quintuple system (2 + 2 + 1).", '4. Summary': 'We have gathered the periods of the third bodies in DLMCBs systems and plotted their relationship with the period of the DLMCBs in Figure 3. As seen from this figure, longer period DLMCBs tend to have farther third bodies. The periods of the third bodies vary from several years to decades, with most having a period greater than 10 years. This suggests that it typically takes more than 10 years of data coverage to detect their third bodies. Most newly discovered DLMCBs do not have long observational history, which makes it di ffi cult to detect the third body based on the light-travel time e ff ect. Additionally, many DLMCBs exhibit variable light curves, which are often attributed to the magnetic activity of the binary component. In such cases, the cyclic changes detected in the O-C curve may be explained by magnetic activity cycles (Applegate 1992). However, both NY Boo and V410 Aur consist of active stars, and their periodic changes detected in the O-C curves are confirmed to be due to the third body through their spectral features. Therefore, the presence rate of additional companions in DLMCB systems may be underestimated. \nIn addition to the O-C curve, the presence of a third body can also be detected through spectroscopic data, as seen in the cases of NY Boo and V410 Aur. Since the period of DLMCBs is much shorter than that of the third body, the spectral lines of the binary components will exhibit blue or red shift due to the Doppler e ff ect, while the spectral lines of the third body remain relatively motionless. LAMOST has initiated a time-domain spectroscopic survey, which will observe each of the 200,000 stars an average of 60 times. This data will allow for the detection of additional bodies orbiting around DLMCBs in various forms in the future. The high proportion of third bodies present indicates that they will be an essential factor to consider when studying the merger of contact binaries. \nAcknowledgments. This work is supported by the International Cooperation Projects of the National Key R&D Program (No. 2022YFE0127300), the Young Talent Project of "Yunnan Revitalization Talent Support Program" in Yunnan Province and CAS "Light of West China" Program. We are grateful to all sta ff members of ASP for their support.', 'References': 'Applegate, J. H. 1992, ApJ, 385, 621 Bond, H. E., Henden, A., Levay, Z. G., et al. 2003, Nature, 422, 405 Christopoulou P.-E., Lalounta E., Papageorgiou A. et al., 2022, MNRAS, 512, 1244. Eggen, O. J. 1967, MmRAS, 70, 111 Guo D., Li K., Liu F., Li H., Liu X., Chen X., 2023, PASP, 135, 044201. \nHo ff man, D. I., Harrison, T. E., & McNamara, B. J. 2009, AJ, 138, 466 \nFigure 3. The relation between the period of DLMCBs and the periods of the third bodies. \n<!-- image --> \nHut, P. 1980, A&A, 92, 167 \nKjurkchieva, D. P., Popov, V. A., Vasileva, D. L., & Petrov, N. I. 2017, RMxAA, 53, 235 \nKorhonen, H., Berdyugina, S. V., Hackman, T., et al. 2007, A&A, 476, 881 \nLi, K., Gao, X., Liu, X.-Y., et al. 2022, AJ, 164, 202 \nLiao, W. P., Qian, S. B., Soonthornthum, B., et al. 2017, PASP, 129, 124204 \nLiao, W. -P., Qian, S. -B., Shi, X. -D. et al., 2022, ApJ, 927, 183 \nMason, E., Diaz, M., Williams, R. E., Preston, G., & Bensby, T. 2010 A&A, 516, A108 \nMartini, P., Wagner, R. M., Tomaney A, et al. 1999, AJ, 118, 1034 \nMould, J., Cohen, J., Graham, J. R., et al. 1990, ApJ, 353, L35 \nMunari, U., Henden, A., Kiyota, S., et al. 2002, A&A, 389, L51 \nMeng F., Zhu L., Qian S., Liu N., Li L., Matekov A., 2023, ApJ, 954, 111 \nNakano, S. 2008, IAU Circ., 8972 \nPribulla, T., Rucinski, S. M., DeBond, H., et al. 2009, AJ, 137, 3646 \nQian, S. B., & Yang, Y. G. 2004, AJ, 128, 2430 \nQian, S. B., Yang, Y. G., Soonthornthum, B., et al. 2005a, AJ, 130, 224 \nQian, S. B., Zhu, L. Y., Soonthornthum, B., et al. 2005b, AJ, 130, 1206 \nQian, S.-B., He, J.-J., Zhang, J., et al. 2017, RAA, 17, 087 \nQian, S.-B., Zhu, L.-Y., Liu, L., et al. 2020, RAA, 20, 163 \nQian, S. B., Liu, L., Soonthornthum, B., Zhu, L. Y., & He, J. J. 2006, AJ, 131, 3028 \nQian, S. B., Liu, L., Soonthornthum, B., Zhu, L. Y., & He, J. J. 2007, AJ, 134, 1475 \nQian, S. B., Liu, L., Zhu, L. Y., et al. 2011, AJ, 141, 151 \nRucinski, S. M. 1998, AJ, 116, 2998 \nRicker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, JATIS, 1, 014003 \nRucinski, S. M., Capobianco, C. C., Lu, W., et al. 2003, AJ, 125, 3258 \nSarotsakulchai, T., Qian, S. B., Soonthornthum, B., et al. 2018, AJ, 156, 199 \nTylenda, R., Hajduk, M., Kamiski , T., et al. 2011, A&A, 528, A114 \n- Watson, C. L., Henden, A. A., & Price, A. 2006, Society for Astronomical Sciences Annual Symposium, 25, 47\n- Yang, Y. G., Qian, S. B., & Zhu, L. Y. 2005, AJ, 130, 2252\n- Yang, Y. G., Qian, S. B., Zhu, L. Y., & He, J. J. 2009, AJ, 138, 540\n- Yang, Y. G., Qian, S. B., Zhang, L. Y., Dai, H. F., & Soonthornthum, B. 2013, AJ, 146, 35\n- Zola, S., Baran, A., Debski, B., & Jableka, D. 2017, MNRAS, 466, 2488\n- Zhu, L. Y., Qian, S. B., Soonthornthum, B., & Yang, Y. G. 2005, AJ, 129, 2806\n- Zhu, L. Y., Qian, S. B., Soonthornthum, B., He, J. J., & Liu, L. 2011, AJ, 142, 124\n- Zhu, L.-Y., Zhao, E.-G., & Zhou, X. 2016, RAA, 16, 68'}

2024arXiv240909227F

Highresolution stellar spectra offer valuable insights into atmospheric parameters and chemical compositions. However their inherent complexity and highdimensionality present challenges in fully utilizing the information they contain. In this study we utilize data from the Apache Point Observatory Galactic Evolution Experiment APOGEE within the Sloan Digital Sky Survey IV SDSSIV to explore latent representations of chemical abundances by applying five dimensionality reduction techniques PCA tSNE UMAP Autoencoder and VAE. Through this exploration we evaluate the preservation of information and compare reconstructed outputs with the original 19 chemical abundance data. Our findings reveal a performance ranking of PCA lt UMAP lt tSNE lt VAE lt Autoencoder through comparing their explained variance under optimized MSE. The performance of nonlinear Autoencoder and VAE algorithms has approximately 10 improvement compared to linear PCA algorithm. This difference can be referred to as the nonlinearity gap. Future work should focus on incorporating measurement errors into extension VAEs thereby enhancing the reliability and interpretability of chemical abundance exploration in astronomical spectra.

2024-09-01T00:00:00Z

['arXiv:2409.09227', '10.48550/arXiv.2409.09227', '2024arXiv240909227F']

['Astrophysics - Instrumentation and Methods for Astrophysics', 'Statistics - Applications']

Exploring Dimensionality Reduction of SDSS Spectral Abundances

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.09227.pdf

{'Exploring Dimensionality Reduction of SDSS Spectral Abundances': 'Qianyu Fan 1 \n1 Department of Statistical Sciences, University of Toronto', 'ABSTRACT': "High-resolution stellar spectra offer valuable insights into atmospheric parameters and chemical compositions. However, their inherent complexity and high-dimensionality present challenges in fully utilizing the information they contain. In this study, we utilize data from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) within the Sloan Digital Sky Survey IV (SDSS-IV) to explore latent representations of chemical abundances by applying five dimensionality reduction techniques: PCA, t-SNE, UMAP, Autoencoder, and VAE. Through this exploration, we evaluate the preservation of information and compare reconstructed outputs with the original 19 chemical abundance data. Our findings reveal a performance ranking of PCA < UMAP < t-SNE < VAE < Autoencoder, through comparing their explained variance under optimized MSE. The performance of non-linear (Autoencoder and VAE) algorithms has approximately 10% improvement compared to linear (PCA) algorithm. This difference can be referred to as the 'non-linearity gap.' Future work should focus on incorporating measurement errors into extension VAEs, thereby enhancing the reliability and interpretability of chemical abundance exploration in astronomical spectra. \nKeywords: Dimensionality Reduction - Chemical Abundance - APOGEE - Neural Network", '1. INTRODUCTION': 'Stellar spectra play a vital role in revealing the fundamental properties of stars, including atmospheric parameters and chemical compositions, shedding light on the formation and evolution of galaxies within the Milky Way. The availability of an enormous amount of high-resolution stellar spectra today is attributed to many past and ongoing extensive spectroscopic surveys. Among these, the Apache Point Observatory Galactic Evolution Experiment (APOGEE) within the Sloan Digital Sky Survey IV (SDSS-IV) stands out for its comprehensive collection of high-resolution stellar spectra. APOGEE (Majewski et al. 2017) has amassed spectra from hundreds of thousands of stars in the inner Galaxy, leveraging near-infrared observations at high resolution ( R = 22 , 500) and high signal-to-noise ratio ( S/N ≥ 100); these provide high-precision kinematical properties and chemical information. \nHigh-resolution stellar spectra contain a wealth of information essential for understanding stellar properties and evolutionary processes. However, this information is often embedded in high-dimensional spaces where complex correlations exhibit non-linear patterns, posing significant challenges for comprehensive analysis. The issue of high dimensionality is not unique to stellar spectra but is pervasive across various domains including biomedical, web, education, medicine, business, and social media (Ayesha et al. 2020). In genomics, for instance, gene expression data collected from microarray experiments typically involve thousands of genes, each representing a dimension in the dataset. Similarly, in neuroscience, brain imaging techniques generate high-dimensional data representing neural activity across multiple regions of the brain. \nThese high-dimensional datasets present challenges in visualization, interpretation, and analysis. The curse of dimensionality exacerbates computational complexity and hinders the exploration of underlying structures and patterns. Consequently, dimensionality reduction techniques have emerged as indispensable tools for navigating high-dimensional spaces and extracting meaningful information. \nThere are many different dimensionality reduction techniques, including linear techniques and non-linear ones. In Nanga et al. (2021) research, it reviewed 14 dimensionality reduction techniques: the linear techniques considered were Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Singular Value Decomposition (SVD), Latent Semantic Analysis (LSA), Locality Preserving Projections (LPP), Independent Component Analysis (ICA) \nand Project Pursuit (PP); and non-linear techniques considered were Kernel Principal Component Analysis (KPCA), Multi-dimensional Scaling (MDS), Isomap, Locally Linear Embedding (LLE), Self-Organizing Map (SOM), Latent Vector Quantization (LVQ), t-Stochastic neighbor embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP). \nAmong these, LDA always requires labels, SVD and LPP have high computational complexity, LSA is primarily designed for text data, ICA assumes independent data distributions, and PP can be inefficient for large-scale data. Though PCA also struggles with nonlinear data, it is widely used in astronomy (Harinder et al. 1998). In contrast to the traditional linear techniques, the non-linear techniques have the ability to deal with complex non-linear data. In particular, the real-world data, such as astronomy data is typically formed a highly non-linear manifold. To assess the improvement brought by nonlinear techniques in dimensionality reduction, so we adopted PCA as a baseline for comparison. For non-linear techniques, t-SNE and UMAP were opted since they offer robust performance compared with others. To pursuing more accuracy, astronomers also used neural networks (NN), like Autoencoder and variational autoencoder (VAE) for dimensionality reduction (Tagliaferri et al. 2003). They can effectively handle high-dimensional data, capturing intricate patterns and non-linear data representations. \nIn our study, we implemented five dimensionality reduction techniques, ranging from linear to non-linear methods, to uncover underlying structures and patterns in 2-dimensional space. Our goal is to explore latent representations of chemical abundances, evaluate the preservation of information during the dimensionality reduction process, and compare reconstruction results with the original data. \nThe remainder of the paper is structured as follows. We describe data and data preprocessing in Section 2. Then in Section 3, we introduce dimensionality reduction techniques and neural network approach. Section 4 summarizes the results and Section 5 discusses these techniques and the next steps before concluding in Section 6.', '2. DATA': "The data used in our study is from APOGEE Data Release 17 (DR17), which encompasses 19 chemical abundances and stellar parameters for 370,060 stars. This data was obtained through automated analysis of spectra using APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) software (Garc'ıa P'erez et al. 2016). We removed poor quality stars by ASPCAPFLAG and STARFLAG bitmasks to mitigate potential quality warnings and also removed outliers and applied standard scale, resulting in a final dataset comprising 133,891 stars. \nWe focused on chemical abundances, which refer to the relative abundance of different chemical elements in the atmosphere of stars. To quantify the relative amounts of individual elements, it uses the abundance ratio as the logarithm of the ratio of two metallic elements in a star relative to their ratio in the sun. For instance, if a star has a [C/FE] value of -0.08, it means its carbon abundance is 83% that of the Sun. We are using 19 dimensions related to chemical abundances, including the metallicity [FE/H] and other 18 element abundances.", '3.1. Dimensionality Reduction Techniques': 'Our study aims to investigate latent representations of chemical abundances in a 2-dimensional space. We utilized five dimensionality reduction techniques to achieve this goal.', '3.1.1. Principle Component Analysis': "Principle Component Analysis (PCA) (Ma'ckiewicz & Ratajczak 1993) is one of the oldest and most widely used techniques which is an unsupervised linear dimension reduction technique. PCA identifies the directions (principal components) in which the data varies the most and projects the data onto these directions, thus reducing the dimensionality while preserving the most important information. The first principal component captures the direction of maximum variance in the data, and each subsequent component captures the remaining variance orthogonal to the previous ones. Since we want to reduce dimension into 2, it identifies two orthogonal directions that capture the maximum variance in the data.", '3.1.2. t-Distributed Stochastic Neighbor Embedding': 't-Distributed Stochastic Neighbor Embedding (t-SNE) (Van der Maaten & Hinton 2008) is a non-linear algorithm that finds similarities between data points in high-dimensional spaces by modeling pairwise similarities using a Gaussian distribution. Then, t-SNE aims to find a low-dimensional representation of the data points in such a way that the \nsimilarities between points in the high-dimensional space are preserved as closely as possible. Since some points in the high dimension will be compressed into one point in the low dimension due to a short tail in the distribution, it uses t-distribution to create low-dimensional space. The algorithm minimizes the divergence between the probability distributions of pairwise similarities in the high-dimensional and low-dimensional spaces using a gradient descent optimization approach. By iteratively adjusting the positions of data points in the low-dimensional space, t-SNE seeks to create a mapping that reveals the underlying structure and clusters within the data.', '3.1.3. Uniform Manifold Approximation and Projection': 'Uniform Manifold Approximation and Projection (UMAP) (McInnes et al. 2018) is similar to t-SNE but with some key differences. It constructed a weighted graph and built a fuzzy topological representation to approximate the manifold structure of the high-dimensional data and preserve both local and global structure. Then, it optimizes a lowdimensional representation of the data through stochastic gradient descent, efficiently adjusting the low-dimensional embedding to best capture the essential features of the data. Additionally, it utilizes a constraint, where each data point is connected to at least its nearest neighbors, to ensure the whole graph is connected. This feature distinguishes UMAP, enabling it to better preserve the global structure of the chemical abundances.', '3.1.4. Autoencoder': 'Autoencoder (Bank et al. 2020) is a type of unsupervised feedforward neural network (Abdi 1994). It consists of two main components: an encoder and a decoder. The encoder takes the input data and compresses it into a lower-dimensional representation, typically called the latent space. This compressed representation contains the most important features of the input data, effectively reducing its dimensionality. The decoder then takes this compressed representation and attempts to reconstruct the original input data from it. The goal of the autoencoder is to learn an encoding-decoding process that minimizes the difference between the input and the reconstructed output. In other words, it aims to learn a compact representation of the data that retains as much relevant information as possible. Autoencoders can be trained using various optimization techniques, such as gradient descent, to minimize a reconstruction loss function, such as mean squared error. In our case, it takes 19-dimensional chemical abundances as input and compresses it to produce a 2-dimensional latent representation, followed by a decoder that takes the latent representation and decompresses it to produce a reconstruction of the original data.', '3.1.5. Variational Autoencoder': 'Variational Autoencoder (VAE) (Kingma & Welling 2013) is an extension of Autoencoder that encodes input as a distribution over the latent space rather than a single point, and the distribution is typically represented by the mean and log variance (used to guarantee variance is positive) parameters of a Gaussian distribution. This is achieved by adding a regularization term to the loss function, which is typically the KL divergence (Kullback & Leibler 1951) between the distribution output by the encoder and the standard Gaussian distribution. To train the VAE using backpropagation, we need to compute gradients with respect to the parameters of the encoder and decoder networks. However, since the encoder outputs a probability distribution, we cannot directly backpropagate through sampling operations. The reparameterization trick addresses this issue by decoupling the sampling process from the parameters of the distribution. Instead of sampling directly from the distribution, we sample from a standard Gaussian distribution and then transform the samples using the parameters of the distribution output by the encoder.', '3.2. Statistical Measure': 'To evaluate the preservation of information during dimensionality reduction, we applied explained variance as a metric to measure the performance of these techniques. Explained variance denotes the proportion of the total variance in the original 19-dimensional data that is captured in its 2-dimensional representation. \nExplained Variance = 1 -Unexplained Variation Total Variation \nAmong this formula, the total variance of 19-dimensional data can be computed by the trace of the variancecovariance matrix. \ntrace(S) = m ∑ j =1 S 2 = m ∑ j =1 ( 1 n n ∑ i =1 ( X ij -¯ X mn,ij ) 2 ) \nwhere m represents the number of dimensions, i.e., 19 dimensions, n represents the number of objects, X ij denotes the value of the j -th dimension for the i -th object, and ¯ X mn,ij denotes the mean value of the j -th dimension across all objects. \nUnexplained variation refers to the portion of the total variability in the data that remains unaccounted, while Mean Squared Error (MSE) captures the difference between predicted reconstructed outputs and original data. They are related to each other, and so we can use MSE as a measure of unexplained variation.', '3.3. Neural Network Architectures': "3.3.1. A Fully-connected Neural Network for t-SNE and UMAP \nWhile PCA provides inherent explained variance, we lack an explicit reconstruction function f ( Z i ; θ ) for t-SNE and UMAP. \nMSE ( θ ) = 1 n n ∑ i =1 ( X i -f ( Z i ; θ ) 2 ) \nwhere X i represents original data and Z i represents the data after dimension reduction. \nTo address this, we trained a fully-connected neural network to predict the reconstructed values for t-SNE and UMAP respectively. The neural network model was constructed using the PyTorch Sequential module, featuring a series of linear layers interspersed with Batch Normalization (Ioffe & Szegedy 2015) and LeakyReLU activation functions. The architecture in Figure 1 begins with a linear layer taking 2-dimensional input data (i.e., data after dimensionality reduction), which is then transformed into a 64-dimensional representation. This is followed by another linear layer reducing the dimension to 32, and finally, the output layer generates the expected 19-dimensional reconstructed output. Each linear transformation is followed by Batch Normalization to normalize the input and improve the stability and performance of the model by reducing internal covariate shift. The LeakyReLU activation is then applied to introduce non-linearity into the model and capture complex patterns in the data. The LeakyReLU activation function is defined as \nLeakyReLU( x ) = { x for x ≥ 0 ax for x < 0 , where a = 0 . 01 \nThis introduces a small slope when input is negative, rather than setting the activation function to zero, helping to alleviate the vanishing gradient problem and enhance the model's representational capacity. \nWe employed an Adam optimizer (Kingma & Ba 2015) with a learning rate of 1 × 10 -4 to minimize the Mean Squared Error (MSE) loss function, with hyperparameters tuned using the Optuna framework. The chemical abundance dataset consists of n objects across m dimensions, and the MSE is calculated as follows: \nMSE = 1 m 1 n n ∑ i =1 m ∑ j =1 ( X ij -X mn,ij ) 2 \nwhere m represents the number of dimensions, i.e., 19 dimensions, n represents the number of objects, X ij denotes the value of the j -th dimension for the i -th object, and X mn,ij denotes the predicted value of the j -th dimension across all objects. \nThe model was trained for 3 , 000 epochs using mini-batches of data. Each epoch involved forward and backward passes, with model weights updated using the Adam optimizer. To ensure convergence, early stopping was implemented with a convergence threshold set to 1 × 10 -4 . If the average loss on the test set showed no significant improvement for 5 consecutive epochs, the model was considered to have converged. This mechanism effectively prevented overfitting by halting the training process when the model's performance on the test set no longer improves. Finally, the model with the lowest MSE on the test set was retained as the optimal model.", '3.3.2. Autoencoder and VAE Neural Networks': "The architecture of both Autoencoder and VAE in Figure 2 is symmetric, with the encoder and decoder being mirror images of each other. To ensure consistency and facilitate comparative analysis with t-SNE and UMAP, both models utilized their decoder parts to reconstruct the 2-dimensional latent representations back to the original 19-dimensional \nFigure 1 Structure of a fully-connected neural network with a 2-64-32-19 architecture. Each linear transformation is followed by Batch Normalization and LeakyReLU activation. We trained this neural network for t-SNE and UMAP respectively to predict the reconstructed values since they don't have explicit reconstruction function.Then, we use MSE to measure difference between reconstructed values and original data and calculate explained variance. \n<!-- image --> \n(a) Structure of a autoencoder with a 19-32-64-2-64-32-19 architecture. \n<!-- image --> \n(b) Structure of a VAE with a 19-32-64-2-64-32-19 architecture. \n<!-- image --> \nFigure 2 To ensure consistency and facilitate comparative analysis with t-SNE and UMAP, both models utilized their decoder parts to reconstruct the 2-dimensional latent representations back to the original 19-dimensional space. Then we measure MSE and explained variance to evaluate their performances. \nspace, so the decoder follows a 2-64-32-19 architecture. Given their symmetry, both models have a 19-32-64-2-64-32-19 architecture. \nFurthermore, it's worth noting that other aspects of the Autoencoder and VAE models, such as the optimization algorithm, the number of epochs, the implementation of early stopping, and the use of the MSE loss function, also align with the neural network used for t-SNE and UMAP. The only difference lies in optimizing the total loss during the training process of VAE, which includes MSE and KL divergence terms, but the optimization still focuses on minimizing the MSE on the test set. This consistency ensures a fair comparison between the different dimensionality reduction techniques.", '3.4. Visualization and Comparison': 'After dimensionality reduction, we visualized the distribution of objects in the latent space generated by the five techniques, coloring them based on mean metallicity [Fe/H]. We also compared the explained variance through boxplots obtained from the reconstruction by neural networks. Additionally, we selected a sample from the dataset and plotted line charts to compare the reconstructed results of these algorithms with the original data.', '4.1. 2-dimensional Representation of Five Algorithms': "Figure 3 depicts 2-dimensional representations of our 19-dimensional chemical abundances. PCA and Autoencoder exhibit two clusters, while t-SNE and UMAP reveal three clusters, but it's hard to recognize in VAE representation. \nThe structure of PCA is connected, since it operates under the assumption of linear relationships, and it seeks to find a linear transformation of the data that captures the most variance. In contrast, UMAP does not rely on linear relationships and can capture more complex, nonlinear relationships in the data. Therefore, it is not typically connected as PCA does. UMAP has the best 2-dimensional visualization as it balanced both local and global structure. \nThe circular or brain-like shape observed in two-dimensional t-SNE visualizations is attributed to its utilization of t-distribution resembling Gaussian distribution, and this algorithm's tendency emphasizes local structure in the data. t-SNE is designed to preserve local relationships between data points, meaning that nearby points in the highdimensional space are encouraged to remain close together in the low-dimensional representation. Notably, PCA and UMAP outperform t-SNE in preserving the global structure of the data, resulting in more coherent graphs. \nHowever, the representation of Autoencoder looks weird. This is because we don't know regularity of the latent space. Specifically, encoding 19-dimensional chemical abundances into a single point in latent space, each unique combination of chemical abundances is mapped to a single point in the latent space. This compression provides a high degree of freedom, meaning that multiple different combinations of chemical abundances can potentially map to the same point in the latent space, leading to overfitting. So, when visualizing the representation generated by the Autoencoder, points in the latent space appears random. Conversely, VAE encodes input data as a distribution approximating Gaussian distribution, manifesting as circular patterns. \nBy measuring the strength of absorption lines in stellar spectra, it is possible to infer the abundance of different elements in the stellar atmosphere. Among them, the abundance of iron (Fe) (typically represented by [Fe/H]) is one of the important indicators of interest to astronomers because iron plays a significant role in the formation and evolution processes of stars, and the abundance of iron is closely related to the age and chemical evolutionary history of stars. Consequently, metallicity [Fe/H] serves as a primary contributor to the variation in the data, evident in the trend observed when coloring latent representations.", '4.2. Explained Variance Comparison': "The boxplots in Figure 4 display the performance of these algorithms and Table 1 provides detailed results, including epoch when model converged, optimized MSE, and explained variance. PCA, with its linear assumption, achieves an explained variance of 0.7689. For t-SNE, the explained variance is 0.8535 with an MSE of 0.1465, while UMAP achieves an explained variance of 0.8356 with an MSE of 0.1644. UMAP's performance is slightly worse than t-SNE, possibly because UMAP preserves more global structure at the expense of explained variance in our case. The Autoencoder achieves an explained variance of 0.8632 with an MSE of 0.1368, while VAE achieves an explained variance of 0.8606 with an MSE of 0.1394. The performances of the Autoencoder and VAE are quite close. VAE incorporates KL divergence in the loss function, but this only increases model stability, with no improvement in explained variance. \nSince we found best MSE for each algorithm, so we can compare their explained variance. The performance ranking of these algorithms is PCA < UMAP < t-SNE < VAE < Autoencoder. PCA achieves the lowest explained variance \n<!-- image --> \n<!-- image --> \nFigure 3 2-dimensional representations for five algorithms \n<!-- image --> \namong all algorithms, indicating that it may not capture as much of the variability in the data compared to the other techniques. UMAP and t-SNE perform better than PCA in terms of explained variance, suggesting that they are more effective at preserving the overall structure of the data.Autoencoder and VAE achieve the highest explained variance, indicating that they are able to capture a larger proportion of the variance in the data compared to the other algorithms. The performance gap between linear (PCA) and non-linear (Autoencoder and VAE) algorithms is approximately 10%, which suggests that non-linear techniques are more effective at capturing the underlying structure and complexity of the data and we can call it as non-linearity gap. Additionally, Figure 5 displays the training and testing loss to demonstrate that our models have converged, which aligns with their performance. \nFigure 4 Explained variance comparison for five algorithms \n<!-- image --> \n<!-- image --> \n<!-- image --> \n(c) UMAP \nTable 1 Overview of five algorithms \n(a) Training Loss Comparison \n<!-- image --> \nFigure 5 Training and testing loss comparison for t-SNE, UMAP, Autoencoder, and VAE. \n<!-- image --> \n(b) Testing Loss Comparison \nFigure 6 illustrates the comparison between the original 19 chemical abundances and the reconstructed outputs from the five algorithms. There is a noticeable gap for each abundance between the PCA reconstructed data and the original data. For t-SNE, the gap gradually narrows down, and some abundances, such as [O/Fe], [S/Fe], and [Ti/Fe], are matched more closely. Similarly, UMAP exhibits a narrowing gap, though discrepancies remain for abundances like [C/Fe], [Cl/Fe], [N/Fe], and [P/Fe]. Autoencoder and VAE show further improvement, with abundances like [C/Fe], [Cl/Fe], [N/Fe], and [P/Fe] becoming even closer to the original data. \n(a) Original vs. PCA Recon \n<!-- image --> \n(b) Original vs. t-SNE Recon \n(c) Original vs. UMAP Recon \n<!-- image --> \nFigure 6 Comparison between original 19 chemical abundances and reconstructed outputs from the five algorithms. \n<!-- image --> \n- (d) Original vs. Autoencoder Recon \n(e) Original vs. VAE Recon", '5. DISCUSSION': "Although our study explored the latent representations of chemical abundances and evaluated the preservation of information, there still exists limitations will affect results. We only evaluated five dimensionality reduction techniques, and more techniques could be considered in the future work. Also, we only used 2-64-32-19 architecture neural network to predict reconstructed values for t-SNE and UMAP respectively. If we try to utilize other architectures and use different hyperparameters, the explained variance might change as result of optimized MSE changes. \nAdditionally, we didn't consider uncertainties in measurements and the potential impact of intrinsic scatter within these relations. So, in the future, we need to investigate ways to incorporate these uncertainties into deep learningbased approaches, such as extension VAEs, which is the variations of the basic VAE framework including modifications to the architecture, loss functions, or training procedures of VAEs.", '6. CONCLUSION': "Efficiently exploring large astronomical data is an important problem that can be addressed with dimensionality reduction techniques. In this study, we investigated latent representations of chemical abundances using five such techniques and evaluated their performance based on explained variance. Our findings revealed a performance ranking of PCA < UMAP < t-SNE < VAE < Autoencoder by comparing their explained variance under optimized MSE. The performance of non-linear (Autoencoder and VAE) algorithms has approximately 10% improvement compared to linear (PCA) algotirhm. This difference can be referred to as the 'non-linearity gap.' Additionally, we compared reconstructed outputs with the original 19 chemical abundances which provided more explicit differences for each \n<!-- image --> \n<!-- image --> \ndimensionality reduction techniques Future work should focus on incorporating measurement errors into extension VAEs, thereby enhancing the reliability and interpretability of chemical abundance exploration in astronomical spectra.", '7. ACKNOWLEDGEMENT': 'I would like to express my sincere appreciation to Prof. Joshua S. Speagle for his guidance and support throughout this research course.', 'REFERENCES': "Abdi, H. 1994, Journal of Biological Systems, 2(3), 247 Ayesha, S., Hanif, M. K., & Talib, R. 2020, Information Fusion, 59, 44-58, doi: 10.1016/j.inffus.2020.01.005 Bank, D., Koenigstein, N., & Giryes, R. 2020, arXiv: https://arxiv.org/abs/2003.05991 \nGarc'ıa P'erez, A. E., Prieto, C. A., Holtzman, J. A., et al. \n2016, The Astrophysical Journal, 151(6), 144, \ndoi: 10.3847/0004-6256/151/6/144 \nHarinder, P., Ravi, K., & Ranjan, G. 1998, Monthly Notices of the Royal Astronomical Society, 295, 312-318, doi: 10.1046/j.1365-8711.1998.01255.x \nIoffe, S., & Szegedy, C. 2015, International Conference on Machine Learning \nKingma, D. P., & Ba, J. 2015, arXiv: \nhttps://arxiv.org/abs/1412.6980 \nKingma, D. P., & Welling, M. 2013, arXiv: \nhttps://arxiv.org/abs/1312.6114 \nKullback, S., & Leibler, R. A. 1951, The Annals of Mathematical Statistics, doi: 10.1214/aoms/1177729694 Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, The Astrophysical Journal, 154(3), 94, doi: 10.3847/1538-3881/aa784d Ma'ckiewicz, A., & Ratajczak, W. 1993, Computers & Geosciences, 19, 303, doi: 10.1016/0098-3004(93)90090-r McInnes, L., Healy, J., & Melville, J. 2018, arXiv: https://arxiv.org/abs/1802.03426 Nanga, S., Bawah, A., Acquaye, B., et al. 2021, Journal of Data Analysis and Information Processing, 9, 189, doi: 10.4236/jdaip.2021.93013 Tagliaferri, R., Longo, G., Milano, L., et al. 2003, Neural Netw., 16, 297, doi: 10.1016/s0893-6080(03)00028-5 \nVan der Maaten, L., & Hinton, G. 2008, Journal of machine \nlearning research, 9(11)"}

2024arXiv240907153B

We present the algebraic classification of the gravitational field in fourdimensional general metricaffine geometries thus extending the current results of the literature in the particular framework of WeylCartan geometry by the presence of the traceless nonmetricity tensor. This quantity switches on four of the eleven fundamental parts of the irreducible representation of the curvature tensor under the pseudoorthogonal group in such a way that three of them present similar algebraic types as the ones obtained in WeylCartan geometry whereas the remaining one includes thirty independent components and gives rise to a new algebraic classification. The latter is derived by means of its principal null directions and their levels of alignment obtaining a total number of sixteen main algebraic types which can be split into many subtypes. As an immediate application we determine the algebraic types of the broadest family of static and spherically symmetric black hole solutions with spin dilation and shear charges in MetricAffine Gravity.

2024-09-01T00:00:00Z

['arXiv:2409.07153', '2024arXiv240907153B', '10.48550/arXiv.2409.07153']

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

Algebraic classification of the gravitational field in general metricaffine geometries

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07153.pdf

{'Algebraic classification of the gravitational field in general metric-affine geometries': 'Sebastian Bahamonde, 1, ∗ Jorge Gigante Valcarcel, 2, 3, † and Jos´e M.M. Senovilla 4, 5 \n1 Kavli Institute for the Physics and Mathematics of the Universe (WPI), \nThe University of Tokyo Institutes for Advanced Study (UTIAS), \nThe University of Tokyo, Kashiwa, Chiba 277-8583, Japan. \n2 Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Korea. \n3 Department of Physics, Tokyo Institute of Technology, \n1-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan. \n4 Departamento de F´ısica, Facultad de Ciencia y Tecnolog´ıa, \nUniversidad del Pa´ıs Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain. \n5 EHU Quantum Center, Universidad del Pa´ıs Vasco UPV/EHU, Bilbao, Spain. \nWe present the algebraic classification of the gravitational field in four-dimensional general metricaffine geometries, thus extending the current results of the literature in the particular framework of Weyl-Cartan geometry by the presence of the traceless nonmetricity tensor. This quantity switches on four of the eleven fundamental parts of the irreducible representation of the curvature tensor under the pseudo-orthogonal group, in such a way that three of them present similar algebraic types as the ones obtained in Weyl-Cartan geometry, whereas the remaining one includes thirty independent components and gives rise to a new algebraic classification. The latter is derived by means of its principal null directions and their levels of alignment, obtaining a total number of sixteen main algebraic types, which can be split into many subtypes. As an immediate application, we determine the algebraic types of the broadest family of static and spherically symmetric black hole solutions with spin, dilation and shear charges in Metric-Affine Gravity.', 'I. Introduction': "Algebraic classification has certainly played a significant role in the development of General Relativity (GR). Indeed, as featured in the Einstein's field equations, our current understanding of the gravitational interaction is based on the physical correspondence between the space-time curvature and the energy-momentum tensor of matter, which naturally leads to the study of the algebraic properties of these quantities, in order to find out, analyse and interpret different classes of solutions [1, 2]. \nFrom a mathematical point of view, some tensor quantities on a Lorentzian manifold can be recast as linear maps acting on a vector space, which lays the foundations of algebraic classification via the resolution of an eigenvalue problem [3]. In the framework of GR, the gravitational field is fully ascribed to the Riemann curvature tensor, whose irreducible decomposition under the pseudo-orthogonal group expresses it as a linear combination of the Ricci scalar and the completely traceless Weyl and Ricci tensors; the latter presenting nontrivial eigenvalue problems that lead to the so called Petrov and Segre classifications, respectively [4, 5]. \nThese classifications have numerous applications in the study of black holes, cosmology and gravitational waves. Of particular interest is the formulation of the Goldberg-Sachs theorem, which states that any vacuum solution of the Einstein's field equations admits a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special [6]. Such a congruence defines a null vector field that is, at each point, multiply aligned with the algebraic structure of the Weyl tensor, which is a manifestation of its 'speciality'. In fact, the consideration of an algebraically special Type D Weyl tensor -in which case there are two such doubly aligned null vector fields- turned out to be crucial to find the first known rotating black hole solution in GR, namely the stationary and axially symmetric Kerr solution [7]. Thereby, despite of the cumbersome form of the field equations for a stationary and axially symmetric configuration, the Kerr solution possesses a significant degree of symmetry, which is actually realised by the existence of a closed nondegenerate conformal Killing-Yano tensor [8] -actually, all type D vacuum solutions in GR admit a conformal Killing tensor as proven by Walker and Penrose [1]. These objects provide a separability structure for the wave and geodesic equations defined on the space-time, which in turn implies the complete integrability of causal geodesics and the algebraic Type D of the Weyl tensor [1, 9-14]. Likewise, a simple example of gravitational radiation is described by the plane-fronted waves with parallel rays ('pp waves'), which include exact vacuum solutions of \nthe Einstein's field equations and correspond to an algebraically special Type N Weyl tensor [1, 15], that is, with a unique multiply aligned null vector field. On the other hand, different matter sources of physical interest, such as the electromagnetic field, pure radiation matter field and the perfect fluid, also lead in the framework of GR to algebraically special types of the Ricci tensor that are included in the Segre classification [1]. \nGiven the relationship between the algebraic classification of the Weyl tensor and the existence of aligned null directions such as those discovered by Goldberg and Sachs, alternative classifications have been investigated relying exclusively on the existence of aligned null vectors. Here the concept of alignment is somewhat complicated and depends on the particular properties of the target tensor, but they can be rigorously defined in general by the vanishing of well-defined contractions and exterior products of the target tensor with the null vector. These are referred to as the principal null directions (PNDs) of the tensor. Such alternative classifications are then based on the number of different PNDs and their multiplicities, or their level of alignment. In four dimensions, this approach provides an algebraic classification for the Weyl tensor that is fully equivalent to the Petrov classification [16, 17] (see [18, 19] for further generalisations in higher dimensions), whereas a richer algebraic classification is obtained for the traceless Ricci tensor, in comparison with the Segre classification [20]. \nThe PNDs of any target tensor can also be characterised, in a much direct and simpler manner, as the null vectors whose contraction on all indices with the superenergy tensor [21] of the target tensor vanishes, see [19, 22, 23] and references therein. This generally leads to a refined classification [22] depending on the number of contractions of the the null vector with the superenergy tensor needed to get a vanishing result. The superenergy tensor is the (basically) unique tensor quadratic on the target tensor that satisfies a generalised dominant energy condition, and can be seen as a mathematical generalisation of the traditional energy-momentum tensor. The paradigmatic example is the famous Bel-Robinson tensor, which is just the superenergy tensor of the Weyl tensor [21]. \nTherefore, the problem of the algebraic classification of the gravitational field in GR and other theories of gravity based on Riemannian geometry is settled, but the presence of additional degrees of freedom in the geometry requires an extension of these results. In particular, a post-Riemannian description of the space-time in the presence of torsion and nonmetricity leads to the formulation of Metric-Affine Gravity (MAG), which constitutes a viable extension of GR and provides a diverse phenomenology at astrophysical and cosmological scales [24-74]. Thereby, a gauge invariant Lagrangian can be constructed from the generalised field strength tensors of this framework, in order to introduce the dynamics of the gravitational field enhanced by torsion and nonmetricity. A complete algebraic classification requires then to classify all the field strength tensors of torsion and nonmetricity, which naturally appear in the irreducible decomposition of the curvature tensor under the pseudo-orthogonal group [75]. Indeed, this problem has been recently addressed in the particular case of Weyl-Cartan geometry [76], while the case of general metric-affine geometries remains unsolved. \nIn this work we perform a complete algebraic classification of general metric-affine geometries by means of PNDs. Such a case is modeled by an affinely connected metric space-time that is characterised by completely general curvature, torsion and nonmetricity tensors. In particular, in contrast with a Weyl-Cartan space-time, it includes a traceless nonmetricity tensor, whose dynamics is described in the gravitational action of MAG by four field strength tensors; all of them obeying their own algebraic classifications. Indeed, even though the methods of algebraic classification are well-known in the literature, only a small number of tensors, including the Weyl, Ricci and Faraday tensors, have been formally classified. Therefore, the purpose of this work is twofold: we aim to obtain a full algebraic classification of the gravitational field in general metric-affine geometries, which on the other hand demands to obtain a new algebraic classification for a completely traceless and cyclic tensor that constitutes one of the field strengths of the traceless nonmetricity tensor. In comparison with the Weyl, Ricci and Faraday tensors, which carry ten, nine and six independent components in four dimensions, this field strength tensor carries thirty independent components, giving rise to a more complicated problem and in fact to a much richer algebraic classification. Apart from that, it is worthwhile to stress that, although our study refers to the framework of metric-affine geometry, the results are valid for any tensor quantities presenting the same algebraic properties as the ones considered in this work. \nThis paper is organised as follows. In Sec. II, we introduce the irreducible decomposition of the curvature tensor in metric-affine geometry, which is determined by eleven building blocks that provide the dynamics of the gravitational field with curvature, torsion and nonmetricity. Taking into account the algebraic symmetries of the mentioned building blocks, they can be sorted into four different categories, each one characterised by its own type of algebraic classification. In fact, it was recently shown that three of these types appear in the framework of Weyl-Cartan geometry [76], hence we briefly revisit them in Sec. III, IV and V. The main study is then addressed in Sec. VI, where we obtain the last type of algebraic classification that can take place in general metric-affine geometries. This requires a thorough analysis on the algebraic structure of one of the field strength tensors of the traceless nonmetricity tensor, for which we find its PNDs and their respective levels of alignment in Sec. VI A. We then apply, in Sec. VI B, the refinements derived by using the more elaborated classification using the superenergy tensor of this field strength. Once the algebraic classification is settled, in Sec. VII we determine the algebraic types of all of the field strength tensors of torsion and nonmetricity for the broadest family of static and spherically symmetric black hole solutions \nwith spin, dilation and shear charges in MAG, finding that the gravitational field of the solution is indeed algebraically special. Finally, we present the conclusions in Sec. VIII, while some technical details are relegated to the appendices. \nWe work in natural units c = G = 1 and consider the metric signature (+ , -, -, -). On the other hand, we use a tilde accent to denote those quantities that are defined from the general affine connection, in contrast to their unaccented counterparts constructed from the Levi-Civita connection. In addition, we denote with a diagonal arrow the traceless and pseudotraceless pieces of tensors (e.g. ↗ Q λ µν and ↗ ˜ R λ [ ρµν ] ). Latin and Greek indices run from 0 to 3, referring to anholonomic and coordinate bases, respectively.", 'II. Irreducible decomposition of the curvature tensor in metric-affine geometry': 'An independent affine connection includes the torsion and nonmetricity tensors \nT λ µν = 2 ˜ Γ λ [ µν ] , Q λµν = ˜ ∇ λ g µν , (1) \nas its antisymmetric part and as the covariant derivative of the metric tensor, which gives rise to a general curvature tensor that can be expressed as the sum of the Riemann tensor and further post-Riemannian corrections \n˜ R λ ρµν = R λ ρµν + ∇ µ N λ ρν -∇ ν N λ ρµ + N λ σµ N σ ρν -N λ σν N σ ρµ , (2) \nwith \nN λ µν = 1 2 ( T λ µν -T µ λ ν -T ν λ µ ) + 1 2 ( Q λ µν -Q µ λ ν -Q ν λ µ ) . (3) \nThereby, whereas in Riemannian geometry the irreducible decomposition of the curvature tensor into irreducible pieces under the pseudo-orthogonal group simply expresses this tensor as a linear combination of the Ricci scalar and the completely traceless Weyl and Ricci tensors, its general form in metric-affine geometry turns out to present a much richer structure [75]. Specifically, it includes eleven irreducible pieces, which can be grouped into antisymmetric and symmetric components \n˜ R λρµν = ˜ W λρµν + ˜ Z λρµν , (4) \nwith \n˜ W λρµν := ˜ R [ λρ ] µν , ˜ Z λρµν := ˜ R ( λρ ) µν . (5) \nIn general, the antisymmetric component includes both Riemannian and post-Riemannian contributions, whereas the symmetric one is switched on only in the presence of nonmetricity. In fact, the nonmetricity tensor can also be separated as the sum of two trace and traceless parts \nQ λµν = 1 4 g µν Q λρ ρ + ↗ Q λµν , (6) \nin such a way that each of these parts provides its own contribution in the aforementioned components. \nThe decomposition can then be expressed in a convenient way by the definition of the following building blocks [76]: \n↗ ˜ R ( µν ) = ↗ R µν + ∇ λ T ( µν ) λ -∇ ( µ T λ ν ) λ + 1 2 g µν ∇ λ T ρλ ρ -∇ λ Q ( µν ) λ + 1 2 ∇ ( µ Q ν ) λ λ + 1 2 ∇ λ Q λ µν + 1 4 g µν ( ∇ λ Q ρ ρ λ -∇ λ Q λ ρ ρ ) + 1 2 T ρ λ ( µ T ν ) λ ρ + T ρλ ρ T ( µν ) λ + 1 4 T µλρ T ν λρ + 1 4 g µν ( T λ λσ T ρ ρ σ -1 2 T λρσ T ρλσ -1 4 T λρσ T λρσ ) + Q λµρ Q [ λ ν ρ ] + 1 2 Q λρ ρ Q ( µν ) λ -1 4 ( Q λµν Q λρ ρ + Q µλρ Q ν λρ ) + 1 16 g µν ( 2 Q λρσ Q ρλσ + Q ρλ λ Q ρ σ σ -Q λρσ Q λρσ -2 Q σλ λ Q ρ ρσ ) + 1 2 ( T λ ( µ ρ Q ν ) λρ -Q λρ ( µ T λ ν ) ρ -2 Q ( µν ) λ T ρλ ρ + Q λρ ( µ T ρ ν ) λ -2 Q λρ ( µ T ν ) λρ -Q λ ρ ρ T ( µν ) λ + Q λµν T ρλ ρ ) + 1 4 g µν ( T λ λρ Q ρσ σ -T λρσ Q σλρ -T λ λρ Q σρ σ ) , (7) \n˜ R ( T ) [ µν ] = ˜ ∇ [ µ T λ ν ] λ + 1 2 ˜ ∇ λ T λ µν -1 2 T λ ρλ T ρ µν , ˜ R λ λµν = ∇ [ ν Q µ ] λ λ , (8) \nλ \n[ \nρ \n| \nλσ \n[ \nρ \n↗ ˆ R ( Q ) ( µν ) = ˜ ∇ λ ↗ Q ( µν ) λ -˜ ∇ ( µ ↗ Q λ ν ) λ + ↗ Q λρ λ ↗ Q ( µν ) ρ -↗ Q λρ ( µ ↗ Q ν ) λρ -T λρ ( µ ↗ Q λρ ν ) , (9) \n˜ \nR \nˆ R ( Q ) [ µν ] = ˜ ∇ [ µ ↗ Q λ ν ] λ -˜ ∇ λ ↗ Q [ µν ] λ -1 2 ˜ ∇ [ µ ↗ Q ν ] λ λ + ↗ Q [ νµ ] λ ↗ Q ρ λρ -↗ Q ρλ [ µ ↗ Q ν ] ρλ + ↗ Q λρ [ µ T λ ν ] ρ + 1 4 ↗ Q λρ ρ T λµν , (10) \n( \nT \n) \nλ \n[ \nρµν \n] \n= \n1 \n2 \ng \n+ \nT \n˜ \n∇ \nσ \nT \nT \nσ \nσ \n| \nµν \n] \nµν \n] \n- \n+ \ng \n1 \n2 \ng \nλ \n[ \nρ \nλ \n[ \nρ \n˜ \n∇ \nT \nµ \nσ \nT \nµν \n] \nσ \nν \n] \nσ \nT \n- \nσω \n, \n(11) \n↗ ˜ R ( Q ) λ [ ρµν ] = 3 2 ( g λ [ ρ | ˜ ∇ σ ↗ Q | µν ] σ -g λ [ ρ ˜ ∇ µ ↗ Q σ ν ] σ -2 ˜ ∇ [ ρ ↗ Q µν ] λ + g λ [ ρ ↗ Q µν ] σ ↗ Q ω σω + g λ [ ρ ↗ Q σ µ ω ↗ Q ν ] σω + ↗ Q σλ [ ρ T σ µν ] + g λ [ ρ | ↗ Q σ | µ | ω T σ ω | ν ] + 1 2 Q [ ρ | σ σ ↗ Q | µν ] λ ) , (12) \n(1) ˜ Z λρµν = ˜ R ( λρ ) µν -1 4 ( ↗ ˜ R ( Q ) λ [ ρµν ] + ↗ ˜ R ( Q ) ρ [ λµν ] ) -1 6 ( g λν ˆ R ( Q ) [ ρµ ] + g ρν ˆ R ( Q ) [ λµ ] -g λµ ˆ R ( Q ) [ ρν ] -g ρµ ˆ R ( Q ) [ λν ] + g λρ ˆ R ( Q ) [ µν ] ) -1 4 g λρ ˜ R σ σµν -1 8 ( g λν ↗ ˆ R ( Q ) ( ρµ ) + g ρν ↗ ˆ R ( Q ) ( λµ ) -g λµ ↗ ˆ R ( Q ) ( ρν ) -g ρµ ↗ ˆ R ( Q ) ( λν ) ) , (14) \n(1) ˜ W λρµν = ˜ R [ λρ ] µν -3 4 ( ↗ ˜ R ( T ) λ [ ρµν ] + ↗ ˜ R ( T ) ν [ λρµ ] -↗ ˜ R ( T ) ρ [ λµν ] -↗ ˜ R ( T ) µ [ λρν ] ) -1 2 ( ↗ ˜ R ( Q ) µ [ λρν ] -↗ ˜ R ( Q ) ν [ λρµ ] ) + 1 24 ∗ ˜ R ε λρµν -1 4 [ g λµ ( 2 ↗ ˜ R ( ρν ) + ↗ ˆ R ( Q ) ( ρν ) ) + g ρν ( 2 ↗ ˜ R ( λµ ) + ↗ ˆ R ( Q ) ( λµ ) ) -g λν ( 2 ↗ ˜ R ( ρµ ) + ↗ ˆ R ( Q ) ( ρµ ) ) -g ρµ ( 2 ↗ ˜ R ( λν ) + ↗ ˆ R ( Q ) ( λν ) )] -1 4 [ g λµ ( 2 ˜ R ( T ) [ ρν ] + ˆ R ( Q ) [ ρν ] ) + g ρν ( 2 ˜ R ( T ) [ λµ ] + ˆ R ( Q ) [ λµ ] ) -g λν ( 2 ˜ R ( T ) [ ρµ ] + ˆ R ( Q ) [ ρµ ] ) -g ρµ ( 2 ˜ R ( T ) [ λν ] + ˆ R ( Q ) [ λν ] ) + ˜ R σ σλ [ µ g ν ] ρ -˜ R σ σρ [ µ g ν ] λ ] -1 6 ˜ Rg λ [ µ g ν ] ρ , (13) \n˜ \nR \n= \nR \n- \n2 \n∇ \nµ \nT \nν \n+ \n∇ \nµ \nQ \nν \n-∇ \nµ \nQ \nν \n+ \n1 \n4 \nT \nλµν \nT \n+ \n1 \n2 \nT \nλµν \nT \n- \nT \nλν \nT \nµ \n+ \nT \nλµν \nQ \n+ T λ λν Q µν µ -T λ λν Q νµ µ + 1 4 Q λµν Q λµν -1 2 Q λµν Q µλν + 1 2 Q νλ λ Q µ µν -1 4 Q νλ λ Q ν µ µ , (15) \n∗ ˜ R = ε λρµν ( ∇ λ T ρµν + 1 2 T σ λρ T σµν -Q λσρ T σ µν ) , (16) \nwhich gives rise to six irreducible parts ˜ W λρµν = 6 ∑ i =1 ( i ) ˜ W λρµν in the antisymmetric component: \n(1) ˜ W λρµν = ˜ W λρµν -6 ∑ i =2 ( i ) ˜ W λρµν , (17) \n(2) ˜ W λρµν = 3 4 ( ↗ ˜ R ( T ) λ [ ρµν ] + ↗ ˜ R ( T ) ν [ λρµ ] -↗ ˜ R ( T ) ρ [ λµν ] -↗ ˜ R ( T ) µ [ λρν ] ) + 1 2 ( ↗ ˜ R ( Q ) µ [ λρν ] -↗ ˜ R ( Q ) ν [ λρµ ] ) , (18) \n(3) ˜ W λρµν = -1 24 ∗ ˜ R ε λρµν , (19) \n(4) ˜ W λρµν = 1 4 [ g λµ ( 2 ↗ ˜ R ( ρν ) + ↗ ˆ R ( Q ) ( ρν ) ) + g ρν ( 2 ↗ ˜ R ( λµ ) + ↗ ˆ R ( Q ) ( λµ ) ) -g λν ( 2 ↗ ˜ R ( ρµ ) + ↗ ˆ R ( Q ) ( ρµ ) ) -g ρµ ( 2 ↗ ˜ R ( λν ) + ↗ ˆ R ( Q ) ( λν ) )] , (20) \n(5) ˜ W λρµν = 1 4 [ g λµ ( 2 ˜ R ( T ) [ ρν ] + ˆ R ( Q ) [ ρν ] ) + g ρν ( 2 ˜ R ( T ) [ λµ ] + ˆ R ( Q ) [ λµ ] ) -g λν ( 2 ˜ R ( T ) [ ρµ ] + ˆ R ( Q ) [ ρµ ] ) -g ρµ ( 2 ˜ R ( T ) [ λν ] + ˆ R ( Q ) [ λν ] ) + ˜ R σ σλ [ µ g ν ] ρ -˜ R σ σρ [ µ g ν ] λ ] , (21) \n(6) ˜ W λρµν = 1 6 ˜ Rg λ [ µ g ν ] ρ , (22) \nas well as to five ˜ Z λρµν = 5 ∑ i =1 ( i ) ˜ Z λρµν in the symmetric one: \n(1) ˜ Z λρµν = ˜ Z λρµν -5 ∑ i =2 ( i ) ˜ Z λρµν , (23) \nνµ \nµ \nν \nν \nµ \nλµν \nµλν \nλ \nµ \nν \nνλµ \nω \ng \n˜ \n∇ \nT \nσ \n+ \n1 \n24 \nε \nε \nαβγ \n( \n˜ \n∇ \nT \nσ \n+ \nT \nT \nωσ \n) \n↗ \nλσ \n[ \nρ \nµν \n] \nλρµν \nσ \nγ \nβα \nβωγ \nα \n(2) ˜ Z λρµν = 1 4 ( ↗ ˜ R ( Q ) λ [ ρµν ] + ↗ ˜ R ( Q ) ρ [ λµν ] ) , (24) \n(4) ˜ Z λρµν = 1 4 g λρ ˜ R σ σµν , (26) \n(3) ˜ Z λρµν = 1 6 ( g λν ˆ R ( Q ) [ ρµ ] + g ρν ˆ R ( Q ) [ λµ ] -g λµ ˆ R ( Q ) [ ρν ] -g ρµ ˆ R ( Q ) [ λν ] + g λρ ˆ R ( Q ) [ µν ] ) , (25) \n(5) ˜ Z λρµν = 1 8 ( g λν ↗ ˆ R ( Q ) ( ρµ ) + g ρν ↗ ˆ R ( Q ) ( λµ ) -g λµ ↗ ˆ R ( Q ) ( ρν ) -g ρµ ↗ ˆ R ( Q ) ( λν ) ) . (27) \nThe resulting eleven irreducible parts of the curvature tensor can then be included in the general action of MAG to provide the dynamics of the gravitational field enhanced by torsion and nonmetricity [24]. Thereby, it is clear that these parts play a crucial role in MAG, which merits the study of their algebraic structure, in line with the analyses carried out for the Weyl and Ricci tensors in GR. In any case, further studies can also be focused on the torsion and nonmetricity tensors per se, which has already found applications in the particular framework of teleparallelism [77, 78]. \nThus, in order to perform the algebraic classification of the building blocks involved in the irreducible decomposition of the curvature tensor, it is first essential to take into account their algebraic symmetries. Specifically, the building block (1) ˜ W λρµν represents the Weyl tensor in the presence of torsion and nonmetricity, fulfilling the following algebraic symmetries: \n(1) ˜ W λρµν = -(1) ˜ W ρλµν = -(1) ˜ W λρνµ , (28) \n(1) ˜ W λ [ ρµν ] = (1) ˜ W λ µλν = 0 . (29) \nOn the other hand, the antisymmetrised building blocks ↗ ˜ R ( T ) λ [ ρµν ] and ↗ ˜ R ( Q ) λ [ ρµν ] are both completely traceless and pseudotraceless tensors: \ng λρ ↗ ˜ R ( T ) λ [ ρµν ] = g λρ ↗ ˜ R ( Q ) λ [ ρµν ] = 0 , (30) \nε λρµν ↗ ˜ R ( T ) λ [ ρµν ] = ε λρµν ↗ ˜ R ( Q ) λ [ ρµν ] = 0 , (31) \nthe symmetric building blocks ↗ ˜ R ( µν ) and ↗ ˆ R ( Q ) ( µν ) are also traceless: \ng µν ↗ ˜ R ( µν ) = g µν ↗ ˆ R ( Q ) ( µν ) = 0 , (32) \nwhereas ˜ R ( T ) [ µν ] , ˆ R ( Q ) [ µν ] and ˜ R λ λµν are simply antisymmetric. Finally, the building block (1) ˜ Z λρµν also constitutes a traceless tensor, which additionally satisfies a cyclic property: \n(1) ˜ Z λ λµν = (1) ˜ Z λ µλν = 0 , (33) \n(1) ˜ Z λ [ ρµν ] = 0 . (34) \nAs is clear, the aforementioned algebraic symmetries constrain the number of independent components of the building blocks, which for the case of a four-dimensional affinely connected metric space-time can be collected in Table I. Indeed, it turns out that the sets {↗ ˜ R ( T ) λ [ ρµν ] , ↗ ˜ R ( Q ) λ [ ρµν ] , ↗ ˜ R ( µν ) , ↗ ˆ R ( Q ) ( µν ) } and { ˜ R ( T ) [ µν ] , ˆ R ( Q ) [ µν ] , ˜ R λ λµν } contain building blocks with 9 and 6 independent components, which already suggests their respective building blocks may obey the same type of algebraic classification. \nFollowing these lines, in the next sections we shall see there exist in general four different types of algebraic classification in metric-affine geometry.', 'III. Algebraic classification of (1) ˜ W λρµν': "The fact that the tensor (1) ˜ W λρµν represents the Weyl part of the curvature tensor in the presence of torsion and nonmetricity, fulfilling the algebraic symmetries (28) and (29), clearly points out that this tensor must obey the Petrov classification. Indeed, this classification can be derived by means of its PNDs, which requires to express the tensor in terms of a set of null vectors l µ , k µ , m µ and ¯ m µ that satisfy the following pseudo-orthogonality and normalisation conditions: \nk µ l µ = -m µ ¯ m µ = 1 , (35) \nTABLE I: Number of independent components of the building blocks. \nk µ m µ = k µ ¯ m µ = l µ m µ = l µ ¯ m µ = 0 , (36) k µ k µ = l µ l µ = m µ m µ = ¯ m µ ¯ m µ = 0 . (37) \nThereby, the 10 independent components of the tensor (1) ˜ W λρµν can be described by five complex scalars { Σ i } 4 i =0 as \n(1) ˜ W λρµν = -1 2 ( Σ 2 + ¯ Σ 2 ) ( { l λ k ρ l µ k ν } + { m λ ¯ m ρ m µ ¯ m ν } ) + ( Σ 2 -¯ Σ 2 ) { l λ k ρ m µ ¯ m ν } -1 2 ( ¯ Σ 0 { k λ m ρ k µ m ν } +Σ 0 { k λ ¯ m ρ k µ ¯ m ν } ) -1 2 ( Σ 4 { l λ m ρ l µ m ν } + ¯ Σ 4 { l λ ¯ m ρ l µ ¯ m ν } ) + ( Σ 2 { l λ m ρ k µ ¯ m ν } + ¯ Σ 2 { l λ ¯ m ρ k µ m ν } ) -¯ Σ 1 ( { l λ k ρ k µ m ν } + { k λ m ρ m µ ¯ m ν } ) -Σ 1 ( { l λ k ρ k µ ¯ m ν } + { k λ ¯ m ρ ¯ m µ m ν } ) +Σ 3 ( { l λ k ρ l µ m ν } - { l λ m ρ m µ ¯ m ν } ) + ¯ Σ 3 ( { l λ k ρ l µ ¯ m ν } - { l λ ¯ m ρ ¯ m µ m ν } ) , (38) \nwhere \nΣ 0 = -(1) ˜ W λρµν l λ m ρ l µ m ν , Σ 1 = -(1) ˜ W λρµν l λ k ρ l µ m ν , Σ 2 = -(1) ˜ W λρµν l λ m ρ ¯ m µ k ν , (39) \nΣ 3 = -(1) ˜ W λρµν l λ k ρ ¯ m µ k ν , Σ 4 = -(1) ˜ W λρµν k λ ¯ m ρ k µ ¯ m ν , (40) \nand \n{ w λ x ρ y µ z ν } = w λ x ρ y µ z ν -w λ x ρ z µ y ν -x λ w ρ y µ z ν + x λ w ρ z µ y ν + y λ z ρ w µ x ν -y λ z ρ x µ w ν -z λ y ρ w µ x ν + z λ y ρ x µ w ν . (41) \nThe PNDs can then be found by performing a rotation along the null vector k µ , given by a complex function /epsilon1 : \nk ' µ = k µ , m ' µ = m µ + /epsilon1 k µ , ¯ m ' µ = ¯ m µ +¯ /epsilon1 k µ , l ' µ = l µ +¯ /epsilon1 m µ + /epsilon1 ¯ m µ + /epsilon1 ¯ /epsilon1 k µ , (42) \nwhich transforms the complex scalars as \nΣ ' 4 = Σ 4 , Σ ' 3 = Σ 3 + /epsilon1 Σ 4 , Σ ' 2 = Σ 2 +2 /epsilon1 Σ 3 + /epsilon1 2 Σ 4 , (43) \nΣ ' 1 = Σ 1 +3 /epsilon1 Σ 2 +3 /epsilon1 2 Σ 3 + /epsilon1 3 Σ 4 , (44) \nΣ ' 0 = Σ 0 +4 /epsilon1 Σ 1 +6 /epsilon1 2 Σ 2 +4 /epsilon1 3 Σ 3 + /epsilon1 4 Σ 4 . (45) \nThus, the different roots of the quartic polynomial equation Σ ' 0 = 0 and their multiplicities provide the PNDs and their levels of alignment, respectively, which determines the algebraic types of the classification. In particular, it is possible to find a rotated null tetrad where they satisfy the following constraints 1 : \nl [ σ (1) ˜ W λ ] ρµ [ ν l ω ] l ρ l µ = 0 ⇐⇒ Σ 0 = 0 , (46) \nTABLE II: Algebraic types for the tensor (1) ˜ W λρµν . \n(1) ˜ W λρµ [ ν l ω ] l ρ l µ = 0 ⇐⇒ Σ 0 = Σ 1 = 0 , (47) \n(1) ˜ W λρµ [ ν l ω ] l µ = 0 ⇐⇒ Σ 0 = Σ 1 = Σ 2 = 0 , (49) \n(1) ˜ W λρµ [ ν k ω ] k ρ k µ = (1) ˜ W λρµ [ ν l ω ] l ρ l µ = 0 ⇐⇒ Σ 0 = Σ 1 = Σ 3 = Σ 4 = 0 , (48) \n(1) ˜ W λρµν l µ = 0 ⇐⇒ Σ 0 = Σ 1 = Σ 2 = Σ 3 = 0 . (50) \nThe algebraic classification of the tensor (1) ˜ W λρµν can then be outlined in Table II. \nIV. Algebraic classification of ↗ ˜ R ( T ) λ [ ρµν ] , ↗ ˜ R ( Q ) λ [ ρµν ] , ↗ ˜ R ( µν ) and ↗ ˆ R ( Q ) ( µν ) \nIn order to classify the tensors ↗ ˜ R ( T ) λ [ ρµν ] , ↗ ˜ R ( Q ) λ [ ρµν ] , ↗ ˜ R ( µν ) and ↗ ˆ R ( Q ) ( µν ) , it is first worthwhile to stress that ↗ ˜ R ( T ) λ [ ρµν ] and ↗ ˜ R ( Q ) λ [ ρµν ] can be expressed as second order symmetric and traceless tensors as follows: \n↗ ˜ M µν = 1 6 ε ( µ λρσ ↗ ˜ R ( T ) ν )[ λρσ ] , (51) \n↗ ˜ K µν = 1 6 ε ( µ λρσ ↗ ˜ R ( Q ) ν )[ λρσ ] . (52) \nAccordingly, all of the tensors ↗ ˜ R ( T ) λ [ ρµν ] , ↗ ˜ R ( Q ) λ [ ρµν ] , ↗ ˜ R ( µν ) and ↗ ˆ R ( Q ) ( µν ) can be ascribed to a set of second order symmetric and traceless tensors {↗ ˜ B ( i ) µν } 4 i =1 . The algebraic classification can then be directly derived from the eigenvalue equations \n↗ ˜ B ( i ) a b v b = λv a , (53) \nin such a way that the corresponding characteristic polynomials are determined by the invariants \n˜ U ( i ) = ↗ ˜ B ( i ) a b ↗ ˜ B ( i ) b a , ˜ V ( i ) = ↗ ˜ B ( i ) a b ↗ ˜ B ( i ) b c ↗ ˜ B ( i ) c a , ˜ W ( i ) = ↗ ˜ B ( i ) a b ↗ ˜ B ( i ) b c ↗ ˜ B ( i ) c d ↗ ˜ B ( i ) d a , (54) \nyielding \nλ 4 -˜ U ( i ) 2 λ 2 -˜ V ( i ) 3 λ + 1 8 [( ˜ U ( i ) ) 2 -2 ˜ W ( i ) ] = 0 . (55) \nThe multiplicities of the roots of the characteristic equation (55) turn out to be determined by different combinations of signs for the subsequent invariants [5, 79]: \n˜ U ( i ) ∗ = ( ˜ W ( i ) ∗ ) 3 -{ 3 ˜ U ( i ) ˜ W ( i ) ∗ +4 [ 3 ( ˜ V ( i ) ) 2 -( ˜ U ( i ) ) 3 ]} 2 , ˜ V ( i ) ∗ = 2 ˜ U ( i ) -| ˜ W ( i ) ∗ | 1 / 2 , ˜ W ( i ) ∗ = 7 ( ˜ U ( i ) ) 2 -12 ˜ W ( i ) , (56) which provides the well-known Segre classification described in Table III. \nV. Algebraic classification of ˜ R ( T ) [ µν ] , ˆ R ( Q ) [ µν ] and ˜ R λ λµν \nThe tensors ˜ R ( T ) [ µν ] , ˆ R ( Q ) [ µν ] and ˜ R λ λµν can be ascribed to a set of antisymmetric tensors { ˜ X ( i ) [ µν ] } 3 i =1 , which in turn can be expressed in terms of the null vectors as \n˜ X ( i ) [ µν ] = 2 [ Ω 2 k [ µ m ν ] + ¯ Ω 2 k [ µ ¯ m ν ] -Ω 0 l [ µ ¯ m ν ] -¯ Ω 0 l [ µ m ν ] -( Ω 1 + ¯ Ω 1 ) k [ µ l ν ] + ( Ω 1 -¯ Ω 1 ) m [ µ ¯ m ν ] ] , (57) \nwhere \nΩ ( i ) 0 = k [ µ m ν ] ˜ X ( i ) [ µν ] , Ω ( i ) 1 = 1 2 ( k [ µ l ν ] -m [ µ ¯ m ν ] ) ˜ X ( i ) [ µν ] , Ω ( i ) 2 = -l [ µ ¯ m ν ] ˜ X ( i ) [ µν ] , (58) \nconstitute three sets of complex scalars, each set { Ω ( i ) 1 , Ω ( i ) 2 , Ω ( i ) 3 } 3 i =1 encoding the six independent components of the associated tensor. \nThereby, a rotation of the form (42) transforms the complex scalars as \nΩ ( i ) ' 0 = Ω ( i ) 0 , Ω ( i ) ' 1 = Ω ( i ) 1 +¯ /epsilon1 Ω ( i ) 0 , Ω ( i ) ' 2 = Ω ( i ) 2 +2¯ /epsilon1 Ω ( i ) 1 +¯ /epsilon1 2 Ω ( i ) 0 , (59) \nwhich allows the PNDs of the respective tensors to be found by obtaining the roots of the quadratic polynomial equations Ω ( i ) ' 2 = 0, namely \nΩ ( i ) 2 +2¯ /epsilon1 Ω ( i ) 1 +¯ /epsilon1 2 Ω ( i ) 0 = 0 . (60) \nThe different multiplicities of the PNDs give rise to the algebraic types of the classification, which in this case can be characterised by the following constraints: \n( ˜ X ( i ) [ µν ] l λ -˜ X ( i ) [ µλ ] l ν ) l µ = 0 ⇐⇒ Ω ( i ) 2 = 0 , (61) \n˜ X ( i ) [ µν ] l µ = 0 ⇐⇒ Ω ( i ) 1 = Ω ( i ) 2 = 0 . (62) \nThe algebraic classification of the tensor ˜ X ( i ) [ µν ] can then be summarised in Table IV. \nTABLE III: Algebraic types for the tensor ↗ ˜ B ( i ) µν . \nTABLE IV: Algebraic types for the tensor ˜ X ( i ) [ µν ] . \nwhere", 'VI. Algebraic classification of (1) ˜ Z λρµν': "As pointed out in Sec. II, the tensor (1) ˜ Z λρµν constitutes one of the irreducible parts of the symmetric component of the curvature tensor in the presence of torsion and nonmetricity. Therefore, besides being symmetric in the first pair of indices and antisymmetric in the last pair, it fulfils the algebraic symmetries (33) and (34). In terms of null vectors l µ , k µ , m µ and ¯ m µ , the 30 dof of such a tensor can then be distributed into 15 complex scalars { ∆ i } 14 i =0 as \n2 \n(1) ˜ Z λρµν = -2∆ 0 [ k λ k ρ k µ ¯ m ν ] -2 ¯ ∆ 0 [ k λ k ρ k µ m ν ] + 3 ( ∆ 1 + ¯ ∆ 1 )( 3 [ k λ ¯ m ρ k µ m ν ] -[ k λ k ρ l µ k ν ] ) +2 ( ∆ 1 -¯ ∆ 1 )( [ k λ k ρ m µ ¯ m ν ] + [ k λ ¯ m ρ k µ m ν ] ) +2∆ 2 [ k λ ¯ m ρ k µ ¯ m ν ] + 2 ¯ ∆ 2 [ k λ m ρ k µ m ν ] +2∆ 3 ( [ k λ m ρ l µ k ν ] + [ k λ m ρ ¯ m µ m ν ] ) +2 ¯ ∆ 3 ( [ k λ ¯ m ρ l µ k ν ] + [ k λ ¯ m ρ m µ ¯ m ν ] ) -2∆ 4 ( 2[ l λ k ρ k µ ¯ m ν ] + [ k λ ¯ m ρ m µ ¯ m ν ] + [ ¯ m λ ¯ m ρ k µ m ν ] ) -2 3 ∆ 5 [ ¯ m λ ¯ m ρ k µ ¯ m ν ] -2 ¯ ∆ 4 ( 2[ l λ k ρ k µ m ν ] + [ k λ m ρ ¯ m µ m ν ] + [ m λ m ρ k µ ¯ m ν ] ) -2 3 ¯ ∆ 5 [ m λ m ρ k µ m ν ] -2 3 ∆ 6 ( 3[ m λ m ρ l µ k ν ] + [ m λ m ρ ¯ m µ m ν ] ) -2 3 ¯ ∆ 6 ( 3[ ¯ m λ ¯ m ρ l µ k ν ] + [ ¯ m λ ¯ m ρ m µ ¯ m ν ] ) +2∆ 7 ( 2[ l λ k ρ m µ ¯ m ν ] + 2[ l λ ¯ m ρ k µ m ν ] + [ m λ ¯ m ρ m µ ¯ m ν ] -[ l λ k ρ l µ k ν ] ) + 2 3 ∆ 9 [ m λ m ρ l µ m ν ] -2 ¯ ∆ 7 ( 2[ l λ k ρ m µ ¯ m ν ] + 2[ m λ ¯ m ρ l µ k ν ] + [ m λ ¯ m ρ m µ ¯ m ν ] + [ l λ k ρ l µ k ν ] ) + 2 3 ¯ ∆ 9 [ ¯ m λ ¯ m ρ l µ ¯ m ν ] + 2 3 ∆ 8 ( 3[ l λ ¯ m ρ k µ ¯ m ν ] + [ ¯ m λ ¯ m ρ m µ ¯ m ν ] ) + 2 3 ¯ ∆ 8 ( 3[ l λ m ρ k µ m ν ] + [ m λ m ρ ¯ m µ m ν ] ) +2∆ 10 ( 2[ l λ m ρ l µ k ν ] + [ l λ m ρ ¯ m µ m ν ] + [ m λ m ρ l µ ¯ m ν ] ) +2 ¯ ∆ 10 ( 2[ l λ ¯ m ρ l µ k ν ] + [ l λ ¯ m ρ m µ ¯ m ν ] + [ ¯ m λ ¯ m ρ l µ m ν ] ) -2∆ 11 ( [ l λ l ρ k µ ¯ m ν ] + [ l λ ¯ m ρ m µ ¯ m ν ] ) -2 ¯ ∆ 11 ( [ l λ l ρ k µ m ν ] + [ l λ m ρ ¯ m µ m ν ] ) -2∆ 12 [ l λ m ρ l µ m ν ] + 2 ( ∆ 13 -¯ ∆ 13 )( [ l λ ¯ m ρ l µ m ν ] + [ l λ l ρ m µ ¯ m ν ] ) +2∆ 14 [ l λ l ρ l µ m ν ] -2 ¯ ∆ 12 [ l λ ¯ m ρ l µ ¯ m ν ] -2 ( ∆ 13 + ¯ ∆ 13 )( [ l λ l ρ l µ k ν ] + [ l λ ¯ m ρ l µ m ν ] ) +2 ¯ ∆ 14 [ l λ l ρ l µ ¯ m ν ] , (63) \n∆ 0 = (1) ˜ Z λρµν l λ l ρ l µ m ν , (64) \n∆ 1 = 1 2 (1) ˜ Z λρµν ( l λ l ρ l µ k ν -l λ l ρ m µ ¯ m ν ) , (65) \n2 \n∆ \n= \n(1) \n˜ \nZ \nλρµν \nl \nλ \nm \nρ \nl \nµ \nm \nν \n, \n(66) \n∆ 3 = 1 2 (1) ˜ Z λρµν ( l λ ¯ m ρ l µ k ν + l λ ¯ m ρ ¯ m µ m ν ) , (67) \n∆ \n∆ 4 = 1 2 (1) ˜ Z λρµν ( l λ m ρ l µ k ν -l λ m ρ m µ ¯ m ν ) , (68) \n5 \n= \n(1) \n˜ \nZ \nλρµν \nm \nλ \nm \nρ \nl \nµ \nm \nν \n, \n(69) \n∆ 6 = 1 2 (1) ˜ Z λρµν ( ¯ m λ ¯ m ρ ¯ m µ m ν + ¯ m λ ¯ m ρ l µ k ν ) , (70) \n∆ 8 = -1 2 (1) ˜ Z λρµν ( m λ m ρ m µ ¯ m ν + m λ m ρ k µ l ν ) , (72) \n∆ 7 = 1 2 (1) ˜ Z λρµν ( m λ ¯ m ρ l µ k ν -m λ ¯ m ρ m µ ¯ m ν ) , (71) \n∆ 9 = -(1) ˜ Z λρµν ¯ m λ ¯ m ρ k µ ¯ m ν , (73) \n∆ 11 = -1 2 (1) ˜ Z λρµν ( k λ m ρ k µ l ν + k λ m ρ m µ ¯ m ν ) , (75) \n∆ 10 = -1 2 (1) ˜ Z λρµν ( k λ ¯ m ρ k µ l ν -k λ ¯ m ρ ¯ m µ m ν ) , (74) \n(1) \n˜ \nλ \nρ \nµ \nν \n, \n(76) \n∆ 14 = -(1) ˜ Z λρµν k λ k ρ k µ ¯ m ν . (78) \n∆ 12 = -Z λρµν k ¯ m k ¯ m ∆ 13 = -1 2 (1) ˜ Z λρµν ( k λ k ρ k µ l ν -k λ k ρ ¯ m µ m ν ) , (77) \nand \n[ x λ y ρ z µ w ν ] = x ( λ y ρ ) z [ µ w ν ] -y ( λ w ρ ) x [ µ z ν ] -x ( λ w ρ ) y [ µ z ν ] . (79) \nNote that, by the interchange l µ ↔ k µ (and m µ ↔ ¯ m µ ), the scalars ∆ i are in direct correspondence with -∆ 14 -i for all i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Thereby, any alignment property referred to l µ based on the first set of complex scalars has a replica as an alignment property of k µ by replacing with the corresponding ones of the second set. \nUnder the null rotation (42), defined by a complex function /epsilon1 that keeps the null vector k µ fixed, the scalars transform as \n∆ ' 0 = ∆ 0 +4 /epsilon1 ∆ 1 +2¯ /epsilon1 ∆ 2 +6 /epsilon1 2 ∆ 3 +8 /epsilon1 ¯ /epsilon1 ∆ 4 +¯ /epsilon1 2 ∆ 5 +4 /epsilon1 3 ∆ 6 +12 /epsilon1 2 ¯ /epsilon1 ∆ 7 +4 /epsilon1 ¯ /epsilon1 2 ∆ 8 + /epsilon1 4 ∆ 9 +8 /epsilon1 3 ¯ /epsilon1 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 4 ¯ /epsilon1 ∆ 12 +4 /epsilon1 3 ¯ /epsilon1 2 ∆ 13 + /epsilon1 4 ¯ /epsilon1 2 ∆ 14 , \n(80) \n∆ ' 1 = ∆ 1 +3 /epsilon1 ∆ 3 +2¯ /epsilon1 ∆ 4 +3 /epsilon1 2 ∆ 6 +6 /epsilon1 ¯ /epsilon1 ∆ 7 +¯ /epsilon1 2 ∆ 8 + /epsilon1 3 ∆ 9 +6 /epsilon1 2 ¯ /epsilon1 ∆ 10 +3 /epsilon1 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 3 ¯ /epsilon1 ∆ 12 +3 /epsilon1 2 ¯ /epsilon1 2 ∆ 13 + /epsilon1 3 ¯ /epsilon1 2 ∆ 14 , (81) \n∆ ' 2 = ∆ 2 +4 /epsilon1 ∆ 4 +¯ /epsilon1 ∆ 5 +6 /epsilon1 2 ∆ 7 +4 /epsilon1 ¯ /epsilon1 ∆ 8 +4 /epsilon1 3 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 ∆ 11 + /epsilon1 4 ∆ 12 +4 /epsilon1 3 ¯ /epsilon1 ∆ 13 + /epsilon1 4 ¯ /epsilon1 ∆ 14 , (82) \n∆ ' 3 = ∆ 3 +2 /epsilon1 ∆ 6 +2¯ /epsilon1 ∆ 7 + /epsilon1 2 ∆ 9 +4 /epsilon1 ¯ /epsilon1 ∆ 10 +¯ /epsilon1 2 ∆ 11 +2 /epsilon1 2 ¯ /epsilon1 ∆ 12 +2 /epsilon1 ¯ /epsilon1 2 ∆ 13 + /epsilon1 2 ¯ /epsilon1 2 ∆ 14 , (83) \n∆ ' 4 = ∆ 4 +3 /epsilon1 ∆ 7 +¯ /epsilon1 ∆ 8 +3 /epsilon1 2 ∆ 10 +3 /epsilon1 ¯ /epsilon1 ∆ 11 + /epsilon1 3 ∆ 12 +3 /epsilon1 2 ¯ /epsilon1 ∆ 13 + /epsilon1 3 ¯ /epsilon1 ∆ 14 , (84) \n∆ ' 5 = ∆ 5 +4 /epsilon1 ∆ 8 +6 /epsilon1 2 ∆ 11 +4 /epsilon1 3 ∆ 13 + /epsilon1 4 ∆ 14 , (85) \n∆ ' 6 = ∆ 6 + /epsilon1 ∆ 9 +2¯ /epsilon1 ∆ 10 +2 /epsilon1 ¯ /epsilon1 ∆ 12 +¯ /epsilon1 2 ∆ 13 + /epsilon1 ¯ /epsilon1 2 ∆ 14 , (86) \n∆ ' 7 = ∆ 7 +2 /epsilon1 ∆ 10 +¯ /epsilon1 ∆ 11 + /epsilon1 2 ∆ 12 +2 /epsilon1 ¯ /epsilon1 ∆ 13 + /epsilon1 2 ¯ /epsilon1 ∆ 14 , (87) \n∆ ' 8 = ∆ 8 +3 /epsilon1 ∆ 11 +3 /epsilon1 2 ∆ 13 + /epsilon1 3 ∆ 14 , (88) \n' \n9 \n∆ \n= ∆ \n9 \n+2¯ /epsilon1 \n∆ \n12 \n+¯ /epsilon1 \n∆ \n14 \n, \n(89) \n∆ ' 10 = ∆ 10 + /epsilon1 ∆ 12 +¯ /epsilon1 ∆ 13 + /epsilon1 ¯ /epsilon1 ∆ 14 , (90) \n∆ ' 11 = ∆ 11 +2 /epsilon1 ∆ 13 + /epsilon1 2 ∆ 14 , (91) \n∆ ' 12 = ∆ 12 +¯ /epsilon1 ∆ 14 , (92) \n∆ ' 13 = ∆ 13 + /epsilon1 ∆ 14 , (93) \n' \n14 \n∆ \n= ∆ \n14 \n. \n(94) \nAn algebraic classification of the tensor (1) ˜ Z λρµν can then be obtained by defining its PNDs and their levels of alignment. An alternative, which leads to a more refined classification, can also be achieved by establishing the levels of alignment with its superenergy tensor [21, 22]. In this sense, we shall first apply in Sec. VI A the method of PNDs for the tensor (1) ˜ Z λρµν alone, in order to derive the basic algebraic classification for this tensor, whereas in Sec. VI B we shall show the main refinements that arise when using its superenergy tensor.", 'A. Algebraic classification by means of the PNDs and their levels of alignment': "For any arbitrary tensor there is a well established definition of PND [23], also called aligned null direction or AND [19, 20, 80], that depends on the index-symmetry properties of the tensor. For the case of the tensor (1) ˜ Z λρµν , this definition reads \nl µ is a PND ⇐⇒ (1) ˜ Z λρµ [ ν l σ ] l λ l ρ l µ = 0 , (95) \nfor a (necessarily) null l µ . This implies that l µ is somehow aligned with the structure of the tensor (1) ˜ Z λρµν , and it can be seen equivalent to the vanishing of the scalar ∆ 0 defined in Expression (64): \nl µ is a PND ⇐⇒ ∆ 0 = 0 . (96) \nHowever, one immediately notices that the above PND condition (96) is also satisfied for null vectors l µ that comply with stricter conditions, such as for instance \n(1) ˜ Z λρµν l λ l ρ l µ = 0 , (97) \nor even \n2 \n(1) ˜ Z λρµν l λ = 0 . (98) \nThese stricter conditions entail a higher-order alignment of l µ with the tensor (1) ˜ Z λρµν . To provide a measure of the several levels of alignment, one introduces the concept of boost order associated to any null direction [19, 20]. For a given null l µ and introducing the null tetrad { l µ , k µ , m µ , ¯ m µ } , one can associate an integer number bo to each of the complex scalars (64)-(78) by counting each appearance of l µ with a +1 and each appearance of k µ with a -1. Concretely, ∆ 0 has bo = 3, ∆ 1 and ∆ 2 have bo = 2, ∆ 3 , ∆ 4 and ∆ 5 have bo = 1 and so on until ∆ 14 with bo = -3. The boost order of any null l µ , say bo ( l ), is then given by the maximum bo of the ∆'s in the null tetrad { l µ , k µ , m µ , ¯ m µ } . This is independent of the choice of k µ . \nOne immediately realises that, for a general l µ , bo ( l ) = 3; but, if l µ happens to be a PND, then bo ( l ) < 3. And thus the higher orders of alignment can be simply defined by all the possibilities for bo ( l ) in order. Hence, l µ is said to be a PND of multiplicity m ∈ { 1 , 2 , 3 , 4 , 5 , 6 } if bo ( l ) = 3 -m . This leads to the classification of PNDs, as follows: \nAlignment Class VI: bo ( l ) = -3; m = 6 . \nObserve that the Roman numerals denoting the alignment class agree with the value of the multiplicity m of l µ . Each of the alignment classes can be expressed by invariant conditions involving only l µ and the tensor (1) ˜ Z λρµν , which can be deduced by taking into account the corresponding complex scalars that vanish for each case. Thus, we have the following equivalences 2 : \nAlignment Class I: (1) ˜ Z λρµ [ ν l σ ] l λ l ρ l µ = 0 ⇐⇒ ∆ 0 = 0 . (99) Alignment Class II: l [ ω (1) ˜ Z λ ] ρµ [ ν l σ ] l ρ l µ = 0 ⇐⇒ ∆ 0 = ∆ 1 = ∆ 2 = 0 . (100) Alignment Class III: l [ τ l [ ω (1) ˜ Z λ ] ρ ] µ [ ν l σ ] l µ = 0 ⇐⇒ { ∆ i } i =0 ,..., 5 = 0 . (101) Alignment Class IV: l [ ω (1) ˜ Z λ ] ρµ [ ν l σ ] l µ = l λ l [ ω (1) ˜ Z λ ρ ] µν = 0 ⇐⇒ { ∆ i } i =0 ,..., 8 = 0 . (102) Alignment Class V: l [ τ l [ ω (1) ˜ Z λ ] ρ ] µν = 0 ⇐⇒ { ∆ i } i =0 ,..., 11 = 0 . (103) Alignment Class VI: l [ σ (1) ˜ Z λ ] ρµν = 0 ⇐⇒ { ∆ i } i =0 ,..., 13 = 0 . (104) \nThe classification of the tensor (1) ˜ Z λρµν is then given by two natural numbers - that we will express in Roman numerals- in conjunction: the first one is the maximal alignment class of any null vector, and the second one is the next to maximal alignment class. In other words, the first natural number gives the multiplicity of the maximal aligned null vector, while the second one is the multiplicity of the next to maximal aligned null vector. Thus, the first number is always greater than or equal to the second one. But there is a further important restriction: the sum of the two numbers cannot be greater than 6. This follows because, by choosing the null tetrad with l µ the maximally aligned and k µ the next to maximally aligned null vectors, and using the property mentioned above of the symmetry interpretation between ∆ i and -∆ 14 -i for all i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } , if the sum of the two numbers were greater than 6, the tensor (1) ˜ Z λρµν would necessarily vanish identically -because all the ∆-scalars would be zero. \nThereby, besides the trivial case where the tensor vanishes, the fundamental classification ends up having 11 main different types, though some of them have special situations where the second PND does not exist, in which case we will denote them with a star * added to the main type, so that in total there are 15 nontrivial types as follows 3 : \nType N or null: \n(VI , - ); \nType L: \n(V , I); \nType L*: \n(V , - ); \nType F: \n(IV , II); \nType H: \n(IV , I); \nType H*: \n(IV , - ); \nType D: \n(III , III); \nType M: \n(III , II); \nType K: \n(III , I); \nType K*: \n(III , - ); \nType B: \n(II , II); \nType S or special: (II \n, II , II); \nType C: \n(II , I); \nType C*: \n(II , - ); \nType I: \n(I , I) . \nThe left numeral is always uniquely fixed except for Type D, where both III can be interchanged, and analogously for Types B and I. However, the second Roman numeral can be, in some of the types, chosen in many different ways. This will be analysed in the next section. However, the type S (that can be seen as a Type B-special) is of a different kind, because there are three different PNDs of Class II, and thereby there are three different choices for (II,II). This is why this case, though it could be considered as a subcase of Type B, is included as a different one in the classification. Let us remark that all the cases with * are actually very peculiar, for being uncommon, and they require extremely specific relations between some of the scalars ∆ n . This will be made plain in Sec. VI A 2. \nEach of the types has its own different properties, as well as several subcases and various possibilities. This is partly discussed in Appendix A, where several refinements arise naturally, and also in the next subsections where we discuss how many choices for the second numeral can be, as well as how many PNDs there can be in general.", '1. On the number and multiplicities of the PNDs': "The previous part of the classification deals with the alignment classes of given PNDs and takes care of their level of alignment with the tensor (1) ˜ Z λρµν . However, in order to get a complete view of the algebraic classification of this tensor, the possible number of PNDs should be known, as well as their alignment classes. \nTo that end, one needs to see how many possible null directions satisfy the relation (95) or, equivalently, (195). This can be achieved by choosing any null tetrad { l µ , k µ , m µ , ¯ m µ } , then performing an arbitrary null rotation of type (42) so that the new l ' µ , that depends on /epsilon1 , represents any possible null direction -except k µ -, and then finding which of them, that is to say, for which values of /epsilon1 this new l ' µ is a PND. In other words, one needs to ascertain the number of solutions for the complex parameter /epsilon1 of the equation ∆ ' 0 = 0. According to Expression (80), this equation reads explicitly \n∆ 0 +4 /epsilon1 ∆ 1 +2¯ /epsilon1 ∆ 2 +6 /epsilon1 2 ∆ 3 +8 /epsilon1 ¯ /epsilon1 ∆ 4 +¯ /epsilon1 2 ∆ 5 +4 /epsilon1 3 ∆ 6 +12 /epsilon1 2 ¯ /epsilon1 ∆ 7 +4 /epsilon1 ¯ /epsilon1 2 ∆ 8 + /epsilon1 4 ∆ 9 +8 /epsilon1 3 ¯ /epsilon1 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 4 ¯ /epsilon1 ∆ 12 +4 /epsilon1 3 ¯ /epsilon1 2 ∆ 13 + /epsilon1 4 ¯ /epsilon1 2 ∆ 14 = 0 . (105) \nThis is a polynomial relation of total degree 6 involving the complex variable /epsilon1 and its complex conjugate ¯ /epsilon1 . There does not seem to be any general result in the mathematical literature for such types of polynomial equations 4 . A possible way to proceed consists of considering Eq. (105), together with its own complex conjugate equation ¯ ∆ ' 0 = 0, \nwith z arbitrary, and also by \n( /epsilon1, z = 0) , (110) \nwith /epsilon1 arbitrary. Among such a huge number of solutions, only those with z = 0 in the first case, and only those with /epsilon1 = 0 in the second case, satisfy the constraint that z = ¯ /epsilon1 . Thus, the solution of the original equation is unique, given by /epsilon1 = 0. Its multiplicity can be guessed by noting that Eq. (106) can also be written as \n| /epsilon1 | 6 e 2 iφ ∆ 14 = 0 , (111) \nwhere (here and later on) φ is the phase of /epsilon1 , which leads to | /epsilon1 | = 0 six times, ergo multiplicity 6. \nAs a second and more interesting example, let the case be such that only ∆ 0 = 0 = ∆ 14 are nonzero, all other ∆-scalars vanish. Using the renaming (107), Eq. (105) collapses to simply \n/negationslash \n/negationslash \n∆ 0 + /epsilon1 4 z 2 ∆ 14 = 0 , (112) \nwhile the complex conjugate of (105) reads \n¯ ∆ 0 + /epsilon1 2 z 4 ¯ ∆ 14 = 0 . (113) \nThe solutions to this pair of equations can be easily obtained by the method of substitution, and they are \n/epsilon1 k = ∣ ∣ ∣ ∆ 0 ∆ 14 ∣ ∣ ∣ 1 / 6 e i ( φ 0 -φ 14 ) / 2+ i (2 k +1) π/ 6 , z k = ± ∣ ∣ ∣ ∆ 0 ∆ 14 ∣ ∣ ∣ 1 / 6 e i ( φ 14 -φ 0 ) / 2 -i (4 k +5) π/ 6 , (114) \n∣ ∣ ∣ ∣ for all k ∈ { 0 , 1 , 2 , 3 , 4 , 5 } , where φ n will denote the phase of any ∆-scalar: \n∆ n = | ∆ n | e iφ n , n ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 } . (115) \nThus, there are 12 solutions in total, but only two proper solutions of Eq. (112) (i.e. with z k = ¯ /epsilon1 k ), which are given by k = 1 for the -sign and by k = 4 for the + sign. Hence, both solutions have simple multiplicity and read \n/epsilon1 1 = i ∣ ∣ ∣ ∣ ∆ 0 ∆ 14 ∣ ∣ ∣ ∣ 1 / 6 e i ( φ 0 -φ 14 ) / 2 , /epsilon1 4 = -i ∣ ∣ ∣ ∣ ∆ 0 ∆ 14 ∣ ∣ ∣ ∣ 1 / 6 e i ( φ 0 -φ 14 ) / 2 . (116) \nas a system of two polynomial equations in the variables { /epsilon1, ¯ /epsilon1 } . Unfortunately, all relevant results about such systems provide the number of solutions, counted with its multiplicity, for the case where the two variables are considered to be fully independent [81-83]. For cases as ours, where they are mutually complex conjugate, not even a definition of multiplicity is available. \nThat said, obviously the number of actual solutions will always be less or equal than the total number of solutions with the two independent complex variables. Hence, the number of the latter will provide a bound for the number of solutions of interest. Concerning the multiplicity, even though there is no mathematical accepted definition for such, we will consider in our case that the multiplicity will be the same as that arising as solutions of the system when the variables are assumed to be independent. \nTo fix ideas, let us consider a few simple examples. Imagine the situation is such that all the scalars are zero, except ∆ 14 = 0. Then, Eq. (105) reduces to simply \n/negationslash \n/epsilon1 4 ¯ /epsilon1 2 ∆ 14 = 0 . (106) \nIn that case, renaming \n¯ /epsilon1 -→ z , (107) \nand considering the pair { /epsilon1, z } as two independent complex variables, Eq. (106) and its complex conjugate are rewritten as \n/epsilon1 4 z 2 ∆ 14 = 0 , z 4 /epsilon1 2 ¯ ∆ 14 = 0 , (108) \nwhich have an infinite number of solutions, given by \n( /epsilon1 = 0 , z ) , (109) \nNote that the only vanishing scalars in the null tetrads provided by the complex rotations of value /epsilon1 1 and /epsilon1 4 are ∆ ' 0 ( /epsilon1 1 ) and ∆ ' 0 ( /epsilon1 4 ), respectively, but we can always choose a different null tetrad where both rotated scalars ∆ ' 0 and ∆ ' 14 vanish: \nL µ = l ' µ ( /epsilon1 1 ) , K µ = 1 4 ∣ ∣ ∣ ∆ 14 ∆ 0 ∣ ∣ ∣ 1 / 3 l ' µ ( /epsilon1 4 ) , (117) \n∣ ∣ M µ = m ' µ ( /epsilon1 4 ) + l ' µ ( /epsilon1 4 ) ¯ /epsilon1 1 -¯ /epsilon1 4 , ¯ M µ = ¯ m ' µ ( /epsilon1 4 ) + l ' µ ( /epsilon1 4 ) /epsilon1 1 -/epsilon1 4 , (118) \nin such a way that L µ and K µ would constitute the corresponding PNDs under that choice. \nAnother example, which will be of interest later for the Type L in the classification, arises when the only nonzero complex scalars are ∆ 12 , ∆ 13 and ∆ 14 . Eq. (105) then collapses to \n/epsilon1 3 ¯ /epsilon1 ( 2 /epsilon1 ∆ 12 +4¯ /epsilon1 ∆ 13 + /epsilon1 ¯ /epsilon1 ∆ 14 ) = 0 . (119) \nPassing to the independent variables by (107), we have for Eq. (119) and its complex conjugate \n/epsilon1 3 z ( 2 /epsilon1 ∆ 12 +4 z ∆ 13 + /epsilon1z ∆ 14 ) = 0 , (120) \nThere are obvious solutions ( /epsilon1 = 0 , z ) with arbitrary z , as well as ( /epsilon1, z = 0) with arbitrary /epsilon1 . Furthermore, there are also solutions of the system \nz 3 /epsilon1 ( 2 z ¯ ∆ 12 +4 /epsilon1 ¯ ∆ 13 + z/epsilon1 ¯ ∆ 14 ) = 0 . (121) \n2 /epsilon1 ∆ 12 +4 z ∆ 13 + /epsilon1z ∆ 14 = 0 , 2 z ¯ ∆ 12 +4 /epsilon1 ¯ ∆ 13 + z/epsilon1 ¯ ∆ 14 = 0 . (122) \nThese can be computed easily and are given by ( /epsilon1 = 0 , z = 0) and by \nz = ¯ /epsilon1 = 2 4∆ 13 ¯ ∆ 13 -∆ 12 ¯ ∆ 12 ∆ 14 ¯ ∆ 12 -2∆ 13 ¯ ∆ 14 . (123) \n/negationslash \nNotice that this solution is a proper solution of the original Eq. (119). In summary, there is a solution with /epsilon1 = 0, providing a PND l µ of Class V, plus a (generally) unique second PND given by l ' µ in (42) with /epsilon1 as in (123). However, there is a special situation, because the above unique extra solution is defined only if ∆ 14 ¯ ∆ 12 -2∆ 13 ¯ ∆ 14 = 0. For the very special case where \n∆ 14 ¯ ∆ 12 -2∆ 13 ¯ ∆ 14 = 0 , (124) \nthis condition readily entails | ∆ 12 | = 2 | ∆ 13 | and the numerator on (123) vanishes too. In that case, there will be no further solutions in general, leading to Type L*. One can check that, nevertheless, in this special situation there are extra solutions whenever ∆ 12 = ¯ ∆ 12 , given by \n/epsilon1 = -8 ∆ 13 ∆ 14 ae i arccos a , (125) \nwhere a is a nonzero real number satisfying -1 ≤ a ≤ 1. Thus, the infinite values that the parameter a can take within the interval [ -1 , 1] provide an infinite number of solutions. Thereby, we find a behaviour that will arise in some other extremely special situations: there may be an infinite number of PNDs . These very exceptional cases will be generally termed as 'exceptional' within the corresponding algebraic type, and will carry a subindex 'e'. For instance, the case just analysed belongs to the Type L and will be denoted by Type L e or, alternatively, by (V,I ∞ ). \nIn a general and generic situation, however, the scalars ∆ n will take arbitrary complex nonzero values without any relation between them, so that the original Eq. (105) has to be considered. By the renaming (107), we can rearrange Eq. (105) as \n∆ ' 0 = p 0 ( /epsilon1 ) + zp 1 ( /epsilon1 ) + z 2 p 2 ( /epsilon1 ) = 0 , (126) \nwith \np 0 ( /epsilon1 ) = ∆ 0 +4∆ 1 /epsilon1 +6∆ 3 /epsilon1 2 +4∆ 6 /epsilon1 3 +∆ 9 /epsilon1 4 , (127) \np 1 ( /epsilon1 ) = 2∆ 12 /epsilon1 4 +8∆ 10 /epsilon1 3 +12∆ 7 /epsilon1 2 +8∆ 4 /epsilon1 +2∆ 2 , (128) \np 2 ( /epsilon1 ) = ∆ 14 /epsilon1 4 +4∆ 13 /epsilon1 3 +6∆ 11 /epsilon1 2 +4∆ 8 /epsilon1 +∆ 5 , (129) \nwhile its complex conjugate equation yields \n¯ ∆ ' 0 = q 0 ( /epsilon1 ) + q 1 ( /epsilon1 ) z + q 2 ( /epsilon1 ) z 2 + q 3 ( /epsilon1 ) z 3 + q 4 ( /epsilon1 ) z 4 , (130) \nwhere \nq 0 ( /epsilon1 ) = ¯ ∆ 0 +2 ¯ ∆ 2 /epsilon1 + ¯ ∆ 5 /epsilon1 2 , (131) \nq 1 ( /epsilon1 ) = 4 ¯ ∆ 1 +8 ¯ ∆ 4 /epsilon1 +4 ¯ ∆ 8 /epsilon1 2 , (132) \nq 2 ( /epsilon1 ) = 6 /epsilon1 2 ¯ ∆ 11 +12 ¯ ∆ 7 /epsilon1 +6 ¯ ∆ 3 , (133) \nq 3 ( /epsilon1 ) = 4 ¯ ∆ 6 +8 ¯ ∆ 10 /epsilon1 +4 ¯ ∆ 13 /epsilon1 2 , (134) \nq 4 ( /epsilon1 ) = ¯ ∆ 9 +2 ¯ ∆ 12 /epsilon1 + ¯ ∆ 14 /epsilon1 2 . (135) \nThen, by taking ∆ ' 0 = ¯ ∆ ' 0 = 0 as a system of two equations for the variable z , we can define the Sylvester matrix as [83]: \nM ( /epsilon1 ) = q 0 0 p 0 0 0 0 q 1 q 0 p 1 p 0 0 0 q 2 q 1 p 2 p 1 p 0 0 q 3 q 2 0 p 2 p 1 p 0 q 4 q 3 0 0 p 2 p 1 0 q 4 0 0 0 p 2 , (136) \nwhose determinant, called the resultant, becomes \ndet( M ( /epsilon1 )) = p 2 2 [ p 2 0 ( 2 q 0 q 4 -2 q 1 q 3 + q 2 2 ) + p 0 p 1 (3 q 0 q 3 -q 1 q 2 ) + p 2 1 q 0 q 2 ] + p 2 [ p 3 0 ( q 2 3 -2 q 2 q 4 ) + p 2 0 p 1 (3 q 1 q 4 -q 2 q 3 ) + p 0 p 2 1 ( q 1 q 3 -4 q 0 q 4 ) -p 3 1 q 0 q 3 ] + q 4 ( p 4 0 q 4 -p 3 0 p 1 q 3 + p 2 0 p 2 1 q 2 -p 0 p 3 1 q 1 + p 4 1 q 0 ) + p 3 2 [ p 0 ( q 2 1 -2 q 0 q 2 ) -p 1 q 0 q 1 ] + p 4 2 q 2 0 . (137) \nThis is a polynomial in the variable /epsilon1 of degree 20. It is known that the solutions of the system are included in the solutions of the resultant equated to zero. Therefore, we have derived an upper bound for the number of solutions -counted with its multiplicity- of the system: 20. This number is actually exact in generic situations due to the Bernstein's theorem [81-83], which is applied to the above system in Appendix B. \nNevertheless, this bound of 20 solutions applies to the case where the two variables /epsilon1 and z are fully independent. We need to extract the solutions with ¯ z = /epsilon1 , and there is no known way to quantify this. An important remark is in order here: in the analysis of Eq. (80) in terms of the two independent variables /epsilon1 and z , if ( /epsilon1 0 , z 0 ) happens to be a solution of this equation, then ( /epsilon1 = ¯ z 0 , z = ¯ /epsilon1 0 ) is also necessarily a solution. Hence, the solutions that do not satisfy the constraint ¯ z = /epsilon1 come in pairs, and thus the number of proper solutions that satisfy this constraint in generic situations will be 20 minus an even number. Note however, that depending on their multiplicities, the total number of different solutions may well be odd. \nFor the definition of nongeneric situations, as well as their general characterisations in terms of the scalars { ∆ i } 14 i =0 , consult Appendix B. Let us remark that the property of substracting an even number will also apply to these nongeneric situations. More on this in the next subsection, where the different cases arising for each type in the algebraic classification are identified and studied. \n- 2. The full classification including exceptional cases and the possible number of PNDs for each type \nThe best way to complete the algebraic classification by finding the possibilities for the second Roman numeral is to analyse the different cases in order, from maximum to minimum alignment. Thus, we start with Type N.", 'Type N': 'In this case, there is a PND of Class VI and, therefore, all the complex scalars vanish, except ∆ 14 . This case was already studied in the previous subsection by analysing Eq. (106), where we proved that such PND is unique. Thus, only Type \n(VI , -) , \nexists.', 'Types L and L*': 'These are defined by the existence of a PND of Class V. Choosing this PND as l µ in a null tetrad implies that ∆ n = 0 for all n = 0 , . . . , 11. Again this situation was already analysed in the previous subsection, see Eq. (119), where we proved that generally there is a second simple PND given by (123). This provides the case \n(V , I) . \nHowever, we also proved that under the constraint (124) there are two special situations in which either there is no second PND, or there is an infinite number of them; the latter if the constraint ∆ 12 = ¯ ∆ 12 holds as well. These types are then \n(V , -) , (V , I ∞ ) , \nrespectively.', 'Type F': "This type is given by the existence of a PND of Class IV, and a second PND of Class II. We can choose a preferred tetrad { l µ , k µ , m µ , ¯ m µ } , with l µ being the Class-IV PND and k µ the Class-II PND. In such a preferred tetrad, one has \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = ∆ 12 = ∆ 13 = ∆ 14 = 0 . (138) \nTo see if there can be any other PNDs, we can consider Eq. (105) with the previous restrictions, that is \n/epsilon1 2 ( /epsilon1 2 ∆ 9 +8 /epsilon1 ¯ /epsilon1 ∆ 10 +6¯ /epsilon1 2 ∆ 11 ) = 0 , (139) \nwhich, removing the factor /epsilon1 2 , can be rewritten as either \n/epsilon1 2 ∆ 9 +8 /epsilon1 ¯ /epsilon1 ∆ 10 +6¯ /epsilon1 2 ∆ 11 = 0 , (140) \nor \n| /epsilon1 | 2 ( e 2 iφ ∆ 9 +8∆ 10 +6 e -2 iφ ∆ 11 ) = 0 . (141) \nIt is easily seen that there are no solutions for this equation with /epsilon1 = 0, unless very specific restrictions exist between the scalars ∆ 9 , ∆ 10 and ∆ 11 . This general case, characterised solely by the trivial solution /epsilon1 = 0, constitutes the Type \n(IV \n, \nII) \n. \nHowever, there are exceptional situations with solutions of Eq. (141) for the phase φ , whereas | /epsilon1 | remains free, giving rise to an infinite number of extra solutions. In particular, two possibilities arise, depending on whether the extra infinite PNDs are of Class II or of Class I. The former takes place when the left-hand side of Eq. (140) has the form of a perfect square of type ( A/epsilon1 + B ¯ /epsilon1 ) 2 , with | A | = | B | . This can happen only if \n8∆ 2 10 = 3∆ 9 ∆ 11 , | ∆ 9 | = 6 | ∆ 11 | . (142) \nIn this case, the form of the solution is \n/epsilon1 = | /epsilon1 | e i [ φ 11 -φ 9 +2(2 k +1) π ] / 4 , where k = 0 , 1 , (143) \nso that the phase of /epsilon1 is fixed but | /epsilon1 | remains free, leading then to an infinite number of PNDs. To check that they are of Class II, we perform the transformation (42) with these solutions for /epsilon1 , in order to verify that ∆ ' 0 = ∆ ' 1 = ∆ ' 2 = 0. Using the formulas (80)-(94) for the rotated tetrad, we certainly find \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (144) \n∆ ' n = 0 , for all n = 3 , ..., 11 . (145) \n/negationslash \nThis exceptional type can then be denoted as F e , or as \n(IV , II ∞ ) . \n/negationslash \nthe previous equation reduces to \nOn the other hand, the second possibility that gives rise to an infinite number of PNDs, but now of Class I, arises for example for the particular case \n∆ 11 = 0 , | ∆ 9 | = 8 | ∆ 10 | , (146) \nwhich leads to the nontrivial solutions \n/epsilon1 = | /epsilon1 | e i [ φ 10 -φ 9 +(2 k +1) π ] / 2 , where k = 0 , 1 , (147) \nand, once again, arbitrary | /epsilon1 | . This is an infinite number of extra PNDs, providing another special Type F e , which will be denoted by \n(IV \n, \nII) \nI \n∞ \n. \nIn summary, there exist three different cases within the Type F: the generic case (IV , II) with one PND of Class IV and another one of Class II, as well as two exceptional cases (IV , II ∞ ) and (IV , II) I ∞ ; the first one with one PND of Class IV and infinite of Class II, and the second one with one PND of Class IV, another one of Class II and infinite of Class I. Hence, it is worthwhile to stress that in the cases (IV , II) and (IV , II) I ∞ the PNDs of Class IV and II are uniquely defined.", 'Types H and H*': 'For these types, there is a PND of Class IV, but there is no PND of Class II. We choose a preferred tetrad { l µ , k µ , m µ , ¯ m µ } with l µ being the PND of Class IV, so that \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = 0 . (148) \nWith these restrictions, Eq. (105) reads \n/epsilon1 2 ( /epsilon1 2 ∆ 9 +8 /epsilon1 ¯ /epsilon1 ∆ 10 +6¯ /epsilon1 2 ∆ 11 +2 /epsilon1 2 ¯ /epsilon1 ∆ 12 +4 /epsilon1 ¯ /epsilon1 2 ∆ 13 + /epsilon1 2 ¯ /epsilon1 2 ∆ 14 ) = 0 . (149) \nFirst of all, we notice that there are cases devoid of nontrivial solutions for /epsilon1 ; for instance, if \n∆ 9 = ∆ 11 = ∆ 12 = ∆ 13 = 0 , (150) \n/epsilon1 2 | /epsilon1 | 2 (8∆ 10 + | /epsilon1 | 2 ∆ 14 ) = 0 , (151) \nwhich does not have any nontrivial solution for /epsilon1 if ∆ 10 ¯ ∆ 14 is not real. Thus, this particular case constitutes a Type H* or \n(IV \n, \n- \n) \n. \nIn other situations, there is at least a nontrivial solution for /epsilon1 of Eq. (149). One can then adapt the tetrad such that k µ is one PND so that, without any loss of generality, Eq. (149) becomes \n/epsilon1 2 ∆ 9 +8 /epsilon1 ¯ /epsilon1 ∆ 10 +6¯ /epsilon1 2 ∆ 11 +2 /epsilon1 2 ¯ /epsilon1 ∆ 12 +4 /epsilon1 ¯ /epsilon1 2 ∆ 13 = 0 , (152) \nwhere we have removed the /epsilon1 2 factor in the equation. Now the question remains on whether there can be more nontrivial solutions of this equation for /epsilon1 , and whether or not there is a finite or infinite number of them. To answer these questions, we shall show explicit examples below. \nFirst of all, subcases with infinite and also with none /epsilon1 = 0 solutions arise for \n/negationslash \n∆ 9 = ∆ 11 = 0 , ∆ 12 = 2∆ 13 . (153) \nConsequently, any nontrivial solution must satisfy the equation \n4∆ 10 +( /epsilon1 +¯ /epsilon1 ) ∆ 13 = 0 , (154) \nwhich has no solution if ∆ 10 ¯ ∆ 13 is not real, but it has an infinite number of solutions if ∆ 10 ¯ ∆ 13 is real -because the imaginary part of /epsilon1 remains free. The latter leads to Type H e or \n(IV \n, \nI \n∞ \n) \n. \nOn the other hand, different subcases with a finite number of nontrivial solutions for /epsilon1 are \n- · ∆ 10 = ∆ 11 = ∆ 13 = 0, with the unique solution 2 /epsilon1 = -¯ ∆ 9 / ¯ ∆ 12 ;\n- · ∆ 9 = ∆ 11 = ∆ 12 = 0, also with a unique solution /epsilon1 = -2 ¯ ∆ 10 / ¯ ∆ 13 ;\n- · ∆ 10 = ∆ 11 = ∆ 12 = 0, with three distinct solutions \n/epsilon1 k = 1 4 ∣ ∣ ∣ ∆ 9 ∆ 13 ∣ ∣ ∣ e i [ φ 13 -φ 9 +(2 k -1) π ] / 3 , where k = 0 , 1 , 2 . (155) \n∣ ∣ Therefore, all these subcases lead to a general Type H, given by \n(IV , I) , \nand one should keep in mind that in some situations there are several choices for the secondary I.', 'Type D': 'This type is defined by the existence of two PNDs of Class III. Choosing the preferred tetrad { l µ , k µ , m µ , ¯ m µ } with both l µ and k µ of Class III, we have \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 9 = ∆ 10 = ∆ 11 = ∆ 12 = ∆ 13 = ∆ 14 = 0 . (156) \nAccordingly, Eq. (105) with the previous restrictions reads \n4 /epsilon1 ( /epsilon1 2 ∆ 6 +3 /epsilon1 ¯ /epsilon1 ∆ 7 +¯ /epsilon1 2 ∆ 8 ) = 0 , (157) \nwhich, removing the 4 /epsilon1 factor, can be rewritten as \n| /epsilon1 | 2 ( e 2 iφ ∆ 6 +3∆ 7 + e -2 iφ ∆ 8 ) = 0 . (158) \nAs is shown, Eq. (158) acquires the same form as Eq. (141), leading to no solutions in general, or to an infinite number of PNDs of Class I (e.g. ∆ 8 = 0 and | ∆ 6 | = 3 | ∆ 7 | ), or to an infinite number of Class II PNDs (if 9∆ 2 7 = 4∆ 6 ∆ 8 and | ∆ 6 | = | ∆ 8 | ) with arbitrary | /epsilon1 | in these last two exceptional situations. \nThese cases with infinite extra solutions are the special Type D e and denoted by \n(III , III) II ∞ , (III , III) I ∞ . \nThe general case with no extra PNDs is the generic Type D denoted as \n(III , III) . \nIn all cases, (III , III), (III , III) II ∞ and (III , III) I ∞ , the two PNDs of Class III are uniquely defined.', 'Type M': 'This type is defined by the existence of a unique PND of Class III, and a second PND of Class II. We choose a preferred tetrad { l µ , k µ , m µ , ¯ m µ } with l µ being the PND of Class III, and k µ a Class-II one. In such a preferred tetrad one has \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 12 = ∆ 13 = ∆ 14 = 0 , (159) \nand Eq. (105) becomes \n/negationslash \nAgain, besides the trivial solution /epsilon1 = 0, one can exhibit cases with an infinite number of /epsilon1 = 0 solutions giving PNDs of Class I (∆ 6 = ∆ 8 = ∆ 9 = 0, 4∆ 10 = 3∆ 11 , ∆ 7 ¯ ∆ 11 = ¯ ∆ 7 ∆ 11 ) or with no such solutions (∆ 6 = ∆ 8 = ∆ 9 = 0, 4∆ 10 = 3∆ 11 , ∆ 7 ¯ ∆ 11 = ¯ ∆ 7 ∆ 11 ) and also with a finite number of them. They give, respectively, Types M e and M, denoted by \n(III , II) I ∞ , (III , II) . \nNote that both PNDs of Class III and Class II are uniquely determined in these cases. \nThe question remains if there can be an infinite number of PNDs of Class II. This will happen if the left-hand side of Eq. (160) can be factorised as /epsilon1 ( A/epsilon1 + B ¯ /epsilon1 ) 2 ( a + b/epsilon1 ) with | A | = | B | . Specifically, this occurs if \n8∆ 2 10 = 3∆ 9 ∆ 11 , 4∆ 6 ∆ 8 = 9∆ 2 7 , 6∆ 6 ∆ 11 = ∆ 8 ∆ 9 , 2 | ∆ 6 | = 3 | ∆ 7 | , (161) \nwhere, apart from the infinite number of PNDs of Class II, there exists one extra PND. Thereby, this is another exceptional Type M e , denoted by \n(III , II ∞ ) . \n/epsilon1 ( 4 /epsilon1 2 ∆ 6 +12 /epsilon1 ¯ /epsilon1 ∆ 7 +4¯ /epsilon1 2 ∆ 8 + /epsilon1 3 ∆ 9 +8 /epsilon1 2 ¯ /epsilon1 ∆ 10 +6 /epsilon1 ¯ /epsilon1 2 ∆ 11 ) = 0 . (160) \n/negationslash', 'Types K and K*': 'Now there is a unique PNDof maximal Class III, but no PND of Class II. We choose a preferred tetrad { l µ , k µ , m µ , ¯ m µ } with l µ being the PND of Class III, so that in this tetrad Eq. (105) reads \n/epsilon1 ( 4 /epsilon1 2 ∆ 6 +12 /epsilon1 ¯ /epsilon1 ∆ 7 +4¯ /epsilon1 2 ∆ 8 + /epsilon1 3 ∆ 9 +8 /epsilon1 2 ¯ /epsilon1 ∆ 10 +6 /epsilon1 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 3 ¯ /epsilon1 ∆ 12 +4 /epsilon1 2 ¯ /epsilon1 2 ∆ 13 + /epsilon1 3 ¯ /epsilon1 2 ∆ 14 ) = 0 . (162) \nThe first thing to know is whether there are cases without nonzero solutions for /epsilon1 while keeping ∆ 14 and at least one of ∆ 6 , ∆ 7 , ∆ 8 different from zero. By setting ∆ 8 = ∆ 9 = ∆ 10 = ∆ 11 = ∆ 12 = ∆ 13 = 0 and ∆ 6 = 3∆ 7 , this equation can be written as \n/epsilon1 2 [4∆ 6 ( /epsilon1 +¯ /epsilon1 ) + /epsilon1 2 ¯ /epsilon1 2 ∆ 14 ] = 0 , (163) \nwhich does not provide any nontrivial solution if ∆ 6 ¯ ∆ 14 = ¯ ∆ 6 ∆ 14 . This case then represents a Type K*, or \n/negationslash \n(III , -) . \nOtherwise, if there are solutions of Eq. (162) leading to PNDs different from l µ , we can choose one of these extra PND as the k µ in the null tetrad, so that without any loss of generality ∆ 14 can be set to zero in Eq. (162): \n/epsilon1 ( 4 /epsilon1 2 ∆ 6 +12 /epsilon1 ¯ /epsilon1 ∆ 7 +4¯ /epsilon1 2 ∆ 8 + /epsilon1 3 ∆ 9 +8 /epsilon1 2 ¯ /epsilon1 ∆ 10 +6 /epsilon1 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 3 ¯ /epsilon1 ∆ 12 +4 /epsilon1 2 ¯ /epsilon1 2 ∆ 13 ) = 0 . (164) \n/negationslash \nNevertheless, now we must keep at least one of the complex scalars ∆ 12 and ∆ 13 different from zero; otherwise, this will belong to the previous Type M. Now, following the same ideas as in previous cases, it becomes rather easy to find cases with an infinite number of extra solutions (e.g. all ∆ n = 0 except ∆ 6 and ∆ 12 , with ∆ 6 ¯ ∆ 12 = ¯ ∆ 6 ∆ 12 ) and with no extra solutions (e.g. all ∆ n = 0 except ∆ 6 and ∆ 12 , with ∆ 6 ¯ ∆ 12 = ¯ ∆ 6 ∆ 12 ), or with a finite number of them. These lead respectively to the Types K e and K, or equivalently \n(III , I ∞ ) , (III , I) , \nwhere in the latter case there may be several different choices for the PND of Class I.', 'Types B and S': "Now there are two alignment PNDs of Class II, so that we can choose a tetrad with both l µ and k µ of Class II. This implies ∆ 0 = ∆ 1 = ∆ 2 = ∆ 12 = ∆ 13 = ∆ 14 = 0 and the main equation (105) written in this preferred tetrad reads \n6 /epsilon1 2 ∆ 3 +8 /epsilon1 ¯ /epsilon1 ∆ 4 +¯ /epsilon1 2 ∆ 5 +4 /epsilon1 3 ∆ 6 +12 /epsilon1 2 ¯ /epsilon1 ∆ 7 +4 /epsilon1 ¯ /epsilon1 2 ∆ 8 + /epsilon1 4 ∆ 9 +8 /epsilon1 3 ¯ /epsilon1 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 2 ∆ 11 = 0 , (165) \nwhere one must keep at least one of ∆ 3 , ∆ 4 , ∆ 5 , and at least one of ∆ 9 , ∆ 10 , ∆ 11 different from zero. \n/negationslash \nThe first question to elucidate is the possible existence of a third PND of Class II. Intuition developed so far tells us that this may be the case if there are /epsilon1 = 0 solutions of the previous equation with multiplicity 2. Keeping the double solution for /epsilon1 = 0, this may happen if the left-hand side in Eq. (165) can be factorised in any of the following forms: A/epsilon1 2 ( /epsilon1 + B ) 2 , A/epsilon1 2 (¯ /epsilon1 + B ) 2 , A ¯ /epsilon1 2 ( /epsilon1 + B ) 2 , A/epsilon1 ¯ /epsilon1 ( /epsilon1 + B ) 2 , A/epsilon1 2 ( /epsilon1 + B )(¯ /epsilon1 + ¯ B ), A/epsilon1 ¯ /epsilon1 ( /epsilon1 + B )(¯ /epsilon1 + ¯ B ), or ( a/epsilon1 + b ¯ /epsilon1 ) 2 ( A + B/epsilon1 + C/epsilon1 2 ), for some A,B,C,a,b ∈ C and with | a | = | b | . These lead to the following seven possibilities respectively: \n- i) All ∆ n = 0 except for ∆ 3 , ∆ 6 and ∆ 9 with 2∆ 2 6 = 3∆ 3 ∆ 9 . Then, Eq. (165) has a double solution given by \n/epsilon1 = -2 ∆ 6 ∆ 9 = -3 ∆ 3 ∆ 6 . (166) \nii) All ∆ n = 0 except for ∆ 3 , ∆ 7 and ∆ 11 with ∆ 2 7 = ∆ 3 ∆ 11 . Then, Eq. (165) has a double solution given by \n¯ /epsilon1 = -∆ 7 ∆ 11 = -∆ 3 ∆ 7 . (167) \niii) All ∆ n = 0 except for ∆ 5 , ∆ 8 and ∆ 11 with 2∆ 2 8 = 3∆ 5 ∆ 11 . Then, Eq. (165) has a double solution given by \n/epsilon1 = -∆ 8 3∆ 11 = -∆ 5 2∆ 8 . (168) \niv) All ∆ n = 0 except for ∆ 4 , ∆ 7 and ∆ 10 with 9∆ 2 7 = 16∆ 4 ∆ 10 . Then, Eq. (165) has a double solution given by \n/epsilon1 = -3∆ 7 4∆ 10 = -4∆ 4 3∆ 7 . (169) \nv) All ∆ n = 0 except for ∆ 3 , ∆ 6 , ∆ 7 and ∆ 10 with ∆ 7 ∆ 6 = ∆ 3 ∆ 10 and 3∆ 7 ¯ ∆ 10 = ¯ ∆ 6 ∆ 10 . Then, Eq. (165) has a double solution given by \n/epsilon1 = -3∆ 7 2∆ 10 = -¯ ∆ 6 2 ¯ ∆ 10 . (170) \nvi) All ∆ n = 0 except for ∆ 4 , ∆ 7 , ∆ 8 and ∆ 11 with ∆ 7 ∆ 8 = ∆ 4 ∆ 11 and ∆ 8 ¯ ∆ 11 = 3 ¯ ∆ 7 ∆ 11 . Then, Eq. (165) has a double solution given by \n/epsilon1 = -2∆ 8 3∆ 11 = -2 ¯ ∆ 7 ¯ ∆ 11 . (171) \nvii) Conditions (161), together with \n8∆ 2 4 = 3∆ 3 ∆ 5 , 6∆ 8 ∆ 3 = ∆ 6 ∆ 5 , (172) \nall hold. \nTo check whether or not these solutions define a PND of Class II, we need to verify that in the new tetrad (42) the corresponding scalars ∆ ' 0 , ∆ ' 1 and ∆ ' 2 vanish. Starting from the last possibility vii), Eq. (165) factorises as ( e iα = 3∆ 7 / (2∆ 6 )) \n( /epsilon1 + e iα ¯ /epsilon1 ) 2 ( ∆ 9 /epsilon1 2 +6∆ 7 /epsilon1 +∆ 5 ) = 0 , (173) \nleading to an infinite number of solutions \n/epsilon1 = ± i | /epsilon1 | e iα/ 2 , (174) \nwith arbitrary modulus. Using now formulas (80)-(94) with the above restrictions on the ∆ n , one easily gets \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = 0 , ∆ ' 3 = 0 , (175) \n/negationslash \nfor arbitrary values of | /epsilon1 | and thus this case has an infinite number of PNDs of Class II. This is one of the Types B e , denoted by \n(II , II ∞ ) ≡ (II ∞ ) . \nFrom Expression (173), one sees that there are two further PNDs of Class I (or another one of Class II). Concerning the other cases i) - vi), the formulas (80)-(94) we have, for the first case i) \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = 0 , ∆ ' 3 = ∆ 3 = 0 . (176) \n/negationslash \nhence this gives indeed another PND of Class II. For completeness, the rest of scalars in the new tetrad are \n∆ ' 4 = ∆ ' 5 = ∆ ' 7 = ∆ ' 8 = ∆ ' 10 = ∆ ' 11 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , ∆ ' 6 = -∆ 6 , ∆ ' 9 = ∆ 9 , (177) \nwhich keeps the property 2∆ ' 6 2 = 3∆ ' 3 ∆ ' 9 . It is easy to check then that, starting with the new null tetrad { l ' µ , k µ , m ' µ , ¯ m ' µ } and getting the third PND of Class II by solving the primed version of Eq. (165) one gets back to the original PND given by l µ . \nConcerning possibility ii), formulas (80)-(94) lead to \n/negationslash \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = 0 , ∆ ' 3 = ∆ ' 4 = 0 , ∆ ' 5 = 6∆ 11 ¯ ∆ 2 3 ¯ ∆ 2 7 = 0 , (178) \nso that this gives indeed another PND of Class II. For completeness, the rest of scalars in the new tetrad are \n∆ ' 6 = ∆ ' 7 = ∆ ' 9 = ∆ ' 10 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (179) \n∆ ' 8 = -3∆ 11 ¯ ∆ 3 ¯ ∆ 7 , ∆ ' 11 = ∆ 11 . (180) \nNotice that 2∆ ' 8 2 = 3∆ ' 5 ∆ ' 11 leading to possibility iii). One expects, therefore, that possibility iii) will also define a PND of Class II, which should actually provide new ∆ ' n defining the possibility ii). Indeed, using formulas (80)-(94) under possibility iii) we obtain \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = ∆ ' 4 = ∆ ' 5 = ∆ ' 6 = ∆ ' 8 = ∆ ' 9 = ∆ ' 10 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (181) \n/negationslash \n∆ ' 3 = ¯ ∆ 2 5 4 ¯ ∆ 2 8 ∆ 11 = 0 , ∆ ' 7 = -¯ ∆ 5 2 ¯ ∆ 8 ∆ 11 , ∆ ' 11 = ∆ 11 , (182) \nwith ∆ ' 7 2 = ∆ ' 3 ∆ ' 11 as required. \nConsider now possibility iv). A similar calculation provides \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = ∆ ' 4 = ∆ ' 5 = ∆ ' 8 = ∆ ' 9 = ∆ ' 11 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (183) \n∆ ' 3 = 3 4 ∆ 7 ¯ ∆ 7 ¯ ∆ 10 , ∆ ' 6 = -3 ¯ ∆ 7 2 ¯ ∆ 10 ∆ 10 , ∆ ' 7 = -1 2 ∆ 7 , ∆ ' 10 = ∆ 10 , (184) \nwhich satisfies ∆ ' 7 ∆ ' 6 = ∆ ' 3 ∆ ' 10 and 3∆ ' 7 ¯ ∆ ' 10 = ¯ ∆ ' 6 ∆ ' 10 leading to possibility v). Again, we expect then that possibility v) will lead to a new PND of possibility iv). This can be checked as before, because in possibility v) the new ∆ ' n are \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = ∆ ' 3 = ∆ ' 5 = ∆ ' 6 = ∆ ' 8 = ∆ ' 9 = ∆ ' 11 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (185) \n∆ ' 4 = 9 4 ∆ 2 7 ¯ ∆ 10 , ∆ ' 7 = -2∆ 7 , ∆ ' 10 = ∆ 10 , (186) \nhaving 16∆ ' 4 ∆ ' 10 = 9∆ ' 7 2 , that is, the properties defining case iv). \nFinally, in possibility vi) we compute \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = ∆ ' 3 = ∆ ' 5 = ∆ ' 6 = ∆ ' 9 = ∆ ' 10 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , (187) \n∆ ' = ∆ 4 , ∆ ' = ∆ 7 , ∆ ' = ∆ 8 , ∆ ' = ∆ 11 , (188) \n4 7 -8 11 \ndefining an extra PND and keeping the type possibility vi). \nSummarising, for all possibilities i)-vi) there is always a third PND of Class II -and no more. This is a special situation, because the notation (II , II) would be ambiguous, as we do not know which two PNDs, among the three ones of Class II, are there. And there are three different possible choices for two Class-II PNDs among three. This is the reason that we introduce a special notation for this particular type, called Type S, given by \n(II \n, \nII \n, \nII) \n. \nFurthermore, the six possibilities studied may be different, in the sense that the three PNDs in each possibility may be of different kinds in the refined classification based on the superenergy tensor developed in Appendix A. One can easily see, however, that possibilities i), ii) and iii) are equivalent, and possibilities , iv), v) and vi) are also equivalent between them. The former has two PNDs of Class IIa and one of Class IId 5 , while the latter has two PNDs of Class II (with ∆ 3 = 0) and one PND of Class IIa. Thus, there exist two different types (II , II , II), given by \n(IIa , IIa , IId) and (II ∆ 3 =0 , II ∆ 3 =0 , IIa) . \nGoing back to Eq. (165), when the above possibilities i) - vii) do not hold, there are only two PNDs of Class II. Thus the only remaining question is to discern if there can be an infinite number of extra PNDs (all necessarily of Class I). For this task, it is enough to show an example where this can happen. In particular, assume that all ∆ n = 0, except for ∆ 3 , ∆ 10 . Then, Eq. (165) collapses to simply \n2 /epsilon1 2 (3∆ 3 +4 /epsilon1 ¯ /epsilon1 ∆ 10 ) = 0 , (189) \nwhich has an infinite number of solutions for /epsilon1 if ∆ 3 / ∆ 10 is real and negative, as the phase of /epsilon1 remains free. In consequence, there is another Type B e and the general Type B, or also \n(II , II) I ∞ , (II , II) , \nwhere in the latter case there may be a finite number of extra PNDs of Class I. \nrespectively. \nWith the algebraic classification of the tensor (1) ˜ Z λρµν settled, we display in Table V a summary of all of the algebraic types obtained in this section, while we show their possible degenerations in Figure 1.", 'Types C and C*': 'Types C are defined by having a unique PND of Class II, and this is the maximal alignment for all PNDs. Thus, all other PNDs, if they exist, can only be of Class I. Choosing the null tetrad with l µ along the PND of Class II, Eq. (105) reads \n6 /epsilon1 2 ∆ 3 +8 /epsilon1 ¯ /epsilon1 ∆ 4 +¯ /epsilon1 2 ∆ 5 +4 /epsilon1 3 ∆ 6 +12 /epsilon1 2 ¯ /epsilon1 ∆ 7 +4 /epsilon1 ¯ /epsilon1 2 ∆ 8 + /epsilon1 4 ∆ 9 +8 /epsilon1 3 ¯ /epsilon1 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 4 ¯ /epsilon1 ∆ 12 +4 /epsilon1 3 ¯ /epsilon1 2 ∆ 13 + /epsilon1 4 ¯ /epsilon1 2 ∆ 14 = 0 . (190) \nCases devoid of nontrivial solutions for /epsilon1 are easily found. For instance, set all ∆ n = 0, except for ∆ 3 and ∆ 14 . Then the above equation is reduced to \n/negationslash \n/epsilon1 2 ( 6∆ 3 + /epsilon1 2 ¯ /epsilon1 2 ∆ 14 ) = 0 , (191) \nwithout /epsilon1 = 0 solution if ∆ 3 / ∆ 14 is not real and negative. Thus, such cases lead to Type C* or \n(II , -) . \nOn the other hand, in the same situation, if ∆ 3 / ∆ 14 is real and negative then there exists an infinite number of solutions for /epsilon1 as only the norm | /epsilon1 | is fixed. This leads to Type C e or \n(II , I ∞ ) . \nThe remaining case with a finite number of nontrivial solutions of Eq. (190) are simply called Type C, or denoted by \n(II , I) .', 'Type I': '/negationslash \nOn the other hand, by following the same ideas as in previous types, it is quite straightforward to find cases with an infinite number of extra PNDs, or with only a finite number (an example given by Eq. (114), for the couple of solutions (116)). These are the cases I e and I; namely, \nNow, all possible PNDs are of the basic Class I, and there is at least one of these. Choosing such a PND as l µ in the null tetrad, we have ∆ 0 = 0 with | ∆ 1 | 2 + | ∆ 2 | 2 = 0, and this is the unique restriction on Eq. (105). In this regard, we have been unable to find any possibility with just a unique PND, or with none, and we believe that they do not exist. Hence, Types I* and ∅ , corresponding to (I , -) and to ( -, -), respectively, are missing. \n(I , I ∞ ) , (I , I) , \n/negationslash \n/negationslash \n/negationslash \n/negationslash \nTABLE V: Algebraic types for the tensor (1) ˜ Z λρµν . The complex scalars are shown in the preferred null tetrad chosen such that in general the left and right numerals refer to null vectors l µ and k µ , respectively, while for simplicity in the presentation the extra constraints related to the exceptional cases with infinite PNDs are not shown in the table, but they can be found for each case in Sec. VI A 2. \n/negationslash \n/negationslash \n/negationslash \n/negationslash \n/negationslash \n/negationslash \n/negationslash \nFIG. 1: Flow diagram of the algebraic classification of the tensor (1) ˜ Z λρµν . The null tetrad is chosen as in Table V, while for simplicity in the presentation the extra constraints related to the cases marked with ∗ are not shown in the diagram, but they can be found for each case in Sec. VI A 2. \n<!-- image -->', 'B. Refined classification based on the superenergy tensor': 'As previously stressed, the tensor (1) ˜ Z λρµν is symmetric in the first pair of indices and antisymmetric in the last one, while it additionally fulfils the algebraic symmetries (33) and (34). Therefore, its superenergy tensor is given by (formula (19) in [21], conveniently adapted): \nT αβλµτν ( (1) ˜ Z ) = (1) ˜ Z αλτρ (1) ˜ Z βµν ρ + (1) ˜ Z βλτρ (1) ˜ Z αµν ρ + (1) ˜ Z αµτρ (1) ˜ Z βλν ρ + (1) ˜ Z αλνρ (1) ˜ Z βµτ ρ -g αβ ( (1) ˜ Z σλτρ (1) ˜ Z σ µν ρ + (1) ˜ Z σλνρ (1) ˜ Z σ µτ ρ ) -g λµ ( (1) ˜ Z αστρ (1) ˜ Z β σ ν ρ + (1) ˜ Z ασνρ (1) ˜ Z β σ τ ρ ) -1 2 g τν ( (1) ˜ Z αλσρ (1) ˜ Z βµ σρ + (1) ˜ Z αµσρ (1) ˜ Z βλ σρ ) + g αβ g λµ (1) ˜ Z σγτρ (1) ˜ Z σγ ν ρ + 1 2 g αβ g τν (1) ˜ Z σλγρ (1) ˜ Z σ µ γρ + 1 2 g λµ g τν (1) ˜ Z ασγρ (1) ˜ Z β σγρ -1 4 g αβ g λµ g τν (1) ˜ Z δσγρ (1) ˜ Z δσγρ . (192) \nThis tensor has the following direct properties \nT αβλµτν ( (1) ˜ Z ) = T ( αβ )( λµ )( τν ) ( (1) ˜ Z ) = T λµαβτν ( (1) ˜ Z ) , T αβλµρ ρ ( (1) ˜ Z ) = 0 . (193) \nA more refined algebraic classification of the tensor (1) ˜ Z λρµν can then be achieved by using this superenergy tensor, as outlined in [22, 23]. Any superenergy tensor has the dominant property , meaning in our case that \nT αβλµτν ( (1) ˜ Z ) u α 1 u β 2 u λ 3 u µ 4 u ν 5 u τ 6 ≥ 0 (194) \nfor arbitrary future pointing u µ a ( a ∈ { 1 , 2 , 3 , 4 , 5 , 6 } ), in such a way that the equality can only occur if at least one of the u α a is null. Thereby, one defines the PND of (1) ˜ Z λρµν as the null l µ such that \nT αβλµτν ( (1) ˜ Z ) l α l β l λ l µ l τ l ν = 0 . (195) \nThese PNDs are sometimes called aligned null directions (AND), and the classification using the superenergy tensor (192) is greatly related to the one based on null alignment (see e.g. [20]), as the aligned null directions are the PNDs. It must be stressed that relation (195) is fully equivalent to either (95) or (96). \nThe refined classification simply analyses the level of alignment of any particular PND by finding the actual number of contractions with l µ needed to get the zero on the right-hand side of (195). This is efficiently achieved by removing, in an orderly manner, instances of the given PND from the original equation (195) step by step. For simplicity in the presentation, we derive in detail the aforementioned classification in Appendix A, choosing l µ as the given PND, which allows us to find seventeen different alignment classes. The main results can be summarised in Table VI, while in Figure 2 we show a flow diagram, specifying how all of these classes are related. \nOnce the refined classes have been identified, a more elaborate classification can be achieved. The basic idea is to consider each of the 15 main types and particularise the two (or exceptionally three) Roman numerals to the different possibilities arising in Table VI. The full classification considers all combinations of possibilities derived from that table, is too long but straightforward to get, and thus we will just explain how to derive it by exhibiting illustrative examples. \nThe most obvious refinement arises for Type S, and has already been identified leading to the more specific cases \n(IIa , IIa , IId) and (II ∆ 3 =0 , II ∆ 3 =0 , IIa) . \nTypes N and L* cannot be refined, but Type L can as \n(V , I) , (V , Ia) , (V , Ib) . \nType F, for instance, will lead to 18 subtypes by combining the three classes IV, IVa and IVb with the classes II, IIa, IIb, IIc, IId, and IIe. And the type with more subcases is Type B, with a total of 36 subpossibilities. And so on and so forth. The notation for each case is also obvious. \nTABLE VI: Alignment classes for the tensor (1) ˜ Z λρµν derived from its superenergy tensor. \nFIG. 2: Flow diagram of the alignment classes of the tensor (1) ˜ Z λρµν derived from its superenergy tensor. \n<!-- image --> \n/negationslash', 'VII. Algebraic types of Reissner-Nordstrom-like solutions with dynamical torsion and nonmetricity': "Once the algebraic classification in general metric-affine geometries is clear, it is possible to characterise any solution of the field equations of MAG according to its algebraic types. Hence, we consider Reissner-Nordstrom-like solutions endowed with dynamical torsion and nonmetricity, which in fact represent the broadest family of static and spherically symmetric black hole solutions with spin, dilation and shear charges in MAG. \nThe MAG model associated with the solutions is described by the gravitational action [59]: \nS = 1 64 π ∫ [ -4 R -6 d 1 ˜ R λ [ ρµν ] ˜ R λ [ ρµν ] -9 d 1 ˜ R λ [ ρµν ] ˜ R µ [ λνρ ] +2 d 1 ( ˜ R [ µν ] + ˆ R [ µν ] )( ˜ R [ µν ] + ˆ R [ µν ] ) +18 d 1 ˜ R λ [ ρµν ] ˜ R ( λρ ) µν -3 d 1 ˜ R ( λρ ) µν ˜ R ( λρ ) µν +6 d 1 ˜ R ( λρ ) µν ˜ R ( λµ ) ρν +2(2 e 1 -f 1 ) ˜ R λ λµν ˜ R ρ ρ µν +8 f 1 ˜ R ( λρ ) µν ˜ R ( λρ ) µν -2 f 1 ( ˜ R ( µν ) -ˆ R ( µν ) )( ˜ R ( µν ) -ˆ R ( µν ) ) +3(1 -2 a 2 ) T [ λµν ] T [ λµν ] ] d 4 x √ -g . (196) \nAs can be seen, the model constitutes an extension of GR in the presence of dynamical torsion and nonmetricity, whose field strength tensors are given by deviations from the first and third Bianchi identities of GR. In terms of building blocks of the curvature tensor, the action reads \nS = 1 64 π ∫ [ -4 R -9 d 1 ↗ ˜ R ( T ) λ [ ρµν ] ↗ ˜ R ( T ) λ [ ρµν ] +2 d 1 ˜ R ( T ) [ µν ] ˜ R ( T )[ µν ] -d 1 8 ∗ ˜ R 2 + 1 8 ( d 1 +32 e 1 ) ˜ R λ λµν ˜ R ρ ρ µν +8 f 1 (1) ˜ Z λρµν (1) ˜ Z λρµν + 1 3 (4 f 1 -3 d 1 ) ↗ ˜ R ( Q ) λ [ ρµν ] ↗ ˜ R ( Q ) λ [ ρµν ] + 1 6 (3 d 1 +16 f 1 ) ˆ R ( Q ) [ µν ] ˆ R ( Q )[ µν ] + d 1 ˜ R ( T ) [ µν ] ˜ R λ λ µν +6 d 1 ↗ ˜ R ( T ) λ [ ρµν ] ↗ ˜ R ( Q ) λ [ ρµν ] +2 d 1 ˜ R ( T ) [ µν ] ˆ R ( Q )[ µν ] + 1 2 d 1 ˆ R ( Q ) [ µν ] ˜ R λ λ µν +3(1 -2 a 2 ) T [ λµν ] T [ λµν ] ] d 4 x √ -g . (197) \nThereby, it introduces {↗ ˜ R ( T ) λ [ ρµν ] , ˜ R ( T ) [ µν ] , ∗ ˜ R, ˜ R λ λµν , (1) ˜ Z λρµν , ↗ ˜ R ( Q ) λ [ ρµν ] , ˆ R ( Q ) [ µν ] } as field strength tensors for torsion and nonmetricity, the latter including nontrivial trace and traceless parts. \nBy setting the form of the metric, torsion and nonmetricity tensors relative to a static and spherically symmetric space-time [84]: \nds 2 = Ψ 1 ( r ) dt 2 -dr 2 Ψ 2 ( r ) -r 2 dϑ 2 -r 2 sin 2 ϑdϕ 2 , (198) \nwe can consider null vectors \nl µ = { 1 √ 2 ( Ψ 2 ( r ) Ψ 3 1 ( r ) ) 1 / 4 , -1 √ 2 ( Ψ 3 2 ( r ) Ψ 1 ( r ) ) 1 / 4 , 0 , 0 } , m µ = { 0 , 0 , i √ 2 r , csc ϑ √ 2 r } , (199) 1 / 4 (200) \nk µ = { 1 √ 2(Ψ 1 ( r )Ψ 2 ( r )) 1 / 4 , (Ψ 1 ( r )Ψ 2 ( r )) √ 2 , 0 , 0 } , ¯ m µ = { 0 , 0 , -i √ 2 r , csc ϑ √ 2 r } , \nwhere l µ and k µ correspond to radially ingoing and outgoing null geodesics of the static and spherically symmetric space-time, respectively. Then, given the fact that the tensor (1) ˜ Z λρµν obeys a completely new algebraic classification, in comparison with the rest of the field strength tensors of the model, it is worthwhile to study its algebraic structure in a static and spherically symmetric space-time. \nFirst of all, it turns out that the only nontrivial complex scalars of the tensor (1) ˜ Z λρµν in a general static and spherically symmetric space-time are ∆ 1 , ∆ 7 and ∆ 13 . This immediately tells us that l µ and k µ constitute PNDs of Class I for this tensor, unless some of the mentioned complex scalars vanishes. To see if there are any other PNDs, we simply analyse Eq. (105) for the rotated principal scalar, which is reduced to \n∆ ' 0 = 4 /epsilon1 ( ∆ 1 +3 /epsilon1 ¯ /epsilon1 ∆ 7 + /epsilon1 2 ¯ /epsilon1 2 ∆ 13 ) = 0 . (201) \nSpecifically, there will be no /epsilon1 = 0 solutions, and therefore no further PNDs for the tensor (1) ˜ Z λρµν , unless one of the following conditions hold: \n/negationslash \n± ( 9∆ 2 7 -4∆ 1 ∆ 13 ) 1 / 2 -3∆ 7 2∆ 13 ∈ R + , ∆ 13 = 0 , (202) \n/negationslash \n-∆ 1 ∆ 7 ∈ R + , ∆ 13 = 0 , ∆ 7 = 0 . (203) \nThereby, if neither (202) nor (203) holds, then the tensor (1) ˜ Z λρµν in a general static and spherically symmetric space-time is of Type I, with only two PNDs of Class I (i.e. case (I , I)). \nBy contrast, if (202) holds, there exist infinite nontrivial solutions of Eq. (105), where the modulus | /epsilon1 | is fixed but the phase remains arbitrary. In this case, on top of the PNDs l µ and k µ associated with the trivial solution /epsilon1 = 0, there is then an infinite number of different PNDs. In general, all of them will be of Class I, thus leading to a Type I e of the kind \n(I , I ∞ ) , \nunless the further constraint \n9∆ 2 7 -4∆ 1 ∆ 13 = 0 , -∆ 1 ∆ 7 ∈ R + , (204) \nis satisfied, in which case there is double solution for the norm \n| /epsilon1 | 2 = -2∆ 1 3∆ 7 = -3∆ 7 2∆ 13 . (205) \nIn this particular case, it is straightforward to check by formulas (80)-(94) that the above value of | /epsilon1 | implies \n∆ ' 0 = ∆ ' 1 = ∆ ' 2 = 0 , ∆ ' 3 = -¯ /epsilon1 ∆ 7 = 0 , (206) \n/negationslash \nwhich means that the infinite PNDs are of Class II. Thus, in this case the tensor (1) ˜ Z λρµν is of Type B e , version \n(II , II ∞ ) ≡ (II ∞ ) , \nwith two extra PNDs of Class I. \n/negationslash \nOn the other hand, if (203) holds, then ∆ ' 8 = ∆ ' 9 = ∆ ' 10 = ∆ ' 11 = ∆ ' 12 = ∆ ' 13 = ∆ ' 14 = 0 , ∆ ' 7 = 0, which means that k µ is a PND of Class III, and for any value of /epsilon1 such that \n| /epsilon1 | = + √ -∆ 1 3∆ 7 , (207) \nthen l ' µ defines an infinite number of extra PNDs of Class I. This is a Type K e , or \n(III , I ∞ ) . \nIf ∆ 13 = 0, but (203) does not hold, then the PND k µ is of Class III, and the only different PND is l µ . In this case, the tensor (1) ˜ Z λρµν is of Type K, or \n(III , I) . \nFinally, if ∆ 1 = ∆ 13 = 0, there are no PNDs different from l µ and k µ , but both of them are of Class III, leading to Type D, or \n(III , III) , \nwhereas, if ∆ 7 = ∆ 13 = 0, then k µ is actually of Class V and the tensor (1) ˜ Z λρµν becomes Type L, that is \n(V , I) . \nOnce the algebraic structure of the tensor (1) ˜ Z λρµν in a static and spherically symmetric space-time is clear, it is then straightforward to determine its algebraic type for the Reissner-Nordstrom-like solutions of the model. In this case, the metric functions read \nΨ( r ) ≡ Ψ 1 ( r ) = Ψ 2 ( r ) = 1 -2 m r + d 1 κ 2 s -4 e 1 κ 2 d -2 f 1 κ 2 sh r 2 , (208) \nand \nwhere, on top of the mass m , the constants κ s , κ d and κ sh represent the spin, dilation and shear charges of the solution. On the other hand, the complex scalar ∆ 13 vanishes, whereas ∆ 1 and ∆ 7 acquire the following values: \n/negationslash \n∆ 1 = -iκ s [ 2 κ sh d 1 + c 2 ( d 1 -8 f 1 ) r +2 c 3 ( d 1 -8 f 1 ) r -( d 1 -8 f 1 ) ( d 1 +8 f 1 ) ] 2 ( d 1 -8 f 1 ) r 2 Ψ( r ) , if d 1 = ± 8 f 1 ; -iκ s [ κ sh (1 + log ( r )) + c 2 r +2 c 3 ] 2 r 2 Ψ( r ) , if d 1 = 8 f 1 ; -iκ s ( κ sh + c 2 r ) 2 r 2 Ψ( r ) , if d 1 = -8 f 1 ; (209) \n∆ 7 = κ sh 6 r 2 , ∀ d 1 , f 1 ∈ R . (210) \nTherefore, the algebraic type of the tensor (1) ˜ Z λρµν for the Reissner-Nordstrom-like solutions is Type K e = (III , I ∞ ), except at the points where the complex scalar ∆ 1 in Expression (209) vanishes; at those points, the algebraic type becomes Type D. Similarly, if the spin charge κ s vanishes, then ∆ 1 = 0 and the algebraic type is always Type D, provided that the shear charge is nonzero. For a vanishing shear charge, but nonzero spin charge, the complex scalar ∆ 7 vanishes and the algebraic type is Type L = (V , I), except at the points where ∆ 7 also vanishes, which corresponds to the trivial Type O. \nIn addition, for the Reissner-Nordstrom-like solutions, the Riemannian Weyl and traceless Ricci tensors fulfil the constraints \n(1) W λρµ [ ν k ω ] k ρ k µ = (1) W λρµ [ ν l ω ] l ρ l µ = 0 , (211) \nU (3) ∗ = V (3) ∗ = 0 , W (3) ∗ = 64 ( d 1 κ 2 s -4 e 1 κ 2 d -2 f 1 κ 2 sh ) 4 r 16 , (212) \ndescribing, respectively, algebraic types [(1 1) 1] and [(1 , 1) (1 1)], since the traceless Ricci tensor can be described by a diagonal matrix with two eigenvalues λ ± = ± ( d 1 κ 2 s -4 e 1 κ 2 d -2 f 1 κ 2 sh ) /r 4 and four eigenvectors { (1 , 0 , 0 , 0) , (0 , 1 , 0 , 0) , (0 , 0 , 1 , 0) , (0 , 0 , 0 , 1) } . \nFurthermore, for the field strength tensors ↗ ˜ R ( T ) λ [ ρµν ] and ↗ ˜ R ( Q ) λ [ ρµν ] , we have \n˜ U (1) ∗ = ˜ V (1) ∗ = 0 , ˜ W (1) ∗ = 1024 κ 4 sh 81 r 8 , (213) \n˜ U (2) ∗ = ˜ V (2) ∗ = ˜ W (2) ∗ = 0 , (214) \nleading to algebraic types [2 (1 1)] and [(2 1 1)], respectively, since the former is characterised by two eigenvalues λ ± = ± 2 κ s / (3 r 2 ) and the latter only by λ = 0, but both of them give rise to three eigenvectors { (1 , 1 , 0 , 0) , (0 , 0 , 1 , 0) , (0 , 0 , 0 , 1) } . \nFinally, the field strength tensors ˜ R ( T ) [ µν ] , ˜ R λ λµν and ˆ R ( Q ) [ µν ] satisfy \n( ˜ R ( T ) [ µν ] l λ -˜ R ( T ) [ µλ ] l ν ) l µ = ( ˜ R ( T ) [ µν ] k λ -˜ R ( T ) [ µλ ] k ν ) k µ = 0 , (215) \n( ˜ R ρ ρµν l λ -˜ R ρ ρµλ l ν ) l µ = ( ˜ R ρ ρµν k λ -˜ R ρ ρµλ k ν ) k µ = 0 , (217) \n( ˆ R ( Q ) [ µν ] l λ -ˆ R ( Q ) [ µλ ] l ν ) l µ = ( ˆ R ( Q ) [ µν ] k λ -ˆ R ( Q ) [ µλ ] k ν ) k µ = 0 , (216) \nso that they are doubly aligned with the PNDs l µ and k µ .", 'VIII. Conclusions': 'In this work, we have derived the algebraic classification of the gravitational field in general metric-affine geometries, which are characterised by the presence of curvature, torsion and nonmetricity. For this task, we have considered \nthe irreducible decomposition of the curvature tensor under the pseudo-orthogonal group, which in general displays eleven fundamental parts: three of them constituting the generalisations of the Ricci scalar and of the Weyl and Ricci tensors in metric-affine geometry, as well as eight additional quantities that represent field strength tensors for torsion and nonmetricity. Thereby, a study on the algebraic structure of all of these quantities has a relevant interest in the search and analysis of solutions of the field equations of MAG, which in turn can describe a wide variety of systems, such as black holes and stars with intrinsic hypermomentum, gravitational waves and cosmological scenarios. \nTaking into account the algebraic symmetries of the eleven fundamental parts of the curvature tensor, they can be sorted into four different categories, each one characterised by its own type of algebraic classification. Specifically, three of these categories match the well-known algebraic classifications of the Weyl, Ricci and Faraday tensors (see Tables II, III and IV), whereas the last one is related to one of the field strengths of the traceless nonmetricity tensor and provides a completely new algebraic classification. Then, we formally classify this quantity by means of its PNDs and their levels of alignment, finding a total of sixteen algebraic types, whose main properties and possible degenerations are shown in Table V and Figure 1. In fact, as pointed out in [21], several refinements can also arise when establishing the alignment classes of the PNDs from the superenergy tensor of this quantity, which are displayed in detail in Table VI and Figure 2. \nAs an immediate application, we determine the algebraic types for the Reissner-Nordstrom-like solutions of MAG, showing that indeed the aforementioned field strength of the traceless nonmetricity tensor presents a rich algebraic structure, in contrast with the Riemannian Weyl and Ricci tensors, as well as with the rest of field strenghts of the torsion and nonmetricity tensors of the solution. In any case, despite of the complexity of the solution, the gravitational field turns out to be algebraically special, which could be relevant to address the corresponding extension to stationary and axisymmetric space-times, by providing a significant simplification of the field equations of the model in such space-times. Further research in this direction will be addressed in future works.', 'Acknowledgements': "This research was initiated during a visit by JMMS to the Tokyo Institute of Technology. The authors would like to thank Mart'ın Sombra for helpful discussions. S.B. is supported by 'Agencia Nacional de Investigaci'on y Desarrollo' (ANID), Grant 'Becas Chile postdoctorado al extranjero' No. 74220006. The work of J.G.V. is supported by the Institute for Basic Science (IBS-R003-D1). J.G.V. also acknowledges the JSPS Postdoctoral Fellowships for Research in Japan and KAKENHI Grant-in-Aid for Scientific Research No. JP22F22044. JMMS is supported by Basque Government grant IT1628-22, and by Grant PID2021-123226NB-I00 funded by the Spanish MCIN/AEI/10.13039/501100011033 together with 'ERDF A way of making Europe'.", 'Appendix A. Explicit computations of the alignment classes of (1) ˜ Z λρµν based on its superenergy tensor': 'In this appendix, we carry out all the computations for the alignment classification of the tensor (1) ˜ Z λρµν using its superenergy tensor (192). In general, the main alignment classes based on this method arise by considering all the possible contractions of the null vector l µ and the tensor (1) ˜ Z λρµν . For this reason, we shall divide the presentation into six different subsections.', '1. Contraction of T αβλµτν ( (1) ˜ Z ) with six copies of l µ : Class I': "The first possible contraction is the superenergy tensor contracted with 6 copies of l µ , which simply becomes \nleading to \nT αβλµτν l α l β l λ l µ l ν l τ = 4 ( (1) ˜ Z αλτρ l α l λ l τ )( (1) ˜ Z βµν ρ l β l µ l ν ) = -8∆ 0 ¯ ∆ 0 = 0 , (A1) \n∆ 0 = 0 . (A2) \nThen, if such a PND exists, the tensor (1) ˜ Z λρµν is said to be of Class I . For this case, the maximum bo ( l ) is 2. \nLet us analyse Eq. (A1) further. This condition implies that the vector (1) ˜ Z βµτ ρ l β l µ l τ is null, and as it is also orthogonal to l ρ , it must be proportional to it yielding (95). Conversely, in general one has \nso that the combination \n(1) ˜ Z λρµν l λ l ρ l µ = ( ∆ 1 + ¯ ∆ 1 ) l ν -¯ ∆ 0 m ν -∆ 0 ¯ m ν . (A3) \n(1) ˜ Z λρµ [ ν l σ ] l λ l ρ l µ = ∆ 0 l [ ν ¯ m σ ] + ¯ ∆ 0 l [ ν m σ ] = 0 (A4) \nis equivalent to Eq. (A1) or to (A2) and represents the intrinsic characterisation of PND for the tensor (1) ˜ Z λρµν . µ µ \nBy the 'symmetry' mentioned in Sec. (VI A) between the null vectors l and k , one immediately knows that \n∆ 14 = 0 , (A5) \nis the corresponding characterisation for k µ to be a PND, that is to say \n(1) ˜ Z λρµ [ ν k σ ] k λ k ρ k µ = 0 . (A6)", '2. Contraction of T αβλµτν ( (1) ˜ Z ) with five copies of l µ : Class Ia and Class II': 'The next step consists of removing one null vector l µ from Expression (195). By doing that, there are two independent possibilities, which we shall explain and categorise separately. \na. Class Ia \nThe first possible contraction is \nT αβλµτν l α l β l λ l µ l τ = -l α l β l λ l µ l ν (1) ˜ Z αβ τσ (1) ˜ Z λµτσ +4 l α l τ l β l λ l µ (1) ˜ Z αβνσ (1) ˜ Z λµτ σ = 0 , (A7) \nfrom where one can notice that ∆ 0 = 0 (by contracting it with l ν ). Then, by assuming this, the above expression becomes \nT αβλµτν l α l β l λ l µ l τ = -8∆ 1 ¯ ∆ 1 l ν = 0 . (A8) \nTherefore, this case (even though it does not imply that l µ is a multiple PND) will be labeled as Class Ia and is equivalent to having \n∆ 0 = ∆ 1 = 0 , (A9) \n¯ \nm \nµ \n, \n(A19) \nand f (2) σρ is a tensor depending on ∆ 0 , ∆ 1 , ∆ 2 and their conjugates. Then, from (A18) one can intrinsically write the equivalent form of (A17) as \nP µ = ( ∆ 7 + ¯ ∆ 7 ) l µ -¯ ∆ 4 m µ -∆ 4 Q µ = ( ∆ 7 + ¯ ∆ 7 ) l µ -( ∆ 3 + ¯ ∆ 4 ) m µ -( ¯ ∆ 3 +∆ 4 ) ¯ m µ , (A20) \nl λ l [ α (1) ˜ Z σ ] λτ [ ρ l β ] l τ = ∆ 0 l [ α k σ ] l [ ρ ¯ m β ] +∆ 1 m [ α l σ ] l [ ρ ¯ m β ] +∆ 2 ¯ m [ α l σ ] l [ ρ ¯ m β ] +c.c. = 0 , (A21) \nwhere c.c. stands for complex conjugate.', '3. Contraction of T αβλµτν ( (1) ˜ Z ) with four copies of l µ : Class Ib, Class IIa, Class IIb and Class III': 'The next step consists of removing two null vectors l µ from Expression (195). In that case, there are four independent possibilities which again we shall explain and categorise separately. Notice that, for all of these cases, the condition ∆ 0 = ∆ 1 = 0 always holds. This can be straightforwardly seen by taking all the possible independent contractions with four copies and contracting them with l µ . \nmeaning that the corresponding bo is 2. Since Eq. (A7) includes the specific contraction (1) ˜ Z αλσρ l α l λ , it is advantageous to compute the explicit dependence on ∆ 3 for this term using the representation of the tensor (1) ˜ Z λρµν in terms of complex scalars and null vectors given by (63): \n(1) ˜ Z αλσρ l α l λ = ( l σ u ρ -l ρ u σ ) + f (1) σρ , u ρ = ∆ 3 m ρ + ¯ ∆ 3 ¯ m ρ , l σ u σ = 0 , (A10) \nwhere f (1) σρ is a tensor depending on ∆ 0 , ∆ 1 and their conjugates. Then, it turns out that the further contractions (1) ˜ Z βµνρ l β l µ l ν and (1) ˜ Z αλ [ σρ l β ] l α l λ depend solely on these two scalars as \n(1) ˜ Z βµνρ l β l µ l ν = ( ∆ 1 + ¯ ∆ 1 ) l ρ -¯ ∆ 0 m ρ -∆ 0 ¯ m ρ , (A11) \nin such a way that Expression (A9) is equivalent to vanishing these two independent contractions. \n(1) ˜ Z αλ [ σρ l β ] l α l λ = -2∆ 0 l [ σ k ρ ¯ m β ] -2 ¯ ∆ 0 l [ σ k ρ m β ] +2 ( ∆ 1 -¯ ∆ 1 ) l [ σ m ρ ¯ m β ] , (A12) \nTherefore, the intrinsic characterisation of this class is simply given by the constraints \n(1) \n˜ \nZ \nβµνρ \nl \nβ \nl \nµ \nl \nν \n= 0 \n, \n(A13) \n(1) ˜ Z αλ [ σρ l β ] l α l λ = 0 . (A14) \nb. Class II \nThe second possible contraction with five copies of l µ is \nT αβλµτν l β l λ l µ l τ l ν = 4 l τ l σ l β l λ l µ (1) ˜ Z αβλω (1) ˜ Z µτσ ω -2 l α l τ l β l λ l µ (1) ˜ Z β σ λ ω (1) ˜ Z µστω = 0 , (A15) \nwhere by contracting it with l α one gets ∆ 0 = 0 and then the above expression becomes \nT αβλµτν l β l λ l µ l τ l ν = -2 l α (1) ˜ Z σλτρ l λ l τ (1) ˜ Z σ µν ρ l µ l ν = -4 ( ∆ 1 ¯ ∆ 1 +∆ 2 ¯ ∆ 2 ) l α = 0 , (A16) \nwhich clearly means ∆ 1 = ∆ 2 = 0. This is the second intermediate case of Type I, and now it does state the multiplicity of the PND l µ and the maximum bo ( l ) is 1. We will call this Class II and its ∆-scalar characterisation would then read as \n∆ 0 = ∆ 1 = ∆ 2 = 0 . (A17) \nNow, by using Expression (63) and assuming (A17), one finds \n(1) ˜ Z σλτρ l λ l τ = l σ P ρ + Q σ l ρ + f (2) σρ , l ρ P ρ = 0 , l σ Q σ = 0 , (A18) \nwhere \nThe first contraction with four copies of l µ is \nT αβλµτν l α l β l λ l µ = -( l α l β (1) ˜ Z αβ ρσ )( l λ l µ (1) ˜ Z λµρσ ) g ντ +4 ( l α l β (1) ˜ Z αβνρ )( l λ l µ (1) ˜ Z λµτ ρ ) = 0 , (A22) \nfrom where one easily notices that ∆ 0 = ∆ 1 = 0 by contracting it with l τ l ν and l τ , respectively. By using those conditions, we arrive at \nT αβλµτν l α l β l λ l µ = -8∆ 3 ¯ ∆ 3 l ν l τ = 0 . (A23) \nThus, this case requires \n∆ 0 = ∆ 1 = ∆ 3 = 0 , (A24) \nand it will be labelled as Class Ib . The maximum bo ( l ) (boost order of l \n) is 2 once again. \nFrom Eq. (A22), one notices that the intrinsic characterisation of this case just simplifies as \n(1) ˜ Z αλσρ l α l λ = -2∆ 0 k [ σ ¯ m ρ ] +2∆ 1 ( k [ σ l ρ ] + m [ σ ¯ m ρ ] ) + 2∆ 3 l [ σ m ρ ] +c.c. = 0 , (A25) \nwhich is equivalent to the condition (A24). \nb. Class IIa \nThe next possible contraction with four copies of l µ reads \nT αβλµτν l α l β l τ l ν = -2 ( l α l β (1) ˜ Z ω αβ ν )( l ρ l σ (1) ˜ Z ωρσν ) g λµ +4 ( l α l β (1) ˜ Z λαβω )( l ρ l σ (1) ˜ Z µρσ ω ) = 0 , (A26) \nwhere again we notice that ∆ 0 = 0 (by contracting it with l λ l µ ) and also ∆ 1 = ∆ 2 = 0 (by contracting it with l µ ). With these conditions, the above equation reduces to \nT αβλµτν l α l β l τ l ν = -8∆ 4 ¯ ∆ 4 l λ l µ = 0 . (A27) \nThen, putting all of the conditions together, we find that \n∆ \n0 \n= ∆ \n1 \n= ∆ \n2 \n= ∆ \n4 \n= 0 \n. \n(A28) \nWe will denote this case as Class IIa . For this case the maximum bo ( l ) is 1. \nFrom Eq. (A26), we notice that the quantity needed to characterise the previous conditions intrinsically is \n(1) ˜ Z σλτρ l λ l τ = J σ l ρ + f (3) σρ , l ρ J ρ = 0 , (A29) \nwhere \nJ µ = ( ∆ 7 + ¯ ∆ 7 ) l µ -∆ 3 m µ -¯ ∆ 3 ¯ m µ , (A30) \nand f (3) σρ is a tensor depending on ∆ 0 , ∆ 1 , ∆ 2 , ∆ 4 and their conjugates. Then, from Expression (A29), we find the intrinsic characterisation \nl λ l τ (1) ˜ Z σλτ [ ρ l β ] = -∆ 0 k σ ¯ m [ ρ l β ] +∆ 1 m σ ¯ m [ ρ l β ] -∆ 2 ¯ m σ l [ ρ ¯ m β ] -∆ 4 l σ ¯ m [ ρ l β ] +c.c. = 0 , (A31) \nwhich is equivalent to the condition (A28). \nc. Class IIb \nThe next possibility is given by \nT αβλµτν l α l β l λ l τ = l α l β ( 1 2 l µ l ν (1) ˜ Z α λωρ (1) ˜ Z βλωρ -2 l λ l µ (1) ˜ Z αωνρ (1) ˜ Z β ω λ ρ -l λ l ν (1) ˜ Z βλ ωρ (1) ˜ Z µαωρ \n+2 l ω l λ (1) ˜ Z λων ρ (1) ˜ Z µαβρ +2 l ω l λ (1) ˜ Z βλω ρ (1) ˜ Z µανρ ) = 0 . (A32) \nBy contracting this expression with l µ l ν , one notices that ∆ 0 = 0, whereas by contracting it with l µ and l ν , one finds ∆ 1 = 0 and ∆ 2 = 0, respectively. Then, by assuming these three conditions, the expression becomes \nT αβλµτν l α l β l λ l τ = -4 ( ∆ 3 ¯ ∆ 3 +∆ 4 ¯ ∆ 4 ) l µ l ν = 0 , (A33) \nwhere one notices that ∆ 3 = ∆ 4 = 0. Hence, the corresponding ∆-scalar version of Eq. (A32) reads \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0 , (A34) \nand we will labelled this case as Class IIb . The maximum bo ( l ) is 1 now. \nLet us now find the intrinsic characterisation of this case. From Eq. (A32), we notice that we need to write down the quantity l α (1) ˜ Z αβλµ . Thus, by considering as in the previous cases the explicit form (63) for the tensor (1) ˜ Z λρµν , we find \nl α (1) ˜ Z αβλµ = Al β ( l λ k µ -l µ k λ ) + l β ( l λ R µ -l µ R λ ) + 2 l β h [ λµ ] + h βµ l λ -h βλ l µ + f (1) βλµ , (A35) \nwhere \nh µν = -∆ 6 m µ m ν -∆ 7 m ν ¯ m µ +c.c. , R µ = ∆ 10 m µ +c.c. , A = -( ∆ 7 + ¯ ∆ 7 ) . (A36) \nare two quantities fully orthogonal to both l µ and k µ , while f (1) βλµ is a tensor depending on the complex scalars ∆ 0 , ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 and their conjugates. Then, the following contractions solely depend on the aforementioned scalars and provide the intrinsic characterisation of this case: \nl \nl α l β (1) ˜ Z αβµν = -2∆ 0 k [ µ ¯ m ν ] +2∆ 1 ( k [ µ l ν ] -¯ m [ µ m ν ] ) -2∆ 3 m [ µ l ν ] +c.c. = 0 , (A37) (1) ˜ \n[ \nγ \nl \nα \nZ \nβ \n] \nα \n[ \nλµ \nl \nν \n] \n= 2∆ \nwhich together are therefore equivalent to Expression (A34). \n0 k [ γ l β ] ¯ m [ λ l µ k ν ] +2∆ 1 ( l [ γ m β ] ¯ m [ λ l -) -+ ( ∆ 3 -¯ ∆ 4 ) l [ γ m β ] ¯ m [ λ m µ l ν ] + ( ¯ ∆ 3 -∆ 4 ) l [ γ ¯ m β ] m [ λ ¯ m µ l ν ] +c.c. = 0 , (A38)', 'd. Class III': 'The final possibility with four copies of l µ is defined by \nT αβλµτν l α l λ l τ l ν = l α ( 2 l ϕ l λ l µ (1) ˜ Z (1) βρωα ˜ Z λ ρ ϕ ω -l β l λ l µ (1) ˜ Z ϕρω α (1) ˜ Z ϕρλω -2 l ϕ l ρ l λ (1) ˜ Z βµωα (1) ˜ Z λϕρ ω +2 l ϕ l β l λ (1) ˜ Z µρωα (1) ˜ Z λ ρ ϕ ω -2 l ϕ l ρ l ω (1) ˜ Z βϕλα (1) ˜ Z µρω λ +2 l ϕ l ρ l ω (1) ˜ Z βλϕα (1) ˜ Z µρω λ ) = 0 . (A39) \nIt is easy to see that, by contracting the above expression with l µ l β , one finds ∆ 0 = 0. Then, by contracting it with l µ , one gets ∆ 1 = ∆ 2 = 0. Thereby, by replacing these three conditions in Eq. (A39), we arrive at \nT αβλµτν l α l λ l τ l ν = -2 ( ∆ 3 ¯ ∆ 3 +2∆ 4 ¯ ∆ 4 +∆ 5 ¯ ∆ 5 ) l β l µ = 0 , (A40) \nmeaning that ∆ 3 = ∆ 4 = ∆ 5 = 0. Putting all together, this case is represented by \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = 0 . (A41) \nThe maximum bo ( l ) is now zero, so this will mean Class III in the terminology of ANDs. \nBy looking into Eq. (A39), one notices that we need to compute the form of l τ (1) ˜ Z σγρτ , in order to find the intrinsic characterisation of this case: \nwhere \nl µ (1) ˜ Z αβλµ = l λ [ A ( l α k β + l β k α ) + c αβ + l α v β + l β v α ] + l α y βλ + l β y αλ + f (2) αβλ , (A42) \nc µν = -( ∆ 6 + ¯ ∆ 8 ) m µ m ν -( ∆ 7 + ¯ ∆ 7 ) m ν ¯ m µ +c.c. , (A43) \nµ \nk \nν \n] \nl \n[ \nγ \nk \nβ \n] \nm \n¯ \n[ \nλ \nm \nµ \nl \nν \n] \n2∆ \n2 \nl \n[ \nγ \n¯ \nm \nβ \n] \n¯ \nm \n[ \nλ \nk \nµ \nl \nν \n] \ny µν = -∆ 7 m µ ¯ m ν -∆ 8 ¯ m µ ¯ m ν + 1 2 ∆ 11 l µ ¯ m ν +c.c. , (A44) \nv µ = ( ∆ 10 + ¯ ∆ 11 ) m µ -1 2 ∆ 13 l µ +c.c. , (A45) \nare three quantities fully orthogonal to l µ and k µ , A is given by Expression (A36) and f (2) αβλ is a tensor depending on the complex scalars ∆ 0 , ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 and their conjugates. Then, it turns out that the intrinsic characterisation of this case is given by \nl [ γ l [ δ (1) ˜ Z α ] β ] λ [ µ l λ l ν ] = -∆ 0 k [ γ l α ] k [ δ l β ] ¯ m [ µ l ν ] +∆ 1 m [ γ l α ] k [ δ l β ] ¯ m [ µ l ν ] -∆ 1 k [ γ l α ] l [ δ m β ] ¯ m [ µ l ν ] +∆ 2 ¯ m [ γ l α ] k [ δ l β ] ¯ m [ µ l ν ] -∆ 2 k [ γ l α ] l [ δ ¯ m β ] ¯ m [ µ l ν ] -∆ 3 l [ δ m β ] l [ γ m α ] ¯ m [ µ l ν ] -∆ 4 l [ γ ¯ m α ] l [ δ m β ] ¯ m [ µ l ν ] -∆ 4 l [ γ m α ] l [ δ ¯ m β ] ¯ m [ µ l ν ] -∆ 5 l [ γ ¯ m α ] l [ δ ¯ m β ] ¯ m [ µ l ν ] +c.c. = 0 , (A46) \nwhich is equivalent to the condition (A41).', '4. Contraction of T αβλµτν ( (1) ˜ Z ) with three copies of l µ : Class IIc, Class IId, Class IIIa and Class IV': 'We remove now three null vectors l µ from Expression (195) and, by doing that, there are four independent possibilities that we shall explain and categorise separately. Notice that all of these cases always satisfy ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0 plus other extra conditions. \na. Class IIc \nThe first possibility with three copies of l µ is \nT αβλµτν l α l β l τ = l α ( 1 2 l β l ν g λµ (1) ˜ Z α σρω (1) ˜ Z βσρω -2 l σ l β g λµ (1) ˜ Z αρνω (1) ˜ Z β ρ σ ω +2 l σ l β (1) ˜ Z αµνρ (1) ˜ Z λβσ ρ +2 l σ l β (1) ˜ Z αλνρ (1) ˜ Z µβσ ρ -l β l ν (1) ˜ Z αλσρ (1) ˜ Z µβ σρ ) = 0 . (A47) \nBy contracting this equation with l λ l µ , l µ l ν and l λ , gives ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0, respectively. Putting all together, the above condition is just \n-8∆ 7 ¯ ∆ 7 l λ l µ l ν = 0 , (A48) \nwhich implies the following ∆ characterisation: \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 7 = 0 . (A49) \nThus, the maximum bo ( l ) is again 1. We will labelled this as Class IIc . \nBy looking into Eq. (A47), one immediately notices that the contraction l α (1) ˜ Z αβλµ obtained in (A35) is also important to find the intrinsic characterisation of this case, which arises by the further contraction \nl α (1) ˜ Z αλβµ l β = -∆ 0 k λ ¯ m µ +∆ 1 ( k λ l µ + m λ ¯ m µ ) +∆ 2 ¯ m λ ¯ m µ -∆ 3 l µ m λ -2∆ 4 ¯ m ( λ l µ ) + 1 2 ( ∆ 7 + ¯ ∆ 7 ) l λ l µ +c.c. = 0 , (A50) \nand \nl α (1) ˜ Z αλ [ βµ l σ ] = -2∆ 0 k λ ¯ m [ β l µ k σ ] -2∆ 1 ( m λ ¯ m [ β k µ l σ ] + k λ ¯ m [ β m µ l σ ] ) +2∆ 2 ¯ m λ ¯ m [ β l µ k σ ] +2∆ 3 m λ ¯ m [ β m µ l σ ] -2∆ 4 ( l λ ¯ m [ β l µ k σ ] -¯ m λ ¯ m [ β m µ l σ ] ) + ( ¯ ∆ 7 -∆ 7 ) l λ ¯ m [ β m µ l σ ] +c.c. = 0 , (A51) \nboth representing an equivalent result to the condition (A49). \nThe second case with three null vectors l µ is \nT αβλµτν l α l β l λ = l α ( 1 2 l β l µ g ντ (1) ˜ Z α λϕι (1) ˜ Z βλϕι -2 l β l µ (1) ˜ Z αλνϕ (1) ˜ Z β λ τ ϕ -l β l λ g ντ (1) (1) ˜ Z αµϕι ˜ Z βλ ϕι +2 l β l λ (1) ˜ Z αµνϕ (1) ˜ Z βλτ ϕ +2 l β l λ (1) ˜ Z αµτϕ (1) ˜ Z βλν ϕ ) = 0 . (A52) \nIt is easy to see from this equation that ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0, since its contractions with l µ l τ l ν , l µ l τ , l ν l τ , l µ and l τ , respectively, give rise to such conditions. Then, Eq. (A52) simply provides \nT αβλµτν l α l β l λ = -4 ( ∆ 6 ¯ ∆ 6 +∆ 7 ¯ ∆ 7 ) l µ l ν l τ = 0 , (A53) \nand therefore the ∆-characterisation reads: \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 6 = ∆ 7 = 0 . (A54) \nThe maximum bo ( l ) is again 1 and we will label this as Class IId . \nOnce again, it is clear from Eq. (A52) that the contraction l α (1) ˜ Z αβλµ is essential to find an intrinsic characterisation for this case, which turns out to read \nl α l [ ρ (1) ˜ Z α λ ] βµ = 2∆ 0 l [ λ k ρ ] k [ β ¯ m µ ] +2∆ 1 ( l [ λ k ρ ] l [ β k µ ] -m [ λ l ρ ] ¯ m [ β k µ ] + l [ λ k ρ ] ¯ m [ β m µ ] ) -2∆ 2 ¯ m [ λ l ρ ] ¯ m [ β k µ ] +2∆ 3 ( l [ λ k ρ ] m [ β l µ ] -m [ λ l ρ ] k [ β l µ ] -l [ λ m ρ ] ¯ m [ β m µ ] ) +2∆ 4 ( ¯ m [ λ l ρ ] l [ β k µ ] + ¯ m [ λ l ρ ] ¯ m [ β m µ ] ) +2∆ 6 m [ λ l ρ ] m [ β l µ ] -2∆ 7 ¯ m [ λ l ρ ] l [ β m µ ] +c.c. = 0 , (A55) \nand, as expected, is equivalent to the condition (A54). \nc. Class IIIa \nThe third possible contraction with three copies of l µ is \nT αβλµτν l λ l τ l ν = l λ ( 2 l σ l µ (1) ˜ Z ανωλ (1) ˜ Z β ν σ ω -l σ l µ g αβ (1) ˜ Z νωρ λ (1) ˜ Z νωσρ -2 l σ l ν (1) ˜ Z αµωλ (1) ˜ Z βσν ω -2 l σ l ν (1) ˜ Z ασν ω (1) ˜ Z βµωλ ) +2 l σ l ν l λ g αβ (1) ˜ Z µωρλ (1) ˜ Z σ ω ν ρ = 0 . (A56) \nThen, by contracting the above equation with l α l β l µ , l α l β and l α , one finds ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = 0. By using those conditions, the above equation is reduced to \nfrom where we conclude \nT αβλµτν l λ l τ l ν = -4 ( ∆ 7 ¯ ∆ 7 +∆ 8 ¯ ∆ 8 ) l α l β l µ = 0 , (A57) \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 7 = ∆ 8 = 0 . (A58) \nThe maximum bo ( l ) is then zero and we will name this case as Class IIIa . \nFrom Eq. (A56), we notice that this time the contraction l µ (1) ˜ Z αβλµ obtained in (A42) is important to provide an intrinsic characterisation for this case \nl [ γ (1) ˜ Z α ] βλ [ µ l λ l ν ] = ∆ 0 k [ γ l α ] k β ¯ m [ µ l ν ] -∆ 1 ( k [ γ l α ] m β ¯ m [ µ l ν ] -l [ γ m α ] k β ¯ m [ µ l ν ] ) +∆ 2 ( l [ γ k α ] ¯ m β ¯ m [ µ l ν ] -¯ m [ γ l α ] k β ¯ m [ µ l ν ] ) +∆ 3 m [ γ l α ] m β ¯ m [ µ l ν ] -∆ 4 ( l [ γ k α ] l β ¯ m [ µ l ν ] + l [ γ ¯ m α ] m β ¯ m [ µ l ν ] + l [ γ m α ] ¯ m β ¯ m [ µ l ν ] ) +∆ 5 l [ γ ¯ m α ] ¯ m β l [ µ ¯ m ν ] -∆ 7 m [ γ l α ] l β ¯ m [ µ l ν ] +∆ 8 l [ γ ¯ m α ] l β ¯ m [ µ l ν ] +c.c. = 0 , (A59) \nwhich is equivalent to the condition (A58). \nThe last possibility with three copies of l µ is \nT αβλµτν l α l λ l τ = -1 4 l β l µ l ν (1) ˜ Z αλστ (1) ˜ Z αλστ + l α ( -l σ l λ (1) ˜ Z βµτα (1) ˜ Z λσν τ + l λ l µ (1) ˜ Z βστα (1) ˜ Z λ σ ν τ -l β l µ (1) ˜ Z λστ α (1) ˜ Z λσντ + l β l λ (1) ˜ Z µστα (1) ˜ Z λ σ ν τ ) + l α ( -l λ l µ (1) ˜ Z α σ λ τ (1) ˜ Z βσντ + 1 2 l µ l ν (1) ˜ Z α λστ (1) ˜ Z βλστ -1 2 l λ l ν (1) ˜ Z αλ στ (1) ˜ Z βµστ + l σ l λ (1) ˜ Z αλσ τ (1) ˜ Z βµντ + l σ l λ (1) ˜ Z αµντ (1) ˜ Z βλσ τ -l β l λ (1) ˜ Z α σ λ τ (1) ˜ Z µσντ + 1 2 l β l ν (1) ˜ Z α λστ (1) ˜ Z µλστ + l σ l λ (1) ˜ Z αβντ (1) ˜ Z µλσ τ -1 2 l λ l ν (1) ˜ Z αβστ (1) ˜ Z µλ στ ) = 0 . (A60) \nIt is easy to see, by contracting this equation with l β l µ l ν , l β l µ , l β l ν , l β , l ν , that ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = 0, respectively. Then, the equation acquires the simple form \nT αβλµτν l α l λ l τ = -2 ( ∆ 6 ¯ ∆ 6 +2∆ 7 ¯ ∆ 7 +∆ 8 ¯ ∆ 8 ) l β l µ l ν = 0 , (A61) \nwhich means that the ∆-characterisation becomes \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = 0 . (A62) \nAccordingly, the maximum bo ( l ) in this case is -1 and thus we will name this case as Class IV . The intrinsic characterisation for this case can be written as \nl [ γ (1) ˜ Z α ] βλ [ µ l λ l ν ] = ∆ 0 k [ γ l α ] k β ¯ m [ µ l ν ] +∆ 1 ( l [ γ m α ] k β ¯ m [ µ l ν ] -k [ γ l α ] m β ¯ m [ µ l ν ] ) +∆ 2 ( ¯ m [ γ l α ] k β l [ µ ¯ m ν ] -l [ γ k α ] ¯ m β l [ µ ¯ m ν ] ) -∆ 3 l [ γ m α ] m β ¯ m [ µ l ν ] +∆ 4 ( k [ γ l α ] l β ¯ m [ µ l ν ] -l [ γ m α ] ¯ m β ¯ m [ µ l ν ] + ¯ m [ γ l α ] m β ¯ m [ µ l ν ] ) +∆ 5 l [ γ ¯ m α ] ¯ m β l [ µ ¯ m ν ] +∆ 7 l [ γ m α ] l β ¯ m [ µ l ν ] +∆ 8 l [ γ ¯ m α ] l β ¯ m [ µ l ν ] +c.c. = 0 , (A63) l α l [ ρ (1) ˜ Z α λ ] βµ = 2∆ 0 l [ λ k ρ ] k [ β ¯ m µ ] +2∆ 1 ( l [ λ k ρ ] l [ β k µ ] -m [ λ l ρ ] ¯ m [ β k µ ] + l [ λ k ρ ] ¯ m [ β m µ ] ) -2∆ 2 ¯ m [ λ l ρ ] ¯ m [ β k µ ] +2∆ 3 ( l [ λ k ρ ] m [ β l µ ] -m [ λ l ρ ] k [ β l µ ] -l [ λ m ρ ] ¯ m [ β m µ ] ) +2∆ 4 ( ¯ m [ λ l ρ ] l [ β k µ ] + ¯ m [ λ l ρ ] ¯ m [ β m µ ] ) +2∆ 6 m [ λ l ρ ] m [ β l µ ] -2∆ 7 ¯ m [ λ l ρ ] l [ β m µ ] +c.c. = 0 . (A64) \nClearly, these two conditions together are equivalent to Eq. (A61).', '5. Contraction of T αβλµτν ( (1) ˜ Z ) with two copies of l µ : Class IIe, Class IIIb, Class IVa and Class V': 'In this step, we remove four null vectors l µ from Expression (195), which gives rise to four independent possibilities. Notice that the different contractions of the resulting expression with null vectors l µ always lead at least to ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0. Since the equations in this step become cumbersome and we have already explained in detail how the computation works, in the following we will omit explicit equations and just present the important results. \na. Class IIe \nThe first possible contraction with two copies of l µ is \nT αβλµτν l α l β = 0 , (A65) \nwhich gives ∆ 0 = ∆ 1 = ∆ 3 = 0 by contracting it with l λ l µ l τ l ν , l λ l µ l τ and l λ l µ , respectively. By taking into account these conditions and contracting Eq. (A65) with l λ l τ l ν and l λ l τ , we also find ∆ 2 = ∆ 4 = 0, respectively, whereas the contraction with l λ provides then ∆ 6 = ∆ 7 = 0. Finally, by replacing all of these conditions in the equation, we find T αβλµτν l α l β = -8∆ 10 ¯ ∆ 10 l λ l µ l ν l τ = 0, which implies ∆ 10 = 0. In summary, we find the ∆-characterisation \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 6 = ∆ 7 = ∆ 10 = 0 . (A66) \nThe maximum bo ( l ) in this case is 1, so that we will name this case as Class IIe . \nOne can notice that Eq. (A65) always depends on l α (1) ˜ Z αβλµ , in such a way that the intrinsic characterisation for this case is \nl α (1) ˜ Z αβλµ = 2∆ 0 k β ¯ m [ λ k µ ] +2∆ 1 ( m β k [ λ ¯ m µ ] + k β m [ λ ¯ m µ ] -k β l [ λ k µ ] ) +2∆ 2 ¯ m β k [ λ ¯ m µ ] +2∆ 3 ( m β l [ λ k µ ] -k β m [ λ l µ ] + m β ¯ m [ λ m µ ] ) +2∆ 4 ( ¯ m β l [ λ k µ ] + l β ¯ m [ λ k µ ] + ¯ m β ¯ m [ λ m µ ] ) +2∆ 6 m β m [ λ l µ ] +2∆ 7 ( l β k [ λ l µ ] + ¯ m β m [ λ l µ ] + l β m [ λ ¯ m µ ] ) +2∆ 10 l β l [ λ m µ ] +c.c. = 0 , (A67) \nthat is equivalent as (A66).', 'b. Class IIIb': 'The second possible case with two null vectors l µ is \nT αβλµτν l τ l ν = 0 , (A68) \nwhich provides ∆ 0 = ∆ 1 = ∆ 2 = ∆ 4 = 0 by contracting it with l α l β l λ l µ , l α l β l λ and l α l β , respectively. Moreover, if we use these conditions, further contractions with l α l λ and l µ lead to ∆ 3 = ∆ 5 = ∆ 7 = ∆ 8 = 0. By using all of these conditions, we find T αβλµτν l τ l ν = -8∆ 11 ¯ ∆ 11 l α l β l λ l µ = 0, which means ∆ 11 = 0. Putting all of these conditions together, we find the ∆-characterisation \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 7 = ∆ 8 = ∆ 11 = 0 . (A69) \nThe maximum bo ( l ) is zero and thus we will name this case as Class IIIb . \nIn addition, the intrinsic characterisation reads \n(1) ˜ Z αβλ [ µ l λ l ν ] = ∆ 0 k α k β l [ µ ¯ m ν ] -2∆ 1 k ( α m β ) l [ µ ¯ m ν ] -2∆ 2 ¯ m ( α k β ) l [ µ ¯ m ν ] +∆ 3 m α m β l [ µ ¯ m ν ] +2∆ 4 ( k ( α l β ) l [ µ ¯ m ν ] + ¯ m ( α m β ) l [ µ ¯ m ν ] ) +∆ 5 ¯ m α ¯ m β l [ µ ¯ m ν ] -2∆ 7 l ( α m β ) l [ µ ¯ m ν ] -2∆ 8 ¯ m ( α l β ) l [ µ ¯ m ν ] +∆ 11 l α l β l [ µ ¯ m ν ] +c.c. = 0 , (A70) \nwhich is equivalent to the condition (A69).', 'c. Class IVa': 'The third possible case with two copies of l µ reads \nT αβλµτν l λ l τ = 0 , (A71) \nwhich gives us ∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = 0 if we contract it with l α l β l µ l ν , l α l β l µ and l α l β , respectively. Furthermore, by using these conditions and contracting the previous equation with l α , we find ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = 0, respectively. Then, using all these conditions together, the equation becomes T αβλµτν l λ l τ = -4 ( ∆ 10 ¯ ∆ 10 + ∆ 11 ¯ ∆ 11 ) l α l β l µ l ν = 0, implying ∆ 10 = ∆ 11 = 0. Therefore, this case gives us the ∆-characterisation \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = ∆ 10 = ∆ 11 = 0 , (A72) \nand then, the maximum bo ( l ) is -1 and we name this case as Class IVa . \n- +∆ 3 ( l [ γ m α ] k β l λ -k [ γ l α ] m β l λ + l [ γ m α ] m β ¯ m λ ) -∆ 4 ( k [ γ l α ] ¯ m β l λ + ¯ m [ γ l α ] k β l λ + k [ γ l α ] l β ¯ m λ \nOn the other hand, the intrinsic characterisation for this case can be given by two expressions. The first one reads l [ γ (1) ˜ Z α ] βλµ l µ = -∆ 0 k [ γ l α ] k β ¯ m λ +∆ 1 ( k [ γ l α ] k β l λ -l [ γ k α ] m β ¯ m λ + m [ γ l α ] k β ¯ m λ ) +∆ 2 ( k [ γ l α ] ¯ m β ¯ m λ -l [ γ ¯ m α ] k β ¯ m λ ) \n- -l [ γ ¯ m α ] m β ¯ m λ -l [ γ m α ] ¯ m β ¯ m λ ) +∆ 5 l [ γ ¯ m α ] ¯ m β ¯ m λ +∆ 6 m [ γ l α ] l λ m β \n-∆ 8 ( l [ γ ¯ m α ] ¯ m β l λ + l [ γ ¯ m α ] l β ¯ m λ ) -1 2 ( ∆ 10 + ¯ ∆ 11 ) m [ γ l α ] l β l λ -1 2 ( ¯ ∆ 10 +∆ 11 ) ¯ m [ γ l α ] l β l λ +c.c. = 0 , \n- +∆ 7 ( k [ γ l α ] l β l λ -l [ γ ¯ m α ] m β l λ -l [ γ m α ] ¯ m β l λ + m [ γ l α ] l β ¯ m λ ) \nwhereas the second one is simply \nl [ γ (1) ˜ Z α ] β [ λµ l ν ] = 4∆ 10 l [ γ m α ] l β ¯ m [ λ m µ l ν ] -4 ¯ ∆ 10 l [ γ ¯ m α ] l β ¯ m [ λ m µ l ν ] = 0 . (A74) \nIt is then clear that (A73) and (A74), together, are equivalent to the condition (A72). \n(A73) \nThe last case with two copies of l µ is \nT αβλµτν l α l λ = 0 , (A75) \nand gives us ∆ 0 = ∆ 1 = 0 by contracting with l β l µ l τ . Thus, by using these conditions and contracting the previous equation with l β l µ and l β l ν , we find ∆ 2 = ∆ 3 = ∆ 4 = 0. A further contraction with l β and l ν then yields ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = 0. Finally, if we use all of these conditions in Eq. (A75), we find T αβλµτν l α l λ = -2 ( ∆ 9 ¯ ∆ 9 +2∆ 10 ¯ ∆ 10 + ∆ 11 ¯ ∆ 11 ) l β l µ l ν l τ = 0 and then ∆ 9 = ∆ 10 = ∆ 11 = 0. Putting it all together, the ∆-characterisation is \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = ∆ 9 = ∆ 10 = ∆ 11 = 0 . (A76) \nTherefore, the maximum bo ( l ) is -2 and we name this case as Class V . The intrinsic characterisation for this case reads \nl [ γ (1) ˜ Z α ] [ β λµ l ρ ] = 2∆ 0 k [ γ l α ] l [ β ¯ m [ λ k µ ] k ρ ] -2∆ 1 ( k [ γ l α ] l [ β k [ µ l λ ] k ρ ] -l [ γ m α ] l [ β ¯ m [ λ k µ ] k ρ ] + k [ γ l α ] l [ β ¯ m [ λ m µ ] k ρ ] -l [ γ k α ] l [ β ¯ m [ λ k µ ] m ρ ] ) +2∆ 2 ( l [ γ ¯ m α ] l [ β ¯ m [ λ k µ ] k ρ ] -k [ γ l α ] l [ β ¯ m [ λ k µ ] ¯ m ρ ] ) -2∆ 3 ( l [ γ m α ] l [ β l [ λ k µ ] k ρ ] + l [ γ k α ] l [ β l [ λ m µ ] k ρ ] + k [ γ l α ] l [ β k [ λ l µ ] m ρ ] + k [ γ l α ] m [ β ¯ m [ λ m µ ] l ρ ] -m [ γ l α ] m [ β k [ λ ¯ m µ ] l ρ ] -l [ γ m α ] l [ β m [ λ ¯ m µ ] k ρ ] ) +2∆ 4 ( ¯ m [ γ l α ] l [ β ¯ m [ λ m µ ] k ρ ] -l [ γ ¯ m α ] l [ β l [ λ k µ ] k ρ ] -k [ γ l α ] ¯ m [ β l [ λ k µ ] l ρ ] + l [ γ ¯ m α ] l [ β k [ λ ¯ m µ ] m ρ ] -l [ γ k α ] ¯ m [ β m [ λ ¯ m µ ] l ρ ] -m [ γ l α ] l [ β k [ λ ¯ m µ ] ¯ m ρ ] ) +2∆ 5 l [ γ ¯ m α ] ¯ m [ β ¯ m [ λ k µ ] l ρ ] -2∆ 6 ( l [ γ m α ] m [ β l [ λ k µ ] l ρ ] + m [ γ l α ] l [ β l [ λ m µ ] k ρ ] -k [ γ l α ] l [ β m [ λ l µ ] m ρ ] -m [ γ l α ] l [ β m [ λ ¯ m µ ] m ρ ] ) -2∆ 7 ( l [ γ ¯ m α ] m [ β l [ λ k µ ] l ρ ] + l [ γ m α ] ¯ m [ β l [ λ k µ ] l ρ ] + ¯ m [ γ l α ] l [ β l [ λ m µ ] k ρ ] -m [ γ l α ] ¯ m [ β ¯ m [ λ m µ ] l ρ ] + ¯ m [ γ l α ] l [ β ¯ m [ λ m µ ] m ρ ] + l [ γ k α ] l [ β m [ λ l µ ] ¯ m ρ ] ) -2∆ 8 ( l [ γ ¯ m α ] ¯ m [ β l [ λ k µ ] l ρ ] + ¯ m [ γ l α ] ¯ m [ β m [ λ ¯ m µ ] l ρ ] ) -2∆ 9 l [ γ m α ] m [ β m [ λ l µ ] l ρ ] -2∆ 10 ( l [ γ ¯ m α ] m [ β m [ λ l µ ] l ρ ] -m [ γ l α ] l [ β l [ λ m µ ] ¯ m ρ ] ) -2∆ 11 l [ γ ¯ m α ] ¯ m [ β m [ λ l µ ] l ρ ] +c.c. = 0 , (A77) \nwhich indeed is equivalent to Eq. (A76).', '6. Contraction of T αβλµτν ( (1) ˜ Z ) with one copy of l µ : Class IVb and Class VI': 'The last step is obtained by removing five null vectors l µ in Expression (195), which allows only two independent possibilities. \na. Class IVb \nThe first possible case with one null vector l µ reads \nT αβλµτν l ν = 0 . (A78) \nFirst, it is clear that this condition gives ∆ i = 0 with i = 1 , ..., 5. By applying this in Eq. (A78) and contracting the resulting expression with l α l β , one finds ∆ 7 = 0. Then, by contracting with l α l µ , one also finds ∆ 6 = ∆ 8 = 0. Furthermore, by replacing all of these conditions and contracting Eq. (A78) with l α , one finds ∆ 10 = ∆ 11 = 0, which ends up reducing the equation itself to ∆ 13 = 0. Hence, the ∆-characterisation is \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = ∆ 10 = ∆ 11 = ∆ 13 = 0 . (A79) \nIn other words, in this case only the complex scalars ∆ 9 , ∆ 12 and ∆ 14 are nonvanishing. The maximum bo ( l ) is -1 and, thus, we will name this case as Class IVb . \nOn the other hand, it is possible to find an intrinsic characterisation in terms of two different conditions. The first one is \nl ν (1) ˜ Z αβµν = ∆ 0 k α k β ¯ m µ -∆ 1 ( k α k β l µ +2 m ( α k β ) ¯ m µ ) -2∆ 2 ¯ m ( α k β ) ¯ m µ \nand the second one is \n+∆ 3 ( 2 m ( α k β ) l µ + m α m β ¯ m µ ) +2∆ 4 ( ¯ m ( α k β ) l µ + k ( α l β ) ¯ m µ + ¯ m ( α m β ) ¯ m µ ) +∆ 5 ¯ m α ¯ m β ¯ m µ -∆ 6 m α m β l µ -2∆ 7 ( l ( α k β ) l µ + ¯ m ( α m β ) l µ + m ( α l β ) ¯ m µ ) -∆ 8 ( ¯ m α ¯ m β l µ +2¯ m ( α l β ) ¯ m µ ) +2∆ 10 l ( α m β ) l µ +∆ 11 ( 2 ¯ m ( α l β ) l µ + l α l β ¯ m µ ) -1 2 ( ∆ 13 + ¯ ∆ 13 ) l α l β l µ +c.c. = 0 , (A80) \n(1) ˜ Z αβ [ λµ l ν ] = -2 ( ∆ 13 -¯ ∆ 13 ) l α l β ¯ m [ λ m µ l ν ] + f (1) αβλµν = 0 , (A81) \nwhere for simplicity we have introduced the tensor f (1) αβλµν that depends on ∆ 0 , ∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 , ∆ 5 , ∆ 6 , ∆ 7 , ∆ 8 , ∆ 11 and their conjugates. Notice that the first condition does not directly imply ∆ 13 = 0, but the second condition is needed to vanish it. Hence, these two expressions together provide the condition (A79). \nb. Class VI \nThe last possibility remaining in the classification is \nT αβλµτν l α = 0 . (A82) \nAs the previous case, clearly we first have ∆ i = 0, with i = 1 , ..., 5. By taking into account this condition in the explicit expression of Eq. (A82) and contracting it with l τ l µ , one then finds ∆ 6 = ∆ 7 = ∆ 8 = 0. Then, a further contraction of Eq. (A82) with l λ and l τ leads to ∆ 9 = 0 and ∆ 10 = ∆ 11 = 0. Finally, by replacing all of these conditions in Eq. (A82), one straightforwardly finds ∆ 12 = ∆ 13 = 0. In summary, the ∆-characterisation of this case reads \n∆ 0 = ∆ 1 = ∆ 2 = ∆ 3 = ∆ 4 = ∆ 5 = ∆ 6 = ∆ 7 = ∆ 8 = ∆ 9 = ∆ 10 = ∆ 11 = ∆ 12 = ∆ 13 = 0 , (A83) \nin such a way that the only nonvanishing complex scalar is ∆ 14 . The maximum bo ( l ) is -3, so that this case will be named as Class VI . \nFrom Expression (63), it is clear that the tensor (1) ˜ Z αβλµ acquires the form \n(1) ˜ Z αβλµ = 2 l α l β ( l λ Y µ -l µ Y λ ) + f (2) αβλµ , (A84) \nwhere Y µ is orthogonal to l µ and k µ , while f (2) αβλµ depends on all of the complex scalars and their conjugates, except on ∆ 14 and ¯ ∆ 14 . In our tetrad, we have Y µ = (1 / 2)(∆ 14 m µ + ¯ ∆ 14 ¯ m µ ). \nThereby, the intrinsic characterisation for this case reads \n- l [ γ (1) ˜ Z α ] βλµ = -2∆ 0 k [ γ l α ] k β ¯ m [ λ k µ ] +2∆ 1 ( k [ γ l α ] k β l [ λ k µ ] -l [ γ m α ] k β ¯ m [ λ k µ ] + k [ γ l α ] m β ¯ m [ λ k µ ] + k [ γ l α ] k β ¯ m [ λ m µ ] ) -2∆ 2 ( l [ γ ¯ m α ] k β ¯ m [ λ k µ ] + l [ γ k α ] ¯ m β ¯ m [ λ k µ ] ) +2∆ 3 ( l [ γ m α ] k β l [ λ k µ ] -k [ γ l α ] m β l [ λ k µ ] + k [ γ l α ] k β m [ λ l µ ] + l [ γ m α ] m β ¯ m [ λ k µ ] + l [ γ m α ] k β ¯ m [ λ m µ ] -k [ γ l α ] m β ¯ m [ λ m µ ] ) -2∆ 4 ( l [ γ ¯ m α ] k β k [ λ l µ ] -k [ γ l α ] ¯ m β k [ λ l µ ] -l [ γ ¯ m α ] m β ¯ m [ λ k µ ] -l [ γ ¯ m α ] k β ¯ m [ λ m µ ] + m [ γ l α ] ¯ m β ¯ m [ λ k µ ] -l [ γ k α ] ¯ m β ¯ m [ λ m µ ] -k [ γ l α ] l β k [ λ ¯ m µ ] ) +2∆ 5 ¯ m [ γ l α ] ¯ m β k [ λ ¯ m µ ] -2∆ 6 ( l [ γ m α ] m β l [ λ k µ ] -l [ γ k α ] m β m [ λ l µ ] -m [ γ l α ] k β l [ λ m µ ] + l [ γ m α ] m β ¯ m [ λ m µ ] ) +2∆ 7 ( k [ γ l α ] l β l [ λ k µ ] -l [ γ ¯ m α ] m β l [ λ k µ -m [ γ l α ] ¯ m β k [ λ l µ ] + l [ γ k α ] ¯ m β m [ λ l µ ] -¯ m [ γ l α ] k β m [ λ l µ ] + k [ γ l α ] l β ¯ m [ λ m µ ] -l [ γ ¯ m α ] m β ¯ m [ λ m µ ] -l [ γ m α ] ¯ m β ¯ m [ λ m µ ] -m [ γ l α ] l β k [ λ ¯ m µ ] ) -2∆ 8 ( l [ γ ¯ m α ] ¯ m β l [ λ k µ ] + l [ γ ¯ m α ] l β ¯ m [ λ k µ ] -l [ γ ¯ m α ] ¯ m β m [ λ ¯ m µ ] ) -2∆ 9 m [ γ l α ] m β l [ λ m µ ] +2∆ 10 ( l [ γ m α ] l β l [ λ k µ ] + k [ γ l α ] l β m [ λ l µ ] -l [ γ ¯ m α ] m β m [ λ l µ ] -l [ γ m α ] ¯ m β m [ λ l µ ] + l [ γ m α ] l β ¯ m [ λ m µ ] ) -2∆ 11 ( l [ γ ¯ m α ] ¯ m β m [ λ l µ ] + ¯ m [ γ l α ] l β l [ λ k µ ] -l [ γ ¯ m α ] l β ¯ m [ λ m µ ] ) +2∆ 12 l [ γ m α ] l β m [ λ l µ ] +2∆ 13 l [ γ ¯ m α ] l β m [ λ l µ ] +c.c. = 0 , (A85) \nwhich is indeed equivalent to the condition (A83).', "Appendix B. Bernstein's theorem applied to ∆ ' 0 = 0": "Consider the main equation for the rotated complex scalar ∆ ' 0 : \n∆ ' 0 =∆ 0 +4 /epsilon1 ∆ 1 +2¯ /epsilon1 ∆ 2 +6 /epsilon1 2 ∆ 3 +8 /epsilon1 ¯ /epsilon1 ∆ 4 +¯ /epsilon1 2 ∆ 5 +4 /epsilon1 3 ∆ 6 +12 /epsilon1 2 ¯ /epsilon1 ∆ 7 +4 /epsilon1 ¯ /epsilon1 2 ∆ 8 + /epsilon1 4 ∆ 9 +8 /epsilon1 3 ¯ /epsilon1 ∆ 10 +6 /epsilon1 2 ¯ /epsilon1 2 ∆ 11 +2 /epsilon1 4 ¯ /epsilon1 ∆ 12 +4 /epsilon1 3 ¯ /epsilon1 2 ∆ 13 + /epsilon1 4 ¯ /epsilon1 2 ∆ 14 = 0 , (B1) \nwhere we want to solve for /epsilon1 and ¯ /epsilon1 . In the following, we will assume that both quantities are independent of each other. \nTo determine the maximum number of solutions, we can use the Bernstein's theorem, which involves calculating the areas of certain polytopes associated with the equation [82]. \nFig. 3 shows the points corresponding to the terms of the equation where /epsilon1 and ¯ /epsilon1 appear with different powers. These points are: \n(0 , 0) , (1 , 0) , (0 , 1) , (2 , 0) , (1 , 1) , (0 , 2) , (3 , 0) , (2 , 1) , (1 , 2) , (4 , 0) , (2 , 2) , (3 , 1) , (4 , 1) , (3 , 2) , (4 , 2) \nand they form a polygon with an area of 8 where the polygon is drawn with red lines in Fig. 3. \nFIG. 3: Polygon generated from the powers of /epsilon1 and ¯ /epsilon1 \n<!-- image --> \nFig. 4 represents the conjugate area of the same points. For the conjugate terms, the powers of /epsilon1 and ¯ /epsilon1 are swapped. The same points form a different polygon with an area of 8, also with red lines. \nFIG. 4: Polygon generated from the conjugate of powers of /epsilon1 and ¯ /epsilon1 . \n<!-- image --> \nFinally, Fig. 5 shows the overall polytope which includes all combinations of /epsilon1 and ¯ /epsilon1 . To obtain such a figure, all the points from the first drawing are added to all the points from the second drawing (see Figs. 3 and 4). This means \neach ( i, j ) from the first drawing is added to each ( i ' , j ' ) from the second drawing, where i, j, i ' , j ' are the powers of /epsilon1 and ¯ /epsilon1 . The sum is performed as ( i, j ) + ( i ' , j ' ) = ( i + i ' , j + j ' ). \nWhen summing, many points will appear multiple times. However, this repetition is not important. What matters is the resulting polytope and the convex polygon that encompasses it. \nThe points that form the large square are: \n```\n(0 , 0) , (1 , 0) , (2 , 0) , (3 , 0) , (4 , 0) , (5 , 0) , (6 , 0) , (0 , 1) , (1 , 1) , (2 , 1) , (3 , 1) , (4 , 1) , (5 , 1) , (6 , 1) , (0 , 2) , (1 , 2) , (2 , 2) , (3 , 2) , (4 , 2) , (5 , 2) , (6 , 2) , (0 , 3) , (1 , 3) , (2 , 3) , (3 , 3) , (4 , 3) , (5 , 3) , (6 , 3) , (0 , 4) , (1 , 4) , (2 , 4) , (3 , 4) , (4 , 4) , (5 , 4) , (6 , 4) , (0 , 5) , (1 , 5) , (2 , 5) , (3 , 5) , (4 , 5) , (5 , 5) , (6 , 5) , (0 , 6) , (1 , 6) , (2 , 6) , (3 , 6) , (4 , 6) , (5 , 6) , (6 , 6) .\n``` \nEach point represents the sum of the corresponding powers of /epsilon1 and ¯ /epsilon1 from the original polygons. The resulting polytope is the convex hull that includes all these points, forming a square with an area of 36. \nFIG. 5: Total polygon generated from the possible existing powers of /epsilon1 and ¯ /epsilon1 . \n<!-- image --> \nTotal Area: 36 \nUsing the Bernstein's theorem, the maximum number of solutions is given by the total area minus the areas of the individual polygons. In this case, the calculation is 36 -8 -8 = 20. Therefore, the maximum number of solutions for the equation, when considering /epsilon1 and ¯ /epsilon1 as two independent complex variables, is 20. This result comes from subtracting the areas of the primary and conjugate polygons from the total area, effectively accounting for the overlap and ensuring the count of unique solutions. \nIn order to analyse the nongeneric cases within the context of the Bernstein's theorem, we need to consider the (sub)polynomials associated with the edges that form the final polygon and determine when these, considered jointly, have solutions. The edges of the first polygon correspond to combinations of powers of /epsilon1 and ¯ /epsilon1 from the terms in the original equation. Similarly, the edges of the second polygon correspond to combinations of powers of /epsilon1 and ¯ /epsilon1 from the conjugate terms. When summing the edges of the two original polygons, we obtain the edges of the final polygon. This is done by summing the corresponding points from the edges of each polygon: \n( i, j ) + ( i ' , j ' ) = ( i + i ' , j + j ' ) . \nEach edge in the original polygons is associated with a subpolynomial. The nongeneric cases arise in general when the sum of these subpolynomials, considered jointly, has solutions. \nThen, for our equation, there are only two independent combinations. The first set would be \n0 = ∆ 0 +2¯ /epsilon1 ∆ 2 +¯ /epsilon1 2 ∆ 5 , (B2) \n0 = ¯ ∆ 0 +4¯ /epsilon1 ¯ ∆ 1 +6¯ /epsilon1 2 ¯ ∆ 3 +4¯ /epsilon1 3 ¯ ∆ 6 +¯ /epsilon1 4 ¯ ∆ 9 , (B3) \nwhile the second set related to particular cases is \n0 =∆ 5 +4 /epsilon1 ∆ 8 +6 /epsilon1 2 ∆ 11 +4 /epsilon1 3 ∆ 13 + /epsilon1 4 ∆ 14 , (B4) \n0 = ¯ ∆ 9 +2 /epsilon1 ¯ ∆ 12 + /epsilon1 2 ¯ ∆ 14 . (B5) \nLet us start by solving the system (B2)-(B3). We can first proceed isolating ¯ /epsilon1 in Eq. (B2) and, then, replacing it into Eq. (B3). If ∆ 5 = 0 , this leads to the following constraint: \n/negationslash \n¯ ∆ 0 = 4∆ 2 ¯ ∆ 1 ∆ 5 ± 4 ¯ ∆ 1 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 5 -12∆ 2 2 ¯ ∆ 3 ∆ 2 5 + 6∆ 0 ¯ ∆ 3 ∆ 5 ∓ 12∆ 2 ¯ ∆ 3 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 2 5 + 16∆ 3 2 ¯ ∆ 6 ∆ 3 5 -12∆ 0 ∆ 2 ¯ ∆ 6 ∆ 2 5 ∓ 16∆ 2 2 ¯ ∆ 6 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 3 5 ± 4∆ 0 ¯ ∆ 6 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 2 5 -8∆ 4 2 ¯ ∆ 9 ∆ 4 5 + 8∆ 0 ∆ 2 2 ¯ ∆ 9 ∆ 3 5 -∆ 2 0 ¯ ∆ 9 ∆ 2 5 ± 8∆ 3 2 ¯ ∆ 9 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 4 5 ∓ 4∆ 0 ∆ 2 ¯ ∆ 9 √ ∆ 2 2 -∆ 0 ∆ 5 ∆ 3 5 , (B6) \nwhereas, if ∆ 5 = 0 and ∆ 2 = 0, we find: \n/negationslash \n¯ ∆ 0 = 2∆ 0 ¯ ∆ 1 ∆ 2 -3∆ 2 0 ¯ ∆ 3 2∆ 2 2 + ∆ 3 0 ¯ ∆ 6 2∆ 3 2 -∆ 4 0 ¯ ∆ 9 16∆ 4 2 . (B7) \nFinally, if ∆ 5 = ∆ 2 = 0, one has: \n4¯ /epsilon1 ¯ ∆ 1 +6¯ /epsilon1 2 ¯ ∆ 3 +4¯ /epsilon1 3 ¯ ∆ 6 +¯ /epsilon1 4 ¯ ∆ 9 = 0 , ∆ 0 = 0 . (B8) \nNow, let us solve the second system composed by (B4)-(B5). Similarly, if ∆ 14 = 0, we find the following constraint: \n/negationslash \n∆ 5 = -8∆ 14 ¯ ∆ 4 12 ¯ ∆ 4 14 + 8∆ 14 ¯ ∆ 9 ¯ ∆ 2 12 ¯ ∆ 3 14 + 16∆ 13 ¯ ∆ 3 12 ¯ ∆ 3 14 -∆ 14 ¯ ∆ 2 9 ¯ ∆ 2 14 -12∆ 13 ¯ ∆ 9 ¯ ∆ 12 ¯ ∆ 2 14 -12∆ 11 ¯ ∆ 2 12 ¯ ∆ 2 14 + 6∆ 11 ¯ ∆ 9 ¯ ∆ 14 + 4∆ 8 ¯ ∆ 12 ¯ ∆ 14 ∓ 8∆ 14 ¯ ∆ 3 12 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 4 14 ± 4∆ 14 ¯ ∆ 9 ¯ ∆ 12 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 3 14 ± 16∆ 13 ¯ ∆ 2 12 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 3 14 ∓ 4∆ 13 ¯ ∆ 9 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 2 14 ∓ 12∆ 11 ¯ ∆ 12 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 2 14 ± 4∆ 8 √ ¯ ∆ 2 12 -¯ ∆ 9 ¯ ∆ 14 ¯ ∆ 14 , (B9) \nand, if ∆ 14 = 0, but ∆ 12 = 0: \n/negationslash \n∆ 5 = ∆ 13 ¯ ∆ 3 9 2 ¯ ∆ 3 12 -3∆ 11 ¯ ∆ 2 9 2 ¯ ∆ 2 12 + 2∆ 8 ¯ ∆ 9 ¯ ∆ 12 . (B10) \nThe last possibility is given by ∆ 14 = ∆ 12 = 0, which from (B4)-(B5) gives rise to: \n4 /epsilon1 3 ∆ 13 +6 /epsilon1 2 ∆ 11 +4 /epsilon1 ∆ 8 -∆ 5 = 0 , ∆ 9 = 0 . 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2024arXiv240905326C

The SmallCorrelatedAgainstLarge Estimator SCALE for smallscale lensing of the cosmic microwave background CMB provides a novel method for measuring the amplitude of CMB lensing power without the need for reconstruction of the lensing field. In our previous study we showed that the SCALE method can outperform existing reconstruction methods to detect the presence of lensing at small scales ell gg 3000. Here we develop a procedure to include information from SCALE in cosmological parameter inference. We construct a precise neural network emulator to quickly map cosmological parameters to desired CMB observables such as temperature and lensing power spectra and SCALE cross spectra. We also outline a method to apply SCALE to fullsky maps of the CMB temperature field and construct a likelihood for the application of SCALE in parameter estimation. SCALE supplements conventional observables such as the CMB power spectra and baryon acoustic oscillations in constraining parameters that are sensitive to the smallscale lensing amplitude such as the neutrino mass mnu. We show that including estimates of the smallscale lensing amplitude from SCALE in such an analysis provides enough constraining information to measure the minimum neutrino mass at 4sigma significance in the scenario of minimal mass and higher significance for higher mass. Finally we show that SCALE will play a powerful role in constraining models of clustering that generate scaledependent modulation to the distribution of matter and the lensing power spectrum as predicted by models of warm or fuzzy dark matter.

2024-09-01T00:00:00Z

['2024arXiv240905326C', '10.48550/arXiv.2409.05326', 'arXiv:2409.05326']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

SCALE at Scale Cosmological applications of smallscale CMB lensing

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05326.pdf

{'SCALE at Scale: Cosmological applications of small-scale CMB lensing': "Victor C. Chan, 1, 2 Ren'ee Hloˇzek, 3, 1 Joel Meyers, 2 and Alexander van Engelen 4 \n1 David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada 2 Department of Physics, Southern Methodist University, Dallas, TX 75275, USA 3 Dunlap Institute for Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada 4 School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA \nThe Small-Correlated-Against-Large Estimator (SCALE) for small-scale lensing of the cosmic microwave background (CMB) provides a novel method for measuring the amplitude of CMB lensing power without the need for reconstruction of the lensing field. In our previous study, we showed that the SCALE method can outperform existing reconstruction methods to detect the presence of lensing at small scales ( ℓ ≫ 3000). Here we develop a procedure to include information from SCALE in cosmological parameter inference. We construct a precise neural network emulator to quickly map cosmological parameters to desired CMB observables such as temperature and lensing power spectra and SCALE cross spectra. We also outline a method to apply SCALE to full-sky maps of the CMB temperature field, and construct a likelihood for the application of SCALE in parameter estimation. SCALE supplements conventional observables such as the CMB power spectra and baryon acoustic oscillations in constraining parameters that are sensitive to the small-scale lensing amplitude such as the neutrino mass m ν . We show that including estimates of the small-scale lensing amplitude from SCALE in such an analysis provides enough constraining information to measure the minimum neutrino mass at 4 σ significance in the scenario of minimal mass, and higher significance for higher mass. Finally, we show that SCALE will play a powerful role in constraining models of clustering that generate scale-dependent modulation to the distribution of matter and the lensing power spectrum, as predicted by models of warm or fuzzy dark matter.", 'I. INTRODUCTION': 'Forthcoming high-resolution, low-noise observations of the cosmic microwave background (CMB) will allow analysis of its gravitational lensing features to greatly exceed the precision of current measurements. Gravitational lensing is a particularly useful probe of cosmological density fluctuations, as it provides an unbiased tracer of the total mass density. CMB lensing, in particular, utilizes a very well-characterized source plane at a known redshift, and thereby is especially powerful in this regard. \nIn Ref. [1, hereafter C24], we developed the SmallCorrelated-Against-Large Estimator (SCALE) for measuring the small-scale ( ℓ ≫ 3000) lensing power in the CMB. We showed that at the sensitivity of upcoming CMBexperiments, SCALE can exceed the signal-to-noise with which small-scale lensing can be measured when compared with lensing reconstruction based on the standard quadratic estimator (QE) [2-4]. SCALE is a novel method to quantify small-scale CMB lensing in highresolution temperature maps. The basic flow of the method is to pre-process a temperature map into a map λ of the large-scale ( ℓ L < 3000) gradient power which is dominated by primary CMB features, and a map ς of small-scale ( ℓ S ≫ 3000) gradient power which is dominated by lensing features. The cross-spectrum of these two maps Ψ ˇ L is directly related to the amplitude of the lensing power spectrum C κκ L of the small-scale regimes associated with ς . We refer the reader to C24 for a complete description of the principles of SCALE as well as comparisons to measurements with quadratic estimator reconstructions [3, 5]. Throughout the paper, we will typically use ℓ to index multipole moments of the CMB \ntemperature, L for those of the CMB lensing field, and ˇ L for the SCALE observables. \nThe value added by using SCALE compared to only utilizing conventional quadratic estimators lies in a few key areas. First, it is a simple method to quickly measure the amplitude of the small-scale CMB lensing power spectrum without the need for a full reconstruction of the lensing field. We established in C24 that SCALE outperforms quadratic estimators in terms of signal-tonoise of recovered lensing signal at small-scales in upcoming experiments. Quadratic estimators remain highly effective, tested, and well-understood tools for estimating CMB lensing on larger scales, while SCALE is presented with an opportunistic, complementary role in the realm of small-scale lensing. If one wishes to use quadratic estimators at small-scales, the reconstruction bias N (1) ,κκ L grows to similar amplitude as the lensing power spectrum C κκ L by L ∼ 1000, and dominates at higher L [6, 7]. The small-scale lensing regime will become ever-more important as the experimental sensitivity improves, with surveys like the upcoming Simons Observatory (SO) [8] and CMB-S4 [9, 10], as well as proposed future surveys like CMB-HD [11]. Improvements in foreground characterization and mitigation also suggest that we will be able to extract a significant amount of cosmology on these scales with appropriate statistical estimators. The maximum likelihood, maximum a posteriori, Gradient Inversion, and Bayesian techniques [12-16] are also currently being developed with the aim to tackle these challenges with small-scale lensing reconstruction and go beyond what the QE is capable of. \nSmall-scale lensing of the CMB is a particularly interesting laboratory for probing the dark matter and clus- \ntering properties of the universe (see Ref. [17] for a review of CMB lensing). The effect of massive neutrinos on the small-scale lensing power spectrum is a nearly scaleindependent suppression, and an accurate measurement of the small-scale lensing amplitude will provide a valuable probe of the total neutrino mass [18, 19]. Cosmological measurement of neutrino mass is complementary to lab-based probes of neutrino mass [20] and allows for insights into physics beyond the Standard Model [21, 22]. CMBlensing probes of neutrino mass have recently taken on additional importance given hints of tighter than expected upper bounds on neutrino mass with existing cosmological data [23-27]. Some non-standard models of warm or fuzzy dark matter also predict a suppression of clustering on small-scales, which in turn leads to a phenomenological, scale-dependent suppression of lensing power [28-34]. These scale-dependent effects could be constrained with multiple measurements of the lensing amplitude at different scales [35, 36]. \nIn this work, we build on the established SCALE method, and extend its application to full-sky maps. We also study its constraining power when applied to practical cosmological parameter estimation, particularly in the context of small-scale lensing. We do not include a characterization of CMB foregrounds in this work, as small-scale power added by foregrounds is not expected to correlate with the large-scale features of the primary CMB temperature field; this is still an interesting hypothesis to test, and we will leave further study of foregrounds for future work. Sampling cosmological parameters quickly requires fast predictions of theoretical spectra to compare to within the likelihood. We construct a neural network emulator and present its performance in § II. The emulator provides significant speed benefits when mapping a set of cosmological parameters to expected band-powers of angular power spectra with only a small penalty to precision ( ≲ 0 . 5%). We present our suite of full-sky simulations for the lensed CMB in § III. We construct a likelihood for CMB temperature power spectra in a ΛCDM parameter estimation, and then extend the model to include conventional lensing reconstruction information as well as SCALE cross spectra in § IV, and present the results in § V. We additionally include an analysis with a model including a general, scale-dependent suppression of lensing at extremely small scales ( L ∼ 10 000), and we show that SCALE can be configured to accurately detect exotic models of dark matter. Finally, we discuss the results and conclude in § VI.', 'II. EMULATION OF CMB AND SCALE SPECTRA': 'We begin with the development of a set of emulators to quickly predict theoretical observables including the CMB lensed temperature power spectra C TT ℓ , lensing power spectra C κκ L , lensing reconstruction bias N (1) ,κκ L , \nas well as analytic SCALE products A ˇ L and Ψ ˇ L (whose values are defined below in Eqs. (1) and (2)). The posterior sampling process typically requires upwards of O (10 3 to 10 4 ) steps per chain in order to reasonably explore a hyperspace of several cosmological parameters, and that necessitates a quick mapping from the set of parameters at each step to the relevant observables. While CMB angular power spectra can be computed quickly with existing software, repeatedly computing the expected SCALE cross-spectra with Eqs. (1) and (2) from different sets of parameters requires a significant speedup over conventional numerical integration methods. 1 We show that neural network (NN) emulators can predict SCALE observables at the speed required for quick posterior sampling without a significant penalty in terms of accuracy. The emulators can also be trained to predict lensed CMB temperature power spectra ˜ C TT ℓ and lensing power spectra C κκ L faster than Boltzmann codes, while maintaining high accuracy.', 'A. Timing of calculations without emulators': "Given a set of cosmological parameters, there are now Boltzmann codes that quickly and accurately compute primary CMB power spectra C TT ℓ , lensing power spectra C ϕϕ L , and lensed CMB power spectra ˜ C TT ℓ ( CAMB 2 , [37]; CLASS 3 , [38]). We opt to use CAMB in this work to keep consistency with C24, but the applications should be comparable to outputs from CLASS . Consider computing example power spectra in CAMB with lens potential accuracy=8 as the only non-default parameter to ensure lensing accuracy at highℓ [39]. CAMB is able to compute and return the power spectra in O (1 s), with some mild dependence on the requested lmax . This is fast enough to be used in parameter estimation, but it is possible to speed up the process further by using an emulator. This is especially true if one is interested in running modified versions such as axionCAMB 4 which take extra computational steps to include non-standard physics which could affect the small-scale lensing power spectrum and can extend the computational time by factors of anywhere from two to ten [40]. \nA stronger motivation for the construction of emulators for theoretical spectra comes from the application of SCALE in a cosmological likelihood. The analytic forms for SCALE products were presented in C24, and are re- \npeated here: \nA ˇ L = [ 2 ∫ d 2 ℓ S (2 π ) 2 W ς ( ℓ S ) W ς ( ˇ L -ℓ S ) × ( ℓ S · ( ℓ S -ˇ L ) ) 1 C TT, obs ℓ S 1 C TT, obs | ˇ L -ℓ S | × ∫ d 2 ℓ L (2 π ) 2 W λ ( ℓ L ) W λ ( ˇ L -ℓ L ) ( ℓ L · ( ℓ L -ℓ S )) × ( ( ˇ L -ℓ L ) · ( ℓ S -ℓ L ) ) ( ℓ L · ( ℓ L -ˇ L ) ) × C TT ℓ L C TT, fid ℓ L C TT, obs ℓ L C TT | ˇ L -ℓ L | C TT, fid | ˇ L -ℓ L | C TT, obs | ˇ L -ℓ L | ] -1 , (1) \n⟨ Ψ ˇ L ⟩ = 2 A ˇ L ∫ d 2 ℓ S (2 π ) 2 W ς ( ℓ S ) W ς ( ˇ L -ℓ S ) × ( ℓ S · ( ℓ S -ˇ L ) ) 1 C TT, obs ℓ S 1 C TT, obs | ˇ L -ℓ S | × ∫ d 2 ℓ L (2 π ) 2 W λ ( ℓ L ) W λ ( ˇ L -ℓ L ) ( ℓ L · ( ℓ L -ℓ S )) × ( ( ˇ L -ℓ L ) · ( ℓ S -ℓ L ) ) ( ℓ L · ( ℓ L -ˇ L ) ) × C TT ℓ L C TT, fid ℓ L C TT, obs ℓ L C TT | ˇ L -ℓ L | C TT, fid | ˇ L -ℓ L | C TT, obs | ˇ L -ℓ L | C ϕϕ | ℓ S -ℓ L | , (2) \nwhere A ˇ L is the normalization for a cross spectrum C λς ˇ L between large-scale gradient power λ and small-scale gradient power ς such that Ψ ˇ L = A ˇ L C λς ˇ L . These equations deviate slightly from their presentation in C24; we generalize them here to include the expected SCALE response with respect to changes in cosmology (reflected in the expected primary temperature power C TT ℓ ) while holding our choice in filters fixed (reflected by the total observed power C TT, obs ℓ = ˜ C TT, fid ℓ + N TT ℓ and the expected primary temperature power at a fiducial cosmology C TT, fid ℓ ). These integrals are constructed in a 4-dimensional Fourier space, and an implementation in which the integrals are numerically computed with the mid-point rule is provided in our publicly available package cmbpix 5 . Consider an example computation of analytic SCALE Ψ ˇ L for multipoles 2 < ˇ L < 2002 such that the width of the small-scale window for ς is ℓ S, max -ℓ S, min = 2000, the large-scale window for λ is 0 < ℓ L < 3000, and each Ψ ˇ L is evaluated with a Riemann sum on 2-dimensional grids of ∆ ℓ S = 75 and ∆ ℓ L = 100. This takes O (10 4 s) or O (10 min) to compute. The numerical accuracy and computational speed of Eq. (1)-(2) is dependent on the resolution of the grid(s) \non which it is evaluated. Regardless, numerically integrating the analytic SCALE products with the midpoint rule is slow enough that one would desire considerable speedups. One approach to speeding up the evaluation of Eq. (1)-(2) is to consider Monte Carlo (MC) integration. We find that a Monte Carlo integration offers a good balance of speed and accuracy and is implemented in cmbpix . One drawback of Monte Carlo integration is its non-deterministic nature, manifesting as an inherent imprecision which is dependent on the number of samples with which the integral is evaluated. We find that MC integration of Eq.(1)-(2) produces approximately ∼ 1% scatter (68% region) when evaluated with N samples ∼ O (10 5 ) samples. This is the accuracy for the evaluation at one ˇ L mode, and when we bin SCALE powers into bands of width ∆ ˇ L = 71, we expect the scatter to be reduced by a factor of √ ∆ ˇ L ≈ 8 . 4. Evaluating Eq.(1)-(2) with N samples = 2 × 10 5 within the MC integration takes O (10 s). This is a significant speed-up, but a further increase in speed is desired for parameter estimation. \nEmulation is further motivated if one wishes to include conventional lensing reconstruction information ( L ≲ 1250) from a quadratic estimator. Typically, an 'observed' lensing power spectrum ˆ C κκ L estimated from a QE reconstruction requires the subtraction of biases N (0) ,κκ L and N (1) ,κκ L such that [3, 6, 7, 41-43] \nC κκ L ∼ ˆ C κκ L -N (0) ,κκ L -N (1) ,κκ L . (3) \nThe zeroth-order reconstruction bias N (0) ,κκ L includes contributions from the disconnected four-point function which is non-zero even in the absence of lensing. It is, in practice, dependent on the particular observed realization of the CMB [44], and the realization-dependent N (0) ,κκ, RD L can be estimated with a combination of the observed CMB temperature power spectrum ˆ C TT ℓ of the particular realization along with the theoretical power spectrum C TT ℓ used in the QE filters [41-43, 45, 46]. The first-order reconstruction bias N (1) ,κκ L includes nonGaussian contributions from the connected four-point function that are not included in the quadratic estimator. For a configuration with isotropic noise, it can be analytically computed with an integral constructed in 4dimensional Fourier space similar to Eq. (1)-(2). Two equivalent representations are derived in Refs. [6, 7]. The version from Ref. [6] for the auto-correlation of the reconstructed lensing potential field ˆ ϕ with the standard temperature-temperature ( TT ) quadratic estimator to estimate the lensing potential power ˆ C ϕϕ L is repeated \nhere: \nN (1) ,ϕϕ TT,TT ( L ) = A 2 TT ( L ) L 2 ∫ d 2 ℓ 1 (2 π ) 2 ∫ d 2 ℓ ' 1 (2 π ) 2 × F TT ( ℓ 1 , ℓ 2 ) F TT ( ℓ ' 1 , ℓ ' 2 ) × { C ϕϕ | ℓ 1 -ℓ ' 1 | f TT ( -ℓ 1 , ℓ ' 1 ) f TT ( -ℓ 2 , ℓ ' 2 ) + C ϕϕ | ℓ 1 -ℓ ' 2 | f TT ( -ℓ 1 , ℓ ' 2 ) f TT ( -ℓ 2 , ℓ ' 1 ) } . (4) \nThe normalization A TT as well as the lensing weight functions f TT and F TT are the same as those presented in Ref. [3, 6] such that L = ℓ 1 + ℓ 2 = ℓ ' 1 + ℓ ' 2 (we are considering contributions to the connected four-point function). We repeat the definitions of A TT , f TT , and F TT here: \nA TT ( L ) = L 2 [ ∫ d 2 ℓ 1 (2 π ) 2 f TT ( ℓ 1 , ℓ 2 ) F TT ( ℓ 1 , ℓ 2 ) ] , (5) \nf TT ( ℓ A , ℓ B ) = C T ∇ T ℓ A ( L · ℓ A ) + C T ∇ T ℓ B ( L · ℓ B ) , (6) \nF TT ( ℓ A , ℓ B ) = f TT ( ℓ A , ℓ B ) 2 C TT, obs ℓ A C TT, obs ℓ B , (7) \nwhere ℓ A and ℓ B can be substituted with any of { ℓ 1 , ℓ 2 , ℓ ' 1 , ℓ ' 2 } . f TT is defined with respect to the expected lensing response C T ∇ T ℓ [47] which is readily computed in CAMB , and it is entirely determined by the effects of lensing on the temperature field. F TT represents the filter used in the quadratic estimator, and it is defined with respect to the expected lensing response C T ∇ T ℓ weighted by the total observed power C TT, obs ℓ . This distinction is relevant for computing the expected bias spectrum for different sets of cosmological parameters while holding the filtering fixed at a fiducial cosmology. Repeatedly computing the expected N (1) ,κκ L during parameter estimation using Eq. (4) is prohibitively slow for the same reasons as for the SCALE observables. In practice, N (1) ,κκ L is typically estimated through Monte Carlo simulations of observational noise at a fiducial cosmology, and its dependence on cosmological parameters is usually included as an approximation in the likelihood with its dependence (both directly and indirectly) on C TT ℓ and C κκ L [42, 43]. These approximations rely on firstorder derivatives of Eq. (4) with respect to C TT ℓ and C κκ L which require calculations similar to Eq. (4). An emulator can quickly predict N (1) ,κκ L as computed with Eq. (4) for a wide range of cosmological parameters, which removes the need for its approximation. This approach works well for our simple models/analyses, though the Monte Carlo computation of N (1) ,κκ L and the associated approximations are well-suited for observed data which is contaminated with foregrounds, masking, etc. \nTABLE I. The set of cosmological parameters chosen for the fiducial model.", 'B. Cosmological model': "Consider a simple cosmological model for which conventional CMB observables can be computed with CAMB or CLASS , including the six ΛCDM parameters { Ω c , Ω b , A s , n s , h, τ } and single massive neutrino species with mass m ν . The chosen fiducial values are listed in Table I. This assumed base model is a good test of the power of SCALE to constrain the small-scale lensing power, which is particularly sensitive to m ν [18, 19, 22]. In addition to this base model where the lensing power spectrum is largely scaled by m ν , we consider a phenomenological model for scale-dependent lensing power suppression that is a feature of many dark matter models beyond the standard picture. These models are easily modeled with CAMB 's included get partially lensed cls method, which applies a lensing amplitude function A lens ( L ) to the non-suppressed lensing power C κκ L such that the suppressed lensing power spectrum C κκ, sup L is \nC κκ, sup L = A lens ( L ) C κκ L . (8) \nWe choose to model the suppression of small-scale lensing using a function similar to a logistic function, parameterized by a suppression scale L 0 , lensing amplitude decay rate B , and suppression amplitude A min : \nA lens ( L ) = 1 -A min 1 + exp ( B ( L -L 0 )) + A min . (9) \nThis function asymptotes to unity for L ≪ L 0 , and to A min for L ≫ L 0 , and the decay rate parameter B sets the steepness of the transition. The shape of the suppression model is reminiscent of those predicted by fuzzy dark matter models as shown in e.g. Ref. [32]. Our choice \nFIG. 1. Top : The lensing convergence power spectrum C κκ L for the fiducial cosmology as computed from CAMB is compared to a model with a suppression of lensing power applied at small scales. Also shown are the three small-scale filter ranges for the applications of SCALE chosen to recover information about the lensing suppression. Bottom : The lensing amplitude A lens applied to C κκ L in our suppression model is shown in dark purple (see parameters in Table I). A version with lower L 0 = 7000 is shown in yellow. A version with higher B = 0 . 005 is shown in red. A version with higher A min = 0 . 5 is shown in green. Also shown is the fiducial model with no suppression obtained by setting A min = 1. \n<!-- image --> \nin the fiducial values for the suppression model are listed in Table I, and the corresponding A lens is illustrated in FIG. 1. This specific model exhibits a modulation in the CMB lensing power at small-scales where SCALE is particularly effective. Our emulator is capable of handling input lensing suppression parameters in each prediction, but in our base analysis we fix A min = 1 to achieve effectively no suppression for all L . In other words, we predict power spectra with standard lensing by passing in A min = 1 into our emulator. We later explicitly turn on lensing suppression for a separate analysis by including A min as a free parameter, and generating a new simulation with an underlying value of A min = 0 . 25, which we describe later. A visualization of the chosen suppression model is shown in Figure 1, along with a few examples where the parameters L 0 , B , and A min are varied individually.", 'C. Construction of emulators': 'We construct a single emulator to predict theoretical CMB (partially) lensed TT power spectra ˜ C TT ℓ , lensing convergence power spectra C κκ L , lensing reconstruction bias N (1) ,κκ L , and SCALE spectra Ψ ˇ L for a wide range of cosmological parameters. The emulators are created in the COMSOPOWER 6 framework [48], which is a Python \nTABLE II. The set of cosmological parameters and their prior ranges used to generate training data for the emulators. The ΛCDM parameters are chosen to be centered on the Planck 2018 best-fit cosmology with ± 4 . 5 σ on either side [50]. Note that the emulators are trained on parameters Ω b , Ω c , and ln(10 10 A s ), but are sampled as shown as direct inputs for CAMB .a Only affects SCALE band-powers. \npackage for the construction of emulators for cosmological observables. It is based on TensorFlow 7 [49], and it provides a structure for training and using neural network (NN) emulators.', '1. Training data': "The training data consist of N train = 8192 sets of cosmological parameters sampled from a Latin hypercube for uniform priors in the ranges set by Table II. This is a relatively small number of training points compared to a characteristic N train = O (10 5 ) [48], but we show later that it is sufficient to train the emulators to a satisfactory degree of accuracy. The range of ΛCDM parameters is chosen to be centered on the reported Planck 2018 values with width ± 4 . 5 σ [50]. We choose a training range of [0 , 2] for the lensing suppression parameter A min , to ensure that the emulator is well-trained to recover the cosmology in the 'vanilla' case where A min = 1 and thus where the suppression is effectively turned off as well as the case where suppression is explicitly included. For each set of parameters in our training space, we compute the following: \n- 1. The unlensed TT power spectrum C TT ℓ from CAMB for 2 ≤ ℓ ≤ 20 000.\n- 2. The lensing response C T ∇ T ℓ from CAMB for 2 ≤ ℓ ≤ 8000. \nTABLE III. Summary of binning for CMB observables. \n- 3. The suppressed lensing power spectrum C ϕϕ, sup L = A lens ( L ) C ϕϕ L from CAMB for 2 ≤ L ≤ 20 000.\n- 4. The partially lensed TT power spectrum ˜ C TT ℓ from CAMB for 2 ≤ ℓ ≤ 20 000.\n- 5. The quadratic estimator normalization A TT from pytempura 8 [51] for 2 ≤ L ≤ 1208.\n- 6. The expected lensing reconstruction bias N (1) ,κκ L from Eq. (4) with Monte Carlo integration in cmbpix for 2 ≤ ℓ ≤ 1208.\n- 7. The SCALE normalization A ˇ L from Eq. (1) with Monte Carlo integration in cmbpix for 2 ≤ ˇ L ≤ 1989.\n- 8. The SCALE observables ⟨ Ψ ˇ L ⟩ from Eq. (2) with Monte Carlo integration in cmbpix for 2 ≤ ˇ L ≤ 1989. \nWe compute the (partially) lensed CMB TT power spectrum out to lmax=20000 with CAMB for all N train = 8192 sets of parameters in the training range, remembering to set lens potential accuracy=8 for highℓ accuracy [39]. The highℓ regime is required for the calculation of the SCALE quantities shown Eq. (1)-(2), which depend on the small-scale power spectra. We bin the TT power spectra into band-powers following the Planck prescription [45] 9 . The 30 largest-scale powers between 2 ≤ ℓ ≤ 31 are kept unbinned, and the powers between 32 ≤ ℓ ≤ 3002 are binned into 99 band-powers of width ∆ ℓ = 30. The bin weights are given by: \nw ℓ b ℓ = ℓ ( ℓ +1) ∑ ℓ ∈ b ℓ ( ℓ +1) , (10) \nwith band-powers ˜ C TT ℓ b = ∑ ℓ ∈ b w ℓ b ℓ ˜ C TT ℓ and bin centers ℓ b = ∑ ℓ ∈ b w ℓ b ℓ ℓ . The binning procedure is summarized in Table III. \nWe also compute the CMB lensing power spectrum out to L = 20000, as the small-scale C ϕϕ L is required for the SCALE calculations. We compute the expected N (1) ,ϕϕ L \nfor each set of cosmological parameters using a Monte Carlo integration method for Eq. (4) included in cmbpix . Since the filter weight F TT is dependent on our choice in QE filtering, we hold it fixed in our training spectra by setting the lensing response C T ∇ T ℓ and the total observed spectrum C TT, obs ℓ = ˜ C TT ℓ + N TT ℓ at the fiducial cosmology parameters described in Table I. The dependence of N (1) ,ϕϕ L on the cosmology of the training space is through the suppressed C ϕϕ, sup L , along with a contribution from the lensing response C T ∇ T ℓ in f TT (which, as stated below Eq. 6, reflects the expected effects of lensing on the temperature field). Finally, we also compute the quadratic estimator normalization A TT for every set of parameters in the training space with Eq. 5. We wish to include information from conventional CMB lensing observables in our likelihood, so we save the expected 'reconstructed' spectrum C κκ, rec L as band-powers with a bin width of ∆ L = 71 between 2 ≤ L ≤ 1208: \nC κκ, rec L = A 2 TT A 2 TT, fid C κκ, sup L + N (1) ,κκ L . (11) \nThe conversion between lensing potential and lensing convergence power is C κκ L = [ L ( L +1)] 2 C ϕϕ L / 4, and D dd L = [ L ( L +1)] 2 C ϕϕ L / 2 π is the default power spectrum returned by CAMB . The 'reconstructed' spectrum reflects what the lensing convergence power spectrum C κκ L is expected to be for a given underlying cosmology given a chosen set of fiducial parameters for filtering (after subtracting N (0) ,κκ L ), hence the renormalization with respect to the fiducial normalization A TT, fid in the first term. We further discuss the physical meaning of C κκ, rec L in § IV. \nWe compute expected SCALE observables for each set of parameters in our training space with Monte Carlo integration of Eq. (1)-(2) included in cmbpix . Similar to N (1) ,κκ L , we hold the filter weights fixed at the fiducial cosmology: i.e., all factors of C TT, fid ℓ and C TT, obs ℓ in Eq. (1)-(2) are held fixed with respect to Table I. The dependence of SCALE observables on the cosmological parameters in the training space (and by extension during parameter estimation) is through the lensing power spectrum C ϕϕ L along with one factor each of C TT ℓ L and C TT | ˇ L -ℓ L | in the numerator of the innermost integrals of Eq. (1)-(2), which together reflect changes in the expected temperature trispectrum from lensing. \nDifferent parts of the highL lensing power are probed by altering the filtering scheme within SCALE: we include in the emulator the dependence of SCALE observables on the small-scale filter width ∆ ℓ S and center ℓ S , which is equivalent to altering the limits of the outer ℓ S integral in Eq. (1)-(2). As an example, ℓ S ∈ [8 000 , 10 000] corresponds to ∆ ℓ S = 2000 and ℓ S = 9000. These 'parameters' allow for flexibility when applying the likelihood for different SCALE data vectors constructed from the same map(s), but with different ranges of small-scale filtering. We further discuss this in § IV. We save the expected Ψ ˇ L for our whole training \nset as band-powers with a bin width of ∆ ˇ L = 71 between 2 ≤ ˇ L ≤ 1989. The band-powers of all CMB observables that we include in our likelihood are summarized in Table III.", '2. Emulator performance': "The emulator we construct contains 4 hidden layers, each with 512 nodes, following the default configuration for a neural network emulator in COSMOPOWER [48]. The input layer contains the 12 parameters in Table II, and outputs are the combined 174 band-powers listed in Table III. The relevant parameters for the emulator are converted to Ω b , Ω c , and ln(10 10 A s ) as inputs rather than their counterparts in Table II. We include 87.5% of the full training set N train = 8192 during training, with the remaining 1024 reserved for validation. The emulator takes approximately 1 min 30 s to train 10 , and it only needs to be trained once for a given set of training spectra computed with a chosen cosmological model. The trained model can be saved and loaded for future use without retraining. \nWe find that the emulator performs much more quickly than the original counterparts used to compute each set of band-powers, providing predictions of their respective observables in O (10 -3 s). FIG. 2 shows that the lensed CMB TT emulator is extremely precise, with a prediction error of ≲ 0 . 05% per band-power. Similarly, FIG. 2 shows that the emulator predicts expected reconstructed band-powers with an error of ≲ 0 . 1%, and the SCALE observables with a precision of ≲ 0 . 5% per band-power. The values of each band-power from the emulator are predicted independently and do not correlate with one another. This applies to their scatter as well; i.e., the bin-to-bin scatter of the emulator's predictions is uncorrelated. We attribute the lower precision of the reconstruction and SCALE band-powers to the inherent scatter of the input spectra computed with Monte Carlo integration related to the number of samples. We also find that the emulator is more accurate/precise when trained on the band-powers rather than their unbinned counterparts and then binning the full predicted spectra (a factor of ∼ 13 . 5 × more scatter for TT band-powers and ∼ 1 . 5 × more scatter for the other band-powers). The scatter from predicting the full SCALE spectra multipole-bymultipole is at approximately the same level as the precision of the MC integration itself ( ∼ 1% before binning). Finally, the emulator exhibits an insignificant bias in its predictions, as the prediction errors effectively scatter around zero in FIG. 2. A summary of the computational speedup provided by the emulator is provided in Table IV. \nTABLE IV. A comparison of computation speed for expected CMB observables. The computation time of C κκ L is comparable to that of ˜ C TT ℓ . The computation times of N (1) ,κκ L and A ˇ L are comparable to that of Ψ ˇ L . The emulator predicts all band-powers summarized in Table III at once for a given set of parameters with a significant speedup for all observables.", 'III. SIMULATIONS': "We compute a suite of full-sky simulations of the lensed CMB temperature field with the lenspyx 11 package [52, 53], which wraps around methods from DUCC 12 (Distinctly Useful Code Collection) and allows for efficient and accurate lensing and de-lensing operations with spherical harmonics transforms [53]. A notebook with instructions to simulate the lensed CMB temperature and polarization is provided in the lenspyx repository. We choose to simulate maps with HEALPix resolution NSIDE=8192 , as the lenspyx accuracy is good out to ℓ ≈ 2 × NSIDE . The simulations are constructed with the same fiducial cosmological parameters as shown in Table I. We additionally apply the quadratic estimator with so-lenspipe 13 as well as SCALE with the following procedure: \n- A. Generate a lensed temperature field, and estimate the total 'observed' temperature power ˆ C TT ℓ .\n- 1. Compute the fiducial unlensed CMB TT power spectrum C TT ℓ and lensing potential power spectrum C ϕϕ L with CAMB out to lmax=20000 with lens potential accuracy=8 .\n- 2. Generate spherical harmonic coefficients T ℓm and ϕ ℓm for both an unlensed temperature T using C TT ℓ and lensing potential ϕ field using C ϕϕ L with synalm .\n- 3. Transform the lensing potential field into a spin-1 deflection field d with lenspyx 's almxfl method.\n- 4. Compute the lensed temperature field ˜ T using the unlensed temperature T and deflection d coefficients with alm2lenmap . This returns a lensed temperature field ˜ T in map-space at NSIDE=8192 . \nFIG. 2. A validation of the emulator for our CMB observables. The % error represented here is computed as the difference between the emulator prediction and the computed output (with CAMB for { C TT ℓ , C κκ L } , and Monte Carlo integration with CAMB spectra for { N (1) ,κκ L , Ψ ˇ L } ) divided by the computed output. We perform training using 7168 out of a set of 8192 band-powers spanning a large range of cosmological parameters outlined in Table II. We perform validation tests using the remaining 1024 sets of band-powers unseen by the emulator during training. The filled rectangles indicate the expected 68-percentile precision centered on the median, and the bin-to-bin scatter of emulator predictions is uncorrelated. The precision of each predicted C TT ℓ band-power is generally within 0 . 05%, and these rectangles are not visible on this y -scale. Error bars indicate the expected observational variance of band-powers, comparable to the diagonal of the covariance matrix underlying FIG. 3. There is no significant bias from the emulator's predictions within our chosen range of cosmological parameter space, and the precision is generally better than the expected observational variance of all band-powers at our chosen level of noise. \n<!-- image --> \n- 5. Generate spherical harmonic coefficients N ℓm for a Gaussian noise temperature field N with synalm (noise parameters also in Table I), convert to map-space with alm2map , and add to the lensed temperature field T obs = ˜ T + N .\n- 6. Convert the observed temperature field T obs to spherical harmonic coefficients T obs ℓm using map2alm , and estimate the total 'observed' TT power spectrum ˆ C TT ℓ with alm2cl , which is saved.\n- B. Apply the quadratic estimator, and estimate the reconstructed lensing power spectrum ˆ C κκ, rec L = ˆ C κκ L + N (0) ,κκ, RD L .\n- 1. Apply an isotropic filter to T obs ℓm with the expected lensing response C T ∇ T ℓ such that 2 ≤ ℓ ≤ 3000 to get T filt ℓm .\n- 2. Reconstruct the lensing potential field ϕ rec ℓm from T filt ℓm with so-lenspipe , and estimate the reconstructed lensing power spectrum ˆ C κκ L with alm2cl .\n- 3. Compute the realization dependent N (0) ,κκ, RD L using the observed temperature power spectra ˆ C TT ℓ with so-lenspipe .\n- 4. Save the total reconstructed lensing power spectrum ˆ C κκ, rec L = ˆ C κκ L -N (0) ,κκ, RD L .\n- C. Apply SCALE, and estimate the cross-spectrum between the large-scale temperature gradient, and the small-scale temperature gradient Ψ ˇ L = A ˇ L C λς ˇ L .\n- 1. Apply a low-pass filter such that 0 < ℓ L < 3000 along with a Wiener filter to the observed temperature field T obs ( W λ ( ℓ ), shown below as Eq. (12)) with almxfl . The product is a set of spherical harmonic coefficients, that when converted to map space with a spin-1 inverse transform alm2map spin , produces the two large-scale gradient components [ ∇ θ T L , ∇ ϕ T L / sin θ ] that make up the λ = ( ∇ θ T L ) 2 + ( ∇ ϕ T L / sin θ ) 2 map of largescale temperature gradient power required for one half of SCALE. Note that the filter is constructed with the theoretical spectra from Step A1 using CAMB , and follow the fiducial cosmology in Table I. \nW λ ( ℓ ) = { √ ℓ ( ℓ +1) C TT ℓ ˜ C TT ℓ + N TT ℓ , ℓ < 3000 0 , ℓ ≥ 3000 . (12) \n- 2. Convert the λ map into spherical harmonic space with map2alm .\n- 3. Apply a high-pass filter such that ℓ S, min < ℓ S < ℓ S, max along with an inverse variance filter to the observed temperature field T obs (shown below as Eq. (13)) with almxfl . The product is a set of spherical harmonic coefficients, that when converted to map space with a spin-1 inverse transform alm2map spin , produces the two small-scale gradient components [ ∇ θ T S , ∇ ϕ T S / sin θ ] that make up the ς = ( ∇ θ T S ) 2 + ( ∇ ϕ T S / sin θ ) 2 map of smallscale temperature gradient power required for \n} \n' \n' \n' \n/lscript \n{ \n} \n{ \nFIG. 3. The correlation matrix between simulated CMB temperature, reconstructed lensing, and SCALE band-powers at the fiducial cosmology (see Table I). The correlations are computed from 600 simulations. The band-powers are binned similarly to the Planck scheme for ˆ C TT ℓ [45]: i.e., unbinned for 2 ≤ ℓ ≤ 31 (sectioned with dashed lines) and with width ∆ ℓ = 30 for 32 ≤ ℓ ≤ 3002. The reconstructed lensing band-powers ˆ C κκ, rec L are binned with width ∆ L = 71 for 2 ≤ L ≤ 1208. SCALE bandpowers ˆ Ψ ˇ L are binned with width ∆ ˇ L = 71 for 2 ≤ ˇ L ≤ 1989. Combinations of contributing components { ˆ C TT ℓ , ˆ C κκ, rec L , ˆ Ψ ˇ L } and their underlying covariance matrix are used in our likelihoods combining conventional CMB observables and SCALE bandpowers. \n<!-- image --> \nthe other half of SCALE. Note that the filter is constructed with the theoretical spectra from Step A1 using CAMB , and follow the fiducial cosmology in Table I. \nW ς ( ℓ ) = {√ ℓ ( ℓ +1) 1 ˜ C TT ℓ + N TT ℓ , ℓ 1 , min < ℓ 1 < ℓ 1 , max 0 , else . (13) \n- 4. Convert the ς map into spherical harmonic space with map2alm .\n- 5. Estimate the cross spectrum ˆ C λς ˇ L between λ and ς with alm2cl , and multiply by A ˇ L to get ˆ Ψ ˇ L then save. \nWe show in § IV that, in principle, Part C may be \nrepeated on the same realization with various choices in the small-scale ℓ S, min < ℓ S < ℓ S, max filter limits, and included in the same likelihood. We include three applications of SCALE for each realization: 8 000 < ℓ S < 10 000, 9 000 < ℓ S < 11 000, 10 000 < ℓ S < 12 000, and apply their appropriate normalization A ˇ L from Eq. (1). We repeat the above procedure for 600 simulations, and we save our CMB observables as bandpowers of { ˆ C TT ℓ , ˆ C κκ, rec L , ˆ Ψ 8k-10k ˇ L , ˆ Ψ 9k-11k ˇ L , ˆ Ψ 10k-12k ˇ L } following Table III. The correlation matrix between { ˆ C TT ℓ , ˆ C κκ, rec L , ˆ Ψ 8k-10k ˇ L } is shown in FIG. 3. The correlation between SCALE observables and the other CMB observables is generally smaller than about 5%. This suggests that SCALE includes unique cosmological information from the small-scale CMB lensing. Note that this in- \n' \nFIG. 4. The correlation matrix between SCALE bandpowers at the fiducial cosmology (see Table I). The correlations between three applications of SCALE are depicted here, with shared large-scale filters 2 ≤ ℓ L ≤ 3000 and different small-scale filters indicated by superscripts. The correlations are computed from 600 simulations. The strongest off-diagonal entries indicate that SCALE band-powers at the same ˇ L are approximately 20-50% correlated between applications of small-scale filters which share half of their multipole coverage. Correlations between SCALE with the other ℓ S filters { ˆ Ψ 9k-11k ˇ L , ˆ Ψ 10k-12k ˇ L } and the other CMB observables { ˆ C TT ℓ , ˆ C κκ, rec L } are similar to those of ˆ Ψ 8k-10k ˇ L in FIG. 3. \n<!-- image --> \nformation is intentionally excluded from our quadratic estimator configuration because we showed in C24 that SCALE will outperform the QE at small-scales, and we wish to study how well SCALE can fulfill its role there. \nWe find that the SCALE observables exhibit very low levels of covariance between band-powers, as we found with the flat-sky simulations in C24. However, bandpowers at the same ˇ L across small-scale filters with overlapping ℓ S ranges (e.g., ˆ Ψ 8k-10k ˇ L and ˆ Ψ 9k-11k ˇ L for at the same ˇ L ) are highly correlated, with off-diagonal entries of approximately 20-50%. This is shown in FIG. 4, and it implies that applying SCALE with different small-scale filter ranges indeed characterizes different parts of the lensing power spectrum. \nWe additionally produce one more set of CMB observables with a separate realization following the above steps. This extra realization serves as the data vector that will be used in subsequent sections for cosmological inference, and is not included in the construction of the covariance/correlation matrices. A comparison of this realization's observable band-powers with the theoretical values at the fiducial cosmology is shown in FIG. 5. We produce one final set of CMB observables with a realization that includes a lensing suppression C κκ, sup L in Step A2 with A min = 0 . 25, and all other parameters following Table I. This realization is used later to test SCALE's ability to constrain more exotic small-scale clustering phenomena.", 'IV. CONSTRUCTING A LIKELIHOOD WITH SCALE': "Consider a model defined by ΛCDM cosmology with the addition of one neutrino mass eigenstate parameterized with m ν . The general suppression of small-scale lensing due to the massive neutrino offers a simple test of SCALE's constraining power. Our initial vector of parameters is ⃗ θ = { m ν , Ω c , Ω b , ln(10 10 A s ) , n s , h, τ } . The theoretical model that our emulator is trained on generally includes the lensing suppression at small scales, but for our initial analysis, we fix the relevant parameters ⃗ θ sup = { L 0 , B, A min } to their fiducial values in Table I. In particular, setting A min = 1 effectively turns off the lensing suppression. There is an implicitly constrained parameter Ω Λ such that Ω c +Ω b +Ω ν +Ω Λ = 1. Our initial data vector ⃗ d = { ˆ C TT ℓ , ˆ C κκ, rec L , ˆ Ψ 8k-10k ˇ L } is composed of TT , QE reconstruction, and SCALE band-powers. We have constructed an emulator in § II to predict the theoretical expected band-powers given a set of cosmological parameters: ⃗ t ( ⃗ θ ). Finally, we have empirical estimates of the covariance between all band-powers of the data vector from our set of simulations in § III: C . This allows us to construct a multivariate normal log-likelihood for the data vector ⃗ d given a set of parameters ⃗ θ : \nlog p ( ⃗ d | ⃗ θ ) ∼ log N ( ⃗ d | ⃗ t ( ⃗ θ ) , ˆ C -1 ) ∼ -1 2 ( ⃗ d -⃗ t ( ⃗ θ )) T ˆ C -1 ( ⃗ d -⃗ t ( ⃗ θ )) . (14) \nNote that the covariance C is estimated from N sims = 600 realizations, and the unbiased estimator for the inverse covariance must include the Hartlap factor [54]: \nˆ C -1 = N sims -P -2 N sims -1 C -1 , (15) \nwhere P is the number of band-powers included in the data vector. There are P TT = 129 TT band-powers, P QE = 17 QE band-powers, and P SCALE = 28 bandpowers for each application (with different small-scale ℓ S filters) of SCALE. It is recommended that there is a minimum of realizations N sims ≳ 2 P to estimate the inverse covariance, which we have satisfied [54]. \nThe likelihood we apply compares the estimated QE reconstructed spectrum (when included) as \nC κκ, rec L = A 2 TT A 2 TT, fid C κκ L + N (1) ,κκ L ∼ ˆ C κκ L -N (0) ,κκ, RD L = ˆ C κκ, rec L , (16) \nwith C κκ, rec L predicted by our emulator as a function of a given set of parameters ⃗ θ , and ˆ C κκ, rec L estimated directly from each realization. In practice (e.g., with the ACT and Planck lensing analyses [42, 43]), the QE-relevant quantities are directly compared to the expected CMB lensing power spectrum C κκ L as expressed in Eq. (3). Our \n25 \nFIG. 5. A comparison between predicted CMB observable band-powers from the emulator at the fiducial cosmology with a set of band-powers computed from simulation. The simulated band-powers are overlaid with error-bars corresponding to the diagonal of FIG. 3. Predicted band-powers at the best fit cosmology from combining all three observables in § V are also shown. The residuals with respect to the fiducial band-powers are shown in the bottom panels. The 'observed' band-powers from simulation match well with both the fiducial and best fit values. \n<!-- image --> \nconstruction includes the dependence of N (1) ,κκ L on parameters directly on the theory side of the likelihood because the emulator allows us to quickly determine the expected reconstruction bias for each given set of parameters. This is in contrast to a more practical likelihood, which includes N (1) ,κκ L and its dependence on cosmological parameters as first-order corrections (with respect to the dependence of N (1) ,κκ L on C TT ℓ and C κκ L ) to the observed quantities due to challenges with repeatedly computing N (1) ,κκ L [42, 45, 46]. Our construction is equivalent to the practical likelihood in the limit of well-controlled, isotropic noise, and the usual treatment of N (1) ,κκ L is preferred for real data analysis (notably with the presence of foregrounds). \nWe set broad and uniform priors p Uni ( ⃗ θ ) for every parameter in ⃗ θ following Table II except for τ , for which we impose a Gaussian prior about the fiducial value (Table I): \np ( τ ) ∼ N (0 . 06 , σ τ ) , (17) \nwhere we choose either σ τ = 0 . 007 set by the value reported by Planck 2018 [50], or σ τ = 0 . 002 set by the cosmic variance limit [55]. We also include an additional likelihood which includes forecasted constraints from Baryon Acoustic Oscillation (BAO) information from the 5-year survey of the Dark Energy Spectroscopic Instrument (DESI, [56]), which mainly constrains the matter density Ω m . We follow the steps in Appendix V of Ref. [57] to construct a Fisher matrix F for BAO observables and the covariances between our other parameters, including m ν , Ω c , Ω b , and h . The BAO log-likelihood is constructed as follows: \nlog p BAO ( ⃗ θ | F ) ∼ -1 2 ( ⃗ θ -⃗ θ fid ) T F ( ⃗ θ -⃗ θ fid ) , (18) \nwhere ⃗ θ fid is the vector of fiducial parameters from Table I. Our final posterior is then expressed by the following: \np ( ⃗ θ | ⃗ d ) ∼ N ( ⃗ d | ⃗ t ( ⃗ θ ) , ˆ C -1 ) N ( τ | 0 . 06 , σ τ ) p BAO ( ⃗ θ | F ) p Uni ( ⃗ θ ) . (19) \nIn principle, the data vector ⃗ d , covariance matrix C , and theory vector constructed from emulators ⃗ t ( ⃗ θ ) can contain any combination of TT , QE, and SCALE bandpowers as long as all three are constructed consistently. We present in § V results from several combinations of observable band-powers. We only consider one application of SCALE with ℓ S, min = 8000 and ℓ S, max = 10000 for our initial analysis. The model including a massive neutrino simply shifts the amplitude of the lensing potential power in the small-scale regime that we consider, so a single SCALE estimator is sufficient. \nIncluding multiple applications of SCALE with different ℓ S, min and ℓ S, max in a single fit should allow for constraints on models which change the shape of the lensing power spectrum C κκ L . To test this, we perform a separate analysis including the other two applications of SCALE in our likelihood with a full covariance combining FIG. 3 and FIG. 4. A separate data vector from the realization in § III with lensing suppression following Table I with A min = 0 . 25 is used, and we allow the relevant parameters ⃗ θ sup to be free and sampled. We sample the logarithm of the decay rate parameter ln( B ) to allow a larger dynamic range. \nWe construct and sample our probabilistic model with the Python implementation of Markov Chain Monte Carlo (MCMC) techniques in emcee 14 [58]. We choose to \nFIG. 6. The sampled posterior distribution of the probabilistic model described in Eq. (19) using the TT band powers along with a cosmic variance τ prior σ τ = 0 . 002, and BAO likelihood is shown in brown . The resulting sampled posterior distribution with the addition of lensing information from both lensing reconstruction and SCALE is shown in green . The contours indicate the 68% (solid), 95% (dashed) and 99.7% (dotted) regions of a kernel density estimate for each 2-dimensional marginalized posterior. The fiducial values are indicated with black lines and × symbols. The addition of CMB lensing observables makes the difference in a detection of the minimum neutrino mass m ν (see FIG. 7). We also see that the covariances between matter clustering related parameters such as m ν , Ω c and Ω b tighten up with the addition of lensing observables. \n<!-- image --> \nuse emcee rather than more commonly-used software designed specifically for cosmology such as CosmoMC 15 [59] \nor cobaya 16 [60, 61] because it offers a simple way to construct log-probabilities with the added flexibility of \nTABLE V. Summary of cosmological parameter constraints from combinations of the observed TT , QE reconstructed, and SCALE band-powers, along with τ and BAO priors. Fiducial values from Table I are also shown for comparison. The cosmic variance τ prior σ CV τ = 0 . 002 [55] is used here. Reported values are the median and 68% credibility interval of each marginalized posterior. The median values of the posterior slightly shift around the fiducial values depending on the realization, but in general the shifts are comfortably within the 68% credibility intervals. The widths of the 68% regions do not change appreciably between realizations. \nallowing for the use of black-box functions in the model. The latter point is essential in order to use the emulator constructed in § II at each step of the chain. We use 14 walkers, or chains if holding ⃗ θ sup fixed, and 20 walkers if ⃗ θ sup are free and sampled. Each chain is run for 50 000 steps each. The first 1 000 steps are discarded as burn-in steps that have yet to converge, and we further thin the chains by a factor of 50 to reduce the auto-correlation between samples. We find that the chains converge (satisfying the Gelman-Rubin ratio requirement R < 1 . 1) after approximately 1 000 steps post-burn-in and before thinning. The results of each model are presented in § V.", 'V. RESULTS': 'The results for our base model fits using the realization of observables presented in FIG. 5 are summarized in Table V and FIG. 6-7. The band-powers as predicted by our emulator for the best-fit including all three observables is also included in FIG. 5. We find that the size of the 68% region for the best fit results do not change appreciably if we choose a different realization for the data vector. The center of the best fit can vary slightly between realizations, but the change is generally well within the 68% range. The base model allows the TT bandpowers, combined with the BAO likelihood and τ prior, to constrain most of the parameters ⃗ θ to high precision in the presence of noise levels similar to that expected from CMB-S4 [9, 10]. The exception is m ν , for which the TT -only model with either a Planck 2018 prior or a cosmic variance prior on τ is not able to detect (see FIG. 7). We also see in FIG. 6 and FIG. 7 that the marginalized posterior for m ν using only the TT band-powers as observables causes the distribution to hit the edge of the prior at 0 eV. \nThe addition of SCALE into the data vector affects the parameters most sensitive to lensing (see FIG. 6): m ν , Ω c , and Ω b . We see that the addition of SCALE alters the degeneracies between these parameters to become more constraining. Perhaps the most salient effect of including \nTABLE VI. Summary of cosmological parameter constraints on a model with small-scale lensing suppression from the observed TT , QE reconstructed, and SCALE band-powers, along with τ and BAO priors. Fiducial values from Table I are also included. The cosmic variance τ prior σ CV τ = 0 . 002 [55] is used here. Reported values are the median and 68% credibility interval of each marginalized posterior. The median values of the fit slightly shift around the fiducial values depending on the realization, but in general the shifts are comfortably within the 68% credibility intervals. The widths of the 68% regions do not change appreciably between realizations. \nSCALE is the added ability to provide evidence for nonzero m ν at 2 . 4 σ with the Planck τ prior. The effect is more prominent if we swap the τ prior to the cosmic variance limit σ τ = 0 . 002 (see Table V), which is forecasted to be achievable with upcoming data from the LiteBIRD satellite mission [55] or the CLASS ground-based survey [62]. In this case the detection jumps to a signalto-noise of 4 σ with the inclusion of SCALE. We have thus demonstrated that the addition of small-scale lensing information with SCALE provides extra constraining power for parameters which alter the lensing amplitude at ℓ ≫ 3000. \nFinally, we present the results of our lensing suppression analysis in FIG. 8 and Table VI. We find that the three sets of SCALE band-powers with different smallscale filtering regimes provides sufficient information to \nFIG. 7. The marginalized posteriors (with the cosmic variance τ prior σ τ = 0 . 002) for m ν with various combinations of CMB observables. The median of each result is indicated with solid vertical lines of their respective colours, and coloured, dashed lines indicate their 68% credible intervals. A vertical black line indicates the fiducial value at m ν = 0 . 06 eV. The marginalized posteriors using the Planck 2018 τ prior σ τ = 0 . 007 are shown in each respective panel with step histograms (medians and 68% intervals not shown). With the condition of a precise τ estimate, the addition of lensing band-powers in general tightens the distribution of the marginalized posterior enough for a 4 σ detection of minimum mass. The physical effect of neutrino mass is a mostly scale-independent change to the lensing amplitude, so this is true regardless of whether this lensing information comes from the QE reconstruction (top right), SCALE (bottom left), or both (bottom right). \n<!-- image --> \nwell-constrain the three parameters of our general lensing suppression model. This is the case even though the small-scale regimes we consider are overlapping, and technically share similar information about the smallscale lensing power spectrum. We find that the three lensing suppression parameters ⃗ θ sup are not well constrained if we perform the same analysis with a data vector consisting of only two of the applications of SCALE: for example with ⃗ d = { ˆ C TT ℓ , ˆ C κκ, rec L , ˆ Ψ 8k-10k ˇ L , ˆ Ψ 10k-12k ˇ L } . This is expected due to our three-parameter model requiring at least three measured amplitudes of the smallscale lensing power spectrum to constrain. In a similar vein, the same analysis with only conventional CMB observables { ˆ C TT ℓ , ˆ C κκ, rec L } leaves the lensing suppression \nparameters ⃗ θ sup completely unconstrained, as they do not provide any information about the small-scale lensing power spectrum by construction.Therefore, with the choice of simulated data employed here, the SCALE observables provide unique constraining power in the smallscale lensing regime.', 'VI. DISCUSSION AND CONCLUSIONS': "In this paper we explored applications of the SCALE estimator to cosmological parameter estimation, and how much information SCALE provides on top of more conventional methods for CMB lensing measurement. We originally developed SCALE in C24 as a novel estima- \nFIG. 8. The partially marginalized, sampled posterior for a simulation/model with suppressed lensing at small scales as depicted in FIG. 1. All six ΛCDM parameters were also sampled with results similar to FIG. 6, so they are not shown here. Including just a lensing reconstruction at large-scales provides sufficient information to constrain m ν . Only the full set of 3 SCALE band-powers (small-scales filtered for ℓ S ∈ { 8k-10k , 9k-11k , 10k-12k } ) provides constraining information about this particular 3-parameter suppression model. \n<!-- image --> \ntor for the amplitude of CMB lensing power at smallscales ℓ ≫ 3000. The small-scale lensing regime has yet been untouched by conventional lensing reconstruction methods alone due to limits in instrument sensitivity and concerns with foreground contamination. We showed in C24 that SCALE will outperform conventional quadratic estimators in this regime for upcoming/future experiments in terms of the signal-to-noise of a lensing amplitude measurement. While we expect that small-scale foregrounds do not correlate directly with the large-scale CMB primary temperature field, we reserve a study on the effects of foreground contamination on SCALE for future study. Using SCALE is quick and simple to implement because of its nature as the cross-spectrum of the same temperature map with two different filters applied. SCALE's outputs directly inform us about the CMB lensing power at small scales, but they do not estimate the underlying lensing field. In this work, we extended the study of SCALE by providing a framework for its application in a practical cosmological parameter estimation with other CMB observables. We further demonstrate that SCALE fills a beneficial niche by providing useful information that is complementary to wellunderstood and high-performing lensing reconstruction techniques applied to larger scales L ≲ 2000. \nThe effect of massive neutrinos on the lensing power spectrum is a nearly scale-independent decrease in amplitude, so it is a useful gauge for SCALE's effectiveness in comparison with more established QE reconstructions. We confirmed that the inclusion of SCALE observables provides sufficient information about the lensing amplitude at small-scales to provide significant evidence for non-zero m ν with a CMB-S4-like experiment. In terms of m ν , the results with SCALE in the absence of a conventional lensing reconstruction with a QE (bottom left panel of FIG. 7) have a comparable performance to the results using a lensing reconstruction without SCALE (top-right panel of FIG. 7), with both measurements depending on a well-constrained τ . This places SCALE in an interesting position for upcoming studies as a highly effective cross-check of standard CMB lensing estimation methods. This is particularly interesting given recent cosmological measurements of neutrino mass that prefer values smaller than expected from flavor oscillations, and even favor negative neutrino masses [23-27]. \nWe further established SCALE's role in future cosmological analysis of small-scale lensing with a model including a phenomenological suppression of lensing at high L ∼ 10 000. The (mostly) scale-independent nature of the effects of massive neutrinos on CMB lensing means that including one application of SCALE with a single small-scale filter ℓ S is sufficient for its estimation. We showed that including multiple applications of SCALE with different ℓ S filters would allow for additional constraints on models which alter the shape of the lensing potential power spectrum C ϕϕ L , particularly at small-scales. The lensing suppression model we chose to focus on in this work was tailored to be easily identified with our choice/configuration of ℓ S SCALE filtering. The generalization and optimization of SCALE filtering configurations is a natural path for future work. This will open a wide window of opportunity for SCALE to build on the foundation provided by conventional CMB analysis, and place constraints on exotic forms of dark matter or clustering models which are predicted to have non-trivial effects on the shape of the lensing potential power spectrum. \nIn summary, we: \n- · Constructed a neural network emulator for the lensed TT , QE reconstruction, and SCALE bandpowers. The emulator provides quick mapping from cosmological parameters to our expected observable band-powers at ≲ 0 . 5% precision (FIG. 2 and Table IV).\n- · Presented a procedure to simulate a large sample of high-resolution ( NSIDE=8192 ), full-sky simulations of the lensed CMB with the lenspyx package. We also presented a procedure to compute SCALE observables from these full-sky HEALPix representations, and SCALE observables from this procedure match well with the flat-sky results in C24.\n- · Developed a likelihood which includes SCALE in parameter estimation with conventional CMB observables accounting for covariance between the different observable spectra. We find that SCALE band-powers ˆ Ψ ˇ L exhibit low levels of correlation with ˆ C TT ℓ and ˆ C ϕϕ L band-powers, but they share strong correlations with band-powers (only at the same multipole ˇ L ) from other applications of SCALE with overlapping small-scale ℓ S filters. We applied this to a standard ΛCDM model with the addition of a massive neutrino m ν .\n- · Demonstrated that SCALE can directly provide constraining information in the estimation of parameters, such as m ν , which affect the amplitude of small-scale lensing beyond measurements of the lensed CMB power spectrum.\n- · Constructed a phenomenological model (motivated by warm and/or fuzzy dark matter clustering models) for small-scale lensing suppression and demonstrated that multiple applications of SCALE with different small-scale ℓ S filtering regimes provide sufficient information to constrain non-trivial modulation to the shape of the lensing power spectrum. This, together with the above result, establishes SCALE's role in future cosmological analyses by providing complementary information about smallscale lensing in addition to conventional lensing reconstruction methods at large-scales. \nThe upcoming era contains a lineup of highly sensitive CMB surveys that will map large fractions of the sky to unprecedented depths. SCALE and related methods will be critical tools to extract maximal information on the \n- [1] V. C. Chan, R. Hloˇzek, J. Meyers, and A. van Engelen, 'Small-correlated-against-large estimator for the lensing of the cosmic microwave background,' Phys. Rev. D 109 no. 4, (Feb., 2024) 043527, arXiv:2302.13350 [astro-ph.CO] .\n- [2] W. Hu, 'Mapping the dark matter through the cmb damping tail,' Astrophys. J. Lett. 557 (2001) L79-L83, arXiv:astro-ph/0105424 .\n- [3] W. Hu and T. Okamoto, 'Mass reconstruction with cmb polarization,' Astrophys. J. 574 (2002) 566-574, arXiv:astro-ph/0111606 .\n- [4] T. Okamoto and W. Hu, 'CMB lensing reconstruction on the full sky,' Phys. Rev. D 67 (2003) 083002, arXiv:astro-ph/0301031 .\n- [5] W. Hu, S. DeDeo, and C. Vale, 'Cluster Mass Estimators from CMB Temperature and Polarization Lensing,' New J. Phys. 9 (2007) 441, arXiv:astro-ph/0701276 .\n- [6] M. Kesden, A. Cooray, and M. Kamionkowski, 'Lensing reconstruction with CMB temperature and polarization,' Phys. Rev. D 67 no. 12, (June, 2003) 123507, arXiv:astro-ph/0302536 [astro-ph] . \nnature of dark matter and the evolution of structures in our Universe.", 'ACKNOWLEDGMENTS': "The authors would like to thank Kendrick Smith for early discussions that inspired this work. The authors would also like to thank Keir Rogers for help in getting started with cosmological emulators. Canadian coauthors acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC). RHis supported by Natural Sciences and Engineering Research Council of Canada Discovery Grant Program and the Connaught Fund. JM is supported by the US Department of Energy under Grant DE-SC0010129 and by NASA through Grant 80NSSC24K0665. AvE acknowledges support from NASA grants 22-ADAP22-0149 and 22-ADAP22-0150. Some computational resources for this research were provided by SMU's Center for Research Computing. The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto. The authors at the University of Toronto acknowledge that the land on which the University of Toronto is built is the traditional territory of the Haudenosaunee, and most recently, the territory of the Mississaugas of the New Credit First Nation. They are grateful to have the opportunity to work in the community, on this territory. Computations were performed on the SciNet supercomputer at the SciNet HPC Consortium. 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2024arXiv240911469D

We study the implications of relaxing the requirement for ultralight axions to account for all dark matter in the Universe by examining mixed dark matter MDM cosmologies with axion fractions f leq 0.3 within the fuzzy dark matter FDM window 1025 eV lesssim m lesssim 1023 eV. Our simulations using a new MDM gravity solver implemented in AxiREPO capture wave dynamics across various scales with high accuracy down to redshifts zapprox 1. We identify halos with Rockstar using the CDM component and find good agreement of inferred halo mass functions HMFs and concentrationmass relations with theoretical models across redshifts z110. This justifies our halo finder approach a posteriori as well as the assumptions underlying the MDM halo model AxionHMcode. Using the inferred axion halo mass cold halo mass relation MtextaMtextc and calibrating a generalised smoothing parameter alpha to our MDM simulations we present a new version of AxionHMcode. The code exhibits excellent agreement with simulations on scales klt 20 h cMpc1 at redshifts z13.5 for fleq 0.1 around the fiducial axion mass m 1024.5 eV 3.16times 1025 eV with maximum deviations remaining below 10. For axion fractions fleq 0.3 the model maintains accuracy with deviations under 20 at redshifts zapprox 1 and scales klt 10 h cMpc1 though deviations can reach up to 30 for higher redshifts when f0.3. Reducing the runtime for a single evaluation of AxionHMcode to below 1 minute these results highlight the potential of AxionHMcode to provide a robust framework for parameter sampling across MDM cosmologies in Bayesian constraint and forecast analyses.

2024-09-01T00:00:00Z

['arXiv:2409.11469', '2024arXiv240911469D', '10.48550/arXiv.2409.11469']

['Astrophysics - Cosmology and Nongalactic Astrophysics']

Improved Halo Model Calibrations for Mixed Dark Matter Models of Ultralight Axions

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11469.pdf

{'Improved Halo Model Calibrations for Mixed Dark Matter Models of Ultralight Axions': "Tibor Dome 1 , 2 ⋆ , Simon May 3 , 4 , Alex Laguë 5 , David J. E. Marsh 7 , Sarah Johnston 6 , Sownak Bose 6 , Alex Tocher 1 , 2 , Anastasia Fialkov 1 , 2 \n1 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK \n- 2 Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK\n- 3 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada\n- 4 Department of Physics, North Carolina State University, Raleigh, NC, 27695-8202, USA\n- 5 Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA 19104\n- 6 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\n- 7 King's College London, Strand, London, WC2R 2LS, United Kingdom \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': 'We study the implications of relaxing the requirement for ultralight axions to account for all dark matter in the Universe by examining mixed dark matter (MDM) cosmologies with axion fractions f ≤ 0 . 3 within the fuzzy dark matter (FDM) window 10 -25 eV ≲ m ≲ 10 -23 eV. Our simulations, using a new MDM gravity solver implemented in A xi REPO, capture wave dynamics across various scales with high accuracy down to redshifts z ≈ 1. We identify halos with R ockstar using the CDM component and find good agreement of inferred halo mass functions (HMFs) and concentration-mass relations with theoretical models across redshifts z = 1 -10. This justifies our halo finder approach a posteriori as well as the assumptions underlying the MDM halo model A xion HM code . Using the inferred axion halo mass-cold halo mass relation M a( M c) and calibrating a generalised smoothing parameter α to our MDM simulations, we present a new version of A xion HM code . The code exhibits excellent agreement with simulations on scales k < 20 h cMpc -1 at redshifts z = 1 -3 . 5 for f ≤ 0 . 1 around the fiducial axion mass m = 10 -24 . 5 eV = 3 . 16 × 10 -25 eV, with maximum deviations remaining below 10 %. For axion fractions f ≤ 0 . 3, the model maintains accuracy with deviations under 20 % at redshifts z ≈ 1 and scales k < 10 h cMpc -1 , though deviations can reach up to 30 % for higher redshifts when f = 0 . 3. Reducing the run-time for a single evaluation of A xion HM code to below 1 minute, these results highlight the potential of A xion HM code to provide a robust framework for parameter sampling across MDMcosmologies in Bayesian constraint and forecast analyses. \nKey words: dark matter - cosmology: large-scale structure of Universe - methods: numerical', '1 INTRODUCTION': "Ideal candidates for ultralight bosonic dark matter (DM) - also called fuzzy dark matter (FDM) - from particle physics are ultralight axions which can emerge from both field theory and string theory. In field theory, these candidates are typically axion-like particles (ALPs), which share their conceptual origins with quantum chromodynamics (QCD) axions (Preskill et al. 1983; Abbott & Sikivie 1983; Dine & Fischler 1983) but can have much lighter masses with couplings to the Standard Model typically taken as free parameters (although see Dias et al. 2014; Kim & Marsh 2016). In string theory, pseudoscalar fields with axion-like properties arise in compactifications as Kaluza-Klein (KK) zero modes of antisymmetric tensor fields, with their masses and couplings determined by the topology \nand geometry of the compact manifold (Svrcek & Witten 2006; Arvanitaki et al. 2010; Visinelli & Vagnozzi 2019; Gendler et al. 2023). \nIn both field theory and string theory, the lightest of these axions could have a mass lower than 10 -19 eV, which implies the existence of particles with de Broglie wavelengths on galactic or even cosmological scales. In the last decade, several extensions of the simplest one-field FDM model have been explored in the literature. Some notable studies include vector FDM where FDM is a higher-spin field (Amin et al. 2022), multi-field FDM (Gosenca et al. 2023; Luu et al. 2024), and self-interacting FDM (Mocz et al. 2023) where the quartic self-coupling term is included. In the self-interacting model, soliton cores can undergo a phase transition from dilute to denser states, and small-scale structure formation may be enhanced. \nString theory compactifications typically yield a plenitude of N = O (100) axion fields with logarithmically distributed masses (Mehta et al. 2021). Importantly, the relic density of ultralight parti- \ncles, Ω FDM, does not necessarily match the total DM density of the Universe, Ω m -Ω b. For example, Bachlechner et al. (2018) show that in theories with N = O (100) axions and a lightest axion of mass m ∼ 10 -22 eV, axion misalignment can generate a DM abundance of Ω FDM ∼ 0 . 1. Models with a small number N = O (1) of axions can also give rise to Ω FDM ∼ 0 . 1 via axion misalignment (Cicoli et al. 2022) if the product S f a of the instanton action S (giving rise to the axion potential) and the axion decay constant f a satisfies S f a ≳ 1, slightly violating the Weak Gravity Conjecture (Hebecker et al. 2018; Alonso & Urbano 2017). This makes phenomenological multi-field FDM models a natural extension within the framework of string theory. \nThe phenomenology of the two-field FDM model can be complex, novel, and compelling, as shown by several studies. Huang et al. (2023) conducted the first two-field cosmological simulations with cold dark matter (CDM) initial conditions and showed that minor admixtures of heavier axions to a lighter-mass major component can neither significantly disturb stable major-component solitons nor form minor-component solitons. Gosenca et al. (2023) studied stellar heating in FDM halos with two or more fields, while Glennon et al. (2023) simulated soliton collision of two-field FDM with inter-field and self-interactions. Jain et al. (2024) investigated Bose-Einstein condensation (BEC) formation in the kinetic regime, and Luu et al. (2024) showed that in the regime where the mass ratio m 2 / m 1 is in the range 2 -7, a minimal model with uncoupled fields predicts a central soliton with a nested structure, distinguishable from the typical flat-core soliton found in one-field halos. Importantly, multi-field FDM circumvents the Catch 22 problem that challenges the use of the simplest non-interacting one-field FDM as a solution to the small scale problems of CDM (Marsh & Silk 2014). \nIn the context of mixed dark matter (MDM) models, we make the simplifying assumption that only one of the DM particle species is ultralight while the others have negligible de Broglie wavelengths. In this case, we group the combined relic density of the (near-)collisionless components into Ω CDM since the species can be modelled as CDM by virtue of the correspondence between the Schrödinger-Poisson and Vlasov-Poisson equations (Mocz et al. 2018). How do observations constrain MDM models? While the FDM constraints compiled in Dome et al. (2022) assume a pure FDM cosmology, many constraints were recently reported assuming axions do not constitute all of the DM in the Universe. In the mass range 10 -32 eV ≲ m ≲ 10 -26 eV, FDM can only comprise a few percent of the total DM (Hlozek et al. 2015; Hložek et al. 2018; Laguë et al. 2022; Rogers et al. 2023). For higher values of m , the bound becomes weaker since the FDM power suppression moves to smaller scales, while for smaller values of m , FDM behaves as dark energy and is strongly degenerate with ΩΛ . \nAt the higher-mass end, Lyα forest and ultra-faint dwarf (UFD) data indicates that DM cannot be fully described by pure FDM models in the mass range 10 -21 eV ≲ m ≲ 3 × 10 -19 eV (Armengaud et al. 2017; Marsh & Niemeyer 2019; Zimmermann et al. 2021; Dalal & Kravtsov 2022; Iršiˇc et al. 2024), with a stronger bound for 10 -23 eV ≲ m ≲ 10 -21 eV where FDM must not comprise more than O (10 %) of the total DM (Kobayashi et al. 2017). The Dark Energy Survey year 1 was used by Dentler et al. (2022) to search for shear-correlation suppressions caused by FDM (assuming pure FDM), ruling out the existence of pure FDM in the mass range 10 -25 eV ≲ m ≲ 10 -23 eV (also see Preston et al. 2024). However, current constraints suggest that it is possible for FDM to exist in fairly large portions (albeit not the entirety of the DM) in the region 10 -25 eV ≲ m ≲ 10 -23 eV, referred to as the FDM window . \nRecent studies have demonstrated the potential of 21 cm inter- \nferometers, particularly the Hydrogen Epoch of Reionisation Array (HERA), to detect FDM fractions of f ≳ O (1 %) within the FDM window and beyond (Jones et al. 2021; Flitter & Kovetz 2022). This potential was further supported by Lazare et al. (2024), who in addition established an upper bound on FDM with a particle mass m = 10 -23 eV, restricting it to 16 % of the total DM. Their constraints are based on a range of observations, including UV luminosity function (LF) data from the Hubble Space Telescope, constraints on the neutral hydrogen fraction from high-redshift quasar spectroscopy, cosmic microwave background (CMB) optical depth measurements from Planck, and upper bounds on the 21 cm power spectrum from HERA. These constraints tighten for smaller masses, reaching down to 1 % for m = 10 -26 eV. However, their machine learning-based emulator does neither account for the 'quantum pressure' term in the evolution of the axion field, nor the impact of FDM on star formation, and 21 cm FAST relies on the excursion set formalism (Mesinger et al. 2011), which may not adequately capture small-scale physics. Using UVLF and Planck CMB data only, Winch et al. (2024) report slightly weaker constraints, restricting FDM to less than ≈ 22 % of the DM at m = 10 -23 eV and to less than ≈ 5 % at m = 10 -26 eV. \nThe future Square Kilometre Array (SKA) is expected to provide strong constraints on FDM and MDM models, at few per cent level for masses m ∼ 10 -22 eV, and improving to below 1 % for lighter masses (Bauer et al. 2021; Dome et al. 2024). For pure FDM, Hotinli et al. (2022) showed that by means of velocity acoustic oscillations in the large-scale 21 cm power spectrum, HERA may be sensitive to axions with masses up to m ≈ 10 -18 eV (see also Marsh 2015, for FDM relative velocity e ff ects). \nShevchuk et al. (2023) argued that FDM with a particle mass of m ≲ 10 -24 eV is inconsistent with the observed Einstein radii of several strong lensing systems. However, their analysis is based on simplified models, such as expressing total DM density profiles as ρ DM( r ) = f ρ FDM( r ) + (1 -f ) ρ CDM( r ), and assuming idealised solitons derived solely from FDM simulations, without accounting for contributions from cold dark matter or baryons. \nWe thus believe that the FDM window remains an appealing regime to explore in MDM simulations, for a number of reasons. First, Blum & Teodori (2021) proposed that O (10 %) of DM in the form of FDM with particle mass in the window, m ∼ 10 -25 eV, could explain the tension between inferences of H 0, the current expansion rate of the Universe, which are based on the time delay in lensed quasar measurements, and those based on CMB observations. Note that both Shevchuk et al. (2023) and Blum & Teodori (2021) assume a soliton core in the (lens) density profile for FDM fractions of O (10 %), the validity of which we will address in more detail later (see Secs. 4.2 and 4.3). Second, the 5 σ tension between Planck CMB and Lyα forest data might prefer a fraction ≈ (1 -5) % of FDM with particle mass m ∼ 10 -25 eV, which is very close to the axion mass that we probe (Rogers et al. 2023). Third, a fraction f ∼ 10 % of FDMwas suggested to explain the suppressed amplitude of the matter power spectrum at late times (also known as the σ 8 / S8 tension, Allali et al. 2021; Ye et al. 2023). \nIn this work, our aim is to push both the numerical and (semi-) analytical frontiers of MDM modelling. Alongside implementing a new MDM gravity solver and conducting a suite of simulations within the FDM window for a particle mass of m = 10 -24 . 5 eV = 3 . 16 × 10 -25 eV, we refine the halo model framework A xion HMcode (Vogt et al. 2023), calibrating some of its key parameters to the MDM simulations. We will achieve this mainly by modelling the axion halo mass-cold halo mass relation M a( M c) as a broken power law below the linear Jeans mass and calibrating a generalised transition smoothing parameter α . \nThis paper is organised as follows. We start with the physics of MDM in Sec. 2.1 and describe the numerical methodology in Sec. 2.5, which we implement in A xi REPO (May & Springel 2021, 2023). Key findings regarding large-scale structure and halo statistics are presented in Sec. 3. We detail our improvements to MDM halo model calibrations in Sec. 4 before we conclude in Sec. 5.", '2.1 Mixed Dark Matter Physics': 'We define the FDM fraction (of the total matter density) as \nf ≡ Ω FDM Ω FDM + Ω CDM + Ω b , (1) \nwhere we adopt Planck Collaboration et al. (2016) cosmological parameters. In this paper, the simulation-based analyses (up to Sec. 4) rely on DM-only modelling, where baryons are e ff ectively absorbed into the CDM component and assumed to behave similarly to CDM, such that Ω CDM ← Ω CDM + Ω b and Ω b ← 0. In other words, for the purposes of our simulations, we set Ω b = 0 and increase Ω CDM correspondingly. \nIn DM-only models (as in our simulations) this definition of f is standard (e. g. Laguë et al. 2023), and f max = 1 corresponds to pure FDM. In models that distinguish between the CDM component and baryons, our definition (1) di ff ers from the more commonly used ˜ f = Ω FDM / ( Ω FDM +Ω CDM), which is also employed in the Vogt et al. (2023) MDM halo model central to this paper. In such cases, the maximum FDM fraction is f max = 0 . 843. Converting between the more common definition and Eq. (1) simply involves downscaling ˜ f by the factor ( Ω FDM + Ω CDM) / ( Ω FDM + Ω CDM + Ω b). \nWe consider axions and axion-like particles, characterised by a periodicity defined by the energy scale f a and represented as an angular variable ϕ . The initial, random value of the misalignment angle θ a = ϕ/ f a (or the distribution thereof in our Hubble patch) determines the FDM abundance Ω FDM. In case of the QCD axion, the associated global U(1) symmetry is the Peccei-Quinn (PQ) symmetry whose angular pseudoscalar field ϕ couples to the strong sector and via non-perturbative QCD e ff ects (instantons) dynamically relaxes the θ term in the QCD Lagrangian to zero, thus \'solving\' the strong-CP problem (Peccei & Quinn 1977). Similar nonperturbative e ff ects also exist in string-theoretical realisations of axions (brane instantons, Svrcek & Witten 2006; Svrcek 2006; Blumenhagen et al. 2009) and in both cases lead to the spontaneous breaking of the exact shift symmetry ϕ → ϕ + const. at scale Λ ≪ f a, rendering axions and axion-like particles massive (hence the term pseudo-Nambu-Goldstone boson). The potential produced by the non-perturbative e ff ects respects the periodicity of ϕ and is usually written as \nV ( ϕ ) = Λ 4 " 1 -cos ϕ f a !# , (2) \nwhere the axion mass can be expressed as m = Λ 2 / f a in case of string theory models and using temperature-dependent terms in case of QCD axions (Niemeyer 2020). \nIn the limit of small field displacements away from the potential minimum, ϕ ≪ f a, and ignoring the Hubble drag in a late-time Friedmann-Lemaître-Robertson-Walker (FLRW) background cosmology, the axion satisfies the Klein-Gordon equation (in natural units), \n( -∂µ∂ µ + m 2 ) ϕ = 0 . (3) \nHere we are interested in the case where ϕ has both spatial and temporal fluctuations. On scales much larger than the Compton wavelength (in natural units) \nλ c ≡ 2 π m , (4) \nbut much smaller than the particle horizon, one can employ a nonrelativistic approximation to the dispersion relation and a Newtonian approximation to the gravitational interaction embedded in the covariant derivatives (since ∂µ∂ µ = g µν ∂µ∂ν ) of the field equation. \nIt is then convenient to define a complex scalar field \nψ = √ ρ FDM e i γ , (5) \nthe \'classical\' wavefunction, constructed from the amplitude and phase of the field ϕ , \nϕ = √ ρ FDM cos( mt -γ ) . (6) \nThe wavefunction then obeys \ni ∂ t + 3 2 ˙ a a ! ψ = -1 2 m ∇ 2 + m Ψ ! ψ, (7a) \n∇ 2 Ψ = 4 π G ( ρ CDM + ρ FDM -¯ ρ tot) , (7b) \nwhere Ψ is the Newtonian gravitational potential, the FDM density can be expressed as ρ FDM = | ψ | 2 and ¯ ρ tot is the mean of the total DM density. This is simply the non-linear Schrödinger-Poisson (SP) system of equations for a self-gravitating many-body field in a potential well, embedded in an expanding Universe. Note that the right hand side of the first equation of system (7) vanishes for the unperturbed background and the energy density in the axion field ρ FDM = | ψ | 2 ∝ a -3 redshifts like matter. In terms of comoving coordinates x ≡ r / a , the SP equations become \ni ∂ψ c ∂ t = -a -2 1 2 m ∇ 2 c ψ c + a -1 m Ψ c ψ c , (8a) \n∇ 2 c Ψ c = 4 π G ( ρ CDM,c + ρ FDM,c -¯ ρ m,c) , (8b) \nwhere we have defined comoving quantities, relating to physical quantities as \nρ i , c ≡ a 3 ρ i , ψ c ≡ a 3 / 2 ψ, ∇ c ≡ a ∇ , Ψ c ≡ a Ψ , (9) \nfor i ∈ { CDM , FDM , m } , where \'m\' denotes total matter.', '2.2 Characteristic Scales in MDM': "Afundamental length scale in the context of MDM is the linear Jeans scale (Hu et al. 2000; Marsh & Silk 2014), \nk J = GLYPH<16> 16 π Ga 4 ¯ ρ m GLYPH<17> 1 / 4 GLYPH<18> m ℏ GLYPH<19> 1 / 2 ≃ 66 . 5 (1 + z ) -1 / 4 Ω m h 2 0 . 12 ! 1 / 4 GLYPH<18> m 10 -22 eV GLYPH<19> 1 / 2 cMpc -1 . (10) \nThis scale emerges from the non-vanishing e ff ective sound speed of scalar fields, which introduces a Jeans-like dispersion relation for perturbation modes. Notably, the linear Jeans scale exhibits minimal dependence on redshift and is una ff ected by the axion fraction f . The corresponding linear Jeans mass can be written as \nM J = 4 3 π π k J ! 3 ¯ ρ m ≈ (1 -2) × 10 11 M ⊙ / h for z = 1 -4 , (11) \nwhere we have assumed spherical basis functions, and as usual we have taken m = 3 . 16 × 10 -25 eV. In pure FDM cosmologies ( f = f max), no halos can form with mass M < M J due to the pressure \nTable 1. Scaling of several characteristic MDM scales with axion mass m , redshift z , and axion fraction f . The best-fit results for the scaling of the cuto ff scale k cut with axion mass, redshift, and axion fraction were obtained by evaluating Eq. (15) and comparing it to the virial radius in the redshift range z = 1 -4, axion mass range m = 10 -26 -10 -23 eV, and axion fraction range f = 0 . 01 -0 . 3, respectively. These length scales show no dependence on the axion fraction f and are useful scales to consider for both FDM and MDM cosmologies. \nsupport from the scalar field. Some residual structures (not halos) may still form via fragmentation. However, this estimate is based on a purely linear e ff ect and indeed the growth rate is enhanced at second order (Li et al. 2019), hence heuristic reasoning based on experience from the baryonic Jeans length can only be applied with caution. Schive et al. (2014b) derived a slightly more accurate minimum halo mass from the properties of FDM solitons that can form, but their estimate is in close agreement with the linear Jeans mass M J. \nThe Jeans mass can be generalised to non-linear structures, by adding the dependence on the halo profile ρ NFW of the CDM halo. Hu et al. (2000) and Marsh & Silk (2014) showed that in a pure axion cosmology, no virialised halo is expected to form if this socalled halo Jeans scale r hJ is larger than the virial radius R vir. The corresponding mass scale is called the cut-o ff mass M cut. In comoving units, the halo Jeans wavenumber is given by (cf. Eq. 10) \nk hJ = 66 . 5 (1 + z ) -1 / 4 Ω m h 2 0 . 12 ! 1 / 4 GLYPH<18> m 10 -22 eV GLYPH<19> 1 / 2 ρ NFW( r hJ) ¯ ρ m ! 1 / 4 cMpc -1 . (12) \nHere, k hJ is calculated by converting r hJ = π/ k hJ as in Marsh & Silk (2014) instead of r hJ = 2 π/ k hJ as in Hu et al. (2000). Since we assume that r hJ ≤ R vir, the Navarro-Frenk-White (NFW) density at r hJ can be written as (Navarro et al. 1997) \nρ NFW( r hJ) = ¯ ρ m ∆ vir c 2 3 f ( c ) R vir r hJ(1 + r hJ c / R vir) 2 , (13) \nwith f ( x ) = -x / (1 + x ) + ln(1 + x ) and c = R vir / r -2 being the halo concentration. The virial mass definition we adopt is \nM vir = M tot = 4 π 3 ¯ ρ R 3 vir ∆ vir( z ) , (14) \nwhere ∆ vir is the virial overdensity (Bryan & Norman 1998). Substituting Eq. (13) back into Eq. (12), we obtain \nr 3 hJ = GLYPH<18> π 66 . 5 GLYPH<19> 4 (1 + z ) GLYPH<18> m 10 -22 eV GLYPH<19> -2 Ω m h 2 0 . 12 ! -1 3 f ( c ) ∆ vir c 2 (1 + r hJ c / R vir) 2 R vir , (15) \nwhere r hJ and R vir are measured in ckpc and we have assumed spherical basis functions as we did for Eq. (11). Eq. (15) can then be solved for r hJ. The dependence on axion fraction f and redshift z only enters via the concentration c which we take to be the mean cosmic concentration at mass M vir as per Eqs. (43). In fact, after comparing to the virial radius, we find that the cut-o ff mass is e ff ectively independent of f , and attains values M cut ≈ 1 . 7 × 10 12 -3 . 1 × 10 13 M ⊙ / h for our MDM cosmologies across redshifts z = 1 -4. \nHow do we interpret the cut-o ff mass M cut? In pure FDM, while the Jeans mass M J provides a fundamental lower bound on the mass \nof any halo that can form based on the balance between gravity and 'quantum pressure' in the linear regime, the cut-o ff mass is a halospecific lower bound that describes the minimum mass of virialised halos that can form. In MDM cosmologies, virialised halos can form below M cut, but they will be primarily CDM-dominated. \nAnother important scale is the characteristic mass at which the non-linear matter power spectra and HMFs begin to deviate from the predictions of the CDM model. This characteristic mass, denoted as M 0, is given by (Schive et al. 2016) \nM 0 = 1 . 6 × 10 10 ( m / 10 -22 eV) -4 / 3 M ⊙ (16) \nThe scaling of M 0 is expected to be almost independent of redshift, as it is primarily set during the radiation-dominated epoch (Hu et al. 2000). \nFinally, we highlight a fourth mass scale, M conc ≈ 20 -50 M 0, at which the mean halo concentration-mass relation c ( M ) is significantly a ff ected by axion physics. This was demonstrated by Bose et al. (2016); Ludlow et al. (2016); Dentler et al. (2022) and can be attributed to the fact that halo concentration is determined when only a small fraction ( ≈ 0 . 01) of the mass has accumulated (Navarro et al. 1997; Bullock et al. 2001). These four mass scales are illustrated in Fig. 1, using a fiducial MDM cosmology with axion mass m = 3 . 16 × 10 -25 eV and axion fraction f = 0 . 1 at redshift z = 1. \nFor reference, we show the scalings of these four characteristic length scales with axion mass m , redshift z and axion fraction f in Table 1. We translate between mass and inverse length scales via M = 4 π ¯ ρ R 3 / 3. Among the four scales, the cut-o ff scale k cut shows the strongest dependence on axion mass, scaling as k cut ∝ m 0 . 66 . Even though these scales were first introduced for pure FDM cosmologies and thus have no dependence on the axion fraction (while M cut is e ff ectively independent of f ), they remain valuable to consider in the context of MDM cosmologies as well. Specifically, we will show in Sec. 3.1 that despite its weak dependence on the axion fraction, M 0 serves as an e ff ective scale for characterising the suppression of the halo mass function (HMF) at the low-mass end. In Sec. 3.3, we will show that the weak dependence of M conc on the axion fraction is corroborated by inferred concentration-mass relations. In addition, we will establish in Sec. 4.3 that M cut provides a useful approximate scale below which halos are predominantly CDM-dominated.", '2.3 Initial Conditions': "The goal of an initial conditions (ICs) generator for a cosmological simulation is to faithfully reproduce the statistical properties of the density field in the early universe with a finite number of point particles (for the CDM component) or grid cells (for the FDM component). Given a pre-IC particle distribution representing a completely homogeneous universe (we use grid pre-ICs, see Dome et al. 2022), the next task is to perturb the particles to produce a density field that reproduces the cosmologically relevant expected statistical properties. Let us consider a (Gaussian) over-density field δ ( r ) that is completely described by its power spectrum P ( k ) = ⟨ δ k δ ∗ k ⟩ . It is customary to express the amplitude of density fluctuations in terms of the transfer function T ( k , a ), \nP ( k , a ) ≈ Bk ns T ( k , a ) 2 , (17) \nwhere n s is the spectral index, and B can be expressed in terms of the normalisation A s and the pivot scale k piv (Planck Collaboration et al. 2016). \nSetting up ICs for cosmological simulations at a certain scale factor a thus involves generating a white noise sample of random val- \nFigure 1. Characteristic scales in the MDM framework. We illustrate the linear Jeans mass M J (see Eq. 10), the cut-o ff mass M cut (see Eq. 15 and associated text), and the characteristic mass M 0, where deviations from CDM become apparent in the matter transfer function (TF) and HMF (see Eq. 16). Additionally, we identify M conc, the mass scale at which the concentration-mass relation c ( M ) shows notable changes. Results are presented for a fiducial MDM cosmology with axion mass m = 3 . 16 × 10 -25 eV and axion fraction f = 0 . 1 at redshift z = 1. \n<!-- image --> \n/circledot \nues µ ( r ) (typically sampled from a Gaussian distribution with zero mean and unit variance) and requiring that their amplitudes follow a specific power spectrum P ( k , a ). This is achieved by multiplying the Fourier transformed white noise field µ k with the square root of the power spectrum, i. e. for all k representable on a grid of given resolution set \nδ k = p P ( k , a ) µ k = √ Bk ns / 2 T ( k , a ) µ k . (18) \nThe real-space over-density field δ ( r ) is then obtained by inverse Fourier transformation, and this procedure is typically called ' k -space sampling'. \nNote that a product in Fourier space simply corresponds to a convolution in real space, i. e. Eq. (18) is equivalent to \nδ ( r ) = T ( r , a ) ⋆ µ ( r ) , (19) \nwhere T ( r , a ) is the real-space counterpart of T ( k , a ) = √ Bk ns / 2 T ( k , a ), and ' ⋆ ' denotes a convolution. It is hence mathematically equivalent whether Eq. (18) is evaluated in Fourier space, followed by an inverse transform ( k -space sampling), or whether Eq. (19) is evaluated using an inverse transform of T ( k , a ) followed by the convolution (real-space sampling). Most cosmological IC codes follow the first approach (see e. g. Bertschinger 2001), while e. g. Pen (1997) and Sirko (2005) use the second or variations thereof. \nHowever, the discrete realisations of the density fields derived with the two approaches will have significant di ff erences. This has been demonstrated conclusively by Sirko (2005), who showed amongst others that employing Eq. (18) imposes periodicity of the real-space transfer function on box scales and leads to an underestimation of the two-point correlation function on large (sub-box) scales. This discrepancy arises because, with a finite number of particles in a finite box, achieving high accuracy for estimates of both the correlation function and the power spectrum is challenging due to finite box e ff ects and the limitations in resolving all relevant scales. Even with an infinite number of particles, the finite volume of the box introduces periodic boundary conditions and aliasing e ff ects that can distort the two-point correlation function, particularly at large scales. Importantly, in this work we use the public code M usic to generate ICs (Hahn & Abel 2011), which is likewise based on a convolution of Gaussian white noise with a real-space transfer function \nkernel. The linear MDM power spectrum which serves as an input to M usic is calculated using A xion C amb (Hlozek et al. 2015). \nHaving generated the Eulerian-grid seed density field δ ( r ) using real-space sampling while ensuring its consistency with both the input correlation function and the input power spectrum, we need to calculate the initial positions and velocities of the IC macroparticles. This is typically achieved using first (1LPT) or second order Lagrangian perturbation theory (2LPT), where δ ( r ) is used as the source field for Lagrangian perturbation theory. The displacement field in 1LPT only contains contributions from the gravitational potential, which is often called the Zel'dovich approximation (Zel'dovich 1970). For all the simulations analysed in this work, we employ a more accurate representation by incorporating secondorder e ff ects using M usic . \nHow do we take account of the multi-fluid nature of MDM beyond calculating separate density transfer functions for each component and coupling them gravitationally in the 2LPT? The growth of density perturbations in a two-component fluid can only be correctly reproduced if besides the di ff erent initial amplitudes of density perturbations also the di ff erence in initial velocities between the two components are respected (Yoshida et al. 2003). ICs for the twocomponent fluid thus ought to reflect these important di ff erences between the two components. 1 The correct growth of fluctuations in both components consistent with the predictions from linear perturbation theories can only be achieved with a proper modelling of (relative) velocities (Somogyi & Smith 2010; Laguë et al. 2021). \nTo obtain a velocity transfer function, recall that the curl of a peculiar velocity field, ∇× v ∝ a -1 , drops o ff with the expansion of the Universe and can be neglected at late times since there is no source for vorticity (although see Mocz et al. 2017; Hui et al. 2021). Hence we can write v as the gradient of a velocity potential, v = ∇V , and so \nvk = i k V k . (20) \nLinear Newtonian perturbation theory applied to a pure DM cosmol- \n1 \nFigure 2. Velocity transfer functions (squared) in an MDM cosmology consisting of a majority CDM component with an f = 0 . 1 axion admixture of boson mass m = 10 -24 . 5 eV at z = 127. The unapproximated FDM (blue solid) and CDM (green solid) velocity transfer functions in Newtonian gauge (NG) are calculated using a modified version of A xion C amb . The vertical dashed line marks the FDM linear Jeans scale k J (Eq. 10). The approximated Newtonian gauge FDM velocity, derived from MDM matter transfer functions (see Eq. 24), is represented by the orange solid line. This L ( k ) model provides a poor approximation to the blue solid line on small scales k > 5 cMpc -1 . For completeness, we also show the (unapproximated) FDM velocity transfer function in synchronous gauge (SG), calculated similarly using a modified version of A xion C amb . Note that CDM SG is zero by definition, and so does not appear. \n<!-- image --> \nogy (either CDM or FDM) thus implies that \nvk = ia k k 2 d δ k d t . (21) \nFor the growing mode of δ , we obtain \nvk = ia k k 2 Ha δ k F ( Ω m) , (22) \nwhere \nF ( Ω m) ≡ d ln( D + ( k , a )) d ln( a ) . (23) \nNote that Eq. (22) remains valid even in MDM cosmologies, where we have one such equation for each DM component. In contrast to pure CDM, the FDM growth rate depends on scale k , which arises due to the non-vanishing sound speed of the axion fluid (Hwang & Noh 2009; Marsh 2016). \nIt is nonetheless possible to approximately separate out the dependence on scale k following Laguë et al. (2021), \nD FDM + ( k , a ) ≈ L ( k ) D CDM + ( a ) . (24) \nNote that we can express L ( k ) as \nL ( k ) ≈ D FDM + ( k , a ) D CDM + ( a ) = s ∆ FDM ( k , a ) ∆ CDM ( k , a ) , (25) \nwhere ∆ ( k , a ) ≡ T ( k , a ) 2 is the squared transfer function of the respective DM component. In CDM, recall that T ( k , a ) = T ( k ) D + ( a ). \nWe can now approximate the velocity field of the axion component in MDM cosmologies using the CDM velocity field as \nv FDM k ≈ L ( k ) v CDM k . (26) \nTo approximate the squared velocity transfer function ∆ FDM v ( k , a ) \nfor the axion component in an MDM cosmology, it should thus be su ffi cient to calculate ∆ CDM v ( k , a ) in the corresponding pure CDM cosmology and construct the squared ratio of FDM-to-CDM matter transfer functions ∆ FDM ( k , a ) / ∆ CDM ( k , a ). Assuming separability of D FDM + ( k , a ) based on Eq. (25), we have \n∆ FDM v ( k , a ) ≈ ∆ FDM ( k , a ) ∆ CDM ( k , a ) ∆ CDM v ( k , a ) . (27) \nNow, both ∆ CDM v ( k , a ) and the squared transfer function ratio can be generated easily using the public version of A xion C amb (Hlozek et al. 2015). We call Eq. (27) the L ( k ) model since it relies on the assumption of separability. To assess the fidelity of the L ( k ) model, we have modified A xion C amb such that we can retrieve the unapproximated FDM velocity transfer function ∆ FDM v ( k , a ) in Newtonian gauge directly from the Boltzmann code. In Fig. 2, we show velocity transfer functions in an MDM cosmology with a 10 % admixture of a m = 10 -24 . 5 eV axion field. We see that the L ( k ) model provides a poor approximation for ∆ FDM v ( k , a ) on small scales k > 5 cMpc -1 in Newtonian gauge, 2 justifying our more reliable, unapproximated approach to velocity transfer functions. \nEquipped with the velocity and matter transfer functions of each component, we use the public code M usic to generate ICs (Hahn & Abel 2011). M usic takes the transfer functions at a particular redshift (in our case, z = 127) as input, and generates initial positions and velocities for macroparticles of each component.", '2.4 Madelung Formulation': "How do we initialise the FDM wavefunction on a grid from macroparticle positions and velocities? We use the Madelung change of variables (Madelung 1927), which can aid with physical intuition. We start with the decomposition (5) of the wavefunction into its amplitude and phase, ψ c = √ ρ FDM,c e i γ , and define the Madelung velocity as the gradient of the phase, \nv M ≡ ∇ γ m . (28) \nOn linear scales before shell crossing, the fluid velocity is a gradient flow, and it resembles that of a superfluid. The Schrödinger equation can then be written as \n∂ρ FDM,c + ∇ c · ( ρ FDM,c v M) = 0 , \n∂ t (29a) ∂ v M ∂ t + a -2 v M · ∇ c v M = -a -1 ∇ c Ψ c + a -2 1 2 m 2 ∇ c ∇ 2 √ ρ FDM,c √ ρ FDM,c ! . (29b) \nThe Schrödinger equation possesses a U(1) symmetry, which amounts to the rotation of ψ by a phase. With the identification of the fluid velocity, what is normally understood as probability conservation (i. e. conservation of the associated Noether current) in quantum mechanics is recast as mass conservation in Eq. (29a). The last term in the Euler equation (29b) is often referred to as the 'quantum pressure' term. It is a misnomer since we have a classical system. In addition, the term arises from a stress tensor rather than mere pressure: \nΣ i j = 1 4 m 2 GLYPH<16> ρ -1 ∂ i ρ∂ j ρ -∂ i ∂ j ρ GLYPH<17> = -ρ 4 m 2 ∂ i ∂ j ln( ρ ) , (30) \ni. e. ∂ i ( ∇ 2 √ ρ / √ ρ ) / (2 m 2 ) = -ρ -1 ∂ j Σ i j . We have dropped the subscripts on the FDM density field in Eq. (30), ρ = ρ FDM,c. The stress tensor Σ i j represents how the fluid description accounts for the underlying wave dynamics. It shows how the particle limit is obtained: for large m , the Euler equation reduces to that for a pressureless fluid, as is appropriate for particle DM. The insight that the wave formulation in the large m limit can be used to model particle CDM was exploited by Widrow & Kaiser (1993). The wave description e ff ectively reshu ffl es information in a phase-space Boltzmann distribution into a position-space wavefunction and o ff ers a number of insights that might otherwise be obscured (Uhlemann et al. 2019; Garny et al. 2020). This correspondence can be formalised (Mocz et al. 2018). It is worth noting that A xion C amb also employs the Madelung description. \nTo initialise the FDM wavefunction, we calculate the FDM phase γ by constructing the Madelung velocity field v M and solving Eq. (28) in Fourier space. Equipped with CDM macroparticle positions and velocities as well as the FDM wavefunction, we can now evolve the joint MDM field.", '2.5 Pseudo-Spectral Method': 'We use a spectral method to simulate MDM structure formation implemented in the A xi REPO code (May & Springel 2021, 2023). The system of equations (8) is solved using a second-order symmetrised split-step pseudo-spectral Fourier method, colloquially called a \'kick-drift-kick\' leapfrog-like scheme. For a small time step ∆ t , the time evolution can be simplified using the following approximation (Edwards et al. 2018; May & Springel 2021): \nψ c( t + ∆ t , x ) \n= T exp " -i Z t +∆ t t -ℏ 2 m 1 a ( t \' ) 2 ∇ 2 c + m ℏ 1 a ( t \' ) Ψ c( t \' , x ) ! d t \' # ψ c( t , x ) ≈ exp " i ∆ t 2 -ℏ m 1 a ( t ) 2 ∇ 2 c -m ℏ 1 a ( t ) Ψ c( t + ∆ t , x ) -m ℏ 1 a ( t ) Ψ c( t , x ) !# × ψ c( t , x ) ≈ exp " -i m ℏ 1 a ( t ) ∆ t 2 Ψ c( t + ∆ t , x ) # exp " i ℏ m 1 a ( t ) 2 ∆ t 2 ∇ 2 c # × exp " -i m ℏ 1 a ( t ) ∆ t 2 Ψ c( t , x ) # ψ c( t , x ) , (31) \nwhere T is the time ordering operator and, using the BakerCampbell-Hausdor ff formula, the time evolution operator has been split into three unitary parts which do not mix functions of the position and derivative operators. This makes it natural to automatically couple the method to particle-based N -body techniques that evolve collisionless components such as CDM and stellar particles on the same sub-time step spacing. Coupling to gas cells is also straightforward and is achieved via the full gravitational potential Ψ c (including the baryonic contribution) in both the SP equations and the forces evolving the gas cells. Simulations involving mixed ultralight and baryonic physics are left for future work. \nThe fields ψ c and Ψ c are discretised on a uniform Cartesian grid with N 3 mesh points in a periodic box of length L to allow for e ffi cient numerical computations using the Fast Fourier Transform (FFT). The pseudo-spectral method is summarised in Algorithm 1. The choice of the time step ∆ t is determined by the requirement that the phase di ff erence in the exponentials must not exceed 2 π , at which point the time step would be incorrectly \'aliased\' to a smaller time step corresponding to the phase di ff erence subtracted by a multiple of 2 π due to the periodicity of the exponential function. The kicks \nAlgorithm 1 Pseudo-Spectral Method for MDM \nand the drift yield separate constraints for ∆ t , both of which must be simultaneously fulfilled. The resulting time step criterion is \n∆ t < min 4 3 π m ℏ a 2 ∆ x 2 , 2 π ℏ m a 1 | Ψ c , max | ! , (32) \nwhere ∆ x = L / N is the spatial resolution and Ψ c , max is the maximum value of the potential. Note that Eq. (32) is essentially a CourantFriedrichs-Lewy (CFL) condition. The dependence ∆ t ∝ ∆ x 2 can be viewed as a reflection of the relation of the Schrödinger equation to di ff usion problems. Since N -body codes for gravity and Eulerian fluid solvers scale as ∝ ∆ x , this adds computational cost to the simulations. However, Algorithm (1) allows for machine precision control of the total kinetic energy and achieves spectral (exponential) convergence in space. \nWhen considering the velocity field v c = v M = ℏ ∇ c γ/ m , another constraint on the validity of the discretisation becomes apparent. Since the di ff erence in the gradient of the phase between two points can be at most 2 π , it follows that the discretised velocity field cannot exceed a maximum value (depending on the concrete form of the discretised gradient operator) of about \nv max = ℏ m 2 π ∆ x . (33) \nVelocities v ≥ v max cannot be represented in a simulation with resolution ∆ x , which translates into a constraint on resolution, which should be good enough to resolve the de Broglie wavelength λ dB of the largest velocities: \n∆ x < π ℏ mv max ≡ 1 2 λ dB( v max) . (34) \nRequirements (32) and (34) exemplify why FDM simulations are computationally much more costly than traditional particle-based CDM simulations, where resolution can be set independent of velocities and time step constraints are less restrictive (e. g. Springel 2005).', '2.6 Simulation Setup': 'As mentioned, in the MDM model only one of the DM particle species - occupying a fraction f of the total matter content - is ultralight while all others are assumed to have negligible de Broglie wavelengths and are thus modelled as CDM. Schwabe et al. (2020) pointed out that a soliton may not form in cosmological simulations with f < 0 . 1. This result was confirmed by Laguë et al. (2023) who investigated the impact of a mixture of CDM and FDM in various proportions f = [0 , 1 , 10 , 50 , 100] % and for ultralight particle masses ranging over five orders of magnitude (2 . 5 × 10 -25 eV - 2 . 5 × 10 -21 eV) using A xio N yx , albeit mostly for relatively small box sizes, L box = 1 cMpc / h . The authors also implemented a modified friends-of-friends (FOF) halo finder and found good agreement between the inferred halo abundance and the predictions from the adapted halo model A xion HM code in a narrow mass range. Expanding upon their findings, we aim to identify distinctive characteristics \nTable 2. Overview of the MDM simulation suite: (1) FDM fraction f ; (2) side length of the simulation box L box; (3) number of CDM N -body particles; (4) number of FDM grid cells; (5) mass per CDM N -body particle. The softening scale for CDM is fixed to ϵ = 1 . 78 ckpc / h in comoving units and capped at ϵ = 0 . 89 pkpc / h in physical units following Power et al. (2003). All simulations are DM-only and assume an axion mass of m = 10 -24 . 5 eV = 3 . 16 × 10 -25 eV. \nthat set MDM apart from single-particle models, thereby enhancing our understanding of its cosmological implications. \nFor our MDM simulations, we adopt an axion mass of m = 10 -24 . 5 eV = 3 . 16 × 10 -25 eV and vary the FDM fraction in the set f ∈ [0 . 01 , 0 . 1 , 0 . 2 , 0 . 3]. Note that due to the presence of a dominant CDM component ( f < 0 . 5), the potential wells in which the wavefunction evolves become steeper, which increases the FDM velocity dispersion and decreases its de Broglie wavelength λ dB, Eq. (34). MDM simulations thus require a higher resolution than their pure FDM ( f = f max) counterparts. Consequently, conducting large-scale cosmological simulations involving the solution of the full Schrödinger-Poisson system within the MDM framework is challenging. While we ascertain the necessary CDM and FDM resolution requirements according to the criteria outlined in Sec. 2.5 and Dome et al. (in prep), we thus ensure at the same time that our selection of simulation parameters is conservative. The final specifications of our MDM simulation suite are given in Table 2. \nTo ensure the fidelity of the high redshift evolution of the simulations, we performed several tests: we verified that across several orders of magnitude the partial (FDM and CDM components) and total power spectra of the first snapshot ( z = 127) replicate the target power spectra obtained using the modified version of A xion -C amb (see Sec. 2.3); we also tested the linear growth prediction D + ( k , a ) ∝ a (from linear theory) which holds on large scales for both partial and total power spectra. Finally, we exclude snapshots with redshifts below z ≲ 1, as even with our conservative selection of simulation parameters, there is a risk that their highest velocity dispersions may remain unresolved by the FDM solver (see Eq. 34). \nWe visualise the MDM density distribution for an axion fraction f = 0 . 1 in Fig. 3, juxtaposing the FDM and CDM fields. The clustering of the former is visibly suppressed below the axion Jeans scale k J, Eq. (10). The smallest-mass halos can remain completely CDMdominated. As known from the pure FDM case ( f = f max), the wave dynamics are reflected in interference fringes along filaments as well as granule structures in the DM halos.', '3 HALO MASS DISTRIBUTION AND DENSITY PROFILES': 'In the following, we present a study of HMFs and halo density profiles in MDM cosmologies. To identify halos, we use the R ockstar halo finder (Behroozi et al. 2013), which is based on adaptive hierarchical refinement of FOF groups in six dimensions (position and momentum space). This method provides robust tracking of substructure, being grid-independent, orientation-independent, and resilient to noise. However, R ockstar is a particle-based halo finder and thus \ncan only be applied to particle distributions. Although it is theoretically possible to convert FDM grid cells into particles by concentrating the mass into points at the centre of each cell, we found 3 that this approach leads to unreliable FOF groupings due to artifacts introduced by the grid features in the particle distribution. \nGiven that the dominant component in our MDM simulations is CDM ( f < 0 . 5), we base our halo identification exclusively on the CDM component. This is achieved by running R ockstar on the (equal-mass) CDM particle distribution after correcting the CDM particle mass m CDM such that the total mass of CDM particles present within the simulation box aligns with the anticipated total DM mass. 4 We will justify the usage of R ockstar on the CDM component a posteriori.', '3.1 Halo Mass Functions': "To measure the halo abundance in the simulations, we choose logarithmic mass bins of width ∆ log( M tot) = 0 . 03 and count the number of R ockstar halos that fall into each bin. We show the resulting HMFs in Fig. 4 at redshifts z = 1 -4. While CDM and the f = 0 . 01 cosmology follow the bottom-up structure formation paradigm in which small-mass halos form first and thus the small-mass end of their HMF barely changes from z = 4 to z = 1, MDM cosmologies with f ≳ 0 . 1 violate this picture and many small-mass halos are assembled at low redshift, with corresponding changes in the smallmass amplitude of their HMF. \nWe now compare against theoretical predictions for the HMF by employing the Sheth-Tormen approach for the halo mass function (Press & Schechter 1974; Sheth & Tormen 2002), \n1 M d n d ln( M ) = 1 2 ¯ ρ ( z ) M 2 f ( ν ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> d ln( σ 2 ) d ln( M ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> , (35) \nwhere n is the halo number density, ν = δ crit /σ ( M , z ) is the peak height with δ crit = 1 . 686 the critical linear density threshold for halo collapse, and the multiplicity function (for ellipsoidal collapse) is given by \nf ST( ν ) = A r 2 π √ q ν (1 + ( √ q ν ) -2 p ) e -q ν 2 2 , (36) \nwith A = 0 . 3222, p = 0 . 3 and q = 0 . 707. We adopt a spherical top hat window function W in real space when calculating the variance of the linear MDM power spectrum, 5 \nσ ( R , z ) 2 = 1 2 Z ∞ P L ( k , z ) ˜ W ( Rk ) 2 k 2 d k , \n˜ W ( x ) = 3 3 (sin x -x cos x ) . \n2 π 0 (37) x (38) \nThe variance can be transformed into a function of the halo mass via M = 4 π ¯ ρ R 3 / 3. \nWe show these model predictions as dashed curves in Fig. 4 and find a reasonable agreement between the model and the simulations, \nFigure 3. MDMstructure formation for m = 3 . 16 × 10 -25 eV and f = 0 . 1. We plot projected (comoving) densities of the FDM (left column) and CDM (right column) components along the line of sight on a logarithmic scale. The redshift spacings (di ff erent rows) correspond to equal logarithmic spacings in the scale factor. The axion Jeans scale prevents FDM from clustering around small-mass halos, which thus remain CDM-dominated. The wave dynamics of FDM is reflected in interference fringes along filaments as well as the presence of granule structures in the DM halos, most visible towards lower redshift. \n<!-- image --> \nFigure 4. MDM halo mass function (HMF) across redshifts z = 1 -4. We show results for a m = 3 . 16 × 10 -25 eV MDM cosmology at various FDM fractions f (see legend) and the reference CDM cosmology. Solid lines with markers show the HMF inferred from the simulation. The dashed curves trace the Sheth-Tormen HMF (based on linear MDM power spectra), which show good agreement with the inferred HMF. Dotted curves are best-fit results against Eq. (39), where we keep α 1 = 1 . 1 and α 2 = 2 . 2 fixed during the fit. The vertical dashed line corresponds to the mass resolution 50 × m ' CDM , where m ' CDM is the rescaled CDM particle mass (see text). \n<!-- image --> \n/circledot \n/circledot \nTable 3. Best-fit values of β in Eq. (39) for our MDM simulation suite across a range of redshifts z = 1 -8. Note that the dependence on f is substantial at higher redshift z ≳ 3. For f = 0 . 01 the exact value of β is less relevant since only the combination β f enters the parametrisation (39). \nat all redshifts shown z = 1 -4 and in fact out to z = 10 (not shown). The model curves for f ≳ 0 . 1 branch o ff from the CDM curves at the characteristic mass M 0 (see Eq. 16) below which the FDM HMF \nis suppressed with respect to the CDM case. While M 0 has been introduced for pure FDM ( f = f max), our MDM HMFs also branch o ff at approximately M 0 = 2 . 34 × 10 13 M ⊙ / h , suggesting that M 0 remains a useful and accurate mass scale even when f < f max. \nWe also aim to provide an analytic form for the HMFs as a function of the FDM fraction f . To that end, we generalise the twoparameter model of Schive et al. (2016) and write \nn ( M tot) = n ST ( P c L ) 1 -β f + β f 1 + M tot M 0 ! -α 1 ! -α 2 , (39) \nwhere n ST ( P c L ) is the Sheth-Tormen HMF in CDM. The steepness of the suppression is controlled by the parameter α 2 (but also β for M significantly smaller than M 0) while the sharpness of the transition at M ≈ M 0 is controlled by α 1. Fitting this model to N -body simulations of pure FDM ( f = f max) results in α 1 = 1 . 1, α 2 = 2 . 2 (Schive et al. 2016). Schive et al. (2016) have taken special care when removing spurious halos and have found that the redshift-independent suppression (1 + ( M h / M 0) -1 . 1 ) -2 . 2 provides a good fit across a range of redshifts z > 4. Subsequent works simulating bona fide FDM including the axion wave dynamics (May & Springel 2023) have found that this parametrisation remains robust down to at least z = 3. Our \nFigure 5. Spherically averaged density profiles in MDM at redshift z = 1. The mass bins range from 10 9 -10 10 M ⊙ / h in the top left to 10 12 -10 13 M ⊙ / h in the bottom right. The shaded areas delineate the standard error of the median while the solid curves trace the median profile in each mass bin. Arrows at the bottom of the panels indicate inner convergence radii r conv = 5 ϵ . The r -1 and r -2 power laws delineate two characteristic regimes of an NFW-like profile. Einasto best-fits are shown by circular markers, and we find that three-parameter Einasto model provides a good parametrisation of MDM density profiles across the entire range of FDM fractions f = 0 . 0 -0 . 3. \n<!-- image --> \nMDMHMFsarealso well fit using these reference values for α 1 and α 2, hence we adopt them throughout this work rather than refitting them. \nThe only parameter that requires tuning is thus β . This parameter is crucial for achieving higher levels of suppression, 1 -β f , to the left of the characteristic mass M 0 than would be allowed by the value of f alone. Best-fit values of β are presented in Table 3. Our results indicate that β typically decreases as f increases and as z decreases. The dependence on f is notably weaker at low redshift ( z ≲ 3). Specifically, the values β = 3 . 1, β = 2 . 9 and β = 2 . 4 provide a good fit at redshifts z = 3, z = 2 and z = 1, respectively. The best-fit results at z = 1 -4 are depicted as dotted curves in Fig. 4. The overall agreement with inferred HMFs is good across axion fractions f = 0 . 01 -0 . 3, indicating that the mass scale M 0 accurately characterises the suppression of the HMF despite being independent of f . The deviation from the inferred HMF at the high-mass end in case of f = 0 . 3 reveals limitations of parametrisation (39) as well as a potential dependence of M 0 on f .", '3.2 Density Profiles': 'A well-known prediction of pure FDM models ( f = f max) is the formation of solitonic cores at the centre of halos (Schive et al. 2014a). The situation is more complex in MDM, where in case of CDMdominated models ( f < 0 . 5) we expect the soliton to be buried beneath the CDM central density cusp. Before we verify this intuitive picture, we first quantify the total DM density profile averages in various mass bins. To better sample the FDM field in the centre of halos, we interpolate FDM density values (which are defined on a Eulerian grid in the simulation code) trilinearly to inter-grid points, while CDMdensity profiles are obtained using simple C osmic P rofiles routines (Dome 2023). The density profiles are calculated around halo centres as identified by R ockstar , which neglects the FDM component. This approach is justified in our CDM-dominated models ( f < 0 . 5), where the CDM cusp dominates the gravitational potential in the centre and shapes its minimum. Note that the FDM core exhibits time-dependent behaviour with no stable centre, undergoing random walks by an amount of the order of the soliton radius on \ntimescales τ ∝ ( m σ 2 FDM ) -1 , driven by perturbations away from the perfect stationary state (Chiang et al. 2021; Li et al. 2021). \nThe resulting density profiles in four mass bins from M tot = 10 9 -10 13 M ⊙ / h are shown in Fig. 5, tracing the two characteristic powerlaw regimes of an NFW-like profile (Navarro et al. 1997). The solid curves show the median density profile in logarithmic radial bins of width ∆ log( r ) = 0 . 05 while the shaded area delineates the standard error of the median. Note that in CDM the scale radius rs ≡ r -2 at which the logarithmic slope has the isothermal value of -2, i. e. d ln( ρ ) / d ln( r ) | r -2 = -2, migrates towards larger normalised radii as the halo mass grows. In other words, the concentration \nc ≡ R vir r -2 (40) \nof CDM halos decreases as the halo mass increases. This wellknown result reflects the higher background density at earlier epochs when smaller mass halos form (Navarro et al. 1997; Bullock et al. 2001; Ludlow et al. 2014). \nTo approximate the halo density profiles using an analytic form and to determine their concentration, we invoke the Einasto profile (Einasto 1965) \nln ρ E( r ) ρ -2 ! = -2 ζ r r -2 ! ζ -1 (41) \nand assess the fidelity thereof a posteriori. Best-fit Einasto profiles are determined by adjusting the three parameters ( ζ , r -2 and ρ -2) of Eq. (41) in order to minimise a figure-of-merit defined as \nψ 2 = 1 N bin N bin X i = 1 GLYPH<2> ln( ρ i ) -ln( ρ E( ri ; ρ -2; r -2; ζ )) GLYPH<3> 2 . (42) \nWe choose an inner convergence radius of r conv = 5 ϵ , where ϵ is the gravitational softening length. This choice is based on our finding that the 3 ϵ recommendation by Power et al. (2003) is not su ffi ciently conservative, particularly for small-mass bins, potentially leading to an underestimation of halo concentration. In addition, radial bins that exceed the outer limit of 0 . 8 R vir are also discarded in the fit since they might correspond to radii where halos are not fully relaxed (Ludlow et al. 2016). Best-fit results are shown in Fig. 5 and reproduce MDM density profiles very well across the entire range of FDM fractions f = 0 . 0 -0 . 3 at the redshift shown ( z = 1) but in fact out to z = 10 (not shown), albeit at much lower statistical significance.', '3.3 Concentration-Mass Relation': 'We now turn to the concentration-mass relation c ( M , z ) which follows trivially from the Einasto fits. The MDM concentration-mass relation inferred from the simulations at z = 1 is shown in Fig. 6 for logarithmic mass bins of width ∆ log( M tot) = 0 . 2. While the CDM concentration follows the characteristic decrease toward higher mass as mentioned before, this decrease is weaker at higher FDM fractions f and even seems to reverse for f = 0 . 3. To model the concentration mass relation, we invoke the Ludlow et al. (2016) approach based on extended Press-Schechter (EPS) theory. It stipulates that in order to estimate the mean concentration of halos at a given redshift z 0, one needs to solve the following system of coupled non-linear equations, \n⟨ ρ -2 ⟩ ρ 0 = C H ( z -2) H ( z 0) ! 2 , (43a) \nM -2 M 0 = erfc δ sc( z -2) -δ sc( z 0) p 2( σ 2 ( f coll M 0) -σ 2 ( M 0)) . (43b) \nc \nFigure 6. Concentration-mass relation c ( M , z ) in MDM cosmologies at redshift z = 1. Square markers indicate median concentrations obtained from Einasto best-fits. Analytic model predictions from Ludlow et al. (2016) for parameters f coll = 0 . 02 and C = 650 (see text) are traced by dashed curves, showing good agreement across f = 0 . 0 -0 . 3. \n<!-- image --> \n/circledot \nThe first equation relates the mean inner density within the scale radius, ⟨ ρ -2 ⟩ , with the critical density of the Universe at the collapse redshift, z -2. The second equation expresses the collapse mass fraction using EPS theory (Lacey & Cole 1993), encapsulating the physical meaning of the collapse redshift as the redshift at which the characteristic mass, M -2, was first contained in progenitors more massive than a fraction f coll of the final halo mass M 0. \nFollowing the recommendations of Ludlow et al. (2016), we set f coll = 0 . 02, C = 650 and solve Eqs. 43 in the respective MDM cosmologies. Note that the linear variance σ 2 entering the equations depends on the linear MDM matter power spectrum. We show the analytic model predictions in Fig. 6 and find good agreement with the inferred c ( M , z = 1) relation across the entire range of FDM fractions f = 0 . 0 -0 . 3. \nAccording to the model, the concentration c exhibits a very flat U -shaped profile as a function of mass M tot for cosmologies with high values of f . It indicates that the strict bottom-up structure formation picture in which smaller-mass halos have necessarily higher concentration is invalidated in MDM cosmologies. Note that this result is well-known in pure FDM and warm DM (WDM) cosmologies (Schneider et al. 2012; Dome et al. 2022). Even though we cannot directly resolve the concentration mass scale M conc discussed around Fig. 1, a visual extrapolation of the inferred concentration-mass relation c ( M , z ) to higher masses suggests that the curves converge around M conc ≈ 4 . 7 × 10 14 M ⊙ / h . This implies that M conc ≈ 20 -50 M 0 represents a characteristic scale, independent of the axion fraction, at which the c ( M , z ) relation in MDM diverges from that in CDM.', '4.1 Halo Model Details': "Having established that identifying halos based solely on the CDM component produces reliable HMFs and c ( M , z ) relations, we now focus on calibrating the halo model framework A xion HM code (Vogt \net al. 2023) to the MDM simulations and compare the resulting nonlinear power spectra. A xion HM code , an adaptation of HMC ode -2020 (Mead et al. 2021), incorporates ultralight particles as part of the DM. It builds on the halo model originally implemented in Dentler et al. (2022) to constrain axions using weak lensing shear statistics. The code takes as input a range of cosmological and DM parameters: Ω m, Ω b, f , m , H 0, ns , As , k piv. Here, the axion density parameter is Ω a ≡ Ω FDM = f Ω m, while the combined density of CDM and baryons ('cold' matter) is Ω c = Ω m -Ω a = Ω CDM + Ω b. Using this input, A xion HM code calculates the non-linear power spectrum at a desired redshift z . The linear FDM power spectrum, which is integral to the model, is calculated using the public version of A x -ion C amb . The total matter overdensity in the MDM cosmology can be decomposed into a sum of the cold matter, δ c, and axions, δ a, \nδ m = Ω c Ω m δ c + Ω a Ω m δ a . (44) \nThe non-linear power spectrum in A xion HM code is constructed as \nP ( k ) = Ω c Ω m ! 2 P c( k ) + 2 Ω c Ω a Ω 2 m P c,a( k ) + Ω a Ω m ! 2 P a( k ) , (45) \nwhere P c, P c,a( k ) ∝ δ c δ a and P a are the cold, cross and axion power spectrum, respectively. For P c( k ), the standard halo model (see Sec. 4.4 for improvements on top of this) is adopted, splitting the power spectrum into the one-halo term P 1h and two-halo term P 2h , \nP c( k ) = P 1h c ( k ) + P 2h c ( k ) . (46) \nFor axions we adopt the biased tracer formalism (Massara et al. 2014) which assumes that a sub-component, δ L, cannot cluster and evolves approximately linearly while the remaining fraction \nF h = 1 ¯ ρ a Z ∞ M cut d M c n ( M c) b ( M c) M a( M c) ∈ [0 , 1] (47) \nis in halos, i. e. \nδ a = F h δ h + (1 -F h) δ L . (48) \nHere, n ( M c) denotes the cold HMF, and we have introduced the cold halo bias, b ( M c), as well as the axion halo mass to cold halo mass relation, M a( M c). The biased tracer formalism assumes that axion halos only form in and around cold matter halos and thus the halo mass function for axions is the same as for the cold field, n ( M a)d M a = n ( M c)d M c, and the linear axion halo bias corresponds to the cold halo bias, b ( M a) = b ( M c), i. e. M a is itself a function of M c, a relation we need to specify (see below). In total there are three new quantities we have to provide to complete the MDM halo model: the cut-o ff mass, M cut, the axion halo mass relation, M a( M c), and the axion halo density profile ρ a( r , M a , z ). For details see Vogt et al. (2023). \nTo construct the cold halo density profile, we retain the Bullock et al. (2001)-inspired model for the concentration parameter of cold halos of mass M c, \nc c( M c , z ) = B 1 + z f( M c , z ) 1 + z ! , (49) \nwhere z f denotes the formation redshift, which is defined by \nD + ( z f) D + ( z ) σ c(0 . 01 M c , z ) = δ crit , (50) \nwith D + ( z ) denoting the total MDM linear growth rate (depending only on cosmology and redshift). The minimum halo concentration B = 5 . 196 is attained when the solution to Eq. (50) yields z f < z . \nFigure 7. Spherically averaged density profile of the FDM component in the f = 0 . 1 MDM cosmology at redshift z = 1. The top panel covers a total halo mass range of M tot = 10 9 -10 13 M ⊙ / h while the lower panel focuses on the range M tot = 10 10 . 4 -10 12 . 4 M ⊙ / h with a finer mass resolution of ∆ log( M tot) = 0 . 2. Shaded areas delineate the standard error of the median, and solid curves trace the median profile in each mass bin. Arrows at the bottom of the panels indicate the FDM grid resolution scale of ∆ x / 2 = 14 . 6 ckpc / h , rescaled by the mean virial radius in each mass bin. For masses M tot > M cut = 1 . 7 × 10 12 M ⊙ / h , the axion profile displays a distinct core-like feature, indicating the presence of a soliton. The steep decline in central densities between neighboring mass bins becomes pronounced for M tot < M J = 1 . 0 × 10 11 M ⊙ / h , reflecting a diminished influence of the FDM component on structure formation. \n<!-- image --> \nEven though the Bullock et al. (2001) model has been shown to predict a sharp decline in the concentration at high mass that is inconsistent with N -body simulations and to insu ffi ciently capture e ff ects in non-CDM cosmologies such as WDM, the more accurate Ludlow et al. (2016) approach introduced in Sec. 3.2 is less suitable for the halo model since it typically underpredicts CDM small-scale power around k ≳ 5 cMpc -1 in both pure CDM and MDM simulations. Shifting to Ludlow et al. (2016) would also necessitate recalibrating HMC ode -2020 parameters (most notably the halo bloating parameter η ) without gaining in precision.", '4.2 Cut-O ff Mass': "We first perform a sanity check of the cut-o ff mass M cut adopted by A xion HM code . It is obtained by invoking the concept of the halo \nFigure 8. Axion halo mass-cold halo mass relation M a( M c) in MDM cosmologies at various axion fractions f = 0 . 0 -0 . 3 across redshifts z = 1 -4. The mass resolution is ∆ log( M tot) = 0 . 2 as in Fig. 7 and we estimate the axion halo mass M a = 4 π R R vir 0 d r r 2 ρ FDM( r ) by integrating the axion density profile out to the virial radius R vir. For consistency we estimate the cold halo mass M c likewise via integration since the direct estimate M c = Ω c / Ω m M tot from the R ockstar virial mass can deviate therefrom at the high-mass end due to resolution e ff ects. Dashed curves indicate the cosmic mean relation M a = Ω a Ω c M c, currently implemented in A xion HM code . Colored arrows at the bottom of the panels indicate the cut-o ff mass M cut = 5 × 10 11 -10 12 M ⊙ / h while the black arrow denotes the Jeans mass M J. Note that we infer a steeper M a( M c) relation than the cosmic average for M c ≲ M J, with steepness increasing toward lower f . The transition range widens toward high axion fractions, covering M J -M cut in some cases. We fit a broken power law, Eq. (52), for M a > 10 7 M ⊙ / h (grey arrow) and show results as dotted curves. \n<!-- image --> \n/circledot \n/circledot \nJeans scale r hJ from Sec. 2.2. To verify the accuracy of Eq. (15) as the length scale below which axions fail to cluster inside DM halos, we calculate density profiles of the axion component in the MDM simulations. Fig. 7 illustrates these profiles for the f = 0 . 1 MDM cosmology at redshift z = 1 across various mass bins. Our analysis, which includes simulations across axion fractions f = 0 . 0 -0 . 3 and redshifts up to z = 10, consistently shows that halos with mass M tot ≳ M cut exhibit a distinct soliton core. Below the cut-o ff mass, and particularly below the Jeans mass M J, the central density of the core declines sharply, resulting in featureless axion profiles for halos with M tot ≪ M J. \nThis steep decline in profile density supports M cut as an e ff ective estimate of the mass scale below which halos (slowly) transition to being predominantly CDM-dominated in MDM. It is important to note that in the transitional mass range M J -M cut, 'quantum pressure' continues to play a role but is insu ffi cient to fully counteract gravitational e ff ects (recall definition of M J in Sec. 2.2). The transition to CDM-dominated halos can occur over a large mass range exceed- \ning 0.5 dex, and is not always complete for mass M tot ≈ M J. In the following, we study integrated axion density profiles (i. e. the axion halo mass relation) and we will see that the Jeans mass and cut-o ff mass can help describe the transition toward CDM-dominated halos.", '4.3 Axion Halo Mass Relation': 'We now turn to the relationship between the axion halo mass, M a, and the cold halo mass, M c. In the public version of A xion HM code , it is assumed that the axion mass M a follows the cosmic abundance fraction relative to the cold halo mass M c down to the cut-o ff mass M cut, i. e. that M a = ( Ω a / Ω c) M c for M c > M cut. However, as seen in Sec. 4.2, there are strong indications that this simplistic approach may not accurately capture the axion halo mass-cold halo mass relation found in simulations. Fig. 8 shows the M a( M c) relation inferred from our MDM simulations at various axion fractions f = 0 . 0 -0 . 3 across redshifts z = 1 -4. We estimate both the axion and cold halo \nTable 4. Best-fit values of β 2 in Eq. (52) for our MDM simulation suite across a range of redshifts z = 1 -8. In the fitting process, we fix β 1 at 1, as allowing β 1 to vary only results in minor improvements. Note that β 2 increases with higher redshift and lower values of f . \nmass by integrating the respective density profile out to the virial radius R vir, \nM a,c = 4 π Z R vir 0 d rr 2 ρ a,c( r ) . (51) \nThe cold halo mass could also be obtained directly from the R ock -star virial mass, M c = ( Ω c / Ω m) M tot, but the two estimates can deviate by more than 0 . 5 dex (at the high-mass end) due to resolution e ff ects, hence for consistency we determine M c via integration. \nAs shown in Fig. 8, the cosmic average relation, M a = ( Ω a / Ω c) M c, is a valid approximation at the high-mass end. 6 Toward lower mass, the simulated M a( M c) relation becomes significantly steeper. Deviations from the cosmic mean exceed 1 dex in several mass bins. These deviations become particularly pronounced below the Jeans mass, M J, consistent with the axion density profiles discussed in Sec. 4.2. The break can span a wide mass range, especially at higher axion fractions, typically encompassing M J -M cut and exceeding 0.5 dex for high values of f . To better capture this behaviour, we propose to parametrise the axion halo mass-cold halo mass relation as a broken power law: \nM a = 1 + M c M J ! -β 1 -β 2 Ω a Ω c M c , (52) \nwhere the steepness of the suppression is controlled by the parameter β 2 while the sharpness of the transition at M ≈ M J is controlled by β 1, similar to Eq. (39). The linear Jeans mass is given by Eq. (11). Letting β 1 vary during the fit along with β 2 leads to only minor improvements, hence we fix β 1 = 1. Best-fit values of β 2 for M a > 10 7 M ⊙ / h are presented in Table 4, and the corresponding best-fit curves are illustrated in Fig. 8. We observe a weak dependence of β 2 on redshift, but a notable decrease of β 2 as the axion fraction f increases, indicating a weaker suppression. This analysis shows that MDM cosmologies dominated by CDM f < 0 . 5 are different from pure axion cosmologies ( f = f max), where axion halos do not exist below M J and virialised axion halos only form above M cut (see Sec. 2.2). \nAnother implication of the M a( M c) relation deviating from the cosmic mean, where M a = ( Ω a / Ω c) M c, is that expressing total DM density profiles as ρ DM( r ) = f ρ FDM( r ) + (1 -f ) ρ CDM( r ) (see e. g. Shevchuk et al. 2023) can at best be a good approximation at \nhigh masses above the cut-o ff mass M cut. A more detailed analysis of axion density profiles, including their parametrisation via soliton + NFWprofiles, their (non-)formation at low axion fractions (see e. g. Schwabe et al. 2020), and their use in axion forecasts and constraints, is beyond the scope of this paper and will be addressed in future work.', '4.4 A xion HM code Parameters': 'The updated version of A xion HM code has several improvements over the original implementation by Vogt et al. (2023), which we now summarise. As in Vogt et al. (2023), we adopt the HMC ode -2020 parameters (Mead et al. 2021), which have been introduced to improve the model in its fit to Λ CDM simulations over the standard halo model. The HMC ode -2020 parameters were calibrated using the M ira T itan matter power spectrum emulator of Heitmann et al. (2016); Lawrence et al. (2017). This cosmic emulator encompasses eight cosmological parameters and provides an accuracy of 4 % for k < 7 h cMpc -1 . In turn, the accuracy of HMC ode -2020 when compared to simulated Λ CDMdata is excellent with a RMS error of less than 2.5 % for k < 10 h cMpc -1 and z < 2 (Mead et al. 2021). \nThe HMC ode -2020 parameters are only calibrated up to z = 2 and it is a priori not guaranteed that a non-linear power spectrum with these parameters at z > 2 is more accurate than without the parameters (i. e. standard halo model). We adopt the HMC ode -2020 parameters up to z = 3 . 5 and will show in Sec. 4.5 that not only is the agreement with simulations within the 10 % margin for pure Λ CDM, but that the improvements to A xion HM code (including generalising and recalibrating one of the HMC ode -2020 parameters) yield good agreement with simulated non-linear power spectra up to at least z = 3 . 5 for axion fractions that we can assess, f < 0 . 3. \nOne of the HMC ode -2020 parameters is the one-halo term damping. In the standard halo model approach (see Eq. 46), the one-halo term is typically constant on large scales. However, this does not accurately reflect mass and momentum conservation. It was demonstrated by Smith et al. (2003) that the one-halo term should increase as P 1h( k ) ∝ k 4 at small k (i. e., it should dampen compared to a constant at small k ). To address this, HMC ode -2020 implements a modification: \nP 1h c ( k ) → P 1h c ( k ) ( k / k ∗ ) 4 1 + ( k / k ∗ ) 4 . (53) \nThis adjustment ensures that the one-halo term grows as expected and is suppressed on large scales. Consequently, on large scales, the non-linear power spectrum is primarily determined by the two-halo term, which aligns with the (perturbed) linear power spectrum. The suppression e ff ect is controlled by the free parameter k ∗ , which was fitted as: \nk ∗ = 0 . 05618 × σ 8 , c( z ) -1 . 013 h cMpc -1 . (54) \nWe can now list the improvements made in the new version of A xion HM code : \n- · To construct the two-halo term for the cold matter component, we apply a perturbative damping to the linear power spectrum, so that \nP 2h c ( k ) = P L c ( k ) 1 -q ( k / k d) n d 1 + ( k / k d) n d ! . (55) \nThis formulation aligns with the recommendations of Mead et al. (2021), who suggest that, given the precision of modern halo models, it is crucial to account for the largest-scale non-linear e ff ects. \nTable 5. A xion HM code parameters, built on top of the standard halo model. For each parameter, we provide its description, the equation defining it, the default value in the standard halo model, the fitted functional form or value, and an example of this function evaluated at z = 1 for a standard MDM cosmology with f = 0 . 1, m = 3 . 16 × 10 -25 eV and Planck Collaboration et al. (2016) cosmological parameters. \nSpecifically, perturbation theory indicates that the most significant non-linear e ff ect on large scales is a small damping of power. The term introduced in Eq. (55) is designed to capture this e ff ect, with the three parameters fitted and given by Mead et al. (2021), \nk d = 0 . 05699 × σ 8 , c( z ) -1 . 089 h cMpc -1 , q = 0 . 2696 × σ 8 , c( z ) 0 . 9403 , (56) n d = 2 . 853 . \nNote that the di ff erence between perturbatively corrected linear theory and the standard two-halo term is tiny but significant gains in computational time are made when replacing the integral expression of the standard two-halo term by Eq. (55). \n- · We achieve additional significant speed-ups through modularisation and in-memory storage of arrays. For example, the formation redshift z f for a given cold halo mass M c is computed once and subsequently stored. As a result, we reduce the runtime on a single-core machine to under 1 minute for a single evaluation of A xion HM code , with only a minor increase in memory overhead.\n- · We fix a bug in the implementation of the halo bloating e ff ect mediated by the parameter η which scales the wavenumber k in the Fourier transformation of the NFW profile of cold halos as \n˜ u ( k , M , z ) → ˜ u ( ν η k , M , z ) . (57) \nThe halo bloating parameter was fitted to (Mead et al. 2021) \nη = 0 . 1281 × σ 8 , c( z ) -0 . 3644 , (58) \nwhich we also adopt in A xion HM code . \n- · We adjust the cosmic axion halo mass-cold halo mass relation M a( M c) = ( Ω a / Ω c) M c by incorporating a broken power law below the linear Jeans mass M J, in accordance with the findings from our MDM simulations which reveal a strong decrement in the inferred axion halo mass M a compared to the cosmic mean relation (see Eq. 52). Consequently, lower integration limits for various quantities, such as the clustered fraction of Eq. (47), are adjusted from M cut down to 0.\n- · Wecontinue to use the smoothing parameter α , which allows us to overcome the simplistic assumption of a purely additive behaviour \nof one- and two-halo terms by modelling \nP c( k ) = ( P 1h c ( k ) α + P 2h c ( k ) α ) 1 /α . (59) \nMead et al. (2021) fitted parameter α to a general form, \nα = 1 . 875 × (1 . 603) n e ff ,c ( z ) , (60) \nwhere the e ff ective spectral index at the non-linear length scale is \nn e ff ,c( z ) = -d ln( σ 2 c ( R , z )) d ln( R ) GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> σ c = δ crit -3 . (61) \nWe first generalise the smoothing parameter to the cold matter component of MDM. To prevent strong smoothing e ff ects to spill over from the quasi-linear regime into the large-scale regime, the coldcold smoothing is modeled using a logistic function between α 1 , c and α 2 , c: \nα c( k , z ) = α 2 , c( z ) + α 1 , c( z ) -α 2 , c( z ) 1 + exp(( k piv -k ) / ∆ k ) . (62) \nThe width of the transition is controlled by ∆ k = 0 . 1 h cMpc -1 . In the small-scale limit, we use the following expression: \nα 1 , c( z ) = 1 . 875 × (1 . 603) n e ff ,c ( z ) 1 + a GLYPH<16> 10 -24 m GLYPH<17> b GLYPH<16> Ω a Ω m GLYPH<17> d (1 + z ) e . (63) \nThis parametrisation reduces to the default smoothing parameter for Λ CDM in the limit of a high axion mass m → ∞ or a low axion fraction Ω a → 0, given that b , d > 0. Due to our lack of MDM simulation for axion mass values other than m = 3 . 16 × 10 -25 eV, we fix the exponent b = 0 . 0450 based on the expectation of a weak dependence on m . In the large-scale limit, where k < k piv = 0 . 5 h cMpc -1 , the smoothing is well fit by \nα 2 , c( z ) = α 1 , c( z ) for f < 0 . 01 max GLYPH<16> α 1 , c( z ) , 1 . 10 1 + z GLYPH<17> for f ≥ 0 . 01 . (64) \nWe fit parameters a , d , e against the simulated cold-cold power spectrum P c (see Eq. 59) using the Nelder & Mead (1965) simplex algorithm with equal logarithmic weights in k ∈ [0 . 2 -30] h cMpc -1 and \nFig. 10 displays the non-linear matter power spectrum predictions from the updated version of A xion HM code at redshifts z = 1 -3 . 5. We use the default configuration, which includes large-scale damping of the one-halo term, perturbative damping of the two-halo term, halo bloating, and transition smoothing between one- and twohalo terms in the quasi-linear regime. These predictions are compared with MDM power spectra derived from our simulations using P ylians (Villaescusa-Navarro 2018). \n<!-- image --> \n2 \nFigure 9. A xion HM code vs MDM simulations. We compare the non-linear power spectrum for our MDM simulation (solid) with the halo model implementation using the new version of A xion HM code (dashed). Results are shown for f = 0 . 1 and m = 3 . 16 × 10 -25 eV at redshifts z = 1. We show the full power spectrum (blue) and the three contributing terms from Eq. (45) (green, orange, and red). For comparison, we show the total matter power spectrum in a pure Λ CDM cosmology (black solid). The dashed vertical line indicates the fundamental frequency of the simulation box k f = 2 π/ L box while the Nyquist frequency k Ny = π N 1 / 3 CDM / L box is shown as a vertical dotted line. \nusing data from z = 1 , 2 , 3 and f = 0 . 0 , 0 . 01 , 0 . 1 , 0 . 2 , 0 . 3 with equal weight, minimising the figure-of-merit \nψ 2 = 1 N X z , f , k | log(AxHM( k , z , f )) -log(Sim( k , z , f )) | 2 . (65) \nIn addition, we find better agreement with simulations if we add an independent smoothing parameter to the cold-axion cross power spectrum P c,a (which depends on P c): \nP c,a( k ) = F h GLYPH<16> P 1h c,a ( k ) α c,a + P 2h c,a ( k ) α c,a GLYPH<17> 1 /α c,a + (1 -F h) q P c( k ) P L a ( k ) (66) \nWe parametrise α c,a in the exact same way as the small-scale coldcold smoothing term α 1 , c( z ) in Eq. (63), and fit parameters a , d , e against the simulated P c,a spectrum. Best-fit values for α 1 , c , α 2 , c and α c,a, along with all other A xion HM code parameters, are provided in Table 5. \n· In the new version of A xion HM code , we activate the halo bloating parameter η (for the cold halo density profile), along with the one-halo damping (see Eq. 53), the two-halo damping (see Eq. 55), and the smoothing parameter (see Eqs. 63 and 62) by default. This configuration significantly enhances the agreement with MDM simulations, both qualitatively and quantitatively. These improvements validate the necessity of HMC ode -2020-like parameters for accurate MDMsimulations, as initially suggested by Vogt et al. (2023).', '4.5 Non-Linear Power Spectra': "We now validate the updated version of A xion HM code . Recall that the M a( M c) relation was fitted over the redshift range z = 1 -8, the smoothing parameter was fitted for z = 1 -3, and the remaining HMC ode -2020 parameters are calibrated up to z = 2. As a re- \nsult, A xion HM code should be used with caution outside the range z ≈ 1 -3 and f ≈ 0 -0 . 3. In Fig. 9, we illustrate its predictions for the non-linear power spectrum, compared to results from our MDM simulations for f = 0 . 1 at redshift z = 1. We find very good agreement between the total predicted and simulated matter power spectrum (blue solid and dashed). The moderate suppression of power relative to Λ CDM on scales k > 1 h cMpc -1 is well captured. The agreement between A xion HM code and simulations extends to the cold matter power spectrum (green), though there is a slight discrepancy for the cross and FDM-FDM power spectra (orange and red) at k > 10 h cMpc -1 . Note that beyond k > 10 h cMpc -1 we expect strong e ff ects from baryonic physics (see e.g. Mead et al. 2021), which are not accounted for in either our MDM simulations or A x -ion HM code . \nA xion HM code also captures the enhanced non-linear power on scales k ≈ (1 -10) h cMpc -1 relative to Λ CDM. The e ff ect has been reported by Vogt et al. (2023) who traced it to an enhancement in the cross and FDM-FDM power spectra, which in turn is caused by the coherence of the soliton in the axion halo density profile, increasing the correlation function of the axion field on such scales. The enhancement is stronger for axion masses around m ≈ 10 -22 eV (shifting toward higher k ) and was also observed in pure FDM cosmologies when accounting for 'quantum pressure' (Nori et al. 2018; May & Springel 2023). \nAs anticipated, A xion HM code demonstrates excellent agreement (deviations less than 10 %) with simulations for pure CDM ( f = 0 . 0) across all scales and redshifts examined. For an axion fraction of f = 0 . 1, the code maintains high accuracy on both large (small k ) and small scales (large k ) at all considered redshifts, with maximum deviations also remaining below 10 %. When the axion fraction is increased to f = 0 . 2, maximum deviations rise to approximately 20 % for scales with k < 10 h cMpc -1 ; however, at redshifts around z ≈ 1, deviations on these scales remain under 10 %. For f = 0 . 3, the maximum deviations increase to about 30 % for scales with k < 10 h cMpc -1 , though at redshifts around z ≈ 1, deviations do not exceed approximately 20 % on these scales. \nWe are unable to assess the agreement with MDM simulations for redshifts z < 1 due to the potential for unresolved high velocity dispersions in the FDM solver (see Sec. 2.6). Similarly, neither HMC ode -2020 nor A xion HM code are anticipated to perform well at redshifts above z ≈ 3 . 5, which defines the upper limit of our redshift window. Note that our MDM simulation data is limited to FDM fractions f ≤ 0 . 3. However, this limitation is not critical, as ultralight axions are permitted to exist in substantial portions within the FDM window but not at extremely high fractions close to f max, which are increasingly ruled out with greater significance beyond the FDM window. Additionally, our fitting of A xion HMcode parameters is constrained to Λ CDM and MDM cosmologies with axion mass m = 3 . 16 × 10 -25 eV, which lies within the FDM window 10 -25 eV ≲ m ≲ 10 -23 eV. Due to the lack of MDM simulation data for other axion mass values, we cannot validate the accuracy of A xion HM code for axion masses significantly di ff erent from m = 3 . 16 × 10 -25 eV, except in the limit as m → ∞ . The exponent value b = 0 . 0450 in Eq. (63) is thus an educated guess based on the \nFigure 10. Comparison of non-linear matter power spectrum predictions from the updated version of A xion HM code and MDM simulations at across redshifts z = 1 -3 . 5. Solid lines represent the new model, while dotted lines show the previous version from Vogt et al. (2023). Horizontal dashed and dotted lines indicate 10 % and 20 % deviations, respectively. Arrows at the bottom mark comoving Jeans scale k J at each redshift (see Eq. 10). The axion fraction f increases from the top left panel ( f = 0 . 0) to the bottom right ( f = 0 . 3) along with a decrease in overall accuracy of A xion HM code . \n<!-- image --> \nexpectation of a weak dependence on m . Future MDM simulations will provide the opportunity to refine this estimate and validate the parameter b .", '5 CONCLUSIONS': 'In view of tighter constraints being put on pure, single-field ultralight axion DM (see Dome et al. 2022, for a compilation), it is promising to relax the requirement that ultralight axions must comprise all of the DM in the Universe. In this work, we focused on the FDM window 10 -25 eV ≲ m ≲ 10 -23 eV in which ultralight axions are allowed to exist in large portions (albeit not f = f max) (although see Shevchuk et al. 2023; Lazare et al. 2024; Winch et al. 2024). \nMethods: We implemented an MDM gravity solver (see Sec. 2.5) and ran state-of-the-art simulations of mixed ultralight axion cosmologies dominated by CDM ( f < 0 . 5). Our MDM simulations were designed to capture the wave dynamics across small and intermediate length scales, with particular emphasis on achieving numerical convergence and resolution thresholds. ICs were set up carefully via second order Lagrangian perturbation theory (2LPT) using M u -sic (Hahn & Abel 2011) based on matter and velocity power spectra \ncalculated using A xion C amb (Hlozek et al. 2015). By rigorously enforcing criteria such as velocity resolution and resolving the axion half-mode scale k 1 / 2, while also ensuring accurate representation of halo populations, our simulations faithfully reproduce internal halo structures above redshifts z ≈ 1, providing a robust platform for evaluating common (semi-)analytical techniques such as halo models. \nHalo Mass Distribution: We identified halos using the R ockstar particle-based halo finder applied on the (equal-mass) CDM distribution and report total halo mass distributions ( M tot = M c + M a) in MDM. We found good agreement between the Sheth-Tormen model based on the linear MDM matter power spectrum and the inferred HMF across a very wide range of redshifts z = 1 -10 and axion fractions f = 0 . 0 -0 . 3, justifying the usage of R ockstar on the CDM component a posteriori. The HMF in MDM cosmologies branches o ff from the CDM one at the characteristic mass M 0 = 1 . 6 × 10 10 ( m / 10 -22 eV) -4 / 3 M ⊙ , but instead of plateauing toward low mass as in WDM or turning over as in pure FDM, the HMFcontinues to increase in a power-law fashion as M tot decreases. By providing best-fit results to a one-parameter model of the MDM HMF, Eq. (39), we hope to facilitate parameter sampling across MDMcosmologies in Bayesian constraint and forecast analyses. \nDensity profiles: We fit the Einasto model against total DM density profiles and found the Einasto parametrisation to be reliable across the entire range of FDM fractions f = 0 . 0 -0 . 3 and redshifts z = 1 -10 studied. The resulting median concentration-mass relation c ( M tot) is in good agreement with the Ludlow et al. (2016) model based on extended Press-Schechter theory, suggesting that instead of turning over at around two decades above the half-mode mass, 100 × M 1 / 2, as in pure WDM and FDM, the concentration exhibits a decrease before recovering and increasing toward smaller halo mass M tot. This results in an e ff ective flat U-shaped c ( M tot) relation for high values of the axion fraction f , and is in agreement with insights from the halo mass distribution. \nCalibrating AxionHMcode: We aimed at improving the calibration of the halo model code A xion HM code based on a biased tracer approach using insights from our MDM simulations. The aforementioned success of reproducing analytical total halo mass distributions and concentration-mass relations based on halos identified solely using the CDM component lends additional credence to the viability of the biased tracer approach. The modifications we introduce (apart from minor ones) are threefold: First, we model the axion halo mass-cold halo mass relation M a( M c) as a broken power law below the Jeans mass M J and retain the cosmic mean relation M a = ( Ω a / Ω c) M c above M J. Second, we generalise the transition smoothing parameter α to MDM with a dependence on m and Ω a while heeding the spill-over e ff ects of strong smoothing on the largescale regime using a logistic function for the wavenumber-dependent smoothing α c( k ). Third, we introduce various speed-ups by making sure numerical functions are not evaluated too often, leading to a slight increase in memory requirements while reducing run-time on a single-core machine to below 1 minute for a single evaluation of A xion HM code . The code exhibits excellent agreement with simulations for pure Λ CDM, with deviations under 10 % on scales below k < 20 h cMpc -1 and redshifts z = 1 -3 . 5. For axion fractions f ≤ 0 . 3, the model maintains accuracy with deviations under 20 % at redshifts z ≈ 1 and scales k < 10 h cMpc -1 , though deviations can reach up to 30 % for higher redshifts when f = 0 . 3. \nOutlook: Mixed ultralight axion cosmologies dominated by CDM ( f < 0 . 5) have their clustering properties determined to first order by the CDM component. More precisely, the peak height distribution is largely shaped by the cold component as first suggested by Massara et al. (2014), motivating not only the very assumptions underlying A xion HM code but also the identification of halos based solely on the CDM component. MDM models might have the potential to reconcile observational constraints while providing a semiphenomenological route to understanding the nature of DM. In upcoming forecast and constraint analyses based on observational data, having good control of non-linear predictions in MDM models is key, and this work aims to contribute to that undertaking.', '6 ACKNOWLEDGEMENTS': 'We acknowledge useful discussions with Sophie Vogt. TD acknowledges support from the Isaac Newton Studentship and the UK Research and Innovation (UKRI) Science and Technology Facilities Council (STFC) under Grant No. ST / V50659X / 1. SM acknowledges support by the National Science Foundation under Grant No. 2108931. SB is supported by the UKRI Future Leaders Fellowship (Grant No. MR / V023381 / 1). AL acknowledges support from NASA grant 21-ATP21-0145. DJEM is supported by an Ernest Rutherford Fellowship from the STFC, Grant No. ST / T004037 / 1. This work used the DiRAC@Durham facility managed by the Insti- \ntute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility, with equipment funded by BEIS capital funding via STFC capital grants ST / K00042X / 1, ST / P002293 / 1, ST / R002371 / 1 and ST / S002502 / 1, Durham University and STFC operations grant ST / R000832 / 1.', '7 DATA AVAILABILITY': 'MDMsnapshot data and post-processing scripts are made available upon reasonable request.', 'REFERENCES': 'Heitmann K., et al., 2016, ApJ, 820, 108'}

2024arXiv240911245L

In the milliHertz frequency band stochastic gravitationalwave background can be composed of both astronomical and cosmological sources both can be anisotropic. Numerically depicting these anisotropies can be critical in revealing the underlying properties of their origins. For the first time we perform a theoretical analysis of the constraining ability of TianQin on multiple moments of the stochastic background. First we find that with a oneyear operation for a background with a signaltonoise ratio of 16 TianQin can recover the multiple moments up to l4. We also identified a unique feature of the stochastic background sky map which is the mirror symmetry along the fixed orbital plane of TianQin. Thirdly we explain the difference in anisotropy recovering ability between TianQin and LISA by employing the criteria of the singularity of the covariance matrix which is the condition number. Finally we find that since the different data channel combinations correspond to different singularities certain combinations might have an advantage in stochastic background mapmaking. We believe that the findings of this work can provide an important reference to future stochastic background analysis pipelines. It can also serve as a guideline for designing better gravitationalwave detectors aiming to decipher anisotropies in the stochastic background.

2024-09-01T00:00:00Z

['arXiv:2409.11245', '2024arXiv240911245L', '10.48550/arXiv.2409.11245']

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

Mapping Anisotropies in the Stochastic GravitationalWave Background with TianQin

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.11245.pdf

{'Mapping Anisotropies in the Stochastic Gravitational-Wave Background with TianQin': 'Zhi-Yuan Li, 1 Zheng-Cheng Liang, 1, ∗ En-Kun Li, 1 Jian-dong Zhang, 1 and Yi-Ming Hu 1, † 1 MOE Key Laboratory of TianQin Mission, TianQin Research Center for Gravitational Physics & School of Physics and Astronomy, Frontiers Science Center for TianQin, CNSA Research Center for Gravitational Waves, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China (Dated: September 18, 2024) \nIn the milli-Hertz frequency band, stochastic gravitational-wave background can be composed of both astronomical and cosmological sources, both can be anisotropic. Numerically depicting these anisotropies can be critical in revealing the underlying properties of their origins. For the first time, we perform a theoretical analysis of the constraining ability of TianQin on multiple moments of the stochastic background. First, we find that with a one-year operation, for a background with a signal-to-noise ratio of 16, TianQin can recover the multiple moments up to l = 4. We also identified a unique feature of the stochastic background sky map, which is the mirror symmetry along the fixed orbital plane of TianQin. Thirdly, we explain the difference in anisotropy recovering ability between TianQin and LISA, by employing the criteria of the singularity of the covariance matrix (which is the condition number). Finally, we find that since the different data channel combinations correspond to different singularities, certain combinations might have an advantage in stochastic background map-making. We believe that the findings of this work can provide an important reference to future stochastic background analysis pipelines. It can also serve as a guideline for designing better gravitational-wave detectors aiming to decipher anisotropies in the stochastic background.', 'I. INTRODUCTION': "A stochastic gravitational-wave (GW) background (SGWB) consists of GWs that, when individually undetectable, collectively form a background [1]. Various mechanisms contribute to the SGWB, which can be categorized into astrophysical and cosmological origins [2-4]. Astrophysical origins primarily include double white dwarfs (DWDs) [5, 6], massive binary black holes (BBHs) [7], stellar-mass BBHs [8, 9], binary neutron stars (BNSs) [10, 11], extreme-mass-ratio inspirals (EMRIs) [12-14], and GW bursts [15]. Cosmological origins mainly contain inflation [16], first-order phase transitionss (PTs) [17], and cosmic defects [18, 19]. SGWB from either astrophysical origin [5, 7, 12, 20] or cosmological origin [21-27] can exhibit the potential for anisotropy [28]. The anisotropy can offer insights into the distribution of compact binaries within our Galaxy, the history of galaxy mergers [29-31], as well as the priceless primordial information of the early Universe [32-37]. For example, the structure of our Galaxy can be mapped using the lower multipole moments with l ≤ 4 [38, 39]. Conversely, the higher multipole moments, with l ≳ 100, are more suitable for detecting the subtle imprints of primordial fluctuations that are preserved in cosmological backgrounds [4, 34]. \nThe diverse origins of the SGWB allow it to span multiple frequency bands. At frequencies in the hundreds of hertz, ground-based detectors, despite not yet detecting SGWBs, have set an upper limit on the dimensionless energy density [40]. In the nano-hertz (nHz) frequency band, recent findings from Pulsar Timing Ar- \nrayss (PTAs) have provided compelling evidence for the existence of SGWB [41-44]. Moving to the milli-hertz (mHz) frequency band, space-borne GW detectors, such as TianQin [45] and Laser Interferometer Space Antenna (LISA) [46], are expected to detect a Galactic foreground [47, 48]. \nWhen conducting GW detection, space-borne detectors face the challenge of canceling laser noise due to their motion. This issue can be addressed through time delay interferometry (TDI) techniques, which enable the formation of various types of TDI channels [49-53]. Among all the TDI channels, the unequal-arm Michelson, typically represented by the XYZ channel set, is the most widely utilized. This set can be further recombined to form the orthogonal AET channel set, with the T channel specifically designed to be insensitive to GWs. By employing the T channel as a noise monitor, the SGWB signal can be extracted through the correlation of data from other channels, which is often referred to as the null-channel method [54-62]. \nFrom the SGWB signal, one can derive a distribution of GW power across the sky, know as a sky map, to uncover the anisotropy of the SGWB. This process is termed 'map-making'. Initially, the SGWB signal produces a dirty map, which reflects the GW sky map as observed by detectors. The dirty map is susceptible to alterations due to variations in the detector's beam pattern. To refine the dirty map into a pure GW sky map, or clean map, it is essential to solve the deconvolution problem [63-65]. Thrane et al. are among the first to tackle this challenge [66], employing maximum likelihood estimates for the dirty map and then deconvolving it to recover a clean map [67-71]. Building on this foundation, Renzini et al. have introduced a frequentist maximum likelihood method to map GW power [72], conducting an all-sky analysis based on the complete likelihood of the \ndata. In addition to the maximum likelihood method, a Bayesian spherical harmonic method has recently been proposed [73-75]. \nIn this paper, we harness maximum likelihood estimation to perform map-making with TianQin. In order to quantify the ability of TianQin, we simulate a Gaussian, stationary, and unpolarized point source into the data, both in the absence and presence of noise. To furnish a thorough analysis of the map-making process, we also examine the influence of several key factors, including the detector design, the spectral characteristics of the SGWB, the selected frequency range, and the specific TDI channel sets employed for the data analysis. \nThe structure of the paper is organized as follows: we review the formalism of the SGWB detection in Sec. II. The maximum likelihood method and the deconvolution process for map-making are detailed in Sec. III. Based on the knowledge from the previous sections, we present the set up and results of map-making in IV. Finally, we give a brief summary in Sec. V.", 'A. Energy spectral density of stochastic background': "An SGWB is composed of a large number of GWs. Consequently, in transverse-traceless coordinates, the metric perturbations h ( t, ⃗x ) associated with the SGWB can be represented as a superposition of sinusoidal plane waves, characterized by frequency f and the unit vector ˆ n indicating the line-of-sight direction on the sky: \nh ij ( t, ⃗x ) = ∑ P =+ , × ∫ ∞ -∞ d f ∫ S 2 d ˆ Ω ˆ n e P ij (ˆ n ) ˜ h P ( f, ˆ n ) × e i2 πf [ t +ˆ n · ⃗x ( t ) /c ] , (1) \nwhere ⃗x denotes the location at which the GW measurement is conducted, c is the speed of light. The symbol e P ij (ˆ n ) signifies the GW polarization tensor corresponding to the polarization P , while ˜ h P ( f, ˆ n ) denotes the Fourier amplitude. \nIn this work, we focus on the anisotropic, Gaussian, stationary, and unpolarized SGWB. Thus, the Fourier amplitude is treated as a zero-mean random variable, characterized by the one-sided power spectral density (PSD) P h : \n⟨ ˜ h P ( f, ˆ n ) ˜ h ∗ P ' ( f, ˆ n ' ) ⟩ = 1 4 δ ( f -f ' ) δ PP ' δ 2 (ˆ n -ˆ n ' ) P h ( f, ˆ n ) , (2) \nwhere the symbol * denotes the conjugate. The factor of 1/4 pertains to the definition of one-sided PSD, encompassing the summation over both polarizations. Given the assumption that the direction and intensity of the SGWB are independent, the P h ( f, ˆ n ) can be factorized \ninto the spectral shape ¯ H ( f ) and angular distribution P h (ˆ n ): \nP h ( f, ˆ n ) = ¯ H ( f ) P h (ˆ n ) . (3) \nThe angular distribution P h (ˆ n ) can be expanded in terms of a set of basis: \nP h (ˆ n ) = ∑ α P α e α (ˆ n ) , (4) \nwhere e α (ˆ n ) can be spherical harmonic or pixel basis. With the definitions provided above, the dimensionless energy density spectrum Ω gw can be derived as follows [66]: \nΩ gw ( f ) = 1 ρ c d ρ gw d(ln f ) = 2 π 2 f 3 3 H 2 0 ∫ S 2 d ˆ Ω ˆ n P h ( f, ˆ n ) , (5) \nwhere the critical energy density ρ c = 3 H 2 0 c 2 / 8 πG , incorporating the gravitational constant G and the Hubble constant H 0 . The term d ρ gw represents the GW energy density contained within the frequency interval [ f , f +d f ]. Generally speaking, the energy density spectrum can be characterized by a power-law form with an index α and a reference frequency f r : \nΩ gw ( f ) = Ω r ( f f r ) α , (6) \nwhere Ω r denotes the reference energy density. \nThus far, we has focused solely on the intrinsic anisotropy of the SGWB. However, the SGWB can also exhibit a degree of kinematic Doppler anisotropy, analogous to observations made in the cosmic microwave background (CMB) [76-79]. The Doppler effect can cause shifts in both the frequency and direction of the SGWB. By defining the unit vector ˆ v to indicate the direction of relative motion between the SGWB rest frame and the moving frame, and ˆ n D to signify the direction of the SGWB within the moving frame, the coefficient of frequency change due to the Doppler effect can be determined by \nD = √ 1 -β 2 1 -β ξ , (7) \nand the direction shift can be expressed as \nˆ n = ˆ n D + ˆ v [( γ -1) ξ -γβ ] γ (1 -βξ ) . (8) \nHere, β = | ⃗ v | /c , ξ = ˆ n D · ˆ v . Based on these definitions, the transformation of the energy density between the two frames can be written as: \nΩ D GW ( f ) = ∫ S 2 d ˆ Ω ˆ n D 4 Ω GW ( D -1 f, ˆ n + ˆ v [( γ -1) ξ -γβ ] γ (1 -βξ ) ) . (9) \nFurther details regarding this topic can be found in Ref. [77].", 'B. Detector noise': "Among the various space missions proposed for GW detection, this paper primarily focuses on TianQin [45] and LISA [46]. TianQin will consist of three identical drag-free satellites arranged in an equilateral triangular constellation orbiting the Earth. The orbital plane of TianQin is designed to always direct toward to J0806 ( θ s = -4 . 7 · , ϕ s = 120 . 5 · ), with each satellite separated approximately 1 . 7 × 10 5 km. Operating on a 'three months on + three months off' mode, TianQin will have an operational duration of six months each year. LISA, on the other hand, is designed to orbit the Sun, trailing Earth by about 20 · , with three satellites separated about 2 . 5 × 10 6 km apart. For the sake of convenience, shorthand notations will be employed in figures and equations: TQ for TianQin, and LS for LISA. \nGiven the motion of space-borne detectors, phase noise becomes the dominant noise in GW detection. Fortunately, the phase noise can be depressed by several orders of magnitude through the implementation of TDI channels [59, 80-83]. Despite such alleviation, other types of noise, including position noise and acceleration noise, persist as significant considerations. For the conventional TDI channel set XYZ, which utilizes each satellite in harmony with its neighboring arm, the auto- and cross-PSD manifest as follows [84, 85]: \nN II = 4 L 2 sin 2 [ f f ∗ ][ S p ( f ) + 2 ( cos 2 [ f f ∗ ] +1 ) S a ( f ) (2 πf ) 4 ] N IJ = -2 L 2 sin 2 [ f f ∗ ] cos [ f f ∗ ]( S p ( f ) + 4 S a ( f ) (2 πf ) 4 ) , (10) \n̸ \nwhere I , J label the X/Y/Z channels, with I = J . S p and S a represent the position noise and the acceleration noise, respectively. Given the arm length L , the characteristic frequency f ∗ = c/ 2 πL . For TianQin and LISA, the characteristic frequencies f TQ ∗ and f LS ∗ are estimated to be approximately 0.28 Hz and 0.019 Hz. \nBased on the channel set XYZ, one can further construct channel set AET by \nA = 1 √ 2 (Z -X) , E = 1 √ 6 (X -2Y + Z) , T = 1 √ 3 (X + Y + Z) . (11) \nWith the above construction, the auto-PSD of channel set AET can be derived from the the auto- and cross-PSD of channel set XYZ, as shown in Eq. (10) [85]. Furthermore, the cross-PSD of channel set AET is identically zero, thereby ensuring that the three channels are noiseorthogonal.", 'C. Detector response': 'In addition to detector noise, detector response is another important aspect to GW detection. The detector motion can lead to variations in the response to GWs over time. Hence, a long data stream needs to be divided into numerous segments. These segments should be long enough to cover the sensitive bands of the detector after transformation from the time domain to the frequency domain. Meanwhile, they should also be short enough to ensure that the response within each segment can be regarded as stationary, allowing the application of a short-term Fourier transform [1]. \nWithin a time interval τ wherein the response of channel I can be considered stationary, the frequency domain SGWB signal \n˜ h I ( f, τ ) = ∑ P ∫ S 2 d ˆ Ω ˆ k F P I ( f, τ, ˆ n ) ˜ h P ( f, ˆ n ) e i2 πf ˆ n · ⃗x ( τ ) /c , (12) \nwhere F P I denotes the response function. For the channel set XYZ, \nF P I ( f, τ, ˆ n ) =2 sin 2 ( f f ∗ )[ ˆ u a ( τ )ˆ u b ( τ ) T ( f, ˆ n , ˆ u ( τ )) -ˆ v a ( τ )ˆ v b ( τ ) T ( f, ˆ n , ˆ v ( τ )) ] e P ab (ˆ n ) , (13) \nwhere ˆ u and ˆ v are the orientations of arms associated with the channel I . The transfer function \nT [ f, ˆ u ( τ ) , ˆ n ] = 1 2 [ sinc [ f 2 f ∗ (1 + ˆ n · ˆ u ( τ )) ] e -i f 2 f ∗ [3 -ˆ n · ˆ u ( τ )] +sinc [ f 2 f ∗ (1 -ˆ n · ˆ u ( τ )) ] e -i f 2 f ∗ [1 -ˆ n · ˆ u ( τ )] ] , (14) \nwith sinc( x ) = sin x/x . \nGiven the rotational symmetry of the X/Y/Z channels, an inherent equivalence exists among them in the SGWB detection. However, the A/E/T channels exhibit different responses: the A/E channels are responsive to GWs, whereas the T channel is typically less responsive unless it captures high-frequency GWs. Consequently, the T channel serves as a tool for monitoring detector noise [55, 56]. Furthermore, the response for A/E/T channels is not merely a linear superposition of the response for X/Y/Z channels; it also contains some exponential terms that significantly impact the response for the T channel. The specific form can be found in Eq. (18) of Ref. [85]. \nBy incorporating the response function and the varying arrival times of the signal at each channel, one can further define the antenna pattern [86]: \nγ IJ ( f, τ, ˆ n ) = 1 2 ∑ P F P I ( f, τ, ˆ n ) F P ∗ J ( f, τ, ˆ n ) e i2 πf ˆ n ∆ ⃗x/c , \n(15) \nFIG. 1. ORF of single channels for TianQin. The blue, orange, and green lines are denote the X/Y/Z channels, the A/E channels, and the T channel, respectively. \n<!-- image --> \nwhere the factor of 1 / 2 arises from the average of polarization. The separation vector between channels ∆ ⃗x = ⃗x I -⃗x J . Notably, when I = J , the antenna pattern refers to the auto-correlation of individual channels. Integrating the antenna pattern over all sky yields the overlap reduction function (ORF), which quantifies the reduction in correlation due to the non-parallel alignment and time delay between channels: \nΓ IJ ( f, τ ) = 1 4 π ∫ S 2 d 2 ˆ Ω ˆ n γ IJ ( f, τ, ˆ n ) , (16) \nwhere the factor of 1 / 4 π ensures that the ORF is properly scaled. \nIn Fig. 1, we present the ORF for individual channels of TianQin. Given the symmetry inherent in the channel set, the ORFs for the X/Y/Z channels and the A/E channels are identical, respectively. For the detection channels X/Y/Z and A/E, the ORF scales with f 2 , while the ORF of the null channel T scales with f 8 . At lower frequencies, the ORF of the null channel is markedly inferior compared to that of the detection channels. Nonetheless, as the frequency increases, the ORF of the null channel can dramatically rise, thereby transitioning into a detection role. Fig. 2 further illustrates the corresponding antenna pattern when TianQin is positioned at perihelion, within the ecliptic coordinates [87]. For clarity, the antenna pattern is standardized relative to its apex, termed the hot spot. At the low-frequency threshold ( f = f TQ ∗ / 3), the hot spots of the antenna pattern for detection channels converge with the fixed orientation of TianQin, designated by the coordinates (lon, lat)= (120 . 5 · , -4 . 7 · ). As the frequency ascends to 3 f TQ ∗ , the hot spots migrate towards the lateral directions where TianQin is oriented. For the null channel, the antenna pattern demonstrates its minimal intensity in the direction TianQin is facing. Moreover, the overall intensity \ndistribution across the celestial sphere remains relatively consistent despite the frequency surge.', 'III. MAP-MAKING ANALYSIS': "This section is bifurcated into two pivotal aspects: the initial part explores the utilization of the maximum likelihood method to ascertain the optimal estimator for the SGWB's dirty map, while the subsequent part delves into the application of singular-value decomposition (SVD) to transform a dirty map into a clean map.", 'A. Maximum likelihood method': "The data d I is the linear addition of the SGWB signal h I and the noise n I . In the frequency domain, we have \n˜ d I ( f, τ ) = ˜ h I ( f, τ ) + ˜ n I ( f, τ ) . (17) \nThe correlation measurement can be constructed by correlating two data: \nD IJ ( f, τ ) = 2 τ d I ( f, τ ) d ∗ J ( f, τ ) , (18) \nThe factor of 2 is consistent with the definition of onesided PSD. Combined with Eqs. (2), (3), and (12), the expectation of the correlation measurement is given by [85, 88]: \n⟨ D IJ ( f, τ ) ⟩ = ¯ H ( f ) ∫ d 2 ˆ Ω ˆ n γ IJ ( f, τ, ˆ n ) P h (ˆ n )+ N IJ ( f, τ ) . (19) \nTo ensure the expectation value attributes pure signals, one can further define the measurement as follows [89]: \nC IJ ( f, τ ) = D IJ ( f, τ ) -N IJ ( f, τ ) . (20) \nIn this work, we utilize the pixel basis grounded in HEALPix [90]. HEALPix, a grid-based methodology, is employed for the analysis and visualization of celestial data. It is uniquely adept at managing spherical data, characterized by its capacity to uniformly segment the sphere into equal-area and iso-latitude regions. Within this paradigm, the angular distribution \nP h (ˆ n ) = P ˆ n ' δ (ˆ n , ˆ n ' ) . (21) \nThen, the expectation of and variance of the above measurement C IJ [48] \nµ IJ ( f, τ ) = ⟨ C IJ ( f, τ ) ⟩ = ∑ ˆ n ¯ H ( f ) γ IJ ( f, τ, ˆ n ) P ˆ n σ 2 IJ ( f, τ ) = ⟨ C 2 IJ ( f, τ ) ⟩ - ⟨ C IJ ( f, τ ) ⟩ 2 = 1 + δ IJ 2 T ∆ f N II ( f, τ ) N JJ ( f, τ ) W IJ ( f, τ ) , (22) \nFIG. 2. Antenna pattern of different channels for TianQin, which is plotted on a Mollweide projection of the sky in ecliptic coordinates. In this representation, the blue star signifies the pointing direction of TianQin ([lon, lat]= [120 . 5 · , -4 . 7 · ]). Top and bottom panels correspond to frequencies of f TQ ∗ / 3 and 3 f TQ ∗ , respectively. \n<!-- image --> \nwhere ∆ f denotes the frequency resolution, the correction function W IJ is specifically designed for scenarios involving a strong SGWB: \nW IJ ( f, τ ) = 1 + µ II ( f, τ ) N JJ ( f, τ ) + µ JJ ( f, τ ) N II ( f, τ ) N II ( f, τ ) N JJ ( f, τ ) + µ II ( f, τ ) µ JJ ( f, τ ) + (1 -δ IJ ) µ 2 IJ ( f, τ ) N II ( f, τ ) N JJ ( f, τ ) . \n(23) \nIn terms of Eq. (22), the signal-to-noise ratio (SNR) is expressed as follows [86]: \nρ IJ = √ √ √ √ ∑ f,τ µ 2 IJ ( f, τ ) σ 2 IJ ( f, τ ) , (24) \nand the log-likelihood for the P ˆ n is given by \nln [ L ( n )] = const . -∑ fτ ∣ ∣ C IJ ( f, τ ) -¯ H ( f ) γ IJ ( f, τ, ˆ n ) P ˆ n ∣ ∣ 2 2 σ 2 IJ ( f, τ ) . (25) \nThe process of maximizing the likelihood function is fundamentally aligned with optimizing the SNR. Consequently, the estimator that maximizes the likelihood for the clean map P ˆ n can be derived as [66]: \nˆ P ˆ n = ( F -1 ) IJ ˆ n ˆ n ' X IJ ˆ n ' , (26) \nwhere the dirty map represents the GW sky as observed by the detectors: \nX IJ ˆ n ' = ∑ fτ γ ∗ IJ ( f, τ, ˆ n ' ) ¯ H ( f ) σ 2 IJ ( f, τ ) C IJ ( f, τ ) , (27) \nand the covariance matrix of the dirty map \nF IJ ˆ n ˆ n ' = ∑ fτ γ IJ ( f, τ, ˆ n ) ¯ H 2 ( f ) σ 2 IJ ( f, τ ) γ ∗ IJ ( f, τ, ˆ n ' ) . (28) \nIt is crucial to note that the inverse of F IJ ˆ n ˆ n ' serves as the covariance matrix for the clean map: \n⟨ ˆ P ˆ n ˆ P ∗ ˆ n ' ⟩ - ⟨ ˆ P ˆ n ⟩⟨ ˆ P ∗ ˆ n ' ⟩ ≈ ( F -1 ) IJ ˆ n ˆ n ' . (29) \nHence, F IJ ˆ n ˆ n ' is commonly referred to as the Fisher information matrix (FIM). Furthermore, when considering multiple channel pairs { IJ } , the aggregation of these individual contributions leads to the formation of the total dirty map and total FIM: \nX tot ˆ n = ∑ IJ X IJ ˆ n F tot ˆ n ˆ n ' = ∑ IJ F IJ ˆ n ˆ n ' . (30)", 'B. Deconvolution': "Eq. (26) indicates the possibility of recovering the clean map by inverting the FIM F IJ ˆ n ˆ n ' . However, the FIM is often a singular matrix, necessitating regularization for inversion. A common method to invert the FIM involves a SVD regularization scheme. Given the Hermitian nature of the FIM, its SVD representation takes the form \nF = U Σ U ∗ , (31) \nwhere U is a unitary matrix, and Σ is a diagonal matrix containing positive real eigenvalues s i of the FIM. When arranging the diagonal elements of Σ in descending order, a threshold s min can be selected to condition the matrix. Values below this threshold can be typically replaced with infinity or the smallest eigenvalue above the cutoff, resulting in the matrix Σ ' . Following this procedure, one can directly derive the regularized F ' using the resulting matrix Σ ' : \nF ' = U Σ ' U ∗ , (32) \nwith its inverse \nF '-1 = U Σ '-1 U ∗ . (33) \nBy multiplying the inverted-regularized FIM with the dirty map, the clean map can be derived: \nˆ P ˆ n = ( F '-1 ) tot ˆ n ˆ n ' X tot ˆ n ' . (34) \nThe process of discarding eigenvalues during SVD can lead to the loss of information associated with pixels of weak intensity, which can result in the generated clean map differing from the true sky map. In addition, when solving the equation represented by Eq. (34), round-off errors are introduced. These errors are inherently related to the condition number of the FIM. \nWithin the framework of a system of equations denoted as Ax = b , the condition number κ ( A ) associated with the coefficient matrix A functions as an indicator of singularity, illustrating its susceptibility to perturbations in the input data. It is defined as follows [69, 91]: \nκ ( A ) = ∥ A ∥∥ A -1 ∥ , (35) \nwhere ∥ ... ∥ denotes the norm of matrix. In numerous applications, the 2-norm is often preferred due to its direct relationship with the eigenvalues of the matrix: \n∥ A ∥ 2 = √ λ max ( A ∗ A ) , (36) \nwhere λ max labels the maximum eigenvalue of the matrix. Given that the FIM usually exhibits off-diagonal elements with negligible imaginary parts, it can be accurately approximated as a symmetric, positive definite matrix. Under this assumption, the condition number κ ( A ) of the matrix simplifies to the ratio of its maximum to minimum eigenvalues: \nκ ( A ) = λ max ( A ) /λ min ( A ) . (37) \nThe condition number measures the extent to which the resulting clean map ˆ P ˆ n fluctuates in response to changes in the dirty map X . A high condition number suggests that the clean map is highly sensitive to even slight alterations in the dirty map, making it difficult to obtain accurate results. Conversely, a low condition number implies that changes in the dirty map have a minimal impact on the clean map, indicating that the FIM is 'well-behaved' and thus more effective at recovering the clean map.", 'IV. SET UP AND RESULTS': 'In this section, we will demonstrate the capability of TianQin in recovering the sky map for the SGWB. We narrow our attention to the pixel-based decomposition, with a specific emphasis on point sources. \nFIG. 3. Auto-correlation data for the X channel of TianQin, with the top and bottom panels involving spectral indices α for SGWB at 2/3 and 3. Within each panel, the unfolded data is symbolized by the gray line, whereas the folded data is illustrated by the red line. Additionally, the black and green lines denote the noise and signal PSDs, respectively, corresponding to an SNR of approximately 16 over the course of 1-year operation. \n<!-- image -->', 'A. Data simulation': "We start by generating random Gaussian SGWB signals, of which the covariance is obtained by summing contributions from each pixel across the all sky: \nS in IJ ( f, τ ) = 4 π N pix ∑ ˆ n γ IJ ( f, τ, ˆ n ) ¯ H in ( f ) P in ˆ n , (38) \nwhere the intensity of SGWB is determined by the spectral shape ¯ H in ( f ) and the angular distribution P in ˆ n . N pix denotes the number of pixels. Increasing the N pix enhances resolution but also escalates computational demands. To strike a balance, we opt for N in pix = 768 for both injecting and subsequently outputting the signal. In addition, to handle the computational load associated with analyzing the real-time response of the detector, which is subject to temporal variations, we divide the data into discrete time segments. We assume that within each brief interval, the detector's response \nremains consistent. To encapsulate the full segment's response, we employ the response at the midpoint of the interval as a transient representation. For TianQin, we select segments of 3600 seconds, where the directional change in response is minimal [92]. This allows a compact period of 3.64 days to be divided into 87 segments. On the other hand, we simulate the random Gaussian frequency-domain noise based on the noise PSD provided in Eq. (10). By combining the generated signal with the noise, we derive the correlation measurement D IJ ( f ) as described in Eq. (18). \nNext, we examine data gathered during the operational cycle of TianQin, which entails a three-month on phase followed by a subsequent three-month off phase. Incorporating a year's worth of data thus yields 50 periods, and these 50 periods can be then aggregated into a single period for analysis. By aggregating multiple periods into a single unit, one can significantly reduce computational effort. While this method may result in the loss of some low-frequency data, it is deemed acceptable as the data lies outside the sensitive frequency range of TianQin. The aforementioned data folding [93, 94] process is equivalent to rearranging the time summation in Eq. (27), maintaining the formulaic expression without alteration. Fig. 3 presents a sample of the data from the auto-correlated X channel for SGWB signal with different spectral indices α . The gray line represents the unfolded data from a single period, while the red line showcases the folded data after averaging across the 50 periods. For comparison purposes, we also plot the PSDs of the noise and SGWB signal using black and green lines, respectively. The SNRs for the signals with α values of 2/3 and 3 are approximately 16, achieved over the course of TianQin's 1-year operation.", 'B. Map-making': 'The process now pivots towards map-making. Unless otherwise specified, we will employ a Gaussian, stationary, unpolarized point source expanded up to multipole moments of l up to 4, with the signal PSD illustrated in Fig. 3. As depicted in Fig. 4, we initiate our analysis by injecting a pure signal with α = 2 / 3 into the XYZ channel set for data simulations. The left panel displays the resulting map after injection, with SGWB signal located at [lon, lat]= [5 π/ 3 , π/ 6]. It is noteworthy that, to ensure a non-negative distribution of spherical harmonics, Clebsch-Gordan coefficients are employed [73, 95]. The middle and right panels exhibit the clean maps recovered by truncating the data to f TQ ∗ / 3 and 3 f TQ ∗ , respectively. For straightforward comparison, these maps have been normalized to the brightest point of the injected map. \nThe fixed orbital orientation of TianQin renders it incapable of distinguishing between two signals that are symmetrically aligned with the orbital plane. As a result, upon injecting a point signal, two point signals are symmetrically recovered relative to the orbital plane (marked \nby the orange dashed lines), with the intensity of these recovered signals approximating half of the original injected signal. This mirror symmetry poses a substantial hurdle in precisely ascertaining the true orientation of the signal. We will further explore this issue in Sec. IV C by analyzing it through the lens of the Doppler effect. \nFurthermore, a noteworthy trend in the antenna pattern γ IJ manifests as the increasing prominence of the imaginary component with rising frequency. This evolution is notably significant as it amplifies the process of inverting the FIM, a pivotal procedure for capturing finescale details 1 . As a result, determining the optimal frequency truncation becomes imperative. Under the noisefree premise, as depicted in the middle and right panels of Fig. 4, the distinction between a cutoff frequency of f TQ ∗ / 3 and 3 f TQ ∗ is negligible. To rigorously evaluate the effects of frequency truncation, we utilized the JensenShannon divergence (JSD) [96], a metric that quantifies the divergence between two distributions. A minimum value for JSD of 0 signifies that the two distributions are precisely identical, while a maximum value of ln 2 indicates that the two distributions are entirely distinct. With a computed JSD value of 0.017 nat, we infer that truncating the frequency to f TQ ∗ / 3 suffices for noise-free conditions. \nThe effect of noise on map-making is a substantial concern. In Fig. 5, we further compare the results of mapmaking with and without noise in the top and bottom panels, where the data is truncated at 3 f TQ ∗ . For comparative analysis, the left and right panels refer to two different spectral shapes, α = 2 / 3 and α = 3, respectively. In the presence of noise, the capability to map the SGWB with α = 2 / 3 is significantly hindered, resulting in an increased JSD of 0.127 nat. Conversely, when α = 3, the recovered map becomes more accurate, with a JSD of 0.062 nat. This trend aligns with the observations in Fig. 4, where an increase in the cutoff frequency led to the gradual absorption of the high-frequency components of the signal. The transition from α = 2 / 3 to α = 3 noticeably intensifies the high-frequency components, which in turn facilitates the results for map-making.', 'C. Doppler-induced effects': "As previously discussed, TianQin faces a challenge due to mirror symmetry in map-making, as shown in Fig. 4 and Fig. 5. Eq. (9) implies that Doppler shifts can serve as a useful complementary tool for investigating the frequency profile of the SGWB and its inherent rest-frame anisotropies. We will explore strategies to tackle this symmetry issue in the context of Doppler effects. \nCoupled with the Earth's velocity of 3 × 10 4 m / s, the space-borne detector's motion towards the signal direction leads to a frequency increase in the real signal and a \nFIG. 4. Normalized sky map of a pure SGWB signal. The left panel illustrates the injected map for the SGWB signal which corresponds to the top panel of Fig. 3, positioned at [lon, lat]= [5 π/ 3 , π/ 6]. The middle and right panels refer to the recovered map, with the data truncated at f TQ ∗ / 3 and 3 f TQ ∗ , respectively. The orange dashed line indicates the orbital plane of TianQin, serving as a visual reference to highlight the symmetry observed in the recovered maps. \n<!-- image --> \nFIG. 5. Recovered sky map for the injected SGWB signal, as referenced in Fig. 3. The top and bottom panels correspond to the pure-signal and signal-plus-noise scenarios, respectively. In both cases, the cutoff frequency of the data is set to 3 f TQ ∗ . The orange dashed line signifies the orbit plane of TianQin. \n<!-- image --> \ndecrease in the frequency of the symmetric signal. This variation in frequency is instrumental in determining the precise location of a point source. Considering the point source shown in Fig. 4, we calculate the Doppler shift using Eq. (7). Our results indicate that the Doppler effect can cause frequency alterations and directional shifts of approximately 6 × 10 -5 and 2 × 10 -5 , respectively. However, when comparing the top and bottom panels of Fig. 5, it becomes evident that noise-induced errors reach a level of 10 -1 . Consequently, we conclude that the effects caused by the Doppler shift are not strong enough to overcome the mirror symmetry for the mapmaking process.", 'D. Singular matrix analysis': "The condition number of the FIM plays a critical role in the map-making process. Given that variations in the FIM can occur across different channels, it is essential to conduct a comparative analysis. To this end, we plan to utilize both the XYZ and AET channel sets for this study. Our goal is to evaluate and understand the differences in their performance and effectiveness within the mapmaking process. \nIn Fig. 6, we calculate the condition number of the FIM using Eq. (37). The eigenvalues of the matrix are plotted in descending order, and we vary the number of eigenvalues retained for comparative analysis. It is important to recognize that while retaining a higher count \n=2/3 \nFIG. 6. Condition number of FIM for different channel sets of TianQin and LISA. In the top panel, we display the condition number while holding the spectral index α constant, investigating the influence of altering the cutoff frequency and selecting different channel sets. The bottom panel maintains a constant cutoff frequency at 3 f ∗ , which allows us to examine the impact of different spectral indices on the condition number. \n<!-- image --> \nof eigenvalues incorporates more information, it generally inflate the condition number of the FIM. Therefore, our analysis focuses on the condition numbers associated with the first 192 eigenvalues. \nThe top panel compares the condition numbers for TianQin and LISA at cutoff frequencies of f ∗ / 3 and 3 f ∗ , with a fixed spectral index α of 2/3. The condition number for LISA consistently remains lower than that for TianQin, especially at higher eigenvalue counts. The discrepancy can be attributed to the differing operational designs of the detectors: LISA is capable of surveying various directions in the sky throughout the year, offering dynamic coverage that enhances its condition number performance. In contrast, TianQin's operational focus remains fixed on J0806, leading to a stable and less varying antenna pattern. This stability significantly inflates the condition number of the FIM. Despite these differences, both detectors share certain characteristics in the \nmap-making process. As expected, there is a notable correlation between an increase in the cutoff frequency and a decrease in the condition number, regardless of whether the XYZ or AET channel set is used. Interestingly, when the cutoff frequency is set at f ∗ / 3, the XYZ channel set typically exhibits a higher condition number compared to the AET channel set. However, the result dynamic changes when the cutoff frequency is elevated to 3 f ∗ , with the AET channel set then showing a higher condition number. The selection of both cutoff frequency and channel set plays a crucial role in influencing the condition number of the FIM for map-making. \nThe results presented above are predicated on a specific spectral index, α = 2 / 3. Should this index vary, the outcomes would necessarily adjust to accommodate such changes. To illustrate this point, the bottom panel keeps the cutoff frequency constant at 3 f ∗ , and compares the results for α = 2 / 3 with those for α = 3. A distinct pat- \nis observed: with α = 2 / 3, both TianQin and LISA tend to show a higher condition number for the XYZ channel set compared to the AET channel set; however, the situation is reversed when α = 3. This observation underscores that the spectral shape of the SGWB has a significant impact on the accuracy and reliability of the map-making process.", 'V. SUMMARY': 'In this paper, we have explored the map-making for the anisotropic SGWB using TianQin. We utilized the maximum likelihood method to construct maps from simulated data. By using a point source as a case study, we reconstructed its clean map up to multipole moments of order l ≤ 4 both in the absence and presence of noise, achieving an SNR of 16. We observed that the inclusion of noise necessitates the use of higher-frequency data to achieve satisfactory map-making results. A notable limitation of TianQin lies in distinguishing between a point source and its mirror position relative to the orbital plane. To tackle this challenge, we investigated the potential of harnessing Doppler-induced effects to break this symmetric degeneracy. However, the theoretical deviation in the signal between two symmetrical directions, induced by the Doppler effect, is considerably smaller than the statistical error introduced by the noise. Distinguishing signals originating from symmetrical directions remains a tough challenge. \nRegarding both TianQin and LISA, we further investigated various factors that could influence the mapmaking process by analyzing the condition number of the FIM with respect to three key aspects: the cutoff frequency of the data, the choice of channel set, and the spectral profile of the SGWB. Our results indicated that adopting a sufficiently high cutoff frequency, specifically \n- [1] J. D. Romano and N. J. Cornish, Living Rev. Rel. 20 , 2 (2017), arXiv:1608.06889 [gr-qc].\n- [2] N. Christensen, Rept. Prog. Phys. 82 , 016903 (2019), arXiv:1811.08797 [gr-qc].\n- [3] J. D. 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Notably, at this cutoff frequency, the XYZ channel set can outperform the AET channel set in terms of map-making efficacy when the spectral index α is set at 2/3. Additionally, the map-making process for an SGWB with a spectral index α of 3 consistently yielded better results compared to an index α of 2/3. \nIt is noteworthy that the injection of a point source represents an idealized scenario, selected because it provides a straightforward example to analyze the detection capabilities for anisotropic SGWB. Furthermore, we focused on single space-borne detectors, underscoring the critical necessity for accurate noise level estimation to facilitate effective SGWB detection. Failure to do so can significantly undermine the accuracy of detection [97]. A strategy to address this issue entails crosscorrelating data from multiple detectors [92, 98]. Unlike ground-based detectors, the complex relative motion of space-borne detectors substantially amplifies the computational demands for determining antenna patterns during the map-making process. Research into this topic is ongoing and will be expounded upon in forthcoming publications.', 'ACKNOWLEDGMENTS': "This work has been supported by the National Key Research and Development Program of China (No. 2020YFC2201400), the Natural Science Foundation of China (Grants No. 12173104), the Natural Science Foundation of Guangdong Province of China (Grant No. 2022A1515011862), and the Guangdong Basic and Applied Basic Research Foundation(Grant No. 2023A1515030116). Z.C.L. is supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515111184). \n- [10] M. Surace, K. D. Kokkotas, and P. Pnigouras, Astron. Astrophys. 586 , A86 (2016), arXiv:1512.02502 [astroph.CO].\n- [11] D. Talukder, E. Thrane, S. Bose, and T. Regimbau, Phys. Rev. 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2024EPJP..139..821G

The star cluster surrounding the supermassive black hole in the center of Milky Way is probed using the data on the S2 star. The value of precession found at the physicsinformed neural networks PINN analysis of the S2 data is used to consider the role of the scattering of S2 star on stars of the cluster described by a random force given by the Holtsmark distribution. The critical value for the star density of the core cluster for which the observed precession value by PINN lies inside 70 confidence interval between 15 and inlineformula idIEq1mmlmathmmlmrowmmlmn85mmlmnmmlmommlmommlmrowmmlmathinlineformula quantiles around the median of precession due to scattering is obtained as inlineformula idIEq2mmlmathmmlmrowmmlmsubmmlminmmlmimmlmtextcritmmlmtextmmlmsubmmlmommlmommlmn8.3mmlmnmmlmommlmommlmsupmmlmn10mmlmnmmlmn6mmlmnmmlmsupmmlmspace width0.166667emmmlmspacemmlmsupmmlmtextpcmmlmtextmmlmrowmmlmommlmommlmn3mmlmnmmlmrowmmlmsupmmlmrowmmlmathinlineformula that is at higher star densities the perturbation of the orbit of S2 would exceed the observed one.

2024-09-01T00:00:00Z

['arXiv:2409.10567', '10.48550/arXiv.2409.10567', '10.1140/epjp/s13360-024-05619-9', '2024arXiv240910567G', '2024EPJP..139..821G']

['Astrophysics - Astrophysics of Galaxies']

S2star dynamics probing the galaxy core cluster

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

https://arxiv.org/pdf/2409.10567.pdf

{'S2-star dynamics probing the Galaxy core cluster': 'N.Galikyan 1 , 2 , Sh.Khlghatyan 2 , A.A.Kocharyan 3 , V.G.Gurzadyan 2 , 4 \n- 1 National Research Nuclear University MEPhI, Moscow, Russia\n- 2 Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia\n- 3 School of Physics and Astronomy, Monash University, Clayton, Australia\n- 4 SIA, Sapienza Universita di Roma, Rome, Italy \nReceived: date / Revised version: date \nAbstract. The star cluster surrounding the supermassive black hole in the center of Milky Way is probed using the data on the S2 star. The value of precession found at the physics-informed neural networks (PINN) analysis of the S2 data is used to consider the role of the scattering of S2 star on stars of the cluster, described by a random force given by the Holtsmark distribution. The critical value for the star density of the core cluster for which the observed precession value by PINN lies inside 70% confidence interval (between 15% and 85% quantiles) around the median of precession due to scattering, is obtained as n crit ≈ 8 . 3 × 10 6 pc -3 , that is at higher star densities the perturbation of the orbit of S2 would exceed the observed one. \nPACS. 98.80.-k Cosmology', '1 Introduction': "Galactic centers have long been among the major goals of intense observational and theoretical investigations. The presence of a central massive black hole surrounded by a dense star cluster, the accretion onto the black hole, the stellar tidal disruption events and the diversity of linked physical effects are traced at observational dedicated surveys. The revealing of the black hole shadows in the centers of the galaxy M87 and of the Milky Way had enabled to probe the properties of the metric in the vicinity of the central black hole [1,2]. \nThe monitoring of S-stars motion around the supermassive black hole (SMBH) SgrA* in the center of the Galaxy represents a unique window to test the General Relativity (GR), i.e. the motion of S2 star enabled to detect the precession consistent with the predictions of GR for the Schwarzschild metric [3]. The data on the S-stars have been used to constrain modified theories of gravity, the dark matter models as scalar and vector clouds, the extended mass component in the gravitational potential, etc, (see [4,5,6,7,8,9,10,11,12]), thus complementing the other tests of gravity theories, e.g. [13,14,15,16,17,18,19]. \nThe observational data on the S-stars [20] have been analyzed in [21,22] by means of the physics-informed neural networks (PINN) [23,24]. In [21] the weak-field modified GR [25,26,27] involving the cosmological constant has been constrained. The interest to that modified gravity version was due to its fitting the data on the dynamics of groups and clusters of galaxies [26,27,28], providing an explanation to the Hubble tension as a result of two flows, local and global ones [29,30], to the properties of the filaments in the local Universe [31,32,33]. In [22] the PINN was used to constrain the extended dark matter contribution to the S2 precession and the conclusion on the absence of a signature of extended mass up to 0.01% of the central mass inside the apocenter of S2 was drawn. \nIn the present study we use the S2 data and the PINN to probe the parameters of the star cluster surrounding the central black hole. The estimates on the star density in the Galactic core e.g. at a distance of 0 . 01 pc from center were around n ≃ 2 . 6 ± 0 . 3 × 10 7 pc -3 [34] , hence, it is of particular interest to probe the central stellar cluster role using the S2 data. We used the Holtsmark distribution approach H ( β ) on the statistics of the gravitational force acting on a star within uniformly distributed point masses [35,36] \nH ( β ) = 2 πβ ∫ ∞ 0 x sin xe -( x β ) 3 / 2 dx, (1) \nwhere H ( β ) dβ is the probability for the relative force β = ε/ε mean ; ε mean within the interval dβ is given below, Eq.(14). This approach assumes the random character of the scatter of the monitored star S2 on a stars of the star \ncluster. The details of the scatter of the gravitating bodies are discussed in [35]. Thus, analysing the possible influence (scatter) of galactic core stars on the precession of S2-star, we reveal the critical i.e. the maximal star concentration n crit ≈ 8 . 3 × 10 6 pc -3 which could result in the observed precession, while at higher core's star concentration the orbit of S2 would have been disturbed more than allowed by the observations.", '2 PINN': "In this section we briefly describe the PINN and highlight the previous results [21] needed for this study. The PINN architecture consists of two parts as shown in Fig.(1): \n- -A neural network with dense layers in which the input are the polar angles φ and output is the inverse radius u := 1 r . The output is compared with the observed data and minimizes the mean squared error between them.\n- -The physical part includes differential equations corresponding to our physical model and provides an additional contribution to the loss function. The physical process is described by Eq.(2) \nF ( x, y, y ' x , y '' x , . . . , y ( n ) x ) = 0 , (2) \nwhich may have train parameters. Then the physical loss is given by the following loss function Eq.(3) \nL phys ( f ( x ) , x ) = F 2 ( x, f ( x ) , f ' ( x ) , f '' ( x ) , . . . , f ( n ) ( x ) ) . (3) \nFig. 1: PINN basic scheme \n<!-- image --> \nThe physical process can be described by different gravity models. In this paper we use the results of the training based on GR and Darwin's equations of motion [38] \nd 2 χ dφ 2 = µe sin χ, (4) \n( dχ dφ ) 2 = 1 -2 µ (3 + e cos χ ) , (5) \nu = µ M · (1 + e cos χ ) , µ := M · p , (6) \nwhere u := 1 r , M · is the SMBH mass, e and p are respectively the eccentricity of S2 orbit and the focal parameter of the elliptical orbit, and χ is a variable called a relativistic anomaly [38]. The e, p , and M · are the trainable parameters. Using this scheme Fig.(2) for training of the S2 data we found the following values for precession, orbital parameters and SMBH mass (the value of mass was fixed after some point during the training) [21]: \nδφ Reg = 11 . 84 ' ± 0 . 03 ' , ˆ e = 0 . 88512 ± 0 . 00001 , ˆ p = 219 . 2 ± 0 . 2 au, ˆ M · = 0 . 04 au ≡ 4 . 05 × 10 6 M ⊙ . (7) \nFig. 2: Darwin PINN scheme \n<!-- image -->", '3 Scattering': 'To find the influence of a cluster of stars in the Galactic center on the motion of S2 star, we first investigate small angle scattering of S2 on the cluster stars. \nConsider the Keplerian motion of the S2 star with parameters of orbit p and e , \nr = p 1 + e cos φ , (8) \nand a star with mass M = M ⊙ at a point ( x 0 , y 0 , z 0 ) that acts as a scattering center for S2 star. In that case, we assume that a single act of scattering happens at the point ( x ∗ , y ∗ , 0), where ( x ∗ , y ∗ ) are the coordinates of the intersection of S2 orbit and a line from (0 , 0 , 0) to ( x 0 , y 0 ). The scattering happens with impact parameter b \nb = √ z 2 0 +( x 0 -x ∗ ) 2 +( y 0 -y ∗ ) 2 , (9) \nand momentum ∆⃗ρ is transferred to S2, where ∆⃗ρ points from ( x ∗ , y ∗ , 0) to ( x 0 , y 0 , z 0 ). Due to the small angle scattering ∆ρ can be estimated as \n∆ρ = -b ρ ( φ ∗ ) ∫ ∞ -∞ GM ( x 2 + b 2 ) 3 / 2 dx = 2 GM ρ ( φ ∗ ) b . (10) \nThe energy is increased by ∆E = ⃗ ρ · ∆⃗ρ and the angular momentum is increased by ∆L = ∆ρ ang r ∗ , where ∆ρ ang is the non-radial component of transferred momentum. Substituting the momentum ρ ( φ ∗ ) at point ( r ∗ , φ ∗ ), we get \n∆ρ = -2 pGM bL √ 1 + e 2 +2 e cos φ ∗ . (11) \nAfter the scattering, we recalculate the values of the focal parameter ˜ p and eccentricity ˜ e and get an updated law of motion \nr \n= \n˜ \np \n. \n1 + ˜ \ne \ncos( \nφ \n+ \nδφ \ns \n) \nThen we find the value of the additional precession due to scattering δφ s by equating the radius of the orbit at point of scattering, i.e. \np 1 + e cos φ ∗ = ˜ p 1 + ˜ e cos( φ ∗ + δφ s ) . (12) \n<!-- image --> \n<!-- image --> \n(a) N=1 \n<!-- image --> \nFig. 3: The histograms of 10000 experiments for N = 1 , 17 , 33 , 40. The green solid line is the mean value of δφ Reg , green dashed line denotes the Schwarzschild precession calculated via PINN predicted parameters Eq.(7), orange line is the median value of the resulting distribution. The solid red and dashed red lines are the boundaries of the 15% -85% and 2 . 5% -97 . 5% quantile intervals, respectively. \n<!-- image -->', '3.1 Critical concentration': 'To find the critical concentration n crit at which the effect of precession reaches the value δφ Reg , we consider the following procedure. Within two apocenters 2ˆ a , where ˆ a = ˆ p 1 -ˆ e 2 , N point masses are generated randomly on which S2 is scattered. For every generated point, the procedure described in Sec. 3 is repeated, and the total precession is taken as the sum of δφ s Eq.(12) for every scattering and the precession of the first-order GR correction in the Schwarzchild metric (Schwarzchild precession) before the scattering δφ SP . The initial values of focal parameter and eccentricity are taken from PINN Eq.(7), then after each scattering are updated according to Sec. 3. For N from 1 to 8 this scheme is repeated 10000 times and histograms Fig.(3) are obtained, which show the distribution of the precession value for every N . From the histograms it is clear that starting from N = 17 δφ Reg is inside the 95% confidence interval and starting from N crit = 33 is inside the 70% confidence interval The confidence intervals are defined as follows: 95% confidence interval when the median of the precession is within the interval containing 97 . 5% of the distribution and 2 . 5% is outside the interval, i.e. is inside between 5% and 95% quantiles; 70% confidence interval when the median of the precession is within the interval of 85% of the distribution and 15% is outside the interval, i.e. is within 15% and 85% quantiles (i.e. the region closest to the peak of the distribution). The value of critical concentration n crit within 70% confidence interval is \nn crit ≈ N crit ( 2 5 3 π ˆ a 3 ) -1 ≈ 8 . 3 × 10 6 pc -3 . (13)', '4 Holtsmark distribution': 'According to Eq.(1) in an isotropic system the mean force acting on a star vanishes, while the distribution for a non-vanishing mean square force ⟨ ϵ 2 ⟩ is given as [35,36,37], \n⟨ ε 2 ⟩ = ca 4 3 (14) \na = 4 15 (2 πGm ) 3 2 n, (15) \nwhere m is the mean star mass, c is a dimensionless coefficient \nc = ∫ β max 0 β 2 H ( β ) dβ, (16) \nwhere β as the dimensionless per unit mass force is ε = βa 2 3 [35] and correspondingly ε max = β max a 2 3 is the maximum interaction force during the interaction. \nThus, we assume that at a distance equal to the impact parameter the force acting on S2 is given by the Holtsmark distribution Eq.(14) \nb H := ( Gm ε ) 1 2 (17) \nε := √ ⟨ ε 2 ⟩ . (18) \nFor this impact parameter Eq.(11) transforms to the following \n∆ρ H = -2 pGm L √ 1 + e 2 +2 e cos φ ∗ √ 2 π ( 4 15 ) 2 / 3 c 1 / 4 n 1 / 3 . (19) \nTo find the transformed momentum ∆p H we must limit the maximum dimensionless force β max to find the coefficient c ( n ). As such a force we take the one arising at a distance of αR ⊙ , i.e. (R in [m]): \nβ max ( n ) ≈ 7 . 8 α -2 n -2 / 3 × 10 -19 . \nThe evaluation of integral (16) yields \nc ( n ) ≈ 10 π Γ ( 5 2 ) sin ( 3 π 4 ) β 1 2 max ( n ) ≈ 3 β 1 2 max ( n ) . \nHence, \n∆ρ H ≈ -0 . 007 pGM L √ 1 + e 2 +2 e cos φ ∗ α -1 4 n 1 4 . (20) \nCalculating the total precession using Eq.(20) for concentration values n = 10 4 ÷ 10 8 pc -3 and assuming that physically significant are those values for n for which the mean number of stars N inside 2ˆ a is N = 1 ÷ 50 we can check whether the obtained values are in agreement with the Monte-Carlo approach in Section 3.1. Fig.(4) shows the 70% and 95% bounds of precession value with respect to n and Fig.(5) shows the distribution of precession for n = 8 × 10 6 pc -3 . \nFig. 4: The dependence of the median value (blue solid line) of precession and its 70% and 95% confidence intervals from the concentration n . The green solid line is the mean value of δφ Reg , green dashed line is the Schwarzschild precession calculated via PINN predicted parameters Eq.(7). The solid black lines show the interval of concentrations for which N = 1 ÷ 50 \n<!-- image --> \nFig. 5: The histograms of precession values obtained by scattering according to the Holtsmark distribution. The green solid line is the mean value of δφ Reg , green dashed line is the Schwarzschild precession calculated via PINN predicted parameters Eq.(7), orange line is the median value of the resulting distribution. The solid red and dashed red lines are the boundaries of the 15% -85% and 2 . 5% -97 . 5% quantile intervals, respectively. \n<!-- image --> \nFig. (4 and 5) show that the procedure described in Section 3.1 fully agrees with the disturbance of the orbit according to the Holtsmark distribution. \nThe massive black hole situated in a star cluster is predicted to lead to a redistribution of the density run of the cluster as n ( r ) ∝ r -7 / 4 [39] due to the loss of stars at their tidal disruption [40,41]. The constraint obtained here on the star density near the vicinity of the black hole may support more accurate consideration of the evolutionary effects in the Galactic center core cluster.', '5 Conclusions': "We used the S2-star's orbital data to probe the parameters of the star cluster surrounding the supermassive black hole Sgr A* in the center of the Galaxy. We considered the influence of scatterings on the value of the S2-star precession and the precession value previously found using physics-informed neural networks (PINN). Based on residual precession δφ S , we found the maximal star density n crit at which the scattering effect will not exceed the precession previously obtained within PINN analysis δφ Reg . \nWe applied the small-angle scattering approximation to find the residual precession δφ S . Within this approximation and for Monte Carlo random positions inside 2ˆ a , the role of S2 and N star scatterings has been analysed. The total precession was calculated as residual to the Schwarzchild precession vs the cumulative effect of the angle shifts at N scatterings. We obtained the critical number of the scatterings, N crit = 33, with the mean value of δφ Reg inside the 70% confidence interval (between 15% and 85% quantiles), and from there obtained the maximal star density n crit ≈ 8 . 3 × 10 6 pc -3 . The results correspond to the stationary distribution of the force per unit mass, i.e Holtsmark distribution. The performed analysis confirmed the efficiency of PINN in the analysis of the S-star data in further probing of the physical processes in the Galactic center.", '6 Acknowledgments': 'We are thankful to the referee for helpful comments. Sh.K. is acknowledging the ANSEF grant 23AN:PS-astroth2922.', '7 Data Availability Statement': 'Data sharing not applicable to this article as no datasets were generated or analysed during the current study.', 'References': "- 1. The Event Horizon Telescope Collaboration, ApJ, 875, L1 (2019)\n- 2. The Event Horizon Telescope Collaboration, ApJ, 930, L12 (2022)\n- 3. R. Abuter, et al, A&A, 636, L5 (2020)\n- 4. S. Capozziello, D.Borka, P.Jovanovi'c, V.Borka Jovanovi'c, Phys. Rev. D90, 044052 (2014)\n- 5. E.A. Becerra-Vergara, et al, A&A, 641, A34 (2020)\n- 6. D. Borka, S. Capozziello, V. Borka Jovanovi'c, A.F. Zakharov and P. Jovanovi'c, Universe, 7 407 (2021)\n- 7. A.F. Zakharov, MNRAS, 511, L35 (2022)\n- 8. A.F. Zakharov, Moscow Univ. Phys. Bull. 77, 341 (2022)\n- 9. P. Jovanovi'c, V. Borka Jovanovi'c. D. Borka and A.F. Zakharov, JCAP, 2023(03), 056 (2023)\n- 10. R.F. Fernandez, R. Della Monica, I. de Martino, arXiv:2306.06937 (2023)\n- 11. A. Foschi, et al, MNRAS, 524, 1075 (2023)\n- 12. A. Foschi, et al, MNRAS, 528, 3549 (2023)\n- 13. A.A. Nucita, F. De Paolis, G. Ingrosso, A. Qadir, A.F. Zakharov, PASP 119, 349 (2007)\n- 14. A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso, Phys. Rev. D 76, 062001 (2007)\n- 15. I. Ciufolini, et al, Eur. Phys. J. C 76, 120 (2016)\n- 16. I. Ciufolini, et al, Eur. Phys. J. Eur. Phys. J. C 79, 872 (2019)\n- 17. I. Ciufolini, et al, Eur. Phys. J. Eur. Phys. J. Plus, 132, 336 (2017)\n- 18. I. Ciufolini, R. Matzner, V. Gurzadyan, R. Penrose, Eur. Phys. J. C 77, 819 (2017)\n- 19. A. Stepanian, Sh. Khlghatyan, V.G. Gurzadyan, Eur. Phys. J. Plus, 136, 127 (2021)\n- 20. S. 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2024AnPhy.47069803S

In this article we study the black hole evaporation process and shadow property of the TangherliniReissnerNordstrm TRN black holes. The TRN black holes are the higherdimensional extension of the ReissnerNordstrm RN black holes and are characterized by their mass mmlmath altimgsi1.svg displayinline idd1e479mmlmiMmmlmimmlmath charge mmlmath altimgsi172.svg displayinline idd1e484mmlmiqmmlmimmlmath and spacetime dimensions mmlmath altimgsi173.svg displayinline idd1e489mmlmiDmmlmimmlmath. In higherdimensional spacetime the black hole evaporation occurs rapidly causing the black holes horizon to shrink. We derive the rate of mass loss for the higherdimensional charged black hole and investigate the effect of higherdimensional spacetime on charged black hole shadow. We derive the complete geodesic equations of motion with the effect of spacetime dimensions mmlmath altimgsi173.svg displayinline idd1e495mmlmiDmmlmimmlmath. We determine impact parameters by maximizing the black holes effective potential and estimate the critical radius of photon orbits. The photon orbits around the black hole shrink with the effect of the increasing number of spacetime dimensions. To visualize the shadows of the black hole we derive the celestial coordinates in terms of the black hole parameters. We use the observed results of M87 and Sgr Ammlmath altimgsi206.svg displayinline idd1e500mmlmsupmmlmrowmmlmrowmmlmrowmmlmommlmommlmrowmmlmsupmmlmath black hole from the Event Horizon Telescope and estimate the angular diameter of the charge black hole shadow in the higherdimensional spacetime. We also estimate the energy emission rate of the black hole. Our finding shows that the angular diameter of the black hole shadow decreases with the increasing number of spacetime dimensions mmlmath altimgsi173.svg displayinline idd1e508mmlmiDmmlmimmlmath.

2024-11-01T00:00:00Z

['10.1016/j.aop.2024.169803', '10.48550/arXiv.2409.07951', '2024AnPhy.47069803S', 'arXiv:2409.07951', '2024arXiv240907951P']

['Black holes in higher dimensions', 'Null geodesics', 'Black hole shadow', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']

Black hole evaporation process and TangherliniReissnerNordstrm black holes shadow

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

https://arxiv.org/pdf/2409.07951.pdf

{'Tangherlini-Reissner-Nordstrom Black Holes Shadow': 'Balendra Pratap Singh a ∗ \na Department of Physics, Applied Science Cluster, SOAE, UPES, Energy Acres, Bidholi, Via Prem Nagar, Dehradun, Uttarakhand, 248007, India', 'Abstract': "In this article, we study the black hole evaporation process and shadow property of the Tangherlini-Reissner-Nordstrom (TRN) black holes. The TRN black holes are the higherdimensional extension of the Reissner-Nordstrom (RN) black holes and are characterized by their mass M , charge q , and spacetime dimensions D . In higher-dimensional spacetime, the black hole evaporation occurs rapidly, causing the black hole's horizon to shrink. We derive the rate of mass loss for the higher-dimensional charged black hole and investigate the effect of higher-dimensional spacetime on charged black hole shadow. We derive the complete geodesic equations of motion with the effect of spacetime dimensions D . We determine impact parameters by maximizing the black hole's effective potential and estimate the critical radius of photon orbits. The photon orbits around the black hole shrink with the effect of the increasing number of spacetime dimensions. To visualize the shadows of the black hole, we derive the celestial coordinates in terms of the black hole parameters. We use the observed results of M87 and Sgr A ∗ black hole from the Event Horizon Telescope and estimate the angular diameter of the charge black hole shadow in the higher-dimensional spacetime. We also estimate the energy emission rate of the black hole. Our finding shows that the angular diameter of the black hole shadow decreases with the increasing number of spacetime dimensions D . \nKeywords: Black Holes in Higher Dimensions, Null Geodesics, Black hole shadow. \nPACS numbers:", 'I. INTRODUCTION': "The higher-dimensional spacetimes are the extension of our four-dimensional world. The theories of gravity and their properties get modified in the higher-dimensional spacetime. After the discovery of general relativity, Kaluza [1] and Klein [2] added another extra dimension to Einstein's general theory of relativity. Examining black holes in higher-dimensional spacetime is intriguing due to their peculiarity. String theory is the theoretical framework that supports the study of black holes in higher dimensions [3]. There is a chance of producing higher-dimensional micro black holes during particle collisions in the Large Hadron Collider at TeV scale [4, 5]. The equations of motion of particles around the black hole are significantly altered in higher dimensions [6, 7]. The spectroscopic study of hydrogen atoms in the higher-dimensional spacetime commits the existence of the extra number of spacetime dimensions [8-13]. Maxwell's electrodynamics equations also get modified in the extradimensional spacetime [14]. In this context, the study of higher-dimensional black holes has attracted the attention of various researchers for decades. In 1963, Tangherlini initiated the study of black holes in higher-dimensional spacetime by extending the Schwarzschild solution [15]. In 1986, Myers and Perry found the exact solution for higher-dimensional rotating spacetime [16]. The properties of black holes are strongly affected by the spacetime dimensions and have been extensively researched [17-32]. \nHawking posited that a black hole emits particles like neutrinos and photons, known as Hawking radiation. [33, 34]. The measurements of these particles can be done by including the quantum gravitational effects. With the help of Hawking radiation people are finding the relations between general relativity and quantum mechanics [35]. A black hole emits thermal radiation, causing it to lose its hairs such as mass, charge, and spin. The rate of change of the mass of a black hole depends on the surface gravity of the black hole. The black hole keeps losing its mass until it completely evaporates. The lifetime of a four-dimensional Schwarzschild black hole is t ≈ M 3 0 , where M 0 is the initial mass of the black hole [36]. The static-charged black hole loses its charge as well as its mass, and over time the charged black hole evolves into the static black hole [37]. The four-dimensional rotating black hole loses its angular momentum faster than its mass and with time it evolves toward the Schwarzschild black hole [36, 38]. In higher-dimensional spacetime, the black hole horizon shrinks and this increases the surface gravity which causes the emission rate to fast [39], and hence a black \nhole in higher dimensions evaporates faster than the four-dimensional spacetime [40]. \nThe physical appearance of the black hole comes in nature due to its shadow property. Black holes, although dark, can be visualized by their shadows. The black hole shadow is a dark disc surrounded by bright photon rings. Event Horizon Telescope (EHT) collaborators published the observed shadow images of the M87 black hole in [41-46] and the Sgr A ∗ black hole in [47-54]. The study of black hole shadow in higher-dimensional spacetime has been the subject of interest for the last few years. Papnoi et al. investigate the shadow property of 5 D rotating Myers-Perry black holes [55]. The shadow property for GaussBonnet gravity rotating black holes in six-dimensional spacetime has been done by [56]. Amir et al. investigate the shadow properties of five-dimensional EMCS black holes in [57]. Belhaj et al. extended the study of the quintessential black hole shadow in arbitrary dimensions [58]. The authors of [59] investigate the shadow properties of AdS higher-dimensional black holes of Einstein-Horndeski- Maxwell gravity. Banerjee et al. study the extra-dimensional spacetime using EHT observations in [60]. \nIn our previous work, we highlight the shadow properties of Schwarzschild's black hole in higher dimensions [61] and its rotating counterpart [62]. Our previous finding shows that in higher dimensional spacetime the size of black hole shadow decreases with the increasing number of spacetime dimensions. In this work, we study the cause of this effect. This study aims to investigate the process of black hole evaporation, generalize shadow properties for the TRN class of black holes, and determine their angular diameter. Generalizing the null geodesic equations of motion for photons into the arbitrary number of spacetime dimensions, we find the effective potential of the black hole. We maximize the effective potential to obtain photon orbits around the charged black hole in higher dimensions. The spacetime dimensions significantly affect the charged black hole shadow. In our study, we show that the effective size and the angular diameter of the black hole shadow decrease with the increasing number of spacetime dimensions, and also the effect of the charge is dominated by the increasing number of spacetime dimensions. \nThis paper is arranged in the following manner: In section two (II), we describe the black hole metric and its thermodynamical properties. We derive the complete null geodesic equations in section three (III). We calculate the equation of rate of mass loss in (IV) and in section (V), we study the black hole shadow with the increasing number of spacetime dimensions. We check the consistency of the higher-dimensional black hole with the M87 \nand Sgr A ∗ black hole shadow in (VI). We calculate the energy emission rate of the higherdimensional charged black hole in (VII), and finally, we conclude our results in section (VIII).", 'II. GENERAL FORMALISM OF THE BLACK HOLE METRIC': 'The action for an asymptotically flat higher-dimensional charged black hole can be written as \nS = 1 16 πG ∫ M d d x √ -g ( R -F µν F µν ) , (1) \nwhere R is the Ricci scalar and F µν is the Maxwell tensor. The spherically symmetric charged black hole metric in D -dimensional spacetime with natural units ( G = c = /planckover2pi1 = 1) is given by [40, 63-66] \nds 2 = -( 1 -µ r D -3 + ν 2 r 2( D -3) ) dt 2 + ( 1 -µ r D -3 + ν 2 r 2( D -3) ) -1 dr 2 + r 2 d Ω 2 ( D -2) (2) \nd Ω 2 D -2 = dθ 1 2 +sin 2 θ 1 dθ 2 2 + , ..., + D -3 ∏ i =1 sin 2 θ i dθ 2 D -2 , (3) \nµ = 16 πM ( D -2)Ω D -2 , (4) \nwhere \nis the line element on ( D -2)-dimensional unit sphere [67], and \nwith \nand the parameter ν is associated with black hole charge q [67] via \nΩ D -2 = 2 π D -1 2 Γ ( D -1 2 ) , (5) \nν = √ 8 π ( D -2)( D -3) q Ω D -2 . (6) \nThe parameters µ and ν are the ADM mass and electric charge of the black hole respectively [16]. The black hole metric (2) is the higher-dimensional extension of the RN black hole solution. In the absence of the charge parameter q = 0, the black hole metric (2) reduces to the Schwarzschild-Tangherlini black hole [61]. In four-dimensional spacetime, the black hole mass is 2 M and, the black hole charge is q , which is the RN black hole [68] and for q = 0 it reduces to the Schwarzschild black hole [68]. One can find the roots of the black hole metric \n- (2) by simply solving f ( r ) = 0, as \nr ± = ( µ 2 ± µ 2 √ 1 -4 ν 2 µ 2 ) 1 / ( D -3) , (7) \nwhere r + corresponds to the outer horizon and r -is the inner or Cauchy horizon of the black hole. Through horizon equation (7), one can find \nr D -3 + + r D -3 -= µ and r D -3 + r D -3 -= ν 2 . (8) \nThe TRN black hole solution develops naked singularity when ν 2 > µ 2 / 4 with the central singularity r = 0. The area of the event horizon of the TRN black hole is given by [64]: \nA = Ω ( D -2) r ( D -2) + . (9) \nIn this context, [69] the entropy of the black hole follows the given expression \nS = 1 4 Ω ( D -2) r ( D -2) + , (10) \nhere the entropy of the black hole varies with the black hole horizon and the spacetime dimension D . The temperature of the black hole depends on the surface gravity at the horizon of the black hole and is obtained as \nT = ( D -3) ( M 2 -Q 2 + M √ M 2 -Q 2 ) 2 π ( M + √ M 2 -Q 2 ) 2 D -5 D -3 . (11) \nAs we go to the higher-dimensional spacetime the horizon of the black hole decreases and the temperature of the black hole increases.', 'III. BASIC EQUATIONS FOR NULL GEODESICS': 'In this section, we derive the null geodesic equations of motion to find the orbits of photons around the TRN black holes. The required equations of motion can be obtained with the help of the associated Lagrangian function. The Lagrangian of the black hole takes the following form [68] \nL = 1 2 g µν ˙ x µ ˙ x ν , (12) \nwhere g µν is the metric tensor of the black hole which can be found via the black hole metric Eq. (2), and over dot represents the derivative with respect to the affine parameter τ . We calculate the canonically conjugate momentum via the Lagrangian function of the black hole \nas [68] \nP t = ( 1 -µ r D -3 + ν 2 r 2( D -3) ) ˙ t = E , (13) \nP θ i = r 2 D -3 ∑ i =1 i -1 ∏ n =1 sin 2 θ n ˙ θ D -3 , i = 1 , ..., D -3 (15) \nP r = ( 1 -µ r D -3 + ν 2 r 2( D -3) ) -1 ˙ r, (14) \nP φ = r 2 D -3 ∏ i =1 sin 2 θ i ˙ θ D -2 = L, (16) \nwhere P φ = P θ D -2 and the parameter E and L represent the energy and the angular momentum respectively. In four-dimensional spacetime, the above momentum equations (13-16) reduce for the RN black hole and take the form \nP θ 1 = r 2 ˙ θ 1 , P φ = r 2 sin 2 θ 1 ˙ θ 2 . \nNext, we apply the Hamilton-Jacobi method to obtain our complete equations of motion. In higher-dimensional spacetime, the Hamilton-Jacobi equation reads [68, 70] \n∂S ∂τ = H = -1 2 g µν ∂S ∂x µ ∂S ∂x ν , (17) \nwhere S represents the Jacobian action, and g µν is the inverse of the metric tensor. Using black hole metric (2) in Eq. (17), one can obtain \n-2 ∂S ∂τ = -1 ( 1 -µ r D -3 + ν 2 r 2( D -3) ) ( ∂S t ∂t ) 2 + ( 1 -µ r D -3 + ν 2 r 2( D -3) )( ∂S r ∂r ) 2 + D -3 ∑ i =1 1 r 2 i -1 ∏ n =1 sin 2 θ n ( ∂S θ i ∂θ i ) 2 + 1 r 2 D -3 ∏ i =1 sin 2 θ i ( ∂S φ ∂φ ) 2 . (18) \nThe above Eq. (18) is the higher-dimensional partial differential equations with a set of coordinates. To separate these coordinates, we choose an additive separable solution [70] \nS = 1 2 m 2 τ -E t + Lφ + S r ( r ) + D -3 ∑ i =1 S θ i ( θ i ) , (19) \nwhere m is the mass of the test particle, which is zero for the case of a photon, S r ( r ) is the function of radial coordinate, and S θ i ( θ i ) is the function of angular coordinate. In the \nFIG. 1: Plot showing the variation of black hole shadow with charge q and spacetime dimensions D . \n<!-- image --> \nabove Eq. (19), the second term E t corresponds to the energy conservation, and the third term Lφ corresponds to the angular momentum conservation. Using the separable solution (19) in the Hamilton-Jacobi Eq. (17), we obtain separate integrable equations for the radial \nTABLE I: Variation of critical radius r c and η + ξ 2 with the black hole charge q and spacetime dimension D . \nand the angular coordinates as [70] \nr 4 ( 1 -µ r D -3 + ν 2 r 2( D -3) ) 2 ( ∂S r ∂r ) 2 = E 2 r 4 -r 2 ( 1 -µ r D -3 + ν 2 r 2( D -3) ) ( K + L 2 ) , (20) D -3 ∑ i =1 1 i -1 ∏ n =1 sin 2 θ i ( ∂S θ i ∂θ i ) 2 = KD -3 ∏ i =1 L 2 cot 2 θ i , (21) \nwhere K is the separable constant usually known as Carter constant and first introduced by Carter in [71]. We get the complete null geodesics equations for the TRN black hole by substituting Eqs. (13)-(16) in Eqs. (20) and (21) in the given form \ndt dτ = E 1 -µ r D -3 + ν 2 r 2( D -3) , (22) \nr 2 dr dτ = ± √ R , (24) \ndφ dτ = L r 2 D -3 ∏ i =1 sin 2 θ i , (23) \nr 2 D -3 ∑ i =1 i -1 ∏ n =1 sin θ i dθ i dτ = ± √ Θ i , (25) \nwhere the parameter R ( r ) and Θ i ( θ i ) are the functions of radial and angular coordinates \nFIG. 2: Plot showing the variation of the black hole shadow with the increasing number of spacetime dimensions D . \n<!-- image --> \nrespectively, and take the following form \nR ( r ) = E 2 r 4 -r 2 [ 1 -µ r D -3 + ν 2 r 2( D -3) ] [ K + L 2 ] , (26) \nΘ i ( θ i ) = KD -3 ∏ i =1 L 2 cot 2 θ i . (27) \nThe above equations from (22) to (25) describe the motion of photons in the context of the TRN black hole. Now we define two parameters in terms of constants E and L as: ξ = L/ E and η = K / E 2 . These impact parameters characterize the motion of photons around the black hole. The incoming photons towards the black hole may have two possibilities: they may fall inside the black hole or form bound orbits. The innermost unstable circular bound orbit around the black hole defines the black hole shadow boundary. The effective potential of the black hole can be used to analyze the circular bound orbits. To get the expression of \nthe effective potential of the black hole, we redefine the radial equation of motion as \n( dr dτ ) 2 + V eff ( r ) = 0 . (28) \nHere, V eff is the effective potential of the black hole and can be expressed in terms of impact parameters as \nV eff = E 2 [ 1 r 2 ( 1 -µ r D -3 + ν 2 r 2( D -3) ) ( η + ξ 2 ) -1 ] . (29) \nIn the limit of D → 4, this effective potential reduces for the RN black hole, and in the absence of the black hole charge, the effective potential recovers its shape for the Schwarzschild black hole [61]. Our interest is to find the innermost unstable circular orbits by maximizing the effective potential, which follows a given equation \nV eff = 0 and ∂V eff ∂r = 0 . (30) \nBy maximizing the effective potential of the black hole, we obtain numerical values for the critical radius r c and the impact parameters η + ξ 2 with the increasing number of spacetime dimensions D . We show the variation of r c and η + ξ 2 with D in Table (I). In four-dimensional spacetime, which is the case of Schwarzschild black hole ( q = 0), the critical radius of innermost unstable circular orbits is 3 M D (cf. Table I) while in the presence of charge parameter q , it further decreases from 3 M D to 2 M D with the increasing values of charge parameter q from 0 to 1 M D . As we go to the higher-dimensional spacetime, the value of critical radius r c decreases, and in five-dimensional spacetime, it approaches 1 . 2917 M D with q = 0 and further decreases as we consistently increase the number of spacetime dimensions (cf. Table I). The critical radius r c also decreases with the increasing values of charge parameter q in higher-dimensional spacetime. In higher-dimensional spacetime, the effect of charge parameter q on the critical radius of the innermost circular orbits is small as compared to the four-dimensional spacetime.', 'IV. BLACK HOLE EVAPORATION OF TRN BLACK HOLE': 'In this section, we study the black hole evaporation process for the TRN black hole. The black hole evaporates and loses its mass so the black hole mass decreases with time. Here in our study, we consider the emitted particles to be massless. Let us consider a ( D -1)- \nspatial dimensional cavity and the number of modes for this cavity is given by [72, 73] \ndN = dx 1 dx 2 . . . dx D -1 dp 1 dp 2 . . . dp D -1 h D -1 = V Ap D -2 dp h D -1 , (31) \nwhere h is the Planck constant and V is the ( D -1)-dimensional cavity volume and A = ( D -1) π D -1 2 / Γ( D +1 2 ), is the area of the ( D -1)-dimensional sphere of unit radius. In fourdimensional spacetime, it becomes 4 π . The emitted radiation has ( D -2) independent polarization so we multiply this factor by the above equation [74], and we obtain \ndN ( ω ) = V ( D -2) Aω D -2 dω (2 π ) D -1 , (32) \nwhere ω is the frequency of the emitted radiation. One can obtain the expression of ( D -1)dimensional photon gas energy by multiplying the photon energy /planckover2pi1 ω and the Bose-Einstein factor 1 / ( e ( /planckover2pi1 ω/T ) -1) in Eq. (32) as \nU = V ( D -2) A (2 π ) D -1 ∫ ∞ 0 /planckover2pi1 ω ω D -2 e /planckover2pi1 ω T -1 dω = V ( D -2) AT D (2 π ) D -1 /planckover2pi1 D -1 ∫ ∞ 0 x D -1 e x -1 dx. (33) \nThe energy density is proportional to the T D . The expression of the Stefan-Boltzmann constant in higher-dimensional spacetime is given by \nk D = ( D -2) A (2 π ) D -1 /planckover2pi1 D -1 ∫ ∞ 0 x D -1 e x -1 dx. (34) \nIn four-dimensional spacetime, the expression of the energy density takes the following form as \nU = π 2 15 /planckover2pi1 3 V T 4 . (35) \nIn the geometrical optics approximation, the emitted massless quanta particle will follow the path of the null geodesics [75]. The radial null geodesics equation of motion of the emitted particle from the black hole reads \n( dr dτ ) 2 = E 2 -J 2 ( 1 -µ r D -3 + ν 2 r 2( D -3) ) r 2 (36) \nwhere E and J are the energy and the angular momentum of the emitted quanta particle from the black hole. The condition for emitted radiation to reach infinity rather than falling inside the black hole is \nE 2 J 2 ≥ ( 1 -µ r D -3 + ν 2 r 2( D -3) ) r 2 . (37) \nThe ratio of J and E is the impact parameter and the critical value of this impact parameter is given by [75] \nξ c = r p √ ( 1 -µ r D -3 p + ν 2 r 2( D -3) p ) . (38) \nAs we obtained the critical value of the impact parameter, now according to the D -dimensional Stefan-Boltzmann law [76, 77], the Hawking emission power is \ndM dt = -C ξ D -2 c T D , (39) \nwhere C = ( D -2) π D 2 -1 Ω D -2 Γ( D ) Γ( D 2 ) ζ ( D ). The black hole mass loss rate is dimension-dependent and as we move towards higher-dimensional spacetime the black hole mass decreases faster and the black hole evaporates rapidly.', 'V. TRN BLACK HOLE SHADOW': 'In this section, we estimate the geometrical radius of the photon orbits around the TRN black hole. The rotating black hole shadow in the higher-dimensional spacetime has an oblate shape due to the angular momentum of the black hole and spacetime dimensions D [62]. The TRN is a nonrotating higher-dimensional class of black holes, so the observed photon orbits are circular. To visualize these photon orbits, we define celestial coordinates X and Y i . These celestial coordinates can be defined via \nX = lim r 0 →∞ ( r 0 P ( φ ) P ( t ) ) , (40) \nY i = lim r 0 →∞ ( r 0 P ( θ i ) P ( t ) ) , i = 1 , ..., D -3 , (41) \nwhere P ( φ ) , P ( t ) and P ( θ i ) are the vi-tetrad component of momentum and r 0 is the distance between the black hole and the far observer [68]. Substituting Eqs. (13)-(16) and (22)-(25) in the above celestial coordinate Eqs. (40)-(41), we find celestial coordinates in terms of impact parameters and inclination angle θ i , which reads \nX = -ξ D -3 sin θ i , \nY i = ± √ η -ξ 2 D -3 i =1 cot 2 θ i . \n∏ i =1 (42) √ √ √ ∏ (43) \nFIG. 3: Plot showing the variation of shadow radius with spacetime dimensions D . \n<!-- image --> \nFor simplicity, we choose equatorial plane where θ i = π/ 2 and our equations of celestial coordinate reduces to simpler form \nX = -ξ, Y = ± √ η, (44) \nand the above Eq. (44) must follow the condition \nX 2 + Y 2 = η + ξ 2 . (45) \nThe contour plot of the above equation traces the photon orbits around the TRN black hole. In four-dimensional spacetime, the above Eq. (45) reduces to the RN black hole, and in the absence of charge parameter q → 0, it recovers the results for Schwarzschild black holes in higher-dimensional spacetime [61]. We trace several contour plots of black hole shadow with the variation of spacetime dimension D and charge parameter q (cf. Fig. 1). In higherdimensional spacetime, the black hole shadow rapidly decreases (cf. Fig. 1 and Fig. 2). In Fig. (1), the inner dashed black circular ring shows the shapes of the TRN black hole and the outer orange circular ring shows the shapes of Schwarzschild-Tangherlini black holes. In Fig. (3), we plot the shadow radius as a function of the spacetime dimensions D . The \nshadow radius R s decreases monotonically with increasing spacetime dimensions. In fourdimensional spacetime, the radius of the black hole shadow R s varies from (5 . 19 to 4 . 31 M D ) for ( q = 0 to 0 . 9 M D ). Similarly, for seven-dimensional spacetime, the shadow radius varies from (1 . 15 to 1 . 14 M D ) for ( q = 0 to 0 . 9 M D ). From Fig. (1), (2) and (3), it is noticeable that the impact of the charge parameter on the size of the black hole shadow decreases as we move towards higher-dimensional spacetime. The effect of spacetime dimensions becomes more dominant than the effect of charge as we move toward higher-dimensional spacetime.', 'VI. CONSISTENCY WITH M87 AND SGR A ∗ BLACK HOLE OBSERVATIONS': "The recent observations from Very Long Baseline Interferometry provide new insights into gravity in strong gravitational regimes. Near the horizon of the supermassive black hole, the deflection angle is unbound, and the photons are trapped into the indefinite orbits. These indefinite orbits are called photon shells. In this section, we find the consistency of the higher-dimensional charged black hole with the astrophysical black hole M87 and Sgr A ∗ . Here we calculate the angular diameter of the charged black hole shadow in the higher-dimensional spacetime with the observed data of astrophysical black holes from EHT collaborators. The angular diameter of the black hole shadow can be calculated using the following equation [78] \nθ D = 2 r 0 × √ A 0 π (46) \nwhere A 0 is the area of the black hole shadow and r 0 is the observer's distance from the black hole shadow. We use Eq. (46) to estimate the angular diameter of the higher-dimensional charged black hole shadow.", 'A. M87 black hole': 'In April 2019, EHT published the first image of the M87 black hole with its astrophysical properties in [41-46]. The M87 black hole has mass 6 . 5 × 10 9 M /circledot and is 16 . 8 Mps distant from the earth. We analytically estimate the angular diameter of the higher-dimensional charged black hole shadow with the M87 black hole observations. The estimated angular diameter in four-dimensional spacetime varies from θ D ≈ (39 to 33) µas for q → (0 to 0 . 9) M D . The angular diameter of the black hole shadow decreases monotonically in the higher- \nFIG. 4: Variation of angular diameter θ D with the spacetime dimension D and charge q with M87 black hole observations. \n<!-- image --> \nonal spacetime. In D = 7, the angular diameter varies from θ D ≈ (13 to 7) µas for q → (0 to 0 . 9) M D (cf. Fig. 4).', 'B. Sgr A ∗ black hole': 'The observed results of Sgr A ∗ black hole was published by EHT collaborators in [49-54]. The Sgr A ∗ black hole is situated at the center of our galaxy Milky Way. The observed mass of this black hole is 4 × 10 6 M /circledot and its 8 kps distance far from the Earth. In four-dimensional spacetime, the estimated angular diameter of Sgr A ∗ black hole is (50 to 40) µas for q → (0 to 0 . 9) M D and as we go to the higher-dimensional spacetime the angular diameter of the black hole shadow decreases. For D = 7, the angular diameter take values from θ D ≈ (11 to 9) µas for q → (0 to 0 . 9) M D (cf. Fig. 5). \nFIG. 5: Variation of angular diameter θ D with the spacetime dimension D and charge q with Sgr A ∗ black hole observations. \n<!-- image --> \nThe angular diameter of the charged black hole shadow decreases in the higherdimensional spacetime. The TRN black hole shadow in the higher-dimensional spacetime appears smaller in comparison with four-dimensional spacetime.', 'VII. ENERGY EMISSION RATE': 'In our previous sections (IV) and (V), we have discussed that the black hole loses its mass in higher-dimensional spacetime, and the size of the black hole shadow monotonically decreases with the increasing spacetime dimensions. In that sequence, we aim to determine how the energy emission rate varies as the number of spacetime dimensions increases. The emission rate of the black hole in higher-dimensional spacetime is given by [61] \nd 2 E dωdt = 2 π 2 σ lim ( exp( ω T ) -1 ) ω ( D -1) (47) \nFIG. 6: Variation of the energy emission rate with spacetime dimensions and black hole charge. \n<!-- image --> \nwhere ω is the frequency of the emitted radiation, σ lim is the limiting constant value. The shadow of the black hole corresponds to the high energy absorption cross-section for a fardistant observer [79]. The absorption cross sections oscillate to a limiting constant value σ lim for a spherically symmetric black hole and can be estimated via the area of the photon sphere [80, 81]. In higher-dimensional spacetime, the σ lim can be defined as \nσ lim ≈ π D -2 2 R ( D -2) s Γ( D 2 ) , (48) \nwhere R s is the black hole shadow radius. The limiting constant value is reduced to πR 2 s in four-dimensional spacetime. The expression of energy emission rate in higher dimensional spacetime is expressed by \nd 2 E dωdt = 2 π ( D +2 2 ) ( ωR s ) ( D -2) ( e ω/T -1)Γ( D/ 2) ω. (49) \nThe variation of the energy emission rate of the charged black hole with the increasing number of spacetime dimensions has been shown in Fig. (6). In the first plot of Fig. (6), we vary the charge of the black hole in the four-dimensional spacetime. As the charge parameter q increases, the rate of energy emission also increases. The second plot of Fig. (6) shows \nthe emission rate in five-dimensional spacetime. Here, we can see the energy emission rate increases compared to the four-dimensional spacetime. The third and the fourth plots of Fig. (6) show the variation of energy emission rate in sixth and seventh-dimensional spacetime. The emission rate increases as we move to higher-dimensional spacetime. However, the effect of the charge on the emission rate is effectively small compared to four-dimensional spacetime.', 'VIII. CONCLUDING REMARKS': 'Our study reveals that the shadow of a charged black hole decreases in size when observed in spacetimes with a higher number of dimensions. The black hole evaporation process occurs very fast in the higher dimensional spacetime. Due to this evaporation process, the black hole radiates more and loses its mass [82]. The horizon of the black hole shrinks and the effective size of the black hole shadow decreases. The key results of our study are given below \n- · The effective potential of the higher-dimensional charged black hole has been originally derived, and the numerical study of the critical radius of photon orbits has been done with the effect of spacetime dimensions.\n- · With the definition of the impact parameters, the geometry of the photon orbits has been studied for the increasing number of spacetime dimensions.\n- · We derived the equation of the rate of mass loss and concluded that the higherdimensional charged black hole evaporates rapidly.\n- · The size of the black hole shadow decreases with spacetime dimensions D and we have shown several plots of black hole shadow for the increasing number of spacetime with the effect of charge.\n- · The angular diameter of the charged black hole shadow has been estimated with the given data of M87 and Sgr A ∗ black holes.\n- · The angular diameter of the charged black hole shadow rapidly decreases in higherdimensional spacetime in comparison to the Schwarzschild-Tangherlini black hole \nwhich emphasizes that the higher-dimensional charged black hole shadow appears much smaller in comparison to the higher-dimensional Schwarzschild black hole [61]. \nIt is now clear from EHT observations that black holes exist in our Universe, however, we are still missing the signature of higher-dimensional astrophysical black holes. 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2024arXiv240905956D

We perform an observational study of modified gravity considering a potential inflationary interpretation of pulsar timing arrays PTA. We use a motivated model known as no slip in which the gravitational wave propagation is modified. Specifically by using two different parametrizations for the model we find the approximate transfer functions for tensor perturbations. In this way we obtain the spectral energy density of gravitational waves and use NANOGrav and IPTA second data release to constrain parameters of the model. We find that there is degeneracy between the model parameters xi and cM. For cM we only get an upper bound on the parameter. Thus it is difficult to constrain them with percent level accuracy with the current PTA data.

2024-09-01T00:00:00Z

['10.48550/arXiv.2409.05956', 'arXiv:2409.05956', '2024arXiv240905956D']

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

Testing No slip model with pulsar timing arrays NANOGrav and IPTA

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.05956.pdf

{'Mohammadreza Davari a Alireza Allahyari a, 1 Shahram Khosravi a': 'a Department of Astronomy and High Energy Physics, Kharazmi University, 15719-14911, Tehran, Iran \nE-mail: m.davari@khu.ac.ir, alireza.al@khu.ac.ir, khosravi sh@khu.ac.ir \nAbstract. We perform an observational study of modified gravity considering a potential inflationary interpretation of pulsar timing arrays (PTA). We use a motivated model known as no slip in which the gravitational wave propagation is modified. Specifically, by using two different parametrizations for the model, we find the approximate transfer functions for tensor perturbations. In this way, we obtain the spectral energy density of gravitational waves and use NANOGrav and IPTA second data release to constrain parameters of the model. We find that there is degeneracy between the model parameters ξ and c M . For c M , we only get an upper bound on the parameter. Thus, it is difficult to constrain them with percent level accuracy with the current PTA data.', '1 Introduction': "Generation of the stochastic gravitational waves background are predicted by various astronomical and early universe models. Recently, there are also hints towards their existence by various observations like NANOGrav [1], Parkers PTA [2], European PTA [3] and the China PTA [4]. There are different mechanisms which could generate a stochastic background. Two prevailing views for the sources of these gravitational waves (GW) [5] are the superpositions of GW signals from the merging of supermassive black hole binaries [1, 6-17] and the primordial GWs from early universe [18-22]. Other mechanisms which could contribute to this background include induced gravitational waves from scalar perturbations [23-32], first order cosmological phase transitions [33-37] and domain walls or cosmic strings [38-46]. Focusing for concreteness on the NANOGrav signal, this stochastic GW background has a blue-tilted tensor spectrum, with the spectral index n T = 1 . 8 ± 0 . 3 [26]. The primordial production of GW background is possible as there are some mechanisms to generate blue-tilted power spectrum, for instance within 'phantom' inflationary models [47]. In these models ω < -1 and one expects n T > 0. Thereby, a blue-tilted spectrum is achieved [26]. There are also mechanisms which produce a modified power law spectrum [22, 48, 49]. The primordial origin for the GW background is investigated in [50-52]. \nPTAs provide an arena to look for modifications to gravity. The study of GWs in modifications of gravity has also been the subject of focused studies [53-71]. The spectrum of primordial GWs is not only determined by the evolution of the background cosmology, but also it can be significantly affected by modifications to General Relativity. \nThe Horndeski theories of gravity [72-76] are the most general scalar-tensor theories with second-order equations of motion. Modified gravities predict different generation and propagation mechanisms for GWs. Additionally, we need to consider the fact that the recent \nmeasurement of the speed of propagation of GWs from GW170817 relative to its electromagnetic counterpart GRB170817A [77] severely limits the deviations from the speed of light. In the subclass of Horndeski theories, a motivated model named no slip model does not alter the speed of GWs. Moreover, the slip between metric potentials, should vanish and yet the gravity theory does not reduce to general relativity [78]. \nIn this paper, we will constrain the parameters of no slip gravity model as a subclass of Horndeski theories, using NANOGrav 15-year data set (NG15) [1] and International PTA second data release (IPTA2) [79, 80]. We assume that GW background has a primordial origin from an inflationary era and after the inflationary era, universe is described by no slip model [50]. We study two different parametrizations of the model, taking a phenomenological approach. In both parametrizations, we assume one extra parameter to constrain. We compare our results with our previous study using the standard sirens approach combined with cosmic microwave background. For various recent works on GWs in modified gravity models as an explanation of the PTA observations see [81-93]. \nThis paper is organized as follows. Section (2) is devoted to no slip model. We adopt two phenomenological parametrizations. Additionally, we consider the propagation of GWs in no slip model. In Section (3), we derive the approximate transfer functions for tensor perturbations for this model and find the spectral energy density. By using the NANOGrav and IPTA data, we constrain the model. Section (4) is summary and discussion.", '2 No slip model': 'Let us assume a flat FRLW background. The scalar and tensor perturbations on this background are are given by \nds 2 = a ( η ) 2 { -(1 + 2Ψ) dη 2 +(1 -2Φ) δ ij dx i dx j + h ij dx i dx j } , (2.1) \nwhere Φ and Ψ are scalar perturbations in conformal Newtonian gauge and h ij denotes tensor perturbations. In no slip model, the effects of gravity on observations of matter and light in the universe can be suitably described by modified Poisson equations relating the time-time metric potential Ψ and space-space metric potential Φ (in Newtonian gauge) to the matter perturbations as [78] \n∇ 2 Ψ = 4 πG N δρ × G matter , ∇ 2 (Ψ + Φ) = 8 πG N δρ × G light , (2.2) \nwhere the first equation governs the growth of structures, with a gravitational strength G matter , and the second governs the deflection of light, with a gravitational strength G light . The offset between G matter and G light , or Ψ and Φ, is referred to as the gravitational slip [78] with \n¯ η ≡ G matter G light , (2.3) \nwhere ¯ η = 1 corresponds to vanishing slip. The expressions for G matter , G light , and slip in Horndeski gravity, or the equivalent effective field theory (EFT) approach, are given in [94-96].', '2.1 Modified gravitational wave propagation': 'The evolution of linear, transverse-traceless perturbations for the tensor modes due to modifications of the gravity theory are generally described by the following equation [97] \nh ij +(3 + ν ) H ˙ h ij +( c 2 T k 2 /a 2 + µ 2 ) h ij = Γ ij , (2.4) \nwhere h ij is the metric tensor perturbation and dot denotes derivative with respect to comoving time dt = a ( η ) dη . The four time dependent parameters are as follows. c T is the GW propagation speed, µ is the effective graviton mass, ν is related to the running of the effective Planck mass, and Γ ij denotes the anisotropic stress generating GWs. Assuming there is no anisotropy and adapting the above equation in the context of the Horndeski gravity, we have [54] \nh ij +(3 + α M ) H ˙ h ij +(1 + α T ) k 2 a 2 h ij = 0 , (2.5) \nwhere we have identified ν = α M , c 2 T = 1 + α T , µ = 0, and also α M and α T are two dimensionless functions given by [54]. We have \nα M = 1 HM 2 ∗ dM 2 ∗ dt , (2.6) \nthe definitions for M ∗ can be found in [54]. The no slip model has the property that the speed of propagation of gravitational waves equals to the speed of light, that is c 2 T = 1 or α T = c 2 T -1 = 0 [78]; So we have \nh ij +(3 + α M ) H ˙ h ij + k 2 a 2 h ij = 0 . (2.7) \nAlso comparing with the equation in [98], we have q t = 2 G 4 = M 2 ⋆ where the definition for G 4 is given in [54]. We take a phenomenological approach and parameterize the theory in terms of well motivated parameters [78, 98] used in a standard siren study of GWs with LISA and cosmic microwave background study [78]. In this work, we apply these two common parameterizations for no slip model: \nParameterization I : According to the parametrization introduced in [99] for no slip model, the parameter α M can be expressed as \nα M = -2 δ ( z ) = d ln q t d ln a = -2 n (1 -ξ ) 1 -ξ + ξ (1 + z ) n , (2.8) \nwhere ξ and n are constants and z is the redshift. More specifically, to relate these parameters with M ∗ , we have [100] \nξ = lim z →∞ M ∗ (0) M ∗ ( z ) , (2.9) \nn ≃ α M 0 2( ξ -1) , (2.10) \nwith α M 0 = -2 δ (0). This parametrization is applicable to models in which we expect some simple properties in low and high redshifts. More explicitly, we find that at the high redshifts z → ∞ , we have δ ( z ) → 0. Moreover, when z → 0, this yields δ ( z ) → n (1 -ξ ), where for ξ = 1 we have δ ( z ) → 0. Thus, we recover the standard expression for the luminosity distance at high and low redshifts [98]. \nParameterization II : According to the parametrization introduced in [78] and used in [101] to investigate the effects of no slip gravity for CMB lensing and B-modes; we have \nα M = 4 c M ( a/a t ) τ [( a/a t ) τ +1] 2 , (2.11) \nwhere c M is the amplitude of the running of the Planck mass, a t is the scale factor at the transition time and τ is the rapidity. For this form the stability condition requires c M ≥ 0 and 0 < τ ≤ 3 / 2 [78].', '3 Primordial GWs and PTAs': "In this section, we derive the expressions for the spectral energy density Ω GW . Moreover, we find tensorial transfer function and relate Ω GW to primordial power spectrum. We use this expressions to find constraints on the relevant parameters. \nThe relevant quantity in PTA observations is the spectral energy density. The spectral energy density of GWs can be derived as [18] \nΩ GW ( k, η ) = 1 12 H 2 a 2 [ T ' ( k, η )] 2 P t ( k ) , (3.1) \nwhere T ( k, η ) is the transfer function. Transfer function describes the evolution of GW modes after the modes re-enter the horizon. The quantity P t ( k ) is the primordial power spectrum of GWs at the end of the inflationary period. We may write this in terms of a tilt n t and a tensor amplitude A as \nP t ( k ) = A ( k k ⋆ ) n t , (3.2) \nwhere k ⋆ is a pivot scale set as k ⋆ = 0 . 05 Mpc -1 . The transfer function in modified gravity can be approximated as a product of a correction factor and a general relativity part. For no slip model we have [54] \nT ( k, η ) MG = exp ( -1 2 ∫ η α M H d η ' ) T ( k, η ) GR = e D T ( k, η ) GR , (3.3) \nwhere H is the Hubble parameter in terms of conformal time. The exponential term in equation 3.3 is the correction factor. \nWe plot this factor as a function of redshift using the consistent values for the parameters given in section 3 from NG15 in our analysis. This factor reaches unity asymptotically in higher redshifts. \n<!-- image --> \nFigure 1 . The correction factor in terms of redshift z. Left : parametrization I. Right : parametrization II. \n<!-- image --> \nThe transfer function and its time derivative in context of GR, are numerically computed in [102, 103] and are given by \nT ( k, η 0 ) GR = 3 j 1 ( kη 0 ) kη 0 √ 1 . 0 + 1 . 36( k k eq ) + 2 . 50( k k eq ) 2 , (3.4) \nT ' ( k, η 0 ) GR = -3 j 2 ( kη 0 )Ω m kη 0 √ 1 . 0 + 1 . 36( k k eq ) + 2 . 50( k k eq ) 2 , (3.5) \nwhere j 1 , j 2 are Bessel functions and η 0 is the present conformal time. For GWs in PTA scales, we can use the approximation ( k ≫ k eq ) [26]. Then from equations 3.3, the transfer functions for both parametrizations is obtained as \nT ( k, η ) I = ( a n (1 -ξ ) + ξ ξ ) T ( k, η ) GR , (3.6) \nT ( k, η ) II = exp ( 2 c M τ [ a τ + a τ t a τ t (1 + ( a a t ) τ ) 2 -1 ]) T ( k, η ) GR , (3.7) \nwhere we assumed n > 0 and τ > 0 for integration. Subscript I (II) denotes the parametrization I (II), respectively. \nIn pulsar timing experiments, it is convenient to express wavenumbers k in terms of frequencies f as [26] \nf ≃ 1 . 54 × 10 -15 ( k Mpc -1 ) Hz . (3.8) \nWe find that f ⋆ ≃ 7 . 7 × 10 -17 Hz. Also in PTAs, the GW spectral energy density is rather written in terms of the power spectrum of the GW strain h c given by \nΩ PTA gw ( f ) = 2 π 2 3 H 2 0 f 2 h 2 c ( f ) . (3.9) \nh c ( f ) is supposed to take a powerlaw form with respect to a reference frequency f yr and can be expressed as \nh c ( f ) = A ( f f yr ) α , (3.10) \nwhere f yr = 1yr -1 ≈ 3 . 17 × 10 -8 Hz. \nFinally using α = 3 -γ 2 , the present GWs spectral energy density can be found [26]. We have the spectral energy density of GWs as \nΩ GW ( f ) I = A 2 2 π 2 3 H 2 0 ( 1 ξ + nH 0 (1 -ξ ) 2 πfξ ) 2 ( f 5 -γ yr γ -3 ) , (3.11) \nΩ GW ( f ) II = A 2 2 π 2 3 H 2 0 exp ( 4 c M τ [ 1 (1 + ( 1 a t ) τ ) 2 -1 ])( -c M H 0 πfa τ t (1 + ( 1 a t ) τ ) 2 +1 ) 2 ( f 5 -γ yr γ -3 ) , (3.12) \nγ = 5 -n t . \nwhere we consider H 0 = 67 . 36 from Planck 2018 [104].", '3.1 Constraints by PTAs': 'We set to constrain the parameters of theory by PTA observations. We use NANOGrav 15-year data (NG15) and IPTA second data release (IPTA2). To constrain the parameters ξ and c M , we use the python package PTArcade [105], that is a wrapper of ENTERPRISE [106] and ceffyl [107] to provide an accessible way to perform Bayesian analyses with PTA data. In this work we use ceffyl with NG15 and IPTA2 and use uniform priors on the parameters as ( -18 < log 10 A < -6), (0 < γ < 6) for both parameterizations. For parameterization I , we set ( n = 0 . 2) from [98] and (0 < ξ < 2); and for parameterization II we set ( τ = 1) from [101] and (0 < c M < 0 . 5). \nThe results for parameterization I using NG15 and IPTA2 are illustrated in figure 2. The green lines show NG15 and blue lines show IPTA2. The results from both observations are consistent given the uncertainties on the parameters. We find that there exists a strong correlation between ξ and log 10 A . This degeneracy will hinder our ability to constrain them precisely. The constraints are summarized in table 1. To compare our study with other observations in table 1, we have provided results from our previous study in [98] where we generated three mock standard sirens catalogs based on the merger of massive black hole binaries which are expected to be observed with LISA [108, 109]. We combined the mock \nFigure 2 . The posterior plots and marginal posteriors of no slip model parameters for parameterization I . The contours show at 68% and 95% confidence levels for International PTA second data release (IPTA2) in blue and NANOGrav 15-year data set (NG15) in green. \n<!-- image --> \ncatalog and CMB observations to constrain the parameters of no slip model. The second row in table 1, shows the precision of different studies. \nThe results for parameterization II are provided in figure 3. The green lines show NG15 and blue lines show IPTA2. We find that c M and log 10 A are also correlated. Increasing c M has the effect of increasing the amplitude. In this case we only find an upper bound as c M < 0 . 2. We compare the results for c M in table 2 with results from the CMB study [101]. It is seen that the current PTAs data can not perform better in constraining the model. \nIn figure 4 and 5, for concreteness, we show h 2 Ω GW as a function of frequency in logarithmic scales using the best fit of values of the parameters obtained from NG15 and IPTA2 for parameterization I and parameterization II , respectively. h is the reduced Hubble constant. The violin plots are from NG 15-year and IPTA second data release. \nFigure 3 . The posterior plots and marginal posteriors of no slip model parameters for parameterization II . The contours show at 68% and 95% confidence levels for International PTA second data release (IPTA2) in blue and NANOGrav 15-year data set (NG15) in green. \n<!-- image --> \nTable 1 . Constraints at 68% confidence level from (NG15), (IPTA2), (Pop III + CMB), (Delay + CMB) and (No Delay + CMB) [98], and their relative errors for parameterization I .', '4 Summary and discussion': 'In this work, we constrained a modified gravity model in PTA scales. We studied no slip model as a subclass of Horndeski models. In this model GWs have a different damping term compared to the standard general relativity case. The speed of GWs is equal to the speed of light in this model. Using two different parametrizations, we derived approximate transfer \nTable 2 . Constraints at 68% confidence level from NG15 and IPTA2 for parameterization II . \nFigure 4 . The current spectral energy density of GWs as a function of frequency in logarithmic scales for no slip model for parameterization I . The violin plots show NANOGrav 15-year data set (NG15) and International PTA second data release (IPTA2). \n<!-- image --> \nfunctions for GWs in this model. Moreover, we used this transfer function to find Ω GW in our model. \nWe constrained the model parameters by using NANOGrav and IPTA second data release. Two parameters ξ and c M are correlated and degenerate with other parameters, so it is hard to constrain them with percent level accuracy. \nIn our work, we only considered PTAs. One extension is to use multi probes of GWs by combining the existing GW detectors or future detectors like Einstein Telescope to estimate the constraints on the parameters [110, 111]. Additionally, more thorough investigations could take into account various effects which could contaminate PTA signals. Effects like the interstellar medium, the solar wind and solar system ephemeris errors contribute to the noise [112]. \nFigure 5 . The current spectral energy density of GWs as a function of frequency in logarithmic scales for no slip model for parameterization II . 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2024Sci...385.1318J

The Heisenberg uncertainty principle dictates that the position and momentum of an object cannot be simultaneously measured with arbitrary precision giving rise to an apparent limitation known as the standard quantum limit SQL. Gravitationalwave detectors use photons to continuously measure the positions of freely falling mirrors and so are affected by the SQL. We investigated the performance of the Laser Interferometer GravitationalWave Observatory LIGO after the experimental realization of frequencydependent squeezing designed to surpass the SQL. For the LIGO Livingston detector we found that the upgrade reduces quantum noise below the SQL by a maximum of three decibels between 35 and 75 hertz while achieving a broadband sensitivity improvement increasing the overall detector sensitivity during astrophysical observations.

2024-09-01T00:00:00Z

['arXiv:2404.14569', '2024Sci...385.1318M', '2024Sci...385.1318J', '10.1126/science.ado8069', '10.48550/arXiv.2404.14569', '2024arXiv240414569J']

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

Squeezing the quantum noise of a gravitationalwave detector below the standard quantum limit

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

https://arxiv.org/pdf/2404.14569.pdf

{'LIGO operates with quantum noise below the Standard Quantum Limit': "W. Jia , 1, ∗ V. Xu , 1, † K. Kuns , 1 M. Nakano , 2 L. Barsotti , 1 M. Evans , 1 N. Mavalvala , 1 R. Abbott, 3 I. Abouelfettouh, 4 R. X. Adhikari , 3 A. Ananyeva, 3 S. Appert, 3 K. Arai , 3 N. Aritomi, 4 S. M. Aston, 2 M. Ball, 5 S. W. Ballmer, 6 D. Barker, 4 B. K. Berger , 7 J. Betzwieser , 2 D. Bhattacharjee, 8 G. Billingsley , 3 N. Bode , 9, 10 E. Bonilla , 7 V. Bossilkov, 2 A. Branch, 2 A. F. Brooks , 3 D. D. Brown, 11 J. Bryant, 12 C. Cahillane , 6 H. Cao, 13 E. Capote, 6 Y. Chen, 14 F. Clara, 4 J. Collins, 2 C. M. Compton, 4 R. Cottingham, 2 D. C. Coyne , 3 R. Crouch, 4 J. Csizmazia, 4 T. Cullen, 3 L. P. Dartez, 4 N. Demos, 1 E. Dohmen, 4 J. C. Driggers , 4 S. E. Dwyer, 4 A. Effler , 2 A. Ejlli , 15 T. Etzel, 3 J. Feicht, 3 R. Frey , 5 W. Frischhertz, 2 P. Fritschel, 1 V. V. Frolov, 2 P. Fulda, 16 M. Fyffe, 2 D. Ganapathy , 1 B. Gateley, 4 J. A. Giaime , 2, 17 K. D. Giardina, 2 J. Glanzer, 17 E. Goetz , 18 A. W. Goodwin-Jones , 19 S. Gras, 1 C. Gray, 4 D. Griffith, 3 H. Grote , 15 T. Guidry, 4 E. D. Hall , 1 J. Hanks, 4 J. Hanson, 2 M. C. Heintze, 2 A. F. Helmling-Cornell , 5 H. Y. Huang , 20 Y. Inoue, 20 A. L. James , 15 A. Jennings, 4 S. Karat, 3 M. Kasprzack , 3 K. Kawabe, 4 N. Kijbunchoo , 21 J. S. Kissel , 4 A. Kontos , 22 R. Kumar, 4 M. Landry, 4 B. Lantz , 7 M. Laxen , 2 K. Lee , 23 M. Lesovsky, 3 F. Llamas, 24 M. Lormand, 2 H. A. Loughlin, 1 R. Macas , 25 M. MacInnis, 1 C. N. Makarem, 3 B. Mannix, 5 G. L. Mansell , 6 R. M. Martin , 26 N. Maxwell, 4 G. McCarrol, 2 R. McCarthy, 4 D. E. McClelland , 21 S. McCormick, 2 L. McCuller, 3 T. McRae, 21 F. Mera, 4 E. L. Merilh, 2 F. Meylahn , 9, 10 R. Mittleman, 1 D. Moraru, 4 G. Moreno, 4 M. Mould , 1 A. Mullavey, 2 T. J. N. Nelson, 2 A. Neunzert, 4 J. Oberling, 4 T. O'Hanlon, 2 C. Osthelder, 3 D. J. Ottaway , 11 H. Overmier, 2 W. Parker , 2 A. Pele , 3 H. Pham, 2 M. Pirello, 4 V. Quetschke, 24 K. E. Ramirez , 2 J. Reyes, 26 J. W. Richardson , 13 M. Robinson, 4 J. G. Rollins , 3 J. H. Romie, 2 M. P. Ross , 27 T. Sadecki, 4 A. Sanchez, 4 E. J. Sanchez, 3 L. E. Sanchez, 3 R. L. Savage , 4 D. Schaetzl, 3 M. G. Schiworski , 11 R. Schnabel , 28 R. M. S. Schofield, 5 E. Schwartz , 15 D. Sellers, 2 T. Shaffer, 4 R. W. Short, 4 D. Sigg , 4 B. J. J. Slagmolen , 21 S. Soni , 1 L. Sun , 21 D. B. Tanner, 16 M. Thomas, 2 P. Thomas, 4 K. A. Thorne, 2 C. I. Torrie, 3 G. Traylor, 2 G. Vajente , 3 J. Vanosky, 3 A. Vecchio , 12 P. J. Veitch , 11 A. M. Vibhute , 4 E. R. G. von Reis, 4 J. Warner, 4 B. Weaver, 4 R. Weiss, 1 C. Whittle , 3 B. Willke , 9, 10 C. C. Wipf, 3 H. Yamamoto , 3 H. Yu, 1, 29 L. Zhang, 3 and M. E. Zucker 1, 3 1 LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 LIGO Livingston Observatory, Livingston, LA 70754, USA 3 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA 4 LIGO Hanford Observatory, Richland, WA 99352, USA 5 University of Oregon, Eugene, OR 97403, USA 6 Syracuse University, Syracuse, NY 13244, USA 7 Stanford University, Stanford, CA 94305, USA 8 Kenyon College, Gambier, Ohio 43022, USA 9 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany 10 Leibniz Universitat Hannover, D-30167 Hannover, Germany 11 OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia 12 University of Birmingham, Birmingham B15 2TT, United Kingdom 13 University of California, Riverside, Riverside, CA 92521, USA 14 TAPIR, California Institute of Technology, Pasadena, California 91125, USA 15 Cardiff University, Cardiff CF24 3AA, United Kingdom 16 University of Florida, Gainesville, FL 32611, USA 17 Louisiana State University, Baton Rouge, LA 70803, USA 18 University of British Columbia, Vancouver, BC V6T 1Z4, Canada 19 OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia 20 National Central University, Taoyuan City 320317, Taiwan 21 OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia 22 Bard College, Annandale-On-Hudson, NY 12504, USA 23 Sungkyunkwan University, Seoul 03063, Republic of Korea 24 The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA 25 University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom 26 Montclair State University, Montclair, NJ 07043, USA 27 University of Washington, Seattle, WA 98195, USA 28 \nUniversitat Hamburg, D-22761 Hamburg, Germany \n29 University of Vienna, Faculty of Physics & Research Network Quantum \nAspects of Space Time (TURIS), Boltzmanngasse 5, 1090 Vienna, Austria \nPrecision measurements of space and time, like those made by the detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO), are often confronted with fundamental limitations imposed by quantum mechanics. The Heisenberg uncertainty principle dictates that the position and momentum of an object cannot both be precisely measured, giving rise to an apparent limitation called the Standard Quantum Limit (SQL). Reducing quantum noise below the SQL in gravitational-wave detectors, where photons are used to continuously measure the positions of freely falling mirrors, has been an active area of research for decades. Here we show how the LIGO A+ upgrade reduced the detectors' quantum noise below the SQL by up to 3 dB while achieving a broadband sensitivity improvement, more than two decades after this possibility was first presented. \nIntroduction One of the most profound consequences of quantum mechanics is the Heisenberg uncertainty principle, which posits that the product of the measurement noises of conjugate observables (i.e, position and momentum) cannot be less than ℏ / 2. Measuring the position x of an object with an uncertainty ∆ x inevitably perturbs its momentum by ∆ p ≥ ℏ / (2∆ x ). After a time τ , the massive object m will freely evolve with additional position uncertainty ∆ x ' from the momentum perturbation ∆ x ' = τ ∆ p/m = ℏ τ/ (2 m ∆ x ). An extremely precise measurement (∆ x → 0) will make the next position measurement totally unpredictable (∆ x ' → ∞ ) due to quantum back action [1]. The minimal possible uncertainty can be achieved with ∆ x = ∆ x ' = √ ℏ τ/ (2 m ), which is known as the Standard Quantum Limit (SQL) [2]. While the SQL applies to measurements of microscopic particles, it is also a limiting factor for the measurements made by the LIGO interferometric detectors, which probe attometer-scale displacements of macroscopic mirrors [3]. \nIn the 1980s, it was suggested that the SQL could be surpassed by introducing quantum correlations between the interferometer's laser light and the mirrors [46]. In the early 2000s, proposed designs to convert the LIGO into 'quantum nondemolition interferometers' 1 emerged [11]; one approach, referred to as 'squeezedinput' interferometer, suggested that it is possible to break the SQL at a particular frequency by injecting a non-classical state of light, known as squeezed vacuum state, to the LIGO interferometer. Furthermore, the addition of a detuned Fabry-P'erot 'filter cavity' would produce a frequency-dependent phase shift on the squeezed vacuum states reflected from it and therefore enable a broadband quantum enhancement below the SQL. \nIn the first proof-of-principle demonstration in 2020, we injected squeezed vacuum states in LIGO to demonstrate quantum correlations and surpass the SQL in a narrow frequency region of the detection band (3050 Hz) [12]. However, since this was achieved without \na filter cavity, it led to a quantum noise increase at frequencies outside the sub-SQL dip and an overall decrease of the astrophysical sensitivity, as predicted in [11]. \nAs part of the LIGO A+ upgrade that started in 2022, a 300-m long filter cavity was added to both LIGO Livingston (L1) and Hanford (H1) interferometers to achieve broadband reduction of quantum noise [13]. Here we present the first modeling and analysis of quantum noise in the LIGO interferometer operating with a filter cavity. We show that LIGO's quantum noise surpasses the SQL by up to 3 dB between 35 Hz and 75 Hz in astrophysical operation, with enhanced sensitivity in most of the detection band, thereby realizing the goal first set out over two decades ago [11]. \nTheory Gravitational-wave modulations of spacetime are quantified by strain h . The LIGO gravitationalwave interferometer converts these modulations into a measurable differential displacement between two pairs of suspended mirrors. The dimensionless gravitationalwave strain and the interferometer differential displacement ∆ x are related by h = ∆ x/L arm , where L arm = 4 km is the length of the each interferometer arm. At relevant measurement frequencies, the interferometer mirrors move freely, and the SQL for these mirrors can be expressed in units of gravitational-wave strain noise amplitude spectral density as \nh SQL (Ω) = ∆ x SQL (Ω) L arm = √ 2 ℏ ( m/ 4)Ω 2 1 L arm (1) \nwhere ℏ is the reduced Planck's constant, and Ω is the measurement frequency. Notably, this limit depends on the mass of the object rather than the number of photons used to probe the object (i.e., the laser power). In LIGO, the mass of the object is the reduced mass ( m/ 4) of the differential motion of each pair of arm cavity mirrors with mass m = 40kg \nh 40kg SQL (Ω) ≈ 1 . 8 × 10 -24 ( 2 π × 100 Hz Ω ) 1 √ Hz . \nIn [13], we presented a simplified model of quantum noise in the LIGO interferometers. In the ideal lossless case, the power spectral density (PSD) of quantum noise can be expressed in units of strain as: \nS (Ω) = h 2 SQL (Ω) 2 ( K (Ω) + 1 K (Ω) ) (2) \nwhere the first term represents noise due to quantum back action and the second term represents imprecision noise. The optomechanical coupling strength K (Ω) increases with the circulating laser power in arm cavities P arm as \nK (Ω) = 16 k 0 P arm mγ 0 L arm 1 Ω 2 ( 1 + Ω 2 γ 2 0 ) -1 . (3) \nwhere k 0 = 2 π/ (1064 nm) is the laser wavenumber, and γ 0 ≈ 2 π × 450 Hz is the detector's signal bandwidth. \nTogether, these two forms of quantum noise enforce the so-called SQL for displacement sensing (Eq. (2)), which arises from the use of uncorrelated photons to probe mirror positions. Eq. (2) enforces the SQL because it is an incoherent superposition of quantum back action and imprecision noise. In the presence of quantum correlations between light and mirrors, Eq. (2) no longer holds, allowing the SQL to be surpassed. \nAt frequencies below 100 Hz, measurement back action dominates due to the strong opto-mechanical coupling ( K (Ω) ≫ 1). At frequencies above γ 0 , the optomechanical coupling is weak ( K (Ω) ≪ 1), and the measurement imprecision from photon shot noise dominates. These two forms of quantum noise contribute equally to the total quantum noise at the SQL frequency Ω SQL , defined by K (Ω SQL ) = 1. Note Ω SQL scales with the square root of the laser power; for a circulating power of P arm = 260kW, Ω SQL = 2 π × 37 Hz. \nSqueezed vacuum is a non-classical state of light which uses quantum correlations between photon pairs to reduce one form of quantum noise (e.g. imprecision noise) at the expense of the other (e.g. quantum back action noise), in the way allowed by the Heisenberg uncertainty principle [14]. The injection of squeezed vacuum into the output port of an interferometer [15] modifies its quantum noise relative to Eq. (2) to produce \nS SQZ (Ω) = S (Ω) [ e -2 r cos 2 ( ϕ -θ (Ω)) + e 2 r sin 2 ( ϕ -θ (Ω)) ] (4) \nwhere e -2 r is the factor by which the injected quantum noise is squeezed relative to vacuum noise, ϕ is the relative phase between the input squeezed field and the interferometer field (i.e. the 'squeeze angle'), and θ (Ω) = tan -1 K (Ω) is the squeeze angle rotation due to the optomechanical response of the interferometer. \nFrequency-dependent squeezed states, where the squeeze angle varies as a function of frequency ϕ → ϕ (Ω), can be prepared by reflecting the frequency-independent squeezed state from a detuned and overcoupled FabryP'erot cavity [13, 16-18]. When the filter cavity linewidth is well-matched to Ω SQL , it imparts the phase rotation ϕ (Ω) ≈ θ (Ω) = tan -1 K (Ω) upon the reflected squeezed vacuum and enables quantum noise reduction of e -2 r at all frequencies [19]: \nS FDSQZ (Ω) = h 2 SQL (Ω) 2 ( K (Ω) + 1 K (Ω) ) e -2 r . (5) \nFIG. 1. Simplified schematic of the LIGO A+ interferometer as of the fourth astrophysical observing run (O4) started in 2023. The squeezing system is shown overlaying the shaded main interferometer, which is a dual-recycled Michelson with 4-km long arm cavities. All optical components shown, except for the main laser, are suspended in ultra-high vacuum. Frequency-independent squeezed vacuum is generated by an optical parametric amplifier ('squeezer'), which consists of a nonlinear optical crystal in a dually-resonant bowtie cavity. The outgoing squeezed beam is reflected from a 300-m long filter cavity to produce frequency-dependent squeezing, injected via the Faraday isolator, and then propagated through the full LIGO interferometer. \n<!-- image --> \nIn particular, around Ω SQL , quantum noise is reduced below the SQL by a factor of e -2 r \nS FDSQZ (Ω SQL ) = h 2 SQL (Ω SQL ) e -2 r . (6) \nExperimental setup Fig. 1 shows a simplified diagram of the LIGO interferometer [20], which includes Fabry-P'erot arm cavities formed by a pair of 40-kg mirrors to resonantly enhance strain sensitivity, input power recycling to increase the circulating laser power (and thus K (Ω)), and output signal extraction to broaden the detection bandwidth. Components of the squeezing system, comprising the squeezed vacuum source ('squeezer') and the filter cavity, are highlighted in the figure. \nSqueezed vacuum is injected at the output port of the interferometer to reduce quantum noise [21]. The LIGO squeezer generates frequency-independent squeezed vacuum via spontaneous parametric down-conversion in a bowtie optical parametric amplifier cavity containing a nonlinear PPKTP crystal [22, 23]. As described in [11, 13], the 300-m filter cavity is controlled on the resonance at a detuned frequency with respect to the carrier frequency of the main laser, thus producing frequency- \n√ \nFIG. 2. Strain sensitivity of the LIGO L1 interferometer. The squeezed quantum noise surpasses the standard quantum limit h SQL (Ω) by up to 3 dB in the shaded region between 35-75 Hz. Error bars indicate the total 1σ uncertainty. This configuration is representative of the nominal detector noise during O4, demonstrating the use of quantum correlations to directly improve astrophysical sensitivity. The total detector noise spectrum is an incoherent sum of the classical and quantum noise. The unsqueezed reference total noise (solid black) is measured without squeezing injection. An unsqueezed quantum noise model (dashed black) is subtracted from the measured reference total noise to obtain an estimate of the underlying classical noise (gray). The inferred detector quantum noise with squeezing (purple dots) is obtained by subtracting the classical noise estimate (gray) from the measured squeezed total noise spectra (solid purple). The dashed purple trace shows a fitted model of frequencydependent squeezed noise spectra, given our best knowledge of the detector and squeezer parameters. \n<!-- image --> \ndependent squeezing ( ϕ → ϕ (Ω)) before injection into the interferometer. \nResults Fig. 2 shows the first detailed quantum noise analysis of the LIGO L1 detector operating with frequency-dependent squeezing. Beyond our previous work [13] that shows only the total detector noise reduction with frequency-dependent squeezing, here we demonstrate quantum noise that surpasses the SQL between 35-75 Hz, by as much as 3 dB near 50 Hz, as highlighted in the purple shaded regions. While a complete analysis was done only for the L1 interferometer data, qualitatively similar results were observed in H1. \nAccurate estimation of squeezed quantum noise below 100 Hz is complicated by the presence of non-quantum ('classical') noises that are a factor of 2 higher in amplitude. In this work, we performed further measurements and extensive analysis to accurately infer the squeezed quantum noise from total noise measurements. \nThere are two steps to inferring quantum noise. First, we infer the classical noise (gray) by subtracting an unsqueezed quantum noise model (dashed black) from measurements of the total unsqueezed detector noise (solid black). An accurate model of unsqueezed quantum noise \nis crucial to determine classical noise from subtraction. The model is known to have degenerate parameters. For example, the circulating power and optical loss in the readout path affect the imprecision noise in the same way phenomenologically. To constrain the parameter space, we experimentally set a few constant squeezing angles ϕ and find a set of interferometer parameters that accurately models the measured noise for each ϕ , since the quantum noise S SQZ heavily depends on ϕ (Eq. (4)). We perform a Markov Chain Monte Carlo inference to find a set of parameters that make a common fit to all 11 different squeeze angle datasets (with a subset of these data shown in Fig. 3). These parameters include key experimental non-idealities such as squeezing phase noise, optical loss and mode-mismatches across the interferometer, as described in [24-26]. Second, we subtract this classical noise estimate from subsequent measurements of the total detector noise with squeezing (purple) to infer the squeezed quantum noise (purple dots), representing our measure of √ S SQZ (Ω) from Eq. (4). Detailed discussion of inferred parameters and model residuals are contained in the Supplemental Material. \nThe two-step noise subtraction process assumes that \nclassical noise remains identical across unsqueezed and squeezed modes of operation. Variations in classical noise between these modes will then appear as estimation uncertainties. Here we use the same uncertainty propagation methods as [12] to estimate the total error budget, including statistical uncertainties from detector noise PSD estimation and non-stationary classical noise, and the systematic uncertainties from calibration and residual model errors. \nStatistical uncertainties limit our estimation of low frequency quantum noise. This includes uncertainties from PSD estimation (requiring long averaging times) and non-stationary classical noise (requiring technical detector improvements) [27]. To reduce the statistical uncertainty, the total noise measurements in Fig. 2 were obtained by averaging the detector noise over 0 . 5-1 hour in each configuration, and by alternating between unsqueezed and squeezed configurations to control for time variations of the classical noise. We find that differences in the classical noise between segments (non-stationarity) was comparable to the total uncertainty from one hour of PSD estimation with optimal frequency binning. \nThe main systematic uncertainty arises from the realtime calibration process, where we apply a known force to the mirror to actively modulate the strain and measure the instrument's response [28, 29]. For the data shown here, the systematic uncertainties are less than 5%. Full derivations of total uncertainty budget can be found in Supplemental Materials. \nFig. 3 shows L1 measurements of the inferred quantum noise with frequency-dependent squeezing (purple traces) and frequency-independent squeezing at two injected squeeze angles ϕ , in decibels of quantum noise reduction compared to no squeezing (i.e., 20 log 10 [ √ S SQZ (Ω) /S (Ω) ] ). Dashed traces show numerical quantum noise models that include best-fit experimental parameters for the full interferometer, squeezer, and filter cavity. Strong agreements between model curves (dashed traces) and measured spectra (dots with error bars) support the unsqueezed quantum noise model used for subtraction and experimental parameters for the squeezer. This model is then extended to include the filter cavity parameters, initially described in [13]. The quantum noise models with frequency-dependent squeezing (dashed purple curve) agree well with the inferred quantum noise spectra (purple dots). \nWhile the current frequency-dependent squeezing configuration achieves quantum-noise suppression above 35 Hz (see the dashed purple 'current FC' curve in Fig. 3), frequency-independent squeezing models and measurements all suggest that an optimal filter cavity would yield significantly greater quantum noise reduction at astrophysically-important low frequencies (solid purple 'optimal FC' curve). The discrepancy between the current and optimal filter cavity arises from the mis- \nFIG. 3. Quantum noise reduction in units of decibels. Dots show the inferred quantum noise from measurements of the total detector noise in various configurations. Dashed traces are the quantum noise models. The input filter cavity rotates the injected squeezing angle ϕ as a function of frequency to produce frequency-dependent squeezing ( ϕ → ϕ (Ω)) [13]. Blue and olive traces show the inferred quantum noise with frequency-independent squeezing injected at two different ϕ . They outline the minimum quantum noise achievable at particular frequencies, given detector losses. The three purple traces show the quantum noise with (i) frequency-dependent squeezing using the current filter cavity (dashed purple, same as Fig. 2), (ii) an optimal filter cavity (solid purple) with a cavity linewidth well-matched to the current circulating laser power and 60-ppm round-trip optical loss, and (iii) a lossless optimal filter cavity (dotted purple). \n<!-- image --> \nmatch between the current SQL frequency and the filter cavity linewidth. In the lossless case, the optimal filter cavity would have an equal half-width-half-maximum linewidth γ FC and detuning both determined by the SQL frequency, γ FC = Ω SQL / √ 2 [19]. The current filter cavity was designed to have γ FC = 2 π × 42 Hz, using an input coupler power transmissivity of T in ≈ 1000 ppm [13] and assuming 60 ppm optical loss, to approximately match Ω SQL = √ 2 γ FC = 2 π × 59 Hz. However, the current SQL frequency is at Ω SQL = 2 π × 37 Hz. Since Ω SQL is proportional to the square root of arm power as in Eq. (3), the optimal filter cavity curve in Fig. 3 could be approached by either reducing the filter cavity linewidth (reducing T in , solid purple), or increasing the current arm power from 260 kW to 500 kW, as shown in Fig. 9. This is because a higher circulating laser power couples back action into the measurement over a larger bandwidth, requiring a higher bandwidth filter cavity to compensate. \nCompared to frequency-independent squeezing spectra (blue and olive traces), the lossless optimal filter cavity \nrotates the injected squeezing angle as a function of frequency, ϕ → ϕ (Ω) ≈ tan -1 K (Ω), to approach the minimum quantum noise at all frequencies simultaneously (dotted purple). Ideally, frequency-dependent squeezing is a single configuration that reaches the envelope of minimal quantum noises achievable by all frequencyindependent spectra. \nConclusions With frequency-dependent squeezing, the LIGO A+ detectors now operate with quantumlimited sensitivity surpassing the SQL, as envisioned for the first time over two decades ago [11]. The methods described here enabled us to accurately model quantum noise through the complex optical systems of the LIGO interferometers, with important insights that inform the next steps toward the A+ target of 6 dB of broadband squeezing enhancement. \nConcepts for future upgrades in the LIGO facilities and the next generation of gravitational-wave detectors like Cosmic Explorer [30] and Einstein Telescope [31] include the ambitious goal of 10 dB squeezing enhancement. Techniques and methods presented here are fundamental to achieving this goal and further enhancing the scientific potential of gravitational-wave observatories.", 'ACKNOWLEDGMENTS': "The authors would like to thank Vivishek Sudhir, Dennis Wilken, and Harald Pfeiffer for helpful discussions. Funding: The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, and the Max-Planck-Society (MPS) for support of the construction of Advanced LIGO. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors acknowledge the LIGO Scientific Collaboration Fellows program for additional support. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under cooperative agreement PHY-2309200. Advanced LIGO was built under award PHY-18680823459. The A+ Upgrade to Advanced LIGO is supported by US NSF award PHY-1834382 and UK STFC award ST/S00246/1, with additional support from the Australian Research Council. Author contributions: L.B., M.E., and N.M. conceived and supervised the project. W.J. and M.N. optimized and configured the LIGO Livingston detector to collect the data. W.J., V.X., and K.K. analyzed the data with the model developed by L.M. and K.K.. W.J., V.X., L.B., and M.E. prepared the manuscript. Competing interests: No conflict of interest. Data and materials availability: All data are available in the manuscript or the supple- \nmentary materials. \n- ∗ wenxuanj@mit.edu\n- † victoriaa.xu@ligo.org\n- [1] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science 209 , 547 (1980).\n- [2] V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement (Cambridge University Press, Cambridge, 1992).\n- [3] The LIGO Scientific Collaboration, Classical and Quantum Gravity 32 , 074001 (2015).\n- [4] W. G. Unruh, in Quantum Optics, Experimental Gravity, and Measurement Theory , NATO Advanced Science Institutes Series, edited by P. Meystre and M. O. 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Evans et al. , Cosmic Explorer: Science, Observatories, Community , Tech. Rep. CE-P2100003 (Cosmic Explorer, 2021).\n- [31] M. Punturo et al. , Classical and Quantum Gravity 27 , 194002 (2010).", 'Uncertainty Analysis': 'The argument that the LIGO detector operates beyond the standard quantum limit (SQL) requires a high statistical significance of the inferred quantum noise below the SQL. We follow our previous work [12] to estimate the total uncertainty of the inferred quantum noise. The formalism is briefly summarized here. The strain noise power spectral density (PSD) of inferred quantum noise Q (Ω) is obtained from measured reference (unsqueezed) total noise D r (Ω), measured squeezed total noise D s (Ω), and model of the reference quantum noise M (Ω): \nQ (Ω) = D s (Ω) -( D r (Ω) -M r (Ω)) . (7) \nThe total uncertainty of Q (Ω) is \n∆ Q 2 (Ω) = Q 2 (Ω) δG 2 cal (Ω) + [ ∆ D 2 s (Ω) + ∆ D 2 r (Ω) + ∆ M 2 r (Ω) + ( D r (Ω) -M r (Ω)) 2 ( δN 2 t (Ω) + δN 2 m (Ω)) ] (8) \nwhere \n- · δG cal (Ω) is the reported combined calibration error and uncertainty estimate [28],\n- · ∆ D (Ω) is the statistical uncertainty due to PSD estimation,\n- · ∆ M r (Ω) is the uncertainty of the unsqueezed reference quantum noise model, and\n- · δN (Ω) describes the non-stationary changes in the classical noise contributions, where δN t (Ω) is timenonstationarity and δN m (Ω) is the operating mode nonstationarity between unsqueezed and squeezed operating modes. \nIn this paper, we follow the convention in [12] and use ∆ to describe the 1σ uncertainty of the variable, and use δ for the relative uncertainty δD = ∆ D/D . We plot the noise spectrum in units of amplitude spectral density (ASD) q (Ω) = √ Q (Ω). The relative error in ASD is \nδq (Ω) = 1 2 δQ (Ω) = √ δG 2 cal (Ω) 4 + 1 4 Q 2 (Ω) [∆ D 2 s (Ω) + ∆ D 2 r (Ω) + ∆ M 2 r (Ω) + C 2 (Ω)( δN 2 t (Ω) + δN 2 m (Ω))] . (9)', 'Re-binning Power Spectral Density': 'The statistical uncertainty δD of the PSD scales inversely with the square root of the number of averages, which is proportional to the product of the duration T of the time series and the frequency bin width f \nδD = 1 √ Tf . (10) \nWe first take the linear FFT of the raw time series to estimate the total noise PSD. For each frequency bin, we take the median statistics to indirectly remove potential glitches in the time series, as described in our previous work [12]. The linearly spaced PSD has the constant frequency bin width, for which we choose a frequency resolution of 0 . 0625 Hz. To reduce the statistical uncertainty and fit the model, we re-bin the PSD into a log-spaced frequency bins. Each new frequency bin collects all the energy of the old frequency bins that falls into the bin so that the total spectral energy is conserved. The statistical uncertainty of the new PSD with log-spaced and larger bin width still follows the relation of Eq. (10). \nThe raw PSD measures the total differential displacement between the two pairs of arm cavity mirrors, which contain many peaks and resonances including harmonics of the 60-Hz power line and 500-Hz violin mechanical modes of test masses suspensions, etc. These peaks would inflate the energy of our re-binned PSD. Therefore, we remove all the known noise peaks before re-binning. \nFIG. 4. Comparison of the unsqueezed noise stationarity between two PSDs measured in the same unsqueezed operating mode (e.g. two segments with the squeezer beam diverter open), or measured in two different unsqueezed operating modes (e.g. with squeezer beam diverter open and beam diverter closed). Both uncertainties are the same, suggesting the squeezer system does not introduce excess technical noise in the full detectors. \n<!-- image -->', 'Non-stationarity Verification': "The stationarity uncertainty has two contributing terms: time-nonstationarity δN t (Ω) that captures slow thermal drifts of the interferometer, and mode-nonstationarity δN m (Ω) that contains changes introduced by different operating modes of the interferometer, namely with and without squeezing. \nTo measure the unsqueezed total noise as closely as the configuration with frequency-dependent squeezing, we set up the squeezing configuration but without squeezed vacuum generated. Specifically, we leave both the squeezer and filter cavity locked on resonance but without the nonlinear parametric down-conversion process. As seen in Fig. 1, we only send auxiliary control sidebands to the squeezer cavity for lock acquisition [13], but not the 532-nm pump laser. The squeezer is locked on the resonance to allow transmission of the control field to filter cavity. The filter cavity is also locked on resonance with the auxiliary field to mimic the nominal operation with frequency-dependent squeezing. If there is any extra technical noise introduced with frequency-dependent squeezing, for example backscatter noise driven by filter cavity length fluctuations, the interferometer would sense it in the total noise spectra in both configurations. \nTo confirm if there are any excessive noises including backscatter, we compare the total unsqueezed interferometer noise with the following two operating modes. The first one is to open the squeezer beam diverter to mimic the frequency-dependent squeezing case as mentioned above, and the second one is to close the squeezer beam diverter on the injection path such that no backscattered light can be transmitted between interferometer and squeezing system. We follow Eq.(13) in [12] to estimate the uncertainties. We have two PSD of each mode and calculate δN (Ω) between PSDs of the same and different operating modes. \nFig. 4 shows that the stationarity uncertainty curves are nearly identical between two PSD taken at the same operating mode or different operating mode, confirming that the mode-nonstationarity contribution to the total stationarity uncertainty is negligible. \nThe time-nonstationarity occurs due to thermal drifts of the interferometer. For the faster averaging timescales used in our measurements, slow drifts can be reduced with longer averaging times, similarly to statistical PSD estimation. Therefore, both of drifts and statistical uncertainties are reduced after the re-binning process. \nCalibration uncertainty δG cal (Ω) are estimated in the same way as [12]. Note that it is a form of systematic error instead of statistical error. Therefore, the calibration error is added to the total uncertainty after re-binning, since it can not be reduced by averaging. The contributions of aforementioned uncertainties to the total uncertainty are shown in Fig. 7. \nTABLE I. Parameters of the LIGO Livingston detector inferred using Markov Chain Monte Carlo (MCMC) methods. Fixed and chosen parameters are input parameters for the MCMC, which infers the 'common' and 'independent' parameters. Common parameters are shared across squeeze angle measurements, whereas independent parameters are allowed to change across squeeze angle measurements. See [3, 20] for more detailed parameters of LIGO, including e.g. optic transmissivities.", 'Full Quantum Noise Model': 'The only remaining source of uncertainty to be discussed is the modeling uncertainty δM (Ω). In this paper, we use a novel method to estimate and constrain model parameters with Markov Chain Monte Carlo (MCMC) inference. Before discussing details of the inference method, we briefly explain the latest model of quantum noise. \nThe LIGO detector is essentially an assembly of individual optical cavities. The core optics of the interferometer is composed of two 4-km long arm cavities as two Michelson arms. The two input ports of the Michelson have two partially-reflective mirrors to boost arm power and increase signal bandwidth separately at bright and dark port (see Fig. 1). The squeezing system generates squeezed vacuum using an optical parametric amplifer cavity, performs frequency-dependent rotation with a detuned filter cavity, and couples into the interferometer at the dark port. Each cavity in the system has degredations like optical losses and off-resonance detunings. In addition, there are non-zero mismatches between the fundamental spatial modes of two consecutive cavities. The full model captures all of these non-idealities based on our latest theoretical work [26]. \nWeuse Gravitational Wave Interferometer Noise Calculator (GWINC) to numerically compute the detector quantum noise. It is a phenomenological and analytical model that is derived from input-output relations [24]. It extends the optical fields of fundamental (TEM 00 ) spatial mode to one higher-order (TEM 20 ) mode [24, 26]. The model includes \nall the decoherences, degradations, and dephasings of the squeezed vacuum [25]. The full sets of parameters can be found in Table I. \nFor parameter estimation, we rely on external measurements as our priors whenever possible. But, we do not have external measures of several quantities, and for others, external measurements do not have the necessary accuracy and precision, and may vary if not measured in-situ. We use MCMC, informed by external measurements whenever possible, to estimate experimental parameters. The full model of interferometer with frequency-dependent squeezing has a total of 20 impactful parameters. We reduce the problem by isolating the interferometer from the squeezing system first. Then we introduce the squeezer parameters to model frequency-independent squeezing measurements, and finally the filter cavity to model frequency-dependent squeezing measurements.', 'Inferring Interferometer Parameters': "LIGO employs an active calibration system, known as the Photon Calibrator [29], to calibrate the measured optical power into meters of differential arm length. The system actively modulates the differential arm length by sending an amplitude-modulated laser beam on the test mass. Therefore, we can directly measure the interferometer's transfer function (in units of meters/milliAmp, often called the 'sensing function') by sweeping the Photon Calibrator laser frequency. \nAt the dark port of the interferometer, LIGO has an additional mirror, known as signal recycling mirror with 32.5% power transmission, to effectively broaden the sensitivity bandwidth to the differential arm length signal. The cavity formed by signal recycling mirror and the interferometer has parameters such as loss, mode-mismatch, and off-resonance detuning, which directly impact the measured sensing function. Therefore, we can isolate and infer these parameters by fitting the sensing function with MCMC. The total parameter space is reduced after we successfully infer parameters of the signal recycling cavity from the sensing function.", 'Inferring Frequency-Independent Squeezing Parameters': "After inferring the parameters of the signal recycling cavity, we feed them into the model that describes the squeezed interferometer. We simplify the squeezing system by bypassing the filter cavity first. It is often difficult to infer model parameters when many parameters of the model are degenerate. For example, mode-mismatch and loss between interferometer and output mode cleaner cavity are degenerate when this mode-mismatch is the only mismatch in the optical path [26]. If we introduce multiple mismatches to break the degeneracy, there are redundant parameters that provide more than one solution to satisfy measurements. \nTo constrain the quantum noise model, we change the squeezing angle ϕ to alter the quantum noise S SQZ (Ω) in Eq. (4) while keeping the filter cavity end mirror misaligned to simplify the system ( ϕ is frequency-independent in this case). Since we have only changed the squeezing parameter, there should exist a set of model parameters that can fit all of the measurements by only altering the squeezing angle, if the model fully captures the physics. Assuming such a set of common parameters should break certain degeneracies in the model and constrain the parameter space. \nExperimentally, we misalign the filter cavity and change the the squeezing angle ϕ by adjusting the offset of locking point of the phase-locking-loop between frequency-independent squeezing and the local oscillator field of the interferometer. We operate the interferometer in an unsqueezed mode (pump laser blocked so no squeezed photons are being generated) and frequency-independently squeezed mode at various ϕ . 20-minute time series data is taken in each operating mode. Assuming the classical noise C (Ω) is stationary across different configurations, we can take the difference of two measured total noise PSDs and model the quantum noise differences (Eq. (4)), \nS diff ( r, ϕ ) = D s ( r, ϕ ) -D r ( r = 0) = S ( r, ϕ ) -S ( r = 0) (11) \nwhere the total measured noise is D (Ω) = S (Ω) + C (Ω). Although we can not directly measure the classical noise, we can still model the quantum noise difference that is measurable. \nLIGO reads out optical power fluctuations that are transmitted (cleaned) by the output mode cleaner cavity. The transmitted beam is divided onto two photodetectors using a 50/50 beam splitter, and we read out the summation of the photocurrent signals. The sum reads out the squeezed quantum noise we observe, and the difference of the two photocurrents, known as the null channel, subtracts all of the correlated noise and only leave the uncorrelated noises of the two photodiodes, namely quantum shot noise and dark noise of the detector. The null channel provides a simultaneous monitoring of the calibrated quantum shot noise, which is computed by dividing the flat quantum shot noise in milliAmps by the sensing function. \nFIG. 5. Inference results on the difference of total noise between frequency-independent squeezed and unsqueezed interferometer at various squeezing angles. The negative PSD difference means that the quantum noise is being squeezed. The residual between model and measurements are normalized by the 1σ uncertainty and shown in the bottom plot. \n<!-- image --> \nWe find the best inference of the parameters with MCMC. We use a Gaussian likelihood with a set of Gaussian priors for each parameter. For each measurement with certain squeezing angle, we fit both the noise difference and the quantum shot noise, the latter of which is used to infer the readout loss. The initial walkers are distributed with a flat probability in a bounded interval. As a result, the method is able to find a set of common parameters that minimize the residual of all squeeze angle measurements, as presented in Fig. 5 and Table I. \nThere are four types of parameters in our inference methods: \n- · 'Fixed' parameters are fixed across all squeeze angle datasets. For example, the signal recycling cavity param- \ners we inferred earlier are assumed to be the same for all. \n- · 'Chosen' parameters are selected and different for each squeeze angle dataset. For example, the squeezing angle is actively changed to obtain different squeezing PSD.\n- · 'Common' parameters are shared degrees of freedom that MCMC infers a single value across all squeeze angle datasets. For example, the power within arm cavity should be the same across measurements, and we use MCMC to infer its exact number.\n- · 'Independent' parameters are degrees of freedom of MCMC infers differently for each squeeze angle dataset. \nIn Table I, we set the squeezing angle and phase noise as 'chosen parameters'. It is known that the residual phase noise error of the aforementioned phase-locking-loop depends on the control offset and therefore the squeezing angle. To be able to fit the PSD difference, we still need to set the mode-mismatch phasing between interferometer and output mode cleaner (Fig. 1) as an 'independent parameter', which is an extra phase in the optical path calculated from 2-dimensional overlap integral of the wavefronts of two eigenmodes of two cavities [26]. This mode-mismatch phasing only helps us fit the model phenomenologically, and is not expected to physically depend on the squeezing angle. Instead, the MCMC adjusts this phasing to mimic certain physics that is not fully captured in the latest model in order to fit the measurements. \nFig. 5 validates our quantum noise model as we successfully fit all of the measurements at various squeezing angles by independently tuning a minimal set of parameters. The model uncertainty δM (Ω) is obtained by taking the 16th and 84th percentile of the model curve computed from the parameters of the MCMC chain (after burning in). Now we collect all sources of uncertainties (Fig. 7) and compute the inferred quantum noise with frequency-dependent squeezing. \n√ \nFIG. 6. Inference of the quantum noise with frequency-dependent squeezing. \n<!-- image -->", 'Inferring Filter Cavity Parameters': 'Using the interferometer and squeezer quantum noise models obtained in the previous subsections, we can compute the inferred quantum noise ASD with frequency-dependent squeezing from Eq. (7). We perform a final MCMC to infer the remaining filter cavity parameters. \nSince LIGO is currently operating at a lower arm power than the designed value, the filter cavity is not operating in the optimal configuration [19]. This is the reason why the current frequency-dependent squeezed quantum noise does not trace the sub-SQL dips of each frequency-independent measurements, in addition to a noise bump near 80 Hz due to scattered light. In the MCMC, we assumed the filter cavity finesse to be 7000 in order to fit external cavity ringdown and linewidth measurements of the filter cavity. The inferred parameters are shown in Table I.', 'Total Uncertainty Budget': 'Now that we have collected all sources of the uncertainties δq (Ω) of the inferred quantum noise amplitude spectral density q (Ω), we can add these independent noises together in quadrature to obtain the final 1σ uncertainty. The contributions of each uncertainty is shown in Fig. 7. \nFIG. 7. Total uncertainty budget of inferred quantum noise from various error sources. \n<!-- image --> \nThe statistical uncertainty dominates both positive and negative error bars at low frequency due to the small frequency bin width (Eq. (10)). At high frequencies above 500 Hz, the statistical error decreases as there are more averages available per bin width. Both statistical and stationarity error are symmetrical, whereas the calibration error and modeling error are not. The calibration error, obtained from the calibration pipeline [28], dominates at high frequency above 200 Hz. \nConsidering all measurement uncertainties, the LIGO detector operates with sub-SQL quantum noise at more than 3σ statistical confidence, as enabled by frequency-dependent squeezing (Fig. 2). \n√', 'Sub-SQL Performance': '20 \nFIG. 8. Quantum noise reduction in strain amplitude spectral density. Blue, olive, lime, and teal traces show the inferred quantum noise with frequency-independent squeezing injected at four different squeeze angles ϕ . The three purple traces show the quantum noise with three frequency-dependent squeezing configurations, same as Fig. 3. \n<!-- image --> \nFig. 8 compares the sub-SQL performance with frequency-independent squeezing (constant squeezing angle ϕ ) and frequency-dependent squeezing ( ϕ = ϕ (Ω)). The sub-SQL dip can be produced by sending squeezing at a fixed angle, as previously observed [12]. However, the dip has a very narrow frequency range. Although we can move the dip frequency by changing squeezing angle, it is not an optimal configuration for maximum sensitivity at all frequencies. As mentioned in the main text, frequency-dependent squeezing can theoretically achieve the sub-SQL envelope that covers all dips that frequency-independent squeezing can achieve (dotted purple). The current and optimal filter cavity are more realistic configurations, and they are the same as Fig. 3.', 'Future Filter Cavity Upgrade': 'While we demonstrate that the optimal lossless filter cavity is able to simultaneously achieve all sub-SQL dips that frequency-independent squeezing can do, we have to acknowledge the fact that a realistic filter cavity has a non-zero loss. The designed round-trip loss of the filter cavity is 60 ppm, compared to the loss of 100 ppm suggested by our MCMC. A few different filter cavity configurations are shown in Fig. 9. \n10 \n0 \nFIG. 9. Comparison of the quantum noise with various filter cavity configurations. \n<!-- image --> \nIn Fig. 9, the relative quantum noise curves with current filter cavity (dashed purple) and optimal filter cavity (solid purple) are identical to Fig. 3. If we achieve the designed loss of 60 ppm with current filter cavity, the squeezing will improve from dashed purple to the orange curve. It is only possible to achieve squeezing at all frequencies when we adjust the filter cavity linewidth γ FC to approach Ω SQL / √ 2, for example, reducing the filter cavity input coupler transmission to 584 ppm (purple curve) or increasing the arm cavity power to 500 kW (blue curve). The lossless filter cavity is shown in the dotted purple trace. Note that the squeezing in the lossless case is not flat because we have a nonzero phase difference between the local oscillator field and the signal field, known as the readout angle. For each trace in Fig. 9, the detuning frequency of the filter cavity is optimized to maximize sensitivity to binary neutron star inspirals - a standard figure of merit for gravitational wave detectors. \n√ \nFIG. 10. Sub-budget of contributions to the total quantum noise. \n<!-- image --> \nFig. 10 shows the contributions of the total quantum noise plotted in Fig. 2. At low frequencies below 40 Hz, quantum noise is mostly limited by misrotation of the squeezed state due to the non-optimal filter cavity. At high frequencies above 200 Hz, squeezing is limited by the losses due to injection, readout, and mode-mismatches along the optical path. Reducing these major noise sources is the key to further quantum enhancement in the LIGO detectors.'}

2024arXiv240907410P

We present a kinematic and spectroscopic analysis of 40 red giant branch stars in 9 fields exquisitely delineating the lower segment of the North West Stream NWK2 which extends for sim80 kpc from the centre of the Andromeda galaxy. We measure the streams systemic velocity as 439.34.13.8 kms with a velocity dispersion 16.45.63.8 kms that is in keeping with its progenitor being a dwarf galaxy. We find no detectable velocity gradient along the stream. We determine 1.3pm0.1 le ltFeHrm specgt le 1.2pm0.8 but find no metallicity gradient along the stream. We are able to plausibly associate NWK2 with the globular clusters PandAS04 PandAS09 PAndAS10 PAndAS11 PandAS12 but not with PandAS13 or PandAS15 which we find to be superimposed on the stream but not kinematically associated with it.

2024-09-01T00:00:00Z

['2024arXiv240907410P', '10.48550/arXiv.2409.07410', 'arXiv:2409.07410']

['Astrophysics - Astrophysics of Galaxies']

Properties of the Lower Segment of M31s North West Stream

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2409.07410.pdf

{"Properties of the Lower Segment of M31's North West Stream": "Janet Preston, 1 ∗ Denis Erkal, 1 Michelle L.M. Collins, 1 Rodrigo Ibata, 2 R. Michael Rich 3 \n- 1 Department of Physics, University of Surrey, Guildford, GU2 7XH, Surrey, UK. ★\n- 2 Observatoire de Strasbourg, 11, rue de l'Université, F-67000, Strasbourg\n- 3 Department of Physics and Astronomy, UCLA, 430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547, USA \nAccepted XXX. Received YYY; in original form ZZZ", 'ABSTRACT': "We present a kinematic and spectroscopic analysis of 40 red giant branch stars, in 9 fields, exquisitely delineating the lower segment of the North West Stream (NW-K2), which extends for ∼ 80 kpc from the centre of the Andromeda galaxy. We measure the stream's systemic velocity as -439.3 + 4 . 1 -3 . 8 km s -1 with a velocity dispersion = 16.4 + 5 . 6 -3 . 8 km s -1 that is in keeping with its progenitor being a dwarf galaxy. We find no detectable velocity gradient along the stream. We determine -1.3 ± 0.1 ≤ <[Fe/H] spec > ≤ -1.2 ± 0.8 but find no metallicity gradient along the stream. We are able to plausibly associate NW-K2 with the globular clusters PandAS-04, PandAS-09, PAndAS-10, PAndAS-11, PandAS-12 but not with PandAS-13 or PandAS-15 which we find to be superimposed on the stream but not kinematically associated with it. \nKey words: galaxies: dwarf - galaxies: interactions - Local Group", '1 INTRODUCTION': "The Andromeda Galaxy (M31) is host to numerous stellar streams (Ibata et al. 2001, 2004, 2007, 2014, Martin et al. 2014a, Ferguson & Mackey 2016, McConnachie et al. 2018). These spectacular structures are the ghostly remnants of dwarf galaxies and globular clusters accreted by M31 over billions of years. They provide tangible evidence for the Λ CDM paradigm that larger galaxies grow by devouring smaller ones (Press & Schechter 1974, Springel et al. 2006, Frenk & White 2012) and enable us to explore the gravitational potentials required to produce them (e.g. Ibata et al. 2004, Chapman et al. 2006, Koposov et al. 2010, Fardal et al. 2013, Lux et al. 2013, Ibata et al. 2014). \nThe North West (NW) stream in the outer halo of M31 has an estimated length of ∼ 200 kpc and comprises two segments: an upper segment, labelled K1 (hereafter NW-K1) in Figure 12 of McConnachie et al. (2018); and a lower segment labelled K2 (hereafter NW-K2). NW-K1 was discovered by Richardson et al. (2011) using data obtained from the 3.6 m Canada-France-Hawaii Telescope (CFHT) for the Pan-Andromeda Archaeological Survey (PAndAS, McConnachieetal. 2009). They found this segment of the NW stream to be ∼ 3 · in length at a distance of ∼ 50-80 kpc from the centre of M31. Later work by McConnachie et al. (2018) found NW-K1 to have a luminosity of M 𝑉 = -10.5 ± 0.5 and a stellar mass of M ∗ = 9.4 × 10 5 M ⊙ . \nTwo years earlier, NW-K2 had been discovered by McConnachie et al. (2009) who determined that it had a projected length of ∼ 6 · and lay 50 ≤ R proj kpc ≤ 120 from the centre of M31. McConnachie et al. (2018) found NW-K2 to have a luminosity of M 𝑉 = -12.3 ± 0.5 and a stellar mass of M ∗ = 8.5 × 10 6 M ⊙ and noted that it was \nas bright as the NGC147 stream ( M 𝑉 = -12.2, McConnachie et al. 2018 ) and Andromeda II ( M 𝑉 = -12.6 ± 0.2, Martin et al. 2016). \nWork by Mackey et al. (2010), Veljanoski et al. (2013) and Huxor et al. (2014) found NW-K2 was co-located with the globular clusters (GCs) PAndAS-04, PAndAS-09, PAndAS-10, PAndAS11, PAndAS-12, PAndAS-13 and PAndAS-15. Subsequently, Veljanoski et al. (2014) found spatial and kinematic associations between the stream and PAndAS-04, PAndAS-09, PAndAS-10, PAndAS-11, PAndAS-12 and PAndAS-13 as well as detecting that the radial velocities of the GCs became increasingly negative the nearer a GC was to the centre of M31. However, they found that PAndAS-15, the GC closest to M31, did not follow this trend and concluded that it was unlikely to be associated with NW-K2 despite seeming to lie directly on top of it. Analysis by Sakari & Wallerstein (2022) determined metallicities for three of the GCs: PAndAS-04 [Fe/H] = -2.07 ± 0.2, PAndAS-09 [Fe/H] = -1.56 ± 0.2 and PAndAS-11 [Fe/H] = -2.16 ± 0.2. Work by Mackey et al. (2018) determined the specific frequency (which connects the number of GCs hosted by a galaxy to its total luminosity) of the NW-K2 GCs to be ∼ 70-85, which they noted was much higher than values found for other dwarf galaxy progenitors of streams around M31. This, they concluded, could be due to a much higher luminosity galaxy, probably totally destroyed with any residual debris lying within M31's inner halo, being the progenitor of NW-K2. \nDespite NW-K1 and NW-K2 being discovered separately, their morphology, as understood at the time, led Richardson et al. (2011) to conclude that they were part of a single structure wrapped around M31. Corroborating evidence was reported by Ibata et al. (2014) who detected similar metallicities ( -1.7 <[Fe/H] < -1.1) in both segments. Meanwhile, noting similar density variations and gaps in the two segments Carlberg et al. (2011) modelled the stream as a single structure finding it to be ∼ 10 Gyrs old and ∼ 5 kpc wide. They \nestimated the luminosity of NW-K2 to be 7.4 × 10 5 L ⊙ and noted that it was nearly intact while NW-K1 had some significant gaps possibly due to interactions with dark matter sub-haloes (Yoon et al. 2011). \nOn discovering the dwarf spheroidal (dSph) galaxy Andromeda XXVII (And XXVII) to be co-located with NW-K1, Richardson et al. (2011) postulated that it was likely to be the progenitor of the whole of the NW Stream. This view was sustained by Carlberg et al. (2011), based on the number of GCs associated with NW-K2 being consistent with the progenitor being a dwarf galaxy, and by Kirihara et al. (2017b) who noted that NW-K2's progenitor would need to have a stellar mass ∼ 10 6 -8 M ⊙ and a minimum r ℎ ≥ 30 pc, which did not rule out And XXVII as being a plausible progenitor for both it and NW-K1. \nHowever, Preston et al. (2019) cast doubt on the NW stream being a single structure with their findings of a velocity gradient of 1.7 ± 0.3 km s -1 degree -1 along NW-K1 that could be indicative of an infall trajectory onto M31. When considered in conjunction with the detection of a similar velocity gradient along the NW-K2 GCs, also most likely on an infall trajectory onto M31, by Veljanoski et al. (2013, 2014) and findings that both NW-K1 (heliocentric distance = 827 ± 47 kpc, Richardson et al. 2011, Preston et al. 2019) and NW-K2 (distance modulus ∼ 24.63 ± 0.19, Komiyama et al. 2018) lie behind M31, Preston et al. concluded that this could indicate that the two segments were not parts of a single structure. \nIn this work we aim to analyse the kinematic and spectroscopic properties of NW-K2 such that we can compare them with those of NW-K1 and see if there is any association between the two streams. We also aim to confirm, or otherwise, the association of the GCs PAndAS-04, PAndAS-09, PAndAS-10, PAndAS-11, PAndAS-12, PAndAS-13 and PAndAS-15 with NW-K2. This information will then further inform modelling of the trajectories of the two streams to more fully understand the nature of the enigmatic NW stream (Preston et al, in preparation). \nThe paper is structured as follows: Sections 2 describe our observational data and approach to data reduction. Section 3 includes determination of candidate NW-K2 stars and our kinematic and spectroscopic analyses. We present a discussion of our results in Section 4 and our conclusions in Section 5.", '2 OBSERVATIONS': 'Our photometric data were obtained within the Pan-Andromeda Archaeological Survey (PAndAS, McConnachie et al. 2009) using the 3.6m Canada-France-Hawaii Telescope (CFHT). Equipped with the MegaPrime/MegaCam, that comprises 36, 2048 x 4612, CCDs with a pixel scale of 0.185"/pixel, it was able to deliver ∼ 1 degree 2 field of view McConnachie et al. (2009). g -band (4140Å- 5600Å) and i -band (7020Å-8530Å) filters were used to facilitate good colour discernment of Red Giant Branch (RGB) stars. Seeing of < 0".8 enabled stellar resolution to a depth of g = 26.5 and i = 25.5 with a signal to noise ratio of ∼ 10 (McConnachie et al. 2009, Collins et al. 2013, Martin et al. 2014b). \nTo determine the photometric zero points, de-bias, flat-field and fringe correct the data they were first processed by the Elixir system, Magnier & Cuillandre (2004), at CFHT. The data were then further reduced using a specifically constructed pipeline at the Cambridge Astronomical Survey Unit, Irwin & Lewis (2001). Finally, morphological classifications (e.g. point source, non-stellar, noise) of the data were identified and stored along with g and i values on the \nPAndAS catalogue, Richardson et al. (2011). For this work we select point source objects. \nAdditional observations (see Table 1) along the NW-K2 stream were taken with the Keck II telescope fitted with the DEep-Imaging Multi-Object Spectrograph (DEIMOS) using the OG550 filter with the 1200 lines/mm grating with a resolution of ∼ 1.1Å-1.6Å at FWHM. The masks were observed as follows: NWS3 and NWS6 were observed for a total of 1 hour 40 mins split into 5 x 20 minute integrations; 506HaS, 507HaS and 704HaS were observed for 1 hour 30 mins (3 x 30 minutes); NWS1 and NWS5 had a total observation time of 1 hour 20 minutes (4 x 20 minutes) and 606HaS had 3 x 15 minute integrations with a total observation time of 45 minutes. \nWe selected our target stars based on their location within the colour magnitude diagram (CMD). Our highest priority targets were bright stars lying on the NW-K2 RGB with 20.3 < i 0 < 22.5, where i 0 is the de-reddened i -band magnitude, given by i 0 = i - 2.086E(BV), obtained using extinction maps and correction coefficient, from Table 6, in Schlegel et al. (1998). Our next priority were fainter stars on the RGB, i.e. 22.5 < i 0 < 23.5. The remainder of the mask was filled with stars in the field with 20.5 < i 0 < 23.5 and 0.0 < g-i < 4.0. \nThese data were reduced using the pipeline, described in Ibata et al. (2011), that corrected for: scattered light, flat-fields, the slit function and illumination within the telescope as well as calibrating the wavelength of each pixel. The pipeline also determined velocities and associated uncertainties for the stars by: (1) creating model spectra comprising a continuum and the absorption profiles of the Calcium Triplet (CaT) lines (at 8498Å, 8542Å and 8662Å); (2) crosscorrelating these models with non-resampled stellar spectra using a Markov Chain Monte Carlo (MCMC) approach to obtain the optimum Doppler shift and CaT line widths; and (3) correcting the velocities and associated uncertainties to the heliocentric frame. \nThroughout this work we assume a radial velocity of -300 ± 4 km s -1 and an heliocentric distance of 783 ± 25 kpc for M31 (McConnachie 2012). With respect to this latter value we note that it precedes current findings for M31\'s heliocentric distance of 761 ± 11 kpc by Li et al. (2021) and 798 ± 28 kpc by Beasley et al. (2023) but decide to continue using this value so our results are comparable to earlier works. We also assume an heliocentric distance of 827 ± 47 kpc for And XXVII Richardson et al. (2011), Preston et al. (2019).', '3.1 Kinematics': 'Todetermine the properties of the NW-K2 stream we first identify and confirm members of its stellar population. As work by Mackey et al. (2018) and Komiyama et al. (2018) noted a kinematic association between NW-K2 and the co-located GCs, we assume that NW-K2 stream stars will have velocities consistent with those of the GCs (see Table 2) , i.e. between ∼ -397 km s -1 for PAndAS-04 and ∼ -570 km s -1 for PAndAS-13. We therefore select stars from each mask with velocities in the range -600 km s -1 to 0 km s -1 that have velocity uncertainties < 20 km s -1 . \nWe plot an initial set of velocity histograms for each mask and, taking the above criteria into account, find no NW-K2 candidate stars on mask 508HaS. We note that this mask lies close to the edge of the stream (see Figure 1) and conclude that, as it was targeted on the same basis as the other masks, there is either a lower density of stream stars at this location or there is a gap in the stream. This is consistent with findings by Carlberg et al. (2011) and Komiyama et al. (2018) who noted density variations along NW-K2. \nTable 1: Properties for the observed fields along NW-K2, including: mask name; date observations were made; observing PI; Right Ascension and Declination of the centre of the field, the projected radius of the mask centre from the centre of M31 and the number of confirmed members in the stellar populations for each field. The 𝛼 and 𝛿 for the centre of each mask are determined by taking the mean of the coordinates for all stars on the mask. The masks are listed in order of increasing distance from M31. ( 𝑎 ) are stars that probably belong to M31 or MW stellar populations, but as there were no candidate NW-K2 stars on this mask, these data were not analysed. ( 𝑏 ) are stars in the NW-K2 velocity range but do not lie on its RGB. \n<!-- image --> \n<!-- image --> \nFigure 1: On sky locations of the masks and GCs along NW-K2. The left hand panel shows the locations of all stars on each mask and the GCs plotted in stream coordinates. M31 is not shown in this panel as it lies well below the left hand corner, out of range of both the x and y-axes, with coordinates of (-5.13 -2.97). The middle panel shows NW-K1 (open cyan polygon) and NW-K2 (open orange polygon) super-imimposed over a stellar density map of M31. The right hand panel shows the locations of all stars on each mask and the GCs, plotted in tangent plane coordinates, along with other features, such as And XXVII. The solid blue line represents the M31 halo (taking a semi-major axis of 55 kpc with a flattening of 0.6, Ferguson & Mackey 2016). The dotted, magenta, lines outlining the inner and outer edges of an ellipse tracing the possible track of the NW Stream, assuming it to be a single feature (following the approach by Carlberg et al. 2011). The purple, diamond indicates the centre of And XXVII and the star icons represent the GC\'s, PAndAS-04, PAndAS-09, PAndAS-10, PAndAS-11, PAndAS-12, PAndAS-13 and PAndAS-15. The small circular icons represent observed stars, colour-coded to show their respective masks. \n<!-- image --> \nFor the remaining masks we note that there are only a small number of potential NW-K2 stars on each and decide to proceed by combining the data from all masks for further analysis. To enable us to obtain a consistent set of NW-K2 stars, determine any possible velocity gradient and create simple orbital models across the observed fields, we transform our data to an NW-K2 frame of reference. Using techniques described in, for example, Koposov et al. (2010, 2019, 2023) and Erkal et al. (2017, 2018, 2019) we convert the stellar coordinates from ( 𝛼 , 𝛿 ) to NW-K2 centric coordinates ( 𝜙 1 , \n𝜙 2 ) by rotating the celestial equator to a great circle where the pole, ( 𝛼 pole , 𝛿 pole ) = (-64.01, -21.05) with a zero point azimuthal angle, 𝜙 0 = 138.58 · , in the new coordinates. Our rotation matrix is shown at Appendix A. We also perform the same rotation on the co-located GCs. \nTo determine which stellar population (NW-K2, M31 or MW) a given star of velocity, 𝑣 i and velocity uncertainty of 𝑣 err , i is most likely to belong to, we define a single Gaussian function for each of \nTable 2: Properties for GCs, co-located on sky with NW-K2, from Veljanoski et al. (2014), Huxor et al. (2014) and Mackey et al. (2018), with metallicities from Sakari & Wallerstein (2022) . \nthem of the form: \n𝑃 struc = 1 √︃ 2 𝜋 ( 𝜎 2 𝑣, struc + 𝑣 2 err ,𝑖 + 𝜎 2 sys ) × exp " -1 2 GLYPH<18> 𝑣 𝑟, struc -𝑣 𝑟,𝑖 √︃ 𝜎 2 v , struc + 𝑣 2 err ,𝑖 + 𝜎 2 sys ) GLYPH<19> 2 # , (3.1) \nwhere: 𝑃 struc is the resulting probability distribution function (pdf); 𝑣 r , struc km s -1 is the systemic velocity of the structure (i.e. NWK2, M31 or MW); 𝜎 v , struc km s -1 is the velocity dispersion of the structure, 𝑣 r , i km s -1 is the velocity of each star on the masks, 𝑣 err , i km s -1 the associated velocity uncertainty and 𝜎 sys is a systematic uncertainty component of 2.2 km s -1 , determined by Simon & Geha (2007), Kalirai et al. (2010) and Tollerud et al. (2012) and applicable to our observations. The likelihood function for membership of NWK2, based on velocity, is then defined as: \nlog [L( 𝑣 𝑟 , 𝜎 𝑟 )] = 𝑁 ∑︁ 𝑖 = 1 log ( 𝜂 M31 𝑃 𝑖, M31 + 𝜂 MW 𝑃 𝑖, MW + 𝜂 K2 𝑃 𝑖, K2 ) , (3.2) \nwhere 𝜂 M31 , 𝜂 MW and 𝜂 K2 are the fraction of stars within each stellar population, v 𝑟 includes v 𝑟 K2 , v 𝑟 M31 and v 𝑟 MW and 𝜎 𝑟 includes 𝜎 𝑟 K2 , 𝜎 𝑟 M31 and 𝜎 𝑟 MW . \nWeincorporate the above equations, tailored for each stellar population into an MCMC analysis, using the /e.pc/m.pc/c.pc/e.pc/e.pc software algorithm, Goodman & Weare (2010), Foreman-Mackey et al. (2013). We base the initial values of the fraction parameters 𝜂 M31 , 𝜂 MW and 𝜂 K2 on the distribution of stars plotted on a velocity histogram for the combined data. We set the initial values of the systemic velocities to previously published values, with the initial values for the velocity dispersions based on the spread of velocities in each of the potential stellar populations. For M31 we know that the velocity dispersion changes with the on-sky projected distance, R, from the centre of the galaxy in accordance with Equation 3.3 from Chapman et al. (2006) and Mackey et al. (2013) (which we adopt for consistency with and comparison to our previous work in Preston et al. 2019, while recognising that there are more recent findings for the velocity dispersion across the M31 halo by Gilbert et al. 2018). So we calculate M31 velocity dispersions for each star based on the projected radius of their respective masks from the centre of M31and provide these values to our MCMC analysis. \n𝜎 𝑣 ( 𝑅 ) = GLYPH<18> 152 -0 . 9 𝑅 1 kpc GLYPH<19> kms -1 kpc -1 , (3.3) \nWe set broad priors for each stellar populations on the masks i.e. : \n- · systemic velocities are -600 ≤ 𝑣 K2 / km s -1 ≤ -350, -350 ≤ 𝑣 M31 / km s -1 ≤ -290 and -170 ≤ 𝑣 MW / km s -1 ≤ 0.\n- · velocity dispersions are 0 ≤ 𝜎 𝑣 K2 / km s -1 ≤ 50 and 0 ≤ 𝜎 𝑣 MW / km s -1 ≤ 150\n- · the fraction parameters are 0 ≤ 𝜂 ≤ 1 with 𝜂 K2 + 𝜂 M31 + 𝜂 MW = 1. \nWeset our Bayesian analysis to run for 100 walkers taking 100,000 steps and a burn-in of 50,000. We use the /e.pc/m.pc/c.pc/e.pc/e.pc algorithm to fit Gaussians and derive posterior distributions for the systemic velocity, velocity dispersion and fraction parameters for each stellar population. To ensure that the chains have converged, we check the autocorrelation time ( 𝜏 ) and find it to be in the range 50 < 𝜏 < 110. This indicates that the number of steps is well above the recommended 10 𝜏 (Hogg & Foreman-Mackey 2018) and sufficient to deliver a robust number of independent samples and well constrained parameters. We also review the posteriors, see example at Appendix B, to visualise the distribution and covariance of the various parameters. \nHaving obtained a Gaussian pdf for each of the three stellar populations, we derive the probabilities for each star belonging to a given population using: \n𝑃 vel = 𝑃 K2 𝑃 M31 + 𝑃 MW + 𝑃 K2 , (3.4) \nwith the probability of being a contaminant given by: \n𝑃 contam = 𝑃 M31 + 𝑃 MW 𝑃 M31 + 𝑃 MW + 𝑃 K2 , (3.5) \nWe plot a velocity histogram, Figure 2, which reveals three kinematically distinct stellar populations: stars likely to be members of the MW ( v 𝑟 = ∼ -80 km s -1 , Collins et al. 2013), stars likely to be members of the M31 halo (systemic velocity ∼ -300 km s -1 , Ibata et al. 2005) and stars likely to be members of NW-K2 ( ∼-600 ≤ v K2 km s -1 ∼≤ -400). Also on this plot, we see clear indication that the GC, PAndAS-13, (orange vertical line) is unlikely to be associated with NW-K2, which is consistent with findings by Veljanoski et al. (2014) and that PAndAS-15 (pink vertical line) and PAndAS-04 (red vertical line) are also on the periphery of association with the stream. \nWe then refine the pool of possible NW-K2 stars by examining \nFigure 2: Kinematic analysis of NW-K2 fields showing the velocity histograms fields overlaid with membership probability distribution function for each of the three stellar populations - shown in blue for the NW-K2, red for M31 and green for the MW. The plots also include the systemic velocities (the vertical dotted lines) for the globular clusters co-located with NW-K2. \n<!-- image --> \nFigure 3: CMD for NW-K2 candidate stars with an extinction and distance ( D ⊙ = 843 kpc) corrected array of isochrones aged 12 Gyrs, [ 𝛼 /Fe] = 0.0 and metallicities of -2.0 ≤ [Fe/H] ≤ -0.8 (moving from left to right across the plot) lying along the main spine of the RGB. The small grey dots show stars from the main PAndAS catalogue that lie within 15 arcmins of one of the masks(NWS6).Thestarsarecolourcodedbytheirstrengthofassociationwith their nearest isochrone. The dashed line indicates the limits of the bounding box. Stars outside the box are those with velocities similar to that of NW-K2 but which do not lie on the RGB and are, therefore, to be excluded from the NW-K2 stellar population. The red-dotted delineated box indicates the extent of the TRGB using a distance correction of (m -M) = 24.58 ± 0.19, which is the closest heliocentric distance reported for NW-K2 by Komiyama et al. (2018). \n<!-- image --> \ntheir proximity to the NW-K2 RGB. Following an approach by Ibata et al. (2007) and Gilbert et al. (2009), we overlay the NW-K2 RGB with an array of isochrones whose metallicities cover the range of metallicities for the GCs i.e. -2.0 ≤ [Fe/H] ≤ -0.8 and that lie along the spine of the RGB. We use the Dartmouth Stellar Evolution Database (Dotter et al. 2008) to generate isochrones appropriate for the CFHT-MegaCam ugriz filter, aged 12 Gyrs and with [ 𝛼 /Fe]= 0.0 to form our array. We use the heliocentric distance of NW-K2 (843 ± 77 kpc, Komiyama et al. 2018) for the distance correction of the isochrones, which we also correct for reddening using E(B-V) = 0.08 as interpolated from the extinction maps in Schlegel et al. (1998) by Richardson et al. (2011). We then plot the NW-K2 candidate stars (i.e. those with P vel ≥ 50%) onto the array of isochrones surrounded by a bounding box. Stars within the boundary of the box are very likely to be NW-K2 stars, but we cannot say definitively that they are. However, we can have confidence that those lying outside the bounding box, further away from the NW-K2 RGB, are very unlikely to be members of NW-K2. \nWe follow a technique used by Tollerud et al. (2012) to determine each star\'s probability of membership of NW-K2, and also to determine its photometric metallicity, based on its proximity to the nearest isochrone using: \n𝑃 iso = exp GLYPH<18> -Δ ( g -i ) 2 2 𝜎 c -Δ ( i ) 2 2 𝜎 m GLYPH<19> , (3.6) \nwhere Δ ( 𝑔 -𝑖 ) and Δ ( 𝑖 ) are distances from the isochrone nearest to the star, 𝜎 𝑐 is a free parameter that takes into account the range of colours of the stars on the CMD and 𝜎 𝑚 is a free parameter addressing distance and photometric errors. As this technique was used by Preston et al. (2019), we adopt their values of 𝜎 𝑐 = 0.15 and 𝜎 𝑚 = 0.45 as our initial values and find that they deliver the appropriate results i.e. that stars lying well away from the NW-K2 RGB have a low probability of association with the isochrones. \nWe note that star number 30 from mask 505HaS lies close to the top of the RGB just above the upper limit of our bounding box. To see if this star could be a member of NW-K2, we re-run the isochrone analysis using using a distance modulus of 24.58 ± 0.19, this being the nearest heliocentric distance for NW-K2 reported by Komiyama et al. (2018). We denote the new location of the top of the bounding box with red dotted lines (see Figure 3) and see that star number 30 is now included as a possible member of NW-K2. \nOur results also indicate that there are 5 candidate stars to be excluded from further analysis. These include: \n- · star number 74 on mask 505HaS. As it lies close to the top of the bounding box it could be a very bright stream star. The mask lies close to the GCs PAndAS-09 and PAndAS-10 but this star, with (g-i) 0 ∼ 2.4, is too red to be a member of either cluster where (g-i) 0 ∼ 0.62 and 0.75 respectively, Huxor et al. (2014). So it is more likely that this is an M31 halo star.\n- · star number 9 on mask NWS3. This star lies further away from the NW-K2 RGB so is unlikely to be a stream member. The mask lies close to PAndAS-11 but, again, with (g-i) 0 ∼ 2.8 the star is too red to be a member of this globular cluster where (g-i) 0 ∼ 0.67, Huxor et al. (2014). So it, too, is more likely to be an M31 halo star.\n- · star numbers 86, 44 and 3 on masks 506HaS, 606HaS NWS1, respectively, could be M31 halo stars. Integrating under the M31 gaussian we obtain an expectation ∼ 5-6 stars in the velocity range of all of our excluded stars (i.e. between ∼ -480 km s -1 and ∼ -400 km s -1 ) so it is not implausible for all of them to belong to the M31 halo. It is entirely possible that they were acquired inadvertently by the selection function that targeted RGB stars using colour selection \nFigure 4: The left hand plot shows the trajectory of NW-K2, in stream coordinates, created using a leapfrog integrator to generate orbits backward and forward from the stream\'s progenitor, which we assume to be the GC PAndAS-12. The right hand plot shows the radial velocity of the stream relative to M31 together with the velocity gradient = 16 + 3 . 2 -3 . 3 km s -1 degree -1 across the GCs (magenta line). The velocity gradient along the stream through the observed fields is found to be -1.2 + 1 . 9 -1 . 8 km s -1 degree -1 while the velocity gradient along the section of the orbit traversing these fields is determined to be 4.7 ± 0.004 km s -1 degree -1 . In both plots we show the data for 100 random orbits (red lines) and the best fit orbit (black line) of the NW-K2 stream. \n<!-- image --> \nFigure 5: Velocity gradient across the NW-K2 observed fields with respect to the centre of M31. The solid black line shows the velocity gradient determined by our MCMC analysis across the observed fields. It has a slope of -1.2 + 1 . 9 -1 . 8 km s -1 degree -1 that is consistent with zero at 1 𝜎 . The shaded area, bounded by dotted lines, indicates the velocity dispersion of the stream. The magenta lines indicate the velocity gradient and dispersion across the GCs. \n<!-- image --> \n\' \nboxes on the CMD, as described in McConnachie et al. (2008), Martin et al. (2009) and Richardson et al. (2011). \nWe re-run our Bayesian analysis, this time excluding the stars that do not lie on the NW-K2 RGB. We re-set our systemic velocity priors to -650 ≤ 𝑣 K2 / km s -1 ≤ -370, -450 ≤ 𝑣 M31 / km s -1 ≤ -290 and -170 ≤ 𝑣 MW / km s -1 ≤ 0. We retain the same values for the velocity dispersions and fraction parameters as described earlier. To ascertain if there is a velocity gradient across the stream, we use techniques described by Martin & Jin (2010) and Collins et al. (2017), and \namend Equation 3.1 to include a velocity gradient ( 𝑑𝑣 𝑑𝜙 1 ) as shown in 3.7. \n𝑃 struc = 1 √︃ 2 𝜋 ( 𝜎 2 𝑣, struc + 𝜎 2 sys ) × exp " -1 2 GLYPH<18> Δ 𝑣 𝑟, i √︃ 𝜎 2 𝑣, struc + 𝜎 2 sys ) GLYPH<19> 2 # , (3.7) \nTable 3: Results of the kinematic analysis of NW-K2 fields. The table shows the mean velocity and the number of confirmed NW-K2 stars for each mask. \nwhere Δ 𝑣 𝑟, i ( km s -1 ) is the velocity difference between the i 𝑡 ℎ star and a velocity gradient, 𝑑𝑣 𝑑𝜙 1 ( km s -1 degree -1 ) acting along the angular distance of the star\'s location on NW-K2, 𝜙 1 , given by: \nΔ 𝑣 𝑟, i = 𝑣 𝑟, i -⟨ 𝑣 𝑟 ⟩ + 𝑑𝑣 𝑑𝜙 1 𝜙 1 ! , (3.8) \nWe use the same number of walkers, steps and burn-in described previously and the same equations, 3.4 and 3.5, to determine probability of membership based on velocity. We then determine the overall probability each star\'s membership of NW-K2 by combining their probabilities of membership from the CMD and velocity analyses as follows: \n𝑃 K2 = 𝑃 vel × 𝑃 iso , (3.9) \nOur MCMC analysis returns a systemic velocity for NWK2 = -439.3 + 4 . 1 -3 . 8 km s -1 with a velocity dispersion = 16.4 + 5 . 6 -3 . 8 km s -1 . It also identifies an insignificant velocity gradient of -1.2 + 1 . 9 -1 . 8 km s -1 degree -1 that is consistent with zero at 1 𝜎 . This is not consistent with findings by Veljanoski et al. (2014) who detected a velocity gradient along the stream, based on the properties of the GCs that they associated with it, of 1.0 ± 0.1 km s -1 kpc -1 , which equates to 14.4 ± 1.4 km s -1 degree -1 in the same units as our gradient. \nTo explore this further, we plot the relative velocities of points along a model orbit of the NW-K2 stream, see Figure 4. The model orbits are generated following an approach described by Erkal et al. (2018, 2019) that converts the data, 𝛼 , 𝛿 and velocity, to galactocentric values and then uses a leapfrog integrator to generate orbits backward and forward from the stream\'s progenitor, which, following the approach by Kirihara et al. (2017b), we assume to be the GC PAndAS-12. The coordinates along the model orbits are then converted to stream-centric values. The potential for M31 is modelled, as described by Preston et al (in preparation), using parameters reported by Fardal et al. (2012, 2013). \nWe see a similar disparity between the slope of the velocity gradient of this orbital model and that for the GCs. We plot the velocities of the stars in each stellar population as a function of position along the stream ( 𝜙 1 ) and show the results in Figure 5. We then separate the data back into the component masks to determine the number of confirmed stars in each stellar population; to calculate a mean velocity for each mask and to undertake our spectroscopic analysis. We present the results of our kinematic analysis in Table 3.', '3.2 Metallicities': 'We measure the spectroscopic metallicities of the stars in NW-K2 using the CaT lines between 8400Å-8700Å. As the relationship between equivalent width and [Fe/H] has been long established by Rutledge et al. (1997), Battaglia et al. (2008) and Starkenburg et al. (2010), we fit a Gaussian function to the three CaT lines to obtain estimates of their equivalent widths and, following the approach by Starkenburg et al. (2010) and Collins et al. (2013), substitute our derived equivalent widths into: \n[ Fe / H ] = 𝑎 + 𝑏𝑀 + 𝑐𝐸𝑊 + 𝑑𝐸𝑊 -1 . 5 + 𝑒𝐸𝑊𝑀, (3.10) \nwhere: 𝑎 , 𝑏 , 𝑐 , 𝑑 and 𝑒 are taken from the calibration to the JohnsonCousins M 𝐼 values and equal to -2.78, 0.193, 0.442, -0.834 and 0.0017 respectively; EW = 0.5EW 8498 + EW 8542 + 0.6EW 8662 and 𝑀 is the absolute magnitude of the star given by: \n𝑀 = 𝑖 -5 × log 10 ( D ⊙ ) + 5 , (3.11) \nwhere: 𝑖 is the 𝑖 -magnitude of the star and 𝐷 ⊙ is the heliocentric distance for the star, which, in keeping with the hypothesis that NWK1 and NW-K2 are parts of a single structure, we assume to be the heliocentric distance for And XXVII. The uncertainties on the equivalent widths are determined from the covariance matrix produced by the fitting process and these are combined in quadrature to yield the uncertainties on the metallicity. \nIn some cases not all of the CaT lines are well resolved so we examine each spectrum by eye to determine which CaT lines are clearest for use in the determination of the metallicity. In the cases where we see too much noise around a CaT peak or one that has been affected by skylines following the approach by Collins et al. (2013), we ignore those lines and determine the value for EW using the more reliable lines, e.g. where the first CaT line is affected we take EW = EW 8542 + EW 8662 , where only the second line is reliable we use EW=1.7EW 8542 and where the third line is affected we assume EW = 1.5EW 8498 + EW 8542 . \nHaving obtained metallicity values for the individual stars we then determine an average metallicity per mask and an average metallicity for NW-K2. We show these results in the right hand column of Table 4. \nAs the S/N < 3 for most of the NW-K2 stars, we note that using the individual spectra to determine metallicity values may not deliver robust results. As the spread of metallicities on the CMD is not that large we stack the spectra (following the approaches by Ibata et al. 2005, Chapman et al. 2005, 2007, Collins et al. 2010, 2011 and Preston et al. 2019) as combining them could provide a more accurate estimate of the mean metallicity for each mask. \nWe prepare the individual spectra following the approach described by Collins et al. (2013), by correcting for the velocity of the individual star, smoothing the spectrum and normalising the data using a median filter. We weight each spectrum by the S/N of its star and interpolate to return the spectrum to the original lambda scale while retaining the velocity corrected position of CaT lines. We derive a value for M for the co-added spectrum by finding the average of the sum of the absolute magnitudes of the stream stars, each weighted by their S/N. We simultaneously fit a Gaussian function to the CaT lines of the co-added spectra, as per the example shown at Figure 6, to obtain the equivalent widths and take the mean of the absolute magnitude values for the NW-K2 stars for use in the metallicity calculations described above. Finally, we derive a mean metallicity for the stream and present our results in left hand column of Table 4. \nFigure 6: Example of co-added spectra for mask NWS5. The normalised spectrum is overlaid with a best fit curve (blue line). The vertical dotted lines indicate the positions of the CaT lines. This example is representative of the results for the other masks. \n<!-- image --> \nTable 4: Metallicities obtained from co-added spectra weighted by S/N for stars on each mask and an overall mean metallicity for the stream (left hand column) and from the average of the individual metallicities on the mask (right hand column).', '4 DISCUSSION': 'Our kinematic and spectroscopic analyses have confirmed a secure stellar population of 40 RGB stars for the NW-K2 segment of the NW stream. We present our results in Table 5, with details of the properties of all the observed stars on the masks provided on-line in Appendix C.', '4.1 Kinematics': 'We find NW-K2 to have a systemic velocity of -439.3 + 4 . 1 -3 . 8 km s -1 with a velocity dispersion of 16.4 + 5 . 6 -3 . 8 km s -1 . This is in keeping with the progenitor of NW-K2 being a dwarf galaxy and is consistent with other streams thought to have dwarf galaxy progenitors, see Table 6. With a current working assumption that the NW stream \nis a single structure, we also note these results are consistent with findings for NW-K1, Preston et al. (2019). \nWe find plausible associations of the GCs PAndAS-04, PAndAS09, PAndAS-10, PAndAS-11 and PAndAS-12 with NW-K2, which is in-keeping with findings by Mackey et al. (2010, 2018), Veljanoski et al. (2014) and Komiyama et al. (2018) who reported that most of the GCs in the M31 halo are closely associated with the underlying tidal streams. We also find that PAndAS-13 and PAndAS-15 (v 𝑟 = -570 ± 45 km s -1 and v 𝑟 = -385 ± 6 km s -1 respectively) are unlikely to be associated with NW-K2 despite their very clear co-locations. \nWeobtain a velocity gradient of -1.2 + 1 . 9 -1 . 8 km s -1 degree -1 , that is consistent with zero at 1 𝜎 , along the stream with velocities becoming increasingly negative in the direction of M31. This is not consistent with findings from Veljanoski et al. (2014) who detected a stronger velocity gradient (across the GCs they associated with NW-K2) of 1.0 ± 0.1 km s -1 kpc -1 (which equates to 14.4 ± 1.4 km s -1 degree -1 in the same units as our gradient) which they believed indicated that the stream progenitor was on an infall trajectory towards M31. \nTo explore the disparity in the velocity gradients, we fit the gradient model to only the GCs. From this we obtain a systemic velocity of -452 + 6 . 6 -6 . 4 km s -1 and a velocity dispersion of 7.1 + 11 . 1 -5 . 1 km s -1 which is indicative of a very cold system with the progenitor most likely to be a globular cluster (e.g: 300S, 𝜎 𝑣 ∼ 2.5 km s -1 and Ophicus, 𝜎 𝑣 ∼ 2.4 km s -1 , Li et al. 2022). However, to have created a stellar stream containing, at least, 5 GCs, any progenitor of NW-K2 would have to be a significantly larger object with a much larger velocity dispersion and is, therefore, more likely to be a dwarf galaxy such as the dSph galaxies Sagittarius and Fornax, which have 𝜎 𝑣 ∼ 11.4 km s -1 and ∼ 11.8 km s -1 and host 8 and 5 GCs respectively, or NGC185 where 𝜎 𝑣 ∼ 24 km s -1 and which hosts 8 GCs, Forbes et al. (2018). As a result, it is unlikely that the velocity gradient of 16 + 3 . 2 -3 . 3 km s -1 degree -1 that we find from this analysis (and which is consistent with that obtained by Veljanoski et al. 2014) is representative of the velocity gradient of NW-K2. \nHowever, it is surprising that a stream of such scale has virtually no discernible velocity gradient, so we review our results, focusing on the data set for mask NWS6. This mask has a number of NWK2 candidate stars clustered around the overall systemic velocity for the stream, but it also has 4 outliers with velocities ∼ 500 km s -1 (see Table 7), which is significant given the total number of stars in our sample. Given the proximity of this mask to the M31 halo it is possible that these outliers are the "real" stream stars while the other candidate NW-K2 stars belong to the M31 halo. These outliers have metallicities of -1.9 ≤ [Fe/H] ≤ -1.2 that are consistent with those of confirmed NW-K2 stars on other masks and they also lie on the NW-K2 RGB so it is possible that they are stream stars. However, our data analysis consistently identified them as non-NW-K2 stars. The only way to force them to be associated with the stream was to preferentially select them by tightening boundary conditions and defining rather than fitting data parameters (such as the percentage of stream stars in a given data set). This did increase the systemic velocity on this mask, which in turn increased the velocity gradient to ∼ 5 km s -1 degree -1 , which is still much shallower than that for the GCs. While intriguing, this is not a robust result since it requires artificially tight priors. For our results, we instead use broad priors for all masks. Figure 5 shows the results of this analysis and indicates the stellar populations for NW-K2, M31 and the MW. The leftmost line of the plot shows the results for NWS6 with the small group of outliers (in the bottom left hand corner) ostensibly colour coded as M31 halo stars. \nSo what do these 4 stars represent? The possibilities include: \nTable 5: Table showing properties of NW-K2 candidate stars. The columns include: (1) Mask name/Star number; (2) Right Ascension in J2000; (3) Declination in J2000; (4) i -band magnitude; (5) g -band magnitude; (6) signal to noise ratio; (7) line of sight heliocentric velocity; (8) Metallicity. Photometric metallicity values, derived from proximity to a fiducial isochrone, are marked * where the spectrum was incomplete and ** where there was no spectrum for the star. All other values are spectroscopic metallicities; (9) Probability of membership of NW-K2 based on velocity, (10) Probability of membership of NW-K2 based on proximity to a fiducial isochrone and (11) Probability of membership of NW-K2 based on velocity and isochrone proximity. \nTable 6: Streams likely to have dwarf galaxy progenitors. Data for the table is sourced from (1) Preston et al. (2019), (2) Ibata et al. (1997), (3) Gilbert et al. (2009), Ibata et al. (2007), (4) Li et al. (2022) and (5) this work. \nTable 7: Properties of the outlying stars on field NWS6 including line of sight heliocentric velocity, v ( km s -1 ) and spectroscopic metallicity. Star number 7, which has an incomplete spectrum, shows the photometric metallicity derived from its proximity to a fiducial isochrone \nFigure 7: Metallicity distributions of M31, NW-K2 and the NWS6 outlier stars. \n<!-- image --> \n(i) Despite the fitting process excluding these outliers from NWK2, given their proximity to M31 and the consistency of their spectroscopic metallicities with those of NW-K2, see Figure 7, they could be high velocity stream stars and actually be members of that stellar population. \n(ii) Extrapolating from Figures 6 and 7 from Opitsch et al. (2018) and Figure 5 from Dey et al. (2023), none of which extend as far out as NWS6, it is possible that these outliers are part of M31\'s outer halo and could be members of the M31\'s thin or thick disk stellar \npopulations. Collins et al. (2011) produced a contour map of expected velocities of stars in circular orbits around M31 which, at a similar projected radius are lower than those of the outliers. Collins et al. also report a velocity dispersion of 50.8 ± 1.9 km s -1 and 35.7 ± 1.0 km s -1 for the thick disk and the thin disk respectively both of which of which are higher than the velocity dispersion across the outliers, where 𝜎 𝑣 ∼ 9 km s -1 . This velocity dispersion is also inconsistent with velocity dispersion profiles reported by Gilbert et al. (2018) at a similar projected radius in the M31 halo, making it unlikely that the outliers are members of this stellar population. \n(iii) TheycouldbeWesternShelfstars.SimulationsoftheWestern Shelf, considered to have formed during the merger of M31 with a progenitor that created the Giant Stellar Stream in the south west of M31\'s halo, show a broad diffused structure to its outer edge, see Fardal et al. (2012), Kirihara et al. (2017a) and Milošević et al. (2023), that could be intersecting NW-K2. However, Kirihara et al. (2017a) find a metallicity distribution for the Western Shelf of -0.8 ≤ [Fe/H] -0.5, which is more metal rich than the 4 outliers, and heliocentric velocities greater than those of the outliers at a similar projected radius, so it is unlikely that these outliers are members of the Western Shelf. \n(iv) Given their low velocity dispersion, they could be the cold component of a previously undiscovered substructure within the M31 halo. \nThis leaves us with an intriguing conundrum as to the nature of the 4 outliers, with the only way to determine what they really do represent being to obtain additional data, e.g. perhaps by extension of DESI coverage, as suggested by Dey et al. (2023), and/or by the Subaru Prime Focus Spectrograph which is in the final stages of development and scheduled to become a Subaru Strategic Programme from 2024 Takada et al. (2014), Tamura et al. (2022). \nWith no discernible gradient along NW-K2, it is unwise to postulate whether it and NW-K1 form a single structure or are separate streams. However, as these gradients are based on line of sight velocities we recognise that there could be stronger, undetected, velocity components acting along the streams in other directions and that could affect their overall directions of motion. Further modelling of the 3-d trajectories of both segments of the NW stream will be required ascertain the structure of the NW stream (Preston et al, in preparation).', '4.2 Metallicities': 'We determine -1.3 ± 0.1 ≤ <[Fe/H] spec > ≤ -1.2 ± 0.8 for the stream, with most stars in the NW-K2 stellar population having -1.8 <[Fe/H] spec < -0.9, which is consistent with work by Ibata et al. (2014). This is also consistent with findings of the metallicities of many Local Group dwarf galaxies McConnachie (2012) who used the mass-metallicity relationship to estimate that the stellar mass of the NW stream progenitor could be ∼ 10 6 -8 M ⊙ , indicative of it being a dwarf galaxy. However, we do not find a metallicity gradient along the stream which is somewhat surprising since most dwarf galaxies have been found to have a range of metallicities that become increasingly metal-poor with distance from their centres (Mercado et al. 2021) so we would expect to see a similar trend in their stellar streams. However, Mercado et al. also found flatter gradients in galaxies with younger stellar populations (0-5 Gyrs) which aligns with findings by Mackey et al. (2019) who classified the NW-K2 GCs in a sub-group of GCs accreted by M31 more recently (e.g. 2-3 billion years) than the other M31 GCs. Koleva et al. (2011) found flatter metallicity gradients in dwarf irregular and late-type spiral \ngalaxies, so it is possible that one of these galaxies, or a dwarf galaxy with a very small innate metallicity gradient, is the progenitor of NW-K2.', '5 CONCLUSIONS': "In this work we present the results of our kinematic and spectroscopic analyses of 40 RGB stars from 9 fields spanning the length of the NWK2 segment of the NW Stream. We have identified secure members of the NW-K2 stellar population based on significant similarities in velocities and strong ( ≥ 1 𝜎 ) association with a grid of fiducial isochrones. \nOur results do not conclusively indicate whether or not NW-K2 and NW-K1 are elements of the same, single, structure. If they are, then it is very likely that And XXVII is the progenitor of both of them. However, if they are separate streams then, while it is generally accepted that And XXVII is the progenitor of NW-K1, the progenitor of NW-K2 is, at present, undetected. Our findings indicate that this progenitor could be a dwarf irregular or late-type spiral galaxy, accreted by M31 within the last ∼ 5 Gyrs. Other possibilities are that the progenitor could be: \n- (i) a more massive system that collided with M31 on a radial orbit creating some of the other substructures (e.g. The East Cloud and South West Cloud) in the M31 halo. Such an event would have expelled significant volumes of stars and GCs out to large radii and could have created tangential streams, such as NW-K2. This would also provide a plausible explanation as to how this faint feature is associated with so many GCs, McConnachie et al. (2018). \n(ii) an ultra diffuse galaxy (UDG), some of which have been found to host significant GC system, van Dokkum et al. (2017), Amorisco et al. (2018), Lim et al. (2018), Forbes et al. (2020) and Carleton et al. (2021). Indeed Forbes et al., in their analysis of UDGs in the Coma system, postulate that UDGs with large GC abundances may have formed their GCs and fields stars early on and very quickly, giving rise to metal-poor stellar populations. Failure of these UDGs to undergo subsequent star formation would yield the high specific frequencies (i.e. the number of GCs per unit galaxy luminosity) associated with these galaxies. \nNW-K2 stretches out 100 kpc, and possibly beyond, to the north west of M31's halo. It is indeed an intriguing feature that could yet provide more insights into its own formation and that of M31. Further detections of the stream, both to the southeast and northwest of M31, might locate its progenitor and yield the opportunity to better constrain the shape and size of the M31 potential.", '6 ACKNOWLEDGEMENTS': "The authors wish to thank the anonymous reviewer for their insightful comments and advice. JP wishes to thank Stuart Sullivan, Joan Sullivan and Barry Sullivan for their inspiration. JP also wishes to thank Mark Fardal and Dougal Mackey for their generous donations of data and private communications. \nThis work used the community-developed software packages: Matplotlib (Hunter 2007), NumPy (van der Walt et al. 2011) and Astropy (The Astropy Collaboration, et al. 2013, 2018, 2022). \nMost of the observed data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. \nThe Observatory was made possible by the generous financial support of the W.M. Keck Foundation. Data were also used from observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The authors wish to recognise and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.", '7 DATA AVAILABILITY': 'The data used in this paper are available herein, in Preston et al. (2019) and in their associated on-line supplementary materials. The raw DEIMOS data are available via the Keck archive.', 'REFERENCES': "- Amorisco N. C., Monachesi A., Agnello A., White S. D. M., 2018, Mon. Not. R. Astron. Soc., 4251, 4235\n- Battaglia G., Irwin M., Tolstoy E., Hill V., Helmi A., Letarte B., Jablonka P., 2008, Mon. Not. R. Astron. Soc., 383, 183\n- Beasley M. A., Fahrion K., Gvozdenko A., 2023, Mon. Not. R. Astron. Soc., 9, 1\n- Carlberg R. G., et al., 2011, Astrophys. J., 731, 124\n- Carleton T., Guo Y., Munshi F., Tremmel M., Wright A., 2021, Mon. Not. R. Astron. Soc., 502, 398\n- Chapman S. C., Ibata R., Lewis G. F., Ferguson A. M. N., Irwin M., McConnachie A., Tanvir N., 2005, Astrophys. J., 632, L87\n- Chapman S. C., Ibata R., Lewis G. F., Ferguson A. M. N., Irwin M., McConnachie A., Tanvir N., 2006, Astrophys. J., 653, 255\n- Chapman S. C., et al., 2007, Astrophys. J., 662, L79\n- Collins M. L. M., et al., 2010, Mon. Not. R. Astron. Soc., 407, 2411\n- Collins M. L. M., et al., 2011, Mon. Not. R. Astron. Soc., 413, 1548\n- Collins M. L. M., et al., 2013, Astrophys. J., 768, 172\n- Collins M. L. M., Tollerud E. J., Sand D. J., Bonaca A., Willman B., Strader J., 2017, Mon. Not. R. Astron. Soc., 467, 573\n- Dey A., Najita J. R., Koposov S. E., Josephy-Zack J., Maxemin G., Bell E. F., Poppett C., Patel E., 2023, The Astrophysical Journal, 944\n- Dotter A., Chaboyer B., Jevremović D., Kostov V., Baron E., Ferguson J. W., 2008, Astrophys. J. Suppl. Ser., 178, 89\n- Erkal D., Koposov S. E., Belokurov V., 2017, Mon. Not. R. Astron. Soc., 470, 60\n- Erkal D., Li T. S., Koposov S. E., Belokurov V., Balbinot E., 2018, Mon. Not. R. Astron. Soc., 481, 3148\n- Erkal D., et al., 2019, Mon. Not. R. Astron. Soc., 487, 2685\n- Fardal M. A., et al., 2012, Mon. Not. R. Astron. Soc., 423, 3134\n- Fardal M. A., et al., 2013, Mon. Not. R. Astron. Soc., 434, 2779\n- Ferguson A. M. N., Mackey A. D., 2016, Astrophys. Sp. Sci. Libr., 420, 191 Forbes D. A., Read J. I., Gieles M., Collins M. L. M., 2018, Monthly Notices of the Royal Astronomical Society, 481, 5592\n- Forbes D. A., Alabi A., Romanowsky A. J., Brodie J. P., Arimoto N., Forbes D. A., 2020, Mon. Not. R. Astron. Soc., 10, 4874\n- Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, Publ. Astron. Soc. Pacific, 125, 306\n- Frenk C. S., White S. D. M., 2012, Ann. Phys., 524, 507\n- Geehan J. J., Fardal M. A., Babul A., Guhathakurta P., 2006, Mon. Not. R. Astron. Soc., 366, 996\n- Gilbert K. M., et al., 2009, Astrophys. J., 705, 1275\n- Gilbert K. M., et al., 2018, Astrophys. J., 825, 128\n- Goodman J., Weare J., 2010, Commun. Appl. Math. Comput. Sci., 5, 65 \n-0 . 0548755604 \n-0 . 8734370902 \n-0 \n. \n4448296300 \n-0 \n. \n1980763734 \n' \n› \n-0 . 4838350155 \n« \n+ 0 . 7469822445 \n+ \n0 \n. \n4559837762 \n‹ \n' › cos ( 𝛼 ) cos ( 𝛿 ) sin ( 𝛼 ) cos ( 𝛿 ) sin ( 𝛿 ) ' fi \n« \n‹", 'APPENDIX B: POSTERIORS FOR THE STELLAR POPULATIONS OF NW-K2, M31 AND MW': 'APPENDIX C: PROPERTIES OF OBSERVED STARS FROM THE MASKS \nAppendix C is available as a separate on-line document. \n0 \n. \n4941094279 \n-0 \n. \n8676661490 \n+', 'APPENDIX A: CELESTIAL COORDINATES ( 𝛼 , 𝛿 ) TO STREAM COORDINATES ( 𝜙 1 , 𝜙 2 ) TRANSFORMATION': "' › cos ( 𝜙 1 ) cos ( 𝜙 2 ) sin ( 𝜙 1 ) cos ( 𝜙 2 ) sin ( 𝜙 2 ) ' fi = \n« \n‹ \n' \nfi \n· \n(A1) \nFigure B1: Posteriors for the stellar populations of NW-K2, M31 and MW where: v 1 , v 2 and v 3 are the systemic velocities of each stellar population (NW-K2, M31 and MW respectively); s 1 and s 3 are the velocity dispersions for NW-K2 and MW respectively (NB: the velocity dispersion for M31 was not fitted as it varied with the distance of each star from the centre of M31); 𝜂 1 , 𝜂 2 and 𝜂 3 are the fraction of stars within the NW-K2, M31 and MW stellar populations respectively and 𝑑𝑣 𝑑𝜙 1 is the velocity gradient along the stream. In each of the diagonal posteriors we indicate the median value for each parameter together with the 1𝜎 uncertainties. \n<!-- image -->"}

2024arXiv240210190K

In this paper we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely we construct smooth oneparameter families of smooth spherically symmetric solutions to the EinsteinMaxwellVlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically these solutions can be understood as beams of gravitationally selfinteracting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion or undergoes gravitational collapse. At the critical value of the parameter an extremal ReissnerNordstrm black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.

2024-02-01T00:00:00Z

['2024arXiv240210190K', 'arXiv:2402.10190', '10.48550/arXiv.2402.10190']

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

Extremal black hole formation as a critical phenomenon

['EPRINT_HTML', 'EPRINT_PDF']

https://arxiv.org/pdf/2402.10190.pdf

{'Extremal black hole formation as a critical phenomenon': 'Christoph Kehle ∗1 and Ryan Unger †2 \n1 Institute for Theoretical Studies & Department of Mathematics, ETH Zürich, Clausiusstrasse 47, 8092 Zürich, Switzerland \n2 Department of Mathematics, Princeton University, Washington Road, Princeton NJ 08544, United States of America \nFebruary 15, 2024', 'Abstract': 'In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordström black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.', 'Contents': 'References \n91', '1 Introduction': "One of the most spectacular predictions of general relativity is the existence and formation of black holes. Solutions of the Einstein field equations, \nRic( g ) -1 2 R ( g ) g = 2 T , (1.1) \ncan undergo gravitational collapse to form a black hole dynamically, starting from regular, one-ended Cauchy data. Building on earlier work of Lemaître [Lem33], Oppenheimer and Snyder [OS39] produced the first mathematical example of gravitational collapse with the collapse of a homogeneous dust cloud. In contrast, for reasonable matter models, solutions with small initial data disperse without a black hole forming. It is a fundamental problem in classical general relativity to understand how these different classes of spacetimescollapsing and dispersing-fit together in the moduli space of solutions. The interface between collapse and dispersion is known as the black hole formation threshold and families of solutions crossing this threshold are said to exhibit critical collapse . Spacetimes lying on the threshold are called critical solutions . \nCritical collapse has been extensively studied numerically, starting with the influential work of Choptuik [Cho93] on the spherically symmetric Einstein-scalar field model, in a regime where the critical solutions are believed to be naked singularities. The Einstein-Vlasov system is believed to have static 'star-like' critical solutions [RRS98; OC02], but critical collapse involving naked singularities has so far not been observed. These numerical studies on critical collapse (see also the survey [GM07]) have yet to be made rigorous. \nAt first glance, the Reissner-Nordström family of metrics (indexed by the mass M > 0 and charge e ) appears to exhibit a type of critical behavior: the solution contains a black hole when | e | < M ( subextremal ) or | e | = M ( extremal ) and does not contain a black hole when | e | > M ( superextremal ). However, the ReissnerNordström black holes are eternal and arise from two-ended Cauchy data, while the superextremal variants contain an eternal 'naked singularity' that has historically caused much confusion. Moreover, it was long thought that extremal black holes could not form dynamically (a consideration closely related to the third law of black hole thermodynamics [BCH73; Isr86]). Were this true, it would seem to rule out extremal ReissnerNordström as the late-time behavior of any critical solution. In [KU22], the present authors disproved the third law in the Einstein-Maxwell-charged scalar field model and showed that an exactly extremal ReissnerNordström domain of outer communication can indeed form in gravitational collapse. \nIn this paper, we continue our investigation of extremal black hole formation by showing that extremal Reissner-Nordström does arise as a critical solution in gravitational collapse for the Einstein-Maxwell-Vlasov model, giving an example of a new phenomenon that we call extremal critical collapse . \n̸ \nTheorem 1. There exist extremal black holes on the threshold between collapsing and dispersing smooth configurations of charged matter. More precisely, for any mass M > 0 , fundamental charge e = 0 , and particle mass 0 ≤ m ≤ m 0 , where 0 < m 0 ≪ 1 depends only on M and e , there exists a smooth one-parameter family of smooth, spherically symmetric, one-ended asymptotically flat Cauchy data { Ψ λ } λ ∈ [0 , 1] for the EinsteinMaxwell-Vlasov system for particles of fundamental charge e and mass m , such that the maximal globally hyperbolic development of Ψ λ , denoted by D λ , has the following properties. \n- 1. D 0 is isometric to Minkowski space and there exists λ ∗ ∈ (0 , 1) such that for λ < λ ∗ , D λ is future causally geodesically complete and disperses towards Minkowski space. In particular, D λ does not contain a black hole or naked singularity. If λ < λ ∗ is sufficiently close to λ ∗ , then for sufficiently large advanced times and sufficiently small retarded times, D λ is isometric to an appropriate causal diamond in a superextremal Reissner-Nordström solution.\n- 2. D λ ∗ contains a nonempty black hole region BH . = M\\ J -( I + ) and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + . = ∂ ( BH ) , is isometric to that of an extremal Reissner-Nordström solution of mass M . The spacetime contains no trapped surfaces.\n- 3. For λ > λ ∗ , D λ contains a nonempty black hole region BH and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + , is isometric to that of a subextremal Reissner-Nordström solution. The spacetime contains an open set of trapped surfaces. \nIn addition, for every λ ∈ [0 , 1] , D λ is past causally geodesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time. \nFigure 1: Penrose diagrams of the one-parameter family {D λ } from Theorem 1 in the case of massive particles. The dark gray region depicts the physical space support of the Vlasov matter beam. The region of spacetime to the left of the beam is exactly Minkowski space and the region to the right of the beam is exactly Reissner-Nordström with the parameter ratio as depicted. In every case, the beam 'bounces' before it hits the center { r = 0 } due to the repulsive effects of angular momentum and the electromagnetic field. When λ < λ ∗ , the beam bounces before a black hole is formed. We note already that the beams actually have more structure than is depicted here in these 'zoomed out' pictures. See already Fig. 11. \n<!-- image --> \nIn the proof of Theorem 1, we construct one-parameter families of charged Vlasov beams coming in from past timelike infinity (if m > 0 , cf. Fig. 1) or from past null infinity (if m = 0 , cf. Fig. 2). In the dispersive case λ < λ ∗ , the area-radius r of the beam grows linearly in time as the matter expands towards the future. Moreover, the macroscopic observables of the Vlasov matter (the particle current N and energy momentum tensor T ) decay at the sharp t -3 rate in the massive case and at the sharp t -2 rate in the massless case (with faster decay for certain null components), see already (5.20)-(5.24) in Proposition 5.5. In fact, this same dispersive behavior happens in the past for every λ ∈ [0 , 1] . \nAs a direct consequence of Theorem 1, we obtain \nCorollary 1. The very 'black hole-ness' of an extremal black hole arising in gravitational collapse can be unstable: There exist one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-Vlasov system, leading to the formation of an extremal black hole, such that an arbitrarily small smooth perturbation of the data leads to a future causally geodesically complete, dispersive spacetime. \nThis is in stark contrast to the subextremal case, where formation of trapped surfaces behind the event horizon-and hence stable geodesic incompleteness [Pen65]-is expected. Despite this inherent instability of the critical solution, we expect extremal critical collapse itself to be a stable phenomenon: We conjecture that there exists a telelologically determined 'hypersurface' B crit in moduli space which consists of asymptotically extremal black holes, contains D λ ∗ , and locally delimits the boundary in moduli space between future complete and collapsing spacetimes. This 'codimension-one' property is expected to hold for other variants of critical collapse and will be discussed in detail in Section 1.4. \nWe further expect extremal critical collapse to be a more general phenomenon: we conjecture it to occur in the spherically symmetric Einstein-Maxwell-charged scalar field model and also for the Einstein vacuum equations, where extremal Kerr is the model critical solution. In this paper, we also prove (see already Theorem 4 in Section 4.3) that extremal critical collapse already occurs in the simpler-but singularbouncing charged null dust model , which was first introduced by Ori in [Ori91]. The proof of Theorem 1, which will be outlined in Section 5.1, can be viewed as a global-in-time desingularization of these extremal critical collapse families in dust. \nBesides the Einstein-Maxwell-Vlasov and bouncing charged null dust models, it turns out that the thin \nFigure 2: Penrose diagrams of the one-parameter family {D λ } from Theorem 1 in the massless case. In the ingoing (resp., outgoing) phases, the massless beams are entirely contained in slabs of finite advanced (resp., retarded) time. Therefore, for sufficiently early advanced times and sufficiently late retarded times, the solutions are vacuum and isometric to Minkowski space. \n<!-- image --> \ncharged shell model [Isr66; DI67] also exhibits extremal critical collapse: Prószyński observed in [Pró83] that if a thin charged shell is injected into Minkowski space (so the interior of the shell is always flat), the parameters can be continuously varied so that the exterior of the shell goes from forming a subextremal Reissner-Nordström black hole, to forming an extremal Reissner-Nordström black hole, to forming no black hole or naked singularity at all: the shell 'bounces' off to future timelike infinity. Because the thin shell model is quite singular (the energy-momentum tensor is merely a distribution and the metric can fail to be C 1 across the shell), it seems to have been discounted as a serious matter model. We refer to the discussion in [KU22, Section 1.4.1] in reference to the thin charged shell counterexample to the third law by Farrugia and Hajicek [FH79]. Theorem 1 can be viewed as a vindication of [Pró83], since our smooth Einstein-Maxwellmassive Vlasov solutions exhibit all of the qualitative features of Prószyński's dust shells. In particular, Fig. 10 below is strikingly similar to Fig. 3 in [Pró83].", '1.1 The Einstein-Maxwell-Vlasov system': 'In this paper, we consider the Einstein-Maxwell-Vlasov system, which models a distribution of collisionless, self-gravitating charged particles with mass m ≥ 0 and fundamental charge e ∈ R \\{ 0 } . The model consists of a quadruple ( M 4 , g, F, f ) , where ( M 4 , g ) is a spacetime, F is a closed 2-form representing the electromagnetic field strength, and f = f ( x, p ) , called the distribution function of the Vlasov matter, is a smooth nonnegative function defined on the mass shell \nP m . = { ( x, p ) ∈ T M : p is future-directed causal and g ( p, p ) = -m 2 } . \nThe equations of motion are \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (1.2) \n∇ µ F µν = -e N ν , (1.3) \nXf = 0 , (1.4) \nwhere T EM µν . = F µ α F να -1 4 g µν F αβ F αβ is the energy-momentum tensor of the electromagnetic field, N and T are the number current and energy-momentum tensor of the Vlasov matter, defined by \nN µ [ f ]( x ) . = ∫ P m x p µ f ( x, p ) dµ m x ( p ) , T µν [ f ]( x ) . = ∫ P m x p µ p ν f ( x, p ) dµ m x ( p ) , (1.5) \nand X ∈ Γ( TT M ) is the electromagnetic geodesic spray vector field, defined relative to canonical coordinates ( x µ , p µ ) on T M by \nX . = p µ ∂ ∂x µ -( Γ µ αβ p α p β -e F µ α p α ) ∂ ∂p µ . (1.6) \nFor the definition of the family of measures dµ m x on P m and a proof of the consistency of the system (1.2)(1.4), we refer to Section 3.1. \nThe integral curves of the vector field X consist of curves of the form s ↦→ ( γ ( s ) , p ( s )) ∈ T M , where p = dγ/ds and p satisfies the Lorentz force equation \nDp µ ds = e F µ ν p ν . \nWe refer to such curves γ as electromagnetic geodesics . The vector field X is tangent to P m for any m ≥ 0 , and the Vlasov equation (1.4) implies that f is conserved along electromagnetic geodesics. Since f ≥ 0 , N is a future-directed causal vector field on M and the model satisfies the dominant energy condition. \nWhen m > 0 , the system (1.2)-(1.4) is locally well-posed outside of symmetry, which can be seen as a special case of results in [BC73] or by applying the general methods of [Rin13]. Well-posedness when m = 0 is conditional and is a delicate issue that we will return to in Section 3.3. We emphasize at this point that Theorem 1 produces examples of extremal critical collapse for any sufficiently small positive particle mass, where well-posedness is unconditional and valid outside of spherical symmetry.', '1.2 The problem of critical collapse': "We would like to place Theorem 1 into the larger picture of critical collapse , the general study of the black hole formation threshold. In particular, we conjecture that our examples in Theorem 1 have a suitable codimension-one property as is expected to hold for other, so far only numerically observed, critical phenomena in gravitational collapse. \nIn order to discuss the general concept of critical collapse, it is very helpful to have a notion of 'phase space' or moduli space for initial data (or maximal Cauchy developments) for the Einstein equations. Consider, formally, the set M of one-ended asymptotically flat Cauchy data for the Einstein equations with a fixed matter model (or vacuum) and perhaps with an additional symmetry assumption. We will be intentionally vague about what regularity elements of M have, what decay conditions to impose, or what topology to endow M with. We will also not discuss gauge conditions, which could be viewed as taking specific quotients of M . These questions are related to several fundamental issues in general relativity, see for instance [Chr94; Chr99b; Chr02; DS18; LO19; Keh22; RSR23; Keh23; KM24; Sin24]. 1 Indeed, it seems likely that there is no single 'correct' definition-it is doubtful that a single moduli space will capture every interesting phenomenon. \nNevertheless, we will pretend in this section that a 'reasonable' definition of M exists. At the very least, M ought to consist of initial data possessing a well-posed initial value problem. For each element Ψ = (¯ g, ¯ k, . . . ) ∈ M (where ¯ g is a Riemannian metric on R 3 , ¯ k the induced second fundamental form, and . . . denotes possible matter fields), we have a unique maximal globally hyperbolic development D = ( M , g, . . . ) of Ψ , where M ∼ = R 4 [Fou52; CG69; Sbi16]. 2 We assume that ( M , g ) is asymptotically flat. In particular, we assume that we have a well-defined notion of future null infinity I + and past null infinity I -. \nRemark 1.1 . In the proof of Theorem 1, we define a 'naive moduli space' M ∞ consisting of all smooth solutions of the Einstein-Maxwell-Vlasov constraint equations on R 3 , equipped with the C ∞ loc topology, and with no identifications made. See already Definition 5.29. This topology is inadequate for addressing asymptotic stability questions but since our families are electrovacuum outside a fixed large compact set anyway, they will be continuous in any 'reasonable' topology that respects asymptotic flatness. \n̸ \nLet C ⊂ M denote the subset of initial data with future causally geodesically complete developments. We also highlight the special class D ⊂ C of initial data with dispersive developments, i.e, those solutions whose geometry asymptotically converges to Minkowski space in the far future and matter fields decay suitably. 3 Nontrivial stationary states, if they exist, lie in C \\ D since they do not decay. 4 Let B ⊂ M denote the set of initial data leading to the formation of a nonempty black hole region , i.e., BH . = M\\ J -( I + ) = ∅ . The question of critical collapse is concerned with the study of phase transitions between C , D , and B , that is, the structure of the boundaries ∂ C , ∂ D , and ∂ B , how they interact, and characterizing solutions lying on the threshold. \nA natural way of exploring this phase transition is by studying continuous paths of initial data interpolating between future complete and black hole forming solutions. \nDefinition 1.2. An interpolating family is a continuous one-parameter family { Ψ λ } λ ∈ [ -1 , 1] ⊂ M such that Ψ 0 ∈ C and Ψ 1 ∈ B . Given such a family, we may define the critical parameter λ ∗ and the critical solution D λ ∗ (the development of Ψ λ ∗ ) by \nλ ∗ . = sup { λ ∈ [0 , 1] : Ψ λ ∈ C } . \nThe prototypical critical collapse scenario consists of a spherically symmetric self-gravitating massless scalar field pulse with fixed profile and 'total energy' ∼ λ . At λ = 0 , the solution is Minkowski space and for λ very close to 0 , the solution disperses and is future complete [Chr86]. As λ approaches 1 , a trapped surface forms in evolution, signaling the formation of a black hole [Chr91]. This is precisely the scenario first studied numerically by Christodoulou in his thesis [Chr71] and then later by Choptuik in the influential work [Cho93]. Based on numerical evidence, it is believed that the critical solutions for these types of families are naked singularities that form a codimension-one 'submanifold' in moduli space. For discussion of Choptuik's results we refer to the survey [GM07]. \nRemark 1.3 . A codimension-one submanifold of naked singularities is nongeneric and therefore compatible with the weak cosmic censorship conjecture, which has been proved in this model by Christodoulou [Chr99b]. \nRemark 1.4 . A rigorous understanding of Choptuik's critical collapse scenario would in particular give a construction of naked singularities in the Einstein-scalar field system starting from smooth initial data, in contrast to Christodoulou's examples in [Chr94]. It already follows from work of Christodoulou [Chr91] that a critical solution cannot be a black hole in this model and from work of Luk and Oh that a critical solution cannot 'scatter in BV norm' [LO15]. This leaves the possibility of either a first singularity along the center not hidden behind an event horizon 5 or a solution in C \\ D which 'blows up at infinity.' Ruling out this latter case is an interesting open problem. \nWhen massive fields are introduced, such as in the spherically symmetric Einstein-massive Klein-Gordon or Einstein-massive Vlasov systems, then static 'star-like' critical solutions can be observed numerically [BCG97; RRS98; OC02; AR06; AAR21]. These static solutions are nonsingular and lie in C \\ D . It is interesting to note that while Einstein-Klein-Gordon also displays Choptuik-like naked singularity critical solutions, there is no numerical evidence for the existence of naked singularities in the Einstein-Vlasov system. We again refer to [GM07] for references and would also like to point out the new development [Bau+23] on numerical critical collapse in vacuum.", '1.3 Extremal critical collapse': "So far, all numerically observed critical solutions are believed to be either naked singularities or complete and nondispersive. It follows at once from Penrose's incompleteness theorem [Pen65] and Cauchy stability that a critical solution cannot contain a trapped surface. While a generic black hole is expected to contain trapped surfaces, 6 members of the extremal Kerr-Newman family do not. In view of this, we raise the question of whether extremal black holes can arise on the black hole formation threshold: \nDefinition 1.5. An interpolating family { Ψ λ } λ ∈ [0 , 1] exhibits extremal critical collapse if the critical solution D λ ∗ asymptotically settles down to an extremal black hole. \nOur main result, Theorem 1, proves that the Einstein-Maxwell-Vlasov system exhibits extremal critical collapse, with critical solution D λ ∗ exactly isometric to extremal Reissner-Nordström in the domain of outer communication at late advanced times. As shown by Prószyński [Pró83] and the present authors in Theorem 4, the fundamentally singular thin charged shell and charged null dust models, respectively, exhibit extremal critical collapse, also with extremal Reissner-Nordström as the critical solution. We expect this phenomenon to also occur in the spherically symmetric Einstein-Maxwell-charged scalar field system and even for the Einstein vacuum equations, where the critical solution is expected to be based on the extremal Kerr solution. Note that we only require the asymptotic geometry of the critical solution to be an extremal black hole in Definition 1.5, which is a much weaker condition than being exactly extremal as in Theorem 1. \nRemark 1.6 . Because black holes in the spherically symmetric Einstein-scalar field model always contain trapped surfaces [Chr91], this model does not exhibit extremal critical collapse. In particular, since the presence of a trapped surface in this model already implies completeness of null infinity and the existence of a black hole [Daf05b], B is open in the spherically symmetric Einstein-scalar field model . \nRemark 1.7 . It is not possible for a Kerr solution with nonzero angular momentum (i.e., not Schwarzschild) to appear as the asymptotic state in axisymmetric vacuum gravitational collapse. This is because the Komar angular momentum (16 π ) -1 ∫ S ⋆dZ ♭ , where Z is the axial Killing vector field, is independent of the sphere S , which is nullhomologous. Similarly, it is not possible for a Kerr-Newman solution with nonzero charge (i.e., not Kerr) to appear as the asymptotic state in gravitational collapse for the Einstein-Maxwell system. This is because the charge (4 π ) -1 ∫ S ⋆F is independent of the sphere S , which is nullhomologous. The presence of charged matter is essential in Theorem 1. \nRemark 1.8 (Stationary solutions and the extremal limit) . In the 1960s and '70s, it was suggested that astrophysical black holes could form through quasistationary accretion processes. In a landmark work, Bardeen and Wagoner [Bar70; BW71] numerically studied axisymmetric stationary states of the Einsteindust system (modeling accretion disks) and found that a 'black hole limit' was only possible in the 'extremal limit' of the dust configuration. 7 In this limit, the exterior metric of the disk converges, in a certain sense, to the metric of the domain of outer communication of extremal Kerr. \nHowever, the event horizon of a stationary black hole is necessarily a Killing horizon and therefore an exactly stationary black hole solution cannot admit a one-ended asymptotically flat Cauchy hypersurface. It follows that a sequence of one-ended stationary states cannot actually smoothly converge to a black hole spacetime up to and including the event horizon, and that the black hole threshold cannot be directly probed by studying limits of stationary states-black hole formation is a fundamentally dynamical process. \nNevertheless, there is a substantial body of numerical and heuristic literature exploring 'extremal black hole limits' of stationary solutions in dust models [NM95; Mei06; Mei+08; MH11; KLM11] and using Einstein-Yang-Mills-Higgs magnetic monopoles [LW99; LW00]; see also references therein. In particular, we refer the reader to [MH11] for a cogent explanation of the exact nature of the convergence of these stationary states to extremal Reissner-Nordström/Kerr exteriors and throats. It would be interesting to see if perturbing these 'near-extremal' non-black hole stationary states can provide another route to extremal critical collapse (and also perhaps to new examples of third law violating solutions), but this seems to be a difficult and fully dynamical problem. \nRemark 1.9 (Overcharging and overspinning) . Extremal critical collapse should not be confused with the attempt to overcharge or overspin a black hole, i.e., the attempt to destroy the event horizon and create a 'superextremal naked singularity' by throwing charged or spinning matter into a (near-)extremal black hole. The fear of forming such a naked singularity provided some impetus for the original formulation of the third law in [BCH73] 8 and many arguments for and against have appeared in the literature, see [Wal74; Hub99; \n8 With this in mind, the formulation of the third law in [BCH73] can be thought of as simply outright forbidding the formation of extremal black holes. The formulation in Israel's work [Isr86] is more refined and specifically refers to subextremal black holes 'becoming' extremal in a dynamical process. In any case, both formulations are false as shown in [KU22] and again in the present paper. \nJS09; SW17] and references therein. Overcharging has been definitively disproved in spherical symmetry for the class of 'weakly tame' matter models [Daf05b; Kom13], which includes the Einstein-Maxwell-Vlasov system considered in this paper. We expect overcharging and overspinning to be definitively disproved with a positive resolution of the black hole stability problem for extremal black holes, to be discussed in Section 1.4 below.", '1.4 Stability of extremal critical collapse': "Before discussing the stability of our interpolating families in Theorem 1, we must first address the expected notion of stability for the domain of outer communication of the extremal Reissner-Nordström solution. \nFirstly, since the asymptotic parameter ratio of the black hole is inherently unstable, we can at most expect a positive codimension stability statement for extremal Reissner-Nordström. This should be compared with the codimension-three nonlinear stability theorem of the Schwarzschild solution by Dafermos, Holzegel, Rodnianski, and Taylor [DHRT]: Only a codimension-three 'submanifold' of moduli space can be expected to asymptote to Schwarzschild, which has codimension three in the Kerr family (parametrized by the mass and specific angular momentum vector). In the case of Reissner-Nordström, the set of extremal solutions has codimension one in the full family. Indeed, any fixed parameter ratio subfamily of the Reissner-Nordström family has codimension one. See already Remark 1.12. \nSecondly, and far less trivially, the stability problem for extremal black holes is complicated by the absence of the celebrated redshift effect , which acts as a stabilizing mechanism for the event horizon of subextremal black holes. The event horizon of extremal Reissner-Nordström (and axisymmetric extremal black holes in general) suffers from a linear instability known as the Aretakis instability [Are11a; Are11b; Are15; Ape22], which causes ingoing translation invariant null derivatives of solutions to the linear wave equation to (generically) either not decay, or to blow up polynomially along the event horizon as v → ∞ . Weissenbacher has recently shown that a similar instability (non-decay of the first derivative of the energymomentum tensor) occurs for the linear massless Vlasov equation on extremal Reissner-Nordström [Wei23]. \nHowever, the Aretakis instability is weak and does not preclude asymptotic stability and decay away from the event horizon . Including the horizon, we expect a degenerate type of stability, with decay in directions tangent to it, and possible non-decay and growth transverse to it (so-called horizon hair ). This behavior has been shown rigorously for a semilinear model problem on a fixed background [Ang16; AAG20] and numerically for the coupled spherically symmetric nonlinear Einstein-Maxwell-(massless and neutral) scalar field system [MRT13]. \nTo further complicate matters, the massive and massless Vlasov equations behave fundamentally differently and we state two separate conjectures. In these statements, we consider characteristic data posed on a bifurcate null hypersurface C out ∪ C in , where C out is complete and C in penetrates the event horizon in the case of trivial data. Solutions of the linear massless Vlasov equation decay exponentially on subextremal Reissner-Nordström black holes [Big23; Wei23] and Velozo Ruiz has proved nonlinear asymptotic stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system [Vel23]. Based on this, [MRT13; Ang16; AAG20], and [DHRT, Conjecture IV.2], we make the \nConjecture 1. The extremal Reissner-Nordström solution is nonlinearly asymptotically stable to spherically symmetric perturbations in the Einstein-Maxwell-massless Vlasov model in the following sense: Given sufficiently small characteristic data posed on a bifurcate null hypersurface C out ∪ C in and lying on a 'codimensionone submanifold' M stab (which contains the trivial solution) of the moduli space of such initial data, the maximal Cauchy development contains a black hole which asymptotically settles down to the domain of outer communication of an extremal Reissner-Nordström solution, away from the event horizon H + . Moreover, along the horizon, the solution decays towards extremal Reissner-Nordström in tangential directions, with possibly growing 'Vlasov hair' transverse to the horizon. \nRemark 1.10 . There exist nontrivial spherically symmetric static solutions of the Einstein-massless Vlasov system containing a black hole which are isometric to a Schwarzschild solution in a neighborhood of the event horizon [And21]. However, these are not small perturbations of Schwarzschild (as the structure of trapping for null geodesics has to be significantly modified) and their existence is therefore consistent with [Vel23] and Conjecture 1. \nFigure 3: A cartoon depiction of the conjectured structure of a neighborhood of moduli space near an interpolating family { Ψ λ } from Theorem 1. We have suppressed infinitely many dimensions and emphasize the codimension-one property of the critical 'submanifold' B crit which consists of asymptotically extremal black holes in accordance with Conjectures 1 and 2. The interpolating family { Ψ ' λ } is a small perturbation of { Ψ λ } which also crosses B crit and exhibits extremal critical collapse. Locally, B is foliated by 'hypersurfaces' B ( r ) consisting of black hole spacetimes with asymptotic parameter ratio r close to 1 . \n<!-- image --> \nThe massive Vlasov equation admits many nontrivial stationary states on black hole backgrounds, which is an obstruction to decay and we do not expect a general asymptotic stability statement to hold, even in the subextremal case. In fact, it has been shown that there exist spherically symmetric static solutions of Einstein-massive Vlasov bifurcating off of Schwarzschild [Rei94; Jab21]. We refer to [Vel23] for a characterization of the 'largest' region of phase space on which one can expect decay for the massive Vlasov energy-momentum tensor on a Schwarzschild background. However, one might still hope for orbital stability of the exterior, with a non-decaying Vlasov matter atmosphere , and that the horizon itself decays to that of extremal Reissner-Nordström: \nConjecture 2. The extremal Reissner-Nordström solution is nonlinearly orbitally stable to spherically symmetric perturbations in the Einstein-Maxwell-massive Vlasov model in the following sense: Given sufficiently small characteristic data posed on a bifurcate null hypersurface C out ∪ C in and lying on a 'codimension-one submanifold' M stab of the moduli space of such initial data, the maximal Cauchy development contains a black hole which remains close to an extremal Reissner-Nordström solution in the domain of outer communication and asymptotically settles down to extremal Reissner-Nordström tangentially along the horizon, with possibly growing 'Vlasov hair' transverse to the horizon. \nRemark 1.11 . We emphasize that this type of nonlinear orbital stability for massive Vlasov has not yet been proven even in the subextremal case, where we do not expect horizon hair to occur. \nWith the conjectured description of the stability properties of the exterior of the critical solution at hand, we are now ready to state our conjecture for the global stability of the extremal critical collapse families in Theorem 1. Refer to Fig. 3 for a schematic depiction of this conjecture. \nConjecture 3. Extremal critical collapse is stable in the following sense: Consider the moduli space M of the spherically symmetric Einstein-Maxwell-Vlasov system for particles of mass m . Let { Ψ λ } be one of the interpolating families given by Theorem 1. Then there exists a 'codimension-one submanifold' B crit of M such that Ψ 0 ∈ B crit ⊂ B , which has the following properties: \n- 1. B crit is critical in the sense that B and C locally lie on opposite sides of B crit .\n- 2. If m = 0 and Ψ ∈ B crit , the domain of outer communication of the maximal Cauchy development of Ψ asymptotically settles down to an extremal Reissner-Nordström black hole as in Conjecture 1.\n- 3. If m > 0 and Ψ ∈ B crit , the domain of outer communication of the maximal Cauchy development of Ψ remains close to an extremal Reissner-Nordström black hole and the event horizon asymptotically settles down to an extremal Reissner-Nordström event horizon as in Conjecture 2. \nTherefore, any nearby interpolating family { Ψ ' λ } also intersects B crit and exhibits extremal critical collapse. \nRemark 1.12 . We further conjecture that given r ∈ [1 -ε, 1] for some ε > 0 , there exists a one-parameter family of disjoint 'codimension-one submanifolds' B ( r ) ⊂ B , varying 'continuously' in r , such that B (1) = B crit and if Ψ ∈ B ( r ) , then the maximal Cauchy development of Ψ contains a black hole which asymptotes to a Reissner-Nordström black hole with parameter ratio r = e f /M f , where M f is the final renormalized Hawking mass and e f is the final charge. One can then interpret equation (5.136) below as saying that the families { Ψ λ } in Theorem 1 intersect the foliation { B ( r ) } transversally , as depicted in Fig. 3. \nWhile one should think that B crit in Conjecture 3 corresponds to M stab in Conjectures 1 and 2, Part 1 of Conjecture 3 is also a highly nontrivial statement about the interiors of the black holes arising from B crit . In particular, by the incompleteness theorem, it would imply that there are no trapped surfaces in the maximal developments of any member of B crit ; see [GL19, Remark 1.8] and the following remark. \nRemark 1.13 . Using arguments from [LO19, Appendix A], one can show the following statement in the spherically symmetric Einstein-Maxwell-(neutral and massless) scalar field model: If the maximal Cauchy development of a partial Cauchy hypersurface 9 with ∂ u r < 0 contains a black hole with asymptotically extremal parameter ratio, then the development does not contain trapped symmetry spheres. The argument uses crucially the constancy of charge and absence of T uv in this model.", '1.5 Extremal critical collapse of a charged scalar field and in vacuum': "It is natural to conjecture the analog of Theorem 1 for a massless charged scalar field in spherical symmetry: \nConjecture 4. Extremal critical collapse occurs in the spherically symmetric Einstein-Maxwell-charged scalar field model and there exist critical solutions which are isometric to extremal Reissner-Nordström in the domain of outer communication after sufficiently large advanced time. \nIn [KU22], the present authors showed that a black hole with an extremal Reissner-Nordström domain of outer communication and containing no trapped surfaces can arise from regular one-ended Cauchy data in the spherically symmetric charged scalar field model (see Corollary 3 of [KU22]). The proof is based on a characteristic gluing argument, in which we glue a late ingoing cone in the interior of extremal ReissnerNordström to an ingoing cone in Minkowski space. The desired properties of the spacetime are obtained softly by Cauchy stability arguments. In particular, the method is inadequate to address whether the solution constructed in [KU22, Corollary 3] is critical. \nIt is also natural to conjecture the analog of Theorem 1 for the Einstein vacuum equations, \nRic( g ) = 0 , (1.7) \nwhere the role of extremal Reissner-Nordström is played by the rotating extremal Kerr solution. 10 \nConjecture 5. Extremal critical collapse occurs in vacuum gravitational collapse and there exist critical solutions which are isometric to extremal Kerr in the domain of outer communication after sufficiently large advanced time. \nIn [KU23], the present authors constructed examples of vacuum gravitational collapse which are isometric to Kerr black holes with prescribed mass M and specific angular momentum a , where M and a are any Kerr parameters satisfying 0 ≤ | a | /M ≤ a 0 for some small positive constant a 0 . Like in [KU22], the proof is by characteristic gluing and uses the perturbative and obstruction-free gluing results of AretakisCzimek-Rodnianski [ACR21] and Czimek-Rodnianski [CR22] as a black box. In particular, this provided a fundamentally new proof of Christodoulou's seminal theorem on black hole formation in vacuum [Chr09]. Our proof does not work for large values of a and whether extremal Kerr black holes can form in gravitational collapse remains open. \nRemark 1.14 . In [KU23], the Cauchy data (¯ g, ¯ k ) are constructed with regularity H 7 / 2 -loc × H 5 / 2 -loc , which is well above the threshold for classical existence and uniqueness for the Einstein vacuum equations [HKM76; \nFigure 4: Penrose diagram of a counterexample to the third law of black hole thermodynamics in the Einstein-Maxwell-Vlasov model from Theorem 2. The broken curve A ' is the outermost apparent horizon of the spacetime. This view is zoomed in on the Vlasov beam that charges up the subextremal black hole to extremality. We refer to Fig. 15 in Section 6.1 for diagrams of the entire spacetime. \n<!-- image --> \nPR07; Chr13]. This limited regularity is because the characteristic gluing results [ACR21; CR22] which we use as a black box are limited to C 2 regularity of transverse derivatives in the non-bifurcate case. Using the more recent spacelike gluing results of Mao-Oh-Tao [MOT23], it is possible to construct suitable Cauchy data in H s loc × H s -1 loc for any s . \nIf extremal critical collapse involving the Kerr solution does occur, then one may also ask about stability as in Section 1.4. In this case, the question hangs on the stability properties of extremal Kerr, which are more delicate than for extremal Reissner-Nordström. While extremal Kerr is mode-stable [TdC20], axisymmetric scalar perturbations have been shown to exhibit the same non-decay and growth hierarchy as general scalar perturbations of extremal Reissner-Nordström [Are12; Are15]. In light of the newly discovered azimuthal instabilities of extremal Kerr by Gajic [Gaj23], in which growth of scalar perturbations already occurs at first order of differentiability, the full (in)stability picture of extremal Kerr may be one of spectacular complexity!", '1.6 The third law of black hole thermodynamics and event horizon jumping at extremality': "The techniques used to prove Theorem 1 can also be immediately used to disprove the third law in the Einstein-Maxwell-Vlasov model, which complements our previous disproof in the Einstein-Maxwell-charged scalar field model [KU22]. The present method has the advantage of constructing counterexamples which are past causally geodesically complete, like the spacetimes in Theorem 1. \nTheorem 2. There exist smooth solutions of the Einstein-Maxwell-Vlasov system for either massless or massive particles that violate the third law of black hole thermodynamics: a subextremal Reissner-Nordström apparent horizon can evolve into an extremal Reissner-Nordström event horizon in finite advanced time due to the incidence of charged Vlasov matter. \nMore precisely, there exist smooth, spherically symmetric, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-Vlasov system for either massive or massless particles such that the maximal globally hyperbolic development D has the following properties. \n- 1. D contains a nonempty black hole region and for sufficiently large advanced times, the domain of outer communication, including the event horizon H + , is isometric to that of an extremal ReissnerNordström solution.\n- 2. D contains a causal diamond which is isometric to a causal diamond in a subextremal ReissnerNordström black hole, including an appropriate portion of the subextremal apparent horizon. This subextremal region contains an open set of trapped surfaces. \n- 3. The outermost apparent horizon A ' of D has at least two connected components. One component of A ' coincides in part with the subextremal apparent horizon and the last component (with respect to v ) coincides with the extremal event horizon.\n- 4. D is past causally goedesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time. \nRefer to Fig. 4 for a Penrose diagram of one of these solutions. Note the disconnectedness of the outermost apparent horizon A ' , which is necessary in third law violating spacetimes-see the discussion in Section 1.4.3 of [KU22]. It is striking that the Vlasov beams we construct in the proof of Theorem 2 do not even touch the subextremal apparent horizon, which should be compared with the hypothetical situation depicted in Fig. 1 of [Isr86]. As with Theorem 1, Theorem 2 is proved by desingularizing suitable bouncing charged null dust spacetimes which we construct in Section 4.4. \nIt is now very natural to ask if some critical behavior can be seen in the examples from Theorem 2. They are clearly not candidates for critical collapse because they contain trapped surfaces. Nevertheless, by tuning the final charge to mass ratio of the outermost beam in Theorem 2 (subextremal to superextremal as in Theorem 1), we construct one-parameter families of solutions satisfying the following \nTheorem 3. There exist smooth one-parameter families of smooth, spherically symmetric, one-ended asymptotically flat Cauchy data { Ψ λ } λ ∈ [ -1 , 1] for the Einstein-Maxwell-Vlasov system for either massive or massless particles with the following properties. Let D λ be a choice of maximal globally hyperbolic development 11 of Ψ λ for which the double null gauge ( u, v ) is continuously synchronized as a function of λ . (See already Assumption 6.1 and Remark 6.2 for the definition of continuous synchronization.) Then the following holds: \n̸ \n- 1. For λ = 0 , D λ contains a black hole whose domain of outer communication is isometric to that of a subextremal Reissner-Nordström black hole with mass M λ and charge | e λ | < M λ for sufficiently large advanced times.\n- 2. D 0 contains a black hole whose domain of outer communication is isometric to that of an extremal Reissner-Nordström black hole with mass M 0 and charge | e 0 | = M 0 for sufficiently large advanced times.\n- 3. The location of the event horizon is discontinuous as a function of λ : Let u λ, H + denote the retarded time coordinate of the event horizon H + λ of D λ with respect to the continuously synchronized gauge ( u, v ) . Then λ ↦→ u λ, H + is continuous from the left but discontinuous from the right, and \nlim λ → 0 + u λ, H + > lim λ → 0 -u λ, H + . (1.8) \n4. The functions λ ↦→ M λ and λ ↦→ e λ are continuous from the left but discontinuous from the right, and \nlim λ → 0 + M λ < lim λ → 0 -M λ , lim λ → 0 + | e λ | < lim λ → 0 -| e λ | , (1.9) \nlim λ → 0 + | e λ | M λ < lim λ → 0 -| e λ | M λ = 1 , lim λ → 0 + r λ, H + < lim λ → 0 -r λ, H + , where r λ, H + . = M λ + √ M 2 λ -e 2 λ . \n(1.10) \nIn addition, for every λ ∈ [ -1 , 1] , D λ is past causally geodesically complete, possesses complete null infinities I + and I -, and is isometric to Minkowski space near the center { r = 0 } for all time. \nFrom the perspective of the dynamical extremal black hole D 0 , an arbitrarily small perturbation to D λ with λ > 0 causes the event horizon to jump by a definite amount in u (i.e., not o (1) in λ ) and the parameter ratio to drop by a definite amount. The proof of Theorem 3 relies crucially on the absence of trapped surfaces in a double null neighborhood of the horizon in the solutions of Theorem 2, cf. Fig. 4. In the asymptotically subextremal case, trapped surfaces are expected to asymptote towards future timelike infinity i + . In this \ncase, we prove in Proposition 6.4 below that the location of the event horizon is continuous as a function of initial data, under very general assumptions in spherical symmetry. Therefore, (1.8) is a characteristic feature of extremal black hole formation. \nWe expect this 'local critical behavior' to be stable in the sense of Section 1.4 and to play a key role in the general stability problem for extremal black holes. \nRemark 1.15 . By a suitable modification of the characteristic gluing techniques in [KU22], Theorem 3 can be proved for the spherically symmetric Einstein-Maxwell-charged scalar field model, but past completeness of the solutions does not follow immediately from our methods. It is also natural to conjecture analogs for Theorem 3 in (electro)vacuum; see in particular [DHRT, Section IV.2]. 12", '1.7 Outline of the paper': "Section 2. We introduce basic definitions and properties of general Einstein-matter systems and electromagnetic fields in spherical symmetry. \nSection 3. We first recall the Einstein-Maxwell-Vlasov system in Section 3.1 and derive its spherically symmetric formulation in double null coordinates in Section 3.2. We also prove local well-posedness in Section 3.3 (the main iteration argument being deferred to Appendix A) and the generalized extension principle in Section 3.4. Finally, we define time-symmetric seed data sets and their developments in Section 3.5 which will play an important role in the paper. \nSection 4. Before turning to the proof of our main result Theorem 1 in Section 5, we show in Section 4 that a singular toy model-Ori's bouncing charged null dust model-exhibits extremal critical collapse. We first recall the definition of the model in Section 4.1. We then introduce a radial parametrization of bouncing charged null dust spacetimes in Section 4.2 in which we teleologically prescribe a regular, spacelike, totally geodesic bounce hypersurface. These spacetimes consist of an explicit ingoing charged Vaidya metric pasted along the radially parametrized bounce hypersurface to an outgoing charged Vaidya metric through a physically motivated surgery procedure. In Sections 4.3 and 4.4, we use the radial parametrization to construct new examples of bouncing charged null dust spacetimes. In Section 4.3, we show that Ori's model exhibits extremal critical collapse (Theorem 4) and in Section 4.4, we show that the third law of black hole thermodynamics is false in Ori's model (Theorem 5). We then discuss the fundamental flaws of Ori's model in Section 4.5: the ill-posedness across the bounce hypersurface, the singular nature of the solutions, and the ill-posedness near the center. In Section 4.6, we conclude Section 4 with the formal radial charged null dust system in double null gauge which will be important for the setup of our initial data in Section 5. \nSection 5. This section is devoted to the proof of our main result, Theorem 1. The proof relies crucially on a very specific teleological choice of Cauchy data which aims at desingularizing the dust examples of extremal critical collapse in Theorem 4, globally in time. In Section 5.1, we give a detailed guide to the proof of Theorem 1. In Section 5.2, we define the hierarchy of beam parameters and state the key ingredient in the proof of Theorem 1: the existence and global structure of outgoing charged Vlasov beams (Proposition 5.5). These beams arise from data posed on a Cauchy hypersurface which is analogous to the 'bounce hypersurface' associated to Ori's model in Section 4. In Section 5.3, we solve the constraint equations and prove estimates for the solution along the initial data hypersurface. Section 5.4 is devoted to estimates for the 'near region' establishing the bouncing character of our Vlasov beams. To overcome certain difficulties associated with low momenta, our construction features an 'auxiliary beam' which is treated in Section 5.5. Section 5.6 is concerned with the 'far region' and in Section 5.7 we prove the sharp dispersive estimates in the case of massive particles. In Section 5.8 we conclude the proof of Proposition 5.5. This proposition is then used to prove Theorem 1 in Section 5.9. Finally, in Section 5.10 we show that in a certain hydrodynamic limit of our parameters, the family of solutions constructed in Theorem 1 converge in a weak* sense to the family constructed in the charged null dust model in Section 4. This result rigorously justifies Ori's bouncing charged null dust construction from [Ori91]. \nSection 6. In this final section, we disprove the third law of black hole thermodynamics for the EinsteinMaxwell-Vlasov model (Theorem 2) in Section 6.1. Section 6.2 is concerned with the phenomenon of event horizon jumping at extremality. We first show in Proposition 6.4 that for a general class of (so-called weakly tame ) spherically symmetric Einstein-matter systems the retarded time coordinate of the event horizon is lower semicontinuous as a function of initial data. Secondly, we show by example (Theorem 3) that event horizon jumping can occur in the Einstein-Maxwell-Vlasov system for extremal horizons, which proves the sharpness of semicontinuity in Proposition 6.4.", 'Acknowledgments': 'The authors would like to express their gratitude to Mihalis Dafermos for many stimulating discussions. They would also like to thank Carsten Gundlach, István Kádár, Georgios Moschidis, Amos Ori, Frans Pretorius, Harvey Reall, Igor Rodnianski, Andrew Strominger, and Robert Wald for their interest, helpful discussions, and comments. C.K. acknowledges support by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. R.U. thanks the University of Cambridge and ETH Zürich for hospitality as this work was being carried out.', '2.1 Double null gauge': "Let ( M , g ) be a smooth, connected, time-oriented, four-dimensional Lorentzian manifold. We say that ( M , g ) is spherically symmetric with (possibly empty) center of symmetry Γ ⊂ M if M\\ Γ splits diffeomorphically as ˚ Q× S 2 with metric \ng = g Q + r 2 γ, \nwhere ( Q , g Q ) , Q = ˚ Q∪ Γ , is a (1+1)-dimensional Lorentzian spacetime with boundary Γ (possibly empty), γ . = dϑ 2 + sin 2 ϑdφ 2 is the round metric on the unit sphere, and r is a nonnegative function on Q which can be geometrically interpreted as the area-radius of the orbits of the isometric SO(3) action on ( M , g ) . In a mild abuse of notation, we denote by Γ both the center of symmetry in M and its projection to Q . Moreover, if Γ is non-empty, we assume that the SO(3) action fixes Γ and that Γ consists of one timelike geodesic along which r = 0 . We further assume that ( Q , g Q ) admits a global double-null coordinate system ( u, v ) such that the metric g takes the form \ng = -Ω 2 dudv + r 2 γ (2.1) \nfor a positive function Ω 2 . = -2 g Q ( ∂ u , ∂ v ) on Q and such that ∂ u and ∂ v are future-directed. Our conventions are so that u = t -r and v = t + r give a double null coordinate system on (3 + 1) -dimensional Minkowski space, with r = 1 2 ( v -u ) and Ω 2 ≡ 1 . We will also use the notation / g . = r 2 γ . The constant u and v curves are null in ( Q , g Q ) and correspond to null hypersurfaces 'upstairs' in the full spacetime ( M , g ) . We further assume that along the center Γ , the coordinate v is outgoing and u is ingoing, i.e., ∂ v r | Γ > 0 , ∂ u r | Γ < 0 . We will often refer interchangeably to ( M , g ) and the reduced spacetime ( Q , r, Ω 2 ) . \nRecall the Hawking mass m : M→ R , which is defined by \nm . = r 2 (1 -g ( ∇ r, ∇ r )) \nand can be viewed as a function on Q according to \nm = r 2 ( 1 + 4 ∂ u r∂ v r Ω 2 ) . (2.2) \nWe will frequently use the formula \nΩ 2 = 4( -∂ u r ) ∂ v r 1 -2 m r (2.3) \nto estimate Ω 2 when 1 -2 m r > 0 . \nThe isometric action of SO(3) on ( M , g ) extends to the tangent bundle T M as follows: Let ϱ : SO(3) → Diff( M ) be the representation of SO(3) given by the spherically symmetric ansatz, so that the group action is given by \nR · x . = ϱ ( R )( x ) \nfor R ∈ SO(3) and x ∈ M . For ( x, p ) ∈ T M , we define \nR · ( x, p ) . = ( ϱ ( R )( x ) , ϱ ( R ) ∗ p ) , (2.4) \nwhere of course ϱ ( R ) ∗ p lies in T ϱ ( R )( x ) M . \nFinally, we note that the double null coordinates ( u, v ) above are not uniquely defined and for any strictly increasing smooth functions U, V : R → R , we obtain new global double null coordinates (˜ u, ˜ v ) = ( U ( u ) , V ( v )) such that g = -˜ Ω 2 d ˜ ud ˜ v + / g , where ˜ Ω 2 (˜ u, ˜ v ) = ( U ' V ' ) -1 Ω 2 ( U -1 (˜ u ) , V -1 (˜ v )) and r (˜ u, ˜ v ) = r ( U -1 (˜ u ) , V -1 (˜ v )) .", '2.2 Canonical coordinates on the tangent bundle': "Given local coordinates ( ϑ 1 , ϑ 2 ) on a (proper open subset of) S 2 , the quadruple ( u, v, ϑ 1 , ϑ 2 ) defines a local coordinate system on the spherically symmetric spacetime ( M , g ) . Given p ∈ T x M , we may expand \np = p u ∂ u | x + p v ∂ v | x + p 1 ∂ ϑ 1 | x + p 2 ∂ ϑ 2 | x . \nThe octuple ( u, v, ϑ 1 , ϑ 2 , p u , p v , p 1 , p 2 ) defines a local coordinate system on T M , and is called a canonical coordinate system on T M dual to ( u, v, ϑ 1 , ϑ 2 ) . One is to think of p as the 'momentum coordinate' and x as the 'position coordinate.' The tangent bundle of Q trivializes globally as Q× R 2 , with coordinates p u and p v on the second factor. We let \nπ : T Q → Q \ndenote the canonical projection. \nOn a spherically symmetric spacetime, we define the angular momentum function by \nℓ : T M→ [0 , ∞ ) ( x, p ) ↦→ √ r 2 / g AB p A p B , \nwhere summation over A,B ∈ { 1 , 2 } is implied. This function is independent of the angular coordinate system chosen and is itself spherically symmetric as a function on T M . \nGiven a double null gauge ( u, v ) , it will be convenient to define a 'coordinate time' function \nτ . = 1 2 ( v + u ) . \nAssociated to this time function is the τ -momentum coordinate \np τ . = 1 2 ( p v + p u ) .", '2.3 The Einstein equations and helpful identities in double null gauge': "A tensor field on a spherically symmetric spacetime is said to be spherically symmetric if it is itself invariant under the SO(3) -action of the spacetime. If ( M , g ) is a spherically symmetric spacetime satisfying the Einstein equations (1.1), then the energy-momentum tensor T is a spherically symmetric, symmetric (0 , 2) -tensor field. We may decompose \nT = T uu du 2 + T uv ( du ⊗ dv + dv ⊗ du ) + T vv dv 2 + S / g, \nwhere \nS . = 1 2 / g AB T AB = 1 2 tr g T + 2 Ω 2 T uv . \nIt will be convenient to work with the contravariant energy momentum tensor, which takes the form \nT ♯♯ = T uu ∂ u ⊗ ∂ u + T uv ( ∂ u ⊗ ∂ v + ∂ v ⊗ ∂ u ) + T vv ∂ v ⊗ ∂ v + S / g -1 , \nwhere \nT uu = 1 4 Ω 4 T vv , T uv = 1 4 Ω 4 T uv , T vv = 1 4 Ω 4 T uu . \nThe Christoffel symbols involving null coordinates are given by \nΓ u uu = ∂ u log Ω 2 , Γ v vv = ∂ v log Ω 2 , Γ u AB = 2 ∂ v r Ω 2 r / g AB , Γ v AB = 2 ∂ u r Ω 2 r / g AB , Γ A Bu = ∂ u r δ A B , Γ A Bv = ∂ v r δ A B , \nr r \nand the totally spatial Christoffel symbols Γ A BC are the same as for γ in the coordinates ( ϑ 1 , ϑ 2 ) . \nFor a spherically symmetric metric g written in double null gauge (2.1), the Einstein equations (1.1) separate into the wave equations for the geometry, \n∂ u ∂ v r = -Ω 2 2 r 2 m + 1 4 r Ω 4 T uv , (2.5) \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -1 2 Ω 4 T uv -Ω 2 S , (2.6) \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (2.7) \n∂ v ( ∂ v r Ω 2 ) = -1 4 r Ω 2 T uu . (2.8) \nThe Hawking mass (2.2) satisfies the equations \n∂ u m = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) , (2.9) \n∂ v m = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) . (2.10) \nIf X is a spherically symmetric vector field, then \nX \n= \nX \n∂ \nu \n+ \nX \n∂ \nv \nand X satisfies div g X = 0 if and only if \n∂ u ( r 2 Ω 2 X u ) + ∂ v ( r 2 Ω 2 X v ) = 0 . (2.11) \nThe contracted Bianchi identity, \ndiv g T = 0 , \nwhich follows from the Einstein equations (1.1), implies the following pair of identities: \n∂ u ( r 2 Ω 4 T uu ) + ∂ v ( r 2 Ω 4 T uv ) = ∂ v log Ω 2 r 2 Ω 4 T uv -4 r∂ v r Ω 2 S , (2.12) \n∂ v ( r 2 Ω 4 T vv ) + ∂ u ( r 2 Ω 4 T uv ) = ∂ u log Ω 2 r 2 Ω 4 T uv -4 r∂ u r Ω 2 S . (2.13) \nIf α is a spherically symmetric two-form which annihilates TS 2 , then it may be written as \nα = -Ω 2 f 2 r 2 du ∧ dv, \nwhere f : Q → R is a smooth function. We then have \n∇ µ α uµ = ∂ u f r 2 , ∇ µ α vµ = -∂ v f r 2 . (2.14) \nu \nv \nand Raychaudhuri's equations", '2.4 Spherically symmetric electromagnetic fields': "We will additionally assume that our spherically symmetric spacetime ( M , g ) carries a spherically symmetric electromagnetic field with no magnetic charge. The electromagnetic field is represented by a closed two-form \nF , which takes the Coulomb form \nF = -Ω 2 Q 2 r 2 du ∧ dv, (2.15) \nfor a function Q : Q → R . The number Q ( u, v ) is the total electric charge enclosed in the ( u, v ) -symmetry sphere S u,v ⊂ M , which can be seen from the gauge-invariant formula \nQ ( u, v ) = 1 4 π ∫ S u,v ⋆F, (2.16) \nwhere ⋆ is the Hodge star operator and we orient M by du ∧ dv ∧ dϑ ∧ dφ . \nThe electromagnetic energy momentum tensor is defined by \nT EM µν . = F µ α F να -1 4 g µν F αβ F αβ , (2.17) \nand relative to a double null gauge is given by \nT EM = Ω 2 Q 2 4 r 4 ( du ⊗ dv + dv ⊗ du ) + Q 2 2 r 4 / g \nin spherical symmetry. If F satisfies Maxwell's equation \n∇ α F µα = J µ \nfor a charge current J , then the divergence of the electromagnetic energy momentum tensor satisfies \n∇ µ T EM µν = -F να J α . (2.18) \nIn spherical symmetry, Maxwell's equations read (see (2.14)) \n∂ u Q = -1 2 r 2 Ω 2 J v , ∂ v Q = 1 2 r 2 Ω 2 J u . \nFinally, we will utilize the renormalized Hawking mass \nϖ . = m + Q 2 2 r (2.19) \nto account for the contribution of the electromagnetic field to the Hawking mass m .", '2.5 The Lorentz force': 'We next briefly recall the Lorentz force law for the motion of a charged particle. Let ( M , g, F ) be a charged spacetime, where F is a closed 2-form representing the electromagnetic field. If γ is the worldline of a particle of mass m > 0 and charge e ∈ R , then it satisfies the Lorentz force equation \nm Du µ dτ = e F µ ν u ν , \nwhere τ is proper time and u . = dγ/dτ (so that g ( u, u ) = -1 ). Defining the momentum of γ by p . = m u and rescaling proper time to s = m -1 τ (so that p = dγ/ds ), we can rewrite the Lorentz force equation as \nDp µ ds = e F µ ν p ν . (2.20) \nThis equation, which we call the electromagnetic geodesic equation , makes sense for null curves as well, and can be taken as the equation of motion for all charged particles [Ori91]. \nRemark 2.1 . Because the Lorentz force equation (2.20) is not quadratic in p , s is not an affine parameter, which has fundamental repercussions for the dynamics of the electromagnetic geodesic flow. In particular, trajectories of the Lorentz force with parallel, but not equal, initial velocities will in general be distinct, even up to reparametrization. \nThe electromagnetic geodesic flow has two fundamental conserved quantities which will feature prominently in this work. \nLemma 2.2. Let γ : I → M be a causal electromagnetic geodesic in a spherically symmetric charged spacetime, where I ⊂ R is an interval. Let p = dγ/ds . Then the rest mass \nm [ γ ] . = √ -g γ ( p, p ) \nand the angular momentum \nℓ [ γ ] . = ℓ ( γ, p ) \nare conserved quantities along γ . \nProof. To show that m is constant, we compute \nd ds g ( p, p ) = 2 g ( Dp ds , p ) = 2 e F ( p, p ) = 0 , \nwhere the final equality follows from the antisymmetry of F . \nSince ℓ is independent of the coordinates chosen on S 2 , we may assume that ( ϑ 1 , ϑ 2 ) are normal coordinates on S 2 at the point ( γ 1 ( s 0 ) , γ 2 ( s 0 )) . To show that ℓ is constant, we then compute \nd ds r 4 γ AB p A p B ∣ ∣ ∣ ∣ s = s 0 = 4 r 3 dr ds γ AB ˙ γ A ˙ γ B -2 r 4 γ AB ( -2Γ A Cu ˙ γ u -2Γ A Cv ˙ γ v ) ˙ γ B ˙ γ C = 0 , \nwhere we used the formulas for the Christoffel symbols in spherical symmetry from Section 2.3. \nFor an electromagnetic geodesic γ ( s ) with angular momentum ℓ = ℓ [ γ ] and mass m = m [ γ ] , the Lorentz force law can be written as \nd ds p u = -∂ u log Ω 2 ( p u ) 2 -2 ∂ v r r Ω 2 ℓ 2 r 2 -e Q r 2 p u , (2.21) \nd ds p v = -∂ v log Ω 2 ( p v ) 2 -2 ∂ u r r Ω 2 ℓ 2 r 2 + e Q r 2 p v , (2.22) \nΩ 2 p u p v = ℓ 2 r 2 + m 2 , (2.23) \nwhere p u . = dγ u /ds , p v . = dγ v /ds , and the third equation, known as the mass shell relation , is directly equivalent to the definition of mass and angular momentum. In this work, it will not be necessary to follow the angular position of the electromagnetic geodesics in the (3 + 1) -dimensional spacetime. It is very convenient to rewrite (2.21) and (2.22) as \nd ds (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 -e Q r 2 (Ω 2 p u ) , (2.24) \nd ds (Ω 2 p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 + e Q r 2 (Ω 2 p v ) . (2.25)', '3 The Einstein-Maxwell-Vlasov system': 'In this section, we introduce the main matter model considered in this paper, the Einstein-Maxwell-Vlasov system. This model describes an ensemble of collisionless self-gravitating charged particles which are either massive or massless. We begin by defining the general system outside of symmetry in Section 3.1 and then specialize to the spherically symmetric case in Section 3.2. In Sections 3.3 and 3.4, we formulate the fundamental local theory for this model, local existence and a robust continuation criterion known as the generalized extension principle . Finally, in Section 3.5, we define a procedure for solving the constraint equations for the spherically symmetric Einstein-Maxwell-Vlasov system.', '3.1.1 The volume form on the mass shell': 'Let ( M 4 , g ) be a spacetime. For x ∈ M and m ≥ 0 , we define the (future) mass shell at x by \n̸ \nP m x . = { p ∈ T x M : p is future directed and g x ( p, p ) = -m 2 , p = 0 } \nand the associated smooth fiber bundle \nP m . = ⋃ x ∈M P m x , \nwith projection maps π m : P m →M . \nFix x ∈ M and let ( x µ ) be normal coordinates at x so that \ng x = -( dx 0 ) 2 +( dx 1 ) 2 +( dx 2 ) 2 +( dx 3 ) 2 . \nLet p µ be dual coordinates to x µ on T M , defined by p µ = dx µ ( p ) for p ∈ T x M . For m ≥ 0 , the restrictions of p 1 , p 2 , p 3 to P m x , denoted by ¯ p 1 , ¯ p 2 , ¯ p 3 , define a global coordinate system on P m x , with p 0 determined by \np 0 = √ m 2 + | ¯ p 1 | 2 + | ¯ p 2 | 2 + | ¯ p 3 | 2 . (3.1) \nDefinition 3.1. Let m ≥ 0 and x ∈ M . The canonical volume form µ m x ∈ Ω 3 ( P m x ) is defined by \nµ m x = ( p 0 ) -1 d ¯ p 1 ∧ d ¯ p 2 ∧ d ¯ p 3 , \nin normal coordinates at x , where p 0 is given by (3.1). \nOne can show that this form is independent of the choice of normal coordinates. When m > 0 , P m x is a spacelike hypersurface in T x M if it is equipped with the Lorentzian metric g x . In this case, µ m x = m -1 ω m x , where ω m x is the induced Riemannian volume form on P m x . The factor of m -1 is needed to account for the degeneration of ω m x as m → 0 , since P 0 x is a null hypersurface. For more information about the volume form on the mass shell, see [SW77; Rin13; SZ14]. The canonical volume form is uniquely characterized by the following property, which can be found in [SW77, Corollary 5.6.2]. \nLemma 3.2. The form µ m x defined above does not depend on the choice of local coordinates on M . Moreover, it is uniquely characterized by the following property. Let H ( p ) . = 1 2 g x ( p, p ) . If α is a 3-form in T x M defined along P m x such that \ndH ∧ α = √ -det g ( x ) dp 0 ∧ dp 1 ∧ dp 2 ∧ dp 3 , \nthen i ∗ m α = µ m x , where i m : P m x → T x M denotes the inclusion map. \nWe denote the integration measure associated to µ m x by dµ m x . A distribution function is a nonnegative function f ∈ C ∞ ( P m ) which decays sufficiently quickly on the fibers so that the relevant integrals are welldefined and are smooth functions of x . Given a distribution function f we may now define the number current N and the energy momentum tensor T of f by \nN µ ( x ) . = ∫ P m x p µ f ( x, p ) dµ m x , T µν ( x ) . = ∫ P m x p µ p ν f ( x, p ) dµ m x . (3.2) \nThese are readily verified to be tensor fields on M . Taking divergences, we have [Rin13, Appendix D] \n∇ µ N µ = ∫ P m x X 0 ( f ) dµ m x , ∇ µ T µν = ∫ P m x p ν X 0 ( f ) dµ m x , (3.3) \nwhere X 0 . = p µ ∂ x µ -Γ µ νρ p ν p ρ ∂ p µ ∈ Γ( TT M ) is the geodesic spray vector field.', '3.1.2 The equations': "Definition 3.3. The Einstein-Maxwell-Vlasov system for particles of mass m ∈ R ≥ 0 and fundamental charge e ∈ R \\ { 0 } consists of a charged spacetime ( M , g, F ) and a distribution function f : P m → [0 , ∞ ) satisfying the following equations: \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (3.4) \n∇ α F µα = e N µ , (3.5) \nXf = 0 , (3.6) \nwhere T EM is the electromagnetic energy momentum tensor defined in (2.17), T µν and N µ are the Vlasov energy-momentum tensor and number current, respectively, defined in (3.2), and \nX . = p µ ∂ ∂x µ -( Γ µ αβ p α p β -e F µ α p α ) ∂ ∂p µ ∈ Γ( TT M ) (3.7) \nis the electromagnetic geodesic spray vector field. \nThe vector field X is easily seen to be tangent to the mass shell P m , which means that the Vlasov equation (3.6) is indeed a transport equation on P m . The vector field F µ α p α ∂ p µ is itself tangent to P m and we have the integration by parts formulas \n∫ P m x F µ α p α ∂ p µ f dµ m x = 0 , ∫ P m x F µ α p ν p α ∂ p µ f dµ m x = -F ν α N α , \nwhich are easily verified in normal coordinates. Combined with (3.3) and the transport equation (3.6), we obtain the fundamental conservation laws \n∇ µ N µ = 0 , (3.8) \n∇ µ T µν = e N α F να . (3.9) \nWe now see that the Einstein-Maxwell-Vlasov system is consistent: (3.8) implies that Maxwell's equation (3.5) is consistent with antisymmetry of F and (3.9) implies (using also (2.18)) the contracted Bianchi identity \n∇ µ ( T EM µν + T µν ) = 0 . (3.10) \nfor the total energy-momentum tensor of the system.", '3.1.3 Relation to the relativistic Maxwell-Vlasov system': "The system (1.2)-(1.4) includes gravity and thus generalizes the special relativistic Maxwell-Vlasov system which is typically written in the form 13 (cf. [Gla96]) \n∂ t f K + ˆ v · ∂ x f K + e ( E + ˆ v × B ) · ∂ v f K = 0 , ∂ t E -∇× B = -j K , ∂ t B + ∇× E = 0 , ∇· E = ρ K , ∇· B = 0 , \nwhere f K ( t, x, v ) ≥ 0 is the distribution function, ( t, x, v ) ∈ R × R 3 × R 3 , E is the electric field, B is the magnetic field, ˆ v . = ( m 2 + | v | 2 ) -1 / 2 v is the 'relativistic velocity' and has modulus smaller than unity, and \nρ K ( t, x ) . = e ∫ R 3 f ( t, x, v ) dv, j K ( t, x ) . = e ∫ R 3 ˆ vf ( t, x, v ) dv. \nThis system is equivalent to the covariant equations (3.5) and (3.6) in Minkowski space under the identifications ( √ m 2 + | v | 2 , v ) = p , f K ( t, x, v ) = f ( t, x, √ m 2 + | v | 2 , v ) , E i = F i 0 , B i = 1 2 ε i jk F jk , ρ K = e N 0 , and ( j K ) i = e N i .", '3.2 Spherically symmetric definitions and equations': "Let ( Q , r, Ω 2 ) be the (1+1) -dimensional reduced spacetime associated to a spherically symmetric spacetime. For m ≥ 0 , we define the reduced mass shell by \nP m red . = { ( u, v, p u , p v ) ∈ T Q : Ω 2 ( u, v ) p u p v ≥ m 2 , p τ > 0 } , (3.11) \nwhere the second condition in the definition forbids Ω 2 p u p v = 0 in the m = 0 case. The angular momentum function from Section 2.2 can be defined on the reduced mass shell by \nℓ : P m red → R ≥ 0 ( u, v, p u , p v ) ↦→ r √ Ω 2 p u p v -m 2 . \nNote that ℓ > 0 on P 0 red . The definition of ℓ can be rewritten as the fundamental mass shell relation \nΩ 2 p u p v = ℓ 2 r 2 + m 2 . (3.12) \nDefinition 3.4. A spherically symmetric distribution function of massive (if m > 0 ) or massless (if m = 0 ) particles is a smooth function \nf : P m red → R ≥ 0 . \nWe say that f has locally compact support in p if for every compact set K ⊂ Q there exists a compact set K ' ⊂ R 2 such that spt( f ) ∩ P m red | K ⊂ K × K ' . We say that f has locally positive angular momentum if for every K ⊂ Q compact there exists a constant c K > 0 such that ℓ ≥ c K on spt( f ) ∩ P m red | K . \nIn order to define appropriate moments of a distribution function f on a spherically symmetric spacetime ( Q , r, Ω 2 ) , we require that f decays in the momentum variables p u and p v . For σ > 0 , k ≥ 0 an integer, and K ⊂ Q compact, we define the norm \n∥ f ∥ C k σ ( P m red | K ) . = ∑ 0 ≤ i 1 + i 2 ≤ k sup P m red | K ⟨ p τ ⟩ σ + i 2 | ∂ i 1 x ∂ i 2 p f | , (3.13) \nwhere ∂ i 1 x ∂ i 2 p f ranges over all expressions involving i 1 derivatives in the ( u, v ) -variables and i 2 derivatives in the ( p u , p v ) -variables. If the norm (3.13) is finite for all compact sets K ⊂ Q , we say that f ∈ C ∞ σ, loc ( P m red ) . If f has locally compact support in p , then it lies in C ∞ σ, loc ( P m red ) . \nRemark 3.5 . Our well posedness theory for the Einstein-Maxwell-Vlasov system requires that p -derivatives of f decay faster, which is the reason for the i 2 weight in (3.13). \nGiven a spherically symmetric spacetime ( Q , r, Ω 2 ) with distribution function f , we define the Vlasov number current by \nN u ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p u f ( u, v, p u , p v ) dp u dp v , (3.14) \nN v ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p v f ( u, v, p u , p v ) dp u dp v (3.15) \nand the Vlasov energy momentum tensor by \nT uu ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 ( p u ) 2 f ( u, v, p u , p v ) dp u dp v , (3.16) \nT uv ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 p u p v f ( u, v, p u , p v ) dp u dp v , (3.17) \nT vv ( u, v ) . = π Ω 2 ∫ Ω 2 p u p v ≥ m 2 ( p v ) 2 f ( u, v, p u , p v ) dp u dp v , (3.18) \nS ( u, v ) . = π 2 Ω 2 ∫ Ω 2 p u p v ≥ m 2 (Ω 2 p u p v -m 2 ) f ( u, v, p u , p v ) dp u dp v ≤ Ω 2 2 T uv ( u, v ) . (3.19) \nIf f ∈ C ∞ σ, loc ( P m red ) with σ > 4 , then these moments are well defined smooth functions of u and v . \nDefinition 3.6. The spherically symmetric Einstein-Maxwell-Vlasov model for particles of mass m ∈ R ≥ 0 and fundamental charge e ∈ R \\{ 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and a smooth distribution function f ∈ C ∞ σ, loc ( P m red ) for a decay rate σ > 4 fixed. When m = 0 , we require that f also has locally positive angular momentum. To emphasize that the distribution functions we consider in this paper have these regularity properties in p , we say that such a solution has admissible momentum . \nThe system satisfies the wave equations \n∂ u ∂ v r = -Ω 2 2 r 2 ( m -Q 2 2 r ) + 1 4 r Ω 4 T uv , (3.20) \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -Ω 2 Q 2 r 4 -1 2 Ω 4 T uv -Ω 2 S, (3.21) \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (3.22) \n∂ v ( ∂ v r Ω 2 ) = -1 4 r Ω 2 T uu , (3.23) \n∂ u Q = -1 2 e r 2 Ω 2 N v , (3.24) \n∂ v Q = + 1 2 e r 2 Ω 2 N u , (3.25) \nwhere N u , N v , T uu , T uv , T vv , and S are defined by equations (3.14)-(3.19). Finally, f satisfies the spherically symmetric Vlasov equation \nXf = 0 , (3.26) \nwhere X ∈ Γ( TP m red ) is the reduced electromagnetic geodesic spray \nX . = p u ∂ u + p v ∂ v -( ∂ u log Ω 2 ( p u ) 2 + 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u ) ∂ p u -( ∂ v log Ω 2 ( p v ) 2 + 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v ) ∂ p v . (3.27) \nRemark 3.7 . Since f ≥ 0 for a solution of the Einstein-Maxwell-Vlasov system, the components N u and N v of the number current are nonnegative. It follows from the Maxwell equations (3.24) and (3.25) that e Q is decreasing in u and increasing in v , unconditionally. This monotonicity property is a fundamental feature of the spherically symmetric Einstein-Maxwell-Vlasov system and will be exploited several times in this paper. \nRemark 3.8 . Both the electromagnetic energy-momentum tensor T EM and the Vlasov energy-momentum tensor T of the Einstein-Maxwell-Vlasov system satisfy the dominant energy condition. \nRemark 3.9 . As an abuse of notation, we have denoted the spray (3.7) on T M and the spray (3.27) on T Q by the same letter X . It will always be clear from the context which vector field we are referring to. They are related by the pushforward along the natural projection map P m → P m red . \nFor a solution of the Einstein-Maxwell-Vlasov system, the Hawking mass m satisfies the equations \n∂ u m = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) + Q 2 2 r 2 ∂ u r, (3.28) \n∂ v m = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) + Q 2 2 r 2 ∂ v r, (3.29) \nwhich can be derived from (2.9) and (2.10). The modified Hawking mass ϖ can then be seen to satisfy \n∂ u ϖ = 1 2 r 2 Ω 2 ( T uv ∂ u r -T vv ∂ v r ) -1 2 e r Ω 2 QN v , (3.30) \n∂ v ϖ = 1 2 r 2 Ω 2 ( T uv ∂ v r -T uu ∂ u r ) + 1 2 e r Ω 2 QN u . (3.31) \nthe Raychaudhuri equations \nand the Maxwell equations \nThe particle current N satisfies the conservation law \n∂ u ( r 2 Ω 2 N u ) + ∂ v ( r 2 Ω 2 N v ) = 0 (3.32) \nby (2.11) and (3.8). Alternatively, it can be directly derived from the spherically symmetric Vlasov equation, which we will do the proof of Proposition 3.14 in Appendix A. Finally, for the Einstein-Maxwell-Vlasov system, the Bianchi identities (2.12) and (2.13) read \n∂ u ( r 2 Ω 4 T uu ) + ∂ v ( r 2 Ω 4 T uv ) = ∂ v log Ω 2 r 2 Ω 4 T uv -4 r∂ v r Ω 2 S -e Ω 4 QN u , (3.33) \n∂ v ( r 2 Ω 4 T vv ) + ∂ u ( r 2 Ω 4 T uv ) = ∂ u log Ω 2 r 2 Ω 4 T uv -4 r∂ u r Ω 2 S + e Ω 4 QN v . (3.34) \nAgain, this follows either from (3.10) or directly from the spherically symmetric equations.", '3.2.1 The spherically symmetric reduction': 'Proposition 3.10. Let ( Q , r, Ω 2 , Q, f sph ) be a solution of the spherically symmetric Einstein-MaxwellVlasov system as defined by Definition 3.6. Then ( M , g, F, f ) solves the Einstein-Maxwell-Vlasov system if we lift the solution according to \nM \n. \n= \nQ× \nS \n2 \n, \ng . = -Ω 2 2 ( du ⊗ dv + dv ⊗ du ) + r 2 γ, (3.35) \nF . = -Q 2 r 2 du ∧ dv, (3.36) \nf ( u, v, ϑ 1 , ϑ 2 , p u , p v , p 1 , p 2 ) . = f sph ( u, v, p u , p v ) , (3.37) \nwhere ( ϑ 1 , ϑ 2 ) is a local coordinate system on S 2 . \nNote that f in (3.37) is SO(3) -invariant as a function on T M as defined in Section 2.1. \nProof. As the equations (3.20)-(3.25) are equivalent to the Einstein equations and Maxwell equations, we must only check that f , defined by (3.37), satisfies Xf = 0 , where X is given by (3.7), and that the spherically symmetric formulas (3.14)-(3.19) appropriately reconstruct the (3 + 1) -dimensional number current and energy-momentum tensor. \nLet γ ( s ) be an electromagnetic geodesic. Then Xf = 0 at ( γ ( s 0 ) , ˙ γ ( s 0 )) ∈ P m is equivalent to \nd ds ∣ ∣ ∣ ∣ s = s 0 f ( γ ( s ) , ˙ γ ( s )) = 0 . (3.38) \nUsing the chain rule \nd ds ∣ ∣ ∣ ∣ s = s 0 f sph ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) = p u ∂ u f sph + p v ∂ v f sph + dp u ds ∂ p u f sph + dp v ds ∂ p v f sph , \nthe spherically symmetric Lorentz force equations (2.21) and (2.22), the mass shell relation (2.23), and the spherically symmetric Vlasov equation (3.26), we see that (3.38) holds. \nTo compute the energy-momentum tensor in spherical symmetry, we use Lemma 3.2. Let x ∈ M and take ( ϑ 1 , ϑ 2 ) to be local coordinates on S 2 which are normal at the spherical component of x , so that γ AB = δ AB . We have \nTherefore, if we define \nthen \ndH = -Ω 2 2 p v dp u -Ω 2 2 p u dp v + r 2 p 1 dp 1 + r 2 p 2 dp 2 . \nα . = -r 2 ( p v ) -1 dp v ∧ dp 1 ∧ dp 2 , \ndH ∧ α = 1 2 Ω 2 r 2 dp u ∧ dp v ∧ dp 1 ∧ dp 2 . \nTherefore, by Lemma 3.2, \ndµ m x = r 2 ( p v ) -1 dp v dp 1 dp 2 \nas measures on (0 , ∞ ) × R 2 ( p 1 ,p 2 ) . The remaining momentum variable p u is obtained from p v , p 1 , and p 2 via the mass shell relation (3.12). If we set tan β = p 2 /p 1 and use that ℓ 2 = r 4 (( p 1 ) 2 +( p 2 ) 2 ) , then \ndµ m x = r -2 ( p v ) -1 dp v ℓ dℓ dβ. \nFor any weight function w = w ( p u , p v ) , we therefore have \n∫ (0 , ∞ ) × R 2 wf dµ m x = r -2 ∫ 2 π 0 ∫ ∞ 0 ∫ ∞ 0 w ( ℓ 2 + m 2 r 2 r 2 Ω 2 p v , p v ) f sph ( u, v, ℓ 2 + m 2 r 2 r 2 Ω 2 p v , p v ) dp v p v ℓ dℓ dβ. (3.39) \nIntegrating out β and applying the coordinate transformation ( p v , ℓ ) ↦→ ( p u , p v ) reproduces the formulas (3.14)-(3.19) for N and T . \nRemark 3.11 . Other works on the spherically symmetric Einstein-Vlasov system in double null gauge, such as [DR16; Mos18; Mos23], represent the distribution function f differently, opting to (at least implicitly) eliminate either p u or p v in terms of ℓ using the mass shell relation (3.12). This leads to different formulas for N µ and T µν , as these will then involve an integral over ℓ , as in (3.39). To make this precise, we can define the outgoing representation 14 of the spherically symmetric f by \nf ↗ ( u, v, p v , ℓ ) . = f ( u, v, ℓ 2 + r 2 m 2 r 2 Ω 2 p v , p v ) , (3.40) \nand (3.39) implies, for instance, \nT uv = 2 π r 4 Ω 2 ∫ ∞ 0 ∫ ∞ 0 ℓ 2 + r 2 m 2 p v f ↗ ( u, v, p v , ℓ ) dp v ℓ dℓ. \nThe outgoing representation may be taken as an alternative definition of the spherically symmetric Vlasov system. We have chosen the formulation here in terms of p u and p v because of its explicit symmetry, which is key for constructing time-symmetric initial data in the proof of Theorem 1. We have also chosen to always write N µ and T µν in contravariant form, so that N u is associated with p u , etc. This causes extra factors of g uv = -1 2 Ω 2 to appear in various formulas, compared to [DR16; Mos18; Mos23].', '3.2.2 Previous work on the Einstein-Maxwell-Vlasov system': 'Besides the general local existence result of [BC73], the Einstein-Maxwell-Vlasov model does not seem to have been extensively studied. Dispersion for small data solutions of the Einstein-Maxwell-massive Vlasov model (stability of Minkowski space) in spherical symmetry was proved by Noundjeu [Nou05]. See also [NN04] for local well-posedness in Schwarzschild coordinates and [NNR04] for the existence of nontrivial solutions of the constraints. Many static solutions are known to exist for the massive system, first studied numerically by Andréasson-Eklund-Rein [AER09] and proved rigorously by Thaller [Tha19] in spherical symmetry. Thaller has also shown that stationary and axisymmetric (but not spherically symmetric) solutions exist [Tha20].', '3.3 Local well-posedness in spherical symmetry': "Electromagnetic geodesics, in contrast to ordinary geodesics, can have limit points in M . By standard ODE theory, this can only happen if p ( s ) → 0 as s → ±∞ . 15 On a fixed spherically symmetric background, one can show that an electromagnetic geodesic γ cannot have a limit point if either m [ γ ] > 0 or ℓ [ γ ] > 0 . However, even in the massless case, an electromagnetic geodesic with initially positive momentum will still have positive momentum for a short (coordinate) time. Therefore, one can show that local well-posedness in double null gauge holds in the case of massive particles or massless particles with momentum initially supported away from zero. \nRemark 3.12 . Bigorgne has shown that the relativistic Maxwell-massless Vlasov system is classically illposed if the initial data are allowed to be supported near zero momentum [Big22]. We expect a similar result to hold for the Einstein-Maxwell-massless Vlasov system. \nWenow state our fundamental local well-posedness result for the spherically symmetric Einstein-MaxwellVlasov system. We formulate this in terms of the characteristic initial value problem , though the techniques used apply to the Cauchy problem as well. Note that we work in function spaces that allow for noncompact support in the momentum variables, although this is not needed for the applications in this paper (but is useful in the context of cosmic censorship [DR16]). The proof of local existence is deferred to Appendix A. \nGiven U 0 < U 1 and V 0 < V 1 , let \nC ( U 0 , U 1 , V 0 , V 1 ) . = ( { U 0 } × [ V 0 , V 1 ]) ∪ ([ U 0 , U 1 ] ×{ V 0 } ) , R ( U 0 , U 1 , V 0 , V 1 ) . = [ U 0 , U 1 ] × [ V 0 , V 1 ] . \nWe will consistently omit ( U 0 , U 1 , V 0 , V 1 ) from the notation for these sets when the meaning is clear. A function ϕ : C → R is said to be smooth if it is continuous and ϕ | { U 0 }× [ V 0 ,V 1 ] and ϕ | [ U 0 ,U 1 ] ×{ V 0 } are C ∞ single-variable functions. This definition extends naturally to functions f : P m red | C → R ≥ 0 . \nDefinition 3.13. A smooth (bifurcate) characteristic initial data set for the spherically symmetric EinsteinMaxwell-Vlasov system with parameters m , e , and σ consists of smooth functions ˚ r, ˚ Ω 2 , ˚ Q : C → R with ˚ r and ˚ Ω 2 positive, and a smooth function ˚ f : P m red | C → R ≥ 0 , where P m red | C is defined using ˚ Ω 2 . Moreover, we assume that the norms \n∥ ˚ f ∥ C k σ ( P | C ) . = ∑ 0 ≤ i 1 + i 2 ≤ k ( sup P κ | { U 0 }× [ V 0 ,V 1 ] ⟨ p τ ⟩ σ + i 2 | ∂ i 1 v ∂ i 2 p ˚ f | + sup P κ | [ U 0 ,U 1 ] ×{ V 0 } ⟨ p τ ⟩ σ + i 2 | ∂ i 1 u ∂ i 2 p ˚ f | ) \nare finite for every k ≥ 0 . In the case m = 0 , we also assume that ˚ f has locally positive angular momentum. Finally, we assume that Raychaudhuri's equations (3.22) and (3.23), together with Maxwell's equations (3.24) and (3.25) are satisfied to all orders in directions tangent to C . \nProposition 3.14. For any m ≥ 0 , e ∈ R , σ > 4 , B > 0 , and c ℓ > 0 , there exists a constant ε loc > 0 with the following property. Let (˚ r, ˚ Ω 2 , ˚ Q, ˚ f ) be a characteristic initial data set for the spherically symmetric Einstein-Maxwell-Vlasov system on C ( U 0 , U 1 , V 0 , V 1 ) . If U 1 -U 0 < ε loc , V 1 -V 0 < ε loc , \n∥ log˚ r ∥ C 2 ( C ) + ∥ log ˚ Ω 2 ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P m red | C ) ≤ B, \nand in the case m = 0 , ℓ ≥ c ℓ on spt( ˚ f ) , then there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-Maxwell-Vlasov system on R ( U 0 , U 1 , V 0 , V 1 ) which extends the initial data. If ˚ f has locally compact support in p , then so does f . Moreover, the norms \n∥ log r ∥ C k ( R ) , ∥ log Ω 2 ∥ C k ( R ) , ∥ Q ∥ C k ( R ) , ∥ f ∥ C k σ ( P m red | R ) \nare finite for any k and can be bounded in terms of appropriate higher order initial data norms. \nThe proof of the proposition is given in Appendix A.2.", '3.4 The generalized extension principle': "Recall that a spherically symmetric Einstein-matter model is said to satisfy the generalized extension principle if any 'first singularity' either emanates from a point on the spacetime boundary with r = 0 , or its causal past has infinite spacetime volume. This property has been shown to hold for the Einstein-massless scalar field system by Christodoulou [Chr93], for the Einstein-massive Vlasov system by Dafermos and Rendall [DR16], and for the Einstein-Maxwell-charged Klein-Gordon system by Kommemi [Kom13]. In this paper, we extend the generalized extension principle of Dafermos-Rendall to the Einstein-Maxwell-Vlasov system: \nProposition 3.15 (The generalized extension principle) . Let ( Q , r, Ω 2 , Q, f ) be a smooth solution of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum, which is defined on an open set Q ⊂ R 2 u,v . If Q contains the set R ' . = R ( U 0 , U 1 , V 0 , V 1 ) \\{ ( U 1 , V 1 ) } and the following two conditions are satisfied:", "1. R ' has finite Lorentzian volume, i.e.,": "∫∫ R ' Ω 2 dudv < ∞ , (3.41) \n- 2. the area-radius is bounded above and below, i.e., \nsup R ' | log r | < ∞ , (3.42) \nthen the solution extends smoothly, with admissible momentum, to a neighborhood of ( U 1 , V 1 ) . \nTherefore, since this system satisfies the dominant energy condition (Remark 3.8), the Einstein-MaxwellVlasov system is strongly tame in Kommemi's terminology [Kom13], under the admissible momentum assumption. This is an important 'validation' of the Einstein-Maxwell-Vlasov model over the charged null dust model and means the model enjoys Kommemi's a priori boundary characterization [Kom13], which will be used in Section 6.2.1 below. \nProposition 3.15 is also used crucially in the proof of Theorem 1 because it provides a continuation criterion at zeroth order. This allows us to avoid commutation when treating the singular 'main beam' in the construction of bouncing charged Vlasov beams. \nWe now give the proof of Proposition 3.15, assuming the 'fundamental local spacetime estimate' to be stated and proved in Section 3.4.2 below. The proof of the local estimate is based on a streamlining of the ideas already present in [DR16] together with the monotonicity of charge inherent to the Einstein-MaxwellVlasov system and a quantitative lower bound on the 'coordinate time momentum' p u + p v obtained from the mass shell relation. \nProof of Proposition 3.15. By Lemma 3.16 below, (3.41) and (3.42) imply that \nB . = ∥ log r ∥ C 2 ( R ' ) + ∥ log Ω 2 ∥ C 2 ( R ' ) + ∥ Q ∥ C 1 ( R ' ) + ∥ f ∥ C 1 σ ( P m red | R ' ) < ∞ . \nLet U ' 1 > U 1 and V ' 1 > V 1 be such that the segments [ U 1 , U ' 1 ] ×{ V 0 } and { U 0 } × [ V 1 , V ' 1 ] lie inside of Q and let c ℓ > 0 be a lower bound for ℓ on spt( f ) ∩ P m red | C ( U 0 ,U ' 1 ,V 0 ,V ' 1 ) if m = 0 . Let ε loc > 0 be the local existence time for the spherically symmetric Einstein-Maxwell-Vlasov system with parameters ( m , e , σ, 2 B,c ℓ ) given by Proposition 3.14. Fix ( ˜ U, ˜ V ) ∈ R ' with U 1 -˜ U < ε loc and V 1 -˜ V < ε loc . \nObserve that if U 2 > U 1 is sufficiently close to U 1 and V 2 > V 1 is sufficiently close to V 1 , then \nB . = ∥ log r ∥ C 2 ( ˜ C ) + ∥ log Ω 2 ∥ C 2 ( ˜ C ) + ∥ Q ∥ C 1 ( ˜ C ) + ∥ f ∥ C 1 σ ( P m red | ˜ C ) ≤ 2 B \nand ℓ ≥ c ℓ on spt( f ) ∩ P m red | ˜ C if m = 0 , where ˜ C . = C ( ˜ U,U 2 , ˜ V , V 2 ) . Indeed, this is clear for log r , log Ω 2 , and Q by smoothness of these functions on Q . For f , we can also easily show this using the mean value theorem and the finiteness of ∥ f ∥ C 2 σ ( P m red | K ) on compact sets K ⊂ Q . For ℓ , this follows immediately from conservation of angular momentum and the domain of dependence property if U 2 ≤ U ' 1 and V 2 ≤ V ' 1 . \nTherefore, by Proposition 3.14, the solution extends smoothly, with admissible momentum, to the rectangle R ( ˜ U,U 2 , ˜ V , V 2 ) , which contains ( U 1 , V 1 ) . This completes the proof.", '3.4.1 Horizontal lifts and the commuted Vlasov equation': 'Local well-posedness for ( r, Ω 2 , Q, f ) takes place in the space C 2 × C 2 × C 1 × C 1 , since the Christoffel symbols and electromagnetic field need to be Lipschitz regular to obtain a unique classical solution of the Vlasov equation (3.26). In order to estimate ∂ 2 u Ω 2 and ∂ 2 v Ω 2 , one has to commute the wave equation for Ω 2 , (3.21), with ∂ u and ∂ v . This commuted equation contains terms such as ∂ u T uv , which can only be estimated by first estimating ∂ u f and ∂ v f . On the other hand, naively commuting the spherically symmetric Vlasov equation, (3.26), with spatial derivatives introduces highest order nonlinear error terms such as ∂ 2 u Ω 2 ∂ p u f . 16 Therefore, it would appear that the system does not close at this level of regularity. \nHowever, as was observed by Dafermos and Rendall in [DR05a] in the case of Einstein-Vlasov (see also the erratum of [RR92]), the horizontal lifts \nˆ ∂ u f . = ∂ u f -p u ∂ u log Ω 2 ∂ p u f, \nˆ ∂ v f . = ∂ v f -p v ∂ v log Ω 2 ∂ p v f \nof ∂ u f and ∂ v f with respect to the Levi-Civita connection satisfy a better system of equations without these highest order errors. In the case of Einstein-Maxwell-Vlasov, we directly commute (3.26) with { ˆ ∂ u , ˆ ∂ v , ∂ u , ∂ v } to obtain \nX ( ˆ ∂ u f ) = p u ∂ u log Ω 2 ˆ ∂ u f + ( p u ∂ u log Ω 2 ∂ p u ζ u -∂ u ζ u -∂ u log Ω 2 ζ u ) ∂ p u f + ( ∂ u ∂ v log Ω 2 ( p u ) 2 -∂ u ζ v + p u ∂ u log Ω 2 ∂ p u ζ v ) ∂ p v f, \n(3.43) \nX ( ˆ ∂ v f ) = p v ∂ v log Ω 2 ˆ ∂ v f + ( ∂ u ∂ v log Ω 2 ( p v ) 2 -∂ v ζ u + p v ∂ v log Ω 2 ∂ p v ζ u ) ∂ p u f + ( p v ∂ v log Ω 2 ∂ p v ζ v -∂ v ζ v -∂ v log Ω 2 ζ v ) ∂ p v f, (3.44) \nX ( ∂ p u f ) = -ˆ ∂ u f +(3 p u ∂ u log Ω 2 -∂ p u ζ u ) ∂ p u f -∂ p u ζ v ∂ p v f, (3.45) \nX ( ∂ p v f ) = -ˆ ∂ v f -∂ p v ζ u ∂ p u f +(3 p v ∂ v log Ω 2 -∂ p v ζ v ) ∂ p v f, (3.46) \nζ u . = 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u , ζ v . = 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v . \nUpon using the wave equation (3.21), we see that the right-hand sides of (3.43)-(3.46) do not contain second derivatives of Ω 2 .', '3.4.2 The fundamental local spacetime estimate': "Lemma 3.16. For any m ≥ 0 , e ∈ R , σ > 4 , and C 0 > 0 , there exists a constant C ∗ < ∞ with the following property. Let ( r, Ω 2 , Q, f ) be a solution of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum for particles of charge e , mass m , and momentum decay rate σ defined on R ' . = R ( U 0 , U 1 , V 0 , V 1 ) \\ { ( U, V ) } . Assume U 1 -U 0 ≤ C 0 , V 1 -V 0 ≤ C 0 , the initial data estimates \n∥ log r ∥ C 2 ( C ) + ∥ log Ω 2 ∥ C 2 ( C ) + ∥ Q ∥ C 1 ( C ) + ∥ f ∥ C 1 σ ( P m red | C ) ≤ C 0 , (3.47) \nthe global estimates \n∫∫ R ' Ω 2 dudv ≤ C 0 , (3.48) \nsup R ' | log r | ≤ C 0 , (3.49) \ninf spt( f ) ∩ P m red | C ℓ ≥ C -1 0 . (3.50) \nThen we have the estimate \n∥ log r ∥ C 2 ( R ' ) + ∥ log Ω 2 ∥ C 2 ( R ' ) + ∥ Q ∥ C 1 ( R ' ) + ∥ f ∥ C 1 σ ( P m red | R ' ) ≤ C ∗ . \nProof. In this proof, we use the notation A ≲ 1 to mean that A ≤ C , where C is a constant depending only on m , e , and σ , and C 0 . When writing area integrals, we will also make no distinction between R and R ' , though the integrands are strictly speaking only assumed to be defined on R ' . \nFrom (3.47) and the monotonicity properties of Maxwell's equations (3.24) and (3.25), it follows that \nsup R ' | Q | ≲ 1 . (3.51) \nRewriting (3.20), we obtain \nwhere \nand in the case m = 0 , assume also that \nr 2 Ω 4 T uv = 2 ∂ u ∂ v r 2 +Ω 2 ( 1 -Q 2 r 2 ) . (3.52) \nIntegrating over R ' and using (3.47), (3.48), and (3.49) yields \n∫∫ R Ω 4 T uv dudv ≲ ∫∫ R r 2 Ω 4 T uv dudv ≲ ∫∫ R ∂ u ∂ v r 2 dudv + ∫∫ R Ω 2 dudv ≲ 1 . (3.53) \nRewriting (3.52) slightly, we obtain \n∂ u ( r∂ v r ) = -Ω 2 4 ( 1 -Q 2 r 2 ) + 1 4 r 2 Ω 4 T uv . (3.54) \nIntegrating this in u and using (3.47), (3.49), and (3.51), we have \nsup [ U 0 ,U 1 ] ×{ v } | r∂ v r | ≲ 1 + ∫ U 1 U 0 Ω 2 ( u, v ) du + ∫ U 1 U 0 Ω 4 T uv ( u, v ) du \nfor any v ∈ [0 , V ] . Integrating this estimate in v and using (3.49), (3.48), and (3.53) yields \n∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | ∂ v r | dv ≲ ∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | r∂ v r | dv ≲ 1 . (3.55) \nBy Raychaudhuri's equation (3.22), ∂ u r changes signs at most once on each ingoing null cone. Therefore, by the fundamental theorem of calculus and (3.49), \nsup v ∈ [ V 0 ,V 1 ] ∫ U 1 U 0 | ∂ u r | ( u, v ) du ≤ 2 ( sup R ' r -inf R ' r ) ≲ 1 . (3.56) \nCombining (3.55) and (3.56) yields \n∫∫ R | ∂ u r∂ v r | dudv ≤ ( ∫ V 1 V 0 sup [ U 0 ,U 1 ] ×{ v } | ∂ v r | dv )( sup v ∈ [ V 0 ,V 1 ] ∫ U 1 U 0 | ∂ u r | ( u, v ) du ) ≲ 1 . (3.57) \nUsing the definition of the Hawking mass (2.2), (3.49), and (3.48), we readily infer \n∫∫ R Ω 2 | m | dudv ≲ 1 (3.58) \nBy the wave equation (3.21), the fundamental theorem of calculus, and the estimates (3.19), (3.48), (3.49), (3.51), (3.53), and (3.58), we have \nsup R ' | log Ω 2 | ≲ 1 + ∣ ∣ ∣ ∣ ∫∫ R ∂ u ∂ v log Ω 2 dudv ∣ ∣ ∣ ∣ ≲ 1 + ∫∫ R ( Ω 2 +Ω 2 | m | +Ω 4 T uv ) dudv ≲ 1 . \nWe now prove estimates for the electromagnetic geodesic flow. Let γ : [0 , S ] →R ' be an electromagnetic geodesic such that ( γ (0) , p (0)) ∈ spt( f ) ∩ P m red | C . We aim to prove that \n∣ ∣ ∣ ∣ log ( Ω 2 p u ( s ) Ω 2 p u (0) )∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ log ( Ω 2 p v ( s ) Ω 2 p v (0) )∣ ∣ ∣ ∣ ≲ 1 (3.59) \nfor s ∈ [0 , S ] , where the implied constant does not depend on γ . \nIt suffices to prove this estimate for p u , as the proof of the estimate for p v differs only in notation. Following [Mos18] (see also [Daf06]), we write an integral formula for log(Ω 2 p u ) , which can then be estimated using our previous area estimates. Rewriting the mass shell relation (3.12) as \nℓ 2 r 2 = ( ℓ 2 ℓ 2 + m 2 r 2 ) Ω 2 p u p v , \nwe deduce from the Lorentz force equation (2.24) that \nd ds log(Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) p v -e Q r 2 . \nIntegrating in s and changing variables yields \nlog ( Ω 2 p u ( s ) Ω 2 p u (0) ) = ∫ γ ([0 ,s ]) ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv -∫ s 0 e Q r 2 ∣ ∣ ∣ ∣ γ ( s ' ) ds ' . \nWe use the fundamental theorem of calculus on the first integral to obtain \n∫ γ ([0 ,s ]) ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv = ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ∂ u [( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 )] dudv + ∫ { 0 }× [0 ,γ v ( s )] ( ∂ v log Ω 2 -2 ∂ v r r )( ℓ 2 ℓ 2 + m 2 r 2 ) dv, \nwhere s v ∈ [0 , S ] is defined by γ v ( s v ) = v . Using now the wave equations (3.20) and (3.21), we arrive at \nlog ( Ω 2 p u ( s ) Ω 2 p u (0) ) = ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ( 3Ω 2 m r 3 -3Ω 2 Q 2 2 r 4 -Ω 2 2 r 2 -Ω 4 T uv -Ω 2 S )( ℓ 2 ℓ 2 + m 2 r 2 ) dudv + ∫ γ v ( s ) γ v (0) ∫ γ u ( s v ) 0 ( 2 ∂ u r∂ v r -∂ u r∂ v log Ω 2 ) 2 m 2 ℓ 2 ( ℓ 2 + m 2 r 2 ) 2 dudv + ∫ { 0 }× [0 ,γ v ( s )] ( ∂ v log Ω 2 -2 ∂ v r r ) dv -∫ s 0 e Q r 2 ∣ ∣ ∣ ∣ γ ( s ' ) ds ' . (3.60) \nWe now bound each of the four terms in (3.60). Using (3.48), (3.49), (3.51), (3.53), and (3.58), the first double integral in (3.60) is readily seen to be uniformly bounded. To estimate the second double integral, we integrate the wave equation (3.21) in u to obtain \nsup [0 ,U ] ×{ v } | ∂ v log Ω 2 | ≲ 1 + ∫ U 0 ( Ω 2 | m | +Ω 2 +Ω 4 T uv ) ( u, v ) du \nfor any v ∈ [0 , V ] . Integrating this estimate in v and using the previous area estimates yields \n∫ V 0 sup [0 ,U ] ×{ v } | ∂ v log Ω 2 | dv ≲ 1 \nwhich when combined with (3.56) gives \n∫∫ R | ∂ u r || ∂ v log Ω 2 | dudv ≲ 1 . (3.61) \nCombined with (3.57), we now readily see that the second double integral in (3.60) is uniformly bounded. The integral along initial data is clearly also bounded by assumption. \nUsing (3.49) and (3.51), we estimate \n∫ s 0 e | Q | r 2 ds ≲ S. (3.62) \nTo estimate S , define the function τ ( s ) . = τ | γ ( s ) , which is strictly increasing and satisfies 0 ≤ τ ( s ) ≲ 1 for every s ∈ [0 , S ] . Using the mass shell relation (3.12), we have \n√ 4 Ω 2 ( ℓ 2 r 2 + m 2 ) ≤ p τ = dτ ds . (3.63) \nSince either m > 0 or (3.50) holds, it follows that dτ/ds is uniformly bounded away from zero, which implies S ≲ 1 for any electromagnetic geodesic in the support of f . Combined with (3.62), this uniformly bounds the final term in (3.60) and completes the proof of (3.59). \nSince f is constant along ( γ ( s ) , p ( s )) , we therefore have ⟨ p τ ( s ) ⟩ σ f ( γ ( s ) , p ( s )) ≲ ⟨ p τ (0) ⟩ σ f ( γ (0) , p (0)) , which implies \n∥ f ∥ C 0 σ ( P m red | R ' ) +sup R ' ( N u + N v + T uu + T uv + T vv + S ) ≲ 1 . \nBy (3.24) and (3.25), \nsup R ' ( | ∂ u Q | + | ∂ v Q | ) ≲ 1 . \nBy integrating (3.54) and also using that ∂ u ( r∂ v r ) = ∂ v ( r∂ u r ) , we now readily estimate \nsup R ' ( | ∂ u r | + | ∂ v r | ) ≲ 1 . \nAs this bounds the Hawking mass pointwise, we can now estimate \nsup R ' ( | ∂ u log Ω 2 | + | ∂ v log Ω 2 | + | ∂ u ∂ v log Ω 2 | ) ≲ 1 \nusing (3.21). It then follows immediately from (3.20), (3.22), and (3.23) that \nsup R ' ( | ∂ 2 u r | + | ∂ u ∂ v r | + | ∂ 2 v r | ) ≲ 1 . \nAlong an electromagnetic geodesic γ lying in the support of f , we have that \n| X ( ˆ ∂ u f ) | ≲ p τ | ˆ ∂ u f | +( p τ ) 2 | X ( ∂ p u f ) | +( p τ ) 2 | X ( ∂ p v f ) | , | X ( ˆ ∂ v f ) | ≲ p τ | ˆ ∂ v f | +( p τ ) 2 | X ( ∂ p u f ) | +( p τ ) 2 | X ( ∂ p v f ) | , | X ( ∂ p u f ) | ≲ | ˆ ∂ u f | + p τ | X ( ∂ p u f ) | + p τ | X ( ∂ p v f ) | , | X ( ∂ p v f ) | ≲ | ˆ ∂ v f | + p τ | X ( ∂ p u f ) | + p τ | X ( ∂ p v f ) | \nby (3.43)-(3.46) and all of the estimates obtained so far. It follows that, defining \nF ( s ) . = ( ( p τ ) σ ˆ ∂ u f, ( p τ ) σ ˆ ∂ v f, ( p τ ) σ +1 ∂ p u f, ( p τ ) σ +1 ∂ p v f ) ( s ) \nalong γ ( s ) , we have \n∣ ∣ ∣ ∣ d ds F ∣ ∣ ∣ ∣ ≲ p τ |F| . \nBy Grönwall's inequality and (3.63), it follows that |F| ≲ 1 along γ . Recovering ∂ u f and ∂ v f from ˆ ∂ u f and ˆ ∂ v f , we now readily bound \n∥ f ∥ C 1 σ ( P m red | R ' ) +sup R ' ( | ∂ u T uv | + | ∂ v T uv | + | ∂ u S | + | ∂ v S | ) ≲ 1 . \nCommuting the wave equation (3.21) with ∂ u and ∂ v , we obtain the final estimates \nsup R ' ( | ∂ 2 u log Ω 2 | + | ∂ 2 v log Ω 2 | ) ≲ 1 , \nwhich completes the proof.", '3.4.3 Local existence in characteristic slabs': "The spacetime estimate Lemma 3.16 can also be used to improve Proposition 3.14 to local existence in a full double null neighborhood of a bifurcate characteristic hypersurface of arbitrary length: \nProposition 3.17. For any m ≥ 0 , e ∈ R , σ > 4 , B > 0 , and c ℓ > 0 , there exists a constant ε slab > 0 with the following property. Let (˚ r, ˚ Ω 2 , ˚ Q, ˚ f ) be a characteristic initial data set for the spherically symmetric Einstein-Maxwell-Vlasov system on C ( U 0 , U 1 , V 0 , V 1 ) with admissible momentum. If \n∥ log˚ r ∥ C 2 ( C ) + ∥ log ˚ Ω 2 ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P m red | C ) ≤ B \nand either m > 0 or m = 0 and ℓ ≥ c ℓ on spt( ˚ f ) , then there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-Maxwell-Vlasov system with admissible momentum on \nR ( U 0 , U 0 +min { ε slab , U 1 -U 0 } , V 0 , V 1 ) ∪ R ( U 0 , U 1 , V 0 , V 0 +min { ε slab , V 1 -V 0 } ) \nwhich extends the initial data. \nProof. We prove existence in the slab which is thin in the u -direction, the proof in the other slab being identical. Let C 0 = 10 B and let C ∗ be the constant obtained from the fundamental local spacetime estimate Lemma 3.16 with this choice. Let A ⊂ [ V 0 , V 1 ] denote the set of ˜ V such that the solution exists on R ( U 0 , U ' 1 , V 0 , ˜ V ) , where U ' 1 . = U 0 +min { ε slab , U 1 -U 0 } and satisfies the estimates \nsup R ( U 0 ,U ' 1 ,V 0 , ˜ V ) | log r | + sup R ( U 0 ,U ' 1 ,V 0 , ˜ V ) | log Ω 2 | ≤ C 0 . (3.64) \nWe will show that if ε slab = min { ε loc ( C ∗ ) , B ( U 1 -U 0 ) -1 C -1 ∗ } , then A is nonempty, closed, and open. Nonemptyness follows from Proposition 3.14 and closedness by continuity of the bootstrap assumptions. Let now ˜ V ∈ A . To improve the bootstrap assumptions, we note that | ∂ u ∂ v log r | ≤ C ∗ on R ( U 0 , U ' 1 , V 0 , ˜ V ) by Lemma 3.16, whence | log r | ≤ ε slab ( U 1 -U 0 ) C ∗ + 3 B ≤ 1 2 C 0 by the fundamental theorem of calculus. A similar argument applies for log Ω 2 . Therefore, by applying Proposition 3.14 again, a simple continuity argument shows that ˜ V + η ∈ A for η > 0 sufficiently small.", '3.5 Time-symmetric seed data and their normalized developments': "In the proof of Theorem 1, we will pose data for the Einstein-Maxwell-Vlasov system on a mixed spacelikenull hypersurface, with the Vlasov field f supported initially on the spacelike hypersurface and away from the center. The initial data is given by a compactly supported distribution function ˚ f on the spacelike hypersurface, a numerical parameter that fixes the location of the initial outgoing null cone, together with the mass and charge of the particles. As we will only consider time-symmetric initial configurations, these data are sufficient to uniquely determine the solution. \nDefinition 3.18. A time-symmetric seed data set S . = ( ˚ f, r 2 , m , e ) for the spherically symmetric EinsteinMaxwell-Vlasov system consists of a real numbers r 2 ∈ R > 0 , m ∈ R ≥ 0 , and e ∈ R , together with a compactly supported nonnegative function ˚ f ∈ C ∞ ( (0 , ∞ ) r × (0 , ∞ ) p u × (0 , ∞ ) p v ) which is symmetric in the second and third variables, ˚ f ( · , p u , p v ) = ˚ f ( · , p v , p u ) , and satisfies spt ( ˚ f ( · , p u , p v ) ) ⊂ (0 , r 2 ] for every p u , p v ∈ (0 , ∞ ) . \nGiven a seed data set S = ( ˚ f, r 2 , m , e ) and r ∈ [0 , r 2 ] , we define \n˚ ϱ ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 ˚ f ( v, p u , p v ) dp u dp v , (3.65) \n˚ N u ( r ) . = ˚ N v ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 p u ˚ f ( r, p u , p v ) dp u dp v , (3.66) \n˚ T uu ( r ) . = ˚ T vv ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 ( p u ) 2 ˚ f ( r, p u , p v ) dp u dp v , (3.67) \n˚ T uv ( r ) . = π ∫ ∞ 0 ∫ ∞ 0 p u p v ˚ f ( r, p u , p v ) dp u dp v . (3.68) \nRemark 3.19 . These formulas are missing a factor of Ω 2 compared to (3.14)-(3.19). This is because Ω 2 is not explicitly known on the initial data hypersurface and is accounted for by extra factors of Ω 2 in the constraint system (3.69)-(3.70) below. \nFigure 5: Penrose diagram of a normalized development U of a time symmetric seed S . The spacelike hypersurface { τ = 0 } is totally geodesic, i.e., time symmetric, and the outgoing cone { u = -r 2 } has f = 0 . To the left of the support of f , the spacetime is vacuum: both the Hawking mass m and charge Q vanish identically. For the significance of the cone { v = r 0 } , see already Remark 3.24. \n<!-- image --> \nLet the functions ˚ m = ˚ m ( r ) and ˚ Q = ˚ Q ( r ) be the unique solutions of the first order system \nd dr ˚ m = r 2 4 ( 1 -2˚ m r ) -2 ( ˚ T uu +2 ˚ T uv + ˚ T vv ) + ˚ Q 2 2 r 2 , (3.69) \nd dr ˚ Q = 1 2 e r 2 ( 1 -2˚ m r ) -2 ( ˚ N u + ˚ N v ) (3.70) \nwith initial conditions ˚ m (0) = 0 and ˚ Q (0) = 0 . If \nsup r ∈ [0 ,r 2 ] 2˚ m r < 1 , \nthen ˚ m and ˚ Q exist on the entire interval [0 , r 2 ] and we say that S is untrapped . We also define \n˚ Ω 2 . = ( 1 -2˚ m r ) -1 , ˚ ϖ . = ˚ m + ˚ Q 2 2 r . \nFinally, we say that S is consistent with particles of mass m if ˚ Ω 2 ( r ) p u p v ≥ m 2 for every ( r, p u , p v ) ∈ spt ˚ f . Remark 3.20 . We have not attempted to formulate the most general notion of seed data for the spherically symmetric Einstein-Maxwell-Vlasov Cauchy problem here as it is not needed for our purposes. \nAssociated with time-symmetric seed data as in Definition 3.18, we will introduce normalized developments of such data in the following. For r 2 > 0 , let \nC r 2 . = { τ ≥ 0 } ∩ { v ≥ u } ∩ { u ≥ -r 2 } ⊂ R 2 u,v \nand let U r 2 denote the collection of connected relatively open subsets U ⊂ { v ≥ u } ⊂ R 2 u,v for which there exists a (possibly empty) achronal curve ζ ⊂ C r 2 , extending from the center { u = v } and reaching the cone { u = -r 2 } , such that U = C r 2 ∩ { u + v < ζ u + ζ v } and { τ = 0 } ∩ { 0 ≤ v ≤ r 2 } ⊂ U . See Fig. 5. \nWe also define the cones \nC u 0 . = U ∩ { u = u 0 } , C v 0 . = U ∩ { v = v 0 } . \nDefinition 3.21. Let S = ( ˚ f, r 2 , m , e ) be an untrapped time-symmetric seed data set which is consistent with particles of mass m . A normalized development of S consists of a domain U ∈ U r 2 and a spherically symmetric solution ( r, Ω 2 , Q, f ) of the Einstein-Maxwell-Vlasov system for particles of mass m and fundamental charge e defined over U \\ { u = v } such that the following holds. \n- 1. For every ( v, p u , p v ) ∈ (0 , r 2 ] × (0 , ∞ ) × (0 , ∞ ) , \nr ( -v, v ) = v, (3.71) \n∂ v r ( -v, v ) = 1 2 , (3.72) \n∂ u r ( -v, v ) = -∂ v r ( -v, v ) , (3.73) \nΩ 2 ( -v, v ) = ˚ Ω 2 ( v ) , (3.74) \n∂ v Ω 2 ( -v, v ) = 1 2 ( d dr ˚ Ω 2 ) ( v ) , (3.75) \n∂ u log Ω 2 ( -v, v ) = -∂ v log Ω 2 ( -v, v ) , (3.76) \nf ( -v, v, p u , p v ) = ˚ f ( v, p u , p v ) . (3.77) \n- 2. Along the initial outgoing null cone C -r 2 , \nr ( -r 2 , v ) = 1 2 r 2 + 1 2 v, (3.78) \nΩ 2 = ˚ Ω 2 ( r 2 ) , (3.79) \nQ = ˚ Q ( r 2 ) , (3.80) \nf = 0 . (3.81) \n- 3. The functions r , Ω 2 , Q , and m extend smoothly to the center Γ . = U ∩ { u = v } and satisfy there the boundary conditions \nr = m = Q = 0 , (3.82) \n∂ u r < 0 , ∂ v r > 0 . (3.83) \n- 4. Let γ : [0 , S ) →U\\ Γ be a future-directed electromagnetic geodesic such that r ( γ ( s )) → 0 as s → S . 17 Then ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) attains a limit on Γ , say ( u ∗ , v ∗ , p u ∗ , p v ∗ ) , and there exists a unique electromagnetic geodesic γ ' : ( S, S + ε ) →U\\ Γ for some ε > 0 such that ( γ ' u ( s ) , γ ' v ( s ) , p ' u ( s ) , p ' v ( s )) → ( u ∗ , v ∗ , p u ∗ , p v ∗ ) as s → S . We then require that \nlim s ↗ S f ( γ u ( s ) , γ v ( s ) , p u ( s ) , p v ( s )) = lim s ↘ S f ( γ ' u ( s ) , γ ' v ( s ) , p ' u ( s ) , p ' v ( s )) . \nWe use the adjective 'normalized' to emphasize the choice of a development with double null gauge anchored to the data as in points 1. and 2. above. \nRemark 3.22 . The 'time-symmetric' aspect of the development is captured by the first equalities in (3.67) and (3.68), and the equations (3.73) and (3.76). One can moreover easily verify, using (3.73), (3.76), and the formulas for the Christoffel symbols in Section 2.3, that { τ = 0 } ∩ U is a totally geodesic spacelike hypersurface with respect to the (3 + 1) -dimensional metric (2.1). \nFor a normalized development of seed data, we clearly have \nN u = ˚ Ω 2 ˚ N u , N v = ˚ Ω 2 ˚ N v , T uu = ˚ Ω 2 ˚ T uu , T vv = ˚ Ω 2 ˚ T vv , T uv = ˚ Ω 2 ˚ T uv , S = ˚ Ω 4 2 ˚ T uv -˚ Ω 2 m 2 ˚ ϱ, Q = ˚ Q, ϖ = ˚ ϖ \nalong { τ = 0 } ∩ U . \nProposition 3.23. Let S be an untrapped time-symmetric seed data set which is consistent with particles of mass m . Then there exists a δ > 0 and a unique normalized development ( r, Ω 2 , Q, f ) of S defined on { 0 ≤ τ < δ } ∩ C r 2 . \nProof. Using essentially the same methods as the proof of Proposition 3.14 in Appendix A, we obtain a unique local smooth solution ( r, Ω 2 , Q, f ) to the system of equations (3.20), (3.21), (3.24), and (3.26), with initial data given by (3.71)-(3.81). It remains to show that the constraints (3.22), (3.23), and (3.25) hold. \nBy the same calculation as in the proof of Proposition 3.14, equation (3.26) implies the conservation law (3.32) for N . Let v ∈ (0 , r 2 ) . By integration of (3.24), \nQ ( u, v ) = ˚ Q ( v ) -∫ u -v 1 2 e r 2 Ω 2 N v du ' \nfor u ≥ -v . Differentiating in v , using (3.70), (3.32), and the fundamental theorem of calculus yields (3.25) at ( u, v ) . \nTo prove that (3.22) and (3.23) hold, we argue as in the proof of Proposition 3.14. Therefore, it suffices to show that (3.23) holds on initial data (the corresponding argument for (3.22) being the same). By (3.78) and (3.81), (3.23) clearly holds on the initial outgoing cone. By taking the absolute v -derivative of ∂ v r ( -v, v ) = 1 2 , we obtain ∂ 2 v r ( -v, v ) = ∂ u ∂ v r ( -v, v ) . Therefore, using (3.20), (3.69), and (3.75), we readily compute \n∂ 2 v r -∂ v r∂ v log Ω 2 + 1 4 r Ω 4 T vv = ∂ u ∂ v r -1 4Ω 2 d dr ( 1 -2˚ m r ) -1 + 1 4 r Ω 4 T vv = -Ω 2 m 2 r 2 + Ω 2 Q 2 4 r 3 + 1 4 r Ω 4 T uv -1 4Ω 2 ( -2Ω 4 ˚ m r 2 + r Ω 6 2 ( T uu +2 T uv + T vv ) + Ω 4 Q 2 r 3 ) + 1 4 r Ω 4 T vv = 0 , \nwhere every function is being evaluated at ( -v, v ) . This is equivalent to (3.23) and completes the proof. \nRemark 3.24 . Let r 0 ∈ (0 , r 2 ) be such that ˚ f ( r, p u , p v ) = 0 if r ∈ (0 , r 0 ] . Since ˚ f is assumed to be compactly supported, such an r 0 necessarily exists. Then if ( U , r, Ω 2 , Q, f ) is a normalized development of S , the portion of the triangle { v ≤ r 0 } inside of U is identically Minkowskian in the sense that \nr = 1 2 ( v -u ) , Ω 2 = 1 , Q = f = 0 \non U ∩ { v ≤ r 0 } . In fact, we may therefore assume that any normalized development of S contains the full corner C r 2 ∩ { v ≤ r 0 } . \nRemark 3.25 . One can verify that a normalized development as in Definition 3.21 defines a solution of the constraint equations associated to the (3 + 1) -dimensional Einstein-Maxwell-Vlasov system after applying the correspondence of Proposition 3.10. In particular, the lift of τ = 0 will be totally geodesic in the (3 + 1) -dimensional spacetime. \nRemark 3.26 . One can 'maximalize' Proposition 3.23 to show the existence of a maximal globally hyperbolic development of S , but this requires treating the local existence and uniqueness problem for the spherically symmetric Einstein-Maxwell-Vlasov system at the center of symmetry, which we do not address in this paper. 18 Indeed, since our charged Vlasov beams spacetimes will always be vacuum near r = 0 , existence and uniqueness near the center will be completely trivial in our specific construction and is established in the following lemma. \nLemma 3.27. Let u 0 ≤ v 0 < v 1 , r 0 ≥ 0 , λ 0 > 0 , and α : [ u 0 , v 0 ] → R > 0 and β : [ v 0 , v 1 ] : R > 0 be smooth functions satisfying the relations \nα ( u 0 ) = β ( v 0 ) , α ( v 0 ) = 4 λ 2 0 , r 0 = 1 4 λ 0 ∫ v 0 u 0 α ( u ' ) du ' . \nThen there exists a unique smooth solution ( r, Ω 2 , Q, f ) of the spherically symmetric Einstein-MaxwellVlasov system on \n[ u 0 , v 0 ] × [ v 0 , v 1 ] ∪ ( { v ≥ u } ∩ { u ≥ v 0 } ∩ { v ≤ v 1 } ) \nwith Q and f identically vanishing, satisfying the boundary conditions of Definition 3.21 along { u = v } , together with \nr ( u 0 , v 0 ) = r 0 , ∂ v r ( u 0 , v 0 ) = λ 0 , Ω 2 | [ u 0 ,v 0 ] ×{ v 0 } = α, Ω 2 | { u 0 }× [ v 0 ,v 1 ] = β. \nThe solution is given by the explicit formulas \nr ( u, v ) = r 0 + λ 0 β ( v 0 ) ∫ v v 0 β ( v ' ) dv ' -1 4 λ 0 ∫ u u 0 α ( u ' ) du ' , Ω 2 ( u, v ) = α ( u ) β ( v ) β ( v 0 ) \nfor ( u, v ) ∈ [ u 0 , v 0 ] × [ v 0 , v 1 ] and \nr ( u, v ) = λ 0 β ( v 0 ) ∫ v u β ( v ' ) dv ' , Ω 2 ( u, v ) = 4 λ 0 β ( v 0 ) 2 β ( u ) β ( v ) \nfor ( u, v ) ∈ { v ≥ u } ∩ { u ≥ v 0 } ∩ { v ≤ v 1 } . \nRemark 3.28 . From the last formula, it follows that \n∂ u Ω 2 ( u, u ) = ∂ v Ω 2 ( u, u ) . (3.84)", '4 A singular toy model: bouncing charged null dust': "In this section, we introduce Ori's bouncing charged null dust model [Ori91]. We then show that Ori's model exhibits extremal critical collapse and can be used to construct counterexamples to the third law of black hole thermodynamics. Later, in Section 5, we will then show that these dust solutions can be (in an appropriate sense) globally desingularized by passing to smooth solutions of the Einstein-Maxwell-Vlasov system. The constructions in this section are crucial to motivate the choice of initial data in the proof of Theorem 1 in Section 5.", "4.1 Ori's bouncing charged null dust model": "We begin by recalling the general notion of charged null dust from [Ori91]: \nDefinition 4.1. The Einstein-Maxwell-charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a charged spacetime ( M , g, F ) , a future-directed null vector field k representing the momentum of the dust particles, and a nonnegative function ρ which describes the energy density of the dust. The equations of motion are \nR µν -1 2 Rg µν = 2 ( T EM µν + T µν ) , (4.1) \n∇ α F µα = e ρk µ , (4.2) \nk ν ∇ ν k µ = e F µ ν k ν , (4.3) \n∇ µ ( ρk µ ) = 0 , (4.4) \nwhere T EM µν was defined in (2.17) and T µν . = ρk µ k ν is the energy-momentum tensor of a pressureless perfect fluid. By the forced Euler equation (4.3), the integral curves of k are electromagnetic null geodesics. \nAny two functions ϖ in , Q in ∈ C ∞ ( R ) determine a spherically symmetric solution to the system (4.1)-(4.4) by the formulas \ng in [ ϖ in , Q in ] . = -D ( V, r ) dV 2 +2 dV dr + r 2 γ, (4.5) \nF . = -Q in r 2 dV ∧ dr, (4.6) \nk . = e ˙ Q in ( ˙ ϖ in -Q in ˙ Q in r ) ( -∂ r ) , ρ . = ( ˙ Q in ) 2 e 2 r 2 ( ˙ ϖ in -Q in ˙ Q in r ) -1 , (4.7) \nwhere · denotes differentiation with respect to V and \nD ( V, r ) . = 1 -2 ϖ in ( V ) r + Q in ( V ) 2 r 2 . \nThe metric (4.5) is known as the ingoing charged Vaidya metric [PS68; BV70] and describes a 'time dependent' Reissner-Nordström spacetime in ingoing Eddington-Finkelstein-type coordinates ( V, r, ϑ, φ ) . The spacetime is time oriented by -∂ r . The metric (4.5) and Maxwell field (4.6) are spherically symmetric and may therefore be considered as a spherically symmetric charged spacetime in the framework of Section 2. One easily sees that D = 1 -2 m r , Q = Q in , and ϖ = ϖ in . \nWe will always make the assumption ˙ ϖ in ≥ 0 so that T µν = ρk µ k ν satisfies the weak energy condition for r sufficiently large. We also assume that e > 0 and impose the condition ˙ Q in ≥ 0 on the seed function Q in , which just means that positively charged particles increase the charge of the spacetime. (If e < 0 , we would instead assume ˙ Q in ≤ 0 and the discussion would otherwise remain unchanged.) \nWe define a function r b = r b ( V ) , called the bounce radius , by \nr b . = Q in ˙ Q in ˙ ϖ in \nwhenever ˙ ϖ in ( V ) > 0 . The reason for this terminology will become clear shortly. By inspection of (4.7), we observe the following: For r > r b ( V ) , ( g in , F, k, ρ ) defines a solution of the Einstein-Maxwell-charged null dust system, k is future-directed null, and ρ ≥ 0 . If r b ( V ) > 0 and r ↘ r b ( V ) , then k and T dust vanish. If also ˙ Q in ( V ) > 0 , then ρ blows up at r = r b ( V ) , but ρk is nonzero and bounded. Finally, for r < r b ( V ) , k is past-directed null and ρ < 0 , so T dust violates the weak energy condition. \nPhysically, the ingoing Vaidya metric and (4.7) describe an ingoing congruence of radial charged massless dust particles which interact with the electromagnetic field that they generate. One can interpret the vanishing of k as the dust being 'stopped' by the resulting repulsive Lorentz force. Integral curves of k are ingoing radial electromagnetic null geodesics γ ( s ) with limit points on the bounce hypersurface Σ b . = { r = r b } as s → ∞ . The charged null dust system is actually ill-posed across Σ b since the transport equation (4.3) breaks down there. Because of this, Ori argued in [Ori91] that the ingoing charged Vaidya metric (4.5) (and the associated formulas in (4.7)) should only be viewed as physical to the past of Σ b and must be modified if we wish to continue the solution beyond Σ b . \nRemark 4.2 . The divergence of ρ along Σ b does not seem to have been explicitly mentioned by Ori, but it is one of the fundamentally singular features of charged null dust. One can also see that ρ can blow up if ˙ Q in /Q in blows up as a function of V , which occurs if the dust is injected into Minkowski space. \nRemark 4.3 . Before Ori's paper [Ori91], the 'standard interpretation' [SI80; LZ91] of the ingoing Vaidya metric (4.5) did not actually involve Maxwell's equation and the fluid equation was simply taken to be the standard geodesic equation. The set { r < r b } was included in the ingoing solution and the dust was thought to violate the weak energy condition in this region. We refer to [Ori91] for discussion. \nIn order to continue the dust solution across Σ b , we must make some further (nontrivial!) assumptions on the seed functions ϖ in and Q in . In order to not trivially violate causality, we must demand that Σ b is spacelike , so that the 'other side' { r < r b } of Σ b does not intersect the past of Σ b . This is equivalent to \nD -2˙ r b < 0 on Σ b . (4.8) \nWe further assume that Σ b does not contain trapped symmetry spheres, which is equivalent to \nD > 0 on Σ b . (4.9) \nBy examining the behavior of almost-radial electromagnetic null geodesics in Reissner-Nordström, Ori proposed the following bouncing continuation of the solution through Σ b : it should be as an outgoing charged Vaidya metric. This metric takes the form \ng out [ ϖ out , Q out ] . = -D ( U, r ) dU 2 -2 dUdr + r 2 γ, (4.10) \nFigure 6: Penrose diagram of Ori's bouncing charged null dust model. The geometry of the beams is described by the ingoing and outgoing Vaidya metrics, g in and g out , which are related by the gluing map G . The spacetime to the left and right of the bouncing beam is described by the Reissner-Nordström solution, with parameters ( ϖ 1 , Q 1 ) and ( ϖ 2 , Q 2 ) with ϖ 1 < ϖ 2 and Q 1 < Q 2 . The endpoints of Σ b correspond to symmetry spheres in these Reissner-Nordström spacetimes with radii r 1 < r 2 . In this diagram, the V coordinate is normalized according to the ingoing solution. We have depicted here the case of a totally geodesic bounce hypersurface Σ b and the outgoing beam is exactly the time-reflection of the ingoing beam. \n<!-- image --> \nwhere \nD ( U, r ) . = 1 -2 ϖ out ( U ) r + Q out ( U ) 2 r 2 , \nfor free functions ϖ out and Q out . The coordinates ( U, r, ϑ, φ ) are now outgoing Eddington-Finkelstein-like. Ori defined a procedure for gluing an outgoing Vaidya metric to the ingoing Vaidya metric along Σ b by demanding continuity of the second fundamental form of Σ b from both sides. One sets \n( ϖ out , Q out )( U ) = ( ϖ in , Q in ) · G -1 ( U ) , \nwhere the gluing map G = G ( V ) is determined by \nd G dV = D ( V, r b ( V )) -2˙ r b ( V ) D ( V, r b ( V )) , (4.11) \nup to specification of the (unimportant) initial condition. Notice that G is strictly monotone decreasing on account of (4.8) and (4.9). It turns out that this continuation preserves the weak energy condition through Σ b . We formalize this choice of extension of the Vaidya metric with the following \nDefinition 4.4. Let ϖ in and Q in be nondecreasing charged Vaidya seed functions such that spt( ˙ ϖ in ) = spt( ˙ Q in ) = [ V 1 , V 2 ] and r b is well-defined and positive on [ V 1 , V 2 ] . Assume also the conditions (4.8) and (4.9). Ori's bouncing charged null dust model consists of the ingoing charged Vaidya metric g in [ ϖ in , Q in ] on M in . = { V ∈ spt( ˙ ϖ in ) , r ≥ r b ( V ) } × S 2 with spacelike, untrapped bounce hypersurface Σ in b . = { V ∈ spt( ˙ ϖ in ) , r = r b ( V ) }× S 2 glued to the outgoing charged Vaidya metric g out [ ϖ in ·G -1 , Q in ·G -1 ] on M out . = { U ∈ spt( ˙ ϖ out ) , r ≥ r b · G -1 ( U ) } × S 2 with spacelike, untrapped bounce hypersurface Σ out b . = { U ∈ spt( ˙ ϖ out ) , r = r b ( G -1 ( U )) } × S 2 along the map G × id r : Σ in b → Σ out b defined by (4.11). Outside the support of the dust, Ori's bouncing charged null dust model extends by attaching two Reissner-Nordström solutions with parameters ( ϖ 1 , Q 1 ) . = ( ϖ in , Q in )( V 1 ) and ( ϖ 2 , Q 2 ) . = ( ϖ in , Q in )( V 2 ) as depicted in Fig. 6. \nThe model can be generalized to allow for multiple beams of dust by iterating the above definition in the obvious manner.", '4.2 The radial parametrization of bouncing charged null dust spacetimes': "It is not immediately clear that interesting seed functions ϖ in and Q in satisfying the requirements of Definition 4.4 exist. Therefore, it is helpful to directly prescribe the geometry of Σ b and the dust along it in the following sense. Given a spacetime as in Fig. 6, we can parametrize Σ b by the area-radius function r . Then the renormalized Hawking mass ϖ and charge Q , which are gauge-invariant quantities, can be viewed as functions of r on Σ b , and we wish to prescribe these functions. We will also prescribe Σ b to be totally geodesic . While not essential, this condition greatly simplifies Proposition 4.7 below and will later play a key role in our Vlasov construction in Section 5. \nDefinition 4.5. Let P denote the set of points ( r 1 , r 2 , ϖ 1 , ϖ 2 , Q 1 , Q 2 ) ∈ R 6 ≥ 0 subject to the conditions \n0 < r 1 < r 2 , Q 1 < Q 2 , ϖ 1 < ϖ 2 (4.12) \n2 r 1 ϖ 1 ≥ Q 2 1 , (4.13) \n2 r 1 ( ϖ 2 -ϖ 1 ) < Q 2 2 -Q 2 1 < 2 r 2 ( ϖ 2 -ϖ 1 ) , (4.14) \nmin r ∈ [ r 1 ,r 2 ] ( r 1 r 2 -2 ϖ 1 r 1 r + Q 2 1 r -Q 2 2 r + Q 2 2 r 1 ) > 0 . (4.15) \nElements of P will typically be denoted by the letter α and are called admissible parameters . Let V denote the set of triples ( α, ˇ ϖ, ˇ Q ) ∈ P × C ∞ ( [0 , ∞ ) ) × C ∞ ( [0 , ∞ ) ) such that the functions ˇ ϖ = ˇ ϖ ( r ) and ˇ Q = ˇ Q ( r ) are monotone increasing and satisfy \nspt( ˇ ϖ ' ) = spt( ˇ Q ' ) = [ r 1 , r 2 ] , (4.16) \nd dr ˇ Q 2 ( r ) = 2 r d dr ˇ ϖ ( r ) , (4.17) \nˇ ϖ ( r 1 ) = ϖ 1 , ˇ Q ( r 1 ) = Q 1 , ˇ ϖ ( r 2 ) = ϖ 2 , ˇ Q ( r 2 ) = Q 2 , (4.18) \nwhere ' denotes differentiation with respect to r . \nRemark 4.6 . In the proof of Theorem 1 we will employ the regular center parameter space P Γ , consisting of those α ∈ P with ϖ 1 = Q 1 = 0 . \nProposition 4.7 (Radial parametrization of bouncing charged null dust) . Let ( α, ˇ ϖ, ˇ Q ) ∈ V , and define strictly monotone functions V , U : [ r 1 , r 2 ] → R by \nV ( r ) = -U ( r ) = ∫ r r 1 D ( r ' ) -1 dr ' , \nwhere D ( r ) . = 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 . Then: \n- 1. The seed functions ( ϖ in , Q in ) . = (ˇ ϖ, ˇ Q ) · V -1 and ( ϖ out , Q out ) . = (ˇ ϖ, ˇ Q ) · U -1 define a bouncing charged null dust spacetime as in Definition 4.4 with gluing map G ( V ) = -V and bounce radius r b ( V ) = V -1 ( V ) .\n- 2. The bounce hypersurface Σ b is spacelike and untrapped. With the setup as in Fig. 6, the left edge of Σ b has area-radius r 1 and Reissner-Nordström parameters ( ϖ 1 , Q 1 ) and the right edge has area-radius r 2 and Reissner-Nordström parameters ( ϖ 2 , Q 2 ) . The Hawking mass m is nonnegative on Σ b .\n- 3. The bounce hypersurface Σ b is totally geodesic with respect to g in and g out . \nProof. We must check that ϖ in . = ˇ ϖ · V -1 and Q in . = ˇ Q · V -1 satisfy the assumptions of Definition 4.4. Using the chain rule and (4.17), we compute \nr b ( V ) = ˇ Q ( V -1 ( V )) ˇ Q ' ( V -1 ( V )) ˇ ϖ ' ( V -1 ( V )) = V -1 ( V ) . \nDifferentiating, we obtain \n˙ r b ( V ) = D ( V -1 ( V )) = D ( V, r b ( V )) , (4.19) \nwhich implies that d G /dV = -1 . To prove (4.9), we show that D ( r ) > 0 for r ∈ [ r 1 , r 2 ] . Integrating (4.17) in r and integrating by parts yields \nˇ ϖ ( r ) = ϖ 1 + 1 2 ∫ r r 1 1 r ' d dr ' ˇ Q 2 ( r ' ) dr ' = ϖ 1 + ˇ Q 2 ( r ) 2 r -Q 2 1 2 r 1 + 1 2 ∫ r r 1 ˇ Q 2 ( r ' ) r ' 2 dr ' . (4.20) \nUsing condition (4.15) and ˇ Q ≤ Q 2 , we then find \nD ( r ) = 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 = 1 -2 ϖ 1 r + Q 2 1 r 1 r -1 r ∫ r r 1 ˇ Q 2 ( r ' ) r ' 2 dr ' > 1 r 1 r 2 ( r 1 r 2 -2 ϖ 1 rr 1 + Q 2 1 r -Q 2 2 r + Q 2 2 r 1 ) > 0 (4.21) \nfor r ∈ [ r 1 , r 2 ] . This proves (4.9) and since D -2˙ r b = -D , also proves (4.8). Condition (4.13) implies that the Hawking mass is nonnegative at r 1 . Finally, that Σ b is a totally geodesic hypersurface is shown by directly computing its second fundamental form and using (4.19). \nThe definition of V involves many more conditions than just (4.8) and (4.9) alone, but it turns out that these are relatively easy to satisfy. In particular, we have: \nProposition 4.8. The natural projection map V →P admits a smooth section ς : P → V . In other words, given any smooth family of parameters in P we may associate a smooth family of bouncing charged null dust spacetimes attaining those parameters, with totally geodesic bounce hypersurfaces. \nRemark 4.9 . In the remainder of the paper, we fix the choice of section to be the one constructed in the proof below. \nProof. Define a smooth, surjective function ψ : R 2 → (0 , 1) by \nψ ( x, ξ ) = 1 1 + exp [ -( x -ξ ) e ( x -ξ ) 2 ] . \nis strictly monotone increasing and surjective. \nNote that for each fixed ξ ∈ R the function x ↦→ ψ ( x, ξ ) Moreover, for x ∈ R we have ψ ( x, ξ ) → 0 as ξ →∞ and ψ ( x, ξ ) → 1 as ξ →-∞ . We now define the function ˇ Q : ( r 1 , r 2 ) × R ×P → R by \nˇ Q ( r, ξ, α ) . = Q 1 +( Q 2 -Q 1 ) ψ ( log ( r -r 1 r 2 -r ) , ξ ) . \n(4.22) \nBy construction of ψ , the function ˇ Q extends smoothly to [0 , ∞ ) × R × P by setting ˇ Q ( r, ξ, α ) = Q 1 for 0 ≤ r ≤ r 1 and ˇ Q ( r, ξ, α ) = Q 2 for r ≥ r 2 . \nWith our family of candidate ˇ Q 's at hand, we aim to satisfy the constraint ˇ ϖ ( r 2 ) = ϖ 2 , where ˇ ϖ ( r ) is defined by (4.20). Consider the smooth map Π : ( ξ, α ) ∈ R ×P → R defined by \nΠ( ξ, α ) . = ϖ 1 + Q 2 2 2 r 2 -Q 2 1 2 r 1 + ∫ r 2 r 1 ˇ Q 2 ( r ' , ξ, α ) 2 r ' 2 dr ' . \nSince ψ satisfies ∂ψ ∂ξ < 0 on R 2 , we have that ∂ Π ∂ξ < 0 on R ×P . Moreover, using the pointwise limits of ψ , a direct computation gives \nlim ξ →∞ Π( α, ξ ) = ϖ 1 + Q 2 2 2 r 2 -Q 2 1 2 r 2 , lim ξ →-∞ Π( α, ξ ) = ϖ 1 + Q 2 2 2 r 1 -Q 2 1 2 r 1 . \nBy condition (4.14), this implies that \nlim ξ →∞ Π( α, ξ ) < ϖ 2 < lim ξ →-∞ Π( α, ξ ) . \nThus, the intermediate value theorem and the fact that ∂ Π ∂ξ < 0 show that there exists a unique ξ ( α ) such that Π( α, ξ ( α )) = ϖ 2 . Moreover, a direct consequence of the implicit function theorem is that the assignment P ∋ α ↦→ ξ ( α ) ∈ R is smooth. The above construction shows that the functions ˇ Q ( r, ξ ( α ) , α ) and ˇ ϖ satisfy all required properties. \nFigure 7: Penrose diagrams of extremal critical collapse in Ori's bouncing charged null dust model. Compare with Fig. 2. In Theorem 4, λ ∗ = 1 . \n<!-- image --> \nThe set P is defined by simple polynomial relations and includes many interesting examples as we will see in the next two sections.", "4.3 Extremal critical collapse in Ori's model": "The first application of Propositions 4.7 and 4.8 is the construction of one-parameter families of bouncing charged null dust spacetimes exhibiting extremal critical collapse . We first show that the regular center parameter space P Γ contains elements with arbitrary final Reissner-Nordström parameters: \nLemma 4.10. Let ϖ 2 , Q 2 > 0 . Then there exist 0 < r 1 < r 2 such that ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ . If ϖ 2 ≥ Q 2 , then r 2 can moreover be chosen so that r 2 < ϖ 2 -√ ϖ 2 2 -Q 2 2 . \nProof. Let \nr 1 . = Q 2 ( Q 2 2 ϖ 2 -ε ) , r 2 . = Q 2 ( Q 2 2 ϖ 2 + ε ) , \nwhere ϵ > 0 is a small parameter to be determined. With this choice, (4.14) is clearly satisfied. Let p ( r ) . = r 1 r 2 -Q 2 2 r + Q 2 2 r 1 and observe that \nlim ε → 0 p ( Q 2 2 2 ϖ 2 ) = Q 6 2 8 ϖ 3 2 > 0 . \nIt follows that (4.15) is satisfied for ε sufficiently small. If x ≥ 1 , then (2 x ) -1 < x -√ x 2 -1 , so taking ε perhaps smaller ensures that r 2 < ϖ 2 -√ ϖ 2 2 -Q 2 2 . \nUsing this, we can show that Ori's model exhibits extremal critical collapse. Compare the following theorem with Theorem 1 and refer to Fig. 7 for Penrose diagrams. \nTheorem 4. For any M > 0 and fundamental charge e ∈ R \\{ 0 } , there exist a small parameter δ > 0 and a smooth two-parameter family of regular center parameters { α λ,M ' } ⊂ P Γ for λ ∈ (0 , 2] and M ' ∈ [ M -δ, M + δ ] such that the two-parameter family of bouncing charged null dust spacetimes {D λ,M ' } , obtained by applying Proposition 4.7 to ς ( α λ,M ' ) , has the following properties: \n- 1. For 0 < λ < 1 , D λ,M ' is isometric to Minkowski space for all sufficiently late retarded times u and hence future causally geodesically complete. In particular, it does not contain a black hole or naked singularity, and for λ < 1 sufficiently close to 1 , sufficiently large advanced times v ≥ v 0 and sufficiently small \nretarded times u ≤ u 0 , the spacetime is isometric to an appropriate causal diamond in a superextremal Reissner-Nordström solution. Moreover, D λ,M ' converges smoothly to Minkowski space as λ → 0 . \n- 2. λ = 1 is critical: D 1 ,M ' contains a nonempty black hole region BH and for sufficiently large advanced times v ≥ v 0 , the domain of outer communication, including the event horizon H + , is isometric to that of an extremal Reissner-Nordström solution of mass M ' . The spacetime contains no trapped surfaces.\n- 3. For 1 < λ ≤ 2 , D λ,M ' contains a nonempty black hole region BH and for sufficiently large advanced times v ≥ v 0 , the domain of outer communication, including the event horizon H + , is isometric to that of a subextremal Reissner-Nordström solution. The spacetime contains an open set of trapped surfaces. \nIn addition, for every λ ∈ [0 , 2] , D λ,M ' is isometric to Minkowski space for sufficiently early advanced time and near the center { r = 0 } for all time, and possesses complete null infinities I + and I -. \nProof. Using Lemma 4.10, choose 0 < r 1 < r 2 < r -(4 M, 2 M ) such that ( r 1 , r 2 , 0 , 4 M, 0 , 2 M ) ∈ P Γ . We consider \nα λ,M ' . = ( r 1 , r 2 , 0 , λ 2 M ' , 0 , λM ' ) (4.23) \nand note that α λ,M ' lies in P Γ for | λ -2 | sufficiently small and | M -M ' | ≤ δ sufficiently small by the openness of the conditions defining P Γ . Moreover, from the scaling properties of (4.14) and the monotonicity of (4.15), we observe that α λ,M ' ∈ P Γ for all 0 < λ ≤ 2 and | M -M ' | ≤ δ . \nAfter applying Proposition 4.7 to ς ( α λ,M ' ) for λ > 0 , it remains only to show that D λ,M ' extends smoothly to Minkowski space as λ → 0 . Indeed, a direct inspection of the proof of Proposition 4.8 shows that ξ ( α λ,M ' ) is independent of λ , so that the function r ↦→ ˇ Q ( r, ξ ( α λ,M ' ) , α λ,M ' ) defined in (4.22) converges smoothly to the function ˇ Q ≡ 0 as λ → 0 . Therefore, ˇ ϖ also converges smoothly to the zero function and hence D λ,M ' converges smoothly to Minkowski space as λ → 0 . \nThe construction in the proof of Theorem 1 can be thought of as a global-in-time desingularization of this family of dust solutions. In fact, we will make essential use of the one-parameter family { ς ( α λ,M ' ) } when constructing initial data for the Einstein-Maxwell-Vlasov system.", "4.4 A counterexample to the third law of black hole thermodynamics in Ori's model": "Using the radial parametrization, we can now give a very simple disproof of the third law: \nTheorem 5. There exist bouncing charged null dust spacetimes that violate the third law of black hole thermodynamics: a subextremal Reissner-Nordström apparent horizon can evolve into an extremal ReissnerNordström event horizon in finite advanced time due to the incidence of charged null dust. \nProof. Apply Propositions 4.7 and 4.8 to ( r 1 , r 2 , ϖ 1 , ϖ 2 , Q 1 , Q 2 ) ∈ P satisfying r 2 < ϖ 2 , Q 1 < ϖ 1 , and Q 2 = ϖ 2 . For example, one may take (0 . 85 , 0 . 88 , 0 . 56 , 1 , 0 . 5 , 1) ∈ P . See Fig. 8. \nRemark 4.11 . Since the energy-momentum tensor remains bounded in Ori's model and the weak energy condition is satisfied, this is indeed a counterexample to Israel's formulation of the third law [Isr86]. \nThe counterexample in Theorem 5 explicitly displays the disconnectedness of the outermost apparent horizon which is also present in our charged scalar field counterexamples to the third law [KU22]. Note that the bouncing dust beam does not cross the subextremal apparent horizon, as is required by (4.9). \nRemark 4.12 . In the example depicted in Fig. 8, the parameters ˇ ϖ and ˇ Q satisfy ˇ ϖ ( r ) < ˇ Q ( r ) for r ∈ ( r 2 -ε, r 2 ) and some ε > 0 . Indeed, the ODE (4.17) implies \nˇ Q ' ( r ) = r ˇ Q ( r ) ˇ ϖ ' ( r ) < ˇ ϖ ' ( r ) \nnear r 2 , where we have used r 2 < Q 2 . The possibility (in fact, apparent inevitability) of the Vaidya parameters being superextremal right before extremality is reached seems to have been overlooked in the literature [SI80; FGS17]. 19 \nFigure 8: Penrose diagram of a counterexample to the third law of black hole thermodynamics in Ori's charged null dust model, Theorem 5. Note that the bounce Σ b lies behind the extremal event horizon H + since r 2 < ϖ 2 and ϖ 2 is the area-radius of the extremal horizon. The broken curve A ' is the outermost apparent horizon of the spacetime. The disconnectedness of A ' is necessary in third law violating spacetimes-see the discussion in Section 1.4.3 of [KU22]. A crucial feature of this counterexample is that Σ b lies strictly between the (initially) subextremal apparent horizon and the (eventually) extremal event horizon. Compare with Fig. 4. \n<!-- image --> \nRemark 4.13 . If one applies the old 'standard interpretation' of the ingoing Vaidya metric from [SI80; LZ91] to the seed functions ϖ in ( V ) and Q in ( V ) constructed in the proof of Theorem 5, one sees that the beam will hit the subextremal apparent horizon with a negative energy density, which is consistent with [SI80]. \nUsing methods from the proof of Theorem 1, the dust spacetimes in Theorem 5 can be 'desingularized' to smooth Einstein-Maxwell-Vlasov solutions. The desingularized solutions can also be chosen to have the property that the matter remains strictly between the subextremal apparent horizon and the event horizon and we refer back to Section 1.6.", '4.5 Issues with the bouncing charged null dust model': 'While Proposition 4.7 allows us to construct these interesting examples, the bouncing charged null dust model is unsatisfactory and we should seek to replace it for several reasons: \n- 1. The model does not arise as a well-posed initial value problem for a system of PDEs. Pasting the ingoing and outgoing Vaidya solutions together is a deliberate surgery procedure that only works for seed functions ϖ in and Q in satisfying several nontrivial and nongeneric conditions.\n- 2. The solutions are generally not smooth along Σ b , nor along any cone where Q = 0 (recall Remark 4.2). The fluid density ρ is unbounded along Σ b and the number current N = ρk is discontinuous across Σ b .\n- 3. Null dust is ill-posed once the dust reaches the center of symmetry [Mos17]. \nNevertheless, we will show in the course of this paper that the bouncing charged null dust model can be well-approximated (near Σ b ), in a precise manner, by smooth solutions of the Einstein-Maxwell-Vlasov system. See already Section 5.10.', '4.6 The formal radial charged null dust system in double null gauge': "In order to precisely phrase the manner in which Einstein-Maxwell-Vlasov approximates bouncing charged null dust, as well as to motivate the choice of Vlasov initial data, we now reformulate Ori's model in double null gauge. Following Moschidis [Mos17; Mos20], we reformulate the system by treating N and T as the fundamental variables. By eliminating the fluid variables k and ρ , we can view the ingoing and outgoing phases as two separate well-posed initial value problems, with data posed along the bounce hypersurface. This helpfully suppresses the issue of blowup of ρ on Σ b . \nDefinition 4.14. The spherically symmetric formal outgoing charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and two nonnegative smooth functions N v and T vv on Q . \nThe system satisfies the wave equations \n∂ u ∂ v r = -Ω 2 2 r 2 ( m -Q 2 2 r ) , (4.24) \n∂ u ∂ v log Ω 2 = Ω 2 m r 3 -Ω 2 Q 2 r 4 , (4.25) \n∂ u ( ∂ u r Ω 2 ) = -1 4 r Ω 2 T vv , (4.26) \n∂ v ( ∂ v r Ω 2 ) = 0 , (4.27) \n∂ u Q = -1 2 e r 2 Ω 2 N v , (4.28) \n∂ v Q = 0 . (4.29) \nThe number current satisfies the conservation law \n∂ v ( r 2 Ω 2 N v ) = 0 (4.30) \nand the energy-momentum tensor satisfies the Bianchi equation \n∂ v ( r 2 Ω 4 T vv ) = + e Ω 4 QN v . (4.31) \nIn the outgoing model, we may think of N u , T uu , and T uv to just be defined as identically zero. From (4.24) and (4.26)-(4.29) one easily derives \n∂ u m = -1 2 r 2 Ω 2 T vv ∂ v r + Q 2 2 r 2 ∂ u r, ∂ v m = Q 2 2 r 2 ∂ v r, (4.32) \n∂ u ϖ = -1 2 r 2 Ω 2 T vv ∂ v r -1 2 e r Ω 2 QN v , ∂ v ϖ = 0 . (4.33) \nFurthermore, if we set k v . = T vv /N v , then \nk v ∂ v k v + ∂ v log Ω 2 ( k v ) 2 = + e Q r 2 k v , (4.34) \nwhich is the spherically symmetric version of (4.3) for the vector field k . = k v ∂ v . The energy density of the fluid is defined by ρ . = ( N v ) 2 /T vv whenever the denominator is nonvanishing. \nDefinition 4.15. The spherically symmetric formal ingoing charged null dust model for particles of fundamental charge e ∈ R \\ { 0 } consists of a smooth spherically symmetric charged spacetime ( Q , r, Ω 2 , Q ) and two nonnegative smooth functions N u and T uu on Q . The system satisfies the same equations as the ingoing system with u ↔ v and the opposite sign in front of N u . \nIn the ingoing case, k u . = T uu /N u and ρ . = ( N u ) 2 /T uu . \nRemark 4.16 . By (4.34), these formal systems define solutions of the Einstein-Maxwell-charged null dust system (see Definition 4.1) whenever k and ρ are well-defined. \nRemark 4.17 . Inspection of (4.31) reveals that T vv can reach zero in finite backwards time. If one were to continue the solution further, T vv could become negative, which shows that the formal system actually reproduces the old 'standard interpretation' of the charged Vaidya metric discussed in [Ori91]. As we will see, because the dominant energy condition holds in the Einstein-Maxwell-Vlasov model, only dust solutions with T uu , T vv ≥ 0 will arise as limiting spacetimes, confirming Ori's heuristic picture discussed in [Ori91]. \nthe Raychaudhuri equations \nand the Maxwell equations", '4.6.1 The Cauchy problem for outgoing formal charged null dust': "Mirroring the treatment of time-symmetric 20 seed data for the Einstein-Maxwell-Vlasov system in Section 3.5, we make the following definition: \nDefinition 4.18. A time-symmetric seed data set S d . = ( ˚ N v , ˚ T vv , r 2 , e ) for the spherically symmetric formal outgoing charged null dust system consists of real numbers r 2 ∈ R > 0 and e ∈ R \\ { 0 } , together with nonnegative compactly supported smooth functions ˚ N v and ˚ T vv with support contained in (0 , r 2 ] . \nIn the dust case, we define ˚ m and ˚ Q on [0 , r 2 ] with ˚ m (0) = ˚ Q (0) = 0 by solving \nd dr ˚ m = r 2 4 ( 1 -2˚ m r ) -2 ˚ T vv + ˚ Q 2 2 r 2 , (4.35) \nd dr ˚ Q = 1 2 e r 2 ( 1 -2˚ m r ) -2 ˚ N v , (4.36) \nprovided 2˚ m < r on [0 , r 2 ] . The remaining definitions from the Vlasov case, in particular Definition 3.21, can be carried over to dust with the obvious modification that N v = T vv = 0 along Γ . 21 \nProposition 4.19. Let S d be an untrapped time-symmetric seed data set for outgoing dust. Then there exists a unique global smooth solution of the formal outgoing charged null dust system ( r, Ω 2 , Q, N v , T vv ) on C r 2 attaining the seed data. \nRemark 4.20 . Let r 1 . = inf(spt ˚ N v ∪ spt ˚ T vv ) . Then ( r, Ω 2 , Q ) is isometric to Minkowski space for u ≥ -r 1 . \nProof. This can be proved by applying a suitable coordinate transformation to a suitable outgoing charged Vaidya metric. However, it is instructive to give a direct proof using the evolution equations. \nWe pose initial data \n˚ r ( r ) = r, ˚ Ω 2 ( r ) = ( 1 -2˚ m r ) -1 , ˚ Q ( r ) = ∫ r 0 1 2 e r 2 ( 1 -2˚ m r ) -2 ˚ N v dr ' , \nand for derivatives according to Definition 3.21, for the equations (4.24), (4.25), and (4.29). By a standard iteration argument, this determines the functions ( r, Ω 2 , Q ) uniquely. The existence of a global development is strictly easier than the corresponding proof in Proposition 5.5 once the rest of the system has been derived and is omitted. We now define \nN v . = -2 e r 2 Ω 2 ∂ u Q, T vv . = -4 r Ω 2 ∂ u ( ∂ u r Ω 2 ) \nand aim to show that the rest of the equations in Definition 4.14 are satisfied. \nTo prove (4.30), simply rearrange the definition of N v and use (4.29). Note that the definition of N v is consistent with ˚ N v = ˚ Ω -2 ˚ N v by (4.36). \nUsing (4.24), (4.25), and (4.29), a tedious calculation yields \n∂ u ( r∂ 2 v r -r∂ v r∂ v log Ω 2 ) = 0 . (4.37) \nArguing as in Proposition 3.23, we see that (4.27) holds on initial data and is therefore propagated by (4.37). This proves the evolution equation ∂ v m = ∂ v rQ 2 / (2 r 2 ) and by using (4.24) once more, we see that \n∂ u m = -2 r∂ v r∂ u ( ∂ u r Ω 2 ) + Q 2 2 r 2 ∂ u r. \nComparing this with (4.35) and the definition of T vv yields ˚ T vv = ˚ Ω -2 ˚ T vv , as desired. Finally, (4.31) is proved by directly differentiating the definition of T vv and using (4.24), (4.25), and (4.28). \nΣ b totally geodesic \n<!-- image --> \nFigure 9: An outgoing charged null dust beam obtained by applying Proposition 4.19 to the seed S d ,α for parameters α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) . The electrovacuum boundary C ⋆ can be attached to a ReissnerNordström spacetime with parameters ϖ 2 and Q 2 .", '4.6.2 Outgoing charged Vaidya as formal outgoing dust': "We now want to represent the outgoing portion of a regular center bouncing charged null dust beam given by Proposition 4.7 in terms of the outgoing formal system. Let α ∈ P Γ , ς ( α ) = ( α, ˇ ϖ, ˇ Q ) be given by Proposition 4.8, and consider the time-symmetric dust seed data S d ,α . = ( ˚ N v d , 0 , r 2 , e ) , where \n˚ N v d . = 2 e r 2 ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 ˇ Q ' . (4.38) \nFor this choice of seed, the constraints (4.35)-(4.36) read \nd dr ˚ m = ˚ Q 2 2 r 2 , (4.39) \nd dr ˚ Q = ( 1 -2˚ m r ) -2 ( 1 -2 ˇ m r ) 2 ˇ Q ' . (4.40) \nTherefore, by (4.17), ˚ m = ˇ m and ˚ Q = ˇ Q , where ˇ m . = ˇ ϖ -ˇ Q 2 / (2 r ) . It follows that the outgoing formal dust solution ( r d , Ω 2 d , Q d , N v d , T vv d ) provided by Proposition 4.19 on C r 2 is indeed the same as the outgoing charged Vaidya metric provided by the radial parametrization method, Proposition 4.7. See Fig. 9.", '5 The construction of bouncing charged Vlasov beams and the proof of the main theorem': 'In this section, we prove Theorem 1 by constructing bouncing charged Vlasov beams as in Fig. 1 and Fig. 2 with prescribed final parameters. This is achieved by a very specific choice of time-symmetric Vlasov seed data and global estimates for the resulting developments. We give a detailed outline of the proof in Section 5.1 and the proof itself occupies Sections 5.2 to 5.9. Finally, in Section 5.10, we show that these bouncing charged Vlasov beams weak* converge to the bouncing charged null dust spacetimes of Proposition 4.7 in a hydrodynamic limit of the beam parameters.', '5.1.1 The heuristic picture': "The essential idea in the proof of Theorem 1 is to 'approximate' the bouncing radial charged null dust solutions from Theorem 4 and Fig. 7 by smooth families of smooth Einstein-Maxwell-Vlasov solutions. Indeed, at least formally, dust can be viewed as Vlasov matter f ( x, p ) concentrated on a single momentum p = k ( x ) at each spacetime point x . One is faced with having to perform a global-in-time desingularization of families of dust solutions which are singular in both the space and momentum variables. \nAssuming that this can be done, the heuristic picture is that of a focusing beam of Vlasov matter coming in from infinity with particles of mass m = 0 or m ≪ 1 (so that the particles look almost massless for very large time scales) and very small angular momentum 0 < ℓ ≪ 1 , which are decelerated by the electromagnetic field that they generate. Then, along some 'approximate bounce hypersurface,' the congruence smoothly 'turns around' and becomes outgoing, escaping to infinity if a black hole has not yet formed. Along the way, the particles do not hit the center of symmetry. By appropriately varying the beam parameters, we can construct families of spacetimes as depicted in Fig. 1 or Fig. 2. \nAs should be apparent from the treatment of the Cauchy problem for the Einstein-Maxwell-Vlasov system in Section 3.5 and for charged null dust in Section 4.6, we want to pose Cauchy data on (what will be) the approximate bounce hypersurface for the desingularized Vlasov solutions. We will choose the initial data for f to be supported on small angular momenta ℓ ∼ ε and so that the charge ˚ Q and Hawking mass ˚ m profiles closely approximate the initial data for dust as described in Section 4.6.2. The Vlasov beam which is intended to approximate charged null dust is called the main beam . \nAs we will see, desingularizing bouncing charged null dust requires an ansatz for ˚ f which necessarily degenerates in ε . Closing estimates in the region of spacetime where Q ≲ ε is then a fundamental issue because the repulsive effect of the electromagnetic field is relatively weak there. We overcome this issue by adding an auxiliary beam to the construction, which stabilizes the main beam by adding a small amount of charge on the order of η ≫ ε . This beam is not dust-like, consists of particles with angular momentum ∼ 1 , and is repelled away from the center by the centrifugal force. \nThe goal will be to construct a smooth family of Vlasov seeds λ ↦→ S λ for λ ∈ [ -1 , 2] such that S -1 is trivial (i.e., evolves into Minkowski), S 2 forms a subextremal Reissner-Nordström black hole with charge to mass ratio ≈ 1 2 , and λ ∗ ≈ 1 is the critical parameter for which an extremal Reissner-Nordström black hole with mass M forms. For λ ∈ [0 , 2] , the Vlasov development D λ closely approximates the dust developments from Theorem 4 (in a sense to be made precise in Section 5.10 below) and λ ∈ [ -1 , 0] smoothly 'turns on' the auxiliary beam. At the very end of the proof, λ is simply rescaled to have range [0 , 1] . \nIn fact, our methods allow us to desingularize any bouncing charged null dust beam given by Proposition 4.8. Adding dependence on λ is then essentially only a notational hurdle. We now highlight specific aspects of the construction in more detail.", '5.1.2 Time symmetry and reduction to the outgoing case': "The starting point of the construction of bouncing charged Vlasov beams is the prescription of Cauchy data on an approximate bounce hypersurface Σ b , using the radial parametrization of bouncing charged null dust as a guide. We can now see the utility of the time-symmetric ansatz in Section 3.5: it reduces the problem to constructing an outgoing beam, which is then reflected and glued to maximally extended ReissnerNordström to construct a time symmetric spacetime as depicted in Fig. 10 below. These 'maximal timesymmetric spacetimes' are constructed in Section 5.9.1. The globally hyperbolic developments in Theorem 1 are obtained by taking appropriate subsets and identifying suitable Cauchy hypersurfaces. \nThe problem now reduces to constructing the region bounded to the past by C ⋆ , Σ b , and the center in Fig. 10. In this region, the solution is always dispersive . Therefore, we can actually treat the subextremal, extremal, and superextremal cases at once. Detection of whether a black hole forms in the doubled spacetime takes place on the level of initial data and we heavily exploit the global structure of the Reissner-Nordström family itself in this process. Note that while we prescribe data in the black hole interior when λ ≤ 0 , there clearly exist Cauchy surfaces lying entirely in the domain of outer communication. In fact, the solutions are always past complete and disperse to the past. See already Proposition 5.30. \nFigure 10: Penrose diagrams of the 'maximal time-symmetric doubled spacetimes' used in the proof of Theorem 1 when m > 0 . When λ ≤ 0 , these spacetimes are evidently not globally hyperbolic, but one can easily observe that the globally hyperbolic spacetimes depicted in Fig. 1 when λ ≤ 0 are simply the above spacetimes restricted to the past of CH + ∪ { i + } ∪ I + . The exterior region is isometric to a subset of the maximally extended Reissner-Nordström solution with parameters depending on λ . \n<!-- image -->", '5.1.3 The choice of seed data': 'We now describe our desingularization procedure for bouncing charged null dust on the level of initial data. Consider the outgoing portion of a charged null dust beam ( r d , Ω 2 d , Q d , N v d , T vv d ) as in Section 4.6.2, with Cauchy data posed along the bounce hypersurface Σ b . The geometry of the outgoing dust beam is entirely driven by the choice of renormalized number current ˚ N v d in (4.38). Importantly, the energy-momentum tensor of dust vanishes identically along Σ b . \nSince radial charged null dust has ℓ = 0 , we wish to approximate dust with Vlasov matter consisting of particles with angular momentum ℓ ∼ ε , where 0 < ε ≪ 1 is a small parameter to be chosen. We want to choose the initial distribution function ˚ f so that \n˚ N u + ˚ N v = ˚ N v d = 1 e r 2 ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 d ˇ Q dr , (5.1) \n˚ Q ≈ ˇ Q, ˚ ϖ ≈ ˇ ϖ, ˚ T uu , ˚ T uv , ˚ T vv ≈ 0 (5.2) \non Σ b , as ε → 0 . These conditions are satisfied if we choose \n˚ f α,ε main ( r, p u , p v ) . = c e r 2 ε ( 1 -2 ˇ ϖ r + ˇ Q 2 r 2 ) 2 d ˇ Q dr δ ε ( p u ) δ ε ( p v ) (5.3) \nfor r ∈ [ r 1 , r 2 ] , where δ ε are approximations of the identity with support [ ε, 2 ε ] and c is a normalization constant that depends on the precise choice of the family δ ε . In order for the mass shell inequality Ω 2 p u p v ≥ m 2 to hold on the support of ˚ f α,ε main , (5.3) forces us to choose m ∈ [0 , m 0 ] with 0 < m 0 ≪ ε . \nFigure 11: Penrose diagram of outgoing charged Vlasov beams (evolution of the seed data S α,η,ε ). Note that the beams do not intersect when m = 0 . When m > 0 , one can show that they do, but it is not necessary to do so for our purposes here. \n<!-- image --> \nRemark 5.1 . In the full bouncing null dust model, N is discontinuous across Σ b . Indeed, to the past of Σ b , N points in the u -direction and has a nonzero limit along Σ b , but to the future points in the v -direction and also has a nonzero limit. In the Vlasov case, time symmetry demands N be smooth across, and orthogonal to, Σ b . By comparing (3.70) with (4.36), we see that ˚ N u + ˚ N v in Vlasov takes the role of ˚ N v in dust. \nObserve directly from (5.3) that ˚ f α,ε main behaves pointwise like ε -3 and therefore pointwise estimates for N and T in evolution must utilize precise estimates of the electromagnetic flow to cancel factors of ε . Closing estimates independently of ε is the main challenge of this scheme and we directly exploit the null structure of the spherically symmetric Einstein-Maxwell-Vlasov system in the proof. The main mechanisms ensuring boundedness of N and T in the main beam are: \n- 1. The angular momentum ℓ is conserved, so that ℓ ∼ ε throughout the main beam.\n- 2. If γ is an electromagnetic geodesic arising from the support of ˚ f α,ε main , then p v should rapidly increase due to electromagnetic repulsion. Dually, p u should rapidly decrease, which ought to suppress the ingoing moments N u , T uu , and T uv . This should be compared with the vanishing of N u , T uu , and T uv in outgoing null dust. We say that the main beam bounces due to electromagnetic repulsion . \nAs is apparent from (2.22), the magnitude of the repulsive effect is proportional to Q . If we were to evolve the seed ˚ f α,ε main on its own, the inner edge of the beam would experience less electromagnetic repulsion since Q is potentially quite small in the inner region. \nIn order to reinforce the repulsive effect of the electric field in the main beam and get a consistent hierarchy of powers of ε , we introduce an auxiliary beam on the inside of the main beam which bounces due to the centrifugal force associated to electromagnetic geodesics with large angular momentum. The initial data for the auxiliary beam is chosen to be \n˚ f r 1 ,η aux ( r, p u , p v ) . = η φ ( r, p u , p v ) , (5.4) \nwhere η ≫ ε is a constant determining the amplitude, φ is a cutoff function supported on the set [ 1 3 r 1 , 2 3 r 1 ] × [Λ -1 , Λ+1] × [Λ -1 , Λ+1] , and Λ is a fixed large constant that determines the strength of the centrifugal force felt by the auxiliary beam. The auxiliary beam ensures that the main beam always interacts with an electric field of amplitude ≳ η , which acts as a crucial stabilizing mechanism.', '5.1.4 The near and far regions and the hierarchy of scales': "The total seed for an outgoing Vlasov beam is taken to be S α,η,ε . = ( ˚ f α,η,ε tot , r 2 , m , e ) , where \n˚ f α,η,ε tot . = ˚ f r 1 ,η aux + ˚ f α,ε main , (5.5) \nthe fundamental charge e > 0 is fixed, the mass m lies in the interval [0 , m 0 ] , and η, ε , and m 0 need to be chosen appropriately small. \nTo study the evolution of S α,η,ε , depicted in Fig. 11, we distinguish between the near region { v ≤ ˘ v } and the far region R ˘ v, ∞ far = { v ≥ ˘ v } , where ˘ v is a large advanced time to be determined. Roughly, the ingoing cone { v = ˘ v } is chosen so that the geometry is very close to Minkowskian and the Vlasov field is 'strongly outgoing' and supported far away from the center, i.e., \np u p v ≲ r -2 ≪ 1 (5.6) \n̸ \nfor every p u and p v such that f ( · , ˘ v, p u , p v ) = 0 . The near region is further divided into the main and auxiliary regions, corresponding to the physical space support of the main and auxiliary beams and denoted by R ˘ v main and R ˘ v aux , respectively. 22 \nThe beam parameters η , ε, m 0 and the auxiliary parameter ˘ v satisfy the hierarchy \n0 < m 0 ≪ ε ≪ η ≪ ˘ v -1 ≪ 1 . (5.7) \nTo prove the sharp rate of dispersion when m > 0 , we augment this hierarchy with \n0 < v -1 # ≪ m , \nwhere v # is a very large time after which the additional dispersion associated to massive particles kicks in. \nThe proof of Theorem 1 proceeds by showing that if (5.7) holds, then the solution exists, with certain properties, in the regions R ˘ v main , R ˘ v aux , and R ˘ v, ∞ far , in that order. The sharp decay rates of N and T are then shown a posteriori by re-analyzing the electromagnetic geodesic flow in the far region. \nRemark 5.2 . The final Reissner-Nordström parameters of the total Vlasov seed (5.5) depend on the approximation parameters η and ε , but are O ( η ) -close to ( ϖ 2 , Q 2 ) . Therefore, in order to reach any fixed set of parameters, the background dust seed has to be appropriately modulated. See already Section 5.9.5.", '5.1.5 Outline of the main estimates': "The main beam in the near region: In this region, the main goal is proving smallness (in ε ) of N u , T uu , and T uv , which are identically zero in the background dust solution. We define the phase space volume function V : Q → R ≥ 0 by \n̸ \nV ( u, v ) . = Ω 2 ( u, v ) |{ ( p u , p v ) : f ( u, v, p u , p v ) = 0 }| , (5.8) \nwhere | · | is the Lebesgue measure on R 2 p u ,p v . The function V is invariant under gauge transformations of u and v . Using the mass shell relation (3.12) and the change of variables formula, we find \nV ( u, v ) = 2 r 2 ∫ ∞ 0 ∫ { p v : f ( u,v,p u ,p v ) =0 } dp v p v ℓ dℓ, (5.9) \n̸ \nwhere we view p u as a function of p v and ℓ . Because of the addition of ˚ f r 1 ,η aux to the seed data and the good monotonicity properties of (3.24) and (3.25), it holds that Q ≳ η in R ˘ v main . Under relatively mild bootstrap assumptions, any electromagnetic geodesic γ in the main beam is accelerated outwards at a rate ≳ η , i.e., \np v ≳ ε + η min { τ, 1 } , p u ≲ ε 2 r 2 ( ε + η min { τ, 1 } ) , \nwhere τ . = 1 2 ( u + v ) is a 'coordinate time.' We also show that if γ 1 and γ 2 are two electromagnetic geodesics in the main beam which reach the same point ( u, v ) ∈ R ˘ v main , then \n| p v 1 -p v 2 | ≲ ε η 2 \nat ( u, v ) . Using these estimates, conservation of angular momentum, and the hierarchy (5.7), we show that \nV ( u, v ) ≲ η ε 3 ε + η min { τ, 1 } , \nwhere the notation A ≲ η B means A ≤ CB , where C is a constant depending on η . Then, simply using the transport nature of the Vlasov equation, we obtain the estimates \nT uu ( u, v ) ≲ ε 1 / 2 , T uv ( u, v ) ≲ ε 1 / 2 , ∫ v -u N u ( u, v ' ) dv ' ≲ ε 1 / 2 , \nwhich capture the fundamental characteristic of outgoing null dust. These estimates allow us to control the geometry at C 1 order, which is more than enough to use the generalized extension principle, Proposition 3.15, to extend the solution. For details, see Section 5.4. When λ ∈ [ -1 , 0] and the main beam has not yet been turned on, constructing the solution in this region is trivial since the solution is electrovacuum. \nThe auxiliary beam in the near region: Since the auxiliary beam is genuinely weak ( ˚ f r 1 ,η aux ≲ η pointwise), the bootstrap argument in R ˘ v aux is a standard Cauchy stability argument, perturbing off of Minkowski space. We use explicit knowledge of the impact parameter and asymptotics of null geodesics with angular momentum ∼ Λ on Minkowski space and treat the charge as an error term in this region. For details, see Section 5.5. \nExistence in the far region: The argument in this region is a refinement of Dafermos' proof of the stability of Minkowski space for the spherically symmetric Einstein-massless Vlasov system [Daf06] (see also [Tay15, Chapter 4]). Because of the singular nature of f main in powers of ε , it seems difficult to obtain uniform in ε pointwise decay estimates for T uv by the usual method of estimating decay of the phase space volume of the support of f at late times. Fortunately, we are able to exploit the a priori energy estimates \n∫ r 2 Ω 2 T uv ∂ u r du ' ≲ 1 , ∫ r 2 Ω 2 T uv ∂ v r dv ' ≲ 1 (5.10) \ncoming from the monotonicity of the Hawking mass when ∂ v r > 0 and ∂ u r < 0 (see [Daf05b]). It is important to note that these energy estimates are independent of initial data and are a fundamental feature of the spherically symmetric Einstein equations. Under the bootstrap assumption that the electromagnetic geodesics making up the support of f are 'outgoing' for v ≥ ˘ v , the energy estimates (5.10) imply decay for the unweighted fluxes of T uv . This shows that the geometry remains close to Minkowski in C 1 and recovering the bootstrap assumption on the support of f follows from good monotonicity properties of the electromagnetic geodesic flow when close to Minkowski. We also note that this approach using energy estimates allows us to treat the cases m = 0 and m > 0 simultaneously. For details, see Section 5.6. \nDispersion in the far region: Once the solution has been shown to exist globally, we prove sharp (in coordinate time τ ) pointwise decay statements for N and T (see [RR92; Nou05; Tay15]). As the decay rates differ when m = 0 or m > 0 , these two cases are treated separately. \nThe massless case. It follows immediately from the mass shell relation (3.12) that p u ≲ r -2 in the far region. Since this is integrable, the beams are confined to null slabs and can even be shown to be disjoint as depicted in Fig. 11. Since each p u contributes a factor of r -2 and our solutions have bounded angular momentum, we obtain the sharp dispersive hierarchy \nN v + T vv ≤ C (1 + τ ) -2 , N u + T uv ≤ C (1 + τ ) -4 , T uu ≤ C (1 + τ ) -6 , \nwhere the constant C depends on α, η , and ε . For details, see Section 5.7. \nThe massive case. When m > 0 , p u does not decay asymptotically. After a very late time v # ≫ m -1 , p u ∼ η m 2 , which drives additional decay of the phase space volume. We prove this by a change of variables \nargument, turning volume in p u at later times v ≥ v # into physical space volume of the support of f at time v = v # . This leads to the sharp isotropic decay rate \nM≤ C (1 + τ ) -3 \nfor any moment M of f , where C depends on α, η, ε , and a lower bound for m . For details, see Section 5.8.", '5.2.1 The beam parameters, fixed constants, and conventions': "First, we fix once and for all the fundamental charge e ∈ R \\ { 0 } . Without loss of generality, we may take e > 0 , as all of the arguments and definitions in the remainder of Section 5 require only minor cosmetic modifications to handle the case e < 0 . Next, we fix an even function φ ∈ C ∞ c ( R ) satisfying spt φ = [ -1 , 1] , φ ≥ 0 , and \n∫ 1 -1 φdx = 1 . \nLet θ ∈ C ∞ ( R ) be a nondecreasing function such that θ ( λ ) = 0 for λ ≤ -1 and θ ( λ ) = 1 for λ ≥ 0 . Let ζ ∈ C ∞ ( R ) be a nondecreasing function such that ζ ( λ ) = 0 for λ ≤ 0 , ζ ( λ ) = λ for λ ≥ 1 2 , and ζ ' ( λ ) > 0 for λ ∈ (0 , 1 2 ] . Finally, we fix a large 23 number Λ ≥ 1 , such that \nmin p 1 ,p 2 ∈ [Λ -1 , Λ+1] p 1 p 2 ( p 1 + p 2 ) 2 ≥ 81 400 . (5.11) \nWe emphasize that: \nThe quintuple ( e , φ, θ, ζ, Λ) is fixed for the remainder of the paper. \nRecall the set P Γ of regular center admissible parameters of the form α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) which was defined in Section 4.2. Let η, ε , and m 0 be positive real numbers. In the course of the proofs below, the particle mass m will be restricted to satisfy 0 ≤ m ≤ m 0 . \nDefinition 5.3. Let α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ , η > 0 , and ε > 0 . The time-symmetric outgoing charged Vlasov beam seed S α,η,ε is given by ( ˚ f r 1 ,η aux + ˚ f α,ε main , r 2 , m , e ) , where \n˚ f r 1 ,η aux ( r, p u , p v ) . = η φ ( 6 r 1 r -3 ) φ ( p u -Λ) φ ( p v -Λ) , (5.12) \nand \n˚ f α,ε main ( r, p u , p v ) . = 8 3 π e ε 3 r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) φ ( 2 p u ε -3 ) φ ( 2 p v ε -3 ) , (5.13) \nwhere ˇ ϖ and ˇ Q are taken from ς ( α ) , where the map ς was defined in Proposition 4.8 (cf. Remark 4.9). \nDefinition 5.4. Let M > 0 , let 0 < r 1 < r 2 , and let { α λ,M ' } (with | M -M ' | ≤ δ ) be as in (4.23) in the proof of Theorem 4. For λ ∈ [ -1 , 2] , η > 0 , ε > 0 , m 0 , we define \nS λ,M ' ,η,ε . = S α ζ ( λ ) ,M ' ,θ ( λ ) η,ε (5.14) \nfor particles of mass 0 ≤ m ≤ m 0 . For λ ≤ 0 , the ˇ ϖ and ˇ Q components of ς ( α ζ ( λ ) ,M ' ) are interpreted as identically zero, in correspondence with the proof of Theorem 4. \nThroughout Section 5, the notation A ≲ B means that there exists a constant C > 0 , which only depends on e , φ , θ , ζ , Λ , M , r 1 , and r 2 such that A ≤ CB . The notation A ≳ B is defined similarly and A ∼ B means A ≲ B and A ≳ B . Moreover, we use the convention that all small (large) constants in 'sufficiently small (large)' may depend on e , φ , θ , ζ , Λ , M , r 1 , and r 2 . In Section 5.7, we will also use the notation \nA ≲ η B (resp., A ≲ η,ε B ), in which we allow the constants to also depend on η (resp., η and ε ). The relations A ∼ η B and A ∼ η,ε B are defined in the obvious way. \nFor the evolution problem, we will introduce a large parameter ˘ v to separate C r 2 into the 'near' and 'far' regions. We will always assume that the parameter hierarchy \n0 < m 0 ≪ ε ≪ η ≪ ˘ v -1 ≪ 1 (5.15) \nholds, by which we mean that any given statement holds for ˘ v sufficiently large, η sufficiently small depending on ˘ v , ε sufficiently small depending on ˘ v and η , and m 0 sufficiently small depending on ˘ v , η , and ε . To prove dispersion in the massive case, we introduce an even larger parameter v # satisfying \n0 < v -1 # ≪ m ≤ m 0 , \nso that v # is chosen sufficiently large depending on ˘ v, η, ε , and m .", '5.2.2 The global structure of outgoing charged Vlasov beams': "Proposition 5.5. Fix a fundamental charge e > 0 , cutoff functions φ, θ , and ζ as in Section 5.2.1, a number Λ ≥ 1 satisfying (5.11) , M > 0 , and 0 < r 1 < r 2 < r -(4 M, 2 M ) . Let δ > 0 be as in the statement of Theorem 4 and define S λ,M ' ,η,ε as in Definition 5.4 for λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , η > 0 , ε > 0 , and for particles of mass 0 ≤ m ≤ m 0 , where m 0 > 0 . \nIf η is sufficiently small, ε is sufficiently small depending on η , and m 0 is sufficiently small depending on η and ε , then for any λ ∈ [ -1 , 2] and | M ' -M | ≤ δ , the seed S λ,M ' ,η,ε is untrapped and consistent with particles of mass m . There exists a unique maximal normalized development ( U , r, Ω 2 , Q, f ) of S λ,M ' ,η,ε for particles of charge e and mass m with the following properties. 24 If m > 0 : \n- 1. The development is global in the normalized double null gauge, i.e., U = C r 2 .\n- 2. The (3 + 1) -dimensional spacetime obtained by lifting ( U , r, Ω 2 ) is future causally geodesically complete and satisfies globally the estimates \n∂ v r ∼ 1 , | ∂ u r | ≲ 1 , Ω 2 ∼ 1 , (1 + u 2 ) | ∂ u Ω 2 | +(1 + v 2 ) | ∂ v Ω 2 | ≲ 1 . (5.16) \n- 3. Define the final Reissner-Nordström parameters ˜ M and ˜ e of S λ,M ' ,η,ε to be the constant values of ϖ and Q , respectively, on the cone C -r 2 . Then ˜ M and ˜ e are smooth functions of ( λ, M ' , η, ε, m ) , satisfy the estimate \n| ˜ M -ζ ( λ ) 2 M ' | + | ˜ e -ζ ( λ ) M ' | ≲ η, (5.17) \nand extend smoothly to η = ε = 0 , where they equal ζ ( λ ) 2 M ' and ζ ( λ ) M ' , respectively. The spacetime ( U , r, Ω 2 ) contains antitrapped surfaces (symmetry spheres where ∂ u r ≥ 0 ) if and only if ˜ e ≤ ˜ M and r 2 < r -, where r ± . = ˜ M ± √ ˜ M 2 -˜ e 2 . In this case, we nevertheless have ∂ u r ∼ -1 for v sufficiently large and the antitrapped surfaces are restricted to lie in the slab { 2 r --r 2 ≤ v ≤ 2 r + -r 2 } . \n- 4. The Vlasov distribution function f is quantitatively supported away from the center, \ninf π (spt f ) r ≥ 1 6 r 1 , (5.18) \nand the beam asymptotes to future timelike infinity i + in the sense that \nπ (spt f ) ⊂ { C 1 v ≤ u ≤ C 2 v } , (5.19) \nwhere C 1 and C 2 are positive constants that may additionally depend on η, ε, m , and λ . The connected component of U \\ π (spt f ) containing the center is isometric to Minkowski space. The connected component of U \\ π (spt f ) containing future null infinity I + is isometric to an appropriate neighborhood of future null infinity in the Reissner-Nordström solution with parameters M and e . \n- 5. The Vlasov matter disperses in the sense that the macroscopic observables decay pointwise: \nM≤ C (1 + τ ) -3 , (5.20) \nwhere M∈{ N u , N v , T uu , T uv , T vv , S } and the constant C may additionally depend on η, ε, m , and λ . \nThe same conclusions hold if m = 0 , but points 4. and 5. are improved to: \n- 4. ' The estimate (5.18) still holds but (5.19) is improved to \nπ (spt f ) ⊂ {-r 2 ≤ u ≤ 1 3 r 1 } , (5.21) \ni.e., the beam is confined to a null slab. The spacetime is isometric to Minkowski space for u ≥ r 1 . \n- 5. ' The Vlasov matter disperses in the sense that the macroscopic observables decay pointwise: \nN v + T vv ≤ C (1 + τ ) -2 , (5.22) \nN u + T uv ≤ C (1 + τ ) -4 , (5.23) \nT uu ≤ C (1 + τ ) -6 , (5.24) \nwhere the constant C may additionally depend on η, ε , and λ . \nRemark 5.6 . An analogous version of Proposition 5.5 may be proved for any α = ( r 1 , r 2 , 0 , ϖ 2 , 0 , Q 2 ) ∈ P Γ or α = ( r 1 , r 2 , 0 , 0 , 0 , 0) by evolving the seed data S α,η,ε given by Definition 5.3. In that case, (5.17) becomes \n| ˜ M -ϖ 2 | + | ˜ e -Q 2 | ≲ η. (5.25) \nRemark 5.7 . The decay rate τ -3 in (5.20) is sharp for massive particles [RR92; Nou05]. The hierarchy of decay rates in (5.22)-(5.24) is sharp for massless particles [Tay15].", '5.3 Estimates on the initial data': "For the remainder of Section 5, we assume the notation and hypotheses of Proposition 5.5. \nLemma 5.8. For η , ε , and m 0 sufficiently small and any λ ∈ [ -1 , 2] and | M ' -M | ≤ δ , S λ,M ' ,η,ε is untrapped and consistent with particles of mass 0 ≤ m ≤ m 0 . Let ( U , r, Ω 2 , Q, f ) be a development of S λ,M ' ,η,ε , such as the one obtained from Proposition 3.23. Then the following holds: \n- 1. The set U can be assumed to contain the corner C r 2 ∩ { v ≤ 1 3 r 1 } . The solution is equal to Minkowski space in this region in the sense that \nr = 1 2 ( v -u ) , 2 \nΩ = 1 , \nQ = f = 0 \non C r 2 ∩ { v ≤ 1 3 r 1 } . \n- 2. Estimates on the initial data for the auxiliary beam: \n˚ f r 1 ,θ ( λ ) η aux ≲ θ ( λ ) η 1 [ 1 3 r 1 , 2 3 r 1 ] × [Λ -1 , Λ+1] × [Λ -1 , Λ+1] , (5.26) \nsup v ∈ [ 1 3 r 1 ,r 1 ] | (Ω 2 -1 , ∂ u log Ω 2 , ∂ v log Ω 2 , Q, m )( -v, v ) | ≲ θ ( λ ) η, (5.27) \nQ ( -2 3 r 1 , 2 3 r 1 ) ≳ θ ( λ ) η, (5.28) \nℓ ( -v, v, p u , p v ) ≈ 1 for every ( v, p u , p v ) ∈ spt( ˚ f r 1 ,θ ( λ ) η aux ) . (5.29)", '3. Estimates on the initial data for the main beam:': "˚ f α ζ ( λ ) ,M ' ,θ ( λ ) η,ε main ≲ ε -3 1 [ r 1 ,r 2 ] × [ ε, 2 ε ] × [ ε, 2 ε ] · { 1 if λ > 0 0 if λ ≤ 0 , (5.30) \nsup v ∈ [ 2 3 r 1 ,r 2 ] | (Ω 2 , ∂ u log Ω 2 , ∂ v log Ω 2 , Q, m )( -v, v ) -( ˇ Ω 2 , ˇ ω, ˇ ω, ˇ Q, ˇ m )( v ) | ≲ θ ( λ ) η, (5.31) \ninf v ∈ [ 2 3 r 1 ] Q ( -v, v ) ≳ θ ( λ ) η, (5.32) \nℓ ( -v, v, p u , p v ) ≈ ε for every ( v, p u , p v ) ∈ spt( ˚ f α ζ ( λ ) ,M ' ,θ ( λ ) η,ε main ) , (5.33) \nwhere \nˇ m ( v ) . = ˇ ϖ ( v ) -ˇ Q 2 ( v ) 2 v , ˇ Ω 2 ( v ) . = ( 1 -2 ˇ m ( v ) v ) -1 , ˇ ω . = -ˇ ω . = 1 2 d dv log ˇ Ω 2 ( v ) . (5.34)", '4. Estimates on the initial outgoing cone C -r 2 :': "ϖ ( -r 2 , v ) = ϖ ( -r 2 , r 2 ) , (5.35) \nQ ( -r 2 , v ) = Q ( -r 2 , r 2 ) , (5.36) \n0 ≤ m ( -r 2 , v ) ≤ 10 M, (5.37) \nr ( -r 2 , v ) = 1 2 v + 1 2 r 2 (5.38) \nfor v ≥ r 2 . \nProof. Consistency with particles of mass m ≤ m 0 follows immediately from Definition 5.3 and the estimates (5.27) and (5.31) by taking m 0 sufficiently small. We therefore focus on proving the estimates and as a byproduct infer the untrapped property of S λ,M ' ,η,ε . \nPart 1. This is a restatement of Remark 3.24. \nPart 2. The estimate (5.26) follows immediately from the definition (5.12). Inserting the ansatz (5.26) into (3.66)-(3.68), we find \n˚ N u ( r ) = ˚ N v ( r ) = πθ ( λ ) η Λ φ ( 6 r 1 r -3 ) , ˚ T uu ( r ) = ˚ T vv ( r ) = πθ ( λ ) η ( Λ 2 + ∫ 1 -1 x 2 φ ( x ) dx ) φ ( 6 r 1 r -3 ) , ˚ T uv ( r ) = πθ ( λ ) η Λ 2 φ ( 6 r 1 r -3 ) \nfor r ∈ [ 1 3 r 1 , 2 3 r 1 ] . For η sufficiently small, it then follows readily from the system (3.69) and (3.70) that ˚ Q and ˚ m are nonnegative, nondecreasing functions, and \n0 ≤ ˚ Q ( r ) + ˚ m ( r ) ≲ θ ( λ ) η \nfor r ∈ [ 1 3 r 1 , 2 3 r 1 ] and \n˚ Q ( 2 3 r 1 ) ≳ θ ( λ ) η. \nUsing the definition of ˚ Ω 2 , we infer | ˚ Ω 2 -1 | ≲ θ ( λ ) η , and to estimate | ∂ v log Ω 2 ( -v, v ) | , we observe that \n| ∂ v log Ω 2 ( -v, v ) | = ∣ ∣ ∣ ∣ d dv log ˚ Ω 2 ( v ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ˚ Ω -2 ( 2˚ m ( v ) v 2 -2 v d dv ˚ m ( v ) )∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ 2˚ m ˚ Ω 2 v 2 -2 ˚ Ω 2 v ( ˚ Ω 4 v 2 4 ( ˚ T uu +2 ˚ T uv + ˚ T vv ) + ˚ Q 2 2 v 2 )∣ ∣ ∣ ∣ ∣ ≲ θ ( λ ) η. \nThe estimate for ∂ u log Ω 2 ( -v, v ) follows from (3.76). This establishes (5.27) and (5.28). Finally, (5.29) follows from the mass shell relation and (5.27), provided m 0 is chosen sufficiently small. \nPart 3. The estimate (5.30) follows immediately from the definition (5.13). Inserting the ansatz (5.13) into (3.66)-(3.68), we find \n˚ N u ( r ) = ˚ N v ( r ) = 1 e r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.39) \n˚ T uu ( r ) = ˚ T vv ( r ) = ε 6 e r 2 ( 9 + ∫ 1 -1 x 2 φ ( x ) dx )( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.40) \n˚ T uv ( r ) = 3 ε 4 e r 2 ( 1 -2 ˇ ϖ ( r ) r + ˇ Q 2 ( r ) r 2 ) 2 ˇ Q ' ( r ) , (5.41) \nwhere ˇ ϖ and ˇ Q are obtained from ς ( α λ,M ' ) . Inserting (5.39)-(5.41) into (3.69) and (3.70) yields \nd ˚ m = ˚ Q 2 2 +Err \ndr 2 r , \nd dr ˚ Q = ( 1 -2˚ m r ) -2 ( 1 -2 ˇ m r ) 2 ˇ Q ' , \n| Err | ≲ ( 1 -2˚ m r ) -2 ε \nwhere \nand ˚ m ( r 1 ) , ˚ Q ( r 1 ) ≲ θ ( λ ) η . Therefore, by (4.17), (4.21), and a simple Grönwall and bootstrap argument, ˚ m and ˚ Q exist on [ r 1 , r 2 ] and satisfy \nsup r ∈ [ r 1 ,r 2 ] | (˚ m -ˇ m, ˚ Q -ˇ Q )( r ) | ≲ θ ( λ ) η. \nThis implies the same estimate for | ˚ Ω 2 -ˇ Ω 2 | by definition. To estimate the other quantities, we may now argue as in the proof of Part 2. \nPart 4. Equations (5.35), (5.36), and (5.38) follow immediately from the definitions. Inequality (5.37) follows from (5.31) provided η is chosen sufficiently small.", '5.4 The main beam in the near region': "For τ 0 > 0 , let \nR τ 0 main . = { 0 ≤ τ ≤ τ 0 } ∩ {-2 3 r 1 ≤ u ≤ -r 2 } ⊂ C r 2 . (5.42) \nLemma 5.9. For any ˘ v , η , ε , and m 0 satisfying (5.15) , the following holds. Any normalized development ( U , r, Ω 2 , Q, f ) of S λ,M ' ,η,ε , such as the one obtained from Proposition 3.23, can be uniquely extended to R 1 2 ˘ v -1 3 r 1 main . Moreover, the solution satisfies the estimates \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.43) \nr ∼ v, Ω 2 ∼ 1 , (5.44) \n∂ v r ∼ 1 , | ∂ u r | ≲ 1 , (5.45) \n| ∂ u Ω 2 | ≲ 1 , | ∂ v Ω 2 | ≲ v -3 (5.46) \non R 1 2 ˘ v -1 3 r 1 main and (5.47) \n1 3 ≤ ∂ v r ≤ 2 3 , ∂ u r ∼ -1 \non R 1 2 ˘ v -1 3 r 1 main ∩ C ˘ v . Finally, the support of the distribution function satisfies \nπ (spt f ) ∩ R 1 2 ˘ v -1 3 r 1 main ⊂ {-5 6 r 1 ≤ u ≤ -r 2 } (5.48) \nFigure 12: Penrose diagram of the bootstrap region R τ 0 main used in the proof of Lemma 5.9. \n<!-- image --> \n̸ \nand if u ∈ [ -r 2 , -5 6 r 1 ] , p u , and p v are such that f ( u, ˘ v, p u , p v ) = 0 , then \np u p v ≲ ˘ v -2 , ℓ 2 p v ≲ 1 . (5.49) \nWhen λ ≤ 0 , (5.43) reads instead \n0 ≤ m ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η. \nThe proof of Lemma 5.9 will be given on Page 61. We will make use of a bootstrap argument in the regions R τ 0 main , where τ 0 ranges over [0 , 1 2 ˘ v -1 3 r 1 ] . For the basic geometric setup of the lemma and its proof, refer to Fig. 12. As the proof is much simpler when λ ≤ 0 (the main beam is absent), we focus only on the case λ > 0 , in which case θ ( λ ) η = η . \nWe first make some definitions that will be used to define the bootstrap assumptions. Let C 1 > 0 be a constant such that \nC -1 1 ≤ ( 1 -2 ˇ m ( v ) v ) -1 ≤ C 1 \nfor v ∈ [ 2 3 r 1 , r 2 ] , where ˇ m is given by (5.34). (Recall that ˇ m ( v ) = 0 for v ≤ r 1 .) We then define \nC 2 . = 8 C 1 ( 3 r 2 2 r 1 -1 )( 5 M + 18 M 2 r 1 ) , \nC 3 . = 2 max v ∈ [ 2 3 r 1 ,r 2 ] | ˇ ω ( v ) | +100 C 1 e C 2 ( 5 M + 27 M 2 r 1 )∫ ∞ 2 3 r 1 dv ( 2 3 (1 -1 6 e -C 2 ) r 1 + 1 6 e -C 2 v ) 3 , \nThe constants C 1 , C 2 , and C 3 do not depend on η , ε , or m 0 . \nThe quantitative bootstrap assumptions for the proof of Lemma 5.9 are \n1 6 e -C 2 ≤ ∂ v r ≤ 3 2 e C 2 , (5.50) \n1 8 C -1 1 ≤ Ω -2 ∂ v r ≤ C 1 , (5.51) \n| ∂ u log Ω 2 | ≤ C 3 , (5.52) \nϖ ≤ 5 M, (5.53) \nN v ≤ Ae Bτ , (5.54) \non R τ 0 main where A ≥ 1 and B ≥ 1 are constants to be determined which may depend on ˘ v and η , but not on ε . We now derive some consequences of the bootstrap assumptions for the geometry of the solution. \nLemma 5.10. If (5.15) holds, τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] , R τ 0 main ⊂ U , and the bootstrap assumptions (5.50) -(5.53) hold on R τ 0 main , then \nη ≲ Q ≤ 6 M, (5.55) \non R τ 0 main . \nWe will frequently use that (5.56) implies \n2 3 (1 -1 6 e -C 2 ) r 1 + 1 6 e -C 2 v ≤ r ≤ r 2 + 3 2 e C 2 v, (5.56) \nΩ 2 ≈ 1 , (5.57) \n| ∂ u r | ≲ 1 (5.58) \nr \n∼ \nv \non R τ 0 main without further comment. \nProof. For η and ε sufficiently small, η ≲ Q ≤ 6 M on { τ = 0 } ∩ {-r 2 ≤ u ≤ -2 3 r 1 } and { τ ≥ 0 } ∩ { v = r 2 } by Lemma 5.8. Since N v ≥ 0 by definition, Maxwell's equation (3.24) implies the upper bound in (5.55). The lower bound also follows from Maxwell's equation (3.25) and N u ≥ 0 . The inequality (5.56) follows from integrating the bootstrap assumption (5.50). The inequality (5.57) follows directly by multiplying the bootstrap assumptions (5.50) and (5.51). To estimate ∂ u r , we rewrite the definition of the Hawking mass (2.2) and the renormalized Hawking mass (2.19) as \n∂ u r = -1 4 ( 1 -2 ϖ r + Q 2 r 2 ) Ω 2 ∂ v r . (5.59) \nNow (5.58) follows immediately from (5.51), (5.53), and (5.56). \nWe now use the basic geometric control obtained in Lemma 5.10 to obtain crucial control of the electromagnetic geodesic flow. It is convenient to first introduce some notation. Let Γ f denote the set of maximally extended electromagnetic geodesics γ : I →U , where I is an interval, such that ( γ, p )( I ) ⊂ spt f , where p = dγ/ds . If γ passes through the point ( u, v ) , we denote by s u,v the parameter value such that γ ( s u,v ) = ( u, v ) . Let Γ f ( u, v ) denote the subset of Γ f consisting of curves passing through ( u, v ) . Note that every curve in Γ f intersects C r 2 ∩ { τ = 0 } . \nLemma 5.11. If (5.15) holds, A is sufficiently large depending only on α , τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] , R τ 0 main ⊂ U , the bootstrap assumptions (5.50) -(5.54) hold on R τ 0 main , and ( u, v ) ∈ R τ 0 main , then \nV ( u, v ) ≲ ε 3 η ( ε + η min { τ, 1 } ) ( 1 + A B e Bτ ) , (5.60) \nwhere V is the phase space volume function defined by (5.9) . Furthermore, if γ ∈ Γ f ( u, v ) , then \n0 < u -u 0 ≲ η -1 ε, (5.61) \nε + η min { τ, 1 } ≲ p v ( s u,v ) ≲ ε +min { τ, 1 } . (5.62) \nwhere u 0 is the retarded time coordinate of the intersection of γ with { τ = 0 } . \nProof. Let γ ∈ Γ f ( u, v ) . We will use the Lorentz force written in the form of equation (2.25) to estimate p v . Since ( u, v ) ∈ R τ 0 main , γ intersects { τ = 0 } in spt( ˚ f α,ε main ) and therefore has angular momentum ℓ ∼ ε . By the bootstrap assumptions and Lemma 5.10, it holds that \n∣ ∣ ∣ ∣ ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 ∣ ∣ ∣ ∣ ≲ ε 2 v 2 (5.63) \nalong the entire length of γ . Let ( u 0 , v 0 ) be the coordinate of the intersection of γ with { τ = 0 } . Using (5.55)-(5.57) and the fact that p v ( s u 0 ,v 0 ) ∈ [ ε, 2 ε ] , we have \ne Q r 2 (Ω 2 p v ) ∣ ∣ ∣ ∣ s =0 ≳ ηε. \nIf ε is sufficiently small (independent of γ , but depending on η ), these estimates show that \nd ds (Ω 2 p v ) ≳ ηε > 0 \nalong γ . Using (5.57), we see that \np v ≳ ε (5.64) \nalong γ . \nIt is now convenient to parametrize γ by the advanced time coordinate v of the spacetime. The Lorentz force equation then becomes \ndγ u dv = p u p v = ℓ 2 r -2 + m 2 Ω 2 ( p v ) 2 , (5.65) \nd dv (Ω 2 p v ) = 1 p v ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 + e Q r 2 Ω 2 , (5.66) \nwhere however p v is still given by dγ v /ds and we have used the mass shell relation in (5.65). By (5.63) and (5.64), \n∣ ∣ ∣ ∣ 1 p v ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 ∣ ∣ ∣ ∣ ≲ ε v 2 \nalong γ . This implies, using v 0 ≥ r 1 and hence ∫ ∞ v 0 v '-2 dv ' ≲ 1 , that \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε. (5.67) \nUsing Lemma 5.10, we readily deduce that \n∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ≲ 1 (5.68) \nand \n∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ≳ η ( 1 v 0 -1 v ) ≳ η min { 1 , v -v 0 } . \nCombining this with (5.64) and (5.67), we deduce \nε + η min { 1 , v -v 0 } ≲ p v ( v ) ≲ 1 (5.69) \nalong γ . \nWe are now able to prove (5.61). Since r ≲ v ≲ ˘ v , r 2 m 2 ≲ ε 2 ≲ ℓ 2 for m 0 sufficiently small while respecting the hierarchy (5.15). Therefore, using also Lemma 5.10 and (5.69), we find \ndγ u dv ≲ ε 2 v 2 ( ε + η min { 1 , v -v 0 } ) 2 . (5.70) \nIf v ∈ [ v 0 , v 0 +1] , we compute \n∫ v v 0 ε 2 v ' 2 ( ε + η ( v ' -v 0 )) 2 dv ' ≲ ε ( v -v 0 ) ε + η ( v -v 0 ) ≲ η -1 ε \nand if v ∈ [ v 0 +1 , ∞ ) , we compute \n∫ v v 0 +1 ε 2 v ' 2 ( ε + η ) 2 dv ' ≲ η -2 ε 2 ≲ η -1 ε, \nfor ε ≤ η . Therefore, integrating (5.70), we find \nu -u 0 = γ u ( v ) -γ u ( v 0 ) = ∫ v v 0 dγ u dv dv ' ≲ η -1 ε, (5.71) \nwhich proves (5.61). \nNow u 0 = -v 0 , so (5.71) implies u + v 0 ≲ η -1 ε . Therefore, we have \n2 τ = v + u = ( v -v 0 ) + ( u + v 0 ) ≲ η -1 ε + v -v 0 \nand \nε + ητ ≲ ε + η ( v -v 0 ) ≲ p v ( v ) \nfor τ ≲ 1 . This, together with (5.67) and (5.68), proves (5.62). \nTo prove (5.60), we use the approximate representation formula (5.67) for p v and the change of variables formula (5.9). Using the bootstrap assumptions, Lemma 5.10, the mean value theorem, Maxwell's equation (3.24), and the estimate (5.61), we have \n∣ ∣ ∣ ∣ ∣ Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) -Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) ∣ ∣ ∣ ∣ ∣ ≲ ( 1 + sup [ γ u ( v ' ) ,u ] ×{ v ' } | ∂ u Q | ) ( γ u ( v ' ) -u ) ≲ Aη -1 εe B ( u + v ' ) / 2 \nfor every v ' ∈ [ v 0 , v ] and A sufficiently large depending only on α . Using this and (5.67), we find \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( γ u ( v ) ,v ) -∫ v v 0 e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ + ∫ v v 0 ∣ ∣ ∣ ∣ ∣ Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( γ u ( v ' ) ,v ' ) -Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) ∣ ∣ ∣ ∣ ∣ dv ' ≲ ε + ∫ v v 0 Aη -1 εe B ( u + v ' ) / 2 dv ' ≤ ε ( 1 + A Bη e B ( u + v ) / 2 ) . (5.72) \nNext, we estimate \n0 ≤ ∫ v 0 -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ v 0 + u = u -u 0 ≲ η -1 ε. (5.73) \nCombining (5.72) and (5.73) yields \n∣ ∣ ∣ ∣ ∣ Ω 2 p v ∣ ∣ ( u,v ) -∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε η ( 1 + A B e B ( u + v ) / 2 ) . (5.74) \nTherefore, if γ 1 , γ 2 ∈ Γ f ( u, v ) and we parametrize both by advanced time, and denote the v -momentum of γ i by p v i for i = 1 , 2 , we find \n| p v 1 ( v ) -p v 2 ( v ) | ≤ ∣ ∣ ∣ ∣ ∣ p v 1 ( v ) -1 Ω 2 ( u, v ) ∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ p v 2 ( v ) -1 Ω 2 ( u, v ) ∫ v -u e Q r 2 Ω 2 ∣ ∣ ∣ ∣ ( u,v ' ) dv ' ∣ ∣ ∣ ∣ ∣ ≲ ε η ( 1 + A B e B ( u + v ) / 2 ) . \nInserting this estimate, (5.62), and ℓ ∼ ε in (5.9) yields (5.60), as desired. \nProof of Lemma 5.9. The proof is a bootstrap argument based on the bootstrap assumptions (5.50)-(5.54) and continuation criterion given by the extension principle Proposition 3.15. Let \nA . = { τ 0 ∈ [0 , 1 2 ˘ v -1 3 r 1 ] : the solution extends uniquely to R τ 0 main and (5.50)-(5.54) hold on R τ 0 main } . \nThe set A is nonempty by Proposition 3.23 if A is chosen sufficiently large and η , ε , and m 0 are sufficiently small. It is also manifestly connected and closed by continuity of the bootstrap assumptions. We now show that if A and B are sufficiently large depending on η and (5.15) holds, then A is also open. \nLet τ 0 ∈ A . First, we use Lemmas 5.10 and 5.11 to estimate N v and improve (5.54). Since f is transported along electromagnetic geodesics, we have f ( u, v, p u , p v ) ≲ ε -3 for ( u, v ) ∈ R τ 0 main . Using (5.60) and (5.62), we infer directly from the definition of N v that \nN v ( u, v ) ≲ ( ε +min { τ, 1 } ) ε -3 V ( u, v ) ≲ ε +min { τ, 1 } η ( ε + η min { τ, 1 } ) ( 1 + A B ) e Bτ ≲ η -2 ( 1 + A B ) e Bτ . \nLetting C ∗ = C ∗ ( e , φ, Λ , α ) denote the implicit constant in this inequality, and choosing A = 4 C ∗ η -2 and B ≥ A , we see that \nN v ( u, v ) ≤ 1 2 Ae Bτ , \nwhich improves (5.54). \nTo continue, we now estimate N u , T uu , and T uv in the same fashion, making use now of the strong decay of p u . If γ ∈ Γ f ( u, v ) , then \np u ( s u,v ) = ℓ 2 r -2 + m 2 Ω 2 p v ( s u,v ) ≲ ε 2 ε + η min { τ, 1 } \nby (5.62). Using (5.60), we therefore find \nN u ≲ ε 2 η 1 ( ε + η min { 1 , τ } ) 2 Ae Bτ , (5.75) \nT uv ≲ ε 2 η ε +min { τ, 1 } ε + η min { 1 , τ } Ae Bτ , (5.76) \nT uu ≲ ε 4 η 1 ( ε + η min { 1 , τ } ) 3 Ae Bτ . (5.77) \nUsing the hierarchy (5.15), these estimates imply \nr 2 T uv + r 2 T uu + ∫ v -u r 2 N u ( u, v ' ) dv ' ≲ ε 1 / 2 (5.78) \nfor any ( u, v ) ∈ R τ 0 main . With these final estimates in hand, we may begin to improve the remaining bootstrap assumptions (5.50)-(5.53). We will then carry out the rest of the continuity argument and prove all of the stated conclusions of the lemma. \nImproving (5.50): The wave equation (3.20) can be rewritten as \n∂ u ∂ v r = -1 2 r 2 Ω 2 ∂ v r ( ϖ -Q 2 r ) ∂ v r + r 4 Ω 4 T uv . \nUsing an integrating factor, we find \n∂ u [ exp (∫ u -r 2 1 2 r Ω 2 ∂ v r ( ϖ -Q 2 r ) du ' ) ∂ v r ] = r 4 Ω 4 T uv exp (∫ u -r 2 1 2 r Ω 2 ∂ v r ( ϖ -Q 2 r ) du ' ) , (5.79) \nwhere the integral is taken over fixed v . The bootstrap assumptions imply \n∫ -2 3 r 1 -r 2 1 2 r Ω 2 ∂ v r ∣ ∣ ∣ ∣ ϖ -Q 2 r ∣ ∣ ∣ ∣ du ' ≤ C 2 . \nFor ε sufficiently small, the right-hand side of (5.79) is pointwise ≤ 1 10 on R τ 0 main , so integrating this equation yields \n2 5 e -C 2 ≤ ∂ v r ( u, v ) ≤ 3 5 e C 2 \nfor any ( u, v ) ∈ R τ 0 main , which improves (5.50). \nImproving (5.51): Raychaudhuri's equation (3.23) can be rewritten as \n∂ v log ( ∂ v r Ω 2 ) = -r 4 ∂ v rT uu . (5.80) \nTo improve the upper bound in (5.51), note that the right-hand side of (5.80) is nonpositive by (5.50) and hence \n∂ v r Ω 2 ( u, v ) ≤ ∂ v r Ω 2 ( -v, v ) = 1 2 ( 1 -2 ˇ m ( v ) v ) ≤ C 1 2 . \nTo improve the lower bound, note that the right-hand side of (5.80) is bounded by log 2 in absolute value for ε sufficiently small and hence \n∂ v r Ω 2 ( u, v ) ≥ 1 2 ∂ v r Ω 2 ( -v, v ) ≥ 1 4 C 1 , \nwhich improves (5.51). \nImproving (5.52): The wave equation (3.21) can be rewritten as \n∂ v ∂ u log Ω 2 = Ω 2 r 3 ( ϖ -3 Q 2 2 r ) -1 2 Ω 4 T uv -Ω 2 S. (5.81) \nIntegrating this equation in v and using the bootstrap assumptions, (5.31), and (5.78) yields \n| ∂ u log Ω 2 | ≤ 3 4 C 3 \nfor η and ε sufficiently small, which improves (5.52). \nImproving (5.53): Integrating the evolution equation for the renormalized Hawking mass (3.31) and using (5.78), we have \n| ϖ ( u, v ) -ϖ ( -v, v ) | ≲ ε 1 / 2 , \nwhich improves (5.53). \nWe have thus improved the constants in all of the bootstrap assumptions (5.50)-(5.54). Using the local existence theory Proposition 3.14 and generalized extension principle Proposition 3.15, there exists a τ ' 0 > τ 0 such that U ⊂ R τ ' 0 main . Choosing τ ' 0 > τ 0 perhaps smaller, the bootstrap assumptions (5.50)-(5.54) extend to R τ ' 0 main by continuity. Therefore, A is open and the bootstrap argument is complete. \nWe now prove the remaining conclusions of the lemma. First, m ( u, v ) ≥ 0 for every ( u, v ) ∈ R 1 2 ˘ v -1 3 r 1 main because either ∂ u r ( u, v ) < 0 (and by Raychaudhuri (3.22) also for any u ' > u ) or ∂ u r ( u, v ) ≥ 0 and then m ( u, v ) ≥ 0 directly from the definition (2.2). In the first case, the evolution equation (3.29) implies m is nondecreasing along the outgoing cone terminating at ( u, v ) . Since this cone either intersects { τ = 0 } , where m ≥ 0 , or a sphere where ∂ u r ≤ 0 (and hence m ≥ 0 ), we conclude m ( u, v ) ≥ 0 . Now integrating the wave equation (3.21) in u and using (5.78), we see that | ∂ v log Ω 2 | ≲ v -3 . Together with the bootstrap assumptions and Lemma 5.10, this proves (5.43)-(5.46). Next, (5.47) follows from integrating the wave equation (3.20) in u along C ˘ v and taking ˘ v ∼ r sufficiently large and similarly in (5.59). The inclusion (5.48) follows immediately from the u -deflection estimate (5.61) for electromagnetic geodesics in Γ f and the hierarchy (5.15). Finally, to prove (5.49) we use the mass shell relation, (5.62), and the parameter hierarchy to estimate \nr 2 p u p v = ℓ 2 + r 2 m 2 Ω 2 ( p v ) 2 ≲ ε 2 η 2 ≲ 1 , ℓ 2 p v ≲ ε 2 η ≲ 1 , \nwhich completes the proof. \n<!-- image --> \nFigure 13: Penrose diagram of the bootstrap region R v 0 aux used in the proof of Lemma 5.12. \n<!-- image -->", '5.5 The auxiliary beam in the near region': "For v 0 > 0 , let \nR v 0 aux . = { v ≥ u } ∩ { τ ≥ 0 } ∩ {-2 3 r 1 ≤ u ≤ 1 3 r 1 } ∩ { 1 3 r 1 ≤ v ≤ v 0 } , (5.82) \n˜ R v 0 aux . = { v ≥ u } ∩ { τ ≥ 0 } ∩ { u ≥ -2 3 r 1 } ∩ { v ≤ v 0 } . (5.83) \nLemma 5.12. For any ˘ v , η , ε , and m 0 satisfying (5.15) , the following holds. The development of S λ,M ' ,η,ε obtained in Lemma 5.9 can be uniquely extended to ˜ R ˘ v aux . The spacetime is vacuum for u ≥ 1 3 r 1 and v ≤ ˘ v . Moreover, the solution satisfies the estimates \n0 ≤ m ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η, Ω 2 ∼ 1 , ∂ v r ∼ -∂ u r ∼ 1 , (1 + u 2 ) | ∂ u Ω 2 | +(1 + v 2 ) | ∂ v Ω | 2 ≲ 1 \non ˜ R ˘ v aux and \n1 4 ≤ ∂ v r ≤ 3 4 \non ˜ R ˘ v aux ∩ C ˘ v . Finally, the support of the distribution function satisfies \nπ (spt f ) ∩ ˜ R ˘ v aux ⊂ {-2 r 1 ≤ u ≤ 1 r 1 } , \ninf spt( f ) ∩ ˜ R ˘ v aux r ≥ 1 6 r 1 \n3 6 , \n̸ \nand if u ∈ [ -2 3 r 1 , 1 3 r 1 ] , p u , and p v are such that f ( u, ˘ v, p u , p v ) = 0 , then \np u p v ≲ ˘ v -2 , ℓ 2 p v ≲ 1 . (5.84) \nThe proof of Lemma 5.12 will be given on Page 66. We will make use of a bootstrap argument in the regions R v 0 aux , where v 0 ranges over [ 1 3 r 1 , ˘ v ] . The triangle { v ≥ u }∩{ u ≥ 1 3 r 1 }∩{ v ≤ ˘ v } is Minkowskian and can simply be attached at the very end of the argument, cf. Lemma 3.27. For the basic geometric setup of the lemma and its proof, refer to Fig. 13. \nFor ( u, v ) ∈ R ˘ v aux , let \nˆ r ( u, v ) = r 1 6 -u 2 + 1 2 ∫ v 1 3 r 1 β ( v ' ) dv ' , ˆ Ω 2 ( u, v ) = β ( v ) , \nwhere \nβ ( v ) . = { 1 if v < 2 3 r 1 Ω 2 ( -2 3 r 1 , v ) if v ≥ 2 3 r 1 . \nIt is easily verified that (ˆ r, ˆ Ω 2 ) is a solution of the spherically symmetric Einstein vacuum equations and matches smoothly in v with ( r, Ω 2 ) from R 1 2 ˘ v -1 3 r 1 main along C -2 3 r 1 . \nThe first bootstrap assumption is \n| Q | + | ϖ | + | ∂ v r -∂ v ˆ r | + | ∂ u r -∂ u ˆ r | + | Ω 2 -ˆ Ω 2 | + | ∂ v Ω 2 -∂ v ˆ Ω 2 | + | ∂ u Ω 2 | ≤ Aθ ( λ ) ηe Bτ , (5.85) \nwhere A ≥ 1 and B ≥ 1 are constants to be determined that may depend on ˘ v but not η . We also make the following assumption on the electromagnetic geodesic flow. For ( v ' 0 , p u 0 , p v 0 ) ∈ spt( ˚ f r 1 ,θ ( λ ) η aux ) , let γ be an electromagnetic geodesic of mass m for ( r, Ω 2 , Q ) starting at ( -v ' 0 , v ' 0 , p u 0 , p v 0 ) , and let ˆ γ be a null geodesic for (ˆ r, ˆ Ω 2 ) starting at ( -v ' 0 , v ' 0 , p u 0 , p v 0 ) . Then, assuming both γ and ˆ γ remain within R v 0 aux , we assume that \n| Ω 2 p u -ˆ Ω 2 ˆ p u | + | Ω 2 p v -ˆ Ω 2 ˆ p v | ≤ Aθ ( λ ) ηe B ( γ v -v ' 0 ) . (5.86) \nFirst, we note the following immediate consequences of the first bootstrap assumption: \nLemma 5.13. If (5.15) holds, v 0 ∈ [ 1 3 r 1 , ˘ v ] , R v 0 aux ⊂ U , the bootstrap assumption (5.85) holds on R v 0 aux , and η is sufficiently small depending on A and B , then on R v f aux it holds that \n0 ≤ ϖ ≲ θ ( λ ) η, 0 ≤ Q ≲ θ ( λ ) η, | log ∂ v r | + | log( -∂ u r ) | + | log Ω 2 | ≲ 1 , | ∂ v Ω 2 | + | ∂ u Ω 2 | ≲ 1 . \nNext, we use the second bootstrap assumption to obtain \nLemma 5.14. If (5.15) holds, v 0 ∈ [ 1 3 r 1 , ˘ v ] , R v 0 aux ⊂ U , the bootstrap assumptions (5.85) and (5.86) hold on R v 0 aux , B is sufficiently large, and η is sufficiently small depending on A and B , then the following holds. Let γ : [0 , S ] →R v 0 aux be a future-directed electromagnetic geodesic starting in spt( ˚ f r 1 ,θ ( λ ) η aux ) , then \nr ≥ 1 6 r 1 , u ≤ 1 3 r 1 , r 2 p u p v ≲ 1 , ℓ 2 p v ≲ 1 \nalong γ . \nProof. Upon making the coordinate transformation \n˜ u = u, ˜ v = 1 3 r 1 + ∫ v 1 3 r 1 β ( v ' ) dv ' , (5.87) \nthe metric (ˆ r, ˆ Ω 2 ) is brought into the standard Minkowski form ( 1 2 (˜ v -˜ u ) , 1) . If ˜ t . = 1 2 (˜ v + ˜ u ) , ˆ γ is a null geodesic in R ˘ v aux with respect to (ˆ r, ˆ Ω 2 ) intersecting { τ = 0 } with momentum ( p u 0 , p v 0 ) = ( p ˜ u 0 , p ˜ v 0 ) at an area-radius of ˆ r 0 , then it is easy to check that \nˆ r 2 = ˜ t +sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 ˆ E 2 2 + ˆ ℓ 2 ˆ E 2 , (5.88) \np ˜ u = ˆ E + √ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t < -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 ˆ E -√ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t ≥ -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 , (5.89) \np ˜ v = ˆ E -√ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t < -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 ˆ E + √ ˆ E 2 -ˆ ℓ 2 / ˆ r 2 if ˜ t ≥ -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 / ˆ E 2 (5.90) \nalong ˆ γ , where ˆ ℓ 2 . = ˆ r 2 p ˜ u p ˜ v and ˆ E . = 1 2 ( p ˜ v + p ˜ u ) are conserved quantities. If ( p u 0 , p v 0 ) ∈ [Λ -1 , Λ+1] 2 and Λ satisfies (5.11), then \n81 100 ˆ r 2 0 ≤ ˆ ℓ 2 ˆ E 2 ≤ ˆ r 2 0 . \nFrom (5.88) it is apparent that \nmin ˆ γ r ≥ ˆ ℓ ˆ E ≥ 9 10 ˆ r 0 , (5.91) \nsup ˆ γ ˜ u = -sign( p v 0 -p u 0 ) √ ˆ r 2 0 -ˆ ℓ 2 ˆ E 2 ≤ 45 100 ˆ r 0 , (5.92) \nand by inspection of (5.89) and (5.90) that \nˆ r 2 p ˜ u p ˜ v ≲ 1 , ˆ ℓ 2 p ˜ v ≲ 1 (5.93) \nalong ˆ γ . \nLet γ and ˆ γ be as defined before (5.86). Parametrize γ and ˆ γ by v as in the proof of Lemma 5.11. Then \n∣ ∣ ∣ ∣ d dv ( γ u -ˆ γ u ) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ p u p v -ˆ p u ˆ p v ∣ ∣ ∣ ∣ ≲ Aθ ( λ ) ηe B ( v -v ' 0 ) \nby (5.86) and the observation that p v ≳ 1 and p ˜ v ≳ 1 , so that \n| γ u -ˆ γ u | ≤ θ ( λ ) ηe B ( v -v ' 0 ) (5.94) \nfor B chosen sufficiently large. Therefore, the conclusions of the lemma follow from the estimates (5.91)(5.93) and the fact that ˆ r 0 ∈ [ 1 3 r 1 , 2 3 r 1 ] after undoing the coordinate transformation (5.87) and applying the bootstrap assumptions. \nProof of Lemma 5.12. The proof is a bootstrap argument based on the bootstrap assumptions (5.85) and (5.86), the continuation criterion Proposition 3.15, and Lemma 3.27. Define the bootstrap set \nA . = { v 0 ∈ [ 1 3 r 1 , ˘ v ] : the solution extends uniquely to R v 0 aux and (5.85), (5.86) hold on R v 0 aux } . \nThe set A is nonempty by Propositions 3.17 and 3.23 if A is chosen sufficiently large and is manifestly closed and connected. We now show that if the parameters satisfy (5.15), then A is also open. \nLet v 0 ∈ A . Taking m 0 sufficiently small and using the formula (5.9), the initial data estimate Lemma 5.8, and Lemmas 5.13 and 5.14, we immediately find \nN u + N v + T uu + T uv + T vv + S ≲ r -2 θ ( λ ) η 1 { r ≥ 1 6 r 1 } (5.95) \non R v 0 aux . By (5.27) and the observation that f = 0 along C -2 3 r 1 ∩ R v 0 aux , we have \n| Q | + | ϖ | + | ∂ v r -∂ v ˆ r | + | Ω -2 ∂ v r -ˆ Ω -2 ∂ v ˆ r | + | Ω 2 -ˆ Ω 2 | + | ∂ v Ω 2 -∂ v ˆ Ω 2 | + | ∂ u Ω 2 | ≲ θ ( λ ) η (5.96) \nalong { τ = 0 } and C -2 3 r 1 in R v 0 aux . Then, using (5.95) and the Einstein-Maxwell-Vlasov system, we see that (5.96) holds on R v 0 aux . This improves (5.85) for an appropriate choice of A (independent of η ). \nWe now improve (5.86). Using the Lorentz force equations (2.24) and (2.25), we have \nd dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 p v ∣ ∣ ∣ ∣ γ -( ∂ v log ˆ Ω 2 -2 ∂ v ˆ r ˆ r ) ˆ ℓ 2 ˆ r 2 ˆ p v ∣ ∣ ∣ ∣ ∣ ˆ γ -e Ω 2 Q r 2 p u p v , d dv (Ω 2 p v -ˆ Ω 2 ˆ p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 p v ∣ ∣ ∣ ∣ γ + 2 ∂ u ˆ r ˆ r ˆ ℓ 2 ˆ r 2 ˆ p v ∣ ∣ ∣ ∣ ∣ ˆ γ + e Ω 2 Q r 2 . \nwhere \nFigure 14: Penrose diagram of the bootstrap region R ˘ v,v f far used in the proof of Lemma 5.15. \n<!-- image --> \nUsing the parameter hierarchy (5.15), the bootstrap assumptions (5.85) and (5.86), Lemmas 5.13 and 5.14, and the bound (5.94), we can estimate \n∣ ∣ ∣ ∣ d dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ d dv (Ω 2 p u -ˆ Ω 2 ˆ p u ) ∣ ∣ ∣ ∣ ≲ θ ( λ ) η + Aθ ( λ ) ηe B ( v -v 0 ) . \nIntegrating and choosing the constants A and B sufficiently large in terms of the implied constants and ˘ v improves (5.86) and shows that the solution extends to R ˘ v aux . \nThe solution is at once extended to ˜ R ˘ v aux by Lemma 3.27. The rest of the conclusions of the lemma follow immediately from Lemmas 3.27, 5.9, 5.13, and 5.14 and (5.95).", '5.6 The far region': "Lemma 5.15. For any ˘ v , η , ε , and m 0 satisfying (5.15) , there exists a constant C ν > 0 such that the following holds. The development of S λ,M ' ,η,ε obtained in Lemma 5.12 can be uniquely extended globally to C r 2 . Moreover, the solution satisfies the estimates \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.97) \nΩ 2 ∼ 1 , ∂ v r ∼ -∂ u r ∼ 1 , (5.98) \n(1 + u 2 ) | ∂ u Ω 2 | ≲ 1 , v 2 | ∂ v Ω 2 | ≲ 1 (5.99) \non C r 2 ∩ { v ≥ ˘ v } , and the distribution function satisfies \nπ (spt f ) ∩ { v ≥ ˘ v } ⊂ { 6 C ν u ≤ v } . \nWe will make use of a bootstrap argument in the regions \nR ˘ v,v f far . = { v ≥ u } ∩ { τ ≥ 0 } ∩ { u ≥ -r 2 } ∩ { ˘ v ≤ v ≤ v f } , \nwhere v f ≥ ˘ v . Refer to Fig. 14. The bootstrap assumptions are \n-C ν ≤ ∂ u r ≤ -C -1 ν , (5.100) \n1 5 ≤ ∂ v r ≤ 1 , (5.101) \nπ (spt f ) ∩ R ˘ v,v f far ⊂ W , (5.102) \nW . = { 6 C ν u ≤ v } ∩ { v ≥ ˘ v } \nand the constant 10 ≤ C ν ≲ 1 is chosen so that -1 2 C ν ≤ ∂ u r ≤ -2 C -1 ν on C ˘ v . Such a constant exists by Lemmas 5.9 and 5.12. \nLemma 5.16. If (5.15) holds, v f ≥ ˘ v , R ˘ v,v f far ⊂ U , and the bootstrap assumptions (5.100) -(5.102) hold on R ˘ v,v f far , then \n0 ≤ m ≤ 10 M, 0 ≤ Q ≤ 6 M, (5.103) \nΩ 2 ∼ 1 , (5.104) \n∂ v log Ω 2 ≲ v -2 , (5.105) \n1 2 ≤ 1 -2 m r ≤ 1 (5.106) \nr ∼ v, (5.107) \n∂ v log Ω 2 < 2 ∂ v r r (5.108) \nm = Q = 0 (5.109) \non R ˘ v,v f far \\ W . \nProof. Proof of (5.103) and (5.109): The bootstrap assumptions (5.100) and (5.101) imply ∂ u r < 0 and ∂ v r > 0 . Therefore, (5.103) follows from the monotonicity properties of the Einstein-Maxwell-Vlasov system, Lemmas 5.8, 5.9, and 5.12, and the boundary condition (3.82). \nProof of (5.107): By the bootstrap assumption (5.100), \nr ( u, v ) -r ( -r 2 , v ) ≥ -C ν ( u + r 2 ) ≥ -1 6 v -C ν r 2 \nfor ( u, v ) ∈ W . By (5.38), the lower bound in (5.107) easily follows if ˘ v is taken sufficiently large. Since ∂ u r < 0 , r ( u, v ) ≤ r ( -r 2 , v ) ≲ v for v ≥ ˘ v and ˘ v sufficiently large, which proves the upper bound in (5.107). Proof of (5.106): This is immediate for ˘ v chosen sufficiently large in light of (5.103) and the fact that \ninf W r ≳ ˘ v, (5.110) \nwhich follows from (5.107). \nProof of (5.104): This follows from (2.3) by combining the bootstrap assumptions (5.100) and (5.101) with (5.106). \nProof of (5.105): Let ( u, v ) ∈ W . We will show that \n∫ u -r 2 T uv ( u ' , v ) du ' ≲ v -2 , (5.111) \nwhich together with (3.21), (5.103), (5.104), and (5.107), readily implies (5.105). To prove (5.111), we observe that by the bootstrap assumptions (5.100) and (5.101) and the evolution equation (3.28), ∂ u m ≤ 0 . Using that m ≥ 0 in R ˘ v,v f aux , we therefore infer \n∫ u -r 2 [ 1 2 r 2 Ω 2 ( T uv ( -∂ u r ) + T vv ∂ v r ) + Q 2 2 r 2 ( -∂ u r ) ]∣ ∣ ∣ ∣ ( u ' ,v ) du ' = m ( -r 2 , v ) -m ( u, v ) ≤ 2 ϖ 2 . \nSince all three integrands are nonnegative, this energy estimate, taken together with (5.100), (5.104) and (5.107) imply \nv 2 ∫ u -r 2 T uv ( u ' , v ) du ' ≲ ∫ u -r 2 r 2 Ω 2 T uv ∂ v r ∣ ∣ ( u ' ,v ) du ' ≲ 1 \nfor ( u, v ) ∈ W , which proves (5.111). \nProof of (5.108): This follows from the fact that ∂ v r/r ≳ v -1 in W by (5.101) and (5.107). \non R ˘ v,v f far and \non W . Furthermore, \nProof of Lemma 5.15. The proof is a bootstrap argument based on the bootstrap assumptions (5.100)(5.102). Let \nA . = { v f ∈ [˘ v, ∞ ) : R ˘ v,v f aux ⊂ U and (5.100)-(5.102) hold on R ˘ v,v 0 aux } . \nThe set A is nonempty by Proposition 3.23, Lemma 5.9, and Lemma 5.12. It is also manifestly closed by continuity of the bootstrap assumptions. We now show that if (5.15) holds, then A is also open. Let v f ∈ A . ˘ v,v ' \nImproving (5.101): Let ( u, v ) ∈ R f aux . Integrating the wave equation (3.20) in u starting at u = -r 2 and using the estimates of Lemma 5.16 yields \n| ∂ v r ( u, v ) -1 2 | ≤ ∫ u -r 2 ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 4 T uv )∣ ∣ ∣ ∣ ( u ' ,v ) du ' ≲ v -1 ≤ ˘ v -1 , \nwhich improves (5.101) for ˘ v sufficiently large. \nImproving (5.100): Using (3.29), ∂ v m ≥ 0 , and (5.110), we obtain (similarly to (5.111)) \n∫ v 2 v 1 rT uv | ( u,v ' ) dv ' ≲ ˘ v -1 (5.112) \nfor any ( u, v 1 ) , ( u, v 2 ) ∈ R ˘ v,v f aux . Let ( u, v ) ∈ R ˘ v,v f aux . We integrate the wave equation (3.20) in v starting at C ˘ v if u ≤ ˘ v and at ( u, u ) ∈ Γ if u > ˘ v . In the former case, \n| ∂ u r ( u, v ) -∂ u r ( u, ˘ v ) | ≤ ∫ v ˘ v ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 2 T uv )∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ ˘ v -1 , \nwhich improves (5.100) for ˘ v sufficiently large by the definition of C ν . In the latter case, the boundary condition (3.82) implies ∂ u r ( u, u ) = -∂ v r ( u, u ) , so \n| ∂ u r ( u, v ) + ∂ v r ( u, u ) | ≤ ∫ v u ( Ω 2 2 r 2 ( m + Q 2 2 r ) + 1 4 r Ω 2 T uv )∣ ∣ ∣ ∣ ( u,v ' ) dv ' ≲ ˘ v -1 , \nwhich by (5.101) improves (5.100) for ˘ v sufficiently large by the definition of C ν . \nImproving (5.102): Let γ : [0 , S ) →R ˘ v,v f aux be an electromagnetic geodesic in the support of f starting at C ˘ v at s = 0 . By (2.24) and (5.108), \nd ds (Ω 2 p u ) ≤ 0 , \nso by (5.104), \np u ( s ) ≲ p u (0) . (5.113) \nUsing (2.22), the signs of ∂ u r and Q , and parametrizing γ by v yields \nd dv log p v ≥ -∂ v log Ω 2 . (5.114) \nBy (5.105), it is easy to see that \nand therefore \n∫ v ˘ v | ∂ v log Ω 2 ( γ u ( v ' ) , v ' ) | dv ' ≲ ˘ v -1 \nexp ( -∫ v ˘ v ∂ v log Ω 2 ( γ u ( v ' ) , v ' ) dv ' ) ≥ 1 2 \nfor ˘ v sufficiently large. It follows from (5.114) that \np v ( s ) ≳ p v (0) . (5.115) \nCombining (5.113) and (5.115) yields \np u p v ( s ) ≤ 1 7 C ν , \nfor ˘ v sufficiently large by (5.49) and (5.84). It is then easy to show that γ ( s ) stays in { 7 C ν u ≤ v } for every s ∈ [0 , S ) , which quantitatively improves (5.102). The rest of the existence and uniqueness proof now follows a standard continuity argument using Proposition 3.17 and Lemma 3.27. \nTo estimate | ∂ u Ω 2 | , note that | ∂ u Ω 2 | ≲ (1 + u 2 ) -1 along C ˘ v ∪ Γ by Lemma 5.12 and (3.84). Observe that \ninf C u ∩W∩R ˘ v, ∞ far r ∼ inf C u ∩W∩R ˘ v, ∞ far v = max { ˘ v, 6 C ν u } ≳ 1 + u, \nso the energy estimate (5.112) can be improved to \n∫ v 2 v 1 T uv ( u, v ' ) dv ' ≲ (1 + u ) -2 \nfor any u ≤ v 1 ≤ v 2 . Therefore the desired estimate can be propagated to the interior by integrating the wave equation (3.21) in v . Together with the bootstrap assumptions and Lemma 5.16, this completes the proof of the estimates (5.97)-(5.99).", '5.7 The dispersive estimate in the massive case': "Let m > 0 and consider the solution ( r, Ω 2 , Q, f ) given by Lemma 5.15, defined globally on C r 2 . We augment the hierarchy (5.15) with a large parameter v # satisfying \n0 < v -1 # ≪ m ≤ m 0 (5.116) \nand aim to prove the following \nLemma 5.17. For any ˘ v, η, ε, m 0 , m , and v # satisfying (5.15) and (5.116) , we have the decay \nM≤ Cv -3 \nfor v ≥ v # and any M∈{ N u , N v , T uu , T uv , T vv , S } , where C may depend on η, ε , m , and v # . \nThe proof is based on a bootstrap argument for the dispersion of ingoing momentum p u along spt f as v →∞ , which leads to cubic decay of the phase space volume V , which was defined in (5.8). Using the mass shell relation (3.12) and the change of variables formula, we have \nV ( u, v ) = 2 r 2 ∫ ∞ 0 ∫ { p u : f ( u,v,p u ,p v ) =0 } dp u p u ℓ dℓ, (5.117) \n̸ \nwhere we view p v as a function of p u and ℓ . Compare with (5.9). \nLemma 5.18. If (5.15) and (5.116) hold and ( u, v, p u , p v ) ∈ spt f , then p v ≲ 1 if v ≥ ˘ v and p u ∼ η m 2 if v ≥ v # . \nRemark 5.19 . The estimate p v ≲ 1 also holds in the massless case. The non-decay of p u for massive particles drives the decay rate v -3 , but only at very late times. \nProof. Let γ ∈ Γ f and v ≥ ˘ v . Parametrizing γ by v and using (5.49), (5.66), (5.84), and the estimates in Lemma 5.15, we infer \n∣ ∣ ∣ ∣ d dv (Ω 2 p v ) ∣ ∣ ∣ ∣ ≲ v -2 , \nwhich is integrable and hence shows that p v ≲ 1 . By (5.49) and (5.84), we have p v ≳ η 1 . We then obtain \np u = ℓ 2 r -2 + m 2 Ω 2 p v ∼ η ℓ 2 r 2 + m 2 ∼ m 2 \nfor r ≳ v # sufficiently large. \nUsing this lemma and (5.65), we immediately infer: \n<!-- image --> \nLemma 5.20. If (5.15) and (5.116) hold, ( u, v, p u , p v ) ∈ spt f , and v ≥ v # , then u ∼ η m 2 v . If γ 1 , γ 2 ∈ Γ f ( u, v ) , then we have | γ u 1 ( s 1 v # ) -γ u 2 ( s 2 v # ) | ( s v # ) ≲ η m 2 ( v -v # ) , where s i v # is the parameter time for which γ v i ( s i # ) = v # . \nLet ( u 0 , v 0 ) ∈ C r 2 and let γ ∈ Γ f ( u 0 , v 0 ) have ingoing momentum p u 0 and angular momentum ℓ at ( u 0 , v 0 ) . We parametrize γ by v going backwards in time and denote this by \nγ u ( v ) . = γ u ( v ; u 0 , v 0 , p u 0 , ℓ ) , p u ( v ) . = p u ( v ; u 0 , v 0 , p u 0 , ℓ ) . \nWe readily derive the equations \nd dv γ u = (Ω 2 p u ) 2 Ω 2 ( ℓ 2 r -2 + m 2 ) , (5.118) \nd dv (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 Ω 2 p u r 2 m 2 + ℓ 2 -e Q r 2 m 2 + ℓ 2 (Ω 2 p u ) 2 . (5.119) \nNext, we define the variational quantities \nu ( v ; u 0 , v 0 , p u 0 , ℓ ) . = ∂ ∂p u 0 γ u ( v ; u 0 , v 0 , p u 0 , ℓ ) , p ( v ; u 0 , v 0 , p u 0 , ℓ ) . = ∂ ∂p u 0 (Ω 2 p u )( v ; u 0 , v 0 , p u 0 , ℓ ) , \nwhere we emphasize that the derivative in p u 0 is taken with ℓ fixed. From (5.118) and (5.119) we obtain \nd dv u = 2 p u ℓ 2 r -2 + m 2 p -( p u ) 2 ( ℓ 2 r -2 + m 2 ) 2 [ ∂ u Ω 2 ( ℓ 2 r 2 + m 2 ) -2Ω 2 ℓ 2 ∂ u r r 3 ] u , (5.120) d dv p = [( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 m 2 + ℓ 2 -2 e Q Ω 2 p u r 2 m 2 + ℓ 2 ] p + [( ∂ u ∂ v log Ω 2 -2 ∂ u ∂ v r r +2 ∂ v r∂ u r r 2 ) ℓ 2 Ω 2 p u r 2 m 2 + ℓ 2 -( ∂ v log Ω 2 -2 ∂ v r r ) 2 m 2 r∂ u rℓ 2 Ω 2 p u ( r 2 m 2 + ℓ 2 ) 2 -e (Ω 2 p u ) 2 r 2 m 2 + ℓ 2 ∂ u Q + 2 e Q m 2 r∂ u r (Ω 2 p u ) 2 ( r 2 m 2 + ℓ 2 ) 2 ] u . (5.121) \nNote that u ( v 0 ) = 0 and p ( v 0 ) = Ω 2 ( u 0 , v 0 ) ∼ 1 . \nLemma 5.21. If (5.15) and (5.116) hold, and v ≥ v # , then \nV ( u, v ) ≲ η,ε v # v 3 . (5.122) \nProof. We claim that there exists a constant C ∗ , depending on η and ε , such that \nC -1 ∗ ≤ u ( v ) v -v 0 ≤ C ∗ (5.123) \nfor any v # ≤ v ≤ v 0 . To see how this proves (5.122), let Φ u 0 ,v 0 ,ℓ ( p u 0 ) . = γ u ( v # ; u 0 , v 0 , p u 0 , ℓ ) and observe that Φ ' u 0 ,v 0 ,ℓ ( p u 0 ) = u ( v # ) < 0 . Changing variables in the p u integral in (5.117) to γ u ( v # ) and using Lemma 5.20 to estimate the u -dispersion along C v # , we find \nV ( u 0 , v 0 ) ≲ η,ε 1 r 2 m 2 C ∗ | v # -v 0 | min { m 2 v # , m 2 ( v 0 -v # ) } ≲ C ∗ v # v 3 0 . (5.124) \nWe prove (5.123) by a bootstrap argument as follows. Let v f ≥ v # and assume (5.123) holds for all ( v, u 0 , v 0 , p u 0 , ℓ ) with v # ≤ v ≤ v 0 ≤ v f . The assumption is clearly satisfied for some choice of C ∗ for v f sufficiently close to v # on account of u ( v 0 ) = 0 . \nWe now show that for m sufficiently small and v # sufficiently large, we can improve the constant in (5.123). Using (5.122), we estimate \nN v ≲ η,ε C ∗ v # v 3 , T uv ≲ η,ε C ∗ m 2 v # v 3 \nfor v ∈ [ v # , v f ] . Using this, as well as (3.21), (3.24), Lemma 5.15, and p u ∼ η m 2 in (5.121) yields \n∣ ∣ ∣ ∣ d dv p ∣ ∣ ∣ ∣ ≲ η,ε ( 1 m 2 v 3 + 1 v 2 ) | p | + ( 1 v 4 + m 2 v 3 + C ∗ m 2 v # v 3 ) | u | . \nWe use the bootstrap assumption (5.123) and (5.116) to infer \n∣ ∣ ∣ ∣ d dv ( p -p ( v 0 )) ∣ ∣ ∣ ∣ ≲ η,ε 1 v 2 | p -p ( v 0 ) | + 1 v 2 + C 2 ∗ m 2 v # v 3 | v -v 0 | . \nand then use Grönwall's inequality to obtain \n| p ( v ) -p ( v 0 ) | ≲ η,ε v -1 # + C 2 ∗ m 2 v # v 2 | v -v 0 | . \nUsing p ( v 0 ) ∼ 1 , we therefore have \n∫ v 0 v p ( v ' ) dv ' ∼ η,ε v 0 -v + O η,ε ( C 2 ∗ m 2 | v -v 0 | ) ∼ v 0 -v (5.125) \nfor m sufficiently small. Finally, we use (5.99) and Lemma 5.20 to estimate \n∫ v 0 v | ∂ u Ω 2 | ( γ u ( v ' ) , v ' ) ( ℓ 2 r 2 + m 2 ) dv ' ≲ η ∫ γ u ( v 0 ) γ u ( v ) 1 1 + u ' 2 du ' ≲ 1 γ u ( v # ) ≲ 1 m 2 v # . (5.126) \nIntegrating (5.120) and using (5.125) and (5.126) improves the constant in (5.123) for m sufficiently small and v # sufficiently large, which completes the proof. \nProof of Lemma 5.17. \nImmediate from (5.122).", '5.8 Proof of Proposition 5.5': "Proof. Part 1. This follows immediately from combining Lemmas 5.9, 5.12, and 5.15. \nPart 2. The estimates in (5.16) follow directly from the estimates in Lemmas 5.9, 5.12, and 5.15. We will now prove causal geodesic completeness of the spacetime. Let γ be a future-directed causal geodesic in the (3 + 1) -dimensional spacetime. Then the projection of γ to the reduced spacetime, still denoted γ , satisfies \nd ds (Ω 2 p u ) = ( ∂ v log Ω 2 -2 ∂ v r r ) ℓ 2 r 2 , (5.127) \nd ds (Ω 2 p v ) = ( ∂ u log Ω 2 -2 ∂ u r r ) ℓ 2 r 2 , (5.128) \nΩ 2 p u p v = ℓ 2 r 2 + m 2 , (5.129) \nwhere s is an affine parameter and γ is continued through the center according to Part 4 of Definition 3.21. We will show for any future-directed causal geodesic γ : [0 , S ) →C r 2 , p τ is uniformly bounded in any compact interval of coordinate time 0 ≤ τ ≤ τ 0 along γ . This implies that γ can be extended to [0 , ∞ ) by the normal neighborhood property of the geodesic flow in Lorentzian manifolds [ONe83]. \nBy (5.127)-(5.129) and (5.16), \n∣ ∣ ∣ ∣ d ds (Ω 2 p τ ) ∣ ∣ ∣ ∣ ≲ ( 1 + ∣ ∣ ∣ ∣ ∂ v r + ∂ u r r ∣ ∣ ∣ ∣ ) (Ω 2 p τ ) 2 . (5.130) \nWhen r ≥ 1 6 r 1 , the term in the absolute value on the right-hand side of this estimate is clearly bounded by (5.16). When r ≤ 1 6 r 1 , the spacetime is Minkowski and the formulas in Lemma 3.27 can be used to show that \n∂ v r + ∂ u r ≲ v -u ≲ r. (5.131) \nTherefore, passing to a τ parametrization of γ , (5.130) implies \n∣ ∣ ∣ ∣ d dτ (Ω 2 p τ ) ∣ ∣ ∣ ∣ ≲ Ω 2 p τ , \nand the proof is completed by an application of Grönwall's lemma. 25 \nPart 3. The estimate (5.17) for the final parameters follows from (5.31). The remaining claims in this part follow from the proof of Lemma 5.8, (5.59) on C -2 3 r 1 , and Lemma 5.15. \nPart 4. The estimate (5.18), the upper bound in (5.19), and the claim about the neighborhood of the center follow from Lemma 5.15. The lower bound in (5.19) follows from Lemma 5.18, which implies that γ u grows linearly in v at very late times for any electromagnetic geodesic γ in the support of f . Since a neighborhood of I + is then electrovacuum, it is isometric to an appropriate Reissner-Nordström solution by Birkhoff's theorem. \nPart 5. This follows immediately from Lemma 5.17. \nPart 4 ' . We now take m = 0 and seek to prove the upper bound in (5.21), the rest following immediately from Birkhoff's theorem. Let γ ( s ) be an electromagnetic geodesic lying in the support of f with γ v (0) = ˘ v . When m = 0 , the mass shell relation, together with (5.49), (5.84), and (5.115) gives the estimate \np u p v ( s ) = r 2 (0) r 2 ( s ) Ω 2 (0) Ω 2 ( s ) p u (0) p v ( s ) p v (0) p v ( s ) ≲ v -2 , \nwhich has integral ≲ ˘ v -1 . Therefore, the upper bound in (5.21) follows from (5.65) and taking ˘ v sufficiently large. \nPart 5 ' . When m = 0 , the mass shell relation implies p u ≲ v -2 for any electromagnetic geodesic lying in the support of f . The estimates (5.22)-(5.24) follow from this and the formula (5.9).", '5.9.1 The maximal time-symmetric doubled spacetime': "Let α ∈ P Γ or be of the form ( r 1 , r 2 , 0 , 0 , 0 , 0) . Let η and ε be beam parameters for which the conclusion of Proposition 5.5 holds, recalling also Remark 5.6. Let ˜ M and ˜ e denote the final Reissner-Nordström parameters of S = S α,η,ε . We say that S is subextremal if ˜ e < ˜ M , extremal if ˜ e = ˜ M , and superextremal if ˜ e > ˜ M . 26 If S is not superextremal, we may define \nr ± = ˜ M ± √ ˜ M 2 -˜ e 2 , \nwhich is the formula for the area radii of the outer and inner horizons in Reissner-Nordström. \nThe following lemma is an easy consequence of the well-known structure of the maximally extended Reissner-Nordström solution: \nLemma 5.22. For any ˜ M, ˜ e > 0 and 0 < r 2 < r -(if ˜ e ≤ ˜ M ), there exists a relatively open set \nE ˜ M, ˜ e,r 2 ⊂ { u ≤ -r 2 } ∩ { v ≥ r 2 } ⊂ R 2 u,v \nand an analytic spherically symmetric solution ( r, Ω 2 , Q ) of the Einstein-Maxwell equations on E ˜ M, ˜ e,r 2 with the following properties: The renormalized Hawking mass ϖ = ˜ M globally, the charge Q = ˜ e globally, r ( -r 2 , r 2 ) = r 2 , ∂ v r ( -r 2 , r 2 ) = -∂ u r ( -r 2 , r 2 ) = 1 2 , and Ω 2 is constant along the cones { u = -r 2 }∩{ v ≥ r 2 } and { v = r 2 } ∩ { u ≤ -r 2 } . Moreover, we may choose ( E ˜ M, ˜ e,r 2 , r, Ω 2 , Q ) to be maximal with these properties, and it will then be unique. \n̸ \nRemark 5.23 . The (3 + 1) -dimensional lift of ( E ˜ M, ˜ e,r 2 , r, Ω 2 , Q ) is isometric to a subset of the maximally extended Reissner-Nordström solution with parameters ˜ M and ˜ e . The hypersurface ( { τ = 0 }∩E ˜ M, ˜ e,r 2 ) × S 2 is then totally geodesic. \nRemark 5.24 . If ˜ M < ˜ e , then E ˜ M, ˜ e,r 2 = { u ≤ -r 2 } ∩ { v ≥ r 2 } . In the case ˜ e ≤ ˜ M , E ˜ M, ˜ e,r 2 is a strict subset of { u ≤ -r 2 } ∩ { v ≥ r 2 } with this choice of gauge. \nDefinition 5.25. Let ( C r 2 , r, Ω 2 , Q, f ) be the maximal normalized development of S = S α,η,ε for particles of mass m with final Reissner-Nordström parameters ( ˜ M, ˜ e ) for which the conclusion of Proposition 5.5 holds, recalling also Remark 5.6. Let \n˜ C r 2 . = { τ ≤ 0 } ∩ { v ≥ u } ∩ { v ≤ r 2 } \nbe the time-reflection of C r 2 . For ( u, v ) ∈ ˜ C r 2 , define \n˜ r ( u, v ) = r ( -v, -u ) , ˜ Ω 2 ( u, v ) = Ω 2 ( -v, -u ) , ˜ Q ( u, v ) = Q ( -v, -u ) , ˜ f ( u, v, p u , p v ) = f ( -v, -u, p v , p u ) . \nLet M . = C r 2 ∪ ˜ C r 2 ∪E ˜ M, ˜ e,r 2 and define ( r, Ω 2 , Q, f ) on M by simply gluing the corresponding functions across the boundaries of the sets C r 2 , ˜ C r 2 , and E ˜ M, ˜ e,r 2 . (We therefore now drop the tilde notation on the solution in C r 2 , except for in the proof of Lemma 5.26 below.) The tuple ( M , r, Ω 2 , Q, f ) is called the maximal time-symmetric doubled spacetime associated to S . \nLemma 5.26. ( M , r, Ω 2 , Q, f ) is a smooth solution of the spherically symmetric Einstein-Maxwell-Vlasov system. The hypersurface { τ = 0 }∩ M is a totally geodesic hypersurface once lifted to the (3+1) -dimensional spacetime by Proposition 3.10. \nProof. This is immediate except perhaps across { τ = 0 } ∩ { v ≤ r 2 } . We only have to show that the solution is C 2 × C 2 × C 1 × C 1 regular across this interface by the regularity theory of Proposition 3.14. By Definition 3.18 and Definition 3.21, the solution is clearly C 1 × C 1 × C 0 × C 0 regular. Using Raychaudhuri's equations (3.22), (3.23) and Maxwell's equations (3.24), (3.25), it is easy to see that r is C 2 and Q is C 1 across { τ = 0 } . To check second derivatives of Ω 2 , differentiate ( ∂ u + ∂ v )Ω 2 ( -v, v ) = 0 in v to obtain ∂ 2 u Ω 2 ( -v, v ) = ∂ 2 v Ω 2 ( -v, v ) , which together with the wave equation (3.21) implies C 2 matching. The p u and p v derivatives of f are also continuous by inspection and continuity of spatial derivatives can be proved as follows: On { τ = 0 } , ∂ v f can be eliminated in terms of ∂ u f and d dv ˚ f by (3.77). Then the Maxwell-Vlasov equation (3.26) can be solved for ∂ u f . Performing the same calculation for ˜ f shows that ∂ u f is continuous across { τ = 0 } . The same argument applies for ∂ v f and the proof is complete.", '5.9.2 The anchored Cauchy hypersurface': "In order to define the family of Cauchy data in Theorem 1, we need to identify an appropriate Cauchy hypersurface in each M . Let \nv ∞ . = { 2 r --r 2 if ˜ e ≤ ˜ M ∞ if ˜ e > ˜ M (5.132) \nand let Σ u : [ -50 ˜ M,v ∞ ) → R be the smooth function defined by \nΣ u ( v ) = -100 ˜ M -v \nfor v ∈ [ -50 ˜ M,r 1 ] , by solving the ODE \nd dv Σ u ( v ) = ∂ v r ∂ u r ∣ ∣ ∣ ∣ (Σ u ( v ) ,v ) (5.133) \nwith initial condition Σ u ( r 2 ) = -100 ˜ M -r 2 for v ∈ [ r 2 , v ∞ ) , and by applying the following easy consequence of Borel's lemma for v ∈ [ r 1 , r 2 ] : \nLemma 5.27. Given 0 < r 1 < r 2 , ˜ M > 0 , and a sequence of real numbers a 1 , a 2 , . . . with a 1 < 0 , there exists a smooth function Σ u : [ r 1 , r 2 ] → [ -100 ˜ M -r 2 , -100 ˜ M -r 1 ] such that d dv Σ u ( v ) < 0 for v ∈ [ r 1 , r 2 ] , Σ u has Taylor coefficients ( -100 ˜ M -r 1 , -1 , 0 , 0 , . . . ) at r 1 , and Taylor coefficients ( -100 ˜ M -r 2 , a 1 , a 2 , . . . ) at r 2 . Moreover, if ˜ M and each a j are smooth functions of the parameters ϖ 2 , Q 2 , ε, η , then f can be chosen to depend smoothly on ϖ 2 , Q 2 , ε, η . \nRemark 5.28 . For v ≥ r 2 , the curve Σ : v ↦→ (Σ u ( v ) , v ) ∈ E ˜ M, ˜ e,r 2 lies in the domain of outer communication if ˜ e ≤ ˜ M and is contained in a constant time hypersurface in Schwarzschild coordinates. Indeed, the timetranslation Killing vector field in E ˜ M, ˜ e,r 2 is given by the Kodama vector field \nK . = 2Ω -2 ∂ v r ∂ u -2Ω -2 ∂ u r ∂ v , \nwhich is clearly orthogonal to Σ .", '5.9.3 Cauchy data for the Einstein-Maxwell-Vlasov system': 'Let ( M , g, F, f ) be a solution of the (3 + 1) -dimensional Einstein-Maxwell-Vlasov system as defined in Section 3.1. Let i : R 3 → ˜ Σ ⊂ M be a spacelike embedding with future-directed unit timelike normal n to ˜ Σ . As usual, we may consider the induced metric ¯ g . = i ∗ g and second fundamental form k (pulled back to R 3 along i ) of ˜ Σ . The electric field is defined by ¯ E . = i ∗ F ( · , n ) and the magnetic field is defined by ¯ B . = ( ⋆ ¯ g i ∗ F ) ♯ , where ♯ is taken relative to ¯ g . Since the domain of f is the spacetime mass shell P m which is not intrinsic to ˜ Σ , one first has to define a projection procedure to T ˜ Σ ∼ = T R 3 , after which the restriction of f to ˜ Σ can be thought of as a function ¯ f : T R 3 → [0 , ∞ ) . Similarly, the volume forms in the Vlasov energy momentum tensor T and number current N have to be written in terms of ¯ g , after which ¯ ρ T . = i ∗ T ( n, n ) , ¯ j T . = i ∗ T ( n, · ) and ¯ ρ N . = i ∗ N ( n ) can be evaluated on ˜ Σ only in terms of ¯ g and ¯ f . For details of this procedure, we refer to [Rin13, Section 13.4]. \nDefinition 5.29. A Cauchy data set for the Einstein-Maxwell-Vlasov system for particles of mass m and fundamental charge e consists of the tuple Ψ = (¯ g, ¯ k, ¯ E, ¯ B, ¯ f ) on R 3 satisfying the constraint equations 27 \nR ¯ g -| ¯ k | 2 ¯ g +(tr ¯ g ¯ k ) 2 = | ¯ E | 2 ¯ g + | ¯ B | 2 ¯ g +2¯ ρ T [ ¯ f ] , div ¯ g ¯ k -d tr ¯ g ¯ k = -( ⋆ ¯ g ( ¯ E ♭ ∧ ¯ B ♭ )) ♯ + ¯ j T [ ¯ f ] div ¯ g ¯ E = e ¯ ρ N [ ¯ f ] , div ¯ g ¯ B = 0 . \nWe denote by M ∞ ( R 3 , m , e ) the set of solutions Ψ of the Einstein-Maxwell-Vlasov constraint system on R 3 with the C ∞ loc subspace topology.', '5.9.4 The globally hyperbolic region': 'By Proposition 5.5, Remark 5.6, their time-reversed ( u ↦→ -v , v ↦→ -u ) versions, Remark 5.28, and the structure of the Reissner-Nordström family, we have: \nProposition 5.30. Let S and ( M , r, Ω 2 , Q, f ) be as in Definition 5.25 and let Σ u ( v ) be the function defined in Section 5.9.2. Let \nX . = { v ≥ u } ∩ { v < v ∞ } ⊂ M \nand let Σ = { (Σ u ( v ) , v ) : v ∈ [ -50 ˜ M,v ∞ ) } . Then the following holds: \n- 1. The manifold M . = (( X \\ Γ) × S 2 ) ∪ Γ with metric g = -Ω 2 dudv + r 2 γ , electromagnetic field, and Vlasov field lifted according to Proposition 3.10 is a globally hyperbolic, asymptotically flat spacetime, free of antitrapped surfaces, with Cauchy hypersurface ˜ Σ . = ((Σ \\ Γ) × S 2 ) ∪ (Σ ∩ Γ) .\n- 2. ( M , g ) possesses complete null infinities I ± and is past causally geodesically complete.\n- 3. If ˜ e > ˜ M , then ( M , g ) is future causally geodesically complete. \n- 4. If ˜ e ≤ ˜ M , there are two options. (If S α,η,ε is to be untrapped, then necessarily r 2 / ∈ [ r -, r + ] .) \n̸ \n- (a) If r 2 < r -, then ( M , g ) is future causally geodesically incomplete. The spacetime contains a nonempty black hole region, i.e., BH . = M\\ J -( I + ) = ∅ . The Cauchy hypersurface ˜ Σ is disjoint from BH . The event horizon H + = ∂ ( BH ) is located at u = r 2 -2 r + .\n- (b) If r 2 > r + , then ( M , g ) is future causally geodesically complete.', '5.9.5 Proof of the main theorem': "Proof of Theorem 1. Fix a fundamental charge e > 0 , cutoff functions φ , θ , and ζ as in Section 5.2.1, and a number Λ ≥ 1 satisfying (5.11). Fix the extremal black hole target mass M > 0 and let 0 < r 1 < r 2 and δ > 0 be as in Theorem 4. Let η 0 > 0 be such that Proposition 5.5 applies to the multiparameter family of seed data S λ,M ' ,η,ε (which was defined in Definition 5.4) for λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , 0 < η ≤ η 0 , ε > 0 sufficiently small depending on η , and particle mass 0 ≤ m ≤ m 0 , where m 0 is sufficiently small depending on η and ε . \nLet F : [0 , ∞ ) 2 → [0 , ∞ ) 2 be defined by F ( λ, M ' ) = ( λ 2 M ' , λM ' ) , which is easily verified to be smoothly invertible on (0 , ∞ ) 2 . Define the function F η,ε ( λ, M ' ) = ( ˜ M, ˜ e ) , the final Reissner-Nordström parameters of S λ,M ' ,η,ε . By (5.17), we find \n| F η,ε ( λ, M ' ) -F ( ζ ( λ ) , M ' ) | ≲ η (5.134) \nfor λ ∈ [ -1 , 2] , | M ' -M | ≤ δ , and 0 < η ≤ η 0 . There exists a constant 0 < λ 0 ≪ 1 depending on η 0 such that if λ ∈ [ -1 , λ 0 ] , then | F η,ε ( λ, M ' ) | < 1 2 r 1 . Also, (5.134) implies \nsup λ 0 ≤ λ ≤ 2 , | M ' -M |≤ δ ∣ ∣ ∣ ∣ ˜ e ˜ M -1 ζ ( λ ) ∣ ∣ ∣ ∣ ≲ λ 0 η. (5.135) \nFrom smooth convergence of F η,ε to F as η → 0 , we obtain \nsup λ 0 ≤ λ ≤ 2 , | M ' -M |≤ δ ∣ ∣ ∣ ∣ d dλ ( ˜ e ˜ M ) + ζ ' ( λ ) ζ ( λ ) 2 ∣ ∣ ∣ ∣ ≲ λ 0 η. (5.136) \nIt follows that the charge to mass ratio ˜ e/ ˜ M is strictly decreasing as a function of λ , for λ ≥ λ 0 . \nOn any fixed neighborhood of (1 , M ) ∈ R 2 , F -1 · F η,ε converges uniformly to the identity map by (5.134). Therefore, by a simple degree argument 28 we find an assignment ( η, ε ) → ( λ ( η, ε ) , M ' ( η, ε )) such that F η,ε ( λ ( η, ε ) , M ' ( η, ε )) = ( M,M ) . We claim that for η sufficiently small, the family of seed data λ ↦→ S λ . = S λ,M ' ( η,ε ) ,η,ε gives rise to the desired family of spacetimes and Cauchy data: \n- 1. For λ ∈ [ -1 , λ 0 ] , the final Reissner-Nordström parameters of S λ are < 1 2 r 1 by definition of λ 0 and hence the globally hyperbolic spacetime D λ associated to S λ by Proposition 5.30 is future causally geodesically complete and dispersive. At λ = -1 , the seed data is trivial and hence the development is isometric to Minkowski space.\n- 2. For λ ∈ [ λ 0 , λ ( η, ε )) , the final charge to mass ratio ˜ e/ ˜ M is strictly decreasing towards 1 by (5.136). Therefore, D λ is future causally geodesically complete and dispersive. A neighborhood of spatial infinity i 0 is isometric to a superextremal Reissner-Nordström solution.\n- 3. λ = λ ( η, ε ) is, by construction, critical, with parameter ratio ˜ e/ ˜ M = 1 . D λ contains a nonempty black hole region and for sufficiently large advanced time, the domain of outer communication and event horizon are isometric to an appropriate portion of an extremal Reissner-Nordström black hole.\n- 4. For λ ∈ ( λ ( η, ε ) , 2] , ˜ e/ ˜ M decreases away from 1 by (5.136). By definition of r 2 , r -> r 2 for η sufficiently small and hence D λ contains a nonempty black hole region and for sufficiently large advanced time, the domain of outer communication and event horizon are isometric to an appropriate portion of a subextremal Reissner-Nordström black hole. By (5.135), the charge to mass ratio at λ = 2 can be made arbitrarily close to 1 2 . \nTo complete the proof, we assign a smooth family of Cauchy data to D λ . Let i λ : [0 , ∞ ) → Σ λ be the arc length parametrization (with respect to the metric g λ ) of the Cauchy surface associated to D λ by Proposition 5.30. Then we may define the embedding ˜ i λ = i λ × id S 2 / ∼ : R 3 → ˜ Σ λ ⊂ M λ (where the central sphere is collapsed to a point). The natural map λ ↦→ Ψ λ , where Ψ λ is the Cauchy data induced on ˜ Σ λ by pullback along ˜ i λ , is smooth. This completes the proof of the theorem.", '5.10 Weak* convergence to dust': "In this section, we show that the spacetimes constructed in Proposition 5.30 weak* converge in an appropriate sense to the bouncing charged null dust spacetimes given by Proposition 4.7. First, we show convergence of an outgoing Vlasov beam to the underlying outgoing formal dust beam: \nProposition 5.31. Let α ∈ P Γ , let { η i } and { ε j } be decreasing sequences of positive numbers tending to zero, let ( r, Ω 2 , Q, f ) be the solution of the Einstein-Maxwell-Vlasov system associated to ( α, η i , ε j ) by Proposition 5.5 for j ≫ i and arbitrary allowed mass, and let ( r d , Ω 2 d , Q d , N v d , T vv d ) be the outgoing formal dust solution from Section 4.6.2 on C r 2 . Then the following holds for any relatively compact and relatively open set U ⊂ C r 2 : \n- 1. We have the following strong convergence: \nlim i →∞ j ≫ i ( ∥ r -r d ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ∥ C 1 ( U ) + ∥ Q -Q d ∥ C 0 ( U ) + ∥ T uu ∥ C 0 ( U ) + ∥ T uv ∥ C 0 ( U ) ) = 0 , (5.137) \nwhere the limit is to be understood as taking i → ∞ while keeping j sufficiently large for each i such that the conclusion of Proposition 5.5 applies for S α,η i ,ε j . \n- 2. If U ' ⊂ U is disjoint from a neighborhood of { τ = 0 } , then \nlim i →∞ j ≫ i ∥ N u ∥ C 0 ( U ' ) = 0 . (5.138) \n- 3. We have the following weak* convergence: For any φ ∈ C 1 c ( R 2 ) with spt φ ∩ C r 2 ⊂ U , \nlim i →∞ j ≫ i ∫ U ( N u , N v , T vv ) φdudv = ∫ U (0 , N v d , T vv d ) φdudv. (5.139) \n- 4. We have the following weak* convergence: For any φ ∈ L 1 ( U ) , \nlim i →∞ lim j →∞ ∫ U ( N u , N v , T vv ) φdudv = ∫ U (0 , N v d , T vv d ) φdudv. (5.140) \nProof. It is clear from the estimates used in the proof of Lemma 5.12 that the Vlasov solutions converge strongly to Minkowski space in the regions R ˘ v aux . It therefore suffices to prove the proposition only for the case U = R 1 2 ˘ v -1 3 r 1 main , where we use the estimates of Lemma 5.9. \nPart 1. Using Lemma 5.8, (3.25), and (5.78), we already obtain \n| Q ( u, v ) -ˇ Q ( u ) | ≲ η \nfor any ( u, v ) ∈ U . = R 1 2 ˘ v -1 3 r 1 main and note that ˇ Q ( u ) = Q d ( u, v ) by (4.29). Using this estimate, Lemma 5.8, (5.78), and Grönwall's inequality on the differences of (3.20), (3.21) and (4.24), (4.25), we readily infer \n| r -r d | + | Ω 2 -Ω 2 d | ≲ η \non U , which completes the proof of (5.137). \nPart 2. This follows immediately from (5.75) since τ is bounded below on U ' . \nPart 3. Since φ is bounded, \n∣ ∣ ∣ ∣ ∫ U N u φdudv ∣ ∣ ∣ ∣ ≲ ∫ -2 3 r 1 -r 2 ∫ ˘ v -u N u dvdu ≲ ε 1 / 2 \nby (5.78). Next, let ˜ φ . = φ/ ( -e r 2 Ω 2 ) , use Maxwell's equation (3.24), and integrate by parts: \n∫ U N v φdudv = ∫ U ∂ u Q ˜ φdudv = -∫ U Q∂ u ˜ φdudv + ∫ ∂U Q ˜ φ (5.141) \nBy Part 1, Q∂ u ˜ φ → Q d ∂ u ˜ φ d and Q ˜ φ → Q d ˜ φ d uniformly as i → ∞ and j ≪ i , where ˜ φ d . = φ/ ( -e r 2 d Ω 2 d ) . Therefore, passing to the limit, we have \nlim i →∞ j ≫ i ∫ U N v φdudv = -∫ U Q d ∂ u ˜ φ d dudv + ∫ U Q d ˜ φ d = ∫ U ∂ u Q d ˜ φ d = ∫ U N v d dudv, \nwhere we have again integrated by parts and used (4.28). A similar argument applies for the convergence of T vv using the Raychaudhuri equations (3.22) and (4.26). This completes the proof of (5.139). \nPart 4. We first fix i and take j →∞ . By (5.75), N u ≲ i 1 , for every j ≪ i , where we use the notation A ≲ i B to denote A ≤ CB , where C may depend on i . By the Banach-Alaoglu theorem, there exists a subsequence j n and an L ∞ ( U ) function h such that N u ∗ ⇀h in L ∞ ( U ) . However, by (5.138), it is clear that h = 0 almost everywhere. Since the subsequential limit is unique, N u ∗ ⇀ 0 as j → ∞ . This completes the proof of (5.140) for N u . \nLet ˚ m V ,i and ˚ Q V ,i be the values of ˚ m and ˚ Q at r = 2 3 r 1 for the Vlasov seed S α,η i ,ε j . Note that these numbers do not depend on j . Let ˚ m i and ˚ Q i be the solutions of the system (4.35) and (4.36) on [ 2 3 r 1 , r 2 ] with initial conditions ˚ m V ,i and ˚ Q V ,i at r = 2 3 r 1 , current ˚ N v given by (4.38), and identically vanishing energy-momentum tensor ˚ T vv . Following the proof of Proposition 4.19, we obtain a unique smooth solution ( r d ,i , Ω 2 d ,i , Q d ,i , N v d ,i , T vv d ,i ) of the formal outgoing charged null dust system on U which attains the initial data just described. \nBy repeating the proof of Part 1 of the present proposition, we see that \nlim j →∞ ( ∥ r -r d ,i ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ,i ∥ C 1 ( U ) + ∥ Q -Q d ,i ∥ C 0 ( U ) ) = 0 \nfor fixed i . Arguing as in Part 2, we then see that \nlim j →∞ ∫ U ( N v , T vv ) φdudv = ∫ U ( N v d ,i , T vv d ,i ) φdudv (5.142) \nfor any φ ∈ C 1 c ( R 2 ) . Since N v , T vv ≲ i 1 , a standard triangle inequality argument shows that φ can be replaced by any L 1 function in (5.142). Now it follows by construction that \n| N v d ,i -N v d | + | T vv d ,i -T vv d | ≲ η i \non U , so we can safely take i →∞ \nin (5.142), which completes the proof of (5.140). \nIn order to globalize this, we must first define bouncing charged null dust in the formal system in double null gauge. So consider again the outgoing solution ( r d , Ω 2 d , Q d , N v d , T vv d ) on C r 2 with seed data ( ˚ N v d , 0 , r 2 , e ) given by (4.38). As in Definition 5.25, we extend r d , Ω 2 d , and Q d to ˜ C r 2 by reflection, and set \nN u d ( u, v ) = N v d ( -v, -u ) , T uu d ( u, v ) = T vv d ( -v, -u ) \nfor ( u, v ) ∈ ˜ C r 2 \\ { τ = 0 } . We extend N v d and T vv d to zero in ˜ C r 2 \\ { τ = 0 } and similarly extend N u d and T uu d to zero in C r 2 . Using Lemma 5.22, we attach a maximal piece of Reissner-Nordström with parameters ϖ 2 and Q 2 to C r 2 ∪ ˜ C r 2 . Let v ∞ be as defined in (5.132). \nDefinition 5.32. The globally hyperbolic bouncing formal charged null dust spacetime associated to a set of parameters α ∈ P Γ is the tuple ( X d , r d , Ω 2 d , Q d , N u d , N v d , T uu d , T vv d ) , where X d . = { v ≥ u } ∩ { v < v ∞ } . \nFor τ ≥ 0 , ( r d , Ω 2 d , N v d , T vv d ) solves the outgoing formal dust system and for τ < 0 , ( r d , Ω 2 d , N u d , T uu d ) solves the ingoing formal dust system. The functions r d and Ω 2 d are C 1 across { τ = 0 } and Q d , T uu d , and T vv d are C 0 across { τ = 0 } . The currents N u d and N v d are discontinuous across { τ = 0 } (since we extended by zero), but of course N u d = N v d at { τ = 0 } . \nUsing this definition and Proposition 5.31, we immediately obtain the following \nTheorem 6. Let α ∈ P Γ , let { η i } and { ε j } be decreasing sequences of positive numbers tending to zero, let ( X , r, Ω 2 , Q, f ) be the globally hyperbolic solution of the Einstein-Maxwell-Vlasov system associated to ( α, η i , ε j ) by Proposition 5.30 for j ≪ i and arbitrary allowed mass, and let ( X d , r d , Ω 2 d , Q d , N u d , N v d , T uu d , T vv d ) be the globally hyperbolic bouncing formal charged null dust spacetime associated to α by Definition 5.32. Then the following holds for any relatively compact open set U ⊂ X d :", '1. We have the following strong convergence:': 'lim i →∞ j ≫ i ( ∥ r -r d ∥ C 1 ( U ) + ∥ Ω 2 -Ω 2 d ∥ C 1 ( U ) + ∥ Q -Q d ∥ C 0 ( U ) + ∥ T uv ∥ C 0 ( U ) ) = 0 . \n- 2. We have the following weak* convergence: For any φ ∈ C 1 c ( U ) , \nlim i →∞ j ≫ i ∫ U ( N u , N v , T uu , T vv ) φdudv = lim i →∞ j ≪ i ∫ U ( N u d , N v d , T uu d , T vv d ) φdudv. \n- 3. We have the following weak* convergence: For any φ ∈ L 1 ( R 2 ) with spt φ ⊂ U , \nlim i →∞ lim j →∞ ∫ U ( N u , N v , T uu , T vv ) φdudv = lim i →∞ j ≪ i ∫ U ( N u d , N v d , T uu d , T vv d ) φdudv. \nRemark 5.33 . The globally hyperbolic region X depends on η i and ε j , but it always holds that U ⊂ X for i sufficiently large and j ≫ i .', '6 The third law and event horizon jumping at extremality': 'Using the technology of bouncing Vlasov beams developed in the proof of Theorem 1, we are now able to quickly prove Theorems 2 and 3. As complete proofs would require more lengthy setup, we only sketch the proofs of these results. We refer the reader back to Section 1.6 for the theorem statements and discussion.', '6.1 Counterexamples to the third law': "Refer to Fig. 15 for global Penrose diagrams. \nProof of Theorem 2. Let α ∈ P be third law violating dust parameters as in Theorem 5. We desingularize this dust beam as in the proof of Theorem 1, noting that the charge on the inner edge of the beam is bounded below and hence no auxiliary beam is required. In order to achieve extremality we must modulate α slightly as in Theorem 1, but all required inequalities for this construction are strict, so this can be done. By this procedure we obtain the time-symmetric solution D ext depicted in Fig. 16 below. \nLet ( ϖ 1 , Q 1 ) be the initial Reissner-Nordström parameters of α , which will also be the initial parameters of D ext . Using the same methods as Theorem 1 and Lemma 4.10, we can construct a solution D sub of Einstein-Maxwell-Vlasov collapsing to a subextremal Reissner-Nordström black hole with parameters ϖ 1 and Q 1 . In the case of massless particles, the desired third law violating spacetime is then obtained by deleting an appropriate double null rectangle from D sub and gluing in an appropriate piece of D ext . In the case of massive particles, the beams from D ext and D sub will possibly interact in the early past, but as is clear from the proof of Proposition 5.5, this happens in the dispersive region and the proof of Lemma 5.15 can be repeated to show global existence and causal geodesic completeness in the past. \nFigure 15: Penrose diagrams of counterexamples to the third law of black hole thermodynamics in the Einstein-Maxwell-Vlasov model. The disconnected thick black curve denotes the outermost apparent horizon A ' , which jumps outward as the black hole becomes extremal. This behavior is necessary in third law violating spacetimes which obey the weak energy condition, see [KU22, Proposition 1.1] and [Isr86]. \n<!-- image -->", '6.2.1 General results for spherically symmetric event horizons': "We now consider general weakly tame spherically symmetric Einstein-matter systems, i.e., those satisfying the dominant energy condition and the weak extension principle. We refer to [Daf05b; Kom13] for the precise definition of the weak extension principle, but note that it is a strictly weaker condition than the generalized extension principle as formulated in Proposition 3.15 and therefore holds for the Einstein-Maxwell-KleinGordon, Einstein-Higgs, and Einstein-Maxwell-Vlasov systems [Chr93; Daf05a; Kom13; DR16]. \nLet Ψ = (Σ , ¯ g, ¯ k, . . . ) be a spherically symmetric, asymptotically flat Cauchy data set on R 3 with a regular center in the given matter model. We will assume that Ψ contains no spherically symmetric antitrapped surfaces, i.e., that ∂ u r < 0 on Σ . This assumption is physically motivated by the observation that if the maximal Cauchy development D = ( Q , r, Ω 2 , . . . ) does not contain a white hole, then there are no antitrapped surfaces in the spacetime. Furthermore, by Raychaudhuri's equation (2.7) and the dominant energy condition, ∂ u r < 0 is propagated to the future of the Cauchy hypersurface Σ . \nUnder these assumptions, a very general a priori characterization of the boundary of ( Q , r, Ω 2 ) is available due to the work of Dafermos [Daf05b] and we refer to Kommemi [Kom13] for a detailed account. We will utilize the following two facts: \n̸ \nFact 1. If ( Q , r, Ω 2 ) contains a trapped or marginally trapped surface, i.e., ∂ v r ( u 0 , v 0 ) ≤ 0 for some ( u 0 , v 0 ) ∈ Q , then the black hole region is nonempty ( BH . = Q\\ J -( I + ) = ∅ ), future null infinity is complete in the sense of Christodoulou [Chr99a], and ( u 0 , v 0 ) ∈ BH . \nFact 2. The Hawking mass m extends to a (not necessarily continuous) nonincreasing and nonnegative function on future null infinity I + , called the Bondi mass . If the Hawking mass of Σ is bounded, 29 then the Bondi mass is also bounded and the final Bondi mass M f . = inf I + m is finite. Then the 'event horizon Penrose inequality' sup H + r ≤ 2 M f holds and by the no antitrapped surfaces condition we also obtain \nsup BH r ≤ 2 M f . (6.1) \nWe wish to consider sequences of initial data and their developments. In order to compare them, we have to ensure that the double null gauges are synchronized in an appropriate sense. We consider only \nFigure 16: Penrose diagram of the time symmetric Einstein-Maxwell-Vlasov solution D ext interpolating between subextremality and extremality. This diagram is valid for both massive and massless particles. \n<!-- image --> \ndevelopments D = ( Q , r, Ω 2 , . . . ) for which the center Γ is a subset of { u = v } ⊂ R 2 u,v and if i : [0 , ∞ ) → Σ denotes the embedding map of the Cauchy hypersurface into Q , we demand that i (0) ∈ Γ . Clearly, these conditions can always be enforced by an appropriate transformation of the double null gauge. \nAssumption 6.1. Let { Ψ j } be a sequence of spherically symmetric asymptotically flat Cauchy data for a weakly tame Einstein-matter system. Let D j = ( Q j , r j , Ω 2 j , . . . ) denote the maximal development of Ψ j with Cauchy hypersurface Σ j ⊂ Q j and embedding map i j : [0 , ∞ ) → Σ j normalized as above. We assume that Ψ j converges to another data set Ψ ∞ and the developments converge in the following sense: \n- 1. Gauge condition: Let D ∞ denote the maximal development of Ψ ∞ with Cauchy hypersurface and embedding map i ∞ : [0 , ∞ ) → Σ ∞ . We assume that D j and D ∞ have continuously synchronized gauges in the sense that ( u, v ) · i j : [0 , ∞ ) → R 2 converges uniformly on compact sets as j →∞ .\n- 2. Cauchy stability of the area-radius: If U ⊂ Q ∞ is a relatively compact open set, then U ⊂ Q j for j sufficiently large and r j → r ∞ in C 1 ( U ) . \nRemark 6.2 . The notion of continuous synchronization also makes sense for continuous one-parameter families of Cauchy data λ ↦→ Ψ λ . In this case we require the maps ( u, v ) · i λ ( x ) to be jointly continuous in λ and x ∈ [0 , ∞ ) . \nRemark 6.3 . The initial data and developments given by Proposition 5.30 are continuously synchronized as functions of the beam parameters ( α, η, ε, m ) . \nProposition 6.4. Let Ψ j → Ψ ∞ be a convergent sequence of one-ended asymptotically flat Cauchy data for a weakly tame spherically symmetric Einstein-matter system, containing no spherically symmetric antitrapped surfaces, and satisfying Assumption 6.1. Assume that the sequence has uniformly bounded Bondi mass and that the development D j of Ψ j contains a black hole for each j ∈ N . Let r j, H + denote the limiting area-radius of the event horizon H + j of D j and u j, H + its retarded time coordinate (also for j = ∞ if D ∞ contains a black hole). Then the following holds: \n- 1. If future timelike infinity i + ∞ is a limit point of the center Γ in D ∞ (in particular, D ∞ does not contain a black hole), then \nlim j →∞ u j, H + = sup Q ∞ u. (6.2) \n- 2. If D ∞ contains a black hole, then\n- (a) The retarded time of the event horizon is lower semicontinuous: \nlim inf j →∞ u j, H + ≥ u ∞ , H + . (6.3) \n- (b) Assume further that there are trapped surfaces asymptoting to i + ∞ in the following sense: Let ( u ∞ ,i + , v ∞ ,i + ) denote the coordinates 30 of i + ∞ and suppose there exist sequences u 1 > u 2 > · · · → u ∞ ,i + and v 1 < v 2 < · · · → v ∞ ,i + such that ∂ v r ∞ ( u i , v i ) < 0 for every i ≥ 1 . Then (6.3) is upgraded to \nlim j →∞ u j, H + = u ∞ , H + (6.4) \nand it additionally holds that \nlim inf j →∞ r j, H + ≥ r ∞ , H + . (6.5) \nProof. Part 1: The inequality ≤ in (6.2) follows directly from Cauchy stability, which implies the stronger statement \nlim sup j →∞ sup Q j u ≤ sup Q ∞ u. \nWe now prove the inequality ≥ . Let u 0 < sup Q ∞ u and let M be an upper bound for the Bondi mass of the sequence {D j } . Since the cone C u 0 in Q ∞ reaches future null infinity, we can choose v 0 such that r ∞ ( u 0 , v 0 ) > 2 M . By Cauchy stability, r j ( u 0 , v 0 ) > 2 M for j sufficiently large, so ( u 0 , v 0 ) / ∈ BH j by (6.1). This implies u j, H + ≥ u 0 which completes the proof. \nPart 2 (a): The argument to prove (6.3) is the same as the proof of Part 1, since C u 0 reaches future null infinity in Q ∞ for any u 0 < u ∞ , H + . \nPart 2 (b): By (6.3) we must now show that for u 0 > u ∞ , H + , u j, H + ≤ u 0 for j sufficiently large. Let ( u i , v i ) be a trapped sphere for D ∞ as in the statement, with u 0 > u i > u ∞ , H + . By Cauchy stability, ( u i , v i ) is then trapped in D j for j sufficiently large and therefore u i > u j, H + , which completes the proof of (6.4). Using this, we now have by Cauchy stability and monotonicity of r j along H + j that \nlim inf j →∞ r j, H + ≥ lim j →∞ r j ( u j, H + , v 0 ) = r ∞ ( u ∞ , H + , v 0 ) \nfor any v 0 < v ∞ ,i + . Letting v 0 → v ∞ ,i + completes the proof. \nRemark 6.5 . It is natural to ask if the 'reverse' of (6.5), i.e., \nlim sup j →∞ r j, H + ≤ r ∞ , H + , (6.6) \nholds at this level of generality (even assuming trapped surfaces asymptoting towards i + ∞ ). It turns out that (6.6) is false without additional assumptions. On the one hand, assuming additionally a strict inequality in (6.3), a minor modification of the arguments used to show (6.3) can be used to show (6.6). 31 On the other hand, without the assumption of a strict inequality in (6.3), using the ingoing (uncharged) Vaidya metric, one can construct a counterexample to (6.6) which moreover satisfies the asymptotic trapped surface assumption of Part 2 (b). One is to imagine inflating the event horizon of a Schwarzschild black hole by injecting a fixed dust packet at later and later advanced times v ∼ j . In the limit j → ∞ , the dust disappears, and lim sup r j, H + > r ∞ , H + . Curiously, since black holes in the Vaidya model always have trapped surfaces behind the horizon, Part 2 (b) of the proposition implies that u j, H + is actually continuous in this process. This is because injecting a fixed amount of matter at later and later times causes the horizon to move outwards less and less (in u ), causing it to converge back to the original Schwarzschild horizon as j → ∞ . Therefore, in order for (6.6) to hold, one must assume that Ψ j converges to Ψ ∞ in a norm that sufficiently respects the asymptotically flat structure. We emphasize at this point that the conclusions of Proposition 6.4 hold only under an assumption of local Cauchy stability -no asymptotic stability or weighted convergence is required.", '6.2.2 Proof of event horizon jumping in the Einstein-Maxwell-Vlasov model': 'Proof of Theorem 3. This is proved by following the proof of Theorem 2 and varying the final parameters of D ext as in the proof of Theorem 1. \nThis theorem shows that it is not always possible to have equality in (6.3) when D ∞ is extremal: the event horizon can and does jump as a function of the initial data.', 'A The characteristic initial value problem for spherically symmetric nonlinear wave-transport systems': "In this appendix, we prove local well-posedness for the spherically symmetric Einstein-Maxwell-Vlasov system in small characteristic rectangles away from the center. In fact, we consider the general system of equations \n∂ u ∂ v Ψ = F (Ψ , ∂ Ψ , Q, M [ f ] , M uv [ f ]) , (A.1) \n∂ u Q = K (Ψ , M v [ f ]) , (A.2) \nX ( p u , p v , Ψ , ∂ Ψ , Q ) f = 0 , (A.3) \nwhere Ψ : R 2 u,v → R N is a vector-valued function taking the role of the 'wave-type' variables r and Ω 2 , Q : R 2 u,v → R is the charge, f : T R 2 u,v → R ≥ 0 is the distribution function, \nM [ f ] . = ∫ ∞ 0 ∫ ∞ 0 f dp u dp v , M v [ f ] . = ∫ ∞ 0 ∫ ∞ 0 p v f dp u dp v , M uv [ f ] . = ∫ ∞ 0 ∫ ∞ 0 p u p v f dp u dp v \nare moments of f , F = ( F 1 , . . . , F N ) and K are smooth functions of their variables, and X is a vector field on R 2 u,v of the form \nX ( p u , p v , Ψ , ∂ Ψ , Q ) = p u ∂ u + p v ∂ v + ξ u ( p u , p v , Ψ , ∂ Ψ , Q ) ∂ p u + ξ v ( p u , p v , Ψ , ∂ Ψ , Q ) ∂ p v , \nwhere ξ u and ξ v are smooth functions of their variables. Letting a ∈ { u, v } , we can write X using Einstein notation as \nX = p a ∂ a + ξ a ∂ p a . \nWe assume that there exist functions G k : R ≥ 0 → R ≥ 0 for k ≥ 0 such that \n| D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p ξ u ( p u , p v , Ψ , ∂ Ψ , Q ) | + | D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p ξ v ( p u , p v , Ψ , ∂ Ψ , Q ) | ≤ G k ( M ) ⟨ p τ ⟩ 2 -i 2 (A.4) \nif | Ψ | + | ∂ Ψ | + | Q | ≤ M and i 1 + i 2 = k , where ⟨ s ⟩ . = √ 1 + s 2 and p τ . = 1 2 ( p u + p v ) . Here D i 1 Ψ ,∂ Ψ ,Q ∂ i 2 p denotes any expression involving i 1 derivatives in the (Ψ , ∂ Ψ , Q ) -variables and i 2 derivatives in the ( p u , p v ) -variables. We also assume that there exists a constant m ≥ 0 such that \nξ u p v + ξ v p u = 0 (A.5) \nwhenever p u p v = m 2 . These structural assumptions are verified for a renormalized version of the spherically symmetric Einstein-Maxwell-Vlasov system. \nGiven U 0 < U 1 and V 0 < V 1 , let \nC ( U 0 , U 1 , V 0 , V 1 ) . = ( { U 0 } × [ V 0 , V 1 ]) ∪ ([ U 0 , U 1 ] ×{ V 0 } ) , R ( U 0 , U 1 , V 0 , V 1 ) . = [ U 0 , U 1 ] × [ V 0 , V 1 ] . \nWe will consistently omit ( U 0 , U 1 , V 0 , V 1 ) from the notation for these sets. We also define \nP m . = { ( u, v, p u , p v ) ∈ T R 2 : p u ≥ 0 , p v ≥ 0 , p τ > 0 , p u p v ≥ m 2 } , H κ . = { ( u, v, p u , p v ) ∈ T R 2 : p u ≥ 0 , p v ≥ 0 , p τ > κ } \nand set P . = P 0 . A function ϕ : C → R is said to be smooth if it is continuous and ϕ | { U 0 }× [ V 0 ,V 1 ] and ϕ | [ U 0 ,U 1 ] ×{ V 0 } are C ∞ single-variable functions. This definition extends naturally to functions f : P m | C → R . A smooth characteristic initial data set for the system (A.1)-(A.3) consists of a triple ( ˚ Ψ , ˚ Q, ˚ f ) and numbers κ > 0 , σ > 4 , where ˚ Ψ : C → R N , ˚ Q : { U 0 }× [ V 0 , V 1 ] → R , and ˚ f : P m | C → R ≥ 0 are smooth. We additionally assume that spt( ˚ f ) ⊂ H κ | C for some κ > 0 (which is only an extra assumption when m = 0 ) and that \n∥ ˚ f ∥ C k σ ( P | C ) . = ∑ 0 ≤ i 1 + i 2 ≤ k ( sup P m | { U 0 }× [ V 0 ,V 1 ] ⟨ p τ ⟩ σ + i 2 | ∂ i 1 v ∂ i 2 p ˚ f | + sup P m | [ U 0 ,U 1 ] ×{ V 0 } ⟨ p τ ⟩ σ + i 2 | ∂ i 1 u ∂ i 2 p ˚ f | ) (A.6) \nis finite for every k ≥ 0 . For f : P m | R → R ≥ 0 and k ≥ 0 , we define the norms \n∥ f ∥ C k σ ( P | R ) . = ∑ 0 ≤ i 1 + i 2 ≤ k sup P m | R ⟨ p τ ⟩ σ + i 2 | ∂ i 1 x ∂ i 2 p f | , \nwhere ∂ i 1 x denotes i 1 derivatives in the ( u, v ) -variables. \nProposition A.1. For any B > 0 , κ > 0 , and σ > 4 there exists a constant ε > 0 (depending also on F , K , and X ) with the following property. Let ( ˚ Ψ , ˚ Q, ˚ f ) be a smooth characteristic initial data set for the system (A.1) -(A.3) on C ( U 0 , U 1 , V 0 , V 1 ) . If U 1 -U 0 < ε , V 1 -V 0 < ε , and \n∥ ˚ Ψ ∥ C 2 ( C ) + ∥ ˚ Q ∥ C 1 ( C ) + ∥ ˚ f ∥ C 1 σ ( P | C ) ≤ B, (A.7) \nthen there exists a unique smooth solution (Ψ , Q, f ) of (A.1) -(A.3) on R ( U 0 , U 1 , V 0 , V 1 ) which extends the initial data. Moreover, the distribution function f is supported in H κ/ 2 and for any k ≥ 0 , the norm \n∥ Ψ ∥ C k ( R ) + ∥ Q ∥ C k ( R ) + ∥ f ∥ C k σ ( P | R ) \nis finite and can be bounded in terms of initial data norms. \nRemark A.2 . While we assume that the initial data ( ˚ Ψ , ˚ Q, ˚ f ) are smooth (and that ˚ f satisfies the nontrivial bound (A.6) at any order), the existence time ε in the proposition depends only on the estimate (A.7). \nRemark A.3 . If we assume that ˚ f has compact support in the momentum variables, then (A.6) is automatic \nby smoothness.", 'A.1 Proof of Proposition A.1': "In this section, we assume the hypotheses and setup of Proposition A.1. We also set U 0 = V 0 = 0 and define τ . = 1 2 ( v + u ) . Therefore, 0 ≤ τ ≤ ε on R . \nWe will construct the solution (Ψ , Q, f ) as the limit of an iteration scheme. \nLemma A.4. There exist sequences of constants { ˜ C k } and { C k } such that the following holds. For any ε sufficiently small and every n ≥ 1 , there exist functions (Ψ n , Q n , f n ) ∈ C ∞ ( R ) × C ∞ ( R ) × C ∞ ( P m | R ) solving the iterative system \n∂ u ∂ v Ψ n = F (Ψ n -1 , ∂ Ψ n -1 , Q n -1 , M [ f n -1 ] , M uv [ f n -1 ]) , (A.8) \n∂ u Q n = K (Ψ n -1 , M v [ f n -1 ]) , (A.9) \nX ( p u , p v , Ψ n -1 , ∂ Ψ n -1 , Q n -1 ) f n = 0 , (A.10) \nwhere we set (Ψ 0 , Q 0 , f 0 ) to be identically zero, with initial conditions \nΨ n | C = ˚ Ψ , Q n | C = ˚ Q, f n | P | C = ˚ f. (A.11) \nMoreover, spt( f n ) ⊂ H κ/ 2 and these functions satisfy the bounds \n∥ Ψ ∥ C k ( R ) ≤ ˜ C k e C k τ , (A.12) \n∥ Q n ∥ C k ( R ) ≤ ˜ C k +1 e C k +1 τ , (A.13) \n∥ f n ∥ C k σ ( P | R ) ≤ ˜ C k +1 e C k +1 τ . (A.14) \nIt is convenient to set \nF n . = F (Ψ n , ∂ Ψ n , Q n , M [ f n ] , M uv [ f n ]) , K n . = K (Ψ n , M v [ f n ]) , X n . = X ( p u , p v , Ψ n , ∂ Ψ n , Q n ) \nfor n ≥ 0 . We first require a preliminary lemma about integral curves of the vector field X n . \nLemma A.5. For n ≥ 0 , let Γ m ,κ n denote the set of maximal integral curves ˜ γ = ( γ, p ) : I → T R (where I is a closed interval containing 0 ) of the vector field X n subject to the condition that ˜ γ (0) ∈ H κ ∩ P m . Assume that Ψ n satisfies (A.12) for k = 1 and Q n satisfies (A.13) for k = 0 . Then for ε sufficiently small (depending in particular on κ ) and any ˜ γ ∈ Γ n , γ is a future-directed causal curve in R connecting C with the future boundary of R , ˜ γ ( s ) ∈ P m ∩ H κ/ 2 for every s ∈ I , and \n1 2 p τ (0) ≤ p τ ( s ) ≤ 2 p τ (0) (A.15) \nfor every s ∈ I . \nProof. Let ˜ γ = ( γ, p ) ∈ Γ n and set τ 0 . = γ τ (0) . By definition, \ndγ a ds = p a , dp a ds = ξ a n \nfor a ∈ { u, v } , where ξ a n . = ξ a ( p u , p v , Ψ n , ∂ Ψ n , Q n ) . Observe that X n is tangent to the the boundary of P m , { ( u, v, p u , p v ) ∈ T R : p u p v = m 2 } , by (A.5), so ˜ γ remains within P m . Reparametrizing ˜ γ by τ gives \nd dτ ( p τ ) 2 = 2( ξ u n + ξ v n ) . \nUsing (A.4), the assumptions on Ψ n and Q n , and Grönwall's inequality, we have \np τ ( τ ) 2 ≤ e O ( ε ) ( p τ ( τ 0 ) 2 + O ( ε ) ) ≤ 2 p τ ( τ 0 ) 2 \nfor τ in the domain of ˜ γ and for ε sufficiently small. This proves the second inequality in (A.15). To prove the first inequality, we observe that by the estimate we have just proved, \n| ( p τ ( τ )) 2 -( p τ ( τ 0 )) 2 | ≲ ε sup ˜ γ | ξ u n + ξ v n | ≲ εp τ ( τ 0 ) 2 \nChoosing ε perhaps even smaller proves (A.15) and completes the proof of the lemma. \nProof of Lemma A.4. The proof is by induction on n and induction on k for each fixed n . As the existence and estimates for the base case n = 0 are trivial, we assume the existence of (Ψ n -1 , Q n -1 , f n -1 ) satisfying (A.12)-(A.14), where the constants are still to be determined. We will choose the constants to satisfy ˜ C k ≤ C k ≤ ˜ C k +1 , which we use without comment in the sequel. By (A.12)-(A.14) for (Ψ n -1 , Q n -1 , f n -1 ) and iterating the chain rule, it is easy to see that \n| ∂ k M n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 0) , (A.16) \n| F n -1 | ≤ C ( C 1 ) , (A.17) \n| ∂ k F n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 1) , (A.18) \n| K n -1 | ≤ C ( C 1 ) , (A.19) \n| ∂ k K n -1 | ≤ C ( ˜ C k +1 ) e C k +1 τ ( k ≥ 1) , (A.20) \nwhere M n -1 ∈ { M [ f n -1 ] , M v [ f n -1 ] , M uv [ f n -1 ] } . We also define the number \nB ' . = | F ( ˚ Ψ , ∂ ˚ Ψ , ˚ Q,M [ ˚ f ] , M uv [ ˚ f ]) | (0 , 0) + | K ( ˚ Ψ , M v [ ˚ f ]) | (0 , 0) . \nStep 1. The function Ψ n is defined by the explicit formula \nΨ n ( u, v ) . = ∫ u 0 ∫ v 0 F n -1 ( u ' , v ' ) dv ' du ' + ˚ Ψ( u, 0) + ˚ Ψ(0 , v ) -˚ Ψ(0 , 0) . (A.21) \nIt follows by inspection of this representation formula and (A.17) that \n∥ Ψ n ∥ C 1 ( R ) ≤ 10 B (A.22) \nif ε is sufficiently small depending on C 1 . We estimate k -th order derivatives ( k ≥ 2) of the form ∂ k u Ψ n , ∂ k -2 x ∂ u ∂ v Ψ n , and ∂ k v Ψ n separately. For the first type, we simply differentiate the representation formula (A.21) k times to obtain \n| ∂ k u Ψ n | ≤ (data) + ∫ v 0 | ∂ k -1 u F n -1 | dv ' ≤ C + C ( ˜ C k ) C k e C k τ ≤ ˜ C k e C k τ \nfor appropriate choices of ˜ C k and C k . For mixed derivatives, we differentiate the wave equation to obtain \n| ∂ k -2 x ∂ u ∂ v Ψ n -1 | ≤ sup { 0 }× [0 ,V 0 ] | ∂ k -2 x F n -1 | + ∫ u 0 | ∂ u ∂ k -2 F n -1 | du ' ≤ C ( C k -1 ) + C ( ˜ C k ) C k e C k τ ≤ ˜ C k e C k τ \nfor appropriate choices of ˜ C k and C k . The estimate for ∂ k v Ψ n is similar to ∂ k u Ψ n . For later use, we derive a slightly improved estimate for ∂ 2 x Ψ n . Using the mean value theorem and (A.18), we have \n| F n -1 -B ' | ≤ C ( C 2 ) τ. \nWe also have \n| ∂ 2 u Ψ n -(data) | + | ∂ 2 v Ψ n -(data) | ≤ C ( C 2 ) τ. \nTherefore, for ε sufficiently small depending on C 2 , combined with (A.22), we infer \n∥ Ψ n ∥ C 2 ( R ) ≤ 20( B + B ' ) . (A.23) \nThis completes Step 1. \nStep 2. The function Q n is defined by the explicit formula \nQ n ( u, v ) = ∫ u 0 K n -1 ( u ' , v ) du ' + ˚ Q (0 , v ) . \nIt follows by inspection of this representation formula and (A.19) that \n∥ Q ∥ C 0 ( R ) ≤ 10 B \nfor ε sufficiently small depending on C 1 . For k ≥ 1 , we estimate \n| ∂ k v Q n | ≤ (data) + ∫ u 0 | ∂ k v K n -1 | du ' ≤ C + C ( ˜ C k +1 ) C k +1 e C k +1 τ ≤ ˜ C k +1 e C k +1 τ , \n| ∂ k -1 ∂ u Q n | ≤ sup { 0 }× [0 ,V 0 ] | ∂ k -1 K n -1 | + ∫ u 0 | ∂ k K n -1 | du ' ≤ C ( C k ) + C ( ˜ C k +1 ) C k +1 e C k +1 τ ≤ ˜ C k +1 e C k +1 τ \nfor appropriate chices of ˜ C k +1 and C k +1 . Arguing as in Step 1, for ε sufficiently small depending on C 2 , we also infer \n∥ Q n ∥ C 1 ( R ) ≤ 20( B + B ' ) . (A.24)", 'This completes Step 2.': "Step 3. For f n we do not have an explicit representation formula and must instead infer its existence from general properties of flows of vector fields. A slight technical issue is that the 'initial data hypersurface' P | C is not smooth because of the corner in C . Therefore, in order to prove the existence of a smooth f n , we will construct it as a smooth limit of smooth solutions to the X n -1 transport equation, corresponding to initial data where we smooth out the corner in C . To carry out this idea, we first extend ˚ f to C ∞ σ ( P m | R ) according to \nE ˚ f ( u, v, p u , p v ) . = ˚ f ( u, 0 , p u , p v ) + ˚ f (0 , v, p u , p v ) -˚ f (0 , 0 , p u , p v ) \nand set, for j ≥ 1 , \nS . = { ( u, v ) ∈ R : uv = 2 -j } \nj , R j . = { ( u, v ) ∈ R : uv ≥ 2 -j . } \n̸ \nFor any j ≥ j 0 sufficiently large that S j = ∅ , there exists a unique function f n,j ∈ C ∞ ( P m | R j ) such that \nX n -1 f n,j = 0 (A.25) \nin R j , with initial data f n,j = E ˚ f on P m | S j . The existence follows immediately from the flowout theorem (see [Lee13, Theorem 9.20]), the fact that X n -1 is transverse to P m | S j , and Lemma A.5. It also follows from Lemma A.5 that spt( f n,j ) ⊂ H κ/ 2 . We will use the fact that p τ ≥ κ/ 2 for any ˜ γ ∈ Γ m ,κ n -1 often and without further comment in the sequel. \nWe now claim that we can choose ˜ C k and C k such that \n∥ f n,j ∥ C k σ ( P | R j ) ≤ ˜ C k +1 e C k +1 τ (A.26) \nfor every n ≥ 0 , j ≥ j 0 , and k ≥ 0 . Let F n,j,k denote the vector with ( k +3 k ) components of the form \n( p τ ) σ + i 2 ∂ i 1 x ∂ i 2 p f n,j , (A.27) \nwhere i 1 + i 2 = k . We will show inductively that ˜ C k +1 and C k +1 can be chosen so that \nsup P m | R j |F n,j,k | ≤ ˜ C k +1 e C k +1 τ , (A.28) \nwhich would imply (A.26). \nOrders k = 0 and k = 1 are slightly anomalous in our scheme and we handle them first. We require the estimate \n⟨ p τ ⟩ -2 | ∂ ≤ 1 x ξ a n -1 | + ⟨ p τ ⟩ -1 | ∂ p ξ a n -1 | ≲ 1 , (A.29) \nwhich follows from (A.4), (A.23), and (A.24). Using (A.25), we compute \nX n -1 (( p τ ) σ f n,j ) = σ 2 ( p τ ) σ -1 ( ξ u n -1 + ξ v n -1 ) f n,j . \nFrom (A.29) we then infer \n| X n -1 (( p τ ) σ f n,j ) | ≲ ( p τ ) σ +1 f n,j . \nLet ˜ γ ∈ Γ m ,κ n -1 be parametrized by coordinate time τ . Then along ˜ γ we have \n∣ ∣ ∣ ∣ d dτ F n,j, 0 ∣ ∣ ∣ ∣ ≲ |F n,j, 0 | , \nwhence by Grönwall's inequality \n|F n,j, 0 | ≲ ∥E ˚ f ∥ C 0 σ ( P | R ) ≲ ∥ ˚ f ∥ C 0 σ ( P | C ) . \nNext, we compute \n| X n -1 (( p τ ) σ ∂ x f n,j ) | ≲ ( p τ ) σ +1 | ∂ x f n,j | +( p τ ) σ | [ X n -1 , ∂ x ] f n,j | . \nThe commutator is estimated using (A.29) by \n| [ X n -1 , ∂ x ] f n,j | ≲ | ∂ x ξ a n -1 || ∂ p f n,j | ≲ ( p τ ) 2 | ∂ p f n,j | . \nThe p derivative of f n,j satisfies \n| X n -1 (( p τ ) σ +1 ∂ p f n,j ) | ≲ ( p τ ) σ +2 | ∂ p f n,j | +( p τ ) σ +1 | [ X n -1 , ∂ p ] f n,j | , \nwhere the commutator is now estimated by \n| [ X n -1 , ∂ p ] f n,j | ≲ | ∂ x f n,j | + | ∂ p ξ a n -1 || ∂ x f n,j | ≲ | ∂ x f n,j | + p τ | ∂ p f n,j | . \nPutting these estimates together, we find that \n∣ ∣ ∣ ∣ d dτ F n,j, 1 ∣ ∣ ∣ ∣ ≲ |F n,j, 1 | , \nwhence again by Grönwall's inequality we conclude a uniform bound \n|F n,j, 1 | ≲ ∥ ˚ f ∥ C 1 σ ( P | C ) . (A.30) \nHaving now established cases k = 0 and k = 1 of (A.28), we now assume (A.28) up to order k -1 . Let φ i 1 ,i 2 be a component of F n,j,k . We adopt the convention that if either i 1 or i 2 are negative, then φ i 1 ,i 2 is interpreted as identically zero. Using (A.4) and (A.29), we estimate \n| X n -1 ( φ i 1 ,i 2 ) | ≲ ( p τ ) σ + i 2 -1 | ξ u n -1 + ξ v n -1 || ∂ i 1 x ∂ i 2 p f n,j | +( p τ ) σ + i 2 | [ X n -1 , ∂ k ] f n,j | ≲ p τ | φ i 1 ,i 2 | +( p τ ) σ + i 2 | [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j | +( p τ ) σ + i 2 | [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f n,j | . (A.31) \nThe first commutator, [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j , vanishes unless i 1 ≥ 1 and therefore consists of terms of the form ∂ i 1 +1 x ∂ i 2 -1 p f n,j , which implies \n( p τ ) σ + i 2 | [ p a ∂ a , ∂ i 1 x ∂ i 2 p ] f n,j | ≲ p τ | φ i 1 +1 ,i 2 -1 | ≲ p τ |F n,j,k | . (A.32) \nThe second commutator can be estimated by \n| [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f | ≲ ∑ 1 ≤ j 1 + j 2 j 1 ≤ i 1 ,j 2 ≤ i 2 | ∂ j 1 x ∂ j 2 p ξ a n -1 || ∂ i 1 -j 1 x ∂ i 2 +1 -j 2 p f n,j | . \nBy inspection, ∂ j 1 x ∂ j 2 p ξ a n -1 is linear in ∂ j 1 +1 x Ψ n -1 , which is the worst behaved term in our inductive hierarchy. Therefore, using again (A.4), we may estimate \n| ∂ j 1 x ∂ j 2 p ξ a n -1 | ≲ ( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) ( p τ ) 2 -j 2 \nand therefore infer \n( p τ ) σ + i 2 | [ ξ a n -1 ∂ p a , ∂ i 1 x ∂ i 2 p ] f n,j | ≲ p τ ∑ 1 ≤ j 1 + j 2 j 1 ≤ i 1 ,j 2 ≤ i 2 ( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) |F n,j,k +1 -j 1 -j 2 | . \nIf j 1 < k , then ˜ C j 1 +1 e C j 1 +1 τ ≤ C ( C k ) . If j 1 = k , then j 2 = 0 and \n( C ( C k ) + ˜ C j 1 +1 e C j 1 +1 τ ) |F n,j,k +1 -j 1 -j 2 | = ( C ( C k ) + ˜ C k +1 e C k +1 τ ) |F n,j, 1 | ≲ C ( C k ) ˜ C k +1 e C k +1 τ , (A.33) \nwhere we have used (A.30). Therefore, the sum in (A.1) can be estimated by \n≲ C ( C k ) ( |F n,j,k | + ˜ C k +1 e C k +1 τ ) . \nPutting (A.31), (A.32), (A.1), and (A.33) together, we arrive at \n| X n -1 F n,j,k | ≲ C ( C k ) p τ ( |F n,j,k | + ˜ C k +1 e C k +1 τ ) . \nAs before, a simple Grönwall argument now establishes (A.26) for appropriate choices of ˜ C and C . \nk +1 k +1 Having now established the boundedness of the sequence f n,j , we may take the limit j →∞ (after perhaps passing to a subsequence). This shows the existence of a function f n ∈ C ∞ ( P m | R ) with spt( f n ) ⊂ H κ/ 2 , satisfying the estimates (A.14), and attaining ˚ f on P m | C . Finally, uniqueness of f n is immediate since it is constant along the integral curves of X n -1 . \nProof of Proposition A.1. We prove first that the sequence iterative sequence (Ψ n , Q n , f n ) constructed in Lemma A.4 is Cauchy in C 2 × C 1 × C 1 σ . We claim that if ε is sufficiently small, then the following estimate holds for every n ≥ 2 : \n∥ Ψ n -Ψ n -1 ∥ C 2 ( R ) + ∥ Q n -Q n -1 ∥ C 1 ( R ) + ∥ f n -f n -1 ∥ C 1 σ ( P | R ) ≤ 1 ( ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) ) . (A.34) \n2 \nUsing the mean value theorem and the boundedness of the iterative sequence, we immediately estimate \n∥ F n -1 -F n -2 ∥ C 1 ( R ) ≲ ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) . (A.35) \nUsing the formula \n(Ψ n -Ψ n -1 )( u, v ) = ∫ u 0 ∫ v 0 ( F n -1 -F n -2 ) dv ' du ' \nwe readily infer that for ε sufficiently small, Ψ n -Ψ n -1 and ∂ i a (Ψ n -Ψ n -1 ) for a ∈ { u, v } and i ∈ { 1 , 2 } are bounded by an arbitrarily small multiple of the right-hand side of (A.34). To estimate the mixed second partial derivative, we simply use the fundamental theorem of calculus to bound \n∥ Ψ n -1 -Ψ n -2 ∥ C 1 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 0 ( R ) + ∥M n -1 -M n -2 ∥ C 0 ( R ) \nby an arbitrarily small multiple of the right-hand side of (A.34) and then use the mean value theorem to estimate \n| ∂ u ∂ v (Ψ n -Ψ n -1 ) | ≤ ∥ F n -1 -F n -2 ∥ C 0 ( R ) ≲ ∥ Ψ n -1 -Ψ n -2 ∥ C 1 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 0 ( R ) + ∥M n -1 -M n -2 ∥ C 0 ( R ) . \nThe argument for bounding ∥ Q n -Q n -1 ∥ C 1 ( R ) is essentially the same and is omitted. To estimate f n -f n -1 , we examine the quantity \nF n . = ( p τ ) σ ( f n -f n -1 , ∂ x ( f n -f n -1 ) , p τ ∂ p ( f n -f n -1 )) \n̸ \nalong integral curves of X n -1 . First, note that F n = 0 only along curves in Γ m ,κ n -1 ∪ Γ m ,κ n -2 . By Lemma A.5 (and choosing ε perhaps smaller), any value ˜ γ n -2 ( s ) for a curve ˜ γ n -2 ∈ Γ m ,κ n -2 can be realized as an initial value ˜ γ n -1 (0) for a curve ˜ γ n -1 ∈ Γ m ,κ/ 2 n -1 . Therefore, it suffices to observe that the following estimate holds along any curve ˜ γ n -1 ∈ Γ m ,κ/ 2 n -1 : \n∣ ∣ ∣ ∣ d dτ F n ∣ ∣ ∣ ∣ ≲ |F n | + ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) , \nwhich is obtained by simply differentiating F n and using the estimates proved in Lemma A.4. Therefore, since F n vanishes along P m | C , Grönwall's inequality implies \n|F n | ≲ ε ( ∥ Ψ n -1 -Ψ n -2 ∥ C 2 ( R ) + ∥ Q n -1 -Q n -2 ∥ C 1 ( R ) + ∥ f n -1 -f n -2 ∥ C 1 σ ( P | R ) ) . \nAfter choosing ε sufficiently small, the proof of (A.35) is complete. \nTherefore, (Ψ n , Q n , f n ) converges to a solution (Ψ , Q, f ) of the system (A.1)-(A.3) in C 2 × C 1 × C 1 , which is moreover C ∞ smooth by the higher order estimates proved in Lemma A.4. Uniqueness of the solution can be proved along the same lines as the proof of the estimate (A.34) and is omitted.", 'A.2 Proof of local well posedness for the Einstein-Maxwell-Vlasov system': "In this section, we prove Proposition 3.14, local well-posedness for the Einstein-Maxwell-Vlasov system in small characteristic rectangles. The proof has 3 steps: In the first step, we solve the wave equations (3.20) and (3.21), the ingoing Maxwell equation (3.24), and the Maxwell-Vlasov equation (3.26) using Proposition A.1. In order to directly quote Proposition A.1, we in fact consider a renormalized system that fixes the location of the mass shell to the fixed family of hyperbolas ˜ p u ˜ p v = m 2 . In step 2, we show that the outgoing Maxwell equation holds as a result of conservation law (3.32), which follows from the Maxwell-Vlasov equation (3.26). Finally, in step 3, we show that Raychaudhuri's equations (3.22) and (3.23) hold, using now the Bianchi identities (3.33) and (3.34), which again follow from (3.26). Steps 2 and 3 may be thought of as propagation of constraints , as they require the relevant equations to hold on initial data. \nProof of Proposition 3.14. Step 1. Consider the wave-transport system \n∂ u ∂ v r = -Ω 2 4 r -∂ u r∂ v r r + πr Ω 2 4 ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ p v ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.36) \n∂ u ∂ v log Ω 2 = Ω 2 2 r 2 + 2 ∂ u r∂ v r r 2 + π Ω 2 2 ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ p v ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v \n-π Ω 2 2 ∫ ˜ p u ˜ p v ≥ m 2 (˜ p u ˜ p v -m 2 ) ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.37) \n∂ u Q = -π 2 e r 2 Ω ∫ ˜ p u ˜ p v ≥ m 2 ˜ p u ˜ f ( u, v, ˜ p u , ˜ p v ) d ˜ p u d ˜ p v , (A.38) \n˜ X ˜ f = 0 , (A.39) \nwhere \n˜ X . = ˜ p u ∂ u + ˜ p v ∂ v -( ∂ u log Ω(˜ p u ) 2 -∂ v log Ω˜ p u ˜ p v + 2 ∂ v r r (˜ p u ˜ p v -m 2 ) + e Ω Q r 2 ˜ p u ) ∂ ˜ p u , \n-( ∂ v log Ω(˜ p v ) 2 -∂ u log Ω˜ p u ˜ p v + 2 ∂ u r r (˜ p u ˜ p v -m 2 ) -e Ω Q r 2 ˜ p v ) ∂ ˜ p v \nwith initial data ˚ Ψ = (log˚ r, log ˚ Ω 2 ) , ˚ Q , and ˚ ˜ f ( u, v, ˜ p u , ˜ p v ) = ˚ f ( u, v, ˚ Ω -1 ˜ p u , ˚ Ω -1 ˜ p v ) . Since log ˚ Ω 2 is bounded on C and either m > 0 or ℓ ≥ c ℓ on spt( ˚ f ) , there exists a κ > 0 such that spt( ˚ ˜ f ) ⊂ H κ . Furthermore, the structural conditions (A.4) and (A.5) are easily verified for ˜ X , so Proposition A.1 produces a unique local smooth solution to (A.36)-(A.39) if ε loc is chosen sufficiently small. Making the change of variables ˜ p u ↦→ Ω p u and ˜ p v ↦→ Ω p v , defining f ( u, v, p u , p v ) = ˜ f ( u, v, Ω p u , Ω p v ) , and observing that \nXf = Ω -1 ˜ X ˜ f = 0 , \nwe have obtained a unique local smooth solution ( r, Ω 2 , Q, f ) for the system (3.20), (3.21), (3.24), and (3.26) on R which extends the initial data. \nStep 2. We first aim to prove (3.32) using only (3.26). At this point, one could apply Proposition 3.10 and (2.11) to derive (3.32), but we give now a direct proof. \nFirst, using the definitions (3.14) and (3.15), we have \n∂ u ( r 2 Ω 2 π N u ) + ∂ v ( r 2 Ω 2 π N v ) = ∂ u ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 p u f dp u dp v + ∂ v ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 p v f dp v dp u = ∫ Ω 2 p u p v ≥ m 2 ( 2 r∂ u r Ω 4 p u f +2 r 2 Ω 4 ∂ u log Ω 2 p u f + r 2 Ω 4 p u ∂ u f ) dp u dp v + ∫ Ω 2 p u p v ≥ m 2 ( 2 r∂ v r Ω 4 p v f +2 r 2 Ω 4 ∂ v log Ω 2 p v f + r 2 Ω 4 p v ∂ v f ) dp u dp v + ∫ ∞ 0 m 4 r 2 ( p v ) 2 ∂ u log Ω 2 f dp v + ∫ ∞ 0 m 4 r 2 ( p u ) 2 ∂ v log Ω 2 f dp u . (A.40) \nAdding both terms involving spatial derivatives of f and using (3.26) yields \n∫ Ω 2 p u p v ≥ m 2 r 2 Ω 4 ( p u ∂ u f + p v ∂ v f ) dp u dp v = ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 Ξ u ∂ p u f dp u dp v + ∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 Ξ v ∂ p v f dp v dp u , (A.41) \nwhere \nΞ u . = ∂ u log Ω 2 ( p u ) 2 + 2 ∂ v r r Ω 2 (Ω 2 p u p v -m 2 ) + e Q r 2 p u , Ξ v . = ∂ v log Ω 2 ( p v ) 2 + 2 ∂ u r r Ω 2 (Ω 2 p u p v -m 2 ) -e Q r 2 p v . \nIntegrating the first term on the right-hand side of (A.41) by parts, we find \n∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p v ) r 2 Ω 4 Ξ u ∂ p u f dp u dp v = -∫ Ω 2 p u p v ≥ m 2 ( 2 r 2 Ω 4 ∂ u log Ω 2 p u +2 r∂ v r Ω 4 p v + e Ω 4 Q ) f dp u dp v -∫ ∞ 0 ( m 4 r 2 ( p v ) 2 ∂ u log Ω 2 + e m 2 Ω 2 p v Q ) f dp v , (A.42) \nwhere f is evaluated at p u = m 2 / (Ω 2 p v ) , and for the second term, \n∫ ∞ 0 ∫ ∞ m 2 / (Ω 2 p u ) r 2 Ω 4 Ξ v ∂ p v f dp v dp u = -∫ Ω 2 p u p v ≥ m 2 ( 2 r 2 Ω 4 ∂ v log Ω 2 p v +2 r∂ u r Ω 4 p u -e Ω 4 Q ) f dp u dp v -∫ ∞ 0 ( m 4 r 2 ( p u ) 2 ∂ v log Ω 2 -e m 2 Ω 2 p u Q ) f dp u , (A.43) \nwhere f is evaluated at p v = m 2 / (Ω 2 p u ) . Combining (A.40)-(A.43) yields (3.32) after noting that \n∫ ∞ 0 e m 2 Ω 2 p v Qf dp v = ∫ ∞ 0 e m 2 Ω 2 p u Qf dp u . \nWe can now derive the ingoing Maxwell equation (3.25). By (3.24), we have \nQ ( u, v ) = Q ( U 0 , v ) -∫ u U 0 1 2 e r 2 Ω 2 N v du ' \non R . We then derive \n∂ v Q ( u, v ) = ∂ v Q ( U 0 , v ) -∫ u U 0 ∂ v ( 1 2 e r 2 Ω 2 N v ) du ' = ∂ v Q ( U 0 , v ) + ∫ u U 0 ∂ u ( 1 2 e r 2 Ω 2 N u ) du ' = 1 2 e r 2 Ω 2 N v ( u, v ) , \nwhere in the final equality we used the fundamental theorem of calculus and the assumption that (3.25) holds on { U 0 } × [ V 0 , V 1 ] . \nStep 3. By a lengthy calculation which is very similar to the one performed in step 2, one may use (3.26) to derive the Bianchi identities in the form (3.33) and (3.34). Using the Maxwell equations (3.24) and (3.25), this implies the Bianchi identities in the form (2.12) and (2.13), where \nT uu = T uu , T uv = T uv + Q 2 Ω 2 r 4 , T vv = T vv , S = S + Q 2 2 r 4 . \nBy another lengthy calculation, using now also the wave equations (3.20) and (3.21), one can derive the pair of identities \n∂ v ( r∂ 2 u r -r∂ u r∂ u log Ω 2 + 1 4 r 2 Ω 4 T vv ) = 0 , ∂ u ( r∂ 2 v r -r∂ v r∂ v log Ω 2 + 1 4 r 2 Ω 4 T uu ) = 0 . \nThese identities, together with the assumption that (3.22) holds on [ U 0 , U 1 ] × { V 0 } and (3.23) holds on { U 0 } × [ V 0 , V 1 ] , prove that (3.22) and (3.23) hold throughout R .", 'References': "[AAR21] E. Ames, H. Andréasson, and O. Rinne. 'Dynamics of gravitational collapse in the axisymmetric Einstein-Vlasov system'. Classical Quantum Gravity 38.10 (2021), Paper No. 105003, 34. \n[AH23] X. An and T. He. 'Dynamics of apparent horizon and a null comparison principle' (2023). arXiv: 2312.15666 ."}