Scalable and sequential inference of the neutron star equation of state with the Einstein Telescope

Author(s)

Wouters, Thibeau, Ng, Thomas C.K., Koehn, Hauke, Gittins, Fabian, Pang, Peter T.H., Dietrich, Tim, Van Den Broeck, Chris

Abstract

Future gravitational-wave observatories such as the Einstein Telescope will detect tens of thousands of binary neutron star mergers per year, making gravitational waves a dominant probe of the neutron-star equation of state in the coming decades. However, extracting this information requires hierarchical inference across a rapidly growing catalog of events, and existing methods must restart from scratch whenever a new event arrives, making them impractical at this scale. In this Letter, we introduce a hybrid sequential Monte Carlo algorithm that combines data and likelihood tempering to adaptively update the equation-of-state posterior in batches, reusing previous inference results instead of restarting from a wide prior. Accelerated by GPU hardware, our method infers the equation of state from a simulated month of around 1500 binary neutron star mergers in a few hours on a single GPU, and we project that a full year of detections could be processed in a few days on current hardware. This establishes sequential Monte Carlo as a practical route for real-time hierarchical inference with future gravitational-wave detectors.

Figures

Illustration of our algorithm processing a growing catalog. Each iteration maintains a set of particles distributed according to the partial posterior conditioned on the $n$ events observed so far (\textit{upper left}). Observations that are not informative enough to significantly affect the posterior are collected in a queue (data tempering, \textit{upper right}). If, after accumulating $q$ events, the batch of new events is sufficiently informative, we update the partial posterior using likelihood tempering (\textit{lower}). Afterwards, we update our current posterior to be informed by $n+q$ events, and continue the loop.
Caption Illustration of our algorithm processing a growing catalog. Each iteration maintains a set of particles distributed according to the partial posterior conditioned on the $n$ events observed so far (\textit{upper left}). Observations that are not informative enough to significantly affect the posterior are collected in a queue (data tempering, \textit{upper right}). If, after accumulating $q$ events, the batch of new events is sufficiently informative, we update the partial posterior using likelihood tempering (\textit{lower}). Afterwards, we update our current posterior to be informed by $n+q$ events, and continue the loop.
Diagnostic plots of our \ac{SMC} sampler. \textit{Top panel}: Sampling efficiency when reweighting the current set of posterior samples with respect to the new events. The inverted triangles show iterations where the ESS falls below the $50\%$ threshold (dashed line), and we update the partial posterior with likelihood tempering. \textit{Bottom panel}: Cumulative wall time as a function of the number of events.
Caption Diagnostic plots of our \ac{SMC} sampler. \textit{Top panel}: Sampling efficiency when reweighting the current set of posterior samples with respect to the new events. The inverted triangles show iterations where the ESS falls below the $50\%$ threshold (dashed line), and we update the partial posterior with likelihood tempering. \textit{Bottom panel}: Cumulative wall time as a function of the number of events.
Constraints on the \ac{EOS} (top panel) and tidal deformability (bottom panel) after processing each batch, shown as $95\%$ credible bands. The injected \ac{EOS} is shown with a black dashed line, while the median of the posterior in the final batch (i.e., with all events processed) is shown in white. The median is taken at each value for the density or \ac{NS} mass.
Caption Constraints on the \ac{EOS} (top panel) and tidal deformability (bottom panel) after processing each batch, shown as $95\%$ credible bands. The injected \ac{EOS} is shown with a black dashed line, while the median of the posterior in the final batch (i.e., with all events processed) is shown in white. The median is taken at each value for the density or \ac{NS} mass.
Extrapolation of the cumulative wall time as a function of the number of events. The markers show the runtimes recorded at the batches shown in Fig.~\ref{fig:smc-updating-diagnostics}, while the curve shows a fit based on the data points from $n>400$ onwards (solid line). The vertical lines show the expected number of detections for a catalog starting from a lower frequency of $5$ Hz, and analyzed with the \texttt{IMRPhenomXAS\_NRTidalv3} waveform.
Caption Extrapolation of the cumulative wall time as a function of the number of events. The markers show the runtimes recorded at the batches shown in Fig.~\ref{fig:smc-updating-diagnostics}, while the curve shows a fit based on the data points from $n>400$ onwards (solid line). The vertical lines show the expected number of detections for a catalog starting from a lower frequency of $5$ Hz, and analyzed with the \texttt{IMRPhenomXAS\_NRTidalv3} waveform.
\textit{Left}: Constraints from NICER (blue) and GW170817 (orange) used to determine the maximum likelihood sample (green). \textit{Right}: Sound-speed profile of the maximum likelihood \ac{EOS} sample compared to the $95\%$ credible band of the posterior (pink).
Caption \textit{Left}: Constraints from NICER (blue) and GW170817 (orange) used to determine the maximum likelihood sample (green). \textit{Right}: Sound-speed profile of the maximum likelihood \ac{EOS} sample compared to the $95\%$ credible band of the posterior (pink).
Results for the \ac{SMC} diagnostics (\textit{left panels}) and the \ac{EOS} constraints (\textit{right panels}) when analyzing the catalog of one month of \ac{BNS} sources, simulated without detector noise. The \ac{ESS} decreases more smoothly, and the slight bias in the recovered mass--tidal-deformability curve observed in Fig.~\ref{fig:eos-constraints} is removed.
Caption Results for the \ac{SMC} diagnostics (\textit{left panels}) and the \ac{EOS} constraints (\textit{right panels}) when analyzing the catalog of one month of \ac{BNS} sources, simulated without detector noise. The \ac{ESS} decreases more smoothly, and the slight bias in the recovered mass--tidal-deformability curve observed in Fig.~\ref{fig:eos-constraints} is removed.
Results for the \ac{SMC} diagnostics (\textit{left panels}) and the \ac{EOS} constraints (\textit{right panels}) when analyzing the catalog of one month of \ac{BNS} sources, simulated without detector noise. The \ac{ESS} decreases more smoothly, and the slight bias in the recovered mass--tidal-deformability curve observed in Fig.~\ref{fig:eos-constraints} is removed.
Caption Results for the \ac{SMC} diagnostics (\textit{left panels}) and the \ac{EOS} constraints (\textit{right panels}) when analyzing the catalog of one month of \ac{BNS} sources, simulated without detector noise. The \ac{ESS} decreases more smoothly, and the slight bias in the recovered mass--tidal-deformability curve observed in Fig.~\ref{fig:eos-constraints} is removed.
References
  • [1] J. M. Lattimer and M. Prakash, The physics of neutron stars, Science 304, 536 (2004), arXiv:astro-ph/0405262.
  • [2] F. Özel and P. Freire, Masses, Radii, and the Equation of State of Neutron Stars, Ann. Rev. Astron. Astrophys. 54, 401 (2016), arXiv:1603.02698 [astro-ph.HE].
  • [3] G. F. Burgio, H. J. Schulze, I. Vidana, and J. B. Wei, Neutron stars and the nuclear equation of state, Prog. Part. Nucl. Phys. 120, 103879 (2021), arXiv:2105.03747 [nucl-th].
  • [4] K. Chatziioannou, H. T. Cromartie, S. Gandolfi, I. Tews, D. Radice, A. W. Steiner, and A. L. Watts, Neutron stars and the dense matter equation of state, Rev. Mod. Phys. 97, 045007 (2025), arXiv:2407.11153 [nucl-th].
  • [5] H. Koehn et al., From existing and new nuclear and astrophysical constraints to stringent limits on the equation of state of neutron-rich dense matter, Phys. Rev. X 15, 021014 (2025), arXiv:2402.04172 [astro-ph.HE].
  • [6] G. Hobbs, R. Edwards, and R. Manchester, Tempo2, a new pulsar timing package. 1. overview, Mon. Not. Roy. Astron. Soc. 369, 655 (2006), arXiv:astro-ph/0603381.
  • [7] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Shapiro Delay Measurement of A Two Solar Mass Neutron Star, Nature 467, 1081 (2010), arXiv:1010.5788 [astroph.HE].
  • [8] K. C. Gendreau, Z. Arzoumanian, P. W. Adkins, C. L. Albert, J. F. Anders, A. T. Aylward, C. L. Baker, E. R. Balsamo, W. A. Bamford, S. S. Benegalrao, et al., The neutron star interior composition explorer (nicer): design and development, in Space telescopes and instrumentation 2016: Ultraviolet to gamma ray, Vol. 9905 (2016) pp. 420–435.
  • [9] S. Bogdanov et al., Constraining the Neutron Star Mass-Radius Relation and Dense Matter Equation of State with NICER. III. Model Description and Verification of Parameter Estimation Codes, Astrophys. J. Lett. 914, L15 (2021), arXiv:2104.06928 [astro-ph.HE].
  • [10] A. L. Watts, Constraining the neutron star equation of state using Pulse Profile Modeling, AIP Conf. Proc. 2127, 020008 (2019), arXiv:1904.07012 [astro-ph.HE].
  • [11] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral, Phys. Rev. D 81, 123016 (2010), arXiv:0911.3535 [astro-ph.HE].
  • [12] T. Damour and A. Nagar, Relativistic tidal properties of neutron stars, Phys. Rev. D 80, 084035 (2009), arXiv:0906.0096 [grqc].
  • [13] T. Damour, A. Nagar, and L. Villain, Measurability of the tidal polarizability of neutron stars in late-inspiral gravitational-wave signals, Phys. Rev. D 85, 123007 (2012), arXiv:1203.4352 [gr-qc].
  • [14] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc].
  • [15] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi GBM, INTEGRAL, IceCube, AstroSat Cadmium Zinc Telluride Imager Team, IPN, Insight-Hxmt, ANTARES, Swift, AGILE Team, 1M2H Team, Dark Energy Camera GW-EM, DES, DLT40, GRAWITA, Fermi-LAT, ATCA, ASKAP, Las Cumbres Observatory Group, OzGrav, DWF (Deeper Wider Faster Program), AST3, CAASTRO, VINROUGE, MASTER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, TZAC Consortium, KU, Nordic Optical Telescope, ePESSTO, GROND, Texas Tech University, SALT Group, TOROS, BOOTES, MWA, CALET, IKI-GW Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre Auger, ALMA, Euro VLBI Team, Pi of Sky, Chandra Team at McGill University, DFN, ATLAS Telescopes, High Time Resolution Universe Survey, RIMAS, RATIR, SKA South Africa/MeerKAT), Multi-messenger Observations of a Binary Neutron Star Merger, Astrophys. J. Lett. 848, L12 (2017), arXiv:1710.05833 [astro-ph.HE].
  • [16] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL), Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017), arXiv:1710.05834 [astroph.HE].
  • [17] B. P. Abbott et al. (LIGO Scientific, Virgo), GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett. 121, 161101 (2018), arXiv:1805.11581 [gr-qc].
  • [18] B. P. Abbott et al. (LIGO Scientific, Virgo), Model comparison from LIGO–Virgo data on GW170817’s binary components and consequences for the merger remnant, Class. Quant. Grav. 37, 045006 (2020), arXiv:1908.01012 [gr-qc].
  • [19] M. Punturo et al., The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav. 27, 194002 (2010).
  • [20] S. Hild et al., Sensitivity Studies for Third-Generation Gravitational Wave Observatories, Class. Quant. Grav. 28, 094013 (2011), arXiv:1012.0908 [gr-qc].
  • [21] D. Reitze et al., Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc. 51, 035 (2019), arXiv:1907.04833 [astro-ph.IM].
  • [22] M. Evans et al., A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, (2021), arXiv:2109.09882 [astro-ph.IM].
  • [23] M. Evans et al., Cosmic Explorer: A Submission to the NSF MPSAC ngGW Subcommittee, (2023), arXiv:2306.13745 [astro-ph.IM].
  • [24] M. Maggiore et al. (ET), Science Case for the Einstein Telescope, JCAP 03, 050, arXiv:1912.02622 [astro-ph.CO].
  • [25] A. Abac et al. (ET), The Science of the Einstein Telescope, JCAP 03, 081, arXiv:2503.12263 [gr-qc].
  • [26] J. Aasi et al. (LIGO Scientific), Advanced LIGO, Class. Quant. Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc].
  • [27] F. Acernese et al. (VIRGO), Advanced Virgo: a secondgeneration interferometric gravitational wave detector, Class. Quant. Grav. 32, 024001 (2015), arXiv:1408.3978 [gr-qc].
  • [28] T. Akutsu et al. (KAGRA), Overview of KAGRA: Detector design and construction history, PTEP 2021, 05A101 (2021), arXiv:2005.05574 [physics.ins-det].
  • [29] P. Couvares et al., Gravitational Wave Data Analysis: Computing Challenges in the 3G Era, (2021), arXiv:2111.06987 [gr-qc].
  • [30] Q. Hu and J. Veitch, Costs of Bayesian parameter estimation in third-generation gravitational wave detectors: An assessment of current acceleration methods, Phys. Rev. D 112, 084039 (2025), arXiv:2412.02651 [gr-qc].
  • [31] E. Thrane and C. Talbot, An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models, Publ. Astron. Soc. Austral. 36, e010 (2019), [Erratum: Publ.Astron.Soc.Austral. 37, e036 (2020)], arXiv:1809.02293 [astro-ph.IM].
  • [31] E. Thrane and C. Talbot, An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models, Publ. Astron. Soc. Austral. 36, e010 (2019), [Erratum: Publ.Astron.Soc.Austral. 37, e036 (2020)], arXiv:1809.02293 [astro-ph.IM].
  • [32] I. Mandel, W. M. Farr, and J. R. Gair, Extracting distribution parameters from multiple uncertain observations with selection biases, Mon. Not. Roy. Astron. Soc. 486, 1086 (2019), arXiv:1809.02063 [physics.data-an].
  • [33] A. G. Abac et al. (LIGO Scientific, VIRGO, KAGRA), GWTC5.0: Population Properties of Merging Compact Binaries, (2026), arXiv:2605.27226 [astro-ph.HE].
  • [34] A. G. Abac et al. (LIGO Scientific, VIRGO, KAGRA), GWTC5.0: Methods for Identifying and Characterizing Gravitationalwave Transients, (2026), arXiv:2605.27224 [gr-qc].
  • [35] A. G. Abac et al. (LIGO Scientific, VIRGO, KAGRA), GWTC5.0: Tests of General Relativity, (2026), arXiv:2607.19293 [gr-qc].
  • [36] W. Del Pozzo, T. G. F. Li, M. Agathos, C. Van Den Broeck, and S. Vitale, Demonstrating the feasibility of probing the neutron star equation of state with second-generation gravitational wave detectors, Phys. Rev. Lett. 111, 071101 (2013), arXiv:1307.8338 [gr-qc].
  • [37] M. Agathos, J. Meidam, W. Del Pozzo, T. G. F. Li, M. Tompitak, J. Veitch, S. Vitale, and C. Van Den Broeck, Constraining the neutron star equation of state with gravitational wave signals from coalescing binary neutron stars, Phys. Rev. D 92, 023012 (2015), arXiv:1503.05405 [gr-qc].
  • [38] T. Regimbau et al., A Mock Data Challenge for the Einstein Gravitational-Wave Telescope, Phys. Rev. D 86, 122001 (2012), arXiv:1201.3563 [gr-qc].
  • [39] M. Branchesi et al., Science with the Einstein Telescope: a comparison of different designs, JCAP 07, 068, arXiv:2303.15923 [gr-qc].
  • [40] R. Tania and J. Suresh, Mock data challenge for next-generation detectors, Phys. Rev. D 113, 063028 (2026), arXiv:2506.12237 [gr-qc].
  • [41] C. Talbot, R. Smith, E. Thrane, and G. B. Poole, Parallelized Inference for Gravitational-Wave Astronomy, Phys. Rev. D 100, 043030 (2019), arXiv:1904.02863 [astro-ph.IM].
  • [42] B. Edelman, B. Farr, and Z. Doctor, Cover Your Basis: Comprehensive Data-driven Characterization of the Binary Black Hole Population, Astrophys. J. 946, 16 (2023), arXiv:2210.12834 [astro-ph.HE].
  • [43] C. Talbot, A. Farah, S. Galaudage, J. Golomb, and H. Tong, GWPopulation: Hardware agnostic population inference for compact binaries and beyond, J. Open Source Softw. 10, 7753 (2025), arXiv:2409.14143 [astro-ph.IM].
  • [44] M. Qazalbash, M. Zeeshan, and R. O’Shaughnessy, Implementation to identify the properties of multiple populations of gravitational wave sources, Phys. Rev. D 113, 103003 (2026), arXiv:2509.13638 [gr-qc].
  • [45] M. Mancarella and D. Gerosa, Sampling the full hierarchical population posterior distribution in gravitational-wave astronomy, Phys. Rev. D 111, 103012 (2025), arXiv:2502.12156 [gr-qc].
  • [46] M. Mould, N. E. Wolfe, and S. Vitale, Rapid inference and comparison of gravitational-wave population models with neural variational posteriors, Phys. Rev. D 111, 123049 (2025), arXiv:2504.07197 [astro-ph.IM].
  • [47] K. Leyde, S. R. Green, M. Dax, M. Mould, C. M. Fabbri, and J. Gair, End-to-End Population Inference from GravitationalWave Strain using Transformers, (2026), arXiv:2605.11274 [gr-qc].
  • [48] T. Wouters, P. T. H. Pang, H. Koehn, H. Rose, R. Somasundaram, I. Tews, T. Dietrich, and C. Van Den Broeck, Leveraging differentiable programming in the inverse problem of neutron stars, Phys. Rev. D 112, 043037 (2025), arXiv:2504.15893 [astro-ph.HE].
  • [49] A. Papadopoulos, C. E. A. Chapman-Bird, R. Gray, C. Messenger, and T. Bertheas, Scalable Dark Siren Cosmology with gwcosmo: GPU Acceleration, Validation and Systematics, (2026), arXiv:2605.23538 [astro-ph.CO].
  • [50] K. Leyde and E. Colangeli, CosmoPyro: Gradients for Gravitational-Wave Cosmology, (2026), arXiv:2608.18281 [astro-ph.CO].
  • [51] K. Leyde, S. R. Green, A. Toubiana, and J. Gair, Gravitational wave populations and cosmology with neural posterior estimation, Phys. Rev. D 109, 064056 (2024), arXiv:2311.12093 [gr-qc].
  • [52] S. Mastrogiovanni, D. Laghi, R. Gray, G. C. Santoro, A. Ghosh, C. Karathanasis, K. Leyde, D. A. Steer, S. Perries, and G. Pierra, Joint population and cosmological properties inference with gravitational waves standard sirens and galaxy surveys, Phys. Rev. D 108, 042002 (2023), arXiv:2305.10488 [astro-ph.CO].
  • [53] S. Mastrogiovanni, G. Pierra, S. Perriès, D. Laghi, G. Caneva Santoro, A. Ghosh, R. Gray, C. Karathanasis, and K. Leyde, ICAROGW: A python package for inference of astrophysical population properties of noisy, heterogeneous, and incomplete observations, Astron. Astrophys. 682, A167 (2024), arXiv:2305.17973 [astro-ph.CO].
  • [54] M. Tagliazucchi, M. Moresco, N. Borghi, and M. Fiebig, Accelerating the standard siren method: Improved constraints on modified gravitational-wave propagation with future data, Astron. Astrophys. 702, A244 (2025), arXiv:2504.02034 [astroph.CO].
  • [55] H. Koehn, T. Wouters, G. Sadeh, P. T. H. Pang, M. Bulla, C. Van Den Broeck, M. W. Coughlin, and T. Dietrich, Binary neutron stars in the next-generation era: Multi-messenger detection prospects and constraints on the equation of state, mass distribution, and cosmology, (2026), arXiv:2607.28438 [astro-ph.HE].
  • [56] N. E. Wolfe, M. Mould, J. Veitch, and S. Vitale, Neural Bayesian updates to populations with growing gravitationalwave catalogs, (2026), arXiv:2602.20277 [astro-ph.IM].
  • [57] G. Demasi and W. Del Pozzo, Massively parallel transdimensional sampling with data assimilation: an application to LISA galactic binaries, (2026), arXiv:2609.17674 [gr-qc].
  • [58] M. Prathaban, H. Bevins, and W. Handley, Accelerated nested sampling with posterior repartitioning and 𝛽-flows for gravitational waves, Mon. Not. Roy. Astron. Soc. 541, 200 (2025), arXiv:2411.17663 [astro-ph.IM].
  • [59] N. Williams, A. Puecher, G. Grams, C. V. Flores, and T. Dietrich, Investigating the impact of quasiuniversal relations on neutron star constraints in third-generation detectors, Phys. Rev. D 114, 024063 (2026), arXiv:2602.14659 [gr-qc].
  • [60] M. Prathaban, C. Hoy, and M. J. Williams, Leveraging rapid parameter estimates for efficient gravitational-wave Bayesian inference via posterior repartitioning, (2026), arXiv:2601.21630 [gr-qc].
  • [61] R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. 55, 364 (1939).
  • [62] J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev. 55, 374 (1939).
  • [63] M. á. Ferreira and C. Providência, Unveiling the nuclear matter EoS from neutron star properties: a supervised machine learning approach, JCAP 07, 011, arXiv:1910.05554 [nucl-th].
  • [64] A. Thete, K. Banerjee, and T. Malik, Realizing the potential of deep neural network for analyzing neutron star observables and dense matter equation of state, Phys. Rev. D 108, 063028 (2023), arXiv:2208.13163 [nucl-th].
  • [65] B. T. Reed, R. Somasundaram, S. De, C. L. Armstrong, P. Giuliani, C. Capano, D. A. Brown, and I. Tews, Toward Accelerated Nuclear-physics Parameter Estimation from Binary Neutron Star Mergers: Emulators for the Tolman–Oppenheimer–Volkoff Equations, Astrophys. J. 974, 285 (2024), arXiv:2405.20558 [astro-ph.HE].
  • [66] R. Somasundaram, I. Svensson, S. De, A. E. Deneris, Y. Dietz, P. Landry, A. Schwenk, and I. Tews, Inferring threenucleon couplings from multi-messenger neutron-star observations, Nature Commun. 16, 9819 (2025), arXiv:2410.00247 [nucl-th].
  • [67] F. Di Clemente, M. Scialpi, and M. Bejger, Explainable autoencoder for neutron star dense matter parameter estimation, Mach. Learn. Sci. Tech. 6, 025044 (2025), arXiv:2501.15222 [physics.comp-ph].
  • [68] H. Rose, H. Koehn, T. Wouters, P. T. H. Pang, M. Bulla, M. W. Coughlin, and T. Dietrich, nmma: An extended Bayesian framework for Nuclear Multimessenger Astronomy in the Era of Next-Generation Detectors, (2026), arXiv:2607.03045 [astro-ph.IM].
  • [69] J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, Y. Katariya, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018).
  • [70] R. Frostig, M. J. Johnson, and C. Leary, Compiling machine learning programs via high-level tracing, Systems for Machine Learning 4 (2018).
  • [71] P. Kidger, On Neural Differential Equations, Ph.D. thesis, University of Oxford (2021).
  • [72] I. Kobyzev, S. J. D. Prince, and M. A. Brubaker, Normalizing Flows: An Introduction and Review of Current Methods, IEEE Trans. Pattern Anal. Machine Intell. 43, 3964 (2021), arXiv:1908.09257 [stat.ML].
  • [73] G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, Normalizing Flows for Probabilistic Modeling and Inference, J. Machine Learning Res. 22, 2617 (2021), arXiv:1912.02762 [stat.ML].
  • [74] D. Ruhe, K. Wong, M. Cranmer, and P. Forré, Normalizing Flows for Hierarchical Bayesian Analysis: A Gravitational Wave Population Study, (2022), arXiv:2211.09008 [astroph.IM].
  • [75] C. Adamcewicz, H. McDougall, P. D. Lasky, and E. Thrane, Fortifying gravitational-wave population inference with normalizing flows, Phys. Rev. D 114, 043057 (2026), arXiv:2606.14229 [astro-ph.HE].
  • [76] P. Del Moral, A. Doucet, and A. Jasra, Sequential monte carlo samplers, Journal of the Royal Statistical Society Series B: Statistical Methodology 68, 411 (2006).
  • [77] N. Chopin and O. Papaspiliopoulos, An Introduction to Sequential Monte Carlo, Springer Series in Statistics (Springer Cham, 2020).
  • [77] N. Chopin and O. Papaspiliopoulos, An Introduction to Sequential Monte Carlo, Springer Series in Statistics (Springer Cham, 2020).
  • [78] C. Dai, J. Heng, P. E. Jacob, and N. Whiteley, An invitation to sequential monte carlo samplers, Journal of the American Statistical Association 117, 1587 (2022).
  • [79] A. Cabezas, A. Corenflos, J. Lao, and R. Louf, Blackjax: Composable Bayesian inference in JAX (2024), arXiv:2402.10797 [cs.MS].
  • [80] H. Wiegand, Kish, l.: Survey sampling. john wiley & sons, inc., new york, london 1965, ix + 643 s., 31 abb., 56 tab., preis 83 s., Biometrische Zeitschrift 10, 88 (1968), https://onlinelibrary.wiley.com/doi/pdf/10.1002/bimj.19680100122.
  • [81] M. Karamanis, F. Beutler, J. A. Peacock, D. Nabergoj, and U. Seljak, Accelerating astronomical and cosmological inference with preconditioned Monte Carlo, Mon. Not. Roy. Astron. Soc. 516, 1644 (2022), arXiv:2207.05652 [astro-ph.IM].
  • [82] M. Karamanis, D. Nabergoj, F. Beutler, J. A. Peacock, and U. Seljak, pocoMC: A Python package for accelerated Bayesian inference in astronomy and cosmology, J. Open Source Softw. 7, 4634 (2022), arXiv:2207.05660 [astro-ph.IM].
  • [83] M. Karamanis and U. Seljak, Persistent sampling: Enhancing the efficiency of sequential monte carlo, Statistics and computing 35, 144 (2025).
  • [84] M. J. Williams, M. Karamanis, Y. Luo, and U. Seljak, Validating sequential Monte Carlo for gravitational-wave inference, Mon. Not. Roy. Astron. Soc. 543, 1479 (2025), arXiv:2506.18977 [astro-ph.IM].
  • [85] M. J. Williams, Accelerated Sequential Posterior Inference via Reuse for Gravitational-Wave Analyses, Phys. Rev. Lett. 137, 021410 (2026), arXiv:2511.04218 [hep-ex].
  • [86] G. Demasi et al., The Sequential Monte Carlo goes NUTS: boosting gravitational-wave inference, Eur. Phys. J. C 86, 686 (2026), arXiv:2601.02336 [gr-qc].
  • [87] N. Chopin, A sequential particle filter method for static models, Biometrika 89, 539 (2002).
  • [88] P. Madau and M. Dickinson, Cosmic Star Formation History, Ann. Rev. Astron. Astrophys. 52, 415 (2014), arXiv:1403.0007 [astro-ph.CO].
  • [89] E. Fonseca et al., Refined Mass and Geometric Measurements of the High-mass PSR J0740+6620, Astrophys. J. Lett. 915, L12 (2021), arXiv:2104.00880 [astro-ph.HE].
  • [90] T. E. Riley et al., A 𝑁𝐼𝐶𝐸𝑅 View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation, Astrophys. J. Lett. 887, L21 (2019), arXiv:1912.05702 [astro-ph.HE].
  • [91] M. C. Miller et al., PSR J0030+0451 Mass and Radius from 𝑁𝐼𝐶𝐸𝑅 Data and Implications for the Properties of Neutron Star Matter, Astrophys. J. Lett. 887, L24 (2019), arXiv:1912.05705 [astro-ph.HE].
  • [92] M. C. Miller et al., The Radius of PSR J0740+6620 from NICER and XMM-Newton Data, Astrophys. J. Lett. 918, L28 (2021), arXiv:2105.06979 [astro-ph.HE].
  • [93] T. Salmi et al., The Radius of the High-mass Pulsar PSR J0740+6620 with 3.6 yr of NICER Data, Astrophys. J. 974, 294 (2024), arXiv:2406.14466 [astro-ph.HE].
  • [94] D. Choudhury et al., A NICER View of the Nearest and Brightest Millisecond Pulsar: PSR J0437–4715, Astrophys. J. Lett. 971, L20 (2024), arXiv:2407.06789 [astro-ph.HE].
  • [95] L. Mauviard et al., A NICER View of the 1.4 M⊙ Edgeon Pulsar PSR J0614-3329, Astrophys. J. 995, 60 (2025), arXiv:2506.14883 [astro-ph.HE].
  • [96] B. P. Abbott et al. (KAGRA, LIGO Scientific, Virgo), Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA, Living Rev. Rel. 19, 1 (2016), arXiv:1304.0670 [gr-qc].
  • [97] J. Skilling, Nested Sampling, AIP Conf. Proc. 735, 395 (2004).
  • [98] J. Skilling, Nested sampling for general Bayesian computation, Bayesian Analysis 1, 833 (2006).
  • [99] J. S. Speagle, dynesty: a dynamic nested sampling package for estimating Bayesian posteriors and evidences, Mon. Not. Roy. Astron. Soc. 493, 3132 (2020), arXiv:1904.02180 [astroph.IM].
  • [100] S. Koposov, J. Speagle, K. Barbary, G. Ashton, E. Bennett, J. Buchner, C. Scheffler, C. Talbot, B. Cook, J. Guillochon, P. Cubillos, A. A. Ramos, M. Dartiailh, Ilya, E. Tollerud, D. Lang, B. Johnson, jtmendel, E. Higson, T. Vandal, T. Daylan, R. Angus, ruskin23, P. Cargile, P. Sheehan, M. Pitkin, M. Kirk, L. Xu, J. Leja, and joezuntz, joshspeagle/dynesty: v3.1.0 (2026).
  • [101] G. Ashton et al., BILBY: A user-friendly Bayesian inference library for gravitational-wave astronomy, Astrophys. J. Suppl. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM].
  • [102] I. M. Romero-Shaw et al., Bayesian inference for compact binary coalescences with bilby: validation and application to the first LIGO–Virgo gravitational-wave transient catalogue, Mon. Not. Roy. Astron. Soc. 499, 3295 (2020), arXiv:2006.00714 [astro-ph.IM].
  • [103] A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors, Phys. Rev. D 80, 084043 (2009), arXiv:0907.0700 [gr-qc].
  • [104] L. Blanchet, Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel. 17, 2 (2014), arXiv:1310.1528 [gr-qc].
  • [105] P. G. Krastev and B.-A. Li, Imprints of the nuclear symmetry energy on the tidal deformability of neutron stars, J. Phys. G 46, 074001 (2019), arXiv:1801.04620 [nucl-th].
  • [106] C. Mondal and F. Gulminelli, Can we decipher the composition of the core of a neutron star?, Phys. Rev. D 105, 083016 (2022), arXiv:2111.04520 [nucl-th].
  • [107] F. Iacovelli, M. Mancarella, C. Mondal, A. Puecher, T. Dietrich, F. Gulminelli, M. Maggiore, and M. Oertel, Nuclear physics constraints from binary neutron star mergers in the Einstein Telescope era, Phys. Rev. D 108, 122006 (2023), arXiv:2308.12378 [gr-qc].
  • [108] Z. Zhu and R. O’Shaughnessy, Limitations in constraining neutron star radii and nuclear properties from inspiral gravitational wave detections, Phys. Rev. D 113, 103037 (2026), arXiv:2508.11875 [astro-ph.HE].
  • [109] C. A. Raithel and E. R. Most, Degeneracy in the Inference of Phase Transitions in the Neutron Star Equation of State from Gravitational Wave Data, Phys. Rev. Lett. 130, 201403 (2023), arXiv:2208.04294 [astro-ph.HE].
  • [110] C. A. Raithel and E. R. Most, Tidal deformability doppelgänger: Implications of a low-density phase transition in the neutron star equation of state, Phys. Rev. D 108, 023010 (2023), arXiv:2208.04295 [astro-ph.HE].
  • [111] G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. RamosBuades, H. Estelles, and R. Jaume, Setting the cornerstone for a family of models for gravitational waves from compact binaries: The dominant harmonic for nonprecessing quasicircular black holes, Phys. Rev. D 102, 064001 (2020), arXiv:2001.11412 [gr-qc].
  • [112] A. Abac, T. Dietrich, A. Buonanno, J. Steinhoff, and M. Ujevic, New and robust gravitational-waveform model for high-massratio binary neutron star systems with dynamical tidal effects, Phys. Rev. D 109, 024062 (2024), arXiv:2311.07456 [gr-qc].
  • [113] L. Negr et al., Impact of the Einstein Telescope’s duty cycle on the estimation of binary black holes parameters, (2026), arXiv:2606.19201 [gr-qc].
  • [114] P. Landry and J. S. Read, The Mass Distribution of Neutron Stars in Gravitational-wave Binaries, Astrophys. J. Lett. 921, L25 (2021), arXiv:2107.04559 [astro-ph.HE].
  • [115] J. Golomb and C. Talbot, Hierarchical Inference of Binary Neutron Star Mass Distribution and Equation of State with Gravitational Waves, Astrophys. J. 926, 79 (2022), arXiv:2106.15745 [astro-ph.HE].
  • [116] J. Golomb, I. Legred, K. Chatziioannou, and P. Landry, Interplay of astrophysics and nuclear physics in determining the properties of neutron stars, Phys. Rev. D 111, 023029 (2025), arXiv:2410.14597 [astro-ph.HE].
  • [117] R. Gamba, M. Breschi, S. Bernuzzi, M. Agathos, and A. Nagar, Waveform systematics in the gravitational-wave inference of tidal parameters and equation of state from binary neutron star signals, Phys. Rev. D 103, 124015 (2021), arXiv:2009.08467 [gr-qc].
  • [118] N. Kunert, P. T. H. Pang, I. Tews, M. W. Coughlin, and T. Dietrich, Quantifying modeling uncertainties when combining multiple gravitational-wave detections from binary neutron star sources, Phys. Rev. D 105, L061301 (2022), arXiv:2110.11835 [astro-ph.HE].
  • [119] K. Chatziioannou, Uncertainty limits on neutron star radius measurements with gravitational waves, Phys. Rev. D 105, 084021 (2022), arXiv:2108.12368 [gr-qc].
  • [120] A. B. Yelikar, R. O. Shaughnessy, J. Lange, and A. Z. Jan, Waveform systematics in gravitational-wave inference of signals from binary neutron star merger models incorporating higher-order modes information, Phys. Rev. D 110, 064024 (2024), arXiv:2404.16599 [gr-qc].
  • [121] A. B. Yelikar, R. O’Shaughnessy, D. Wysocki, and L. Wade, Effects of waveform systematics on inferences of neutron star population properties and the nuclear equation of state, (2024), arXiv:2410.14674 [gr-qc].
  • [122] G. Pratten, P. Schmidt, and N. Williams, Impact of Dynamical Tides on the Reconstruction of the Neutron Star Equation of State, Phys. Rev. Lett. 129, 081102 (2022), arXiv:2109.07566 [astro-ph.HE].
  • [123] S. Ghosh, S. Mukherjee, S. Bose, and D. Chatterjee, Tidal dissipation in binary neutron star inspirals from hyperon bulk viscosity: phase modelling and parameter estimation bias, Mon. Not. Roy. Astron. Soc. 543, 2987 (2025), arXiv:2503.14606 [gr-qc].
  • [124] F. Gittins, H. Narola, T. Wouters, P. T. H. Pang, T. Hinderer, and C. Van Den Broeck, Detecting Tidal Resonances in Binary Neutron Stars, (2026), arXiv:2606.06376 [gr-qc].
  • [125] A. Samajdar, J. Janquart, C. Van Den Broeck, and T. Dietrich, Biases in parameter estimation from overlapping gravitationalwave signals in the third-generation detector era, Phys. Rev. D 104, 044003 (2021), arXiv:2102.07544 [gr-qc].
  • [126] A. Antonelli, O. Burke, and J. R. Gair, Noisy neighbours: inference biases from overlapping gravitational-wave signals, Mon. Not. Roy. Astron. Soc. 507, 5069 (2021), arXiv:2104.01897 [gr-qc].
  • [127] J. Janquart, T. Baka, A. Samajdar, T. Dietrich, and C. Van Den Broeck, Analyses of overlapping gravitational wave signals using hierarchical subtraction and joint parameter estimation, Mon. Not. Roy. Astron. Soc. 523, 1699 (2023), arXiv:2211.01304 [gr-qc].
  • [128] C. Pankow et al., Mitigation of the instrumental noise transient in gravitational-wave data surrounding GW170817, Phys. Rev. D 98, 084016 (2018), arXiv:1808.03619 [gr-qc].
  • [129] T. Narikawa, N. Uchikata, K. Kawaguchi, K. Kiuchi, K. Kyutoku, M. Shibata, and H. Tagoshi, Discrepancy in tidal deformability of GW170817 between the Advanced LIGO twin detectors, Phys. Rev. Research. 1, 033055 (2019), arXiv:1812.06100 [astro-ph.HE].
  • [130] N. Sarin, H. V. Peiris, D. J. Mortlock, J. Alsing, S. M. Nissanke, and S. M. Feeney, Measuring the nuclear equation of state with neutron star-black hole mergers, Phys. Rev. D 110, 024076 (2024), arXiv:2311.05689 [gr-qc].
  • [131] V. Dubey, R. Kashyap, and Y. Bhoge, Constraining equation of state of neutron star using neutron star-black hole mergers, (2026), arXiv:2608.30995 [gr-qc].
  • [132] P. T. H. Pang, S. M. Brown, T. Wouters, and C. Van Den Broeck, Revealing tensions in neutron star observations with pressure anisotropy, Phys. Rev. D 113, 063047 (2026), arXiv:2507.13039 [astro-ph.HE].
  • [133] S. M. Brown, B. Krishnan, R. Somasundaram, and I. Tews, Can Neutron Star Tidal Effects Obscure Deviations from General Relativity?, Astrophys. J. 982, 133 (2025), arXiv:2411.19129 [gr-qc].
  • [134] T. Wouters and P. T. H. Pang, JESTER: JAX-based EoS and TOV solvER.
  • [135] T. Wouters, Dataproducts for scalable and sequential inference of the neutron star equation of state with the einstein telescope, 10.5281/zenodo.22300363 (2026).
  • [136] I. C. F. Wong, F. Cireddu, S. Kumar, L. Negri, and H. Narola, gwmock: A python package for generating mock data challenge datasets for the gravitational-wave community. (2026).
  • [137] D. Ward, Flowjax: Distributions and normalizing flows in jax (2026).
  • [138] P. Landry and R. Essick, Nonparametric inference of the neutron star equation of state from gravitational wave observations, Phys. Rev. D 99, 084049 (2019), arXiv:1811.12529 [gr-qc].
  • [139] R. Essick, P. Landry, and D. E. Holz, Nonparametric Inference of Neutron Star Composition, Equation of State, and Maximum Mass with GW170817, Phys. Rev. D 101, 063007 (2020), arXiv:1910.09740 [astro-ph.HE].
  • [140] P. Landry, R. Essick, and K. Chatziioannou, Nonparametric constraints on neutron star matter with existing and upcoming gravitational wave and pulsar observations, Phys. Rev. D 101, 123007 (2020), arXiv:2003.04880 [astro-ph.HE].
  • [141] I. Legred, K. Chatziioannou, R. Essick, and P. Landry, Implicit correlations within phenomenological parametric models of the neutron star equation of state, Phys. Rev. D 105, 043016 (2022), arXiv:2201.06791 [astro-ph.HE].
  • [142] R. Essick, I. Legred, K. Chatziioannou, S. Han, and P. Landry, Phase transition phenomenology with nonparametric representations of the neutron star equation of state, Phys. Rev. D 108, 043013 (2023), arXiv:2305.07411 [astro-ph.HE].
  • [143] E. Finch, I. Legred, K. Chatziioannou, R. Essick, S. Han, and P. Landry, Unified nonparametric equation-of-state inference from the neutron-star crust to perturbative-QCD densities, Phys. Rev. D 112, 103023 (2025), arXiv:2505.13691 [nucl-th].
  • [144] A. Boudon, H. Qi, J.-F. Coupechoux, P. Landry, and V. Sordini, A Realistic Projection for Constraining the Neutron Star Equation of State with the LIGO–Virgo–KAGRA Detector Network in the A+ Era, Astrophys. J. 1002, 141 (2026), arXiv:2501.11585 [gr-qc].
  • [145] S. Ng, I. Legred, L. Suleiman, P. Landry, L. Traylor, and J. Read, Inferring the neutron star equation of state with nuclear-physics informed semiparametric models, Class. Quant. Grav. 42, 205008 (2025), arXiv:2507.03232 [astroph.HE].
  • [146] L. Suleiman, A. F. Fantina, F. Gulminelli, and J. Read, Nuclear parameter inference with semi-agnostic priors 10.1051/00046361/202558416 (2025), arXiv:2512.05315 [astro-ph.HE].
  • [147] D. Wysocki, R. O’Shaughnessy, L. Wade, and J. Lange, Inferring the neutron star equation of state simultaneously with the population of merging neutron stars, (2020), arXiv:2001.01747 [gr-qc].
  • [148] S. Vitale, D. Gerosa, W. M. Farr, and S. R. Taylor, Inferring the properties of a population of compact binaries in presence of selection effects 10.1007/978-981-15-4702-7_45-1 (2020), arXiv:2007.05579 [astro-ph.IM].
  • [149] R. Essick and M. Fishbach, Ensuring Consistency between Noise and Detection in Hierarchical Bayesian Inference, Astrophys. J. 962, 169 (2024), arXiv:2310.02017 [gr-qc].
  • [150] A. Kedia, R. O’Shaughnessy, L. Wade, and A. Yelikar, Astrophysical assumptions and equation of state framework have larger impact on equation of state inference than individual neutron star observations, (2024), arXiv:2405.17326 [astroph.HE].
  • [151] T. Ghosh, B. Biswas, S. Bose, and S. J. Kapadia, Joint Inference of Population, Cosmology, and Neutron Star Equation of State from Gravitational Waves of Dark Binary Neutron Stars, Astrophys. J. Suppl. 281, 11 (2025), arXiv:2407.16669 [gr-qc].
  • [152] A. Toubiana et al., Comparing astrophysical models to gravitational-wave data in the observable space, Phys. Rev. D 113, 083006 (2026), arXiv:2507.13249 [gr-qc].
  • [153] F. Douchin and P. Haensel, A unified equation of state of dense matter and neutron star structure, Astron. Astrophys. 380, 151 (2001), arXiv:astro-ph/0111092.
  • [154] J. Margueron, R. Hoffmann Casali, and F. Gulminelli, Equation of state for dense nucleonic matter from metamodeling. I. Foundational aspects, Phys. Rev. C 97, 025805 (2018), arXiv:1708.06894 [nucl-th].
  • [155] J. Margueron, R. Hoffmann Casali, and F. Gulminelli, Equation of state for dense nucleonic matter from metamodeling. II. Predictions for neutron star properties, Phys. Rev. C 97, 025806 (2018), arXiv:1708.06895 [nucl-th].
  • [156] I. Tews, J. Margueron, and S. Reddy, Critical examination of constraints on the equation of state of dense matter obtained from GW170817, Phys. Rev. C 98, 045804 (2018), arXiv:1804.02783 [nucl-th].
  • [157] R. Somasundaram, I. Tews, and J. Margueron, Investigating signatures of phase transitions in neutron-star cores, Phys. Rev. C 107, 025801 (2023), arXiv:2112.08157 [nucl-th].
  • [158] P. Neal and G. Roberts, Optimal scaling of random walk metropolis algorithms with non-gaussian proposals, Methodology and Computing in Applied Probability 13, 583 (2011).
  • [159] M. C. Miller et al., The Radius of PSR J0437–4715 from NICER Data, Astrophys. J. Lett. 1000, L48 (2026), arXiv:2512.08790 [astro-ph.HE].
  • [160] Y. Kini et al., A NICER View of PSR J0030+0451: Updated Constraints from 6 yr of NICER Observations, Astrophys. J. 1005, 201 (2026), arXiv:2602.23743 [astro-ph.HE].
  • [161] M. C. Miller et al., The Radius of the Neutron Star PSR J06143329 from NICER Data, (2026), arXiv:2609.00965 [astroph.HE].
  • [162] L. Mauviard et al., A NICER view of PSR J1614−2230: a massive and compact millisecond pulsar, (2026), arXiv:2609.00172 [astro-ph.HE].
  • [163] D. González-Caniulef et al., A NICER view of the millisecond pulsar PSR J2124−3358: evidence for a helium atmosphere, (2026), arXiv:2607.03721 [astro-ph.HE].
  • [164] T. Wouters, P. T. H. Pang, T. Dietrich, and C. Van Den Broeck, Incorporating neutron star physics into gravitational wave inference with physics-informed priors using normalizing flows, Phys. Rev. D 114, 043005 (2026), arXiv:2511.22987 [astroph.HE].
  • [165] C. García-Quirós, S. Husa, M. Mateu-Lucena,and A.Borchers, Accelerating the evaluation of inspiral–merger–ringdown waveforms with adapted grids, Class. Quant. Grav. 38, 015006 (2021), arXiv:2001.10897 [gr-qc].
  • [166] S. Morisaki, Accelerating parameter estimation of gravitational waves from compact binary coalescence using adaptive frequency resolutions, Phys. Rev. D 104, 044062 (2021), arXiv:2104.07813 [gr-qc].
  • [167] G. Papamakarios, T. Pavlakou, and I. Murray, Masked Autoregressive Flow for Density Estimation, (2017), arXiv:1705.07057 [stat.ML].
  • [168] C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, Neural Spline Flows (2019) arXiv:1906.04032 [stat.ML].
  • [169] R. Smith et al., Bayesian Inference for Gravitational Waves from Binary Neutron Star Mergers in Third Generation Observatories, Phys. Rev. Lett. 127, 081102 (2021), arXiv:2103.12274 [gr-qc].
  • [170] M. Dax, S. R. Green, J. Gair, N. Gupte, M. Pürrer, V. Raymond, J. Wildberger, J. H. Macke, A. Buonanno, and B. Schölkopf, Real-time inference for binary neutron star mergers using machine learning, Nature 639, 49 (2025), arXiv:2407.09602 [grqc].
  • [171] Q. Hu, J. Irwin, Q. Sun, C. Messenger, L. Suleiman, I. S. Heng, and J. Veitch, Decoding Long-duration Gravitational Waves from Binary Neutron Stars with Machine Learning: Parameter Estimation and Equations of State, Astrophys. J. Lett. 987, L17 (2025), arXiv:2412.03454 [gr-qc].
  • [172] R. Tenorio and D. Gerosa, Scalable data-analysis framework for long-duration gravitational waves from compact binaries using short Fourier transforms, Phys. Rev. D 111, 104044 (2025), arXiv:2502.11823 [gr-qc].
  • [173] D. Yallup, M. Prathaban, J. Alvey, T. C. K. Ng, T. Wouters, N. Sarin, and W. Handley, Ab Initio Real-Time GravitationalWave Parameter Estimation, (2026), arXiv:2607.28265 [grqc].
  • [174] N. Guttman, A. M. Baker, P. D. Lasky, and E. Thrane, Licence to Bin: Accurate and Scalable Inference for Binary Neutron Stars in Next-Generation Gravitational-Wave Detectors, (2026), arXiv:2606.14197 [astro-ph.HE].
  • [175] A. S. Sengupta, Fast Bayesian Inference for Long-Duration Gravitational-Wave Signals in 3G detectors, (2026), arXiv:2609.05375 [gr-qc].
  • [176] R. O’Shaughnessy, A. Jan, J. Lange, and R. Mechum, Scaling RIFT 1: Extending RIFT to third-generation gravitationalwave analyses, (2026), arXiv:2609.37393 [gr-qc].
  • [177] A. Virtuoso and E. Milotti, Beyond the long wavelength approximation: Next-generation gravitational-wave detectors and frequency-dependent antenna patterns, Phys. Rev. D 111, 064058 (2025), arXiv:2412.01693 [gr-qc].
  • [178] P. Baral, S. Morisaki, I. Gupta, and J. Creighton, Parameter estimation of gravitational-wave signals with frequency-dependent antenna responses and higher modes, Class. Quant. Grav. 42, 215007 (2025), arXiv:2503.09627 [astro-ph.IM]. SUPPLEMENTAL MATERIAL HIERARCHICAL BAYESIAN INFERENCE OF THE EQUATION OF STATE The matter inside all neutron stars is assumed to follow a single, universal EOS. One option to model it is through nonparametric methods, such as Gaussian processes [138–146]. Here, we focus instead on sampling a flexible EOS parametrization 𝜃EOS (described in more detail below). The hyperparameters 𝜃EOS are constrained by a collection of 𝑁 observations of BNS mergers D = {𝑑1, . . . , 𝑑𝑁 } through Bayes’ theorem: P(𝜃EOS|D) = L(D|𝜃EOS)𝜋(𝜃EOS) Z , (1) where P(𝜃EOS|D) is the hyperposterior, L(D|𝜃EOS) the hyperlikelihood, 𝜋(𝜃EOS) the prior and Z the Bayesian evidence. When the observations are independent, the hyperlikelihood is given by [31, 32] L(D|𝜃EOS) = 𝑝det(𝜃EOS)−𝑁 𝑁 Ö 𝑖=1 ∫ L (𝑑𝑖 |𝜃𝑖) 𝑝 (𝜃𝑖 |𝜃EOS) d𝜃𝑖 , (2) where 𝑝det(𝜃EOS) encodes selection effects, and L (𝑑𝑖 |𝜃𝑖) is the single-event likelihood. In this study, our observed catalog is based on an SNR threshold. Since varying the EOS mainly impacts the tidal deformabilities, which have a negligible effect on SNR, we neglect selection effects in this study.