- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Equation of state extrapolation systematics: Parametric vs nonparametric inference of neutron star structure

Bhaskar Biswas

PDF

Phys. Rev. D 113, 123032 – Published 11 June, 2026

DOI: https://doi.org/10.1103/x12t-29d9

Abstract

The equation of state (EOS) of cold dense matter remains one of the central open problems in nuclear astrophysics. Its inference is complicated by the lack of ab initio theoretical control above about twice nuclear saturation density, where the EOS must be extrapolated. Parametric schemes such as piecewise polytropes (PP) are computationally efficient but impose restrictive functional forms, while nonparametric approaches such as Gaussian processes (GP) offer greater flexibility at the cost of larger prior volumes. In this work, we extend our hybrid EOS framework by replacing the high-density polytropic extension with a Gaussian process representation of the squared sound speed, anchored at low densities by the SLy crust EOS and a nuclear meta-model constrained by χEFT and laboratory measurements. Using a hierarchical Bayesian analysis, we jointly constrain the EOS and the neutron star mass distribution with multimessenger data, including NICER radius measurements, GW170817 and GW190425 tidal deformabilities, 2M⊙ pulsars, and neutron skin experiments. We explore four scenarios defined by the choice of high-density extrapolation (PP vs GP) and hot spot geometry in the NICER modeling of PSR J0030+0451 (ST+PDT vs PDT-U). We find that GP extrapolations generally yield softer EOS posteriors with broader uncertainty bands. Hot spot geometry assumptions also play an important role, leading to systematic shifts in the inferred mass-radius relations. Bayesian evidence strongly favors the ST+PDT geometry over PDT-U across both extrapolation schemes, while the GP extension is consistently preferred over PP, with substantial support. Taken together, these results underscore the importance of both observational modeling choices and EOS extrapolation strategies in shaping neutron star EOS inferences, and demonstrate that a GP-based extension provides a robust framework for quantifying systematic uncertainties in high-density matter.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Observation of gravitational waves from a binary neutron star inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  2. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Properties of the binary neutron star merger GW170817, Phys. Rev. X 9, 011001 (2019).
  3. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett. 121, 161101 (2018).
  4. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW190425: Observation of a compact binary coalescence with total mass ∼3.4M⊙, Astrophys. J. Lett. 892, L3 (2020).
  5. T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E. Strohmayer, A NICER view of PSR J0030+0451: Nested samples for millisecond pulsar parameter estimation, 10.5281/zenodo.3386449 (2019).
  6. Thomas E. Riley et al., A NICER view of the massive pulsar PSR J0740+6620 informed by radio timing and XMM-Newton spectroscopy, Astrophys. J. Lett. 918, L27 (2021).
  7. 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).
  8. M. C. Miller et al., The radius of PSR J0740+6620 from NICER and XMM-Newton data, Astrophys. J. Lett. 918, L28 (2021).
  9. Devarshi Choudhury et al., A NICER view of the nearest and brightest millisecond pulsar: PSR J0437 − 4715, Astrophys. J. Lett. 971, L20 (2024).
  10. Lucien Mauviard et al., A NICER view of the 1.4 solar-mass edge-on pulsar PSR J0614 − 3329, arXiv:2506.14883.
  11. John Antoniadis et al., A massive pulsar in a compact relativistic binary, Science 340, 6131 (2013).
  12. H. Thankful Cromartie et al., Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar, Nat. Astron. 4, 72 (2019).
  13. Brendan T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Implications of PREX-II on the equation of state of neutron-rich matter, arXiv:2101.03193.
  14. D. Adhikari et al. (CREX Collaboration), Precision determination of the neutral weak form factor of Ca48, Phys. Rev. Lett. 129, 042501 (2022).
  15. Evgeny Epelbaum, Hans-Werner Hammer, and Ulf-G. Meissner, Modern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009).
  16. R. Machleidt and D. R. Entem, Chiral effective field theory and nuclear forces, Phys. Rep. 503, 1 (2011).
  17. Hans-Werner Hammer, Andreas Nogga, and Achim Schwenk, Three-body forces: From cold atoms to nuclei, Rev. Mod. Phys. 85, 197 (2013).
  18. Kai Hebeler, Three-nucleon forces: Implementation and applications to atomic nuclei and dense matter, Phys. Rep. 890, 1 (2021).
  19. C. Drischler, J. W. Holt, and C. Wellenhofer, Chiral effective field theory and the high-density nuclear equation of state, Annu. Rev. Nucl. Part. Sci. 71, 403 (2021).
  20. Tyler Gorda, Aleksi Kurkela, Risto Paatelainen, Saga Säppi, and Aleksi Vuorinen, Soft interactions in cold quark matter, Phys. Rev. Lett. 127, 162003 (2021).
  21. Tyler Gorda, Oleg Komoltsev, and Aleksi Kurkela, Ab-initio QCD calculations impact the inference of the neutron-star-matter equation of state, Astrophys. J. 950, 107 (2023).
  22. Oleg Komoltsev and Aleksi Kurkela, How perturbative QCD constrains the equation of state at neutron-star densities, Phys. Rev. Lett. 128, 202701 (2022).
  23. Jocelyn S. Read, Benjamin D. Lackey, Benjamin J. Owen, and John L. Friedman, Constraints on a phenomenologically parameterized neutron-star equation of state, Phys. Rev. D 79, 124032 (2009).
  24. Lee Lindblom, Spectral representations of neutron-star equations of state, Phys. Rev. D 82, 103011 (2010).
  25. S. K. Greif, G. Raaijmakers, K. Hebeler, A. Schwenk, and A. L. Watts, Equation of state sensitivities when inferring neutron star and dense matter properties, Mon. Not. R. Astron. Soc. 485, 5363 (2019).
  26. Isaac Legred, Katerina Chatziioannou, Reed Essick, and Philippe Landry, Implicit correlations within phenomenological parametric models of the neutron star equation of state, Phys. Rev. D 105, 043016 (2022).
  27. Gabriele Montefusco, Marco Antonelli, and Francesca Gulminelli, Frozen and β-equilibrated f and p modes of cold neutron stars: Nuclear metamodel predictions, Astron. Astrophys. 694, A150 (2025).
  28. Silvia Traversi, Prasanta Char, and Giuseppe Pagliara, Bayesian inference of dense matter equation of state within relativistic mean field models using astrophysical measurements, Astrophys. J. 897, 165 (2020).
  29. Luca Passarella, Jerome Margueron, and Giuseppe Pagliara, Relativistic mean-field predictions for the dense-matter equation of state and application to neutron stars, Phys. Rev. C 112, 035805 (2025).
  30. Bhaskar Biswas, Prasanta Char, Rana Nandi, and Sukanta Bose, Towards mitigation of apparent tension between nuclear physics and astrophysical observations by improved modeling of neutron star matter, Phys. Rev. D 103, 103015 (2021).
  31. Philippe Landry and Reed Essick, Nonparametric inference of the neutron star equation of state from gravitational wave observations, Phys. Rev. D 99, 084049 (2019).
  32. Philippe Landry, Reed Essick, and Katerina Chatziioannou, Nonparametric constraints on neutron star matter with existing and upcoming gravitational wave and pulsar observations, Phys. Rev. D 101, 123007 (2020).
  33. Nikhil Sarin, Hiranya V. Peiris, Daniel J. Mortlock, Justin Alsing, Samaya M. Nissanke, and Stephen M. Feeney, Measuring the nuclear equation of state with neutron star-black hole mergers, Phys. Rev. D 110, 024076 (2024).
  34. Bhaskar Biswas, Rana Nandi, Prasanta Char, Sukanta Bose, and Nikolaos Stergioulas, GW190814: On the properties of the secondary component of the binary, Mon. Not. R. Astron. Soc. 505, 1600 (2021).
  35. Bhaskar Biswas, Impact of PREX-II and combined radio/NICER/XMM-Newton’s mass–radius measurement of PSR J0740+6620 on the dense-matter equation of state, Astrophys. J. 921, 63 (2021).
  36. I. Bombaci and U. Lombardo, Asymmetric nuclear matter equation of state, Phys. Rev. C 44, 1892 (1991).
  37. Bhaskar Biswas and Stephan Rosswog, Simultaneously constraining the neutron star equation of state and mass distribution through multimessenger observations and nuclear benchmarks, Phys. Rev. D 112, 023045 (2025).
  38. F. Douchin and P. Haensel, A unified equation of state of dense matter and neutron star structure, Astron. Astrophys. 380, 151 (2001).
  39. I. Bombaci and U. Lombardo, Asymmetric nuclear matter equation of state, Phys. Rev. C 44, 1892 (1991).
  40. Gabriele Montefusco, Marco Antonelli, and Francesca Gulminelli, An asymptotically causal metamodel for neutron star equations of state, arXiv:2604.00196.
  41. M. Oertel, M. Hempel, T. Klähn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89, 015007 (2017).
  42. Robert E. Kass and Adrian E. Raftery, Bayes factors, J. Am. Stat. Assoc. 90, 773 (1995).
  43. Bhaskar Biswas and Prasanta Char, Systematics from NICER pulse profiles drive uncertainty in multi-messenger inference of the neutron star equation of state, Phys. Rev. D 112, 063049 (2025).
  44. M. Fortin, C. Providencia, A. R. Raduta, F. Gulminelli, J. L Zdunik, P. Haensel, and M. Bejger, Neutron star radii and crusts: Uncertainties and unified equations of state, Phys. Rev. C 94, 035804 (2016).
  45. Camille Ducoin, Jerome Margueron, Constanca Providencia, and Isaac Vidana, Core-crust transition in neutron stars: Predictivity of density developments, Phys. Rev. C 83, 045810 (2011).
  46. Pietro Klausner, Marco Antonelli, and Francesca Gulminelli, Properties of the neutron star crust informed by nuclear structure data, Phys. Rev. C 113, 025808 (2026).
  47. S. Burrello, F. Gulminelli, M. Antonelli, M. Colonna, and A. F. Fantina, Bayesian inference of neutron star crust properties using an ab-initio–benchmarked metamodel, Phys. Rev. C 112, 035802 (2025).
  48. Reed Essick, Philippe Landry, and Daniel E. Holz, Nonparametric inference of neutron star composition, equation of state, and maximum mass with GW170817, Phys. Rev. D 101, 063007 (2020).
  49. Serena Vinciguerra et al., An updated mass-radius analysis of the 2017–2018 NICER data set of PSR J0030+0451, arXiv:2308.09469.
  50. Serena Vinciguerra, Tuomo Salmi, Anna L. Watts, Devarshi Choudhury, Thomas E. Riley, Paul S. Ray, Slavko Bogdanov, Yves Kini, Sebastien Guillot, Deepto Chakrabarty, Wynn C. G. Ho, Daniela Huppenkothen, Sharon M. Morsink, Zorawar Wadiasingh, and Micheal T. Wolff, An updated mass-radius analysis of the 2017-2018 NICER data set of PSR J0030+0451, 10.5281/zenodo.8239000 (2023).
  51. Tuomo Salmi et al., The radius of the high mass pulsar PSR J0740+6620 with 3.6 years of NICER data, arXiv:2406.14466.
  52. Tuomo Salmi, Devarshi Choudhury, Yves Kini, Thomas Riley, Serena Vinciguerra, Anna L. Watts, Michael T. Wolff, Zaven Arzoumanian, Slavko Bogdanov, Deepto Chakrabarty, Keith Gendreau, Sebastien Guillot, Wynn C. G. Ho, Daniela Huppenkothen, Renee M. Ludlam, Sharon M. Morsink, and Paul S. Ray, Data and software for: “The radius of the high- mass pulsar PSR J0740+6620 with 3.6 yr of NICER data”, 10.5281/zenodo.10519473 (2024).
  53. Devarshi Choudhury, Tuomo Salmi, Vinciguerra Serena, Thomas Riley, Yves Kini, Anna L. Watts, Bas Dorsman, Slavko Bogdanov, Sebastien Guillot, Paul S. Ray, Daniel Reardon, Ronald A. Remillard, Anna Bilous, Daniela Huppenkothen, James Lattimer, Nathan Rutherford, Zaven Arzoumanian, Keith Gendreau, Sharon Morsink, and Wynn C. G. Ho, Reproduction package for: “A NICER view of the nearest and brightest millisecond pulsar: PSR J0437 − 4715”, 10.5281/zenodo.13766753 (2024).
  54. BUQEYE Collaboration, buqeye software repository, https://buqeye.github.io/software/ (2020), Bayesian uncertainty quantification for equation of state.
  55. C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, How well do we know the neutron-matter equation of state at the densities inside neutron stars? A Bayesian approach with correlated uncertainties, Phys. Rev. Lett. 125, 202702 (2020).
  56. D. Adhikari et al. (PREX Collaboration), An accurate determination of the neutron skin thickness of Pb208 through parity-violation in electron scattering, arXiv:2102.10767.
  57. Yi-Zhong Fan, Ming-Zhe Han, Jin-Liang Jiang, Dong-Sheng Shao, and Shao-Peng Tang, Maximum gravitational mass MTOV=2.25−0.07+0.08M⊙ inferred at about 3% precision with multimessenger data of neutron stars, Phys. Rev. D 109, 043052 (2024).
  58. Justin Alsing, Hector O. Silva, and Emanuele Berti, Evidence for a maximum mass cut-off in the neutron star mass distribution and constraints on the equation of state, Mon. Not. R. Astron. Soc. 478, 1377 (2018).
  59. Tuomo Salmi et al., A NICER view of PSR J1231−1411: A complex case, Astrophys. J. 976, 58 (2024).
  60. J. Buchner, A. Georgakakis, K. Nandra, L. Hsu, C. Rangel, M. Brightman, A. Merloni, M. Salvato, J. Donley, and D. Kocevski, X-ray spectral modelling of the AGN obscuring region in the CDFS: Bayesian model selection and catalogue, Astron. Astrophys. 564, A125 (2014).
  61. Sunny Ng, Isaac Legred, Lami Suleiman, Philippe Landry, Lyla Traylor, and Jocelyn Read, Inferring the neutron star equation of state with nuclear-physics informed semiparametric models, arXiv:2507.03232.

Outline

Information

Phys. Rev. D 113, 123032– Published 11 June, 2026
  • Received 23 September 2025
  • Revised 23 February 2026
  • Accepted 14 May 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation