Open accessJournal Article

# Bayesian inference of the dense-matter equation of state encapsulating a first-order hadron-quark phase transition from observables of canonical neutron stars

02 Mar 2021-Physical Review C (American Physical Society)-Vol. 103, Iss: 3, pp 035802
Abstract: Background: The remarkable progress in recent multimessenger observations of both isolated neutron stars (NSs) and their mergers has provided some of the much needed data to improve our understanding about the equation of state (EOS) of dense neutron-rich matter. Various EOSs with or without some kinds of phase transitions from hadronic to quark matter (QM) have been widely used in many forward modelings of NS properties. Direct comparisons of these predictions with observational data sometimes also using ${\ensuremath{\chi}}^{2}$ minimizations have provided very useful constraints on the model EOSs. However, it is normally difficult to perform uncertain quantifications and analyze correlations of the EOS model parameters involved in forward modelings especially when the available data are still very limited.Purpose: We infer the posterior probability distribution functions (PDFs) and correlations of nine parameters characterizing the EOS of dense neutron-rich matter encapsulating a first-order hadron-quark phase transition from the radius data of canonical NSs reported by LIGO/VIRGO, NICER, and Chandra Collaborations. We also infer the QM mass fraction and its radius in a $1.4\phantom{\rule{4pt}{0ex}}{\mathrm{M}}_{\ensuremath{\bigodot}}$ NS and predict their values in more massive NSs.Method: Metamodelings are used to generate both hadronic and QM EOSs in the Markov-Chain Monte Carlo sampling process within the Bayesian statistical framework. An explicitly isospin-dependent parametric EOS for the $npe\ensuremath{\mu}$ matter in NSs at $\ensuremath{\beta}$ equilibrium is connected through the Maxwell construction to the QM EOS described by the constant speed of sound (CSS) model of Alford, Han, and Prakash with and without using the Seidov stability condition for first-order phase transitions.Results: In the default calculation with the Seidov stability condition, we find that (i) The most probable values of the hadron-quark transition density ${\ensuremath{\rho}}_{t}/{\ensuremath{\rho}}_{0}$ and the relative energy density jump there $\mathrm{\ensuremath{\Delta}}\ensuremath{\varepsilon}/{\ensuremath{\varepsilon}}_{t}$ are ${\ensuremath{\rho}}_{t}/{\ensuremath{\rho}}_{0}=1.{6}_{\ensuremath{-}0.4}^{+1.2}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\varepsilon}/{\ensuremath{\varepsilon}}_{t}=0.{4}_{\ensuremath{-}0.15}^{+0.20}$ at 68% confidence level, respectively. The corresponding probability distribution of QM fraction in a $1.4\phantom{\rule{4pt}{0ex}}{\mathrm{M}}_{\ensuremath{\bigodot}}$ NS peaks around 0.9 in a 10 km sphere. Strongly correlated to the PDFs of ${\ensuremath{\rho}}_{t}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\varepsilon}/{\ensuremath{\varepsilon}}_{t}$, the PDF of the QM speed of sound squared ${c}_{\mathrm{QM}}^{2}/{c}^{2}$ peaks at $0.{95}_{\ensuremath{-}0.35}^{+0.05}$, and the total probability of being less than 1/3 is very small. (ii) The correlations between PDFs of hadronic and QM EOS parameters are very weak. While the most probable values of parameters describing the EOS of symmetric nuclear matter remain almost unchanged, the high-density symmetry energy parameters of neutron-rich matter are significantly different with or without considering the hadron-quark phase transition. Removing the Seidov condition, while there are appreciable and interesting changes in the PDFs of quark matter EOS parameters, the qualitative conclusions remain the same.Conclusions: The available astrophysical data considered together with all known EOS constraints from theories and terrestrial nuclear experiments prefer the formation of a large volume of QM even in canonical NSs.

Topics: Mass fraction (67%), ,  ... read more
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Open accessJournal Article
Abstract: Taking the format of two hardbound volumes a year, this series discusses new developments in the field at a level suitable for the general nuclear and particle physicist, and also, in greater technical depth, to explore the most important advances in those areas. Although the majority of the articles will be in one or other of the fields of the title of the series, a particular effort is made to treat topics which combine both particle and nuclear physics, in particular the role of mesons, isobars and quarks in nuclear structure and the use of complex nuclei for probing fundamental symmetries.

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453 Citations

Open accessJournal Article
Abstract: PSR J0740$+$6620 has a gravitational mass of $2.08\pm 0.07~M_\odot$, which is the highest reliably determined mass of any neutron star. As a result, a measurement of its radius will provide unique insight into the properties of neutron star core matter at high densities. Here we report a radius measurement based on fits of rotating hot spot patterns to Neutron Star Interior Composition Explorer (NICER) and X-ray Multi-Mirror (XMM-Newton) X-ray observations. We find that the equatorial circumferential radius of PSR J0740$+$6620 is $13.7^{+2.6}_{-1.5}$ km (68%). We apply our measurement, combined with the previous NICER mass and radius measurement of PSR J0030$+$0451, the masses of two other $\sim 2~M_\odot$ pulsars, and the tidal deformability constraints from two gravitational wave events, to three different frameworks for equation of state modeling, and find consistent results at $\sim 1.5-3$ times nuclear saturation density. For a given framework, when all measurements are included the radius of a $1.4~M_\odot$ neutron star is known to $\pm 4$% (68% credibility) and the radius of a $2.08~M_\odot$ neutron star is known to $\pm 5$%. The full radius range that spans the $\pm 1\sigma$ credible intervals of all the radius estimates in the three frameworks is $12.45\pm 0.65$ km for a $1.4~M_\odot$ neutron star and $12.35\pm 0.75$ km for a $2.08~M_\odot$ neutron star.

40 Citations

Open accessJournal Article
Thomas E. Riley1, Anna L. Watts1, Paul S. Ray2, Slavko Bogdanov3  +31 moreInstitutions (22)
Abstract: We report on Bayesian estimation of the radius, mass, and hot surface regions of the massive millisecond pulsar PSR J0740$+$6620, conditional on pulse-profile modeling of Neutron Star Interior Composition Explorer X-ray Timing Instrument (NICER XTI) event data. We condition on informative pulsar mass, distance, and orbital inclination priors derived from the joint NANOGrav and CHIME/Pulsar wideband radio timing measurements of arXiv:2104.00880. We use XMM European Photon Imaging Camera spectroscopic event data to inform our X-ray likelihood function. The prior support of the pulsar radius is truncated at 16 km to ensure coverage of current dense matter models. We assume conservative priors on instrument calibration uncertainty. We constrain the equatorial radius and mass of PSR J0740$+$6620 to be $12.39_{-0.98}^{+1.30}$ km and $2.072_{-0.066}^{+0.067}$ M$_{\odot}$ respectively, each reported as the posterior credible interval bounded by the 16% and 84% quantiles, conditional on surface hot regions that are non-overlapping spherical caps of fully-ionized hydrogen atmosphere with uniform effective temperature; a posteriori, the temperature is $\log_{10}(T$ [K]$)=5.99_{-0.06}^{+0.05}$ for each hot region. All software for the X-ray modeling framework is open-source and all data, model, and sample information is publicly available, including analysis notebooks and model modules in the Python language. Our marginal likelihood function of mass and equatorial radius is proportional to the marginal joint posterior density of those parameters (within the prior support) and can thus be computed from the posterior samples.

Topics: Pulsar (61%), Millisecond pulsar (60%)

36 Citations

Open accessJournal Article
Thomas E. Riley1, Anna L. Watts1, Paul S. Ray2, Slavko Bogdanov3  +31 moreInstitutions (22)
Abstract: We report on Bayesian estimation of the radius, mass, and hot surface regions of the massive millisecond pulsar PSR J0740$+$6620, conditional on pulse-profile modeling of Neutron Star Interior Composition Explorer X-ray Timing Instrument (NICER XTI) event data. We condition on informative pulsar mass, distance, and orbital inclination priors derived from the joint NANOGrav and CHIME/Pulsar wideband radio timing measurements of arXiv:2104.00880. We use XMM European Photon Imaging Camera spectroscopic event data to inform our X-ray likelihood function. The prior support of the pulsar radius is truncated at 16 km to ensure coverage of current dense matter models. We assume conservative priors on instrument calibration uncertainty. We constrain the equatorial radius and mass of PSR J0740$+$6620 to be $12.39_{-0.98}^{+1.30}$ km and $2.072_{-0.066}^{+0.067}$ M$_{\odot}$ respectively, each reported as the posterior credible interval bounded by the 16% and 84% quantiles, conditional on surface hot regions that are non-overlapping spherical caps of fully-ionized hydrogen atmosphere with uniform effective temperature; a posteriori, the temperature is $\log_{10}(T$ [K]$)=5.99_{-0.06}^{+0.05}$ for each hot region. All software for the X-ray modeling framework is open-source and all data, model, and sample information is publicly available, including analysis notebooks and model modules in the Python language. Our marginal likelihood function of mass and equatorial radius is proportional to the marginal joint posterior density of those parameters (within the prior support) and can thus be computed from the posterior samples.

Topics: Pulsar (61%), Millisecond pulsar (60%)

35 Citations

Open accessJournal Article
04 Jun 2021-Universe

Topics: Nuclear astrophysics (62%), Nuclear matter (60%), Neutron star (57%) ... read more

17 Citations

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Journal Article
J. R. Oppenheimer1, G. M. Volkoff1Institutions (1)
15 Feb 1939-Physical Review
Abstract: It has been suggested that, when the pressure within stellar matter becomes high enough, a new phase consisting of neutrons will be formed. In this paper we study the gravitational equilibrium of masses of neutrons, using the equation of state for a cold Fermi gas, and general relativity. For masses under $\frac{1}{3}\ensuremath{\bigodot}$ only one equilibrium solution exists, which is approximately described by the nonrelativistic Fermi equation of state and Newtonian gravitational theory. For masses $\frac{1}{3}\ensuremath{\bigodot}lml\frac{3}{4}\ensuremath{\bigodot}$ two solutions exist, one stable and quasi-Newtonian, one more condensed, and unstable. For masses greater than $\frac{3}{4}\ensuremath{\bigodot}$ there are no static equilibrium solutions. These results are qualitatively confirmed by comparison with suitably chosen special cases of the analytic solutions recently discovered by Tolman. A discussion of the probable effect of deviations from the Fermi equation of state suggests that actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely, although more and more slowly, never reaching true equilibrium.

Topics: , , Fermi gas (51%)

2,512 Citations

Open accessJournal Article
B. P. Abbott1, Richard J. Abbott1, T. D. Abbott2, Fausto Acernese3  +1235 moreInstitutions (132)
Abstract: On 17 August 2017, the LIGO and Virgo observatories made the first direct detection of gravitational waves from the coalescence of a neutron star binary system. The detection of this gravitational-wave signal, GW170817, offers a novel opportunity to directly probe the properties of matter at the extreme conditions found in the interior of these stars. The initial, minimal-assumption analysis of the LIGO and Virgo data placed constraints on the tidal effects of the coalescing bodies, which were then translated to constraints on neutron star radii. Here, we expand upon previous analyses by working under the hypothesis that both bodies were neutron stars that are described by the same equation of state and have spins within the range observed in Galactic binary neutron stars. Our analysis employs two methods: the use of equation-of-state-insensitive relations between various macroscopic properties of the neutron stars and the use of an efficient parametrization of the defining function pðρÞ of the equation of state itself. From the LIGO and Virgo data alone and the first method, we measure the two neutron star radii as R1 ¼ 10.8 þ2.0 −1.7 km for the heavier star and R2 ¼ 10.7 þ2.1 −1.5 km for the lighter star at the 90% credible level. If we additionally require that the equation of state supports neutron stars with masses larger than 1.97 M⊙ as required from electromagnetic observations and employ the equation-of-state parametrization, we further constrain R1 ¼ 11.9 þ1.4 −1.4 km and R2 ¼ 11.9 þ1.4 −1.4 km at the 90% credible level. Finally, we obtain constraints on pðρÞ at supranuclear densities, with pressure at twice nuclear saturation density measured at 3.5 þ2.7 −1.7 × 1034 dyn cm−2 at the 90% level.

Topics: Neutron star (62%), LIGO (60%), Gravitational wave (58%) ... read more

1,097 Citations

Journal Article
John W. Negele1, D. Vautherin1Institutions (1)
12 Jun 1973-Nuclear Physics
Abstract: An extremely simple form for the energy density of a nuclear many-body system derived from the two-body nucleon-nucleon interaction is used to determine the ground state configuration of matter at sub-nuclear density. As the baryon density is increased, nuclei become progressively more neutron rich until neutrons eventually escape, yielding a Coulomb lattice of bound neutron and proton clusters surrounded by a dilute neutron gas. The clusters enlarge and the lattice constant decreases with increasing density, approaching a completely uniform state near nuclear density.

Topics: Nuclear density (69%), Neutron (65%), Neutron scattering (64%) ... read more

621 Citations

Open accessJournal Article
Abstract: Neutron stars are not only of astrophysical interest, but are also of great interest to nuclear physicists because their attributes can be used to determine the properties of the dense matter in their cores. One of the most informative approaches for determining the equation of state (EoS) of this dense matter is to measure both a star’s equatorial circumferential radius R e and its gravitational mass M. Here we report estimates of the mass and radius of the isolated 205.53 Hz millisecond pulsar PSR J0030+0451 obtained using a Bayesian inference approach to analyze its energy-dependent thermal X-ray waveform, which was observed using the Neutron Star Interior Composition Explorer (NICER). This approach is thought to be less subject to systematic errors than other approaches for estimating neutron star radii. We explored a variety of emission patterns on the stellar surface. Our best-fit model has three oval, uniform-temperature emitting spots and provides an excellent description of the pulse waveform observed using NICER. The radius and mass estimates given by this model are km and (68%). The independent analysis reported in the companion paper by Riley et al. explores different emitting spot models, but finds spot shapes and locations and estimates of R e and M that are consistent with those found in this work. We show that our measurements of R e and M for PSR J0030+0451 improve the astrophysical constraints on the EoS of cold, catalyzed matter above nuclear saturation density.