Astrophysical measurement of the equation of state of neutron star matter (original) (raw)
Related papers
Low‐Mass Neutron Stars and the Equation of State of Dense Matter
The Astrophysical Journal, 2003
Neutron-star radii provide useful information on the equation of state of neutron rich matter. Particularly interesting is the density dependence of the equation of state (EOS). For example, the softening of the EOS at high density, where the pressure rises slower than anticipated, could signal a transition to an exotic phase. However, extracting the density dependence of the EOS requires measuring the radii of neutron stars for a broad range of masses. A "normal" 1.4M⊙ (M⊙=solar mass) neutron star has a central density of a few times nuclear-matter saturation density (ρ0). In contrast, low mass (≃ 0.5M⊙) neutron stars have central densities near ρ0 so its radius provides information on the EOS at low density. Unfortunately, low-mass stars are rare because they may be hard to form. Instead, a precision measurement of nuclear radii on atomic nuclei may contain similar information. Indeed, we find a strong correlation between the neutron radius of 208 Pb and the radius of a 0.5M⊙ neutron star. Thus, the radius of a 0.5M⊙ neutron star can be inferred from a measurement of the the neutron radius of 208 Pb. Comparing this value to the measured radius of a ≃ 1.4M⊙ neutron star should provide the strongest constraint to date on the density dependence of the equation of state.
Neutron stars: Cosmic laboratories for matter under extreme conditions
EPJ Web of Conferences, 2016
The true nature and the internal constitution of the compact stars known as neutron stars (NSs) is one of the most fascinating enigma in modern astrophysics. We discuss some of the present models for the internal structure of NSs and the connection with the properties of ultra dense hadronic matter. In particular, we discuss the role of strangeness on the equation of state and the implications of the measurement of 2 solar mass NSs in PSR J1614-2230 and PSR J0348+0432. Neutron stars (NSs) are incomparable natural laboratories that allow us to investigate the fundamental constituents of matter and their interactions under extreme conditions that cannot be reproduced in terrestrial laboratories. The bulk properties and the internal constitution of NSs primarily depend on the equation of state (EoS) of strong interacting matter [1], i.e. on the thermodynamical relation between the matter pressure, energy density and temperature. Determining the correct EoS model describing NS is a fundamental problem of nuclear physics and astrophysics, and a major effort has been made during the last few decades to solve it by measuring different NS properties using the data collected by various generations of X-ray and γ-ray satellites and by ground-based radio telescopes. The rather recent accurate measurement of the masses, M = 1.97 ± 0.04 M sun [2] and M = 2.01 ± 0.04 M sun [3], of the neutron stars in PSR J1614-2230 and PSR J0348+0432 respectively, has ruled out all the EoS models which cannot support such high values of stellar masses.
The Dense Matter Equation of State from Neutron Star Radius and Mass Measurements
The Astrophysical Journal, 2016
We present a comprehensive study of spectroscopic radius measurements of twelve neutron stars obtained during thermonuclear bursts or in quiescence. We incorporate, for the first time, a large number of systematic uncertainties in the measurement of the apparent angular sizes, Eddington fluxes, and distances, in the composition of the interstellar medium, and in the flux calibration of X-ray detectors. We also take into account the results of recent theoretical calculations of rotational effects on neutron star radii, of atmospheric effects on surface spectra, and of relativistic corrections to the Eddington critical flux. We employ Bayesian statistical frameworks to obtain neutron star radii from the spectroscopic measurements as well as to infer the equation of state from the radius measurements. Combining these with the results of experiments in the vicinity of nuclear saturation density and the observations of ∼ 2 M neutron stars, we place strong and quantitative constraints on the properties of the equation of state between ≈ 2 − 8 times the nuclear saturation density. We find that around M = 1.5 M , the preferred equation of state predicts radii between 10.1 − 11.1 km. When interpreting the pressure constraints in the context of high density equations of state based on interacting nucleons, our results suggest a relatively weak contribution of the three-body interaction potential.
Neutron star interiors and the equation of state of ultra-dense matter
2006
Neutron stars contain matter in one of the densest forms found in the Universe. This feature, together with the unprecedented progress in observational astrophysics, makes such stars superb astrophysical laboratories for a broad range of exciting physical studies. This paper gives an overview of the phases of dense matter predicted to make their appearance in the cores of neutron stars. Particular emphasis is put on the role of strangeness. Net strangeness is carried by hyperons, K-mesons, H-dibaryons, and strange quark matter, and may leave its mark in the masses, radii, moment of inertia, dragging of local inertial frames, cooling behavior, surface composition, and the spin evolution of neutron stars. These observables play a key role for the exploration of the phase diagram of dense nuclear matter at high baryon number density but low temperature, which is not accessible to relativistic heavy ion collision experiments.
Dense Stellar Matter and Structure of Neutron Stars
Eprint Arxiv Astro Ph 9609074, 1996
After giving an overview of the history and idea of neutron stars, I shall discuss, in part one of my lectures, the physics of dense neutron star matter. In this connection concepts like chemical equilibrium and electric charge neutrality will be outlined, and the baryonic and mesonic degrees of freedom of neutron star matter as well as the transition of confined hadronic matter into quark matter at supernuclear densities are studied. Special emphasis is put onto the possible absolute stability of strange quark matter. Finally models for the equation of state of dense neutron star matter, which account for various physical possibilities concerning the unknown behavior of superdense neutron star matter, will be derived in the framework of relativistic nuclear field theory. Part two of my lectures deals with the construction of models of static as well as rapidly rotating neutron stars and an investigation of the cooling behavior of such objects. Both is performed in the framework of Einstein's theory of general relativity. For this purpose the broad collection of models for the equation of state, derived in part one, will be used as an input. Finally, in part three a comparison of the theoretically computed neutron star properties (e.g., masses, radii, moments of inertia, red and blueshifts, limiting rotational periods, cooling behavior) with the body of observed data will be performed. From this conclusions about the interior structure of neutron stars-and thus the behavior of matter at extreme densities-will be drawn.
Properties of high-density matter in neutron stars
This short review aims at giving a brief overview of the various states of matter that have been suggested to exist in the ultra-dense centers of neutron stars. Particular emphasis is put on the role of quark deconfinement in neutron stars and on the possible existence of compact stars made of absolutely stable strange quark matter (strange stars). Astrophysical phenomena, which distinguish neutron stars from quark stars, are discussed and the question of whether or not quark deconfinement may occur in neutron stars is investigated. Combined with observed astrophysical data, such studies are invaluable to delineate the complex structure of compressed baryonic matter and to put firm constraints on the largely unknown equation of state of such matter.
GW170817: Measurements of Neutron Star Radii and Equation of State
Physical Review Letters, 2018
GW170817: Measurements of neutron star radii and equation of state The LIGO Scientific Collaboration and The Virgo Collaboration On August 17, 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 parameterization of the defining function p(ρ) 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 R 1 = 10.8 +2.0 −1.7 km for the heavier star and R 2 = 10.7 +2.1 −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 R 1 = 11.9 +1.4 −1.4 km and R 2 = 11.9 +1.4 −1.4 km at the 90% credible level. Finally, we obtain constraints on p(ρ) at supranuclear densities, with pressure at twice nuclear saturation density measured at 3.5 +2.7 −1.7 × 10 34 dyn cm −2 at the 90% level.
Neutron-rich nuclei and the equation of state of stellar matter
Physica Scripta, 2013
In this contribution we will review our present understanding of the matter equation of state in the density and temperature conditions where it can be described by nucleonic degrees of freedom. At zero temperature, all the information is contained in the nuclear energy functional in its isoscalar and isovector channels. At finite temperature, particular emphasis will be given to the specificity of the thermodynamics in the nucleonic regime, with the simultaneous presence of long range electromagnetic and short range nuclear interactions. The astrophysical implications of the resulting phase diagram, as well as of different observables of exotic nuclei on the neutron-rich side, will be touched upon.
The Equation of State of Nuclear Matter and Neutron Stars Properties
Journal of Modern Physics, 2014
The equation of state (EOS) of symmetric nuclear and pure neutron matter has been investigated extensively by adopting the non-relativistic Brueckner-Hartree-Fock (BHF). For more comparison, the extended BHF approaches using the self-consistent Green's function approach or by including a three-body force will be done. The EOS will be studied for different approaches at zero temperature. We can calculate the total mass and radius of neutron stars using various equations of state. A comparison with relativistic BHF calculations will be done. Relativistic effects are known to be important at high densities, giving an increased repulsion. This leads to a stiffer EOS compared to the EOS derived with a non-relativistic approach.
Constraints on the Equation-of-State of neutron stars from nearby neutron star observations
Journal of Physics: Conference Series, 2012
We try to constrain the Equation-of-State (EoS) of supra-nuclear-density matter in neutron stars (NSs) by observations of nearby NSs. There are seven thermally emitting NSs known from X-ray and optical observations, the so-called Magnificent Seven (M7), which are young (up to few Myrs), nearby (within a few hundred pc), and radio-quiet with blackbody-like X-ray spectra, so that we can observe their surfaces. As bright X-ray sources, we can determine their rotational (pulse) period and their period derivative from X-ray timing. From XMM and/or Chandra X-ray spectra, we can determine their temperature. With precise astrometric observations using the Hubble Space Telescope, we can determine their parallax (i.e. distance) and optical flux. From flux, distance, and temperature, one can derive the emitting area-with assumptions about the atmosphere and/or temperature distribution on the surface. This was recently done by us for the two brightest M7 NSs RXJ1856 and RXJ0720. Then, from identifying absorption lines in X-ray spectra, one can also try to determine gravitational redshift. Also, from rotational phase-resolved spectroscopy, we have for the first time determined the compactness (mass/radius) of the M7 NS RBS1223. If also applied to RXJ1856, radius (from luminosity and temperature) and compactness (from X-ray data) will yield the mass and radius-for the first time for an isolated single neutron star. We will present our observations and recent results.
Nuclear matter in neutron stars
Physics Subject Headings (PhySH)
Neutron stars are the most dense objects in the observable Universe and conventionally one uses nuclear theory to obtain the equation of state (EOS) of dense hadronic matter and the global properties of these stars. In this work, we review various aspects of nuclear matter within an effective Chiral model and interlink fundamental quantities both from nuclear saturation as well as vacuum properties and correlate it with the star properties.
Neutron stars and the equation of state
Journal of Astrophysics and Astronomy, 2018
The interior of neutron stars consists of the densest, although relatively cold, matter known in the universe. Here, baryon number densities might reach values close to ten times the nuclear saturation density. These suggest that the constituents of neutron star cores not only consist of nucleons, but also of more exotic baryons like hyperons or a phase of deconfined quarks. We discuss the consequences of such exotic particles on the gross properties and phenomenology of neutron stars. In addition, we determine the general phase structure of dense and also hot matter in the chiral parity-doublet model and confront model results with the recent constraints derived from the neutron star merger observation.
Neutron Star Interiors and the Equation of State of Superdense Matter
Astrophysics and Space Science Library, 2009
Neutron stars contain matter in one of the densest forms found in the Universe. This feature, together with the unprecedented progress in observational astrophysics, makes such stars superb astrophysical laboratories for a broad range of exciting physical studies. This paper gives an overview of the phases of dense matter predicted to make their appearance in the cores of neutron stars. Particular emphasis is put on the role of strangeness. Net strangeness is carried by hyperons, K-mesons, H-dibaryons, and strange quark matter, and may leave its mark in the masses, radii, moment of inertia, dragging of local inertial frames, cooling behavior, surface composition, and the spin evolution of neutron stars. These observables play a key role for the exploration of the phase diagram of dense nuclear matter at high baryon number density but low temperature, which is not accessible to relativistic heavy ion collision experiments.
Neutron Star Matter and Neutron Star Models
Zeitschrift für Naturforschung A, 1974
Various methods to study the ground state of neutron star matter are compared and the corresponding neutron star models are contrasted with each other. In the low density region ρ < 1014gr cm-3 the nuclear gas is treated here by means of a Thomas Fermi method and the nuclei are described by the droplet model of Myers and Swiatecki. For ρ > 1014 gr cm-3 both standard Brueckner theory with more realistic interaction (one-boson-exchange) potentials and the semiphenomenological theory of Fermi liquids (together with the standard Reid softcore potential) are applied to neutron star matter. It is shown that while the high mass limit of neutron stars is hardly affected, some properties of lowmass neutron stars such as their binding depend sensitively on these refinements. Various tentative (but unreliable) extensions of the equation of state into high density regime ρ > 1015 gr cm-3 are investigated and it is shown that the mass limit for heavy neutron stars lies around 2.5 solar ...
Neutron stars and the high density equation of state
AIP Conference Proceedings, 2009
One of the key ingredients to understand the properties of neutrons stars 1 (NS) is the equation of state at finite densities far beyond nuclear saturation. Investigating the phase structure of quark matter that might be realized in the core of NS inspires theory and observation. We discuss recent results of our work to point out our view on challenges and possibilities in this evolving field by means of a few examples.
Equation of state of nucleon matter and neutron star structure
Physical Review C, 1998
Properties of dense nucleon matter and the structure of neutron stars are studied using variational chain summation methods and the new Argonne v 18 two-nucleon interaction, which provides an excellent fit to all of the nucleonnucleon scattering data in the Nijmegen data base. The neutron star gravitational mass limit obtained with this interaction is 1.67M ⊙ . Boost corrections to the two-nucleon interaction, which give the leading relativistic effect of order (v/c) 2 , as well as three-nucleon interactions, are also included in the nuclear Hamiltonian. Their successive addition increases the mass limit to 1.80 and 2.20 M ⊙ . Hamiltonians including a three-nucleon interaction predict a transition in neutron star matter to a phase with neutral pion condensation at a baryon number density of ∼ 0.2 fm −3 . Neutron stars predicted by these Hamiltonians have a layer with a thickness on the order of tens of meters, over which the density changes rapidly from that of the normal to the condensed phase. The material in this thin layer is a mixture of the two phases. We also investigate the possibility of dense nucleon matter having an admixture of quark matter, described using the bag model equation of state. Neutron stars of 1.4M ⊙ do not appear to have quark matter admixtures in their cores. However, the heaviest stars are predicted to have cores consisting of a quark and nucleon matter mixture. These admixtures reduce the maximum mass of neutron stars from 2.20 to 2.02 (1.91) M ⊙ for bag constant B = 200 (122) MeV/fm 3 . Stars with pure quark matter in their cores are found to be unstable. We also consider the possibility that matter is maximally incompressible above an assumed density, and show that realistic models of nuclear forces limit the maximum mass of neutron stars to be below 2.5M ⊙ . The effects of the phase transitions on the composition of neutron star matter and its adiabatic index Γ are discussed.
EOS of Neutron Matter and Neutron Star Properties
Journal of Environmental Studies, 2014
The Equation of State (EOS) of pure neutron matter at zero temperature is calculated up to five saturation densities within the Brueckner theory with the inclusion of three-body forces. Three different realistic and accurate two-body forces are considered to evaluate the G-matrix effective interaction for nuclear matter. These models are the chiral N 3 LO, the CD-Bonn and the Argonne V 18 , which give quite different EOS. Two types of three-body forces are included to the effective interaction, which might be important at densities several times that of nuclear matter density. Using a microscopic EOS for pure neutron matter, static properties of non-rotating neutron stars such as masses and radii are evaluated. The resulting maximum masses of neutron stars using different interactions near 2M ʘ are found to be in reasonable agreement with the measured ones PSR J1614-2230 (with M max = 1.97 ± 0.04 M ʘ ) and PSR J0348+0432 (with M max = 2.01 ± 0.04 M ʘ ).
Related topics
Cited by
Physical Review D, 2011
We discuss the phase structure of dense matter, in particular the nature of the transition between hadronic and quark matter. Calculations within a Ginzburg-Landau approach show that the axial anomaly can induce a critical point in this transition region. This is possible because in three-flavor quark matter with instanton effects a chiral condensate can be added to the color-flavor locked (CFL) phase without changing the symmetries of the ground state. In (massless) two-flavor quark matter such a critical point is not possible since the corresponding color superconductor (2SC) does not break chiral symmetry. We study the effects of a nonzero but finite strange quark mass which interpolates between these two cases. Since at ultra-high density the first reaction of CFL to a nonzero strange quark mass is to develop a kaon condensate, we extend previous Ginzburg-Landau studies by including such a condensate. We discuss the fate of the critical point systematically and show that the continuity between hadronic and quark matter can be disrupted by the onset of a kaon condensate. Moreover, we identify the mass terms in the Ginzburg-Landau potential which are needed for the 2SC phase to occur in the phase diagram.
Monthly Notices of the Royal Astronomical Society, 2012
Using deep Chandra observations of the globular cluster M28, we study the quiescent X-ray emission of a neutron star in a low-mass X-ray binary in order to constrain the chemical composition of the neutron star atmosphere and the equation of state of dense matter. We fit the spectrum with different neutron star atmosphere models composed of hydrogen, helium or carbon. The parameter values obtained with the carbon model are unphysical and such a model can be ruled out. Hydrogen and helium models give realistic parameter values for a neutron star, and the derived mass and radius are clearly distinct depending on the composition of the atmosphere. The hydrogen model gives masses/radii consistent with the canonical values of 1.4 M ⊙ and 10 km, and would allow for the presence of exotic matter inside neutron stars. On the other hand, the helium model provides solutions with higher masses/radii, consistent with the stiffest equations of state. Measurements of neutron star masses/radii by spectral fitting should consider the possibility of heavier element atmospheres, which produce larger masses/radii for the same data, unless the composition of the accretor is known independently.
Further stable neutron star models from f ( R ) gravity
Journal of Cosmology and Astroparticle Physics, 2013
Neutron star models in perturbative f (R) gravity are considered with realistic equations of state. In particular, we consider the FPS, SLy and other equations of state and a case of piecewise equation of state for stars with quark cores. The mass-radius relations for f (R) = R + R(e −R/R 0 − 1) model and for R 2 models with logarithmic and cubic corrections are obtained. In the case of R 2 gravity with cubic corrections, we obtain that at high central densities (ρ > 10ρns, where ρns = 2.7 × 10 14 g/cm 3 is the nuclear saturation density), stable star configurations exist. The minimal radius of such stars is close to 9 km with maximal mass ∼ 1.9M⊙ (SLy equation). A similar situation takes place for AP4 and BSK20 EoS. Such an effect can give rise to more compact stars than in General Relativity. If observationally identified, such objects could constitute a formidable signature for modified gravity at astrophysical level. Another interesting result can be achieved in modified gravity with only a cubic correction. For some EoS, the upper limit of neutron star mass increases and therefore these EoS can describe realistic star configurations (although, in General Relativity, these EoS are excluded by observational constraints).
Equation of state of neutron star matter, and the nuclear symmetry energy
Physical Review C, 2011
The nuclear mean-field potentials obtained in the Hartree-Fock method with different choices of the in-medium nucleon-nucleon (NN) interaction have been used to study the equation of state (EOS) of the neutron star (NS) matter. The EOS of the uniform NS core has been calculated for the npeµ composition in the β-equilibrium at zero temperature, using version Sly4 of the Skyrme interaction as well as two density-dependent versions of the finite-range M3Y interaction (CDM3Yn and M3Y-Pn), and versions D1S and D1N of the Gogny interaction. Although the considered effective NN interactions were proven to be quite realistic in numerous nuclear structure and/or reaction studies, they give quite different behaviors of the symmetry energy of nuclear matter at supranuclear densities that lead to the soft and stiff scenarios discussed recently in the literature. Different EOS's of the NS core and the EOS of the NS crust given by the compressible liquid drop model have been used as input of the Tolman-Oppenheimer-Volkov equations to study how the nuclear symmetry energy affects the model prediction of different NS properties, like the cooling process as well as the gravitational mass, radius, and moment of inertia.
The transient gravitational-wave sky
Classical and Quantum Gravity, 2013
Interferometric detectors will very soon give us an unprecedented view of the gravitational-wave sky, and in particular of the explosive and transient Universe. Now is the time to challenge our theoretical understanding of short-duration gravitational-wave signatures from cataclysmic events, their connection to more traditional electromagnetic and particle astrophysics, and the data analysis arXiv:1305.0816v1 [gr-qc]
Matter effects on binary neutron star waveforms
Physical Review D, 2013
Using an extended set of equations of state and a multiple-group multiple-code collaborative effort to generate waveforms, we improve numerical-relativity-based data-analysis estimates of the measurability of matter effects in neutron-star binaries. We vary two parameters of a parameterized piecewise-polytropic equation of state (EOS) to analyze the measurability of EOS properties, via a parameter Λ that characterizes the quadrupole deformability of an isolated neutron star. We find that, to within the accuracy of the simulations, the departure of the waveform from point-particle (or spinless double black-hole binary) inspiral increases monotonically with Λ, and changes in the EOS that did not change Λ are not measurable.
Implementation of a simplified approach to radiative transfer in general relativity
Physical Review D, 2013
ABSTRACT We describe in detail the implementation of a simplified approach to radiative transfer in general relativity by means of the well-known neutrino leakage scheme (NLS). In particular, we carry out an extensive investigation of the properties and limitations of the NLS for isolated relativistic stars to a level of detail that has not been discussed before in a general-relativistic context. Although the numerous tests considered here are rather idealized, they provide a well-controlled environment in which to understand the relationship between the matter dynamics and the neutrino emission, which is important in order to model the neutrino signals from more complicated scenarios, such as binary neutron-star mergers. When considering nonrotating hot neutron stars we confirm earlier results of one-dimensional simulations, but also present novel results about the equilibrium properties and on how the cooling affects the stability of these configurations. In our idealized but controlled setup, we can then show that deviations from the thermal and weak-interaction equilibrium affect the stability of these models to radial perturbations, leading models that are stable in the absence of radiative losses, to a gravitational collapse to a black hole when neutrinos are instead radiated.
Structure and Cooling of Neutron and Hybrid Stars
Exciting Interdisciplinary Physics, 2013
The study of neutron stars is a topic of central interest in the investigation of the properties of strongly compressed hadronic matter. Whereas in heavy-ion collisions the fireball, created in the collision zone, contains very hot matter, with varying density depending on the beam energy, neutron stars largely sample the region of cold and dense matter with the exception of the very short time period of the existence of the proto-neutron star. Therefore, neutron star physics, in addition to its general importance in astrophysics, is a crucial complement to heavy-ion physics in the study of strongly interacting matter. In the following, model approaches will be introduced to calculate properties of neutron stars that incorporate baryons and quarks. These approaches are also able to describe the state of matter over a wide range of temperatures and densities, which is essential if one wants to connect and correlate star observables and results from heavy-ion collisions. The effect of exotic particles and quark cores on neutron star properties will be considered. In addition to the gross properties of the stars like their masses and radii their expected inner composition is quite sensitive to the models used. The effect of the composition can be studied through the analysis of the cooling curve of the star. In addition, we consider the effect of rotation, as in this case the particle composition of the star can be modified quite drastically.
Testing general relativity with present and future astrophysical observations
Classical and Quantum Gravity, 2015
One century after its formulation, Einstein's general relativity has made remarkable predictions and turned out to be compatible with all experimental tests. Most (if not all) of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that general relativity should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of general relativity. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einstein's theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime.
Improved mass and radius constraints for quiescent neutron stars in ω Cen and NGC 6397
2014
We use Chandra and XMM-Newton observations of the globular clusters ω Cen and NGC 6397 to measure the spectrum of their quiescent neutron stars (NSs), and thus to constrain the allowed ranges of mass and radius for each. We also use Hubble Space Telescope photometry of NGC 6397 to identify a potential optical companion to the quiescent NS, and find evidence that the companion lacks hydrogen. We carefully consider a number of systematic problems, and show that the choices of atmospheric composition, interstellar medium abundances, and cluster distances can have important effects on the inferred NS mass and radius. We find that for typical NS masses, the radii of both NSs are consistent with the 10 − 13 km range favored by recent nuclear physics experiments. This removes the evidence suggested by Guillot and collaborators for an unusually small NS radius, which relied upon the small inferred radius of the NGC 6397 NS.
Constraining the mass and radius of neutron stars in globular clusters
Monthly Notices of the Royal Astronomical Society
We analyze observations of eight quiescent low-mass X-ray binaries in globular clusters and combine them to determine the neutron star mass-radius curve and the equation of state of dense matter. We determine the effect that several uncertainties may have on our results, including uncertainties in the distance, the atmosphere composition, the neutron star maximum mass, the neutron star mass distribution, the possible presence of a hotspot on the neutron star surface, and the prior choice for the equation of state of dense matter. We find that the radius of a 1.4 solar mass neutron star is most likely from 10 to 14 km and that tighter constraints are only possible with stronger assumptions about the nature of the neutron stars, the systematics of the observations, or the nature of dense matter. Strong phase transitions are preferred over other models and interpretations of the data with a Bayes factor of 8 or more, and in this case, the radius is likely smaller than 12 km. However, radii larger than 12 km are preferred if the neutron stars have uneven temperature distributions.
Constraints on perturbative f ( R ) gravity via neutron stars
Journal of Cosmology and Astroparticle Physics, 2011
We study the structure of neutron stars in perturbative f (R) gravity models with realistic equations of state. We obtain the mass-radius relation in gravity models of the form f (R) = R + αR 2 and f (R) = R + βR 3. Using the recent observational constraints on the mass-radius relation, we find that |α| 10 10 cm 2 and |β| 10 21 cm 4. This implies, for such gravity models, that deviations from Einstein's general relativity in the strong gravity regime should be within a curvature scale of 10 −10 cm −2. Contents
Physical Review D, 2014
Neutron stars are thought to be excellent laboratories for determining the equation of state (EoS) of cold dense matter. Their strong gravity suggests that they can also be used to constrain gravity models. The two observables of neutron stars-mass and radius (M-R)-both depend on the choice of EoS and relativistic gravity, meaning that neutron stars cannot be simultaneously good laboratories for both of these questions. A measurement of mass and/or radius would constrain the less well known physics input. The most common assumption-namely, that M-R measurements can be used to constrain EoS-presumes general relativity (GR) is the ultimate model of gravity in the classical regime. We calculate the radial profile of compactness and curvature (square root of the full contraction of the Weyl tensor) within a neutron star and determine the domain not probed by the Solar System tests of GR. We find that, except for a tiny sphere of radius less than a millimeter at the center, the curvature is several orders of magnitude above the values present in Solar System tests. The compactness is beyond the solar surface value for r > 10 m, and increases by 5 orders of magnitude towards the surface. With the density being only an order of magnitude higher than that probed by nuclear scattering experiments, our results suggest that the employment of GR as the theory of gravity describing the hydrostatic equilibrium of the neutron stars is a rather remarkable extrapolation from the regime of tested validity, as opposed to that of EoS models. Our larger ignorance of gravity within neutron stars suggests that a measurement of mass and/or radius constrains gravity rather than the EoS, and given that the EoS has yet to be determined by nucleon scattering experiments, M-R measurements cannot tightly constrain the gravity models either. Near the surface the curvature and compactness attain their largest values, while the EoS in this region is fairly well known. This renders the crust as the best site to look for deviations from GR.
Relativistic stars in Starobinsky gravity with the matched asymptotic expansions method
Physical Review D
We study the structure of relativistic stars in R + αR 2 theory using the method of matched asymptotic expansion to handle the higher order derivatives in field equations arising from the higher order curvature term. We find solutions, parametrized by α, for uniform density stars. We obtain the mass-radius relations and study the dependence of maximum mass on α. We find that Mmax is almost linearly proportional to α. For each α the maximum mass configuration has the biggest compactness parameter (η = GM/Rc 2), and we argue that the general relativistic stellar configuration corresponding to α = 0 is the least compact among these.
Anisotropic compact stars in the Buchdahl model: A comprehensive study
Physical Review D
In this article we present a class of relativistic solutions describing spherically symmetric and static anisotropic stars in hydrostatic equilibrium. For this purpose, we consider a particularized metric potential, namely, Buchdahl ansatz [Phys. Rev. D 116, 1027 (1959).] which encompasses almost all the known analytic solutions to the spherically symmetric, static Einstein equations with a perfect fluid source, including, in particular, the Vaidya-Tikekar and Finch-Skea. We developed the model by considering an anisotropic spherically symmetric static general relativistic configuration that has a significant effect on the structure and properties of stellar objects. We have considered eight different cases for generalized Buchdahl dimensionless parameter K and analyzed them in a uniform manner. As a result it turns out that all the considered cases are valid at every point in the interior spacetime. In addition to this, we show that the model satisfies all the energy conditions and maintains the hydrostatic equilibrium equation. In the frame work of anisotropic hypothesis, we consider analogue objects with similar mass and radii, such as LMC X-4, SMC X-1, EXO 1785-248 etc. to restrict the model parameter arbitrariness. Also, establishing a relation between pressure and density in the form of P ¼ PðρÞ, we demonstrate that equation of state (EoS) can be approximated to a linear function of density. Despite the simplicity of this model, the obtained results are satisfactory.
2021
We report exact black hole solutions in asymptotically flat or (A)dS four-dimensional spacetime with a conformally coupled self-interacting scalar field in f (R) gravity. We first consider the asymptotically flat model f (R) = R − 2α √ R and derive an exact black hole solution. Then, we consider the asymptotically (A)dS model f (R) = R − 2Λ − 2α √ R − 4Λ and derive an exact black hole solution. In both cases the modified gravity parameter α, which has the dimension of the inverse mass, cannot be set to zero and the self-interacting potential is determined from the Klein-Gordon equation, preserving the conformal invariance. The thermodynamics of the solutions is also studied.
Physical Review C
Employing a recently proposed metamodeling for the nucleonic matter equation of state we analyze neutron star global properties such as masses, radii, momentum of inertia, and others. The impact of the uncertainty on empirical parameters on these global properties is analyzed in a Bayesian statistical approach. Physical constraints, such as causality and stability, are imposed on the equation of state and different hypotheses for the direct Urca (dUrca) process are investigated. In addition, only metamodels with maximum masses above 2M ⊙ are selected. Our main results are the following: the equation of state exhibits a universal behavior against the dUrca hypothesis under the condition of charge neutrality and β-equilibrium; neutron stars, if composed exclusively of nucleons and leptons, have a radius of 12.7±0.4 km for masses ranging from 1 up to 2M ⊙ ; a small radius lower than 11 km is very marginally compatible with our present knowledge of the nuclear empirical parameters; and finally, the most important empirical parameters which are still affected by large uncertainties and play an important role in determining the radius of neutrons stars are the slope and curvature of the symmetry energy (L sym and K sym) and, to a lower extent, the skewness parameters (Q sat/sym).
The Astrophysical Journal
This paper presents a new analysis of the thermal emission from the neutron star (NS) surface to constrain the dense matter equation of state. We employ an empirical parameterization of the equation of state with a Markov-Chain Monte Carlo approach to consistently fit the spectra of quiescent lowmass X-ray binaries in globular clusters with well-measured distances. Despite previous analyses predicting low NS radii, we show that it is possible to reconcile the astrophysical data with nuclear physics knowledge, with or without including a prior on the slope of the symmetry energy L sym. With this empirical parameterization of the equation of state, we obtain radii of the order of about 12 km without worsening the fit statistic. More importantly, we obtain the following values for the slope of the symmetry energy, its curvature K sym , and the isoscalar skewness parameter Q sat : L sym = 37.2 +9.2 −8.9 MeV, K sym = −85 +82 −70 MeV, and Q sat = 318 +673 −366 MeV. These are the first measurements of the empirical parameters K sym and Q sat. Their values are only weakly impacted by our assumptions, such as the distances or the number of free empirical parameters, provided the latter are taken within a reasonable range. We also study the weak sensitivity of our results to the set of sources analyzed, and we identify a group of sources that dominates the constraints. The resulting masses and radii obtained from this empirical parameterization are also compared to other measurements from electromagnetic observations of NSs and gravitational wave signals from the NS-NS merger GW 170817.
Does dark matter admixed pulsar exist?
The European Physical Journal Plus
In this paper, we have considered a twofluid model assuming that the pulsars are made of ordinary matter admixed with dark matter.Contribution of dark matter comes from the fitting of the rotation curves of the SPARC sample of galaxies[95]. For this we have investigated the dark matter based on the Singular Isothermal Sphere (SIS) dark matter density profile in the galactic halo region. Considering this twofluid model, we have studied the physical features of the pulsars present in different galaxy in details. Here, we compute the probable radii, compactness (u) and surface red-shift (Z s
The scenario of two families of compact stars
The European Physical Journal A, 2016
We will follow the two-families scenario described in the accompanying paper, in which compact stars having a very small radius and masses not exceeding about 1.5M⊙ are made of hadrons, while more massive compact stars are quark stars. In the present paper we discuss the dynamics of the transition of a hadronic star into a quark star. We will show that the transition takes place in two phases: a very rapid one, lasting a few milliseconds, during which the central region of the star converts into quark matter and the process of conversion is accelerated by the existence of strong hydrodynamical instabilities, and a second phase, lasting about ten seconds, during which the process of conversion proceeds as far as the surface of the star via production and diffusion of strangeness. We will show that these two steps play a crucial role in the phenomenological implications of the model. We will discuss the possible implications of this scenario both for long and for short Gamma Ray Bursts, using the proto-magnetar model as the reference frame of our discussion. We will show that the process of quark deconfinement can be connected to specific observed features of the GRBs. In the case of long GRBs we will discuss the possibility that quark deconfinement is at the origin of the second peak present in quite a large fraction of bursts. Also we will discuss the possibility that long GRBs can take place in binary systems without being associated with a SN explosion. Concerning short GRBs, quark deconfinement can play the crucial role in limiting their duration. Finally we will shortly revisit the possible relevance of quark deconfinement in some specific type of Supernova explosions, in particular in the case of very massive progenitors. PACS. PACS-key describing text of that key-PACS-key describing text of that key
Relativistic stars in bigravity theory
Physical Review D
Assuming static and spherically symmetric spacetimes in the ghost-free bigravity theory, we find a relativistic star solution, which is very close to that in general relativity. The coupling constants are classified into two classes: Class [I] and Class [II]. Although the Vainshtein screening mechanism is found in the weak gravitational field for both classes, we find that there is no regular solution beyond the critical value of the compactness in Class [I]. This implies that the maximum mass of a neutron star in Class [I] becomes much smaller than that in GR. On the other hand, for the solution in Class [II], the Vainshtein screening mechanism works well even in a relativistic star and the result in GR is recovered.
Modified gravity with logarithmic curvature corrections and the structure of relativistic stars
Physical Review D, 2013
We consider the effect of a logarithmic f (R) theory, motivated by the form of the one-loop effective action arising from gluons in curved spacetime, on the structure of relativistic stars. In addition to analysing the consistency constraints on the potential of the scalar degree of freedom, we discuss the possibility of observational features arising from a fifth force in the vicinity of the neutron star surface. We find that the model exhibits a chameleon effect that completely suppresses the effect of the modification on scales exceeding a few radii, but close to the surface of the neutron star, the deviation from General Relativity can significantly affect the surface redshift that determines the shift in absorption (or emission) lines. We also use the method of perturbative constraints to solve the modified Tolman-Oppenheimer-Volkov equations for normal and self-bound neutron stars (quark stars).
Living Reviews in Relativity, 2012
We review the current status of studies of the coalescence of binary neutron star systems. We begin with a discussion of the formation channels of merging binaries and we discuss the most recent theoretical predictions for merger rates. Next, we turn to the quasi-equilibrium formalisms that are used to study binaries prior to the merger phase and to generate initial data for fully dynamical simulations. The quasi-equilibrium approximation has played a key role in developing our understanding of the physics of binary coalescence and, in particular, of the orbital instability processes that can drive binaries to merger at the end of their lifetimes. We then turn to the numerical techniques used in dynamical simulations, including relativistic formalisms, (magneto-)hydrodynamics, gravitational-wave extraction techniques, and nuclear microphysics treatments. This is followed by a summary of the simulations performed across the field to date, including the most recent results from both fully relativistic and microphysically detailed simulations. Finally, we discuss the likely directions for the field as we transition from the first to the second generation of gravitational-wave interferometers and while supercomputers reach the petascale frontier.
Frontiers in Astronomy and Space Sciences
The appearance of strangeness in the form of hyperons within the inner core of neutron stars is expected to affect its detectable properties, such as its global structure or gravitational wave emission. This work explores the parameter space of hyperonic stars within the framework of the Relativistic Mean Field model allowed by the present uncertainties in the state-of-the-art nuclear and hypernuclear experimental data. We impose multi-physics constraints at different density regimes to restrict the parameter space: Chiral effective field theory, heavy-ion collision data, and multi-messenger astrophysical observations of neutron stars. We investigate possible correlations between empirical nuclear and hypernuclear parameters, particularly the symmetry energy and its slope, with observable properties of neutron stars. We do not find a correlation for the hyperon parameters and the astrophysical data. However, the inclusion of hyperons generates a tension between the astrophysical and...
Vacuum revealed: The final state of vacuum instabilities in compact stars
Physical Review D, 2011
Quantum fields in compact stars can be amplified due to a semiclassical instability. This generic feature of scalar fields coupled to curvature may affect the birth and the equilibrium structure of relativistic stars. We point out that the semiclassical instability has a classical counterpart, which occurs exactly in the same region of the parameter space. For negative values of the coupling parameter the instability is equivalent to the well-known "spontaneous scalarization" effect: the plausible end-state of the instability is a static, asymptotically flat equilibrium configuration with nonzero expectation value for the quantum fields, which is compatible with experiments in the weak-field regime and energetically favored over stellar solutions in general relativity. For positive values of the coupling parameter the new configurations are energetically disfavored, and the endpoint of the instability remains an open and interesting issue. The vacuum instability may provide a natural mechanism to produce spontaneous scalarization, leading to new experimental opportunities to probe the nature of vacuum energy via astrophysical observations of compact stars.
Ranking Love Numbers for the Neutron Star Equation of State: The Need for Third-Generation Detectors
Physical Review Letters, 2022
Gravitational-wave measurements of the tidal deformability in neutron-star binary coalescences can be used to infer the still unknown equation of state (EoS) of dense matter above the nuclear saturation density. By employing a Bayesian-ranking test we quantify the ability of current and future gravitational-wave observations to discriminate among families of nuclear-physics based EoS which differ in particle content and ab-initio microscopic calculations. While the constraining power of GW170817 is limited, we show that even twenty coalescences detected by LIGO-Virgo at design sensitivity are not enough to discriminate between EoS with similar softness but distinct microphysics. However, just a single detection with a third-generation detector such as the Einstein Telescope or Cosmic Explorer will rule out several families of EoS with very strong statistical significance, and can discriminate among models which feature similar softness, hence constraining the properties of nuclear matter to unprecedented levels.
Using iron line reverberation and spectroscopy to distinguish Kerr and non-Kerr black holes
Journal of Cosmology and Astroparticle Physics, 2015
The iron Kα line commonly observed in the X-ray spectrum of both stellar-mass and supermassive black hole candidates is produced by the illumination of a cold accretion disk by a hot corona. In this framework, the activation of a new flaring region in the hot corona imprints a time variation on the iron line spectrum. Future X-ray facilities with high time resolution and large effective areas may be able to measure the so-called 2-dimensional transfer function; that is, the iron line profile detected by a distant observer as a function of time in response to an instantaneous flare from the X-ray primary source. This work is a preliminary study to determine if and how such a technique can provide more information about the spacetime geometry around the compact object than the already possible measurements of the time-integrated iron line profile. Within our simplified model, we find that a measurement of iron line reverberation can improve constraints appreciably given a sufficiently strong signal, though that most of the information is present in the timeintegrated spectrum. Our aim is to test the Kerr metric. We find that current X-ray facilities and data are unable to provide strong tests of the Kerr nature of supermassive black hole candidates. We consider an optimistic case of 10 5 iron line photons from a next-generation data set. With such data, the reverberation model improves upon the spectral constraint by an order of magnitude.
Future prospects for constraining nuclear matter parameters with gravitational waves
Physical Review D, 2019
The gravitational wave emission from the merging binary neutron star system GW170817 arrived full of tidal information which can be used to probe the fundamental ultra-dense nuclear physics residing in these stars. In previous work, we used two-dimensional correlations between nuclear matter parameters and tidal deformabilities of neutron stars applying specifically to GW170817 to derive constraints on the former. Here, we extend this analysis by finding similar correlations for varying chirp masses, the dominant determining factor in the frequency evolution of the inspiral, such that one can apply the same method to future detections. We estimate how accurately one can measure nuclear parameters with future gravitational wave interferometers and show how such measurements can be improved by combining multiple events. We find that bounds on the nuclear parameters with future observations can improve from the current one with GW170817 only by ∼ 30% due to the existence of systematic errors caused mainly by the remaining uncertainty in the equation of state near and just above the nuclear saturation density. We show that such systematic errors can be reduced by considering multidimensional correlations among nuclear parameters and tidal deformabilities with various neutron star masses.
Constraining three-nucleon forces with multimessenger data
Physical Review C, 2021
We report the results of a study aimed at inferring direct information on the repulsive threenucleon potential V R ijk-driving the stiffness of the nuclear matter equation of state at supranuclear densities-from astrophysical observations. Using a Bayesian approach, we exploit the measurements of masses, radii and tidal deformabalities performed by the NICER satellite and the LIGO/Virgo collaboration, as well as the mass of the heaviest observed pulsar, to constrain the strength of V R ijk. The baseline of our analysis is the widely employed nuclear Hamiltonian comprising the Argonne v18 nucleon-nucleon potential and the Urbana IX model of three-nucleon potential. The numerical results, largely determined by the bound on the maximum mass, suggest that existing and future facilities have the potential to provide valuable new insight into microscopic nuclear dynamics at supranuclear densities.
Imposing multi-physics constraints at different densities on the neutron Star Equation of State
The European Physical Journal A, 2022
Neutron star matter spans a wide range of densities, from that of nuclei at the surface to exceeding several times normal nuclear matter density in the core. While terrestrial experiments, such as nuclear or heavy-ion collision experiments, provide clues about the behaviour of dense nuclear matter, one must resort to theoretical models of neutron star matter to extrapolate to higher density and finite neutron/proton asymmetry relevant for neutron stars. In this work, we explore the parameter space within the framework of the Relativistic Mean Field model allowed by present uncertainties compatible with state-of-the-art experimental data. We apply a cutoff filter scheme to constrain the parameter space using multi-physics constraints at different density regimes: chiral effective field theory, nuclear and heavy-ion collision data as well as multi-messenger astrophysical observations of neutron stars. Using the results of the study, we investigate possible correlations between nuclear and astrophysical observables.
Physical Review C, 2015
Using a solitonic model of nuclear matter, the BPS Skyrme model, we compare neutron stars obtained in the full field theory, where gravitational back reaction is completely taken into account, with calculations in a mean-field approximation using the Tolman-Oppenheimer-Volkoff approach. In the latter case, a mean-field-theory equation of state is derived from the original BPS field theory. We show that in the full field theory, where the energy density is non-constant even at equilibrium, there is no universal and coordinate independent equation of state of nuclear matter, in contrast to the mean-field approximation. We also study how neutron star properties are modified by going beyond mean field theory, and find that the differences between mean field theory and exact results can be considerable. Further, we compare both exact and mean-field results with some theoretical and phenomenological constraints on neutron star properties, demonstrating thus the relevance of our model even in its most simple version.
Physical Review D, 2011
Using the solutions of the gap equations of the magnetic-color-flavor-locked (MCFL) phase of paired quark matter in a magnetic field, and taking into consideration the separation between the longitudinal and transverse pressures due to the field-induced breaking of the spatial rotational symmetry, the equation of state (EoS) of the MCFL phase is self-consistently determined. This result is then used to investigate the possibility of absolute stability, which turns out to require a field-dependent "bag constant" to hold. That is, only if the bag constant varies with the magnetic field, there exists a window in the magnetic field vs. bag constant plane for absolute stability of strange matter. Implications for stellar models of magnetized (self-bound) strange stars and hybrid (MCFL core) stars are calculated and discussed.
PossibleExistence of Dark-Matter-Admixed Pulsar in the Disk Region of the Milky Way Galaxy
Universe
In our previous study, (Eur Phys J Plus 135:362, 2020 & Eur Phys J Plus 135:637, 2020), we have discussed the possible existence of the dark-matter-admixed pulsars, located in dwarf as well as in massive spiral galaxies (based on Singular Isothermal Sphere dark-matter density profile) and in the Milky Way galaxy (based on Universal Rotational Curve dark-matter density profile). In this article, we use the Navarro–Frenk–White (NFW) dark-matter density profile to get analogous results for the pulsars in the disk region of the Milky Way galaxy. These findings may be treated as valuable complements to the previous findings. We conclude from our findings that there is a unique possibility of the presence of dark-matter-admixed pulsars in all the regions of the galaxies.
Acceptability conditions and relativistic barotropic equations of state
European Physical Journal C, 2021
We sketch an algorithm to generate exact anisotropic solutions starting from a barotropic EoS and setting an ansatz on the metric functions. To illustrate the method, we use a generalization of the polytropic equation of state consisting of a combination of a polytrope plus a linear term. Based on this generalization, we develop two models which are not deprived of physical meaning as well as fulfilling the stringent criteria of physical acceptability conditions. We also show that some relativistic anisotropic polytropic models may have singular tangential sound velocity for polytropic indexes greater than one. This happens in anisotropic matter configurations when the polytropic equation of state is implemented together with an ansatz on the metric functions. The generalized polytropic equation of state is free from this pathology in the tangential sound velocity.
Neutron stars in a perturbativef(R) gravity model with strong magnetic fields
Journal of Cosmology and Astroparticle Physics, 2013
We investigate the effect of a strong magnetic field on the structure of neutron stars in a model with perturbative f (R) gravity. The effect of an interior strong magnetic field of about 10 17∼18 G on the equation of state is derived in the context of a quantum hadrodynamics (QHD) model. We solve the modified spherically symmetric hydrostatic equilibrium equations derived for a gravity model with f (R) = R + αR 2 . Effects of both the finite magnetic field and the modified gravity are detailed for various values of the magnetic field and the perturbation parameter α along with a discussion of their physical implications. We show that there exists a parameter space of the modified gravity and the magnetic field strength, in which even a soft equation of state can accommodate a large (> 2 M ⊙ ) maximum neutron star mass through the modified mass-radius relation.
Mapping neutron star data to the equation of state using the deep neural network
Physical Review D, 2020
The densest state of matter in the universe is uniquely realized inside central cores of the neutron star. While first-principles evaluation of the equation of state of such matter remains as one of the longstanding problems in nuclear theory, evaluation in light of neutron star phenomenology is feasible. Here we show results from a novel theoretical technique to utilize deep neural network with supervised learning. We input up-to-date observational data from neutron star X-ray radiations into the trained neural network and estimate a relation between the pressure and the mass density. Our results are consistent with extrapolation from the conventional nuclear models and the experimental bound on the tidal deformability inferred from gravitational wave observation.
Journal of High Energy Physics, 2021
We discuss deep learning inference for the neutron star equation of state (EoS) using the real observational data of the mass and the radius. We make a quantitative comparison between the conventional polynomial regression and the neural network approach for the EoS parametrization. For our deep learning method to incorporate uncertainties in observation, we augment the training data with noise fluctuations corresponding to observational uncertainties. Deduced EoSs can accommodate a weak first-order phase transition, and we make a histogram for likely first-order regions. We also find that our observational data augmentation has a byproduct to tame the overfitting behavior. To check the performance improved by the data augmentation, we set up a toy model as the simplest inference problem to recover a double-peaked function and monitor the validation loss. We conclude that the data augmentation could be a useful technique to evade the overfitting without tuning the neural network arc...
Physical Review C, 2013
Using a set of model equations of state satisfying the latest constraints from both terrestrial nuclear experiments and astrophysical observations as well as state-of-the-art nuclear many-body calculations of the pure neutron matter equation of state, the tidal polarizability of canonical neutron stars in coalescing binaries is found to be a very sensitive probe of the high-density behavior of nuclear symmetry energy which is among the most uncertain properties of dense neutron-rich nucleonic matter. Moreover, it changes less than ±10% by varying various properties of symmetric nuclear matter and symmetry energy around the saturation density within their respective ranges of remaining uncertainty.