Pure Neutron Matter Research Articles

Neutron Star Matter as a Dilute Solution of Protons in Neutrons.

Neutron stars contain neutron-rich matter with around 5% protons at nuclear saturation density. In this Letter, we consider equilibrium between bulk phases of matter based on asymmetric nuclear matter calculations using chiral effective field theory interactions rather than, as has been done in the past, by interpolation between the properties of symmetric nuclear matter and pure neutron matter. Neutron drip (coexistence of nuclear matter with pure neutrons) is well established, but from earlier work it is unclear whether proton drip (equilibrium between two phases, both of which contain protons and neutrons) is possible. We find that proton drip is a robust prediction of any physically reasonable equation of state, but that it occurs over a limited region of densities and proton fractions. An analytical model based on expanding the energy in powers of the proton density, rather than the neutron excess, is able to account for these features of the phase diagram.

Analysis of Neutron Star f-mode Oscillations in General Relativity with Spectral Representation of Nuclear Equations of State

We study quasi-normal f-mode oscillations in neutron star (NS) interiors within a linearized general relativistic formalism. We utilize approximately 9000 nuclear equations of state (EOSs) using spectral representation techniques, incorporating constraints on nuclear saturation properties, chiral effective field theory for pure neutron matter, and perturbative quantum chromodynamics for densities pertinent to NS cores. The median values of the f-mode frequency, ν f (damping time, τ f ) for NSs with masses ranging from 1.4 to 2.0 M ⊙ lie between 1.80 and 2.20 kHz (0.13–0.22 s) for our entire EOS set. Our study reveals a weak correlation between f-mode frequencies and individual nuclear saturation properties, prompting the necessity for more intricate methodologies to unveil multiparameter relationships. We observe a robust linear relationship between the radii and f-mode frequencies for different NS masses. Leveraging this correlation alongside NICER observations of PSR J0740+6620 and PSR J0030+0451, we establish constraints that exhibit partial and minimal overlap for observational data from Riley et al. and Miller et al., respectively, with our nucleonic EOS data set. Moreover, NICER data align closely with the radius and frequency values for a few hadron–quark hybrid EOS models. This indicates the need to consider additional exotic particles such as deconfined quarks at suprasaturation densities. We conclude that future observations of the radius or f-mode frequency for more than one NS mass, particularly at the extremes of the viable NS mass scale, would either rule out nucleon-only EOSs or provide definitive evidence in its favor.

Bayesian Survey of the Dense Matter Equation of State Built upon Skyrme Effective Interactions

The nonrelativistic model of nuclear matter (NM) with zero-range Skyrme interactions is employed within a Bayesian approach in order to study the behavior of the neutron star (NS) equation of state (EOS). A minimal number of constraints from nuclear physics and ab initio calculations of pure neutron matter (PNM) are imposed together with causality and a lower limit on the maximum mass of an NS to all our models. Our key result is that accounting for correlations among the values that the energy per neutron in PNM takes at various densities and that are typically disregarded efficiently constrains the behavior of the EOS at high densities. A series of global NS properties, e.g., maximum mass, central density of the maximum mass configuration, minimum NS mass that allows for direct URCA, and radii of intermediate and massive NSs, appear to be correlated with the value of effective neutron mass in PNM at 0.16 fm−3. Together with similar studies in the literature our work contributes to a better understanding of the NS EOS as well as its link with the properties of dense NM.

Tabulated equations of state from models informed by chiral effective field theory

Abstract We construct four equation of state (EoS) tables, tabulated over a range of temperatures, densities, and charge fractions, relevant for neutron star applications such as simulations of neutron star mergers. The EoS are computed from a relativistic mean-field theory constrained by the pure neutron matter EoS from chiral effective field theory, inferred properties of isospin-symmetric nuclear matter, and astrophysical observations of neutron star structure. To model nuclear matter at low densities, we attach an EoS that models inhomogeneous nuclear matter at arbitrary temperatures and charge fractions. The four EoS tables we develop are available from the CompOSE EoS repository https://compose.obspm.fr/eos/297 and gitlab.com/ahaber/qmc-rmf-tables.

Frequencies of f - and p -oscillation modes in cold and hot compact stars

A large collection of equations of state (EOSs) built within the covariant density functional theory of hadronic matter and allowing for density dependent couplings is employed to study polar $f$- and $p$-oscillations of cold and hot compact stars. Correlations between oscillation frequencies of cold purely nucleonic neutron stars (NSs), their global parameters as well as properties of nuclear matter (NM) are investigated by considering a set of models from Beznogov and Raduta, [Phys. Rev. C 107, 045803 (2023)], where a number of constraints on the saturation properties of NM, pure neutron matter and the lower bound of the maximal NS mass were imposed within a Bayesian framework. The roles of finite temperature and exotic particle degrees of freedom, e.g., hyperons, $\mathrm{\ensuremath{\Delta}}$-resonances, antikaon condensates or a hadron to quark phase transition, are addressed by employing a family of models publicly available on compose and assuming idealized profiles of temperature or entropy per baryon and charge fraction. We find that finite temperature effects reduce the oscillation frequencies of nucleonic stars while the opposite effect is obtained for stars with exotic particle degrees of freedom. When the $\mathrm{\ensuremath{\Gamma}}$-law is employed to build finite temperature EOSs, errors in estimating oscillation modes frequencies are of the order of 10% to 30%, depending on the mass. Throughout this work the Cowling approximation is used.

Probing nuclear equation of state with the cdm3y version of B3y-fetal effective interaction

This paper is a study of the nuclear Equation of State (EOS) of cold nuclear matter with the B3Y-Fetal effective interaction in its CDM3Y density-dependent version within the framework of Hartree-Fock approximation. The well-known saturation properties of both symmetric and asymmetric nuclear matter are well-reproduced in this work. Using the CDM3Y-K approach, this study has evolved a new set of user interactions, some of which are CDB3Y1-, CDB3Y2-, CDB3Y3-, CDB3Y4-, CDB3Y5-, CDB3Y6-Fetal interactions with corresponding incompressibilities K0 = 188, 204, 217,228, 241 and 252 MeV respectively, in excellent agreement with those of the M3Y-Paris and M3Y-Reid effective interactions. For asymmetric nuclear matter, the new set of interactions has produced the symmetry energy Esym = 32.00 MeV with an associated slope parameter L = 55 MeV at a saturation density ρ = 0.17fm−3 and asymmetry parameter δ = 1.00 (pure neutron matter) in good agreement with the standard values obtained from coupled channel analysis of charge exchange reactions, statistical multifragmentation model and terrestrial Nuclear Physics experimental analyses. Furthermore, the new set of interactions has been found to have bright prospects in a nuclear reaction as the real folded potential computed with the CDB3Y6-Fetal interaction within the framework of double folding potential has been found to be good and similar to that of CDB3Y6-Paris whose optical potential has a repulsive direct component.

Systematic analysis of the impacts of symmetry energy parameters on neutron star properties

The impacts of various symmetry energy parameters on the properties of neutron stars (NSs) have been recently investigated, and the outcomes are at variance, as summarized in Table III of Phys. Rev. D 106, 063005 (2022). We have systematically analyzed the correlations of slope and curvature parameters of symmetry energy at the saturation density ($\rho_0=0.16 \text{fm}^{-3}$) with the tidal deformability and stellar radius of non-spinning neutron stars in the mass range of $1.2 - 1.6 M_\odot$ using a large set of minimally constrained equations of state (EoSs). The EoSs at low densities correspond to the nucleonic matter and are constrained by empirical ranges of a few low-order nuclear matter parameters from the finite nuclei data and the pure neutron matter EoS from chiral effective field theory. The EoSs at high densities ($\rho > 1.5 - 2\rho_0$) are obtained by a parametric form for the speed of sound that satisfies the causality condition. Several factors affecting the correlations between the NS properties and the individual symmetry energy parameters usually encountered in the literature are considered. These correlations are quite sensitive to the choice of the distributions of symmetry energy parameters and their interdependence. But, variations of NS properties with the pressure of $\beta -$ equilibrated matter at twice the saturation density remain quite robust which maybe due to the fact that the pressure depends on the combination of multiple nuclear matter parameters that describe the symmetric nuclear matter as well as the density dependence of the symmetry energy. Our results are practically insensitive to the behavior of EoS at high densities.

Spanning the full range of neutron star properties within a microscopic description

The high-density behavior of nuclear matter is analyzed within a relativistic mean-field description with non-linear meson interactions. To assess the model parameters and their output, a Bayesian inference technique is used. The Bayesian setup is limited only by a few nuclear saturation properties, the neutron star maximum mass larger than 2 M$_\odot$, and the low-density pure neutron matter equation of state (EOS) produced by an accurate N$^3$LO calculation in chiral effective field theory. Depending on the strength of the non-linear scalar vector field contribution, we have found three distinct classes of EOSs, each one correlated to different star properties distributions. If the non-linear vector field contribution is absent, the gravitational maximum mass and the sound velocity at high densities are the greatest. However, it also gives the smallest speed of sound at densities below three times saturation density. On the other hand, models with the strongest non-linear vector field contribution predict the largest radii and tidal deformabilities for 1.4 M$_\odot$ stars, together with the smallest mass for the onset of the nucleonic direct Urca processes and the smallest central baryonic densities for the maximum mass configuration. These models have the largest speed of sound below three times saturation density, but the smallest at high densities, in particular, above four times saturation density the speed of sound decreases approaching approximately $\sqrt{0.4}c$ at the center of the maximum mass star. On the contrary, a weak non-linear vector contribution gives a monotonically increasing speed of sound. A 2.75 M$_\odot$ NS maximum mass was obtained in the tail of the posterior with a weak non-linear vector field interaction. We found that pQCD favors models with a large non-linear vector field contribution or hyperons.

Impact of symmetry energy on sound speed and spinodal decomposition in dense neutron-rich matter

Using a meta model for nuclear Equation of State (EOS) with its parameters constrained by astrophysical observations and terrestrial nuclear experiments, we examine effects of nuclear EOS especially its symmetry energy \esym term on the speed of sound squared $C^2_s(\rho)$ and the critical density $\rho_t$ where $C^2_s(\rho_t)$ vanishes (indicating the onset of spinodal decomposition) in both dense neutron-rich nucleonic matter relevant for relativistic heavy-ion collisions and the cold $n+p+e+\mu$ matter in neutron stars at $\beta$-equilibrium. Unlike in nucleonic matter with fixed values of the isospin asymmetry $\delta$, in neutron stars with a density dependent isospin profile $\delta(\rho)$ determined consistently by the $\beta$ equilibrium and charge neutrality conditions, the $C^2_s(\rho)$ almost always show a peak and then vanishes at $\rho_t$. The latter strongly depends on the high-density behavior of \esym if the skewness parameter $J_0$ characterizing the stiffness of high-density symmetric nuclear matter (SNM) EOS is not too far above its currently known most probable value of about $-190$ MeV inferred from recent Bayesian analyses of neutron star observables. Moreover, in the case of having a super-soft \esym that is decreasing with increasing density above about twice the saturation density of nuclear matter, the $\rho_t$ is significantly lower than the density where the \esym vanishes (indicating the onset of isospin-separation instability and pure neutron matter formation) in neutron star cores.

Perturbed nuclear matter studied within density functional theory with a finite number of particles

Nuclear matter is studied within the Density Functional Theory (DFT) framework. Our method employs a finite number of nucleons in a box subject to periodic boundary conditions, in order to simulate infinite matter and study its response to an external static potential. We detail both the theoretical formalism and its computational implementation for pure neutron matter and symmetric nuclear matter with Skyrme-like Energy Density Functionals (EDFs). The implementation of spin-orbit, in particular, is carefully discussed. Our method is applied to the problem of the static response of nuclear matter and the impact of the perturbation on the energies, densities and level structure of the system is investigated. Our work is a crucial step in our program of ab initio-based nuclear EDFs [Phys. Rev. C 104, 024315 (2021)] as it paves the way towards the goal of constraining the EDF surface terms on ab initio calculations.

Spontaneous scalarization in proto-neutron stars

Proto-neutron stars are born when a highly evolved and massive star collapses under gravity. In this paper, we investigate the spontaneous scalarization in proto-neutron stars. Based on the scalar tensor theory of gravity as well as the physical conditions in proto-neutron star, we examine the structure of proto-neutron star. To describe the fluid in proto-neutron star, we utilize SU(2) chiral sigma model and the finite temperature extension of the Brueckner–Bethe–Goldstone quantum many-body theory in the Brueckner–Hartree–Fock approximation. Here, we apply the equation of state of proto-neutron stars considering different cases i.e. hot pure neutron matter and hot beta -stable neutron star matter without neutrino trapping as well as with neutrino trapping. The effects of temperature and entropy of proto-neutron stars on the star structure are also studied. Our results confirm that the spontaneous scalarization is affected by different physical conditions in proto-neutron stars.

Bayesian inference of the dense matter equation of state built upon covariant density functionals

A modified version of the density dependent covariant density functional model proposed in [T. Malik, M. Ferreira, B. K. Agrawal and C. Provid\^encia, ApJ 930, 17 (2022)] is employed in a Bayesian analysis to determine the equation of state (EOS) of dense matter with nucleonic degrees of freedom. Various constraints from nuclear physics and microscopic calculations of pure neutron matter (PNM) along with a lower bound on the maximum mass of neutron stars (NSs) are imposed on the EOS models to investigate the effectiveness of progressive incorporation of the constraints, their compatibility as well as correlations among parameters of nuclear matter and properties of NSs. Our results include the different roles played by pressure and energy per particle of PNM in constraining the isovector behavior of nuclear matter; tension with the values of Dirac effective mass extracted from spin-orbit splitting; correlations between the radius of the canonical mass NS and second and third order coefficients in the Taylor expansion of energy per particle as a function of density; correlation between the central pressure of the maximum mass configuration and Dirac effective mass of the nucleon at saturation. For some of our models the tail of the NS maximum mass reaches $2.7~\mathrm{M}_{\odot}$, which means that the secondary object in GW190814 could have been a NS.

Bayesian refinement of covariant energy density functionals

The last five years have seen remarkable progress in our quest to determine the equation of state of neutron rich matter. Recent advances across the theoretical, experimental, and observational landscape have been incorporated in a Bayesian framework to refine existing covariant energy density functionals previously calibrated by the properties of finite nuclei. In particular, constraints on the maximum neutron star mass from pulsar timing, on stellar radii from the NICER mission, on tidal deformabilities from the LIGO-Virgo collaboration, and on the dynamics of pure neutron matter as predicted from chiral effective field theories have resulted in significant refinements to the models, particularly to those predicting a stiff symmetry energy. Still, even after these improvements, we find it challenging to reproduce simultaneously the neutron skin thickness of both $^{208}\mathrm{Pb}$ and $^{48}\mathrm{Ca}$ recently reported by the PREX/CREX collaboration.

Effect of the σ -cut potential on the properties of neutron stars with or without a hyperonic core

Motivated by the recent observation of high-mass pulsars ($M \simeq 2 M_{\odot}$), we employ the $\sigma$-cut potential on the equation of state (EOS) of high-density matter and the properties of neutron stars within the relativistic mean-field (RMF) model using TM1$^{*}$ parameter set. The $\sigma$-cut potential is known to reduce the contributions of the $\sigma$ field, resulting in a stiffer EOS at high densities and hence leading to larger neutron star masses without affecting the properties of nuclear matter at normal saturation density. We also analyzed the effect of the same on pure neutron matter and also on the neutron star matter with and without hyperonic core and compared it with the available theoretical, experimental, and observational data. The corresponding tidal deformability ($\Lambda_{1.4}$) is also calculated. With the choice of meson-hyperon coupling fixed to hypernuclear potentials, we obtain $\approx 10~\%$ increase in mass by employing the $\sigma$-cut potential for $f_{s} = 0.6$. Our results are in good agreement with various experimental constraints and observational data, particularly with the GW170817 data.

Relativistic mean-field theories for neutron-star physics based on chiral effective field theory

We describe and implement a procedure for determining the couplings of a Relativistic Mean-Field Theory (RMFT) that is optimized for application to neutron star phenomenology. In the standard RMFT approach, the couplings are constrained by comparing the theory's predictions for symmetric matter at saturation density with measured nuclear properties. The theory is then applied to neutron stars which consist of neutron-rich matter at densities ranging up to several times saturation density, which allows for additional astrophysical constraints. In our approach, rather than using the RMFT to extrapolate from symmetric to neutron-rich matter and from finite-sized nuclei to uniform matter, we fit the RMFT to properties of uniform pure neutron matter obtained from chiral effective field theory. Chiral effective field theory incorporates the experimental data for nuclei in the framework of a controlled expansion for nuclear forces valid at nuclear densities and enables us to account for theoretical uncertainties when fitting the RMFT. We construct four simple RMFTs that span the uncertainties provided by chiral effective field theory for neutron matter, and are consistent with current astrophysical constraints on the equation of state. Our RMFTs can be used to model the properties of neutron-rich matter across the vast range of densities and temperatures encountered in neutron stars and their mergers.

Inferring the nuclear symmetry energy at suprasaturation density from neutrino cooling

An ambitious goal of the astrophysical community is not only to constrain the equation of state (EOS) of neutron star (NS) matter by confronting it with astrophysics observations, but ultimately also to infer the NS composition. Nevertheless, the composition of the NS core is likely to remain uncertain unless we have an accurate determination of the nuclear symmetry energy at supra saturation density ($\rho>\rho_0$). We investigate how the nucleonic direct Urca (dUrca) processes can be used as an effective probe to constraint the high density nuclear symmetry energy. A large number of minimally constrained EOSs has been constructed by applying a Bayesian approach to study the correlations of the symmetry energy at different densities with a few selected properties of a NS. The nuclear symmetry energy above the baryon density 0.5 fm$^{-3}$ ($\sim 3 \rho_0$) is found to be strongly correlated with NS mass at which the onset of nucleonic dUrca neutrino cooling takes place in the core. This allows us to constrain the high density behavior of nuclear symmetry energy within narrow bounds. {The pure neutron matter pressure constraint from chiral effective field theory rules out the onset of nucleonic dUrca in stars with a mass $\lesssim$ 1.4 $M_\odot$.} The onset of dUrca inside 1.6 M$_\odot$ to 1.8 M$_\odot$ NS implies a slope of the symmetry energy $L$ at $\sim 2.5~\rho_0$, respectively, between 54 and 48 MeV.

Hot dense nuclear matter with the Thomas-Fermi approximation

We analyze the thermal effects on the EOS of dense nuclear matter, using a statistical approach where a semi-classical mean-field (MF) model based on the Thomas-Fermi (TF) approximation is introduced. Within this density-functional-based theory, the equation of state (EOS) of hot dense nuclear matter for different compositions of baryonic matter such as symmetric nuclear matter (SNM), pure neutron matter (PNM), and neutrino-free (neutrino-trapped) matter in β-equilibrium is investigated, while the contribution of photons is taken into account for only the β-equilibrated cases. The Landau effective mass, by which the thermal effects on dense matter can be judged, is also extracted from the cold single-particle (SP) potential of neutrons in phase space for both SNM and PNM. We show that the characteristic behavior of the nuclear symmetry free energy (SFE), especially at supra-saturation densities, has a clear relevance to the general influence of the temperature on the EOS. Based on the Γ-law approximation, a simple parametrization for the thermal properties of the EOS is provided by the thermal index Γth, while the presence of trapped neutrinos in β-equilibrated matter gives rise to significantly more consistency with the original astrophysical range 1.5≤Γth≤2. Our EOSs are then employed in an isentropic scheme to investigate the properties of a hot compact object under the assumption of the conserved baryonic mass, while complying with the current observational constraints on the structure of neutron stars (NSs). Within the present model, we cannot predict the possibility that the lighter component of GW190814 [1] is a massive cold neutron star (NS).

From neutron skins and neutron matter to the neutron star crust

We present the first Bayesian inference of neutron star crust properties to incorporate neutron skin data, including the recent PREX measurement of the neutron skin of 208Pb, combined with recent chiral effective field theory predictions of pure neutron matter with statistical errors. Using a compressible liquid drop model with an extended Skyrme energy-density functional, we obtain the most stringent constraints to date on the transition pressure Pcc=0.33−0.07+0.07 MeV fm−3 and chemical potential μcc=12.6−1.9+1,8 MeV (which control the mass, moment of inertia and thickness of a neutron star crust), the proton fractions that bracket the pasta phases yp=0.115−0.017+0.016 and ycc=0.041−0.006+0.007, as well as the relative mass and moment of inertia ΔMp/ΔMc≈ΔIp/ΔIc=0.54−0.09+0.05 and thickness ΔRp/ΔRc=0.129−0.030+0.019 of the layers of non-spherical nuclei (nuclear pasta) in the crust.

A new approach for the equation of state of pure neutron matter using the quantum second virial coefficient with the Reid-93 soft-core potential

In this work, a new approach is presented for determining the equation of state for pure neutron matter. This is valid in the temperature- and density-regime where the system behaves like a nonideal gas. Further, the calculations involved are confined to the nonrelativistic and low-energy ([Formula: see text] 150 MeV) regime. The approach is based on evaluating the quantum second virial coefficient [Formula: see text] of the system, where the input potential is the Reid-93 soft-core potential for [Formula: see text], together with the one-pion-exchange potential (OPEP) for higher [Formula: see text]. The many-body phase shifts are determined within the framework of a generalized scattering theory, taken here as the Galitskii–Migdal–Feynman formalism. The integral equations involved are solved using a highly-accurate matrix-inversion method. These medium phase shifts are then inserted in the Beth–Uhlenbeck formula to determine [Formula: see text]. Once this coefficient has been calculated, other thermophysical properties of the system can be readily computed according to standard expressions. Specifically, these properties include the equation of state (pressure–temperature–density relations), Helmholtz (free) energy, entropy, mean internal energy, specific heat capacity and chemical potential. Our results are compared, whenever possible, to those of previous calculations. The agreement is, on the whole, fair considering that the available calculations vary considerably among themselves.

Nearly model-independent constraints on dense matter equation of state in a Bayesian approach

We apply a Bayesian approach to construct a large number of minimally constrained equations of state (EOSs) and study their correlations with a few selected properties of a neutron star (NS). Our set of minimal constraints includes a few basic properties of saturated nuclear matter and low-density pure neutron matter EOS which is obtained from a precise next-to-next-to-next-to-leading-order (${\mathrm{N}}^{3}\mathrm{LO}$) calculation in chiral effective field theory. The tidal deformability and radius of a NS with mass $1--2\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ are found to be strongly correlated with the pressure of $\ensuremath{\beta}$-equilibrated matter at densities higher than the saturation density (${\ensuremath{\rho}}_{0}=0.16\text{ }\text{ }{\mathrm{fm}}^{\ensuremath{-}3}$) in a nearly model-independent manner. These correlations are employed to parametrize the pressure for $\ensuremath{\beta}$-equilibrated matter, around $2{\ensuremath{\rho}}_{0}$, as a function of neutron star mass and the corresponding tidal deformability. The maximum mass of the neutron star is also found to be strongly correlated with the pressure of $\ensuremath{\beta}$-equilibrated matter at densities $\ensuremath{\sim}4.5{\ensuremath{\rho}}_{0}$.