Equations of state for supernovae and compact stars

Modelling compact stars is a complex task which depends on many ingredients, among others the properties of dense matter. In this contribution models for the equation of state (EoS) of dense matter will be discussed, relevant for the description of core-collapse supernovae, compact stars and compact star mergers. Such EoS models have to cover large ranges in baryon number density, temperature and isospin asymmetry. The characteristics of matter change dramatically within these ranges, from a mixture of nucleons, nuclei, and electrons to uniform, strongly interacting matter containing nucleons, and possibly other particles such as hyperons or quarks. Some implications for compact star astrophysics will be highlighted, too.

Neutron stars are born when massive stars run out of their nuclear fuel and undergo gravitational collapse. Neutron stars belong to the most compact objects in the observable Universe. Macroscopic properties of neutron stars like their masses and radii are sensitive to the microscopic properties of the nuclear equation of state of dense matter. The equation of state is determined by the strong interaction among the constituents. The underlying theory is quantum chromodynamics that is, however, highly non-perturbative in the physics regime relevant for neutron stars. Moreover, neutron stars provide an interplay between nuclear physics and astrophysics. Astrophysical observations like the detection of 2 solar mass neutron stars have a major impact on the equation of state. Radii are, however, inherently difficult to measure due to systematic uncertainties. Other observables like the moment of inertia or the tidal deformability present promising alternatives. The double neutron star system PSR J0737-3039 constitutes an outstanding system as it provides the prospect of a moment of inertia measurement for the first time. A new era stated with the pioneering observation of gravitational waves from a binary neutron star merger. The analysis of the gravitational wave signal of GW170817 provides a range for the tidal deformability of typical neutron stars. Moreover, the current NICER mission will provide simultaneous mass-radius measurements. In this thesis, we use state-of-the-art chiral effective field theory interactions to describe the equation of state at nuclear densities. In the high-density regime beyond nuclear saturation density, we use different extrapolation approaches. First, we utilize the established ansatz of piecewise polytropic equations of state which provides a direct parametrization. However, piecewise polytropic equations of state possess unphysical behavior such as discontinuities in the speed of sound. Second, we use a physically motivated parametrization of the speed of sound inside the neutron star from which we derive the equation of state. Both methods allow us to probe the equation of state over a large range of densities. We further impose general constraints on the equation of state such as the requirement of causality at all densities and the support of at least 2 solar mass neutron stars. From the equations of state compatible with the constraints, we determine diverse neutron star observables. We begin with non-rotating neutron stars and focus on their masses and radii. We study correlations among properties of the equation of state at nuclear densities and observables of typical neutron stars. Moreover, we explore the impact of hypothetical, simultaneous measurements of masses and radii of neutron stars on the equation of state. Applying both simple compatibility cuts and the framework of Bayesian statistics, we investigate the sensitivity of the inference on the chosen parametrization of the equation of state. We extend then our considerations to slowly rotating neutron stars and study the moment of inertia. Assuming hypothetical moment of inertia measurements, we determine constraints for the radius of neutron stars and thus the equation of state. In addition, we extend our considerations of isolated neutron stars to binary neutron star systems. In particular, we treat the tidal field of the companion as a small perturbation. This allows us to determine the tidal deformability. By applying higher orders in the metric perturbation, we calculate the quadrupole moment of neutron stars. Although the structure of neutron stars is sensitive to the equation of state, relations between the moment of inertia, the tidal deformability, and the quadrupole moment are remarkably insensitive. We investigate the properties of neutron stars in binary systems and ultimately confront the results of our models with the gravitational wave constraints from a binary neutron star merger.

The properties of compact stars and their formation processes depend on many physical ingredients. The equation of state (EoS) of the involved matter, describing its thermodynamic properties is one of them. It is not an obvious task to construct such an EoS, first of all because very large ranges in baryon number density, temperature and asymmetry have to be covered. Within these ranges the characteristics of matter change dramatically, from an ideal gas of nuclei to uniform strongly interacting matter, containing in the most simple case just free nucleons and potentially many other components such as hyperons, mesons or even quarks. I will summarize existing constraints on the EoS and its composition by terrestrial experiments, astrophysical observations and theoretical considerations. Then I will discuss some recent EoS developments and improvements, with a particular emphasis on the hyperon puzzle, i. e. the fact that many EoSs predict the onset of hyperons at about twice nuclear saturation density and at the same time maximum neutron star masses well below the observed ones.

The Research Topic "Strong and Weak Interactions in Compact Stars" provides a broad overview of recent advances in the study of the physics of compact stars. The contributions span nuclear and quark matter equations of state, weak interaction rates in dense matter, nucleosynthesis, rotational and thermal effects in neutron stars, and the interpretation of observational constraints. Combined they shed light on the progress achieved and the challenges that remain in constructing a coherent, multi-scale description of compact stars.Several articles focus on the equation of state (EoS) of dense matter, which remains a central uncertainty in neutron star physics. Tong et al. provide a concise review of relativistic Brueckner-Hartree-Fock theory formulated in full Dirac space, emphasizing recent technical advances beyond common angleaveraging approximations and their implications for neutron star structure. Reinforcing this microscopic perspective, Sammaruca and Ajagbonna argue for the use of state-of-the-art ab initio nuclear and neutron matter calculations as a robust baseline for high-density extrapolations. By combining these with causality, maximum-mass constraints, and speed-of-sound-guided parametrizations, they delineate allowed regions of the EoS and present associated predictions for neutron star cooling.The connection between nuclear experiments, astrophysical observations, and dense-matter theory is explored further by Burgio et al., who investigate correlations between the density dependence of the symmetry energy and neutron skin thickness measurements in finite nuclei, in light of recent CREX and PREX results. By analyzing a broad ensemble of microscopic and phenomenological EoS models consistent with neutron star mass and tidal deformability constraints, this work highlights emerging tensions between laboratory data and current theoretical descriptions of the nuclear EoS.Strong interactions at even higher densities, where deconfined quark matter may appear, are addressed in several contributions. Alford et al. study the bulk viscosity of warm, dense, neutrino-transparent quark matter in the two-flavor color-superconducting (2SC) phase driven by weak interaction β-decays (Urca reactions). Using an extended SU(3) Nambu-Jona-Lasinio model, they demonstrate a pronounced sensitivity of bulk viscosity and damping timescales to vector interactions, with important implications for the dissipation of density oscillations in merging compact stars. In a complementary phenomenological approach, Kourmpetis et al. explore whether color-flavor locked (CFL) quark matter, modeled by the MIT bag model, can explain the observed properties of two compact stars, which have similar low masses but potentially different radii. The study explores two scenarios: absolutely stable strange quark matter and hybrid stars, determining acceptable ranges for the superconducting gap and bag parameter in each case. This work illustrates how observational constraints can discriminate between different realizations of quark matter in compact stars.Weak interactions play a crucial role in shaping the thermal and chemical evolution of compact stars and their progenitors. Kabir et al. investigate β-decay properties of medium-mass nuclei relevant for stellar environments, combining relativistic mean-field calculations of nuclear deformation with pn-QRPA evaluations of Gamow-Teller strength and stellar weak rates. The resulting rates, systematically larger than those obtained in alternative models, are of direct relevance for simulations of late-stage stellar evolution and nucleosynthesis. On a much larger astrophysical scale, Blaschke et al. address the long-standing puzzle of the near-universality of heavy-element abundances. Using a nonequilibrium freeze-out framework and a phenomenological characterization of r-process distributions, they show how weak-interaction-driven dynamics and density fluctuations can naturally account for both the typical abundance pattern and its observed variations.The macroscopic manifestations of dense-matter microphysics are further explored through studies of gravity and rotation. Cai and Li derive equation-of-state-independent constraints on supradense matter by analyzing the scaled Tolman-Oppenheimer-Volkoff equations in general relativity. This work reveals tight bounds on the pressure-energy-density ratio and establishes direct links between observable neutron star properties and the dense matter EoS, without reliance on specific nuclear models. Farrell et al. examine the effects of differential rotation and finite temperature on neutron star structure and stability, using finite-temperature relativistic Brueckner-Hartree-Fock equations of state. The results demonstrate that differential rotation has a significant impact on maximum masses and rotational instabilities, while temperature plays a comparatively minor role within the explored range. These findings are particularly relevant for interpreting post-merger remnants.Combined, the articles collected in this Research Topic illustrate the rich and multifaceted role of strong and weak interactions in compact stars, from the microphysics of nuclei and quarks to the global structure and dynamics of general-relativistic, rotating objects. They underscore the necessity of combining microscopic theory, phenomenological modeling, and observational input to make progress in this field. We hope that this collection will serve both as a highlight of current advances and as a stimulus for future work aimed at unraveling the physics of matter under extreme conditions.

In contrast to heavy ion collisions, matter in astrophysical systems such as neutron stars, compact star mergers and supernova environments can be highly isospin asymmetric. We focus on core collapse supernova matter where temperatures reach tens of MeV. Both conditions, the high temperatures and isospin asymmetry, can favour an early phase transition to quark matter already close to nuclear saturation density. We examine the QCD phase transition during the early postbounce phase of core collapse supernovae. We discuss the microphysical input, i.e. the modelling of the phase transition to strange quark matter, and the consequences on the dynamical evolution as well as the observable neutrino signal from the phase transition. The equation of state for strange quark matter is based on the MIT bag model. The phase transition is constructed applying the Gibbs criterion which results in an extended coexistence region in the phase diagram between the hadronic and the quark phases, i.e. the mixed phase. The supernovae are simulated via general relativistic radiation hydrodynamics based on three-flavour Boltzmann neutrino transport in spherical symmetry. The dynamical evolution of the phase transition to quark matter is determined by an adiabatic collapse due to the softening of the equation of state in the mixed phase. The equation of state for the pure quark phase stiffens again which causes the collapse to halt and a shock wave forms at the boundary between the mixed and the pure hadronic phases. This shock accelerates and launches an explosion, which releases a burst of neutrinos dominated by electron anti-neutrinos due to the lifted degeneracy of the shock-heated hadronic material.

Heavy baryons in compact stars

The equation of state (EoS) of hot and dense matter is a fundamental input to describe static and dynamical properties of neutron stars, core-collapse supernovae and binary compact-star mergers. We review the current status of the EoS for compact objects, that have been studied with both ab-initio many-body approaches and phenomenological models. We limit ourselves to the description of EoSs with purely nucleonic degrees of freedom, disregarding the appearance of strange baryonic matter and/or quark matter. We compare the theoretical predictions with different data coming from both nuclear physics experiments and astrophysical observations. Combining the complementary information thus obtained greatly enriches our insights into the dense nuclear matter properties. Current challenges in the description of the EoS are also discussed, mainly focusing on the model dependence of the constraints extracted from either experimental or observational data (specifically, concerning the symmetry energy), the lack of a consistent and rigorous many-body treatment at zero and finite temperature of the matter encountered in compact stars (e.g. problem of cluster formation and extension of the EoS to very high temperatures), the role of nucleonic three-body forces, and the dependence of the direct URCA processes on the EoS.

We present a comprehensive analysis of hot and dilute isospin-asymmetric nuclear matter employing the temperature-dependent effective-relativistic mean-field theory (E-RMF). The E-RMF is applied to study the effect of $\delta$ and $\omega-\rho$ meson cross-coupling on the thermal properties of asymmetric nuclear matter using two recently developed IOPB-I and G3 parameter sets. These sets are known to reproduce the nuclear matter properties in agreement with various experimental and observational constraints. We consider the nuclear matter to be homogeneous and study the equation of state (EoS) for densities, temperature and asymmetry which are relevant for astrophysical simulations such as supernovae explosion. The effect of temperature is investigated in reference to the density-dependent free symmetry energy and its higher-order derivatives using the well known parabolic approximation. The larger value of $\lambda_\omega$ cross-coupling in G3 in addition to the $\delta$ meson coupling in G3 smoothen the free symmetry energy. Thermal effects on various state variables are examined at fixed temperature and isospin asymmetry by separating their T=0 and the finite-T expressions. The thermal effects are mainly governed by effective mass with larger effective mass estimating larger thermal contribution. The effect of temperature on isothermal and isentropic incompressibility is discussed which is in harmony with various available microscopic calculations. The liquid-gas phase transition properties are examined in asymmetric matter with two conserved charges in the context of different slope parameter and comparable symmetry energy in IOPB-I and G3 set. The spinodal instability, binodal curve and critical properties are found to be influenced by the slope parameter $L_{sym}$.

Nucleon short-range correlations (SRCs) and their associated high-momentum tails (HMTs) in the single-nucleon momentum distribution $$n_{\textbf{k}} = n(k)$$ n k = n ( k ) have emerged as key manifestations of strong, short-range dynamics in nuclear many-body systems. Despite substantial recent progress, our understanding of these correlations and their implications for finite nuclei, nuclear reactions, and dense matter remains incomplete and continues to evolve. In this review, we offer a necessarily selective overview of several aspects of SRC physics that directly influence the equation of state (EOS) of dense matter, particularly in regimes of large isospin asymmetry and high baryon density. We first summarize the empirical and theoretical features of the momentum distribution $$n_{\textbf{k}} = n(k)$$ n k = n ( k ) , including its isospin dependence, microscopic origins, and representative parameterizations. Special emphasis is placed on the strong neutron–proton (np) tensor force at intermediate momenta, which drives the dominance of correlated np pairs and enhances the minority-species HMT in asymmetric nuclei and nuclear matter. We further discuss connections to nucleon effective masses, quasi-deuteron components, and orbital entanglement entropy, providing a broader microscopic foundation that links SRCs to single-particle and two-body structure. We then examine how SRC-induced HMTs modify the EOS of asymmetric nuclear matter within both non-relativistic and relativistic frameworks. The depletion of low-momentum states and the repopulation of high-momentum components alter kinetic and potential contributions to the EOS. We additionally consider generalizations to arbitrary spatial dimensions and estimates involving very high-momentum components, which help clarify the sensitivity of the EOS to the detailed structure of n ( k ). Particular attention is devoted to the softening of the kinetic symmetry energy and to deviations from the standard parabolic approximation of isospin-asymmetric nuclear-matter EOS, effects that grow increasingly important with isospin asymmetry. In the context of heavy-ion reactions, we summarize the influence of SRCs on isospin-sensitive observables including particle yields, nuclear collective flows, and neutron–proton bremsstrahlung gamma rays. These effects arise from both modified initial momentum distributions and the increased availability of high relative-momentum np pairs, which can strongly affect threshold behavior and transport dynamics. We also briefly comment on experimental probes of high-momentum nucleon components, including electron- and proton-induced knockout reactions and meson production channels in heavy-ion reactions. Finally, we discuss implications for neutron-star matter, wherein extreme densities and large isospin asymmetries amplify many SRC-induced effects known from finite nuclei. Topics include consequences for mass–radius relations, tidal deformabilities, proton fractions, Migdal–Luttinger Z -factors, cooling processes, and the core–crust transition. We also highlight potential connections between SRC-modified nucleon momentum distributions and dark-matter interactions in dense astrophysical environments.

The present article investigates the role of heavy nuclear clusters and weakly bound light nuclear clusters based on a newly developed equation of state for core collapse supernova studies. A novel approach is brought forward for the description of nuclear clusters, taking into account the quasiparticle approach and continuum correlations. It demonstrates that the commonly employed nuclear statistical equilibrium approach, based on noninteracting particles, for the description of light and heavy clusters becomes invalid for warm nuclear matter near the saturation density. This has important consequences for studies of core collapse supernovae. To this end, we implement this nuclear equation of state provided for arbitrary temperature, baryon density, and isospin asymmetry, to spherically symmetric core collapse supernova simulations to study the impact on the dynamics as well as on the neutrino emission. For the inclusion of a set of weak processes involving light clusters the rate expressions are derived, including medium modifications at the mean-field level. A substantial impact from the inclusion of a variety of weak reactions involving light clusters on the post bounce dynamics and on the neutrino emission could not be found.

Correlations are an essential feature of interacting many-body systems such as nuclear and stellar matter. They cause the formation of clusters, i.e. nuclei and scattering resonances, changing the chemical composition and affecting the thermodynamical properties of the system. The equation of state has to be known in a wide range of density, temperature and isospin asymmetry for astrophysical simulations of core-collapse supernovae and compact stars. In a generalized relativistic density functional approach, the properties of dense stellar matter can be described with the correct limits at low densities, the model-independent virial equation of state, and at high densities using a quasiparticle mean-field approach. Effects of correlations and changes of the cluster properties in dilute matter can be studied experimentally in heavy-ion collisions.

We present a Bayesian analysis to constrain the equation of state of dense nucleonic matter by exploiting the available data from symmetric nuclear matter at saturation, observations of compact X-ray sources, and the gravitational wave event GW170817. For the first time, such an analysis is performed by using a class of models, the relativistic mean field models, that allow one to consistently construct an equation of state in a wide range of densities, isospin asymmetries, and temperatures. The selected class of models contains five nuclear physics empirical parameters at saturation for which we construct the joint posterior distributions. By exploring different types of priors, we find that the equations of state with the largest evidence are the ones featuring a strong reduction of the effective mass of the nucleons in dense matter, which can be interpreted as an indication of a phase transition to a chiral symmetry restored phase. Those equations of state, in turn, predict R 1.4 ∼ 12 km. Finally, we present a preliminary investigation of the effect of including Λ hyperons, showing that they appear in stars more massive than about 1.6 M ⊙ and lead to radii larger than about R 1.4 ∼ 14 km. Within the model explored here, the formation of such particles provides poor agreement with the constraints from GW170817.

The properties of compact stars and their formation processes depend on many physical ingredients. The composition and the thermodynamics of the involved matter is one of them. We will investigate here uniform strongly interacting matter at densities and temperatures, where potentially other components than free nucleons appear such as hyperons, mesons or even quarks. In this paper we will put the emphasis on two aspects of stellar matter with non-nucleonic degrees of freedom. First, we will study the phase diagram of baryonic matter with strangeness, showing that the onset of hyperons, as that of quark matter, could be related to a very rich phase structure with a large density domain covered by phase coexistence. Second, we will investigate thermal effects on the equation of state (EoS), showing that they favor the appearance of non-nucleonic particles. We will finish by reviewing some recent results on the impact of non-nucleonic degrees freedom in compact star mergers and core-collapse events, where thermal effects cannot be neglected.

The composition and equation of state (EoS) of dense matter relevant to compact stars are quite inconclusive. However, certain observational constraints on the structural properties of compact stars help us constrain the EoS to a fair extent. Moreover, gravitational asteroseismology gives a notion of the composition and EoS of compact stars. The next generation gravitational wave (GW) detectors are likely to detect several oscillation mode frequencies of the GWs. In this work we compute the fundamental (f) and the first pressure (p1) mode frequencies (ff and fp1, respectively) with different compositions viz., hadronic, quark, and hybrid star (HS) matter. For HSs, we also study the gravity (g) mode frequency (fg). For each phase we also study the correlation between the oscillation frequencies of 1.4 M⊙ and 2.01 M⊙ compact stars with other different properties. We find that various possible composition of compact stars substantially affects the oscillation frequencies. However, the mass-scaled angular f mode frequency (ωfM) varies universally with compactness (C) for all hadronic, quark and hybrid stars. The f mode frequency (ff1.4) of the canonical 1.4 M⊙ compact star, obtained with different composition, is quite correlated with the canonical radius (R1.4) and tidal deformability (Λ1.4) while fp1.4 is well correlated with slope parameter of the symmetry energy. We also show that fg1.4 of the HSs varies almost linearly with Λ1.4. Should g modes be detected, they could not only support the existence of HSs, but fg could be useful to understand the strength of quark repulsion in HSs.

Dense nuclear matter equation of state from heavy-ion collisions