A review is given of various theoretical approaches for the equation of state (EoS) of dense matter, relevant for the description of core-collapse supernovae, compact stars, and compact star mergers. The emphasis is put on models that are applicable to all of these scenarios. 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. As the development of an EoS requires joint efforts from many directions, different theoretical approaches are considered and relevant experimental and observational constraints which provide insights for future research are discussed. Finally, results from applications of the discussed EoS models are summarized.

We investigate nuclear matter at a finite temperature and density, including the formation of light clusters up to the $\alpha${} particle (1$\leqslant${}A$\leqslant${}4). The novel feature of this work is to include the formation of clusters as well as their dissolution due to medium effects in a systematic way using two many-body theories: a microscopic quantum statistical (QS) approach and a generalized relativistic mean-field (RMF) model. Nucleons and clusters are modified by medium effects. While the nucleon quasiparticle properties are determined within the RMF model from the scalar and vector self-energies, the cluster binding energies are reduced because of Pauli blocking shifts calculated in the QS approach. Both approaches reproduce the limiting cases of nuclear statistical equilibrium (NSE) at low densities and cluster-free nuclear matter at high densities. The treatment of the cluster dissociation is based on the Mott effect due to Pauli blocking, implemented in slightly different ways in the QS and the generalized RMF approaches. This leads to somewhat different results in the intermediate density range of about 10-3 to 10-1$\hskip0.3em${}fm-3, which gives an estimate of the present accuracy of the theoretical predictions. We compare the numerical results of these models for cluster abundances and thermodynamics in the region of medium excitation energies with temperatures T$\leqslant${}20\,MeV and baryon number densities from zero to a few times saturation density. The effects of cluster formation on the liquid-gas phase transition and on the density dependence of the symmetry energy are studied. It is demonstrated that the parabolic approximation for the asymmetry dependence of the nuclear equation of state breaks down at low temperatures and at subsaturation densities because of cluster formation. Comparison is made with other theoretical approaches, in particular, those that are commonly used in astrophysical calculations. The results are relevant for heavy-ion collisions and astrophysical applications.

A new scheme for testing nuclear matter equations of state (EoSs) at high densities using constraints from neutron star (NS) phenomenology and a flow data analysis of heavy-ion collisions is suggested. An acceptable EoS shall not allow the direct Urca process to occur in NSs with masses below $1.5M_{\odot }$, and also shall not contradict flow and kaon production data of heavy-ion collisions. Compact star constraints include the mass measurements of $2.1\pm 0.2M_{\odot }$ ($1\sigma$ level) for PSR J0751+1807 and of $2.0\pm 0.1M_{\odot }$ from the innermost stable circular orbit for 4U 1636–536, the baryon mass—gravitational mass relationships from Pulsar B in J0737–3039 and the mass-radius relationships from quasiperiodic brightness oscillations in 4U 0614+09 and from the thermal emission of RX J1856–3754. This scheme is applied to a set of relativistic EoSs which are constrained otherwise from nuclear matter saturation properties. We demonstrate on the given examples that the test scheme due to the quality of the newly emerging astrophysical data leads to useful selection criteria for the high-density behavior of nuclear EoSs.