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.

A statistical model for the equation of state and the composition of supernova matter is presented with focus on the liquid-gas phase transition of nuclear matter. It consists of an ensemble of nuclei and interacting nucleons in nuclear statistical equilibrium. A relativistic mean field model is applied for the nucleons. The masses of the nuclei are taken from nuclear structure calculations which are based on the same nuclear Lagrangian. For known nuclei experimental data is used directly. Excluded volume effects are implemented in a thermodynamic consistent way so that the transition to uniform nuclear matter at large densities can be described. Thus the model can be applied at all densities relevant for supernova simulations, i.e. ρ = 10 5 − 10 15 g/cm 3 , and it is possible to calculate a complete supernova equation of state table. The model allows to investigate the role of shell effects, which lead to narrow-peaked distributions around the neutron magic numbers for low temperatures. At larger temperatures the distributions become broad. The significance of the statistical treatment and the nuclear distributions for the composition is shown. We find that the contribution of light clusters is very important and is only poorly represented by α-particles alone. The results for the equation of state are systematically compared to two commonly used models for supernova matter which are based on the single nucleus approximation. Apart from the composition, in general only small differences of the different equations of state are found. The differences are most pronounced around the (low-density) liquid-gas phase transition line where the distribution of light and intermediate clusters has an important effect. Possible extensions and improvements of the model are discussed.

Many of the currently available equations of state for core-collapse supernova simulations give large neutron star radii and do not provide large enough neutron star masses, both of which are inconsistent with some recent neutron star observations. In addition, one of the critical uncertainties in the nucleon-nucleon interaction, the nuclear symmetry energy, is not fully explored by the currently available equations of state. In this article, we construct two new equations of state which match recent neutron star observations and provide more flexibility in studying the dependence on nuclear matter properties. The equations of state are also provided in tabular form, covering a wide range in density, temperature and asymmetry, suitable for astrophysical simulations. These new equations of state are implemented into our spherically symmetric core-collapse supernova model, which is based on general relativistic radiation hydrodynamics with three-flavor Boltzmann neutrino transport. The results are compared with commonly used equations of state in supernova simulations of 15 and 40 M ⊙ progenitors. We do not find any simple correlations between individual nuclear matter properties at saturation and the outcome of these simulations. However, the new equations of state lead to the most compact neutron stars among the relativistic mean-field models which we considered. The new models also obey the previously observed correlation between the time to black hole formation and the maximum mass of an s = 4 neutron star.

We explore the implications of the QCD phase transition during the postbounce evolution of corecollapse supernovae. Using the MIT bag model for the description of quark matter and assuming small bag constants, we find that the phase transition occurs during the early postbounce accretion phase. This stage of the evolution can be simulated with general relativistic three-flavor Boltzmann neutrino transport. The phase transition produces a second shock wave that triggers a delayed supernova explosion. If such a phase transition happens in a future galactic supernova, its existence and properties should become observable as a second peak in the neutrino signal that is accompanied by significant changes in the energy of the emitted neutrinos. In contrast to the first neutronization burst, this second neutrino burst is dominated by the emission of anti-neutrinos because the electrondegeneracy is lifted when the second shock passes through the previously neutronized matter. In search of the phase transition from hadronic to deconfined matter, heavy ion experiments at RHIC and at LHC at CERN explore the QCD phase diagram for large temperatures and small baryochemical potentials. For these conditions, which were also present in the early universe, lattice QCD calculations predict a crossover transition between the deconfined chirally symmetric phase and the confined phase with broken chiral symmetry. For high chemical potentials and low temperatures a first order chiral phase transition is expected and will be tested at the FAIR facility at GSI Darmstadt.Due to their large central densities, compact stars can also serve as laboratories for nuclear matter beyond saturation density and may contain quark matter [1]. The formation of quark matter in compact stars is mainly discussed in two scenarios, in protoneutron stars (PNS) after the supernova explosion [2] and in old accreting neutron stars [3,4]. For the first case, deleptonization leads to the loss of lepton pressure and therefore to an increase in the central density so that the phase transition takes place. Possible observables are the emission of gravitational waves [3,4] due to the contraction of the neutron star or delayed γ-ray bursts [5].In this article we want to follow a third and less discussed case. The phase transition from hadronic to quark matter can already occur in the early postbounce phase of a core-collapse supernova [6,7,8,9,10]. This requires a phase transition onset close to saturation density, which can be realized for high temperatures and low proton fractions. For such a scenario Ref. [8] found the formation of a second shock as a direct consequence of the phase transition. However, the lack of neutrino transport in their model allowed them to investigate the dynamics only for a few ms after bounce. Very recently, a quark matter phase transition has been considered with Boltzmann neutrino transport for a 100 M ⊙ progenitor [11]. The appearance of quark matter shortened the time until black hole formation due to the softening of the equation o...

We discuss three new equations of state (EOS) in core-collapse supernova simulations. The new EOS are based on the nuclear statistical equilibrium model of Hempel and Schaffner-Bielich (HS), which includes excluded volume effects and relativistic mean-field (RMF) interactions. We consider the RMF parameterizations TM1, TMA, and FSUgold. These EOS are implemented into our spherically symmetric core-collapse supernova model, which is based on general relativistic radiation hydrodynamics and three-flavor Boltzmann neutrino transport. The results obtained for the new EOS are compared with the widely used EOS of H. Shen et al. and Lattimer & Swesty. The systematic comparison shows that the model description of inhomogeneous nuclear matter is as important as the parameterization of the nuclear interactions for the supernova dynamics and the neutrino signal. Furthermore, several new aspects of nuclear physics are investigated: the HS EOS contains distributions of nuclei, including nuclear shell effects. The appearance of light nuclei, e.g., deuterium and tritium is also explored, which can become as abundant as alphas and free protons. In addition, we investigate the black hole formation in failed core-collapse supernovae, which is mainly determined by the high-density EOS. We find that temperature effects lead to a systematically faster collapse for the non-relativistic LS EOS in comparison to the RMF EOS. We deduce a new correlation for the time until black hole formation, which allows to determine the maximum mass of proto-neutron stars, if the neutrino signal from such a failed supernova would be measured in the future. This would give a constraint for the nuclear EOS at finite entropy, complementary to observations of cold neutron stars.