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Hot Dense Matter in The Quark-Meson-Coupling Model (QMC): Equation of State and Composition of Proto-Neutron Stars

We report the first results of the extension of the QMC model for asymmetric dense matter at finite temperature. The effects of temperature on particle composition (including the full baryon octet content) of the core of (proto-)neutron stars, as well as on the equation of state, are studied. We consider both dense matter in chemical equilibrium and matter in which neutrinos are trapped. In order to simulate stellar temperature profiles that increase with density and stellar radius, the entropy per baryon is fixed. Under these conditions, the model predicts that proto-neutron stars are already born with hyperons present at about the threshold density for their appearance in cold neutron stars, reaching about 20% of the baryon content in the center of the most massive star produced.


I. INTRODUCTION
Properties of young proto-neutron stars (PNS's) born in core-collapse supernova have been one of the main topics of interest in observation and theoretical modeling for a long time [1][2][3]. More recently, the topic has resurfaced in the context of the possible emission of detectable gravitational waves (e.g. [4]). It has been generally accepted that, in addition to the nucleons, heavy baryons exist in the cores of proto-neutron stars subject to weak interactions. Dexheimer et al. [5] [7] reported the appearance the hyperons in supernovae appearing ∼0.5-0.7 s after the bounce to trigger a recollapse into a black hole. And, again in the context of gravitational waves, Sekiguchi et al. [8] and Radice et al. [9] have shown how neutron star mergers can be influenced by the appearance of hyperons.
Several variants of the QMC model, differing somewhat from the formulation adopted in this work, have also been applied to model neutron stars and dense matter. They are listed and briefly discussed in Refs. [10] and [11]. Pertinent to this work, Panda et al. [12] studied neutrino-free stellar matter and matter with trapped neutrinos at fixed temperatures and fixed entropies per baryon and compared their results to the outcome of a non-linear Walecka model. They calculated the hyperon population in the core of a neutron star at T=0 and 10 MeV and obtained results close to those reported here. However, they did not study stellar particle population at higher temperatures. Their model predicted an increase in pressure with density in matter with trapped neutrinos (as compared to matter without neutrinos) and a shift of the threshold for appearance of strangeness to lower density (at T=10 MeV when compared to T=0). It was also demonstrated that, in neutrino-free chemically equilibrated matter, the EoS softens due to the onset of hyperons but stiffens again when a higher temperature is accounted for.
Models assuming deconfined quarks in addition to hadrons have been extensively reported in the literature and the hadron-quark phase transition and its consequence for the protoneutron stars have been studied (e.g. [13][14][15][16][17][18][19][20]). Very recently Roark et al. [21] explored the role of hyperons and quarks in proton-neutron stars using the CMF model at finite temperatures and fixed entropy per baryon, with and without neutrino trapping. Hyperons and quarks were found in the cores of large-mass stars and their interplay and the possibility of mixtures of phases was taken into account and analyzed. Despite the great variety of models and approaches, no convergence to a general consensus on the EoS and structure of high density matter in the core of neutron stars has been achieved as yet [22], although the presence of hyperons at finite temperatures has been repeatedly predicted in different formalisms.
In this paper we study hot and dense hadronic matter in the framework of the latest version of the QMC model [10], extended to finite temperature with the aim to follow detailed evolution of baryon and lepton populations as a function of temperature and entropy per baryon and its consequences for the EoS of (proto)neutron stars. The QMC model of Guichon and collaborators [23][24][25][26] was created in order to explore the connection between nuclear binding and the modification of the structure of a particle embedded in a nuclear medium. It was shown that, when the quarks in one nucleon interact self-consistently with the quarks in surrounding nucleons by exchanging a σ meson, the effective mass of the bound nucleon is no longer linear in the scalar mean field (σ): where the coefficient d is known as the "scalar polarizability". The appearance of this term, a natural consequence of the quark structure of the nucleon, is sufficient to lead to nuclear saturation. The QMC model has been applied successfully to nuclear matter at zero temperature, predicting the appearance of Λ, Ξ − , and Ξ 0 hyperons in the interior of cold neutron stars [27]. It has also led to impressive results when applied to finite nuclei [22,28,29] (for a recent review, see Ref. [10]). The full derivation of the finite temperature formalism for the QMC model will appear in a separate publication. Nevertheless, the main expressions relevant to this paper can be found in the supplementary material in QMC-finite-T.pdf at https://www2.physics.ox.ac.uk/contacts/people/stonej.

A. Particle composition
Predictions of the QMC model for the particle population distribution of protons, neutrons, Λ, Ξ − , and Ξ 0 hyperons, electrons, and ν e are shown in Figs. 1 -4 at selected temperatures of 10, 20, 40, 70 MeV, respectively. Only populations higher than 1% of the total are shown. The range of temperatures was selected to cover both the proto-neutron stars born in core-collapse supernovae, which are likely to reach temperature of several tens of MeV [3,30,31] and remnants of neutron star mergers, which could reach even higher temperatures [32]. Charge neutrality was always imposed and two extreme regimes were considered, one with trapped neutrinos (imposed through a large fixed lepton fraction) and the other in which neutrinos were allowed to escape (chemical equilibrium), throughout the whole temperature range. In reality, models suggest that matter is opaque to neutrinos in core-collapse supernovae and, only after some time (∼1 min), does it start to cool via URCA type processes, when neutrinos diffuse to the surface and the stars become neutrino transparent. This time evolution has a significant effect on the distribution of baryonic and leptonic constituents in the star interior, in dependence on both density and temperature, but is beyond the scope of the current work. in addition, predicts the baryon density at which the deleptonization happens to be around 1 fm −3 at T=10 MeV.
The calculations of particle populations performed at fixed temperature, however, may not be telling the full story because they assume, somewhat schematically, that the whole neutron star core is at the same temperature. Calculations of the particle population at fixed entropy per baryon or entropy density per baryon density S/A are more realistic, as they yield a temperature variation with baryon number density. Figs. 5 and 6 show the entropy per baryon profile in the fixed temperature case and the temperature profile in the fixed entropy per baryon case. We present results for the latter case in Figs. 7 -9, again, up to temperatures relevant for neutron-star mergers. It can be seen that, even in the fixed entropy per baryon scenario, the hyperon population is always significant.
In particular, the temperature range about 10 -40 MeV expected for the cores of protoneutron stars from core-collapse supernovae corresponds to the range of S/A = 2-3 k B , being almost constant within 0.4 ≤ n B ≤ 1.0 fm −3 in both the neutrino free and trapped scenario. Most of the baryon octet is predicted to be present under these conditions and further cooling of stars should not appreciably change the already developed particle makeup of the core [33]. Even the higher temperature regime, expected to be reached during mergers and by the merger remnants, is predicted not to be significantly affected by the presence of neutrinos. Fig. 5 shows that S/A becomes more density dependent and decreases more rapidly with increasing density in the chemically equilibrated case. Fig. 6 shows that temperature becomes more density dependent and grows more rapidly with increasing density in the chemically equilibrated case.
We note that the hyperon composition in all of the scenarios discussed above does not predict Σ hyperons to appear at baryon densities below 1. Moreover, the QMC prediction of the absence Σ hyperons is supported by the fact that no bound Σ-hypernucleus at medium or high mass has been found as yet, despite dedicated search [38,39]. The appearance of the Ξ hyperons at rather low densities indicates the existence of a bound Ξ-hypernucleus. This prediction is in line with recent results of Nakazawa et al., who reported observation of a bound state of the Ξ− 14 N system [40]. As shown in Figs. 1 -4, the absence of Σ hyperons in cold dense nuclear persists in hot matter.

B. The Equation of State
The composition of neutron stars significantly affects their EoS. The QMC EoS is presented in Fig. 10 Fig. 11, for which similar conclusions can be drawn.

C. Compact objects
As pointed by Lattimer and Swesty [41], in stars with finite temperature/entropy a crust of high entropy should be used, since the shock wave created during supernova explosions leaves the outer regions with a much higher entropy than the rest of the star. The crust remains warmer for a longer time serving as an insulating blanket for the core, delaying the star from coming to a complete thermal equilibrium with the interstellar medium. In this case, the crust can be stiff enough to generate massive stars for small central densities, resulting in large radii. In both scenarios, the maximum mass of the object is above the currently observed gravitational mass limit for neutron stars (considered cold) [42]. Rezzolla et al [43], combining the GW observations of merging systems of binary neutron stars and quasi-universal relations, set constraints on the maximum mass that can be attained by nonrotating stellar models of neutron stars, implying the the maximum mass of a non-rotating neutron star is between 2.01 +0.04 −0.04 and 2.16 +0. 17 −0.15 .The maximum mass of a cold neutron star, predicted in the QMC model, 2.00 M , lies within these limits. However, in these estimates, no effects of finite temperature were taken into account. The QMC model predicts higher stellar maximum mass for the cases with larger entropies per baryon and trapped neutrinos (when compared with the cold beta-equilibrated case). This is consistent with the EoS behavior shown in Fig. 11.
The results of the QMC model shown in this work, consistent with results of other models [6,12], predict a substantial presence of hyperons in massive compact objects with temperatures equal to and above 10 MeV. Previous calculation at T=0 MeV also yielded a hyperon presence in cold massive neutron star cores [27]. As shown by Roark et al. [21], even calculations including a phase transition to deconfined quark matter predict that hyperons, specially the Λ, can be present in different stages of neutron star evolution.
Finally, note that our results re-open the question of the existence of r-modes in rotating neutron stars [44,45]. Jones [46,47] reported that the bulk viscosity of hyperonic matter in neutron stars would produce a serious damping of the r-modes. Lidblom and Owen [48] argued that the cooling of the proto-neutron star is too rapid to influence the r-modes.
Damping arising from different phases of quark matter in strange quark stars and hybrid stars have been also studied [49]. It will be interesting to pursue the connection between r-modes and the internal composition of neutron stars in the future.

III. SUMMARY
In this work we presented for the first time results for the QMC model for matter at different stages of stellar evolution. These included trapped neutrinos and chemically-equilibrated matter and were shown for constant temperatures up to 70 MeV, using an entropy per baryon prescription that allows the temperature to increase with baryon density. In all cases, a substantial amount of hyperons were found in the core of massive neutron stars, which were predicted to be considerably larger and even more massive right after they are created in core-collapse supernova explosions. This has to do with the amount of hyperons (which usually soften the equation of state) being larger when larger temperatures / entropies per baryon are considered, but being suppressed when trapped neutrinos are included.
The absence of Σ hyperons, already reported in Ref. [10,27] for the zero-temperature case, persisted in this work, both with the inclusion of temperature and trapped neutrinos.
This novel result is in agreement with the fact that no bound Σ-hypernucleus has been found as yet [38,39]. In the future, we intend to use the QMC model to generate equation of state tables as a function of density, temperature, and lepton fraction in order to study the role played by hyperons in core-collapse supernova explosions.