2.1. Small nuclear reaction networks

In the regime of low densities and low temperatures (T < 0.5 MeV), small nuclear reaction networks are commonly used which include about 14–50 nuclear species as explained in Thielemann et al. (Reference Thielemann, Brachwitz, Höflich, Martinez-Pinedo and Nomoto2004) (the implementation of the network into our supernova model is discussed in Fischer et al. Reference Fischer, Whitehouse, Mezzacappa, Thielemann and Liebendörfer2010). Even though they cannot accurately account for the evolution of Y _e—matter is nearly isospin symmetric with Y _e ≃ 0.5 [see the region below the horizontal dash-dotted line in Figure 1(a)]—they are sufficient for the nuclear energy generation. This domain, where time-dependent nuclear processes determine the evolution, corresponds to the outer core of the stellar progenitor, with the nuclear composition of dominantly silicon, sulphur as well as carbon and oxygen. In some cases, even parts of the hydrogen-rich helium envelope are taken into account, in particular in simulations of supernova explosions in order to be able to follow the shock evolution for tens of seconds through parts of the stellar envelope. However, during the early post-bounce evolution prior to the supernova explosion onset, the stellar envelope remains nearly unaffected by the dynamics in the supernova core [see Figure 1(b) above 10³ km].

2.2. NSE

Towards the stellar core, the temperature increases above T = 0.5 MeV where NSE is fulfilled and where the relation μ_{(A, Z)} = Zμ_p + (A − Z)μ_n between the chemical potential of nucleus μ_{(A, Z)}, with atomic mass A and charge Z, and the chemical potentials of neutron μ_n and proton μ_p holds. The NSE conditions found in the collapsing stellar core feature a broad distribution of nuclei with a pronounced peak around the iron-group, at low densities [see Figure 2(a)]. These nuclei can be classified within the NSE average, including nuclear shell effects as discussed in Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010). This method extends beyond the commonly used single-nucleus approximation, which is marked by the crosses in Figure 2. Note also that with increasing density, the nuclear distribution shifts towards heavier nuclear species, moreover, it broadens with increasing temperature as illustrated in Figure 2(b). At high temperatures, around T ≃ 5 − 10 MeV, heavy nuclei dissolve via photodisintegration and (in)homogeneous nuclear matter forms, as shown in Figure 2(c). It will be discussed further in Section 3.

Figure 2. Nuclear composition in the chart of nuclides (neutron number N vs. proton number Z) based on the modified NSE approach of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010), obtained from the central conditions of the core-collapse evolution of Hempel et al. (Reference Hempel, Fischer, Schaffner-Bielich and Liebendörfer2012).

2.3. Weak interactions with heavy nuclei

Heavy nuclei with nuclear charge Z and mass A, are subject to fast electron captures, that are described collectively via the average composition as follows:

This process deleptonises stellar matter as the final-state neutrinos escape freely. As a consequence, the entropy per particle remains low during the entire stellar core collapse (cf. Bethe et al. Reference Bethe, Brown, Applegate and Lattimer1979; van Riper & Lattimer Reference van Riper and Lattimer1981), as illustrated in Figure 1(b). Electron capture rates commonly employed in supernova studies were developed by Bruenn (Reference Bruenn1985) with a crudely simplified description of the Gamow window. Improved rates were provided by Juodagalvis et al. (Reference Juodagalvis, Langanke, Hix, Martínez-Pinedo and Sampaio2010), based on large-scale nuclear shell-model calculations including several 1 000 nuclear species. A detailed comparison of both electron-capture rates and their impact on the collapse dynamics and neutrino signal can be found in Langanke et al. (Reference Langanke2003) and Hix et al. (Reference Hix, Messer, Mezzacappa, Liebendörfer, Sampaio, Langanke, Dean and Martínez-Pinedo2003). The rates of Juodagalvis et al. (Reference Juodagalvis, Langanke, Hix, Martínez-Pinedo and Sampaio2010) are averaged over the NSE composition and provided to the community as a table.

In addition to electron captures, coherent neutrino-nucleus scattering is taken into account following Bruenn (Reference Bruenn1985):

for all flavours. This channel is essential for neutrino trapping once neutrinos are being produced with sufficiently high energies via (1). Inelastic neutrino-nucleus scattering rates were provided in Langanke et al. (Reference Langanke2008). Moreover, heavy nuclei in the collapsing stellar core can exist at excited states, due to temperatures reached on the order of 0.5 MeV up to few MeV. The subsequent nuclear de-excitation process via the emission of neutrino-antineutrino pairs,

can be understood in a similar fashion as the neutral-current process (2). The original idea, pointed out by Fuller & Meyer (Reference Fuller and Meyer1991), would potentially contribute to the losses during stellar collapse. In fact, in Fischer, Langanke, & Martínez-Pinedo (Reference Fischer, Langanke and Martínez-Pinedo2013), it was confirmed that process (3) is the leading source of heavy lepton-flavour neutrinos as well as $\bar{\nu }_{\text{e}}$ during stellar core collapse. However, the neutrino fluxes remain small, compared to those of ν_e (see Figure 3 top panel), and the neutrino energies are low, on the order of few MeV (see Figure 3 bottom panel). Hence, the overall impact of nuclear de-excitations is negligible on the collapse dynamics and the neutrino signal. Instead, the stellar core collapse is dominated by nuclear electron captures and losses associated with ν_e (see Figure 3).

Figure 3. Supernova neutrino signal, luminosities (top panel) and mean energies (bottom panel) for all flavours, sampled in the co-moving frame of reference at 1 000 km. The supernova simulations were published in Fischer (Reference Fischer2016a), launched from the 18 M_⊙ progenitor of Woosley, Heger, & Weaver (Reference Woosley, Heger and Weaver2002).

2.4. Heavy nuclear structures at high density

With increasing density, nuclei become heavier, as long as the temperatures are not too high which would enable efficient photodisintegration. This situation as well as the relevant density range is illustrated in Figure 4, for matter in β-equilibrium at two selected temperatures, based on the Thomas–Fermi approximation of Shen et al. (Reference Shen, Toki, Oyamatsu and Sumiyoshi1998). A detailed comparison between different nuclear approaches was performed in Shen, Horowitz, & Teige (Reference Shen, Horowitz and Teige2011). Comparing the Thomas–Fermi approximation of Shen et al. (Reference Shen, Toki, Oyamatsu and Sumiyoshi1998), compressible liquid drop model with Skyrme interactions by Lattimer & Swesty (Reference Lattimer and Swesty1991) and the virial EOS with nucleons and nuclei of Shen, Horowitz, & Teige (Reference Shen, Horowitz and Teige2010b) (combined with the RMF (RMF) EOS of Shen, Horowitz, & Teige Reference Shen, Horowitz and Teige2010a), qualitative agreement was found for the gross properties, e.g., pressure, entropy, nuclear abundances, average nuclear mass, and charge, comparing the three models.

Figure 4. Composition of heavy nuclear structures, average mass number as well as charge (top panel), and mass fraction, for matter in β-equilibrium at two selected temperatures, based on the Thomas–Fermi approximation of Shen et al. (Reference Shen, Toki, Oyamatsu and Sumiyoshi1998). The increasing neutron excess visible is due to the continuously decreasing Y _e with increasing density in β-equilibrium.

Note that at the density range where these nuclear structures appear (see Figure 4), all protons in the system are consumed into heavy nuclei, such that effectively only free neutrons exist besides heavy and light nuclei. The latter aspect will be further discussed below in Section 4. The situation illustrated in Figure 4 corresponds to the liquid–gas phase transition at finite temperatures and large isospin asymmetry (note the very low proton abundances of Y _p = 0.01 − 0.1 in β-equilibrium in this density domain).

It has long been realised that the formation of the heavy nuclear structures sketched in Figure 4 via spherical heavy nuclei, are due to the competition of the attractive long-range nuclear force and Coulomb repulsion (cf. Watanabe et al. Reference Watanabe, Sato, Yasuoka and Ebisuzaki2003, Reference Watanabe, Sonoda, Maruyama, Sato, Yasuoka and Ebisuzaki2009; Newton & Stone Reference Newton and Stone2009; Giménez Molinelli et al. Reference Molinelli, Nichols, López and Dorso2014; Schneider et al. Reference Schneider, Berry, Briggs, Caplan and Horowitz2014, and references therein). Due to surface effects, these structures form shapes, e.g., spaghetti, lasagna, and meat balls, denoted collectively as ‘nuclear pasta’. The conditions where nuclear pasta phases exist are listed in Table 1 at two selected values of Y _e, based on the detailed three-dimensional Skyrme–Hartree–Fock calculations of Newton & Stone (Reference Newton and Stone2009). Comparing the density ranges of Table 1 for Y _e = 0.05 and Figure 4, it becomes clear that only towards high density nuclear pasta appears, where at low densities spherical heavy nuclei exist. The reason why the heavy structures dissolve already below ρ = 10¹⁴ g cm⁻³ in Figure 4 is due to the even lower Y _e in β-equilibrium, which is assumed in Figure 4. This points to the very sensitive dependence of nuclear pasta phases on temperature and Y _e. Moreover, it has been realised that the neutrino mean free path is modified in nuclear pasta. Detailed molecular dynamics simulations of the neutrino response from coherent neutrino scattering were conduced in Horowitz, Pérez-García, & Piekarewicz (Reference Horowitz, Pérez-García and Piekarewicz2004a) and Horowitz et al. (Reference Horowitz, Pérez-García, Carriere, Berry and Piekarewicz2004b). An alternative approach has been developed in Alcain, Giménez Molinelli, & Dorso (Reference Alcain, Giménez Molinelli and Dorso2014).

Note that this phase is relevant for the post-bounce supernova evolution prior to the explosion onset, since temperatures in this density range exceed T = 5 MeV and hence pasta melts (see Figure 4). However, the situation changes during the later PNS deleptonisation, after the supernova explosion onset when the temperature decreases continuously. Even though the structure of the PNS is not affected by the presence of such heavy nuclear structures, neutrino interactions may well modify the timescale on which neutrinos diffuse out of the PNS interior. Therefore, the very first detailed supernova simulations with sophisticated neutrino transport and an effective description of coherent neutrino-pasta scattering have been presented recently in Horowitz et al. (Reference Horowitz, Berry, Caplan, Fischer, Lin, Newton, O’Connor and Roberts2016). These results show qualitatively the role of nuclear pasta, i.e. an extended deleptonisation and cooling time of the PNS, once pasta phases form.

3.1. Weak processes

Here we distinguish electronic charged-current processes,

neutral-current elastic scattering on nucleons (N),

inelastic scattering on electrons and positrons,

and pair processes,

where $\nu \in \lbrace \nu _{\text{e}},\bar{\nu }_{\text{e}},\nu _{\mu /\tau },\bar{\nu }_{\mu /\tau }\rbrace$ and N ∈ {n, p} else notified otherwise. In Buras et al. (Reference Buras, Janka, Keil, Raffelt and Rampp2003), additional inelastic scattering processes have been considered, in analogy to the process (7b),

which are relevant at high densities and temperatures where a large trapped ν _e component exists. Recently, in Fischer (Reference Fischer2016b), the inverse neutron decay has been implemented,

Reactions (4) and (10) are known as Urca processes. Together with (5), they are typically treated in supernova simulations within the zero-momentum transfer approximation of Bruenn (Reference Bruenn1985). Inelastic contributions as well as corrections from weak magnetism are taken into account effectively in present supernova studies, following Horowitz (Reference Horowitz2002), which also takes into account the strangeness contents in the baryons via a strangeness axial-vector coupling constant, g _S , which effectively reduces the axial-vector coupling constant, g _A − g _S . The currently accepted value for the nucleon strangeness contents, deduced from deep-inelastic proton-scattering experiments, relates to values of g _S ≃ 0.1 (cf. Hobbs, Alberg & Miller Reference Hobbs, Alberg and Miller2016, and references therein). Note that weak magnetism enhances the opacity for ν while it suppresses the opacity for $\bar{\nu }$ . It leads to the non-negligible enhancement of spectral differences between ν and $\bar{\nu }$ , in particular for the heavy lepton flavour neutrinos where it is the leading cause, besides neutrino electron/positron scattering (6). The pair processes (7a) and (7b) do not distinguish between ν and $\bar{\nu }$ , i.e., both are produced with identical spectra.

For the processes (4) and (10), it is important to treat these weak interactions consistently with the underlying nuclear EOS, which was pointed out by Martínez-Pinedo et al. (Reference Martínez-Pinedo, Fischer, Lohs and Huther2012) and Roberts, Reddy, & Shen (Reference Roberts, Reddy and Shen2012a) based on the mean-field description of Reddy, Prakash, & Lattimer (Reference Reddy, Prakash and Lattimer1998). The associated medium modification, △U = U _n − U _p, defines the difference between neutron and proton single particle potentials (i.e. vector self-energies within the RMF framework as was discussed in, e.g., Hempel Reference Hempel2015). They depend on the nuclear symmetry energy △U∝E _sym(T, ρ) which has a strong density dependence. A detailed comparison between neutron matter and symmetric matter EOS can be found in Typel et al. (Reference Typel, Wolter, Röpke and Blaschke2014). Moreover, it was confirmed in detailed supernova simulations that E _sym determines the spectral difference between ν_e and $\bar{\nu }_{\text{e}}$ ; however, with relevance only during the PNS deleptonisation after the supernova explosion onset has been launched. This is related to the energy scales involved. Note the Q-values for processes (4): Q = ±Q ₀ ± △U, for ν_e (+) and $\bar{\nu }_{\text{e}}(-)$ . Q ₀ denotes the vacuum Q-value of the processes, Q ₀ = m _n − m _p = 1.2935 MeV, being the neutron-to-proton rest mass difference. At low densities, the energetics of the processes (4) is determined by Q ₀, since △U ≪ Q ₀. With increasing density, the medium modifications start to dominate when △U ≳ Q ₀.

In order to determine (U _n, U _p), it is essential for supernova simulations to employ model EOS with a ‘good’ low-density behaviour. Here, the ab-initio approach is chiral EFT of dilute neutron matter. Figure 5 illustrates the chiral EFT results from Krüger et al. (Reference Krüger, Tews, Hebeler and Schwenk2013) together with a selection of RMF model EOS (DD2–IUFSU) and the non-relativistic EOS (LS180 and LS220), which were and still are commonly used in supernova simulations. Details about these EOS and a table that lists a selection of nuclear matter properties can be found in Fischer et al. (Reference Fischer, Hempel, Sagert, Suwa and Schaffner-Bielich2014). Important here is the maximum neutron star mass constraint of 2 M_⊙ (not fulfilled by FSUgold, IUFSU, and LS180) and the constraint of the nuclear symmetry energy and its slope parameter at nuclear saturation density (fulfilled by DD2 and SFHo, for details cf. Lattimer & Lim Reference Lattimer and Lim2013). Moreover, from Figure 5, it becomes evident that the RMF model with density-dependent couplings DD2 of Typel et al. (Reference Typel, Röpke, Klähn, Blaschke and Wolter2010) is in quantitative agreement with chiral EFT. The other two EOS in good agreement with chiral EFT are the EOS of Steiner et al. (Reference Steiner, Hempel and Fischer2013), SFHo and SFHx, which were developed in accordance with neutron star radii deduced from the analysis of low-mass X-ray binaries by Steiner, Lattimer, & Brown (Reference Steiner, Lattimer and Brown2010). All other EOS, including the quark matter EOS of Fischer et al. (Reference Fischer, Hempel, Sagert, Suwa and Schaffner-Bielich2014) based on the thermodynamic bag model (QB139α _S 0.7—we will come back to quark matter EOS in more details in Section 5), violate the chiral EFT constraint, besides the aforementioned conflicts with the other constraints.

Figure 5. Neutron matter energy per particle for a selection of supernova model EOS, in comparison to the chiral EFT constraint of Krüger et al. (Reference Krüger, Tews, Hebeler and Schwenk2013). See text for details. (Figure adopted from Fischer et al. Reference Fischer, Hempel, Sagert, Suwa and Schaffner-Bielich2014).

In addition to the mean-field effects, which modify the charged-current processes, nuclear many-body correlations suppress the charged-current absorption rates and neutral-current neutrino scattering processes (5) with increasing density, for which the expressions of Burrows & Sawyer (Reference Burrows and Sawyer1998) and Burrows & Sawyer (Reference Burrows and Sawyer1999) are commonly employed in supernova studies. Recently, Horowitz et al. (Reference Horowitz, Caballero, Lin, O’Connor and Schwenk2017) reviewed many-body correlations for the neutral-current neutrino-nucleon scattering processes. The authors provide a useful fit for the vector response function, in combination with the Random Phase Approximation at high densities (see also Reddy et al. Reference Reddy, Prakash, Lattimer and Pons1999; Roberts & Reddy Reference Roberts and Reddy2017, and references therein) and the virial EOS for the low-density part.

Reaction rates for pair processes are provided in Bruenn (Reference Bruenn1985), where as N–N-bremsstrahlung rates were developed in Hannestad & Raffelt (Reference Hannestad and Raffelt1998) based on the vacuum 1π-exchange framework developed by Friman & Maxwell (Reference Friman and Maxwell1979). Recently, Fischer (Reference Fischer2016b) extended this treatment of the vacuum 1π-exchange for N–N-bremsstrahlung by taking into account the leading-order medium modifications, i.e., dressing of the πNN-vertex. Based on the Fermi-liquid theory, expressions have been derived that can be implemented into supernova simulations. In analogy, Bartl et al. (Reference Bartl, Bollig, Janka and Schwenk2016) describe such medium modifications at the level of chiral EFT.

The annihilation of trapped $\nu _{\text{e}}\bar{\nu }_{\text{e}}$ pairs processes of (7b) couples electron and heavy lepton flavour neutrinos, at high temperatures and densities. This channel reduces the difference of the luminosities and average energies of both flavours. Reaction rates were implemented in Buras et al. (Reference Buras, Janka, Keil, Raffelt and Rampp2003) and Fischer et al. (Reference Fischer, Whitehouse, Mezzacappa, Thielemann and Liebendörfer2009). Moreover, the highly inelastic neutrino-electron(positron) scattering (6) thermalises the neutrino spectra. Expressions for weak rates can be found, e.g., in Mezzacappa & Bruenn (Reference Mezzacappa and Bruenn1993c).

3.2. Post-bounce supernova dynamics and neutrino emission

Reaction (4) is responsible for the launch of the ν_e-burst (top central panel in Figure 3), which is associated with the propagation of the bounce shock across the neutrinospheres of last scattering [see Figure 1(b)] between 5–20 ms after core bounce. Weak equilibrium is re-established as matter is shock heated, associated with the sudden rise of the temperature. This highly non-equilibrium phenomenon is essential for the following supernova evolution as it determines a major source of losses, several 10⁵³ erg s⁻¹, being partly responsible for the dynamic bounce shock turning into a standing accretion front [see Figure 1(b)]. Moreover, only slightly before core bounce, when positrons exist, all other neutrino flavours are being produced mainly via electron-positron annihilation as well as via nucleon–nucleon bremsstrahlung [pair processes (7a)]. The luminosities of $\bar{\nu }_{\text{e}}$ and heavy-lepton flavour neutrinos rise accordingly (see Figure 3). The luminosities of all heavy lepton flavour neutrinos rise somewhat faster than those of $\bar{\nu }_{\text{e}}$ , since the latter are coupled more strongly to matter via the charged-current channel [see the second process of (4)]. This feature will eventually allow us to probe the neutrino mass hierarchy via the neutrino signal rise time from the neutrino observation of the next galactic supernova event (details can be found in Serpico et al. Reference Serpico, Chakraborty, Fischer, Hüdepohl, Janka and Mirizzi2012).

The later post-bounce evolution is determined by mass accretion onto the standing shock [see Figure 1(b)], during which the average neutrino energy hierarchy is determined by the different coupling strengths to matter (see bottom panel in Figure 3). Consequently, each neutrino species has their own neutrinosphere radius R _ν of last (in)elastic collision where the following hierarchy holds: $R_{\nu _{\text{e}}}>R_{\bar{\nu }_{\text{e}}}>R_{\nu _{\mu /\tau }}\gtrsim R_{\bar{\nu }_{\mu /\tau }}$ . The electron (anti)neutrinos decouple in a thick layer of low-density material accumulated at the PNS surface, powered by the charged-current processes (4). Consequently, their luminosity can be approximated by the accretion luminosity as follows (cf. Janka et al. Reference Janka, Langanke, Marek, Martínez-Pinedo and Müller2007; Janka Reference Janka2012):

with a typical mass enclosed inside the PNS M and radius associated with the neutrinospehere $R_{\nu _{\text{e}}}$ as well as mass accretion rate $\dot{m}$ . On the other hand, the heavy lepton neutrino flavours are determined by diffusion in the absence of charged-current processes.

The process of neutrino decoupling from matter is neutrino transport problem. Accurate three-flavour Boltzmann neutrino transport has been developed for spherically symmetric supernova models in Mezzacappa & Bruenn (Reference Mezzacappa and Bruenn1993a) and Liebendörfer et al. (Reference Liebendörfer2004). It leads to the establishment of a large cooling layer towards high densities at the PNS surface, where dQ/dt < 0Footnote ³ , as illustrated in Figure 6. It corresponds to the domain where high energy neutrinos decouple from matter, while the low energy spectrum is still thermalised with the medium. At low densities, between the standing shock at around 10⁹ g cm⁻³, these low energy neutrinos deposit parts of their energy via absorption processes into the medium. There, a heating layer establishes where dQ/dt > 0. However, since most weak interaction rates have a strong dependence on the neutrino energy, the integrated heating rates are significantly smaller than the cooling at higher density, besides the smaller mass enclosed in the heating layer than in the cooling layer. These are the two main reasons why spherically symmetric supernova explosions could not be obtained for the massive progenitor stars that develop an extended mass accretion period, typically for stars with initial mass above around 10 M_⊙. The success of the neutrino-heating mechanism in multi-dimensional simulations is attributed to the development of convection which allows material to remain effectively longer in the heating region, which increased the neutrino-heating efficiency. However, it should be mentioned that up to now only neutrino transport approximation schemes have been employed in multi-dimensional supernova studies. Note further that the situation is different for the low progenitor mass range, between 8 and10 M_⊙. Such stars develop either oxygen-neon cores (cf. Nomoto Reference Nomoto1987; Jones et al. Reference Jones2013), leading to electron-capture supernovae as explored in Kitaura, Janka, & Hillebrandt (Reference Kitaura, Janka and Hillebrandt2006) and Fischer et al. (Reference Fischer, Whitehouse, Mezzacappa, Thielemann and Liebendörfer2010), or “tiny” iron-cores as was explored recently in Melson et al. (Reference Melson, Janka and Marek2015). In both cases, the special structure of the stellar core, i.e., sharp density gradient separating core and envelope, leads to fast shock expansions and explosions even in spherical symmetric supernova simulations with low explosion energies ~10⁵⁰ erg and small amount of nickel ejected (for details, see Wanajo et al. Reference Wanajo, Nomoto, Janka, Kitaura and Müller2009). Similar core structures are obtained from binary stellar evolution. This was explored in Tauris et al. (Reference Tauris, Langer, Moriya, Podsiadlowski, Yoon and Blinnikov2013) and Tauris, Langer, & Podsiadlowski (Reference Tauris, Langer and Podsiadlowski2015), leading to so-called ultra-stripped progenitors of the secondary star that has undergone major mass transfer during the common-envelope evolution.

Figure 6. Integrated neutrino heating (dQ/dt > 0) and cooling (dQ/dt < 0) rates of the different channels charged-current (cc) processes (4), neutrino-electron and positron scattering (νe ^±) processes (6), and the sum of all pair reactions (pair) processes (7a). The data are from the reference supernova simulation of Fischer (Reference Fischer2016a) as illustrated in Figure 1(b) at about 300 ms post bounce, and the density domain corresponds to the region between PNS surface at around 15–20 km and the standing bounce shock around 80 km.

4.1. EOS with light clusters

The consistent description of the nuclear medium with light clusters as explicit degrees of freedom reduces the abundance of the free nucleons and ⁴He, in the domain where light clusters are abundant. This is illustrated in Figure 7 (middle panel) at selected conditions found during the early PNS deleptonisation shortly after the supernova explosion onset—temperature and Y _e profiles are shown in the top panel, with respect to the baryon density. Here, we compare the medium-modified NSE approach of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010) including the complete abundances of all nuclear clusters (red lines) with those of the same approach where only ⁴He is considered as light nuclear cluster (blue lines). Note that the latter case corresponds to the ‘classical’ supernova EOS composition that was commonly employed in numerous supernova simulations. It becomes evident that not only the abundance of ⁴He is largely overestimated, also the abundances of neutrons and protons are overestimated. Note further that the region where light clusters and free protons are equally abundant, corresponds to the supernova cooling region (see Figure 6). Hence, this urges the need for the systematic comparison of EOS based on the ‘full’ composition and only simplified nuclear composition (n, p, α, 〈A, Z〉), in supernova simulations in much greater detail, especially within multi-dimensional framework. In particular, we find that weak reactions with protons, most relevant for the charged-current $\bar{\nu }_{\text{e}}$ -opacity, are off by a factor greater than two, since the reaction rates scale with the number density of protons. The magnitude of the differences in spherically symmetric supernova simulations is illustrated in Figure 8, where we compare the modified NSE approach of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010) with ‘all’ nuclear clusters included with the simplified composition (n, p, α, 〈A, Z〉). In particular, the luminosity and average energy of $\bar{\nu }_e$ are overestimated when considering the simplified composition. Moreover, the rise time of the neutrino signal (for details about the role of the neutrino rise time can be found in Serpico et al. Reference Serpico, Chakraborty, Fischer, Hüdepohl, Janka and Mirizzi2012), in particular for $\bar{\nu }_{\text{e}}$ and heavy-lepton flavour neutrinos, is suppressed with (n, p, α, 〈A, Z〉), being related to the suppression of N–N-bremsstrahlung processes. This may have implications for the appearance of prompt convection, which occurs on a short timescale on the order of few tens of milliseconds after core bounce. The potential impact remains to be explored in multi-dimensional simulations.

Figure 7. Abundances of the supernova composition of neutrons and protons (b--c) and selected light clusters (d--e). The thermodynamic conditions in terms of temperature and electron fraction Y_e, shown in graph (a), correspond to the early PNS deleptonization phase shortly after the supernova explosion onset has been launched when the abundance of light nuclear clusters with A = 2 − 3 is maximum relative to those of protons (data obtained from Fischer et al. Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016).

Figure 8. Neutrino luminosities and average energies sampled in the co-moving frame of reference at 1 000 km, comparing simulations where ‘all’ nuclear clusters are included based on the modified NSE approach of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010) (same as shown in Figure 3) with the simplified composition (n, p, α, 〈A, Z〉).

In Figure 7, we also compare the modified NSE of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010) with the more sophisticated approaches for the description of in-medium nuclear clusters, i.e. the generalised RMF approach (gRDF) of Typel et al. (Reference Typel, Röpke, Klähn, Blaschke and Wolter2010). The latter is based on in-medium nuclear properties, e.g., binding energies obtained within first-principle quantum statistical calculations of Röpke (Reference Röpke2009) and Röpke (Reference Röpke2011). There it becomes evident that the modified NSE approach provides a sufficient description of the growth properties, such as the particle densities, of light nuclear clusters as illustrated in Figure 7 (c). However, the caveat is at high densities, ρ > 10¹⁴ g cm⁻³, where the geometric excluded volume of the modified NSE fails to properly describe the dissolving of nuclear states into the mean field. Modified NSE provides an inaccurate description of the phase transition to homogeneous matter with over- and underestimated abundances of the light clusters, depending on density and temperature.

4.2. Weak processes with light clusters

The inclusion of self-consistent weak rates with these light clusters is a much more subtle problem. Unlike for nuclear electron captures (1), where average rates are employed, here rates with individual nuclei must be taken into account. It is common to focus on the most abundant species, ²H, ³H, and ³He. In Table 2, we provide a list of all weak reactions with these light nuclei A = 2 − 3, that were considered in the study of Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016).

Cross-sections $\sigma _{\nu ^2\text{H}}$ for spallation reactions with ²H, (1) and (2) in Table 2, are provided by Nakamura et al. (Reference Nakamura, Sato, Gudkov and Kubodera2001). They also provide cross-sections for inelastic neutrino scattering on deuteron, (12) in Table 2. In Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016), it has been realised that these cross-sections are related to the electron and positron capture reactions (3) and (4) in Table 2 via the following replacements:

regarding initial-state and final-state phase space occupations of nucleons N and deuteron, as well as the following transformations for the differential cross-sections,

assuming relativistic electrons/positrons. Moreover, cross-sections for the spallation reactions with ³H, (5) and (6) in Table 2, were calculated in Arcones et al. (Reference Arcones, Martínez-Pinedo, O’Connor, Schwenk, Janka, Horowitz and Langanke2008) based on the random phase approximation. In Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016), we realised that the processes (7) and (8) in Table 2 are significantly more important than the spallation reactions, with three nucleons in the final-state. Cross-sections can be given as follows:

where

with Fermi constant G _F and B(GT) = 5.97, known experimentally from the triton decay. Electron and positron energies are related to the ν_e and $\bar{\nu }_{\text{e}}$ energies via, $E_{\text{e}^-}=E_{\nu _{\text{e}}} + Q_0$ and $E_{\text{e}^+}=E_{\bar{\nu }_{\text{e}}} - Q_0$ . The vacuum Q-value, Q ₀ = 0.529 MeV, is the restmass difference between ³He and ³H. Figure 9 compares the cross-sections of all these charged-current weak processes, for A = 2 (top panel) and A = 3 (bottom panel) in comparison to those of the Urca processes (4), where

with vector and axial vector coupling constants g _V = 1.0 and g _A = 1.26. Here we relate electron(positron) and ν_e( $\bar{\nu }_{\text{e}}$ ) energies with the vacuum Q-value Q ₀ = 1.2935 MeV.

Figure 9. Charged current cross sections for ν_e- and $\bar{\nu }_{\text{e}}$ -absorption on light nuclei with A = 2 (top panel) and A = 3 (bottom panel), in comparison to those of the Urca processes (4) for charged current reactions.

When turning these cross-sections into reaction rates, the following two aspects are essential: (i) the vacuum cross-sections, including those of Nakamura et al. (Reference Nakamura, Sato, Gudkov and Kubodera2001), introduced above have to be ‘mapped’ into medium-modified cross-sections. The procedure therefore has been introduced in Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016) based on the mean-field treatment with single-particle energies and effective nucleon masses, (ii) the phase space of the contributing particles has to be taken into account properly (detailed expressions are provided as well in Fischer et al. Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016). If this is done accurately, then detailed balance is fulfilled (unlike was done in Furusawa et al. Reference Furusawa, Nagakura, Sumiyoshi and Yamada2013), and the impact from weak interactions with light nuclear clusters on the supernova neutrino signal and dynamics was found to be negligible in Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016). The reason for this is illustrated at the example of the mean-free paths in Figure 10—neutral-current scattering (top panel) and the charged-current absorption processes (bottom panel)Footnote ⁴ , comparing processes with free nucleons and reactions with light clusters with A = 2 − 4. Note that elastic scattering with light clusters is based on the coherent description of Bruenn (Reference Bruenn1985). The mean free paths, including the neutrino distribution functions, are obtained in detailed core-collapse supernova simulations with Boltzmann neutrino transport of Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016), including all these reactions with light clusters (for definitions of mean free path as well as neutrinospheres of last scattering, cf. Fischer et al. Reference Fischer, Martínez-Pinedo, Hempel and Liebendörfer2012).

Figure 10. Mean-free paths for ν_e (left panel) and $\bar{\nu }_{\text{e}}$ -reactions (right panel) with light nuclei with A = 2 − 4, for neutral-current scattering (top) and charged-current absorption (bottom). The conditions are shown in Figure 7. (Figure adopted from Fischer et al. Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016).

For the ν_e, scattering and absorption reactions with free neutrons are the dominating channels by orders of magnitude over those with any light nucleus. This is due to the high abundance of free neutrons in comparison with any other nuclear species (see Figure 7). Moreover, the total opacity is dominated by charged-current absorption on neutrons (see Figure 10). The situation is similar for $\bar{\nu }_{\text{e}}$ , for which the neutral current channel is also dominated by scattering on neutrons. Nevertheless, $\bar{\nu }_e$ -absorption on protons dominates less strictly over the absorption on ²H, however, at increasing density. There, the mean-free path for $\bar{\nu }_e$ -absorption on ²H is comparable to other inelastic processes, e.g., $\bar{\nu }_{\text{e}}$ scattering on electrons and positrons as well as N–N-bremsstrahlung. Note that in Figure 10 the labels $\nu _{\text{e}}\bar{\nu }_{\text{e}}$ correspond to the sum of all pair processes (7a).

Weak reactions with A = 3 have in general a negligible role. In Figure 10, we show only processes (7) and (8) of Table 2, which exceed the break-up reactions with ³H by orders of magnitude. The difference to Arcones et al. (Reference Arcones, Martínez-Pinedo, O’Connor, Schwenk, Janka, Horowitz and Langanke2008) may be due to the lack of final-state blocking contributions. Even though ν _e -absorption on ³H exceeds ν _e -absorption on ²H, it still lacks short by at least one order of magnitude ν _e -absorption on neutrons. The largely suppressed $\bar{\nu }_e$ -absorption on ³He is due to the low abundance of ³He. Note also the region of relevance here, illustrated by black vertical dashed lines in Figure 10, which mark the locations of the average neutrinospheres of last inelastic (bottom panels) and the effective neutrinospheres (top panels).

The conditions of Figure 10 correspond to the early PNS deleptonisation phase at about 1 s after the supernova explosion onset has been launched, when the thick layer of accumulated material at the PNS surface from the mass accretion phase falls into the gravitational potential of the PNS. This is the moment of maximum impact of weak processes with light clusters, when the abundance of ²H and ³H exceed the one of protons maximally. However, the temperatures are already somewhat lower than during the post-bounce mass accretion period prior to the supernova explosion onset. Therefore, Fischer et al. (Reference Fischer, Martínez-Pinedo, Hempel, Huther, Röpke, Typel and Lohs2016) performed supernova simulations based on three-flavour Boltzmann neutrino transport, including in addition to the standard weak processes (1)–(10) also all weak processes with light clusters with A = 2 − 3 shown in Table 2. It was found that the impact of weak processes with light clusters on the overall supernova dynamics and the neutrino signal is negligible during the mass accretion phase as well as during the PNS deleptonisation phase.

Finally, when comparing the density domain where nuclear pasta may exists in Figure 4 with the one where light nuclei are abundant in Figure 7, it becomes evident that both overlap towards high densities. It is therefore important to develop sophisticated models for nuclear pasta that take the presence of light nuclear clusters into account consistently.

5.1. Excluded volume approach

In order to explore the uncertainties of the supersaturation density EOS in simulations of core-collapse supernovae, the geometric excluded volume mechanism has been employed in Fischer (Reference Fischer2016a) based on the formalism developed in Typel (Reference Typel2016). There, the available volume of the nucleons V _N is suppressed via V _N = Vϕ, where V is the total volume of the system. Thereby, the density functional ϕ has the following Gaussian type:

for both neutrons and protons, in order to ensure a smooth behaviour above saturation density. Based on the choice of the excluded volume parameter, v, additional stiffening is provided to the EOS for v > 0 or softening for v < 0 at supersaturation density. The reference case corresponds to v = 0, for which the modified NSE approach of Hempel & Schaffner-Bielich (Reference Hempel and Schaffner-Bielich2010) is selected together with the RMF parameterisation DD2 of Typel et al. (Reference Typel, Röpke, Klähn, Blaschke and Wolter2010), henceforth denoted as HS(DD2), while HS(DD2-EV) denote EOS with excluded volume modifications. Even though DD2 is already rather stiff at supersaturation density, in particular, in view of the symmetric matter flow constraint obtained from the detailed analysis of heavy-ion collisions by Danielewicz, Lacey, & Lynch (Reference Danielewicz, Lacey and Lynch2002), an even stiffer EOS cannot be ruled out a priori for supernova matter (i.e., at large isospin asymmetry).

It is important to note that within the approach (18), nuclear saturation properties remain unmodified, i.e., the saturation density n ₀ and the symmetry energy at n ₀ is J = 31.67 MeV. On the other hand, quantities which relate to derivatives are modified, e.g., the (in)compressibility modulus (a summary of present constraints are given in Stone, Stone, & Moszkowski Reference Stone, Stone and Moszkowski2014) varies from K ≃ 541 MeV to K ≃ 201 MeV for the two selected values of v = +8.0 fm³ and v = -3.0 fm³, respectively, compared to the reference case K ≃ 243 MeV (for v = 0). Further details are given in Typel (Reference Typel2016) and Fischer (Reference Fischer2016a).

Figure 11 shows the resulting pressures as a function of density (restmass density ρ at the bottom scale and baryon density n _B at the top scale) for the two extreme choices, v = +8.0 fm⁻³ and v = -3.0 fm⁻³ (units are skipped in the Figure’s legend for simplicity). These values are selected such that causality is obtained (see the sound speed in the inlay of Figure 11) for the relevant supernova densities and maximum neutron star masses are in agreement with the current constraint of about 2 M_⊙. Only for v = +8.0 fm⁻³, the sound speed exceeds the speed of light, however, at densities far above the central densities reached in the core-collapse simulations which will be further discussed below.

Figure 11. High density behaviour of the supernova EOS, at selected temperature of T = 5 MeV and electron fraction Y _e = 0.3, comparing the reference treatment (v = 0) with the modified excluded volume approach with additional stiffening for v = +8.0 fm⁻³ and softening for v = -3.0 fm⁻³ above supersaturation density (ρ > ρ₀). (Figure adopted from Fischer Reference Fischer2016a).

The central conditions found during the supernova evolutions are illustrated in Figure 12, in terms of density and temperature. In comparison to the reference case, significantly higher(lower) central densities and temperatures are obtained for the soft(stiff) modification of HS(DD2) with v = -3.0 fm⁻³(v = +8.0 fm⁻³). The central density reaches 4.25 × 10¹⁴ g cm⁻³ for v = -3.0 fm⁻³ in comparison to only 3.0 × 10¹⁴ g cm⁻³ for v = +8.0 fm⁻³, at about 500 ms post bounce. However, it was realised in Fischer (Reference Fischer2016a) that despite such large variation of the central conditions, the overall impact on the supernova dynamics and neutrino signal is negligible (see Figures (4) and (5) in Fischer Reference Fischer2016a). This is because the post-bounce dynamics is determined dominantly at low densities (inhomogeneous nuclear matter, region of high entropy in Figure 1(b)) and the supersaturation density domain of the PNS is generally small, ~0.05–0.1 M_⊙ of the total enclosed baryonic mass of PNS with ~1.4–1.8 M_⊙ that result from supernova simulations launched from progenitors in the mass range of ~10–30 M_⊙ (cf. Ugliano et al. Reference Ugliano, Janka, Marek and Arcones2012).

Figure 12. Supernova evolution of central density and temperature comparing reference EOS HS(DD2) and variations due the excluded volume approach (see text for details). The dynamical evolution of the reference case is illustrated in Figure 1 and the neutrino signal is shown in Figure 3. (Figure adopted from Fischer Reference Fischer2016a).

5.2. Phase transition to quark matter

Another similarly uncertain aspect of the EOS at supersaturation density is the question of a possible phase transition from nuclear matter, with hadrons as degrees of freedom, to the deconfined quark gluon plasma with quarks and gluons as the new degrees of freedom. This has long been explored in the context of cold neutron stars. In general, medium properties of quark matter have long been studied (cf. Bender et al. Reference Bender, Poulis, Roberts, Schmidt and Thomas1998; Roberts & Schmidt Reference Roberts and Schmidt2000; Buballa Reference Buballa2005; Alford, Blaschke, & Drago Reference Alford2007; Klähn et al. Reference Klähn, Blaschke, Sandin, Fuchs, Faessler, Grigorian, Röpke and Trümper2007; Ayriyan et al. Reference Ayriyan, Alvarez-Castillo, Benic, Blaschke, Grigorian and Typel2017a; Pagliara & Schaffner-Bielich Reference Pagliara and Schaffner-Bielich2008; McLerran & Pisarski Reference McLerran and Pisarski2007; Sagert et al. Reference Sagert2009; Pagliara, Hempel, & Schaffner-Bielich Reference Pagliara, Hempel and Schaffner-Bielich2009; Blaschke et al. Reference Blaschke, Sandin, Klähn and Berdermann2009; Klähn et al. Reference Klähn, Roberts, Chang, Chen and Liu2010b; Klähn, Blaschke, & Lastowiecki Reference Klähn, Blaschke and Lastowiecki2012; Chen et al. Reference Chen, Baldo, Burgio and Schulze2011; Fischer et al. Reference Fischer2011; Weissenborn, Chatterjee, & Schaffner-Bielich Reference Weissenborn, Chatterjee and Schaffner-Bielich2012; Bonanno & Sedrakian Reference Bonanno and Sedrakian2012; Blaschke et al. Reference Blaschke, Buballa, Dubinin, Röpke and Zablocki2014; Klähn, Łastowiecki, & Blaschke Reference Klähn, Łastowiecki and Blaschke2013; Kurkela et al. Reference Kurkela, Fraga, Schaffner-Bielich and Vuorinen2014; Beisitzer, Stiele, & Schaffner-Bielich Reference Beisitzer, Stiele and Schaffner-Bielich2014, and references therein). First-principle calculations of Quantum Chromodynamics (QCD)—the theory of strongly interacting matter—are possible by means of conducting large-scale numerical studies (cf. Fodor & Katz Reference Fodor and Katz2004; Ratti, Thaler, & Weise Reference Ratti, Thaler and Weise2006), however, only at vanishing density. They predict a smooth cross-over transition from hadronic matter to deconfined quark matter, for a temperature of T = 154 ± 9 MeV at μ_B ≃ 0 (cf. Borsányi et al. Reference Borsányi, Fodor, Katz, Krieg, Ratti and Szabó2012; Bazavov et al. Reference Bazavov2012a, Reference Bazavov2012b; Borsányi et al. Reference Borsányi, Fodor, Hoelbling, Katz, Krieg and Szabó2014, and references therein). Consequently, in astrophysics studies associated with high baryon densities, the two-phase approach is commonly used based on a hadronic EOS and a different EOS for quark matter at high density. It results in first-order phase transition for which Maxwell or Gibbs constructions are commonly employed. Alternatively, pasta-phases arise when taking into account finite-size effects (Yasutake et al. Reference Yasutake, Łastowiecki, Benić, Blaschke, Maruyama and Tatsumi2014). To this end, Ayriyan et al. (Reference Ayriyan, Alvarez-Castillo, Benic, Blaschke, Grigorian and Typel2017a) studied the appearance of such phases and the associated stability of hybrid stars.

It is important to note that perturbative QCD is valid only in the limit of asymptotic freedom where quarks are no longer strongly coupled (Kurkela et al. Reference Kurkela, Fraga, Schaffner-Bielich and Vuorinen2014), which automatically excludes astrophysical applications. Hence, in the interior of neutron stars and supernovae, where generally densities are encountered far below the asymptotic limit, effective quark matter models have been commonly employed, e.g., the thermodynamic bag model of Farhi & Jaffe (Reference Farhi and Jaffe1984) and models based on the Nambu–Jona–Lasinio (NJL) approach developed by Nambu & Jona-Lasinio (Reference Nambu and Jona-Lasinio1961), see also Klevansky (Reference Klevansky1992) and Buballa (Reference Buballa2005).

In simulations of core-collapse supernovae, the thermodynamic bag model has been employed in the detailed studies of Nakazato, Sumiyoshi, & Yamada (Reference Nakazato, Sumiyoshi and Yamada2008) and Sagert et al. (Reference Sagert2009). The latter study assumed low onset densities for the quark-hadron phase transition—tuned via the bag constant as free parameter—which are realised in the core of canonical supernovae launched from progenitors with initial masses in the range of 10–20 M_⊙. It was found that the thermodynamically unstable region between the stable hadron and quark phases results in the collapse of the PNS, which launches a shock wave that in turn triggers the supernova explosion onset in otherwise non-exploding models in spherical symmetry (for details, see Fischer et al. Reference Fischer2011). It also releases a millisecond neutrino burst of all neutrino flavours, which was found to be observable with the current generation of neutrino detectors (details can be found in Dasgupta et al. Reference Dasgupta2010). Unfortunately, these detailed studies violate several ‘solid’ constraints, e.g., maximum neutron star masses are far below the current value of 2 M_⊙, and chiral physics is largely violated. All this urges the need to develop more elaborate phenomenological quark matter EOS, being consistent with current constraints. This marks a major task due to the generally three-dimensional dependencies of supernova matter in terms of temperature, baryon, and isospin densities. Moreover, also the development of weak processes in deconfined quark matter, consistent with the underlying quark-matter EOS, is a major undertaken (for some recent progress, cf., Berdermann et al. Reference Berdermann, Blaschke, Fischer and Kachanovich2016, and references therein).

It has been demonstrated recently in Klähn & Fischer (Reference Klähn and Fischer2015) that the thermodynamic bag model and the NJL approach can be understood as limiting solutions of QCD’s in-medium Dyson–Schwinger gap equations (see also Bashir et al. Reference Bashir2012; Cloet & Roberts Reference Cloet and Roberts2014; Chang, Roberts, & Tandy Reference Chang, Roberts and Tandy2011; Roberts Reference Roberts2012; Roberts & Schmidt Reference Roberts and Schmidt2000; Chen et al. Reference Chen2008, Reference Chen, Baldo, Burgio and Schulze2011, Reference Chen, Wei, Baldo, Burgio and Schulze2015; Klähn et al. Reference Klähn, Roberts, Chang, Chen and Liu2010a). Based on the newly developed model vBag of Klähn & Fischer (Reference Klähn and Fischer2015), it has been realised that repulsive vector interaction provides the necessary stiffness for the EOS at high densities in order to yield massive hybrid stars with maximum masses in agreement with the current constraint of 2 M_⊙. Higher order vector repulsion terms were explored in the non-local NJL model of Benic et al. (Reference Benic, Blaschke, Alvarez-Castillo, Fischer and Typel2015). This work has been complemented recently in Kaltenborn, Bastian, & Blaschke (Reference Kaltenborn, Bastian and Blaschke2017), providing a relativistic density functional approach to quark matter that implements a confinement mechanism for quarks and allows extensions to finite temperatures. Both aforementioned articles studied the physical case of massive hybrid stars which lead to the ‘twin’ phenomenon (see also Haensel, Potekhin, & Yakovlev Reference Haensel, Potekhin and Yakovlev2007; Read et al. Reference Read, Lackey, Owen and Friedman2009; Zdunik & Haensel Reference Zdunik and Haensel2013; Alford, Han, & Prakash Reference Alford, Han and Prakash2013, and references therein). It relates to the existence of two families of compact stellar objects with similar-to-equal masses but different radii, linked in the mass-radius diagram via a disconnected (i.e., thermodynamically unstable) branch (cf. Alvarez-Castillo & Blaschke Reference Alvarez-Castillo and Blaschke2017). This interesting situation is part of the science case of the NICER (Neutron Star Interior Composition Explorer)Footnote ⁶ NASA mission, which aims at deducing radii of massive neutron stars at high precision of 0.5 km. See also Alvarez-Castillo et al. (Reference Alvarez-Castillo, Ayriyan, Benic, Blaschke, Grigorian and Typel2016a) and Ayriyan et al. (Reference Ayriyan, Alvarez-Castillo, Benic, Blaschke, Grigorian and Typel2017b) for a recent Bayesian analysis of constraints on EOS with high-mass twin property using fictitious radius data, that could be provided in the near future, e.g., by NICER and similar missions. Moreover, recent developments point towards a universality condition for the hadron-quark phase transition (cf. Alvarez-Castillo et al. Reference Alvarez-Castillo, Benic, Blaschke, Han and Typel2016b).

In Klähn, Fischer, & Hempel (Reference Klähn, Fischer and Hempel2017a), vBag was extended to finite temperature and arbitrary isospin asymmetry. Figure 13 illustrates the resulting phase diagram for matter in β-equilibriumFootnote ⁷ , for a well selected model parameter B _χ (chiral bag constant) which defines chiral symmetry restoration—marked in Figure 13 by ρ_χ—being in agreement with pion mass and decay constant. Onset densities for the first-order phase transition from the selected underlying hadronic EOS, here HS(DD2), is marked as ρ_H in Figure 13. Moreover, in contrast to the standard NJL-approach, here (de)confinement is taken into account via a phenomenological parameter (B _dc) which is determined by the underlying hadronic EOS, being equal to the hadronic pressure at the chiral transition (for details, see Klähn et al. Reference Klähn, Fischer and Hempel2017a). Hence, it becomes medium dependent, B _dc(T, μ _Q ), where T and μ _Q are temperature and charge chemical potential (the latter referring to isospin asymmetry), respectively. Consequently, additional terms appear which add to the EOS in order to ensure thermodynamic consistency, related to derivatives of B _dc with respect to T and μ _Q .

Figure 13. vBag phase diagram for matter in β-equilibrium for the quark matter model developed in Klähn & Fischer (Reference Klähn and Fischer2015), with the two parameters B _χ and B _dc, in comparison to the standard NJL approach (B _dc = 0). (Figure adopted from Klähn et al. Reference Klähn, Fischer, Cierniak and Hempel2017b).

The canonical approach for constructing phase transitions based on the two-phase approach, i.e., setting B _dc = 0 (also illustrated in Figure 13) corresponds to the standard NJL approach commonly used in neutron star studies. It results in significantly higher transition densities for the quark-hadron phase transition. Moreover, this concept leads to the problem that chiral symmetry is (at least partly) restored while matter remains in the confined phase. At present, one can only speculate about the existence of such phase as the currently running heavy-ion collision experiment at NICA (cf. the NICA white paper Blaschke et al. Reference Blaschke2016, and selected contributions therein) in Dubna (Russia) has not reported any scientific results yet, the low-energy fixed-target experiment at the Relativistic Heavy-Ion Collider (RHIC) in Brookhaven (USA) is expected to produce first results in 2018/2019, and the future Facility for Antiproton and Ion research (FAIR) at the GSI in Darmstadt (Germany) is still under construction. The science runs, in particular, experiments related to the compressed baryonic matter physics (cf. the CBM physics book Friman et al. Reference Friman, Höhne, Knoll, Leupold, Randrup, Rapp and Senger2011), will help to shed light onto the phase structure of the entire phase diagram at high densities and lower temperatures.