Towards generating a new supernova equation of state: A systematic analysis of cold hybrid stars

The hadron-quark phase transition in core-collapse supernovae (CCSNe) has the potential to trigger explosions in otherwise nonexploding models. However, those hybrid supernova equations of state (EOS) shown to trigger an explosion do not support the observational $2{\mathrm{M}}_{\odot }$ neutron star maximum mass constraint. In this work, we analyze cold hybrid stars by the means of a systematic parameter scan for the phase transition properties, with the aim to develop a new hybrid supernova EOS. The hadronic phase is described with the state-of-the-art supernova EOS HS(DD2), and quark matter by an EOS with a constant speed of sound (CSS) of $c_{\mathrm{QM}}^2=1/3$. We find promising cases which meet the $2{\mathrm{M}}_{\odot }$ criterion and are interesting for CCSN explosions. We show that the very simple CSS EOS is transferable into the well-known thermodynamic bag model, important for future application in CCSN simulations. In the second part, the occurrence of reconfinement and multiple phase transitions is discussed. In the last part, the influence of hyperons in our parameter scan is studied. Including hyperons no change in the general behavior is found, except for overall lower maximum masses. In both cases (with and without hyperons) we find that quark matter with $c_{\mathrm{QM}}^2=1/3$ can increase the maximum mass only if reconfinement is suppressed or if quark matter is absolutely stable.

Figure 1

This illustration was published in [7] and shows the classification of hybrid stars by means of their $M\text{−}R$ curve. The two important criteria are the presence of a third family branch and the stability of hybrid stars at the onset of quark matter.

Figure 2

Calculated hybrid star configurations, colored to distinguish the four cases A (absent), B (both), C (connected) and D (disconnected). The lines in blue show the maximum mass contours for 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, and $2.8{\mathrm{M}}_{\odot }$. The thick black dashed line shows the analytic criterion from Seidov [48], above which neutron stars are unstable at the onset of quark matter.

Figure 3

Similar to Fig. 2, but with red lines showing the solutions of the bag model from Eq. (16) for varying $B$ with increasing values from left to right, different values of ${\alpha }_s$ (as indicated in the figure), and $m_s=0$.

Figure 4

Dependency of the speed of sound on the energy density for four different values of $m_s$ (0, 100, 200 and 300 MeV), $B^{1/4}=155\mathrm{MeV}$ and ${\alpha }_s=0.3$. The phase-transition points are indicated with triangles.

Figure 5

Parameter scan extended to higher $\mathrm{\Delta }\epsilon /{\epsilon }_{\text{trans}}$ and logarithmic scale for $p_{\text{trans}}/{\epsilon }_{\text{trans}}$. Additionally, the phase transition parameters of the hybrid EOSs of Table 1 are plotted with yellow squares (explosions found) and green triangles (no explosions found). Note that the hadronic part of these EOSs is based on STOS, whereas the results in the figure (maximum mass and classification) are based on HS(DD2), for details see the main text. Marked with a yellow circle is an example case whose $M\text{−}R$ relation is shown in Fig. 6.

Figure 6

$M\text{−}R$ relation of an example hybrid EOS which might be interesting for CCSNe. The quark matter parameters are $m_s=100\mathrm{MeV}$, ${\alpha }_s=0.7$, and $B^{1/4}=138.5\mathrm{MeV}$.

Figure 7

Example case with three phase transitions. The inlay shows a zoom-in of the first two phase transitions.

Figure 8

This figure shows the number of phase transitions that occur for given $\mathrm{\Delta }\epsilon$ and $p_{\text{trans}}$. The yellow dots represent cases with one phase transition (HQ), red dots with three phase transitions (HQHQ), and grey dots with two phase transitions (QHQ), where quark matter is absolutely stable. The black dots represent QHQ cases, where even negative energy densities occur.

Figure 9

Results for the parameter scan taking into account the occurrence of multiple phase transitions. Red dots show parameter combinations resulting in three phase transitions (HQHQ), grey in two phase transitions (QHQ) and absolutely stable strange quark matter. The other points have only one phase transition (HQ), for which the Alford classification (A, B, C, or D) can be done. The empty triangular region in the lower right corner corresponds to unphysical EOSs with negative energy density, for which the $M\text{−}R$ relation has not been calculated.

Figure 10

As in Fig. 2, but including hyperons by using the $\mathrm{BHB}\mathrm{\Lambda }\varphi$ EOS instead of HS(DD2).

Figure 11

As in Fig. 9, but including hyperons by using the $\mathrm{BHB}\mathrm{\Lambda }\varphi$ EOS instead of HS(DD2).