New temperature dependent hyperonic equation of state: Application to rotating neutron star models and $I\text{−}Q$ relations

In this work we present a newly constructed equation of state (EoS), applicable to stellar core collapse and neutron star mergers including the entire baryon octet. Our EoS is compatible with the main constraints from nuclear physics and, in particular, with a maximum mass for cold $\beta$-equilibrated neutron stars of $2M_{\odot }$ in agreement with recent observations. As an application of our new EoS, we compute numerical stationary models for rapidly (rigidly) rotating hot neutron stars. We consider maximum masses of hot stars, such as protoneutron stars or hypermassive neutron stars in the postmerger phase of binary neutron star coalescence. The universality of $I\text{−}Q$ relations at nonzero temperature for fast rotating models, comparing a purely nuclear EoS with its counterparts containing $\mathrm{\Lambda }$ hyperons or the entire baryon octet, respectively, is discussed, too. We find that the $I\text{−}Q$ universality is broken in our models when thermal effects become important, independent on the presence of entropy gradients. Thus, the use of $I\text{−}Q$ relations for the analysis of protoneutron stars or merger remnant data, including gravitational wave signals from the last stages of binary neutron star mergers, should be regarded with care.

Figure 1

The lines delimit the regions in temperature and baryon number density for which the overall hyperon fraction exceeds ${10}^{-4}$, which are situated above the lines. The dark thick purple line corresponds to the $\mathrm{BHB}\mathrm{\Lambda }\varphi$ model and light thin red line to the DD2Y model. Different charge fractions are shown as indicated within the panels.

Figure 2

Values of $E_{\mathrm{sym}}$ and $L$ in different nuclear interaction models. The two gray rectangles correspond to the range for $E_{\mathrm{sym}}$ and $L$ derived in Ref. [80] (light gray) and Ref. [79] (dark gray) from nuclear experiments and some neutron star observations.

Figure 3

Pressure (left panel) and energy per baryon (right panel) of pure neutron matter as functions of baryon number density within different nuclear interaction models compared with the ab initio calculations of Ref. [83], indicated by the blue band.

Figure 4

Gravitational mass vs circumferential equatorial radius for cold spherically symmetric neutron stars within different EoS models. The two horizontal bars indicate the two recent precise NS mass determinations, PSR $\mathrm{J}1614-2230$ [41, 43] (hatched gray) and PSR J0348+0432 [42] (green).

Figure 5

Total strangeness fraction as function of baryon number density for different values of fixed temperature and electron fraction $Y_e$ within $\mathrm{BHB}\mathrm{\Lambda }\varphi$ (dotted lines) and DD2Y (solid lines) EoS. In (a)–(c), $Y_e=0.3$ and in (d)–(f), $T=30$ MeV. For information, the $\mathrm{\Lambda }$ fraction in model DD2Y is indicated, too (dash-dotted lines).

Figure 6

Pressure (d)–(f) and normalized free energy per baryon (a)–(c) as function of baryon number density for different values of fixed temperature and $Y_e=0.3$ within the three different EoS, HS(DD2), $\mathrm{BHB}\mathrm{\Lambda }\varphi$, and DD2Y. For information, the pressure in the classical models LS220 and STOS is displayed, too.

Figure 7

Same as Fig. 6, but for $T=30$ MeV and different fixed values of $Y_e$.

Figure 8

Left: $T\left(n_B\right)$ relations used for the computation of hot star models. For comparison, the profiles obtained for different values of constant $s_B$ within the DD2Y EoS are shown, too. Right: $s_B\left(n_B\right)$ resulting from the two chosen $T\left(n_B\right)$ relations within the DD2Y EoS.

Figure 9

Gravitational mass vs central baryon number density for nonrotating stars for three different EoSs, without hyperons (left), with $\mathrm{\Lambda }$ hyperons (middle), and the complete baryon octet (right). Different values of constant total entropy $S$ have been used, indicated in units of solar masses. $\beta$ equilibrium has been assumed. For comparison, the cold result is shown, too. For DD2Y, at $S=3M_{\odot }$ and $S=5M_{\odot }$, the curves end at some central density above which no longer any $\beta$-equilibrated solution is found. The reason is that the electron fraction becomes lower than the limiting value of the EoS table ($Y_e=0.01$). No maximum could be determined in this case.

Figure 10

Same as Fig. 9, comparing stars with $Y_L=0.4$ at different constant total entropy values with the $\beta$-equilibrated result for cold stars.

Figure 11

Particle fractions in hot neutron star matter for the three different EoSs discussed here. The lepton fraction has been fixed to $Y_L=0.4$ and the entropy per baryon corresponds to the value of the respective maximum mass configuration with the total entropy indicated in each panel. The vertical lines show the central density of the respective maximum mass configurations.

Figure 12

Normalized moment of inertia vs quadrupole moment for different EoSs. The color coding corresponds to different EoS models whereas different symbols indicate rotation frequencies and entropy per baryon of the stars. In panel (a) different nucleonic and hyperonic EoSs are shown for cold stars, in $\beta$ equilibrium, for the slow as well as fast rotating case, respectively. The LS220 EoS and the counterpart with $\mathrm{\Lambda }$ hyperons are thereby shown by red symbols, STOS EoS with and without hyperons by green symbols, and the three EoSs based on DD2, HS(DD2), $\mathrm{BHB}\mathrm{\Lambda }\varphi$, and DD2Y by violet symbols. For the latter three EoSs in addition the results for slowly rotating configurations with $s_B=4$ are displayed. In the three other panels the cold reference case with the HS(DD2) EoS is displayed by black symbols and different situations are considered with the three EoSs, HS(DD2), $\mathrm{BHB}\mathrm{\Lambda }\varphi$, and DD2Y, indicated by violet symbols: constant $Y_L=0.4$ (b), profile $T_2$ (c), profile $T_1$ (d).

Figure 13

Relative difference between Eq. (29)—the fitted results for cold slowly rotating stars—and the results at different constant $s_B$ values for the DD2Y EoS, assuming neutrinoless $\beta$ equilibrium.