Hybrid equation of state with pasta phases, and third family of compact stars

The effect of pasta phases on the quark-hadron phase transition is investigated for a set of relativistic mean-field equations of state for both hadron and quark matter. The results of the full numerical solution with pasta phases are compared with those of an interpolating construction used in previous works, for which we demonstrate an adequate description of the numerical results. A one-to-one mapping of the free parameter of the construction to the physical surface tension of the quark-hadron interface is obtained for which a fit formula is given. For each pair of quark and hadron matter models the critical value of the surface tension is determined, above which the phase transition becomes close to the Maxwell construction. This result agrees well with earlier theoretical estimates. The study is extended to neutron star matter in beta equilibrium with electrons and muons and is applied to investigate the effect of pasta phases on the structure of hybrid compact stars and the robustness of a possible third family solution.

Figure 1

Pressure as a function of the density for all the combinations of models under consideration with the MC (thin lines) and the GC with $\sigma =0$ and no electrostatic contribution (thick lines) for the case without muons.

Figure 2

Pressure as a function of the chemical potential for the H2-Q1 model for the surface tension $\sigma =[0,40,80]\mathrm{MeV}/{\mathrm{fm}}^2$ (upper panel) and the H1-Q2 model with $\sigma =[0,16,32]\mathrm{MeV}/{\mathrm{fm}}^2$ (lower panel). Solid lines denote the quark and hadron EoSs, symbols denote the numerical results with structure types indicated in the legend, and the dashed lines show the best-fit curves using Eq. (13).

Figure 3

Pressure as a function of the baryon density for the H2-Q1 model for the surface tension $\sigma =[0,40,80]\mathrm{MeV}/{\mathrm{fm}}^2$ (upper panel) and the H1-Q2 model with $\sigma =[0,16,32]\mathrm{MeV}/{\mathrm{fm}}^2$ (lower panel). Solid lines denote the quark and hadron homogeneous matter EoSs, symbols denote the numerical results with the structure type indicated in the legend, and the dashed lines show the best-fit curves using Eq. (13). For comparison, we show also the MC by a dotted line. In the rightmost panels where $\sigma >{\sigma }_c$ we use the symbols for homogeneous phase also in the coexistence region because the space is divided just into two phases and no structures can be identified.

Figure 4

Upper panel: Relative pressure shift ${\mathrm{\Delta }}_P$ at the MC chemical potential ${\mu }_c$ as a function of the surface tension $\sigma$ for all the combinations of models. Solid lines show the results of the fit with use of Eq. (17), and the dashed lines are plotted with use of the mean value of $\beta$ (18) and the best-fit ${\mathrm{\Delta }}_P^{\left(0\right)}$ and ${\sigma }_c$ for each model. Lower panel: The dimensionless function $S\left(x\right)$ (17) for our models. The solid line with dots shows $S\left(x\right)$ for the mean value $\beta$, and the shaded area indicates the change of $S\left(x\right)$ for the variation of $\beta$ with $\mathrm{\Delta }\beta =0.18$.

Figure 5

Critical surface tension ${\sigma }_c$ as a function of the pressure on the MC line $P_c$ for all the combinations of models considered. Full symbols denote the results of the numerical calculation, and open symbols are the analytical estimates using Eq. (20). The dashed line shows a linear fit given by Eq. (21) and the gray area denotes its estimated uncertainty.

Figure 6

Same as Fig. 4 but with inclusion of muons.

Figure 7

Same as Fig. 5 but with inclusion of muons.

Figure 8

$M-R$ relationship for the four pairs of EoSs discussed in this work. Solid lines denote the results for the Maxwell construction ($\sigma >{\sigma }_c$), and the dashed lines denote the results for $\sigma =0$. Unstable CS configurations are shown by thin dotted lines. The horizontal bands denote the CS mass measurements for PSR J0348+432 [1] and for PSR J0740+6620 [2]. The magenta cross-hatched regions are excluded by the GW observations from GW170817 with arguments for the left region provided in [30] and for the right region in [31], respectively.

Figure 9

Moment of inertia as a function of the compact star gravitational mass. Solid lines show the results for the MC case ($\sigma >{\sigma }_c$), and dashed lines for $\sigma =0$. Thin dotted lines denote unstable CS configurations. The vertical line indicates the precisely measured mass of star (A) in the binary CS system J0737-3039 [33] for which a determination of the moment of inertia with high precision is under way.

Figure 10

Relation between tidal deformabilities ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ of the two compact stars that merged in the event GW170817 with the chirp mass $\mathcal{M}=1.188M_{\odot }$. Solid lines show the results for the MC case ($\sigma >{\sigma }_c$), and dashed lines for $\sigma =0$. The dotted line shows the NS-HS branch, possible within the H1-Q2 model. For a comparison, the 90% and 50% confidence level regions from the analysis of the GW signal by the LIGO-Virgo Collaboration [35] are also shown.