Universal Relations for Elastic Hybrid Stars and Quark Stars

TL;DR

This work tests whether the well-known universal relations among the moment of inertia $I$, tidal deformability $\lambda_2$, spin-induced quadrupole moment $Q$, and compactness $C$ extend to elastic hybrid stars and quark stars with a crystalline color superconducting (CCS) quark matter core. By modeling background shear as pressure anisotropy within a relativistic elasticity framework and using the CSS parametrization for the quark matter equation of state, the authors compute $I$, $\lambda_2$, and $Q$ under slow rotation and weak tidal deformations, across variations in the CCS parameters. They find that the $I$-$\lambda_2$-$Q$-$C$ relations remain robust to within about $2\%$ for elastic HSs and about $3\%$ for QSs, with $C$-related relations staying consistent with fluid-model uncertainties; the presence of QM rigidity introduces systematic shifts but preserves universality. The results are shown to be reasonably insensitive to the NM equation of state (APR vs STOS) within the CSS framework and to CSS parameter variations. This supports the use of universal relations as reliable probes of the internal composition of compact stars, including those with elastic crystalline quark matter cores.

Abstract

Some compact stars may contain deconfined quark matter, forming hybrid stars or quark stars. If the quark matter forms an inhomogeneous condensate in the crystalline color superconducting phase, its rigidity may be high enough to noticeably alter the stellar properties. In this paper, we investigate whether these elastic stars follow the universal relations, i.e., relations insensitive to equations of state, that have been well established for fluid stars. We improve upon previous studies by allowing quark matter in the background, static, and spherically symmetric configuration to be sheared. Such background shear can be treated in the form of an effective pressure anisotropy. We then calculate the moment of inertia $I$, tidal deformability $λ_2$, and spin-induced quadrupole moment $Q$ of these models with pressure anisotropy. The $I$-$λ_2$-$Q$ universal relations for the elastic hybrid (quark) star models are valid up to a variation of $\approx2\,(3)\%$, larger than that for typical fluid star models, when the maximal magnitude of quark matter shear modulus is considered in the crystalline color superconducting phase from realistic calculations. The uncertainty in universal relations related to the stellar compactness for these elastic star models, on the other hand, remain comparable to those for typical fluid star models. Our results demonstrate the validity of universal relations for hybrid stars and quark stars with a realistic degree of pressure anisotropy due to the crystalline color superconducting quark matter.

Figures (10)

• Figure 1: $M$-$R$ relations for the APR EoS model (black dashed curve) and the canonical HS model (see the main text for the definition) with different values of $\kappa$. The red dashed-dotted curve represents the fluid limit, and the blue (green) solid curves represent the elastic models with $\kappa = 7\times10^{25} (10^{26})$ cm$^{1/2}$ g$^{1/2}$ s$^{-2}$. The curves transit from translucent to opaque above $1$ M$_\odot$, and terminate at their TOV limits. The HS models satisfy the $M$-$R$ relations for the gravitational-wave event GW170817 (orange error bar) at the $90\%$ credible interval LIGOScientific:2018cki, as well as the pulsars PSR J0030+0451 and PSRJ0740+6620 (purple error bars) measured by NICERMiller:2019cacSalmi:2024aum at the $68\%$ credible intervals. The subplot in the upper-right corner zooms in the $M$-$R$ relations for the HS models near the TOV limits.
• Figure 2: $C$, $\bar{I}$, $\bar{\lambda}_2$, and $\bar{Q}$ versus $p_c$ for the models displayed in Fig. \ref{['fig: M-R, HS, vary kappa']}. The curves disperse starting from $p_\text{trans} = 6\times10^{33}{\text{\,erg\,cm$^{-3}$}}$ at which the QM cores appear. The canonical HS model with $p_c = p_\text{trans}$ has a mass of $\approx0.2$ M$_\odot$. Therefore, all the HS models above $1$ M$_\odot$ in the plots consist of a QM core and a NM envelope.
• Figure 3: Upper panels: universal relations for the models displayed in Fig. \ref{['fig: M-R, HS, vary kappa']}. Lower panels: fractional relative difference in the universal relations between the models and the fitting formulae Yagi:2016bkt. The brown solid curves represent the fitting formulae, and the maximal variability of the fitting formulae for fluid NSs ($I$-${\lambda}_2$ relations: $0.7\%$; $I$-$Q$ relations: $2\%$; $Q$-${\lambda}_2$ relations: $2\%$; $I$-$C$ relations: $10\%$; $C$-${\lambda}_2$ relations: $7\%$; $Q$-$C$ relations: $10\%$) is further shaded in the lower panels. Unlike most of the other literature, we do not take the absolute value of the fractional relative difference in the lower panels, so that the systematic shifts due to QM shear modulus can be read off more easily.
• Figure 4: $M$-$R$ relations (left plot) and $I$-$Q$ relations (right plot) for the APR EoS model (black dashed curve) and the HS model with different values of $p_\text{trans}$. The blue curve indicates the canonical model, and the red (green) curve is a model softer (stiffer) than the canonical one by tuning the value of $p_\text{trans}$. The dashed-dotted curves represent the fluid limits, and the solid curves represent the elastic models with $\kappa=7\times10^{26}$ cm$^{1/2}$ g$^{1/2}$ s$^{-2}$. The brown solid curves and the shaded region in the lower panel of the right plot carry the same meaning as described in Fig. \ref{['fig: uni, HS, vary kappa']}.
• Figure 5: Same as Fig. \ref{['fig: uni, HS, vary ptrans']}, but for the HS model with different values of $\Delta\rho$.