Two-layer compact stars with crystalline quark matter: Screening effect on the tidal deformability

It is well known that the tidal deformability of a compact star carries important information about the interior equation of state (EOS) of the star. The first gravitational-wave event GW170817 from a binary compact star merger observed by the LIGO/VIRGO detectors has already put limits on the tidal deformability and provided constraints on the ultra-high nuclear density EOS. In view of this ground-breaking discovery, we revisit and extend our previous work [Phys. Rev. D 95, 101302(R) (2017)] which found that taking the effect of elasticity into account in the calculation of the tidal deformability of compact star models composed of crystalline color-superconducting (CCS) quark matter can break the universal I-Love relation discovered for fluid compact stars. In this paper, we present our formulation in detail and provide more analysis to complement our previous findings. We focus and extend the study of the screening effect on the tidal deformability, which we found previously for hybrid star models, to various theoretical two-layer compact star models. Besides solid quark stars and hybrid stars, we also consider 1) solid quark stars dressed in a thin nuclear-matter crust and 2) quark stars with a fluid quark-matter core in the color-flavor-locked phase surrounded by a solid CCS quark-matter envelope. We show that the screening effect of these two-layer models in general depends on the thickness of the envelope and the ratio between the density gap and the core density at the core-envelope interface. However, for models with a fluid envelope and a vanishingly small density gap, the screening effect remains strong even as the thickness of the envelops tends to zero if the quark-matter core has a fairly uniform density. The relevance of our study to GW170817 is also discussed. We find that quark star models which are ruled out by the observation limits on the tidal deformability can be revived if the entire quark star is in a CCS phase instead of a fluid phase, thus complicating putting constraints on the quark star EOSs. In contrast, the screening effect causes the tidal deformability of a hybrid star with a CCS quark-matter core to agree with that of a corresponding stellar model with a fluid core to within less than 1% if the core size is less than about 70% of the stellar radius. The implication is that if a hybrid star EOS model is ruled out by the observation limits on the tidal deformability, the conclusion will hold no matter whether the quark matter is in a fluid or solid state, assuming that a large solid core comparable to the stellar radius is not favored in nature. Our study advocates that the tidal deformability not only provides us with information on the EOS, but may also give insights into the multilayer structure and elastic properties of compact star models composed of CCS quark matter.

Figure 1

The density profiles of the hybrid star models of $1.4M_{\odot }$. All the models have a density gap of around 20% of the central densities at the core-envelope interface.

Figure 2

The fractional deviation in the normalized tidal deformability against the gap parameter, $\mathrm{\Delta }$, of the hybrid stars HS1, HS2, HS3 and solid quark star SQS. The parameter, ${\overline{\lambda }}_{\text{fluid}}$, here is the normalized tidal deformability of the fluid counterpart of that model with the same EOS but composed entirely of fluid as defined in the main text. The theoretical range of $\mathrm{\Delta }$ is bounded by two vertical dashed lines.

Figure 3

The I-Love relation of the hybrid star models (HS1–HS3) and solid quark star model (SQS) with gap parameter $\mathrm{\Delta }=25\mathrm{MeV}$ are shown together with the fitting formula of the universal relation in fluid compact stars given by Yagi and Yunes [57, 58].

Figure 4

The density profile of the dressed quark star model. The CCS quark-matter core (solid lines) occupies more than 95% of the total radius. The nuclear-matter crust is indicated by dotted lines.

Figure 5

The normalized tidal deformabilities of the dressed quark stars (DQS) and bare solid quark stars (SQS) with different gap parameters $\mathrm{\Delta }$ are plotted. The point with $\mathrm{\Delta }=0\mathrm{MeV}$ corresponds to the models with a fluid quark-matter core.

Figure 6

The fractional deviation in the normalized tidal deformability against the gap parameter, $\mathrm{\Delta }$, of two-layer quark stars (CFL-CCS1, CFL-CCS2, CFL-CCS3) and a solid quark star (SQS). The curve for the solid quark star is represented by dots and it nearly overlaps with that of CFL-CCS3. The models are all fixed at $1.4M_{\odot }$. The range of $\mathrm{\Delta }$ we consider is bounded by two dashed lines.

Figure 7

The I-Love relation of the two-layer quark star models tabulated in Table 3 and solid quark star models (SQS) with gap parameter $\mathrm{\Delta }=25\mathrm{MeV}$ are shown together with the fitting formula of the universal relation in fluid compact stars given by Yagi and Yunes [57, 58]. The two-layer quark star models with mass $1.4M_{\odot }$ presented in Table 3 are indicated with red crosses.

Figure 8

The screening factor plotted against the fractional core radius of the two-layer models with a fluid core and a solid envelope. The models contain no density gap at the interface.

Figure 9

The screening factor plotted against the fractional core radius of the two-layer models with a solid core and a fluid envelope. The models contain no density gap at the interface.

Figure 10

The screening factor, $\delta \overline{\lambda }/(\delta \overline{\lambda }{)}_{\max }$, against $\mathrm{\Delta }\rho /{\rho }_c$ of two-layer incompressible models with solid core densities of ${10}^{15}\mathrm{g}{\mathrm{cm}}^{-3}$ and fluid envelope densities ${\rho }_f$. The fluid envelope is set to be very thin so that we can freely adjust the density gap without altering the overall background profile.

Figure 11

The screening factor, $\delta \overline{\lambda }/(\delta \overline{\lambda }{)}_{\max }$, against the fractional core radius of incompressible stars with different compactness.

Figure 12

The normalized tidal deformability, $\overline{\lambda }$, of a $1.4M_{\odot }$ solid quark star with $a_4=0.7$, $a_2^{1/2}=0$ and $B_{\mathrm{eff}}^{1/4}=135\mathrm{MeV}$. As the gap parameter $\mathrm{\Delta }=0\mathrm{MeV}$, i.e., for a fluid quark star, $\overline{\lambda }$ exceeds the upper bound of 800 [1]. For $\mathrm{\Delta }>9\mathrm{MeV}$, $\overline{\lambda }$ is within the upper bound.