Compact Stars with Sequential QCD Phase Transitions

Compact stars may contain quark matter in their interiors at densities exceeding several times the nuclear saturation density. We explore models of such compact stars where there are two first-order phase transitions: the first from nuclear matter to a quark-matter phase, followed at a higher density by another first-order transition to a different quark-matter phase [e.g., from the two-flavor color-superconducting (2SC) to the color-flavor-locked (CFL) phase]. We show that this can give rise to two separate branches of hybrid stars, separated from each other and from the nuclear branch by instability regions, and, therefore, to a new family of compact stars, denser than the ordinary hybrid stars. In a range of parameters, one may obtain twin hybrid stars (hybrid stars with the same masses but different radii) and even triplets where three stars, with inner cores of nuclear matter, 2SC matter, and CFL matter, respectively, all have the same mass but different radii.

Figure 1

Schematic plot showing how we parametrize the EOS of dense matter with two phase transitions to two quark-matter phases. For convenience and specificity, we call the first quark-matter phase 2SC and the second CFL.

Figure 2

The stellar mass as a function of the star’s central pressure for four different values of $\mathrm{\Delta }{ϵ}_2$. The other parameters of the EOS are fixed at $P_1=1.7\times {10}^{35}\mathrm{dyn}{\mathrm{cm}}^{-2}$, $s_1=0.7$, $\mathrm{\Delta }{ϵ}_{2\mathrm{SC}}/{ϵ}_1=0.27$, $\mathrm{\Delta }{ϵ}_1/{ϵ}_1=0.6$, and $s_2=1$. The vertical dotted lines mark the two phase transitions at $P_1$ and $P_2$. Stable branches are solid lines; unstable branches are dashed lines. We see the emergence of separate 2SC and CFL hybrid branches along with the occurrence of triplets.

Figure 3

The $M\text{−}R$ relations for the parameter values defined in Fig. 2. We have fixed the properties of the $\mathrm{nuclear}\to 2\mathrm{SC}$ transition and the speed of sound in 2SC and CFL matter. For the $2\mathrm{SC}\to \mathrm{CFL}$ transition, we have fixed the critical pressure, and we vary the energy-density discontinuity $\mathrm{\Delta }{ϵ}_2$. The separate 2SC and CFL hybrid branches are clearly visible, along with the occurrence of triplets.

Figure 4

The profiles (here the log of pressure as a function of the internal radius) of the three members of a triplet with masses $M=1.975M_{\odot }$. Here “$N$” means the nuclear phase. The parameter values are the same as in Fig. 2, with $\mathrm{\Delta }{ϵ}_2/\mathrm{\Delta }{ϵ}_1=0.23$.

Figure 5

The $M\text{−}R$ relation for a less stiff 2SC phase ($s_1=0.5$) with four different values of $\mathrm{\Delta }{ϵ}_2$, keeping $\mathrm{\Delta }{ϵ}_1/{ϵ}_1=0.6$. The 2SC branch is shorter, but there can still be separate 2SC and CFL hybrid branches and triplets (the corresponding region is magnified in the figure inset).