Constraining color flavor locked strange stars in the gravitational wave era

We perform a detailed analysis of the fundamental mode of nonradial pulsations of color flavor locked strange stars. Solving the general relativistic equations for nonradial pulsations for an equation of state derived within the MIT bag model, we calculate the frequency and the gravitational damping time of the fundamental mode for all the parametrizations of the equation of state that lead to self-bound matter. Our results show that color flavor locked strange stars can emit gravitational radiation in the optimal range for present gravitational wave detectors and that it is possible to constrain the equation of state's parameters if the fundamental oscillation mode is observed and the stellar mass is determined. We also show that the $f$-mode frequency can be fitted as a function of the square root of the average stellar density $\sqrt{M/R^3}$ by a single linear relation that fits quite accurately the results for all parametrizations of the equation of state. All results for the damping time can also be fitted as a function of the compactness $M/R$ by a single empirical relation. Therefore, if a given compact object is identified as a color flavor locked strange star these two relations could be used to determine the mass and the radius from the knowledge of the frequency and the damping time of gravitational waves from the $f$ mode.

Figure 1

Three stability windows of CFL quark matter, for $\mathrm{\Delta }=50,100,150$ MeV. The vertical solid line is the limit imposed by requiring instability of two-flavor quark matter. For each of the indicated values of $\mathrm{\Delta }$, CFL quark matter is absolutely stable if the strange quark mass $m_s$ and the bag constant $B$ lie inside the corresponding bounded region. For example, a model with $B=100$ MeV ${\mathrm{fm}}^{-3}$ and $m_s=150$ MeV is inside the stability window for $\mathrm{\Delta }=100,150$ MeV but is outside the stability window for $\mathrm{\Delta }=50$ MeV.

Figure 2

The equation of state of CFL strange quark matter for all the parametrizations presented in Table 1. The slashed line corresponds to the ultrarelativistic limit $p=\epsilon /3$.

Figure 3

Mass of CFL stars as a function of the radius $R$ and the surface gravitational redshift $Z$.

Figure 4

Frequency of the fundamental oscillation mode as a function of the stellar mass $M$. In (a) and (b) we fixed the bag constant $B$ to 60 and 100 MeV ${\mathrm{fm}}^{-3}$, respectively, and varied the pairing gap $\mathrm{\Delta }$. In (c) and (d) we fixed the $\mathrm{\Delta }$ parameter to 100 and 150 MeV, and varied the bag constant $B$. The curves are identified with the label introduced in Table 1 followed by the value of the three parameters of the EOS: ($B$ [MeV ${\mathrm{fm}}^{-3}$], $\mathrm{\Delta }$ [MeV], $m_s$ [MeV]). For comparison we show the results for hadronic stars using the relativistic mean field model EOSs GM1 and NL3 with nucleons and electrons (see, e.g., [28] and references therein).

Figure 5

The damping time of the fundamental oscillation mode as a function of the stellar mass $M$ for the same models shown in Fig. 4.

Figure 6

The frequency of the fundamental mode as a function of the square root of the average density for the different parametrizations of the EOS considered in this paper. The full yellow line is the fitting given in Eq. (30) and the dotted line is the fitting for hadronic stars given in Ref. [5].

Figure 7

The damping time of the fundamental mode as a function of the compactness $M/R$ for the different parametrizations of the EOS considered in this paper. The full yellow line is the fitting given in Eq. (31) and the dotted line is the fitting for hadronic stars given in Ref. [5].

Figure 8

Comparison with the Cowling approximation. The frequency $f_{\mathrm{exact}}$ is obtained through the full calculations of the present paper while $f_{\mathrm{Cowling}}$ is obtained within the Cowling approximation in which the metric perturbations are neglected (see [12, 32]).