Gravitational wave echoes from interacting quark stars

We show that interacting quark stars (IQSs) composed of interacting quark matter (IQM), including the strong interaction effects such as perturbative QCD corrections and color superconductivity, can be compact enough to feature a photon sphere that is essential to the signature of gravitational wave echoes. We utilize an IQM equation of state unifying all interacting phases by a simple reparametrization and rescaling, through which we manage to maximally reduce the number of degrees of freedom into one dimensionless parameter $\bar{\lambda }$ that characterizes the relative size of strong interaction effects. It turns out that gravitational wave echoes are possible for IQSs with $\bar{\lambda }\gtrsim 10$ at large center pressure. Rescaling the dimension back, we illustrate its implication on the dimensional parameter space of effective bag constant $B_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and the superconducting gap $\mathrm{\Delta }$ with variations of the perturbative QCD parameter $a_4$ and the strange quark mass $m_s$. We calculate the rescaled GW echo frequencies ${\bar{f}}_{\text{echo}}$ associated with IQSs, from which we obtain a simple scaling relation for the minimal echo frequency $f_$\text{echo}$^{\rm min}\approx 5.76 {\sqrt{B_{\rm eff}/$\text{(100 MeV)}$^4}} ,,, \rm kHz$atthelarge$\bar{\lambda}$ limit.