Mutual Friction in a Cold Color-Flavor-Locked Superfluid and $r$-Mode Instabilities in Compact Stars

Dissipative processes acting in rotating neutron stars are essential in preventing the growth of the $r$-mode instability. We estimate the damping time of $r$ modes of a hypothetical compact quark star made up by color-flavor-locked quark matter at a temperature $T\lesssim 0.01\mathrm{MeV}$. The dissipation that we consider is due to the mutual friction force between the normal and the superfluid component arising from the elastic scattering of phonons with quantized vortices. This process is the dominant one for temperatures $T\lesssim 0.01\mathrm{MeV}$, where the mean free path of phonons due to their self-interactions is larger than the radius of the star. We find that $r$-mode oscillations are efficiently damped by this mechanism for pulsars rotating at frequencies of the order of 1 Hz at most. Our analysis rules out the possibility that cold pulsars rotating at higher frequencies are entirely made up by color-flavor-locked quark matter.