Instability windows of relativistic r-modes

The detectability of the gravitational-wave signal from $r$-modes depends on the interplay between the amplification of the mode by the CFS instability and its damping due to dissipative mechanisms present in the stellar matter. The instability window of $r$-modes describes the region of stellar parameters (angular velocity, $\Omega$, and redshifted stellar temperature, $T^\infty$), for which the mode is unstable. In this study, we reexamine this problem in nonbarotropic neutron stars, taking into account the previously overlooked nonanalytic behavior (in $\Omega$) of relativistic $r$-modes and enhanced energy dissipation resulting from diffusion in superconducting stellar matter. We demonstrate that at slow rotation rates, relativistic $r$-modes exhibit weaker amplification by the CFS instability compared to Newtonian ones. However, their dissipation through viscosity and diffusion is significantly more efficient. In rapidly rotating neutron stars within the framework of general relativity, the amplification of $r$-modes by the CFS mechanism and their damping due to shear viscosity become comparable to those predicted by Newtonian theory. In contrast, the relativistic damping of the mode by diffusion and bulk viscosity remains significantly stronger than in the nonrelativistic case. Consequently, account for diffusion and general relativity leads to a substantial modification of the $r$-mode instability window compared to the Newtonian prediction. This finding is important for the interpretation of observations of rotating neutron stars, as well as for overall understanding of $r$-mode physics.