Long-term evolution of CFS-unstable neutron stars and the role of differential rotation on short time-scales

I consider differential rotation, associated with radiation-driven Chandrasekhar–Friedman–Schutz (CFS) instability, and respective observational manifestations. I focus on the evolution of the apparent spin frequency, which is typically associated with the motion of a specific point on the stellar surface (e.g. polar cap). I start from long-term evolution (on the time-scale when instability significantly changes the spin frequency). For this case, I reduce the evolution equations to one differential equation and I demonstrate that it can be directly derived from energy conservation law. This equation governs the evolution rate through a sequence of thermally equilibrium states and it provides linear coupling for the cooling power and rotation energy losses via gravitational wave emission. In particular, it shows that differential rotation does not affect long-term spin-down. In contrast, on short time-scales, differential rotation can significantly modify the apparent spin-down, if we examine a strongly unstable star with a very small initial amplitude for the unstable mode. This statement is confirmed by considering a Newtonian non-magnetized perfect fluid and dissipative stellar models as well as a magnetized stellar model. For example, despite the fact that the widely applied evolution equations predict effective spin to be constant in the absence of dissipation, the CFS-unstable star should be observed as spinning-down. However, the effects of differential rotation on apparent spin-down are negligible for realistic models of neutron star recycling, unless the neutron star is non-magnetized, the r-mode amplitude is modulated faster than the shear viscosity dissipation time-scale, and the amplitude is large enough that spin-down can be measured on a modulation time-scale.