# Oscillations and instabilities of fast and differentially rotating relativistic stars

## TL;DR

This study investigates how differential rotation affects the oscillation modes and stability of relativistic stars, revealing that small degrees of differential rotation can increase the stability threshold against the CFS instability.

## Contribution

It provides new insights into the impact of differential rotation on the stability and oscillation properties of relativistic stars, using relativistic polytrope models in the Cowling approximation.

## Key findings

- Small differential rotation raises the critical rotation for CFS instability.
- Differential rotation increases the critical T/|W| at mass-shedding limit.
- High differential rotation can lower the neutral point of CFS instability for highly compact stars.

## Abstract

We study non-axisymmetric oscillations of rapidly and differentially rotating relativistic stars in the Cowling approximation. Our equilibrium models are sequences of relativistic polytropes, where the differential rotation is described by the relativistic $j$-constant law. We show that a small degree of differential rotation raises the critical rotation value for which the quadrupolar f-mode becomes prone to the CFS instability, while the critical value of $T/|W|$ at the mass-shedding limit is raised even more. For softer equations of state these effects are even more pronounced. When increasing differential rotation further to a high degree, the neutral point of the CFS instability first reaches a local maximum and is lowered afterwards. For stars with a rather high compactness we find that for a high degree of differential rotation the absolute value of the critical $T/|W|$ is below the corresponding value for rigid rotation. We conclude that the parameter space where the CFS instability is able to drive the neutron star unstable is increased for a small degree of differential rotation and for a large degree at least in stars with a higher compactness.