Unstable quasi-normal modes in anisotropic neutron stars

Stellar pulsation modes carry information about the interior structure of a star and serve as a way to probe the properties of matter under extreme conditions, like in the neutron star core. In general relativity, the non-radial modes lose energy over time due to the emission of gravitational waves and are therefore described as quasi-normal modes, i.e., confined waves in an open system. These modes are characterized by complex frequencies, where the sign of the imaginary part determines the stability of the mode: either an exponential decay (stable) or an exponential growth (unstable). It has been shown that in an isotropic star, all unstable modes have no oscillations and cannot cause outgoing gravitational radiation. However, this is not proven for the anisotropic case, in which the fluid pressure has a directional dependence. In this talk, I will present our work on a numerical computation of the quasi-normal mode frequencies of an anisotropic neutron star. We derive the pulsation formalism from perturbed Einstein field equations, which we verify with existing results in various known limits. In particular, the real part of the f-mode frequencies deviates from that of the relativistic Cowling approximation by 10%, while the p-mode deviations decrease below 1% as we go higher in the mode order. The trend of this deviation agrees with the known isotropic cases. We then apply the formalism to several anisotropic neutron star models. Our results show that some p-modes become unstable as the anisotropy increases, while they still have non-zero oscillation frequencies. This suggests we have oscillatory unstable modes. I will further discuss the possible implications of such instabilities.