Numerical analysis of quasinormal modes in nearly extremal Schwarzschild-de Sitter spacetimes

TL;DR

This paper investigates high-order quasinormal modes in nearly extremal Schwarzschild-de Sitter spacetimes, revealing the limitations of existing analytical formulas and oscillatory behaviors in highly damped modes for electromagnetic and gravitational perturbations.

Contribution

It provides a detailed numerical analysis of high-order quasinormal modes, highlighting the inaccuracy of extremal limit formulas for highly damped modes and discovering oscillatory behaviors.

Findings

Analytical approximation formulas are accurate for low-order modes with small $r_1\kappa_1$.

High-order modes with large imaginary parts deviate from analytical predictions.

Real parts of frequencies oscillate in highly damped modes, with different behaviors for electromagnetic and gravitational perturbations.

Abstract

We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasinormal frequencies with large imaginary parts, on the other hand, this formula becomes inaccurate even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes, which are similar to those observed in the case of a Reissner-Nordstr{\" o}m black hole. The amplitude of oscillating ${\rm Re(\omega)}$ as a function of ${\rm Im}(\omega)$ approaches a non-zero constant value for gravitational perturbations and zero for electromagnetic perturbations in the limit of highly damped modes, where $\omega$ denotes the quasinormal frequency. This means that for gravitational perturbations, the real part of quasinormal modes of the nearly extremal Schwarzschild-de Sitter spacetime appears not to approach any constant value in the limit of highly damped modes. On the other hand, for electromagnetic perturbations, the real part of frequency seems to go to zero in the limit.