Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes

TL;DR

This paper derives a full asymptotic description of quasi-normal modes for Kerr-de Sitter black holes, revealing mode splitting and confirming their interpretation as oscillation frequencies and decay rates.

Contribution

It establishes a Bohr-Sommerfeld type condition for these modes and demonstrates their asymptotic accuracy, including mode splitting due to broken spherical symmetry.

Findings

Asymptotic description of quasi-normal modes for Kerr-de Sitter black holes.

Observation of Zeeman-like splitting of modes at zero rotation.

Validation of mode-based expansion for wave solutions and decay rates.

Abstract

We establish a Bohr-Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr-de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a=0 (Schwarzschild-de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.