Analytic estimates of quasi-normal mode frequencies for black holes in General Relativity and beyond

TL;DR

This paper reviews and extends the eikonal technique for analytically estimating black hole quasi-normal mode frequencies within General Relativity and scalar Gauss-Bonnet gravity, demonstrating accuracy especially at small coupling constants.

Contribution

It introduces an extension of the eikonal method to theories beyond General Relativity, including scalar Gauss-Bonnet gravity, accounting for coupled metric and scalar perturbations.

Findings

Analytic estimates match numerical results well at small coupling.

The method captures the breaking of isospectrality in extended theories.

Extension of the eikonal technique to complex theories is validated.

Abstract

In this chapter, we review the eikonal technique to analytically derive approximate quasi-normal mode frequencies of black holes. We first review the procedure in General Relativity and extend it to theories beyond General Relativity. As an example of the latter, we focus on scalar Gauss-Bonnet gravity in which a scalar field can couple to the Gauss-Bonnet invariant in an arbitrary way in the action. In this theory, metric and scalar perturbations are coupled in the polar sector, forcing the isospectrality to break and making the eikonal calculation more complex. We show that the analytic estimates can accurately capture the numerical behavior of the quasi-normal mode frequencies, especially when the coupling constant of the theory is small.