An analytic approach to quasinormal modes for coupled linear systems

TL;DR

This paper develops an analytic approximation method for calculating quasinormal modes in coupled linear systems, extending previous approaches and validated through a toy model with high accuracy.

Contribution

It introduces a novel analytic scheme for quasinormal mode spectra in coupled systems, generalizing Schutz and Will's method beyond WKB analysis.

Findings

Analytic approximation agrees with numerical results within sub-percent accuracy for the fundamental mode.

The method is effective even for strong coupling with mixing parameter of order one.

Validated using a controllable toy model demonstrating high precision.

Abstract

Quasinormal modes describe the ringdown of compact objects deformed by small perturbations. In generic theories of gravity that extend General Relativity, the linearized dynamics of these perturbations is described by a system of coupled linear differential equations of second order. We first show, under general assumptions, that such a system can be brought to a Schr\"odinger-like form. We then devise an analytic approximation scheme to compute the spectrum of quasinormal modes. We validate our approach using a toy model with a controllable mixing parameter $\varepsilon$ and showing that the analytic approximation for the fundamental mode agrees with the numerical computation when the approximation is justified. The accuracy of the analytic approximation is at the (sub-) percent level for the real part and at the level of a few percent for the imaginary part, even when $\varepsilon$ is of order one. Our approximation scheme can be seen as an extension of the approach of Schutz and Will to the case of coupled systems of equations, although our approach is not phrased in terms of a WKB analysis, and offers a new viewpoint even in the case of a single equation.