# Parametrized black hole quasinormal ringdown. II. Coupled equations and quadratic corrections for nonrotating black holes

## TL;DR

This paper extends previous analyses of black hole quasinormal modes by including second-order perturbative corrections and coupled wave equations, providing a practical method to compute modes in modified gravity theories.

## Contribution

It introduces a framework for second-order and coupled perturbation corrections in black hole quasinormal mode calculations, applicable to various modified gravity models.

## Key findings

- Derived second-order corrections for quasinormal frequencies.
- Addressed coupled wave equations with degenerate spectra.
- Demonstrated applicability to multiple gravity theories.

## Abstract

Linear perturbations of spherically symmetric spacetimes in general relativity are described by radial wave equations, with potentials that depend on the spin of the perturbing field. In previous work we studied the quasinormal mode spectrum of spacetimes for which the radial potentials are slightly modified from their general relativistic form, writing generic small modifications as a power-series expansion in the radial coordinate. We assumed that the perturbations in the quasinormal frequencies are linear in some perturbative parameter, and that there is no coupling between the perturbation equations. In general, matter fields and modifications to the gravitational field equations lead to coupled wave equations. Here we extend our previous analysis in two important ways: we study second-order corrections in the perturbative parameter, and we address the more complex (and realistic) case of coupled wave equations. We highlight the special nature of coupling-induced corrections when two of the wave equations have degenerate spectra, and we provide a ready-to-use recipe to compute quasinormal modes. We illustrate the power of our parametrization by applying it to various examples, including dynamical Chern-Simons gravity, Horndeski gravity and an effective field theory-inspired model.