Mode coupling of Schwarzschild perturbations: Ringdown frequencies

TL;DR

This study investigates whether nonlinearities affect black hole ringdown frequencies and finds that second-order perturbations decay at the same frequencies as linear theory, simplifying gravitational wave modeling.

Contribution

The paper demonstrates through high-accuracy simulations that nonlinear perturbations do not alter the fundamental decay frequencies of black hole ringdowns, confirming the adequacy of linear quasinormal modes.

Findings

Second-order perturbation decay frequencies match linear theory.

Nonlinear effects do not modify the fundamental ringdown frequencies.

Standard quasinormal modes suffice for modeling nonlinear gravitational wave signals.

Abstract

Within linearized perturbation theory, black holes decay to their final stationary state through the well-known spectrum of quasinormal modes. Here we numerically study whether nonlinearities change this picture. For that purpose we study the ringdown frequencies of gauge-invariant second-order gravitational perturbations induced by self-coupling of linearized perturbations of Schwarzschild black holes. We do so through high-accuracy simulations in the time domain of first and second-order Regge-Wheeler-Zerilli type equations, for a variety of initial data sets. We consider first-order even-parity $(\ell=2,m=\pm 2)$ perturbations and odd-parity $(\ell=2,m=0)$ ones, and all the multipoles that they generate through self-coupling. For all of them and all the initial data sets considered we find that ---in contrast to previous predictions in the literature--- the numerical decay frequencies of second-order perturbations are the same ones of linearized theory, and we explain the observed behavior. This would indicate, in particular, that when modeling or searching for ringdown gravitational waves, appropriately including the standard quasinormal modes already takes into account nonlinear effects.