Computing nonlinearity ratios using second order black hole perturbation theory

TL;DR

This paper develops an analytical approximation for computing nonlinearity ratios involving quadratic quasinormal modes in black hole perturbation theory, validating it against numerical simulations and exploring its limitations and applications.

Contribution

The paper introduces a new analytical scheme for calculating nonlinearity ratios involving quadratic quasinormal modes, extending previous methods and analyzing their accuracy and limitations.

Findings

Excellent match with numerical simulations for certain channels.

Method can fail or give crude estimates for some channels.

Nonlinearity ratios are insensitive to regularization choices at horizon and infinity.

Abstract

We revisit an analytical approximation scheme for computing nonlinearity ratios involving quadratic quasinormal modes (QQNMs). We compute these ratios for the general case when the QQNM is not one of the linear QNMs, for the $(l,m)$ channel $(2,2) \times (2,2) \to (4,4)$. We find an excellent match with numerical simulations. We also discuss where and why the method can fail, for example, for the channel $(2,0) \times (2,0) \to (2,0)$ where we can only get crude estimates for the nonlinearity ratio. Motivated by recent studies on nonlinear ringdown at the horizon, we also compute the nonlinearity ratios at the horizon. We find that the ratio both at the horizon and infinity is insensitive to different choices of regularization of the source term in the second order perturbations. We also discuss amplitudes of QQNMs sourced by linear overtones. Finally, we discuss the issues that must be resolved within this method to do precision analysis of nonlinear ringdown.