Ringdown nonlinearities in the eikonal regime

TL;DR

This paper investigates nonlinear effects in black hole quasinormal modes at high multipole numbers, revealing that quadratic mode amplitudes tend to a finite ratio with linear modes, challenging recent theoretical proposals.

Contribution

It provides the first numerical analysis of quadratic quasinormal modes for large multipole numbers, clarifying their behavior and impact on black hole perturbation theory.

Findings

Quadratic to linear mode amplitude ratio converges for large b5.

The b5 d7 b5 d7 2b5 channel converges to a finite value.

No breakdown of perturbation theory for realistic initial data.

Abstract

The eikonal limit of black hole quasinormal modes (the large multipole limit $\ell \gg 1$) can be realized geometrically as a next-to-leading order solution to the geometric optics approximation, and also as linear fluctuations about the Penrose limit plane wave adapted to the lightring. Extending this interpretation beyond the linear order in perturbation theory requires a robust understanding of quadratic quasinormal modes for large values of $\ell$. We analyze numerically the relative excitation of quadratic to linear quasinormal modes of Schwarzschild black holes, with two independent methods. Our results suggest that the ratio of quadratic to linear amplitudes for the $\ell \times \ell \to 2\ell$ channel converges towards a finite value for large $\ell$, in sharp contrast with a recent proposal inspired by the Penrose limit perspective. On the other hand, the $2 \times \ell \to \ell + 2$ channel seems to have a linearly growing ratio. Nevertheless, we show that there is no breakdown of black hole perturbation theory for physically realistic initial data.