Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole

TL;DR

This paper investigates quadratic quasi-normal modes of Schwarzschild black holes, focusing on their amplitudes and frequencies, with implications for testing gravity through gravitational wave observations.

Contribution

Introduces regular variables and computes amplitudes for quadratic modes using the Leaver algorithm, including new low-frequency modes and matching numerical relativity results.

Findings

Nonlinear ratio for the $ ext{l}=4$ mode is approximately 0.154e^{-0.068i}

Amplitudes of quadratic modes are computed and match numerical relativity

Predicts new low-frequency $ ext{l}=2$ quadratic modes

Abstract

Quadratic quasi-normal modes, generated at second order in black hole perturbation theory, are a promising target for testing gravity in the nonlinear regime with next-generation gravitational wave detectors. While their frequencies have long been known, their amplitudes remain poorly studied. We introduce regular variables and compute amplitudes for Schwarzschild black holes with the Leaver algorithm. We find a nonlinear ratio $\mathcal{R}\simeq0.154e^{-0.068i}$ for the most excited $\ell=4$ mode, matching results from Numerical Relativity. We also predict new low-frequency $\ell=2$ quadratic modes.