Amplitudes and Polarizations of Quadratic Quasi-Normal Modes for a Schwarzschild Black Hole

TL;DR

This paper calculates the amplitudes and polarizations of quadratic quasi-normal modes for Schwarzschild black holes, revealing their dependence on linear modes and fundamental parameters, with implications for gravitational wave analysis.

Contribution

It provides the first detailed computation of quadratic mode amplitudes and polarizations, showing their relation to linear modes and introducing four fundamental parameters.

Findings

Quadratic modes are determined by linear mode amplitudes and polarizations.

The ratio of quadratic to linear amplitudes depends on four fundamental numbers.

Two selection rules govern the vanishing of certain quadratic modes.

Abstract

General Relativity predicts the existence of quadratic quasi-normal modes at second order in perturbation theory. Building on our recent work, we compute the amplitudes and polarizations of these modes for non-rotating black holes, showing that they are completely determined by the amplitudes and polarizations of linear modes. We obtain the ratio of quadratic to linear amplitudes, which still depends on the initial conditions of the merger through the polarization of linear modes. However, we demonstrate that this dependence is captured by four fundamental numbers, independent of initial conditions, representing four different combinations of linear modes parities. Additionally, we prove two selection rules regarding the vanishing of classes of quadratic modes. Our results are available online as a package which provides the ratio of amplitudes across a broad spectrum of angular momenta.