Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole

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 $\mathcal{R}\simeq 0.154e^{-0.068i}$ for the most excited $\mathrm{\ell }=4$ mode, matching results from Numerical Relativity. We also predict new low-frequency $\mathrm{\ell }=2$ quadratic modes.