The Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole

Black hole spectroscopy probes the spectrum of quasi-normal modes (QNMs) during the ringdown phase of a merger. Future detectors will be sensitive not only to linear QNMs, but also to non-linear excitations known as quadratic QNMs (QQNMs). These contributions must be understood for accurate waveform modelling and precision tests of gravity. We present a perturbative framework that yields QQNM amplitudes and phases for arbitrary angular momentum, parity, and overtone number in the Schwarzschild case. The relative amplitude of a quadratic mode to its linear parents is a universal prediction of black hole perturbation theory, independent of the initial perturbation up to a known dependence on parity, and in some cases large enough to be detectable. We reproduce known results and predict new quadratic modes of potential observational interest. In the eikonal limit, the structure of these ratios simplifies, as does the well studied linear frequency spectrum.