Quadratic Mode Couplings in Rotating Black Holes and Their Detectability

Quadratic quasinormal modes encode fundamental properties of black hole spacetimes. They are also one of the key ingredients of nonlinearities of general relativity in the ringdown stage of binary black hole coalescence. In this Letter, we classify all possible quadratic coupling channels of quasinormal modes for a generic Kerr black hole and use a frequency-domain pseudospectral code with hyperboloidal slicing to calculate these couplings. After accounting for all the channels in systems with reflection symmetry, our results become consistent with those extracted from numerical simulations and time-domain fits. This agreement provides a compelling example demonstrating the success of black hole second-order perturbation theory. We also explore potential applications of our calculations in future ringdown data analysis by carrying out a detectability survey for various quadratic modes. We find that a few of them are observationally relevant for third-generation ground-based detectors like Cosmic Explorer, as well as the spaceborne Laser Interferometer Space Antenna.