Quadratic Gravitational-Wave Scattering by Kerr Black Holes

Abstract: Second-order black-hole perturbation theory will be an important component of precision gravitational-wave modeling and tests of strong-field gravity with next-generation detectors. Beyond quasinormal modes (QNMs), whose quadratic interactions have been the focus of recent work in black-hole spectroscopy, a generic retarded solution consists of a continuum of real-frequency scattering states whose nonlinear interactions in Kerr remain comparatively less explored. We use numerical relativity to study the quadratic response of Kerr black holes to nearly monochromatic $(\ell,m)=(2,\pm2)$ incident gravitational waves, measuring the complex self-coupling to the outgoing $(\ell,m)=(4,4)$ daughter at twice the parent frequency for black-hole spins up to $a=0.95$. At low frequencies, the coupling is strongly suppressed and exhibits opposite spin dependence for prograde and retrograde scattering. At higher prograde frequencies, we identify a daughter-mode resonance whose frequency and width extracted from the nonlinear response track the fundamental $(\ell,m,n)=(4,4,0)$ QNM. Near the fundamental $(2,2,0)$ QNM frequency, the in-mode coupling grows with spin, in contrast to the decreasing quadratic QNM coupling, demonstrating that the nonlinear response depends on the full parent scattering state rather than on its frequency alone. A complementary semi-analytic second-order Teukolsky calculation reproduces the nonlinear response from our numerical relativity simulations. Our numerical-relativity results and their semi-analytic extension provide a basis for generic homogeneous radiative perturbations of Kerr at second order, with applications to dynamical tides, near-extremal dynamics, and second-order gravitational self-force theory.