Nonlinearities in Black Hole Ringdowns

The gravitational wave strain emitted by a perturbed black hole (BH) ringing down to a steady state is typically modeled analytically using first-order BH perturbation theory. We show, however, that second-order effects are necessary for accurately modeling ringdowns from BH merger simulations. Focusing on the strain's $(ell,m)=(4,4)$ angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic $(4,4)$ mode's amplitude exhibits a quadratic scaling with the fundamental $(2,2)$ mode---its parent mode. The nonlinear mode's amplitude is comparable to or even larger than that of the linear $(4,4)$ mode. Therefore correctly modeling the ringdown of the strain's higher harmonics---improving mode mismatches by two orders of magnitude---requires the inclusion of nonlinearities.