Nonlinear Quasi-Normal Modes: Uniform Approximation

Abstract

Recent works have suggested that nonlinear (quadratic) effects in black hole perturbation theory may be important for describing a black hole ringdown. We show that the technique of uniform approximations can be used to accurately compute 1) nonlinear amplitudes at large distances in terms of the linear ones, 2) linear (and nonlinear) quasi-normal mode frequencies, 3) the wavefunction for both linear and nonlinear modes. Our method can be seen as a generalization of the WKB approximation, with the advantages of not losing accuracy at large overtone number and not requiring matching conditions. To illustrate the effectiveness of this method we consider a simplified source for the second-order Zerilli equation, which we use to numerically compute the amplitude of nonlinear modes for a range of values of the angular momentum number.

17 pages + appendices, v2: minor improvements, added eq. (4.20), matches journal version

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• Quasinormal modes and their excitation beyond general relativity
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• Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole
• Parametrized Black Hole Quasinormal Ringdown Formalism for Higher Overtones
• Amplitudes and Polarizations of Quadratic Quasi-Normal Modes for a Schwarzschild Black Hole
• Black-hole spectroscopy: quasinormal modes, ringdown stability and the pseudospectrum
• Iterative extraction of overtones from black hole ringdown
• Bounds on tidal charges from gravitational-wave ringdown observations
• Systematic bias in LISA ringdown analysis due to waveform inaccuracy
• The second-order quasi-normal modes for AdS black branes
• Numerical computation of electromagnetically sourced nonlinear tails