Systematic High Overtone Expansion of Kerr Quasinormal Modes

Abstract

We develop a systematic method to compute the high overtone expansion of Kerr quasinormal modes on normal angular branches to arbitrary order. We first obtain the expansion of the angular eigenvalue at large frequencies by a simple polynomial recursion. Using this result in an alternative form of the radial Teukolsky equation, we find that all the WKB corrections are defined on a fixed genus-one spectral curve, and that the quantum periods at the second and higher orders are expressed in terms of only two elliptic periods. This allows us to determine the expansion coefficients order by order. We confirm that the truncated expansions reproduce quasinormal frequencies obtained by solving the angular and radial equations directly, for scalar, electromagnetic and gravitational fields. We also compare the scalar results with the exact WKB analysis of extremal Kerr black holes.