Parametrized quasinormal mode framework for modified Teukolsky equations

Modifications to general relativity lead to effects in the spectrum of quasinormal modes of black holes. In this paper, we develop a parametrized formalism to describe deviations from general relativity in the Teukolsky equation, which governs linear perturbations of spinning black holes. We do this by introducing a correction to the effective potential of the Teukolsky equation in the form of a 1/r expansion controlled by free parameters. The method assumes that a small deviation in the effective potential induces a small modification in the spectrum of modes and in the angular separation constants. We isolate and compute the universal linear contribution to the quasinormal mode frequencies and separation constants in a set of coefficients and test them against known examples in the literature (massive scalar field, Dudley-Finley equation and higher-derivative gravity). We make the coefficients publicly available for relevant overtone, angular momentum, and azimuthal numbers of modes and different values of the black hole spin.