Scalar fields and parametrized spherically symmetric black holes: Can one hear the shape of space-time?

Abstract

In this work we study whether parametrized spherically symmetric black hole solutions in metric theories of gravity can appear to be isospectral when studying perturbations. From a theory agnostic point of view, the test scalar field wave equation is the ideal starting point to approach the quasinormal mode spectrum in alternative black hole solutions. We use a parametrization for the metric proposed by Rezzolla and Zhidenko, as well as the higher order WKB method in the determination of the quasinormal mode spectra. We look for possible degeneracies in a tractable subset of the parameter space with respect to the Schwarzschild quasinormal modes. Considering the frequencies and damping times of the expected observationally most relevant quasinormal modes, we find such degeneracies. We explicitly demonstrate that the leading Schwarzschild quasinormal modes can be approximated by alternative black hole solutions when their mass is treated as free parameter. In practice, we conclude that the mass has to be known with extremely high precision in order to restrict the leading terms in the metric expansion to currently known limits coming from the PPN expansion. Possible limitations of using the quasinormal mode ringdown to investigate black hole space-times are discussed.