Scalar Fields and Parametrized Spherically Symmetric Black Holes: Can one hear the shape of space-time?

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 quasi-normal mode spectrum in alternative black hole solutions. We use a parametrization for the metric proposed by Rezzolla and Zhidenko, as well as the the higher order WKB method in the determination of the quasi-normal mode spectra. We look for possible degeneracies in a tractable subset of the parameter space with respect to the Schwarzschild quasi-normal modes. Considering the frequencies and damping times of the expected observationally most relevant quasi-normal modes, we find such degeneracies. We explicitly demonstrate that the leading Schwarzschild quasi-normal 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 quasi-normal mode ringdown to investigate black hole space-times are being discussed.