Superradiance and quasinormal modes of the gravitational perturbation around rotating hairy black hole

The No Hair theorem in classical general relativity predicts that rotating black holes are specified by the Kerr metric, which is uniquely identified by the mass and spin. However, as a pioneering study beyond general relativity, the rotating hairy black hole has been proposed, which encompasses the Kerr black hole as a special case. In these black holes, there are extra hair which could appear due to the additional surrounding sources such as dark matter or dark energy. In this work, we study the phenomenology of the rotating hairy black hole in terms of gravitational perturbations. In particular, the superradiance and the quasinormal modes. Using the matching-asymptotic method, we derive the amplification factor and the superradiance conditions. We also calculate the quasinormal modes using the continued fraction method. The results are in very good agreement with previous studies in the Kerr limit. We also show how the amplification and quasinormal modes will shift in response to variations in the hairy parameters, black hole spin, and quantum numbers.