Bumpy black holes in alternative theories of gravity

We generalize the bumpy black hole framework to allow for alternative theory deformations. We construct two model-independent parametric deviations from the Kerr metric: one built from a generalization of the quasi-Kerr and bumpy metrics and one built directly from perturbations of the Kerr spacetime in Lewis-Papapetrou form. We find the conditions that these ``bumps'' must satisfy for there to exist an approximate second-order Killing tensor so that the perturbed spacetime still possesses three constants of the motion (a deformed energy, angular momentum and Carter constant) and the geodesic equations can be written in first-order form. We map these parametrized metrics to each other via a diffeomorphism and to known analytical black hole solutions in alternative theories of gravity. The parametrized metrics presented here serve as frameworks for the systematic calculation of extreme mass-ratio inspiral waveforms in parametrized non-general relativity theories and the investigation of the accuracy to which space-borne gravitational wave detectors can constrain such deviations.