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 parameterized metrics to each other
via a diffeomorphism and to known analytical black hole solutions in
alternative theories of gravity. The parameterized metrics presented here serve
as frameworks for the systematic calculation of extreme-mass ratio inspiral
waveforms in parameterized non-GR theories and the investigation of the
accuracy to which space-borne gravitational wave detectors can constrain such
deviations.Comment: 17 pages, replaced with version published in Phys. Rev.
