Evolving extreme mass-ratio inspirals in a perturbed Schwarzschild spacetime

In this work, we develop the modified Teukolsky formalism that describes the GW radiation from a point mass orbiting around a perturbed Schwarzschild BH. This perturbation of the background spacetime induces a secular change in the orbital phase of the point mass. In turn, this causes a modification in the GW flux, which can be used to probe the background spacetime. We explicitly apply this formalism to a bumpy Schwarzschild spacetime as a proof of principle. The results pave the way for the description of EMRIs in generic perturbed Kerr spacetime in future developments.