General formalism for dirty extreme-mass-ratio inspirals

Detecting the environment around the supermassive black holes and tests of general relativity are important applications of extreme-mass-ratio inspirals (EMRIs). There is still a challenge to efficiently describe various “dirty” impacts on the inspirals, such as dark matter, gas, dipole radiation, and electromagnetic interaction. In this study, we find the inherent linearity of the asymptotic solution of the inhomogeneous Teukolsky equation. On the basis of this property, we completely decouple the factors of the perturber and the background spacetime in the energy fluxes and waveforms. With the new decoupling form, the waveforms of EMRIs with non-geodesic motion in Kerr spacetime can be conveniently calculated. This will help to resolve the environment (including gas, field, dark matter, electromagnetic interaction) around supermassive black holes and test general relativity.