Modeling extreme mass-ratio inspiral within an ultralight scalar cloud

Extreme mass-ratio inspirals (EMRIs), with their hundreds of thousands of orbital cycles, offer an important probe of the spacetime geometry and the astrophysical environment surrounding supermassive black holes. Modeling EMRI waveforms in such scenarios was limited to non-rotating black holes in the past. Recent advances have enabled a perturbative treatment of these effects via a two-parameter expansion in the mass ratio and the amplitude of modified-gravity or environmental effects. Within this framework, a modified Teukolsky formalism (MTF) has been developed to describe EMRI dynamics in non-vacuum environments, such as ultralight scalar clouds and thin matter shells. In our previous work, we applied the MTF to compute the scalar radiation emitted by a complex ultralight scalar cloud—formed through superradiance around a spinning supermassive black hole, when perturbed by a secondary in circular, equatorial orbit. In this talk, we present the first complete treatment of this system by extending the MTF to calculate the additional gravitational radiation induced by the cloud. We show how to extract the corresponding energy and angular-momentum fluxes at the horizon and at null infinity from the modified Teukolsky equations, and clarify how this procedure differs from that in vacuum general relativity. By combining the scalar and gravitational fluxes, we quantify the resulting EMRI waveform dephasing and assess the detectability of this effect with future space-based gravitational-wave detectors.