Postadiabatic waveforms from extreme mass ratio inspirals in the presence of dark matter

Extreme mass-ratio inspirals (EMRIs), in which a solar mass compact object is whirling around a supermassive black hole, act as precise tracers of the spacetime geometry and astrophysical environment around the supermassive black hole. These systems are highly sensitive to even the smallest deviations from the vacuum general relativity scenario. However, detecting these signals requires highly accurate waveform modeling up to the first post-adiabatic order, incorporating self-force effects, system parameters, and environmental influences. In this paper, we focus on the impact of dark matter on gravitational waveforms. Cold dark matter in galactic centers can be redistributed by the gravitational pull of a supermassive black hole, forming a dense, spike-like profile. When an EMRI evolves in such an environment, the interaction between the binary and the surrounding dark matter can leave distinctive imprints on the emitted waveform, and thus offer a novel way to probe the nature and distribution of dark matter. We specifically examine how dark matter modifies the background spacetime. By treating these modifications perturbatively, we present a framework to incorporate dark matter environmental effects into gravitational waveform modeling at the first post-adiabatic order.