We investigate the influence of dark matter halos surrounding supermassive black holes on the gravitational waves emitted by extreme mass ratio inspirals (EMRIs). Focusing on circular orbits, we model the orbital evolution by incorporating both gravitational-wave radiation reaction and dynamical friction induced by the dark matter distribution, including possible density spikes near the black hole. Using frequency-domain waveform analysis, we compute the phase evolution of gravitational waves and quantify the dephasing caused by different halo parameters, including slope, density, and mass ratio. We further explore the distinguishability of dark matter models with annihilation, non-annihilation, and p-wave velocity dependence, as well as the potential to differentiate between astrophysical and primordial black holes. Our results show that even small variations in the dark matter properties lead to observable phase differences over a four-year EMRI evolution, making space-based detectors such as LISA sensitive probes of central dark matter distributions. Finally, we employ the Fisher matrix formalism to estimate the precision with which key parameters, such as halo slope and density, can be constrained, demonstrating that EMRI observations provide a promising avenue to probe both the nature of dark matter and the formation history of supermassive black holes.
Wang et al. (Sat,) studied this question.