Velocity Distribution of Dark Matter Spike around Schwarzschild Black Holes and Effects on Gravitational Waves from EMRIs

Dark matter (DM) constitutes the predominant portion of matter in our universe. Despite compelling evidence, the precise characteristics of DM remain elusive. Among the leading DM candidates are weakly-interacting massive particles, which may clump into steep concentrations around the central black holes of galaxies. However, DM profiles of the resulting dense spikes remain uncertain. Here we employ the relativistic dynamics in Schwarzschild geometry and first evaluate the velocity distributions of DM within such spikes. Through variations in black hole masses and dark halo parameters, we identify universal features in DM profiles and fit them with Gaussian distributions. Additionally, we illustrate with the impact of dynamical friction on gravitational waves generated by extreme-mass-ratio inspirals (EMRIs) within DM spikes, taking into account the velocity distribution of DM in the relativistic regime. Our findings demonstrate the phase shifts in the time-domain waveform, potentially providing useful insights for probing DM in galactic centers by gravitational-wave experiments.