Recent work has highlighted the importance of a fully relativistic treatment of the dephasing of gravitational waves induced by dark-matter overdensities in extreme mass-ratio inspirals (EMRIs). However, a general-relativistic description of the dark matter phase-space distribution is currently available only for the case of a dark matter “spike” arising from adiabatic black hole growth. Here we develop a fully general-relativistic formalism for the more realistic scenario in which a supermassive stellar progenitor collapses to a black hole and produces a shallower dark matter overdensity, or “mound.” We follow self-consistently the evolution of the supermassive star, its collapse, and the subsequent growth of the resulting black hole, together with the collisionless dark matter orbits. We find that in the regime where the collapse becomes nonadiabatic, the dark matter distribution function is significantly reshaped, with a clear depletion in the low-binding-energy region of phase space. Our results provide a more realistic prediction for the dark matter phase-space distribution around supermassive black holes, which is an essential step in using future EMRI observations to extract information about their formation history and the nature of dark matter.
