Detecting dark matter around black holes with gravitational waves: Effects of dark-matter dynamics on the gravitational waveform

A dark matter overdensity around a black hole may significantly alter the dynamics of the black hole's merger with another compact object. We consider here intermediate mass-ratio inspirals of stellar-mass compact objects with intermediate-mass black holes dressed with dark matter. We first demonstrate that previous estimates based on a fixed dark-matter dress are unphysical for a range of binaries and dark-matter distributions by showing that the total energy dissipated by the compact object through dynamical friction, as it inspirals through the dense dark matter environment toward the black hole, is larger than the gravitational binding energy of the dark-matter dress itself. We then introduce a new formalism that allows us to self-consistently follow the evolution of the dark-matter dress due to its gravitational interaction with the binary. We show that the dephasing of the gravitational waveform induced by dark matter is smaller than previously thought, but is still potentially detectable with the LISA space interferometer. The gravitational waves from such binaries could provide powerful diagnostics of the particle nature of dark matter.