Mass and spin coevolution of black holes inspiralling through dark matter

In extreme/intermediate-mass-ratio inspirals (E/IMRIs) embedded in dark-matter (DM) spikes, the secondary black hole can accrete collisionless particles from the surrounding halo. We study how the companion's spin controls this process, and the ensuing back-reaction on the magnitude and direction of the companion's spin vector. We find that higher spin suppresses the mass accretion rate but enhances the accretion-induced torques, driving spin-down and secular alignment of the companion's spin with the orbital plane. Collisionless DM accretion generically imprints a near-universal mass-spin correlation characterized by a spin-evolution parameter $s \simeq 2.8$, much larger than is the case for typical astrophysical environments, and largely independent of the local DM density and the spike slope. The associated spin-down proceeds on astrophysically relevant timescales, thus observations of rapidly spinning IMRI companions would disfavor the presence of dense DM environments, providing constraints complementary to those arising from dynamical friction.