The Depletion of Collisionless Dark Matter Spikes
Dense concentrations of dark matter (DM) surrounding black holes provide a compelling opportunity to probe the nature of DM. In the classic Gondolo-Silk model, the adiabatic growth of a massive black hole (MBH) in a DM cusp produces a steep density spike ($\rho \propto r^{-7/3}$), potentially inducing measurable gravitational-wave dephasings in intermediate and extreme mass-ratio inspirals (IMRIs/EMRIs). We challenge this paradigm by considering a collisionless spike embedded in a realistic nuclear star cluster (NSC). Using 1D orbit-averaged Fokker-Planck (FP) simulations of isotropic NSCs, we show that mass segregation in a multi-mass stellar cusp accelerates relaxation, relative to single-mass models, thereby driving the DM to the lower density $r^{-3/2}$ Bahcall-Wolf profile within $\lesssim 1 \mathrm{Gyr}$. In the inner regions, where the FP description breaks down, we model strong triple interactions between DM particles and EMRIs using post-Newtonian 3-body simulations. We show that EMRIs eject DM particles via slingshots, depleting the inner spike over a few Gyrs. Because EMRI number densities are too low to drive two-body relaxation, and collisionless DM cannot efficiently repopulate the depleted phase space, this depletion is irreversible. While the extent of EMRI-induced depletion depends on the EMRI rate and mass, we find reductions in DM densities by several orders of magnitude. Hence, DM-induced dephasings for EMRIs may fall below the detectability threshold of LISA for MBHs at $z = 3$ (2.14 Gyr) with masses $\lesssim 10^{5}\,M_\odot$ (for an $\mathcal{O}(10) \, \mathrm{Gyr}^{-1}$ EMRI rate), extending to $\lesssim 10^6\,M_\odot$ for more optimistic rates of $\mathcal{O}(300-1000) \, \mathrm{Gyr}^{-1}$. Our findings substantially reduce the parameter space over which MBHs can host detectable collisionless DM spikes.