The Depletion of Collisionless Dark Matter Spikes
Dense concentrations of dark matter surrounding black holes provide a compelling opportunity to probe the nature of dark matter. In the classic Gondolo--Silk model, the adiabatic growth of a massive black hole in a dark matter cusp produces a steep density spike ($ρ\propto r^{-7/3}$), potentially inducing measurable gravitational-wave dephasings in intermediate and extreme mass-ratio inspirals (IMRIs/EMRIs). We challenge this paradigm by considering a collisionless dark matter spike embedded in a realistic nuclear star cluster. Using Fokker--Planck models of isotropic nuclear clusters, we show that mass segregation in a multi-mass stellar cusp accelerates relaxation relative to single-mass models, thereby driving the dark matter to the lower density $r^{-3/2}$ Bahcall--Wolf profile within 1 Gyr. In the inner regions, where the Fokker--Planck description breaks down, we model strong triple interactions between dark matter particles and EMRIs using post-Newtonian 3-body simulations. We show that EMRIs eject dark matter particles via gravitational slingshots, depleting the inner spike over a few Gyr. Because EMRI number densities are too low to drive two-body relaxation, and replenishment by DM self-relaxation is negligible, this depletion is irreversible. While the extent of EMRI-induced DM depletion depends on the EMRI rate and mass, we find reductions in densities by several orders of magnitude. As a result, the dark-matter-induced dephasings for EMRIs may fall below the LISA detectability threshold for massive black holes at $z = 3$ (2.14 Gyr) with masses $\lesssim 10^{5}\,M_\odot$ (for a low $\mathcal{O}(10) \, \mathrm{Gyr}^{-1}$ EMRI rate), extending to $\lesssim 10^6\,M_\odot$ for more realistic rates of $\mathcal{O}(100 - 300)\,$Gyr$^{-1}$. Our findings substantially reduce the parameter space over which massive black holes can host detectable collisionless dark matter spikes.