The Irreversible Depletion of Dark Matter Spikes by Extreme-Mass-Ratio Inspirals

Dense dark matter spikes around massive black holes (MBHs) have been widely predicted to produce gravitational-wave dephasing in extreme-mass-ratio inspirals (EMRIs) that are potentially observable by LISA. We show that EMRIs, the very objects used to probe these spikes, also destroy them. Combining Fokker-Planck models of nuclear star clusters with post-Newtonian three-body simulations of EMRI-dark matter encounters, we demonstrate that repeated gravitational slingshots irreversibly eject dark matter particles from the cluster’s loss cone over gigayear time-scales. For MBHs at $z=3$ with masses $\lesssim {10}^5{M}_{\odot }$, even conservative EMRI rates of $\mathcal{O}(1-10){\mathrm{G}\mathrm{y}\mathrm{r}}^{-1}$ suffice to deplete the spike density by orders of magnitude; for more realistic rates of $\mathcal{O}(100-300){\mathrm{G}\mathrm{y}\mathrm{r}}^{-1}$, the mass range extends to $\lesssim {10}^6{M}_{\odot }$. As a result, the gravitational-wave dephasing expected from such systems is greatly reduced, narrowing the prospects of detecting DM through this channel. Only heavy, high-redshift systems with sufficiently low EMRI rates – for example, a $\sim {10}^6{M}_{\odot }$ MBH at $z\gtrsim 5$ with $\lesssim 100$ EMRIs per Gyr – can induce dephasing that exceeds one radian.