Accretion of self-interacting dark matter onto supermassive black holes

Dark matter (DM) spikes around supermassive black holes (SMBHs) may lead to interesting physical effects such as enhanced DM annihilation signals or dynamical friction within binary systems, shortening the merger time and possibly addressing the `final parsec problem'. They can also be promising places to study the collisionality of DM because their velocity dispersion is higher than in DM halos allowing us to probe a different velocity regime. We aim to understand the evolution of isolated DM spikes for self-interacting dark matter (SIDM) and compute the BH accretion rate as a function of the self-interaction cross-section per unit DM mass ($\sigma/m_\chi$). We have performed the first $N$-body simulations of SIDM spikes around supermassive black holes (SMBH) and studied the evolution of the spike with an isolated BH starting from profiles similar to the ones that have been shown to be stable in analytical calculations. We find that the analytical profiles for SIDM spikes remain stable over the time-scales of hundreds of years that we have covered with our simulations. In the long-mean-free-path (LMFP) regime, the accretion rate onto the BHs grows linearly with the cross-section and flattens when we move towards the short-mean-free-path (SMFP) regime. In both regimes, our simulations match analytic expectations, which are based on the heat conduction description of SIDM. A simple model for the accretion rate allows us to calibrate the heat conduction in the gravothermal fluid prescription of SIDM. Using this prescription, we determine the maximum allowed accretion rate which occurs when $r_{\rm isco} \rho(r_{\rm isco}) \sigma/m_\chi \sim 1$, where $r_{\rm isco}$ the radius of the innermost stable orbit. Our calibrated DM accretion rates could be used for statistical analysis of SMBH growth and incorporated into subgrid models to study BH growth in cosmological simulations.