Self-interacting Dark Scalar Spikes around Black Holes via Relativistic Bondi Accretion

We consider the spike mass density profile in a dark halo by self-consistently solving the relativistic Bondi accretion of dark matter onto a non-spining black hole of mass $M$. We assume that the dominant component of the dark matter in the halo is a Standard model gauge-singlet scalar. Its mass $m\simeq 10^{-5}{\rm eV}$ and quartic self-coupling $\lambda\lesssim10^{-19}$ are constrained to be compatible with the properties of galactic dark halos. In the hydrodynamic limit, we find that the accretion rate is bounded from below, $\dot{M}_{\rm min}=96\pi G^2M^2 m^4/\lambda\hbar^3$. Therefore, for $M=10^6~{\rm M}_\odot$ we have $\dot{M}_{\rm min}\simeq1.41\times 10^{-9}~{\rm M}_\odot~{\rm yr}^{-1}$, which is subdominant compared to the Eddington accretion of baryons. The spike density profile $\rho_0(r)$ within the self-gravitating regime cannot be fitted well by a single-power law but a double-power one. Despite that, we can fit $\rho_0(r)$ piecewise and find that $\rho_0(r) \propto r^{-1.20}$ near the sound horizon, $\rho_0(r) \propto r^{-1.00}$ towards the Bondi radius and $\rho_0(r) \propto r^{-1.08}$ for the region in between. This contrasts with more cuspy $\rho_0(r) \propto r^{-1.75}$ for dark matter with Coulomb-like self-interaction.