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-consinstently solving the relativisitic 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 with mass $m\simeq 10^{-5}{\rm eV}$ and quartic self-coupling
$\lambda\gtrsim10^{-20}$ to be compatible with the properties of a typical dark
halo. 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 $\dot{M}_{\rm min}\simeq1.41\times 10^{-10}~{\rm
M}_\odot~{\rm yr}^{-1}$, 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 the dark matter with Coulomb-like self-interaction.
