  The relativistic accretion rate of dark matter by a black hole is revisited.
Under the assumption that the phase space density indicator,
$Q=\rho_{\infty}/\sigma^3_{\infty}$, remains constant during the inflow, the
derived accretion rate can be higher up to five orders of magnitude than the
classical accretion formula, valid for non-relativistic and non-interacting
particles, when typical dark halo conditions are considered. For these typical
conditions, the critical point of the flow is located at distances of about
30-150 times the horizon radius. Application of our results to black hole seeds
hosted by halos issued from cosmological simulations indicate that dark matter
contributes to no more than ~10% of the total accreted mass, confirming that
the bolometric quasar luminosity is related to the baryonic accretion history
of the black hole.
