Horizon growth of supermassive black hole seeds fed with collisional
  dark matter
  We present the accretion of collisional dark matter on a supermassive black
hole seed. The analysis is based on the numerical solution of the fully coupled
system of Einstein-Euler equations for spherically symmetric flow, where the
dark matter is modeled as a perfect fluid that obeys an ideal gas equation of
state. As the black hole actually grows, the accretion rate of dark matter
corresponds to the black hole apparent horizon growth rate. We analyse cases
with infall velocity as high as $0.5c$ and an environment density of
$100M_{\odot}/\mathrm{pc}^3$, which are rather extreme conditions. Being the
radial flux the maximum accretion case, our results show that the accretion of
an ideal gas, eventually collisional dark matter, does not contribute
significantly to SMBH masses. This result favors models predicting SMBHs were
formed already with supermasses. We show that despite the fact that we are
solving the full general relativistic system, for the parameter space studied
our results are surprisingly similar to those obtained using the Bondi formula,
which somehow certifies its use as a good approximation of a fully evolving
space-time with spherical symmetry at short scales at least for dark matter
densities. Additionally, we study the density profile of the gas and find that
the presence of SMBHs redistributes the gas near the event horizon with a cuspy
profile, whereas beyond a small fraction of a parsec it is not-cuspy anymore.
