Collisional dark matter density profiles around supermassive black holes
  We solve the spherically symmetric time dependent relativistic Euler
equations on a Schwarzschild background space-time for a perfect fluid, where
the perfect fluid models the dark matter and the space-time background is that
of a non-rotating supermassive black hole. We consider the fluid obeys an ideal
gas equation of state as a simple model of dark matter with pressure. Assuming
out of equilibrium initial conditions we search for late-time attractor type of
solutions, which we found to show a constant accretion rate for the non-zero
pressure case, that is, the pressure itself suffices to produce stationary
accretion regimes. We then analyze the resulting density profile of such
late-time solutions with the function $A/r^{\kappa}$. For different values of
the adiabatic index we find different slopes of the density profile, and we
study such profile in two regions: a region one near the black hole, located
from the horizon up to 50$M$ and a region two from $\sim 800M$ up to $\sim
1500M$, which for a black hole of $10^{9}M_{\odot}$ corresponds to $\sim
0.1$pc. The profile depends on the adiabatic index or equivalently on the
pressure of the fluid and our findings are as follows: in the near region the
density profile shows values of $\kappa <1.5$ and in the limit of the
pressure-less case $\kappa \rightarrow 1.5$; on the other hand, in region two,
the value of $\kappa<0.3$ in all the cases we studied. If these results are to
be applied to the dark matter problem, the conclusion is that, in the limit of
pressure-less gas the density profile is cuspy only near the black hole and
approaches a non-cuspy profile at bigger scales within 1pc. These results show
on the one hand that pressure suffices to provide flat density profiles of dark
matter and on the other hand show that the presence of a central black hole
does not distort the density profile of dark matter at scales of 0.1pc.
