Exploring the effects of pressure on the radial accretion of dark matter
  by a Schwarzschild supermassive black hole
  Based on the numerical solution of the time-dependent relativistic Euler
equations onto a fixed Schwarzschild background space-time, we estimate the
accretion rate of radial flow toward the horizon of a test perfect fluid
obeying an ideal gas equation of state. We explore the accretion rate in terms
of the initial density of the fluid for various values of the inflow velocity
in order to investigate whether or not sufficiently arbitrary initial
conditions allow a steady state accretion process depending on the values of
the pressure. We extrapolate our results to the case where the fluid
corresponds to dark matter and the black hole is a supermassive black hole
seed. Then we estimate the equation of state parameters that provide a steady
state accretion process. We found that when the pressure of the dark matter is
zero, the black hole's mass grows up to values that are orders of magnitude
above $10^{9}M_{\odot}$ during a lapse of 10Gyr, whereas in the case of the
accretion of the ideal gas dark matter with non zero pressure the accreted mass
can be of the order of $\sim 1M_{\odot}/10Gyr$ for black holes of
$10^{6}M_{\odot}$. This would imply that if dark matter near a supermassive
black hole acquires an equation of state with non trivial pressure, the
contribution of accreted dark matter to the supermassive black hole growth
could be small, even though only radial accretion is considered.
