Exploring the effects of pressure on the radial accretion of dark matter by a Schwarzschild supermassive black hole

Abstract

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.