Black Hole in a Radiation-Dominated Universe

We study a black hole in an expanding Universe during the radiation-dominated stage. In the case when the black hole radius is much smaller than the cosmological horizon, we present a solution of the Einstein equations for the metric, the matter density and velocity distributions. At distances much smaller than the cosmological horizon the solution features quasi-stationary accretion, while at large distances the solution asymptotes the homogeneous radiation dominated FLRW. We discuss how our solution is related to the McVittie solution. The obtained results can be applied, in particular, for the formation of dark matter density spikes around primordial black holes, and for the evolution of dark matter clumps during the radiation-dominated stage.