General Relativistic Self-Similar Solutions in Cosmology

Abstract

We present general relativistic solutions for self-similar spherical perturbations in an expanding cosmological background of cold pressure-less gas. We focus on solutions having shock discontinuities propagating in the surrounding cold gas. The pressure, $p$, and energy-density, $μ$, in the shock-heated matter are assumed to obey $p=wμ$, where $w$ is a positive constant. Consistent solutions are found for shocks propagating from the symmetry center of a region of a positive density excess over the background. In these solutions, shocks exist outside the radius marking the event horizon of the black hole which would be present in a shock-less collapse. For large jumps in the energy-density at the shock, a black hole is avoided altogether and the solutions are regular at the center. The shock-heated gas does not contain any sonic points, provided the motion of the cold gas ahead of the shock deviates significantly from the Hubble flow. For shocks propagating in the uniform background, sonic points always appear for small jumps in the energy-density. We also discuss self-similar solutions without shocks in fluids with $w<-1/3$.