Breakdown of self-similar evolution in homogeneous perfect fluid collapse

TL;DR

This paper analyzes the stability of self-similar solutions in homogeneous perfect fluid collapse, revealing that low-pressure conditions lead to instability and the development of inhomogeneous density profiles.

Contribution

It provides an analytical study of perturbations in self-similar collapse, identifying an unstable mode that causes departure from self-similarity in low-pressure fluids.

Findings

Existence of an unstable normal mode in low-pressure collapse.

Self-similar behavior terminates due to instability.

Inhomogeneous density profiles develop over time.

Abstract

The stability analysis of self-similar solutions is an important approach to confirm whether they act as an attractor in general non-self-similar gravitational collapse. Assuming that the collapsing matter is a perfect fluid with the equation of state $P=\alpha\rho$, we study spherically symmetric non-self-similar perturbations in homogeneous self-similar collapse described by the flat Friedmann solution. In the low pressure approximation $\alpha \ll 1$, we analytically derive an infinite set of the normal modes and their growth (or decay) rate. The existence of one unstable normal mode is found to conclude that the self-similar behavior in homogeneous collapse of a sufficiently low pressure perfect fluid must terminate and a certain inhomogeneous density profile can develop with the lapse of time.