A critical threshold for the cosmological Euler-Poisson system

TL;DR

This paper investigates the behavior of solutions to the cosmological Euler-Poisson system in an expanding universe, identifying a critical expansion rate at which the system transitions from stability to shock formation.

Contribution

It establishes a critical threshold at the matter-dominated expansion rate, showing global stability for ta>2/3 and shock formation for ta2/3, highlighting fundamental differences from relativistic models.

Findings

Solutions are globally stable for ta>2/3 with small initial data.

Numerical evidence shows shock formation for ta2/3 or less.

The stability threshold is independent of the fluid's speed of sound.

Abstract

We consider the gravitational Euler-Poisson system with a linear equation of state on an expanding cosmological model of the Universe. The expansion of the spatial sections introduces an additional dissipating effect in the Euler equation. We prescribe the expansion rate of space by a scale factor $a(t)=t^\alpha$ with $\alpha\in(0,1)$, which describes the growth of length scales over time. This model is regularly applied in cosmology to study classical fluids in an expanding Universe. We study the behaviour of solutions to this system arising from small, near-homogeneous initial data and discover a \emph{critical} change of behaviour near the expansion rate $\alpha=2/3$, which corresponds to the matter-dominated regime in cosmology. In particular, we prove that for $\alpha>2/3$ the fluid variables are global in time and remain small provided they are sufficiently small in a suitable norm initially. In the complementary regime $\alpha\leq2/3$, we present numerical evidence for shock formation of solutions to the Euler equation for arbitrarily small initial data. In combination, this establishes the existence of a critical stability threshold for barotropic fluids in expanding domains. In contrast to our previous work on the corresponding relativistic system, the threshold in the classical system considered here is independent of the speed of sound of the fluid. This establishes that fluids in cosmology behave fundamentally different in the non-relativistic regime than in the relativistic one.