The Nonlinear Future-Stability of the FLRW Family of Solutions to the Euler-Einstein System with a Positive Cosmological Constant

TL;DR

This paper proves that small perturbations of certain cosmological solutions to the Euler-Einstein system with a positive cosmological constant remain stable and complete in the future, even with non-zero vorticity, extending previous irrotational results.

Contribution

It extends previous stability results to include fluids with non-zero vorticity using new energy current techniques.

Findings

Background solutions are globally future-stable.

Perturbed solutions are future causally geodesically complete.

Results hold for fluids with 0 < c_s^2 < 1/3.

Abstract

In this article, we study small perturbations of the family of Friedmann-Lema\^itre-Robertson-Walker cosmological background solutions to the 1 + 3 dimensional Euler-Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state pressure = c_s^2 * energy density, with 0 < c_s^2 < 1/3, the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0,infty) x T^3, are future causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably-defined energy currents.