The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant

Abstract

In this article, we study small perturbations of the family of Friedmann–Lemaître–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 p=c[2 over s]ρ, 0 < c[2 over s] < 1/3 , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0,∞) × T[superscript 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.