  In this article, we study small perturbations of the family of
Friedmann-Lema\^itre-Robertson-Walker cosmological background solutions to the
Euler-Einstein system with a positive cosmological constant in 1 + 3
dimensions. The background solutions describe an initially uniform quiet fluid
of positive energy density evolving in a spacetime undergoing accelerated
expansion. We show that under the equation of state p = c_s^2*(energy density),
0 < c_s^2 < 1/3, the background solutions are globally future asymptotically
stable under small irrotational perturbations. In particular, we prove that the
perturbed spacetimes, which have the topological structure [0,infinity) x T^3,
are future causally geodesically complete.
