In the paradigm of inflation, at some time the early universe behaves roughly as the de Sitter solution of the relativistically covariant (i.e. hyperbolic) vacuum Einstein field equations with a large positive cosmological constant. Here, we consider what happens if a metric g(t) on a compact 3-manifold evolves according to R. Hamilton 's parabolic Ricci flow equation g'(t) = -2k Ric&(t)) + + Dg(t), with an addedpositive source term Dg(t) which is analogous to the cosmological term in Einstein's equation. In spite of the fact that this Ricci-flowequation is parabolic, and hence nonrelativis tic, we show that if the diffusion constant k in the Ricci-flow equation is related to the cosmological constant A by A = 3/16 (c/k) 2 and D = 1/2 c2/k, then the Ricci flow solution agrees with the de Sitter solution, modulo exponentially decaying terms, in the case where spherical symmetly is assumed. In the case where the initial metric on the spatial 3manifold has positive (but not necessarily nearly constant) Ricci curvature, we show that under the Ricci flow, the universe not only expands, but becomes as round and de Sitter-like as desired, by choosing k appropriately. In contrast, similar "no hair" theorems in the relativistic case, not only assume a homogeneous spatial metric,but also have weaker conclusions. Moreover, in the relativistic setting, most attempts to reduce local inhomogeneities, in order to conform with observations, are rather contrived, as their authors admit.