  Yes, if the universe has compact topology. Inflation is currently the most
elegant explanation of why the universe is old, large, nearly flat, homogeneous
on large scales and structured on small scales. One of the weaknesses of the
inflationary paradigm is the problem of initial conditions for inflation: the
pre-inflationary universe must be somewhat old, somewhat large and somewhat
homogeneous. These initial condition requirements are even more severe in Omega
< 1 inflationary models: if the universe does not inflate enough to appear
flat, then it does not inflate enough to appear homogeneous. One solution is to
have two inflationary epochs. Here, we propose another solution to the problem
of pre-inflationary homogeneity: if the universe is compact, then during the
pre-inflationary period, there is sufficient time to homogenize the universe as
chaotic mixing smoothes out primordial fluctuations. Gradients in the
energy-density are reduced as e^(-kappa d), where kappa is the
Kolomogorov-Sinai entropy of the flow, and d is the distance the flow travels.
We explore this homogenization process, outline why compact negatively-curved
universes are the most natural, and conclude with a discussion of the
implications of living in such a universe.
