Does Chaotic Mixing Facilitate Omega<1 Inflation?
  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. 
Neil J. Cornish, David N. Spergel, Glenn D. Starkman