"Soft bang" instead of "big bang": model of an inflationary universe   without singularities and with eternal physical past time
  The solution for an inflationary universe without singularities is derived from the Einstein-Lemaitre equations. The present state of the universe evolved from a steady state solution for a tiny, but classical micro-universe with large cosmological constant or large equivalent vacuum energy density and with an equal energy density of radiation and/or some kind of relativistic primordial matter in the infinite past. An instability of this state outside the quantum regime caused a "soft bang" by triggering an expansion that smoothly started with zero expansion rate, continuously increased, culminated in an exponentially inflating phase and ended through a phase transition, the further evolution being a Friedmann-Lemaitre evolution as in big bang models. As a necessary implication of the model the universe must be closed. All other parameters of the model are very similar to those of big bang models and comply with observational constraints. 
E. Rebhan (Heinrich-Heine-Universitaet Duesseldorf)