“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-Lemaı̂tre 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-Lemaı̂tre 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.