On the past-completeness of inflationary spacetimes

We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate \(H_{avg}>0\)) must be incomplete in null and timelike past directions. This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic. As stated, we show this claim is an open issue. We show that the calculation of \(H_{avg}\) yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for \(H_{avg}\) and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having \(H_{avg}>0\). We discuss a standardized definition for inflationary spacetimes as well as quantum (semi-classical) cosmological concerns over physically reasonable scale factors.