Past-completeness of inflationary spacetimes
Joseph E. Lesnefsky, Damien A. Easson, P. C. W. Davies
Abstract
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 ${H}_{\mathrm{avg}}>0$) 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 ${H}_{\mathrm{avg}}$ yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for ${H}_{\mathrm{avg}}$ and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having ${H}_{\mathrm{avg}}>0$. We discuss a standardized definition for inflationary spacetimes as well as quantum (semiclassical) cosmological concerns over physically reasonable scale factors.