Maximal extensions and singularities in inflationary spacetimes

Extendibility of inflationary spacetimes with flat spatial geometry is investigated. We find that the past boundary of an inflationary spacetime becomes a so-called parallely propagated curvature singularity if the ratio diverges at the boundary, where and a represent the time derivative of the Hubble parameter and the scale factor, respectively. On the other hand, if the ratio converges, then the past boundary is regular and continuously extendible. We also develop a method to judge the continuous (C ⁰) extendibility of spacetime in the case of slow-roll inflation driven by a canonical scalar field. As applications of this method, we find that Starobinsky inflation has a C ⁰ parallely propagated curvature singularity, but a small field inflation model with a Higgs-like potential does not. We also find that an inflationary solution in a modified gravity theory with limited curvature invariants is free of such a singularity and is smoothly extendible.