Geodesic Completeness of Inflationary Spacetimes

In this opus, I challenge the claim that inflationary spacetimes must be past geodesi-cally incomplete. To do this, I utilize the warped product formalism of Bishop and O’Neill and build upon the venerable Friedmann Robertson Walker (FRW) space- time formalism to the Generalized Friedmann Robertson Walker (GFRW) spacetime formalism, where the achronal spacelike sections can be any geodesically complete Riemannian manifold (Σ, gΣ ). I then solve the GFRW geodesic equation in generality as a functional of the scale factor f , and derive a main theorem, which characterizes the geodesic completeness in GFRW spacetimes. After offering a definition of infla- tion which enumerates the topological requirements which permit a local foliation of a scale factor, I discuss a cohort of geodesically complete inflationary GFRWs which have averaged expansion quantity Havg > 0, proving that classical counter-examples to the theorem of Borde, Guth, and Vilenkin do exist. I conclude by introducing conjectures concerning the relationship between geodesic completeness and inflation: in particular, I speculate that if a spacetime is geodesically complete and non-trivial, it must inflate!