Geodesic completeness, cosmological bounces and inflation

TL;DR

This paper revises the Borde-Guth-Vilenkin theorem to better diagnose geodesic completeness in cosmological models, showing that certain complete universes involve bounces or accelerated expansion phases, independent of gravity theories.

Contribution

It proposes a simple amendment to the BGV theorem, providing new proofs linking geodesic completeness with cosmological bounces and accelerated expansion without relying on Einstein's equations.

Findings

Revised BGV theorem addresses loopholes in geodesic completeness diagnosis.

Geodesic completeness implies cosmological bounces or emergent phases.

Accelerated expansion with increasing Hubble rate is linked to completeness.

Abstract

The question of geodesic completeness of cosmological spacetimes has recently received renewed scrutiny. A particularly interesting result is the observation that the well-known Borde-Guth-Vilenkin (BGV) theorem may misdiagnose geodesically complete cosmologies. We propose a simple amendment to the BGV theorem which addresses such loopholes while retaining much of its generality. We give straightforward proofs of some recently offered conjectures concerning (generalized) Friedmann-Lema\^itre-Robertson-Walker spacetimes: geodesic completeness implies (i) the existence of a bounce, loitering phase or an emergent cosmology, and (ii) a phase of accelerated expansion with strictly increasing Hubble rate. Our results are purely kinematic and do not assume general relativity or energy conditions.