A simplified proof of a cosmological singularity theorem

TL;DR

This paper offers a simplified proof of a cosmological singularity theorem using the virtual positive first Betti number conjecture, enhancing the understanding of spacetime incompleteness under certain conditions.

Contribution

It presents a more streamlined proof of a singularity theorem by leveraging the virtual positive first Betti number conjecture instead of the surface subgroup conjecture.

Findings

The theorem applies to cosmological models with positive cosmological constant.

The proof is simplified and unified using the virtual positive first Betti number conjecture.

Examples illustrate the theorem and its rigidity under null geodesic completeness.

Abstract

In a previous paper [9], we proved the following singularity theorem applicable to cosmological models with a positive cosmological constant: if a four-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface which is expanding in all directions, then the spacetime is past null geodesically incomplete unless the Cauchy surface is topologically a spherical space. The proof in [9] made use of the positive resolution of the surface subgroup conjecture [15]. In this note, we demonstrate how the less-broadly-known positive resolution of the virtual positive first Betti number conjecture [1] provides a more streamlined and unified approach to the proof. We illustrate the theorem with some examples and analyze its rigidity under null geodesic completeness.