A simplified proof of a cosmological singularity theorem

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.