A singularity theorem based on spatial averages

TL;DR

This paper proves a new singularity theorem showing that non-rotating, expanding universes with non-zero spatial averages of matter variables are geodesically incomplete in the past, distinguishing singular from non-singular cosmologies.

Contribution

It introduces a novel singularity theorem based on spatial averages, extending Raychaudhuri's work to characterize conditions for cosmological singularities.

Findings

Non-rotating, expanding universes with non-zero spatial averages are past geodesically incomplete.

Singularity-free models must have zero spatial average of energy density and related variables.

Provides a clear criterion to differentiate between singular and non-singular cosmological models.

Abstract

Inspired by Raychaudhuri's work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating everywhere expanding universes with non-vanishing spatial average of the matter variables are severely geodesically incomplete to the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies.