A singularity theorem based on spatial averages

Inspired by Raychaudhuri’s work, and using the equation named after him as a basic ingredient, I prove a new singularity theorem: open non-rotating everywhere expanding universes with non-vanishing spatial average of the matter variables are totally geodesically incomplete to the past. Another way of stating the same 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.