  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.
