Cyclic cosmology and geodesic completeness

We consider recently proposed bouncing cosmological models for which the Hubble parameter is periodic in time, but the scale factor grows from one cycle to the next as a mechanism for shedding entropy. Since the scale factor for a flat universe is equivalent to an overall conformal factor, it has been argued that this growth corresponds to a physically irrelevant rescaling, and such bouncing universes can be made perfectly cyclic, extending infinitely into the past and future. We show that any bouncing universe which uses growth of the scale factor to dissipate entropy must necessarily be geodesically past-incomplete, and therefore cannot be truly cyclic in time.