  Conformal Cyclic Cosmology (CCC) is a cyclic model of the universe put
forward by Sir Roger Penrose. A conformal invariance assumption in the
neighbourhood of the crossover region between cycles (which Penrose calls
aeons) allows successive space-times to be related by a conformal rescaling. A
major open problem is how to choose the conformal factor in a unique way, and
is a fundamental hurdle to further study. Proposals have been put forward by
Newman, Tod and Nurowski, but they disagree in one way or another with
Penrose's original assumptions as well as each other. In this paper we compare
these different models in detail and rule out certain choices for the conformal
factor that have been put forward by Penrose. We extend the results of Newman
and fix inconsistencies that arose in his calculations. A new class of
solutions are put forward which agree with Penrose's assumptions exactly so
long as a certain additional relation is satisfied.
