Penrose has proposed an interesting cyclic cosmology (see this, , and this) in which two subsequent cosmologies are glued along conformal boundary together. The metric of the next cosmology is related to that of previous by conformal scaling factor, which approaches zero at the 3-D conformal boundary. The physical origin of this kind of distance scaling is difficult to understand. The prediction is the existence of concentric circles of cosmic size interpretable as kind of memories about previous cosmic cycles.
In TGD framework zero energy ontology (ZEO) inspired theory of consciousness suggest an analogous sequence of cosmologies. Now the cycles would correspond to life cycles of cosmic size serving as a conscious entity having causal diamond (CD) as imbedding space correlate. The arrow of geometric time is defined as the time direction to which the temporal distance between the ends of CD increases in sequence of state function reductions leaving passive boundary of CD unaffected and having interpretation as weak measurements. The arrow of time changes "big" state function reductions changing the roles of the boundaries of CD and meaning the death and re-incarnation of self with opposite arrow of time. Penrose's gluing procedure would be replaced with "big" state function reduction in TGD framework. This proposal is discussed in some detail and the possibility that also now concentric low variance circles in CMB could carry memories about the previous life cycles of cosmos. This picture applies to all levels in the hierarchy of cosmologies (hierarchy of selves) giving rise to a kind of Russian doll cosmology.
See the chapter TGD based cosmology or the article Conformal cyclic cosmology of Penrose and zero energy ontology based cosmology.