Is Penrose's CCC consistent with Penrose's singularity theorem? According to Penrose's Conformal Cyclic Cosmology (CCC), there were universes prior to ours, prior to the singularity of our universe.
But how is this claim compatible with his famous singularity theorem, according to which spacetime geodesics cannot be extended beyond a singularity?
I believe Penrose doesn't deny the big bang singularity. Then how does he make sense of 'spacetime prior to the big bang singularity' in CCC?
 A: Long story short, the Big Bang is a singularly unique singularity which is mathematically no different than the massively expanded universe in the far, far future.  Because they are the same, one infinitely expanded universe becomes the infinitely small start of the next.
The mathematics he uses to demonstrate this comparison is called conformal geometry, a math that remains consistent despite working with the cosmically infinite be it infinitely huge or small.  Conformal geometry has some advantages, apparently, in that it "squashes" the infinities at the beginning and end of the universe into quantised concepts.  It also has some advantages because it allows the Big Bang to occur without the need for Inflation in the very early moments of the universe.
In fact, one of the main theoretical arguments in favour of CCC is that it overcomes some issues with Inflation which the cosmic background radiation map presents, namely some ripples which should not exist with Inflation.  CCC explains those ripples as the gravity outcomes of collisions between super massive blackholes or the final 'pops' as blackholes eventually evaporate in the previous universe.  Gravity, the mysterious non-force force, traverses from one universe to the next and makes itself apparent in the ripples of the microwave background.  He makes predictions about what those ripples would looks like and has a few people supporting him with claims that they see the suspected ripples.
A: My reading of the paper is that there is a singularity, in the sense of the singularity theorems, at the beginning of each cycle, but it's physically irrelevant because physics is precisely scale invariant there. The FLRW metric in terms of conformal time is $ds^2 = a(η)^2 (dη^2 - d\mathbf Σ^2)$, where the "coordinate" $\mathbf Σ$ ranges over a sphere*. If the scale factor is physically irrelevant then the space is effectively a sphere at the big bang even though $a(η)\to 0$. (In particular see the top half of the left column of page 2761.)
* I think space has to be positively curved in this model because the future infinity of de Sitter space is spherical, but he doesn't seem to say that in the paper unless I missed it, so I'm worried that I'm missing something.
