A couple things on conformal cyclic cosmology (CCC) for smooth 3+1 dimensional spacetimes:
> it restarts once there's nothing left but energy
The "nothing left but energy" part is a requirement that matter (in the broadest sense) must be conformally invariant so that its active gravitation as a source allows for the CCC conformal rescaling of the FLRW metric, which is how a predecessor "aeon" is connected mathematically to its successor.
Of the Friedmann dusts, only lightlike radiation can be rescaled this way. Equivalently, all the matter left in the universe at the end of an aeon must be on null geodesics.
Because local Lorentz invariance is fully baked into the Standard Model of Particle Physics (SM), and because the SM has particles with nonzero rest masses (which in a Lorentzian patch cannot couple to null geodesics), NOBODY knows how to do this in a consistent way (let alone in a way that matches actual evidence from particle physics).
(If we restrict to QED we need to convert all electrons and positrons into photons and impose some unknown mechanism to suppress statmech fluctuations back to a non-conformally-invariant condition. In reverse order, two-photon physics are maaaaaaybe supressible adiabatically. However, there just aren't enough positrons to find and annihilate every electron (adiabatic expansion makes that even less likely!), and electrons on their own don't decay into photons. Of course, once we add in the weak and strong forces, we are far beyond quantum electrodynamics, with all sorts of new ways in which reaching a conformally invariant stress-energy state becomes implausible. Oh yeah, and now do dark matter.)
WRT previous comments in this thread, any black hole (BH) at the end of an "aeon" must radiate only massless bosons. This puts a pretty strong lower limit on the mass of Hawking-radiating BHs crossing an boundary between "aeons", or destroys the mathematics of the hypothesized conformal rescaling by virtue of having incompatible non-conformally-invariant field theories on both sides of the Einstein Field Equations.
Binary BHs don't fit cleanly into the FLRW rescaling picture either: among other things they source a metric that isn't locally isotropic and homogeneous (a dust of high mass isolated singleton Birkhoff-theorem BHs is mostly fine though, and in principle you could get through through BH mergers and a stronger cosmological coupling (some types of "fifth force"/quintessence, for instance, to break apart wide BH binaries/triples/multiples)).
There are maybe escapes from some of these constraints in extra dimensions and lattices, but I've been under the impression that one of the attractions of CCC is that it's compatible with the FLRW metric of the standard cosmology.
Does CCC work with small perturbations on the boundary between aeons? Who knows. However, it probably works with small perturbations near that boundary, because we do perturbative FLRW routinely these days. So maybe there's some mechanism (e.g. in dark energy) that makes small deviations from conformally invariant field configurations entirely vanish at the aeon boundary. But we have no astrophysical or particle physics evidence for that at all.
Finally, the article at the top is about inhomogeneous cosomology -- i.e., Timescapes metric is not FLRW metric -- and whatever one's position on CCC, it rests on the standard view that at large scales the universe is homogeneous and that any inhomogenities, anisotropies, and backreactions vanish at smaller scales (and so admit perturbation theory).
I haven't read the literature, but if what you've said is close to right I don't buy CCC. There's too many alternative explanations that fit the data at least as well.
Infinity is a big number though, if stat mech lets it go through it will go through given enough time.
Again, CCC is one of those "feel good" theories in my eyes. It's not better than normal heat death from a prediction standpoint, right now. The universe doesn't end for good, it restarts. Yeah right.
Edit: Is CCC on a positive cosmological constant? If so and you accept a few assumptions anything that can go through will. Hawking radiation off the event horizon will get you anything you need to annihilate through nucleation.
Edit 2: looking at your history I suppose you know that because it's relevant to Boltzmann brains. I'm confused about your reference to fluctuations 14 days ago though. Nucleation and fluctuation Boltzmann brains aren't the same thing.
Me neither (to say the least), but my comment was an attempt to fairly characterize CCC in this thread's context.
> Is CCC on a positive cosmological constant?
AFAIK it's not a requirement of Penrose's approach, but with a positive CC the aeon/future_aeon conformal boundary is spacelike, so you can have global hyperbolicity (and importantly a(t) is always positive and describable by classical EOMs, and the overall structure is just flat de Sitter with a radiation fluid).
[For clarity, here I'm the one extrapolating from the radiation fluid, and afair I'm not summarizing someone's attempt to do this rigorously.]
> it restarts
The conformal rescaling isn't exactly a restart; a Eulerian observer A crossing into its future aeon could encounter a highly similar scaled-up obsever B running much more slowly (and made from fundamental particles with much longer de Broglie wavelengths compared to A's, those having gotten there by freezing out of A's cold sparse ultra-long-wavelength massless boson gas, which to B is B's hot dense ultra-short-wavelength early conditions).
> Infinity is a big number though
Yeah, I think that's why CCC proponents are OK with past-aeon's (pseudo) heat death managing to be a rescaled future-aeon's low entropy initial conditions.
On the other hand, they seem to think in merely large-but-finite timescales when they talk about final BH evaporations leaving a lower-entropy mark on the pseudo heat death that can carry across the boundary (and rescaled to possibly sub-horizon perturbations from B's perspective) be detectable in future aeon's structure formation.
> confused
Sorry about that. Perhaps I lost some train of thought that if made explicit would have made it clearer why I might have been talking about both nucleated and fluctuated structure. I don't remember and am unlikely to re-check the context.
> it restarts once there's nothing left but energy
The "nothing left but energy" part is a requirement that matter (in the broadest sense) must be conformally invariant so that its active gravitation as a source allows for the CCC conformal rescaling of the FLRW metric, which is how a predecessor "aeon" is connected mathematically to its successor.
Of the Friedmann dusts, only lightlike radiation can be rescaled this way. Equivalently, all the matter left in the universe at the end of an aeon must be on null geodesics.
Because local Lorentz invariance is fully baked into the Standard Model of Particle Physics (SM), and because the SM has particles with nonzero rest masses (which in a Lorentzian patch cannot couple to null geodesics), NOBODY knows how to do this in a consistent way (let alone in a way that matches actual evidence from particle physics).
(If we restrict to QED we need to convert all electrons and positrons into photons and impose some unknown mechanism to suppress statmech fluctuations back to a non-conformally-invariant condition. In reverse order, two-photon physics are maaaaaaybe supressible adiabatically. However, there just aren't enough positrons to find and annihilate every electron (adiabatic expansion makes that even less likely!), and electrons on their own don't decay into photons. Of course, once we add in the weak and strong forces, we are far beyond quantum electrodynamics, with all sorts of new ways in which reaching a conformally invariant stress-energy state becomes implausible. Oh yeah, and now do dark matter.)
WRT previous comments in this thread, any black hole (BH) at the end of an "aeon" must radiate only massless bosons. This puts a pretty strong lower limit on the mass of Hawking-radiating BHs crossing an boundary between "aeons", or destroys the mathematics of the hypothesized conformal rescaling by virtue of having incompatible non-conformally-invariant field theories on both sides of the Einstein Field Equations.
Binary BHs don't fit cleanly into the FLRW rescaling picture either: among other things they source a metric that isn't locally isotropic and homogeneous (a dust of high mass isolated singleton Birkhoff-theorem BHs is mostly fine though, and in principle you could get through through BH mergers and a stronger cosmological coupling (some types of "fifth force"/quintessence, for instance, to break apart wide BH binaries/triples/multiples)).
There are maybe escapes from some of these constraints in extra dimensions and lattices, but I've been under the impression that one of the attractions of CCC is that it's compatible with the FLRW metric of the standard cosmology.
Does CCC work with small perturbations on the boundary between aeons? Who knows. However, it probably works with small perturbations near that boundary, because we do perturbative FLRW routinely these days. So maybe there's some mechanism (e.g. in dark energy) that makes small deviations from conformally invariant field configurations entirely vanish at the aeon boundary. But we have no astrophysical or particle physics evidence for that at all.
Finally, the article at the top is about inhomogeneous cosomology -- i.e., Timescapes metric is not FLRW metric -- and whatever one's position on CCC, it rests on the standard view that at large scales the universe is homogeneous and that any inhomogenities, anisotropies, and backreactions vanish at smaller scales (and so admit perturbation theory).