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A brief summary from wikipedia

Penrose's basic construction is to connect a countable sequence of open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes, each representing a big bang followed by an infinite future expansion. Penrose noticed that the past conformal boundary of one copy of FLRW spacetime can be "attached" to the future conformal boundary of another, after an appropriate conformal rescaling.

Furthermore, there does seem to be some experimental evidence. Specifically:

In 2010, Penrose and Vahe Gurzadyan published a preprint of a paper claiming that observations of the cosmic microwave background made by the Wilkinson Microwave Anisotropy Probe and the BOOMERanG experiment showed concentric anomalies which were consistent with the CCC hypothesis

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    $\begingroup$ Completely hypothetical, although I have to give it two thumbs up. I like the basic idea. $\endgroup$ – CuriousOne Mar 6 '16 at 10:41
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    $\begingroup$ Regarding the close vote as primarily opinion-based: I do not see what about asking about the current status of a theory established in peer-reviewed articles constitutes a "primarily opinion-based" question. Sure, there are possible opinion-based answers, but there's also the obvious non-opinion-based one: The answer surveying the status of emprical confirmation as well as recent theoretical developments in this direction. $\endgroup$ – ACuriousMind Mar 6 '16 at 11:08
  • $\begingroup$ @CuriousOne Surprised at the thumbs up - I thought you hated multiverse theories $\endgroup$ – user56903 Mar 6 '16 at 11:12
  • $\begingroup$ But it's not a multiverse theory. @ACuriousMind: A theoretical model for the long term future of the universe (and a past that is shrouded by an event horizon, as far as we can tell), can never be anything else than opinion. I do admit that I like it, that I might even be evangelizing among layman for the advantages of something like this model, but I will never pretend that I have any hopes of bringing it down from the level of an opinion to even the level of a working scientific theory for the present era within my lifetime. The younger ones among us may be more optimistic... $\endgroup$ – CuriousOne Mar 6 '16 at 11:26
  • $\begingroup$ @CuriousOne Cyclic universe models are multiverse models as they cycle through all possible states. Also, if you read the text, "(and a past that is shrouded by an event horizon, as far as we can tell), can never be anything else than opinion." is not true and they offer evidence $\endgroup$ – user56903 Mar 6 '16 at 15:06

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