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I have heard about a number called the Poincare recurrence time on Numberphile. It did not seem that exact in the way that it calculated this number, which was a power tower of 10s.

What is the Poincare recurrence of the observable universe, which is finite? (And assuming this universe is a Hamiltonian finite system.)

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  • $\begingroup$ The paper in "Numberphile" ("Information loss in black holes and/or conscious beings") dates from 1994, and the acceleration of spatial expansion seems to have been discovered in the supernovae 1a studies of the late 1990's, so that Page's paper might not have taken that acceleration into account, perhaps leaving open that possibility of Poincare recurrence which was mentioned by Anders Sandberg, in his answer and comments: On the other hand, it's difficult to conceive of any post-1927 cosmological model that would have excluded some acceleration of spatial expansion.) $\endgroup$
    – Edouard
    Commented Apr 7, 2022 at 19:55

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The problem is that this is not a closed system. The observable universe is growing as time goes by and more and more photons arrive from remote parts of the past. So in a sense it doesn't have a recurrence time. Also, since the universe is expanding it is not clear phase space volume is conserved, so the theorem may not apply.

The recurrence time for a finite region of radius R and total mass M is bounded by the Bekenstein bound $I=k R M$ which tells us the number of bits needed to describe it. $\tau_1 \sim 2^I=2^{k MR}$ is the number of states the region could be in, and if we assume it is moving between distinguishable states this gives a timescale until there is some repeat - there has to be a return to one earlier state since there are no more untested states. If we exponentiate it again to $\tau_2 \sim 2^{2^{kMR}}$ we get a rough estimate of how long it takes to run through all the different evolutions randomly and end up at a particular starting point.

The time unit, whether Planck seconds or kalpas, does not matter much since the number of bits in the exponent is 10 to the power of something for any macroscopic system; double exponentials dominate over whatever time measure you use. This is also why it doesn't matter if the base is 10, e or 2 ("All e's become tens" as the video says).

This is much less than in Page's paper if you just use the observable universe, but in practice the magnitude does not matter. These kinds of magnitudes are interesting only when comparing to things of a far different magnitude. The details do not matter as much as the principles.

The longest timescales I have seen in physics papers that are not doubly exponential are some estimates of vacuum decay timescales running up to $10^{1000}$ years; this is much much smaller than $\tau_1$, which is much much much smaller than $\tau_2$.

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  • $\begingroup$ -Am I right in assuming that phase space would, at least on average, be more apt to be conserved in a multiverse of causally-separated, indefinitely-expanding local universes (as in Poplawski's torsion-based model) than in Penrose's Conformal Cycle Cosmology, although averaging of the expansions between the different temporal iterations of Penrose's model might achieve the same result (albeit with with varying and unpredictable effects on the recurrence time)? Perhaps naively, I'm guessing that pi, in the dimensionality of the phase tube, would tend to conserve phase space. $\endgroup$
    – Edouard
    Commented Jul 28, 2021 at 9:42
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    $\begingroup$ The phase space of two independent universes is just the product of their phase spaces, so I think it would change unless you had some special model. But this depends a lot on which style of multiverse you use. $\endgroup$
    – Anders Sandberg
    Commented Jul 28, 2021 at 16:40
  • $\begingroup$ In Poplawski's past- & future-eternal model (which is a version of Black Hole Genesis, with each of its local universes germinating within one of the black holes formed by collapse of a rotating star in an older & larger one), the spatial and temporal scales would decrease sequentially between each LU & its parent, so that the combined products of the phase spaces in the progenitors of the newest would eventually equal the physical space of a subsequent one, giving his model a unitarity-preserving potential that Penrose's might lack. Comments on this possibility would be appreciated. $\endgroup$
    – Edouard
    Commented Jul 28, 2021 at 17:59
  • $\begingroup$ Actually, the past eternality of Poplawski's model would imply that its physical space would've asymptotically reached its phase space in an unobservable past. (Past eternality is, of course, as unproveable as divine creation, although hypotheses for it may tend to be simpler.) $\endgroup$
    – Edouard
    Commented Jul 28, 2021 at 18:15

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