I've read the paper more closely and I think I understand it somewhat

The paradox in the paper is actually closely related to the comments with
which I concluded my earlier message.  What they are saying is that if
we are part of a Poincare recurrence, it is overwhelmingly likely that
the past is false.  And worse, it is overwhelmingly likely that the past
is inconsistent; that we were or are on a trajectory that could not be
continued all the way back to the Big Bang.

That is because the entropy at the time of the Big Bang was very low
compared to today, so the vast majority of today-like worlds have no
plausible continuation back to a Big Bang.

The specific example they give is, what if the cosmic microwave
background was 10 degrees rather than 2.7 degrees?  If we think about a
Poincare recurrence in the process of forming, time running backwards,
a hotter than usual microwave background will not cause any problems.
You could still get planets being un-destroyed, life becoming un-extinct
and running backwards, etc.  Eventually you get back to the present day.
And we look around and see a universe that's at 10 degrees.

That's a big problem.  There's no way our universe could have formed from
a Big Bang if the microwave background was that hot.  It means the Big
Bang was completely different, we wouldn't see the elemental abundances we
see today, galaxies wouldn't have formed properly, and so on.  We would
see a world which was inconsistent with its putative past.

Yet we don't.  Broadly speaking we seem to live in a universe to which we
can ascribe a reasonably consistent model of the past.  This contradicts
what we would predict if we lived in a Poincare recurrence, hence we
probably don't.

Now you might say, so what, the whole idea that we formed in this way
was so absurd that no one would ever take it seriously anyway.  But the
authors of this paper seem to be saying that if you assume that there is
a positive cosmological constant (as the cosmological evidence seems to
show), eventually we will get into this de Sitter state, and based on
some assumptions (which I didn't follow) we really should see Poincare
recurrences.  Then by the anthropic principle we should be overwhelmingly
likely to be living in one.  Hence there may be something wrong with
our cosmological theories.

Another point, Tim is of course right that the time for one of these
recurrences to happen is enormous.  The formula they give is
t = exp(S1-S2), where S1 is the entropy of the equilibrium state, which
they estimate as 10^120 for the de Sitter universe, and S2 is the entropy
of the state we are going to chance into, which they say would be about
10^10 for time near the Big Bang.  So this means that the time until
something interesting happens is about exp(10^120).  The authors comment,
"This seems like an absurdly big time between interesting events, which by
comparison last for a very short time. Nevertheless dismissing such long
times as 'unphysical' may be a symptom of extreme temporal provincialism."

As far as the de Sitter model, the references I have found agree that
it has rapid, exponential expansion, with no beginning and no ending,
and that it is a "steady state" model of the universe, which looks the
same to all observers at all times.  However some authors say that it has
no matter, and others say that it has a constant density of mass-energy,
and I'm not sure how to reconcile that.  In this paper the authors do a
coordinate change where the de Sitter model can be considered to have a
constant volume and density, so that sooner or later recurrences should,
well, recur.

Hal Finney

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