Guest blog by Paul Frampton
Dear Luboš, thank you for the kind invitation to contribute as a guest on your remarkable blog. My subject is cyclic cosmology and will be based on a recent paper archived at 1411.7887 [grqc] although I will provide only a nontechnical description without many equations and will begin with the interesting history of cyclic model building.
One surprising and interesting output is that no inflation is required to explain the observed flatness and homogeneity of the universe.
Let's briefly summarize the history. In 1917 Einstein applied his general relativity to the cosmos, then in 1922 Friedmann derived his famous equations. It was evident that there was a big bang singularity where the density and temperature become infinite at a finite time in the past, now known to be within 0.5% of 13.8 billion years. Friedmann suggested a cyclic model to avoid this. An infinitely cyclic model proceeding by expansion → turnaround → contraction → bounce → etc. was advocated in the 1920s by, in alphabetical order, De Sitter, Einstein, Friedmann, LeMaitre and Tolman.
Then something extraordinary happened: Tolman proved a nogo theorem based on considering the entropy of the universe showing that a cyclic universe is totally impossible! His paper is Phys Rev 38, 1758 (1931) and he explained it more fully in his book (one of my top10ever books on theoretical physics) "Relativity, Thermodynamics and Cosmology", Oxford (1934). Grossly simplifying, his argument was that because entropy increases by the second law of thermodynamics the cycles grow larger and longer in the future while in the past they are smaller and shorter leading inevitably back to a big bang.
This discouraged research in cyclic cosmology for the remainder of the twentieth century! The Tolman Entropy Conundrum (TEC) must, nevertheless, be solved in any viable cyclic model. The innocentseeming assumptions implicit in the nogo theorem must somehow be violated.
In 1965 the discovery of the Cosmic Microwave Background (CMB) by Penzias and Wilson resolved the dichotomy between the big bang and steadystate models in favor of the former. Since, aside from the nogo theorem, there was the third possible theory of a cyclic model, it really reduced a trichotomy to a dichotomy between bang and bounce which is the present situation.
Going forward 75 years from Tolman, a first effort to solve the TEC was in hepth/0610213 with my student Lauris Baum (Phys. Rev. Lett 98, 071301 (2007)). This made use of the observed accelerated superluminal expansion first observed in 1998 which led to a 2011 Nobel prize for those experimentalists. Tolman had implicitly assumed a decelerating expansion. However, that BF model used phantom dark energy which is better avoided because it violates sacred energy conditions.
Let \(\Omega\) denote the total density divided by the critical density so that a flat universe has \(\Omega = 1\). If we divide the expansion Friedmann equation by the square of the Hubble parameter there arises the useful equation of \(\Omega  1\) equaling curvature divided by the square of the expansion velocity. In a decelerating universe the velocity becomes very small and the departure from flatness grows very rapidly.
In an unadorned big bang (no inflation) extrapolation back to, say, the Planck time requires fine tuning of \(\Omega = 1\) by 60 orders of magnitude. Inflation resolves this by injecting an era of superluminal accelerated expansion which makes the velocity so large that extreme flatness appears, enough that the subsequent decelerating expansion does not remove it.
In order for the entropy conundrum to be solved, the entropy of the universe must be reset to zero at some point in each cycle. The turnaround is, I believe, the only possible time for resetting of entropy to zero. This was implemented in the BF model by the Come Back Empty (CBE) assumption that the contracting universe is empty of any matter and is much smaller than the expanding one. A fraction \(f\) is the relative size of the contracting to the expanding universe and \(f\) will play an important role in the calculation.
The universe contracts adiabatically to the bounce with zero entropy leading to an immediate explanation of homogeneity. To be clear, zero entropy means a dimensionless entropy (divided by the Boltzmann constant) that is negligibly small compared to the present entropy which is more than a googol. An entropy of 10, 20 or 100 is thus "zero" for these purposes.
What about the initial condition on flatness? Here there is a very pleasant surprise. Unlike the BF model of 2007, I avoid phantom dark energy and assume instead a conventional LambdaCDM model. The turnaround is assumed to be at time \(t(T)\), 150 billion years in the future. The cosmological constant causes superluminal accelerating expansion so that the scale factor increases exponentially from \(a=1\) at present to \(a(T)=57,000\).
Consider now the visible universe whose present radius is 44 billion light years. It expands (but much less than the superluminal expansion of space) to an asymptotic radius 57 billion light years, governed by the speed of light. At turnaround the space occupied by the present visible universe has meanwhile stretched to 2.5 quadrillion light years or 5 orders of magnitude larger.
In selecting the visible universe to contract we must be very careful: it must have no matter only dark energy (with no entropy) and an extremely tiny amount of curvature and radiation. In particular one must resist the Ptolemaic thought that it contain the galaxy we presently inhabit.
The fraction \(f\) is the ratio of the two radii, giving \(f = 0.000023\). The next step is to calculate the flatness at the bounce and the expression \(\Omega  1\) turns out to be suppressed by the fourth power of \(f\) and, at the Planck time, is now down by 80 orders of magnitude! After reexpansion, this leads to the prediction that \(\Omega = 1\) at the present time to many (15 or 20) decimal places, very challenging to confirm but any departure would refute the model.
Nevertheless, from this viewpoint, the observed homogeneity and flatness are evidence for a bounce, not a bang.
I have avoided technical details which are available in the papers mentioned that readers may consult.
I have focused on my own approach to cyclic cosmology. Other groups such as Steinhardt and Turok, and Penrose, have very interesting and useful approaches but do not address directly the TEC issue.
Luboš, that is my guestblog in which I hope to have provided a useful explanation of this development in cyclic cosmology. I am excited because such models have always interested me since reading Tolman's book as a student many years ago.
With my very best regards, to you and The Reference Frame,
Paul
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Dear Paul, thanks for the inspiring text and good to have you back outside Latin America, I guess. It's good that you have worked to address the Tolman problem because it's one thing that I understand and immediately think of when someone says "cyclic cosmology".
Although I tried to ask you to explain how the CBE (come back empty) assumption differs from HAMO (here a miracle occurs), i.e. from an explicit and cutely semimasked violation of the second law of thermodynamics (and perhaps other laws), I am still not seeing what makes this apparent violation of the second law defensible. Could you please give me a special bonus sentence on that?
The BigRiplike superexpansions would violate all the energy conditions and if they were needed, that would itself reduce my subjective probability that the model may be correct at least by 6 orders of magnitude.
But even if you avoid this thing, I am still not getting how you can escape more general forms of the Tolman problem.
If there are infinitely many cycles, they are either getting shorter, or longer, aren't they? It is infinitely unlikely  and needs some infinite finetuning somewhere  for the cycles to be exactly the same. The exact constancy of the length cycle is probably inevitably an unstable point, isn't it?
So one either sees that the cycles in the past are shorter, and probably  if similar enough to the geometric series  the total proper time from the true beginning is finite, just like Tolman's case. Or one gets the opposite situation and needs ever longer cycles in the past, with the prediction of some doom after a certain number of cycles in the future. The latter, while pessimistic, could perhaps be acceptable.
It would be really surprising if those CBE scenarios could also predict
polarization of the CMB as already measured by BICEP2, isn't it?
"In selecting the visible universe to contract we must be very careful:
it must have no matter only dark energy (with no entropy) and an
extremely tiny amount of curvature and radiation."
So parts of our current universe contract separately and become new ones? At least that would explain what happens to the entropy; it doesn't go anywhere, only pieces without entropy create new universes.
Nice and interesting article Paul. I have two questions which may have been answered in your papers! Anyway,
(1) Does the CC have the same sign and value during the reversal? In that case what causes the reversal?
(2) This is similar to Lubos' question about second law. During the current phase of expansion, the second law is in contradiction with all the microscopic laws which are time reversal invariant. Are these microscopic laws forcing somehow agreement with second law in the reversal? Somehow you have to reduce entropy to zero before the subsequent bounce.
The second law is not in contradiction with QM because it is statistical in nature. Entropy almost always increases but it may decrease though improbable.
Well. It is still an open question.One can argue that every single molecular collision is time reversal invariant. In the Boltzmann derivation, the hidden assumption is the chaos in molecular collisions. I believe, no one has been able to derive second law from microscopic laws in a clean manner. By the time you go to macroscopic system, the second law (miraculously!!) appears. Some people relate it to the fact of expanding universe. Let us see what Frampton says.
I should have stated that I understand, Feynman and others including Lubos believe that there is no contradiction between time reversal invariance and second law. The only point I am wondering in my question is whether Frampton agrees with this or not and what happens to the statistical argument during the reversal.
Well, according to Wiki
http://en.wikipedia.org/wiki/Second_law_of_thermodynamics
The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently, the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics. This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequenceof the irreversibility or singular nature of the general transition matrix.
Gosh! That derivation is from Everett.
OK! i will try to understand this little better. But then are you ruling out reversal and bounce? You will have to believe that our universe is just one time affair and that is it!
OH MY!! Many worlds again?
His dissertation no less. Well, I guess I can (try to) read it as well, since there is a link to PDF.
No, I'm like you, just trying to understand things better. I am certainly not ruling out anything yet :)
I think I'm missing something here...how does the empty space contract if there's no gravity to drive the contraction?
also, from the point of view of an observer in the superluminaly expanding sphere you mention, they would say that our region of space is superluminaly moving away from them (relativity)...so their sphere is going to contract...how do you resolve this?
I'm sorry, but after an hour or so of speed reading, jumping over a lot of material, I'm not able to understand if it is, or is not, a bona fide derivation of the second law from the first principles of QM, regardless of any worlds interpretation.
It may be a good exercise for somebody to try figure it out, but in my case it would take months, I'm afraid.
One would expect it to be better known, if the above is indeed the case.
Note also this article that relates it to Heisenberg uncertainty relations:
http://www.nature.com/ncomms/journal/v4/n4/full/ncomms2665.html
and these musings by Wootters:
http://cqi.inf.usi.ch/qic/94_Wootters.pdf
I agree with everybody who says the second law of thermodynamics is consistent with the timereversal invariance of microscopic processes. PHF
Dear Tony, this Everett excitement of yours is silly. The relationship between Hermiticity and unitarity in QM was known from the mid 1920s and in mathematics, in the 19th century, and all the proofs of the 2nd law that may be talked about are just obvious variations of Boltzmann's proof and other old known arguments.
The term "doubly stochastic" misleadingly suggests that the stochasticity comes from2 sources. But it doesn't. It means that the matrix and its transpose are leftstochastic  i.e.the sum of rows as well as columns is always one.
The increase of entropy always comes from the same source as the basic logic in classical statistical physics, and quantum mechanics doesn't add any "new source" of the increase of entropy.
Why don't you look e.g. at this proof in Weinberg's book?
http://motls.blogspot.com/2009/08/arrowoftimeunderstoodfor100years.html
It's the scanned image in that blog post.
Thank you, Paul. I will only memorize the order of letters in BCE or CBE or EBC when I get sufficient reasons to think that it is a good investment of the memory. So far, I used BCE, probably like in "before Common Epoch", i.e. before Christ.
I have no problem with saying that the big expansion of the Universe creates some large regions that are almost completely empty.
Whoa! Easily one of the best articles on this blog that I read recently, IMHO.
Thanks Paul. So both of my speculations in the comments are wrong. Sorry. I should not have made comments before reading and understanding your paper! The intriguing question about, how with positive CC and zero gravity collapse can take place, still remains.
The only features that ultimate reality must have, in order to provide me with the peripheral portion of my state of atheistic enlightenment %>, is a timeless energy producing tendency combined with a perhaps smaller infinity of impossible energy patterning outcomes (or trends) than possible ones.
These features are to me no harder to accept than is the aspect of What Is going on that consists of the proven 'quantumweird' substrate of everything.
Thanks. I've stopped writing articles about topics that are "too identical" to some previous ones which guarantees that the best texts about some welldefined important enough issues were published years ago even if the average quality is kept constant.
But by continuing the blog as a nonrepetitive thing, one must obvious address "less fundamental" and more special issues or newer, less important angles to look at things.
The second law appears for macroscopic systems because you have to "coarse grain" and therefore irreversibly lose (smaller scale) information.
Improbable is an understatement! An example I give in teaching statistical mechanics is that
it would be consistent with the laws of physics for a student to be deprived of oxygen molecules near his or her mouth and nose for five minutes and hence die of asphyxiation. Why are not ambulances waiting outside? I then write on the blackboard the probability in the form 0.000000000........ and given each zero occupies one inch ask students where is the first nonzero entry? Is it at the other end of the blackboard? At the other side of campus? In the next nearby city?
An estimate shows that the correct answer to the question is not in the Solar System, but far away in the Milky Way.
Another example of violating the second law is to run in reverse a video of somebody diving into a swimming pool. This would merely reverse momenta of all the relevant molecules and violate no law of physics but is no more probable than the student dying of asphyxiation.
The genius of Boltzmann was to realize that
such a statistical law should be regarded as a "law" of physics although it can be violated. Another giant of 19th century theoretical physics, Maxwell, never accepted this. Now over a century later, I believe 90% of physicists would agree that Maxwell was wrong and Boltzmann was correct.
Lubos, A useful mnemonic for CBE can be the honorific Commander of the British Empire.
Your kind comments all serve to underline the CBE hypothesis that the contracting universe must be empty of baryonic matter, dark matter and black holes. Black holes are a nono because they will proliferate and grow thereby prompting a premature bounce in a failed visible universe.
Similarly any matter will clump as you imply. Only nonclumping contents like dark energy, curvature and radiation can be present.
By the way, and this may answer other commenters, the contraction with an FLRW metric is described by the Friedmann equation (which is time reversal invariant) just like expansion.
I hope that the community, which includes you my good friend Lubos, will quickly understand what I did with CBE and accept such a bounce as a more natural explanation of flatness and homogeneity than inflation.
"What did the first drawer unlocked contain?" – James Joyce
Thank you, Paul. BTW ECB is the European Central Bank, also a permutation.
A simple question: Can one assume the FRW geometry and Friedmann equations in your question? If he can, isn't it obvious that the negative pressure (and therefore effectively inflation) is the only term that has the ability to push Omega closer to one?
Omega is also being pushed closer to one in the "modest" positive cosmological constant case, when the Universe is getting empty, but that de Sitter Universe can't "recollapse" e.g. because of the second law, can it?
Inflation differs from empty dS because the positive cosmological constant is temporary, and it may drop and reheating begins at Omega near one. But one must decide, either the cosmological constant is the real minimum, or just a temporary state. If it is the real minimum,there's no interesting future with dense matter,and if it is a temporary state, then it is inflation. Is that wrong?
You're substitute for Uncle Al when he is not around?
Ah, cheeky bastards! I can't say I expected it but I did wonder what are you going to say  thus Gosh!
Actually, Uncle Al is a paragon of clarity compared to Peter F.
Eh, there is a lot for me yet to learn on this blog.
Both contentwise and readershipwise.
Yes, I realize I should start browsing through older posts, among other things to avoid asking dumb questions that have already been well explained elsewhere.
Takes time.
OTOH, with time some people will begin to recognize when I'm joking without me typing :)
Congratulations on getting sprung, Peter F.
Watch out for Cynholt. (rumor has it she's quite a fine stargazer)
The speed of sound can be breached with jets, lightning strikes, or the crack of a whip, but there was a time before Chuck Yeager when people seriously doubted that humans could survive the stress of traveling faster than sound. Not that it was impossible.
OTOH faster than light has no observable example that can be pointed to to say, yeah such a thing is possible. Superluminal is an optical illusion. A laser can be swung across an arc of the sky so that the beam appears to move faster than light. I can imagine the point of light moving from the moon to Jupiter with a flick of the wrist, but that doesn't mean the actual beam moved that incredible distance. The individual photons move at a more leisurely pace, and it's a different set that impacts the moon than impacts Jupiter. Neither set exceeds lightspeed.
That's my understanding of the thing.
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