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 [gr-qc] although I will provide only a non-technical 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 no-go 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 top-10-ever 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 innocent-seeming assumptions implicit in the no-go 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 steady-state models in favor of the former. Since, aside from the no-go 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 hep-th/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 re-set to zero at some point in each cycle. The turnaround is, I believe, the only possible time for re-setting 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 re-expansion, 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,