A question about a proposed "big bounce" This may be a silly question, but it popped into mind while reading this article on Scientific American: Did the Universe Boot Up with a “Big Bounce?”
One of the early bits of evidence for Mirror Symmetry in string theory, as I understand, is that a compact circle of radius $R$ produces identical physics to one of radius $1/R$. So suppose that the macroscopic part of the universe were nevertheless compact (say for the sake of argument a 4-torus). Would one be able to produce a big bounce by having the universe shrink asymptotically to zero in size, with the bounce occurring as the radii reach $R = 1$?
(This is perhaps a silly question; I'm mixing pop-science articles with pop-science takes on string theory, which is probably bound to produce mostly poppycock. Still, I'm curious if there is any sense in what came to mind.)
Edit: My question is really more about whether or not this would provide a possible model for a big bounce, not whether or not the big bounce is a valid theory.
 A: Here is the paper by Turok and Geilen, if you want to get the first source. This looks like a fairly complicated paper. It is not long but it appears there are a lot of steps in between the quoted equations.
The FLRW equation can be derived with Newtonian mechanics for the zero total energy,
$$
H~=~\left(\frac{\dot a}{a}\right)^2~=~\frac{8\pi G\rho}{3}.
$$
The density of mass-energy has various components $\rho~=~\rho_m~+~\rho_r~+~\rho_{vac}$. These are matter, radiation and the dark energy or vacuum energy density. These vary with the scale factor $a~=~a(t)$ as
$$
\rho(t)~=~\frac{\mu}{a^3}~+~\frac{\epsilon}{a^4}~+~\rho_{vac},
$$
and the mass and energy constants $\mu,~\epsilon$ determine the omega factor with $1~=~\Omega_{tot}$ $=~\Omega_m~+~\Omega_r~+~\Omega_{vac}$, where these vary with time. 
The cosmology we can observe appears to have large $\Omega_{vac}~\simeq~.73$ currently, and with the vacuum energy $\rho_{vac}~=~wp$ for $w~=~-1$ this is going to asymptote to one as the cosmology expands indefinitely. However, for certain initial conditions it is possible that $\Omega_r$ or $\Omega_m$ is large enough so another cosmology may recollapse. Our observable cosmos was matter dominated until around $6$ or $7$ billion years ago, when dark matter dominated its dynamics. There appears nothing that forbids however a cosmology in the multiverse scenario that can recollapse.
Will a recollapsing cosmology then bounce? In this paper the argument appears to be that the fundamental degrees of freedom in the cosmology, or equivalently the qubits of the cosmology, survive the recollapse. I have no particular qualm over that. Whether these emerge in another expanding cosmology is unknown. For all we know these degrees of freedom or qubits may be reabsorbed by the inflationary de Sitter or anti-de Sitter spacetime that "spin off" spacetime cosmologies with vacuum energy of lower magnitude. 
