mustang19 said:
...
I wonder what happens when S approaches infinity, the recession rate must have started off at some finite level.
That's a really interesting question and it's getting increased attention these days.
The work being done, several lines of research being pursued, is not entirely speculative, since there appear to be ways to test different models by observation.
At extreme density one expects
quantum effects which might overwhelm the normal attraction of gravity. In some models only a finite density is reached. The start of expansion is actually a rebound from a prior contraction. The highest density achieved then depends on the model,
how the quantum effects of extreme density are calculated.
The crucial thing is to be able to
test quantum cosmology models of the start of expansion by deriving predictions about potentially observable features of the ancient light background (the CMB, its polarization, its spectrum of temperature fluctuations) that can be looked for. If a model predicts stuff that is not found, then it can be ruled out. Falsifiability is an important merit which some models have and others, as yet, do not. So there is a struggle in progress to construct testable models of start of expansion.
Not all the models involve a quantum bounce at extreme density. However in case anyone is curious here is a rather atypical idea which the authors (Steinhardt and Lehners) say is testable and describe in a paper that just came out.
http://arxiv.org/abs/1304.3122
Planck 2013 results support the simplest cyclic models
Jean-Luc Lehners, Paul J. Steinhardt
(Submitted on 10 Apr 2013)
We show that results from the Planck satellite reported in 2013 are consistent with the simplest cyclic models for natural parameter ranges i.e., order unity dimensionless coefficients, assuming the standard entropic mechanism for generating curvature perturbations. With improved precision, forthcoming results from Planck and other experiments should be able to test the parameter ranges by confirming or refuting the core predictions - i.e., no observable primordial B-mode polarization and detectable local non-gaussianity. A new prediction, given the Planck 2013 constraints on the bispectrum, is a sharp constraint on the local trispectrum parameter g
NL; namely, the simplest models predict it is negative, with g
NL < -1700.
Comments: 5 pages
A model like this, if it passes tests and gains credence, would answer your question about the very early history of the expansion rate. The rate would be negative (contraction) and rise thru zero ( at bounce time) and then have an humongous spike of very high rate expansion.
To be honest, Steinhardt and Lehners model is not a favorite of mine. I am not sufficiently familiar with it to say much, but I admire and respect how up front they are about stating specific ways their idea can be falsified by practical observation (e.g. Planck mission).
I guess the moral of the story is that nobody can answer your question about the early history of the expansion rate because there are competing models of what was going on right around start of expansion and they all need to be tested and some need to be ruled out--falsified--before the dust settles and smoke clears
