MTd2, I think you know something of the Loop cosmology picture so you might be interested in what Hmax
is during the brief pre-inflation phase associated with the cosmological bounce.
The model can accommodate a conventional slow-roll inflation that occurs later. However the bounce itself is governed by the quantum-corrected Friedmann equation, and does not require any exotic matter or "inflaton" field. The Hubble parameter H is given directly by the quantum-corrected Friedmann equation. As you know, H is expressed in units of reciprocal time and it is, of course, negative during contraction. As one would expect, H = 0 at the exact moment of the bounce.
In the Loop Friedmann model, H then grows extremely rapidly (the repulsion effect of quantum gravity at near-Planck-scale densities) in a phase called "super-inflation".
has been calculated in a paper by Ashtekar and Sloan. In the case they worked out numerically, it reached 0.93 of Planck.
But this effect depends on high energy density and the density slacks off very rapidly, so this "superinflation" phase (although involving a very high expansion rate H) is extremely brief. It is a pre-inflation "spike" in H.
It's common in Loop cosmology to consider a subsequent episode of the usual inflation, with an "inflaton" field, which superinflation prepares the way for.
Inflation is a period where H is gradually declining but nearly constant. So the scale factor grows exponentially.
Superinflation is a period where H is increasing
so you get faster
than exponential growth. That is the reason for the terminology.
I'll get a link to that Ashtekar Sloan paper. Their Hmax
is less than Planck scale, but still roughly on the order of 1043
per second. (The reciprocal of Planck time which is roughly 10-43
Probability of Inflation in Loop Quantum Cosmology
Abhay Ashtekar, David Sloan
(Submitted on 12 Mar 2011)
... success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy ...We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure.
The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA?
We show that the probability is very close to 1...
34 pages, 3 figures