Highest expansion rate during inflation.

[MTd2]

What is the highest expansion rate during inflation?

 

[bapowell]

The expansion rate is given by the Hubble parameter, which is related through the Friedmann equation to the energy density: $H^2 \propto \rho$. The upper limit to the inflationary energy scale is in principle the Planck scale. However, we have experimental bounds on the energy for the general class of models known as slow roll inflation. In these models, at sufficiently high energy the inflationary expansion generates gravitational waves with an amplitude that would have been measured by CMB telescopes. If I recall correctly, the upper bound is somewhere around $\rho \sim 10^{16}$ GeV.

 

[Chronos]

In simpler models of inflation, the early universe should have expanded by a factor exceeding ten to the ten millionth power in a fraction of a second.

 

[alexg]

If I'm remembering correctly, in the Guth-Linde version of inflation, the universe expanded by a factor of 10⁸⁰ in the period between 10^-45 and 10^-35 seconds.

 

[MTd2]

In simpler models of inflation, the early universe should have expanded by a factor exceeding ten to the ten millionth power in a fraction of a second.

That much? I thought models that considered a duration of 10^-32s with an expansion of e^60 ~ 10^26, which would mean 10^54 for each second... That is on average. I'd like to know the peak rate.

 

[marcus]

MTd2, I think you know something of the Loop cosmology picture so you might be interested in what H_max 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".
H_max 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 H_max 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 second.

http://arxiv.org/abs/1103.2475
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

 

[marcus]

http://arxiv.org/abs/1103.2475
Probability of Inflation in LQC
==quote page 17==
At the end of super-inflation, the Hubble parameter takes its maximum value, H_max = 0.93 m_Pl.
==endquote==

There is a figure on page 22 (Fig. 2) where they plot the curve of ordinary inflation. The time-scale is 1 = 10⁷ Planck time units.
The vertical scale is in "e-folds". It shows how the scalefactor a(t) grows, from about 1 e-fold at the end of superinflation, at a gradually declining rate, to about 60-e-folds, where it levels off.

 

[MTd2]

Hi Marcus!

I am more interested in the 2nd inflation, the usual one. I am wondering if you can trigger a de inflation and "release it" for a mean of propulsion. 1 nanometer de inflated and re inflated can move you 10 thousand light years.

 

[bapowell]

That much? I thought models that considered a duration of 10^-32s with an expansion of e^60 ~ 10^26, which would mean 10^54 for each second... That is on average. I'd like to know the peak rate.

The amount of expansion is commonly given in terms of the number of e-folds of expansion, N, where $a(t) = a(t_i)e^{N}$. From this expression the rate of change is seen to be
$$\frac{dN}{dt} = H$$