I Inflation: meaning of χ (chi) for time scales?

[DoobleD]

This is another question related to an undergrad video lecture on inflation by Guth. In the course, he derives the inflationary expansion rate :

a(t) ~ e^χt, with χ = sqrt(8πGρ_f/3), and ρ_f being the mass density of the false vacuum energy.

Later in the lecture, around 51:35, Guth gives sample numbers associated to inflation. The starting point is based on the idea that inflation started at energies where Grand Unified Theories were valid, which is on the order of E_GUT ~ 10¹⁶ GeV. From that, he deduces ρ_f by dimensional analysis. He then computes the inverse of the expansion rate, χ^-1 ~ 10^-38 seconds, and what I think is the event horizon length before inflation : c/χ ~ 10^-28 cm.

Here is a screenshot (he forgot the the ^-1 exponent for χ ~ 2.8 * 10^-38 s, he corrects it later) :

I'm ok with E_GUT and ρ_f. What I'm not sure is what do to with χ^-1. Does this represent the time at which inflation might have started ? Or the duration of inflation ? And if it is one of those, why ?

Alternatively, in those http://www.emu.dk/sites/default/files/guth_inflation.pdf, he says at some point :

The basic inflationary scenario begins by assuming that at least some patch of the early universe was in this peculiar false vacuum state. To begin inflation, the patch must be approximately homogeneous on the scale of χ⁻¹ , as defined by Equation (1.10).

Equation (1.10) being a(t) ~ e^χt, with χ = sqrt(8πGρ_f/3).​

And I'm not really sure what that phrase means either ?

 

[mfb]

It is the time needed for the universe to increase all lengths by a factor e. Inflation lasts much longer than that.

 

[DoobleD]

It is the time needed for the universe to increase all lengths by a factor e. Inflation lasts much longer than that.

Thank you. Is there a (simple) way we can evaluate how long inflation lasts ? And when it starts ?

 

[mfb]

If there was something before inflation, then inflation should have started at its typical energy scale.
Unfortunately we don't know how long it lasts, only lower limits can be found. The answer could be forever (eternal inflation).

 

[DoobleD]

If there was something before inflation, then inflation should have started at its typical energy scale.

Which might be E_GUT ? Is there a way to put a time on that energy scale ? I read poeple saying inflation might have started at 10^-30something s but I can't find the sources of those affirmations.

Unfortunately we don't know how long it lasts, only lower limits can be found. The answer could be forever (eternal inflation).

Do you know how we can find those lower limits ? Same thing I read durations of the order of 10^-30something s but no proof is given.

 

[mfb]

Which might be E_GUT ?

Sure. The time scale for the start (if there was a start at all) depends on unknown physics.

Do you know how we can find those lower limits ? Same thing I read durations of the order of 10^-30something s but no proof is given.

In the standard literature on inflation, they should be cited in the Wikipedia article about inflation.

 

[DoobleD]

Ok, I have found some stuff. But there is still something I can't find : why is the end of the GUT epoch is estimated to be 10^-36 s (or sometimes, 10^-35 s) ?

If we assume inflation happened at a GUT energy scale, it must have started after the Planck time (10^-44 s) and before the GUT breaks into the strong and electroweak forces. Which is estimated to have happen at around 10^-36 s, for some reason nobody cares to give.

I have checked web pages, courses notes, and even cosmology books, and none explains how this estimation is made. I feel like it must come from one or two papers I can't get my hands on.

Should I create a new thread for that specific question ("why do we think the GUT epoch ended at around 10^-36s ?" ) ?

 

[mfb]

A new thread is probably better.