This is another question related to an undergrad video lecture on inflation by Guth. In the course, he derives the inflationary expansion rate :(adsbygoogle = window.adsbygoogle || []).push({});

a(t) ~ e^{χt}, with χ = sqrt(8πGρ_{f}/3), and ρ_{f}being the mass density of the false vaccum 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}~ 10^{16}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 very similar PDF notes from Guth, 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 ?

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

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