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During inflation, expansion happens at a very rapid rate.

How many years of expansion did it fast forward through?

Meaning, if it weren't for inflation, how many years would it take for the universe to expand at its basic rate from the big bang to the size of the universe after inflation?

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    $\begingroup$ The expansion that the Universe went through during inflation isn't usually measured in 'years-at-normal-rate', presumably because it doesn't make any sense to talk about a normal rate when that's simply not what happened. It is measured in e-folds; experimental data requires that inflation lasted for about $60$ e-folds, aka the Universe expanded by a factor $e^{60}$ or more. $\endgroup$
    – Danu
    Mar 6, 2014 at 6:53
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    $\begingroup$ There are a lot of different models though, and the number of e-folds they are capable of producing can easily go up to $100$ or maybe even more: we are unable to observationally tell the difference anyhow because of the limited size of the observable Universe. $\endgroup$
    – Danu
    Mar 6, 2014 at 7:20

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Inflation is what produces the homogeneous, flat universe that we observe, so if you leave it out, the universe never reaches anything like its present state in any amount of time. If it managed to reach its present size (by whatever metric), it would be far more irregular and probably wouldn't have galaxies, etc.

You could write down a model in which the universe was already homogeneous and flat before inflation, so the universe doesn't "need" inflation to get to its present state. The effect of adding inflation to that model is actually that it takes longer for the universe to reach its present state. The scale factor may increase by a factor of $e^{60}$ or more during inflation, but the scale factor is only meaningful up to an overall rescaling anyway. Physically meaningful quantities like energy density and the Hubble parameter are roughly equal before and after inflation (at least, they change less than during an equivalent period of ordinary expansion), so inflation just "pauses" the expansion for a while. If inflation lasts $10^{-33}\text{ s}$, then it only takes $10^{-33}\text{ s}$ longer to get to the present era, but it does technically take longer.

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Assuming we have the correct value for the cosmological constant the doubling time, that is the time it will take for the universe to double in size, is around 11.4 billion years.

We have few hard theories about inflation, but suppose the universe expanded by $e^{60}$ as Danu suggests in his comment, then the number of doubling times is $60/ln(2) \approx 87$. The time it will take the current universe to double in size 87 times is about 990 billion years.

You can obviously adapt this sum for whatever number of $e$-foldings your preferred theory of inflation predicts.

Footnote: the doubling time really only applies once the universe has expanded enough to make the density of matter negligable compared to the cosmological constant. However we aren't far off that stage and given how approximate this sum is I don't think it really matters.

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  • $\begingroup$ So... inflation fast forwards through 990 billion years of expansion? $\endgroup$
    – MikeHelland
    Mar 10, 2014 at 13:54
  • $\begingroup$ @John Rennie : I am not so sure about this. $\\$ The Hubble Time, 11.4Gyrs that you refer to, is the time that it would take the universe to expand by one e-fold today. This is not a constant number, but a timescale that changes with the scale of the universe. Namely, the time that it would take the universe to expand by one e-fold tomorrow will be (slightly) bigger. So your calculation is incorrect I think. Please correct me if I am wrong... $\endgroup$
    – Flint72
    May 9, 2014 at 15:06
  • $\begingroup$ @Flint72: because of dark energy the universe is asymptotically approaching a de Sitter geometry. I probably ought to sit down and think hard about this, but off the top of my head I believe this means the doubling time is actually decreasing slightly with time, but tending to a constant value. You are correct to say the doubling time will change a bit in the future, but given how rough the above calculation is I don't think we are committing any great scientific sins by approximating it as a constant. $\endgroup$
    – John Rennie
    May 9, 2014 at 15:34

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