cosmology - Inflation factor and doubling time

I get the general idea about cosmic inflation, but the numbers associated with it seem to be pulled out at random.

For example, in The Elegant Universe, Brian Greene says that the universe doubled every 10^(-34) seconds about a hundred times to an overall factor of 10^30.

Another astronomy text I read recently gives the doubling time as 10^(-35) secs and the overall expansion factor as 10^50.

Where are all these numbers coming from ? For example, why couldn't the universe have doubled every 10^(-40) seconds many billions of times during inflation ?

Answer

Dear Cosmonut, the overall factor is usually expressed as a number of e-foldings. You need at least 60 e-foldings or so to explain why the current Universe is apparently so flat and homogeneous. And $e^{60}\approx 10^{26}$ or so. That's where the $10^{30}$ estimate comes from. Less than 60 e-foldings of inflation would mean that inflation doesn't solve a significant part of the problems it is supposed to solve.

However, there is nothing wrong with inflation that lasts for a much longer time. There could have been 50, 100, or 10,000 e-foldings as well and there are papers that find circumstantial evidence why such a very long inflation could be preferred. We don't know how many e-foldings have taken place but if inflation is good for the things we think it is, it should be more than 60 e-foldings.

The time scale when the inflation occurred is also unknown but it had to be a tiny fraction of a second after the true beginning of our Universe. The higher energy scale - typical "inflaton mass" etc. - you use, the shorter figure for the time after the big bang you get. The typical figure is that inflation began $10^{-36}$ seconds after the big bang and ended around $10^{-33}$ or $10^{-32}$ seconds after the big bang. That's the high-scale, GUT-scale inflation. Note that the shorter time scale is just a little bit longer than the Planck time $10^{-43}$ seconds. It's unlikely that inflation occurred earlier, closer to the Planck scale, because that would predict bigger-than-observed non-uniformities of the cosmic microwave background temperature (they're relatively small because the time when inflation occurred is larger than the Planck time).

However, inflation could have occurred later than that. Of course, it had to occur before the epoch when the temperatures were close to the electroweak scale, e.g. before $10^{-15}$ seconds or so, because we know that the inflaton isn't too light (lighter than the electroweak scale) and because of other reasons.

What inflation achieves is mostly of a qualitative character - making the space flat and uniform, diluting bad exotic objects, and so on - and this qualitative job may be done at many different time scales and with many different durations of the cosmic inflation. The inflaton couldn't have been directly detected so far so you shouldn't be surprised that all the numerical details about inflation remain largely unknown even though they're constrained by inequalities and they become sharper in particular models.