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It seems that in a lot of lecture courses and notes, including mine and those online, seem to state that the number of e-foldings required is of the order of 50-60.

Perhaps I'm looking in the wrong places, or my understanding is a little off but this value often seems to be plucked out of thin air. I'm sure there is a good reason for it.

So, could anyone give me a very brief and consise reason why the number of e-foldings in the current inflationary model needs to be of the order of 50-60...?

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60 e-folds is more or less what you need in order to solve the horizon problem, i.e. the fact that the universe appears to be extremely homogeneous despite the fact that different parts of the universe would not have been in causal contact under the usual Big Bang evolution.

Inflation allows for the whole of the observable universe to have originated in a single causally connected region. Suppose inflation begins at $t_i$ and that the Hubble rate stays constant throughout inflation, $a(t) = a_ie^{H_{\mathrm{inf}}(t-t_i)}$. The comoving causal horizon during inflation is

$$d_p^c = \int_{t_i}^t\frac{\mathrm d t}{a(t)} = (a_iH_{\mathrm{inf}})^{-1}(1-e^{-H_{\mathrm{inf}}(t-t_i)}) \simeq (a_iH_{\mathrm{inf}})^{-1}.$$

If the comoving scales correponding to the observable universe today, $\lambda_0 \sim (a_0H_0)^{-1}$ originated inside this causal horizon we must have

$$\frac{\lambda_0}{d_p^c} < 1 \qquad \Leftrightarrow \qquad \frac{a_iH_{\inf}}{a_0H_0} = \frac{a_i}{a_{\mathrm{end}}}\frac{a_{\mathrm{end}}H_{\mathrm{inf}}}{a_0H_0} = e^{-N}\frac{a_{\mathrm{end}}H_{\mathrm{inf}}}{a_0H_0}<1.$$

The subscript 'end' refers to the end of inflation and $N$ is the number of e-foldings of inflation. Assume for simplicity that the universe is radiation dominated ($a\propto H^{-1/2}$) from the end of inflation until today. Then we must have

$$N > \ln\left(\frac{a_{\mathrm{end}}H_{\mathrm{inf}}}{a_0H_0}\right)=\frac{1}{2}\ln\left(\frac{H_{\mathrm{inf}}}{H_0}\right).$$

Since $H_0\sim 10^{-42}$ GeV, if for example the scale of inflation is $H_{\mathrm{inf}}\sim 10^{14}$ GeV this amounts to about $N>64$ being required for the observable universe to have originated in a single causally connected region. One can refine this estimate by taking into account that $H$ evolves during inflation and that $a$ evolved differently during reheating, radiation domination era and matter domination era but this is the basic idea. Nothing prevents inflation from lasting longer than this, however.

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