124. Talk in Munich. Regularizing correlators of curvature perturbation

This post is hopefully the last one in the series devoted to my seminar in Munich Last time I have explained why correlation functions of the scalar field on de Sitter background should be actually infrared finite. This time, using similar trick, I will argue that the correlation functions of the curvature perturbation (by curvature perturbation, as usual, I mean curvature of the 3-dimensional slice) should be also infrared finite due to the effects of eternal inflation.

1. formalism

The idea of  formalism is quite simple. If we take into account that curvature perturbation gives the following contribution into the overall metric of spacetime:

( includes all three spatial coordinates) and recall the definition of the number of efoldings

,

we immediately find that

, (1)

where is the inflaton and prime denotes derivative w.r.t. the value of the inflaton field.

Simple formula (1) allows us to easily construct correlation functions of the curvature perturbation, if correlation functions of the inflaton are known. Namely, keeping only the first order on the expansion (1), one has

(2)

(here we have used the logarithmic  divergence of the inflaton correlator discussed in details in one of the previous posts). What is the scale ? What is the physical effect responsible for its value? Here I would like to argue that this effect is eternal inflation.

2. How to calculate curvature perturbation from stochastic formalism

This is a bit complicated, but let me still try to explain it. Trivially, one has

, (3)

where is the total number of efoldings achieved in a given inflationary model.

What is not so trivial is the question how to calculate (3) in the stochastic formalism. Inflation will surely come to the end in a given Hubble patch when the inflaton expectation value drops below the boundary corresponding to the scale of selfreproduction, and the evolution of the inflaton field becomes deterministic. However, although the motion of the inflaton is nearly deterministic, there are still stochastic fluctuations of the inflaton field which may lead to a sudden change of in a given Hubble patch from a value to the value , where the slow roll conditions break down.

The probability distribution for the stochastic moment of the end of inflation or, in other words, the total number of e-folds can be determined from the probability distribution in the Fokker-Planck equation according to

(4)

The reason is that probability is conserved along the path

(5)

as long as inflation became deterministic, so we actually can calculate distribution functions like – they are not completely meaningless

Generally, it is very hard to calculate the distribution function (4) directly. Instead, one can calculate the moments

(6)

directly related to this distribution function. The moments (6) satisfy the following Fokker-Planck-type recursive set of equations:

(7)

while the very first equation in this set is

(8)

It is actually possible to find the solution of the equations (7), (8) in the closed form (please see our paper on the subject), but I don’t want to present it here (it is enormously long).

The bottom line is that all the integrals in this solution I did not present are well behaved since the probability distribution is well-behaved normalizable function of and , as we agreed in the post about correlators of the inflaton. Therefore, all the correlation functions of the total number of efoldings (3) and, automatically, correlators of the curvature perturbation are infrared finite.

Actually, to get a reasonable answer (or at least order of magnitude estimation) for the curvature perturbation correlation function (2) in the formalism, one just needs to keep the first term in the expansion and put the scale of eternal inflation as the IR cutoff scale present in the logarithm

3. So what?

Ok, all these correlation functions (both inflaton and curvature perturbation) are infrared finite, so what? The message is that eternal inflation generally predicts very high level of non-gaussianity. Indeed, we can use the formalism developed above to calculate , etc. just to find that they are all nonzero and generally very large. Is there a chance to see this non-gaussianity? The answer is positive only if you long enough to approach the scale of eternal inflation For us, human beings, there is maybe only a chance (also, very low) to see the running in the correlation functions of non-gaussianity that will give a hint about the scale of eternal inflation.

This is somewhat similar to what happens in the continuum limit of the lattice QFT. At large scales, only renormalizable terms survive and we are unable to get any information about physics that shows up at the scales of the order of the lattice spacing.