54. Eternal inflation: stochastic approach 2 (Inflationary perturbations 7)

In the previous post we have started to discuss the regime of eternal inflation realized when classical displacement of the inflaton field becomes comparable with the average amplitude of fluctuations generated at super-Hubble scale. The latter in practice means that the gravitational perturbations become of the same order as the background. How to treat theory in this regime?

Let us for simplicity consider a scalar field in the potential such that . We will focus on what an observer leaving within a given Hubble patch sees. It is convenient to divide the quantum field into subhubble and superhubble scales

(1)

where the second term satisfies the free scalar field equation, i.e.,

(2)

and is conformal time. The Heisenberg field satisfies classical equation of motion

(3)

and after substituting the decomposition (1) into it we have for the IR part

(4)

where

(5)

Although both and f are complicated composites of operators , one can immediately check that all terms in the Eq. (4) are commuting with each other and therefore both and can be considered classical quantitites.

Exercise 7.1. Check it explicitly.

On the other hand, calculating the correlation function of f in the Bunch-Davies vacuum we have

(6)

Therefore, equation (4) is nothing but a Langevin equation describing random walk (Brownian motion) of the variable under the action of the random Gaussian force with correlation properties (6).

The physical picture related to this random walk is the following. As long as the classical displacement of the inflaton becomes of the same order of magnitude as the average quantum fluctuation amplitude , deterministic classical description of the inflaton dynamics breaks down. Quantum modes constantly leave the Hubble scale and their amplitude becomes classical at . The quantum phase of the modes also freezes, and its value after leaving the Hubble scale is essentially random. The averaging in (6) is essentially averaging over this random phase. After modes leave the Hubble scale, they start to contribute into (it is simple impossible for an onserver inside a given Huble patch to make a distinction between superhubble fluctuation and the fluctuation of the background \phi).

As a result of this randomness, the overall spacetime becomes divided into Hubble patches with own dynamics of the IR inflaton field inside each Hubble patch.