Consider an inflationary domain (or mini-universe in the context of Exercise 16.10) of initial radius H−1, in which the value of the scalar field ϕ≫1. In a time interval Δt=H−1, show that classically, in the slow-roll approximation, the value of ϕ will change by
Δϕ≈−ϕ2.
Assuming that the typical amplitude of quantum fluctuations in the scalar field is δϕ≈H/(2π), show that
δϕ≈2π13V(ϕ).
Hence, for the case V(ϕ)=21m2ϕ2, show that the decrease in the value of the scalar field due to its classical motion is less than changes due to quantum fluctuations generated in the same time interval, provided that
ϕ≫m6.
Assuming that the typical wavelength of the quantum fluctuation is δϕ is H−1, show that, after a time interval Δt=H−1, the original domain becomes effectively divided into e3∼20 domains of radius H−1, each containing a roughly homogeneous scalar field ϕ+Δϕ+δϕ. Thus, on average, the volume of the universe containing a growing ϕ-field increases by a factor ∼10 after every time interval Δt=H−1.
Note: This is the mechanism underlying stochastic inflation.