MODERN COSMOLOGY

Quantum fluctuations in the inflationary universe 165

δφ(x)because of the large friction term 3Hφ ̇in the equation of motion of the field
φ. The amplitude of this fluctuation then remains almost unchanged for a very
long time, whereas its wavelength grows exponentially. Therefore, the appearance
of such a frozen fluctuation is equivalent to the appearance of a classical field
δφ(x)that does not vanish after averaging over macroscopic intervals of space
and time.
Because the vacuum contains fluctuations of all wavelengths, inflation
leads to the continuous creation of new perturbations of the classical field with
wavelengths greater thanH−^1 , i.e. with momentumksmaller thanH.One
can easily understand on dimensional grounds that the average amplitude of
perturbations with momentumk∼HisO(H). A more accurate investigation
shows that the average amplitude of perturbations generated during a time interval
H−^1 (in which the universe expands by a factor of e) is given by [7]

|δφ(x)|≈

H

2 π

. (4.7)

Some of the most important features of inflationary cosmology can be
understood only with an account taken of these quantum fluctuations. That is
why in this section we will discuss this issue. We will begin this discussion on a
rather formal level, and then we will suggest a simple interpretation of our results.
First of all, we will describe inflationary universe with the help of the metric
of a flat de Sitter space,
ds^2 =dt^2 −e^2 Htdx^2. (4.8)

We will assume that the Hubble constantHpractically does not change during
the process, and for simplicity we will begin with investigation of a massless field
φ.
To quantize the massless scalar fieldφin de Sitter space in the coordinates
(4.8) in much the same way as in Minkowski space [11]. The scalar field operator
φ(x)can be represented in the form

φ(x,t)=( 2 π)−^3 /^2

∫

d^3 p[a+pψp(t)eipx+a−pψ∗p(t)e−ipx], (4.9)

whereψp(t)satisfies the equation

ψ ̈p(t)+ 3 Hψ ̇p(t)+p^2 e−^2 Htψp(t)= 0. (4.10)

The term 3Hψ ̇p(t)originates from the term 3Hφ ̇in equation (4.1), the last term
appears because of the gradient term in the Klein–Gordon equation for the field
φ. Note, thatpis a comoving momentum, which, just like the coordinatesx, does
not change when the universe expands.
In Minkowski space,ψp(t)√^12 pe−ipt,wherep=

√

p^2. In de Sitter space

(4.8), the general solution of (4.10) takes the form

ψp(t)=

√

π

2

Hη^3 /^2 [C 1 (p)H 3 (^1 /) 2 (pη)+C 2 (p)H 3 (^2 / 2 )(pη)], (4.11)