MODERN COSMOLOGY

Quantum fluctuations in the inflationary universe 167

Indeed, the spectrum of vacuum fluctuations (4.15) strongly differs from the
spectrum in Minkowski space whenkH. If the fluctuation spectrum before
inflation has a cut-off atk ≤k 0 ∼Tresulting from high-temperature effects,
or atk ≤k 0 ∼Hdue to a small initial size∼H−^1 of an inflationary region,
then the spectrum will change at the time of inflation, due to exponential growth
in the wavelength of vacuum fluctuations. The spectrum (4.15) will gradually
be established, but only at momentak≥k 0 e−Ht. There will then be a cut-off
in the integral (4.14). Restricting our attention to contributions made by long-
wave fluctuations withk≤H, which are the only ones that will subsequently be
important for us, and assuming thatk 0 =O(H), we obtain

〈φ^2 〉≈

H^2

2 ( 2 π)^3

∫H

He−Ht

d^3 k

k

=

H^2

4 π^2

∫ 0

−Ht

dln

k

H

≡

H^2

4 π^2

∫ Ht

0

dln

p

H

=

H^3

4 π^2

t. (4.17)

A similar result is obtained for a massive scalar fieldφ. In that case, long-
wave fluctuations withm^2 H^2 behave as

〈φ^2 〉=

3 H^4

8 π^2 m^2

[

1 −exp

(

−

2 m^2

3 H

t

)]

. (4.18)

Whent≤ 3 H/m^2 ,theterm〈φ^2 〉grows linearly, just as in the case of the massless
field (4.17), and it then tends to its asymptotic value

〈φ^2 〉=

3 H^4

8 π^2 m^2

. (4.19)

Let us now try to provide an intuitive physical interpretation of these results.
First, note that the main contribution to〈φ^2 〉(4.17) comes from integrating over
exponentially smallk(withk∼Hexp(−Ht)). The corresponding occupation
numbersnk(4.16) are then exponentially large. One can show that for large
l=|x−y|eHt, the correlation function〈φ(x)φ(y)〉for the massless fieldφ
is

〈φ(x,t)φ(y,t)〉≈〈φ^2 (x,t)〉

(

1 −

1

Ht

lnHl

)

. (4.20)

This means that the magnitudes of the fieldsφ(x)andφ(y)will be highly
correlated out to exponentially large separationsl ∼ H−^1 exp(Ht),andthe
corresponding occupation numbers will be exponentially large. By all these
criteria, long-wave quantum fluctuations of the fieldφwithkH−^1 behave like
a weakly inhomogeneous (quasi)classical fieldφgenerated during the inflationary
stage.
Analogous results also hold for a massive field withm^2  H^2 .There,
the principal contribution to〈φ^2 〉comes from modes with exponentially small