MODERN COSMOLOGY

166 Inflationary cosmology and creation of matter in the universe

whereη=−H−^1 e−Htis the conformal time, and theH 3 (/i) 2 are Hankel functions:

H 3 (/^2 ) 2 (x)=[H 3 (/^1 ) 2 (x)]∗=−

√

2

πx

e−ix

(

1 +

1

ix

)

. (4.12)

Quantization in de Sitter space and Minkowski space should be identical in the
high-frequency limit, i.e. C 1 (p)→0, C 2 (p)→−1asp→∞. In particular,
this condition is satisfied† for C 1 ≡0, C 2 ≡−1. In that case,

ψp(t)=

iH

p

√

2 p

(

1 +

p

iH

e−Ht

)

exp

(

ip

H

e−Ht

)

. (4.13)

Note that at sufficiently larget(whenpe−Ht<H),ψp(t)ceases to oscillate, and
becomes equal to iH/p

√

2 p.

The quantity〈φ^2 〉may be simply expressed in terms ofψp:

〈φ^2 〉=

1

( 2 π)^3

∫

|ψp|^2 d^3 p=

1

( 2 π)^3

∫ (

e−^2 Ht

2 p

+

H^2

2 p^3

)

d^3 p. (4.14)

The physical meaning of this result becomes clear when one transforms from the
conformal momentump, which is time-independent, to the conventional physical
momentumk=pe−Ht, which decreases as the universe expands:

〈φ^2 〉=

1

( 2 π)^3

∫

d^3 k

k

(

1

2

+

H^2

2 k^2

)

. (4.15)

The first term is the usual contribution of vacuum fluctuations in Minkowski space
withH=0. This contribution can be eliminated by renormalization. The second
term, however, is directly related to inflation. Looked at from the standpoint of
quantization in Minkowski space, this term arises because of the fact that de Sitter
space, apart from the usual quantum fluctuations that are present whenH=0,
also containsφ-particles with occupation numbers

nk=

H^2

2 k^2

. (4.16)

It can be seen from (4.15) that the contribution to 〈φ^2 〉 from long-wave
fluctuations of theφfield diverges.
However, the value of〈φ^2 〉for a massless fieldφis infinite only in eternally
existing de Sitter space withH=constant, and not in the inflationary universe,
which expands (quasi)exponentially starting at some timet=0 (for example,
when the density of the universe becomes smaller than the Planck density).

† It is important that if the inflationary stage is long enough, all physical results are independent of
the specific choice of functions C 1 (p)and C 2 (p)if C 1 (p)→0, C 2 (p)→−1asp→∞.