168 Inflationary cosmology and creation of matter in the universe
momentak ∼ Hexp(− 3 H^2 / 2 m^2 ), and the correlation length is of order
H−^1 exp( 3 H^2 / 2 m^2 ).
Later on we will develop a stochastic formalism which will allow us to
describe various properties of the motion of the scalar field.
4.4 Quantum fluctuations and density perturbations
Fluctuations of the fieldφlead to adiabatic density perturbationsδρ∼V′(φ)δφ,
which grow after inflation. The theory of inflationary density perturbations
is rather complicated, but one can make an estimate of their post-inflationary
magnitude in the following intuitively simple way: Fluctuations of the scalar field
lead to a local delay of the end of inflation by the timeδt∼δφ/φ ̇. Density of the
universe after inflation decreases ast−^2 , so the local time delayδtleads to density
contrast|δρ/ρ|∼| 2 δt/t|. If one takes into account thatδφ∼H/ 2 πand that at
the end of inflationt−^1 ∼H, one obtains an estimate
δρ
ρ∼
H^2
2 πφ ̇. (4.21)
Needless to say, this is a very rough estimate. Fortunately, however, it gives a
very good approximation to the correct result which can be obtained by much
more complicated methods [2–4, 7]:
δρ
ρ=C
H^2
2 πφ ̇, (4.22)
where the parameterCdepends on equation of state of the universe. For example,
C= 6 /5 for the universe dominated by cold dark matter [4]. Then equations
3 Hφ ̇=V′andH^2 = 8 πV/ 3 MP^2 imply that
δρ
ρ=
16
√
6 π
5V^3 /^2
V′
. (4.23)
Hereφis the value of the classical fieldφ(t)(4), at which the fluctuation
we consider has the wavelengthl ∼k−^1 ∼ H−^1 (φ)and becomes frozen in
amplitude. In the simplest theory of the massive scalar field withV(φ)=^12 m^2 φ^2
one has
δρ
ρ
=
8
√
3 π
5mφ^2. (4.24)Taking into account (4.4) and also the expansion of the universe by about
1030 times after the end of inflation, one can obtain the following result for
the density perturbations with the wavelengthl(cm) at the moment when these
perturbations begin growing and the process of the galaxy formation starts:
δρ
ρ∼mlnl(cm). (4.25)