210
Inflationary cosmology
Thus. we require a flat region of width A~ with
2
H3
H 2 Nt
(il~) > - 1 " = - -
(7.90)
4rrl
471'2
so that we require
A~ > HN~/2
(7.91)
211' •
We shall see in the example in section 7.8 that there is often a stronger constraint
from the rquirement of obtaining density perturbations of the size found by
CO SE.
7.7 Density perturbations
The quantum fluctuations in the inflaton field discussed in the previous section
result in density perturbations [14-16] in the post-inflationary universe. which
may be responsible for galaxy formation. The density perturbations arise because
the quantum fluctuations in ;, give ~. i.e. the expectation value of ;,. slightly
different values in different regions of space. This results in perturbations to the
value of the vacuum energy density.
Central to the discussion of the formation of density perturbations is the fact
that a given comoving wavelength (Le. a wavelength in units of the scale factor
R(I) of the universe) can start inside the horizon before inflation begins. cross
outside the horizon at some time during inflation. and then cross back inside the
horizon after inflation has ended and a radiation-dominated universe has been
established. (By 'horizon' we shall mean here not the particle horizon but the
comoving Hubble length H -1/ R (I). This is a measure of the distance light travels
during an appreciable amount of expansion of the universe. Inside of a comoving
Hubble length. causal processes do not feel the expansion of the universe.» This
behaviour is a consequence of the fact that when R(I) is increasing as a power t P
of 1 with p < I, the comoving Hubble length increases with time. whereas when
R(I) is increasing exponentially with time, the comoving Hubble length decreases
with time, while the comoving wavelength is, by definition, constant. (See
figure 7.2.) The (classical) inflaton field perturbation &~(I. x) may be expressed
in terms ofperturbations&~(t. k) of momentum k as
&~(t.x) = f d3keii·%&~(t.k)
(7.92)
where we have written the complete inflaton field ~ (I. x) as
~(I.X) = t/Jo(l) +&~(I.X)
(7.93)
with t/Jo(l) the homogeneous classical field. Quantum fluctuations 8~(t, k)
develop when the comoving scale Ikl- I / R(t) is inside the horizon and become
Inflationary cosmology
Thus. we require a flat region of width A~ with
2
H3
H 2 Nt
(il~) > - 1 " = - -
(7.90)
4rrl
471'2
so that we require
A~ > HN~/2
(7.91)
211' •
We shall see in the example in section 7.8 that there is often a stronger constraint
from the rquirement of obtaining density perturbations of the size found by
CO SE.
7.7 Density perturbations
The quantum fluctuations in the inflaton field discussed in the previous section
result in density perturbations [14-16] in the post-inflationary universe. which
may be responsible for galaxy formation. The density perturbations arise because
the quantum fluctuations in ;, give ~. i.e. the expectation value of ;,. slightly
different values in different regions of space. This results in perturbations to the
value of the vacuum energy density.
Central to the discussion of the formation of density perturbations is the fact
that a given comoving wavelength (Le. a wavelength in units of the scale factor
R(I) of the universe) can start inside the horizon before inflation begins. cross
outside the horizon at some time during inflation. and then cross back inside the
horizon after inflation has ended and a radiation-dominated universe has been
established. (By 'horizon' we shall mean here not the particle horizon but the
comoving Hubble length H -1/ R (I). This is a measure of the distance light travels
during an appreciable amount of expansion of the universe. Inside of a comoving
Hubble length. causal processes do not feel the expansion of the universe.» This
behaviour is a consequence of the fact that when R(I) is increasing as a power t P
of 1 with p < I, the comoving Hubble length increases with time. whereas when
R(I) is increasing exponentially with time, the comoving Hubble length decreases
with time, while the comoving wavelength is, by definition, constant. (See
figure 7.2.) The (classical) inflaton field perturbation &~(I. x) may be expressed
in terms ofperturbations&~(t. k) of momentum k as
&~(t.x) = f d3keii·%&~(t.k)
(7.92)
where we have written the complete inflaton field ~ (I. x) as
~(I.X) = t/Jo(l) +&~(I.X)
(7.93)
with t/Jo(l) the homogeneous classical field. Quantum fluctuations 8~(t, k)
develop when the comoving scale Ikl- I / R(t) is inside the horizon and become