5.1. Linear evolution of perturbations
When the perturbations are small, one can ignore the second term in the
right hand side of (65) and replace
(1 + ) by unity in the
first term on the right hand side. The resulting
equation is valid in the linear regime and hence will be satisfied by
each of the Fourier modes
k(t) obtained by Fourier
transforming
(t, x) with
respect to x. Taking
(t, x) =
D(t) f (x), the D(t) satisfies
the equation
![]() |
(66) |
The power spectra P(k, t) = <
|k(t)|2 > at two
different redshifts in the linear regime are related by
![]() |
(67) |
where T (called transfer function) depends only on the parameters
of the background universe (denoted by `bg') but
not on the initial power spectrum and can be computed by solving
(66). It is now clear that the only new input which structure
formation scenarios require is the specification of the initial
perturbation at all relevant
scales, which requires one arbitrary function of the wavenumber
k = 2 /
.
Let us first consider the transfer function.
The rate of growth of small perturbations is essentially decided by two
factors: (i)
The relative magnitudes of the proper wavelength of perturbation
prop(t)
a(t)
and the Hubble radius dH(t)
H-1(t) =
(
/ a)-1
and (ii) whether the universe is radiation dominated or matter dominated.
At sufficiently early epochs, the universe will be radiation dominated
and dH(t)
t will be
smaller than the proper wavelength
prop(t)
t1/2. The density contrast of such modes, which are
bigger than the Hubble radius, will grow as a2 until
prop =
dH(t).
[When this occurs, the perturbation at a given wavelength is said to
enter the Hubble radius. One can use (66) with the right hand side
replaced by
4
(1 + w)(1 + 3w)
G
in
this case; this leads to
D
t
a2.]
When
prop <
dH and the universe is
radiation dominated, the perturbation does not grow significantly and
increases at best only logarithmically
[193].
Later on, when the universe becomes matter dominated for
t > teq, the perturbations again
begin to grow. It follows from this result that modes with wavelengths
greater than
deq
dH(teq) - which enter the Hubble
radius only in the matter dominated epoch - continue to grow at all
times while modes with wavelengths smaller than
deq suffer
lack of growth (in comparison with longer wavelength modes) during the
period tenter < t < teq.
This fact leads to a distortion of the shape of the
primordial spectrum by suppressing the growth of small wavelength modes
in comparison with longer ones.
Very roughly, the shape of T2(k) can be
characterized by the behaviour T2(k)
k-4
for k > keq and
T2
1 for k < keq. The wave number
keq corresponds to the length scale
![]() |
(68) |
(eg., [44],
p.75). The spectrum at wavelengths
>>
deq is undistorted by
the evolution since T2 is essentially unity at these
scales. Further evolution
can eventually lead to nonlinear structures seen today in the universe.
At late times, we can ignore the effect of radiation in solving (66). The linear perturbation equation (66) has an exact solution (in terms of hyper-geometric functions) for cosmological models with non-relativistic matter and dark energy with a constant w. It is given by
![]() |
(69) |
Figure 15 shows the growth factor for different values of w including the one for cosmological constant (corresponding to w = - 1) and an open model (with w = -1/3.)
![]() |
Figure 15. The growth factor for different values of w including the one for cosmological constant (corresponding to w = - 1) and an open model (with w = - 1/3). |
For small values of a,
D a
which is an exact result for
= 0,
NR = 1
model. The growth rate slows down in the cosmological constant dominated
phase (in models with
NR +
= 1
with w = - 1) or in the curvature dominated phase
(open models with
NR < 1
corresponding to w = - 1/3). Between the two cases, there is
less growth in open models compared to models with cosmological constant.
It is possible to rewrite equation (65) in a different form to find an approximate solution for even variable w(a). Converting the time derivatives into derivatives with respect to a (denoted by a prime) and using the Friedmann equations, we can write (65) as
![]() |
(70) |
In a universe populated by only non relativistic matter and dark energy
characterized by an equation of state function w(a), this
equation can be recast in a different manner by introducing a time dependent
[as in equation
(43)] by the relation
Q(t) = (8
G / 3) [
NR(t) / H2(t)]
so that (dQ/d ln a) = 3wQ(1 - Q).
Then equation (65) becomes in terms of the variable
f
(d
ln
/ d ln a)
![]() |
(71) |
Unfortunately this equation is not closed in terms of f and
Q since it also involves
=
exp[
(da /
a)f]. But in the linear regime, we can ignore the second
term on the right hand side and replace
(1 +
) by unity in the
first term thereby getting a closed equation:
![]() |
(72) |
This equation has approximate power law solutions [189] of the form f = Qn when |dw / dQ| << 1 / (1 - Q). Substituting this ansatz, we get
![]() |
(73) |
[Note that Q(t)
1 at high
redshifts, which is anyway the domain of validity of the linear
perturbation theory].
This result shows that n is weakly dependent on
NR;
further, n
(4/7)
for open Friedmann model with non relativistic matter and
n
(6/11)
0.6
in a k = 0 model with cosmological constant.
It is possible to provide simple analytic fitting functions for the transfer function, incorporating all the above effects. For models with a cosmological constant, the transfer function is well fitted by [194]
![]() |
(74) |
where p = k /
( h
Mpc-1) and
=
NR
h
exp[-
B(1
+ (2 h)1/2 /
NR)] is
called the `shape factor'.
Let us next consider the initial power spectrum P(k, zi). The following points need to be emphasized regarding the initial fluctuation spectrum.
(1) It can be proved that known local physical phenomena, arising
from laws tested in the laboratory in a medium with
(P/)
> 0, are incapable producing the initial
fluctuations of required magnitude and spectrum (eg.,
[9],
p. 458). The initial fluctuations,
therefore, must be treated as arising from physics untested at the moment.
(2) Contrary to claims sometimes made in the literature, inflationary models are not capable of uniquely predicting the initial fluctuations. It is possible to come up with viable inflationary potentials ([196], chapter 3) which are capable of producing any reasonable initial fluctuation.
A prediction of the initial fluctuation spectrum was indeed made by Harrison [197] and Zeldovich [198], who were years ahead of their times. They predicted - based on very general arguments of scale invariance - that the initial fluctuations will have a power spectrum P = Akn with n = 1. Considering the simplicity and importance of this result, we shall briefly recall the arguments leading to the choice of n = 1.
If the power spectrum is
P
kn
at some early epoch, then the power per logarithmic band of wave numbers
is
2
k3P(k)
k(n+3). Further, when the wavelength of the mode is
larger than the Hubble radius,
dH(t) =
(
/
a)-1, during the radiation dominated phase,
the perturbation grows as a2 making
2
a4k(n+3). We need to determine how
scales with
k when the mode enters the Hubble
radius dH(t). The epoch
aenter at which this occurs is determined by the
relation
2
aenter /
k = dH. Using
dH
t
a2 in the radiation dominated phase, we get
aenter
k-1
so that
![]() |
(75) |
It follows that the amplitude of fluctuations is independent of scale
k at the time of entering the Hubble radius, only if n =
1. This is the essence of
Harrison-Zeldovich and which is independent of the inflationary paradigm.
It follows that verification of n = 1 by any observation is
not a verification of inflation. At best it verifies a
far deeper principle of scale invariance. We also note that the power
spectrum of gravitational potential
P scales as
P
P /
k4
k(n-4). Hence
the fluctuation in the gravitational potential (per decade in k)
2
k3
P
is proportional to
2
k(n-1). This fluctuation in the gravitational
potential is also independent of k for n = 1 clearly
showing the special nature of this choice.[It is not possible to take
n strictly equal
to unity without specifying a detailed model; the reason has to do with
the fact that scale invariance is always broken at some level and this
will lead to a small difference between n and unity].
Given the above description, the basic model of cosmology is based on
seven parameters. Of these 5 parameters (H0,
B,
DM,
,
R)
determine the background universe and the two parameters
(A, n) specify the initial fluctuation spectrum.
The presence of dark energy, with a constant w, will also affect the transfer function and hence the final power spectrum. An approximate fitting formula can be given along the following lines [195]. Let the power spectrum be written in the form
![]() |
(76) |
where AQ is a normalization, TQ is
the modified transfer function and gQ = (D /
a) is the ratio between linear growth factor in the presence of
dark energy compared to that in
= 1 model. Writing
TQ as the product
TQ
T
where
T
is given by (74), numerical work shows that
![]() |
(77) |
where k is in Mpc-1, and
is a scale-independent but
time-dependent coefficient well approximated by
=
(-w)s with
![]() |
(78) |
where the matter density parameter is
NR(a)
=
NR /
[
NR + (1
-
NR)
a-3w].
Similarly, the relative growth factor can be expressed in the form
gQ
(gQ /
g
) = (- w)t with
![]() |
(79) |
Finally the amplitude AQ can be expressed in the form
AQ = H2(c /
H0)n+3 /
(4
), where
![]() |
(80) |
and 0 =
(a = 1) of
equation (78), and
![]() |
(81) |
This fit is valid for
-1 w
- 0.2.