and Cosmology

7.4 Evolution of Density Fluctuations
i.e., the integral over the correlation function with
a weight factor depending on k ∼ 2π/L. This relation
can also be inverted, and thus ξ(r) can be computed
from P(k).
In general, knowing the power spectrum is not
sufficient to unambiguously describe the statistical
properties of any random field – in the same way as
the correlation function ξ(r) only provides an incomplete
characterization. However, random fields do exist,
so-called Gaussian random fields, which are uniquely
characterized by P(k). Such Gaussian random fields
play an important role in cosmology because it is assumed
that at very early epochs, the density field obeyed
Gaussian statistics.
7.4 Evolution of Density Fluctuations
P(k) and ξ(r) both depend on cosmological time or redshift
because the density field in the Universe evolves
over time. Therefore, the dependence on t is explicitly
written P(k, t) and ξ(r, t). Note that P(k, t) is linearly
related to ξ(r, t), according to (7.20), and ξ in turn depends
quadratically on the density contrast δ. Ifwe
interpret x as a comoving separation vector, from (7.14)
we then know the time dependence of the density fluctuations,
δ(x, t) = D + (t)δ 0 (x). Thus, within the scope
of the validity of (7.14),
ξ(x, t) = D 2 + (t)ξ(x, t 0), (7.21)
and accordingly
P(k, t) = D 2 + (t) P(k, t 0) =: D 2 + (t) P 0(k), (7.22)
where k is a comoving wave number. We shall stress
once again that these relations are valid only in the
framework of Newtonian, linear perturbation theory in
the matter dominated era of the Universe, to which we
had restricted ourselves in Sect. 7.2.2. Equation (7.22)
states that the knowledge of P 0 (k) is sufficient to obtain
the power spectrum P(k, t) at any time, again within the
framework of linear perturbation theory.
7.4.1 The Initial Power Spectrum
The Harrison–Zeldovich Spectrum. Initially it may
seem as if P 0 (k) is a function that can be chosen arbi-
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trarily, but one objective of cosmology is to calculate
this power spectrum and to compare it to observations.
More than thirty years ago, arguments were already
developed to specify the functional form of the initial
power spectrum.
At early times, the expansion of the Universe follows
apowerlaw,a(t) ∝ t 1/2 in the radiation-dominated era.
At that time, no natural length-scale existed in the Universe
to which one might compare a wavelength. The
only mathematical function that depends on a length
but does not contain any characteristic scale is a power
law; 4 hence for very early times one should expect
P(k) ∝ k n s
. (7.23)
Many years ago, Harrison, Zeldovich, Peebles and others
argued, based on scaling relations, that it should
be n s = 1. For this reason, the spectrum (7.23) with
n s = 1iscalledHarrison–Zeldovich spectrum. With
such a spectrum, we may choose a time t i after the
inflationary epoch and write
P(k, t i ) = D+ 2 (t i) Ak n s
, (7.24)
where A is a normalization constant that cannot be determined
from theory but has to be fixed by observations.
Assuming the validity of (7.22),
P 0 (k) = Ak n s
would then apply.
The Transfer Function. This relation above needs
to be modified for several reasons. In linear perturbation
theory, which led to δ(x, t) = D + (t)δ 0 (x), we
assumed the validity of Newtonian dynamics, considered
only the matter-dominated epoch of the Universe,
and disregarded any pressure terms. The evolution of
perturbations in the radiation-dominated cosmos proceeds
differently though, also depending on the scale
of the perturbations in comparison to the length of the
horizon, so that a correction term of the form
P 0 (k) = Ak n s
T 2 (k) (7.25)
4 You can convince yourself of this by trying to find another type of
function of a scale that does not involve a characteristic length; e.g.,
sin x does not work if x is a length, since the sine of a length is not
defined; one thus needs something like sin(x/x 0 ), hence introducing
a length-scale. The same arguments apply to other functions, such
as the logarithm, the exponential, etc. Also note that the sum of two
power laws, e.g., Ax α + Bx β defines a characteristic scale, namely
that value of x where the two terms become equal.