MODERN COSMOLOGY

60 An introduction to the physics of cosmology

Alternatively, this sum can be obtained without replacing〈δδ〉by〈δδ∗〉, from the
relation between modes with opposite wavevectors that holds for any real field:
δk(−k)=δ∗k(k). Now, by the periodic boundary conditions, all the cross terms
withk′=kaverage to zero. Expressing the remaining sum as an integral, we
have

ξ(r)=

V

( 2 π)^3

∫

|δk|^2 e−ik·rd^3 k.

In short, the correlation function is the Fourier transform of thepower spectrum.
This relation has been obtained by volume averaging, so it applies to the specific
mode amplitudes and correlation function measured in any given realization of
the density field. Taking ensemble averages of each side, the relation clearly
also holds for the ensemble average power and correlations—which are really the
quantities that cosmological studies aim to measure. We shall hereafter often use
the alternative notation
P(k)≡〈|δk|^2 〉

for the ensemble-average power.
In an isotropic universe, the density perturbation spectrum cannot contain
a preferred direction, and so we must have an isotropic power spectrum:
〈|δk|^2 (k)〉=|δk|^2 (k). The angular part of thek-space integral can therefore be
performed immediately: introduce spherical polars with the polar axis alongk,
and use the reality ofξso that e−ik·x→cos(krcosθ). In three dimensions, this
yields

ξ(r)=

V

( 2 π)^3

∫

P(k)

sinkr

kr

4 πk^2 dk.

We shall usually express the power spectrum in dimensionless form, as the
variance per lnk(^2 (k)=d〈δ^2 〉/dlnk∝k^3 P[k]):

^2 (k)≡

V

( 2 π)^3

4 πk^3 P(k)=

2

π

k^3

∫∞

0

ξ(r)

sinkr

kr

r^2 dr.

This gives a more easily visualizable meaning to the power spectrum than does
the quantityVP(k), which has dimensions of volume:^2 (k)=1 means that
there are order-unity density fluctuations from modes in the logarithmic bin
around wavenumberk.^2 (k)is therefore the natural choice for a Fourier-space
counterpart to the dimensionless quantityξ(r).
This shows that the power spectrum is a central quantity in cosmology,
but how can we predict its functional form? For decades, this was thought to
be impossible, and so a minimal set of assumptions was investigated. In the
absence of a physical theory, we should not assume that the spectrum contains
any preferred length scale, otherwise we should then be compelled to explain this
feature. Consequently, the spectrum must be a featureless power law:

〈|δk|^2 〉∝kn.

The indexngoverns the balance between large-and small-scale power.