MODERN COSMOLOGY

Quantifying large-scale structure 59

properties ofδshould also be homogeneous, even though it is a field that describes
inhomogeneities.
We will often need to use the〈···〉symbol, that denotes averaging over an
ensemble of realizations of the statisticalδprocess. In practice, this will usually
be equated to the spatial average over a sufficiently large volume. Fields that
satisfy this property, whereby

volume average↔ensemble average

are termedergodic.

2.7.1 Fourier analysis of density fluctuations

It is often convenient to consider building up a general field by the superposition
of many modes. For a flat comoving geometry, the natural tool for achieving this
is via Fourier analysis. How do we make a Fourier expansion of the density field
in an infinite universe? If the field were periodic within some box of sideL,then
we would just have a sum over wave modes:

F(x)=

∑

Fke−ik·x.

The requirement of periodicity restricts the allowed wavenumbers to
harmonic!boundary conditions

kx=n

2 π

L

, n= 1 , 2 ...,

with similar expressions forkyandkz. Now, if we let the box become arbitrarily
large, then the sum will go over to an integral that incorporates the density of
states ink-space, exactly as in statistical mechanics. The Fourier relations inn
dimensions are thus

F(x)=

(

L

2 π

)n∫

Fk(k)exp(−ik·x)dnk

Fk(k)=

(

1

L

)n∫

F(x)exp(ik·x)dnx.

As an immediate example of the Fourier machinery in action, consider the
important quantity
ξ(r)≡〈δ(x)δ(x+r)〉,

which is the autocorrelation function of the density field—usually referred to
simply as thecorrelation function. The angle brackets indicate an averaging over
the normalization volumeV. Now expressδas a sum and note thatδis real, so
that we can replace one of the twoδ’s by its complex conjugate, obtaining

ξ=

〈∑

k

∑

k′

δkδ∗k′ei(k

′−k)·x

e−ik·r

〉

.