Physical Foundations of Cosmology

8.1 Characterizing perturbations 323

8.1 Characterizing perturbations

At a given moment in time small inhomogeneities can be characterized by the spatial
distribution of the gravitational potential or by the energy density fluctuations
δε/ε 0. It turns out to be convenient to treat them asrandom fields,for which we
will use the common notationf(x). Subdividing an infinite universe into a set of
large spatial regions we can consider a particular configurationf(x)within some
region as a realization of a random process. This means that the relative number of
regions where a given configurationf(x)occurs can be described by a probability
distribution function. Then averaging over the statistical ensemble is equivalent to
averaging over the volume of the whole infinite universe.
It is convenient to describe the random process using Fourier methods. The
Fourier expansion of the function f(x)in a given region of volumeVcan be
written as

f(x)=

1

√

V

∑

k

fkeikx. (8.1)

In the case of a dimensionless function fthe complex Fourier coefficients, fk=
ak+ibk, have dimension cm^3 /^2. The reality offrequiresf−k=fk∗and hence the
real and imaginary parts offkmust satisfy the constraintsa−k=akandb−k=−bk.
The coefficientsakandbktake different values in different spatial regions. Given a
very large numberNof such regions, the definition of the probability distribution
functionp(ak,bk)tells us that in

dN=Np

(

a′k,b′k

)

dakdbk (8.2)

of them the value ofaklies betweena′kandak′+dakand that ofbkbetweenbk′and
b′k+dbk. Inflation predicts onlyhomogeneous and isotropic Gaussianprocesses,
for which:

p(ak,bk)=

1

πσk^2

exp

(

−

a^2 k

σk^2

)

exp

(

−

bk^2

σk^2

)

, (8.3)

where the variance depends only onk=|k|; it is the same for bothindependent
variablesakandbkand is equal toσk^2 /2. This variance characterizes the corre-
sponding Gaussian process entirely and all correlation functions can be expressed
in terms ofσk^2. For example, for the expectation value of the product of Fourier
coefficients one finds

〈fkfk′〉=〈akak′〉+i(〈akbk′〉+〈ak′bk〉)−〈bkbk′〉=σk^2 δk,−k′, (8.4)

where we have taken into account thata−k=akandb−k=−bk. Hereδk,−k′= 1
fork=−k′and is otherwise equal to zero.