Physical Foundations of Cosmology

324 Inflation II: origin of the primordial inhomogeneities

In the continuous limit, asV→∞, the sum in (8.1) is replaced by the integral

f(x)=

∫

fkeikx

d^3 k

( 2 π)^3 /^2

(8.5)

and (8.4) becomes

〈fkfk′〉=σk^2 δ

(

k+k′

)

, (8.6)

whereδ

(

k+k′

)

is the Dirac delta function. Note that the Fourier coefficients in
(8.5) are related to the Fourier coefficients in (8.1) by a factor of

√

Vand have
dimension cm^3. In contrast to the dimensionlessδk,−k′, the Dirac delta function
has dimension cm^3. The dimension ofσk^2 does not change in the transition to the
continuous limit and this quantity does not acquire any extra volume factors.
Alternatively, a Gaussian random field can be characterized by the spatial two-
point correlation function

ξf(x−y)≡〈f(x)f(y)〉. (8.7)

This function tells us how large the field fluctuations are on different scales. In
the homogeneous and isotropic case, the correlation function depends only on the
distance between the pointsxand y,that is,ξf=ξf(|x−y|). Substituting (8.5)
into (8.7) and averaging over the ensemble with the help of (8.6), we find

ξf(|x−y|)=

∫

σk^2 k^3

2 π^2

sin(kr)

kr

dk

k

, (8.8)

wherer≡|x−y|. In deriving this relation we have taken into account isotropy
and performed the integration over angles. Thedimensionless variance

δ^2 f(k)≡

σk^2 k^3

2 π^2

(8.9)

is roughly the typical squared amplitude of the fluctuations on scalesλ∼ 1 /k.

Problem 8.1Verify that the typical fluctuations of f, averaged over the volume
V∼λ^3 , can be estimated as

〈⎛

⎝^1

V

∫

V

fd^3 x

⎞

⎠

2 〉^1 /^2

∼O( 1 )δf(k∼λ). (8.10)

When does this estimate fail?

Problem 8.2Show that different variances forakandbkcontradict the assumption
of homogeneity and that the dependence ofσk^2 on the direction ofkis in conflict
with isotropy.