MODERN COSMOLOGY

94 An introduction to the physics of cosmology

whereqˆis a unit vector that specifies direction on the sky. Since the spherical
harmonics satisfy the orthonormality relation

∫

Y"mY"∗′m′d^2 q = δ""′δmm′,the
inverse relation is

a"m=

∫

δT

T

Y"∗md^2 q.

The analogues of the Fourier relations for the correlation function and power
spectrum are

C(θ)=

1

4 π

∑

"

m∑=+"

m=−"

|a"m|^2 P"(cosθ)

|a"m|^2 = 2 π

∫ 1

− 1

C(θ)P"(cosθ)dcosθ.

These are exact relations, governing the actual correlation structure of the
observed sky. However, the sky we see is only one of infinitely many possible
realizations of the statistical process that yields the temperature perturbations; as
with the density field, we are more interested in theensemble average power.A
common notation is to defineC"as the expectation value of|a"m|^2 :

C(θ)=

1

4 π

∑

"

( 2 "+ 1 )C"P"(cosθ), C"≡〈|am"|^2 〉,

where nowC(θ)is the ensemble-averaged correlation. For smallθand large",
the exact form reduces to a Fourier expansion:

C(θ)=

∫∞

0

T^2 (K)J 0 (Kθ)

dK

K

,

T^2 (K="+^12 )=

("+^12 )( 2 "+ 1 )

4 π

C".

The effect of filtering the microwave sky with the beam of a telescope may
be expressed as a multiplication of theC", as with convolution in Fourier space:

CS(θ)=

1

4 π

∑

"

( 2 "+ 1 )W"^2 C"P"(cosθ).

When the telescope beam is narrow in angular terms, the Fourier limit can be
used to deduce the appropriate"-dependent filter function. For example, for a
Gaussian beam ofFWHM(full-width to half maximum) 2. 35 σ, the filter function
isW"=exp(−"^2 σ^2 / 2 ).
For the large-scale temperature anisotropy, we have already seen that
what matters is the Sachs–Wolfe effect, for which we have derived the spatial
anisotropy power spectrum. The spherical harmonic coefficients for a spherical
slice through such a field can be deduced using the results for large-angle galaxy