MODERN COSMOLOGY

70 An introduction to the physics of cosmology

correlation function in the usual way in terms of an angular power spectrum^2 θ
and angular wavenumberK:

w(θ)=

∫∞

0

^2 θ(K)J 0 (Kθ)

dK

K

^2 θ(K="+^12 )=

2 "+ 1

8 π

∑

m

|a"m|^2.

An important relation is that between the angular and spatial power spectra.
In outline, this is derived as follows. The perturbation seen on the sky is

δ(qˆ)=

∫∞

0

δ(y)y^2 φ(y)dy,

whereφ(y)is theselection function, normalized such that

∫

y^2 φ(y)dy =1,
andyis comoving distance. The functionφis the comoving density of objects
in the survey, which is given by the integrated luminosity function down to
the luminosity limit corresponding to the limiting flux of the survey seen at
different redshifts; a flat universe (
= 1) is assumed for now. Now write
down the Fourier expansion ofδ. The plane waves may be related to spherical
harmonics via the expansion of a plane wave inspherical Bessel functions
j"(x)=(π/ 2 x)^1 /^2 Jn+ 1 / 2 (x)(see chapter 10 of Abramowitz and Stegun (1965)
or section 6.7 of Presset al(1992)):

eikrcosθ=

∑∞

0

( 2 "+ 1 )i"P"(cosθ)j"(kr),

plus the spherical harmonic addition theorem

P"(cosθ)=

4 π

2 "+ 1

m∑=+"

m=−"

Y"∗m(qˆ)Y"m(qˆ′);ˆq·ˆq′=cosθ.

These relations allow us to take the angular correlation functionw(θ) =
〈δ(qˆ)δ(qˆ′)〉and transform it to give the angular power spectrum coefficients. The
actual manipulations involved are not as intimidating as they may appear, but they
are left as an exercise and we simply quote the final result (Peebles 1973):

〈|a"m|^2 〉= 4 π

∫

^2 (k)

dk

k

[∫

y^2 φ(y)j"(ky)dy

] 2

.

What is the analogue of this formula for small angles? Rather than
manipulating large-" Bessel functions, it is easier to start again from the
correlation function. By writing as before the overdensity observed at a particular