MODERN COSMOLOGY

Quantifying large-scale structure 69

be the usualHankel transformpair:

w(θ)=

∫∞

0

^2 θJ 0 (Kθ)dK/K,

^2 θ=K^2

∫∞

0

w(θ)J 0 (Kθ)θdθ.

For power-law clustering,w(θ)=(θ/θ 0 )−,thisgives

^2 θ(K)=(Kθ 0 ) 21 −

( 1 −/ 2 )

(/ 2 )

,

which is equal to 0. 77 (Kθ 0 )for = 0 .8. At large angles, these relations
are not quite correct. We should really expand the sky distribution inspherical
harmonics:
δ(qˆ)=

∑

am"Y"m(qˆ),

whereqˆis a unit vector that specifies direction on the sky. The functionsY"m
are the eigenfunctions of the angular part of the∇^2 operator: Y"m(θ,φ) ∝
exp(imφ)P"m(cosθ),whereP"m are theassociated Legendre polynomials(see
e.g. section 6.8 of Presset al1992). Since the spherical harmonics satisfy the
orthonormality relation

∫

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

am"=

∫

δ(qˆ)Y"∗md^2 q.

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

w(θ)=

1

4 π

∑

"

m∑=+"

m=−"

|a"m|^2 P"(cosθ)

|a"m|^2 = 2 π

∫ 1

− 1

w(θ)P"(cosθ)dcosθ.

For smallθand large", these go over to a form that looks like a flat sky, as
follows. Consider the asymptotic forms for the Legendre polynomials and theJ 0
Bessel function:

P"(cosθ)

√

2

π"sinθ

cos

[(

"+

1

2

)

θ−

1

4

π

]

J 0 (z)

√

2

πz

cos

[

z−

1

4

π

]

,

for respectively"→∞,z →∞; see chapters 8 and 9 of Abramowitz and
Stegun (1965). This shows that, for"1, we can approximate the small-angle