MODERN COSMOLOGY

Quantifying large-scale structure 71

direction on the sky as a radial integral over the spatial overdensity, with a
weighting ofy^2 φ(y), we see that the angular correlation function is

〈δ(qˆ 1 )δ(qˆ 2 )〉=

∫∫

〈δ(y 1 )δ(y 2 )〉y 12 y 22 φ(y 1 )φ(y 2 )dy 1 dy 2.

We now change variables to the mean and difference of the radii,y≡(y 1 +y 2 )/2;
x≡(y 1 −y 2 ). If the depth of the survey is larger than any correlation length, we
only get a signal wheny 1 y 2 y. If the selection function is a slowly varying
function, so that the thickness of the shell being observed is also of order of the
depth, the integration range onxmay be taken as being infinite. For small angles,
we then obtainLimber’s equation:

w(θ)=

∫∞

0

y^4 φ^2 dy

∫∞

−∞

ξ

(√

x^2 +y^2 θ^2

)

dx

(see sections 51 and 56 of Peebles 1980). Theory usually supplies a prediction
about the linear density field in the form of the power spectrum, and so it is
convenient to recast Limber’s equation:

w(θ)=

∫∞

0

y^4 φ^2 dy

∫∞

0

π^2 (k)J 0 (kyθ)dk/k^2.

This power-spectrum version of Limber’s equation is already in the form required
for the relation to the angular power spectrum, and so we obtain the direct small-
angle relation between spatial and angular power spectra:

^2 θ=

π

K

∫

^2 (K/y)y^5 φ^2 (y)dy.

This is just a convolution in log space, and is considerably simpler to evaluate and
interpret than thew–ξversion of Limber’s equation.
Finally, note that it is not difficult to make allowance for spatial curvature in
this discussion. Write the RW metric in the form

c^2 dτ^2 =c^2 dt^2 −R^2

[

dr^2

1 −kr^2

+r^2 θ^2

]

;

fork = 0, the notationy = R 0 rwas used for comoving distance, where
R 0 = (c/H 0 )| 1 −
|−^1 /^2. The radial increment of comoving distance was
dx=R 0 dr, and the comoving distance between two objects was(dx^2 +y^2 θ^2 )^1 /^2.
To maintain this version of Pythagoras’s theorem, we clearly need to keep the
definition ofyand redefine radial distance: dx = R 0 drC(y),whereC(y)=
[ 1 −k(y/R 0 )^2 ]−^1 /^2. The factorC(y)appears in the non-Euclidean comoving
volume element, dV ∝y^2 C(y)dy, so that we now require the normalization
equation forφto be ∫
∞

0

y^2 φ(y)C(y)dy= 1.