MODERN COSMOLOGY

Cosmic background fluctuations 91

Here, the Bessel function comes from the angular part of the Fourier transform:
∫
exp(ixcosφ)dφ= 2 πJ 0 (x).

Now, in order to predict the observed anisotropy of the microwave
background, the problem we must solve is to integrate the temperature
perturbation field through thelast-scattering shell. In order to do this, we assume
that the sky is flat; we also neglect curvature of the 3-space, although this is only
strictly valid for flat models withk =0. Both these restrictions mean that the
results are not valid for very large angles. Now, introducing the Fourier expansion
of the 3D temperature perturbation field (with coefficientsTk3D) we can construct
the observed 2D temperature perturbation field by integrating overkspace and
optical depth:
δT
T

=

V

( 2 π)^3

∫∫

Tk3De−ik·rd^3 ke−τdτ.

A further simplification is possible if we approximate e−τdτby a Gaussian in
comoving radius:

exp(−τ)dτ∝exp[−(r−rLS)^2 / 2 σr^2 ]dr.

This says that we observe radiation from a last-scattering shell centred at
comoving distancerLS(which is very nearly identical torH, since the redshift
is so high). The thickness of this shell is of the order of the mean free path to
Compton scattering at recombination, which is approximately

σr= 7 (
h^2 )−^1 /^2 Mpc

(see p 287 of Peacock 1999).
The 2D power spectrum is thus a smeared version of the 3D one: any
feature that appears at a particular wavenumber in 3D will cause a corresponding
feature at the same wavenumber in 2D. A particularly simple converse to this
rule arises when there arenofeatures: the 3D power spectrum is scale-invariant
(T3D^2 =constant). In this case, for scales large enough that we can neglect the
radial smearing from the last-scattering shell,

T2D^2 =T3D^2

so that the pattern on the CMB sky is also scale invariant. To apply this machinery
for a general spectrum, we now need quantitative expressions for the spatial
temperature anisotropies.

Sachs–Wolfe effect.To relate to density perturbations, use Poisson’s equation
∇^2 δk= 4 πGρδk. The effect of∇^2 is to pull down a factor of−k^2 /a^2 (a^2
becausekis a comoving wavenumber). Eliminatingρin terms of
andzLS
gives

Tk=−


( 1 +zLS)

2

(

H 0

c

) 2

δk(zLS)

k^2

.