MODERN COSMOLOGY

90 An introduction to the physics of cosmology

2.8.3 The temperature power spectrum

The statistical treatment of CMB fluctuations is very similar to that of spatial
density fluctuations. We have a 2D field of random fluctuations in brightness
temperature, and this can be analysed by the same tools that are used in the case
of 2D galaxy clustering.
Suppose that the fractional temperature perturbations on a patch of sky of
sideLare Fourier expanded:

δT

T

(X)=

L^2

( 2 π)^2

∫

TKexp(−iK·X)d^2 K

TK(K)=

1

L^2

∫

δT

T

(X)exp(iK·X)d^2 X,

whereXis a 2D position vector on the sky, andKis a 2D wavevector. This is
only a valid procedure if the patch of sky under consideration is small enough to
be considered flat; we give the full machinery later. We will normally take the
units of length to be angle on the sky, although they could also in principle be
h−^1 Mpc at a given redshift. The relation between angle and comoving distance
on the last-scattering sphere requires the comoving angular-diameter distance to
the last-scattering sphere; because of its high redshift, this is effectively identical
to the horizon size at the present epoch,RH:

RH=

2 c


mH 0

(open)

RH

2 c


^0 m.^4 H 0

(flat);

the latter approximation for models with
m+
v=1 is due to Vittorio and Silk
(1991).
As with the density field, it is convenient to define a dimensionless power
spectrum of fractional temperature fluctuations,

T^2 ≡

L^2

( 2 π)^2

2 πK^2 |TK|^2 ,

so thatT^2 is the fractional variance in temperature from modes in a unit range
of lnK. The corresponding dimensionless spatial statistic is the two-point
correlation function

C(θ)=

〈

δT

T

(ψ)

δT

T

(ψ+θ)

〉

,

which is the Fourier transform of the power spectrum, as usual:

C(θ)=

∫

T^2 (K)J 0 (Kθ)

dK

K

.