Physical Foundations of Cosmology

368 Cosmic microwave background anisotropies

represents pristine information about the primordial inhomogeneities. In this section
we show that for perturbations predicted by inflation, the spectrum of temperature
fluctuations on large angular scales has a nearly flat plateau, the height and slope of
which are mainly determined by the amplitude and spectral index characterizing the
primordial inhomogeneities and practically independent of the other cosmological
parameters.
The Hubble scale at recombinationHr−^1 = 3 tr/2 spans 0. 87 ◦on the sky today
(see (2.73)). Therefore, the results derived in this section refer to the anglesθ 1 ◦,
or to the multipoleslπ/θH∼ 200.
As shown in Section 7.4, for adiabatic perturbations withkηr 1 ,the rela-
tive energy density fluctuations in the radiation component itself can be expressed
through the gravitational potential as

δk(ηr)−

8

3

(^) k(ηr),δ′k(ηr) 0. (9.41)
According to (9.32), the resulting temperature fluctuation due to large-scale imho-
mogeneities is
δT
T
(η 0 ,x 0 ,l)

1

3

(ηr,x 0 −lη 0 ). (9.42)

That is, the fluctuation amplitude is equal to one third of the gravitational potential
at the point on the last scattering surface from which the photons emanated. In this
estimate, we neglect the contribution of radiation to the gravitational potential at
recombination and both integrated Sachs–Wolfe effects, which are subdominant.
After matter–radiation equality, the potential on superhorizon scales drops by a
factor of 9/10. Taking this into account, substituting (9.41) into (9.38), and calcu-
lating the integral with the help of the identity

∫∞

0

sm−^1 jl^2 (s)ds= 2 m−^3 π

( 2 −m)

(

l+

m

2

)

^2

(

3 −m

2

)



(

l+ 2 −

m

2

), (9.43)

we find for a scale-invariant initial spectrum with|

(

(^0) k

) 2

k^3 |=B,the plateau:

l(l+ 1 )Cl

9 B

100 π

=const, (9.44)

on large angular scales or forl200. Since the main contribution to large an-
gular scales comes from superhorizon inhomogeneities, we have neglected here
the modification of the spectrum for subhorizon modes. In actuality, eachClis a
weighted integral over allk, including near horizon and subhorizon scales, where
the fluctuation amplitude rises and falls. The above result is a good approximation