Physical Foundations of Cosmology

384 Cosmic microwave background anisotropies

where the integrals

Im

(

l/lf

)

≡

∫∞

1

(lnx)m

x^2

√

x^2 − 1

e−(l/lf)

(^2) x 2
dx (9.105)
can be calculated in terms of hypergeometric functions. However, the resulting
expressions are not very transparent and therefore it makes sense to find a numerical
fit for them. The final result is
N 1  0. 063 ξ^2

[

P− 0. 22

(

l/lf

) 0. 3

− 2. 6

] 2

1 + 0. 65

(

l/lf

) 1. 4 e−(l/lf)

2

. (9.106)

Similarly, we obtain

N 2 

0. 037

( 1 +ξ)^1 /^2

[

P− 0. 22 (l/lS)^0.^3 + 1. 7

] 2

1 + 0. 65 (l/lS)^1.^4

e−(l/lS)

2

. (9.107)

The Doppler contribution to the nonoscillating part of the spectrum is comparable
toN 2 and is equal to

N 3 

0. 033

( 1 +ξ)^3 /^2

[

P− 0. 5 (l/lS)^0.^55 + 2. 2

] 2

1 + 2 (l/lS)^2

e−(l/lS)

2

. (9.108)

The numerical fits forNreproduce the exact result for multipoles 200<l< 1000
to within a few percent accuracy for a wide range of cosmological parameters.
The ratio of the value ofl(l+ 1 )Clforl>200 to its value for low multipole
moments (the flat plateau) is

l(l+ 1 )Cl

(l(l+ 1 )Cl)lowl

=

100

9

(O+N 1 +N 2 +N 3 ), (9.109)

whereO,N 1 ,N 2 ,N 3 are given by (9.102), (9.106), (9.107) and (9.108) respectively.

The result in the case of theconcordance model( (^) m= 0. 3 ,
= 0. 7 ,
b= 0. 04 ,
(^) tot=1 andH=70 km s−^1 Mpc−^1 ) is presented in Figure 9.2, where we have
shown the total nonoscillating contribution and the total oscillatory contribution by
the dashed lines. Their sum is the solid line.
Our results are in good agreement with numerical calculations for a rather wide
range of cosmological parameters around the concordance model. Although nu-
merical codes are more precise, the analytic expressions enable us to understand
how the main features in the anisotropy power spectrum arise and how they depend
on cosmological parameters.