Physical Foundations of Cosmology

9.7 Anisotropies on small angular scales 377

9.7.2 Multipole moments

To calculate the multipole momentsCl, we have to substitute (9.63) and (9.64) into

(9.38) with an extra exp(− 2 (σkηr)^2 ) factor inside the integrand to account for the

finite thickness effect. The resulting integral expressions are rather complicated,

but can be very much simplified forl1. We first remove the derivatives of the

spherical Bessel function in (9.38) using the identity

(

djl(y)

dy

) 2

=

[

1 −

l(l+ 1 )

y^2

]

jl^2 (y)+

1

2 y

d^2

(

yjl^2 (y)

)

dy^2

, (9.69)

which can be verified using the Bessel function equation. Substituting (9.69) into

(9.38) and integrating by parts, we obtain

Cl=

2

π

∫ [∣∣

∣

∣^ +

δ

4

∣∣

∣

∣

2

k^2 +

9

∣

∣δ′

∣

∣^2

16

(

1 −

l(l+ 1 )

(kη 0 )^2

)]

×( 1 +)e−^2 (σkηr)

2

jl^2 (kη 0 )dk, (9.70)

wheredenotes corrections of orderηr/η 0 and(kη 0 )−^1 ,which were estimated

taking into account (9.63) and (9.64) for the source functions. Recall thatηr/η 0 

zr−^1 /^2 ∼ 1 /30 is small. Asl→∞, we can approximate the Bessel functions as

jl(y)→

⎧

⎪⎨

⎪⎩

0 , y<ν,

1

y

12 (

y^2 −ν^2

) 14 cos

[√

y^2 −ν^2 −νarccos

(

ν

y

)

−

π

4

]

,y>ν,

(9.71)
whereν/y=1 is held fixed andν≡l+ 1 / 2 .Only those modes for whichy=
kη 0 >lcontribute to the integral and therefore the corrections of order(kη 0 )−^1 <
l−^1 1 can also be neglected. Using the approximation (9.71) in the integrand of
(9.70), and bearing in mind that the argument ofjl^2 (kη 0 )changes withkmuch more
rapidly than the argument in the oscillating part of the source functions (9.63) and
(9.64), we can replace the cosine squared coming from (9.71) by its average value
1 / 2 .The result is

Cl

1

16 π

∫∞

lη− 01

[

| 4 +δ|^2 k^2

(kη 0 )

√

(kη 0 )^2 −l^2

+

9

√

(kη 0 )^2 −l^2

(kη 0 )^3

∣∣

δ′

∣∣ 2

]

e−^2 (σkηr)

2

dk, (9.72)

where for largelwe have setl+ 1 ≈l.

Let us consider the case of a scale-invariant spectrum of initial density pertur-

bations,|

(

(^0) k

) 2

k^3 |=B,whereBis constant. Substituting (9.63) and (9.64) into

(9.72) and changing the integration variable tox≡kη 0 /l, we get a sum of integrals