Physical Foundations of Cosmology

9.7 Anisotropies on small angular scales 383

Problem 9.13Using the stationary (saddle) point method, verify that for a slowly
varying functionf(x)

∫∞

1

f(x)cos(bx)

√

x− 1

dx

≈

f( 1 )

(

1 +B^2

) 1 / 4

√

π

b

cos

(

b+

π

4

+

1

2

arcsin

B

√

1 +B^2

)

, (9.100)

where

B≡

(

dlnf

bdx

)

x= 1

.

For largebwe can setB≈0 and the above formula becomes

∫∞

1

f(x)cos(bx)

√

x− 1

dx≈f( 1 )

√

π

b

cos

(

b+

π

4

)

(9.101)

(HintMake the substitutionx=y^2 +1 in (9.100).)

Using (9.101) to estimate the integrals in (9.75) and (9.76), we obtain

O

√

π

!l

(A 1 cos(l!+π/ 4 )+A 2 cos( 2 l!+π/ 4 ))e−(l/lS)

2

, (9.102)

with the coefficients

A 1  0. 1 ξ

((P− 0. 78 )^2 − 4 .3)

( 1 +ξ)^1 /^4

e

( 1

2

(

l−S^2 −l−f^2

)

l^2

)

,

A 2  0. 14

( 0. 5 + 0. 36 P)^2

( 1 +ξ)^1 /^2

, (9.103)

which are slowly varying functions ofl.In deriving this expression we used (9.97)
and (9.98) for the transfer functions, which are valid in the range of multipoles
200 <l< 1000 .Forl>1000 the fluctuations are strongly suppressed and this
effect is roughly taken into account by the exponential factor in (9.102). However,
the expected accuracy in this region is not as good as for 200<l< 1000.
We note that the contribution of the Doppler term toOis equal to zero in this
approximation. A precise numerical evaluation reveals that the Doppler contribution
to the oscillating integrals is small, only a few percent or less of the total, for multi-
polesl>200.
Substituting (9.97) in (9.80) for the nonoscillating contributionN 1 ,we obtain

N 1 ξ^2

[

( 0. 74 − 0. 25 P)^2 I 0 −( 0. 37 − 0. 125 P)I 1 +( 0. 25 )^2 I 2

]

, (9.104)