Physical Foundations of Cosmology

378 Cosmic microwave background anisotropies

with “oscillating”(O)and “nonoscillating” functions(N)in the integrands, so that

l(l+ 1 )Cl

B

π

(O+N). (9.73)

Because of the cross-term that arises when the expression in (9.63) is squared in
the integrand of (9.72), the oscillating contribution tol(l+ 1 )Clcan be written as
a sum of two integrals:

O=O 1 +O 2 , (9.74)

where

O 1 = 2

√

cs

(

1 −

1

3 cs^2

)∫∞

1

TpToe

(

−^12

(

l−f^2 +l−S^2

) 2

l^2 x^2

)

cos(l!x)

x^2

√

x^2 − 1

dx (9.75)

and

O 2 =

cs

2

∫∞

1

To^2

(

1 − 9 c^2 s

)

x^2 + 9 c^2 s

x^4

√

x^2 − 1

e−(l/lS)

(^2) x 2
cos( 2 l!x)dx. (9.76)
Note that the periods of the cosines enteringO 1 andO 2 differ by a factor of 2.
As we will soon see, the acoustic peaks and valleys in the spectrum forl(l+ 1 )Cl
result from the constructive and deconstructive interference of these two terms. The
parameter

!≡

1

η 0

∫ηr

0

cs(η)dη (9.77)

determines the locations of the peaks.
The scaleslfandlS, which characterize the damping due to the finite thickness
and Silk dissipation effects, are equal to

l−f^2 ≡ 2 σ^2

(

ηr

η 0

) 2

; l−S^2 ≡ 2

(

σ^2 +(kDη)r−^2

)(ηr

η 0

) 2

, (9.78)

whereσis given in (9.60) andkDηris given roughly by (9.62).
Likewise, the nonoscillating contribution tol(l+ 1 )Clis a sum of three integrals:
N=N 1 +N 2 +N 3 , (9.79)

where

N 1 =

(

1 −

1

3 c^2 s

) 2 ∫∞

1

Tp^2 e−(l/lf)

(^2) x 2
x^2

√

x^2 − 1

dx (9.80)