Physical Foundations of Cosmology

382 Cosmic microwave background anisotropies

recombination. This is why the parameter!and the location of the acoustic peaks
in a flat universe are sensitive to the cosmological constant.
The transfer functions TpandTodepend only on

kηeq=

ηeq

η 0

lx 0. 72

(

(^) mh^275

)− 1 / 2

Il 200 x, (9.95)

wherel 200 ≡l/200 andx≡kη 0 /l.

Problem 9.12Verify that

ηeq

η 0

=

(√

2 − 1

) I

√

zeq

 3. 57 × 10 −^3

(

(^) mh^275

)− 1 / 2

I. (9.96)

As we will see, for the most interesting range 1000>l>200, the dominant
contribution to the integrals in (9.73) comes fromxclose to unity for which 10>
kηeq>1. Therefore, we can use for the transfer functions the approximations (9.67)
and (9.68), which can be rewritten in terms ofxand cosmological parameters as

Tp(x) 0. 74 − 0. 25 (P+lnx) (9.97)

and

To(x) 0. 5 + 0. 36 (P+lnx), (9.98)

where the function

P(l,

m,h 75 )≡ln

⎛

⎝√Il^200

(^) mh^275

⎞

⎠ (9.99)

tells us how the transfer function determining the fluctuations in multipolelscales
depends on the cosmological term and cold matter energy density. The physical
reason for the dependence of the transfer functions on the matter density is explained
in Section 9.7.1.

9.7.4 Calculating the spectrum

We will now proceed to calculate the multipole spectruml(l+ 1 )Cl.The main
contribution to the integralsO 1 andO 2 in (9.75) and (9.76) arises in the vicinity of
the singular pointx= 1.