Physical Foundations of Cosmology

376 Cosmic microwave background anisotropies

10 100 1 10 100 1000

0

0.5

1.0

0.1 1

0

0.5

1.0

1.5

2.0

0.1

kηeq

Tp To

kηeq

Fig. 9.1.

few acoustic peaks and therefore are most interesting. Unfortunately, the transfer

functions in the intermediate region between two asymptotics can be calculated

only numerically. In general,TpandToshould depend onkandηeq,which by

dimensional analysis must enter in the combinationkηeq,and the baryon density. To

simplify the analysis, we will restrict ourselves to the case where the baryon density

is small compared with the total density of dark matter, (^) b (^) m,a practical limit
since this condition is satisfied by the real universe. With this assumption, we can
neglect the baryon contribution to the gravitational potential compared with the
contribution from cold dark matter, and the (^) b-dependence of the transfer functions
can be ignored. The result of the numerical calculation ofTpandToin the limit of
negligible baryon density is presented in Figure 9.1.
For intermediate range scales 10>kηeq>1, which give the leading contribu-
tions to the first acoustic peaks of the microwave background anisotropy, one can
approximateTpby
Tp 0 .25 ln

(

14

kηeq

)

, (9.67)

andToby

To 0 .36 ln

(

5. 6 kηeq

)

. (9.68)

The transfer functions are monotonic and approach their asymptotic values given

by (9.65) and (9.66) in the appropriate limits.