Physical Foundations of Cosmology

7.4 Baryon–radiation plasma and cold dark matter 319

Transfer functionsFor long-wavelength perturbations withkηr 1 ,the ampli-
tudes of the metric and radiation fluctuations after equality in terms of 0 are given
in (7.120) and (7.121). To find the transfer functions for short-wavelength inhomo-
geneities we have to express

(

η>ηeq

)

andAk,entering (7.127), in terms of (^0) k.
This can be done analytically in two limiting cases: for perturbations which enter
the horizonlong enough afterandwell beforeequality, that is, for the modes with
kηeq1 andkηeq 1.
The perturbation withkηeq1 enters the horizon when the cold matter already
dominates and determines the gravitational potential. Therefore the gravitational
potential does not change and it is given by
(^) k

(

η>ηeq

)

= 1090 k. (7.133)

The solution (7.127) forδγis applicable even when the wavelength of the perturba-
tion exceeds the curvature scale (see Problem 7.17). After equality atηηeq,the
amplitude ofδγfor the supercurvature modes withkη 1 ,according to (7.127),

should be equal toδγ− (^8) k/ 3 =const. Assuming that the baryons have a neg-
ligible effect on the speed of sound at this time, so thatc^2 s→ 1 /3, we find that
A= (^4) k/ 33 /^4. As a result, after equality but before recombination, we have
δγ(η)=

⎡

⎣−^4

3 c^2 s

+

4

√

cs

33 /^4

cos

⎛

⎝k

∫η

0

csdη ̃

⎞

⎠e−(k/kD)^2

⎤

⎦

(

9

10

(^0) k

)

(7.134)

for the modes withη−eq^1 kη−r^1.
Now we consider perturbations withkηeq 1 ,which enter the horizon well be-
fore equality. Atηηeq, radiation dominates over dark matter and baryons. There-
fore, andδγare well approximated by (7.61) and (7.62), describing perturbations
in the radiation-dominated universe. Neglecting the decaying mode on supercurva-

ture scales and expressingC 1 in terms of (^0) k, we find that atηeqηk−^1
δγ (^60) kcos

(

kη/

√

3

)

,k(η)−

(^90) k
(kη)^2
cos

(

kη/

√

3

)

. (7.135)

To determine the fluctuations in the cold dark matter component, we integrate
(7.108) to obtain

δd(η)= 3 (η)+

∫η

dη ̃

a

∫η ̃

adη. ̄ (7.136)

This is an exact relation which is always valid for anyk.During the radiation-domi-
nated epoch, the main contribution to the gravitational potential is due to radiation,
and, therefore, we can treat in (7.136) as an external source given by (7.61).
The two constants of integration can be fixed by substituting (7.61) in (7.136) and