Physical Foundations of Cosmology

7.4 Baryon–radiation plasma and cold dark matter 321

whereη∗=ηeq/(

√

2 −1).Neglecting the decaying mode and using the relation
between the gravitational potential andδd, verify that

(^) k

(

η>ηeq

)



ln

(

0. 15 kηeq

)

(

0. 27 kηeq

) 2

0

k. (7.142)

The fluctuations in the radiation component after equality continue to behave as
sound waves in the external gravitational potential given by (7.142). The integration
constantAin (7.127) can be fixed by comparing the oscillating part of this solution
to the result in (7.135) atη∼ηeq. Then we find that atη>ηeq,

δγ

⎡

⎣−^4

3 c^2 s

ln

(

0. 15 kηeq

)

(

0. 27 kηeq

) 2 +^35 /^4

√

4 cscos

⎛

⎝k

∫η

0

csdη

⎞

⎠e−(k/kD)^2

⎤

⎦ (^0) k (7.143)
for modes withkηeq1.
It follows from (7.134) and (7.143) that, fork>η−r^1 ,the spectrum ofδγat
recombination is partially modulated by the cosine. This is because all sound waves
with the samek=|k|enter the horizon and begin to oscillate simultaneously. As
we will see in the next chapter, this leads to peaks and valleys in the spectrum of
the temperature fluctuations of background radiation.
In summary, the results obtained in this section allow us to express the gravi-
tational potential and the radiation energy density fluctuations at recombination in
terms of the basic cosmic parameters and the primordial perturbation spectrum. The
primordial spectrum is described by the gravitational potential (^0) k, characterizing
a perturbation with comoving wavenumberkat very early times when its size still
exceeds the curvature scale. For modes whose wavelength exceeds the curvature
scale at recombination, the spectrum of remains unchanged except that its am-
plitude drops by a factor of 9/10 after matter–radiation equality, and the amplitude
of radiation fluctuations is given in (7.121). For perturbations whose wavelength is
less than the Hubble scale, we have derived asymptotic expressions for the modes
which enter the curvature scalewell beforeequality (see (7.142), (7.143)), and
long enough afterequality (see (7.133), (7.134)). For these perturbations the initial
spectrum is substantially changed as a result of evolution.