Physical Foundations of Cosmology

7.4 Baryon–radiation plasma and cold dark matter 317

To find the general solution of the homogeneous equation (7.122) we employ the
WKB approximation. Let us consider a plane wave perturbation with wavenumber
kand introduce the new variable

yk(x)≡δγ(k,x)exp

(

2

5

k^2

∫

τγ

a

dx

)

. (7.124)

Then, it follows from (7.122) that the variableyksatisfies the equation

d^2 yk

dx^2

+

k^2

cs^2

[

1 −

4 c^2 s

25

(

kτγ

a

) 2

−

2 c^2 s

5

(τγ

a

)′]

yk= 0. (7.125)

First note that, for a perturbation whose physical wavelength (λph∼a/k) is much
larger than the mean free path of the photons (∼τγ),the second term in the square
brackets is negligible. The third term is roughlyτγ/aη∼τγ/t1 and can also
be skipped. With these simplifications, the WKB solution foryis

ykAk

√

cscos

(

k

∫

dx

cs

)

, (7.126)

whereAkis a constant of integration and the indefinite integral implies that the
argument of the cosine function includes an arbitrary phase. Given the definition
ofykin (7.124) and combining this solution with (7.123), we obtain the general
approximate solution of (7.122), valid atη>ηeq:

δγ(k,η)−

4

3 cs^2

(^) k

(

η>ηeq

)

+Ak

√

cscos

(

k

∫

csdη

)

e−(k/kD)

2

. (7.127)

Here we have converted back fromxto conformal timeηand introduced the
dissipation scale corresponding to the comoving wave number:

kD(η)≡

⎛

⎝^2

5

∫η

0

c^2 s

τγ

a

dη ̃

⎞

⎠

− 1 / 2

. (7.128)

In the limit of constant speed of sound and vanishing viscosity, the solution is
exact and valid not only for the short-wavelength perturbations but also for the
perturbations withkη1.

Problem 7.17Find the corrections to (7.127) and determine when the WKB solu-
tion is applicable.

Silk damping From (7.127), it is clear that the viscosity efficiently damps perturba-
tions on comoving scalesλ≤ 1 /kD. Viscous damping is due to the scattering and
mixing of the photons. Therefore, the scale where the viscous damping is important