Physical Foundations of Cosmology

302 Gravitational instability in General Relativity

√Perturbations on scales smaller than the Jeans lengthλJ∼cst,for which
wkη 1 ,behave as sound waves with decaying amplitude

(^) k∝η−ν−
(^12)
exp

(

±i

√

wkη

)

. (7.60)

In aradiation-dominateduniverse(w= 1 / 3 ),the order of the Bessel functions
isν= 3 /2 and they can be expressed in terms of elementary functions. We have

(^) k=

1

x^2

[

C 1

(

sinx

x

−cosx

)

+C 2

(cosx

x

+sinx

)]

, (7.61)

wherex≡kη/

√

3. The corresponding energy density perturbations are

δε

ε 0

= 2 C 1

[(

2 −x^2

x^2

)(

sinx

x

−cosx

)

−

sinx

x

]

+ 4 C 2

[(

1 −x^2

x^2

)(

cosx

x

+sinx

)

+

sinx

2

]

. (7.62)

General case Unfortunately, (7.51) cannot be solved exactly for an arbitrary equa-
tion of statep(ε).However, it turns out to be possible to derive asymptotic solu-
tions for both long-wavelength and short-wavelength perturbations. To do this, it
is convenient to recast the equation in a slightly different form. The “friction term”
proportional to ′can be eliminated if we introduce the new variable

u≡exp

(

3

2

∫(

1 +c^2 s

)

Hdη

)

(7.63)

=exp

(

−

1

2

∫(

1 +

p 0 ′

ε 0 ′

)

ε′ 0

(ε 0 +p 0 )

dη

)

=

(ε 0 +p 0 )^1 /^2

,

where we have usedc^2 s=p 0 ′/ε′ 0 and expressedHin terms ofεandpvia the
conservation lawε′=− 3 H(ε+p).After some tedious calculations, and using
the background equations (see (1.67), (1.68))

H^2 =

8 πG

3

a^2 ε 0 , H^2 −H′= 4 πGa^2 (ε 0 +p 0 ), (7.64)

the equation forucan be written in the form

u′′−c^2 su−

θ′′

θ

u= 0 , (7.65)

where

θ≡

1

a

(

1 +

p 0

ε 0

)− 1 / 2

=

1

a

(

2

3

(

1 −

H′

H^2

))− 1 / 2

. (7.66)