Physical Foundations of Cosmology

7.3 Hydrodynamical perturbations 301

Hubble radius. The behavior of the energy density perturbations, however, depends
crucially on the scale.
Let us consider a plane wave perturbation with comoving wavenumberk≡|k|,
for whichC 1 , 2 ∝exp(ikx).If the physical scaleλph∼a/kis much larger than the
Hubble scaleH−^1 ∼aηor, equivalently,kη 1 ,then to leading order

δε

ε 0

− 2 C 1 +

3 C 2

η^5

. (7.55)

Neglecting the decaying mode, the relation between the energy density fluctuations
and the gravitational potential on superhorizon scales becomesδε/ε 0 − 2.
For shortwave perturbations withkη 1 ,we have
δε
ε 0

−

k^2

6

(

C 1 η^2 +C 2 η−^3

)

=C ̃ 1 t^2 /^3 +C ̃ 2 t−^1 , (7.56)

in agreement with the Newtonian result (6.56)

Problem 7.8Determine the behavior of the peculiar velocity for nonrelativistic
matter.

Problem 7.9Substituting (7.53) and (7.54) in (7.29) and (7.30 ), calculate the met-
ric and energy density perturbations in the synchronous coordinate system. Analyze
the behavior of the long- and short-wavelength perturbations. Is the Newtonian limit
explicit in this coordinate system?

Ultra-relativistic matterLet us now study the behavior of adiabatic perturbations
(δS= 0 )in the universe dominated by relativistic matter with equation of state
p=wε,wherewis a positive constant. In this case the scale factor increases as
a∝η^2 /(^1 +^3 w)(see Problem 1.18). Withc^2 s=w, and for a plane wave perturbation

= (^) k(η)exp(ikx),(7.51) becomes
′′k+
6 ( 1 +w)
1 + 3 w

1

η

′k+wk^2 k= 0. (7.57)

The solution of this equation is

(^) k=η−ν

[

C 1 Jν

(√

wkη

)

+C 2 Yν

(√

wkη

)]

,ν≡

1

2

(

5 + 3 w

1 + 3 w

)

, (7.58)

whereJνandYνare Bessel functions of orderν.
Considering long-wavelength inhomogeneities, for which

√

wkη 1 ,and us-
ing the small-argument expansion of the Bessel functions, we see that in this limit
the nondecaying mode of is constant. It follows from (7.47) that

δε/ε 0 − 2 . (7.59)