Physical Foundations of Cosmology

272 Gravitational instability in Newtonian theory

The Hubble velocityV 0 depends explicitly onxand therefore the Fourier transform
with respect to the Eulerian coordinatesxdoes not reduce these equations to a
decoupled set of ordinary differential equations. This is why it is more convenient
to use the Lagrangian (comoving with the Hubble flow) coordinatesq,which are
related to the Eulerian coordinates via

x=a(t)q, (6.39)

wherea(t) is the scale factor.The partial derivative with respect to time taken at
constantxis different from the partial derivative taken at constantq.For a general
functionf(x,t)we have
(
∂f(x=aq,t)
∂t

)

q

=

(

∂f

∂t

)

x

+aq ̇ i

(

∂f

∂xi

)

t

(6.40)

and therefore
(
∂
∂t

)

x

=

(

∂

∂t

)

q

−(V 0 ·∇x). (6.41)

The spatial derivatives are more simply related:

∇x=

1

a

∇q. (6.42)

Replacing the derivatives in (6.36)–(6.38) and introducing the fractional ampli-
tude of the density perturbationsδ≡δε/ε 0 , we finally obtain
(
∂δ
∂t

)

+

1

a

∇δv= 0 , (6.43)

(

∂δv

∂t

)

+Hδv+

c^2 s

a

∇δ+

1

a

∇δφ= 0 , (6.44)

δφ= 4 πGa^2 ε 0 δ, (6.45)

where∇≡∇qandare now the derivatives with respect to the Lagrangian coor-
dinatesqand the time derivatives are taken at constantq.In deriving (6.43) we have
used (6.33) for the background and noted that∇xV 0 = 3 Hand (δv·∇x)V 0 =Hδv.
Taking the divergence of (6.44) and using the continuity and Poisson equations to
express∇δvandδφin terms ofδ, we derive the closed form equation

̈δ+ 2 H ̇δ−c

2

s

a^2

δ− 4 πGε 0 δ= 0 , (6.46)

which describes gravitational instability in an expanding universe.