MODERN COSMOLOGY

48 An introduction to the physics of cosmology

Now, for sufficiently small perturbations, terms containing a product of
perturbations such asδv·∇δρmust be negligible in comparison with the first-
order terms. Remembering thatρ 0 is homogeneous leaves the linearized equation

[∂/∂t+v 0 ·∇]δρ=−ρ 0 ∇·δv−δρ∇·v 0.

It is straightforward to perform the same steps with the other equations; the
results look simpler if we define the fractional density perturbation

δ≡

δρ

ρ 0

.

As before, when dealing with time derivatives of perturbed quantities, the full
convective time derivative D/Dtcan always be replaced by d/dt≡∂/∂t+v 0 ·∇,
which is the time derivative for an observer comoving with the unperturbed
expansion of the universe. We then can write

d

dt

δv=−

∇δp

ρ 0

−∇δ−(δv·∇)v 0

d

dt

δ=−∇·δv

∇^2 δ= 4 πGρ 0 δ.

There is now only one complicated term to be dealt with:(δv·∇)v 0 on the right-
hand side of the perturbed Euler equation. This is best attacked by writing it in
components:
[(δv·∇)v 0 ]j=[δv]i∇i[v 0 ]j=H[δv]j,

where the last step follows becausev 0 = Hx 0 ⇒∇i[v 0 ]j = Hδij.This
leaves a set of equations of motion that have no explicit dependence on the global
expansion speedv 0 ; this is only present implicitly through the use of convective
time derivatives d/dt.
These equations of motion are written in Eulerian coordinates: proper length
units are used, and the Hubble expansion is explicitly present through the velocity
v 0. The alternative approach is to use the comoving coordinates formed by
dividing the Eulerian coordinates by the scale factora(t):

x(t)=a(t)r(t)

δv(t)=a(t)u(t).

The next step is to translate spatial derivatives into comoving coordinates:

∇x=

1

a

∇r.

To keep the notation simple, subscripts on∇will normally be omitted hereafter,
and spatial derivatives will be with respect to comoving coordinates. The