MODERN COSMOLOGY

Dynamics of structure formation 47

solution to a potential of unexplained form may seem a retrograde step. However,
it is at least a testable step: the prediction of figure 2.7 is thatw− 0 .8 today, so
that the quintessence density scales asρ∝a−^0.^6. This is a significant difference
from the classicalw=−1 vacuum energy, and it should be detectable as the
SNe data improve. The existing data already require approximatelyw<− 0 .5,
so there is the entrancing prospect that the equation of state for the vacuum will
soon become the subject of experimental study.

2.6 Dynamics of structure formation

The overall properties of the universe are very close to being homogeneous; and
yet telescopes reveal a wealth of detail on scales varying from single galaxies
tolarge-scale structuresof size exceeding 100 Mpc. This section summarizes
some of the key results concerning how such structure can arise via gravitational
instability.

2.6.1 Linear perturbations

The study of cosmological perturbations can be presented as a complicated
exercise in linearized general relativity; fortunately, much of the essential physics
can be extracted from a Newtonian approach. We start by writing down the
fundamental equations governing fluid motion (non-relativistic for now):

Euler:

Dv

Dt

=−

∇p

ρ

−∇

energy:

Dρ

Dt

=−ρ∇·v

Poisson: ∇^2 = 4 πGρ,

where D/Dt=∂/∂t+v·∇is the usual convective derivative. We now produce the
linearized equations of motionby collecting terms of first order in perturbations
about a homogeneous background:ρ=ρ 0 +δρetc. As an example, consider the
energy equation:

[∂/∂t+(v 0 +δv)·∇](ρ 0 +δρ)=−(ρ 0 +δρ)∇·(v 0 +δv).

For no perturbation, the zero-order equation is

(∂/∂t+v 0 ·∇)ρ 0 =−ρ 0 ∇·v 0 ;

sinceρ 0 is homogeneous andv 0 =Hxis the Hubble expansion, this just says
ρ ̇ 0 =− 3 Hρ 0. Expanding the full equation and subtracting the zeroth-order
equation gives the equation for the perturbation:

(∂/∂t+v 0 ·∇)δρ+δv·∇(ρ 0 +δρ)=−(ρ 0 +δρ)∇·δv−δρ∇·v 0.