MODERN COSMOLOGY

Dynamics of structure formation 49

linearized equations for conservation of momentum and matter as experienced
by fundamental observers moving with the Hubble flow then take the following
simple forms in comoving units:

u ̇+ 2

a ̇

a

u=

g

a

−

∇δp

ρ 0

δ ̇=−∇·u,

where dots stand for d/dt. The peculiar gravitational acceleration∇δ/ais
denoted byg.
Before going on, it is useful to give an alternative derivation of these
equations, this time working in comoving length units right from the start. First
note that the comoving peculiar velocityuis just the time derivative of the
comoving coordinater:

x ̇= ̇ar+a ̇r=Hx+a ̇r,

where the right-hand side must be equal to the Hubble flow Hx,plusthe
peculiar velocityδv= au. In this equation, dots stand for exact convective
time derivatives—i.e. time derivatives measured by an observer who follows a
particle’s trajectory—rather than partial time derivatives∂/∂t. This allows us to
apply the continuity equation immediately in comoving coordinates, since this
equation is simply a statement that particles are conserved, independent of the
coordinates used. The exact equation is

D

Dt

ρ 0 ( 1 +δ)=−ρ 0 ( 1 +δ)∇·u,

and this is easy to linearize because the background densityρ 0 is independent of
time when comoving length units are used. This gives the first-order equation
δ ̇ =−∇·uimmediately. The equation of motion follows from writing the
Eulerian equation of motion asx ̈ = g 0 +g,whereg = ∇δ/ais the
peculiar acceleration defined earlier, andg 0 is the acceleration that acts on a
particle in a homogeneous universe (neglecting pressure forces, for simplicity).
Differentiatingx=artwice gives

x ̈=au ̇+ 2 a ̇u+

a ̈

a

x=g 0 +g.

The unperturbed equation corresponds to zero peculiar velocity and zero peculiar
acceleration: (a ̈/a)x = g 0 ; subtracting this gives the perturbed equation of
motionu+ 2 (a ̇/a)u=g, as before. This derivation is rather more direct than the
previous route of working in Eulerian space. Also, it emphasizes that the equation
of motion is exact, even though it happens to be linear in the perturbed quantities.
After doing all this, we still have three equations in the four variablesδ,u,
δandδp. The system needs an equation of state in order to be closed; this may