MODERN COSMOLOGY

54 An introduction to the physics of cosmology

equation:ρ ̇=−∇·(ρv)in proper units, which obviously takes the same form
ρ ̇=−∇·(ρu)if lengths and densities are in comoving units. If we express the
density asρ=ρ 0 ( 1 +δ)(where in comoving unitsρ 0 is just a number independent
of time), the continuity equation takes the form

δ ̇=−∇·[( 1 +δ)u],

which becomes just
∇·u=−δ ̇

in linear theory when bothδanduare small. This states that it is possible to have
vorticity modes with∇·u=0, for whichδ ̇vanishes. We have already seen that
δeither grows or decays as a power of time, so these modes require zero density
perturbation, in which case the associated peculiar gravity also vanishes. These
vorticity modes are thus the required homogeneous solutions, and they decay as
v=au∝a−^1 , as with the kinematic analysis for a single particle. For any
gravitational-instability theory, in which structure forms via the collapse of small
perturbations laid down at very early times, it should therefore be a very good
approximation to say that the linear velocity field must be curl-free.
For the growing modes, we want to try looking for a solutionu=F(t)g.
Then using continuity plus Gauss’s theorem,∇·g= 4 πGaρδ,givesus

δv=

2 f(
)

3 H

g,

where the functionf(
)≡(a/δ)dδ/da. A very good approximation to this
(Peebles 1980) isg
^0.^6 (a result that is almost independent of;Lahavet
al1991). Alternatively, we can work in Fourier terms. This is easy, asgandk
are parallel, so that∇·u=−ik·u=−iku. Thus, directly from the continuity
equation,

δvk=−

iHf(
)a

k

δkkˆ.

The 1/kfactor shows clearly that peculiar velocities are much more sensitive
probes of large-scale inhomogeneities than are density fluctuations. The existence
of large-scale homogeneity in density requiresn > −3, whereas peculiar
velocities will diverge unlessn>−1 on large scales.

2.6.4 Transfer functions

We have seen that power spectra at late times result from modifications of any
primordial power by a variety of processes: growth under self-gravitation; the
effects of pressure; dissipative processes. We now summarize the two main ways
in which the power spectrum that exists at early times may differ from that which
emerges at the present, both of which correspond to a reduction of small-scale
fluctuations (at least, for adiabatic fluctuations; we shall not consider isocurvature
modes here):