Dynamics of structure formation 55(1) Radiation effects. Prior to matter–radiation equality, we have already
seen that perturbations inside the horizon are prevented from growing by
radiation pressure. Oncezeqis reached, if collisionless dark matter dominates,
perturbations on all scales can grow. We therefore expect a feature in the transfer
function aroundk∼ 1 /rH(zeq). In the matter-dominated approximation, we get
dH=2 c
H 0(z)−^1 /^2 ⇒deq= 39 (h^2 )−^1 Mpc.The exact distance–redshift relation is
R 0 dr=c
H 0dz
( 1 +z)√
1 +mz+( 1 +z)^2 r,
from which it follows that the correct answer for the horizon size including
radiation is a factor
√
2 −1 smaller:deq= 16. 0 (h^2 )−^1 Mpc.
(2) Damping. In addition to having their growth retarded, very small-scale
perturbations will be erased entirely, which can happen in one of two ways.
For collisionless dark matter, perturbations are erased simply byfree-streaming:
random particle velocities cause blobs to disperse. At early times (kT>mc^2 ), the
particles will travel atc, and so any perturbation that has entered the horizon will
be damped. This process switches off when the particles become non-relativistic;
for massive particles, this happens long beforezeq(resulting in cold dark matter
(CDM)). For massive neutrinos, however, it happensat zeq: only perturbations
on very large scales survive in the case of hot dark matter (HDM). In a purely
baryonic universe, the corresponding process is calledSilk damping: the mean
free path of photons due to scattering by the plasma is non-zero, and so radiation
can diffuse out of a perturbation, convecting the plasma with it.
The overall effect is encapsulated in thetransfer function, which gives the
ratio of the late-time amplitude of a mode to its initial value:
Tk≡δk(z= 0 )
δk(z)D(z),
whereD(z)is the linear growth factor between redshiftzand the present. The
normalization redshift is arbitrary, so long as it refers to a time before any scale
of interest has entered the horizon.
It is invaluable in practice to have some accurate analytic formulae that fit the
numerical results for transfer functions. We give below results for some common
models of particular interest (illustrated in figure 2.8, along with other cases where
a fitting formula is impractical). For the models with collisionless dark matter,
Bis assumed, so that all lengths scale with the horizon size at matter–
radiation equality, leading to the definition
q≡k
h^2 Mpc−^1