Structure Formation 231exist nonlinear modes they are outside it (the Jeans wavelength is greater than the
horizon). A mass scale푀is said to enter the Hubble radius when푀=푀H.Wellinside
the Hubble radius, the fluctuations may start to grow as soon as the Universe becomes
matter dominated, which occurs at time푡eq≈54 500yr.
Upon recombination, the baryonic Jeans mass falls dramatically. If the fluid is com-
posed of some nonbaryonic particle species (cold dark matter), the Jeans wavelength
is small after radiation–matter equality, allowing subhorizon perturbations to grow
from this time. After matter–radiation equality, nonbaryonic matter can form poten-
tial wells into which baryons can fall after recombination.
Matter can have two other effects on perturbations. Adiabatic fluctuations lead to
gravitational collapse if the mass scale is so large that the radiation does not have time
to diffuse out of one Jeans wavelength within the time푡eq. As the Universe approaches
decoupling, the photon mean free path increases and radiation can diffuse from over-
dense regions to underdense ones, thereby smoothing out any inhomogeneities in
the plasma. For wavelengths below the Jeans wavelength,collisional dissipationor
Silk damping(afterJ. Silk) erases perturbations in the matter (baryon) radiation field
through photon diffusion. This becomes most important around the time of recombi-
nation. Random waves moving through the medium with the speed of sound푐serase
all perturbations with wavelengths less than푐s푡eq. This mechanism sets a lower limit
to the size of the structures that can form by the time of recombination: they are
not smaller than rich clusters or superclusters. But, in the presence of nonbaryonic
matter, Silk damping is of limited importance because nonbaryonic matter does not
couple with the radiation field.
The second effect is free streaming of weakly interacting relativistic particles such
as neutrinos. This erases perturbations up to the scale of the horizon, but this also
ceases to be important at the time of matter–radiation equality.
The situation changes dramatically at recombination, when all the free electrons
suddenly disappear, captured into atomic Bohr orbits, and the radiation pressure
almost vanishes. This occurs at time 400000yr after Big Bang (see Figure 6.5).
Now the density perturbations which have entered the Hubble radius can grow with
full vigour.
Sunyaev–Zel’dovich Effect (SZE). At some stage the hydrogen gas in gravitation-
ally contracting clouds heats up enough to become ionized and to re-ionize the CMB:
the Sunyaev–Zel’dovich effect. We refer to the Planck result in Equation (8.46) that
such re-ionization clouds occur at a half-reionization redshift푧r≈^11 .15.
The free electrons and photons in the ionized clouds build up a radiation pres-
sure, halting further collapse. The state of such clouds today depends on how much
mass and time there was available for their formation. Small clouds may shrink
rapidly, radiating their gravitational binding energy and fragmenting. Large clouds
shrink slowly and cool by the mechanism of electron Thomson scattering. As the
recombination temperature is approached the photon mean free paths become larger,
so that radiation can diffuse out of overdense regions. This damps the growth of
inhomogeneities.