Previously we analyzed the growth of perturbations in an expanding universe via the use of fluid dynamical equations. The physical stability criteria for fluctuation growth is a competition between self-gravity within a fluid element and the internal pressure.
A critical length scale can be identified, over which self-gravity overcomes the internal pressure, and the solution to the fluid dynamics equations can be treated in a pressurless case.
The major result of this analysis is the identification of the time dependence of the growth of density fluctuations. In units of redshift this is specified as
Now we can perform a consistency check:
Recombination occurs at z = 1100 11003/2 is approximately 36,000. Since we know that there are factors of two overdensity in the Universe on large scales (we directly observe those), then a perturbation as small as 36,000-1 could have been present at recombination and been amplified by gravitational instability to produce the factor of two overdensity at z =0.
These leads to the simple expectation that the fluctuation level in the CBR should be a few x 10-5 which is essentially what is observed. The 4 year COBE data set has produced a measurement of 1.5 x 10-5 which is 66,000.
This can be taken as good evidence that the gravitational instability paradigm for structure formation must be correct at some fundmental level because it satisfies the COBE anisotropy constraints. However, the constraint is just barely satisified. For instance, if we observed density constrasts in excess of 4 on some large scale then that would compute back to an initial anistropy at recombination, which is too large.
Note, that the COBE anisotropy is measured over many horizons:
This is just barely consistent with what we measure today
In a Friedmann Universe (\eg homogenous and isotropic) the matter density as a function of redshift is given by:
where OMEGAo is the present value. From the observations previously discussed, OMEGAo appears to be in the range 0.1--0.3. In this case, at z = 10, OMEGA was at least 0.55 and at z = 100 it would be nearly 1. Hence, even in a low density Universe, the major time of perturbation growth from z =1100 to z = 10 would have occurred in the domain of OMEGA ~ 1 in which case the growth rate goes as
Note that in an open universe, there will be some redshift at which OMEGA does begin to significantly deviate from 1 leading to a much slower growth rate. Recall that for OMEGA=0, there is no time dependence on the growth of structure. Hence, structure formation in an open Universe effectively is over when OMEGA(z) approaches 0.
This condition is satisfied by
Thus if OMEGAo = 0.1, then structure formation by this process should be over at redshift z = 9. The fact that clusters of galaxies show substructure at z = 0 has been used as an argument that structure formation has not yet frozen out and therefore OMEGA = 1.
Recall that the Jeans mass at anytime in the Universe is a function of its temperature and density. For our case of an ideal gas, the Jeans mass for a spherical perturbation is