As in Eq. 3 this ratio could be evaluated at redshift zg if the galaxies were present then.
Gravity is scale-invariant at distances small compared with the Hubble length, and the expansion of the universe is scale-invariant or close to it. If the initial conditions were scale-invariant and we could ignore nongravitational interactions then structure formation would follow a scaling law where the length scale as a function of redshift is set by the spectrum of primeval mass density fluctuations. Eqs. 3–5 show that galaxies at redshift
resemble scaled versions of present-day clusters of galaxies.
We might pause to consider the slope of the power spectrum of density fluctuations that would scale protogalaxies to clusters. The ratio of comoving wavenumbers that enclose the masses defined by Eqs. 1 and 2 is
The ratio of the mean square primeval mass density fluctuations on these two scales is
The power spectrum is assumed to vary as P∝kn. The last expression is the ratio of the gravitational growth factor up to the present for clusters and up to redshift zg for protogalaxies, where D(Ω) is the departure from the Einstein-de Sitter growth factor. In a cosmologically flat model D(Ω) is close to unity, and Eqs. 7 and 8 with zg~10 require n~−1.3. This is not far from the measured slope of the power spectrum on larger scales where the present galaxy distribution is thought to be linearly related to the primeval mass distribution. That is, the scaling picture seems to be consistent with what is known about the present mass distribution.
Substructure shows that there is considerable ongoing merging and relaxation in present-day clusters of galaxies. Within this scaling picture one similarly would expect substantial merging and relaxation at z~10 in young galaxies that are considerably more irregular than they are now. The main point is that because nature manifestly is capable of producing bound mass concentrations on the scale of clusters now it is capable of doing about the same on the scale of galaxies at z=10.
The value for zg from this argument depends on the choice of rg in Eq. 2. A larger radius would yield a smaller value for zg, which in this picture would be interpreted as a characteristic epoch for the assembly of the extended dark halos of spiral galaxies. A substantially smaller value for rg would probe the baryon-dominated parts of galaxies, which certainly can behave differently from the CDM that is assumed to dominate clusters and the outer parts of individual spirals.
The scaling picture would fail if the mass concentration within the Abell radius of a cluster typically were evolving on the scale of the Hubble time, but this is inconsistent with the evidence that clusters are close to dynamical equilibrium at the Abell radius, and that many clusters at z=0.5 have central masses comparable to present-day systems [as discussed in this issue of the Proceedings by Mushotzky (3)]. The picture would fail if the spectrum of primeval density fluctuations were cut off at about the mass of a galaxy: pancake collapse at the cutoff might allow a larger collapse factor than is seen in clusters of galaxies. The point is worth exploring but not relevant to the adiabatic CDM model or the proposed isocurvature variant. I turn now to an issue that could be very relevant, the breaking of scaling by dissipation by gaseous baryons.
If protogalaxies were assembled well after redshift z=10 they would have to have collapsed by a considerable factor to reach the densities characteristic of the bright parts of galaxies. Gaseous baryons can be strongly dissipative, and dissipation can result in strong settling. This is a commonly held basis for the assumption of relatively late galaxy assembly, but some issues should be considered.
Fig. 1 shows models for galaxy assembly at relatively early and late epochs. The curves contain baryonic and total masses
MB=1×1011M⊙, M=6×1011M⊙. 
The former is typical of an L· spiral like the Milky Way. The ratio of baryonic to CDM mass is in the range now under discussion (4, 5); the uncertainty in the ratio does not affect these considerations much. The background cosmological model has density and Hubble parameters
Ω=0.3, h=0.7. 
If the mass distribution may be modeled as spherical and homogeneous (to avoid orbit crossing) then the relation between the nominal redshift zc at collapse to zero radius and the radius at maximum expansion may be approximated as
The second expression uses the numbers in Eqs. 9 and 10. The curves in Fig. 1 follow the spherical model through maximum expansion and nearly free collapse to a radius typical of a large spiral galaxy. In a spherical and homogeneous mass distribution the kinetic energy vanishes at maximum expansion, and collapse by a factor of two in radius produces enough kinetic energy for virial equilibrium at density contrast ~100. A real protogalaxy or protocluster likely has appreciable kinetic energy at nominal maximum expansion, which would tend to reduce the collapse factor. Thus it is not surprising that clusters are close to equilibrium at least as far out as the Abell radius