In summary, the available data on the properties of high redshift galaxies and QSO absorption lines suggest the following:

The initial stage of galaxy formation ,defined as when the first generation of stars is formed, occurred prior to z = 3 and is best identified with the formation of spheroids (either elliptical galaxies or spiral bulges). The formation of extended disks clearly takes a longer time and was apparently very active between z = 1-2. Vogt et al. (1996) show convincing evidence that objects with normal disk kinematics are in place by z = 1. The presence of these high redshift structures severely limits the amount of matter that can be obtained in any HDM model.

At z = 5 the universe is 7% of its present age or .7 - 1.4 billion years. QSOs have been detected at this redshift so we know that small-scale structure formation can occur on the 1 Gyr time scale. Its possible that these distant QSOs are the manifestation of galaxy formation and the formation of the first generation of stars. To generate the QSO activity requires the presence of a massive black hole. Possibly it is these massive black holes that have acted as the seeds to attract additional baryonic material. In fact, the origin of these massive black holes, 1 billion years after the birth of the universe is really quite interesting. If they are the evolved remnants of massive star clusters, then they obviously formed much earlier than z = 5.

The simple idea that a protogalaxy would form the bulk of its stars during the initial collapse is probably incorrect. Over a dynamical timescale (a few x 10⁸ years for galactic potentials), if most of the gas turns into stars then a star formation rate of 100-1000 M per year would result. While such a large star formation rate has been observed in some Ultraluminous IRAS galaxies (see Sanders et al. 1988), which are most likely the merger of two well formed galaxies, there are no objects at high redshift yet identified that exhibit this behavior. This is a strong argument that galaxy formation is not a quick process, marked by a very large star formation rate (and a very large supernova and metal-enrichment rate ), but perhaps is a far more quiescent and longer process. Indeed, detailed studies of elliptical galaxies at z = 0 now strongly suggest that there is a range of ages in their stellar populations and that their full formation occurred over several billion years (see Rose et al. 1995).

The role of feedback to the galaxy formation process either through supernova or the formation of QSOs is not yet well understood. If the Universe has been completely re-ionized by QSOs, the observations indicate that this occurred at z 5. Possibly this event served to further delay the general process of galaxy formation.

The observations of Steidel et al. that star formation in galaxies was well in place by z = 3.5 is difficult to understand in CDM models as this implies there was already small scale power by this redshift. Mo et al. (1997) demonstrate that the presence of small scale power at this redshift is greatly aided by non-zero as the time per unit redshift interval is greater in this case.

The morphology of objects in the HDF gives the strong visual impression that galaxy formation is occurring via an assembly line process in which small sub-units are being accreted into a larger entity. However, these sub-units are already composed of gas and stars so some process had to produce them at a much earlier epoch. Possibly, this process is the one physical process that we understand - simple Jeans mass collapse at high redshift. These (baryonic) sub-units then produce galaxies, via merging, as they respond to the underlying mass distribution which is dominated by dark matter. This is a potentially complex physical process that will challenge our understanding.

Pairwise and Peculiar Velocities

The final small scale constraint which can be considered is the average velocity and/or spatial separation between two random galaxies. Peculiar velocities that might arise from gravitational interactions between galaxies or between a galaxy and an overdense region such as a cluster cause deviations from Hubble flow but do not alter the position of the galaxy on the plane of the sky. Thus spatial correlation functions that are performed in physical space which may be isotropic become anisotropic when mapped onto redshift space (see discussion in Kaiser 1987). The amount of anisotropy in redshift space can be measured through the lower order moments of the peculiar velocity distribution. For galaxy pairs, the first moment of the distribution, v₁₂ is sensitive to the growth of the spatial or two-point correlation function. The second moment ₁₂ provides a direct measurement of the kinetic energy of any random motions. In the equilibrium gas, ₁₂ balances the gravitational potential and hence can be used to measure the effective mass. This is the situation in a cluster of galaxies in hydrostatic equilibrium.

For standard CDM, normalized to give the observed power on small scales, ₁₂ is predicted to be 1000 km s^-1. Open models in which ₀ 0.2 predict ₁₂ 500 km s^-1. The most recent determination of ₁₂ is based on a sample of 12,800 galaxies that comprise a well-defined subset of the Northern and Southern Sky Redshift surveys. The results (see Marzke et al. 1996) of this analysis are unfortunately ambiguous:

The measured ₁₂ is 540 ± 180 km s^-1. While this is larger than the 1983 measurement of 340 ± 40 km s^-1 (see Davis and Peebles 1983), it still does not effectively discriminate between open and closed CDM models.

The samples are "contaminated" by the presence of rich clusters where ₁₂ reflects the cluster velocity dispersion which is significantly higher than ₁₂ for field galaxies. This "contamination" is severe. When galaxies which are thought to be members of rich clusters are removed from the sample ₁₂ lowers significantly to 295 ± 100 km/s. In essence, this removal is accounting for the most non-linear structures that are present and these aren't necessarily a good probe of CDM structure formation scenarios. In this case, it would seem that the open Universe CDM models are strongly favored.

The amount of "contamination" depends on the volume of the redshift survey. Local samples are biased against selecting galaxies that are members of rich clusters and hence ₁₂ is biased to low values. This explains the low value originally measured by Davis and Peebles. If one uses the observed distribution of cluster velocity dispersions (see Zabludoff et al. 1993), it is possible to estimate how big a volume must be obtained in order for this "contamination" to not be a dominant effect in the sample. Marzke et al. (1996) estimate the required volume exceeds the volume of the existing redshift sample and therefore no fair sample yet exists to properly measure ₁₂. Nevertheless, the indications are that ₁₂ is relatively low and the small scale velocity field is therefore mostly quiescent.

This quiesence would appear to rule out most of the explosion models and the large scale hydrodynamic models of Cen and Ostriker (1994) . Since those models introduce a non-gravitational component to the peculiar velocity, they necessarily produce high ₁₂. However, one way to reduce ₁₂ is via galaxy-galaxy interactions and dynamical friction. Accounting for the possible role of mergers appears to make the ₁₂ measurements consistent with the predictions of high resolution hydrodynamic simulation such as those of Zurek et al. (1994). If this is true, however, the small scale clustering and dynamical properties of galaxies would then be probing the evolution of galaxy merging more than structure formation scenarios. A recent analysis of the Canadian deep redshift survey finds strong evidence for increased merger activity out to z 0.3 and derives a merging rate that goes as (1 + z)^{2.9 ± 0.9}, consistent with the expected (1 + z)³ dependence (see Patton et al. 1996).

In fact, the effects of merging, which means that the number density of galaxies as a function of redshift is not conserved, has serious implications on the use of small scale structure to constrain structure formation scenarios. This is because merging greatly modifies what is observed on small scales and leads to an overall decrease in the galaxy density on small scales if numerous small galaxies merger into one larger galaxy. Hence the pairwise velocity dispersion, the two point correlation function, and the amount of power on small scales could all be modified, by merging, from their original values. Such a signature, however, would be clearly detected as a strong redshift evolution of these quantities. At present, insufficient data exists to search for this signature.