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Over the last three or four years, the thoughts of theorists have narrowed on the birth and evolution of galaxies - including dark matter halos, plus the baryons we observe more directly. These adventuresome researchers have bi-modally attacked the problems with a pair of model types. Most contemporary modeling assumes, ab initio, the Lambda Cold Dark Matter cosmology (LCDM).

Following Weinberg et al., (1999), we note that the current (broad) theory of galaxy formation and early evolution follows White & Rees (1978) and their "successors" - gravitational collapse of a dark matter halo, gas falling into the potential well so defined, and then gas astrophysics (cooling, contracting, and eventually forming stars in a dense baryonic core). Now we often add inflationary cosmological parameters and thus demand Omegam + OmegaLambda = 1.

The "technology" for modeling often takes one of two paths. The first is hierarchical numerical simulations (with a realistic treatment of the collapse) including additional gas-phase physics and plenty of computational effort to cover the wide size range of non-spherical assemblies (the "roots" of the assembly "tree") that appear.

The second tool, deemed the semi-analytical approach, assumes again LCDM halos. The proto-galaxies contract within, and then we find small sub-galactic systems (or fragments?) with the specific physically-motivated "stories" given by the strengths of their star-burst mergers. The mergers obviously increase the model masses, and also modify the relative numbers of luminous stars and the amount of residual gas. Even before that step, the semi-analytic models utilize the Press-Schecter (Press & Schechter 1974) formalism to describe the number of halos as a function of their mass. This approach allows conventional and mature use of population synthesis and even chemical evolution schemes in conjunction with the mergers demanded to build up galaxies of reasonable mass with moderate star-formation rates.

One of the strengths of the direct numerical simulations is to utilize the non-spherical distribution of dark matter and baryons to produce a more realistic treatment of the model's gravitation. Then the more "astrophysical" computations can proceed; Weinberg et al., (1999) predict the surface densities of galaxies as a function of their star-formation rate (SFR) over a relevant range of redshifts.

The semi-analytic models (cf. Baugh et al. 1999; Somerville & Primak 1999) have now been amplified to include a range of interesting physical processes, hopefully relevant to early galaxy evolution. For example, the central baryons and the outer dark matter (DM) halo interact to change the halo structure and foster further contraction of the model galaxy. Baugh et al.(1998) mention that the main constraining property of local galaxies they favor for comparison with semi-analytic modeling is the field galaxy luminosity function. The agreement is good; one can then easily visualize the effect of omitting or including various individual physical processes, like star-formation (SF) feedback.

The SFR in the early Universe (say, to z = 3 or 4) of these models is also well-matched by observations of the SFR per unit volume. (Madau et al. 1998).

Weinberg et al. (1999) also show the numerical simulation's cumulative distribution of galaxies (with the parameter = surface density / box' / unit z) as a function of their SF rate from z = 10 to z = 0.5. At the moderately large galaxian SFR = 10 Modot yr-1 for z = 5, the predicted surface density of galaxies is nearly 5 / box'. This surface density is rather higher (by a factor of ~ 3) than observed by Spinrad and collaborators (although some of this observational statistic is derived from the Lyalpha - SFR correlation, which may be suspect). The best unpublished observational estimate for the SFR surface density at z ~ 5 is now 2±1 / box'. However, this surface density for Lyalpha emitters is uncertain because their continua are often very weak and thus not necessarily sampled consistently in terms of galaxy luminosity. The theoretical simulations and follow-up astrophysical scaling may, of course, be systematically over-efficient in, for example, converting cooling gas to massive star births.

The numerical simulations with LCDM may have one flaw: they over-predict the number of small galaxies near larger ones (which are countable) and thus the number of stars at low redshifts. We are not positive that a real problem exists; it may be that dark halos with coupled non-stellar baryons (e.g., high velocity clouds (Klypin et al., 1999) are being "counted" as observable systems.

The potential problems of early galaxy evolution from the theoretical side may well change, increasing or decreasing as their confrontations with empirical "facts" or new understandings go forward. The general outlines of the theory and relevant observations are probably fairly firm.

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