Annu. Rev. Astron. Astrophys. 2000. 38: 667-715
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5.1. Galaxy counts vs. simple models

The time-honored method of comparing cosmological models to field-galaxy surveys has been through the classical number-magnitude, size-magnitude, magnitude-redshift, etc. relations [Sandage 1988]. Much of this effort, including early results from the HDF, has been reviewed in [Ellis 1997]. There are three important changes to the scientific landscape that we must acknowledge before proceeding to discuss the more recent interpretation of the HDF number counts. The first change is the transition (motivated by high-z SNe Ia, cluster baryon fractions, etc.) from a favored cosmology with OmegaM = Omegatot = 1 to one with OmegaM, OmegaLambda, Omegatot = 0.3, 0.7, 1. The latter cosmology comes closer to matching the galaxy counts without extreme amounts of evolution. The second change is the realization that galaxy counts at HDF depths push to depths where the details of galaxy formation become extremely important. Models that posit a single "epoch of galaxy formation" were never realistic, and are no longer very useful. Finally, it has become more popular to model quantities such as the metal enrichment history dot{rho}Z(z), than it is to model galaxy counts. Consequently, to our knowledge there are no published papers that compare the overall N(m) predictions of hierarchical semi-analytic models with the currently favored cosmology to the HDF galaxy counts.

5.1.1. No-evolution (NE) models

The assumption of no evolution is not physically reasonable, but provides a useful fiducial for identifying how much and what kinds of evolution are required to match faint-galaxy data. Traditional no-evolution models are based on estimates of the z = 0 luminosity functions for different types of galaxies. [Bouwens et al. 1997] construct a non-evolving model from the HDF itself, using 32 galaxies brighter than I814 = 22.3 to define a fiducial sample. They construct Monte-Carlo realizations of the HDF that might be seen from a universe uniformly populated with such galaxies, shifted to different redshifts and k-corrected on a pixel-by-pixel basis. This kind of simulation automatically incorporates the selection and measurement biases of the HDF at faint magnitudes. It also normalizes the model by fiat to match the counts at I814 = 22.3, where traditional no-evolution models, normalized to the local luminosity function, already see significant discrepancies for the Einstein-de Sitter model. The resulting models underpredict the HDF counts at I814 = 27 by factors of 4 and 7 for models with OmegaM = 0.1 and 1.0 (with OmegaLambda = 0), respectively. The angular sizes of galaxies in the models are also too big at faint magnitudes, with a median half-light radius about a factor of 1.5 larger than that observed for galaxies with 24 < I < 27.5. Because the typical redshift of the template galaxies in this model is z ~ 0.5, this model comparison suggests that much of the evolution in galaxy number densities, sizes, and luminosities occurs at higher redshift. The typical small sizes of faint galaxies essentially rule out low-surface-brightness galaxies [Ferguson & McGaugh 1995, McLeod & Rieke 1995] as a significant contributor to the counts at magnitudes I814 > 20 [Ferguson 1999].

5.1.2. Pure-Luminosity Evolution models

Many of the models that have been compared to the HDF number counts are variants of pure luminosity evolution (PLE) models [Tinsley 1978], wherein galaxies form at some redshift zf, perhaps varying by type, with some star-formation history psi(t). There is no merging. [Metcalfe et al. 1996] compare several different models to the counts and colors of galaxies in the HDF and in deep images taken at the William Herschel Telescope. For low Omega a reasonable fit to I approx 26 is achieved, but the model progressively underpredicts the counts to fainter magnitudes, and the long star-formation timescales and heavily dwarf-dominated IMF adopted for ellipticals in this model seems inconsistent with the fossil evidence in local ellipticals. By including a simple prescription for dust attenuation, [Campos & Shanks 1997] are able to achieve a reasonable fit to the counts for low OmegaM without resorting to a peculiar IMF. Another set of PLE models was considered by [Pozzetti et al. 1998], with emphasis on the near-UV counts and the effect of UV attenuation by intergalactic neutral hydrogen. From color-magnitude relations and a study of the fluctuations in the counts in the different HDF bands, [Pozzetti et al. 1998] conclude that at B450 = 27 roughly 30% of the sources in the HDF are at z > 2. The PLE model considered has OmegaM = 0.1 with no cosmological constant, and with a redshift of formation zf = 6.3. This model matches the counts quite well but predicts that about 80 objects brighter than V606 = 28 should disappear from the F450W band because of their high redshift, although [Madau et al. 1996] identify only about 15 such sources. [Ferguson & Babul 1998] considered another low- Omega PLE model and encountered similar problems, predicting roughly 400 B-band Lyman-break objects where only 15 or so were observed. The utility of PLE models clearly breaks down for z gtapprox 2, where the details of galaxy formation become critical.

5.1.3. Models with Additional Galaxy Populations

The great difficulty in achieving a fit with OmegaM = Omegatot = 1, even to ground-based galaxy counts, motivated investigations into different kinds of galaxies that might be missed from the census of the local universe but could contribute to the counts of galaxies at faint magnitudes [Ferguson & McGaugh 1995, Babul & Ferguson 1996, Koo et al. 1993, McLeod & Riek1995]. Perhaps the most physically motivated of these more exotic possibilities is the idea that the formation of stars in low-mass galaxy halos could be inhibited until low redshifts z ltapprox 1 because of photoionization by the metagalactic UV radiation field [Efstathiou 1992, Babul & Rees 1992]. [Ferguson & Babul 1998] compared the predictions of such an OmegaM = 1 "disappearing dwarf" model in detail to the HDF. They found that the simplest version of the model (a) overpredicts the counts at faint magnitudes, and (b) overpredicts the sizes of very faint galaxies. These problems are caused by the fact that, for a Salpeter IMF, the dwarfs fade too slowly and would still be visible in great numbers in the HDF at redshifts z < 0.5. [Campos 1997] considered a model with much milder evolution, with each dwarf undergoing a series of relatively long (a few × 108 yr) star-formation episodes. Acceptable fits to the counts and colors of galaxies are achieved both for high and low values of Omega. Both the [Ferguson & Babul 1998] and the [Campos 1997] models predict that the HDF sample at I > 25 is dominated by galaxies with z < 1, a result inconsistent with existing photometric-redshift measurements. Using a volume-limited photometric redshift sample to construct the bivariate brightness distribution of galaxies with 0.3 < z < 0.5, [Driver 1999] concludes that the volume density of low-luminosity, low-surface-brightness galaxies is not sufficient to explain the faint-blue excess either by themselves or as faded remnants. Further constraints on the low-redshift, low-luminsity population can be expected from the HDF STIS UV observations.

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