Annu. Rev. Astron. Astrophys. 1992. 30: 613-52
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3.3 A Conventional Model to Fit the Observations

We have constructed a new model for faint galaxy statistics to evaluate some of the claims described earlier concerning the distribution of counts, colors, and redshifts. In particular, this model allows us to evaluate the faint K-band counts on the same basis as the counts at shorter wavelengths. The galaxy spectral energy distributions are those of Bruzual (1988 version of the models), and are basically similar to those of Guiderdoni and Rocca-Volmerange (1987). The modelling procedure is elaborated in Bruzual (1988, 1990), except that we now constrain the model with the observed redshifts (Figure 5).

We adjust the intrinsic colors and luminosities to match bright-galaxy data. Valuable guides in this work have been the (K, B - K, z) distribution of a sample of bright galaxies by Mobasher et al. (1986), and the luminosity function parameters suggested by Shanks (1990; see also Metcalfe et al. 1991) according to categories of rest-frame B - V. An important spectral energy distribution is that which results from a constant rate of star formation and a Salpeter initial mass function (which can be thought of as a kind of ``default'' model galaxy). It has B - V = 0.56 at the present time, and the adopted luminosity function corresponding to this spectral energy distribution has a steep slope (alpha = -1.5 in the Schechter formulation) to fainter luminosities. Bluer galaxies, which are essential to match both the bright and faint color distributions, are modelled as having also a constant star formation rate, but with a mass function that is relatively enhanced in massive stars. Five such components were added to reproduce at least qualitatively the color-luminosity relations of Huchra (1977) and Bershady et al. (1990), i.e., the bluest galaxies appear selectively at the lowest luminosities. Redder galaxies at the present epoch are the ``µ-models'' of Bruzual with a flat luminosity function (alpha = -1) at faint luminosities. Four such components were added to reproduce the redder galaxies; as expected, these are especially important in modelling of the K-band data. In Bruzual's scheme for luminosity evolution (rate of star formation depending on the available gas), these redder components become more important at higher redshifts because of their enhanced luminosities. Our no-evolution calculation uses the spectral energy distributions as they are at an age of 16 Gyr, and the bluer classes are generally of greater significance because of their smaller K-corrections. In the following we explore just the results of this no-evolution case, which provides a working null hypothesis with the fewest extra parameters.

The spectral energy distributions predict V - K colors for elliptical galaxies that are too blue by 0.3 mag compared to direct measurements. A potentially more serious problem is the funnelling of all models to a common very blue observed color, bJ - rF ~ 0, at z > 1. The colors of galaxies at z > 1 may actually be spread over a redder range by internal extinction, which would explain the absence of an observed spike at bJ - rF ~ 0 in the faintest intervals shown in Figure 2. The ability of the models to match the observed color-redshift relation (Figure 3) is however some testimony to the overall accuracy of the spectral energy distributions, at least for z < 0.7

With respect to models of other investigators, ours is relatively complex regarding the specification of the input galaxy parameters, but we find this complexity to be necessary, since the broad range of observed colors at all magnitudes suggests that the counts are not dominated by any small number of categories of galaxies.

We have adopted q0 = 0 because the enhanced volume element with respect to q0 = 0.5 provides the needed higher faint counts. As found by Guiderdoni and Rocca-Volmerange (1990) and others, low q0 and H0, and correspondingly large ages for present-day galaxies, make the predicted evolutionary effects at a given redshift smaller, another desirable property.

In order to keep the results general, we have not modelled photometric errors nor the incompleteness of image detection, although as we stressed earlier, any detailed comparison to data should do so.

The specific model components - principally, the density and luminosity normalizations for the individual luminosity functions assigned to the respective spectral energy distributions - were determined interactively. [In principle, the procedure could be automated - see Bruzual (1990).] Of prime importance were the observed bJ - rF color distributions (Figure 2) and redshift distributions (Figure 5). We allowed the no-evolution model to under-predict the redshifts observed in faint samples (Figure 5) in anticipation that mild luminosity evolution would provide a better fit. The color distributions at bright magnitudes are fit by construction. The success of the model to fit the color distribution at faint magnitudes checks that the spectral energy distributions, and their assigned distribution over luminosity, are at least adequate. The key unconstrained prediction is the galaxy counts, shown in Figure 1.

Our no-evolution model for the blue counts curves gradually from the Euclidean expectation at bright magnitudes to flatter slopes at fainter magnitudes. When normalized to galaxy counts at B = 18, it falls a factor of 2 to 3 short of the data some 8 magnitudes fainter (and most of this excess is already apparent by bJ ~ 23, i.e., within reach of spectroscopic investigation). A similar situation pertains to the red counts, whereas the no-evolution model fits the K-band counts very well. Considering all of the uncertainties in both the data and the model, we regard these simultaneous fits to be remarkably good. Any enhancement of star formation rates in the past will presumably serve to raise selectively the faint blue and red counts with respect to the K-band counts. Since even the no-evolution model predicts that at B = 25.5 the median redshift is appreciable, z = 0.9, any evolutionary effects are evidently mild so as not to predict too many high-redshift galaxies at brighter magnitudes.

As indicated earlier, the model spectral energy distributions, if taken at face value, produce a spike in color at bJ - rF ~ 0 for bJ > 25. Even if this feature were smeared out by observational error, the predicted color distribution at faint limits is still far bluer than actually observed, and the problem is aggravated if the star formation rate is higher at earlier epochs.

Maddox et al. (1990) suggested that there was evidence for evolution at low redshifts because their counts were steeper than a no-evolution model in the range 17 < B < 19. Figure 1 shows that in this range of magnitudes the data are steeper than even our model, which contains very blue and very low-luminosity galaxies. In general one would be surprised to see strong evolutionary effects at low redshifts, because it implies that the bulk properties of galaxies have been changing rapidly near the present epoch. There is no a priori reason to consider model galaxies that pass through such a phase late in their lives, and in particular, no compelling reason to believe that galaxies should be properly phased such that they show a strong collective effect close to the present epoch.

Sharp features in the count distribution are difficult to model since a wide range of absolute magnitudes and galaxy types contribute at each apparent magnitude. An increase by a factor of 1.6 in a 2-magnitude interval, as proposed by Maddox et al. (1990), would represent such a sharp feature, but neither the color distribution nor the redshift distribution are unusual at these magnitudes. The evolutionary effects proposed by Maddox et al. (1990) would be expected in the counts of galaxies in the r-band. These counts also show a moderately steep slope in the range 16 < r < 19, but the difference between the data and the no- evolution model appears to be smaller. Galaxy luminosity evolution of the kind conventionally assumed does make some difference even at these low redshifts (z < 0.1). We attribute the effect noted by Maddox et al. (1990) to some combination of model uncertainties and conventional evolution. In principle there could also be systematic errors in the data or distortion due to very large-scale structure in the galaxy distribution, but such would have to be true also for the other galaxy-count studies that have covered the same range of magnitudes with similar results.

The predicted no-evolution K-band counts are also shown in Figure 1. It is important to note that these predicted counts, like the predicted r-band counts, are plotted without any extra degree of freedom, since for each galaxy of magnitude B, the K magnitude follows from its apparent B - K color. The R - K and B - K colors in our model approximately match the distributions of Elston, Rieke, and Rieke (1990) and Cowie et al. (1992), respectively, especially after the offset of 0.3 mag in V - K mentioned earlier is taken into account. The predicted K-band counts initially rise more steeply than the blue counts because the K-corrections are smaller, and the counts then flatten to a shallower slope as the relatively local blue galaxies make up an increasing proportion of the faint counts in the bJ band. This expected behavior is just what the data show - the very faint K-band counts of Cowie et al. (1992) are slightly lower, and have a shallower slope, with respect to the blue counts at an equivalent faint limit.

These features were interpreted by Cowie et al. (1992) to indicate that Omega ~ 1 on the assumption that the K-band counts are a more robust measure of volume than the B-band counts, because galaxy spectral energy distributions at long wavelengths are dominated by older stars. If Omega = 1 is assumed, the apparent excess of blue galaxies with respect to the corresponding B-band count prediction is greater, which they attribute to a strongly evolving, lower luminosity, blue population that has since faded beyond recognition in local samples. According to our model, it is in fact not necessary to invoke ad hoc evolutionary processes to understand the K-band counts and the B - K colors, nor to conclude that Omega = 1.

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