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7.2. Biasing

If the formation of galaxies depends on environment, then both the fraction of mass in galaxies and the mass function of galaxies may in the end vary over large scales. The concept of biased galaxy formation emerged in the 1980's (Kaiser 1984; Rees 1985; Davies et al. 1985; Bardeen et al. 1986) as a way to reconcile the theoretically attractive ``flat universe'' with Omega = 1 to the lower Omega inferred from the virial analysis of rich clusters and from studies of large-scale structure.

There are many plausible ways to introduce bias into the distribution of galaxies (Rees 1985; Dekel 1986). Spatial variations in the efficiency of galaxy formation could arise from the nature of the dark matter (e.g. if a significant fraction of Omega is in hot neutrinos), from the existence of a universal threshold in the density required for galaxy formation (see below), or from the coupling of galaxy formation to the feedback of energy from the first generations of objects (e.g. via photoionization of the IGM from early starbursts or AGN, or via bulk flows introduced by early supernovae). Because of the uncertainties in the physics, it has been convenient to parametrize biasing by a single parameter b that is the ratio of the rms density fluctuations in luminous matter to the underlying rms mass fluctuations. Because ellipticals and spirals cluster differently, the value of b obviously depends on galaxy type. The COBE results suggest that b also depends on the scale over which the rms fluctuations are measured (Efstathiou et al. 1992; Padmanabhan and Narasimha 1992), at least for CDM-like power spectra.

Some type of biasing mechanism is clearly necessary to explain the clustering properties of dE galaxies (Sect. 6). Perhaps the most thoroughly developed is the concept of statistical biasing, based on the idea that galaxies form above some global threshold nu in the Gaussian random field of primordial fluctuations (nu = delta / sigma where sigma is the rms density variation on a given scale). A standard hypothesis (Dekel and Silk 1986; White et al. 1987; Schaeffer and Silk 1988) has been that giant galaxies form only at nu approx 2-3, while dwarfs can form at any peak (and so the vast majority of them should come from typical peaks of nu = 1). Such a model predicts that giant galaxies should be more correlated than dwarfs. The enhancement in the giant-galaxy two-point correlation function xig relative to the mass two-point correlation function xi m is approximated by (Kaiser 1984; Politzer and Wise 1984)

Equation 11 (11)

which corresponds to roughly an order of magnitude at nu/sigma = 3. Although an explicit estimate of xig for dE galaxies has not been made, the available data clearly rule out any appreciable decrease in xig from giant E's to dE's (see Sect. 6). (The data are still ambiguous for the comparison of field Irr's to field spirals - (Eder et al. 1989; Salzer et al. 1990; Alimi 1994.)

West (1993) has argued that the same sort of statistical bias may account for different specific globular cluster frequencies of spiral and elliptical galaxies, and the higher globular cluster frequencies of cD galaxies. If one assumes that globular clusters can only form at density peaks higher than some global critical value nu, the enhancement in the globular cluster frequency in galaxy and cluster halos can be calculated from the formalism described above. For reasonable choices of the power spectrum and the density thresholds for spirals, ellipticals, and rich galaxy clusters, the observed variation of the globular cluster specific frequency can be reproduced. West (1993) speculates that the same reasoning might apply to dwarf galaxies, which for the same threshold nu ought then to be more strongly clustered than globular clusters.

As a mechanism for forming globular clusters, West's proposal faces serious difficulties. For example, there is no evidence that globular clusters have dark-matter halos, as they must in this model. Furthermore, the CDM power spectrum, which produces the best match to the trends in globular-cluster specific frequencies, would predict a globular-cluster mass function steeply rising to low masses, and extending to higher masses, contrary to observation. Finally, it appears that at least some globular clusters, for example the relatively young ones observed in the LMC and NGC 1275 (Holtzman et al. 1992), form at late epochs, probably not through collapse onto primordial density fluctuations.

On the other hand, West's proposal is perhaps more attractive for explaining the clustering of dE galaxies, which show strong evidence for dark matter and have a steeply rising mass function (although not as steep as CDM models predict). The idea that statistical bias could enhance the clustering of dE galaxies runs contrary to all previous statements about the influence of statisical bias on the clustering properties of dwarf galaxies, but nevertheless has the positive feature that it could potentially account for the variation of the dwarf/giant ratio with cluster richness (Sect. 6), and it may provide a natural explanation for the high central dark-matter densities inferred for some dwarf ellipticals (Sect. 3). However, interpretation of the clustering of dE galaxies in the context of statistical biasing requires that their formation threshold nu was comparable to that of giant elliptical galaxies. The condition that giant ellipticals form at high sigma peaks arises naturally from the standard cooling time argument: the density of their halos must be high enough that the cooling time will be shorter than the free-fall time (Sect. 7.3). However, the cooling time decreases rapidly with decreasing halo velocity dispersion (tcool propto V2), so this mechanism cannot be invoked to argue for a high nu for dE galaxies. While other possible mechanisms exist (West 1993), missing from the argument is a compelling physical reason why dE's (or globular clusters) should form only above some global density threshold.

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