5.2.2. Biasing in the Universe

The concept of linear biasing between the distribution of light and mass was introduced in Chapter 3. Since the structure formation models that we are considering all predict the distribution of mass, the role of biasing is pivotal if mapping the observed light distribution back into the model. One of the first indicators that biasing does exist came from comparing the galaxy-galaxy correlation function to the cluster-cluster correlation function. The observations indicate that while both correlation functions have the same power-law slope, the cluster-cluster correlation function has an amplitude about 20 times larger. If galaxies and clusters have both arisen due to gravitational instability and amplification of fluctuations in the primordial density field, then both scales should trace large scale structure equally well. The large difference in correlation amplitude, however, is quite inconsistent with this expectation.

To reconcile this discrepancy, Nick Kaiser (1984) shaped the following physical argument. Very rich clusters of galaxies are obviously the most massive objects that have collapsed by the present epoch into some kind of equilibrium state. These clusters (e.g. the Coma cluster) are also extremely rare as the number density of rich clusters is orders of magnitude smaller than the number density of galaxies. In the gravitational instability scenario, at fixed mass scale, the first objects to collapse are the densest. For simplicity we assume that the primordial spectrum of density fluctuations was Gaussian in nature. This assumption arises naturally out of our original hypothesis that the phases of _k are random. Furthermore, analysis of the 4-year COBE anisotropy data show virtually no departure from Gaussininity (Hinshaw et al. 1995). For Gaussian fluctuations it follows that these high-density rich clusters are rare because their initial density was several above the mean value.

A general statistical property of Gaussian random fields is that the rare high- peaks tend to occur near other high peaks. Quantitatively, the relatively weak correlation on large scales is amplified with respect to the background matter density by a factor of ² where equals the number of that corresponds to the density fluctuation. To account for the significantly stronger correlation function among clusters, relative to galaxies, requires that ² is 20 or app 4; such events in the Universe would be quite rare thus accounting for the very low space density of rich clusters. This is known as statistical biasing. Clusters form at very high peaks and these peaks are near other such peaks. Density fluctuations of lower amplitude would collapse on longer timescales and be distributed in a way that is more representative of the overall mass distribution.

The same argument of statistical biasing in the formation of rich clusters can also be applied on a smaller scale to the formation of galaxies. In a landmark study, Alan Dressler (1980) presented strong, almost indisputable evidence, that galaxy morphological type correlates with the local galaxy density. In particular, he found that elliptical galaxies favored regions of high local galaxy density (e.g. clusters of galaxies) and spirals favored lower density regions. With the discovery of LSB galaxies and a study of their environments, we can extend the Dressler morphology-density relation to a more general statement which says that the phase-space density within a galaxy correlates with local galaxy density. The very low density LSB galaxies are relatively isolated. If rich clusters of galaxies are formed in a biased manner and they mostly contain ellipticals, then this suggests that galaxy formation is similarly biased. Dense galaxies (e.g., ellipticals) form from rare 3-4 peaks and lower density galaxies form from the more common, less biased 1-2 peaks.

The idea of biased galaxy formation is at the heart of the first-generation CDM models in the mid 80's. In the numerical simulations of Davis et al. (1984) a value of b = 2.4 was needed to explain the small scale correlations of galaxies. This is a rather high bias factor. Since that time, a great effort has been undertaken to produce fair samples of galaxies in redshift surveys in order to measure b more robustly. This is highly relevant to any structure formation scenarios for if it were possible to predict b from theory, then strong observational constraints would exist. Estimations of the amount of biasing have greatly changed in the last 10 years with some recent studies (e.g., Lin et al. 1996; Benoist et al. 1996) now showing that b = 1 (no bias) or b is slightly less than one (indicating an anti-bias -> mass is more strongly clustered than light). In terms of our structure formation scenarios the least biased sample would correspond to those objects that formed at low density (e.g., 1-2 peaks) fluctuations. While these are thought to be mostly spiral galaxies, its not clear that there exists a unique morphological signature of a galaxy which formed from a low density fluctuation and so its difficult to a priori select the most unbiased sample.

The above scenario for biasing however explicitly assumes that galaxy formation is the result of the collapse of individual potentials whose density distribution Gaussian. But this simplified idea of galaxy formation may be somewhat inconsistent with recent HST images of high-redshift galaxies. In particular, the Hubble Deep Field (HDF) (see Figure 5-3) provides a strong indication that galaxies are essentially assembled from smaller sub-units (see Windhorst et al. 1996) instead of forming from the smooth collapse of gas inside a potential. If this is indeed the case, then the morphology-density relation may not reflect statistical biasing at all but instead is produced by environmental processes that determine the rate at which galaxies are assembled. Gas removal processes during this assembly phase then may determine the kind of galaxy that ultimately forms. At the very least, the HDF data indicate that the formation of galaxies is likely to be a complex and extended process (see more below).