The fact that galaxies of different types cluster differently [23, 41, 51, 43, 32, 27] implies that many of them are biased tracers of the underlying mass distribution. Without such biasing, it is hard to reconcile the existence of large volumes void of galaxies  and the spiky distribution of galaxies on ~ 100 h-1 Mpc scales, today  and at high redshifts [56, 55], with the standard theory of gravitational instability theory (GI). There is partial theoretical understanding of the origin of biasing [35, 14, 3, 22, 21, 8, 1, 42, 46], supported by cosmological simulations which confirm the existence of biasing [10, 37, 6, 54] and show that it becomes stronger at high redshifts [2, 34, 59, 54].
The biasing is interesting as a constraint on galaxy formation, but it is also of great importance when estimating the cosmological density parameter . If one assumes linear and deterministic biasing and applies the linear approximation for GI, . v = -f() where f() 0.6 , the observables g and . v are related via the degenerate combination f() / b. Thus, one cannot pretend to have determined by measuring without a detailed knowledge of the biasing scheme.
It turns out that different methods lead to different estimates of in the range 0.4 1.1 [15, 57, 18, 16]. The methods include: (a) comparisons of local moments of g (from redshift surveys) and (from peculiar velocities) or the corresponding power spectra or correlation functions; (b) linear regressions of the fields g and , or the corresponding velocity fields; and (c) analyses of redshift distortions in redshift surveys. In order to sharpen our determination of , it is important that we understand this scatter in . Some of it is due to the different types of galaxies involved and some may be due to the effects of non-linear gravity and perhaps other sources of systematic errors. Here we investigate the possible contribution of nontrivial properties of the biasing scheme such as stochasticity and non-linearity.
The theory of density peaks in a Gaussian random field [35, 3] predicts that the linear galaxy-galaxy and mass-mass correlation functions are related via gg(r) = b2 mm(r), where the biasing parameter b is a constant independent of scale r. However, a much more demanding linear biasing model is often assumed, in which the local density fields are related deterministically via the relation g(x) = b (x). This is not a viable model because (a) it has no theoretical motivation, (b) if b > 1 it must break down in deep voids because values of g below -1 are forbidden, and (c) conservation of galaxy number implies that the linear biasing relation is not preserved during fluctuation growth. Thus, non-linear biasing, where b varies with , is inevitable. Indeed, the theoretical analysis of the biasing of collapsed halos by Mo & White , using the extended Press-Schechter approximation , predicts that the biasing is non-linear. It provides a useful approximation for its behavior as a function of scale, time and mass. N-body simulations, which provide a more accurate description (see Figure 1; ), show that this model is indeed useful.
Figure 1. Biasing of galactic halos versus mass in a cosmological N-body simulation, demonstrating non-linearity and stochasticity. The conditional mean [< g | > = b()] (solid curve) and scatter [< 2 | > = b2() 2] (error bars) are marked. The fields smoothed with a top-hat window of radius 8 h-1 Mpc are plotted at the points of a uniform grid. The halos are selected above a mass threshold of 2 x 1012 M. Left: at the time when 8 = 0.6 (e.g., z = 0). Right: at an earlier time when 8 = 0.3 (e.g., z = 1). (Based on .)
Note that once the biasing is non-linear at one smoothing scale, the biasing at any other smoothing scale must obey a different functional form of b() and is non-deterministic. Thus, any deviation from the simplified linear biasing model must also involve scale-dependence and scatter. Another inevitable source of scatter is physical scatter in the efficiency of galaxy formation as a function of , because the mass density at a certain smoothing scale (larger than the scale of galaxies) cannot be the sole quantity determining galaxy formation. For example, the random variations in the density on smaller scales and the local geometry of the background structure must play a role too. These hidden parameters would show up as scatter in the density-density relation. A third obvious source of scatter is the shot noise. One can try to remove it a priori, but this is sometimes difficult because of the small-scale anti-correlations introduced by the finite extent of galaxies. The alternative is to treat the shot noise as an intrinsic part of the local stochasticity of the biasing relation. The scatter arising from all the above is clearly seen for halos in simulations including gravity alone (Section 4) even before the complex processes of gas dynamics, star formation and feedback affect the biasing.