3.2 Biasing and Large-Scale Velocities

The relation between the mass density parameter and the gravitational motions of the galaxies is an issue rich enough for a separate category in Table 1. It has been known for the past decade that if galaxies were fair tracer of mass then the small-scale relative velocities of the galaxies would imply that is well below unity [3]. If the mass distribution were smoother than that of the galaxies, the smaller mass fluctuations would require a larger mean mass density to gravitationally produce the observed galaxy velocities. Davis, Efstathiou, Frenk & White [31] were the first to show that such a biased distribution of galaxies relative to mass readily follows in numerical N-body simulations of the growth of structure, and the demonstration has been repeated in considerable detail ([32], [33], and references therein). This is a serious argument for the biasing effect. But here are three arguments for the proposition that galaxies are fair tracers of mass for the purpose of estimating .

First, in many numerical simulations dwarf galaxies are less strongly clustered than giants. This is reasonable, for if much of the mass were in the voids defined by the giant galaxies, as required if = 1, then surely there would be remnants of the suppressed galaxy formation in the voids, irregular galaxies that bear the stigmata of a hostile early environment. The first systematic redshift survey showed that the distributions of low and high luminosity galaxies are strikingly similar [34]. No survey since, in 21-cm, infrared, ultraviolet, or low surface brightness optical, has revealed a void population. There is a straightforward interpretation: the voids are nearly empty because they contain little mass.

Second, one can use the galaxy two-point correlation function in Eq. (1) and the mass autocorrelation function from a numerical simulation of structure formation to define the bias function

(15)

In numerical simulations b typically varies quite significantly with separation and redshift [32]. That is, the galaxies give a biased representation of the statistical character of the mass distribution in a typical numerical simulation. The issue is whether the galaxies, or the models, or both, are biased representations of the statistical character of the real mass distribution. What particularly strikes me is the observation that the low order galaxy correlation functions have some simple properties. The galaxy two-point function is close to a power law over some three orders of magnitude in separation (Eq. 2). The value of the power law index changes little back to redshift z ~ 1. Within the clustering length r₀ the higher order correlation functions are consistent with a power law fractal. A reasonable presumption is that the regularity exhibited by the galaxies reflects a like regularity in the mass, because galaxies trace mass. I am impressed by the power of the numerical simulations, and believe they reflect important aspects of reality, but do not think we should be surprised if they do not fully represent other aspects, such as relatively fine details of the mass distribution.

Figure 2. The density parameter derived from galaxy peculiar motions on the assumption galaxies trace mass. From the left, the estimates are based on the Local Group of galaxies, clusters of galaxies, the peculiar infall toward the Virgo Cluster [35], and the analyses in references [36] to [38].

The third argument deals with the idea that blast waves or radiation from the formation of a galaxy may have affected the formation of nearby galaxies, producing scale-dependent bias. In this case the apparent value of the density parameter derived from gravitational motions within systems of galaxies on the assumption galaxies trace mass would be expected to vary with increasing scale, approaching the true value when derived from relative motions on scales larger than the range of influence of a forming galaxy. Fig. 2 shows a test. The abscissa at the entry for clusters of galaxies is the comoving radius of a sphere that contains the mass within the Abell radius. The estimates at larger scales are plotted at approximate values of the radius of the sample. If it were not for the last two points at the right-hand side of Fig. 2, one might conclude that the apparent density parameter is increasing to the true value ~ 1 at R ~ 50h^-1 Mpc. But considering the last two points, and the sizes of the error flags, it is difficult to see any evidence for scale-dependent bias.

I assign a strongly negative grade for the Einstein-de Sitter model in line 2a in Table 1, based on galaxy motions on relatively small scales, because biasing certainly is required if = 1 and I have argued there is no evidence for it. The more tentative grade in line 2b is based on Fig. 2: the apparent value of the density parameter does not seem to scale with depth.