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Astron. Astrophys. 337, 96-104 (1998)

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5. The bias coefficient

A model of the Universe in which light traces the mass distribution accurately (unbiased model) is subject to several problems. As pointed out by Davis et al. (1985) an unbiased CDM produces a galaxy correlation function which is steeper than that observed and a pairwise velocity dispersion larger than that deduced from redshift surveys. A remedy to this problem can be found if we do not assume that light traces mass and adopt the biasing concept, i.e., galaxies are more clustered than the distribution of matter in agreement with the concept of biasing inspired by Kaiser (1984). The observations show that clusters of galaxies cluster more strongly than galaxies, in the sense that the dimensionless two-point correlation function, [FORMULA], is much larger than the galaxy two-point function, [FORMULA]. The galaxy two-point correlation function [FORMULA] is a power-law:

[EQUATION]

with a correlation length [FORMULA] Mpc and a slope [FORMULA] for [FORMULA] Mpc (Davis & Peebles 1983; Davis et al. 1985; Shanks et al. 1989), (some authors disagree with this values; for example Strauss et al. 1992 and Fisher et al. 1993 find [FORMULA] Mpc and [FORMULA]). As regards the clusters of galaxies the form of the two-point correlation function, [FORMULA], is equal to that given by Eq. (29). Only the correlation length is different. In the case of clusters of galaxies the value of [FORMULA] is uncertain (see Bahcall & Soneira 1983; Postman et al. 1986; Sutherland 1988; Bahcall 1988; Dekel et al. 1989; Olivier et al. 1990 and Sutherland & Efstathiou 1991) however it lies in the range [FORMULA] Mpc in any case larger than [FORMULA]. One way of defining the bias coefficient of a class of objects is that given by (BBKS):

[EQUATION]

where [FORMULA] is:

[EQUATION]

We want to remember that, as shown by Coles (1993), the biasing parameter can be also estimated by means of the ratio of the amplitudes of the correlation function, [FORMULA], and the matter auto-covariance function, [FORMULA]:

[EQUATION]

or by means of the ratio of the cumulative integral of the two-points correlation function ([FORMULA]) and that of the auto-covariance function ([FORMULA]):

[EQUATION]

or finally the ratio of galaxy, [FORMULA], to mass, [FORMULA], power spectra:

[EQUATION]

As stressed by Coles (1993) a local bias generally produces a different response in each of these descriptors. Even the qualitative behaviour of the limit of large scales can be different, i.e. [FORMULA] can increase or decrease. So one should decide very carefully which one of these definitions must be used, when discussing the behaviour of the biasing parameter. Being conscious of these difficulties we have chosen one of the most popular descriptor of the biasing parameter (see also Lilje 1990, Liddle & Lith 1993, Croft & Efstathiou 1994, CAD) in order to make comparisons with other models. From Eq. (31) it is clear that the bias parameter can be calculated once a spectrum, [FORMULA], is fixed. The bias parameter depends on the shape and normalization of the power spectrum. A larger value is obtained for spectra with more power on large scale (Kauffmann et al. 1996). In this calculation we continue to use the standard CDM spectrum ([FORMULA], [FORMULA]) normalized imposing that the rms density fluctuations in a sphere of radius [FORMULA] is the same as that observed in galaxy counts, i.e. [FORMULA]. The calculations have been performed for three different values of the filtering radius ([FORMULA] 3, 4 [FORMULA]). The values of b, that we have obtained, are respectively, in increasing order [FORMULA], 1.6, 1.93 and 2.25.

As shown, the value of the bias parameter tends to increase with [FORMULA] due the filter effect of [FORMULA]. As shown [FORMULA] acts as a filter, increasing the filtering radius, [FORMULA], the value of [FORMULA] at which [FORMULA] increases. In other words when [FORMULA] increases, [FORMULA] selects density peaks of a larger height. The reason for this behavior must be searched in the smoothing effect that the increasing of the filtering radius produces on density peaks. When [FORMULA] increases the density field smooths and [FORMULA] has to shift towards a higher value of [FORMULA] in order to select a class of object of fixed mass M.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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