3. Biasing problem
A possible way to save the CHDM model is to assume that there is a very high biasing factor to raise the correlation function of halos in the CHDM model. From Fig. 2c, if this factor is as large as 5, the power of two-point correlation functions mass would be able to fit with QSO's clustering. However, biasing is actually not a free parameter which can be arbitrarily chosen (Einasto et al. 1994).
Generally speaking, the correlation functions of objects identified from a mass field will be different from the mass itself. A different method in the identification procedure can lead to a different biasing. The question now is: can we choose other QSO identifications to produce larger biasing, and then to give higher amplitude of the two-point correlation functions of QSOs in the model?
As an example, let us first consider the QSO identification by velocity dispersion. It has been known for a decade that QSOs with low redshift are preferentially located in small groups of galaxies. This is in evidence from QSO-galaxy correlation function (Yee & Green 1987), CIV-associated absorption in high redshift radio-loud QSOs (Flotz et al. 1988), clustering analyses of QSO distribution (Bahcall & Chokshi 1991) and the galaxy environments around QSOs (Ellingson et al. 1991a). It has been also shown that the velocity dispersion of galaxies around QSOs is 400 km s-1 (Ellingson et al. 1991b). Therefore the environment suitable of QSOs formation seems to be small groups of velocity dispersion . The strength of the QSO correlation function, is intermediate between galaxies and rich clusters, so once again it is similar to galaxy groups.
If we assume that high redshift QSOs formed in the same environment as that in the low redshift, the QSOs should be identified as collapsed halos with 3-dimensional velocity dispersion 700 km s-1 between and .
In the PS formalism, the comoving number density of halos with velocity dispersion can be calculated by
As in Eq. (4), in Eq.(11) is the critical overdensity for collapsed at redshift z. The meaning of R here is the same as in § 2. The relationship between and R is given by (Narayan & White 1988)
for Einstein-de-Sitter universe, and
for open universe or the LCDM universe. The coefficient has been determined by comparing Eqs. (11) and (12) with N-body simulation (Jing & Fang 1994). It found . For most calculations, is a preferred value. Hence, one can safely use Eqs.(11)- (13) even when the 3-D velocity dispersion is as large as about 800 km s-1.
The total number density of the collapsed halos with the velocity dispersion greater than a certain value, say , is
Using Eq. (14), we found that the SCDM and LCDM models can produce sufficient number of 700 km s-1 halos to fit with the number density of QSOs. But the number of such halos in the CHDM model is too small. To have enough number of halos, we should use 300 km s-1 or less in the CHDM model.
The mass of the -selected halos can be determined from Eqs.(12) or (13). One can then calculate the two-point correlation functions of them by Eq. (10). Here we get almost the same results as before because is one-to-one related to the mass or the radius of the top-hat windows. Similar to Fig. 2c, the 300 km s-1 halos still lack of correlation power on all scales larger than 2 h-1 Mpc in the CHDM model.
This result should be expected. An identification of objects from a density field is a sampling. As one knows, a sampled field will not be different from the original field on scales much larger than the characteristic scale, , of the sampling (Vanmarke 1983). This means that, in principally, no biasing on scales larger than the characteristic scale can be introduced by the identification. For our question here, it is not easy to given the characteristic scale of the identification. However, it is probably reasonable to choose , where is the average of inside R (Hamilton et al. 1991). So the scale is actually the coherence length of the density field smoothed by window R. The distribution of halos should not be biased from mass distribution on scales larger than . The correlation function can effectively be amplified by linear geometrical biasing only for large halos (Kaiser, 1984). Fig. 2 show a break in the correlation functions at , which is roughly equal to . When , the correlation functions of the collapsed halos should approach to the mass correlations, while when , the biasing leads to larger correlation functions.
All identifications based on gravitational parameters, such as circular velocity, virial temperature etc. are essentially equal to mass identifications, because for a gravitational collapsed systems these parameters are one-to-one related. Therefore, they can be expressed as an equivalence to M. Biases in these parameters cannot give significant different results from those using or M.
Obviously, any non-gravitational identifications are not constrained by the gravitational characteristic scale. For instance, if gas processes in the formation of QSOs play the role of biasing, the characteristic scale will not equal to . However, the characteristic scale of gas processes should be much less than , because the velocity of gaseous component is too small to segregate the QSO halos on scales of 5 h-1 Mpc.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998