## 3. Biasing problemA 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 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 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 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 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 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 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 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 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 © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |