## 2. Two-point correlation functions of QSOsThe spatial number density of uncollapsed spherical regions can
also be calculated in PS formalism. We define
to be the number density of regions with mass
and radius at where is defined by .
is the overdensity, for which a region
will evolve into a sphere with radius The mass correlation function is determined by the relative enhancement of mass density in the spherical shell around area . Only the spheres with radius can contribute to this enhancement. The mean enhancement of each sphere is approximately described by its mass variance . Therefore, the mass correlation function can be estimated by The factors and are the mean mass in the spherical shell and the volume , respectively. In deriving Eq. (7), we implicitly assumed that there is no correlation among the initial spheres, so the mass fluctuation in the shell is simply given by the sum of the individual components. Because it is impossible that a collapsed halo with mass where the constant is introduced to maintain the normalization condition We have then . Like calculating Eq. (7), let's consider a typical spherical shell
. Only the spheres with radius
can contribute to the mass enhancement in this
shell. The total number of uncollapsed regions with radius
and masses in a volume
is . The total number of
collapsed halos Eq. (10) implicitly assumed that in uncollapsed regions, the
collapsed halos have the same linear variance as the mass. The
approximation given by Eq. (10) is found to be in good agreement with
the linear approximation on scales larger than Using Eq. (10), we have calculated the correlation function at redshift . The results are shown in Fig. 2, in which 2a and 2b are for the models of SCDM and LCDM, and 2c for CHDM. The eight curves in Figs. 2a and 2b are corresponding to masses , from left to right, respectively. In Fig. 2c, it is and .
The two-point correlation function of QSOs is found to be obey the
same power law as galaxies , and the amplitude
(or the correlation length
Mpc) at when
is taken to be 0.5 (Mo & Fang 1993). This
gives for 5 - 10
h Fig. 2a and 2b show that the correlation function of halos in the SCDM and LCDM models are consistent with the observational data. These halos have also proper spatial number density as the observed QSOs. The difference between the two models are very small. In fact, the top-hat evolutions of a spherical mass in SCDM and LCDM are indistinguishable, because it has almost the same dynamic trajectory in all flat universes with . On the other hand, Fig. 2c shows that the two-point correlation
functions of the CHDM model are very small on large scales. For halos
with mass , the amplitude of the correlation
function is well less than 0.1 on scales of 5
-10 h © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |