2. Two-point correlation functions of QSOs
The 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 z. Similar to Eq. (4), we have
where is defined by . is the overdensity, for which a region will evolve into a sphere with radius r at z. Function is also calculated in Bi & Fang (1996). The total number of such spherical regions in an arbitrary volume should on average be given by .
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 M formed in a uncollapsed region with , the number density of M collapsed halos in an uncollapsed sphere of mass should be zero if M is greater than . Thus, it is reasonable to assume that the number density of collapsed halos in each PS sphere r at z on average is
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 M in each sphere is , and the variance of the number is . Therefore, the correlation function of collapsed halos with mass larger than M can be approximated as (Bi & Fang 1996)
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 R, but higher than the empirical formalism of Hamilton et al. (1991) on scales less than R, here R is the scale within which the non-linear effects are significant. Therefore, Eq. (10) can be used at least as a upper limit to the correlation function.
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-1 Mpc. The clustering of QSOs on scales of Mpc is also found to be significant (Deng et al. 1994). Some observations and statistics have even indicated the possible existence of groups of QSOs with comoving sizes as large as about Mpc (e.g. Clowes & Camppusano 1991, Komberg et al. 1996.) Despite is decreasing for , the amplitude is still larger than 0.1 (Mo & Fang 1993; Komberg et al. 1994.)
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-1 Mpc. Therefore, the CHDM model seems to be unfavored under the abundance-plus-correlation test. According to the abundance, the mass of QSO hosts in the CHDM model should be as small as . On the other hand, the two-point correlation function of such halos is much less than what is observed.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998