Recently, we have developed a method to test popular dark-matter models of structure formation using the number density and the two-point correlation function of high redshift objects (Bi & Fang 1996). We found an approximate expression of two-point correlation function of mass and collapsed halos in the Press-Schechter formalism (Press & Schechter 1974, hereafter PS). In this approximation, the nonlinear gravitational interaction was treated as the sum of various individual spherical top-hat clustering. These top-hat spheres consist of both collapsed PS halos and uncollapsed regions. Moreover, the bias that massive PS halos have stronger correlation than the background mass can naturally be introduced by considering that no collapsed halo of mass M exists in initial regions (or top-hat spheres) of mass less than M.
This method was applied to CIV absorption systems in QSO spectra. Because CIV systems should be hosted by collapsed halos, one can obtain an lower limit to the spatial number density of these host halos from the observed CIV number density. This requires that the hosts of CIV should consist of halos with mass as low as . On the other hand, the two-point correlation of halos is stronger when the mass of the halos is larger. The observed two-point correlation functions set a lower limit, , to the mass of host halos. Obviously, a reasonable model has to give . The standard cold dark matter (SCDM) model normalized to and the low density flat cold dark matter (LCDM) model can pass this test. However, for the cold-plus-hot dark matter (CHDM) models with parameters and , or and , the two-point correlation functions of halos with mass are too small to explain the observed correlation functions. In order to have enough number of collapsed halos to host CIV systems, should not be larger than . But the observed two-point correlation function on the scales of km/s indicates that should not be less than .
In this paper, we study the same problem, but using QSOs as the discriminator. The topic of finding constraints on dark matter models from QSOs is not new. As early as 1980s, the highest redshift of QSOs was used to rule out the model of hot dark matter (HDM), because the HDM predicted that collapsed halos cannot form at redshift larger than 3, which is much less than the redshifts of many existing QSOs. Consequently, the formation of high redshift QSOs cannot be explained in the HDM model (e.g. Efstathiou & Rees 1988). This redshift test is passed for current dark matter models. For instance, the tilted CDM, scaled CDM and CHDM models all are able to produce with the observed abundance of QSOs (Nusser & Silk 1993). However, the QSO abundance test alone is not free from the uncertainties of discriminating among the models like SCDM, LCDM and CHDM. While some works claimed that the CHDM model is consistent with the QSO abundance, other works found oppositely (Ma & Bertschinger 1994). We will show that adding a clustering test to the abundance fitting will reduce the number of free parameters in the discrimination, and gives much better results.
1.1. Abundance of QSOs
The spatial number density of collapsed halos can be calculated from the standard PS theory. We define to be the 3-D density fluctuation field of dark matter extrapolated to redshift z assuming linear evolution. A density field , representing the smoothed fluctuation on scale R, can be derived from by
where the function is the top-hat window for the comoving volume . The total mass within on average is , where is the mean density at the present if the scaling factor of the universe is set to be unity at .
For Gaussian perturbations, the mass fluctuation within a top-hat window of radius R is described by variance , which is determined by the initial density spectrum and normalization factor , i.e. h-1 Mpc). In the case of Einstein-de Sitter universe, the linear evolution of the variance is . Thus, the fraction of the total mass having fluctuations larger than a given in an arbitrary spatial domain is
If we take to be the critical overdensity for the collapse of the spherical mass M at z, should be identified as the sum of masses of all collapsed halos, each of which is massive greater than M. For the Einstein-de Sitter universe, we have . For the flat universe, function is calculated by Bi & Fang (1996). The differential gives the total mass of collapsed halos in the mass range M to . Hence, if is defined as the spatial number density of halos between M and at z, we have
where we use the subscript c in to emphasize that it is for collapsed halos. Considering the cloud-in-cloud problem, the above defined should be multiplied by a factor of 2. The normalization can then be fulfilled. Therefore, the spatial number density of halos with mass between M and at redshift z is given by
where erfc(x) is the complementary error function.
We will study models of SCDM, LCDM and CHDM, for which the density parameter , the cosmological constant , and , are listed in Table 1 (the Hubble constant is taken to be h km s-1 Mpc-1.) It is worth to point out that the of the CHDM and LCDM models are compatible with the COBE-DMR observation, but that of the SCDM model is not. The value gives a good fit for the SCDM model to all data except for the COBE result. We assumed that the initial primordial power spectrum is the Harrison-Zel'dovich type. The linear transfer functions of the SCDM and the LCDM models are taken from Bardeen et al. (1986) and that of the CHDM model from Klypin et al. (1995).
Table 1. Parameters of the cosmological models
The cumulative number density, , of all halos with mass greater than M should be
The abundance of halos calculated from Eq. (5) has been verified by a number of N-body simulations (e.g. Lacey & Cole 1994).
The redshift evolution of the number density of collapsed halos, , are shown in Figs. 1a, 1b and 1c for the SCDM, LCDM and CHDM models, respectively. The eight curves in Figs. 1a and 1b correspond to , with from top to bottom. In Fig. 1c, it is , with .
Pairs and multiples of QSOs are not common to see. Therefore, it is reasonable to assume that each collapsed halo hosts only one, or at most a few QSOs. The number density of QSOs with magnitude is plotted as crosses in Fig. 1. The data points are taken from Pei (1995) whose result is based on the observations of Hewett et al. (1993) and Schmidt et al. (1992). In Pei's paper, is measured in the Einstein-de-Sitter cosmological model (, and h = 0.5). When comparing the observations to the LCDM model, we have made corrections of the cosmological effect on due to the non-zero .
Fig. 1 shows that for all the models, the lower mass limits are mainly determined by the QSO abundance at . The possible hosts of QSOs in the SCDM and LCDM models can be provided by halos with masses above , and , respectively. In the CHDM model, the number densities of collapsed halos at these mass range are much fewer. Therefore, for the CHDM, we have to include lower mass halos to host QSOs. The mass of possible QSO halos will be as low as about , which is more than two orders of magnitude below those in the models of SCDM and LCDM.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998