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Astron. Astrophys. 345, 329-362 (1999)

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6. Quasar number density

As quasars are among the first objects to form they provide an interesting probe of the Universe at high redshifts [FORMULA]. Moreover, they constrain the models of gravitational clustering to form objects of large mass at high redshifts. We shall now see whether our formalism can satisfy this requirement. We first assume that the quasar mass [FORMULA] is proportional to the mass of gas [FORMULA] available in the inner parts of galaxies:

[EQUATION]

For galaxies which have not already transformed most of their gas content into stars we can use (D5). With (B5) this means that the quasar mass is proportional to the stellar mass [FORMULA]. We write the bolometric luminosity [FORMULA] of the quasar as:

[EQUATION]

where [FORMULA] is the quasar radiative efficiency (fraction of central rest mass energy converted into radiation) and [FORMULA] is the quasar lifetime. If quasars radiate at most at the Eddington limit at which radiation pressure on free electrons balances gravity (Efstathiou & Rees 1988; Nusser & Silk 1993) one has [FORMULA] where [FORMULA] yr. Finally we use a bolometric correction factor [FORMULA]. Since we only consider here very massive and rare quasars we write the quasar comoving number density as:

[EQUATION]

in a fashion similar to Efstathiou & Rees (1988) and Nusser & Silk (1993). Here [FORMULA] is the galaxy comoving multiplicity function and we assumed a fraction [FORMULA] of galaxies actually contains a quasar. Thus we only have two parameters: [FORMULA] which sets the quasar mass, from (43) and (44), and [FORMULA] which enters the multiplicity function in (45). Thus a first estimate of the comoving number density of quasars brighter than [FORMULA] ([FORMULA]) is:

[EQUATION]

where [FORMULA] is the number of galactic halos more massive than [FORMULA] at the redshift z. The minimum galaxy mass [FORMULA] is given by (44). However, the mass of gas available in very massive galaxies does not increase linearly with the dark matter mass of the parent halo because in these very dense galaxies star formation was very efficient so that they have already consumed most of their initial gas content, as can be seen from (B6) and (B7). As a consequence, the mass of gas which may power the central black hole reaches a finite maximum for galaxies somewhat more massive than the transition between [FORMULA] and [FORMULA]. We can note that the maximum quasar mass (or luminosity) obtained in this way decreases as time goes on, along with the decline in the mass of gas which is progressively turned into stars. This very simple effect will obviously have important consequences, at variance with the predictions of models like (46) where the typical quasar mass keeps increasing with time as larger scales become non-linear. Thus we write a second estimate of the quasar comoving number density as:

[EQUATION]

where [FORMULA] is three times larger than the mass which corresponds to the intersection of the curves [FORMULA] and [FORMULA] at the redshift z. This takes care of the upper bound on the possible quasar mass, as described above. When [FORMULA] (which is always the case at high redshifts) the cutoff at [FORMULA] has no influence and we recover the number density of quasars given by the previous calculation (46).

6.1. [FORMULA]

In the case [FORMULA] with a CDM power-spectrum, Fig. 26 shows the comoving number density of bright quasars [FORMULA] as a function of redshift for both prescriptions: [FORMULA] and [FORMULA]. We use [FORMULA] and [FORMULA]. This corresponds for instance to [FORMULA], [FORMULA], [FORMULA] (Eddington luminosity) and [FORMULA]. Then, the minimum dark matter mass for [FORMULA] is [FORMULA]. Note that for nearby galaxies [FORMULA] (Magorrian et al. 1998) where [FORMULA] is the stellar mass of the bulge (assumed to have formed at the same time as the central black hole) and that [FORMULA] may be smaller than [FORMULA] by a factor 2 for the bright galaxies we consider here, see (D4).

[FIGURE] Fig. 26. The comoving number density of bright quasars [FORMULA] and [FORMULA]. These two estimates superpose at high redshift and [FORMULA] shows a sharper cutoff at low redshift. The observational points are from Pei (1995).

As explained above, both prescriptions [FORMULA] and [FORMULA] superpose at high redshift and match the data for [FORMULA]. We recover the broad maximum between [FORMULA] obtained by observations (see also Hartwick & Schade 1990 and Warren et al. 1994). At low redshifts, the number density of bright quasars given by [FORMULA] declines too slowly as compared with observations. On the other hand, [FORMULA] predicts a very fast decrease so that there are no more bright quasars at [FORMULA]. Of course, observations show some quasars at [FORMULA] but this smoother cutoff could certainly be obtained with a more detailed model which would include some scatter in the mass-luminosity relation (and in the properties of galaxies). Moreover, the physics of quasars is certainly much more complex than the modelisation we used here. Hence we think that these results show that our description is consistent with observations (a better agreement with the data would require a more refined model of quasars themselves to be meaningful) and provides a very natural simple model for quasars and galaxies. Note also that our prescription for quasars is a straightforward by-product of our model for galaxies: it is its simplest possible extension and we did not have to introduce an ad-hoc redshift dependance in the mass-luminosity relation to obtain a low redshift decline contrary to Haehnelt & Rees (1993). Note that the counts at high z are very sensitive to the normalization of the power-spectrum and to the mass of the dark matter halos associated to quasars.

6.2. [FORMULA]

Fig. 27 shows the comoving quasar number density of bright quasars [FORMULA] as a function of redshift in the case [FORMULA] for a CDM power-spectrum. We now use [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. This leads to [FORMULA]. Thus the mass of the dark matter halo which corresponds to a given quasar luminosity is lower because the fraction of baryonic matter in this universe is larger. As we can see in the figure we obtain a behaviour similar to the case [FORMULA]. However, the evolution with redshift of the number density of virialized halos is much slower which implies that the decline of the quasar comoving number density at high z is slower than for a critical universe (the same effect also appeared for the galaxy luminosity function).

[FIGURE] Fig. 27. The comoving number density of bright quasars [FORMULA] and [FORMULA]. These two estimates superpose at high redshift and [FORMULA] shows a sharper cutoff at low redshift. The observational points are from Pei (1995).

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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