## 6. Quasar number densityAs quasars are among the first objects to form they provide an interesting probe of the Universe at high redshifts . 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 is proportional to the mass of gas available in the inner parts of galaxies: 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 . We write the bolometric luminosity of the quasar as: where is the quasar radiative efficiency (fraction of central rest mass energy converted into radiation) and 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 where yr. Finally we use a bolometric correction factor . Since we only consider here very massive and rare quasars we write the quasar comoving number density as: in a fashion similar to Efstathiou & Rees (1988) and Nusser & Silk (1993). Here is the galaxy comoving multiplicity function and we assumed a fraction of galaxies actually contains a quasar. Thus we only have two parameters: which sets the quasar mass, from (43) and (44), and which enters the multiplicity function in (45). Thus a first estimate of the comoving number density of quasars brighter than () is: where is the number of galactic
halos more massive than at the
redshift where is three times larger than
the mass which corresponds to the intersection of the curves
and
at the redshift ## 6.1.In the case with a CDM power-spectrum, Fig. 26 shows the comoving number density of bright quasars as a function of redshift for both prescriptions: and . We use and . This corresponds for instance to , , (Eddington luminosity) and . Then, the minimum dark matter mass for is . Note that for nearby galaxies (Magorrian et al. 1998) where is the stellar mass of the bulge (assumed to have formed at the same time as the central black hole) and that may be smaller than by a factor 2 for the bright galaxies we consider here, see (D4).
As explained above, both prescriptions
and
superpose at high redshift and
match the data for . We recover the
broad maximum between obtained by
observations (see also Hartwick & Schade 1990 and Warren et al.
1994). At low redshifts, the number density of bright quasars given by
declines too slowly as compared
with observations. On the other hand,
predicts a very fast decrease so
that there are no more bright quasars at
. Of course, observations show some
quasars at 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 ## 6.2.Fig. 27 shows the comoving quasar number density of bright quasars
as a function of redshift in the
case for a CDM power-spectrum. We
now use ,
,
and . This leads to
. 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
. 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
© European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |