6.5 Quasar Luminosity Function
Even before the discovery of the first gravitational lens it was
recognized that lensing has the potential to modify our view of
cosmologically distant parts of the universe
(Barnothy 1965,
1966;
Barnothy &
Barnothy 1968,
1986;
De Silva 1970).
One particular question that has been discussed often is whether the
observed quasar luminosity function could be significantly affected by lensing
(Turner 1980,
Avni 1981,
Peacock 1982,
Setti & Zamorani
1983,
Vietri &
Ostriker 1983,
Vietri 1985,
Ostriker &
Vietri 1986,
Isaacson &
Canizares 1989).
Flux conservation demands that the average magnification of a randomly
selected source in an inhomogeneous universe must be unity compared to
a smooth universe of the same 
0
(Weinberg 1976,
Ehlers &
Schneider 1986).
However, in the former universe one has a
distribution of magnifications, P (µ), while in
the latter all sources have unit magnification. Consequently, the observed
luminosity functions can differ in the two cases. Typically,
gravitational macrolensing has a P (µ) that is peaked
around µ ~ 1 with a power-law tail extending to large
µ. The exponent
in the tail is -3 if large magnifications are dominated by fold
caustics and -7/2 if due to the exterior single-image region of cusp
caustics. If the intrinsic source differential luminosity function
(L) is flatter than
L-3, then lensing has only a minor
effect on the observed luminosity function. This is true for optical
quasars fainter than B ~ 19
(Boyle et
al. 1988).
However, if there is any range of L for which the intrinsic counts are
steeper than -3, e.g. quasars brighter than B ~ 19, then the
observed population could potentially have a large contribution from
highly-magnified intrinsically-weak sources (because of magnification
bias).
Quantitative estimates based on the known populations of galaxies and
clusters in the universe indicate that lensed quasars are unlikely to
dominate in any magnitude range where there are substantial
source counts. Nevertheless, even a modest influence due to lensing,
say at bright magnitudes, is of interest since it implies that
muliply-imaged quasars will be particularly common in such a
population. Based on this line of thinking, many lens searches have
been confined to high redshift, high apparent luminosity quasars (cf
Section 7.2). This strategy has had
some success
(Magain et
al. 1988),
but the lack of a greater success rate does suggest that macrolensing
has only a weak effect on the quasar luminosity function even at the
brightest end (B 
17).
The additional effect due to microlensing has been considered by some
authors
(Vietri 1985,
Ostriker &
Vietri 1986,
Schneider
1987a,
b,
Bartelmann &
Schneider 1990).
Once again, at very large µ,
P (µ) has a power-law character due to the effect of caustics.
However, the power-law tail is cut off above a critical µ that
depends on the size of the source and on the mass distribution of the
microlenses. It is our opinion that microlensing has only a marginal
influence on the quasar luminosity function even at the brightest
quasar magnitudes. However, it is possible that microlensing does
play a role in some of the quasar-galaxy associations discussed in
Section 5.2.2.
Given a model of P (µ) and an observed luminosity function
(L), it is possible in
principle to obtain the true luminosity
function true (L)
(Schneider 1992).
Several authors have investigated whether the observed anisotropy of
the cosmic microwave background could be significantly modified by
gravitational lensing
(Dyer 1976;
Mitrofanov 1981;
Nottale 1984;
Chitre et
al. 1986;
Blanchard &
Schneider 1987;
Linder 1988,
1990a,
b;
Cole &
Efstathiou 1989;
Sasaki 1989;
Durrer & Kovner
1990;
Watanabe &
Tomita 1991).
Since a stationary lens does not alter the surface
brightness of a source, all that a population of gravitational lenses
will do is to distort the brightness fluctuations on the sky. This
can modify the angular power spectrum of the microwave anisotropy and
shift the scale on which these fluctuations are observed. In
principle, for extremely strong lensing, the fluctuations can be so
badly scrambled as to be wiped out entirely at the resolution of the
observations. However, given the present limits on the number density
and masses of lenses, this appears to be very unlikely.
It has been pointed out that surface brightness is not preserved if
the lens is moving across the line-of-sight
(Mitrofanov 1981,
Birkinshaw &
Gull 1983,
Kaiser & Stebbins 1984,
Gurvits &
Mitrofanov 1981,
Khmil' 1988).
The temperature ahead of the lens is
greater than that behind by ~ T 
v
/ c, where T is
the mean temperature,
is the deflection angle at the lens,
and v is the perpendicular velocity. To detect this effect
with present-day techniques, one either needs a large v/ c
(e.g. a relativistically moving cosmic string) or a large (e.g.
a supercluster-scale lens).
