Annu. Rev. Astron. Astrophys. 1992. 30: 311-358
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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 Omega0 (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 phi (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 ltapprox 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 phi (L), it is possible in principle to obtain the true luminosity function phitrue (L) (Schneider 1992).

6.6 Microwave Background

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 alpha vperp/ c, where T is the mean temperature, alpha is the deflection angle at the lens, and vperp is the perpendicular velocity. To detect this effect with present-day techniques, one either needs a large vperp/ c (e.g. a relativistically moving cosmic string) or a large alpha (e.g. a supercluster-scale lens).

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