Annu. Rev. Astron. Astrophys. 1992. 30: 499-542 Copyright © 1992 by Annual Reviews. All rights reserved

4.6 Gravitational Lensing

As described in Section 3.6 and illustrated in Figure 9, gravitational lensing frequencies are potentially sensitive indicators of a non-zero , especially along the fashionable _k = 0 line. This fact, implicit in the lensing statistics analysis of Gott et al (1989), was pointed out explicitly by Fukugita et al (1990a) and by Turner (1990). It is an effect that has the potential for making a decisive test of the possibility of an -dominated universe. Earlier work on lensing with non-zero values (Paczynski & Gorski 1981, Alcock & Anderson 1986) concentrated on quantities such as image angular separations which are quite insensitive indicators (Fukugita et al 1992) and thus gave little hope for a useful test.

Whether or not currently available data on, and understanding of, gravitational lens statistics yet allows any clear conclusion is a somewhat controversial question. Turner (1990) found that a naive calculation of the expected lensing rates in flat -dominated models predicted far more lens systems in known quasar samples than have been observed and concluded that the data excluded large values (with various caveats). Fukugita & Turner (1991) reexamined the issue attempting to take into account more carefully both observational and theoretical uncertainties and concluded that although the strength of the conclusion was weakened, models as dominated as model C in Table 1 could only be accommodated by stretching both sorts of uncertainties to their plausible limits (i.e. that it was only marginally allowed). A yet more elaborate treatment by Fukugita et al (1992) reached a similar conclusion.

The principal difficulties in calculating lensing frequencies and comparing the results to observational determinations include: (a) characterizing the mass distributions of the low-redshift galaxy population accurately enough to allow a determination of its lensing effectiveness (the critical issues being the space density of galaxies, the distribution of their potential well depths, their mass core radii, and their ellipticities); (b) accounting for possible evolution of the galaxy population (note that here one need only consider evolution of the galaxies' mass distributions without regard to any possible luminosity evolution); (c) determining the selection biases in specific quasar and lens surveys, particularly those which might cause lens systems to be entirely omitted from the sample (e.g. by the rejection of objects with nonstellar images) or to go unrecognized (e.g. by lack of sufficient resolution to detect the multiple images); and () adjusting the predictions for the effect of amplification biases - the sometimes strong tendency of lens systems to be preferentially included in flux limited samples due to the boosting of their brightnesses (Turner 1980, Turner et al 1984).

These are a formidable set of complications, which cannot yet be dealt with precisely: however, the uncertainties in accounting for them amount to a factor of 1.5 or perhaps 2, while the differences associated with substantial variations of are substantially larger, typically an order of magnitude (see Figure 9). Furthermore, the effect of increasing is to make some of these uncertainties smaller; for example, lensing cross sections become less sensitive to galaxy core radii, and significant galaxy evolution at redshifts that dominate the total lensing integrated probabilities become less astrophysically plausible (because the universe is vacuum rather than mass dominated). It is also important that large values tend to predict too many lensing events; a prediction of too few events would be far easier to explain away by invoking an otherwise unknown population of lenses or by supposing that physical multiples were being mistaken for lens systems. It is these considerations which give some reason for confidence in the upper limits on (now typically about 0.9) in _k = 0 cosmologies that have been adduced from available calculations and observations.

On the other hand, Kochanek (1991) and Mao (1991) have emphasized these possible sources of systematic error, and believe that firm conclusions are premature. Since both improved theoretical (numerical) predictions are possible (Kochanek 1991) and since a variety of carefully controlled quasar surveys (in which lensing events may be found with predictable efficiencies) are becoming available or are in progress (Crampton 1991, Hartwick & Schade 1990), rapid progress should be possible for this test. In the end, its value may be limited by our understanding of galaxy properties (i.e. the lens population) and their evolution (Mao 1991), just as for several of the other tests already discussed.

Recently, Kochanek (1992) has suggested a new test of . He considers the expected lens redshift distribution for systems with given source redshift and image separations (i.e. angular diameters of the lens Einstein ring) and shows that flat, zero models predict much lower typical lens redshifts than do -dominated flat models (like model C). Comparing this to the data for the small number of known lens systems for which all of the required data is available, he concludes that the results significantly favor the = 0 model. This technique is extremely promising, although it too needs to be examined for possible systematic problems (e.g closer lenses are easier to detect and have their redshifts measured more readily) and for possible worries about its sensitivity to details of the lens (galaxy) properties and their evolution.

On balance, it is probably fair to conclude that gravitational lens statistics (of both sorts discussed above) currently offer the biggest empirical challenge to cosmological models with significant terms, and that they are perhaps the most immediately promising area for further study, both observational and theoretical. However, no conclusions strong enough to deter either theoretical enthusiasts nor the pursuit of other observational tests are yet in hand.