3.1 Gravitational Lens Statistics
Fukugita, Futamase & Kasai (1990) and Turner (1990) suggested that a statistical study of the number density of gravitational lenses could provide a powerful test of a non-zero . Subsequently a number of studies have been undertaken (e.g. Fukugita & Turner 1991; Bahcall et al. 1992; Maoz et al. 1993; Kochanek 1993, 1996). The basic idea behind this method is simple: the number of gravitationally lensed objects is a very sensitive function of . For larger values of , there is a greater probability that a quasar will be lensed because the volume over a given redshift interval is increased. In a flat universe with a value of = 1, approximately an order of magnitude more gravitational lenses are predicted than in a universe with = 0 (Turner 1990). Thus, simply counting the numbers of gravitationally lensed quasars can provide a very powerful limit on the value of . In practice, however, there are a number of complications: galaxies evolve (and perhaps merge) with time, even elliptical galaxies contain dust, the properties of the lensing galaxies are not well-known (in particular, the dark matter velocity dispersion is unknown), and the numbers of lensing systems known at present is very small (~ 20). Moreover, while the predicted effects are very large for = 1, because the numbers are such a sensitive function of , it is very difficult to provide limits below a value of about 0.6, given these complicating effects.
Kochanek (1996) has recently discussed these various effects in some detail, and investigated the sensitivity of the results to different lens models and extinction. His best estimated limits to date are : < 0.66 (95% confidence) for m + = 1, and m = 0.15 (90% confidence) if = 0. Significant improvements to these limits could be made by increasing the size of the current lens samples.