**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.