3.3. Gravitational lensing and the cosmological constant
Consider a distant source at redshift z which is lensed by some intervening object. The lensing is most effective if the lens is located midway between the source and the observer (see, eg., page 196 of ). This distance will be (rem/2) if the distance to the source is rem. (To be rigorous, one should be using angular diameter distances rather than rem for this purpose but the essential conclusion does not change.) To see how this result depends on cosmology, let us consider a source at redshift z = 2, and a lens, located optimally, in: (a) = 1 matter dominated universe, (b) a very low density matter dominated universe in the limit of 0, (c) vacuum dominated universe with = tot. In case (a), dH H-1(z) (1 + z)-3/2, so that
The lens redshift is determined by the equation
For z = 2, this gives zL = 0.608. For case (b), a t giving dH (1 + z)-1 and rem(z) ln(1 + z). The equation to be solved is (1 + zL) = (1 + z)1/2 which gives zL = 0.732 for z = 2. Finally, in the case of (c), dH is a constant giving rem(z) z and zL = (1/2)z. Clearly, the lens redshift is larger for vacuum dominated universe compared to the matter dominated universe of any . When one considers a distribution of lenses, this will affect the probability for lensing in a manner which depends on . From the observed statistics of lensing, one can put a bound on .
More formally, one can compute the probability for a source at redshift zs being lensed in a + NR = 1 universe (relative to the corresponding probability in a NR = 1, = 0 model). This relative probability is nearly five times larger at zs = 1 and about thirteen times larger for zs = 2 in a purely cosmological constant dominated universe [90, 91, 92, 93, 94, 2, 3]. This effect quantifies the fact that the physical volume associated with unit redshift interval is larger in models with cosmological constant and hence the probability that a light ray will encounter a galaxy is larger in these cases.
Current analysis of lensing data gives somewhat differing constraints based on the assumptions which are made [95, 96, 97, 98]; but typically all these constraints are about 0.7. The result  from Cosmic Lens All Sky Survey (CLASS), for example, gives NR = 0.31+0.27-0.14 (68%) +0.12-0.10 (systematic) for a k = 0 universe.