**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
[46]).
This distance will be (*r*_{em}/2)
if the distance to the source is *r*_{em}.
(To be rigorous, one should be using angular diameter distances rather than
*r*_{em} 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),
*d*_{H}
*H*^{-1}(*z*)
(1 +
*z*)^{-3/2}, so that

(38) |

The lens redshift is determined by the equation

(39) |

For *z* = 2, this gives *z*_{L} = 0.608.
For case (b),
*a* *t*
giving *d*_{H}
(1 +
*z*)^{-1} and *r*_{em}(*z*)
ln(1 + *z*). The
equation to be solved is
(1 + *z*_{L}) = (1 + *z*)^{1/2} which gives
*z*_{L} = 0.732 for *z* = 2.
Finally, in the case of (c), *d*_{H} is a constant giving
*r*_{em}(*z*)
*z* and
*z*_{L} = (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 *z*_{s} 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
*z*_{s} = 1 and about thirteen
times larger for *z*_{s} = 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 [99]
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.