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

**3.6 Gravitational Lens Probabilities**

One effect of a non-zero cosmological constant is to change, in some
cases drastically, the probability that quasars are gravitationally
lensed by intervening galaxies
(Fukugita et al 1990a,
Turner 1990).
While the absolute lens probability obviously depends on the absolute
density and gravitational potential of the lensing galaxies, a useful
statistic is the probability for lensing by a population of isothermal
spheres of constant comoving density *relative* to the fiducial
case _{M} = 1,
_{} = 0, given by the integral

(Fukugita et al 1992).
Here *z*_{s} is
the redshift of the source (quasar). The prefactor normalizes the
fiducial value to unity. The function *d*(*z*_{1},
*z*_{2}) is the angular diameter
distance from redshift *z*_{1} to redshift
*z*_{2}, given by the generalization of Equation 25,

Equation 33 quantifies the geometrical differences affecting ray paths
and volumetric factors among different
_{M} and
_{} models.
Figure 9
plots the value of *P*_{lens} in the
(_{M},
_{tot}) plane for the
specific (but
reasonable) choice *z*_{s} = 2. Along the diagonal line
_{} = 0, one sees
that lens probabilities increase as the universe becomes emptier, but
only by a modest factor ~ 2. By contrast, as the matter density is
decreased along the line
_{tot} = 1 (that is,
compensated by increasing
_{}), the lens probability
rises dramatically, by a factor ~ 10. We
will see below that gravitational lensing, because it distinguishes so
sharply between low
_{M} universes of
differing _{}, holds great promise
for putting firm limits on _{}.