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

33.

(Fukugita et al 1992). Here zs is the redshift of the source (quasar). The prefactor normalizes the fiducial value to unity. The function d(z1, z2) is the angular diameter distance from redshift z1 to redshift z2, given by the generalization of Equation 25,

34.

Equation 33 quantifies the geometrical differences affecting ray paths and volumetric factors among different M and models. Figure 9 plots the value of Plens in the (M, tot) plane for the specific (but reasonable) choice zs = 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 .

 Figure 9. Probability for observing a gravitational lens, as contours in the (M, tot) plane, normalized to unity for the case M = 1, = 0. Gravitational lens statistics are the most promising method for ruling out -dominated models along the tot = 1 line, and already give the best bounds on to the right of that line.