6. The gravitational lensing rate

The probability of production of multiple images of a quasar or a radio source by gravitational lensing by a foreground galaxy, or of strongly lensed images of a galaxy by a foreground cluster of galaxies, adds the relativistic expression for the deflection of light to the physics of the homogeneous cosmological model. Fukugita, Futamase, and Kasai (1990) and Turner (1990) point out the value of this test: at small _M0 the predicted lensing rate is considerably larger in a flat model with than in an open model with = 0 (as illustrated in Fig 13.12 in Peebles, 1993).

The measurement problem for the analysis of quasar lensing is that quasars that are not lensed are not magnified by lensing, making them harder to find and the correction for completeness of detection harder to establish. Present estimates (Falco, Kochanek, and Muñoz, 1998; Helbig et al., 1999) do not seriously constrain _M0 in an open model, and in a flat model (_K0 = 0) suggest _M0 > 0.36 at 2. This is close to the upper bound in Eq. (59). Earlier indications that the lensing rate in a flat model with constant requires a larger value of _M0 than is suggested by galaxy dynamics led Ratra and Quillen (1992) and Waga and Frieman (2000) to investigate the inverse power-law potential dark energy scalar field case. They showed this can significantly lower the predicted lensing rate at _K0 = 0 and small _M0. The lensing rate still is too uncertain to draw conclusions on this point, but advances in the measurement certainly will be followed with interest.

The main problem in the interpretation of the rate of strong lensing of galaxies by foreground clusters as a cosmological test is the sensitivity of the lensing cross section to the mass distribution within the cluster (Wu and Hammer, 1993); for the present still somewhat uncertain state of the art see Cooray (1999) and references therein.