Critical lines in gravitational lenses and the determination of cosmological parameters

We investigate the cosmological test recently done by Fort, Mellier, & Clantel-Fort (f;MD), in which the observed location of the critical line in gravitational lensing is used to determine the cosmological parameters Omega and lambda. Applying this method to the cluster of galaxies Cl 0024 + 1654, FMT) obtained a bound on the cosmological constant, lambda > 0.6, assuming the spatially Bat universe. The fact that the angular diameter distance-redshift relation depends on the cosmological models through the cosmological parameters plays a crucial role in this method. Using the angular diameter distance in the Friedmann-Lemaitre-Robertson-Walker universe for a wide range of cosmological parameters, we show that it is difficult to determine Omega by this method without the assumption of the spatially Bat universe. Moreover, the uncertainty of the lens model is also mentioned, which is regarded as one of the most serious problems in this lest. As a result, this method cannot be taken as a clear cosmological test to determine lambda, although one may put a bound on lambda by the use of many clusters of galaxies.