Critical Lines in Gravitational Lenses and the Determination of Cosmological Parameters

We investigate the cosmological test recently proposed by B. Fort, Y. Mellier and M. Dantel-Fort (FMD), where 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 Cl0024+1654, FMD obtained a constraint on the cosmological constant, $\lambda > 0.6$, assuming the spatially flat universe. It plays a crucial role in this method that the angular diameter distance-redshift relation depends on the cosmological models through the cosmological parameters. First, using the angular diameter distance in the Friedmann-Lemaitre- Robertson-Walker universe, we show that one can hardly determine $\Omega$ by this method without the assumption of the spatially flat universe. We also investigate the effect of inhomogeneities of the universe by using the Dyer-Roeder angular diameter distance. It is shown that the effect of inhomogeneities can become too large to be ignored, particularly for a high density universe. As a result, this method cannot be taken as a clear cosmological test to determine $\Omega$ and $\lambda$, though it may provide a bound on $\Omega$ and $\lambda$. Moreover, we mention the uncertainty of the determination of the velocity dispersion, which is regarded as one of the most serious problems in this test.