Constraining the Cosmological Parameters using Strong Lensing

We investigate the potentiality of using strong lensing clusters to constrain the cosmological parameters Omega_m and Omega_lambda. The existence of a multiple image system with known redshift allows, for a given (Omega_m, Omega_lambda) cosmology, absolute calibration of the total mass deduced from lens modelling. Recent Hubble Space Telescope (HST) observations of galaxy clusters reveal a large number of multiple images, which are predicted to be at different redshifts. If it is possible to measure spectroscopically the redshifts of many multiple images then one can in principle constrain (Omega_m,Omega_lambda) through ratios of angular diameter distances, independently of any external assumptions. For a regular/relaxed cluster observed by HST with 3 multiple image systems, each with different spectroscopic redshifts, we show by analytic calculation that the following uncertainties can be expected: Omega_m=0.30 +/- 0.11, Omega_lambda=0.70 +/- 0.23 or Omega_m=1.00 +/- 0.17, Omega_lambda=0.00 +/- 0.48 for the two most popular world models. Numerical tests on simulated data confirm these good constraints, even in the case of more realistic cluster potentials, such as bimodal clusters, or when including perturbations by galaxies. To investigate the sensitivity of the method to different mass profiles, we also use an analytic ``pseudo-elliptical'' Navarro, Frenk & White profile in the simulations. These constraints can be improved if more than 3 multiple images with spectroscopic redshifts are observed, or by combining the results from different clusters. Some prospects on the determination of the cosmological parameters with gravitational lensing are given.