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 \pm 0.11 , \Omega _ { \lambda } = 0.70 \pm 0.23 or \Omega _ { m } = 1.00 \pm 0.17 , \Omega _ { \lambda } = 0.00 \pm 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 .