1 Introduction

A new "standard cosmological model'' has arisen in the last few years, favoring a flat Universe with $\Omega_m\sim 0.3$ and $\Omega_\lambda\sim 0.7$. This is mainly based on the results of two experiments which give roughly orthogonal constraints in the $(\Omega _m, \Omega _\lambda )$plane (see Fig. 1 for a recent update). The first one is obtained by considering type Ia supernovae (SNIa) as standard candles. The detection of a sample of high redshift SNIa (up to $z \sim$ 1) by two groups favours a non-vanishing cosmological constant (Perlmutter et al. 1998; Riess et al. 1998), large enough to produce an accelerating expansion. However, evidence for a non-zero cosmological constant is still controversial, since supernovae might evolve with redshift and/or may be dimmed by intergalactic dust (Aguirre 1999). The fundamental assumption of a homogeneous Universe and its implication for a non-zero cosmological constant are also discussed (Célérier 2000; Kolatt & Lahav 2001). The second constraint is derived from the location of features in the cosmic microwave background (CMB) anisotropy spectrum, particularly the first Doppler peak. The most recent results obtained with the Boomerang and MAXIMA experiments favor a flat Universe (Balbi et al. 2000; Melchiorri et al. 2000). However, there still remains a degeneracy in the combination of $\Omega _m$ and $\Omega _\lambda$ because CMB experiments are primarily sensitive to the total curvature of the Universe. Even with the accuracy of the future MAP and Planck missions, the constraint issued from the CMB alone will be degenerate.

The combination of these two sets of constraints has led to the currently favored model of low matter density and a non-zero cosmological constant, preserving a flat geometry (e.g. White 1998; Efstathiou et al. 1999; Freedman 2000; Sahni & Starobinsky 2000; Jaffe et al. 2001). Although these recent results are quite spectacular, there still remain many sources of uncertainties with both methods. Thus any other independent test to constrain the large scale geometry of the Universe is important to investigate. Gravitational lensing, an effect involving large distance scales, has been considered as a very promising tool for such determinations. Indeed, the statistics of gravitational lenses depend on the cosmological parameters via angular size distances and the comoving spherical volume (e.g. Turner et al. 1984; Turner 1990; Kochanek 1996; Falco et al. 1998). This technique has provided an upper limit on $\Omega _\lambda$ using different surveys of galaxy lens systems: multiple quasar statistics (Kochanek 1996; Chiba & Yoshii 1999), lensed radio sources (Cooray 1999), lensed galaxies in the Hubble Deep Field (Cooray et al. 1999). Although most authors favor a lambda-dominated flat Universe, there remain some uncertainties in the mass distribution of the galaxy lenses and on the luminosity function of the sources. Evolutionary effects may also play a role in these statistics.

Another application of gravitational lensing to constrain the cosmological parameters is to use the statistics of the "cosmic'' shear variance. Van Waerbeke et al. (1999) showed that it is related to the power spectrum of the large scale mass fluctuations, and then to $\Omega _m$. The first results of deep wide field imaging surveys favor $\Omega _m$ in the range [0.2-0.5] (Maoli et al. 2001; Van Waerbeke et al. 2001; Bacon et al. 2000; Wittman et al. 2000). Imaging surveys with the next generation of panoramic CCD cameras will reinforce this very promising technique. In the case of weak lensing by clusters of galaxies, Lombardi & Bertin (1999) and Gautret et al. (2000) suggested methods to constrain the geometry of the Universe. These methods need however to recover the mass distribution and/or to know acurately the redshift of a huge number of distant galaxies, making this method not practical in the near future.

In this paper, we focus on a measurement technique of $(\Omega _m, \Omega _\lambda )$ using gravitational lensing as a purely geometrical test of the curvature of the Universe, since the lens equation depends on the ratio of angular size distances which is sensitive to the cosmological parameters. In the most favorable case, a massive cluster of galaxies can lens several background galaxies, splitting the images into several families of multiple images. The existence of one family of multiple images, at known redshift, allows to calibrate the total cluster mass in an absolute way. In the case of several sets of multiple images with known redshifts, it is possible in principle to constrain the geometry of the Universe. This method was pointed out by Blandford & Narayan (1992), and earlier suggested by Paczynski & Gorski (1981), but the uncertainties in any lens studies were considered too large compared to the small variations induced by the cosmological parameters. More recently, Link & Pierce (1998) (hereafter LP98) re-analysed the question in the light of the typical accuracy reached with HST images of clusters of galaxies. Following their method, which inspired our work, we try to quantify in this paper what can be reasonably obtained on $(\Omega _m, \Omega _\lambda )$ from accurate lens modeling of realistic cluster-lenses.

The paper is organized as follows. In Sect. 2 we summarize the main lensing equations and we introduce the relevant angular size distance ratio which contains the dependence on the cosmological parameters. The variation of this ratio is then compared to that of other variables (lens potential parameters and redshifts) to derive the expected uncertainties on $\Omega _m$ and $\Omega _\lambda$. In Sect. 3 we present the method in detail and the results from simulations of various types of families of images and of different types of lens potentials. Some conclusions and prospects for the application to real data are discussed in Sect. 4. Throughout this paper we assume H₀=65 km s^-1 Mpc^-1 (note however that the proposed method and results are independent of H₀).

Figure 1: Constraints on $(\Omega _m, \Omega _\lambda )$ derived from the most recent results of the BOOMERANG and MAXIMA-I experiments on the CMB fluctuations and the last results from the SNIa analysis (from Jaffe et al. 2001). Overplotted is the ratio of the geometrical factor of the lens equation for two source redshifts, $e(z_{\rm S1}, z_{\rm S2}) = E(z_{\rm S1})/ E(z_{\rm S2})$ as discussed in the text. In this example, the redshifts are chosen as representative of a typical lens configuration: $z_{\rm L} = 0.3$, $z_{\rm S1} = 0.7$ and $z_{\rm S2} = 2$.