2 Influence of $\mathsf\Omega_\mathsf{m}$ and $\mathsf\Omega_\mathsf{\lambda}$ on image formation

2.1 Cosmology dependent lensing relations

We first define the basic mathematical framework, following the formalism presented in Schneider et al. (1992). We consider a lens at a redshift $z_{\rm L}$ with a two-dimensional projected mass distribution $\Sigma(\vec{\theta})$ and a projected gravitational potential $\phi(\vec{\theta})$, where $\vec{\theta}$ is a two-dimensional vector representing the angular position. A source galaxy with redshift $z_{\rm S}$is located at position $\vec{\theta}_{\bf S}$ in the absence of a lens, and its image is at position $\vec{\theta}_{\bf I}$. In the lens equation

$$\left\lbrace \begin{array}{l} \vec{\theta}_{\bf S} =\vec{\the... ...}}{D_{\rm OL}D_{\rm OS}}}\phi (\vec\theta), \end{array}\right.$$ (1)

$D_{\rm OL}$, $D_{\rm LS}$ and $D_{\rm OS}$ are respectively the angular diameter distances from the Observer to the Lens, from the Lens to the Source and from the Observer to the Source (Peebles 1993). $\varphi$ is the reduced gravitational potential which satisfies $\nabla^2 \varphi=2 \ \Sigma / \Sigma_{\rm crit}$ with the critical density

$$\Sigma_{\rm crit}=\displaystyle{\frac{c^{2}}{4\pi G}}\displaystyle{\frac{D_{\rm OS}}{D_{\rm LS}D_{\rm OL}}}\cdot$$ (2)

In these equations the dependence on the cosmological parameters appears only through the angular diameter distances ratios $F=\displaystyle{\frac{D_{\rm OL} \ D_{\rm LS}}{D_{\rm OS}}}$ and $E=\displaystyle{\frac{D_{\rm LS}}{D_{\rm OS}}}$ (E as "efficiency'') for a fixed cluster redshift. They correspond to a scaling of the lens equation, reflecting the geometrical properties of the Universe.

In the general case, we can scale the potential gradient as:

$$\vec\nabla \phi(\vec{\theta}_{\bf I})= \sigma_0^2 \ D_{\rm OL} \ \vec f(\vec{\theta}_{\bf I},\theta_{\rm C},\alpha, \ldots)$$ (3)

where $\sigma _0$ is the central velocity dispersion and $\vec f$ a dimensionless function that describes the mass distribution of the cluster. It can be represented by fiducial parameters such as a core radius $\theta_{\rm C}$ or a mass profile gradient $\alpha$. The lens equation reads

 

$$\vec{\theta}_{\bf S} \vec{\theta}_{\bf I}-\frac{\sigma_0^2}{c^2} \ \frac{D_{\rm LS}}{D_{\rm OS}} \ \vec f(\vec{\theta}_{\bf I},\theta_{\rm C},\alpha, \ldots )$$ = (4)

$$\vec{\theta}_{\bf I} - \frac{\sigma_0^2}{c^2} \ \vec f(\vec{\thet... ...C},\alpha, \ldots ) \times E(\Omega_m, \Omega_\lambda , z_{\rm L}, z_{\rm S} ).$$ =

We will focus on the E-term which entirely contains the dependence on $\Omega _m$ and $\Omega _\lambda$ and which is independent of H₀.

2.2 The E-term

For a given lens plane $z_{\rm L}$, the ratio E increases rapidly as the source redshift increases, and then flattens at $z_{\rm S}\sim1.5$. There are also small but significant changes with the cosmological parameters (Fig. 2). The dependence of E with respect to $\Omega _m$ is weak, whereas the variation with  $\Omega _\lambda$ is larger.

We now consider fixed redshifts for the lens and sources. Assuming a fixed world model, a single family of multiple images can in principle constrain the total cluster mass as well as the shape of the potential, removing the unknown position of the source $\vec{\theta}_{\bf S}$ using Eq. (4). In practice, good constraints on the shape of the potential $\vec f$are obtained with triple, quadruple or quintuple image systems. However the absolute normalization $\sigma _0$ of the mass is degenerate with the E-term, that is with respect to $\Omega _m$ and $\Omega _\lambda$.

Figure 2: Variation of the lens efficiency $E(z_{\rm S})$ as a function of the source redshift for different sets of cosmological parameters. Left: $\Omega _\lambda = 0$ and $\Omega _m$ varies from 0.1 to 1.0. Right: $\Omega _m = 0.3$ and $\Omega _\lambda$ varies from 0 to 0.7. In both cases, the lens redshift is $z_{\rm L} = 0.3$. For clarity, above each main plot, we also plotted the same curves, normalised with the $E(z_{\rm S})$ function for $(\Omega _m,\Omega _\lambda )=(0.3,0.)$.

   
2.3 Ratio of E-terms for 2 sets of source redshifts

To break this degeneracy a second family of multiple images is needed. To get rid of the strong dependence on $\sigma _0$, it is useful to consider the ratio of the positions:

$$\displaystyle \frac{\Vert\vec{\theta}_{\bf I_1^1} - \vec{\the... ...I^2_1},\ldots) -\vec f(\vec{\theta}_{\bf I^2_2},\ldots )\Vert}$$ (5)

(here and hereafter, the superscript refers to a family and the subscript to a particular image within a family). This ratio is plotted in Fig. 1, highlighting the influence of $\Omega _m$ and $\Omega _\lambda$ through the ratio $$e(z_{\rm S1},z_{\rm S2})=E(z_{\rm S1})/E(z_{\rm S2})$$.

Note that the discrepancy between the different cosmological parameters is not very large, less than 3% between the Einstein-de Sitter model (EdS) and a flat, low matter density one. Moreover, a characteristic degeneracy appears in the $(\Omega _m, \Omega _\lambda )$ plane, which is roughly orthogonal to the one given by the detection of high redshift supernovae, and quite different from the CMB constraints. A similar degeneracy was also found in the analog weak lensing analyses, (Lombardi & Bertin 1999) or by Gautret et al. (2000).

Another approach to quantify the dependence of a given lens configuration on $\Omega _m$ and $\Omega _\lambda$ is to fix the lens redshift and to search for two source redshifts $z_{\rm S1}$ and  $z_{\rm S2}$ which give the largest variation of the E-term when scanning the $(\Omega _m, \Omega _\lambda )$ plane. For illustration we arbitrarily choose two sets of cosmological parameters, for which the relative variation of Eis large: CP1 $(\Omega _m=0.3, \Omega _\lambda =0)$ and CP2 ( $\Omega _m=1, \Omega _\lambda =0$, i.e. the EdS model). Varying $z_{\rm S1}$ and $z_{\rm S2}$, the function $\varepsilon(z_{\rm S1},z_{\rm S2})=e_{\rm CP2}(z_{\rm S1},z_{\rm S2})/e_{\rm CP1}(z_{\rm S1},z_{\rm S2})-1$represents the percentage of discrepancy between CP1 and CP2 for $z_{\rm S1}\geq z_{\rm S2} \ (\geq z_{\rm L})$ (Fig. 3).

For a given high-redshift $z_{\rm S2}$ source the best lowest source redshift is $z_{\rm S1} \simeq z_{\rm L}$, and for a given $z_{\rm S1}$ the best  $z_{\rm S2}$ is the highest redshift, the difference between cosmological models increasing with $z_{\rm S2}$. In all cases, this relative difference is of the order of a few %, meaning that the lens mass distribution must be known to the same degree of accuracy to get further constraints on the cosmological parameters. Hence, for 2 systems of images, the best configuration is one background source close to the lens, in the rising part of $E(z_{\rm S})$ and another one at high redshift, to take into account the asymptotic value of the ratio. Note however that for a source redshift close to the lens, the E-term becomes very small. Also, the location of the images is very close to the lens center which makes the detection of multiple images quite improbable, as small caustic sizes imply small cross sections.

Figure 3: Relative difference between the ratio $e(z_{\rm S1}, z_{\rm S2}) = E(z_{\rm S1})/ E(z_{\rm S2})$ for two extreme cosmological models: $(\Omega _m=0.3, \Omega _\lambda =0)$ and ( $\Omega _m=1, \Omega _\lambda =0$). This function $\varepsilon (z_{\rm S1},z_{\rm S2})$ is plotted in the $(z_{\rm S1}, z_{\rm S2})$ plane, assuming $z_{\rm S1}\le z_{\rm S2}$ and $z_{\rm L} = 0.3$.

  
2.4 Relative influence of the lens parameters

2.4.1 Physical assumptions

In order to quantify the expected uncertainties on $\Omega _m$ and  $\Omega _\lambda$, it is possible to analytically estimate the influence of the different lens models parameters. We use a model of the potential derived from the mass density described by Hjorth & Kneib (2001), hereafter HK. It is based on a physical scenario of violent relaxation in galaxies, also valid in clusters of galaxies. The mass density is characterized by a core radius a and a cut-off radius s:

$$\rho(r)=\frac{\rho_{0}}{\left (1+\frac{r^{2}}{a^{2}}\right ) \left (1+\frac{r^{2}}{s^{2}}\right )}\cdot$$ (6)

Then the projected mass density $\Sigma(\theta)$ is represented by:

$$\Sigma(\theta)=\Sigma_0\ \frac{\theta_a\theta_s}{\theta_s-\th... ...a^2+\theta_a^2}}- \frac{1}{\sqrt{\theta^2+\theta_s^2}}\right),$$ (7)

where $\theta$ is the angular coordinate, $\theta_a = a/D_{\rm OL}$ and $\theta_s = s/D_{\rm OL}$. $\Sigma_0$ is a normalization factor, related to the cluster parameters:

$$\Sigma_0 = \pi \rho_0 D_{\rm OL} \ \frac{\theta_a\theta_s}{\theta_s+\theta_a}\cdot$$ (8)

Finally, the mass inside the projected angular radius $\theta$ is:

 

$$M(\theta) = 2\pi\Sigma_0 D_{\rm OL}^2 \ \frac{\theta_a\theta_s}{\... ...sqrt{\theta^2+\theta_a^2}-\sqrt{\theta^2+\theta_s^2}+ \theta_s-\theta_a\right].$$ (9)

The velocity dispersion $\sigma(\theta)$ is related to the mass density and the gravitational potential via the Jeans equation. Assuming an isotropic velocity dispersion and retaining terms up to first order in $\theta_a/\theta_s$, we get the relation between the central velocity dispersion $\sigma_0=\sigma(0)$ and $\rho_0$:

$$\sigma_0^2 = \frac{\pi^3Ga^2}{2} \ \rho_0 s.$$ (10)

Finally we compute the expression of the deviation angle between the positions of the source and of the image due to the lens: $D_{\theta_{\rm I}}=\Vert\vec{\theta}_{\bf I}-\vec{\theta}_{\bf S}\Vert$, neglecting second order terms in $\theta_a/\theta_s$ (we suppose here $s\gg a$):

$$D_{\theta_{\rm I}}\! =\! \frac{16}{\pi}\frac{\sigma_0^2}{c^2}... ..._a^2} -\sqrt{\theta^2+\theta_s^2}+ \theta_s-\theta_a\right]\!.$$ (11)

2.4.2 The single multiple-image configuration

The central velocity dispersion (or equivalently the mass normalization of the cluster core) is obviously the predominant factor in any lens configuration. With a single family of images we can only constrain the combination $\sigma_0^2E$ and cannot disentangle the influence of the cosmological parameters and the absolute normalization of the mass (Fig. 4). If we were able to measure the total mass within the Einstein radius independently from lensing techniques and with an accuracy better than a few %, we could in principle put some constraints on $\Omega _\lambda$. Observationally, there are 2 situations where it is likely that we could disentangle the effect of cosmology and absolute mass:
1) in the case of a cluster-lens with extremely good X-ray data, particularly in estimating the temperature distribution of the gas (under the assumption of hydrostatic equilibrium),
2) in the case of a multiple system around a single galaxy, for which one is able to measure accurately the stellar velocity dispersion of the lensing galaxy (Tonry & Franx 1999).
However in both cases this represents some observational challenge and requires the most powerful instruments to achieve this goal.

Figure 4: Variation of the central velocity dispersion $\sigma _0$ in the ( $\Omega _m,\Omega _\lambda$) plane, assuming that the product $\sigma _0^2 E(z_{\rm S})$ is constant. $\sigma _0$ has been fixed to 1000 km s-1 for $\Omega _m=1$, $\Omega _\lambda = 0$ while $z_{\rm L} = 0.3$ and $z_{\rm S}=1$.

Although the error budget in the image positions is dominated by the error on the total cluster mass (or equivalently the velocity dispersion), we can determine the relative influence of the other parameters to infer the importance of $\Omega _m$ and $\Omega _\lambda$in the image formation. The relative error on the deviation angle $D_{\theta_{\rm I}}$ depends on $\sigma _0$, $\theta _a$, $\theta _s$ and $\theta_{\rm I}$ for the gravitational potential, and $z_{\rm L}$, $z_{\rm S}$, $\Omega _m$ and $\Omega _\lambda$ for the E-term (Eq. (11)). Therefore we can write:

$$\frac{{\rm d}D_{\theta_{\rm I}}}{D_{\theta_{\rm I}}}\!=\!\fra... ...\alpha_{\theta_I}\frac{{\rm d}\theta_{\rm I}}{\theta_{\rm I}},$$ (12)

with

$$\frac{{\rm d}E}{E} (z_{\rm S})\! =\! \alpha_{\Omega_m}\frac{{... ... L}}+\alpha_{z_{\rm S}}\frac{{\rm d}z_{\rm S}}{z_{\rm S}}\cdot$$ (13)

$\alpha_{\theta_a}$, $\alpha_{\theta_s}$ and $\alpha_{\theta_{\rm I}}$ can be computed analytically while $\alpha_{\sigma_0}=2$ is the largest factor. Since the angular diameter distances do not have an analytic expression if $\Omega _\lambda$ is non-zero, the coefficients $\alpha_{\Omega_m}$, $\alpha_{\Omega_\lambda}$, $\alpha_{z_{\rm L}}$ and $\alpha_{z_{\rm S}}$ must be computed numerically. In practise, they are computed for a given set of parameters $(z_{\rm L}, \Omega_m, \Omega_\lambda)$ as their variation with $\Omega _m$ and $\Omega _\lambda$ is of higher order. For a reasonable set of lens parameters, the $\alpha$-coefficients are of the same order of magnitude, except that $\alpha_{z_{\rm S}}$ and $\alpha_{z_{\rm L}}$ can dominate the error budget if the source redshift is close to the lens (an unlikely case). On the contrary, $\alpha_{\Omega_m}$ and $\alpha_{\Omega_\lambda}$ are of second order, and $\alpha_{\Omega_\lambda}$ is somewhat larger than $\alpha_{\Omega_m}$. This reflects again the fact that E-term is more sensitive to $\Omega _\lambda$ than to $\Omega _m$.

To quantify the relative influence of all the parameters in the case of a single family of images, we computed explicitely ${\rm d}D_{\theta_{\rm I}}/D_{\theta_{\rm I}}$ in two cases, for a cluster-lens and for a galaxy-lens.

1) For a cluster of galaxies, we take the following parameters: $z_{\rm L} = 0.3$, $z_{\rm S}=4$, $\Omega _m = 0.3$, $\Omega_\lambda=0.7$, $\theta_s/\theta_a=10$and $\theta_{\rm I}/\theta_a=4$. We thus find from Eq. (11):

 

$$\frac{{\rm d}D_{\theta_{\rm I}}}{D_{\theta_{\rm I}}} = 2 ~ \frac{... ...m}}{\Omega_{m}} + 0.14 ~ \frac{{\rm d}\Omega_{\lambda}}{\Omega_{\lambda} }\cdot$$ (14)

Let us assume a perfectly known mass profile (i.e. d $\theta_a={\rm d} \theta_s=0.$). Neglecting the influence of $\Omega _m$, we ask what precision would be required on $\sigma _0$ to derive an error of 50% on $\Omega _\lambda$. The accuracy of the position of the center of the images is calculated using the first moment of the flux $f(\theta)$ on a given image: $\theta_{\rm I}=\int \theta f(\theta){\rm d}\theta^2$/ $\int f(\theta){\rm d}\theta^2$ which yields an error ${\rm d} {\theta_{\rm I}}$ of a fraction of the spatial resolution. HST observations are then required to reach ${\rm d}\theta_{\rm I}=0.1\arcsec$ (LP98) or better. To reduce the uncertainty on the redshift measurements, we assume spectroscopic determinations, so that d $z \simeq0.001$. Finally we have to compute the relative errors on $D_{\theta_{\rm I}}= \Vert \vec \theta_{\bf I} - \vec \theta_{\bf S} \Vert$, so the position of the source is in principle required. But as we are in the strong lensing regime, we assume that $\theta_{\rm S}\ll\theta_{\rm I}$, so that both quantities $D_{\theta_{\rm I}} \simeq \theta_{\rm I}$ and $${\frac{{\rm d}D_{\theta_{\rm I}}} {D_{\theta_{\rm I}}} \simeq \frac{{\rm d}\theta_{\rm I}}{\theta_{\rm I}}}$$ are directly related to observable ones. Taking these values into account, we need to know $\sigma _0$with 3.6% accuracy to get the expected constraint on $\Omega _\lambda$. Such an accuracy is out of reach with observations of clusters of galaxies.

2) For a single galaxy, we consider typically: $z_{\rm L} = 0.3$, $z_{\rm S}=4$, $\Omega _m = 0.3$, $\Omega_\lambda=0.7$, $\theta_s/\theta_a=200$and $\theta_{\rm I}/\theta_a=200$ (ratios taken from the modelisation of the lens HST 14176+5226 by Hjorth & Kneib 2001), leading to:

 

$$\frac{{\rm d}D_{\theta_{\rm I}}}{D_{\theta_{\rm I}}} = 2 ~ \frac{... ...{m}}{\Omega_{m}} + 0.14 ~ \frac{{\rm d}\Omega_{\lambda}}{\Omega_{\lambda}}\cdot$$ (15)

Taking the same values for the observational errors and considering a perfectly known mass profile, we require an accuracy of 6.4% on $\sigma _0$ to derive a 50% error on $\Omega _\lambda$. For a typical galaxy, this represents about 15 km s^-1. Warren et al. (1998) measured the velocity dispersion in the deflector of the Einstein ring 0047-2808 with an error of 30 km s^-1. A better accuracy could be obtained by looking at particular strong absorption features with 10 m class telescope observations. This could be sufficient to confirm an accelerating Universe.

2.4.3 Configuration with 2 multiple-image systems

With a second system of multiple images another region of the $E(z_{\rm S})$curve is probed while the cluster parameters are the same. In that case, the relevant quantity becomes the ratio of the deviation angles for 2 images $\theta_{\rm I^1}$ and $\theta_{\rm I^2}$ belonging to 2 different families at redshifts $z_{\rm S1}$ and $z_{\rm S2}$. We define $R_{\theta_{\rm I^1},\theta_{\rm I^2}}=\displaystyle{\frac{D_{\theta_{\rm I^2}}} {D_{\theta_{\rm I^1}}}}$. This function has the advantage that it does not depend on $\sigma _0$ anymore. Following our previous definitions, we can write

 

$$\frac{{\rm d}R_{\theta_{\rm I^1},\theta_{\rm I^2}}}{R_{\theta_{\rm I^1},\theta_{\rm I^2}}} \frac{{\rm d}E(z_{\rm S2})}{E(z_{\rm S2})} - \frac{{\rm d}E(z_{\r... ...left(\theta_{\rm I^1}\right) ~ \frac{{\rm d}\theta_{\rm I^1}}{\theta_{\rm I^1}}$$ =

$$+ \Big( \alpha_{\theta_a}\left(\theta_{\rm I^2}\right) - \alpha_{... ..._s} \left(\theta_{\rm I^1}\right) \Big) ~ \frac{{\rm d}\theta_s}{\theta_s}\cdot$$ (16)

Numerically, we chose a typical configuration to compute $${\frac{{\rm d}R_{\theta_{\rm I^1},\theta_{\rm I^2}}} {R_{\theta_{\rm I^1},\theta_{\rm I^2}}}}$$: $\theta_s/\theta_a=10$, $\theta_{\rm I^2} / \theta_{a}=4$, $\theta_{\rm I^2} / \theta_{\rm I^1}=2$, $z_{\rm L} = 0.3$, $z_{\rm S1}=0.6$, $z_{\rm S2}=5$, assuming $\Omega _m = 0.3$ and $\Omega_\lambda=0.7$. This gives the following error budget:

$$\frac{{\rm d}R_{\theta_{\rm I^1},\theta_{\rm I^2}}}{R_{\theta_{\r... ...}}{\Omega_{m}} + 0.037 ~ \frac{{\rm d}\Omega_{\lambda}}{\Omega_{\lambda}} \cdot$$ (17)

The contribution of the physical lens parameters in this error budget is strongly attenuated comprared to the single family case. There is no more dependence on $\sigma _0$ and the dependence on the mass profile ( $\theta_a, \theta_s$) is reduced by about a factor of 2 compared to a single family of images. This corresponds to the variation of the potential between  $\theta_{\rm I1}$and $\theta_{\rm I2}$, the absolute normalization being removed. Anyhow, this can still represent the main source of error because we cannot expect to constrain $\theta _a$ to better than 1.5% and $\theta _s$ to better than 2% typically (see Sect. 3.3.3).

For the source redshifts, we have selected one of the sources at $z_{\rm S1}=0.6$, which means that its $\alpha$-coefficient is quite large. The accurate value of the redshifts is thus fundamental, and a spectroscopic determination is essential (d $z \simeq0.001$). A photometric redshift estimate would not be satisfactory, because we cannot expect an accuracy better than 10% in most cases ( ${\rm d}z \simeq 0.1 {-} 0.2$, Bolzonella et al. 2000). We keep ${\rm d}\theta_{\rm I}=0.1\arcsec$. The strong lensing regime approximation leads to $R_{\theta_{\rm I^1},\theta_{\rm I^2}}=\displaystyle{\frac{\Vert \vec \theta_{\b... ...{\bf S^1} \Vert}}\simeq\displaystyle{\frac{\theta_{\rm I^2}}{\theta_{\rm I^1}}}$ and $${\frac{{\rm d}R_{\theta_{\rm I^1},\theta_{\rm I^2}}} {R_{\theta_{... ...^2}}{\theta_{\rm I^2}}- \frac{{\rm d}\theta_{\rm I^1}}{\theta_{\rm I^1}}}\cdot$$

We can then separate the contributions of the parameters that do not depend on $\Omega _m$ or $\Omega _\lambda$ from those which depend on them and re-write Eq. (16):

$$A_{\Omega_m} \frac{{\rm d}\Omega_m}{\Omega_m} + A_{\Omega_\lambda} \frac{{\rm d}\Omega_\lambda}{\Omega_\lambda} = \sqrt{{\rm Err1}^2 \left( \theta_{\rm I^1},\theta_{\rm I^2}, \the... ...^2 \left( \Omega_m, \Omega_\lambda, z_{\rm L}, z_{\rm S1}, z_{\rm S2} \right)}.$$ (18)

$A_{\Omega_m}$ and $A_{\Omega_\lambda}$ depend on $\Omega_m, \Omega_\lambda, z_{\rm L}, z_{\rm S1}, z_{\rm S2}$ while Err1² and Err2² are the quadratic sums of the errors, with a separation between the geometrical parameters and those depending on the cosmology. For each set of cosmological parameters we then compute all these coefficients numerically. In addition, we also need a calculation of the "degeneracy'' $\partial \Omega_m/ \partial\Omega_{\lambda}$ to obtain either ${\rm d}\Omega_m$ or ${\rm d}\Omega_{\lambda}$. This is the slope of the degeneracy curves of the E-terms ratio plotted in Fig. 1. Indeed considering 2 points $(\Omega _m, \Omega _\lambda )$ and $(\Omega_m+{\rm d}\Omega_m,\Omega_\lambda+ {\rm d}\Omega_\lambda)$ on such a curve (for a given set of $z_{\rm L}$, $z_{\rm S1}$, $z_{\rm S2}$), we have $e(\Omega_m,\Omega_\lambda)=e(\Omega_m+{\rm d}\Omega_m,\Omega_\lambda+ {\rm d}\Omega_\lambda)$ so that we get $\partial \Omega_m/ \partial \Omega_{\lambda}=-\partial_{\Omega_{\lambda}} e(z_{\rm S1},z_{\rm S2})/\partial_{\Omega_m}e(z_{\rm S1},z_{\rm S2})$. The final expected errors on ${\rm d}\Omega_m$ and ${\rm d}\Omega_{\lambda}$ are plotted in Fig. 5 for a continuous set of world models. The method is in general far more sensitive to the matter density than to the cosmological constant, for which the error bars are larger. This apparent contradiction with the general statement that lensing is more sensitive to the cosmological constant than to the matter density is due to the fact that we analysed the ratio of two E-terms and this ratio varies more rapidly with $\Omega _m$ when scanning the $(\Omega _m, \Omega _\lambda )$ plane (Fig. 1). For illustration, we quantitatively obtain the following errors for the corresponding cosmological models:

$$\begin{eqnarray*}\Lambda{\rm CDM\!\!:} & \qquad & \delta \Omega_m=0.11 \qquad \... ...qquad & \delta \Omega_m=0.17 \qquad \delta \Omega_\lambda=0.48. \end{eqnarray*}$$

Figure 5: Final errors d$\Omega _m$ (left) and d $\Omega _\lambda$ (right) for a given $(\Omega _m, \Omega _\lambda )$ in the lens configuration discussed in the text (Sect. 2.4), for two source redshifts $z_{\rm S1}=0.6$ and $z_{\rm S2}=5$.

This analysis shows that the expected results are quite encouraging, and the constraints we could get are similar to the ones currently obtained by other methods. Note however that these typical values require both HST imaging of cluster lenses and deep spectroscopic observations for the redshift determination of multiple arcs. They may depend on the choice of the lens parameters and on the potential model chosen to describe the lens, a problem that we will now investigate.