3 Constraints on the cosmological parameters from strong lensing

   
3.1 Existence of multiple systems of lensed images

More and more cluster-lenses are known to show several systems of multiple images (with spectroscopic or photometric redshifts). Lens modelling is then performed with a good accuracy and allows the prediction of extra families of images and their expected redshifts. When these images are later identified and if their redshift can be measured spectroscopically, an iterative process brings the lens model to a high level of accuracy, where most of the parameters which characterise the mass profile are strongly constrained. This full process has been applied successfully in a few clusters such as A2218 (Kneib et al. 1996), A370 (Kneib et al. 1993a; Smail et al. 1996; Bézecourt et al. 1999a) or AC114 (Natarajan et al. 1998; Campusano et al. 2001).

In order to apply more systematically the method proposed here, we may ask whether the few cited cluster-lenses are representative of some generic cluster or if they correspond to very peculiar configurations. To answer this question, we simulated a typical cluster at redshift z=0.2 with the following characteristics. A main clump is described with the potential of Eq. (6), the so-called HK mass density, with a=50 kpc and s=500 kpc. These values are typical of cluster-lenses at this redshift (Smith et al. 2001). The central velocity dispersion is varied from 800 to 1400 km s^-1 to allow a variation of the Einstein radius. In addition, 12 individual galaxies are added in the mass distribution, following the prescription used by Natarajan & Kneib (1996) and for a total contribution of 30% of the total mass. Their individual masses are scaled with respect to their luminosity L_i by the Faber-Jackson relation:

$$\sigma_i=\sigma_0^{\rm G} \left(\frac{L_i}{L_0}\right)^{\frac{1}{4}}$$ (19)

with $\sigma_0^{\rm G}=150$ km s^-1 (following Faber et al. 1997), and with a cut-off radius:

$$\theta_{{\rm S}_i}=\theta_{\rm S_0}^{\rm G} \left(\frac{L_i}{L_0}\right)^{\frac{1}{2}}$$ (20)

providing a constant ratio M/L (Natarajan & Kneib 1997).

To simulate the background sources, we used the Hubble Deep Field (HDF) image acquired by the HST (Williams et al. 1996). From Fernández-Soto et al. (1999), 946 galaxies were extracted from the deepest zone of the F814W image, up to a magnitude limit AB(8140)=28.0 and over an angular area of 3.92 arcmin². These authors also provide a catalog of photometric redshifts for all these objects. In addition, for about 10% of them, a spectroscopic redshift is available. We used this redshift distribution (spectroscopic redshift preferably used when available) as a sample of galaxy-sources to be lensed by the simulated clusters. In order to increase the statistical significance of this simulation, we generated a source catalogue with 10 times the number of galaxies extracted from the HDF image. We then distributed these sources at random angular positions over the central inner $40\times 40$ arcsec². We checked that this region includes the external radial caustic line, so that no multiple images are lost. The increase in the galaxy density is then corrected for in the final results.

 

 

Table 1: Number of systems of images obtained for simulated cluster potentials with different values of the central velocity dispersion $\sigma _0$ (the corresponding Einstein radius $R_{\rm E}$ is given for z=1). The redshift distribution of the sources is assumed from the HDF data. We take random positions for the sources over the central inner $40\times 40$ arcsec². n_j is the number of systems of j images. n_j^* is the number of systems of j images with AB(8140)<24.5, corresponding to "observable'' ones. Then, each system is counted both in n_i and in n_j^* with $j\le i$ (if only j images among i are detectable). Systems counted in n₀ show no "observable'' image. So $n_1+n_{\rm tot}=n_0+ n_1^*+n_{\rm tot}^*$, which is the number of galaxies in the selected field.

$\sigma _0$	(km s-1)		800	1000	1200	1400
$R_{\rm E}$	(arcsec)		5	14	28	40
	0	$n_0^{\ ~(1)}$	78	73	69	65
		n0*	0	0	0	0
	1	$n_1^{\ ~(2)}$	107	107	99	66
		n1* (3)	29	34	34	26
	2	n2	0	0.068	0.14	0.10
		n2*	0.11	0.41	2.3	13
	3	n3	0.12	0.60	8.0	41
		n3*	0.034	0.068	1.7	3.9
j	4	n4	0.034	0.011	0.034	0.057
		n4*	0.011	0.011	0.034	0.011
	5	n5	0.022	0.011	0.11	0.011
		n5*	0.011	0	0	0.011
	6	n6	0.011	0	0	0.011
		n6*	0	0	0	0.011
	7	n7	0.011	0	0.011	0.011
		n7*	0	0.011	0	0
	8	n8	0	0.011	0	0
		n8*	0	0	0	0
total	(j>2)	$n_{\rm tot}$	0.18	0.70	8.3	41
		$n_{\rm tot}^*$	0.17	0.50	4.0	17

(1) Including 0.6 galaxies at $z\le 0.2$.
(2) Including 5.2 galaxies at $z\le 0.2$.
(3) Including 4.6 galaxies at $z\le 0.2$.

Table 1 presents for each value of the central velocity dispersion the number of systems found with their image multiplicity. We also determined the number of systems in which each image could be observed (with a magnitude AB(8140)<24.5, corresponding to typical HST integration time of 10 ksec). Objects with AB(8140)>28.0 could be observed due to the lens effect if the magnification exceeds a factor of 25. This very rare configuration is neglected in our simulations for simplicity. For a cluster massive enough ( $\sigma_0\ge 1200$ km s^-1, corresponding to $M_{\rm tot}\ge 2 \times 10^{14}~M_{\odot}$ for our potential model), numerous systems of multiple images (mainly triple images) are formed and a significant fraction could be observable. Although these simulations are quite simple and cannot be used for realistic statistics of image formation, it gives us confidence that the use of multiple image families for the determination of the cosmological parameters is achievable and should be applied on a large number of rich clusters.

3.2 Method and algorithm for numerical simulations

In most cases, clusters of galaxies present a global ellipticity in their light distribution or in their gas distribution traced by X-ray isophotes. It is generally believed that this is related to an ellipticity in the mass distribution. This has indeed been recognized several times by the modeling of cluster lenses such as MS2137-23 (Mellier et al. 1993) or Abell 2218 (Kneib et al. 1996). So we include such an ellipticity in our modeling of cluster potentials. The basic distribution of matter we consider is again the HK one, with, in addition, a substitution of the radial distance r by R defined as:

$$R = \left(\frac{X\cos\theta+Y\sin\theta}{1+\epsilon}\right)^2 + \left(\frac{-X\sin\theta+Y\cos\theta}{1-\epsilon}\right)^2$$ (21)

where X=(x-x₀) and Y=(y-y₀). The potential $\phi$ is then characterized by 7 parameters, namely: $x_0, y_0, \epsilon, \theta$for the geometry of the lens and $\sigma_0, \theta_a, \theta_s$ for the shape of the mass profile.

Another popular density profile to be tested is the so-called Navarro, Frenk & White (NFW) density profile found in many simulations of dark matter and cluster formation (Navarro et al. 1997):

$$\rho(r)=\frac{\rho_{\rm c}}{(r/r_{\rm s}) (1+r/r_{\rm s})^2}$$ (22)

where $\rho_{\rm c}$ is a characteristic density and $r_{\rm s}$ a scale radius. No analytic developments have been proposed so far for the corresponding ellipsoidal profile. In a companion paper (Golse & Kneib 2002) we propose a new "pseudo-elliptical'' NFW profile and compute its lensing properties. The corresponding potential is characterized by 6 parameters: $x_0, y_0, \epsilon, \theta$ for the geometry of the lens and $v_{\rm c}, \theta_{\rm s}$ for the shape of the mass profile. The characteristic velocity $v_{\rm c}$ is defined by

$$v_{\rm c}^2=\frac{8}{3}{\rm G}r_{\rm s}^2\rho_{\rm c}$$ (23)

as explained in Golse & Kneib (2002).

To create a simulated lens configuration we need to fix some arbitrary values of the cosmological parameters $(\Omega_m^0$, $\Omega_\lambda^0)$ as well as the cluster lens redshift $z_{\rm L}$. The numerical code LENSTOOL developed by one of us (Kneib 1993) can then trace back the source of a given image or determine the images of an elliptical source galaxy at a redshift $z_{\rm S}$. The initial data are several sets of multiple images at different redshifts. In all cases we do not take into account the central de-magnified images, which are generally not detected. With these observables, we can recover some parameters of the potential while we scan a grid in the $(\Omega_m$, $\Omega_\lambda)$ plane. The likelihood of the result is obtained via a $\chi ^2$-minimization (with a parabolic or a Monte Carlo method), where $\chi ^2$ is computed in the source plane as:

$$\chi^2=\displaystyle {\sum_{i=1}^n \sum_{j=1}^{n^i} \frac {[\... ...\theta}_{{\bf SG}\vec{^i}})]^2} {\sigma_{{\rm I}_j^i}^2}}\cdot$$ (24)

The superscript i refers to a given family of multiple images and the subscript j to the images inside a family of n_i images. There is a total of $\sum_{i=1}^n n_i=N$ images, and $\sum_{i=1}^n 2(n_i-1)=N_{\rm C}$ constraints on the models assuming that only the position of the images are fitted. $\vec{\theta}_{{\bf S}\vec{_j^i}}$ is the source position associated with the image $\vec{\theta}_{{\bf I}\vec{_j^i}}$ in the lens inversion. $\vec{\theta}_{{\bf SG}\vec{^i}}$ is the barycenter of all the $\vec{\theta}_{{\bf S}\vec{_j^i}}$ belonging to the same family i. $\mathcal A_j^i$ is the magnification matrix for a particular image and $\sigma_{{\rm I}_j^i}$ is the error on the position of the center of image $\vec{\theta}_{{\bf I}\vec{_j^i}}$. Quantitatively we will take $\sigma_{\rm I} = 0.1\arcsec$ for all images, assuming that their positions are measured on HST images.

$\chi ^2$ computed from Eq. (24) in the source plane is mathematically equivalent to $\chi ^2$ computed in the image plane, written as:

$$\chi^2=\displaystyle{\sum_{i=1}^n \sum_{j=1}^{n^i} \frac{(\ve... ...vec{\theta}_{{\bf IG}\vec{_j^i}})^2}{\sigma_{{\rm I}_j^i}^2}},$$ (25)

where $\vec{\theta}_{{\bf IG}\vec{_j^i}}$ is the image of $\vec{\theta}_{{\bf SG}\vec{^i}}$close to $\vec{\theta}_{{\bf I}\vec{_j^i}}$. Indeed $\vec{\theta}_{{\bf S}\vec{_j^i}}-\vec{\theta}_{{\bf SG}\vec{^i}}\equiv\vec{\delta} {\bf S}\vec{^i_j}$ and $\vec{\theta}_{{\bf I}\vec{_j^i}}-\vec{\theta}_{{\bf IG}\vec{_j^i}}\equiv\vec{\delta} {\bf I}\vec{^i_j}$are assumed to be small quantities compared to the variation scale of the elements of the magnification matrix $\mathcal A_j^i$. Therefore the local transformation from the image plane to the source plane is written as $\vec{\delta} {\bf I}\vec{^i_j}=\mathcal A_j^i~\vec{\delta} {\bf S}\vec{^i_j}$. The main motivation for working in the source plane is numerical simplicity because the mapping from the source to the image plane is not a one-to-one mapping and we may not recover all the images when solving the lens equation.

If $M_{\rm p}$ is the number of fitted parameters for the potential, there is a total of $M=M_{\rm p}+2$ adjustable parameters (including $\Omega _m$and $\Omega _\lambda$) and $N_{\rm C}$ independant data points. We compute a $\chi ^2$-distribution for $\nu=N_{\rm C}-M$ degrees of freedom. In practice, in our simulation we try to recover only the most important parameters, like $\sigma _0$ (or $\sigma_{\rm c}$), $\theta _a$ or $\theta _s$, to limit the number of degrees of freedom. This would be the case in a real application.

3.3 Numerical simulations in different configurations

To recover the most important parameters of the potential, we generated 3 families of multiple images (2 tangential ones and a radial one for a total number of constraints $\nu=16$, see Fig. 6 and Table 2) with the pseudo-elliptical NFW profile developed in Golse & Kneib (2002). We also chose regularly distributed source redshifts (Table 2). The 4 geometrical parameters of the cluster lens were left fixed during the minimization x₀=y₀=0, $\theta=0 \degr$ and $\epsilon =0.1$), while the 2 parameters of the potential ($v_{\rm c}$ and $\theta _s$) were allowed to vary. The initial values for these parameters, used to create the set of images, correspond to reasonable values found in cluster lenses: $\theta_{s} = 31.3\arcsec$ (i.e. 150 kpc) and $v_{\rm c}=2000$ km s^-1. This last value corresponds to a "classical'' central velocity dispersion $\sigma_0=1230$ km s^-1for a HK model (see Sect. 3.3.2). $(\Omega_m^0,\Omega_\lambda^0)$ were fixed to the $\Lambda$CDM values (0.3,0.7).

 

 

Table 2: Details on the 3 sets of multiple images used in the simulations in Sects. 3.3.1 and 3.3.2. n_i represents the number of images used for each family. It does not include the central de-magnified image created for tangential images. $N_{\rm C}$ is the number of constraints in the lens modeling for each family. $N_{\rm C}=2\times n_i-2$ for x and y position. The unknown position of the source $(x_{\rm S},y_{\rm S})$ is then removed, reducing $N_{\rm C}$ by 2 units.

Family	Type	ni	$z_{\rm S}$	$N_{\rm C}$
i=1	Tangential	4	0.6	6
i=2	Radial	3	1.	4
i=3	Tangential	4	4.	6

   
3.3.1 Simple cluster potential

In this case, the number of degrees of freedom is $\nu = 16 - 4 = 12$as 2 cluster parameters are fitted. The confidence levels of the minimization are plotted in Fig. 7. The trajectory of the minimum includes the initial point (0.3,0.7) in the $(\Omega _m, \Omega _\lambda )$ plane with $\chi ^2_{\rm min}=0$. The degeneracy in the cosmological parameters is found as expected in Fig. 1. Tighter constraints can be deduced on $\Omega _m$than on $\Omega _\lambda$. We also recover the cluster parameters quite satisfactorily with: $v_{\rm c}=2000^{+90}_{-90}$ km s^-1 (Fig. 8) and $\theta_a=31.3 \arcsec ^{+1.2}_{-1.3}$. Note that these errors represent only the variations of the fitted parameters when we scan the $(\Omega _m, \Omega _\lambda )$plane during the optimisation process.

This preliminary step corresponds to the "ideal'' case where we recover the same type of potential we used to generate the images. Moreover, the morphology of the cluster is regular without substructure, and we included one radial system among the families of multiple images. These images are known to probe the cluster core efficiently. Finally, the redshift distribution of the sources is wide and the selected redshifts are well separated, for an optimal sampling of the E-term. One could ask whether any such lens configuration has already been detected among the known cluster lenses. It seems that the case of MS2137.3-2353 ( $z_{\rm L} = 0.31$) is quite close to this type of configuration (Mellier et al. 1993) with at least 3 families of multiple images, including a radial one. Uunfortunately, no spectroscopic redshift has been determined for any of the images so far.

Figure 6: Multiple images generated by a pseudo-elliptical NFW cluster at $z_{\rm L} = 0.3$ with the lens parameters: $v_{\rm c}=2000$ km s-1, $\theta _{s}=31.3$ (rs=150 kpc) and $\epsilon =0.1$. Close to their respective critical lines, 3 families of multiple images are identified: a tangential one (# 1, $z_{\rm S1}=0.6$), a radial one (# 2, $z_{\rm S2}=1$) and another tangential one (# 3, $z_{\rm S3}=4$). Units are given in arcseconds.

Figure 7: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 6. The 2 main cluster parameters $v_{\rm c}$ and $\theta _s$ were recovered with $\chi ^2_{\rm min}=0$. The cross (+) represents the original values $(\Omega _m^0,\Omega _\lambda ^0)=(0.3,0.7)$. Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

Figure 8: Solid lines: distribution of the best-fit velocity dispersion $v_{\rm c}$ from the optimisation of the lens configuration shown in Fig. 6, for each cosmological model. The cross (+) represents the original value for $(\Omega _m^0,\Omega _\lambda ^0)=(0.3,0.7)$: $v_{\rm c}=2000$ km s-1. Dashed lines correspond to the 1$\sigma$ confidence level contours from Fig. 7.

   
3.3.2 Changing the shape of the mass profile

To test the sensitivity of the method to the chosen fiducial mass profile, we tried to recover the lens with another potential, namely an elliptical HK profile, keeping the same simulated lens. $\sigma _0$, $\theta _a$ and $\theta _s$ were left free for the optimization. We first optimized the geometrical parameters for an arbitrary choice of cosmological parameters. The best values found are: $x_0=0.059\arcsec$, $y_0=0.063\arcsec$, $\theta=-0.063\degr$, and $\epsilon=0.280$. These values are close to the generating ones ( $x_0=y_0=0\arcsec$, $\theta=0 \degr$), except for the ellipticity which does not correspond to the same physical meaning in the pseudo-elliptical NFW profile (Golse & Kneib 2002). They were then kept fixed for the rest of the optimization. For the lens parameters, we found $\sigma_0=1230^{+50}_{-50}$ km s^-1, $\theta_a=4.6\arcsec^{+0.2}_{-0.1}$ and $\theta_s=190\arcsec^{+20}_{-10}$. The confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane are displayed in Fig. 9.

Although the reconstruction with a potential model different from the initial "real'' one does not perfectly fit the data, the results are quite satisfactory. The confidence levels are even tighter than in the previous case, but the HK-type potential is characterised by one additional parameter or equivalently one degree of freedom less ($\nu =11$), compared to the pseudo-elliptical NFW profile. Nevertheless we find a minimum reduced $\chi^2=5$ rather far from 0.

Figure 9: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 6. In this plot, the potential was fitted with a model different from the initial one (an elliptical HK profile instead of a pseudo-elliptical NFW). Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

Several other mass profiles were tested as we wanted to discriminate between the different families of density profiles and test their sensitivity in the estimate of the cosmological parameters after the lens reconstruction. We used 5 types of profiles, namely:
i) the pseudo-elliptical NFW profile (Golse & Kneib 2002),
ii) the singular isothermal ellipsoid (SIE) with $\rho(R)=\rho_0 / R^2$, R being the elliptical coordinate (Eq. (21)),
iii) the isothermal ellipsoid with core radius (CIE), obtained by replacing R by $\sqrt{R^2+a^2}$ in the previous expression (see Kovner 1989),
iv) the HK profile (Eq. (6)),
v) and the King profile characterised by

$$\rho(R)=\rho_0\ \frac{1+ \frac{1-2\alpha}{3} ~ R^2 / a^2} {\left(1+R^2/a^2 \right)^{2+\alpha}}\cdot$$ (26)

The first 2 profiles are cusped, while the latter have a core radius and then an additional parameter. For each mass model, we generated the system of images defined in Table 2 (except for the SIE for which the radial system consists only of 2 images). We then fitted these images with the other 4 models. All the lens parameters were left free in this optimisation to get the minimum reduced $\chi ^2$. We did not change the cosmological parameters in these recoveries. The results are presented in Table 3. We note that the "core-radius'' profiles (especially the HK and King ones) can easlily recover the systems generated by any other models. Indeed in the fit of cusped lens images by shallower profiles, the core radius can be reduced to very small values to mimic a large density slope near the center. This is not the case for the cusped models which cannot mimic images given by a finite core radius lens model.

 

 

Table 3: Results of the lens reconstruction using a mass model different from the one used to generate the systems of images. The minimum reduced $\chi ^2$ is given for each simulation.

Input profile	HK	King	CIE	NFW	SIE
Fitted profile
HK ($\nu =11$)	0.	23.	72.	460.	4500.
King ($\nu =11$)	33.	0.	33.	150.	1500.
CIE ($\nu =11$)	23.	0.26	0.	87.	2800.
NFW ($\nu=12$)	6.2	21.	18.	0.	680.
SIE ($\nu=12$)	0.14	0.011	0.28	76.	0.

   
3.3.3 Influence of the number of multiple systems

Figure 10: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration described in Table 5. Left: 2 systems and $\nu = 10-4=6$ degrees of freedom. Middle: 3 systems and $\nu = 16-5=11$ degrees of freedom. Right: 4 systems and $\nu = 20-5=15$ degrees of freedom. The cross (+) represents the original values $(\Omega _m^0,\Omega _\lambda ^0)=(0.3,0.7)$. Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

In the preceding sections we considered 3 systems of multiple images. As the method proposed is based on the difference of angular distance ratios for different redshift planes, we now investigate the influence of the number of image families. The potential model is again an HK-type profile at $z_{\rm L} = 0.3$ with $\sigma _0=1400$ km s^-1, $\theta_a=13.5$ (i.e. 65 kpc), $\theta_s=146$ (i.e. 700 kpc) and $\epsilon=0.2$. With   systems of images, we consider only 2 free parameters for the cluster, because there are not enough observables to yield results for more parameters, while in the other cases, 3 parameters are fitted. In all cases, these parameters are strongly constrained by the fit. Table 4 reports the errors on the fitted parameters in the optimisation process, for the different sets of multiple images detailed in Table 5. The differences in the fitted parameters between the different cases are small, as they are already well constrained with a single multiple images system.

 

 

Table 4: Recovering of the free parameters of the lens potential for the Table 5 different systems of images. The errors represent the variation of the fitted parameters at 1-$\sigma$ level when scanning the $(\Omega _m, \Omega _\lambda )$ plane in the optimisation process.

Nb of systems	$\sigma _0$ (km s-1)	$\theta _a$ ()	$\theta _s$ ()
2	1400+60-60	13.5+0.25-0.15	-
3	1400+70-60	13.5+0.3-0.2	146+2-2
4	1400+60-60	13.5+0.3-0.2	146+14-6

The expected constraints on $(\Omega _m, \Omega _\lambda )$ tighten when the number of families of multiple images increases (Fig. 10), especially when their redshift distribution is wide. 2 families would only provide marginal information on the cosmological parameters whereas 4 spectroscopically measured systems would give very tight error bars, provided they are well distributed in redshift.

 

 

Table 5: Sets of multiple images used in the simulations to test the influence of their number. n_i represents the number of images used for each family. It does not include the central de-magnified image created for tangential images. $N_{\rm C}$ is the number of constraints in the lens modeling for each family. $N_{\rm C}=2\times n_i-2$.

Nb of systems	Family	Type	ni	$z_{\rm S}$	$N_{\rm C}$
2	i=1	Tangential	4	0.6	6
	i=2	Radial	3	1.	4
	i=1	Tangential	4	0.6	6
3	i=2	Radial	3	1.	4
	i=3	Tangential	4	2.	6
	i=1	Tangential	4	0.6	6
4	i=2	Radial	3	1.	4
	i=3	Tangential	4	2.	6
	i=4	Radial	3	4.	4

3.3.4 Influence of additional galaxy masses

In the previous parts, we considered only a main cluster potential with a regular morphology. We now test the contribution of individual galaxies, following the prescription used by Natarajan & Kneib (1996) as in Sect. 3.1. We generated 3 systems of multiple images formed by the sum of a main cluster with the mass density (HK-type) characterised by $\sigma _0=1400$ km s^-1, $\theta_a=13\arcsec$ and $\theta_s=150\arcsec$ and 12 individual galaxies which represent 30% of the total cluster mass (Fig. 11).

Figure 11: Multiple images generated by a cluster at $z_{\rm L} = 0.3$ with an elliptical HK lens profile and the parameters: $\sigma _0=1400$ km s-1, $\theta _a=13.54$(a=65 kpc) and $\theta _s=145.8$ (s=700 kpc). 12 individual galaxies are added in the potential. 3 families of multiple images are identified (see Table 2 for details). We represent the radial (inside) and tangential (outside) critical lines corresponding to the multiple images redshifts. Their characteristic radii are increasing with redshift. Units are given in arcseconds.

The images were reconstructed using a main cluster potential with the same kind of shape as the initial one and the contribution of the galaxies scaled with $\sigma_0^{\rm G}$. In addition, we fixed $\sigma_0^{\rm G}$proportional to $\sigma _0$ to avoid an increase of the number of free parameters. Consequently, any variation in $\sigma _0$ means a rescaling of the total mass of the cluster. So at first order we find that $\sigma_0^2E$ is constant when we scan the $(\Omega _m, \Omega _\lambda )$ plane. Keeping the geometrical parameters fixed ( $x_0=y_0=0\arcsec$, $\theta=0 \degr$, and $\epsilon=0.2$), we obtain the confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane plotted in Fig. 12 and the following constraints on the potential parameters: $\sigma_0=1400^{+60}_{-65}$ km s^-1, $\theta_a=13\arcsec^{+0.3}_{-0.3}$ and $\theta_s=151\arcsec^{+1}_{-1}$.

Figure 12: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 11. For the individual galaxies, we assumed that their mass is scaled with the total mass with $\sigma _0^{\rm G}\propto \sigma _0$. The 3 main cluster parameters $\sigma _0$, $\theta _a$ and $\theta _s$ were recovered with $\chi ^2_{\rm min}=0$ and $\nu =11$ degrees of freedom. Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

To test the influence of the individual galaxies, we tried a reconstruction without their contribution. For the geometrical parameters first optimised we obtain $x_0=0.227\arcsec$, $y_0=0.060\arcsec$, $\theta=-0.748\degr$ and $\epsilon=0.193$, still close to the generating values. The confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane are plotted in Fig. 13. The contours are slightly shifted and widened compared to the "good'' ones (Fig. 12) but not significantly different. The minimum reduced $\chi ^2$ is 17. So we are able to correctly retrieve the cluster potential, even without the individual galaxies ( $\sigma_0=1380^{+70}_{-60}$ km s^-1, $\theta_a=11.9\arcsec^{+0.3}_{-0.2}$ and $\theta_s=180\arcsec^{+3}_{-3}$). Adding their contribution is nevertheless useful to determine precisely the minimum region and to tighten the confidence levels. It becomes quite critical in more complex cases or when a single galaxy strongly perturbs the location of an image.

Figure 13: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 11. Here, we did not introduce the individual galaxies when recovering the global potential. The 3 main cluster parameters $\sigma _0$, $\theta _a$ and $\theta _s$ were recovered but with a non-zero reduced $\chi ^2_{\rm min}$ ( $\chi ^2_{\rm min}=17$). Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

3.3.5 Bi-modal cluster mass distribution

Up to this point, we have considered simple clusters, dominated by a single massive component. In reality, most clusters are not fully virialised and present sub-structure as the result of accretion processes or merging phases. With these more complex mass distributions, the lensing configurations are more widely distributed. Therefore we examine how the cosmological parameters can be constrained with this type of realistic mass distribution. We thus generated a bi-modal cluster consisting of two clumps of equal mass and 3 families of multiple images probing each part of the lens (Fig. 14). The total potential is axisymmetric and each clump is characterised by an HK-type elliptical mass profile. As the number of multiple images is rather small, we limited the number of parameters to recover and chose $\sigma _0$ and $\theta _a$ for each clump as adjustable variables. Therefore we fixed $x_{01}=-34\arcsec$, $x_{02}=34\arcsec$, $y_{01}=y_{02}=0\arcsec$, $\theta_1=-45\degr$, $\theta_2=+45\degr$, $\epsilon_1=\epsilon_2=0.2$ and $\theta_{s1}=\theta_{s2}=167\arcsec$.

Figure 14: Multiple images generated by a bimodal cluster at $z_{\rm L} = 0.3$ with the lens parameters: $\sigma _{01}=\sigma _{02}=1100$ km s-1, $\theta_{a1}=\theta_a{2}=12\arcsec$ (58 kpc) and $\theta_{s1}=\theta_{s2}=167\arcsec$ (800 kpc). 3 families of multiple images are identified at $z_{\rm S1} = 0.7$, $z_{\rm S2}=1$ and $z_{\rm S3}=2$. Units are given in arcseconds.

Figure 15: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 14. The main cluster parameters $\sigma _{01}$, $\sigma _{02}$, $\theta _{a1}$ and $\theta _{a2}$ were recovered with $\chi ^2_{\rm min}=0$ and a number of degrees of freedom $\nu =6$. The cross (+) represents the original values $(\Omega _m^0,\Omega _\lambda ^0)=(0.3,0.7)$. Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

Fixing again the initial values of $(\Omega_m^0,\Omega_\lambda^0)$ to the $\Lambda$CDM model (0.3,0.7), we obtain the confidence levels plotted in Fig. 15. The contours are widened compared to the case of a single potential (in this case, the number of degrees of freedom is reduced from 11 to 6, but they still give reasonable constraints). Moreover we note that there is little variation in the fitted parameters: $\sigma_{01}=1100^{+55}_{-50}$ km s^-1, $\sigma_{02}=1100^{+55}_{-45}$ km s^-1, $\theta_{a1}=12.1\arcsec^{+0.1}_{-0.1}$, and $\theta_{a2}=12.1\arcsec^{+0.3}_{-0.2}$. This configuration is close to the case of the cluster Abell 370, modeled with a bi-modal mass distribution (Kneib et al. 1993b; Bézecourt et al. 1999b) needed to reproduce the peculiar shape of the central multiple-image system. Unfortunately, up to now only two redshifts are known for the multiple images identified in A370!

Figure 16: Multiple images generated by a cluster at $z_{\rm L} = 0.3$ consisting of a main clump ( $\sigma _0=1400$ km s-1, $\theta_a=13.54\arcsec$-65 kpc - and $\theta_s=145.8\arcsec$-700 kpc) and a smaller one ( $\sigma _0=500$ km s-1, $\theta_a=5.2\arcsec$-25 kpc - and $\theta_s=45.9\arcsec$-220 kpc) located $102\arcsec$ from the main one. Close to their respective critical lines, 3 families of multiple images are identified: a tangential one (# 1, $z_{\rm S1}=0.6$), a radial one (# 2, $z_{\rm S2}=1$) and another tangential one (# 3, $z_{\rm S3}=2$). Units are given in arcseconds.

Figure 17: $\chi ^2$ confidence levels in the $(\Omega _m, \Omega _\lambda )$ plane obtained from the optimisation of the lens configuration shown in Fig. 16. The 3 main cluster parameters $\sigma _0$, $\theta _a$ and $\theta _s$ were recovered with a reduced $\chi ^2_{\rm min}=9$ and a number of degrees of freedom $\nu =11$. The cross (+) which represents the original values $(\Omega _m^0,\Omega _\lambda ^0)=(0.3,0.7)$ is now outside the 3-$\sigma$ confidence levels. Dark to light colors delimit the confidence levels (from 1-$\sigma$ to 4-$\sigma$).

Last, we generated another system of 3 families of multiple images produced by a cluster consisting of a main clump ( $\sigma _0=1400$ km s^-1) and a smaller one ( $\sigma _0=500$ km s^-1) representing 22% of the total mass (Fig. 16). We chose to miss the small clump in the mass recovery as this may happen when dealing with some "dark clumps''. Fitting the configuration with a single main cluster, we found in a first round the geometrical parameters, which then remain constant in the $\chi ^2$-optimisation: $x_0=0.348\arcsec$, $y_0=0.189\arcsec$, $\theta=1.880\degr$ and $\epsilon=0.259$. We note in particular that the ellipticity is larger than the one used to generate the main clump ( $\epsilon=0.2$). This seems to be the response of the fitting process in order to mimic the missing second clump.

The parameters left free are again $\sigma _0$, $\theta _a$ and $\theta _s$. The confidence contours are shown in Fig. 17. We found the following values of the parameters: $\sigma_0=1400^{+40}_{-70}$ km s^-1, $\theta_a=12.8\arcsec^{+0.2}_{-0.2}$ and $\theta_s=169\arcsec^{+2}_{-2}$. However in this case, we do not recover correctly the set of cosmological parameters used to generate the system: $(\Omega_m,\Omega_\lambda)=(0.3,0.7)$ is excluded at the 3-$\sigma$ level. Moreover the shape of the contours is not the one expected from the lensing degeneracy. This could be considered to be a signature of an incorrect fiducial mass distribution due to a missing clump in the mass reconstruction. This example demonstrates that the initial guess and the modeling of the different components of a cluster are very sensitive elements. They need to be carefully determined if one wants to test further constraints on the cosmological parameters