The present paper should be considered as a further step in building up the ground for nonlinear cluster reconstruction techniques. In contrast to Papers I & II we have taken into account a redshift distribution of the sources. Our main results can be summarized as follows:
(1) We have related the local expectation values of the moments of the image ellipticities to the local lens parameters and in Eq. (3.5). These moments depend in a complicated way on the redshift distribution of the sources as shown in Eqs. (3.5) and (3.6). In Sect. 4.3 we have shown that in the case of non-critical clusters the local expectation values depend only on the local lens parameters and and on few moments of the effective distance w of the sources from the lens, weighted with the redshift distribution of the sources [see Eqs. (4.12), (4.15), and (4.16)].
(2) In Sect. 4.2, we have generalized the inversion method developed in Paper II such that the redshift distribution of the sources is accounted for. This is important if the cluster is at a high redshift (), because then the effective lensing strength for sources with is significantly different from that for sources with . For the reconstruction the redshift distribution of the sources has to be assumed and the inversion equation is solved iteratively. In Sect. 6 we have applied the reconstruction technique to synthetic data and shown that the reconstructed mass density is in good agreement with the original one. Compared to the mass reconstruction in the case of a single source redshift, the noise is slightly increased if the same number density of galaxy images is used.
(3) For non-critical clusters we have simplified the inversion method in Sect. 4.3 such that only one integration is necessary. Then the inversion equation is very similar to that developed previously for a single redshift of the sources (see Sect. 4.1). For the reconstruction of a non-critical cluster, one only has to assume the first two moments and of the effective distance w weighted with the redshift distribution. The weaker the cluster, the less information on the redshift distribution has to be available for the reconstruction of the mass distribution. For a weak cluster it is sufficient to know the first moment .
(4) We have shown that the mean surface mass density is still a free variable in the inversion equation (4.9), i.e., there still remains a global invariance transformation for the surface mass density field which leaves the field of unchanged. For non-critical clusters, this invariance transformation is given explicitly in Eq. (4.20) and it is similar to that derived previously for a single redshift of the sources [see Eq. (4.5)].
(5) In Sect. 5 we have discussed some ideas to break the mass invariance: maximizing the likelihood for the observed image ellipticities, using the second moments of image ellipticities or the magnification effect on the position of the galaxy images. For all these cases, the mass invariance is broken only in the presence of a redshift distribution of the sources. We find that the invariance is broken too weakly to make use of it in practice.
(6) Using the magnification effect on the number density of the galaxy images (see Broadhurst, Taylor & Peacock 1995) is the most promising way to break the mass degeneracy. In contrast to Broadhurst et al. (1995) and Broadhurst (1995), we do not use the observed local number density to derive the local surface mass density (because this local estimate has a considerably larger noise) but we use the total number of observed images to determine the mean mass density: we assume a value for , perform the mass reconstruction, then calculate the local magnification and from that the number of expected galaxy images in the field. Comparing this with the number of `observed' galaxy images via a -analysis gives a confidence interval for . This method works only if the distribution of the unlensed sources with flux larger than S has a slope , because for the number density of lensed objects is not changed compared to the unlensed one. The confidence interval obtained for is the narrower, the larger is, because the magnification bias (anti-bias) increases with increasing (decreasing ).
(7) In Sect. 6 we have used the same numerically modelled cluster as in Paper II for testing the reconstruction technique. We reconstruct the mass density for the case that (i) the assumed redshift distribution is the true one, (ii) the mean redshift is overestimated and (iii) underestimated. The mean mass density is adjusted such that the minimum of the reconstructed mass density is zero. We find that the substructure of the cluster is nicely recovered regardless of the assumed redshift distribution. The total mass detected is within of the true mass for (i), underestimated for (ii) and overestimated for (iii). Using the method as describe above in (6), we find that reliable limits on the mean mass density can be derived with a significance increasing with increasing number density of galaxy images and increasing . We note that for a successful application of this method on real data it is essential (a) to know the number density of the unlensed sources which would have been detected in the absence of lensing but under the same observing conditions, (b) to choose a colour which gives an appropriate slope of the faint galaxies in flux ( large !) and (c) to know the mean redshift of the sources considered.
Finally, we should mention that although the cluster reconstruction method developed in this and our earlier papers is considerably more complicated than straight application of the original Kaiser & Squires (1993) method, these modifications are essential if applied to WFPC2 observations of a cluster center. For the one case we considered, i.e., the cluster Cl 0939+4713 (Seitz et al. 1996), the small field-of-view of the WFPC2, the lensing strength of the cluster center, and the fairly high redshift of the cluster make it absolutely necessary to apply an unbiased finite-field inversion technique, to use a non-linear reconstruction method, and to account for a redshift distribution of the galaxies.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998