5.2 Distance of Faint Arclets

Deep multi-color photometry in subarcsecond seeing conditions gave preliminary indications that most of the faintest arclets are at a redshift between 0.7 and 1.5 (Fort 1992). But the distance of the arclets could be more securely investigated with the study of the geometry and distribution of arclets around cluster centers, since these also depend on the redshift distribution of the sources. The relationship between image position, image orientation and ellipticity and the source position and geometry is dependent on the lens modeling. It is relatively complex so that very little work has been done at present to infer the redshift of the background sources in this way.

Kochanek (1990) was in fact the first to discuss the possibility of obtaining the average redshift of the arclets by measuring their ellipticity and orientation relative to the shear axis. He argued that in practice, it is difficult to recover the mean redshift distribution from the statistics of arclets unless we add a priori information about the shape of the distribution (for instance: unimodal redshift distribution, concave-down distribution, magnitude-redshift relation, etc...). Kochanek also drew attention to selection effects which could modify the true distribution of the lensed sources in the image plane. In addition, Miralda-Escudé (1991a) emphasized that the intrinsic ellipticity distribution of the sources makes it harder to recover their redshifts. Fortunately, the dependence on cosmological parameters is small, at least with a null cosmological constant.

Kneib et al. (1994) and Longaretti (1994) rediscussed the relation between the source redshift distribution and the observed lensed images. They proposed a slightly different method from Kochanek's technique in the sense that they define for each individual arclet a probability distribution with an average redshift and a dispersion parameter. They applied this technique to about 60 arclets detected in A370 (Fort 1992, Kneib et al. 1994) with sub-arcsecond seeing CCD images (< 0.7") and for which a good lens modeling is possible (Kneib et al. 1993). They found that most of the arclets are at redshift below 1, in good agreement with the predictions based on the color-redshift relations for various galaxy evolution models (Fort 1992, Kneib et al. 1994). Their sample also included three sets of gravitational pairs which are strongly constrained by the modeling within the redshift space (Kneib et al 1993). Thanks to the gravitational telescope, the arclets population is intrinsically much fainter than the objects in the faintest spectroscopic surveys. However, the position of the arclets in a B_J - z diagram (Fig. 21) extends surprisingly well the results of the deep spectroscopic surveys. These data seem to favor non-evolution models up to z = 1 (see Koo & Kron 1992 and references therein). Note that the Kneib et al. analysis is only based on one well-known cluster and that it should be pursued with other clusters with well known and more regular potentials.

Bonnet et al. (1994) recently extended this work to the weak shear field they found around Cl0024+1654. They calculated the average redshift of the faintest background sources detectable on their images (V = 27.5). They show that the shear pattern out to 10 arcminutes is compatible with an isothermal model similar to the one found for the inner part of the cluster from the modeling of the triple arc by Kassiola et al. (1992). With the observed velocity dispersion, they conclude that sources should be at a redshift of about 0.8. They claim that the shear pattern is less compatible with a de Vaucouleurs model which would require the average source redshift to increase to z 1.2, and a velocity dispersion marginally compatible with the velocity dispersion of cluster galaxies. But this result needs to be confirmed.

From all these studies it seems that 80% of the faint distant galaxies with B < 27 are at a redshift below 1. This result could be checked by using rich clusters of galaxies of similar velocity dispersion regularly distributed from z = 0 up to z = 1, or even higher. If faint galaxies are at very large distances, distorted sources should be observed even in the most distant clusters. Conversely, if all faint galaxies are instead nearby dwarf systems, the number of distorted galaxies should decrease with cluster redshift with a complete cut-off when the cluster is at the redshift of the sources. By using this technique, one could imagine finally determining the redshift distribution of faint sources. Therefore Smail, Ellis & Fitchett (1994) analyzed the redshift distributions of faint sources from the arclets distribution around three distant clusters with redshifts of 0.26, 0.55 and 0.89. They reached a similar limiting magnitude as Kneib et al. (1994), but with a seeing of 1". Using maximum likelihood techniques, they found the best cluster parameters and analyzed three redshift distributions, deducing which one was compatible with the shear observed. The deficiency of arclets in the most distant cluster, as well as the small number observed in the medium distant cluster allowed Smail et al. to conclude that their data are more compatible with a non-evolving population, with most faint galaxies at redshifts lower than 1. They also excluded models dominated by nearby dwarf galaxies, a possibility which was not completely excluded by the deep spectroscopic surveys (Tresse et al. 1993).

Figure 21. The redshift versus magnitude diagram obtained from the deepest redshift surveys in B and I, the spectroscopic redshifts of giant arcs, and the arclet redshift distribution.

In fact, the methods used by Kneib et al. (1994) and Bonnet et al. (1994), and Smail et al. (1994) confirm a result which could be guessed from the data of the first arcs survey: many arclets were observed in rich clusters with z in the range 0.15-0.3 while a small number were detected in clusters with a redshift larger than 0.5 (Fort et al. 1992). If it is established that most of the faint arclets are at a redshift below one, their actual redshift distribution is still uncertain because both methods suffer from their limitations. The first method uses a cluster with a well known velocity dispersion (or arc redshift) and a good modeling of the potential from arc systems (A370 and Cl0024+1654), but the cluster is single and located at a unique redshift. The second method of Smail et al. uses three clusters that sample fairly well the redshift space up to z = 1 but with a poor knowledge of the lens models. Since clusters are rather young gravitational systems, it is quite possible that their critical density strongly evolves with redshift so that the decrease of the number of distorted objects with cluster redshift reflects the evolution of clusters as well as the effect of the distance of the sources. Henry et al. (1992) claimed that the X-ray luminous clusters strongly evolve with redshift. Therefore, it seems absolutely necessary that Smail et al.'s approach be used with clusters with several multiple gravitational images which allows a similar modeling as that described for A370, MS2137-23, Cl0024+1654 (cf part 4.1).

From an observational and technical point of view, such work is really a challenge: first we need to detect at least one multiple arc system and to properly define the geometry of arclets with deep subarsecond CCD images. Second, we need the redshift of one arc and a comparison of the corresponding angular scale parameter with the observed velocity dispersion of the cluster. Third, we need to achieve very good modeling of the potential for each cluster. Fourth, the observation of a reference blank field for each cluster under the same seeing conditions is necessary to correct the data for additional uncertainties that come from the intrinsic dispersion of ellipticity and size of the sources.

Furthermore, it seems absolutely necessary to verify the arclets method by testing it on a range of galaxy redshifts that overlaps with the deepest redshift surveys. This idea was considered by Smail et al. (1994) who compared the redshifts of their brightest arclets with the redshift from the faintest redshift surveys of Lilly (1993) and Tresse et al. (1993). Kneib et al. (1994) also argued that their brightest arclets have redshift compatible with the faintest redshifts from Colles et al. (1993) and also with the redshifts of most giant arcs. This ``boot-strap'' method should be done with nearby cluster lenses like A2218 (z = 0.13). In any case, whatever the price to be paid for the observational and technical work needed to derive the exact redshift distribution of arclets, it is a crucial issue because we do not see any other way to study the distribution and evolution of very faint distant galaxies, even with the next generation of very large telescopes.