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Astron. Astrophys. 361, 415-428 (2000)

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6. Conclusions

The gravitational magnification by cluster lenses represents an effective tool to probe the distant universe and the mass distribution in the lenses. The first aim of this paper was to study the effects of lens mass profile model parameters on the typical features of depletion curves. We have also attempted to characterize some features associated with the background redshift distribution of the galaxies. Models were constructed in three bands covering a large spectral range and by using five different lens models.

Our simulations agree well with very deep and high quality images of the cluster MS1008-1224 obtained with the VLT and FORS. The depletion effect is clearly seen in this cluster, and we have fitted its radial variation with several sets of mass profiles. The results are quite satisfying as we are able to constrain the mass profile up to a reasonable distance from the center (about 200", or equivalently 1.1 [FORMULA] Mpc). Our results marginally favor the NFW mass profile over the isothermal profile, and more significantly reject a power-law distribution, essentially because of the steep rise of the depletion curve just after its minimum. Note that this region corresponds to the "intermediate" lensing regime, where shear measurements are more difficult to relate to the mass distribution. We have also studied the shape of the depletion area, which is easy to relate to the ellipticity of the mass distribution. We were then able to constrain the ellipticity and orientation of the potential of MS1008-1224 with a good accuracy.

This preliminary study highlights the need for additional exploration of several issues not fully explored in the present paper. For example, the question of clustering of the background sources still remains, although in the case of MS1008-1224, we have shown how it can be partly eliminated. Schneider et al. (2000) also mention this problem and insist on the fact that, at deep magnitudes, the two point correlation function of the sources is still quite uncertain, but a positive signal does not seem to extend much above [FORMULA] 1 ". More quantitatively, we can use the most recent measures of the two point correlation function from the HDF-South (Fynbo et al., 2000). At the magnitude limits used in our paper, this correlation function does not exceed a few percent for an angular separation of 10", typical of our bin size. Previous measurements were less optimistic in the sense that their values attained 8 to 10% at 10", even at very faint magnitudes. But although there is still only poor knowledge of the amplitude of the correlation function at the faintest levels, we can hope that the effect will not be dramatic in our studies of the magnification bias, provided we retain a reasonable bin size at several arcseconds, to wash out most of the inhomogeneities.

Following preliminary approaches from the observational point of view (Taylor et al., 1998; Athreya et al., 2000) or a more theoretical one (Schneider et al., 2000), one now clearly needs to extensively and quantitatively compare the weak lensing approach with the magnification bias. In particular, with the new facilities of deep wide-field imaging presently available, most of the difficulties related to a small field-of-view can be overcome: for weak lensing measurement, a complete mass reconstruction requires shear measurements up to the "no-shear" region in the outer parts of the cluster to integrate the mass inwards. The absolute normalization of the field number counts for the magnification bias can also be estimated outside the cluster, in exactly the same observing conditions (filter, magnitude limit, seeing, ...), giving an absolute calibration of the depletion effect.

Moreover, the full 2D mass reconstruction of the cluster from the depletion signal alone should be tested. As the signal is directly related to the magnification [FORMULA], and then to [FORMULA] and [FORMULA], it should be in principle feasible to invert the depletion map to produce a non-parametric mass map. In practice, the reconstruction is simpler as soon as the shear [FORMULA] becomes small, because in that case µ and [FORMULA] are simply linearly related. In any case, we have shown in this paper that it is rather easy to reach the outer parts of the cluster up to Mpc scales. This is quite similar to what can be done with deep X-ray maps for intermediate redshift clusters (Soucail et al., 2000). We want to insist on the fact that the magnification bias effect is easy to detect from the observational point of view because it is less sensitive to seeing conditions or to geometrical distortions of the instruments than are shear measurements (Broadhurst et al., 1995; Fort et al., 1997). It only requires deeper observations, not necessarily in photometric conditions, provided one is able to reach the outer parts of the cluster to normalise the number counts. In order to improve the mass reconstruction, one may progress with the help of photometric redshifts, quite useful for faint objects, to constrain the redshift distribution of the sources (Pello et al., 1999; Bolzonella et al., 2000). This approach might also be useful in order to limit the influence of large scale over-density fluctuations when evaluating the magnification and the asymptotic limit of the depletion curves. This of course requires deep multi-color photometry, such as the one obtained on MS1008-1224. Extending this study up to large-scale structures (LSS) would probably be more difficult to implement as compared to the search for cosmic shear (Van Waerbeke et al., 2000), and the depletion effect should be restricted to cluster scales, at least with the simple method used in this paper.

Finally, one of our initial prospects was to try to constrain the background redshift distribution with multi-wavelength observations of the depletion effect. We have shown that this is quite a difficult task as the wavelength dependence of the depletion curves is a kind of second order effect. Nevertheless, this may be an interesting point to explore at other wavelengths, where the background redshift distribution is quite different from that in the optical. For example, deep ISO observations in the mid-IR of a few cluster lenses (Altieri et al., 1999) may represent an extension of our analysis, as well as submm observations with SCUBA, provided enough sources are detected behind the lenses for a statistical analysis. Another possibility would be to address the question of the nature of the X-ray background sources and their redshift distribution through deep and high resolution observations of clusters with the new X-ray satellites Chandra and XMM (Refregier & Loeb, 1997).

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Online publication: October 2, 2000