Annu. Rev. Astron. Astrophys. 1999. 37: 127-189
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7.3. Magnification Bias

The depletion of the galaxy number density as a function of radial distance from the cluster center can potentially provide information on the cosmological constant. The reason for this is ultimately the same as for giant arcs - namely, the ratios of angular distances which strongly depend on the cosmological constant. Therefore, if the redshift distribution of the sources and the mass distribution of the lensing cluster are known, the shape of the depletion curve - in particular, its extension at a large radius - is constrained by lambda.

Fort et al (1997) have used this property in order to constrain the cosmological constant. They used ultra-deep images of the lensing clusters Cl0024+1654 and A370 which permit a good signal to noise ratio of the depletion curves. These clusters have giant arcs with known redshift so the mass at a given critical line can be scaled. The method provides jointly the redshift of the sources and the cosmological parameters. Fort et al (1997) found that the location of this high redshift critical line rather favors a flat cosmology with lambda greater than 0.6.

It is remarkable that from these two clusters only the method predicts a value of lambda compatible with other independent approaches (see White 1998 and references therein). Since it needs a good model for the lens, this method has still many uncertainties and can be significantly improved with a large sample of arc clusters, in particular by using a maximum likelihood analysis applied to the probabilities of reproducing their observed local shears and convergences. A strong improvement can come from the new possibility of using the redshift distribution found independently. This should be possible using photometric redshifts. Even more promising, Gautret et al (1998) proposed to use triplets of neighbouring arclets at a different redshift. Because they are close together, the positions of these arclets are independent of the mass profile but only depend on lambda. This in principle breaks the degeneracy and solves this problem.

All the methods described above do not yet provide convincing results on lambda mainly because they use simultaneously different quantities which are degenerate without external information: mass distribution of the lensing-cluster, redshift distribution of the sources, cosmological parameters, and evolution scenarios of clusters and of sources. The approach using statistics of arc(let)s looks promising but demands very good simulations and a good understanding of selection functions of cluster samples which are used for comparison with observations. The method using lens modeling requires very good lens models and information on the redshift distribution of galaxies, in particular for the most distant ones, since they contain the population which depends the most on lambda. This approach can use the redshift distribution obtained from photometric redshifts, and should focus on regular lensing clusters containing giant arcs with known redshift. As emphasized by Fort et al (1997), Lombardi & Bertin (1999), Gautret et al (1998), significant results cannot be expected until many clusters have been investigated. This should be done within the next few years, in particular using 10-meter class telescopes. However, it is remarkable that the Fort et al limit corresponds to the value given by Im et al (1997) from the measurement of strong lensing produced by elliptical galaxies, and to the upper limit given by Kochanek (1996) from the statistics of lensed quasars.

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