Introduction

The physical mechanism that induces gravitational distortions is well known since the foundation of the General Relativity: photons are deflected by any mass concentration placed in the vicinity of their path, and the deflecting angle is simply proportional to the gradient of the gravitational potential. The most spectacular consequence of such an effect is the apparition of multiple images and giant arcs observed through a peaked mass concentration.

Until very recently the gravitational effects have been mainly used to probe the mass distribution of galaxy clusters. Since the historical discovery of Soucail et al. (1987) of the first gravitational arc at CFHT, many others have been observed. Cosmologists are now rich with a large variety of robust data on the mass content and profile of clusters. And it is worthwhile to note that it provides unquestionable proofs of the existence of a dark component in galaxy clusters.

Although foreground masses distort all the background sources, when the induced distortion is too small it cannot be seen in a single object: intrinsic fluctuations of the shape of background galaxies dominate. However, the gravitational effect is betrayed by the large scale angular correlation it induces in the orientation of the objects. At scales up to one degree, mass fluctuations of the large scale structures of the Universe are expected to produce shear correlation at the level of a few percents. Given the number of background galaxies for which the shape can be measured in deep galaxy surveys, such a detection is now feasible. The MECAGAM project represents a unique way to complete such a program because it allies a large field with a very good image quality.

Cosmology with distortion maps

In this section we summarize the conclusions of recent theoretical investigations by Villumsen (1996), Jain & Seljak (1997), Kaiser (1996) and more particularly by Bernardeau, van Waerbeke & Mellier (1997) on the use of the weak lensing effects as a cosmological probe. In the weak lensing regime, the observable quantities are all related to the local 2D deformation matrix whose elements are the second order derivatives of the projected gravitational potential .

(1)

The information which is directly observable is the amplitude and direction of the deformation effect (₁ and ₂ in Equation 1). The magnification effect, given by the convergence , provides the global change in size of the background objects, but is not directly measurable. This is unfortunate, since it is proportional to the projected mass and is therefore the quantity observers are more particularly aiming at. It can however be reconstructed through a differential equation taking advantage of the fact that ₁, ₂ and are second order derivatives of a unique scalar field (see Kaiser 1995). This has an important consequence. Large reliable mass maps can only be done with connected and as compact as possible deformation map in order to reduce the boundary effects.

What then could be learn from such projected mass maps? The local convergence () at the angular position is directly related to the mass fluctuations _mass(,D) along the line of sight in the direction at different angular distances D up to the distance of the source plane D_s. In case of a flat Universe we have

(2)

where a is the expansion factor and is the usual density of the Universe in units of the critical density. Assuming that we have at our disposal a large compact projected mass map of a fair fraction of the universe, the Equation 2 shows that we can directly relate the statistical properties of the mass fluctuations to the ones of via , the cosmological constant (that intervenes in the expression of the angular distances) and D_s.

For instance, for any given smoothing angular scale, , the histograms of the filtered local convergence determined at different lines of sight contain information of cosmological interest. Its width is proportional to the amplitude of the mass fluctuations. Expressing the latter in terms of ₈ we have,

(3)

where n is the power spectrum index at the filtering scale. Note that the result strongly depends on the redshift of the sources, z_s. The Figure 5 shows the reason for such a dependence: the more distant the sources are, the larger the number of lenses can be, and the greater their gravitational effect.

The shape of such histograms (Figure 4) is also of great cosmological interest. It indeed characterizes the level of non-linearities that has been reached by the gravitational dynamics. In the quasi-linear regime, one simple quantity to consider is the reduced skewness which is independent of the width of the distribution. We have here,

(4)

Provided the redshift of the sources is well known it would be a unique way to determine .

Beyond the intrinsic statistical properties of the local convergence, it will be of great interest to confront such maps with other maps such as the X-ray survey that is planned with the satellite XXM. All the deep potential wells detected with the weak lensing survey should appear as strong X-ray sources.

The observational requirements

We have already mentioned that it is necessary to have a survey as compact as possible to reduce the boundary effects. This is not only important for the reconstruction of the mass map, but also for reliable estimations of any statistical quantities. More precisely, we have shown that a map of about 25 square degrees would provide a good determination of the amplitude of the power spectrum and of (to the 10% level). However, it is possible only if the redshift distribution of the sources is known to sufficient precision. It is obviously not necessary to know the redshift of all the sources and we think that the photometric redshifts should be able to provide the required information. A multi band observation is thus necessary on a small fraction of the survey (about one square degree).

To summarize, the completion of this program would require 25 nights for the I band observations, and 5 nights for the UBVR bands in a subsample.

References

Fort, B. & Mellier, Y., 1994, A&AR, 5, 239

Jain, B. & Seljak, U. 1997, ApJ, 484, 560

Kaiser, N. 1995, ApJ, 439, L1

Kaiser, N. 1996, 1996, submitted to ApJ, astro-ph/9610120

Soucail, G., Mellier, Y., Fort, B, Mathez, G. & Hammer, F. 1987, A&A, 184, 7L

Villumsen, J. V. 1996, MNRAS, 281, 369