One of the most interesting applications of gravitational lenses is the determination of the projected mass distribution from weak lensing observations. As noted, among others, by Webster (1985), the mean orientation of a large number of distant galaxies gives a measure of the shear associated with the lens. The observed shear can then be used to derive the two-dimensional mass distribution of the lens responsible for the deformation induced on the background. This last step can be carried out in two different ways. The easier route is to use a specific model for the lens with a number of free parameters that will be determined by a comparison between the observed and the predicted shear (see, e.g., Kneib et al. 1996). A more general procedure is the so called "parameter-free reconstruction" (Kaiser & Squires 1993; see also Bartelmann et al. 1996). In this latter method the mass distribution can be directly determined from the shear map, provided that the shear is known with sufficient accuracy and detail, which requires the existence of a large number of source galaxies.
Such reconstruction techniques are, of course, a powerful tool to study the matter distribution in clusters (see e.g. Tyson, Valdes, Wenk 1990, Fahlman et al. 1994, Smail et al. 1994) and for large scale structures. It is then important to optimize the reconstruction process in order to make the best use of the observations. For this purpose, we have to assess the expected error of a specific reconstruction method, which is the main goal of the present paper.
In this article we focus our attention on the parameter-free method, mainly because this is more general and does not depend on the particular lens under consideration. In a previous paper (Lombardi & Bertin 1998, hereafter Paper I) we have provided expressions for the error involved in the local measurements of the shear (or the reduced shear) of the lens as a function of the parameters characterizing the distribution of source galaxies. Here we extend the statistical analysis to the inferred global mass distribution.
The text is organized as follows. In Sect. 2 we introduce the spatial weight function and we briefly describe various reconstruction methods used to infer the lens mass distribution. In Sect. 3 we calculate the expected error on the measured shear in the regime of weak lensing and in the more general case as a function of position in a given field of the sky; here the formulae of Paper I are generalized to the two-point correlation function for the shear map (see Eq. (25)). This important result is then used in Sect. 4 to calculate the expected errors on the mass distribution associated with the various reconstruction methods. The results are then compared, in Sect. 5, to the simulations by Seitz & Schneider (1996).
The main result of the paper is contained in Eq. (35) (together with Eq. (26)) that describes the two-point correlation function for the mass density obtained from weak lensing analysis. This proves that, in order to optimize the reconstruction process for observations in a finite area of the sky, a curl-free kernel should be used (see Eq. (36)). This behavior is confirmed by numerical simulations.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998