Astron. Astrophys. 342, 15-33 (1999)

1. Introduction

Mass reconstruction from gravitational distortion inversion is a promising technique to probe the mass distribution and the clustering on very large scales, regardless of the nature and the dynamical state of the dark and luminous matter. Pioneering theoretical work done by Gunn (1967), Jaroszyski et al. (1990), Blandford et al. (1991), Miralda-Escudé (1991) and Kaiser (1992) has shown that the expected distortion amplitude of weak lensing effects produced by large scale mass fluctuations ( 1Mpc) is roughly at the percent level. This low level of distortion is observable due to the large number density of galaxies observed in deep surveys (Kaiser 1992). Needless is to say that the observation of such distortion fields provides a unique way to build a picture of the large scale mass distribution, independent on any biasing and/or dynamical prescriptions.

The scientific impact on the determination of cosmological parameters from weak lensing surveys has been underlined by recent theoretical work. Most of the papers quoted above show that the distortion two-point correlation function can be used to constrain the mass power spectrum. Villumsen (1996) remarked that the amplitude of the local distortion is proportional to the amplitude of the 3D density fluctuations and roughly to the density parameter . Bernardeau et al. (1997, hereafter BvWM) extended these calculations in the plane and show that the amplitude is also slightly dependent on the cosmological parameter . In the same paper the authors also show that the shape of the convergence probability distribution function can be used to disentangle the and dependence for models of large-scale structure formation with Gaussian initial conditions. In particular they demonstrate that the skewness ¹, third moment expressed in terms of the square of the second, is roughly inversely proportional to the total mass density of the Universe , but independent of the amplitude of the fluctuations, as well as the shape of the power spectrum. This result, obtained by means of perturbation theory, is expected to be exact at large enough scale (see Gaztañaga & Bernardeau, 1998). The skewness should be enhanced in the nonlinear regime at small scales (Gaztañaga & Bernardeau 1998, see also Colombi et al. 1997, and Jain et al. 1998) which amplifies the differences between open and flat cosmologies. So, even if definitive quantitative predictions for such a quantity cannot be given from our present knowledge, it is clear that the skewness can accurately discriminate between different cosmological models.

Recently many theoretical and observational aspects have been investigated in detail in order to converge towards an unbiased measurement of such small distortions (Bonnet & Mellier 1995, Kaiser et al. 1995, and Van Waerbeke et al. 1997). The technical limitations that have been recognized so far come from systematics due to the correction of spurious distortions generated by the optic defects, fuzzy-shaped point spread function (PSF), and pixel convolution (sampling). Available image analysis techniques and image quality ensure that such systematics can now be reduced to a one percent level, which is necessary for these weak lensing applications, but beyond the need for cluster mass reconstruction. The ultimate limitation for the use of weak lensing surveys as a cosmological probe depends therefore on the accuracy with which the correction of the spurious distortions can be corrected. On the other hand, if the systematics can be reduced to the sub-percent level, the weak lensing analyses are limited by the intrinsic ellipticities of galaxies which acts as a shot noise for the gravitational distortion effect.

In this paper we investigate the effects of the shot noise and finite size survey (cosmic variance) on the determination of the cosmological parameters and the power spectrum. In particular, two goals are looked for,

• the precision on the determination of some cosmological parameters that can be expected from such maps;

• the maximum level of observational systematics which is acceptable in order to achieve these theoretical precisions.

This study is made in particular in the perspective of ongoing and future wide field deep imaging surveys devoted to weak lensing analysis (for instance the MEGACAM project, Boulade et al. 1998) at CFHT  ² or the SDSS project, Stebbins 1996). In such a perspective, there are still open issues concerning the optimal observing strategy that should be adopted:

• Wide and shallow survey versus deep and narrow.

• The determination of the optimal shape and size of the survey depends on the noise properties, and on the correlation properties of the signal. The investigations that have been done so far (Kaiser 1992, 1998 and Seljak 1997) assume that the projected mass follows a Gaussian statistics. If such an assumption had to be dropped it may significantly change the conclusions that have been reached.

In addition, since the reconstruction of the projected mass from the shape of the galaxies is neither local nor linear in terms of the distortion field (Kaiser 1995) and of the intrinsic ellipticities of the galaxies, it is essential to understand how the noise propagates in the reconstructed mass maps. In order to investigate how these different effects may couple together, we built a series of projected mass maps that contain a realistic amount of non-Gaussianity. The associated distortion field is then derived on the basis of the full non-linear lens equation. A noise is added on the distortion maps, and the convergence is finally reconstructed. This paper presents the statistical analysis of those reconstructed maps for different cosmological models and different observational contexts.

The details of mass map generation and useful definitions are presented in Sect. 2. Sect. 3 presents the power spectrum analysis, the noise properties in the reconstructed mass maps, and the cosmic variance on the estimated power spectrum. Sect. 4 repeats similar analysis on moments in real space, where a comparison between top hat and compensated filter is done. It contains also some highlights about other possible statistical quantities that could be used to measure , and a comparison with results obtained for different power spectra. We finally summarize our results and discuss the best observational strategies.

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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