The detection of weak gravitational lensing effects (weak shear and magnification bias) at large distance from cluster centers is now well established (see Fort & Mellier 1994 and Narayan & Bartelmann 1996 for reviews). With the present development of wide field CCD detectors mounted on the best telescopes it becomes possible to start the investigation of large scale structures from lensing effect observations in an angular area of significant size. The shear and the magnification are expected to be imposed on the background galaxies by the density fluctuations along the line-of-sight. Compared to usual determinations of the density fluctuations in the local universe with galaxy counts (in fact, the light distribution), this approach is extremely attractive since it is directly sensitive to the global mass fluctuations regardless of any biases associated with the light distribution. The detection of weak lensing has thus naturally been suggested as a possible means to measure the power spectrum of the large scale density fluctuations (Blandford et al. 1991).
Theoretical studies done by Blandford et al. (1991), Miralda-Escudé (1991a), Kaiser (1992) and more recently by Villumsen (1996) proposed various approaches and discussed the feasibility of observational strategies for some specific cosmological scenarios. In all cases, they concluded that weak lensing induced by large scale structures is detectable with the present techniques. They claimed that the projected power spectrum should be recovered provided some crucial observational issues, namely the correction of image degradation, are solved. However, even if the observational aspects are certainly the main difficulties in the future, different theoretical problems have not been addressed yet. In particular, the dependence of the physical quantities (such as the convergence, see hereafter) on the cosmological parameters have not been discussed in complete detail. Moreover the errors associated with those quantities for realistic scenarios of large scale structure formation. This paper thus is a theoretical study in which we explore the dependence of a priori physical quantities on the cosmological parameters. We will focus our analysis on the second and third moment of the local filtered convergence at large angular scale, i.e. in a regime where Perturbation Theory results are expected to be valid. We take here advantage of many results that have been obtained in the last few years in this regime, showing that the Perturbation Theory calculations give extremely accurate results for the behavior of the high order moments of the cosmic density at large scale (many papers should be quoted, see Juszkiewicz & Bouchet 1995 or Bernardeau 1996b for short review papers on the subject).
Our aim is to present quantitative predictions on the behavior of the moments of the one-point probability distribution function of the local convergence as a function of the matter power spectrum, the cosmological parameters and . As long as one is interested in its second moment in the linear regime, results can easily be extrapolated to other quantities such as the magnification or the distortion (see Sect. 3 for more details on these quantities and their relationships). For the third moment however, the choice of the local convergence (or any other scalar quantity) is crucial. The distortion is intrinsically irrelevant since its third moment, for obvious symmetry reasons, should vanish. The first non-trivial moment would then be its kurtosis, the connected part of the fourth moment, but this is then more intricate to calculate, and also would be more difficult to measure. Other scalar quantities could have been considered, such as the local magnification, but it is not (at the second order) directly proportional to the local density, as it is the case for the convergence. This is why we have preferred this latter quantity.
The paper is structured as follows. In Sect. 2 we present the basic equations, valid for any cosmological model, from which we derive the quantities of interest. In Sect. 3 we explore the dependence of the variance of the convergence on the cosmological parameters. In Sect. 4 we consider third order moments. In the last section we present a study of the expected errors due to the use of a finite sample that are expected to affect the measurements of the second and third moment. The errors have been quantified for a realistic scenario (APM like power spectrum in an Einstein-de Sitter Universe). We eventually propose, in view of the previous results, an adapted strategy to do such observations. Throughout the paper we adopt the convention , the physical distances have thus to be multiplied by 3000 for /Mpc to be expressed in Mpc.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998