Recent reports of cosmic shear detection (van Waerbeke et al. 2000a; Bacon et al. 2000; Wittman et al. 2000; Kaiser et al. 2000) have underlined the interest that such observations can have for exploring the large-scale structures of the Universe. Previous papers have stressed that not only it could be possible to measure the projected power spectrum (Blandford et al. 1991; Miralda-Escudé 1991; Kaiser 1992) of the matter field, but also that non-linear effects could be significant and betray the value of the density parameter of the Universe. More specifically, Bernardeau et al. (1997) have shown that the skewness, third order moment of the local convergence field, when properly expressed in terms of the second moment can be a probe of the density parameter independently of the amplitude of the density fluctuations. This result can be extended to higher order moments, to the nonlinear regime (Jain & Seljak 1997; Hui 1999; Munshi & Coles 2000; Munshi & Jain 1999b) and the whole shape of the one-point PDF (Valageas 2000a,b; Munshi & Jain 1999a,b).
In case of weak lensing surveys it appears however that it is more convenient to consider the so called aperture mass statistics that corresponds to filtered convergence fields with a compensated filter, that is with a filter of zero spatial average (Kaiser et al. 1994; Schneider 1996). Indeed, it is possible to relate the local aperture mass to the observed shear field only, whereas, in contrast, convergence maps require the resolution of a non-local inversion problem and are only obtained to a mass sheet degeneracy. The aperture mass statistics have proved valuable in particular for cosmic variance related issues (Schneider et al. 1998). Thus, in this article we present a method to compute the one-point PDF of the aperture mass, both for the quasilinear and strongly non-linear regimes. In the case of the quasilinear regime we can use rigorous perturbative methods while in the highly non-linear regime we have to use a specific hierarchical tree model (which has been seen to agree reasonably well with numerical simulations). Although the details of the calculations are specific to each case we point out the general pattern common to both regimes which is brought about by the projection effects. In particular, our methods are quite general and actually apply to any filters, though we are restricted to axisymmetric filters for the quasi-linear regime. Our results for the non-linear regime, where there is not such a restriction, can also be extended to multivariate statistics (point PDFs).
In Sect. 2 we recall the definitions of the local convergence and aperture mass and how they are related to the cosmic 3D density fluctuations. In particular we present the shape of the compensated filters that we use for the explicit computations we present in the following. In Sect. 3 we describe the relationship between the PDF and the cumulant generating function of the 3D density field and we show how this extends to the projected density. The details of the calculations are presented in Sect. 4, for the quasilinear theory, and in Sect. 5 for the nonlinear theory. Numerical results are presented in Sect. 6.
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000