In this article we have described methods that allow exact reconstructions of the one-point PDF of the local aperture mass in weak lensing maps. These methods do take into account the projection effects but not all the nonlinear couplings between the local density field and the observed distortions field such as lens-lens coupling effects, or departure from the Born approximation.
In the course of this paper we have examined both the quasi-linear and non-linear regimes. In particular, although the details of the calculations are specific to each case we have pointed out the generic properties which are common to both regimes and the features brought about by the projection effects. For instance, in both quasi-linear and non-linear domains the PDF should show two asymmetric exponential tails. Our methods are quite general and can be extended in a straightforward fashion to other statistics. In the quasi-linear regime our approach can be applied to any filter which is axisymmetric while in the non-linear regime there are no restrictions. In particular, in this latter case our results can be extended to multivariate statistics (which can be obtained from filters which consist of several disconnected parts).
We have briefly investigated the dependence of the PDF with the shape of the filter. Thus, we have checked that for filters with a large compensation radius we recover approximatly the shape of the PDF which is relevant for the top-hat filtered convergence.
Finally, we have checked that our predictions agree reasonably well with the results of available numerical simulations (although we have not included any noise effect at this level) at scale about 4' where the data should provide the largest signal to noise ratio (e.g. Jain & Seljak 1997). In particular, we recover the asymmetric shape of the PDF. Moreover, our approach provides a prediction for the full shape of the while earlier models were restricted to the positive tail of the PDF. We have also shown that the difference between the PDFs obtained for two cosmologies (CDM and OCDM) is larger than the inaccuracy of our predictions (due to parameterization we need to introduce to describe the underlying 3D density field). This suggests that our results could be used to estimate the cosmological parameters. Thus, in addition to the skewness which is traditionally used to this purpose one could take advantage of the expected shape of the PDF to build other statistics which would maximize the dependence on the seeked parameters. Such a study is left for further work.
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000