Tomography and Weak lensing Statistics

Abstract: Extending previous studies, we derive generic predictions for lower order cumulants and their correlators for individual tomographic bins as well as between two different bins. We derive the corresponding one- and two-point joint probability distribution function for the tomographic convergence maps from different bins as a function of angular smoothing scale. The modelling of weak lensing statistics is obtained by adopting a detailed prescription for the underlying density contrast. In this paper we concentrate on the convergence field $\kappa$ and use top-hat filter; though the techniques presented can readily be extended to model the PDF of shear components or to include other windows such as the compensated filter. The functional form for the underlying PDF and bias is modelled in terms of the non-linear or the quasilinear form depending on the smoothing angular scale. Results from other semi-analytical models e.g. the lognormal distribution are also presented. Introducing a reduced convergence for individual bins, we are able to show that the tomographic PDFs and bias for each bin sample the same functional form of the underlying PDF of density contrast but with varying variance. The joint probability distribution of the convergence maps that correspond to two different tomographic bins can be constructed from individual tomographic PDF and bias. We study their dependence on cosmological parameters for source distributions corresponding to the realistic surveys such as LSST and DES. We briefly outline how photometric redshift information can be incorporated in our computation of cumulants, cumulant correlators and the PDFs. Connection of our results to the full 3D calculations is elucidated. Analytical results for inclusion of realistic noise and finite survey size are presented in detail.