From the early days of detection the weak lensing (Munshi et al. 2008) studies have matured to a point where weak lensing results from Euclid are expected to constrain the cosmological parameters to sub-per cent accuracy. However, weak lensing at smaller angular scales probes the non-linear regime of gravitational clustering, and is thus key to understanding the non-Gaussianity induced by the non-linearity and fullly exploiting in the weak lensing maps. The higher order statistics are also useful for the breaking of parameter degeneracies in studies involving the power spectrum analysis alone and they are also important in understanding the variance or error of estimation of lower order statistics. These higher order statistics including the cumulants (Bernardeau 1994b) and their correlators (Bernardeau 1996a; Calabrese et al. 2010; Munshi et al. 2011; Riquelme & Spergel 2012) are among the best-known diagnostics of the deviation from Gaussianity that characterizes the non-linear regime (Bartolo et al. 2004), with a long history analytical modelling (Bernardeau et al. 2002a). Most of these studies use extensions of perturbative results in the quasi-linear regime valid at large smoothing angular scales or variants of halo models (Cooray & Sheth 2002). Early studies concentrated on measurements of higher order correlation hierarchy in the angular space due to small survey size (Bernardeau, Mellier & van Waerbeke 2002b; Bernardeau, Waerbeke & Mellier 2003). However, the near all-sky coverage of future surveys such as Euclid will let us estimate higher order statistics in the harmonic domain with unprecedented accuracy (Amendola et al. 2013). While measurements of real space correlations are much simpler in the presence of complicated survey design, the measurements for different angular scales can be highly correlated (Munshi 2000; Munshi & Jain 2000). In comparison measurements in the harmonic domain are less correlated and each mode contains (nearly) independent information in the limit of all-sky coverage. The primary motivation of this study is to develop analytical predictions for one such statistic called the skew-spectrum, and test them against numerical simulations. We will borrow the concepts developed for constructing skew-spectra for the study of non-Gaussianity in the context of CMBR observations (Planck Collaboration XVII2016b). However, we also include gravity-induced secondary non-Gaussianity. The skew-spectrum is the lowest order member in the family of higher order spectra (Munshi et al. 2011a, 2020). In a series of papers, the one-point statistics such as the skewness and kurtosis were generalized to two-point cumulant correlator, e.g. the two-to-one correlator and its higher order generalizations. These can be represented in the harmonic domain by their associated spectra such as the skew-spectrum (Munshi & Heavens 2010) and its higher order generalizations (Munshi et al. 2011a, 2020). These spectra have already been used to analyse WMAP9 (Smidt et al. 2010) as well as Planck data (Planck Collaboration XVII2016b). They are useful tools to separate individual contributions and estimate systematics. In this paper, we will concentrate on the projected skew-spectrum and kurt-spectrum in the context of weak lensing surveys (Munshi et al. 2011c).
Citation: Bernardeau, Waerbeke & Mellier 2003
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