Higher-order Statistics of Weak Lensing Shear and Flexion
  Owing to their more extensive sky coverage and tighter control on systematic
errors, future deep weak lensing surveys should provide a better statistical
picture of the dark matter clustering beyond the level of the power spectrum.
In this context, the study of non-Gaussianity induced by gravity can help
tighten constraints on the background cosmology by breaking parameter
degeneracies, as well as throwing light on the nature of dark matter, dark
energy or alternative gravity theories. Analysis of the shear or flexion
properties of such maps is more complicated than the simpler case of the
convergence due to the spinorial nature of the fields involved. Here we develop
analytical tools for the study of higher-order statistics such as the
bispectrum (or trispectrum) directly using such maps at different source
redshift. The statistics we introduce can be constructed from cumulants of the
shear or flexions, involving the cross-correlation of squared and cubic maps at
different redshifts. Typically, the low signal-to-noise ratio prevents recovery
of the bispectrum or trispectrum mode by mode. We define power spectra
associated with each multi- spectra which compresses some of the available
information of higher order multispectra. We show how these can be recovered
from a noisy observational data even in the presence of arbitrary mask, which
introduces mixing between Electric (E-type) and Magnetic (B-type) polarization,
in an unbiased way. We also introduce higher order cross-correlators which can
cross-correlate lensing shear with different tracers of large scale structures.
