Optimal capture of non-Gaussianity in weak lensing surveys: power
  spectrum, bispectrum and halo counts
  We compare the efficiency of weak lensing-selected galaxy clusters counts and
of the weak lensing bispectrum at capturing non-Gaussian features in the dark
matter distribution. We use the halo model to compute the weak lensing power
spectrum, the bispectrum and the expected number of detected clusters, and
derive constraints on cosmological parameters for a large, low systematic weak
lensing survey, by focusing on the $\Omega_m$-$\sigma_8$ plane and on the dark
energy equation of state. We separate the power spectrum into the resolved and
the unresolved parts of the data, the resolved part being defined as detected
clusters, and the unresolved part as the rest of the field. We consider four
kinds of clusters counts, taking into account different amount of information :
signal-to-noise ratio peak counts; counts as a function of clusters' mass;
counts as a function of clusters' redshift; and counts as a function of
clusters' mass and redshift. We show that when combined with the power
spectrum, those four kinds of counts provide similar constraints, thus allowing
one to perform the most direct counts, signal-to-noise peaks counts, and get
percent level constraints on cosmological parameters. We show that the weak
lensing bispectrum gives constraints comparable to those given by the power
spectrum and captures non-Gaussian features as well as clusters counts, its
combination with the power spectrum giving errors on cosmological parameters
that are similar to, if not marginally smaller than, those obtained when
combining the power spectrum with cluster counts. We finally note that in order
to reach its potential, the weak lensing bispectrum must be computed using all
triangle configurations, as equilateral triangles alone do not provide useful
information.
