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We compare the efficiency of weak lensing-selected galaxy clusters counts andof the weak lensing bispectrum at capturing non-Gaussian features in the darkmatter distribution. We use the halo model to compute the weak lensing powerspectrum, the bispectrum and the expected number of detected clusters, andderive constraints on cosmological parameters for a large, low systematic weaklensing survey, by focusing on the $\Omega_m$-$\sigma_8$ plane and on the darkenergy equation of state. We separate the power spectrum into the resolved andthe unresolved parts of the data, the resolved part being defined as detectedclusters, and the unresolved part as the rest of the field. We consider fourkinds 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 ofclusters' mass and redshift. We show that when combined with the powerspectrum, those four kinds of counts provide similar constraints, thus allowingone to perform the most direct counts, signal-to-noise peaks counts, and getpercent level constraints on cosmological parameters. We show that the weaklensing bispectrum gives constraints comparable to those given by the powerspectrum and captures non-Gaussian features as well as clusters counts, itscombination with the power spectrum giving errors on cosmological parametersthat are similar to, if not marginally smaller than, those obtained whencombining the power spectrum with cluster counts. We finally note that in orderto reach its potential, the weak lensing bispectrum must be computed using alltriangle configurations, as equilateral triangles alone do not provide usefulinformation.

Optimal capture of non-Gaussianity in weak lensing surveys: power spectrum, bispectrum and halo counts