The kurtosis of the cosmic shear field

We study the fourth‐order moment of the cosmic shear field using the dark matter halo approach to describe the non‐linear gravitational evolution of structure in the Universe. Since the third‐order moment of the shear field vanishes because of symmetry, non‐Gaussian signatures in its one‐point statistics emerge at the fourth‐order level. We argue that the shear kurtosis parameter S γ,4 ≡〈γ 4 i 〉 c /〈γ 2 i 〉 3 may be more directly applicable to realistic data than the well‐studied higher‐order statistics of the convergence field, since obtaining the convergence requires a non‐local reconstruction from the measured shear field. We compare our halo model predictions for the variance, skewness and kurtosis of lensing fields with ray‐tracing simulations of cold dark matter models and find good agreement. The shear kurtosis calculation is made tractable by developing approximations for fast and accurate evaluations of the eight‐dimensional integrals necessary to obtain the shear kurtosis. We show that on small angular scales, θ≲ 5 arcmin , more than half of the shear kurtosis arises from correlations within massive dark matter haloes with M ≳ 10 14 M ⊙ . The shear kurtosis is sensitive to the matter density parameter of the Universe, Ω m0 , and has relatively weak dependences on other parameters. Therefore, a detection of the shear kurtosis can be used to break degeneracies in determining Ω m0 and the power spectrum amplitude σ 8 so far provided from measurements of the two‐point shear statistics. The approximations we develop for the third‐ and fourth‐order moments allow for accurate halo model predictions for the three‐dimensional mass distribution as well. We demonstrate their accuracy in the small‐scale regime, below 2 Mpc, where analytical approaches used in the literature so far cease to be accurate.