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 nonlinear 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 Sγ,4 ≡ 〈γ 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 8-dimensional integrals necessary to obtain the shear kurtosis. We show that on small angular scales, θ < ∼ 5 , more than half of the shear kurtosis arises from correlations within massive dark matter halos with M > ∼ 10 14M⊙. The shear kurtosis is sensitive to the matter density parameter of the universe, Ωm0, and has relatively weak dependences on other parameters. Therefore, a detection of the shear kurtosis can be used to break degeneracies in determining Ωm0 and the power spectrum amplitude σ8 so far provided from measurements of the two-point shear statistics. The approximations we develop for the thirdand fourth-order moments allow for accurate halo model predictions for the 3-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.