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 S_4 = <gamma^4>/<gamma^2>^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 needed to obtain the kurtosis. We show that on small scales it is dominated by correlations within halos more massive than 10^14 solar masses. The shear kurtosis is sensitive to the mass density parameter of the universe, Omega, and has relatively weak dependences on other parameters. The approximations we develop for the third- and 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.