Non-Gaussian statistics of critical sets in 2 and 3D: Peaks, voids, saddles, genus and skeleton

The formalism to compute the geometrical and topological one-point statistics of mildly non-Gaussian 2D and 3D cosmological fields is developed. Leveraging the isotropy of the target statistics, the Gram-Charlier expansion is reformulated with rotation invariant variables. This formulation allows us to track the geometrical statistics of the cosmic field to all orders. It then allows us to connect the one point statistics of the critical sets to the growth factor through perturbation theory, which predicts the redshift evolution of higher order cumulants. In particular, the cosmic non-linear evolution of the skeleton's length, together with the statistics of extrema and Euler characteristic are investigated in turn. In 2D, the corresponding differential densities are analytic as a function of the excursion set threshold and the shape parameter. In 3D, the Euler characteristics and the field isosurface area are also analytic to all orders in the expansion. Numerical integrations are performed and simple fits are provided whenever closed form expressions are not available. These statistics are compared to estimates from N-body simulations and are shown to match well the cosmic evolution up to root mean square of the density field of ~0.2. In 3D, gravitational perturbation theory is implemented to predict the cosmic evolution of all the relevant Gram-Charlier coefficients for universes with scale invariant matter distribution. The one point statistics of critical sets could be used to constrain primordial non-Gaussianities and the dark energy equation of state on upcoming cosmic surveys; this is illustrated on idealized experiments.