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The formalism to compute the geometrical and topological one-point statisticsof mildly non-Gaussian 2D and 3D cosmological fields is developed. Leveragingthe isotropy of the target statistics, the Gram-Charlier expansion isreformulated with rotation invariant variables. This formulation allows us totrack the geometrical statistics of the cosmic field to all orders. It thenallows us to connect the one point statistics of the critical sets to thegrowth factor through perturbation theory, which predicts the redshiftevolution of higher order cumulants. In particular, the cosmic non-linearevolution of the skeleton's length, together with the statistics of extrema andEuler characteristic are investigated in turn. In 2D, the correspondingdifferential densities are analytic as a function of the excursion setthreshold and the shape parameter. In 3D, the Euler characteristics and thefield isosurface area are also analytic to all orders in the expansion.Numerical integrations are performed and simple fits are provided wheneverclosed form expressions are not available. These statistics are compared toestimates from N-body simulations and are shown to match well the cosmicevolution up to root mean square of the density field of ~0.2. In 3D,gravitational perturbation theory is implemented to predict the cosmicevolution of all the relevant Gram-Charlier coefficients for universes withscale invariant matter distribution. The one point statistics of critical setscould be used to constrain primordial non-Gaussianities and the dark energyequation of state on upcoming cosmic surveys; this is illustrated on idealizedexperiments.

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