The local theory of the cosmic skeleton

The local theory of the critical lines of two- and three-dimensional random fields that underline the cosmic structures is presented. In the context of cosmological matter distribution, the subset of critical lines of the three-dimensional density field serves to delineate the skeleton of the observed filamentary structure at large scales. A stiff approximation used to quantitatively describe the filamentary skeleton shows that the flux of the skeleton lines is related to the average Gaussian curvature of the (N - 1) dimensional sections of the field. The distribution of the length of the critical lines with threshold is analysed in detail, while the extended descriptors of the skeleton - its curvature and singular points - are introduced and briefly described. Theoretical predictions are compared to measurements of the skeleton in realizations of Gaussian random fields in two and three dimensions. It is found that the stiff approximation accurately predicts the shape of the differential length, allows for analytical insight and explicit closed form solutions. Finally, it provides a simple classification of the singular points of the critical lines: (i) critical points; (ii) bifurcation points and (iii) slopping plateaux.