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A method to compute the full hierarchy of the critical subsets of a density field is presented. It is based on a Watershed technique and uses a probability propagation scheme to improve the quality of the segmentation by circumventing the discreteness of the sampling. It can be applied within spaces of arbitrary dimensions and geometry. This recursive segmentation of space yields, for a  d ‐dimensional space, a   d − 1  succession of  n ‐dimensional subspaces that fully characterize the topology of the density field. The final one‐dimensional manifold of the hierarchy is the fully connected network of the primary critical lines of the field: the skeleton. It corresponds to the subset of lines linking maxima to saddle points, and provides a definition of the filaments that compose the cosmic web as a precise physical object, which makes it possible to compute any of its properties such as its length, curvature, connectivity etc.  When the skeleton extraction is applied to initial conditions of cosmological  N ‐body simulations and their present‐day non‐linear counterparts, it is shown that the time evolution of the cosmic web, as traced by the skeleton, is well accounted by the Zel'dovich approximation (ZA) provided the small‐scale artificial smoothing introduced by the displacement field computation is taken into account. This scale is shown to be well modelled as  in units of the non‐linear scale,  a  being the expansion factor. Comparing this skeleton to the initial skeleton undergoing the Zel'dovich mapping shows that two effects are competing during the formation of the cosmic web: a general dilation of the larger filaments that is captured by a simple deformation of the skeleton of the initial conditions on the one hand, and the shrinking, fusion and disappearance of the more numerous smaller filaments on the other hand. The net result corresponds to a decrease of the cosmic skeleton's length with time.  Other applications of the  N ‐dimensional skeleton and its peak patch hierarchy are discussed.

The fully connected N ‐dimensional skeleton: probing the evolution of the cosmic web