Skeleton as a probe of the cosmic web:: the two-dimensional case

We discuss the skeleton as a probe of the filamentary structures of a two-dimensional random field. It can be defined for a smooth field as the ensemble of pairs of field lines departing from saddle points, initially aligned with the major axis of local curvature and connecting them to local maxima. This definition is thus non-local and makes analytical predictions difficult, so we propose a local approximation: the local skeleton is given by the set of points where the gradient is aligned with the local curvature major axis and where the second component of the local curvature is negative. We perform a statistical analysis of the length of the total local skeleton, chosen for simplicity as the set of all points of space where the gradient is either parallel or orthogonal to the main curvature axis. In all our numerical experiments, which include Gaussian and various non-Gaussian realizations such as chi(2) fields and Zel'dovich maps, the differential length f of the skeleton is found within a normalization factor to be very close to the probability distribution function (pdf) of the smoothed field, as expected and explicitly demonstrated in the Gaussian case where semi-analytical results are derived. As a result of the special nature of the skeleton, the differences between f and the pdf are small but noticeable. We find in the Gaussian case that they increase with the coherence parameter 0 <= gamma <= 1 of the field: f(x, gamma) equivalent to 1/L-tot partial derivative L/partial derivative x similar or equal to 1/root 2 pi e(-x2/2)[1+0.15 gamma(2)(x(2)-1)-0.015y(4)(x(4)-6x(2)+3)] Here, L-tot is the total length of the skeleton and L(x) is the length of the skeleton in the excursion rho > sigma x where sigma is the variance of the density field. This result makes the skeleton an interesting alternative probe of non-Gaussianity. Our analyses furthermore assume that the total length of the skeleton is a free, adjustable parameter. This total length could in fact be used to constrain cosmological models, in cosmic microwave background maps but also in three-dimensional galaxy catalogues, where it estimates the total length of filaments in the Universe. Making the link with other works, we also show how the skeleton can be used to study the dynamics of large-scale structure.