The persistent cosmic web and its filamentary structure - I. Theory and implementation

We present DisPerSE, a novel approach to the coherent multiscale identification of all types of astrophysical structures, in particular the filaments, in the large-scale distribution of the matter in the Universe. This method and the corresponding piece of software allows for a genuinely scale-free and parameter-free identification of the voids, walls, filaments, clusters and their configuration within the cosmic web, directly from the discrete distribution of particles in N-body simulations or galaxies in sparse observational catalogues. To achieve that goal, the method works directly over the Delaunay tessellation of the discrete sample and uses the Delaunay tessellation field estimator density computed at each tracer particle; no further sampling, smoothing or processing of the density field is required. The idea is based on recent advances in distinct subdomains of the computational topology, namely the discrete Morse theory which allows for a rigorous application of topological principles to astrophysical data sets, and the theory of persistence, which allows us to consistently account for the intrinsic uncertainty and Poisson noise within data sets. Practically, the user can define a given persistence level in terms of robustness with respect to noise (defined as a 'number of sigma') and the algorithm returns the structures with the corresponding significance as sets of critical points, lines, surfaces and volumes corresponding to the clusters, filaments, walls and voids - filaments, connected at cluster nodes, crawling along the edges of walls bounding the voids. From a geometrical point of view, the method is also interesting as it allows for a robust quantification of the topological properties of a discrete distribution in terms of Betti numbers or Euler characteristics, without having to resort to smoothing or having to define a particular scale. In this paper, we introduce the necessary mathematical background and describe the method and implementation, while we address the application to 3D simulated and observed data sets in the companion paper (Sousbie, Pichon & Kawahara, Paper II).