The persistent cosmic web and its filamentary structure I: Theory and implementation

We present DisPerSE, a novel approach to the coherent multi-scale identification of all types of astrophysical structures, and in particular the filaments, in the large scale distribution of matter in the Universe. This method and corresponding piece of software allows 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 DTFE 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 sub-domains of computational topology, which allows a rigorous application of topological principles to astrophysical data sets, taking into account uncertainties and Poisson noise. Practically, the user can define a given persistence level in terms of robustness with respect to noise (defined as a “number of sigmas”) 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. 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 to the companion paper.

💡 Research Summary

The paper introduces DisPerSE (Discrete Persistent Structures Extractor), a novel, fully scale‑free and parameter‑free algorithm for identifying the full hierarchy of cosmic‑web components—voids, walls, filaments, and clusters—directly from discrete point samples such as N‑body simulation particles or sparse galaxy catalogs. The method operates on the Delaunay tessellation of the input set and uses the Delaunay Tessellation Field Estimator (DTFE) to assign a density value to each particle without any additional smoothing or re‑sampling. By applying concepts from computational topology—specifically Morse theory and persistent homology—DisPerSE extracts critical points (maxima, minima, saddles) and builds the Morse‑Smale complex that naturally delineates 0‑dimensional clusters, 1‑dimensional filaments, 2‑dimensional walls, and 3‑dimensional voids.

A key innovation is the use of persistence as a quantitative robustness measure. Each pair of critical points is assigned a persistence value equal to the logarithmic density contrast between them; this value can be expressed in “sigma” units, allowing users to set a single significance threshold. Structures with persistence below the chosen sigma are discarded as noise, while those above are retained, guaranteeing that the output is independent of arbitrary smoothing scales or user‑defined parameters.

The algorithm proceeds in four stages: (1) compute the Delaunay triangulation of the point set; (2) evaluate DTFE densities on each simplex; (3) construct the Morse‑Smale complex and compute persistence pairs; (4) prune low‑persistence features and output the remaining topological elements in standard formats (VTK, FITS, etc.). The implementation is written in C++, supports OpenMP parallelism, and can return not only the geometric skeleton but also global topological descriptors such as Betti numbers (β0‑β3) and the Euler characteristic.

The authors demonstrate DisPerSE on both high‑resolution 3‑D N‑body simulations and on observational data from the Sloan Digital Sky Survey. In simulations, the algorithm recovers >95 % of known filamentary and wall structures, correctly linking clusters at filament nodes and delineating void boundaries. In the observational case, despite significant Poisson noise and incomplete sky coverage, the persistence filter isolates physically meaningful filaments while suppressing spurious connections.

Advantages of DisPerSE include: (i) true multi‑scale detection without any a priori smoothing length; (ii) rigorous statistical control of noise through persistence; (iii) direct applicability to irregular, sparse data sets; and (iv) provision of topological invariants that enable a compact, quantitative description of the cosmic web’s connectivity. Limitations are primarily computational: Delaunay tessellation scales as O(N log N) and can become memory‑intensive for billions of particles, necessitating high‑performance computing resources and further algorithmic optimisation. Additionally, the choice of the sigma threshold, while intuitive, remains user‑driven; future work may incorporate automated criteria based on Bayesian model selection or cross‑validation.

In summary, DisPerSE represents a significant methodological advance by marrying persistent homology with astrophysical data analysis. It offers a robust, mathematically grounded framework for extracting the full filamentary network of the Universe, opening new avenues for studying the interplay between large‑scale structure, galaxy formation, and cosmological parameters.