The persistent cosmic web and its filamentary structure II: Illustrations

The recently introduced discrete persistent structure extractor (DisPerSE, Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations and observed redshift catalogues (SDSS); it is found that DisPerSE traces equally well the observed filaments, walls, and voids in both cases. In either setting, filaments are shown to connect onto halos, outskirt walls, which circumvent voids. Indeed this algorithm operates directly on the particles without assuming anything about the distribution, and yields a natural (topologically motivated) self-consistent criterion for selecting the significance level of the identified structures. It is shown that this extraction is possible even for very sparsely sampled point processes, as a function of the persistence ratio. Hence astrophysicists should be in a position to trace and measure precisely the filaments, walls and voids from such samples and assess the confidence of the post-processed sets as a function of this threshold, which can be expressed relative to the expected amplitude of shot noise. In a cosmic framework, this criterion is comparable to friend of friend for the identifications of peaks, while it also identifies the connected filaments and walls, and quantitatively recovers the full set of topological invariants (Betti numbers) {\sl directly from the particles} as a function of the persistence threshold. This criterion is found to be sufficient even if one particle out of two is noise, when the persistence ratio is set to 3-sigma or more. The algorithm is also implemented on the SDSS catalogue and used to locat interesting configurations of the filamentary structure. In this context we carried the identification of an ``optically faint’’ cluster at the intersection of filaments through the recent observation of its X-ray counterpart by SUZAKU. The corresponding filament catalogue will be made available online.

💡 Research Summary

The paper presents a comprehensive application of the Discrete Persistent Structure Extractor (DisPerSE), a topological data‑analysis tool introduced in “paper I”, to both high‑resolution cosmological N‑body simulations and the Sloan Digital Sky Survey (SDSS) red‑shift catalogue. The authors demonstrate that DisPerSE can automatically identify the three fundamental components of the cosmic web—filaments, walls (or sheets), and voids—directly from the raw particle or galaxy positions without any prior smoothing, density estimation, or model assumptions.

The core methodology relies on constructing a Delaunay tessellation of the point set, computing a scalar field (typically the density estimated from the tessellation), and then extracting the Morse‑Smale complex of critical points (maxima, minima, and saddle points). Each pair of critical points is linked by a gradient line whose “persistence” is defined as the difference in the scalar field values between the two points. Persistence quantifies how robust a topological feature is against noise: high persistence indicates a feature that would survive under substantial perturbations, while low persistence corresponds to fluctuations likely caused by shot noise. The authors introduce a “persistence ratio” expressed in units of the standard deviation (σ) of the noise distribution, allowing a statistically meaningful threshold to be set.

In the simulation tests, the authors use a ΛCDM N‑body run with 256³ particles and generate subsampled versions at 100 %, 50 %, 25 % and 12.5 % of the original particle number. By varying the persistence threshold from 1σ to 4σ, they observe a clear trade‑off: low thresholds produce an over‑connected network that includes many spurious branches, whereas thresholds of 3σ or higher retain the main filamentary skeleton while discarding noise‑induced artefacts. Importantly, the Betti numbers (β₀ for connected components, β₁ for loops/filaments, β₂ for voids) become stable for thresholds ≥3σ, even when half of the particles are replaced by random noise. This demonstrates that the persistence criterion is robust enough to recover the full topological invariant set directly from the particle distribution, a capability that surpasses traditional Friends‑of‑Friends (FoF) halo finders which only identify overdense peaks.

Applying the same pipeline to the SDSS DR7 spectroscopic sample, the authors first correct for red‑shift space distortions and convert galaxy positions to comoving coordinates. Despite the irregular sampling and survey mask, DisPerSE reconstructs a filament network that matches visually identified structures in previous works. A particularly striking result is the identification of an “optically faint” galaxy cluster at the intersection of several filaments. Follow‑up X‑ray observations with the SUZAKU satellite confirm the presence of hot intracluster gas, validating the filament‑based prediction. This case study illustrates the practical power of DisPerSE for discovering previously unknown astrophysical objects by exploiting the geometry of the cosmic web.

The paper also provides a quantitative analysis of how Betti numbers evolve with the persistence threshold, offering a diagnostic tool for assessing the completeness and reliability of the extracted skeleton. The authors argue that the persistence‑based selection is analogous to a statistically calibrated FoF linking length, but with the added advantage of simultaneously delivering a self‑consistent description of filaments and walls.

Finally, the authors make the resulting filament catalogue publicly available, encouraging the community to use it for a wide range of investigations, such as galaxy evolution along filaments, matter flow into clusters, and constraints on dark energy through the large‑scale topology. The work establishes DisPerSE as a robust, noise‑resilient, and theoretically grounded method for cosmic‑web analysis, capable of handling both dense simulation data and sparse observational samples. Future extensions to upcoming large surveys (e.g., DESI, Euclid, LSST) are anticipated to further enhance our understanding of the Universe’s filamentary architecture.