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

Thierry Sousbie

arXiv:1009.4015v1astro-ph.COmath-phphysics.comp-ph

TL;DR

DisPerSE addresses the difficulty of identifying cosmic-web structures in discrete, sparse, and noisy astrophysical data. It applies discrete Morse theory and persistent homology directly to Delaunay tessellations with DTFE densities, producing scale-free, parameter-free structure identifications. Persistence filtering can isolate significant filaments while preserving sampling-scale resolution, although some critical-point cancellations are impossible in special configurations.

  • Problem

    Cosmic-web structures remain difficult to define and identify consistently, especially in sparse data where noise and the discrete nature of galaxy distributions conflict with smooth Morse theory.

  • Method

    DisPerSE combines Morse theory, discrete Morse theory, and persistent homology to compute a discrete Morse complex directly from DTFE densities on a Delaunay tessellation.

  • Results

    At 2-σ, the probability that a topological arc is caused by Poisson noise is ∼5%, falling to ∼0.006% at 4-σ.

  • Takeaways & Limitations

    The method provides coherent, scale-free, parameter-free identification of cosmic-web structures in potentially sparse simulations and observational galaxy catalogues.

  • Takeaways & Limitations

    Some critical-point cancellations are impossible when the points share no arc or share multiple arcs, because gradient reversal would create a forbidden loop.

Abstract

from arXiv · show

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.

1 INTRODUCTION

The paper introduces DisPerSE as a rigorous, coherent framework for identifying cosmic-web structures directly from discrete simulations or galaxy catalogues. It combines computational-topology methods with Delaunay-based density estimation to handle discreteness, sparse sampling, and noise.

  • 1 INTRODUCTION: DisPerSE aims to define and identify voids, walls, filaments, and haloes consistently in simulated and observational data.The framework addresses both the physical definitions of these structures and their numerical identification.
  • 1 INTRODUCTION: Applying classical Morse theory to astrophysical data is difficult because measurements are discrete or sampled rather than smooth Morse functions.Sparse data also make it harder to distinguish noise features from genuine structures.
  • 1 INTRODUCTION: DisPerSE overcomes these incompatibilities using discrete Morse theory and persistent homology from computational topology.These tools provide the mathematical basis for analyzing non-ideal astrophysical data while accounting for persistence and noise.
  • 1 INTRODUCTION: The algorithm computes a discrete Morse complex from DTFE densities on the Delaunay tessellation of a discrete sample.The paper describes this computation for data such as galaxy distributions, while implementation issues include boundary conditions and post-identification smoothing.
  • 1 INTRODUCTION: Ascending 3-, 2-, and 1-manifolds represent voids, walls, and filaments in a cosmological simulation.Figure 1 displays the density field alongside these manifold classes.

2 MORSE THEORY FOR SMOOTH MANIFOLDS

Morse theory links the geometry and topology of a scalar density field to critical points, gradient-flow manifolds, and their intersections. In the cosmic-web setting, these constructions organize voids, walls, filaments, and their connectivity.

  • 2 MORSE THEORY FOR SMOOTH MANIFOLDS: Morse theory studies smooth scalar functions through their gradients, critical points, and gradient flow.Critical points are classified by the number of negative Hessian eigenvalues, distinguishing minima, saddles, and maxima.
  • 2 MORSE THEORY FOR SMOOTH MANIFOLDS: An integral line follows the gradient flow and has critical points as its origin and destination.Integral lines through distinct non-critical points are either identical or non-intersecting.
  • 2 MORSE THEORY FOR SMOOTH MANIFOLDS: Ascending and descending manifolds classify regions reached from a critical point or ending at one, with dimensions determined by the point’s order.The Morse complex is the set of these ascending or descending manifolds.
  • 2 MORSE THEORY FOR SMOOTH MANIFOLDS: A Morse-Smale function requires ascending and descending manifolds to intersect transversely, enabling well-defined cells from their intersections.These cells include arcs, quads, and crystals.

3 DISCRETE MORSE THEORY

Discrete Morse theory adapts Morse-theoretic ideas to functions on simplicial complexes, replacing smooth critical points and gradient flows with critical simplexes, gradient pairs, and V-paths. These constructions provide the combinatorial basis for discrete Morse-Smale complexes.

  • Discrete Morse functions: Discrete Morse theory applies to intrinsically discrete functions defined on simplicial complexes, avoiding the smoothness requirements of classical Morse theory.It is a combinatorial adaptation suited to discrete or sampled astrophysical data.
  • Simplicial complexes: A simplicial complex is a collection of simplexes closed under faces, with simplex intersections required to be empty or lower-dimensional simplexes.The Delaunay tessellation is a relevant astrophysical example.
  • Discrete Morse functions: A discrete Morse function assigns values to simplexes so each simplex has at most one exceptional lower-valued facet and at most one exceptional higher-valued cofacet.This restriction ensures a locally well-defined preferential direction for the discrete gradient.
  • Critical simplexes: Critical k-simplexes have no qualifying facet or cofacet relationship, making them the discrete counterparts of smooth critical points of order k.In 2D, critical vertices, segments, and triangles correspond respectively to minima, saddles, and maxima.
  • Discrete gradients and V-paths: Gradient vector fields pair adjacent simplexes satisfying the discrete conditions, while unpaired simplexes remain critical; V-paths trace alternating sequences through these gradient pairs.The discrete gradient arrow points from the lower-valued simplex toward the higher-valued one.
  • Discrete Morse-Smale complexes: Ascending and descending manifolds collect simplexes connected to critical simplexes by V-paths, and their intersections define the discrete Morse-Smale complex.Extended manifolds recursively include relevant cofaces and ascending manifolds.

4 TOPOLOGICAL PERSISTENCE

Topological persistence measures how long topological features exist during a filtration, providing a significance criterion and a way to remove noise-related structures. The framework extends from smooth sub-level sets to discrete functions on simplicial complexes and tracks components, loops, and shells.

  • Persistence: Persistence quantifies the lifetime of topological features and was developed to assess their robustness in the presence of noise.It also supports topological simplification by removing less significant features.
  • Sub-level sets and filtrations: Sub-level sets track regions of a function above a threshold as the threshold changes, while filtrations provide the analogous nested construction for discrete simplicial complexes.As the filtration grows, components, loops, and shells can appear and disappear.
  • Noise robustness: Unlike a fixed density threshold, persistence distinguishes genuine peaks from noise in sampled functions, identifying one persistent peak in A′ and two in B′.The comparison is presented against the corresponding noiseless functions A and B.
  • Topological cycles: In three dimensions, k-cycles represent independent components, loops, or shells for k=0, 1, or 2, respectively.A 2-cycle is a set of simplexes bounding a three-dimensional empty region.
  • Persistence pairs: Persistence pairs critical simplexes that create and destroy the same topological feature, with persistence determined by the separation of those events in the filtration.The value measures how much the function would need to change to remove the feature.
  • Topological simplification: Persistence supplies an objective criterion for critical-point significance and permits local cancellation of nonpersistent pairs to remove topological noise.The simplification process removes selected pairs without affecting the other critical points in the illustrated case.

5 DISCRETE MORSE COMPLEX

DisPerSE constructs a discrete Morse complex directly from a Delaunay tessellation and a DTFE-derived density field. It orders and pairs simplexes to obtain critical structures, then traces their manifolds to form the Morse-Smale complex.

  • Method overview: DisPerSE computes a discrete Morse complex from a discrete density field obtained with DTFE on the Delaunay tessellation of sampled data.The approach is scale adaptive and parameter free because the complex is computed directly on the tessellation.
  • Discrete gradient: The discrete Morse function Fρ assigns simplex values from facet values and vertex densities, with infinitesimal ε terms enforcing a valid, unique ordering.Only comparisons are needed in implementation, so ε need not receive an explicit numerical value.
  • Discrete gradient: The algorithm processes simplexes in increasing dimension and increasing Fρ value, pairing each available simplex with the closest eligible cofacet or leaving it unpaired.Unpaired simplexes are critical and represent the discrete analogues of critical points.
  • Morse complex computation: Ascending and descending manifolds are computed by iteratively following gradient pairs through cofacets or facets until the relevant simplex sets are exhausted.The two procedures differ by replacing cofacets with facets for descending manifolds.
  • Morse-Smale complex: The resulting Morse-Smale complex is formed by intersecting ascending and descending manifolds, producing cells that connect critical structures.Figure 8 illustrates manifolds of critical saddles, minima, and maxima and their connecting arcs.

6 DEALING WITH NOISE: PERSISTENCE AND TOPOLOGICAL SIMPLIFICATION

DisPerSE uses persistence in a discrete Morse framework to distinguish significant cosmic-web structures from noise and simplify the resulting topology. The method pairs critical simplexes by feature lifetime, filters low-significance pairs, and preserves meaningful filamentary structure while reducing spurious complexity.

  • 6 DEALING WITH NOISE: PERSISTENCE AND TOPOLOGICAL SIMPLIFICATION: The discrete Morse-Smale complex can identify filaments, walls, and voids directly on a Delaunay tessellation, but noise produces many spurious structures.In one example, 12,771 maxima and 32,457 type-1 saddles suggested that only about 6% of detected structures were cosmologically significant.
  • 6 DEALING WITH NOISE: PERSISTENCE AND TOPOLOGICAL SIMPLIFICATION: Persistence measures the lifetime of topological features as the filtration evolves through density levels.Features are created and destroyed by critical simplexes, with components, loops, and shells corresponding to 0-, 1-, and 2-cycles.
  • 6.1 Pairing critical simplexes and persistence: The algorithm pairs surviving critical simplexes that create and destroy the same cycles after the discrete gradient removes ε-persistent arcs.Critical simplexes are tagged positive or negative according to whether they create or destroy cycles, and persistence is expressed through the difference between their filtration values.
  • 6.2 Simplification: Simplification cancels persistence pairs below a chosen significance threshold, removing their associated critical simplexes, manifolds, and arcs.The choice of ascending or descending filtration order is arbitrary, while cancellations can be impossible when critical simplexes share no arc or more than one arc.
  • 6.3 Filtering Poisson noise: Monte Carlo estimates of persistence-pair probabilities support significance levels expressed in sigma units for filtering Poisson noise.The paper reports that an arc has approximately 5% Poisson-noise probability at 2-σ and approximately 0.006% at 4-σ.
  • 6.4 Illustration in 2D: At 4-σ, simplification reduces an intricate network around a dark-matter halo to neat filaments branching from a central clump while preserving resolution.The procedure also permits identification of the merger of two relatively noisy filaments.

7 BOUNDARY CONDITIONS AND TECHNICALITIES

DisPerSE handles finite observational boundaries through a hybrid compactification and simplifies discrete Morse structures with boundary-aware filtering and ordered persistence-pair cancellations. Its filament representation remains resolution-limited, while some special 3D configurations cannot be canceled rigorously.

  • Boundary conditions: Periodic simulations use periodic boundaries, whereas observational catalogues require special treatment because their domains are not periodic.Compactification transforms the definition domain into a boundaryless domain for applying Morse-theoretic methods.
  • Boundary conditions: The implementation mirrors a user-selected boundary fraction, applies sphere compactification with a minus-infinite point, and tags affected Delaunay simplices as boundary.A mirrored fraction around 10–15% of the initial distribution size is reported to work well.
  • Boundary conditions: Boundary simplices require special handling because critical pairs involving them have spurious persistence ratios and the infinite vertex can form infinite-persistence pairs.The method prevents boundary simplices with potentially incorrect DTFE densities from affecting the resulting discrete Morse complex.
  • Technical implementation: Finite sampling limits the resolution of ascending and descending manifolds, making structure shapes arbitrary below the initial sampling resolution.The implemented features are subsets of the initial Delaunay or dual Voronoi tessellations.
  • Technical implementation: Filament geometry is represented by sequences of Delaunay-simplex centers, then smoothed iteratively with a naturally adaptive length over the simplicial complex.The same approach can be adapted to ascending and descending manifolds representing voids and walls.
  • Technical implementation: Cancellation order controls computational time and memory, so the implementation cancels pairs in an order minimizing the number of newly created arcs.The number of created arcs is defined as N = Nc − Nd using the critical-point connectivity counts.

8 CONCLUSION

DisPerSE provides a scale-free, parameter-free framework for coherently identifying 3D cosmic-web structures directly in discrete data. Its discrete-topological construction supports noise-aware structure selection, object-level relationships, and topology measurements without smoothing scales.

  • Conclusion: DisPerSE identifies voids, walls, filaments, and clusters coherently from discretely sampled density fields, including sparse simulations or galaxy catalogues.The method applies directly to Delaunay tessellations and is designed for individual structures and their relationships.
  • Conclusion: The method is scale free because it operates directly on the sample’s Delaunay tessellation, defining structures down to the sample’s resolution limit.A regular-grid implementation is also available.
  • Conclusion: Discrete Morse theory makes the formalism rigorous for astrophysical data sets, unlike approaches based on smooth Morse theory such as watershed methods.The resulting numerically identified structures inherit the well-studied Morse-theory formalism.
  • Conclusion: Voids, walls, filaments, and clusters correspond to volumes, surfaces, curves, and points, with individual filaments associated with saddle points.This coherent decomposition supports recovering relationships such as the clusters at a filament’s extremities.
  • Conclusion: Persistence theory incorporates sampling and Poisson noise by letting users specify a detection confidence level in number of sigmas.The corresponding simplification of the discrete Morse complex is controlled by that confidence level.
  • Conclusion: The topological foundation enables robust computation of Betti numbers and Euler characteristics despite shot noise and without defining a smoothing scale.This supports analysis of the cosmic web’s multi-scale nature.

APPENDIX A: APPLICABILITY OF MORSE THEORY TO PRACTICAL DATA-SETS

The appendix explains why watershed-based recovery of the Morse complex can misidentify filaments on sampled or noisy density fields. It motivates a discrete, mathematically consistent alternative by showing that sampled field lines and basin boundaries can generate spurious critical lines.

  • Watershed approach: Watershed methods reconstruct voids, walls, and filaments from basins and their boundaries, but produce only a pseudo Morse complex.Filaments are approximated by boundaries shared by three watershed basins.
  • Watershed approach: A multi-scale probabilistic watershed was implemented on Delaunay tessellations with DTFE vertex densities to assess these identification problems.The transform propagates probabilities through the natural neighborhood defined by dual Voronoi cells.
  • Identification failure: In sampled non-Morse fields, field lines can cross, creating spurious critical lines where basin patches intersect rather than where the true field lines end.The illustration distinguishes spurious red and yellow identifications from the actual blue critical line.
  • Identification failure: The tendency of void and peak patches to become interleaved follows from sampling itself, not merely from the selected sampling method.The appendix notes that this behavior may be especially relevant in cosmological dark-matter density fields.
  • Identification failure: Because sampled or noisy boundaries do not trace the cosmic network correctly, watershed methods cannot reliably count filaments attached to a halo or measure individual-filament properties.The appendix presents this as motivation for a more mathematically consistent approach.

APPENDIX B: SIMPLICIAL HOMOLOGY

Simplicial homology represents topological structure through chains, boundaries, cycles, and equivalence classes. Betti numbers quantify the independent topological features of a complex, including components and holes.

  • Chains and boundaries: A k-chain is a collection of k-dimensional simplexes in a simplicial complex, such as vertices, segments, facets, or tetrahedrons in 3D.Chains use simplex coefficients modulo 2 in the presented formulation.
  • Chains and boundaries: The boundary operator maps each k-chain to its (k−1)-dimensional boundary, retaining faces belonging to exactly one k-simplex.Shared faces cancel under addition modulo 2, and the boundary of a boundary is void.
  • Cycles and homology: Cycles are chains with no boundary, while boundaries are cycles generated as boundaries of higher-dimensional chains.Thus Bk is included in Zk.
  • Betti numbers: In the 2D filtration illustration, holes appear as white regions, while different contours distinguish cycles that may or may not bound sets of facets.The figure demonstrates how holes affect boundaries and cycle equivalence.
  • Cycles and homology: Two cycles are homologous when they differ by a bounding cycle, creating equivalence classes that form the homology groups Hk = Zk/Bk.Holes prevent some cycles from being related through boundaries and therefore produce distinct homology classes.
  • Betti numbers: The kth Betti number βk is the rank of the free part of Hk and gives the minimum number of homology classes needed to generate all k-cycles.Betti numbers quantify and enable comparison of a space’s topology.

APPENDIX C: PERSISTENCE AND BETTI NUMBERS IN A SIMPLICIAL COMPLEX

Persistence tracks how topological features appear and disappear as simplices enter an ordered filtration. Betti numbers record the changing numbers of components and holes during this process.

  • Filtrations: A filtration is an ordered sequence of growing sub-complexes, with simplex entry determined by values of a discrete function.Each Ki is a simplicial complex and corresponds to a discrete sub-level-set sequence.
  • Betti numbers through a filtration: β0 counts connected components, while β1 counts holes or independent non-homologous 1-cycles in each filtration stage.The figure labels each stage with its index and Betti numbers.
  • Betti numbers through a filtration: As simplices enter the filtration, vertices can create components, segments can merge components, and facets can create or fill holes.The example follows these changes across successive complexes.
  • Persistence pairs: Each k-simplex is labeled positive when it creates a k-cycle and negative when it destroys one during the filtration.This labeling pairs the simplices responsible for feature creation and destruction.
  • Implementation boundary: The appendix omits the procedure for deciding whether a newly entered simplex belongs to a cycle, referring readers to prior work and the implementation section.This is a presentation boundary rather than a stated limitation of the method.
  • Persistence pairs: Persistence is the index or value difference between paired creation and destruction events, representing a cycle’s lifetime in the filtration.The procedure extends to arbitrary dimensions, not only the illustrated 2D case.

TERMINOLOGY

The terminology defines the discrete geometric and topological objects used to describe cosmic-web structure. It connects simplicial complexes, gradient flows, Morse-Smale cells, and persistence concepts.

  • Discrete Morse structures: A discrete gradient pairs a simplex with one of its facets; unpaired simplexes are critical, and the lower-valued simplex is the tail.These pairs define discrete gradient arrows.
  • Simplicial structures: A filtration is a growing sequence of sub-complexes defined by thresholding a discrete function over the simplexes.It is the discrete counterpart of growing sub-level sets.
  • Cosmic-web structures: Ascending 1-manifolds can represent filaments and ascending 3-manifolds can represent voids in three-dimensional data.A peak patch is a descending 3-manifold, whereas a void patch is an ascending 3-manifold.
  • Cosmic-web structures: The Morse-Smale complex consists of intersections of ascending and descending manifolds, partitioning space into cells of uniform gradient flow.Its cells include arcs, quads, and in 3D crystals.
  • Persistence: Persistence measures the value difference within a persistence pair, while persistence ratio is preferred for strictly positive density fields.Both quantities characterize the lifetime or contrast of paired features.
  • Simplicial structures: A simplicial complex is a set of simplexes containing every face of each simplex, with simplex intersections also belonging to the complex.A k-simplex is the k-dimensional analogue of a triangle and has k + 1 vertices.
  • Discrete Morse structures: A V-path is the discrete counterpart of an integral line, while an arc is a 1-cell connecting critical points whose orders differ by one.Integral lines follow the gradient of a scalar function.
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