Cosmic Web Geometry

• Cosmic web geometry is the study of the universe’s large-scale structure, defined by voids, walls, filaments, and nodes formed via gravitational collapse.
• It employs advanced computational and topological methods, such as Hessian analysis and skeleton extraction, to classify and quantify complex morphological features.
• Quantitative measures like mass fractions, filament widths, and connectivity provide actionable insights into the evolution and dynamics of cosmic structures.

The cosmic web is the multiscale network of voids, walls (sheets), filaments, and dense nodes that constitutes the large-scale structure of the universe. Its geometry arises from gravitational instability amplifying small primordial Gaussian density fluctuations, resulting in a highly anisotropic, non-uniform distribution of baryons and dark matter. The geometric classification and quantification of this cosmic web has become central to cosmology, connecting nonlinear structure formation theory, advanced computational geometry, and topological analysis, and underpinning observations and simulations of the galaxy distribution and large-scale flows.

1. Geometric Elements and Mathematical Definitions

The cosmic web is built from four archetypal morphological elements with sharply distinct geometries and topologies:

• Voids: Maximal single-stream regions; convex or nearly spherical underdense domains tessellating space. In geometric models, voids are large convex polyhedra meeting at flat walls, and can be quantitatively described as connected regions where all principal deformation tensor (or tidal/velocity shear Hessian) eigenvalues are below a collapse threshold (Cautun et al., 2015, Hidding et al., 2013, Aragon-Calvo, 2023).
• Walls (Sheets/Pancakes): Planar or sheet-like boundaries between voids, generated by the local collapse along one axis. Geometrically, they are two-stream regions bounded by pairs of parallel caustic (crease) planes, or, equivalently, flat facets in a tessellation (Neyrinck, 2014, Neyrinck, 2014, Hidding et al., 2012).
• Filaments: 1D multi-stream ridges, usually interpreted as elongated thread-like structures at the intersection of walls, channeling matter toward dense knots. Filaments have polygonal (2D) or polyhedral (3D) cross sections and are characterized by their connectivity, linear mass density, and width (Cautun et al., 2015, Cautun et al., 2014).
• Nodes (Clusters/Knots): Compact, high-density regions where three or more filaments intersect, often corresponding to galaxy clusters or massive dark matter haloes. Nodes are defined by anisotropic collapse along all three axes, manifesting topologically as tetrahedra or higher-order polyhedra at the junctions of the network (Hidding et al., 2013, Neyrinck, 2014).

Mathematically, these elements can be demarcated by the eigenvalues of the (density, potential, or velocity) Hessian tensor. For a threshold $\lambda_{\mathrm{th}}$ and ordered eigenvalues $\lambda_1\ge\lambda_2\ge\lambda_3$:

Element	Collapse Directions	Eigenvalue Condition
Void	0	$\lambda_1<\lambda_{\mathrm{th}}$
Wall	1	$\lambda_1\ge\lambda_{\mathrm{th}}$, $\lambda_2<\lambda_{\mathrm{th}}$
Filament	2	$\lambda_1\ge\lambda_{\mathrm{th}}$, $\lambda_2\ge\lambda_{\mathrm{th}}$, $\lambda_3<\lambda_{\mathrm{th}}$
Node	3	$\lambda_3\ge\lambda_{\mathrm{th}}$

(Cautun et al., 2015, Cautun et al., 2014, Aragon-Calvo, 2023)

2. Morphological Classification Frameworks

Several methodologies have been developed to assign environmental labels (void, wall, filament, node) to points in observed or simulated cosmic structures:

• Hessian/Scale-space Classifiers: Methods such as MMF, NEXUS+, and related multiscale Hessian filters use the sign and magnitude patterns of Hessian eigenvalues of the (smoothed) density, potential, or velocity field, scanning across a range of scales and selecting the strongest morphological signal at each point (Cautun et al., 2014, Cautun et al., 2015, 0912.3448, Libeskind et al., 2017).
• Skeleton/Ridge Extraction: Topological methods like the persistent skeleton (as in DisPerSE), identify the ridge set (network of maximum-persistence topological connections) of the density scalar field, defining filaments by discrete Morse–Smale complexes and connecting critical points (minima, saddles, maxima) (Codis et al., 2018, Libeskind et al., 2017).
• Watershed and SpineWeb: Void-centred approaches segment the density landscape into basins (voids) using watershed transforms; filaments arise as separatrices (intersections of void boundaries), and nodes as points where multiple filaments meet (0912.3448).
• Phase-space Sheet and Multistream Analysis: Methods such as ORIGAMI and multistream field analysis classify regions by the number of phase-space streams at each location, revealing the complex folding of the dark matter sheet and yielding a highly parameter-free geometric segmentation; e.g., single-stream (void), two-stream (wall), three-stream (filament), four-stream or higher (node) (Ramachandra et al., 2016, Libeskind et al., 2017).
• Alpha-shape Topology and Homology: Alpha shapes, parameterized by a scale $\alpha$, probe multiscale topological features by tracking the numbers of connected components $\lambda_1\ge\lambda_2\ge\lambda_3$0, tunnels $\lambda_1\ge\lambda_2\ge\lambda_3$1, and cavities $\lambda_1\ge\lambda_2\ge\lambda_3$2 in the point set or density field, with Betti number curves as robust invariants distinguishing morphological types (Weygaert et al., 2010, 0912.3448).
• Fractal and Multifractal Analysis: The cosmic web exhibits a non-lacunary multifractal structure, with the singularity spectrum $\lambda_1\ge\lambda_2\ge\lambda_3$3 encoding the hierarchy of mass concentrations and voids. This framework unifies web geometry, hierarchical clustering, and halo structure (Gaite, 2018).

3. Quantitative Characterization of Geometry

The quantification of cosmic web geometry proceeds by direct measurement of morphological element properties in data and simulations.

• Mass and Volume Fractions: At $\lambda_1\ge\lambda_2\ge\lambda_3$4, filaments typically hold $\lambda_1\ge\lambda_2\ge\lambda_3$550% of the mass in $\lambda_1\ge\lambda_2\ge\lambda_3$66–10% of the volume, walls hold $\lambda_1\ge\lambda_2\ge\lambda_3$720–25% mass over $\lambda_1\ge\lambda_2\ge\lambda_3$818–25% volume, voids occupy $\lambda_1\ge\lambda_2\ge\lambda_3$977–80% of the volume at $\lambda_1<\lambda_{\mathrm{th}}$015% of the mass, and nodes contain the remainder in negligible volume (Cautun et al., 2015, Cautun et al., 2014, Libeskind et al., 2017).
• Filament Linear Mass Density: The peak of the filament linear mass density distribution shifts from $\lambda_1<\lambda_{\mathrm{th}}$1Mpc at $\lambda_1<\lambda_{\mathrm{th}}$2 to $\lambda_1<\lambda_{\mathrm{th}}$3Mpc at $\lambda_1<\lambda_{\mathrm{th}}$4, reflecting hierarchical merging into thicker, more massive filaments (Cautun et al., 2015).
• Filament and Wall Width/Thickness: Filament widths typically range from 0.5 to 5 $\lambda_1<\lambda_{\mathrm{th}}$5 Mpc (80% $\lambda_1<\lambda_{\mathrm{th}}$6 5 $\lambda_1<\lambda_{\mathrm{th}}$7 Mpc); wall thicknesses from 0.5 to 8 $\lambda_1<\lambda_{\mathrm{th}}$8 Mpc (Cautun et al., 2014). Surface densities for walls decrease from $\lambda_1<\lambda_{\mathrm{th}}$9/Mpc$\lambda_1\ge\lambda_{\mathrm{th}}$0 at $\lambda_1\ge\lambda_{\mathrm{th}}$1 to $\lambda_1\ge\lambda_{\mathrm{th}}$2/Mpc$\lambda_1\ge\lambda_{\mathrm{th}}$3 at $\lambda_1\ge\lambda_{\mathrm{th}}$4 (Cautun et al., 2015).
• Voids: Voids exhibit a universal shell-crossing profile when density is plotted versus boundary distance (not radius), rising sharply at the boundary, with median void size doubling from $\lambda_1\ge\lambda_{\mathrm{th}}$5 to $\lambda_1\ge\lambda_{\mathrm{th}}$6. The void effective radius distribution follows $\lambda_1\ge\lambda_{\mathrm{th}}$7 at $\lambda_1\ge\lambda_{\mathrm{th}}$8 (Cautun et al., 2015).
• Connectivity: Typical nodes are globally connected to $\lambda_1\ge\lambda_{\mathrm{th}}$9 filaments in 3D Gaussian random field cosmologies. The local filament multiplicity emerging from a peak is $\lambda_2<\lambda_{\mathrm{th}}$0 on average in 3D, depending on the height of the node (mass) (Codis et al., 2018).
• Local Dimension: The local dimension $\lambda_2<\lambda_{\mathrm{th}}$1, extracted from the scaling $\lambda_2<\lambda_{\mathrm{th}}$2, provides a scale-dependent classifier: $\lambda_2<\lambda_{\mathrm{th}}$3 (filament), $\lambda_2<\lambda_{\mathrm{th}}$4 (sheet), $\lambda_2<\lambda_{\mathrm{th}}$5 (cluster), with SDSS and Millennium Simulation analyses finding filaments dominant at $\lambda_2<\lambda_{\mathrm{th}}$6–$\lambda_2<\lambda_{\mathrm{th}}$7 Mpc, sheets at $\lambda_2<\lambda_{\mathrm{th}}$8–$\lambda_2<\lambda_{\mathrm{th}}$9 Mpc, and clusters at $\lambda_1\ge\lambda_{\mathrm{th}}$0–$\lambda_1\ge\lambda_{\mathrm{th}}$1 Mpc (Sarkar et al., 2011).

4. Topological and Multiscale Measures

Advanced topological analyses provide scale-resolved fingerprints of the web’s connectivity and complexity.

• Alpha Shapes and Betti Numbers: Alpha shape analysis produces scale-dependent curves $\lambda_1\ge\lambda_{\mathrm{th}}$2 distinguishing connectivity ($\lambda_1\ge\lambda_{\mathrm{th}}$3), tunnels ($\lambda_1\ge\lambda_{\mathrm{th}}$4), and cavities ($\lambda_1\ge\lambda_{\mathrm{th}}$5) for point-set or galaxy data. Wall-dominated, filament-dominated, and cluster-dominated morphologies exhibit distinct signatures in the order, width, and amplitude of Betti peaks as $\lambda_1\ge\lambda_{\mathrm{th}}$6 is varied (Weygaert et al., 2010).
• Minkowski Functionals: The four 3D Minkowski functionals—volume ($\lambda_1\ge\lambda_{\mathrm{th}}$7), surface area ($\lambda_1\ge\lambda_{\mathrm{th}}$8), integrated mean curvature ($\lambda_1\ge\lambda_{\mathrm{th}}$9), and Euler characteristic ($\lambda_2\ge\lambda_{\mathrm{th}}$0)—quantify the global shape of isodensity surfaces derived from Delaunay tessellation or marching tetrahedra. Filaments present high filamentarity $\lambda_2\ge\lambda_{\mathrm{th}}$1 and low planarity $\lambda_2\ge\lambda_{\mathrm{th}}$2, while voids maximize $\lambda_2\ge\lambda_{\mathrm{th}}$3 and exhibit positive $\lambda_2\ge\lambda_{\mathrm{th}}$4 (Aragon-Calvo et al., 2010).
• Percolation and Multistream Geometry: Excursion set percolation applied to the multistream field reveals that the percolating filament network at the percolation threshold ($\lambda_2\ge\lambda_{\mathrm{th}}$5) is a factor $\lambda_2\ge\lambda_{\mathrm{th}}$6 thinner (by volume fraction) than filaments defined by density thresholds, and that a single void percolates nearly all ($\lambda_2\ge\lambda_{\mathrm{th}}$793%) of the single-stream volume (Ramachandra et al., 2016).
• Fractal Spectrum: The measured $\lambda_2\ge\lambda_{\mathrm{th}}$8 spectrum from simulations exhibits $\lambda_2\ge\lambda_{\mathrm{th}}$9, $\lambda_3<\lambda_{\mathrm{th}}$0 (mass concentrate), $\lambda_3<\lambda_{\mathrm{th}}$1 (full support for underdense regions), and $\lambda_3<\lambda_{\mathrm{th}}$2–$\lambda_3<\lambda_{\mathrm{th}}$3 (weakest voids), indicating a non-lacunary multifractal with all voids containing nested structure (Gaite, 2018).

5. Dynamical and Geometric Models

The formation and geometry of the cosmic web can be modeled in several frameworks with distinctive geometrical implications:

• Zel’dovich and Adhesion Approximations: In the Zel’dovich approximation, the cosmic web arises from anisotropic, gradient-driven displacements, with caustics emerging where the map becomes singular. The adhesion model adds viscosity, forming shocks that become the walls and filaments of the geometric web. In this model, the cosmic web is a weighted Voronoi tessellation (Eulerian) dual to a weighted Delaunay triangulation (Lagrangian), with polyhedral cells, faces, and edges corresponding to voids, walls, filaments, and nodes, respectively (Hidding et al., 2012, Hidding et al., 2012).
• Origami Geometry: The origami approximation restricts the dark-matter sheet to piecewise-isometric folds without local stretching. The resulting web features convex polyhedral voids tessellated by caustic planes (creases), straight walls and filaments, and polyhedral nodes (e.g., tetrahedral collapse), with node spins and filament connections constrained by rigid face-rotations and Euler characteristic closure (Neyrinck, 2014, Neyrinck, 2014).
• Catastrophe Theory and Caustic Skeletons: Catastrophe theory classifies the singularities of the Lagrangian–Eulerian map into folds (A₂), cusps (A₃, marking proto-filament formation), swallowtails (A₄, mediating filament mergers), and umbilics (D₄, filament nodes). The full cosmic skeleton arises as a percolating network of these singularities, whose topology and geometry define the web’s backbone (Hidding et al., 2013, Feldbrugge et al., 2022).
• Hierarchical Spine Reconstruction: The H-Spine method builds a three-level hierarchy of voids, walls, and filaments using Hessian eigen_analysis, watershed segmentation, and explicit adjacency matrices, capturing both connectivity and nesting properties at every scale (Aragon-Calvo, 2023).

6. Scale Dependence, Environmental Trends, and Observables

The web’s geometry evolves with scale, redshift, and environment, influencing and reflecting the dynamics of galaxy assembly and dark matter accretion.

• Scale Dependence: The dominant morphological element depends on the spatial scale; at small scales ($\lambda_3<\lambda_{\mathrm{th}}$45 $\lambda_3<\lambda_{\mathrm{th}}$5 Mpc), filaments and sheets dominate; at larger scales ($\lambda_3<\lambda_{\mathrm{th}}$610 $\lambda_3<\lambda_{\mathrm{th}}$7 Mpc), sheets and clusters become dominant, and filaments vanish above $\lambda_3<\lambda_{\mathrm{th}}$850 $\lambda_3<\lambda_{\mathrm{th}}$9 Mpc (Sarkar et al., 2011).
• Environmental Preferences: Filaments preferentially reside in low-to-moderate density regions, sheets at intermediate to high densities, clusters at high-density peaks. The cross-over density threshold between morphologies decreases as one moves to larger scales due to the growing prominence of supervoids and superclusters (Sarkar et al., 2011).
• Connectivity–Mass Scaling: The number of filaments connecting to a node scales approximately as $\lambda_3\ge\lambda_{\mathrm{th}}$0; high-mass clusters sit at highly connected nodes whereas low-mass haloes are fed by few filaments (Codis et al., 2018).
• Spin and Angular Momentum Alignments: In origami and caustic-skeleton frameworks, the closure of face-rotation angles in polyhedral collapse constrains filament angular momenta, leading to explicit predictions for spin–spin correlation functions between nodes connected by filaments (Neyrinck, 2014).
• Observational Proxies: Explicit identification of caustic surfaces, spin alignments, and convexity of voids in N-body simulations allows tests of geometric predictions. Filament and wall properties (linear/surface mass density, width, and connectivity) are measurable in redshift and lensing surveys using appropriate geometric classifiers (Cautun et al., 2015, Pomarede et al., 2017, Zhang et al., 2022).

7. Methodological Comparisons and Unified Picture

The diversity of geometric and topological methodologies converges on key quantitative and structural agreements:

• Independent classification approaches (multiscale Hessian, phase-space multistreaming, Morse–Smale skeletons, stochastic point-process) consistently identify the same overall hierarchy of cosmic web features, with knots in high-density nodes, filaments as interconnecting bridges, sheets as planar divides, and voids as residual, underdense interiors (Libeskind et al., 2017).
• Quantitative overlaps exist in the volume and density ranges assigned to each environment; all methods find that knots occupy <1% of volume but contain $\lambda_3\ge\lambda_{\mathrm{th}}$1 mass, filaments up to $\lambda_3\ge\lambda_{\mathrm{th}}$2 volume and $\lambda_3\ge\lambda_{\mathrm{th}}$3–$\lambda_3\ge\lambda_{\mathrm{th}}$4 mass, sheets $\lambda_3\ge\lambda_{\mathrm{th}}$5–$\lambda_3\ge\lambda_{\mathrm{th}}$6 volume, and voids $\lambda_3\ge\lambda_{\mathrm{th}}$7–$\lambda_3\ge\lambda_{\mathrm{th}}$8 volume but only $\lambda_3\ge\lambda_{\mathrm{th}}$9–$\alpha$0 mass (Libeskind et al., 2017).
• Recent frameworks such as SCONCE provide novel means of filament extraction in spherical and conic geometries relevant to realistic survey data, ensuring geometric invariance and robustness against projection and redshift-space distortions (Zhang et al., 2022).

Across methodologies, the cosmic web emerges as a maximally anisotropic, hierarchically nested, multiscale and multifractal arrangement of convex voids, flat or gently curved walls, spatially extended filaments, and highly connected nodes, with sharp topological and geometric invariants available for direct computation in both simulations and observations.