The Topology of the Cosmic Web in Terms of Persistent Betti Numbers

Abstract

We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence offer a powerful means of describing the rich connectivity structure of the cosmic web and of its multiscale arrangement of matter and galaxies. Emanating from algebraic topology and Morse theory, Betti numbers and persistence diagrams represent an extension and deepening of the cosmologically familiar topological genus measure, and the related geometric Minkowski functionals. In addition to a description of the mathematical background, this study presents the computational procedure for computing Betti numbers and persistence diagrams for density field filtrations. The field may be computed starting from a discrete spatial distribution of galaxies or simulation particles. The main emphasis of this study concerns an extensive and systematic exploration of the imprint of different weblike morphologies and different levels of multiscale clustering in the corresponding computed Betti numbers and persistence diagrams. To this end, we use Voronoi clustering models as templates for a rich variety of weblike configurations, and the fractal-like Soneira-Peebles models exemplify a range of multiscale configurations. We have identified the clear imprint of cluster nodes, filaments, walls, and voids in persistence diagrams, along with that of the nested hierarchy of structures in multiscale point distributions. We conclude by outlining the potential of persistent topology for understanding the connectivity structure of the cosmic web, in large simulations of cosmic structure formation and in the challenging context of the observed galaxy distribution in large galaxy surveys.

31 pages, 24 figures, Published in MNRAS in original version. This version updates the reference list

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Cited by in corpus (65)

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• Finding cosmic voids and filament loops using topological data analysis
• Persistent homology in cosmic shear: constraining parameters with topological data analysis
• The Persistence of Large Scale Structures I: Primordial non-Gaussianity
• Measuring the topology of reionization with Betti numbers
• Homological Percolation and the Euler Characteristic
• Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs towards Betti Numbers and Persistence Diagrams
• Persistent homology of the cosmic web. I: Hierarchical topology in $\Lambda$CDM cosmologies
• Quantum algorithm for persistent Betti numbers and topological data analysis
• Persistent Homology and Non-Gaussianity
• Modeling and replicating statistical topology, and evidence for CMB non-homogeneity
• Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background
• Persistent homology in cosmic shear II: A tomographic analysis of DES-Y1
• Persistent topology of the reionisation bubble network. I: Formalism & Phenomenology
• Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?
• Constraining Cosmology with Big Data Statistics of Cosmological Graphs
• When do cosmic peaks, filaments or walls merge? A theory of critical events in a multi-scale landscape
• Persistent homology of unweighted complex networks via discrete Morse theory
• Finding self-similar behavior in quantum many-body dynamics via persistent homology
• Higher order statistics of shear field: a machine learning approach
• Beta-Skeleton Analysis of the Cosmic Web
• Hierarchical structure of the cosmic web and galaxy properties
• Topological signatures of interstellar magnetic fields - I. Betti numbers and persistence diagrams
• Large-scale structures in the $\Lambda$CDM Universe: network analysis and machine learning
• Persistent topology of the reionization bubble network. II: Evolution & Classification
• Replacing dark energy by silent virialisation
• Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Expansion History using the Genus Amplitude as a Standard Ruler
• The Tessellation-Level-Tree: characterising the nested hierarchy of density peaks and their spatial distribution in cosmological N-body simulations
• Persistent Homology of Fractional Gaussian Noise
• Topological data analysis and diagnostics of compressible MHD turbulence
• Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation
• Confinement in non-Abelian lattice gauge theory via persistent homology
• Applications of Persistent Homology in Nuclear Collisions
• Network analysis of the COSMOS galaxy field
• Statistics of Two-point Correlation and Network Topology for Lyman Alpha Emitters at $z\approx 2.67$
• Imprint of massive neutrinos on Persistent Homology of large-scale structure
• Differentiating small-scale subhalo distributions in CDM and WDM models using persistent homology
• Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
• Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the $\text{NPIPE}$ and $\text{FFP10}$ data releases
• How complex is the cosmic web?
• Topological data analysis reveals differences between simulated galaxies and dark matter haloes
• The topology of data hides in quantum thermal states
• Forman-Ricci curvature and Persistent homology of unweighted complex networks
• Probing cosmology via the clustering of critical points
• Quantum topological data analysis via the estimation of the density of states
• Geometric Interpretations of the $k$-Nearest Neighbour Distributions
• Topological Echoes of Primordial Physics in the Universe at Large Scales
• Topological Data Analysis for the String Landscape
• Study on Base Station Topology in Cellular Networks: Take Advantage of Alpha Shapes, Betti Numbers, and Euler Characteristics
• The Complexity and Information Content of Simulated Universes
• Homology reveals significant anisotropy in the cosmic microwave background
• On the evolution of Betti curves in the Cosmic web
• Impact of Redshift Space Distortion on Persistent Homology of cosmic matter density field
• Cosmology with Persistent Homology: Parameter Inference via Machine Learning
• A universal and physically motivated threshold for Hessian-based cosmic web identification: V-Web case
• Nonparametric Estimation of Surface Integrals on Level Sets
• Detecting and analysing the topology of the cosmic web with spatial clustering algorithms I: Methods
• Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
• More than a void? The detection and characterization of cavities in a simulated galaxy's interstellar medium
• Quantifying Weighted Morphological Content of Large-Scale Structures via Simulation-Based Inference
• Holey graphs: very large Betti numbers are testable
• Extremal life times of persistent loops and holes
• Fundamentals on Base Stations in Cellular Networks: From the Perspective of Algebraic Topology