Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals

Abstract

This study presents a numerical analysis of the topology of a set of cosmologically interesting three-dimensional Gaussian random fields in terms of their Betti numbers ${\beta }_0$, ${\beta }_1$ and ${\beta }_2$. We show that Betti numbers entail a considerably richer characterization of the topology of the primordial density field. Of particular interest is that Betti numbers specify which topological features - islands, cavities or tunnels - define its spatial structure. A principal characteristic of Gaussian fields is that the three Betti numbers dominate the topology at different density ranges. At extreme density levels, the topology is dominated by a single class of features. At low levels this is a \emph{Swiss-cheeselike} topology, dominated by isolated cavities, at high levels a predominantly \emph{Meatball-like} topology of isolated objects. At moderate density levels, two Betti number define a more \emph{Sponge-like} topology. At mean density, the topology even needs three Betti numbers, quantifying a field consisting of several disconnected complexes, not of one connected and percolating overdensity. A {\it second} important aspect of Betti number statistics is that they are sensitive to the power spectrum. It reveals a monotonic trend in which at a moderate density range a lower spectral index corresponds to a considerably higher (relative) population of cavities and islands. We also assess the level of complementary information that Betti numbers represent, in addition to conventional measures such as Minkowski functionals. To this end, we include an extensive description of the Gaussian Kinematic Formula (GKF), which represents a major theoretical underpinning for this discussion.

arXiv comment44 pages, 27 Figures, 2 Tables, accepted for publication in MNRAS in the original version. This version updates references and fixes typos

References in corpus (55)

• Planck 2015 results. XIII. Cosmological parameters
• Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data
• Non-Gaussianity from Inflation: Theory and Observations
• Power Spectra for Cold Dark Matter and its Variants
• The persistent cosmic web and its filamentary structure I: Theory and implementation
• Primordial Non-Gaussianities from Inflation Models
• Planck 2015 results. XVII. Constraints on primordial non-Gaussianity
• A hierarchy of voids: Much ado about nothing
• TASI Lectures on Inflation
• The persistent cosmic web and its filamentary structure II: Illustrations
• Beyond genus statistics: a unifying approach to the morphology of cosmic structure
• Constraining Peaks in Gaussian Primordial Density Fields: an Application of the Hoffman-Ribak Method
• The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
• Disentangling the Cosmic Web I: Morphology of Isodensity Contours
• Minkowski Functionals of SDSS galaxies I : Analysis of Excursion Sets
• Morphology of the supercluster-void network in LCDM cosmology
• The Structure of Structure Formation Theories
• On the connectivity of the cosmic web: theory and implications for cosmology and galaxy formation
• Analytic Minkowski Functionals of the Cosmic Microwave Background: Second-order Non-Gaussianity with Bispectrum and Trispectrum
• Morphological fluctuations of large-scale structure: the PSCz survey
• Persistent Homology for Random Fields and Complexes
• Felix: A Topology based Framework for Visual Exploration of Cosmic Filaments
• Measuring the Geometry and Topology of Large Scale Structure using SURFGEN: Methodology and Preliminary Results
• The Topology of the IRAS Point Source Catalogue Redshift Survey
• Non-Gaussian Minkowski functionals & extrema counts in redshift space
• Cosmic Web Reconstruction through Density Ridges: Method and Algorithm
• Topology of structure in the Sloan Digital Sky Survey: model testing
• Betti numbers of Gaussian fields
• Large-Scale Structure of the Universe as a Cosmic Standard Ruler
• The DTFE public software: The Delaunay Tessellation Field Estimator code
• A Gaussian kinematic formula
• The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D
• Galaxy Clustering Topology in the Sloan Digital Sky Survey Main Galaxy Sample: a Test for Galaxy Formation Models
• Direct Minkowski Functional analysis of large redshift surveys: a new high--speed code tested on the luminous red galaxy Sloan Digital Sky Survey-DR7 catalogue
• Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski Functionals
• Morphometry of Spatial Patterns
• Persistent Homology and Non-Gaussianity
• Modeling and replicating statistical topology, and evidence for CMB non-homogeneity
• A global descriptor of spatial pattern interaction in the galaxy distribution
• Euler Integration of Gaussian Random Fields and Persistent Homology
• Persistent topology of the reionisation bubble network. I: Formalism & Phenomenology
• Search for anomalous alignments of structures in Planck data using Minkowski Tensors
• Tensor Minkowski Functionals: first application to the CMB
• Filaments and Pancakes in the IRAS 1.2 Jy Redshift Catalogue
• Betti number signatures of homogeneous Poisson point processes
• Tensor Minkowski Functionals for random fields on the sphere
• Hot and cold spots counts as probes of non-Gaussianity in the CMB
• Minkowski Tensors in Three Dimensions - Probing the Anisotropy Generated by Redshift Space Distortion
• Topological signatures of interstellar magnetic fields - I. Betti numbers and persistence diagrams
• Estimating thresholding levels for random fields via Euler characteristics
• Rotation and scale space random fields and the Gaussian kinematic formula
• Topological data analysis and diagnostics of compressible MHD turbulence

Cited by in corpus (40)

• Networks beyond pairwise interactions: structure and dynamics
• Persistent homology in cosmic shear: constraining parameters with topological data analysis
• 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
• Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background
• Neural Network-based Top Tagger with Two-Point Energy Correlations and Geometry of Soft Emissions
• Morphology of 21cm brightness temperature during the Epoch of Reioinization using Contour Minkowski Tensor
• Persistent topology of the reionisation bubble network. I: Formalism & Phenomenology
• Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?
• Probing massive neutrinos with the Minkowski functionals of large-scale structure
• Higher order statistics of shear field: a machine learning approach
• The HI-halo mass relation at redshift $z\sim 1$ from the Minkowski functionals of 21 cm intensity maps
• Clustering of Local Extrema in Planck CMB maps
• Persistent topology of the reionization bubble network. II: Evolution & Classification
• Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Expansion History using the Genus Amplitude as a Standard Ruler
• Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Shape of the Matter Power Spectrum
• Morphology of CMB fields -- effect of weak gravitational lensing
• Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation
• Morphology of Weak Lensing Convergence Maps
• Imprint of massive neutrinos on Persistent Homology of large-scale structure
• When Gas Dynamics Decouples from Galactic Rotation: Characterizing ISM Circulation in Disk Galaxies
• Differentiating small-scale subhalo distributions in CDM and WDM models using persistent homology
• Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
• Topological data analysis reveals differences between simulated galaxies and dark matter haloes
• Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the $\text{NPIPE}$ and $\text{FFP10}$ data releases
• Detecting structured sources in noisy images via Minkowski maps
• Probing cosmology via the clustering of critical points
• Large-scale geometry and topology of gas fields: Effects of AGN and stellar feedback
• Galaxy Superclusters and Their Complexes in the Cosmic Web
• Improving SDSS cosmological constraints through $\beta$-skeleton weighted correlation functions
• Homology reveals significant anisotropy in the cosmic microwave background
• Cosmic Strings-induced CMB anisotropies in light of Weighted Morphology
• Impact of Redshift Space Distortion on Persistent Homology of cosmic matter density field
• Testing General Relativity with Cosmological Large Scale Structure
• Detecting and analysing the topology of the cosmic web with spatial clustering algorithms I: Methods
• Non-Gaussian Magnetic Structures in the Small-Scale Turbulent Dynamo
• Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields
• Cosmological constraints from the density gradient weighted correlation function