Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs towards Betti Numbers and Persistence Diagrams

Abstract

The topology and geometry of random fields - in terms of the Euler characteristic and the Minkowski functionals - has received a lot of attention in the context of the Cosmic Microwave Background (CMB), as the detection of primordial non-Gaussianities would form a valuable clue on the physics of the early Universe. The virtue of both the Euler characteristic and the Minkowski functionals in general, lies in the fact that there exist closed form expressions for their expectation values for Gaussian random fields. However, the Euler characteristic and Minkowski functionals are summarizing characteristics of topology and geometry. Considerably more topological information is contained in the homology of the random field, as it completely describes the creation, merging and disappearance of topological features in superlevel set filtrations. In the present study we extend the topological analysis of the superlevel set filtrations of two-dimensional Gaussian random fields by analysing the statistical properties of the Betti numbers - counting the number of connected components and loops - and the persistence diagrams - describing the creation and mergers of homological features. Using the link between homology and the critical points of a function - as illustrated by the Morse-Smale complex - we derive a one-parameter fitting formula for the expectation value of the Betti numbers and forward this formalism to the persistent diagrams. We, moreover, numerically demonstrate the sensitivity of the Betti numbers and persistence diagrams to the presence of non-Gaussianities.

References in corpus (11)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Observational Signatures and Non-Gaussianities of General Single Field Inflation
• Primordial Non-Gaussianities from Inflation Models
• The three dimensional skeleton: tracing the filamentary structure of the universe
• Limits on Primordial Non-Gaussianity from Minkowski Functionals of the WMAP Temperature Anisotropies
• Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background
• Betti number signatures of homogeneous Poisson point processes
• Primordial non-Gaussian signatures in CMB polarization
• Topological signatures of interstellar magnetic fields - I. Betti numbers and persistence diagrams
• Statistical imprints of CMB B-type polarization leakage in an incomplete sky survey analysis
• Non Gaussian Minkowski functionals and extrema counts for 2D sky maps

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