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

## TL;DR

This paper extends the topological analysis of Gaussian and non-Gaussian random fields by studying Betti numbers and persistence diagrams, revealing their sensitivity to non-Gaussian features in the context of cosmology.

## Contribution

It introduces a new formalism linking homology, Betti numbers, and persistence diagrams to the critical points of random fields, enhancing topological analysis methods.

## Key findings

- Betti numbers and persistence diagrams are sensitive to non-Gaussianities.
- Derived a fitting formula for the expectation of Betti numbers.
- Numerical demonstrations show topological measures detect non-Gaussian features.

## 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.