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

## TL;DR

This paper analyzes the topology of three-dimensional Gaussian random fields using Betti numbers, revealing their dependence on density levels and power spectrum, and comparing their information content to Minkowski functionals.

## Contribution

It introduces Betti numbers as a richer topological descriptor for Gaussian fields, detailing their density-dependent behavior and spectral sensitivity, with theoretical insights from the Gaussian Kinematic Formula.

## Key findings

- Betti numbers characterize different topological features at various density levels.
- Topology varies from cavities to isolated objects, depending on density.
- Betti numbers are sensitive to the power spectrum, affecting the topology.

## 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.