Betti numbers of Gaussian fields

TL;DR

This paper explores the relationship between Betti numbers and the genus in Gaussian fields, revealing how Betti numbers provide detailed topological insights into cosmic structures and vary with the power spectrum.

Contribution

It introduces the use of Betti numbers to analyze the topology of Gaussian fields, showing their dependence on the power spectrum and their potential for cosmological applications.

Findings

Betti numbers correspond to specific topological features at different thresholds.

Betti number curves depend on the power spectrum slope, unlike the genus.

Betti numbers offer a more detailed topological characterization of large-scale structures.

Abstract

We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; $\beta_0$ is the number of connected regions, $\beta_1$ is the number of circular holes, and $\beta_2$ is the number of three-dimensional voids. Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. $\beta_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). $\beta_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and $\beta_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum $n$ in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as $n$ decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.