Statistical Topology and the Random Interstellar Medium

TL;DR

This paper uses topological data analysis to challenge the Gaussian random field assumption in modeling the interstellar medium, revealing significant local differences across regions.

Contribution

It introduces a topological approach to analyze interstellar medium data, providing a non-parametric method to detect distribution differences beyond traditional analyses.

Findings

Topological methods detect differences where conventional analyses do not.

Strong evidence against Gaussian models for the studied regions.

Local topological differences distinguish between interstellar medium regions.

Abstract

Current astrophysical models of the interstellar medium assume that small scale variation and noise can be modelled as Gaussian random fields or simple transformations thereof, such as lognormal. We use topological methods to investigate this assumption for three regions of the southern sky. We consider Gaussian random fields on two-dimensional lattices and investigate the expected distribution of topological structures quantified through Betti numbers. We demonstrate that there are circumstances where differences in topology can identify differences in distributions when conventional marginal or correlation analyses may not. We propose a non-parametric method for comparing two fields based on the counts of topological features and the geometry of the associated persistence diagrams. When we apply the methods to the astrophysical data, we find strong evidence against a Gaussian random field model for each of the three regions of the interstellar medium that we consider. Further, we show that there are topological differences at a local scale between these different regions.