Statistical Topology and the Random Interstellar Medium

Abstract We use topological methods to investigate the small-scale variation and local spatial characteristics of the interstellar medium (ISM) in three regions of the southern sky. We demonstrate that there are circumstances where topological methods can identify differences in distributions when conventional marginal or correlation analyses may not. We propose a nonparametric method for comparing two fields based on the counts of topological features and the geometry of the associated persistence diagrams. We investigate the expected distribution of topological structures quantified through Betti numbers under Gaussian random field (GRF) assumptions, which underlie many astrophysical models of the ISM. When we apply the methods to the astrophysical data, we find strong evidence that one of the three regions is both topologically dissimilar to the other two and not consistent with an underlying GRF model. This region is proximal to a region of recent star formation whereas the others are more distant. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.