  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.
