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

Understand

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.