Topology and geometry of Gaussian random fields I: On Betti numbers, Euler characteristic, and Minkowski functionals

This study presents a numerical analysis of the topology of a set of cosmologically interesting 3D Gaussian random fields in terms of their Betti numbers β₀, β₁, and β₂. We show that Betti numbers entail a considerably richer characterization of the topology of the primordial density field. Of particular interest is that the Betti numbers specify which topological features - islands, cavities, or tunnels - define the spatial structure of the field. A principal characteristic of Gaussian fields is that the three Betti numbers dominate the topology at different density ranges. At extreme density levels, the topology is dominated by a single class of features. At low levels this is a Swiss-cheeselike topology dominated by isolated cavities, and, at high levels, a predominantly Meatball-like topology composed of isolated objects. At moderate density levels, two Betti numbers define a more Sponge-like topology. At mean density, the description of topology even needs three Betti numbers, quantifying a field consisting of several disconnected complexes, not of one connected and percolating overdensity. A secondimportant aspect of Betti number statistics is that they are sensitive to the power spectrum. They reveal a monotonic trend, in which at a moderate density range, a lower spectral index corresponds to a considerably higher (relative) population of cavities and islands. We also assess the level of complementary information that the Betti numbers represent, in addition to conventional measures such as Minkowski functionals. To this end, we include an extensive description of the Gaussian Kinematic Formula, which represents a major theoretical underpinning for this discussion.