Finding cosmic voids and filament loops using topological data analysis

We present a method called Significant Cosmic Holes in Universe (SCHU) for identifying cosmic voids and loops of filaments in cosmological datasets and assigning their statistical significance using techniques from topological data analysis. In particular, persistent homology is used to find different dimensional holes. For dark matter halo catalogs and galaxy surveys, the 0-, 1-, and 2-dimensional holes can be identified with connected components (i.e. clusters), loops of filaments, and voids. The procedure overlays dark matter halos/galaxies on a three-dimensional grid, and a distance-to-measure (DTM) function is calculated at each point of the grid. A nested set of simplicial complexes (a filtration) is generated over the lower-level sets of the DTM across increasing threshold values. The filtered simplicial complex can then be used to summarize the birth and death times of the different dimension homology group generators (i.e., the holes). Persistent homology summary diagrams, called persistence diagrams, are produced from the dimension, birth times, and death times of each homology group generator. Using the persistence diagrams and bootstrap sampling, we explain how p-values can be assigned to each homology group generator. The homology group generators on a persistence diagram are not, in general, uniquely located back in the original dataset volume so we propose a method for finding a representation of the homology group generators. This method provides a novel, statistically rigorous approach for locating informative generators in cosmological datasets, which may be useful for providing complementary cosmological constraints on the effects of, for example, the sum of the neutrino masses. The method is tested on a Voronoi foam simulation, and then subsequently applied to a subset of the SDSS galaxy survey and a cosmological simulation. Lastly, we calculate Betti functions for two of the MassiveNuS simulations and discuss implications for using the persistent homology of the density field to help break degeneracy in the cosmological parameters.