Finding cosmic voids and filament loops using topological data analysis

Abstract

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 2dimensional 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.

Topological data analysis for the string landscape

“…simplicial complexes. Persistent homology has proven useful in diverse fields including sensor networks [43], image processing [44], bioinformatics [45], genomics [46], protein structure [47], neuroscience [48], cosmology [49][50][51][52][53][54], and many more. For example, persistent homology has been used to study the Cosmic Web of large-scale structure, composed of interlocking voids, filaments, and sheets at a variety of scales [51].…”

EmT: Locating empty territories of homology group generators in a dataset

“…Cosmic voids are considered useful in studying the universe structure and galaxy evolution [27,29,28]. Many methods have been developed to detect cosmic voids in the universe [26,30,32,22,20,38]. In this section, we use persistent homology to find cosmic voids similarly to the work in [38].…”

“…Many methods have been developed to detect cosmic voids in the universe [26,30,32,22,20,38]. In this section, we use persistent homology to find cosmic voids similarly to the work in [38]. While [38] used a smoothed distance function to construct the filtration and the reported representations are not necessarily empty inside, this work builds the filtration using the VR complex as introduced in Sec.…”

“…11a displays a Voronoi foam simulation dataset, which is used as an approximation of the universe structures [ 21,33]. A Voronoi foam simulation dataset is generated by a set of randomly sampled void seeds (which specify the locations of the voids) and their Voronoi tessellation; the simulation procedure is explained in detail in [38]. Since the void location is known, the matching of H 2 generators to void seeds can be calculated by comparing distances of the H 2 generators to each void seed.…”

“…Persistent homology detects homology group generators and summarizes them in a persistence diagram, but there is not a rigorous framework to locate a desirable representation of the homology group generators back in the original dataset, which can be useful information (e.g., [38]). One major issue with finding representations of homology group generators in original dataset is that there is not a unique representation.…”