Tessellation analysis of the cosmic web

The large scale distribution of matter and galaxies features a complex network of interconnected filamentary galaxy associations. This network, which has become known as the Cosmic Web, contains structures from a few megaparsecs up to tens and even hundreds of Megaparsecs of size. Galaxies and mass exist in a wispy weblike spatial arrangement consisting of dense compact clusters, elongated filaments, and sheetlike walls, amidst large near-empty void regions. An important additional aspect of this mass distribution is that it is marked by substructure over a wide range of scales and densities. The vast Megaparsec cosmic web is one of the most striking examples of complex geometric patterns found in nature, and certainly the largest in terms of sheer size.

The overwhelming complexity of both the individual structures as well as their connectivity, the lack of structural symmetries, the intrinsic multiscale nature and the wide range of densities that one finds in the cosmic matter distribution has prevented the use of simple and straightforward instruments. In this lecture, I describe the considerable advances that have been made over the past decade towards unravelling the structure of the Cosmic Web, enabled by a range of tools and concepts from computational geometry and computational topology. This will include our own work, in which Voronoi and Delaunay tessellations figure prominently through their high sensitivity to density and local shape of the local galaxy distribution, or particle distribution in the case of computer simulations of cosmic structure formation. It has led to the development of the Delaunay Tessellation Field Estimator (DTFE) formalism, which forms the basis of a range of techniques to identify different aspects of the Cosmic Web [weyschaap2009]. Examples are the Watershed Void Finder to trace voids, the Nexus multiscale morphology formalism and the Morse-based SpineWeb formalism to find walls, filaments and clusters. Recently, we used alpha shapes to study the multiscale topology of the Cosmic Web, in terms of Betti numbers and persistence diagrams. I will also review a number of other astronomical applications of tesssellations, motivated by their quickly proliferaating use in astrophysics and cosmology.