The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
  We introduce a multiscale topological description of the Megaparsec weblike
cosmic matter distribution. Betti numbers and topological persistence offer a
powerful means of describing the rich connectivity structure of the cosmic web
and of its multiscale arrangement of matter and galaxies. Emanating from
algebraic topology and Morse theory, Betti numbers and persistence diagrams
represent an extension and deepening of the cosmologically familiar topological
genus measure, and the related geometric Minkowski functionals. In addition to
a description of the mathematical background, this study presents the
computational procedure for computing Betti numbers and persistence diagrams
for density field filtrations. The field may be computed starting from a
discrete spatial distribution of galaxies or simulation particles. The main
emphasis of this study concerns an extensive and systematic exploration of the
imprint of different weblike morphologies and different levels of multiscale
clustering in the corresponding computed Betti numbers and persistence
diagrams. To this end, we use Voronoi clustering models as templates for a rich
variety of weblike configurations, and the fractal-like Soneira-Peebles models
exemplify a range of multiscale configurations. We have identified the clear
imprint of cluster nodes, filaments, walls, and voids in persistence diagrams,
along with that of the nested hierarchy of structures in multiscale point
distributions. We conclude by outlining the potential of persistent topology
for understanding the connectivity structure of the cosmic web, in large
simulations of cosmic structure formation and in the challenging context of the
observed galaxy distribution in large galaxy surveys.
