Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
An incremental algorithm for Betti numbers of simplicial complexes
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Shapes and implementations in three-dimensional geometry
Shapes and implementations in three-dimensional geometry
Handbook of discrete and computational geometry
On the definition and the construction of pockets in macromolecules
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
Voronoi Tessellations and the Cosmic Web: Spatial Patterns and Clustering across the Universe
ISVD '07 Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering
Topology for Computing
Alpha Shape Topology of the Cosmic Web
ISVD '10 Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and Engineering
Geometry of the Cosmic Web: Minkowski Functionals from the Delaunay Tessellation
ISVD '10 Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and Engineering
Computing Robustness and Persistence for Images
IEEE Transactions on Visualization and Computer Graphics
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We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, α. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of α, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the Cosmic Web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field. Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the Cosmic Web.