Multi-scale Connectivity and Hierarchical Topology of the Cosmic Web

On the largest scales, matter in the Universe is arranged in a complex and highly connected cosmic web, consisting of high density clusters, elongated filaments, flat walls, and large, empty voids. The characteristics of this web-like structure, such as its shape and connectivity, are determined by the interplay between dark matter and dark energy. What these consist of is among the fundamental questions in cosmology. In this thesis, I try to find answers by using topological data analysis. In particular, I analyse how connections in the Universe are formed through the process of gravitational structure formation, and how the resulting network is influenced by dark energy.In Chapter 2, I determine the baseline connectivity by analysing structure formation in five N-body simulations following the standard model of cosmology (LCDM), where dark energy takes the form of a cosmological constant. In Chapter 3, I compare the LCDM connectivity to that of two additional cosmological models (both with a dynamic dark energy), and find significant differences. In Chapter 4, I prepare for identifying these differences in cosmic observations by investigating the topological structure of dark matter haloes. I identify a topological bias, where the mass of a halo influences its geometric environment (whether it is located in a filament or wall). In Chapter 5, I focus on the emergence of cosmic connections as predicted from the initial conditions using caustic theory and the Zeldovich approximation. I find remarkable similarities between this predicted connectivity and the connectivity emerging under fully non-linear gravitational evolution in N-body simulations. The successful application of topological methods in these four chapters is highly promising for their use in upcoming large galaxy surveys