Using a set of ΛCDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We follow the development of the cosmic web topology in terms of the evolution of Betti number curves and feature persistence diagrams of the three (topological) classes of structural features: matter concentrations, filaments and tunnels, and voids. The Betti curves specify the prominence of features as a function of density level, and their evolution with cosmic epoch reflects the changing network connections between these structural features. The persistence diagrams quantify the longevity and stability of topological features. In this study we establish, for the first time, the link between persistence diagrams, the features they show, and the gravitationally driven cosmic structure formation process. By following the diagrams’ development over cosmic time, the link between the multiscale topology of the cosmic web and the hierarchical buildup of cosmic structure is established. The sharp apexes in the diagrams are intimately related to key transitions in the structure formation process. The apex in the matter concentration diagrams coincides with the density level at which, typically, they detach from the Hubble expansion and begin to collapse. At that level many individual islands merge to form the network of the cosmic web and a large number of filaments and tunnels emerge to establish its connecting bridges. The location trends of the apex possess a self-similar character that can be related to the cosmic web’s hierarchical buildup. We find that persistence diagrams provide a significantly higher and more profound level of information on the structure formation process than more global summary statistics like Euler characteristic or Betti numbers.

We trace the overall connectivity of the cosmic web as defined by haloes in the Planck-Millennium simulation using the persistence and Betti curve analysis developed in our previous papers. We consider the presence of clustering in excess of the second-order correlation function, and investigate the extent to which the dark matter haloes reflect the intricate weblike pattern of the underlying dark matter distribution. With our systematic topological analysis we correlate local information and properties of haloes with the multi-scale geometrical environment of the cosmic web, delineated by elongated filamentary bridges and sheetlike walls that establish the connections between compact clusters at the nodes and define the boundaries of near-empty voids.We capture the multi-scale topology traced by the discrete spatial halo distribution through filtering the distance field of the corresponding Delaunay tessellation. The adaptive character of the tessellation is sensitive to the local density and perfectly outlines the local geometry. The resulting nested alpha shapes contain the complete information on the multiscale topology.After normalising second-order clustering, we find a remarkable linear relationship between halo masses and topology: haloes of different mass trace environments with different topological signature. This is topological bias, a bias that is independent of the halo clustering bias associated with the two-point correlation function. Topological bias can be viewed as an environmental structure bias. We quantify this bias through a linear relation that accounts for selection effects in the analysis and interpretation of the spatial distribution of galaxies. Using this mass-dependent scaling relation allows us to take clustering into account and determine the overall connectivity on the basis of a limited sample of galaxies. This is of particular relevance with the large upcoming galaxy surveys such as DESI, Euclid, and the Vera Rubin telescope surveys.

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