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Using a set of $\Lambda$CDM simulations of cosmic structure formation, westudy the evolving connectivity and changing topological structure of thecosmic 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 theevolution of Betti number curves and feature persistence diagrams of the three(topological) classes of structural features: matter concentrations, filamentsand tunnels, and voids. The Betti curves specify the prominence of features asa function of density level, and their evolution with cosmic epoch reflects thechanging network connections between these structural features. The persistencediagrams quantify the longevity and stability of topological features. In thisstudy we establish, for the first time, the link between persistence diagrams,the features they show, and the gravitationally driven cosmic structureformation process. By following the diagrams' development over cosmic time, thelink between the multiscale topology of the cosmic web and the hierarchicalbuildup of cosmic structure is established. The sharp apexes in the diagramsare intimately related to key transitions in the structure formation process.The apex in the matter concentration diagrams coincides with the density levelat which, typically, they detach from the Hubble expansion and begin tocollapse. At that level many individual islands merge to form the network ofthe cosmic web and a large number of filaments and tunnels emerge to establishits connecting bridges. The location trends of the apex possess a self-similarcharacter that can be related to the cosmic web's hierarchical buildup. We findthat persistence diagrams provide a significantly higher and more profoundlevel of information on the structure formation process than more globalsummary statistics like Euler characteristic or Betti numbers.

Persistent homology of the cosmic web. I: Hierarchical topology in $\Lambda$CDM cosmologies