Topological bias: How haloes trace structural patterns in the cosmic web

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 web-like pattern of the underlying dark matter distribution. With our systematic topological analysis we correlate local information and halo properties with the multi-scale geometrical environment of the cosmic web, delineated by elongated filamentary bridges and sheetlike walls that connect 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 tessellation is naturally adaptive to the local density, perfectly outlining the local geometry. The resulting nested alpha shapes contain the complete information on the multi-scale topology. 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 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 it through a linear relation accounting for selection effects in the analysis and interpretation of the spatial distribution of galaxies. This mass-dependent scaling relation allows us to take clustering into account and determine the overall connectivity based on a limited sample of galaxies. This is of particular relevance with large upcoming galaxy surveys such as DESI, Euclid, and the Vera Rubin telescope surveys.