The cosmic foam:Stochastic geometry and spatial clustering across the universe

Galaxy redshift surveys have uncovered the existence of a salient and pervasive foamlike pattern in the distribution of galaxies on scales of a few up to more than a hundred Megaparsec. The significance of this frothy morphology of cosmic structure has been underlined by the results of computer simulations. These suggest the observed cellular patterns to be a prominent and natural aspect of cosmic structure formation for a large variety of scenarios within the context of the gravitational instability theory of cosmic structure formation.We stress the importance of stochastic geometry as a branch of mathematical statistics particularly suited to model and investigate nontrivial spatial patterns. One of its key concepts, Voronoi tessellations, represents a versatile and flexible mathematical model for foamlike patterns. Based on a seemingly simple definition, Voronoi tessellations define a wealthy stochastic network of interconnected anisotropic components, each of which can be identified with the various structural elements of the cosmic galaxy distribution. The usefulness of Voronoi tessellations is underlined by the fact that they appear to represent a natural asymptotic situation for a range of gravitational instability scenarios of structure formation in which void-like regions are prominent.Here,we describe results of an ongoing thorough investigation of a variety of aspects of cosmologically relevant spatial distributions and statistics within the framework of Voronoi tessellations. Particularly enticing is the recent finding of a profound scaling of both clustering strength and clustering extent for the distribution of tessellation nodes, suggestive for the clustering properties of galaxy clusters. This is strongly suggestive of a hitherto unexpected fundamental and profound property of foamlike geometries. In a sense, cellular networks may be the source of an intrinsic "geometrically biased" clustering.