The Cosmic Foam and the Self-Similar Cluster Distribution

Voronoi Tessellations form an attractive and versatile geometrical asymptotic model for the foamlike cosmic distribution of matter and galaxies. In the Voronoi model the vertices are identified with clusters of galaxies. For a substantial range out to a scale in the order of the cellsize, their spatial two-point correlation function is a power-law with a slope $\gamma \approx 1.95$. This study presents recent results showing that subsets of vertices selected on the basis of their richness'', i.e. inflow volume, retain this power-law correlation behaviour. Interestingly, they do so with a clustering length’’ $r_{\mathrm{o}}$ that is exactly linearly proportional to the average inter-vertex distance in the sample, thus forming a realization of the Szalay-Schramm prescription. For the geometry and structural patterns even more significant is the finding tessellation vertices display a similar linear increase for their correlation length $r_{\mathrm{a}}$, the coherence length at which $\xi (r_{\mathrm{a}})=0$. Such patterns therefore exhibit positive correlations out to distances considerably in excess of the cellsize. Most intriguing is the implication of self-similar scaling, while these results may be regarded as the presentation of a ``geometrical bias’’ effect. A seemingly rigid structure appears to represent a flexible and useful geometric model for exploring the statistical and dynamical repercussions of the nontrivial cellular patterns in Megaparsec scale cosmic structure.