33 - Efficiency of gravitational clustering

Summary

Some non-linear descriptions, such as the cosmic virial theorem, can be derived directly from first principles, but so far this has not been done for clustering efficiency. Here the technique is to construct a model, simple enough to solve analytically, which retains the essential physics of the problem.

Let us suppose there is a hierarchy of clustering on different well-defined levels. Consider the interaction of two adjacent levels of this hierarchy. For example, these levels could represent the clustering of globular clusters (or subgalaxies if these are formed from many globular clusters) to form galaxies. They could also represent the clustering of galaxies to form clusters of galaxies. Objects at the lower level are presumed to be tightly bound; we shall call them particles and represent them by point masses. Objects at the higher level are more loosely bound; they are the clusters.

Our analysis starts in a region which has already undergone its phase transition. The density of a cluster is large compared to the average background density. Not all the particles are in clusters. We shall consider the fate of an arbitrary vagrant particle moving through a field of clusters, and ask: ‘What is the probability that vagrant particles will be captured by clusters?’ This probability is a measure of the clustering efficiency.

The vagrant particle may be one which has been ejected from a forming cluster, or it may never have been bound.