The Helmholtz Hierarchy: Phase Space Statistics of Cold Dark Matter

Abstract

We present a new formalism to study large-scale structure in the universe. The result is a hierarchy (which we call the "Helmholtz Hierarchy") of equations describing the phase space statistics of cold dark matter (CDM). The hierarchy features a physical ordering parameter which interpolates between the Zel'dovich approximation and fully-fledged gravitational interactions. The results incorporate the effects of stream crossing. We show that the Helmholtz hierarchy is self-consistent and obeys causality to all orders. We present an interpretation of the hierarchy in terms of effective particle trajectories.

43 pages; added sections 7 and 8, which include interpretation as particle trajectories

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• Projection of the gravitational dynamics on a subspace of probability distributions: curl-free Gaussian ansatz