Finding closure: Approximating Vlasov-Poisson using finitely generated cumulants

Abstract

Since dark matter almost exclusively interacts gravitationally, the phase-space dynamics is described by the Vlasov-Poisson equation. A key characteristic is its infinite cumulant hierarchy, a tower of coupled evolution equations for the cumulants of the phase-space distribution. While on large scales the matter distribution is well described as a fluid and the hierarchy can be truncated, smaller scales are in the multi-stream regime in which all higher-order cumulants are sourced through nonlinear gravitational collapse. This regime is crucial for the formation of bound structures and the emergence of characteristic properties such as their density profiles. We present a novel closure strategy for the cumulant hierarchy that is inspired by finitely generated cumulants and hence beyond truncation. This constitutes a constructive approach for reducing nonlinear phase-space dynamics of Vlasov-Poisson to a closed system of equations in position space. Using this idea, we derive Schr\"odinger-Poisson as approximate quantal method for solving classical dynamics of Vlasov-Poisson with cold initial conditions. Our deduction complements the common reverse inference of the Schr\"odinger-Vlasov relation using a semi-classical limit of quantum mechanics and provides a clearer picture of the correspondence between classical and quantum dynamics. Our framework outlines an essential first step towards constructing approximate methods for Vlasov-like systems in cosmology and plasma physics with different initial conditions and potentials.