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
