Projection of the gravitational dynamics on a subspace of probability
  distributions: curl-free Gaussian ansatz
  We present a new approach to model the gravitational dynamics of large-scale
structures. Instead of solving the equations of motion up to a finite
perturbative order or building phenomenological models, we follow the evolution
of the probability distribution of the displacement and velocity fields within
an approximation subspace. Keeping the exact equations of motion with their
full nonlinearity, this provides a nonperturbative scheme that goes beyond
shell crossing. Focusing on the simplest case of a curl-free Gaussian ansatz
for the displacement and velocity fields, we find that truncations of the power
spectra on nonlinear scales directly arise from the equations of motion. This
leads to a truncated Zeldovich approximation for the density power spectrum,
but with a truncation that is not set a priori and with different power spectra
for the displacement and velocity fields. The positivity of their auto power
spectra also follows from the equations of motion. Although the density power
spectrum is only recovered up to a smooth drift on BAO scales, the predicted
density correlation function agrees with numerical simulations within $2\%$
from BAO scales down to $7 h^{-1} {\rm Mpc}$ at $z \geq 0.35$, without any free
parameter.
