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 $2\mathrm{\%}$ from BAO scales down to $7h^{-1}\mathrm{M}\mathrm{p}\mathrm{c}$ at $z\ge 0.35$, without any free parameter.