On the statistical evolution of the large-scale structure and the effective dynamics

In this paper, we study the statistical evolution of large-scale structure (LSS), focusing on joint probability distribution function (PDF) coarse-grained cosmic field and its role in constructing effective dynamics. As most comprehensive statistics, PDF encodes all cosmological information modes, therefore, could serve as basis LSS modelling. Following so-called PDF-based method from turbulence, write down PDF's equation, which describes conservation. We show that conservation equation's characteristic curves follow same history be considered an dynamics field. Unlike EFT LSS, conceptually would work at both realization statistics level, is valid only statistically. However, `statistical equivalence' also provides valuable insight into scale interactions level. It enables predicting a wide variety beyond typical N-point polyspectra, including, e.g. topologies, density non-linear covariance matrices etc. Our formula expresses small-scale effect ensemble average their conditional modes. This suggests interesting way to measure terms directly simulation. By applying Gram-Charlier expansion, demonstrate different these terms. formalism natural framework for discussing properties alternative view understanding relationship between general standard perturbation theory.