Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies

The cosmic large scale structure encodes the formation and evolution of a weblike network of dark matter and galaxies within the Universe. The cosmological information is wrapped up in non-Gaussian statistics requiring characterisation beyond two-point correlations. Accurate modelling of these non-Gaussian statistics and the underlying non-linear dynamics of gravitational collapse are key to extracting maximal information from ongoing and upcoming cosmological surveys. This thesis centres on questions relating to clustering statistics, dynamics, and fundamental physics: A. How can we efficiently characterise the statistics of the late time matter field? B. How can we capture the non-linear phase-space dynamics of gravitational collapse? C. How do changes to fundamental physics impact those clustering statistics and dynamics? Specifically we present four aspects addressing these questions: 1. We demonstrate the probability distribution function (PDF) of the matter density can be accurately predicted in modified gravity and dynamical dark energy models, and that it provides good complementarity to standard two-point analyses for detecting these features. 2. We demonstrate the joint PDF of densities in two cells can be used to predict the covariance of the one-point PDF in simple clustering models, providing estimates of the density dependent clustering and super-sample covariance missed in cosmological simulations. 3. We use a wave-based forward model of dark matter to demonstrate its capability to encode the full phase-space dynamics beyond a perfect fluid and determine certain universal scaling features in such models. 4. Using the wave dark matter forward model we analyse one-point statistics to complement existing analytic and numerical approaches in studying fundamentally wavelike dark matter.