The evolution of the large-scale structure of the universe: beyond the linear regime

These lecture notes introduce analytical tools, methods and results describing the growth of cosmological structure beyond the linear regime. The presentation is focused on the single flow regime of the Vlasov-Poisson equation describing the development of gravitational instabilities in a pressureless fluid. These notes include the introduction of diagrammatic representations of the standard perturbation theory with applications to the calculation of the so-called loop contributions to the power spectra. A large part of these notes is devoted to the exploration of the convergence properties of these terms from the contribution of both the long-wave modes and the short-wave modes. The resulting performances of the two-loop corrections of the power spectra are then presented. Finally other avenues that use different methods are explored. In particular it is shown how joint density and profile probability distribution functions can be constructed out of the multiple-variable cumulant generating functions computed at tree order.