Non-linear regime of cosmic structure formation: Imprints on the covariance of the power spectrum and dynamics of Dark Energy inhomogeneities

In the near future numerous galaxy surveys are going to provide a cartography of the distribution of matter in the universe, with which it will be possible to constrain the cosmological parameters with an unprecedented precision. This means that theoretical predictions of the observables at issue need to attain a high level of precision, so that we can extract the maximum cosmological information when comparing them to observations. Upcoming surveys have been conceived to understand the nature of dark energy, a component of which we have postulated the existence to explain the acceleration of the expansion of the universe observed at low redshift. In the standard cosmological model this component is represented as a cosmological constant Λ that dominate the energy budget of the universe today. Even if the ΛCDM model remains in good agreement with observations, a number of theoretical problems regarding the nature of the cosmological constant justify the study of a great number of alternative models, that now have to be put to the test with observations. To do so, it is necessary to take into account the effects of the non-linear dynamics of the gravitational collapse process. This thesis is placed in this context, using N-body simulations as a tool to link observations to the theory of cosmic structure formation in non-linear regime. On one side we are going to explore the impact of non-linear effects on the estimation of cosmological parameters in a statistical framework. This requires the computation of the covariance matrix of the matter power spectrum. The estimation of this matrix using simulations allows the quantification of the importance of non-linear effects that, if neglected, can bias the results or lead to an under-estimation of the statistical errors on the cosmological parameters. On the other side, we are going to present the numerical methods to solve the evolution of a dark energy fluid with perturbations that have a very low speed of propagation with respect to the speed of light. Collapsing dark energy perturbations leave a distinctive imprint on the distribution of matter in the universe. In this case, dark energy cannot be treated as a homogeneous fluid and its evolution has to be followed in the non-linear regime together with the dark matter component. These numerical methods, once coupled to an N-body code, will allow the production of the first simulations of the evolution of an inhomogeneous dark energy model.