Semi-Analytical Methods beyond Standard Perturbation Theory for the Large-Scale Structure of the Universe

The main aim of this thesis is to investigate new methods beyond standard perturbation theory (SPT) to study the statistical properties of the Large Scale Structure (LSS) of the Universe. In particular, we will focus on the advantages that the use of the linear response function can bring to the evaluation of the matter power spectrum at scales presenting subdominant but crucial non-linear effects. In order to pursue this end, after an introductory part where we recall the importance of the studies on the LSS for contemporary Cosmology, as well as an excursus from the first analytical and observative attempts trying to understand its features until the state of the art in the field, we develop SPT in a typical field theory fashion, that is by means of generating functionals and Feynman rules, and we derive constraints between correlators arising from the galilean invariance of the dynamical system. Then, we define the linear response function as a tracker of the coupling between different cosmological modes, a genuine non-linear effect, we give a diagrammatic representation for it and we compute this object at the lowest order in SPT, comparing our result with N-body simulations. Moreover, we present two applications of the linear response function. The first consists in an improvement of the predictions of the Gamma-expansion method, based on multi-point propagators, on the power spectrum at slight non-linear scales: in particular, we show that, by restoring galilean invariance, which is broken by most resummation methods, we can increase the maximum wavenumber at which the non-linear power spectrum can be trusted by 20%, and by 50% with respect to SPT. The second consists in the possibility to use the linear response function as an interpolator between different cosmologies at slight non-linear scales: in particular, it can be seen as an object able to encode the variations between the power spectrum of a reference cosmology and the one related to a small modification of a cosmological parameter with respect the reference configuration; we obtain that the modified power spectra generally differ from the corresponding simulated ones within about the 2% by changing the parameters within an enhancement or reduction of about 3σ even if the exact values depend on the specific considered modified parameter: this procedure is particularly interesting as it provides a tool to limit the number of heavy N-body simulations in the study of the LSS of our Universe.