Relativistic Cosmology from the linear to the non-linear regime

Cosmology is a data-driven science and in the past decades we have seen the Lambda-Cold-Dark-Matter (Lambda-CDM) model establishing itself as the standard model for cosmology. However, the standard model still presents several open problems; among them, explaining the mechanism of the current cosmic acceleration. In the near future, Large Scale Structure (LSS) surveys are expected to play a pivotal role in shedding light on this subject. Galaxy surveys will be able to map the galaxies distribution on very large volumes with greatly improved statistics, while new techniques such as the intensity mapping applied to the 21 cm neutral hydrogen emission promise to explore the evolution of the Universe until the epoch of reionization. Therefore, it is of crucial importance to combine this huge effort from the observational side with a correct model for the observable quantities. During my PhD I worked on several topics in the field of Large Scale Structure (LSS), which is considered here as a laboratory to test fundamental aspects of gravity and cosmology. The thesis is divided in three parts. In part 1 we will introduce the basics of modern cosmology. This introduction does not aim to be all-encompassing, but it will specifically address the topics that are relevant in understanding the main body of the thesis, i.e. part 2 and part 3. Part 2 of the thesis concerns the largest scales that we will be able to test with the future generation of galaxy surveys. These scales can be studied within the framework of linear perturbation theory. However, a full relativistic treatment is necessary in order to take into account all the horizon effects that are predicted by General Relativity and in general by any metric theory of gravity. The fact that photons travel from the source to the observer in an inhomogeneous universe introduces volumes and redshift correction to the observed quantities. In the linear regime they include the standard redshift-space distortions (RSD) also known as the Kaiser effect, the gravitational lensing magnification, the Doppler effect and gravitational redshift, Shapiro time-delay, Sachs-Wolfe and integrated Sachs-Wolfe effects. The standard LSS analysis includes the Kaiser effect, while the other relativistic effects are often neglected. In part 2 we investigated the relevance of the relativistic effects for future planned galaxy survey, focusing on their impact on cosmological tests and parameter estimation (chapter 4 and chapter 5) and their detectability (chapter 6). Part 3 of the thesis focuses on the highly non-linear regime of the LSS. The vector degrees of freedom are often neglected in modelling the cosmic velocities. This approximation works fairly well on large scales, i.e. where linear theory applies. However, on non-linear scales it is well-known that vorticity is generated. In chapter 7 we investigated numerically the generation of vorticity with the recently-born relativistic N-body code gevolution. Even if the generation of vorticity is a purely newtonian effect, a relativistic treatment extends the newtonian approach: for example, it allows to investigate the interplay between the vorticity and the vector degrees of freedom in the metric. Finally, in chapter 8 we summarize the results and draw possible extensions of the work presented in this manuscript.