Large-scale structure of the Universe in General Relativity and beyond

The large-scale structure of the Universe and the nature of dark energy are beginning to be comprehensively understood, as we have recently entered the era of precision cosmology. The standard LCDM cosmological model, which includes a cosmological constant L and a cold dark matter component, successfully explains the evolution and composition of our Universe. However, recent measurements with tighter constraints on cosmological parameters have revealed several serious anomalies, such as the Hubble constant tension, which could be a signal of a possible theoretical model crisis. Therefore, it is necessary to reassess the fundamental pillars of the LCDM model, i.e., 1) General Relativity is the underlying gravitational theory that governs cosmological dynamics; 2) the Universe is homogeneous and isotropic on scales greater than approximately 100 Mpc (cosmological principle). This thesis aims to discuss the robustness of these two pillars by exploring possible close scenarios with respect to the LCDM model, such as modified gravity theories and inhomogeneous cosmologies. We are interested in addressing the following questions: how can cosmological data enable us to distinguish between the LCDM cosmological model and modified gravity theories? What is the impact of local inhomogeneities on cosmological observables? Firstly, modified gravity theories predict deviations from LCDM, providing alternatives to the cosmological constant to ensure the present cosmic acceleration. Secondly, the large-scale structure of the Universe and local deviations from spatial homogeneity could impact our cosmological measurements. All these alternative proposals allow us to investigate unresolved cosmological issues.