Fundamental physics from the cross-correlation of the Cosmic Microwave Background with large scale structure observables

Over the last 60 years, cosmology has achieved substantial progress in understanding the fundamental properties of the Universe, significantly driven by the discovery and analysis of the Cosmic Microwave Background (CMB) radiation. This invaluable observable has provided unique insights into the physics of the early Universe, including the mechanisms that shaped its initial conditions. Moreover, the CMB has also opened numerous windows onto the history and evolution of our Universe, however there remain subtle features of this radiation that are yet to be revealed. Among them, the detection of Primordial Gravitational Waves through polarization B-modes, which is one of the main targets of the observational campaign for the coming decade. Another interesting example is represented by the Integrated Sachs-Wolfe (ISW) effect, along with its nonlinear counterpart the Rees-Sciama (RS), which offers a way to test the late-time evolution of the Universe. The ISWRS effect arises from the interaction of CMB photons with evolving Large Scale Structure (LSS) of the Universe along their travel path, and can be probed only through the cross-correlation of the CMB with LSS tracers. While the linear ISW has already been measured, the nonlinear RS has yet to be detected. In an era where cross-correlation methods are increasingly central to cosmological analysis, the study of the ISWRS effect is both timely and highly relevant. In preparation for future datasets, that will allow for a more robust measurement of the linear ISW effect and for a possible detection of the nonlinear RS effect, it is critical to develop theoretical frameworks and computational tools that can effectively characterize the ISWRS signal across different cosmological scenarios. It is essential to refine both simulation models and analysis pipelines to estimate the statistical significance of future measurements and optimize the extraction of cosmological information from the ISWRS cross-correlations. This thesis addresses all of these challenges by developing novel methodologies for simulating the ISWRS signal, with an emphasis on accurately modelling both the large and the small scales. We explore the cross-correlation between the CMB and three distinct LSS tracers: Galaxy Clustering (GC), Cosmic Shear (CS), and CMB-Lensing (CMBL) potential. We analyse the potential of these cross-correlation techniques for extracting cosmological information at different angular scales. Concerning the linear regime, we develop dedicated pipelines for the scientific exploitation of ISW measurements through the cross-correlation with the Euclid GC sample, focusing specifically on the constraining power that this effect has on Dark Energy (DE) and Primordial non-Gaussianity (PNG). Whereas for the nonlinear scales, we use state-of-the-art simulations to verify that current semi-analytic models do not provide the accuracy necessary to model the nonlinearities in structure evolution, and consequently are ineffective in representing the RS and its cross-correlation with LSS probes. We develop our analytical method to overcome this limitation and, once validated against N-body simulations, we use it to test the detectability of the RS and its constraining power on the neutrino mass. We demonstrate that with future LSS surveys (such as Euclid and LSST) and CMB experiments (such as CMB-S4, CMB-HD and PICO), it will be possible to measure the RS effect with high significance, and potentially detect its dependency on the neutrino mass.