Exploring dark energy and modified gravity with Euclid-CMB cross-correlation

In this thesis we study the synergy between observations of the cosmic microwave background (CMB) and large scale structure of the Universe, in the context of the ESA mission Euclid. The Euclid mission aims at building the largest galaxy catalogue to date, observing galaxies in about a third of the sky, and providing high quality data for the study of dark energy, dark matter and gravity. In this context we first discuss modified gravity theories that can be targeted by Euclid and provide original cosmological constraints with existing datasets on a subset of scalar-tensor theories of gravity, discussing also their implications for the Hubble tension. We then forecast for Euclid capabilities in further constraining this class of models in combination with CMB experiments. In the forecasts we initially focus on the combination of CMB lensing with Euclid observables showing its relevance for extended cosmological models, and then provide forecasts for the full CMB-Euclid joint analysis. This study shows the striking complementarity of these datasets, which breaks degeneracies between cosmological parameters, guaranteeing the largest constraining power, both on the standard cosmological parameters and on the parameters of the extended models, such as the sum of the neutrino masses or the modified gravity parameters. In order to provide these constraints and fulfill the forecasts it is necessary to build an end-to-end pipeline for the joint analysis of Euclid and CMB data, we explore this topic as well by presenting the results of the validation and implementation of the likelihood module for the cross-correlation observables. In particular we delve into the details of the likelihood for the detection of the integrated Sachs-Wolfe effect through the correlation of the CMB temperature field and galaxy number counts; and also discuss the implementation of CMB lensing in the official Euclid likelihood code.