Title: Towards a robust exploration of the Dark Sector with Euclid and Stage-IV surveys

Abstract: The next generation of galaxy surveys—Euclid, LSST, DESI, Nancy Roman, SKAO—will deliver unprecedented precision in mapping the large-scale structure (LSS) of the Universe. These datasets will probe the expansion rate, non-linear structure growth, and the cosmic matter-energy budget, providing new insights into Dark Matter, Dark Energy, and persistent tensions in parameters like H0 and S8. Modified Gravity models, including f(R) and scalar field theories, will be rigorously tested, and next-generation galaxy surveys, in combination with current and upcoming CMB experiments, will reveal details about the composition of cosmic components and primordial fluctuations. Cross-correlations, such as those between optical and radio cosmology via 21cm intensity mapping, will enhance these efforts, providing better constraints but also creating new challenges in the modeling of systematics. Achieving robust results in the study of the Dark Sector requires addressing degeneracies such as massive neutrinos and baryonic feedback. I will discuss how the Euclid collaboration is integrating these factors into its photometric and spectroscopic likelihoods, using both machine learning methods like emulators and semi-analytic models like effective field theory of LSS to perform detailed forecasts for upcoming data analyses. I will also discuss ongoing work in provable computations and differentiable programming, which can improve the robustness and efficiency of our analysis.