Probing the Early Universe with CMB polarisation: challenges and insights for future experiments

The polarisation of the Cosmic Microwave Background (CMB) carries imprints of the physical conditions that prevailed long before its emission. In particular, the detection of primordial $B$ modes represents one of the central challenges in modern cosmology, as it could provide strong evidence for the primordial gravitational waves predicted by cosmic inflation. Beyond inflation, CMB polarisation may contain signatures of parity-violating phenomena such as cosmic birefringence, providing a powerful test of new physics. However, achieving the sensitivity required to detect such signals demands exquisite control of Galactic foregrounds and instrumental systematic effects. This thesis explores two complementary research directions, sharing the common goal of extracting robust cosmological information from CMB polarisation measurements. The first developed a methodology to translate an allocated error budget on the tensor-to-scalar ratio $r$ into quantitative requirements on instrumental systematic effects for the \emph{LiteBIRD} satellite. In two parallel analyses, we assessed the impact of gain calibration uncertainties and imperfect beam far sidelobe characterisation on the estimation of $r$, using the blind Needlet Internal Linear Combination (NILC) algorithm as component separation method. For gain calibration, the requirements range from $0.16$ to $3.22\%$, with the most stringent constraints at CMB frequencies, reflecting a clear dependence on the NILC weights. These requirements differ from those obtained with a parametric component separation method, demonstrating their dependence on the adopted foreground-cleaning technique. Extending the analysis to more complex sky morphologies showed that, in the most realistic configuration, the requirements must be tightened by a factor of $\sim 1.8$ to remain within the allocated error budget. For beam far sidelobes, the most sensitive channels correspond to dust-dominated frequencies, in particular the $402$-GHz channel, with the requirements depending on both the NILC weights and foreground amplitude. Overall, these analyses demonstrate that instrumental requirements cannot be defined independently of the analysis pipeline, but depend jointly on the component separation method, sky complexity, and the nature of the systematic effect itself. The second research direction explored the origin of cosmic birefringence through the reionisation bump of the $EB$ angular power spectrum, which could help distinguish between dark energy and dark matter interpretations of the signal. At low multipoles, however, the $EB$ statistics become analytically intractable, making conventional likelihood-based inference challenging. We therefore adopted a simulation-based inference (SBI) framework to constrain the $EB$ signal at low multipoles, without requiring an explicit analytical likelihood. In an idealised noise- and cosmic variance-limited setup, SBI recovers characteristic constraints on the mass of the underlying pseudo-scalar field. However, robustly distinguishing between the two physical interpretations is limited by polarisation angle miscalibration and gravitational lensing, highlighting the need for accurate angle calibration and effective delensing. Overall, this thesis approaches CMB polarisation from two complementary perspectives, assessing the instrumental performance required for robust measurements and developing advanced statistical inference methods to extract and interpret fundamental physics from future CMB observations.