Emulating the first billion years of the Universe

The first detection of the 21-cm signal is now within reach with the upcoming Square Kilometre Array (SKA). The SKA is expected to deliver tomographic maps of the 21-cm signal across more than half of the observable Universe, providing transformative constraints on the astrophysical and cosmological processes that shaped its evolution. At present, however, such measurements remain out of reach, primarily due to challenges in data analysis and interpretation. Instead, we only have upper limits on the 21- cm power spectrum, reported by several SKA precursors. In this thesis, I employ machine learning to facilitate and enhance the interpretation of both current and upcoming cosmic dawn and epoch of reionisation observations within a Bayesian inference framework. Current upper limits on the 21-cm power spectrum become informative only when combined with complementary observations. To enable such synergistic analyses, I developed a machine-learning emulator of six summary statistics, which accelerates inference by more than four orders of magnitude compared to direct simulation. Despite the increasing realism of simulations, current Bayesian inferences of the 21-cm power spectrum remain limited by the small simulation volumes required for computational feasibility. Small box sizes restrict the number of large-scale modes that can be probed, setting a fundamental lower bound on the precision of forward-modelled power spectra at large scales — a limitation known as sample variance. To address this limitation, I developed a simulator-independent framework that mitigates sample variance with a score-based diffusion model and thereby significantly improves the constraining power of the inference. Put together, the work presented in this thesis is aimed at squeezing the most juice out of current upper limits and upcoming first detections of the 21-cm power spectrum, thereby extracting the maximum possible amount of information out of them.