The work presented in this thesis focuses on data analysis of sky-averaged (global) 21-cm experiments and is based on seven first author papers. The topic is introduced in Part I, with the scientific background of the global 21-cm signal, the challenges faced by experiments aiming to detect the signal and current experimental techniques and results being outlined. Part II introduces a series of data analysis tools that have been developed over the past few years, and Part III illustrates the application of these tools to real data sets with the aim to constrain the astrophysics of the early universe.

In Part II, chapter 2 is adapted from [Bevins et al. 2021a] and discusses the development of an efficient algorithm for fitting functions with constrained derivatives, and consequently smooth properties, to data from global 21-cm experiments as a model for the smooth synchrotron and free-free Galactic and extragalactic emission. We demonstrate the application of Maximally Smooth Functions (MSFs) to data from two different global 21-cm experiments, EDGES and LEDA. Chapter 3 is based on [Bevins et al. 2021b] and outlines a novel 21-cm signal emulator that significantly improves on the accuracy and runtime of the previous state-of-the-art. Finally, chapter 4 is adapted from [Bevins et al. 2022b] and [Bevins et al. 2022c] which outline a framework for performing marginal Bayesian analysis and efficiently combining the constraining power of different data sets. In this chapter, we demonstrate the application of the code margarine to the combination of real data from Planck and DES.

In Part III, chapter 5 details the application of a variant of MSFs and the neural network signal emulator, globalemu, described in chapter 3, to data from SARAS2 and is based on [Bevins et al. 2022a]. It illustrates the application of Bayesian Analysis to data from 21-cm experiments, a concept which is furthered in the following chapters, and an approach to deal with unknown systematics in data sets. This was the first time that systematics have been modelled alongside the sky-averaged 21-cm signal to derive constraints on the first galaxies. The work showed that systematics do not have to be a limiting factor in our data analysis, and pioneered concepts key to the Cambridge-based REACH experiment. Chapter 6 is adapted from [Bevins et al. 2022d] and is concerned with the derivation of constraints on the properties of some of the first galaxies to form in the history of the Universe at much higher redshifts than in the previous chapter using Bayesian analysis and globalemu. This work produced the first limits on the properties of galaxies and star formation in galaxies at 𝑧 = 20. This is followed by the application of chapter 4 to data from the global 21-cm signal experiment SARAS3 and 21-cm power spectrum experiment HERA adapted from [Bevins et al. 2023] in chapter 7. This was the first time a joint analysis of interferometric and global signal 21-cm data has been performed, and through the combination of the constraints from the different data sets the limits in the previous chapter are further advanced.

In Part IV, I detail future work plans including the application of Bayesian techniques to data from upcoming moon based missions for the derivation of constraints on key cosmological parameters, the future analysis of data from REACH and extensions to the work in chapter 4. I follow this with conclusions.