A skewed perspective on the universe: advancements, challenges and prospects in the hunt for primordial non-Gaussianity

The most well-established paradigm in cosmology, the study of the universe on the largest scales, is that the universe underwent a brief period of rapid expansion, known as cosmic inflation. Such an expansion explains the homogeneity and isotropy observed in the universe on large scales, for example, in the nearly identical temperature of the Cosmic Microwave Background (CMB) radiation in every direction in the sky. At the same time, it provides a mechanism for generating tiny density fluctuations across the universe at the end of inflation, that serve as the seeds of structure formation, thus setting the initial conditions for our universe. Nevertheless, the precise physical mechanism responsible for driving this early expansion remains unknown, but a better understanding of the universe's initial conditions can provide invaluable insights into the physics at play during inflation. We hope to achieve this understanding through ever more precise cosmological observations, such as those of the CMB or the distribution of galaxies, but our ability to learn about the initial conditions relies on developing accurate modeling of these observables, and statistical techniques to optimally extract information from large sets of data. In this thesis, I investigate new cosmological observables and methods for analyzing such data, using both analytical and state-of-the-art machine learning techniques. My findings can help in improving our understanding of the evolution of the universe and in particular the inflationary expansion, with future cosmological surveys.