• adiabatic renormalisation
• cosmological inflation
• cosmological perturbation theory
• primordial black holes
• second order tensor modes
• stochastic inflation

According to the inflationary paradigm, the seeds of all cosmic structures developed from small quantum fluctuations, which were stretched to cosmological scales by the expansion of spacetime. This mechanism ruffled the spacetime’s canvas in a seemingly random, though correlated manner: these correlations, which preserve information on their primordial origin, are an invaluable source of information on the inflationary universe. Remarkably, the underlying inflationary framework makes theoretical predictions that are supported by cosmological observations to a surprising degree. Despite the great success of the inflationary paradigm, however, many fundamental questions remain, which prevent us from reconstructing a complete picture of the early universe. To gain new insights is therefore necessary to go further and consider new observables. This is a joint effort, which must combine both the growth of observational perspectives, and the theoretical investigation of new effects.
This thesis is set in this challenging context, and considers different aspects of the inflationary scenario, combining different methods (quantum field theory in curved spacetime, stochastic inflation, cosmological perturbation theory beyond linear order) and focusing on different scenarios (particular inflationary models,
specific features in the inflationary potentials), which give rise to interesting phenomenology, among which primordial black holes and primordial gravitational waves.
The thesis is organised as follows: in Chapter 1, we give an overview of the Hot Big Bang model of cosmology and we present its shortcomings. In Chapter 2, we introduce cosmological inflation and we address some of its open questions, devoting part of the Chapter to the theory of cosmological perturbations. In Chapter 3 we introduce quantum field theory in curved spacetime and we focus on a model of inflation driven by a pseudo-scalar field coupled to an abelian gauge field. In this context, we show how the standard adiabatic renormalisation seems problematic, and we propose how to modify it according to a new prescription. The latter is employed for the study of useful quantities in the estimate of the backreaction in the model. In Chapter 4, we review the stochastic dN formalism and how it can be used to incorporate quantum diffusion into the properties of cosmological perturbations. With this formalism, we study the case of stochastic tunnelling, where the inflaton escapes from a false vacuum thanks to the effect of quantum diffusion, and we consider the production of primordial black holes in such a scenario. Chapter 5 is dedicated to the topic of second-order tensor modes generated by mode coupling during inflation: we focus on the issue of their gauge dependence and we propose a new direction to tackle the gauge problem.