  In this thesis, we discuss several instances in which non-linear behaviour
affects cosmological evolution in the early Universe. We begin by reviewing the
standard cosmological model and the tools used to understand it theoretically
and to compute its observational consequences. This includes a detailed
exposition of cosmological perturbation theory and the theory of inflation. We
then describe the results in this thesis, starting with the non-linear
evolution of the curvature perturbation in the presence of vector and tensor
fluctuations, in which we identify the version of that variable that is
conserved in the most general situation. Next, we use second order perturbation
theory to describe the most general initial conditions for the evolution of
scalar perturbations at second order in the standard cosmological model. We
compute approximate solutions valid in the initial stages of the evolution,
which can be used to initialize second order Boltzmann codes, and to compute
many observables taking isocurvature modes into account. We then move on to the
study of the inflationary Universe. We start by analysing a new way to compute
the consequences of a sudden transition in the evolution of a scalar during
inflation. We use the formalism of quantum quenches to compute the effect of
those transitions on the spectral index of perturbations. Finally, we detail
the results of the exploration of a multi-field model of inflation with a
non-minimal coupling to gravity. We study popular attractor models in this
regime in both the metric and the Palatini formulations of gravity and find all
results for both the power spectrum and bispectrum of fluctuations to closely
resemble those of the single-field case. In all systems under study we discuss
the effects of non-linear dynamics and their importance for the resolution of
problems in cosmology.
