  Quantum effects play an essential role in modern cosmology. Perhaps the most
striking example comes from large-scale structures, generally assumed to
originate from vacuum quantum fluctuations and stretched by an expansion phase.
Inflation is the leading paradigm in explaining this process. The various
observational successes of inflationary models drive the scientific community
into elaborating more and more stringent tests, which can simultaneously be
used to probe beyond the simple slow-roll, single field inflation. However,
inflation is not a theory, and going beyond inflation is a necessity. Various
alternatives and/or complementary mechanisms to inflation have been invoked in
the literature. The best-known cosmological models endowed with the capacity of
explaining large-scale observations while avoiding the singularity form a class
called non-singular bouncing models. The main features of these models are the
presence of a contraction phase before expansion, and a never-vanishing scale
factor. A non-singular bounce generally appears when quantum effects are part
of the model, playing the role of a regulator leading to the avoidance of
singularities. This thesis focuses on a Hamiltonian formulation of quantum
effects in cosmology. We first explore stochastic perturbations in a collapsing
universe. Then, we show that quantum cosmology with Bohmian mechanics resolves
the initial singularity. Adding a non-minimal coupling of gravity with
electromagnetism, we show that the generation of magnetic fields is possible.
Finally, we apply the affine quantisation on the Brans-Dicke Theory, the
prototype of modified gravity theories, and we discuss the quantum equivalence
of the Jordan and Einstein frames within this framework. We show that in both
frames a smooth bounce is expected, and that equivalence between frames holds
at the quantum level.
