  The focus of the thesis is to obtain a universal formalism to evaluate the
perturbations during inflation at all orders that can be applied to any theory
of gravity and matter source in the early universe. We first look at the
equivalence of two approaches --- action and order-by-order gravity equation
approach --- for cosmological perturbation theory, and establish that both lead
to equivalent results for any gravity models at any order of perturbations. We
then focus on Hamiltonian formalism that has not been studied extensively in
the literature. We provide a generalized Hamiltonian approach for cosmological
perturbations which is equivalent to the Lagrangian approach. We show that, the
approach can be applied to any model at any order of perturbations. Using this
approach, we show that evaluating interaction Hamiltonian is simpler and
efficient than earlier approach. In the next work, we concentrate on
generalized non-canonical scalar field and by introducing a new variable
provide a technique to write Hamiltonian for generalized non-canonical scalar
field. We then implement our Hamiltonian approach for non-canonical scalar
field and evaluate interaction Hamiltonian without slow-roll approximations.
Finally, for the first time, we construct a vector Galileon model in curved
space-time in which, the field equations do not contain any higher-derivative
terms, yet, preserving U(1) gauge-invariance. Conformal invariance is broken in
this model which leads to primordial magnetogenesis and we compare the
predictions of our model with observations.
