Hamiltonian formalism of cosmological perturbations and higher derivative theories

Abstract

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.