  Recent and upcoming experimental data as well as the possibility of rich
phenomenology have spiked interest in studying the quantum effects in cosmology
at low (inflation-era) energy scales. One of the approaches to find covariant
quantum corrections is the DeWitt-Vilkovisky's (DV) covariant effective action
formalism that is gauge invariant and background field invariant. We use the
DeWitt-Vilkovisky method to study formal and cosmological aspects of quantum
fields in curved spacetime, and take initial steps towards studying quantum
gravitational corrections in cosmological setting. The thesis comprises of
mainly two parts. We first study the formal aspects of rank-2 antisymmetric
tensor field which appear in the low energy limit of superstring models and are
thus relevant in the early universe, in particular the quantization and quantum
equivalence properties, for the case with and without spontaneous Lorentz
violation. The effective action is generalized for gauge theories whose gauge
parameters possess additional symmetries. When used in the case of
spontaneously Lorentz violating antisymmetric tensor field model, it is found
that classical equivalence with a vector theory breaks down at one-loop level
due to the presence of Lorentz violating terms. The final chapter of this
thesis is devoted to taking first steps towards exploring applications of DV
method in early universe cosmology. We calculate perturbatively the covariant
one-loop quantum gravitational effective action for a scalar field model
inspired by the recently proposed nonminimal natural inflation model. The
effective potential is evaluated taking into account the finite corrections,
and an order-of-magnitude estimate of the one-loop corrections reveals that
gravitational and non-gravitational corrections have same or comparable
magnitudes.
