  One essence in capturing the history of primordial fluctuations that arise
during inflation and eventually lead to formation of large scale structures in
the universe, relies on quantizing general relativity coupled to a scalar
field. This is a system that respects diffeomorphism invariance, the freedom of
choosing the coordinate system to work with without changing the physics. Hence
the way to handle its quantization requires an understanding of how to quantize
diffeomorphisms.Deciding on suitable coordinate choices and making sure that
this gauge fixing, which is unavoidable in any calculation, does not effect the
end result, is tricky. In this thesis we make full use of the effects of
diffeomorphism invariance of the theory on the primordial fluctuations to
pursue two different approaches that focus on treating perturbations after
gauge fixing. On one hand we work towards developing our understanding of how
to handle quantization in terms of Dirac quantization and Becchi, Rouet, Stora,
Tyutin (BRST) quantization. On another, we focus on how to generalize the
allowed interactions and understand the scales they bring in the era of
preheating that follows inflation, with effective field theory (EFT) methods on
cosmological backgrounds.
