Ultimately, the degree of freedom dubbed the "inflaton", is the expectation value of some fundamental or composite degree of freedom, "slow-rolling" in a quantum effective potential. Rather than simply postulating such a potential, as is often done, it would be satisfying to be able to derive it from some underlying renormalizable theory. This brings about a number of questions related to quantum field theory in curved space-time.
I will briefly introduce cosmological perturbations and basic QFT in curved spaces, and try to clarify a number of concepts and approaches that enter in the story. Then I will show one attempt to calculate quantum corrections to the inflaton potential, including also the effect of scalar metric perturbations.