No one loop back reaction in chaotic inflation

We use an invariant operator to study the quantum gravitational back reaction to scalar perturbations during chaotic inflation. Our operator is the inverse covariant d’Alembertian expressed as a function of the local value of the inflaton. In the slow roll approximation this observable gives $-{1/(2H}^2)$ for an arbitrary homogeneous and isotropic geometry; hence it is a good candidate for measuring the local expansion rate even when the spacetime is not perfectly homogeneous and isotropic. Corrections quadratic in the scalar creation and annihilation operators of the initial value surface are included using the slow-roll and long wavelength approximations. The result is that all terms which could produce a significant secular back reaction cancel from the operator, before one even takes its expectation value. Although it is not relevant to the current study, we also develop a formalism for using stochastic samples to study back reaction.