An effective field theory during inflation II: stochastic dynamics and power spectrum suppression

We obtain the non-equilibrium effective action of an inflaton like scalar field --the system-- by tracing over sub Hubble degrees of freedom of "environmental" light scalar fields. The effective action is stochastic leading to effective Langevin equations of motion for the fluctuations of the inflaton-like field, with self-energy corrections and stochastic noise correlators that obey a de Sitter space-time analog of a fluctuation dissipation relation. We solve the Langevin equation implementing a dynamical renormalization group resummation of the leading secular terms and obtain the corrections to the power spectrum of super Hubble fluctuations of the inflaton field, $\mathcal{P}(k;\eta) = \mathcal{P}_0(k)\,e^{-\gamma(k;\eta)}$ where $\mathcal{P}_0(k)$ is the nearly scale invariant power spectrum in absence of coupling. $\gamma(k;\eta)>0$ describes the suppression of the power spectrum, it features Sudakov-type double logarithms and entails violations of scale invariance. We also obtain the effective action for the case of a heavy scalar field of mass $M \gg H$, this case yields a local "Fermi" limit with a very weak self-interaction of the inflaton-like field and dissipative terms that are suppressed by powers of $H/M$. We conjecture on the possibility that the large scale anomalies in the CMB may originate in dissipative processes from inflaton coupling to sub-Hubble degrees of freedom.