Effective field theory during inflation. II. Stochastic dynamics and power spectrum suppression

We obtain the nonequilibrium effective action of an inflatonlike 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 inflatonlike 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 inflatonlike field and dissipative terms that are suppressed by powers of $H/M$. We conjecture on the possibility that the large scale anomalies in the cosmic microwave background may originate in dissipative processes from inflaton coupling to sub-Hubble degrees of freedom.

Figure 1

The correlation functions $G^{>}(\overrightarrow{x}-\overrightarrow{y},\eta ,{\eta }^{\prime })$, $G^{<}(\overrightarrow{x}-\overrightarrow{y},\eta ,{\eta }^{\prime })$.

Figure 2

Effective Fermi vertex in the case for $J[\chi ]={\chi }^2:\mathcal{O}[\psi ]=\psi /\eta$ for $\mu =M_{\phi }/H\gg 1$.