Backreaction and Stochastic Effects in Single Field Inflation

The formalism of stochastic inflation is a powerful tool for analyzing the backreaction of cosmological perturbations, and making precise predictions for inflationary observables. We demonstrate this with the simple $m^2\phi^2$ model of inflation, wherein we obtain an effective field theory for IR modes of the inflaton, which remains coupled to UV modes through a classical noise. We compute slow-roll corrections to the evolution of UV modes (i.e. quantum fluctuations), and track this effect from the UV theory to the IR theory, where it manifests as a correction to the classical noise. We compute the stochastic correction to the spectral index of primordial perturbations, finding a small effect, and discuss models in which this effect can become large. We extend our analysis to tensor modes, and demonstrate that the stochastic approach allows us to recover the standard tensor tilt $n_T$, plus corrections.