Quantum Fluctuations of a Self-interacting Inflaton

We present a method to analytically compute the quantum corrected two-point correlation function of a scalar field in leading order at each loop in a homogeneous, isotropic and spatially flat spacetime where the expansion rate is time dependent and express the quantum corrected power spectrum $\Delta^2(k)$ as a time derivative of the coincident correlation function evaluated at time $t_k$ of the first horizon crossing of a mode with comoving wave number $k$. To facilitate the method, we consider the simplest version of inflation driven by a massive, minimally coupled inflaton endowing a quartic self-interaction---with positive or negative self-coupling. We compute the quantum corrected two-point correlation function, power spectrum, spectral index $n(k)$ and the running of the spectral index $\alpha(k)$ for the inflaton fluctuations at one-loop order. Numerical estimates of the $n(k)$ and $\alpha(k)$ and the cosmological measurements are in agreement, within reasonable ranges of values for the physical parameters in the model.