Quantum effects of mass on scalar field correlations, power spectrum, and fluctuations during inflation

Abstract

We consider an infrared truncated massive minimally coupled scalar field with a quartic self-interaction $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ in the locally de Sitter background of an inflating universe. We compute the quantum-corrected two-point correlation function of the scalar analytically at one- and two-loop order. The one-loop correlator at a fixed comoving separation asymptotes to zero in the massive case but grows like $\ensuremath{-}\ensuremath{\lambda}{\mathrm{ln}}^{2}(a)$ at late times in the massless limit, where $a$ is the cosmic scale factor. For a fixed physical distance, on the other hand, it grows like $\ensuremath{-}\ensuremath{\lambda}{\mathrm{ln}}^{3}(a)$ at late times in the massless limit. This growth is severely suppressed in the massive case. In fact, the one-loop correlator asymptotes effectively to zero for masses larger than half the expansion rate. We use our quantum-corrected correlation function to compute the stochastic contributions to the power spectrum, spectral index, and running of the spectral index. The spectrum of fluctuations of a massive scalar is red tilted at tree and one-loop order. As the mass decreases, so does the tilt. In the massless limit, although the tilt vanishes at tree order, the one-loop correction still induces a red tilt. Thus, the amplitudes of scalar field fluctuations---massive or massless---grow toward the larger scales. We also compute the variance of the field variation at tree and one-loop order. The one-loop variance implies that the tree-order result decreases when quantum corrections are included. Hence, the actual effect that any local observer perceives in the field strength as fluctuations happen does not deviate from the average effect as much as the tree-order variance implies.