Quantum corrections to inflaton dynamics: The semiclassical approach and the semiclassical limit

Computations of quantum corrections to the cosmic microwave background spectrum and to scalar field dynamics during inflation very often take advantage of the “semiclassical” approach, where the metric fluctuations are simply omitted. On the other hand, a complete computation ought to take into account that the matter field perturbation and scalar metric perturbation together constitute a single physical degree of freedom. The question then naturally arises, in which sense the semiclassical approach is an approximation to the complete calculation, and whether there are specific limits where this is also a good approximation. We demonstrate this by explicitly computing the leading quantum radiative corrections to the evolution equation of the mean field (“condensate”) and the Friedmann equations taking into account scalar perturbations of both the matter field and the metric, and when omitting the latter. We find that the two agree in the limit $H\ll M_{\text{pl}}$, but one is not a limit of the other. We also find that in simple models of inflation, $H/M_{\text{pl}}$ is not small enough that the two approaches can be said to agree.