Quantum corrections to inflaton dynamics, the semi-classical approach and the semi-classical limit

Computations of quantum corrections to the CMB spectrum and to scalar field dynamics during inflation very often take advantage of the "semi-classical" 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 semi-classical approach is an approximation to the complete calculation, and whether there are specific limits where this is also a good approximation. We consider the quantum corrected dynamics of interacting scalar fields in an expanding inflationary background. 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 << M_pl , but one is not a limit of the other. We also find that in simple models of inflation, H/M_pl is not small enough that the two approaches can be said to agree. By direct comparison, we demonstrate how to interpret the "semi-classical" approach often employed in more complex computations as a well-defined approximation, and quantify its validity.