Evolution of cosmological perturbations in a stage dominated by an oscillatory scalar field

In the investigation of the evolution of cosmological perturbations in inflationary universe models, the behavior of perturbations during the reheating stage is the most unclear point. In particular in the early reheating phase in which a rapidly oscillating scalar field dominates the energy density, the behavior of perturbations is not well known because their evolution equation expressed in terms of the curvature perturbation becomes singular. In this paper it is shown that, in spite of this singular behavior of the evolution-equation, the Bardeen parameter stays constant to good accuracy during this stage for superhorizon-scale perturbations except for a sequence of negligibly short intervals around the zero points of the time derivative of the scalar field. This justifies the conventional formula relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the Friedmann stage, except for possible corrections produced by the energy transfer from the scalar field to radiation in the late stage of reheating. It is further shown that outside the above sequence of time intervals, the behavior of the perturbations coincides to good accuracy with that for a perfect fluid system obtained from the original scalar held system by the WKB approximation and a spacetime averaging over a Hubble horizon scale.