Parametric Amplification of Density Perturbation in the Oscillating   inflation
  We study the adiabatic density perturbation in the {\it oscillating inflation}, proposed by Damour and Mukhanov. The recent study of the cosmological perturbation during reheating shows that the adiabatic fluctuation behaves like as the perfect fluid and no significant amplification occurs on super-horizon scales. In the oscillating inflation, however, the accelerated expansion takes place during the oscillating stage and there might be a possibility that the parametric amplification on small scales affects the adiabatic long-wavelength perturbation. We analytically show that the density perturbation neglecting the metric perturbation can be amplified by the parametric resonance and the instability band becomes very broad during the oscillating inflation. We examined this issue by solving the evolution equation for perturbation numerically. We found that the parametric resonance is strongly suppressed for the long wave modes comparable to the Hubble horizon. The result indicates that the metric perturbation plays a crucial role for the evolution of scalar field perturbation. Therefore, in the single field case, there would be no significant imprint of the oscillating inflation on the primordial spectrum of the adiabatic perturbation. However, it could be expected that the oscillating inflation in the multi-field system gives the enormous amplification on large scales, which may lead to the production of the primordial black holes. 
A.Taruya