Reheating in the higher-derivative inflationary models

In both the Starobinsky and the $R^2$ models of the inflationary universe, after the inflationary phase the Universe enters into a period in which the scalar curvature oscillates rapidly. The rapid oscillations in the geometry result in particle production when conformally noninvariant quantum fields are present. By solving the semiclassical back-reaction equations the temperature to which the Universe reheats due to the particle production is computed. It is also shown that the oscillations are damped essentially exponentially with the result that the Universe evolves into a classical radiation-dominated Friedmann phase. Both analytical and numerical solutions to the back-reaction equations are obtained, with the results in accurate agreement with each other. The numerical schemes presented here for solving the semiclassical back-reaction equations can be used for scalar quantum fields with arbitrary curvature couplings and arbitrary masses. They are expected to be useful for many future calculations of the evolution of the early Universe. The analytical analysis presented provides some insight into the coupling between the quantum field and the higher-derivative terms in the back-reaction equations. The implications of this are discussed.