Perturbation approach to the Hill equation with a slowly varying parameter with application to the inflationary cosmology

Abstract

We develop a general perturbative approach to solving the Hill equation with a slowly varying parameter based on the Floquet theory and asymptotic expansions in the vicinity of the exact solutions with a “frozen” parameter. Equations of this type describe parametric resonance in a wide class of physical systems being under the influence of slowly varying factors. In particular, such equations describe the parametric instability of the fluctuations of the inflaton scalar field oscillating near a minimum of the effective potential in an expanding universe. We give a general procedure for constructing asymptotic solutions of the Hill equation and write out explicit formulas of the zero-order and first-order approximations. As an example, we consider the ϕ2–ϕ4 inflaton potential and construct the approximate solutions of the corresponding Lamé equation with the energy density of the oscillating scalar background as a slowly varying parameter. The obtained solutions are found to be in good agreement with the results of direct numerical integration. Based on these, we find the shape and characteristic size of a single scalar field fluctuation.