Analytical study of the parametric instability of an oscillating scalar field in an expanding universe

We investigate the dynamics of the perturbations of the inflaton scalar field oscillating around a minimum of its effective potential in an expanding universe. With the assumption of smallness of the ratio of the Hubble parameter to the oscillation frequency we apply the technique of separation of fast and slow motions. Considering the oscillation phase and the energy density as fast and slow variables we derive the Hill equation for the fluctuation modes in which the energy density is treated as a slowly varying parameter. We develop a general perturbative approach to solving the equations of this type, which is based on the Floquet theory and asymptotic expansions in the vicinity of the solutions with the "frozen" parameters. As an example, we consider the $\phi ^{2}-\phi ^{4}$ potential and construct the approximate solutions of the corresponding Lam\'{e} equation. The obtained solutions are found to be in a good agreement with the results of the direct numerical integration.