Preheating and locked inflation: an analytic approach towards parametric resonance

We take an analytic approach towards the framework of parametric resonance and apply it on preheating and locked inflation. A two-scalar toy model is analytically solved for the \lambda\phi^2\chi^2 coupling for the homogenous modes. The effects of dynamic universe background and backreaction are taken into account. We show the average effect of parametric resonance to be that \chi's amplitude doubles for each cycle of \phi. Our framework partly solves the preheating scenario, showing two distinct stages of preheating and making the parameters of preheating analytically calculable. It is demonstrated for slowroll inflation models, preheating is terminated, if by backreaction, typically in the 5th e-fold. Under our framework, a possible inhomogeneity amplification effect is also found during preheating, which both may pose strong constraints on some inflationary models and may amplify tiny existing inhomogeneities to the desired scale. For demonstration, we show it rules out the backreaction end of preheating of the quadratic slowroll inflation model with mass m\sim10^{-6}. For locked inflation, parametric resonance is found to be inhibited if \phi{} has more than one real component.