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.
