Reheating and Preheating after Inflation

It is assumed that during inflation, all energy was contained in a slow-rolling inflaton field $\varphi$. The particles constituting the Universe are created due to interactions with the field $\varphi$ coherently oscillating after inflation. The leading channel of the particle production would be the non-perturbative regime of parametric resonance, preheating. Some of the recent developments in the theory of preheating are briefly reviewed. As a prototype we use the model ${\lambda\over 4}\phi^4 + {m^2 \over 2}\phi^2+ {g^2\over 2}\phi^2\chi^2$ in an expanding universe. In different domains of parameters the character of preheating $(\varphi \to \chi )$ or $(\varphi \to \delta \varphi )$ is different, ranging from regular Lam'{e} to stochastic parametric resonances. Fortunately, in many important cases simple analytic resonant solutions are found. Eventually, a picture which unifies the different regimes emerges.