Structure of resonance in preheating after inflation

We consider preheating in the theory $\frac{1}{4}\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}+\frac{1}{2}{g}^{2}{\ensuremath{\varphi}}^{2}{\ensuremath{\chi}}^{2}$, where the classical oscillating inflaton field $\ensuremath{\varphi}(t)$ decays into $\ensuremath{\chi}$ particles and $\ensuremath{\varphi}$ particles. The parametric resonance which leads to particle production in this conformally invariant theory is described by the Lam\'e equation. It significantly differs from the resonance in the theory with a quadratic potential. The structure of the resonance depends in a rather nontrivial way on the parameter ${g}^{2}/\ensuremath{\lambda}$. We find an ``unnatural selection'' rule: the most efficient creation of particles occurs not for particles which have the strongest coupling to the inflaton field, but for those which have the greatest characteristic exponent \ensuremath{\mu}. We construct the stability-instability chart in this theory for arbitrary ${g}^{2}/\ensuremath{\lambda}$. We give simple analytic solutions describing the resonance in the limiting cases ${g}^{2}/\ensuremath{\lambda}\ensuremath{\ll}1$ and ${g}^{2}/\ensuremath{\lambda}\ensuremath{\gg}1$, and in the theory with ${g}^{2}=3\ensuremath{\lambda}$, and with ${g}^{2}=\ensuremath{\lambda}$. From the point of view of parametric resonance for $\ensuremath{\chi}$, the theories with ${g}^{2}=3\ensuremath{\lambda}$ and with ${g}^{2}=\ensuremath{\lambda}$ have the same structure, respectively, as the theory $\frac{1}{4}\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$, and the theory $(\ensuremath{\lambda}/4N)({\ensuremath{\varphi}}_{i}^{2}{)}^{2}$ of an $N$-component scalar field ${\ensuremath{\varphi}}_{i}$ in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We show that in some of the conformally invariant theories such as the simplest model $\frac{1}{4}\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$, the resonance can be terminated by the back reaction of produced particles long before $〈{\ensuremath{\chi}}^{2}〉$ or $〈{\ensuremath{\varphi}}^{2}〉$ become of the order ${\ensuremath{\varphi}}^{2}$. We analyze the changes in the theory of reheating in this model which appear if the inflaton field has a mass $m$. In this case the conformal invariance is broken, and the resonance may acquire the features of stochasticity and intermittancy even if the mass is very small, so that ${(m}^{2}/2){\ensuremath{\varphi}}^{2}\ensuremath{\ll}(\ensuremath{\lambda}/4){\ensuremath{\varphi}}^{4}$. We give a classification of different resonance regimes for various relations between the coupling constants, masses, and the amplitude of the oscillating inflaton field $\ensuremath{\varphi}$ in a general class of theories $\ifmmode\pm\else\textpm\fi{}{(m}^{2}/2){\ensuremath{\varphi}}^{2}+(\ensuremath{\lambda}/4){\ensuremath{\varphi}}^{4}{+(g}^{2}/2){\ensuremath{\varphi}}^{2}{\ensuremath{\chi}}^{2}$.