Structure of resonance in preheating after inflation

We consider preheating in the theory $\frac{1}{4}\lambda {\phi }^4+\frac{1}{2}{g}^2{\phi }^2{\chi }^2$, where the classical oscillating inflaton field $\phi \mathrm{}\left(t\right)\mathrm{}$ decays into $\chi$ particles and $\phi$ particles. The parametric resonance which leads to particle production in this conformally invariant theory is described by the Lamé 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 $g^2/\lambda$. 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 μ. We construct the stability-instability chart in this theory for arbitrary $g^2/\lambda$. We give simple analytic solutions describing the resonance in the limiting cases $g^2/\lambda \ll 1$ and $g^2/\lambda \gg 1$, and in the theory with $g^2=3\mathrm{}\lambda$, and with $g^2=\lambda$. From the point of view of parametric resonance for $\chi$, the theories with $g^2=3\mathrm{}\lambda$ and with $g^2=\lambda$ have the same structure, respectively, as the theory $\frac{1}{4}\lambda {\phi }^4$, and the theory $(\lambda /4N)({\phi }_i^2{)}^2$ of an $N$-component scalar field ${\phi }_i$ in the limit $N\to \infty$. We show that in some of the conformally invariant theories such as the simplest model $\frac{1}{4}\lambda {\phi }^4$, the resonance can be terminated by the back reaction of produced particles long before $〈{\chi }^2〉$ or $〈{\phi }^2〉$ become of the order ${\phi }^2$. 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 ${(m}^2/2){\phi }^2\ll (\lambda /4){\phi }^4$. We give a classification of different resonance regimes for various relations between the coupling constants, masses, and the amplitude of the oscillating inflaton field $\phi$ in a general class of theories $\pm {(m}^2/2){\phi }^2+(\lambda /4){\phi }^4{+(g}^2/2){\phi }^2{\chi }^2$.