Structure of Resonance in Preheating after Inflation

Abstract

We consider preheating in the theory $1/4 λϕ^4 + 1/2 g^2ϕ^2χ^2 $, where the classical oscillating inflaton field $ϕ$ decays into $χ$-particles and $ϕ$-particles. The parametric resonance which leads to particle production in this conformally invariant theory is described by the Lame 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 $g^2/λ$. We construct the stability/instability chart in this theory for arbitrary $g^2/λ$. We give simple analytic solutions describing the resonance in the limiting cases $g^2/λ\ll 1$ and $g^2/λ\gg 1$, and in the theory with $g^2=3λ$, and with $g^2 =λ$. From the point of view of parametric resonance for $χ$, the theories with $g^2=3λ$ and with $g^2 =λ$ have the same structure, respectively, as the theory $1/4 λϕ^4$, and the theory $λ/(4 N) (ϕ^2_i)^2$ of an N-component scalar field $ϕ_i$ in the limit $N \to \infty$. We show that in some of the conformally invariant theories such as the simplest model $1/4 λϕ^4$, the resonance can be terminated by the backreaction of produced particles long before $<χ^2>$ or $<ϕ^2 >$ become of the order $ϕ^2$. We analyze the changes in the theory of reheating in this model which appear if the inflaton field has a small mass.