Structure of Resonance in Preheating after Inflation

TL;DR

This work analyzes preheating after inflation in a conformally invariant model with ${\lambda\over4}\phi^4+{g^2\over2}\phi^2\chi^2$, showing that resonance is governed by the ratio ${g^2\over\lambda}$ and is described by a Lame equation. By transforming to conformal variables, the expansion of the universe drops out and the fluctuation dynamics reduce to a Minkowski-space problem, enabling a detailed stability chart in the $(\kappa^2, {g^2\over\lambda})$ plane and analytic solutions for special ratios ${g^2\over\lambda}= {n(n+1)\over2}$, including ${g^2=\lambda}$ and ${g^2=3\lambda}$. The authors derive closed-form solutions for these cases, identify limiting behaviors for ${g^2\over\lambda}\ll1$ (Mathieu-like) and ${g^2\over\lambda}\gg1$ (parabolic-scattering), and examine how backreaction reshapes and eventually terminates resonance, leading to a restructuring of instability bands. They also extend the discussion to a massive inflaton, showing that the presence of a mass can alter the resonance from regular to stochastic regimes, with important implications for the early thermal history of the universe. Overall, the paper provides a comprehensive, parameter-sensitive map of preheating dynamics in conformal theories and connects analytic results to broader nonperturbative reheating phenomena.

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.

Figures (15)

• Figure 1: Oscillations of the field $\phi$ after inflation in the theory ${\lambda\phi^4\over 4}$. The value of the scalar field here and in all other figures in this paper is measured in units of $M_p$, time is measured in units of $(\sqrt\lambda M_p)^{-1}$.
• Figure 2: The exact solution (\ref{['elliptic']}) for the oscillations of the inflaton field after inflation in the conformally invariant theory ${1 \over 4} \lambda \phi^4$. We show the field in rescaled conformal field and time variables, $f(x)= {\varphi\over\tilde{\varphi}}$ (solid curve) and the first term, $\cos(0.8472x)$, in its harmonic expansion (\ref{['series']}) (dotted curve).
• Figure 3: The typical resonant production of particles at the particular choice of rescaled comoving momentum $\kappa^2 = 1.6$, and the parameter $\frac{g^2}{\lambda} = 3$. The upper plot shows the amplification of the real part of the eigenmode $X_k(x)$. The lower plot shows the logarithm of the comoving particle number density, $n_k$, calculated with formula (\ref{['number']}). The number of particles grows exponentially, $\log n_k \approx {2 \mu_k x}$. In this case, $\mu_k \approx 0.035$.
• Figure 4: The stability/instability chart for the Lame equation for fluctuations $X_k(x)$ in the variables $(\kappa^2, {g^2 \over \lambda})$, obtained from the numerical solution of equation (\ref{['fluc3']}). Shaded (unshaded) areas are regions of instability (stability). For instability bands, the darker shade implies a larger characteristic exponent $\mu_k$. Altogether, there are $10$ color steps. One color step corresponds to the increment $\Delta \mu_k=0.0237$, so the darkest shade corresponds to maximal $\mu_k=0.237$, the least dark shade in the instability bands corresponds to $\mu_k=0.009$. For positive $\kappa^2$, there is only one instability band for the particular values of the parameter $\frac{g^2}{\lambda} = 1$ and $3$. This occurs because the higher bands shrink into nodes as $\frac{g^2}{\lambda}$ approaches $1$ and $3$.
• Figure 5: Slices of the stability/instability chart, Fig. \ref{['inst']} , reveal the dependence of the characteristic exponent, $\mu_k$, on $\kappa^2$ for several particular values of $\frac{g^2}{\lambda}$. For top panel $\frac{g^2}{\lambda}= 1.0$, $1.5$, $2.0$, $2.5$, and $3.0$, labeled $a$ through $e$ respectively. The numerical curves $a$ and $e$ for $\frac{g^2}{\lambda} = 1$ and $\frac{g^2}{\lambda} = 3$ are identical to the analytic predictions (\ref{['mu2']}) of Section VI and (\ref{['mu3']}) of Section VII. For lower panel $\frac{g^2}{\lambda} = 6.0$, $7.0$, $8.0$, $9.0$, and $10$, labeled $a$ through $e$ respectively.