Parametric Amplification of Metric Fluctuations During Reheating in Two Field Models

TL;DR

This work demonstrates that parametric resonance during reheating can exponentially amplify super-Hubble metric fluctuations in certain two-field and hybrid-inflation models, driven by resonant excitations in the isocurvature sector that feed the adiabatic curvature perturbation $\zeta$. By deriving Lamé-type equations for the Sasaki–Mukhanov variables and identifying regimes where long-wavelength modes resonate, the authors show that $\zeta$ and the non-adiabatic pressure $p\Gamma$ can grow exponentially when background fields remain nonzero. They articulate three practical rules governing efficient resonance and validate them with negative-coupling and hybrid-inflation-inspired constructions, revealing that massless or light isocurvature modes and nonzero background values are key. The findings imply that reheating dynamics can meaningfully alter large-scale perturbations and potentially influence inflationary predictions, with relevance to hybrid and string-inspired models where massless modes abound.

Abstract

We study the parametric amplification of super-Hubble-scale scalar metric fluctuations at the end of inflation in some specific two-field models of inflation, a class of which is motivated by hybrid inflation. We demonstrate that there can indeed be a large growth of fluctuations due to parametric resonance and that this effect is not taken into account by the conventional theory of isocurvature perturbations. Scalar field interactions play a crucial role in this analysis. We discuss the conditions under which there can be nontrivial parametric resonance effects on large scales.

Figures (10)

• Figure 1: Evolution of $(1+w) \zeta$ for the model of Eq. (3) as a function of the a-dimensional time $z=\sqrt{\lambda} M_{\rm pl} t$ for $\chi_0 = \dot \chi_0 = 0$ and $\phi_0 = 3.5 M_{\rm pl} \, , \, \dot \phi_0 = - .1 M_{\rm pl}$ as initial conditions for the background. The fluctuation $Q_\phi$ and $Q_\chi$ start in the adiabatic vacuum $40$ e-foldings before inflation ends. The wavenumber is $k = 10^2$, which corresponds to five times the Hubble radius at the beginning of the simulation. Note that the mode is far outside the Hubble radius at the end of inflation.
• Figure 2: Evolution in logarithmic scale of $(1 + w) \zeta$ for $\chi_0 = 2 \times 10^{-2} M_{\rm pl} \,, \, \dot \chi_0 = 0$ (top) and $\chi_0 = 2 \times 10^{-8} M_{\rm pl} \,, \, \dot \chi_0 \sim \lambda M_{\rm pl} \chi_0$ (bottom) as initial value for $\chi$. The initial condition for $\phi$ and $\dot \phi$ are the same as in Fig. 1 in both of the panels. The fluctuations $Q_\phi$ and $Q_\chi$ start in the adiabatic vacuum $40$ e-foldings before inflation ends. The wavenumber is $k = 10^2$, which corresponds to five times the Hubble radius at the beginning of the simulation. The growth of $\zeta$ is delayed in the second case because the background field, and consequently the mixing terms in Eq. (\ref{['phieq']}), are smaller than in the first case: in this way $Q_\chi$ takes longer to feed the growth of $Q_\phi$ and $\zeta$. The initial conditions for the second case correspond to the values obtained through renormalization arguments.
• Figure 3: Evolution in logarithmic scale of $Q_\phi$ (above) and $Q_\chi$ (below) for the second set of initial conditions of Fig. 2.
• Figure 4: Evolution in logarithmic scale of the total non-adiabatic pressure $p \Gamma$ for the first set of initial conditions of Fig. 2.
• Figure 5: Evolution in logarithmic scale of $(1 + w) \zeta$ for the model (\ref{['GPRmodel']}) for $g^2 = \lambda$, $\lambda_\chi=10^2 \lambda$ (top) and $\lambda = \lambda_\chi \,, g^2 = \lambda/2 (\sqrt{5} - 1)$ (bottom). The initial conditions for the background are $\phi_0 = 3.5 M_{\rm pl} \,, \, \dot \phi_0 = -.1 \sqrt{\lambda} M_{\rm pl} \,, \, \chi_0 = 2 10^{-8} \times M_{\rm pl}$ and $\dot \chi_0 =0$. The fluctuations $Q_\phi$ and $Q_\chi$ start in the adiabatic vacuum $40$ e-foldings before inflation ends. The wavenumber is $k = 10^2$, which roughly corresponds to five times the Hubble radius at the beginning of the simulation.