Preheating in Hybrid Inflation

TL;DR

The paper systematically analyzes the onset of preheating in hybrid inflation by examining post-inflation oscillations of the fields $φ$ and $σ$ across three coupling regimes: $g^2 \ll λ$, $λ \ll g^2$, and $g^2 \sim λ$. It shows that direct preheating of $φ$ and $σ$ is typically inefficient, but explosive production of a coupled light field $χ$ is possible when the resonance conditions are met, described by Mathieu-type equations and modulated by cosmic expansion. In the extreme cases, energy transfer can be dominated by either $φ$- or $σ$-oscillations, while in the mixed regime chaotic dynamics can still permit resonant channels; a second-stage inflation scenario further suppresses reheating, potentially leaving a PBH-dominated early universe that reheats via PBH evaporation. These results delineate the parameter windows for efficient preheating and highlight novel cosmological consequences, such as primordial black hole formation, and underscore the need for numerical lattice studies to fully capture backreaction effects.

Abstract

We investigate the possibility of preheating in hybrid inflation. This scenario involves at least two scalar fields, the inflaton field $φ$, and the symmetry breaking field $σ$. We found that the behavior of these fields after inflation, as well as the possibility of preheating (particle production due to parametric resonance), depends crucially on the ratio of the coupling constant $λ$ (self-interaction of the field $σ$) to the coupling constant $g^2$ (interaction of $φ$ and $σ$). For $λ\gg g^2$, the oscillations of the field $σ$ soon after inflation become very small, and all the energy is concentrated in the oscillating field $φ$. For $λ\sim g^2$ both fields $σ$ and $φ$ oscillate in a rather chaotic way, but eventually their motion stabilizes, and parametric resonance with production of $χ$ particles becomes possible. For $λ\ll g^2$ the oscillations of the field $φ$ soon after inflation become very small, and all the energy is concentrated in the oscillating field $σ$. Preheating can be efficient if the effective masses of the fields $φ$ and $σ$ are much greater than the Hubble constant, or if these fields are coupled to other light scalar (or vector) fields $χ$. In the recently proposed hybrid models with a second stage of inflation after the phase transition, both preheating and usual reheating are inefficient. Therefore for a very long time the universe remains in a state with vanishing pressure. As a result, density contrasts generated during the phase transition in these models can grow and collapse to form primordial black holes. Under certain conditions, most of the energy density after inflation will be stored in small black holes, which will later evaporate and reheat the universe.

Figures (11)

• Figure 1: The trajectory in field space $(\sigma,\phi)$ after the end of inflation, along the ellipse $(\phi/\phi_c)^2 + (\sigma/\sigma_0)^2 = 1$. Oscillations occur around $\phi=0$ and $\sigma=\sigma_0$. This figure corresponds to $\lambda=1, g=8\times10^{-4}, M=10^{-3}\,M_{\rm P}, m=1.5\times10^{-7}\, M_{\rm P}$.
• Figure 2: The evolution after the end of inflation of $H/H_0$, $\phi/\phi_c$ and $\sigma/\sigma_0$, as a function of the number of oscillations of the field $\phi$, $N=\bar{m}_\phi t/2\pi$.
• Figure 3: The exponential growth of the occupation number $n_k$ of $\chi$-particles, for $k=\bar{m}_\phi$, as a function of the number of oscillations of the field $\phi$, $N=\bar{m}_\phi t/2\pi$. One can distinguish here the broad stochastic resonance regime. The typical value of the growth parameter is $\mu \simeq 0.13$ during the last stages of preheating.
• Figure 4: The evolution after the end of inflation of $H/H_0$, $\phi/\phi_c$ and $\sigma/\sigma_0$, as a function of the number of oscillations of the field $\sigma$, $N=\bar{m}_\sigma t/2\pi$. This figure corresponds to $\lambda=2\times10^{-16}, g =1.4\times10^{-5}, M=m=10^{-8}\, M_{\rm P}$.
• Figure 5: The evolution after the end of inflation of $H/H_0$, $\phi/\phi_c$ and $\sigma/\sigma_0$, as a function of $N=\bar{m}_\sigma t/2\pi$, in the case $M \gg H$. Note that the number of oscillations of the field $\sigma$ is about $N/3$, while the amplitude of oscillations of both fields is large. This figure corresponds to $\lambda=10^{-2}, g=1, m=10^3\,{\rm GeV}, m=1.3\times10^{11}\,{\rm GeV}$.