Towards the Theory of Reheating After Inflation

TL;DR

This work develops a comprehensive analytical framework for reheating after inflation that goes beyond perturbation theory to treat broad, stochastic parametric resonance (preheating) in an expanding universe. By introducing an adiabatic/Bogoliubov formulation and a parabolic-scattering interpretation, the authors quantify stochastic resonance, backreaction, and rescattering, showing how large initial amplitudes yield rapid, nonperturbative particle production and how the system transitions to a perturbative reheating regime. Key results include the identification of stochastic resonance with typical growth rates around $\mu_k\sim\mathcal{O}(0.1)$, a two-stage preheating scenario, and the possibility of producing superheavy particles ($M\gg m$) under certain couplings, with significant implications for nonthermal phase transitions and baryogenesis. The study highlights that preheating can substantially alter the early-universe energy budget and thermal history, but that the final reheating temperature emerges from the later, perturbative decay of remnants rather than from the peak of resonance.

Abstract

Reheating after inflation occurs due to particle production by the oscillating inflaton field. In this paper we describe the perturbative approach to reheating, and then concentrate on effects beyond the perturbation theory. They are related to the stage of parametric resonance called preheating. It may occur in an expanding universe if the initial amplitude of oscillations of the inflaton field is large enough. We investigate a simple model of a massive inflaton field coupled to another scalar field X. Parametric resonance in this model is very broad. It occurs in a very unusual stochastic manner, which is different from the parametric resonance in the case when the expansion of the universe is neglected. Quantum fields interacting with the oscillating inflaton field experience a series of kicks which occur with phases uncorrelated to each other. We call this process stochastic resonance. We develop the theory of preheating taking into account the expansion of the universe and backreaction of produced particles, including the effects of rescattering. The process of preheating can be divided into several distinct stages. At the first stage the backreaction of created particles is not important. At the second stage backreaction increases the frequency of oscillations of the inflaton field, which makes the process even more efficient than before. Then the effects related to scattering of X-particles terminate the resonance. We calculate the density of X-particles and their quantum fluctuations with all backreaction effects taken into account. This allows us to find the range of masses and coupling constants for which one has efficient preheating. In particular, under certain conditions this process may produce particles with a mass much greater than the mass of the inflaton field.

Figures (12)

• Figure 1: Oscillations of the field $\phi$ after inflation in the theory ${m^2\phi^2\over 2}$. 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 $m^{-1}$.
• Figure 2: Narrow parametric resonance for the field $\chi$ in the theory ${m^2\phi^2\over 2}$ in Minkowski space for $q \sim 0.1$. Time is shown in units of $m/2\pi$, which is equal to the number of oscillations of the inflaton field $\phi$. For each oscillation of the field $\phi(t)$ the growing modes of the field $\chi$ oscillate one time. The upper figure shows the growth of the mode $\chi_k$ for the momentum $k$ corresponding to the maximal speed of growth. The lower figure shows the logarithm of the occupation number of particles $n_k$ in this mode, see Eq. (\ref{['number']}). As we see, the number of particles grows exponentially, and $\ln n_k$ in the narrow resonance regime looks like a straight line with a constant slope. This slope divided by $4\pi$ gives the value of the parameter $\mu_k$. In this particular case $\mu_k \sim 0.05$, exactly as it should be in accordance with the relation $\mu_k \sim q/2$ for this model.
• Figure 3: Broad parametric resonance for the field $\chi$ in Minkowski space for $q \sim 2\times10^2$ in the theory ${m^2\phi^2\over 2}$. For each oscillation of the field $\phi(t)$ the field $\chi_k$ oscillates many times. Each peak in the amplitude of the oscillations of the field $\chi$ corresponds to a place where $\phi(t) = 0$. At this time the occupation number $n_k$ is not well defined, but soon after that time it stabilizes at a new, higher level, and remains constant until the next jump. A comparison of the two parts of this figure demonstrates the importance of using proper variables for the description of preheating. Both $\chi_k$ and the integrated dispersion $\langle\chi^2\rangle$ behave erratically in the process of parametric resonance. Meanwhile $n_k$ is an adiabatic invariant. Therefore, the behavior of $n_k$ is relatively simple and predictable everywhere except the short intervals of time when $\phi(t)$ is very small and the particle production occurs. In our particular case, the average rate of growth of $n_k$ is close to the maximal possible rate for our model, $\mu_k \sim 0.3$.
• Figure 4: Early stages of parametric resonance in the theory ${m^2\phi^2\over 2}$ in an expanding universe with scale factor $a \sim t^{2/3}$ for $g = 5\times 10^{-4}$, $m = 10^{-6} M_p$. According to our conventions (\ref{['Q']}), initial value of the parameter $q$ in this process is $q_0 \sim 3\times 10^3$. Note that the number of particles $n_k$ in this process typically increases, but it may occasionally decrease as well. This is a distinctive feature of stochastic resonance in an expanding universe. A decrease in the number of particles is a purely quantum mechanical effect which would be impossible if these particles were in a state of thermal equilibrium.
• Figure 5: The same process as in Fig. \ref{['fig4']} during a longer period of time. The parameter $q = {g^2\Phi^2\over 4 m^2}$ decreases as $t^{-2}$ during this process, which gradually makes the broad resonance more and more narrow. As before, we show time $t$ in units of ${2\pi\over m}$, which corresponds to the number of oscillations of the inflaton field.