Gravity, Parametric Resonance and Chaotic Inflation

TL;DR

This work addresses whether nonlinear gravitational dynamics modify preheating after inflation beyond scalar-field nonlinearities. It advances the study by solving the full Einstein–scalar system in a planar 1+1D setup for a $V(\phi)=\frac{\lambda}{4}\phi^4$ potential, comparing fully relativistic results to perturbative gauge-invariant analyses and to nonlinear field evolution in a rigid background. The key findings show gravity introduces measurable effects, notably enhanced growth of long-wavelength modes when all perturbation modes are initially excited, but does not reproduce primordial black hole formation; the resonance picture largely aligns with rigid-background expectations. The study provides a robust nonperturbative framework for examining gravitational backreaction in preheating and paves the way for extending to more complex resonant models and potential gravitational-wave implications.

Abstract

We investigate the possibility that nonlinear gravitational effects influence the preheating era after inflation. Our work is based on numerical solutions of the inhomogeneous Einstein field equations, and is free of perturbative approximations. The one restriction we impose is to limit the inhomogeneity to a single spatial direction. We compare our results to perturbative calculations and to solutions of the nonlinear field equations in a rigid (unperturbed) spacetime, in order to isolate gravitational phenomena. We consider two types of initial conditions: where only one mode of the field perturbation has a non-zero initial amplitude, and where all the modes begin with a non-zero amplitude. Here we focus on preheating following inflation driven by a scalar field with a quartic potential. We confirm the broad picture of preheating obtained from the nonlinear field equations in a rigid background, but gravitational effects have a measurable impact on the dynamics for both sets of initial data. The rigid spacetime results predict that the amplitude of a single initially excited mode drops rapidly after resonance ends, whereas in the relativistic case the amplitude is roughly constant. With all modes initially excited, the longest modes in the simulation grow much more rapidly in the relativistic calculation than with a rigid background. However, we see no evidence for the sort of gravitational collapse associated with the formation of primordial black holes. The numerical codes described here are easily extended to more complicated resonant models, which we will examine in the future.

Figures (11)

• Figure 1: The evolution of a single resonant mode is plotted as a function of $\eta$. The top panel shows the evolution of $Q_\phi$ predicted by perturbation theory. The second plot shows the evolution of $Q_\phi$ derived from the full Einstein field equations. The third panel superimposes the perturbative and non-perturbative results at the moment when perturbation theory breaks down.
• Figure 2: We plot the evolution of $Q_\phi$ ($\chi/a$) derived from the nonlinear field equations with a static background. The specific mode plotted is the same as that shown in fig. (\ref{['onemode1']}), but the normalization is different since we dropped the $a"/a$ term before solving equation (\ref{['chieofm']}).
• Figure 3: The power spectrum, $|\Phi_k|$ of the gauge-invariant metric perturbation is plotted against $k$. A single resonant mode is initially excited, but since we specify the initial perturbation in a specific gauge, all modes of $\Phi$ have some initial power. The plots show $|\Phi_k|$ at $\eta = 0.22, 414.8$ and $902.8$. The $x$-axis is scaled to that a mode with a wavelength equal to the initial Hubble radius has $k=1$.
• Figure 4: The power spectrum, $|\Phi_k|$, of the gauge-invariant metric perturbation is plotted against $k$, with all modes initially excited. The spectra are plotted at $\eta=$0.0183, 274.5, 439.2, 750.3 and 915, respectively.
• Figure 5: The power spectrum of $|Q_k|$, is plotted against $k$, with all modes initially excited. The spectra are plotted at the same times as in fig. (\ref{['manymode1']}).