The non-linear dynamics of axion inflation: a detailed lattice study

TL;DR

This work develops and validates a lattice framework for fully inhomogeneous axion inflation with a shifted-symmetric inflaton coupled to a dark $U(1)$ gauge field via $\phi F_{\mu\nu}\tilde{F}^{\mu u}$, capturing local non-linear backreaction across weak, mild, and strong regimes. It reveals that local backreaction drives a novel electromagnetically slow-rolling phase with magnetic energy dominating the gauge sector and generates scale-dependent chirality and a longitudinal gauge mode, effects invisible to homogeneous backreaction or gradient-expansion formalisms. The authors provide detailed convergence studies, UV-coverage requirements, and practical criteria for switching from linear to non-linear evolution, including an intermediate BD cutoff to manage vacuum tails. The results have important implications for predicting gravitational waves and primordial black holes from axion inflation and demonstrate the necessity of fully local simulations to obtain reliable phenomenology. The methodology and parametrizations laid out here lay the groundwork for robust, first-principles predictions in axion-inflation scenarios with gauge-field backreaction.

Abstract

We study in detail the fully inhomogeneous non-linear dynamics of axion inflation, identifying three regimes: weak-, mild-, and strong-backreaction, depending on the duration of inflation. We use lattice techniques that explicitly preserve gauge invariance and shift symmetry, and which we validate against other computational methods of the linear dynamics and of the homogeneous backreaction regime. Notably, we demonstrate that the latter fails to accurately describe the truly local dynamics of strong backreaction. We investigate the convergence of simulations of local backreaction, determining the requirements to achieve an accurate description of the dynamics, and providing useful parametrizations of the delay of the end of inflation. Additionally, we identify key features emerging from a proper local treatment of strong backreaction: the dominance of magnetic energy against the electric counterpart, the excitation of the longitudinal mode, and the generation of a scale-dependent chiral (im)balance. Our results underscore the necessity to accurately capture the local nature of the non-linear dynamics of the system, in order to correctly assess phenomenological predictions, such as e.g. the production of gravitational waves and primordial black holes.

Figures (25)

• Figure 1: Evolution of the tachyonic excitation of the gauge field for the coupling $\alpha_{\Lambda}=18$. Solid coloured lines correspond the mean value from averaging over 5 random realizations on the lattice, with shaded bands representing the $1\sigma$ deviations. The mode by mode numerical solution in the 1D method is shown in dashed. The colour gradient represents the evolution from $\mathcal{N}_{\text{start}}$ (blue) to $\mathcal{N}=-0.3$ (orange), with gaps of $0.7$ e-foldings.
• Figure 2: Comparison for $\alpha_{\Lambda}=18$ in the linear regime between the different energy contributions: $\rho_{\rm K}$ in red, $\rho_{\rm V}$ in black, and $\rho_{\rm EM}$ in purple, between the lattice simulations (solid lines), and the numerical solutions of the mode-by-mode approach on a 1D grid (dashed lines). For the lattice simulations we include the mean over 5 random realizations and the corresponding $1\sigma$ deviations as a shaded band. We show the total energy density in the top panel, and the relative difference of the electromagnetic energy density between both methods in the bottom panel. The dashed vertical line corresponds to $\mathcal{N}_{\rm switch}$.
• Figure 3: Comparison for $\alpha_{\Lambda} = 15$ (left) and $18$ (right) of the terms related to the backreaction used to establish the time $\mathcal{N}_{\text{switch}}$. Top: the evolution of $\rho_{\rm K}$ (red), $\rho_{\rm V}$ (black) and $\rho_{\rm EM}$ (purple). Bottom: the evolution of $|\dot{\pi}_{\phi}|$ (red), $3H|\pi_{\phi}|$ (orange), and $m^2|\phi|$ (black) of Eq. (\ref{['eqn:eom1']}), together with $\frac{\alpha_{\Lambda}}{a^3 m_p} |\langle\vec{E} \cdot \vec{B}\rangle|$ (purple). The different purple lines represent different integration ranges: the dark solid purple lines are the contribution integrated for the range $[k_{\text{IR}}, k_{\text{BD}}]$, the light purple the integral for $[k_{\text{IR}}, k_{\text{UV}}]$, the dashed line is for only the excited IR contribution for $[k_{\rm IR}, k_{\rm cut}(\mathcal{N})]$, and the dotted purple line is the contribution of the excitation considering the whole cosmic history, i.e.$[k_{\rm min}, k_{\rm cut}(\mathcal{N})]$ (see text). The vertical dashed gray lines correspond to the times $\mathcal{N}_{\text{switch}}$.
• Figure 4: Comparison of the evolution of the gauge power spectra for simulations with different initialisation techniques: BD vacuum solution (solid lines) and already excited (solid). We include two realizations for each initialization method with the same colour and line style. The spectra are extracted from $\mathcal{N}_{\rm start}$ until the end of non-linear inflation with $0.7$ e-folds between lines.
• Figure 5: Comparison between the predictions of the homogeneous backreaction scheme on the lattice (solid) and of the gradient expansions formalism Gorbar:2021rlt (dashed) for $\alpha_{\Lambda}=15$ (left) and $\alpha_{\Lambda}=18$ (right). Top: the evolution of the different energy components normalised with respect the total energy density. Middle: the evolution of the inflationary parameter $\epsilon_H$. Bottom: the relative difference in $\epsilon_H$ (\ref{['eq:reldiff_epsilon']}). The vertical lines represent the end of inflation for each method.