Nonlinear physics of axion inflation

Abstract

An axion-like field coupled to an Abelian gauge field provides one of the simplest inflationary models that is free from the eta problem and possesses an efficient reheating mechanism. For sufficiently large coupling, this system enters a regime of strong gauge-field backreaction, exhibiting rich and intricate dynamics. In this work, we employ a semi-analytical method, the gradient-expansion formalism, to perform a comprehensive parameter scan and determine the precise conditions under which backreaction sets in. Previous studies have shown that the Anber-Sorbo solution, in which the potential-gradient force acting on the axion is balanced by Hubble friction and gauge-field backreaction, is unstable. Here, we broaden the parameter space and identify a new region in which the Anber-Sorbo solution remains stable despite strong backreaction. Although our analysis is restricted to a homogeneous axion field and to perturbations that depend only on time, we expect that this stability property can be extrapolated to generic time- and space-dependent perturbations. This newly identified region therefore represents a distinct type of backreaction - stable backreaction - which may not be accompanied by the rapid growth of perturbations. We further investigate the nonlinear behavior of solutions in the backreaction regime in a toy model (de Sitter, constant potential slope, no axion gradients), identifying a supercritical Hopf bifurcation at the onset of instability, a nontrivial limit cycle in the unstable regime, and burst-like oscillatory dynamics. Finally, we present a more stringent criterion for the onset of (unstable) backreaction, based on crossing the instability threshold, and apply this criterion to two benchmark inflationary models.

Figures (7)

• Figure 1: Stability diagram in the LGEF approach with $n_\mathrm{cut}=70$. The background color shows the largest real part among all Lyapunov exponents, $\operatorname{Re}\zeta_1$, as a function of the axion--vector coupling in Hubble units, $b=\beta H/M_p$, and the potential gradient in Hubble units, $v = -V'/H^3$. The solid and dashed parts of the red line correspond to $\operatorname{Re}\zeta_1=0$ and mark the boundary of the unstable region when approached from the ordinary SR regime and from the stable backreaction regime, respectively. The red dot at the merger of the solid and dashed red curves denotes the codimension-2 bifurcation point (see the discussion in Sec. \ref{['subsec:bifurcation']}). The solid black lines show contours of constant $\overline{\xi}$. The solid green segment denotes the crossing of the instability band for which the bifurcation diagram is constructed in Sec. \ref{['subsec:bifurcation']} (cf. Fig. \ref{['fig:bifurcation']}). The purple stars indicate the parameter points studied in the strong-backreaction regime in Sec. \ref{['subsec:bursting']} (cf. Fig. \ref{['fig:bursting']}).
• Figure 2: Parameter-space regions corresponding to negligible backreaction (white), unstable backreaction (yellow shaded), and stable backreaction (green shaded). The blue solid line indicates the conventional backreaction criterion $\delta_{\mathrm{KG}}=1$. The vertical dashed black line marks the value of $v$ fixed by the CMB normalization. The gray region corresponds to excessively large vacuum scalar perturbations, $\mathcal{P}_{\mathcal{R}}>10^{-2}$, where the approximation of a homogeneous inflaton field breaks down even in the absence of gauge fields. The rest of the notation is the same as in Fig. \ref{['fig:instability1']}.
• Figure 3: Panel (a): Threshold value of the instability variable, $\overline{\xi}_\mathrm{thr}$, at which the system enters the strong-backreaction regime, shown as a function of the axion--vector coupling $b=\beta H/M_{\mathrm{P}}$. The blue and red curves have the same meaning as in Fig. \ref{['fig:backreaction']}. The blue solid line corresponds to the old backreaction criterion, $\delta_{\mathrm{KG}}=1$, defined by the relative magnitude of the gauge-field contribution in the Klein--Gordon equation. The red solid line represents the new criterion, $\operatorname{Re}\zeta_1 = 0$, based on the stability analysis of linear perturbations. The red dashed curve also corresponds to $\operatorname{Re}\zeta_1 = 0$, but occurring within the strong-backreaction regime; it therefore marks the boundary between the SB and UB regimes. The yellow dashed-dotted line shows the approximate expression given in Eq. \ref{['eq:fit-xi-old-BR']}. The black dashed-dotted line corresponds to the fit function in Eq. \ref{['eq:fit-xi-new-BR']}, while the black dotted line represents the fit in Eq. \ref{['eq:fit-xi-stable-BR']}. Panel (b): Threshold value of the potential-gradient parameter, $v_\mathrm{thr}$, as a function of $b$. The blue curve and the solid and dashed red curves have the same interpretation as in panel (a). The yellow dashed-dotted line shows the approximate relation given in Eq. \ref{['eq:fit-v-old-BR']}, the black dashed-dotted line corresponds to the fit function in Eq. \ref{['eq:fit-v-new-BR']}, and the black dotted line represents the fit in Eq. \ref{['eq:fit-v-stable-BR']}.
• Figure 4: Panel (a): Bifurcation diagram illustrating the transformation of a stable solution $\xi=\overline{\xi}(b,v)$ into an unstable one, followed by the emergence of a new stable solution, as the axion--vector coupling $b=\beta H/M_{\mathrm{P}}$ is gradually increased for a fixed gradient parameter $v=200$ (the parameters vary along the green segment shown in Figs. \ref{['fig:instability1']} and \ref{['fig:backreaction']}). The solid green line shows the dependence $\overline{\xi}(b,v=200)$ when it corresponds to a stable solution, while the dashed green line indicates the same branch when it is unstable. The solid red curves show the maximum (upper) and minimum (lower) values of $\xi$ attained along the limit cycle within the unstable region. Red dots denote values extracted directly from numerical solutions of the full GEF system with truncation order $n_{\mathrm{tr}}=99$, while the red straight segments between them are obtained by interpolation. The figure demonstrates two supercritical Hopf bifurcations facing each other on opposite sides of the instability region. The two vertical solid lines mark $b=0.11$ and $b=0.17$, for which the phase-space trajectories are shown in panels (b) and (c), respectively. Panels (b) and (c): Phase-space trajectories in the three-dimensional space spanned by $(\xi,\, d\xi/dN,\, \rho_E/\rho_{E,0}-1)$, starting from initial conditions slightly displaced from the stationary solution $\overline{\xi}(b,v)$. As time increases, the trajectories are followed in a clockwise direction (color changing from red to purple). Panel (b) shows convergence toward a stable fixed point, whereas panel (c) shows convergence toward a limit cycle after the trajectory escapes the unstable fixed point.
• Figure 5: Column (a): Time evolution of the instability variable $\xi$ during one period of motion on the late-time attractor. Column (b): Projection of the late-time attractor (limit manifold) in the three-dimensional phase space spanned by $(\xi,\,d\xi/dN,\,\rho_E/\rho_{E,0})$; the color coding matches column (a). Column (c): Fourier spectra of the periodic solutions shown in column (a), with the fundamental frequency $\omega_1 = 2\pi/T$. Rows 1 to 4 correspond to the four parameter points marked by purple stars in Figs. \ref{['fig:instability1']} and \ref{['fig:backreaction']}, for $v=3.6\times 10^7$ and $b \in \{1.4,\,1.9,\,3.3,\,24\}\times10^{-6}$, respectively.