Perturbative limits on axion-SU(2) gauge dynamics during inflation from the energy density of spin-2 particles

TL;DR

The paper addresses whether perturbation theory remains valid for backreaction from spin-2 gauge perturbations in an axion-SU(2) inflation model, using the energy-density ratio $|\delta\rho_A|/\bar{\rho}_A$ as the breakdown criterion. It solves the full coupled system including tensor modes without slow-roll simplifications and regularizes tensor integrals to obtain $\mathcal{P}_\chi$, $\mathcal{J}_A$, and $\delta\rho_A$, finding that perturbativity generally breaks near the strong backreaction boundary, though in some configurations perturbation theory fails earlier. The results show a strong dependence on background configuration (cases (a) and (b)) and initial conditions, with BR and perturbative limits aligning for case (a) but potentially diverging for case (b). The work implies that reliable analyses of the BR regime require non-perturbative treatments, such as lattice simulations, and suggests that GW signals from BR could be enhanced in such non-perturbative dynamics, potentially affecting observational prospects like PTA interpretations.

Abstract

We investigate the conditions under which the perturbative treatment of the backreaction of spin-2 particles on the dynamics of an axion-SU(2) gauge field system breaks down during cosmic inflation. This condition is based on the ratio of the energy density of spin-2 particles from the SU(2) gauge field to that of the background field. The perturbative treatment breaks down when this ratio exceeds unity. We show that this occurs within a parameter space nearly identical to the strong backreaction regime identified in previous studies. However, in some cases, the ratio exceeds unity even before the system enters the strong backreaction regime. Our results suggest that attempts to study the strong backreaction regime using perturbation theory are necessarily limited. Reliable calculations require non-perturbative treatments, such as three-dimensional lattice simulations.

Figures (11)

• Figure 1: Time evolution of the background fields, $m_Q$ (top-left), $\tilde{\chi}$ (top-right), the backreaction term $\tilde{\cal J}_A$ divided by $2m_Q(1+m^2_Q)$ (middle-left), the backreaction term $\tilde{\cal P}_\chi$ divided by $\beta/\lambda$ (middle-right), and $\delta \rho_A/\bar{\rho}_A$ (bottom), as a function of $y=\ln aH$. The values of the parameters are $\lambda=100$, $\kappa=10^{-2}$, and $g_A=10^{-3}$. The input values of $m_Q^{\rm in}$ are indicated in the legend. The initial conditions are given by initial condition (I), in which $m_Q^{\rm in}=m_Q^{\rm st}(\tilde{\chi}=\tilde{\chi}_i)$, $\tilde{\chi}_i=0.3\pi$, and their derivatives are given by Eqs. \ref{['eq:dxiA']} and \ref{['eq:dchi']}.
• Figure 2: Same as Figure \ref{['fig:kappa001_i1_BG']}, but for $\kappa=10^{2}$.
• Figure 3: Same as Figure \ref{['fig:kappa100_i1_BG']}, but under initial condition (II), in which the initial conditions for the derivatives of the background fields are given by $m'_Q=\tilde{\chi}'=0$.
• Figure 4: Perturbative limit on the $g_A$-$m_{Q}^{\rm in}$ plane for case (a) with $\lambda=10^2$ under initial condition (I). The perturbative calculation breaks down, i.e. $\delta \rho_A>\bar{\rho}_A$, in the shaded regions above the solid lines for $\kappa=10^{-2}$ and $10^{-1}$, as indicated in the legend. The dashed line shows the upper limit of the stable solution found in Ref. Ishiwata:2021yne. Below this line, the backreaction terms have a negligible effect on the differential equations.
• Figure 5: Same as Figure \ref{['fig:limit_a_i1']}, but for case (b) under initial conditions (I) (left) and (II) (right). The shaded regions correspond to the breakdown of the perturbative calculation for $\kappa=10$ and $10^2$, as indicated in the legend.