Non-Abelian Gauge Field Inflation

TL;DR

This paper develops gauge-flation, an inflationary scenario where slow-roll is driven by a non-Abelian gauge field coupled to gravity and stabilized to preserve isotropy via an SU(2) triad, with a background A^a_{i}=\\psi(t)e^a_{i}. A phenomenological $F^4$ term provides an effective negative pressure ($P=-\\rho$) that enables sustained inflation, characterized by two parameters $g$ and $\\kappa$. The authors derive a consistent reduced background dynamics, analyze slow-roll analytically, and confirm robustness through numerical simulations, showing ample e-folds over a wide range of initial conditions. They formulate a complete gauge-invariant perturbation theory, revealing distinctive features such as a nonzero scalar anisotropic stress and parity-violating tensor modes, and compute the primordial spectra: ${\\cal P}_{\\mathcal R}$ with $n_s-1\\approx -2(\\epsilon-\\eta)$ and tensor spectra with $n_T\\approx -2\\epsilon$, including a chiral GW signal. Fitting to data yields predictions like $r>0.05$ and sub-Planckian gauge-field values, highlighting a natural, testable link between inflation and beyond-Standard-Model physics, while outlining avenues for embedding, reheating, and Planck-era tests.

Abstract

In [arXiv:1102.1513] we introduced an inflationary scenario, Non-Abelian Gauge Field Inflation or gauge-flation for short, in which slow-roll inflation is driven by non-Abelian gauge field minimally coupled to gravity. We present a more detailed analysis, both numerical and analytical, of the gauge-flation. By studying the phase diagrams of the theory, we show that getting enough number of e-folds during a slow-roll inflation is fairly robust to the choice of initial gauge field values. In addition, we present a detailed analysis of the cosmic perturbation theory in gauge-flation which has many special and interesting features compared the standard scalar-driven inflationary models. The specific gauge-flation model we study in this paper has two parameters, a cutoff scale Lambda and the gauge coupling g. Fitting our results with the current cosmological data fixes Λ\sim 10 H \sim 10^{15} GeV (H is the Hubble parameter) and g\sim 10^{-4}, which are in the natural range of parameters in generic particle physics beyond standard models. Our model also predicts a tensor-to-scalar ratio r>0.05, in the range detectable by the Planck satellite.

Figures (8)

• Figure 1: The classical trajectory for ${\psi_i=0.035,\dot\psi_i=-10^{-10};\ g=2.5\times10^{-3}, \kappa=1.733\times 10^{14}}$. These values correspond to a slow-roll trajectory with $H_i=3.4\times 10^{-5},\ \gamma_i=6.62,\ \epsilon_i=9.3\times 10^{-3},\ \delta_i=8.4\times 10^{-5}$. These are the values very close to the range for which the gauge-flation is compatible with the current cosmological and CMB data (cf. discussions of section \ref{['testing-the-model']}). Note that $\kappa,\ H_i$ and $\psi_i$ are given in the units of $M_{\rm pl}$.
• Figure 2: The classical trajectory for ${\psi_i=0.025,\dot\psi_i=-10^{-10};\ g=2.507\times10^{-3}, \kappa=1.3\times 10^{15}}$. These values correspond to a slow-roll trajectory with $H_i=3.63\times 10^{-5},\ \gamma_i=2.98,\ \epsilon_i=2.5\times 10^{-3},\ \delta_i=1.1\times 10^{-4}$. These figures show that it is possible to get arbitrarily large numbers of e-folds within the slow-roll phase of our gauge-flation model.
• Figure 3: Classical trajectory for ${\psi_i=8.0\times 10^{-2} ,\dot\psi_i=-10^{-4};\, g=4.004\times 10^{-4}\ , \kappa=4.73\times 10^{13}}$. These values correspond to a non-slow-roll trajectory with $\delta\sim 2$, $H_i=6.25\times 10^{-4},\ \epsilon_i=6.4\times 10^{-3}$. We start far from the slow-roll regime for which $\delta\sim \epsilon^2\ll 1$. This latter is also seen from the phase diagram (bottom left figure). Despite starting far from slow-roll regime, as we see from the top left figure, after an abrupt oscillation the field $\psi$ loses its momentum and falls into the standard slow-roll trajectory. As shown in the bottom right figure, for this case we get a large number of e-folds. Getting a large enough number of e-folds seems to be a fairly robust result not depending much on the initial value of $\delta$.
• Figure 4: $\tilde{h}_{R}$ undergoes a tachyonic growth phase in $\tilde{\tau}_2(\gamma)\geq -k\tau \geq 1$ (cf. \ref{['tilde-h-simple']} and \ref{['tilde-h-omega']}). In this figure, we have depicted $\tilde{\tau}_2$ vs. $\gamma$. The minimum is $\tilde{\tau}=5$ which is at $\gamma\simeq0.6$.
• Figure 5: This figure presents the tensor modes solution for $\psi=5\times 10^{-2}$, $\gamma=10$ and $H_0=10^{-6}$. In the top-left panel, we have the tensor field values $\frac{h_{_{R}}}{aH}$ and $\frac{\tilde{h}_{_{R}}}{aH}$ versus $-k\tau$, where Re and Im denote read and imaginary parts of the corresponding quantity. The small box presented the superhorizon behavior of the fields. The top-right panel shows $\frac{\pi^T_{_{R}}}{aH^3}$. In the bottom panels we presented the left-handed polarizations.