Gauge-flation and Cosmic No-Hair Conjecture

TL;DR

This work investigates whether the gauge-flation model, driven by a non-Abelian SU(2) gauge field, respects the cosmic no-hair conjecture. Using analytic arguments and numerical simulations in a Bianchi type-I background, it shows that isotropic FLRW inflation is a global attractor and that anisotropies are damped within a few e-folds, with a bounded shear Σ on the order of the slow-roll parameter ε. The findings imply negligible statistical anisotropy in the CMB for gauge-flation and extend Wald's no-hair ideas to vector-field-driven inflation. The results bolster the viability of gauge-flation as an inflationary framework compatible with current observational bounds and motivate further generalization to broader anisotropic backgrounds.

Abstract

Gauge-flation, inflation from non-Abelian gauge fields, was introduced in [1,2]. In this work, we study the cosmic no-hair conjecture in gauge-flation. Starting from Bianchi-type I cosmology and through analytic and numeric studies we demonstrate that the isotropic FLRW inflation is an attractor of the dynamics of the theory and that the anisotropies are damped within a few e-folds, in accord with the cosmic no-hair conjecture.

Figures (5)

• Figure 1: The phase diagram in the $\lambda'-\lambda$ plane, the vertical axis is $\lambda'$ and the horizontal is $\lambda$. Both figures show existence of attractor at $|\lambda|=1$, corresponding to the isotropic FLRW background. The left figure shows the phase diagram over a large range of values for $\lambda$, while the right figure shows the phase diagram for $\lambda$ in the vicinity of the attractor solution $\lambda=1$. The left figure explicitly exhibits the $\lambda'(\lambda)=-\lambda'(-\lambda)$ symmetry.
• Figure 2: The classical trajectory for ${\kappa = 1.733\times10^{14},\ g = 2.5\times10^{-3},\ \psi_0 = 0.034,\ \dot\psi_0 = -10^{-10},}$${\lambda_0 = 1.05,\ \dot\lambda_0 = 1.5\times10^{-10}}$. These values corresponds to a trajectory with $\dot\alpha_0=7.5\times 10^{-5},\ \epsilon_0=3\times 10^{-3}$ and $\dot\sigma_0=-8.6\times10^{-9}$.
• Figure 3: The classical trajectory for ${\kappa=0.43\times10^{14},\ g = 5\times10^{-3},\ \psi_0 = 0.033,\ \dot\psi_0 = -0.8\times10^{-10},}$${\lambda_0 = 0.9,\ \dot\lambda_0 = 6\times10^{-10}}$. These values give a trajectory with $\dot\alpha_0=7\times 10^{-5},\ \epsilon_0=8\times 10^{-3}$, and $\dot\sigma_0=1.8\times10^{-8}$.
• Figure 4: The classical trajectory for ${\kappa = 3.77\times10^{15},\ g = 10^{-1},\ \psi_0 = 0.6\times 10^{-3},\ \dot\psi_0 = 10^{-10},}$${\lambda_0 = 10,\ \dot\lambda_0 = -3.6}$. Here $\dot\lambda=-\dot\alpha\lambda$ which corresponds to $A_1=0$ case in \ref{['lam2']} and as expected by our analytical calculations has a period in which $\Sigma$ is positive and increasing. These values lead to a trajectory with $\dot\alpha_0=4\times10^{-4},\ \epsilon_0=0.24$ and $\dot\sigma_0=-5\times10^{-7}$. The initial value of $\epsilon$ is rather large, but with in one number of e-folds it decreases and reaches $10^{-2}$. Note that value of $\epsilon$ at the point of maximum $\Sigma$ is equal to $0.05$ ($\Sigma\simeq\frac{1}{3}\epsilon$), almost saturating our upper bound for anisotrpy $\Sigma$.
• Figure 5: The classical trajectory for ${\kappa = 1.99\times10^{10},\ g = 10^{-2},\ \psi_0 = 0.099,\ \dot\psi_0 = 10^{-10},}$${\lambda_0 = 0.1,\ \dot\lambda_0 = 1.288\times10^{-3}}$. Here $\dot\lambda=\dot\alpha\lambda$ which corresponds to the $A_2=0$ case in \ref{['lam1']}. As has been predicted by the analytical calculations, $\Sigma$ is negative and monotonically damped. These values give a trajectory with $\dot\alpha_0=4\times 10^{-3},\ \epsilon_0=0.16$ which is a rather large initial $\epsilon$ value. As we learn from the top right figure, $\epsilon$ decreases within a few number of $e$-folds to $0.5\times10^{-2}$. For these values $\dot\sigma_0=-0.8\times10^{-4}$.