$\mathrm{SU}(N)$ natural inflation

We study the $\mathrm{SU}(N)$ gauge fields coupled with the inflaton through the Chern-Simons coupling and propose a general procedure for constructing homogeneous, isotropic, and attractor solutions of the gauge fields during inflation. Gauge fields develop various vacuum expectation values corresponding to different spontaneous symmetry-breaking patterns of $\mathrm{SU}(N)$, where embedded SU(2) subgroups are broken with the spatial rotation SO(3) symmetry. As specific examples, we construct the stable solutions for $N=3$ and 4. We numerically solve the gauge field dynamics and confirm that our analytic solutions are complete and act as attractors. Notably, the proposed approach is applicable to the other simple Lie groups.

Figure 1

Effective potential $V_{\mathrm{eff}}(\sigma )$ for $\xi =2.3$ with ${\lambda }^{-1}=1$ (purple), $\sqrt{2}$ (violet), 2 (orange), and $\sqrt{10}$ (green), which are solutions for the case of SU(3) or SU(4). The origin $\sigma =0$ is metastable, ${\sigma }_{-}/\lambda$ is the local maximum, and ${\sigma }_{+}/\lambda$ is the minimum with depth proportional to ${\lambda }^{-2}$.

Figure 2

Time evolution of gauge field mean amplitude $\tilde{\sigma }(t)$ with $\xi =3$ in the SU(3) (top) and SU(4) (bottom) cases. The initial conditions range over $\tilde{\sigma }(0)\simeq [0,10]$ with random amplitudes and orientations of the gauge fields. The black horizontal lines represent the analytic stable solutions in Eqs. (21) and (22), which act as attractors.

Figure 3

Typical time evolution of gauge field amplitudes with $\xi =3$ in the SU(3) case. Only the components for $a=1$, 2, and $i=x$, $y$, $z$ are shown as examples.

Figure 4

Errors in gauge constraints, Eq. (a5), under the same dynamics as in Fig. 3. Eight lines corresponding to the gauge constraints from $A_0^a$ are plotted. The violation of gauge constraints is always smaller than ${10}^{-8}$.

Figure 5

Convergence of gauge fields with $\xi =3$ in SU(3). The top panel depicts the convergence of the amplitude and the coefficient of the commutation relation in Eqs. (a6) and (a7). We also present the criterion on convergence for isotropy and the SU(2) subgroup in the bottom panel.

Figure 6

Dynamics of the mean-field amplitude with the extended initial conditions for $\xi =3$. Each green line represents the time evolution of the gauge fields under different initial conditions. The initial mean amplitudes range from $\tilde{\sigma }(t_0)\simeq [0,{10}^2]$. The black horizontal lines represent the stable solutions derived in this work. No additional attractor solution is identified.