Universality of linear perturbations in $\mathrm{SU}(N)$ natural inflation

We prove the universality of predictions for linear perturbations from the entire class of models of inflation driven by a pseudoscalar field coupled to an $\mathrm{SU}(N)$ gauge boson, where SU(2) subgroups in the $\mathrm{SU}(N)$ crossed with the background spatial SO(3) spontaneously break into a single SO(3). The effect of which SU(2) subgroup in $\mathrm{SU}(N)$ acquires a vacuum expectation value through spontaneous symmetry breaking can be quantified by a single parameter $\lambda$, which always appears in combination with the gauge coupling constant $g$. In the linear perturbations, as well as the background system, the same dynamics and predictions as in the chromo-natural inflation hold for its $\mathrm{SU}(N)$ extension by replacing $g\to g\lambda$. The latter class of models thereby draws the same prediction curve on the $n_s–r$ plane as the former at the tree level as long as $g\lambda$ stays constant during inflation. We briefly discuss possible transitions from one value of $\lambda$ to another during inflation and the observational prospects.

Figure 1

Effective potential for the gauge field VEV $m_Q$ defined in Eq. (12) for $\xi =1$ (blue), $3/\sqrt{2}$ (orange), and 2.4 (green). One can observe that a nontrivial stable solution at $m_Q=m_{Q+}$ becomes the true vacuum when $\xi >3/\sqrt{2}$. In the marginal case with $\xi =3/\sqrt{2}$, we have a degenerated vacuum at $m_{Q+}=\sqrt{2}$ (vertical dashed line). Thus, the nontrivial true vacuum always appears at $m_{Q+}>\sqrt{2}$. Note that $V_{\mathrm{eff}}$ is an effective potential only for the dynamics of $Q$ and the apparent negative value in the green curve has no physical consequence for the expansion.