perturbation analysis for Chromo-natural inflation was first carried out in [@Dimastrogiovanni:2012st] in the limit of heavy vector field (see below), in which the gauge field can be integrated out, leading to an effective single-scalar field $P[(\partial\varphi)^{2},\varphi]$ model of inflation with a non-canonical kinetic term. A full analysis at linear order in perturbation theory was later presented in [@Dimastrogiovanni:2012ew], where the model was found to be highly unstable in the sub-horizon regime for: $$\frac{g^2 \, Q^2}{H^2} < 2 \;\;\;,\;\;\; {\rm instability \; in \;   Chromo\mbox{-}natural \; inflation} \;\;. 
\label{cn-instab}$$ In this relation $Q$ is the SU(2) vev, related to (\[gF\]) by $Q = {\hat \phi } / a$ (where $a$ is the scale factor of the universe), while $H$ is the Hubble rate. In Chromo-natural inflation, the quantity $m_g \equiv \sqrt{2} \, g \, Q$ coincides with the mass of the vector field fluctuations in the $m_g \gg H$ regime [@Dimastrogiovanni:2012st], which is the regime where the computation of [@Dimastrogiovanni:2012st] applies.

Ref. [@Dimastrogiovanni:2012ew], also noted that the model can lead to a large production of the vacuum gravity wave mode, in excess of the standard Lyth bound [@Lyth:1996im] $r > 16 \epsilon$ (see eqs. (\[epsilon\]) and (\[def-r\]) for the definition of 