Does the detection of primordial gravitational waves exclude low energy inflation?

We show that a detectable tensor-to-scalar ratio $(r\ge {10}^{-3})$ on the CMB scale can be generated even during extremely low energy inflation which saturates the BBN bound ${\rho }_{\inf }\approx {\left(30\text{MeV}\right)}^4$. The source of the gravitational waves is not quantum fluctuations of graviton but those of $SU\left(2\right)$ gauge fields, energetically supported by coupled axion fields. The curvature perturbation, the backreaction effect and the validity of perturbative treatment are carefully checked. Our result indicates that measuring r alone does not immediately fix the inflationary energy scale.