ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}.$$

The energy-momentum tensor of the system, $$T_{\mu \nu} =  F^{a}_{\mu \alpha} F^a_{\nu \beta} g^{\alpha \beta} - g_{\mu \nu} \left[  \frac{1}{4} F^2 + \frac{\kappa}{96} \left( F {\tilde F} \right)^2 \right] \;\;, 
\label{Tdd}$$ once evaluated on the background, assumes the standard form $$\label{s2two}
T_{\;\; \nu}^{\mu}=diag\left(-\rho,P,P,P\right) \;\;.$$ In this expression, the energy density and the pressure can be written as the sum of two contributions, the first one arising from the Yang-Mills terms and the second one from the $(F\tilde{F})^{2}$ interaction $$\label{s2three}
\rho=\rho_{YM}+\rho_{\kappa},\quad\quad\quad P=\frac{1}{3}\rho_{YM}-\rho_{\kappa} \;\;,$$ where $$\label{s2four}
\rho_{YM}\equiv\frac{3}{2}\left(\frac{\dot{{\hat \phi } }^{2}}{a^{2}}+g^{2}\frac{{\hat \phi } ^{4}}{a^{4}}\right),\quad\quad\quad \rho_{\kappa}\equiv\frac{3}{2}\kappa g^{2}\frac{\dot{{\hat \phi } }^{2}{\hat \phi } ^{4}}{a^{6}} \;\;.$$ We can immediately see that the standard Yang-Mills contribution leads to the equation of state of radiation, while the second contribution to an effective equation of state of a cosmological constant. The $(F\tilde{F})^{2}$ term thus turns out to be a convenient choice for building a model of inflation. Despite being higher order, this term has only two time derivatives, due to its anti-symmetric structure, and the theory can be stable for some choice of parameters (as our study below shows). However, when this term dominates over the Yang-Mills one, it is natural to wonder whether higher order operators (which 