Perturbations of the Yang-Mills field in the universe

It has been suggested that the Yang-Mills (YM) field can be a kind of candidate for the inflationary field at high energy scales or the dark energy at very low energy scales, which can naturally give the equation of state $-1<\omega<0$ or $\omega<-1$. We discuss the zero order and first order Einstein equations and YM field kinetic energy equations of the free YM field models. From the zero order equations, we find that $\omega+1\propto a^{-2}$, from which it follows that the equation of state of YM field always goes to -1, independent of the initial conditions. By solving the first order Einstein equations and YM field equations, we find that in the YM field inflationary models, the scale-invariant primordial perturbation power spectrum cannot be generated. Therefore, only this kind of YM field is not enough to account for inflationary sources. However, as a kind of candidate of dark energy, the YM field has the `sound speed' $c_s^2=-1/3<0$, which makes the perturbation $\phi$ have a damping behavior at the large scale. This provides a way to distinguish the YM field dark energy models from other kinds of models.Comment: 5 pages, no figur