On the Quintessence with Abelian and Non-abelian Symmetry

We study the perturbations on both “radial” and “angular” components of the quintessence with an internal abelian and non-abelian symmetry. The properties of the perturbation on the “radial” component depend on the specific potential of the model and is similiar for both abelian and non-abelian case. We show that the consine-type potential is very interesting for the O($\mathit{\text{N}}$) quintessence model and also give a critical condition of instability for the potential. While the properties of perturbations on “angular” components depend on whether the internal symmetry is abelian or non-abelian, which we have discussed respectively. In the non-abelian case, the fluctuation of the “angular” component will increase rapidly with time while in the abelian case it will not.