The Noether symmetry approach plays a crucial role for symmetries in physics especially in general relativity and cosmology. We have found that serious problems arise in the research literatures when such an approach is applied to vector field cosmology. The central point is that all the symmetries of the systems should and could be solved once along the Noether approach, this is the key of it. But in the precedent works in the literatures, only part of the symmetries may be obtained, the left symmetries are added into the system by hands. This is a serious misunderstanding on the Noether approach, meanwhile other errors are also existent. So we should present a correct picture on such an approach to vector field cosmology.
If we include vector field to the cosmological models, we should study the statistical anisotropy, cosmic isotropization, instability, non-Gaussianity, and gravitational wave, etc. The field fluctuation can also provide the seeds of curvature perturbation, which is related to the structure formation. The anisotropic hair associated with the so-called cosmic no-hair theorem and the mechanism of magnetogenesis can also be discussed on this stage. The cosmic dynamical equations are integrated numerically and the behaviors of cosmological evolution are presented certainly along the Noether approach. We find that our results can give a clear picture on the cosmic scenarios certainly. It shows that the universe is in the period of early-time decelerating expansion firstly and then developed into the period of late-time accelerating expansion afterwards. Such a picture of decelerating-accelerating expansion is in agreement with the observations definitely. 
Fig. 1 The scale factor a(t) versus time t . The universe evolutes from the decelerating regime to the accelerating regime qualitatively
The variation of the scale factor a versus time t is shown in the figure. It can be seen that the universe is expanding since the first order derivative of the scale factor is positive with respect to time t. In the early time, the second order derivative of the scale factor is negative, that is, the universe is in a phase of decelerating expansion. While in the late time, the second order derivative of the scale factor is positive, that is, the universe is in a phase of accelerating expansion.
Such a picture can also be expressed by the deceleration parameter as shown in the above figure. The deceleration parameter is positive in the early time giving a decelerating expansion universe, while it is negative in the late time giving an accelerating expansion one. It is also clear that the universe is in the accelerating stage now (t = 1). All these are in agreement with the observations.
Furthermore, the anisotropic expansion could approach an isotropization case during evolution. Through the study of isotropization, the universe will approach an isotropic Robertson-Walker one. That is, a de Sitter type one should arise in the late time.
This work is published by the journal of Astrophysics and Space Science ，359 (2015) 16. by Sun Zhang and Yi-Qiao Dong,
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