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In such approximation we obtain the new equations of motion 

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which , ( ... ) ′ , denotes derivative of ( ... ), with respect to T . The solutions to the above system for the scale factor a and its conjugate momenta are 

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where a$_{0}$ = a | T $_{=0}$, P$_{0}$ = P | T =0 and C is a constant. A positive C with positive a$_{0}$ provides an accelerating expanded universe, with a positive constant Hubble, H = C . We have recovered a monotonic dependence of our new variable T with respect to the isotropic variable of the Universe a and therefore T shows to be a relational time variable for the gravitational dynamics. In this case, the universe expands exponentially according to the cosmological time variable, and naturally accelerating without a beginning singularity. 

In the presence of the potential term, i.e., far from the singularity, the equations of motion are 

<img src='content_image/349006.jpg'>

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Although, it is hard to solve these set of coupled nonlinear differential equations analytically, a numerical solution of the equations shows, in particular, the dynamic of the scale factor in 

