In such approximation we obtain the new equations of motion which , ( ... ) ′ , denotes derivative of ( ... ), with respect to T . The solutions to the above system for the scale factor a and its conjugate momenta are 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 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 a ′ = 1 12 a $^{2}$P$_{a}$ , (25) P$_{a}$ ′ = − 1 12 aP$_{a}$ $^{2}$, (26) P$_{+}$ ′ = P − ′ = 0 , (27) 1 β$_{+}$ ′ = − 12 P$_{+}$, (28) β − ′ = − 1 12 P − , (29) a = a$_{0}$e $^{CT}$, (30) P$_{a}$ = P$_{0}$e − $^{CT}$, (31) P − ′ = − $_{4}$√$_{3}$$_{a}$ $^{4}$[ − e − 2 β$_{+}$ $_{sinh(2}$√$_{3}$$_{β}$$_{−}$$_{) +}$$_{e}$ 4 β$_{+}$ $_{sinh(4}$√$_{3}$$_{β}$$_{−}$$_{)]}$$_{.}$ (37) P$_{+}$ ′ = − 4 a $^{4}$[ − e − 8 β$_{+}$ + e − 2 β$_{+}$ $_{cosh(2}$√$_{3}$$_{β}$$_{−}$$_{)}$ + e 4 β$_{+}$ $_{(cosh(4}$√$_{3}$$_{β}$$_{−}$$_{)}$ − 1)] , (36) β − ′ = − 1 12 P − , (35) β$_{+}$ ′ = − 1 12 P$_{+}$, (34) + 2 e 4 β$_{+}$ $_{(cosh(4}$√$_{3}$$_{β}$$_{−}$$_{)}$ − 1)] , (33) a ′ = 1 12 a $^{2}$P$_{a}$ , (32) P$_{a}$ ′ = − 1 12 aP$_{a}$ 2 − 2 a $^{3}$[ e − 8 β$_{+}$ − 4 e − 2 β$_{+}$ $_{cosh(2}$√$_{3}$$_{β}$$_{−}$$_{)}$ 6