field ¯ ϕ : where ¯ ρ$_{ϕ}$ = 1 2 ˙ ¯ ϕ 2 + V ( ¯ ϕ ) and ¯ ρ$_{Λ}$ = Λ( t ) 8 πG ( t $_{)}$, with the improved conservation equation (because of the improved Einstein equation G$_{µν}$ = − 8 πG ( t ) T$_{µν}$ ): where ¯ ρ = ¯ ρ$_{ϕ}$ + ¯ ρ$_{Λ}$ and ¯ p = ¯ p$_{ϕ}$ + ¯ p$_{Λ}$ are the total density and power, respectively. Note that the dynamics of the background field is assumed to be not affected by the improvement and the effect of RG improvement on the perturbations would be investigated. Eliminating the potential of the scalar field using (21) and (22), the time evolution of the scalar non–perturbed field would be given by: In order to get the evolution equation of the gauge invariant quantity, we have to fix the gauge. We take the same gauge as in [18] and consider δϕ$_{q}$ = 0 and B$_{q}$ = 0 to find R$_{q}$ . Hence δu vanishes. Indeed, perturbations of pressure and density in this gauge are δρ$_{ϕ}$ = δp$_{ϕ}$ = − 1 $_{2}$E ˙ ¯ ϕ $^{2}$. Therefore energy-momentum conservation (19) leads to: Substituting E = ˙ A H from (16) and using (14), the evolution equation of A can be written It is worth noting that the improvement given by the solutions of RGE, do not change the equations (14) and (16), but the conservation equation of the background field changes, as given by (24) since the scalar field exchanges energy with the quantum improved part. This energy exchange can be seen more explicitly in the (22) where the dynamic of scalar field is under impression of the improvement term 1 ¯ ˙ ϕ ( ˙ Λ( t ) 8 πG ( t ) − Λ( t ) ˙ G ( t ) 8 πG ( t ) 2 $^{)}$. − 2 a H ∇ $^{2}$F − ¨ A + ˙ A ˙ G ( t ) 2 G ( t ) ( − G ( t ) 2 d dt$^{(}$ ˙ G ( t ) G ( t ) 2 ) + G ( t ) d dt$^{(}$ ˙ H HG ( t )$^{)}$ + 3 ˙ H − 3 2 H ˙ G ( t ) G ( t )$^{)}$ = 0 . (24) 10 ˙ ¯ ϕ 2 = 1 4 πG ( t )$^{(}$ H ˙ G ( t ) 2 G ( t ) − ˙ H$^{)}$ . (23) ˙ ¯ ρ = − 3 H (¯ ρ + ¯ p ) − ˙ G ( t ) G ( t ) ¯ ρ (22) H 2 = 8 πG$_{0}$ 3 (¯ ρ$_{ϕ}$ + ¯ ρ$_{Λ}$ ) (21)