behavior for times that we are interested in. This is what is expected, since all the inflationary models are built to purge the effects of the initial conditions, although they have different scale factors at the start of the inflation era. As a result of the above discussion, one can conclude that although the details of the improved gauge invariant quantity (47) depends on the inflationary model used, but the general behavior is the same. Because of this, we confine ourselves to relation (47) in the next sections. For a Λ-inflation model we consider the running gauge coupling, Λ( t ) and not the ϕ -field, as the agent that produces exponential expansion in the inflation era. Although this not a good model (because of giving a non–improved spectral index n$_{s}$ = − 2), but we investigate it to see how the improvement affects such a model. For simplicity, here we use the IMSE in terms of time t , instead of the cosmological time: in which If we assume that the inflation field only produce the scalar perturbations and does not affect the expansion rate of universe, we can neglect ρ$_{ϕ}$ with respect to ρ$_{Λ}$ . Substituting H = √ Λ( t ) 3 and G ( t ) and Λ( t ) from (6) and (7) in (52), we get: S ( t ) ≃ + a$_{0}$e$^{√}$ Λ( t ) 3 t t $^{[}$α$_{1}$ β$_{2}$$^{(}$ t$_{pl}$ t ) − 2 + ( α$_{1}$γ$_{2}$ + β$_{1}$β$_{2}$ ) + ( α$_{1}$δ$_{2}$ + β$_{1}$γ$_{2}$ + γ$_{1}$β$_{2}$ )$^{(}$ t$_{pl}$ t ) 2 + ( α$_{1}$ Σ$_{2}$ + β$_{1}$δ$_{2}$ + γ$_{1}$γ$_{2}$ + δ$_{1}$β$_{2}$ )$^{(}$ t$_{pl}$ t ) 4 $^{]}$+ $^{√}$3Λ$_{0}$ − 5 t + O$^{(}$$^{(}$ t t$_{pl}$$^{)}$$^{−}$$^{5}$$^{)}$ (53) S ( t ) = 3 H − 2 ˙ H H + ¨ H ˙ H + ˙ G ( t ) 2 G ( t ) H ˙ H d dt ln$^{(}$ G ( t ) 2 ˙ H ˙ G ( t ) H 3 ) . (52) d $^{2}$R$_{q}$ dt 2 + S ( t ) dR$_{q}$ dt + q 2 a $_{2}$R$_{q}$ = 0 (51) B Case II: Λ -inflation 17