content_image/310578.jpg content_image/310581.jpg content_image/310582.jpg content_image/310583.jpg content_image/310584.jpg
we see that it can be decomposed as: 

<img src='content_image/310578.jpg'>

in which S$_{0}$ ( τ ) = d dτ ln ( aH ′ H 2 $^{)}$, and S$_{I}$ ( τ ) = G ′ 2 G H H ′ d dτ ln ( G $^{2}$H ′ G $^{′}$H 3 ) is the improved part. It can be seen that in the non–improved limit, where G and Λ are constants, MSE (8) is recovered. 

## IV Solutions of IMSE 

In this section the solution of the IMSE for two cases will be obtained. First, for a model in which Λ$_{0}$ = 0 and it remains zero as time passing. Second, for the Λ-inflation model. We shall see that IMSE can be solved iteratively for both cases. 

## A Case I: Λ$_{0}$ = 0 

The Fridemann equation and conservation equation in this case are 

<img src='content_image/310581.jpg'>

<img src='content_image/310582.jpg'>

Considering the well–known exponential potential V ( ϕ ) = ge − λϕ , where g and λ are arbitrary real constants, the perturbed H = H$_{0}$ + ˜ ωH$_{1}$ and ϕ = ϕ$_{0}$ + ˜ ωϕ$_{1}$ seem to be appropriate solutions for (29) and (30) up to first order. Substituting G ( t ) from (6), gives H$_{1}$ and its’ time evolution, ˙ H$_{1}$ , up to O (1) : 

<img src='content_image/310583.jpg'>

<img src='content_image/310584.jpg'>

