content_image/304969.jpg content_image/304970.jpg content_image/304971.jpg content_image/304972.jpg
The gauge invariant quantity with this initial condition becomes 

<img src='content_image/304969.jpg'>

where η = − i K ′ 8 $_{π}$Γ( 3 $_{2}$) is a constant with the dimension of [ L ] − 1 and C 3 2 = − 4 i 6 $_{π}$. Clearly the second term in the parenthesis is the consequence of improvement. 

Before continuing, we have to note that, here the IMSE (26), for inflation with exponential potential V ( ϕ ) = ge − λϕ was solved. One may be wondering about how the results depends on the specific inflationary model we used. Therefore, it seems appropriate to take a look at the other models such as Starobinsky [20] or chaotic inflation [21]. The former, which seems to have a compatible description of CMB, is the result of a specific f ( R ) gravity (with f ( R ) = R + R $^{2}$/ 6 M $^{2}$). The Starobinsky model in the Einstein frame (obtained by the conformal transformation g$_{µν}$ → ˜ g$_{µν}$ = e √$_{16}$$_{πG}$ 3 $^{ϕ}$g$_{µν}$ ) is described by the action 

<img src='content_image/304970.jpg'>

where V ( ϕ ) = 3 M $^{2}$(1 − e $_{−}$√$_{16}$$_{πG}$ 3 $^{ϕ}$) $^{2}$/ 32 πG . For the FRW background this leads to an inflation- ary scale factor a ( t ) ∼ e t $^{2}$/ 12 t 2 pl . Since we are interested in the effects of improvement at the horizon exit time , and expanding t ≃ t$_{0}$ + δt (with t$_{0}$ much larger than t$_{pl}$ ), the behavior of the scale factor at times that we are interested in is given by 

<img src='content_image/304971.jpg'>

This is a similar behavior as the one obtained by the exponential potential 

<img src='content_image/304972.jpg'>

A similar treatment holds for the chaotic inflation models with potential V ( ϕ ) = gϕ $^{n}$. For example the scale factor for typical n = 2 is given as a ( t ) ∼ e − α ( β + t/t$_{pl}$ ) $^{2}$, leading to the same 

