The gauge invariant quantity with this initial condition becomes 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}$$_{πG}$ 3 $^{ϕ}$g$_{µν}$ ) is described by the action where V ( ϕ ) = 3 M $^{2}$(1 − e $_{−}$√$_{16}$$_{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 This is a similar behavior as the one obtained by the exponential potential 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 16 R o $_{q}$( τ ) = ηq − 3 / 2 ( 1 − K 4 π C 3 2 $^{[}$( − qτ ) 2 − ( − qτ$_{0}$ ) 2 ] ) (47) S = ∫ d $^{4}$x $^{√}$− ˜ g ( 1 16 πG ˜ R − 1 2 ˜ g $^{µν}$∂$_{µ}$ ϕ∂$_{ν}$ϕ − V ( ϕ )) (48) a ( t ) ∼ exp$^{(}$ t$_{0}$δt 6 t 2 pl ) (49) a ( t ) ∼ ( t/t$_{pl}$ ) 1 /ϵ ∼ exp$^{(}$ ¯ t$_{0}$δt ϵt 2 pl ) (50)