¨ A + ˙ A$^{(}$ 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 )) − a − $^{2}$∇$^{2}$A = 0 . (25) in the form: As mentioned previously, working with Fourier modes is easier, and thus in what follows we use Fourier transformed equations. It is clear that the scalar perturbations of metric would not be invariant under transfor- mation of x α → x α + ξ $^{α}$. The mentioned transformation changes the perturbation A$_{q}$ to ˜ A$_{q}$ = A$_{q}$ + H ( t ) ξ $^{0}$. To have a covariant description, usually in the non–improved theory, the gauge invariant quantity R$_{q}$ ≡ − A$_{q}$ 2 + Hδu$_{q}$ is defined. Since in this special gauge, the coeffi- cients of G ( t ) in the (14) and (16) vanish, the corrections due to running couplings do not affect the scalar perturbations. The corrections manifest themselves in the conservation equation of the background flow. The Hubble parameter H ( t ), has a correction term in R$_{q}$ which leads to a second order perturbation in gauge invariant quantity and can be ignored. Therefore, the non–improved form of the gauge invariant quantity saves its invariance after improvement of couplings . On the other hand the assumption − qτ ≪ 1 at the late time of inflation, causes the ignorance of the term Hδu$_{q}$ with respect to the scalar metric perturbation, A$_{q}$ 2 in R$_{q}$ . Finally considering the conformal time as an independent variable, R$_{q}$ evolves as: Here the prime denotes derivative with respect to the conformal time. This is the improved Mukhanov–Sassaki equation (IMSE). Denoting the coefficient of R ′ q by S ( τ ): 11 S ( τ ) = d dτ ln$^{(}$ aH ′ H 2 ) + G ′ 2 G H H ′ d dτ ln$^{(}$ G $^{2}$H ′ G $^{′}$H 3 ) , (27) R ′′ q +$^{(}$ d dτ ln$^{(}$ aH ′ H 2 ) + G ′ 2 G H H ′ d dτ ln$^{(}$ G $^{2}$H ′ G $^{′}$H 3 )) R ′ q + q $^{2}$R$_{q}$ = 0 . (26)