is where Γ$_{k}$ integrates out all the fluctuations at scale k and connects the admissible fundamental action at Γ$_{k}$$_{→∞}$ = S to the conventional effective action at Γ$_{k}$$_{→}$$_{0}$ = Γ. This average–like effective action at tree–level describes all gravitational phenomena for each momentum of order k [11]. Indeed, the IR–cutoff R$_{k}$ ( p $^{2}$) which appears in the definition of Γ$_{k}$ , eliminates the effects of fluctuations of p 2 < k 2 on RG flow and is defined by an arbitrary smooth function R$_{k}$ ( p $^{2}$) ∝ k $^{2}$R$^{(0)}$( p 2 k $_{2}$) where R $^{(0)}$( ψ ) satisfies the conditions R $^{(0)}$(0) = 1 and R $^{(0)}$( ψ → ∞ ) → 0. The exponential form R $^{(0)}$( ψ ) = ψ exp( ψ ) − 1 is a common chosen form in the literatures [12]. Since the multiplicity of couplings in the effective action makes the β -function intricate, truncation would project the RG flow into the finite dimensional subspace spanned by the essential couplings. This method gives a finite number of ordinary differential equations. The Enistein–Hilbert truncation, is a common truncation for cosmological models. It is shown numerically in [13] that for the small values of cutoff k → 0, at perturbative regime , the solutions of β -functions for this model lead to the following power–series dimen- sionful couplings where ω = 1 6 $_{π}$[24Φ 2 $_{2}$(0) − Φ 1 $_{1}$(0)] , ν = 1 4 $_{π}$Φ 1 $_{2}$(0) and Φ p $_{n}$( w ) is the threshold function, which depends on the IR–cutoff as: G ( k ) = G$_{0}$ $^{[}$1 − ωG$_{0}$k 2 + O ( G 2 $_{0}$k$^{4}$) ] (3) Λ( k ) = Λ$_{0}$ + νG$_{0}$k 4 $^{[}$1 + O ( G$_{0}$k $^{2}$) ] (4) Φ p $_{n}$( w ) = 1 Γ( n ) ∫ ψ n − $_{1}$R $^{(0)}$( ψ ) − ψ R (0) $^{′}$( ψ ) [ ψ + R $^{(0)}$( ψ ) + w ] p dψ 4 Γ$_{k}$[ g$_{αβ}$ ] = 1 16 πG$_{k}$ ∫ d $^{4}$x √ g ( − R ( g ) + 2Λ$_{k}$) (2) k∂$_{k}$ Γ$_{k}$ = 1 2 T r [(Γ (2) k + R$_{k}$ ) − $^{1}$k∂$_{k}$ R$_{k}$ ] (1)