content_image/307244.jpg content_image/307243.jpg content_image/307239.jpg content_image/307240.jpg content_image/307241.jpg
is 

<img src='content_image/307244.jpg'>

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. 

<img src='content_image/307243.jpg'>

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 

<img src='content_image/307239.jpg'>

<img src='content_image/307240.jpg'>

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: 

<img src='content_image/307241.jpg'>

