a critical density universe, if we assume that the primordial spectrum can be approximated by a power law (ignoring the gravitational waves and the k − dependence of the power n$_{s}$ ) [13] where n$_{s}$ is called the spectral index (or spectral tilt), the reference scale is: k$_{0}$ = 7 . 5 a$_{0}$H$_{0}$ ∼ 0 . $_{002 Mpc}$− $^{1}$, and the error bar on the normalization is given at 1 σ . The spectral index n ( k ) is defined as This definition is equivalent to the power law behavior if n ( k ) is close to a constant quantity over a range of k of interest. One particular value of interest is n$_{s}$ ≡ n ( k$_{0}$ ). If n$_{s}$ = 1, the spectrum is flat and known as Harrison-Zeldovich spectrum [200, 201]. For n$_{s}$ = 1, the spectrum is tilted, and n$_{s}$ > 1 ( n$_{s}$ < 1) is known as a blue (red) spectrum. In the slow-roll approximation, this tilt can be expressed in terms of the slow-roll parameters and at first order 8 The running of these parameters are given by [156] Slow-roll inflation requires that ϵ ≪ 1 , | η | ≪ 1, and therefore naturally predicts small variation in the spectral index within Δ ln k ≈ 1 [202] Independently of slow-roll considerations or of the number of fields involved in the dynamics of inflation, a new set of parameters, known as the Hubble-flow parameters, were discussed 8 At second order, where C is a numerical constant C = ln 2 + γ$_{E}$ − 2 ≃ − 0 . 7296 [180]. P $_{ζ}$( k ) ≃ (2 . 445 ± 0 . 096) × $_{10}$− 9 ( k k$_{0}$ ) n$_{s}$ − 1 , (35) n ( k ) − 1 ≡ d ln P ζ d ln k . (36) n$_{s}$ − 1 = − 6 ϵ + 2 η + O ( ϵ $^{2}$, η$^{2}$, ϵη, ξ$^{2}$) . (37) d ϵ d ln k = 2 ϵη − 4 ϵ 2 , d η d ln k = − 2 ϵη + ξ 2 , d ξ 2 d ln k = − 2 ϵξ 2 + ηξ 2 + σ 3 , (38) ξ 2 ≡ M 4 P V ′ (d $^{3}$V / d φ $^{3}$) V 2 , σ 3 ≡ M 6 P V ′ $^{2}$(d$^{4}$V / d φ $^{4}$) V 3 . (39) d n ( k ) d ln k = − 16 ϵη + 24 ϵ 2 + 2 ξ 2 . (40) where n$_{s}$ − 1 = 2 [ − 3 ϵ + η − ( 5 3 + 12 C ) ϵ 2 + (8 C − 1) ϵη + 1 3 η 2 − ( C − 1 3 ) ξ 2 ] , 19