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Along the D -flat direction, the VEVs are φ$_{1}$ = φ$_{2}$ = φ$_{3}$ ≪ M ≪ M$_{P}$ . One can further make an assumption such that κ$_{3}$ < − 1 / 3, so that the S field becomes heavy compared to the Hubble expansion rate, and therefore the field relaxes to its minimum VEV, S = 0, within one e-folding. By assuming | φ$_{i}$ | = $_{ϕ/}$√$_{2,}$ β = ( κ$_{2}$ − 1), λ = ( β ( β + 1) + 1 / 2 + κ$_{1}$/ 12) and γ = 2 / (6) 3 n/ 2 < ∼ 0 . 14, the potential along ϕ is given by [85] 2 4 3 n 

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One needs to suppress the quartic term in the potential, i.e. ϕ 4 term, otherwise for | γϕ 3 $^{n}$/M 3 $^{n}$|  ≪ | ( λ/ 4)( ϕ $^{4}$/M4 $_{P}$) | , it turns out that M ≥ M$_{P}$ . For a specific parameter space, M ≈ 10 $^{15}$− 10 16 GeV and µ ≈ 10 $^{13}$− 10 14 GeV and n = 2 , 3 , 4 and β ≤ 0 . 03, it is possible to match the amplitude of the perturbations and the spectral tilt within n$_{s}$ = 0 . 96 ± 0 . 014 for N = 60 e-foldings of inflation. The model requires pre-inflationary phase to set the initial conditions for ϕ ≈ 0 to be realizable. 

Another example of flavon is to consider a vacuum alignment potential as studied in the SU(3) family symmetry model of [85, 709]. It was assumed that < φ$_{23}$ > ∝ (0 , 1 , 1) T and < Σ > = diag( a, a, − 2 a ) are already at their minima, and that the relevant part of the superpotential which governs the final step of family symmetry breaking is given by [709] 

<img src='content_image/369303.jpg'>

where a singlet, S , is the driving superfield for the flavon φ$_{123}$ and the non-minimal K¨ ahler potential is given by: 

<img src='content_image/369304.jpg'>

During inflation the fields with larger VEVs Y$_{123}$, φ$_{123}$, ¯ φ$_{123}$ do not evolve, the inflationary potential is dominated by 

where | S | = $_{σ/}$√$_{2,}$ | Z$_{123}$ | = $_{ξ/}$√$_{2 and}$ γ = κ$_{SZ}$ − 1. Inflation can happen if the coefficients in front of the mass terms for σ and/or ξ are sufficiently small. 

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The inflaton could be σ if γ < − 1 / 3, the mass of ξ becomes heavy compared to the Hubble scale during inflation. For κ$_{S}$ ≈ (0 . 005 − 0 . 01) and κ ≈ (0 . 001 − 0 . 05), the spectral index is 

