can mimic assisted inflation [286] $^{79}$. Beyond MSSM : A gauge invariant inflationary model has been proposed sometime ago in in Ref. [746]. The idea is that ( SU (3)) 3 gauge group is spontaneously broken down to SU (3)$_{c}$ × SU (2)$_{L}$ × SU (2)$_{R}$ × U $^{(1)$_{B}$}$− $_{L}$. The quartic contribution to the superpotential is given by a D -flat direction of ( SU (3)) $^{3}$, Φ, which is invariant under 27, W ∼ λ M$_{P}$ (2727) $^{2}$, (373) where λ is a small number determined by matching the amplitude of the CMB observations, and Φ is the monomial representing, N, ¯ N or ν $^{c}$, ¯ ν $^{c}$. The potential along such a D -flat direction is given by [746]: where M$_{S}$ ∼ 10 3 GeV, denotes the soft SUSY breaking scale. It was argued that the negative mass squared term would appear due to running in presence of strong dynamics [746]. Inflation happens near φ ∼ 0, and ends with a VEV, φ ∼ M ∼ λ − 1 / 2 √$_{M}$$_{P}$$_{M}$$_{S}$ GeV. In order to match the CMB temperature anisotropy, Δ T/T ≈ 0 . 023 λN 2 $_{Q}$, where N$_{Q}$ = (2 π/ 3)( φ/M$_{P}$ ) 2 is the number of e-foldings before the end of inflation. We require M ∼ 10 15 GeV and λ ∼ 10 − 7 [746]. The spectral index for the scalar perturbations tend to be small n ≃ 0 . 92 − 0 . 88, while the ratio of the tensor to the scalar ratio is given by; r ≈ 0 . 4 − 0 . 7. 79 Let us consider M fields H$_{i}$ and N − 1 fields G$_{j}$ in the fundamental representation N of the gauge group SU ( N ). Note that the matter content is also enhanced, which has a total N − 1 + M degrees of freedom. Then there exists a D -flat direction described by a gauge invariant polyno- mial I = ∑ M j $_{=1}$α$_{j}$ $^{ϵ$_{d}$$_{1}$}$··· d$_{N}$$_{−}$ 1 $_{e}$H d 1 1 · · · H d$_{N}$$_{−}$ 1 N − 1 G e $_{j}$, which after solving the constraint equations ∂I/∂H a j = CH a ∗ j , ∂I/∂G a i = CG a ∗ i produces a vacuum configuration: H a j = δ a $_{N}$φ$_{j}$ , for j = 1 , · · · , M , and G a i = δ a i √ M j =1 | φ$_{j}$ | $^{2}$, for i = 1 , · · · , N − 1. When one substitutes these into D -terms, one finds that all D -terms vanish. The Lagrangian for the flat direction is given by L = c ∑ M j =1 | D$_{µ}$H$_{j}$ | $^{2}$+ c ∑ N − 1 i =1 | D$_{µ}$G$_{i}$ | $^{2}$− V ( { H$_{i}$, G$_{j}$ } ), where c = 1 / 2 for the real fields, and c = 1 for the complex fields [286, 743]. The lagrangian reduces to: L = 1 $_{2}$∂$_{µ}$ Φ † [1 + ( N − 1) P ] ∂ $^{µ}$Φ − V (Φ) where P = ΦΦ † / (Φ † Φ) is the projection operator, and the field configurations of the real fields are: Φ = ( φ$_{1}$, . . . , φ$_{M}$ ) $^{T}$, for φ$_{i}$ ∈ R . Similar generalizations can be made for N × N non-commutative hermitian matrices, see [744, 745]. 172 V ( φ ) ≈ − M 2 S | φ | 2 + λ 2 3 | φ | 6 M 2 P , (374)