where a gauge singlet S is the inflaton, 16$_{H}$ , 16$_{H}$ , additional singlets are P , P , Q , and Q are displaced from the present values, for which two eigenvalues can potentially act like a moduli, while two of them are heavy as the GUT scale with a large VEV. The dimensionless constants are κ, κ$_{i}$, ρ [76]. where all scalar fields except S contribute to φ$_{k}$ . The factor (1 + ∑ k | φ$_{k}$ | $^{2}$/M 2 P + · · · ) in front originates from e K/M 2 P . Provided, | D$_{φ}$$_{k}$W | /M$_{P}$ are much smaller than the Hubble scale ( ∼ κM 2 $_{eff}$/M$_{P}$ ), the flatness of S will be guaranteed even after including SUGRA corrections. This can be realized by the choice κ ≪ 1, and κ/ρ ≪ 1. The U (1) R -symmetry ensures the absence of terms proportional to S $^{2}$, S $^{3}$, etc. in the superpotential, which otherwise could spoil the slow-roll conditions. The spectral tilt arising from this model is very close to one, i.e. n$_{s}$ = 0 . 99 ± 0 . 01 and | dn$_{s}$/d ln k | ≪ 0 . 001. The smooth hybrid inflation model was also originally built within SO (10) [63, 692] though to our knowledge this model has not be embedded into a specific SUSY GUTs model. Its proximity to the shifted model allows us to guess that there should not be any problem as long as an additional singlet is assumed to play the role of the inflaton. As mentioned earlier in this chapter, these two models can also be realized without the non-renormalizable superpotential terms leading to the new shifted [70], and the new smooth hybrid inflation [83] models, at the cost of adding more fields in the picture, see Sec.IV E 7. • Flipped SU (5): Inflation in flipped SU (5) (= SU (5) × U (1)$_{X}$), which is a maximal subgroup of SO (10) with a chiral superfield in the spinorial representation 16 per family, was also consid- ered in [78, 693]. The breaking of SU (5) × U (1)$_{X}$ to the MSSM gauge group happens V$_{F}$ ≈ ( 1 + ∑ k | φ$_{k}$ | 2 M 2 P + · · · )[ κ $^{2}$M 4 eff ( 1 + | S | 4 2 M 4 P ) + ( 1 + | S | 2 M 2 P + | S | 4 2 M 4 P ) ∑ k | D$_{φ}$$_{k}$W | 2 ] , (295) It is assumed that initially, |〈 S 〉| 2 ≈ M 2 B − $_{L}$[1 / (4 ζ ) − 1] / 2, with 1 / 4 < ζ < 1 / 7 . 2, and 〈 16$_{H}$ 〉 , 〈 16$_{H}$ 〉 , 〈 P 〉 , 〈 P 〉 , 〈 Q 〉 , 〈 Q 〉 = 0, where M 2 eff ∼ M 2 B − $_{L}$. With D$_{S}$W ≈ − κM 2 $_{eff}$(1 + | S | $^{2}$/M 2 $_{P}$), the F -term part of the potential is given by: 138