is driving inflation in order to obtain an effective potential similar to Eq. (283) with an appropriate level of CMB temperature anisotropies. The theory is based on the following potential (Φ still represents the GUT Higgs in the adjoint, and H$_{5}$ represents the Higgs in the fundamental representation which is realizing the electro-weak breaking) [682] (see also [494, 686, 687] for recent reviews) The inflaton develops a Coleman-Weinberg potential due to its coupling to Φ and H$_{5}$ . Its precise expression is obtained by minimizing the above potential for Φ which settles the system in the inflationary valley. Indeed, the breaking SU (5) → G$_{S}$$_{M}$ is realized in the usual T$_{24}$ ∝ Diag(1 , 1 , 1 , − 3 / 2 , − 3 / 2) direction, the VEV of Φ being a function of that of χ because of the coupling λ$_{2}$ , ( λ$_{c}$ ≡ a +7 b/ 15 represents the mixture of the Φ 4 terms in V .) Discarding the pure H$_{5}$ sector (relevant at the EW scale) and computing the masses of the triplet and doublet in H$_{5}$ that enter the Coleman-Weinberg formula, one can reduce the potential to V ( φ, χ ) and then to the effective inflationary potential using Eq. (285) [682] where χ$_{0}$ is the position of the minimum of V new eff ( χ ) and A is a function of the couplings λ$_{2}$ , λ$_{c}$ and the gauge coupling g$_{5}$ . The system after inflation is trapped in the global minimum at χ = χ$_{0}$ and φ = φ$_{0}$ = √ 2 λ$_{2}$/λ$_{c}$ . The mass of the superheavy gauge bosons inducing the proton decay is proportional to φ$_{0}$ , Thus the phase of inflation take place at an energy close to the mass scale involved in the proton decay, M$_{X}$ ∼ 2 V 1 / 4 0 , and its stability constrains the inflationary scale. The predictions for SU (5) singlet inflation [3, 5, 682, 683, 685] is similar to that of the potential; V = V$_{0}$ [1 − λ$_{χ}$ ( χ/µ ) $^{4}$], with λ$_{χ}$ = A ln χ/χ$_{0}$ . The predictions depend on A or alternatively on V$_{0}$ , and for V 1 / 4 0 ∈ [2 × 10 $^{15}$, 4 × 10 $^{16}$] GeV, they are found in the range: V $^{new}$( χ, Φ , H$_{5}$ ) = 1 4 a (tr Φ $^{2}$)2 + 1 2 b tr Φ 4 − α ( $_{H}$† $_{5}$H$_{5}$ )tr Φ 2 + γ 4 ( $_{H}$† $_{5}$H$_{5}$ ) 2 − $_{βH}$† $_{5}$Φ$^{2}$H$_{5}$ + λ$_{1}$ 4 χ 4 − λ$_{2}$ 2 χ $^{2}$tr Φ2 + λ$_{3}$ 2 χ 2 $_{H}$† $_{5}$H$_{5}$ . (284) 〈 Φ 〉 = √ 2 15 φ Diag(1 , 1 , 1 , − 3 / 2 , − 3 / 2) , with φ 2 = (2 λ$_{2}$/λ$_{c}$ ) χ 2 . (285) V new eff ( χ ) = Aχ 4 [ ln ( χ χ$_{0}$ ) − 1 4 ] + Aχ 4 0 4 , (286) M$_{X}$ = √ 5 3 gφ$_{0}$ 2 , (287) 130