handed sneutrinos. During inflation N can take large VEVs as Φ is stuck at the zero VEV. The K¨ ahler potential was given by [79]: K = | S | $^{2}$+ | φ | $^{2}$+ | N | 2 + κ$_{S}$ | S | 4 4 M 2 P + κ$_{N}$ | N | 4 4 M 2 P + κ$_{φ}$ | φ | 4 4 M 2 P + κ$_{Sφ}$ | S | $^{2}$| φ | 2 M 2 P + κ$_{SN}$ | S | $^{2}$| N | 2 M 2 P + κ$_{Nφ}$ | N | $^{2}$| φ | 2 M 2 P + . . . , (299) V = κ 2 ( | φ | 4 M ′ 2 − M 2 ) 2 ( 1 + (1 − κ$_{Sφ}$ ) | φ | 2 M 2 P + (1 − κ$_{SN}$ ) | N | 2 M 2 P − κ$_{S}$ | S | 2 M 2 P ) + 4 λ 2 N M 2 ∗ ( | N | $^{4}$| φ | 2 + | N | $^{2}$| φ | $^{4}$) + . . . , (300) V ≈ κ $^{2}$M4 ( 1 + (1 − κ$_{SN}$ ) | N | 2 2 M 2 P + δ | N | 4 4 M 4 P ) + . . . , (301) W = λφH$_{u}$H$_{d}$ + κφN $^{2}$, (302) V = V$_{0}$ + λA$_{λ}$φH$_{u}$H$_{d}$ + κA$_{κ}$φN 2 + h . c . + m 2 $_{0}$( | H$_{u}$ | 2 + | H$_{d}$ | 2 + | N | $^{2}$) + m$_{φ}$φ $^{2}$, (303) 141 and the F -term potential is given by: Inflation is driven by the singlet field, S , where φ has a zero VEV. By virtue of the coupling between φ and N , the N field remains massless during inflation and therefore subject to random fluctuations during inflation. The authors assumed that the last 60 e-foldings of inflation arises when the sneutrino field is rolling slowly down the potential given by: where δ = 0 . 5 + κ 2 SN − κ$_{SN}$κ$_{N}$ + 1 . 25 κ$_{N}$ + . . . . The predictions of the model is typical of a hybrid inflation with a nearly flat spectrum. The WMAP data constraints parameters such as | (1 − κ$_{SN}$ ) | ≤ 0 . 02, also the running in the spectral tilt and tensor to scalar ratio are negligible. The model also generates isocurvature fluctuations, since both S and N in principle can be light during inflation, but they have assumed that S obtains a heavy mass and therefore settled down to its minimum in one Hubble time or so during inflation [79]. Hybrid inflation model with Dirac neutrinos was constructed in Refs. [80, 705, 706], in a specific Type-I string theory with the help of anisotropic compactification [706]. The model explains the smallness of the µ -term, strong-CP and the Dirac nature of neutrinos. The relevant part of the superpotential for inflation is given by: where λ ∼ κ ∼ 10 − $^{10}$. Including the soft SUSY breaking terms, the F -term potential is given by [80, 705]