content_image/364911.jpg content_image/364912.jpg content_image/364913.jpg content_image/364914.jpg content_image/364915.jpg
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]: 

<img src='content_image/364911.jpg'>

and the F -term potential is given by: 

<img src='content_image/364912.jpg'>

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: 

<img src='content_image/364913.jpg'>

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: 

<img src='content_image/364914.jpg'>

where λ ∼ κ ∼ 10 − $^{10}$. Including the soft SUSY breaking terms, the F -term potential is given by [80, 705] 

<img src='content_image/364915.jpg'>

