Unfortunately, in these models the value of f$_{NL}$ is very small and undetectable by the future experiments. The expression is also a generalization of [575], derived for the two field case. Similar exercise has been performed for models of multi-field inflation with non-canonical kinetic terms in [576, 577], we will discuss such models of inflation in Sec. VIII B. Non-Gaussianities can also be created after inflation through tachyonic preheating at the end of inflation [245, 570, 578]. Preheating of light fields after inflation has also been considered in [243, 244, 246], and in δN formalism [247, 579]. It was found that the λφ 4 model can lead to levels of non-Gaussianities that are already on the verge of being excluded by current observations [243]. Also, very recently, it was pointed out that the waterfall part of the dynamics in a generic non-SUSY hybrid type inflation (or hilltop inflation) with a potential of the generic form could generate a large amount of non-Gaussian fluctuations given by [580] $^{54}$. where the index e denotes the end of inflation and γ is a semi-analytic parameter in the range [0 . 03 − 0 . 1]. For non-SUSY models of inflation there are two more challenges related at the classical and the quantum level. At the classical level inflaton can couple to other light scalar fields during inflation. In any of the effective potentials considered so far, there is no symmetry argument which • Effective couplings and symmetries: 54 A word of caution when we use δN formalism to estimate non-Gaussianity during parametric resonance or during tachyonic prehetaing. Note that prehetaing is a violent and non-adiabatic process, which can happen at time scales much shorter than one Hubble time. Especially, during tachyonic prehetaing, the field displacement can be very negligible as compared to one Hubble time, during which the δN ≈ 0, therefore in a separate universe approach, where the Hubble patches evolve homogeneously and smoothly, one can not trust the δN formalism. 92 6. Challenges for non-SUSY models V ( φ, ψ ) ∼ V$_{0}$ + ηV$_{0}$ 2 M 2 P φ 2 − λφψ 2 + O ( φ $^{3}$, . . . ) , (204) f$_{NL}$ 1 . 3 × 10 4 ∼ ( γ 10 − 2 ) 3 / 2 ( V 1 / 4 e 10 − $^{3}$M$_{P}$ ) 4 ( 10 − $^{2}$M$_{P}$ λ )( 10 − 2 ϵ$_{e}$ ) 1 / 2 , (205)