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By using ζ = (1 / 3) δ$_{σ}$ and Eq. (56), the non-Gaussianity parameter can be determined by (for a quadratic potential) for f$_{NL}$ ≫ 1 [131] $^{12}$: 

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Another interesting proposal is that the perturbations could be generated from the fluc- tuations of the inflaton coupling to the SM degrees of freedom [253–257]. It has been argued that the coupling strength of the inflaton to ordinary matter or the inflaton mass, instead of being a constant, could depend on the VEV of various fields in the theory. If these fields are light during inflation their quantum fluctuations will lead to spatial fluctuations in the inflaton decay rate. As a consequence, when the inflaton decays, adiabatic density pertur- bations will be created because fluctuations in the decay rate translate into fluctuations in the reheating temperature. 

This can be understood intuitively from the fact that fluctuations in the inflaton de- cay rate leads to fluctuations in the reheat temperature of the universe, given by T$_{rh}$ ∼ λ √ m$_{φ}$M$_{P}$ , where m$_{φ}$ is the mass of the inflaton. The fluctuations in the decay rate, Γ ∼ m$_{φ}$λ 2 can be translated into fluctuations in the energy density of a thermal bath with δρ$_{γ}$/ρ$_{γ}$ = − (2 / 3) δ Γ / Γ [254, 255]. The factor 2 / 3 appears due to red-shift of the modes during the decay of the inflaton whose coherent oscillations still dominates the energy den- sity of the universe. The inflaton decay rate fluctuates if either λ or m$_{φ}$ is a function of a fluctuating light field. 

The fluctuation in the decay rate for the various cases is given by: 

Various examples do predict non-Gaussianity within a range of f$_{NL}$ ∼ 5 [254, 255]. 

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