the inflaton evolves according to the formula Eq. (381). Thus, there exists a final time after which | φ | < m$_{ψ}$/h , and the total mass no longer vanishes, then the resonant production of fermion ends. The Dirac equation (in conformal time η ) for a fermionic field is given by: ( i a γ $^{µ}$∂$_{µ}$ + i 3 2 Hγ 0 − m ( η ) ) ψ = 0 , (390) where a is the scale factor of the universe, H = a ′ /a 2 the Hubble rate and ′ denotes derivative w.r.t. η , and m ( η ) = m$_{ψ}$ + hφ ( η ), where m$_{ψ}$ is the bare mass of the fermion. The particle density per physical volume V = a 3 at time η is given by: n ( η ) ≡ 〈 0 | N V | 0 〉 = 1 π $^{2}$a3 ∫ dk k 2 | β$_{k}$ | $^{2}$, (391) ρ$_{ψ}$ ∝ m$_{ψ}$N$_{ψ}$ ∝ qm 1 / 2 ψ (392) λ M$_{P}$ φ ( H ¯ q$_{l}$ ) q$_{R}$ , (393) λ M$_{P}$ φF$_{µν}$F µν , (394) where α$_{k}$, β$_{k}$ are the Bugolyubov’s coefficients satisfying: | α$_{k}$ | 2 + | β$_{k}$ | 2 = 1. The occupation number of created fermions is thus given by n$_{k}$ = | β$_{k}$ | $^{2}$, and the above condition ensures that the Pauli limit n$_{k}$ < 1 is respected. One important physical quantity is the scaling of the total energy which is linear in q = h 2 ˆ φ $^{2}$/m2 $_{φ}$, as generally expected [338, 774–776], but also note that m$_{ψ}$ ( t ) ∝ q 1 / $^{2}$. In a realistic case, since the SM fermions are chiral, if the inflaton is a SM gauge singlet, then it can only couple via dimension-5 operators, i.e. where λ ∼ O (1), H is the SM Higgs doublet and q$_{l}$, q$_{R}$ are the SU (2)$_{l}$ doublet and the right handed SM fermions, respectively. As a result preheating of SM fermions from a gauge singlet inflaton becomes less important due to weak coupling. In Ref. [338], it was argued that an inflaton coupling to right handed neutrino, hφ ¯ NN , where N is right handed neutrino, will induce non-thermal leptogenesis, where the right handed neutrinos were treated gauge singlets. Anyway, if we embed the right handed neu- trinos in a gauge sector, where they get their masses via some Higgs mechanism, then one requires non-renormalizable couplings like Eq. (393). Similar argument holds for coupling to the SM gauge bosons, where the inflaton can only couple via non-renormalizable operator, i.e. 183