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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: 

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: 

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 

<img src='content_image/333595.jpg'>

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<img src='content_image/333598.jpg'>

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. 

<img src='content_image/333599.jpg'>

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. 

<img src='content_image/333600.jpg'>

