Our focus is on bosonic preheating which acts most efficiently in transferring the energy density from the inflaton oscillations. We consider models of large field inflation, such as chaotic inflation or hybrid models, for which bosonic preheating is most pronounced. The relevant renormalizable couplings between the inflaton φ and a scalar field χ will read from the following potential: where we have considered φ and χ to be real. Here σ is a coupling which has a [mass] dimension. The only scalar field in the SM is the Higgs doublet. Therefore in a realistic case χ denotes the real and imaginary parts of the Higgs components. The cubic interaction term is required for a complete inflaton decay. The quartic self-coupling of χ is required to bound the potential from below along the χ direction. The dimensionless couplings σ/m$_{φ}$ and h (as well as λ ) are not related to each other, hence either of the cubic or the quartic terms can dominate at the beginning of inflaton oscillations (i.e. when the Hubble expansion rate is H ( t ) ≃ m$_{φ}$ and the amplitude of oscillations is ˆ φ ∼ O ( M$_{P}$ )). In a non-SUSY case efficient preheating happens over a narrow window 3 × 10 − 4 ≤ h ≤ 10 − $^{3}$. The reason is that the h $^{2}$φ$^{2}$χ2 term yields a quartic self-coupling for the inflaton at a one-loop level which is constrained by the CMB normalization of the density perturbations, i.e. λ ≤ 10 − $^{12}$. However, in SUSY this correction is canceled out by that from fermionic partner of χ , so in principle one could expect a rather broader range of parameter space. Neglecting the self interaction for χ field, the equation of motion for χ$_{k}$ quanta is given by: It is assumed that the inflaton oscillations are homogeneous, φ ( t ) = ˆ φ ( t ) sin( m$_{φ}$t ), where ˆ φ ( t ) ≈ ( M$_{P}$$_{/}$√$_{3}$$_{πm}$$_{φ}$$_{t}$$_{), for chaotic inflation with mass}$ m$_{φ}$ . The occupation number for the excited χ$_{k}$ is given by: It was observed in Refs. [97–100], that in general one can have a narrow resonance , when expansion of the universe and the trilinear interaction are neglected, then the evolution for χ$_{k}$ yields a Mathieu equation, which has well known instability bands, during which the mode grows exponentially, χ$_{k}$ ∝ exp( µ n $_{k}$z ), where µ n k is set by the instability band Δ n k labeled by an integer n , and z = m$_{φ}$t . The resonance occurs for k = 0 . 5 m$_{φ}$ (1 ± q/ 2), where 177 V = 1 2 m 2 $_{φ}$φ 2 + 1 2 m 2 $_{χ}$χ 2 + σφχ 2 + h $^{2}$φ$^{2}$χ2 + λχ 4 , (378) ¨ χ$_{k}$ + 3 ˙ a a ˙ χ$_{k}$ + ( k 2 a 2 + m 2 χ + 2( σφ + h $^{2}$φ$^{2}$) ) χ$_{k}$ = 0 . (379) n$_{k}$ = ω$_{k}$ 2 ( | ˙ χ$_{k}$ | 2 ω 2 k + | χ$_{k}$ | 2 ) − 1 2 , (380)