µ$_{k}$ vanishes at the edges and takes the maximum value µ$_{k}$ = q/ 2, where q = g $^{2}$( ˆ φ $^{2}$/ 4 m 2 $_{φ}$). Thus the occupation number grows exponentially. The situation changes quite dramatically when one switches the expansion rate of the universe, the evolution of the scalar field during the first 10 − 50 oscillations modifies to: φ ( t ) ≃ M$_{P}$ √$_{3}$$_{π}$ cos ( m$_{φ}$t ) m$_{φ}$t , (381) where t is the physical time. The presence of the t at the denominator shows the damping of the oscillations due to the expansion of the universe. During this period the stochastic resonance come into the picture [100], where there are resonance bands as well as decrease in the particle number due to quantum effects. In either case (expanding or non-expanding background), based on initial VEV of σ there would be two distinct cases. • σ ≪ h $^{2}$M$_{P}$ : In this regime the h $^{2}$φ$^{2}$χ2 term is dominant at the beginning of the inflaton oscilla- tions. This case has been studied in detail in first two references of [99, 100]. For a nominal value of the inflaton mass, i.e. m$_{φ}$ = 10 13 GeV in chaotic inflation case, non-perturbative χ production during every oscillations of φ field, with a physical momentum, k < ∼ ( hm$_{φ}$ ˆ φ ) 1 / 2 (where ˆ φ ∼ M$_{P}$ ), takes place if h > 10 − $^{6}$. Particle pro- duction is particularly efficient if h > 3 × 10 − $^{4}$, and results in an explosive transfer of energy to χ quanta. The number density of χ$_{k}$ quanta increases exponentially. The parametric resonance ends when re-scatterings destroy the inflaton condensate. The whole process happens over a time scale ∼ 150 $_{m}$− 1 φ , which depends logarithmically on h [100, 102]. • σ ≫ h $^{2}$M$_{P}$ : In this regime the cubic term σφχ 2 dominates. This case was recently considered in Refs. [103, 806], where the the χ field becomes tachyonic during half of each oscillation. For σ > m 2 $_{φ}$/M$_{P}$ (which amounts to σ > 10 7 GeV for m$_{φ}$ = 10 13 GeV) this tachyonic instability transfers energy from the oscillating condensate very efficiently to the χ quanta with a physical momentum k < ∼ ( σ ˆ φ ) 1 / $^{2}$. Particle production ceases when the back-reaction from χ self-coupling induces a mass-squared > ∼ σ ˆ φ . Depending on the size of λ , most of the energy density may or may not be in χ quanta by the time back reaction becomes important [807]. 178