X j $_{k}$( t ) = α j k √$_{2}$$_{ω}$$_{k}$ e i ∫ ω$_{k}$dt + β j k √$_{2}$$_{ω}$$_{k}$ e − i ∫ ω$_{k}$$^{dt}$, (26) ( α j +1 k β j +1 k ) = ( 1 D$_{k}$ R k D ∗ $_{k}$e − 2 iθ j k R$_{k}$ $_{D$_{k}$}$e 2 iθ j k 1 D ∗ k ) ( α j k β j k ) (28) = ( √$_{1 +}$ e − πκ $^{2}$e iϕ$_{k}$ ie − π $_{2}$κ $^{2}$− 2 iθ j k − ie − π $_{2}$κ$^{2}$+2 iθ j k √$_{1 +}$ e − πκ $^{2}$e − iϕ$_{k}$ ) ( α j k β j k ) , R$_{k}$ = − ie iϕ$_{k}$ √$_{1 +}$ e πκ 2 (29) D$_{k}$ = e − iϕ$_{k}$ √$_{1 +}$ e − πκ 2 , (30) ϕ$_{k}$ = arg { Γ ( 1 + i κ 2 2 )} + κ 2 2 ( 1 + ln 2 κ 2 ) . (31) ¨ X$_{k}$ + ω 2 $_{k}$X$_{k}$ = 0 (24) ω 2 k = k 2 a $^{2}$( t ) + m 2 $_{χ}$+ g $^{2}$Φ$^{2}$( t ) sin $^{2}$( mt ) − 9 4 H $^{2}$− 3 2 ˙ H. (25) k 2 a 2 ≤ 2 $_{3}$√$_{3}$ gm Φ( t ) − m 2 $_{χ}$. (23) ¨ χ$_{k}$ + 3 H ˙ χ$_{k}$ + ( k 2 a 2 + m 2 χ + g $^{2}$Φ( t ) 2 sin $^{2}$( mt ) χ$_{k}$ = 0 . (22) d $^{2}$X$_{k}$ dτ 2 + ( κ 2 + τ $^{2}$) X$_{k}$ = 0 . (27) C. Preheating in an Expanding Background 5 So far, we have studied preheating in a toy model in which Standard Model matter is modeled by a scalar field χ . However, in principle we are interested in the produc- tion of SM fermions. Such fermions could be produced after preheating into a scalar field χ which then in turn couples to fermions. In particular, in supersymmetric theories to be discussed in a later section there are many channels for this to happen. However, it turns out that preheating into fermions is also effective, in spite of the fact that the occupation number of any fixed state can- not be greater than one (because of the Pauli exclusion principle). This is discussed in detail in [28, 29]. Note that in the matter-dominated background which we are considering the last two terms on the right-hand side cancel. The equation of motion (24) represents a harmonic os- cillator equation with a time-dependent frequency. The evolution of the solution will be described by the WKB approximation (which entails the absence of particle pro- duction) unless the adiabaticity condition is violated. This will happen during short time intervals around the instances t = t$_{j}$ when φ = 0, as discussed in the previous section. We label the intervals of adiabatic evolution by an integer j . In the j’th interval (lasting from $^{t$_{j}$}$− 1 to t$_{j}$ ), the adiabatic evolution of X$_{k}$ is given by where the coefficients α j k and β j k (the “Bogoliubov coeffi- cients”) are constant and satisfy the normalization con- dition | α j k | 2 − | β j k | 2 = 1 (derived from the Heisenberg uncertainty principle). During the brief time periods when φ is close to zero, we can use the approximation φ $^{2}$( t ) ≃ Φ $^{2}$m$^{2}$( t − t$_{j}$ ) $^{2}$. Introducing the new time variable τ = g Φ m ( t − t$_{j}$ ) and a rescaled momentum κ 2 = ( k $^{2}$/a$^{2}$)+ m 2 χ g Φ m the equation of motion for X becomes This equation corresponds to scattering from a parabolic potential. The non-adabatic evolution of X$_{k}$ during the short in- tervals when φ crosses the origin leads to a transforma- tion of the coefficients of Bogoliubov type The adiabaticity condition is now violated for momenta satisfying: where we defined θ j k = ∫ t$_{j}$ 0 dtω$_{k}$ which is the phase ac- cumulated at t$_{j}$ . The reflection and transmission coeffi- cients R$_{k}$ and D$_{k}$ are given by where the phase ϕ$_{k}$ is: Here Γ stands for the complex Gamma function. In ac- cordance with the general theory of particle production Note that the expansion of space makes broad resonance more effective since more k modes are red-shifted into the instability band as time proceeds. We will see below that the improved analysis yields the same expression for the resonance band except for the exact value of the numerical coefficient of the first term on the r.h.s.. Broad parametric resonance ends when q ≤ 1 / 4. Let us now move on to a quantitative analysis of this problem, building on the comprehensive study of [10]. It proves convenient to eliminate the Hubble friction term in the equation of motion by rescaling the field variable. We consider the variable X$_{k}$ ( t ) = a 3 / $^{2}$( t ) χ$_{k}$ ( t ) in terms of which the equation of motion (22) becomes: with We will first give a qualitative analysis of broad reso- nance in an expanding background characterized by the cosmological scale factor a ( t ) which is increasing in time as given by the Friedmann equations. The equation of motion for χ is Provided that the Floquet exponent is not much smaller than unity, the parametric resonance instability leads to an energy transfer from the inflaton to matter particles which is rapid on the scale of the Hubble time. Thus, an analysis neglecting the expansion of the uni- verse is self-consistent. However, as discussed in detail in [10], it is not too difficult to include the expansion of space.