3. Tachyonic preheating The second scenario is known as tachyonic preheating . Let us consider a simple example of tachyonic potential V = V$_{0}$ − 1 2 m $^{2}$χ2 + λ 4 χ 4 (388) The rolling of a tachyon in itself results in an exponential instability in the perturbations of χ with physical momenta smaller than the mass. The tachyonic growth takes place within a short time interval, $^{t}$∗ ∼ (1 / 2 m ) ln( π $^{2}$/λ ) (see [725]). During this short period the occupation number of χ quanta grows exponentially for modes k < m up to n$_{k}$ ∼ exp(2 $^{mt}$∗ ) ∼ exp(ln( π $^{2}$/λ )) ∼ π $^{2}$/λ . For very small self-coupling, which is required for a successful inflation, the occupation number, which depends inversely on the coupling constant, can become much larger than one. First, the number density of the produced particles in x -space is given by n$_{χ}$ ∼ m $^{3}$/ (8 πλ ). Hence the total energy density stored in produced χ quanta is given by [725] The plasma from the non-perturbative inflaton decay eventually reaches full thermal equi- librium, though, at time scales much longer than that of preheating itself [102, 104]. The occupation number of particles is f$_{k}$ ≫ 1 in the meantime. This implies that dangerous relics (such as gravitino and moduli) can be produced much more copiously in the aftermath of preheating than in full thermal equilibrium. This is a negative aspect of an initial stage of preheating. One usually seeks a late stage of entropy release, in order to dilute the excess of relics. As we shall show, SUSY naturally provides us a tool to undo preheating [123], see Sec. VI D 1 The resonant and instant preheating calculations can be reanalyzed for a fermionic cou- pling, hφ ¯ ψψ . In both the cases one would expect a large exponential growth in particle creation. It is also possible to excite superheavy fermions from resonant preheating [338, 774– 776]. However note that both φ and ψ are SM gauge singlets. For an inflaton field coherently oscillating about the minimum of the potential V = 1 $_{2}$m 2 $_{φ}$φ $^{2}$, If one neglects the back reaction of the created particles, then after few oscillations, 4. Fermionic preheating 182 ρ$_{χ}$ ∼ 1 2 ( δ (1) ˙ χ ) 2 ∼ mn$_{χ}$ ∼ 1 8 π m 4 λ . (389)