VI. INFLATON DECAY, REHEATING AND THERMALIZATION A. Perturbative decay and thermalization For a plasma which is in full thermal equilibrium, the energy density, ρ , and the number density, n , of relativistic particles are given by [26, 96] ρ = ( π $^{2}$/ 30 ) T 4 , n = ( ζ (3) /π 2 ) T 3 , (Boson) , ρ = (7 / 8) ( π $^{2}$/ 30 ) T 4 , n = (3 / 4) ( ζ (3) /π 2 ) T 3 , (Fermion) , (375) where T is the temperature of a thermal bath. Note that in a full equilibrium the relation- ships, 〈 E 〉 ∼ ρ 1 / $^{4}$, and n ∼ ρ 3 / 4 hold, with 〈 E 〉 = ( ρ/n ) ≃ 3 T being the average particle energy. On the other hand, right after the inflaton decay has completed, the energy density of the universe is given by: ρ ≈ 3 (Γ$_{d}$ M$_{P}$ ) $^{2}$. For a perturbative decay, which generates en- tropy, we have 〈 E 〉 ≈ m$_{φ}$ ≫ ρ 1 / $^{4}$. Then, from the conservation of energy, the total number density is found to be, n ≈ ( ρ/m$_{φ}$ ) ≪ ρ 3 / $^{4}$. Hence the complete inflaton decay results in a dilute plasma which contains a small number of very energetic particles. This implies that the universe is far from full thermal equilibrium initially [93, 94, 97–100, 120, 122, 747]. Reaching full equilibrium requires re-distribution of the energy among different particles, kinetic equilibrium , as well as increasing the total number of particles, chemical equilibrium . Therefore both the number-conserving and the number-violating reactions must be involved. f$_{1}$ f$_{2}$ f$_{1}$ f$_{2}$ f$_{1}$ f$_{2}$ f$_{1}$ f$_{2}$ Fig. A Typical scattering diagram which builds kinetic equilibrium in the reheat plasma. Note that the t − channel singularity which results in a cross-section ∝ | t | − $^{1}$. The second panel shows a typical scattering diagram which increase the number of particles. • Kinetic equilibrium among SM fermions: The most important processes are 2 → 2 scatterings with gauge boson exchange in 173