123 Substituting these equations into Eq. (B.9) gives The argument of the delta function is only =0 if p ′ = p . Therefore we have We have From the 4-momenta of p$_{2}$ , Eq. (B.2), we know that E$_{2}$ = p . In addition, given that the energy of a χ particle is T , we therefore have Let us consider the time of primordial inflation reheating. We have Therefore σ ≈ − g 4 64 π $^{2}$E$_{1}$ E$_{3}$ ∫ 1 p δ (√ p $^{2}$+ m 2 φ, $_{eff}$+ p − √ p ′ $^{2}$+ m 2 φ, eff − p ′ ) p ′$_{d}$ p ′ ∫ Ω dΩ (B.12) σ ≈ − g 4 64 π $^{2}$E 2 1 ∫ Ω dΩ (B.13) E 2 1 = p $^{2}$+ m 2 φ, eff (B.14) E 2 1 = T $^{2}$+ m 2 φ, eff (B.15) m$_{φ,}$$_{eff}$ ∼ gT (B.16) E 2 1 ∼ ( 1 + g 2 ) T 2 (B.17) ∼ T 2 (B.18)