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inelastic scatterings as a thermalization rate of the universe 

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Since the inflaton decay products have SM gauge interactions, the universe reaches full thermal equilibrium immediately after the inflaton decay. The reason is that the 2 → 3 scatterings with gauge boson exchange in the t − channel are very efficient, see [120–122]. 

Even before all inflatons decay, the decay products form a plasma can very quick ther- malize, and the plasma has the instantaneous temperature given by Eq. (76). The plasma can reach its maximum T$_{max}$ soon after the inflaton field starts to oscillate around the min- imum of its potential, which happens for a Hubble parameter H$_{I}$ ≤ m$_{φ}$ . During this era the energy density of the universe is still dominated by the (non–relativistic) inflatons that haven’t decayed yet. The scale factor of the universe a then varies as a ∝ T − 8 / 3 [96]. The universe remains in this phase as long as H > Γ$_{d}$. During this phase one can produce massive long–lived or stable weakly interacting massive particles (WIMPS) [121, 752–759], see also gravitational production of particles [760, 761]. There are three possible scatterings which have been discussed in the literature. 

Particle creation via Soft-soft scatterings were investigated in detail in Refs. [752–756, 759], where the relevant Boltzmann equations governing the production and annihilation of stable particles, χ ’s, are solved both numerically and analytically. In Refs. [752–754, 759], out of equilibrium production of χ from scatterings in the thermal bath were studied and the final result is found to be (the superscript “ss” stands for χ production from “soft–soft” scatterings) $^{80}$: 

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Here Ω$_{χ}$ is the χ mass density in units of the critical density, and h is the Hubble constant in units of 100 km / (s · Mpc). The cross section for χ pair production or annihilation is given by: σ ≃ α 2 $_{χ}$/m 2 $_{χ}$. Most χ particles are produced at T ≃ m$_{χ}$/ 4. The density of earlier produced particles is strongly red–shifted, while χ production at later times is suppressed by the Boltzmann factor. 

