content_image/355045.jpg content_image/355052.jpg
already before the end of reheating [96]. If the inflaton decays with a rate Γ$_{φ}$, then the instantaneous plasma temperature is found to be [96]: 

The temperature of the universe reaches its maximum T$_{max}$ soon after the inflaton field starts oscillating around the minimum. Once the maximum temperature is reached, then ρ$_{ψ}$ ∼ a − 3 / $^{2}$, and T ∼ a − 3 / 8 until reheating and thermalization is completely over. Thermalization is achieved when both kinetic and chemical equilibrium are reached, for a review see [125]. 

<img src='content_image/355045.jpg'>

Note that the above analysis is solely based on energetic argument. It claims that T$_{R}$ ≤ ρ 1 / 4 $_{inf}$, but ignored the microphysical aspects such as what degrees of freedom are excited after the end of inflation. For a successful cosmology one needs to ask how the inflaton energy gets converted into the SM degrees of freedom. This will be discussed in chapters V and VI. 

For large reheat temperatures, T$_{R}$ ∼ 10 9 GeV, the universe could abundantly create thermal relics of unstable gravitinos with a mass of order 100 − 1000 GeV, which could spoil the success of BBN [309–314]. For extremely low reheat temperatures, i.e. T$_{R}$ ∼ O (1 − 10) MeV, it becomes a great challenge to obtain matter-anti-matter asymmetry and the right abundance for the dark matter. Only a few particle physics scenarios can successfully create baryons and dark matter at such a low temperature, see for instance [315–318]. 

## G. Requirements for a successful inflation 

The success of inflation is closely tied to the success of BBN [26, 27, 29] as the epoch constraints the number of relativistic degrees of freedom beyond the SM, and the bary- onic asymmetry. Furthermore the CMB data and galaxy formation also constraints the abundance of cold dark matter and baryons. 

## 1. Baryons and nucleosynthesis 

For a successful BBN, which takes place within the first few hundred seconds, the abun- dances of light elements $^{2}$H, $^{3}$He, $^{4}$He and $^{7}$Li crucially depends on the baryon-to-photon ratio: 

<img src='content_image/355052.jpg'>

