content_image/328641.jpg content_image/328642.jpg
## 3. Thermal corrections to the curvaton 

The flat direction VEV naturally induces a mass ∼ λϕ$_{inf}$ to the field which are coupled to it, where λ is a gauge or Yukawa coupling. Note that ϕ$_{inf}$ is the initial VEV of the curvaton, after the end of inflation slides down gradually. If there is a thermal bath with a temperature T then there is a thermal corrections to the flat direction depending on whether λϕ$_{inf}$ ≤ T or λϕ$_{inf}$ > T , different situations arise. 

• λϕ$_{inf}$ ≤ T : 

Fields in the plasma which have a mass smaller than temperature are kinematically accessible to the thermal bath. They will reach full equilibrium and result in a thermal correction V$_{th}$ to the flat direction potential 

<img src='content_image/328641.jpg'>

The flat direction then starts oscillating, provided that λT > H [122, 123, 136, 749]. 

• λϕ$_{inf}$ > T : 

Fields which have a mass larger than temperature will not be in equilibrium with the thermal bath. For this reason they are also decoupled from the running of gauge couplings (at finite temperature). This shows up as a correction to the free energy of gauge fields, which is equivalent to a logarithmic correction to the flat direction potential [136]: 

where α is a gauge fine structure constant. Decoupling of gauge fields (and gauginos) results in a positive correction, while decoupling of matter fields (and their super- partners) results in a negative sign. The overall sign then depends on the relative contribution of decoupled fields. As an example, let us consider the H$_{u}$H$_{d}$ flat direc- tion. This direction induces large masses for the top (s)quarks which decouples them from the thermal bath, while not affecting gluons and gluinos. Therefore this leads to a positive contribution from the free energy of the gluons. 

<img src='content_image/328642.jpg'>

Obviously only corrections with a positive sign can lead to flat direction oscillations around the origin. Oscillations begin when the second derivative of the potential exceeds the Hubble rate-squared which, from Eq. (453), reads ( αT $^{2}$/ϕ$_{inf}$ ) > H . 

