4.1 Thermal Inflation 44 of the interaction between φ and the thermal bath. This results in a thermal correction to the effective mass of m 2 T ≃ g $^{2}$T $^{2}$. Within the context of statisti- cal mechanics, the interpretation of V$_{eff}$ is that of the free energy of φ when the field is in thermal equilibrium with the thermal bath at temperature T , with the minima in V$_{eff}$ defining the equilibrium states, with 〈 φ 〉 representing the thermal average, as opposed to the vacuum expectation value (VEV). Let us take the following potential Initially, the temperature of the Universe will be sufficiently high that the temperature term in the brackets will be greater than the mass term in the brackets. This will have the affect of holding the φ field at φ = 0 . When the energy density of the Universe falls below the value of V$_{0}$ in the thermal inflation potential, the V$_{0}$ term will come to dominate the energy density of the Universe and thermal inflation will begin. It will continue until very shortly after the point when the temperature term has become smaller than the mass term, at which point spontaneous symmetry breaking will occur and so φ will start to roll down the potential towards either the positive or negative VEV. 1 The shape of the potential in this scenario is displayed in Fig. 4.1. This scenario is quite general and would not be particularly unexpected in the early Universe. However, Thermal Inflation was originally proposed as a solution to the moduli problem [47,48]. Moduli are scalar fields that arise $^{1}$For ease of visualization and calculation, it is usually assumed that a scalar field rolls down to the positive VEV. V ( φ, T ) = V$_{0}$ + ( g $^{2}$T 2 − 1 2 m 2 0 ) φ $^{2}$+ λ φ 6 M 2 P (4.1)