of the waterfall field, assumed complex. This modification was proposed as a new way to generate a baryonic asymmetry. Finally, the thermal hybrid inflation model has also been considered [162]. In this model, thermal corrections are assumed to generate part of the hybrid potential 50 V ( φ, ψ ) = V$_{0}$ + T 4 + T $^{2}$ψ2 − 1 2 m 2 $_{ψ}$ψ 2 + T $^{2}$φ2 . (187) However, as it is well known, the rapid expansion due to inflation dilute the content of the universe, exponentially rapidly driving the temperature to 0. This model therefore requires that a period of hot big-bang evolution takes place immediately before the phase of inflation. This condition can be considered rather fine tuned. In addition, inflation starts with T ∼ V 1 / 4 0 and the temperature triggers the waterfall transition taking place at T ∼ m$_{ψ}$ . With T ∼ 1 /a during inflation, inflation will last only 10 e-foldings and to be acceptable, this mechanism should be invoked many times. Another way temperature effects could influence inflation has been proposed in a number of recent articles (see [544, 545]), known as thermal inflation . This mechanism is based on the assumption that the couplings between the inflaton field and other particles (inde- pendently required for a successful reheating) can generate a constant decay of the inflaton during inflation (assuming Γ ∼ H$_{infl}$ ). This particle production would induce a thermal bath with a finite temperature that back-reacts on the inflaton dynamics and induces finite tem- perature effects on the potential. In particular this mechanism introduces a new viscosity term C$_{w}$ ˙ φ in the field dynamics Eq. (2), which slows down the rolling of the inflaton $^{51}$. This idea was used in a number of articles to realize inflation with potentials that would be too steep for inflation without temperature effects, for example in string theory (see for exam- ple [546]). This possibility is however ad-hoc and still debated and some authors [100, 547] argue that these effects are unlikely to take place, as the viscosity term is expected to be 50 In order to estimate the coefficients of thermal corrections one has to understand the exact particle contents. However note that φ belongs to a hidden sector, therefore exact particle contents are model dependent and sometimes chosen just to meet the desired results. 51 To our knowledge the viscosity term has never been introduced consistently in the equations of motion in an expanding background. Note that the decay term usually introduced phenomenologically during the inflaton oscillations is valid only when the frequency of the inflaton oscillations is larger than the Hubble expansion rate. Therefore one can safely use the decay of the inflaton as in the case of a flat space-time [100]. 87