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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 

<img src='content_image/365742.jpg'>

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 

