content_image/332719.jpg content_image/332720.jpg
only valid in the large field limit φ ≫ $^{φ}$∗ , since in the small field limit, the potential is not bounded from below and should be completed. 

## C. Non-SUSY models involving several fields 

## 1. Original hybrid inflation 

The most studied multi-field inflation model is the hybrid inflation first discussed in Ref. [147] (and studied extensively in [512]) as a model that differs from chaotic inflation on two main properties; it ends inflation with a waterfall triggered by a Higgs (not necessarily the SM Higgs) field coupled to the inflaton and it does not necessarily require an extremely small coupling to account for the normalization of the power-spectrum. The model is based on the potential given by [147] 

where φ is the inflaton and ψ is the Higgs-type field. λ and λ ′ are two positive coupling constants, m and M are two mass parameters. It is the most general form (omitting a quartic term λ ′′ φ $^{4}$) of renormalizable potential satisfying the symmetries: ψ ↔ − ψ and φ ↔ − φ . Inflation is assumed to be realized in the false-vacuum along the ψ = 0 valley and ends with a tachyonic instability for the Higgs-type field. The critical point of instability below which the potential develops non-vanishing minimum is at 

<img src='content_image/332719.jpg'>

<img src='content_image/332720.jpg'>

The system then evolves toward its true minimum at V = 0, 〈 φ 〉 = 0, and 〈 ψ 〉 = ± M $^{48}$. 

The inflationary valley, for 〈 ψ 〉 = 0, is usually assumed to be where the last 60 e-foldings take place. This is supported by numerical and analytical simulations [206, 514, 536–538], where the fine-tuning of the initial conditions were discussed. In Ref. [206] it was found that when the initial VEV of the inflaton, φ , is sub-Planckian, a subdominant but non-negligible part of the initial conditions for the phase space leads to a successful inflation, i.e. around less than 15% depending on the model parameters. Initial conditions with super-Planckian 

