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. 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 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 48 The hybrid inflation models were also considered in Refs. [163, 164, 316, 317, 529–534] in the context of large extra dimensions at TeV scale [535]. C. Non-SUSY models involving several fields 1. Original hybrid inflation V ( φ, ψ ) = 1 2 m $^{2}$φ2 + λ 4 ( ψ 2 − M 2 ) 2 + λ ′ 2 φ $^{2}$ψ2 , (176) φ$_{c}$ = M √ λ λ ′ . (177) 83