D. SM Higgs as the inflaton It is natural to study if the SM Higgs can play the role of the inflaton field. This question has been discussed long ago [586], and has regained interest in the last few years [86, 290, 587– 590]. 1. Dynamics of the SM Higgs inflation It has been proposed long ago to improve this situation by abandoning the universal coupling to gravity [86, 586, 591, 592] $^{56}$, in order to flatten the Higgs potential at high energies. The lagrangian is extended to contain a non-minimal coupling to gravity (only) for a Higgs field H L = L SM − M 2 2 R − $_{ξH}$†$_{HR .}$ (210) S J = ∫ d 4 $_{x}$√ − g { M 2 + ξh 2 2 R + 1 2 ∂$_{µ}$h∂ $^{µ}$h − λ 4 ( h 2 − v 2 ) 2 } . (211) Ω 2 = 1 + ξh 2 , g$_{µν}$ → ˆ g$_{µν}$ = Ω $^{2}$g$_{µν}$ , h → χ with d χ d h = √ Ω 2 + 6 ξ $^{2}$h$^{2}$/M 2 P Ω 4 , (212) This non-minimal coupling can be motivated by the renormalizability of the λφ 4 potential [606]. The very form of this Lagrangian might represent the first challenge of the model, since the equivalence principal is lost; all particles are not coupled in the same way to gravity. The above Lagrangian has been studied in the past [586, 607], where H is a GUT Higgs field and was applied in [86] to the SM Higgs field. If h is the Higgs field in the unitary gauge, the resulting action for h , in the Jordan frame, reads Since the coupling to gravity is not minimal in S$_{J}$ , studying the phenomenology of the model is simplified once the conformal transformation is applied [86, 586] 56 Such attempts were previously made in connection with scalar tensor theories of inflation to flatten the inflationary potential [586, 593–602], or to exit the false vacuum inflation [593, 603–605]. Typically the gravitational part of the action is given by: S = ∫ d $^{4}$x √ − g [ 1 $_{2}$M$^{2}$f ( φ ) R − 1 $_{2}$∂$_{µ}$ φ∂ $^{µ}$φ ]. The action is dynamically equivalent to a theory in which the gravitational action is the usual one, via the conformal transformation: ˆ g$_{µν}$ = f ( φ ) g$_{µν}$ , where we use the bar to indicate a quantity in the new frame. The new action looks like: S$_{E}$ = 1 2 ∫ d 4 $_{x}$√ − ˆ g [ M 2 ˆ R − K ( φ )( ˆ ∂φ ) $^{2}$], where, K ( φ ) ≡ 2 f ( φ )+3 M $^{2}$f ′ $^{2}$( φ ) 2 f $^{2}$( φ ) . 95