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## 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 2 

<img src='content_image/330323.jpg'>

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] 

<img src='content_image/330324.jpg'>

<img src='content_image/330325.jpg'>

