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$^{which can play the role of the waterfall fields Φ}$±. For the D -term model, the presence of a constant FI term would require that the group U (1)$_{ξ}$ is not the subset of a non-abelian group [64]. As a consequence, if the SM is embedded in a model based on the group G$_{GUT}$ , the whole theory is based on G$_{GUT}$ × U (1)$_{ξ}$ and inflation takes place in the chain [64], 

<img src='content_image/333838.jpg'>

where “Infl” identify the symmetry breaking that triggers the waterfall at the end of inflation. This represents the lowest possible level of embedding of inflation within SUSY GUTs, since both the inflaton field and the Higgs fields are introduced in addition to the field content motivated by the particle physics. 

The F -term inflation model does not contain the restriction due to the presence of a FI term, therefore its embedding withing SUSY GUT has a richer phenomenology. The superfields Φ and Φ are assumed to belong to a non-trivial representation and the complex conjugate representation of some group G$_{infl}$ , whereas S is a singlet of G$_{infl}$ , in such a way that W = κS (ΦΦ) is invariant under G$_{infl}$ . 

The general picture is when inflation takes place, within the cascade of spontaneous symmetry breaking induced by the Higgs sector is therefore [64], 

<img src='content_image/333839.jpg'>

Cosmological observations can constrain what G$_{infl}$ can and cannot be, i.e. one of the motivation to introduce inflation in GUT was to solve the monopole problem. Therefore, the breaking of G$_{infl}$ , and all subsequent SUSY breaking cannot give rise to the formation of monopoles. Following the argument of Ref. [65] for SO(10), a systematic study has been done in Ref. [444] for all possible models that can be constructed using SUSY GUTs based on all possible GUT group of rank lower than 8 (including SO (10), SU ( n ), E$_{6}$ ). It was assumed that a discrete Z$_{2}$ symmetry is left unbroken at low energies to protect the proton from a too rapid decay. It was shown that generically the waterfall at the end of inflation gives rise to the formation of cosmic strings, since almost all possible ways to break G$_{GUT}$ down to G$_{S}$$_{M}$ involve the generation of a U (1) symmetry (leading to monopoles) or its breaking (leading to cosmic strings). 

In addition, for an SO (10) or E$_{6}$ based models, the waterfall accompanies generically the breaking of the U $^{(1)$_{B}$}$− L symmetry [444]. Consequently, the Higgs coupled to the inflaton 

