n$_{s}$ ∈ [0 . 93 , 0 . 96] , r ∈ [10 − 5 − 10 − $^{1}$] , α$_{s}$ ∈ [0 . 6 , 1 . 3] × 10 − $^{3}$. These predictions are found within the 2 σ limit of the WMAP data provided that V$_{0}$ is large, [687, 688], though this requires that inflation taking place for super Planckian VEVs [687]. A large scale V$_{0}$ ∼ 10 16 GeV also implies a proton lifetime roughly estimated in the range τ ( p → π $^{0}$e$^{+}$) ∈ [10 $^{34}$, 10 $^{38}$] years [687]. It was also proposed in Ref. [687] to realize a chaotic-like inflation with the same potential but at large field χ > χ$_{0}$ , but this also imposes super Planckian VEVs. Taking the VEV above the Planck scale does not make any physical meaning as it would be hard to rely on the Coleman-Weinberg one-loop corrected potential away from the renormalization scale. 

Other challenges for this model makes it unappealing or unrealistic. The analysis of the parameter space ( a, b, g ) in [689] revealed that the potential possess a local minimum with symmetries SU (4) × U (1) in which the system gets trapped, even when starting close to the SM minimum (with symmetries SU (3) × SU (2) × U (1)). Obtaining the standard model at low energy is therefore only possible after another first order transition that leads to the problems of old inflation. Other problems were found related to the reheating and the generation of baryon asymmetry (see for e.g. [494]) $^{71}$. 

## 2. Hybrid inflation within SUSY GUTs and topological defects 

The embedding of inflation in SUSY GUTs has mostly been studied for hybrid inflation as those models consider a coupling between the inflaton sector and the GUT Higgs sector. It was therefore natural to first inquire if some GUT Higgs field already present in theories 

