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]) $^{71}$. 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 2. Hybrid inflation within SUSY GUTs and topological defects 71 One can also think of realizing a brief period of thermal inflation close to the electroweak scale with the help of the GUT singlet [161, 162]. The idea is that thermal effects will keep the singlet scalar field close to the symmetric phase, but once inflation begins, the universe would cool down and the field would roll down the potential and settle in its minimum, which is close to the GUT scale. The simplest potential will be like: V ∼ V$_{0}$ − m $^{2}$φ$^{2}$. The idea is appealing, but challenging to realize. The model requires a very light GUT singlet, i.e. m ∼ 100 GeV to be kept in thermal equilibrium with other light degrees of freedom to get + T $^{2}$φ2 correction. However the singlet should break the GUT group down to the SM when it develops a VEV and excite the SM degrees of freedom alone at the electroweak scale. If V 1 / 4 0 ∼ 10 16 GeV, then the initial temperature of the universe prior to such a phase transition ought to be very high fairly close to the GUT temperatures and final temperature should be close to the electroweak scale to get sufficient inflation, i.e. roughly 7 − 10 e-foldings of inflation, which is sufficient to dilute the unwanted relics such as excess gravitons or damping the moduli oscillations, etc. To our knowledge there is no explicit light GUT singlet which has been constructed to execute this idea. 131