theories [505–507] (see Sec. IV E 1) as well as in certain classes of brane-world models [508], where it has been claimed to give rise to a larger expansion rate due to modification in the Hubble equation rate. The necessity of super Planckian VEVs represents though a challenge to such embedding in particle physics. Variants of these simple models have been constructed, based on the radiative corrections appearing at the loop level. First, the radiative corrections due to the inflaton self-interaction can induce a running of the mass of the inflaton with its VEV (see Sec. III B). For example for α = 2, a “running mass” potential of the form V ( φ ) = m $^{2}$(1 + α ln( φ/µ )) φ 2 has been considered in [509]. Though the dynamics is not significantly affected by this running, it was found to be able to affect the decay of the inflaton during reheating via fragmentation. In [510], the chaotic inflation potential was extended to include a Yukawa coupling ( h/ 2) φ ¯ N$_{R}$N$_{R}$ to the right-handed neutrino N$_{R}$ . This introduces a one-loop correction to the inflation potential of the form $^{V$_{1}$}$− loop ≃ κφ $^{4}$ln( hφ/µ ), where κ = h $^{4}$/ 16 π 2 and µ is a renormalization scale. It was found that the radiative corrections can affect significantly the predictions of the chaotic models, namely by reducing the level of tensor-to-scalar ratio. For example, for the quartic potential, h = 0 gives rise to n$_{s}$ ≃ 0 . 95 and r ≃ 0 . 25, whereas for h ≃ 1 . 7 × 10 − $^{3}$, n$_{s}$ ≃ 0 . 95 and r ≃ 0 . 084 which is within the 1 σ contour of the WMAP data. Note that these results are very sensitive to the value of h . It would give rise to a power law expansion a t ) ∝ t $^{p}$, so that inflation occurs when p > 1. The case p = 2 corresponds to the exactly de Sitter evolution and a never ending accelerated expansion. Even for p = 2, violation of slow-roll never takes place, since ϵ ( φ ) = 1 /p and inflation has to be ended by a phase transition or gravitational production of particles [10, 511]. The confrontation to the CMB data yields: n$_{s}$ = 1 − 2 /p and r = 16 /p ; the model predicts a hight tensor to scalar ratio and it is within the one sigma contour-plot of WMAP (with non-negligible r ) for p ∈ [73 − 133]. Multiple exponentials with differing slopes give rise to what has been dubbed as assisted inflation [157]. Such potentials might arise in string theories and theories with extra dimensions. b. Exponential potential: An exponential potential also belongs to the large field mod- els: V ( φ ) = V$_{0}$ exp ( − √ 2 p φ M$_{P}$ ) . (165) 79