content_image/362715.jpg content_image/362717.jpg content_image/362719.jpg
## 1. Large field models 

a. Power-law chaotic inflation: The simplest inflation model by the number of free parameters is perhaps the chaotic inflation [4] with the potential dominated by only one of the terms in the above series 

<img src='content_image/362715.jpg'>

with α a positive integer. The first two slow-roll parameters are given by 

<img src='content_image/362717.jpg'>

Inflation ends when ϵ = 1, reached for φ$_{e}$ = αM$_{P}$$_{/}$√$_{2.}$ The largest cosmological scale becomes super-Hubble when φ$_{Q}$ = √ 2 N$_{Q}$αM$_{P}$ , which is super Planckian; this is the first challenge for this class of models (see discussion Sec. II F). The spectral index for the scalar and tensor to scalar ratio read: 

<img src='content_image/362719.jpg'>

The amplitude of the density perturbations, if normalized at the COBE scale, yields to extremely small coupling constants; λ$_{α}$ ≪ 1 (for e.g. λ$_{4}$ ≃ 3 . 7 × 10 − $^{14}$). The smallness of the coupling, λ$_{α}$/M α − 4 P , is often considered as an unnatural fine-tuning. Even when dimension full, for example if α = 2, the generation (and the stability) of a mass scale, √$_{λ}$$_{2}$$_{M}$$_{P}$ ≃ 10 13 GeV, is a challenge in theories beyond the SM, as they require unnatural cancellations. These class of models have an interesting behavior for initial conditions with a large phase space distribution where there exists a late attractor trajectory leading to an end of inflation when the slow-roll conditions are violated close to the Planck scale [4, 280, 283, 503]. 

Recently, these models also suffer from an observational challenge which comes from the prediction of a high tensor to scalar ratio that is under tension by the most recent CMB+BAO+SN data [13]. The cases α ≥ 4 lie outside the 2 σ region, though the case α = 2 is still in the 1 σ region for N$_{Q}$ > ∼ 60 (see Sec. II E). 

Note that the above mentioned monomial potential can be a good approximation to describe in a certain field range for various models of inflation proposed and motivated from particle physics; natural inflation when the inflaton is a pseudo-Goldstone boson [504], or the Landau-Ginzburg potential when the inflaton is a Higgs-type field [86]. Some of these potentials will be discussed below. Chaotic inflation was also found to emerge from SUGRA 

