content_image/375458.jpg content_image/375463.jpg
to scalar in the slow-roll approximations read 

Let us first discuss the small field limit. For p > 1 and p ̸ = 2, the model is considered slightly disfavored by the recent data, since n$_{s}$ > ∼ 1 with negligible tensor to scalar ratio, and running of the spectral index is predicted ( r ∝ ( φ/µ ) 2 $^{p}$, α$_{s}$ ∝ ( φ/µ ) 2 $^{p}$). It was pointed out however that to confront this model with CMB data, one could include the concordance model parameters and a weight for cosmic strings, since this model in its two-field version can produce cosmic strings. In that case, n$_{s}$ ∼ 1 is in agreement with observations at 1 σ [263]. In the case p = 2, η is constant and n$_{s}$ is predicted to be: 

<img src='content_image/375458.jpg'>

<img src='content_image/375463.jpg'>

which is well above unity unless µ ≫ M$_{P}$ . This is equivalent to having an extreme fine-tuning on a coupling constant, which is considered unnatural. The model is therefore disfavored at more than 2 σ if the data are analyzed with the minimal 6-parameter model. However, if µ > ∼ 7 M$_{P}$ then it predicts a spectral index around 1 < n$_{s}$ < ∼ 1 . 1, which is still in agreement with current observations if cosmic strings have a non-negligible contribution to the CMB anisotropies (see [263] and Sec. III G). 

The case p = 2 has been studied away from the small field regime and/or without the slow-roll approximations [206, 512, 518]. It was shown that the model is in agreement with the CMB data if inflation is realized in the large field regime. This can be achieved by some mechanism (such as a waterfall triggered by an external field) independently of the parameter µ [206, 512, 518] or when µ < ∼ 0 . 32 M$_{P}$ [206]. In the last case, the Hubble-flow parameter ϵ$_{1}$ differs from ϵ at small VEVs and violation of slow-roll is responsible for forcing inflation to take place in the large field model without any condition on a waterfall parameter [206]. In both cases, inflation is necessarily realized for inflaton VEVs large compared to µ , which reduces the appeal of the model. 

An inverted hybrid (or hiltop) inflation model has also been proposed where the curvature 

