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Non-slow-roll dynamics in $α-$attractors

In this paper we consider the $α-$attractor model and study inflation under a non-slow-roll dynamics. More precisely, we follow the approach recently proposed by Gong and Sasaki \cite{Gong:2015ypa} by means of assuming $N=N\left(ϕ\right)$. Within this framework we obtain a family of functions describing the local shape of the potential during inflation. We study a specific model and find an inflationary scenario predicting an attractor at $n_{s}\approx0.967$ and $r\approx5.5\times10^{-4}$. We further show that considering a non-slow-roll dynamics, the $α-$attractor model can be broaden to a wider class of models that remain compatible with value of $r<0.1$. We further explore the model parameter space with respect to large and small field inflation and conclude that the inflaton dynamics is connected to the $ α- $ parameter, which is also related to the Kähler manifold curvature in the supergravity (SUGRA) embedding of this model. We also comment on the stabilization of the inflaton's trajectory.