Non-slow-roll dynamics in $\alpha-$attractors

In this paper we consider the $\alpha-$attractor model and study inflation under a generalization of slow-roll dynamics. We follow the recently proposed Gong \& Sasaki approach \cite{Gong:2015ypa} by assuming $N=N\left(\phi\right)$. We relax the requirement of inflaton potential flatness and allow a sufficiently steep one to support 60-efoldings. In addition, we obtain a family of functions describing the local shape of the potential during inflation. We derive spectral indices for scalar and tensor power spectrum in slow-roll parameters higher orders. We find that this type of inflationary scenario predicts an attractor at $n_{s}\approx0.967$ and $r\approx5.5\times10^{-4}$ which are very close to the predictions of the first chaotic inflationary model in supergravity (Goncharov-Linde model) \cite{Goncharov:1983mw}. We show that under a non-slow-roll dynamics, the $\alpha-$attractor model remains compatible with any 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 $ \alpha- $ parameter, which is also related to the K\"ahler manifold curvature in the supergravity (SUGRA) embedding of this model. We also comment on the stabilization of the inflaton's trajectory.