A novel way of constraining the $\alpha$-attractor chaotic inflation through Planck data

Defining a scale of $k$-modes of the quantum fluctuations during inflation through the dynamical horizon crossing condition $k = aH$ we go from the physical $t$ variable to $k$ variable and solve the equations of cosmological first-order perturbations self consistently, with the chaotic $\alpha$-attractor type potentials. This enables us to study the behaviour of $n_{s}$, $r$, $n_{t}$ and $N$ in the $k$-space. Comparison of our results in the low-$k$ regime with the Planck data puts constraints on the values of the $\alpha$ parameter through microscopic calculations. Recent studies had already put model-dependent constraints on the values of $\alpha$ through the hyperbolic geometry of a Poincar\'{e} disk: consistent with both the maximal supergravity model $\mathcal{N}=8$ and the minimal supergravity model $\mathcal{N}=1$, the constraints on the values of $\alpha$ are $\frac{1}{3}$, $\frac{2}{3}$, 1, $\frac{4}{3}$, $\frac{5}{3}$, 2, $\frac{7}{3}$. The minimal $\mathcal{N}=1$ supersymmetric cosmological models with $B$-mode targets, derived from these supergravity models, predicted the values of $r$ between $10^{-2}$ and $10^{-3}$. Both in the $E$-model and the $T$-model potentials, we have obtained, in our calculations, the values of $r$ in this range for all the constrained values of $\alpha$ stated above, within $68\%$ CL. Moreover, we have calculated $r$ for some other possible values of $\alpha$ both in low-$\alpha$ limit, using the formula $r=\frac{12\alpha}{N^{2}}$, and in the high-$\alpha$ limit, using the formula $r=\frac{4n}{N}$, for $n=2$ and $4$. With all such values of $\alpha$, our calculated results match with the Planck-2018 data with $68\%$ or near $95\%$ CL.