Constraining the quintessential $\alpha$-attractor inflation through dynamical horizon exit method

In the present paper, we perform a sub-Planckian quantum mode analysis of linear cosmological perturbation in the inflaton field over a classical quasi de-Siter metric background by dynamical horizon exit (DHE) method. In this way, we probe the inflationary regime of a quintessential $\alpha$-attractor model by analysing the COBE/Planck normalized power spectra, spectral indices, tensor to scalar ratio, number of e-folds, running of the spectral index and inflationary Hubble parameter in $k$-space. We compare our results with ordinary $\alpha$-attractor $E$ and $T$ models and with that of Planck-2018 results. Our estimated values of $n_s$ and $r$ lie within $68\%$ CL with respect to Planck data for $k=0.001 - 0.009$ Mpc$^{-1}$ for all values of $\alpha$. The $\alpha$ values, obtained in our calculations satisfy various post inflationary constraints regarding preheating and reheating, reported in current literature. We observe that quintessence sets an upper bound of $\alpha=4.3$ and thereby restricts the model from becoming of the power law type, making it more efficacious than ordinary $\alpha$-attractors in explaining both inflation and dark energy. A striking observation in our analyses is that, unlike in our previous study, we find a continuous values of $\alpha$ within $\frac{1}{10}\leq \alpha\leq 4.3$ for the specified $k$ range. At the end, we have shown that the model parameters constrained in this work give a very small vacuum density $\sim 10^{-117}-10^{-115} M_P^4$ which is an essential criterion for current and future dark energy observations of the universe.