Constraining an Early Dark Energy Motivated Quintessential $\alpha$-Attractor Inflaton Potential

We construct a new model of quintessential $\alpha$ attractor inflation in conjunction with the features of non-oscillating early dark energy (EDE). Slow-roll plateau of this model is obtained, and analyzed in $k$-space, through the inflaton field and its first-order perturbation over a quasi de-Sitter metric fluctuation in the range $k=0.001-0.009$ Mpc$^{-1}$. The estimated cosmological parameters are found to obey Planck+BICEP2/Keck bounds with $68\%$ CL with the required trend of spectral tilts in the $n_s-r$ parametric space. We verify that, the inclusion of the EDE does not significantly affect the observed parameters. Its presence manifests in obtaining \textit{improved values} of the energy scale of inflation ($M$) and the present-day vacuum density ($V_{\Lambda}$). They are found to be $M=5.58\times 10^{-4}-4.57\times 10^{-3} M_P$ and $V_{\Lambda}=1.042\times 10^{-119}-4.688\times 10^{-116} M_P^4$. However, the $\alpha$-parameter is drastically constrained in two ways. Its lower end is fixed by the consistency analysis of the $k$-mode equations, while the upper end is evaluated as a derived expression of $\alpha$-cut-off through the aspects of EDE \textit{viz.,} the effects of \textit{Enhanced Symmetry Point} (ESP) in the potential during inflation. Improvised range of $\alpha$ is found to be $0.001\leq\alpha<0.1$ for the model parameters $\gamma$ and $n$ lying within $0.01\leq\gamma\leq 0.09$ and $8\leq n\leq 10$ respectively. These ranges are shown to be essential for satisfying the COBE/Planck normalized energy scale of inflation and the Planck-value of present-day vacuum density. If we choose $\gamma=0.0818$ and $n=8$, then we get $0.001\leq\alpha\leq 0.0186$. Thus, the lower and upper limits of $\alpha$ are diminished substantially, compared with those in the earlier studies.