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We have carried out detailed $k$-mode analysis of single-inflaton slow-rollinflationary perturbations including quantum fluctuations by setting upnon-linear coupled differential equations of inflaton field ($\phi$), itsperturbation ($\delta\phi$) and the metric perturbation (the Bardeen potential,$\Phi_B$), and calculated the number of e-folds ($N$), scalar spectral index($n_s$), tensor spectral index ($n_h$), scalar power spectrum ($\Delta_s$),tensor power spectrum ($\Delta_h$), tensor-to-scalar ratio ($r$) and the Hubbleparameter ($H$) for different $k$ values at the horizon crossing. In thesecalculations we have employed the $E$-model $\alpha$-attractor potentials whichare found to display slow-roll behaviour. The values of $n_s$ and $r$ obtainedby us are consistent with those given by the well-known universal$\alpha$-attractor formulae. We got $n_s = 0.956908$ and $r= 0.005571 $ at $k =10^6$ Planck unit for the value of the potential parameter $n = 1$. These canbe compared with the Planck-2018 data viz., $n_s = 0.9649\pm 0.0042$ at $68\%$CL, $r<0.064$ at $95\%$ CL and ACT-2020 data viz., $n_s = 0.9691\pm 0.0041$ at$68\%$ CL.

Mode analysis of cosmological perturbations with the $E$-model $\alpha$-attractor inflaton potentials