Analyzing $\mu$ Parameter Range and the Duration of Reheating in the Polynomial $\alpha$-attractor Models using CMB Observations

Cosmological $\alpha$-attractor models are renowned for their universal predictions for inflationary observables, largely independent of the detailed form of the inflaton potential. In this work, we investigate the extent to which this universality persists for polynomial $\alpha$-attractor models, with particular emphasis on the scalar spectral index ($n_s$) and the post-inflationary reheating history. We derive an extended analytical expression for the scalar spectral index ($n_s$) and the tensor-to-scalar ratio ($r$), retaining sub-leading contributions beyond the commonly used universal approximation. Interestingly, we find that these corrections introduce an explicit dependence of $n_s$ on the parameter characterizing the polynomial attractor potential $\mu$, a dependence that is absent in the leading universal expression. We systematically investigate how this model dependence modifies the predictions in the $(n_s-r)$ plane and affects the model parameter ($\mu$). We further incorporate the Duration of reheating ($\mathrm{\Delta }{N}_{rh}$) and effective equation of state parameter of the reheating epoch ($w_{rh}$), accounting for their impact on the number of e-folds between horizon crossing and the end of inflation ($\mathrm{\Delta }{N}_{\mathrm{C}\mathrm{M}\mathrm{B}}$). By confronting the resulting predictions with current CMB observations, we identify the viable parameter space and examine the interplay between the model parameter ($\mu$) and reheating. Our results demonstrate that sub-leading corrections can provide a window into the underlying structure of polynomial attractors, offering a possible way to distinguish different members of the attractor class observationally.