Viability of generalized $\alpha$-inflation from Planck, ACT, and DESI Data

Abstract

We study inflationary constraints from reheating on two classes of single-field inflationary models: a generalized $\alpha$-attractor with potential $V(\phi) = V_0 \!\left(1 - \sech^p\!\left[\phi/(\sqrt{6\alpha}\,M_{Pl})\right]\right)$ and the $\alpha$-Starobinsky model with potential $V(\phi) = V_0 \!\left(1 - e^{- \sqrt{2/(3\alpha)}\, \phi / M_{Pl}} \right)^2$. Using a semi-analytical relation that connects inflationary dynamics to reheating, we solve for the horizon-crossing field value and calculate the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$, and the reheating temperature $T_{re}$. We consider three perturbative decay channels of the inflaton: gravitational, Yukawa (fermionic), and scalar. The parameter space spanned by $(\alpha,p,\omega_{re})$ for the first model and $(\alpha,\omega_{re})$ for the second is explored and compared with recent measurements from Planck and ACT DR6, as well as BAO from DESI DR2. For clarity, we emphasize that ACT DR6 alone is fully compatible with Planck regarding $n_s$. The upward shift toward $n_s \simeq 0.975$ appears only when DESI DR2 BAO are combined with CMB datasets (Planck+DESI, ACT+DESI, SPT-3G+DESI), and is likewise seen in the P--ACT--LB2+DESI combination. CMB-only combinations (e.g., Planck+ACT or Planck+ACT+SPT-3G) primarily refine constraints and do not by themselves raise $n_s$. Our comparisons and model assessments are therefore made with this distinction in mind: CMB-only constraints versus CMB+DESI combinations. We conclude that both models remain consistent with current CMB-only data in restricted regions of parameter space, and that residual tension with the higher $n_s$ favored by CMB+DESI persists.