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

TL;DR

This work examines how reheating physics links inflationary dynamics to observable signals in two single-field models: a generalized $oldsymbol{\alpha}$-attractor with a sech$^p$ potential and the $\boldsymbol{\alpha}$-Starobinsky model. Using a semi-analytical framework that connects horizon-crossing field values to the reheating EOS $\omega_{re}$, reheating temperature $T_{re}$, and decay channels (gravitational, Yukawa, and scalar), the authors compute $n_s$, $r$, and $T_{re}$ across parameter spaces $(\alpha,p,\omega_{re})$ and $(\alpha,\omega_{re})$ and compare with Planck, ACT DR6, and DESI DR2 BAO data. They find that decreasing $p<1$ or increasing $\alpha$ can push the predicted $n_s$ toward the higher values favored by CMB$+$DESI combinations, but these shifts often come at the cost of larger $r$, potentially violating current bounds. The analysis highlights that the upward shift in $n_s$ arises specifically when DESI BAO data are included with CMB measurements, and that reconciling high $n_s$ with low $r$ remains challenging, pointing to the need for improved reheating modeling and data to robustly test these inflationary scenarios.

Abstract

We study inflationary constraints from reheating on two classes of single-field inflationary models: a generalized $α$-attractor with potential $V(φ) = V_0 \!\left(1 - \sech^p\!\left[φ/(\sqrt{6α}\,M_{Pl})\right]\right)$ and the $α$-Starobinsky model with potential $V(φ) = V_0 \!\left(1 - e^{- \sqrt{2/(3α)}\, φ/ M_{Pl}} \right)^2$. 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 $(α,p,ω_{re})$ for the first model and $(α,ω_{re})$ 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.

Figures (7)

• Figure 1: Predictions of the generalized $\mathop{\mathrm{sech}}\nolimits^p$ potential under gravitational reheating. Shown are trajectories in the $r$--$n_s$ plane for $p=1/2$, $p=1/4$, and $p=1/10$. The solid curves correspond to different values of $\alpha$, with colors denoting ranges of the reheating equation-of-state parameter: red for $-1/3 \leq \omega_{re} < 0$, blue for $0 \leq \omega_{re} < 1/3$, and green for $1/3 \leq \omega_{re} \leq 1$. The shaded regions indicate the $1\sigma$ intervals favored by current observations ACT:2025fju: Planck (light red, $n_s=0.9651\pm0.0044$), ACT DR6 (light brown, $n_s=0.9666\pm0.0077$), P+ACT (light yellow, $n_s=0.9709\pm0.0038$), and P-ACT-LB2 (light blue, $n_s=0.9752\pm0.0030$). Decreasing $p$ shifts the curves toward the P-ACT-LB2-preferred region but simultaneously increases $r$. For very small $p$ the predictions exceed the current upper bound on $r$.
• Figure 2: Reheating predictions for the $\mathop{\mathrm{sech}}\nolimits^p$ potential with $p=1/4$ assuming gravitational decay. The plot shows $T_{re}$ as a function of $n_s$, obtained by solving Eq. (\ref{['principal']}) with $T_{re}$ given by Eq. (\ref{['Tregra']}). Curves correspond to different $\alpha$ values, color-coded by $\omega_{re}$ as in Fig. \ref{['sechregiones']}. For $\alpha \sim 1$, the trajectories approach the P-ACT - LB2 region, though only for $\omega_{re}$ close to the stiff limit $\omega_{re}\to1$ (green color).
• Figure 3: Reheating predictions for the $\mathop{\mathrm{sech}}\nolimits^p$ potential with $p=1/4$ assuming Yukawa (fermionic) decay. The plot shows $\log_{10} T_{re}$ versus $n_s$ obtained from Eq. (\ref{['Trey']}). The Yukawa coupling $y$ is varied from $10^{-17}$ to unity. Larger $y$ values yield higher reheating temperatures and reduce the allowed range of $n_s$. For $\omega_{re}=1/3$, the dependence on $T_{re}$ cancels, and the transition between the blue and green segments occurs at a fixed $n_s$, independent of $y$.
• Figure 4: Reheating predictions for the $\mathop{\mathrm{sech}}\nolimits^p$ potential with $p=1/4$ assuming scalar decay. The vertical axis shows $\log_{10} T_{re}$ computed from Eq. (\ref{['Treg']}), with the effective coupling $\tilde{g}=g/M_{Pl}$ varied between $10^{-23}$ and $10^{-5}$. Increasing $\tilde{g}$ raises the reheating temperature and narrows the allowed $n_s$ interval, more strongly than in the Yukawa case (Fig. \ref{['LogTreYuka']}). As in other decay scenarios, the blue–green transition corresponds to $\omega_{re}=1/3$ and lies at fixed $n_s$ for a given $\alpha$.
• Figure 5: Predictions of the $\alpha$-Starobinsky potential under gravitational decay. Left: trajectories in the $r$--$n_s$ plane. Right: corresponding reheating predictions in the $T_{re}$--$n_s$ plane, obtained by solving Eq. (\ref{['principal']}) with $T_{re}$ from Eq. (\ref{['Tregra']}). Curves represent different $\alpha$ values, color-coded for $\omega_{re}$ as in Fig. \ref{['sechregiones']}. Larger $\alpha$ shifts the curves toward the P-ACT-LB2-preferred $n_s$ region but simultaneously increases $r$. For $\alpha \gtrsim 10$, $r$ exceeds the current upper bounds, setting an approximate upper limit for this scenario.