Refined Predictions for Starobinsky Inflation and Post-inflationary Constraints in Light of ACT

Abstract

Recent measurements from the Atacama Cosmology Telescope (ACT), combined with Planck and DESI data, suggest a higher value for the spectral index $n_s$. This places Starobinsky inflation at the edge of the $2\sigma$ constraints for a number of e-folds $N_\star$ around $60$ when using the usual analytical approximations. We present refined predictions for Starobinsky inflation that go beyond the commonly used analytical approximations. By evaluating the model with these improved expressions, we show that for $N_\star \gtrsim 58$ it remains consistent with current observational constraints at the $2\sigma$ level. Additionally, we examine the implications of the ACT results for post-inflationary reheating parameters. Specifically, we find a lower bound on the effective equation of state parameter during reheating of approximately $\omega \gtrsim 0.462$; this excludes purely perturbative reheating, which leads to $\omega \simeq 0$. We also show that the reheating temperature is constrained to be $T_{\text{rh}} \lesssim 2 \times 10^{10}~\text{GeV}$, assuming $\omega \leq 1$. Furthermore, we find that the predictions for the spectral index and tensor-to-scalar ratio can lie within $1\sigma$ of the recent ACT constraints if the reheating temperature satisfies $4~\text{MeV} \lesssim T_{\text{rh}} \lesssim 10~\text{GeV}$ for $0.8 \lesssim \omega \leq 1$.