Constraining the Potential Index $n$ of the Early Dark Energy Model Using Cosmic Birefringence from Planck and ACT

Abstract

Cosmic birefringence and the Hubble tension represent compelling challenges to the standard $Λ$CDM model. The early dark energy (EDE) model with potentials $V(ϕ) \propto [1-\cos(ϕ/f)]^n$ offer a unified framework to address both anomalies through energy injection near matter-radiation equality and parity-violating Chern--Simons coupling to photons. While previous studies have focused on $n=3$, the dependence of the birefringence signal on the potential index $n$ remains largely unexplored. We perform a comprehensive statistical analysis of axion-like EDE models with $n=2$, $n=3$, and $n=\infty$, using $EB$ cross-polarization data from Planck-$EB$ and ACT-$EB$. The $n=2$ model is severely disadvantaged, displaying extreme coupling values ($gM_{\rm pl} \approx 69.912$ for Planck, $-40.726$ for ACT), large $χ^2_{\rm min}$ (144.52 and 86.93), and $Δχ^2<1$ with many local minials. Conversely, $n=3$ achieves the best fits ($χ^2_{\rm min} = 65.70$ and $48.08$) with consistent couplings ($gM_{\rm pl} = -0.210 \pm 0.024$ and $-0.158 \pm 0.025$) that accurately reproduce observations across all angular scales. We checked that the $n=3$ configuration represents the optimal choice for simultaneously addressing the Hubble tension and cosmic birefringence within a theoretically viable framework.