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 $\hat{I}›$CDM model. The early dark energy (EDE) model with potentials $V(\ddot{I}•)\propto [1-\cos (\ddot{I}•/f){]}^n$ 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 ($g{M}_{\mathrm{pl}}\approx 69.912$ for Planck, $-40.726$ for ACT), large $\ddot{I}{‡}_{\min }^2$ (144.52 and 86.93), and $\hat{I}”\ddot{I}{‡}^2<1$ with many local minials. Conversely, $n=3$ achieves the best fits ($\ddot{I}{‡}_{\min }^2=65.70$ and $48.08$) with consistent couplings ($g{M}_{\mathrm{pl}}=-0.210\pm 0.024$ 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.