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

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

Problem Statement
The standard $\Lambda$CDM cosmological model faces two persistent observational challenges: the Hubble tension (a significant discrepancy between late-time $H_0$ measurements and early-universe inferences) and the detection of cosmic birefringence (a non-zero rotation of the CMB polarization plane, indicating parity violation). The Early Dark Energy (EDE) model, featuring axion-like scalar fields with potentials $V(\phi) \propto [1 - \cos(\phi/f)]^n$, offers a unified framework to address both anomalies. While previous literature has predominantly focused on the cubic potential case ($n=3$), the dependence of the cosmic birefringence signal on the potential index $n$ remains largely unexplored. This paper investigates whether different potential indices ($n=2, 3, \infty$) can simultaneously satisfy constraints from the Hubble tension and cosmic birefringence observations.

Methodology
The authors perform a comprehensive statistical analysis using Markov Chain Monte Carlo (MCMC) methods to constrain the Chern-Simons coupling constant ($g_{M_{pl}}$) for axion-like EDE models. The study utilizes EB cross-polarization data from two distinct datasets:

1. Planck 2018: Space-based observations providing full-sky coverage and constraints on large angular scales (multipoles $\ell \approx 51$ to $1491$).
2. ACT DR6: Ground-based observations from the Atacama Cosmology Telescope, offering superior angular resolution and constraints on small scales (multipoles $\ell \approx 600$ to $3325$).

The analysis compares three specific potential configurations:

• $n=2$ (Quadratic)
• $n=3$ (Cubic)
• $n=\infty$ (Approximated numerically as $n=31$ to represent the asymptotic limit)

The goodness of fit is evaluated using the $\chi^2$ statistic, comparing theoretical power spectra ($D^{EB}_\ell$) against observational data. The study examines the behavior of the coupling parameter $g_{M_{pl}}$, the resulting rotation angle $\beta$, and the stability of the likelihood landscape for each potential index.

Key Results
The analysis yields distinct outcomes for each potential index, revealing a strong statistical preference for specific configurations:

• $n=3$ (Cubic Potential): This configuration achieves the best fit to both datasets.

  • Planck-EB: $\chi^2_{min} = 65.70$ with $g_{M_{pl}} = -0.210 \pm 0.024$.
  • ACT-EB: $\chi^2_{min} = 48.08$ with $g_{M_{pl}} = -0.158 \pm 0.025$.
  • The theoretical power spectrum for $n=3$ accurately reproduces the observed EB correlations across all angular scales, capturing both the large-scale envelope (Planck) and the high-$\ell$ oscillatory structures (ACT). The coupling values are consistent with the perturbative regime.

• $n=\infty$ (Steep Potential): This model provides statistically acceptable fits but is slightly less preferred than $n=3$.

  • Planck-EB: $\chi^2_{min} = 68.38$ with $g_{M_{pl}} = -0.017 \pm 0.002$.
  • ACT-EB: $\chi^2_{min} = 49.67$ with $g_{M_{pl}} = -0.012 \pm 0.002$.
  • The coupling strength is approximately an order of magnitude smaller than in the $n=3$ case, reflecting the rapid decay of the EDE component in steeper potentials. The authors note that increasing the steepness beyond $n=3$ offers no additional benefit to the fit.

• $n=2$ (Quadratic Potential): This configuration is severely disfavored and effectively ruled out.

  • Planck-EB: $\chi^2_{min} = 144.52$ with an unphysically large coupling $g_{M_{pl}} \approx 69.912$.
  • ACT-EB: $\chi^2_{min} = 86.93$ with $g_{M_{pl}} \approx -40.726$.
  • The likelihood landscape for $n=2$ exhibits pathological features, including multiple local minima and a failure to satisfy the $\Delta\chi^2 = 1$ criterion for a reliable $1\sigma$ confidence interval. The model predicts a $D^{EB}_\ell$ spectrum with frequent zero-crossings and sign flips that are inconsistent with the smooth polarization measurements. The required coupling constants exceed perturbative limits, indicating a fundamental incompatibility with observational data.

Significance and Claims
The paper concludes that the $n=3$ potential represents the optimal configuration for the axion-like EDE model within the current observational framework. It successfully balances the requirements for early-time energy injection (to address the Hubble tension) and rapid late-time dilution (to preserve $\Lambda$CDM successes) while generating a cosmic birefringence signal consistent with both Planck and ACT data.

The study emphasizes the complementary nature of the datasets: Planck's full-sky coverage constrains the integrated birefringence effect, while ACT's high resolution resolves fine-scale oscillatory structures. The robustness of the $n=3$ preference is reinforced by its consistent performance across both independent datasets. The authors assert that their analysis provides strong evidence that the cubic potential is the preferred realization of early dark energy for explaining the observed cosmic birefringence, while the quadratic ($n=2$) and excessively steep ($n=\infty$) potentials are either incompatible or suboptimal. Future experiments (AliCPT, LiteBIRD, Simons Observatory) are noted as necessary for further tightening these constraints, but the current analysis establishes $n=3$ as the theoretically viable and observationally favored choice.