Axion Early Dark Energy (AEDE)

• Axion Early Dark Energy (AEDE) is a cosmological model where an axion-like scalar field transiently contributes to dark energy before recombination.
• The model employs a periodic scalar potential, causing the field's dynamics to shift from a cosmological constant (w ≃ –1) to w = 1/2, affecting the sound horizon.
• Observational constraints from CMB and BAO data limit the AEDE energy fraction to around 0.1, offering only modest relief to the Hubble tension.

Axion Early Dark Energy (AEDE) refers to a class of cosmological models in which an axion-like scalar field produces a transient, non-negligible dark energy component at high redshift—specifically, before recombination, around the epoch of matter-radiation equality. Unlike the cosmological constant scenario of the standard ΛCDM model (where dark energy is dynamically unimportant at early times), AEDE models introduce a scalar field with a periodic potential that becomes dynamically relevant, briefly contributing to the total energy density before redshifting away rapidly. This modifies the expansion history of the Universe, the sound horizon at recombination, and thereby the inference of the Hubble constant from early-universe probes. The AEDE scenario is motivated in part by axion physics and string theory, and is studied extensively as a potential solution to the Hubble tension.

1. Theoretical Framework of AEDE

In AEDE, the early dark energy is modeled as an axion-like field ϕ with a potential of the form

$V(\phi) = m^2 f^2 [1 - \cos(\theta)]^n,$

where θ = ϕ/f, m is the axion mass, f is the decay constant, and n (typically n = 3) determines the rapidity of post-dynamical dilution (Khalife et al., 31 Jul 2025, Karwal et al., 2016, McDonough et al., 2022). The critical epoch when ϕ becomes dynamical (i.e., when H ≈ m) occurs at redshift z_c, and the fraction of energy density in AEDE is maximal at this redshift.

The axion remains frozen at early times due to Hubble friction, acting as a cosmological constant (w ≃ –1). Once H ≲ m, the field rolls and oscillates rapidly, and its equation of state transitions to

$$\langle w_{\mathrm{EDE}} \rangle = \frac{n-1}{n+1}.$$

For n = 3, this yields w = 1/2, ensuring that the energy density dilutes faster than radiation.

Key parameterizations employed in AEDE analyses are:

• Fractional EDE at the critical epoch:

$$f_{\mathrm{EDE}} = [\rho_{\mathrm{EDE}}/\rho_{\mathrm{tot}}]_{z = z_c}$$

• Critical redshift z_c (or equivalently a critical scale factor a_c)
• Initial field displacement θ_i

The background and perturbative evolution of the field is governed by the Klein–Gordon equation

$\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0.$

2. Observational Signatures and Parameter Constraints

AEDE modifies the cosmic expansion history, most notably the sound horizon at recombination, r_s. A reduced r_s leads to an increased inference of H₀ from CMB data when interpreted within the model, providing a possible resolution to the Hubble tension between early- and late-universe measurements (Khalife et al., 31 Jul 2025, Hill et al., 2021, Efstathiou et al., 2023).

Ground-based CMB experiments provide stringent constraints on AEDE. The latest SPT-3G D1 data (Khalife et al., 31 Jul 2025) gives

$f_{\mathrm{EDE}} < 0.12 \quad \text{(95\% CL)},$

tightening to

$f_{\mathrm{EDE}} < 0.091 \quad \text{(95\% CL)}$

when combining SPT, ACT, and Planck data. BAO data from DESI further tightens constraints and produces mild “detections”, e.g., for SPT + DESI: $$f_{\mathrm{EDE}} = 0.081^{+0.037}_{-0.052} \quad \text{(68\% CL)}.$$ CMB-only AEDE models typically do not show statistically significant evidence for EDE and provide at most moderate reduction in Hubble tension.

Combined fits (CMB + DESI) find that residual Hubble tension is reduced from roughly 6.4σ (ΛCDM) to 3.3σ (AEDE) or as low as 1.5–2.3σ for certain data combinations. However, this improvement is not robustly statistically significant once model complexity is properly included.

A summary of AEDE bounds from recent data combinations is shown below:

Data Set	AEDE Fraction $f_{\mathrm{EDE}}$ (95% CL)	H₀ Tension (σ)
SPT-3G D1	< 0.12	2.3
cmball (SPT+ACT+Planck)	< 0.091	3.3
SPT+DESI	$0.081^{+0.037}_{-0.052}$ (68% CL)	1.5
CMB+DESI	$0.071^{+0.035}_{-0.038}$ (68% CL)	2.3

3. Impact on the Hubble Tension and Cosmological Inference

By increasing the expansion rate prior to recombination, AEDE models reduce the sound horizon r_s, allowing the inferred Hubble constant H₀ from early-universe probes to shift upward, partially alleviating the Hubble tension (Khalife et al., 31 Jul 2025, Hill et al., 2021, Karwal et al., 2016). However, the extent of this shift is strictly limited by constraints on f_EDE from CMB and BAO data, with joint analyses rarely reducing the tension below ≈2σ unless strong external priors are applied.

The improvement in χ² and information criteria (AIC) are modest; e.g., Δχ² ≃ –10.8 for cmball + DESI, but when accounting for three extra model parameters (N = 3), the improvement is only weak (ΔAIC ≃ –4.8). This weak statistical preference is insufficient for strong Bayesian evidence.

Moreover, the parameter shift induced by DESI BAO data reflects an existing BAO–CMB tension under ΛCDM rather than robust new evidence in favor of AEDE per se (Khalife et al., 31 Jul 2025).

4. Model Discrimination, Data Concordance, and Statistical Issues

Different experiments exhibit somewhat inconsistent constraints and hints of AEDE:

• Planck yields the tightest upper limits, typically disfavors nonzero AEDE, and restricts H₀ (Efstathiou et al., 2023, Smith et al., 2023).
• ACT (and to some extent SPT) data show mild preference for nonzero f_EDE, particularly in polarization spectra, but joint analyses with Planck diminish the significance (Hill et al., 2021, Khalife et al., 31 Jul 2025).
• Addition of DESI BAO data increases the preferred f_EDE and reduces H₀ tension; this reflects discrepancies between BAO and CMB under ΛCDM.

No combination to date produces compelling statistical significance for AEDE when model complexity penalties are taken into account. The results are potentially affected by prior-volume effects, which are controlled for in some analyses by comparing MCMC posteriors to profile likelihoods (Efstathiou et al., 2023).

5. Mathematical Structure and Model Parameters

The AEDE model is specified by the axion potential, choice of n, and three primary parameters:

• $f_{\mathrm{EDE}}$: maximum fractional energy contribution at $z_c$
• $z_c$: redshift of dynamical onset
• θ_i: initial misalignment

Statistical analysis metrics include:

• Q_DMAP: square root of the χ² difference between fits with and without SH0ES
• ΔAIC: χ² improvement penalized by 2N extra parameters

Key formulae include: $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$

$$f_{\mathrm{EDE}}=\left.\frac{\rho_{\mathrm{EDE}}}{\rho_{\mathrm{tot}}}\right|_{z=z_c}$$

$\Delta \mathrm{AIC} = \Delta \chi^2 + 2N$

6. Future Prospects and Ongoing Challenges

The success of AEDE in reducing the Hubble tension and reconciling BAO–CMB discrepancies depends critically on future improvements in CMB (SPT, Simons Observatory, CMB-S4) and BAO (DESI) data. Analyses free from prior-volume bias (e.g., profile likelihoods), increased statistical power, and improved modeling of systematics will be essential (Khalife et al., 31 Jul 2025, Smith et al., 2023).

Persistent or growing discrepancies between BAO and CMB in ΛCDM, as highlighted by the impact of DESI data, might point toward new physics or systematic uncertainties. While AEDE is among the best-motivated new-physics candidates, its statistical preference remains weak unless future data deliver significantly tighter constraints or stronger evidence for a nonvanishing early dark energy fraction.

7. Comparison to Related Early Dark Energy Scenarios

AEDE is a representative member of the broader EDE class but differs in its microphysical realization and phenomenological flexibility (e.g., reliance on a specific axion potential raised to the third power for n = 3). In contrast to purely phenomenological models or “acoustic” EDE, AEDE is UV-motivated and admits explicit embeddings in string theory and two-field models with screening mechanisms (McDonough et al., 2022, Smith et al., 8 May 2025, Cicoli et al., 2023).

Data strongly disfavor EDE scenarios with larger or more persistent contributions, and Planck/BAO constraints require any early dark energy component to be transient, with $f_{\mathrm{EDE}} \lesssim 0.1$ near z_c ≈ 3000–5000. Failure to obtain sufficient dilution results in conflicts with the observed CMB acoustic structure and the clustering of large-scale structure.

Summary Table: AEDE Constraints and Hubble Tension Impact

Data Combination	$f_{\mathrm{EDE}}$ Upper Limit / Detection	H₀ Tension (σ)	Preference vs. ΛCDM (ΔAIC)
SPT-3G D1	< 0.12 (95% CL)	~2.3	Weak
SPT+ACT+Planck (“cmball”)	< 0.091 (95% CL)	~3.3	Weak
SPT+DESI	$0.081^{+0.037}_{-0.052}$ (68% CL)	1.5	Weak
cmball+DESI	$0.071^{+0.035}_{-0.038}$ (68% CL)	2.3	Weak

In conclusion, AEDE is an axion-motivated extension of ΛCDM that produces a transient, early-universe energy injection potentially capable of easing the Hubble tension. High-precision CMB and BAO data robustly constrain the allowed AEDE parameter space, with only weak statistical evidence for nonzero EDE. The tension between BAO and CMB data in ΛCDM is not yet fully resolved by AEDE, though future experimental advances may further test its viability.