Axion-Like Early Dark Energy: Theory, Methods, Results

• Axion-like EDE is a pre-recombination model where a frozen pseudo–Nambu–Goldstone field transitions dynamically near matter–radiation equality, temporarily enhancing cosmic expansion.
• The methodology involves solving the Klein–Gordon and Friedmann equations with axion-like potentials using advanced Boltzmann solvers and neural-network emulators to accurately capture perturbative effects.
• Observational analyses using CMB data from ACT, Planck, and SPT reveal parameter tensions and highlight challenges in reconciling the model with H0 estimates and large-scale structure data.

Axion-like early dark energy (EDE) is a class of pre-recombination dark-energy models in which a pseudo-Nambu–Goldstone scalar field is initially frozen by Hubble friction, contributes a transient vacuum-like energy component near matter–radiation equality, and then becomes dynamical at a critical redshift $z_c$, after which its energy density redshifts away faster than matter. In the standard implementation, the field is governed by an axion-like potential $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$, usually with $n=3$, so that after oscillations the effective equation of state is $w\simeq (n-1)/(n+1)=1/2$ (Jiang et al., 2022). Its cosmological role is to raise the pre-recombination expansion rate, reduce the sound horizon $r_s$, and thereby permit a larger CMB-inferred $H_0$ at fixed acoustic angle. Subsequent work has treated axion-like EDE both as a candidate resolution of the Hubble tension and as a target of increasingly stringent tests from Planck, ACT, SPT, BAO, supernovae, CMB lensing, and full-shape large-scale-structure likelihoods, while also extending the framework to dark-sector triggers, cosmic birefringence, and multi-field realizations (Efstathiou et al., 2023).

1. Field-theoretic structure and background dynamics

A canonical formulation writes the total action as

$$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$

with the axion-like potential

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

and $n=3$ in the main observational studies (Yao et al., 2023). The homogeneous field obeys the Klein–Gordon equation

$\ddot\phi+3H\dot\phi+V_{,\phi}=0,$

while the Friedmann equation includes $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$0 and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$1 alongside the standard radiation, baryon, dark-matter, and late-time dark-energy sectors (Qu et al., 2024).

The standard phenomenological picture is that $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$2 is frozen at early times when $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$3, so $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$4 is approximately constant. The field unfreezes when $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$5, or more precisely $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$6 when the initial misalignment angle is important, and then rolls and oscillates. For $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$7, the post-oscillation energy density redshifts faster than matter; this rapid dilution is one of the reasons the $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$8 potential is repeatedly adopted in data analyses (Jiang et al., 2022).

Operationally, most cosmological fits trade the microscopic parameters $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$9 for the phenomenological triplet $n=3$0, where $n=3$1 is the initial displacement and

$n=3$2

This parameterization is designed to isolate the quantities that directly control the transient energy injection, its timing, and its perturbation dynamics (Yao et al., 2023).

2. Cosmological mechanism and perturbative signatures

The principal effect of axion-like EDE is a temporary enhancement of $n=3$3 before recombination. Because

$n=3$4

a larger pre-recombination expansion rate reduces the sound horizon. At fixed measured acoustic angle $n=3$5, this permits a larger inferred $n=3$6 than in $n=3$7CDM (Hill et al., 2021). This is the core mechanism by which EDE addresses the Hubble tension.

The same injection also perturbs the diffusion scale, damping tail, and early-ISW contribution. Accordingly, axion-like EDE is not a purely background-level modification. In synchronous gauge, the field perturbation satisfies

$n=3$8

and these perturbations feed back into the CMB TT, TE, and EE spectra (Simon, 2023). Several analyses therefore emphasize that accurate constraints require solving the full field dynamics rather than using a purely effective fluid description (Hill et al., 2021).

Jiang and Piao found that the compensating parameter shifts induced by pre-recombination injection empirically satisfy

$n=3$9

so that increasing $w\simeq (n-1)/(n+1)=1/2$0 toward $w\simeq (n-1)/(n+1)=1/2$1 km/s/Mpc pushes $w\simeq (n-1)/(n+1)=1/2$2 toward unity (Jiang et al., 2022). In their combined CMB runs, $w\simeq (n-1)/(n+1)=1/2$3 becomes consistent with $w\simeq (n-1)/(n+1)=1/2$4 at approximately the $w\simeq (n-1)/(n+1)=1/2$5 level. This connection ties axion-like EDE not only to the Hubble tension but also to the question of whether a Harrison–Zeldovich primordial spectrum can be made compatible with modern data.

The detailed observational leverage comes disproportionately from the small-scale CMB. SPT-3G and ACT analyses stress the role of the damping scale $w\simeq (n-1)/(n+1)=1/2$6, the shape of the high-$w\simeq (n-1)/(n+1)=1/2$7 TT spectrum, and the interplay between TT and polarization. One SPT-3G study reports that ACT DR4 prefers $w\simeq (n-1)/(n+1)=1/2$8, whereas Planck gives $w\simeq (n-1)/(n+1)=1/2$9, while SPT-3G 2018 agrees with Planck with $r_s$0 (Smith et al., 2023). This suggests that the viability of the model is controlled at least as much by damping-tail and polarization residuals as by the overall shrinkage of $r_s$1.

3. Inference pipelines and data combinations

Axion-like EDE has been constrained with several closely related but non-identical inference pipelines. The dominant pattern is the use of modified Boltzmann solvers derived from CLASS, including CLASS_EDE and AxiCLASS, occasionally with CAMB-based implementations, coupled to samplers such as Cobaya, MontePython, or CosmoMC (Jiang et al., 2022). Reported convergence thresholds vary across analyses, including Gelman–Rubin criteria $r_s$2, $r_s$3, $r_s$4, and $r_s$5, reflecting differing posterior geometries and precision targets (Smith et al., 2023).

The basic EDE priors are typically flat in

$r_s$6

with representative choices such as $r_s$7 or $r_s$8, $r_s$9 or $H_0$0, and $H_0$1 (Kochappan et al., 2024). Some analyses also vary neutrino mass, dark-matter properties, or late-time dark-energy parameters, but the core parameterization remains the same.

The data combinations span a wide range of CMB and low-redshift inputs: Planck 2018 or Planck NPIPE primary spectra, Planck lensing, ACT DR4 or DR6 TT/TE/EE or lensing, SPT-3G 2018 temperature and polarization, BAO from 6dFGS, SDSS MGS, BOSS DR12, eBOSS DR16, or DESI Y1, Pantheon or Pantheon+ supernovae, full-shape BOSS likelihoods within the EFT of LSS, and optional local-distance-ladder priors from SH0ES or H0DN (Efstathiou et al., 2023).

Methodological details materially affect the resulting posteriors. Jiang and Piao truncated Planck high-$H_0$2 TT to $H_0$3 to avoid the known over-smoothing anomaly at $H_0$4, and found best-fit points with BOBYQA and posterior visualizations with GetDist (Jiang et al., 2022). By contrast, the NPIPE analysis used a new high-$H_0$5 CamSpec-style TT, TE, EE likelihood cleaned with a 545 GHz dust template and explicitly compared Bayesian posteriors with profile likelihoods to assess prior-volume effects (Efstathiou et al., 2023).

A notable methodological development is the use of neural-network emulators. One DESI-era analysis employed the TensorFlow package “cosmopower,” trained on $H_0$6 CLASS_EDE runs, to reproduce observables at sub-percent accuracy while running $H_0$7 faster than full Boltzmann solutions, enabling convergence in $H_0$8 hours rather than weeks (Qu et al., 2024).

4. Observational status and dataset dependence

The empirical status of axion-like EDE is not uniform across datasets. Some ACT-centered or Pantheon$H_0$9-inclusive analyses find a nonzero EDE fraction and $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$0 in the low $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$1s, whereas SPT-, NPIPE-, DESI-, and EFT-BOSS-centered analyses pull the model much closer to the $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$2CDM limit.

Analysis	Representative constraint	Interpretation
ACT DR4 + Planck $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$3 + lensing + BAO (Hill et al., 2021)	$$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$4, $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$5 km/s/Mpc	Preference for EDE at $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$6 CL
Planck2018 $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$7 + ACT DR4 + SPT-3G Y1 (Jiang et al., 2022)	$$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$8, $$S=\int d^4x\sqrt{-g}\,\Bigl[\tfrac12R-\tfrac12(\partial\phi)^2-V(\phi)\Bigr]+S_{\rm m}[g_{\mu\nu},\Psi_{\rm m}],$$9, $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$0	$V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$1, mainly ACT-driven
SPT-3G 2018 TT/TE/EE (Smith et al., 2023)	$V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$2 at $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$3 CL	No preference for axion-like EDE
Planck NPIPE + BAO + Pantheon + lensing (Efstathiou et al., 2023)	$V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$4, $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$5 km/s/Mpc	Residual $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$6 tension with SH0ES
Planck + CMB lensing + DESI BAO (Qu et al., 2024)	$V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$7 at $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$8 CL, $V(\phi)=m^2f^2\bigl[1-\cos(\phi/f)\bigr]^n,$9 km/s/Mpc	No significant preference for EDE
Planck 2018 + EFT-BOSS (Ivanov et al., 2020)	$n=3$0 at $n=3$1 CL, $n=3$2 km/s/Mpc	Full-shape LSS sharply tightens constraints

A further Pantheon$n=3$3 and EFTofLSS reassessment found that axion-like EDE remained comparatively competitive under the combination Planck+ext-BAO+EFTofBOSS+EFTofeBOSS+Pantheon$n=3$4+$n=3$5, reporting $n=3$6, $n=3$7, $n=3$8 km/s/Mpc, $n=3$9, $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$0, and a residual tension $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$1 (Simon, 2023). This stands in marked contrast to the tighter NPIPE and DESI-era exclusions.

Taken together, the literature exhibits a recurring tension between ACT-favoring analyses and those dominated by Planck high-$\ddot\phi+3H\dot\phi+V_{,\phi}=0,$2, SPT, DESI BAO, or EFT-based galaxy-clustering information. Several papers identify the decisive modes as high-$\ddot\phi+3H\dot\phi+V_{,\phi}=0,$3 TT or TE/EE multipoles, especially around $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$4–$\ddot\phi+3H\dot\phi+V_{,\phi}=0,$5, the damping tail, and the detailed shape of the polarization spectra. The disagreement has been discussed as a possible physical preference, a systematic effect, or a statistical fluctuation, but no single interpretation is established across the cited studies.

A related large-scale-structure conclusion is comparatively stable: when full-shape information is used rather than compressed BAO+$\ddot\phi+3H\dot\phi+V_{,\phi}=0,$6 summaries, the parameter space that allows the $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$7 injection needed for a full Hubble-tension resolution contracts substantially (Ivanov et al., 2020).

5. Extensions, variants, and associated phenomena

A number of extensions have tested whether the core axion-like EDE mechanism can be made more concordant with other observables. Yao and Meng replaced CDM with dark matter having a constant equation of state $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$8, so that $\ddot\phi+3H\dot\phi+V_{,\phi}=0,$9 and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$00. For the combined CMB+BAO+Pantheon+$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$01+$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$02 dataset they obtained $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$03, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$04, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$05, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$06, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$07 km/s/Mpc, and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$08. They found that the Hubble tension is reduced to $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$09, but $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$10 is driven close to zero, the $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$11 tension remains at $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$12, and AIC mildly favors standard EDE over the $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$13 extension (Yao et al., 2023).

Lin et al. proposed a dark-matter-triggered early dark sector in which $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$14, so that the rise of dark matter near equality itself triggers the rolling of the EDE field. In this trigger EDS model the coincidence problem is addressed at the background level without fine tuning of the coupling or the initial conditions, while the best-fit model reaches $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$15 km/s/Mpc and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$16 with $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$17. The same work emphasizes that Planck selects a specific range of initial field positions, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$18, because of perturbative effects in the high-$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$19 acoustic structure (Lin et al., 2022).

A different extension couples the axion-like EDE field to electromagnetism through a Chern–Simons term,

$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$20

producing cosmic birefringence. In a joint fit to Planck TT/EE, Planck lensing, a measured EB spectrum, SDSS DR12 BAO, and a local $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$21 prior, the $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$22 model yielded a marginalized mean $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$23, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$24, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$25, and derived $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$26 km/s/Mpc, with the best-fit rotation angle $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$27 and a reduced $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$28 for 72 degrees of freedom in the EB fit (Kochappan et al., 2024). The same analysis notes that current EB measurements are noise-limited and identifies LiteBIRD, PICO, CMB-S4, and AliCPT as decisive future tests.

The most direct modification of the axion-like sector is to add more fields. A 2026 analysis of a two-field model with $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$29 reported that splitting the EDE injection into two smoother peaks improves the fit to Planck NPIPE high-$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$30 data, raises the best-fit $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$31 by $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$32 relative to the one-field case, and reduces the residual tension with H0DN to $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$33, with no substantial improvement beyond two fields (Bella et al., 15 Apr 2026). This suggests that the sharpness of the single-field injection history is itself part of the difficulty.

By contrast, modifications of the late-time dark-energy equation of state appear to have little leverage within axion-like EDE. In analyses that replaced $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$34 by CPL or oscillating $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$35 ansätze, the best fit remained compatible with $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$36, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$37, and the change in $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$38 was $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$39 km/s/Mpc relative to the pure $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$40 case (Wang et al., 2022).

6. UV realizations, theoretical constraints, and unresolved issues

The microphysical status of axion-like EDE has been investigated most explicitly in string-theoretic constructions. McDonough and Scalisi showed in a supergravity toy model that a potential of the form

$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$41

can arise from a delicate balance of separate non-perturbative effects. In a KKLT-type compactification with the EDE scalar realized as a two-form axion, however, they found that successful embedding with stabilized moduli requires restrictive assumptions on the Pfaffians and on the exponents of the non-perturbative terms that generate the EDE harmonics (McDonough et al., 2022).

A more detailed Type IIB construction with full closed-string moduli stabilization considers both $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$42 and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$43 axions in KKLT and in the Large Volume Scenario. That analysis argues that the most promising candidates are $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$44 axions in LVS with three non-perturbative corrections to the superpotential generated by gaugino condensation on D7-branes with non-zero world-volume fluxes. It reports that the phenomenologically desired values $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$45, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$46, and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$47 eV can be achieved without tuning the microscopic prefactors if the relevant axion violates the mild axionic weak gravity conjecture, whereas WGC-respecting candidates require fine tuning (Cicoli et al., 2023).

These UV studies align with a broader theoretical concern already visible at the effective level: the coincidence problem. Standard axion-like EDE assumes that the field becomes dynamical near equality even though it is not, in its minimal form, explicitly linked to the matter or radiation densities. Trigger models seek to address this by coupling the field to dark matter (Lin et al., 2022), while string constructions seek a controlled origin for the required ultra-light mass scale and harmonic structure (McDonough et al., 2022).

Three empirical issues remain persistent across the literature. First, the model is not uniformly favored by current data: some analyses find $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$48, but others set $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$49 or $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$50 at $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$51 confidence and recover $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$52 values close to $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$53CDM (Efstathiou et al., 2023). Second, standard axion-like EDE generally raises $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$54, $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$55, and $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$56, so relief of the Hubble tension does not automatically imply improved global concordance (Yao et al., 2023). Third, late-time freedom in $V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$57 does not materially change this conclusion within the standard axion-like setup (Wang et al., 2022).

The resulting picture is technically well defined but observationally unsettled. Axion-like EDE remains one of the most extensively developed pre-recombination solutions to the Hubble tension, with a clear Lagrangian description, identifiable perturbative signatures, and active UV model-building. At the same time, its status is increasingly controlled by the consistency of high-$V(\phi)=m^2f^2[1-\cos(\phi/f)]^n$58 CMB spectra, full-shape large-scale-structure data, and the degree to which additional dark-sector structure is admitted.