Early Dark Energy Models

• Early Dark Energy (EDE) models are characterized by a non-negligible dark energy fraction during pre-recombination epochs, altering the expansion rate near matter–radiation equality.
• They are implemented through scalar-field dynamics, modified gravity, or fluid-based approaches that produce a temporary surge in dark energy at critical cosmic times.
• Observational constraints from the CMB, BAO, and large-scale structure help limit the EDE fraction while offering potential resolutions to the Hubble and S8 tensions.

Early Dark Energy (EDE) models propose that the Universe contained a non-negligible fraction of dark energy during epochs preceding recombination, often peaking near matter–radiation equality (redshift $z\sim 3000$), before decaying away to subdominant levels at later times. The main motivation for EDE has been its capacity to alter the pre-recombination expansion rate, thereby reducing the comoving sound horizon, and potentially reconciling discrepancies—such as the Hubble tension—between early- and late-Universe determinations of cosmological parameters. Theoretical realisations and phenomenological parameterisations have proliferated, with recent developments focusing on scalar-field dynamics, modified gravity, coupled dark sectors, and fluid-based approaches. Constraints on the EDE fraction and dynamics are now derived from a combination of CMB, large-scale structure, baryon acoustic oscillations (BAO), supernovae, and direct measurements of $H_0$.

1. General Parametrizations and Theoretical Motivation

EDE models postulate departures from $\Lambda$CDM by positing a small, persistent dark energy density fraction at early epochs,

$$\Omega_e \equiv \lim_{z \to \infty} \Omega_{\rm DE}(z) > 0$$

Such behavior can arise from scaling or tracking solutions of scalar fields with appropriate potentials (e.g., exponential or inverse power-law) or from barotropic fluids with early-time $w \sim 0$ ($c_s^2 \sim 0$), mimicking cold dark matter in both background and perturbations (Bielefeld et al., 2014).

Empirical parameterizations—such as the Doran–Robbers form,

$$\Omega_{\rm DE}(a) = \frac{\Omega_{\rm DE,0} - \Omega_e (1 - a^{-3w_0})}{\Omega_{\rm DE,0} + \Omega_{m,0} a^{3w_0}} + \Omega_e (1 - a^{-3w_0})$$

enable direct identification of an early dark energy “floor” $\Omega_e$ and allow for sharp or smooth transitions to $\Lambda$ domination at late times (Pettorino et al., 2013, Shi et al., 2015). Generalizations introduce additional parameters to control transition redshift and rapidity, or incorporate time-varying $w(z)$ ansätze for continuous evolution between radiation-like and cosmological constant phases (García et al., 2020, Pu et al., 2014).

2. Scalar-Field and Modified Gravity Realisations

Canonical Scalar-Field Models

Canonical scalar fields, often minimally coupled, can realise EDE through potentials that enforce a brief or sustained period of negative pressure in the pre-recombination Universe. Examples include:

• Exponential potentials: $V(\phi) = V_0 e^{-\alpha \phi/M_p}$ produce constant early energy fractions $\Omega_\phi = n k^2/\alpha^2$ during radiation or matter dominance (Chamings et al., 2019).
• Axion-like or oscillatory potentials: Potentials of the form $V(\phi) = m^2 f^2 [1-\cos(\phi/f)]^n$ naturally lead to a triggered onset of field oscillations at $z_c \sim 3000$, yielding a transient EDE “bump” with a peak fractional energy contribution $f_{\rm EDE}$ and a late-time cosmological constant phase (Niedermann et al., 2021).
• Quintessential EDE: Potentials such as the “modified steep exponential” $$V(\phi) = V_0 \exp\left[-\gamma (\phi/M_p)^n/(\phi_0+(\phi/M_p)^n)\right]$$ can interpolate between a nearly constant early plateau and $\Lambda$-like behavior today, unifying EDE and late DE in a single field (Sohail et al., 2024).

Modified Gravity and Coupled Scenarios

Generalized Brans-Dicke (BD) EDE: The action

$$S_J = \int d^4x \sqrt{-g} [\phi R - \frac{\omega_{BD}}{\phi} (\nabla\phi)^2 - 2V(\phi) + 2\mathcal{L}_r(\phi)]$$

with nonminimal coupling and a potential $V(\phi)$, when translated to the Einstein frame and coupled to radiation, produces a natural EDE plateau. Under specific conditions, the BD scalar can mimic a cosmological constant (slow-roll regime: $w_{\rm eff} \to -1$) or quintessence ($-1 < w_{\rm eff} < 0$), with the early EDE fraction controlled by the coupling parameter $\alpha$ and the Brans-Dicke parameter $\omega_{BD}$ ($\Omega_\varphi^{\rm early} \sim 1\%$) (Bisabr, 2024).

Screening and Symmetry-Breaking Mechanisms: Conformally coupled quintessence, particularly with chameleon- or symmetry-breaking potentials, can dynamically reduce a large early cosmological constant to subdominant values at late times. In such scenarios, the scalar remains trapped at $\varphi=0$ for large matter densities, releasing its vacuum energy only when $\rho_{dm}$ drops below a critical value, often coinciding with matter–radiation equality. The parameter $M$ controls the coupling strength, with typical early EDE plateaus of $\Omega_\phi \sim 3$–$5\%$ allowed to address the $H_0$ tension (Sadjadi et al., 2022, Trodden, 2022).

Couplings to Neutrinos and Dark Matter: Models where the scalar couples to the neutrino sector (neutrino mass–varying EDE) or to dark matter, with the onset of EDE controlled by neutrino mass thresholds or matter–radiation equality triggerings, address coincidence and fine-tuning problems without requiring ultralight masses or severe UV tuning (Trodden, 2022).

3. Observational Signatures and Key Constraints

CMB and Pre-Recombination Imprints

• CMB Angular Scale and Sound Horizon: A nonzero EDE fraction at $z_{*} \sim 1100$ directly alters the sound horizon $r_s$ and the angular-diameter distance to last scattering, shifting the acoustic peaks in the CMB TT/TE/EE spectra. The strongest constraints arise from models in which EDE is active at or before recombination (Pettorino et al., 2013, Shi et al., 2015).
• Best-Fit Bounds: For persistent or plateau EDE models, WMAP9+SPT and Planck analyses yield $95\%$ C.L. limits of $\Omega_e < 0.014$ for constant EDE; for scenarios where EDE is switched off after $z\lesssim 300$, the bound relaxes, but plateaus prior to $z \sim 100$ remain tightly constrained ($\Omega_e<0.02$) (Pettorino et al., 2013, Pu et al., 2014).
• Special EDE Forms: For freezing EDE models with a transient $w=+1$ spike, data require that the transition occurs no later than $z_c \gtrsim 400$ (at $3\sigma$), else significant CMB and large-scale structure distortions result (Bielefeld et al., 2013).

Large-Scale Structure, Lensing, and Cluster Abundances

• Growth Suppression and Power Spectrum: EDE generically suppresses the growth of matter perturbations during its active phase, leading to shifts in the matter power spectrum turnover (up to $10$–$15\%$ for large allowed $\Omega_e$), and alters the halo mass function at high redshift ($z\gtrsim 1$), potentially doubling galaxy-mass halo counts at $z=4$ in canonical axion or plateau EDE scenarios (Klypin et al., 2020, Shi et al., 2015).
• BAO and Correlation Function: The BAO peak position in $\xi(R)$ or $P(k)$ is shifted by $1$–$2\%$ for EDE fractions $f_{\rm EDE}\sim 0.1$ at $z\sim 3000$, a signature robust to nonlinear effects. Forthcoming DESI and Euclid datasets are sensitive to subpercent BAO shifts (Klypin et al., 2020).
• CMB Lensing: The amplitude and scale-dependence of the lensing power spectrum is sensitive to both background EDE and to residual A$_L$ anomalies. CMB lensing constrains $f_{\rm EDE} \lesssim 0.1$ at $95\%$ C.L., and degeneracies with Hubble and $S_8$ parameters imply that EDE alone cannot fully resolve the Planck lensing amplitude tension without additional ad hoc rescaling (Haridasu et al., 2022).

4. Phenomenological, Fluid, and Alternative Implementations

A range of phenomenological “fluid-based” models parameterize EDE without explicit microphysical models:

• Statefinder/Fluid Models: Three-parameter EDE models evolve smoothly from radiation-like ($w=1/3$) at $z \gg z_*$ to $w \rightarrow -1$ at $z \rightarrow 0$, with transitions parametrized by a “steepness” factor $m$ and present-day normalization $\Omega_{\phi,0}$ (García et al., 2020).
• Nonlinear Electrodynamics: EDE can arise from a generalized nonlinear electromagnetic Lagrangian, producing a radiative equation of state ($w=1/3$) at early times and interpolating naturally to $w\approx-1$ at late times. Bayesian fits to Planck+BBN+BAO+SNe+SH0ES yield $H_0=70.2\pm 0.9~\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $\sigma_8=0.798\pm0.007$, potentially alleviating both $H_0$ and $S_8$ tensions (Benaoum et al., 2023).
• Zero-Point Quantum Fluctuations: Some approaches derive a time-dependent $\rho_Z(t)\propto H^2(t)$ component arising from subtracted vacuum fluctuations. This automatically generates an EDE plateau during radiation/matter domination and transitions naturally to late-time acceleration (Maggiore et al., 2011).
• Late-Time Drag Models: Interaction terms that couple EDE to dark matter via post-recombination momentum exchange ($\Gamma(a)$) can compensate for the increase in small-scale power usually exacerbated by EDE, allowing simultaneous mitigation of both $H_0$ and $S_8$ tensions when the drag term acts at $z < 10$ (Simon et al., 2024).

5. Unified Models, Coincidence, and Theoretical Viability

Theoretical challenges for EDE include technical naturalness (the need to avoid radiative instability for ultralight scalar masses), coincidence (why EDE turns on near matter–radiation equality), and microphysical justification for potential and coupling choices.

• Unified Scalar Models: Potentials that interpolate from flat plateaus (EDE phase) to cosmological-constant-like tails (late DE) have been proposed as unifying frameworks but currently show no strong statistical preference over $\Lambda$CDM in multi-probe cosmological fits (Sohail et al., 2024).
• Coupled Triggered Models: Neutrino–assisted and chameleon-coupled EDE address both coincidence and technical naturalness by triggering EDE injection via neutrino mass thresholds or dark-matter dominance, allowing heavier natural masses for the scalar and predictive alignment of the EDE onset epoch (Trodden, 2022, Sadjadi et al., 2022).
• Screening and Z$_2$ Symmetry Breaking: Screening-based EDE provides a minimal symmetry rationale for the timing of the EDE phase, though the detailed suppression of the dark energy fraction prior to recombination is constrained by structure growth and cosmic history requirements.

6. Current Status and Observational Prospects

• Viable Early Energy Fractions: Current cosmological constraints consistently require $\Omega_e \lesssim 0.01$–$0.02$ for persistent or plateau EDE models at recombination. Sharp or transient EDE “bumps” peaking at $\sim5$–$10\%$ are allowed within a narrow window, contingent on sufficiently rapid decay before recombination and compatibility with CMB, BAO, and large-scale structure data (Pettorino et al., 2013, Klypin et al., 2020).
• Phenomenological Advantages: EDE models offer routes to mitigating the $H_0$ and possibly $S_8$ tensions, but at the expense of increased model complexity and parameter freedom. Only models in which EDE decays swiftly and does not persist at late times match CMB, structure, and lensing constraints well.
• Future Tests: High-precision CMB lensing (Simons Observatory, CMB-S4), high-redshift galaxy/cosmic shear surveys (Euclid, LSST, JWST), and improved BAO constraints (DESI, Euclid) are projected to reduce the allowed EDE window by an order of magnitude, probing the subpercent level of $\Omega_e$ and potentially confirming or falsifying current models (Klypin et al., 2020, Haridasu et al., 2022).

7. Summary Table: Core EDE Model Classes and Observational Features

Model Type/Mechanism	Characteristic EDE epoch/fraction	Unique Observational Signature(s)	Leading Constraint(s)
Plateau/tracking scalar	Sustained $\Omega_e$ ($\sim$0.01) pre-recombination	$\sim$1–2% CMB peak/BAO shifts; suppressed late-time growth	CMB, BAO, lensing: $\Omega_e\lesssim0.01$ (1301.52791511.00692)
Axion/triggered EDE	Transient “bump” at $z_c\sim3000$ ($f_{\rm EDE}\sim$0.05–0.10)	Temporally localized $r_s$ reduction, no late DE	Peak allowed $f_{\rm EDE}\lesssim0.1$, rapid decay required (Niedermann et al., 2021)
Modified gravity (BD, screening)	Sustained or triggered by matter/radiation sector properties	Plateau from nonminimal coupling; natural decay	Planck+Solar: $\omega_{BD}\gtrsim10^3$ (Bisabr, 2024Sadjadi et al., 2022)
Fluid/Chaplygin/nonlinear electrodynamics	$w(z)$ interpolates $+1/3 \to -1$; EDE via background evolution	Smooth interpolation across eras; SNIa, BAO fit	Bayesian analyses: $H_0$ and $S_8$ tensions partly alleviated (Benaoum et al., 2023)
Late-time DM–EDE drag	Post-recombination momentum-exchange	Simultaneous $H_0$, $S_8$ tension alleviation	CMB+LSS: drag allowed only post-recombination (Simon et al., 2024)

Continued investigation of the microphysical origin, trigger mechanisms, and distinct signatures of EDE remains a central theme in modern cosmology. State-of-the-art analyses indicate that, although EDE models effectively address puzzles such as the Hubble tension in certain settings, their viability is progressively constrained by the synergy of CMB, BAO, large-scale structure, and local measurements. Future cosmological surveys will further clarify the status and structure of early dark energy.