When Dark Energy Turns On: Constraints on a Critical Emergence Model

1. Problem Statement

The standard cosmological model, $\Lambda$CDM, faces two major challenges:

• The Hubble Constant ($H_0$) Tension: A statistically significant discrepancy (currently $\sim 7.1\sigma$) exists between the expansion rate inferred from early-Universe Cosmic Microwave Background (CMB) data (assuming $\Lambda$CDM) and direct late-Universe measurements (e.g., SH0ES, H0DN).
• Large-Scale Structure (LSS) Tensions: Recent data from the Dark Energy Spectroscopic Instrument (DESI) and other surveys suggest a preference for dynamical dark energy (DDE) over a strictly constant cosmological constant ($\Lambda$).

While many extensions to $\Lambda$CDM exist, few successfully address both the $H_0$ tension and the indications of late-time dynamical dark energy simultaneously. This paper investigates a specific class of models where dark energy (DE) is effectively absent in the early Universe and "emerges" dynamically only after a critical cosmic epoch via a phase transition.

2. Methodology

Theoretical Framework: Critically Emergent Dark Energy (CEDE)

The authors analyze the Critically Emergent Dark Energy (CEDE) model. In this scenario:

• Phase Transition: Dark energy is zero for redshifts $z \geq z_c$ (early Universe) and becomes non-zero for $z < z_c$.
• Parametrization: The dimensionless DE density is defined as:
  $$\tilde{\Omega}_{DE}(z) = \begin{cases} \Omega_{DE,0} \left( \frac{z_c - z}{z_c} \right)^{1/2} & z < z_c \\ 0 & z \geq z_c \end{cases}$$
  where $z_c$ is the critical redshift of the transition.
• Equation of State (EoS): This density parametrization implies an EoS parameter $w_{DE}(z) = -1 - \frac{1+z}{6(z_c-z)}$. Notably, once the transition occurs, the DE enters a phantom regime ($w_{DE} < -1$), leading to a faster late-time expansion rate compared to standard $\Lambda$CDM.
• Perturbations: The study includes linear cosmological perturbations in the synchronous gauge, assuming a sound speed $c_{s,DE}^2 = 1$ and neglecting anisotropic stress.

Observational Datasets

The authors constrain the model using a comprehensive combination of datasets:

1. CMB: Planck 2018 temperature/polarization data (plikTTTEEE+lowl+lowE) + ACT lensing.
2. Baryon Acoustic Oscillations (BAO): Two independent compilations:
   • SDSS (Sloan Digital Sky Survey).
   • DESI DR2 (Dark Energy Spectroscopic Instrument Data Release 2), including BGS, ELG, LRG, QSO, and Ly-$\alpha$ forest data.
3. Type Ia Supernovae (SNIa): Two independent compilations:
   • PantheonPlus.
   • Union3 (2,087 supernovae).

Statistical Analysis

• Parameter Estimation: Markov Chain Monte Carlo (MCMC) analysis using the Cobaya sampler and modified CAMB code.
• Free Parameters: 7 parameters (6 standard $\Lambda$CDM parameters + 1 transition scale factor $a_c = 1/(1+z_c)$).
• Model Comparison:
  • $\Delta\chi^2$: Difference in minimum chi-square between CEDE and $\Lambda$CDM.
  • Bayesian Evidence: Calculation of the Bayes factor ($B_{ij}$) and log-evidence ($\ln B_{ij}$) to assess statistical preference, interpreted via the Jeffreys' scale.

3. Key Results

Constraints on the Transition Epoch ($a_c$ or $z_c$)

The evidence for a phase transition is highly dependent on the dataset combination:

• CMB Only: Strongly favors a late transition with $z_c \sim 1.4$ ($a_c \approx 0.41$). This yields a high $H_0 \approx 80.2$ km/s/Mpc but with large uncertainties.
• CMB + SDSS: The preferred transition shifts to an earlier epoch ($z_c \sim 6.2$, $a_c \approx 0.14$). The $H_0$ tension is only mildly reduced ($H_0 \approx 68.85$).
• CMB + DESI: Favors a transition at $z_c \sim 5.5$ ($a_c \approx 0.15$). Due to DESI's preference for lower matter density ($\Omega_m$) compared to SDSS, this combination yields a higher $H_0 \approx 69.56$ km/s/Mpc, reducing the tension to $\sim 3.9\sigma$.
• Supernova Impact: When SNIa data (PantheonPlus or Union3) are added to CMB+BAO, the evidence for a transition often vanishes (only upper bounds on $a_c$ are found), except for the CMB+DESI+Union3 combination, which constrains an early transition ($z_c \sim 9.3$).

The Hubble Tension

• The CEDE model does not fully resolve the Hubble tension.
• While CMB-only analysis allows for high $H_0$, adding BAO data pulls the value down toward the $\Lambda$CDM Planck value.
• The best reduction in tension is achieved with CMB+DESI, raising $H_0$ to $69.56 \pm 0.61$ km/s/Mpc, but the discrepancy with local measurements ($H_0 \approx 73.5$) remains significant ($\sim 3.9\sigma$).

Model Comparison Statistics

• CMB Only: $\Delta\chi^2 = -5.0$ and $\ln B_{ij} = 2.01$. Indicates a preference for CEDE over $\Lambda$CDM, though the evidence is "Weak" according to the Jeffreys' scale.
• CMB + SDSS: $\Delta\chi^2 = -3.9$ and $\ln B_{ij} = 1.0$. Weak preference for CEDE.
• CMB + DESI: $\Delta\chi^2 = -3.2$ and $\ln B_{ij} = 1.6$. Weak preference for CEDE.
• Combinations with SNIa: Most combinations (e.g., CMB+SDSS+PantheonPlus) show no significant preference for CEDE, with $\Delta\chi^2 \geq 0$ and $\ln B_{ij} \leq 0$, favoring $\Lambda$CDM.

Growth of Structure

The evolution of the growth rate observable $f\sigma_8(z)$ in the CEDE model closely tracks $\Lambda$CDM over the redshift range considered, with only mild deviations at low redshift. The model does not introduce significant anomalies in structure formation compared to the standard model.

4. Significance and Conclusions

• Viability of Emergent DE: The study demonstrates that a dark energy model featuring a late-time phase transition is not ruled out by current data. In fact, specific dataset combinations (CMB alone, CMB+SDSS, CMB+DESI) show a statistical preference for CEDE over $\Lambda$CDM.
• Dataset Dependence: The inferred epoch of the dark energy "turn-on" is highly sensitive to the BAO dataset used. DESI data, favoring lower $\Omega_m$, pushes the transition to later times (lower $z_c$) compared to SDSS, which in turn allows for a higher $H_0$.
• Limitations: The model fails to fully resolve the $H_0$ tension when constrained by the full suite of modern data (especially when including SNIa). The statistical evidence for CEDE remains "weak" rather than "strong" or "decisive."
• Future Outlook: The authors conclude that while CEDE is a phenomenologically interesting extension, upcoming high-precision surveys are essential to determine if emergent dark energy represents a genuine physical departure from $\Lambda$CDM or merely an effective description of current data anomalies.

In summary, the paper provides a rigorous test of the CEDE model, finding it to be a viable, albeit not definitive, alternative to $\Lambda$CDM that offers a partial, dataset-dependent mitigation of the Hubble tension.