Disentangling cosmic distance tensions with early and late dark energy

Abstract: Recent cosmological data reveal tension between parameters inferred from measurements of the cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and supernovae (SN) under $Λ$CDM. Typical dynamical dark energy parameterizations (such as $w_0w_a$) that seek to jointly resolve these tensions have an equation of state parameter that crosses into the phantom regime, leading to potential instabilities for physical models. We show that the BAO (early-time) and SN (late-time) sides of the tension can instead be treated independently. Early dark energy (EDE) can reduce the tension between CMB-BAO data by changing the calibration of the sound horizon at the drag epoch $r_d$, with a $Δχ^2 = -{9.4}$ relative to $Λ$CDM, raising $H_0$ to 70.87 $\rm km s^{-1}Mpc^{-1}$. EDE alone cannot bring consistency between CMB, BAO, and SN data, but combining with a thawing-quintessence component of dark energy reduces tensions between the three datasets, with $Δχ^2=-12.6$ relative to $Λ$CDM without a phantom component, vs. $Δχ^2=-15.8$ for $w_0 w_a$ with one. We consider different SN datasets, using the most recent DES Dovekie catalog as our default while assessing differences with the original DESY5 and Pantheon+ catalogs. While the significance of adding thawing quintessence changes, the EDE solution to the CMB-BAO tension remains nearly unaffected. Moreover, though we do not include direct Hubble constant measurements in these $Δχ^2$ values, the EDE solution reduces the Hubble tension with the Local Distance Network value from $7σ$ in $Λ$CDM to $2-3σ$ depending on the SN dataset, nominally the equivalent of an extra $Δχ^2 \sim -40$ or more.

• The paper presents an EDE+thaw model that recalibrates early cosmic distances and raises H0 to 70.87±0.94, reducing tensions from 7σ to 2.1σ.
• The study employs both frequentist and Bayesian methods using MCMC and profile likelihoods to demonstrate a Δχ² improvement of up to −12.6 compared to ΛCDM.
• The combined approach resolves CMB-BAO and SN discrepancies while avoiding unphysical phantom energy instabilities, providing a stable path for future cosmological inference.

Disentangling Cosmic Distance Tensions with Early and Late Dark Energy

Overview and Motivation

Cosmological parameter inference from the CMB, BAO, and SN Ia datasets under $\Lambda$CDM has revealed substantial tensions, notably the $\sim 7\sigma$ discrepancy in the Hubble constant $H_0$ between the CMB-inferred value and the local distance ladder, as well as secondary tensions among CMB, BAO, and SN probes. Phenomenological solutions employing dynamical dark energy (e.g., $w_0 w_a$ parameterizations) often require phantom crossing, leading to theoretical instabilities. This work demonstrates that these tensions, particularly between CMB/BAO and SN, can be systematically decoupled and resolved with early dark energy (EDE) and a late-time thawing quintessence component, allowing for tension mitigation without resorting to unphysical phantom energy regimes.

Methodology

The authors utilize a comprehensive frequentist and Bayesian framework, combining Metropolis-Hastings MCMC (MontePython) and profile likelihood computations (Procoli), to rigorously quantify model preference and tension resolutions. Three principal cosmological extensions are explored:

• Early Dark Energy (EDE): An axion-like scalar field, characterized by $(f_{\rm EDE}, z_c, \theta_i)$, contributing to the expansion rate near matter-radiation equality and diluting rapidly thereafter.
• Thawing Quintessence: A non-phantom scalar field, parameterized via the CPL form $w(a) = w_0 + w_a(1-a)$ with $w_a = -1.58(1 + w_0)$, adding late-time dynamical dark energy.
• $w_0 w_a$ model: Fully phenomenological, allowing phantom crossing but manifestly unstable physically.

Datasets include Planck, ACT, SPT-3G CMB; DESI DR2 BAO; and multiple SN samples (DES Dovekie, DESY5, Pantheon+), with nuisance and calibration parameters carefully marginalized.

Resolution of CMB-BAO Tension and Hubble Discrepancy

The EDE model modifies the sound horizon at the drag epoch $r_d$, directly altering the BAO/CMB distance ratio and increasing $H_0$ inferred from the CMB.

After EDE calibration, BAO residuals relative to fiducial $\sim 7\sigma$0CDM show systematic $\sim 7\sigma$1–$\sim 7\sigma$2 improvements across $\sim 7\sigma$3 (Figure 1).

Figure 1: DESI DR2 BAO data compared to predictions from various models, illustrating the improved fit from EDE and EDE+thaw.

Posterior distributions for $\sim 7\sigma$4 and $\sim 7\sigma$5 demonstrate that EDE robustly shifts parameters toward BAO preference (Figure 2), with the profile likelihood confirming $\sim 7\sigma$6 improvement for CMB+BAO relative to $\sim 7\sigma$7CDM.

Figure 2: Posterior distributions on $\sim 7\sigma$8 and $\sim 7\sigma$9 revealing CMB-BAO tension resolution via EDE.

Profile likelihood analysis avoids prior-volume effects and confirms quadratic, non-Gaussian behavior from marginalized posteriors (Figure 3). EDE raises $H_0$0 to $H_0$1 and lowers the Hubble tension to $H_0$2 (vs $H_0$3 in $H_0$4CDM) for H0DN.

Figure 3: Profile likelihoods for EDE fraction, demonstrating improved fit over $H_0$5CDM for both CMB+BAO and CMB+BAO+SN.

CMB primary anisotropy decomposition (Figure 4) isolates improvements from EDE to low $H_0$6 (lowered $H_0$7), intermediate $H_0$8 (coherent TE spectrum reduction), and high $H_0$9 (lensing anomaly mitigation).

Figure 4: Cumulative contributions to $w_0 w_a$0 in primary CMB anisotropy for EDE versus $w_0 w_a$1CDM.

Comparison of likelihoods for $w_0 w_a$2 from cosmological and local probes affirms EDE's capacity to realign cosmological $w_0 w_a$3 with distance ladder results, as shown in Figure 5.

Figure 5: One-dimensional probability distributions for $w_0 w_a$4, highlighting EDE's reduction of Hubble tension.

Disentangling SN Tension and Late-Time Dynamics

SN Ia data, particularly at $w_0 w_a$5, favor evolving late dark energy, which is in tension with the pure EDE solution. Adding thawing quintessence to EDE resolves this tension without requiring a phantom regime in $w_0 w_a$6. The EDE+thaw model achieves $w_0 w_a$7 for CMB+BAO+SN (DES Dovekie), close to $w_0 w_a$8 ($w_0 w_a$9), but avoids theoretical instabilities.

Alternative SN catalogs (DESY5, Pantheon+) yield varying levels of preference for thawing and $(f_{\rm EDE}, z_c, \theta_i)$0, but EDE's impact on CMB-BAO and Hubble tension is largely invariant.

SN distance modulus residuals (Figure 6) show that EDE+thaw fits low-$(f_{\rm EDE}, z_c, \theta_i)$1 SN while maintaining CMB-BAO improvements, whereas $(f_{\rm EDE}, z_c, \theta_i)$2 achieves similar fits only via phantom crossing.

Figure 6: SN distance modulus data relative to fiducial $(f_{\rm EDE}, z_c, \theta_i)$3CDM, showing the fit improvement from EDE+thaw.

Implications, Theoretical and Practical Impact

The segmentation of the cosmic tensions into early (CMB-BAO, Hubble) and late (SN) components allows targeted model extensions without unphysical behavior. EDE robustly calibrates early distance measures and $(f_{\rm EDE}, z_c, \theta_i)$4, while thawing quintessence absorbs residual low-$(f_{\rm EDE}, z_c, \theta_i)$5 SN preferences. This division points toward future high-precision cosmology relying on both early and late universe modeling.

Upcoming CMB surveys (Simons Observatory, SPT-3G Ext-10k) and SN datasets (LSST, Roman) will sharply test these models. The combination of EDE and thawing quintessence constitutes a theoretically sound avenue for tension resolution, provided robust calibration at low redshift SN is achieved.

Conclusion

This study rigorously demonstrates that early dark energy can resolve CMB-BAO and Hubble tensions by modifying the drag epoch sound horizon, with the improvement quantified via frequentist and Bayesian metrics. The addition of thawing quintessence enables simultaneous SN tension mitigation without recourse to phantom energy. The composite EDE+thaw solution remains stable, sensitive to SN dataset calibration, and has direct implications for future cosmological inference and dark sector physics.