A 2% determination of $N_{\rm eff}$ from primordial element abundance, cosmic microwave background, and baryon acoustic oscillation measurements

1. Problem Statement

The effective number of relativistic species, $N_{\text{eff}}$, is a critical cosmological parameter quantifying the radiation energy density in the early universe relative to photons. In the Standard Model (SM), $N_{\text{eff}}$ is predicted to be 3.044, a value slightly higher than the three neutrino flavors due to non-instantaneous neutrino decoupling and electron-positron annihilation effects.

However, many Beyond-Standard-Model (BSM) scenarios predict additional light degrees of freedom (e.g., sterile neutrinos, axions, dark photons) that would increase $N_{\text{eff}}$. Conversely, a value lower than 3.044 could indicate photon heating after decoupling or low reheating temperatures. Previous constraints on $N_{\text{eff}}$ have been limited by:

• Degeneracies between cosmological parameters in Cosmic Microwave Background (CMB) data.
• Inconsistencies between different datasets (specifically between CMB and Baryon Acoustic Oscillation (BAO) data when including large-scale polarization).
• The precision of primordial element abundance measurements (Helium-4 and Deuterium).

The goal of this work is to derive the tightest possible constraint on $N_{\text{eff}}$ by combining the latest high-precision datasets while rigorously addressing dataset consistency.

2. Methodology

The authors performed a Bayesian parameter estimation using the Cobaya sampler and CAMB (with CosmoRec and PRyMordial codes) for theoretical predictions. They combined four distinct classes of observational data:

• Primordial Abundances:
  • Helium-4 ($Y_p$): Used the new measurement from the Large Binocular Telescope (LBT) YP Project: $Y_p = 0.2458 \pm 0.0013$. This is more than twice as constraining as the 2024 PDG value.
  • Deuterium (D/H): Used the consensus value from the 2024 PDG: $(D/H) \times 10^5 = 25.47 \pm 1.04$.
• Cosmic Microwave Background (CMB):
  • Planck 2018: High-$\ell$ temperature and polarization spectra, plus lensing.
  • ACT DR6 & SPT-3G D1: High-resolution temperature and polarization data.
  • Crucial Decision: The authors excluded Planck low-$\ell$ E-mode polarization data ($\ell < 30$) from their baseline analysis. They found that including this data creates a $\sim 3\sigma$ tension with DESI BAO data regarding the matter density ($\Omega_m$) and Hubble constant ($H_0$). Excluding it allows for a fully consistent combination of CMB and BAO data without significantly degrading the $N_{\text{eff}}$ constraint.
• Baryon Acoustic Oscillations (BAO):
  • Used the combined DESI DR2 sample (galaxies, quasars, Lyman-$\alpha$ forest) spanning $0.295 \leq z \leq 2.33$.

Priors and Parameters:

• Fixed neutrino mass sum: $\sum m_\nu = 0.06$ eV.
• Uniform priors on standard $\Lambda$CDM parameters ($\Omega_b h^2$, $\Omega_c h^2$, $\theta_{MC}$, $A_s$, $n_s$, $\tau$).
• Uniform prior on $N_{\text{eff}}$: $0.1 \leq N_{\text{eff}} \leq 6.0$.

3. Key Contributions

1. Resolution of Dataset Tension: The paper identifies that the mild tension between CMB and BAO in the $(H_0 r_d, \Omega_m, N_{\text{eff}})$ parameter space is driven by Planck low-$\ell$ polarization data. By excluding this specific data subset, the authors demonstrate that CMB and BAO data become statistically consistent ($< 1.6\sigma$), enabling a robust joint analysis.
2. Robustness of $N_{\text{eff}}$: They show that current constraints on $N_{\text{eff}}$ are essentially insensitive to the inclusion or exclusion of large-scale polarization data (which primarily constrains the optical depth $\tau$). This makes $N_{\text{eff}}$ a highly robust parameter for future cosmological studies.
3. Synthesis of Multi-Messenger Data: This is the first work to combine the new LBT helium measurement, primordial deuterium, the full suite of modern CMB data (Planck + ACT + SPT), and DESI DR2 BAO data into a single coherent constraint.

4. Key Results

The primary result of the paper is the most precise determination of $N_{\text{eff}}$ to date:

$$N_{\text{eff}} = 2.990 \pm 0.070 \quad (68\% \text{ C.L.})$$

Specific Findings:

• Agreement with Standard Model: The result is in excellent agreement with the SM prediction of $3.044$, differing by less than $1\sigma$.
• Constraint on Excess Radiation: The bound on excess relativistic degrees of freedom is $\Delta N_{\text{eff}} < 0.107$ (95% C.L.).
  • This excludes any new light particle that decoupled after the QCD phase transition at $>5\sigma$.
  • It nearly reaches the sensitivity required to detect a light spin-3/2 particle decoupling at any time after inflation ($>2\sigma$).
• Comparison with Previous Work: This constraint improves upon the previous tightest bound (LBT + Planck) by approximately 15%.
• Hubble Constant Tension: In the $\Lambda$CDM+$N_{\text{eff}}$ cosmology, the combined data yields $H_0 = 68.16 \pm 0.50$ km/s/Mpc. This remains in $5\sigma$ tension with local distance ladder measurements (e.g., SH0ES, $H_0 \approx 73$ km/s/Mpc). The authors conclude that "free-streaming dark radiation" (increasing $N_{\text{eff}}$) is not a viable solution to the Hubble tension.

5. Significance

• New Physics Constraints: The result places stringent limits on a broad class of BSM models involving light particles (axions, sterile neutrinos, dark photons) and strongly interacting dark radiation.
• Hubble Tension Implications: By ruling out simple $N_{\text{eff}}$ increases as a solution to the Hubble tension, the paper forces the community to look toward more complex models (e.g., early dark energy, modified gravity, or late-time interactions) to resolve the discrepancy.
• Methodological Benchmark: The paper establishes a rigorous protocol for combining CMB and BAO data by identifying and excluding specific inconsistent polarization data subsets, a strategy likely to be adopted in future analyses (e.g., with Simons Observatory or CMB-S4 data).
• Future Outlook: The authors note that upcoming improvements in primordial helium measurements and next-generation surveys (DESI-II, Euclid) will further tighten these bounds, potentially providing the first evidence for new light particles in the early universe.