Pinning N_eff to 2.99: How ACT, SPT & Planck Ruled Out Dark Radiation

The Theoretical Framework of Relativistic Degrees of Freedom

Constraining Dark Radiation with CMB and BBN

1. The Damping Tail and Silk Scale

   The CMB provides an exquisite laboratory for measuring N_eff via the phenomenon of Silk damping. As the universe expands, photons diffuse out of overdense regions, exponentially suppressing temperature anisotropies at small angular scales (high multipoles, ℓ). An increase in N_eff enhances the Hubble expansion rate prior to recombination, which decreases the physical size of the sound horizon.

   r_d² = ∫_0^t_rec (1 / (6 n_e σ_T a)) (R² / (1 + R)²) dt

   Crucially, the diffusion length—the characteristic scale of Silk damping—scales as the square root of time, whereas the sound horizon scales linearly with time. Consequently, an elevated N_eff increases the ratio of the diffusion damping scale to the sound horizon, resulting in enhanced suppression of the CMB power spectrum damping tail. High-resolution surveys, particularly the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), are explicitly designed to map this high-ℓ tail with arcminute precision, breaking the degeneracy between the primordial scalar spectral index and the primordial helium mass fraction.

2. Primordial Nucleosynthesis (BBN)

   Complementary to the CMB, BBN operates precisely during the epoch when the universe is highly sensitive to the expansion rate driven by N_eff. The freeze-out of neutron-to-proton conversion is governed by the competition between the weak interaction rate and the Hubble rate. A larger N_eff leads to earlier freeze-out, resulting in a higher neutron-to-proton ratio and, subsequently, a larger primordial helium-4 mass fraction, Y_p.

   The recent LBT Y_p Project leverages observations of metal-poor HII regions to constrain helium alongside ultra-precise measurements of deuterium. Deuterium serves as an exceptional baryometer. When combined with ACT, SPT, and Planck CMB data, it anchors the baryon density independently of late-time astrophysical systematics, drastically narrowing the allowed window for the relativistic energy budget and acting as a vital cross-check for high-ℓ CMB constraints.

3. Goldstein & Hill 2026 Constraints

   Synthesizing these disparate datasets, Goldstein & Hill (Phys. Rev. D 114, L021305, 17 July 2026) performed a comprehensive Markov Chain Monte Carlo (MCMC) analysis. By jointly fitting Planck PR4 data, ACT DR6, SPT-3G, DESI Baryon Acoustic Oscillations (BAO), and the latest LBT Y_p Project results, they achieved unprecedented precision, pinpointing N_eff = 2.990 ± 0.070 at the 68% confidence level.

   This global measurement firmly bounds any deviation to ΔN_eff < 0.107 (95% CL). The significance of this result cannot be overstated: it perfectly brackets the Standard Model prediction of 3.044 while imposing severe theoretical limits on any additional relativistic species that might have populated the thermal bath prior to recombination.

Phenomenological Implications for Light Relics

1. Sterile Neutrinos and Hot Axions

   The immediate consequence of the stringent ΔN_eff < 0.107 bound is the exclusion of numerous dark sector models that invoke light relics. Fully thermalized sterile neutrinos, which would contribute ΔN_eff ≈ 1, are definitively ruled out. Similarly, hot axions or axion-like particles (ALPs) that decouple subsequent to the quantum chromodynamics (QCD) phase transition are heavily constrained.

   For an ALP coupling exclusively to photons or gluons, the decoupling temperature must now exceed several hundred GeV to ensure its contribution to the relativistic degrees of freedom remains below the 0.107 threshold. This effectively closes the window on a vast swath of parameter space for thermal dark radiation, forcing model builders to invoke extremely weak couplings or very early decoupling epochs that push beyond the reach of near-term observable signatures.

2. Deficits in N_eff and the Escudero Scenarios

   While most dark radiation models predict ΔN_eff > 0, recent anomalies in ground-based CMB observations sparked interest in scenarios where N_eff is actively depleted. Early analyses of ACT DR6 data originally suggested N_eff = 2.86 ± 0.13, a mild deficit relative to the Standard Model. Escudero et al. demonstrated that N_eff < 3 could be generated by the presence of thermal electrophilic relics with masses between 8 and 13 MeV, or through the late-time decay of heavy particles that preferentially heat the photon bath relative to the neutrino background.

   ℒ_int = − g_χe χ ē e − (1/2) m_χ² χ²

   In such a model, the relic χ annihilates directly into electrons and positrons after neutrino decoupling, thereby injecting entropy exclusively into the electromagnetic sector. However, the subsequent combined analysis by Goldstein & Hill, which shifted the ACT-only value toward 2.89 ± 0.11 and the global value to 2.990 ± 0.070, largely evaporates the statistical significance of this deficit. The electrophilic relic scenario, while theoretically elegant, is no longer observationally necessitated by the high-ℓ damping tail.

The Hubble Tension and Early Universe Modifications

1. Extra Radiation and H_0

   The persistent discrepancy between local measurements of the Hubble constant (H_0) and its value inferred from the early universe remains the most pressing crisis in modern cosmology. A popular class of resolutions involves increasing the early-universe expansion rate to reduce the sound horizon at recombination, thereby raising the inferred H_0 to match late-time supernovae data. This is typically achieved by injecting dark radiation (ΔN_eff > 0) or introducing early dark energy.

   Δr_s / r_s ≈ − (1/2) [ ΔN_eff / (N_eff + (8/7) (11/4)^4/3 (ρ_m / ρ_γ)) ]

   Because a higher H_0 requires a substantially smaller sound horizon r_s, any extra-radiation solution typically demands ΔN_eff ≈ 0.2 to 0.4. The Goldstein & Hill bound of ΔN_eff < 0.107 strictly prohibits these extra-radiation models from functioning as viable solutions to the Hubble tension. By sealing off this avenue, the cosmological community is increasingly directed toward late-time modifications of gravity or acknowledging unrecognized systematics in the local distance ladder.

2. Future Horizons with the Simons Observatory

   Despite the remarkable precision achieved by the combination of Planck, ACT, and SPT, the ultimate theoretical limit of CMB-derived N_eff constraints has not yet been reached. The forthcoming data releases from the Simons Observatory (SO) will blanket the microwave sky with unprecedented sensitivity and multifrequency coverage.

   Forecasts indicate that SO will achieve a 1-sigma sensitivity of σ(N_eff) = 0.045, sufficient to probe the very threshold of the Grand Unified Theory (GUT) decoupling limit. If SO centralizes its measurement precisely at the Standard Model value of 3.044, it will conclusively rule out any thermal light relic decoupling after the electroweak phase transition, fundamentally mapping the thermal history of the universe.

Conclusion