N_eff: How Cosmology Counts Neutrino Species

The cosmic radiation budget at the time of nucleosynthesis is set by the Standard Model. Photons contribute most of it. Three neutrino flavors — having decoupled from the photon-electron plasma at temperatures around 1 MeV — contribute the rest. The total radiation energy density is conventionally written as: ${\rho }_R={\rho }_{\gamma }\left(1+\frac{7}{8}{\left(\frac{T_{\nu }}{T_{\gamma }}\right)}^4{N}_{\text{eff}}\right)$ where $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$ accounts for the heating of photons (but not neutrinos) by electron-positron annihilation, and $N_{\text{eff}}$ — the effective number of neutrino species — is the dimensionless parameter that captures how many fully-thermalised relativistic degrees of freedom contribute beyond photons.

The Standard Model, with three light neutrino flavors and no other relativistic species, predicts $N_{\text{eff}}=3.044$. This is not a clean integer because of small corrections: neutrino decoupling is not instantaneous, electron-positron annihilation continues during decoupling, and QED finite-temperature effects redistribute energy among the species. The full calculation, refined over the past two decades, converges on 3.044 with about 0.002 theoretical uncertainty.

If anything else were in the early universe — a fourth neutrino, axions, dark photons, gravitons, sterile species — it would contribute additional relativistic energy and push the measured $N_{\text{eff}}$ above 3.044. The observable is, in this sense, a precision counter of relativistic Standard Model species and a constraint on physics beyond.

This post is about how $N_{\text{eff}}$ is measured, what the current values are, and what they imply for the structure of the early universe.

Two independent probes

$N_{\text{eff}}$ affects the early universe in two regimes that are observed with very different techniques.

Big Bang Nucleosynthesis (BBN), at $T\sim 1$ MeV, sets the primordial light-element abundances. The relevant variable is the Hubble rate $H\propto \sqrt{{\rho }_R}$ at the time of neutron-proton freeze-out and at the subsequent nucleosynthesis epoch. A higher $N_{\text{eff}}$ increases $H$, shortens the time available for the n/p ratio to relax toward equilibrium, and yields a higher final ${}^4\text{He}$ abundance. The dominant constraint comes from primordial helium-4, $Y_p$, with smaller contributions from deuterium, lithium-7, and helium-3. Current observations give $Y_p\approx 0.245\pm 0.003$ from quasar absorption-system spectroscopy and metal-poor extragalactic HII region data. This translates into $N_{\text{eff}}=2.85\pm 0.28$ from BBN alone.

Cosmic Microwave Background (CMB), at $T\sim 0.3$ eV (recombination epoch, redshift $\sim 1100$). $N_{\text{eff}}$ affects the CMB through three independent observables: the position of the acoustic peaks (sensitive to the epoch of matter-radiation equality), the damping scale (the Silk damping at small angular scales), and the helium fraction (which affects the recombination history). The Planck 2018 result is $N_{\text{eff}}=2.99\pm 0.17$, consistent with the Standard Model expectation of 3.044.

The two measurements are sensitive to different physics: BBN is sensitive to $N_{\text{eff}}$ at ~1 MeV; CMB at ~0.3 eV. Any change in the relativistic content of the universe between these epochs would affect the two measurements differently. Because both are consistent with the Standard Model expectation, additional thermalised species at any time between $T\sim 1$ MeV and $T\sim 0.3$ eV are constrained.

What is and isn’t constrained

The $N_{\text{eff}}$ measurement constrains certain hypothetical species:

Additional thermalised neutrinos. Each fully-thermalised additional flavor would contribute approximately $\Delta N_{\text{eff}}=1$. The 0.17 uncertainty on Planck excludes a fully-thermalised fourth neutrino with high confidence ($\Delta N_{\text{eff}}=1$ is approximately $5\sigma$ above the central value).

Sterile neutrinos partially thermalised through small mixing. If the sterile is mixed with active flavors at the level needed to explain the LSND anomaly, it would partially thermalise in the early universe before BBN. The expected $\Delta N_{\text{eff}}$ depends on the mixing, but for the LSND region it is approximately 0.5, conflicting with both BBN and CMB measurements.

Axion-like particles that thermalise. An axion that thermalises before BBN would contribute approximately $\Delta N_{\text{eff}}=1/2$ if it is a single boson, or 1 if it is two species. The constraints exclude axion-photon couplings strong enough to thermalise the axion above a certain mass threshold.

Dark photons in certain coupling regimes that thermalise the dark sector with the visible sector early enough.

Right-handed neutrinos that come into thermal equilibrium with the Standard Model thermal bath.

What is NOT constrained:

• Light particles that never thermalised in the first place (decoupled from the visible sector at all times)
• Particles that decoupled before $T\sim 100$ GeV (they would have been diluted by entropy production from QCD-phase-transition era and other effects)
• Particles that contribute as dark matter rather than as relativistic species (CDM doesn’t contribute to $N_{\text{eff}}$)
• Particles that decay before BBN (no contribution at the relevant epoch)

The value 3.044 in detail

The classical pre-1990s calculation gave $N_{\text{eff}}=3.000$, treating neutrino decoupling as instantaneous and electron-positron annihilation as fully complete only after decoupling. This is a useful pedagogical baseline but ignores corrections:

Non-instantaneous decoupling: the neutrinos do not decouple at a sharp temperature. There is a finite range over which they fall out of equilibrium, during which some of the electron-positron annihilation energy is delivered to neutrinos as well as photons. This correction increases $N_{\text{eff}}$ by approximately $0.04$.

Finite-temperature QED corrections: The photon-electron interactions at finite temperature modify the equation of state slightly, redistributing energy. Correction is approximately $0.01$.

Flavor oscillations among neutrinos during decoupling can also slightly redistribute energy. Correction is at the $0.005$ level.

The total: $3.044\pm 0.002$, with the theoretical uncertainty dominated by the precise calculation of the non-instantaneous decoupling. Recent reviews (de Salas et al. 2016, Bennett et al. 2021) converge on this value.

Combined constraints

The most precise current constraint on $N_{\text{eff}}$ uses Planck CMB combined with BAO (baryon acoustic oscillation) data and BBN abundance measurements. The resulting joint constraint is: $N_{\text{eff}}=2.99\pm 0.17(95\%\text{ C.L.})$

The measurement is dominated by Planck. BAO contributes by breaking the degeneracy between $N_{\text{eff}}$ and the matter density. BBN provides an independent check at a different epoch.

The agreement of the central value (2.99) with the Standard Model prediction (3.044) is striking. Within the 0.17 uncertainty, the data show no evidence for any additional relativistic species during the radiation-dominated era.

Tension and reanalyses

Earlier WMAP data suggested a value of $N_{\text{eff}}$ slightly above 3, generating considerable theoretical interest. The Planck 2013 release gave $N_{\text{eff}}=3.30\pm 0.27$, which is approximately 1$\sigma$ above the Standard Model. By Planck 2015 the central value had moved to 3.04 with smaller error bars; by Planck 2018 it was 2.99.

The interpretation is that the early Planck preference for slightly elevated $N_{\text{eff}}$ was a statistical fluctuation that resolved as the data accumulated and analysis matured. Standard-Model consistency, with $N_{\text{eff}}$ near 3.044, is the current picture.

Some authors continue to point out that the central value is slightly below 3.044, raising the (unphysical) possibility of $N_{\text{eff}}<3$. This is generally interpreted as a 1-sigma fluctuation rather than a real effect.

Future prospects

Forthcoming CMB measurements will tighten the $N_{\text{eff}}$ constraint substantially.

CMB-S4 is a planned next-generation ground-based CMB observatory that aims for $\sigma (N_{\text{eff}})\approx 0.03$ — about a factor of 6 better than Planck. This would provide a 5σ test of the Standard Model prediction at the level needed to detect even minimal thermalised species (axions partially thermalised, very-mildly-mixed sterile neutrinos, dark photons in specific coupling regimes).

LiteBIRD (a Japanese-led satellite mission targeting CMB B-mode polarization) will not directly improve $N_{\text{eff}}$ but will provide complementary constraints on early-universe physics through inflationary gravitational waves.

Stage IV galaxy surveys like LSST and Euclid will improve the BAO constraints that break degeneracies with $N_{\text{eff}}$, indirectly tightening the joint analysis.

By 2030, $N_{\text{eff}}$ will be measured to approximately 0.03 precision. If the result remains consistent with 3.044, the Standard Model relativistic content of the early universe will be confirmed at sub-percent precision. Any deviation would be a major signal of new physics — most likely a partially thermalised light boson or a fourth neutrino with very small mixing.

What N_eff confirms

The measurement of $N_{\text{eff}}$ ties together early-universe cosmology and particle physics in a remarkably direct way. The precision with which we can count relativistic species in the early universe — at temperatures from 1 MeV to 0.3 eV, redshifts from $\sim 10^{10}$ to $\sim 10^3$ — is now better than 6%, and improving.

What this confirms:

1. Three light neutrino flavors and no fourth, fully-thermalised flavor. Combined with the LEP measurement of the Z-boson invisible width (which confirms three flavors at the electroweak scale), this is now confirmed at multiple energy scales.

2. No surprise relativistic species in the early universe between BBN and recombination. Whatever exists beyond the Standard Model either does not thermalise, decoupled before BBN, or decays before recombination.

3. Standard cosmological framework is self-consistent. The CMB, BBN, and BAO measurements all point to the same parameter space — including the same value of $N_{\text{eff}}$ — without significant tensions.

The deeper questions — neutrino mass, dark matter content, dark energy properties, primordial perturbation spectrum — remain. But the count of relativistic species is now nailed down. Three flavors. No more, no less. The early universe was, in this sense, exactly what the Standard Model predicts.

That is itself a non-trivial result. The early universe is a different physical regime — different temperatures, different energy scales, different processes — and yet the counting of fundamental species is consistent across the regimes. The unity of the picture is one of cosmology’s quieter achievements.