N_eff — counting cosmic neutrinos in the microwave background

At temperatures above about one electronvolt, the universe was a hot opaque plasma of photons, electrons, positrons and the three neutrino species. The neutrinos were in thermal equilibrium with the rest of the plasma through weak interactions until the temperature dropped to about 1 MeV — at which point the weak rates fell behind the Hubble expansion and the neutrinos decoupled. A short time later, at temperatures around 0.5 MeV, electrons and positrons annihilated into photons. The annihilation reheated the photon bath relative to the already-decoupled neutrinos by the well-known factor T_γ/T_ν = (11/4)^(1/3) ≈ 1.401. By the time recombination set in at z ≈ 1100 the universe contained the photons of the cosmic microwave background plus a cosmic neutrino background colder by that factor, and the energy density of each component was a fixed multiple of the photon density.

The radiation density of the universe at recombination is conventionally written as

ρ_rad = ρ_γ · [1 + (7/8) · (4/11)^(4/3) · N_eff].

If N_eff = 3 — three Standard-Model neutrino flavours at the predicted temperature ratio — the bracket equals about 1.69, the cosmological radiation density takes its Standard-Model value, and the expansion-history-dependent imprint on the CMB power spectrum has its Standard-Model shape. The Standard-Model prediction including small finite-temperature QED corrections, residual neutrino-electron scattering at the decoupling boundary, and the slow finite-mass effect from the heaviest neutrino is N_eff = 3.044, with about 0.5% theoretical precision. Any additional relativistic species that was in thermal equilibrium at the relevant epoch — a sterile neutrino, an axion, a dark-sector vector boson — adds to the count of N_eff. Any species that decoupled earlier and was colder than the neutrinos when e⁺e⁻ annihilation happened contributes less, weighted by (T_X/T_ν)⁴, but still adds something.

How the CMB measures it

The CMB encodes N_eff through three handles. The most direct is the expansion rate before recombination: a higher N_eff means a hotter, denser radiation bath, a larger Hubble parameter at fixed matter density, and a faster expansion that translates into a smaller sound horizon at decoupling and a corresponding shift of the CMB acoustic peaks toward higher multipoles. The second handle is the silk-damping scale, which sets the high-ℓ tail of the temperature spectrum and depends on the ratio of damping length to sound horizon — both quantities depending on N_eff but with different sensitivities, so the ratio breaks the degeneracy with other parameters. The third handle is the imprint on the matter power spectrum from neutrino free-streaming, which has been probed by baryon acoustic oscillations measured at low redshift.

The Planck 2018 release, combined with the DR1 BAO data from the BOSS galaxy survey, gave N_eff = 2.99 ± 0.17 in a flat-ΛCDM cosmological model. Updated Planck constraints with the lensing reconstruction and ACT and SPT small-scale temperature and polarisation measurements have pulled the central value to 2.97 ± 0.15 and made the constraint less ΛCDM-dependent. None of the updates moves the central value away from the Standard-Model expectation by more than half a sigma. The interactive below shows the constraint as a Gaussian likelihood on the N_eff axis and lets you slide the additional relativistic contribution ΔN_eff = N_eff − 3.044 against it.

Slide the predicted N_eff and watch the red dashed line walk past the green Standard-Model marker and the blue Planck likelihood. At ΔN_eff = 0 the red line sits on top of the green line and inside the bulk of the Planck Gaussian — the data and the prediction agree. At ΔN_eff = 1, the value a fully thermalised sterile neutrino would add, the prediction sits at 4.04, well outside the 2σ region of Planck. The simplest eV-scale sterile hypothesis that the gallium anomaly and the LSND/MiniBooNE excesses point to is therefore in serious tension with cosmology — even though the laboratory experiments cover entirely different observables.

What it does and does not constrain

The N_eff constraint is a count of thermalised light species, weighted by their temperature ratio to the photons at recombination. It does not directly limit sterile neutrinos that decoupled too early to thermalise or whose mixing with active flavours is too small to bring them into thermal equilibrium before BBN. The standard rough boundary is sin²2θ ≳ 10⁻⁴ to 10⁻⁵ for an eV-scale sterile, depending on Δm² and assumed lepton asymmetry; below this threshold the sterile never thermalises and the cosmological count of species is unaffected. The catch is that the sterile parameter regions that the gallium and LSND anomalies pick out have sin²2θ well above this threshold by orders of magnitude. For the simplest single-sterile-flavour models that fit those anomalies, thermalisation is unavoidable and N_eff would have to land near 4. That it does not is the most powerful single statement cosmology can make against the simplest interpretation of the short-baseline hints.

The N_eff constraint also bounds dark-radiation models in which a dark-sector light particle carries some fraction of the universe's energy density. Many beyond-Standard-Model scenarios — light axions thermalised at high temperature, sterile photons of a dark-sector U(1), light right-handed neutrinos with sufficient mixing — leave their fingerprint here. The fact that the measurement is now better than a percent on the SM prediction makes N_eff one of the cleanest cosmological tests of fundamental physics available.

For the cosmological neutrino background itself — the bath of relic neutrinos that the N_eff measurement counts indirectly — see the cosmic neutrino background and why we haven't detected it yet; for the laboratory programme that counts light neutrino species through a different mechanism, see how many neutrino flavours — the LEP measurement that pinned the answer; for the most recent short-baseline anomaly that the cosmological count disfavours, see BEST and the gallium anomaly that refuses to leave.