The Cosmic Neutrino Background

About one second after the Big Bang, when the universe cooled to roughly 1 MeV, neutrinos decoupled from the thermal plasma. They have propagated freely ever since and now fill space with a thermal distribution at an effective temperature of $T_{\nu }\approx 1.95$ K — just slightly colder than the cosmic microwave background (CMB) photons at 2.725 K.

This cosmic neutrino background (CνB or CNB) is the second-oldest electromagnetic signal from the early universe, predating the CMB by nearly 400,000 years. It has never been directly detected. Its existence is, however, confirmed with high significance by the imprint it leaves on the CMB and on large-scale structure.

Temperature relation

After neutrino decoupling, electrons and positrons annihilated into photons, heating the photon bath but not the already-decoupled neutrinos. Entropy conservation in the photon sector gives $T_{\nu }={\left(\frac{4}{11}\right)}^{1/3}{T}_{\gamma }$ With the CMB temperature known to seven significant figures from COBE/FIRAS, this yields $T_{\nu }=1.9454\pm 0.0001$ K today.

The relic number density per flavor (summed over neutrinos and antineutrinos) is $n_{\nu }=\frac{3}{11}{n}_{\gamma }\approx 112{\text{ cm}}^{-3}\text{ per flavor}$ so about 336 neutrinos per cubic centimetre across the three flavors. Globally, relic neutrinos outnumber photons by a small factor and outnumber baryons by about $10^9$.

Effective number of species $N_{\text{eff}}$

The CMB is sensitive to the radiation energy density at recombination. Any relativistic species contributes to the total, and the effective number of species is defined by ${\rho }_{\text{rad}}={\rho }_{\gamma }\left[1+\frac{7}{8}{\left(\frac{4}{11}\right)}^{4/3}{N}_{\text{eff}}\right]$ Standard Model decoupling gives $N_{\text{eff}}=3.044$ (slightly above 3 because of partial re-heating and finite-temperature QED corrections).

Planck’s measurement yields $N_{\text{eff}}=2.99\pm 0.17$, in excellent agreement with the Standard Model prediction. This is the strongest indirect evidence that three thermally produced neutrino species exist in cosmology — and one of the tightest constraints on the existence of additional light relics, be they sterile neutrinos, axions, or other beyond-Standard-Model degrees of freedom.

Imprint on structure

Relic neutrinos are hot dark matter at decoupling but cool as the universe expands. For masses below about 0.1 eV they remain relativistic until late times; for heavier masses they become non-relativistic earlier. Either way, their thermal velocities prevent them from clustering on small scales and thereby suppress the growth of large-scale structure on scales below the free-streaming length.

This suppression is quantified by the CMB power spectrum and by galaxy surveys. The resulting combined constraints on $\sum m_{\nu }$ are now approaching $\sim 0.1$ eV, as discussed in neutrino mass.

Direct detection: the PTOLEMY proposal

Direct laboratory detection of relic neutrinos is the holy grail. The only process with no kinematic threshold is neutrino capture on beta-unstable nuclei: ${\nu }_e+{}^3\text{H}\to {}^3\text{He}+e^{-}$ This reaction has no threshold because the parent nucleus is already beta-unstable; any relic neutrino, however cold, can induce it. The signature is an electron with energy slightly above the natural beta endpoint — separated by twice the neutrino mass.

The PTOLEMY project proposes to deploy ~100 g of tritium deposited on graphene, with a cyclotron-radiation emission spectroscopy readout, to detect a handful of such capture events per year. The experimental challenges — tritium handling, energy resolution at the eV scale, overwhelming beta-decay background — are enormous. A detection would be the cosmological equivalent of the CMB discovery in 1965.

Physics potential of a detection

Direct relic-neutrino detection would:

• Probe the early universe at a much earlier epoch than the CMB (1 second vs. 380,000 years)
• Fix the absolute neutrino mass scale through the exact offset of the capture peak from the beta endpoint
• Test lepton number conservation through the rate ratio of capture to beta decay (finite only for Dirac; double for Majorana at equal local density)
• Constrain neutrino clustering by comparing local to mean cosmic densities

In the absence of detection, cosmology will continue to shape our understanding of this relic background through the CMB and large-scale structure, with upcoming surveys (CMB-S4, Simons Observatory, DESI, Euclid) tightening $N_{\text{eff}}$ and $\sum m_{\nu }$ significantly over the next decade.