The cosmic neutrino background (CνB) is the universe's background particle radiation composed of neutrinos.

Like the cosmic microwave background radiation (CMB), the CνB is a relic of the big bang, and while the CMB dates from when the universe was 380,000 years old, the CνB decoupled from matter when the universe was 2 seconds old. It is estimated that today the CνB has a temperature of roughly 1.95 K. Since low-energy neutrinos interact only very weakly with matter, they are notoriously difficult to detect and the CνB might never be observed directly. There is, however, compelling indirect evidence for its existence.

Derivation of the temperature of the CνB

Given the temperature of the CMB, the temperature of the CνB can be estimated. Before neutrinos decoupled from the rest of matter, the universe primarily consisted of neutrinos, electrons, positrons and photons, all in thermal equilibrium with each other. Once the temperature reached approximately 1 MeV, the neutrinos decoupled from the rest of matter. Despite this decoupling, neutrinos and photons remained at the same temperature as the universe expanded. However, when the temperature dropped below the mass of the electron, most electrons and positrons annihilated, transferring their heat and entropy to photons, and thus increasing the temperature of the photons. So the ratio of the temperature of the photons before and after the electron-positron annihilation is the same as the ratio of the temperature of the photons and the neutrinos today. To find this ratio, we assume that the entropy of the universe was approximately conserved by the electron-positron annihilation. Then using

where $\backslash sigma$ is the entropy, $g$ is the effective number of degrees of freedom and $T$ is the temperature, we find that

where the subscript 0 denotes before the electron-positron annihilation and 1 denotes after. To find $g\_0$, we add the degrees of freedom for electrons, positrons and photons:

• 2 for photons, since they are massless bosons

• 2(7/8) each for electrons and positrons, since they are fermions

$g\_1$ is just 2 for photons. So

Given the current value of $T\_\backslash gamma\; =\; 2.725\; \backslash rm\; K$, it follows that $T\_\backslash nu\; \backslash approx\; 1.95\; \backslash rm\; K$.

The above discussion is valid for massless neutrinos, which are always relativistic. For neutrinos with a non-zero rest mass, the description in terms of a temperature is no longer appropriate after they become non-relativistic, i.e., when their thermal energy $3/2\; kT\_\backslash nu$ falls below the rest mass energy $m\_\backslash nu\; c^2$. Instead, in this case one should rather track their energy density, which remains well-defined.

Indirect evidence for the CνB

Relativistic neutrinos contribute to the radiation energy density of the Universe $\backslash rho$_R, typically parameterized in terms of the effective number of neutrino species $N$_ν:

where $z$ denotes the redshift. The first term in the square brackets is due to the CMB, the second comes from the CνB. The Standard Model with its three neutrino species predicts a value of $N\_\{\backslash rm\; \backslash nu\}\; \backslash simeq\; 3.046$, including a small correction caused by a non-thermal distortion of the spectra during e⁺-e^--annihilation. The radiation density had a major impact on various physical processes in the early Universe, leaving potentially detectable imprints on measurable quantities, thus allowing us to infer the value of $N\_\backslash nu$ from observations.

Big Bang Nucleosynthesis

Due to its effect on the expansion rate of the Universe during Big Bang nucleosynthesis (BBN), the theoretical expectations for the primordial abundances of light elements depend on $N\_\backslash nu$. Astrophysical measurements of the primordial ⁴He and Deuterium abundances lead to a value of $N\_\{\backslash rm\; \backslash nu\}\; =\; 3.14^\{+0.70\}\_\{-0.65\}$ at 68% c.l., in very good agreement with the Standard Model expectation.

CMB anisotropies and structure formation

The presence of the CνB affects the evolution of CMB anisotropies as well as the growth of matter perturbations in two ways: due to its contribution to the radiation density of the Universe (which determines for instance the time of matter-radiation equality), and due to the neutrinos' anisotropic stress which dampens the acoustic oscillations of the spectra. Additionally, free-streaming massive neutrinos suppress the growth of structure on small scales. The WMAP satellite's five-year data combined with type Ia Supernova data and information about the baryon acoustic oscillation scale yield $N\_\{\backslash rm\; \backslash nu\}\; =\; 4.4\; \backslash pm\; 1.5$ at 68% c.l., providing an independent confirmation of the BBN constraints. In the near future, probes such as the Planck satellite will likely improve present errors on $N\_\backslash nu$ by an order of magnitude.

See also

• Cosmic microwave background