# Cosmic neutrino background

The cosmic neutrino background (CNB or CνB[1]) is the universe's background particle radiation composed of neutrinos. They are sometimes known as relic neutrinos.

The CNB is a relic of the Big Bang; while the cosmic microwave background radiation (CMB) dates from when the universe was 379,000 years old, the CNB decoupled (separated) from matter when the universe was just one second old. It is estimated that today, the CNB has a temperature of roughly 1.95 K.

As neutrinos rarely interact with matter, these neutrinos still exist today. They have a very low energy, around 10−4 to 10−6 eV.[1] Even high energy neutrinos are notoriously difficult to detect, and the CνB has energies around 1010 times smaller, so the CνB may not be directly observed in detail for many years, if at all.[1] However, Big Bang cosmology makes many predictions about the CνB, and there is very strong indirect evidence that the CνB exists.[1]

## Derivation of the CνB temperature

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 dropped to approximately 2.5 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 neutrinos and the photons today. To find this ratio, we assume that the entropy of the universe was approximately conserved by the electron-positron annihilation. Then using

${\displaystyle \sigma \propto gT^{3},}$

where σ is the entropy, g is the effective degrees of freedom and T is the temperature, we find that

${\displaystyle \left({\frac {g_{0}}{g_{1}}}\right)^{\frac {1}{3}}={\frac {T_{1}}{T_{0}}},}$

where T0 denotes the temperature before the electron-positron annihilation and T1 denotes after. The factor g0 is determined by the particle species:

• 2 for photons, since they are massless bosons[2]
• 2 × (7/8) each for electrons and positrons, since they are fermions.[2]

g1 is just 2 for photons. So

${\displaystyle {\frac {T_{\nu }}{T_{\gamma }}}=\left({\frac {2}{2+2\times 7/8+2\times 7/8}}\right)^{\frac {1}{3}}=\left({\frac {4}{11}}\right)^{\frac {1}{3}}.}$

Given the current value of Tγ = 2.725 K,[3] it follows that Tν1.95 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 kTν falls below the rest mass energy mνc2. Instead, in this case one should rather track their energy density, which remains well-defined.