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
where σ is the entropy, g is the effective degrees of freedom and T is the temperature, we find that
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 × (7/8) each for electrons and positrons, since they are fermions.
g1 is just 2 for photons. So
Given the current value of Tγ = 2.725 K, it follows that Tν ≈ 1.95 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.