Cosmic neutrino background

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The Cosmic Neutrino Background (CNB) is the background particle radiation composed of neutrinos.

Like the CMB, the CNB is a relic of the big bang, whereas the CMB dates from when the universe was 300,000 years old, the CNB decoupled from matter when the universe was 2 seconds old. It is estimated that the CNB has a temperature of 1.9 kelvins or lower. Neutrinos are notoriously difficult to detect, and because these particles are so cold, the CNB may never be observed directly.

Given the temperature of the CMB, the temperature of the CNB 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 below the masses of the W and Z bosons, the neutrinos decoupled from the rest of matter. At this point, neutrinos and photons still had the same temperature. When the temperature dropped below the mass of the electron, most electrons and positrons annihilated, transfering their heat and entropy to photons. So the ratio of the temperature of the photons before and after the electron-positron annihiliation 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

$\sigma \propto gT^3$ ,

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

$\left(\frac{g_0}{g_1}\right)^{1/3} = \frac{T_1}{T_0}$ ,

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

$\frac{T_\nu}{T_\gamma} = \left(\frac{4}{11}\right)^{1/3}$ .

Given the current value of $T γ = 2.73K$ , it follows that $T_\nu \approx 1.9 \rm K$ .

The above discussion is valid for massless neutrinos, which are always relativistic. If neutrinos have a positive rest mass, they become non-relativistic when the thermal energy $3 / 2 k T ν$ falls well below the rest mass energy $m ν c 2$ . Non-relativistic matter cools faster than relativistic matter as the Universe expands. Precise calculations, keeping the entropy of each fermion constant, give for today's neutrino temperature $T_\nu \approx 1.6 \cdot 10^{-4} \left(m_\nu / 1 \rm eV \right)^{-1} \rm K$ .