## Wednesday, 20 June 2007

### Cosmic Neutrino Background

The cosmic neutrino background (CNB, CνB) is the universe's background particle radiation composed of neutrinos. They are sometimes known as relic 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 379,000 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 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 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 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 T
0
denotes the temperature before the electron-positron annihilation and T
1
denotes after. The factor g
0
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
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
= 2.725 K, it follows that T
ν
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 kT
ν
falls below the rest mass energy m
ν
c2
. Instead, in this case one should rather track their energy density, which remains well-defined.

http://en.wikipedia.org/wiki/Cosmic_neutrino_background