Cosmic neutrino background

Given the temperature of the cosmic microwave background (CMB) the temperature of the cosmic neutrino background (CνB) can be estimated. It involves a change between two regimes:

Regime 1

The original state of the universe is a thermal equilibrium, the final stage of which has photons and leptons freely creating each other through annihilation (leptons create photons) and pair production (photons create leptons). This was the very brief state, right after the Big Bang. Its last stage involves only the lowest-mass possible fermions that interact with photons: electrons and positrons.

Regime 2

Once the universe has expanded enough that the photon+lepton plasma has cooled to the point that Big Bang photons no longer have enough energy for pair production of the lowest mass/energy leptons, the remaining electron–positron pairs annihilate. The photons they create are cool, and are then unable to create new particle pairs. This is the current state of most of the universe.[lower-alpha 2]

At very high temperatures, 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 (${\displaystyle 17.4\times 10^{9}}$ K), the neutrinos decoupled from the rest of matter, and for practical purposes, all lepton and photon interactions with these neutrinos stopped.[lower-alpha 3]

Despite this decoupling, neutrinos and photons remained at the same temperature as the universe expanded as a "fossil" of the prior Regime 1, since both are cooled in the same way by the same process of cosmic expansion, from the same starting temperature. However, when the temperature dropped below double the mass of the electron, most electrons and positrons annihilated, transferring their heat and entropy to photons, 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 in the current Regime 2. To find this ratio, we assume that the entropy  s  of the universe was approximately conserved by the electron–positron annihilation. Then using

${\displaystyle s\propto g\,T^{3}~,}$

where  g  is the effective number of degrees of freedom and T is the plasma or photon temperature. Once reactions cease, the entropy  s  should remain approximately "stuck" for all temperatures below the cut-off temperature, and we find that

${\displaystyle {\frac {\;T_{1}\;}{T_{2}}}=\left({\frac {\;g_{2}\;}{g_{1}}}\right)^{\tfrac {1}{3}}~,}$

Here ${\displaystyle \;T_{1}\propto T_{\mathrm {\nu } }\;}$ denotes the lowest temperature where pair production and annihilation were in equilibrium; and ${\displaystyle \;T_{2}\propto T_{\mathrm {\gamma } }\;}$ denotes the temperature after the temperature fell below the regime-shift temperature ${\displaystyle \;T_{1}\;}$, after the remaining, but no longer refreshed, electron–positron pairs had annihilated and contributed to the total photon energy. The related temperatures ${\displaystyle \;T_{\mathrm {\gamma } }\;}$ and ${\displaystyle \;T_{\mathrm {\nu } }\;}$ are the simultaneous temperatures of the photons (γ) and neutrinos (ν) respectively, whose ratio stays "stuck" at the same value indefinitely, after ${\displaystyle \;T_{\mathrm {\gamma } }<T_{1}\;.}$

The factor ${\displaystyle \;g_{1}\;}$ is determined by a sum, based on the particle species engaged in the original equilibrium reaction:

+   2   for each photon (or other massless bosons, if any).[3]

+   7/4   for each electron, positron, or other fermion.[3]

Whereas the factor ${\displaystyle \;g_{2}\;}$ is simply   2,   since the present regime only concerns photons, in thermal equilibrium with at most themselves.[3]

So

${\displaystyle {\frac {\;T_{\mathrm {\nu } }\;}{T_{\mathrm {\gamma } }}}={\frac {\;T_{1}\;}{T_{2}}}=\left({\frac {\;g_{2}\;}{g_{1}}}\right)^{\tfrac {1}{3}}=\left({\frac {2}{\;2+{\tfrac {7}{4}}+{\tfrac {\,7\,}{4}}\;}}\right)^{\tfrac {\,1\,}{3}}=\left({\frac {4}{\;11\;}}\right)^{\tfrac {\,1\,}{3}}\approx {0.714}~.}$

Since the cosmic photon background temperature at present has cooled to ${\displaystyle \;T_{\mathrm {\gamma } }=2.725\,\mathrm {K} ~,}$[4] it follows that the neutrino background temperature is currently

${\displaystyle \;T_{\mathrm {\nu } }\approx {1.95}\,\mathrm {K} ~.}$

The above discussion is technically valid for massless neutrinos, which are always relativistic. For neutrinos with a non-zero rest mass, at low temperature where the neutrinos become non-relativistic, a description in terms of a temperature is not appropriate. In other words, when the neutrinos' thermal energy ${\displaystyle \;{\frac {3}{2}}\,k\,T_{\mathrm {\nu } }\;}$ (k is the Boltzmann constant) falls below the rest mass energy ${\displaystyle \;m_{\mathrm {\nu } }\,c^{2}\;;}$ in a low-temperature case one should instead speak of the neutrinos' collective energy density, which remains both relevant and well-defined.

Relativistic neutrinos contribute to the radiation energy density of the universe ρ_R, typically parameterized in terms of the effective number of neutrino species N_ν :

${\displaystyle \rho _{\mathrm {R} }={\frac {\pi ^{2}}{15}}T_{\mathrm {\gamma } }^{4}(1+z)^{4}\left[1+{\frac {7}{8}}N_{\mathrm {\nu } }\left({\frac {4}{11}}\right)^{\frac {4}{3}}\right],}$

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_ν ≃ 3.046,[5] 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_ν from observations.

Confirmation of the existence of these relic neutrinos may only be possible by directly detecting them using experiments on Earth. This will be difficult as the neutrinos which make up the CνB are non-relativistic, in addition to interacting only weakly with normal matter, and so any effect they have in a detector will be hard to identify. One proposed method of direct detection of the CνB is to use the capture of cosmic relic neutrinos on tritium i.e. ³H, leading to an induced form of beta decay.[9]

The neutrinos of the CνB would lead to the production of electrons via the reaction

${\displaystyle \mathrm {\nu } +{}^{3}\mathrm {H} \rightarrow {}^{3}\mathrm {He} +e^{-}~,}$

while the main background comes from electrons produced via natural beta decay

${\displaystyle {}^{3}\mathrm {H} \rightarrow {}^{3}\mathrm {He} +e^{-}+\mathrm {\bar {\nu }} ~.}$

These electrons would be detected by the experimental apparatus in order to measure the size of the CνB. The latter source of electrons is far more numerous, however their maximum energy is smaller than the average energy of the CνB-electrons by twice the average neutrino mass. Since this mass is tiny, of the order of a few eVs or less, such a detector must have an excellent energy resolution in order to separate the signal from the background. One such proposed experiment is called PTOLEMY, which will be made up of 100 g of tritium target.[10] The detector demonstrator (with about 0.2 g of tritium) should be ready by 2025.[11]

• Cosmic background radiation
• Dark matter
• Diffuse supernova neutrino background
• Gravitational wave background