Cosmic neutrino background

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 6987400544121750000♠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:

g₁ is just 2 for photons. So

Given the current value of T_γ = 7000272500000000000♠2.725 K, it follows that T_ν ≈ 7000195000000000000♠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_νc². Instead, in this case one should rather track their energy density, which remains well-defined.

Indirect mathematical evidence for the CνB

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:

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 ≃ 7000304600000000000♠3.046, 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.

Big Bang nucleosynthesis

Due to its effect on the expansion rate of the universe during Big Bang nucleosynthesis (BBN), the theoretical expectations for the primordial abundances of light elements depend on N_ν. Astrophysical measurements of the primordial 4
He
and 2
D
abundances lead to a value of N_ν = 7000314000000000000♠3.14+0.70
−0.65 at 68% c.l., in very good agreement with the Standard Model expectation.

CMB anisotropies and structure formation

The presence of the CνB affects the evolution of CMB anisotropies as well as the growth of matter perturbations in two ways: due to its contribution to the radiation density of the universe (which determines for instance the time of matter-radiation equality), and due to the neutrinos' anisotropic stress which dampens the acoustic oscillations of the spectra. Additionally, free-streaming massive neutrinos suppress the growth of structure on small scales. The WMAP spacecraft's five-year data combined with type Ia supernova data and information about the baryon acoustic oscillation scale yielded N_ν = 7000434000000000000♠4.34+0.88
−0.86 at 68% c.l., providing an independent confirmation of the BBN constraints. The Planck spacecraft collaboration has published the tightest bound to date on the effective number of neutrino species, at N_ν = 7000315000000000000♠3.15±0.23.

Prospects for the direct detection of the CνB

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 capture of cosmic relic neutrinos on tritium i.e. 3 H , leading to an induced form of beta decay. The neutrinos of the CνB would lead to the production of electrons via the reaction ν + 3 H → 3 H e + e − , while the main background comes from electrons produced via natural beta decay 3 H → 3 H e + e − + ν ¯ . 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 100g of tritium target.