As described in the previous section neutrinos decouple with electrons, positrons and photons at temperature T~ a few MeV and remain as such until today. Soon after the neutrino decoupling, at about T = 0.5 MeV, e^± pair annihilate into photons and transfer their entropy to these photons. This transfer of entropy to photons effectively slows down the rate of decrease in photon temperature compared with that in neutrino temperature as the Universe expands [3]. Since the entropy of neutrinos is conserved, T_ν = (4/11)^1/3T_γ and, since T^now_γ = 2.7 K, T^now_ν = 1.9 K = 1.7 × 10⁻⁴ eV. Therefore the number density of neutrinos is given by n_ν = (3/22)n_γ = 56 cm⁻³ per flavor. In the standard Big Bang model it is assumed that n_ν = n_ν and neutrinos' chemical potentials are zero. However, in this review it is not assumed, in which case it is convenient to introduce an asymmetry parameter for neutrino with flavor α, ν_α: η_α = (n_να − n_να)/n_γ = [π²/12ζ(3)](ξ_α + ξ³_α/π²)(T_ν/T_γ)³ where ξ_α = μ_α/T with μ_α being the chemical potential of ν_α [5]. A recent study of the primordial He abundance data and WMAP data finds that the sum of the asymmetries of all neutrino flavors, taking into account the effect of neutrino oscillation with the neutrino mixing angle sin²θ₁₃ = 0.04 (0.00), is −0.071 (− 0.064) ≤ η_ν ≤ 0.054 (0.072) (90% C.L.) [6]. At the time of neutrino decoupling (T_ν,dec ~ a few MeV), since we know that neutrino masses are much less than a few MeV, all neutrinos are relativistic (for the status of neutrino masses refer to the review by the Particle Data Group) [7]. However, as the current neutrino temperature is 1.9 K (1.7 × 10⁻⁴ eV), some of neutrinos may be non-relativistic, in which case whether neutrinos are Dirac or Majorana type is important.

From a recent analysis by the Planck collaboration using the CMB and Baryon Acoustic Oscillation (BAO) data, the effective number of relativistic active neutrinos at the time of decoupling of the CMB N_eff is found to be 3.30^+0.51_−0.51, consistent with the existence of three active neutrinos [8]. However, according to the analysis using the Planck, WMAP and BICEP2 data by Giusarma et al., N_eff is found to be 4.00 ± 0.41 [9].

As we know that neutrino has mass, it is possible that CνB can be trapped in gravitational potential well of some structures in the Universe such as large galaxies and clusters of galaxies when CνB has velocity smaller than the escape velocity. For a large galaxy like Milky Way or a large cluster of galaxies the escape velocities are ~600 or ~2000 km/s, respectively. From the Maxwell–Boltzmann distribution, the average velocity of non-relativistic neutrino of mass m_ν at temperature Tν,  < |βν| >  =8kTν/(πmν)=4.3×10−4eV/mν. Therefore < |v_ν| > = 19,600 (6200) km/s for m_ν = 0.1 (1.0) eV [10]. It seems that only a small fraction of neutrinos can be gravitationally trapped in a large cluster of galaxies. However, detailed simulation that takes into account either the current static mass distribution of the Milky Way (MWnow) or an estimated halo mass distribution before the formation of Milky Way through baryonic compression (NFWhalo) reveals that there may be local neutrino overdensity effect at the position of our solar system [11]. The simulation consists of several types of weakly interacting, self-gravitating particles (cold dark matter and neutrinos) modeled as a multi-component collisionless gas with the Vlasov equation and uses the density profile of cold dark matter (CDM) proposed by Navarro, Frank and White (NFWhalo) [12]. The neutrino overdensity n_ν/ < n_ν >, where < n_ν > is the average neutrino density predicted by the standard Big Bang model, is with the NFWhalo model, 12 and 1.4 for m_ν = 0.6 and 0.15 eV, respectively, and with the MWnow model, 20 and 1.6 for m_ν = 0.6 eV and 0.15 eV, respectively. In the both cases (NFWhalo and MWnow) when m_ν < 0.1 eV, there is no overdensity. Overdensity values 20 and 1.4 correspond to values for the neutrino asymmetry parameter η_ν, 2.7 and 0.19, respectively [11].

In this category, there are two possible methods: use of order G_F effect and of G²_F effect where G_F is the Fermi constant.

The first suggestion to use neutrino capture by β-decaying nuclei (NCB) was by Weinberg in 1962 to detect CνB [27]. Electron (anti)neutrino capture by a nucleus N that naturally undergoes beta (positron) decay to the daughter nucleus N′ as ν_e/ν_e + N → N′ + e⁺/e⁻ has no energy threshold on the incident (anti)neutrino energy. In β-decay in which the energy conservation requires M(N) − M(N′) = Q_β > 0 where M(N) and M(N′) are the masses of nucleus of N and N', respectively, and Q_β is the kinetic energy of electron/positron. For massive neutrino of mass m_ν, the electron kinetic energy in NCB is E_e = Q_β + E_ν ≥ Q_β + m_ν, while, neglecting the nucleus recoil energy, E_e ≤ Q_β − m_ν for electrons from β-decay. Thus there is a gap of 2m_ν around Q_β in electron kinetic energy between β-decay and neutrino capture event. Cocco et al. find that the NCB cross section times neutrino velocity can be written as σ_NCBv_ν = 2π²ln2/(At_1/2) where A is a function of E_ν only, when the target nucleus characterized by Q_β and Z is given, and t_1/2 is the half-life of the target nucleus [28]. As β-decay events are the major background source to the NCB, larger ratio of the NCB to β-decay events ~ σ_NCB(v_ν/c)t_1/2 is preferable. Among β-decaying nuclei isotopes ³H and ¹⁸⁷Re have the large ratios 3.0 × 10⁵ and 5.9 × 10⁷, respectively. However, the fact that σ_NCB(v_ν/c) for ³H and ¹⁸⁷Re are 7.8 × 10⁻⁴⁵ cm² and 4.3 × 10⁻⁵² cm², respectively, ³H seems the best choice. Although there is a gap of 2m_ν in the electron kinetic energy between the endpoint of the β-decay and the NCB events, the success of this method depends on the energy resolution whose effect may fill this gap. Cocco et al. estimate the signal (NCB) to the background (β-decay) event ratio R in the outgoing electron kinetic energy region W₀ − Δ < E_e < W₀ where Δ is the energy resolution and W₀ is the endpoint energy. They find that for R = 3 and m_ν = 0.7 (0.3) eV the energy resolution should be better than 0.2 (0.1) eV. The event rate per mole per year is calculated to be 2.85 × 10⁻²[σ_NCB(v_ν/c)/10⁻⁴⁵]cm² yr⁻¹ mol⁻¹. With 100 g of ³H as the target, the event rate per year is, for m_ν = 0.6 eV, 7.5, 90, and 150 events using the standard Fermi-Dirac (FD) distribution, Navarro-Frank-White profile (NFWhalo), and the present day mass distribution of the Milky Way (MWnow), respectively. For m_ν = 0.3 (0.15) eV, the rate is 7.5 (7.5), 23 (10), and 23 (12), respectively, with FD, NFWhalo and MWnow distribution [11].

A calculation by Faessler et al. [29] finds that the NCB rate per year for the ³H target (50 μg) to be used for the KATRIN experiment [30] is 4.2 × 10⁻⁶ n_ν/ < n_ν > which is consistent with the result by Cocco et al. above for the FD distribution with n_ν/ < n_ν > = 1. A similar calculation for 760 g of ¹⁸⁷Re to be used for the MARE calorimetric experiment [31] finds the event rate per year 6.7 × 10⁻⁸ n_ν/ < n_ν >, which is too small for a likely value for n_ν/ < n_ν >.

According to Lazauskas et al. [32], the event rate per year per Mcu (mega-curries = 3.7 × 10¹⁶ decays ~ 2.1 × 10^{25 3}H atoms) for approximately 100 g of ³H is 6.5 × n_ν/ < n_ν > yr⁻¹ Mcu⁻¹, which is also consistent with the result by Cocco et al. As for the required energy resolution to have a reasonable signal to noise ratio of unity, they conclude that the resolution should be a factor of two or more smaller than the neutrino mass m_ν, which is similar to the conclusion by Cocco et al.

Note that the KATRIN experiment's goal for the energy resolution is 0.93 eV and the mass of the ³H source is only 50 μg [30] and to increase the mass of the ³H source or to improve the energy resolution it is necessary to seek a new way to perform this type of experiment. Toward this goal of improvement the Project 8 experiment has started [33]. This experiment utilizes detection of cyclotron radiation from β-decay electron to achieve an improvement in the energy resolution. Another proposal to improve the neutrino mass resolution is to use cold atomic tritium rather than molecular tritium used for the KATRIN experiment to detect both the electron and ³He in the final state of β-decay [34]. Finally the PTOLEMY experiment (Princeton Tritium Observatory for Light-Early Universe Massive-neutrino Yield) is most ambitious [35]. It is based on relic neutrino capture on tritium. It uses, in addition to MAC-E Filter (Magnetic Adiabatic Collimation combined with an Electrostatic Filter) adopted in the KATRIN experiment, a large surface-deposition tritium target, cryogenic calorimeter, RF tracking similar to the Project 8 experiment, and time-of-flight systems to achieve the required background suppression and enough event rate. PTOLEMY plans to use 100-g of atomic tritium as the target which can provide enough event rate.

It is argued that the Universe is opaque to electrons, nucleons and photons at energies higher than 10²³ eV [36]. A more careful consideration finds that the interaction between cosmic ray proton and CMB photon p + γ_CMB → π + N imposes an energy threshold known as Greisen-Zatsepin-Kuzmin (GZK) cutoff E_GZK ≈ 5 × 10¹⁹ eV, beyond which cosmic ray protons do not survive [37, 38]. However, Weiler argues that the only particle that can survive the GKZ cutoff is neutrinos and interactions of ultra-high energy cosmic ray neutrinos with CνB through the Z resonance ν + ν → Z → p + any introduces a dip at certain energy in ultra-high energy neutrino flux. If these neutrinos come from sources with the redshift z = 3.5, this dip occurs at 9 × 10¹⁹ eV. Depending on the three neutrino masses and the value of redshift parameter z of the sources of these ultra-high energy neutrinos, there may be possibly three dips in energy spectrum of cosmic ray neutrinos if these sources exist. The existence of these neutrino sources can produce cosmic rays beyond the GZK cutoff through the Z resonance (Z-burst). However, although the AGASA experiment claimed that they detected cosmic ray events above the GZK cutoff [39], neither the HiRes Experiment [40] nor the Auger experiment [41] confirmed such events.

As for detailed theoretical analyses on the CνB spectroscopy using ultra-high energy cosmic rays, refer to the papers by Barenboim et al. [42] and by D'Olivo et al. [43]. A detailed theoretical analysis on the Z-burst can be found, for example, in the paper by Fargion et al. [44].

Another interaction of cosmic rays with CνB to detect CνB was proposed by Wigman [45]. He proposes to explain, in terms of inverse β-decay interactions such as p + ν_e → n + e⁺, the changes in the power index of cosmic rays spectrum around 10^15.3 eV (the first knee) observed by the CASA-BLANCA experiment [46] from n = 2.72 ± 0.02 to n = 2.95 ± 0.02, and also at the second knee around 10^17.5 eV from n = 3.01 ± 0.06 to n = 3.27 ± 0.02 observed by the Fly's Eye experiment [47]. In the interaction p + ν_e→ n + e⁺ where ν_e is CνB, the center of mass energy ECM≈mp2+2Epmν should be greater than the sum of proton and neutron masses m_p + m_n. This leads to the threshold proton energy for the interaction at 1.695 × 10¹⁵/(m_ν/(1 eV)) eV. If the energy of the knee is, combining several experimental data, at (3 ± 1) × 10¹⁵ eV and the cause of the first knee is the inverse β-decay interaction by proton, it is consistent with the neutrino mass m_νe = 0.5 ± 0.2 eV/c².

Although evidence of high energy cosmic ray neutrinos has finally emerged as the IceCube experiment shows [48], there is only one experimental result that suggests that there are cosmic rays beyond the GZK cutoff, which has not been confirmed by the HiRes and Auger experiment. Various mechanisms in addition to the inverse β-decay theory have been proposed to explain the existence of the knees in cosmic ray spectrum [49]. However, it is not clear which explanation is the correct model for the knees.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.