Proposal for Resonant Detection of Relic Massive Neutrinos

We present a novel method for detecting the relic neutrino background that takes advantage of structured quantum degeneracy to amplify the drag force from neutrinos scattering off a detector. Developing this idea, we present a characterization of the present day relic neutrino distribution in an arbitrary frame, including the influence of neutrino mass and neutrino reheating by $e^+e^-$ annihilation. We present explicitly the neutrino velocity and de Broglie wavelength distributions for the case of an Earthbound observer. Considering that relic neutrinos could exhibit quantum liquid features at the present day temperature and density, we discuss the impact of neutrino fluid correlations on the possibility of resonant detection.

We study the present day relic neutrino distribution in the Earth frame and characterize the dependence on neutrino mass and freeze-out conditions in early Universe. We present explicitly the neutrino velocity and de Broglie wavelength distributions. We characterize the expected neutrino drag force O(G 2 F ) of a spherical detector in the hard sphere scattering limit, as would be experienced by a macroscopic (millimeter sized) coherent quantum detector moving at velocity 300 ± 30 km/s in this neutrino background. One of the great challenges of this century is to develop the experimental capability to detect cosmic background neutrinos. For this objective it is important to precisely understand the expected neutrino signal, both in terms of the neutrino number density and velocity distribution, and to evaluate the effect on a hypothetical detector [1][2][3][4][5][6][7]. We recently characterized precisely the evolution of relic neutrinos in the free-streaming period [8], laying the groundwork for a characterization of the present day spectrum which we explore now from the perspective of an observer located on Earth. Beyond consideration of neutrino distributions, we evaluate the O(G 2 F ) mechanical force originating in the anisotropy of the neutrino flux.
In [8][9][10] the Fermi-Dirac-Einstein-Vlasov (FDEV) neutrino distribution after freeze-out was derived. By casting it in a relativistically invariant form we can then make a transformation to the Earth rest frame and obtain the neutrino distribution The 4-vector characterizing the rest frame of the neutrino FDEV distribution is where we have chosen coordinates so that the relative motion is in the z-direction. The neutrino freeze-out temperature T k is red shifted as the universe expands Υ ν is the so called fugacity factor that describes the underpopulation of neutrino phase space that was frozen into the neutrino FDEV distribution in the process of decoupling from the background e ± , photon plasma at a temperature T k = O(1) MeV. It allows neutrinos to be less degenerate which reduces the quantum character of the distribution. Though relatively unknown in the realm of neutrino physics, its presence is necessary [8].
The fugacity Υ (resp. reheating ratio T ν /T γ ) are uniquely determined by the freezeout temperature. In fact they are strictly decreasing (resp. increasing) functions of T k [8]. The residual Earth velocity relative to the CMB is measured as V ⊕ = 300 ± 30 km/s. Since the neutrino background and CMB were in equilibrium, with decoupling occuring only at T k ≃ O(MeV), one surmises that relative to the relic neutrino background the Earth has the same velocity and the relative velocity of the two frames is V ⊕ . In this and subsequent formulas, we will write velocities in units of c, though our specific results will be presented in km/s. The FDEV distribution Eq. (1) has a thermal shape only for m ν = 0 and only in that case is Eq. (3) strictly a temperature. However, speaking in the more general case of T ν as a temperature is reasonable since, apart from the reheating ratio, T ν tracks the photon background temperature.
The amount of entropy that is transfered from e ± into neutrinos before freeze-out can be characterized using the effective number of neutrinos, a measure of the energy density in relativistic neutrinos as compared to the energy density of a massless fermion with two degrees of freedom and standard reheating ratio. By definition, any transfer of entropy from e ± into neutrinos results in N ν > N f ν = 3, the number of physical neutrino flavors. N ν can be measured by seeking consistency in the Universe dynamics. A numerical computation based on the Boltzmann equation with two body scattering [11] gives to N th ν = 3.046. However, recent experimental PLANCK CMB results contain several fits [12] which suggest that N ν ≃ (3.30-3.62) ± 0.25. PLANCK CMB and lensing observations [13] lead to N ν = 3.45 ± 0.23. The relations derived in [8] determine T ν /T γ and Υ in terms of the measured value of N ν and show that N ν increases as T k decreases. In particular, a value of N ν ≃ 3.5 can be interpreted in terms of a delayed neutrino freeze-out during the e ± annihilation era. In the following we treat N ν as a variable model parameter within this general experimental range and use the above mentioned relations to characterize our results in terms of N ν .
There are several available bounds on neutrino masses. Neutrino energy and pressure components are important before photon freeze-out and thus m ν impacts Universe dynamics. The CMB data alone leads to m ν < 0.66eV and including Baryon Acoustic Oscillation (BAO) gives m ν < 0.23eV [12]. PLANCK CMB with lensing observations [13] lead to m ν = 0.32 ± 0.081 eV. Upper bounds have been placed on the electron neutrino mass in direct laboratory measurements mν e < 2.05eV [14,15]. In the subsequent analysis we will focus on the range neutrino mass range 0.05eV to 2eV in order to show that direct measurement sensitivity allows to explore a wide mass range.
Using Eq. (1), the normalized FDEV velocity distribution for a Earth bound observer has the form We show f v in figure 1 for several values of the neutrino mass, V ⊕ = 300 km/s, and N ν = 3.046 (solid lines) and N ν = 3.62 (dashed lines). As expected, the lighter the neutrino the more f v is weighted towards higher velocities with the velocity becoming visibly peaked about V ⊕ for m ν = 2 eV. In figure  5 of Ref. [8] the absolute normalization 58-60 cm −3 per flavor and ν,ν depending on N ν can be read off. A similar procedure produces the normalized FDEV energy distribution f E . In Eq. (4) we replace dp/dv → dp/dE where it is understood that We show f E in figure 2 for several values of the neutrino mass, V ⊕ = 300 km/s, and N ν = 3.046 (solid lines) and N ν = 3.62 (dashed lines). The width of the FDEV energy distribution is on the micro-eV scale and the kinetic energy T = E − m ν is peaked  about T = 1 2 m ν V 2 ⊕ , implying that the relative velocity between the Earth and the CMB is the dominant factor for m ν > 0.1 eV.
Using λ = 2π/p we find in the Earth rest frame the normalized FDEV de Broglie wavelength distribution shown in figure 3 for V ⊕ = 300 km/s and for several value of N ν and m ν . While the FDEV distributions are peaked at the millimeter wavelength scale, to dis- criminate neutrino masses it is necessary to consider somewhat larger wavelengths.
The result presented in figure 3 is in principle what will govern future efforts to detect relic neutrinos in the laboratory. The de Broglie wavelength characterizes the quantum size of the neutrino, directly impacting its capability to interact with a coherent quantum target, akin to the proposed detection of supernovae MeV energy scale neutrinos [16] by means of collisions with the entire atomic nucleus. In the following, and without offering a practical realization, we imagine that neutrinos interact with a coherent millimeter-centimeter size quantum target just in this fashion -for a charge neutral target the interaction with protons and neutrons cancels and what remains is coherent interaction with all neutrons found within the quantum coherence volume. We focus on non-relativistic neutrinos and hence characterize the drag using Newtonian force F = ∆p/∆t and the non-relativistic relation between momentum and velocity.
We now evaluate precisely the mechanical drag force on a spherical (coherent quantum) detector of radius R due to neutrino back-scattering with probability P (p). For a single scattering event where the component of the momentum normal to the sphere is p ⊥ = (p ·r)r, the change in particle momentum is ∆p = −2p ⊥ . The differential number of scatterings per unit surface area, per unit time, per unit volume in momentum space at a point r on a radius R sphere S 2 R and inward pointing momentum p (i.e.
where the factor of two comes from combining neutrinos and anti-neutrinos of a given flavor. Therefore the recoil change in detector momentum per unit time is The only angular dependence in f is through p·ẑ so by symmetry, thex andŷ components integrate to 0. Therefore we have We perform this integration in spherical coordinates for r and in the spherical coordinate vector field basis for p = p rr + p θrθ + p φrφ , p r < 0, where we recall r = cos θ sin φx + sin θ sin φŷ + cos φẑ, r θ = − sin θx + cos θŷ, r φ = cos θ cos φx + sin θ cos φŷ − sin φẑ.
We have considered two cases, pure hard sphere scattering where P (p) = 1 as well as P (p) = 1 + cos θ s where θ s is the scattering angle, the latter factor being found in the cross sections of V − A weak process. However, numerically in the latter case the total drag is simply reduced by a factor ≃ 2/3 and thus we refrain from describing the procedure, only noting that if we denote the momentum of the scattered neutrino by p s = p − 2p ⊥ then cos θ s = p · p s /|p||p s | = 1 − 2(p ·r) 2 /|p| 2 .
For hard sphere scattering, the magnitude of the drag force per unit surface area for a single neutrino flavor due to motion of the detector with respect to the CMB reference frame is shown by combining the forces for each flavor with their distinct mass values. Considering the current frontier of precision force measurements is on the order of yocto-Newton (yN= 10 −24 N ) [17], the order of zepto-Newton (zN= 10 −21 N ) magnitude obtained here is encouraging if a macroscopic quantum detectors could be constructed capable of reflecting the relic neutrinos. We performed a least squares fit over the range 270 km/s ≤ V ⊕ ≤ 330 km/s; 3 ≤ N ν ≤ 3.7; 0.05 eV ≤ m ν ≤ 2 eV. We define δN ν = N ν − 3, δV ⊕ = V ⊕ /300 km/s − 1 and recalling the CMB temperature T CMB = 0.235 meV, x = m ν /10 4 T CMB . Using these variable we achieve approximately 1% relative error in the above fit region with the following fitting function In summary, we have characterized the relic cosmic neutrinos and their velocity, energy, and de Broglie wavelength distributions in the Earth frame of reference. We have shown explicitly the mass m ν dependence and the dependence on neutrino reheating expressed by N ν , choosing a range within the experimental constraints. Given all the assumptions made, our study of the neutrino spectrum is precise but there are considerable improvements possible which are particullarly important for heavy eVneutrinos: each relic massive neutrino entering solar system will follow a Keplerian orbit modulated by large gravitating masses near to the Earth. The velocity distribution figure 1 shows a highly significant variation of the spectra as a function of neutrino mass and as result we can expect that a study of neutrino planetary motion will produce a highly mass dependent result, just like it is the case with neutrino clustering due to galactic gravitational field [20], an effect we also did not explore.
Under the assumption of hard sphere scattering, we have computed drag force on a spherical detector originating in the dipole anisotropy induced by Earth motion and have shown its dependence on cosmic neutrino physical parameters explicitly for the total net force. We have evaluated seasonal variation due to the change in relative velocity of Earth and CMB considering the time dependence of the Earth orbital velocity vector. This effect is often discussed in the context of dark matter detection [18,19].
Assuming that a macroscopic coherent quantum detector allowing hard sphere scattering could be constructed, we find that the measurement of neutrino drag force could be within experimental reach. We note here an earlier, more penetrating and less optimistic order of magnitude discussion of the drag force in Ref. [6].