Local number density and momentum distribution of the cosmic neutrino background at the Solar System

Direct-detection forecasts for the cosmic neutrino background (CνB) — for example neutrino capture on tritium — assume that relic neutrinos populate the Solar-System volume with a thermal momentum spectrum, possibly modified by gravitational clustering in the Milky Way. The predicted local clustering enhancement is halo-model and neutrino-mass dependent, and beyond-Standard-Model neutrino–dark-matter interactions could repel relic neutrinos from the Galactic neighborhood altogether, so that a nondetection would not discriminate cosmology from local microphysics. The problem is to determine the local relic-neutrino phase-space distribution — the number-density enhancement and the momentum/angular distribution, including the dipolar wind and its annual modulation — through validated prediction and, ultimately, direct measurement, since every CνB detection claim and rate forecast depends on it.

Background

About one second after the Big Bang, neutrinos decoupled from the primordial plasma and have since free-streamed as the cosmic neutrino background (CνB), a relic sea with a thermal spectrum at temperature $T_\nu \simeq 1.95\,\mathrm{K}$ and an average number density of about 56 cm$^{-3}$ for each mass eigenstate and antiparticle (336 cm$^{-3}$ in total). Neutrino oscillations fix the mass-squared differences, so at least two mass eigenstates are heavier than roughly 0.009 and 0.05 eV; since typical relic momenta today are $p \sim 3 k_B T_\nu \approx 5\times10^{-4}\,\mathrm{eV}$, relic neutrinos heavier than about $0.05$ eV are non-relativistic today. Detecting them is a long-standing goal of neutrino physics. The leading proposal is neutrino capture on beta-decaying nuclei, e.g. $\nu + {}^3\mathrm{H} \to {}^3\mathrm{He} + e^-$, the strategy of the proposed PTOLEMY experiment: capture is exothermic and thresholdless, producing a peak in the electron spectrum displaced by $2m_\nu$ above the tritium beta-decay endpoint, with a rate proportional to the local neutrino number density (of order 4–8 events per year for a 100 g tritium target under standard assumptions). This is where the local question enters: the capture rate and any spectral or angular signature depend on the phase-space distribution of relic neutrinos at the Solar System, not on the cosmic average.

Massive, non-relativistic relic neutrinos are gravitationally deflected and trapped by the Milky Way's potential, producing a local clustering enhancement $f_c = n_\nu^{\rm loc}/n_\nu^0$ relative to the homogeneous prediction $n_\nu^0$. Calculations of $f_c$ — from semi-analytic linear-response solutions of the collisionless Boltzmann (Vlasov) equation to test-particle ('N-one-body') and back-tracking simulations in a fixed or reconstructed Galactic potential — agree that clustering is negligible for minimal neutrino masses but grows steeply with mass. Long, Lunardini and Sabancilar (2014) tabulated $f_c$ for two standard Milky-Way halo models (the Navarro–Frenk–White profile, a standard parametrization of dark-matter halos, and a multi-component Milky-Way model), finding $f_c \simeq 1.4$–$1.6$ at $m_\nu = 0.15$ eV but $12$–$20$ at $0.6$ eV: an order-unity halo-model uncertainty in the regime where clustering matters. Clustering also modifies the momentum distribution (bound neutrinos carry higher-than-thermal momenta), and the Earth's motion relative to the CνB rest frame produces a dipolar anisotropy in the flux (the 'CνB wind') with a corresponding annual modulation.

A logically distinct possibility is local microphysics. Davoudiasl, Mohlabeng and Sullivan (2018) proposed that neutrinos are massless in vacuum and instead acquire an 'environmental' (or 'refractive') mass from a feeble scalar force sourced by the Galactic dark-matter density. In such models the position-dependent induced mass acts as a repulsive potential barrier: relic neutrinos, with kinetic energies of order $10^{-4}$ eV, cannot penetrate the Solar neighborhood, so the local CνB would be strongly suppressed and a nondetection by a direct-detection experiment would reflect local microphysics rather than cosmology. More broadly, a measured capture rate is degenerate among gravitational clustering, a primordial lepton asymmetry (a neutrino–antineutrino imbalance that can enhance the relic density), neutrino decay, late-time entropy injection, and sterile-neutrino components, so a single rate-plus-peak-energy measurement cannot disentangle them. The unresolved target is therefore the local relic-neutrino phase-space distribution itself: the number density and the momentum (speed and angular) distribution at the Solar System.

Problem Statement

Determine the phase-space distribution of relic neutrinos in the Solar-System neighborhood — the local number density $n_\nu^{\rm loc}$, equivalently the clustering enhancement $f_c = n_\nu^{\rm loc}/n_\nu^0$ relative to the homogeneous-cosmology value $n_\nu^0$, and the momentum distribution $f(\mathbf{p})$, including its speed spectrum, dipolar anisotropy (the CνB wind), and annual modulation — on which the event rate and the spectral and angular signatures of any cosmic-neutrino-background direct-detection experiment depend. This comprises a prediction of $f_c$ and $f(\mathbf{p})$ from gravitational clustering in the Milky Way and the local large-scale structure, with quantified uncertainties that bracket the halo-model, method, and initial-condition dependence, across the range of neutrino masses and orderings still allowed by oscillation, laboratory (KATRIN), and cosmological data; an assessment of whether beyond-Standard-Model neutrino–dark-matter interactions — in particular the scalar-mediated environmental (refractive) neutrino mass mechanism, in which the Galactic dark-matter density acts as a repulsive potential barrier that non-relativistic relic neutrinos of kinetic energy $\mathcal{O}(10^{-4}\,\mathrm{eV})$ cannot overcome — appreciably suppress or enhance the local density relative to gravitational clustering alone; and, ultimately, validation by a direct measurement of the local relic-neutrino overdensity or momentum distribution (for example via neutrino capture on tritium). The question also covers whether relic neutrinos are present in the Solar-System volume at all, so that a direct-detection signal or its absence can be interpreted unambiguously.

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