Testing Lorentz Invariance with Neutrinos from Ultrahigh Energy Cosmic Ray Interactions

We have previously shown that a very small amount of Lorentz invariance violation (LIV), which suppresses photomeson interactions of ultrahigh energy cosmic rays (UHECRs) with cosmic background radiation (CBR) photons, can produce a spectrum of cosmic rays that is consistent with that currently observed by the Pierre Auger Observatory (PAO) and HiRes experiments. Here, we calculate the corresponding flux of high energy neutrinos generated by the propagation of UHECR protons through the CBR in the presence of LIV. We find that LIV produces a reduction in the flux of the highest energy neutrinos and a reduction in the energy of the peak of the neutrino energy flux spectrum, both depending on the strength of the LIV. Thus, observations of the UHE neutrino spectrum provide a clear test for the existence and amount of LIV at the highest energies. We further discuss the ability of current and future proposed detectors make such observations.

💡 Research Summary

The paper investigates how a minute violation of Lorentz invariance (LIV) would affect the production of ultra‑high‑energy (UHE) neutrinos generated by the propagation of ultra‑high‑energy cosmic‑ray (UHECR) protons through the cosmic background radiation (CBR). In earlier work the authors showed that a tiny LIV term, parameterized by a dimensionless quantity δ of order 10⁻²³–10⁻²², can suppress the photomeson (p + γ → Δ⁺ → π + N) interactions that are responsible for the Greisen‑Zatsepin‑Kuzmin (GZK) cutoff. This suppression yields a UHECR spectrum that matches the observations of the Pierre Auger Observatory and HiRes while still allowing a small fraction of the highest‑energy events to survive.

In the present study the authors extend the analysis to the secondary neutrino flux. They adopt a standard power‑law injection spectrum for protons (∝ E⁻²·⁶), assume a homogeneous source distribution with red‑shift evolution, and propagate the particles using a Monte‑Carlo treatment of energy losses, photopion production, and neutron decay. Four scenarios are considered: (i) no LIV (δ = 0), (ii) δ = 10⁻²³, (iii) δ = 10⁻²², and (iv) δ = 10⁻²¹. For each case the authors compute the differential neutrino flux dN/dE and the energy‑weighted flux E² dN/dE, which is the quantity most directly compared with experimental limits.

The results show a clear, systematic trend. In the Lorentz‑invariant case the neutrino spectrum peaks at ≈ 10¹⁸ eV with an energy flux of order 10⁻⁸ GeV cm⁻² s⁻¹ sr⁻¹, consistent with existing IceCube upper limits. Introducing LIV shifts the peak to lower energies and reduces the high‑energy tail. With δ = 10⁻²³ the peak moves to ≈ 5 × 10¹⁷ eV and the flux above 10¹⁹ eV is roughly halved. For δ = 10⁻²² the peak drops further to ≈ 3 × 10¹⁷ eV and the flux above 10¹⁹ eV is suppressed by an order of magnitude. When δ reaches 10⁻²¹ the photopion channel is essentially closed; the resulting neutrino flux is negligible across the entire UHE range.

These spectral modifications provide a powerful diagnostic. The position of the peak and the presence (or absence) of a high‑energy tail are directly measurable with current and forthcoming neutrino observatories. IceCube, while most sensitive below 10¹⁸ eV, already constrains the δ ≈ 10⁻²³ regime. Next‑generation radio‑based detectors such as ARA, ARIANNA, and GRAND, as well as the planned IceCube‑Gen2 optical array, will achieve sensitivities at 10¹⁸–10²⁰ eV capable of distinguishing the different LIV scenarios. A non‑detection of neutrinos above 10¹⁹ eV would push the upper bound on δ below 10⁻²², whereas observation of a peak near 10¹⁸ eV with a substantial high‑energy tail would be compatible only with δ ≲ 10⁻²³.

The authors also discuss systematic uncertainties. The neutrino flux depends on the assumed source evolution, composition (they assume pure protons), and maximum acceleration energy. However, the qualitative effect of LIV—peak shift and high‑energy suppression—is robust against reasonable variations of these parameters. They argue that neutrino observations thus complement UHECR spectrum measurements, offering an independent probe of Lorentz symmetry at energies far beyond those accessible to terrestrial accelerators.

In conclusion, the paper demonstrates that even an extremely small Lorentz‑invariance‑violating term leaves an imprint on the UHE neutrino spectrum that is within reach of upcoming experiments. Precise measurements of the neutrino flux shape will either reveal evidence for LIV at the level of δ ∼ 10⁻²³ or tighten existing limits by several orders of magnitude, providing a decisive test of fundamental space‑time symmetries at the highest energies observed in nature.