Ultra-High-Energy Cosmic Ray Boosted Relic Neutrinos

This paper systematically calculates the diffuse flux of ultra-high-energy cosmic ray-boosted relic neutrinos by modeling various Standard Model scattering channels and cosmic ray compositions, ultimately deriving upper limits on the cosmic neutrino background overdensity using current IceCube and Pierre Auger Observatory data.

The Gist

The Big Picture: The Ghostly Ocean and the Speeding Bullet

Imagine the entire universe is filled with a calm, invisible ocean of "ghost particles" called relic neutrinos. These are the leftovers from the Big Bang, floating everywhere, but they are so cold and slow that they are practically invisible to our detectors. They are like dust motes floating in a sunbeam—everywhere, but too small to see.

Now, imagine a Ultra-High-Energy Cosmic Ray (UHECR). This is a subatomic particle (like a proton or a heavy nucleus) that has been accelerated to speeds almost as fast as light by some violent event in the cosmos. Think of this cosmic ray as a massive, speeding bullet train.

The Paper's Core Idea:
The authors ask: What happens if this speeding bullet train crashes into the calm ocean of ghost particles?

Usually, the ghost particles are too lazy to be noticed. But if a cosmic ray hits one, it can transfer some of its massive energy to the ghost, "boosting" it to high speeds. Suddenly, that ghost particle becomes a high-energy neutrino that our telescopes (like IceCube in Antarctica) might actually be able to catch.

The paper is essentially a detailed manual on how to calculate exactly how many of these "boosted" ghosts we should expect to see, depending on what kind of bullet train is hitting them.

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The Different Ways the Crash Can Happen (The Scattering Channels)

The authors realized that when the cosmic ray hits a relic neutrino, the interaction isn't just one simple thing. It depends on how hard the hit is and what the cosmic ray is made of. They broke this down into five different "collision modes," like different ways a car can crash into a wall:

1. Elastic Scattering (The Bumper Car): The cosmic ray hits a single proton or neutron inside a nucleus, and they bounce off each other without breaking anything. This happens at moderate speeds.
2. Coherent Scattering (The Whole Building): If the cosmic ray is moving slowly (relatively speaking) and the target is a heavy nucleus (like Iron), the neutrino hits the entire nucleus as if it were one giant object. It's like throwing a pebble at a whole building; the building shakes as one unit. This creates a huge "signal boost" at low energies.
3. Incoherent Scattering (The Brick-by-Brick): As the cosmic ray speeds up, it stops seeing the nucleus as a whole and starts seeing the individual bricks (protons and neutrons) inside. The "whole building" effect disappears, and the neutrino starts hitting the bricks one by one.
4. Resonance Production (The Spring-Loaded Trap): At higher speeds, the collision is so energetic that it temporarily excites the particle, making it wobble or vibrate like a spring before settling down. This is a specific "sweet spot" of energy where the interaction gets very strong.
5. Deep Inelastic Scattering (The Smash-Up): At the highest, most violent energies, the cosmic ray smashes the nucleus so hard that it shatters the internal structure, revealing the tiny quarks inside. It's like a car crash so severe that the engine explodes.

The Paper's Finding: The authors mapped out exactly which of these five "crash modes" dominates at different speeds. They found that for heavy cosmic rays, the "Whole Building" (Coherent) mode is king at low speeds, but as they get faster, the "Brick-by-Brick" and "Spring-Loaded" modes take over.

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The Bullet Train Models (Cosmic Ray Flux)

To make their predictions, the authors had to guess what the "bullet trains" (cosmic rays) actually look like. They didn't just guess; they used three different detailed maps (models) of the universe:

1. The PriNCe Model: This is a complex simulation that tracks cosmic rays as they travel through the universe, losing energy and changing composition along the way. It's like a GPS that accounts for traffic jams and roadblocks.
2. The H3a Model: A theoretical map where the highest-speed cosmic rays are mostly heavy, mixed-up nuclei (like a train made of heavy cargo containers).
3. The H4a Model: A theoretical map where the highest-speed cosmic rays are almost entirely pure protons (like a train made of lightweight, high-speed racing cars).

The Paper's Finding: The type of train matters immensely.

• If the universe is full of heavy cargo trains (H3a), the "Whole Building" collisions dominate, and we see a lot of low-energy boosted neutrinos.
• If the universe is full of racing cars (H4a), the collisions are much more violent, creating a huge number of high-energy boosted neutrinos that can reach the "Deep Inelastic" (smash-up) regime.

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The Detective Work (Constraints on Overdensity)

The universe is supposed to have a specific, standard amount of these relic neutrinos. However, some theories suggest there might be more of them in our local neighborhood (an "overdensity"), perhaps because dark matter decayed into them or they got trapped in a cosmic cloud.

The authors used their calculations to play detective:

• They calculated how many boosted neutrinos we should see if the density is normal.
• They compared this to what the IceCube and Pierre Auger Observatory telescopes have actually seen so far.
• Since we haven't seen a massive flood of these boosted neutrinos, the authors can set an upper limit on how many extra neutrinos could be hiding in our neighborhood.

The Paper's Conclusion:

• The current data tells us that the local density of these relic neutrinos cannot be more than about 10 million to 10 billion times the standard amount (depending on the specific model used).
• This is a much stricter limit than previous lab experiments (like KATRIN) could set, proving that looking at the sky with cosmic rays is a powerful new way to hunt for these ghost particles.

Summary in a Nutshell

This paper is a comprehensive guide on how to turn the invisible, slow "ghosts" of the Big Bang into detectable "high-speed ghosts" by crashing them into cosmic rays. The authors built a complete toolkit to calculate this process, accounting for every type of collision and every possible type of cosmic ray. They found that the answer depends heavily on what the cosmic rays are made of, and by comparing their math to real telescope data, they successfully narrowed down how many of these ghost particles could be hiding in our cosmic neighborhood.

Technical Summary

Technical Summary: Ultra-High-Energy Cosmic Ray Boosted Relic Neutrinos

Problem Statement
The Cosmic Neutrino Background (CνB), a relic of the Big Bang, remains undetected directly due to its extremely low energy scale ($T_{\nu,0} \approx 1.95$ K) and weak interaction cross-sections. While various indirect probes have been proposed, such as resonant annihilation or neutrino capture on tritium, they face significant experimental challenges. A complementary approach involves using Ultra-High-Energy Cosmic Rays (UHECRs) to boost relic neutrinos to detectable energies via Standard Model (SM) neutral-current interactions. Previous studies have addressed this mechanism but often relied on simplified assumptions, such as purely protonic UHECR compositions or restricted sets of scattering channels. This work addresses the need for a systematic, comprehensive treatment that accounts for the full hierarchy of SM scattering processes, mixed UHECR compositions, and source evolution models to accurately predict the diffuse boosted CνB flux.

Methodology
The authors perform a systematic calculation of the diffuse UHECR-boosted CνB flux by integrating three primary components:

1. Scattering Channels: The study constructs a unified SM neutral-current framework covering five distinct scattering regimes based on momentum transfer ($Q^2$) and hadronic invariant mass ($W$):

   • Elastic Scattering (ES): Neutrino-nucleon scattering.
   • Coherent Elastic Scattering (COH): Neutrino-nucleus scattering where the nucleus acts as a single coherent unit (dominant at low $Q^2$).
   • Incoherent Scattering (INCOH): Neutrino-nucleus scattering where individual nucleons are resolved.
   • Resonance Production (RES): Excitation of baryon resonances (e.g., $\Delta(1232)$).
   • Deep Inelastic Scattering (DIS): Scattering off quark and antiquark constituents.
     The authors utilize the GENIE neutrino event generator for resonance cross-sections (summing 17 baryon states) and nCTEQ15 parton distribution functions (PDFs) for DIS, ensuring a consistent transition between the resonance and DIS regions ($W_{cut} = 1.7$ GeV).

2. UHECR Flux Models: To quantify the dependence on composition and propagation, three distinct UHECR flux descriptions are employed:

   • PriNCe: A propagation-based code solving coupled transport equations for mixed nuclear species (p, He, N, Si, Fe) including energy losses and fragmentation in cosmic photon backgrounds.
   • Hillas H3a: A phenomenological model with mixed-composition populations extending to high energies.
   • Hillas H4a: A variant where the highest-energy population is proton-dominated, serving as a benchmark for proton-rich scenarios.
     These models are combined with three source evolution scenarios: Star Formation Rate (SFR), Quasi-Stellar Objects (QSO), and Gamma-Ray Bursts (GRB).

3. Flux Calculation and Constraints: The boosted neutrino flux is calculated by integrating the differential cross-sections over the UHECR flux and redshift ($0 \le z \le 6$) within a flat $\Lambda$CDM cosmology. The resulting flux is proportional to the CνB overdensity factor $\eta$. The authors derive upper limits on $\eta$ by comparing predicted event rates with current data from the IceCube Neutrino Observatory and the Pierre Auger Observatory (PAO), utilizing Feldman-Cousins upper limits on background events.

Key Contributions

• Comprehensive Channel Treatment: This work is the first to consistently include and match all five relevant SM neutral-current channels (ES, COH, INCOH, RES, DIS) within a single framework for boosted CνB calculations.
• Resonance Inclusion: A significant extension is the explicit inclusion of the Resonance (RES) channel, demonstrating its role as a critical bridge between the elastic and deep inelastic regimes.
• Composition and Rigidity Dependence: The study systematically quantifies how the boosted flux depends on UHECR composition (proton vs. heavy nuclei) and maximum rigidity, revealing distinct spectral signatures for different flux models.
• Updated Constraints: The paper provides updated, model-dependent upper limits on the local CνB overdensity $\eta$, surpassing current direct laboratory bounds from KATRIN in specific parameter spaces.

Results

• Scattering Hierarchy: A clear hierarchy of scattering channels emerges based on the boosted neutrino energy and UHECR composition.
  • For nuclear UHECRs, COH dominates the low-energy flux. As energy increases, coherence is suppressed, and INCOH becomes dominant. RES contributes significantly to the high-energy tail, while DIS appears only at the highest energies.
  • For proton-dominated UHECRs (e.g., PriNCe and H4a models), ES dominates at lower energies, with RES and DIS becoming increasingly important at higher energies.
• Flux Sensitivity: The predicted boosted flux is highly sensitive to the UHECR model. The H4a model (proton-dominated at high energies) yields the highest flux and extends the spectrum to higher energies compared to the PriNCe and H3a models. The PriNCe model, dominated by protons at lower energies but including heavy nuclei, shows a different spectral shape where nuclear components are subdominant.
• Overdensity Limits: Using IceCube and PAO data, the authors derive constraints on $\eta$.
  • For a lightest neutrino mass $m_1 = 0.1$ eV and the H4a model with GRB evolution, the limit is $\eta < 2.52 \times 10^5$ (IceCube) and $\eta < 1.72 \times 10^6$ (PAO).
  • These limits are significantly tighter than the current KATRIN bound ($\eta < 9.7 \times 10^{10}$) for the same mass range.
  • Constraints are strongest for the H4a model and weakest for the PriNCe model.
  • For $m_1 \lesssim 0.01$ eV, constraints are nearly flat as the signal is dominated by heavier mass eigenstates ($m_2, m_3$). For $m_1 \gtrsim 0.1$ eV, constraints become mass-independent due to degeneracy.
• Event Interpretation: The predicted spectra for PriNCe and H3a models peak in the hundreds of PeV range, overlapping with the reconstructed energy of the KM3-230213A event, suggesting a potential interpretation for this observation.

Significance
The paper establishes that reliable predictions of the UHECR-boosted CνB signal require a combined treatment of SM scattering channels, UHECR composition, source evolution, and the neutrino mass spectrum. By demonstrating that the Resonance channel is non-negligible and that the choice of UHECR flux model (specifically composition and maximum rigidity) drastically alters the predicted signal, the work provides a more robust theoretical foundation for future indirect searches. The derived constraints on CνB overdensity represent the most stringent limits to date for specific neutrino mass scenarios, offering a powerful indirect probe of the relic neutrino background that complements direct detection efforts.