High-Energy, Cosmic-Ray, and Multimessenger Constraints

High-energy photons, neutrinos, charged cosmic rays, and multimessenger transients combine enormous energies or baselines with severe source and propagation uncertainty. They can tightly constrain modified dispersion, forbidden decays, shifted thresholds, and interaction operators. The inference remains an operator-level exclusion unless a source-independent correlated signal survives composition, background, and calibration alternatives.

Required background. Quantum-Gravity Observables and Test Taxonomy fixes the measurement-to-claim chain, and Causality, Growth, and Analytic Domains supplies the conventional analytic baseline. Helpful background. Lorentz-Invariance and Modified-Dispersion Tests fixes coefficient conventions, while Massless Exchange and Infrared Subtractions explains why gravitational infrared structure complicates naive UV inference.

Why thresholds are sensitive

Consider a photon-sector dispersion

$$E_\gamma^2=p_\gamma^2 +\eta_n\frac{p_\gamma^{n+2}}{M^n},$$

with standard electrons and additive four-momentum. For head-on pair production $\gamma(E)+\gamma(\epsilon)\to e^+e^-$, the conventional threshold is $E\epsilon\simeq m_e^2$. With the stated convention, the leading correction has the schematic form

$$E\epsilon-\frac{\eta_n}{4} \frac{E^{n+2}}{M^n}\simeq m_e^2.$$

The factor and sign change if electron dispersion or composition laws change. The sensitivity scale follows from

$$\frac{E^{n+2}}{M^n}\sim m_e^2,$$

so a Planck-suppressed term can compete at energies far below $M$. Photon decay $\gamma\to e^+e^-$ or vacuum Čerenkov radiation $e\to e\gamma$ may become kinematically allowed. Observing a stable particle then excludes the relevant coefficient region—provided its energy, identity, and propagation history are secure.

First application: a population threshold analysis

Choose one coefficient basis for photons, electrons, and background photons. Compute the modified reaction rate, not only the threshold:

$$\tau(E,z)= \int_0^z dz'\, \frac{dl}{dz'} \int d\epsilon\,n(\epsilon,z') \int d\mu\, (1-\mu)\, \sigma_{\gamma\gamma}(s_{\mathrm{mod}}).$$

The observed flux from source $j$ is

$$F_j^{\mathrm{obs}}(E) =R_j(E;\nu_{\mathrm{det}}) F_j^{\mathrm{int}}(E;\nu_j) e^{-\tau(E,z_j;\eta_n,\nu_{\mathrm{EBL}})},$$

where $R_j$ denotes the detector response, $F_j^{\mathrm{int}}$ the intrinsic spectrum, and $\nu_{\mathrm{EBL}}$ the background-light model. A population analysis fits $\eta_n$ jointly with source spectra, redshifts, energy scale, and background density. A sharp intrinsic cutoff can otherwise imitate anomalous attenuation.

For cosmic rays, replace the photon background and reaction network with photopion or photodisintegration processes. Nuclear composition, hadronic interaction models, magnetic deflection, source evolution, and energy calibration dominate the interpretation. A spectral suppression alone does not say whether propagation or source acceleration caused it.

Multimessenger clocks and rulers

Photons, neutrinos, and gravitational waves from a common transient can compare propagation laws. The measured lag is

$$\Delta t_{ab} =\Delta t_{ab}^{\mathrm{emit}} +\Delta t_{ab}^{\mathrm{prop}} +\Delta t_{ab}^{\mathrm{det}}.$$

The second term is the target; the first is often much less certain. A credible likelihood uses an emission model or a conservative time window rather than assuming simultaneous production. Directional and temporal coincidence must also account for the number of searched sources and windows.

LHAASO’s high-energy light curve for GRB 221009A yielded stringent null constraints on linear and quadratic photon dispersion LHAASO Collaboration 2024. PeV photon observations also sharply constrain photon decay in specified Lorentz-violating models LHAASO Collaboration 2022. These results are powerful precisely because the long baseline and high energy amplify tiny coefficients. Their ceiling remains model-specific exclusion.

Adversarial control: source and composition mimicry

Generate a source population whose maximum acceleration energy evolves with luminosity and redshift. Propagate it with standard kinematics through uncertain backgrounds, then fit it with a fixed source model plus modified thresholds. If a nonzero coefficient appears, source misspecification can imitate the signal. Repeat the exercise over proton-rich, mixed, and heavy compositions and over independent hadronic interaction models.

For transient timing, inject an energy-dependent intrinsic lag correlated with luminosity. A genuine propagation term should follow the redshift integral and remain common across source classes after their intrinsic distributions are allowed to differ. For threshold effects, consistency should extend across photon survival, charged-particle stability, and polarization channels with the coefficient correlations predicted by the EFT.

Common pitfalls

Modifying one particle in isolation. Reaction thresholds depend on every species and on the conservation law. A photon-only bound cannot be quoted as model independent.

Using the highest-energy event without selection modeling. Energy migration, background probability, exposure, and trial factors can dominate a tail event. Population likelihoods and calibration replicas are safer than a threshold argument based on a single reconstructed value.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• LHAASO Collaboration. “Exploring Lorentz Invariance Violation from Ultrahigh-Energy $\gamma$ Rays Observed by LHAASO.” Physical Review Letters 128, 051102 (2022). DOI.
• LHAASO Collaboration. “Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A.” Physical Review Letters 133, 071501 (2024). DOI.
• Liberati, S. “Tests of Lorentz Invariance: A 2013 Update.” Classical and Quantum Gravity 30, 133001 (2013). DOI.