⚛️ general relativity

LQG-modified dispersion relations and the problem of cosmic photons threshold anomalies

This paper investigates how combining Lorentz invariance violation with loop quantum gravity-modified photon dispersion relations explains cosmic photon threshold anomalies, specifically addressing gamma-ray attenuation by background radiation to bridge LQG theory with ultra-high-energy astrophysical observations.

Original authors: P. A. L. Mourão, G. L. L. W. Levy, J. A. Helayël-Neto

Published 2026-07-24

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: P. A. L. Mourão, G. L. L. W. Levy, J. A. Helayël-Neto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean. For over a century, physicists have believed this ocean is perfectly smooth, like a sheet of glass. This idea, called "Lorentz invariance," is a cornerstone of modern physics. It tells us that the rules of how things move and interact don't change, no matter how fast you are going or which direction you face. It's the reason a baseball thrown on a speeding train behaves the same way as one thrown on a quiet field. But, just like a calm lake might hide ripples or currents beneath the surface, some scientists suspect that at the tiniest, most microscopic scales, this "glass" might actually be bumpy, granular, or even made of tiny, discrete chunks. This is the realm of "Loop Quantum Gravity," a theory suggesting that space itself isn't a smooth continuum but is woven from tiny loops.

Now, imagine shooting a super-fast photon (a particle of light) across this ocean. If the ocean is perfectly smooth, the photon travels in a straight line forever. But if the ocean is bumpy, the photon might bump into the "grain" of space, slowing down or changing its path. This paper asks a thrilling question: What if these tiny bumps in space are real? Specifically, the authors investigate how these potential bumps affect the most energetic photons in the universe. Normally, when a high-energy gamma-ray photon meets a low-energy photon from the background glow of the universe, they smash together to create an electron and a positron (a process called pair production). This acts like a cosmic speed limit or a foggy wall, preventing very high-energy light from traveling far. But what if the "bumpy space" changes the rules of the smash-up? This paper explores whether the graininess of space could make the universe transparent to light that should be blocked, potentially explaining some of the most mysterious high-energy signals we've recently detected.

The Cosmic Speed Bumps and the 1.42 PeV Mystery

In this paper, the authors, P. A. L. Mourão, G. L. L. W. Levy, and J. A. Helayël-Neto, take a ride through the theoretical landscape of Loop Quantum Gravity (LQG) to solve a puzzle about cosmic photons. They start with a specific idea: that space isn't smooth, but has a "granular" texture made of tiny loops. They use a mathematical tool called an "effective Hamiltonian" (think of it as a rulebook for how energy behaves in this bumpy space) to write down a new set of rules for how light travels. These new rules are called "Modified Dispersion Relations" (MDR).

Usually, light travels at a constant speed. But in this bumpy space, the authors suggest that the speed of a photon might depend on its energy, and the relationship between its energy and momentum gets a little twist. They found that this twist introduces a new factor, which they call $\xi_{LQG}$ . This factor is like a "bumpiness meter" for space. If space is perfectly smooth, this meter reads zero. If space is bumpy, it reads something else.

The authors then applied these new rules to a famous cosmic event: the Breit-Wheeler effect. This is the process where a high-energy gamma-ray photon crashes into a low-energy background photon (like the afterglow of the Big Bang, known as the Cosmic Microwave Background, or CMB) and turns into matter (an electron and a positron). In standard physics (Special Relativity), this crash is inevitable if the gamma-ray is too energetic. It creates a "fog" or an "opacity window" that stops ultra-high-energy photons from traveling across the universe. If you send a photon with too much energy, it gets eaten by the background light before it reaches Earth.

However, the authors discovered something fascinating when they plugged their "bumpy space" rules into this crash scenario. They derived a cubic threshold equation. Imagine this equation as a balance scale. On one side, you have the energy of the collision trying to create matter. On the other side, you have the "bumpiness" of space fighting back.

They found three possible outcomes based on how "bumpy" the space is (the value of $\xi$ ) compared to a critical tipping point ( $\xi_c$ ):

The Opacity Window: If the space is just right (but not too bumpy), the crash still happens, but only within a specific energy range. It's like a toll booth that only lets cars through if they are between a certain weight.
The Critical Threshold: At a precise point, the toll booth closes completely.
Anomalous Transparency: If the space is very bumpy (meaning $\xi$ is larger than the critical point), the crash is kinematically forbidden. The "bumpiness" of space acts like a force field that prevents the high-energy photon from turning into matter, even if it hits a background photon. The universe becomes "transparent" to these super-energetic particles.

The 1.42 PeV Smoking Gun

The authors then turned their attention to a real-world mystery. In 2021, a detector in China called LHAASO spotted a photon with an energy of 1.42 PeV (that's $1.42 \times 10^{15}$ electronvolts) coming from a star-forming region in our galaxy called the Cygnus Cocoon.

According to standard physics (Special Relativity), a photon with this much energy should have been stopped dead in its tracks by the Cosmic Microwave Background (CMB) long before it reached Earth. The "fog" should have been too thick. The fact that LHAASO saw it suggests that either the photon is a fluke, or our understanding of the "fog" is wrong.

The authors ran the numbers using their Loop Quantum Gravity model. They calculated the "bumpiness" factor ( $\xi_{LQG}$ ) predicted by their theory and compared it to the critical threshold needed to stop the photon.

For the Cosmic Microwave Background (CMB), their calculated $\xi_{LQG}$ is about $5.50 \times 10^{-27}$ eV $^{-1}$ .
The critical threshold for the CMB to block the photon is roughly $2.50 \times 10^{-33}$ eV $^{-1}$ .

Here is the kicker: The predicted "bumpiness" is about one million times larger ( $10^6$ ) than the critical threshold needed to stop the crash.

Because the "bumpiness" is so strong, the authors suggest that the Breit-Wheeler crash is effectively blocked. The photon doesn't turn into an electron-positron pair; it sails right through the cosmic fog. This explains the "anomalous transparency." The universe isn't opaque to these photons; it's clear as glass, allowing the 1.42 PeV photon to travel 1.4 kiloparsecs (about 4,500 light-years) and reach our detectors without being eaten.

What This Means (and What It Doesn't)

The paper suggests that the Loop Quantum Gravity model provides a consistent explanation for why we can see these record-breaking photons. It proposes that the "granularity" of space acts as a shield, preventing the high-energy photons from interacting with the background light.

However, the authors are careful to note that this is a theoretical explanation based on a specific model. They acknowledge a bit of tension with other experiments. For instance, observations of a different cosmic event (GRB 221009A) have placed a very strict limit on how much the speed of light can change with energy, suggesting the "bumpiness" might be smaller than what this specific LQG model predicts. The authors argue that this isn't a contradiction because different physical processes (like the speed of light vs. the ability to create matter) might be sensitive to different aspects of the theory.

In short, the paper doesn't claim to have proved that space is bumpy. Instead, it shows that if space is bumpy in the way Loop Quantum Gravity predicts, then the strange detection of the 1.42 PeV photon makes perfect sense. It opens the door to a universe where the "fog" of space is actually a "clearing" for the most energetic travelers, extending the "gamma-ray horizon" far beyond what we thought was possible. The authors hope that future, more sensitive telescopes will be able to test these ideas and perhaps finally catch a glimpse of the quantum grain of space itself.

Technical Summary: LQG-Modified Dispersion Relations and Cosmic Photon Threshold Anomalies

Problem Statement
The paper addresses the "threshold anomaly" problem in ultra-high-energy (UHE) astrophysics. According to Special Relativity (SR), the Universe should be opaque to photons with energies exceeding specific thresholds due to the Breit-Wheeler effect ( $\gamma\gamma \to e^+e^-$ ), where high-energy gamma rays interact with background photons from the Extragalactic Background Light (EBL) and the Cosmic Microwave Background (CMB). This interaction creates an "opacity window" that should prevent the free propagation of photons above $\sim 0.4$ PeV (for CMB interactions). However, recent observations by the Large High Altitude Air Shower Observatory (LHAASO) have detected photons with energies up to 1.42 PeV from the Cygnus Cocoon, challenging the SR-predicted opacity limits. The authors investigate whether Loop Quantum Gravity (LQG)-induced Lorentz Invariance Violation (LIV) can provide a consistent explanation for this anomalous transparency.

Methodology
The authors employ a semiclassical effective field theory approach within the electromagnetic sector of LQG.

Derivation of Modified Dispersion Relation (MDR): Starting from an effective Hamiltonian for the electromagnetic field extended by LQG parameters (specifically referencing the work of Alfaro et al.), the authors derive modified Maxwell-like equations. By applying plane wave solutions and linearizing the equations in the Planck scale ( $\ell_p$ ), they obtain a cubic MDR: $\omega^2 = k^2 + 2\theta_7 \ell_p k^3$ . This is simplified to $\omega^2 = k^2 - \xi k^3$ , where $\xi$ is a non-arbitrary LIV parameter fixed by the theory ( $\xi = -2\theta_7 \ell_p$ ).
Kinematic Analysis: The authors analyze the Breit-Wheeler pair production process ( $\gamma(\omega, k) + \gamma(\epsilon_b) \to e^+e^-$ ) using the derived MDR for the gamma photon while treating the background photons and the resulting electron-positron pair within standard Special Relativity.
Cubic Threshold Equation: By applying energy and momentum conservation laws, the authors deduce a cubic threshold equation for the gamma photon momentum $k$ : $\xi k^3 - 4\epsilon_b k + 4m_e^2 = 0$ .
Discriminant Analysis: The behavior of the roots of this cubic equation is analyzed via its discriminant ( $\Delta$ $Δ$ ). The sign of $\Delta$ $Δ$ determines the kinematic regime:
- $\Delta > 0$ : Two real positive roots, defining an "opacity window" where pair production is allowed.
- $\Delta = 0$ : A critical threshold where the window closes.
- $\Delta < 0$ : No real positive roots, implying "anomalous transparency" where pair production is kinematically forbidden.
Quantitative Comparison: The authors calculate the critical LIV parameter ( $\xi_c$ ) required to close the opacity window for the Cosmic Optical Background (COB), Cosmic Infrared Background (CIB), and CMB. These are compared against the theoretical LQG value ( $\xi_{LQG}$ ) derived from the effective Hamiltonian.

Key Contributions

Theoretical Derivation: The paper derives a specific, non-arbitrary MDR directly from the LQG effective Hamiltonian, rather than assuming a generic phenomenological form.
Critical Threshold Determination: It establishes a method to determine critical LIV thresholds ( $\xi_c$ ) for different cosmic backgrounds by setting the discriminant of the cubic threshold equation to zero.
Resolution of the 1.42 PeV Anomaly: The study quantitatively demonstrates that the LQG-derived LIV parameter ( $\xi_{LQG} \approx 5.50 \times 10^{-27} \text{ eV}^{-1}$ ) is significantly larger than the critical threshold for the CMB ( $\xi_{CMB}^c \approx 2.50 \times 10^{-33} \text{ eV}^{-1}$ ).
Gamma-Ray Horizon Extension: The analysis shows that for $\xi > \xi_c$ , the mean free path of UHE photons diverges ( $\lambda \to \infty$ ), effectively removing the SR-imposed opacity limit for PeV-scale photons.

Results

Opacity Window (COB): For the Cosmic Optical Background, the condition $\xi_{LQG} < \xi_{COB}^c$ holds. This results in a positive discriminant, creating an opacity window between approximately 230 GeV and 29 TeV. Within this range, pair production is kinematically allowed.
Anomalous Transparency (CIB and CMB): For the CIB and CMB, the condition $\xi_{LQG} > \xi_{CIB/CMB}^c$ holds. This results in a negative discriminant, meaning the cubic equation has no real positive roots. Consequently, the Breit-Wheeler process is kinematically forbidden for photons interacting with these backgrounds.
Survival Probability: Applying these results to the LHAASO J2032+4102 source (1.42 PeV photon), the paper calculates that while SR predicts a survival probability of $\approx 83\%$ (due to attenuation), the LQG model predicts a survival probability of $\approx 100\%$ because the kinematic cost of pair production exceeds the available energy.
Parameter Tension: The authors acknowledge a tension between their derived $\xi_{LQG}$ and constraints from GRB 221009A (which limit linear-order LIV based on time-of-flight). They argue this is not a contradiction because their model addresses pair-production thresholds (cubic momentum terms), whereas GRB constraints address propagation speed (linear terms), suggesting different physical processes are sensitive to different LIV parameters.

Significance
The paper claims to provide a robust, theory-driven explanation for the detection of 1.42 PeV photons by LHAASO, which would otherwise be suppressed by the CMB according to Special Relativity. By linking the LIV parameter directly to LQG theory rather than treating it as an arbitrary phenomenological constant, the authors argue that the observed "anomalous transparency" is a natural consequence of spacetime granularity at the Planck scale. The work suggests that the gamma-ray horizon is not limited to $\sim 0.4$ PeV but extends indefinitely for UHE photons, allowing for the detection of extragalactic sources at these energies. The authors conclude that their LQG framework offers a consistent interpretation of threshold anomalies and propose extending this methodology to other Standard Model particles, such as protons (to reinterpret the GZK limit) and neutrinos.

The Cosmic Speed Bumps and the 1.42 PeV Mystery

The 1.42 PeV Smoking Gun

What This Means (and What It Doesn't)

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