Modified Dispersion and Lorentz Tests

Energy-dependent propagation can accumulate over cosmological baselines, but a measured lag is interpretable only after identifying the operator, source emission history, polarization response, and detector calibration. Explicit Lorentz violation, spontaneous breaking, and deformed relativity have different conservation laws and radiative behavior; one time-delay formula cannot represent them all.

Required background. Equivalence Principles and Gravitational Measurement supplies the operational distinction between local propagation, clock calibration, and tidal or source effects.

Helpful background. Flat-Spacetime Quantum Field Theory supplies EFT operators and loop stability; Einstein Equations supplies cosmological propagation; Quantum Information supplies polarization and channel tests; Quantum Gravity supplies claim boundaries; and Gravitational Radiation and Waveforms supplies independent dispersion tests.

From an operator to a delay

In a specified cosmological rest frame, use a rotationally invariant phenomenological dispersion relation with integer $n\ge1$ and energy scale $M\gt0$,

$$E^2=p^2+m^2+\eta_n\frac{p^{n+2}}{M^n},$$

and retain only the leading correction when $p/M\ll1$ and $|\eta_n|(p/M)^n\ll1$. For a massless particle, expansion gives

$$v_g=\frac{\partial E}{\partial p} \approx1+\frac{n+1}{2}\eta_n\left(\frac{E}{M}\right)^n.$$

For photons observed with energy difference $\Delta E^n=E_h^n-E_l^n$ from redshift $z$, the leading propagation lag is

$$\Delta t_{\mathrm{prop}} \approx-\frac{n+1}{2}\frac{\eta_n\Delta E^n}{M^n} \int_0^z\frac{(1+z')^n\,dz'}{H(z')}.$$

The sign follows the declared dispersion convention. The emitted energies are larger by $1+z'$, which produces the numerator. An intrinsic source lag adds to this term and need not have the same sign.

In EFT, polarization dependence and operator dimension matter. CPT-odd photon operators can produce opposite phase velocities for the two helicities, rotating or depolarizing linear polarization. Polarimetry may then beat time-of-flight bounds. Loop corrections can generate lower-dimension Lorentz-violating operators unless a symmetry protects them; an isolated high-dimension coefficient is not automatically radiatively stable.

Pinned event data and calibration

The concrete benchmark is Fermi GRB 090510, using the published Gamma-ray Burst Monitor and Large Area Telescope event sample. The legacy P6_V3_TRANSIENT event selection and response are documented in the collaboration’s detailed burst analysis Ackermann et al. 2010, § 4. The key high-energy event is the reported 30.5 GeV photon, with uncertainty from energy reconstruction, associated with the short burst at redshift $z=0.903$. The collaboration used several conservative emission-time assumptions to set lower bounds on a linear suppression scale Fermi LAT and GBM Collaborations 2009, pp. 331–333 and table 1.

Selections must reproduce the published time windows, energy thresholds, region of interest, event class, zenith or quality cuts, and association probability. Calibration includes spacecraft clock, GBM and LAT relative timing, LAT effective area, point-spread function, energy dispersion, and the exact legacy response. Substituting a modern Pass 8 response creates a new analysis and requires redefining selections and simulations rather than claiming reproduction of the 2009 likelihood.

The result is a constraint, not a detection. A single highest-energy photon combined with a conservative lower bound on its emission time constrains only lags larger than that bound. It cannot identify a positive propagation delay.

Likelihood, source nuisance, and selections

A modern multi-burst likelihood may write observed arrival time as

$$t_{ij}=t^{\mathrm{src}}_{ij} +\tau_n K_n(z_j)E_{ij}^n +\delta t^{\text{detector}}_{ij},$$

where $K_n$ is the cosmological integral and $\tau_n$ is the propagation coefficient. Here $t^{\mathrm{src}}_{ij}$ is already in the observer frame: a source-frame emission interval contributes $(1+z_j)\Delta t^{\mathrm{em}}_{ij}$. Source pulse widths, spectral evolution, and intrinsic energy lags are nuisance parameters. Priors or hierarchical population distributions for them must be declared. Detector response should forward-fold true energies and times into reconstructed events.

Analysis component	Estimator or data choice	Dominant uncertainty or degeneracy
Single-photon bound	Conservative emission-time interval for the 30.5 GeV GRB 090510 event	Unknown emission time and association probability
Burst light-curve likelihood	Profile or posterior for $\tau_n$ with pulse-shape nuisance parameters	Spectral evolution, pulse overlap, and detector response
Multi-redshift population	Common propagation coefficient plus source-lag distribution	Source-class evolution and selection bias
Polarization test	Rotation or depolarization across an energy band	Intrinsic polarization angle and instrument calibration

The selection function favors bright bursts with measured redshift and high-energy photons. A population analysis must model that truncation; otherwise an apparent redshift trend can be selection bias. Trials over time windows, energy cuts, source subclasses, and both signs require correction.

Results and independent checks through 2026-08-20

This discussion reflects results available through 2026-08-20. Time-of-flight, polarization, cosmic-ray threshold, neutrino, and gravitational-wave observations strongly constrain selected Lorentz-violating coefficients, but no modified propagation is detected. The review by Addazi and collaborators compares the operator and source-systematic regimes Addazi et al. 2022, §§ 3–4.

Evidence channel	Statistical status	Scope and limitation
GRB 090510 timing	Conservative lower limits on selected linear and quadratic scales	One burst and uncertain intrinsic emission time
Multi-source photon timing	Null constraints from accumulated lags	Hierarchical source-lag models are essential
Astrophysical polarimetry	Strong null constraints on birefringent photon operators	Applies to polarization-dependent combinations, not every dispersion model
GW and multimessenger propagation	Constraints on tensor speed and dispersion	Maps to gravitational operators under waveform and emission assumptions

Reproduction status: the legacy Fermi likelihood was not rerun for this page. Reproduction requires the public GBM and LAT event files, spacecraft data, exact P6 response and calibration, published selections, redshift, cosmology, and code implementing each emission-time assumption. If the legacy response cannot be restored, quote the published constraint and label a Pass 8 reanalysis as new. Archive event lists, response versions, source models, simulation seeds, and profile-likelihood curves.

Systematic tests vary intrinsic-lag models, association cuts, energy scale, cosmology, event classes, and source populations. Independent birefringence and lower-dimension EFT bounds must be applied to the same operator coefficients. A result should be reviewed if a source model absorbs the redshift trend or if radiative mixing makes the chosen single coefficient inconsistent.

Exercise

For $n=1$, $\eta_1=-1$, negligible low energy, $E_h=30$ GeV, $M=10^{19}$ GeV, and an approximate propagation integral $K_1=5\times10^{17}$ s, estimate the delay.

Solution

For $n=1$,

$$\Delta t_{\mathrm{prop}} =-\eta_1\frac{E_h}{M}K_1 =\frac{30}{10^{19}}(5\times10^{17}\,\mathrm{s}) =1.5\,\mathrm{s}.$$

The chosen negative $\eta_1$ makes the high-energy photon subluminal in this convention and therefore late. A measured 1.5 s lag would not establish propagation because an intrinsic emission delay of the same size is degenerate. Multiple redshifts and source-model marginalization are needed.

References

• Ackermann, M., et al. (2010), “Fermi Observations of GRB 090510: A Short–Hard Gamma-Ray Burst with an Additional, Hard Power-Law Component from 10 keV to GeV Energies,” The Astrophysical Journal 716, 1178–1190, DOI: 10.1088/0004-637X/716/2/1178.
• Addazi, Andrea, et al. (2022), “Quantum Gravity Phenomenology at the Dawn of the Multi-Messenger Era—A Review,” Progress in Particle and Nuclear Physics 125, 103948, arXiv:2111.05659.
• Fermi LAT and GBM Collaborations (2009), “A Limit on the Variation of the Speed of Light Arising from Quantum Gravity Effects,” Nature 462, 331–334, DOI: 10.1038/nature08574.