Hierarchical Test of Lorentz Invariance with Gamma-Ray Burst Spectral-Lag Measurements
Gamma-ray bursts (GRBs) are among the most potent probes of Lorentz invariance violation (LIV), offering direct constraints on the quantum gravity energy scale ($E_{\rm QG}$) based on observations of energy-dependent time lags. Individual GRBs with well-defined positive-to-negative lag transitions have been used to set lower limits on $E_{\rm QG}$, but they suffer from uncertainties of spectral-lag measurements and systematics due to theoretical modeling of each burst. Here, we combine observations of 32 GRBs with positive-to-negative lag transitions to derive a statistically robust constraint on $E_{\rm QG}$ through hierarchical Bayesian inference. We find that the dominant systematic uncertainty in LIV constraints arises from the intrinsic lag modeling. Accounting for this uncertainty with cubic spline interpolation, we derive robust limits of $E_{\rm QG,1} \ge 4.37 \times 10^{16}$~GeV for linear LIV and $E_{\rm QG,2} \ge 3.02 \times 10^{8}$~GeV for quadratic LIV. We find that the probability for LIV, i.e., $E_{\rm QG,1}$ being below the Planck scale, is estimated to be around 90\%, which we conclude as no significant evidence for LIV signatures in current GRB spectral lag observations. Our hierarchical approach provides a rigorous statistical framework for future LIV searches and can be extended to incorporate multi-messenger observations.