Hierarchical Test of Lorentz Invariance with Gamma-Ray Burst Spectral-Lag Measurements

TL;DR

Gamma-ray bursts enable tests of Lorentz invariance violation (LIV) via energy-dependent spectral lags. The authors develop a hierarchical Bayesian framework to jointly analyze 32 GRBs with positive-to-negative lag transitions, marginalizing over intrinsic-lag uncertainties and comparing intrinsic-lag models. They derive robust lower limits on the LIV energy scales $E_{ m QG,1}$ and $E_{ m QG,2}$, finding no statistically significant LIV signal, with the dominant systematics arising from intrinsic-lag modeling. The framework provides a rigorous, extensible method for future LIV and multi-messenger tests, offering a clear path to incorporate additional GRB data and other astrophysical probes.

Abstract

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.

Figures (6)

• Figure 1: Schematic illustration of the hierarchical Bayesian framework for inferring the distribution of $E_{\rm QG}$ from $N$ GRBs. For each burst, posterior samples of $E_{\rm QG}$ are inferred from the single-event likelihood $L_0(d_i \mid E_{\rm QG})$ as adopting a uniform prior with large range. The population-level distribution of $E_{\rm QG}$ is modeled by a probabilistic model $P(E_{\rm QG} \mid \lambda)$ parametrized by $\lambda$. Hierarchical inference is performed using the total likelihood $L_{\rm tot}(\{d_i\} \mid \lambda)$, which marginalizes the single-event likelihoods over the posterior samples of $E_{\rm QG}$. This marginalization can be evaluated by importance-sampling the posteriors of $E_{\rm QG}$. The posterior population distribution of $E_{\rm QG}$ is finally calculated by marginalizing over the posterior samples of $\lambda$.
• Figure 2: One- and two-dimensional marginal posterior probability distributions of hyper-parameters. $A$ ($C$) and $B$ ($D$) are the parameters for Gaussian (log-normal) distribution model. Upper and lower panels show the results with analyzing the posterior samples of $\log_{10} E_{\rm QG,1}$ and $\log_{10} E_{\rm QG,2}$, respectively. In each panel, we compare the results inferred from Sample I and Sample II.
• Figure 3: Inferred posterior population distribution of quantum gravity energy scale. Top and bottom panels correspond to the results inferred from Sample I and Sample II, respectively. The mean PPDs are shown by solid and dashed lines, and the shaded areas present the $1\sigma$ (dark) and $2\sigma$ (light) credible intervals. The black dotted line denotes the Planck energy scale.
• Figure 4: Inferred marginal posterior distribution of linear quantum gravity energy scale from each GRB. Gray and green histograms correspond to Sample I and Sample II, respectively.
• Figure 5: Same as in Figure \ref{['fig:FigC']}, but for the quadratic quantum gravity energy scale.