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Constraints on Lorentz Invariance Violation from Gamma-ray Burst rest-frame spectral lags using Profile Likelihood

T0 review · 3 major / 8 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Analyzing 56 gamma-ray burst spectral lags with profile likelihood, the paper finds no Lorentz invariance violation and sets a 95% lower limit of 2.07 × 10^14 GeV on the linear LIV energy scale.

desk verdict Profile-likelihood reanalysis of W17's GRB spectral-lag data confirms the Bayesian limits, but the one-sided 95% calibration is off. read the letter →

arxiv 2502.00805 v2 pith:57XC4CGY submitted 2025-02-02 astro-ph.HE astro-ph.IM

Vyaas Ramakrishnan , Shantanu Desai This is my paper · ORCID
classification astro-ph.HEastro-ph.IM
keywords gamma-rayburstsspectrallagsLorentzinvarianceviolationprofilelikelihoodfrequentistinferencequantumgravityenergyscalerest-frameanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reanalyzes the rest-frame spectral lags of 56 gamma-ray bursts to test whether the speed of light depends on photon energy, as some quantum-gravity models predict. It uses profile likelihood to separate the LIV energy scale $E_{QG}$ from two nuisance parameters: a constant astrophysical time lag and an unknown intrinsic scatter. The fit never reaches a global minimum in $\chi^2$ below the Planck scale, so the data only give one-sided lower limits: $E_{QG} \geq 2.07 \times 10^{14}$ GeV for linear LIV and $E_{QG} \geq 3.71 \times 10^{5}$ GeV for quadratic LIV. These limits match the earlier Bayesian analysis of the same dataset, showing that the statistical framework does not change the conclusion. The result matters because it provides a seamless frequentist route to one-sided LIV bounds, complementing the usual Bayesian treatment.

What carries the argument

The central mechanism is the profile likelihood: for each fixed $E_{QG}$, a Gaussian likelihood in the rest-frame lag is maximized over the two nuisance parameters, the average intrinsic lag $\langle b \rangle$ and the intrinsic scatter $\sigma_{\rm int}$ added in quadrature to the measurement errors. The resulting $\chi^2(E_{QG})$ is compared with its value at the Planck scale, and, through Wilks' theorem, $\Delta\chi^2 = 4$ marks the one-sided 95% confidence boundary. The LIV delay itself comes from the standard cosmological LIV time-delay integral (Eq. 2), evaluated with the same cosmological parameters and fixed rest-frame energy bands as the earlier analysis; the minimization is performed with a downhill-simplex algorithm.

What would settle it

Measure spectral lags for a single GRB in multiple rest-frame energy bands and check whether the residuals around the best-fit constant-lag model correlate monotonically with energy; a coherent energy-dependent residual pattern, not attributable to measurement noise, would falsify the constant intrinsic-lag assumption and with it the $E_{QG}$ limits.

Watch

Extended reading notes

Core claim

The paper claims that a profile-likelihood reanalysis of the same 56-GRB rest-frame spectral-lag dataset confirms the earlier Bayesian lower bound on the quantum-gravity energy scale. Each burst's rest-frame lag is modeled as $\Delta t_{\rm obs}/(1+z) = a_{\rm LIV}K + \langle b \rangle$, where $a_{\rm LIV}K$ is the LIV delay computed from the redshift integral of Eq. (2) and $\langle b \rangle$ is a constant intrinsic lag. Profiling out $\langle b \rangle$ and the intrinsic scatter $\sigma_{\rm int}$ gives a $\chi^2(E_{QG})$ that decreases monotonically up to the Planck scale, so Wilks' theorem yields one-sided limits rather than a bounded interval: $E_{QG} \geq 2.07 \times 10^{14}$ GeV for linear ($n=1$) and $E_{QG} \geq 3.71 \times 10^{5}$ GeV for quadratic ($n=2$) LIV. The best-fit nuisance parameters agree with the Bayesian posteriors, and the reduced fit $\chi^2$ is close to one.

Load-bearing premise

The load-bearing premise is that each GRB's intrinsic astrophysical time lag is a single constant value that does not depend on photon energy, redshift, or burst-to-burst properties; if intrinsic lags vary with energy, the fitted LIV term could absorb or mimic the effect and bias the $E_{QG}$ limits.

Editorial extensions

If this is right

  • If these limits are right, GRB rest-frame spectral lags contain no evidence for linear LIV at the $2 \times 10^{14}$ GeV scale, in agreement with the earlier Bayesian result.
  • The much weaker quadratic limit, $E_{QG} \geq 3.71 \times 10^{5}$ GeV, shows that the same data are far less sensitive to energy-squared delays.
  • Because $\chi^2$ has no minimum below the Planck scale, better data will tighten the one-sided bound rather than produce a detection unless a genuine turnover in the lag-energy relation appears.
  • The profile-likelihood treatment offers a template for turning lower-bound-only problems in LIV searches into one-sided frequentist confidence intervals without prior choices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension implied but not pursued in the paper is a hierarchical model with a separate intrinsic lag per burst; if the inferred lag distribution is broad, the single-$\langle b \rangle$ assumption could be a larger systematic than the statistical uncertainty.
  • The appendix's single reshuffling of measurement uncertainties shifts the linear limit from $2.07$ to $3.1 \times 10^{14}$ GeV, so a full bootstrap over uncertainty reassignment would quantify how strongly the bound depends on the reported error bars.
  • With future GRB detectors that measure lags in more than two rest-frame energy bands per burst, the same profile-likelihood machinery could add an intrinsic lag spectral index as a nuisance parameter, directly testing the constant-lag assumption.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The paper reanalyzes the rest-frame spectral lags of 56 Swift-BAT GRBs from Bernardini et al. (2015) to constrain the Lorentz-invariance-violating energy scale E_QG, using the same model and data as Wei & Wu (2017) but with frequentist profile likelihood instead of Bayesian inference. The model treats the intrinsic astrophysical lag as a constant nuisance parameter <b> and adds an intrinsic scatter sigma_int, both profiled over for each fixed E_QG. The authors scan E_QG on a logarithmic grid up to the Planck scale, find no interior minimum of the profile chi-square, and quote one-sided 95% lower limits E_QG >= 2.07e14 GeV (linear) and E_QG >= 3.71e5 GeV (quadratic), close to the W17 Bayesian result. An appendix reports a single shuffled-uncertainty realization with both frequentist and Bayesian limits.

Significance. If the statistical calibration is corrected, the paper provides a useful frequentist cross-check of an existing Bayesian LIV analysis, with the advantage of transparent handling of nuisance parameters and openly available code. The numerical agreement with W17 supports the robustness of the earlier conclusion under a different inference framework. However, the novelty is modest: the data, likelihood, and astrophysical-lag model are inherited from W17, and the central quantitative claim depends on a confidence-level calibration that is currently not statistically self-consistent. The paper is therefore of interest to the LIV and GRB spectral-lag community, but its main numerical limits need revision before the stated confidence levels can be taken at face value.

major comments (3)
  1. [Sec. 3, Eq. (5)] The paper sets one-sided 95.4% lower limits by finding the EQG at which Delta-chi^2 = 4, but for a one-parameter profile likelihood a 95% one-sided lower limit corresponds to Delta-chi^2 = 2.71, not 4; Delta-chi^2 = 4 is a two-sided 95.4% interval (equivalently a one-sided limit at about 97.7%). Moreover, because chi^2_min is attained at the upper grid boundary E_pl rather than at an interior point, the usual regularity conditions for Wilks' theorem do not hold; under the standard boundary mixture (Chernoff) the one-sided 95% threshold is again 2.71. The quoted numbers are therefore not exactly 95% profile-likelihood limits, and the stated confidence level is mislabeled. Please recalibrate the limits (for example using Delta-chi^2 = 2.71 for one-sided 95%) or report the correct confidence level, and ideally validate the coverage with Monte Carlo simulations of the null model.
  2. [Sec. 3, Eq. (5)] This is a load-bearing point because the entire limit-setting procedure depends on the reference chi^2_min.
  3. [Sec. 2, Eq. (1)] This is the main astrophysical assumption and the weakest point of the analysis.
minor comments (8)
  1. [Table 1] The text defines DOF as N minus the number of free parameters (three), which for 56 GRBs gives 53, but Table 1 reports DOF = 54. Please correct this inconsistency.
  2. [Appendix, Figs. 5-6] The Bayesian lower limits quoted in the appendix text (E_QG >= 3.1e14 GeV for linear and >= 1.8e5 GeV for quadratic) disagree with the figure captions (3.66e13 GeV and 2.01e5 GeV, respectively). Please reconcile the text and the captions.
  3. [Appendix] The shuffled-uncertainty robustness test uses only one realization, which the authors acknowledge; the paper should state explicitly that this is illustrative and carries no statistical significance. The quadratic shuffled case is also unclear: if Delta-chi^2 never reaches 4 after its minimum, it is not obvious how a 95% lower limit is derived from the stated procedure.
  4. [Sec. 3] The PDG cosmological parameters are given as 'H0 = 67.4 km/sec and Omega_m = 0.315'; the units for H0 should be km/s/Mpc.
  5. [Appendix] The uniform priors are stated as U(1,19), U(-1,1), and U(0,1) on log(E_QG), b, and sigma_int; please specify that the first is uniform on log10(E_QG/GeV), not on E_QG itself.
  6. [Sec. 4] The paper says the analysis codes are available on Github but does not provide the repository URL; please include it.
  7. [Eq. (5) and Table 1] The same symbol chi^2 is used for -2 ln L in Eq. (5) and for the goodness-of-fit chi^2_fit in Eq. (6); the notation should be distinguished more clearly throughout.
  8. [Abstract and Sec. 3] The abstract says 95% c.l. while Sec. 3 says 95.4% c.l. and then abbreviates it to 95%; please standardize the terminology and, more importantly, use the correct one-sided threshold.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the target E_QG is scanned, not fitted, and the limits are one-sided constraints from an external dataset and model.

full rationale

The paper's derivation chain is not circular. The observable model (Eq. 1) and data (Bernardini et al. 2015) are inherited from W17, but inheritance of data and model is not circularity because the parameter of interest E_QG is never defined in terms of the output limits and is not fitted to the data: the profile likelihood scans E_QG on a fixed grid (Sec. 3) while profiling the nuisance parameters (<b>, sigma_int). The quoted limits are obtained as the x-intercept of Delta-chi^2 = 4, not by plugging a fitted E_QG back into the model, so none of the enumerated circularity patterns applies. The only self-citation is Ref. [1] (Desai 2024), a background review cited for the statement that spectral lags are widely used as a LIV probe; it is not load-bearing for the likelihood, the profile construction, or the limit. The appendix's caveat that the shuffled-uncertainty check uses only one realization is a robustness limitation, not a circular step. The skeptical concern that a one-sided 95% profile-likelihood bound should use Delta-chi^2 = 2.71 rather than 4, and that chi^2_min is evaluated at the Planck-grid endpoint rather than at a true interior minimum, is a statistical calibration or coverage issue, not a self-referential reduction: it does not show that any fitted quantity is being relabeled as a prediction or that the result is equivalent to its inputs by construction. The analysis is self-contained against an external dataset and an external Bayesian comparison, so the circularity score is low.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The analysis rests on the W17 model and dataset, with nuisance parameters <b> and σ_int fitted by profile likelihood. The main unstated assumptions are the constancy of the intrinsic lag and the applicability of Wilks' theorem at the boundary of the E_QG grid.

free parameters (2)
  • <b> (constant intrinsic lag) = -0.035 (linear), -0.011 (quadratic)
    Fitted nuisance parameter representing the average astrophysical time lag in the rest frame; values from Table 1.
  • σ_int (intrinsic scatter) = 0.023 (both models)
    Fitted nuisance parameter added in quadrature to observed lag uncertainties; value from Table 1.
assumptions (4)
  • standard math Wilks' theorem applies to the profile likelihood ratio for E_QG despite the global minimum lying at the Planck-scale boundary of the grid.
    Invoked in Sec. 2 after Eq. (5); the theorem assumes interior MLE and regularity, which is not checked at the boundary.
  • domain assumption The astrophysical contribution to the rest-frame spectral lag is a single constant <b> for all 56 GRBs.
    Stated in Sec. 2, Eq. (1), following W17; energy-dependent intrinsic lags would bias the LIV term.
  • domain assumption Observed spectral lag uncertainties are Gaussian and independent, with intrinsic scatter added in quadrature.
    Used in the likelihood Eq. (3); asymmetric errors from Bernardini et al. are averaged into symmetric values in Sec. 3.
  • domain assumption Cosmological parameters H0=67.8 km/s/Mpc and Ωm=0.308 are fixed as in W17.
    Used in Eq. (2); the authors check PDG values and find negligible changes in the limits.

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Cite this review

Pith. "Pith review of Constraints on Lorentz Invariance Violation from Gamma-ray Burst rest-frame spectral lags using Profile Likelihood." pith.science (2026). https://pith.science/paper/57XC4CGY

@misc{pith2026250200805,
  author       = {Pith},
  title        = {Pith review of: Constraints on Lorentz Invariance Violation from Gamma-ray Burst rest-frame spectral lags using Profile Likelihood},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/57XC4CGY}},
  note         = {Machine review of arXiv:2502.00805}
}
abstract

We reanalyze the spectral lag data for 56 Gamma-Ray Bursts (GRBs) in the cosmological rest frame to search for Lorentz Invariance Violation (LIV) using frequentist inference. For this purpose, we use the technique of profile likelihood to deal with the nuisance parameters, corresponding to a constant time lag in the GRB rest frame and an unknown intrinsic scatter, while the parameter of interest is the energy scale for LIV ($E_{QG}$). With this method, we do not obtain a global minimum for $\chi^2$ as a function of $E_{QG}$ up to the Planck scale. Thus, we can obtain one-sided lower limits on $E_{QG}$ in a seamless manner. Therefore, the 95\% c.l. lower limits which we thus obtain on $E_{QG}$ are then given by: $E_{QG}\geq 2.07 \times 10^{14}$ GeV and $E_{QG}\geq 3.71\times 10^{5}$ GeV, for linear and quadratic LIV, respectively.

Figures

Figures reproduced from arXiv: 2502.00805 by the authors.

Figure 1
Figure 1. ∆χ 2 , defined as (χ 2 − χ 2 Epl ), plotted against EQG for a linearly dependent LIV, corresponding to n = 1 , in Eq. 2. The horizontal magenta dashed line represents ∆χ 2 = 4, and the vertical magenta dashed line provides us the x-intercept, the 95% confidence level lower limit for EQG = 2.07 × 1014 GeV [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. ∆χ 2 , defined as (χ 2 − χ 2 Epl ), plotted against EQG for a quadratically dependent LIV, cor￾responding to n = 2, in Eq. 2. The horizontal magenta dashed line represents ∆χ 2 = 4, and the vertical magenta dashed line provides us the x-intercept, the 95% confidence level lower limit for EQG = 3.71 × 105 GeV. 4. Conclusions In this work, we have reanalyzed the data for spectral lags of 56 GRBs between two fixed ener… view at source ↗
Figure 3
Figure 3. ∆χ 2 , defined as (χ 2 − χ 2 Epl ), plotted against EQG for a linearly dependent LIV, corresponding to n = 1 , in Eq. (2), after shuffling the uncertainties in the spectral lags among the GRBs. The horizontal magenta dashed line represents ∆χ 2 = 4, and the vertical magenta dashed line provides us the x-intercept, the 95% confidence level lower limit for EQG = 3.1 × 1014 GeV. 10 1 10 4 10 7 10 10 10 13 10 16 10 19 E… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: ∆χ 2 , defined as (χ 2 − χ 2 Epl ), plotted against EQG for a quadratically dependent LIV, cor￾responding to n = 2, in Eq. (2), after shuffling the uncertainties in the spectral lags among the GRBs. The horizontal magenta dashed line represents ∆χ 2 = 4, and the vertic…
Figure 5
Figure 5. Figure 5: The marginalized contours for EQG, b and σint at 68% and 95% credible intervals for linear model of LIV, corresponding to n = 1, in Eq. (2). The corresponding 95% lower limit for EQG is given by EQG = 3.66 × 1013 GeV. 0.09 0.12 0.15 0.18 0.21 int 0.24 0.16 0.08 0.00 0.…
Figure 6
Figure 6. Figure 6: The marginalized contours for EQG, b and σint at 68% and 95% credible intervals for quadratic model of LIV, corresponding to n = 2, in Eq. (2) The corresponding 95% lower limit for EQG is given by EQG = 2.01 × 105 GeV [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Forward citations

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