# In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts

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## Abstract

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## 1. Introduction

## 2. Analysis

## 3. Closing Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Values of $|\Delta t|$ versus ${E}^{*}$ for the IceCube GRB-neutrino candidates discussed in Refs. [18,20] (black points) and for the GRB photons discussed in Refs. [20,23] (blue points). The photon points in figure also factor in the result of a one-parameter fit estimating the average magnitude of intrinsic time lags (details in Refs. [20,23]).

**Figure 2.**Normalized distribution of ${\eta}_{\gamma}^{\left[pair\right]}$ for all pairs of photons (from the same GRB) within our data set. For bins where the observed population is higher than expected we color the bar in purple up to the level expected, showing then the excess in red. For bins where the observed population is lower than expected the bar height gives the expected population, while the blue portion of the bar quantifies the amount by which the observed population is lower than expected.

**Figure 3.**As in Figure 1 blue points here are for the GRB photons discussed in Refs. [20,23] (with energy at emission greater than 40 GeV). Here black points give the ${E}_{pair}^{*}$ and the $\Delta {t}_{pair}$ for our pairs of GRB photons, including only cases in which both photons have energy at emission lower than 40 GeV and the associated value of ${\eta}_{\gamma}^{\left[pair\right]}$ is rather sharp (relative error of less than 30%) and between 10 and 100. The gray lines characterize the range of values of ${\eta}_{\gamma}$ favored by the blue points, which is also the region where black points are denser. The violet line is for ${\eta}_{\gamma}=34$ and intends to help the reader notice the similarity of statistical properties between the distribution of black and blue points, that goes perhaps even beyond the quantitative aspects exposed in our histograms.

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**MDPI and ACS Style**

Amelino-Camelia, G.; D’Amico, G.; Fiore, F.; Puccetti, S.; Ronco, M.
In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts. *Symmetry* **2021**, *13*, 541.
https://doi.org/10.3390/sym13040541

**AMA Style**

Amelino-Camelia G, D’Amico G, Fiore F, Puccetti S, Ronco M.
In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts. *Symmetry*. 2021; 13(4):541.
https://doi.org/10.3390/sym13040541

**Chicago/Turabian Style**

Amelino-Camelia, Giovanni, Giacomo D’Amico, Fabrizio Fiore, Simonetta Puccetti, and Michele Ronco.
2021. "In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts" *Symmetry* 13, no. 4: 541.
https://doi.org/10.3390/sym13040541