Microphysical origin of the cool-synchrotron gamma-ray burst prompt emission: which dissipation and acceleration mechanism yields the soft injection index, mixed cooling regimes, and required radiative efficiency?

Time-resolved spectral fits of single-peaked gamma-ray bursts (GRBs) with a physically computed synchrotron model show that about 95% of the time-resolved Fermi/GBM spectra of such single-peaked bursts can be described by synchrotron radiation from electrons that are often not fully cooled ('cool synchrotron'), with a soft electron injection index $p\approx3.5$ and a mix of cooling regimes (Burgess et al. 2020). This interpretation survives later consistency checks but leaves its cause unidentified: no known dissipation and particle-acceleration process in the moderately magnetized, ultra-relativistic outflow has been shown, analytically or in kinetic (particle-in-cell) simulations, to simultaneously produce the soft injection index, the observed incomplete cooling (effective reheating), and the radiative efficiency needed to power the observed luminosities without violating opacity and deceleration-radius constraints.

Background

Gamma-ray bursts (GRBs) are flashes of $10^{42}$–$10^{47}$ J of gamma radiation marking the collapse of massive stars or compact-object mergers; the flash itself, lasting milliseconds to minutes, is called the prompt emission. Its physical origin has been debated for fifty years. The observed prompt spectrum is usually described by the empirical Band function, a smoothly broken power law; its low-energy photon index $\alpha$ long argued against synchrotron emission — the radiation of relativistic electrons spiraling in magnetic fields — because elementary synchrotron theory forbids photon indices above the 'line of death' $\alpha=-2/3$, while many observed spectra appeared harder. Synchrotron models come in two cooling regimes: in fast cooling, electrons radiate away their energy almost immediately and the photon index below the spectral peak is fixed near $-3/2$; in slow or partial cooling, electrons retain energy, giving $\alpha=-(p-1)/2$, where $p$ is the index of the injected electron energy distribution $n_e(\gamma)\propto\gamma^{-p}$. Standard diffusive shock acceleration in relativistic shocks predicts $p\approx2.2$, and shocks are inefficient at dissipation, so they require complete (fast) cooling. The magnetization $\sigma$ (ratio of magnetic to particle energy flux) measures how magnetically dominated the outflow is; Burgess et al. (2020) reparameterize the problem in terms of the fractional magnetization $\xi_B = U_B/U_e$, the ratio of the magnetic energy density $U_B=B^2/(8\pi)$ to the energy density $U_e$ in the heated electrons, scanning $\xi_B$ from $10^{-2}$ to $1$ and finding the fits favor $\xi_B\sim1$, i.e. moderate magnetization; in strongly magnetized jets the dissipation must instead come from plasma processes such as magnetic reconnection or turbulence, possibly with internal mini-jets moving with Lorentz factor $\Gamma_{\rm em}$ relative to the flow. Particle-in-cell (PIC) simulations — kinetic plasma simulations resolving electron scales — are the standard tool for computing the electron index $p$ that a given acceleration process actually produces; second-order Fermi acceleration is stochastic reheating by plasma turbulence. Burgess et al. (2020, Nature Astronomy 4, 174; preprint arXiv:1810.06965) fitted proper (non-empirical), time-dependent synchrotron spectra to time-resolved Fermi Gamma-ray Burst Monitor (GBM) data of single-peaked GRBs, letting the electron distribution cool during the observation interval. They found synchrotron fits ~95% of time-resolved spectra, so the line-of-death argument had been misleading; but the electrons are injected with a very soft $p\approx3.5$ (versus the shock prediction 2.24), many spectra are not fast-cooling, and the implied energetics are challenging: reconciling the absence of a photospheric component with the pair-opacity constraint (transparency of the outflow to photon–photon pair production, which sets a lower limit on the bulk Lorentz factor) requires a magnetized jet with $\xi_B\sim1$ and mini-jet $\Gamma_{\rm em}>10$ emitting at large radii that still lie below the deceleration radius, where the jet begins to slow down by sweeping up the surrounding medium, and the high observed luminosities are hard to reach when electrons are not fully cooled. The model also fixed the injection Lorentz factor $\gamma_{\rm inj}=10^{5}$ (leaving all scales degenerate), assumed a single pulse and a single dissipation episode, and neglected particle escape, inverse-Compton and adiabatic cooling, geometry, and the evolution of the magnetic field $B$.

Problem Statement

Identify a physically plausible dissipation and particle-acceleration process operating in the moderately magnetized ($\xi_B\sim1$ up to high magnetization $\sigma$) ultra-relativistic outflow of gamma-ray bursts, and demonstrate quantitatively — through analytic theory or kinetic (PIC) simulations in the magnetization regime inferred for the emitting region — that it simultaneously reproduces what the cool-synchrotron spectral fits of Burgess et al. (2020) of time-resolved Fermi/GBM spectra of single-peaked bursts infer: (1) a soft electron injection index $p\approx3.5$, markedly softer than the canonical relativistic-shock value $p\approx2.2$; (2) the observed mix of cooling regimes, i.e. many time-resolved spectra showing incomplete cooling or effective reheating rather than universal fast cooling; and (3) radiatively efficient, energetically self-consistent emission — a combination of electron energy fraction $\varepsilon_e$, re-acceleration, emitting-region Lorentz factor $\Gamma_{\rm em}$, and magnetization $\xi_B$ that yields the observed luminosities without violating pair-opacity limits (transparency to photon–photon pair production, which bounds the bulk Lorentz factor from below) or the deceleration-radius constraint (the emission radius must remain below the radius at which the jet begins to decelerate as it sweeps up the surrounding medium).

References

1. Burgess, J. M., Bégué, D., Greiner, J., Giannios, D., Bacelj, A., Berlato, F. 'Gamma-ray bursts as cool synchrotron sources.' Nature Astronomy 4, 174–179 (2020). DOI 10.1038/s41550-019-0911-z, https://doi.org/10.1038/s41550-019-0911-z (preprint arXiv:1810.06965, https://arxiv.org/abs/1810.06965)
2. Wang, D.-Z., Zhao, X.-H., Zhang, Z., Zhang, B.-B., Peng, Z.-Y. 'A Comprehensive Consistency Check between Synchrotron Radiation and the Observed Gamma-Ray Burst Spectra.' The Astrophysical Journal 926, 178 (2022). DOI 10.3847/1538-4357/ac4782, https://doi.org/10.3847/1538-4357/ac4782
3. Yan, Z.-Y., Yang, J., Zhao, X.-H., Meng, Y.-Z., Zhang, B.-B. 'One Fits All: A Unified Synchrotron Model Explains GRBs with FRED-shape Pulses.' The Astrophysical Journal 962, 85 (2024). DOI 10.3847/1538-4357/ad14fb, https://doi.org/10.3847/1538-4357/ad14fb
4. Liu, K., Lin, D.-B., Li, J., Li, Y.-F., Lu, R.-J., Liang, E.-W. 'Electron Spectrum for the Prompt Emission of Gamma-Ray Bursts in the Synchrotron Radiation Scenario.' The Astrophysical Journal 911, 13 (2021). DOI 10.3847/1538-4357/abe76c, https://doi.org/10.3847/1538-4357/abe76c
5. Chen, J.-M., Zhu, K.-R., Peng, Z.-Y., Zheng, Y.-G., Gong, Y.-L., Chang, S., Tian, S.-T., Zhang, L. 'Fast-cooling Synchrotron in Decaying Magnetic Fields: Implications for the Gamma-Ray Burst Spectral Distribution.' The Astrophysical Journal 1003, 55 (2026). DOI 10.3847/1538-4357/ae633c, https://doi.org/10.3847/1538-4357/ae633c
6. Chen, J.-M., Zhu, K.-R., Chang, S., Peng, Z.-Y., Zheng, Y.-G., Zhang, L. 'Modeling Gamma-Ray Burst Spectra with Convolutional Neural Networks: Fast-cooling Synchrotron Emission in a Decaying Magnetic Field.' The Astrophysical Journal 1005, 76 (2026). DOI 10.3847/1538-4357/ae78c7, https://doi.org/10.3847/1538-4357/ae78c7
7. Comisso, L., Sironi, L. 'Pitch-Angle Anisotropy Controls Particle Acceleration and Cooling in Radiative Relativistic Plasma Turbulence.' Physical Review Letters 127, 255102 (2021). DOI 10.1103/PhysRevLett.127.255102, https://doi.org/10.1103/PhysRevLett.127.255102
8. Comisso, L., Jiang, B. 'Pitch-angle Anisotropy Imprinted by Relativistic Magnetic Reconnection.' The Astrophysical Journal 959, 137 (2023). DOI 10.3847/1538-4357/ad1241, https://doi.org/10.3847/1538-4357/ad1241 (preprint arXiv:2310.17560, https://arxiv.org/abs/2310.17560)
9. Macera, S., Banerjee, B., Mei, A., Tiwari, P., Oganesyan, G., Branchesi, M. 'Gamma-ray burst prompt emission spectra at high energies.' Astronomy & Astrophysics 700, A88 (2025). DOI 10.1051/0004-6361/202553801, https://doi.org/10.1051/0004-6361/202553801
10. Gompertz, B. P., Ravasio, M. E., Nicholl, M., Levan, A. J., Metzger, B. D., Oates, S. R., Lamb, G. P., Fong, W., Malesani, D. B., Rastinejad, J. C., Tanvir, N. R., Evans, P. A., Jonker, P. G., Page, K. L., Pe'er, A. 'The case for a minute-long merger-driven gamma-ray burst from fast-cooling synchrotron emission.' Nature Astronomy 7, 67–79 (2022). DOI 10.1038/s41550-022-01819-4, https://doi.org/10.1038/s41550-022-01819-4
11. Yang, J., Zhao, X.-H., Yan, Z., Wang, X. I., Zhang, Y.-Q., An, Z.-H., Cai, C., Li, X.-Q., Li, Z., Liu, J.-C., Liu, Z.-K., Ma, X., Meng, Y.-Z., Peng, W.-X., Qiao, R., Shao, L., Song, L.-M., Tan, W.-J., Wang, P., Wang, C.-W., Wen, X.-Y., Xiao, S., Xue, W.-C., Yang, Y.-H., Yin, Y.-H. I., Zhang, B., Zhang, F., Zhang, S., Zhang, S.-N., Zheng, C., Zheng, S.-J., Xiong, S.-L., Zhang, B.-B. 'Synchrotron Radiation Dominates the Extremely Bright GRB 221009A.' The Astrophysical Journal Letters 947, L11 (2023). DOI 10.3847/2041-8213/acc84b, https://doi.org/10.3847/2041-8213/acc84b
12. Tan, W.-J., Wang, C.-W., Xiong, S.-L., Zheng, S.-J., He, J., Zhao, X.-Y., Huang, Y., Yi, S.-X., Li, B., Gao, H., Wu, B.-B., Zhang, B., Daigne, F., Bernardini, M.-G., Zhang, B.-B., Basa, S., Cordier, B., Deng, J.-S., Dong, Y.-W., Dornic, D., Godet, O., Han, X.-H., Huang, M.-H., Lachaud, C., Li, H.-L., Maggi, P., Piron, F., Qiu, Y.-L., Schanne, S., Song, L.-M., Turpin, D., Vergani, S., Wang, J., Wang, P., Wei, J.-Y., Wu, C., Xin, L.-P., Xu, Y., Zhang, L., Zhang, S.-N. 'GRB 240715A: Revealing a Novel Intrinsic Mechanism by Different Individual Pulses.' The Astrophysical Journal Letters 995, L16 (2025). DOI 10.3847/2041-8213/ae2020, https://doi.org/10.3847/2041-8213/ae2020 (preprint arXiv:2511.18371, https://arxiv.org/abs/2511.18371)
13. Li, L., Ryde, F., Pe'er, A., Yu, H.-F., Acuner, Z. 'Bayesian Time-resolved Spectroscopy of Multipulse GRBs: Variations of Emission Properties among Pulses.' The Astrophysical Journal Supplement Series 254, 35 (2021). DOI 10.3847/1538-4365/abee2a, https://doi.org/10.3847/1538-4365/abee2a
14. Vereshchagin, G., Li, L., Bégué, D. 'Is magnetically dominated outflow required to explain GRBs?' Monthly Notices of the Royal Astronomical Society 512, 4846–4851 (2022). DOI 10.1093/mnras/stac757, https://doi.org/10.1093/mnras/stac757 (preprint arXiv:2201.05062, https://arxiv.org/abs/2201.05062)
15. Zhang, B., Wang, Y., Li, L. 'Dissecting the Energy Budget of a Gamma-Ray Burst Fireball.' The Astrophysical Journal Letters 909, L3 (2021). DOI 10.3847/2041-8213/abe6ab, https://doi.org/10.3847/2041-8213/abe6ab
16. Iyyani, S. 'Recent advances in the study of the prompt emission of gamma-ray bursts.' Journal of Astrophysics and Astronomy 43, 37 (2022). DOI 10.1007/s12036-022-09817-8, https://doi.org/10.1007/s12036-022-09817-8
17. Pe'er, A. 'Gamma-Ray Bursts: What Do We Know Today That We Did Not Know 10 Years Ago?' Galaxies 13(1), 2 (2024). DOI 10.3390/galaxies13010002, https://doi.org/10.3390/galaxies13010002
18. Gupta, S., Sahayanathan, S., Shanu, S., Nath, R. 'Are Single-zone Emission Models Sufficient to Explain GRB 220426A and GRB 230812B?' The Astrophysical Journal 1003, 197 (2026). DOI 10.3847/1538-4357/ae6809, https://doi.org/10.3847/1538-4357/ae6809
19. Bordoloi, P. P., Mittal, S., Iyyani, S. 'From Empirical to Physical Model: Direct Fits of Optically Thin Inverse Compton Scattering to Prompt GRB Spectra.' The Astrophysical Journal 1003, 58 (2026). DOI 10.3847/1538-4357/ae623d, https://doi.org/10.3847/1538-4357/ae623d
20. Florou, I., Petropoulou, M., Mastichiadis, A. 'A marginally fast-cooling proton–synchrotron model for prompt GRBs.' Monthly Notices of the Royal Astronomical Society 505, 1367–1381 (2021). DOI 10.1093/mnras/stab1285, https://doi.org/10.1093/mnras/stab1285