Recovering unmodelled physical emission components in Gaussian-process-marginalized gamma-ray burst afterglow inference

Aksulu, Wijers, van Eerten & van der Horst (2020) introduced a Bayesian framework for fitting gamma-ray burst afterglow light curves in which a forward-shock model (scalefit) forms the mean function and a Gaussian process (GP) absorbs the systematic deviations between model and data — instrumental calibration effects, propagation effects, and any physics missing from the model — with all hyperparameters marginalized by nested sampling. The authors acknowledged that known physical effects (reverse-shock emission, synchrotron self-Compton, late-time energy injection) are absent from the mean function and can be partially absorbed into the GP residual, and left their explicit recovery unresolved. Their own 2022 follow-up resolved most of the paper's other acknowledged limitations — extending to wind-like circumburst media, marginalizing the frequency length-scale hyperparameter, settling the $\xi_N$ degeneracy as fundamental to afterglow light curves, and withdrawing the preliminary common-energy-reservoir clustering on a 26-burst sample — but the mean function remains forward-shock-only and the GP is still designed to soak up unmodelled physical structure. The open problem is therefore the disentangling question: incorporate the missing physical components into the GP-marginalized framework and demonstrate — for example through injection–recovery tests on synthetic data and application to well-sampled afterglows such as GRB 970508 and GRB 210702A — that genuine physical emission is recovered explicitly rather than marginalized as nuisance systematics.

Background

Gamma-ray bursts (GRBs) are brief flashes of $\gamma$ -rays from ultra-relativistic, collimated cosmic explosions, followed by afterglow emission that fades over days to years as the ejecta drives a relativistic blast wave into the gas surrounding the progenitor. In the standard picture, electrons accelerated at the forward shock — the shock propagating into the external medium — radiate synchrotron emission (radiation from relativistic electrons spiralling in magnetic fields) across the whole spectrum. The afterglow flux as a function of observer time and frequency is predicted by a handful of parameters: the jet opening angle $\theta_j$ , the isotropic-equivalent kinetic energy $E_{K,\mathrm{iso}}$ , the circumburst density (number density $n_0$ for a uniform interstellar medium, or the mass-loss normalization of a stellar wind), the viewing angle, the power-law index $p$ of the accelerated-electron energy distribution, the fractions of post-shock energy carried by accelerated electrons and magnetic fields ( $\epsilon_e$ and $\epsilon_B$ ), and $\xi_N$ , the fraction of electrons that are accelerated. Numerical implementations of this model exist at two cost levels: boxfit solves radiative transfer on pre-computed two-dimensional relativistic hydrodynamic simulations, while scalefit reproduces boxfit's spectral breaks and peak fluxes through scaling relations, making it fast enough for repeated evaluation inside a sampling algorithm.

Real afterglow data deviate from any such model: interstellar scintillation (radio variability induced by propagation through ionized Galactic gas), dust extinction and absorption in the host galaxy, calibration offsets between instruments, and physical effects the model omits. Aksulu et al. (2020) therefore model the residual between data and the scalefit mean function as a Gaussian process — a probability distribution over functions, specified here by a kernel whose time and frequency length scales, $l_1$ and $l_2$ in log-space, set how correlated the residual is across a light curve — and marginalize the GP hyperparameters together with the physical parameters using nested sampling (a Monte Carlo algorithm for Bayesian evidences and posteriors, implemented via MultiNest). On synthetic data containing systematics, this GP-marginalized approach yields better-calibrated posterior distributions than plain $\chi^2$ fitting; applied to five well-observed long GRBs, it recovered explosion and microphysical parameters and a preliminary clustering of beaming-corrected kinetic energies (the isotropic-equivalent energy corrected for the jet's solid angle, $E_K \approx E_{K,\mathrm{iso}}(1-\cos\theta_j)$ ) near $1.7\times10^{51}\,\mathrm{erg}$ , with $\xi_N$ fixed to 0.1 because it is exactly degenerate with the other parameters: $(E_{K,\mathrm{iso}}, n_0)$ rescale as $1/\xi_N$ and $(\epsilon_B, \bar\epsilon_e)$ as $\xi_N$ , where $\bar\epsilon_e = \epsilon_e\,(p-2)/(p-1)$ .

The same authors' follow-up (Aksulu et al. 2022) applied the framework to 26 bursts, added wind-like circumburst media selected per burst by Bayesian evidence, marginalized both length-scale hyperparameters instead of fixing the frequency one, fixed $\xi_N$ to the canonical value of 1 while stating that it cannot be determined independently from afterglow light curves, and withdrew the preliminary common-energy-reservoir clustering ('this no longer holds'). Both papers nevertheless keep the mean function strictly forward-shock-only, and the 2022 paper states that the GP is intended to absorb 'physical effects that are shorter in time and frequency scale than is included in the model, such as radio scintillation, minor flares'. This is the tension behind this problem: the flexibility that makes the framework robust against systematics also lets it soak up genuine physics — for example, late-time energy injection reshapes a light-curve decay in a way a correlated residual can imitate — so recovering physical components explicitly requires more than appending them to the model. The need is concrete in current data: the millimetre rebrightening of GRB 210702A is attributed to energy injection or a late-time reverse shock (the shock that runs backward into the ejecta as the blast wave decelerates, rather than forward into the surrounding medium) but could only be probed with ad hoc closure-relation tests (predicted relations between spectral and temporal power-law indices) outside the GP framework (de Wet et al. 2024); Bayesian fits of GRB 241030A require synchrotron self-Compton (SSC — inverse-Compton scattering of synchrotron photons by the same electron population) contributions and still yield extreme parameters (Ducoin et al. 2026); a dedicated SSC code with exact Klein–Nishina cross-sections and pair production exists for the TeV-band afterglow of GRB 190114C, but outside any systematics-marginalizing framework (Derishev & Piran 2021); and an evolving magnetic-field strength has been inferred from multi-band spectral-break evolution, again outside the GP framework (Varela et al. 2025). Methodological support for marginalizing GP hyperparameters with nested sampling is available (Simpson et al. 2020), but no published work embeds these physical components in a GP-marginalized mean function or demonstrates their recovery.

Problem Statement

The problem, as left open by Aksulu et al. (2020, 2022): extend the GP-marginalized afterglow inference framework so that the physical components its mean function currently omits — reverse-shock emission, synchrotron self-Compton, and late-time energy injection — are modelled explicitly rather than partially absorbed into the Gaussian-process residual, and demonstrate that they are thereby recovered. A demonstration of recovery could take forms such as injection–recovery tests on synthetic afterglows built from the extended mean function plus realistic systematics (synthetic data generated with known parameter values, analysed by the full inference pipeline, and the recovered posteriors compared against those truth values), or application to real, well-sampled afterglows displaying such structure — for example the optical re-brightening of GRB 970508, attributed to late-time energy injection, or the millimetre rebrightening of GRB 210702A; the requirement is satisfied only when all three components have been added to the mean function and each is shown to be recovered as an explicit model component rather than absorbed into the Gaussian-process residual, the minimum evidence for this being a synthetic injection–recovery test that reports injected-versus-recovered posterior agreement for each added component on at least one run, with the real-data applications above serving as optional corroboration rather than a completion criterion. Constructing extended models without the recovery demonstration does not complete the problem.

References

M. D. Aksulu, R. A. M. J. Wijers, H. J. van Eerten, A. J. van der Horst (2020), 'A new approach to modelling gamma-ray burst afterglows: using Gaussian processes to account for the systematics', Monthly Notices of the Royal Astronomical Society 497, 4672-4683, DOI: 10.1093/mnras/staa2297, https://doi.org/10.1093/mnras/staa2297 (arXiv:2004.04166, https://arxiv.org/abs/2004.04166)
M. D. Aksulu, R. A. M. J. Wijers, H. J. van Eerten, A. J. van der Horst (2022), 'Exploring the GRB population: robust afterglow modelling', Monthly Notices of the Royal Astronomical Society 511, 2848-2867, DOI: 10.1093/mnras/stac246, https://doi.org/10.1093/mnras/stac246 (arXiv:2106.14921, https://arxiv.org/abs/2106.14921)
F. Simpson, V. Lalchand, C. E. Rasmussen (2020), 'Marginalised Gaussian Processes with Nested Sampling', arXiv:2010.16344, https://arxiv.org/abs/2010.16344
E. Derishev, T. Piran (2021), 'GRB Afterglow Parameters in the Era of TeV Observations: The Case of GRB 190114C', The Astrophysical Journal 923, 135, DOI: 10.3847/1538-4357/ac2dec, https://doi.org/10.3847/1538-4357/ac2dec (arXiv:2106.12035, https://arxiv.org/abs/2106.12035)
S. de Wet, T. Laskar, P. J. Groot, R. Barniol Duran, E. Berger, S. Bhandari, T. Eftekhari, C. Guidorzi, S. Kobayashi, D. A. Perley, R. Sari, G. Schroeder (2024), 'A Millimeter Rebrightening in GRB 210702A', The Astrophysical Journal 974, 279, DOI: 10.3847/1538-4357/ad77bb, https://doi.org/10.3847/1538-4357/ad77bb (arXiv:2408.14641, https://arxiv.org/abs/2408.14641)
K. Varela, J. Greiner, P. Schady, H. J. van Eerten (2025), 'The GROND gamma-ray burst sample - II. Fireball parameters for four gamma-ray burst afterglows', Astronomy & Astrophysics 701, A11, DOI: 10.1051/0004-6361/202554710, https://doi.org/10.1051/0004-6361/202554710 (arXiv:2507.09002, https://arxiv.org/abs/2507.09002)
J.-G. Ducoin, C. Pellouin, V. Aivazyan, D. Akl, F. Alvarez, C. Andrade, et al. (2026), 'GRB 241030A: A bright afterglow challenging forward shock emission', Astronomy & Astrophysics 711, A79, DOI: 10.1051/0004-6361/202558838, https://doi.org/10.1051/0004-6361/202558838 (arXiv:2603.18956, https://arxiv.org/abs/2603.18956)