Supplementary light-curve fitting products (fits, residuals, corner plots and cumulative count correlations) for "X-ray Analysis of Gamma-Ray Burst Flares and Underlying Afterglows: Insights into the Origin of Flares"

This dataset provides the supplementary light-curve fitting products associated with the manuscripts:

“X-ray Analysis of Gamma-Ray Burst Flares and Underlying Afterglows: Insights into the Origin of Flares” (submitted to ApJ) and the article “Unraveling the Origins of Gamma-Ray Burst X-Ray Plateaus through a Study of X-Ray Flares” (published in ApJ).

It includes the best-fit X-ray light curves, residuals, posterior distributions, corner plots showing the posterior distributions and parameter covariances  and cumulative observed-simulated count comparison for all 89 GRBs analyzed in these studies. These products are generated using physically motivated synchrotron afterglow models with optional flares and jet-break components, and were used to perform the Bayesian model comparison described in the manuscripts.

This dataset allows readers to inspect the fits, residuals, and inferred parameter distributions for each GRB.

Description for Figures:

To illustrate the figure format used throughout the dataset, I will use the same example GRB 190719C as in Dereli-Bégué et al. (2025), see  Appendix Figure A1.

Figure 1: X-ray count-rate light curve of GRB 190719C obtained with the Swift-XRT instrument. The black crosses represent the XRT-WTSLEW, XRT-WT, and XRT-PC mode data, with $1\sigma$ uncertainties. The data are fitted in logarithmic space; therefore, for consistency, they are shown in logarithmic scale.

The data are overlaid with the best-fit model (blue), selected through Bayesian model comparison. In this example, a three-segment broken power-law (two break times) combined with two Norris functions is used to model the underlying afterglow and two flares. This corresponds to case C/F in a wind environment, including steep decay, plateau phase, and late-time afterglow, without a jet break (Model C4 in Table C1 of Dereli-Bégué et al. 2025 and Table A1 in Dereli-Bégué et al. 2025b).

Bottom panel: Residuals,  defined as the difference between the observed data and the best-fit model, divided by the data uncertainty (data - model) / \sigma, showing the agreement between the model and the data. In the majority of cases, the residuals remain consistent with noise, indicating a satisfactory agreement between the data and the model with deviations typically within a few standard deviations (|residual| \lesssim 2--3 \sigma). In this example, we see localized negative deviations in the residuals at the \sim 5\sigma level at early times. This may indicate the presence of additional unresolved variability (e.g., a possible third flare) that is not included in the model.

Figure 2: Corner plot showing the posterior probability distributions of the model parameters and their covariances, obtained from the Bayesian inference for GRB 190719C. The contours represent the 68% and 95% credible regions, illustrating parameter uncertainties and degeneracies.

Figures 3–4: Forward and reverse cumulative-count correlation diagnostics for GRB 190719C. Figure 3 shows the correlation between the observed and simulated cumulative counts obtained by integrating the count rate from early to late times, while Figure 4 shows the corresponding reverse cumulative counts obtained by integrating from late to early times. Each point represents the total accumulated counts associated with a given time bin in the light curve.

The dashed black line indicates the one-to-one relation corresponding to perfect agreement between the model and the data. The blue curve shows the median posterior predictive realization, while the shaded regions correspond to the 68% and 95% predictive intervals derived from the posterior distribution.

This diagnostic provides a global test of the model performance by assessing whether the total accumulated counts are consistently reproduced over the duration of the burst. Since cumulative counts are weighted toward the early, bright phases of the emission, they primarily probe the overall count budget rather than localized deviations. To complement this representation, we also provide a reverse cumulative-count comparison, obtained by integrating the count rate from late to early times, which is more sensitive to discrepancies at late times.

This representation is less sensitive to localized deviations (e.g., individual flares or rapid transitions), but provides a robust consistency check of the overall model behavior.

To further quantify the overall quality of the fit, we provide the normalized root mean square error (NRMSE), defined as NRMSE = RMSE / \mathrm{cum\_obs}_{\max}. This metric measures the average deviation of the model from the one-to-one relation, normalized by the total accumulated observed counts. 

In the example of GRB 190719C, an NRMSE of 1.04% for the cumulative-count comparison and a reverse NRMSE of 3.34% for the reverse cumulative-count comparison indicate that the model reproduces the cumulative count budget with high precision over the full duration of the observation. The slightly higher reverse NRMSE is expected because the reverse cumulative representation is more sensitive to discrepancies at late times.

As a practical guide, NRMSE values of a few percent indicate excellent agreement between the observed and simulated cumulative counts, while values above $\sim10%$ suggest noticeable discrepancies and values above $\sim20%$ warrant closer inspection. In our sample, only a small number of GRBs reach NRMSE values above $20%$, and these generally correspond to cases where localized structures or deviations are already apparent in the residual analysis. Thus, the cumulative-count diagnostics are largely consistent with the conclusions drawn from the residual inspection.

These products are provided for all 89 GRBs analyzed in the study, following a uniform fitting and analysis procedure. For each GRB, a set of three panels is presented: (1) light curve and residuals, (2) posterior parameter distributions (corner plot), and (3) cumulative observed vs. simulated counts for global model validation.