The intrinsic luminosity-decay correlation in subsamples of GRB X-ray afterglows

Abstract: The intrinsic luminosity-decay correlation in gamma-ray burst (GRB) afterglows, between the early-time luminosity and the average rate of decay past this time, has previously been observed in the radio, optical/UV, X-ray and GeV wavebands and quantitatively shows that more luminous afterglows tend to have higher average rates of decay. We have compiled an updated sample of 427 X-ray afterglows with measured redshifts, observed with Swift/XRT over 20 years. For each GRB, we measure the luminosity at 200 seconds in the rest frame, $L_{\mathrm{X,200s}}$, and the average rate of decay from this time, $α<em>{X,&gt;200s}$. We find these parameters are correlated with a Spearman's rank coefficient ($R</em>\mathrm{sp}$) of $0.54\pm0.04$ at a significance of $\geq3σ$ and a linear regression slope of $0.18\pm0.02$. We separate our sample into subsamples, including 395 long GRBs (LGRBs) and 32 short GRBs (SGRBs) and find evidence of the $L_{\mathrm{X,200s}}$-$α<em>{X,&gt;200s}$ correlation at a significance of $\geq3σ$ in LGRBs but not in SGRBs, consistent with previous studies. In a subsample of 102 LGRBs with well-sampled light curves and late end times, we find that scatter in the correlation is significantly reduced and the strength increases to $R</em>\mathrm{sp}=0.80\pm0.04$ whilst the slope remains consistent with the full sample. We discuss our results and, briefly, their potential implications on constraining the cause of the correlation. Possible causes include geometric effects due to the angle between the observer and the jet-axis, or some mechanism that regulates the rate at which energy is released by the GRB central engine.

• The paper confirms the intrinsic luminosity-decay correlation in X-ray afterglows of GRBs, analyzed with a sample of 427 X-ray afterglows.
• The average decay index and the luminosity at 200 seconds post-burst are substantially correlated ($R_ ext{sp} = 0.80 ext{p} ext{m} 0.04$), especially in well-sampled light curves.
• The correlation is not found in low-luminosity LGRBs or SGRBs, but would need a larger sample size for a conclusive analysis in SGRBs.

The luminosity–decay correlation in gamma-ray burst (GRB) X-ray afterglows — whereby more luminous afterglows decay on average more rapidly — has previously been established in LGRBs across radio, optical/UV, X-ray and GeV wavebands. This paper by Shilling et al. compiles an updated sample of 427 X-ray afterglows with measured redshifts from Swift/XRT observations spanning December 2004 to November 2024, doubling the mission baseline of the previous X-ray study (Racusin et al., 2016). The analysis confirms the correlation at $\geq3\sigma$ significance in the full sample ($R_\mathrm{sp}=0.54\pm0.04$, regression slope $0.18\pm0.02$), demonstrates that its strength is strongly dependent on light-curve sampling quality, and finds no evidence for it in SGRBs or low-luminosity LGRBs. The paper also applies the correlation as a diagnostic tool for GRB classification questions, including whether radio-loud and radio-quiet LGRBs constitute distinct populations.

Sample construction and measurement methodology

The parent sample derives from Swift/BAT-triggered GRBs with XRT detections and literature redshifts (N=552). Light curves are converted from count rate to k-corrected rest-frame luminosity density at 1 keV using time-averaged photon indices and counts-to-flux conversion factors from the XRT repository, with times transformed via $t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$.

Two parameters are measured per burst: the luminosity at $t_\mathrm{rest}=200$ s ($L_{200}$) and the average decay index $\alpha_{\mathrm{avg}}$ fitted as a single power-law from 200 s onward. A central methodological choice is the systematic removal of prompt-emission contamination: flare intervals identified by the automated XRT pipeline are excluded from fits, and steep-decay segments are flagged using two criteria ($\alpha > 2.5$ at $t < 5000$ s, or a break transition of $\alpha_{n+1}-\alpha_n \leq -1$). In the 95 cases where a steep-decay segment extends past 200 s, $R_\mathrm{sp}=0.54\pm0.04$0 is recovered by extrapolating the post-steep power-law segment back to 200 s, and $R_\mathrm{sp}=0.54\pm0.04$1 is fit only from the end of the segment onward. After manual inspection of potentially misidentified flares (8 exclusions) and removal of bursts with large parameter uncertainties ($R_\mathrm{sp}=0.54\pm0.04$2 dex in $R_\mathrm{sp}=0.54\pm0.04$3 or $R_\mathrm{sp}=0.54\pm0.04$4 in $R_\mathrm{sp}=0.54\pm0.04$5), the final sample comprises N=427.

Correlation results across subsamples

Subsamples are constructed using prompt emission properties: long versus short classification via $R_\mathrm{sp}=0.54\pm0.04$6 (with literature-driven reassignments of SGRBs misclassified by extended emission); regular versus low-luminosity LGRBs via $R_\mathrm{sp}=0.54\pm0.04$7 erg; SGRBs with and without extended emission; and well-sampled regular LGRBs requiring $R_\mathrm{sp}=0.54\pm0.04$8 data points in $R_\mathrm{sp}=0.54\pm0.04$9 logarithmic time bins ("dex bins").

Subsample	N	$0.18\pm0.02$0	p-value	Slope
Full sample	427	$0.18\pm0.02$1	$0.18\pm0.02$2	$0.18\pm0.02$3
All LGRBs	395	$0.18\pm0.02$4	$0.18\pm0.02$5	$0.18\pm0.02$6
Regular LGRBs	382	$0.18\pm0.02$7	$0.18\pm0.02$8	$0.18\pm0.02$9
Low-luminosity LGRBs	13	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$0	0.43	—
Well-sampled regular LGRBs	102	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$1	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$2	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$3
All SGRBs	32	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$4	0.72	—
Regular SGRBs	24	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$5	$t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$6	—

The headline result concerns the well-sampled regular LGRB subsample: restricting to light curves with $t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$7 points in five or more dex bins raises the Spearman coefficient from 0.58 to $t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$8 while substantially reducing scatter, with the slope unchanged. The authors interpret this as evidence that all regular LGRBs intrinsically follow the correlation tightly, and that observed scatter arises primarily from incomplete temporal coverage causing $t_\mathrm{rest} = t_\mathrm{obs}/(1+z)$9 to be weighted toward individual morphological phases rather than representative of the full decay. A KS test on the $t_\mathrm{rest}=200$0 distributions between well-sampled and poorly sampled groups yields no brightness bias at $t_\mathrm{rest}=200$1, supporting this interpretation, although a subtle energetic bias towards higher $t_\mathrm{rest}=200$2 in the well-sampled group is present (KS statistic 0.21, p = $t_\mathrm{rest}=200$3).

Consistent with prior work (Racusin et al., 2016), no correlation is recovered in any SGRB subsample, nor in the low-luminosity LGRB subsample — though the latter contains only 13 events, insufficiently reliable for interpretation.

Instrumental biases and robustness checks

Several potential systematics are examined. Re-classifying long/short with the BATSE-motivated boundary $t_\mathrm{rest}=200$4 s instead of 2 s changes both $t_\mathrm{rest}=200$5 values by $t_\mathrm{rest}=200$6. Comparison of BAT-measured against GBM-measured $t_\mathrm{rest}=200$7 for 76 common LGRBs shows BAT systematically overestimates isotropic energy (owing to extrapolation without measured $t_\mathrm{rest}=200$8) and underestimates fluence; the resulting possible contamination of the regular LGRB sample is judged unlikely to affect results given the large subsample size. Redshift-binned analyses show the correlation weakens and becomes less robust at $t_\mathrm{rest}=200$9, which the authors attribute plausibly to Malmquist bias truncating the high-redshift luminosity distribution — though they cannot exclude physical evolution of the correlation with redshift. Morphology tests within the well-sampled group show similar correlation strength for canonical and non-canonical light curves, indicating morphology does not drive the strengthened correlation.

Origin of the correlation

Because the correlation holds consistently across wavebands spanning 15 orders of magnitude in frequency, any causal mechanism must be achromatic. Two candidates remain viable: geometric effects from viewing angle relative to the jet axis, and central-engine energy-regulation mechanisms such as time-varying microphysical parameters ($L_{200}$0, $L_{200}$1, $L_{200}$2). The authors note a discriminating prediction: viewing-angle effects operate independently of progenitor class, so if geometry causes the correlation it should appear in SGRBs as well as LGRBs, provided emission is jetted through comparable mechanisms. Its apparent absence in SGRBs would therefore disfavour the geometric explanation — but the current SGRB statistics are insufficient to draw this conclusion.

A Monte Carlo simulation assuming the correlation exists in regular SGRBs estimates that approximately nine additional regular SGRBs (at the current detection rate of roughly one per year, i.e. $L_{200}$3 further years of XRT operation) are required for a >90% probability of recovering the correlation at $L_{200}$4. The simulation exploits the systematically lower and narrower $L_{200}$5 distribution of SGRBs (mean $L_{200}$6 versus $L_{200}$7 for LGRBs), consistent with their lower-energy explosions in lower-density environments.

Applications

Two applications exploit the tightest correlation available, in well-sampled regular LGRBs. First, re-measuring $L_{200}$8 and $L_{200}$9 at later epochs shows the correlation remains significant up to 10 ks but drops below $\alpha_{\mathrm{avg}}$0 at 72 ks, establishing practical limits for including late-detected bursts (e.g. Einstein Probe or SVOM targets-of-opportunity) in future correlation analyses. Second, dividing bright ($\alpha_{\mathrm{avg}}$1 erg) well-sampled LGRBs into radio-loud (N=24) and radio-quiet (N=17) populations yields consistent correlation strengths ($\alpha_{\mathrm{avg}}$2 and $\alpha_{\mathrm{avg}}$3 respectively) and slopes within $\alpha_{\mathrm{avg}}$4. This argues against radio-loud and radio-quiet LGRBs being physically distinct classes, supporting the selection-effect interpretation of Osborne et al. over progenitor-difference claims.

Limitations and open questions

Several caveats bear directly on the conclusions. The SGRB non-detection cannot be distinguished from a true absence of the correlation given N=32 total events, and the simulation's conclusion depends on the assumption that SGRBs do follow the relation. The low-luminosity LGRB subsample is too small for meaningful inference, and the threshold itself rests on BAT-measured energies subject to systematic bandpass-dependent offsets. The reduced scatter in well-sampled LGRBs could not be fully separated from a possible confounding effect of observing strategy on morphology, though direct tests found morphology does not drive the result. Whether the weakened correlation at $\alpha_{\mathrm{avg}}$5 reflects Malmquist bias or genuine physical evolution remains unresolved. Finally, the underlying cause — viewing-angle geometry versus engine-regulation physics — remains undetermined; resolving it requires either a definitive SGRB correlation measurement or numerical afterglow simulations incorporating physically constrained time-varying microphysics.

Conclusion

Using a sample twice the size of previous X-ray work, this paper confirms the intrinsic luminosity–decay correlation in GRB X-ray afterglows at $\alpha_{\mathrm{avg}}$6 and demonstrates that careful exclusion of prompt-emission features combined with adequate temporal sampling recovers a substantially tighter relation ($\alpha_{\mathrm{avg}}$7 in well-sampled regular LGRBs). The correlation is absent from all SGRB subsamples at current significance levels, and the paper quantifies the additional observing time needed to test its existence there rigorously. Applications to late-time follow-up limits and the radio-loud/radio-quiet population question illustrate the correlation's utility as a diagnostic, while leaving the identification of its physical cause — jet geometry or central-engine regulation — as the principal open question.