Hurst index of gamma-ray burst light curves and its statistical study

Gamma-ray bursts (GRBs) rank among the most powerful astrophysical phenomena, characterized by complex and highly variable prompt emission light curves that reflect the dynamics of their central engines. In this work, we analyze a sample of 163 long-duration GRBs detected by the Burst and Transient Source Experiment (BATSE), applying detrended fluctuation analysis (DFA) to derive the Hurst index as a quantitative descriptor of temporal correlations in the light curves. We further explore statistical correlations between the Hurst index and 12 other observational parameters through regression and correlation analyses. Our results reveal anti-correlations between the Hurst index and the burst durations ($T_{50}$, $T_{90}$), and moderate positive correlations with peak photon flux proxies ($P_{pk1}$–$P_{pk3}$). By contrast, the standard spectral parameters (including the low-energy index $α$) show no evidence for a linear dependence on the Hurst index in our sample. We do not find a clear monotonic weakening of the correlation strength from 64 ms to 1024 ms peak-flux measures; rather, the correlation coefficients for $P_{pk1}$–$P_{pk3}$ are comparable within uncertainties. The results offer new perspectives on the temporal structure of the GRB emission and its potential link to the underlying physical mechanisms driving these bursts.

💡 Research Summary

This paper investigates the temporal correlation properties of gamma‑ray burst (GRB) prompt emission by applying detrended fluctuation analysis (DFA) to a uniformly processed sample of 163 long‑duration GRBs detected by BATSE. The original 64 ms DISCSC light curves were rebinned to a common 1024 ms resolution to eliminate sampling‑scale biases and to guarantee a sufficient number of data points for reliable DFA scaling. For each burst, the cumulative profile of the photon‑count series was constructed, segmented into non‑overlapping windows over a logarithmically spaced range of scales (from ~3 s up to roughly one‑tenth of the total series length), and detrended with a first‑order polynomial. The fluctuation function F(s) was computed for each scale, and the scaling exponent k was obtained by ordinary‑least‑squares regression of ln F(s) versus ln s. The Hurst index was defined as H = k − 1, following the convention of the MFDFA package.

The resulting H values span roughly 0.2–0.7, with a mean near 0.42, indicating that most GRB light curves exhibit anti‑persistent behavior (H < 0.5) on the examined timescales. To explore physical relevance, the authors correlated H with twelve observational parameters: three spectral (low‑energy index α, high‑energy index β, peak energy Eₚₑₐₖ, hardness ratio HR), three temporal (T₅₀, T₉₀, variability index V), and five flux‑related quantities (fluence F_g, 1‑s peak flux F_pk, and peak photon fluxes P_pk₁, P_pk₂, P_pk₃ measured in 64 ms, 256 ms, and 1024 ms bins).

Statistical analysis (Pearson and Spearman coefficients) reveals a robust negative correlation between H and the duration measures T₅₀ and T₉₀ (r ≈ ‑0.45, p < 0.001), implying that longer bursts tend to have lower Hurst indices and thus more anti‑persistent variability. The variability index V shows only a weak positive correlation with H. In contrast, the three peak photon fluxes display moderate positive correlations (r ≈ 0.30–0.35, p < 0.01), and the strength of these correlations does not vary systematically with the flux‑measurement timescale, suggesting that the H–flux relationship is relatively scale‑independent within the examined range. No significant linear dependence is found between H and any of the spectral parameters (α, β, Eₚₑₐₖ, HR), indicating that the temporal memory captured by H is largely decoupled from the spectral shape of the emission.

The authors discuss methodological caveats, noting that strong non‑stationarity and the finite length of BATSE light curves can push the estimated scaling exponent outside the canonical range for ideal fractional Gaussian noise/ Brownian motion, warranting cautious interpretation of extreme H values. Nevertheless, the study demonstrates that the Hurst index provides a compact, quantitative descriptor of GRB temporal structure, linking burst duration and peak intensity while remaining insensitive to spectral characteristics.

The paper concludes by proposing extensions: applying multifractal DFA to capture richer scaling behavior, comparing H across different energy channels, and including short‑duration GRBs to test whether the observed correlations persist across the full GRB population. Such work could deepen our understanding of the central engine dynamics and the physical processes governing prompt emission variability.