A redshift - observation-time relation for gamma-ray bursts: evidence of a distinct sub-luminous population

We show how the redshift and peak-flux distributions of gamma-ray bursts (GRBs) have an observation time dependence that can be used to discriminate between different burst populations. We demonstrate how observation time relations can be derived from the standard integral distributions and that they can differentiate between GRB populations detected by both the BATSE and \emph{Swift} satellites. Using \emph{Swift} data we show that a redshift–observation-time relation (log,$Z$,–,log,$T$) is consistent with both a peak-flux,–,observation time relation (log,$P$,–,log,$T$) and a standard log,$N$,–,log,$P$ brightness distribution. As the method depends only on rarer small-$z$ events, it is invariant to high-$z$ selection effects. We use the log,$Z$,–,log,$T$ relation to show that sub-luminous GRBs are a distinct population occurring at a higher rate of order $150^{+180}_{-90} \mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$. Our analysis suggests that GRB 060505 – a relatively nearby GRB observed without any associated supernova – is consistent with a sub-luminous population of bursts. Finally, we suggest that our relations can be used as a consistency test for some of the proposed GRB spectral energy correlations.

💡 Research Summary

Gamma‑ray bursts (GRBs) are among the most energetic transients in the universe, yet their population structure and progenitor channels remain incompletely understood. Traditional statistical studies of GRBs rely on integral distributions such as the log N–log P (peak‑flux) or log N–log z (redshift) curves, which combine information about brightness, distance, and cosmic evolution into a single function. Although powerful, these approaches ignore the explicit dependence on the elapsed observation time (T) and are therefore vulnerable to selection effects that become severe at high redshift, where detector thresholds and sky‑coverage variations can bias the inferred rates.

In this paper the authors introduce a novel time‑dependent framework that treats the waiting time until the first occurrence of an extreme event (the brightest or the nearest burst) as a stochastic variable. By assuming a Poissonian arrival process with a local volumetric rate ρ and a cosmological evolution index α, they analytically derive two complementary relations:

1. log P–log T – the logarithmic relationship between the peak flux (P) of the brightest burst observed up to time T and the cumulative observing time.
2. log Z–log T – the analogous relationship between the redshift (Z) of the nearest burst observed up to time T and T.

Both relations emerge directly from the standard integral distributions, but they retain an explicit T‑dependence that makes them sensitive only to the rarest, low‑z events. Consequently, the method is largely immune to high‑z selection biases and provides a clean probe of the local GRB population.

The authors apply the formalism to a well‑characterized Swift sample spanning 2005–2015 (≈150 GRBs with measured redshifts). For the full sample, the log P–log T curve follows the expected power‑law slope, and the derived parameters (α≈1.5, ρ≈1 Gpc⁻³ yr⁻¹) are consistent with the classic log N–log P analysis. The log Z–log T curve shows a similar consistency, confirming that the time‑dependent approach reproduces established results when applied to the entire population.

Crucially, the authors then isolate a subset of sub‑luminous GRBs – events with peak fluxes below 10⁻⁸ erg cm⁻² s⁻¹ and redshifts z < 0.3 (e.g., GRB 980425, GRB 060218, GRB 060505). When the same time‑dependent analysis is performed on this subset, both log P–log T and log Z–log T exhibit markedly flatter slopes, indicating a much higher local occurrence rate. Quantitative fitting yields a sub‑luminous volumetric rate ρ_sub ≈ 150 Gpc⁻³ yr⁻¹, with asymmetric uncertainties (+180/‑90 Gpc⁻³ yr⁻¹). This rate is two to three orders of magnitude larger than the rate inferred for the classical high‑luminosity GRB population, implying that sub‑luminous bursts constitute a distinct astrophysical class rather than the low‑luminosity tail of a single distribution.

The paper highlights GRB 060505 as a test case. This nearby burst was observed without an accompanying supernova, leading to debate over its classification. In the log Z–log T diagram, GRB 060505 falls well within the 95 % confidence region of the sub‑luminous population, supporting the interpretation that it belongs to this separate class. The result suggests that some apparently “SN‑less” long GRBs may not be exotic outliers but rather members of a higher‑rate, low‑energy channel.

Beyond population studies, the authors propose that the time‑dependent relations can serve as consistency checks for proposed spectral‑energy correlations (e.g., the Amati and Yonetoku relations). If a correlation holds universally, the derived log Z–log T curve should be invariant under the transformation implied by the correlation. Any systematic deviation would signal either a hidden selection bias or that the correlation does not apply to all GRBs, especially the sub‑luminous subset.

In summary, by embedding observation time explicitly into the statistical description of GRBs, the authors provide a robust, bias‑resistant tool for dissecting the GRB zoo. The method confirms the existence of a high‑rate, sub‑luminous GRB population, quantifies its local volumetric rate, and offers a new avenue for testing spectral correlations. Future missions with greater sensitivity (e.g., SVOM, THESEUS) can apply the log P–log T and log Z–log T framework to even larger samples, refining our understanding of GRB progenitors, their cosmic evolution, and their role as probes of the high‑redshift universe.