Discovery of a Universal Correlation for Long and Short GRBs and Its Application for the Study of Luminosity Function and Formation Rate

Gamma-ray bursts (GRBs) are known to be the most violent explosions in the universe, and a variety of correlations between observable GRB properties have been proposed in the literature, but none of these correlations are valid for both long and short GRBs. In this paper we report the discovery of a universal correlation that is suitable for both long and short GRBs using three prompt emission properties of GRBs, i.e., the isotropic peak luminosity L _iso, the peak energy of the time-integrated prompt emission spectrum E _peak, and the "high signal" timescale T _0.45, $L_{iso}\propto E_{peak}^{1.94}{T}_{0.45}^{0.37}$. This universal correlation involves properties of GRB prompt emission and does not require knowledge of the afterglow phase, which can be used as a relatively unbiased redshift estimator. Here we use this correlation to estimate the pseudoredshifts for short GRBs and then use the Lynden–Bell method to obtain a nonparametric estimate of their luminosity function and formation rate. The luminosity function is $\psi \left(L_0\right)\propto L_0^{-0.63\pm 0.07}$ for dim SGRBs and $\psi \left(L_0\right)\propto L_0^{-1.96\pm 0.28}$ for bright SGRBs, with the break point ${6.95}_{-0.76}^{+0.84}\times {10}^{50}erg{s}^{-1}$. The local formation rate of SGRBs is about 15 events Gpc⁻³ yr⁻¹ . This universal correlation may have important implications for GRB physics, implying that the long and short GRBs should share similar radiation processes.