Comparison of the Origin of Short Gamma-Ray Bursts with or without Extended Emission

Abstract: The merger of compact binary stars produces short gamma-ray bursts (sGRBs), involving channels such as neutron star - neutron star (BNS) and neutron star - black hole (NS-BH). The association between sGRB 170817A and gravitational wave GW 170817 provides reliable evidence for the BNS channel. Some speculations suggest that sGRBs with extended emission (EE) may represent another distinct population. The offset is the distance between the GRB sky localization and the host galaxy center. We compared the offset distributions of these two types of samples (46 sGRBs with EE and 9 without EE samples) and found that they follow the same distribution. Utilizing non-parametric methods, we examined the luminosity function and formation rate of sGRBs without any assuming. The luminosity function can be described as $\psi(L_{0}) \propto L_{0}^{-0.12 \pm 0.01}$ for $L_{0} < L_0^b$ ($\psi(L_{0}) \propto L_{0}^{-0.73 \pm 0.02}$ for $L_{0} > L_0^b$) for sGRB without EE and $\psi(L_{0}) \propto L_{0}^{-0.13 \pm 0.003}$ for $L_{0} < L_0^b$ ($\psi(L_{0}) \propto L_{0}^{-0.61 \pm 0.01}$ for $L_{0} > L_0^b$) for sGRBs with EE. The formation rate is characterized as $\rho(z) \propto (1 + z)^{-3.04 \pm 0.10}$ for $z < 1$ and $\rho(z) \propto (1 + z)^{-0.29 \pm 0.38}$ for $1 < z < 3$ for sGRB without EE, while for sGRBs with EE, it is $\rho(z) \propto (1 + z)^{-3.85 \pm 0.15}$ for $z < 1$ and $\rho(z) \propto (1 + z)^{-0.40 \pm 1.11}$ for $1 < z < 3$. Our findings suggest no significant difference in the progenitors of sGRBs with and without EE when considered in terms of spatial offsets, formation rates, and luminosity function.