The Formation Rate of Short Gamma-Ray Bursts and Gravitational Waves

In this paper, we study the luminosity function and formation rate of short gamma-ray bursts (sGRBs). First, we derive the $E_p-L_p$ correlation using 16 sGRBs with redshift measurements and determine the pseudo redshifts of 284 Fermi sGRBs. Then, we use the Lynden-Bell c- method to study the luminosity function and formation rate of sGRBs without any assumptions. A strong evolution of luminosity $L(z) \propto (1+z)^4.47$ is found. After removing this evolution, the luminosity function is $\Psi(L)\propto L_0^{-0.29 \pm 0.01}$ for dim sGRBs and $\psi (L)\propto L_0^{-1.07\pm 0.01}$ for bright sGRBs, with the break point $8.26× 10^{50}$erg s$^{-1}$. We also find that the formation rate decreases rapidly at $z< 1.0$, which is different from previous works. The local formation rate of sGRBs is 7.53 events Gpc$^{-3}$ yr$^{-1}$. Considering the beaming effect, the local formation rate of sGRBs including off-axis sGRBs is $203.31{-135.54}^{+1152.09}$ events Gpc$^{-3}$ yr$^{-1}$. We also estimate that the event rate of sGRBs detected by the advanced LIGO and Virgo is $0.85{-0.56}^{+4.82}$ events yr$^{-1}$ for an NS-NS binary.