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). Firstly, 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} $fordimsGRBsand$ \psi (L) \propto L_0 ^ {- 1.07 \pm 0.01} $ for bright sGRBs, with the break point $8.26\times {10}^{50}$ erg s${}^{-1}$. We also find that the formation rate decreases rapidly at $z<1.0$, which is different with 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 NS-NS binary.