Short Gamma Ray Burst Formation Rate from BATSE data using E_p-L_p correlation and the minimum gravitational wave event rate of coalescing compact binary

Using 72 Short Gamma Ray Bursts (SGRBs) with well determined spectral data observed by BATSE, we determine their redshift and the luminosity by applying $E_p$–$L_p$ correlation for SGRBs found by \cite{tsutsui13}. For 53 SGRBs with the observed flux brighter than $4\times {10}^{-6}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, the cumulative redshift distribution up to $z=1$ agrees well with that of 22 {\it Swift}~SGRBs. This suggests that the redshift determination by the $E_p$–$L_p$ correlation for SGRBs works well. The minimum event rate at $z=0$ is estimated as $\rho_{SGRB}(0) = 6.3_{-3.9}^{+3.1} \times 10^{-10}~{\rm events~Mpc^{-3}yr^{-1}}$sothattheminimumbeamingangleis$0.6^\circ-7.8^\circ$assumingthemergingrateof$10^{-7}-4\times 10^{-6}~{\rm events~Mpc^{-3}yr^{-1}}$ suggested from the binary pulsar data. Interestingly, this angle is consistent with that for SGRB130603B of $\sim 4^\circ-8^\circ$\citep{fong13b}. On the other hand, if we assume the beaming angle of $\sim 6^{\circ }$ suggested from four SGRBs with the observed value of beaming angle, the minimum event rate including off-axis SGRBs is estimated as $\rho_{SGRB,all}^{min}(0)=1.15_{-0.71}^{+0.57}\times 10^{-7}~{\rm events~Mpc^{-3}yr^{-1}}$. If SGRBs are induced by coalescence of binary neutron stars (NSs) and/or black holes (BHs), this event rate leads to the minimum gravitational-wave detection rate of $\rm 3.9_{-2.4}^{+1.9} (152_{-94}^{+75})~events~y^{-1}$ for NS-NS (NS-BH) binary, respectively, by a worldwide network with KAGRA, advanced-LIGO, advanced-Virgo, and GEO.