Correlations between Event Rates of Short Gamma-Ray Bursts and Star Formation Rates with/without Time Delay

In this paper, we systematically investigate the redshift and luminosity distributions as well as the event rates of short Gamma-Ray Bursts (SGRBs) detected by Swift, Fermi, Konus-wind satellites. It is found that the distributions of redshift and luminosity of Fermi and Konus-wind SGRBs are identical and they obviously differ from those of Swift/BAT SGRBs. The luminosity distributions of SGRBs detected by diverse detectors can be uniformly fitted by a smoothly broken power-law function. The median luminosity of Swift SGRBs is about one order of magnitude smaller than that of Fermi/GBM or Konus-wind SGRBs. We also compare the local event rates of Swift/BAT, Fermi/GBM and Konus-wind SGRBs and find that the local rate of Swift SGRBs is around two orders of magnitude larger than that of either Fermi or Konus-wind SGRBs, while the latter two rates are comparable. The observed SGRB rates can be successfully fitted by a power-law plus Gauss function. The SGRB rates of three kinds of detectors matches the delayed/undelayed SFRs well except the delayed Lognormal and/or Gaussian SFRs at higher redshift and exceed all types of SFRs at lower redshift of $z<1$. After deducting the diverse SFR components from the SGRB rates, we surprisingly notice that the remaining SGRB rates steeply decline with redshift in a power-law-like form, indicating that these SGRBs could emerge from the old star populations or compact binary star mergers.