Cosmological Evolution of the Formation Rate of Short Gamma-Ray Bursts with and without Extended Emission

Originating from neutron star-neutron star or neutron star-black hole mergers, short gamma-ray bursts (SGRBs) are the first electromagnetic emitters associated with gravitational waves (GWs). This association makes the determination of SGRB formation rate (FR) a critical issue. We determine the true SGRB FR and its relation to the cosmic star formation rate (SFR). This can help in determining the expected GW rate involving small mass mergers. We present nonparametric methods for the determination of the evolutions of the luminosity function (LF) and the FR using SGRBs observed by Swift, without any assumptions. These are powerful tools for small samples, such as our sample of 68 SGRBs. We combine SGRBs with and without extended emission (SEE), assuming that both descend from the same progenitor. To overcome the incompleteness introduced by redshift measurements we use the Kolmogorov-Smirnov (KS) test to find flux thresholds yielding a sample of sources with a redshift drawn from the parent sample including all sources. Using two subsamples of SGRBs with flux limits of 4.57 x 10(-7) and 2.15 x 10(-7) erg cm(-2) s(-1) with respective KS p = (1, 0.9), we find a 3 sigma evidence for luminosity evolution (LE), a broken power-law LF with significant steepening at L similar to 10(50) erg s(-1), and an FR evolution that decreases monotonically with redshift (independent of LE and the thresholds). Thus, SGRBs may have been more luminous in the past with an FR delayed relative to the SFR as expected in the merger scenario.