The isotropic energy function and formation rate of short gamma-ray bursts

Gamma-ray bursts (GRBs) are brief, intense, gamma-ray flashes in the universe, lasting from a few milliseconds to a few thousand seconds. For short gamma-ray bursts (sGRBs) with duration less than 2 seconds, the isotropic energy (E-iso) function may be more scientifically meaningful and accurately measured than the luminosity (L-p) function. In this work we construct, for the first time, the isotropic energy function of sGRBs and estimate their formation rate. First, we derive the L-p - E-p correlation using 22 sGRBs with known redshifts and well-measured spectra and estimate the pseduo redshifts of 334 Fermi sGRBs. Then, we adopt the Lynden-Bell c(-) method to study isotropic energy functions and formation rate of sGRBs without any assumption. A strong evolution of isotropic energy E-iso proportional to (1+z)(5.79) is found, which is comparable to that between L-p and z. After removing effect of the cosmic evolution, the isotropic energy function can be reasonably fitted by a broken power law, which is phi( E-iso,E- 0) proportional to E-iso, 0(-0.45) for dim sGRBs and phi( E-iso,E- 0) proportional to E-iso, 0(-1.11) for bright sGRBs, with the break energy 4.92 x 10(49) erg. We obtain the local formation o,o rate of sGRBs is about 17.43 events Gpc(-3) yr(-1). If assuming a beaming angle is 6 degrees to 26 degrees, the local formation rate including off-axis sGRBs is estimated as rho(0, all) = 155.79 - 3202.35 events Gpc(-3) yr(-1).