Abstract
We derive the luminosity function phi(L) and redshift distribution Psi(z) of short gamma-ray bursts (SGRBs) using all the available observer-frame constraints (i.e. peak flux, fluence, peak energy and duration distributions) of the large population of Fermi SGRBs and the rest-frame properties of a complete sample of SGRBs detected by Swift. We show that a steep phi(L) ∝ L^- alpha with alpha >= 2.0 is excluded if the full set of constraints is considered. We implement a Markov chain Monte Carlo method to derive the phi(L) and Psi(z) functions assuming intrinsic E_p-L_iso and E_p-E_iso correlations to hold or, alternatively, that the distributions of intrinsic peak energy, luminosity, and duration are independent. To make our results independent from assumptions on the progenitor (NS-NS binary mergers or other channels) and from uncertainties on the star formation history, we assume a parametric form for the redshift distribution of the population of SGRBs. We find that a relatively flat luminosity function with slope ~0.5 below a characteristic break luminosity ~3 x 10^52 erg s^-1 and a redshift distribution of SGRBs peaking at z ~ 1.5-2 satisfy all our constraints. These results also hold if no E_p-L_iso and E_p-E_iso correlations are assumed and they do not depend on the choice of the minimum luminosity of the SGRB population. We estimate, within ~200 Mpc (i.e. the design aLIGO range for the detection of gravitational waves produced by NS-NS merger events), that there should be 0.007-0.03 SGRBs yr^-1 detectable as gamma-ray events. Assuming current estimates of NS-NS merger rates and that all NS-NS mergers lead to a SGRB event, we derive a conservative estimate of the average opening angle of SGRBs ⟨ theta_jet ⟩ ~ 3deg.-6deg.. The luminosity function implies a prompt emission average luminosity ⟨L⟩ ~ 1.5 x 10^52 erg s^-1, higher by nearly two orders of magnitude than previous findings in the literature, which greatly enhances the chance of observing SGRB 'orphan' afterglows. Effort should go in the direction of finding and identifying such orphan afterglows as counterparts of GW events.