The host galaxy of the short GRB 111117A at

It is notoriously difficult to localize short γ-ray bursts (sGRBs) and their hosts to measure their redshifts. These measurements, however, are critical for constraining the nature of sGRB progenitors, their redshift distribution, and the r-process element enrichment history of the universe. Here we present spectroscopy of the host galaxy of GRB 111117A and measure its redshift to be z = 2.211. This makes GRB 111117A the most distant high-confidence short duration GRB detected to date. Our spectroscopic redshift supersedes a lower, previously estimated photometric redshift value for this burst. We use the spectroscopic redshift, as well as new imaging data to constrain the nature of the host galaxy and the physical parameters of the GRB. The rest-frame X-ray derived hydrogen column density, for example, is the highest compared to a complete sample of sGRBs and seems to follow the evolution with redshift as traced by the hosts of long GRBs. From the detection of Lyα emission in the spectrum, we are able to constrain the escape fraction of Lyα in the host. The host lies in the brighter end of the expected sGRB host brightness distribution at z = 2.211, and is actively forming stars. Using the observed sGRB host luminosity distribution, we find that between 43% and 71% of all Swift-detected sGRBs have hosts that are too faint at z ~ 2 to allow for a secure redshift determination. This implies that the measured sGRB redshift distribution could be incomplete at high redshift. The high z of GRB 111117A is evidence against a lognormal delay-time model for sGRBs through the predicted redshift distribution of sGRBs, which is very sensitive to high-z sGRBs. From the age of the universe at the time of GRB explosion, an initial neutron star (NS) separation of a0 < 3.1 R⊙ is required in the case where the progenitor system is a circular pair of inspiralling NSs. This constraint excludes some of the longest sGRB formation channels for this burst