A Short GRB "No-Host'' Problem? Investigating Large Progenitor Offsets for Short GRBs with Optical Afterglows

[abridged] We investigate the afterglow properties and large-scale environments of several short-duration gamma-ray bursts (GRBs) with sub-arcsecond optical afterglow positions but no bright coincident host galaxies. The purpose of this joint study is to robustly assess the possibility of significant offsets, a hallmark of the compact object binary merger model. Five such events exist in the current sample of 20 short bursts with optical afterglows, and we find that their optical, X-ray, and gamma-ray emission are systematically fainter. These differences may be due to lower circumburst densities (by about an order of magnitude), to higher redshifts (by dz~0.5-1), or to lower energies, although in the standard GRB model the smaller gamma-ray fluences cannot be explained by lower densities. To study the large-scale environments we use deep optical observations to place limits on underlying hosts and to determine probabilities of chance coincidence for galaxies near each burst. In 4 of the 5 cases the lowest probabilities of chance coincidence (P(<dR)~0.1) are associated with bright galaxies at separations of dR~10", while somewhat higher probabilities of chance coincidence are associated with faint galaxies at separations of ~2". By measuring redshifts for the brighter galaxies in three cases (z=0.111, 0.473, 0.403) we find physical offsets of ~30-75 kpc, while for the faint hosts the assumption of z>1 leads to offsets of ~15 kpc. Alternatively, the limits at the burst positions (>26 mag) can be explained by typical short GRB host galaxies (L~0.1-1 L*) at z>2. Thus, two possibilities exist: (i) ~1/4 of short GRBs explode ~50 kpc or ~15 kpc from the centers of z~0.3 or z>1 galaxies, respectively, and have fainter afterglows due to the resulting lower densities; or (ii) ~1/4 of short GRBs occur at z>2 and have fainter afterglows due to their higher redshifts.