Galaxy clusters associated with short GRBs. I. The fields of GRBs 050709, 050724, 050911, and 051221a

We present a search for galaxy clusters in the fields of three bona fide short gamma-ray bursts ( 050709, 050724, and 051221a) and the putative short-burst GRB 050911, using multislit optical spectroscopy. These observations are part of a long-term program to constrain the progenitor age distribution based on the fraction of short GRBs in galaxy clusters and early-type galaxies. We find no evidence for cluster associations at the redshifts of the first three bursts, but we confirm the presence of the cluster EDCC 493 within the error circle of GRB 050911 and determine its redshift, z = 0.1646, and velocity dispersion, sigma approximate to 660 km s(-1). In addition, our analysis of Swift XRT observations of this burst reveals diffuse X-ray emission coincident with the optical cluster position, with luminosity L-X approximate to 4.9 x 10(42) ergs s(-1) and temperature kT approximate to 0.9 keV. The inferred mass of the cluster is 2.5 x 10(13) M-circle dot, and the probability of chance coincidence is about 0.1%-1%, indicating an association with GRB 050911 at the 2.6-3.2 sigma confidence level. A search for diffuse X-ray emission in coincidence with the 15 other short GRBs observed with XRT and Chandra reveals that, with the exception of the previously noted cluster ZwCl 1234.0+02916 likely associated with GRB 050509b, no additional associations are evident to a typical limit of 3 x 10(-14) ergs s(-1) cm(-2), or M less than or similar to 5 x 10(13) M-circle dot, assuming a typical z = 0.3. The estimated fraction of short GRBs hosted by galaxy clusters of about 5%-20% is in rough agreement with the fraction of stellar mass in clusters of sigma 10%-20%.