Constraints on long-lived remnants of neutron star binary mergers from late-time radio observations of short duration gamma-ray bursts

The coalescence of a binary neutron star (NS) system (an 'NS merger' or NSM) may in some cases produce a massive NS remnant that is long lived and, potentially, indefinitely stable to gravitational collapse. Such a remnant has been proposed as an explanation for the late-X-ray emission observed following some short-duration gamma-ray bursts (GRBs) and as possible electromagnetic counterparts to the gravitational wave chirp. A stable NS merger remnant necessarily possesses a large rotational energy greater than or similar to 10(52) erg, the majority of which is ultimately deposited into the surrounding circumburst medium (CBM) at mildly relativistic velocities. We present Very Large Array radio observations of seven short GRBs, some of which possessed temporally extended X-ray emission, on time-scales of similar to 1-3 yr following the initial burst. No radio sources were detected, with typical upper limits similar to 0.3 mJy at nu = 1.4 GHz. A basic model for the synchrotron emission from the blast wave is used to constrain the presence of a long-lived NSM remnant in each system. Depending on the GRB, our non-detections translate into upper limits on the CBM density n less than or similar to 3 x 10(-2)-3 cm(-3) required for consistency with the remnant hypothesis. Our upper limits rule out a long-lived remnant in GRB 050724 and 060505, but cannot rule out such a remnant in other systems due to their lower inferred CMB densities based on afterglow modelling or the lack of such constraints.