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 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 > 1e52 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 7 short GRBs, some of which possessed temporally extended X-ray emission, on timescales of ~1-3 years following the initial burst. No radio sources were detected, with typical upper limits ~0.3 mJy at 1.4 GHz. A basic model for the synchrotron emission from the blast wave is used to constrain the presence of a long-lived NS merger remnant in each system. Depending on the GRB, our non-detections translate into upper limits on the CBM density n < 3e-2 - 3e-3 particles/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 CBM densities based on afterglow modeling or the lack of such constraints. Additional VLA observations in the near future could place tighter limits on the presence of merger remnants in these system. The population of long-lived NS merger remnants will also be constrained by their (non-)detection with upcoming radio transient surveys.