  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.
