The merger of a neutron star binary may result in the formation of a rapidly-spinning magnetar .
The magnetar can potentially survive for seconds or longer as a supramassive neutron star before collapsing to a black hole if , indeed , it collapses at all .
During this process , a fraction of the magnetar ’ s rotational energy of \sim 10 ^ { 53 } erg is transferred via magnetic spin-down to the surrounding ejecta .
The resulting interaction between the ejecta and the surrounding circumburst medium powers a \gtrsim year-long synchrotron radio transient .
We present a search for radio emission with the Very Large Array following nine short-duration gamma-ray bursts ( GRBs ) at rest-frame times of \approx 1.3 - 7.6 years after the bursts , focusing on those events which exhibit early-time excess X-ray emission that may signify the presence of magnetars .
We place upper limits of \lesssim 18 - 32 \mu Jy on the 6.0 GHz radio emission , corresponding to spectral luminosities of \lesssim ( 0.05 - 8.3 ) \times 10 ^ { 39 } erg s ^ { -1 } .
Comparing these limits to the predicted radio emission from a long-lived remnant and incorporating measurements of the circumburst densities from broad-band modeling of short GRB afterglows , we rule out a stable magnetar with an energy of 10 ^ { 53 } erg for half of the events in our sample .
A supramassive remnant that injects a lower rotational energy of 10 ^ { 52 } erg is ruled out for a single event , GRB 050724A .
This study represents the deepest and most extensive search for long-term radio emission following short GRBs to date , and thus the most stringent limits placed on the physical properties of magnetars associated with short GRBs from radio observations .