  The magnetar model has been proposed to explain the apparent energy injection
in the X-ray light curves of short gamma-ray bursts (SGRBs), but its
implications across the full broadband spectrum are not well explored. We
investigate the broadband modelling of four SGRBs with evidence for energy
injection in their X-ray light curves, applying a physically motivated model in
which a newly formed magnetar injects energy into a forward shock as it loses
angular momentum along open field lines. By performing an order of magnitude
search for the underlying physical parameters in the blast wave, we constrain
the characteristic break frequencies of the synchrotron spectrum against their
manifestations in the available multi-wavelength observations for each burst.
The application of the magnetar energy injection profile restricts the
successful matches to a limited family of models that are self-consistent
within the magnetic dipole spin-down framework.We produce synthetic light
curves that describe how the radio signatures of these SGRBs ought to have
looked given the restrictions imposed by the available data, and discuss the
detectability of these signatures with present-day and near-future radio
telescopes. Our results show that both the Atacama Large Millimetre Array and
the upgraded Very Large Array are now sensitive enough to detect the radio
signature within two weeks of trigger in most SGRBs, assuming our sample is
representative of the population as a whole. We also find that the upcoming
Square Kilometre Array will be sensitive to depths greater than those of our
lower limit predictions.
