Did GW170817 harbor a pulsar?

If the progenitor of GW170817 harbored a pulsar, then a Poynting flux dominated bow-shock cavity would have been expected to form around the traveling binary. The characteristic size of this evacuated region depends strongly on the spin-down evolution of the pulsar companion, which in turn depends on the merging timescale of the system. If this evacuated region is able to grow to a sufficiently large scale, then the deceleration of the jet, and thus the onset of the afterglow, would be noticeably delayed. The first detection of afterglow emission, which was uncovered 9.2 days after the $\gamma$-ray burst trigger, can thus be used to constrain the size of a pre-existing pulsar-wind cavity. We use this information, together with a model of the jet to place limits on the presence of a pulsar in GW170817 and discuss the derived constraints in the context of the observed double neutron star binary population. We find that the majority of Galactic systems that are close enough to merge within a Hubble time would have carved a discernibly large pulsar-wind cavity, inconsistent with the onset timescale of the X-ray afterglow of GW170817. Conversely, the recently detected system J1913+1102, which host a low-luminosity pulsar, provides a congruous Milky Way analog of GW170817's progenitor model. This study highlights the potential of the proposed observational test for gaining insight into the origin of double neutron star binaries, in particular if the properties of Galactic systems are representative of the overall merging population.