One of the most exciting near-term prospects in physics is the potential discovery of gravitational waves by the advanced LIGO and Virgo detectors. To maximise both the confidence of the detection and the science return, it is essential to identify an electromagnetic counterpart. This is not trivial, as the events are expected to be poorly localised, particularly in the near-term, with error regions covering hundreds or even thousands of square degrees. In this paper we discuss the prospects for finding an X-ray counterpart to a gravitational wave trigger with the Swift X-ray Telescope, using the assumption that the trigger is caused by a binary neutron star merger which also produces a short gamma-ray burst. We show that it is beneficial to target galaxies within the GW error region, highlighting the need for substantially complete galaxy catalogues out to distances of 300 Mpc. We also show that nearby, on-axis short GRBs are either extremely rare, or are systematically less luminous than those detected to date. We consider the prospects for detecting afterglow emission from an an off-axis GRB which triggered the GW facilities, finding that the detectability, and the best time to look, are strongly dependent on the characteristics of the burst such as circumburst density and our viewing angle.
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