The Swift short gamma-ray burst rate density: implications for binary
  neutron star merger rates
  Short gamma-ray bursts (SGRBs) observed by {\it Swift} are potentially
revealing the first insight into cataclysmic compact object mergers. To
ultimately acquire a fundamental understanding of these events requires
pan-spectral observations and knowledge of their spatial distribution to
differentiate between proposed progenitor populations. Up to April 2012 there
are only some 30% of SGRBs with reasonably firm redshifts, and this sample is
highly biased by the limited sensitivity of {\it Swift} to detect SGRBs. We
account for the dominant biases to calculate a realistic SGRB rate density out
to $z\approx0.5$ using the {\it Swift} sample of peak fluxes, redshifts, and
those SGRBs with a beaming angle constraint from X-ray/optical observations. We
find an SGRB lower rate density of $8^{+5}_{-3}$
$\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$ (assuming isotropic emission), and a
beaming corrected upper limit of $1100^{+700}_{-470} $
$\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$. Assuming a significant fraction of binary
neutron star mergers produce SGRBs, we calculate lower and upper detection rate
limits of $(1-180)$ yr$^{-1}$ by an aLIGO and Virgo coincidence search. Our
detection rate is similar to the lower and realistic rates inferred from
extrapolations using Galactic pulsar observations and population synthesis.
