A Pixelated Approach to Galaxy Catalogue Incompleteness: Improving the
  Dark Siren Measurement of the Hubble Constant
  The use of gravitational wave standard sirens for cosmological analyses is
becoming well known, with particular interest in measuring the Hubble constant,
$H_0$, and in shedding light on the current tension between early- and
late-time measurements. The current tension is over $4\sigma$ and standard
sirens will be able to provide a completely independent measurement. Dark
sirens (binary black hole or neutron star mergers with no electromagnetic
counterparts) can be informative if the missing redshift information is
provided through the use of galaxy catalogues to identify potential host
galaxies of the merger. However, galaxy catalogue incompleteness affects this
analysis, and accurate modelling of it is essential for obtaining an unbiased
measurement of $H_0$. Previously this has been done by assuming uniform
completeness within the sky area of a gravitational wave event. This paper
presents an updated methodology in which the completeness of the galaxy
catalogue is estimated in a directionally-dependent matter, by pixelating the
sky and computing the completeness of the galaxy catalogue along each line of
sight. The $H_0$ inference for a single event is carried out on a
pixel-by-pixel basis, and the pixels are combined for the final result. A
reanalysis of the events in the first gravitational wave transient catalogue
(GWTC-1) leads to an improvement on the measured value of $H_0$ of
approximately 5% compared to the 68.3% highest-density interval of the
equivalent LIGO and Virgo result, with $H_0= 68.8^{+15.9}_{-7.8}$ km s$^{-1}$
Mpc$^{-1}$.
