Gravitational wave observations, distance measurement uncertainties, and
  cosmology
  Gravitational waves from the coalescence of compact binaries, together with
an associated electromagnetic counterpart, are ideal probes of cosmological
models. As demonstrated with GW170817, such multimessenger observations allow
one to use the source as a standard siren, analog of standard candles in
conventional astronomy, in order to measure cosmological parameters such as the
Hubble constant. No cosmological ladder is needed to estimate the source
luminosity distance from the detected gravitational waves. The error on the
luminosity distance plays a crucial r\^ole in the error budget for the
inference of the Hubble constant. In this paper, we provide analytic
expressions for the statistical errors on the luminosity distance inferred from
gravitational wave data as a function of the sky position and the detector
network. In particular, we take into account degeneracy in the parameter space
of the gravitational waveform showing that in certain conditions on the
gravitational-wave detector network and the source sky position it may not be
possible to estimate the luminosity distance of the source. Our analytic
approximants shows a good agreement with the uncertainties measured with
Bayesian samplers and simulated data. We also present implications for the
estimation error on the Hubble constant.
