Covariant Decomposition of The Non-Linear Galaxy Number Counts and Their
  Monopole
  We present a fully non-linear and relativistically covariant expression for
the observed galaxy density contrast. Building on a null tetrad tailored to the
cosmological observer's past light cone, we find a decomposition of the
non-linear galaxy over-density into manifestly gauge-invariant quantities, each
of which has a clear physical interpretation as a cosmological observable. This
ensures that the monopole of the galaxy over-density field (the mean galaxy
density as a function of observed redshift) is properly accounted for. We
anticipate that this decomposition will be useful for future work on
non-linearities in galaxy number counts, for example, deriving the relativistic
expression for the galaxy bispectrum. We then specialise our results to
conformal Newtonian gauge, with a Hubble parameter either defined globally or
measured locally, illustrating the significance of the different contributions
to the observed monopole of the galaxy density.
