Understanding the relativistic overdensity of galaxy surveys
  The main goal of galaxy surveys is to map the distribution of the galaxies,
for the purpose of understanding the properties of this distribution and its
implications for the content and the evolution of the universe. However, in
order to realise the full potential of these surveys, we need to ensure that we
are using the correct analysis: a relativistic analysis, which has been widely
studied recently. In this work, the known relativistic overdensity of galaxy
surveys is re-examined. Unlike in previous works, a consistent approach for
incorporating both the relativistic number-count overdensity and the
relativistic cosmic magnification over-density in the total observed
overdensity of a generic survey, is presented. Since in practice, analyses are
often done for specific sample types (flux-limited or volume-limited) the
approach in this work allows for the total observed overdensity to be easily
reduced to either of the individual overdensities by applying the same limiting
conditions as for extracting the corresponding data samples. This is not
obvious, or in some cases not possible, with the results of previous works.
Thus, the calculations in this work serve to generalise the expression for the
total observed overdensity. However, care must be taken to apply it
appropriately: the type of the data sample in consideration needs to be taken
into account.
