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 overdensity 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.