Covariant Decomposition of The Non-Linear Galaxy Number Counts and Their Monopole

TL;DR

This paper develops a fully non-linear, gauge-invariant, relativistically covariant framework for analyzing galaxy number counts, including the monopole, which is crucial for accurate cosmological measurements.

Contribution

It introduces a novel decomposition of galaxy over-density into gauge-invariant quantities using a null tetrad, enabling precise analysis of non-linear effects and the monopole in galaxy surveys.

Findings

Provides a covariant expression for galaxy density contrast.

Clarifies the physical interpretation of gauge-invariant components.

Highlights the importance of monopole accounting in cosmological observations.

Abstract

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.