Observed galaxy number counts on the lightcone up to second order: II. Derivation

TL;DR

This paper derives the observed galaxy number over-density up to second order in perturbation theory, including all relativistic effects, to improve cosmological parameter estimation and test gravity models.

Contribution

It provides a comprehensive derivation applicable to all dark energy and modified gravity models, accounting for relativistic projection effects.

Findings

Derivation of galaxy over-density up to second order.

Inclusion of all relativistic effects on the lightcone.

Applicability to various dark energy and gravity models.

Abstract

We present a detailed derivation of the observed galaxy number over-density on cosmological scales up to second order in perturbation theory. We include all relativistic effects that arise from observing on the past lightcone. The derivation is in a general gauge, and applies to all dark energy models (including interacting dark energy) and many modified gravity models. The result will be important for accurate cosmological parameter estimation, including non-Gaussianity, since all projection effects need to be taken into account. It also offers the potential for new probes of General Relativity, dark energy and modified gravity. This paper accompanies Paper I which presents the key results for the concordance model in Poisson gauge.