# General relativistic effects in the galaxy bias at second order

## TL;DR

This paper derives the second-order general relativistic correction to galaxy bias, showing it results from volume distortions caused by long-wavelength modes rather than new bias parameters.

## Contribution

It provides a detailed method to obtain local coordinates that remove mode coupling and derives the relativistic correction to galaxy bias at second order.

## Key findings

- The correction arises from volume distortions, not mode modulation.
- No new bias parameters are introduced by the correction.
- The method clarifies the relativistic effects in galaxy bias modeling.

## Abstract

The local galaxy bias formalism relies on the energy constraint equation at the formation time to relate the metric perturbation to the matter density contrast. In the Newtonian approximation, this relationship is linear, which allows us to specify the initial galaxy density as a function of local physical operators. In general relativity however, the relationship is intrinsically nonlinear and a modulation of the short-wavelength mode by the long-wavelength mode might be expected. We describe in detail how to obtain local coordinates where the coupling of the long- to the short-wavelength modes is removed through a change of coordinates (in the absence of the primordial non-Gaussianity). We derive the general-relativistic correction to the galaxy bias expansion at second order. The correction does not come from the modulation of small-scale clustering by the long-wavelength mode; instead, it arises from distortions of the volume element by the long-wavelength mode and it does not lead to new bias parameters.