# Long gradient mode and large-scale structure observables

## TL;DR

This paper demonstrates that long gradient mode perturbations do not affect large-scale structure observables like galaxy counts or halo bias when analyzed within a relativistic framework, ensuring their effects vanish.

## Contribution

It extends the analysis of long mode perturbations to various large-scale observables and establishes their effects vanish in a relativistic setting, providing a consistency condition for halo bias.

## Key findings

- Gradient mode effects on observables vanish in a relativistic framework.
- Matter variance at fixed physical scale is unaffected by long gradient modes.
- Long gradient modes do not modulate halo bias when the consistency condition is satisfied.

## Abstract

We extend the study of long mode perturbations to other large scale observables such as cosmic rulers, galaxy number counts and halo bias. The long mode is a pure gradient mode that is still outside observer's horizon. We insist that gradient mode effects on observables vanish. It is also crucial that the expressions for observables are relativistic. This allows us to show that the effects of a gradient mode on the large scale observables vanishes identically in a relativistic frame work. To study the potential modulation effect of the gradient mode on halo bias, we derive a consistency condition to the first order in gradient expansion. We find that the matter variance at a fixed physical scale is not modulated by the long gradient mode perturbations when the consistency condition holds. This shows that the contribution of long gradient modes to bias vanishes in this frame work.