Non-linear general relativistic effects in the observed redshift

TL;DR

This paper derives and verifies the second-order relativistic corrections to observed redshift, ensuring gauge invariance and clarifying discrepancies in previous calculations, advancing precision in cosmological observations.

Contribution

It provides the first explicit verification of gauge transformation properties for second-order redshift calculations in general relativity.

Findings

Derived the second-order expression for observed redshift.

Verified gauge invariance of the second-order redshift expression.

Clarified origins of discrepancies in previous second-order calculations.

Abstract

We present the second-order expression for the observed redshift, accounting for all the relativistic effects from the light propagation and from the frame change at the observer and the source positions. We derive the generic gauge-transformation law that any observable quantities should satisfy, and we verify our second-order expression for the observed redshift by explicitly checking its gauge transformation property. This is the first time an explicit verification is made for the second-order calculations of observable quantities. We present our results in popular gauge choices for easy use and discuss the origin of disagreements in previous calculations.