Large-Scale-Structure Observables in General Relativity Validated at Second Order

TL;DR

This paper develops second-order relativistic calculations of large-scale-structure observables in cosmology, validating the formulas across various metrics and showing their significance on mildly non-linear scales.

Contribution

It provides the first fully non-linear second-order formulas for cosmic rulers and clock, validated across multiple metrics, advancing precision in relativistic galaxy number density modeling.

Findings

Second-order observables are suppressed by ~10^{-4} on large scales.

They become comparable to linear contributions on mildly non-linear scales.

Validated formulas across different space-time metrics.

Abstract

We present a second-order calculation of relativistic large-scale-structure observables in cosmological perturbation theory, specifically the "cosmic rulers and clock", which are the building-blocks of any other large-scale-structure observable, including galaxy number counts, on large scales. We calculate the scalar rulers (longitudinal perturbation and magnification) and the cosmic clock to second order, using a fully non-linear covariant definition of the observables. We validate our formulae on three non-trivial space-time metrics: two of them are null tests on metrics which are obtained by applying a gauge transformation to the background space-time, while the third is the "separate universe" curved background, for which we can also compute the observables exactly. We then illustrate the results by evaluating the second-order observables in a simplified symmetric setup. On large scales, they are suppressed over the linear contributions by $\sim 10^{-4}$, while they become comparable to the linear contributions on mildly non-linear scales. The results of this paper form a significant (and the most complicated) part of the relativistic galaxy number density at second order.