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

TL;DR

This work derives the second-order relativistic corrections to the cosmic clock and cosmic rulers, foundational covariant quantities that govern large-scale structure observables such as galaxy number counts. By solving the photon geodesic equation to second order in a perturbed FLRW spacetime and expressing results in observer-inferred coordinates, the authors obtain explicit second-order expressions for the clock $\mathcal{T}$ and the rulers $\mathcal{C}, \mathcal{M}$ (and their 1-form counterparts). They validate these results with three non-trivial tests: a constant-potential null test, a second-order gradient-mode null test, and a curved-background separate-universe test, and provide numerical results illustrating the scale dependence of the second-order terms. The findings show that while second-order contributions are suppressed on the largest scales (by about $10^{-4}$ relative to linear order), they become comparable to linear terms on mildly non-linear scales, and these second-order pieces constitute the most intricate part of the relativistic galaxy density at second order. Overall, the paper delivers a robust, covariant, and highly validated framework for incorporating second-order GR corrections into relativistic LSS analyses, with a public Mathematica notebook for reproducing the tests.

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.

Figures (2)

• Figure 1: Plots of $\mathbb{E}(X_{l0}X_{l0})$, for $X \in\left\{\mathcal{T},\mathfrak{C},\mathfrak{M}\right\}$, for potentials described by a pure sinusoidal mode. Here, $l=0$, $\tilde{z} = 1$ and $\sigma(k) = 10^{-4}$. The left panel shows low $k$ and the right panel shows larger values of $k$, where the second-order correction starts to dominate over the first order. Dashed lines show the pure first-order contribution in the top row, and the pure second-order part is plotted as full lines.
• Figure 2: Plots of $\mathbb{E}(X_{l0}X_{l0})$, for $X \in\left\{\mathcal{T},\mathfrak{C},\mathfrak{M}\right\}$, for potentials described by a pure sinusoidal mode. Here, $l=2$, $\tilde{z} = 1$ and $\sigma(k) = 10^{-4}$. The left panel shows low $k$ and the right panel shows larger values of $k$, where the second-order correction starts to dominate over the first order. Dashed lines show the pure first-order contribution in the top row, and the pure second-order part is plotted as full lines.