Theoretical and numerical perspectives on cosmic distance averages

TL;DR

This paper reviews theoretical predictions and tests them against high-resolution simulations to understand how small-scale inhomogeneities bias cosmological distance measurements, finding good agreement and quantifying biases up to 1% at high redshift.

Contribution

It provides a comprehensive comparison of second-order perturbation theory predictions with ray-tracing simulations for cosmic distance measures, including non-linear effects.

Findings

Numerical results agree well with theoretical predictions in the super-sample variance limit.

Small-scale inhomogeneities do not cause unexpectedly large biases in distance measures.

Biases on certain distance measures can reach just under 1% at high redshift.

Abstract

The interpretation of cosmological observations relies on a notion of an average Universe, which is usually considered as the homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker (FLRW) model. However, inhomogeneities may statistically bias the observational averages with respect to FLRW, notably for distance measurements, due to a number of effects such as gravitational lensing and redshift perturbations. In this article, we review the main known theoretical results on average distance measures in cosmology, based on second-order perturbation theory, and we fill in some of their gaps. We then comprehensively test these theoretical predictions against ray tracing in a high-resolution dark-matter $N$-body simulation. This method allows us to describe the effect of small-scale inhomogeneities deep into the non-linear regime of structure formation on light propagation up to $z=10$. We find that numerical results are in remarkably good agreement with theoretical predictions in the limit of super-sample variance. No unexpectedly large bias originates from very small scales, whose effect is fully encoded in the non-linear power spectrum. Specifically, the directional average of the inverse amplification and the source-averaged amplification are compatible with unity; the change in area of surfaces of constant cosmic time is compatible with zero; the biases on other distance measures, which can reach slightly less than $1\%$ at high redshift, are well understood. As a side product, we also confront the predictions of the recent finite-beam formalism with numerical data and find excellent agreement.