Theoretical and numerical perspectives on cosmic distance averages

¹ Aix Marseille Univ., CNRS, CNES, LAM, Marseille, France
e-mail: michel-andres.breton@lam.fr
² Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
e-mail: pierre.fleury@uam.es

Received: 16 December 2020
Accepted: 15 August 2021

Abstract

The interpretation of cosmological observations relies on a notion of an average Universe, which is usually considered as the homogeneous and isotropic Friedmann-Lemaître-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.