Michel-Andrès Breton, Pierre Fleury

The interpretation of cosmological observations relies on a notion of average Universe, which is usually taken 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 fill 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 the 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 $\sim 10^{-3}$ 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.

Sandra Baumgartner, Jaiyul Yoo

The standard theoretical description $\Theta(\hat n)$ of the observed CMB temperature anisotropies is gauge-dependent. It is, however, well known that the gauge mode is limited to the monopole and the higher angular multipoles $\Theta_l$ ($l\geq1$) are gauge-invariant. Several attempts have been made in the past to properly define the monopole fluctuation, but the resulting values of the monopole power $C_0$ are infinite due to the infrared divergences. The infrared divergences arise from the contribution of the uniform gravitational potential to the monopole fluctuation, in violation of the equivalence principle. Here we present the gauge-invariant theoretical description of the observed CMB temperature anisotropies and compute the monopole power $C_0=1.66\times10^{-9}$ in a $\Lambda$CDM model. While the gauge-dependence in the standard calculations originates from the ambiguity in defining the hypersurface for the background CMB temperature $\bar T$ today, it is in fact well defined and one of the fundamental cosmological parameters. Adopting simple approximations for the anisotropy formation, we derive a gauge-invariant analytical expression for the observed CMB temperature anisotropies to study the CMB monopole fluctuation and the cancellation of the uniform gravitational potential contributions on large scales.