Cosmological distances with general-relativistic ray tracing: framework and comparison to cosmographic predictions

In this work we present the first results from a new ray-tracing tool to calculate cosmological distances in the context of fully nonlinear general relativity. We use this tool to study the ability of the general cosmographic representation of luminosity distance, as truncated at third order in redshift, to accurately capture anisotropies in the "true" luminosity distance. We use numerical relativity simulations of cosmological large-scale structure formation which are free from common simplifying assumptions in cosmology. We find the general, third-order cosmography is accurate to within 1% for redshifts to z\approx 0.034 when sampling scales strictly above 100 Mpc/h, which is in agreement with an earlier prediction. We find the inclusion of small-scale structure generally spoils the ability of the third-order cosmography to accurately reproduce the full luminosity distance for wide redshift intervals, as might be expected. For a simulation sampling small-scale structures, we find a +/- 5% variance in the monopole of the ray-traced luminosity distance at z \approx 0.02. Further, all 25 observers we study here see a 9--20% variance in the luminosity distance across their sky at z \approx 0.03, which reduces to 2--5% by z \approx 0.1. These calculations are based on simulations and ray tracing which adopt fully nonlinear general relativity, and highlight the potential importance of fair sky-sampling in low-redshift isotropic cosmological analysis.