Distance measures in cosmology
David W. Hogg
TL;DR
This note compiles explicit formulae for a broad set of cosmological distance measures—including line-of-sight and transverse comoving distances, angular diameter distance, luminosity distance, parallax distance, comoving volume, lookback time, and intersection probabilities—within the FLRW cosmology framework. It shows interrelations among measures via $H(z)$ and curvature, e.g., $D_L=(1+z)D_M$ and $D_A=D_M/(1+z)$, and provides practical expressions for k-corrections and distance moduli to aid observational analysis. The work serves as a compact, algorithm-friendly reference (with minimal code guidance) for cosmography computations and model comparisons. Overall, it offers a reusable toolkit for converting redshift observations into distances, volumes, and evolutionary inferences across cosmological models.
Abstract
Formulae for the line-of-sight and transverse comoving distances, proper motion distance, angular diameter distance, luminosity distance, k-correction, distance modulus, comoving volume, lookback time, age, and object intersection probability are all given, some with justifications. Some attempt is made to rationalize disparate terminologies, or at least abuse bad usage.