Comoving distance in cosmology

Definition and relation to the scale factor

In practical terms, comoving distance is a coordinate distance that multiplies by the cosmic scale factor to give the proper (physical) distance at a chosen cosmic time. If a(t) denotes the scale factor, then proper distance = a(t) × comoving distance. At the present epoch the scale factor is often normalized to a(0)=1, so current proper distance and comoving distance coincide.

How it is computed

For a source at redshift z, the comoving distance is obtained by integrating the inverse Hubble parameter over redshift. In words: integrate the speed of light divided by the expansion rate H(z) from redshift zero to z. The exact value depends on the cosmological parameters (matter, radiation, dark energy and curvature) that determine H(z).

Uses and examples

• Mapping large-scale structure and counting objects per comoving volume, which remains meaningful across cosmic time.
• Defining the particle horizon and comoving horizon used in early-universe theory and inflationary models.
• Interpreting features such as baryon acoustic oscillations and correlation lengths in a way that removes the trivial effect of expansion.

Distinctions and notable facts

Comoving distance differs from several other cosmological distances: luminosity distance (relevant for observed brightness), angular diameter distance (relates physical size to observed angular size), and light-travel (lookback) time. These distances are related but emphasize different observational or theoretical quantities. For further background on cosmological distance measures see cosmology references.