Comoving and proper distances

Comoving distance removes the effect of cosmic expansion from a separation, while proper distance is the physical separation at a specified cosmic time. Their relation depends on the universe’s scale factor.

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Scale factor: A dimensionless measure of how the universe’s expansion changes distances relative to a chosen reference time. Multiplying comoving distance by the scale factor gives proper distance at that time.
Metric tensor: A mathematical object that specifies distances, durations, and angles in a spacetime or curved space. The cosmological metric defines spatial distance on constant-time slices.
Comoving distance: The present-day-equivalent separation between locations after factoring out the universe’s expansion. It is the expansion-adjusted distance measure at the center of this distinction.
Lookback time: The time elapsed between the emission of observed light and its detection. It measures a temporal interval, not the spatial separation between locations.
Hubble's law: The relation between the recession speed of distant galaxies and their distance in an expanding universe. Its distance depends on which cosmic-time measure is used.
Friedmann–Lemaître–Robertson–Walker metric: A spacetime metric describing a homogeneous and isotropic universe with evolving spatial scale. Its spatial line element distinguishes comoving coordinates from physical distance.
Proper distance: The physical spatial separation between locations measured on a specified cosmic-time slice. It is the time-dependent physical counterpart to comoving distance.
Light-travel distance: The distance light travels in a specified time interval, often used as a cosmological distance measure. It does not generally equal comoving or proper distance in an expanding universe.
Hubble parameter: The fractional expansion rate of the universe at a given cosmic time. It describes how rapidly proper distances change as the scale factor grows.
Comoving coordinates: Coordinates that remain fixed for objects carried along with the average cosmic expansion. They keep the positions of ideal Hubble-flow objects fixed while proper separations grow.