Comoving Distance - Comoving Distance and Proper Distance

Comoving Distance and Proper Distance

Comoving distance is the distance between two points measured along a path defined at the present cosmological time. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula:

where a(t′) is the scale factor, t_e is the time of emission of the photons detected by the observer, t is the present time, and c is the speed of light in vacuum.

Despite being an integral over time, this does give the distance that would be measured by a hypothetical tape measure at fixed time t, i.e. the "proper distance" as defined below, divided by the scale factor a(t) at that time. For a derivation see (Davis and Lineweaver, 2003) "standard relativistic definitions".

Definitions

• Many textbooks use the symbol for the comoving distance. However, this must be distinguished from the coordinate distance r in the commonly-used comoving coordinate system for a FLRW universe where the metric takes the form

.

In this case the comoving coordinate distance is related to by if k=0 (a spatially flat universe), by if k=1 (a positively-curved 'spherical' universe), and by if k=-1 (a negatively-curved 'hyperbolic' universe).

• Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them proper distance. On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted, the proper distance at an arbitrary time is simply given by where is the scale factor. (e.g. Davis and Lineweaver, 2003) The proper distance between two galaxies at time t is just the distance that would be measured by rulers between them at that time.