Definitions

# Comoving distance

In standard cosmology, 'comoving' distance and 'proper distance' are two closely related distance measures used by cosmologists to define distances between objects.

## Comoving coordinates

While general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are natural choices with which it is easier to work. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as isotropic. Such observers are called "comoving" observers because they move along with the Hubble flow.

A comoving observer is the only observer that will perceive the universe, including the cosmic microwave background radiation, to be isotropic. Non-comoving observers will see regions of the sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local frame of reference called the comoving frame. The velocity of an observer relative to the local comoving frame is called the peculiar velocity of the observer. Most large lumps of matter, such as galaxies, are nearly comoving, i.e., their peculiar velocities are low.

The comoving time coordinate is the elapsed time since the Big Bang according to a clock of a comoving observer and is a measure of cosmological time. The comoving spatial coordinates tell us where an event occurs while cosmological time tells us when an event occurs. Together, they form a complete coordinate system, giving us both the location and time of an event.

Space in comoving coordinates is (on the average) static, as most bodies are comoving, and comoving bodies have static, unchanging comoving coordinates.

The expanding Universe has an increasing scale factor which explains how constant comoving coordinates are reconciled with distances that increase with time.

## Comoving 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:

$chi = int_\left\{t_e\right\}^\left\{t\right\} \left\{ c ; mbox\left\{d\right\} t\text{'} over a\left(t\text{'}\right)\right\}$

where $a$($t\text{'}$) is the scale factor.
$t_e$ is the time of emission of the photons detected by the observer
$t$ is the time "now".
Despite being an integral over time, this does give the distance that would be measured by a hypothetical tape measure at fixed time t. For a derivation see (Davis and Lineweaver, 2003) "standard relativistic definitions".Symbols and substitute names for comoving distance

• Some textbooks use the symbol $chi$.
• proper distance is the name used by (Weinberg, 1972) for comoving distance.
• 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. (e.g. Davis and Lineweaver, 2003)

### Uses of the comoving distance

Cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, that is, in the local comoving frame. Comoving distance is also equal to the locally measured distance in the comoving frame for nearby objects. To measure the comoving distance between two distant objects, one imagines that one has many comoving observers in a straight line between the two objects, so that all of the observers are close to each other, and form a chain between the two distant objects. All of these observers must have the same cosmological time. Each observer measures his distance to the nearest observer in the chain, and the length of the chain, the sum of distances between nearby observers, is the total comoving distance. It is important to the definition of comoving distance that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or geodesic between the two points, one would not be correctly measuring comoving distance. Comoving distance is not quite the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a nearly empty universe, where both sorts of distance can be measured. In this thought experiment the value of comoving distance is not equal to the value of the distance as defined by special relativity. (Wright)

If one divides a comoving distance by the present cosmological time (the age of the universe) and calls this a "velocity", then the resulting "velocities" of "galaxies" near the particle horizon or further than the horizon can be above the speed of light. This apparent superluminal expansion is not in conflict with special or general relativity, and is a consequence of the particular definitions used in cosmology. Note that the cosmological definitions used to define the velocities of distant objects are coordinate dependent - there is no general coordinate independent definition of velocity between distant objects in general relativity (Baez and Bunn, 2006) The issue of how to best describe and popularize the apparent superluminal expansion of the universe has caused a minor amount of controversy. One viewpoint is presented in (Davis and Lineweaver, 2003)

## Proper distance vs. comoving distance from small galaxies to galaxy clusters

Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (i.e. the comoving distance measured using the scale factor of the universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.