Comoving distance

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In standard cosmology, 'comoving' distance or 'proper distance' is one of several distance measures used by cosmologists to define distances between objects.

1 Comoving coordinates
2 Comoving distance
- 2.1 Uses of the comoving distance
3 Other distances used in standard cosmology
4 Proper distance vs. comoving distance from small galaxies to galaxy clusters
5 See also
6 External links

[edit] Comoving coordinates

While general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are natural choices which are easy to work with. 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 elapsed time since the Big Bang according to a clock of a comoving observer 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.

see also metric expansion of space.

[edit] Comoving distance

Comoving distance is the distance between two points measured along a path of constant cosmological time. It can be computed by the following formula:

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

where

a

(t') is the scale factor.

t e

is the time of emission

t is the time "now".

For a derivation see (Lineweaver and Davis, 2003) "standard relativistic definitions".

Symbols and substitute names for comoving distance

Some textbooks use the symbol $χ$ .
proper distance is the name used by (Weinberg, 1972) for comoving distance.
(Lineweaver and Davis, 2003) defines 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'. By their 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.

[edit] 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 (Lineweaver and Davis, 2003).

[edit] Other distances used in standard cosmology

It has been suggested that this article or section be merged into Distance measures (cosmology). (Discuss)

light-travel distance - simply the speed of light times the cosmological time interval, i.e. integral of c dt, while the comoving distance is the integral of c dt /a( t ) .
d_L luminosity distance
d_pm proper motion distance
- called the angular size distance by Peebles 1993, but should not be confused with angular diameter distance [1])
- sometimes called the coordinate distance
- sometimes d_pm is called the angular diameter distance
d_a angular diameter distance

The latter three are related by:

d_a = d_pm / (1 + z) = d_L /(1 + z)²

where z is the redshift.

If and only if the curvature is zero, then proper motion distance and comoving distance are identical, i.e. $d pm = χ$ .

For negative curvature,

$d_{\mbox{pm}} = R_C \sinh {\chi \over R_C}$ ,

while for positive curvature,

$d_{\mbox{pm}} = R_C \sin {\chi \over R_C}$ ,

where $R C$ is the (absolute value of the) radius of curvature.

A practical formula for numerically integrating $d p$ to a redshift $z$ for arbitrary values of the matter density parameter $Ω m$ , the cosmological constant $Ω Λ$ , and the quintessence parameter $w$ is

$d_p \equiv \chi(z) = {c \over H_0} \int^{a'=1}_{a'=1/(1+z)} {\mbox{d}a \over a \sqrt{ \Omega_m /a - (\Omega_m + \Omega_\Lambda -1) + \Omega_\Lambda a^{-(1+3w)} } },$

where c is the speed of light and H₀ is the Hubble constant.

By using sin and sinh functions, proper motion distance d_pm can be obtained from d_p.

[edit] 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.

[edit] See also

Distance measures (cosmology), for a comparison to other distance measures
Friedmann-Lemaître-Robertson-Walker, for more about the most popular cosmological model
Shape of the universe for more about a recently proposed observational program

[edit] External links

Distance measures in cosmology
Ned Wright's cosmology tutorial
Weinberg, Steven (1972)
Peebles (1993)
Lineweaver & Davis, Expanding Confusion
Ned Wright's Javascript cosmology calculator
Quick reference guide, Hans R. de Ruiter
General method, including locally inhomogeneous case and Fortran 77 software (Hamburg group)
cosmdist-0.2.0 - command line and/or C or Fortran library, based on GNU Scientific Library, for $d p, d p m, t$ as functions of z and their inverses

Retrieved from "http://en.wikipedia.org../../../c/o/m/Comoving_distance.html"

Comoving distance

From Wikipedia, the free encyclopedia

Contents

[edit] Comoving coordinates

[edit] Comoving distance

[edit] Uses of the comoving distance

[edit] Other distances used in standard cosmology

[edit] Proper distance vs. comoving distance from small galaxies to galaxy clusters

[edit] See also

[edit] External links

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