Comoving distance
From Academic Kids

The comoving distance or conformal distance of two objects in the universe is the distance divided by a timevarying scale factor representing the expansion of the universe. As a result the comoving distances on average are not increasing with time. The scale factor is usually taken to be one at present, so currently the comoving distance is equal to the ordinary distance.
Thinking about the shape of the universe in the context of the standard Big Bang model is simplest using comoving coordinates.
While special relativity states that all inertial reference frames are equivalent, i.e. that there is no favoured set of spacetime coordinates, this is only a local theory.
General relativity is also a local theory, but it is used to constrain the local properties of a Riemannian manifold, which itself is global.
In the context of general relativity, the assumption of Weyl's postulate is that a favoured reference frame in spacetime can be decided. The most common notion of such coordinates is that of comoving coordinates, where the spatial reference frame is attached to the average positions of galaxies (or any large lumps of matter which are at most moving slowly).
With this set of coordinates, both time and expansion of the Universe can be ignored in order to concentrate on the shape of space (formally speaking, of a spatial hypersurface at constant cosmological time).
Space in comoving coordinates is (on average) static. This is perfectly consistent with the fact that the Universe is expanding. A choice of coordinates is just a choice of labels. There happens to be (according to the standard Big Bang model) a choice of these labels which can be used either for formal calculations or for intuition in which the Universe is static. To get back to thinking about an expanding Universe just requires remembering the scale factor.
This way there is also cosmological time, which for an observer at a fixed spatial point in comoving coordinates is identical to her local measurement of time.
Comoving distance is then the distance in comoving coordinates between two points in space, at a single cosmological time:
 <math> \chi = \int_{t}^{t_0} { c \; \mbox{d} t' \over a(t')}<math>
where <math>a(t')<math> is the scale factor.
equivalent names
 Some textbooks use the symbol <math>\chi<math> for comoving distance.
 proper distance is the name used by Weinberg (1972) [1] (http://cdsads.ustrasbg.fr/cgibin/nphbib_query?bibcode=1972gcpa.book.....W&db_key=AST&high=3ece3b4cfd08957) for comoving distance.
Contents 
Is comoving distance a meaningless concept?
Comoving distance and cosmological time definitely exist as part of the standard Big Bang model.
However, while cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, the comoving distance is not, in the general case, identical to a distance as physically experienced by a particle moving slower than or at the speed of light.
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 is the paradox of the ambiguous phrase space expanding faster than the speed of light. An umambiguous rewording of the phrase can now be made:
 For a "galaxy" towards or beyond the horizon, its "velocity", defined as comoving distance from the observer divided by the present cosmological time, can be greater than the speed of light.
This is a correct statement (Is it, really? According to this lecture (http://www.shef.ac.uk/physics/teaching/phy323/current_year/lectures/condensed/lec_10_comoving_coords_summary.pdf) by Edward Daw (http://www.shef.ac.uk/physics/people/edaw/), for quick reference, comoving coordinates system is "a coordinate system in which the coordinates of objects COMOVING with the expansion remain fixed as the expansion proceeds"; and I have seen a number of pages saying exactly the same). What is debatable is the philosophical interpretation.
Problems according to a strictly empirical point of view (according to which something hidden inside a box does not exist, cf. Bertrand Russell) include:
 A distant "galaxy" towards the horizon is seen a long time in the past, when it was essentially just a lump of slightly overdense hydrogen with a bit of helium, so it is unlikely that stars had formed by then, so that the "galaxy" is impossible to observe.
 Moreover, a "galaxy" beyond the horizon can only be observed in the future.
 If you consider a distance to the "galaxy" defined by following the light path from that "galaxy", then this cannot be done for a "galaxy" beyond the horizon: the path does not arrive at the observer. On the other hand, for a "galaxy" inside the horizon, use of this same lighttravel definition of distance, instead of comoving distance, can be done, but it yields a velocity less than the speed of light.
For these reasons, some people consider the comoving distance to be a merely theoretical construct with no physical meaning. However, in doing so, those people assert that the standard Big Bang model has no physical meaning, since comoving coordinates are an intrinsic part of the model.
Other distances useful in cosmology
 lighttravel 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
 (confusingly called the angular size distance by Peebles 1993 [2] (http://cdsads.ustrasbg.fr/cgibin/nphbib_query?bibcode=1993ppc..book.....P&db_key=AST&high=3ece3bb64809032))
 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. <math>d_\mbox{pm} =\chi<math>.
For negative curvature,
 <math>d_\mbox{pm} = R_C \sinh {\chi \over R_C}<math>,
while for positive curvature,
 <math>d_\mbox{pm} = R_C \sin {\chi \over R_C}<math>,
where <math>R_C<math> is the (absolute value of the) radius of curvature.
A practical formula for numerically integrating <math>d_{p}<math> to a redshift <math>z<math> for arbitrary values of the matter density parameter <math>\Omega_m<math>, the cosmological constant <math>\Omega_\Lambda<math>, and the quintessence parameter <math>w<math> is
 <math> 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)} } }, <math>
where c is the speed of light and H_{0} is the Hubble constant.
By using sin and sinh functions, proper motion distance d_{pm} can be obtained from d_{p}.
Distances useful on small — galaxy or cluster of galaxies — scales
The ordinary distance as experienced by particles travelling slower than or at the speed of light is simply the comoving distance multiplied by the value of the scale factor at the cosmological time studied.
Different names for this include
 physical distance  this has the problem that it suggests that comoving distance is less physical than ordinary distance.
 proper distance  this is confusing (see above) though correct if calculated at the cosmological time studied.
Free (as in speech) software
 cosmdist0.2.0 (http://cosmo.torun.pl/GPLdownload/dodec/cosmdist0.2.0.tar.gz)  command line and/or C or fortran library, based on GSL, for <math>d_p, d_{pm}, t<math> as functions of z and their inverses
See also
FriedmannLemaîtreRobertsonWalker.
External links
 Weinberg, Steven (1972) (http://cdsads.ustrasbg.fr/cgibin/nphbib_query?bibcode=1972gcpa.book.....W&db_key=AST)
 Peebles (1993) (http://cdsads.ustrasbg.fr/cgibin/nphbib_query?bibcode=1993ppc..book.....P&db_key=AST)
 quick reference guide, Hans R. de Ruiter (http://www.bo.astro.it/~deruiter/cosmo/Cosfor.htm)
 general method, including locally inhomogeneous case (http://arXiv.org/abs/astroph/9603028) and fortran77 software (Hamburg group)
fr:distance comobile pl:współrzędne współporuszające się simple:comoving distance