Editing Comoving and proper distances (section)

===Definitions===
The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula (derived using the [[Friedmann–Lemaître–Robertson–Walker metric]]):
<math display="block"> \chi = \int_{t_e}^t c \; \frac{\mathrm{d} t'}{a(t')} </math>
where ''a''(''t''&prime;) is the [[Scale factor (cosmology)|scale factor]], ''t''<sub>e</sub> 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 [[time integral|integral over time]], this expression gives the correct distance that would be measured by a set of comoving local rulers at fixed time ''t'', i.e. the "proper distance" (as defined below) after accounting for the time-dependent ''comoving speed of light'' via the inverse scale factor term <math>1 / a(t')</math> in the integrand. By "comoving speed of light", we mean the velocity of light ''through'' comoving coordinates [<math>c / a(t')</math>] which is time-dependent even though ''locally'', at any point along the [[null geodesic]] of the light particles, an observer in an inertial frame always measures the speed of light as <math>c</math> in accordance with special relativity. For a derivation see "Appendix A: Standard general relativistic definitions of expansion and horizons" from Davis & Lineweaver 2004.<ref name="D&L_EC">{{cite journal |author=Davis |first1=T. M. |last2=Lineweaver |first2=C. H. |date=2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |journal=Publications of the Astronomical Society of Australia |volume=21 |issue=1 |pages=97–109 |arxiv=astro-ph/0310808v2 |bibcode=2004PASA...21...97D |doi=10.1071/AS03040 |s2cid=13068122}}</ref> In particular, see ''eqs''. 16–22 in the referenced 2004 paper [note: in that paper the scale factor <math>R(t')</math> is defined as a quantity with the dimension of distance while the radial coordinate <math>\chi </math> is dimensionless.]

Many textbooks use the symbol <math>\chi</math> for the comoving distance. However, this <math>\chi</math> must be distinguished from the coordinate distance <math>r</math> in the commonly used comoving coordinate system for a [[Friedmann–Lemaître–Robertson–Walker metric|FLRW universe]] where the metric takes the form (in reduced-circumference polar coordinates, which only works half-way around a spherical universe):
<math display="block">ds^2 = -c^2 \, d\tau^2 = -c^2 \, dt^2 + a(t)^2 \left( \frac{dr^2}{1 - \kappa r^2} + r^2 \left(d\theta^2 + \sin^2 \theta \, d\phi^2 \right)\right).</math>

In this case the comoving coordinate distance <math>r</math> is related to <math>\chi</math> by:<ref name=Roos-2015>{{cite book
 |title=Introduction to Cosmology
 |edition=4th
 |first1=Matts
 |last1=Roos
 |publisher=[[John Wiley & Sons]]
 |year=2015
 |isbn=978-1-118-92329-0
 |page=37
 |url=https://books.google.com/books?id=RkgZBwAAQBAJ}} [https://books.google.com/books?id=RkgZBwAAQBAJ&pg=PA37 Extract of page 37 (see equation 2.39)]</ref><ref name=Webb-1999>{{cite book
 |title=Measuring the Universe: The Cosmological Distance Ladder
 |edition=illustrated
 |first1=Stephen
 |last1=Webb
 |publisher=[[Springer Science & Business Media]]
 |year=1999
 |isbn=978-1-85233-106-1
 |page=263
 |url=https://books.google.com/books?id=ntZwxttZF-sC}} [https://books.google.com/books?id=ntZwxttZF-sC&pg=PA263 Extract of page 263]</ref><ref name=Lachieze-1999>{{cite book
 |title=The Cosmological Background Radiation
 |edition=illustrated
 |first1=Marc
 |last1=Lachièze-Rey
 |first2=Edgard
 |last2=Gunzig
 |publisher=[[Cambridge University Press]]
 |year=1999
 |isbn=978-0-521-57437-2
 |pages=9–12
 |url=https://books.google.com/books?id=3LO75VmI9BMC}} [https://books.google.com/books?id=3LO75VmI9BMC&pg=PA11 Extract of page 11]</ref>
<math display="block">\chi = \begin{cases}
|\kappa|^{-1/2}\sinh^{-1} \sqrt{|\kappa|} r , & \text{if } \kappa<0 \ \text{(a negatively curved ‘hyperbolic’ universe)} \\
r, & \text{if } \kappa=0 \ \text{(a spatially flat universe)}  \\
|\kappa|^{-1/2}\sin^{-1}  \sqrt{|\kappa|} r , & \text{if } \kappa>0 \ \text{(a positively curved ‘spherical’ universe)}
\end{cases}</math>

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 <math>\chi</math>, the proper distance <math>d(t)</math> at an arbitrary time <math>t</math> is simply given by 
<math display="block">d(t) = a(t) \chi</math> 
where <math>a(t)</math> is the scale factor (e.g. Davis & Lineweaver 2004).<ref name="D&L_EC">{{cite journal |author=Davis |first1=T. M. |last2=Lineweaver |first2=C. H. |date=2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |journal=Publications of the Astronomical Society of Australia |volume=21 |issue=1 |pages=97–109 |arxiv=astro-ph/0310808v2 |bibcode=2004PASA...21...97D |doi=10.1071/AS03040 |s2cid=13068122}}</ref> The proper distance <math>d(t)</math> between two galaxies at time ''t'' is just the distance that would be measured by rulers between them at that time.<ref name=Hogg-1999>{{Cite arXiv |last=Hogg |first=David W. |date=1999-05-11 |title=Distance measures in cosmology |page=4 |eprint=astro-ph/9905116 |language=en}}</ref>