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Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable quantity (such as the luminosity of a distant quasar, the redshift of a distant galaxy, or the angular size of the acoustic peaks in the CMB power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc). The distance measures discussed here all reduce to the naïve notion of Euclidean distance at low redshift.

In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann-Lemaitre-Robertson-Walker solution is used to describe the universe.

Overview

There is a few different definitions of "distance" in cosmology which all coincide for sufficiently small redshifts. The expressions for these distances are most practical when written as functions of redshift, since redshift is always the observable. They can easily be written as functions of scale factor $a=1/(1+z)$ , cosmic $t$ or conformal time $\eta$ as well by performing a simple transformation of variables. By defining the dimensionless Hubble parameter

E(z)={\sqrt {\Omega _{m}(1+z)^{3}+\Omega _{k}(1+z)^{2}+\Omega _{\Lambda }}}

and the Hubble distance $d_{H}=c/H_{0}$ , the relation between the different distances becomes apparent. Here, $\Omega _{m}$ is the total matter density, $\Omega _{\Lambda }$ is the dark energy density, $\Omega _{k}=1-\Omega _{m}-\Omega _{\Lambda }$ , $H_{0}$ is the Hubble parameter today and $c$ is the speed of light. The following measures for distances from the observer to an object at redshift $z$ along the line of sight are commonly used in cosmology^[1]:

Comoving distance:

d_{C}(z)=d_{H}\int _{0}^{z}{\frac {dz'}{E(z')}}

Transverse comoving distance:

d_{M}(z)={\frac {d_{H}}{\sqrt {\Omega _{k}}}}\sinh \left({\sqrt {\Omega _{k}}}d_{C}(z)/d_{H}\right)=\left\{{\begin{array}{ll}{\frac {d_{H}}{\sqrt {\Omega _{k}}}}\sinh \left({\sqrt {\Omega _{k}}}d_{C}(z)/d_{H}\right)&{\text{for }}\Omega _{k}>0\\d_{C}(z)&{\text{for }}\Omega _{k}=0\\{\frac {d_{H}}{\sqrt {-\Omega _{k}}}}\sin \left({\sqrt {-\Omega _{k}}}d_{C}(z)/d_{H}\right)&{\text{for }}\Omega _{k}<0\end{array}}\right.

Angular diameter distance:

d_{A}(z)={\frac {d_{M}(z)}{1+z}}

Luminosity distance:

d_{L}(z)=(1+z)d_{M}(z)

Light-travel distance:

d_{T}(z)=d_{H}\int _{0}^{z}{\frac {dz'}{(1+z')E(z')}}

Note that the comoving distance is recovered from the transverse comoving distance by taking the limit $\Omega _{k}\to 0$ , such that the two distance measures are equivalent in a flat Universe.

Alternative terminology

Peebles (1993, pp 310-320) calls the transverse comoving distance the "angular size distance", which is not to be mistaken for the angular diameter distance [1]. Even though it is not a matter of nomenclature, the comoving distance is equivalent to the proper motion distance, which is defined as the the ratio of the transverse velocity and its proper motion in radians per time. Occasionally, the symbols $\chi$ or $r$ are used to denote both the comoving and the angular diameter distance. Sometimes, the light-travel distance is also called the "lookback distance".

Details

Comoving distance

The comoving distance between fundamental observers, i.e. observers that are comoving with the Hubble flow, does not change with time, as it accounts for the expansion of the Universe. It is obtained by integrating up the proper distances of nearby fundamental observers along the line of sight, where the proper distance is what a measurement at constant cosmic time would yield.

Transverse comoving distance

Two comoving objects at constant redshift $z$ that are separated by an angle $\delta \theta$ on the sky are said to have the distance $\delta \theta d_{M}(z)$ , where the transverse comoving distance $d_{M}$ is defined appropriately.

Angular diameter distance

An object of size $x$ at redshift $z$ that appears to have angular size $\delta \theta$ has the angular diameter distance of $d_{A}(z)=x/\theta$ . This is commonly used to observe so called standard rulers, for example in the context of baryon acoustic oscillations.

Luminosity distance

If the intrinsic luminosity $L$ of a distant object is known, we can calculate its luminosity distance by measuring the flux $S$ and determine ${\sqrt {L/4\pi S}}$ , which turns out to be equivalent to the expression above. This quantity is important for measurements of standard candles like type 1a supernovae, which were first used to discover the acceleration of the expansion of the Universe.

Light-travel distance

This distance is simply the time that it took light to reach the observer from the object multiplied by the speed of light. For instance, the radius of the observable Universe in this distance measure becomes simply the age of the Universe, i.e. 13.7 billion light years. Also see misconceptions about the size of the visible universe.

References

^ David W. Hogg (2000). "Distance measures in cosmology". arXiv:astro-ph/9905116v4.

P. J. E. Peebles, Principles of Physical Cosmology. Princeton University Press (1993)
Scott Dodelson, Modern Cosmology. Academic Press (2003).

External links

'The Distance Scale of the Universe' compares different cosmological distance measures.
'Distance measures in cosmology' explains in detail how to calculate the different distance measures as a function of world model and redshift.
iCosmos: Cosmology Calculator (With Graph Generation ) calculates the different distance measures as a function of cosmological model and redshift, and generates plots for the model from redshift 0 to 20.

[Hogg-1] David W. Hogg (2000). "Distance measures in cosmology". arXiv:astro-ph/9905116v4.

[1]

@@ Line 1: / Line 1: @@
 {{cosmology}}
 '''Distance measures''' are used in [[physical cosmology]] to give a natural notion of the [[distance]] between two objects or events in the [[universe]].  They are often used to tie some ''observable'' quantity (such as the [[luminosity]] of a distant [[quasar]], the [[redshift]] of a distant [[galaxy]], or the angular size of the acoustic peaks in the [[CMB]] power spectrum) to another quantity that is not ''directly'' observable, but is more convenient for calculations (such as the [[comoving coordinates]] of the quasar, galaxy, etc).  The distance measures discussed here all reduce to the naïve notion of [[Euclidean]] distance at low redshift.
-In accord with our present understanding of cosmology, these measures are calculated within the context of [[general relativity]], where the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaitre–Robertson–Walker]] solution is used to describe the universe.
-==Types of distance measures==
+In accord with our present understanding of cosmology, these measures are calculated within the context of [[general relativity]], where the [[Friedmann-Lemaître-Robertson-Walker metric|Friedmann-Lemaitre-Robertson-Walker]] solution is used to describe the universe.
-* '''[[Angular diameter distance]]''' is a good indication (especially in a flat universe) of how near an astronomical object was to us when it emitted the light that we now see.
-* '''[[Luminosity distance]]'''.
-* '''[[Comoving distance]]'''. The distance between two points measured along a path defined at the present [[cosmological time]].
-* '''[[Comoving distance|Cosmological proper distance]]'''. The distance between two points measured along a path defined at a constant cosmological time. The cosmological proper distance should not be confused with the more general [[proper length|proper length or proper distance]].
-* '''Light travel time''' or '''lookback time'''. This is how long ago light left an object of given redshift.
-* '''Light travel distance (LTD)'''. The ''light travel time'' times the [[speed of light]]. For values above 2 billion light years, this value does not equal the ''comoving distance'' or the ''angular diameter distance'' anymore, because of the expansion of the universe. Also see [[Observable_universe#Misconceptions|misconceptions]] about the size of the visible universe.
-* Naive [[Hubble's law]], taking ''z''&nbsp;=&nbsp;''H''<sub>0</sub>''d''/''c'', with ''H''<sub>0</sub> today's [[Hubble constant]], ''z''&nbsp;the [[redshift]] of the object,  c the [[speed of light]], and d the "distance."
+==Overview==
-==Comparison of distance measures==
+There is a few different definitions of "distance" in cosmology which all coincide for sufficiently small redshifts. The expressions for these distances are most practical when written as functions of [[redshift]], since redshift is always the observable. They can easily be written as functions of [[scale factor]] <math>a=1/(1+z)</math>, [[Cosmic time|cosmic]] <math>t</math> or [[conformal time]] <math>\eta</math> as well by performing a simple transformation of variables. By defining the dimensionless [[Hubble parameter]]
+:<math>E(z)=\sqrt{\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda}</math>
+and the Hubble distance <math> d_H = c/H_0 </math>, the relation between the different distances becomes apparent. Here, <math>\Omega_m</math> is the total matter density, <math>\Omega_\Lambda</math> is the [[dark energy]] density, <math>\Omega_k = 1-\Omega_m-\Omega_\Lambda</math>, <math>H_0</math> is the Hubble parameter today and <math>c</math> is the [[speed of light]]. The following measures for distances from the observer to an object at redshift <math>z</math> along the line of sight are commonly used in cosmology<ref name=Hogg>{{cite arxiv|title=Distance measures in cosmology |author=David W. Hogg |year=2000 |arxiv=astro-ph/9905116v4}}</ref>:
+'''Comoving distance:'''
+:<math>  d_C(z)  = d_H \int_0^z \frac{dz'}{E(z')}</math>
+'''Transverse comoving distance:'''
+:<math>  d_M(z) = \frac{d_H}{\sqrt{\Omega_k}} \sinh\left(\sqrt{\Omega_k}d_C(z)/d_H\right) = \left\{ \begin{array}{ll} \frac{d_H}{\sqrt{\Omega_k}} \sinh\left(\sqrt{\Omega_k}d_C(z)/d_H\right) & \text{for } \Omega_k>0\\
+d_C(z) & \text{for }\Omega_k=0\\
+\frac{d_H}{\sqrt{-\Omega_k}} \sin\left(\sqrt{-\Omega_k}d_C(z)/d_H\right) & \text{for }\Omega_k<0\end{array}\right.</math>
+'''Angular diameter distance:'''
+:<math> d_A(z)  = \frac{d_M(z)}{1+z}</math>
+'''Luminosity distance:'''
+:<math>  d_L(z)  = (1+z) d_M(z)</math>
+'''Light-travel distance:'''
+:<math>d_T(z) = d_H \int_0^z \frac{d z'}{(1+z')E(z')} </math>
+Note that the comoving distance is recovered from the transverse comoving distance by taking the limit <math>\Omega_k \to 0</math>, such that the two distance measures are equivalent in a flat Universe.
 [[Image:CosmoDistanceMeasures z to onehalf.png|thumb|right|400px|A comparison of cosmological distance measures, from redshift zero to redshift of&nbsp;0.5. The background cosmology is Hubble parameter 72 km/s/Mpc, Omega_lambda&nbsp;=&nbsp;0.732, Omega_matter&nbsp;=&nbsp;0.266, Omega_radiation&nbsp;=&nbsp;0.266/3454, and Omega_k chosen so that the sum of Omega parameters is&nbsp;1.]]
-* light-travel distance – simply the speed of light times the cosmological time interval, i.e. integral of ''c&nbsp;dt'', while the comoving distance is the integral of ''c''&nbsp;''dt''/''a''(''t'').
-* ''d''<sub>L</sub> [[luminosity distance]]
-* ''d''<sub>pm</sub> [[proper motion distance]]
-**called the ''angular size distance'' by Peebles 1993, but should not be confused with angular diameter distance [http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1993ppc..book.....P&db_key=AST&high=3ece3bb64809032]'')
-**sometimes called the ''coordinate distance''
-**sometimes ''d''<sub>pm</sub> is called the ''angular diameter distance''
-* ''d''<sub>a</sub> [[angular diameter distance]]
+[[Image:CosmoDistanceMeasures z to 1e4.png|thumb|right|400px|A comparison of cosmological distance measures, from redshift zero to redshift of 10,000, corresponding to the epoch of matter/radiation equality. The background cosmology is Hubble parameter 72&nbsp;km/s/Mpc, Omega_lambda&nbsp;=&nbsp;0.732, Omega_matter&nbsp;=&nbsp;0.266, Omega_radiation&nbsp;=&nbsp;0.266/3454, and Omega_k chosen so that the sum of Omega parameters is one.]]
-The latter three are related by:
-:''d''<sub>a</sub> =&nbsp;''d''<sub>pm</sub>/(1&nbsp;+&nbsp;''z'') =&nbsp;''d''<sub>L</sub>/(1&nbsp;+&nbsp;''z'')<sup>2</sup>
-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_\mathrm{pm} = R_C \sinh {\chi \over R_C},</math>
-while for positive curvature,
-:<math>d_\mathrm{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]] density parameter <math>\Omega_\Lambda</math>, and the equation of state 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>
+==Alternative terminology==
-[[Image:CosmoDistanceMeasures z to 1e4.png|thumb|right|400px|A comparison of cosmological distance measures, from redshift zero to redshift of 10,000, corresponding to the epoch of matter/radiation equality. The background cosmology is Hubble parameter 72&nbsp;km/s/Mpc, Omega_lambda&nbsp;=&nbsp;0.732, Omega_matter&nbsp;=&nbsp;0.266, Omega_radiation&nbsp;=&nbsp;0.266/3454, and Omega_k chosen so that the sum of Omega parameters is one.]]
+Peebles (1993, pp 310-320) calls the transverse comoving distance the "angular size distance", which is not to be mistaken for the angular diameter distance [http://cdsads.u-strasbg.fr/cgi-bin/nph-bib_query?bibcode=1993ppc..book.....P&db_key=AST&high=3ece3bb64809032]. Even though it is not a matter of nomenclature, the comoving distance is equivalent to the proper motion distance, which is defined as the the ratio of the transverse velocity and its proper motion in radians per time. Occasionally, the symbols <math>\chi</math> or <math>r</math> are used to denote both the comoving and the angular diameter distance. Sometimes, the light-travel distance is also called the "lookback distance".
+==Details==
+===Comoving distance===
+{{Main|Comoving distance}}
+The comoving distance between fundamental observers, i.e. observers that are comoving with the Hubble flow, does not  change with time, as it accounts for the expansion of the Universe. It is obtained by integrating up the proper distances of nearby fundamental observers along the line of sight, where the proper distance is what a measurement at constant cosmic time would yield.
+===Transverse comoving distance===
+Two comoving objects at constant redshift <math>z</math> that are separated by an angle <math>\delta\theta</math> on the sky are said to have the distance <math>\delta\theta d_M(z)</math>, where the transverse comoving distance <math>d_M</math> is defined appropriately.
+===Angular diameter distance===
+{{Main|Angular diameter distance}}
+An object of size <math>x</math> at redshift <math>z</math> that appears to have angular size <math>\delta\theta</math> has the angular diameter distance of <math>d_A(z)=x/\theta</math>. This is commonly used to observe so called [[Standard ruler|standard rulers]], for example in the context of [[baryon acoustic oscillations]].
+===Luminosity distance===
+{{Main|Luminosity distance}}
+If the intrinsic [[luminosity]] <math>L</math> of a distant object is known, we can calculate its luminosity distance by measuring the flux <math>S</math> and determine <math>\sqrt{L/4\pi S}</math>, which turns out to be equivalent to the expression above. This quantity is important for measurements of [[standard candles]] like [[type 1a supernovae]], which were first used to discover the acceleration of the [[Dark Energy|expansion of the Universe]].
+===Light-travel distance===
+This distance is simply the time that it took light to reach the observer from the object multiplied by the [[speed of light]]. For instance, the radius of the [[observable Universe]] in this distance measure becomes simply the age of the Universe, i.e. 13.7 billion light years. Also see [[Observable_universe#Misconceptions|misconceptions]] about the size of the visible universe.
-where ''c'' is the [[speed of light]] and ''H''<sub>0</sub> is the [[Hubble constant]].
-By using sin and sinh functions, proper motion distance ''d''<sub>pm</sub> can be obtained from&nbsp;''d''<sub>p</sub>.
 ==See also==
 * [[Big Bang]]
 * [[Comoving distance]]
 * [[Friedmann equations]]
 * [[Physical cosmology]]
 * [[Cosmic distance ladder]]
-* [[Friedmann–Lemaître–Robertson–Walker metric]]
+* [[Friedmann-Lemaître-Robertson-Walker metric]]
 ==References==
 <references/>
 * P. J. E. Peebles, ''Principles of Physical Cosmology.'' Princeton University Press (1993)
 * Scott Dodelson, ''Modern Cosmology.'' Academic Press (2003).
 ==External links==
 *[http://www.atlasoftheuniverse.com/redshift.html 'The Distance Scale of the Universe'] compares different cosmological distance measures.
 *[http://arxiv.org/abs/astro-ph/9905116 'Distance measures in cosmology'] explains in detail how to calculate the different distance measures as a function of world model and redshift.
 *[http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )] calculates the different distance measures as a function of cosmological model and redshift, and generates plots for the model from redshift 0 to 20.
 [[Category:Physical cosmology]]
 [[Category:Physical quantities]]
 [[de:Entfernungsmaß]]