Distance measures (cosmology)

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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 common 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 solution is used to describe the universe.


There are 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 z, 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


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 represents the curvature, H_0 is the Hubble parameter today and c is the speed of light. The Hubble parameter at a given redshift is then H(z) = H_0 E(z). 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) = \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{d z'}{(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.

Age of the universe is \lim_{z\rightarrow\infty} d_T(z)/c, and the time elapsed since redshift z until now is

 t(z) = d_T(z)/c

A comparison of cosmological distance measures, from redshift zero to redshift of 0.5. The background cosmology is Hubble parameter 72 km/s/Mpc, \Omega_\Lambda=0.732, \Omega_{\rm matter}=0.266, \Omega_{\rm radiation}=0.266/3454, and \Omega_k chosen so that the sum of Omega parameters is 1.
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 km/s/Mpc, \Omega_\Lambda=0.732, \Omega_{\rm matter}=0.266, \Omega_{\rm radiation}=0.266/3454, and \Omega_k chosen so that the sum of Omega parameters is one.

Alternative terminology[edit]

Peebles (1993) calls the transverse comoving distance the "angular size distance", which is not to be mistaken for the angular diameter distance.[2] Even though it is not a matter of nomenclature, the comoving distance is equivalent to the proper motion distance, which is defined as 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".


Comoving distance[edit]

Main article: 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 (LOS), where the proper distance is what a measurement at constant cosmic time would yield.

Transverse comoving distance[edit]

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[edit]

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/\delta\theta. This is commonly used to observe so called standard rulers, for example in the context of baryon acoustic oscillations.

Luminosity distance[edit]

Main article: 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 d_L(z)=\sqrt{L/4\pi S}, which turns out to be equivalent to the expression above for d_L(z). This quantity is important for measurements of standard candles like type Ia supernovae, which were first used to discover the acceleration of the expansion of the universe.

Light-travel distance[edit]

This distance is the time (in years) 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 the age of the universe multiplied by the speed of light (1 light year/year) i.e. 13.8 billion light years. Also see misconceptions about the size of the visible universe.

Etherington's distance duality[edit]

The Etherington's distance-duality equation [3] is the relationship between the luminosity distance of standard candles and the angular-diameter distance. It is expressed as follows: d_L=(1+z)^2 d_A

See also[edit]


  1. ^ David W. Hogg (2000). "Distance measures in cosmology". arXiv:astro-ph/9905116v4. 
  2. ^ Peebles, P. J. E. (1993). Principles of Physical Cosmology. Princeton University Press. pp. 310–320. Bibcode:1993ppc..book.....P. ISBN 978-0-691-01933-8. 
  3. ^ I.M.H. Etherington, “LX. On the Definition of Distance in General Relativity”, Philosophical Magazine, Vol. 15, S. 7 (1933), pp. 761-773.
  • Scott Dodelson, Modern Cosmology. Academic Press (2003).

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