Distance measures (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 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.

Overview

There are a few different definitions of "distance" in cosmology which are all asymptotic one to another for 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 also be written as functions of scale factor $a=1/(1+z).$

We first give formulas for several distance measures, and then describe them in more detail further down. Defining the "Hubble distance" as

d_{H}={\frac {c}{H_{0}}}\approx 3000h^{-1}{\text{Mpc}}\approx 9.26\cdot 10^{25}h^{-1}{\text{m}}

where $c$ is the speed of light, $H_{0}$ is the Hubble parameter today, and $h$ is the dimensionless Hubble constant, all the distances are asymptotic to $z\cdot d_{H}$ for small $z$ .

We also define a dimensionless Hubble parameter:^[1]

E(z)={\frac {H(z)}{H_{0}}}={\sqrt {\Omega _{r}(1+z)^{4}+\Omega _{m}(1+z)^{3}+\Omega _{k}(1+z)^{2}+\Omega _{\Lambda }}}

Here, $\Omega _{r}$ is the total radiation energy density, $\Omega _{m}$ is the total matter density, $\Omega _{\Lambda }$ is the dark energy density, and $\Omega _{k}=1-\Omega _{r}-\Omega _{m}-\Omega _{\Lambda }$ represents the curvature. The Hubble parameter at a given redshift is then $H(z)=H_{0}E(z)$ .

The formula for comoving distance, which serves as the basis for most of the other formulas, involves an integral. Although for some limited choices of parameters (see below) the comoving distance integral has a closed analytic form, in general—and specifically for the parameters of our Universe—we can only find a solution numerically. Cosmologists commonly use the following measures for distances from the observer to an object at redshift $z$ along the line of sight:^[2]

Comoving distance:

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

There is a closed-form expression for this integral if

\Omega _{r}=\Omega _{m}=0

or, by substituting the scale factor

a

for

1/(1+z)

, if

\Omega _{\Lambda }=0

. Our universe now seems to be closely represented by

\Omega _{r}=\Omega _{k}=0.

In this case, we have:

d_{C}(z)=d_{H}\Omega _{m}^{-1/3}\Omega _{\Lambda }^{-1/6}[f((1+z)(\Omega _{m}/\Omega _{\Lambda })^{1/3})-f((\Omega _{m}/\Omega _{\Lambda })^{1/3})]

where

f(x)\equiv \int _{0}^{x}{\frac {dx}{\sqrt {x^{3}+1}}}

Transverse comoving distance:

d_{M}(z)={\begin{cases}{\frac {d_{H}}{\sqrt {\Omega _{k}}}}\sinh \left({\frac {{\sqrt {\Omega _{k}}}d_{C}(z)}{d_{H}}}\right)&\Omega _{k}>0\\d_{C}(z)&\Omega _{k}=0\\{\frac {d_{H}}{\sqrt {|\Omega _{k}|}}}\sin \left({\frac {{\sqrt {|\Omega _{k}|}}d_{C}(z)}{d_{H}}}\right)&\Omega _{k}<0\end{cases}}

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')}

There is a closed-form solution of this if

\Omega _{r}=\Omega _{m}=0

involving the inverse hyperbolic functions

{\text{arcosh}}

or

{\text{arsinh}}

. If

\Omega _{r}=\Omega _{\Lambda }=0

then there is a closed-form solution for

d_{T}(z)

but not for

z(d_{T}).

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 $\lim _{z\to \infty }d_{T}(z)/c$ , 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

Peebles (1993) 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 transverse 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".

Details

Comoving distance

The comoving distance between fundamental observers, i.e. observers that are both moving with the Hubble flow, does not change with time, as comoving distance accounts for the expansion of the universe. Comoving distance is obtained by integrating 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.

In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to measure distances between objects; the comoving distance is the proper distance at the present time.

Proper distance

Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance factors out the expansion of the universe, which gives a distance that does not change in time due to the expansion of space (though this may change due to other, local factors, such as the motion of a galaxy within a cluster); the comoving distance is the proper distance at the present time.

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 $d_A(z)=x/\delta\theta$ . 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 $d_L(z)=\sqrt{L/4\pi S}$ , 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

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.

Etherington's distance duality

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: $d_{L}=(1+z)^{2}d_{A}$

References

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

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.

[peebles1993-1] 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.

[Hogg-2] David W. Hogg (2000). "Distance measures in cosmology". arXiv:astro-ph/9905116v4.

[3] I.M.H. Etherington, “LX. On the Definition of Distance in General Relativity”, Philosophical Magazine, Vol. 15, S. 7 (1933), pp. 761-773.

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