Distance measure

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 cosmic microwave background (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.

Not to be confused with Distance measurement.

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.

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

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

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 $a=1/(1+z).$

In the remainder of this article, the peculiar velocity is assumed to be negligible unless specified otherwise.

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 $z$ .

According to the Friedmann equations, 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, $\Omega _{r},\Omega _{m},$ and $\Omega _{\Lambda }$ are normalized values of the present radiation energy density, matter density, and "dark energy density", respectively (the latter representing the cosmological constant), and $\Omega _{k}=1-\Omega _{r}-\Omega _{m}-\Omega _{\Lambda }$ determines 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 (LOS):^[2]