Distance measure

Distance measures are used in physical cosmology to generalize the concept of distance between two objects or events in an expanding universe. They may be 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.

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

{\displaystyle z}

, since redshift is always the observable. They can also be written as functions of scale factor

a

=

1

/

(

1

+

z

)

.

{\displaystyle 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

=

c

H

0

≈

3000

h

−

1

Mpc

≈

9.26

⋅

10

25

h

−

1

m

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

where

c

{\displaystyle c}

is the speed of light,

H

0

{\displaystyle H_{0}}

is the Hubble parameter today, and h is the dimensionless Hubble constant, all the distances are asymptotic to

z

⋅

d

H

{\displaystyle z\cdot d_{H}}

for small z.