Cosmological Distance

**Dominic Ford**, Editor

## Introduction

The plots on this page provide a quick conversion between cosmological redshift and the physical distance of objects. However, on cosmological scales the definition of the distance between two points is less clearly defined than on small scales.

#### Proper motion distance

One definition is very simple and intuitive: a hypothetical tape
measure is stretched through space from the Earth to a particular object, and
its length measured. This corresponds to what cosmologists call *proper
motion distance*. But it is arguably the least useful definition of
distance. Such tape measures don't exist, and astronomers are much more
interested in knowing how big and how bright galaxies will appear at different
distances.

#### Luminosity and angular size distances

In classical physics, sources of light become smaller and fainter with distance in a straightforward manner. The angular size of an object scales inversely with its distance, and its brightness scales inversely with the square of its distance. But these relationships assume that the light of an object is spreading out through a flat Euclidean space. In fact, on cosmological length scales, spacetime is curved according to Einstein's General Theory of Relativity. The expressions for how large and how bright objects appear as a function of distance become more complicated.

Redshift is the quantity that cosmologists quote to refer to the distance of an object because it can be directly measured from the object's spectrum. By contrast, the distance of an object has to be estimated by rather contorted means, and with reference to a particular cosmological model, with its associated uncertainties.

But, it is nonetheless useful to be able to convert between the angular size
and brightness of an object on the sky, and its physical size and luminosity.
For this reason, and to conceal the details of General Relativity, two new
quantities with physical units of distance – the *angular size
distance* \(D_\text{A}\) and the *luminosity distance*
\(D_\text{L}\) – are *defined* such that the classical
relationships hold:
\[ \text{Angular size} = \frac{D}{D_\text{A}} \]
\[ \text{Flux} = \frac{ \text{Luminosity} }{ 4\pi D_\text{L}^2 } \]

where \(D\) is the physical diameter of the object. In the former expression, the small angle approximation has been made such that \( \text{arctan}(x) \approx x\).

The plots below show how these quantities vary with redshift. They must
necessarily assume a particular cosmological model for the expansion of the
Universe; I have used a standard Λ-CDM cosmology, where \(H_0=70\),
\(\Omega_\text{m}=0.27\) and \(\Omega_\Lambda=0.73\). Distances are
expressed in Gpc, where 1 Gpc equals 3.26 billion
lightyears. For more information and mathematical details, see David Hogg's
article *Distance measures
in cosmology*; note that the accuracy to which cosmological parameters
are known have dramatically increased since its publication due to the results
of the WMAP probe.

## Plots

#### Lookback time

The black line below shows the time taken for light to travel from an object at redshift \(z\) to us. The red line shows the time elapsed since the Big Bang at the moment when the light reaching us today set out on its journey from an object at redshift \(z\). Put more simply, observations of an object at redshift \(z\) show us how the Universe appeared at the age indicated by the red line. At any given redshift, the black and red lines sum by definition to the current age of the Universe, 13.87 billion years. This figure is also available in pdf format.

#### Proper motion distance

The plot below shows the instantaneous distance that would be measured to an object at redshift \(z\) by a hypothetical tape measure. This figure is also available in pdf format.

#### Angular size distance

The plot below shows how angular size distance \(D_\text{A}\) varies with redshift. This figure is also available in pdf format.

#### Luminosity distance

The plot below shows how luminosity distance \(D_\text{L}\) varies with redshift. This figure is also available in pdf format.

## References

Hogg, D.,*Distance measures in cosmology*, astro-ph 9905116v4