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Distance in an Expanding Universe

We now reach a fundamental problem when we are describing techniques to measure distance in the expanding Universe. What do we mean with distance?

The light we receive at the present day from objects that are very far away, has been travelling for a long time, up to several billion years. During that time the Universe has been expanding, so what do we really mean if we talk about THE distance to those objects?

In cosmology there are different definitions for distance, each of which will give different outcomes for an object that is far away (that has a large redshift).

The most important of these definitions are:
1. The “naive” Hubble distance is assuming that the Hubble parameter is constant over cosmic time, which is not correct. The inverse of the Hubble constant is “Hubble Time” which is a naïve measure for the age of the Universe. Using that with the speed of light and the measured redshift gives this “naïve” measure for distance.
2. Light travel time (or Lookback time) is simply how long ago the light left the object that we now see. This is more a time measure than a distance measure and it requires us to have an estimate of when the light left the source. This is generally found as a fraction of the age of the Universe, derived from Hubble’s Law and observed redshift. According to this measure the edge of the observable Universe is at 13.8 billion years of light travel time. In this context light travel distance is light travel time multiplied by the speed of light. That gives us the distance to the source at the time the light we receive left the source.
3. Comoving distance tells where the object is now when we receive the light. This measure is also called Proper Distance but at the present time. Comoving Distance grows with the expansion of the Universe, but of course very slowly in spite of the fact that the expansion of space at large distances is (much) faster than the speed of light. On this basis the edge of the visible Universe is now at about 46 billion light years. This is probably the most realistic measure of distance in an expanding universe. It is also referred to as Line of Sight (LOS) Comoving distance.
4. Angular diameter distance is a measure of distance as a function of the angular diameter as we observe it. When we have information about the size of the object we can derive this type of distance. This is successfully used in the analysis of the Cosmic Microwave Background briefly discussed below. This is a measure of where the object was with respect to us, when the light left the object, similar to light travel distance.
5. Luminosity distance is the distance derived from the luminosity. It is useful as a standard candle measure in relation to e.g. type Ia Supernovae. At very large distances however, this is not a realistic distance measure, because due to the expanding Universe the travelling light photons are stretched (cosmological red shift), which causes the object to appear dimmer than its actual luminosity would suggest. Therefore the luminosity distance will always give a larger value than any of the other distance measures.

A more detailed description can be found here.

All distance measures above, except the first, depend on assumptions about the cosmological model that is used for the expanding Universe.

 A comparison of cosmological distance measures for different ranges of redshift z.For relatively small values of red shift (left), the differences between these various distance measures is negligible. However for large red shift (z > ~0.1), this cannot be ignored.Source: Wikipedia, reference in link above.