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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 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 many billions of 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 very different outcomes for an object that is far away (has a large redshift).

Here is a simplified list.

1. The “naive” Hubble’s law distance is what we have discussed above, when we use the relativistic Doppler red shift formula and Hubble’s Law. Even if we correct for relativistic effects it gives very large “distances” for large red shift.
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 redshift. According to this measure the edge of the observable Universe is at 13.8 billion years of light travel time. We could say that light travel distance is light travel time multiplied by the speed of light. That would give 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 has a scale that grows with the expansion of the Universe. 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.
4. Angular diameter distance is more or less the opposite of comoving distance. It is a measure of where the object was with respect to us, when the light left the object.
5. Luminosity distance is the distance derived from the luminosity. 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 much dimmer than its actual luminosity would suggest. Therefore the luminosity distance will always give a (much) larger value than any of the other distance measures.

A more detailed list can be found here

All  distance measures above but the first, are precisely defined in cosmology and 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, the differences between these various distance measures is negligible. However for large red shift (z > ~0.1), this cannot be ignored.Source: Wikipedia

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