# Calculating distance to galaxies seen through a telescope

When we see a galaxy through a telescope, say the James Webb telescope, it is said we see a galaxy as much as about 10 billion light years away. But 10 billion years ago we were much closer (because the universe was less than 4 billion years old) and now we are much further apart because of the expansion of the universe.

How do we know what distance the light had to travel to get here? Ten billion years ago the Earth did not even exist. If we use the Hubble relationship to calculate distance, that would be based on the red shift 10 billion years ago. The Earth formed 5.3 billion years later because the Earth is about 4.7 billion years old.

• Space expansion can be modelled with the Friedmann equations. If we see a galaxy 10 billion light years away, then we can evolve the Friedmann equations backwards in time to deduce where it was 10 billion years ago (assuming peculiar velocity is negligible). Commented Dec 11, 2022 at 7:58
• @Allure - This looks like an answer to me? Commented Dec 11, 2022 at 8:22
• @Farcher I don't know the math well and am not confident in writing an answer, unfortunately. Commented Dec 11, 2022 at 8:31

## 1 Answer

The "10 billion light years away" statement is an estimate of how long, on a clock carried on the Earth (or at a position in the universe where the Earth would be), the light has been travelling. Or another way of thinking about it is that the light was emitted from the galaxy 3.7 billion years after the Big Bang and we received it 13.7 billion years after the Big Bang.

This "light travel distance" is defined in the way described above and is estimated by combining the redshift and a mathematical model of the expanding universe.

The mathematics of how that is done is described here. A convenient calculator that will tell you the light travel distance given a redshift is here. It is not estimated by a naive application of Hubble's law, which relates the rate of change of proper distance to the proper distance - where proper distance is the distance the galaxy is from us now (which will be greater than 10 billion light years).

• Thank you, but how do we calculate how long light has been traveling when we see something in a telescope? The answers accept the premise that light has been traveling 10 billion light years, I wanted to know how to arrive at that figure of time. Is there simple way to explain that? Commented Dec 11, 2022 at 8:32
• @SNair you use the Friedmann equations. It's not trivial, the distance depends on the composition of the universe (i.e. how much radiation, matter, dark energy there is). There are several different distances in cosmology as well (angular diameter distance, covariant distance, light travel distance, etc). Try something like this: astro.ucla.edu/~wright/CosmoCalc.html Commented Dec 11, 2022 at 8:38