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I understand that we can observe far away galaxies to get an idea how the universe looked like in the past.

Assuming the universe is 13.82b years old, would it be correct to say that if we looked at a galaxy that was 13.82b ly away, we would see whatever happened just after the big bang? If this is correct, what would we have observed had we looked at the same point 1b years ago?

Is this an application for special relativity?

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Time is kind of funny when you look at such distances. Let's imagine that you are running away from a person throwing a ball at you. The ball will travel further to hit you, based on your speed, than the distance that was present when you started to run.

The universe is expanding. The early Universe expanded very quickly, giving large deviations from the expected position today than what we see through a telescope. Something that we see that is 12 billion years old is actually quite a bit further away today, although we can't know exactly where it is. As I understand, this hypothetical object would be around 40 billion light years away today (See this question), given current theories.

But yes, if you looked at a galaxy that appears in today's distance to be 13 billion light years away, then the universe was 800 million year old, roughly speaking. Galaxies took about a billion years to form, however, so you probably wouldn't see such a galaxy.

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