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A galaxy 13 billion light-years distant is 94% of the way to the big bang. The universe was 800 million years old when the light was emitted. That galaxy today is traveling away from us at 94% of the speed of light roughly. So 13 billion years ago it was 750 million light years away from us. Assuming the expansion is NOT greater than light speed, that photon did NOT go backward. This is also evidenced by the fact that the photon arrived at earth. It seems to me the photon took 13 billion years to cross 750 million light years. What am I doing wrong? Question two. If this is right then please explain how we go from 750 (or 800) million light years to 46.5 billion light years.

David Wilson
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    Just read about co-moving distance. – Mithoron Jul 15 '18 at 20:02
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    There are many misconceptions about the expansion of the Universe in this question (don't worry, they are very common misconceptions). I think you would benefit from reading this, this, this, this, and/or this question. To get you started, the Universe is much larger than 13.8 Glyr because it expands, and expansion is faster than light. – pela Jul 15 '18 at 20:32
  • Thanks for answering so quickly. I'm not sure I am understanding the links you provided. Maybe it would help if you told me what the radius of the visible universe was 13 billion years ago and if the CMB was then also the furthest visible object. – David Wilson Jul 15 '18 at 22:01
  • Actually, this link found on the page one of you provided, while not explaining it in terms I could understand, convinced me to give up on this quest. https://arxiv.org/pdf/astro-ph/0310808.pdf – David Wilson Jul 15 '18 at 22:55
  • Yes, that paper is an absolute classic. Anyway: Briefly, in a static universe, after 13.8 Gyr the farthest you'd be able to see would be 13.8 Glyr, as expected. But since our Universe expands, the objects that we observe to have emitted its light 13.8 Gyr ago, are now much farther away than 13.8 Glyr (in fact 46.3 Glyr). – pela Jul 16 '18 at 12:43
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    13 Gyr ago — i.e. when the Universe was 0.8 Gyr old — the radius of the region that today is our observable Universe was roughly 6 Glyr. The observable Universe at that time, however, was smaller, because light from more distant regions hadn't had the time to reach an observer; in fact it was only 2.3 Glyr. And yes, the CMB was also the furthest visible "object". – pela Jul 16 '18 at 12:48
  • Note that these calculations are not really trivial, but require an integration of the Friedmann equation. If you really want to understand these matters, you should try to understand Figure 1 of the paper you linked to. – pela Jul 16 '18 at 12:50
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    Today that galaxy is receding at much more than the speed of light. – ProfRob Jul 16 '18 at 21:47
  • @RobJeffries Actually a galaxy 13 Glyr away is only a z = 1.25 and hence recedes only at 0.9c. But of course it isn't "94% of the way to Big Bang". Such a galaxy (i.e. at a distance of 0.94 x 46.3 Glyr = 43.5 Glyr) would be receding at 3c. – pela Jul 18 '18 at 15:52
  • @Pela A galaxy that "is 13 billion light years away", in common usage means that the light has taken 13 billion years to reach us. At least that is obviously the meaning attached to it in the question ("that is 94% of the way to the big bang") and is not the co-moving distance. As such, my comment is entirely correct and the galaxy has a redshift of $z\simeq 8$. – ProfRob Jul 18 '18 at 16:13
  • Okay okay @RobJeffries, I meant no offense. You are of course right that that was the intention of the question :) – pela Jul 19 '18 at 09:55

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Neither the amount of time it took for light to reach us emitted long ago from a distant galaxy nor the distance to that galaxy are directly observable. What is observable is the redshift, how much light has been shifted toward longer wavelengths.

The amount of time it took for that redshifted light to reach us can be calculated from the redshift $z$ by assuming various parameters regarding the universe (regular matter vs dark matter, the Hubble constant, whether the universe is open, flat, or closed, etc.), and by assuming that that galaxy was and our galaxy is more or less moving with the local Hubble flow. A distance value can be calculated simply by multiplying the calculated time by the speed of light.

Even though the second calculation is much simpler than the first, it is this second calculation that is rather misleading. There's an underlying assumption in this calculation that the universe is not expanding. The universe is expanding, so it's not quite valid to multiple very long times such as 13 billion years by the speed of light to get 13 billion light years. In fact, that distant galaxy is about 29 billion light years from us now and was only about 3.5 billion light years away when the light was emitted.

David Hammen
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  • Thank you. I am in awe and envious of everyone's ability to understand this. I assume it's General Relativity. – David Wilson Jul 18 '18 at 14:55
  • One thing that is confusing is that Hubble's constant predicts the wrong rate of recession. Or does it? I should get a book. Can you recommend one? Also, you know that the number of megaparsecs in the U x H = c. Is that a coincidence. Is it irrelevant? – David Wilson Jul 18 '18 at 15:02