<p>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.</p>

<p>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.</p>

<p>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.</p>

CC BY-SA 4.0
2018-07-18T14:55:15.460Z
Thank you. I am in awe and envious of everyone's ability to understand this. I assume it's General Relativity.
David Wilson
CC BY-SA 4.0
2018-07-18T15:02:39.377Z
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
CC BY-SA 4.0
2018-07-17T17:50:54.807Z
2018-07-17T17:50:54.807Z
David Hammen
<p>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.</p>

CC BY-SA 4.0
2018-07-15T20:02:57.123Z
Just read about [co-moving distance](https://en.wikipedia.org/wiki/Comoving_and_proper_distances).
Mithoron
CC BY-SA 4.0
2018-07-15T20:32:38.297Z
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](https://astronomy.stackexchange.com/q/23105/5264), [this](https://astronomy.stackexchange.com/q/1948/5264), [this](https://astronomy.stackexchange.com/q/1860/5264), [this](https://astronomy.stackexchange.com/q/1001/5264), and/or [this](https://astronomy.stackexchange.com/q/6514/5264) question. To get you started, the Universe is much larger than 13.8 Glyr because it expands, and expansion _is_ faster than light.
pela
CC BY-SA 4.0
2018-07-15T22:01:57.900Z
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
CC BY-SA 4.0
2018-07-15T22:55:53.457Z
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
CC BY-SA 4.0
2018-07-16T12:43:52.480Z
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
CC BY-SA 4.0
2018-07-16T12:48:10.060Z
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
CC BY-SA 4.0
2018-07-16T12:50:23.657Z
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
CC BY-SA 4.0
2018-07-16T21:47:22.883Z
Today that galaxy is receding at much more than the speed of light.
ProfRob
CC BY-SA 4.0
2018-07-18T15:52:52.797Z
@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
CC BY-SA 4.0
2018-07-18T16:13:21.093Z
@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
CC BY-SA 4.0
2018-07-19T09:55:36.873Z
Okay okay @RobJeffries, I meant no offense. You are of course right that that was the intention of the question :)
pela
<universe><observable-universe><size>
Yet another Question about the expansion of universe
David Wilson
# Yet another Question about the expansion of universe

(Asked by: username_0 on 2018-07-15)

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.

**Tags:** universe, observable-universe, size

> **Comments:**
> **username_1**: Just read about [co-moving distance](https://en.wikipedia.org/wiki/Comoving_and_proper_distances).  
> **username_2**: 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](https://astronomy.stackexchange.com/q/23105/5264), [this](https://astronomy.stackexchange.com/q/1948/5264), [this](https://astronomy.stackexchange.com/q/1860/5264), [this](https://astronomy.stackexchange.com/q/1001/5264), and/or [this](https://astronomy.stackexchange.com/q/6514/5264) question. To get you started, the Universe is much larger than 13.8 Glyr because it expands, and expansion _is_ faster than light.  
> **username_0**: 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.  
> **username_0**: 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  
> **username_2**: 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).  
> **username_2**: 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".  
> **username_2**: 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.  
> **username_3**: Today that galaxy is receding at much more than the speed of light.  
> **username_2**: @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.  
> **username_3**: @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 $z\simeq 8$.  
> **username_2**: Okay okay @RobJeffries, I meant no offense. You are of course right that that was the intention of the question :)  

---

## 1 answers

### Answer by username_4 on 2018-07-17. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you. I am in awe and envious of everyone's ability to understand this. I assume it's General Relativity.  
> **username_0**: 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?  

---

