Yet another Question about the expansion of universe

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.

• 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. Jul 15 '18 at 22:01
• 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
• Today that galaxy is receding at much more than the speed of light. Jul 16 '18 at 21:47

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.