# Distance of distant galaxies

Can two galaxies, one with a redshift of 7 and the other a redshift of 8 have the same angular distance (distance from the Earth during the emission of the observed photons)?

If not why ?

Example: Using Ned Wright's calculator, a galaxy with a redshift of 8 has an angular distance of 3.31 billion light years and its image was emitted 0.646 million years after the big bang.

Logically another galaxy could be at the same angular distance 3.31 but would have emitted the light we observe today later in the history of the universe (when the expansion rate of the universe had decreased) say 0.900 million years after the big bang it would therefore have a different redshift but an identical angular distance. But this is not what I observe in this calculator, as soon as we reduce the redshift it increases the angular distance and the moment when the image we observe was emitted

Here is the calculator: https://www.astro.ucla.edu/~wright/CosmoCalc.html

Redshift, angular-size distance, and time-since big bang are all ways of describing the distance of an object or (since distance and time are related) the time when the light was emitted.

Now, you could measure distance in "miles", "km" or "journey time at 100km/h" It wouldn't make sense to talk about an town that was 200 miles, 500km and at a journey time of 3 hours. Since all those measures of distance are different.

Likewise, it doesn't make sense to talk about a galaxy at z=8 angular size = 3.31 and time after big bang =800 million years. as those all represent different distances.

For nearby objects, the various measures of distance are equivalent. If I hold my hand in front of my face at a distance of 30cm, it has a "proper distance" of 30cm = 1 light-nanosecond, it has a light-travel-time of 1 nanosecond, and it has an angular distance of 30cm, or 1 light-nanosecond. If I double the (proper) distance to my hand, the time light will take to travel will double to 2 nanoseconds, and the angular size of my hand will half (corresponding to an angular-size distance of 60cm)

For fairly nearby objects the three measures are still roughly equal. Eg at 140 million light years, the light travel time is 139 million light years and the angular size distance is 138.7 million light years. So an object that is 140 million light-years away will look 138.7 million times smaller than an object one light year away.

However when looking over cosmological distances and expanding space, these measures of distance diverge. If you double the proper distance, (say from 2 to 4 billion light years) the light travel time doesn't double. Moreover the angular size can actually increase: things that are further away can actually look bigger.

Now, an object that is 29.8 billion light years away (z=8) will look 3.31 billion times smaller than an object that is 1 light year away. This is now a big difference. And an object that is 28.7 billion light years away will look 3.59 billion times smaller. Note that as you move further away, from 28.7 to 29.8 billion light years, the object actually seems to be getting bigger!

In fact, an object that is 4.46 billion light years away (z=0.344) will also look 3.31 billion times smaller, So two identically sized galaxies at distances 29.8 billion light years and 4.46 billion light years, will appear to be the same angular distance in the sky. However the more distant galaxy would be redshifted further and would be fainter.

Both the angular size distance and the time after the big bang related by being functions of "z", so a time of "800 million years after the big bang" corresponds to z=6.8 and an angular size distance of 3.657. You can't have an object at 800 million years after big bang and any other value for angular size distance. (with the default cosmological parameters)

• But for example, what would be the value of z for a galaxy which would have been 3.31 billion light years from Earth when it would have emitted its light (angular distance) and 800 million years after the big Bang ? It would not be possible if I understand correctly? That's what I'm having a little trouble understanding. Sep 11, 2022 at 18:35
• That doesn't make any sense. If something was 800 million years after the big bang, then its z value would be 6.8, its proper distance would be 28.8 billion light years and its "angular size distance" would be 3.657 billion light years (ie it would be 3.657 billion times smaller than something that was 1 light year away) So it is impossible for something to have angular size distance of 3.31 and be 800 million years after the big bang. Sep 11, 2022 at 18:44
• "angular size distance", "time after big bang" and "redshift" are really just different ways to talk about distance. So your comment is like asking "what is the distance in km of a town that is 300 miles away and takes two hours to drive to at 100mph?" Sep 11, 2022 at 19:09
• Angular diameter turnaround: xkcd.com/2622 Sep 11, 2022 at 21:19
• What confuses me is that I thought the time after the big bang and the angular distance were independent of each other (changing one without changing the other), the closer you get to the big bang the faster the expansion becomes but the farther the object is from the Earth the greater the expansion between it and the Earth Sep 11, 2022 at 23:01