<p>In spatially flat cosmologies they are the same, and your argument is one way of showing why.</p>
<p>If the spatial curvature is nonzero then they aren't the same. For example, in the Einstein static universe, the angular-size distance is maximal for galaxies on the spatial &quot;equator&quot;, and decreases back to zero beyond that distance, but the recession speed of galaxies at the equator is not <span class="math-container">$c$</span>, but zero like everywhere else.</p>
<p>Your argument fails when <span class="math-container">$k\ne 0$</span> because the recession speed depends on the comoving distance, while the angular-size distance is the reduced circumference (great-circle radius divided by <span class="math-container">$2π$</span>), and those are only equal when <span class="math-container">$k=0$</span>.</p>

CC BY-SA 4.0
2022-08-11T21:16:50.907Z
So for our universe, ignoring any differences due to the local speed of the Earth and Sun's velocities. and the velocity of the distant galaxy then the maximum angular diameter distance occurs where the expansion of the universe is equal to the speed of light when the light from the distant galaxy was emitted, correct?
Sheldon
CC BY-SA 4.0
2022-08-10T20:44:37.670Z
2022-08-10T20:44:37.670Z
benrg
<p>What is the minimum redshift for galaxies receding <span class="math-container">$\geq c$</span> when light was first transmitted that are visible today from Earth?  Is that value the same as maximum angular size distance?</p>
<p>This question is related to this <a href="https://astronomy.stackexchange.com/questions/21006/understanding-the-turnover-point-of-angular-diameter-distance/21017#21017">stackexchange Turnover point of Angular Diameter</a> question.  You can see the maximum in the angular diameter from this diagram.
<a href="https://i.sstatic.net/s3IM9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/s3IM9.png" alt="enter image description here" /></a></p>
<p>I looked up redshift cosmological calculators and came across <a href="https://www.astro.ucla.edu/%7Ewright/CosmoCalc.html" rel="nofollow noreferrer">Ned Wright's Cosmological redshift</a> calculator.  If I put in a z-factor redshift <span class="math-container">$z\approx 1.62$</span>, then I get the approximate value for the maximum angular size distance of <span class="math-container">$\approx$</span> 5.80 billion light years, for light that was emitted <span class="math-container">$\approx$</span> 9.69 billion years ago.</p>
<p>Does this maximum in the angular diameter curve correspond to light emitted from a distant galaxy that was originally receding from us at exactly the speed of light?  I think that as gravity slows down the expansion of the universe then the light from that galaxy finally begins approaching and makes its way towards Earth; previously having &quot;stalled&quot; at the maximum angular size limit.</p>
<p>The angular size distance is pretty cool in that galaxies with a larger redshift than this maximum begin to have a <em>larger angular</em> diameter with increasing distance until we finally get to the CMB with a redshift <span class="math-container">$\approx$</span> 1100 where the angular size distance diminishes to <span class="math-container">$\approx$</span> 41 million light years when the light was first transmitted.</p>

<redshift><angular-diameter><cosmological-horizon>
Minimum redshift for galaxies receding $\geq c$ when light transmitted
Sheldon
# Minimum redshift for galaxies receding $\geq c$ when light transmitted

(Asked by: username_0 on 2022-08-05)

What is the minimum redshift for galaxies receding $\geq c$ when light was first transmitted that are visible today from Earth? Is that value the same as maximum angular size distance?

This question is related to this [stackexchange Turnover point of Angular Diameter](https://astronomy.stackexchange.com/questions/21006/understanding-the-turnover-point-of-angular-diameter-distance/21017#21017) question. You can see the maximum in the angular diameter from this diagram.
[![enter image description here](https://i.sstatic.net/s3IM9.png)](https://i.sstatic.net/s3IM9.png)

I looked up redshift cosmological calculators and came across [Ned Wright's Cosmological redshift](https://www.astro.ucla.edu/%7Ewright/CosmoCalc.html) calculator. If I put in a z-factor redshift $z\approx 1.62$, then I get the approximate value for the maximum angular size distance of $\approx$ 5.80 billion light years, for light that was emitted $\approx$ 9.69 billion years ago.

Does this maximum in the angular diameter curve correspond to light emitted from a distant galaxy that was originally receding from us at exactly the speed of light? I think that as gravity slows down the expansion of the universe then the light from that galaxy finally begins approaching and makes its way towards Earth; previously having "stalled" at the maximum angular size limit.

The angular size distance is pretty cool in that galaxies with a larger redshift than this maximum begin to have a *larger angular* diameter with increasing distance until we finally get to the CMB with a redshift $\approx$ 1100 where the angular size distance diminishes to $\approx$ 41 million light years when the light was first transmitted.

**Tags:** redshift, angular-diameter, cosmological-horizon

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## 1 answers

### Answer by username_1 on 2022-08-10. Score: 2

In spatially flat cosmologies they are the same, and your argument is one way of showing why.

If the spatial curvature is nonzero then they aren't the same. For example, in the Einstein static universe, the angular-size distance is maximal for galaxies on the spatial "equator", and decreases back to zero beyond that distance, but the recession speed of galaxies at the equator is not $c$, but zero like everywhere else.

Your argument fails when $k\ne 0$ because the recession speed depends on the comoving distance, while the angular-size distance is the reduced circumference (great-circle radius divided by $2π$), and those are only equal when $k=0$.

> **Comments:**
> **username_0**: So for our universe, ignoring any differences due to the local speed of the Earth and Sun's velocities. and the velocity of the distant galaxy then the maximum angular diameter distance occurs where the expansion of the universe is equal to the speed of light when the light from the distant galaxy was emitted, correct?  

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