Understanding The Turnover Point of Angular Diameter Distance

I am trying to get a better understanding of cosmological distances, in particular the angular diameter distance which I have also seen referred to as angular size distance.

What I am looking for is a more detailed explanation of why the angular diameter distance "turns over" in the following graph.

Wikipedia states in their article on angular diameter distance that objects beyond a certain redshift have a smaller angular diameter and as such appear larger on the sky.

I have searched around and have been unable to find a complete explanation of this behaviour and have only found brief answers which state the expansion of the universe or the size of the universe.

I would like if possible for a more detailed explanation as well as the physical significance if any at all of what it means for an object to be at this redshift when they emitted light we are now observing.

Thank you for any feedback or and discussion or suggestions where to look. I would also like to include as it may be of some help that I am currently in my final year of undergraduate studies in physics so please do not hold back any details or technicalities!

• It turns out that Peebles coined the phrase "Angular Size Distance" as a synonym for Transverse Co-moving Distance $D_M$. "Angular Diameter Distance", $D_A$ is another concept entirely and is related to the Co-moving distance by $D_A(z)=\frac{D_M(z)}{1+z}$. You need to read this page en.wikipedia.org/wiki/Distance_measures_(cosmology) carefully including the section on 'Alternate Terminology'. Dec 28, 2018 at 20:07
• xkcd.com/2622 May 21, 2022 at 18:57

On the one hand an object spans a smaller angle the farther away it is, as expected. On the other hand, due to the expansion of the Universe and the finite speed of light, very distant objects were closer to us when they emitted the light we see today. At that time they spanned a larger angle.

Any point in a galaxy emits light in all directions; the light emitted from a galaxy's edges in the particular direction where your eyes happens to be, defines how large you perceive it, i.e. the angle you measure the galaxy to span. If the Universe did not expand, you would measure the angular size of a galaxy as the angle between the two green lines here:

However, the Universe does expand, so when the "green" rays arrive at the point we were located when they were emitted, we are no longer there. Luckily, the galaxy also emitted rays in the direction where we are now, so that's what we see, illustrated here by the blue lines:

The galaxy is now far away. In a "normal" world, we would see it as small, illustrated by the red lines below, but since we perceive its size by looking along the blue lines, we see it as larger:

Here is an animated version:

Further complications

The above explanation is a little simplified because expansion not only occurs along the line of sight, but also perpendicular to it. Hence the rays will not take straight lines toward us, but somewhat curved. Moreover, if the Universe is not geometrically "flat", but has a negative or positive curvature, this will also affect the exact path of the rays.

The exact turnover — the threshold between "looking smaller because far away" and "looking larger because closer in the past" — depends on the expansion rate history of the Universe, as well as on the way that light propagates in the Universe, which in turn depends on the densities of the various constituents of the Universe. The interrelationship of these quantities is given by the Friedmann equation.

However, for all viable cosmologies, you will find that the turnover point around a redshift of $$z \sim 1.5$$, i.e. roughly $$9.5\pm0.25$$ billion years ago.

• Thank you so much for the clear and very easy to understand answer. I was just lacking that understanding about exactly how the angular diameter distance varied with the expansion of the universe. Thanks you :) May 15, 2017 at 10:24
• You're very welcome, @Michael.
– pela
May 15, 2017 at 12:32
• @pela fee; free to elaborate. Perhaps there is something special about the speed of the expansion of the universe relative to the Earth at the time the light from a $z\approx 1.62$ galaxy was emitted that makes the curve turnover and peak at around 5.80 billion light years angular distance. Is it possible that when the light was emitted, the relative velocity of the galaxy was equal to the speed of light??? Aug 3, 2022 at 16:41
• @Sheldon No, because the relative velocity depends on the distance to a given galaxy. At any point in time, sufficiently nearby galaxies recedes slower, and sufficiently distant galaxies faster, than the speed of light. The turnover depends on the cosmology, i.e. on the exact values of the Hubble constant, curvature, and density parameters, but for all viable cosmologies, you will find that it’s around z ~ 1.5. This is in the matter-dominated epoch, much later than radiation dominated, and much before dark matter kicks in, so there isn’t really anything special about the exact value.
– pela
Aug 3, 2022 at 19:06
• @Pela To clarify for future readers since a new question points here: surely you meant "much before dark energy kicks in"? Aug 19 at 19:47

While the angular diameter turnover is often explained in terms of the expansion of the universe, the more direct physical cause is that gravitational lensing by the universe's homogeneous mass distribution converges the light from distant sources, making them appear larger.

For example, in the standard FLRW cosmology, you can check explicitly that this effect diminishes as you bring down the density of the universe (raise $$\Omega_k$$). When $$\Omega_k = 1$$ (negligible energy density), objects' angular sizes never grow with distance.