Comoving distance vs physical distance

Question

In a recent post in Medium.com Ethan Siegel wrote the following:

"... as you look farther away, objects appear smaller until a critical point: a minimum size that objects will appear in our Universe, which occurs for objects that are somewhere around 15 billion light-years away. Beyond that, they start to appear larger again; if something comes from us close by or very far away, they will appear to be the same angular size on the sky."

Is this statement correct? The comoving distance $D_M = 15$ Glyr corresponds to about $z = 1.6$, ok. The author's text refers to the (physical) angular diameter distance $D_A(z)$ from which his conclusion is derived for an object of fixed size. But, from our point of view, since we are comoving with expansion, shouldn't we use the transverse comoving distance $D_M = D_A(1+z)$ instead to get the correct answer, i.e. that the size of an object is always decreasing with cosmic distance?

Some relevant resources: Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe, by Tamara M. Davis and Charles H. Lineweaver. A popular version of the key points of that paper: Misconceptions about the Big Bang. Also see Professor Davis's site. — PM 2Ring, May 19 '22 at 04:57
@PM2Ring -I understand that space (having no mass) can expand at any speed, and that objects that would've left the Hubble sphere would be invisible to us until that sphere's own expansion would've enveloped them again, so, is your impression that Siegel was saying that the increase in the apparent size of stars that had been invisible to us while they had remained outside that sphere would've resulted from the approach of their photons (the photons comprising their image) toward us after its expansion would've enveloped them again? — Edouard, May 21 '22 at 13:19
I have to admit, that seems to provide a clearer resolution to the OP's conundrum than the one I'd had in mind, although maybe the two possibilities (mine and the common sense blog's) have some neurological relation to each other. (The one time I was in Boston there was an ice storm going on, so, on 1st reading of the blog, I didn't realize that it was referring to the depth of modern Boston below the prehistoric ice sheet that once covered its locale a little more solidly: There's a weird relationship between the two possibilities.) — Edouard, May 22 '22 at 10:10

Andrew · Answer 1 · 2022-05-19T01:26:12.590

2

The angular diameter distance is the distance to a standard ruler you would infer based on its angular size on the sky. (By "angular size", I mean the number of degrees an object takes up on the sky). In equations, given the size of the ruler $L$, and the angular size $\theta$, the angular diameter distance is defined as \begin{equation} d_A = \frac{L}{\theta} \end{equation}

It is indeed true that the angular diameter distance increases with redshift for a while, hits a maximum, and then starts to decrease. Because of the way angular diameter distance is defined, an object with a larger angular diameter distance has a smaller angular size. This implies the result that Ethan writes. If you were to move an object further and further away from us, its angular size would decrease, then hit a maximum, and then start to increase.

To make a philosophical comment: the idea behind distance measures in cosmology is that there are many ways to measure distance in a non-expanding Universe that are all equivalent, but that are no longer equivalent in an expanding Universe. So, we can define multiple measures based on different non-expanding ways of measuring distance, that will all capture different ways of thinking about distance in an expanding Universe. The key things to pay attention to are how these measures are defined, and how they are related to each other. You shouldn't necessarily think that angular diameter distance gives you an "intuitive" definition of distance (personally I find comoving distance much more intuitive); it's just the distance you would infer for an object given its angular size.

edited May 19 '22 at 01:26

answered May 19 '22 at 01:13

Andrew

48,573

Thanks for the speedy answer.It is clear that the ruler moved at increasing distance must stay on our past light cone to remain visible to us. My question is, why is the light cone described by DA(z) which has a maximum at z=1.6, and not by DM(z) in comoving coordinates which is steadily increasing? – Rene Kail May 19 '22 at 05:12
@ReneKail The angular diameter distance does not describe the light cone. It describes the apparent distance you would infer to an object based on its angular size at the time you observe it, and the flat space formula relating angle and distance. – Andrew May 20 '22 at 01:12
Andrew: OK, angular diameter distance per se is a general concept. But in order to be able to observe an object or a source it must lie on your past light cone. Therefore the equations for DA or the comoving DM in function of z or cosmic time are also the equations for the past light cone and we can use DA or DM to "describe" the past light cone in expanding flat space. – Rene Kail May 20 '22 at 21:55
@ReneKail I don't understand your comment. Are you asking a question about the lightcone? – Andrew May 20 '22 at 22:01
I think my question about "Comoving distance vs physical distance" can be settled. I am referring to your comment about 21 hours ago, and this precisely refers to the past light cone. So I am asking if my comment addressed to you (Andrew) is correct. – Rene Kail May 20 '22 at 22:34
@ReneKail I guess I don't understand what "the light cone described by $D_A(z)$" means. The light cone is a null hypersurface defined by null geodescis emanating from a point. The light cone itself does not depend on angular diameter distance or comoving distance at all. – Andrew May 20 '22 at 22:45
Yes, I gree with this, there is no dependence. But then, I ask why are the equations and the graphs in f(z) for the angular diameter distance and the light cone the same, e.g. both show the same maximum at z around 1.6? Therefore it seems to me they are at least somehow related. – Rene Kail May 21 '22 at 00:46
@ReneKail Can you point me to the graphs you mean? – Andrew May 21 '22 at 01:11
Yes. Take any graph in the renowned paper of Davis and Lineweaver astro-ph

arXiv:astro-ph/0310808 and look at the graphs in Fig.1 labelled "Light Cone" ignoring the remaining curves. The past light cone curves are given in function of proper physical distance and in function of comoving distance for flat LCDM. Now compare these light cone curves to any standard plot of DA(z) resp. DM(z) for flat space. They will be exactly the same. As a test you can take any value of the scale factor a and infer the distance and compare to the standard value.

– Rene Kail May 21 '22 at 02:21
@ReneKail I still don't understand. You are claiming that the light cone curve in this figure is the same as a curve involving the angular diameter distance, but you haven't shown me a plot where the two curves are compared. I actually don't understand what it would mean to compare the light cone in figure 1 to an angular diameter distance curve. The light cone is a 2 dimensional hypersurface. Normally when angular diameter distance is plotted, it is as a 1-dimensional function of redshift. – Andrew May 21 '22 at 02:25
Andrew, I completely agree with you: As geometrical object the light cone is a 2D hypersurface with zero thickness. When I imagine making a (vertical) cut along the time axis I get the same curve in f(z) as the graphs show. I think now we have resolved our misunderstanding. – Rene Kail May 21 '22 at 02:53
Andrew, I completely agree with you: As geometrical object the light cone is a 2D hypersurface with zero thickness. When I imagine making a (vertical) cut along the time axis I get the same curve in f(z) as the graphs show. I think now we have resolved our misunderstanding. Mathematically this is what the textbook says: FRW light cone equation in comoving coordinates is Chi = Eta(0) - Eta and this exactly is the comoving distance, ie the DM(z) resp. the DA(z)=DM/(1+z) in the flat case! – Rene Kail May 21 '22 at 03:05
Let us continue this discussion in chat. – Andrew May 21 '22 at 15:43
Thank you, Andrew, for the time you spent with me. I think we both can now agree to the following summary: Our past light cone is this nice teardrop rotationally symmetric hypersurface about the time axis tangent to an infinity of concentric hyperspheres in physical coordinates or, respectively a straight tentlike structure in comoving coordinates. Distances DA(z) resp. DM(z) are the one-dimensional outlines ON these hypercones and these "collapsed" curves are labelled "light cone" to simplify things in most of the textbooks. Kind greetings from Switzerland. René. – Rene Kail May 21 '22 at 15:57
I hope I won't be confusing the issue as much as I sometimes do, but the "light cone" to which you're referring wouldn't be the hubble sphere in Lineweaver's & Davis's "Proper Distance" cross-section of space time, would it? I think its teardrop shape, with its point at "now", just delineates that re-entry of photons as astronomical bodies they'd subsequently comprised into the sphere they would've previously left, 1.4 gly after the BB. (The "sphere" in "Hubble sphere" would, in my reading of the situation, just be referring to "sphere of influence", not necessarily to any physical shape.) – Edouard May 22 '22 at 10:36
Of course, not! The light cones drawn in the diagrams of Fig. 1 by D. and L. are very different from the Hubble sphere evolution. The latter has NOT a point of reentry from any photons from the past, except at infinite time in the comoving diagram. In contrast, our past light cone, while gathering photons from the past, reaches us at the present time. And it is also clear that DA(z) and DM(z), where z is the observed cosmological redshift, must trace out the past light cone since pointlike emitting sources (like SNe) are situated at these distances in order to be visible today. – Rene Kail May 22 '22 at 16:50

score 0 · Answer 2 · answered May 19 '22 at 23:31

After some investigation about my question I found that the statements of Ethan is correct and corresponds to the textbooks of cosmology. I realized that the distance DA(z) is the observed cosmological distance to an object of a fixed transverse size at redshift z, whereas the distance DM(z) is the distance to a linear (expanding) transverse separation in space. The latter does not apply for calculation of the angular diameter.

Comoving distance vs physical distance

2 Answers2