# Does "Angular Diameter Turnaround Point" solve the Great Wall Problem?

According to Wikipedia,

The Hercules–Corona Borealis Great Wall is the largest known structure in the observable universe, measuring approximately 10 billion light-years in length.

But since it's very far away (9.612 to 10.538 billion light-years, given the farthest galaxy found is about 13.5 billion light years), the size of the universe should be relatively smaller back then, making the structure scaled or stretched when we look at it in the sky of today's universe.

To elaborate more, say, when the universe was half its current size some N years ago, it must've taken light N years to travel to the present size of the universe. Now, when we look at it today, this light must look like coming from a distance (radius) proportional to N light-years. For N close to the universe's age, this radius $$>>$$ size of universe N years ago. Hence, these galaxies/structures should be scaled by some factor from their original size!

(EDIT: I learnt that this is called Angular Diameter Turnaround from XKCD's Post)

Question 1: Is this effect generally accounted for in Astronomical observations?

Question 2: If yes, does it solve the apparent controversy that the discovery of the Great Wall contradicts the cosmological principle? If not, how does the Great Wall still break the homogeneity of matter distribution after accounting for this effect?

Note: I would appreciate it if someone include the mathematics of accounting it in the calculations and how it still does not solve the Cosmological Principle Problem.

• 1) No - Space has expanded - this does not mean galaxies should be scaled up. So 2) and 3) do not apply Commented Jun 17, 2022 at 12:22
• @RoryAlsop I think you misunderstood. Galaxies will NOT get scaled because space between them has expanded. But galaxies will APPEAR to be scaled, because when light left it, universe was relatively small. Commented Jun 18, 2022 at 5:15
• Hi, the angular diameter turnaround is a very well known effect, which is of course always taken into account when talking about the size of objects at cosmological distances. Therefore unfortunately I am afraid that it cannot resolve the controversy regarding the Great Wall. In other words, the problem with the Great Wall arises after one as already taken the effect into account Commented Jun 20, 2022 at 9:07
• Hi Monster196883, it seems you unaccepted my answer, which you're of course welcome to. But let me know if there's something that is unclear :)
– pela
Commented Apr 30 at 19:51

### Expansion of the Universe, not of small structures

It is true that large "structures", such as unvirialized clusters, and voids, scale with the expansion of the Universe, but smaller structures such as galaxies are held together by gravity, so they don't follow this scaling.

The expansion of "the Wall" isn't as big as you might think, though. First let me explain how the Wall was discovered: It is not, like clusters and other overdensities, seen "permanently" as a larger-than-average number of galaxies. Rather, its existence was inferred by comparing the number of (transient) gamma-ray bursts (GRBs) coming from that region to the average occurrence of GRBs in the Universe (Horváth et al. 2014). The size you quote is based on the redshift of the most nearby, and the most distant, over-abundance of GRBs in the Wall, which are binned to give $$z_1=1.6$$ and $$z_2=2.1$$, respectively. From these redshifts you can calculate the current distances, which are, respectively, $$d_1 = 15.2$$ and $$d_2=17.8$$ billion lightyears (Glyr)$$^\dagger$$.

You can also calculate the lookback time $$t_L$$ to when the light was emitted, as well as the scale factor $$a$$, i.e. the size of the Universe relative to today. When we look at the near and the far end, we look $$t_{L,1}=9.8$$ and $$t_{L,2}=10.7$$ billion years (Gyr) back in time, when the scale factor was $$a_1=0.38$$ and $$a_2=0.32$$, respectively.

In other words, in the 0.9 Gyr (or 900 Mlyr) that went from the "far" light was emitted till the "near" light was emitted, the Universe expanded by a factor of $$a_1/a_2$$, or roughly 19%. Being an overdensity, however, the Universe has expanded somewhat less in this particular region.

### Angular diameter in cosmology

As you say, there is a "turnover" in the apparent size of any object at cosmological distances. I'll write a thorough post on this soon-ish, when I have time, but for now, let me just say that it is slightly less simple than you write, but the point is that light from distant objects has taken a long time to reach us, and at the time they emitted the light, they were closer to us, hence looking bigger.

This effect is well understood, and most definitely taken into account in all cosmological observations.

### Implications for the Cosmological Principle

The Cosmological Principle states that matter in the Universe is distributed homogeneously and isotropically. Obviously, that isn't true on small scales, meaning that the $$\Lambda$$CDM model — our favored cosmological description of the Universe — only works on scales larger than some size. Observationally, the Universe seems to be mostly homogenous on scales larger than ~500 Mlyr, and theoretically, anisotropies larger than ~1 Glyr shouldn't exist (Yadav et al. (2010) estimate ~1.2 Glyr).

Given that the Wall allegedly is many Glyr across ($$17.8-15.2=2.6\,\mathrm{Glyr}$$, and along the longest dimension it's 6–10 Glyr), taken at face value its existence does indeed challenge $$\Lambda$$CDM.

Importantly, however, one should bear in mind that, firstly the statistical significance of the overdensities have been disputed (e.g. Ukwatta & Woźniak; Christian 2020), and secondly it has been argued that, for GRBs, the scale of homogeneity is larger, almost the size of the observable Universe (Li & Lin 2015).

Nevertheless, should the Wall turn out to be real, it would have major implications for our understanding of cosmology and astronomy. I think though it would depend on how much it really violates the Cosmological Principle. The Principle could still be "sufficiently valid" that we can use it in practice. The equations we use to convert observed redshifts to distances etc. would only be approximate, but not completely useless. But even minor corrections "carry through" to many astronomical diagnostics like luminosity functions, time scales, and various scaling relations.

I don't think there's any reason to believe that the Principle holds true for the entire Universe (which may or may not be infinite). We know that, seemingly, it holds true for the observable Universe, and we think that the cause is inflation, but to my knowledge the Universe could easily be inhomogeneous on scales much, much larger than the observable Universe.

So, the question seems to me to be simply a matter of when the Cosmological Principle breaks down, and of course it would be interesting if that scale is smaller than the observable Universe.

$$^\dagger$$The distances you quote are not the physical distances, but the time that the light has been traveling, divided by the speed of light. The physical distances are larger, because the Universe has been expanding since the light was emitted.