I have a couple of questions about that paper.
In section 4, the paragraph bracketed by equations (10) and (11) makes three references to proper distance. Consulting a
Wikipedia article for definitions of the various distance measures, it looks to me as though all three of that paragraph's references to proper distances should have referred to comoving distances. It seems to me that is a matter of some importance, because (in an expanding universe, in the context of that paragraph) proper distances differ from comoving distances by a factor of 1+z. Can you comment on this?
In a simply expanding space, comoving distance is:
http://latex.codecogs.com/gif.latex?d_C = z \frac{c}{H_0}
Proper distance is:
http://latex.codecogs.com/gif.latex?d_p = \frac{d_C}{1+z}
And luminosity distance is:
http://latex.codecogs.com/gif.latex?d_L = d_C(1+z) = d_p(1+z)^2
The author seems to have that straight, unless I'm missing something.
No, the author doesn't have that straight.
From the last equation you wrote above, the luminosity distance is greater than the proper distance by a factor of (1 + z)
2. But the author says:
Redshift and time dilation cause a luminosity distance greater than the proper distance by (1 + z).
Now that might just be another example of careless/sloppy wording, as in Figure 3 and its caption. But the author goes on to say
The expanding metric makes the angular diameter distance smaller than the proper distance by (1 + z).
According to Wikipedia, however:
dA = dM / (1 + z)
where d
M is
not the proper distance, but is instead the transverse comoving distance, which is equal to the comoving distance d
C if space is flat. From your second equation, and assuming flat space, the proper distance is
dP = dM / (1 + z)
so
dA = dP
In other words, the angular distance should be the same as the proper distance (for flat space).
The author's equation (11) is derived from the author's belief (which I quoted above) that (for an expanding universe) the angular distance is "smaller than the proper distance by (1 + z)." As I have just shown, however, using your equations and those of Wikipedia, the angular distance is actually equal to the proper distance (assuming flat space).
If that is the error it appears to be, then the entire paper is worthless.
And it does appear to be some kind of error. After writing the above, I consulted Weinberg's
Cosmology. Weinberg's equation (1.4.11) says the angular distance is
dA = a(t1)r1
For flat space, the right hand side of that equation is the proper distance (as given by Weinberg's equation (1.1.15)).
At the very least, it seems the author of that paper is writing "proper distance" when he should be writing "comoving distance" or "transverse comoving distance". But if he really believes the angular distance differs from the proper distance by a factor of (1 + z), and derives his equation (11) by removing that factor, as appears to be the case, then I think that's a fatal error.
ETA, read it a couple more times. When he says "the proper distance has expanded by a factor of (1+z)", he's referring to the proper distance now, aka comoving distance.
If he's referring to the comoving distance, he should say so. By writing "proper distance" when, on your reading, he means the proper distance today, i.e. the comoving distance, he confuses readers such as myself who know just enough to check what he wrote but do not know enough to intuit what he really meant.
At the very least, it's a sloppy paper. That much is obvious from Figure 3 and its caption.
The author completed his PhD (in astronomy) at Case Western Reserve University in 2020, and his BS (in Engineering) at Huazhong University of Science and Technology in 2012. He is currently a postdoc at Case Western Reserve University. A young researcher who is probably not a native speaker of English can be forgiven some sloppy prose, but there may be a real question here as to whether his sloppiness carried over into his calculations.
