I am having some difficulty wrapping my head around distance measures in cosmology. From my understanding LOS comoving distance is the radial distance from us the observer to the object of interest while taking into account the universe's expansion, thus line of sight comoving distance. However in a paper I am reading (Distance measures in cosmology, 2000, David W. Hogg) transverse comoving distance seems to be equal to LOS comoving distance, assuming the universe is flat (which seems to be an accepted assumption, from my reading), so what is the difference?
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The Transverse Comoving Distance $D_M$ is used to calculate the Comoving Distance $D_{12}$ between two events with the same redshift separated in the sky by an angle $\boldsymbol{\delta \theta}$ $$\boxed{D_{12}=D_M \cdot \delta \theta}$$ If $D_C$ is the Comoving distance between any of the 2 events and us, (LOS)
$$D_C=D_H \int_0^z \frac{dz'}{E(z')}$$
Then,
for $$\Omega_k>0 \Rightarrow D_M=D_H\frac{1}{\sqrt{\Omega_k}}\sinh \left ( \sqrt{\Omega_k} \, D_C/D_H\right )$$
for $$\Omega_k=0 \quad \Rightarrow \quad D_M=D_C$$
and for $$\Omega_k<0 \Rightarrow D_M=D_H\frac{1}{\sqrt{|\Omega_k|}}\sin \left ( \sqrt{|\Omega_k|} \, D_C/D_H\right )$$
If you are not interested in the Comoving Distance between two events with the same redshift separated in the sky by an angle $\delta \theta$, you do not need to use the concept of Transverse Comoving Distance.
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$\begingroup$ You didn't address the question. $D_A$ is used often as a factor for calculating Luminous Distance ($D_L$) as $D_L = (z + 1)^2 D_A$, so I'm not interested in it's use so much as whether it is, in fact, in a flat universe, the same value one would get when calculating $D_C$. $\endgroup$ Commented Apr 10, 2020 at 21:41