Confusion in Angular diameter distance

[Arman777]

I am writing an article about the Hubble Tension and when I was looking through the angular diameter distance I get confused over something.

In many articles the angular diamater distance to the LSS defined as the

$$D_A^* = \frac{r_s^*}{\theta_s^*}$$ where $r_s^*$ is the comoving sound horizon distance to the LSS and $\theta_s^*$ is the angular size of the LSS.

The angular diameter distance can be defined as

$$d_A = \frac{D}{\theta}~~~~Eqn. (1)$$
By using the FLRW metric for $d\chi = d\phi = 0$ we can write

$$ds = a(t)S_k(\chi)d\theta$$

By taking the integral

$$D \equiv \int ds = a(t)S_k(\chi)\theta~~~~Eqn. (2)$$

In this case by combining (1) and (2) we can write

$$d_A = \frac{D}{\theta}= \frac{a(t)S_k(\chi) \theta}{\theta} = \frac{S_k(\chi)}{(1+z)}$$

where $1+z = a^{-1}$

$$r \equiv S_k(\chi) =
\begin{cases}
sinh(\chi) & k= -1 \\
\chi & k = 0 \\
sin(\chi) & k = +1
\end{cases}$$

From these definitions, $r_s^* = D$ must be satisfied. But that also does not make sense. So my question is, Am I mixing the notations or these are different definitions ? Beacause in the article its claimed that

$$r_s^* = \int_0^{t_*} c_s(t)\frac{dt}{a(t)} = \int_ {z_*}^{\infty} c_s(t)\frac{dz}{H(z)}$$

but $D = a(t)S_k(\chi)\theta = r$, which does not make sense

Note: I am using the \begin{equation} ds^2 = -c^2dt^2 + a^2(t)[d\chi^2 + S_k^2(\chi)d\Omega^2]\end{equation} as my metric where

$$\chi = \int_0^r \frac{dr}{\sqrt{1-kr^2}}\equiv \begin{cases}
sinh^{-1}(r) & k= -1 \\
r & k = 0 \\
sin^{-1}(r) & k = +1
\end{cases}$$

For the reference at the sound horizon etc see https://arxiv.org/abs/1908.03663 Page 3

For the derivation of the angular diameter distance see

Daniel Baumann. Cosmology Part III Mathematical Tripos, pages 14–16.
http://theory.uchicago.edu/~liantaow/my-teaching/dark-matter-472/lectures.pdf

Last accessed on 2020-10-1.

 

[kimbyd]

I recommend this paper for grasping different distance measures:
https://arxiv.org/abs/astro-ph/9905116

As noted on that paper the $\theta$ values are the distance on the surface (which is a transverse angular distance), $D$ is a distance to the surface (comoving line of sight distance), and $r$ is a comoving distance.

With this notation, simple geometry given a small angle produces a simple relationship between these: $r = D \theta$. In precise terms, $r$ is the comoving distance across the surface of the sphere, which is fine for this case as the sound horizon is ~1 degree.

You have to correct this result for spatial curvature, if that is non-zero (as described in the Hogg paper), but it's pretty easy to do so.

 

[Arman777]

I recommend this paper for grasping different distance measures:
https://arxiv.org/abs/astro-ph/9905116

I have that article actually, I even printed it.

which is a transverse angular distance

As far as I can see transverse angular distance is the $r \equiv S_k(\chi)$ ?

I did not understand...can you elaborate a little bit more ?

My problem is when I look into the definitions, we have something like $r_s^* = D = a(t) S_k(\chi)\theta$ but that seems impossible ?

It is also annoying that every book is different kind of notation. Its so confusing to read something.

So let me write in this way

$$D_A^* = \frac{r_s^*}{\theta_s^*} = \frac{\chi}{\theta_s^*} = \frac{\int c \frac{dz}{H(z)}}{\theta_s^*}$$ for $k=0$ which is okay.

Notice the similarity between the above equation and Eqn. (1)

But then when we try to define the angular diameter distance we are using

$D = r = a(t)S_k(\chi)\theta$ (See the derivation in the Eqn (2))

So there's something wrong because clearly $D \ne r$

I guess my derivation is wrong.

 

[kimbyd]

$r$ is a distance on the surface, here an integral from $t=0$ to $t=t_{s}$ (i.e., the distance light could have traveled by $t=t_s$ since $t=0$). $D$ is a distance to the surface, here measured as an integral from $t=t_{s}$ to $t=t_{now}$. $\theta$ is the angular diameter of the fluctuation on the surface.

 

[Arman777]

I think the problem is it about the notation. As you said the $r_s^*$ actually the distance on the surface, but its also used as the distance to an object like in the comoving coordinate system where we use $r$ ?

Otherwise $D_A^* =\frac{r_s^*}{\theta_s^*}$ does not make much sense.