Angular Diameter Distance The definition of the angular diameter distance is the ratio of an object's physical transverse size to its angular size. However when I was reading my textbook, Astrophysics in a Nutshell by Dan Maoz pp.220-221, I am having some trouble trying to understand the notion of angular diameter distance to the last scattering surface. The text calculates the angular diameter distance to the last scattering surface $D_A$:


  Consider flat cosmology (k=0) with no cosmological constant. We wish to calculate the angular size on the sky, as it appears today of a region of physical size
  $$D_s=\frac{2ct_{rec}}{\sqrt{3}}=140 kpc$$ from which light was emitted at time $t_{rec}$. Between recombination and the present time, the Universal expansion is matter-dominated, with $R\propto t^{2/3}$ for this model
  $$\frac{R}{R_0}=\left( \frac{t}{t_0} \right)^{2/3} = \frac{1}{1+z}$$
  and hence we can write
  $$D_s=\frac{2ct_0}{\sqrt{3}}(1+z_{rec})^{-3/2}$$
  The angle subtended by the region equals its size, divided by its distance to us at the
  time of emission (since that is when the angle between rays emanating from two
  sides of the region was set).  


I'm not sure what does the last line actually mean..Can someone please elaborate more on this? I just simply take the $D_s$ as the "physical transverse size".


  As we are concerned with observed angles, the type
  of distance we are interested in is the distance which, when squared and multiplied
  by 4π, will give the area of the sphere centered on us and passing through the
  said region. If the comoving radial coordinate of the surface of last scattering is r, the required distance is currently just $r\times R_0$ and is called the proper motion distance. The proper motion distance can be solved using null geodesic in the FRW metric
  $$\int_{t_{rec}}^{t_0} \frac{c dt}{R(t)} = \int_{0}^{r}\frac{dr}{\sqrt{1-kr^2}}$$
  Setting k = 0, and substituting
  $$R(t)=R_0 \left( \frac{t}{t_0} \right)^{2/3}$$ and integrate
  $$rR_0=3ct_0[1-(1+z_{rec})^{-1/2}]$$   


So I take this as the physical distance of the region from us. The next part is what confuse me:


  However, at the time of emission, the scale factor of the Universe was 1 + z times smaller. The so-called angular diameter distance to the last scattering surface is therefore
  $$D_A=\frac{rR_0}{1+z}=3ct_0[(1+z_{rec})^{-1}-(1+z_{rec})^{-3/2}]$$  


How does a physical distance $rR_0$ comes into play in the angular diameter distance, because from its definition it is just $$D_A=\frac{\text{physical transverse size}}{\text{angular size}}$$??
How does a physical distance $rR_0$ comes into play in the angular diameter distance, because from its definition ...


The manner you choose to calculate it is explained on Wikipedia's "Angular diameter distance" webpage:

The angular diameter distance is a distance measure used in astronomy. It is defined in terms of an object's physical size, $x$, and $\theta$ the angular size of the object as viewed from earth.

$$d_{A}={\frac {x}{\theta}}$$

The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, $z$, is expressed in terms of the comoving distance, $r$ as:

$$d_{A}={\frac {S_{k}(r)}{1+z}}$$

Where $S_{k}(r)$ is the FLRW coordinate defined as:

$$S_{k}(r)={\begin{cases}\sin \left({\sqrt {-\Omega _{k}}}H_{0}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&amp;\Omega _{k}&lt;0\\r&amp;\Omega _{k}=0\\\sinh \left({\sqrt {\Omega _{k}}}H_{0}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&amp;\Omega _{k}&gt;0\end{cases}}$$

Where $\Omega _{k}$ is the curvature density and $H_{0}$ is the value of the Hubble parameter today.

In the currently favoured geometric model of our Universe, the "angular diameter distance" of an object is a good approximation to the "real distance", i.e. the proper distance when the light left the object. Note that beyond a certain redshift, the angular diameter distance gets smaller with increasing redshift.

See also Wikipedia's "Distance measures (cosmology)":



$$\begin{array}{l}
\text{Comparison of cosmological distance $\qquad\quad\;\,$ Comparison of cosmological distance} \\
\text{measures, from redshift 0 to redshift 0.5. $\qquad$ measures, from redshift 0 to redshift 10K.} \\
\hline
\qquad\text{The background cosmology is Hubble parameter 72 km/s/Mpc, $\Omega_\Lambda=0.732$,} \\
\qquad\text{$\Omega_{\rm matter}=0.266$, $\Omega_{\rm radiation}=0.266/3454$, and $\Omega_k$ chosen so that the sum of} \\
\qquad\text{Omega parameters is 1.} 
\end{array}$$

Notice how, even at a small $z$, the choice of cosmological model is important for full accuracy.



The book "Astrophysics in a Nutshell: Second Edition" was published Feb 23, 2016, the first edition was published on Dec 4, 2011. The first edition is old (and 25% of the cost of), the second edition is not new.

Wikipedia explains the Lambda-CDM model:


  The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda (Greek Λ), associated with dark energy, and cold dark matter (abbreviated CDM). It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos:
  
  
  the existence and structure of the cosmic microwave background
  the large-scale structure in the distribution of galaxies
  the abundances of hydrogen (including deuterium), helium, and lithium
  the accelerating expansion of the universe observed in the light from distant galaxies and supernovae
  
  
  The model assumes that general relativity is the correct theory of gravity on cosmological scales. It emerged in the late 1990s as a concordance cosmology, after a period of time when disparate observed properties of the universe appeared mutually inconsistent, and there was no consensus on the makeup of the energy density of the universe.
  
  The ΛCDM model can be extended [tweaked to fix it, depending on what you are doing] by adding cosmological inflation, quintessence and other elements that are current areas of speculation and research in cosmology.
  
  Some alternative models challenge the assumptions of the ΛCDM model. Examples of these are modified Newtonian dynamics, modified gravity, theories of large-scale variations in the matter density of the universe, and scale invariance of empty space.


The reason to nitpick about a tiny difference of opinion is because the distances involved are so enormous, and the difference really isn't so small either. Depending on the distance the amount of time that has passed means that the space through which the light is passing changes somewhat throughout it's lifetime.

See also: "Scale invariant cosmology III: dynamical models and comparisons with observations" (19 May 2016) by André Maeder:


  The basic equations of cosmology are modified, showing an acceleration of the expansion after a certain initial period, the duration of which depends on the mean density of the Universe. Another major consequence of the scale
  invariance is that the laws of conservation of matter-energy show some dependence on the cosmic time. This dependence is very weak for models with a non-zero matter density, but at the conceptual level this is not a minor effect.
  
  We do think it is worth to undertake the present exploration for two main reasons. One is that the recent cosmological results suggest that a totally unknown form of matter-energy, the dark energy, dominates the energy content of the Universe. This is a major problem. The other main reason is that scale invariance is not a kind of adjusted trick to make things work. But it is a basic physical change, that responds to the fundamental wish (Dirac 1973) that the equations expressing basic laws should be invariant under the widest group of transformations.


The newer paper "An alternative to the LCDM model: the case of scale invariance" (14 Jan 2017), by André Maeder contains the calculations ($d_M = R_0 r_1$, $d_A = d_M/(1+z)$, on pages 13 and 14) and the following graph:



Figure 5. The angular diameter distance $d_A$ vs. redshift $z$ for flat scale invariant models (continuous red lines) compared to flat ΛCDM
models (broken blue lines). The curves are given for Ωm = 0, 0.1, 0.3, 0.99, from the upper to the lower curve in both cases (at $z$ > 3).

Please refer to that paper for details on the derivation of the calculations.