angular diameter distance

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The angular diameter distance is a distance measure used in astronomy. The angular diameter distance to an object is defined in terms of the object's actual size, $x$ , and $θ$ 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, $χ$ as:
$d_A = \frac{r(\chi)}{1+z}$
Where $r (χ)$ is defined as:
$r(\chi) = \begin{cases} \sin \left( \sqrt{-\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k < 0\\ \chi & \Omega_k=0 \\ \sinh \left( \sqrt{\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k >0 \end{cases}$
Where $Ω k$ 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." However, beyond a certain redshift it gets smaller with increasing redshift. In other words an object "behind" another of the same size, beyond a certain redshift, appears larger on the sky, and would therefore have a smaller "angular diameter distance".

angular diameter distance

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