Angular size redshift relation

The angular diameter distance is a distance measure used in astronomy. It is defined in terms of an object's physical size, ${\displaystyle x}$, and ${\displaystyle \theta }$ the angular size of the object as viewed from earth.
${\displaystyle d_{A}={\frac {x}{\theta }}}$
The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, ${\displaystyle z}$, is expressed in terms of the comoving distance, ${\displaystyle r}$ as:
${\displaystyle d_{A}={\frac {S_{k}(r)}{1+z}}}$
Where ${\displaystyle S_{k}(r)}$ is the FLRW coordinate defined as:
${\displaystyle S_{k}(r)={\begin{cases}\sin \left({\sqrt {-\Omega _{k}}}H_{0}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}<0\\r&\Omega _{k}=0\\\sinh \left({\sqrt {\Omega _{k}}}H_{0}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}>0\end{cases}}}$
Where ${\displaystyle \Omega _{k}}$ is the curvature density and ${\displaystyle H_{0}}$ is the value of the Hubble parameter today.

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