Angular diameter distance

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:

$${\displaystyle S_{k}(r)={\begin{cases}\sin \left(H_{0}{\sqrt {|\Omega _{k}|}}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}<0\\r&\Omega _{k}=0\\\sinh \left(H_{0}{\sqrt {|\Omega _{k}|}}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}>0\end{cases}}}$$ 

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

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.

 

 

The angular size redshift relation describes the relation between the angular size observed on the sky of an object of given physical size, and the object's redshift from Earth (which is related to its distance, ${\displaystyle d}$ , from Earth). In a Euclidean geometry the relation between size on the sky and distance from Earth would simply be given by the equation:

$${\displaystyle \tan \left(\theta \right)={\frac {x}{d}}.}$$ 

where ${\displaystyle \theta }$  is the angular size of the object on the sky, ${\displaystyle x}$  is the size of the object and ${\displaystyle d}$  is the distance to the object. Where ${\displaystyle \theta }$  is small this approximates to:

$${\displaystyle \theta \approx {\frac {x}{d}}.}$$ 

However, in the ΛCDM model, the relation is more complicated. In this model, objects at redshifts greater than about 1.5 appear larger on the sky with increasing redshift.

This is related to the angular diameter distance, which is the distance an object is calculated to be at from ${\displaystyle \theta }$  and ${\displaystyle x}$ , assuming the Universe is Euclidean.

The Mattig relation yields the angular-diameter distance, ${\displaystyle d_{A}}$ , as a function of redshift z for a universe with Ω_Λ = 0.^[1] ${\displaystyle q_{0}}$  is the present-day value of the deceleration parameter, which measures the deceleration of the expansion rate of the Universe; in the simplest models, ${\displaystyle q_{0}<0.5}$  corresponds to the case where the Universe will expand forever, ${\displaystyle q_{0}>0.5}$  to closed models which will ultimately stop expanding and contract, ${\displaystyle q_{0}=0.5}$  corresponds to the critical case – Universes which will just be able to expand to infinity without re-contracting.

$${\displaystyle d_{A}={\cfrac {c}{H_{0}q_{0}^{2}}}{\cfrac {(zq_{0}+(q_{0}-1)({\sqrt {2q_{0}z+1}}-1))}{(1+z)^{2}}}}$$ 

The angular diameter distance ${\displaystyle d_{A}}$  reaches a maximum at a redshift ${\displaystyle z=z_{t}}$  (in the ΛCDM model, this occurs at ${\displaystyle z_{t}\approx 1.5}$ ), such that the slope of ${\displaystyle d_{A}(z)}$  changes sign at ${\displaystyle z=z_{t}}$ , or ${\displaystyle \partial _{z}d_{A}>0~\forall z<z_{t}}$ , ${\displaystyle \partial _{z}d_{A}<0\forall z>z_{t}}$ . In reference to its appearance when plotted, ${\displaystyle z_{t}}$  is sometimes referred to as the turnover point. Practically, this means that if we look at objects at increasing redshift (and thus objects that are increasingly far away) those at greater redshift will span a smaller angle on the sky only until ${\displaystyle z=z_{t}}$ , above which the objects will begin to span greater angles on the sky at greater redshift. The turnover point seems paradoxical because it contradicts our intuition that the farther something is, the smaller it will appear.

The turnover point occurs because of the expansion of the universe and because we observe distant galaxies as they were in the past. Because the universe is expanding, a pair of distant objects that are now distant from each other were closer to each other at earlier times. Because the speed of light is finite, the light reaching us from this pair of objects must have left them long ago when they were nearer to one another and spanned a larger angle on the sky. The turnover point can therefore tell us about the rate of expansion of the universe (or the relationship between the expansion rate and the speed of light if we do not assume the latter to be constant).

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