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A redshift-angular size relation (or angular size-redshift relation) relates the redshift to how large a known-size object looks at that redshift, i.e., it relates redshift to the ratio between the angular diameter of object and its actual diameter. In simple (Euclidean) space, this is straight forward: for small angles, the size is virtually proportional to the angle. But given curvature and the Hubble expansion presumed in currently accepted cosmological models (with general relativity, such as the Lambda-CDM model), the relation gradually diverges from the Euclidean proportion with distance and beyond a certain redshift, additional distance increases their angular size rather than decreasing it. At the point where distance where angular size increases with distant, they appear dimmer: their apparent surface brightness is decreasing, unlike the constant surface brightness associated with the simple inverse square law. These effects are present in the most distant galaxies now observed.
A simplified measure of distance that totally ignores curvature is the angular diameter distance, which is the distance determined from the angular diameter (of an object of known diameter) calculated as if space were Euclidean. As per the above-mentioned effects, it is inaccurate at long distances, but some calculations devised to produce more accurate determinations begin with this measure and apply an adjustment that approximates the difference.
Such redshift-angular size relations can be derived for various Lambda-CDM parameters and Friedmann models; observations and analysis revealing the actual relation can be used to select and tweak such models. A strategy for determining the actual function (through the course of the universe's history) via observation is to compare the brightness with the angular size of distant galaxies, e.g., by measuring the angular size of the brightest galaxies (radio galaxies are suitable) at various redshifts, assuming they would have similar size. Other proxies for size have been tried, including some function of brightness, and also additional galaxy characteristics.