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When items are further away, they look smaller. If we know the size of something nearby, then we can calculate the size of something a known distance far away. So if there is a galaxy whose size (dimensions not mass) we know and an 'identical looking galaxy' much further away, and we know the distances these galaxies/stars are from us, then we can calculate the size of the galaxy far away.

But the further away galaxy is older and the light it emitted happened when the universe was smaller. So the further away galaxy should appear larger because it appears closer. Is this the case?

There is always the counterargument that you cannot measure the expansion of space by looking at gravitationally bound objects.

My question is specifically about the availability of data regarding the size of distant galaxies, not the arguments against.

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Beyond a certain point (the turnover point) galaxies of the same size will indeed look larger. This corresponds to a redshift of $z = 1.5$ in the ΛCDM model cosmological model. The relationship between angular scale and redshift is plotted on the Wikipedia page (Wikipedia: Angular Diameter Distance).

While the size of galaxies has been proposed as a "standard ruler", this may not be as accurate as other techniques. Very distant galaxies correspond to earlier cosmological times. Being earlier in their evolution, they will differ from nearer galaxies.

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