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A comoving reference frame expands along with the universe, factoring out the effect of the Hubble expansion. Suppose a galaxy has a redshift z=1 and its comoving distance DM is 11Gly=const. . This is also the distance we measure today. After 1000 years the distance to that galaxy is measured again:
Do not confuse coordinate differences with proper physical distances. The co-moving coordinates of an object need not be constant, as the object's motion need not be determined by the expansion of the universe only. For example, two galaxies can attract each other and collide.
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