Will the diameter of a cluster of galaxies change with redshift?
If a cluster of galaxies today is observed at a net redshift of, say z=0.20 and the observed diameter of the cluster is x Megaparsecs, way out in the future if the net redshift of the same cluster is measured to be z=0.22 (due to the cosmic expansion) would its observed diameter still be x Megaparsecs? I know that the motions of the member galaxies of a cluster are described by Newtonian Mechanics and the actual diameters will not change but what would we observe from our rest frame here on earth as far as redshift versus apparent cluster diameter?
My inability to find an answer to the above arises from the fact that within a cluster most of the spacetime is void of light emitting baryonic matter, and how would, then, this seemingly "empty" space appear to us in the face of an expanding universe?
The answer to your question is "more or less yes". The reason is that in a cluster of galaxies, the mutual gravity between the various galaxies is able to surmount the cosmic expansion. Thus within the cluster, the galaxies will not be expanding away from each other. Instead their motions will be governed by the complex gravitational potential of the cluster itself.
As a simple example of this, the Andromeda spiral galaxy, which is a member of the local group along with the Milky Way is currently coming towards us rather than receding from us as it should be if its motion is dominated by cosmic expansion. Here again, because the members of a group are gravitationally bound, they will not obey cosmic expansion.
Technically, the speeds of individual galaxies in a cluster other then the joint recession (since the cluster as a whole is receding away from us due to expansion) are called "peculiar velocities". Our own galaxy is falling towards the Virgo cluster at the current moment. So, a cluster will more or less retain its size rather than expand with the expansion of the Universe. As a result, when the cluster gets farther away from us, only its angular size will decrease.
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