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The Wikipedia article on expansion of the universe states that:

expansion is an intrinsic expansion whereby the scale of space itself changes (...) the metric governing the size and geometry of spacetime itself changes in scale

But at the same time it states that it only affects gravitationally unbound parts, which would imply that the metric on the spacetime of the entire universe doesn't actually change everywhere.

Is this statement a simplification due to the small scale of gravitationally bound parts which make the effects too small to be perceived/observed, or does this statement imply that the scaling of the size/geometry metric is actually related to the vicinity of mass?

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Measurements of distances are relative to local rulers. We do not observe space, we observe distance relationships between objects. Locally rulers maintain the same size relative to themselves, meaning, essentially, that the metric in terms of local rulers does not change. The same applies in gravitationally bound systems; the metric in such a system does not change (providing one does not change the definition of coordinates). Expansion concerns the distances between distant galaxies where there is no gravitational binding.

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Is this statement a simplification due to the small scale of gravitationally bound parts which make the effects too small to be perceived/observed, or does this statement imply that the scaling of the size/geometry metric is actually related to the vicinity of mass?

What they're stating is an approximation. There is a continuum of behavior from strongly bound systems like our solar system to structures like superclusters that have gravitational interactions but are not gravitationally bound. At the strongly bound end of the continuum, the approximation they're stating is incredibly good. See Can the Hubble constant be measured locally? .

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