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 This is a major problem that has perplexed me. I discussed it in my 2005 book Super Cosmos. Good to see others are thinking about it.

"Attempts to estimate the influence of global cosmological expansion on local systems are reviewed. Here “local” is taken to mean that the sizes of the considered systems are much smaller than cosmologically relevant scales. For example, such influences can affect orbital motions as well as configurations of compact objects, like black holes. Also discussed are how measurements based on the exchange of electromagnetic signals of distances, velocities, etc. of moving objects are influenced. As an application, orders of magnitude of such effects are compared with the scale set by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is 10−9 m/s2. There is no reason to believe that the latter is of cosmological origin. However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of general relativity."
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