In article , stargene
writes:

Is there a very general equation(s?) for calculating the gravitational
potential experienced by each member of a pair of roughly equal
masses (stars or galaxies, say) at any separation R, whether or not
they are in an actual bound orbit with each other? Eg: two different
stars separated by thousands of light years or two different galaxies
separated by megaparsecs in widely separate parts of the universe?

Unless I'm missing something, neither the distance nor the question
whether they are bound makes much difference.

I know how to calculate the g-potentials felt by each member of
a binary star pair in circular orbit, but those equations don't seem
appropriate in the more general problem.

Why not?

(To be sure, the finite speed of propagation needs to be taken into
account, at least in a non-static situation, but this is, perhaps
somewhat surprisingly, only a higher-order effect, since to first order
it is cancelled by other effects. Simulations of the gravitational
interaction of the universe on large scales is something many people
have done. As far as I know, the standard Newtonian formula is used
(though within the framework of an expanding space).)