cosmology - Two observers initially at rest in the expanding Universe

The standard question asked about the expansion of the Universe sounds roughly like: "Does my pencil expands, together with expansion of the Universe?"

And the standard answer sounds roughly like: "No, expansion of the Universe works at cosmological scales -- everything must be homogeneous and isotropic and be interacting through gravity only. Your pencil is a very non-homogeneous low-scale thing that is bound by electromagnetic forces. So no."

I've noticed that I'm not the only one who feels that the answer is kind of dodging the substance of the question. What I really want to know is if this "space expansion" really "pushes stuff apart" or not. So I wanted do "distill" this idea into a thought experiment that formulates this intuition into a precise setup.

So, here is the setup:

• We have two non-interacting observers $A$ and $B$ in the expanding Universe.


• The distance (say, proper distance) between them is large enough to consider the Universe to be homogeneous and isotropic.


• We make sure that at the start of the experiment the observers do not move with respect to each other (again, in a sense that the proper distance between them is not changing). We can ensure this with, say, requiring no redshift of light signals between them. (You can propose some more intricate Einstein-light-ray-synchronization procedure for that.)

I think this setup captures the substance of the question quite well. If space really expands, then it is natural to expect the observers $A$ and $B$ to start moving apart from each other. If that doesn't happen, on the other hand, then it doesn't sound like "expanding space" at all.

So the question is: what would be the strict and formal solution for the setup above?