> "usable link"

How much technical detail do you want?

Starting with the "gimme hardcore!" end, I was thinking of how to construct an argument using vierbiens and then how to boil it down to something accessible (or at least representable on LaTeX-free HN), and then remembered that it had already been done by Cooperstock et al.: http://xxx.lanl.gov/abs/astro-ph/9803097 The tl;dr is that if the cosmological expansion induces strain on matter, the strain is too small to be measurable.

Retreating from the hardest of answers, Peter Coles has an old moderate-detail article on this at https://telescoper.wordpress.com/2011/08/19/is-space-expandi... and he refers to both Peacock's and Harrison's textbooks which give greater detail (I recommend the latter if you can get your hands on it at a library).

His approach to the question you're asking ("roughly, does the solar system expand with the universe?") is how I'd go about it too, following on from the comment you replied to. My central point would be that in General Relativity we use exact solutions of the Einstein Field Equations because they're well-understood not because they're accurate. Natural systems don't source e.g the exact Schwarzschild spacetime for several reasons including lack of perfect spherical symmetry, lack of perfect vacuum to infinity outside the source, and nonzero angular momentum. Yet we get good approximate results when we use Schwarzschild to model the Earth or the Sun or the Milky Way, and usually the bad results are fixable with linear corrections). But the real picture is that each of these bodies sources an unknown metric that is slightly different from Schwarzschild, and additionally one has to stitch together two metrics sourced by two bodies each sourcing (different, unknown) Schwarszschild-ish metrics into an (unknown) expanding background.

Numerical relativity has opened up the study of approximate solutions which give better results for real physical systems than the toolkit of known exact solutions (plus linear in v/c corrections), so one could argue that the central research programme in classical General Relativity is the study of the mechanisms that generate the (true) metric.

All that said, we can be much more confident (because of analyses under e.g. the parameterized post-Newtonian formalism and experimental data from gravitational probes of many varieties) about the fit of exact solutions to the bodies in our solar system than the fit of any metric to the cosmos-in-the-large. For the bodies in hydrostatic equilibrium that means Schwarzschild to at least the first order in v/c [in linearized gravity]. If you accept that Schwarzschild is an excellent substitute for the unknown real metric, then you must have a very close fit to a static, asymptotically flat spacetime. Around that you can establish a boundary condition. Outside the boundary is the expanding spacetime, inside is asymptotically flat (i.e., not expanding). Coles discusses some of this ( as does Hossenfelder at http://backreaction.blogspot.com/2017/08/you-dont-expand-jus... ).

Alternatively, you can argue that the real metric sourced by e.g. the Earth (or yourself at a distance where you subtend a very small angle on an observer's sky, or one of your molecules) is less close to Schwarzschild. In that case, Coles takes us back to Cooperstock via Ned Wright's Cosmology FAQ: you will get bad results with poor choices of coordinates (so use e.g. Fermi coordinates because then you know exactly where you have valid and comparable results, and you are forced to consider whether and where geodesics drift apart[1]).

Finally, if you were asking about "naive quintessence models" and their problems, ch 3.2 in Elise Jenning's _Simulations of Dark Energy Cosmologies_ (Springer, 2012) is a decent overview (it contains material from her Ph.D. thesis; she is now at KICP/FNAL).

> "actual proof"

This would make an excellent postdoc research project !

Linked with [1] above, on proving the conjecture, the soft underbelly is the behaviour of geodesics: in an expanding universe, comoving geodesics drift apart. The geodesics in Schwarzschild spacetime do not drift apart. Geodesics in the solar system do not drift apart, and haven't over the course of a few billion years. Geodesics at cosmological scales clearly do drift very noticeably apart over the same period of time. Worse, evidence suggests that the Hubble constant isn't constant in time, so where are the matching perturbations in the orbits of various bodies in the solar system? However, I'm not sure this is the right path to a definitive answer, since one is likely to be able to claim that your atoms in general are not following geodesics; their free-fall is interrupted by the surface of the Earth.