gravity - Would order 1AU metric expansion of space begin if the solar system were not inside a galaxy?

In this question I describe the >30 years of laser ranging between lasers on Earth and the retroreflector arrays on the Moon. Amazingly, after comparing this data to simulation of all of the orbital mechanics and tidal effects, the residual is only a few centimeters.

If $\text{H}_D$ is about 70 $\text{km}\ \text{s}^{-1} \text{Mpc}^{-1} $ (or about 2.3E-18 $\text{sec}^{-1}$), then with a semi-major axis of 3.84E+08 meters, over 37 years the effect would be of order 1 meter. Since these measurements are consistent with zero to the level of a few centimeters, this is taken as experimental evidence that metric expansion is not taking place locally compared to the rate seen between galaxies.

If I understand correctly this is "suppression" is due to the large amount of mass in our galaxy, even though it is thousands of light years away. So I am wondering - if a similar experiment were done in a similar solar system associated with an isolated star alone in an intergalactic region, what does current theory predict - would metric expansion be detected, or would the mass of the one star be enough to suppress it?

Then what if it were just a planet and a moon without the mass of the star, or even two smaller masses?

I'm looking for an answer at a level similar to this answer and this answer, where time was taken to note the the specific relevant concepts and work from the paper linked in the first answer.

Answer

These notes put some numbers on @ACuriousMind 's answer: one needs to be looking at length scales of 100 Mpc and greater for the FLRW metric to be a realistic description of reality. That's a staggering distance, and equivalent to timescales amounting to the whole Mesozoic era, comprising the rise and fall of the Dinosaurs! So one cannot expect the scalefactor expansion of spacetime to apply to our Earth-Moon system simply because the system doesn't fulfill the assumptions that justify the FLRW metric.

Perhaps a more addressable variant to your question would be to ask what difference a positive cosmological constant makes to a metric that does describe the Earth-Moon system. This is a question that can be answered, and it is reasonably straightforward to go through the derivation of the Scwharzschild metric but with a positive cosmological constant. One finds that the metric changes as follows:

$$g_{t\,t} =c^2\left( 1-\frac{r_s}{r} -\frac{r^2\,\Lambda}{3}\right)$$ $$g_{r\,r} = \frac{c^2}{g_{t\,t}}$$

and the cosmological constant, if small enough, does not change the basic character of geodesics; it will however shift the radiusses of stable orbits. These notes sketch how to work through the computation; the radially symmetric system with nonzero $\Lambda$ being Problem 23. in Chapter 23 of Moore, Thomas A., "A General Relativity Workbook".

So there is no ongoing spacetime expansion in this system: orbits are just a little bigger than they would be with $\Lambda=0$ and some weakly bound orbits would become unbound with positive $\Lambda$. Therefore, we would not expect the Moon to be drifting away any faster that it is owing to nonrelativistic effects.