Local expansion measured, near zero via Lunar Ranging - what about deep space probes?

Question

All of the discussion around this question and in the comments below this answer, about the local effects of "the expansion of space" (Metric expansion of space) or Cosmological redshift or just simply the Hubble constant has got me wondering about the feasibility of detecting or even measuring the effect directly using spacecraft as a probe of distance.

Most (if not all see edit below) of the data so far come from interpreting measured shifts of spectral features. These include atomic and molecular emission and absorption lines, as well as broader features including Blackbody radiation and ad-hoc averages of stellar populations. They all have one thing in common - they are interpreted measurements of one-way light from things really really far away.

Since we can now actually put complex instruments on the order of ten billion kilometers away and still exchange data with them and perform measurements on them, it's not unreasonable to start thinking of a controlled experimental measurement of the effect. For example this might be done by Doppler ranging using reflection, or better yet reception and simultaneous rebroadcast by a spacecraft.

My question is - have there been institutionally proposed, or peer-reviewed and published discussions of the detection or measurement of the Metric expansion of space using spacecraft as a probe of distance?

edit: I've changed the title to be plural - I don't want to exclude things like (just for example) compensation for the gradient in local gravity field for want of an "s". (imagine six spacecraft exploring +/- x, y, z)

Note - I am not asking if you think it can or can't be done. I'd like to read serious discussions or proposals on the subject that quantitatively address feasibility.

update: This was shared with me in this answer, and it's a good example of something quantitative, though it's data rather than explanation. The main focus of the paper is experimental testing of the Equivalence Principle (EP; gravitational mass vs inertial mass) using the Sun, Earth and Moon.

It seems to be in the negative as far as the present question is concerned. Expansion seems to be quite readily "measureable", and their results seem to show it to be very small and consistent with zero locally. It reports on Laser ranging of the retro-reflector arrays on the moon. In 34 years, the distance between the earth and moon has followed the best predictions to within a scatter of about 2cm. If my arithmetic is correct, expansion of space at the Hubble constant rate would be about 82cm over 34 years.

While this doesn't help me understand why, it certainly suggests in a peer-reviewed, quantitative way that the expansion might also not be seen by ranging spacecraft in distant orbit around the sun.

APOLLO 11:

APOLLO 14:

APOLLO 15:

GAO Laser Ranging Facility at the Goddard Spaceflight Center:

Answer 1

Here are a couple of papers on this topic:

• Cooperstock, http://arxiv.org/abs/astro-ph/9803097v1

• Carrera, http://arxiv.org/abs/0810.2712

I also have a brief treatment that may be more accessible (but still requires knowledge of general relativity) in my GR book, section 8.2.10. The basic result for a system orbiting with angular frequency $\omega_0$ is that cosmological expansion causes the orbit to expand according to the equation

$\frac{\dot{r}}{r_0} = \omega_0^{-2}\frac{d}{dt}\left(\frac{\ddot{a}}{a}\right)$,

where $a$ is the cosmological scale factor, and dots indicate time derivatives.

Note that the effect depends on the acceleration of cosmological expansion, so it's not something you can just interpret naively as if space expands and therefore the system gets bigger.

For purposes of order-of-magnitude estimation, we can take the second factor on the right-hand side to be the cube of the Hubble constant. For a small system like the earth-moon system, $\omega_0$ is high, and therefore the cumulative effect is incredibly small. Plugging in numbers, I find that over the period of time since the Apollo missions, the contribution of cosmological expansion to the recession of the moon is about $10^{-22}$ cm. Cooperstock has a picturesque way of describing the smallness of such effects, which is that the expansion of the earth's orbit since the age of the dinosaurs is less than the diameter of an atomic nucleus.

Even if lunar laser ranging (LLR) had the kind of precision needed in order to measure this effect, it would never see it, because the distance between the earth and the moon is growing for reasons of tidal dynamics. This is the effect that really is seen in LLR, and it's many, many orders of magnitude greater than the cosmological effect.