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 retroreflector array


APOLLO 14 retroreflector array


APOLLO 15 retroreflector array

GAO Laser Ranging Facility at the Goddard Spaceflight Center:

GAO Laser Ranging Facility at the Goddard Spaceflight Center

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    $\begingroup$ Well, you apparently don't want my answer, so I'll just put it in this comment. It can't be done. The expansion is observable only on scales rather larger than galaxies, since it is otherwise completely swamped by the gravitational interactions within galaxies, or even between galaxies in clusters. $\endgroup$
    – Mark Adler
    Apr 21, 2016 at 4:53
  • $\begingroup$ @MarkAdler can you help find some place where this conclusions has been worked out and discussed quantitatively? Sometimes "can't be done"s are really "don't know how yet"s in disguise. $\endgroup$
    – uhoh
    Apr 21, 2016 at 5:05
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    $\begingroup$ I doubt that you'll find the concept of a direct spacecraft measurement of expansion discussed anywhere, since it can be dismissed within the first few seconds of discussion. $\endgroup$
    – Mark Adler
    Apr 21, 2016 at 5:07
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    $\begingroup$ @uhoh I can only second MarkAdler's opinion. This is a question about astrophysics, not engineering. You'd only feel the additional Hubble-acceleration measurably at a significant distance from our galaxy. And even then it would be challenging to distinguish this effect from the regular galactic potential and the dark matter halo of the galaxy. This is also why there won't be any serious case-study on such a concept. $\endgroup$ Apr 21, 2016 at 10:26
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    $\begingroup$ @uhoh: Space does not expand locally. As you can see i.e. in the fact that our Galaxy is being held together by its own gravity. So there will be no measurable effect of space-time expansion. Measurably means that you can distinguish the signal you search for from other signal sources that accelerate your spacecraft. And as basic GR tells us, there is no acceleration due to metric expansion. This is why the Hubble law was discovered relatively late and why we need distant objects to measure it at all. $\endgroup$ Apr 21, 2016 at 14:06

1 Answer 1


Here are a couple of papers on this topic:

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.

  • $\begingroup$ The tidal effects are discussed in the paper and are part of the model. The residual is what's left over after the best estimates for tidal effects are taken into account. The final result is essentially zero, with an uncertainty of a few centimeters, which is - I believe - in agreement with what you are saying. Thanks for your answer! $\endgroup$
    – uhoh
    Jul 20, 2016 at 0:08
  • $\begingroup$ OK your answer has stood the test of time - almost a week! Thanks. $\endgroup$
    – uhoh
    Jul 26, 2016 at 0:24

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