Measuring Hubble expansion in the lab

This is a high-school level physics question. The Hubble expansion is dramatic at the cosmological level, but absurdly tiny at the human scale. $H_0$ is 73 km/s/Mpc and 1Mpc is $3\times 10^{19}$ km. Converting to a human scale, $H_0=2.4\times 10^{-18}$ Hertz. So, if I take two masses, in free-fall, originally at rest with respect to each other, starting out one meter apart, and I wait one second, then I should find that the distance between these two masses will have increased by $2.4\times 10^{-18}$ meters.

Suppose I take two free-falling mirrors, separate them by 5 kilometers, and then wait one second for the Hubble expansion to pull them apart. The distance between them would presumably change by $1.2\times 10^{-14}$ meters. Absurdly tiny, to be sure, but a distance that LIGO is sensitive to. So my question is, is this change actually measurable?

I can attempt to answer my own question. The answer would be "no", because Hubble expansion implies that the wavelength of the laser light heading down the tunnels is also lengthening, thus cancelling the effect.

Presumably, the laser light is also red-shifting, but this effect is ten orders of magnitude smaller then the frequency stability of those lasers. However, there are light-sources that are extremely stable: the optical lattice clocks, the best of which have one part in $10^{18}$ frequency stability. These are extremely dim, so unusable (I guess?) for measuring that 5km distance. However, the Hubble redshift for light at optical frequencies seems .. almost within reach. Blue light is $4\times 10^{14}$ Hz. Over the course of an hour, it should Hubble red-shift by 1 Hz.

A velocity-stabilized clock is needed to cancel out redshift from ordinary Minkowski relative motion. LIGO provides that amount of velocity-stability. It seems that the Hubble redshift would then be measurable, unless it is cancelled out e.g. because of the motion stabilization.

A different variant of above: The Pound-Rebka experiment measured the redshift of falling photons. The precision was possible because of the rather weird Mossbauer effect. If that setup was changed to horizontal, and motion-stabilized a la LIGO, then, again, it seems that Hubble expansion would be lab-measurable. That is, unless I fundamentally mis-understand Hubble expansion and how it interacts with lab devices. Perhaps the motion stabilization precisely cancels out the Hubble expansion?

My question is a kind-of ball of confusion about distance and time dilation effects as experienced by Earth-bound lab instruments. Measuring distance is dicey, if the length of your ruler is changing. I'm very confused.

And by locally, I mean within the local group of galaxies.

The expansion is not uniform across the universe because it is opposed by gravity so when there is a large enough accumulation of mass-energy, such as within a galaxy, there is no expansion happening. The local group is dense enough and compact enough that there’s no expansion between these galaxies either. At even smaller scales, it’s possible for contraction to occur - such as in the formation of stars and solar systems.

So, mirrors inside a galaxy are not going to see any expansion because there isn’t any at that scale. I haven’t done the maths but I suspect that even in intergalactic space this wouldn’t work because the mirrors themselves have enough gravitational interaction to counteract the expansion.

The Hubble expansion is the residual effect of a much greater expansion rate during the earliest times. A certain density was required to halt the expansion in regions that have collapsed. If you start with two objects at rest w.r.t each other, isolated from all external influences, they will not expand regardless of how low the density is between them (ignoring the tiny effect of dark energy). Instead, they will contract due to mutual gravity. Near the Earth, if the objects are aligned horizontally, they will contract. However, if they are aligned vertically and have a low overall density, they will expand. This behavior is caused by the tidal field of the Earth.

Added a few days later: There is some ambiguity in what it means to place two objects "at rest" in an isolated region, free from nearby influences. Specifically, one must clarify whether the objects are at rest relative to each other or relative to the universe as a whole. By "at rest with respect to the universe," we mean that neither object detects an extra dipole effect in the cosmic microwave background (CMB)—essentially, they are at rest relative to the average motion of galaxies.

If the objects are positioned far apart and are at rest relative to the universe, they must have small recessional velocities directed slightly away from each other, consistent with the expansion of the universe. In this case, they will continue to move apart over time with velocity $H_0d$.

On the other hand, if the objects are placed at rest relative to each other (not relative to the universe), they must initially have small "peculiar velocities" (deviations from the cosmic expansion) toward each other. In this scenario, the objects will not continue to move apart; instead, they may move closer together (subject to provisos mentioned earlier).

This distinction is crucial for understanding the expansion of objects not taking part in the general expansion since the beginning.