[Kryptid (OP)]

Would it be possible with current technology (or at least foreseeable technology) to directly detect the expansion of space using a laser set-up somewhat like that used at LIGO to detect gravitational waves?

The Hubble constant is measured as being somewhere between 67 and 73 kilometers per second per megaparsec (I'll use 70 km/s/mpc (70,000 m/s/mpc) as an average value) . A megaparsec is 1 million parsecs, and a parsec is 3.0857 x 10¹⁶ meters, so a megaparsec is 3.0857 x 10²² meters. 70,000 m/s/mpc / 3.0857 x 10²² meters = 2.2685 x 10^-18 meters/second/meter.

Assuming I did that math right, then the space between two objects in free space separated by a meter is expanding at a rate of about 2.6% of the proton charge radius (8.751 x 10^-16 m) every second. At a separation of 1 kilometer, the expansion rate is about 2.6 times the proton charge radius per second. At 23 million kilometers, it's about the radius of a hydrogen atom every second. Since it takes light about 76.7 seconds to travel 23 million kilometers, it should experience an increase of wavelength on the order of 38 hydrogen atom diameters.

I've read that LIGO could detect mirror movements on the order of one-ten-thousandth of the diameter of a proton. Does that mean we meet the detection threshold? I understand that this might be slightly different, as what we are looking for is a change in red-shift instead of mirror movements. Is it not really the same, then?

[Halc]

70,000 m/s/mpc / 3.0857 x 1022 meters = 2.2685 x 10-18 meters/second/meter.

It is a useful exercise to reduce the terms in that figure. You have meters both in the numerator and the denominator. Doing so illustrates why Hubble's constant isn't a constant.

As to the question asked, certainly technology is probably capable of detecting the change in distance you describe, but the question is, how would one distinguish mere motion of the far mirror from an increase of space between here and the far mirror?  If you attach it to a rod, space can expand all it wants and the mirror isn't going to get any further away.

[evan_au]

directly detect the expansion of space using a laser set-up somewhat like that used at LIGO?

LIGO obtains its extreme sensitivity by using destructive interference between two light paths at right-angles to each other.

If the gravitational wave source is optimally placed, one arm will get shorter while the other gets longer (and vice versa). This causes the interference pattern to be cancellation vs not cancelled.

The cosmic expansion of space is uniform for galaxies in all directions, so you don't benefit from this very sensitive oscillation between cancellation/non-cancellation between two arms of the detector. Both arms of the detector are equally affected.

Another problem with the LIGO configuration is that it has a limited frequency range, about 50Hz to 1kHz. It can't detect a 0Hz steady expansion of the universe.

[Kryptid (OP)]

I didn't mean to make it exactly like LIGO, but rather use something similar (because LIGO is extremely sensitive). As an example idea: use the beam splitter to create two laser beams from the original source. One laser beam is sent directly to the detector, whereas the second is sent at a right angle to the first laser beam, bounces off of a mirror so that it travels parallel to the first beam, then bounces off a second mirror that causes it to come back and intersect the first beam at a 90 degree angle. For this reason, the track of the second beam is longer than the first and thus spatial expansion would cause it to red-shift more.

In LIGO's detector, the two beams are set to destructively interfere with each other so that the detector doesn't pick up anything initially. We could use a similar set-up such that destructive interference between two beams of perfectly equal frequency cancel out in the direction of the detector. However, the slight increase in red-shift on the longer arm of the journey would mean that the lasers are no longer perfectly matched and thus a tiny signal should get through equal in intensity to the difference between the strength of the two beams.

A better set-up would be so that each arm starts off as being equal in length, and that the mirrors on one of the arms could be moved in order to make the distance either shorter or longer than the first arm. This would allow for the relative difference between the red-shift to be directly measured as the mirror distance is varied.

[evan_au]

Using 1 arm of LIGO might work: Compare the wavelength of the transmitted laser light to the wavelength of the light after it has passed down the length of one LIGO arm.

The mirrors are very reflective at the wavelength of operation, so the light bounces backwards and forwards several hundred times before being absorbed or lost. So the effective length of the LIGO arms is hundreds of times longer than it appears on Google Earth.

[Bored chemist]

2.2685 x 10-18 meters/second/meter.

I think you got the maths right but, as you say, what you are looking for is a fractional change in wavelength- a red shift.
And , just looking at the dimensions, I think you need to divide that very small number by the speed of light to get a fractional change per metre.
You are looking for wavelength a change of 1 in 10^26
Good luck.
Mössbauer spectroscopy will get you about 1 part in 10^12
This gets you about 1 in 10^16
https://en.wikipedia.org/wiki/NIST-F1
and that's about as good as it gets.
So you would need a path of about 10^11 metres
Can you put a clock on Pluto?

[Kryptid (OP)]

Can you put a clock on Pluto?

I was indeed wondering if we'd need astronomical distances to make this work. The amount of laser divergence from Earth to Pluto might be too extreme for this to work. Unless we use an exceptionally powerful laser. Then the issue of precision over such a distance becomes an issue.

Are perfectly reflective mirrors possible? I thought I read on this very board once that superconductors are capable of perfectly reflecting light (maybe only under specific circumstances). If so, bouncing a laser beam back and force between two perfect mirrors for long enough might make the red-shift detectable.

[Bored chemist]

Things like general relativity would completely screw it up.
The best clocks are so precise that a shift  of a few inches up or down in Earth's gravity affects the frequency.
You are hoping to get the same gravity (and zero relative velocity (WRT Earth) on Pluto. (Or something like that distance).

[Elliott]

Fruit flies respond to panoramic retinal patterns of visual expansion with robust steering maneuvers directed away from the focus of expansion to avoid collisions and maintain an upwind flight posture. Panoramic rotation elicits comparatively weak syndirectional steering maneuvers, which also maintain visual stability.

[Kryptid (OP)]

Fruit flies respond to panoramic retinal patterns of visual expansion with robust steering maneuvers directed away from the focus of expansion to avoid collisions and maintain an upwind flight posture. Panoramic rotation elicits comparatively weak syndirectional steering maneuvers, which also maintain visual stability.

Keep up the spam and you'll get banned.