LIGO Arm Length & Sensitivity: What's the Relation?

[roineust]

[Moderator's note: Spun off from previous thread due to new question.]

I have read here:
http://backreaction.blogspot.com/2016/02/everything-you-need-to-know-about.html?m=1

That there is a proportionality between the size of LIGO arms and the wavelength of gravitational waves that it can detect.

My following question relies on considering also the multliplication of its arms length, by the Fabry–Pérot interferometry method:

What is the relation between LIGO gravitational waves detectable wavelength range and the LIGO sensitivity number and the intensity/distance of the source incident that produced the wave?

Would it be correct to say that there is a need for much longer LIGO arms, in spite of the Fabry–Pérot multliplication, in order to detect at a much lower sensetivity?

How does the wavelength relate to the intensity and distance of the incident and to sensetivity?

Is there no relation between these LIGO properties of length of arms and sensetivity?

 

[Ibix]

I think what the blog says is that gravitational wave wavelengths are related to the size of the emitting system, not the receiver.

You seem to be conflating several things in your question - the emitted strength of the gravitational waves, the received strength (related to the distance from the source) and the wavelength of the waves.

What LIGO detects is differences in the length of its arms induced by a passing gravitational wave. A given amplitude of change is easier to detect with longer arms, because a 10^-22 change in an arm 1m long is 10^-22m, but a 10^-22 change in a million kilometer long arm is 10^-13m - a much larger absolute difference. But longer arms mean that the instrument is less sensitive to rapid changes, ones that occur in less time than it takes light to fly through the arms. So it would be harder to detect the really strong signals at the very end of a black hole merger (they're too fast) but would let you see the earlier stages of the inspiral, or maybe the late stages of really really massive mergers even if they are very very far away.

It's a complicated trade off depending on what you want to look for, in short. And the above only considers the interferometer, not the technical complexities of mirror mounts, laser stability and detector and signal processing issues, so is very much a simplification.

 

[PeterDonis]

I think what the blog says is that gravitational wave wavelengths are related to the size of the emitting system, not the receiver.

Yes, that's correct. The size of the receiver is one factor that affects what range of wavelengths the receiver is sensitive to, but that of course cannot affect what wavelengths are there to be detected.

 

[mfb]

The sensitivity of gravitational wave detectors can be described as function of relative strain and frequency (or, equivalently, wavelength).
Here are plots for all three. It's not directly the strain because longer signals are easier to detect than shorter signals, but let's skip that technical detail here. As you can see LIGO, Virgo and KAGRA are all most sensitive around ~100-500 Hz. At lower frequencies seismic noise is dominant. KAGRA is underground to limit seismic noise, that's why it has a better sensitivity below 100 Hz. Where the sensitivity is highest thermal noise and radiation pressure are most important. At higher frequencies shot noise is more important - basically random chance how many photons you detect at a given time.
Here is an example breakdown of the important noise sources

If you increase the arm length then seismic noise will go down - its absolute length shift stays the same but the relative shift goes down because the denominator (the arm length) grows. You get more sensitive to lower frequencies.
If you improve the photon recycling then the shot noise will go down - you now have more photons per time and become more sensitive at higher frequencies. As downside the thermal noise tends to go up.

A wavelength equal to the arm lengths would have a frequency of ~100 kHz. At these frequencies you would run into other problems, like the contracting and the expanding part of the wave being in the detector at the same time, cancelling each other. But that's far above the frequency range where the detectors have their sensitivity.