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I think I have understood properly the principle of LIGO, however the sensitivity is around $10^{-18}~\rm m$ of accuracy of distortion. That looks pretty small... just starting with simpler phenomena, for example, thermal expansion. In order to avoid thermal expansion, the temperature needs to be control within a variation of $10^{-10}~\rm K$ or $10^{-11}~\rm K$ along the tubes.

I guess that the tubes do not need to be exactly $4~\rm{km}$ equal as the lasers can be synchronized although the distance is not exactly 4km within the range of $\left[10^{-18}~\rm m\right]$

How is the thermal noise avoided currently in LIGO, related to suspension and not complete vacuum?

It is already demonstrated that the frequency cut-off is low, but, how is that achieved? Which materials and isolation supported it?

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Trebia Project.
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The LIGO experiment doesn't measure the length of the tubes, it measures the changes in the length of the tubes.

It also has a lower frequency cut-off of 10Hz, so any perturbation that is slower than 10Hz won't affect the measurements. In particular temperatures changes will happen on a timescale of hours so if they change the length of the tube that doesn't matter. When LIGO measures the $10^{-18}$ m change in the tube length it doesn't matter if that length is a bit longer at midday than at midnight.

The accepted answer to the question suggested as a duplicate, How does LIGO remove the effects of environmental noise?, includes a graph of possible perturbations as a function of frequency, and you'll note that the frequency scale doesn't go below 1Hz because frequencies below that don't matter.

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John Rennie
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