How does one derive theoretically the shot-noise limited sensitivity of LIGO? It is said that LIGO can measure 1/1000 the width of a proton. How does one derive, using the design parameters of LIGO(arm length, laser power, etc.), its sensitivity?
 A: LIGO is an interferometer where a laser is split into two --each one travelling in the two arms of a L shaped interferometer -- and combine back at the photo diode. All you need is a path length of a wavelength to go from full construtive interference to full destructive interference. With 4 km length of arm and the fact that photons meet at the photodiode (on an average) after a round trip of around 50-100 the accurate measurements can be done for (assuming 1000 nm wavelength)
1000e-9/4000/100 = 2.5e-12,
Now we dont want the full wavelength to change and we are happy even if the interference changes a tiny bit. With a 200W laser there are around 1e20 photons and they can give resolution on another 9 to 10 order of magnitude i.e 10e-21 to 10e-23. This happens to be the strain which a GW causes in the detector.
If you want to increase the sensitivety of a detector the direct way is to increase the length of the arm or increase the number of photons. Off course increasing length of arm is not possible so LIGO has been increasing the laser power. This comes at a cost, the shot noise increases -- photons are like rain drops falling on the mirrors -- and the heat dumped on the mirrors also increase.
