gravity - How are the 4 km arms of LIGO measured so accurately?

The arms of the LIGO interferometer are 4 km long. Now, LIGO functions by measuring the phase difference between two beams of light coming (as in a Michelson interferometer) to a sensitivity of $10^{-18}\: \mathrm m$. Now, we know that if the path length $2d_2-2d_1$ ($d_1,d_2$ being the length of the paths from the mirrors to the partially silvered diverging mirror) is anything other than $0$ then there is a phase difference between the two waves. So, how are the mirrors placed at exactly $4$ km apart and not even an error of $0.000\ldots 1\:\mathrm{mm} $ might have crept in which could have caused a phase difference, and false results creep in?

How are the measurements so accurate?

Answer

It is a misconception that LIGO is a very accurate instrument, it has an uncertainty in calibration which is on the order of 10%. This means that the measured strain amplitude of GW150914 of $1.0 \cdot 10^{-21}$ could easily have been $1.1 \cdot 10^{-21}$. Note that this is just a scaling error.

LIGO is however extremely sensitive, it can measure relative length variations on the order of $10^{-22}$, but only in a bandwidth between 10 and 2000 Hz. At lower frequencies, the measurement fluctuates by several orders of magnitude more. You need to do band-pass filtering to reveal the actual signal.

As already mentioned in Chris's answer, a Michelson interferometer can only measure incremental changes in the path length difference. It does not say anything about the absolute length of the arms, and not even about the absolute difference in arm lengths. For a perfect Michelson interferometer, the resulting power on the photodiode is$$P = \frac{P_0}{2} \left(1 + \sin\left(4 \pi \frac{L_1 - L_2}{\lambda}\right)\right)\,,$$which will only tell you how much the difference in the arm lengths changes over time.

Still, there are reasons why you want to have the long arms as equal as possible. For a simple bench-top interferometer with a crappy laser diode, the path lengths need to be reasonably similar, otherwise you run into problems with the coherence length. This is not an issue for LIGO, they use Nd:YAG lasers which already have a coherence length measured in kilometers when running alone. These lasers are further pre-stabilized by locking them on ~16 meter suspended cavities, and finally the laser frequency is locked on the average length of the two 4 km long arms. The resulting line-width of the laser is on the order of 10 mHz, so a coherence length larger than $10^{10}$ meters ...

You still want to make the length of the 4 km arms pretty equal, since any imbalance would couple residual noise of the laser frequency to the differential length measurement. With standard GPS-based surveying methods, the mirrors are positioned with an accuracy on the order of millimeters. There is no need to do this much more accurate, since there are other sources of asymmetry that can couple frequency noise to the differential measurement, such as the differences in absorption and reflection of the mirrors used in the two arms.