Suppose you put a gold sphere inside the LIGO interferometer, not in the path of the laser beams, but sufficiently close to (in the vicinity of) one of the laser beams. Would the space - time distorsion caused by the gold sphere be detectable by the change in the interference patterns of the LIGO detector , or it would be too weak for detection? If the effect is detectable, this would be an experimental way to study the gravitational field of small objects. Some ballpark estimation of the magnitude of the effect would be appreciated.
LIGO doesn't detect gravitational fields, it detects curvature. Specifically, it detects the type of spacetime curvature associated with tidal forces. (This is why each detector has two arms.) Also, LIGO doesn't detect DC effects, it only detects AC signals within a certain range of frequencies. The most difficult thing about the LIGO experiment is that it also serves as the world's most sensitive vibration sensor. This is why they have two (now 3, IIRC?) facilities far apart. They look for correlated signals between the different facilities.
So if you move a dense object close to one of the detectors, as in the Cavendish experiment, what will happen is that that detector will be momentarily disabled by the vibration, just as it would if a truck drove by and stopped suddenly. Once the detector recovered from the insult, there would be no AC signal, probably no signal that would mimic the kind of tidal distortions it's looking for, and no correlation with the other site(s). So you would get no signal.