A blog post by Sabine Hossenfelder at Backreaction reviews a new paper that discusses how the existence of more than four dimensions, even is accessible only to gravitons, as required by string theory (which needs ten or eleven dimensions), could be ruled out with the LIGO gravitational wave detector and two comparable independent gravitational wave detectors.
Current experiments place bounds on the size of any extra dimensions as follows:
The mass of these excitations is inversely proportional to the radius (in natural units). This means if the radius is small, one needs a lot of energy to create an excitation, and this explains why he haven’t yet noticed the additional dimensions. . . .
From the current lack of observation, one can then derive bounds on the size of the extra-dimension. These bounds depend on the number of extra-dimensions and on their intrinsic curvature. For the simplest case – the flat extra-dimensions used in the paper – the bounds range from a few micrometers (for two extra-dimensions) to a few inverse MeV for six extra dimensions (natural units again).
Basically, gravitational waves, even the ordinary ones we can detect with LIGO (as opposed to the excited ones that involve exited gravitons that have excitations in extra dimensions) behave differently in a world with extra dimensions (even compactified dimensions that only gravitons can enter), than they do in general relativity.
If extra dimensions exist, there are excited states of all particles, gravitons have additional polarizations, and they exhibit a kind of oscillation known as a "breathing mode" that does not exist in general relativity.
Absolutely ruling out the possibility is a technically difficult feat, but placing profoundly more strict limits on their number and size is considerably more straight forward. Only a narrow sub-type of extra dimensions is particular hard to detect with this methodology. So, it would be fairly easy to limit any possible string theory to the kind with that narrow subtype of extra dimensions.