Modern physics has invented all sorts of speculative ideas to help reconcile observations about gravity. Physicists have invoked dark energy to help explain accelerated cosmic expansion and they have postulated extra dimensions to explain the weakness of gravity compared with the other fundamental forces.
The required density of dark energy, which would have a repulsive gravitational effect, seems to be at the level of about 3.8 keV/cm3. This corresponds in natural units to about 85 μm, so it could be that something interesting might happen to Newton's law of gravitation around that scale. The idea with extra spatial dimensions is that the extra dimensions are "curled up" or compactified, and so are not normally observed. In these theories, the gravitational force could become strong at distances smaller than the size of the largest compactified dimension.
To investigate these ideas, D J Kapner and colleagues at the University of Washington in Seattle performed three experiments using a very precise torsion balance, testing the inverse-square law over a range of separations from 9.35 mm down to 55 μm. Their results improve constraints on the inverse square law by up to 100, and show that the law holds down to 56 μm, below the dark-energy scale. This also implies that a single extra dimension has to be smaller than 44 μm.
Further reading
D J Kapner 2007 Phys. Rev. Lett. 98 021101.