Philip Ball describes in the New York Times as well as
a recent speculative preprint
in which Bo Qin, Ue-Li Pen, and Joseph Silk study some quantitative features of Spergel and Steinhardt's model of self-interacting cold dark matter - one of the popular models that can explain that the dark matter halos don't seem to be cuspy, which is a wrong prediction of the older models of cold dark matter theories.
It is argued that the observed data are exactly matched if there are three extra large ADD-like dimensions whose size is 1 about nanometer. Below this distance scale, the gravitational force would intensify from the 1/r2 behavior to 1/r5. This would be pretty hard to measure with the tabletop experiments. ;-)
In these extra large dimensions scenarios, you must think about the ADD braneworlds. For example, type IIA on a six-torus with orientifolds and D6-branes - something that you can learn from Barton Zwiebach's textbook - could, in principle, allow you for such a strange arrangement of the dimensions in which three of them are large.
They also advocate the mass of the dark matter particles to be 0.3 microelectronvolts (the Compton wavelength is around a meter), a fine value for axions. You may think that this is very light; but because the sub-nanometer gravity is so strong in their model, the (purely gravitational) self-interaction cross sections are large enough. On the other hand, at galactic distances, the self-interaction becomes weak which solves some difficulties of Spergel and Steinhardt's model to account for the absence of certain gravitational lensing observations, and for some galactic X-ray data. Three large extra dimensions seems to be the unique discrete number in which all things may be resolved, the new guys argue.
Silk et al. explain that they may thus be capable to almost prove string theory by hardcore data which may be more straightforward than the arguments based on uniqueness of quantum gravity and the incredible mathematical beauty of string theory. ;-) Their proposed scenario, if true, would probably imply a lot of new physics to be seen at the LHC, too.
Can you actually construct a stringy model that has such a light axion?