Brane worlds and the cosmic coincidence problem

Brane world models with “large” extra dimensions with radii in the $r_l\sim 10–100\mu \mathrm{m}$ range and smaller ones at $r_s<~(1\mathrm{}\mathrm{TeV}{)}^{-1}$ have the potential to solve the cosmic coincidence problem, i.e., the apparently fortuitous equality between dark matter and dark energy components today. The main ingredient is the assumption of a stabilization mechanism fixing the total volume of the compact submanifold, but allowing for shape deformations. The latter are associated with phenomenologically safe ultralight scalar fields. The bulk field Casimir energy naturally plays the role of dark energy, which decreases in time because of the expanding $r_l.$ Stable Kaluza-Klein states may play the role of dark matter with an increasing, ${O(1/r}_s),$ mass. The cosmological equations exhibit attractor solutions in which the global equation of state is negative, the ratio between dark energy and dark matter is constant, and the observed value of the ratio is obtained for two large extra dimensions. Experimental searches for large extra dimensions should take into account that, due to the strong coupling between dark matter and radii dynamics, the size of the large extra dimensions inside the galactic halo may be smaller than the average value.