Diluting the cosmological constant in infinite volume extra dimensions

We argue that the cosmological constant problem can be solved in a braneworld model with infinite-volume extra dimensions, avoiding no-go arguments applicable to theories that are four dimensional in the infrared. Gravity on the brane becomes higher dimensional at super-Hubble distances, which entails that the relation between the acceleration rate and vacuum energy density flips upside down compared to the conventional one. The acceleration rate decreases with increasing energy density. The experimentally acceptable rate is obtained for an energy density larger than $(1\mathrm{}\mathrm{TeV}{)}^4.$ The results are stable under quantum corrections because supersymmetry is broken only on the brane and stays exact in the bulk of infinite-volume extra space. The consistency of 4D gravity and cosmology on the brane requires the quantum gravity scale to be around ${10}^{-3}\mathrm{eV}.$ Testable predictions emerging within this approach are (i) simultaneous modifications of gravity at submillimeter and the Hubble scales and (ii) Hagedorn-type saturation in TeV energy collisions due to the Regge spectrum with a spacing equal to ${10}^{-3}\mathrm{eV}.$