Noncommutative Cosmology

We show that the large number coincidence can be interpreted as giving the filling factor in a Landau problem. The analogy with the Landau problem leads to noncommutativity between the gravitational and matter degrees of freedom. We present a toy model that supports this view. We present, also, some of the physical consequences this noncommutativity implies like a different insight into the semiclassical approximation of quantum gravity and a different tackling of the cosmological constant problem. The large number coincidence [1] fascinated [2], [3] and still fascinates [4] many people. The aim of focusing upon it was to find the reasons of the link between micro and macro cosmos the coincidence seems to reveal. Our aim here is to show that the large number coincidence hints to a commutation relation. Unlike the noncommutativity in spacetime assumed in many recent papers [5], the noncommutativity suggested by the large number coincidence is in superspace, between the gravitational and matter degrees of freedom. The large number coincidence, expressed in its most usual form ( ~ H G · c )1/3 ∼ m (1) connects the Hubble parameter H = a/a with the fundamental constants h, c,G and a typical hadron mass m. For power law cosmologies, the scale factor varies like a(t) ∝ t and one can replace the Hubble constant with the horizon distance RH