Cosmological constant and noncommutative spacetime

We show that the cosmological constant appears as an integration constant if nature is described by a canonical noncommutative spacetime. It is thus an arbitrary parameter unrelated to the action and thus to vacuum fluctuations. The noncommutative algebra restricts general coordinate transformations to four-volume-preserving noncommutative coordinate transformations. The noncommutative gravitational action is thus a unimodular noncommutative gravity. We show that spacetime noncommutativity provides a very natural justification to a unimodular gravity solution to the cosmological problem. We obtain the right order of magnitude for the critical energy density of the universe if we assume that the scale for spacetime noncommutativity is the Planck scale.