Conformal symmetry of gravity and the cosmological constant problem

In absence of matter Einstein gravity with a cosmological constant Λ can be formulated as a scale-free theory depending only on the dimensionless coupling constant GΛ, where G is Newton constant. We derive the conformal field theory (CFT) and its improved stress-energy tensor that describe the dynamics of conformally flat perturbations of the metric. The CFT has the form of a constrained λφ4 field theory. In the cosmological framework the model describes the usual Friedmann-Robertson-Walker (RW) flat universe. The conformal symmetry of the gravity sector is broken by coupling with matter. The dimensional coupling constants G and Λ are introduced by different terms in this coupling. If the vacuum of quantum matter fields respects the symmetry of the gravity sector, the vacuum energy has to be zero and the ''physical'' cosmological constant is generated by the coupling of gravity with matter. This could explain the tiny value of the observed energy density driving the accelerating expansion of the universe