Conformal symmetry of gravity and the cosmological constant problem

In absence of matter Einstein gravity with a cosmological constant $\Lambda$ can be formulated as a scale-free theory depending only on the dimensionless coupling constant $G \Lambda$, where $G$ is Newton constant. In regimes where only the dynamics of the conformal factor of the metric is relevant, Einstein gravity can be described by a conformal field theory in four dimensions. In the cosmological framework the model describes the usual Friedmann-Robertson-Walker flat universe. The conformal symmetry of the gravity sector is broken by coupling with matter. The dimensional coupling constants $G$ and $\Lambda$ 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