  In absence of matter Einstein gravity with a cosmological constant $\La$ can
be formulated as a scale-free theory depending only on the dimensionless
coupling constant G \Lambda 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 \lambda \phi^{4} field theory. 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.
