Hamiltonian gravity with variable G,Lambda from the renormalization group

Models of gravity with variable G and Lambda have acquired greater relevance after the recent evidence in favour of the Einstein theory being non-perturbatively renormalizable. The present paper applies the Arnowitt-Deser-Misner formalism and the Dirac-Bergmann theory of constrained Hamiltonian systems to such a class of gravitational models. A modified action functional is then built that reduces to the Einstein-Hilbert action when G is constant, and leads to a linear growth of the scale factor when gravity is coupled to a massless self-interacting phi^{4} theory in a Universe with Friedmann-Lemaitre-Robertson-Walker symmetry, in agreement with the recently developed fixed-point cosmology. Interestingly, our modified action functional makes it necessary to consider an external field that decays as the inverse of the sixth power of cosmological time in the early universe, although the relevant solutions are actually independent of the strength of this new field.