One-loop gravitational action and the cosmological constant problem

Abstract

We investigate the one-loop gravitational effective action, focusing on the interplay between the functional measure, diffeomorphism invariance, and the identification of the ultraviolet cutoff. Considering the Fradkin-Vilkovisky measure, that comes from the phase-space formulation of the theory, we show that its apparently non-covariant contributions are essential for recovering a diffeomorphism-invariant effective action, unlike the commonly adopted Fujikawa measure. On a spherical background, this measure renders the fluctuation operators dimensionless, and we calculate the determinants resorting to a numerical cut N on the number of eigenvalues. We stress that the ultraviolet behavior of the vacuum energy depends crucially on how the N is related to the UV cutoff Λ an on-shell de Sitter identification removes power-law divergences, whereas an off-shell prescription generates the standard quartic and quadratic terms. The same pattern is found for the contributions to the vacuum energy due to scalar and Dirac matter fields.