Warping with dilatation symmetry and self-tuning of the cosmological constant

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We investigate consequences of an ultraviolet fixed point in quantum gravity for the cosmological constant. For this purpose we perform dimensional reduction of a general dilatation symmetric effective action Γ in dimension d>4 to an effective four-dimensional theory of gravity with a dilaton field. We find a stable flat phase in the space of extrema of Γ which results in a vanishing four-dimensional cosmological constant Λ. In order to understand the self-tuning mechanism leading to Λ= 0 we discuss in detail the most general warped geometries with maximal four-dimensional symmetry and SO(d-4) isometry of internal space. While the solutions of the d-dimensional field equations admit singular spaces with arbitrary Λ, the extremum condition for Γ imposes additional restrictions which result in Λ= 0. In cosmology, the dilatation symmetric fixed point may only be reached for asymptotic time t → ∞. At finite t dilatation anomalies result in an effective potential and mass for the pseudo-dilaton or cosmon and in dark energy.