Renormalization-group running cosmologies: A scale-setting procedure

For cosmologies including scale dependence of both the cosmological and the gravitational constant, an additional consistency condition dictated by the Bianchi identities emerges, even if the energy-momentum tensor of ordinary matter stays individually conserved. For renormalization-group (RG) approaches it is shown that such a consistency relation ineluctably fixes the RG scale (which may have an explicit as well as an implicit time dependence), provided that the solutions of the RG equation for both quantities are known. Hence, contrary to the procedures employed in the recent literature, we argue that there is no more freedom in identification of the RG scale in terms of the cosmic time in such cosmologies. We carefully set the RG scale for the RG evolution phrased in a quantum-gravity framework based on the hypothetical existence of an infrared fixed point, for the perturbative regime within the same framework, as well as for an evolution within quantum field theory in a curved background. In the latter case, the implications of the scale setting for the particle spectrum are also briefly discussed.