Scale-dependent gravity and covariant scale-setting

A fundamental element of scale-dependent gravity is the scale-setting procedure. We present a new covariant expression to set the scale that arises when examining the field equations. Considering the renormalization group equations and imposing energy-momentum tensor conservation, we arrive at two models of running of the gravitational and cosmological constants. In the cosmological setting, we found that in one model the Big Bang singularity is avoided, while in the other the Hubble tension can be alleviated. At the level of cosmological perturbations, we derived the basic solutions and qualitatively discussed the impacts of this scenario on structure formation.