Cosmological framework for renormalization group extended gravity at the action level

General relativity (GR) extensions based on renormalization group (RG) flows may lead to scale-dependent couplings with nontrivial effects at large distance scales. Here we develop further the approach in which RG effects at large distance scales are fully encoded in an effective action and we apply it to cosmology. In order to evaluate the cosmological consequences, our main assumption is the use of a RG scale such that the (infrared) RG effects only appear at perturbative order (not at the background level). The emphasis here is on analytical results and qualitative understanding of the implied cosmology. We employ commonly used parametrizations for describing modified gravity in cosmology (as the slip parameter). From them, we describe the dynamics of the first order perturbations and estimate bounds on the single dimensionless parameter ($\nu$) introduced by this framework. Possible impacts on dark matter and dark energy are discussed. It is also shown here that the $\nu$ parameter effects to $f\sigma_8$ are stronger at low redshifts ($z<1.5$), while different values for $\nu$ do not appreciably change $f\sigma_8$ at higher redshifts, thus opening a window to alleviate an issue that is currently faced by $\Lambda$CDM.