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.
