Scale-dependent cosmology from effective quantum gravity in the invariant framework

We explore the possibility of a consistent cosmology based on the gauge-fixing independent running of the gravitational and cosmological constants ($G$ and $\mathrm{\Lambda }$) in the framework of effective quantum gravity. In particular, their running in this framework was found to satisfy $G\propto {\mathrm{\Lambda }}^4$. In the cosmological setting, the covariance of the theory provides energy conservation relations, which are impossible to satisfy with the unique scale parameter. However, by introducing the second sub-dominant scale corresponding to the higher-loop corrections and higher-derivative terms, one can close the system of equations for the running of parameters and arrive at the consistent cosmological solutions. This approach yields a change in the cosmological expansion history that affects the ratio of the Hubble parameter today to the Hubble parameter at high redshift.