A new approach for embedding the renormalization group running of Newton’s constant and the cosmological constant in gravity is proposed. This approach is based on a gravitational Lagrangian that gives rise to a new class of modified gravity theories where $G$ and $\mathrm{\Lambda }$ are spacetime-dependent functions. The Lagrangian formulation can be interpreted as an effective gravitational action that encapsulates the scale dependence of $G$ and $\mathrm{\Lambda }$, arising from quantum effects in the early Universe. We show that the new formalism can be discussed partially using the framework and results of Horndeski modified gravity, excluding the equations of motion of the scalar field. The study explores aspects of this new gravity action. We also analyze an interesting nonsingular cosmological solution featuring power-law inflation, and we discuss the generation of scalar and tensor perturbations within this framework.

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• Alternative gravity theories
• Quantum cosmology
• Renormalization group