Abstract
A new set of field equations for a space-time dependent Newton's constant $G(x)$ and cosmological constant $\mathrm{\ensuremath{\Lambda}}(x)$ in the presence of matter is presented. We prove that it represents the most general mathematically consistent, physically plausible, set of evolution equations assuming at most second derivatives in the dynamical variables. In the new Einstein's equations, only $\mathrm{\ensuremath{\Lambda}}$-kinetic terms arise, while in the modified conservation equation, derivative terms of $G$ also appear. As an application, this formalism is applied in the context of the asymptotic safety scenario to the early Universe, assuming a perfect fluid with a radiation equation of state. Cosmological solutions are obtained for all types of spatial curvature, displaying a variety of interesting cosmic evolutions. As an indication of such behaviors, bouncing solutions, recollapsing solutions, or nonsingular expanding solutions with a transient acceleration era are discussed in detail.