Effective field equations and scale-dependent couplings in gravity
  A new set of field equations for a space-time dependent Newton's constant
$G(x)$ and cosmological constant $\Lambda(x)$ 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 $\Lambda$-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
behaviours, bouncing solutions, recollapsing solutions or non-singular
expanding solutions with a transient acceleration era are discussed in details.
