Scale-Dependent Newton's Constant G in the Conformal Newtonian Gauge
  In classical gravity deviations from the predictions of the Einstein theory
are often discussed within the framework of the conformal Newtonian gauge,
where scalar perturbations are described by two potentials $\phi$ and $\psi$.
In this paper we use the above gauge to explore possible cosmological
consequences of a running Newton's constant $ G (\Box) $, as suggested by the
nontrivial ultraviolet fixed point scenario arising from the quantum
field-theoretic treatment of Einstein gravity with a cosmological constant
term. Here we focus on the effects of a scale-dependent coupling on the
so-called gravitational slip functions $\eta = \psi / \phi -1 $, whose
classical general relativity value is zero. Starting from a set of manifestly
covariant but non-local effective field equations derived earlier, we compute
the leading corrections in the potentials $\phi$ and $\psi$ for a
nonrelativistic, pressureless fluid. After providing an estimate for the
quantity $\eta$, we then focus on a comparison with results obtained in a
previous paper on matter density perturbations in the synchronous gauge, which
gave an estimate for the growth index parameter $\gamma$, also in the presence
of a running $G$. Our results indicate that, in the present framework and for a
given $ G (\Box) $, the corrections tend to be significantly larger in
magnitude for the perturbation growth exponents than for the conformal
Newtonian gauge slip function.
