Scale-Dependent Newton's Constant G in the Conformal Newtonian Gauge

Abstract: 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.