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

TL;DR

This paper investigates the cosmological effects of a scale-dependent Newton's constant G in the conformal Newtonian gauge, focusing on its impact on gravitational slip functions and matter perturbation growth within a quantum gravity framework.

Contribution

It introduces a novel analysis of a running G in the conformal Newtonian gauge and compares its effects on slip functions and growth indices with previous synchronous gauge results.

Findings

Corrections to gravitational slip are smaller than those to growth indices.

Scale-dependent G significantly influences matter perturbation growth.

Results support the importance of gauge choice in quantum gravity cosmological models.

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.