Inhomogeneous Gravity

TL;DR

The paper investigates spatial variations of the gravitational constant $G$ in scalar-tensor theories, focusing on Brans-Dicke and related models. It constructs exact inhomogeneous solutions and a refined spherical-collapse framework to compare $G$ and $\dot{G}$ inside overdensities against the expanding background. The results show that $G$ evolves differently in bound structures and the background, with inhomogeneities typically inducing $\delta G/G$ of order $10^{-6}$ and $\dot{G}/G$ suppressed inside virialised clusters, though the qualitative behavior depends on the scalar-tensor theory. The study emphasizes caution in applying local $G$ bounds to cosmological variations and demonstrates the role of inhomogeneities in constraining varying-$G$ scenarios.

Abstract

We study the inhomogeneous cosmological evolution of the Newtonian gravitational 'constant' G in the framework of scalar-tensor theories. We investigate the differences that arise between the evolution of G in the background universes and in local inhomogeneities that have separated out from the global expansion. Exact inhomogeneous solutions are found which describe the effects of masses embedded in an expanding FRW Brans-Dicke universe. These are used to discuss possible spatial variations of G in different regions. We develop the technique of matching different scalar-tensor cosmologies of different spatial curvature at a boundary. This provides a model for the linear and non-linear evolution of spherical overdensities and inhomogeneities in G. This allows us to compare the evolution of G and \dot{G} that occurs inside a collapsing overdense cluster with that in the background universe. We develop a simple virialisation criterion and apply the method to a realistic lambda-CDM cosmology containing spherical overdensities. Typically, far slower evolution of \dot{G} will be found in the bound virialised cluster than in the cosmological background. We consider the behaviour that occurs in Brans-Dicke theory and in some other representative scalar-tensor theories.

Figures (15)

• Figure 1: This graph illustrates the possible space and time variations that can arise in G in inhomogeneous solutions to the Brans--Dicke field equations, normalised at $r=2.1$ and $t=5$.
• Figure 2: Distribution of $\rho$ as a function of $r$, with $\omega =100$, from eq. (\ref{['rho(rt)']}) and ${ {\mathit{c=0.5}}}$.
• Figure 3: Evolution of $G(r,t)$ in space and time in an inhomogeneous matter dominated Universe, with $\omega =100$, from eq. (\ref{['G(rt)']}) and $c=0.5.$
• Figure 4: Evolution of the scale factor $S$ in the perturbed overdense region (dashed line) and in the background $a$(solid line) with respect to the comoving proper time in the flat background.
• Figure 5: Evolution of $G(t)$ in the overdense perturbed overdense region of positive curvature (dashed line) and in the spatially flat background universe (solid line).