Dynamical analysis of the covarying coupling constants in scalar-tensor gravity

TL;DR

This paper analyzes a scalar-tensor gravity model where the gravitational constant and speed of light vary together, showing they settle into a fixed ratio and influence cosmic evolution, including dark energy-like acceleration.

Contribution

It provides a phase space analysis demonstrating the stable relation between G and c and explores their impact on cosmological dynamics in a scalar-tensor framework.

Findings

Existence of an attractor point with ot;G/G = 3ot;c/c

Stable fixed point where ot;G/G = 3ot;c/c after evolution

Cosmological models with varying c and G can produce accelerated expansion

Abstract

A scalar-tensor theory of gravity is considered wherein the gravitational coupling $G$ and the speed of light $c$ are admitted as space-time functions and combine to form the definition of the scalar field $\phi$. The varying $c$ participates in the definition of the variation of the matter part of the action; it is related to the effective stress-energy tensor which is a result of the requirement of symmetry under general coordinate transformations. The effect of the cosmological coupling $\Lambda$ is accommodated within a possible behaviour of $\phi$. We analyze the dynamics of $\phi$ in the phase space, thereby showing the existence of an attractor point for reasonable hypotheses on the potential $V(\phi)$ and no particular assumption on the Hubble function. The phase space analysis is performed both with the linear stability theory and via the more general Lyapunov's method. Either method lead to the conclusion that the condition $\dot{G}/G=\sigma\left(\dot{c}/c\right)$ where $\sigma=3$ must hold for the rest of cosmic evolution after the system gets to the globally asymptotically stable fixed point and the dynamics of $\phi$ ceases. This result provides a physical foundation for the phenomenological model admitting $\left(G/G_{0}\right)=\left(c/c_{0}\right)^{3}$ used recently to interpret cosmological and astrophysical data. The thus co-varying couplings $G$ and $c$ impact the cosmic evolution after the dynamical system settles to equilibrium. This impact is investigated by constructing the generalized continuity equation in our scalar-tensor model and considering two possible regimes for the varying speed of light -- decreasing $c(a)$ and increasing $c(a)$ -- while solving our modified Friedmann equations. The solutions to the latter equations make room for radiation- and matter-dominated eras that progress to a dark-energy-type of accelerated expansion.