Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity

Abstract

A scalar–tensor theory of gravity was considered, wherein the gravitational coupling G and the speed of light c were admitted as space–time functions and combined to form the definition of the scalar field ϕ. 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 in our gravity model. The effect of the cosmological coupling Λ is accommodated within a possible behavior of ϕ. We analyzed the dynamics of ϕ in the phase space, thereby showing the existence of an attractor point for reasonable hypotheses on the potential V(ϕ) and no particular assumption on the Hubble function. The phase space analysis was performed both with the linear stability theory and via the more general Lyapunov method. Either method led to the conclusion that the condition G˙/G=σc˙/c, where σ=3 must hold for the rest of cosmic evolution after the system arrives at the globally asymptotically stable fixed point and the dynamics of ϕ ceases. This result realized our main motivation: to provide a physical foundation for the phenomenological model admitting G/G0=c/c03, used recently to interpret cosmological and astrophysical data. The thus covarying couplings G and c impact the cosmic evolution after the dynamical system settles to equilibrium. The secondary goal of our work was to investigate how this impact occurs. This was performed by constructing the generalized continuity equation in our scalar–tensor model and considering two possible regimes for the varying speed of light—decreasing c and increasing c—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.