Brans--Dicke cosmology does not have the $Λ$CDM phase as an universal attractor

Abstract: In this paper we seek for relevant information on the asymptotic cosmological dynamics of the Brans--Dicke theory of gravity for several self-interaction potentials. By means of the simplest tools of the dynamical systems theory, it is shown that the general relativity de Sitter solution is an attractor of the Jordan frame (dilatonic) Brans--Dicke theory only for the exponential potential $U(\vphi)\propto\exp\vphi$, which corresponds to the quadratic potential $V(\phi)\propto\phi^2$ in terms of the original Brans--Dicke field $\phi=\exp\vphi$, or for potentials which asymptote to $\exp\vphi$. At the stable de Sitter critical point, as well as at the stiff-matter equilibrium configurations, the dilaton is necessarily massless. We find bounds on the Brans--Dicke coupling constant $\omega_\textsc{bd}$, which are consistent with well-known results.