The original Brans-Dicke scalar-tensor gravity is believed to be attracted towards General Relativity with evolution of the Universe. In the presence of explicit scalar-matter direct coupling in the Brans-Dicke gravity, however, the attractor mechanism must be reconsidered. In this paper, the master equation found by Damour and Nordtvedt is rederived. It is shown that such a generalized master equation could modify the attraction to General Relativity and the scalar-matter direct coupling may accelerate or retard it. In original Brans-Dicke scalar-tensor gravity, the agent of the attraction to General Relativity is the gravitational coupling strength. However, in the presence of the scalar-matter direct coupling, the coupling function which characterizes the coupling of scalar field to matter, makes another potential associated with coupling and it produces a new coupling parameter in the existence of the scalar-matter direct coupling, the master equation which describes the evolution of scalar field, comes to have the force term of right hand side an additional term characterizing the scalar-matter coupling, i.e. the coupling parameter is lying in parallel with the strength parameter. Therefore, the attraction mechanism represented by the strength parameter alone comes to be modified by adding the new coupling parameter. The paper considered, in detail, the friction-dominated approximation which is a case of a negligible curvature of the scalar-curvature coupling potential. Then, depending on the sign of a new coupling parameter, the evolution of scalar field is shown to accelerate the attraction or to retard it. Consequently, the attraction towards General Relativity is modified on account of the scalar-matter direct coupling.
Kim et al. (Mon,) studied this question.