The generalized Brans-Dicke theory and its cosmology

Abstract

The generalized Brans-Dicke (GBD) theory is studied in this paper. The GBD theory is obtained by generalizing the Ricci scalar R to an arbitrary function f (R) in the original Brans-Dicke (BD) action. An interesting property was found in the GBD theory, for example, it can naturally solve the problem of the $ \gamma$ value emerging in f (R) modified gravity (i.e. the inconsistent problem between the observational $ \gamma$ value and the theoretical $ \gamma$ value), without introducing the so-called chameleon mechanism. In this paper, we derive the cosmological equations and study the cosmology in the GBD theory. The cosmological solutions show that the GBD model can pass through the test of the observations, such as the observational Hubble data. Compared with other theories, it can be found that the GBD theory has some other interesting properties or solve some problems existing in other theories. 1) It is well known that the f (R) theory is equivalent to the BD theory with a potential (abbreviated as BDV) for taking a specific value of the BD parameter $ \omega$ = 0 , where the specific choice $ \omega$ = 0 for the BD parameter is quite exceptional, and it is hard to understand the corresponding absence of the kinetic term for the field. However, for the GBD theory, it is similar to the coupled scalar field model, where both fields in the GBD theory have the non-disappeared dynamical effect. 2) One knows that in the coupled scalar field quintom model, it is required to include both the canonical quintessence field and the non-canonical phantom field, in order to make the state parameter cross over w = - 1 , while several fundamental problems are associated with the phantom field, such as the problem of negative kinetic term, the fine-tuning problem, etc. On the other hand, in the GBD model, the state parameter of geometrical dark energy can cross over the phantom boundary w = - 1 as in the quintom model, without bearing the problems existing in the quintom model. 3) The GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV, or the two scalar fields quintom model tend to solve physical problems from the viewpoint of matter. It is possible that several special characteristics of scalar fields could be revealed through studies of geometrical gravity in the GBD theory. As an example, we investigate the potential V( $ \phi$ ) of the BD scalar field, and an effective form of V( $ \phi$ ) could be given by studying the GBD theory. Moreover, it seems that a viable condition for the BD theory could be found, i.e. the BD parameter should be $ \omega$ > 0 for f > 0 , if we assume that the effective form of the BD potential can be approximately written as a popular square function of $ \phi$ .