Study on the Mass Determination of Scalar Field in the Brans-Dike Theory of Gravity

The original Brans-Dicke scalar-tensor theory was a scalar massless theory that does not have the self-acting potential of the scalar field. However, further studies are developing the theory of the free mass scalar field with potential. Then the Brans-Dicke coupling parameter is determined by the mass of the scalar field, and Perivolaropoulos and Alsing have obtained for local gravimeters such as solar systems or near-dual systems. The Brans-Dicke parameter which plays an important role in the theory is being as yet determined in a wide range from minus a few hundreds to forty thousands. Such a wide range of the values of the parameter may be associated with how the model used for determination of the parameter is adequate for description of the real Universe. However, in the cosmological context, the scalar field mass dependence of the Brans-Dicke parameter should be expected to be different from that in the local gravimeter. In this paper, the dependence of the Brans-Dicke coupling parameter on the mass of the scalar field from the Brans-Dicke scalar-tensor gravitational cosmological equation is obtained using cosmological observations. We use type Ia Supernovae data from Union 2. We also show that there exists a lower limit of Brans-Dicke coupling parameter, and we obtain that the corresponding mass is an upper limit of the scalar field mass.