Weak field limit of higher dimensional massive Brans-Dicke gravity: Observational constraints

We consider higher-dimensional massive Brans-Dicke theory with Ricci-flat internal space. The background model is perturbed by a massive gravitating source which is pressureless in the external (our space) but has an arbitrary equation-of-state parameter \(\Omega\) in the internal space. We obtain the exact solution of the system of linearized equations for the perturbations of the metric coefficients and scalar field. For a massless scalar field, relying on the fine-tuning between the Brans-Dicke parameter \(\omega\) and \(\Omega\), we demonstrate that (i) the model does not contradict gravitational tests relevant to the parameterized post-Newtonian parameter \(\gamma\), and (ii) the scalar field is not ghost in the case of nonzero \(|\Omega|\sim O(1)\) along with the natural value \(|\omega|\sim O(1)\). In the general case of a massive scalar field, the metric coefficients acquire the Yukawa correction terms, where the Yukawa mass scale \(m\) is defined by the mass of the scalar field. For the natural value \(\omega\sim O(1)\), the inverse-square-law experiments impose the following restriction on the lower bound of the mass: \(m\gtrsim 10^{-11}\,\)GeV. The experimental constraints on \(\gamma\) requires that \(\Omega\) must be extremely close to \(-1/2\).