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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 $Ω$ 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 $ω$ and $Ω$, we demonstrate that (i) the model does not contradict gravitational tests relevant to the parameterized post-Newtonian parameter $γ$, and (ii) the scalar field is not ghost in the case of nonzero $|Ω|\sim O(1)$ along with the natural value $|ω|\sim O(1)$. 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 $ω\sim O(1)$, 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 $γ$ requires that $Ω$ must be extremely close to $-1/2$.