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Modified Brans-Dicke theory with space-time anisotropic parameters

We consider the ADM formalism of the Brans-Dicke theory and propose an anisotropic extension of the theory by introducing five free parameters. We find that the resulting theory reveals many interesting aspects which are not present in the original BD theory. We first discuss the ghost instability and strong coupling problems which are present in the gravity theory without the full diffeomorphism symmetry and show that they can be avoided in a region of the parameter space. We also perform the post-Newtonian approximation and show that the constraint of the Brans-Dicke parameter $ω_{\rm BD}$ being large to be consistent with the solar system observations could be evaded in the extended theory. We also discuss that accelerating Universe can be achieved without the need of the potential for the Brans-Dicke scalar.